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1<!DOCTYPE html>2<html lang="fr" data-theme="light">3<head>4<meta charset="utf-8">5<meta name="viewport" content="width=device-width, initial-scale=1, viewport-fit=cover">6<title>Séance 9 — Concepts de dépréciation · IMM1033 — Méthodes du coût en évaluation immobilière</title>7<meta name="description" content="Séance 9 du cours IMM1033 (Méthodes du coût en évaluation immobilière) : Concepts de dépréciation. Notes de cours complètes, explorateurs interactifs, exercices corrigés et quiz.">8<meta name="author" content="Simon-Pierre Boucher — Université du Québec en Outaouais">9<meta name="theme-color" content="#0f6180">10<meta property="og:site_name" content="IMM1033 — UQO">11<meta property="og:title" content="Séance 9 — Concepts de dépréciation · IMM1033 — Méthodes du coût en évaluation immobilière">12<meta property="og:description" content="Séance 9 du cours IMM1033 (Méthodes du coût en évaluation immobilière) : Concepts de dépréciation. 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style="background:#2976bb"></i>Partie I — Fondements de la méthode du coût</div><a href="/seance/01/" data-ch="01"><span class="pastille" style="background:#2976bb">1</span><span>Introduction à la méthode du coût</span></a><a href="/seance/02/" data-ch="02"><span class="pastille" style="background:#2976bb">2</span><span>Cadre conceptuel et terminologie</span></a><div class="dd-part"><i style="background:#0f8b8d"></i>Partie II — Évaluation du terrain</div><a href="/seance/03/" data-ch="03"><span class="pastille" style="background:#0f8b8d">3</span><span>Méthodes d'évaluation du terrain</span></a><a href="/seance/04/" data-ch="04"><span class="pastille" style="background:#0f8b8d">4</span><span>Analyse du terrain et ajustements</span></a><div class="dd-part"><i style="background:#cca424"></i>Partie III — Coût de construction</div><a href="/seance/05/" data-ch="05"><span class="pastille" style="background:#cca424">5</span><span>Coût de reproduction et coût de remplacement</span></a><a href="/seance/06/" data-ch="06"><span class="pastille" style="background:#cca424">6</span><span>Méthodes d'estimation des coûts</span></a><a href="/seance/07/" data-ch="07"><span class="pastille" style="background:#cca424">7</span><span>Coûts directs de construction</span></a><a href="/seance/08/" data-ch="08"><span class="pastille" style="background:#cca424">8</span><span>Coûts indirects et profit entrepreneurial</span></a><div class="dd-part"><i style="background:#6b4fbb"></i>Partie IV — La dépréciation</div><a href="/seance/09/" data-ch="09"><span class="pastille" style="background:#6b4fbb">9</span><span>Concepts de dépréciation</span></a><a href="/seance/10/" data-ch="10"><span class="pastille" style="background:#6b4fbb">10</span><span>Dépréciation physique</span></a><a href="/seance/11/" data-ch="11"><span class="pastille" style="background:#6b4fbb">11</span><span>Dépréciation fonctionnelle</span></a><a href="/seance/12/" data-ch="12"><span class="pastille" style="background:#6b4fbb">12</span><span>Dépréciation économique (externe)</span></a><div class="dd-part"><i style="background:#107c4e"></i>Partie V — Applications et intégration</div><a href="/seance/13/" data-ch="13"><span class="pastille" style="background:#107c4e">13</span><span>Applications spécialisées</span></a><a href="/seance/14/" data-ch="14"><span class="pastille" style="background:#107c4e">14</span><span>Synthèse et révision</span></a><div class="dd-part">Enregistrements</div><a href="/videos/"><span class="pastille" style="background:var(--rouge)">▶</span><span>Séances vidéo — enregistrements Zoom, transcription synchronisée</span></a><div class="dd-part">Annexes</div><a href="/aide-memoire/"><span class="pastille">A</span><span>Aide-mémoire des formules</span></a><a href="/glossaire/"><span class="pastille">B</span><span>Glossaire</span></a><a href="/rapport/"><span class="pastille" style="background:var(--teal)">R</span><span>Rapport — Le marché de l’habitation au Québec</span></a></div></div><a class="nav-video" href="/videos/"><svg class="" aria-hidden="true"><use href="#i-video"/></svg> Vidéos</a><div class="dd"><button type="button" aria-expanded="false" aria-haspopup="true">Réviser <span class="caret" aria-hidden="true">▾</span></button><div class="dd-menu "><a href="/aide-memoire/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span>Aide-mémoire<span class="dd-desc">Toutes les formules commentées</span></span></a><a href="/glossaire/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-book"/></svg></span><span>Glossaire<span class="dd-desc">Termes du métier, équivalents anglais</span></span></a><a href="/definitions/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-list"/></svg></span><span>Toutes les définitions<span class="dd-desc">Les formulations attendues aux examens</span></span></a><a href="/fiches/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-cards"/></svg></span><span>Fiches de révision<span class="dd-desc">Recto-verso, suivi d’acquisition</span></span></a><a href="/carte/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-network"/></svg></span><span>Carte du cours<span class="dd-desc">Les 14 séances et leurs objectifs en un coup d’œil</span></span></a><a href="/exercices/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-pencil"/></svg></span><span>Banque d’exercices<span class="dd-desc">Tous les exercices corrigés</span></span></a></div></div><div class="dd"><button type="button" aria-expanded="false" aria-haspopup="true">S’exercer <span class="caret" aria-hidden="true">▾</span></button><div class="dd-menu "><a href="/quiz/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-quiz"/></svg></span><span>Quiz et examens blancs<span class="dd-desc">QCM avec rétroaction, chronomètre</span></span></a><a href="/entrainement/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-dumbbell"/></svg></span><span>Entraînement<span class="dd-desc">Problèmes à données aléatoires, corrigés pas à pas</span></span></a><a href="/jeux/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-game"/></svg></span><span>Jeux de révision<span class="dd-desc">Association, vrai ou faux, remise en ordre</span></span></a><a href="/outils/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-calc"/></svg></span><span>Calculateurs<span class="dd-desc">Les formules en interactif</span></span></a><a href="/progression/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-progress"/></svg></span><span>Ma progression<span class="dd-desc">Lecture, quiz, objectifs, notes</span></span></a></div></div><div class="dd"><button type="button" aria-expanded="false" aria-haspopup="true">Ressources <span class="caret" aria-hidden="true">▾</span></button><div class="dd-menu "><a href="/diapositives/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-slides"/></svg></span><span>Diapositives<span class="dd-desc">Les 14 présentations en ligne</span></span></a><a href="/videos/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-video"/></svg></span><span>Séances vidéo<span class="dd-desc">Enregistrements Zoom : chapitres, transcription, sous-titres</span></span></a><a href="/ressources/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-download"/></svg></span><span>Téléchargements<span class="dd-desc">Manuel, plan, TP, gabarits Excel</span></span></a><a href="/rapport/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-report"/></svg></span><span>Rapport marché QC<span class="dd-desc">Indices de prix, macro, abordabilité — 26 sections, 67 figures</span></span></a><a href="/bibliographie/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-bib"/></svg></span><span>Bibliographie<span class="dd-desc">Références citées</span></span></a><a href="/recherche/"><span class="dd-ico"><svg class="" aria-hidden="true"><use href="#i-search"/></svg></span><span>Recherche<span class="dd-desc">Tout le manuel indexé</span></span></a></div></div></nav>87    <div class="topbar-actions">88      <button class="icon-btn search-btn" type="button" data-action="search" aria-label="Rechercher"><svg class="" aria-hidden="true"><use href="#i-search"/></svg><span>Rechercher</span><kbd>⌘K</kbd></button>89      <button class="icon-btn" type="button" data-action="theme" aria-label="Thème clair ou sombre" title="Thème"><svg class="" aria-hidden="true"><use href="#i-sun"/></svg></button>90      <button class="icon-btn menu-btn" type="button" aria-label="Menu" aria-expanded="false"><svg class="" aria-hidden="true"><use href="#i-menu"/></svg></button>91    </div>92  </div>93  <div class="readbar"><div class="readbar-fill"></div></div>94</header>9596<main id="main" class="chapter-page" data-chapter="09" data-title="Séance 9 — Concepts de dépréciation" data-words="10301">97  98<header class="ch-hero">99  <svg class="deco" viewBox="0 0 1200 300" preserveAspectRatio="xMaxYMin slice" aria-hidden="true"><circle cx="1090" cy="20" r="110"/><circle cx="1160" cy="110" r="160"/><circle class="gold" cx="990" cy="170" r="45"/></svg>100  <div class="watermark" aria-hidden="true">09</div>101  <div class="ch-hero-inner">102    <div class="eyebrow"><span class="part-chip">Partie IV</span> La dépréciation · Séance 9 · IMM1033</div>103    <h1>Concepts de dépréciation</h1>104    <div class="ch-sub">Méthodes du coût en évaluation immobilière · Automne 2026 · Simon-Pierre Boucher · UQO</div>105    <div class="ch-meta">106      <span class="chip"><svg class="" aria-hidden="true"><use href="#i-list"/></svg> 8 sections</span>107      <span class="chip"><svg class="" aria-hidden="true"><use href="#i-book"/></svg> 6 définitions</span>108      <span class="chip"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg> 10 formules</span>109      <span class="chip"><svg class="" aria-hidden="true"><use href="#i-bulb"/></svg> 7 exemples</span>110      <span class="chip"><svg class="" aria-hidden="true"><use href="#i-pencil"/></svg> 5 exercices</span>111      <span class="chip"><svg class="" aria-hidden="true"><use href="#i-eye"/></svg> ≈ 57 min de lecture</span>112      <span class="chip"><svg class="" aria-hidden="true"><use href="#i-chart"/></svg> 1 explorateur interactif</span>113      <span class="chip">Remise du TP2</span>114    </div>115    <div class="ch-tools">116      117      <a class="btn btn-ghost" href="/files/slides/seance09.pdf" target="_blank" rel="noopener"><svg class="" aria-hidden="true"><use href="#i-slides"/></svg> Diapositives (PDF)</a>118      <button class="btn btn-ghost" type="button" data-action="revision" aria-pressed="false"><svg class="" aria-hidden="true"><use href="#i-eye"/></svg> Mode révision</button>119      <button class="btn btn-ghost" type="button" data-action="tts"><svg class="" aria-hidden="true"><use href="#i-audio"/></svg> Écouter</button>120      <a class="btn btn-ghost" href="#quiz"><svg class="" aria-hidden="true"><use href="#i-quiz"/></svg> Quiz de la séance</a>121    </div>122  </div>123</header>124<div class="stepper-wrap"><nav class="stepper" aria-label="Sections"><a href="#s-9-1"><span class="n">1</span>Fondements économiques de la dépréciation</a><a href="#s-9-2"><span class="n">2</span>Les trois catégories de dépréciation</a><a href="#s-9-3"><span class="n">3</span>Curabilité : le raisonnement économique</a><a href="#s-9-4"><span class="n">4</span>Âge effectif et durées de vie</a><a href="#s-9-5"><span class="n">5</span>Les quatre méthodes de mesure</a><a href="#s-9-6"><span class="n">6</span>Modélisation mathématique de la dépréciation</a><a href="#s-9-7"><span class="n">7</span>Étude de cas intégratrice : 99, rue de Magog, Gatineau</a><a href="#s-9-8"><span class="n">8</span>Exercices</a><span class="rem" id="read-remaining"></span></nav></div>125  <div class="ch-layout">126    127<aside class="ch-toc" aria-label="Sommaire de la séance">128  <div class="toc-progress"><div class="bar"><i></i></div><span>0 % lu</span></div>129  <h4>Dans cette séance</h4>130  <ol><li class="lvl2"><a href="#s-9-1"><span class="toc-num">9.1</span><span>Fondements économiques de la dépréciation</span></a></li><li class="lvl3"><a href="#s-9-1-1"><span class="toc-num">9.1.1</span><span>Coût, valeur et valeur contributive</span></a></li><li class="lvl3"><a href="#s-9-1-2"><span class="toc-num">9.1.2</span><span>La dépréciation dans le temps : cumul, taux, marge</span></a></li><li class="lvl3"><a href="#s-9-1-3"><span class="toc-num">9.1.3</span><span>Âge, état, rénovation et marché</span></a></li><li class="lvl2"><a href="#s-9-2"><span class="toc-num">9.2</span><span>Les trois catégories de dépréciation</span></a></li><li class="lvl3"><a href="#s-9-2-1"><span class="toc-num">9.2.1</span><span>L'arbre de la dépréciation</span></a></li><li class="lvl3"><a href="#s-9-2-2"><span class="toc-num">9.2.2</span><span>Type 1 — la dépréciation physique</span></a></li><li class="lvl3"><a href="#s-9-2-3"><span class="toc-num">9.2.3</span><span>Type 2 — la dépréciation fonctionnelle</span></a></li><li class="lvl3"><a href="#s-9-2-4"><span class="toc-num">9.2.4</span><span>Type 3 — la dépréciation externe (économique)</span></a></li><li class="lvl3"><a href="#s-9-2-5"><span class="toc-num">9.2.5</span><span>Les conditions d'additivité</span></a></li><li class="lvl2"><a href="#s-9-3"><span class="toc-num">9.3</span><span>Curabilité : le raisonnement économique</span></a></li><li class="lvl3"><a href="#s-9-3-1"><span class="toc-num">9.3.1</span><span>Courbes de coût et de bénéfice</span></a></li><li class="lvl3"><a href="#s-9-3-2"><span class="toc-num">9.3.2</span><span>Sensibilité, arbre de décision et comparaison d'interventions</span></a></li><li class="lvl2"><a href="#s-9-4"><span class="toc-num">9.4</span><span>Âge effectif et durées de vie</span></a></li><li class="lvl3"><a href="#s-9-4-1"><span class="toc-num">9.4.1</span><span>Construire un âge effectif défendable</span></a></li><li class="lvl3"><a href="#s-9-4-2"><span class="toc-num">9.4.2</span><span>La vie économique : le paramètre le plus lourd</span></a></li><li class="lvl2"><a href="#s-9-5"><span class="toc-num">9.5</span><span>Les quatre méthodes de mesure</span></a></li><li class="lvl3"><a href="#s-9-5-1"><span class="toc-num">9.5.1</span><span>Méthode 1 — âge-vie (linéaire)</span></a></li><li class="lvl3"><a href="#s-9-5-2"><span class="toc-num">9.5.2</span><span>Méthode 2 — âge-vie modifiée</span></a></li><li class="lvl3"><a href="#s-9-5-3"><span class="toc-num">9.5.3</span><span>Méthode 3 — ventilation (<em>breakdown method</em>)</span></a></li><li class="lvl3"><a href="#s-9-5-4"><span class="toc-num">9.5.4</span><span>Méthode 4 — extraction du marché</span></a></li><li class="lvl3"><a href="#s-9-5-5"><span class="toc-num">9.5.5</span><span>Ce que disent 655 ventes réelles : le profil de dépréciation de Gatineau</span></a></li><li class="lvl2"><a href="#s-9-6"><span class="toc-num">9.6</span><span>Modélisation mathématique de la dépréciation</span></a></li><li class="lvl3"><a href="#s-9-6-1"><span class="toc-num">9.6.1</span><span>Introduction au modèle hédonique</span></a></li><li class="lvl2"><a href="#s-9-7"><span class="toc-num">9.7</span><span>Étude de cas intégratrice : 99, rue de Magog, Gatineau</span></a></li><li class="lvl3"><a href="#s-9-7-1"><span class="toc-num">9.7.1</span><span>L'immeuble, ses composantes, son coût neuf</span></a></li><li class="lvl3"><a href="#s-9-7-2"><span class="toc-num">9.7.2</span><span>Terrain, âges, entretien différé</span></a></li><li class="lvl3"><a href="#s-9-7-3"><span class="toc-num">9.7.3</span><span>Les trois catégories et les quatre méthodes</span></a></li><li class="lvl3"><a href="#s-9-7-4"><span class="toc-num">9.7.4</span><span>Réconciliation : raisonner, pas moyenner</span></a></li><li class="lvl2"><a href="#s-9-8"><span class="toc-num">9.8</span><span>Exercices</span></a></li></ol>131  <div class="toc-extra">132    <a href="#objectifs"><svg class="" aria-hidden="true"><use href="#i-target"/></svg> Objectifs d’apprentissage</a>133    <a href="#synthese"><svg class="" aria-hidden="true"><use href="#i-clipboard"/></svg> L’essentiel de la séance</a>134    <a href="#quiz"><svg class="" aria-hidden="true"><use href="#i-quiz"/></svg> Testez-vous</a>135    136    <a href="/fiches/?ch=09"><svg class="" aria-hidden="true"><use href="#i-cards"/></svg> Fiches de la séance</a>137    <a href="/entrainement/"><svg class="" aria-hidden="true"><use href="#i-dumbbell"/></svg> Entraînement</a>138    <a href="/aide-memoire/"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg> Aide-mémoire</a>139  </div>140  <h4 style="margin-top:14px">Raccourcis</h4>141  <div style="font-size:.76rem;color:var(--gris)"><kbd>←</kbd> <kbd>→</kbd> séance précédente / suivante · <kbd>⌘K</kbd> recherche · sélectionnez un passage pour le surligner</div>142</aside>143    <div class="ch-body">144      <div class="revision-banner"><svg class="" aria-hidden="true"><use href="#i-eye"/></svg><span><strong>Mode révision</strong> — seuls les définitions, formules, mises en garde, « À retenir », notes marginales et la fiche synthèse sont affichés.</span><button class="btn btn-sm btn-ghost" type="button" data-action="revision-off">Quitter</button></div>145      <div style="display:flex;justify-content:flex-end;margin-bottom:8px"><button class="btn btn-sm btn-ghost" type="button" data-action="toggle-solutions">Révéler toutes les solutions</button></div>146      <article class="prose" id="notes">147        <p id="p9-1">Les blocs 2 et 3 nous ont donné les deux premiers termes de la formule fondamentale : la valeur du terrain (séances 3–4) et le coût de construction à neuf (séances 5–8). Ce chapitre amorce le bloc 4 et l'étude du dernier terme — le plus subtil : la <span class="motcle" data-term="dépréciation">dépréciation</span>. C'est elle qui transforme un coût de construction en une opinion de valeur; c'est aussi elle qui distingue l'évaluateur compétent du simple estimateur de coûts. Nous posons ici l'appareillage complet — les fondements économiques, les trois catégories de dépréciation, le critère de curabilité, les durées de vie, les quatre méthodes de mesure et les modèles mathématiques qui les sous-tendent — puis nous l'appliquons à un plain-pied <em>réel</em> de Gatineau, bâti en 1985 et vendu en janvier 2026, dont le prix payé sert de pierre de touche. Les séances 10 à 12 détailleront ensuite chaque catégorie.</p>148<section class="box box-obj" id="objectifs"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-target"/></svg></span><span class="box-label">Objectifs d’apprentissage</span><span class="obj-progress" aria-live="polite"></span></div><div class="box-body"><ul><li class="obj"><label><input type="checkbox" class="obj-cb" id="obj-9-1" data-obj="obj-9-1"><span class="obj-text">Définir la dépréciation comme un écart entre coût et valeur, la         situer dans <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><msub><mi>V</mi><mi>T</mi></msub><mo>+</mo><msub><mi>C</mi><mi>N</mi></msub><mo>−</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">V = V_T + C_N - D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> et en expliquer les déterminants         économiques;</span></label></li><li class="obj"><label><input type="checkbox" class="obj-cb" id="obj-9-2" data-obj="obj-9-2"><span class="obj-text">Distinguer les trois catégories de dépréciation — physique,         fonctionnelle, externe — et énoncer les conditions qui permettent         de les additionner sans double comptage;</span></label></li><li class="obj"><label><input type="checkbox" class="obj-cb" id="obj-9-3" data-obj="obj-9-3"><span class="obj-text">Appliquer le critère de curabilité <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>N</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>V</mi><mo>−</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">GN = \Delta V - C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">GN</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> et le         distinguer de la faisabilité technique;</span></label></li><li class="obj"><label><input type="checkbox" class="obj-cb" id="obj-9-4" data-obj="obj-9-4"><span class="obj-text">Calculer et défendre un âge effectif, une vie économique totale et         une vie économique restante;</span></label></li><li class="obj"><label><input type="checkbox" class="obj-cb" id="obj-9-5" data-obj="obj-9-5"><span class="obj-text">Mettre en œuvre les quatre méthodes d'estimation — âge-vie,         âge-vie modifiée, ventilation, extraction du marché — et expliquer         leurs écarts;</span></label></li><li class="obj"><label><input type="checkbox" class="obj-cb" id="obj-9-6" data-obj="obj-9-6"><span class="obj-text">Lire les modèles linéaire, exponentiel et non linéaire de la valeur         dans le temps, leurs dérivées et leurs limites;</span></label></li><li class="obj"><label><input type="checkbox" class="obj-cb" id="obj-9-7" data-obj="obj-9-7"><span class="obj-text">Interpréter avec prudence le coefficient de l'âge dans un modèle         hédonique estimé sur des ventes réelles.</span></label></li></ul></div></section>149<h2 id="s-9-1" class="h-section"><span class="h-num">9.1</span><span class="h-text">Fondements économiques de la dépréciation</span><a class="h-anchor" href="#s-9-1" aria-label="Lien vers cette section">#</a></h2>150<h3 id="s-9-1-1" class="h-subsection"><span class="h-num">9.1.1</span><span class="h-text">Coût, valeur et valeur contributive</span><a class="h-anchor" href="#s-9-1-1" aria-label="Lien vers cette section">#</a></h3>151<div class="box box-def" id="def-9-1"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-book"/></svg></span><span class="box-label">Définition 9.1</span><span class="box-title">— dépréciation</span><a class="box-anchor" href="#def-9-1" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-9">La <span class="motcle" data-term="dépréciation">dépréciation</span> est la <strong>perte de valeur</strong> d'une amélioration par rapport à son coût de reproduction ou de remplacement à neuf, mesurée <strong>à la date d'évaluation</strong> :</p>152<div class="eq" id="eq-9-1"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>D</mi><mtext>  </mtext><mo>=</mo><mtext>  </mtext><msub><mi>C</mi><mi>N</mi></msub><mtext>  </mtext><mo>−</mo><mtext>  </mtext><msub><mi>V</mi><mi>B</mi></msub><mo separator="true">,</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.1)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">D \;=\; C_N \;-\; V_B ,\tag{9.1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span><span class="katex-tag"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.1</span></span><span class="mord">)</span></span></span></span></span></span></div>153<p id="p9-10">où <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">C_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> est le coût neuf complet (directs, indirects et profit entrepreneurial compris) et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">V_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> la <span class="motcle" data-term="valeur contributive">valeur contributive</span> des améliorations — ce que leur présence ajoute au prix que le marché paierait pour le terrain nu.</p></div></div>154<p id="p9-11">Replaçons ce terme dans la formule fondamentale<label for="sn-9-1" class="sn-toggle" title="Note marginale"><span class="sn-mark" aria-hidden="true">◆</span></label><input type="checkbox" id="sn-9-1" class="sn-cb" aria-label="Afficher la note"><span class="sidenote sn-repere"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><msub><mi>V</mi><mi>T</mi></msub><mo>+</mo><msub><mi>C</mi><mi>N</mi></msub><mo>−</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">V = V_T + C_N - D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> — le bloc 4 s'attaque au dernier terme.</span> : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><msub><mi>V</mi><mi>T</mi></msub><mo>+</mo><msub><mi>C</mi><mi>N</mi></msub><mo>−</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">V = V_T + C_N - D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>, ou de façon équivalente <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><msub><mi>V</mi><mi>T</mi></msub><mo>+</mo><msub><mi>V</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">V = V_T + V_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> avec <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>B</mi></msub><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_B = C_N(1-d)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span>. Le terrain vient du bloc 2, le coût neuf du bloc 3; la dépréciation, objet des séances 9 à 12, est l'ajustement qui ramène le coût de création au niveau de ce que le marché paie réellement pour un bâtiment qui n'est plus neuf.</p>155<p id="p9-12">Il faut d'abord savoir <em>de quel coût neuf</em> on part. Le coût de <strong>reproduction</strong> réplique le bâtiment à l'identique, défauts compris; le coût de <strong>remplacement</strong> livre une utilité équivalente avec les moyens d'aujourd'hui. La dépréciation fonctionnelle liée aux défauts d'origine disparaît quand on part du coût de remplacement : elle est déjà « remplacée ». Tout ce chapitre raisonne en coût de remplacement, comme la pratique courante.</p>156<div class="box box-form" id="form-9-1"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.1</span><span class="box-title">— les quatre équations de base</span><a class="box-anchor" href="#form-9-1" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-2"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.25em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>V</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>V</mi><mi>T</mi></msub><mo>+</mo><msub><mi>C</mi><mi>N</mi></msub><mo>−</mo><mi>D</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mtext>(valeur de la propri</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>t</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>, $)</mtext></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>D</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mo>−</mo><msub><mi>V</mi><mi>B</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mtext>(d</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>pr</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>ciation cumul</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>e, $)</mtext></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>d</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>D</mi><msub><mi>C</mi><mi>N</mi></msub></mfrac></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mtext>(taux de d</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>pr</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>ciation, sans unit</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>)</mtext></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>V</mi><mi>B</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mtext> </mtext><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>(valeur contributive, $)</mtext></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(9.2)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}V &amp;= V_T + C_N - D &amp;&amp; \text{(valeur de la propriété, \$)}\\157D &amp;= C_N - V_B &amp;&amp; \text{(dépréciation cumulée, \$)}\\158d &amp;= \frac{D}{C_N} &amp;&amp; \text{(taux de dépréciation, sans unité)}\\159V_B &amp;= C_N\,(1 - d) &amp;&amp; \text{(valeur contributive, \$)}\end{aligned}\tag{9.2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:6.6963em;vertical-align:-3.0982em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5982em;"><span style="top:-6.1185em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span><span style="top:-4.6185em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span><span style="top:-2.5982em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span><span style="top:-0.6222em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.0982em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5982em;"><span style="top:-6.1185em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span><span style="top:-4.6185em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.5982em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.6222em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.0982em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5982em;"><span style="top:-6.1185em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span><span style="top:-4.6185em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span><span style="top:-2.5982em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span><span style="top:-0.6222em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.0982em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5982em;"><span style="top:-6.1185em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord">(valeur de la propri</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">t</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">, $)</span></span></span></span><span style="top:-4.6185em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord">(d</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">pr</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">ciation cumul</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">e, $)</span></span></span></span><span style="top:-2.5982em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord">(taux de d</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">pr</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">ciation, sans unit</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">)</span></span></span></span><span style="top:-0.6222em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord">(valeur contributive, $)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.0982em;"><span></span></span></span></span></span></span></span></span><span class="katex-tag"><span class="katex-strut" style="height:6.6963em;vertical-align:-3.0982em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.2</span></span><span class="mord">)</span></span></span></span></span></span></div>160<p id="p9-13"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">C_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> sont des <em>stocks</em> datés, en dollars de la date d'évaluation; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> est borné entre 0 et 1 et seul comparable d'un bâtiment à l'autre; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> est un <em>cumul</em> depuis la construction, pas une perte annuelle.</p></div></div>161<p id="p9-14">Trois mises en garde s'imposent d'entrée de jeu. La dépréciation, ce n'est <strong>pas</strong> l'amortissement comptable (un calcul fiscal à taux prescrits, sur l'âge chronologique, sans regard sur l'état ni le marché); ce n'est <strong>pas</strong> seulement l'usure physique (deux autres familles de causes existent); et elle n'est <strong>pas</strong> mécaniquement liée à l'âge chronologique (c'est l'état réel qui compte).</p>162<div class="box box-ex" id="ex-9-1"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-bulb"/></svg></span><span class="box-label">Exemple 9.1</span><span class="box-title">— mesure de la dépréciation d'un bâtiment commercial</span><a class="box-anchor" href="#ex-9-1" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-15">Bâtiment construit en 2006, évalué en 2026 (20 ans). Coût de remplacement à neuf : <span class="num">1 200 000 $</span>; valeur contributive estimée : <span class="num">780 000 $</span>.</p>163<div class="eq"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>D</mi><mo>=</mo><mn>1</mn><mtext> </mtext><mn>200</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mn>780</mn><mtext> </mtext><mn>000</mn><mo>=</mo><mn>420</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi><mo separator="true">,</mo><mspace width="2em"/><mi>d</mi><mo>=</mo><mfrac><mrow><mn>420</mn><mtext> </mtext><mn>000</mn></mrow><mrow><mn>1</mn><mtext> </mtext><mn>200</mn><mtext> </mtext><mn>000</mn></mrow></mfrac><mo>=</mo><mn>35</mn><mtext> </mtext><mi mathvariant="normal">%</mi><mo separator="true">,</mo><mspace width="2em"/><msub><mi>V</mi><mi>B</mi></msub><mo>=</mo><mn>1</mn><mtext> </mtext><mn>200</mn><mtext> </mtext><mn>000</mn><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>0,35</mn><mo stretchy="false">)</mo><mo>=</mo><mn>780</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">D = 1\,200\,000 - 780\,000 = 420\,000\,\$,164\qquad165d = \frac{420\,000}{1\,200\,000} = 35\,\%,166\qquad167V_B = 1\,200\,000 \times (1 - 0{,}35) = 780\,000\,\$.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">200</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">780</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">420</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">200</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">420</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">35</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">200</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">35</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">780</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$.</span></span></span></span></span></div>168<p id="p9-16">Le bâtiment a perdu 35 % de son coût neuf — toutes causes confondues. Le travail du bloc 4 consiste précisément à <em>expliquer</em> et <em>ventiler</em> ce 35 %.</p></div></div>169<h3 id="s-9-1-2" class="h-subsection"><span class="h-num">9.1.2</span><span class="h-text">La dépréciation dans le temps : cumul, taux, marge</span><a class="h-anchor" href="#s-9-1-2" aria-label="Lien vers cette section">#</a></h3>170<p id="p9-17">Graphiquement, la dépréciation est l'écart vertical entre la droite du coût neuf — qui ne vieillit pas, mais se réévalue chaque année en dollars du jour — et la courbe de valeur contributive, lu à la date d'évaluation. Les figures suivantes sont <strong>calculées</strong> à partir du coût neuf du sujet du chapitre (<span class="num">317 462 $</span>) et d'un taux exponentiel <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1,03</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\lambda = 1{,}03\,\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">03</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span></span></span></span> par an, <em>estimé</em> à la section <a class="xref" href="#s-9-5">9.5</a> sur 655 ventes réelles de plain-pieds de Gatineau. La <em>forme</em> exponentielle est un choix de modèle; le <em>niveau</em> de <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> vient des données : ne confondez jamais les deux.</p>171<div class="center"><div class="tikz"><img src="/assets/fig/f77508d510bb326f.svg" alt="Figure" width="410" height="216" style="width:min(100%,595px)" loading="lazy" decoding="async"></div></div>172<p id="p9-18">Trois grandeurs se lisent sur cette figure. La <strong>dépréciation cumulée</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> est un stock en dollars; le <strong>taux</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><msub><mi>C</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">d(t) = D(t)/C_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> est ce stock rapporté au coût neuf; la <strong>dépréciation marginale</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>d</mi><mi>V</mi><mi mathvariant="normal">/</mi><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">-dV/dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord">/</span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span> est la pente de la courbe de valeur, c'est-à-dire la perte de <em>cette</em> année. Dans le modèle linéaire, la perte annuelle est constante (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><msub><mi>C</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">\delta C_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, soit <span class="num">4233 $</span> par an pour le sujet avec une vie économique de 75 ans); dans le modèle exponentiel elle décroît avec l'âge (<span class="num">3275 $</span> la première année, <span class="num">2145 $</span> à 41 ans), parce qu'il reste moins de valeur à perdre; dans un modèle non linéaire elle est faible au début puis maximale vers 40 ans, quand les composantes lourdes arrivent ensemble à échéance.</p>173<div class="box box-rem" id="remarque-9-19"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-pen"/></svg></span><span class="box-label">trajectoires illustratives et résultats empiriques</span><a class="box-anchor" href="#remarque-9-19" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-20">Les trois trajectoires (linéaire, exponentielle, non linéaire) détaillées à la section <a class="xref" href="#s-9-6">9.6</a> partent du même coût neuf et donnent, pour le même bâtiment de 41 ans, trois valeurs différentes (143 916, 207 971 et 207 971 $ par construction pour les deux dernières). Ce sont des <em>formes fonctionnelles</em>; seule l'estimation sur des ventes ou la ventilation composante par composante dit laquelle approche le marché.</p></div></div>174<h3 id="s-9-1-3" class="h-subsection"><span class="h-num">9.1.3</span><span class="h-text">Âge, état, rénovation et marché</span><a class="h-anchor" href="#s-9-1-3" aria-label="Lien vers cette section">#</a></h3>175<p id="p9-21">L'âge chronologique n'explique pas tout. Considérez deux bungalows de 1996 au même coût neuf de <span class="num">400 000 $</span> et à la même vie économique de 60 ans. Le premier, entretenu avec soin, affiche un âge effectif de 18 ans : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>A</mi></msub><mo>=</mo><mn>400</mn><mtext> </mtext><mn>000</mn><mo>×</mo><mn>18</mn><mi mathvariant="normal">/</mi><mn>60</mn><mo>=</mo><mn>120</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D_A = 400\,000 \times 18/60 = 120\,000\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">400</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">18/60</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">120</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Le second, négligé, en affiche 42 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>B</mi></msub><mo>=</mo><mn>280</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D_B = 280\,000\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">280</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Même calendrier, écart de <span class="num">160 000 $</span>. L'état physique ne se lit pas au calendrier, c'est l'inspection qui le révèle; l'entretien est un investissement dont le rendement est la dépréciation évitée <span class="citation">(<a class="cite" href="/bibliographie/#knight1996" data-cite="knight1996">Knight et Sirmans, 1996</a>; <a class="cite" href="/bibliographie/#harding2007" data-cite="harding2007">Harding et al., 2007</a>)</span>.</p>176<p id="p9-22">Une rénovation majeure ne change pas l'âge chronologique; elle réduit l'âge <em>effectif</em> et la courbe de valeur remonte d'un cran. La vraie courbe d'un bâtiment est donc <em>dentelée</em> — toiture, cuisine, fenêtres sont remplacées à des dates différentes — et les modèles lisses décrivent la moyenne de centaines de bâtiments d'un âge donné. Le gain de valeur procuré par une rénovation n'est généralement pas égal à son coût : s'il lui est supérieur, la correction était « curable » (section <a class="xref" href="#s-9-3">9.3</a>).</p>177<p id="p9-23">Enfin, la dépréciation n'est pas une propriété du bâtiment seul : c'est le <strong>marché</strong> qui décide. Puisque <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>B</mi></msub><mo>=</mo><mi>V</mi><mo>−</mo><msub><mi>V</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">V_B = V - V_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, si les acheteurs paient davantage pour le secteur, la valeur contributive monte sans qu'une planche ait bougé; quand l'offre est rare, les bâtiments anciens mais fonctionnels se vendent près du coût neuf; et si l'<span class="motcle" data-term="utilisation optimale">utilisation optimale</span> du terrain nu diffère de l'usage actuel, la dépréciation devient totale quel que soit l'état (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>≈</mo><msub><mi>V</mi><mi>T</mi></msub><mo>−</mo><msub><mi>C</mi><mrow><mtext>d</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>molition</mtext></mrow></msub></mrow><annotation encoding="application/x-tex">V \approx V_T - C_{\text{démolition}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">d</span><span class="mord katex-accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight">e</span></span><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mtight">molition</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>). D'où la règle : <em>d'abord</em> l'analyse de l'utilisation optimale, <em>ensuite</em> la dépréciation, jamais l'inverse.</p>178<div class="box box-att" id="attention-9-24"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-warning"/></svg></span><span class="box-label">la DPA n'est pas la dépréciation</span><a class="box-anchor" href="#attention-9-24" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-25">La déduction pour amortissement (DPA, <em>CCA</em>) répartit un coût <em>historique</em> selon des taux prescrits par l'ARC <span class="citation">(<a class="cite" href="/bibliographie/#arcDPA" data-cite="arcDPA">Agence du revenu du Canada, 2026</a>)</span> — catégorie 1 : 4 % dégressif, demi-taux la première année — sur l'âge chronologique, sans regard sur l'état ni le marché. Sur vingt-cinq ans le solde fiscal tombe mécaniquement vers 37 % du coût d'origine, alors que la valeur contributive d'un plain-pied de Gatineau descend vers 77 % du coût neuf du jour et que sa valeur marchande nominale peut même monter si les coûts de construction croissent plus vite que le vieillissement. Ne prenez jamais la fraction non amortie d'un bilan pour une valeur contributive.</p></div></div>179<h2 id="s-9-2" class="h-section"><span class="h-num">9.2</span><span class="h-text">Les trois catégories de dépréciation</span><a class="h-anchor" href="#s-9-2" aria-label="Lien vers cette section">#</a></h2>180<h3 id="s-9-2-1" class="h-subsection"><span class="h-num">9.2.1</span><span class="h-text">L'arbre de la dépréciation</span><a class="h-anchor" href="#s-9-2-1" aria-label="Lien vers cette section">#</a></h3>181<p id="p9-26">La dépréciation peut provenir de plusieurs sources simultanées, que la doctrine regroupe en trois types<label for="sn-9-2" class="sn-toggle" title="Note marginale"><span class="sn-mark" aria-hidden="true">◆</span></label><input type="checkbox" id="sn-9-2" class="sn-cb" aria-label="Afficher la note"><span class="sidenote sn-repere"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mtext>totale</mtext></msub><mo>=</mo><msub><mi>D</mi><mtext>physique</mtext></msub><mo>+</mo><msub><mi>D</mi><mtext>fonctionnelle</mtext></msub><mo>+</mo><msub><mi>D</mi><mtext>externe</mtext></msub></mrow><annotation encoding="application/x-tex">D_{\text{totale}} =182D_{\text{physique}} + D_{\text{fonctionnelle}} + D_{\text{externe}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">totale</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">physique</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">fonctionnelle</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">externe</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> <span class="citation">(<a class="cite" href="/bibliographie/#appraisal2026" data-cite="appraisal2026">Appraisal Institute, 2026</a>; <a class="cite" href="/bibliographie/#rattermannCout" data-cite="rattermannCout">Rattermann, 2021</a>)</span> :</p>183<div class="box box-form" id="form-9-2"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.2</span><span class="box-title">— décomposition de la dépréciation totale</span><a class="box-anchor" href="#form-9-2" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-3"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi>D</mi><mtext>totale</mtext></msub><mtext>  </mtext><mo>=</mo><mtext>  </mtext><munder><munder><mrow><msub><mi>D</mi><mrow><mtext>phys.</mtext><mtext> </mtext><mtext>cur.</mtext></mrow></msub><mo>+</mo><msub><mi>D</mi><mrow><mtext>phys.</mtext><mtext> </mtext><mtext>CT</mtext></mrow></msub><mo>+</mo><msub><mi>D</mi><mrow><mtext>phys.</mtext><mtext> </mtext><mtext>LT</mtext></mrow></msub></mrow><mo stretchy="true">⏟</mo></munder><mtext>physique</mtext></munder><mtext>  </mtext><mo>+</mo><mtext>  </mtext><munder><munder><mrow><msub><mi>D</mi><mrow><mtext>fonct.</mtext><mtext> </mtext><mtext>cur.</mtext></mrow></msub><mo>+</mo><msub><mi>D</mi><mrow><mtext>fonct.</mtext><mtext> </mtext><mtext>inc.</mtext></mrow></msub></mrow><mo stretchy="true">⏟</mo></munder><mtext>fonctionnelle</mtext></munder><mtext>  </mtext><mo>+</mo><mtext>  </mtext><msub><mi>D</mi><mrow><mtext>externe (b</mtext><mover accent="true"><mtext>a</mtext><mo>ˆ</mo></mover><mtext>t.)</mtext></mrow></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.3)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">D_{\text{totale}} \;=\; \underbrace{D_{\text{phys.\,cur.}} + D_{\text{phys.\,CT}} + D_{\text{phys.\,LT}}}_{\text{physique}}184\;+\; \underbrace{D_{\text{fonct.\,cur.}} + D_{\text{fonct.\,inc.}}}_{\text{fonctionnelle}}185\;+\; D_{\text{externe (bât.)}}\tag{9.3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">totale</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.4397em;vertical-align:-1.7563em;"></span><span class="minner munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-1.3798em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">physique</span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="minner munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.0659em;"><span class="pstrut" style="height:3em;"></span><span class="katex-stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13186 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688187 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7188-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z"/></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M199572 214189c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14190 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3191 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0192-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z"/></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3193 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237194-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z"/></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">phys.</span><span class="mspace mtight" style="margin-right:0.2381em;"></span><span class="mord mtight">cur.</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">phys.</span><span class="mspace mtight" style="margin-right:0.2381em;"></span><span class="mord mtight">CT</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">phys.</span><span class="mspace mtight" style="margin-right:0.2381em;"></span><span class="mord mtight">LT</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9341em;"><span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7563em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.1674em;vertical-align:-1.4841em;"></span><span class="minner munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-1.5159em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">fonctionnelle</span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="minner munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.202em;"><span class="pstrut" style="height:3em;"></span><span class="katex-stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13195 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688196 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7197-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z"/></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M199572 214198c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14199 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3200 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0201-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z"/></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3202 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237203-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z"/></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">fonct.</span><span class="mspace mtight" style="margin-right:0.2381em;"></span><span class="mord mtight">cur.</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">fonct.</span><span class="mspace mtight" style="margin-right:0.2381em;"></span><span class="mord mtight">inc.</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.798em;"><span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4841em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.0385em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">externe (b</span><span class="mord katex-accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight">a</span></span><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord mtight">ˆ</span></span></span></span></span></span></span><span class="mord mtight">t.)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span><span class="katex-tag"><span class="katex-strut" style="height:2.5063em;vertical-align:-1.7563em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.3</span></span><span class="mord">)</span></span></span></span></span></span></div>204<p id="p9-27">Chaque type se subdivise en <strong>curable</strong> et <strong>incurable</strong> — sauf l'externe, toujours incurable; la physique incurable distingue les composantes à <strong>courte</strong> (CT) et à <strong>longue</strong> (LT) durée de vie.</p></div></div>205<div class="center"><div class="tikz"><img src="/assets/fig/fc14febf85b06cb9.svg" alt="Figure" width="472" height="168" style="width:min(100%,685px)" loading="lazy" decoding="async"></div></div>206<p id="p9-28">Chaque branche se mesure <strong>différemment</strong> : coût de remise en état pour le curable, ratio âge/vie pour la physique incurable, excédent de coût ou perte capitalisée pour la fonctionnelle, paires de ventes ou capitalisation pour l'externe. Cette dualité des unités de mesure — un coût de travaux quand la correction se fera, une perte de valeur quand personne ne la fera — structure tout le bloc 4.</p>207<h3 id="s-9-2-2" class="h-subsection"><span class="h-num">9.2.2</span><span class="h-text">Type 1 — la dépréciation physique</span><a class="h-anchor" href="#s-9-2-2" aria-label="Lien vers cette section">#</a></h3>208<div class="box box-def" id="def-9-2"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-book"/></svg></span><span class="box-label">Définition 9.2</span><span class="box-title">— dépréciation physique</span><a class="box-anchor" href="#def-9-2" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-29">Perte de valeur causée par l'<strong>usure</strong>, le <strong>vieillissement</strong> et la <strong>détérioration</strong> des composantes physiques du bâtiment : intempéries, cycles de gel-dégel, usage, manque d'entretien.</p></div></div>209<p id="p9-30">C'est la plus intuitive des trois : le temps et les éléments travaillent contre le bâtiment dès le jour de sa livraison — au Québec, une cinquantaine de cycles de gel-dégel par hiver, des toitures chargées de neige, des écarts de soixante degrés entre janvier et juillet. Sa portion <strong>curable</strong> est l'<span class="motcle" data-term="entretien différé">entretien différé</span><label for="sn-9-3" class="sn-toggle" title="Note marginale"><span class="sn-mark" aria-hidden="true">◆</span></label><input type="checkbox" id="sn-9-3" class="sn-cb" aria-label="Afficher la note"><span class="sidenote sn-terme"><strong class="sn-term">Entretien différé</strong> Réparations dues mais non faites : peinture, revêtement, fissure à injecter — mesuré au coût de remise en état.</span> : réparations dues et non faites, mesurées par le coût de remise en état. Sa portion <strong>incurable</strong> se répartit entre les <strong>composantes à courte durée de vie</strong> (CT) — toiture, fenêtres, chauffe-eau, cuisine, finitions, remplacées plusieurs fois pendant la vie du bâtiment — dépréciées chacune au ratio âge/vie utile, et les <strong>composantes à longue durée de vie</strong> (LT) — fondations, ossature, isolation, plomberie et électricité brutes — qui durent autant que le bâtiment et se déprécient sur la base résiduelle au ratio âge effectif / vie économique. La séance 10 détaille cette mécanique.</p>210<div class="box box-terrain" id="terrain-9-31"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-hardhat"/></svg></span><span class="box-label">une composante, une seule mesure</span><a class="box-anchor" href="#terrain-9-31" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-32">Une toiture de 13 ans sur une vie de 25 est une composante CT : coût neuf <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> 13/25. Si elle fuit et doit être refaite maintenant, elle devient de l'entretien différé : coût de remise en état — et elle <em>sort</em> de la liste des composantes CT. Une fissure de fondation à injecter est du curable (1 800 $ dans le cas du chapitre); le vieillissement général du béton, lui, est incurable et loge dans la base de longue durée. Règle d'or : chaque composante est mesurée une seule fois — curable, <em>ou</em> CT, <em>ou</em> base LT. Jamais deux.</p></div></div>211<h3 id="s-9-2-3" class="h-subsection"><span class="h-num">9.2.3</span><span class="h-text">Type 2 — la dépréciation fonctionnelle</span><a class="h-anchor" href="#s-9-2-3" aria-label="Lien vers cette section">#</a></h3>212<div class="box box-def" id="def-9-3"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-book"/></svg></span><span class="box-label">Définition 9.3</span><span class="box-title">— dépréciation fonctionnelle</span><a class="box-anchor" href="#def-9-3" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-33">Perte de valeur causée par des <strong>déficiences de conception</strong>, une <strong>inadéquation aux usages actuels</strong> ou une <strong>surqualité</strong> (<span class="motcle" data-term="superadéquation">superadéquation</span>) des améliorations. Le bâtiment n'est pas usé : il est <em>mal adapté</em> aux attentes du marché d'aujourd'hui.</p></div></div>213<p id="p9-34">La dépréciation fonctionnelle mesure un décalage entre le bâtiment et <em>son époque</em> : désuétude des équipements (mazout, fusibles), déficience de conception (plafonds bas, une seule salle de bain pour quatre chambres), suramélioration (finitions que le marché du secteur ne paie pas), évolution des préférences (cuisine fermée), inadéquation au marché (un entrepôt de 16 pi de hauteur libre là où la logistique veut 28 pi — le cas du TP3). Sa version <strong>curable</strong> regroupe les corrections rentables, par <em>ajout</em> (un échangeur d'air absent) ou par <em>substitution</em> (une cuisine de 1985 refaite); sa version <strong>incurable</strong> couvre la déficience qu'aucun budget raisonnable ne corrige (l'absence de garage quand le quartier en a) et la superadéquation, où le propriétaire a payé pour quelque chose que l'acheteur type refuse de rembourser. La mesure — excédent du coût de correction pour le curable, perte de valeur ou de loyer capitalisée pour l'incurable — est l'objet de la séance 11.</p>214<h3 id="s-9-2-4" class="h-subsection"><span class="h-num">9.2.4</span><span class="h-text">Type 3 — la dépréciation externe (économique)</span><a class="h-anchor" href="#s-9-2-4" aria-label="Lien vers cette section">#</a></h3>215<div class="box box-def" id="def-9-4"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-book"/></svg></span><span class="box-label">Définition 9.4</span><span class="box-title">— dépréciation externe</span><a class="box-anchor" href="#def-9-4" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-35">Perte de valeur causée par des <strong>facteurs extérieurs</strong> à la propriété, hors du contrôle du propriétaire : nuisances de localisation (autoroute, ligne à haute tension, contamination, quartier en déclin) ou facteurs économiques et réglementaires (récession locale, suroffre, zonage défavorable, hausse des taux). Elle est <strong class="c-rouge">toujours incurable</strong> : aucune intervention sur la propriété ne peut la corriger.</p></div></div>216<p id="p9-36">L'incurabilité de l'externe découle de sa définition même : le propriétaire peut refaire sa toiture ou ouvrir sa cuisine, mais il ne peut ni déplacer l'autoroute, ni relancer l'économie régionale, ni faire abroger un règlement. L'effet d'une nuisance décroît typiquement avec la distance — une maison à 50 m d'une artère élargie perd plusieurs points de valeur, celle à 400 m presque rien — et le niveau réel se mesure par paires de ventes (séance 12). L'externe n'est pas nécessairement <em>permanente</em> : une suroffre se résorbe, un chantier voisin s'achève. L'évaluateur la mesure <strong>à la date d'évaluation</strong>, sans spéculer sur sa disparition.</p>217<div class="box box-att" id="attention-9-37"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-warning"/></svg></span><span class="box-label">l'externe frappe aussi le terrain — il faut allouer</span><a class="box-anchor" href="#attention-9-37" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-38">La dépréciation externe affecte <strong>tant le terrain que le bâtiment</strong>. Or, dans la méthode du coût, le terrain est évalué séparément (au marché, donc déjà au prix « déprécié »). Pour éviter le double comptage, on isole la portion attribuable au bâtiment, avec un ratio calculé sur les valeurs <em>avant</em> dépréciation externe :</p>218<div class="eq"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>D</mi><mrow><mtext>externe (b</mtext><mover accent="true"><mtext>a</mtext><mo>ˆ</mo></mover><mtext>t.)</mtext></mrow></msub><mo>=</mo><msub><mi>D</mi><mtext>externe (totale)</mtext></msub><mo>×</mo><mfrac><msub><mi>V</mi><mi>B</mi></msub><mrow><msub><mi>V</mi><mi>B</mi></msub><mo>+</mo><msub><mi>V</mi><mi>T</mi></msub></mrow></mfrac><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">D_{\text{externe (bât.)}} = D_{\text{externe (totale)}} \times219\frac{V_B}{V_B + V_T}.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0385em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">externe (b</span><span class="mord katex-accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight">a</span></span><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord mtight">ˆ</span></span></span></span></span></span></span><span class="mord mtight">t.)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.0385em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">externe (totale)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.1963em;vertical-align:-0.836em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">.</span></span></span></span></span></div>220<p id="p9-39">Exemple : perte externe totale de <span class="num">50 000 $</span>, ratio bâtiment/propriété de 70 % <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> part du bâtiment <span class="num">35 000 $</span>; part du terrain <span class="num">15 000 $</span>, déjà reflétée dans la valeur marchande du terrain.</p></div></div>221<h3 id="s-9-2-5" class="h-subsection"><span class="h-num">9.2.5</span><span class="h-text">Les conditions d'additivité</span><a class="h-anchor" href="#s-9-2-5" aria-label="Lien vers cette section">#</a></h3>222<p id="p9-40">On ne peut additionner les composantes de la formule que si cinq conditions sont réunies : (1) chaque composante physique est comptée <strong>une fois</strong> (curable <em>ou</em> CT <em>ou</em> base LT); (2) les bases sont <strong>successives</strong> — la base de longue durée est le coût neuf <em>moins</em> le curable <em>moins</em> le coût neuf des composantes CT; (3) la fonctionnelle par substitution retire le coût « à neuf » de l'élément déjà déprécié physiquement; (4) l'externe est allouée au bâtiment seulement; (5) toutes les pertes sont en dollars de la date d'évaluation et rapportées au <strong>même</strong> coût neuf (reproduction <em>ou</em> remplacement). Ce « sablier des bases » est la discipline anti-double comptage de la ventilation et du TP3.</p>223<figure class="float-table" id="tab-ch09-tab-types"><figcaption><span class="fig-label">Tableau 9.1</span> Les trois types de dépréciation — tableau comparatif</figcaption><div class="table-wrap"><table class="tbl"><thead><tr><th>Critère</th><th>Physique</th><th>Fonctionnelle</th><th>Externe</th></tr></thead><tbody><tr><td>Cause</td><td>usure, temps, éléments, entretien</td><td>conception, adéquation, préférences</td><td>nuisances, conjoncture, réglementation</td></tr><tr><td>Origine</td><td>interne au bâtiment</td><td>interne au bâtiment</td><td>extérieure à la propriété</td></tr><tr><td>Curabilité</td><td>curable (entretien différé) et incurable (CT, LT)</td><td>curable (ajout, substitution) et incurable (déficience, suramélioration)</td><td>toujours incurable</td></tr><tr><td>Affecte</td><td>bâtiment</td><td>bâtiment</td><td>terrain + bâtiment (allouer)</td></tr><tr><td>Mesure</td><td>coût de remise en état; coût <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> âge/vie</td><td>excédent de coût; perte capitalisée</td><td>paires de ventes; capitalisation</td></tr><tr><td>Séance</td><td>10</td><td>11</td><td>12</td></tr></tbody></table></div></figure>224<h2 id="s-9-3" class="h-section"><span class="h-num">9.3</span><span class="h-text">Curabilité : le raisonnement économique</span><a class="h-anchor" href="#s-9-3" aria-label="Lien vers cette section">#</a></h2>225<div class="box box-def" id="def-9-5"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-book"/></svg></span><span class="box-label">Définition 9.5</span><span class="box-title">— curable et incurable</span><a class="box-anchor" href="#def-9-5" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-41">Une composante de dépréciation est <span class="motcle" data-term="curable">curable</span> lorsque sa correction est <strong>économiquement justifiée</strong> : le gain de valeur qu'elle procure est au moins égal à son coût. Elle est <span class="motcle" data-term="incurable">incurable</span> lorsque la correction n'est pas rentable — même si elle est techniquement possible.</p></div></div>226<div class="box box-form" id="form-9-3"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.3</span><span class="box-title">— critère de curabilité</span><a class="box-anchor" href="#form-9-3" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-4"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>G</mi><mi>N</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>V</mi><mo>−</mo><mi>C</mi><mo separator="true">,</mo><mspace width="2em"/><mtext>curable</mtext><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi mathvariant="normal">Δ</mi><mi>V</mi><mo>≥</mo><mi>C</mi><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><mi>C</mi></mfrac><mo>≥</mo><mn>1</mn><mo separator="true">,</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.4)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">GN = \Delta V - C, \qquad227\text{curable} \iff \Delta V \ge C \iff \frac{\Delta V}{C} \ge 1,\tag{9.4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">GN</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">curable</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7073em;vertical-align:-0.024em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span></span><span class="katex-tag"><span class="katex-strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.4</span></span><span class="mord">)</span></span></span></span></span></span></div>228<p id="p9-42">où <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">GN</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">GN</span></span></span></span> est le gain net ($), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">\Delta V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> la hausse de valeur marchande procurée par la correction (mesurée par paires de ventes, loyers capitalisés ou jugement documenté) et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> le coût complet de la correction dans le bâtiment existant (démolition, remise en état et imprévus compris). La dépréciation retenue est alors <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mi>min</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>C</mi><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D = \min(C, \Delta V)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">min</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mclose">)</span></span></span></span> : le coût de correction si la correction se fera, la perte de valeur sinon.</p></div></div>229<p id="p9-43">Le test n'a rien d'abstrait : c'est exactement le calcul que fait l'acheteur averti en visitant. Repeindre l'extérieur coûte <span class="num">6000 $</span> et augmente la valeur de <span class="num">10 000 $</span> : gain net de <span class="num">4000 $</span>, curable. Rehausser un sous-sol coûterait <span class="num">60 000 $</span> pour un gain de <span class="num">12 000 $</span> : perte nette de <span class="num">48 000 $</span>, incurable. Tout est <em>techniquement</em> corrigeable — la curabilité est un verdict de rentabilité, pas de faisabilité.</p>230<h3 id="s-9-3-1" class="h-subsection"><span class="h-num">9.3.1</span><span class="h-text">Courbes de coût et de bénéfice</span><a class="h-anchor" href="#s-9-3-1" aria-label="Lien vers cette section">#</a></h3>231<p id="p9-44">Le gain de valeur d'une rénovation est <strong>concave</strong> dans le budget : les premiers dollars rapportent le plus (peinture, poignées), les derniers presque rien (granit, électroménagers haut de gamme). Modélisons-le, pour la cuisine du cas du chapitre, par <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mtext> </mtext><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>C</mi><mi mathvariant="normal">/</mi><mi>B</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta V(C) = A\,(1 - e^{-C/B})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0715em;">C</span><span class="mord mtight">/</span><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> avec <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mn>48</mn><mtext> </mtext><mn>000</mn></mrow><annotation encoding="application/x-tex">A = 48\,000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">48</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span></span></span></span> et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mn>25</mn><mtext> </mtext><mn>000</mn></mrow><annotation encoding="application/x-tex">B = 25\,000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">25</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span></span></span></span> (hypothèse pédagogique). Le gain net <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>V</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>−</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">GN(C) = \Delta V(C) - C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">GN</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> est maximal là où <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msup><mi>V</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\Delta V&#x27;(C) = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, soit <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>C</mi><mo>∗</mo></msup><mo>=</mo><mi>B</mi><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mi mathvariant="normal">/</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mn>16</mn><mtext> </mtext><mn>304</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">C^\ast = B\ln(A/B) = 16\,304\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">16</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">304</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> pour un gain net de <span class="num">6696 $</span>; il s'annule au <strong>seuil</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi><mo>=</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\Delta V = C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span>, vers <span class="num">40 600 $</span>, au-delà duquel la correction cesse d'être curable <em>à la marge</em>. Le budget de <span class="num">28 000 $</span> retenu dans le cas donne <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi><mo>=</mo><mn>32</mn><mtext> </mtext><mn>339</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">\Delta V = 32\,339\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">32</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">339</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> : curable (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>N</mi><mo>=</mo><mo>+</mo><mn>4</mn><mtext> </mtext><mn>339</mn></mrow><annotation encoding="application/x-tex">GN = +4\,339</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">GN</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">+</span><span class="mord">4</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">339</span></span></span></span>), mais loin de l'optimum.</p>232<div class="center"><div class="tikz"><img src="/assets/fig/14db879b76e44c7e.svg" alt="Figure" width="391" height="222" style="width:min(100%,567px)" loading="lazy" decoding="async"></div></div>233<h3 id="s-9-3-2" class="h-subsection"><span class="h-num">9.3.2</span><span class="h-text">Sensibilité, arbre de décision et comparaison d'interventions</span><a class="h-anchor" href="#s-9-3-2" aria-label="Lien vers cette section">#</a></h3>234<p id="p9-45">Le verdict dépend de la <strong>qualité de la mesure</strong> de <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">\Delta V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span>. Pour la cuisine, il suffit que le gain réel soit 13 % plus faible que l'estimation pour que le gain net devienne négatif; une marge de sécurité (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi><mi mathvariant="normal">/</mi><mi>C</mi><mo>≥</mo><mn>1,2</mn></mrow><annotation encoding="application/x-tex">\Delta V / C \ge 1{,}2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">2</span></span></span></span>) protège contre l'optimisme des rénovateurs. Erreur fréquente : prendre le coût des travaux pour le gain de valeur — « j'ai mis 40 000 $ dans la cuisine, elle vaut 40 000 $ de plus ». Le marché ne rembourse pas les factures.</p>235<p id="p9-46">La décision suit deux questions successives. La première est <strong>technique</strong> : le défaut peut-il être corrigé physiquement? Si non (externe, structure), il est incurable par nature et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">D = \Delta V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span>. La seconde est <strong>économique</strong> : le gain couvre-t-il le coût? Si oui, curable et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">D = C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span>; sinon, incurable et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">D = \Delta V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span>. Dans le cas d'une substitution, on vérifie enfin la base : l'élément remplacé est-il déjà déprécié physiquement? Son coût « à neuf » est alors retiré du coût de correction.</p>236<figure class="float-table" id="tab-ch09-tab-renos"><figcaption><span class="fig-label">Tableau 9.2</span> Comparer plusieurs interventions sur le même bâtiment (hypothèses du cas)</figcaption><div class="table-wrap"><table class="tbl table-wide"><thead><tr><th>Intervention</th><th class="al-r">Coût <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span></th><th class="al-r">Gain <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">\Delta V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span></th><th class="al-r">Gain net</th><th class="al-r"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi><mi mathvariant="normal">/</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">\Delta V/C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span></th><th>Verdict</th></tr></thead><tbody><tr><td>Peinture intérieure complète</td><td class="al-r"><span class="num">6000 $</span></td><td class="al-r"><span class="num">10 000 $</span></td><td class="al-r"><span class="num">4000 $</span></td><td class="al-r">1,67</td><td>curable</td></tr><tr><td>Rénovation de la cuisine</td><td class="al-r"><span class="num">28 000 $</span></td><td class="al-r"><span class="num">32 339 $</span></td><td class="al-r"><span class="num">4339 $</span></td><td class="al-r">1,15</td><td>curable</td></tr><tr><td>Rénovation de la salle de bain du RDC</td><td class="al-r"><span class="num">16 000 $</span></td><td class="al-r"><span class="num">19 000 $</span></td><td class="al-r"><span class="num">3000 $</span></td><td class="al-r">1,19</td><td>curable</td></tr><tr><td>Ajout d'un garage attaché</td><td class="al-r"><span class="num">45 000 $</span></td><td class="al-r"><span class="num">15 000 $</span></td><td class="al-r"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span></span></span></span><span class="num">30 000 $</span></td><td class="al-r">0,33</td><td>incurable</td></tr><tr><td>Rehaussement du sous-sol</td><td class="al-r"><span class="num">60 000 $</span></td><td class="al-r"><span class="num">12 000 $</span></td><td class="al-r"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span></span></span></span><span class="num">48 000 $</span></td><td class="al-r">0,20</td><td>incurable</td></tr><tr><td>Thermopompe centrale</td><td class="al-r"><span class="num">9000 $</span></td><td class="al-r"><span class="num">8000 $</span></td><td class="al-r"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span></span></span></span><span class="num">1000 $</span></td><td class="al-r">0,89</td><td>incurable (limite)</td></tr></tbody></table></div></figure>237<div class="box box-ex" id="ex-9-2"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-bulb"/></svg></span><span class="box-label">Exemple 9.2</span><span class="box-title">— la cuisine de 1985 : physique et fonctionnelle</span><a class="box-anchor" href="#ex-9-2" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-47">Dans le coût de remplacement neuf du sujet, une cuisine équivalente vaut <span class="num">17 086 $</span> (part des indirects et du profit comprise); elle est déjà dépréciée physiquement à 100 % (41 ans, vie de 25). La refaire dans la maison existante coûte <span class="num">28 000 $</span> pour un gain estimé de <span class="num">32 339 $</span>. Test : curable. Fonctionnelle curable par substitution <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> coût de correction <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span></span></span></span> coût « à neuf » de l'élément <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mn>28</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mn>17</mn><mtext> </mtext><mn>086</mn><mo>=</mo><mn>10</mn><mtext> </mtext><mn>914</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">= 28\,000 - 17\,086 =23810\,914\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">28</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">17</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">086</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">914</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Total imputé à la cuisine : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>17</mn><mtext> </mtext><mn>086</mn><mo>+</mo><mn>10</mn><mtext> </mtext><mn>914</mn><mo>=</mo><mn>28</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">17\,086 + 10\,914 =23928\,000\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">17</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">086</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">914</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">28</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> — exactement ce que l'acheteur déduira pour la refaire. Le découpage ne change pas le total; il explique <em>pourquoi</em> : usure (17 086) et conception (10 914).</p></div></div>240<h2 id="s-9-4" class="h-section"><span class="h-num">9.4</span><span class="h-text">Âge effectif et durées de vie</span><a class="h-anchor" href="#s-9-4" aria-label="Lien vers cette section">#</a></h2>241<div class="box box-def" id="def-9-6"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-book"/></svg></span><span class="box-label">Définition 9.6</span><span class="box-title">— les durées de vie</span><a class="box-anchor" href="#def-9-6" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-48">L'<span class="motcle" data-term="âge chronologique">âge chronologique</span> est le nombre d'années écoulées depuis la construction (rôle, Registre foncier). L'<span class="motcle" data-term="âge effectif">âge effectif</span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">AE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span> est l'âge <strong>apparent</strong> du bâtiment compte tenu de son état, de son entretien et de ses rénovations. La <span class="motcle" data-term="vie physique">vie physique</span> est la durée pendant laquelle le bâtiment peut tenir debout; la <span class="motcle" data-term="vie économique totale">vie économique totale</span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">VE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span>, la durée pendant laquelle il <strong>contribue à la valeur</strong> — toujours inférieure ou égale à la vie physique; la <span class="motcle" data-term="vie économique restante">vie économique restante</span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">VER</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>, la contribution qui reste à courir.</p></div></div>242<div class="box box-form" id="form-9-4"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.4</span><span class="box-title">— relations fondamentales</span><a class="box-anchor" href="#form-9-4" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-5"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>V</mi><mi>E</mi><mo>=</mo><mi>A</mi><mi>E</mi><mo>+</mo><mi>V</mi><mi>E</mi><mi>R</mi><mo separator="true">,</mo><mspace width="2em"/><mi>d</mi><mo>=</mo><mfrac><mrow><mi>A</mi><mi>E</mi></mrow><mrow><mi>V</mi><mi>E</mi></mrow></mfrac><mo separator="true">,</mo><mspace width="2em"/><mi>D</mi><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mo>×</mo><mfrac><mrow><mi>A</mi><mi>E</mi></mrow><mrow><mi>V</mi><mi>E</mi></mrow></mfrac><mo separator="true">,</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.5)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">VE = AE + VER, \qquad d = \frac{AE}{VE}, \qquad D = C_N \times \frac{AE}{VE},\tag{9.5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span><span class="katex-tag"><span class="katex-strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.5</span></span><span class="mord">)</span></span></span></span></span></span></div>243<p id="p9-49">en années <em>effectives</em>. La deuxième relation <strong>suppose</strong> une dépréciation linéaire : chaque année effective consomme <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mi>V</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">1/VE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span> du coût neuf. C'est une hypothèse de travail, pas une loi.</p></div></div>244<p id="p9-50">Pourquoi la vie économique s'arrête-t-elle avant la vie physique? Parce que la valeur d'un bâtiment tient à son <em>utilité concurrentielle</em>, pas à son existence : une aile de motel des années 1960 peut durer trente ans de plus; économiquement, elle a cessé de contribuer le jour où le marché a préféré le terrain nu. La vie physique n'entre dans aucun calcul de valeur.</p>245<h3 id="s-9-4-1" class="h-subsection"><span class="h-num">9.4.1</span><span class="h-text">Construire un âge effectif défendable</span><a class="h-anchor" href="#s-9-4-1" aria-label="Lien vers cette section">#</a></h3>246<p id="p9-51">C'est l'âge <strong>effectif</strong> — jamais le chronologique — qui entre dans la méthode âge-vie<label for="sn-9-4" class="sn-toggle" title="Note marginale"><span class="sn-mark" aria-hidden="true">◆</span></label><input type="checkbox" id="sn-9-4" class="sn-cb" aria-label="Afficher la note"><span class="sidenote sn-repere">Le calendrier est un fait; l'âge effectif est un jugement documenté.</span>. Comment ce jugement se forme-t-il? Par l'inventaire daté des composantes et de leur dernier remplacement, puis par une règle explicite. Une règle simple est la pondération par le coût neuf :</p>247<div class="eq" id="eq-9-6"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>A</mi><msub><mi>E</mi><mrow><mtext>pond</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>r</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover></mrow></msub><mo>=</mo><mfrac><mrow><munder><mo>∑</mo><mi>i</mi></munder><mi>A</mi><msub><mi>E</mi><mi>i</mi></msub><mo>×</mo><msub><mi>C</mi><mrow><mi>N</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow><mrow><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>C</mi><mrow><mi>N</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow></mfrac><mo separator="true">,</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.6)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">AE_{\text{pondéré}} = \frac{\sum_i AE_i \times C_{N,i}}{\sum_i C_{N,i}},\tag{9.6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">pond</span><span class="mord katex-accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight">e</span></span><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mtight">r</span><span class="mord katex-accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight">e</span></span><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord mtight">ˊ</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.4254em;vertical-align:-0.9857em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4397em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.6897em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9857em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span><span class="katex-tag"><span class="katex-strut" style="height:2.4254em;vertical-align:-0.9857em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.6</span></span><span class="mord">)</span></span></span></span></span></span></div>248<p id="p9-52">où les composantes lourdes (structure) pèsent davantage que les finitions. Pour le sujet du chapitre — toiture de 2013, fenêtres de 2009, chauffe-eau de 2021, cuisine et salle de bain du rez-de-chaussée d'origine (1985), sous-sol aménagé en 2006, finitions de 2015, structure jugée à 30 ans — la pondération donne 25,0 ans; l'évaluateur retient <strong>30 ans</strong> par prudence, et l'écrit : « âge effectif de 30 ans compte tenu du remplacement de la toiture (2013), des fenêtres (2009) et de l'aménagement du sous-sol (2006), et de la cuisine et de la salle de bain d'origine ». « Environ 30 ans » tout court est indéfendable en contre-interrogatoire; « âge effectif <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> âge chronologique » par défaut revient à nier l'entretien.</p>249<p id="p9-53">L'échelle de condition relie les deux âges : un bâtiment de 30 ans bien entretenu et rénové peut afficher un âge effectif de 18 ans (ratio 0,60); un entretien normal donne un ratio de 1,00; la négligence pousse le ratio à 1,40 et plus. Le ratio n'est pas constant dans la vie d'un bâtiment : il baisse après chaque rénovation et remonte quand l'entretien cesse — l'entretien différé, lui, s'accélère (infiltration, pourriture, structure). Les études empiriques confirment que le taux de dépréciation des maisons dépend du niveau d'entretien <span class="citation">(<a class="cite" href="/bibliographie/#knight1996" data-cite="knight1996">Knight et Sirmans, 1996</a>; <a class="cite" href="/bibliographie/#wilhelmsson2008" data-cite="wilhelmsson2008">Wilhelmsson, 2008</a>)</span> et distinguent dépréciation brute et nette d'entretien <span class="citation">(<a class="cite" href="/bibliographie/#harding2007" data-cite="harding2007">Harding et al., 2007</a>)</span>.</p>250<div class="widget" data-widget="age-vie"></div>251<h3 id="s-9-4-2" class="h-subsection"><span class="h-num">9.4.2</span><span class="h-text">La vie économique : le paramètre le plus lourd</span><a class="h-anchor" href="#s-9-4-2" aria-label="Lien vers cette section">#</a></h3>252<p id="p9-54">Pour le sujet (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>N</mi></msub><mo>=</mo><mn>317</mn><mtext> </mtext><mn>462</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">C_N = 317\,462\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>E</mi><mo>=</mo><mn>30</mn></mrow><annotation encoding="application/x-tex">AE = 30</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">30</span></span></span></span>), passer d'une vie économique de 50 à 100 ans <strong>divise la dépréciation par deux</strong> : 190 477, 158 731, 126 985 et 95 239 $ pour <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi><mo>=</mo><mn>50</mn></mrow><annotation encoding="application/x-tex">VE = 50</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">50</span></span></span></span>, 60, 75 et 100 ans. Le choix de <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">VE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span> pèse plus que tout raffinement du calcul — d'où la règle de l'AoRE : la vie économique doit être <strong>tirée du marché</strong> (extraction, section <a class="xref" href="#s-9-5">9.5</a>), pas d'une table.</p>253<figure class="float-table" id="tab-ch09-tab-durees"><figcaption><span class="fig-label">Tableau 9.3</span> Durées de vie typiques par type de construction (années, ordres de grandeur) — sources : Marshall &amp; Swift, Altus, pratique québécoise</figcaption><div class="table-wrap"><table class="tbl"><thead><tr><th>Type</th><th>Structure</th><th class="al-c">Vie physique</th><th class="al-c">Vie économique</th></tr></thead><tbody><tr><td>Résidentiel unifamilial</td><td>bois</td><td class="al-c">60–100</td><td class="al-c">50–75</td></tr><tr><td>Résidentiel multilogements</td><td>bois / béton</td><td class="al-c">55–75</td><td class="al-c">45–65</td></tr><tr><td>Commercial (bureaux)</td><td>acier / béton</td><td class="al-c">55–75</td><td class="al-c">40–60</td></tr><tr><td>Commercial (détail)</td><td>acier</td><td class="al-c">45–65</td><td class="al-c">35–55</td></tr><tr><td>Industriel léger</td><td>acier préfabriqué</td><td class="al-c">40–55</td><td class="al-c">30–50</td></tr><tr><td>Institutionnel</td><td>béton armé</td><td class="al-c">70–100</td><td class="al-c">60–80</td></tr><tr><td>Entrepôt</td><td>acier préfabriqué</td><td class="al-c">40–55</td><td class="al-c">30–45</td></tr><tr><td>Patrimonial</td><td>maçonnerie</td><td class="al-c">100 et plus</td><td class="al-c">60–100 et plus</td></tr></tbody></table></div></figure>254<p id="p9-55">Ces fourchettes ne sont que des points de départ. La vie s'allonge quand l'entretien est préventif, quand la construction d'origine est de qualité, quand la conception absorbe les changements d'usage; elle se raccourcit sous la négligence, l'usage intensif, l'humidité, une conception datée — et le climat québécois tire les durées physiques vers le bas des fourchettes. Mais l'<strong>utilisation optimale</strong> peut tout arrêter : si le terrain vaut davantage sous un autre usage, la vie économique s'achève aujourd'hui.</p>255<div class="box box-qc" id="quebec-9-56"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-map"/></svg></span><span class="box-label">pourquoi 75 ans pour un plain-pied de Gatineau</span><a class="box-anchor" href="#quebec-9-56" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-57">Nous retenons pour le sujet la borne haute de la fourchette résidentielle. Les plain-pieds des années 1950–1960 de Gatineau se vendent encore, en 2026, avec un bâtiment valant 60 à 65 % du neuf (section <a class="xref" href="#s-9-5">9.5</a>); une vie économique de 50 ans les aurait déclarés sans valeur. L'extraction du marché suggère même une vie économique implicite bien plus longue : 75 ans reste prudent.</p></div></div>256<div class="box box-ex" id="ex-9-3"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-bulb"/></svg></span><span class="box-label">Exemple 9.3</span><span class="box-title">— trois cas numériques</span><a class="box-anchor" href="#ex-9-3" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-58"><strong>Cas 1.</strong> Immeuble commercial construit en 1996, visité en 2026, bien entretenu, rénové en 2016; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi><mo>=</mo><mn>50</mn></mrow><annotation encoding="application/x-tex">VE = 50</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">50</span></span></span></span> ans, vie physique 65 ans, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>E</mi><mo>=</mo><mn>22</mn></mrow><annotation encoding="application/x-tex">AE = 22</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">22</span></span></span></span> ans. Âge chronologique <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2026</mn><mo>−</mo><mn>1996</mn><mo>=</mo><mn>30</mn></mrow><annotation encoding="application/x-tex">2026 - 1996 = 30</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2026</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1996</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">30</span></span></span></span> ans; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi><mi>R</mi><mo>=</mo><mn>50</mn><mo>−</mo><mn>22</mn><mo>=</mo><mn>28</mn></mrow><annotation encoding="application/x-tex">VER = 50 - 22 = 28</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">50</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">22</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">28</span></span></span></span> ans; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>=</mo><mn>22</mn><mi mathvariant="normal">/</mi><mn>50</mn><mo>=</mo><mn>44</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">d = 22/50 = 44\,\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">22/50</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">44</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span></span></span></span>. La vie physique n'intervient pas.  <strong>Cas 2.</strong> Maison de Gatineau : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>N</mi></msub><mo>=</mo><mn>420</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">C_N = 420\,000\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">420</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>, construite en 2002 (24 ans), entretien moyen : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>E</mi><mo>=</mo><mn>26</mn></mrow><annotation encoding="application/x-tex">AE = 26</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">26</span></span></span></span> ans, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi><mo>=</mo><mn>55</mn></mrow><annotation encoding="application/x-tex">VE = 55</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">55</span></span></span></span>, terrain <span class="num">150 000 $</span>. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mn>420</mn><mtext> </mtext><mn>000</mn><mo>×</mo><mn>26</mn><mi mathvariant="normal">/</mi><mn>55</mn><mo>=</mo><mn>198</mn><mtext> </mtext><mn>545</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D = 420\,000 \times 26/55 = 198\,545\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">420</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">26/55</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">198</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">545</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (47,3 %); <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>B</mi></msub><mo>=</mo><mn>221</mn><mtext> </mtext><mn>455</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">V_B = 221\,455\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">221</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">455</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mn>371</mn><mtext> </mtext><mn>455</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">V = 371\,455\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">371</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">455</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. L'âge effectif dépasse ici le chronologique : l'entretien seulement moyen a vieilli la maison plus vite que le calendrier.  <strong>Cas 3.</strong> Même maison, entretien différé de <span class="num">32 000 $</span> (toiture, peinture, fenêtres). Âge-vie modifiée : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mn>32</mn><mtext> </mtext><mn>000</mn><mo>+</mo><mo stretchy="false">(</mo><mn>420</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mn>32</mn><mtext> </mtext><mn>000</mn><mo stretchy="false">)</mo><mo>×</mo><mn>26</mn><mi mathvariant="normal">/</mi><mn>55</mn><mo>=</mo><mn>32</mn><mtext> </mtext><mn>000</mn><mo>+</mo><mn>183</mn><mtext> </mtext><mn>418</mn><mo>=</mo><mn>215</mn><mtext> </mtext><mn>418</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D = 32\,000 + (420\,000 - 32\,000)257\times 26/55 = 32\,000 + 183\,418 = 215\,418\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">32</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">420</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">32</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">26/55</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">32</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">183</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">418</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">215</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">418</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (51,3 %); <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mn>354</mn><mtext> </mtext><mn>582</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">V = 354\,582\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">354</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">582</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. La méthode modifiée capte <span class="num">16 873 $</span> de plus : le différé est déduit en entier, puis le ratio s'applique au solde. Conservez les fractions exactes (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>26</mn><mi mathvariant="normal">/</mi><mn>55</mn></mrow><annotation encoding="application/x-tex">26/55</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">26/55</span></span></span></span>) jusqu'au bout.</p></div></div>258<h2 id="s-9-5" class="h-section"><span class="h-num">9.5</span><span class="h-text">Les quatre méthodes de mesure</span><a class="h-anchor" href="#s-9-5" aria-label="Lien vers cette section">#</a></h2>259<p id="p9-59">Quatre méthodes principales permettent de quantifier la dépréciation. Les méthodes <strong>âge-vie</strong> raisonnent <em>du haut vers le bas</em> — une hypothèse globale de vieillissement appliquée au coût neuf; la <strong>ventilation</strong> raisonne <em>du bas vers le haut</em> — chaque cause identifiée, mesurée et additionnée; l'<strong>extraction du marché</strong> ne raisonne pas sur le bâtiment — elle observe ce que le marché déduit du coût neuf dans les ventes réelles. Aucune n'est supérieure en soi; un rapport solide en combine au moins deux. Nous les appliquons toutes au sujet du chapitre, le 99, rue de Magog (section <a class="xref" href="#s-9-7">9.7</a>), dont le coût neuf est <span class="num">317 462 $</span> et le terrain <span class="num">215 000 $</span>.</p>260<h3 id="s-9-5-1" class="h-subsection"><span class="h-num">9.5.1</span><span class="h-text">Méthode 1 — âge-vie (linéaire)</span><a class="h-anchor" href="#s-9-5-1" aria-label="Lien vers cette section">#</a></h3>261<div class="box box-form" id="form-9-5"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.5</span><span class="box-title">— méthode âge-vie</span><a class="box-anchor" href="#form-9-5" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-7"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>D</mi><mtext>  </mtext><mo>=</mo><mtext>  </mtext><mfrac><mrow><mi>A</mi><mi>E</mi></mrow><mrow><mi>V</mi><mi>E</mi></mrow></mfrac><mo>×</mo><msub><mi>C</mi><mi>N</mi></msub><mo separator="true">,</mo><mspace width="2em"/><msub><mi>V</mi><mi>B</mi></msub><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mrow><mo fence="true">(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>A</mi><mi>E</mi></mrow><mrow><mi>V</mi><mi>E</mi></mrow></mfrac><mo fence="true">)</mo></mrow><mi mathvariant="normal">.</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.7)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">D \;=\; \frac{AE}{VE} \times C_N, \qquad V_B = C_N\left(1 - \frac{AE}{VE}\right).\tag{9.7}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">.</span></span><span class="katex-tag"><span class="katex-strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.7</span></span><span class="mord">)</span></span></span></span></span></span></div></div></div>262<div class="box box-ex" id="ex-9-4"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-bulb"/></svg></span><span class="box-label">Exemple 9.4</span><span class="box-title">— âge-vie sur le sujet</span><a class="box-anchor" href="#ex-9-4" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-60"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mn>317</mn><mtext> </mtext><mn>462</mn><mo>×</mo><mn>30</mn><mi mathvariant="normal">/</mi><mn>75</mn><mo>=</mo><mn>126</mn><mtext> </mtext><mn>985</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D = 317\,462 \times 30/75 = 126\,985\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">30/75</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">126</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">985</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (40,0 %); <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mn>215</mn><mtext> </mtext><mn>000</mn><mo>+</mo><mn>317</mn><mtext> </mtext><mn>462</mn><mo>−</mo><mn>126</mn><mtext> </mtext><mn>985</mn><mo>=</mo><mn>405</mn><mtext> </mtext><mn>477</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">V = 215\,000 + 317\,462 - 126\,985 = 405\,477\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">215</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">126</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">985</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">405</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">477</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Avec l'âge chronologique (41 ans) on obtiendrait <span class="num">173 546 $</span> (54,7 %), soit <span class="num">47 000 $</span> de trop. Trois bâtiments de même coût neuf et d'âges effectifs 15, 30 et 45 ans ont des valeurs contributives de 253 970, 190 477 et 126 985 $ : chaque année d'âge effectif « coûte » <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>N</mi></msub><mi mathvariant="normal">/</mi><mi>V</mi><mi>E</mi><mo>=</mo><mn>4233</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">C_N/VE =2634233\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">4233</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>.</p></div></div>264<div class="box box-att" id="attention-9-61"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-warning"/></svg></span><span class="box-label">les limites de l'âge-vie</span><a class="box-anchor" href="#attention-9-61" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-62">Deux données, une division, un résultat — mais deux hypothèses héroïques : que la dépréciation s'accumule au même rythme chaque année, et qu'un seul ratio capture l'usure, la conception et l'environnement. Son vrai rôle est celui d'un <em>ordre de grandeur</em>. Utilisée seule, elle expose l'évaluateur à la question fatale : « Sur quoi repose votre vie économique de 75 ans, et pourquoi une droite? »</p></div></div>265<h3 id="s-9-5-2" class="h-subsection"><span class="h-num">9.5.2</span><span class="h-text">Méthode 2 — âge-vie modifiée</span><a class="h-anchor" href="#s-9-5-2" aria-label="Lien vers cette section">#</a></h3>266<div class="box box-form" id="form-9-6"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.6</span><span class="box-title">— méthode âge-vie modifiée</span><a class="box-anchor" href="#form-9-6" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-8"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>D</mi><mtext>  </mtext><mo>=</mo><mtext>  </mtext><msub><mi>D</mi><mtext>curable</mtext></msub><mtext>  </mtext><mo>+</mo><mtext>  </mtext><mfrac><mrow><mi>A</mi><mi>E</mi></mrow><mrow><mi>V</mi><mi>E</mi></mrow></mfrac><mo>×</mo><mo fence="true" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>C</mi><mi>N</mi></msub><mo>−</mo><msub><mi>D</mi><mtext>curable</mtext></msub><mo fence="true" stretchy="true" minsize="1.2em" maxsize="1.2em">)</mo><mi mathvariant="normal">.</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.8)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">D \;=\; D_{\text{curable}} \;+\; \frac{AE}{VE} \times \bigl(C_N - D_{\text{curable}}\bigr).\tag{9.8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">curable</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mopen"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">curable</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing size1">)</span></span><span class="mord">.</span></span><span class="katex-tag"><span class="katex-strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.8</span></span><span class="mord">)</span></span></span></span></span></span></div>267<p id="p9-63">On mesure d'abord <strong>directement</strong> l'entretien différé (coût de remise en état), puis on applique le ratio âge-vie au <strong>solde</strong> du coût neuf.</p></div></div>268<div class="box box-ex" id="ex-9-5"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-bulb"/></svg></span><span class="box-label">Exemple 9.5</span><span class="box-title">— âge-vie modifiée sur le sujet</span><a class="box-anchor" href="#ex-9-5" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-64">Entretien différé relevé : <span class="num">18 100 $</span> (revêtement de vinyle 9 800, fissure de fondation 1 800, resurfaçage de l'allée 4 000, peinture extérieure 2 500 — hypothèses d'inspection). <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mn>18</mn><mtext> </mtext><mn>100</mn><mo>+</mo><mo stretchy="false">(</mo><mn>317</mn><mtext> </mtext><mn>462</mn><mo>−</mo><mn>18</mn><mtext> </mtext><mn>100</mn><mo stretchy="false">)</mo><mo>×</mo><mn>30</mn><mi mathvariant="normal">/</mi><mn>75</mn><mo>=</mo><mn>18</mn><mtext> </mtext><mn>100</mn><mo>+</mo><mn>119</mn><mtext> </mtext><mn>745</mn><mo>=</mo><mn>137</mn><mtext> </mtext><mn>845</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D = 18\,100 + (317\,462 - 18\,100) \times 30/75 = 18\,100 + 119\,745 =269137\,845\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">100</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">30/75</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">119</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">745</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">137</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">845</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (43,4 %); <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mn>394</mn><mtext> </mtext><mn>617</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">V = 394\,617\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">394</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">617</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. La version modifiée capte <span class="num">10 860 $</span> de plus que l'âge-vie simple. Si l'on appliquait le ratio à 317 462 <em>puis</em> qu'on ajoutait 18 100, les composantes réparées seraient dépréciées deux fois : l'assiette est le solde.</p></div></div>270<h3 id="s-9-5-3" class="h-subsection"><span class="h-num">9.5.3</span><span class="h-text">Méthode 3 — ventilation (<em>breakdown method</em>)</span><a class="h-anchor" href="#s-9-5-3" aria-label="Lien vers cette section">#</a></h3>271<div class="box box-form" id="form-9-7"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.7</span><span class="box-title">— ventilation</span><a class="box-anchor" href="#form-9-7" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-9"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>D</mi><mo>=</mo><msub><mi>D</mi><mtext>cur.</mtext></msub><mo>+</mo><munder><mo>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>C</mi><mi>T</mi></mrow></munder><msub><mi>C</mi><mrow><mi>N</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mtext> </mtext><mi>min</mi><mo>⁡</mo><mtext> ⁣</mtext><mrow><mo fence="true">(</mo><mfrac><mrow><mi>A</mi><msub><mi>E</mi><mi>i</mi></msub></mrow><mrow><mi>V</mi><msub><mi>U</mi><mi>i</mi></msub></mrow></mfrac><mo separator="true">,</mo><mn>1</mn><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>B</mi><mrow><mi>L</mi><mi>T</mi></mrow></msub><mtext> </mtext><mfrac><mrow><mi>A</mi><msub><mi>E</mi><mrow><mi>L</mi><mi>T</mi></mrow></msub></mrow><mrow><mi>V</mi><mi>E</mi></mrow></mfrac><mo>+</mo><msub><mi>D</mi><mtext>fonct.</mtext></msub><mo>+</mo><msub><mi>D</mi><mtext>ext.</mtext></msub><mo separator="true">,</mo><mspace width="2em"/><msub><mi>B</mi><mrow><mi>L</mi><mi>T</mi></mrow></msub><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mo>−</mo><msub><mi>D</mi><mtext>cur.</mtext></msub><mo>−</mo><munder><mo>∑</mo><mrow><mi>C</mi><mi>T</mi></mrow></munder><msub><mi>C</mi><mrow><mi>N</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo separator="true">,</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.9)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">D = D_{\text{cur.}} + \sum_{i \in CT} C_{N,i}\,\min\!\left(\frac{AE_i}{VU_i}, 1\right)272  + B_{LT}\,\frac{AE_{LT}}{VE} + D_{\text{fonct.}} + D_{\text{ext.}},273\qquad274B_{LT} = C_N - D_{\text{cur.}} - \sum_{CT} C_{N,i},\tag{9.9}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">cur.</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.7717em;vertical-align:-1.3217em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8557em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.0715em;">C</span><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3217em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">fonct.</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">ext.</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">cur.</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.3443em;vertical-align:-1.2943em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8557em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0715em;">C</span><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2943em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span><span class="katex-tag"><span class="katex-strut" style="height:2.7717em;vertical-align:-1.3217em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.9</span></span><span class="mord">)</span></span></span></span></span></span></div>275<p id="p9-65">où <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mrow><mi>N</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{N,i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> est le coût neuf complet de la composante <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><msub><mi>E</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">AE_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><msub><mi>U</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">VU_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> son âge et sa vie utile, et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mrow><mi>L</mi><mi>T</mi></mrow></msub></mrow><annotation encoding="application/x-tex">B_{LT}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> la base résiduelle de longue durée.</p></div></div>276<p id="p9-66">La ventilation est la méthode la plus détaillée : six étages, trois unités de mesure (coût pour le curable, ratio pour la physique incurable, excédent ou perte pour la fonctionnelle, paires pour l'externe). Sa force n'est pas seulement la précision, c'est sa <em>capacité d'explication</em> : le lecteur du rapport voit chaque poste. Sa faiblesse est symétrique : chaque étage repose sur ses hypothèses, et les erreurs s'additionnent comme les composantes. Le tableau <a class="xref" href="#tab-ch09-tab-vent">9.4</a> donne la ventilation complète du sujet; sa construction est détaillée à la section <a class="xref" href="#s-9-7">9.7</a>.</p>277<figure class="float-table" id="tab-ch09-tab-vent"><figcaption><span class="fig-label">Tableau 9.4</span> Ventilation du 99, rue de Magog — composantes à courte durée de vie (coût neuf complet; âges : hypothèses d'inspection)</figcaption><div class="table-wrap"><table class="tbl table-wide"><thead><tr><th>Composante</th><th class="al-r">Coût neuf</th><th class="al-c">Âge</th><th class="al-c">Vie</th><th class="al-r">Ratio</th><th class="al-r">Dépréciation</th></tr></thead><tbody><tr><td>Toiture (bardeaux, soffites, gouttières)</td><td class="al-r"><span class="num">9278 $</span></td><td class="al-c">13</td><td class="al-c">25</td><td class="al-r">52 %</td><td class="al-r"><span class="num">4824 $</span></td></tr><tr><td>Fenêtres et portes extérieures</td><td class="al-r"><span class="num">15 447 $</span></td><td class="al-c">17</td><td class="al-c">30</td><td class="al-r">57 %</td><td class="al-r"><span class="num">8753 $</span></td></tr><tr><td>Chauffe-eau</td><td class="al-r"><span class="num">2482 $</span></td><td class="al-c">5</td><td class="al-c">12</td><td class="al-r">42 %</td><td class="al-r"><span class="num">1034 $</span></td></tr><tr><td>Plinthes électriques</td><td class="al-r"><span class="num">4356 $</span></td><td class="al-c">21</td><td class="al-c">25</td><td class="al-r">84 %</td><td class="al-r"><span class="num">3659 $</span></td></tr><tr><td>Thermopompe murale</td><td class="al-r"><span class="num">7090 $</span></td><td class="al-c">6</td><td class="al-c">15</td><td class="al-r">40 %</td><td class="al-r"><span class="num">2836 $</span></td></tr><tr><td>Cuisine (armoires, comptoirs, évier)</td><td class="al-r"><span class="num">17 086 $</span></td><td class="al-c">41</td><td class="al-c">25</td><td class="al-r">100 %</td><td class="al-r"><span class="num">17 086 $</span></td></tr><tr><td>Appareils de la salle de bain (RDC)</td><td class="al-r"><span class="num">4014 $</span></td><td class="al-c">41</td><td class="al-c">25</td><td class="al-r">100 %</td><td class="al-r"><span class="num">4014 $</span></td></tr><tr><td>Appareils de la salle de bain (sous-sol)</td><td class="al-r"><span class="num">5767 $</span></td><td class="al-c">20</td><td class="al-c">25</td><td class="al-r">80 %</td><td class="al-r"><span class="num">4613 $</span></td></tr><tr><td>Finitions intérieures du RDC</td><td class="al-r"><span class="num">24 730 $</span></td><td class="al-c">11</td><td class="al-c">20</td><td class="al-r">55 %</td><td class="al-r"><span class="num">13 602 $</span></td></tr><tr><td>Finitions intérieures du sous-sol</td><td class="al-r"><span class="num">31 895 $</span></td><td class="al-c">20</td><td class="al-c">25</td><td class="al-r">80 %</td><td class="al-r"><span class="num">25 516 $</span></td></tr><tr><td>Aménagements extérieurs (terrasse, clôture, remise)</td><td class="al-r"><span class="num">25 721 $</span></td><td class="al-c">12</td><td class="al-c">20</td><td class="al-r">60 %</td><td class="al-r"><span class="num">15 432 $</span></td></tr><tr class="rule-above"><td><strong>Total des composantes CT</strong></td><td class="al-r"><span class="num">147 864 $</span></td><td class="al-c"></td><td class="al-c"></td><td class="al-r"></td><td class="al-r"><span class="num">101 369 $</span></td></tr></tbody></table></div></figure>278<p id="p9-67">À ces <span class="num">101 369 $</span> s'ajoutent le curable (<span class="num">18 100 $</span>), la longue durée — base résiduelle <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>317</mn><mtext> </mtext><mn>462</mn><mo>−</mo><mn>18</mn><mtext> </mtext><mn>100</mn><mo>−</mo><mn>147</mn><mtext> </mtext><mn>864</mn><mo>−</mo><mn>5</mn><mtext> </mtext><mn>893</mn></mrow><annotation encoding="application/x-tex">317\,462 - 18\,100 - 147\,864 - 5\,893</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">147</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">864</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">893</span></span></span></span> (vinyle, allée et VRC, traités ailleurs) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mn>145</mn><mtext> </mtext><mn>605</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">= 145\,605\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">145</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">605</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo><mtext> </mtext><mn>30</mn><mi mathvariant="normal">/</mi><mn>75</mn><mo>=</mo><mn>58</mn><mtext> </mtext><mn>242</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">\times\ 30/75 =27958\,242\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">×</span><span class="mspace"> </span><span class="mord">30/75</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">58</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">242</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> — la fonctionnelle curable (cuisine 10 914 et ajout d'un échangeur d'air 1 525 : <span class="num">12 439 $</span>), la fonctionnelle incurable (absence de garage, <span class="num">15 000 $</span>) et l'externe (nulle) : total <strong><span class="num">202 800 $</span></strong>, soit 63,9 % du coût neuf, pour une valeur de <span class="num">329 662 $</span>.</p>280<div class="box box-att" id="attention-9-68"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-warning"/></svg></span><span class="box-label">première alerte</span><a class="box-anchor" href="#attention-9-68" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-69">Le sujet s'est vendu <span class="num">447 000 $</span>. Avec un terrain de <span class="num">215 000 $</span>, le marché n'a déduit que <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>317</mn><mtext> </mtext><mn>462</mn><mo>−</mo><mn>232</mn><mtext> </mtext><mn>000</mn><mo>=</mo><mn>85</mn><mtext> </mtext><mn>462</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">317\,462 - 232\,000 = 85\,462\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">232</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">85</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (26,9 %). La ventilation « de tables » déduit <span class="num">117 338 $</span> de plus que le marché. Pourquoi? Elle mesure ce qu'il <em>en coûterait</em> de ramener chaque composante à neuf, alors que le marché paie l'<em>utilité résiduelle</em> d'une cuisine de 1985 qui fonctionne; les vies utiles de tables sont des cycles de remplacement, pas des durées d'utilité; et en 2026, l'offre rare de plain-pieds fait absorber une partie de l'usure par les acheteurs. La ventilation <strong>explique</strong>; le marché fixe le <strong>niveau</strong> — toute estimation de dépréciation doit être tirée du marché.</p></div></div>281<h3 id="s-9-5-4" class="h-subsection"><span class="h-num">9.5.4</span><span class="h-text">Méthode 4 — extraction du marché</span><a class="h-anchor" href="#s-9-5-4" aria-label="Lien vers cette section">#</a></h3>282<div class="box box-form" id="form-9-8"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.8</span><span class="box-title">— extraction de la dépréciation du marché</span><a class="box-anchor" href="#form-9-8" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-10"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi>D</mi><mi>i</mi></msub><mo>=</mo><msub><mi>C</mi><mrow><mi>N</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo>−</mo><mo fence="true" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>P</mi><mi>i</mi></msub><mo>−</mo><msub><mi>V</mi><mrow><mi>T</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo fence="true" stretchy="true" minsize="1.2em" maxsize="1.2em">)</mo><mo separator="true">,</mo><mspace width="2em"/><msub><mi>d</mi><mi>i</mi></msub><mo>=</mo><mfrac><msub><mi>D</mi><mi>i</mi></msub><msub><mi>C</mi><mrow><mi>N</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mfrac><mo separator="true">,</mo><mspace width="2em"/><msub><mi>τ</mi><mi>i</mi></msub><mo>=</mo><mfrac><msub><mi>d</mi><mi>i</mi></msub><msub><mrow><mover accent="true"><mtext>a</mtext><mo>ˆ</mo></mover><mtext>ge</mtext></mrow><mi>i</mi></msub></mfrac><mo separator="true">,</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.10)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">D_i = C_{N,i} - \bigl(P_i - V_{T,i}\bigr), \qquad283d_i = \frac{D_i}{C_{N,i}}, \qquad284\tau_i = \frac{d_i}{\text{âge}_i},\tag{9.10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mopen"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing size1">)</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.3324em;vertical-align:-0.9721em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9721em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.3016em;vertical-align:-0.9301em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord text"><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˆ</span></span></span></span></span></span></span><span class="mord">ge</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9301em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span><span class="katex-tag"><span class="katex-strut" style="height:2.3435em;vertical-align:-0.9721em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.10</span></span><span class="mord">)</span></span></span></span></span></span></div>285<p id="p9-70">où <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> est le prix de vente du comparable <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>T</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">V_{T,i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> la valeur de son terrain à la date de vente et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mrow><mi>N</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{N,i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> son coût neuf au même devis; le taux annuel <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>τ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\tau_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (ou sa médiane) s'applique ensuite à l'âge du sujet.</p></div></div>286<p id="p9-71">La démarche comporte six étapes : sélectionner des ventes de bâtiments comparables; ajuster les prix à la date d'évaluation; estimer le terrain de chaque comparable; estimer son coût neuf avec le <em>même</em> devis que le sujet; extraire <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">D_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>τ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\tau_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> et examiner la dispersion; appliquer la médiane au sujet. Les hypothèses de comparabilité sont lourdes : même date, même type de coût neuf, terrains évalués de la même façon, états comparables — toute erreur sur le terrain ou le coût neuf se loge <strong>en entier</strong> dans <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">D_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>287<div class="box box-ex" id="ex-9-6"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-bulb"/></svg></span><span class="box-label">Exemple 9.6</span><span class="box-title">— extraction didactique sur quatre ventes</span><a class="box-anchor" href="#ex-9-6" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-72">Quatre ventes hypothétiques de 18 à 22 ans : prix 650 000, 580 000, 720 000 et 610 000 $; terrains 200 000, 180 000, 230 000 et 190 000 $; coûts neufs 600 000, 550 000, 630 000 et 570 000 $. Les ratios <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi mathvariant="normal">/</mi><msub><mi>C</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">D/C_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> sont de 25,0, 27,3, 22,2 et 26,3 % et les taux annuels de 1,25, 1,24, 1,23 et 1,25 % : cette <strong>convergence</strong> donne sa force probante à l'extraction. Le monde réel est bien plus bruyant.</p></div></div>288<figure class="float-table" id="tab-ch09-tab-extraction"><figcaption><span class="fig-label">Tableau 9.5</span> Extraction sur sept ventes réelles de plain-pieds à moins de 400 m du sujet (2026) — terrain <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> rôle <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> 1,19; coût neuf <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> 3 300 $/m<sup>2</sup> (devis du sujet)</figcaption><div class="table-wrap"><table class="tbl table-wide"><thead><tr><th>Vente (réelle)</th><th class="al-c">Âge</th><th class="al-r">Prix</th><th class="al-r">Terrain</th><th class="al-r">Coût neuf</th><th class="al-r"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi mathvariant="normal">/</mi><msub><mi>C</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">D/C_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></th><th class="al-r">Taux/an</th></tr></thead><tbody><tr><td>115, rue du Saguenay (1983)</td><td class="al-c">43</td><td class="al-r"><span class="num">375 000 $</span></td><td class="al-r"><span class="num">214 000 $</span></td><td class="al-r"><span class="num">326 042 $</span></td><td class="al-r">50,6 %</td><td class="al-r">1,18 %</td></tr><tr><td>64, rue de la Yamaska (1987)</td><td class="al-c">39</td><td class="al-r"><span class="num">500 000 $</span></td><td class="al-r"><span class="num">212 000 $</span></td><td class="al-r"><span class="num">326 042 $</span></td><td class="al-r">11,7 %</td><td class="al-r">0,30 %</td></tr><tr><td>25, rue de la Châteauguay (1983)</td><td class="al-c">43</td><td class="al-r"><span class="num">500 000 $</span></td><td class="al-r"><span class="num">246 000 $</span></td><td class="al-r"><span class="num">370 592 $</span></td><td class="al-r">31,5 %</td><td class="al-r">0,73 %</td></tr><tr><td>37, rue de la Châteauguay (1980)</td><td class="al-c">46</td><td class="al-r"><span class="num">510 000 $</span></td><td class="al-r"><span class="num">253 000 $</span></td><td class="al-r"><span class="num">337 592 $</span></td><td class="al-r">23,9 %</td><td class="al-r">0,52 %</td></tr><tr><td>117, rue Sainte-Rose (1979)</td><td class="al-c">47</td><td class="al-r"><span class="num">360 000 $</span></td><td class="al-r"><span class="num">215 000 $</span></td><td class="al-r"><span class="num">325 712 $</span></td><td class="al-r">55,5 %</td><td class="al-r">1,18 %</td></tr><tr><td>75, rue des Sables (1979)</td><td class="al-c">47</td><td class="al-r"><span class="num">415 000 $</span></td><td class="al-r"><span class="num">214 000 $</span></td><td class="al-r"><span class="num">328 682 $</span></td><td class="al-r">38,8 %</td><td class="al-r">0,83 %</td></tr><tr><td>74, rue de la Châteauguay (1983)</td><td class="al-c">43</td><td class="al-r"><span class="num">485 000 $</span></td><td class="al-r"><span class="num">220 000 $</span></td><td class="al-r"><span class="num">332 642 $</span></td><td class="al-r">20,3 %</td><td class="al-r">0,47 %</td></tr><tr class="rule-above"><td><strong>Médiane</strong></td><td class="al-c"></td><td class="al-r"></td><td class="al-r"></td><td class="al-r"></td><td class="al-r"><strong>31,5 %</strong></td><td class="al-r"><strong>0,73 %</strong></td></tr></tbody></table></div></figure>289<p id="p9-73">Les ratios réels vont de 11,7 à 55,5 % au même âge. Cette dispersion n'est pas un échec de la méthode : elle signifie que les comparables n'ont <strong>pas le même état</strong> — entretien, cuisine refaite ou non, sous-sol — exactement ce qu'annonçait la section <a class="xref" href="#s-9-1">9.1</a>. Le praticien documente chaque comparable, en écarte certains, pondère les autres; la <strong>médiane</strong> (0,73 %/an) résiste aux extrêmes là où la moyenne arithmétique (0,74) est un chiffre sans propriétaire. Appliquée au sujet : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mn>41</mn><mo>×</mo><mn>0,73</mn><mtext> </mtext><mi mathvariant="normal">%</mi><mo>×</mo><mn>317</mn><mtext> </mtext><mn>462</mn><mo>=</mo><mn>95</mn><mtext> </mtext><mn>231</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D = 41 \times 0{,}73\,\% \times 317\,462 = 95\,231\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">41</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">73</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">95</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">231</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (30,0 %), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mn>437</mn><mtext> </mtext><mn>231</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">V = 437\,231\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">437</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">231</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Un taux de 0,73 %/an correspond à une vie économique linéaire implicite de 137 ans : le marché de Gatineau déprécie les plain-pieds bien plus lentement que les tables.</p>290<h3 id="s-9-5-5" class="h-subsection"><span class="h-num">9.5.5</span><span class="h-text">Ce que disent 655 ventes réelles : le profil de dépréciation de Gatineau</span><a class="h-anchor" href="#s-9-5-5" aria-label="Lien vers cette section">#</a></h3>291<p id="p9-74">L'approche des <em>prix par millésime</em> <span class="citation">(<a class="cite" href="/bibliographie/#hulten1981" data-cite="hulten1981">Hulten et Wykoff, 1981</a>)</span> généralise l'extraction : au lieu de quelques comparables, on observe la valeur implicite du bâtiment de centaines de ventes à tous les âges. Nous l'avons appliquée aux 655 ventes de plain-pieds détachés de Gatineau enregistrées entre novembre 2025 et août 2026 (Registre foncier, via UQO Éval, <span class="citation">(<a class="cite" href="/bibliographie/#uqoeval2026" data-cite="uqoeval2026">Boucher, 2026</a>)</span>), jointes au rôle 2024-2026 : terrain au rôle <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> 1,19, valeur implicite du bâtiment <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> prix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span></span></span></span> terrain, rapportée à l'aire hors sol.</p>292<figure class="float-table" id="tab-ch09-tab-profil"><figcaption><span class="fig-label">Tableau 9.6</span> Profil empirique — valeur implicite du bâtiment par m<sup>2</sup> hors sol selon l'âge (655 ventes réelles, Gatineau, 2025-2026)</figcaption><div class="table-wrap"><table class="tbl table-wide"><thead><tr><th class="al-c">Âge</th><th class="al-r">n</th><th class="al-r">Bât./m<sup>2</sup> (méd.)</th><th class="al-r">Q1–Q3</th><th class="al-r">Part terrain</th><th class="al-r">Prix médian</th><th>Dépréciation relative</th></tr></thead><tbody><tr><td class="al-c">0–14 ans</td><td class="al-r">14</td><td class="al-r"><span class="num">3134 $</span></td><td class="al-r">2 812–3 544</td><td class="al-r">33 %</td><td class="al-r"><span class="num">609 000 $</span></td><td>0,0 %</td></tr><tr><td class="al-c">15–24 ans</td><td class="al-r">99</td><td class="al-r"><span class="num">2998 $</span></td><td class="al-r">2 673–3 322</td><td class="al-r">38 %</td><td class="al-r"><span class="num">605 000 $</span></td><td>4,3 %</td></tr><tr><td class="al-c">25–34 ans</td><td class="al-r">41</td><td class="al-r"><span class="num">2985 $</span></td><td class="al-r">2 540–3 442</td><td class="al-r">40 %</td><td class="al-r"><span class="num">564 000 $</span></td><td>4,7 %</td></tr><tr><td class="al-c">35–44 ans</td><td class="al-r">60</td><td class="al-r"><span class="num">2527 $</span></td><td class="al-r">2 146–2 971</td><td class="al-r">48 %</td><td class="al-r"><span class="num">500 000 $</span></td><td>19,3 %</td></tr><tr><td class="al-c">45–54 ans</td><td class="al-r">152</td><td class="al-r"><span class="num">2380 $</span></td><td class="al-r">1 967–2 739</td><td class="al-r">51 %</td><td class="al-r"><span class="num">475 000 $</span></td><td>24,0 %</td></tr><tr><td class="al-c">55–64 ans</td><td class="al-r">154</td><td class="al-r"><span class="num">2033 $</span></td><td class="al-r">1 618–2 386</td><td class="al-r">58 %</td><td class="al-r"><span class="num">448 000 $</span></td><td>35,1 %</td></tr><tr><td class="al-c">65–74 ans</td><td class="al-r">107</td><td class="al-r"><span class="num">1923 $</span></td><td class="al-r">1 372–2 264</td><td class="al-r">57 %</td><td class="al-r"><span class="num">440 000 $</span></td><td>38,6 %</td></tr><tr><td class="al-c">75 ans et plus</td><td class="al-r">28</td><td class="al-r"><span class="num">1860 $</span></td><td class="al-r">1 341–2 312</td><td class="al-r">53 %</td><td class="al-r"><span class="num">361 500 $</span></td><td>40,6 %</td></tr></tbody></table></div></figure>293<div class="center"><div class="tikz"><img src="/assets/fig/d6d690d7294d3f90.svg" alt="Figure" width="430" height="254" style="width:min(100%,623px)" loading="lazy" decoding="async"></div></div>294<p id="p9-75">La régression <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mi>B</mi></msub><mi mathvariant="normal">/</mi><msup><mtext>m</mtext><mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>−</mo><mi>λ</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\ln(V_B/\text{m}^2) = a - \lambda t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord text"><span class="mord">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mord mathnormal">t</span></span></span></span> donne <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1,03</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\lambda = 1{,}03\,\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">03</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span></span></span></span> par an (écart-type 0,07), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>a</mi></msup><mo>=</mo><mn>3</mn><mtext> </mtext><mn>739</mn></mrow><annotation encoding="application/x-tex">e^{a} = 3\,739</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">739</span></span></span></span> $/m², <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>=</mo><mn>0,27</mn></mrow><annotation encoding="application/x-tex">R^2295= 0{,}27</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">27</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>643</mn></mrow><annotation encoding="application/x-tex">n = 643</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">643</span></span></span></span> : une demi-vie de 67 ans. Les bâtiments de 65–75 ans valent encore 1 923 $/m² contre 3 134 pour les quasi neufs : 38,6 % de dépréciation relative, soit environ 0,6 % par an. La valeur du bâtiment décroît <strong>lentement</strong> et de façon <strong>non linéaire</strong> (plateau jusqu'à 35 ans, décrochage ensuite), et la part du terrain passe de 33 à 57 % : plus le bâtiment vieillit, plus la propriété « devient un terrain ». Le coût neuf du sujet (3 300 $/m²) dépasse de 5 % la valeur implicite des quasi neufs — cohérence rassurante.</p>296<div class="box box-rem" id="remarque-9-76"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-pen"/></svg></span><span class="box-label">les limites du profil</span><a class="box-anchor" href="#remarque-9-76" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-77">La qualité n'est pas observée (les millésimes récents sont plus grands et mieux équipés : biais de composition); les bâtiments démolis ne se vendent plus (biais de survie, <span class="citation">(<a class="cite" href="/bibliographie/#hulten1981" data-cite="hulten1981">Hulten et Wykoff, 1981</a>)</span>); le terrain est indexé par un ratio unique et l'aire du rôle ignore le sous-sol; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>=</mo><mn>0,27</mn></mrow><annotation encoding="application/x-tex">R^2 = 0{,}27</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">27</span></span></span></span> : l'âge explique le quart de la variance — l'état fait le reste. Ces résultats sont <em>calculés</em> sur des ventes réelles et valent pour ce marché, en 2026, avec ces limites.</p></div></div>297<figure class="float-table" id="tab-ch09-tab-methodes"><figcaption><span class="fig-label">Tableau 9.7</span> Comparaison des quatre méthodes de mesure de la dépréciation</figcaption><div class="table-wrap"><table class="tbl table-wide"><thead><tr><th>Critère</th><th>Âge-vie</th><th>Âge-vie mod.</th><th>Ventilation</th><th>Extraction</th></tr></thead><tbody><tr><td>Logique</td><td>une hypothèse globale</td><td>curable à part, ratio sur le solde</td><td>chaque cause mesurée et additionnée</td><td>observée dans les ventes</td></tr><tr><td>Données</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">AE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">VE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span></span></span></span> entretien différé</td><td>inventaire, vies, coûts</td><td>ventes, terrains, coûts neufs</td></tr><tr><td>Force</td><td>rapidité</td><td>traite le différé</td><td>explique, auditable</td><td>ancrée dans le marché</td></tr><tr><td>Faiblesse</td><td>aveugle aux causes, sensible à <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">VE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span></td><td>reste globale</td><td>hypothèses empilées</td><td>dispersion; dépend du terrain et du coût neuf</td></tr><tr><td>Sujet</td><td>40,0 %</td><td>43,4 %</td><td>63,9 %</td><td>30,0 %</td></tr><tr><td>Usage</td><td>ordre de grandeur</td><td>rapport courant</td><td>rapport détaillé, TP3</td><td>validation, calibration</td></tr></tbody></table></div></figure>298<div class="box box-terrain" id="terrain-9-78"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-hardhat"/></svg></span><span class="box-label">la combinaison gagnante</span><a class="box-anchor" href="#terrain-9-78" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-79">Extraire d'abord (le marché fixe le niveau), ventiler ensuite (les causes expliquent), vérifier par l'âge-vie (l'ordre de grandeur). Présenter une dépréciation ventilée <em>et</em> corroborée par l'extraction est la marque d'un rapport solide.</p></div></div>299<h2 id="s-9-6" class="h-section"><span class="h-num">9.6</span><span class="h-text">Modélisation mathématique de la dépréciation</span><a class="h-anchor" href="#s-9-6" aria-label="Lien vers cette section">#</a></h2>300<p id="p9-80">Les méthodes de la section précédente sont des cas particuliers de modèles : l'âge-vie est le modèle linéaire, l'extraction estime un taux moyen. Nommer les formes permet de <em>tester</em> les hypothèses au lieu de les subir.</p>301<div class="box box-form" id="form-9-9"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.9</span><span class="box-title">— trois formes fonctionnelles et leurs dérivées</span><a class="box-anchor" href="#form-9-9" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-11"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.25em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>lin</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>aire : </mtext></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mtext> </mtext><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>δ</mi><mi>t</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"/><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>δ</mi><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mi>δ</mi><mtext> </mtext><msub><mi>C</mi><mi>N</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>exponentiel : </mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mtext> </mtext><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>t</mi></mrow></msup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mi>λ</mi><mtext> </mtext><msub><mi>C</mi><mi>N</mi></msub><mtext> </mtext><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>t</mi></mrow></msup><mo>=</mo><mo>−</mo><mi>λ</mi><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>non lin</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>aire : </mtext></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mtext> </mtext><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>λ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msup><mi>t</mi><mi>γ</mi></msup></mrow></msup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><msup><mi>λ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>γ</mi><mtext> </mtext><msup><mi>t</mi><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext> </mtext><msub><mi>C</mi><mi>N</mi></msub><mtext> </mtext><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>λ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msup><mi>t</mi><mi>γ</mi></msup></mrow></msup></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(9.11)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\text{linéaire : } &amp; V(t) = C_N\,(1 - \delta t), \quad 0 \le t \le 1/\delta, &amp; \frac{dV}{dt} &amp;= -\delta\,C_N \\302\text{exponentiel : } &amp; V(t) = C_N\,e^{-\lambda t}, &amp; \frac{dV}{dt} &amp;= -\lambda\,C_N\,e^{-\lambda t} = -\lambda V(t) \\303\text{non linéaire : } &amp; V(t) = C_N\,e^{-\lambda&#x27; t^{\gamma}}, &amp; \frac{dV}{dt} &amp;= -\lambda&#x27;\gamma\,t^{\gamma-1}\,C_N\,e^{-\lambda&#x27; t^{\gamma}}\end{aligned}\tag{9.11}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:6.7723em;vertical-align:-3.1362em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6362em;"><span style="top:-5.6362em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord text"><span class="mord">lin</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">aire : </span></span></span></span><span style="top:-3.2787em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord text"><span class="mord">exponentiel : </span></span></span></span><span style="top:-0.9213em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord text"><span class="mord">non lin</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">aire : </span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1362em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6362em;"><span style="top:-5.6362em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1/</span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mpunct">,</span></span></span><span style="top:-3.2787em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-0.9213em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9925em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0556em;">γ</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1362em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6362em;"><span style="top:-5.6362em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.2787em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.9213em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1362em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6362em;"><span style="top:-5.6362em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.2787em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.2222em;">λV</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-0.9213em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0556em;">γ</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9925em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0556em;">γ</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1362em;"><span></span></span></span></span></span></span></span></span><span class="katex-tag"><span class="katex-strut" style="height:6.7723em;vertical-align:-3.1362em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.11</span></span><span class="mord">)</span></span></span></span></span></span></div>304<p id="p9-81"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span></span></span></span> et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> en an<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>λ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">\lambda&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> en an<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">^{-\gamma}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7713em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0556em;">γ</span></span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span></span></span></span> sans unité. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>V</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">\delta = 1/VE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span> est la fraction du coût neuf perdue par an; le domaine du modèle linéaire est borné (au-delà de <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mi>δ</mi></mrow><annotation encoding="application/x-tex">1/\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span></span></span></span> la valeur serait négative). <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> est un taux instantané : la valeur perd chaque année la même proportion de ce qui reste (demi-vie <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mn>2</mn><mi mathvariant="normal">/</mi><mi>λ</mi></mrow><annotation encoding="application/x-tex">\ln 2/\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2/</span><span class="mord mathnormal">λ</span></span></span></span>). <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\gamma &gt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> donne une perte lente au début puis accélérée; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\gamma &lt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> l'inverse; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\gamma = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> redonne l'exponentiel.</p></div></div>305<p id="p9-82">La dérivée est la <strong>dépréciation marginale</strong> : les dollars perdus pendant l'année <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>. Linéaire : constante; exponentiel : proportionnelle à la valeur restante, donc décroissante; non linéaire avec <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\gamma &gt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> : croissante d'abord (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>t</mi><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">t^{\gamma-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0556em;">γ</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>), puis la valeur restante l'emporte. Pour le sujet, avec <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>75</mn></mrow><annotation encoding="application/x-tex">\delta = 1/75</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/75</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>0,0103</mn></mrow><annotation encoding="application/x-tex">\lambda = 0{,}0103</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0103</span></span></span></span> (estimé) et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>=</mo><mn>1,5</mn></mrow><annotation encoding="application/x-tex">\gamma = 1{,}5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>λ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mn>0,00161</mn></mrow><annotation encoding="application/x-tex">\lambda&#x27; = 0{,}00161</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">00161</span></span></span></span> calé pour que <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mn>41</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(41)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mopen">(</span><span class="mord">41</span><span class="mclose">)</span></span></span></span> égale l'exponentiel), les pertes marginales à 41 ans sont de 4 233, 2 145 et 3 218 $/an. Les trois courbes coïncident à <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> et presque à <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>41</mn></mrow><annotation encoding="application/x-tex">t = 41</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">41</span></span></span></span>; elles divergent avant et après : la <strong>courbure</strong> est la vraie différence entre elles, et seule une série de ventes à tous les âges permet de l'estimer.</p>306<div class="center"><div class="tikz"><img src="/assets/fig/52dba8749135ccde.svg" alt="Figure" width="411" height="222" style="width:min(100%,597px)" loading="lazy" decoding="async"></div></div>307<div class="box box-att" id="attention-9-83"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-warning"/></svg></span><span class="box-label">une forme n'est pas une preuve</span><a class="box-anchor" href="#attention-9-83" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-84">À 41 ans, passer de <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>0,6</mn></mrow><annotation encoding="application/x-tex">\lambda = 0{,}6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">6</span></span></span></span> à <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1,5</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">1{,}5\,\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span></span></span></span> fait varier <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi mathvariant="normal">/</mi><msub><mi>C</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">V/C_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> de 78 à 54 % — 76 000 $ sur le sujet; la courbure déplace la valeur de dizaines de milliers de dollars aux âges intermédiaires. Choisir « exponentiel » ou « linéaire » est une hypothèse de modélisation; elle ne devient une connaissance qu'après <strong>estimation</strong> sur des ventes, <strong>validation</strong> (résidus, hors échantillon) et <strong>comparaison</strong> des formes. Nous avons estimé <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>=</mo><mn>0,27</mn></mrow><annotation encoding="application/x-tex">R^2 = 0{,}27</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">27</span></span></span></span>); nous n'avons <em>pas</em> estimé <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span></span></span></span> : la courbe verte reste une illustration.</p></div></div>308<h3 id="s-9-6-1" class="h-subsection"><span class="h-num">9.6.1</span><span class="h-text">Introduction au modèle hédonique</span><a class="h-anchor" href="#s-9-6-1" aria-label="Lien vers cette section">#</a></h3>309<div class="box box-form" id="form-9-10"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-formula"/></svg></span><span class="box-label">Formule 9.10</span><span class="box-title">— modèle hédonique du prix <span class="citation">(<a class="cite" href="/bibliographie/#rosen1974" data-cite="rosen1974">Rosen, 1974</a>)</span></span><a class="box-anchor" href="#form-9-10" aria-label="Lien">#</a></div><div class="box-body"><div class="eq" id="eq-9-12"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>P</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mtext> </mtext><msub><mrow><mover accent="true"><mtext>A</mtext><mo>ˆ</mo></mover><mtext>ge</mtext></mrow><mi>i</mi></msub><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><mtext> </mtext><msub><mtext>Surface</mtext><mi>i</mi></msub><mo>+</mo><msub><mi>β</mi><mn>3</mn></msub><mtext> </mtext><msub><mrow><mtext>Qualit</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover></mrow><mi>i</mi></msub><mo>+</mo><msub><mi>β</mi><mn>4</mn></msub><mtext> </mtext><msub><mrow><mtext>R</mtext><mover accent="true"><mtext>e</mtext><mo>ˊ</mo></mover><mtext>novation</mtext></mrow><mi>i</mi></msub><mo>+</mo><msup><mi>γ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mtext> </mtext><msub><mtext>Localisation</mtext><mi>i</mi></msub><mo>+</mo><msub><mi>ε</mi><mi>i</mi></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9.12)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\ln(P_i) = \alpha + \beta_1\,\text{Âge}_i + \beta_2\,\text{Surface}_i + \beta_3\,\text{Qualité}_i310 + \beta_4\,\text{Rénovation}_i + \gamma&#x27;\,\text{Localisation}_i + \varepsilon_i\tag{9.12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.1909em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord text"><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">A</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˆ</span></span></span></span></span></span></span><span class="mord">ge</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord text"><span class="mord">Surface</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord text"><span class="mord">Qualit</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord text"><span class="mord">R</span><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span><span class="mord">novation</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9963em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord text"><span class="mord">Localisation</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">ε</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="katex-tag"><span class="katex-strut" style="height:1.1968em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9.12</span></span><span class="mord">)</span></span></span></span></span></span></div>311<p id="p9-85">Le prix d'une propriété est la somme des prix <em>implicites</em> de ses caractéristiques; en logarithme, chaque coefficient se lit approximativement comme un effet en pourcentage du prix par unité de la caractéristique.</p></div></div>312<p id="p9-86">Le modèle <strong>contrôle</strong> les autres caractéristiques quand on mesure l'effet de l'âge, fournit des ajustements pour la méthode de comparaison et permet de tester la forme de la dépréciation (âge, âge<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, indicatrices de millésime). Il exige des ventes nombreuses et documentées, une spécification réfléchie (variables omises <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> coefficients biaisés) et des diagnostics <span class="citation">(<a class="cite" href="/bibliographie/#malpezzi1987" data-cite="malpezzi1987">Malpezzi et al., 1987</a>; <a class="cite" href="/bibliographie/#francke2017" data-cite="francke2017">Francke et van de Minne, 2017</a>)</span>.</p>313<div class="box box-ex" id="ex-9-7"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-bulb"/></svg></span><span class="box-label">Exemple 9.7</span><span class="box-title">— un modèle hédonique estimé sur les 655 ventes réelles</span><a class="box-anchor" href="#ex-9-7" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-87">Spécification : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mtext> </mtext><mrow><mover accent="true"><mtext>a</mtext><mo>ˆ</mo></mover><mtext>ge</mtext></mrow><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mtext>aire</mtext><mo stretchy="false">)</mo><mo>+</mo><msub><mi>β</mi><mn>3</mn></msub><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mtext>terrain</mtext><mo stretchy="false">)</mo><mo>+</mo><msub><mi>γ</mi><mn>1</mn></msub><mtext> </mtext><mi mathvariant="normal">Δ</mi><mtext>lat</mtext><mo>+</mo><msub><mi>γ</mi><mn>2</mn></msub><mtext> </mtext><mi mathvariant="normal">Δ</mi><mtext>long</mtext><mo>+</mo><mi>ε</mi></mrow><annotation encoding="application/x-tex">\ln(P) = \alpha + \beta_1\,\text{âge} + \beta_2\ln(\text{aire}) +314\beta_3\ln(\text{terrain}) + \gamma_1\,\Delta\text{lat} + \gamma_2\,\Delta\text{long} + \varepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord katex-accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˆ</span></span></span></span></span></span></span><span class="mord">ge</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord text"><span class="mord">aire</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord text"><span class="mord">terrain</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord text"><span class="mord">lat</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord text"><span class="mord">long</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span>, moindres carrés ordinaires, écarts-types entre parenthèses : âge <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>0,0054</mn></mrow><annotation encoding="application/x-tex">-0{,}0054</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0054</span></span></span></span> (0,0004); <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ln</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mop">ln</span></span></span></span> aire <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0,41</mn></mrow><annotation encoding="application/x-tex">0{,}41</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">41</span></span></span></span> (0,03); <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ln</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mop">ln</span></span></span></span> terrain <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0,03</mn></mrow><annotation encoding="application/x-tex">0{,}03</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">03</span></span></span></span> (0,01); longitude <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>0,69</mn></mrow><annotation encoding="application/x-tex">-0{,}69</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">69</span></span></span></span> (0,12); <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>=</mo><mn>0,59</mn></mrow><annotation encoding="application/x-tex">R^2 = 0{,}59</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">59</span></span></span></span>; écart-type des résidus 0,15. <em>Variables absentes</em> : qualité, rénovations, sous-sol, garage.  Lecture prudente : à surface, terrain et localisation donnés, le <strong>prix de la propriété</strong> baisse d'environ un demi pour cent par année d'âge; la longitude capte l'axe ouest–est de Gatineau (Aylmer plus cher); la moitié de la variance reste inexpliquée — l'état, invisible ici, en est la principale raison.</p></div></div>315<div class="box box-att" id="attention-9-88"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-warning"/></svg></span><span class="box-label">le coefficient de l'âge n'est pas la dépréciation</span><a class="box-anchor" href="#attention-9-88" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-89">Il porte sur le <strong>prix total</strong>, terrain compris — et le terrain ne se déprécie pas; rapporté au seul bâtiment (environ la moitié du prix), il correspondrait à 1 % par an, cohérent avec le <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> estimé, mais ce n'est qu'une approximation. Il mélange en outre terrain, qualité des millésimes, rénovations et survie. Distinguez toujours : la <em>corrélation</em> brute (les maisons plus vieilles sont moins chères, plus petites et mieux situées), l'<em>effet marginal conditionnel</em> (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>β</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\beta_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, à variables observées constantes), l'<em>effet causal</em> (inobservable sans contrôler l'état), la <em>dépréciation physique</em> (l'usure des composantes) et la <em>dépréciation économique totale</em> (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>N</mi></msub><mo>−</mo><msub><mi>V</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">C_N - V_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> à la date d'évaluation). Et ne présentez jamais un résultat non estimé comme empirique.</p></div></div>316<h2 id="s-9-7" class="h-section"><span class="h-num">9.7</span><span class="h-text">Étude de cas intégratrice : 99, rue de Magog, Gatineau</span><a class="h-anchor" href="#s-9-7" aria-label="Lien vers cette section">#</a></h2>317<p id="p9-90">Le cas assemble toute la chaîne sur un bâtiment <strong>réel</strong>. Sont réels : l'adresse, l'année, les aires, le rôle, la vente du 30 janvier 2026, la remise en vente d'août 2026, la fiche et les photos du courtier <span class="citation">(<a class="cite" href="/bibliographie/#centris10348445" data-cite="centris10348445">RE/MAX Vision, 2026</a>)</span>, les prix nus RSMeans 2026 T4 du module Quantity Survey d'UQO Éval, les ventes voisines et les terrains vacants <span class="citation">(<a class="cite" href="/bibliographie/#uqoeval2026" data-cite="uqoeval2026">Boucher, 2026</a>)</span>. Sont des <strong>hypothèses pédagogiques</strong> : les constats d'inspection (âges des composantes, entretien différé), les ordres de grandeur des ajouts et les gains de valeur des corrections.</p>318<h3 id="s-9-7-1" class="h-subsection"><span class="h-num">9.7.1</span><span class="h-text">L'immeuble, ses composantes, son coût neuf</span><a class="h-anchor" href="#s-9-7-1" aria-label="Lien vers cette section">#</a></h3>319<p id="p9-91">Plain-pied détaché de 1985 (41 ans à la date d'évaluation, le 12 novembre 2026), 96,2 m² hors sol (1 035 pi²), terrain de 545,9 m² clôturé, 11 pièces — 5 chambres dont 3 au sous-sol totalement aménagé, 2 salles de bains —, brique et vinyle, fenêtres à manivelle, plinthes électriques et thermopompe murale, allée asphaltée de 4 places, pas de garage. Rôle 2024-2026 : terrain <span class="num">180 400 $</span>, bâtiment <span class="num">198 500 $</span>, total <span class="num">378 900 $</span>. <strong>Vendu <span class="num">447 000 $</span> le 30 janvier 2026</strong> (Registre foncier); remis en vente le 17 août 2026 à <span class="num">455 000 $</span> (« bien entretenue, léger rajeunissement »). Les photos montrent une cuisine et une salle de bain d'origine, un sous-sol aménagé, un revêtement de vinyle fatigué à l'arrière.</p>320<p id="p9-92">Le coût de remplacement neuf part du <strong>Cas 1</strong> du module Quantity Survey d'UQO Éval — un bungalow de 117,5 m² à Gatineau, 65 lignes RSMeans 2026 T4 localisées : directs nus <span class="num">171 613 $</span>, directs et contingences <span class="num">222 762 $</span> — dont les quantités sont mises à l'échelle du sujet (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>96,2</mn><mi mathvariant="normal">/</mi><mn>117,5</mn><mo>=</mo><mn>0,8187</mn></mrow><annotation encoding="application/x-tex">96{,}2/117{,}5 = 0{,}8187</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">96</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">2/117</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">8187</span></span></span></span>), augmentées de l'aménagement du sous-sol (quantités hypothétiques <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> prix nus RSMeans du Cas 1 : <span class="num">36 323 $</span>) et des ajouts du sujet (thermopompe, allée, terrasse, clôture, remise : <span class="num">24 725 $</span>), puis passées dans la chaîne du cours (séance 8, TP2).</p>321<figure class="float-table" id="tab-ch09-tab-chaine"><figcaption><span class="fig-label">Tableau 9.8</span> Chaîne du coût de remplacement neuf du sujet (conventions du cours)</figcaption><div class="table-wrap"><table class="tbl"><thead><tr><th>Ligne</th><th>Base</th><th class="al-r">Montant</th><th class="al-r">Cumul</th></tr></thead><tbody><tr><td>Directs nus (Cas 1 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> 0,819 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span></span></span></span> sous-sol <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span></span></span></span> ajouts)</td><td>quantités <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> prix nus RSMeans</td><td class="al-r"><span class="num">201 494 $</span></td><td class="al-r"><span class="num">201 494 $</span></td></tr><tr><td>Conditions générales</td><td>8 % des nus</td><td class="al-r"><span class="num">16 120 $</span></td><td class="al-r"><span class="num">217 614 $</span></td></tr><tr><td>Frais généraux</td><td>7 % du cumul</td><td class="al-r"><span class="num">15 233 $</span></td><td class="al-r"><span class="num">232 847 $</span></td></tr><tr><td>Profit de l'entrepreneur général</td><td>8 % du cumul</td><td class="al-r"><span class="num">18 628 $</span></td><td class="al-r"><span class="num">251 475 $</span></td></tr><tr><td>Contingences</td><td>5 % des nus</td><td class="al-r"><span class="num">10 075 $</span></td><td class="al-r"><span class="num">261 550 $</span></td></tr><tr><td>Permis (Gatineau, 4,70 $/1 000 $)</td><td>sur D+C</td><td class="al-r"><span class="num">1229 $</span></td><td class="al-r"></td></tr><tr><td>GCR (unifamiliale visée)</td><td>2 656,68 $/unité</td><td class="al-r"><span class="num">2657 $</span></td><td class="al-r"></td></tr><tr><td>Intérêts intercalaires</td><td>D+C <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> 6,75 % <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> 6/12 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> ½</td><td class="al-r"><span class="num">4414 $</span></td><td class="al-r"></td></tr><tr><td>Honoraires professionnels</td><td>4 % des directs du cours</td><td class="al-r"><span class="num">10 059 $</span></td><td class="al-r"></td></tr><tr><td>Assurance chantier</td><td>0,8 % de D+C</td><td class="al-r"><span class="num">2092 $</span></td><td class="al-r"></td></tr><tr><td>Frais juridiques</td><td>forfait</td><td class="al-r"><span class="num">4000 $</span></td><td class="al-r"><span class="num">286 002 $</span></td></tr><tr><td>Profit entrepreneurial</td><td>11 % de (D+C <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span></span></span></span> indirects)</td><td class="al-r"><span class="num">31 460 $</span></td><td class="al-r"><span class="num">317 462 $</span></td></tr><tr class="rule-above"><td><strong>Coût de remplacement neuf complet</strong></td><td></td><td class="al-r"></td><td class="al-r"><strong><span class="num">317 462 $</span></strong></td></tr></tbody></table></div></figure>322<p id="p9-93">Soit 3 300 $/m² ou 307 $/pi² hors sol; le multiplicateur nus <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> complet (1,5755) est réparti au prorata sur chaque composante, de sorte que les composantes somment exactement au coût neuf. Contrôles : la chaîne appliquée au Cas 1 redonne ses <span class="num">222 762 $</span> de directs et contingences; les plain-pieds quasi neufs de Gatineau impliquent un bâtiment de 3 134 $/m², 5 % sous le coût neuf.</p>323<h3 id="s-9-7-2" class="h-subsection"><span class="h-num">9.7.2</span><span class="h-text">Terrain, âges, entretien différé</span><a class="h-anchor" href="#s-9-7-2" aria-label="Lien vers cette section">#</a></h3>324<p id="p9-94">Le rôle (conditions du marché au 1<sup>er</sup> juillet 2022) donne <span class="num">180 400 $</span>. Les huit terrains vacants vendus en 2025-2026 à moins de 5 km (impasse d'Amos, rue Harold) se sont vendus en médiane <strong>1,19 fois</strong> leur valeur au rôle — la méthode du ratio prix/rôle du TP1 — d'où <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><mn>180</mn><mtext> </mtext><mn>400</mn><mo>×</mo><mn>1,19</mn><mo>≈</mo><mn>215</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">V_T = 180\,400 \times 1{,}19 \approx 215\,000\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">180</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">400</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">19</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">215</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Contrôles : affectation (part médiane du terrain dans les ventes du secteur) <span class="num">212 376 $</span>; prix médian au m² des terrains (265 $/m², lots deux fois plus grands) <span class="num">144 408 $</span>, indicateur faible. Prendre la valeur au rôle telle quelle serait une erreur de <span class="num">35 000 $</span> qui irait <em>entièrement</em> dans la dépréciation extraite.</p>325<p id="p9-95">Âge chronologique : 41 ans; âge effectif pondéré : 25,0 ans; retenu : <strong>30 ans</strong>; structure jugée à 30 ans; vie économique totale <strong>75 ans</strong>; vie économique restante 45 ans. Entretien différé (hypothèses d'inspection) : remplacement du revêtement de vinyle 9 800, injection d'une fissure de fondation 1 800, resurfaçage de l'allée 4 000, peinture des soffites et portes 2 500 : <strong><span class="num">18 100 $</span></strong>, déduits en premier; les composantes qu'il couvre (vinyle, allée) sortent de la liste des composantes à courte durée.</p>326<h3 id="s-9-7-3" class="h-subsection"><span class="h-num">9.7.3</span><span class="h-text">Les trois catégories et les quatre méthodes</span><a class="h-anchor" href="#s-9-7-3" aria-label="Lien vers cette section">#</a></h3>327<p id="p9-96"><strong>Physique.</strong> Curable <span class="num">18 100 $</span>; courte durée <span class="num">101 369 $</span> (tableau <a class="xref" href="#tab-ch09-tab-vent">9.4</a>); longue durée : base résiduelle <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>317</mn><mtext> </mtext><mn>462</mn><mo>−</mo><mn>18</mn><mtext> </mtext><mn>100</mn><mo>−</mo><mn>147</mn><mtext> </mtext><mn>864</mn><mo>−</mo><mn>5</mn><mtext> </mtext><mn>893</mn><mo>=</mo><mn>145</mn><mtext> </mtext><mn>605</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">317\,462 -32818\,100 - 147\,864 - 5\,893 = 145\,605\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">147</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">864</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">893</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">145</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">605</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo><mtext> </mtext><mn>30</mn><mi mathvariant="normal">/</mi><mn>75</mn><mo>=</mo><mn>58</mn><mtext> </mtext><mn>242</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">\times\ 30/75 =32958\,242\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">×</span><span class="mspace"> </span><span class="mord">30/75</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">58</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">242</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Physique totale : <span class="num">177 711 $</span>, 56,0 % du coût neuf.</p>330<p id="p9-97"><strong>Fonctionnelle.</strong> Curable : cuisine (substitution) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>28</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mn>17</mn><mtext> </mtext><mn>086</mn><mo>=</mo><mn>10</mn><mtext> </mtext><mn>914</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">28\,000 - 17\,086331= 10\,914\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">28</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">17</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">086</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">914</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>; échangeur d'air absent (ajout) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mtext> </mtext><mn>200</mn><mo>−</mo><mn>2</mn><mtext> </mtext><mn>675</mn><mo>=</mo><mn>1525</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">4\,200 - 2\,675 =3321525\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">200</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">675</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">1525</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Incurable : absence de garage, perte de valeur par paires <span class="num">15 000 $</span> (hypothèse) inférieure au coût d'ajout de <span class="num">45 000 $</span>.</p>333<p id="p9-98"><strong>Externe.</strong> Rue résidentielle sans artère, ligne ni zone inondable; aucun écart par paires dans l'unité de voisinage : <span class="num">0 $</span>. Le TP3 (atelier-entrepôt de Hull, 15,7 % de locaux vides) illustrera le cas contraire.</p>334<figure class="float-table" id="tab-ch09-tab-quatre"><figcaption><span class="fig-label">Tableau 9.9</span> Les quatre méthodes appliquées au 99, rue de Magog</figcaption><div class="table-wrap"><table class="tbl"><thead><tr><th>Méthode</th><th class="al-r">Dépréciation</th><th class="al-r">% du coût neuf</th><th class="al-r">Valeur indiquée</th></tr></thead><tbody><tr><td>Âge-vie (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>317</mn><mtext> </mtext><mn>462</mn><mo>×</mo><mn>30</mn><mi mathvariant="normal">/</mi><mn>75</mn></mrow><annotation encoding="application/x-tex">317\,462 \times 30/75</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">30/75</span></span></span></span>)</td><td class="al-r"><span class="num">126 985 $</span></td><td class="al-r">40,0 %</td><td class="al-r"><span class="num">405 477 $</span></td></tr><tr><td>Âge-vie modifiée (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>18</mn><mtext> </mtext><mn>100</mn><mo>+</mo><mn>299</mn><mtext> </mtext><mn>362</mn><mo>×</mo><mn>30</mn><mi mathvariant="normal">/</mi><mn>75</mn></mrow><annotation encoding="application/x-tex">18\,100 + 299\,362 \times 30/75</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">299</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">362</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">30/75</span></span></span></span>)</td><td class="al-r"><span class="num">137 845 $</span></td><td class="al-r">43,4 %</td><td class="al-r"><span class="num">394 617 $</span></td></tr><tr><td>Ventilation (six étages)</td><td class="al-r"><span class="num">202 800 $</span></td><td class="al-r">63,9 %</td><td class="al-r"><span class="num">329 662 $</span></td></tr><tr><td>Extraction (7 ventes réelles, 0,73 %/an <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> 41 ans)</td><td class="al-r"><span class="num">95 231 $</span></td><td class="al-r">30,0 %</td><td class="al-r"><span class="num">437 231 $</span></td></tr><tr class="rule-above"><td>Moyenne pondérée mécanique (10/15/40/35)</td><td class="al-r"><span class="num">150 378 $</span></td><td class="al-r">47,4 %</td><td class="al-r"><span class="num">380 000 $</span></td></tr><tr><td>Dépréciation implicite de la vente du sujet</td><td class="al-r"><span class="num">85 462 $</span></td><td class="al-r">26,9 %</td><td class="al-r"><span class="num">447 000 $</span> (prix payé)</td></tr><tr><td><strong>Retenue : extraction, corroborée par la vente</strong></td><td class="al-r"><strong><span class="num">95 231 $</span></strong></td><td class="al-r"><strong>30,0 %</strong></td><td class="al-r"><strong><span class="num">435 000 $</span></strong></td></tr></tbody></table></div></figure>335<h3 id="s-9-7-4" class="h-subsection"><span class="h-num">9.7.4</span><span class="h-text">Réconciliation : raisonner, pas moyenner</span><a class="h-anchor" href="#s-9-7-4" aria-label="Lien vers cette section">#</a></h3>336<p id="p9-99">Une moyenne pondérée mécanique donnerait <span class="num">380 000 $</span>, soit 15 % sous le prix payé : un chiffre sans propriétaire, puisque les quatre méthodes ne mesurent pas la même chose. L'extraction (30,0 %) et la dépréciation implicite de la vente du sujet (26,9 %) se corroborent : le marché de ce secteur déduit entre un quart et un tiers du coût neuf pour un plain-pied de 1985. La ventilation « de tables » (63,9 %) et l'âge-vie (40,0 %) mesurent l'usure et l'obsolescence au <em>coût</em>; elles s'écartent du marché tendu de 2026, où les acheteurs absorbent une partie de l'usure. La ventilation n'est pas « fausse » : elle <strong>explique</strong> de quoi la dépréciation est faite — cuisine, finitions, structure — et chiffre ce qu'il en coûterait de tout remettre à neuf; son niveau se <strong>calibre</strong> sur le marché (vies économiques locales, ratio de longue durée) — c'est la discipline qu'imposera le TP3 dans un marché, lui, détendu.</p>337<div class="box box-ret" id="aretenir-9-100"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-star"/></svg></span><span class="box-label">conclusion du cas</span><a class="box-anchor" href="#aretenir-9-100" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-101"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mn>95</mn><mtext> </mtext><mn>231</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D = 95\,231\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">95</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">231</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (30,0 %) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mn>215</mn><mtext> </mtext><mn>000</mn><mo>+</mo><mn>317</mn><mtext> </mtext><mn>462</mn><mo>−</mo><mn>95</mn><mtext> </mtext><mn>231</mn><mo>=</mo><mn>437</mn><mtext> </mtext><mn>231</mn></mrow><annotation encoding="application/x-tex">V = 215\,000 + 317\,462 -33895\,231 = 437\,231</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">215</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">317</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">462</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">95</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">231</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">437</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">231</span></span></span></span>, arrondie à <strong><span class="num">435 000 $</span></strong> : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>2,7</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">-2{,}7\,\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">2</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">7</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span></span></span></span> par rapport au prix payé en janvier 2026 (<span class="num">447 000 $</span>) et <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>4,4</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">-4{,}4\,\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">4</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span></span></span></span> par rapport au prix demandé en août (<span class="num">455 000 $</span>). Les estimateurs indépendants d'UQO Éval encadrent le résultat : estimation hybride <span class="num">441 100 $</span>, méthode du coût calibrée <span class="num">497 400 $</span>, rôle indexé <span class="num">452 200 $</span>. Sensibilité : chaque dollar d'erreur sur le terrain est un dollar d'erreur sur la dépréciation extraite — de 16 % (terrain au rôle de 2022) à 41 % (terrain à <span class="num">260 000 $</span>). D'où l'ordre des opérations : terrain solide d'abord, extraction ensuite, ventilation pour expliquer.</p></div></div>339<h2 id="s-9-8" class="h-section"><span class="h-num">9.8</span><span class="h-text">Exercices</span><a class="h-anchor" href="#s-9-8" aria-label="Lien vers cette section">#</a></h2>340<p id="p9-102">Ces exercices sont formatifs et ne sont pas notés.</p>341<div class="box box-exo" id="exo-9-1"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-pencil"/></svg></span><span class="box-label">Exercice 9.1</span><span class="box-title">— durées de vie</span><a class="box-anchor" href="#exo-9-1" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-103">Un immeuble à bureaux (acier et béton) construit en 1998 est évalué en 2026. Il a été bien entretenu; les systèmes mécaniques ont été remplacés en 2019. L'évaluateur estime la vie économique totale à 55 ans et l'âge effectif à 20 ans. Calculez l'âge chronologique, la vie économique restante et le ratio de dépréciation âge-vie. Que deviendrait ce ratio avec l'âge chronologique?</p></div></div>342<details class="box box-sol"><summary><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-check"/></svg></span><span class="box-label">Solution</span><span class="sol-hint">Cliquer pour révéler</span></summary><div class="box-body"><p id="p9-104">Âge chronologique <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2026</mn><mo>−</mo><mn>1998</mn><mo>=</mo><mrow><mtext mathvariant="bold">28</mtext><mtext> </mtext><mtext mathvariant="bold">ans</mtext></mrow></mrow><annotation encoding="application/x-tex">2026 - 1998 = \textbf{28 ans}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2026</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1998</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord textbf">28 ans</span></span></span></span></span>; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi><mi>R</mi><mo>=</mo><mn>55</mn><mo>−</mo><mn>20</mn><mo>=</mo><mrow><mtext mathvariant="bold">35</mtext><mtext> </mtext><mtext mathvariant="bold">ans</mtext></mrow></mrow><annotation encoding="application/x-tex">VER = 55 - 20 =343\textbf{35 ans}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">55</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">20</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord textbf">35 ans</span></span></span></span></span>; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>=</mo><mn>20</mn><mi mathvariant="normal">/</mi><mn>55</mn><mo>=</mo><mrow><mtext mathvariant="bold">36,4</mtext><mtext> </mtext><mtext mathvariant="bold">%</mtext></mrow></mrow><annotation encoding="application/x-tex">d = 20/55 = \textbf{36,4\,\%}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">20/55</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord textbf">36,4</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord textbf">%</span></span></span></span></span>. Avec l'âge chronologique : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>28</mn><mi mathvariant="normal">/</mi><mn>55</mn><mo>=</mo><mn>50,9</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">28/55 = 50{,}9\,\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">28/55</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">50</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span></span></span></span> — 14,5 points de trop, soit l'entretien et les remplacements niés.</p></div></details>344<div class="box box-exo" id="exo-9-2"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-pencil"/></svg></span><span class="box-label">Exercice 9.2</span><span class="box-title">— âge-vie et âge-vie modifiée</span><a class="box-anchor" href="#exo-9-2" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-105">Plain-pied à Gatineau : coût neuf <span class="num">380 000 $</span>, âge effectif 24 ans, vie économique 70 ans, terrain <span class="num">195 000 $</span>. L'inspection relève <span class="num">14 000 $</span> d'entretien différé (gouttières, peinture, une porte-patio). Calculez la dépréciation et la valeur par les deux méthodes, puis expliquez l'écart.</p></div></div>345<details class="box box-sol"><summary><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-check"/></svg></span><span class="box-label">Solution</span><span class="sol-hint">Cliquer pour révéler</span></summary><div class="box-body"><p id="p9-106">Âge-vie : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mn>380</mn><mtext> </mtext><mn>000</mn><mo>×</mo><mn>24</mn><mi mathvariant="normal">/</mi><mn>70</mn><mo>=</mo><mn>130</mn><mtext> </mtext><mn>286</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D = 380\,000 \times 24/70 = 130\,286\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">380</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">24/70</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">130</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">286</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (34,3 %); <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mn>195</mn><mtext> </mtext><mn>000</mn><mo>+</mo><mn>380</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mn>130</mn><mtext> </mtext><mn>286</mn><mo>=</mo><mn>444</mn><mtext> </mtext><mn>714</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">V = 195\,000 + 380\,000 - 130\,286 = 444\,714\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">195</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">380</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">130</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">286</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">444</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">714</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Modifiée : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mn>14</mn><mtext> </mtext><mn>000</mn><mo>+</mo><mo stretchy="false">(</mo><mn>380</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mn>14</mn><mtext> </mtext><mn>000</mn><mo stretchy="false">)</mo><mo>×</mo><mn>24</mn><mi mathvariant="normal">/</mi><mn>70</mn><mo>=</mo><mn>14</mn><mtext> </mtext><mn>000</mn><mo>+</mo><mn>125</mn><mtext> </mtext><mn>486</mn><mo>=</mo><mn>139</mn><mtext> </mtext><mn>486</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D =34614\,000 + (380\,000 - 14\,000) \times 24/70 = 14\,000 + 125\,486 =347139\,486\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">14</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">380</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">14</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">24/70</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">14</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">125</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">486</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">139</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">486</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (36,7 %); <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mn>435</mn><mtext> </mtext><mn>514</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">V = 435\,514\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">435</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">514</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>. Écart <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn><mtext> </mtext><mn>200</mn><mtext> </mtext><mi mathvariant="normal">$</mi><mo>=</mo><mn>14</mn><mtext> </mtext><mn>000</mn><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>24</mn><mi mathvariant="normal">/</mi><mn>70</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">9\,200~\$ =34814\,000 \times (1 - 24/70)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">200</span><span class="mspace nobreak"> </span><span class="mord">$</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">14</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">24/70</span><span class="mclose">)</span></span></span></span> : le différé est déduit en entier au lieu d'être déprécié à 34,3 % seulement.</p></div></details>349<div class="box box-exo" id="exo-9-3"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-pencil"/></svg></span><span class="box-label">Exercice 9.3</span><span class="box-title">— curabilité et fonctionnelle par substitution</span><a class="box-anchor" href="#exo-9-3" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-107">Une maison de 1979 a un chauffage au mazout d'origine. Le remplacer par une thermopompe centrale coûte <span class="num">14 500 $</span> dans la maison existante; un système équivalent vaut <span class="num">9000 $</span> dans le coût neuf de remplacement, où il est déjà déprécié physiquement à 100 %. Les paires de ventes indiquent un gain de valeur de <span class="num">18 000 $</span>. La déficience est-elle curable? Quelle dépréciation fonctionnelle retenir?</p></div></div>350<details class="box box-sol"><summary><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-check"/></svg></span><span class="box-label">Solution</span><span class="sol-hint">Cliquer pour révéler</span></summary><div class="box-body"><p id="p9-108">Test : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi><mo>=</mo><mn>18</mn><mtext> </mtext><mn>000</mn><mo>≥</mo><mi>C</mi><mo>=</mo><mn>14</mn><mtext> </mtext><mn>500</mn></mrow><annotation encoding="application/x-tex">\Delta V = 18\,000 \ge C = 14\,500</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">14</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">500</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> curable (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>N</mi><mo>=</mo><mo>+</mo><mn>3</mn><mtext> </mtext><mn>500</mn></mrow><annotation encoding="application/x-tex">GN =351+3\,500</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">GN</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">+</span><span class="mord">3</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">500</span></span></span></span>). Fonctionnelle par substitution <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mn>14</mn><mtext> </mtext><mn>500</mn><mo>−</mo><mn>9</mn><mtext> </mtext><mn>000</mn><mo>=</mo><mn>5500</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">= 14\,500 - 9\,000 =3525500\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">14</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">500</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">5500</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (le coût « à neuf » de 9 000 est déjà déprécié dans la physique). Total imputé au chauffage : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn><mtext> </mtext><mn>000</mn><mo>+</mo><mn>5</mn><mtext> </mtext><mn>500</mn><mo>=</mo><mn>14</mn><mtext> </mtext><mn>500</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">9\,000 + 5\,500 = 14\,500\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">500</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">14</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">500</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span>, le coût de correction — exactement ce que l'acheteur déduit.</p></div></details>353<div class="box box-exo" id="exo-9-4"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-pencil"/></svg></span><span class="box-label">Exercice 9.4</span><span class="box-title">— extraction du marché</span><a class="box-anchor" href="#exo-9-4" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-109">Trois ventes de plain-pieds dans un secteur de Gatineau (données hypothétiques) : A — <span class="num">485 000 $</span>, terrain <span class="num">155 000 $</span>, coût neuf <span class="num">450 000 $</span>, 20 ans; B — <span class="num">510 000 $</span>, terrain <span class="num">170 000 $</span>, coût neuf <span class="num">470 000 $</span>, 25 ans; C — <span class="num">465 000 $</span>, terrain <span class="num">145 000 $</span>, coût neuf <span class="num">440 000 $</span>, 18 ans. Extrayez la dépréciation et le taux annuel de chaque vente, puis appliquez la médiane à un sujet de 26 ans et de <span class="num">420 000 $</span> de coût neuf. Commentez la dispersion.</p></div></div>354<details class="box box-sol"><summary><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-check"/></svg></span><span class="box-label">Solution</span><span class="sol-hint">Cliquer pour révéler</span></summary><div class="box-body"><p id="p9-110">A : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>450</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mo stretchy="false">(</mo><mn>485</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mn>155</mn><mtext> </mtext><mn>000</mn><mo stretchy="false">)</mo><mo>=</mo><mn>120</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">450\,000 - (485\,000 - 155\,000) = 120\,000\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">450</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">485</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">155</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">120</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (26,7 %) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> 1,33 %/an. B : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>470</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mn>340</mn><mtext> </mtext><mn>000</mn><mo>=</mo><mn>130</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">470\,000 - 340\,000 = 130\,000\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">470</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">340</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">130</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (27,7 %) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> 1,11 %/an. C : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>440</mn><mtext> </mtext><mn>000</mn><mo>−</mo><mn>320</mn><mtext> </mtext><mn>000</mn><mo>=</mo><mn>120</mn><mtext> </mtext><mn>000</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">440\,000 - 320\,000 =355120\,000\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">440</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">320</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">120</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (27,3 %) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> 1,52 %/an. Médiane 1,33 %; sujet : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mn>26</mn><mo>×</mo><mn>1,33</mn><mtext> </mtext><mi mathvariant="normal">%</mi><mo>×</mo><mn>420</mn><mtext> </mtext><mn>000</mn><mo>=</mo><mn>145</mn><mtext> </mtext><mn>236</mn><mtext> </mtext><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">D = 26 \times 1{,}33\,\% \times 420\,000 = 145\,236\,\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">26</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">33</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">420</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">145</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">236</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">$</span></span></span></span> (34,6 %). Dispersion modérée (1,11–1,52) : ratios très proches, âges différents; la médiane est défendable.</p></div></details>356<div class="box box-exo" id="exo-9-5"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-pencil"/></svg></span><span class="box-label">Exercice 9.5</span><span class="box-title">— classer et ne pas compter deux fois</span><a class="box-anchor" href="#exo-9-5" aria-label="Lien">#</a></div><div class="box-body"><p id="p9-111">Classez chaque constat (catégorie; curable ou incurable; mesure) : (a) bardeaux en fin de vie, remplacement rentable; (b) maison sans garage dans un quartier où 90 % en ont un; (c) autoroute élargie à 50 m; (d) cuisine fermée démodée; (e) comptoirs de marbre dans un quartier modeste; (f) fissures capillaires de fondation; (g) zonage interdisant l'usage le plus rentable; (h) toiture déjà comptée en entretien différé <em>et</em> dans les composantes à courte durée.</p></div></div>357<details class="box box-sol"><summary><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-check"/></svg></span><span class="box-label">Solution</span><span class="sol-hint">Cliquer pour révéler</span></summary><div class="box-body"><p id="p9-112">(a) physique curable — coût de remise en état; (b) fonctionnelle incurable — perte de valeur par paires; (c) externe, incurable — paires de ventes, à allouer au bâtiment; (d) fonctionnelle curable par substitution — excédent du coût de correction; (e) fonctionnelle incurable (suramélioration) — coût neuf moins valeur contributive; (f) physique : curable si l'injection est rentable, sinon dans la base de longue durée; (g) externe (réglementaire), incurable; (h) <strong>double comptage</strong> : retirer la toiture de la liste des composantes CT.</p></div></details>358<section class="box box-synth" id="synthese"><div class="box-head"><span class="box-icon"><svg class="" aria-hidden="true"><use href="#i-clipboard"/></svg></span><span class="box-label">L’essentiel de la séance</span></div><div class="box-body"><ul><li><strong>Dépréciation</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mo>−</mo><msub><mi>V</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">D = C_N - V_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>=</mo><mi>D</mi><mi mathvariant="normal">/</mi><msub><mi>C</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">d = D/C_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>B</mi></msub><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_B = C_N(1-d)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span> :         un écart entre un coût et une valeur, à une date; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>d</mi><mi>V</mi><mi mathvariant="normal">/</mi><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">-dV/dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord">/</span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span> est la         perte de l'année. Ni amortissement fiscal, ni seule usure, ni         fonction mécanique de l'âge chronologique.</li><li><strong>Trois catégories</strong> : physique (curable, CT, LT),         fonctionnelle (curable par ajout ou substitution; incurable par         déficience ou suramélioration), externe (toujours incurable, à         allouer au bâtiment). Additives seulement si les bases sont         successives et chaque composante comptée une fois.</li><li><strong>Curabilité</strong> : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>N</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>V</mi><mo>−</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">GN = \Delta V - C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">GN</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span>; curable si <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi><mo>≥</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\Delta V \ge C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span>;         <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mi>min</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>C</mi><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D = \min(C, \Delta V)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">min</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mclose">)</span></span></span></span>; technique <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">≠</mo></mrow><annotation encoding="application/x-tex">\neq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord katex-vbox"><span class="katex-thinbox"><span class="rlap"><span class="katex-strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="katex-inner"><span class="mord"><span class="mrel"></span></span></span><span class="katex-fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span></span></span></span> économique; retirer le         coût « à neuf » d'un élément déjà déprécié.</li><li><strong>Durées de vie</strong> : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>E</mi><mo>=</mo><mi>A</mi><mi>E</mi><mo>+</mo><mi>V</mi><mi>E</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">VE = AE + VER</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><msub><mi>C</mi><mi>N</mi></msub><mo>⋅</mo><mi>A</mi><mi>E</mi><mi mathvariant="normal">/</mi><mi>V</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">D = C_N \cdot AE/VE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span> en         années effectives; l'âge effectif se construit (inventaire,         pondération) et se défend; la vie économique se tire du marché.</li><li><strong>Quatre méthodes</strong> : âge-vie (ordre de grandeur), âge-vie         modifiée (le différé à part, ratio sur le solde), ventilation         (explique, poste par poste), extraction (fixe le niveau; lire la         dispersion; dépend du terrain et du coût neuf).</li><li><strong>Modèles</strong> : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>δ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_N(1-\delta t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>N</mi></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">C_N e^{-\lambda t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.9991em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span></span>,         <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>N</mi></msub><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>λ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msup><mi>t</mi><mi>γ</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">C_N e^{-\lambda&#x27; t^\gamma}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0925em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9425em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0556em;">γ</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> — même départ, courbures différentes;         estimer avant d'affirmer. Le coefficient hédonique de l'âge n'est         pas la dépréciation.</li><li><strong>Gatineau 2026</strong> (655 ventes réelles) : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1,03</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\lambda = 1{,}03\,\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">03</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span></span></span></span>/an,         plateau jusqu'à 35 ans puis décrochage, part du terrain de 33 à         57 %; le sujet de 1985 s'est vendu avec 26,9 % de dépréciation,         l'extraction donne 30,0 %, la conclusion <span class="num">435 000 $</span> est à         <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>2,7</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">-2{,}7\,\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.9444em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">2</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">7</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">%</span></span></span></span> du prix payé.</li><li><strong>Réconcilier</strong>, c'est raisonner — jamais moyenner à l'aveugle.</li></ul></div></section>359      </article>360    </div>361  </div>362  <section class="ch-quiz" id="quiz"><div class="ch-quiz-inner"><div id="quiz-app"><p class="gl-count">Chargement du quiz…</p></div></div></section>363  364<nav class="ch-pager" aria-label="Navigation entre les séances">365  <a class="prev" href="/seance/08/"><small>← Séance 8</small><b>Coûts indirects et profit entrepreneurial</b></a>366  <a class="next" href="/seance/10/"><small>Séance 10 →</small><b>Dépréciation physique</b></a>367</nav>368  369<div class="read-tools" role="toolbar" aria-label="Outils de lecture">370  <button type="button" data-action="notes" data-tip="Mes notes et surlignages" aria-pressed="false"><svg class="" aria-hidden="true"><use href="#i-note"/></svg></button>371  <button type="button" data-action="tts" data-tip="Écouter la séance"><svg class="" 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