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id="toc-Fixed_income" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fixed_income"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Fixed income</span> </div> </a> <ul id="toc-Fixed_income-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liabilities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liabilities"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Liabilities</span> </div> </a> <ul id="toc-Liabilities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Capital_management" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Capital_management"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Capital management</span> </div> </a> <ul id="toc-Capital_management-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Private_equity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Private_equity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Private equity</span> </div> </a> <ul id="toc-Private_equity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Calculation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Calculation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Calculation</span> </div> </a> <button aria-controls="toc-Calculation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Calculation subsection</span> </button> <ul id="toc-Calculation-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> <li id="toc-Numerical_solution" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Numerical_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Numerical solution</span> </div> </a> <ul id="toc-Numerical_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical_solution_for_single_outflow_and_multiple_inflows" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Numerical_solution_for_single_outflow_and_multiple_inflows"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Numerical solution for single outflow and multiple inflows</span> </div> </a> <ul id="toc-Numerical_solution_for_single_outflow_and_multiple_inflows-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Exact_dates_of_cash_flows" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exact_dates_of_cash_flows"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Exact dates of cash flows</span> </div> </a> <ul id="toc-Exact_dates_of_cash_flows-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Problems_with_use" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Problems_with_use"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Problems with use</span> </div> </a> <button aria-controls="toc-Problems_with_use-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Problems with use subsection</span> </button> <ul id="toc-Problems_with_use-sublist" class="vector-toc-list"> <li id="toc-Comparison_with_NPV_investment_selection_criterion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparison_with_NPV_investment_selection_criterion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Comparison with NPV investment selection criterion</span> </div> </a> <ul id="toc-Comparison_with_NPV_investment_selection_criterion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximizing_NPV" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximizing_NPV"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Maximizing NPV</span> </div> </a> <ul id="toc-Maximizing_NPV-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Practitioner_preference_for_IRR_over_NPV" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Practitioner_preference_for_IRR_over_NPV"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Practitioner preference for IRR over NPV</span> </div> </a> <ul id="toc-Practitioner_preference_for_IRR_over_NPV-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximizing_long-term_return" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximizing_long-term_return"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Maximizing long-term return</span> </div> </a> <ul id="toc-Maximizing_long-term_return-sublist" class="vector-toc-list"> <li id="toc-Example_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1</span> <span>Example</span> </div> </a> <ul id="toc-Example_2-sublist" class="vector-toc-list"> <li id="toc-Solution" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1.1</span> <span>Solution</span> </div> </a> <ul id="toc-Solution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Multiple_IRRs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple_IRRs"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Multiple IRRs</span> </div> </a> <ul id="toc-Multiple_IRRs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitations_in_the_context_of_private_equity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limitations_in_the_context_of_private_equity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Limitations in the context of private equity</span> </div> </a> <ul id="toc-Limitations_in_the_context_of_private_equity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modified_internal_rate_of_return_(MIRR)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modified_internal_rate_of_return_(MIRR)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Modified internal rate of return (MIRR)</span> </div> </a> <ul id="toc-Modified_internal_rate_of_return_(MIRR)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Average_internal_rate_of_return_(AIRR)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Average_internal_rate_of_return_(AIRR)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Average internal rate of return (AIRR)</span> </div> </a> <ul id="toc-Average_internal_rate_of_return_(AIRR)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Mathematics</span> </div> </a> <ul id="toc-Mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_reinvestment_debate" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_reinvestment_debate"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>The reinvestment debate</span> </div> </a> <ul id="toc-The_reinvestment_debate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_personal_finance" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_personal_finance"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>In personal finance</span> </div> </a> <ul id="toc-In_personal_finance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unannualized_internal_rate_of_return" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Unannualized_internal_rate_of_return"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Unannualized internal rate of return</span> </div> </a> <ul id="toc-Unannualized_internal_rate_of_return-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Internal rate of return</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 26 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-26" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">26 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%AF%D9%84_%D8%A7%D9%84%D8%B9%D8%A7%D8%A6%D8%AF_%D8%A7%D9%84%D8%AF%D8%A7%D8%AE%D9%84%D9%8A" title="معدل العائد الداخلي – Arabic" lang="ar" hreflang="ar" data-title="معدل العائد الداخلي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Daxili_g%C9%99lir_d%C9%99r%C9%99c%C9%99si" title="Daxili gəlir dərəcəsi – Azerbaijani" lang="az" hreflang="az" data-title="Daxili gəlir dərəcəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Taxa_interna_de_rendibilitat" title="Taxa interna de rendibilitat – Catalan" lang="ca" hreflang="ca" data-title="Taxa interna de rendibilitat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Interner_Zinsfu%C3%9F" title="Interner Zinsfuß – German" lang="de" hreflang="de" data-title="Interner Zinsfuß" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno" title="Tasa interna de retorno – Spanish" lang="es" hreflang="es" data-title="Tasa interna de retorno" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DB%8C%D8%B2%D8%A7%D9%86_%D8%A8%D8%A7%D8%B2%D8%AF%D9%87_%D8%AF%D8%A7%D8%AE%D9%84%DB%8C" title="میزان بازده داخلی – Persian" lang="fa" hreflang="fa" data-title="میزان بازده داخلی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Taux_de_rentabilit%C3%A9_interne" title="Taux de rentabilité interne – French" lang="fr" hreflang="fr" data-title="Taux de rentabilité interne" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%82%B4%EB%B6%80%EC%88%98%EC%9D%B5%EB%A5%A0" title="내부수익률 – Korean" lang="ko" hreflang="ko" data-title="내부수익률" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%82%E0%A4%A4%E0%A4%B0%E0%A4%BF%E0%A4%95_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%AB%E0%A4%B2_%E0%A4%A6%E0%A4%B0" title="आंतरिक प्रतिफल दर – Hindi" lang="hi" hreflang="hi" data-title="आंतरिक प्रतिफल दर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/IRR" title="IRR – Indonesian" lang="id" hreflang="id" data-title="IRR" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tasso_interno_di_rendimento" title="Tasso interno di rendimento – Italian" lang="it" hreflang="it" data-title="Tasso interno di rendimento" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%99%D7%A2%D7%95%D7%A8_%D7%AA%D7%A9%D7%95%D7%90%D7%94_%D7%A4%D7%A0%D7%99%D7%9E%D7%99" title="שיעור תשואה פנימי – Hebrew" lang="he" hreflang="he" data-title="שיעור תשואה פנימי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Iek%C5%A1%C4%93j%C4%81s_atdeves_koeficients" title="Iekšējās atdeves koeficients – Latvian" lang="lv" hreflang="lv" data-title="Iekšējās atdeves koeficients" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%B0%E0%B4%BE%E0%B4%A6%E0%B4%BE%E0%B4%AF%E0%B4%A4%E0%B5%8B%E0%B4%A4%E0%B5%8D" title="അന്തരാദായതോത് – Malayalam" lang="ml" hreflang="ml" data-title="അന്തരാദായതോത്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Interne-opbrengstvoet" title="Interne-opbrengstvoet – Dutch" lang="nl" hreflang="nl" data-title="Interne-opbrengstvoet" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%85%E9%83%A8%E5%8F%8E%E7%9B%8A%E7%8E%87" title="内部収益率 – Japanese" lang="ja" hreflang="ja" data-title="内部収益率" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Internrente" title="Internrente – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Internrente" data-language-autonym="Norsk bokmål" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Method of calculating an investment's rate of return</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Implied_repo_rate" title="Implied repo rate">Implied repo rate</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Internal_rate_of_return" title="Special:EditPage/Internal rate of return">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Internal+rate+of+return%22">"Internal rate of return"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Internal+rate+of+return%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Internal+rate+of+return%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Internal+rate+of+return%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Internal+rate+of+return%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Internal+rate+of+return%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">April 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><b>Internal rate of return</b> (<b>IRR</b>) is a method of calculating an <a href="/wiki/Investment" title="Investment">investment</a>'s <a href="/wiki/Rate_of_return" title="Rate of return">rate of return</a>. The term <i>internal</i> refers to the fact that the calculation excludes external factors, such as the <a href="/wiki/Risk-free_rate" title="Risk-free rate">risk-free rate</a>, <a href="/wiki/Inflation" title="Inflation">inflation</a>, the <a href="/wiki/Cost_of_capital" title="Cost of capital">cost of capital</a>, or <a href="/wiki/Financial_risk" title="Financial risk">financial risk</a>. </p><p>The method may be applied either <a href="/wiki/Ex-post" class="mw-redirect" title="Ex-post">ex-post</a> or <a href="/wiki/Ex-ante" title="Ex-ante">ex-ante</a>. Applied ex-ante, the IRR is an estimate of a future annual rate of return. Applied ex-post, it measures the actual achieved investment return of a historical investment. </p><p>It is also called the <a href="/wiki/Discounted_cash_flow" title="Discounted cash flow">discounted cash flow</a> <a href="/wiki/Rate_of_return" title="Rate of return">rate of return</a> (DCFROR)<sup id="cite_ref-main269_1-0" class="reference"><a href="#cite_note-main269-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or yield rate.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_(IRR)"><span id="Definition_.28IRR.29"></span>Definition (IRR)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=1" title="Edit section: Definition (IRR)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The IRR of an investment or project is the "annualized effective compounded return rate" or <a href="/wiki/Rate_of_return" title="Rate of return">rate of return</a> that sets the <a href="/wiki/Net_present_value" title="Net present value">net present value</a> (NPV) of all cash flows (both positive and negative) from the investment equal to zero.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Equivalently, it is the <a href="/wiki/Interest_rate" title="Interest rate">interest rate</a> at which the net <a href="/wiki/Present_value" title="Present value">present value</a> of the future cash flows is equal to the initial investment,<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and it is also the interest rate at which the total present value of costs (negative cash flows) equals the total present value of the benefits (positive cash flows). </p><p>IRR represents the return on investment achieved when a project reaches its breakeven point, meaning that the project is only marginally justified as valuable. When NPV demonstrates a positive value, it indicates that the project is expected to generate value. Conversely, if NPV shows a negative value, the project is expected to lose value. In essence, IRR signifies the rate of return attained when the NPV of the project reaches a neutral state, precisely at the point where NPV breaks even.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>IRR accounts for the <a href="/wiki/Time_preference" title="Time preference">time preference</a> of money and investments. A given return on investment received at a given time is worth more than the same return received at a later time, so the latter would yield a lower IRR than the former, if all other factors are equal. A <a href="/wiki/Fixed_income_investment" class="mw-redirect" title="Fixed income investment">fixed income investment</a> in which money is deposited once, <a href="/wiki/Interest" title="Interest">interest</a> on this deposit is paid to the investor at a specified <a href="/wiki/Interest_rate" title="Interest rate">interest rate</a> every time period, and the original deposit neither increases nor decreases, would have an IRR equal to the specified interest rate. An investment which has the same total returns as the preceding investment, but delays returns for one or more time periods, would have a lower IRR. </p> <div class="mw-heading mw-heading2"><h2 id="Uses">Uses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=2" title="Edit section: Uses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Savings_and_loans">Savings and loans</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=3" title="Edit section: Savings and loans"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the context of savings and loans, the IRR is also called the <a href="/wiki/Effective_interest_rate" title="Effective interest rate">effective interest rate</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Profitability_of_an_investment">Profitability of an investment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=4" title="Edit section: Profitability of an investment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The IRR is an indicator of the <a href="/wiki/Profit_(economics)" title="Profit (economics)">profitability</a>, efficiency, quality, or <a href="/wiki/Yield_(finance)" title="Yield (finance)">yield</a> of an investment. This is in contrast with the <a href="/wiki/Net_present_value" title="Net present value">NPV</a>, which is an indicator of the net <a href="/wiki/Value_(economics)" title="Value (economics)">value</a> or <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> added by making an investment. </p><p>To maximize the <a href="/wiki/Value_(economics)" title="Value (economics)">value</a> of a business, an investment should be made only if its profitability, as measured by the internal rate of return, is greater than a <a href="/wiki/Minimum_acceptable_rate_of_return" title="Minimum acceptable rate of return">minimum acceptable rate of return</a>. If the estimated IRR of a project or investment - for example, the construction of a new factory - exceeds the firm's <a href="/wiki/Cost_of_capital" title="Cost of capital">cost of capital</a> invested in that project, the investment is profitable. If the estimated IRR is less than the cost of capital, the proposed project should not be undertaken.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The selection of investments may be subject to budget constraints. There may be <a href="/wiki/Mutual_exclusivity" title="Mutual exclusivity">mutually exclusive</a> competing projects, or limits on a firm's ability to manage multiple projects. For these reasons, corporations use IRR in <a href="/wiki/Capital_budgeting" title="Capital budgeting">capital budgeting</a> to compare the <a href="/wiki/Profit_(economics)" title="Profit (economics)">profitability</a> of a set of alternative <a href="/wiki/Investment" title="Investment">capital projects</a>. For example, a corporation will compare an investment in a new plant versus an extension of an existing plant based on the IRR of each project. To maximize <a href="/wiki/Rate_of_return" title="Rate of return">returns</a>, the higher a project's IRR, the more desirable it is to undertake the project. </p><p>There are at least two different ways to measure the IRR for an investment: the project IRR and the equity IRR. The project IRR assumes that the cash flows directly benefit the project, whereas equity IRR considers the returns for the shareholders of the company after the debt has been serviced.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Even though IRR is one of the most popular metrics used to test the viability of an investment and compare returns of alternative projects, looking at the IRR in isolation might not be the best approach for an investment decision. Certain assumptions made during IRR calculations are not always applicable to the investment. In particular, IRR assumes that the project will have either no interim cash flows or the interim cash flows are reinvested into the project which is not always the case. This discrepancy leads to overestimation of the rate of return which might be an incorrect representation of the value of the project. <sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Fixed_income">Fixed income</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=5" title="Edit section: Fixed income"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>IRR is used to evaluate investments in fixed income securities, using metrics such as the <a href="/wiki/Yield_to_maturity" title="Yield to maturity">yield to maturity</a> and <a href="/wiki/Yield_(finance)" title="Yield (finance)">yield to call</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Liabilities">Liabilities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=6" title="Edit section: Liabilities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Both IRR and net present value can be applied to liabilities as well as investments. For a liability, a lower IRR is preferable to a higher one. </p> <div class="mw-heading mw-heading3"><h3 id="Capital_management">Capital management</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=7" title="Edit section: Capital management"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Corporations use IRR to evaluate share issues and <a href="/wiki/Share_repurchase" title="Share repurchase">stock buyback</a> programs. A share repurchase proceeds if returning capital to shareholders has a higher IRR than candidate capital investment projects or acquisition projects at current market prices. Funding new projects by raising new debt may also involve measuring the cost of the new debt in terms of the <a href="/wiki/Yield_to_maturity" title="Yield to maturity">yield to maturity</a> (internal rate of return). </p> <div class="mw-heading mw-heading3"><h3 id="Private_equity">Private equity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=8" title="Edit section: Private equity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>IRR is also used for <a href="/wiki/Private_equity" title="Private equity">private equity</a>, from the limited partners' perspective, as a measure of the general partner's performance as investment manager.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This is because it is the general partner who controls the cash flows, including the limited partners' draw-downs of <a href="/wiki/Capital_commitment" title="Capital commitment">committed capital</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Calculation">Calculation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=9" title="Edit section: Calculation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a collection of pairs (<a href="/wiki/Time" title="Time">time</a>, <a href="/wiki/Cash_flow" title="Cash flow">cash flow</a>) representing a project, the <a href="/wiki/Net_present_value" title="Net present value">NPV</a> is a function of the <a href="/wiki/Rate_of_return" title="Rate of return">rate of return</a>. The internal rate of return is a rate for which this function is zero, i.e. the internal rate of return is a solution to the equation NPV = 0 (assuming no arbitrage conditions exist). </p><p>Given the (period, cash flow) pairs (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0301812adb392070d834ca2df4ed97f1cf132f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle C_{n}}"></span>) where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is a non-negative integer, the total number of periods <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>, and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ba546e9a9f8abc122032ca665a22c68d214838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.069ex; height:2.176ex;" alt="{\displaystyle \operatorname {NPV} }"></span>, (<a href="/wiki/Net_present_value" title="Net present value">net present value</a>); the internal rate of return is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> in: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} =\sum _{n=0}^{N}{\frac {C_{n}}{(1+r)^{n}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} =\sum _{n=0}^{N}{\frac {C_{n}}{(1+r)^{n}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9292402e75d6e839b978def94f6af91c201b3d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.086ex; height:7.343ex;" alt="{\displaystyle \operatorname {NPV} =\sum _{n=0}^{N}{\frac {C_{n}}{(1+r)^{n}}}=0}"></span></dd></dl> <p><sup id="cite_ref-:0_2-3" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_3-2" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>This rational polynomial can be converted to an ordinary polynomial having the same roots by substituting <i>g</i> (gain) for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa9ecb1aeafa31add2a9de6053566d6e85758bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.052ex; height:2.343ex;" alt="{\displaystyle 1+r}"></span> and multiplying by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b71cd76143d22e85162aa57d120f0e8a125db5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.81ex; height:3.009ex;" alt="{\displaystyle g^{N}}"></span> to yield the equivalent but simpler condition </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{N}C_{n}g^{N-n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{N}C_{n}g^{N-n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62098a26ecbd9e48aa31e401a4adab309c3aff0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.958ex; height:7.343ex;" alt="{\displaystyle \sum _{n=0}^{N}C_{n}g^{N-n}=0}"></span></dd></dl> <p>The possible IRR's are the real values of <i>r</i> satisfying the first condition, and 1 less than the real roots of the second condition (that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=g-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>g</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=g-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fac94e67dd325a445e6d9627697b4a32695c97c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.266ex; height:2.509ex;" alt="{\displaystyle r=g-1}"></span> for each root <i>g</i>). Note that in both formulas, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23da38e31194b9ae0524ec18c8489693f3be5389" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{0}}"></span> is the negation of the initial investment at the start of the project while <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977915dd0fc4be599697ee95efb498350bad48f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.353ex; height:2.509ex;" alt="{\displaystyle C_{N}}"></span> is the cash value of the project at the end, equivalently the cash withdrawn if the project were to be liquidated and paid out so as to reduce the value of the project to zero. In the second condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23da38e31194b9ae0524ec18c8489693f3be5389" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{0}}"></span> is the leading coefficient of the ordinary polynomial in <i>g</i> while <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977915dd0fc4be599697ee95efb498350bad48f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.353ex; height:2.509ex;" alt="{\displaystyle C_{N}}"></span> is the constant term. </p><p>The period <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is usually given in years, but the calculation may be made simpler if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is calculated using the period in which the majority of the problem is defined (e.g., using months if most of the cash flows occur at monthly intervals) and converted to a yearly period thereafter. </p><p>Any fixed time can be used in place of the present (e.g., the end of one interval of an <a href="/wiki/Annuity_(finance_theory)" class="mw-redirect" title="Annuity (finance theory)">annuity</a>); the value obtained is zero if and only if the NPV is zero. </p><p>In the case that the cash flows are <a href="/wiki/Random_variable" title="Random variable">random variables</a>, such as in the case of a <a href="/wiki/Life_annuity" title="Life annuity">life annuity</a>, the <a href="/wiki/Expected_value" title="Expected value">expected values</a> are put into the above formula. </p><p>Often, the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> that satisfies the above equation cannot be found <a href="/wiki/Analytical_solution" class="mw-redirect" title="Analytical solution">analytically</a>. In this case, <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical methods</a> or <a href="/wiki/Plot_(graphics)" title="Plot (graphics)">graphical methods</a> must be used. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=10" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If an investment may be given by the sequence of cash flows </p> <table class="wikitable" align="left"> <tbody><tr> <th>Year (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>) </th> <th>Cash flow (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0301812adb392070d834ca2df4ed97f1cf132f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle C_{n}}"></span>) </th></tr> <tr align="center"> <td>0 </td> <td>-123400 </td></tr> <tr align="center"> <td>1 </td> <td>36200 </td></tr> <tr align="center"> <td>2 </td> <td>54800 </td></tr> <tr align="center"> <td>3 </td> <td>48100 </td></tr> </tbody></table> <div style="clear:both;" class=""></div> <p>then the IRR <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} =-123400+{\frac {36200}{(1+r)^{1}}}+{\frac {54800}{(1+r)^{2}}}+{\frac {48100}{(1+r)^{3}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>=</mo> <mo>−</mo> <mn>123400</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>36200</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>54800</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>48100</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} =-123400+{\frac {36200}{(1+r)^{1}}}+{\frac {54800}{(1+r)^{2}}}+{\frac {48100}{(1+r)^{3}}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/619afb7c43611165f5af34bffbc437f12550466e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:56.633ex; height:6.009ex;" alt="{\displaystyle \operatorname {NPV} =-123400+{\frac {36200}{(1+r)^{1}}}+{\frac {54800}{(1+r)^{2}}}+{\frac {48100}{(1+r)^{3}}}=0.}"></span></dd></dl> <p>In this case, the answer is 5.96% (in the calculation, that is, r = .0596). </p> <div class="mw-heading mw-heading4"><h4 id="Numerical_solution">Numerical solution</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=11" title="Edit section: Numerical solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the above is a manifestation of the general problem of finding the <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">roots</a> of the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} (r)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} (r)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250503704b03d2cfbda42854000cb0f377023422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.188ex; height:2.843ex;" alt="{\displaystyle \operatorname {NPV} (r)=0}"></span>, there are many <a href="/wiki/Root-finding_algorithm" title="Root-finding algorithm">numerical methods</a> that can be used to estimate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. For example, using the <a href="/wiki/Secant_method" title="Secant method">secant method</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n+1}=r_{n}-\operatorname {NPV} _{n}\cdot \left({\frac {r_{n}-r_{n-1}}{\operatorname {NPV} _{n}-\operatorname {NPV} _{n-1}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n+1}=r_{n}-\operatorname {NPV} _{n}\cdot \left({\frac {r_{n}-r_{n-1}}{\operatorname {NPV} _{n}-\operatorname {NPV} _{n-1}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4612385b616b89d478cdbe3942539a56e9648e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.831ex; height:6.176ex;" alt="{\displaystyle r_{n+1}=r_{n}-\operatorname {NPV} _{n}\cdot \left({\frac {r_{n}-r_{n-1}}{\operatorname {NPV} _{n}-\operatorname {NPV} _{n-1}}}\right).}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57784cdf7f49f2c46baad4eb7b6a7d5c14eb5fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.009ex;" alt="{\displaystyle r_{n}}"></span> is considered the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span><sup>th</sup> approximation of the IRR. </p><p>This <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> can be found to an arbitrary degree of <a href="/wiki/Accuracy_and_precision" title="Accuracy and precision">accuracy</a>. Different accounting packages may provide functions for different accuracy levels. For example, <a href="/wiki/Microsoft_Excel" title="Microsoft Excel">Microsoft Excel</a> and <a href="/wiki/Google_Sheets" title="Google Sheets">Google Sheets</a> have built-in functions to calculate IRR for both fixed and variable time-intervals; "=IRR(...)" and "=XIRR(...)". </p><p>The convergence behaviour of by the following: </p> <ul><li>If the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} (i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} (i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c64ed1eabd8c84ea65eea373639ad654ab5072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.681ex; height:2.843ex;" alt="{\displaystyle \operatorname {NPV} (i)}"></span> has a single <a href="/wiki/Real_number" title="Real number">real</a> root <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, then the sequence converges reproducibly towards <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>.</li> <li>If the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} (i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} (i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c64ed1eabd8c84ea65eea373639ad654ab5072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.681ex; height:2.843ex;" alt="{\displaystyle \operatorname {NPV} (i)}"></span> has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> real roots <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle r_{1},r_{2},\dots ,r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle r_{1},r_{2},\dots ,r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c461af4db4d1d3dc8cfc2523be4775c34810e80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.15ex; height:1.509ex;" alt="{\displaystyle \scriptstyle r_{1},r_{2},\dots ,r_{n}}"></span>, then the sequence converges to one of the roots, and changing the values of the initial pairs may change the root to which it converges.</li> <li>If function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} (i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} (i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c64ed1eabd8c84ea65eea373639ad654ab5072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.681ex; height:2.843ex;" alt="{\displaystyle \operatorname {NPV} (i)}"></span> has no real roots, then the sequence tends towards <a href="/wiki/Extended_real_number_line" title="Extended real number line">+∞</a>.</li></ul> <p>Having <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {r_{1}>r_{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {r_{1}>r_{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef523634414f7137cc43ed460df8232412cb4896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.425ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {r_{1}>r_{0}}}"></span> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} _{0}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} _{0}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8893f28f665f4a7892892fc1073b3f4cbba8da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.384ex; height:2.509ex;" alt="{\displaystyle \operatorname {NPV} _{0}>0}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {r_{1}<r_{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {r_{1}<r_{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83e344371034d35885c606b2cd7bddaa843ec7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.425ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {r_{1}<r_{0}}}"></span> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} _{0}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} _{0}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2854546d2531469ba4b56d2ffbf3201a2d342a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.384ex; height:2.509ex;" alt="{\displaystyle \operatorname {NPV} _{0}<0}"></span> may speed up convergence of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57784cdf7f49f2c46baad4eb7b6a7d5c14eb5fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.009ex;" alt="{\displaystyle r_{n}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Numerical_solution_for_single_outflow_and_multiple_inflows">Numerical solution for single outflow and multiple inflows</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=12" title="Edit section: Numerical solution for single outflow and multiple inflows"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Of particular interest is the case where the stream of payments consists of a single outflow, followed by multiple inflows occurring at equal periods. In the above notation, this corresponds to: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}<0,\quad C_{n}\geq 0{\text{ for }}n\geq 1.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>n</mi> <mo>≥</mo> <mn>1.</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{0}<0,\quad C_{n}\geq 0{\text{ for }}n\geq 1.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c0a70504f466206ddaec1190905fcd19bbca00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.112ex; height:2.509ex;" alt="{\displaystyle C_{0}<0,\quad C_{n}\geq 0{\text{ for }}n\geq 1.\,}"></span></dd></dl> <p>In this case the NPV of the payment stream is a <a href="/wiki/Convex_function" title="Convex function">convex</a>, <a href="/wiki/Monotonic_function" title="Monotonic function">strictly decreasing</a> function of interest rate. There is always a single unique solution for IRR. </p><p>Given two estimates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}"></span> for IRR, the secant method equation (see above) with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"></span> always produces an improved estimate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}"></span>. This is sometimes referred to as the Hit and Trial (or Trial and Error) method. More accurate interpolation formulas can also be obtained: for instance the secant formula with correction </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n+1}=r_{n}-\operatorname {NPV} _{n}\left({\frac {r_{n}-r_{n-1}}{\operatorname {NPV} _{n}-\operatorname {NPV} _{n-1}}}\right)\left(1-1.4{\frac {\operatorname {NPV} _{n-1}}{\operatorname {NPV} _{n-1}-3\operatorname {NPV} _{n}+2C_{0}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>1.4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>3</mn> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n+1}=r_{n}-\operatorname {NPV} _{n}\left({\frac {r_{n}-r_{n-1}}{\operatorname {NPV} _{n}-\operatorname {NPV} _{n-1}}}\right)\left(1-1.4{\frac {\operatorname {NPV} _{n-1}}{\operatorname {NPV} _{n-1}-3\operatorname {NPV} _{n}+2C_{0}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d604451361d61913f2cfd03e51430c47d4f1287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:77.523ex; height:6.176ex;" alt="{\displaystyle r_{n+1}=r_{n}-\operatorname {NPV} _{n}\left({\frac {r_{n}-r_{n-1}}{\operatorname {NPV} _{n}-\operatorname {NPV} _{n-1}}}\right)\left(1-1.4{\frac {\operatorname {NPV} _{n-1}}{\operatorname {NPV} _{n-1}-3\operatorname {NPV} _{n}+2C_{0}}}\right),}"></span></dd></dl> <p>(which is most accurate when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0>\operatorname {NPV} _{n}>\operatorname {NPV} _{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>></mo> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0>\operatorname {NPV} _{n}>\operatorname {NPV} _{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/783ee019f6551ae32064e85c3cd6f59b1cb73b0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.035ex; height:2.509ex;" alt="{\displaystyle 0>\operatorname {NPV} _{n}>\operatorname {NPV} _{n-1}}"></span>) has been shown to be almost 10 times more accurate than the secant formula for a wide range of interest rates and initial guesses. For example, using the stream of payments {−4000, 1200, 1410, 1875, 1050} and initial guesses <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}=0.25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}=0.25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a29a26098e6e78a1688fef3cafa3b1195be8b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.336ex; height:2.509ex;" alt="{\displaystyle r_{1}=0.25}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}=0.2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}=0.2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aafd8f891cd9ac564759e8921e70a5634041868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.173ex; height:2.509ex;" alt="{\displaystyle r_{2}=0.2}"></span> the secant formula with correction gives an IRR estimate of 14.2% (0.7% error) as compared to IRR = 13.2% (7% error) from the secant method. </p><p>If applied iteratively, either the secant method or the improved formula always converges to the correct solution. </p><p>Both the secant method and the improved formula rely on initial guesses for IRR. The following initial guesses may be used: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}=\left(A/|C_{0}|\right)^{2/(N+1)}-1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}=\left(A/|C_{0}|\right)^{2/(N+1)}-1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9589ee4b5b62c92bdc7b12f802542cc2092a33f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.031ex; height:3.509ex;" alt="{\displaystyle r_{1}=\left(A/|C_{0}|\right)^{2/(N+1)}-1\,}"></span></dd></dl></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}=(1+r_{1})^{p}-1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}=(1+r_{1})^{p}-1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d205f769915382627c6bf196e4c96882e50823a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.566ex; height:2.843ex;" alt="{\displaystyle r_{2}=(1+r_{1})^{p}-1\,}"></span></dd></dl></dd></dl> <p>where </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\text{ sum of inflows }}=C_{1}+\cdots +C_{N}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> sum of inflows </mtext> </mrow> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\text{ sum of inflows }}=C_{1}+\cdots +C_{N}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5e8deb6bbbef625c76b8f0ba75a1503fccb44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.197ex; height:2.509ex;" alt="{\displaystyle A={\text{ sum of inflows }}=C_{1}+\cdots +C_{N}\,}"></span></dd></dl></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {\log(\mathrm {A} /|C_{0}|)}{\log(\mathrm {A} /\operatorname {NPV} _{1,in})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {\log(\mathrm {A} /|C_{0}|)}{\log(\mathrm {A} /\operatorname {NPV} _{1,in})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0f88bcfed87db36ed54b0f239fe8a2c0821779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:22.048ex; height:6.509ex;" alt="{\displaystyle p={\frac {\log(\mathrm {A} /|C_{0}|)}{\log(\mathrm {A} /\operatorname {NPV} _{1,in})}}.}"></span></dd></dl></dd></dl> <p>Here, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} _{1,in}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} _{1,in}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169d84511a534d50e43b30fd192fdb80bb4d830d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.134ex; height:2.843ex;" alt="{\displaystyle \operatorname {NPV} _{1,in}}"></span> refers to the NPV of the inflows only (that is, set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} _{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {C} _{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0bc9b33140cc2b2158a9ad958a547c85bfacdbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.993ex; height:2.509ex;" alt="{\displaystyle \mathrm {C} _{0}=0}"></span> and compute NPV). </p> <div class="mw-heading mw-heading3"><h3 id="Exact_dates_of_cash_flows">Exact dates of cash flows</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=13" title="Edit section: Exact dates of cash flows"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A cash flow <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0301812adb392070d834ca2df4ed97f1cf132f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle C_{n}}"></span> may occur at any time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271566db7e8ca8616a4dc3efb6c5982a2d987ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.058ex; height:2.343ex;" alt="{\displaystyle t_{n}}"></span> years after the beginning of the project. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271566db7e8ca8616a4dc3efb6c5982a2d987ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.058ex; height:2.343ex;" alt="{\displaystyle t_{n}}"></span> may not be a whole number. The cash flow should still be discounted by a factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(1+r)^{t_{n}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1+r)^{t_{n}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34d6f99cff8de976acac526bcc391acc443c097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.488ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{(1+r)^{t_{n}}}}}"></span>. And the formula is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} =C_{0}+\sum _{n=1}^{N}{\frac {C_{n}}{(1+r)^{t_{n}}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} =C_{0}+\sum _{n=1}^{N}{\frac {C_{n}}{(1+r)^{t_{n}}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c603d30bc63e5090e23a8972321f33db709372c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.215ex; height:7.343ex;" alt="{\displaystyle \operatorname {NPV} =C_{0}+\sum _{n=1}^{N}{\frac {C_{n}}{(1+r)^{t_{n}}}}=0}"></span></dd></dl> <p>For numerical solution we can use <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{k+1}=r_{k}-{\frac {\operatorname {NPV} _{k}}{\operatorname {NPV} '_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>NPV</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{k+1}=r_{k}-{\frac {\operatorname {NPV} _{k}}{\operatorname {NPV} '_{k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd96ac2967f8c195c85551dad56b5d25975c3cc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.308ex; height:6.176ex;" alt="{\displaystyle r_{k+1}=r_{k}-{\frac {\operatorname {NPV} _{k}}{\operatorname {NPV} '_{k}}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>NPV</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa858aac96c7fbe42810092b120db2352a52693b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.754ex; height:2.509ex;" alt="{\displaystyle \operatorname {NPV} '}"></span> is the derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ba546e9a9f8abc122032ca665a22c68d214838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.069ex; height:2.176ex;" alt="{\displaystyle \operatorname {NPV} }"></span> and given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} '=-\sum _{n=1}^{N}{\frac {C_{n}\cdot t_{n}}{(1+r)^{t_{n}+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>NPV</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} '=-\sum _{n=1}^{N}{\frac {C_{n}\cdot t_{n}}{(1+r)^{t_{n}+1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f80ed710c203d992cd204861aed88490e8afb15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.378ex; height:7.343ex;" alt="{\displaystyle \operatorname {NPV} '=-\sum _{n=1}^{N}{\frac {C_{n}\cdot t_{n}}{(1+r)^{t_{n}+1}}}}"></span></dd></dl> <p>An initial value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"></span> can be given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}={\frac {-1}{C_{0}}}\sum _{n=1}^{N}C_{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}={\frac {-1}{C_{0}}}\sum _{n=1}^{N}C_{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c845c248f9b212eff46b93799520430dcbc100b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.02ex; height:7.343ex;" alt="{\displaystyle r_{1}={\frac {-1}{C_{0}}}\sum _{n=1}^{N}C_{n}-1}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Problems_with_use">Problems with use</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=14" title="Edit section: Problems with use"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Comparison_with_NPV_investment_selection_criterion">Comparison with NPV investment selection criterion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=15" title="Edit section: Comparison with NPV investment selection criterion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a tool applied to making an <a href="/wiki/Investment" title="Investment">investment</a> decision on whether a project adds value or not, comparing the IRR of a single project with the required rate of return, in isolation from any other projects, is equivalent to the NPV method. If the appropriate IRR (if such can be found correctly) is greater than the required rate of return, using the required rate of return to discount cash flows to their present value, the NPV of that project will be positive, and vice versa. However, using IRR to sort projects in order of preference does not result in the same order as using NPV. </p> <div class="mw-heading mw-heading3"><h3 id="Maximizing_NPV">Maximizing NPV</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=16" title="Edit section: Maximizing NPV"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One possible investment objective is to maximize the total NPV of projects. </p><p> When the objective is to maximize total value, the calculated IRR should not be used to choose between mutually exclusive projects. </p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Exclusive_investments.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/a/a8/Exclusive_investments.png/231px-Exclusive_investments.png" decoding="async" width="231" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/a/a8/Exclusive_investments.png/347px-Exclusive_investments.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/a/a8/Exclusive_investments.png/462px-Exclusive_investments.png 2x" data-file-width="960" data-file-height="720" /></a><figcaption>NPV vs discount rate comparison for two mutually exclusive projects. Project 'A' has a higher NPV (for certain discount rates), even though its IRR (= <i>x</i>-axis intercept) is lower than for project 'B' (click to enlarge)</figcaption></figure><p>In cases where one project has a higher initial investment than a second mutually exclusive project, the first project may have a lower IRR (expected return), but a higher NPV (increase in shareholders' wealth) and should thus be accepted over the second project (assuming no capital constraints). </p><p>When the objective is to maximize total value, IRR should not be used to compare projects of different duration. For example, the NPV added by a project with longer duration but lower IRR could be greater than that of a project of similar size, in terms of total net cash flows, but with shorter duration and higher IRR. </p> <div class="mw-heading mw-heading3"><h3 id="Practitioner_preference_for_IRR_over_NPV">Practitioner preference for IRR over NPV</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=17" title="Edit section: Practitioner preference for IRR over NPV"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Apparently, managers prefer to compare investments of different sizes in terms of forecast investment performance, using IRR, rather than maximize value to the firm, in terms of NPV. This preference makes a difference when comparing mutually exclusive projects. </p> <div class="mw-heading mw-heading3"><h3 id="Maximizing_long-term_return">Maximizing long-term return</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=18" title="Edit section: Maximizing long-term return"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Maximizing total value is not the only conceivable possible investment objective. An alternative objective would for example be to maximize long-term return. Such an objective would rationally lead to accepting first those new projects within the capital budget which have the highest IRR, because adding such projects would tend to maximize overall long-term return. </p> <div class="mw-heading mw-heading4"><h4 id="Example_2">Example</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=19" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To see this, consider two investors, Max Value and Max Return. Max Value wishes her net worth to grow as large as possible, and will invest every last cent available to achieve this, whereas Max Return wants to maximize his rate of return over the long term, and would prefer to choose projects with smaller capital outlay but higher returns. Max Value and Max Return can each raise <i>up to</i> 100,000 US dollars from their bank at an annual interest rate of 10 percent paid at the end of the year. </p><p>Investors Max Value and Max Return are presented with two possible projects to invest in, called Big-Is-Best and Small-Is-Beautiful. Big-Is-Best requires a capital investment of 100,000 US dollars today, and the lucky investor will be repaid 132,000 US dollars in a year's time. Small-Is-Beautiful only requires 10,000 US dollars capital to be invested today, and will repay the investor 13,750 US dollars in a year's time. </p> <div class="mw-heading mw-heading5"><h5 id="Solution">Solution</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=20" title="Edit section: Solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The cost of capital for both investors is 10 percent. </p><p>Both Big-Is-Best and Small-Is-Beautiful have positive NPVs: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} ({\text{Big-Is-Best}})={\frac {132,000}{1.1}}-100,000=20,000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Big-Is-Best</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>132</mn> <mo>,</mo> <mn>000</mn> </mrow> <mn>1.1</mn> </mfrac> </mrow> <mo>−</mo> <mn>100</mn> <mo>,</mo> <mn>000</mn> <mo>=</mo> <mn>20</mn> <mo>,</mo> <mn>000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} ({\text{Big-Is-Best}})={\frac {132,000}{1.1}}-100,000=20,000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b23a1572bd37cf820817669cad2094634cf8ccd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.874ex; height:5.343ex;" alt="{\displaystyle \operatorname {NPV} ({\text{Big-Is-Best}})={\frac {132,000}{1.1}}-100,000=20,000}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} ({\text{Small-Is-Beautiful}})={\frac {13,750}{1.1}}-10,000=2,500}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Small-Is-Beautiful</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>13</mn> <mo>,</mo> <mn>750</mn> </mrow> <mn>1.1</mn> </mfrac> </mrow> <mo>−</mo> <mn>10</mn> <mo>,</mo> <mn>000</mn> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>500</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} ({\text{Small-Is-Beautiful}})={\frac {13,750}{1.1}}-10,000=2,500}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc8cf46c940cad791ffbecabba5275d3c499c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.453ex; height:5.343ex;" alt="{\displaystyle \operatorname {NPV} ({\text{Small-Is-Beautiful}})={\frac {13,750}{1.1}}-10,000=2,500}"></span></dd></dl> <p>and the IRR of each is (of course) greater than the cost of capital: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {NPV}}({\text{Big-Is-Best}})={\frac {132,000}{1.32}}-100,000=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">N</mi> <mi class="MJX-tex-mathit" mathvariant="italic">P</mi> <mi class="MJX-tex-mathit" mathvariant="italic">V</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Big-Is-Best</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>132</mn> <mo>,</mo> <mn>000</mn> </mrow> <mn>1.32</mn> </mfrac> </mrow> <mo>−</mo> <mn>100</mn> <mo>,</mo> <mn>000</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {NPV}}({\text{Big-Is-Best}})={\frac {132,000}{1.32}}-100,000=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee81cb8cc27992f3a4083a00699a33ad49a1094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.151ex; height:5.343ex;" alt="{\displaystyle {\mathit {NPV}}({\text{Big-Is-Best}})={\frac {132,000}{1.32}}-100,000=0}"></span></dd></dl> <p>so the IRR of Big-Is-Best is 32 percent, and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NPV} ({\text{Small-Is-Beautiful}})={\frac {13,750}{1.375}}-10,000=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>NPV</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Small-Is-Beautiful</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>13</mn> <mo>,</mo> <mn>750</mn> </mrow> <mn>1.375</mn> </mfrac> </mrow> <mo>−</mo> <mn>10</mn> <mo>,</mo> <mn>000</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {NPV} ({\text{Small-Is-Beautiful}})={\frac {13,750}{1.375}}-10,000=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99269db890a1c203ab6109e7ea1e8b4ab3d8d6ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:49.932ex; height:5.509ex;" alt="{\displaystyle \operatorname {NPV} ({\text{Small-Is-Beautiful}})={\frac {13,750}{1.375}}-10,000=0}"></span></dd></dl> <p>so the IRR of Small-Is-Beautiful is 37.5 percent. </p><p>Both investments would be acceptable to both investors, but the twist in the tale is that these are mutually exclusive projects for both investors, because their capital budget is limited to 100,000 US dollars. How will the investors choose rationally between the two? </p><p>The happy outcome is that Max Value chooses Big-Is-Best, which has the higher NPV of 20,000 US dollars, over Small-Is-Beautiful, which only has a modest NPV of 2,500, whereas Max Return chooses Small-Is-Beautiful, for its superior 37.5 percent return, over the attractive (but not as attractive) return of 32 percent offered on Big-Is-Best. So there is no squabbling over who gets which project, they are each happy to choose different projects. </p><p>How can this be rational for both investors? The answer lies in the fact that the investors do not have to invest the full 100,000 US dollars. Max Return is content to invest only 10,000 US dollars for now. After all, Max Return may rationalize the outcome by thinking that maybe tomorrow there will be new opportunities available to invest the remaining 90,000 US dollars the bank is willing to lend Max Return, at even higher IRRs. Even if only seven more projects come along which are identical to Small-Is-Beautiful, Max Return would be able to match the NPV of Big-Is-Best, on a total investment of only 80,000 US dollars, with 20,000 US dollars left in the budget to spare for truly unmissable opportunities. Max Value is also happy, because she has filled her capital budget straight away, and decides she can take the rest of the year off investing. </p> <div class="mw-heading mw-heading3"><h3 id="Multiple_IRRs">Multiple IRRs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=21" title="Edit section: Multiple IRRs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When the sign of the cash flows changes more than once, for example when positive cash flows are followed by negative ones and then by positive ones (+ + − − − +), the IRR may have multiple real values. In a series of cash flows like (−10, 21, −11), one initially invests money, so a high rate of return is best, but then receives more than one possesses, so then one owes money, so now a low rate of return is best. In this case, it is not even clear whether a high or a low IRR is better. </p><p>There may even be multiple real IRRs for a single project, like in the example 0% as well as 10%. Examples of this type of project are <a href="/wiki/Strip_mine" class="mw-redirect" title="Strip mine">strip mines</a> and <a href="/wiki/Nuclear_power" title="Nuclear power">nuclear power</a> plants, where there is usually a large cash outflow at the end of the project. </p><p>The IRR satisfies a polynomial equation. <a href="/wiki/Sturm%27s_theorem" title="Sturm's theorem">Sturm's theorem</a> can be used to determine if that equation has a unique real solution. In general the IRR equation cannot be solved analytically but only by iteration. </p><p>With multiple internal rates of return, the IRR approach can still be interpreted in a way that is consistent with the present value approach if the underlying investment stream is correctly identified as net investment or net borrowing.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>See <sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> for a way of identifying the relevant IRR from a set of multiple IRR solutions. </p> <div class="mw-heading mw-heading3"><h3 id="Limitations_in_the_context_of_private_equity">Limitations in the context of private equity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=22" title="Edit section: Limitations in the context of private equity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the context of <a href="/wiki/Survivorship_bias" title="Survivorship bias">survivorship bias</a> which makes the high IRR of large private equity firms a poor representation of the average, according to <a href="/wiki/Ludovic_Phalippou" title="Ludovic Phalippou">Ludovic Phalippou</a>, </p><p>"...a headline figure that is often shown prominently as a rate of return in presentations and documents is, in fact, an IRR. IRRs are not rates of return. Something large PE firms have in common is that their early investments did well. These early winners have set up those firms' since-inception IRR at an artificially sticky and high level. The mathematics of IRR means that their IRRs will stay at this level forever, as long as the firms avoid major disasters. In passing, this generates some stark injustice because it is easier to game IRRs on LBOs in Western countries than in any other PE investments. That means that the rest of the PE industry (e.g. emerging market growth capital) is sentenced to look relatively bad forever, for no reason other than the use of a game-able performance metric."<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Also, </p><p>"Another problem with the presentation of pension fund performance is that for PE, time-weighted returns...are not the most pertinent measure of performance. Asking how much pension funds gave and got back in dollar terms from PE, i.e. MoM, would be more pertinent. I went through the largest 15 funds websites to collect information on their performance. Few of them post their PE fund returns online. In most cases, they post information on their past performance in PE, but nothing that enables any meaningful benchmarking. <i>E.g.</i>, CalSTRS [a California public pension fund] provide only the net IRR for each fund they invest in. As IRR is often misleading and can never be aggregated or compared to stock-market returns, such information is basically useless for gauging performance."<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Modified_internal_rate_of_return_(MIRR)"><span id="Modified_internal_rate_of_return_.28MIRR.29"></span>Modified internal rate of return (MIRR)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=23" title="Edit section: Modified internal rate of return (MIRR)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Modified_Internal_Rate_of_Return" class="mw-redirect" title="Modified Internal Rate of Return">Modified Internal Rate of Return</a> (MIRR) considers <a href="/wiki/Cost_of_capital" title="Cost of capital">cost of capital</a>, and is intended to provide a better indication of a project's probable return. It applies a discount rate for borrowing cash, and the IRR is calculated for the investment cash flows. This applies in real life for example when a customer makes a deposit before a specific machine is built. </p><p>When a project has multiple IRRs it may be more convenient to compute the IRR of the project with the benefits reinvested.<sup id="cite_ref-cautionary_14-0" class="reference"><a href="#cite_note-cautionary-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Accordingly, MIRR is used, which has an assumed reinvestment rate, usually equal to the project's cost of capital. </p> <div class="mw-heading mw-heading3"><h3 id="Average_internal_rate_of_return_(AIRR)"><span id="Average_internal_rate_of_return_.28AIRR.29"></span>Average internal rate of return (AIRR)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=24" title="Edit section: Average internal rate of return (AIRR)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Magni (2010) introduced a new approach, named AIRR approach, based on the intuitive notion of mean, that solves the problems of the IRR.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> However, the above-mentioned difficulties are only some of the many flaws incurred by the IRR. Magni (2013) provided a detailed list of 18 flaws of the IRR and showed how the AIRR approach does not incur the IRR problems. <sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=25" title="Edit section: Mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematically, the value of the investment is assumed to undergo exponential growth or decay according to some <a href="/wiki/Rate_of_return" title="Rate of return">rate of return</a> (any value greater than −100%), with discontinuities for cash flows, and the IRR of a series of cash flows is defined as any rate of return that results in a <a href="/wiki/Net_present_value" title="Net present value">NPV</a> of zero (or equivalently, a rate of return that results in the correct value of zero after the last cash flow). </p><p>Thus, internal rate(s) of return follow from the NPV as a function of the rate of return. This function is <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>. Towards a rate of return of −100% the NPV approaches infinity with the sign of the last cash flow, and towards a rate of return of positive infinity the NPV approaches the first cash flow (the one at the present). Therefore, if the first and last cash flow have a different sign there exists an IRR. Examples of time series without an IRR: </p> <ul><li>Only negative cash flows — the NPV is negative for every rate of return.</li> <li>(−1, 1, −1), rather small positive cash flow between two negative cash flows; the NPV is a quadratic function of 1/(1 + <i>r</i>), where <i>r</i> is the rate of return, or put differently, a quadratic function of the <a href="/wiki/Annual_effective_discount_rate" title="Annual effective discount rate">discount rate</a> <i>r</i>/(1 + <i>r</i>); the highest NPV is −0.75, for <i>r</i> = 100%.</li></ul> <p>In the case of a series of exclusively negative cash flows followed by a series of exclusively positive ones, the resulting function of the rate of return is continuous and monotonically decreasing from positive infinity (when the rate of return approaches -100%) to the value of the first cash flow (when the rate of return approaches infinity), so there is a unique rate of return for which it is zero. Hence, the IRR is also unique (and equal). Although the NPV-function itself is not necessarily monotonically decreasing on its whole domain, it <i>is</i> at the IRR. </p><p>Similarly, in the case of a series of exclusively positive cash flows followed by a series of exclusively negative ones the IRR is also unique. </p><p>Finally, by <a href="/wiki/Descartes%27_rule_of_signs" title="Descartes' rule of signs">Descartes' rule of signs</a>, the number of internal rates of return can never be more than the number of changes in sign of cash flow. </p> <div class="mw-heading mw-heading2"><h2 id="The_reinvestment_debate">The reinvestment debate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=26" title="Edit section: The reinvestment debate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is often stated that IRR assumes reinvestment of all cash flows until the very end of the project. This assertion has been a matter of debate in the literature. </p><p>Sources stating that there is such a hidden assumption have been cited below.<sup id="cite_ref-cautionary_14-1" class="reference"><a href="#cite_note-cautionary-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-measuring_17-0" class="reference"><a href="#cite_note-measuring-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Other sources have argued that there is no IRR reinvestment assumption.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>To understand the source of this confusion let's consider an example with a 3-year bond of $1000 face value and coupon rate of 5% (or $50). </p> <table class="wikitable"> <caption> </caption> <tbody><tr> <th> </th> <th> </th> <th colspan="4">Future value, <p>coupons redeemed </p> </th> <th> </th> <th colspan="4">Future value, <p>coupons reinvested </p> </th></tr> <tr> <th> </th> <th> </th> <th>Starting balance </th> <th>Interest, 5% </th> <th>Cash in/out </th> <th>Ending balance </th> <th> </th> <th>Starting balance </th> <th>Interest, 5% </th> <th>Cash in/out </th> <th>Ending balance </th></tr> <tr> <td>Investment </td> <td> </td> <td> </td> <td> </td> <td>-1000 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>-1000 </td> <td> </td></tr> <tr> <td>1st year coupon </td> <td> </td> <td>1000 </td> <td>50 </td> <td>50 </td> <td>1000 </td> <td> </td> <td>1000 </td> <td>50 </td> <td>0 </td> <td>1050 </td></tr> <tr> <td>2nd year coupon </td> <td> </td> <td>1000 </td> <td>50 </td> <td>50 </td> <td>1000 </td> <td> </td> <td>1050 </td> <td>52.5 </td> <td>0 </td> <td>1102.5 </td></tr> <tr> <td>3rd year coupon + bond </td> <td> </td> <td>1000 </td> <td>50 </td> <td>1050 </td> <td>0 </td> <td> </td> <td>1102.5 </td> <td>55.125 </td> <td>1157.625 </td> <td>0 </td></tr> <tr> <td><b>Total return</b> </td> <td> </td> <td> </td> <td> </td> <td><b>150</b> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td><b>157.625</b> </td> <td> </td></tr> <tr> <td><b>IRR</b> </td> <td> </td> <td> </td> <td> </td> <td><b>5%</b> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td><b>5%</b> </td> <td> </td></tr></tbody></table> <p>As can be seen, although the total return is different the IRR is still the same. Put in other words, IRR is neutral to reinvestments made at the same rate. No matter whether the cash is taken out early or reinvested at the same rate and taken out late - the rate is the same. </p><p>To understand why, we need to calculate the present value (PV) of our future cash flows, effectively reproducing IRR calculations manually: </p> <table class="wikitable"> <tbody><tr> <th> </th> <th> </th> <th colspan="3">Coupons redeemed </th> <th> </th> <th colspan="3">Coupons reinvested </th></tr> <tr> <th> </th> <th> </th> <th>Cash in/out </th> <th>PV formula </th> <th>PV </th> <th> </th> <th>Cash in/out </th> <th>PV formula </th> <th>PV </th></tr> <tr> <td>Investment </td> <td> </td> <td>-1000 </td> <td>-1000 / (1+5%)^0 </td> <td>-1000 </td> <td> </td> <td>-1000 </td> <td>-1000 / (1+5%)^0 </td> <td>-1000 </td></tr> <tr> <td>1st year coupon </td> <td> </td> <td>50 </td> <td>50 / (1+5%)^1 </td> <td>47.62 </td> <td> </td> <td>0 </td> <td>0 / (1+5%)^1 </td> <td>0 </td></tr> <tr> <td>2nd year coupon </td> <td> </td> <td>50 </td> <td>50 / (1+5%)^2 </td> <td>45.35 </td> <td> </td> <td>0 </td> <td>0 / (1+5%)^2 </td> <td>0 </td></tr> <tr> <td>3rd year coupon + bond </td> <td> </td> <td>1050 </td> <td>1050 / (1+5%)^3 </td> <td>907.03 </td> <td> </td> <td>1157.625 </td> <td>1157.625 / (1+5%)^3 </td> <td>1000 </td></tr> <tr> <td><b>Total return</b> </td> <td> </td> <td><b>150</b> </td> <td> </td> <td><b>0</b> </td> <td> </td> <td><b>157.625</b> </td> <td> </td> <td><b>0</b> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="In_personal_finance">In personal finance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=27" title="Edit section: In personal finance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The IRR can be used to measure the money-weighted performance of financial investments such as an individual investor's brokerage account. For this scenario, an equivalent,<sup id="cite_ref-gtz_freqmath_24-0" class="reference"><a href="#cite_note-gtz_freqmath-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> more intuitive definition of the IRR is, "The IRR is the annual interest rate of the fixed rate account (like a somewhat idealized savings account) which, when subjected to the same deposits and withdrawals as the actual investment, has the same ending balance as the actual investment." This fixed rate account is also called the <i>replicating fixed rate account</i> for the investment. There are examples where the replicating fixed rate account encounters negative balances despite the fact that the actual investment did not.<sup id="cite_ref-gtz_freqmath_24-1" class="reference"><a href="#cite_note-gtz_freqmath-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> In those cases, the IRR calculation assumes that the same interest rate that is paid on positive balances is charged on negative balances. It has been shown that this way of charging interest is the root cause of the IRR's multiple solutions problem.<sup id="cite_ref-teichrow_et_al_1_25-0" class="reference"><a href="#cite_note-teichrow_et_al_1-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-teichrow_et_al_2_26-0" class="reference"><a href="#cite_note-teichrow_et_al_2-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> If the model is modified so that, as is the case in real life, an externally supplied cost of borrowing (possibly varying over time) is charged on negative balances, the multiple solutions issue disappears.<sup id="cite_ref-teichrow_et_al_1_25-1" class="reference"><a href="#cite_note-teichrow_et_al_1-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-teichrow_et_al_2_26-1" class="reference"><a href="#cite_note-teichrow_et_al_2-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> The resulting rate is called the <i>fixed rate equivalent</i> (<i>FREQ</i>).<sup id="cite_ref-gtz_freqmath_24-2" class="reference"><a href="#cite_note-gtz_freqmath-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Unannualized_internal_rate_of_return">Unannualized internal rate of return</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=28" title="Edit section: Unannualized internal rate of return"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the context of investment performance measurement, there is sometimes ambiguity in terminology between the periodic <a href="/wiki/Rate_of_return" title="Rate of return">rate of return</a>, such as the IRR as defined above, and a holding period return. The term <i>internal rate of return</i> (<i>IRR)</i> or <i>Since Inception Internal Rate of Return</i> (<i>SI-IRR)</i> is in some contexts used to refer to the unannualized return over the period, particularly for periods of less than a year.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=29" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Accounting_rate_of_return" title="Accounting rate of return">Accounting rate of return</a></li> <li><a href="/wiki/Capital_budgeting" title="Capital budgeting">Capital budgeting</a></li> <li><a href="/wiki/Discounted_cash_flow" title="Discounted cash flow">Discounted cash flow</a></li> <li><a href="/wiki/Modified_Dietz_method" title="Modified Dietz method">Modified Dietz method</a></li> <li><a href="/wiki/Modified_internal_rate_of_return" title="Modified internal rate of return">Modified internal rate of return</a></li> <li><a href="/wiki/Net_present_value" title="Net present value">Net present value</a></li> <li><a href="/wiki/Rate_of_return" title="Rate of return">Rate of return</a></li> <li><a href="/wiki/Simple_Dietz_method" title="Simple Dietz method">Simple Dietz method</a></li> <li><a href="/wiki/Marginal_efficiency_of_capital" title="Marginal efficiency of capital">Marginal efficiency of capital</a></li> <li><a href="/wiki/Return_on_investment" title="Return on investment">Return on investment</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-main269-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-main269_1-0">^</a></b></span> <span class="reference-text">Project Economics and Decision Analysis, Volume I: Deterministic Models, M.A.Main, Page 269</span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKellison2009" class="citation book cs1">Kellison, Stephen G. (2009). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/182552985"><i>The theory of interest</i></a> (Third ed.). Boston: McGraw-Hill Irwin. pp. 251–252. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-338244-9" title="Special:BookSources/978-0-07-338244-9"><bdi>978-0-07-338244-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/182552985">182552985</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+theory+of+interest&rft.place=Boston&rft.pages=251-252&rft.edition=Third&rft.pub=McGraw-Hill+Irwin&rft.date=2009&rft_id=info%3Aoclcnum%2F182552985&rft.isbn=978-0-07-338244-9&rft.aulast=Kellison&rft.aufirst=Stephen+G.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F182552985&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+rate+of+return" class="Z3988"></span></span> </li> <li id="cite_note-:1-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBroverman2010" class="citation book cs1">Broverman, Samuel A. 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Winsted, CT: ACTEX Publications, Inc. pp. 264–265. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56698-767-7" title="Special:BookSources/978-1-56698-767-7"><bdi>978-1-56698-767-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/651487023">651487023</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+of+investment+and+credit&rft.place=Winsted%2C+CT&rft.pages=264-265&rft.edition=5th&rft.pub=ACTEX+Publications%2C+Inc&rft.date=2010&rft_id=info%3Aoclcnum%2F651487023&rft.isbn=978-1-56698-767-7&rft.aulast=Broverman&rft.aufirst=Samuel+A.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F651487023&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+rate+of+return" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://npvcalculator.net/irr-calculator/">"IRR Calculator: Calculate Internal Rate of Return Online - NPV Calculator"</a>. 2023-05-21<span class="reference-accessdate">. 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(2013) <a rel="nofollow" class="external text" href="http://www.tandfonline.com/doi/full/10.1080/0013791X.2012.745916">"The Internal-Rate-of-Return approach and the AIRR paradigm: A refutation and a corroboration"</a> The Engineering Economist, 58(2), 73‒111.</span> </li> <li id="cite_note-measuring-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-measuring_17-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external autonumber" href="http://people.stern.nyu.edu/adamodar/pdfiles/ovhds/ch5.pdf#page=43">[1]</a> Measuring Investment Returns</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Dudley, C.L., “A note on reinvestment assumptions in choosing between net present value and internal rate of return.” <i>Journal of Finance</i> 27(4), 1972, 907–15.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Keane, S.M., “The internal rate of return and the reinvestment fallacy.” <i>Abacus</i> 15(1), 1979, 48–55.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Lohmann, J.R., “The IRR, NPV and the fallacy of the reinvestment rate assumptions”. <i>The Engineering Economist</i> 33(4), 1988, 303–30.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Keef, S.P., and M.L. Roush, “Discounted cash flow methods and the fallacious reinvestment assumptions: a review of recent texts.” <i>Accounting Education</i> 10(1), 2001, 105-116.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Rich, S.P., and J.T. Rose, “Re-examining an Old Question: Does the IRR Method Implicitly Assume a Reinvestment Rate?” <i>Journal of Financial Education</i> 10(1), 2014, 105-116.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Dudley, Magni, Carlo Alberto and Martin, John D., “The Reinvestment Rate Assumption Fallacy for IRR and NPV: A Pedagogical Note” '<a rel="nofollow" class="external free" href="https://mpra.ub.uni-muenchen.de/83889/'">https://mpra.ub.uni-muenchen.de/83889/'</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged September 2024">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup>, 2017</span> </li> <li id="cite_note-gtz_freqmath-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-gtz_freqmath_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-gtz_freqmath_24-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-gtz_freqmath_24-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.greaterthanzero.com/freqmath">The Mathematics of the Fixed Rate Equivalent</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged September 2024">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup>, a GreaterThanZero White Paper.</span> </li> <li id="cite_note-teichrow_et_al_1-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-teichrow_et_al_1_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-teichrow_et_al_1_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Teichroew, D., Robicheck, A., and Montalbano, M., Mathematical analysis of rates of return under certainty, Management Science Vol. 11 Nr. 3, January 1965, 395–403.</span> </li> <li id="cite_note-teichrow_et_al_2-26"><span class="mw-cite-backlink">^ <a href="#cite_ref-teichrow_et_al_2_26-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-teichrow_et_al_2_26-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Teichroew, D., Robicheck, A., and Montalbano, M., An analysis of criteria for investment and financing decisions under certainty, Management Science Vol. 12 Nr. 3, November 1965, 151–179.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external autonumber" href="http://www.cfapubs.org/doi/pdf/10.2469/ccb.v2010.n5.1#page=49">[2]</a> Global Investment Performance Standards</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=31" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Bruce J. Feibel. <i>Investment Performance Measurement</i>. New York: Wiley, 2003. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-26849-6" title="Special:BookSources/0-471-26849-6">0-471-26849-6</a></li> <li>Ray Martin, <a rel="nofollow" class="external text" href="http://members.tripod.com/~Ray_Martin/DCF/nr7aa003.html">INTERNAL RATE OF RETURN REVISITED</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_rate_of_return&action=edit&section=32" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20041028124340/http://hspm.sph.sc.edu/Courses/Econ/irr/irr.html">Economics Interactive Lecture from University of South Carolina</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Internal_rate_of_return&oldid=1251879374">https://en.wikipedia.org/w/index.php?title=Internal_rate_of_return&oldid=1251879374</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Corporate_finance" title="Category:Corporate finance">Corporate finance</a></li><li><a href="/wiki/Category:Investment" title="Category:Investment">Investment</a></li><li><a href="/wiki/Category:Capital_budgeting" title="Category:Capital budgeting">Capital budgeting</a></li><li><a href="/wiki/Category:Management_cybernetics" title="Category:Management cybernetics">Management cybernetics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:All_articles_with_dead_external_links" title="Category:All articles with dead external links">All articles with dead external links</a></li><li><a href="/wiki/Category:Articles_with_dead_external_links_from_September_2024" title="Category:Articles with dead external links from September 2024">Articles with dead external links from September 2024</a></li><li><a href="/wiki/Category:Articles_with_permanently_dead_external_links" title="Category:Articles with permanently dead external links">Articles with permanently dead external links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_April_2021" title="Category:Articles needing additional references from April 2021">Articles needing additional references from April 2021</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 18 October 2024, at 16:12<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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