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class="vector-toc-list"> </ul> </li> <li id="toc-Risk_measure_and_risk_metric" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Risk_measure_and_risk_metric"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Risk measure and risk metric</span> </div> </a> <ul id="toc-Risk_measure_and_risk_metric-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-VaR_risk_management" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#VaR_risk_management"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>VaR risk management</span> </div> </a> <ul id="toc-VaR_risk_management-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computation_methods" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computation_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Computation methods</span> </div> </a> <ul id="toc-Computation_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Backtesting" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Backtesting"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Backtesting</span> </div> </a> <ul id="toc-Backtesting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Criticism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Criticism"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Criticism</span> </div> </a> <button aria-controls="toc-Criticism-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 Criticism subsection</span> </button> <ul id="toc-Criticism-sublist" class="vector-toc-list"> <li id="toc-VaR,_CVaR,_RVaR_and_EVaR" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#VaR,_CVaR,_RVaR_and_EVaR"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>VaR, CVaR, RVaR and EVaR</span> </div> </a> <ul id="toc-VaR,_CVaR,_RVaR_and_EVaR-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <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" 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risk</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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/Value_at_risk" title="Value at risk – Bulgarian" lang="bg" hreflang="bg" data-title="Value at risk" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Value_at_risk" title="Value at risk – Czech" lang="cs" hreflang="cs" data-title="Value at risk" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Value-at-Risk" title="Value-at-Risk – Danish" lang="da" hreflang="da" data-title="Value-at-Risk" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Value_at_Risk" title="Value at Risk – German" lang="de" hreflang="de" data-title="Value at Risk" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BE%CE%AF%CE%B1_%CF%83%CE%B5_%CE%9A%CE%AF%CE%BD%CE%B4%CF%85%CE%BD%CE%BF" title="Αξία σε Κίνδυνο – Greek" lang="el" hreflang="el" data-title="Αξία σε Κίνδυνο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Valor_en_riesgo" title="Valor en riesgo – Spanish" lang="es" hreflang="es" data-title="Valor en riesgo" 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/%D8%AF%D8%A7%D8%B1%D8%A7%DB%8C%DB%8C_%D8%AF%D8%B1_%D8%AE%D8%B7%D8%B1" title="دارایی در خطر – Persian" lang="fa" hreflang="fa" data-title="دارایی در خطر" 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/Value_at_risk" title="Value at risk – French" lang="fr" hreflang="fr" data-title="Value at risk" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Nilai_pada_risiko" title="Nilai pada risiko – Indonesian" lang="id" hreflang="id" data-title="Nilai pada risiko" 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/Valore_a_rischio" title="Valore a rischio – Italian" lang="it" hreflang="it" data-title="Valore a rischio" 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%A2%D7%A8%D7%9A_%D7%91%D7%A1%D7%99%D7%9B%D7%95%D7%9F" title="ערך בסיכון – Hebrew" lang="he" hreflang="he" data-title="ערך בסיכון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vert%C4%97s_poky%C4%8Dio_rizika" title="Vertės pokyčio rizika – Lithuanian" lang="lt" hreflang="lt" data-title="Vertės pokyčio rizika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Value_at_Risk" title="Value at Risk – Hungarian" lang="hu" hreflang="hu" data-title="Value at Risk" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82_%D0%B8%D0%B7%D0%BB%D0%BE%D0%B6%D0%B5%D0%BD%D0%B0_%D0%BD%D0%B0_%D1%80%D0%B8%D0%B7%D0%B8%D0%BA" title="Вредност изложена на ризик – Macedonian" lang="mk" hreflang="mk" data-title="Вредност изложена на ризик" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nilai_berisiko" title="Nilai berisiko – Malay" lang="ms" hreflang="ms" data-title="Nilai berisiko" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%90%E3%83%AA%E3%83%A5%E3%83%BC%E3%83%BB%E3%82%A2%E3%83%83%E3%83%88%E3%83%BB%E3%83%AA%E3%82%B9%E3%82%AF" title="バリュー・アット・リスク – Japanese" lang="ja" hreflang="ja" data-title="バリュー・アット・リスク" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Value_at_risk" title="Value at risk – Polish" lang="pl" hreflang="pl" data-title="Value at risk" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Value_at_Risk" title="Value at Risk – Portuguese" lang="pt" hreflang="pt" data-title="Value at Risk" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%BE%D0%B8%D0%BC%D0%BE%D1%81%D1%82%D1%8C_%D0%BF%D0%BE%D0%B4_%D1%80%D0%B8%D1%81%D0%BA%D0%BE%D0%BC" title="Стоимость под риском – Russian" lang="ru" hreflang="ru" data-title="Стоимость под риском" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Hodnota_v_riziku" title="Hodnota v riziku – Slovak" lang="sk" hreflang="sk" data-title="Hodnota v riziku" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Value_at_risk" title="Value at risk – Swedish" lang="sv" hreflang="sv" data-title="Value at risk" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A1%E0%B8%B9%E0%B8%A5%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%AA%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%87" title="มูลค่าความเสี่ยง – Thai" lang="th" hreflang="th" data-title="มูลค่าความเสี่ยง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Riske_maruz_de%C4%9Fer" title="Riske maruz değer – Turkish" lang="tr" hreflang="tr" data-title="Riske maruz değer" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/Value_at_risk" title="Value at risk – Ukrainian" lang="uk" hreflang="uk" data-title="Value at risk" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8F%97%E9%9A%AA%E5%83%B9%E5%80%BC" title="受險價值 – Cantonese" lang="yue" hreflang="yue" data-title="受險價值" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%A3%8E%E9%99%A9%E4%BB%B7%E5%80%BC" title="风险价值 – Chinese" lang="zh" hreflang="zh" data-title="风险价值" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet 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searchaux" style="display:none">Estimated potential loss for an investment under a given set of conditions</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/Valuation_risk" title="Valuation risk">Valuation risk</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:VaR_diagram.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/VaR_diagram.JPG/300px-VaR_diagram.JPG" decoding="async" width="300" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/VaR_diagram.JPG/450px-VaR_diagram.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/VaR_diagram.JPG/600px-VaR_diagram.JPG 2x" data-file-width="911" data-file-height="620" /></a><figcaption>The 5% Value at Risk of a hypothetical profit-and-loss probability density function</figcaption></figure> <p><b>Value at risk</b> (<b>VaR</b>) is a measure of the risk of loss of investment/capital. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses. </p><p>For a given <a href="/wiki/Portfolio_(finance)" title="Portfolio (finance)">portfolio</a>, <a href="/wiki/Time_horizon" title="Time horizon">time horizon</a>, and <a href="/wiki/Probability" title="Probability">probability</a> <i>p</i>, the <i>p</i> VaR can be defined informally as the maximum possible loss during that time after excluding all worse outcomes whose combined probability is at most <i>p</i>. This assumes <a href="/wiki/Mark_to_market_accounting" class="mw-redirect" title="Mark to market accounting">mark-to-market</a> pricing, and no trading in the portfolio.<sup id="cite_ref-Jorion_1-0" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For example, if a portfolio of stocks has a one-day 5% VaR of $1 million, that means that there is a 0.05 probability that the portfolio will fall in value by more than $1 million over a one-day period if there is no trading. Informally, a loss of $1 million or more on this portfolio is expected on 1 day out of 20 days (because of 5% probability). </p><p>More formally, <i>p</i> VaR is defined such that the probability of a loss greater than VaR is (at most) <i>(1-p)</i> while the probability of a loss less than VaR is (at least) <i>p</i>. A loss which exceeds the VaR threshold is termed a "VaR breach".<sup id="cite_ref-Holton_2-0" class="reference"><a href="#cite_note-Holton-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>For a fixed <i>p</i>, the <i>p</i> VaR does not assess the magnitude of loss when a VaR breach occurs and therefore is considered by some to be a questionable metric for risk management. For instance, assume someone makes a bet that flipping a coin seven times will not give seven heads. The terms are that they win $100 if this does not happen (with probability 127/128) and lose $12,700 if it does (with probability 1/128). That is, the possible loss amounts are $0 or $12,700. The 1% VaR is then $0, because the probability of any loss at all is 1/128 which is less than 1%. They are, however, exposed to a possible loss of $12,700 which can be expressed as the <i>p</i> VaR for any <i>p ≤ 0.78125% (1/128)</i>.<sup id="cite_ref-Einhorn_II_3-0" class="reference"><a href="#cite_note-Einhorn_II-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>VaR has four main uses in <a href="/wiki/Finance" title="Finance">finance</a>: <a href="/wiki/Risk_management" title="Risk management">risk management</a>, financial <a href="/wiki/Comptroller" title="Comptroller">control</a>, <a href="/wiki/Financial_statements" class="mw-redirect" title="Financial statements">financial reporting</a> and computing <a href="/wiki/Capital_requirement" title="Capital requirement">regulatory capital</a>. VaR is sometimes used in non-financial applications as well.<sup id="cite_ref-McNeil_4-0" class="reference"><a href="#cite_note-McNeil-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> However, it is a controversial risk management tool. </p><p>Important related ideas are economic capital, <a href="/wiki/Backtesting" title="Backtesting">backtesting</a>, <a href="/wiki/Stress_test_(financial)" title="Stress test (financial)">stress testing</a>, <a href="/wiki/Expected_shortfall" title="Expected shortfall">expected shortfall</a>, and <a href="/wiki/Tail_conditional_expectation" class="mw-redirect" title="Tail conditional expectation">tail conditional expectation</a>.<sup id="cite_ref-Dowd_5-0" class="reference"><a href="#cite_note-Dowd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Details">Details</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=1" title="Edit section: Details"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Common parameters for VaR are 1% and 5% probabilities and one day and two week horizons, although other combinations are in use.<sup id="cite_ref-Pearson_6-0" class="reference"><a href="#cite_note-Pearson-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The reason for assuming normal markets and no trading, and to restricting loss to things measured in <a href="/wiki/Financial_statements" class="mw-redirect" title="Financial statements">daily accounts</a>, is to make the loss <a href="/wiki/Observability" title="Observability">observable</a>. In some extreme financial events it can be impossible to determine losses, either because market prices are unavailable or because the loss-bearing institution breaks up. Some longer-term consequences of disasters, such as lawsuits, loss of market confidence and employee morale and impairment of brand names can take a long time to play out, and may be hard to allocate among specific prior decisions. VaR marks the boundary between normal days and extreme events. Institutions can lose far more than the VaR amount; all that can be said is that they will not do so very often.<sup id="cite_ref-Unbearable_7-0" class="reference"><a href="#cite_note-Unbearable-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The probability level is about equally often specified as one minus the probability of a VaR break, so that the VaR in the example above would be called a one-day 95% VaR instead of one-day 5% VaR. This generally does not lead to confusion because the probability of VaR breaks is almost always small, certainly less than 50%.<sup id="cite_ref-Jorion_1-1" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Although it virtually always represents a loss, VaR is conventionally reported as a positive number. A negative VaR would imply the portfolio has a high probability of making a profit, for example a one-day 5% VaR of negative <span style="white-space: nowrap">$1 million</span> implies the portfolio has a 95% chance of making more than <span style="white-space: nowrap">$1 million</span> over the next day.<sup id="cite_ref-Crouhy_8-0" class="reference"><a href="#cite_note-Crouhy-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Another inconsistency is that VaR is sometimes taken to refer to profit-and-loss at the end of the period, and sometimes as the maximum loss at any point during the period. The original definition was the latter, but in the early 1990s when VaR was aggregated across trading desks and time zones, end-of-day valuation was the only reliable number so the former became the <i><a href="/wiki/De_facto" title="De facto">de facto</a></i> definition. As people began using multiday VaRs in the second half of the 1990s, they almost always estimated the distribution at the end of the period only. It is also easier theoretically to deal with a point-in-time estimate versus a maximum over an interval. Therefore, the end-of-period definition is the most common both in theory and practice today.<sup id="cite_ref-Lopez_9-0" class="reference"><a href="#cite_note-Lopez-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Varieties">Varieties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=2" title="Edit section: Varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The definition of VaR is <a href="/wiki/Constructive_proof" title="Constructive proof">nonconstructive</a>; it specifies a <a href="/wiki/Property_(philosophy)" title="Property (philosophy)">property</a> VaR must have, but not how to compute VaR. Moreover, there is wide scope for interpretation in the definition.<sup id="cite_ref-Roundtable_I_10-0" class="reference"><a href="#cite_note-Roundtable_I-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This has led to two broad types of VaR, one used primarily in <a href="/wiki/Risk_management" title="Risk management">risk management</a> and the other primarily for risk measurement. The distinction is not sharp, however, and hybrid versions are typically used in financial <a href="/wiki/Comptroller" title="Comptroller">control</a>, <a href="/wiki/Financial_statements" class="mw-redirect" title="Financial statements">financial reporting</a> and computing <a href="/wiki/Capital_requirement" title="Capital requirement">regulatory capital</a>.<sup id="cite_ref-Brown_11-0" class="reference"><a href="#cite_note-Brown-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>To a <a href="/wiki/Risk_management" title="Risk management">risk manager</a>, VaR is a system, not a number. The system is run periodically (usually daily) and the published number is compared to the computed price movement in opening positions over the time horizon. There is never any subsequent adjustment to the published VaR, and there is no distinction between VaR breaks caused by input errors (including <a href="/wiki/Information_technology" title="Information technology">IT</a> breakdowns, <a href="/wiki/Fraud" title="Fraud">fraud</a> and <a href="/wiki/Rogue_trader" title="Rogue trader">rogue trading</a>), computation errors (including failure to produce a VaR on time) and market movements.<sup id="cite_ref-Wilmott_12-0" class="reference"><a href="#cite_note-Wilmott-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Frequency_probability" class="mw-redirect" title="Frequency probability">frequentist</a> claim is made that the long-term frequency of VaR breaks will equal the specified probability, within the limits of sampling error, and that the VaR breaks will be <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> in time and independent of the level of VaR. This claim is validated by a <a href="/wiki/Backtesting" title="Backtesting">backtest</a>, a comparison of published VaRs to actual price movements. In this interpretation, many different systems could produce VaRs with equally good backtests, but wide disagreements on daily VaR values.<sup id="cite_ref-Jorion_1-2" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For risk measurement a number is needed, not a system. A <a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> claim is made that given the information and beliefs at the time, the <a href="/wiki/Bayesian_probability" title="Bayesian probability">subjective probability</a> of a VaR break was the specified level. VaR is adjusted after the fact to correct errors in inputs and computation, but not to incorporate information unavailable at the time of computation.<sup id="cite_ref-Crouhy_8-1" class="reference"><a href="#cite_note-Crouhy-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In this context, "backtest" has a different meaning. Rather than comparing published VaRs to actual market movements over the period of time the system has been in operation, VaR is retroactively computed on scrubbed data over as long a period as data are available and deemed relevant. The same position data and pricing models are used for computing the VaR as determining the price movements.<sup id="cite_ref-Holton_2-1" class="reference"><a href="#cite_note-Holton-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Although some of the sources listed here treat only one kind of VaR as legitimate, most of the recent ones seem to agree that risk management VaR is superior for making short-term and tactical decisions in the present, while risk measurement VaR should be used for understanding the past, and making medium term and strategic decisions for the future. When VaR is used for <a href="/wiki/Comptroller" title="Comptroller">financial control</a> or <a href="/wiki/Financial_statements" class="mw-redirect" title="Financial statements">financial reporting</a> it should incorporate elements of both. For example, if a <a href="/wiki/Trader_(finance)" title="Trader (finance)">trading desk</a> is held to a VaR limit, that is both a risk-management rule for deciding what risks to allow today, and an input into the risk measurement computation of the desk's risk-adjusted <a href="/wiki/Return_(finance)" class="mw-redirect" title="Return (finance)">return</a> at the end of the reporting period.<sup id="cite_ref-Dowd_5-1" class="reference"><a href="#cite_note-Dowd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_governance">In governance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=3" title="Edit section: In governance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>VaR can also be applied to <a href="/w/index.php?title=Financial_governance&action=edit&redlink=1" class="new" title="Financial governance (page does not exist)">governance</a> of endowments, trusts, and pension plans. Essentially, trustees adopt portfolio Values-at-Risk metrics for the entire pooled account and the diversified parts individually managed. Instead of probability estimates they simply define maximum levels of acceptable loss for each. Doing so provides an easy metric for oversight and adds accountability as managers are then directed to manage, but with the additional constraint to avoid losses within a defined risk parameter. VaR utilized in this manner adds relevance as well as an easy way to monitor risk measurement control far more intuitive than Standard Deviation of Return. Use of VaR in this context, as well as a worthwhile critique on board governance practices as it relates to investment management oversight in general can be found in <i>Best Practices in Governance.</i><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-heading2"><h2 id="Mathematical_definition">Mathematical definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=4" title="Edit section: Mathematical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> be a profit and loss distribution (loss negative and profit positive). The VaR at level <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 \alpha \in (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df576f7940384416d1553ab063704d37bf99420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.496ex; height:2.843ex;" alt="{\displaystyle \alpha \in (0,1)}"></span> is the smallest number <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> such that the probability that <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 Y:=-X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>:=</mo> <mo>−</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y:=-X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d19c031f363aa44697a292d136a5ed9677a7a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.307ex; height:2.343ex;" alt="{\displaystyle Y:=-X}"></span> does not exceed <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> is at least <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-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9afa7876fb8b4fb8c4d8039ebed6cd1cbc4781cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.49ex; height:2.343ex;" alt="{\displaystyle 1-\alpha }"></span>. Mathematically, <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 {VaR} _{\alpha }(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>VaR</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {VaR} _{\alpha }(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a9338ad7ab25257903853f42e33a740dd47728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.69ex; height:2.843ex;" alt="{\displaystyle \operatorname {VaR} _{\alpha }(X)}"></span> is 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 (1-\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dcc25f05dca60e358d4d22e8342fad5ad7affbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.3ex; height:2.843ex;" alt="{\displaystyle (1-\alpha )}"></span>-<a href="/wiki/Quantile" title="Quantile">quantile</a> 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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, i.e., </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 {VaR} _{\alpha }(X)=-\inf {\big \{}x\in \mathbb {R} :F_{X}(x)>\alpha {\big \}}=F_{Y}^{-1}(1-\alpha ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>VaR</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mo>=</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {VaR} _{\alpha }(X)=-\inf {\big \{}x\in \mathbb {R} :F_{X}(x)>\alpha {\big \}}=F_{Y}^{-1}(1-\alpha ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f2d2e87c32176846e5b0974f21be1a11df2666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:54.562ex; height:3.343ex;" alt="{\displaystyle \operatorname {VaR} _{\alpha }(X)=-\inf {\big \{}x\in \mathbb {R} :F_{X}(x)>\alpha {\big \}}=F_{Y}^{-1}(1-\alpha ).}"></span><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><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></dd></dl> <p>This is the most general definition of VaR and the two identities are equivalent (indeed, for any real random variable <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> its <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</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 F_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/062f285db773e329f6c270cb6b65fa076996c941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}"></span> is well defined). However this formula cannot be used directly for calculations unless we assume that <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> has some parametric distribution. </p><p>Risk managers typically assume that some fraction of the bad events will have undefined losses, either because markets are closed or illiquid, or because the entity bearing the loss breaks apart or loses the ability to compute accounts. Therefore, they do not accept results based on the assumption of a well-defined probability distribution.<sup id="cite_ref-Unbearable_7-1" class="reference"><a href="#cite_note-Unbearable-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Nassim_Taleb" class="mw-redirect" title="Nassim Taleb">Nassim Taleb</a> has labeled this assumption, "charlatanism".<sup id="cite_ref-Taleb_II_16-0" class="reference"><a href="#cite_note-Taleb_II-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> On the other hand, many academics prefer to assume a well-defined distribution, albeit usually one with <a href="/wiki/Kurtosis" title="Kurtosis">fat tails</a>.<sup id="cite_ref-Jorion_1-3" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This point has probably caused more contention among VaR theorists than any other.<sup id="cite_ref-Roundtable_I_10-1" class="reference"><a href="#cite_note-Roundtable_I-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Value at risk can also be written as a <a href="/wiki/Distortion_risk_measure" title="Distortion risk measure">distortion risk measure</a> given by the <a href="/wiki/Distortion_function" title="Distortion function">distortion function</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 g(x)={\begin{cases}0&{\text{if }}0\leq x<1-\alpha \\1&{\text{if }}1-\alpha \leq x\leq 1\end{cases}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo><</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)={\begin{cases}0&{\text{if }}0\leq x<1-\alpha \\1&{\text{if }}1-\alpha \leq x\leq 1\end{cases}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47f1c0fd17fe58441b834211aba8eb85dea8d04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.099ex; height:6.176ex;" alt="{\displaystyle g(x)={\begin{cases}0&{\text{if }}0\leq x<1-\alpha \\1&{\text{if }}1-\alpha \leq x\leq 1\end{cases}}.}"></span><sup id="cite_ref-Wirch_17-0" class="reference"><a href="#cite_note-Wirch-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-PropertiesDRM_18-0" class="reference"><a href="#cite_note-PropertiesDRM-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Risk_measure_and_risk_metric">Risk measure and risk metric</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=5" title="Edit section: Risk measure and risk metric"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term "VaR" is used both for a <a href="/wiki/Risk_measure" title="Risk measure">risk measure</a> and a <a href="/wiki/Risk_metric" title="Risk metric">risk metric</a>. This sometimes leads to confusion. Sources earlier than 1995 usually emphasize the risk measure, later sources are more likely to emphasize the metric. </p><p>The VaR risk measure defines risk as <a href="/wiki/Mark_to_market_accounting" class="mw-redirect" title="Mark to market accounting">mark-to-market</a> loss on a fixed portfolio over a fixed time horizon. There are many alternative risk measures in finance. Given the inability to use mark-to-market (which uses market prices to define loss) for future performance, loss is often defined (as a substitute) as change in <a href="/wiki/Intrinsic_value_(finance)" title="Intrinsic value (finance)">fundamental value</a>. For example, if an institution holds a <a href="/wiki/Loan" title="Loan">loan</a> that declines in market price because <a href="/wiki/Interest" title="Interest">interest</a> rates go up, but has no change in cash flows or credit quality, some systems do not recognize a loss. Also some try to incorporate the <a href="/wiki/Economics" title="Economics">economic</a> cost of harm not measured in daily <a href="/wiki/Financial_statements" class="mw-redirect" title="Financial statements">financial statements</a>, such as loss of market confidence or employee morale, impairment of brand names or lawsuits.<sup id="cite_ref-Dowd_5-2" class="reference"><a href="#cite_note-Dowd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Rather than assuming a static portfolio over a fixed time horizon, some risk measures incorporate the dynamic effect of expected trading (such as a <a href="/wiki/Order_(exchange)" title="Order (exchange)">stop loss order</a>) and consider the expected holding period of positions.<sup id="cite_ref-Dowd_5-3" class="reference"><a href="#cite_note-Dowd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The VaR risk metric summarizes the <a href="/wiki/Probability_distribution" title="Probability distribution">distribution</a> of possible losses by a <a href="/wiki/Quantile_function" title="Quantile function">quantile</a>, a point with a specified probability of greater losses. A common alternative metric is <a href="/wiki/Expected_shortfall" title="Expected shortfall">expected shortfall</a>.<sup id="cite_ref-Jorion_1-4" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="VaR_risk_management">VaR risk management</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=6" title="Edit section: VaR risk management"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Financial_risk_management#Banking" title="Financial risk management">Financial risk management § Banking</a></div><p> Supporters of VaR-based risk management claim the first and possibly greatest benefit of VaR is the improvement in <a href="/wiki/Systems" class="mw-redirect" title="Systems">systems</a> and modeling it forces on an institution. In 1997, <a href="/wiki/Philippe_Jorion" title="Philippe Jorion">Philippe Jorion</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170706074507/http://www.derivativesstrategy.com/magazine/archive/1997/0497fea2.asp">wrote</a>:<sup id="cite_ref-Jorion_I_19-0" class="reference"><a href="#cite_note-Jorion_I-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></p><blockquote><p>[T]he greatest benefit of VAR lies in the imposition of a structured methodology for critically thinking about risk. Institutions that go through the process of computing their VAR are forced to confront their exposure to financial risks and to set up a proper risk management function. Thus the process of getting to VAR may be as important as the number itself.</p></blockquote> <p>Publishing a daily number, on-time and with specified <a href="/wiki/Statistics" title="Statistics">statistical</a> properties holds every part of a trading organization to a high objective standard. Robust backup systems and default assumptions must be implemented. Positions that are reported, modeled or priced incorrectly stand out, as do data feeds that are inaccurate or late and systems that are too-frequently down. Anything that affects profit and loss that is left out of other reports will show up either in inflated VaR or excessive VaR breaks. "A risk-taking institution that <i>does not</i> compute VaR might escape disaster, but an institution that <i>cannot</i> compute VaR will not."<sup id="cite_ref-Einhorn_I_20-0" class="reference"><a href="#cite_note-Einhorn_I-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The second claimed benefit of VaR is that it separates risk into two regimes. Inside the VaR limit, conventional <a href="/wiki/Statistical" class="mw-redirect" title="Statistical">statistical</a> methods are reliable. Relatively short-term and specific data can be used for analysis. Probability estimates are meaningful because there are enough data to test them. In a sense, there is no true risk because these are a sum of many <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> observations with a left bound on the outcome. For example, a casino does not worry about whether red or black will come up on the next roulette spin. Risk managers encourage productive risk-taking in this regime, because there is little true cost. People tend to worry too much about these risks because they happen frequently, and not enough about what might happen on the worst days.<sup id="cite_ref-Haug_21-0" class="reference"><a href="#cite_note-Haug-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Outside the VaR limit, all bets are off. Risk should be analyzed with <a href="/wiki/Stress_testing" title="Stress testing">stress testing</a> based on long-term and broad market data.<sup id="cite_ref-Zask_22-0" class="reference"><a href="#cite_note-Zask-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Probability statements are no longer meaningful.<sup id="cite_ref-Roundtable_II_23-0" class="reference"><a href="#cite_note-Roundtable_II-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Knowing the distribution of losses beyond the VaR point is both impossible and useless. The risk manager should concentrate instead on making sure good plans are in place to limit the loss if possible, and to survive the loss if not.<sup id="cite_ref-Jorion_1-5" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>One specific system uses three regimes.<sup id="cite_ref-Size_24-0" class="reference"><a href="#cite_note-Size-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <ol><li>One to three times VaR are normal occurrences. Periodic VaR breaks are expected. The loss distribution typically has <a href="/wiki/Kurtosis" title="Kurtosis">fat tails</a>, and there might be more than one break in a short period of time. Moreover, markets may be abnormal and trading may exacerbate losses, and losses taken may not be measured in daily <a href="/wiki/Financial_statements" class="mw-redirect" title="Financial statements">marks</a>, such as lawsuits, loss of employee morale and market confidence and impairment of brand names. An institution that cannot deal with three times VaR losses as routine events probably will not survive long enough to put a VaR system in place.</li> <li>Three to ten times VaR is the range for <a href="/wiki/Stress_testing" title="Stress testing">stress testing</a>. Institutions should be confident they have examined all the foreseeable events that will cause losses in this range, and are prepared to survive them. These events are too rare to estimate probabilities reliably, so risk/return calculations are useless.</li> <li>Foreseeable events should not cause losses beyond ten times VaR. If they do they should be <a href="/wiki/Hedge_(finance)" title="Hedge (finance)">hedged</a> or insured, or the business plan should be changed to avoid them, or VaR should be increased. It is hard to run a business if foreseeable losses are orders of magnitude larger than very large everyday losses. It is hard to plan for these events because they are out of scale with daily experience.</li></ol> <p>Another reason VaR is useful as a metric is due to its ability to compress the riskiness of a portfolio to a single number, making it comparable across different portfolios (of different assets). Within any portfolio it is also possible to isolate specific positions that might better hedge the portfolio to reduce, and minimise, the VaR.<sup id="cite_ref-PHIRS_25-0" class="reference"><a href="#cite_note-PHIRS-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Computation_methods">Computation methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=7" title="Edit section: Computation methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>VaR can be estimated either parametrically (for example, <a href="/wiki/Variance" title="Variance">variance</a>-<a href="/wiki/Covariance" title="Covariance">covariance</a> VaR or <a href="/wiki/Greeks_(finance)#Delta" title="Greeks (finance)">delta</a>-<a href="/wiki/Greeks_(finance)#Gamma" title="Greeks (finance)">gamma</a> VaR) or nonparametrically (for examples, historical <a href="/wiki/Simulation" title="Simulation">simulation</a> VaR or <a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">resampled</a> VaR).<sup id="cite_ref-Dowd_5-4" class="reference"><a href="#cite_note-Dowd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Unbearable_7-2" class="reference"><a href="#cite_note-Unbearable-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Nonparametric methods of VaR estimation are discussed in Markovich<sup id="cite_ref-Markovich_26-0" class="reference"><a href="#cite_note-Markovich-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and Novak.<sup id="cite_ref-Novak_27-0" class="reference"><a href="#cite_note-Novak-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> A comparison of a number of strategies for VaR prediction is given in Kuester et al.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>A McKinsey report<sup id="cite_ref-McKinsey_29-0" class="reference"><a href="#cite_note-McKinsey-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> published in May 2012 estimated that 85% of large banks were using <a href="/wiki/Historical_simulation_(finance)" title="Historical simulation (finance)">historical simulation</a>. The other 15% used <a href="/wiki/Monte_Carlo_methods_in_finance" title="Monte Carlo methods in finance">Monte Carlo methods</a> (often applying <a href="/wiki/Principal_component_analysis#Quantitative_finance" title="Principal component analysis">a PCA decomposition</a>) . </p> <div class="mw-heading mw-heading2"><h2 id="Backtesting">Backtesting</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=8" title="Edit section: Backtesting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Backtesting is the process to determine the accuracy of VaR forecasts vs. actual portfolio profit and losses. A key advantage to VaR over most other measures of risk such as <a href="/wiki/Expected_shortfall" title="Expected shortfall">expected shortfall</a> is the availability of several <a href="/wiki/Backtesting" title="Backtesting">backtesting</a> procedures for validating a set of VaR forecasts. Early examples of backtests can be found in Christoffersen (1998),<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> later generalized by Pajhede (2017),<sup id="cite_ref-Pajhede_2017_597–613_31-0" class="reference"><a href="#cite_note-Pajhede_2017_597–613-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> which models a "hit-sequence" of losses greater than the VaR and proceed to tests for these "hits" to be independent from one another and with a correct probability of occurring. E.g. a 5% probability of a loss greater than VaR should be observed over time when using a 95% VaR, these hits should occur independently. </p><p>A number of other backtests are available which model the time between hits in the hit-sequence, see Christoffersen and Pelletier (2004),<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Haas (2006),<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Tokpavi et al. (2014).<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> and Pajhede (2017)<sup id="cite_ref-Pajhede_2017_597–613_31-1" class="reference"><a href="#cite_note-Pajhede_2017_597–613-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> As pointed out in several of the papers, the asymptotic distribution is often poor when considering high levels of coverage, e.g. a 99% VaR, therefore the parametric bootstrap method of Dufour (2006)<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> is often used to obtain correct size properties for the tests. Backtest toolboxes are available in Matlab,<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> or <a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/backtest/backtest.pdf">R</a>—though only the first implements the parametric bootstrap method. </p><p>The second pillar of <a href="/wiki/Basel_II" title="Basel II">Basel II</a> includes a <a href="/wiki/Backtesting" title="Backtesting">backtesting</a> step to validate the VaR figures. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=9" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The problem of risk measurement is an old one in <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Economics" title="Economics">economics</a> and <a href="/wiki/Finance" title="Finance">finance</a>. Financial risk management has been a concern of regulators and financial executives for a long time as well. Retrospective analysis has found some VaR-like concepts in this history. But VaR did not emerge as a distinct concept until the late 1980s. The triggering event was the stock market <a href="/wiki/Black_Monday_(1987)" title="Black Monday (1987)">crash of 1987</a>. This was the first major financial crisis in which a lot of academically-trained <a href="/wiki/Quantitative_analyst" class="mw-redirect" title="Quantitative analyst">quants</a> were in high enough positions to worry about firm-wide survival.<sup id="cite_ref-Jorion_1-6" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The crash was so unlikely given standard <a href="/wiki/Statistical" class="mw-redirect" title="Statistical">statistical</a> models, that it called the entire basis of <a href="/wiki/Quantitative_analyst" class="mw-redirect" title="Quantitative analyst">quant</a> finance into question. A reconsideration of history led some quants to decide there were recurring crises, about one or two per decade, that overwhelmed the statistical assumptions embedded in models used for <a href="/wiki/Trader_(finance)" title="Trader (finance)">trading</a>, <a href="/wiki/Investment_management" title="Investment management">investment management</a> and <a href="/wiki/Derivative_(finance)" title="Derivative (finance)">derivative</a> pricing. These affected many markets at once, including ones that were usually not <a href="/wiki/Financial_correlation" title="Financial correlation">correlated</a>, and seldom had discernible economic cause or warning (although after-the-fact explanations were plentiful).<sup id="cite_ref-Roundtable_II_23-1" class="reference"><a href="#cite_note-Roundtable_II-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Much later, they were named "<a href="/wiki/Black_Swan_theory" class="mw-redirect" title="Black Swan theory">Black Swans</a>" by <a href="/wiki/Nassim_Nicholas_Taleb" title="Nassim Nicholas Taleb">Nassim Taleb</a> and the concept extended far beyond <a href="/wiki/Finance" title="Finance">finance</a>.<sup id="cite_ref-Black_Swan_37-0" class="reference"><a href="#cite_note-Black_Swan-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>If these events were included in <a href="/wiki/Quantitative_analysis_(finance)" title="Quantitative analysis (finance)">quantitative analysis</a> they dominated results and led to strategies that did not work day to day. If these events were excluded, the profits made in between "Black Swans" could be much smaller than the losses suffered in the crisis. Institutions could fail as a result.<sup id="cite_ref-Einhorn_I_20-1" class="reference"><a href="#cite_note-Einhorn_I-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Roundtable_II_23-2" class="reference"><a href="#cite_note-Roundtable_II-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Black_Swan_37-1" class="reference"><a href="#cite_note-Black_Swan-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>VaR was developed as a systematic way to segregate extreme events, which are studied qualitatively over long-term history and broad market events, from everyday price movements, which are studied quantitatively using short-term data in specific markets. It was hoped that "Black Swans" would be preceded by increases in estimated VaR or increased frequency of VaR breaks, in at least some markets. The extent to which this has proven to be true is controversial.<sup id="cite_ref-Roundtable_II_23-3" class="reference"><a href="#cite_note-Roundtable_II-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Abnormal markets and trading were excluded from the VaR estimate in order to make it observable.<sup id="cite_ref-Haug_21-1" class="reference"><a href="#cite_note-Haug-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> It is not always possible to define loss if, for example, markets are closed as after <a href="/wiki/September_11_attacks" title="September 11 attacks">9/11</a>, or severely illiquid, as happened several times in 2008.<sup id="cite_ref-Einhorn_I_20-2" class="reference"><a href="#cite_note-Einhorn_I-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Losses can also be hard to define if the risk-bearing institution fails or breaks up.<sup id="cite_ref-Haug_21-2" class="reference"><a href="#cite_note-Haug-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> A measure that depends on traders taking certain actions, and avoiding other actions, can lead to <a href="/wiki/Self_reference" class="mw-redirect" title="Self reference">self reference</a>.<sup id="cite_ref-Jorion_1-7" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>This is risk management VaR. It was well established in <a href="/wiki/Quantitative_analyst" class="mw-redirect" title="Quantitative analyst">quantitative trading</a> groups at several financial institutions, notably <a href="/wiki/Bankers_Trust" title="Bankers Trust">Bankers Trust</a>, before 1990, although neither the name nor the definition had been standardized. There was no effort to aggregate VaRs across trading desks.<sup id="cite_ref-Roundtable_II_23-4" class="reference"><a href="#cite_note-Roundtable_II-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>The financial events of the early 1990s found many firms in trouble because the same underlying bet had been made at many places in the firm, in non-obvious ways. Since many trading desks already computed risk management VaR, and it was the only common risk measure that could be both defined for all businesses and aggregated without strong assumptions, it was the natural choice for reporting firmwide risk. <a href="/wiki/JPMorgan_Chase" title="JPMorgan Chase">J. P. Morgan</a> CEO <a href="/wiki/Dennis_Weatherstone" title="Dennis Weatherstone">Dennis Weatherstone</a> famously called for a "4:15 report" that combined all firm <a href="/wiki/Risk" title="Risk">risk</a> on one page, available within 15 minutes of the market close.<sup id="cite_ref-Roundtable_I_10-2" class="reference"><a href="#cite_note-Roundtable_I-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Risk measurement VaR was developed for this purpose. Development was most extensive at <a href="/wiki/JPMorgan_Chase" title="JPMorgan Chase">J. P. Morgan</a>, which published the methodology and gave free access to estimates of the necessary underlying parameters in 1994. This was the first time VaR had been exposed beyond a relatively small group of <a href="/wiki/Quantitative_analyst" class="mw-redirect" title="Quantitative analyst">quants</a>. Two years later, the methodology was spun off into an independent for-profit business now part of RiskMetrics Group (now part of <a href="/wiki/MSCI" title="MSCI">MSCI</a>).<sup id="cite_ref-Roundtable_I_10-3" class="reference"><a href="#cite_note-Roundtable_I-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In 1997, the <a href="/wiki/U.S._Securities_and_Exchange_Commission" title="U.S. Securities and Exchange Commission">U.S. Securities and Exchange Commission</a> ruled that public corporations must disclose quantitative information about their <a href="/wiki/Derivative_(finance)" title="Derivative (finance)">derivatives</a> activity. Major <a href="/wiki/Bank" title="Bank">banks</a> and dealers chose to implement the rule by including VaR information in the notes to their <a href="/wiki/Financial_statements" class="mw-redirect" title="Financial statements">financial statements</a>.<sup id="cite_ref-Jorion_1-8" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Worldwide adoption of the <a href="/wiki/Basel_II_Accord" class="mw-redirect" title="Basel II Accord">Basel II Accord</a>, beginning in 1999 and nearing completion today, gave further impetus to the use of VaR. VaR is the preferred <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a> of <a href="/wiki/Market_risk" title="Market risk">market risk</a>, and concepts similar to VaR are used in other parts of the accord.<sup id="cite_ref-Jorion_1-9" class="reference"><a href="#cite_note-Jorion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Criticism">Criticism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=10" title="Edit section: Criticism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>VaR has been controversial since it moved from trading desks into the public eye in 1994. A famous 1997 <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170706074507/http://www.derivativesstrategy.com/magazine/archive/1997/0497fea2.asp">debate</a> between <a href="/wiki/Nassim_Nicholas_Taleb" title="Nassim Nicholas Taleb">Nassim Taleb</a> and Philippe Jorion set out some of the major points of contention. Taleb claimed VaR:<sup id="cite_ref-Taleb_Criticism_38-0" class="reference"><a href="#cite_note-Taleb_Criticism-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Ignored 2,500 years of experience in favor of untested models built by non-traders</li> <li>Was charlatanism because it claimed to estimate the risks of rare events, which is impossible</li> <li>Gave false confidence</li> <li>Would be exploited by traders</li></ol> <p>In 2008 <a href="/wiki/David_Einhorn_(hedge_fund_manager)" title="David Einhorn (hedge fund manager)">David Einhorn</a> and <a href="/wiki/Aaron_Brown_(financial_author)" title="Aaron Brown (financial author)">Aaron Brown</a> debated VaR in <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110226150454/http://www.garpdigitallibrary.org/download/GRR/2012.pdf">Global Association of Risk Professionals Review</a>.<sup id="cite_ref-Einhorn_I_20-3" class="reference"><a href="#cite_note-Einhorn_I-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Einhorn_II_3-1" class="reference"><a href="#cite_note-Einhorn_II-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Einhorn compared VaR to "an <a href="/wiki/Airbag" title="Airbag">airbag</a> that works all the time, except when you have a car accident". He further charged that VaR: </p> <ol><li>Led to excessive risk-taking and leverage at financial institutions</li> <li>Focused on the manageable risks near the center of the distribution and ignored the tails</li> <li>Created an incentive to take "excessive but remote risks"</li> <li>Was "potentially catastrophic when its use creates a false sense of security among senior executives and watchdogs."</li></ol> <p><a href="/wiki/The_New_York_Times" title="The New York Times">New York Times</a> reporter <a href="/wiki/Joseph_Nocera" class="mw-redirect" title="Joseph Nocera">Joe Nocera</a> wrote an extensive piece <a rel="nofollow" class="external text" href="https://www.nytimes.com/2009/01/04/magazine/04risk-t.html?pagewanted=1&_r=1">Risk Mismanagement</a><sup id="cite_ref-Nocera_39-0" class="reference"><a href="#cite_note-Nocera-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> on January 4, 2009, discussing the role VaR played in the <a href="/wiki/Financial_crisis_of_2007%E2%80%932008" class="mw-redirect" title="Financial crisis of 2007–2008">Financial crisis of 2007–2008</a>. After interviewing risk managers (including several of the ones cited above) the article suggests that VaR was very useful to risk experts, but nevertheless exacerbated the crisis by giving false security to bank executives and regulators. A powerful tool for professional risk managers, VaR is portrayed as both easy to misunderstand, and dangerous when misunderstood. </p><p>Taleb in 2009 testified in Congress asking for the banning of VaR for a number of reasons. One was that tail risks are non-measurable. Another was that for <a href="/wiki/Anchoring_(cognitive_bias)" class="mw-redirect" title="Anchoring (cognitive bias)">anchoring</a> reasons VaR leads to higher risk taking.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p><p>VaR is not <a href="/wiki/Subadditivity#Finance" title="Subadditivity">subadditive</a>:<sup id="cite_ref-Dowd_5-5" class="reference"><a href="#cite_note-Dowd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> VaR of a combined portfolio can be larger than the sum of the VaRs of its components. </p><p>For example, the average bank branch in the United States is robbed about once every ten years. A single-branch bank has about 0.0004% chance of being robbed on a specific day, so the risk of robbery would not figure into one-day 1% VaR. It would not even be within an order of magnitude of that, so it is in the range where the institution should not worry about it, it should insure against it and take advice from insurers on precautions. The whole point of insurance is to aggregate risks that are beyond individual VaR limits, and bring them into a large enough portfolio to get statistical predictability. It does not pay for a one-branch bank to have a security expert on staff. </p><p>As institutions get more branches, the risk of a robbery on a specific day rises to within an order of magnitude of VaR. At that point it makes sense for the institution to run internal stress tests and analyze the risk itself. It will spend less on insurance and more on in-house expertise. For a very large banking institution, robberies are a routine daily occurrence. Losses are part of the daily VaR calculation, and tracked statistically rather than case-by-case. A sizable in-house security department is in charge of prevention and control, the general risk manager just tracks the loss like any other cost of doing business. As portfolios or institutions get larger, specific risks change from low-probability/low-predictability/high-impact to statistically predictable losses of low individual impact. That means they move from the range of far outside VaR, to be insured, to near outside VaR, to be analyzed case-by-case, to inside VaR, to be treated statistically.<sup id="cite_ref-Einhorn_I_20-4" class="reference"><a href="#cite_note-Einhorn_I-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>VaR is a static measure of risk. By definition, VaR is a particular characteristic of the probability distribution of the underlying (namely, VaR is essentially a quantile). For a dynamic measure of risk, see Novak,<sup id="cite_ref-Novak_27-1" class="reference"><a href="#cite_note-Novak-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> ch. 10. </p><p>There are common abuses of VaR:<sup id="cite_ref-Unbearable_7-3" class="reference"><a href="#cite_note-Unbearable-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Roundtable_I_10-4" class="reference"><a href="#cite_note-Roundtable_I-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Assuming that plausible losses will be less than some multiple (often three) of VaR. Losses can be extremely large.</li> <li>Reporting a VaR that has not passed a <a href="/wiki/Backtesting" title="Backtesting">backtest</a>. Regardless of how VaR is computed, it should have produced the correct number of breaks (within <a href="/wiki/Sampling_error" title="Sampling error">sampling error</a>) in the past. A common violation of common sense is to estimate a VaR based on the unverified assumption that everything follows a <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a>.</li></ol> <div class="mw-heading mw-heading3"><h3 id="VaR,_CVaR,_RVaR_and_EVaR"><span id="VaR.2C_CVaR.2C_RVaR_and_EVaR"></span>VaR, CVaR, RVaR and EVaR</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Value_at_risk&action=edit&section=11" title="Edit section: VaR, CVaR, RVaR and EVaR"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The VaR is not a <a href="/wiki/Coherent_risk_measure" title="Coherent risk measure">coherent risk measure</a> since it violates the sub-additivity property, which 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 \mathrm {If} \;X,Y\in \mathbf {L} ,\;\mathrm {then} \;\rho (X+Y)\leq \rho (X)+\rho (Y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">f</mi> </mrow> <mspace width="thickmathspace" /> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> </mrow> <mspace width="thickmathspace" /> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {If} \;X,Y\in \mathbf {L} ,\;\mathrm {then} \;\rho (X+Y)\leq \rho (X)+\rho (Y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12bdffaccedbc09021f9c9c67cd8593336cdb97a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.399ex; height:2.843ex;" alt="{\displaystyle \mathrm {If} \;X,Y\in \mathbf {L} ,\;\mathrm {then} \;\rho (X+Y)\leq \rho (X)+\rho (Y).}"></span></dd></dl> <p>However, it can be bounded by coherent risk measures like <a href="/wiki/Conditional_Value-at-Risk" class="mw-redirect" title="Conditional Value-at-Risk">Conditional Value-at-Risk</a> (CVaR) or <a href="/wiki/Entropic_value_at_risk" title="Entropic value at risk">entropic value at risk</a> (EVaR). CVaR is defined by average of VaR values for confidence levels between 0 and <span class="texhtml mvar" style="font-style:italic;">α</span>. </p><p>However VaR, unlike CVaR, has the property of being a <a href="/wiki/Robust_statistics" title="Robust statistics">robust statistic</a>. A related class of risk measures is the 'Range Value at Risk' (RVaR), which is a robust version of CVaR.<sup id="cite_ref-RVaR_41-0" class="reference"><a href="#cite_note-RVaR-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>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 X\in \mathbf {L} _{M^{+}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\in \mathbf {L} _{M^{+}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1435c5764f7020c4e44a9fa1c3d95711e77a16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.63ex; height:2.676ex;" alt="{\displaystyle X\in \mathbf {L} _{M^{+}}}"></span> (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 \mathbf {L} _{M^{+}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} _{M^{+}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2892aa8aa7686529479237a23da3974ce300a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.81ex; height:2.676ex;" alt="{\displaystyle \mathbf {L} _{M^{+}}}"></span> the set of all <a href="/wiki/Borel_measure" title="Borel measure">Borel</a> <a href="/wiki/Measurable_function" title="Measurable function">measurable functions</a> whose <a href="/wiki/Moment-generating_function" title="Moment-generating function">moment-generating function</a> exists for all positive real values) we have </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 {\text{VaR}}_{1-\alpha }(X)\leq {\text{RVaR}}_{\alpha ,\beta }(X)\leq {\text{CVaR}}_{1-\alpha }(X)\leq {\text{EVaR}}_{1-\alpha }(X),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>RVaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>CVaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>EVaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{VaR}}_{1-\alpha }(X)\leq {\text{RVaR}}_{\alpha ,\beta }(X)\leq {\text{CVaR}}_{1-\alpha }(X)\leq {\text{EVaR}}_{1-\alpha }(X),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edcbcd0995f3bd123154231df7d83eed38e98538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.373ex; height:3.009ex;" alt="{\displaystyle {\text{VaR}}_{1-\alpha }(X)\leq {\text{RVaR}}_{\alpha ,\beta }(X)\leq {\text{CVaR}}_{1-\alpha }(X)\leq {\text{EVaR}}_{1-\alpha }(X),}"></span></dd></dl> <p>where </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 {\begin{aligned}&{\text{VaR}}_{1-\alpha }(X):=\inf _{t\in \mathbf {R} }\{t:{\text{Pr}}(X\leq t)\geq 1-\alpha \},\\&{\text{CVaR}}_{1-\alpha }(X):={\frac {1}{\alpha }}\int _{0}^{\alpha }{\text{VaR}}_{1-\gamma }(X)d\gamma ,\\&{\text{RVaR}}_{\alpha ,\beta }(X):={\frac {1}{\beta -\alpha }}\int _{\alpha }^{\beta }{\text{VaR}}_{1-\gamma }(X)d\gamma ,\\&{\text{EVaR}}_{1-\alpha }(X):=\inf _{z>0}\{z^{-1}\ln(M_{X}(z)/\alpha )\},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> </munder> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pr</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>CVaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>γ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>RVaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>β</mi> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>γ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>EVaR</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mo fence="false" stretchy="false">{</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>α</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\text{VaR}}_{1-\alpha }(X):=\inf _{t\in \mathbf {R} }\{t:{\text{Pr}}(X\leq t)\geq 1-\alpha \},\\&{\text{CVaR}}_{1-\alpha }(X):={\frac {1}{\alpha }}\int _{0}^{\alpha }{\text{VaR}}_{1-\gamma }(X)d\gamma ,\\&{\text{RVaR}}_{\alpha ,\beta }(X):={\frac {1}{\beta -\alpha }}\int _{\alpha }^{\beta }{\text{VaR}}_{1-\gamma }(X)d\gamma ,\\&{\text{EVaR}}_{1-\alpha }(X):=\inf _{z>0}\{z^{-1}\ln(M_{X}(z)/\alpha )\},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b336236e5bb1c0c5368cdf6793284a9b7bda4ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:43.953ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}&{\text{VaR}}_{1-\alpha }(X):=\inf _{t\in \mathbf {R} }\{t:{\text{Pr}}(X\leq t)\geq 1-\alpha \},\\&{\text{CVaR}}_{1-\alpha }(X):={\frac {1}{\alpha }}\int _{0}^{\alpha }{\text{VaR}}_{1-\gamma }(X)d\gamma ,\\&{\text{RVaR}}_{\alpha ,\beta }(X):={\frac {1}{\beta -\alpha }}\int _{\alpha }^{\beta }{\text{VaR}}_{1-\gamma }(X)d\gamma ,\\&{\text{EVaR}}_{1-\alpha }(X):=\inf _{z>0}\{z^{-1}\ln(M_{X}(z)/\alpha )\},\end{aligned}}}"></span></dd></dl> <p>in which <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 M_{X}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{X}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ec2a5de6689e9e89d14dac2f2a29d48fceac69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.784ex; height:2.843ex;" alt="{\displaystyle M_{X}(z)}"></span> is the moment-generating function of <span class="texhtml mvar" style="font-style:italic;">X</span> at <span class="texhtml mvar" style="font-style:italic;">z</span>. In the above equations the variable <span class="texhtml mvar" style="font-style:italic;">X</span> denotes the financial loss, rather than wealth as is typically the case. </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=Value_at_risk&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Capital_Adequacy_Directive" title="Capital Adequacy Directive">Capital Adequacy Directive</a></li> <li><a href="/wiki/Conditional_value-at-risk" class="mw-redirect" title="Conditional value-at-risk">Conditional value-at-risk</a> / <a href="/wiki/Expected_shortfall" title="Expected shortfall">Expected shortfall</a></li> <li><a href="/wiki/Cyber_risk_quantification" title="Cyber risk quantification">Cyber risk quantification based on cyber value-at-risk or CyVaR</a></li> <li><a href="/wiki/Extended_Mathematical_Programming_(EMP)#EMP_for_stochastic_programming" class="mw-redirect" title="Extended Mathematical Programming (EMP)">EMP for stochastic programming</a>— solution technology for optimization problems involving VaR and CVaR</li> <li><a href="/wiki/Entropic_value_at_risk" title="Entropic value at risk">Entropic value at risk</a></li> <li><a href="/wiki/Financial_risk_management#Banking" title="Financial risk management">Financial risk management § Banking</a></li> <li><a href="/wiki/Profit_at_risk" title="Profit at risk">Profit at risk</a></li> <li><a href="/wiki/Margin_at_risk" title="Margin at risk">Margin at risk</a></li> <li><a href="/wiki/Liquidity_at_risk" title="Liquidity at risk">Liquidity at risk</a></li> <li><a href="/wiki/Risk_return_ratio" class="mw-redirect" title="Risk return ratio">Risk return ratio</a></li> <li><a href="/wiki/Tail_value_at_risk" title="Tail value at risk">Tail value at risk</a></li> <li><a href="/wiki/Valuation_risk" title="Valuation risk">Valuation risk</a></li></ul> <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=Value_at_risk&action=edit&section=13" 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-Jorion-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Jorion_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Jorion_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Jorion_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Jorion_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Jorion_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Jorion_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Jorion_1-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Jorion_1-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Jorion_1-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-Jorion_1-9"><sup><i><b>j</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 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Berlin: <a href="/wiki/Walter_de_Gruyter" class="mw-redirect" title="Walter de Gruyter">Walter de Gruyter</a>. pp. 177–182. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-311-0183467" title="Special:BookSources/978-311-0183467"><bdi>978-311-0183467</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2169807">2169807</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Finance&rft.place=Berlin&rft.series=de+Gruyter+Series+in+Mathematics&rft.pages=177-182&rft.pub=Walter+de+Gruyter&rft.date=2004&rft.isbn=978-311-0183467&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2169807%23id-name%3DMR&rft.aulast=Foellmer&rft.aufirst=Hans&rft.au=Schied%2C+Alexander&rfr_id=info%3Asid%2Fen.wikipedia.org%3AValue+at+risk" class="Z3988"></span></span> </li> <li id="cite_note-Taleb_II-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Taleb_II_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNassim_Taleb1996–1997" class="citation cs2">Nassim Taleb (December 1996 – January 1997), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20000829231106/http://www.derivativesstrategy.com/magazine/archive/1997/1296qa.asp"><i>The World According to Nassim Taleb</i></a>, Derivatives Strategy, archived from <a rel="nofollow" class="external text" href="http://www.derivativesstrategy.com/magazine/archive/1997/1296qa.asp">the original</a> on 2000-08-29</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+World+According+to+Nassim+Taleb&rft.pub=Derivatives+Strategy&rft.date=1996-12%2F1997-01&rft.au=Nassim+Taleb&rft_id=http%3A%2F%2Fwww.derivativesstrategy.com%2Fmagazine%2Farchive%2F1997%2F1296qa.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AValue+at+risk" class="Z3988"></span></span> </li> <li id="cite_note-Wirch-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wirch_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJulia_L._WirchMary_R._Hardy" class="citation web cs1">Julia L. Wirch; Mary R. Hardy. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160705041252/http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf">"Distortion Risk Measures: Coherence and Stochastic Dominance"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf">the original</a> <span class="cs1-format">(PDF)</span> on July 5, 2016<span class="reference-accessdate">. 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(2010). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/hal-00413729/file/robustriskarxiv.pdf">"Robustness and Sensitivity Analysis of Risk Measurement Procedures"</a> <span class="cs1-format">(PDF)</span>. <i>Quantitative Finance</i>. <b>10</b> (6): 593–606. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14697681003685597">10.1080/14697681003685597</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:158678050">158678050</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quantitative+Finance&rft.atitle=Robustness+and+Sensitivity+Analysis+of+Risk+Measurement+Procedures&rft.volume=10&rft.issue=6&rft.pages=593-606&rft.date=2010&rft_id=info%3Adoi%2F10.1080%2F14697681003685597&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A158678050%23id-name%3DS2CID&rft.aulast=Cont&rft.aufirst=Rama&rft.au=Deguest%2C+Romain&rft.au=Giacomo%2C+Giacomo&rft_id=https%3A%2F%2Fhal.archives-ouvertes.fr%2Fhal-00413729%2Ffile%2Frobustriskarxiv.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AValue+at+risk" class="Z3988"></span></span> </li> </ol></div> <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=Value_at_risk&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Discussion</dt></dl> <ul><li><a rel="nofollow" class="external text" href="http://spauldinggrp.com/value-risk/">"Value At Risk"</a>, <a href="/w/index.php?title=Ben_Sopranzetti&action=edit&redlink=1" class="new" title="Ben Sopranzetti (page does not exist)">Ben Sopranzetti</a>, Ph.D., CPA</li> <li><a rel="nofollow" class="external text" href="http://www.wilmott.com/blogs/satyajitdas/enclosures/perfectstorms%28may2007%291.pdf">"Perfect Storms" – Beautiful & True Lies In Risk Management</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090325160717/http://www.wilmott.com/blogs/satyajitdas/enclosures/perfectstorms%28may2007%291.pdf">Archived</a> 2009-03-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="/wiki/Satyajit_Das" title="Satyajit Das">Satyajit Das</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080914192413/http://www.gloriamundi.org/">"Gloria Mundi" – All About Value at Risk</a>, Barry Schachter</li> <li><a rel="nofollow" class="external text" href="https://www.nytimes.com/2009/01/04/magazine/04risk-t.html?dlbk=&pagewanted=all">Risk Mismanagement</a>, <a href="/wiki/Joe_Nocera" title="Joe Nocera">Joe Nocera</a> <a href="/wiki/NY_Times" class="mw-redirect" title="NY Times">NY Times</a> article.</li> <li><a rel="nofollow" class="external text" href="http://www.savvysoft.com/registerpagevarnothard.htm">"VaR Doesn't Have To Be Hard"</a>, Rich Tanenbaum</li> <li><a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1111/1467-9965.00068">"Coherent measures of Risk"</a>, Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath</li></ul> <dl><dt>Tools</dt></dl> <ul><li><a rel="nofollow" class="external text" href="http://tradinginterestrates.com">"The Pricing and Trading of Interest Rate Derivatives"</a>, <a href="/w/index.php?title=J_H_M_Darbyshire&action=edit&redlink=1" class="new" title="J H M Darbyshire (page does not exist)">J H M Darbyshire</a>, MSc.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071015050012/http://cba.ua.edu/~rpascala/VaR/VaRForm.php">Online real-time VaR calculator</a>, Razvan Pascalau, <a href="/wiki/University_of_Alabama" title="University of Alabama">University of Alabama</a></li> <li><a rel="nofollow" class="external text" href="http://simonbenninga.com/wiener/MiER74.pdf">Value-at-Risk (VaR)</a>, Simon Benninga and Zvi Wiener. (Mathematica in Education and Research Vol. 7 No. 4 1998.)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060316220629/http://www.derivativesstrategy.com/magazine/archive/1998/0498fea1.asp">Derivatives Strategy Magazine. "Inside D. E. Shaw"</a> Trading and Risk Management 1998</li> <li><a rel="nofollow" class="external text" href="https://aiolux.com/tools/var-value-at-risk-calculator">Simulate Historical Value at Risk</a> Online Calculator</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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href="/wiki/Sovereign_credit_risk" title="Sovereign credit risk">Sovereign credit risk</a></li> <li><a href="/wiki/Settlement_risk" title="Settlement risk">Settlement risk</a></li> <li><a href="/wiki/Default_(finance)" title="Default (finance)">Default risk</a></li> <li><a href="/wiki/Concentration_risk" title="Concentration risk">Concentration risk</a></li> <li><a href="/wiki/Credit_derivative" title="Credit derivative">Credit derivative</a></li> <li><a href="/wiki/Securitization" title="Securitization">Securitization</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Market_risk" title="Market risk">Market risk</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Commodity_risk" title="Commodity risk">Commodity risk</a> (e.g. <a href="/wiki/Volume_risk" title="Volume risk">Volume risk</a>, <a href="/wiki/Basis_risk" title="Basis risk">Basis risk</a>, <a href="/wiki/Shape_risk" title="Shape risk">Shape risk</a>, <a href="/wiki/Holding_period_risk" title="Holding period risk">Holding period risk</a>, <a href="/wiki/Price_area_risk" class="mw-redirect" title="Price area risk">Price area risk</a>)</li> <li><a href="/wiki/Equity_risk" title="Equity risk">Equity risk</a></li> <li><a href="/wiki/Valuation_risk" title="Valuation risk">Valuation risk</a></li> <li><a href="/wiki/Foreign_exchange_risk" title="Foreign exchange risk">FX risk</a></li> <li><a href="/wiki/Margining_risk" title="Margining risk">Margining risk</a></li> <li><a href="/wiki/Interest_rate_risk" title="Interest rate risk">Interest rate risk</a></li> <li><a href="/wiki/Inflation_risk" class="mw-redirect" title="Inflation risk">Inflation risk</a></li> <li><a href="/wiki/Volatility_risk" title="Volatility risk">Volatility risk</a></li> <li><a href="/wiki/Liquidity_risk" title="Liquidity risk">Liquidity risk</a> (e.g. <a href="/wiki/Refinancing_risk" title="Refinancing risk">Refinancing risk</a>, <a href="/wiki/Deposit_risk" title="Deposit risk">Deposit risk</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operational_risk" title="Operational risk">Operational risk</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Operational_risk_management" title="Operational risk management">Operational risk management</a></li> <li><a href="/wiki/Business_risk" class="mw-redirect" title="Business risk">Business risk</a></li> <li><a href="/wiki/Model_risk" title="Model risk">Model risk</a></li> <li><a href="/wiki/Reputational_risk" class="mw-redirect" title="Reputational risk">Reputational risk</a></li> <li><a href="/wiki/Country_risk" title="Country risk">Country risk</a></li> <li><a href="/wiki/Political_risk" title="Political risk">Political risk</a></li> <li><a href="/wiki/Legal_risk" title="Legal risk">Legal risk</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Execution_risk" class="mw-redirect" title="Execution risk">Execution risk</a></li> <li><a href="/wiki/Profit_risk" title="Profit risk">Profit risk</a></li> <li><a href="/wiki/Systemic_risk" title="Systemic risk">Systemic risk</a></li> <li><a href="/wiki/Non-financial_risk" title="Non-financial risk">Non-financial risk</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Financial_risk_modeling" title="Financial risk modeling">Modeling</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arbitrage_pricing_theory" title="Arbitrage pricing theory">Arbitrage pricing theory</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes model</a></li> <li><a href="/wiki/Replicating_portfolio" title="Replicating portfolio">Replicating portfolio</a></li> <li><a href="/wiki/Cashflow_matching" title="Cashflow matching">Cash flow matching</a></li> <li><a href="/wiki/Expected_shortfall" title="Expected shortfall">Conditional Value-at-Risk (CVaR)</a></li> <li><a href="/wiki/Copula_(probability_theory)" class="mw-redirect" title="Copula (probability theory)">Copula</a></li> <li><a href="/wiki/Drawdown_(economics)" title="Drawdown (economics)">Drawdown</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First-hitting-time model</a></li> <li><a href="/wiki/Immunization_(finance)" title="Immunization (finance)">Interest rate immunization</a></li> <li><a href="/wiki/Market_portfolio" title="Market portfolio">Market portfolio</a></li> <li><a href="/wiki/Modern_portfolio_theory" title="Modern portfolio theory">Modern portfolio theory</a></li> <li><a href="/wiki/Omega_ratio" title="Omega ratio">Omega ratio</a></li> <li><a href="/wiki/Risk-adjusted_return_on_capital" title="Risk-adjusted return on capital">RAROC</a></li> <li><a href="/wiki/Risk-free_interest_rate" class="mw-redirect" title="Risk-free interest rate">Risk-free rate</a></li> <li><a href="/wiki/Risk_parity" title="Risk parity">Risk parity</a></li> <li><a href="/wiki/Sharpe_ratio" title="Sharpe ratio">Sharpe ratio</a></li> <li><a href="/wiki/Sortino_ratio" title="Sortino ratio">Sortino ratio</a></li> <li><a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a> (<a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards model</a>)</li> <li><a href="/wiki/Tracking_error" title="Tracking error">Tracking error</a></li> <li><a class="mw-selflink selflink">Value-at-Risk (VaR)</a> and extensions (<a href="/wiki/Profit_at_risk" title="Profit at risk">Profit at risk</a>, <a href="/wiki/Margin_at_risk" title="Margin at risk">Margin at risk</a>, <a href="/wiki/Liquidity_at_risk" title="Liquidity at risk">Liquidity at risk</a>, <a href="/wiki/Cash_flow_at_risk" class="mw-redirect" title="Cash flow at risk">Cash flow at risk</a>, <a href="/wiki/Earnings_at_risk" title="Earnings at risk">Earnings at risk</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asset_allocation" title="Asset allocation">Asset allocation</a></li> <li><a href="/wiki/Asset_and_liability_management" title="Asset and liability management">Asset and liability management</a></li> <li><a href="/wiki/Asset_pricing" title="Asset pricing">Asset pricing</a></li> <li><a href="/wiki/Bad_debt" title="Bad debt">Bad debt</a></li> <li><a href="/wiki/Capital_asset" title="Capital asset">Capital asset</a></li> <li><a href="/wiki/Capital_structure" title="Capital structure">Capital structure</a></li> <li><a href="/wiki/Corporate_finance" title="Corporate finance">Corporate finance</a></li> <li><a href="/wiki/Cost_of_capital" title="Cost of capital">Cost of capital</a></li> <li><a href="/wiki/Diversification_(finance)" title="Diversification (finance)">Diversification</a></li> <li><a href="/wiki/Economic_bubble" title="Economic bubble">Economic bubble</a></li> <li><a href="/wiki/Enterprise_value" title="Enterprise value">Enterprise value</a></li> <li><a href="/wiki/Environmental,_social,_and_governance" title="Environmental, social, and governance">ESG</a></li> <li><a href="/wiki/Exchange_traded_fund" class="mw-redirect" title="Exchange traded fund">Exchange traded fund</a></li> <li><a href="/wiki/Expected_return" title="Expected return">Expected return</a></li> <li><a href="/wiki/Finance" title="Finance">Financial</a> <ul><li><a href="/wiki/Financial_adviser" title="Financial adviser">adviser</a></li> <li><a href="/wiki/Financial_analysis" title="Financial analysis">analysis</a></li> <li><a href="/wiki/Financial_analyst" title="Financial analyst">analyst</a></li> <li><a href="/wiki/Financial_asset" title="Financial asset">asset</a></li> <li><a href="/wiki/Financial_betting" title="Financial betting">betting</a></li> <li><a href="/wiki/Financial_crime" title="Financial crime">crime</a></li> <li><a href="/wiki/Financial_engineering" title="Financial engineering">engineering</a></li> <li><a href="/wiki/Financial_law" title="Financial law">law</a></li> <li><a href="/wiki/Financial_risk" title="Financial risk">risk</a></li> <li><a href="/wiki/Financial_social_work" title="Financial social work">social work</a></li></ul></li> <li><a href="/wiki/Fundamental_analysis" title="Fundamental analysis">Fundamental analysis</a></li> <li><a href="/wiki/Growth_investing" title="Growth 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