Value-at-Risk (VaR)

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Value-at-Risk</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="0%" align="left" valign="top"><p class="word_list"> <a href="#base currency">base currency</a><p class="word_list"> <a href="#Value-at-risk">value-at-risk</a><p class="word_list"> <a href="#VaR horizon">VaR horizon</a><p class="word_list"> <a href="#VaR measurement">VaR measurement</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> <td width="100%" align="left" valign="top"><p class="word_list"> <br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table5"> <tr> <td align="right" width="9" height="14"></td> <td width="300" height="14"> <font size="2"> </font><font size="4"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body"><b><a name="Value-at-risk">Value-at-risk</a></b> (VaR) is a category of <a href="../articles/risk_metric_and_risk_measure.htm">risk metrics</a> that describe probabilistically the <a href="../articles/market_risk.htm">market risk</a> of a trading portfolio. Value-at-risk is widely used by banks, <a href="../articles/european_financial_regulation.htm">securities firms</a>, commodity merchants, energy merchants, and other trading organizations. Such firms could track their portfolios' market risk by using <a href="../articles/volatility.htm">historical volatility</a> as a <a href="../articles/risk_metric_and_risk_measure.htm">risk metric</a>. They might do so by calculating the historical volatility of their portfolio's <a href="../articles/valuation.htm">market value</a> over a rolling 100 trading days. The problem with doing this is that it would provide a retrospective indication of <a href="../articles/risk.htm">risk</a>. The historical volatility would illustrate how risky the portfolio had been over the previous 100 days. It would say nothing about how much market risk the portfolio was taking today. </p> <p class="body">For institutions to manage risk, they must know about risks while they are being taken. If a trader mis-hedges a portfolio, his employer needs to find out before a loss is incurred. Value-at-risk gives institutions the ability to do so. Unlike retrospective risk metrics, such as historical volatility, value-at-risk is prospective. It quantifies market risk while it is being taken.</p> <p class="body">Measure time in trading days. Let 0 be the current time. We know a portfolio's current market value <img border="0" src="../symbols/0_p.gif" align="absbottom" width="14" height="18">. Its market value <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> in one trading day is unknown. It is a random variable. We may ascribe it a probability distribution. With value-at-risk, we summarize a portfolio's market risk by reporting some parameter of this distribution. For <a name="example">example</a>, we might report the 90%-<a href="../articles/quantile.htm">quantile</a> of the portfolio's single-period <a href="../articles/currency_codes.htm">USD</a> loss. This is called one-day 90% USD VaR. If a portfolio has a one-day 90% USD VaR of, say, USD 5<a href="../articles/mm.htm">MM</a>, it can be expected to lose more than USD 5MM on one trading day our of ten. This is illustrated in Exhibit 1.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="400" cellpadding="0" id="table6"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header"> <font face="Verdana">Example: One-Day 90% USD VaR</font></span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td><img border="0" src="../images/ex1_value-at-risk.gif" width="300" height="149"></td> </tr> <tr> <td> <p class="exhibit_legend">One-day 90% USD VaR is illustrated for a hypothetical portfolio. Shown is the probability density function for the portfolio's value <sup>1</sup><i>P</i> one trading day from now. The  portfolio's current value <sup>0</sup><i>p</i> is known. value-at-risk equals the amount of money such that there is a 90% probability of the portfolio losing less than that amount over the next trading day. This is indicated in the Exhibit.</td> </tr> </table> </center> </div> <p class="body">Value-at-risk can be measured in other ways. For example, bank regulations require that value-at-risk be calculated as a 99%-quantile of loss over a two-week horizon. Still other metrics are possible. We could measure value-at-risk as the <a href="../articles/standard_deviation.htm">standard deviation</a> of portfolio value or the standard deviation of portfolio <a href="../articles/return.htm">return</a>. Essentially, any parameter of the distribution of a portfolio's future value can be used to measure value-at-risk.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table7"> <tr> <td width="160"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script> </td> <td width="11"> </td> </tr> </table> <p class="body">Let's formalize this. value-at-risk is applicable to any <a href="../articles/liquidity.htm">liquid</a> portfolio—that is, any portfolio that can reasonably be marked to market on a regular basis. Value-at-risk is not applicable to illiquid assets, such as real estate or fine art. Value-at-risk considers a portfolio's performance over a specific horizon—a trading day, two weeks, a month, etc. We call this the <b><a name="VaR horizon">VaR horizon</a></b>. Value-at-risk is measured in a particular currency, USD in the <a href="#example"> example</a> above, but any currency can be used. This is called the <b><a name="base currency">base currency</a></b>. Finally, the portfolio's market risk is summarized with a single number. Informally, we called this a parameter of the distribution of portfolio value. More formally, it is any function of both the portfolio's current value <img border="0" src="../symbols/0_p.gif" align="absbottom" width="14" height="18"> and its (random) value <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> at the end of the VaR horizon. In our example, the function was the 90%-quantile of loss. As we mentioned, other functions are possible. A <b><a name="VaR measurement">VaR measurement</a></b> is the value obtained for that function for a specific portfolio at a specific point in time. </p> <p class="body"> We distinguish between a <a href="../articles/var_measure.htm">VaR measure</a> and a <a href="../articles/var_metric.htm">VaR metric</a>. A VaR measure is the procedure by which we arrive at a VaR measurement. It is some computational algorithm, which is typically coded on a computer. A VaR metric is our interpretation of the VaR measurement. In our examples, the VaR metric was one-day 90% USD VaR. Other example of VaR metrics are:</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">two-week 99% <a href="../articles/currency_codes.htm">EUR</a> VaR</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">one-year standard deviation of USD return</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">one-day <a href="../articles/var_metric.htm">semi-variance</a> of <a href="../articles/currency_codes.htm">JPY</a> portfolio value</p> <p class="body">Value-at-risk became popular with trading organizations during the 1990s. It was during this period that the name "value-at-risk" entered the financial lexicon. However, VaR measures had been used long before this. </p> <p class="body">An early user was Harry Markowitz. In his groundbreaking (<a href="#Markowitz">1952</a>) paper "Portfolio Selection", he adopted a VaR metric of single period <a href="../articles/standard_deviation.htm">variance</a> of return and used this to develop techniques of portfolio optimization. In the early 1980s, the United States <a href="../articles/united_states_financial_regulation.htm">Securities and Exchange Commission</a> (SEC) adopted a crude VaR measure for use in assessing the <a href="../articles/capital_allocation.htm">capital</a> adequacy of <a href="../articles/united_states_financial_regulation.htm">broker-dealers'</a> trading <a href="../articles/united_states_financial_regulation.htm">non-exempt</a> <a href="../articles/security.htm">securities</a>. A few years later, Bankers Trust implemented a VaR measure for use with its <a href="../articles/economic_capital.htm">RAROC</a> <a href="../articles/capital_allocation.htm">capital allocation</a> system. During the late 1980s and early 1990s, a number of institutions implemented VaR measures to support <a href="../articles/capital_allocation.htm">capital allocation</a> or <a href="../articles/risk_limits.htm">market risk limits</a>. </p> <p class="body">In the early 1990s, three events popularized value-at-risk as a practical tool for use on trading floors:</p> <p class="body"><img border="0" src="../images/bullet.gif" width="20" height="10">In 1993, the <a href="../articles/group_of_30_report.htm">Group of 30</a> published a groundbreaking report on derivatives practices. It was influential and helped shape the emerging field of <a href="../articles/risk_management.htm">financial risk management</a>. It promoted the use of value-at-risk by <a href="../articles/derivative_instrument.htm">derivatives</a> dealers and appears to be the first publication to use the phrase "value-at-risk."</p> <p class="body"><img border="0" src="../images/bullet.gif" width="20" height="10">In 1994, JP Morgan launched its free <a href="../articles/riskmetrics.htm">RiskMetrics</a> service. This was intended to promote the use of value-at-risk among the firm's institutional clients. The service comprised a technical document describing how to implement a VaR measure and a covariance matrix for several hundred key factors  updated daily on the internet. </p> <p class="body"><img border="0" src="../images/bullet.gif" width="20" height="10">In 1995, the <a href="../articles/basle_committee.htm">Basel Committee on Banking Supervision</a> implemented market risk capital requirements for banks. These were based upon a crude VaR measure, but the committee also approved, as an alternative,  the use of banks' own proprietary VaR measures in certain circumstances.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table8"> <tr> <td align="right" width="9" height="10"></td> <td width="336" height="10"> <font size="2"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body">These three initiatives came during a period of heightened concern about <a href="../articles/capital_asset_pricing_model.htm">systemic risks</a> due to the emerging—and largely unregulated—<a href="../articles/exchange_traded.htm">OTC</a> derivatives market. It was also a period when a number of organizations—including <a href="../articles/risk_management.htm">Orange County</a>, <a href="../articles/barings_debacle.htm">Barings Bank</a>, and <a href="../articles/risk_management.htm">Metallgesellschaft</a>—suffered staggering losses due to speculative trading, failed hedging programs or derivatives. Financial risk management was a priority for firms, and value-at-risk was rapidly embraced as the tool of choice for quantifying market risk. 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Marrison (<a target="_blank" href="http://www.riskbook.com/link/marrison_c_(2002).htm">2002</a>) discusses value-at-risk in the general context of bank risk management. Holton (<a target="_blank" href="http://www.value-at-risk.net">2003</a>) is the definitive text on value-at-risk.</p> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table12"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap354"> <area href="http://www.riskbook.com/link/butler_c_(1999).htm" shape="rect" coords="0, 0, 93, 124" target="_blank"> </map> <img border="0" src="../images/book_butler_125.jpg" usemap="#FPMap354" width="94" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table13"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/butler_c_(1999).htm"> Mastering Value-at-Risk</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Cormac Butler</b></td> </tr> <tr> <td width="10%"><map name="FPMap40"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/star_4.gif" usemap="#FPMap40" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"><map name="FPMap39"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/tech_2.gif" usemap="#FPMap39" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">1999</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap37"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/butler_c_(1999).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap37" width="146" height="20"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table14"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap357"> <area target="_blank" href="http://www.riskbook.com/link/marrison_c_(2002).htm" shape="rect" coords="0, 0, 98, 124"> </map> <img border="0" src="../images/book_marrison_c_(2002)_125.gif" usemap="#FPMap357" width="99" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table15"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/marrison_c_(2002).htm"> Fundamentals of Risk Measurement</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Chris Marrison</b></td> </tr> <tr> <td width="10%"> <map name="FPMap355"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/star_5.gif" usemap="#FPMap355" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"> <map name="FPMap356"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map><img border="0" src="../images/tech_3.gif" usemap="#FPMap356" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">2002</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap24"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/marrison_c_(2002).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap24" width="146" height="20"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap364"> <area target="_blank" href="http://www.value-at-risk.net" shape="rect" coords="0, 0, 81, 124"> </map> <img border="0" src="../images/book_holton_(2003)_125.jpg" usemap="#FPMap364" width="82" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table17"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.value-at-risk.net">Value-at-Risk: Theory and Practice</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Glyn Holton</b></td> </tr> <tr> <td width="10%"> <map name="FPMap362"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"></map> <img border="0" src="../images/star_6.gif" usemap="#FPMap362" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"> <map name="FPMap363"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 1, 89, 15"> </map><img border="0" src="../images/tech_4.gif" usemap="#FPMap363" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">2003</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap361"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.value-at-risk.net"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap361" width="146" height="20"></td> </tr> </table> </td> </tr> </table> </td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Papers</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table18"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Group of 30">Group of 30</a> (1993). <i>Derivatives: Practices and Principles</i>, Washington: Group of 30.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Markowitz">Markowitz</a>, Harry M. (1952). Portfolio selection, <i>Journal of Finance</i>, 7 (1), 77-91<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15">Morgan Guaranty (1994). <i>RiskMetrics Technical Document</i> 2<sup>nd</sup> Edition, <span style="font-size: 12.0pt; font-family: Times New Roman">New York: Morgan Guaranty.</span></td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Sponsored Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" width="75"> <img border="0" src="../images/ad_side_banner_1.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%">  <p class="ad_legend" style="margin-left: 0; margin-bottom:2"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a></p> <p style="margin-top: 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