VaR Metric

<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"> <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">VaR Metric</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="#expected shortfall">expected shortfall</a><p class="word_list"> <a href="#expected tail loss">expected tail loss</a><p class="word_list"> <a href="#Semi-variance">semi-variance</a><p class="word_list"> <a href="#VaR metrics">VaR metric</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"> This article discusses VaR metrics. It assumes familiarity with concepts described in the articles <a href="../articles/value_at_risk.htm"> value-at-risk</a> and <a href="../articles/var_measure.htm">measuring VaR</a>. </p> <p class="body">It is worth distinguishing between three concepts: </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">A <a href="../articles/var_measure.htm">VaR measure</a> is an algorithm with which we calculate a portfolio's VaR. </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">A <a href="../articles/var_measure.htm">VaR model</a> is the financial theory, mathematics, and logic that motivate a VaR measure. It is the intellectual justification for the computations that are the VaR measure. </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">A <b><a name="VaR metric">VaR metric</a></b> is our interpretation for the output of the VaR measure. </p> <p class="body">Examples of VaR metrics are one-day 95% <a href="../articles/currency_codes.htm"> USD</a> VaR or one-week <a href="../articles/standard_deviation.htm">standard deviation</a> of <a href="../articles/return.htm">return</a> <a href="../articles/currency_codes.htm">EUR</a> VaR. A VaR measure is just a bunch of computations. What justifies our interpreting the output of those computations as, say, two-week 99% EUR VaR? The answer is the VaR model. The VaR model is the intellectual link between the computations of a VaR measure and the interpretation of the output of those computations, which is the VaR metric. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table6"> <tr> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> <p class="body">Let's introduce some notation. We measure time in units equal to the length of the <a href="../articles/value_at_risk.htm">VaR horizon</a>. The present time is time 0. The end of the VaR horizon is time 1. </p> <p class="body">To distinguish between known quantities and random quantities, we denote the former with lowercase letters and the latter with capital letters. With this convention, we denote the portfolio's current <a href="../articles/valuation.htm">market value</a> as <sup>0</sup><i>p</i> and its market value at the end of the VaR horizon as <sup>1</sup><i>P</i>. The preceding superscripts 0 and 1 denote time (see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation).</p> <p class="body">Formally, a VaR metric is a <a href="../articles/set.htm">real function</a> of: </p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">the distribution of <sup>1</sup><i>P</i>, conditional on information available at time 0; and <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">the portfolio’s current value <sup>0</sup><i>p</i>. <p><a href="../articles/standard_deviation.htm">Standard deviation</a> of portfolio <a href="../articles/return.htm">simple return</a> <sup>1</sup><i>Z</i>, conditional on information available at time 0, is a VaR metric: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/var_metric_1.gif" width="240" height="56"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body"> <span style="font-size: 12.0pt; font-family: Times New Roman"> <a href="../articles/quantile.htm">Quantiles</a> of p</span>ortfolio loss, 1L = 0p – 1P, make intuitively appealing VaR metrics. If a portfolio's conditional .95-quantile of 1L is USD 12.5<a href="../articles/mm.htm">MM</a>, then such a portfolio can be expected to lose less than USD 12.5MM on 19 days out of 20.</p> <p class="body">An example of a risk metric that is not a VaR metric is standard deviation of cash flow. Because this generally cannot be expressed as a function of 0p and the conditional distribution of 1P, it is not a VaR metric.</p> <p class="body">VaR metrics can be quite elaborate. <b> <a name="Semi-variance">Semi-variance</a></b> of portfolio return <sup>1</sup><i>Z</i> is one example. Define</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table8"> <tr> <td width="100%" align="center"> <p style="margin-top: 3; margin-bottom: 3"> <img border="0" src="../formulas/var_metric_2.gif" width="132" height="54"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">Then the semi-variance of <sup>1</sup><i>Z</i> is simply the variance of <sup>1</sup><i>Z</i><sup>–</sup>.</p> <p class="body">Another VaR metric is <b><a name="expected tail loss"> expected tail loss</a></b> (ETL), which is sometimes called <b> <a name="expected shortfall">expected shortfall</a></b>. This is the average portfolio loss, assuming that the loss exceeds some quantile of loss. For example, a 90% ETL VaR metric indicates the expected loss conditional on that loss exceeding its own .90-quantile.</p> <p class="body">To fully specify a VaR metric, we must indicate three things:</p> <p class="bullet_line_space"> <img border="0" src="../images/bullet.gif" width="20" height="10"> the period of time—1 day, 2 weeks, 1 month, etc.—between time 0 and time 1; this is the <a href="../articles/value_at_risk.htm">VaR horizon</a>;<p class="bullet_line_space"> <img border="0" src="../images/bullet.gif" width="20" height="10">the function of  0p and the conditional distribution of 1P;<p class="bullet_line_space"> <img border="0" src="../images/bullet.gif" width="20" height="10">the currency in which 0p and 1P are denominated; this is the <a href="../articles/value_at_risk.htm">base currency</a>.<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table9"> <tr> <td align="right" width="9"> <p style="margin-top: 0; margin-bottom: 0"> </td> <td width="336">  <p style="margin-top: 8; margin-bottom: 2" class="ad_legend"> <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: 0; margin-bottom: 0"> <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82402;type=iframe" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I1"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82402;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82402"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82402;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </td> </tr> </table> <p class="body" style="margin-left: 0in; margin-right: 0in; "> We adopt a convention for naming VaR metrics:</p> <p class="bullet_line_space"> <img border="0" src="../images/bullet.gif" width="20" height="10"> The metric’s name is given as the horizon, function and currency, in that order, followed by “VaR.”<p class="bullet_line_space"> <img border="0" src="../images/bullet.gif" width="20" height="10"> If the horizon is expressed in days without qualification, these are understood to be trading days.<p class="bullet_line_space"> <img border="0" src="../images/bullet.gif" width="20" height="10">If the function is a quantile of loss, it is indicated simply as a percentage.<p class="body">For example, we may speak of a portfolio’s</p> <p class="bullet_line_space"> <img border="0" src="../images/bullet.gif" width="20" height="10">1-day standard deviation of simple return USD VaR,<p class="bullet_line_space"> <img border="0" src="../images/bullet.gif" width="20" height="10"> 2-week 95% JPY VaR, or<p class="bullet_line_space" style="margin-bottom: 18"> <img border="0" src="../images/bullet.gif" width="20" height="10">1-week 90% ETL GBP VaR, etc.<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table10"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black" style="margin-top: 8">Related Internal 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" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table11"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/market_risk.htm">market risk</a> </strong>Exposure to the uncertain market value of a portfolio.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/var_measure.htm">measuring value-at-risk</a> Describes how VaR measures work.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/risk_metric_and_risk_measure.htm">risk measure and risk metric</a> Concepts related to the quantification of risk.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/value_at_risk.htm">value-at-risk</a> A category of market risk measures.</td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Books</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="table12"> <tr> <td width="100%"> <p class="explanatory_text">Dows (<a target="_blank" href="http://www.riskbook.com/link/dowd_k_(2002).htm">2002</a>) discusses ETL VaR metrics in detail. 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