Monte Carlo Value-at-Risk

<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">Monte Carlo 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="99%" align="left" valign="top"><p class="word_list"> <a href="#Historical transformations">historical transformation</a><p class="word_list"> <a href="#historical VaR measures">historical VaR</a><p class="word_list"> <a href="#Monte Carlo transformation procedures">Monte Carlo transformation</a><p class="word_list"> <a href="#Monte Carlo VaR measure">Monte Carlo VaR</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <p class="body">This article discusses Monte Carlo and historical VaR measures. Exhibit 1 is reproduced from the overview article <a href="../articles/var_measure.htm">measuring value-at-risk</a>. It indicates the three processes that are essential to all practical <a href="../articles/var_measure.htm">VaR measures</a>. This article assumes familiarity with the concepts referenced in that exhibit.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table5"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Schematic of How VaR Measures Work</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_var_measure.gif" width="350" height="549"></td> </tr> <tr> <td> <p class="exhibit_legend">All practical VaR measures accept portfolio data and historical market data as inputs. They process these with a mapping procedure, inference procedure, and transformation procedure. Output comprises the value of a <a href="../articles/var_metric.htm">VaR metric</a>. That value is the VaR measurement.</td> </tr> </table> </center> </div> <p class="body">A transformation procedure accepts two inputs:</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">a <a href="../articles/var_measure.htm">portfolio mapping function</a> obtained from the <a href="../articles/mapping_procedure.htm">mapping procedure</a>, and</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">a characterization of the conditional joint distribution of the <a href="../articles/var_measure.htm">key factors</a> obtained from the <a href="../articles/var_measure.htm">inference procedure</a>.</p> <p class="body">The transformation procedure must somehow combine these to characterize the conditional distribution of the portfolio's value. Based upon that characterization, the transformation procedure values the desired <a href="../articles/var_metric.htm">VaR metric</a>.</p> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table6"> <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">If a portfolio mapping function is a <a href="../articles/polynomial.htm">linear</a> or <a href="../articles/polynomial.htm">quadratic polynomial</a>, then a <a href="../articles/linear_transformation.htm">linear transformation</a> or <a href="../articles/quadratic_transformation.htm">quadratic transformation</a> should generally be used. Both run rapidly and entail little or no error. Monte Carlo and historical transformations take longer to run and introduce the standard error common to all <a href="../articles/monte_carlo_method.htm">Monte Carlo estimators</a>. For this reason, they are reserved for only those portfolios for which portfolio mapping functions are neither linear nor quadratic. Such portfolios generally contain <a href="../articles/derivative_instrument.htm">derivatives</a>, <a href="../articles/mortgage_backed_security.htm">mortgage-backed securities</a> or other complex non-linear instruments.</p> <p class="body">Let time 0 be the current time, and let time 1 be the end of the <a href="../articles/value_at_risk.htm">VaR horizon</a> (see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation). A portfolio mapping <img border="0" src="../formulas/monte_carlo_transformation_portfolio_mapping.gif" align="absbottom" width="63" height="17"> is obtained from a mapping procedure, where</p> <p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10"><img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> is the random variable for the portfolio's value at the end of the VaR horizon,</p> <p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10"><img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> is the portfolio mapping function, and</p> <p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10"><img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17"> is the <a href="../articles/var_measure.htm">key vector</a>. </p> <p class="body">Based upon the characterization of the joint distribution of <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17"> obtained from the inference procedure, a <b><a name="Monte Carlo transformation">Monte Carlo transformation</a></b> procedure generates several thousand pseudorandom realizations <img border="0" src="../formulas/monte_carlo_transformation_1_r_k.gif" align="absbottom" width="27" height="20"> for <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">. Based upon these, it calculates corresponding realizations <img border="0" src="../formulas/monte_carlo_transformation_1_p_k_realizations.gif" align="absbottom" width="98" height="20">. The histogram of these realizations <img border="0" src="../formulas/monte_carlo_transformation_1_p_k.gif" align="absbottom" width="27" height="20"> provides a discrete approximation for the distribution of <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18">. Based on this, any reasonable VaR metric can be valued. Exhibit 2 illustrates a histogram of portfolio value realizations from an actual Monte Carlo VaR analysis.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table7"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Example: Histogram of Portfolio Values from a Monte Carlo VaR Analysis</span><br> <span class="exhibit_subheader">Exhibit 2</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/ex2_monte_carlo_transformation.gif" width="300" height="190"></td> </tr> <tr> <td> <p class="exhibit_legend">Histogram of portfolio values obtained from an actual VaR analysis. A sample size of 5000 realizations was used. The skewness of the distribution suggests positive <a href="../articles/delta_and_gamma.htm">gamma</a>.</td> </tr> </table> </center> </div> <p class="body">Because it depends upon the <a href="../articles/monte_carlo_method.htm">Monte Carlo method</a>, a Monte Carlo transformation procedure entails standard error. The magnitude of the standard error depends upon many things, including:</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the VaR metric, </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the portfolio, and </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the sample size used in the Monte Carlo analysis. </p> <p class="body">A crude rule of thumb is that the standard error will be about 1% of calculated VaR for a typical portfolio if a <a href="../articles/quantile.htm">quantile</a>-of-loss VaR metric is used and the sample size is 10,000. Because valuing a portfolio mapping function 10,000 times can be an enormous computational load, most organizations don't use nearly as large a sample size. Sample sizes as low as 500 are commonly used. Because the standard error of a Monte Carlo analysis is proportional to the square root of the sample size, such low sample sizes introduce standard errors on the order of 4.5%. (I know of several software vendors who obfuscate this fact in their product literature.)</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table8"> <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">A solution to this problem is to employ variance reduction. Techniques based upon <a href="../articles/monte_carlo_method.htm">control variates</a> and <a href="../articles/monte_carlo_method.htm">stratified sampling</a> were published by Cárdenas, et al. (<a href="#Cárdenas">1999</a>). Both methods they propose employ a <a href="../articles/remappings.htm">quadratic remapping</a> <img border="0" src="../formulas/monte_carlo_transformation_1p_tilde.gif" align="absbottom" width="13" height="19"> for <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18">. The remapping <img border="0" src="../formulas/monte_carlo_transformation_1p_tilde.gif" align="absbottom" width="13" height="19"> does not replace <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18">. Instead, it is used to facilitate variance reduction so VaR can be more easily calculated for <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18">.</p> <p class="body">With the method of control variates, <img border="0" src="../formulas/monte_carlo_transformation_1p_tilde.gif" align="absbottom" width="13" height="19"> is used as a control variate for <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18">. For this purpose, we need to calculate the VaR of <img border="0" src="../formulas/monte_carlo_transformation_1p_tilde.gif" align="absbottom" width="13" height="19">, but this is easily accomplished with a  <a href="../articles/quadratic_transformation.htm">quadratic transformation procedure</a>. Variance reduction is excellent for most portfolios and VaR metrics.</p> <p class="body">With the method of stratified sampling, <img border="0" src="../formulas/monte_carlo_transformation_1p_tilde.gif" align="absbottom" width="13" height="19"> is used to construct a stratification. The methodology varies depending upon the VaR metric. For a quantile of loss VaR metric, realizations <img border="0" src="../formulas/monte_carlo_transformation_1rk.gif" align="absbottom" width="26" height="20"> of <sup>1</sup><b><i>R</i></b> are stratified into two regions:</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">one comprising realizations <img border="0" src="../formulas/monte_carlo_transformation_1rk.gif" align="absbottom" width="26" height="20"> such that <img border="0" src="../formulas/monte_carlo_transformation_843hgj.gif" align="absbottom" width="100" height="20"> exceeds the VaR of <img border="0" src="../formulas/monte_carlo_transformation_1p_tilde.gif" align="absbottom" width="13" height="19">, and</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the other comprises realizations <img border="0" src="../formulas/monte_carlo_transformation_1rk.gif" align="absbottom" width="26" height="20"> such that <img border="0" src="../formulas/monte_carlo_transformation_843hgj.gif" align="absbottom" width="100" height="20"> is less than or equal to the VaR of <img border="0" src="../formulas/monte_carlo_transformation_1p_tilde.gif" align="absbottom" width="13" height="19">.</p> <p class="body">Variance reduction is excellent for most portfolios and VaR metrics.</p> <p class="body">To further improve variance reduction, the above two methodologies can be combined. Cárdenas, et al. (<a href="#Cárdenas">1999</a>) also suggested a technique of selective valuation that can be used. While this is technically not a variance reduction technique, it has largely the same effect. See Holton (<a target="_blank" href="http://www.riskbook.com/link/holton_g_(2003).htm">2003</a>) for a detailed discussion of all these methodologies.</p> <p class="body"><b><a name="Historical transformations">Historical transformations</a></b> are identical to Monte Carlo transformations except for one difference. Both employ the Monte Carlo method to construct a histogram of realizations <img border="0" src="../formulas/monte_carlo_transformation_1_p_k_realizations.gif" align="absbottom" width="98" height="20"> of <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18">. The difference lies in how they construct realizations <img border="0" src="../formulas/monte_carlo_transformation_1rk.gif" align="absbottom" width="26" height="20"> for <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">. Monte Carlo transformations randomly generate them based upon a characterization of the distribution of <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">. Historical transformations employ realizations <img border="0" src="../formulas/monte_carlo_transformation_1rk.gif" align="absbottom" width="26" height="20"> constructed from historical market data for <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">. </p> <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"> Historical transformations were popular during the early 1990s because they were intuitively easy to explain to non-technical professionals—VaR was being calculated based on one-day profit or losses that the portfolio would have realized based on market movements that occurred during each of the past 500 or so trading days. There are, however compelling reasons to avoid historical transformations. First, because they are based on the Monte Carlo method, they entail exactly the same standard error as Monte Carlo transformations. While Monte Carlo transformations can minimize standard error by using large sample sizes, this is not possible for historical transformations. Their sample sizes are limited by the availability of relevant historical market data. It is rare that historical transformations have sample sizes grater than 1000, so standard errors is generally significant. Historical transformations are not amenable to many of the powerful methods of variance reduction available for Monte Carlo transformations. Finally, use of historical realizations introduces biases relating to <a href="../articles/heteroskedasticity.htm">conditional heteroskedasticity</a>. See Holton (<a target="_blank" href="http://www.value-at-risk.net">2003</a>).</p> <p class="body">VaR measures are called <b> <a name="Monte Carlo VaR measures">Monte Carlo VaR measures</a></b> or <b> <a name="historical VaR measures">historical VaR measures</a></b> if they employ, respectively, Monte Carlo or historical transformations. With Monte Carlo VaR measures, an inference procedure typically characterizes the distribution of <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17"> by assuming some standard joint distribution—such as the <a href="../articles/joint_normal_distribution.htm">joint-normal distribution</a>—and specifying a <a href="../articles/correlation.htm">covariance matrix</a> and <a href="../articles/mean.htm">mean vector</a> for this. Based on this characterization, the Monte Carlo transformation randomly generates realizations <img border="0" src="../formulas/monte_carlo_transformation_1rk.gif" align="absbottom" width="26" height="20">. Historical VaR measures are somewhat unique in how their inference procedures characterize the distribution of <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">. Literally, they do so with the <a href="../articles/set.htm">set</a> of historical realizations <img border="0" src="../formulas/monte_carlo_transformation_1rk.gif" align="absbottom" width="26" height="20">. The inference procedure directly passes this set of historical realizations to the historical transformation. The set of historical realizations <img border="0" src="../formulas/monte_carlo_transformation_1rk.gif" align="absbottom" width="26" height="20"> is the inference procedure's characterization of the joint distribution of <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">.</p> <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">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"><a href="../articles/linear_transformation.htm">linear value-at-risk</a> A category of VaR measures that are applicable to linear portfolios.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/mapping_procedure.htm">mapping procedure</a> One of the procedures that comprise a VaR measure.<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"><a href="../articles/monte_carlo_method.htm">Monte Carlo method</a> The use of statistical sampling to solve quantitative problems. <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/quadratic_transformation.htm">quadratic value-at-risk</a> Also called "delta-gamma VaR," this is a category of VaR measures that are applicable to quadratic portfolios.<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/stress_testing.htm">stress testing</a> A simple form of scenario analysis typically used to assess market 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.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/var_metric.htm">VaR metric</a> An interpretation of a VaR measure.</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_2.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: 0; margin-bottom: 0"> <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82436;type=iframe" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I2"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82436;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82436"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82436;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </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">Marrison (<a target="_blank" href="http://www.riskbook.com/link/marrison_c_(2002).htm">2002</a>) is an elementary text that introduces Monte Carlo and historical VaR measures in the 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. It covers Monte Carlo transformations and the use of variance reduction in in such transformations. Glasserman (<a target="_blank" href="http://www.riskbook.com/link/glasserman_p_(2003).htm">2003</a>) also discusses variance reduction in Monte Carlo transformations, presenting a different techniques.</p> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table13"> <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="table14"> <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 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