Linear 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"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table2"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table3"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table4"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Linear 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="table5"> <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="#closed-form">closed form VaR</a><p class="word_list"> <a href="#delta-normal">delta-normal VaR</a><p class="word_list"> <a href="#linear transformations">linear transformation</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"> <a href="#linear VaR measures">linear VaR</a><p class="word_list"> <a href="#parametric">parametric VaR</a><p class="word_list"> <a href="#variance-covariance">variance-covariance VaR</a><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="left" id="table6"> <tr> <td width="300" height="14"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14"></td> </tr> <tr> <td width="300"> <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"><a href="../articles/var_measure.htm">VaR measures</a> have traditionally been categorized according to the <a href="../articles/var_measure.htm">transformations procedures</a> they employ. There are four basic forms of transformation in widespread use:</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><b><a name="linear transformations">linear transformations</a></b>,</p> <p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/quadratic_transformation.htm">quadratic transformations</a>,</p> <p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/monte_carlo_transformation.htm">Monte Carlo transformations</a>, and</p> <p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/monte_carlo_transformation.htm">historical transformations</a>.</p> <p class="body">This article discusses <b><a name="linear VaR measures"> linear VaR measures</a></b>, which employ linear transformations. Many names have been used to describe linear VaR measures, so you may hear them referred to as <b><a name="parametric">parametric</a></b>, <b> <a name="variance-covariance">variance-covariance</a></b>, <b> <a name="closed-form">closed-form</a></b>, or <b><a name="delta-normal"> delta-normal</a></b> VaR measures. There are shortcomings with most of these names. While linear VaR measures are parametric, so are most VaR measures. While linear VaR measures use variances and covariances, so do all VaR measures, with the exception of historical VaR measures. While some linear VaR measures employ a <a href="../articles/remappings.htm">delta remapping</a>, most do not. Also, while a <a href="../articles/normal_distribution.htm">normal</a> assumption is common with linear VaR measures, it is by no means universal. </p> <p class="body">I prefer the name "linear" because it describes the one characteristic that is common to all linear transformations: they are applicable to portfolios whose <a href="../articles/var_measure.htm">portfolio mapping function</a> is a <a href="../articles/polynomial.htm">linear polynomial</a>. Such portfolios include portfolios of equities, physical commodities, or <a href="../articles/future.htm">futures</a>. The <a href="../articles/valuation.htm">market value</a> of such portfolios depends linearly upon applicable <a href="../articles/var_measure.htm">key factors</a>. Other portfolios are so nearly linear that they can reasonably be approximated (<a href="../articles/remappings.htm">remapped</a>) with a linear polynomial. These include portfolios of <a href="../articles/forward.htm">forwards</a> (including foreign exchange forwards) and most non-callable debt.</p> <p class="body">Linear VaR measures are generally not applicable to portfolios that hold <a href="../articles/option.htm">options</a> or instruments with embedded options. These include <a href="../articles/callable_bond.htm">callable bonds</a>, <a href="../articles/mortgage_backed_security.htm">mortgage-backed securities</a> and many structured notes.</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">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> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table8"> <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">A transformation procedure accepts two inputs:</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">a portfolio mapping function 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 joint distribution of the <a href="../articles/var_measure.htm">key vector</a> obtained from the <a href="../articles/var_measure.htm">inference procedure</a>.</p> <p class="body">The transformation procedure must combine these to somehow characterize the distribution of the portfolio's value. Based upon that characterization, the transformation procedure then values the desired <a href="../articles/var_metric.htm">VaR metric</a>.</p> <p class="body">Let time 0 correspond to the current time, and let time 1 correspond to the end of the <a href="../articles/value_at_risk.htm">VaR horizon</a>. Mathematically, a portfolio is defined by a current value <img border="0" src="../symbols/0_p.gif" align="absbottom" width="14" height="18">, which is a known constant, and a future value <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18">, which is a random variable (see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation). Typically, <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> is specified as a function <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> of a random vector <sup>1</sup><i><b>R</b></i>, which is the key vector. Its components <img border="0" src="../formulas/quadratic_transformations_1ri.gif" align="absbottom" width="18" height="17">, called key factors, represent market variables such as prices, interest rates, <a href="../articles/spreads.htm">spreads</a> or <a href="../articles/volatility.htm">implied volatilities</a> as of time 1. Current values of key factors are indicated with a constant vector <img border="0" src="../symbols/0_r_bold.gif" align="absbottom" width="13" height="17">. The relationship <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> = <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15">(<img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">) is called a <a href="../articles/var_measure.htm">portfolio mapping</a>, and the function <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> is the <a href="../articles/var_measure.htm">portfolio mapping function</a>.</p> <p class="body">We say that a portfolio is <b><a name="linear">linear</a></b> if its portfolio mapping function <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> is a linear polynomial. Using matrix notation (see any elementary linear algebra text, such as Strang (1988)) this <a name="[1]">means</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/linear_transformation_1.gif" width="151" height="31"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">where <i><b> b</b></i> is a row vector and <i>a</i> is a scalar. </p> <p class="body">Suppose a portfolio comprises 100 <a href="../articles/common_stock.htm">shares</a> of Dell <a href="../articles/common_stock.htm">stock</a>, 200 shares of IBM stock and a short position of 300 shares of Microsoft stock. In this case, we would define</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/var_me2.gif" width="248" height="86"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">Assuming none of the stocks goes <a href="../articles/record_date.htm">ex-dividend</a> during the VaR horizon, the mapping procedure would <a name="[3]">specify</a>:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table11"> <tr> <td width="100%" align="center"> <img border="0" src="../images/var_measure_10.gif" width="273" height="28"></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">If we let <i><b>b</b></i> = (100   200   –300), then [<a href="#[3]">3</a>] has form [<a href="#[1]">1</a>].</p> <p class="body">Linear transformations are based upon an important result from probability theory related to <a href="../articles/linear_polynomial_of_a_random_vecrtor.htm">linear polynomials of random vectors</a>. Let <img border="0" src="../formulas/linear_var_1_0_mu.gif" align="absbottom" width="21" height="17"> and <img border="0" src="../formulas/linear_var_1_0_sigma.gif" align="absbottom" width="23" height="17"> be the <a href="../articles/mean.htm">mean vector</a> and <a href="../articles/correlation.htm">covariance matrix</a> of <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">. (The superscripts <img border="0" src="../symbols/1_bar_0.gif" align="absbottom" width="16" height="16"> indicate that both parameters are for time 1, conditional on information available at time 0.) Let <img border="0" src="../formulas/linear_transformation_001.gif" align="absbottom" width="40" height="17"> and <img border="0" src="../formulas/linear_transformation_002.gif" align="absbottom" width="48" height="17"> be the mean and <a href="../articles/standard_deviation.htm">standard deviation</a> of <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> conditional on information available at time 0. Then probability theory tells <a name="[4]">us</a> <a name="[5]">that</a> (see article <i> <a href="../articles/linear_polynomial_of_a_random_vecrtor.htm"> linear polynomial of a random vector</a></i>):</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table12"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/linear_transformation_4.gif" width="122" height="28"></td> <td width="5%" align="right">[4]</td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/linear_transformation_2.gif" width="133" height="28"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">where a prime <i>'</i>  indicates transposition. Note that these formulas are general. They require that <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> be a linear polynomial of <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">, but they make no assumptions about the distribution of <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">. With these results, it is possible to value a variety of VaR metrics, including standard deviation of loss</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table13"> <tr> <td width="100%" align="center"> <sup>0</sup><i>std</i>(<sup>1</sup><i>L</i>) =  <sup>0</sup><i>std</i>(<sup>0</sup><i>p</i> –<sup> 1</sup><i>P</i>) = <sup>0</sup><i>std</i>(<sup>1</sup><i>P</i>)</td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">and standard deviation of <a href="../articles/return.htm">return</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table14"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/linear_transformation_7.gif" width="177" height="56"></td> <td width="5%" align="right">[7]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table15"> <tr> <td width="336" height="10"><font size="2"> </font></td> <td width="9" height="10"></td> </tr> <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">On their own, results [<a href="#[4]">4</a>] and [<a href="#[5]">5</a>] are not sufficient to value a <a href="../articles/quantile.htm">quantile</a> of loss VaR metric. Without additional information about a distribution, a mean and standard deviation do not determine the distribution's quantiles. </p> <p class="body">A standard solution is to assume <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> is normally distributed. Because a <a href="../articles/normal_distribution.htm">normal distribution</a> is fully determined by its mean and standard deviation, this assumption—together with [<a href="#[4]">4</a>] and [<a href="#[5]">5</a>]—fully specifies the distribution of <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18">. It should be possible to value any VaR metric.</p> <p class="body">For quantile of loss VaR metrics, we use the fact that <a href="../articles/quantile.htm">quantiles</a> of a normal distribution occur a fixed number of standard deviations from that distribution's mean. Formulas for some standard quantile of loss VaR metrics <a name="[8]">are</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table16"> <tr> <td width="100%" align="center"> 90.0% VaR = 1.282 <sup>0</sup><i>std</i>(<sup>1</sup><i>P</i>) – [ <sup>0</sup><i>E</i>(<sup>1</sup><i>P</i>) – <sup>0</sup><i>p</i> ]</td> <td width="5%" align="right">[8]</td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> 95.0% VaR = 1.645 <sup>0</sup><i>std</i>(<sup>1</sup><i>P</i>) – [ <sup>0</sup><i>E</i>(<sup>1</sup><i>P</i>) – <sup>0</sup><i>p</i> ]</td> <td width="5%" align="right">[9]</td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> 97.5% VaR = 1.960 <sup>0</sup><i>std</i>(<sup>1</sup><i>P</i>) – [ <sup>0</sup><i>E</i>(<sup>1</sup><i>P</i>) – <sup>0</sup><i>p</i> ]</td> <td width="5%" align="right">[10]</td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> 99.0% VaR = 2.326 <sup>0</sup><i>std</i>(<sup>1</sup><i>P</i>) – [ <sup>0</sup><i>E</i>(<sup>1</sup><i>P</i>) – <sup>0</sup><i>p</i> ]</td> <td width="5%" align="right">[11]</td> </tr> </table> <p class="body">These formulas are motivated for the 90% VaR case in Exhibit 2:</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="450" cellpadding="0" id="table17"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Graphical Derivation of Formula [8]</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_linear_transformation.gif" width="450" height="224"></td> </tr> <tr> <td> <p class="exhibit_legend">A graphical derivation of formula [<a href="#[8]">8</a>] is provided. The 90% loss occurs at a portfolio value 1.282 standard deviations below the portfolio's expected value (the mean of the distribution). However, loss is calculated relative to the portfolio's current value as opposed to its expected value, which is why formula [<a href="#[8]">8</a>] includes the [<sup>0</sup><i>E</i>(<sup>1</sup><i>P</i>) – <sup>0</sup><i>p</i>] term.</td> </tr> </table> </center> </div> <p class="body">Over a short VaR horizon, such as a day, it is often reasonable to assume the portfolio's expected value equals its current value. In this case, Formulas [<a href="#[8]">8</a>] to [<a href="#[8]">11</a>] simplify to</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table18"> <tr> <td width="100%" align="center"> 90.0% VaR = 1.282 <sup>0</sup><i>std</i>(<sup>1</sup><i>P</i>)</td> <td width="5%" align="right">[12]</td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> 95.0% VaR = 1.645 <sup>0</sup><i>std</i>(<sup>1</sup><i>P</i>)</td> <td width="5%" align="right">[13]</td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> 97.5% VaR = 1.960 <sup>0</sup><i>std</i>(<sup>1</sup><i>P</i>)</td> <td width="5%" align="right">[14]</td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> 99.0% VaR = 2.326 <sup>0</sup><i>std</i>(<sup>1</sup><i>P</i>)</td> <td width="5%" align="right">[15]</td> </tr> </table> <p class="body">Because the computations for a linear transformation are so modest, <a href="../articles/var_measure.htm">implementations</a> typically run in real time. Simple linear VaR measures can even be implemented on a spreadsheet. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table19"> <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="table20"> <tr> <td width="100%"> <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_transformation.htm">Monte Carlo value-at-risk</a> A category of VaR measures that employ the Monte Carlo method.<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"><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="I1"> <![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="table21"> <tr> <td width="100%"> <p class="explanatory_text">Butler (<a target="_blank" href="http://www.riskbook.com/link/butler_c_(1999).htm">1999</a>) is an elementary introduction to value-at-risk that focuses on Linear VaR measures. 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