Quadratic 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="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">Quadratic 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="#delta-gamma VaR measures">delta-gamma VaR measure</a><p class="word_list"> <a href="#Johnson curves">Johnson curves</a><p class="word_list"> <a href="#quadratic">quadratic portfolio</a><p class="word_list"> <a href="#quadratic transformation">quadratic transformation</a><p class="word_list"> <a href="#quadratic VaR measure">quadratic VaR measure</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="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">During the 1990s, practitioners faced a difficult choice in selecting the form of <a href="../articles/var_measure.htm"> transformation procedure</a> to use with <a href="../articles/value_at_risk.htm">value-at-risk</a> (VaR) measures. <a href="../articles/linear_transformation.htm">Linear transformations</a> were exact and ran in real time, but they applied only to linear portfolios. <a href="../articles/monte_carlo_transformation.htm">Monte Carlo</a> and <a href="../articles/monte_carlo_transformation.htm">historical</a> transformations were more generally applicable, but they entailed standard error and ran more slowly. Practitioners sought a compromise—some middle ground between the accuracy, speed and limited applicability of linear transformations and the general applicability, standard error and slowness of Monte Carlo and historical transformations. An obvious avenue of research was to explore VaR measures that applied to quadratic portfolios. Even before transformation procedures were formalized for them, such measures were called <b><a name="delta-gamma VaR measures">delta-gamma VaR measures</a></b>. A better name—one that makes no tacit assumption that a delta-gamma <a href="../articles/remappings.htm">remapping</a> must be used—is <b><a name="quadratic VaR measure">quadratic VaR measures</a></b>.</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 <sup>1</sup><i>P</i>, which is a random variable (see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation). Typically, <sup>1</sup><i>P</i> 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> called a <a href="../articles/var_measure.htm">key vector</a>. Its components <img border="0" src="../formulas/quadratic_transformations_1ri.gif" align="absbottom" width="18" height="17">, called <a href="../articles/var_measure.htm">key factors</a>, 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 <sup>0</sup><i><b>r</b></i>. The relationship <sup> 1</sup><i>P</i> = <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15">(<sup>1</sup><i><b>R</b></i>) 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 called a <a href="../articles/var_measure.htm">portfolio mapping function</a>.</p> <p class="body">We say that a portfolio is <b><a name="quadratic">quadratic</a></b> if its portfolio mapping function <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> is a <a href="../articles/polynomial.htm">quadratic polynomial</a>:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transformation_1.gif" width="206" height="28"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">Here, <i><b>c</b></i> is a symmetric square matrix, <i><b> b</b></i> is a row vector and <i>a</i> is a scalar. A prime <i>'</i>  indicates transposition. </p> <p class="body">Many researchers worked to develop a transformation procedure applicable to quadratic portfolios—what we call a <b> <a name="quadratic transformation">quadratic transformation</a></b>. An obvious solution is to simply apply the <a href="../articles/monte_carlo_method.htm">Monte Carlo method</a>. With <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> a quadratic polynomial, realizations <sup>1</sup><i>p</i><sup>[<i>k</i>]</sup> = <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15">(<sup>1</sup><i><b>r</b></i><sup>[<i>k</i>]</sup>) can be valued rapidly. Because it depends upon the Monte Carlo method, this solution entails standard error. Depending upon the number of realizations used, it imposes a modest tradeoff between run-time and standard error. </p> <p class="body">Wilson (<a href="#Wilson">1994</a>) published an innovative quadratic transformation based upon an optimizing search routine. This limits <sup>1</sup><i><b>R</b></i> to some pre-defined region and searches for the maximum portfolio loss within that region. The <a href="../articles/var_metric.htm">VaR metric</a>—maximum portfolio loss within a pre-specified set of possible values for <sup>1</sup><i><b>R</b></i> —is non-standard and is of limited usefulness.</p> <p class="body">Researchers sought a general quadratic transformation that might offer the accuracy and calculation speed of linear transformations. Such a solution does exist based upon the established mathematics of quadratic polynomials of <a href="../articles/joint_normal_distribution.htm">joint-normal</a> random vectors. By 1996, a number of researchers had found the solution. In January of that year, Fallon (<a href="#Fallon">1996</a>) circulated a working paper with a solution. The description of his quadratic transformation comprised only a small portion of the paper. It attracted little attention. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table7"> <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">Eleven months later, a fourth edition of the <i> RiskMetrics Technical Document</i> was published. Zangari (<a href="#Zangari">1996</a>) contributed a cursory description of a quadratic transformation similar to Fallon’s. By this time, the <a href="../articles/riskmetrics.htm">RiskMetrics</a> group was struggling to justify its existence within JP Morgan. They were earning fees through consulting and would soon be spun off as a separate consulting firm. For one reason or another, the solution Zangari published was incomplete. It offered tantalizing clues that the RiskMetrics group possessed a solution, but a complete derivation or information on how to implement a general solution were not provided. </p> <p class="body">The first complete published solution was by Rouvinez (<a href="#Rouvinez">1997</a>). He detailed how, if <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> is a quadratic polynomial, and <sup>1</sup><i><b>R</b></i> is joint-normal with <a href="../articles/positive_definite_matrix.htm">positive definite</a> covariance matrix, the characteristic function of <sup>1</sup><i>P</i> can be calculated. Since a characteristic function fully specifies the cumulative distribution function (CDF) of a random variable, this makes it theoretically possible to evaluate any VaR metric. </p> <p class="body">Eight months after Rouvinez published his paper, Cárdenas, et al. (<a href="#Cárdenas">1997</a>) published a similar solution. Working papers by Britten-Jones and Schaefer (<a href="#Britten-Jones">1997</a>) and Jahel, Perraudin and Sellin (<a href="#Jahel">1997</a>) offered similar solutions.</p> <p class="body"> <map name="FPMap372"> <area target="_top" href="http://www.value-at-risk.net" shape="rect" coords="0, 0, 324, 249"> </map> <img border="0" src="../images/ad_var_book_250x325.gif" align="left" usemap="#FPMap372" width="325" height="250">The quadratic transformations described in these papers differ in various respects, but they all employ the mathematics of quadratic polynomials of joint-normal random vectors. Their collective solution can reasonably be called <i>the</i> quadratic transformation. </p> <p class="body">Because of its association with RiskMetrics, Zangari’s solution attracted much attention. Being incomplete, it was not useful. The other papers attracted little attention. All were technical papers that were released with little fanfare. A reader would have to have strong mathematical skills and devote several hours to deciphering any of them in order to realize that the sought-after quadratic solution had indeed been found. Apparently, not many people did.</p> <p class="body">The popular literature on value-at-risk largely ignored the new quadratic measures. Discussions of value-at-risk continued to focus on linear, Monte Carlo or historical transformations. Books on value-at-risk—including Best (<a target="_blank" href="http://www.riskbook.com/link/best_p_(1998).htm">1998</a>), Dowd (<a target="_blank" href="http://www.riskbook.com/link/dowd_k_(1998).htm">1998</a>), Butler (<a target="_blank" href="http://www.riskbook.com/link/butler_c_(1999).htm">1999</a>), and Jorion (<a target="_blank" href="http://www.riskbook.com/link/jorion_p_(2000).htm">2000</a>)—either failed to mention quadratic VaR measures or focused on crude solutions published prior to 1997. Six years after Fallon first circulated his solution, the financial risk management community remained largely unaware of the mathematics of quadratic VaR measures.</p> <p class="header_section_green"><span class="header_section_green"> Example: Quadratic Value-at-Risk in One-Dimension<br> </span>Let’s start with a one-dimensional example. This will illustrate a number of important concepts underlying the mathematics of quadratic VaR measures. </p> <p class="body">Measure time in trading days. A Canadian investor is short a <a href="../articles/option.htm">call option</a> on <a href="../articles/currency_codes.htm">JPY</a> 100<a href="../articles/mm.htm">MM</a>. The option is struck at 0.0120 CAD/JPY and expires in one month. With a current exchange rate of 0.0115 CAD/JPY, the option is <a href="../articles/time_value_and_intrinsic_value.htm">out-of-the-money</a>. The position’s current value <sup>0</sup><i>p</i> is –5,386.67 CAD. The investor wants to calculate the position’s one-day 97.5% CAD VaR. sources of risk include: </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the CAD/JPY exchange rate, </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the <a href="../articles/volatility.htm">implied volatility</a> of the JPY/CAD exchange rate, </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">applicable CAD and JPY interest rates. </p> <p class="body">To limit the problem to a single dimension, treat the implied volatility and interest rates as constant. Specifically, assume the implied volatility equals 16% and applicable CAD and JPY interest rates are .05 and .02, respectively. Model only the CAD/JPY exchange rate as a key factor, which is denoted <sup>1</sup><i>R</i><sub>1</sub>. Based upon <a href="../articles/time_series_stochastic_process.htm">time series analysis</a> of recent market data, <sup>1</sup><i>R</i><sub>1</sub> is assumed normal with mean <img border="0" src="../formulas/quadratic_transformation_1_0_mu_1.gif" align="absbottom" width="24" height="17"> equal to the current exchange rate 0.0115 CAD/JPY and standard deviation <img border="0" src="../formulas/quadratic_transformation_1_0_sigma_1.gif" align="absbottom" width="25" height="17"> of 0.00012 CAD/JPY (the superscripts 1|0 indicate that these are parameters for time 1 conditional on information available at the current time 0).</p> <p class="body">A portfolio mapping function <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> is specified based upon the <a href="../articles/garman_kohlhagen_1983.htm">Garman and Kohlhagen (1983)</a> modified Black-Scholes option pricing formula. This is <a href="../articles/remappings.htm">remapped</a> (approximated) by valuing <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> at three different values for <sup>1</sup><i>R</i><sub>1</sub> and <a href="../articles/interpolation.htm">quadratically interpolating</a> between the results. Values used are centered at <img border="0" src="../formulas/quadratic_transformation_1_0_mu_1.gif" align="absbottom" width="24" height="17"> and are spaced <img border="0" src="../formulas/quadratic_transformation_1_0_sigma_1.gif" align="absbottom" width="25" height="17"> apart. </p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table8"> <tr> <td colspan="2"> <p class="exhibit_header"><span class="exhibit_header">Interpolation Points for the Quadratic Remapping</span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2" colspan="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="150"> <p align="center"><sup>1</sup><i>R</i><sub>1</sub> Value<br> CAD/JPY</td> <td width="200"> <p align="center">Option Value<br> (Garman and Kohlhagen)<br> CAD</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="1" width="150"> <img border="0" src="../images/blank_1x1.gif" width="1" height="1"></td> </tr> <tr> <td width="150" align="left" height="30"> <p style="margin-left: 5"> <img border="0" src="../formulas/quadratic_transformation_1_0_mu_1.gif" align="absbottom" width="24" height="17"> – <img border="0" src="../formulas/quadratic_transformation_1_0_sigma_1.gif" align="absbottom" width="25" height="17"> = .01138</td> <td width="200" align="center" height="30"> –3,141.24</td> </tr> <tr> <td width="150" align="left" height="30"> <p style="margin-left: 5"> <img border="0" src="../formulas/quadratic_transformation_1_0_mu_1.gif" align="absbottom" width="24" height="17">           = .01150</td> <td width="200" align="center" height="30"> –5,007.52</td> </tr> <tr> <td width="150" align="left" height="30"> <p style="margin-left: 5"> <img border="0" src="../formulas/quadratic_transformation_1_0_mu_1.gif" align="absbottom" width="24" height="17"> + <img border="0" src="../formulas/quadratic_transformation_1_0_sigma_1.gif" align="absbottom" width="25" height="17"> = .01162</td> <td width="200" align="center" height="30"> –7,633.18</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 colspan="2"> <img border="0" src="../images/blank_1x1.gif" width="1" height="1"></td> </tr> <tr> <td colspan="2"> <p class="exhibit_legend">Interpolation points used to apply a quadratic remapping.</td> </tr> </table> </center> </div> <p class="body">Interpolating between these points yields the quadratic remapping</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="../images/quadratic_transformation_2.gif" width="424" height="28"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p>Assume <img border="0" src="../formulas/quadratic_transformation_0p_tilde.gif" align="absbottom" width="15" height="18"> = <img border="0" src="../formulas/quadrac_transformation_0p.gif" align="absbottom" width="15" height="18"> = –5,386.67. </p> <p>Exhibit 2 compares quadratic polynomial <img border="0" src="../formulas/quadratic_transformation_theta_tilde.gif" align="absbottom" width="8" height="20"> with the Garman and Kohlhagen formula <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15">, which it approximates. The three interpolation points are shown. The light green region indicates plus or minus 2 standard deviations for <sup>1</sup><i>R</i><sub>1</sub>. Events outside that region are unlikely to significantly impact the position’s 97.5% VaR. Inside that region, <img border="0" src="../formulas/quadratic_transformation_theta_tilde.gif" align="absbottom" width="8" height="20"> appears to be a reasonable approximation for the Garman and Kohlhagen formula <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15">.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="400" cellpadding="0" id="table10"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Quadratic Remapping</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/ex1_quadratic_transformation.gif" width="350" height="240"></td> </tr> <tr> <td> <p class="exhibit_legend">In the foreign exchange example, a portfolio mapping function is constructed from the Garman and Kohlhagen option pricing formula. This is remapped (approximated) with a quadratic polynomial, which is obtained by interpolating between the three indicated points. The light green region indicates plus or minus 2 standard deviations in the key factor <sup>1</sup><i>R</i><sub>1</sub>.</td> </tr> </table> </center> </div> <p class="body">Next, introduce a change of variables <img border="0" src="../images/quadratic_transformation_1_r_1_dot.gif" align="absbottom" width="18" height="20"> ~ <a href="../articles/normal_distribution.htm"><i>N</i>(0,1)</a> with</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"> <p class="body"> <sup>1</sup><i>R</i><sub>1</sub> = .00012 <img border="0" src="../images/quadratic_transformation_1_r_1_dot.gif" align="texttop" width="18" height="20"> + .0115</td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">to <a name="[4]">obtain</a></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"> <p class="body"> <img border="0" src="../formulas/quadratic_transformation_4.gif" width="232" height="28"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">This represents <img border="0" src="../formulas/quadratic_transformation_1_p_tilde.gif" align="absbottom" width="14" height="20"> as a <a href="../articles/polynomial.htm">linear polynomial</a> of two random variables:</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"> <img border="0" src="../images/quadratic_transformation_1_r_1_dot.gif" align="absbottom" width="18" height="20"> ~ <i>N</i>(0,1)</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><map name="FPMap7"><area href="../articles/chi-squared_distribution.htm" shape="rect" coords="40, 0, 83, 18"></map> <img border="0" src="../formulas/quadratic_transformation_chi_squared.gif" usemap="#FPMap7" align="texttop" width="84" height="19"></p> <p class="body">Because one is the square of the other, the two random variables are not independent. By "completing the squares," we rewrite [<a href="#[4]">4</a>] as</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"> <p class="body"> <img border="0" src="../formulas/quadratic_transformation_5.gif" width="214" height="29"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">This represents <img border="0" src="../formulas/quadratic_transformation_1_p_tilde.gif" align="absbottom" width="14" height="20"> as a linear polynomial of a single random variable:</p> <p class="body"> <img border="0" src="../images/bullet.gif" align="middle" width="20" height="10"><img border="0" src="../formulas/quadratic_transformation_chi_squared_2.gif" align="middle" width="178" height="26"></p> <p class="body">This fully characterizes the probability distribution of <img border="0" src="../formulas/quadratic_transformation_1_p_tilde.gif" align="absbottom" width="14" height="20"> in terms of a single random variable with a standard probability distribution. Based upon this, any VaR metric for <img border="0" src="../formulas/quadratic_transformation_1_p_tilde.gif" align="absbottom" width="14" height="20">—including the desired 97.5% VaR metric—can be valued. There are various ways to proceed. Before addressing these, lets extend the mathematics illustrated in this example to multiple dimensions.</p> <p class="body"><span class="header_section_green">Quadratic Transformation in Multiple Dimensions</span><br> Consider a quadratic portfolio</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/quadratic_transformation_1.gif" width="206" height="28"></td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">where <sup>1</sup><b><i>R</i></b> is a joint normal random vector with mean vector <img border="0" src="../formulas/remapping_1_0_mu.gif" align="absbottom" width="20" height="18"> and covariance matrix <img border="0" src="../formulas/remapping_1_0_sigma.gif" align="absbottom" width="23" height="18">. Apply change of variables</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table15"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transformation_7.gif" width="108" height="24"></td> <td width="5%" align="right">[7]</td> </tr> </table> <p class="body">where <img border="0" src="../formulas/quadratric_transformation_h_defined.gif" align="absbottom" width="56" height="18">, <b><i>z</i></b> is the <a href="../articles/cholesky_factorization.htm">Cholesky matrix</a> of <img border="0" src="../formulas/remapping_1_0_sigma.gif" align="absbottom" width="23" height="18">, and <b><i>u</i></b> is a matrix whose rows are the orthonormal <a href="../articles/eigenvalue.htm">eigenvectors</a> of <img border="0" src="../formulas/quadratic_transformation_z_prime_c_z.gif" align="absbottom" width="23" height="17">. With this change of variables, we obtain a new expression for <sup>1</sup><i>P</i>:</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"> <img border="0" src="../formulas/quadratic_transformation_remapped1235.gif" width="166" height="23"></td> <td width="5%" align="right">[8]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table17"> <tr> <td width="160">  </td> <td width="11"> </td> </tr> <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">where:</p> <p class="bullet"> <img border="0" src="../images/bullet.gif" align="middle" width="20" height="10"><img border="0" src="../formulas/quadratic_transformation_c_dot_defined.gif" align="texttop" width="72" height="18"></p> <p class="bullet"> <img border="0" src="../images/bullet.gif" align="middle" width="20" height="10"><img border="0" src="../formulas/quadratic_transformation_b.gif" align="texttop" width="120" height="18"></p> <p class="bullet"> <img border="0" src="../images/bullet.gif" align="middle" width="20" height="10"><img border="0" src="../formulas/quadratic_transformation__a.gif" align="texttop" width="140" height="20"></p> <p class="body">It can be shown—see Holton (<a target="_blank" href="http://www.riskbook.com/link/holton_g_(2003).htm">2003</a>)—that this change of variables achieves four conditions:</p> <p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10"><img border="0" src="../formulas/quadratic_transformation_1_r_dot.gif" align="absbottom" width="13" height="18"> is joint normal,</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the mean vector of <img border="0" src="../formulas/quadratic_transformation_1_r_dot.gif" align="absbottom" width="13" height="18"> is the <b>0</b> vector,</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the covariance matrix of <img border="0" src="../formulas/quadratic_transformation_1_r_dot.gif" align="absbottom" width="13" height="18"> is the identity matrix <i><b>I</b></i>, and</p> <p class="bullet"> v<img border="0" src="../formulas/quadratic_transformation_c_dot.gif" align="absbottom" width="7" height="15"> is a diagonal matrix.</p> <p>The fourth item means that <sup>1</sup><i>P</i> can depend upon each of the variables <img border="0" src="../formulas/quadratic_transformation_1_r_i_dot.gif" align="absbottom" width="16" height="18"> in one of four ways: </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">no dependence: <img border="0" src="../formulas/quadratic_transformation_cii_dot_equals_0.gif" align="absbottom" width="40" height="15"> and <img border="0" src="../formulas/quadratic_transformation_bi_dot_equals_0.gif" align="absbottom" width="35" height="19">; </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">linear dependence: <img border="0" src="../formulas/quadratic_transformation_cii_dot_equals_0.gif" align="absbottom" width="40" height="15"> and <img border="0" src="../formulas/quadratic_transformation_bi_dot_not_equal_0.gif" align="absbottom" width="35" height="19">;</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">central quadratic dependence: <img border="0" src="../formulas/quadratic_transformation_cii_dot_not_equals_0.gif" align="absbottom" width="40" height="15"> and <img border="0" src="../formulas/quadratic_transformation_bi_dot_equals_0.gif" align="absbottom" width="35" height="19">; or</p> <p class="bullet">vnon-central quadratic dependence: <img border="0" src="../formulas/quadratic_transformation_cii_dot_not_equals_0.gif" align="absbottom" width="40" height="15"> and <img border="0" src="../formulas/quadratic_transformation_bi_dot_not_equal_0.gif" align="absbottom" width="35" height="19">. </p> <p class="body">In the last case <sup>1</sup><i>P</i> has a dependence of the form <img border="0" src="../formulas/quadratic_transformation_8483476tj.gif" align="absbottom" width="84" height="18">. Completing the squares, this becomes <img border="0" src="../formulas/quadratic_transformation_732fhu84.gif" align="absbottom" width="126" height="21">. </p> <p class="body">Consequently, <sup>1</sup><i>P</i> is a linear polynomial of independent random variables, each of which is either <a href="../articles/normal_distribution.htm">standard normal</a>, <a href="../articles/chi-squared_distribution.htm">central chi-squared</a> with one degree of freedom, or <a href="../articles/chi-squared_distribution.htm">non-central chi-squared</a> with one degree of freedom and non centrality parameter <img border="0" src="../formulas/quadratic_transformation_438jg.gif" align="absbottom" width="61" height="19">. </p> <p class="body">Since a linear polynomial of independent normal random variables is itself normal, all normal terms can be combined into one. A general expression for <sup>1</sup><i>P</i> <a name="[9]">is</a></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"> <img border="0" src="../formulas/quadratic_transformation_8uht4vt.gif" width="163" height="41"></td> <td width="5%" align="right">[9]</td> </tr> </table> <p class="body">where the <img border="0" src="../formulas/quadratic_transform_xk.gif" align="absbottom" width="17" height="15"> are chi-squared with one degree of freedom and non-centrality parameters <img border="0" src="../formulas/quadratic_transform_noncentral_parameter.gif" align="absbottom" width="13" height="17">. <img border="0" src="../formulas/quadratic_transform_x0.gif" align="absbottom" width="16" height="15"> is standard normal. </p> <p class="body">We are now in much the same position we were in with our one-dimensional example. We have characterized the distribution of <sup>1</sup><i>P</i>. We wish to value a desired <a href="../articles/var_metric.htm">VaR metric</a>. How we do this depends, of course, on the specific VaR metric. In this article, I will focus on the popular <a href="../articles/quantile.htm">quantile</a>-of-loss VaR metrics. </p> <p class="body">Various solutions have been proposed. Zangari (<a href="#Zangari">1996</a>) approximates a solution using Johnson (<a href="#Johnson">1949</a>) curves. Fallon (<a href="#Fallon">1996</a>) and Pichler and Selitsch (<a href="#Pichler">2000</a>) recommend approximate solutions based on the <a href="../articles/cornish_fisher.htm">Cornish-Fisher expansion</a>. Rouvinez (<a href="#Rouvinez">1997</a>) uses the trapezoidal rule to invert the characteristic function. Britten-Jones and Schaefer (<a href="#Britten-Jones">1997</a>) use an approximation due to Solomon and Stephens (<a href="#Solomon">1977</a>). Cárdenas et al. (<a href="#Cárdenas">1997</a>) use the <a href="../articles/fast_fourier_transform.htm">fast Fourier transform</a> (FFT).</p> <p class="body">Of these, solutions based upon the Cornish-Fisher expansion, trapezoidal rule and FFT are the most effective. Holton (<a target="_blank" href="http://www.riskbook.com/link/holton_g_(2003).htm">2003</a>) covers the first two in detail, so I will discuss the third here. Because the method of Johnson curves was mentioned in the <i>RiskMetrics Technical Document</i>, it is of some historical interest. The method is inferior to others, but I will describe it briefly at the end of this article.</p> <p class="body"><span class="header_section_green">Quantile of Loss from the Fast Fourier Transform</span><br> The inversion theorem of probability theory provides the following expression for the probability density function <img border="0" src="../formulas/aaa_fi.gif" align="absbottom" width="7" height="15"> of a random variable in terms of its characteristic function <img border="0" src="../formulas/aaa_cap_si.gif" align="absbottom" width="11" height="15">.</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table19"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transformation_inversion_theorem.gif" width="172" height="53"></td> <td width="5%" align="right">[10]</td> </tr> </table> <p class="body"> <span style="font-size: 12.0pt; font-family: Times New Roman">Substituting <i>w</i> = 2<img border="0" src="../formulas/aaa_pi.gif" align="absbottom" width="9" height="13"></span><i><span style="font-size: 12.0pt; font-family: Times New Roman">t</span></i><span style="font-size: 12.0pt; font-family: Times New Roman">, this becomes</span></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table20"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transform_fourier.gif" width="170" height="58"></td> <td width="5%" align="right">[11]</td> </tr> </table> <p class="body">which is a <a href="../articles/fast_fourier_transform.htm">Fourier transform</a>. It can be approximated with the FFT. </p> <p class="body">It can be shown—see Holton (<a target="_blank" href="http://www.riskbook.com/link/holton_g_(2003).htm">2003</a>)—that the characteristic function of a random variable of form [<a href="#[9]">9</a>] is</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table21"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transform_characteritic.gif" width="205" height="49"></td> <td width="5%" align="right">[12]</td> </tr> </table> <p class="body">where</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table22"> <tr> <td width="35%" align="center">  </td> <td width="65%" align="left"> <img border="0" src="../formulas/quadratic_transformation_a.gif" width="196" height="52"></td> <td width="5%" align="right">[12]</td> </tr> <tr> <td width="35%" align="center">  </td> <td width="65%" align="left"> <img border="0" src="../formulas/quadratic_transformation_b_no_dot.gif" width="196" height="66"></td> <td width="5%" align="right">[13]</td> </tr> <tr> <td width="35%" align="center">  </td> <td width="65%" align="left"> <img border="0" src="../formulas/quadratic_transformation_c.gif" width="134" height="49"></td> <td width="5%" align="right">[14]</td> </tr> <tr> <td width="35%" align="center">  </td> <td width="65%" align="left"> <img border="0" src="../formulas/quadratic_transformation_d.gif" width="160" height="53"></td> <td width="5%" align="right">[15]</td> </tr> </table> <p class="body">and tan<sup>–1</sup> denotes the inverse tangent function with output in radians.</p> <p class="body">For example, suppose a quadratic portfolio's value <sup>1</sup><i>P</i> is expressed in form [<a href="#[9]">9</a>] as</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table23"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transform_ex8596.gif" width="294" height="51"></td> <td width="5%" align="right">[16]</td> </tr> </table> <p class="body">which is a linear polynomial of three independent random variables. Each is non-centrally chi-squared with one degree of freedom and non-centrality parameters <img border="0" src="../formulas/quadratic_transform_noncentral_parameter.gif" align="absbottom" width="13" height="17"> of 9.871, 4,773.841 and 7,261.351, respectively. The characteristic function of <sup>1</sup><i>P</i> is </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table24"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transform_ex_945gijr.gif" width="439" height="69"></td> <td width="5%" align="right">[17]</td> </tr> </table> <p class="body">We substitute <i>w</i> = <span style="font-size: 12.0pt; font-family: Times New Roman">2<img border="0" src="../formulas/aaa_pi.gif" align="absbottom" width="9" height="13"></span><i><span style="font-size: 12.0pt; font-family: Times New Roman">t</span></i> and take the FFT, sampling <i>n</i> = 64 values with sample spacing <img border="0" src="../formulas/aaa_cap_delta.gif" align="absbottom" width="9" height="15"> = .00003/64. Note that input values are <a href="../articles/imaginary_numbers.htm">complex numbers</a>. However, output is <a href="../articles/set.htm">real</a>. Results are indicated in Exhibit 3.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table25"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Probability Density Function of Portfolio Value <sup>1</sup><i>P</i> Obtained with the FFT</span><br> <span class="exhibit_subheader">Exhibit 3</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/ex3_quadratic_transformation.gif" width="300" height="240"></td> </tr> <tr> <td> <p class="exhibit_legend">The probability density function for portfolio value <sup>1</sup><i>P</i> was obtained using the FFT. The graph has the shape of a delta-hedged positive gamma portfolio.</td> </tr> </table> </center> </div> <p class="body">Let's use the FFT results to calculate the portfolio's 95% VaR. For this, we need the .05-quantile of <sup>1</sup><i>P</i>. <span style="font-size: 12.0pt; font-family: Times New Roman">We can calculate area under the PDF using Simpson’s rule. The evenly spaced output from the FFT is excellent for this purpose; however</span>, the .05-quantile <span style="font-size: 12.0pt; font-family: Times New Roman"> does not fall precisely at any of the values obtained from the FFT. We employ Simpson’s rule to find four values of the cumulative distribution function (CDF) that bracket the desired quantile and then <a href="../articles/interpolation.htm">interpolate</a> with a cubic polynomial to obtain the .05</span>-quantile of <sup>1</sup><i>P</i>. Subtracting this from the portfolio's current value <sup>0</sup><i>p</i> yields the portfolio's 95% VaR.</p> <p class="body"><span class="header_section_green">Approximating a Quantile-of-Loss with Johnson Curves</span><br> Zangari (<a href="#Zangari">1996</a>) proposed that a quantile-of-loss VaR metric might be approximated using Johnson (<a href="#Johnson">1949</a>) curves. Because other solutions provide superior results, I cover this solution only for historical interest. </p> <p>When faced with a body of statistical data, researchers often try to fit some standard probability distribution to the data. For this purpose, various families of probability distributions—called families of curves—have been defined. These include the Pearson (<a href="#Pearson">1895</a>), Edgeworth (<a href="#Edgeworth">1898</a>) and Johnson (<a href="#Johnson">1949</a>) families. </p> <p><b><a name="Johnson curves">Johnson curves</a></b> are constructed through translation of variables. In this context, a translation of variables specifies a probability distribution for a random variable <i>X</i> through an equation of the form: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table26"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transform_18.gif" width="63" height="23"></td> <td width="5%" align="right">[18]</td> </tr> </table> <p class="body"> <span style="font-size: 12.0pt; font-family: Times New Roman">where <img border="0" src="../formulas/aaa_fi_open.gif" align="absbottom" width="9" height="12"> is a monotone function and <i>Z</i> is standard normal. Because <i>Z</i> is standard normal, <img border="0" src="../formulas/aaa_fi_open.gif" align="absbottom" width="9" height="12"> imparts a probability distribution to <i>X</i>. This is not an unfamiliar concept. Recall that a random variable <i>X</i> is said to be <a href="../articles/lognormal_distribution.htm">lognormal</a> if:</span></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table27"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transform_lognormal.gif" width="109" height="40"></td> <td width="5%" align="right">[19]</td> </tr> </table> <p class="body"> <span style="font-size: 12.0pt; font-family: Times New Roman">is standard normal for some <i>m</i> and <i>s</i>. This defines a family of curves parameterized by <i>m</i> and <i>s</i>. The more general Johnson family of curves is defined with the family of translation functions comprising any of the <a name="[20] [21] [22]">forms</a></span></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table28"> <tr> <td width="30%" align="center">  </td> <td width="70%" align="left"> <img border="0" src="../formulas/quadratic_transform_20.gif" width="261" height="49"></td> <td width="5%" align="right">[20]</td> </tr> <tr> <td width="30%" align="center">  </td> <td width="70%" align="left"> <img border="0" src="../formulas/quadratic_transform_21.gif" width="133" height="46"></td> <td width="5%" align="right">[21]</td> </tr> <tr> <td width="30%" align="center">  </td> <td width="70%" align="left"> <img border="0" src="../formulas/quadratic_transform_22.gif" width="233" height="58"></td> <td width="5%" align="right">[22]</td> </tr> </table> <p class="body">where <img border="0" src="../formulas/quadratic_transforms_johnson_symbols.gif" align="absbottom" width="79" height="15"> are parameters—similar to <i>m</i> and <i>s</i> for the lognormal family. </p> <p>There are various ways to fit a distribution from a family of curves to a particular random variable. Two methods are: </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">find that curve that matches certain moments of the random variable; <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">find that curve that matches certain quantiles of the random variable. <p>Hill, Hill and Holder (<a href="#Hill">1976</a>) extended the family of Johnson curves to include normal <a name="[23]">distributions</a>: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table29"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transforms_23.gif" width="74" height="22"></td> <td width="5%" align="right">[23]</td> </tr> </table> <p>They also provided an algorithm for fitting a Johnson curve based upon the first four moments of a random variable. The algorithm determines the appropriate form of translation function from [<a href="#[20] [21] [22]">20</a>], [<a href="#[20] [21] [22]">21</a>], [<a href="#[20] [21] [22]">22</a>] and [<a href="#[23]">23</a>] and then determines values <img border="0" src="../formulas/quadratic_transforms_johnson_symbols.gif" align="absbottom" width="79" height="15"> for that function. </p> <p>Given a quadratic portfolio of form [<a href="#[9]">9</a>], you can calculate the first four moments—see Holton (<a target="_blank" href="http://www.riskbook.com/link/holton_g_(2003).htm">2003</a>). Fit a Johnson curve to <sup>1</sup><i>P</i> using the Hill, Hill and Holder algorithm to obtain </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table30"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transforms_24.gif" width="74" height="22"></td> <td width="5%" align="right">[24]</td> </tr> </table> <p class="body"> <span style="font-size: 12.0pt; font-family: Times New Roman">where <i>Z</i> ~ <i>N</i>(0,1) and <img border="0" src="../formulas/aaa_fi_open.gif" align="absbottom" width="9" height="12"> is a translation function. Since translation functions are monotone, they are invertible. You obtain the approximation</span></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table31"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/quadratic_transformations_25.gif" width="75" height="24"></td> <td width="5%" align="right">[25]</td> </tr> </table> <p class="body">Because <span style="font-size: 12.0pt; font-family: Times New Roman"> <img border="0" src="../formulas/aaa_fi_open.gif" align="absbottom" width="9" height="12"></span> is monotone, <span style="font-size: 12.0pt; font-family: Times New Roman"> <img border="0" src="../formulas/aaa_fi_open.gif" align="absbottom" width="9" height="12"></span><sup>–1</sup> is also monotone. Monotone functions map quantiles to quantiles, so you can calculate any quantile of <sup>1</sup><i>P</i> from the corresponding quantile of <i>Z</i>. For example, the .05-quantile of a standard normal random variable is –1.645. Accordingly, the .05-quantile of <sup>1</sup><i>P</i> is approximately <span style="font-size: 12.0pt; font-family: Times New Roman"> <img border="0" src="../formulas/aaa_fi_open.gif" align="absbottom" width="9" height="12"></span><sup>–1</sup>(–1.645).</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table32"> <tr> <td height="30" valign="bottom"> <p class="header2_black">Related Internal Links</td> <td align="right" valign="bottom"> </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="table33"> <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/cornish_fisher.htm">Cornish-Fisher expansion</a> A formula for approximating quintiles of a random variable based only on its first few cumulants.</strong><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" 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