Value-at-Risk: Remappings

<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">Value-at-Risk: Remappings</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="0%" align="left" valign="top"><p class="word_list"> <a href="#delta-gamma remapping">delta-gamma remapping</a><p class="word_list"> <a href="#dual remapping">dual remapping</a><p class="word_list"> <a href="#function remapping">function remapping</a><p class="word_list"> <a href="#global remappings">global remapping</a><p class="word_list"> <a href="#holdings remapping">holdings remapping</a><p class="word_list"> <a href="#interpolation remappings">interpolation remapping</a><p class="word_list"> <a href="#least squares remappings">least squares remapping</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 remappings">linear remapping</a><p class="word_list"> <a href="#portfolio remappings">portfolio remapping</a><p class="word_list"> <a href="#principal component remapping">principal component remapping</a><p class="word_list"> <a href="#quadratic remappings">quadratic remapping</a><p class="word_list"> <a href="#remapping">remapping</a><p class="word_list"> <a href="#variables remapping">variables remapping</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 portfolio remappings, which are widely used in production <a href="../articles/var_measure.htm">VaR measures</a>. The article assumes familiarity with concepts discussed in the overview article <a href="../articles/var_measure.htm">measuring value-at-risk</a> and the article <a href="../articles/mapping_procedure.htm">mapping procedures</a>. Exhibit 1 is reproduced from the first of those articles.</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 <a href="../articles/value_at_risk.htm">VaR measurement</a>.</td> </tr> </table> </center> </div> <p class="body">One of the essential tasks a VaR measure must perform in order to quantify the <a href="../articles/market_risk.htm">market risk</a> of a trading portfolio is to characterize the exposures of that portfolio. This is the purpose of the measure's <a href="../articles/mapping_procedure.htm">mapping procedure</a>. Output of the mapping procedure is a <a href="../articles/var_measure.htm"> portfolio mapping</a>, which becomes an input for the measure's <a href="../articles/var_measure.htm">transformation procedure</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">The most direct way to construct a portfolio mapping is to construct a <a href="../articles/mapping_procedure.htm">primary mapping</a>. Based upon the portfolio's holdings, the portfolio's value is expressed as a weighted sum of the values of the assets it holds. Asset values may not be directly observable in the market, but these can be expressed in terms of more fundamental market variables, such as relevant exchange rates, interest rates, commodity prices, etc—what are called <a href="../articles/var_measure.htm">key factors</a>.</p> <p class="body">Let's express this mathematically (see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation). We let time 0 be the current time, and we let time 1 be the end of the <a href="../articles/value_at_risk.htm">VaR horizon</a>. A risk factor is any random variable <img border="0" src="../symbols/1_q_i_cap.gif" align="absbottom" width="18" height="17"> whose value will be realized during the interval (0,1] and will affect the <a href="../articles/valuation.htm">market value</a> of a portfolio at time 1. A risk vector <img border="0" src="../symbols/1_q_cap_bold.gif" align="absbottom" width="16" height="17"> is a random vector of risk factors. </p> <p class="body">One particular risk factor and two risk vectors are of particular interest. These are: </p> <p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10">the portfolio’s value <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> at the end of the VaR horizon;<p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10">the asset vector <img border="0" src="../symbols/1_s_cap_bold.gif" align="absbottom" width="13" height="18">, whose components are values <img border="0" src="../symbols/1_s_i_cap.gif" align="absbottom" width="14" height="17"> of assets held by the portfolio; and<p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10">the <a href="../articles/var_measure.htm">key vector</a> <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">, whose components are key factors <img border="0" src="../symbols/1_R_i_cap_.gif" align="absbottom" width="16" height="17">.<p class="body">The portfolio's <a href="../articles/mapping_procedure.htm">holdings</a> <img border="0" src="../formulas/aaa_bold_omega.gif" align="absbottom" width="10" height="12"> is a row vector indicating the number of units of each asset held by the portfolio. Using vector multiplication, we indicate the first step of constructing a primary mapping <a name="[1]">as</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/mapping_procedure_2.gif" width="275" height="29"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">We represent this mapping schematically <a name="[2]">as</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table8"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/mapping_procedure_schem1.gif" width="78" height="27"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">On its own, formula [<a href="#[1]">1</a>] defines a simple primary mapping. We can leave it in this form, in which case <img border="0" src="../symbols/1_s_cap_bold.gif" align="absbottom" width="13" height="18"> will play the dual role of both asset vector and key vector. More commonly, we express <img border="0" src="../symbols/1_s_cap_bold.gif" align="absbottom" width="13" height="18"> in terms of more fundamental market variables, and these become our key factors. We define mapping </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/mapping_function_7.gif" width="75" height="28"></td> <td width="5%" align="right">[7]</td> </tr> </table> <p class="body">The function <span style="font-size: 12.0pt; font-family: Times New Roman"> φ may be quite complicated. Essentially, it must value each asset based upon the key factors. If the assets are <a href="../articles/derivative_instrument.htm">exotic derivatives</a> or <a href="../articles/mortgage_backed_security.htm">mortgage-backed securities</a>, φ will need to incorporate sophisticated techniques from <a href="../articles/option_pricing_theory.htm">financial engineering</a>. </span></p> <p class="body">Composing <span style="font-size: 12.0pt; font-family: Times New Roman"> <img border="0" src="../formulas/mapping_procedure_1.gif" align="absbottom" width="16" height="14"> with φ, we obtain portfolio mapping function <img border="0" src="../formulas/mapping_procedure_composition.gif" align="absbottom" width="55" height="15">. Our portfolio mapping <a name="[8]">is</a></span></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/mapping_procedure_8.gif" width="73" height="28"></td> <td width="5%" align="right">[8]</td> </tr> </table> <p class="body">We represent it schematically <a name="[9]">as</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="../formulas/mapping_procedure_9.gif" width="141" height="47"></td> <td width="5%" align="right">[9]</td> </tr> </table> <p class="body">This is the most general form of primary mapping. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table12"> <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">Explicitly or implicitly, every mapping procedure constructs a portfolio mapping by first constructing a primary mapping. Some mapping procedures stop at this point. The primary mapping is their output, which is passed to the transformation procedure. A drawback of this approach is the fact that primary mappings can be extremely complicated. If a portfolio holds several thousand exotic derivatives, the function <span style="font-size: 12.0pt; font-family: Times New Roman"> φ could take hours of processing time to value. Many transformation procedures—especially <a href="../articles/monte_carlo_transformation.htm">Monte Carlo transformation procedures</a>—must value a portfolio mapping function numerous times. They need a portfolio mapping function that is relatively easy to value.</span></p> <p class="body">Accordingly, many mapping procedures apply certain approximations to a primary mapping to obtain what is known as a portfolio remapping. For these mapping procedures, output comprises the simpler portfolio remapping, and this is what is passed to the transformation procedure..</p> <p class="body">Formally, a <b><a name="remapping">remapping</a></b> is an approximation of a risk vector <sup>1</sup><i><b>Q</b></i> with some other risk vector <img border="0" src="../formulas/remapping_1q_tilde.gif" align="absbottom" width="16" height="19">. We are primarily interested in <b><a name="portfolio remappings">portfolio remappings</a></b>, which approximate a portfolio's value <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> with some other random variable <img border="0" src="../formulas/remapping_1p_tilde.gif" align="absbottom" width="13" height="19">. </p> <p class="body">If we have a portfolio mapping <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">), portfolio remappings may take three forms: </p> <ol> <li> <p style="margin-top: 6; margin-bottom: 6">A <b><a name="function remapping">function remapping</a></b> approximates <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">) by replacing <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> with an approximate mapping function <img border="0" src="../formulas/remapping_theta_tilde.gif" align="absbottom" width="7" height="18">, so <img border="0" src="../formulas/remapping_function_remapping.gif" align="absbottom" width="63" height="18">. </li> <li> <p style="margin-top: 6; margin-bottom: 6">A <b><a name="variables remapping">variables remapping</a></b> approximates <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">) by replacing <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17"> with alternative key vector <img border="0" src="../formulas/remapping_1r_tilde.gif" align="absbottom" width="14" height="19">, so <img border="0" src="../formulas/remapping_variables_remapping.gif" align="absbottom" width="60" height="18">. </li> <li> <p style="margin-top: 6; margin-bottom: 6">A <b><a name="dual remapping">dual remapping</a></b> approximates <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">) by replacing both <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> and <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">, so  <img border="0" src="../formulas/remapping_dual_remapping.gif" align="absbottom" width="60" height="18">. </li> </ol> <p class="body">The first and third forms are most common. </p> <p class="body">Many function remappings approximate a portfolio mapping function <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> with a <a href="../articles/polynomial.htm">linear</a> or <a href="../articles/polynomial.htm">quadratic polynomial</a> to facilitate use of a <a href="../articles/linear_transformation.htm">linear transformation</a> or <a href="../articles/quadratic_transformation.htm">quadratic transformation</a>. These are called <a name="global remappings"><b>global remappings</b></a>. They also have other names, depending upon the specific nature of the remapping. They may be called <b> <a name="linear remappings">linear remappings</a></b> or <b> <a name="quadratic remappings">quadratic remappings</a></b>, depending upon the type of polynomial constructed. If a quadratic approximation is constructed based upon the portfolio's <a href="../articles/delta_and_gamma.htm">deltas</a> and <a href="../articles/delta_and_gamma.htm">gammas</a>, the result may be called a <b><a name="delta-gamma remapping">delta-gamma remapping</a></b>. Because deltas and gammas are highly localized exposure metrics, better results are generally obtained by using <a href="../articles/interpolation.htm">interpolation</a> or the <a href="../articles/least_squares.htm">method of least squares</a>. With these approaches, the portfolio mapping function <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15"> is valued at several points and interpolation or the method of least squares is used to fit a quadratic polynomial <img border="0" src="../formulas/remapping_theta_tilde.gif" align="absbottom" width="7" height="18"> to the results. These are called <b><a name="interpolation remappings"> interpolation remappings</a></b> or <b><a name="least squares remappings"> least squares remappings</a></b>.</p> <p class="body">We represent global remappings schematically <a name="[10]">as</a></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"> <img border="0" src="../formulas/remapping_10.gif" width="147" height="107"></td> <td width="5%" align="right">[10]</td> </tr> </table> <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="right" usemap="#FPMap372" width="325" height="250">Schematics such as [<a href="#[10]">10</a>] take a bit of getting used to, but they are extremely helpful for understanding specific remappings. They generalize more simple schematics such as [<a href="#[2]">2</a>] and [<a href="#[9]">9</a>]. In them, horizontal arrows indicate mappings (exact relationships). Vertical arrows indicate remappings (approximations). In schematic [<a href="#[10]">10</a>], we see that mapping function <img border="0" src="../formulas/remapping_theta_tilde.gif" align="absbottom" width="7" height="18"> replaces mapping function <img border="0" src="../formulas/aaa_theta.gif" align="absbottom" width="7" height="15">, but key vector <sup>1</sup><b><i>R</i></b> remains unchanged. The result is an approximation <img border="0" src="../formulas/remapping_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">.</p> <p class="body">Another type of function remapping is a <b> <a name="holdings remapping">holdings remapping</a></b>. These remappings replace the assets held by a portfolio with just a handful of assets that, together, exhibit similar exposures. A simple example of a holdings mapping is to replace a large number of fixed cash flows with just a handful maturing on specific (perhaps annual) maturities. A more sophisticated holdings remapping might replace several thousand derivatives positions with just a handful that have the same combined market value, <a href="../articles/delta_and_gamma.htm">delta</a> and <a href="../articles/vega.htm">vega</a>. With a holdings remapping, the only thing that changes is the portfolio's holdings <img border="0" src="../formulas/aaa_bold_omega.gif" align="absbottom" width="10" height="12">. This is illustrate in schematic [11]:</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/remapping_11.gif" width="144" height="133"></td> <td width="5%" align="right">[11]</td> </tr> </table> <p class="body">Dual remappings may be used to reduce the dimensionality of the key vector <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">. <a href="../articles/principal_components.htm">Principal component</a> analysis can be used for this purpose, in which case the remapping may be called a <b> <a name="principal component remapping">principal component remapping</a></b>. </p> <p class="body">Consider a portfolio mapping <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">), where <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17"> is <i>n</i>-dimensional with <a href="../articles/mean.htm">mean vector</a><sup> <img border="0" src="../formulas/remapping_1_0_mu.gif" align="absbottom" width="20" height="18"></sup> and <a href="../articles/multicollinear.htm">multicollinear</a> <a href="../articles/correlation.htm">covariance matrix</a> <img border="0" src="../formulas/remapping_1_0_sigma.gif" align="absbottom" width="23" height="18"> (the superscripts 1|0 indicate that these are parameters for time 1 conditional on information available at the current time 0). We represent <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17"> in terms of its principal <a name="[12]"> components</a>:</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/remapping_12.gif" width="97" height="20"></td> <td width="5%" align="right">[12]</td> </tr> </table> <p class="body"><span style="font-size: 12.0pt; font-family: Times New Roman">where <img border="0" src="../formulas/aaa_bold_nu.gif" align="absbottom" width="6" height="11"> is the matrix whose columns are orthonormal <a href="../articles/eigenvalue.htm">eigenvectors</a> of <img border="0" src="../formulas/remapping_1_0_sigma.gif" align="absbottom" width="23" height="18">, and <img border="0" src="../symbols/1_d_cap_bold.gif" align="absbottom" width="14" height="17"> is an <i>n</i>-dimensional column vector of the principal components of </span> <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17"><span style="font-size: 12.0pt; font-family: Times New Roman">. We convert [<a href="#[12]">12</a>] to an approximate relationship by discarding principal components <img border="0" src="../symbols/1_d_i_cap.gif" align="absbottom" width="17" height="17"> that have <a href="../articles/standard_deviation.htm">variances</a> close to 0. Suppose we retain <i>m</i> principal components and discard <i>n</i> – <i>m</i>. Let </span> <img border="0" src="../formulas/remapping_1r_tilde.gif" align="absbottom" width="14" height="19"><span style="font-size: 12.0pt; font-family: Times New Roman"> be the <i>m</i>-dimensional vector of retained principal components. Let <img border="0" src="../formulas/remapping_nu_tilde.gif" align="absbottom" width="8" height="15"> be the matrix whose columns are the eigenvectors corresponding to those principal components. We obtain</span></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/remapping_13.gif" width="200" height="23"></td> <td width="5%" align="right">[13]</td> </tr> </table> <p> Schematically, the remapping is</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table17"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/remapping_14.gif" width="151" height="111"></td> <td width="5%" align="right">[14]</td> </tr> </table> <p> The above examples indicate just a few of the many forms portfolio remappings take in practice. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table18"> <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="table19"> <tr> <td width="100%"> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/cubic_spline.htm">cubic spline interpolation</a> Any of several methods of interpolating with cubic splines.</p> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/interpolation.htm">interpolation</a> Any procedure for fitting a function to a set of points in such a manner that the function intercepts each of the points.</p> <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"><strong style="font-weight: 400"><a href="../articles/least_squares.htm">method of least squares</a> Any of several techniques for fitting a curve to data so as to minimize the sum of squared differences between the curve and data points.</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/principal_components.htm">principal component analysis</a> A technique for orthogonalizing a random vector.<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/taylor_series.htm">Taylor series expansion</a> In calculus, a power series obtained as a limit of Taylor polynomials that may approximate or equal the function from which it is constructed.<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">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="table20"> <tr> <td width="100%"> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table21"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap364"> <area target="_blank" href="http://www.value-at-risk.net" shape="rect" coords="0, 0, 81, 124"> </map> <img border="0" src="../images/book_holton_(2003)_125.jpg" usemap="#FPMap364" width="82" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table22"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.value-at-risk.net">Value-at-Risk: Theory and Practice</a></td> </tr> <tr> <td colspan="3"> <p class="author">Glyn Holton</td> </tr> <tr> <td width="10%"> <map name="FPMap362"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"></map> <img border="0" src="../images/star_6.gif" usemap="#FPMap362" width="90" height="16"></td> <td width="10%"> <map name="FPMap363"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 1, 89, 15"> </map><img border="0" src="../images/tech_4.gif" usemap="#FPMap363" width="90" height="16"></td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"> <p class="author">2003</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap361"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.value-at-risk.net"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap361" width="146" height="20"></td> </tr> </table> </td> </tr> </table> </td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">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%"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Forum Discussions</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="table23"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_bullhorn_narrow.gif" align="left" width="27" height="22"><a target="_top" href="http://www.contingencyanalysis.com/archive/link_thread/04-1/0000059c.htm">VaR for a large option portfolio</a> <i>06 Feb 2004</i><br> Realities of implementing a production value-at-risk measure.</td> </tr> </table> </td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table24"> <tr> <td width="100%" bgcolor="#FFFFFF"> <p align="center"><map name="FPMap365"> <area target="_top" href="http://www.riskglossary.com" shape="rect" coords="0, 0, 164, 102"> </map><img border="0" src="../images/logo.jpg" usemap="#FPMap365" width="165" height="103"></p> <p align="center"><span class="verdana10"> <a href="http://www.riskglossary.com/main/disclaimer.htm" target="_self"> Disclaimer</a></span></p> <p align="center">website: <span class="verdana10"> <a target="_top" href="http://www.contingencyanalysis.com"> http://www.contingencyanalysis.com</a></span><br> glossary direct link: <span class="verdana10"> <a target="_top" href="http://www.riskglossary.com"> http://www.riskglossary.com</a></span><br> copyright © Contingency Analysis, 2003</td> </tr> </table> </td> </tr> </table> </td> <td>  </td> </tr> </table> </body>