Mapping Procedure

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue"> <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">Mapping Procedure</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="#holdings">holdings</a><p class="word_list"> <a href="#mapping procedure">mapping procedure</a><p class="word_list"> <a href="#primary mappings">primary mapping</a><p class="word_list"> <a href="#risk factor">risk factor</a><p class="word_list"> <a href="#risk vector">risk vector</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> <td width="100%" align="left" valign="top"><p class="word_list"> <br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <p class="body">This article discusses <b><a name="mapping procedure">mapping procedures</a></b>, which are one of three essential components of a VaR measure. The article assumes familiarity with concepts discussed in the overview article <a href="../articles/var_measure.htm">measuring value-at-risk</a>. Exhibit 1 is reproduced from that article.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table5"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Schematic of How VaR Measures Work</span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex1_var_measure.gif" width="350" height="549"></td> </tr> <tr> <td> <p class="exhibit_legend">All practical VaR measures accept portfolio data and historical market data as inputs. They process these with a mapping procedure, inference procedure, and transformation procedure. Output comprises the value of a <a href="../articles/var_metric.htm">VaR metric</a>. That value is the VaR measurement.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table6"> <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">The purpose of a mapping procedure is to characterize a portfolio's exposures. It does so by expressing the portfolio's value as a function of applicable market variables, such as <a href="../articles/common_stock.htm">stock</a> prices, exchange rates, commodity prices or interest rates. Let's define some concepts.</p> <p class="MsoNormal">Let time 0 be the current time, and let time 1 be the end of the <a href="../articles/value_at_risk.htm">VaR horizon</a>. A <b> <a name="risk factor">risk factor</a></b> 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 (see the <i><a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation). A <b><a name="risk vector">risk vector</a></b> <img border="0" src="../symbols/1_a_bold_cap.gif" align="absbottom" width="16" height="17"> is a random vector of risk factors. If a risk vector reflects a future value of some time series, we may speak of its current value <img border="0" src="../symbols/0_q_bold.gif" align="absbottom" width="14" height="17"> or historical values <img border="0" src="../formulas/mappining_procedures_historical_data.gif" align="absbottom" width="124" height="17"></p> <p class="MsoNormal">One particular risk factor and two risk vectors play important roles in VaR measures. We give them special names and notation. These are: </p> <p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10">the portfolio’s future value <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18">;<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">; and<p class="bullet"> <img border="0" src="../images/bullet.gif" width="20" height="10">the key vector <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">.<p class="MsoNormal">The portfolio’s future value <span style="font-size: 12.0pt; font-family: Times New Roman"> <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"></span> represents the market value at time 1 of the portfolio for which VaR is to be measured. The portfolio is assumed fixed in the sense that it will not be traded during the period [0,1], and no assets will be added or withdrawn. We are interested in the portfolio’s current value <img border="0" src="../symbols/0_p.gif" align="absbottom" width="14" height="18"> only if our <a href="../articles/var_metric.htm">VaR metric</a> depends upon it. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table7"> <tr> <td width="336">  <p style="margin-top: 8; margin-bottom: 2" class="ad_legend"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a></p> <p style="margin-top: 0; margin-bottom: 0"> <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82402;type=iframe" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I2"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82402;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82402"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82402;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </td> <td width="9"> </td> </tr> </table> <p class="MsoNormal">Asset vector <img border="0" src="../symbols/1_s_cap_bold.gif" align="absbottom" width="13" height="18"> has asset values <i> <img border="0" src="../symbols/1_s_i_cap.gif" align="absbottom" width="14" height="17"> </i>as components. These represent accumulated values of specific assets the portfolio may hold. Realizations may be negative, so our definition recognizes no accounting distinction between assets and liabilities. Accumulated value is denominated in the <a href="../articles/value_at_risk.htm">base currency</a> employed by the VaR metric. It may reflect such variables as capital gains, <a href="../articles/common_stock.htm">dividends</a>, <a href="../articles/coupon_bond.htm">coupons</a>, margin payments, reinvestment income, storage costs, insurance, financing, changes in exchange rates, leasing income, etc. </p> <p class="MsoNormal">Every VaR measure must directly characterize a conditional probability distribution for some vector of risk factors, such as prices, interest rates, <a href="../articles/spreads.htm">spreads</a>, or <a href="../articles/volatility.htm">implied volatilities</a>. Those risk factors <img border="0" src="../symbols/1_R_i_cap_.gif" align="absbottom" width="16" height="17"> are called <a href="../articles/var_measure.htm">key factors</a>. They are the components of 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">. Occasionally, we use asset values <i> <img border="0" src="../symbols/1_s_i_cap.gif" align="absbottom" width="14" height="17"></i> as key factors. More often, it is convenient to use more basic financial variables as key factors.</p> <p class="MsoToc1"> <span style="font-size: 12.0pt; font-family: Times New Roman">A portfolio’s <b><a name="holdings">holdings</a></b> is a row vector <img border="0" src="../formulas/mapping_procedure_1.gif" align="absbottom" width="16" height="14"> indicating the number <img border="0" src="../symbols/omega_i.gif" align="absbottom" width="13" height="12"> of units held by the portfolio of each asset. We use <img border="0" src="../formulas/mapping_procedure_1.gif" align="absbottom" width="16" height="14"> to define the portfolio's value <img border="0" src="../symbols/1_p_cap.gif" align="absbottom" width="13" height="18"> in terms of the asset vector </span> <img border="0" src="../symbols/1_s_cap_bold.gif" align="absbottom" width="13" height="18"><span style="font-size: 12.0pt; font-family: Times New Roman">:</span></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_2.gif" width="275" height="29"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">Consider a simple example. A portfolio comprises a short <a href="../articles/options_spread.htm">straddle</a>. It is short 10 <a href="../articles/option.htm">call options</a> on a particular <a href="../articles/future.htm">future</a>, and it is short 10 <a href="../articles/option.htm">put options</a> on the same future. All options have the same <a href="../articles/option.htm">strike</a> and <a href="../articles/option.htm">expiration</a>. We define assets with</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_procedure_3.gif" width="330" height="49"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">so holdings are</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_4.gif" width="163" height="22"></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">and we <a name="[4]">define</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_5.gif" width="285" height="56"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">We represent the mapping schematically <a name="[5]">as</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"> <img border="0" src="../formulas/mapping_procedure_schem1.gif" width="78" height="27"></td> <td width="5%" align="right">[5]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table13"> <tr> <td width="336">  <p style="margin-top: 8; margin-bottom: 2" class="ad_legend"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a></p> <p style="margin-top: 0; margin-bottom: 0"> <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82402;type=iframe" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I3"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82402;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82402"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82402;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </td> <td width="9"> </td> </tr> </table> <p class="body"> At this point in our example, we could be done with the mapping procedure. We would let <img border="0" src="../symbols/1_s_cap_bold.gif" align="absbottom" width="13" height="18"> be our key vector, and [<a href="#[4]">4</a>] would be our portfolio mapping, which we would pass to the <a href="../articles/var_measure.htm">transformation procedure</a>. A problem with doing so is that we would be using <a href="../articles/option.htm">option</a> prices as key factors, which means that the <a href="../articles/var_measure.htm">inference procedure</a> would need to characterize their joint distribution. Designing an inference procedure to do this would be difficult. Because of the limited downside risk of options, their prices have <a href="../articles/skewness.htm">skewed</a> price distributions. Also, their <a href="../articles/standard_deviation.htm">standard deviations</a> would be highly dependent upon whether the options were <a href="../articles/time_value_and_intrinsic_value.htm">in-the-money</a> or <a href="../articles/time_value_and_intrinsic_value.htm"> out-of-the-money</a>. A simpler solution is to not employ options prices as key factors, but to use more fundamental risk factors such as the options' underlier prices and implied volatilities.</p> <p class="body">Consider a portfolio comprising call options with various strikes on the first nearby Henry Hub natural gas future. Asset values <sup><img border="0" src="../symbols/1_s_i_cap.gif" align="absbottom" width="14" height="17"></sup> represent the market values at time 1 of the various strike options. We define</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/mapping_procedure_6.gif" width="268" height="86"></td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">For simplicity, we are employing a single implied volatility for all strikes. A more sophisticated model would use multiple implied volatilities to capture <a href="../articles/volatility_skew.htm">volatility skew</a>. Using <a href="../articles/black_1976.htm">Black's (1976)</a> pricing formula for options on futures, we define asset values <i> <img border="0" src="../symbols/1_s_i_cap.gif" align="absbottom" width="14" height="17"></i> as functions of <img border="0" src="../symbols/1_r_cap_bold.gif" align="absbottom" width="14" height="17">. Note that, in applying the options pricing formula, we use day counts as of time 1, not the current time 0. We obtain mapping</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/mapping_function_7.gif" width="75" height="28"></td> <td width="5%" align="right">[7]</td> </tr> </table> <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 is</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/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="table17"> <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> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table18"> <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">Portfolio mappings of form [<a href="#[5]">5</a>] or [<a href="#[9]">9</a>] are called <b><a name="primary mappings">primary mappings</a></b>. Primary mappings are portfolio mappings constructed directly from a portfolio's holdings, as indicated in our example. Mathematically, a primary mapping works by valuing each asset held by the portfolio, multiplying the values by the portfolio's holdings in each asset, and summing.</p> <p class="body">For many VaR measures, output of the mapping procedure is a primary mapping, but not for all. Use of primary mappings can pose certain problems. The most common of these is that primary mappings can be mathematically complicated. This is especially true for large portfolios of instruments such as <a href="../articles/mortgage_backed_security.htm"> mortgage-backed securities</a> or <a href="../articles/derivative_instrument.htm">exotic derivatives</a>. Applying a <a href="../articles/var_measure.htm">transformation procedure</a> to a complicated portfolio mapping can be computationally expensive. For this reason, many mapping procedures replace primary mappings with simpler approximations. Those approximations are called <a href="../articles/remappings.htm"> portfolio remappings</a>.</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/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" 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