Risk Measure and Risk Metric

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Risk Measure and Risk Metric</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="#measure">measure</a><p class="word_list"> <a href="#measurement">measurement</a><p class="word_list"> <a href="#metric">metric</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="#risk measure">risk measure</a><p class="word_list"> <a href="#risk measurement">risk measurement</a><p class="word_list"> <a href="#risk metrics">risk metric</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="left" id="table5"> <tr> <td width="300" height="14"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14"></td> </tr> <tr> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> <p class="body">The distinction between risk measures and risk metrics was first made by Glyn Holton in his book <i> <a target="_top" href="http://www.value-at-risk.net">Value-at-Risk: Theory and Practice</a></i>. Briefly, <a href="../articles/duration_and_convexity.htm"> duration</a>, <a href="../articles/delta_and_gamma.htm">delta</a>, <a href="../articles/beta.htm"> beta</a>, and <a href="../articles/volatility.htm">volatility</a> are all examples of risk metrics. Any procedures by which we calculate these are risk measures. The distinction is important because there is not a one-to-one correspondence between risk measures and risk metrics. There are, for example, different ways that the volatility of a portfolio might be calculated. Each represents a different risk measure for the single risk metric volatility.</p> <p class="body">Generally, a <b><a name="measure">measure</a></b> is an operation for assigning a number to something. A <b><a name="metric">metric</a></b> is our interpretation of the assigned number. Consider an example. </p> <p class="body">Have someone take off his shoes and stand with his back against a wall that is marked with a scale of inches. Note the number of inches correspond to the top of the person's head. This process is a <b>measure</b>. We interpret the number obtained as the person's height. "Height" is a <b>metric</b>.</p> <p class="body">Here is another example. A hollow glass cylinder is a meter long and has an internal diameter of a centimeter. It is open at one end, sealed at the other and graded with millimeter measurements. A bath of mercury is maintained at a temperature of <img src="../formulas/risk_metric_and_risk_measure_0_degrees.gif" align="absBottom" border="0" width="23" height="15">. The cylinder is immersed in the bath, allowing it to fill with mercury. While keeping the open end immersed, the cylinder is held vertically, with the sealed end about 950 centimeters above the surface of the mercury. A column of mercury is suspended within the cylinder. Its height is measured in centimeters and is multiplied by 0.01316. The result is recorded. This procedure is a <b>measure</b>. We interpret the recorded number as air pressure measured in atmospheres. The interpretation is a <b>metric</b>.</p> <p class="body">When we apply a measure, the number obtained is a <b><a name="measurement">measurement</a></b>.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table6"> <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">Measures are employed to quantify many things: height, temperature, aptitude, speed, consumer confidence, etc. All of these notions being quantified are metrics. The operations with which we quantify them are measures. There are many metrics of risk—volatility, delta, gamma, duration, convexity, beta, etc. We call these <b><a name="risk metrics">risk metrics</a></b>. A measure that supports a risk metric is called a <b><a name="risk measure"> risk measure</a></b>. The value obtained from applying a risk measure is a <b><a name="risk measurement">risk measurement</a></b>.</p> <p class="body">Risk measures tend to be categorized according to the risk metrics they support. There are measures of duration, measures of delta, etc. This is an important point. We do not categorize risk measures according to the specific operations they entail. Operationally, there are many different ways we might arrive at a measurement of a portfolio's volatility. Irrespective of the actual operations, all of them are measures of volatility. 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