Intensity Model

<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">Intensity Model</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="#default intensity">default intensity</a><p class="word_list"> <a href="#hazard rate">hazard rate</a><p class="word_list"> <a href="#Intensity models">intensity model</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="#mortality model of default">mortality model</a><p class="word_list"> <a href="#reduced form">reduced form model</a><p class="word_list"> <a href="#survival function">survival function</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"><b><a name="Intensity models">Intensity models</a></b> (also called <b><a name="reduced form">reduced form models</a></b>) are a form of <a href="../articles/default_model.htm">default model</a>. Lets start by considering what are known as <b> <a name="mortality model of default">mortality models of default</a></b>. These are essentially a discrete form of intensity model. Once we have established them, we will take a limit as time intervals go to zero—and out will pop intensity models. </p> <p class="body">Mortality models derive their name from their similarity to actuarial models of human mortality. Define a <b> <a name="survival function">survival function</a></b> <i>s</i>(<i>t</i>). It might indicate the probability of a human surviving until age <i>t</i> or the probability of a <a href="../articles/coupon_bond.htm">bond</a> surviving without default for <i>t</i> years. For the rest of this article, we shall use it to denote the latter.</p> <p class="body">The probability of a bond defaulting in year <i>t</i> + 1 is given <a name="[1]">by</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"> <i>s</i>(<i>t</i>) – <i>s</i>(<i>t</i> + 1)</td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">This is an unconditional probability. It reflects the probability at time 0 (when the bond is issued) of default between time <i> t</i> and time <i>t</i> + 1. If we want the conditional probability of default—that is, the probability of default between time <i>t</i> and time <i>t</i> + 1—we apply Bayes' theorem to <a name="[2]">obtain</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/intensity_model_2.gif" width="96" height="39"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">A survival function can be constructed from historical bond default data. Constructed in this manner, the survival function defines a mortality model of default. Exhibit 1 indicates empirical survival functions by original credit quality. If the numbers were smoothed, it could reasonably be used to specify a mortality model for default.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="450" cellpadding="0" id="table8"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Empirical Survival Functions by Original Credit Quality</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> <table border="0" cellpadding="4" cellspacing="0" width="512" style="border-collapse: collapse; width: 95%" class="explanatory_text" id="table9"> <colgroup> <col width="64" span="8" style="width:48pt"> </colgroup> <tr height="17" style="height:12.75pt"> <td style="height: 34; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: 1px solid #CC9966; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="center" class="explanatory_text" bgcolor="#FFFFCC" rowspan="2"> <b>year</b></td> <td style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 385; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" bgcolor="#FFFFCC" height="25" colspan="7" align="center"> <b>original credit rating</b></td> </tr> <tr height="17" style="height:12.75pt"> <td style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; border-left: medium none; border-right: medium none; border-top: medium none; border-bottom: 1px solid #CC9966; padding-left: 1px; padding-right: 1px; padding-top: 1px" bgcolor="#FFFFCC" align="center"> AAA</td> <td style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; border-left: medium none; border-right: medium none; border-top: medium none; border-bottom: 1px solid #CC9966; padding-left: 1px; padding-right: 1px; padding-top: 1px" bgcolor="#FFFFCC" align="center"> AA</td> <td style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; border-left: medium none; border-right: medium none; border-top: medium none; border-bottom: 1px solid #CC9966; padding-left: 1px; padding-right: 1px; padding-top: 1px" bgcolor="#FFFFCC" align="center"> A</td> <td style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; border-left: medium none; border-right: medium none; border-top: medium none; border-bottom: 1px solid #CC9966; padding-left: 1px; padding-right: 1px; padding-top: 1px" bgcolor="#FFFFCC" align="center"> BBB</td> <td style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; border-left: medium none; border-right: medium none; border-top: medium none; border-bottom: 1px solid #CC9966; padding-left: 1px; padding-right: 1px; padding-top: 1px" bgcolor="#FFFFCC" align="center"> BB</td> <td style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; border-left: medium none; border-right: medium none; border-top: medium none; border-bottom: 1px solid #CC9966; padding-left: 1px; padding-right: 1px; padding-top: 1px" bgcolor="#FFFFCC" align="center"> B</td> <td style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; border-left: medium none; border-right: medium none; border-top: medium none; border-bottom: 1px solid #CC9966; padding-left: 1px; padding-right: 1px; padding-top: 1px" bgcolor="#FFFFCC" align="center"> CCC</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">1</font></td> <td align="right" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 1</td> <td align="right" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 1</td> <td align="right" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 1</td> <td align="right" x:num="0.99880000000000002" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9988</td> <td align="right" x:num="0.99039999999999995" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9904</td> <td align="right" x:num="0.98399999999999999" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9840</td> <td align="right" x:num="0.95650000000000002" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9565</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">2</font></td> <td align="right" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 1</td> <td align="right" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 1</td> <td align="right" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 1</td> <td align="right" x:num="0.99399999999999999" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9940</td> <td align="right" x:num="0.97409999999999997" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9741</td> <td align="right" x:num="0.93540000000000001" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9354</td> <td align="right" x:num="0.82969999999999999" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.8297</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">3</font></td> <td align="right" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 1</td> <td align="right" x:num="0.99650000000000005" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9965</td> <td align="right" x:num="0.99980000000000002" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9998</td> <td align="right" x:num="0.98860000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9886</td> <td align="right" x:num="0.93500000000000005" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9350</td> <td align="right" x:num="0.87970000000000004" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.8797</td> <td align="right" x:num="00.69" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.6900</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">4</font></td> <td align="right" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 1</td> <td align="right" x:num="0.99460000000000004" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9946</td> <td align="right" x:num="0.99909999999999999" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9991</td> <td align="right" x:num="0.98270000000000002" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9827</td> <td align="right" x:num="0.92879999999999996" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9288</td> <td align="right" x:num="0.82150000000000001" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.8215</td> <td align="right" x:num="0.63380000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.6338</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">5</font></td> <td align="right" x:num="0.99970000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9997</td> <td align="right" x:num="0.99460000000000004" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9946</td> <td align="right" x:num="0.99880000000000002" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9988</td> <td align="right" x:num="0.97719999999999996" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9772</td> <td align="right" x:num="0.90880000000000005" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9088</td> <td align="right" x:num="0.77269999999999994" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.7727</td> <td align="right" x:num="0.61470000000000002" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.6147</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">6</font></td> <td align="right" x:num="0.99970000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9997</td> <td align="right" x:num="0.99460000000000004" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9946</td> <td align="right" x:num="0.998" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9980</td> <td align="right" x:num="0.97150000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9715</td> <td align="right" x:num="0.9002" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9002</td> <td align="right" x:num="0.74059999999999993" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.7406</td> <td align="right" x:num="0.5585" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.5585</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">7</font></td> <td align="right" x:num="0.99970000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9997</td> <td align="right" x:num="0.99460000000000004" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9946</td> <td align="right" x:num="0.99750000000000005" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9975</td> <td align="right" x:num="0.96450000000000002" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9645</td> <td align="right" x:num="0.88529999999999998" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.8853</td> <td align="right" x:num="0.71750000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.7175</td> <td align="right" x:num="0.53299999999999992" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.5330</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">8</font></td> <td align="right" x:num="0.99970000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9997</td> <td align="right" x:num="0.99460000000000004" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9946</td> <td align="right" x:num="0.99660000000000004" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9966</td> <td align="right" x:num="0.96299999999999997" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9630</td> <td align="right" x:num="0.88129999999999997" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.8813</td> <td align="right" x:num="0.70239999999999991" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.7024</td> <td align="right" x:num="0.51560000000000006" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.5156</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">9</font></td> <td align="right" x:num="0.99970000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9997</td> <td align="right" x:num="0.99429999999999996" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9943</td> <td align="right" x:num="0.996" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9960</td> <td align="right" x:num="0.96250000000000002" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9625</td> <td align="right" x:num="0.8659" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.8659</td> <td align="right" x:num="0.69079999999999997" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.6908</td> <td align="right" x:num="0.51560000000000006" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.5156</td> </tr> <tr height="21" style="height:15.75pt"> <td height="21" align="center" style="height: 15.75pt; font-size: 12.0pt; font-family: Arial, sans-serif; color: windowtext; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 45; border-left: medium none; border-right: 1px solid #CC9966; border-top: medium none; border-bottom: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" bgcolor="#FFFFCC"> <font size="2">10</font></td> <td align="right" x:num="0.99970000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9997</td> <td align="right" x:num="0.99409999999999998" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9941</td> <td align="right" x:num="0.996" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9960</td> <td align="right" x:num="0.96019999999999994" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.9602</td> <td align="right" x:num="0.83340000000000003" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.8334</td> <td align="right" x:num="0.68490000000000006" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.6849</td> <td align="right" x:num="0.49419999999999997" style="font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; text-align: general; vertical-align: middle; white-space: nowrap; width: 55; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> 0.4942</td> </tr> </table> </td> </tr> <tr> <td> <p class="exhibit_legend">Empirical survival functions constructed from default data for 1971-2000. Source: Standard & Poor's.</td> </tr> </table> </center> </div> <p class="body">Now, instead of considering a one-year time interval, let's consider an arbitrary time interval <img border="0" src="../symbols/delta.gif" align="absbottom" width="9" height="15"><i>t</i>. Generalizing [<a href="#[2]">2</a>], the probability of default between time <i> t</i> and time <i>t</i> + <img border="0" src="../symbols/delta.gif" align="absbottom" width="9" height="15"><i>t</i>, conditional on their being no default by time <i>t</i>, is</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/intensity_model_3.gif" width="101" height="39"></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">We can express this as an average rate of default by dividing by the time <a name="[4]">interval</a> <img border="0" src="../symbols/delta.gif" align="absbottom" width="9" height="15"><i>t</i>:</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/intensity_model_4.gif" width="101" height="39"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">Consider an example. Let <img border="0" src="../symbols/delta.gif" align="absbottom" width="9" height="15"><i>t</i> be three years. Assume <i>s</i>(5) = 0.8921 and <i>s</i>(8) = 0.8609. Then the conditional probability of default between years 5 and 8 is </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/intensity_model_5.gif" width="169" height="41"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">This is a probability of default over a three year period. By [<a href="#[4]">4</a>], we convert it to an average annual rate of default by dividing by 3. The result is an average rate of .0117 defaults per year over the three-year period.</p> <p class="body">To obtain an instantaneous rate of default <i>f</i>(<i>t</i>) at any time <i>t</i>, we take the limit as <img border="0" src="../symbols/delta.gif" align="absbottom" width="9" height="15"><i>t</i> goes to 0 <a name="[8]">in</a> [<a href="#[4]">4</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" rowspan="3"> <img border="0" src="../formulas/intensity_model_6.gif" width="230" height="160"></td> <td width="5%" align="right">[6]</td> </tr> <tr> <td width="5%" align="right">[7]</td> </tr> <tr> <td width="5%" align="right">[8]</td> </tr> </table> <p class="body">where <img border="0" src="../formulas/intensity_model_s_prime_t.gif" align="absbottom" width="26" height="16"> is the first derivative of <i>s</i> with respect to <i>t</i>.</p> <p class="body">The instantaneous rate of default <i>f</i>(<i>t</i>) is called the <b><a name="default intensity">default intensity</a></b> or, to borrow a word from insurance, the <b><a name="hazard rate">hazard rate</a></b>. Intensity models work by assuming some functional form for <i>f</i>(<i>t</i>) and then calibrating that to current <a href="../articles/interest_rate_spreads.htm">interest rate spreads</a>. <i>f</i>(<i>t</i>) can reflect "real" probabilities to support credit risk management applications. It can reflect <a href="../articles/option_pricing_theory.htm">risk neutral</a> probabilities to support <a href="../articles/option_pricing_theory.htm"> financial engineering</a> applications. The survival function is recovered from <i>f</i>(<i>t</i>) by rearranging [<a href="#[8]">8</a>] and <a name="[9]">integrating</a>:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table14"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/intensity_model_9.gif" width="113" height="58"><p> </td> <td width="5%" align="right">[9]<p> </td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table15"> <tr> <td width="336" height="10"><font size="2"> </font></td> <td width="9" height="10"></td> </tr> <tr> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> <p class="body">From this, probabilities of default can be obtained from [<a href="#[1]">1</a>] or [<a href="#[2]">2</a>] as appropriate.</p> <p class="body">Note how <i>f</i>(<i>t</i>) plays a role similar to that of a continuously compounded interest rate in [<a href="#[9]">9</a>]. Use of default intensities tends to simplify mathematics, which is one reason intensity models are popular with <a href="../articles/option_pricing_theory.htm">financial engineers</a>.</p> <p class="body">Altman (<a href="#Altman E. I. (1989)">1989</a>), Asquith, Mullins and Wolff (<a href="#Asquith, Paul, David W. Mullins and Eric D. Wolff (1989)">1989</a>) and Altman and Suggitt (<a href="#Altman, Edward I., & Suggitt, Heather J. (2000)">2000</a>) discuss mortality models of default. The first published intensity model appears to be Jarrow and Turnbull (<a href="#Jarrow, Robert, Stuart Turnbull (1995)">1995</a>). Subsequent research includes Duffie and Huang (<a href="#Duffie, Darrell, and Ming Huang (1996)">1996</a>), Jarrow, Lando and Turnbill (<a href="#Jarrow, Robert, David Lando and Stuart Turnbull (1997)">1997</a>) and Duffie and Singleton (<a href="#Duffie, Darrell and Kenneth Singleton (1997a)">1997a</a>, <a href="#Duffie, Darrell and Kenneth Singleton (1997b)">1997b</a>).</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <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="table17"> <tr> <td width="100%"> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/asset_value_model.htm">asset value model</a> A type of default model.</strong></p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/credit_derivative.htm">credit derivative</a> A derivative instrument designed to transfer credit risk from one party to another.<p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/credit_risk.htm">credit risk</a> Risk</strong> due to uncertainty in a counterparty's ability to meet its obligations.</p> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/default_model.htm">default model</a> A type of model that assess the likelihood of default by an obligor.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/portfolio_credit_risk.htm">portfolio credit risk</a> Credit risk associated with a portfolio of obligations, typically of multiple obligors.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/presettlement_risk.htm">pre-settlement risk</a> Credit risk of default on a derivative instrument prior to final settlement.</td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Sponsored Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" width="75"> <img border="0" src="../images/ad_side_banner_2.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%">  <p class="ad_legend" style="margin-left: 0; margin-bottom:2"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a></p> <p style="margin-top: 0; margin-bottom: 0"> <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82436;type=iframe" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I1"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82436;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82436"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82436;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Books</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table18"> <tr> <td width="100%"> <p class="explanatory_text">Caouette, et al. 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</map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap366" 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">Related Papers</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="table25"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Altman E. I. (1989)">Altman E. I. (1989)</a>. Measuring corporate bond mortality and performance, <i>Journal of Finance</i>, 44 (4), 909-922.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Altman, Edward I., & Suggitt, Heather J. (2000)">Altman, Edward I., & Suggitt, Heather J. (2000)</a>. Default rates in the syndicated bank loan market - a mortality analysis, <i> Journal of Banking and Finance</i>, 24(1-2), 229-253. <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Asquith, Paul, David W. Mullins and Eric D. Wolff (1989)">Asquith, Paul, David W. Mullins and Eric D. Wolff (1989)</a>. 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An Econometric Model of the Term Structure of Interest-Rate Swap Yields, <i>Journal of Finance</i>, 52 (4), 1287-1321.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Jarrow, Robert, Stuart Turnbull (1995)">Jarrow, Robert, Stuart Turnbull (1995)</a>. Pricing derivatives on financial securities subject to credit risk', <i>Journal of Finance</i>, 50 (1), pp. 53-86</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Jarrow, Robert, David Lando and Stuart Turnbull (1997)">Jarrow, Robert, David Lando and Stuart Turnbull (1997)</a>. 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