Structural Credit Risk 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">Structural Credit Risk 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="99%" align="left" valign="top"><p class="word_list"> <a href="#asset value model">asset value model</a><p class="word_list"> <a href="#distance to default">distance to default</a><p class="word_list"> <a href="#KMV model">KMV model</a><p class="word_list"> <a href="#Mertons model">Merton model</a><p class="word_list"> <a href="#structural credit risk model">structural credit risk model</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table5"> <tr> <td align="right" width="9" height="14"></td> <td width="300" height="14"> <font size="2"> </font><font size="4"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body">The <b> <a name="structural credit risk model">structural credit risk model</a></b> is a model for assessing <a href="../articles/credit_risk.htm">credit risk</a>, typically of a <a href="../articles/corporation.htm">corporation's</a> debt. It is also sometimes called the <b> <a name="Merton model">Merton model</a></b> or<b> <a name="asset value model">asset value model</a></b> . It was proposed in Black and Scholes' (<a href="#Black (1973)">1973</a>) seminal paper on <a href="../articles/option_pricing_theory.htm">option pricing</a> and a more detailed paper by Merton (<a href="#Merton, Robert C. (1974)">1974</a>). Merton anticipated the model in Merton (<a href="#Merton, Robert C. (1970)">1970</a>), and he actively supported the work of Black and Scholes, which is why the model is often called the Merton model. A popular implementation of the model is the commercial <b> <a name="KMV model">KMV model</a></b>. KMV was a boutique software firm that is now owned by Moodys.</p> <p class="body">The model considers a corporation financed through a single debt and a single <a href="../articles/common_stock.htm">equity</a> issue. The debt comprises a <a href="../articles/zero_coupon_bond.htm">zero-coupon bond</a> that matures at time <i>t = t</i>*, at which time it is to pay investors <i>b</i> dollars. The equity pays no <a href="../articles/common_stock.htm">dividends</a>. </p> <p class="body">An unobservable process <b><i>V</i></b> describes the firm's value <img border="0" src="../formulas/asset_value_model_tv.gif" align="absbottom" width="15" height="19"> <img border="0" src="../formulas/asset_value_model_greater_equal_to.gif" align="absbottom" width="9" height="14"> 0 at any time <i>t</i>. We ascribe the outstanding debt and equity values <img border="0" src="../formulas/asset_value_model_tf.gif" align="absbottom" width="12" height="19"> and <img border="0" src="../formulas/asset_value_model_te.gif" align="absbottom" width="13" height="19">, respectively. Accordingly, at any time <i>t</i></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/asset_value_model_tv.gif" align="absbottom" width="15" height="19"> = <img border="0" src="../formulas/asset_value_model_tf.gif" align="absbottom" width="12" height="19"> + <img border="0" src="../formulas/asset_value_model_te.gif" align="absbottom" width="13" height="19"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">At time  <i>t</i>*, the firm's debt matures. At that time, either <img border="0" src="../formulas/asset_value_model_tv_star.gif" align="absbottom" width="22" height="20"> will exceed the bond's maturity value <i>b</i>, or it won't. In the former case, the firm will pay off the bondholders. The remaining value of the firm will belong to the equity holders, so</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/asset_value_model_te_star.gif" align="absbottom" width="19" height="20"> = <img border="0" src="../formulas/asset_value_model_tv_star.gif" align="absbottom" width="22" height="20"> – <i>b</i></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">In the latter case, the firm defaults on its debt. The bondholders take ownership of the firm, and the <a href="../articles/common_stock.htm">stockholders</a> are left with nothing:</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/asset_value_model_te_star.gif" align="absbottom" width="19" height="20"> = 0</td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">Combining the above two results, we obtain a general expression for the value of the firm's <a href="../articles/common_stock.htm">stock</a> at the maturity of its debt:</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/asset_value_model_te_star.gif" align="absbottom" width="19" height="20"> = max( <img border="0" src="../formulas/asset_value_model_tv_star.gif" align="absbottom" width="22" height="20"> – <i>b</i>, 0)</td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">Look closely at this formula. It is precisely the <a href="../articles/options_spread.htm">payoff </a>of a <a href="../articles/option.htm">call option</a> on the firm's value <img border="0" src="../formulas/asset_value_model_tv_star.gif" align="absbottom" width="22" height="20"> with <a href="../articles/option.htm">strike</a> price <i>b</i>. Based upon this realization, the asset value model treats the firm's equity as a call option on the value of the firm struck at the maturity value<i> b </i>of its debt. By <a href="../articles/put_call_parity.htm">put-call parity</a>, the firm's debt comprises a risk-free bond that guarantees payment of <i>b</i> plus a short <a href="../articles/option.htm">put option</a> on the value of the firm struck at <i>b</i>. <a name="[5]">Accordingly</a></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/asset_value_model_tf_star.gif" align="absbottom" width="19" height="20"> =<i> b</i> –<i> </i>max( <i>b – </i> <img border="0" src="../formulas/asset_value_model_tv_star.gif" align="absbottom" width="22" height="20">, 0)</td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">The asset value model treats <img border="0" src="../formulas/asset_value_model_tv.gif" align="absbottom" width="15" height="19"> just like any underlier. It assumes <img border="0" src="../formulas/asset_value_model_tv.gif" align="absbottom" width="15" height="19"> follows a geometric <a href="../articles/brownian_motion.htm">Brownian motion</a> with <a href="../articles/volatility.htm">volatility</a> <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12">. Further, it makes all the other simplifying assumptions of the Black-Scholes (<a href="#Black (1973)">1973</a>) option pricing formula. Accordingly, the firm's equity can be valued at any time <i>t</i> as</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/asset_value_model_te.gif" align="absbottom" width="13" height="19"> = <i>c</i>( <img border="0" src="../formulas/asset_value_model_tv.gif" align="absbottom" width="15" height="19">, <i>b, </i> <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12">, <i>r</i>, <i>t</i>* – <i>t</i> )</td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">where <i>c</i> is the <a href="../articles/black_scholes_1973.htm"> Black-Scholes formula</a> for the value of a call option, and <i>r</i> is the risk-free rate. By [<a href="#[5]">5</a>], we can similarly value the firm's debt as</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/asset_value_model_tf.gif" align="absbottom" width="12" height="19"> = <i>b</i><img border="0" src="../symbols/e^-rt.gif" align="absbottom" width="25" height="19"> – <i>p</i>( <img border="0" src="../formulas/asset_value_model_tv.gif" align="absbottom" width="15" height="19">, <i>b, </i> <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12">, <i>r</i>, <i>t</i>* – <i>t</i> )</td> <td width="5%" align="right">[7]</td> </tr> </table> <p class="body">where <i>p</i> is the Black-Scholes formula for the value of a put. Note that we discount the payment <i>b</i> at the risk free rate because that payment is risk-free in formula [<a href="#[5]">5</a>]—we have stripped out the credit risk as a put option.</p> <p class="body">At any time <i>t</i>, the <b> <a name="distance to default">distance to default</a></b> for a the firm's debt is defined as </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table13"> <tr> <td width="100%" align="center"> ( <img border="0" src="../formulas/asset_value_model_tv.gif" align="absbottom" width="15" height="19"> – <i>b</i>)<font size="4"> / </font> <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12"></td> <td width="5%" align="right">[8]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table14"> <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">This is a metric indicating how many standard deviations the equity holders' call option is <a href="../articles/time_value_and_intrinsic_value.htm">in-the-money</a>. The smaller the distance to default, the more likely a default is to occur. The probability of default is precisely the probability of the call option expiring <a href="../articles/time_value_and_intrinsic_value.htm">out-of-the-money</a>. This is approximately equal to one minus the option's normalized <a href="../articles/delta_and_gamma.htm">delta</a> (if investors were <a href="../articles/risk_aversion.htm">risk neutral</a>, equality would be exact). See this glossary's article <a href="../articles/black_scholes_1973.htm">Black-Scholes (1973) option pricing formula</a> for the Black-Scholes formula for delta. To normalize that value, divide the delta by the underlier's value.</p> <p class="body">Three shortcomings of the asset value model are:</p> <p class="body">1. Its assumption that the firm's debt financing consists of a one-year zero-coupon bond is, for most firms, an oversimplification..</p> <p class="body">2. The Black-Scholes (<a href="#Black (1973)">1973</a>) simplifying assumptions are questionable in the context of corporate debt, and</p> <p class="body">3. The firm's value <img border="0" src="../formulas/asset_value_model_tv.gif" align="absbottom" width="15" height="19"> is not observable, which makes assigning values to it and its volatility problematic.</p> <p class="body">Still, the model provides a useful context for considering and modeling credit risk. Practical implementations of the model are used by financial institutions and institutional investors. These extend the model in some manner to facilitate the assigning of values to <img border="0" src="../formulas/asset_value_model_tv.gif" align="absbottom" width="15" height="19"> and <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12">. 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</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_1.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 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Scholes (1973). The pricing of options and corporate liabilities, <i>Journal of Political Economy</i>, 81, 637-654.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Merton, Robert C. (1970)">Merton, Robert C. (1970)</a>. A dynamic general equilibrium model of the asset market and its application to the pricing of the capital structure of the firm, <i>unpublished manuscript</i>. Available in Merton (<a target="_blank" href="http://www.riskbook.com/link/merton_r_(1992).htm">1990</a>).</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Merton, Robert C. (1974)">Merton, Robert C. (1974)</a>. On the pricing of corporate debt: the risk structure of interest rates, Journal of Finance, 29, 449-470. Available in Merton (<a target="_blank" href="http://www.riskbook.com/link/merton_r_(1992).htm">1990</a>).</p> </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="table29"> <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"><span class="verdana10">website:</span> <span class="verdana10"> <a target="_top" href="http://www.contingencyanalysis.com"> http://www.contingencyanalysis.com</a><br> glossary direct link: <a target="_top" href="http://www.riskglossary.com"> http://www.riskglossary.com</a><br> <font size="3" face="Times New Roman">copyright © Contingency Analysis, 200</font><font face="Times New Roman">3</font></span></td> </tr> </table> </td> </tr> </table> </td> <td>  </td> </tr> </table> </div> </body>