Fixed Income Term Structure: Spot Curve, Forward Curve, Discount Curve, Yield Curve

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Fixed Income Term Structure</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="100%" align="left" valign="top"><p class="word_list"> <a href="#cash loan">cash loan</a><p class="word_list"> <a href="#discount curve">discount curve</a><p class="word_list"> <a href="#discount factors">discount factor</a><p class="word_list"> <a href="#discount">discounting</a><p class="word_list"> <a href="#fixed income term structure">fixed income term structure</a><p class="word_list"> <a href="#forward curve">forward curve</a><p class="word_list"> <a href="#forward loan">forward loan</a><p class="word_list"> <a href="#forward rate">forward rate</a><p class="word_list"> <a href="#spot curve">spot curve</a><p class="word_list"> <a href="#spot loan">spot loan</a><p class="word_list"> <a href="#spot interest rate">spot rate</a><p class="word_list"> <a href="#time value of money">time value of money</a><p class="word_list"> <a href="#yield curve">yield curve</a><p class="word_list"> <a href="#zero-coupon curve">zero-coupon curve</a><p class="word_list"> <a href="#zero-coupon rates">zero-coupon rate</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 present value of a dollar to be received in a year is less than the present value of that dollar if it were received today. We call this the <b> <a name="time value of money">time value of money</a></b>. Financial markets use spot curves, forward curves, discount curves and yield curves to describe the time value of money. These are referred to collectively as the <b><a name="fixed income term structure">fixed income term structure</a></b>. This article defines these notions.</p> <p class="body">A <b><a name="cash loan">cash loan</a></b> is a loan that commences immediately. A <b><a name="spot loan">spot loan</a></b> is a loan that commences <a href="../articles/settlement.htm">spot</a>. A <b> <a name="forward loan">forward loan</a></b> is one that commences on some date later than spot. For example, in the Eurodollar markets a three-month spot loan commences in two business days (spot) and matures three months after that. A 2<img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11">7 forward loan commences two months from the spot date and lasts for five months. With either type of loan, interest can be paid periodically or it can be accumulated and paid at maturity.</p> <p class="body">A <b><a name="spot interest rate">spot interest rate</a></b> for maturity <i>m</i> is an interest rate payable on a spot loan of maturity <i>m</i> that accumulates interest to maturity. Spot rates are sometimes called <b><a name="zero-coupon rates">zero-coupon rates</a></b> because they are the rates of interest payable on obligations that accumulate all interest to maturity. <a href="../articles/libor.htm">Libor</a> rates for maturities of a week or more are spot rates. Exhibit 1 indicates <a href="../articles/currency_codes.htm">USD</a> Libor rates for monthly maturities as of March 1, 2004.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="200" cellpadding="0" id="table6"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">March 1, 2004 <br> USD Libor</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> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse;" width="100%" bgcolor="#FFFFCC" id="table7"> <colgroup> <col width="64" span="3" style="width:48pt"> </colgroup> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 1 month</td> <td align="left" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.10000</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 2 month</td> <td align="left" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.11000</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 3 month</td> <td align="left" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.12000</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 4 month</td> <td align="left" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.13000</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 5 month</td> <td align="left" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.15000</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 6 month</td> <td align="left" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.17000</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 7 month</td> <td align="left" x:num="1.1937500000000001" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.19375</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 8 month</td> <td align="left" x:num="1.2212499999999999" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.22125</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 9 month</td> <td align="left" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.25000</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 10 month</td> <td align="left" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.29000</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 11 month</td> <td align="left" x:num="1.3287500000000001" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.32875</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px"> </td> <td height="17" style="height: 12.75pt; font-weight: 700; font-family: Arial, sans-serif; color: windowtext; font-size: 10.0pt; font-style: normal; text-decoration: none; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text" align="left"> 12 month</td> <td align="left" x:num="1.3674999999999999" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px; width:80" class="explanatory_text"> 1.36750</td> </tr> </table> </div> </td> </tr> <tr> <td> <p class="exhibit_legend">Source: <a target="_blank" href="http://www.bba.org.uk">BBA</a></td> </tr> </table> </center> </div> <p class="body">A <b><a name="spot curve">spot curve</a></b> (or <b> <a name="zero-coupon curve">zero-coupon curve</a></b>) is a graph of spot rates as a function of maturity.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="400" cellpadding="0" id="table8"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Spot Curve</span><br> <span class="exhibit_subheader">Exhibit 2</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/ex2_fixed_income_term_structure.gif" width="350" height="240"></td> </tr> <tr> <td> <p class="exhibit_legend">Spot curve constructed from the Libor rates of Exhibit 1.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table9"> <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">An <i>n</i> <img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11"> (<i>n </i>+<i> m</i>) <b><a name="forward rate">forward rate</a></b> is an interest rate payable on a forward loan that </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">commences <i>n</i> months from the spot date,</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">matures <i>m</i> months after that, and </p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">accumulates interest to maturity.</p> <p class="body">If we have a spot curve, we can calculate forward rates. Suppose we want the 3<img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11">5 forward USD Libor rate for March 1, 2004. We can calculate this from the 3-month and 5-month spot Libor rates. Let <i>r</i> denote the desired forward rate. We use the fact that a 5-month spot loan is financially equivalent to a 3-month spot loan combined with a 3<img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11">5 forward loan. With Libor, <a href="../articles/compounding.htm">simple compounding</a> is used. Based on the 3-month and 5-month spot rates and day counts as of March 1, we conclude</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> (1 + .0112(92/360)) (1 + <i>r </i>(61/360)) = (1 + .0115(153/360))</td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body" style="margin-right: 10">Solving for <i>r</i>, we obtain the forward rate as 1.19%. Note that this exceeds both the spot rates, which are 1.12% and 1.15%. This makes sense. If there are to be no <a href="../articles/arbitrage.htm">arbitrage</a> opportunities, the combined interest from the 3-month spot and forward loans must equal the interest earned on the 5-month spot loan. If the rate earned on the 3-month spot loan is <i>lower than</i> that earned on the 5-month spot loan, then the rate earned on the forward loan will have to be <i>greater than</i> that earned on the 5-month spot loan.</p> <p class="body" style="margin-right: 20">A <b><a name="forward curve">forward curve</a></b> is a graph of forward rates all for the same maturity but with different forward periods. For example, a forward curve might indicate rates for 0<img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11">3, 1<img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11">4, 2<img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11">5, 3<img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11">6, 4<img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11">7, ... , 120<img border="0" src="../symbols/multiplication_symbol.gif" align="absbottom" width="7" height="11">123 forward loans. This would be called a 3-month forward curve. Exhibit 3 indicates a 1-month forward curve calculated from the spot rates of Exhibit 1.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="200" cellpadding="0" id="table11"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">March 1, 2004 <br> USD Forward Curve</span><br> <span class="exhibit_subheader">Exhibit 3</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="0" cellspacing="0" style="border-collapse: collapse" bgcolor="#FFFFCC" width="100%" id="table12"> <colgroup> <col width="64" span="3" style="width:48pt"> </colgroup> <tr height="17" style="height:12.75pt"> <td height="17" width="64" style="height: 12.75pt; width: 20; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td width="64" style="width: 80; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>0x1</b></td> <td align="left" width="64" style="width: 80; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.10000</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>1x2</b></td> <td align="left" x:num="1.1192731329490613" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.11927</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>2x3</b></td> <td align="left" x:num="1.1375379001544406" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.13754</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>3x4</b></td> <td align="left" x:num="1.1573540621507838" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.15735</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>4x5</b></td> <td align="left" x:num="1.2240223518242617" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.22402</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>5x6</b></td> <td align="left" x:num="1.2625390179692944" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.26254</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>6x7</b></td> <td align="left" x:num="1.3314545683478229" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.33145</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>7x8</b></td> <td align="left" x:num="1.4011459252072582" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.40115</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>8x9</b></td> <td align="left" x:num="1.4725528605739058" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.47255</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>9x10</b></td> <td align="left" x:num="1.6292813358109153" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.62928</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>10x11</b></td> <td align="left" x:num="1.6926896579011581" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.69269</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text" align="left"> <b>11x12</b></td> <td align="left" x:num="1.8113532725241934" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 1.81135</td> </tr> </table> </td> </tr> <tr> <td> <p class="exhibit_legend">One-month forward rates calculated from the spot Libor rates of Exhibit 1.</td> </tr> </table> </center> </div> <p class="body">These are graphed in Exhibit 4. </p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="400" cellpadding="0" id="table13"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Forward Curve</span><br> <span class="exhibit_subheader">Exhibit 4</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/ex4_fixed_income_term_structure.gif" width="350" height="240"></td> </tr> <tr> <td> <p class="exhibit_legend">One month forward curve constructed from the spot Libor rates of Exhibit 1. The curve is superimposed over the spot curve of Exhibit 2.</td> </tr> </table> </center> </div> <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">  <p style="margin-top: 0; 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="I1"> <![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> </tr> </table> <p class="body">Note that spot and forward curves provide identical information. If you have one, you can construct the other.</p> <p class="body">A third, also equivalent way to indicate the time value of money is <b><a name="discount factors">discount factors</a></b>. When we calculate the present value of some future cash flow, we are said to <b> <a name="discount">discount</a></b> that future cash flow. A discount factor is the factor by which the future cash flow must be multiplied to obtain the present value. For example, if a <a href="../articles/currency_codes.htm">EUR</a> 100 payment to be made at maturity <i>m</i> has present value EUR 89.4, the EUR discount factor for maturity <i>m</i> is .894. Note that present values are often calculated with a <a href="../articles/settlement.htm">spot</a> <a href="../articles/settlement.htm">value date</a>. If this is the case, discount factors reflect discounting to the spot date as opposed to the current date.</p> <p class="body">Discount factors can be calculated from spot or forward rates. As an example, from Exhibit 1, the March 1, 2004 USD spot 6-month Libor rate was 1.17%. We calculate the corresponding discount factor as</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"> 1 / (1 + .0117(184/360)) = .99406</td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">This represents discounting from the date six months after spot back to the spot date.</p> <p class="body">A <b><a name="discount curve">discount curve</a></b> is a graph of discount factors for different maturities. Exhibit 5 indicates discount factors calculated from the spot Libor rates of Exhibit 1. These are graphed in Exhibit 6.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="200" cellpadding="0" id="table16"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">March 1, 2004 <br> USD Discount Curve</span><br> <span class="exhibit_subheader">Exhibit 5</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> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bgcolor="#FFFFCC" width="100%" id="table17"> <colgroup> <col width="64" span="3" style="width:48pt"> </colgroup> <tr height="17" style="height:12.75pt"> <td height="17" width="64" style="height: 12.75pt; width: 20; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td width="64" style="width: 80; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>1 month</b></td> <td align="left" width="64" style="width: 80; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" x:num="0.99905367415864421" class="explanatory_text"> 0.99905</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>2 month</b></td> <td align="left" x:num="0.9981226975596732" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.99812</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>3 month</b></td> <td align="left" x:num="0.99714594671252055" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.99715</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>4 month</b></td> <td align="left" x:num="0.99618516425709769" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.99619</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>5 month</b></td> <td align="left" x:num="0.99513627147317496" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.99514</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>6 month</b></td> <td align="left" x:num="0.99405554782401229" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.99406</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>7 month</b></td> <td align="left" x:num="0.9929538204070627" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.99295</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>8 month</b></td> <td align="left" x:num="0.99175722334148841" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.99176</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>9 month</b></td> <td align="left" x:num="0.99054170249355111" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.99054</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>10 month</b></td> <td align="left" x:num="0.98915392718837958" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.98915</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>11 month</b></td> <td align="left" x:num="0.98771424096338367" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.98771</td> </tr> <tr height="17" style="height:12.75pt"> <td height="17" style="height: 12.75pt; color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 20; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left"> </td> <td style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" align="left" class="explanatory_text"> <b>12 month</b></td> <td align="left" x:num="0.98632467685024061" style="color: windowtext; font-size: 10.0pt; font-weight: 400; font-style: normal; text-decoration: none; font-family: Arial; text-align: general; vertical-align: bottom; white-space: nowrap; width: 80; border: medium none; padding-left: 1px; padding-right: 1px; padding-top: 1px" class="explanatory_text"> 0.98633</td> </tr> </table> </div> </td> </tr> <tr> <td> <p class="exhibit_legend">Discount factors calculated from the spot Libor rates of Exhibit 1.</td> </tr> </table> </center> </div> <p class="body"> </p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="400" cellpadding="0" id="table18"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Discount Curve</span><br> <span class="exhibit_subheader">Exhibit 6</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/ex6_fixed_income_term_structure.gif" width="350" height="240"></td> </tr> <tr> <td> <p class="exhibit_legend">Discount curve constructed from the spot Libor rates of Exhibit 1.</td> </tr> </table> </center> </div> <p class="body">The fourth way the time value of money can be described is with a <b><a name="yield curve">yield curve</a></b>. This is simply a graph of <a href="../articles/coupon_bond.htm">bond</a> <a href="../articles/yield.htm">yields</a> for various maturities. The curve is typically fit in some manner to price data for bonds of various maturities trading close to <a href="../articles/par_value.htm">par</a> and generally of the same credit quality. Yield curves are falling out of use today. Widespread use of computers in finance makes spot curves, forward curves and discount curves easier to construct and use in pricing work. Also, while <a href="../articles/yield.htm">yields</a> continue to be widely quoted for bonds, fixed income markets are increasingly trading instruments other than bonds for which yield is either a meaningless or not useful notion. 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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="table21"> <tr> <td width="100%"> <p class="explanatory_text">Questa (<a target="_blank" href="http://www.riskbook.com/link/questa_g_(1999).htm">1999</a>) is an introductory text on fixed income and foreign exchange mathematics. 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