Bond-Equivalent Yield

<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">Bond-Equivalent Yield</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="#bond equivalent yields">bond-equivalent yield</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> <td width="100%" align="left" valign="top"><p class="word_list"> <br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <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">In the money market, <a href="../articles/discount_instrument.htm">discount instruments</a> are generally quoted with <a href="../articles/discount_yield.htm">discount yields</a>, which are not directly comparable with <a href="../articles/yield.htm"> yields</a> calculated for other types of instruments. To facilitate comparisons, discount yields may be converted to <b> <a name="bond equivalent yields">bond-equivalent yields</a></b>.</p> <p class="body">Conceptually, here is the idea. <a href="../articles/coupon_bond.htm">Bonds</a> have long <a href="../articles/coupon_bond.htm">terms</a> whereas money market instruments have short terms. However, during their last year prior to maturity, bonds have short terms just like money market instruments. When we calculate the bond-equivalent yield of a discount instrument, we calculate a number comparable to the <a href="../articles/yield.htm">yield to maturity</a> for a <a href="../articles/treasury_note_and_bond.htm"> Treasury bond</a> that is in its last year prior to maturity. There are three issues that must be addressed by the formula for bond-equivalent yield:</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">Treasury bond yields are calculated on an actual/actual basis, while discount yields are usually calculated on an actual/360 basis.</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">The formula for <a href="../articles/discount_yield.htm">discount yield</a> differs from the formula for calculating a Treasury bond's yield to maturity because it divides by the instrument's face value as opposed to its price.</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">Bonds with more than a half year to maturity have one more <a href="../articles/coupon_bond.htm">coupon</a> to pay whereas discount instruments pay no coupons.</p> <p class="body">To address the third problem, there are two formulas for bond-equivalent yield. One is for discount instruments with up to 182 days to maturity. The other is for discount instruments with more than 182 days to maturity.</p> <p class="body">The first formula for bond-equivalent yield is</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/bond_equivalent_yield_[1].gif" width="368" height="93"></td> <td width="5%" align="right">[1]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table7"> <tr> <td align="right" width="9" height="14"></td> <td width="336" height="14"> <font size="2"> </font><font size="4"> </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">Where "actual days in year" is either 365 or 366, depending on whether this is a leap year. "Days to maturity" is the actual number of days to maturity. You may see this formula expressed more simply elsewhere. I like to write it as in [1] because this makes it intuitively clear how the formula addresses the first two of the above three issues. By multiplying by the factor:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table8"> <tr> <td width="100%" align="center" height="60"> <img border="0" src="../formulas/bond_equivalent_yield_[2].gif" width="121" height="42"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body" style="margin-bottom: 24">the formula converts the discount yield from an actual/360 basis to an actual/actual basis. This addresses the first issue. Dividing the discount yield by the factor</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/bond_equivalent_yield_[3].gif" width="229" height="45"></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body" style="margin-top: 24">approximately addresses the second issue.</p> <p class="body">If a discount instrument has more than 182 days remaining to maturity, the formula for bond-equivalent yield <a name="[4]">is </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/bond_equivalent_yield_[4].gif" width="481" height="102"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">where the instrument's face value is assumed to be 100, and "price" is its quoted price.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table11"> <tr> <td align="right" width="9" height="14"></td> <td width="336" height="14"> <font size="2"> </font><font size="4"> </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">Formula [4] is derived by assuming that the discount instrument's price is invested for six months to earn <a href="../articles/compounding.htm">simple interest</a> at a rate equal to the unknown bond-equivalent yield. At the end of the six months, the investment will be worth</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table12"> <tr> <td width="100%" align="center" height="60"> <img border="0" src="../formulas/bond_equivalent_yield_[5].gif" width="215" height="47"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">and there will be </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table13"> <tr> <td width="100%" align="center" height="60"> <img border="0" src="../formulas/bond_equivalent_yield_[6].gif" width="147" height="42"></td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">years remaining until the discount instrument's maturity date. Reinvesting the proceeds from [5] for this period, again at a rate equal to the unknown bond-equivalent yield, our investment will have a value on the discount instrument's maturity date of</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/bond_equivalent_yield_[7].gif" width="558" height="50"></td> <td width="5%" align="right">[7]</td> </tr> </table> <p class="body">Setting this equal to the discount instrument's maturity value, which is its face value of 100, we obtain a quadratic equation in the unknown bond-equivalent yield. Applying the <a href="../articles/closed_form_solution.htm">quadratic formula</a>, we then solve for the bond-equivalent yield. The result is formula [<a href="#[4]">4</a>].</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table15"> <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="table16"> <tr> <td width="100%"> <cop yrigh t( c)c ont i ngen cy anal y sis, 2 00 5> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/coupon_bond.htm">bond</a> Securitized debt.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/compounding.htm">compound interest</a> Any of several methods of crediting interest in which interest is earned on interest.</strong><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/discount_instrument.htm">discount instrument</a> A money market instrument that pays no coupons, matures for its face value, and is issued at a discount to its face value.</strong></p> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/discount_yield.htm">discount yield</a> A convention for calculating yield on a discount instrument.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/fixed_income_term_structure.htm">fixed income term structures</a> Overview article.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/par_value.htm">par value</a> A stated value for a security.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/yield.htm">yield</a> Any of several metrics of the income or return to be earned from an investment.</strong></td> </tr> </table> </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="table17"> <tr> <td width="100%"> <p class="explanatory_text">Fabozzi, Mann and Choudhry (<a target="_blank" href="http://www.riskbook.com/link/fabozzi_mann_choudhry_(2002).htm">2002</a>) covers money market instruments in detail.</p> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table18"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap370"> <area target="_blank" href="http://www.riskbook.com/link/fabozzi_mann_choudhry_(2002).htm" shape="rect" coords="0, 0, 82, 124"> </map> <img border="0" src="../images/book_fabozzi_mann_choudhry_(2002)_125.gif" usemap="#FPMap370" width="83" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table19"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/fabozzi_mann_choudhry_(2002).htm"> The Global Money Markets</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>F. 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