Convexity Bias of Eurodollar Futures

<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="table6"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table7"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table8"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Convexity Bias</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="table9"> <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="#convexity_bias">convexity bias</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="table10"> <tr> <td align="right" width="9"> </td> <td width="300">  </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>Cox, Ingersoll and Ross (<a href="#Cox">1981</a>) and Jarrow and Oldfield (<a href="#Jarrow">1981</a>) suggest that daily <a href="../articles/future.htm">margin</a> payments on <a href="../articles/future.htm">futures</a> may cause <a href="../articles/forward.htm">forward</a> and futures prices to diverge. If there is a <a href="../articles/correlation.htm">correlation</a> between daily futures prices and interest rates, one party to a futures contract will tend to receive margin payments on days when interest rates rise and make margin payments on days when interest rates decline. On average, she will invest the margin payments she receives at interest rates that are higher than those at which she finances the margin payments she makes. The other party will experience the opposite situation. This should cause a divergence in forward and futures prices. The effect should increase with the maturity of contracts and with the <a href="../articles/standard_deviation.htm">standard deviation</a> of a future’s price. </p> <p>Empirical studies by Cornell and Reinganum (<a href="#Cornell">1981</a>), French (<a href="#French">1983</a>) and Park and Chen (<a href="#Park">1985</a>) confirm a modest effect in gold, silver, silver coins, platinum, copper and plywood prices but fail to find one for various currencies. </p> <p>For Eurodollar futures (and other Eurocurrency futures) the effect is exasperated. Not only do the <a href="../articles/fixed_income_term_structure.htm">forward rates</a> they are linked to correlate highly with the overnight rates at which margin payments would typically be financed or invested, but there is an issue with how those margin payments are calculated.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table11"> <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>A forward on a <a href="../articles/eurodollar_deposit.htm">deposit</a> has <a href="../articles/duration_and_convexity.htm">convexity</a>. Its <a href="../articles/valuation.htm">market value</a> rises more for a given decline in the forward rate than it would decline for the same sized rise in the forward rate. If this were also true for Eurodollar futures, this convexity effect would partially offset the margining effect identified by Cox, Ingersoll and Ross (<a href="#Cox">1981</a>) and Jarrow and Oldfield (<a href="#Jarrow">1981</a>). But it is not true of Eurodollar futures. Contract specifications for Eurodollar futures set the daily margin payment at USD 25 per <a href="../articles/basis_point.htm">basis point</a> move in the futures rate. While this margining formula couldn't be simpler, by construction, it deprives Eurodollar futures of the convexity possessed by the <a href="../articles/forward_rate_agreement.htm">forward rate agreements</a> (FRAs) they are intended to mimic. </p> <p>Eurodollar futures rates should diverge slightly from corresponding forward rates for FRAs. Burghardt and Hoskins (<a href="#Burghardt">1995</a>) have called the effect a <b><a name="convexity_bias">convexity bias</a></b>. Note that convexity bias is due to a combination—interaction—of both</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">the margining effect identified by Cox, Ingersoll and Ross (1981) and Jarrow and Oldfield (1981), and</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">the unique margining formula for Eurodollar futures (and other Eurocurrency futures).</p> <p>For short-dated Eurodollar futures—those out to a year or eighteen months, the effect is hardly noticeable, a basis point or less. For longer-dated futures, convexity bias can be more pronounced, causing Eurodollar futures rates to exceed corresponding forward rates by ten basis points or more at the longest maturities. The actual magnitude depends on the level and <a href="../articles/volatility.htm">volatility</a> of interest rates.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table12"> <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>Prior to the early 1990s, traders were unaware of the convexity bias. Eurodollar futures were used to <a href="../articles/hedging_and_diversification.htm">hedge</a> <a href="../articles/interest_rate_swap.htm">interest rate swaps</a>, which were priced using the Eurodollar futures rates as if they were forward rates. This caused <a href="../articles/interest_rate_swap.htm"> swap rates</a> to be higher than they should have been. The situation started to change during the early 1990s, as awareness of the convexity bias spread. <a href="../articles/option_pricing_theory.htm">Financial engineering</a> models were developed to calculate the convexity bias, which was then taken into account as swaps were priced or hedged with Eurodollar futures.</p> <p>Today, Eurodollar futures are still widely used for hedging swaps and other fixed income <a href="../articles/derivative_instrument.htm"> derivatives</a>. Traders have software that calculates hedges for them, taking into account the convexity bias. Eurodollar rates are no longer a benchmark for pricing, though. Swap rates have replaced them as the benchmark.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table13"> <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="table15"> <tr> <td width="100%"> <cop yrigh t( c)c ont i ngen cy anal y sis, 2 00 5> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/eurodollar_deposit.htm">Eurodollar deposit</a> A deposit of US dollars held at a bank branch outside the United States.<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 structure</a> Refers collectively to a spot curve, forward curve, discount curve, yield curve or any other curve that describes the time value of money at a particulate point in time.</p> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/forward_rate_agreement.htm">forward rate agreement</a> A cash-settled forward contract on a short-term loan.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/future.htm">future</a> An exchange-traded derivative that is similar to a forward.</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/forward.htm">forward contract</a> A </strong>trade that is agreed to at one point in time but will take place at some later time.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/interest_rate_swap.htm">interest rate swap</a> A swap under which both cash flow streams are in the same currency and are defined as cash flow streams that might be associated with some fixed income obligations. <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/libor.htm">Libor</a> London Interbank Offered Rate.</p> </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="table16"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Burghardt">Burghardt</a>, Galen and Bill Hoskins (1995). A question of bias, <i>Risk</i>, 8 (3), 63-70.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Cornell">Cornell</a>, Bradford and Marc R. Reinganum (1981). Forward and futures prices: Evidence from the foreign exchange markets, <i>Journal of Finance</i>, 36 (12), 1035-1045.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Cox">Cox</a>, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross (1981). The relation between forward prices and futures prices, <i> Journal of Financial Economics</i>, 9, 321-346.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="French">French</a>, Kenneth R. (1983). A comparison of futures and forward prices, <i>Journal of Financial Economics</i>, 12, 311-342.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Jarrow">Jarrow</a>, Robert A. and George S. Oldfield (1981). Forward contracts and futures contracts, <i>Journal of Financial Economics</i>, 9, 373-382.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Park">Park</a>, Hun Y. and Andrew H. Chen (1985). 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