Duration and Convexity

<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">Duration and Convexity</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="98%" align="left" valign="top"><p class="word_list"> <a href="#convexity">convexity</a><p class="word_list"> <a href="#[1]">duration</a><p class="word_list"> <a href="#duration matching">duration matching</a><p class="word_list"> <a href="#duration-convexity matching">duration-convexity matching</a><p class="word_list"> <a href="#Macaulay duration">Macaulay duration</a><p class="word_list"> <a href="#modified duration">modified duration</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <p class="body">D<strong style="font-weight: 400">uration and convexity are factor sensitivities that describe exposure to parallel shifts in the <a href="../articles/fixed_income_term_structure.htm">spot curve</a>. They can be applied to individual fixed income instruments or to entire fixed income portfolios.</strong></p> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="90%" id="table5"> <tr> <td width="5%"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table6"> <tr> <td width="160"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script> </td> <td width="11"> </td> </tr> </table> </td> <td width="95%" align="center"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="275" cellpadding="0" id="table7"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Changes in the Spot Curve</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> <p style="margin-left: 5; margin-top: 6"> <img border="0" src="../images/ex1_gap_analysis.gif" width="250" height="470"></td> </tr> <tr> <td> <p class="exhibit_legend">Duration assesses exposure to parallel shifts in the spot curve. It cannot warn of exposure to more complex movements in the spot curve, including tilts and bends.</td> </tr> </table> </td> </tr> </table> </div> <p class="body"><strong style="font-weight: 400">The idea behind duration is simple. Suppose a portfolio has a duration of 3 years. Then that portfolio's value will decline about 3% for each 1% increase in interest ratesor rise about 3% for each 1% decrease in interest rates. Such a portfolio is less risky than one which has a 10-year duration. That portfolio is going to decline in value about 10% for each 1% rise in interest rates. Convexity provides additional risk information.</strong></p> <p class="body"><strong style="font-weight: 400">Exhibit 2 illustrates how the price of a fixed income portfolio might respond to parallel shifts in the spot <a name="Exhibit 1">curve</a>. </strong></p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table8"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Example: Sensitivity of a Fixed Income Portfolio to Interest Rates</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/ex1_duration.gif" width="350" height="200"></td> </tr> <tr> <td> <p class="exhibit_legend">The fractional change in a fixed income portfolio's value is graphed as a function of parallel shift in the spot curve.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="left" id="table9"> <tr> <td width="300" height="14"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14"></td> </tr> <tr> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> <p class="body">Here, <img border="0" src="../formulas/duration%20_delta_0_r.gif" align="absbottom" width="23" height="18"> represents an immediate parallel shift in interest rates (see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation). For example, <img border="0" src="../formulas/duration%20_delta_0_r.gif" align="absbottom" width="23" height="18"> = .015 corresponds to a 1.5% (or 150 basis point) parallel rise in the spot curve. The variable <img border="0" src="../formulas/duration_delta_0_p.gif" align="absbottom" width="26" height="18"> is the dollar change in the portfolio's value corresponding to the shift in interest rates. Accordingly, <img border="0" src="../formulas/duration_9iurhjt4un.gif" align="absbottom" width="51" height="19"> is the fractional change in the portfolio's value. Note that we use preceding superscripts 0 to indicate quantities are as of the current time 0—shifts in the spot curve are considered instantaneous.</p> <p class="body">Exhibit 2 fully describes the portfolio's sensitivity to parallel shifts in the spot curve. There is no more information that we could add to the picture. What we try to do with duration and convexity is summarize the entire picture of Exhibit 2 with just two numbers. Certainly, two numbers can not describe the wealth of detail contained in a picture, so what we do is take the two most important pieces of information in the picture. Those two pieces of information are duration and convexity. </p> <p class="body">Let's start with duration. The most significant information Exhibit 2 provides us about this particular portfolio is the fact that its value will decline if interest rates rise—and rise if interest rates fall. This is the information that duration conveys, along with the magnitude of such sensitivity. </p> <p class="body">If we fit a tangent line to the curve in Exhibit 2, it will capture the direction and magnitude of the portfolio's sensitivity to interest rates. For small changes in interest rates, the line and the curve almost overlap.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table10"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Duration</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> <img border="0" src="../images/ex2_duration.gif" width="300" height="171"></td> </tr> <tr> <td> <p class="exhibit_legend">A tangent line is fit to the curve of Exhibit 2. Duration is the slope of the curve multiplied by minus one.</td> </tr> </table> </center> </div> <p class="body">Duration is defined to be the slope of that tangent line, multiplied by negative one. For example, in Exhibit 3, the slope of the tangent line is 2.5 (for each .01 shift in <img border="0" src="../formulas/duration%20_delta_0_r.gif" align="absbottom" width="23" height="18">, <img border="0" src="../formulas/duration_9iurhjt4un.gif" align="absbottom" width="51" height="19"> shifts about –.025). The portfolio's duration is 2.5 years.</p> <p class="body">Tangent lines are the province of calculus, so we turn to calculus for the formal definition. <a name="[1]"> <b>Duration</b></a> is a weighted partial derivative:</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/duration_1.gif" width="144" height="54"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p>This leads to the <a name="[2]">approximation</a></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/duration_2.gif" width="171" height="27"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">For example, suppose a portfolio has a duration of 5 years. That portfolio will appreciate about 5% for each 1% decline in rates. It will depreciated about 5% for each 1% rise in rates. It is as simple as that. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table13"> <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">Suppose a portfolio has a duration of –2 years. The portfolio's value will rise about 2% for every 1% rise in rates. It will decline about 2% for each 1% decline in rates. </p> <p class="body">Approximation [<a href="#[2]">2</a>] is the primary reason people use duration. With a single number, it summarizes a <a href="../articles/coupon_bond.htm">bond</a> or a portfolio's sensitivity to changes in interest rates. </p> <p class="body">Typically, a bond's duration will be positive. However, instruments such as <a href="../articles/io_and_po.htm">IO</a> mortgage backed securities have negative durations. You can also achieve a negative duration by shorting fixed income instruments or paying fixed for floating on an interest rate <a href="../articles/swap.htm">swap</a>. <a href="../articles/inverse_floater.htm">Inverse floaters</a> tend to have large positive durations. Their values change significantly for small changes in rates. Highly leveraged fixed-income portfolios tend to have very large (positive or negative) durations. </p> <p class="body">For portfolios whose cash flows are all fixed (for example, a portfolio of <a href="../articles/callable_bond.htm">non-callable</a> bonds) there is a particularly simple way to calculate duration. For such portfolios, duration is just the average maturity of the of the cash flows. Specifically, assume a portfolio has fixed cash flows <img border="0" src="../formulas/duration_and_convexity_c_i.gif" align="absbottom" width="10" height="11">, each occurring at some time <font face="Symbol"> <img border="0" src="../formulas/duration_and_convexity_tau_i.gif" align="absbottom" width="12" height="12"></font> years from time 0. Let <sup>0</sup><i>pv</i>(<img border="0" src="../formulas/duration_and_convexity_c_i.gif" align="absbottom" width="10" height="11">) denote the present value at time 0 of the cash flow <img border="0" src="../formulas/duration_and_convexity_c_i.gif" align="absbottom" width="10" height="11">, then the duration <a name="[3]">is</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table14"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/duration_and_convexity_3.gif" width="225" height="86"></td> <td width="5%" align="right">[3]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table15"> <tr> <td width="336" height="10"><font size="2"> </font></td> <td width="9" height="10"></td> </tr> <tr> <td width="336">  <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> <td width="9"> </td> </tr> </table> <p class="body">Now the name "duration" should make more sense, as should the fact that duration is measured in years! When duration is calculated in this way, it is called <b><a name="Macaulay duration">Macaulay duration</a></b>. One caveat: the Macaulay formula for duration is correct only if interest rates are <a href="../articles/compounding.htm">continuously compounded</a>. </p> <p class="body">Take, for example, a 5-year <a href="../articles/zero_coupon_bond.htm">zero-coupon note</a>. Because it pays no <a href="../articles/coupon_bond.htm">coupons</a>, its average maturity is precisely 5 years. Hence, based on the Macaulay formula for duration, the bond's duration will be 5 years. This means that a 5-year zero will appreciate about 5% in value for each 1% decline in continuously compounded interest rates based on approximation [<a href="#[2]">2</a>]. </p> <p class="body">In formula [<a href="#[3]">3</a>], all present values should be calculated using the <a href="../articles/fixed_income_term_structure.htm">spot interest rate</a> for the maturity of the cash flow it is discounting. In practice, people often calculate all present values with a non-continuously compounded <a href="../articles/yield.htm">yield to maturity</a><sup> 0</sup><i>y</i><sup> </sup>for the entire portfolio. If this is done, formula [<a href="#[3]">3</a>] must be modified slightly. It becomes</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table16"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/duration_and_convexity_4.gif" width="299" height="85"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">where <i>m</i> is the frequency of compounding for the yield to maturity. For example, if the yield to maturity is compounded quarterly, <i>m</i> = 4. This formula is called <b><a name="modified duration">modified duration</a></b>.</p> <p class="body">For portfolios containing instruments that do not pay fixed cash flows, such as callable bonds, mortgage-backed securities or interest rate caps, the Macaulay or modified formulas for duration will not work. For these portfolios, other means must be employed for calculating duration. </p> <p class="body">Now let's consider convexity. If duration summarized the most significant piece of information about a bond or a portfolio's sensitivity to interest rates, convexity summarizes the second-most significant piece of information. Duration captured the fact that the graph in <a href="#Exhibit 1">Exhibit 2</a> was downward sloping. It did not, however, capture its upward curvature. Convexity describes curvature. </p> <p class="body">Exhibit 4 shows the best-fit parabola for the graph of <a name="Exhibit 3" href="#Exhibit 1">Exhibit 2</a>: </p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table17"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Convexity</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/ex3_duration.gif" width="300" height="171"></td> </tr> <tr> <td> <p class="exhibit_legend">Convexity reflects curvature. Here a "best fit" parabola is fit to the graph of <a href="#Exhibit 1"> Exhibit 2</a>.</td> </tr> </table> </center> </div> <p class="body">Note that the best-fit parabola does not exactly overlay the curve in <a href="#Exhibit 3">Exhibit 4</a> because the curve is not itself a parabola. In general, the best-fit parabola will have the form</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table18"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/duration_and_convexity_5.gif" width="197" height="41"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p>where <b><a name="convexity">convexity</a></b> is defined as a weighted second partial derivative</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table19"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/duration_and_convexity_6.gif" width="137" height="49"></td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">The best fit parabola is simply a second order <a href="../articles/taylor_series.htm">Taylor polynomial</a>. Our first-order approximation [<a href="#[2]">2</a>] now becomes a second-order approximation:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table20"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/duration_and_convexity_7.gif" width="273" height="41"></td> <td width="5%" align="right">[7]</td> </tr> </table> <p class="body">The thing to remember about convexity is that it is a metric of curvature. In Exhibit 4, the curvature of the graph bends upward (like a bowl). The convexity is positive. If the curvature bends downward (like an inverted bowl), the convexity is negative.</p> <p class="body">Duration and convexity have traditionally been used as tools for <a href="../articles/asset_liability_management.htm">asset-liability management</a>. To avoid exposure to parallel spot curve shifts, an organization (such as an insurance company or defined benefit pension plan) with significant fixed income exposures might structure its assets so that their duration matches the duration of its liabilities—so the two offset. This technique is called <b><a name="duration matching">duration matching</a></b>. Even more effective (but less frequently practical) is <b><a name="duration-convexity matching">duration-convexity matching</a></b>, in which assets are structured so that durations and convexities match.</p> <p class="body">In closing, it is worth mentioning that terminology associated with the notion of duration is non-standardized. Different people will use terms in different ways. This is due to the history of the notion duration. Macaulay (<a href="#Frederick R. Macaulay">1938</a>) first introduced the notion of duration as simply weighted average maturity. To him, "duration" was what we now call Macaulay duration. Later, people realized that, if Macaulay duration was calculated using continuously compounded interest rates, the result equaled the factor sensitivity [<a href="#[1]">1</a>] that is called "duration" in this article. Because people didn't typically think in terms of continuously compounded rates, this lead to the modification of Macaulay's formula, which is now called "modified duration." Other terms, including "effective duration" and "option-adjusted duration" are also used. 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