Compounding

<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">Compounding</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="99%" align="left" valign="top"><p class="word_list"> <a href="#annual compounding">annual compounding</a><p class="word_list"> <a href="#compound interest">compound interest</a><p class="word_list"> <a href="#continuous compounding">continuously compound interest</a><p class="word_list"> <a href="#monthly compounding">monthly compounding</a><p class="word_list"> <a href="#quarterly compounding">quarterly compounding</a><p class="word_list"> <a href="#semiannual compounding">semiannual compounding</a><p class="word_list"> <a href="#simple interest">simple interest</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="left" id="table5"> <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">When money is placed on deposit to earn interest, the interest can be paid out periodically as it is earned, or it can be left on deposit. If interest left on deposit does not itself earn interest, the deposit is said to earn <b> <a name="simple interest">simple interest</a></b>. </p> <p class="body">Measure time in years. With simple interest, at any time <i>t</i>, the value of the deposit is given by the <a name="[1]">product</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <i> <img border="0" src="../formulas/compounding_[1].gif" align="absbottom" width="25" height="14"></i></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">Where <i>a</i> is the amount of the initial deposit, and <img border="0" src="../formulas/compounding_rs.gif" align="absbottom" width="9" height="11"> is the simple rate of interest. For example, if <a href="../articles/currency_codes.htm">USD</a> 100 is left on deposit to earn an <img border="0" src="../formulas/compounding_rs.gif" align="absbottom" width="9" height="11"> = .06 rate of simple interest, at the end three years, the deposit will be worth</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> 100 (.06) (3) = 118</td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">Formula [<a href="#[1]">1</a>] means that, when a deposit earns simple interest, its value grows linearly with time.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="250" cellpadding="0" id="table8"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Simple Interest</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> <img border="0" src="../images/ex1_compounding.gif" width="200" height="130"></td> </tr> <tr> <td> <p class="exhibit_legend">The value of a deposit earning simple interest grows linearly over time.</td> </tr> </table> </center> </div> <p class="body">In practice, simple interest is rarely used for deposits held more than a year. An alternative is to credit interest, not based upon the initial value of the deposit, but based on its accumulated value. This approach is called <b><a name="compound interest">compound interest</a></b>. With it, interest is earned on both the initial deposit and on any interest that has already been earned but left on deposit—interest is earned on interest. At any time <i>t</i>, the value of the deposit is given <a name="[3]">by</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/compounding_3.gif" width="78" height="51"></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="right" id="table10"> <tr> <td align="right" width="9" height="10"></td> <td width="336" height="10"> <font size="2"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body" style="margin-bottom: 24">where <i>n</i> is the compounding frequencythe number of times per year that interest is credited. The constant<i> r<sub>n</sub> </i>is the interest rate. Typical values for <i>n</i> include</p> <p class="body" style="margin-top: 7; margin-bottom: 7"> <img border="0" src="../images/bullet.gif" width="20" height="10">1 for <b> <a name="annual compounding">annual compounding</a></b>,</p> <p class="body" style="margin-top: 7; margin-bottom: 7"> <img border="0" src="../images/bullet.gif" width="20" height="10">2 for <b> <a name="semiannual compounding">semiannual compounding</a></b>, </p> <p class="body" style="margin-top: 7; margin-bottom: 7"> <img border="0" src="../images/bullet.gif" width="20" height="10">4 for <b> <a name="quarterly compounding">quarterly compounding</a></b>, and</p> <p class="body" style="margin-top: 7; margin-bottom: 7"> <img border="0" src="../images/bullet.gif" width="20" height="10">12 for <b> <a name="monthly compounding">monthly compounding</a></b>.</p> <p class="body" style="margin-top: 24">For example, if USD 100 is left on deposit to earn an <i>r</i><sub>2</sub> = .06 rate of semiannually compounded interest, at the end three years, the deposit will be worth</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/compounding_4.gif" width="170" height="51"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">Formula [<a href="#[3]">3</a>] means that, when a deposit earns compound interest, its value grows exponentially with time.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="250" cellpadding="0" id="table12"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Compound Interest</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_compounding.gif" width="200" height="130"></td> </tr> <tr> <td> <p class="exhibit_legend">The value of a deposit earning compound interest grows exponentially over time.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table13"> <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">With compounding, larger values of <i>n</i> correspond to interest being credited more and more frequently. The limiting case of this is called <b><a name="continuous compounding">continuous compounding</a></b> where interest is credited on a continuous basis. The distinction is like the difference between getting water from a hand pump and getting water from a faucet. With the hand pump, the water flow is broken. With the faucet, it is continuous. The faucet does not necessarily deliver water any faster than the pump. It just delivers it continuously. </p> <p class="body">With continuous compounding, at any time <i>t</i>, the value of a deposit is given <a name="[3]">by</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table14"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/compounding.gif" width="36" height="25"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">where <i>r<sub>c</sub></i> is the continuously compounded interest rate and <i>e</i> =2.71828182845... </p> <p class="body">For example, if USD 100 is left on deposit to earn an <i>r<sub>c</sub></i> = .06 rate of continuously compounded interest, at the end three years, the deposit will be worth</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"> <img border="0" src="../formulas/compounding_6.gif" width="129" height="25"></td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">Interest is rarely compounded continuously in practice. Continuous compounding is more of a theoretical notion. It is used frequently in theoretical finance because it simplifies many calculations.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <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="table17"> <tr> <td width="100%"> <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"><a href="../articles/bond_accrued_interest.htm">bond accrued interest</a> Interest that is earned but not yet paid on a bond.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/duration_and_convexity.htm">duration and convexity</a> Risk metrics employed in fixed income markets.<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 class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/interest_rate_parity.htm">interest rate parity</a> An arbitrage condition that must hold between the spot interest rates of different currencies.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/interest_rate_spreads.htm">interest rate spreads</a> Discusses credit spreads, liquidity spreads, optionality spreads, etc. in the fixed income markets.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/return.htm">return</a> Any of a number of metrics for the change in an asset's or portfolio's accumulated value.<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><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/zero_coupon_bond.htm">zero-coupon bond</a> A bond that pays no coupons, pays its par value at maturity and is issued at a discount.</td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Sponsored Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" width="75"> <img border="0" src="../images/ad_side_banner_1.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </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="table18"> <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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