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352 lines
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<title>Compound Interest - Periodic Compounding</title>
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<h1 align="center">Compound Interest: Periodic Compounding</h1>
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<p align="center"><i>You may like to read about <a href="compound-interest.html">Compound Interest</a> first.<br />
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You </i><i> can skip straight down to <a href="#periodic">Periodic Compounding</a>.</i></p>
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<h2>Quick Explanation of Compound Interest</h2>
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<p>With <a href="compound-interest.html">Compound Interest</a>, you work out the interest for the first period, add it to the total, and <b>then</b> calculate the interest for the next period, and so on ..., like this:</p>
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<div align="center"><img src="images/interest-compound-flow.svg" alt="interest compound $1000, 10%=$100, $1100, 10%=$110, $1210, 10%=$121, etc " />
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</div><p> </p>
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<p><b>But adding 10% interest is the same as multiplying by 1.10</b> <i>(explained <a href="compound-interest-derivation.html">here</a>)</i></p>
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<p>So it also works like this:</p>
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<p align="center"><img src="images/interest-compound-flow2.svg" alt="interest compound $1000 x1.1 $1100 x1.1 $1210 x1.1 ..." />
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<br /> </p>
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<p> </p>
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<p>In fact we can go from the Start to Year 5 if we<b> multiply 5 times</b> using <a href="../exponent.html">Exponents (or Powers)</a>:
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</p>
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<p align="center"><img src="images/fv-example.gif" width="281" height="42" alt="$1000 x 1.10^5 = $1610.51" /> </p>
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<h2>The Formula</h2>
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<p>This is the formula for Compound Interest (like above but <b>using letters instead of numbers</b>):</p>
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<p align="center"><img src="images/fv-formula.svg" alt="PV x (1+r)^n = FV" /></p>
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<div class="example">
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<h3>Example: $1,000 invested at 10% for 5 Years:</h3>
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<p>Present Value <b>PV = $1,000</b></p>
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<p>Interest Rate is 10%, which as a decimal <b>r = 0.10</b></p>
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<p>Number of Periods <b>n = 5</b></p>
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<div class="so">PV × (1 + r)<sup>n</sup> = FV</div>
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<div class="so">$1,000 × (1 + 0.10)<sup>5</sup> = FV</div>
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<div class="so">$1,000 × 1.10<sup>5</sup> = <b>$1,610.51</b></div>
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<div align="center"></div>
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</div>
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<p>Now we can choose different values, such as an interest rate of 6%: </p>
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<div class="example">
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<h3>Example: $1,000 invested at <span class="hilite">6%</span> for 5 Years:</h3>
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<p>Present Value <b>PV = $1,000</b></p>
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<p>Interest Rate is 6%, which as a decimal <b>r = <span class="hilite">0.06</span></b></p>
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<p>Number of Periods <b>n = 5</b></p>
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<div class="so">PV × (1 + r)<sup>n</sup> = FV</div>
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<div class="so">$1,000 × (1 + <span class="hilite">0.06</span>)<sup>5</sup> = FV</div>
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<div class="so">$1,000 × <span class="hilite">1.06</span><sup>5</sup> = <b>$1,338.23</b></div>
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<div align="center"></div>
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</div>
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<h2><a name="periodic" id="periodic"></a>Periodic Compounding (Within The Year)</h2>
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<p>But sometimes interest is charged Yearly ...</p>
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<p class="center">... but it is calculated more than once within the year, with the interest added each time ... </p>
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<p align="center" class="larger">... so there are compoundings <b>within</b> the Year. </p>
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<div class="example">
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<h3>Example: "10%, Compounded Semiannually"</h3>
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<p>Semiannual means twice a year. So the 10% is split into two: </p>
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<ul>
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<li> 5% halfway through the year, </li>
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<li> and another 5% at the end of the year, </li>
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</ul>
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<p>but each time it is <b>compounded</b> (meaning the interest is added to the total):
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<br /> </p>
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</div>
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<p class="center"><img src="images/semiannual-compounding.svg" alt="semiannual compounding" />
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<br><span class="larger">
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10%, Compounded Semiannually</span>
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<br /> </p>
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<div class="example">
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<p>This results in $1,102.50, which is equal to <b>10.25%</b>, not 10%</p>
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</div>
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<h3>Two Annual Interest Rates?</h3>
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<p>Yes, there are two annual interest rates:</p>
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<div class="beach">
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<table border="0" align="center">
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<tr>
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<td align="center">Example</td>
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<td width="20"> </td>
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<td> </td>
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</tr>
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<tr>
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<td align="center" class="large">10%</td>
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<td> </td>
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<td>The <b>Nominal Rate</b> (the rate they mention)</td>
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</tr>
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<tr>
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<td align="center" class="large">10.25%</td>
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<td> </td>
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<td>The <b>Effective Annual Rate</b> (the rate after compounding)</td>
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</tr>
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</table>
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</div>
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<p>The <b>Effective Annual Rate</b> is what actually gets paid!</p>
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<p>When interest is compounded <b>within</b> the year, the Effective Annual Rate is <b>higher</b> than the rate mentioned. </p>
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<p>How much higher depends on the interest rate, and how many times it is compounded within the year.</p>
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<h3>Working It Out</h3>
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<p>Let's come up with a formula to work out the <b>Effective Annual Rate</b> if we know:</p>
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<ul>
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<li>the rate mentioned (the <b>Nominal Rate, "r"</b>)</li>
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<li>how many times it is compounded (<b>"n"</b>)</li>
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</ul>
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<p>Our task is to take an interest rate (like 10%) and chop it up into "n" periods, compounding each time.</p>
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<p>From the Compound Interest formula (shown above) we can compound "n" periods using</p>
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<p class="large" align="center"> FV = PV (1+r)<sup>n</sup> </p>
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<p>But the interest rate won't be "r", because it has to be chopped into "n" periods like this:</p>
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<p align="center" class="large">r / n</p>
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<p>So we change the compounding formula into:</p>
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<div class="def">
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<p>This is the formula for Periodic Compounding:</p>
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<p align="center"><span class="large">FV = PV (1+(r/n))<sup>n</sup></span> </p>
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<p align="center">where <b>FV</b> = Future Value
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<br /> <b>PV</b> = Present Value
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<br /> <b>r</b> = annual interest rate
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<br />
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<b>n</b> = number of periods within the year</p>
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</div>
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<p>Let's try it on our "10%, Compounded Semiannually" example:</p>
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<p class="larger" align="center">FV = $1,000 (1+(0.10/2))<sup>2</sup> = $1,000(1.05)<sup>2</sup> = $1,000 <b>×</b> 1.1025 = $1,102.50</p>
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<p>That worked! But we want to know what the new <b>interest rate</b> is, we don't want the dollar values in there, so let's remove them:</p>
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<p align="center" class="larger">(1+(r/n))<sup>n</sup> = (1.05)<sup>2</sup> = 1.<b>1025</b></p>
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<p>That has the interest rate in there (0.1025 = 10.25%), but we should subtract the extra 1:</p>
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<p align="center" class="larger">(1+(r/n))<sup>n</sup> − 1 = 0.1025 = <b>10.25%</b></p>
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<p>And so the formula is:</p>
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<p class="large" align="center">Effective Annual Rate = (1+(r/n))<sup>n</sup> − 1</p>
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<br />
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<div class="example">
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<h3>Example: what rate do you get when the ad says "6% compounded monthly"?</h3>
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<p><b>r</b> = 0.06 (which is 6% as a decimal)
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<br /> <b>n</b> = 12</p>
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<p class="center">Effective Annual Rate = (1+(r/n))<sup>n</sup> − 1 </p>
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<p class="center">= (1+(0.06/12))<sup>12</sup> − 1 </p>
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<p class="center">= (1.005)<sup>12</sup> − 1 = 0.06168 = <b>6.168%</b></p>
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<p>So you actually get 6.168%</p>
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</div>
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<div class="example">
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<h3>Example: 7% interest, compounded 4 times a year.</h3>
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<p><b>r</b> = 0.07 (which is 7% as a decimal)
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<br /> <b>n</b> = 4</p>
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<p>So:</p>
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<p align="center"><span class="larger">FV = PV (1+(0.07/4))<sup>4</sup></span> </p>
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<p align="center"><span class="larger">FV = PV (1+(0.07/4))<sup>4</sup></span></p>
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<p align="center"><span class="larger">FV = PV (1.0719...)</span></p>
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<p>The effective annual rate is <b>7.19%</b></p>
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</div>
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<p> </p>
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<p>So remember:</p>
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<div class="beach">
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<table width="77%" border="0" align="center">
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<tr>
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<td width="56%" align="right">Chop the interest rate into "n" periods</td>
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<th width="44%" align="center"><span class="large">r / n</span></th>
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</tr>
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<tr>
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<td width="56%" align="right"> </td>
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<th width="44%" class="large" align="center"> </th>
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</tr>
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<tr>
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<td width="56%" align="right">Compound that "n" times:</td>
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<th width="44%" align="center"><span class="large">(1+(r/n))<sup>n</sup></span></th>
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</tr>
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<tr>
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<td width="56%" align="right"> </td>
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<th width="44%" align="center"> </th>
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</tr>
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<tr>
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<td width="56%" align="right">Don't forget to subtract the "1"</td>
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<th width="44%" align="center"><span class="large">(1+(r/n))<sup>n</sup> − 1</span></th>
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</tr>
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</table>
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</div>
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<p> </p>
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<h2>Table of Values</h2>
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<div class="simple">
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<p>Here are some example values. Notice that compounding has a very small effect when the interest rate is small, but a large effect for high interest rates.</p>
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<table width="50%" border="0" align="center">
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<tr align="right">
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<th width="25">Compounding</th>
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<th width="25">Periods</th>
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<th width="25"> </th>
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<th width="25">1.00%</th>
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<th width="25">5.00%</th>
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<th width="25">10.00%</th>
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<th width="25">20.00%</th>
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<th width="25">100.00%</th>
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</tr>
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<tr align="right">
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<th>Yearly</th>
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<td align="center">1</td>
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<td> </td>
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<td>1.00%</td>
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<td>5.00%</td>
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<td>10.00%</td>
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<td>20.00%</td>
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<td>100.00%</td>
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</tr>
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<tr align="right">
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<th>Semiannually</th>
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<td align="center">2</td>
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<td> </td>
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<td>1.00%</td>
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<td>5.06%</td>
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<td>10.25%</td>
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<td>21.00%</td>
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<td>125.00%</td>
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</tr>
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<tr align="right">
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<th>Quarterly</th>
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<td align="center">4</td>
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<td> </td>
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<td>1.00%</td>
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<td>5.09%</td>
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<td>10.38%</td>
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<td>21.55%</td>
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<td>144.14%</td>
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</tr>
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<tr align="right">
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<th>Monthly</th>
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<td align="center">12</td>
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<td> </td>
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<td>1.00%</td>
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<td>5.12%</td>
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<td>10.47%</td>
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<td>21.94%</td>
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<td>161.30%</td>
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</tr>
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<tr align="right">
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<th>Daily</th>
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<td align="center">365</td>
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<td> </td>
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<td>1.01%</td>
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<td>5.13%</td>
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<td>10.52%</td>
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<td>22.13%</td>
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<td>171.46%</td>
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</tr>
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<tr align="right">
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<th>...</th>
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<td align="center">...</td>
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<td> </td>
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<td> </td>
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<td> </td>
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr align="right">
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<th>Continuously</th>
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<td align="center">Infinite</td>
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<td> </td>
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<td>1.01%</td>
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<td>5.13%</td>
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<td>10.52%</td>
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<td>22.14%</td>
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<td>171.83%</td>
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</tr>
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</table>
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</div>
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<h3>Continuously? </h3>
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<p>Yes, if you have smaller and smaller periods (hourly, minutely, etc) you eventually reach a limit, and we even have a formula for it:</p>
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<p align="center"><img src="images/continuous-compounding.gif" width="89" height="42" alt="e^r - 1" /></p>
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<p align="center">Continuous Compounding Formula
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<br /> <i>Note:<b> e</b>=2.71828..., which is <a href="../numbers/e-eulers-number.html">Euler's number</a>.</i></p>
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<div class="example">
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<h3>Example:Continuous Compounding for 20%</h3>
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<p class="center"><i><b>e</b></i><sup>0.20</sup> − 1 = 1.2214... − 1 = <b>0.2214...</b> </p>
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<p>Or about <b>22.14%</b></p>
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</div>
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<h2>Using It</h2>
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|
<p>Now that you can calculate the Effective Annual Rate (for specific periods, or continuous), we can use it in any normal <a href="compound-interest.html">compound interest</a> calculations.</p>
|
|
<div class="example">
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|
<h3>Example: Continuous Compounding of $10,000 for 2 years at 8%</h3>
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|
<p> Continuous Compounding for 8% is: <i><b>e</b></i><sup>0.08</sup> − 1 = 1.08329... − 1 = <b>0.08329...</b></p>
|
|
<p>That is about <b>8.329</b>%</p>
|
|
<p>Over 2 years (see <i><a href="compound-interest.html">Compound Interest</a></i>) we get:</p>
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|
<p align="center" class="larger">FV = PV × (1+r)<sup>n</sup></p>
|
|
<p align="center" class="larger">FV = $10,000 × (1+0.08329)<sup>2</sup></p>
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|
<p align="center" class="larger">FV = $10,000 × 1.173511... = $11,735.11</p>
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|
</div>
|
|
<p> </p>
|
|
<h2>Summary</h2>
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|
<p class="center large">Effective Annual Rate = (1+(r/n))<sup>n</sup> − 1</p>
|
|
<p>Where:</p>
|
|
<ul>
|
|
<li>r = <b>Nominal Rate</b> (the rate they mention)</li>
|
|
<li>n = number of periods that are compounded (example: monthly=12)</li>
|
|
</ul>
|
|
<p> </p>
|
|
<div class="questions">
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<script type="text/javascript">
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getQ(3752, 3753, 3754, 3755, 3756, 3757, 3758, 3759, 3760, 3761);
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</script> </div>
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<div class="related"><a href="interest.html">Introduction to Interest</a> <a href="compound-interest-calculator.html">Compound Interest Calculator</a> <a href="investment-graph.html">Investment Graph</a> <a href="compound-interest.html">Compound Interest</a></div>
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