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264 lines
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<h1 align="center">Compound Interest Formula Derivations</h1>
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<p align="center"><i>Showing how the formulas are worked out, with Examples!</i></p>
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<p>With <a href="compound-interest.html">Compound Interest</a> we 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 " /></div>
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<h2>Make A Formula</h2>
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<p>Let's look at the first year to begin with:</p>
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<p align="center">$1,000.00 + ($1,000.00 × 10%) = <b>$1,100.00</b></p>
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<p>We can rearrange it like this: </p>
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<p class="center"><img src="images/interest-compound1.svg" alt="interest compound step-by-step" /></p>
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<p class="large" align="center">So, adding 10% interest is the same as multiplying by 1.10</p>
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<p><i>(Note: the Interest Rate was turned into a decimal by dividing by 100: <b>10% = 10/100 = 0.10</b>, read <a href="../percentage.html">Percentages</a> to learn more.)</i></p>
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<h3>And that formula works for any year:</h3>
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<ul>
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<li>We could do the next year like this: <b>$1,100 × 1.10 = $1,210</b></li>
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<li>And then continue to the following year: <b>$1,210 × 1.10 = $1,331</b></li>
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<li>etc...</li>
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</ul>
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<p>So it 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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<h3>In fact we could go straight from the start to Year 5 if we<b> multiply 5 times</b>:<br />
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</h3>
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<p align="center" class="larger">$1,000 × 1.10 × 1.10 × 1.10 × 1.10 × 1.10 = <b>$1,610.51</b></p>
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<p>But it is easier to write down a series of multiplies using <a href="../exponent.html">Exponents (or Powers)</a> like this:</p>
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<div align="center"><img src="images/fv-example.gif" width="281" height="42" alt="$1000 x 1.10^5 = $1610.51" /> </div>
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<h2>The Formula</h2>
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<p>We have been using a real example, but let us make it more general by <b>using letters instead of numbers</b>, like this:</p>
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<p align="center"><img src="images/fv-formula.svg" alt="PV x (1+r)^n = FV" />
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<br> </p>
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<p>(Compare this to the calculation above it: PV = $1,000, r = 0.10, n = 5, and FV = $1,610.51)</p>
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<ul>
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<li>When the interest rate is annual, then <b>n</b> is the number of years</li>
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<li>When the interest rate is monthly, then <b>n</b> is the number of months</li>
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<li>and so on</li>
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</ul>
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<p> </p>
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<h3>Examples</h3>
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<p>How about some examples ...
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<br /> ... what if the loan went for <b>15 Years</b>? ... just change the "n" value:</p>
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<p align="center"><img src="images/fv-example2.gif" width="281" height="41" alt="$1000 x 1.10^15 = $4177.25" /></p>
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<p>... and what if the loan was for 5 years, but the interest rate was only 6%? Here:</p>
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<p align="center"><img src="images/fv-example3.gif" width="281" height="41" alt="$1000 x 1.06^5 = $1338.23" /></p>
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<p>(Note that it is <b>1.06</b>, not 1.6)</p>
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<h2>The Four Formulas</h2>
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<p>So, the basic formula for Compound Interest is:</p>
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<p class="large center">FV = PV (1+r)<sup>n</sup></p>
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<ul>
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<li>FV = Future Value, </li>
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<li>PV = Present Value, </li>
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<li>r = Interest Rate (as a decimal value), and </li>
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<li>n = Number of Periods</li>
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</ul>
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<p>With that we can work out the Future Value <b>FV</b> when we know the Present Value <b>PV</b>, the Interest Rate <b>r</b> and Number of Periods <b>n</b></p>
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<p>And we can <b>rearrange</b> that formula to find FV, the Interest Rate or the Number of Periods when we know the other three.</p>
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<p>Here are all four furmulas:</p>
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<table border="0" align="center">
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<tr>
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<td align="center" class="large"><span class="large center">FV = PV (1+r)<sup>n</sup></span></td>
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<td> </td>
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<td>Find the <b>Future Value</b> when we know a Present Value, the Interest Rate and number of Periods.</td>
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</tr>
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<tr>
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<td align="center" class="large"> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td width="240" align="center" class="large">PV = FV / (1+r)<sup>n</sup></td>
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<td> </td>
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<td>Find the <b>Present Value</b> when we know a Future Value, the Interest Rate and number of Periods.</td>
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</tr>
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<tr>
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<td width="240" align="center" class="large"> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td width="240" align="center" class="large">r = ( FV / PV )<sup>1/n</sup> - 1</td>
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<td> </td>
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<td>Find the <b>Interest Rate</b> when we know the Present Value, Future Value and number of Periods.</td>
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</tr>
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<tr>
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<td width="240" align="center" class="large"> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td width="240" align="center" class="large">n = <span class="intbl">
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<em>ln(FV / PV)</em>
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<strong>ln(1 + r)</strong>
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</span></td>
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<td> </td>
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<td>Find the number of <b>Periods</b> when we know the Present Value, Future Value and Interest Rate</td>
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</tr>
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</table>
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<br />
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<p align="center" class="larger"> </p>
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<p align="center" class="larger">How did we get those other three formulas? Read On!</p>
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<h2>Working Out the Present Value</h2>
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<div class="example">
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<h3>Example: Sam wants to reach $2,000 in 5 Years at 10% annual interest. How much should Sam start with?</h3> </div>
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<p>In other words, we know a Future Value, and <b>want to know a Present Value</b>.</p>
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<p>We can just rearrange the formula to suit ... dividing both sides by (1+r)<sup>n</sup> to give us:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">FV = PV (1+r)<sup>n</sup></span></div>
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<div class="row"><span class="left">Swap sides:</span><span class="right">PV (1+r)<sup>n</sup> = FV</span></div>
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<div class="row"><span class="left">Divide both sides by (1+r)<sup>n</sup>:</span><span class="right">PV = <span class="intbl">
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<em>FV</em>
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<strong>(1+r)<sup>n</sup></strong>
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</span></span></div>
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</div>
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<p>So now we can calculate the answer:</p>
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<div class="example">
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<h3>Example (continued):</h3>
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<p class="center"><b>PV</b> = $2,000 / (1+0.10)<sup>5</sup> = $2,000 / 1.61051 = <b>$1,241.84</b></p>
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<p>So Sam should start with <b>$1,241.84</b></p>
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</div>
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<p>It works like this:</p>
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<p align="center"><img src="images/pv-vs-fv.svg" alt="pv vs fv" /></p>
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<div class="example">
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<h3><b>Another Example:</b> How much do you need to invest now, to get $10,000 in 10 years at 8% interest rate?</h3>
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<p align="center">PV = $10,000 / (1+0.08)<sup>10</sup> = $10,000 / 2.1589 = <b>$4,631.93</b></p>
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<p>So, <b>$4,631.93</b> invested at 8% for 10 Years grows to $10,000</p>
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</div>
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<h2>Working Out The Interest Rate</h2>
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<div class="example">
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<h3>Example: Sam has only $1,000, and wants it to grow to $2,000 in 5 Years, what interest rate should Sam be looking for?</h3> </div>
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<p>We need a rearrangement of the first formula to work it out:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">FV = PV (1+r)<sup>n</sup></span></div>
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<div class="row"><span class="left">Swap sides:</span><span class="right">PV (1+r)<sup>n</sup> = FV</span></div>
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<div class="row"><span class="left">Divide both sides by PV:</span><span class="right">(1+r)<sup>n</sup> = <span class="intbl">
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<em>FV</em>
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<strong>PV</strong>
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</span></span></div>
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<div class="row"><span class="left">Take <a href="../numbers/nth-root.html">nth root</a> of both sides:</span><span class="right">1+r = ( <span class="intbl">
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<em>FV</em>
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<strong>PV</strong>
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</span> )<sup>1/n</sup></span></div>
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<div class="row"><span class="left">Subtract 1 from both sides:</span><span class="right">r = ( <span class="intbl">
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<em>FV</em>
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<strong>PV</strong>
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</span> )<sup>1/n</sup> − 1</span></div>
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</div>
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<p><i>(Note: to understand the step "take nth root" please read <a href="../algebra/exponent-fractional.html">Fractional Exponents</a>)</i></p>
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<p>The result is:</p>
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<p class="center larger">r = ( FV / PV )<sup>1/n</sup> − 1 </p>
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<p>Now we have the formula, it is just a matter of "plugging in" the values to get the result:</p>
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<div class="example">
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<h3>Example (continued):</h3>
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<p align="center">r = ( $2,000 / $1,000 )<sup>1/5</sup> − 1
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<br> = ( 2 )<sup>0.2</sup> − 1
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<br> = 1.1487 − 1
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<br> = <b>0.1487</b></p>
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<p>And 0.1487 as a percentage is <b>14.87%</b></p>
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<p>So Sam needs <b>14.87%</b> to turn $1,000 into $2,000 in 5 years.</p>
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</div>
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<div class="example">
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<h3><b>Another Example:</b> What interest rate do you need to turn $1,000 into $5,000 in 20 Years?</h3>
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<p align="center">r = ( $5,000 / $1,000 )<sup>1/20</sup> − 1 = ( 5 )<sup>0.05</sup> − 1 = 1.0838 − 1 = <b>0.0838</b></p>
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<p>And 0.0838 as a percentage is <b>8.38%</b>. So 8.38% will turn $1,000 into $5,000 in 20 Years.</p>
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</div>
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<h2>Working Out How Many Periods</h2>
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<div class="example">
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<h3>Example: Sam can only get a 10% interest rate. How many years will it take Sam to get $2,000?</h3> </div>
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<p>When we want to know how many periods it takes to turn $1,000 into $2,000 at 10% interest, we can rearrange the basic formula.</p>
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<p>But we need to use the natural logarithm function <i><b>ln() </b></i>to do it. </p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">FV = PV (1+r)<sup>n</sup></span></div>
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<div class="row"><span class="left">Swap sides:</span><span class="right">PV (1+r)<sup>n</sup> = FV</span></div>
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<div class="row"><span class="left">Divide both sides by PV:</span><span class="right">(1+r)<sup>n</sup> = <em>FV / PV</em></span></div>
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<div class="row"><span class="left">Use <a href="../algebra/exponents-logarithms.html">logarithms</a>:</span><span class="right">ln(1+r) × n = ln( <em>FV / PV</em> )</span></div>
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<div class="row"><span class="left">Divide both sides by ln(1+r):</span><span class="right">n = <span class="intbl">
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<em>ln(
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FV / PV )</em>
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<strong>ln(1+r)</strong>
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</span></span></div>
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</div>
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<p><i>(Note: to understand the step "use logarithms" please read <a href="../algebra/exponents-logarithms.html">Working with Exponents and Logarithms</a>).</i></p>
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<p>Now let's "plug in" the values:</p>
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<div class="example">
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<h3>Example (continued):</h3>
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<p align="center">n = ln( $2,000 / $1,000 ) / ln( 1 + 0.10 ) = ln(2)/ln(1.10) = 0.69315/0.09531 = <b>7.27</b></p>
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<p>Magic! It will need <b>7.27 years</b> to turn $1,000 into $2,000 at 10% interest.</p>
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<p>Poor Sam will have to wait over 7 years.</p>
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</div>
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<p> </p>
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<div class="example">
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<h3><b>Another Example:</b> How many years to turn $1,000 into $10,000 at 5% interest?</h3>
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<p align="center">n = ln( $10,000 / $1,000 ) / ln( 1 + 0.05 ) = ln(10)/ln(1.05) = 2.3026/0.04879 = <b>47.19</b></p>
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<p>47 Years! But we are talking about a 10-fold increase, at only 5% interest.</p>
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</div>
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<h2>Conclusion</h2>
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<p>Knowing how the formulas are derived and used makes it easier for you to remember them, and to use them in different situations.</p>
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<h2> </h2>
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<div class="related"><a href="interest.html">Introduction to Interest</a> <a href="investment-graph.html">Investment Graph</a> <a href="compound-interest-calculator.html">Compound Interest Calculator</a> <a href="index.html">Money Index</a></div>
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