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434 lines
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434 lines
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<title>Compound Interest</title>
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<meta name="description" content="With Compound Interest, you work out the interest for the first period, add it to the total, and then calculate the interest for the next period" />
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Compound Interest</h1>
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<p align="center"><i>You may wish to read <a href="interest.html">Introduction to Interest</a> first</i></p>
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<p>With Compound Interest, 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 " style="max-width:100%" /></div>
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<p>It grows faster and faster like this:</p>
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<p class="center"><img src="images/compound-graph-10.svg" alt="compound graph 7 years at 10%" /></p>
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<p>Here are the calculations for 5 Years at 10%:</p>
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<div class="beach">
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<table width="90%" border="0" align="center">
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<tr>
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<th width="18%" height="29">
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<div align="center">Year</div> </th>
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<th width="18%" height="29">
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<div align="center">Loan at Start</div> </th>
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<th width="41%" height="29">
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<div align="center">Interest</div> </th>
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<th width="23%" height="29">
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<div align="center">Loan at End</div> </th>
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</tr>
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<tr>
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<td width="18%">
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<div align="center">0 (Now)</div> </td>
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<td width="18%">
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<div align="center">$1,000.00</div> </td>
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<td width="41%">
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<div align="center">($1,000.00 × 10% = ) <b>$100.00</b></div> </td>
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<td width="23%">
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<div align="center">$1,100.00</div> </td>
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</tr>
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<tr>
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<td width="18%">
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<div align="center">1</div> </td>
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<td width="18%">
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<div align="center">$1,100.00</div> </td>
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<td width="41%">
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<div align="center">($1,100.00 × 10% = ) <b>$110.00</b></div> </td>
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<td width="23%">
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<div align="center">$1,210.00</div> </td>
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</tr>
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<tr>
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<td width="18%">
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<div align="center">2</div> </td>
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<td width="18%">
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<div align="center">$1,210.00</div> </td>
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<td width="41%">
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<div align="center">($1,210.00 × 10% = ) <b>$121.00</b></div> </td>
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<td width="23%">
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<div align="center">$1,331.00</div> </td>
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</tr>
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<tr>
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<td width="18%">
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<div align="center">3</div> </td>
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<td width="18%">
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<div align="center">$1,331.00</div> </td>
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<td width="41%">
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<div align="center">($1,331.00 × 10% = ) <b>$133.10</b></div> </td>
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<td width="23%">
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<div align="center">$1,464.10</div> </td>
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</tr>
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<tr>
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<td width="18%">
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<div align="center">4</div> </td>
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<td width="18%">
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<div align="center">$1,464.10</div> </td>
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<td width="41%">
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<div align="center">($1,464.10 × 10% = ) <b>$146.41</b></div> </td>
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<td width="23%">
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<div align="center">$1,610.51</div> </td>
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</tr>
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<tr>
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<td width="18%">
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<div align="center">5</div> </td>
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<td width="18%">
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<div align="center">$1,610.51</div> </td>
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<td width="41%">
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<div align="center"></div> </td>
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<td width="23%">
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<div align="center"></div> </td>
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</tr>
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</table>
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</div>
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<p>Those calculations are done one step at a time:</p>
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<ol>
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<li>Calculate the Interest (= "Loan at Start" × Interest Rate)</li>
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<li>Add the Interest to the "Loan at Start" to get the "Loan at End" of the year</li>
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<li>The "Loan at End" of the year is the "Loan at Start" of the <b>next</b> year</li>
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</ol>
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<p>A simple job, with lots of calculations. </p>
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<p class="large">But there are quicker ways, using some clever mathematics.</p>
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<h2>Make A Formula</h2>
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<p>Let us make a formula for the above ... just looking 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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<br />
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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 class="large" align="center"><img src="images/pct-to-mult.svg" alt="+10% -> x1.10" /></p>
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<table border="0" align="center">
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<tr>
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<td align="right"><i><b> so this:</b></i></td>
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<td align="right"> </td>
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<td align="right">$1,000 + ($1,000 x 10%) = $1,000 + $100 = <b>$1,100</b></td>
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</tr>
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<tr>
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<td align="right"><i><b>is the same as:</b></i></td>
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<td align="right"> </td>
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<td align="right">$1,000 × 1.10 = <b>$1,100</b></td>
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</tr>
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</table>
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<div class="def">
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<p>Note: the Interest Rate was turned into a decimal by dividing by 100:</p>
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<p class="center"> <b>10% = 10/100 = 0.10</b></p>
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<p> Read <a href="../percentage.html">Percentages</a> to learn more, but in practice just move the decimal point 2 places, like this:</p>
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<p class="center">10% → 1.0 → <b>0.10</b></p>
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<p>Or this: </p>
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<p class="center">6% → 0.6 → <b>0.06</b></p>
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</div>
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<p>The result is that we can do a year in one step:</p>
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<ol>
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<li><b> Multiply the "Loan at Start" by (1 + Interest Rate) to get "Loan at End"</b></li>
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</ol>
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<p>Now, here is the magic ...</p>
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<p class="center larger">... the same formula works for any year! </p>
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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><br />
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</li>
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<li>And then continue to the following year: <b>$1,210 × 1.10 = $1,331</b><br />
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</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 ..." style="max-width:100%" /><br />
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</p>
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<p class="larger">In fact we could go from the start straight to Year 5, if we<b> multiply 5 times</b>:
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</p>
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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"></div>
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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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<p align="center">This does all the calculations in the top table in one go.</p>
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<h2>The Formula</h2>
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<p>We have been using a real example, but let's be 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" /></p>
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<p>(This is the same as above, but with PV = $1,000, r = 0.10, n = 5, and FV = $1,610.51)</p>
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<p>Here is is written with "FV" first:</p>
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<div class="def">
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<p align="center"><span class="large">FV = PV × (1+r)<sup>n</sup></span> </p>
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<p align="center">where <b>FV</b> = Future Value<br />
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<b>PV</b> = Present Value<br />
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<b>r</b> = annual interest rate<br />
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<b>n</b> = number of periods</p>
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</div>
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<table width="90%" border="0" align="center">
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<tr>
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<td width="10%"><img src="../images/style/lightbulb.gif" width="75" height="75" alt="lightbulb" /></td>
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<td width="90%"><i>This is the basic formula for Compound Interest. <br />
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<br />
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Remember it, because it is very useful.</i></td>
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</tr>
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</table>
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<h3>Examples</h3>
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<p>How about some examples ... <br />
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...
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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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<table border="0" align="center">
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<tr>
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<td align="right">Did you see how we just put the
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6% into its place like this:</td>
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<td><img src="images/percent-add.svg" alt="6% -> 1.06" /></td>
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</tr>
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</table>
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<p>... and what if the loan was for 20 years at 8%? ... you work it out!</p>
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<h2>Going "Backwards" to Work Out the Present Value</h2>
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<p>Let's say your goal is to have $2,000 in 5 Years. You can get 10%, so <b>how much should you start with</b>?</p>
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<p align="center">In other words, you know a Future Value, and <b>want to know a Present Value</b>.</p>
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<p>We know that <b>multiplying</b> a Present Value (PV) by<span class="large"> </span><span class="larger">(1+r)<sup>n</sup></span> gives us the Future Value (FV), so we can go backwards by <b>dividing</b>, 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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<p>So the Formula is:</p>
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<p class="center large">PV = <span class="intbl"><em>FV</em><strong>(1+r)<sup>n</sup> </strong></span></p>
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<p>And now we can calculate the answer:</p>
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<div class="tbl">
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<div class="row"><span class="left">PV =</span><span class="right"> <span class="intbl"><em>$2,000</em><strong>(1+0.10)<sup>5</sup> </strong></span> </span></div>
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<div class="row"><span class="left">=</span><span class="right"> <span class="intbl"><em>$2,000</em><strong>1.61051</strong></span> </span></div>
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<div class="row"><span class="left">=</span><span class="right"> $1,241.84</span></div>
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</div>
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<p>In other words, $1,241.84 will grow to $2,000 if you invest it at 10% for 5 years.</p>
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<div class="example">
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<p><b>Another Example:</b> How much do you need to invest now, to get $10,000 in 10 years at 8% interest rate?</p>
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<div class="tbl">
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<div class="row"><span class="left">PV =</span><span class="right"> <span class="intbl"><em>$10,000</em><strong>(1+0.08)<sup>10</sup> </strong></span> </span></div>
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<div class="row"><span class="left">=</span><span class="right"> <span class="intbl"><em>$10,000</em><strong>2.1589</strong></span> </span></div>
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<div class="row"><span class="left">=</span><span class="right"> $4,631.93</span></div>
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</div>
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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>Compounding Periods</h2>
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<p>Compound Interest is not always calculated per year, it could be per month, per day, etc. <b>But if it is not per year it should say so!</b></p>
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<div class="example">
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<p>Example: you take out a $1,000 loan for 12 months and it says "<b>1% per month</b>", how much do you pay back?</p>
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<p>Just use the Future Value formula with "n" being the number of months:</p>
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<div class="tbl">
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<div class="row"><span class="left">FV =</span><span class="right"> PV × (1+r)<sup>n</sup> </span></div>
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<div class="row"><span class="left">=</span><span class="right"> $1,000 × (1.01)<sup>12</sup> </span></div>
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<div class="row"><span class="left">=</span><span class="right"> $1,000 × 1.12683 </span></div>
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<div class="row"><span class="left">=</span><span class="right"> <b>$1,126.83</b> to pay back</span></div>
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</div>
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</div>
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<p>And it is also possible to have yearly interest <i>but with several compoundings <b>within</b> the year</i>, which is called <a href="compound-interest-periodic.html">Periodic Compounding</a>.</p>
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<div class="example">
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<p>Example, 6% interest with "<b>monthly compounding</b>" does not mean 6% per month, it means 0.5% per month (6% divided by 12 months), and is worked out like this:</p>
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<div class="tbl">
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<div class="row"><span class="left">FV =</span><span class="right"> PV × (1+r/n)<sup>n</sup> </span></div>
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<div class="row"><span class="left">=</span><span class="right"> $1,000 × (1 + 6%/12)<sup>12</sup> </span></div>
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<div class="row"><span class="left">=</span><span class="right"> $1,000 × (1 + 0.5%)<sup>12</sup> </span> </div>
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<div class="row"><span class="left">=</span><span class="right"> $1,000 × (1.005)<sup>12</sup> </span></div>
|
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<div class="row"><span class="left">=</span><span class="right"> $1,000 × 1.06168... </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> <b>$1,061.68</b> to pay back</span></div>
|
|
</div>
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<p>This is equal to a <b>6.168%</b> <i>($1,000 grew to $1,061.68)</i> for the whole year. </p>
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</div>
|
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<p>So be careful to understand what is meant!</p>
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<h2>APR</h2>
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|
<p style="float:right; margin: 0 0 12px 25px;"><img src="images/home-load-ad.gif" width="122" height="89" alt="home loan ad" /><br>
|
|
<i>This ad looks like 6.25%, <br>
|
|
but <b>is really 6.335%</b></i></p>
|
|
<p>Because it is easy for loan ads to be confusing (sometimes on purpose!), the "<b>APR</b>" is often used. </p>
|
|
<p><b>APR</b> means "<i><b>Annual Percentage Rate</b></i>": it shows how much you will actually be paying for the year (including compounding, fees, etc).</p>
|
|
<p>Here are some examples:</p>
|
|
<div class="example">
|
|
<p>Example 1: "<b>1% per month</b>" actually works out to be <b>12.683% APR</b> (if no fees)<b>.</b></p>
|
|
</div>
|
|
|
|
<div class="example">
|
|
<p>Example 2: "<b>6% interest with monthly compounding</b>" works out to be <b>6.168% </b><b>APR</b> (if no fees)<b>.</b></p>
|
|
</div>
|
|
<p>If you are shopping around, ask for the APR.</p>
|
|
<p> </p>
|
|
|
|
<h2>Break Time!</h2>
|
|
<p>So far we have looked at using <span class="larger">(1+r)<sup>n</sup> </span>to go from a Present Value (PV) to a Future Value (FV) and back again, plus some of the tricky things that can happen to a loan.</p>
|
|
<p>Now is a good time to have a break before we look at two more topics:</p>
|
|
<ul>
|
|
<li>How to work out the <b>Interest Rate</b> if you know PV, FV and the Number of Periods.</li>
|
|
<li>How to work out the <b>Number of Periods</b> if you know PV, FV and the Interest Rate</li>
|
|
</ul>
|
|
<p> </p>
|
|
<h2>Working Out The Interest Rate</h2>
|
|
<p>You can calculate the Interest Rate if you know a Present Value, a Future Value and how many Periods.</p>
|
|
<div class="example">
|
|
<p>Example: you have $1,000, and want it to grow to $2,000 in 5 Years, what <b>interest rate</b> do you need?</p>
|
|
<p>The formula is:</p>
|
|
<p class="large" align="center">r = ( FV / PV )<sup>1/n</sup> - 1</p>
|
|
<p> </p>
|
|
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/calculator-exponent.gif" width="105" height="86" alt="calculator exponent button" /></p>
|
|
<p><i>Note: the little "1/n" is a <a href="../algebra/exponent-fractional.html">Fractional Exponent</a>, first calculate 1/n, then use that as the exponent on your calculator.</i></p>
|
|
<p><i>For example 2<sup>0.2</sup> is entered as <b>2, "x^y", 0, ., 2, =</b></i></p>
|
|
|
|
|
|
<p>Now we can "plug in" the values to get the result:</p>
|
|
|
|
<div class="tbl">
|
|
<div class="row"><span class="left">r =</span><span class="right"> ( $2,000 / $1,000 )<sup>1/5</sup> − 1 </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> (2)<sup>0.2</sup> − 1 </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> 1.1487 − 1 </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> 0.1487</span></div>
|
|
</div>
|
|
|
|
|
|
<p>And 0.1487 as a percentage is <b>14.87%</b>,</p>
|
|
</div>
|
|
<p>So you need <b>14.87%</b> interest rate to turn $1,000 into $2,000 in 5 years.</p>
|
|
<div class="example">
|
|
<p><b>Another Example:</b> What interest rate do you need to turn $1,000 into $5,000 in 20 Years?</p>
|
|
<div class="tbl">
|
|
<div class="row"><span class="left">r =</span><span class="right"> ( $5,000 / $1,000 )<sup>1/20</sup> − 1 </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> (5)<sup>0.05</sup> − 1 </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> 1.0838 − 1 </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> 0.0838</span></div>
|
|
</div>
|
|
<p>And 0.0838 as a percentage is <b>8.38%</b>. </p>
|
|
<p>So <b>8.38%</b> will turn $1,000 into $5,000 in 20 Years.</p>
|
|
</div>
|
|
<h2>Working Out How Many Periods</h2>
|
|
<p>You can calculate how many Periods if you know a Future Value, a Present Value and the Interest Rate. </p>
|
|
<div class="example">
|
|
<h3>Example: you want to know how many periods it will take to turn $1,000 into $2,000 at 10% interest.</h3>
|
|
<p>This is the formula (note: it uses the natural logarithm function <i><b>ln</b></i>):</p>
|
|
<p align="center" class="large">n = <i>ln</i>(FV / PV) / <i>ln</i>(1 + r)</p>
|
|
<table width="95%" border="0" align="center">
|
|
<tr>
|
|
<td><img src="images/calculator-ln.gif" width="107" height="87" alt="calculator ln button" /></td>
|
|
<td width="82%">
|
|
<p><i>The "<i><b>ln"</b></i> function should be on a good calculator. </i></p>
|
|
<p><i>You could also use <i><b>log</b></i>, just don't mix the two.</i></p> </td>
|
|
</tr>
|
|
</table>
|
|
|
|
|
|
|
|
<p>Anyway, let's "plug in" the values:</p>
|
|
<div class="tbl">
|
|
<div class="row"><span class="left">n =</span><span class="right"> ln( $2,000/$1,000 ) / ln( 1 + 0.10 ) </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> ln(2)/ln(1.10) </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> 0.69315/0.09531 </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> 7.27</span></div>
|
|
</div><p>Magic! It will need <b>7.27 years</b> to turn $1,000 into $2,000 at 10% interest.</p>
|
|
</div>
|
|
|
|
|
|
<div class="example">
|
|
<h3>Example: How many years to turn $1,000 into $10,000 at 5% interest?</h3>
|
|
<div class="tbl">
|
|
<div class="row"><span class="left">n =</span><span class="right"> ln( $10,000/$1,000 ) / ln( 1 + 0.05 ) </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> ln(10)/ln(1.05) </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> 2.3026/0.04879 </span></div>
|
|
<div class="row"><span class="left">=</span><span class="right"> 47.19</span></div>
|
|
</div>
|
|
<p>47 Years! But we are talking about a 10-fold increase, at only 5% interest.</p>
|
|
</div><p> </p>
|
|
<table border="0" align="center">
|
|
<tr>
|
|
<td><a href="compound-interest-calculator.html"><img src="images/comp-interest-calc-thumb.gif" width="150" height="142" border="0" alt="compound interest calculator" /></a></td>
|
|
<td> </td>
|
|
<td><h2>Calculator</h2>
|
|
<p>I also made a <a href="compound-interest-calculator.html">Compound Interest Calculator</a> that uses these formulas.</p></td>
|
|
</tr>
|
|
</table>
|
|
|
|
<h2>Summary</h2>
|
|
<p>The basic formula for Compound Interest is:</p>
|
|
<p class="center large">FV = PV (1+r)<sup>n</sup></p>
|
|
<p>Finds the <b>Future Value</b>, where: </p>
|
|
<ul>
|
|
<li>FV = Future Value, </li>
|
|
<li>PV = Present Value, </li>
|
|
<li>r = Interest Rate (as a decimal value), and </li>
|
|
<li>n = Number of Periods</li>
|
|
</ul>
|
|
<p> </p>
|
|
<p>And by rearranging that formula<i> (see <a href="compound-interest-derivation.html">Compound Interest Formula Derivation</a>)</i> we can find any value when we know the other three:</p>
|
|
<p> </p>
|
|
<p class="center large">PV = <span class="intbl"><em>FV</em><strong>(1+r)<sup>n</sup> </strong></span></p>
|
|
<p>Finds the <b>Present Value</b> when you know a Future Value, the Interest Rate and number of Periods.</p>
|
|
<p> </p>
|
|
<p class="center large">r = (FV/PV)<sup>(1/n)</sup> − 1</p>
|
|
<p>Finds the <b>Interest Rate</b> when you know the Present Value, Future Value and number of Periods.</p>
|
|
<p> </p>
|
|
<p class="center large">n = <span class="intbl"><em><i>ln</i>(FV / PV)</em><strong><i>ln</i>(1 + r)</strong></span></p>
|
|
<p>Finds the number of <b>Periods</b> when you know the Present Value, Future Value and Interest Rate (note: <i><b>ln</b></i> is the <a href="../algebra/logarithms.html">logarithm</a> function)</p>
|
|
<h2>Annuities</h2>
|
|
<p>We have now covered what happens to a value as time goes by ... but what if we have a series of values, like <b> <b>regular loan payments</b> </b>or<b> yearly investments</b>? That is covered in the topic of <a href="annuities.html">Annuities</a>.</p>
|
|
<p></p>
|
|
|
|
<div class="questions">
|
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<script type="text/javascript">getQ(2294, 2295, 2296, 2297, 2298, 2299, 2300, 2301, 2302, 2303);</script>
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</div>
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<div class="related"><a href="index.html">Money Index</a> <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="compound-interest-derivation.html">Compound Interest Derivation</a> <a href="compound-interest-periodic.html">Compound Interest: Periodic Compounding</a></div>
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