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<title>Annuities</title>
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<h1 align="center">Annuities </h1>
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<p>An annuity is a <b>fixed income over a period of time</b>. </p>
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<div class="example">
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<h3>Example: You get $200 a week for 10 years.</h3>
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</div>
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<p>How do you get such an income? <b>You buy it!</b></p>
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<p>So: </p>
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<ul>
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<li><b>you pay them</b> one large amount, then</li>
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<li><b>they pay you</b> back a series of small payments over time</li>
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</ul>
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<div class="example">
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<h3>Example: You buy an annuity</h3>
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<p>It costs you <b>$20,000</b></p>
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<p>And in return you get $400 a month for 5 years </p>
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</div>
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<p><b><i>Is that a good deal?</i></b></p>
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<div class="example">
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<h3>Example (continued): </h3>
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<p> $400 a month for 5 years = $400 × 12 × 5 = <b>$24,000</b></p>
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<p>Seems like a good deal ... you get back more than you put in.</p>
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</div>
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<p>Why do you get <b>more</b> income <i>($24,000)</i> than the annuity originally cost <i>($20,000)</i>?</p>
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<p>Because <b>money now</b> is more valuable than money later.</p>
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<p>The people who got your $20,000 can invest it and earn interest, or do other clever things to make more money.</p>
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<p>So how much <i>should</i> an annuity cost? </p>
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<h2>Value of an Annuity</h2>
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<p>First: let's see the effect of an <b>interest rate of 10%</b> (imagine a bank account that earns 10% interest):</p>
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<div class="example">
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<h3>Example: 10% interest on $1,000</h3>
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<p>$1,000 now could earn $1,000 x 10% = <b>$100</b> in a year.</p>
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<p><b>$1,000 now</b> becomes <b>$1,100 in a year's time</b>.</p>
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</div>
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<p class="center"><img src="images/pv-vs-fv-1yr.svg" alt="present value $1000 vs future value $1100" /><br>
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<br />
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So $1,100 next year is the <b>same</b> as $1,000 now (at 10% interest).</p>
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<div class="center80">
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<p class="center">The <b>Present Value</b> of $1,100 next year is <b>$1,000</b></p>
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</div>
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<p>So, at 10% interest:</p>
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<ul>
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<li>to go from <b>now</b> to <b>next year</b>: multiply by 1.10</li>
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<li>to go from <b>next year</b> to <b>now</b>: divide by 1.10</li>
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</ul>
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<p>Now let's imagine an annuity of <b>4 yearly payments of $500</b>. </p>
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<p>Your first payment of $500 is next year ... how much is that worth <b>now</b>?</p>
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<div class="so">$500 ÷ 1.10 = $454.55 now (to nearest cent)</div>
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<p>Your second payment is 2 years from now. How do we calculate that? Bring it back one year, then bring it back another year:</p>
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<div class="so">$500 ÷ 1.10 ÷ 1.10 = $413.22 now </div>
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<p>The third and 4th payment can also be brought back to today's values:</p>
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<div class="so">$500 ÷ 1.10 ÷ 1.10 ÷ 1.10 = $375.66 now</div>
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<div class="so">$500 ÷ 1.10 ÷ 1.10 ÷ 1.10 ÷ 1.10 = $341.51 now</div>
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<p>Finally we add up the 4 payments (in today's value):</p>
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<div class="so">Annuity Value = $454.55 + $413.22 + $375.66 + $341.51</div>
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<div class="so">Annuity Value = $1,584.94</div>
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<p>We have done our first annuity calculation!</p>
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<p class="center larger">4 annual payments of $500 at 10% interest is worth <b>$1,584.94 now</b></p>
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<p>How about another example:</p>
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<div class="example">
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<h3>Example: An annuity of $400 a month for 5 years. </h3>
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<h3>Use a Monthly interest rate of 1%.</h3>
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</div>
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<p>12 months a year, 5 years, that is <b>60 payments</b> ... and a LOT of calculations.</p>
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<p>We need an easier method. Luckily there is a neat formula:</p>
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<div class="def">
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<p class="center large">Present Value of Annuity: PV = P ×
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<span class="intbl">
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<em>1 − (1+r)<sup>−n</sup></em>
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<strong>r</strong>
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</span></p>
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<ul>
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<li><b>P</b> is the value of each payment</li>
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<li><b>r</b> is the interest rate per period, as a decimal, so 10% is 0.10</li>
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<li><b>n</b> is the number of periods</li>
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</ul>
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</div>
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<div class="example">
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<h3>First, let's try it on our $500 for 4 years example.</h3>
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<p>The interest rate per year is 10%, so <b>r = 0.10</b></p>
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<p>There are 4 payments, so <b>n=4</b>, and each payment is $500, so <b>P = $500</b></p>
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<div class="so">PV = $500 × <span class="intbl">
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<em>1 − (1.10)<sup>−4</sup></em>
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<strong>0.10</strong>
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</span></div>
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<div class="so">PV = $500 × <span class="intbl">
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<em>1 − 0.68301...</em>
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<strong>0.10</strong>
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</span></div>
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<div class="so">PV = $500 ×
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3.169865...</div>
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<div class="so">PV = <span class="intbl">$1584.93</span></div>
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</div>
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<p><b>It matches our answer above</b> (and is 1 cent more accurate)</p>
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<div class="example">
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<h3>Now let's try it on our $400 for 60 months example:</h3>
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<p>The interest rate is 1% per month, so <b>r = 0.01</b></p>
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<p>There are 60 monthly payments, so <b>n=60</b>, and each payment is $400, so <b>P = $400</b></p>
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<div class="so">PV = $400 × <span class="intbl">
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<em>1 − (1.01)<sup>−60</sup></em>
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<strong>0.01</strong>
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</span></div>
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<div class="so">PV = $400 × <span class="intbl">
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<em>1 − 0.55045...</em>
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<strong>0.01</strong>
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</span></div>
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<div class="so">PV = $400 ×
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44.95504...</div>
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<div class="so">PV = <span class="intbl">$17,982.02</span></div>
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<p>Certainly easier than 60 separate calculations.</p>
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</div>
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<p><i>Note: use the interest rate <b>per period</b>: for monhtly payments use the monthly interest rate, etc.</i> </p>
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<h2>Going the Other Way</h2>
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<p>What if you know the annuity value and want to work out the payments?</p>
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<div class="example">
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<p>Say you have $10,000 and want to get a monthly income for 6 years, how much do you get each month (assume a monthly interest rate of 0.5%)</p>
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</div>
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<p>We need to change the subject of the formula above </p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">PV = P × <span class="intbl">
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<em>1 − (1+r)<sup>−n</sup></em>
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<strong>r</strong>
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</span></span></div>
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<div class="row"><span class="left">Swap sides:</span><span class="right">P × <span class="intbl">
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<em>1 − (1+r)<sup>−n</sup></em>
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<strong>r</strong>
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</span> = PV</span></div>
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<div class="row"><span class="left">Multiply both sides by <b>r</b>:</span><span class="right">P × (1 − (1+r)<sup>−n</sup>) = PV × r</span></div>
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<div class="row"><span class="left">Divide both sides by <b>1 − (1+r)<sup>−n</sup></b> :</span><span class="right">P
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<em> </em>= PV × <span class="intbl">
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<em>r</em>
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<strong>1 − (1+r)<sup>−n</sup></strong>
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</span></span></div>
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</div>
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<p>And we get this:</p>
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<div class="def">
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<p class="center large">P
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<em> </em>
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= PV × <span class="intbl">
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<em>r</em>
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<strong>1 − (1+r)<sup>−n</sup></strong>
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</span></p>
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<ul>
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<li><b>P</b> is the value of each payment</li>
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<li><b>PV</b> is the Present Value of Annuity</li>
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<li><b>r</b> is the interest rate per period as a decimal, so 10% is 0.10</li>
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<li><b>n</b> is the number of periods</li>
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</ul>
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</div>
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<p> </p>
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<div class="example">
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<h3>Say you have $10,000 and want to get a monthly income for 6 years out of it, how much could you get each month (assume a monthly interest rate of 0.5%)</h3>
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<p>The monthly interest rate is 0.5%, so <b>r = 0.005</b></p>
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<p>There are 6x12=72 monthly payments, so <b>n=72</b>, and <b>PV = $10,000</b></p>
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<div class="so">P
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<em> </em>
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= PV × <span class="intbl">
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<em>r</em>
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<strong>1 − (1+r)<sup>−n</sup></strong>
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</span></div>
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<div class="so">P
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<em> </em>
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= $10,000 × <span class="intbl">
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<em>0.005</em>
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<strong>1 − (1.005)<sup>−72</sup></strong>
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</span></div>
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<div class="so">P = $10,000 ×
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0.016572888...</div>
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<div class="so">P = <span class="intbl"></span>$165.73</div>
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<p>What do you prefer? <b>$10,000 now</b> or 6 years of <b>$165.73 a month</b></p>
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</div>
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<p> </p>
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<div class="def">
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<h3>Footnote:</h3>
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<p>You don't need to remember this, but you may be curious how the formula comes about:</p>
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<p>With <b>n</b> payments of <b>P</b>, and an interest rate of <b>r</b> we add up like this:</p>
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<div class="so">P × <span class="intbl">
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<em>1</em>
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<strong>1+r</strong>
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</span> + P × <span class="intbl">
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<em>1</em>
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<strong>(1+r)×(1+r)</strong>
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</span><span class="intbl"> </span> + P × <span class="intbl">
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<em>1</em>
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<strong>(1+r)×(1+r)×(1+r)</strong>
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</span><span class="intbl"> </span> + ... (n terms)<span class="intbl"></span></div>
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<p>We can use exponents to help. <span class="intbl">
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<em>1</em>
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<strong>1+r</strong>
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</span> is actually (1+r)<sup>−1</sup> and <span class="intbl">
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<em>1</em>
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<strong>(1+r)×(1+r)</strong>
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</span> is
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(1+r)<sup>−2</sup> etc: </p>
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<div class="so">P × (1+r)<sup>−1</sup> + P × (1+r)<sup>−2</sup> + P × (1+r)<sup>−3</sup> + ... (n terms)<span class="intbl"></span></div>
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<p>And we can bring the "P" to the front of all terms:</p>
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<div class="so">P × [ (1+r)<sup>−1</sup> + (1+r)<sup>−2</sup> + (1+r)<sup>−3</sup> + ... (n terms)<span class="intbl"></span> ]</div>
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<p><i>To simplify that further is a little harder! We need some clever work using <a href="../algebra/sequences-sums-geometric.html">Geometric Sequences and Sums</a> b</i><i>ut trust me, it can be done ... and we get this:</i></p>
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<div class="so">PV = P × <span class="intbl">
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<em>1 − (1+r)<sup>−n</sup></em>
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<strong>r</strong>
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</span></div>
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</div>
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<p> </p><p> </p>
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<script type="text/javascript">getQ(11322, 11323, 11324, 11325, 11326, 11327, 11328, 11329, 11330, 11331);</script>
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</div>
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<div class="related"><a href="interest.html">Introduction to Interest</a> <a href="investment-graph.html">Investment Graph</a></div>
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