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<title>Net Present Value (NPV)</title>
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<h1 class="center">Net Present Value (NPV)</h1>
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<p class="center"><img src="images/net-present-value.svg" alt="net present value" /></p>
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<p class="center larger">Money <b>now</b> is more valuable than money <b>later on</b>.</p>
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<p class="center">Why? Because you can use money to make more money!</p>
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<p>You could run a business, or buy something now and sell it later for more, or simply put the money in the bank to earn interest.</p>
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<div class="example">
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<h3>Example: Let us say you can get 10% interest on your money.</h3>
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<p>So $1,000 now can earn $1,000 x 10% = <b>$100</b> in a year.</p>
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<p class="center">Your <b>$1,000 now</b> becomes <b>$1,100 next year</b>. </p>
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<p class="center"><img src="images/coin-stack-grow.svg" alt="coin stack grows" /></p>
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<p class="center">So $1,000 now is the <b>same</b> as $1,100 next year (at 10% interest):</p>
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</div>
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<div class="center80">
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<p class="center">We say that $1,100 next year has a <b>Present Value</b> of<b> $1,000</b>.</p>
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<p class="center"><img src="images/pv-vs-fv-1yr.svg" alt="Present Value $1000 vs Future Value $1100" /></p>
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<p class="center">Because $1,000 can become $1,100 in one year (at 10% interest).
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</p>
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</div>
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<p>If you understand Present Value, you can skip straight to <a href="#NetPresentValue">Net Present Value</a>.</p>
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<p>Now let us extend this idea further into the future ...</p>
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<p> </p>
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<h2>How to Calculate Future Payments</h2>
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<p>Let us stay with <span class="large">10% Interest</span> , which means money grows by 10% every year, like this:</p>
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<p class="center"><img src="images/interest-compound-flow.svg" alt="interest compound $1000, 10%=$100, $1100, 10%=$110, $1210, 10%=$121, etc " /></p>
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<p>So:</p>
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<ul>
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<li><b>$1,100 next year</b> is the same as <b>$1,000 now</b>.</li>
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<li>And <b>$1,210 in 2 years</b> is the same as <b>$1,000 now</b>.</li>
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<li>etc</li>
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</ul>
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<p class="center">In fact <b>all those amounts are the same</b> (considering <b>when</b> they occur and the 10% interest).</p>
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<h2>Easier Calculation</h2>
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<p>But instead of "adding 10%" to each year it is easier to multiply by 1.10 (explained at <a href="compound-interest.html">Compound Interest</a>):</p>
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<p class="center"><img src="images/pct-to-mult.svg" alt="+10% -> x1.10" /></p>
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<p>So we get this (same result as above):</p>
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<p class="center"><img src="images/interest-compound-flow2.svg" alt="interest compound $1000 x1.1 $1100 x1.1 $1210 x1.1 ..." /></p>
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<h2>Future Back to Now</h2>
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<p>And to see what <b>money in the future</b> is worth <b>now</b>, go backwards (dividing by 1.10 each year instead of multiplying):</p>
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<p class="center"><img src="images/interest-compound-flow-back.gif" width="460" height="31" alt="interest compound flow backwards" /></p>
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<div class="example">
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<h3>Example: Sam promises you <b>$500 next year</b>, what is the Present Value?</h3>
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<div class="so"> To take a future payment backwards one year <b>divide by 1.10</b></div>
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<p>So <b>$500 next year</b> is $500 ÷ 1.10 = <b>$454.55 now</b> (to nearest cent).</p>
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<p>The Present Value is <b>$454.55</b></p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: Alex promises you <b>$900 in 3 years</b>, what is the Present Value?</h3>
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<div class="so"> To take a future payment backwards three years <b>divide by 1.10</b> three times</div>
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<p>So <b>$900 in 3 years</b> is:</p>
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<div class="so"> $900 ÷ 1.10 ÷ 1.10 ÷ 1.10 </div>
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<div class="so"> $900 ÷ (1.10 × 1.10 × 1.10)</div>
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<div class="so">$900 ÷ 1.331</div>
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<div class="so"><b>$676.18 now</b> (to nearest cent).</div>
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</div>
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<h2>Better With Exponents</h2>
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<p>But instead of <b>$900 ÷ (1.10 × 1.10 × 1.10)</b> it is better to use <a href="../exponent.html">exponents</a> (the exponent says <b>how many times </b>to use the number in a multiplication). </p>
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<div class="example">
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<h3>Example: (continued)</h3>
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<p>The Present Value of <b>$900 in 3 years</b> (in one go):</p>
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<div class="so">$900 ÷ 1.10<sup>3</sup> = <b>$676.18 now</b> (to nearest cent).</div>
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</div>
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<p>And we have in fact just used the <b>formula</b> for Present Value: </p>
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<div class="center80">
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<p class="center"><span class="largest">PV = FV / (1+r)<sup>n</sup></span></p>
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<ul>
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<li><b>PV</b> is Present Value</li>
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<li><b>FV</b> is Future Value</li>
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<li><b>r</b> is the interest rate (as a decimal, so 0.10, not 10%)</li>
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<li><b>n</b> is the number of years</li>
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</ul></div>
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<div class="example">
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<h3>Example: (continued)</h3>
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<p>Use the formula to calculate Present Value of <b>$900 in 3 years</b>:</p>
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<div class="so">PV = FV / (1+r)<sup>n</sup></div>
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<div class="so">PV = $900 / (1 + 0.10)<sup>3</sup></div>
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<div class="so">PV = $900 / 1.10<sup>3</sup></div>
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<div class="so">PV = <b>$676.18 </b> (to nearest cent).</div>
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</div>
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<table border="0">
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<tr>
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<td><img src="images/calculator-exponent.gif" width="106" height="85" alt="calculator exponent button" /></td>
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<td> </td>
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<td><p>Exponents are easier to use, particularly with a calculator.</p>
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<p>For example 1.10<sup>6</sup> is quicker than 1.10 × 1.10 × 1.10 × 1.10 × 1.10 × 1.10</p></td>
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</tr>
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</table>
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<p> </p>
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<h2><a name="NetPresentValue"></a>Net Present Value (NPV)</h2>
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<p>A <b>Net</b> Present Value is when you add and subtract all Present Values:</p>
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<ul>
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<li>Add each Present Value you receive</li>
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<li>Subtract each Present Value you pay</li>
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</ul>
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<div class="example">
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<h3>Example: A friend needs $500 now, and will pay you back $570 in a year. Is that a good investment when you can get 10% elsewhere?</h3>
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<p>Money Out: $500 now</p>
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<div class="so">You invested $500 now, so PV = <b>-$500.00</b></div>
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<p>Money In: $570 next year</p>
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<div class="so">PV = $570 / (1+0.10)<sup>1</sup> = $570 / 1.10 = <b>$518.18</b> (to nearest cent)</div>
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<p>The Net Amount is:</p>
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<div class="so">Net Present Value = $518.18 - $500.00 = <b>$18.18</b></div>
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<p>So, at 10% interest, that investment is worth <b>$18.18</b></p>
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<p><i>(In other words it is <b>$18.18</b> better than a 10% investment, in today's money.)</i></p>
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</div>
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<p>A Net Present Value (NPV) that is <b>positive is good</b> (and negative is bad).</p>
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<p>But your choice of interest rate can change things!</p>
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<div class="example">
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<h3>Example: Same investment, but try it at <span class="hilite">15%</span>.</h3>
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<p>Money Out: $500 </p>
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<div class="so">You invested $500 now, so PV = <b>-$500.00</b></div>
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<p>Money In: $570 next year:</p>
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<div class="so">PV = $570 / (1+0.<span class="hilite">15</span>)<sup>1</sup> = $570 / 1.<span class="hilite">15</span> = = <b>$495.65</b> (to nearest cent) </div>
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<p>Work out the Net Amount:</p>
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<div class="so">Net Present Value = $495.65 - $500.00 = <b>-$4.35</b></div>
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<p>So, at 15% interest, that investment is worth <b>-$4.35</b></p>
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<p>It is a bad investment. But only because you are demanding it earn 15% (maybe you can get 15% somewhere else at similar risk).</p>
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</div>
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<p>Side Note: the interest rate that makes the NPV <b>zero</b> (in the previous example it is about 14%) is called the <a href="internal-rate-return.html">Internal Rate of Return</a>. </p>
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<p>Let us try a bigger example. </p>
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<div class="example">
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<h3>Example: Invest $2,000 now, receive 3 yearly payments of $100 each, plus $2,500 in the 3rd year. Use 10% Interest Rate.</h3>
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<p>Let us work year by year (remembering to subtract what you pay out):</p>
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<ul>
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<li>Now: PV = <b>−$2,000</b></li>
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<li>Year 1: PV = $100 / 1.10 = <b>$90.91</b></li>
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<li>Year 2: PV = $100 / 1.10<sup>2</sup> = <b>$82.64</b></li>
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<li>Year 3: PV = $100 / 1.10<sup>3</sup> = <b>$75.13</b></li>
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<li>Year 3 (final payment): PV = $2,500 / 1.10<sup>3</sup> = <b>$1,878.29</b></li>
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</ul>
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<p>Adding those up gets: <b>NPV</b> = −$2,000 + $90.91 + $82.64 + $75.13 + $1,878.29 = <b>$126.97</b></p>
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<p>Looks like a good investment.</p>
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</div>
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And again, but an interest rate of 6%
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<div class="example">
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<h3>Example: (continued) at a <span class="hilite">6%</span> Interest Rate.</h3>
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<ul>
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<li>Now: PV = <b>−$2,000</b></li>
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<li>Year 1: PV = $100 / 1.<span class="hilite">06</span> = <b>$94.34</b></li>
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<li>Year 2: PV = $100 / 1.<span class="hilite">06</span><sup>2</sup> = <b>$89.00</b></li>
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<li>Year 3: PV = $100 / 1.<span class="hilite">06</span><sup>3</sup> = <b>$83.96</b></li>
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<li>Year 3 (final payment): PV = $2,500 / 1.<span class="hilite">06</span><sup>3</sup> = <b>$2,099.05</b></li>
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</ul>
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<p>Adding those up gets: <b>NPV</b> = −$2,000 + $94.34 + $89.00 + $83.96 + $2,099.05 = <b>$366.35</b></p>
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<p>Looks even better at 6%</p>
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</div>
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<p> </p>
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<div class="center80">
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<p>Why is the NPV <b>bigger</b> when the interest rate is <b>lower</b>? </p>
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<table width="100%" border="0">
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<tr>
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<td><img src="../images/soccer-teams.jpg" width="209" height="86" alt="soccer teams" /></td>
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<td><p>Because the interest rate is like the team you are playing against, play an easy team (like a 6% interest rate) and you look good, a tougher team (like a 10% interest) and you don't look so good!</p></td>
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</tr>
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</table>
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<p>You can actually use the interest rate as a "test" or "hurdle" for your investments: demand that an investment have a positive NPV with, say, 6% interest. </p>
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</div>
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<p> </p>
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<p>So there you have it: work out the PV (Present Value) of each item, then total them up to get the NPV (Net Present Value), being careful to subtract amounts that go out and add amounts that come in.</p>
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<p>And a final note: when comparing investments by NPV, make sure to use the <b>same interest rate for each</b>.</p>
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<p> </p>
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<div class="related">
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<a href="investing-introduction.html">Investing (Introduction)</a>
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<a href="investment-graph.html">Investment/Loan Graph</a>
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<a href="present-value.html">Present Value</a>
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<a href="internal-rate-return.html">Internal Rate of Return</a>
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<a href="index.html">Money Index</a>
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