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<title>Present Value (PV)</title>
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<meta name="description" content=" Money now is more valuable than money later on." />
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<h1 class="center">Present Value (PV)</h1>
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<p class="center larger"><span class="center"><br />
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</span>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: You can get 10% interest on your money.</h3>
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<p>So <b>$1,000 now</b> can earn $1,000 x 10% = <b>$100</b> in a year.</p>
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<p>Your <b>$1,000 now</b> can become <b>$1,100 in a year's time</b>.</p>
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</div>
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<h2>Present Value</h2>
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<p class="center"><img src="images/pv-vs-fv-1yr.svg" alt="Present Value $1000 vs Future Value $1100" />
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<br /> So $1,000 now is the <b>same</b> as $1,100 next year (at 10% interest).
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<br /> <img src="images/coin-stack-grow.svg" alt="coin stack grows" /> </p>
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<div class="center80">
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<p class="center">We say 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 class="center">Because we could turn $1,000 into $1,100 (if we could earn 10% interest).</p>
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<p>Now let us extend this idea further into the future ...</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>. That means that 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 " style="max-width:100%" /></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 ..." style="max-width:100%" /></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" alt="interest compound flow backwards" style="max-width:100%" /></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>As a <b>formula</b> it is: </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>
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</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> = $900 / 1.10<sup>3</sup> = <b>$676.18 </b> (to nearest cent).</div>
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</div>
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<p> </p>
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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>
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<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>
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</td>
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</tr>
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</table>
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<p>Let us use the formula a little more:</p>
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<div class="example">
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<h3>Example: What is $570 next year worth now, at an interest rate of 10% ?</h3>
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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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</div>
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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: What is $570 next year worth now, at an interest rate of <span class="hilite">15%</span> ?</h3>
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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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</div>
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<p>Or what if you don't get the money for 3 years</p>
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<div class="example">
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<h3>Example: What is $570 in <span class="hilite">3 years</span> time worth now, at an interest rate of 10% ?</h3>
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<div class="so">PV = $570 / (1+0.10)<sup class="hilite">3</sup> = $570 / <span class="hilite">1.331</span> = <b>$428.25</b> (to nearest cent)</div>
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</div>
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<p>One last example:</p>
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<div class="example">
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<h3>Example: You are promised $800 in 10 years time. What is its Present Value at an interest rate of 6% ?</h3>
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<div class="so">PV = $800 / (1+0.06)<sup>10</sup> = $800 / 1.7908... = <b>$446.72</b> (to nearest cent)</div>
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</div>
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<p> </p>
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<div class="related"> <a href="investing-introduction.html">Investing (Introduction)</a> <a href="investment-graph.html">Investment/Loan Graph</a> <a href="net-present-value.html">Net Present Value</a> <a href="index.html">Money Index</a> </div>
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