View source for 022 Exam 2 Sample B, Problem 4

<span class="exam"> '''Set up the equation to solve. You only need to plug in the numbers-not solve for the particular values!'''

<span class="exam">What is the present value of $3000, paid 8 years from now, in an investment that pays 6% interest,

::<span class="exam">(a) compounded quarterly?
::<span class="exam">(b) compounded continuously?

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations: &nbsp;
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|The primary purpose of this problem is to demonstrate that you understand the difference between continuous compounding and compounding on an interval of time, as well as the concept of present and future value. When we compound on an interval, say monthly, the value in the account only changes at the end of each interval.  In other words, there is no interest accrued for a week or a day.  As a result, we use the formula
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::<math>A\,=\,P\left(1+\frac{r}{n}\right)^{nt},</math>
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|where <math style="vertical-align: 0%;">A</math> is the value of the account, <math style="vertical-align: 0%;">P</math> is the principal (original amount invested), <math style="vertical-align: 0%;">r</math> is the annual rate and <math style="vertical-align: 0%;">n</math> is the number of compoundings per year.  The value of <math style="vertical-align: 0%;">n</math> is <math style="vertical-align: 0%;">365</math> for compounding daily, <math style="vertical-align: 0%;">52</math> for compounding weekly, and <math style="vertical-align: -5%;">12</math> for compounding monthly.  As a result, the exponent <math style="vertical-align: 0%;">nt</math>, where <math style="vertical-align: 0%;">t</math> is the time in years, is the number of compounding periods where we actually earn interest. Similarly, <math style="vertical-align: -22%;">r/n</math> is the rate per compounding period (the annual rate divided by the number of compoundings per year).
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|For example, if we compound monthly for <math style="vertical-align: 0%;">7</math> years at a <math style="vertical-align: -5%;">6%</math> rate, we would compound <math style="vertical-align: -5%;">nt\,=\,12 \cdot7\,=\,84</math> times, once per month, at a rate of <math style="vertical-align: -22%;">0.06/12\,=\,0.005</math> per monthly period. Notice that we <u>'''always'''</u> use the decimal version for interest rates when using these equations.
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|On the other hand, interest compounded continuously earns rate in just that way - continuously.  I can have any value I want for time <math style="vertical-align: 0%;">t</math>, and the total amount in the account will change with each and every second.  Therefore, we express the account value as 
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::<math>A\,=\,Pe^{rt}.</math>
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|Notice that in both formulas, the value at the initial time <math style="vertical-align: 0%;">t\,=\,0</math> is just our initial investment <math style="vertical-align: 0%;">P</math>.
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|However, a present value/future value problem requires you to think a little differently.  When we assess these, we are usually given the eventual payout (future value, or <math style="vertical-align: 2%;">A</math> in the equations above), and then are asked to find the present value, which would be <math style="vertical-align: 2%;">P</math> the equations given. As a result, we can rewrite the equations to solve for <math style="vertical-align: 2%;">P</math>.  The equations then become
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::<math>P\,=\,A\left(1+\frac{r}{n}\right)^{-nt},</math>
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|for the non-continuous case, and 
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::<math>P\,=\,Ae^{-rt}</math>
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|for the continuous case.
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!(a): &nbsp;
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|We are given all the pieces required.  We begin with <math style="vertical-align: -5%;">$3000</math> of principal, and compound quarterly, or <math style="vertical-align: -5%;">4</math> times per year. Using the formula in 'Foundations', the equation for the present value is 
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::<math>P\,=\,A\left(1+\frac{r}{n}\right)^{-nt}\,=\,3000\left(1+\frac{0.06}{4}\right)^{-4\cdot 8}.</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!(b): &nbsp;
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|Again, we need to apply the formula from foundations to find the present value is
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::<math>P\,=\,Ae^{-rt}\,=\,3000e^{-0.06\cdot 8}.</math>

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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer: &nbsp;
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::'''(a)''' <math style="vertical-align: -28%;">P\,=\,3000(1.015)^{-32}.</math> 
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::'''(b)''' <math style="vertical-align: -5%;">P\,=\,3000e^{-0.48}.</math>
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