Difference between revisions of "022 Exam 2 Sample B, Problem 4"
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<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"> '''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">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">(a) compounded quarterly? | ||
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!Foundations: | !Foundations: | ||
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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. | + | |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\,=\,Pe^{rt}.</math> | ::<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>. | + | |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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!(a): | !(a): | ||
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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 | + | |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> | + | ::<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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!(b): | !(b): | ||
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| − | |Again, we need to apply the formula from foundations to find | + | |Again, we need to apply the formula from foundations to find the present value is |
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| − | ::<math> | + | ::<math>P\,=\,Ae^{-rt}\,=\,3000e^{-0.06\cdot 8}.</math> |
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| − | ::(a) <math style="vertical-align: - | + | ::'''(a)''' <math style="vertical-align: -28%;">P\,=\,3000(1.015)^{-32}.</math> |
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| − | ::(b) <math style="vertical-align: | + | ::'''(b)''' <math style="vertical-align: -5%;">P\,=\,3000e^{-0.48}.</math> |
|} | |} | ||
[[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']] | [[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']] | ||
Latest revision as of 08:46, 17 May 2015
Set up the equation to solve. You only need to plug in the numbers-not solve for the particular values!
What is the present value of $3000, paid 8 years from now, in an investment that pays 6% interest,
- (a) compounded quarterly?
- (b) compounded continuously?
| Foundations: |
|---|
| 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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| where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is the value of the account, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is the principal (original amount invested), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is the annual rate and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is the number of compoundings per year. The value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 365} for compounding daily, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 52} for compounding weekly, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12} for compounding monthly. As a result, the exponent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle nt} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} is the time in years, is the number of compounding periods where we actually earn interest. Similarly, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r/n} is the rate per compounding period (the annual rate divided by the number of compoundings per year). |
| For example, if we compound monthly for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7} years at a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6%} rate, we would compound Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle nt\,=\,12 \cdot7\,=\,84} times, once per month, at a rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.06/12\,=\,0.005} per monthly period. Notice that we always use the decimal version for interest rates when using these equations. |
| On the other hand, interest compounded continuously earns rate in just that way - continuously. I can have any value I want for time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} , 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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| Notice that in both formulas, the value at the initial time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t\,=\,0} is just our initial investment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} . |
| 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} in the equations above), and then are asked to find the present value, which would be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} the equations given. As a result, we can rewrite the equations to solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} . The equations then become |
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| for the non-continuous case, and |
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| for the continuous case. |
| (a): |
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| We are given all the pieces required. We begin with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $3000} of principal, and compound quarterly, or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} times per year. Using the formula in 'Foundations', the equation for the present value is |
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| (b): |
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| Again, we need to apply the formula from foundations to find the present value is |
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| Final Answer: |
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