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Math Help - Annual payments of a fund

  1. #1
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    Annual payments of a fund

    Annual deposits of $1,500 are placed into a fund for the next 27 years. Beginning 14 years after the last deposit annual payments commence and continue forever. Find the annual payment to the two decimal places if the fund earns 7% effective.



    The answer I got was 21577.84, but I'm unsure of my steps. Can anyone confirm whether that is the answer or not? If not, could you please tell me how to solve it the correct way?
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  2. #2
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    Hello, actsci231!

    My answer is different but in the "same ballpark".
    Please check my reasoning and my work.


    Annual deposits of $1,500 are placed into a fund for the next 27 years.
    Beginning 14 years after the last deposit, annual payments commence and continue forever.
    Find the annual payment to the two decimal places if the fund earns 7% effective.

    The answer I got was $21,577.84

    The Annuity Formula is: . A \;=\;D\,\frac{(1+i)^n-1}{i}

    . . where: . \begin{Bmatrix}A &=& \text{Final value} \\ D &=& \text{periodic deposite} \\ i &=& \text{periodic interest rate} \\ n &=& \text{number of periods} \end{Bmatrix}


    We have: . D = 1500,\;\;i = 0.07,\;\;n = 27

    Then: . A \;=\;1500\,\frac{1.07^{27}-1}{0.07} \;\approx\;\$111,\!725.73 ... total value after 27 years


    This amount is allowed to accrue interest for the next 14 years.

    Then its value will be: . \$111,\!725.73(1.07)^{14} \;\approx\;\$288,\!088.61


    Then we can make annual withdrawls forever.

    The account earns: . 0.07 \times \$288,\!088.61 \;\approx\;\boxed{\$20,\!166.20} per year.

    And that is the amount we can withdraw each year.

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  3. #3
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    If the annual deposits of $1,500 are beginning of year deposits, then the fund value at the end of 27 years is
    <br />
\ddot S = 1,500 \cdot \frac{{\left( {1.07} \right)^{27}  - 1}}<br />
{{.07}} \times \left( {1.07} \right)<br />
    Since the first perpetual payment is due at the end of 14 years (after the 27th year), you must determine the fund value at the beginning of the 14 year (which is the end of the 13th year) by accumulating <br />
{\ddot S}<br />
for 13 years. Thus
    <br />
1,500 \cdot \frac{{\left( {1.07} \right)^{27}  - 1}}<br />
{{.07}} \times \left( {1.07} \right) \times \left( {1.07} \right)^{13} <br />
    The annual perpetual payment is then determined by
    <br />
1,500 \cdot \frac{{\left( {1.07} \right)^{27}  - 1}}<br />
{{.07}} \times \left( {1.07} \right) \times \left( {1.07} \right)^{13}  \times \left( {.07} \right)<br />
    On the other hand, if the annual deposits of $1,500 are end of year deposits, then the fund value at the end of 27 years is
    <br />
S = 1,500 \cdot \frac{{\left( {1.07} \right)^{27}  - 1}}<br />
{{.07}}<br />
    Since the first perpetual payment is due atů.(repeat the same reasoning)
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