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Math Help - Annuity Problem

  1. #1
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    Annuity Problem

    I'm studying for actuarial exam FM/2, and want to verify my calculations with someone. The problem is as follows: "Jerry will make deposits of 450 at the end of each quarter for 10 years. At the end of 15 years, Jerry will use the fund to make annual payments of Y at the beginning of each year for 4 years, after which the fund is exhausted. The effective annual rate of interest is 7%. Determine Y".

    I first found the equivalent quarterly rate of interest to be 0.017058525 (since 1.017058525^4=1.07). Then 450\displaystyle\frac{1.07^{40}-1}{0.07}=25512.23 is the amount in the account after 10 years.

    Multiplying that by 1.07^5 I got 35783.62. This is then equal to the value of the withdrawals at time 15, so 35783.62=1.07Y\displaystyle\frac{1-1.07^{-4}}{0.07}, and solving this gave me Y=9873.21.

    The answer in the back of the book is 9872, which seems a bit far off for roundoff. Even so, I experimented with different reasonable intermediate round offs and couldn't get 9872...the fluctuations for reasonable round offs were only on the order of a few cents. Did I do something wrong? Any help would be appreciated.
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  2. #2
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    Re: Annuity Problem

    Quote Originally Posted by Diamondlance View Post
    I'm studying for actuarial exam FM/2, and want to verify my calculations with someone. The problem is as follows: "Jerry will make deposits of 450 at the end of each quarter for 10 years. At the end of 15 years, Jerry will use the fund to make annual payments of Y at the beginning of each year for 4 years, after which the fund is exhausted. The effective annual rate of interest is 7%. Determine Y".

    I first found the equivalent quarterly rate of interest to be 0.017058525 (since 1.017058525^4=1.07). Then 450\displaystyle\frac{1.07^{40}-1}{0.07}=25512.23 is the amount in the account after 10 years.

    Multiplying that by 1.07^5 I got 35783.62. This is then equal to the value of the withdrawals at time 15, so 35783.62=1.07Y\displaystyle\frac{1-1.07^{-4}}{0.07}, and solving this gave me Y=9873.21.

    The answer in the back of the book is 9872, which seems a bit far off for roundoff. Even so, I experimented with different reasonable intermediate round offs and couldn't get 9872...the fluctuations for reasonable round offs were only on the order of a few cents. Did I do something wrong? Any help would be appreciated.
    25512.23 * (1.07)^5
    25512.23 * 1.4025517307
    $35782.22

    $35782.22 / PVIFAD(7%,4)
    $35782.22 / PVIFAD(7%,4)
    $35782.22 / 3.62431604442
    9872.82

    It's still off by $0.82

    BTW why did you use 7% in calculating future value of $450 per quarter for 10 years. Shouldn't you have used 7%/4 when compounding that figure
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  3. #3
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    Re: Annuity Problem

    Quote Originally Posted by Diamondlance View Post
    I'm studying for actuarial exam FM/2, and want to verify my calculations with someone. The problem is as follows: "Jerry will make deposits of 450 at the end of each quarter for 10 years. At the end of 15 years, Jerry will use the fund to make annual payments of Y at the beginning of each year for 4 years, after which the fund is exhausted. The effective annual rate of interest is 7%. Determine Y".

    I first found the equivalent quarterly rate of interest to be 0.017058525 (since 1.017058525^4=1.07). Then 450\displaystyle\frac{1.07^{40}-1}{0.07}=25512.23 is the amount in the account after 10 years.

    Multiplying that by 1.07^5 I got 35783.62. This is then equal to the value of the withdrawals at time 15, so 35783.62=1.07Y\displaystyle\frac{1-1.07^{-4}}{0.07}, and solving this gave me Y=9873.21.

    The answer in the back of the book is 9872, which seems a bit far off for roundoff. Even so, I experimented with different reasonable intermediate round offs and couldn't get 9872...the fluctuations for reasonable round offs were only on the order of a few cents. Did I do something wrong? Any help would be appreciated.
    I get the same answer as you do

    (1+r)^4 = 1.07
    r = (1.07)^1/4 - 1
    r = (1.07)^1/4 - 1
    r = 0.017058525001811312664557166663114
    r = 0.017058525

    450 * FVIFA(1.7058525%, 40)
    450 * 56.6960717383
    25513.23

    25513.23
    25513.23 * (1.07)^5
    25513.23 * FVIF(7%,5)
    25513.23 * 1.4025517307
    35783.63


    35783.63 = 1.07 Y [1-(1.07)^-4]/0.07
    Y PVIFAD (7%,4) = 35783.63
    Y 3.62431604442 = 35783.63
    Y = 35783.63 / 3.62431604442
    Y = 9873.20
    Last edited by dexteronline; September 2nd 2011 at 12:46 PM. Reason: added reference links
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  4. #4
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    Re: Annuity Problem

    Quote Originally Posted by dexteronline View Post
    BTW why did you use 7% in calculating future value of $450 per quarter for 10 years. Shouldn't you have used 7%/4 when compounding that figure
    You're right, one of my original paragraphs has two errors in it (I'm sorry I didn't proofread more carefully). It should read:

    I first found the equivalent quarterly rate of interest to be 0.017058525 (since 1.017058525^4=1.07). Then 450\displaystyle\frac{1.017058525^{40}-1}{0.017058525}=25513.23 is the amount in the account after 10 years.
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  5. #5
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    Re: Annuity Problem

    Anyway, thank you for verifying my calculations.
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  6. #6
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    Re: Annuity Problem

    Double confirmation: 9873.21 is right on!!
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