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Math Help - Accumulated value of an annuity

  1. #1
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    Accumulated value of an annuity

    A man age 40 wishes to accumulate a fund for retirement by depositing $1000 at the beginning of each year for 25 years. Starting at age 65 he will make 15 annual withdrawals at the beginning of the year. Assuming that all payments are certain to be made, determine the amount of each withdrawal to the nearest dollar if the annual effective interest rate is 4% during the first 25 years but only 3 % thereafter. The answer is give as $3,633.

    Right now I have: (1.04)^25 + 1000 (s double-dot, angle 25).04 = x(a double-dot, angle 24).04
    but I think I have numbers off somewhere because I'm not getting the right answer.
    Hope you can understand the notation I wasn't sure how else to type it. Thanks!
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  2. #2
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    Sorry..I'm not familiar with the notation you have used...

    Future value of the deposits, given he is 40, he makes a deposit now(assuming its the start of the year)..
    S=R\left[ \frac{(1+i)^n-1}{i}\right]+R=1000\left[ \frac{(1+.04)^{24}-1}{.04}\right]+1000=\$40082.6

    So when he is 65, he will have $40082.6 and starts making withdrawls of X..

    40082.6=X\left[\frac{1- \left( \frac{1}{1.035} \right)^{14}}{0.035} \right]+X=X(10.9205)+X
    Therefore,
     X = \frac{40082.6}{10.9205+1}=\$3362.488
    Last edited by Robb; September 13th 2009 at 04:42 PM.
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  3. #3
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    Ok that makes sense except where does the 10.9205 come from?
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  4. #4
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    Quote Originally Posted by tbl9301 View Post
    Ok that makes sense except where does the 10.9205 come from?
    \left[\frac{1- \left( \frac{1}{1.035} \right)^{14}}{0.035} \right]=10.9205

    My answer was slightly different, since i did the calculations in excel so there was no rounding.
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  5. #5
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    Don't think so, Robb. I agree with given answer of 3633; 3633.38

    Age 40 to 64: deposits 1000 annually @ 4%:
    At age 64 (after 25th deposit) : 41,645.91
    Add 1 year's interest at 4% : 43,311.74 (now age 65)

    Account now switches to 3.5%
    3633.38 withdrawn right away, leaving 39,678.36
    14 more withdrawals brings account to zero.
    Last edited by Wilmer; September 13th 2009 at 07:35 PM. Reason: change month to year
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  6. #6
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    Ok so I have:

    1000(1.04)^24 + 1000[((1.04)^24 - 1)/.04] = x[(1-(1/1.035)^15)/.035]

    ... but I'm like barelyy off??
    Last edited by tbl9301; September 13th 2009 at 05:50 PM.
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  7. #7
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    Quote Originally Posted by Wilmer View Post
    Don't think so, Robb. I agree with given asnswe of 3633; 3633.38

    Age 40 to 64: deposits 1000 annually @ 4%:
    At age 64 (after 25th deposit) : 41,645.91
    Add 1 month's interest at 4% : 43,311.95 (now age 65)

    Account now switches to 3.5%
    3633.38 withdrawn right away, leaving 39,678.57
    14 more withdrawals brings account to zero.
    Why are you adding 1 months interest at 4%? do you mean 1 years?

    I took the question as he makes $1000 deposits for 25 years at the start of every year. Then, in his 65th year he withdraws X at the start of the year to have a zero balance after 15 years. So at the start of his 65th year he deposits $1000 and withdraws X.
    Im probably confused as to the year he actually starts depositing, and what happens in the trasition year :P
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  8. #8
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    Quote Originally Posted by tbl9301 View Post
    Ok so I have:
    1000(1.04)^24 + 1000[((1.04)^24 - 1)/.04] = x[(1-(1/1.035)^15)/.035]
    This is incorrect: 1000(1.04)^24 + 1000[((1.04)^24 - 1)/.04]

    Should be 1000(1.04)^25 + 1000[((1.04)^25 - 1)/.04] - 1000
    which equals 43,311.74

    Now divide that by 11.9205 (see Robb's post) to get 3,633.38
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  9. #9
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    got it! thank you both so much for your help!
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  10. #10
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    Quote Originally Posted by Robb View Post
    Why are you adding 1 months interest at 4%? do you mean 1 years?

    I took the question as he makes $1000 deposits for 25 years at the start of every year. Then, in his 65th year he withdraws X at the start of the year to have a zero balance after 15 years. So at the start of his 65th year he deposits $1000 and withdraws X.
    Im probably confused as to the year he actually starts depositing, and what happens in the trasition year :P
    Yes, I meant 1 year; sorry (now edited).

    The deposits are made from age 40 to age 64: that's 25 deposits.
    A year after his 25th deposit (at age 65), he receives interest at 4%
    for 1 year: now the 25 years at 4% are finished, and balance = 43,311.74

    The new interest of 3.5% now starts; at same time, he makes his first
    withdrawal; the withdrawals are 3,633.38.
    He will make 14 more withdrawals, which will terminate the account: zero balance.

    Hope this is CLEAR!
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  11. #11
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    Quote Originally Posted by Wilmer View Post
    The deposits are made from age 40 to age 64: that's 25 deposits.
    A year after his 25th deposit (at age 65), he receives interest at 4%
    for 1 year: now the 25 years at 4% are finished, and balance = 43,311.74

    The new interest of 3.5% now starts; at same time, he makes his first
    withdrawal; the withdrawals are 3,633.38.
    He will make 14 more withdrawals, which will terminate the account: zero balance.

    Hope this is CLEAR!
    Crystal clear indeed Sir Wilmer.
    If I were to improve upon such clarity, methinks I would merely reduce all that typing with a single statement like

    <br />
\left( {1,000s_{\left. {\overline {\, <br />
 {25} \,}}\! \right| .04} } \right)\left( {1.04} \right) = R + Ra_{\left. {\overline {\, <br />
 {14} \,}}\! \right| .035} <br />
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  12. #12
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    Quote Originally Posted by jonah View Post
    <br />
\left( {1,000s_{\left. {\overline {\, <br />
{25} \,}}\! \right| .04} } \right)\left( {1.04} \right) = R + Ra_{\left. {\overline {\, <br />
{14} \,}}\! \right| .035} <br />
    Extremely illustrating Sir Jonah; plus a welcomed space pre and post the equal sign.

    I was being verbiose in order to indirectly lead Mr Robb towards obtaining
    the required equations by himself; you are kinder than yours truly.
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