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Math Help - annuity

  1. #1
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    annuity

    Bank A is donating $ 400000 today to a charity that will establish an annual prize of $ 40000 with the first prize being awarded today. The charity can invests funds at 6% per annum. What amount of money will the banks prize be worth in its final year.

    Working:

    I have calculated the number of years the charity can operate at this rate to be 13.25 years. My confusion is what is the difference between the two ways of calculations below:

    (1) Future Value = 400000 x (1.06)^13.25 = 865690.6

    (2) Treating this as an annuity, where Payment = 40000, i= 6 and n=13.25 so FV = 800602.6

    Which way of doing is correct?
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  2. #2
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    Re: annuity

    $400,000 with $40,000 payment right away is same as $360,000 with
    $40,000 payments beginning 1 year later; the "account" will look like:
    Code:
    YR    PAYMENT    INTEREST       BALANCE
    00                             360000.00
    01  -40000.00    21600.00      341600.00
    02  -40000.00    20496.00      322096.00
    03  -40000.00    19325.76      301421.76
    .....
    12  -40000.00     5071.31       49593.08
    13  -40000.00     2975.58       12568.67
    14  -13322.80      754.13            .00
    The formula to determine n (number of years) will give n = 13.3266....
    This means the final payment is made .3266 * 365 = ~119.2 days after 13th payment:
    that's unrealistic: the final payment is made 1 year later.
    So final payment is calculated using end of 13th year balance:
    12568.67 * .06 = 754.13, which is the interest during final year.

    So the way to do all this by formula is:
    1: calculate FLOOR(n) when using the formula: that'll give you 13;
    2: calculate future value at period 13: that'll give you 12568.67
    3: multiply that by (1 + i): 12568.67 * 1.06 = 13322.80.
    So final payment is $13,322.80
    Thanks from hooke
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  3. #3
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    Re: annuity

    Thanks Wilmer, is there a way to calculate the balance for the 13th year, 12568.67 without doing in the table from year 1 to 13?
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  4. #4
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    Re: annuity

    Yes; look at it as 2 separate "goings-on":
    1: accumulation of $360,000 over 13 years
    2: accumulation of 13 annual payments of $40,000
    The difference of these is your answer.

    360000(1.06^13) = 767854.18
    40000(1.06^13 - 1) / .06 = 755285.51
    $767,854.18 - $755,285.51 = $12,568.67

    C'est bon, mon ami?
    Thanks from hooke
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    Re: annuity

    Quote Originally Posted by Wilmer View Post
    Yes; look at it as 2 separate "goings-on":
    1: accumulation of $360,000 over 13 years
    2: accumulation of 13 annual payments of $40,000
    The difference of these is your answer.

    360000(1.06^13) = 767854.18
    40000(1.06^13 - 1) / .06 = 755285.51
    $767,854.18 - $755,285.51 = $12,568.67

    C'est bon, mon ami?
    What if the accumulation of annual payment is greater than the accumulation of principal, ie 2>1. Can i still take the difference as the balance of payment?
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  6. #6
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    Re: annuity

    Quote Originally Posted by hooke View Post
    What if the accumulation of annual payment is greater than the accumulation of principal, ie 2>1. Can i still take the difference as the balance of payment?
    It's impossible for that to happen...because of the previous calculation to get n; OK?
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  7. #7
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    Re: annuity

    Quote Originally Posted by Wilmer View Post
    It's impossible for that to happen...because of the previous calculation to get n; OK?
    Lets say now i changed the figure the bank donates 500000 to a company which will give out an annual prize of 50000, with the first prize awarded today and it can invest at 5% per annum.

    n=13.25

    FV = 450000/(1.05)^13=848542.11 and FVA=885649.14 so here FVA>FV ??

    What have i gone wrong?
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  8. #8
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    Re: annuity

    Quote Originally Posted by hooke View Post
    Lets say now i changed the figure the bank donates 500000 to a company which will give out an annual prize of 50000, with the first prize awarded today and it can invest at 5% per annum.
    n=13.25
    FV = 450000/(1.05)^13=848542.11 and FVA=885649.14 so here FVA>FV ??
    What have i gone wrong?
    Your n is wrong: should be 12.253227....

    n = LOG[p / (p - ai)] / LOG(1 + i)
    where:
    a = initial amount (450000)
    p = annual payment (50000)
    i = interest rate (.05)
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  9. #9
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    Re: annuity

    Quote Originally Posted by Wilmer View Post
    Your n is wrong: should be 12.253227....

    n = LOG[p / (p - ai)] / LOG(1 + i)
    where:
    a = initial amount (450000)
    p = annual payment (50000)
    i = interest rate (.05)
    Thanks I understood your way, but i was given this formula, the formula for present value annuity due:

    PVA = PMT ((1-(1+i)^-(n-1)/i)+1) and i plucked the value PVA=500000, Pmt = 50000 and i=0.05 to arrive at the result of n=13.25. Why can't i arrive at your answer using this formula?
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  10. #10
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    Re: annuity

    Quote Originally Posted by hooke View Post
    Thanks I understood your way, but i was given this formula, the formula for present value annuity due:
    PVA = PMT ((1-(1+i)^-(n-1)/i)+1) and i plucked the value PVA=500000, Pmt = 50000 and i=0.05 to arrive at the result of n=13.25. Why can't i arrive at your answer using this formula?
    That formula looks kinda weird...gives 13.25 instead of 12.25 (1 month difference);
    12.25 is the correct answer: don't know what else to tell you...
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