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Math Help - Any questions about annuities

  1. #1
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    Unhappy Any questions about annuities

    Hello friends, given a present value of a uniform, how to determine the annual fee payments if, say, within three years, the interest the first year is 3%, the second year of 8% and third 5%?

    Of course, I have to pay an annuity is like:

    Uniform set value given a present value:

    A= P \left[\displaystyle\frac{i(1 + i)^n}{(1 + i)^n - 1} \right]

    But I do not find what to do with the different interest rates for each of the three years.


    Thank you very much. This is not what I could find on the internet or in books I have on hand.

    Greetings.

    PD: Latex show error...
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  2. #2
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    Quote Originally Posted by Dogod11 View Post
    Hello friends, given a present value of a uniform, how to determine the annual fee payments if, say, within three years, the interest the first year is 3%, the second year of 8% and third 5%?
    Unclear. What is a "uniform"? A "fee"?
    Are you (as example) paying off a loan in 3 annual payments, payments are the same each year,
    but the interest changes each year?

    As example; using loan (or present value) of $1000:
    Code:
    YEAR RATE PAYMENT INTEREST BALANCE
     0                         1000.00
     1    3%  -366.84   30.00   663.16
     2    8%  -366.84   53.05   349.37
     3    5%  -366.84   17.47      .00
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  3. #3
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    Hi, yes, I mean the calculation of the share of an annuity, given a present value, say a loan in this case.

    A general formula for this:

    http://s2.wordpress.com/latex.php?latex=A=%20P%20\left[\displaystyle\frac{i%281%20\pm{}%20i%29^n}{%281%20 \pm{}%20i%29^n%20-%201}%20\right]%20&bg=ffffff&fg=000000&s=0

    That is only with the "+" inside the parentheses, put it well so you know what, because Latex will not let me write it here.


    So I do not know if the problem I have, I use that formula, if so, how to deal with the distitnos values ​​of the interest rate?


    Thank you very much.
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  4. #4
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    Quote Originally Posted by Dogod11 View Post
    Hi, yes, I mean the calculation of the share of an annuity, given a present value, say a loan in this case.
    A general formula for this:
    http://s2.wordpress.com/latex.php?latex=A=%20P%20\left[\displaystyle\frac{i%281%20\pm{}%20i%29^n}{%281%20 \pm{}%20i%29^n%20-%201}%20\right]%20&bg=ffffff&fg=000000&s=0
    That is only with the "+" inside the parentheses, put it well so you know what, because Latex will not let me write it here.
    So I do not know if the problem I have, I use that formula, if so, how to deal with the distitnos values ​​of the interest rate?
    You're confusing me again : what does "distitnos" mean?
    Do you agree with the "accuracy" of the example I gave you?

    Not sure what you're asking; anyhow, 2 points:
    1: the 3 rates can be paid in any order; like 3:8:5 is same as 8:5:3 ...
    2: I think this cannot be solved directly; must be solved numerically...
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  5. #5
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    For example:

    Beatriz Pinzon received a loan of 10 million of his friend Marcela Valencia to pay in 3 years,

    in equal semi-annual, determine the value of the fee if the interest rates for each of the years are:

    a. First year: 8% semiannual

    b. Second year: 10% semester

    c. Third year: 22% per quarter

    http://www.artofproblemsolving.com/F...39767#p2239767
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  6. #6
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    the formula at the top appears to be the payment amount of a level annuity payable annually in arrear (A) if the present value is (P).

    For a level payment, variable interest rate case wont it be true that:

    P = A(v_1 + v_1v_2 + v_1v_2v_3) ?

    where v_k = 1/(1+i_k)

    since you claim to have P already, rearrange for A? wilmer has already shown the solution at 3,8,5%.

    Dont quote me on any of that. Your question is not clear.
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  7. #7
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    Hi, sorry if any confusion, my English is bad:

    I speak of contributions to an annuity payment, how to calculate, given a present value P.

    What usually happens is that one applies the formula of this link:

    View topic - Any questions about annuities • Art of Problem Solving

    When the interest rate is constant, but do not know how to find the amount of an annuity, when the rate is different in each period. As in the example below:

    Beatriz Pinzon Receive a loan of 10 million of historical friend Marcela Valencia to pay in 3 years,

    in equal semi-annual, determine the value of the fee if the Interest Rates'for each of the years are:

    a. First year: 8% semiannual

    b. Second year: 10% semester

    c. Third year: 22% per quarter

    I think it would be solved then numerically, as did Wilmer,


    It is right and now I can understand what I mean?


    Thank you very much.

    PD:

    But in the example of Wilmer, and he gives the payment, otherwise I need is to determine the payment, but with such varying rates. Greetings.
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  8. #8
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    Quote Originally Posted by Dogod11 View Post
    When the interest rate is constant, but do not know how to find the amount of an annuity, when the rate is different in each period. As in the example below:
    Beatriz Pinzon Receive a loan of 10 million of historical friend Marcela Valencia to pay in 3 years,
    in equal semi-annual, determine the value of the fee if the Interest Rates'for each of the years are:
    a. First year: 8% semiannual
    b. Second year: 10% semester
    c. Third year: 22% per quarter
    Your rates are not clearly stated...like 22% per quarter means 5.5% ?
    Plus what's a semester lengthwise?

    Anyway, regardless of the frequency of the rates, ALL rates must be converted
    to semi-annual equivalents, to match the payment frequency.
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  9. #9
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    Hi, ok. I'll try to translate it properly:

    Beatriz Pinzon Receive a loan of 10 million of historical friend Marcela Valencia to pay in 3 years,
    to settle the debt with semiannual payments, determine the value of the shares if the Interest Rates'for Each of the years are:

    First year: 8% compounded semiannually.

    Second year: 10% compounded semiannually.

    Third year: 22% compounded quarterly.

    It is to find the amount of each payment to settle the debt, ie quotas.

    This is done with the formula I put in the link of another forum, when the rate is the same along the entire debt.

    I do not know how to work the values ​​of the different rates.

    My teacher told me to be working with three series.


    A greeting.
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  10. #10
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    The 22% compounded quarterly needs to be converted to semi-annual:
    (1 + r)^2 = 1.055^4 ; r = .113025
    Code:
          payment    interest   rate       balance
     0                                   10,000,000
     1  -1,985,862    400,000  .04        8,414,138
     2  -1,985,862    336,566  .04        6,764,842
     3  -1,985,862    338,242  .05        5,117,222
     4  -1,985,862    255,861  .05        3,387,221
     5  -1,985,862    382,842  .113025    1,784,201
     6  -1,985,862    201,661  .113025            0
    Numerically solved, to closest dollar: $1,985,862
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  11. #11
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    Getting back to your original problem (rates 3%, 5% and 8%), it is possible
    to solve directly IF kept to 3 years:
    u = .03, v = .08, w = .05 (1st, 2nd, 3rd year rates)
    a = 1000 (amount of loan, as example)

    Step 1 : calculate future value of loan amount if no payments :
    f = a(1 + u)(1 + v)(1 + w) = 1168.02

    Step 2 : find a payment p that accumulates to f after 3 payments:
    p + p(1 +v) + p + w[p + p(1 + v)] = f ; simplify:
    p[3 + v + w(2 + v)] = f
    So:
    p = f / [3 + v + w(2 + v)] = 366.84

    You theoretically can work out something similar for a greater number of rates,
    BUT after 3 rates, it gets messier than the Folies Bergeres' dressing room!
    Numerical solving much easier...

    By the way, the order of the rates does not matter in calculating f, BUT it does
    matter in calculating p; the 3 rates can be used 6 ways; 6 different payments:
    3,5,8 : 363.42
    3,8,5 : 366.84 (yours)
    5,3,8 : 365.88
    5,8,3 : 371.70
    8,3,5 : 372.99
    8,5,3 : 375.39

    This is what it all looks like:
    Code:
     FUTURE VALUE LOAN AMOUNT  |     FUTURE VALUE OF PAYMENTS
    YEAR RATE INTEREST BALANCE | YEAR PAYMENT RATE INTEREST BALANCE
     0                 1000.00 |  0                               0
     1    3%   30.00   1030.00 |  1   366.84             0   366.84
     2    8%   82.40   1112.40 |  2   366.84   8%    29.35   763.03
     3    5%   55.62   1168.02 |  3   366.84   5%    38.15  1168.02
    Easy to see why order of rates is important in calculating p,
    since only last 2 rates are used...
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  12. #12
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    Quote Originally Posted by Wilmer View Post
    Unclear. What is a "uniform"? A "fee"?
    Are you (as example) paying off a loan in 3 annual payments, payments are the same each year,
    but the interest changes each year?

    As example; using loan (or present value) of $1000:
    Code:
    YEAR RATE PAYMENT INTEREST BALANCE
     0                         1000.00
     1    3%  -366.84   30.00   663.16
     2    8%  -366.84   53.05   349.37
     3    5%  -366.84   17.47      .00
    I understand your answers, but I do not see how you found numerically payment of 366.84. If you explain me please.

    On the other hand, in another exercise that I have reviewed I would not mind if my friend, because I have a hesitation to build amortization table.

    Natalia Paris received a $ 50 million from Banco Popular to buy a new apartment, if the interest is 30% per semester up, and credit must be paid in equal monthly (advance) for 7 years, determining the value of each fee.

    I know I should spend the effective rate to 0.30 to calculate payments, compounded semi annual 0.30 is equivalent to 0.02356 effective monthly rate.

    However, the advance payment of a series, given a present value, is given by:

    ImageShack® - Online Photo and Video Hosting

    Which I think is right, but I do not know when developing the amortization table in excel, the balance is zero quota is 90 ... I do not know what happens. I appreciate your help.


    A greeting ... And thanks!


    Dogod
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  13. #13
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    Quote Originally Posted by Dogod11 View Post
    I understand your answers, but I do not see how you found numerically payment of 366.84. If you explain me please.
    Can't teach here...use google: "solving numerically"; you'll get sites like:
    Math Forum - Ask Dr. Math
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  14. #14
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    Yes, thanks, in fact I have left is difficult to know what is the equation solved. Because the language of this forum is English. But do not worry I am understanding everything.

    Regarding the second problem, you have an opinion?


    Thank you very much ...
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  15. #15
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    Quote Originally Posted by Dogod11 View Post
    Regarding the second problem, you have an opinion?
    > Natalia Paris received a $ 50 million from Banco Popular to buy a new apartment,
    > if the interest is 30% per semester up, and credit must be paid in equal monthly
    > (advance) for 7 years, determining the value of each fee.
    PLEASE start using proper terminology; above should be a bit like:
    Natalia Paris received a $50 million loan.
    The interest is 30% annually, compounded semi-annually.
    The payments are to be made monthly, in advance, for 7 years.
    Determine the monthly payment amount.

    Your calculation of monthly rate (.02356) is correct.
    Your payment calculation is also right: 1,340,437.35

    I don't know what you mean with stuff like "zero quota"...
    Anyhow, the 1st payment is applied RIGHT AWAY (not one month after loan);
    amortization looks like:
    Code:
        LOAN AMOUNT       PAYMENT       INTEREST        BALANCE
     1 50,000,000.00  -1,340,437.35            .00   48,659,562.65
     2                -1,340,437.35   1,146,419.30   48,465,544.60
    ....
    83                -1,340,437.35      60,997.42    1,309,583.54
    84                -1,340,437.35      30,853.81             .00
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