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Math Help - Rate change

  1. #1
    MHF Contributor
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    Rate change

    Code:
    0 rate=9%=u%            1000.00=a
     
    k rate=12%=v%
     
    11=n                    3000.00=b
    Ok, here's the dope:
    1000 bucks deposited in savings account at year 0, at 9% compounded annually.
    12% compounded annually will later be the rate, effective at a time k which will
    result in a future value of 3000 bucks after 11 years.
    For this to happen, k = 5.451118.....

    If opening rate = u and opening deposit = a,
    second rate = v , future value = b and number of years = n,
    (v > u, b > a(1+u)^n)
    what is k in terms of a,b,n,u,v?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Wilmer View Post
    Code:
    0 rate=9%=u%            1000.00=a
     
    k rate=12%=v%
     
    11=n                    3000.00=b
    Ok, here's the dope:
    1000 bucks deposited in savings account at year 0, at 9% compounded annually.
    12% compounded annually will later be the rate, effective at a time k which will
    result in a future value of 3000 bucks after 11 years.
    For this to happen, k = 5.451118.....

    If opening rate = u and opening deposit = a,
    second rate = v , future value = b and number of years = n,
    (v > u, b > a(1+u)^n)
    what is k in terms of a,b,n,u,v?
    After $$ k years at rate $$ r_1, and (N-k) years at rate $$ r_2 you have:

    b=a (1+r_1)^k(1+r_2)^{N-k}

    To solve for $$ k given $$ r_1, $$ r_2, $$ N and $$ a is an excercise in the laws of logarithms.

    CB
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  3. #3
    MHF Contributor
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    Yep...general case formula:

    k = LOG[(1 + u) / (1 + v)] / LOG[b / a / (1 + v)^n]
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