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Math Help - Annuity problem solving for i (interest rate)

  1. #1
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    Annuity problem solving for i (interest rate)

    The present value of a 10-year annuity-immediate with level annual payments and interest rate i is x. The present value of a 20-year annuity-immediate with the same payment and interest rate is 1.5x. Find i.

    So far I have:

    x=\frac{1-v^{10}}{i}
    1.5x=\frac{1-v^{20}}{i}
    where v=\frac{1}{1+i}

    So:

    1.5(\frac{1-v^{10}}{i})=\frac{1-v^{20}}{i}

    From here multiply both sides by i and get:

    1.5-1.5v^{10}=1-v^{20}


    From here I'm not sure what to to do.

    I know that v^{10}=0.5 => v=0.933032992 => i=0.071773462

    I was thinking that I could substitute x^{2}=v^{20} and x=v^{10} to setup and solve quadratic formula.

    Doing so gives x=0.5,1

    Therefore v^{10}=0.5, 1

    But I'm not sure that's a proper way of solving the problem. Thanks
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  2. #2
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    Re: Annuity problem solving for i (interest rate)

    your approach with the quadratic formula is valid (i haven't checked your working though)
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  3. #3
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    Re: Annuity problem solving for i (interest rate)

    Quote Originally Posted by downthesun01 View Post
    From here multiply both sides by i and get:

    1.5-1.5v^{10}=1-v^{20}
    OK; that simplifies to:
    2v^20 - 3v^10 + 1 = 0
    Let x = v^10; then:
    2x^2 - 3x + 1 = 0; solve to get x = 1 or x = 1/2 ; reject x = 1.

    So v^10 = 1/2
    (1 / (1 + i))^10 = 1/2
    1 / (1 + i) = (1/2)^(1/10)
    Solve for i ; OK?
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