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almost done with this perpetuity problem

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  • almost done with this perpetuity problem

    A perpetuity immediate pays $5000 every two months at a nominal interest rate of i^(4) = .06

    Find its present value

    so i first start by converting i^(4) to i^(6)/6 and I get
    j = i^(6)/6 = .0099751652

    Then the present value would be 5000*a(infinity,j)

    where a(infinity,j) is a perpetuity immediate with interest rate j ?

    But then I was thinking that this wouldnt work since it pays out every 2 months, so then would it look like this?

    5000*(v^2+v^4+v^6+...)

    If so, then I am not sure how you sum up (v^2+v^4+v^6+...)

    any help is appreciated, thanks

  • #2
    It looks like you computed a nominal rate for j. I think you should be using an effective rate.
    Whether you are the lion or the gazelle, when the sun comes up, you better be running.

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    • #3
      v^2+v^4+v^6... is a geometric progression equal to
      v^2 (1+v^2+v^4...) so is equal to v^2[1/(1-v^2)]
      Whether you are the lion or the gazelle, when the sun comes up, you better be running.

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      • #4
        Hi ken, thanks for the reply,

        I am not sure why I need to find an effective rate, since it is a perpetuity then wouldnt the effective rate be much higher and thus make the perpetuity really really big?


        to get j, I just solved the following equation for j: [1+(i^4)/4]^4 =
        [1+(j^6)/6]^6

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        • #5
          Originally posted by astnfan
          Hi ken, thanks for the reply,

          I am not sure why I need to find an effective rate, since it is a perpetuity then wouldnt the effective rate be much higher and thus make the perpetuity really really big?


          to get j, I just solved the following equation for j: [1+(i^4)/4]^4 =
          [1+(j^6)/6]^6
          j is your bi-monthly effective rate.

          In other words, suppose your annual effective interest rate is i, and your m-thly nominal interest rate is denoted i(m). Then, (1+i) = (1+i(m)/m)^m, correct?

          So, i(m)/m is your m-thly effective interest rate - the rate that you can grow the money at exponentially. For example, if you wanted to find out how much would be in your bank account after 2 years and i was the effective interest rate, then it would be (current amount in bank account) * (1+i)^2.

          So, if i(12) the nominal rate, then the amount in the bank after two months is (current amount) * (1+(i(12)/12))^2.

          Getting back to your problem - since you've already made j your bi-monthly interest rate, you don't need to use (v^2 + v^4 + ...). You're only discounting one period at a time, but the way you calculated the rate, the periods are two months long, not one month or one year long. If you wanted to use v^2 + v^4 + ..., you'd have to use j = i(12)/12, which is the monthly effective interest rate.

          Was that okay?

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          • #6
            Originally posted by Ken
            v^2+v^4+v^6... is a geometric progression equal to
            v^2 (1+v^2+v^4...) so is equal to v^2[1/(1-v^2)]
            And Ken is correct on shortening the infinite series. If you can't see how that works, remember two things:
            1) 1 + x + x^2 + x^3 + ... = 1/(1-x) as long as |x| < 1 (absolute value of x < 1)
            2) 1 > v > v^2 = | v^2 | for i > 0

            Finally, let x = v^2, and the result follows from 1).

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            • #7
              Hi wat, thanks for the reply. I think now I am a little more confused. I dont have to use (v^2 + v^4 + ...)? so then how would I write the expression?

              I originally had 5000*(1/j)

              Comment


              • #8
                Originally posted by astnfan
                Hi wat, thanks for the reply. I think now I am a little more confused. I dont have to use (v^2 + v^4 + ...)? so then how would I write the expression?

                I originally had 5000*(1/j)
                That should be it. That doesn't give you the answer you want?

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                • #9
                  Originally posted by wat
                  That should be it. That doesn't give you the answer you want?

                  I get a little over $501k which sounds about right to me. I think my classmate was making it more complicated than it should be.

                  thanks for the help

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