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Increasing Perpetuity. Please help

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  • Increasing Perpetuity. Please help

    Smith has 100,000 which she buys a Perpetuity on Jan 1, 2005. Suppose i=.045 and the perpetuity has annual payment beg Jan 1 2006. The first 3 payments are 2000 each, the next three payment are 2000(1+r) each. Increasing forever by a factor of 1+r every 3 years . What is r ?

  • #2
    Is the answer
    r = .078425625 ?


    • #3
      yeh thats correct. Can you please post the solution


      • #4
        I'm not very good at explaining things online but here goes:

        1) you should separate every 3 year period.
        So at year 3, your accumulated value is 2000S_3 (i=.045)
        time 6, (1+r) * 2000 * S_3 (i = .045)
        time 9, (1+r)^2 * 2000 * S_3 (i = .045)
        and goes on forever into perpetuity

        2) now draw another timeline if you want. times 0,3,6,9,12,----
        I found the effective interest of I for 3 years(i used iconv on the calculator for this) And this is now an immediate perpetuity you are familiar seeing:
        PV = PMT/(i-r)

        3) First payment is 2000*S_3.
        Divide it by I-r
        set it equal to the given PV $100000

        wish i could draw you a picture


        • #5
          OK. I did it in a similar fashion, but I'll explain how I did it just incase it helps you to understand it more. I looked at it like a perpetuity every 3 years. Take the future value of payments 1 and 2 and then add them to payment 3: 2000(1.045)^2+2000(1.045)+2000=6274.05.

          Now your effective interest rate at 4.5% converts to 14.12% every 3 years.

          Use the formula for the sum of an infinite geometric series: k/(1-r), k=6274.05/1.1412 and r=(1+r)/1.1412. So, set 100,000=5497.88/(1-((1+r)/1.1412)). This should give you the same answer as above.
          There ain't no easy way out.

          -Tom Petty