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Thread: Problem with Geometric Series Equation.

  1. #1
    Actuary.com - Level I Poster
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    Problem with Geometric Series Equation.

    Before I begin:
    Book Answer: 4%
    My Answer: 4.2%

    Here's the Question:
    Mike buys a perpetuity immediate with varying annual payments. DUring the first 5 years, the payment is constant and equal to 10.. Beginning in year 6, the payemnts start to increase. For year 6 and all future years, the current year's payment is K% larger than the previous year's payment. At an annual effective interest rate of 9.2%, the perpetuity has a present value of 167.50. Calculate K, given K<9.2


    Now it seems simple enough. We start with a 5 year Annuity-immediate and add the perpetuity (starting at year 6 and discounted to time 0 -- So it's like a deferred perpetuity of sorts).

    Thus we have

    Annuity: a[5] @ i = 9.2% --> = 38.6955

    Perpetuity:
    Let J = [(1+k)/(1+i)]^n

    PV = 10v^5 * (1-J)/(i-k)

    Write this on paper if this doesn't seem clear. I didn't bother making these look pretty, but this equation can be proven when showing the payment series.

    Well, with a perpetuity we have infinite payments so we conclude that
    n = INFINITY

    Thus J = 0 (because 1+k < 1+i)

    So we get

    167.5 = 38.6955 + 10v^5 * 1/(.092-k)
    Solve for k...

    k = 4.2%

    Anyone have any ideas? Anyone out there think they can solve it properly? The book takes a different approach that doesn't make sense to me.

    Refer to ASM P. 243 Question #2.

  2. #2
    Actuary.com - Level I Poster
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    Solved!

    Anyone having trouble, pay attention and take notes

    Here is the solution to this question if you are also having trouble. This goes out to any future FM Exam takers or anyone who's curious!

    The equation that I provided actually includes a first payment WITHOUT modification. In other words, when we have 1/(i-K) * PMT, we actually have
    PMT + PMTv + PMTv^2 + PMTv^3 etc.

    This discounts it all back to t = 0.

    Now, this method will be the EASIEST way to solve this problem. The ASM book has a slightly trickier way I believe and Dr. Krys's method left me stumped.

    OK, Sorry for all the back story, here it is:

    Since payments start to increase at t=6, that means that t=5 is the first payment that has the original payment amount.

    Since we are making 5 constant $10 payments we can actually create an annuity of 4 payments.

    a[4] @ i = 9.2%
    Now we can use the equation I provided above of 1/i-K (again, you can prove this yourself. it's a fun time. Believe me!)

    Equation this we get,

    10a[4] + 10v^4 * [1/(.092-K)] = 167.50
    135.245 / 10v^4 = 1/(.092-K)
    19.2314 = 1/(.092-K)
    1/19.2314 = .092 - k
    K = 4
    K/100 = 4%

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