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combo problem: increasing annuity+reinvestment of interest

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  • combo problem: increasing annuity+reinvestment of interest

    This comes from the finan manual, problem 32.7.

    John invests 100 at the end of year one, 200 at the end of year two, etc until he invests 1,000 at the end of year ten. The investment goes into a bank account earning 4%. At the end of each year, the interest is paid into a second bank account earning 3%. Calculate the total amount John will have after 10 years.

    OK, the first part is simple. Just sum 100+200+....+1000.

    After that, I don't get it. So, the interest that gets reinvested looks like this....4, 12, 24, 40....etc..

    The only way I can think of to do this is in a series of increasing annuities, each seperated by a year. Is that right? Or is there a faster way? Seems like a little overkill for a problem.
    Exams Passed:

    P FM

  • #2
    I got the right answer via the brute force method. Interesting.....Is there a faster way than calculation the future vale of 9 increasing annuities?
    Exams Passed:

    P FM


    • #3
      Did you get the same answer as the book? I got 6201.314181 instead. Close but not sure if this is right. Anyway I set up the series for Accumulated Interest then add the Accumulated Interest to the sum of payments.


      • #4
        Yeah, I did get the same answer you got. I did the same approach. Did you email Prof. Finan? He usually isn't much help. Usually, he will just say stuff like "try again" or "look at this...."
        Exams Passed:

        P FM


        • #5
          By the way, the Accumulated Interest (AI) could be simplified to 100*(i/j)*[(Is")angle 9|j - 45]. This was done by the following process.

          AI = 100i * [ (1+j)^8 + 3(1+j)^7 + ... + 36(1+j)^1 + 45 ]
          (1+j) * AI = 100i * [ (1+j)^9 + 3(1+j)^8 + ... + 45(1+j)^1 ]

          Therefore j*AI = 100i * [ (1+j)^9 + 2(1+j)^8 + ... + 9(1+j) - 45 ]

          Recall that (Is")angle 9|j = (1+j)^9 + ... + 9(1+j)

          Hence j*AI = 100i * [ (Is")angle 9|j - 45 ] or AI = 100*(i/j)*[(Is")angle 9|j - 45]

          This is how I worked out my problem. I don't understand what you mean by brute force but I have an impression that you used
          AI = 100i * [ (1+j)^8 + 3(1+j)^7 + ... + 36(1+j)^1 + 45 ] and plug in the numbers to get AI. That is too much work. :-P