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Math Help - Geometric Series or Complex Polynomial

  1. #1
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    Geometric Series or Complex Polynomial

    Before I post the problem I should check that this is being posted in the right forum. Not being a mathematician myself I am not even certain of the area of maths I should be posting the question. Some say the solution would be solved with a complex polynomial and others say it is a variation on a geometric series). Apologies if this is in the wrong forum and would you kindly point me to the right one.

    In non mathematical terms here is the problem. I have clients who ask me to calculate a return on their investment from existing pension portfolios. The inputs I usually receive are the dates of lump sums invested and amounts and the dates of regular monthly sums invested, their respective dates and amounts. Then I look up the current value and want to try to figure out what the return or ever something resembling a best estimate of a return would be.



    example case
    a) Lump sum invested in December Jan 2011 of 55,186 plus
    b) Regular monthly contributions made since Jan 2011 amounting to 64,220. Monthly contribution amounts have varied between 1394 per month and 1878 per month.
    c) Total value of contributions is a+b (120,036) however b has been invested over a period of time.
    d) Value today is 149,000

    I would be happy taking average value of monthly contributions as a guide if that would make the problem easier to solve.

    Any help would be appreciated.
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  2. #2
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    Re: Geometric Series or Complex Polynomial

    To find the "rate of return", I assume you want the average rate of return (as market fluctuations continuously change the instantaneous rate of return). Assume it is a constant interest rate. Assume it is compounded monthly. Now, this is a problem we can solve. Set it up like you are calculating compound interest. Let's say the client invests for n months. Let's say p_k is the amount the client invests in month k. So, p_0 is the principle investment: 55,186. The total value (after compounding interest) would be:

    \sum_{k=0}^n p_k\dfrac{\left(1+\dfrac{i}{12}\right)^{n-k}-1}{\tfrac{i}{12}} = 149,000

    So,
    \sum_{k=0}^n p_k\left(1+\dfrac{i}{12}\right)^{n-k} = \dfrac{149,000i}{12}+120,036

    As some have told you, this would involve solving a complicated polynomial (the variable I am using is i, which is "annual interest", or annual rate of return). However, there are techniques for solving complicated polynomials that take relatively very little time (especially when you know the approximate range for zeros of the polynomial).

    Alternately, if you use the average monthly contribution, then you can simplify the problem slightly to:

    p_0\left(1+\tfrac{i}{12}\right)^n+p_{avg}\sum_{k=1  }^{n}\left(1+\tfrac{i}{12}\right)^{n-k} = \dfrac{149,000i}{12}+120,036
    The LHS simplifies to this:

    p_0\left(1+\tfrac{i}{12}\right)^n+p_{avg}\dfrac{ \left( 1+\tfrac{i}{12} \right)^n - 1 }{ \tfrac{i}{12} } = \dfrac{149,000i}{12}+120,036

    This is a simpler polynomial, but will still require a computer algebra system to solve.
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  3. #3
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    Re: Geometric Series or Complex Polynomial

    I think I need a bit of time to understand what is going on here. thank you for your reply. I was advised to post the question on business maths and am also looking at whether this can be done with excel using IRR formulas.

    If I had to choose the method it would be excel however I have no way of checking the answers once excel comes up with an answer. Here is what I posted in the business maths section

    Date Payments
    Mar-14 1,879
    Feb-14 1,879
    Jan-14 1,879
    Dec-13 1,879
    Nov-13 1,879
    Oct-13 1,879
    Sep-13 1,879
    Aug-13 1,879
    Jul-13 1,879
    Jun-13 1,746
    May-13 1,746
    Apr-13 1,746
    Mar-13 1,746
    Feb-13 1,746
    Jan-13 1,746
    Dec-12 1,514
    Nov-12 1,514
    Oct-12 1,514
    Sep-12 1,514
    Aug-12 1,514
    Jul-12 1,514
    Jun-12 1,514
    May-12 1,514
    Apr-12 1,514
    Mar-12 1,514
    Feb-12 1,514
    Jan-12 1,514
    Dec-11 1,401
    Nov-11 1,401
    Oct-11 1,401
    Sep-11 1,401
    Aug-11 1,401
    Jul-11 1,401
    Jun-11 1,401
    May-11 1,401
    Apr-11 1,401
    Mar-11 1,394
    Feb-11 1,394
    Jan-11 1,394

    I will ask a friend to help me out with the calculations.
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