Q&A Bank 4 QUESTION 4.12

Discussion in 'CT1' started by sujak, Feb 2, 2016.

  1. sujak

    sujak Member

    In the course reading, we have the recursive formulae for E(Ak) and E(A^2k) with amounts 1 only. How do we arrive at the formulae given in the solution for this question 4.12. Is this formula available in the tables, or do we have to memorise them.
     
    sujak, Feb 2, 2016
    #1
  2. John Lee

    John Lee ActEd Tutor Staff Member

    Alas no. Although questions on this area have only really tested the mean.

    But personally I would just remember the recursive relationship idea. Start with the investment at time zero \(A_0 = P_0\) and then you multiply that by the investment return between times 0 and 1 and then add on the next investment at time 1:

    \(A_1 = P_0*(1+i_1) + P_1\)

    and so on.
     
    John Lee, Feb 3, 2016
    #2
  3. suraj

    suraj Member

    You can do it this way
    Let A, B, C be the returns in year 1, 2, 3 respectively. So accumulated amount at the end of year 3 will be?

    \( 50000(1+A)(1+B)(1+C) + 30000(1+B)(1+C) + 20000(1+C) \)

    Now find the Mean and S.D. of the above expression.
    Formulas which'll be used are the usual ones like

    \( Var(X) = E(X^2) - E^2(X) \\
    Var(X+Y+Z) = Var(X) + Var(Y) + Var(Z) ~~ \text{if X, Y, Z are independent} \\
    Var(XY) \ne Var(X).Var(Y) \\

    E(XY) = E(X).E(Y) ~~ \text{if X, Y are independent} \\
    E(X+Y+Z) = E(X)+E(Y)+E(Z) \\
    \)
     
    Last edited by a moderator: Feb 4, 2016
    suraj, Feb 4, 2016
    #3

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