Q&A Bank 4 QUESTION 4.12

[sujak]

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.

 

[John Lee]

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.

 

[suraj]

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?

Spoiler

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

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) \\
\)