September 2018 Q4 iv)

[DMF]

Hi,

In part ii) of this exam question you're asked to calculate the standard deviation on £100 invested in Stock A and Stock B (each with their own parameter values). In Part iii, we're told the investor invests £50 in each stock. Part iv) then asks for the standard deviation of the portfolio. Which relies on the standard formula: Var = Xa^2 * Var(A) + Xb^2 * Var(B) + 2* Correlation * Xa * Xb * SD(A) * SD(B).

When looking at the solution the variance figures used in the solution to (iv) are those that were calculated in part (ii) and relate to an investment of £100. The standard deviation figures using £50 are much lower than those calculated in ii). However, given iii and the commentary question in part v - I would have thought that the portfolio value would be £100 not £200. So, I am wondering why the solution uses the figures calculated from ii) and not revised figures based on £50? What am I missing?

Thanks,

 

[Steve Hales]

Hi
I see what you mean. But in the solution to part (iv) we have:
\(
V[P_3] = 0.5^2 V[A_3] + 0.5^2 V[B_3] + 2\rho 0.5\times 0.5\times SD[A_3] \times SD[B_3]
\)
All of those "\(0.5\)'s" represent the proportions of the portfolio invested in each of the securities. So even though the figures for \(SD[A_3]\) and \(SD[B_3]\) are used from part (ii), they're reduced by 50% in part (iv).
Hope that helps.