Sharpe ratio of sum of random variable

CFA Level I
#1

Hello All,

I am not able to get the OA.

Let X = Random variable denotes the change in the value of stock of Microsoft.

Let Y = Random variable denotes the change in the value of stock of IBM.

Given, E(X) = 0.10; E(Y) = 0.12; Var (X+Y) = 14.64

Find Sharpe Ratio of Z = X+Y, if risk-free return = 1.5%.

Here’s what I did:

Sharpe Ratio = (E(X) + E(Y) - Rf)/ Sqrt (Var (X+Y)) = (0.22 - 0.015)/Sqrt(14.64) = 0.05357.

However, the OA is 0.050. I found out that official answer uses 2*Rf instead Rf in the above equation. Is this how it is supposed to be? Can someone please explain this?

Thanks in advance.

#2

It’s a weird question, with a weird approach to the answer.

It appears that the author wants you to approach this as if, say, you had invested $1.00 in Microsoft and $1.00 in IBM. Then Z = X + Y would be the expected return (in dollars) on a $2.00 investment, and σ²(X + Y) would be the variance (in dollars²) of a $2.00 investment. In that case, you want the risk-free return on a $2.00 investment, which would be $0.03 (= $2.00 × 1.5%).

What makes the approach weird is that you’re computing the Sharpe ratio using dollar amounts everywhere, rather than percentages everywhere.

A more understandable approach would be to consider a portfolio comprising 50% Microsoft and 50% IBM. Then the epxected return is 0.5(0.10) + 0.5(0.12) = 0.11, the risk-free rate is 1.5%, and the standard deviation of the portfolio returns is 0.5(σ(X + Y)).

#3

Hello S2000magician,

Thank you for your insightful comments! Could you explain what’s the weirdness in using numbers, as opposed to percentages? I am curious.

Best regards and thanks in advance.

#4

It’s weird only because the Sharpe ratio is defined using _ rates _ of return, not amounts of return. If using amounts doesn’t bother you (and it shouldn’t), then you’re fine.

#5

Thank you S2000magician for clarifying this to me!

#6

My pleasure.