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?
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)).
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.