Question

There are two risky portfolios on the Efficient Frontier: Portfolio A has an expected return of 0.10 with a variance of returns of 0.0625. Portfolio B has an expected return of 0.12 with a variance of returns of 0.16. If Portfolio B is the optimal risky portfolio on the Efficient Frontier, is the risk-free rate 0.05 or 0.08?

Answer 1

The Sharpe ratio should be highest to be enable on efficient frontier

Now we are given 2 Risk free rates and told that Portfolio B is optimal risk portfolio

so we have to calculate Sharpe ratio for both risk free rate

Risk free rate = Rf = 0.05

variance of portfolio A = 0.0625, so SDp = SQRT(0.0625) =0.25

Sharpe ratio for portfolio A = (Rp -Rf) /SDp = (0.10 -0.05)/0.25 = 0.20

variance of portfolio B = 0.16, so SDp = SQRT(0.16) =0.4

Sharpe ratio for portfolio B = (Rp -Rf) /SDp = (0.12 -0.05)/0.4 = 0.175

Here, Portfolio A has highest Sharpe ratio, so risk free rate is not 5%

NOW

Risk free rate = Rf = 0.08

Sharpe ratio for portfolio A = (Rp -Rf) /SDp = (0.10 -0.08)/0.25 = 0.08

Sharpe ratio for portfolio B = (Rp -Rf) /SDp = (0.12 -0.08)/0.4 = 0.1

Here, Portfolio B has highest Sharpe ratio, so risk free rate is 8%

Answer : 8%