Question

3. Which of the following statements are true? Please Explain. a. A lower allocation to the risky portfolio reduces the Sharpe (reward-to-volatility) ratio. b. The higher the borrowing rate, the lower the Sharpe ratios of levered portfolios. c. With a fixed risk-free rate, doubling the expected return and standard deviation of the risky portfolio will double the Sharpe ratio. d. Holding constant the risk premium of the risky portfolio, a higher risk-free rate will increase the Sharpe ratio of investments with a positive allocation to the risky asset.

Answer 1

Answer-

The correct Option is b. The higher the borrowing rate the lower the Sharpe ratios of levered portfolios

A higher borrowing rate is a result of the risk of the borrowers’ default. In perfect markets with no additional cost incurred for default, this increment would equal the value of the borrower’s option to default, and the Sharpe measure, with appropriate treatment of the default option, would be the same but the costs to default so that this part of the increment lowers the Sharpe ratio.
The option (a) is incorrect. The lower allocation to the risky portfolio does not reduces the Sharpe ratio.

The option  (c) is incorrect because doubling the expected return with a fixed risk-free rate will more than double the risk premium and the Sharpe ratio.
The option (d) The statement is incorrect as holding constant the risk premium of risky portfolio a higher risk free rate will not increase the Sharpe ratio of investments with positive allocation to the risky asset,