Coefficient of Variation as a Measure of Relative Dispersion

Suppose two investments have the same coefficient of variation. Would the investments _most likely_ have the same return reward on each unit of risk based on the Sharpe ratio?

Incorrect. When the risk-free return was subtracted from the numerator, the larger return would be reduced a smaller percentage than the smaller return.

Incorrect. If the subtraction of the risk-free return had a different impact on one of the investments, then the Sharpe ratio would work out to different numbers and would not be equal.

Correct. The risk-free rate would have a smaller impact on the larger return. For example, one investment had a standard deviation of 2 and an average return of 4 while the other investment had a standard deviation of 3 and an average return of 6. The coefficient of variation of each would be 0.5. If the risk-free return was 1%, then the Sharpe ratios would be as follows. $$\displaystyle \frac{4-1}{2} = 1.50$$ $$\displaystyle \frac{6-1}{3} = 1.67$$

No, since if there was a difference in their returns, the subtraction of the risk-free asset return would have a larger impact on the larger return

Yes, since if there was a difference in their returns, the subtraction of the risk-free asset return would have the same impact on both ratios

No, since if there was a difference in their returns, the subtraction of the risk-free asset return would have a smaller impact on the larger return