Question

Could portfolio A show a higher Sharpe ratio than that of B and at the same time a lower M² measure? Explain.

Answer 1

We know M2 value is given by multiplying sharpe ratio with the standard deviation of the benchmark index and then adding risk free rate of return to arrive at the value of M2.

Now deriving M2 for the portfolio A:

Let the Sharpe ratio be SR_A

The standard deviation of the benchmark index be _b  

Let the risk free rate of return be r_f

Let M2 value for A be M2_A

Therefore, M2_A = SR_A * b + r_f  

Now deriving M2 for the portfolio B:

Let the Sharpe ratio be SR_B   

In this case also, we will use the same standard deviation and risk free rate of return.

Let M2 value for B be M2_B

Therefore, M2_B = SR_B * _b + r_f

Now, it is given that M2_A is lower than M2_B

Therefore, M2_A < M2_B

SR_A * b + r_f <  SR_B * _b + r_f  

Now cancelling out the similar terms, we got

SR_A < SR_B

So, Sharpe ratio of A is lower than sharpe ratio of B.

Therefore, it is not possible for a portfolio having higher sharpe ratio and lower M2 value.