Question

The Sharpe ratio and Jensen’s alpha of portfolio A are 0.10 and 0.004, respectively. The risk-free rate is 3%, the average return on the market portfolio is 7%, the variance of the market portfolio is 0.09, and the correlation coefficient between A and the market portfolio is 0.7. What is the expected return and the variance of A?

Answer 1

Sharpe Ratio =(Expected Return-Risk free Rate)/Standard Deviation =0.10
Expected Return =0.10*Standard Deviation +Risk Free Rate
Expected Return -0.10*Standard Deviation =3% (Equation 1)

Beta =Correlation*Standard Deviation of A/Standard Deviation of Market =0.7*Standard Deviation of A/0.09^0.5 =7*Standard Deviation A/3
Jensen alpha =Expected Return -Required Rate using CAPM
0.004=Expected Return -(Risk Free Rate+Beta*(Market return -Risk Free Rate))
0.004 =Expected Return -(3%+7/3*Standard Deviation*(7%-3%))
0.004 =Expected Return -3%-7/3*4%*Standard Deviation
Expected Return -0.093333*Standard Deviation =0.004+3%
expected Return -0.093333 *Standard Deviation =3.4% (equation 2)

Expected Return -0.10*Standard Deviation =3% (Equation 1)
Expected Return -0.093333 *Standard Deviation =3.4% (equation 2)

solving equation 1 and 2 by subtracting both the equations we get
0.10 *Standard Deviation-0.093333*Standard Deviation =3.4%-3%
Standard Deviation =0.4%/(0.10-0.093333) =0.60
Variance =0.60^2 =36%

Using Standard Deviation of 0.60 in equation 1 we get
Expected Return -0.10*Standard Deviation =3%
Expected Return =0.10*0.60+3% =9%