Question

4. Assume that the portfolio expected return for asset one is 0.15 and the expected return for asset two is 0.10. The variance of both returns is one and the correlation between the returns is 0.75. Assume equal weighting for each asset. Compute the expected return and variance of the portfolio. Assume that the risk free rate is 0.05. Compute the Sharpe ratio of the portfolio. Now allocate 0.60 to asset one and 0.40 to asset two. Recompute all the computations. Which portfolio has the higher Sharpe ratio?

Answer 1

Asset Class	Expected return	Weight	Weight * Expected return	SD
1	15.00%	50%	7.5000%	1
2	10.00%	50%	5.0000%	1
		Total	12.5000%
So expected return is 12.5%
Correlation coefficient	0.75
Calculation of standard deviation
The first step is to calculate the covariance:
COVAB = SDA × SDB × rAB, where rAB is the correlation coefficient between securities A and B.
Now, calculate the standard deviation for the portfolio:
[(SDA2 × WA2) + (SDB2 × WB2) + 2 (WA)(WB)(COVAB)]½
Let's calcualte the co-variance	'=1*1*0.75
	0.75
Now lets calculate the SD
SD portfolio=	'((1^2 * 0.5^2)+(1^2*0.5^2)+(2*0.5*0.5*0.75))^(0.5)
SD portfolio=	0.71
Sharpe ratio = (Mean portfolio return - Risk-free rate)/Standard deviation of portfolio return
Risk free rate	5%
Sharpe ratio=	(12.5%-5%)/0.71
Sharpe ratio=	0.106

Asset Class	Expected return	Weight	Weight * Expected return	SD
1	15.00%	60%	9.0000%	1
2	10.00%	40%	4.0000%	1
		Total	13.0000%
So expected return is 13%
Correlation coefficient	0.75
Calculation of standard deviation
The first step is to calculate the covariance:
COVAB = SDA × SDB × rAB, where rAB is the correlation coefficient between securities A and B.
Now, calculate the standard deviation for the portfolio:
[(SDA2 × WA2) + (SDB2 × WB2) + 2 (WA)(WB)(COVAB)]½
Let's calcualte the co-variance	'=1*1*0.75
	0.75
Now lets calculate the SD
SD portfolio=	'((1^2 * 0.6^2)+(1^2*0.4^2)+(2*0.6*0.4*0.75))^(0.5)
SD portfolio=	0.72
Sharpe ratio = (Mean portfolio return - Risk-free rate)/Standard deviation of portfolio return
Risk free rate	5%
Sharpe ratio=	(13%-5%)/0.72
Sharpe ratio=	0.111