Question

Expected Return	Std. Deviation
X	15%	50%
M	10%	20%
T-bills	5%	0%

The correlation coefficient between X and M is 2 .2

a)     Draw the opportunity set of securities X and M.  

b)     Find the optimal risky portfolio ( O ), its expected return, standard deviation, and Sharpe ratio. Compare with the Sharpe ratio of X and M.  

c)      Find the slope of the CAL generated by T-bills and portfolio O.

d)    Suppose an investor places 2/9 (i.e., 22.22%) of the complete portfolio in the risky portfolio O and the remainder in T-bills. Calculate the composition of the complete portfolio, its expected return, SD, and Sharpe ratio.   

Answer 1

Given :

	Expected return	Std Deviation
X	15%	50%
M	10%	20%
T bills	5%	0%
Correlation coefficient b/w X & M	2.2

A) Expected return :

E(Rp) = w1E(R1) + w2E(R2)

The standard deviation of the portfolio :

σ2p = w21σ21 + w22σ22 + 2ρ(R1, R2) w1w2σ1σ2,

where

2ρ(R1, R2) is the correlation between X and M

W is the weight of respective assets

σ1 and σ2 are the standard deviations of X and Y

Portfolio	X portion	M portion	Expected Return	Standard deviation
1	0.1	0.9	10.50%	7%
2	0.2	0.8	11.00%	11%
3	0.3	0.7	11.50%	13%
4	0.4	0.6	12.00%	16%
5	0.5	0.5	12.50%	18%
6	0.6	0.4	13.00%	20%
7	0.7	0.3	13.50%	22%
8	0.8	0.2	14.00%	23%
9	0.9	0.1	14.50%	24%
10	1	0	15.00%	25%

Efficient frontier: X-axis--> risk

Y-axis --> return

Formula to Calculate Sharpe ratio:

S(x)=StdDev(rx​)(rx​−Rf​)​where:x=The investment x​=The average rate of return of xRf​=The best available rate of return of risk-free security (i.e. T-bills)StdDev(x)=The standard deviation of rx​​

Therefore Sharpe ratio for

X: 0.2

M 0.25

B)

					With X	With M
Portfolio	X portion	M portion	Expected Return	Standard deviation	Risk free asset   weight	Expectedreturn	Std deviation	Risk free asset   weight	Expectedreturn	Std deviation
1	0.1	0.9	10.50%	7%	0.9	6%	0%	0.1	14%	3%
2	0.2	0.8	11.00%	11%	0.8	7%	1%	0.2	12%	3%
3	0.3	0.7	11.50%	13%	0.7	8%	2%	0.3	11%	2%
4	0.4	0.6	12.00%	16%	0.6	9%	4%	0.4	9%	1%
5	0.5	0.5	12.50%	18%	0.5	10%	6%	0.5	8%	1%
6	0.6	0.4	13.00%	20%	0.4	11%	9%	0.6	6%	1%
7	0.7	0.3	13.50%	22%	0.3	12%	12%	0.7	5%	0%
8	0.8	0.2	14.00%	23%	0.2	13%	16%	0.8	3%	0%
9	0.9	0.1	14.50%	24%	0.1	14%	20%	0.9	2%	0%
10	1	0	15.00%	25%	0	15%	25%	1	0%

From the above table we can see that we can form an optimal portfolio with asset M and risk free asset.