Question

Assume a market portfolio consisting of two securities, i = 1,2. Further, assume that the weight of security 1 in the market portfolio is W₁ = 0.5 and that the covariance matrix is given below.

a) Compute the variance of the market portfolio, ?_M² when the standard deviations for i = 1,2 are 5 and 8 percent respectively .

	R1	R2
R1	0.2	0.1
R2	0.1	0.2

b) Compute the CAPM-betas of security 1 and security 2, i.e. 1 and 2. [4]

c) Assume that E[rM] = 0.05 (i.e., 5%) and rf = 0.01 (i.e., 1%). Compute the expected return of both securities and their Sharpe ratios.

Answer 1

	R1	R2	DeviationsR1	DeviationsR2
R1	0.2	0.1	0.05	-0.05
R2	0.1	0.2	0.1	0.2
Average	0.15	0.15
Sum			0.15	0.15

cov1,2= 1/n( Summation of deviations of stock1 *deviations of stock2

[-0.05*0.05+0.1*0.2]/2=0.0175/2= 0.00875

Variance of market portfolio=	SD1^2+W1^2+SD2^2+W2^2+2WAWBCOV(A,B)
	0.05*0.05+0.5*0.5+0.08*0.08*0.5*0.5+2*0.5*0.5*0.00875
	0.0066

SD of portfolio= Square root of variance= 0.08124

Cov(a,m)= Wa*Sda^2+Wb*sdab
	0.5*0.05*0.05+0.5*0.08124
	0.04187

Beta A= Cov(a,m)/variance market

0.04187/0.0066

6.343939

Cov(b,m)= Wb*SDb^2+Wa*SDab
	0.5*0.08*0.08+0.5*0.08124
	0.04382

Beta B= Cov(b,m)/variance market

0.04382/0.0066

6.639394

Expected CAPM return of security = Rf+Beta (Rm-Rf)

Security A CAPM return= 1+6.343939*(5-1)

=26.375%

Security B CAPM return = 1+6.639394*(5-1)

=27.557%

Sharpes Ratio=(Rp-Rf)/SD of security

Rp= 15*0.5+15*0.5= 15%

Security A= (0.15-0.01)/0.05= 2.8

Security B= (0.15-0.01) /0.08= 1.75