Question

The standard deviation of Asset A returns is 36%, while the standard deviation of Asset M returns in 24%. The correlation between Asset A and Asset M returns is 0.4. (a) The average of Asset A and Asset M’s standard deviations is (36+24)/2 = 30%. Consider a portfolio, P, with 50% of funds in Asset A and 50% of funds in Asset M. Will the standard deviation of portfolio P’s returns be greater than, equal to, or less than 30%? Explain this answer intuitively. (b) What, specifically, will be the standard deviation of portfolio P returns? (c) Asset M is in fact the “market” portfolio. What is the Beta coefficient for Asset M? For Asset A? For Portfolio P? (d) Assume that the CAPM holds, that the risk free interest rate is 4% and that the expected return on the market is 9.5%. What is the expected return on Asset A? On portfolio P?

Answer 1

.(a)Consider a portfolio, P, with 50% of funds in Asset A and 50% of funds in Asset M.The standard deviation of portfolio P’s returns will be less than 30%

If w1, w2 , w3 …wn are weight in the portfolio for assets 1, 2,3 ….n

Then,w1+w2+w3+……………………+wn=1

S1, S2, S3……Sn are the standard deviation of the assets 1, 2, 3 …n

Portfolio Variance=(w1^2)*(S1^2)+(w2^2)(S2^2)+………….(wn^2)*(Sn^2)+2w1w2*Cov(1,2)+2w1w3*Cov(1,3)+………+w(n-1)wn*Cov(n,(n-1)

Cov(1,2)=Covariance of returns of asset1 and asset2

Portfolio Standard Deviation =Square root of Portfolio variance

In this case,

w1=w2=0.5,

S1=Standard Deviation of Asset A returns=36%

S2=Standard Deviation of Asset M returns=24%

Cov(1,2)=Covariance of returns of asset1 and asset2=(Correlation of Assets1,2)*S1*S2=0.4*36*24=345.6

Portfolio Variance =(0.5^2)*(36^2)+(0.5^2)*(24^2)+2*0.5*0.5*345.6=640.8

Portfolio Standard Deviation =Square Root (640.8)=25.3%

.(c) Beta for market =1

Beta Coefficient for Asset M=1

Beta=(Covariance)/(Variance of the market return)

Covariance of Asset A and M=345.6

Variance of market return=(24^2)=576

Beta Coefficient of asset A=345.6/576=0.6

Beta of the Portfolio =w1*Beta1+w2*Beta2

w1=w2=0.5

Beta1=Beta of Asset A=0.6

Beta2=Beta of Asset M=1

Beta Coefficient of the Portfolio=0.5*0.6+0.5*1=0.8

As per CAPM

Expected Return of asset=Rs=Rf+Beta*(Rm-Rf)

Rf=Risk Free rate=4%

Beta of asset A=0.6

Beta of portfolio=0.8

Rm=Expected market return=9.5%

Expected Return on Asset A=4+0.6*(9.5-4)= 7.3%

Expected Return of the portfolio =4+0.8*(9.5-4)=8.4%