Question

Consider the following statistics of the returns of Stock A, Stock B and the market (m):

s_A = 0.20      corr_A,m = 0.4

s_B = 0.30      corr_B,m = 0.7

s_m = 0.15

E(r_m) = 0.10

Suppose further that the risk-free rate is 5%.

       (a)    According to the Capital Asset Pricing Model, what should be the expected return of Stock     A and of Stock B? [Hint: This is an open-book exam.]                     

       (b) Suppose that the correlation between the return of Stock A and the return of Stock B is 0.5.     What are the expected return and the standard deviation of the return of a portfolio that has        a 40% investment in Stock A and a 60% investment in Stock B?            

       (c)    Assume that the Capital Asset Pricing Model is valid. How could you construct a new     portfolio (P) using the market portfolio (m) and the risk-free asset that has the same expected return as the portfolio you considered in part (b)? What is the standard deviation of the return       of this portfolio (P)?                                                

(d) i)     Suppose that the correlation between the return of Stock A and the return of Stock B is             -0.7. What are the expected return and the standard deviation of the return of a portfolio     that        has a 40% investment in Stock A and a 60% investment in Stock B? Explain the difference in value between the standard deviations of the return of the portfolios in        parts (c) and (d).                                                   

        ii)     Consider an investment that is made up of a combination of the risky portfolio in (i) and the risk-free asset. In this case, would it be advisable to switch to a portfolio made up    of the market portfolio and the risk-free asset? In other words, is the market portfolio mean-variance efficient in this case? [Hint: Consider the measure that captures the risk- return trade-off.]    

Answer 1

		Covariance(Asset1,2) =Correlation (Asset1,2)*Standard Deviation Asset1*Standard Deviation Asset2
		Covariance A,m=corrA,m*SA*Sm
		Covariance A,m=0.4*0.2*0.15=	0.012
		Covariance B,m=corrB,m*SB*Sm
		Covariance B,m=0.7*0.3*0.15=	0.0315
		Beta of Stock A=Covariance A,m/Variance of market Return
		Variance of market return =sm^2=0.15^2=	0.0225
		Beta of Stock A=Ba=0.012/0.0225	0.533333
		Beta of Stock B=Covariance B,m/Variance of market Return
		Variance of market return =sm^2=0.15^2=	0.0225
		Beta of Stock B=Bb=0.0315/0.0225	1.4
	(a)	CAPM equation:
		Rs=Rf+Beta*(Rm-Rf)
		Rs=Expected Return of Stock
		Rm=Expected market return=	0.1
		Rf=Risk Free rate=5%	0.05
		Ra=Expected Return of A=0.05+Ba*(0.1-0.05)
		Ra=Expected Return of A=0.05+0.53333*(0.1-0.05)	0.0767
		Rb=Expected Return of B=0.05+Bb*(0.1-0.05)
		RB=Expected Return of B=0.05+1.4*(0.1-0.05)	0.1200
	(b)	CovarianceA, B,=corrA,B,*SA*SB
		Covariance A,B=0.5*0.2*0.3=	0.03
		Portfolio Return=Rp=wa*Ra+wb*Rb
		wa=Weight of A in the portfolio	0.4
		wb=Weight of B in the portfolio	0.6
		Rp=Portfolio Return =0.4*0.0767+0.6*0.12	0.1027
		Portfolio Variance =Vp= (wa^2)*(SA^2)+(wb^2)*(SB^2)+2*wa*wb*Covariance A,B
		Portfolio Variance =Vp= (0.4^2)*(0.2^2)+(0.6^2)*(0.3^2)+2*0.4*0.6*0.03
		Vp=Portfolio Variance	0.0532
		Sandard Deviation of the portfolio=Sqiuare Root (Variance)	0.2307	(SQRT(0.0532)
	.(c)	wm=Weight of market portfolio
		wf=Weight of Risk Free Asset=1-wm
		Portfolio Return =0.1027=wm*0.1+wf*0.05
		wm*0.1+(1-wm)*0.05=0.1027
		0.05wm=0.1027-0.05=	0.0527
		wm=0.0527/0.05=	1.053333
		wf=1-1.05333=	-0.05333
		This means Take Loan at risk free rate and invest in market
		Standard Deviation of Risk Free rate (Sf)=0
		Covariance f,m=0
		Vp=Portfolio Variance =(wm^2)*(Sm^2)
		Portfolio Standard Deviation=Square Root of Vp=wm*Sm	0.158	(1.05333*0.15)
	(d)	Covariance A,B=-0.7*0.2*0.3=	-0.042
		wa=Weight of A in the portfolio	0.4
		wb=Weight of B in the portfolio	0.6
		Rp=Portfolio Return =0.4*0.0767+0.6*0.12	0.1027
		Portfolio Variance =Vp= (wa^2)*(SA^2)+(wb^2)*(SB^2)+2*wa*wb*Covariance A,B
		Portfolio Variance =Vp= (0.4^2)*(0.2^2)+(0.6^2)*(0.3^2)+2*0.4*0.6*(-0.042)
		Vp=Portfolio Variance	0.01864
		Sandard Deviation of the portfolio=Sqiuare Root (Variance)	0.1365	(SQRT(0.01864)
		Expected Portfolio Return =Rp=wa*Ra+wb*Rb	0.1027
		Return remains the same, but Standard deviation is decreased significantly
	ii)	Reward/Risk Ratio of the portfolio =Rp/Standard Deviation	0.75198	(0.1027/0.1365)
		Reward/Risk Ratio of the Market =Rm/Sm	0.666667	(0.1/0.15
		Reward /Risk ratio of the portfolio is higher
		It will not be advisable to switch to market+riskfree asset