Question

Problem 11-25 Portfolio Returns and Deviations [LO 1, 2]

Consider the following information on a portfolio of three stocks:

State of	Probability of			Stock A			Stock B			Stock C
Economy	State of Economy			Rate of Return			Rate of Return			Rate of Return
Boom		.15			.02			.32			.60
Normal		.55			.10			.12			.20
Bust		.30			.16		−	.11		−	.35

a. If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the portfolio’s expected return, the variance, and the standard deviation? (Do not round intermediate calculations. Round your variance answer to 5 decimal places, e.g., 32.16161. Enter your other answers as a percent rounded to 2 decimal places, e.g., 32.16.)

Expected return		%
Variance
Standard deviation		%

b. If the expected T-bill rate is 3.75 percent, what is the expected risk premium on the portfolio? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Expected risk premium             %

Answer 1

a.

Stock A
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(A)^2* probability
Boom	0.15	2	0.3	-8.6	0.0011094
Normal	0.55	10	5.5	-0.6	0.0000198
Bust	0.3	16	4.8	5.4	0.0008748
	Expected return %=	sum of weighted return =	10.6	Sum=Variance Stock A=	0.002
			Standard deviation of Stock A%	=(Variance)^(1/2)	4.48
Stock B
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(B)^2* probability
Boom	0.15	32	4.8	23.9	0.00856815
Normal	0.55	12	6.6	3.9	0.00083655
Bust	0.3	-11	-3.3	-19.1	0.0109443
	Expected return %=	sum of weighted return =	8.1	Sum=Variance Stock B=	0.02035
			Standard deviation of Stock B%	=(Variance)^(1/2)	14.26
Stock C
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(C)^2* probability
Boom	0.15	60	9	50.5	0.03825375
Normal	0.55	20	11	10.5	0.00606375
Bust	0.3	-35	-10.5	-44.5	0.0594075
	Expected return %=	sum of weighted return =	9.5	Sum=Variance Stock C=	0.10373
			Standard deviation of Stock C%	=(Variance)^(1/2)	32.21
Covariance Stock A Stock B:
Scenario	Probability	Actual return% -expected return% for A(A)	Actual return% -expected return% For B(B)	(A)*(B)*probability
Boom	0.15	-8.6000	23.9	-0.0030831
Normal	0.55	-0.6	3.9	-0.0001287
Bust	0.3	5.40	-19.1	-0.0030942
			Covariance=sum=	-0.006306
		Correlation A&B=	Covariance/(std devA*std devB)=	-0.987491968
Covariance Stock A Stock C:
Scenario	Probability	Actual return% -expected return% for A(A)	Actual return% -expected return% for C(C)	(A)*(C)*probability
Boom	0.15	-8.6	50.5	-0.0065145
Normal	0.55	-0.6	10.5	-0.0003465
Bust	0.3	540.00%	-44.5	-0.007209
			Covariance=sum=	-0.01407
		Correlation A&C=	Covariance/(std devA*std devC)=	-0.975895953
Covariance Stock B Stock C:
Scenario	Probability	Actual return% -expected return% For B(B)	Actual return% -expected return% for C(C)	(B)*(C)*probability
Boom	0.15	23.9	50.5	0.01810425
Normal	0.55	3.9	10.5	0.00225225
Bust	0.3	-19.1	-44.5	0.0254985
			Covariance=sum=	0.045855
		Correlation B&C=	Covariance/(std devB*std devC)=	0.998098586
	Expected return%=	Wt Stock A*Return Stock A+Wt Stock B*Return Stock B+Wt Stock C*Return Stock C
	Expected return%=	0.4*10.6+0.4*8.1+0.2*9.5
	Expected return%=	9.38
	Variance	=w2A*σ2(RA) + w2B*σ2(RB) + w2C*σ2(RC)+ 2*(wA)*(wB)*Cor(RA, RB)*σ(RA)*σ(RB) + 2*(wA)*(wC)*Cor(RA, RC)*σ(RA)*σ(RC) + 2*(wC)*(wB)*Cor(RC, RB)*σ(RC)*σ(RB)
	Variance	=0.4^2*0.04477^2+0.4^2*0.14265^2+0.2^2*0.32206^2+2*(0.4*0.4*0.04477*0.14265*-0.98749+0.4*0.2*0.14265*0.32206*0.9981+0.4*0.2*-0.9759*0.04477*0.32206)
	Variance	0.010793
	Standard deviation=	(variance)^0.5
	Standard deviation=	10.39%

b.

Expected risk premium = Portfolio return-risk free rate = 9.38-3.75=5.63%