Question

The following represents the probability distribution of future returns for stock A and stock B. State of Economy Probability Return on Security A Return on Security B Boom 0.20 18% 4% Normal 0.60 8% 8% Recession 0.20 −4% 12%

a. What is the expected return for Security A and Security B?

b. What is the expected return on a portfolio consisting of 50% investment in Security A and 50% in security B?

c. What is the standard deviation of a portfolio consisting of 50% investment in Security A and 50% in security B?

Answer 1

a

Stock A
Scenario	Probability	Return%	=rate of return% * probability
Boom	0.2	18	3.6
Normal	0.6	8	4.8
Recession	0.2	-4	-0.8
	Expected return %=	sum of weighted return =	7.6
Stock B
Scenario	Probability	Return%	=rate of return% * probability
Boom	0.2	4	0.8
Normal	0.6	8	4.8
Recession	0.2	12	2.4
	Expected return %=	sum of weighted return =	8

b

Expected return%=	Wt Stock A*Return Stock A+Wt Stock B*Return Stock B
Expected return%=	0.5*7.6+0.5*8
Expected return%=	7.8

c

Stock A
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(A)^2* probability
Boom	0.2	18	3.6	10.4	0.0021632
Normal	0.6	8	4.8	0.4	9.6E-06
Recession	0.2	-4	-0.8	-11.6	0.0026912
	Expected return %=	sum of weighted return =	7.6	Sum=Variance Stock A=	0.00486
			Standard deviation of Stock A%	=(Variance)^(1/2)	6.97
Stock B
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(B)^2* probability
Boom	0.2	4	0.8	-4	0.00032
Normal	0.6	8	4.8	0	0
Recession	0.2	12	2.4	4	0.00032
	Expected return %=	sum of weighted return =	8	Sum=Variance Stock B=	0.00064
			Standard deviation of Stock B%	=(Variance)^(1/2)	2.53
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	0.2	10.4	-4	-0.000832
Normal	0.6	0.4	0	0
Recession	0.2	-11.6	4	-0.000928
			Covariance=sum=	-0.00176
		Correlation A&B=	Covariance/(std devA*std devB)=	-0.997529844
	Variance	=( w2A*σ2(RA) + w2B*σ2(RB) + 2*(wA)*(wB)*Cor(RA, RB)*σ(RA)*σ(RB))
	Variance	=0.5^2*0.06974^2+0.5^2*0.0253^2+2*0.5*0.5*0.06974*0.0253*-0.99753
	Variance	0.0005
	Standard deviation=	(variance)^0.5
	Standard deviation=	2.24%