Question

Stocks A and B have the following returns in each of the states given below.

	boom	Nornmal Economy	Recession
Stock A return	12%	10%	-5%
Stock B return	1%	-5%	15%

The probability of the boom is 0.5, the probability of the normal economy is 0.3 and the probability of the recession is 0.2.
(a) Calculate the variance of the returns of A and the variance of the returns of B
(b) What is the covariance between the returns of A and B?

(c) What is the standard deviation of a portfolio of A and B with equal amounts invested in both?

Answer 1

Stock A
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(A)^2* probability
Boom	0.5	12	6	4	0.0008
Normal	0.3	10	3	2	0.00012
Recession	0.2	-5	-1	-13	0.00338
	Expected return %=	sum of weighted return =	8	Sum=a. Variance Stock A=	0.0043
			Standard deviation of Stock A%	=(Variance)^(1/2)	6.56
Stock B
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(B)^2* probability
Boom	0.5	1	0.5	-1	0.00005
Normal	0.3	-5	-1.5	-7	0.00147
Recession	0.2	15	3	13	0.00338
	Expected return %=	sum of weighted return =	2	Sum=a. Variance Stock B=	0.0049
			Standard deviation of Stock B%	=(Variance)^(1/2)	7

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	0.5	4	-1	-0.0002
Normal	0.3	2	-7	-0.00042
Recession	0.2	-13	13	-0.00338
			b. Covariance=sum=	-0.004
		Correlation A&B=	Covariance/(std devA*std devB)=	-0.871420402
	Expected return%=	Wt Stock A*Return Stock A+Wt Stock B*Return Stock B
	Expected return%=	0.5*8+0.5*2
	Expected return%=	5
	Variance	=( w2A*σ2(RA) + w2B*σ2(RB) + 2*(wA)*(wB)*Cor(RA, RB)*σ(RA)*σ(RB))
	Variance	=0.5^2*0.06557^2+0.5^2*0.07^2+2*0.5*0.5*0.06557*0.07*-0.87142
	Variance	0.0003
	Standard deviation=	(variance)^0.5
	c. Standard deviation=	1.73%