Question

PLEASE SHOW WORK

A.What is the standard deviation of returns for Stocks A and B?RA = 23.5%   RB = 14.6%   σA = 13.05%   σB = 4.98%

State	Probability	Expected Return Stock A             Stock B
Expansion	30%	40%	22%
Normal	60%	20%	12%
Recession	10%	-5%	8%

B.What is the covariance of returns between stocks A and B?σA,B = 60.9 %2

C.What is the correlation coefficient?ρA,B = 0.94

Answer 1

A. It is given that

	Pi	RAi	RBi
Expansion	30%	40%	22%
Normal	60%	20%	12%
Recession	10%	-5%	8%

First we will the expected returns of both the stocks

Let E(R_A) and E(R_B) be the expected return of the stock A & B respectively

E(R_A) = 30% x 40% + 60% x 20% + 10% x -5% = = 12% + 12% - 0.5% = 23.5%

E(R_B) = 30% x 22% + 60% x 12% + 10% x 8% = 6.6% + 7.2% + 0.8%= 14.6%

Let σ(A) and σ(B) be standard deviation of stock returns

Calculation of Standard deviation of Stock A
	Pi	RAi	E(RA)	RAi - E(RA)	[RAi - E(RA)]2	Pi [RAi - E(RA)]2
Expansion	30%	40%	23.50%	16.50%	0.027225	0.0081675
Normal	60%	20%	23.50%	-3.50%	0.001225	0.0007350
Recession	10%	-5%	23.50%	-28.50%	0.081225	0.0081225
					Total	0.0170250

σ(A) = 0.130479 = 13.0479% = 13.05%

Calculation of Standard deviation of Stock B
	Pi	RBi	E(RB)	RBi - E(RB)	[RBi - E(RB)]2	Pi [RBi - E(RB)]2
Expansion	30%	22%	14.60%	7.40%	0.005476	0.0016428
Normal	60%	12%	14.60%	-2.60%	0.000676	0.0004056
Recession	10%	8%	14.60%	-6.60%	0.004356	0.0004356
					Total	0.0024840

σ(B) = 0.04983 = 4.983% = 4.98%

B. Let σ(AB) be the covariance of returns of stock A and B

	Pi	RAi	RBi	E(RA)	E(RB)	RAi - E(RA)	RBi - E(RB)	[RAi - E(RA)].[RBi - E(RB)]
Expansion	30%	40%	22%	23.50%	14.60%	16.500%	7.400%	0.0122100
Normal	60%	20%	12%	23.50%	14.60%	-3.500%	-2.600%	0.0009100
Recession	10%	-5%	8%	23.50%	14.60%	-28.500%	-6.600%	0.0188100
							Total	0.0319300

σ(AB) = 30% x 0.122100 + 60% x 0.0009100 + 10% x 0.188100 = 0.003663 + 0.000546 + 0.001881 = 0.006090 = 60.90 %²

C.Correlation coefficient = ρ_AB = σ(AB) / σ(A).σ(B) = 0.006090 / (0.1305 x 0.0498) = 0.006090 / 0.00649890 = 0.93708 = 0.94