Question

State of Economy	Probability of State of Economy	Return on Stock J	Return on stock K
Bear	.23	-.013	.041
Normal	.58	.145	.069
Bull	.19	.225	.099

What is the Convariance and Correlation between the returns of the 2 stocks?

Answer 1

State of Economy	Probability	J Return	K Return
Bear	0.23	-0.13	0.041
Normal	0.58	0.145	0.069
Bull	0.19	0.225	0.099

For Stock J

Expected Return = p₁R₁ + p₂R_{2 +} p₃R₃

where p_i is the probability of i state of the economy and R_i is the return during i state of the economy.

Expected Return of J = E[R_J] = [0.23*(-0.013)] + [0.58*0.145] + [0.19*0.225] = 0.12386

Variance of stock J is calculate using the below formula:

σ_J² = p₁*(R_J1 - E[R_J])² + p₂*(R_J2 - E[R_J])² + p₃*(R_J3 - E[R_J])²​​​​​​​

σ_J² = 0.23*(-0.013-0.12386)² + 0.58*(0.145-0.12386)²​​​​​​​ + 0.19*(0.225-0.12386)²​​​​​​​ = 0.0065108204

Standard deviation of J = σ_J = 0.0065108204^1/2 = 0.080689655

For Stock K

Expected Return on K = p₁R_K1 + p₂R_{K2 +} p₃R_K3

Expected Return of K = E[R_K] = [0.23*(0.041)] + [0.58*0.069] + [0.19*0.099] = 0.06826

Variance of stock K is calculate using the below formula:

σ_K² = p₁*(R_K1 - E[R_K])² + p₂*(R_K2 - E[R_K])²​​​​​​​ + p₃*(R_K3 - E[R_K])²​​​​​​​

σ_K² = 0.23*(0.041-0.06826)² + 0.58*(0.069-0.06826)²​​​​​​​ + 0.19*(0.099-0.06826)²​​​​​​​ = 0.0003507724

Standard deviation of K = σ_K = 0.0016886028^1/2 = 0.018728919

Covariance between J and K

Cov(J, K) = p_1*(R_J1 - E[R_J])*(R_K1 - E[R_K]) + p_2*(R_J2 - E[R_J])*(R_K2 - E[R_K]) + p_3*(R_J3 - E[R_J])*(R_K3 - E[R_K])

Cov(J, K) = 0.23*(-0.013-0.12386)*(0.041-0.06826) + 0.58*(0.145-0.12386)*(0.069-0.06826) + 0.19*(0.225-0.12386)*(0.099-0.06826) = 0.0014578764

The relation between Covariance and correlation is: Cov(J, K) = ρ_J,K*σ_J*σ_K

where ρ_J,K is the correlation between the return of stock J and K

ρ_J,K= Cov(J, K)/(σ_J*σ_K) = 0.0014578764/(0.080689655*0.018728919) = 0.964695251

Covariance = 0.0014578764

Correlation = 0.964695251​​​​​​​