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State of Economy Probability of State of Economy Return on Stock J Return on stock K...

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?

Solutions

Expert Solution

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 + p2R2 + p3R3

where pi is the probability of i state of the economy and Ri is the return during i state of the economy.

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

Variance of stock J is calculate using the below formula:

σJ2 = p1*(RJ1 - E[RJ])2 + p2*(RJ2 - E[RJ])2 + p3*(RJ3 - E[RJ])2​​​​​​​

σJ2 = 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 = σJ = 0.00651082041/2 = 0.080689655

For Stock K

Expected Return on K = p1RK1 + p2RK2 + p3RK3

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

Variance of stock K is calculate using the below formula:

σK2 = p1*(RK1 - E[RK])2 + p2*(RK2 - E[RK])2​​​​​​​ + p3*(RK3 - E[RK])2​​​​​​​

σK2 = 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 = σK = 0.00168860281/2 = 0.018728919

Covariance between J and K

Cov(J, K) = p1*(RJ1 - E[RJ])*(RK1 - E[RK]) + p2*(RJ2 - E[RJ])*(RK2 - E[RK]) + p3*(RJ3 - E[RJ])*(RK3 - E[RK])

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) = ρJ,KJK

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

ρJ,K= Cov(J, K)/(σJK) = 0.0014578764/(0.080689655*0.018728919) = 0.964695251

Covariance = 0.0014578764

Correlation = 0.964695251​​​​​​​


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