Calculation of covariance given a joint probability function

CFA level I / Quantitative Methods: Basic Concepts / Probability Concepts / Calculation of covariance given a joint probability function

The joint probability function of two random variables, X and Y, denoted P(X, Y), gives the probability of joint occurrences of values of X and Y.

Joint probability function of returns of two stocks A and B

	RB=22 percent	RB=14 percent	RB=8 percent
RA=18 percent	0.35	0	0
RA=12 percent	0	0.40	0
RA=6 percent	0	0	0.25

The expected return of the stock A from the above table = 0.35*0.18 + 0.40*0.12 + 0.25*0.06 = 12.6 percent

The expected return of the stock B from the above table = 0.35*0.22 + 0.40*0.14 + 0.25*0.08 = 15.3 percent

Calculation of covariance:

Cov(R_A, R_B) = E{[R_A - E(R_A)] [R_B - E(R_B)]} = * [R_A - E(R_A)] [R_B - E(R_B)] = 0.35*[(0.18-0.126)(0.22-0.153)] + 0.40*[(0.12-0.126)(0.14 -0.153)] + 0.25*[(0.06-0.126)(0.08-0.153)] = 0.002832

Var(R_A) = E{[R_A - E(R_A)]²} = 0.35*(0.18-0.126)² + 0.40*(0.12-0.126)² + 0.25*(0.06-0.126)² = 0.002124

Var(R_B) = E{[R_B - E(R_B)]²}= 0.35*(0.22-0.153)² + 0.40*(0.14-0.153)² + 0.25*(0.08-0.153)² = 0.003801

σ(R_A) = √Var(R_A) = √0.002124 = 0.0461 = 4.61 percent

σ(R_B) = √Var(R_B) = √0.003801 = 0.0617 = 6.17 percent

Corr(R_A, R_B) = Cov(R_A, R_B)/ σ(R_A) σ(R_B) = 0.002832/(0.0461)(0.0617) = 0.9967.

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