Covariance and correlation

Covariance of two assets
Covariance is a measure of the degree to which returns on two risky assets move in same direction. A positive covariance means that asset returns move together. A negative covariance means returns move inversely.

Cov (X, Y) = Σ(P_i (X_i – E(X) (Y_i – E(Y_a)

Covariance between two random variables X and Y is defined as:

• If -ve, when X is above its mean its is likely that Y is below its mean value;
• If zero, on average the values of the two variables are unrelated.
• If +ve, when X is above its mean its is likely that Y is above its mean value.

Note:

• The covariance of a random variable with itself, its own covariance, is equal to its variance.
  Cov (X, X) = Var (X)

Correlation of two assets
Correlation is a measure that determines the degree to which two variable's movements are associated. Unlike covariance, it enables comparisons between different pairs of assets.

Corr(X,Y) =p_xy= Cov(X,Y) / (σ_X σ_Y)

The correlation coefficient will vary from -1 to +1. The higher the coefficient, the more associated the movements of two assets.

• If =1, perfect positve corrrelation
• If = -1, perfect negative correlation
• If =0, no linear relationship

Covariance between two random variables R₁ and R₂ using the joint probability function for R₁ and R₂ is:
Cov(R₁,R₂) = Σ_iΣ_jP(R_1,i, R_1,,j) [R_2,j – E(R_2,j)]

Note:
Expected return of a two asset portfolio
= w₁ x E(R₁) + w₂ x E(R₂).
Variance of a two asset portfolio
= w₁² σ₁²+ w₂²σ₂² + 2 w₁ w₂ Cov(1,2) = w₁²σ₁² + w₂²σ₂² + 2 w₁w₂ σ₁ σ₂ p1,2