Expected value, standard deviation, covariance, and correlation of returns on a portfolio

CFA level I / Quantitative Methods: Basic Concepts / Probability Concepts / Expected value, standard deviation, covariance, and correlation of returns on a portfolio

We have already discussed the expected value and its calculation. The expected value of a random variable is the probability-weighted average of the possible outcomes of the random variable.

E(X) = Expected value of random variable X = ∑P(X_i)X_i

Calculation of portfolio expected return: The expected return of a portfolio with n securities is a weighted average of the expected returns on the component securities.

E(R_P) = w₁E(R₁) + w₂E(R₂) + .....+ w_nE(R_n)

The variance of a random variable is the expected value of squared deviation from the random variable's expected value.

σ²(X) = Var(X) = E{[X - E(X)]²}

A variance is a number greater than or equal to zero because it is the sum of squared terms. When all the values are equal, then the sum of squared terms equal zero, and the variance is zero. Variance measures the risk or dispersion. Its unit is percent squared for returns.

The standard deviation is the positive square root of the variance. The unit of standard deviation is the same as that of the variance, and it is easier to interpret. It also signifies the risk.

Example 5: Calculating variance and standard deviation

The probability distribution for a company's EPS is given below in the table. Probability EPS($) 0.10 0.80 0.15 1.00 0.25 1.10 0.30 1.30 0.20 1.50 Solution: Expected value of EPS = E(X) = 0.10*$0.80 + 0.15*$1.00 + 0.25*$1.10 + 0.30*$1.30 + 0.20*$1.50 = $1.195. Variance = Var(X) = E{[X - E(X)]2} = 0.0454 dollar squared Probability EPS($) (X) X - E(X) [X - E(X)]2 p(X)[X - E(X)]2 0.10 0.80 -0.395 0.156 0.0156 0.15 1.00 -0.195 0.038 0.0057 0.25 1.10 -0.095 0.009 0.0023 0.30 1.30 0.105 0.011 0.0033 0.20 1.50 0.305 0.093 0.0186 E(X)=1.195 0.0455 Standard deviation = Square root of variance = $0.2132

The covariance between two random variables is the probability-weighted average of the cross products of each random variable's deviation from its expected value. It measures how a random variable varies with another random variable.

Cov(R_i, R_j) = E{[R_i - E(R_i)][R_j - E(R_j)]}

Properties of Covariance:

Covariance is symmetric i.e. Cov(X,Y) = Cov(Y,X). It can range from negative infinity to positive infinity. The covariance of a variable with itself is equal to its variance which is always positive.

A negative covariance means that the two variables move together in the opposite direction. A positive covariance means that the two variables move in the same direction. A zero covariance means that the two variables are unrelated.

Limitations of Covariance:

The covariance has comparability issues across data sets having different scales. It is dependent on the data and scaling of data and can take extremely large values. It does not tell anything about the strength of the relationship between two variables. Two variables having a covariance of 100 may not have a strong relationship than two variables having a covariance of 50.

Example 6: Calculating covariance between two variables

The returns of two stocks are given below for three interest rate scenarios. State P(S) RA RB Increase in interest rates 0.20 0.05 0.02 No change in interest rates 0.50 0.12 0.09 Decrease in interest rates 0.30 0.14 0.18 Calculate and interpret the covariance of return for stock A and stock B. Solution: Expected return of stock A, E(RA) = 0.20*0.05 + 0.50*0.12 + 0.30*0.14= 0.112 Expected return of stock B, E(RB) = 0.20*0.02 + 0.50*0.09 + 0.30*0.18 = 0.103 State P(S) RA RB P(S)[RA-E(RA)][RB-E(RB)] Increase 0.20 0.05 0.02 0.20*(0.05-0.112)(0.02-0.103) No change 0.50 0.12 0.09 0.50*(0.12-0.112)(0.09-0.103) Decrease 0.30 0.14 0.18 0.30*(0.14-0.112)(0.18-0.103) The covariance of returns for stock A and stock B = 0.20*(0.05-0.112)(0.02-0.103) + 0.50*(0.12-0.112)(0.09-0.103) + 0.30*(0.14-0.112)(0.18-0.103) = 0.001624 A covariance is a positive number which means that the returns on the two stocks are positively related. It does not tell anything about the strength of the relationship between the two variables.

The correlation coefficient measures the strength and direction of the linear relationship between two variables. It is calculated by dividing the covariance between two variables divided by the product of their standard deviations.

ρ(R_i, R_j) = Corr(R_i, R_j) = Cov(R_i, R_j)/σ(R_i)σ(R_j)

Properties of Correlation:

It measures the strength of the relationship between two random variables. It has no units and lies between -1 to +1. A correlation coefficient of +1 indicates a perfect positive correlation between two random variables. A correlation coefficient of -1 indicates a perfect negative correlation between two random variables. A correlation coefficient of zero indicates no linear relationship between two random variables.

Two variables with a correlation coefficient of -0.80 are more strongly related to each other than the two variables with a correlation coefficient of +0.50. The magnitude matters rather than sign when looking at the strength of the relationship.

Limitations of Correlation:

It does not specify the factors that cause the linear relationship between the two variables. It is not good for measuring the non-linear relationship.

Example 7: Calculating correlation coefficient between two variables

What is the correlation coefficient between the returns of two stocks having a covariance of 0.00546? The variance of the stocks A and B are 0.0039 and 0.0082 respectively. Solution: Corr(RA, RB) = Var(RA, RB)/ σ(RA)σ(RB) σ(RA) = √Var(RA) = √0.0039 = 0.06245 σ(RB) = √Var(RB) = √0.0082 = 0.09055 Corr(RA, RB) = 0.00546/(0.06245)(0.09055) = 0.9655 There is a strong positive correlation between the stock A and stock B. Note that the sign of the correlation coefficient is always dependent on the sign of covariance because the sign of standard deviation is always positive.

Expected return on a portfolio: The expected return on a portfolio is the weighted average of the expected returns on the constituents securities.

Weight of asset i = w_i = Market value of asset i/Market value of portfolio E(R_P) = ∑w_iE(R_i) = w₁E(R₁) + w₂E(R₂) + .... + w_NE(R_N)

Variance of a portfolio: The variance of a portfolio is not only the function of the variance of constituent securities but also the covariances between the constituent securities. Var(R_P) = ∑∑w_iw_jCov(R_i, R_j)

Variance of two-asset portfolio = w_A²σ(R_A)² + w_B²σ(R_B)² + 2w_Aw_BCov(R_A, R_B) = w_A²σ(R_A)² + w_B²σ(R_B)² + 2w_Aw_B σ(R_A) σ(R_B)ρ(R_A, R_B)

Variance of three-asset portfolio = w_A²σ(R_A)² + w_B²σ(R_B)² + w_C²σ(R_C)²  + 2w_Aw_BCov(R_A, R_B) + 2w_Bw_CCov(R_B, R_C) +2w_Cw_ACov(R_C, R_A)

Covariance Matrix: A covariance matrix contains the covariances between the variables.

Covariance matrix of 2 assets

	Stock A	Stock B
Stock A	0.0024	0.0084
Stock B	0.0084	0.0145

In the above matrix, the variance of stock A is 0.0024, the variance of stock B is 0.0145, and the covariance of stock A and B is 0.0084.

Covariance matrix of 3 assets

	Stock A	Stock B	Stock C
Stock A	0.0039	0.0028	-0.0091
Stock B	0.0028	0.0082	0.0013
Stock C	-0.0091	0.0013	0.0130

Note that each covariance will occur twice and variance will occur only once in the covariance matrix.

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