Covariance of Portfolio Returns Given a Joint Probability Distribution

Covariance between variables can be calculated in two ways. One method is the historical sample covariance between two random variables \(X_i\) and \(Y_i\). It is based on a sample of past data of size \(n\) and is given by:

$$\text{Cov}_{X_i,Y_i}=\frac{\sum_{i=1}^{n}{(X_i -\bar{X})(Y_i -\bar{Y})}}{n-1}$$

Alternatively, covariance can be defined as the probability-weighted average of the cross-products of each random variable’s deviation from its own expected value. That is:

$$\text{Cov}_{X_i,Y_i}=E\left[(X_i -\bar{X})(Y_i -\bar{Y})\right]$$

Consider the following example:

Example: Calculating the Covariance #1

Suppose we wish to find the variance of each asset and the covariance between the returns of ABC and XYZ, given that the amount invested in each company is $1,000.

This table is used to calculate the expected returns:

$$ \begin{array}{c|c|c|c} & \textbf{Strong Economy} & \textbf{Normal Economy} & \textbf{Week Economy} \\ \hline \text{Probability} & {15\%} & {60\%} & {25\%} \\ \hline \text{ABC Returns} & {40\%} & {20\%} & {0\%} \\ \hline \text{XYZ Returns} & {20\%} & {15\%} & {4\%} \\ \end{array} $$

Solution

For us to find the covariance, we must calculate the expected return of each asset as well as their variances. The assets’ weights are:

$$ \text W_{\text{ABC}}=\cfrac {1000}{2000} = 0.5 $$

$$ \text W_{\text{XYZ}}=\cfrac {1000}{2000} = 0.5 $$

Next, we should calculate the individual expected returns:

$$ \text E(\text R_{\text{ABC}}) = 0.15 × 0.40 + 0.60 × 0.2 + 0.25 × 0.00 = 0.18 $$

$$ \text E(\text R_{\text{XYZ}}) = 0.15 × 0.2 + 0.60 × 0.15 + 0.25 × 0.04 = 0.13 $$

Finally, we can compute the covariance between the returns of the two assets:

$$ \begin{align*}
\text{Cov}(\text R_{\text{ABC},\text{XYZ}}) &= 0.15(0.40 – 0.18)(0.20 – 0.13) \\
& + 0.6(0.20 – 0.18)(0.15 – 0.13) \\
& + 0.25(0.00 – 0.18)(0.04 – 0.13) \\
& = 0.0066
\end{align*} $$

Example: Calculating the Covariance #2

A portfolio manager is considering the following two possible economic growth of a country and the joint variability of returns on two stocks in a portfolio:

$$\begin{array}{l|c|c}
\textbf {Economic Growth } & \bf {<4 \%} & \bf {>4 \%} \\
\hline \text { Probability } & 40 \% & 60 \% \\
\hline \text { Return of Stock A } & 2.3 \% & 8 \% \\
\hline \text { Return of Stock B } & 6.5 \% & 3 \% \\
\end{array}
$$

What is the covariance between the return of Stock A and Stock B?

Solution

Expected return of Stock A \(= (40\% × 2.3\%) + (60\% × 8\%) = 5.72\%\)

Expected return of Stock B \(= (40\% × 6.5\%) + (60\% × 3\%) = 4.40\%\)

Note: For the rest of the calculation, your curriculum sometimes ditches the percentage signs so that 4.40% becomes 4.40.

The deviations of returns at economic growth of < 4% \(= (2.3 – 5.72) × (6.5-  4.40) = -7.182\)

The deviations of returns at economic growth of >4% \(= (8 -5.72) × (3-4.40) = -3.192\)

The covariance of returns between stock A and stock B is computed as follows:

$$\text{Cov}(\text R_{\text{A},\text{B}}) = (-7.182 ×  0.40) + (-3.192 × 0.60) = -4.788$$

Interpretation: Since covariance is negative, the two returns show some co-movement in opposite signs.

Question

The following table represents the estimated returns for two motor vehicle production brands – TY and Ford, in 3 industrial environments: strong (50% probability), average (30% probability), and weak (20% probability).

$$ \begin{array}{c|c|c|c} {} & \textbf{TY Returns +6%} & \text{TY Returns +3%} & \textbf{TY Returns -1%} \\ \hline {\text{Ford Sales }+10\%} & \text{Strong (0.5)} & {} & {} \\ \hline {\text{Ford Sales }+4\%} & {} & \text{Average (0.3)} & {} \\ \hline {\text{Ford Sales }-4\%} & {} & {} & \text{Weak (0.2)} \\ \end{array} $$

Given the above joint probability function, the covariance between TY and Ford returns is closest to:

A. 0.054.

B. 0.1542.

C. 0.1442.

Solution

The correct answer is C.

First, we must start by calculating the expected return for each brand:

$$ \text{Expected return for TY} = (0.5 × 6\%) + (0.3 × 3\%) + (0.2 × (-1\%)) = 3\% + 0.9\% – 0.2\% = 3.7\% $$

$$ \text{Expected return for Ford} = (0.5 × 10\%) + (0.3 × 4\%) + (0.2 × (-4\%)) = 5\% + 1.2\% – 0.8\% = 5.4\% $$

Next, we can now compute the covariance:

$$ \begin{align*}
\text{Covariance} & = 0.5(6\% – 3.7\%)(10\% – 5.4\%) \\
& + 0.3(3\% – 3.7\%)(4\% – 5.4\%) \\
& + 0.2(-1\% – 3.7\%)(-4\% – 5.4\%) \\
& = 5.29\% + 0.294\% + 8.836\% \\
& = 0.1442 \\
\end{align*} $$

Interpretation: The covariance is positive. This means that the returns for the two brands show some co-movement in the same direction.

Note: This would most likely be the case in real life because the companies are in the same industry, and therefore, the systematic risks affecting them are quite similar.