Calculating Covariance Given a Joint Probability Function

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 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

Assume that we want 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:

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

Interpretation: since covariance is positive, the two returns show some co-movement, though it’s a weak one.

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).

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

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. (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.)

Reading 8 LOS 8m

Calculate and interpret covariance given a joint probability function.