Covariance, Correlation, and Joint Probability

Historical covariance or other techniques, such as market model regression with historical return data, can help us forecast return covariance and correlation. We use the joint probability function of the random variables for this estimation.

The probability that values of the two random variables \(X\) and \(Y\) will occur simultaneously is given by the joint probability function of \(X\) and \(Y\), denoted as \(P(X, Y)\). For instance, \(P(X = 3, Y = 4)\) represents the likelihood that \(X\) and \(Y\) will be equal to 3 and 4, respectively.

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

$$ \begin{align*} Cov_(X_i Y_i )& =E[(X_i-\bar X)(Y_j-\bar Y)] \\
& =\sum_i \sum_j P(X=x_i,Y=y_j) (X_i-\bar X)(Y_j-\bar Y) \end{align*} $$

This formula calculates the covariance between random variables $X$ and $Y$, such as portfolio returns.

To find it, we take the sum of the products of the deviations of $X$ and $Y$ from their expected values for all possible outcomes.

Each product is weighted by the probability of that specific outcome occurring.

Independence and Correlation

Two random variables, \(X\) and \(Y\), are independent if \(P(X,Y)=P(X)\cdot P(Y)\). That is, \(X\) and \(Y\) are independent. We find the product of independent probability to calculate joint probability.

The independence property is stronger than correlation because the correlation coefficient addresses linear relationships.

If random variables \(X\) and \(Y\) are uncorrelated (also holds for independent random variables), then:

$$ E(XY)=E(X) \cdot E(Y) $$

Example: Calculating the Covariance \(\#1\)

Assume 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:

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

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:

$$ \begin{align*}
W_{ABC} & =\frac {1000}{2000}=0.5 \\
W_{XYZ} &=\frac {1000}{2000}=0.5 \end{align*} $$

Next, we should calculate the individual expected returns:

$$ \begin{align*}
E(R_{ABC}) &=0.15 \times 0.40+0.60 \times 0.2+0.25 \times 0.00=0.18 \\
E(R_{XYZ}) & =0.15 \times 0.2+0.60 \times 0.15+0.25 \times 0.04=0.13 \end{align*} $$

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

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

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:

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

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

Solution

$$ \begin{align*}
\text{Expected return of Stock A} & = (40\% \times 2.3\%)+(60\% \times 8\%)=5.72\% \\
\text{Expected return of Stock B} & = (40\% \times 6.5\%)+(60\%\times 3\%)=4.40\% \end{align*} $$

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

The deviations of returns at the economic growth of

$$ < 4\% =(2.3-5.72) \times (6.5-4.40)=?7.182 $$

The deviations of returns at the economic growth of

$$ > 4\% =(8-5.72) \times (3-4.40)=?3.192 $$

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

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

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).

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

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

1. 0.054.
2. 0.1542.
3. 0.1442.

Solution

The correct answer is C.

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

The expected return for TY:

$$ \begin{align*} & =(0.5 \times 6\%)+(0.3 \times 3\%)+(0.2 \times (-1\%)) \\ & =3\%+0.9\%-0.2\%=3.7\% \end{align*} $$

The expected return for Ford:

$$ \begin{align*} & =(0.5 \times 10\%)+(0.3 \times 4\%) +(0.2 \times (-4\%)) \\ & =5\%+1.2\%-0.8\%=5.4\% \end{align*} $$

Next, we can now compute the covariance:

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

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

In real life, this scenario is highly likely because the companies belong to the same industry. As a result, they share similar systematic risks.