Covariance

The covariance is a measure of the degree of co-movement between two random variables. For instance, we could be interested in the degree of co-movement between the rate of interest and the rate of inflation.

• X = interest rate
• Y = inflation

The general formula used to calculate the covariance between two random variables, X and Y, is:

$$COV\ [X,\ Y] =E[(X-E[X]) (Y-E[Y])]$$

The covariance between two random variables can be positive, negative, or zero.

• A positive number indicates co-movement (i.e., the variables tend to move in the same direction);
• A value of zero indicates no relationship; and
• A negative value shows that the variables move in opposite directions.

Moreover, if variables are independent, their covariance is zero, i.e.,

$$COV\left(X,Y\right)=E\left(XY\right)-E\left(X\right)E\left(Y\right)=0$$

Covariances can be represented in a tabular format in a covariance matrix as follows:

• The off-diagonal terms represent variances since Cov(C, C) = Var(C).
• A two-asset portfolio would have a similar 2 × 2 matrix.
• A correlation matrix can also be created to represent the correlations between various assets in a large portfolio.

Example 1: Calculating the covariance of a portfolio of two assets

A portfolio comprises two stocks – 1 and 2. The returns for the last 5 years are as follow:

Stock 1: 5%; 4.5%; 4.8%; 5.5%; 6%.

Stock 2: 6%; 6.2%; 5.7%; 6.1%; 6.5%.

Compute the covariance.

Solution

Step 1: We calculate the weighted sum of each stock to get the expected return on Stock 1 and Stock 2

$$ E(R_1) = \frac{5\%+4.5\%+4.8\%+5.5\%+ 6\%}{5} = 5.2\% $$

$$ E(R_2) = \frac{6\%+ 6.2\%+ 5.7\%+ 6.1\%+ 6.5\%}{5} = 6.1\% $$

Step 2: We subtract each year’s return from the expected return to obtain [R1-E(R1)] and [R2-E(R2)] as follows:

Step 3: We multiply the values obtained in Step 2 and we divide by the number of observations to get a mean observation.

$$Cov\left(R_1,R_2\right)=E\left[R_1-E\left(R_1\right)\right]\left[R_2-E\left(R_2\right)\right]$$
$$=\frac{(-0.2\%\times-0.1\%)+(-0.7\%\times0.1\%)+(-0.4\%\times-0.4\%)+(0.3\%\times0\%)+(0.8\%\times0.4\%)}{5}$$
$$=\frac{0.000043}{5}=0.0000086$$

Example 2: Covariance

Consider a set of a well-diversified portfolios X and Y. Suppose that the mean returns for X and Y are 6.12 and 7.04, respectively, what is the covariance between these portfolios?

Solution

Recall,

\(COV\left(X,Y\right)=E\left(XY\right)-E\left(X\right)E\left(Y\right)\), for independent variables,

\(E\left(XY\right)=E\left(X\right)E\left(Y\right)=6.12\times7.04=43.0848\)

Which implies that,

\(COV\left(X,Y\right)=E\left(XY\right)-E\left(X\right)E\left(Y\right)=43.0848-6.12\left(7.04\right)=0\)

Correlation

Correlation is the ratio of the covariance between two random variables and the product of their two standard deviations i.e.

$$\text{Correlation}\ (X_1,X_2\ )=\frac{Cov(X_1,X_2\ )}{Standard\ deviation\ (X_1\ )\times Standard\ deviation\ (X_2\ )}$$

Correlation measures the strength of the linear relationship between two variables. While the covariance can take on any value between negative infinity and positive infinity, the correlation is always a value between -1 and +1.

A correlation of -1 indicates a perfect inverse relationship (i.e. a unit change in one means that the other will have a unit change in the opposite direction). Secondly, a correlation of +1 indicates a perfect linear relationship (i.e. the two variables move in the same direction with the unit changes being equal). Finally, a correlation of zero implies that there is no linear relationship between the variables.

Example: Calculating the covariance

We anticipate that there is a 15% chance that next year’s stock returns for ABC Corp will be 6%, a 60% probability that they will be 8% and a 25% probability that they will be 10%. We already know the expected value of returns is 8.2% and the standard deviation is 1.249%.

We also anticipate that the same probabilities and states are associated with a 4% return for XYZ Corp, a 5% return, and a 5.5% return. The expected value of returns is then 4.975 and the standard deviation is 0.46%.

$$ \begin{align*}
\text{Covariance}, \text{cov}(\text R_{ \text{ABC}},\text R_{ \text{XYZ}}) & = 0.15(0.06 – 0.082)(0.04 – 0.04975) \\
& + 0.6(0.08 – 0.082)(0.05 – 0.04975) \\
& + 0.25(0.10 – 0.082)(0.055 – 0.04975) \\
& = 0.0000561 \\
\end{align*} $$

$$\text{Correlation}\ (R_i,R_j\ )=\frac{Cov(R_i,R_j\ )}{(Standard\ deviation\ (R_i\ )\times Standard\ deviation\ (R_j\ ))}$$

Thus,

$$\text{Correlation}=0.0000561\left(0.01249\times0.0046\right)=0.976$$

Interpretation: The correlation between the returns of the two companies is very strong (almost +1) and the returns move linearly in the same direction.

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How does Correlation Impact Portfolio Risk?

As mentioned earlier, correlation ranges from -1 to +1

• +1 = returns are perfectly positively correlated.
• 0 = returns of two assets are not correlated.
• -1 = returns are perfectly negatively correlated.
• What happens to portfolio risk (in a portfolio of two risky assets) when the two assets are perfectly correlated?
• Risk is unaffected; no benefit of diversification

In conclusion, using negatively correlated investments to form a portfolio helps to reduce the overall volatility of the portfolio.