Covariance

Covariance is a measure of the directional relationship between the returns on two risky assets. A positive covariance means that asset returns move together while a negative covariance means returns move inversely. Covariance is calculated by analyzing at-return surprises (standard deviations from expected return) or by multiplying the correlation between the two variables by the standard deviation of each variable.

Covariance evaluates how the mean values of two variables move together. If stock A's return moves higher whenever stock B's return moves higher and the same relationship is found when each stock's return decreases, then these stocks are said to have a positive covariance. In finance, covariances are calculated to help diversify security holdings.

When an analyst has a set of data, a pair of x and y values, covariance can be calculated using five variables from that data. They are:

• x_i = a given x value in the data set
• x_m = the mean, or average, of the x values
• y_i = the y value in the data set that corresponds with x_i
• y_m = the mean, or average, of the y values
• n = the number of data points

Given this information, the formula for covariance is: Cov(x,y) = SUM [(x_i - x_m) * (y_i - y_m)] / (n - 1)

It's important to note that while the covariance does measure the directional relationship between two assets, it does not show the strength of the relationship between the two assets. The coefficient of correlation is a more appropriate indicator of this strength.

Covariances have significant applications in finance and modern portfolio theory. For example, in the capital asset pricing model (CAPM), which is used to calculate the expected return of an asset, the covariance between a security and the market is used in the formula for one of the model's key variables, beta. In the CAPM, beta measures the volatility, or systematic risk, of a security in comparison to the market as a whole; it's a practical measure that draws from the covariance to gauge an investor's risk exposure specific to one security.

Meanwhile, portfolio theory uses covariances to statistically reduce the overall risk of a portfolio by protecting against volatility through covariance-informed diversification. Possessing financial assets with returns that have similar covariances does not provide very much diversification; therefore, a diversified portfolio would likely contain a mix of financial assets that have varying covariances.

Assume an analyst in a company has a five-quarter data set that shows quarterly gross domestic product (GDP) growth in percentages (x) and a company's new product line growth in percentages (y). The data set may look like:

• Q1: x = 2, y = 10
• Q2: x = 3, y = 14
• Q3: x = 2.7, y = 12
• Q4: x = 3.2, y = 15
• Q5: x = 4.1, y = 20

The average x value equals 3, and the average y value equals 14.2. To calculate the covariance, the sum of the products of the x_i values minus the average x value, multiplied by the y_i values minus the average y values would be divided by (n-1), as follows:

Cov(x,y) = ((2 - 3) x (10 - 14.2) + (3 - 3) x (14 - 14.2) + ... (4.1 - 3) x (20 - 14.2)) / 4 = (4.2 + 0 + 0.66 + 0.16 + 6.38) / 4 = 2.85

Having calculated a positive covariance here, the analyst can say that growth of the company's new product line has a positive relationship with quarterly GDP growth.