| Daily Return for Two Stocks Using the Closing Prices | ||
|---|---|---|
| Day | ABC Returns | XYZ Returns |
| 1 | 1.1% | 3.0% |
| 2 | 1.7% | 4.2% |
| 3 | 2.1% | 4.9% |
| 4 | 1.4% | 4.1% |
| 5 | 0.2% | 2.5% |
Next, calculate the average return for each stock:
- For ABC, it would be (1.1 + 1.7 + 2.1 + 1.4 + 0.2) / 5 = 1.30.
- For XYZ, it would be (3 + 4.2 + 4.9 + 4.1 + 2.5) / 5 = 3.74.
Then, take the difference between ABC's return and ABC's average returnand multiply it by the difference between XYZ's return and XYZ's average return.
Finally, divide the result by the sample size and subtract one. If it was the entire population, you could divide by the population size.
This is represented by the following equation:
Covariance=(SampleSize)−1∑(ReturnABC−AverageABC)∗(ReturnXYZ−AverageXYZ)
Using our example of ABC and XYZ above, the covariance is calculated as:
- = [(1.1 - 1.30) x (3 - 3.74)] + [(1.7 - 1.30) x (4.2 - 3.74)] + [(2.1 - 1.30) x (4.9 - 3.74)] + …
- = [0.148] + [0.184] + [0.928] + [0.036] + [1.364]
- = 2.66 / (5 - 1)
- = 0.665
In this situation, we are using a sample, so we divide by the sample size (five) minus one.
The covariance between the two stock returns is 0.665. Because this number is positive, the stocks move in the same direction. In other words, when ABC had a high return, XYZ also had a high return.
If the result were negative, then the two stocks would tend to have opposite returns: when one had a positive return, the other would have a negative return.
Finding Covariance With Microsoft Excel
In MS Excel, you use one of the following functions to find the covariance:
- = COVARIANCE.S() for a sample
- = COVARIANCE.P() for a population
You will need to set up the two lists of returns in vertical columns as in Table 1. Then, when prompted, select each column. In Excel, each list is called an "array," and two arrays should be inside the brackets, separated by a comma.
Uses of Covariance
Covariance can tell how the stocks move together, but to determine the strength of the relationship, look at theircorrelation. The correlation should, therefore, be used in conjunction with the covariance, and is represented by this equation:
Correlation=ρ=σXσYcov(X,Y)where:cov(X,Y)=CovariancebetweenXandYσX=StandarddeviationofXσY=StandarddeviationofY
The equation above reveals that the correlation between two variables is the covariance between both variables divided by the product of the standard deviation ofthe variables. While both measures reveal whether two variables are positively or inversely related, the correlation provides additional information by determining the degree to which both variables move together.
The correlation will always have a measurement value between -1 and 1, and it adds a strength value on how the stocks move together.
If the correlation is 1, they move perfectly together, and if the correlation is -1, the stocks move perfectly in opposite directions. If the correlation is 0, then the two stocks move in random directions from each other.
In short, covariance tells you whether two variables change the same way while correlation reveals how a change in one variable affects a change in the other.
You also may use covariance to find the standard deviation of a multi-stock portfolio. The standard deviation is the accepted calculation for risk, which is extremely important when selecting stocks. Most investors would want to select stocks that move in opposite directions because the risk will be lower, though they'll provide the same amount of potential return.
How Does Covariance Differ From Variance?
Variance measures the dispersion of values or returns of an individual variable or data point about the mean. It looks at a single variable. Covariance instead looks at how the dispersion of the values of two variables corresponds with respect to one another.
Where Is Covariance Used in Finance?
If two stocks have share prices with a positive covariance, they are both likely to move in the same direction when responding to market conditions.If they have negative covariance they tend to move in opposite directions. Covariance is used in modern portfolio theory (MPT), when constructing efficient investment portfolios. In order to achieve the optimal risk-return trade-off one should identify assets that have a low or negative correlation.
How Do Covariance and Correlation Differ?
The correlation coefficient of a pair of variables is derived by taking the covariance and dividing it by the product of each variable's standard deviation:
Correlation (ρ) = cov(X,Y)/(σX σY)
Correlation is therefore a normalized or rangebound interpretation of how two variables move together.
The Bottom Line
Covariance is a common statistical calculation that can show how two stocks tend to move together. Investors can use it to lower their portfolio risk by selecting stocks that move in opposite directions.
However, because covariance is calculated using historical returns, it can never provide complete certainty about the future. It should not be used on its own to construct a portfolio. Instead, it shouldbe used in conjunction with other calculations such as correlation or standard deviation.
