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Covariance is a measure of the co-movements in returns between different assets. It is calculated by multiplying the demeaned returns for each asset. Unlike variance and standard deviation, there is one covariance per pair of assets, not one per asset. Covariance is used to forecast portfolio risk.
Synonym: joint variability
For context, recall that the risk-reduction benefit of diversification in a portfolio is higher when the performance of those assets don't move in tandem. In other words, the co-movements, or covariance, is low. The data that sits in risk models includes covariance measures for each individual asset, or in the case of a multifactor model may include the covariance of style or fundamental factors. That's why covariance is such an important measure in portfolio management applications.
Pam: Hey Eve, what's the best measurement
period for calculating covariance?
Eve: Guy says at least 60. I love Guy, the quant guy, but I understand why you asked me.
This video can be accessed in a new window or App , at the YouTube Channel or from below.
Covariance definition for investment modeling (4:42)
The script includes two sections where we visualize and demonstrate the calculation of covariance.
We're sitting here in Excel, and this is a snippet from our boot camp course.
This is one depiction of covariance, from a risk-return scatterplot of returns for one stock Merck versus a basket of stocks, the Market, for 60 periods. Think about each dot here as returns for the Market on the x-axis and Merck on the y-axis, for each month.
So if stocks exhibit co-movements, as they appear to here, then a pattern will look linear. A random shotgun pattern would have low covariance. This is an example of positive covariance because a rise in the Market, corresponded with positive movements in Merck.
The calculation helps us understand covariance, so let's head there now.
Let's walk through a calculation for two stocks, Microsoft and eBay.
We have six monthly returns for each stock from April to September 2003. Column F is the return on Microsoft, eBay is in column G.
Next we compute the average of each, here 2.38% for Microsoft and 3.98% for eBay. Then move those over to columns H and I.
In column J, take the return minus this average which gives us 3.24%. That's 5.62% minus 2.38%. For eBay it is 8.91% minus 3.98%, or 4.93%. Carry that formula down. Columns J and K are called demeaned returns.
Next, in column L, multiply these together. As you notice, when the stocks move together, like in April, the product is positive. And when they move in opposite directions, the product is negative.
Next, using the
=SUM() function, add
up the products to get -0.0037 for the pair of stocks. Next, divide by
6 observations to get the covariance of -0.0006. If you saw our video
on variance then you know it isn't interpretable as the units are in
returns squared. Covariance is similar, so we translate it to
correlation by dividing by the product of the two standard deviations.
So to interpret, these two stocks had negative covariance, meaning as one moved up, the other moved down. These would be examples of diversifying stocks.
Click box for answer.
True. The latter is the interpretable counterpart.
False. The units are returns squared, which isn't easily interpreted.
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