A primer on the variance of an asset and covariance of a pair of assets

One of the basic pillars of finance is “the greater the risk, the greater the return”. In finance, risk is measured by dispersion from the mean. Consider the following two assets, both of which have an average return of 10% over the past twelve months.

If we analyse the data it appears evident that stock B's returns are more erratic than stock A’s. In order to quantify this variability we need to find the variance of each stock's returns. This is found using a population variance:

Applying this to stock prices, we need to find the deviation of each period's return (Ri) from the average return over the period.

It is clear that the stock which we view to be more erratic has greater deviations from the mean. The next step is to square these deviations from the mean and sum them together as follows:

It is evident from the table above that larger deviations from the mean contribute to a greater sum total. If we divided these figures of 200 and 800 respectively by N, which in this case is 12, we arrive at the variance of each asset:

The units of variance are rather unusual, as for stock return data they would be considered as percentages squared. If we take the square root of the variance, we obtain the standard deviation, which is expressed in percentages, making it a more attractive measure, since percentages are a familiar metric in finance, being used in interest rates, growth rates, inflation etc.

The interpretation of the standard deviation of stock returns is: the higher the number, the more variable the return and the greater the risk. At the absolute limit, if a stock did not deviate from its average return, the standard deviation would be zero.