May 24

Three faces of variance

by David Harper, CFA, FRM, CIPM

FRM | Quant |

Variance, a dispersion metric like volatility (its square root), is the key traditional measure of risk. Traditional is nowadays a bit of a dig: unlike the more modern semi-variance, variance is indifferent to direction. Google's stock on a runup has great variance, but in hindsight, steep upward gains don't seem too risky.

In the FRM curriculum, we have single asset (e.g., stock, bond) variance and portfolio variance. In regard to single asset or single variable variance, at least three types exist:

  • population variance
  • sample variance
  • random variable variance

Population variance

Take the difference between each observation (u = periodic return) and the average observation (u-bar = average of all the periodic returns). Square this series of differences and the average is the population variance (or, sum them and divide by the number of observations):

 

Note: if you are sitting for the FRM, you notice we rarely perform the above "proper" calculation. Instead, we use this:

Which is simply, the average of squared returns. Why? Because we are allowed to make a simplifying assumption when the periodicity is short, as in daily returns or intra-daily returns: We assume the average return is zero. (We are making a second simplifying assumption, in addition. That is, we are using a population variance instead of a sample variance)

 

Sample variance

The only difference is that, instead of averaging the squared differences, we compute something slightly greater than the average. We divide the sum by (m-1) instead of (m)

 

Random variable variance

This is really not so different. But worth remembering. For a single random variable X, the variance is the expected value of X-squared minus (-) the expected value of X, quantity squared:

 

To illustrate these three variances, I prepared the EditGrid spreadsheet below. You can open your own read/write copy here. The key inputs are colored. Notice the sample variance is greater than the population variance. Also, to start, under the random variable variance, each outcome is equally likely (20% x 5 = 100%). Under that scenario (equally likely outcomes), the random variable variance equals the population variance. If you tweak the probabilities, they will start to diverge. To check our calculation, I compare our hand-crunched result to the built-in function. Note how the population variance is given by =VARP() and the sample variance is given by =VAR().

EditGrid Spreadsheet by user/davidharper.

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