Aug 12

Variance (Quant: Stat)

by David Harper, CFA, FRM, CIPM

Learning objective: Define, calculate and interpret the variance.

Variance measures dispersion and is also referred to as the second moment about the mean (where the standardized third and fourth moments are skew and kurtosis). The following are variance formulas but, for the FRM candidate, the third is the most useful:

For example,

The variance of a single six-sided die is about 2.92. E(X^2) – E(X)^2 = 15.17 - (3.5)^2 = 2.92
The variance of a coin toss = E(X^2) – E(X)^2 = (0.5) – 0.5^2 = 0.25

Notes for FRM candidates:

The variance of a standard normal distribution is 1.0
Variance (and standard deviation) are good risk metrics when returns are normally distributed. If returns are non-normal (e.g., positively skewed), variance is inferior to measures of downside risk (e.g., semideviation, Sortino)
If returns are normal, independent and identically distributed (i.i.d.), then we can use the square root rule which says: variance scales with time, such that standard deviation and delta normal value at risk (VaR) scale with the square root of time.