A risk analyst peforming PCA wishes to explain 80% of the variance. The first orthogonal factor has a volatility of 100, and the second 40, and the third 30. Assume there are no other factors. Which of the factors will be included in the final analysis?
The total variance of the system is 100^2 + 40^2 + 30^2 = 12500 (as variance = volatility squared). The first factor alone has a variance of 10,000, or 80%. Therefore only the first factor will be included in the final analysis, and the rest will be ignored.
Interestingly, this example highlights one of the limitations of PCA. Obviously, the second and third factors are material when considering volatility, though the effect of squaring them to get the variance makes them appear less important than they are.