Quantitative Risk Management: A Practical Guide to Financial Risk, + Website

10.5 Principal Components and Risk Aggregation

Principal components is a data-reduction technique that can reduce the effective data dimensionality and provide a summary view of risk intermediate between the very granular level of individual trading desks and the very aggregate level of portfolio volatility or VaR. It takes the original returns for assets or risk factors, represented by a vector Y = [y₁,..., y_n]′ (for example the yields for 1-year, 2-year,..., 30-year bonds), and transforms it into a new set of variables, F, by means of a linear transformation:

(10.9)

The trick is that we can choose A so that the new variables f_i are orthogonal (statistically uncorrelated). The orthogonality is particularly nice because it means that different factors in a sense span different and independent dimensions of the risk. The benefit is that the separate f_i will contribute independently to the portfolio variance or VaR (assuming that the original Y are normally distributed or close to normal). Furthermore, the f_i can be ordered in terms of size or contribution to the variance. In many practical cases, the first few fs contribute the lion's share of the variance and also have an easily understood meaning (for example, yield curves, with level, twist, hump). In this case, the new variables ...