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

8.5 Estimating Risk Factor Distributions

The focus of this book is on how to think about risk and how to measure risk—in other words, how to think about and measure the P&L distribution. Most of our attention has been directed toward issues that are particular to risk measurement, such as how to map from securities to risk factors, or the definition of VaR. To use the P&L distribution, however, we need to estimate it, and this means estimating market risk factor distributions (as discussed in Section 8.3, Step 2). This takes us into the field of statistics and time-series econometrics. I do not want to cover econometrics in depth, as there are many good textbooks, but I will give a brief overview.³⁸

We will focus on the parametric approach to estimating the P&L distribution, which means we assume that market risk factors are normally distributed.³⁹ The normal distribution is completely determined by the standard deviation or variance-covariance matrix (and the mean). Thus, at its simplest, estimating the risk factor distribution means estimating the standard deviation (volatility) from risk factor changes {Δrf₁, . . ., Δrf_n}, using the standard formula (given further on).

So far, so simple. There are, however, a host of questions hidden in this simple-sounding approach:

What are the observations on risk factor changes {Δrf₁, . . ., Δrf_n}? Dollar change? Yield change? Percent or ...