Operational Risk: A Guide to Basel II Capital Requirements, Models, and Analysis

CHAPTER 8

Extreme Value Theory

A risk manager is often concerned with the distribution of the losses that are of low frequency and of high severity. Such types of losses lie in the upper tail of the loss distribution. The field of study that treats the distribution of very high quantiles of data is extreme value theory (EVT). The first model, block maxima model, examines the behavior of the maxima in equally spaced time blocks. A common application of EVT in modeling operational risk is using it to analyze the behavior of losses that exceed a certain high threshold (peak over threshold model). In this chapter, we give some theoretical background and properties of distributions used in EVT modeling. Advantages and limitations of EVT as a risk modeling tool will be presented. We then discuss some empirical studies.

BLOCK MAXIMA MODEL

Consider time series of operational loss data divided into independent blocks (e.g., one block equals one year) of the same size. The block maxima model focuses on the distribution of the largest events taken from each block. See Figure 8.1.

For very large extreme loss observations x, the limiting distribution of such normalized maxima is the generalized extreme value (GEV) distribution:

FIGURE 8.1 Block maxima model.

whereand x refers to the maxima, ...