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

CHAPTER 7

Alpha-Stable Distributions

Operational losses due to errors and omissions, physical loss of securities, natural disasters, and internal fraud are infrequent in nature but can have serious financial consequences for an institution. Such low frequency/high severity operational losses can be extreme in magnitude when compared to the rest of the data. If one constructs a histogram of the loss distribution, these events would be located in the far right end. Because of this, they are often classified as tail events. We say that the data are heavy-tailed when such tail events are present.

In Chapter 6 we discussed common loss distributions that can be used to describe the operational loss distribution. According to BIS (2001 Annex 1, p. 18):¹⁷⁷

The internal risk measurement system must capture the impact of infrequent, but potentially severe, operational risk events. That is, the internally generated risk measure must accurately capture the “tail” of the operational risk loss distribution.

Loss distributions, such as lognormal, gamma, and Weibull, are classified as moderately heavy-tailed and thus may not be sufficient to capture the infrequent but potentially severe operational loss events. In Chapter 6 we also emphasized the importance of the tail behavior of loss distributions to modeling heavy-tailed operational losses.

In this chapter we continue the discussion of modeling operational loss severity with heavy-tailed distributions. In particular, we discuss a wide class ...