5.3 Extreme Events

The most difficult and vexing problem in quantitative risk measurement is trying to quantify tail, or extreme, events. Tail events are important because large losses are particularly significant, and can, in truly extreme situations, wipe out a firm.

Measuring tail events is difficult for some fundamental reasons. First, tail, or extreme, events are by their nature rare and thus difficult to measure. By definition, we do not see many rare events, so it is difficult to measure them reliably and to form judgments about them. Second, because of the scanty evidence, we are often pushed toward making theoretical assumptions about the tails of distributions (extreme events). Unfortunately, simple and common assumptions are often not appropriate for tails. Most importantly, the assumption of normality is often not very good far out in the tails.

Although rare events are rare, they do occur, and measurements across different periods, markets, and securities show that in many cases, extreme events occur more often than they would if the P&L behaved according to the normal distribution in the tails. This does not mean the normal distribution is a bad choice when looking at the central part of the distribution, but it does mean that it can be a poor approximation when examining extreme events.

Broadly speaking, three approaches can be taken when dealing with tail events:

1. Simple rules of thumb

2. Alternative but tractable distributional assumptions

3. Extreme value theory, ...

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