Risk Management in Turbulent Times

By Gilles Bénéplanc; Jean-Charles Rochet | Go to book overview

5
Statistical Risk Measures
In practice it is too complex to work with the entire distribution function of losses. This is why several types of risk measures have been designed to summarize relevant information about risks. These measures were developed by mathematicians and actuaries. They are widely used by risk practitioners in finance, insurance, and risk management. The simplest of these measures are:
 • The expected loss, computed as the mean (or expectation) of Click for a larger version • The variance of Click for a larger version • The linear correlation between two risks, Click for a larger version • The Value at Risk of Click for a larger version at level ɑ:Click for a larger version

More recently, statisticians have proposed more sophisticated tools, such as copulas or abstract risk measures. This chapter successively describes these different measures and shows how they can be used to derive some simple rules for risk management. It also illustrates their dangers, when they are used without precaution. A mathematical appendix provides an introduction to extreme value theory.

5.1 THE EXPECTATION OR MEAN

For simple risks, the size of the loss L (if it occurs) is perfectly anticipated. Then the expected loss (denoted by the symbol

equals the probability of loss multiplied by the size of the loss, as illustrated by Figure 5.1.

For example, the simplest credit risk models assume a fixed Loss Given Default on a loan, implying a simple formula for expected loss:

Expected Loss = (Probability of Default) × (Loss Given Default).

Using standard acronyms, this becomes: EL = PD x LGD.

For complex risks, the expected loss is equal to the probability of loss multiplied by the average loss. When the distribution of losses in discrete, the expected loss is the sum of all possible losses multiplied by their respective probabilities.

Example 1: Expected loss of a binomial distribution. Take the binomial distribution B(3, 0.1) considered in Box 4.4 of Chapter 4 (three loans of nominal 1 and probability of default 10%). Because of the assumption of zero recovery rate, the LGD

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