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# CHAPTER 11Correlations and Copulas

Suppose that a company has an exposure to two different market variables. In the case of each variable, it gains \$10 million if there is a one-standard-deviation increase and loses \$10 million if there is a one-standard-deviation decrease. If changes in the two variables have a high positive correlation, the company's total exposure is very high; if they have a correlation of zero, the exposure is less but still quite large; if they have a high negative correlation, the exposure is quite low because a loss on one of the variables is likely to be offset by a gain on the other. This example shows that it is important for a risk manager to estimate correlations between the changes in market variables as well as their volatilities when assessing risk exposures.

This chapter explains how correlations can be monitored in a similar way to volatilities. It also covers what are known as copulas. These are tools that provide a way of defining a correlation structure between two or more variables, regardless of the shapes of their probability distributions. Copulas have a number of applications in risk management. The chapter shows how a copula can be used to create a model of default correlation for a portfolio of loans. This model is used in the Basel II capital requirements.

## 11.1 DEFINITION OF CORRELATION

The coefficient of correlation, ρ, between two variables V1 and V2 is defined as

(11.1)

where E(.) denotes expected value and SD(.) denotes standard ...

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