Probability and Statistics for Finance

Chapter 16. Copula and Dependence Measures

In previous chapters of this book, we introduced multivariate distributions that had distribution functions that could be presented as functions of their parameters and the values x of the state space; in other words, they could be given in closed form.^[176] In particular, we learned about the multivariate normal and multivariate t-distributions. What these two distributions have in common is that their dependence structure is characterized by the covariance matrix that only considers the linear dependence of the components of the random vectors. However, this may be too inflexible for practical applications in finance that have to deal with all the features of joint behavior exhibited by financial returns.

Portfolio managers and risk managers have found that not only do asset returns exhibit heavy tails and a tendency to simultaneously assume extreme values, but assets exhibit complicated dependence structures beyond anything that could be handled by the distributions described in previous chapters of this book. For example, a portfolio manager may know for a portfolio consisting of bonds and loans the constituents' marginal behavior such as probability of default of the individual holdings. However, their aggregate risk structure may be unknown to the portfolio manager.

In this chapter, in response to the problems just mentioned, we introduce the copula as an alternative approach to multivariate distributions that takes us beyond the strict ...