Mixture or Actuarial Models

The independent-default model is deeply flawed. Not only is it fair to argue that dependence is the single most important aspect of credit-risk modelling, but the tails of the associated loss distribution are overly thin and its asymptotic behaviour is simply too well behaved. This chapter offers a family of approaches—generally referred to as mixture or actuarial models—to address each of these shortcomings. The principle idea behind this new methodology is the randomization of the default probability. Practically, a common state variable is introduced, which induces default dependence among all obligors. Conditionally, default events remain independent, but unconditionally they are related through the realization of the systematic state variable. The structure of the default-loss distribution thus depends on the statistical properties of one’s state-variable choice mixed with underlying binomial-default structure. A variety of possible choices are investigated, convergence properties are explored, and our portfolio example is examined from both analytic and numerical perspectives. Through the law of rare events, a separate class of Poisson-mixture models are explored ultimately leading to the celebrated CreditRisk+ model used widely in practice and first suggested by Wilde (1997, CreditRisk+: A credit risk management framework, Credit Suisse First Boston).