The purpose of this chapter

Once the ‘normal’ portion of the data has been identified as suggested in Chapter 14, we have two tasks ahead: the first is the modelling of the underlying univariate distributions; the second is the choice of how to conjoin them using a suitable copula. In this chapter we look at standard approaches to fitting the univariate distributions. In Chapter 19 we describe how they can be conjoined.

When it comes to modelling the univariate distributions, one can follow a parametric or a non-parametric approach (i.e., one can use the empirical cumulative distributions). There are pros and cons to both methods. We prefer a parametric method because it lends itself more easily to exploring the sensitivity of the results to the input parameters of the marginal distributions. See also the discussion in Section 17.1 regarding our ‘scaling’ approach in normal and excited conditions – an approach that naturally lends itself to a parametric formulation of the problem.

Once the choice has been made to pursue a parametric approach, a suitable distribution must be selected to fit the empirical distribution of ‘normal’ returns obtained for each individual risk factor. The task is made easy by the fact that we only have to fit the body of the empirical distributions; as a consequence, relatively ‘tame’ distributions may describe the data well.