The severity of ruin

Abstract

The conditional distribution of the deficit at the time of ruin, given that ruin has occurred, is the subject matter of this chapter. This quantity may be viewed as an expected discounted penalty introduced in section 9.2, where the penalty function w(x) takes a special form. As discussed in section 9.2, we are thus able to express the conditional distribution of the deficit as the solution of a defective renewal equation, which conveniently leads to the use of many of the techniques developed in earlier chapters. As will be demonstrated in what follows, the defective renewal equation associated with the conditional distribution of the deficit is simple and has a desirable structure. This allows for a mixture representation for the conditional distribution in terms of the probability of ruin and thus for construction of bounds and approximations for it. The advantage of the use of a mixture representation becomes evident when an explicit expression for the conditional distribution is derived in the special case when the claim amount distribution is an Erlang mixture or a general Erlang mixture. The approach of this chapter follows that of Willmot and Lin (1998) and Willmot (2000). We shall continue to use the notation and model of sections 7.3 and 9.2.