Some Finite Time Ruin Problems

ABSTRACT In the classical risk model, we use probabilistic arguments to write down expressions in terms of the density function of aggregate claims for joint density functions involving the time to ruin, the deficit at ruin and the surplus prior to ruin. We give some applications of these formulae in the cases when the individual claim amount distribution is exponential and Erlang(2).21

Modern landmarks in actuarial science

Inaugural professorial lectur

A note on some joint distribution functions involving the time of ruin

In a recent paper, Willmot (2015) derived an expression for the joint distribution function of the time of ruin and the deficit at ruin in the classical risk model. We show how his approach can be applied to obtain a simpler expression, and by interpreting this expression by probabilistic reasoning we obtain solutions for more general risk models. We also discuss how some of Willmot’s results relate to existing literature on the probability and severity of ruin

Optimal reinsurance under multiple attribute decision making

We apply methods from multiple attribute decision making (MADM) to the problem of selecting an optimal reinsurance level. In particular, we apply the Technique for Order of Preference by Similarity to Ideal Solution method with Mahalanobis distance. We consider the classical risk model under a reinsurance arrangement – either excess of loss or proportional – and we consider scenarios that have the same finite time ruin probability. For each of these scenarios we calculate three quantities: released capital, expected profit and expected utility of resulting wealth. Using these inputs, we apply MADM to find optimal retention levels. We compare and contrast our findings with those when decisions are based on a single attribute

Some Optimal Dividends Problems

C1 - Refereed Journal ArticleWe consider a situation originally discussed by De Finetti (1957) in which a surplus process is modified by the introduction of a constant dividend barrier. We extend some known results relating to the distribution of the present value of dividend payments until ruin in the classical risk model and show how a discrete time risk model can be used to provide approximations when analytic results are unavailable. We extend the analysis by allowing the process to continue after ruin

The joint distribution of the surplus prior to ruin and the deficit at ruin in some Sparre Andersen models

ISBN 0734028970 Research paper no. 10

Gerber-Shiu analysis of a risk model with capital injections

We consider the risk model with capital injections studied by Nie et al. (Ann Actuar Sci 5:195–209, 2011; Scand Actuar J 2015:301–318, 2015). We construct a Gerber–Shiu function and show that whilst this tool is not efficient for finding the ultimate ruin probability, it provides an effective way of studying ruin related quantities in finite time. In particular, we find a general expression for the joint distribution of the time of ruin and the number of claims until ruin, and find an extension of Prabhu’s (Ann Math Stat 32:757–764, 1961) formula for the finite time survival probability in the classical risk model. We illustrate our results in the case of exponentially distributed claims and obtain some interesting identities. In particular, we generalise results from the classical risk model and prove the identity of two known formulae for that model