On the Joint Distributions of the Time to Ruin,the Surplus Prior to Ruin,and the Deficit at Ruin in the Classical Risk Model |
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| Authors: | David Landriault FSA PhD Gordon E Willmot FSA FCIA PhD |
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| Institution: | 1. Department of Statistics , Actuarial Science at the University of Waterloo, 200 University Ave. W. , Waterloo , Ontario , Canada, N2L 3G1;2. Department of Statistics and Actuarial Science , University of Waterloo, 200 University Ave. W. , Waterloo , Ontario , Canada, N2L 3G1 |
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| Abstract: | Abstract The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that utilize the ideas of Value-at-Risk and Tail Value-at-Risk. As is well known, calculation of these and related quantities requires knowledge of the associated probability distributions. In this paper we derive an explicit expression for the joint (defective) distribution of the time to ruin, the surplus immediately prior to ruin, and the deficit at ruin in the classical compound Poisson risk model. As a by-product, we obtain expressions for the three bivariate distributions generated by the time to ruin, the surplus prior to ruin, and the deficit at ruin. Finally, we consider mixed Erlang claim sizes and show how the joint (defective) distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin can be calculated. |
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