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On the time value of absolute ruin with debit interest

Part of: Mathematical economics Special processes

Published online by Cambridge University Press:  01 July 2016

Jun Cai
Jun Cai*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email address: [email protected]
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Abstract

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Assume that the surplus of an insurer follows a compound Poisson surplus process. When the surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay the debts from her premium income. The negative surplus may return to a positive level. However, when the negative surplus is below a certain critical level, the surplus is no longer able to be positive. Absolute ruin occurs at this moment. In this paper, we study absolute ruin questions by defining an expected discounted penalty function at absolute ruin. The function includes the absolute ruin probability, the Laplace transform of the time to absolute ruin, the deficit at absolute ruin, the surplus just before absolute ruin, and many other quantities related to absolute ruin. First, we derive a system of integro-differential equations satisfied by the function and obtain a defective renewal equation that links the integro-differential equations in the system. Second, we show that when the initial surplus goes to infinity, the absolute ruin probability and the classical ruin probability are asymptotically equal for heavy-tailed claims while the ratio of the absolute ruin probability to the classical ruin probability goes to a positive constant that is less than one for light-tailed claims. Finally, we give explicit expressions for the function for exponential claims.

Keywords

Compound Poisson processdefective renewal equationkey renewal theoremabsolute ruindeficit at absolute ruinsurplus just before absolute ruinGerber–Shiu functiondebit interestsubexponential distributionlong-tailed distribution

MSC classification

Primary: 91B30: Risk theory, insurance
Secondary: 60K05: Renewal theory 91B70: Stochastic models
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 343 - 359
Copyright
Copyright © Applied Probability Trust 2007 

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