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The Covariance Between the Surplus Prior to and at Ruin in the Classical Risk Model

Published online by Cambridge University Press:  09 August 2013

Georgios Psarrakos  and
Konstadinos Politis
Georgios Psarrakos
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Demetriou Street, Piraeus 18534, Greece, E-mail: [email protected]
Konstadinos Politis
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Demetriou Street, Piraeus 18534, Greece, E-mail: [email protected]
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Abstract

For the classical model of risk theory, we consider the covariance between the surplus prior to and at ruin, given that ruin occurs. A general expression for this covariance is given when the initial surplus u is zero, and we show that the covariance (and hence the correlation coefficient) between these two variables is positive, zero or negative according to the equilibrium distribution of the claim size distribution having a coefficient of variation greater than, equal to, or less than one. For positive values of u, the formula for the covariance may not always lead to explicit results and we thus also study its asymptotic behaviour. Our results are illustrated by a number of examples.

Keywords

Ruin probabilityRenewal equationDeficit at ruinSurplus prior to ruinReliability classesCoefficient of variation
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 2 , November 2012 , pp. 631 - 653
Copyright
Copyright © International Actuarial Association 2012

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References

Asmussen, S. (2003) Applied Probability and queues (2nd edition). Springer, New York.Google Scholar
Basu, S.K. and Bhattacharjee, M.C. (1984) On weak convergence within the HNBUE family of life distributions. Journal of Applied Probability 21, 654–660.Google Scholar
Bhattacharjee, A. and Sengupta, D. (1996) On the coefficient of variation of the L- and L -classes. Statistics and Probability Letters 27, 177–180.Google Scholar
Borovkov, K. and Dickson, D.C.M. (2008) On the ruin time distribution for a Sparre Andersen process with exponential claim sizes. Insurance: Mathematics and Economics 42, 1104–1108.Google Scholar
Boxma, O. (1984) Joint distribution of sojourn time and queue length in the M/G/1 queue with (in)finite capacity. European Journal of Operation Research 16, 246–256.Google Scholar
Dickson, D.C.M. and Hipp, C. (2001) On the time to ruin for Erlang(2) risk processes. Insurance: Mathematics and Economics 29, 333–344.Google Scholar
Dufresne, F. and Gerber, H.U. (1989) Three methods to calculate the probability of ruin. ASTIN Bulletin 19, 71–90.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosh, T. (1997) Modelling extremal events for insurance and finance. Springer-Verlag, Berlin.Google Scholar
Feller, W. (1971) An Introduction to probability theory and its applications. Vol. II, 2nd edition. Wiley, New York.Google Scholar
Gerber, H.U., Goovaerts, M.J. and Kaas, R. (1987) On the probability and severity of ruin. ASTIN Bulletin 17, 151–163.Google Scholar
Gerber, H.U. and Shiu, E.S.W. (1997) The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics 21, 129–137.Google Scholar
Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin. North American Actuarial Journal 2, 48–78.Google Scholar
Gupta, R.C. and Keating, J.P. (1986) Relations for reliability measures under length biased sampling. Scandinavian Journal of Statistics 13, 49–56.Google Scholar
Klefsjö, B. (1981) HNBUE survival under some shock models. Scandinavian Journal of Statistics 8, 39–47.Google Scholar
Klefsjö, B. (1982) The HNBUE and HNWUE classes of life distributions. Naval Research Logistics Quarterly 29, 331–344.Google Scholar
Lai, C.D. and Xie, M. (2006) Ageing and Dependence for Reliability, Springer.Google Scholar
Li, S. and Garrido, J. (2002) On the time value of ruin in the discrete time risk model. Working paper 02-18, Business Economics, University Carlos III of Madrid, 1–28.Google Scholar
Lin, X.S. and Willmot, G. (2000) The moments of the time of ruin, the surplus before ruin and the deficit at ruin. Insurance: Mathematics and Economics 27, 19–44.Google Scholar
Rao, B.V. and Feldman, R.M. (2001) Approximations and bounds for the variance of steady-state waiting times in a GI/G/1 queue. Operations Research Letters 28, 51–62.Google Scholar
Rolski, T. (1975) Mean residual life. Bulletin of the International Statistical Institute 46, 266–270.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999) Stochastic Processes for Insurance and Finance. Wiley, New York.Google Scholar
Sundt, B. and dos Reis, A.D.E. (2007) Cramér-Lundberg results for the infinite time ruin probability in the compound binomial model. Bulletin of the Swiss Association of Actuaries, 179–190.Google Scholar
Willmot, G.E. and Lin, X.S. (2001) Lundberg approximations for compound distributions with insurance applications. Springer-Verlag, New York.Google Scholar