Bounds for Stop-Loss Premiums of Stochastic Sums (with Applications to Life Contingencies)

27 Pages Posted: 12 Jan 2006

See all articles by Tom Hoedemakers

Tom Hoedemakers

KU Leuven - Faculty of Business and Economics (FEB)

Grzegorz Darkiewicz

KU Leuven - Faculty of Business and Economics (FEB)

Griselda Deelstra

Université Libre de Bruxelles (ULB)

Jan Dhaene

Katholieke Universiteit Leuven

Michèle Vanmaele

Ghent University - Department of Applied Mathematics, Computer Science and Statistics

Abstract

In this paper we present in a general setting lower and upper bounds for the stop-loss premium of a (stochastic) sum of dependent random variables. Therefore, use is made of the methodology of comonotonic variables and the convex ordering of risks, introduced by Kaas et al. (2000) and Dhaene et al. (2002a, 2002b), combined with actuarial conditioning. The lower bound approximates very accurate the real value of the stop-loss premium. However, the comonotonic upper bounds perform rather badly for some retentions. Therefore, we construct sharper upper bounds based upon the traditional comonotonic bounds. Making use of the ideas of Rogers and Shi (1995), the first upper bound is obtained as the comonotonic lower bound plus an error term. Next this bound is refined by making the error term dependent on the retention in the stop-loss premium. Further, we study the case that the stop-loss premium can be decomposed into two parts. One part which can be evaluated exactly and another part to which comonotonic bounds are applied. As an application we study the bounds for the stop-loss premium of a random variable representing the stochastically discounted value of a series of cash flows with a fixed and stochastic time horizon. The paper ends with numerical examples illustrating the presented approximations. We apply the proposed bounds for life annuities, using Makeham's law, when also the stochastic nature of interest rates is taken into account by means of a Brownian motion.

Keywords: Stop-loss premium, Life annuity, Comonotonicity, Stochastic time horizon

Suggested Citation

Hoedemakers, Tom and Darkiewicz, Grzegorz and Deelstra, Griselda and Dhaene, Jan and Vanmaele, Michèle, Bounds for Stop-Loss Premiums of Stochastic Sums (with Applications to Life Contingencies). Available at SSRN: https://ssrn.com/abstract=875304 or http://dx.doi.org/10.2139/ssrn.875304

Tom Hoedemakers (Contact Author)

KU Leuven - Faculty of Business and Economics (FEB) ( email )

Naamsestraat 69
Leuven, B-3000
Belgium

Grzegorz Darkiewicz

KU Leuven - Faculty of Business and Economics (FEB) ( email )

Naamsestraat 69
Leuven, B-3000
Belgium

Griselda Deelstra

Université Libre de Bruxelles (ULB) ( email )

Boulevard du Triomphe, CP210
Brussels, Brussels 1050
Belgium

Jan Dhaene

Katholieke Universiteit Leuven ( email )

Naamsestraat 69
Leuven, 3000
Belgium

Michèle Vanmaele

Ghent University - Department of Applied Mathematics, Computer Science and Statistics ( email )

Krijgslaan 281
Ghent, B-9000
Belgium

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