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Optimal Approximations for Risk Measures of Sums of Lognormals Based on Conditional Expectations

Journal of Computational and Applied Mathematics, Vol. 221, No. 1, pp. 202-218

22 Pages Posted: 1 Jun 2006 Last revised: 21 Apr 2009

Steven Vanduffel

Vrije Universiteit Brussel (VUB)

Xinliang Chen

KU Leuven - Faculty of Business and Economics (FEB)

Jan Dhaene

Katholieke Universiteit Leuven - Faculty of Economics and Business

M. J. Goovaerts

affiliation not provided to SSRN

Luc Henrard

BNP Paribas

Rob Kaas

University of Amsterdam - Faculty of Economics & Econometrics (FEE)

Date Written: October 18, 2007

Abstract

In this paper we investigate approximations for the distribution function of a sum S of lognormal random variables.

These approximations are obtained by considering the conditional expectation E[S | Lambda ] of S with respect to a conditioning random variable Lambda.

The choice for Lambda is crucial in order to obtain accurate approximations. The different alternatives for Lambda that have been proposed in literature to date are 'global' in the sense that Lambda is chosen such that the entire distribution of the approximation E[S | Lambda ] is 'close' to the corresponding distribution of the original sum S.

In an actuarial or a financial context one is often only interested in a particular tail of the distribution of S. Therefore in this paper we propose approximations E[S | Lambda ] which are only locally optimal, in the sense that the relevant tail of the distribution of E[S | Lambda ] is an accurate approximation for the corresponding tail of the distribution of S. Numerical illustrations reveal that local optimal choices for Lambda can improve the quality of the approximations in the relevant tail significantly.

We also explore asymptotic properties of the approximations E[S | Lambda] and investigate links with results from Asmussen & Royas-Nandayapa (2005). Finally, we briefly adress the sub-optimality of Asian options from the point of view of risk averse decision makers with a fixed investment horizon.

Keywords: Lognormal, random sum, Asian options, conditional expectation, Lower bound, Annuities,

Suggested Citation

Vanduffel, Steven and Chen, Xinliang and Dhaene, Jan and Goovaerts, M. J. and Henrard, Luc and Kaas, Rob, Optimal Approximations for Risk Measures of Sums of Lognormals Based on Conditional Expectations (October 18, 2007). Journal of Computational and Applied Mathematics, Vol. 221, No. 1, pp. 202-218. Available at SSRN: https://ssrn.com/abstract=905200

Steven Vanduffel (Contact Author)

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, Brabant 1050
Belgium

HOME PAGE: http://www.stevenvanduffel.com

Xinliang Chen

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

Naamsestraat 69
Leuven, B-3000
Belgium

Jan Dhaene

Katholieke Universiteit Leuven - Faculty of Economics and Business ( email )

Naamsestraat 69
Leuven, 3000
Belgium

M. J. Goovaerts

affiliation not provided to SSRN

Luc Henrard

BNP Paribas ( email )

Montagne Du Parc 3
Belgium

Rob Kaas

University of Amsterdam - Faculty of Economics & Econometrics (FEE) ( email )

Roetersstraat 11
Amsterdam, 1018 WB
Netherlands

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