Optimal approximations for risk measures of sums of lognormals based on conditional expectations

  • Authors:

  • S. Vanduffel;X. Chen;J. Dhaene;M. Goovaerts;L. Henrard;R. Kaas

  • Affiliations:

  • Faculty of Business and Economics, Catholic University of Leuven, Naamsestraat 69, 3000 Leuven, Belgium;Faculty of Business and Economics, Catholic University of Leuven, Naamsestraat 69, 3000 Leuven, Belgium;Faculty of Business and Economics, Catholic University of Leuven, Naamsestraat 69, 3000 Leuven, Belgium;Faculty of Business and Economics, Catholic University of Leuven, Naamsestraat 69, 3000 Leuven, Belgium;Fortis, Gloucester Road 77-79, Hong Kong, China;Faculty of Economics and Econometrics, Department of Quantitative Economics, Section Actuarial Science, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2008

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Abstract

In this paper we investigate the approximations for the distribution function of a sum S of lognormal random variables. These approximations are obtained by considering the conditional expectation E[S|@L] of S with respect to a conditioning random variable @L. The choice of @L is crucial in order to obtain accurate approximations. The different alternatives for @L that have been proposed in the literature to date are 'global' in the sense that @L is chosen such that the entire distribution of the approximation E[S|@L] 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|@L] which are only locally optimal, in the sense that the relevant tail of the distribution of E[S|@L] is an accurate approximation for the corresponding tail of the distribution of S. Numerical illustrations reveal that local optimal choices for @L can improve the quality of the approximations in the relevant tail significantly. We also explore the asymptotic properties of the approximations E[S|@L] and investigate links with results from [S. Asmussen, Rojas-Nandayapa, Sums of dependent lognormal random variables: Asymptotics and simulation, Stochastic Series at Department of Mathematical Sciences, University of Aarhus, Research Report number 469, 2005]. Finally, we briefly address the sub-optimality of Asian options from the point of view of risk averse decision makers with a fixed investment horizon.