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

In this paper we investigate the approximations for the distribution function of a sum SS of lognormal random variables. These approximations are obtained by considering the conditional expectation E[S∣Λ]E[S∣Λ] of SS with respect to a conditioning random variable ΛΛ.The choice of ΛΛ is crucial in order to obtain accurate approximations. The different alternatives for ΛΛ that have been proposed in the literature to date are ‘global’ in the sense that ΛΛ is chosen such that the entire distribution of the approximation E[S∣Λ]E[S∣Λ] is ‘close’ to the corresponding distribution of the original sum SS.In an actuarial or a financial context one is often only interested in a particular tail of the distribution of SS. Therefore in this paper we propose approximations E[S∣Λ]E[S∣Λ] which are only locally optimal, in the sense that the relevant tail of the distribution of E[S∣Λ]E[S∣Λ] is an accurate approximation for the corresponding tail of the distribution of SS. Numerical illustrations reveal that local optimal choices for ΛΛ can improve the quality of the approximations in the relevant tail significantly.We also explore the asymptotic properties of the approximations E[S∣Λ]E[S∣Λ] 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.