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 S of lognormal random variables. These approximations are obtained by considering the conditional expectation E[SΛ] of S with respect to a conditioning random variable Λ.The choice of Λ is crucial in order to obtain accurate approximations. The different alternatives for Λ that have been proposed in the literature to date are ‘global’ in the sense that Λ is chosen such that the entire distribution of the approximation E[SΛ] 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[SΛ] which are only locally optimal, in the sense that the relevant tail of the distribution of E[SΛ] is an accurate approximation for the corresponding tail of the distribution of S. Numerical illustrations reveal that local optimal choices for Λ can improve the quality of the approximations in the relevant tail significantly.We also explore the asymptotic properties of the approximations E[SΛ] 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.