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

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,