A note on optimal lower bound approximations for risk measures of sums of lognormals

This paper considers different approximations for computing either the distribution function or various risk measures associated with a discrete sum of nonindependentlognormal random variables. In Dhaene et al. (2002a,b), convex upper and lower bounds for sums of non-independent lognormals were proposed.Of particular interest is the lower bound obtained by taking the conditional expectation of the original sum S with respect to a conditioning random variable. This has proven to provide excellent approximations for various risk measures associated with S. The choice of this conditioning variable plays a crucial rolein arriving at a good approximation and intuitively, it could be chosen so that, in some sense, it is 'close' to the original sum S. In the literature then, variouschoices for have been proposed. For example, Dhaene et al (2002b) propose to determine such that it can be viewed as a first-order approximation to theoriginal sum S. In another development, Vanduffel et al (2005) argue that the best is obtained when the variance of the resulting lower bound is as close aspossible to the variance of the original S. In this paper, we notice that these choices for are global in the sense that they lead to lower bounds for whichthe distribution function presents a global goodness-of-fit when compared with the original distribution function. In many financial and actuarial problems, however, we are usually only interested in the tails of the distribution function. Hence, in this paper, we derive lower bounds approximations considered to be'locally' optimal. ispartof: DTEW - AFI_603 pages:1-14 status: published