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

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 | Λ] 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[S | Λ] 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 | Λ] which are only locally optimal, in the sense that the relevant tail of the distribution of E[S | Λ] 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[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.

Comonotonic Approximations of Risk Measures for Variable Annuity Guaranteed Benefits with Dynamic Policyholder Behavior

“…Note that the risk measures VaR p (S l ) and LTVaR p (S l ) only depend on the unspecified vector (λ 1 , · · · , λ M ) through the vector (r 1 , · · · , r M ). Rather than searching for λ i 's, Vanduffel et al (2008a) proposed two methods for selecting optimal r i 's. Their first approach is to maximize the first-order approximation of the variance of S l (globally optimal choice), in which case N k = Z k for k = 1, · · · , nT .…”

“…B Appendix: Choices of the conditioning random variable Λ As discussed in Vanduffel et al (2008a), the globally optimal choice of Λ refers to the set of r i 's that maximizes the linear approximation of the variance of S l :…”

“…In the work of Vanduffel et al (2008b) and Vanduffel et al (2008a), a total of M = nT normal random variables was used with N k = Z k . Under this choice of random variables, the weights {λ k , k = 1, · · · , M } were derived to approximately maximize TVaR p (S l ) or Var(S l ), see Appendix B.…”

Market Value Margin calculations under the Cost of Capital approach within a Bayesian chain ladder framework

“…In [16], Vandu¤el, Chen, Dhaene, Goovaerts, Henrard Kaas alternately propose to choose such that a …rst-order approximation of the p-level Conditional Tail Expectation of S l is as large as possible and therefore the closest to the p-level Conditional Tail Expectation of S. The choices of the parameters i are then given by…”

“…We give an exact analytical formula for the MVM of the liabilities of a speci…c accident year when reserves are not discounted. We propose accurate approximated analytic formulas for the MVM of the liabilities of all accident years (when reserves are discounted or not) by considering convex order techniques and approximations as used in [5], [6], [15], [16].…”