Mathematical and Statistical Methods for Actuarial Sciences and Finance

Fair costs of guaranteed minimum death benefit

contracts

Franc ̧ois Quittard-Pinon and Rivo Randrianarivony

Abstract.The authors offer a new perspective on the domain of guaranteed minimum death

benefit contracts. These products have the particular feature of offering investors a guaranteed

capital upon death. A complete methodology based on the generalised Fourier transform is

proposed to investigate the impacts of jumps and stochastic interest rates. This paper thus

extends Milevsky and Posner (2001).

Key words:life insurance contracts, variable annuities, guaranteed minimum death benefit,

stochastic interest rates, jump diffusion models, mortality models

1 Introduction

The contract analysed in this article is a Guaranteed Minimum Death Benefit contract

(GMDB), which is a life insurance contract pertaining to the class of variable annuities

(VAs). For an introduction to this subject, see Hardy [4] and Bauer, Kling and Russ [2].

The provided guaranty, only in effect upon death, is paid by continuously deducting

small amounts from the policyholder’s subaccount. It is shown in this chapter how

these fees can be endogenously determined. Milevsky and Posner [8] found these fees

overpriced by insurance companies with respect to their model fair price. To answer

this overpricing puzzle, the effects of jumps in financial prices, stochastic interest

rates and mortality are considered. For this purpose, a new model is proposed which

generalises Milevsky and Posner [8].

2 General framework and main notations

2.1 Financial risk and mortality

Financial risk is related to market risk firstly because the policyholder’saccount is

linked to a financial asset or an index, and secondly via interest rates. We denote byr

M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010