Randomization and the Valuation of Guaranteed Minimum Death Benefits

European Journal of Operational Research, 309(3), pp. 1218-1236, 2023.

36 Pages Posted: 7 Jun 2022 Last revised: 29 Feb 2024

See all articles by Griselda Deelstra

Griselda Deelstra

Université Libre de Bruxelles (ULB)

Peter Hieber

Université de Lausanne

Date Written: May 20, 2022

Abstract

In this article, we focus on death-linked contingent claims (GMDBs) paying a random financial return at a random time of death in the general case where financial returns follow a regime switching model with two-sided phase-type jumps. We approximate the distribution of the remaining lifetime by either a series of Erlang distributions or a Laguerre series expansion, whose capability to fit the tail of the observed mortality data turns out to be much better than the commonly used series of exponential distributions.

More precisely, we develop a Laurent series expansion of the discounted Laplace transform of the subordinated process at an Erlang distributed time, which leads to explicit formulae for European type GMDB as well as related risk measures such as the Value-at-Risk (VaR) and the Conditional Tail-Expectation (CTE).

We further concentrate upon path-dependent GMDBs with lookback features like dynamic fund protection or dynamic withdrawal benefits, by relying on a Sylvester equation approach. The advantage of our approaches is that our prices are of semi-closed form, avoiding numerical Fourier inversion or Monte-Carlo simulation, leading to fast evaluation. This is necessary in risk management, in particularly for nested simulation in the framework of Solvency II.

Several numerical experiments are included. Our results have implications beyond life-insurance and GMDBs, namely in all situations where randomization or Erlangization replaces known quantities, like, for example, model parameters, by random variables. In Finance, it is for example well-known that a random maturity time leads to much more convenient valuation formulas that well approximate its non-random counterpart.

Keywords: Erlangization; risk analysis; variable annuities; regime switching; phase-type distributions

Suggested Citation

Deelstra, Griselda and Hieber, Peter, Randomization and the Valuation of Guaranteed Minimum Death Benefits (May 20, 2022). European Journal of Operational Research, 309(3), pp. 1218-1236, 2023., Available at SSRN: https://ssrn.com/abstract=4115505 or http://dx.doi.org/10.2139/ssrn.4115505

Griselda Deelstra

Université Libre de Bruxelles (ULB) ( email )

Boulevard du Triomphe, CP210
Brussels, Brussels 1050
Belgium

Peter Hieber (Contact Author)

Université de Lausanne ( email )

Lausanne
Switzerland

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