Phase-Type Models in Life Insurance:Fitting and Valuation of Equity-Linked Benefits

Søren Asmussen,Patrick Laub,Hailiang Yang

doi:10.3390/risks7010017

Søren Asmussen, Patrick Laub + Show 1 more

Open Access

https://doi.org/10.3390/risks7010017

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Abstract

Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance. The advantage is that PH distributions form a dense class and that problems having explicit solutions for exponential distributions typically become computationally tractable under PH assumptions. In the first part of this paper, fitting of PH distributions to human lifetimes is considered. The class of generalized Coxian distributions is given special attention. In part, some new software is developed. In the second part, pricing of life insurance products such as guaranteed minimum death benefit and high-water benefit is treated for the case where the lifetime distribution is approximated by a PH distribution and the underlying asset price process is described by a jump diffusion with PH jumps. The expressions are typically explicit in terms of matrix-exponentials involving two matrices closely related to the Wiener-Hopf factorization, for which recently, a Lévy process version has been developed for a PH horizon. The computational power of the method of the approach is illustrated via a number of numerical examples.

Highlights

A random variable (r.v.) τ is phase-type (PH) with representation (α, T ) if it is distributed as the lifetime inf{t ≥ 0 : Jt = †} of a killed time-homogenous Markov process J = { Jt }t≥0 with p < ∞ states, initial distribution α, and generator T, where † is an additional absorbing state
The main reason for this is that many calculations that are explicit for exponential distributions are often computationally tractable with PH assumptions and that, in addition, PH
The first step in implementing this for pricing equity-linked benefits is to develop a version of the Wiener-Hopf factorization for the case where τ is PH and independent of X. This has been done in a companion paper (Asmussen and Ivanovs 2019), and the contribution of the present paper is to provide the further steps such as PH fitting of human mortality and computations of prices of equity-linked benefits for Brownian motion (BM) and jump diffusions

Summary

Introduction

Let τ = τx be the remaining lifetime of an insured of age x when signing the contract, the payment of this class of benefits has the form Ψ(Sτ , Sτ ) where St is the price at time t of some stock or stock index and St = maxv≤t St the running maximum. An example of such a benefit is the guaranteed minimum death benefit (GMDB). In the application to valuing equity-linked insurance products, the distribution at the tail part plays an important role, this motivated us to consider using the PH distribution to approximate the future lifetime distribution. This has been done in a companion paper (Asmussen and Ivanovs 2019), and the contribution of the present paper is to provide the further steps such as PH fitting of human mortality and computations of prices of equity-linked benefits for BM and jump diffusions

Preliminaries

PH Fits of Human Mortality Data

Valuation of Benefits

The Factorization

Numerical Examples

Erlangization and Extrapolation

Conclusions