The Valuation of Guaranteed Lifelong Withdrawal Benefit Options in Variable Annuity Contracts and the Impact of Mortality Risk

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent repository link AbstractIn the light of the growing importance of the Variable Annuities market, in this paper we introduce a theoretical model for the pricing and valuation of Guaranteed Lifelong Withdrawal Benefit Options (GLWB) embedded in Variable Annuity products. As the name suggests, this option offers a lifelong withdrawal guarantee; therefore, there is no limit on the total amount that is withdrawn over the term of the policy, because if the account value becomes zero while the insured is still alive he continues to receive the guaranteed amount annually until death. Any remaining account value at the time of death is paid to the beneficiary as a death benefit We offer a specific framework to value the GLWB in a market consistent manner under the hypothesis of a static withdrawal strategy, according to which the withdrawal amount is always equal to the guaranteed amount. The valuation approach is based on the decomposition of the product into living and death benefits. The model makes use of the standard no-arbitrage models of mathematical finance, that extend the BlackScholes framework to insurance contracts, assuming the fund follows a Geometric Brownian Motion and the insurance fee is paid, on an ongoing basis, as a proportion of the assets. We develop a sensitivity analysis, which shows how the value of the product varies with the key parameters, including the age of the policyholder at the inception of the contract, the guaranteed rate, the risk free rate and the fund volatility. We calculate the fair fee, using Monte Carlo simulations under different scenarios. We give special attention to the impact of mortality risk on the value of the option, using a flexible model of mortality dynamics, which allows for the possible perturbations by mortality shock of the standard mortality tables used by practitioners. Moreover, we evaluate the introduction of roll-up and step-up options and the effect of the decision to delay withdrawing. Empirical analyses are performed and numerical results are provided.

Pricing guaranteed minimum/lifetime withdrawal benefits with various provisions under investment, interest rate and mortality risks

“…Piscopo and Haberman (2011) and Fung et al (2013) point out that the issuers of lifetime-payment guarantees may suffer from longevity risk since policyholders on average live longer than expected. Yang and Dai (2013) adopt a deterministic mortality model in pricing the GMWB contracts and didn't address the issue of longevity risk.…”

“…This price deviation problem becomes significant with the increment of the policy's maturity, especially for a lifelong policy such as a GLWB. Recently, a few studies have examined the GLWB rider (see Piscopo and Haberman, 2011;Holz et al, 2012), but they have not considered the interest rate risk.…”

“…In addition, the longevity risk plays a significant role in pricing GMWBs/GLWBs (see Piscopo and Haberman, 2011;Yang and Dai, 2013;Fung et al, 2013), especially for the later one. Piscopo and Haberman (2011) and Fung et al (2013) point out that the issuers of lifetime-payment guarantees may suffer from longevity risk since policyholders on average live longer than expected.…”

“…Piscopo and Haberman (2011), Holz et al (2012) andYang and Dai (2013) also assume that the mortality rate is independent of the investment risk.…”

Regression-based Monte Carlo methods for stochastic control models: variable annuities with lifelong guarantees

“…Shah and Bertsimas (2008) analyze the impact of various risk factors, like market risks and mortality risks, in the fair value of GLWB using the continuous time models. Under the assumption of static withdrawal strategy, Piscopo and Haberman (2011) analyze the impact of mortality risk using a flexible model of mortality dynamics on the fair value of GLWB. Holz et al (2012) price the GLWB for different product design and model parameters under the GBM dynamics of the underlying fund value process.…”