Pricing of guaranteed minimum withdrawal benefits in variable annuities under stochastic volatility, stochastic interest rates and stochastic mortality via the componentwise splitting method

Abstract

This paper values guaranteed minimum withdrawal benefit (GMWB) riders embedded in variable annuities assuming that the underlying fund dynamics evolve under the influence of stochastic interest rates, stochastic volatility, stochastic mortality and equity risk. The valuation problem is formulated as a partial differential equation (PDE) which is solved numerically by employing the operator splitting method. Sensitivity analysis of the fair guarantee fee is performed with respect to various model parameters. We find that (i) the fair insurance fee charged by the product provider is an increasing function of the withdrawal rate; (ii) the GMWB price is higher when stochastic interest rates and volatility are incorporated in the model, compared to the case of static interest rates and volatility; (iii) the GMWB price behaves non-monotonically with changing volatility of variance parameter; (iv) the fair fee increases with increasing volatility of interest rates parameter, and increasing correlation between the underlying fund and the interest rates; (v) the fair fee increases when the speed of mean-reversion of stochastic volatility or the average long-term volatility increases; (vi) the GMWB fee decreases when the speed of mean-reversion of stochastic interest rates or the average long-term interest rates increase. We investigate both static and dynamic (optimal) policyholder's withdrawal behaviours; we present the optimal withdrawal schedule as a function of the withdrawal account and the investment account for varying volatility and interest rates. When incorporating stochastic mortality, we find that its impact on the fair guarantee fee is rather small. Our results demonstrate the importance of correct quantification of risks embedded in GMWBs and provide guidance to product providers on optimal hedging of various risks associated with the contract.

Keywords:

• Variable annuities
• Guaranteed minimum withdrawal benefit (GMWB) riders
• Stochastic mortality
• Stochastic volatility
• Stochastic interest rates
• Componentwise splitting method

JEL Classification:

• C63
• G12
• G22
• G23

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Katja Ignatieva http://orcid.org/0000-0001-7472-5475

Jonathan Ziveyi http://orcid.org/0000-0001-9029-8039

Notes

1 Improvements in mortality observed across the entire developed world lead to a decrease in the old-age support ratio, i.e. the proportion of people aged 15–64 years old to those aged 65 years old and over, which relates the number of individuals that are capable of providing economic support to the number of older people dependent on the support of others (United Nations 2013).

2 Under the static withdrawal strategy, the policyholder behaves ‘passively’ in terms of execution of the policy, that is the withdrawals are constant and equal to the contractually agreed withdrawal amount. Under the dynamic withdrawal strategy, the policyholder adjusts the amount to be withdrawn depending on the market conditions.

3 Guarantees embedded in VA are akin to option contracts in the financial markets.

4 We use the terms ‘guarantee fee’ and ‘insurance fee’ interchangeable throughout of the paper.

5 In what follows, $t_j^{-}$ ($t_j^{+}$) is a moment of time immediately before (after) the withdrawal time $t_j$.

6 For notational convenience, we let $\omega \equiv {\omega }_{\tau }\equiv {\tilde{W}}_{\tau }$, $v\equiv v_{\tau }$ and $r\equiv r_{\tau }$.

7 Alternatively, condition on the Vega (i.e. sensitivity to volatility changes) of an option could be imposed: $\underset{v\to \mathrm{\infty }}{lim}\left(\mathrm{\partial }U\right(\tau ,\omega ,v,r\left)\right)/\mathrm{\partial }v=0$. We refer to Ikonen and Toivanen (2009) who use this condition in the context of pricing of American options under the Heston stochastic volatility model.

8 The coupled system is presented in Appendix A.2 for completeness. Here, we have chosen to express the system in terms of the differential operators in scalar form.

9 In the case of a stochastic variance process, σ will be used to denote the volatility of the variance parameter.

10 Note that 1 basis point (b.p.) is equivalent to 0.01%. Also note that for comparison purposes, we do not incorporate mortality so as to explicitly highlight the impact of stochastic volatility and stochastic interest rates.

11 Liu (2010) finds (see table 2.3) that as the frequency of withdrawals increases, the fair insurance fee increases and approaches the fair fee for the continuous withdrawals case.

12 See figure 4 in Donnelly et al. (2014).

13 SVSI stands for stochastic volatility and stochastic interest rates; SV, stochastic volatility; SI, stochastic interest; DW, dynamic withdrawals; SW, static withdrawals.