A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities

Abstract

Variable Annuities with embedded guarantees are very popular in the US market. There exists a great variety of products with both, guaranteed minimum death benefits (GMDB) and guaranteed minimum living benefits (GMLB). Although several approaches for pricing some of the corresponding guarantees have been proposed in the academic literature, there is no general framework in which the existing variety of such guarantees can be priced consistently. The present paper fills this gap by introducing a model, which permits a consistent and extensive analysis of all types of guarantees currently offered within Variable Annuity contracts. Besides a valuation assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, we are able to price the contract under optimal policyholder behavior. Using both, Monte-Carlo methods and a generalization of a finite mesh discretization approach, we find that some guarantees are overpriced, whereas others, e.g. guaranteed annuities within guaranteed minimum income benefits (GMIB), are offered significantly below their risk-neutral value.

A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB)

“…As discussed in [4], for some variations of GMWB contracts, the balance of the guarantee account can increase at certain points in time if no withdrawals have been made so far. In [8,14] another possibility is discussed whereby an excessive withdrawal may result in a decrease greater than the withdrawal amount in the guarantee account.…”

“…In 2004, sixty-nine percent of all variable annuity contracts sold in the US included a GMWB option [4].…”

“…the anniversary of contract inception). Consequently, it might be supposed that a discrete withdrawal formulation would be preferred [4,8]. In [8], a penalty approach is used to solve the singular control problem.…”

Variable annuities: A unifying valuation approach

“…Mortality is considered, e.g., by Milevsky and Posner (2001), dealing with the pricing of guaranteed minimum death benefits. The valuation of a variable annuity combining death and withdrawal guarantees is the target of Bélanger et al (2009), while a general framework for the pricing of guarantees under the assumption of optimal policyholder behaviour is suggested by Bauer et al (2008). The case of lifelong withdrawals is dealt with by Holz et al (2007) and Piscopo (2009).…”

Evaluation of variable annuity guarantees with the effect of jumps in the asset price process

“…Many studies on evaluation of guarantees are based on guarantees embedded in SPVA (Bacinello, Millossovich, Olivieri, & Pitacco, 2011;Milevsky & Posner, 2001). The implication of the SPVA is that it is mainly meant for adults in their 50s and above who have earned enough from their work to make only one payment (Bacinello, Millossovich, Olivieri, & Pitacco, 2012;Bauer, Kling, & Russ, 2008).…”