Pricing bounds and bang-bang analysis of the Polaris variable annuities

Abstract

This paper studies the no-arbitrage pricing of the ‘Polaris Income Plus Daily’ structured in the ‘Polaris Choice IV’ variable annuities recently issued by the American International Group. Distinct from the withdrawal benefits studied in the literature, Polaris allows the income base to ‘lock in’ the high water mark of the investment account over a certain monitoring period which is related to the timing of the policyholder's first withdrawal. By prudently introducing certain auxiliary state and decision variables, we manage to formulate the pricing model under a Markovian stochastic optimal control framework. By a slight modification of the fee structure, we show the existence of a bang-bang solution to the stochastic control problem: the optimal withdrawal strategy is among a few explicit choices. We consequently design a novel Least Squares Monte Carlo (LSMC) algorithm to approach the optimal solution. Convergence results are established for the algorithm by applying the theory of nonparametric sieve estimation. Compared with existing LSMCs, our algorithm possesses a number of advantages such as memory reduction, preservation of convexity and monotonicity of the continuation value, reducing computational cost of the tuning parameter selection, and evading extrapolation of the value function estimate. Finally, we prove that the obtained pricing result works as an upper bound of the no-arbitrage price of Polaris with the real fee structure. Numerical experiments show that this upper bound is fairly tight.

Keywords:

• Variable annuities
• Bang-bang solution
• Least squares Monte Carlo
• Sieve estimation

JEL Classification:

• G22
• C14
• C63

Acknowledgments

The authors are thankful to Alexander Schied (University of Waterloo), Yue Kuen Kwok (The Hong Kong University of Science and Technology), Yao Tung Huang (Magnum Research Limited), Pingping Zeng (Southern University of Science and Technology), and Pengfei Li (University of Waterloo) for helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

† Specifically, the Polaris VAs are sold by the American General Life Insurance Company beyond New York City. In New York, they are sold by the United States Life Insurance Company; see the cover page of the prospectus (AIG 2016).

† Throughout the paper, the ‘optimality’ of the withdrawal strategy should be understood as the strategy that maximizes the liability of the insurer under the risk-neutral pricing measure. Despite the fact that the policyholder might not behave in accordance with such a strategy, the obtained optimal strategy is helpful for the insurer to create a super-hedging strategy. Therefore, it is instructive to study the pricing problem under such a stochastic optimal control framework; see Forsyth and Vetzal (2014) and Azimzadeh et al. (2014) for detailed discussions.

† Here the maturity refers to the Latest Annuity Date when the contract is converted into a fixed annuity automatically, see the discussion in ‘Terminal payoff’ below.

† To be more precise, in simulating the trajectories of the state process, Huang and Kwok (2016) assumes the PH withdraws at a fixed amount throughout the contract life, or equivalently, the PH initiates the withdrawal at $t_1$ and maintains a constant withdrawal amount thereafter.

‡ See the ‘Supplement to the variable annuity prospectus’ of the prospectus (AIG 2016) for details.

† Here we assume the rider charge percentage is deterministic. In real contract specification, the value of η is tied to the VIX and thus is not a constant. Our framework can easily accommodate this feature by introducing an extra state variable as it is done in Cui et al. (2017). The key point is that the evolution of VIX is not influenced by the PH's decision and thus introducing this state variable would not ruin the argument for proving the existence of a bang-bang solution.

‡ The benefit base is the parallel income base in a GLWB contract. It basically plays the role of determining the guaranteed payment of the insurer.

† The use of historical volatility might be debatable since the risk-neutral pricing framework requires calibrating the model to the market. However, the lack of long-term (more than ten years or even lifetime) and highly liquid derivatives renders the usual model calibration procedure—as it is adopted in the pricing of financial derivatives—also questionable.

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada [NSERC-RGPIN-2016-04001].