A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework

Abstract

Abstract:In this paper, we review pricing of the variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework and review the existing numerical methods. We also discuss pricing under the complete/incomplete financial market models, stochastic mortality and optimal/sub-optimal policyholder behavior, and in the presence of taxes. For numerical valuation of these contracts in the case of simple risky asset process, we develop a direct integration method based on the Gauss-Hermite quadratures with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is easier to implement and faster when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a numerical benchmark for practitioners and researchers developing pricing of variable annuity guarantees to assess the accuracy of their numerical implementation.

A Note on the Impact of Management Fees on the Pricing of Variable Annuity Guarantees

“…i.e., the sum of the cash flow paid by the insurer and the net liability of the remaining term of the contract is maximized. The strategy Γ L given by (24) is called the liability maximization strategy.…”

Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs

“…In case of early surrender let f : [0, T] → R + 0 be a non-increasing function of time such that the surrender benefit at t is equal to: I exp (Y (t) − f (t)). If f is properly chosen, we are able to overcome the problem that an insurance 4 In the suggested modeling approach the mortality intensity can become negative with positive probability. This probability can be calculated analytically, see Appendix A.2.…”

“…Since VAs are usually unit-linked, they allow policyholders to participate in rising stock prices while their guarantees offer protection against the reverse trend. For further reading on VAs and implicit options embedded in general life insurance products, see [2][3][4]. In contrast to the policyholders, for VA providers the GMXBs that are offered may cause severe financial and actuarial risks: First, the minimum benefits could expire in-the-money, i.e., worth more than the corresponding position in stocks.…”

Assessing Guaranteed Minimum Income Benefits and Rationality of Exercising Reset Options in Variable Annuities

“…Guaranteed minimum benefits can come in many forms; however, the four main types are Guaranteed Minimum Withdrawal Benefits (GMWB), Guaranteed Minimum Death Benefits (GMDB), Guaranteed Minimum Accumulation Benefits (GMAB), and Guaranteed Minimum Income Benefits (GMIB). Some works related to the pricing of these specific guarantees in isolation are (Milesvsky & Salisbury, 2006), (Milevsky & Posner, 2001), (Shevchenko & Luo, 2016), and (Marshall, Hardy, & Saunders, 2010), respectively. With a few exceptions, closed-form formulas are usually not available and numerical methods, such as Monte Carlo, have to be used to price these guarantees.…”