A unified pricing of variable annuity guarantees under the optimal stochastic control framework

In this paper, we review pricing of 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. For numerical valuation of these contracts, we develop a direct integration method based on Gauss-Hermite quadrature 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 very efficient 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 also present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a benchmark for practitioners and researchers developing pricing of variable annuity guarantees.Comment: Keywords: variable annuity, guaranteed living and death benefits, guaranteed minimum accumulation benefit, optimal stochastic control, direct integration metho

Regulation of Business - Securities Act of 1933 - SEC Loses Fight to Regulate Variable Annuity

The defendant, Variable Annuity Life Insurance Company, regulated as a life insurance company by the District of Columbia, issued a contract which it termed an annuity, but which differed from a conventional annuity in certain important respects. Ordinary annuity premiums are invested in debt securities while the premiums paid on the variable annuity are invested in common stocks. Further, instead of benefit payments in fixed dollar amounts, the variable annuity\u27s benefits fluctuate since the value of the fund from which they are paid is affected by changing stock prices and dividend policies. The SEC, claiming these provisions brought the contract within the definition of a security in the Securities Act of 1933 and the company, within the definition of an investment company in the Investment Company Act of 1940, sought to enjoin the issuance of policies until the defendant complied with the provisions of the acts. Held, complaint dismissed. Because of the novelty of the agreement the court is unable to classify it either as a security or an annuity. Congress must decide whether there is to be federal regulation of the securities aspect of this contract. SEC v. Variable Annuity Life Insurance Compan

Variable Annuity with GMWB: surrender or not, that is the question

Under the optimal withdrawal strategy of a policyholder, the pricing of variable annuities with Guaranteed Minimum Withdrawal Benefit (GMWB) is an optimal stochastic control problem. The surrender feature available in marketed products allows termination of the contract before maturity, making it also an optimal stopping problem. Although the surrender feature is quite common in variable annuity contracts, there appears to be no published analysis and results for this feature in GMWB under optimal policyholder behaviour - results found in the literature so far are consistent with the absence of such a feature. Also, it is of practical interest to see how the much simpler bang-bang strategy, although not optimal for GMWB, compares with optimal GMWB strategy with surrender option. In this paper we extend our recently developed algorithm (Luo and Shevchenko 2015a) to include surrender option in GMWB and compare prices under different policyholder strategies: optimal, static and bang-bang. Results indicate that following a simple but sub-optimal bang-bang strategy does not lead to significant reduction in the price or equivalently in the fee, in comparison with the optimal strategy. We observed that the extra value added by the surrender option could add very significant value to the GMWB contract. We also performed calculations for static withdrawal with surrender option, which is the same as bang-bang minus the "no-withdrawal" choice. We find that the fee for such contract is only less than 1% smaller when compared to the case of bang-bang strategy, meaning that th "no-withdrawal" option adds little value to the contract.Comment: arXiv admin note: substantial text overlap with arXiv:1410.860