Optimal behavior strategy in the GMIB product

  • 632 Views
  • 273 Downloads

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License

Guaranteed Minimum Income benefit are variable annuities contract, which offer the policyholder the possibility to con- vert the guarantee level into an annuities income for life. This paper focuses on the optimal customer behavior assuming the maximization of the discounted expected future cash flows over the full life of the contract duration. Using convenient scaling properties of the contract value enables to reduce the complexity (dimension) of the problem and to characterize the policyholder’s decision as a function of the contract moneyness across four main choices: zero withdrawals, guaranteed withdrawals, lapse and the income period election. Sensitivities to key drivers such as the market volatility, the interest rate and the roll-up rate illustrate how crucial are not only the environment, but also the product design features, in order to ensure a fair and robust pricing for both customer and life insurer. In particular, the authors find that most empirical contracts are usually underpriced compared to mean optimal behavior pricing, which empirically translated into multiple updates of behavior assumptions and re-reserving by life insurers in the recent years.

view full abstract hide full abstract
    • Fig. 1. Behavior of the account value and the different options for the GMIB benefit base when no withdrawals have been performed
    • Fig. 2. The withdrawals strategy γ ̃=γ/G as a function of time tand moneyness A/G
    • Fig. 3. The withdrawals strategy as a function of time t and moneyness A/G
    • Fig. A1. Policyholder optimal withdrawals amount as a function of the account value A and benefit base G for Product A and A-DB
    • Fig. A2. The withdrawals strategy as a function of time t and moneyness A/G
    • Fig. A3. The withdrawals strategy as a function of time t and moneyness A/G
    • Fig. A4. The withdrawals strategy as a function of time tand moneyness A/G for σ = 10%
    • Fig. A5. The withdrawals strategy as a function of time tand moneyness A/G for σ = 30%
    • Fig. A6. The contract value at inception as a function of the total fees α┴- for Product A with and without DB for r = 2%, 4%
    • Fig. A7. The contract value at inception as a function of the roll-up rate η for Product A and B with and without DB for r= 2%
    • Table 1. GMIB benefit base evolution for an allocation of $100,000 with partial withdrawals that does not affect the value of the guaranteed account
    • Table 2. Protected account value and GMIB behavior given a static withdrawals strategy and lifetime payments
    • Table 3. GMIB and GMDB behavior given policyholder’s withdrawals when the account value is less than the GMIB benefit base at the time of the first withdrawal
    • Table 4. GMIB and GMDB behavior given policy holder’s withdrawals when the account value is greater than the GMIB benefit base at the time of the first withdrawal
    • Table 5. Model parameters
    • Bacinello, R., Millossovich, P., Olivieri, A., & Pitacco, E. (2011). Variable annuities: A unifying valuation approach. Insurance: Mathematics and Economics, 49(3), 285–297.
    • Bauer, D., Kling, A., & Russ, J. (2008). A universal pricing framework for guaranteed minimum benefits in variable annuities. Astin Bulletin, 38(2), 621–651.
    • Boyle P., & Tian, W. (2008). The design of equity-indexed annuities. Insurance: Mathematics and Economics, 43(3), 303–315.
    • Chen, Z., Vetzal, K., & Forsyth, P. A. (2008). The effect of modelling parameters on the value of gmwb guarantees. Insurance: Mathematics and Economics, 42(1), 165–173.
    • Dai, M., Kuen Kwok, Y., and Zong, J. (2008). Guaranteed minimum withdrawal benefit in variable annuities. Mathematical Finance, 18(4), 595–611.
    • Dai, M., Kwok, Y. K., & Zong, J. (2007). Guaranteed minimum withdrawal benefit in variable annuities.Mathematical Finance, 18(4), 595–611.
    • Deelstra, G., & Rayée, G. (2013). Pricing variable annuity guarantees in a local volatility framework. Insurance: Mathematics and Economics, 53(3), 650–663.
    • Duffy, D. J. (2004). A critique of the Crank-Nicolson scheme, strengths and weakness for financial instrument pricing. Wilmott, 4, 68–76.
    • Fung, M. C., Ignatieva, K., & Sherris, M. (2014). Systematic mortality risk: An analysis of guaranteed life-time withdrawal benefits in variable annuities. Insurance: Mathematics and Economics, 58, 103–115.
    • Gao, J., & Ulm, E. R. (2012). Optimal consumption and allocation in variable annuities with guaranteed minimum death benefits. Insurance: Mathematics and Economics, 51, 586–598.
    • Gerber, H. U., & Shiu, E. S. (2003). Pricing lookback options and dynamic guarantees. North American Actuarial Journal, 7(1), 48–66.
    • Graf, S., Kling, A., & Ruß, J. (2011). Risk analysis and valuation of life insurance contracts: Combining actuarial and financial approaches. Insurance: Mathematics and Economics, 49(1), 115–125.
    • Holz, D., Kling, A., & Ruß, J. (2012). Gmwb for life an analysis of lifelong withdrawal guarantees. Zeitschrift für die gesamte Versicherungswissenschaft, 101(3), 305–325.
    • Huang, Y., Forsyth, P., & Labahn, G.. (2012). Iterative methods for the solution of a singular control formulation of a GMWB pricing problem. Numerische Mathematik, 122,133–167.
    • Lee, R. D., & Carter, L. R. (1992). Modeling and forecasting u.s. mortality. Journal of the American Statistical Association, 87(419), 659–671.
    • Milevsky, M. A., & Posner, S. E. (2001). The titanic option: Valuation of the guaranteed minimum death benefit in variable annuities and mutual funds. Journal of Risk and Insurance, 68(1), 93–128.
    • Milevsky, M. A., & Salisbury, T. S. (2005). Financial valuation of guaranteed minimum withdrawal benefits. Insurance : Mathematics and Economics.
    • Milevsky, M. A., & Salisbury, T. S. (2006). Financial valuation of guaranteed minimum withdrawal ben- efits. Insurance: Mathematics and Economics, 38(1), 21–38.
    • Piscopo, G., & Haberman, S. (2011). The valuation of guaranteed lifelong withdrawal benefit options in variable annuity contracts and the impact of mortality risk. North American Actuarial Journal, 15(1), 59–76.
    • Shah, P., & Bertsimas, D. (2008). An analysis of the guaranteed withdrawal benefits for life option.
    • Shevchenko, P. V., & Luo, X. (2016). A unified pricing of variable annuity guarantees under the optimal stochastic control framework.
    • Yang, S. S., & Dai, T. S. (2013). A flexible tree for evaluating guaranteed minimum withdrawal bene- fits under deferred life annuity contracts with various provisions. Insurance: Mathematics and Economics, 52(2), 231–242.
Download Preview
Close
Insurance Markets and Companies
Insurance Markets and Companies