×

Comonotonic approximations of risk measures for variable annuity guaranteed benefits with dynamic policyholder behavior. (English) Zbl 1354.91066

Summary: The computation of various risk metrics is essential to the quantitative risk management of variable annuity guaranteed benefits. The current market practice of Monte Carlo simulation often requires intensive computations, which can be very costly for insurance companies to implement and take so much time that they cannot obtain information and take actions in a timely manner. In an attempt to find low-cost and efficient alternatives, we explore the techniques of comonotonic bounds to produce closed-form approximation of risk measures for variable annuity guaranteed benefits. The techniques are further developed in this paper to address in a systematic way risk measures for death benefits with the consideration of dynamic policyholder behavior, which involves very complex path-dependent structures. In several numerical examples, the method of comonotonic approximation is shown to run several thousand times faster than simulations with only minor compromise of accuracy.

MSC:

91B30 Risk theory, insurance (MSC2010)
60E15 Inequalities; stochastic orderings
65C50 Other computational problems in probability (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bauer, D.; Kling, A.; Russ, J., A universal pricing framework for guaranteed minimum benefits in variable annuities, Astin Bull., 38, 2, 621-651, (2008) · Zbl 1274.91399
[2] Bacinello, A. R.; Millossovich, P.; Olivieri, A.; Pitacco, E., Variable annuities: a unifying valuation approach, Insurance Math. Econom., 49, 3, 285-297, (2011)
[3] Farr, I.; Mueller, H.; Scanlon, M.; Stronkhorst, S., (Economic Capital for Life Insurance Companies, SOA Monograph, (2008))
[4] Koursaris, A., A least sqaures Monte Carlo approach to liability proxy modelling and capital calculation. technical report, (2011), Barrie & Hibbert
[5] Bauer, D.; Bergmann, D.; Kiesel, R., On the risk-neutral valuation of life insurance contracts with numerical methods in view, Astin Bull., 40, 1, 65-95, (2010) · Zbl 1230.91066
[6] Hardy, M. R., Investment guarantees: modeling and risk management for equity-linked life insurance, (2003), John Wiley & Sons, Inc. New Jersey · Zbl 1092.91042
[7] Ulm, E. R., Analytic solution for return of premium and rollup guaranteed minimum death benefit options under some simple mortality laws, Astin Bull., 38, 2, 543-563, (2008) · Zbl 1256.91035
[8] Ulm, E. R., Analytic solution for ratchet guaranteed minimum death benefit options under a variety of mortality laws, Insurance Math. Econom., 58, 14-23, (2014) · Zbl 1304.91133
[9] Chi, Y.; Lin, S. X., Are flexible premium variable annuities under-priced?, ASTIN Bull., 42, 2, 559-574, (2012) · Zbl 1277.91078
[10] Costabile, M., Analytical valuation of periodical premiums for equity-linked policies with minimum guarantee, Insurance Math. Econom., 53, 3, 597-600, (2013) · Zbl 1290.91157
[11] Marshall, C.; Hardy, M.; Saunders, D., Valuation of a guaranteed minimum income benefit, N. Am. Actuar. J., 14, 1, 38-58, (2010)
[12] Bernard, C.; MacKay, A.; Muehlbeyer, M., Optimal surrender policy for variable annuity guarantees, Insurance Math. Econom., 55, 116-128, (2014) · Zbl 1296.91144
[13] Feng, R.; Volkmer, H. W., Analytical calculation of risk measures for variable annuity guaranteed benefits, Insurance Math. Econom., 51, 3, 636-648, (2012) · Zbl 1285.91055
[14] Feng, R.; Volkmer, H. W., Spectral methods for the calculation of risk measures for variable annuity guaranteed benefits, Astin Bull., 44, 3, 653-681, (2014)
[15] Feng, R.; Shimizu, Y., Applications of central limit theorems for equity-linking insurance, Insurance Math. Econom., 69, 138-148, (2016) · Zbl 1369.91082
[16] Feng, R., A comparative study of risk measures for guaranteed minimum maturity benefits by a PDE method, N. Am. Actuar. J., 18, 4, 445-461, (2014)
[17] Gorski, L. M.; Brown, R. A., Recommended approach for setting regulatory risk-based capital requirements for variable annuities and similar products. technical report, (2005), American Academy of Actuaries Life Capital Adequacy Subcommittee Boston
[18] Dhaene, J.; Vanduffel, S.; Goovaerts, M. J.; Kaas, R.; Tang, Q.; Vyncke, D., Risk measures and comonotonicity: a review, Stoch. Models, 22, 4, 573-606, (2006) · Zbl 1159.91403
[19] Marin-Solano, J.; Dhaene, J.; Ribas, C.; Bosch-Princep, M.; Vanduffel, S., Buy-and-hold strategies and comonotonic approximations, Belg. Actuar. Bull., 9, 1, 17-28, (2010)
[20] Dhaene, J.; Vanduffel, S.; Goovaerts, M.; Kaas, R.; Vyncke, D., Comonotonic approximations for optimal portfolio selection problems, J. Risk Insurance, 72, 2, 253-301, (2005)
[21] Simon, S.; Goovaerts, M.; Dhaene, J., An easy computable upper bound for the price of an arithmetic Asian option, Insurance Math. Econom., 26, 2-3, 175-184, (2000) · Zbl 0964.91021
[22] Albrecher, H.; Dhaene, J.; Goovaerts, M.; Schoutens, W., Static hedging of Asian options under Lévy models: the comonotonicity approach, J. Deriv., 12, 3, 63-72, (2005)
[23] Hobson, D.; Laurence, P.; Wang, T., Static-arbitrage upper bounds for the prices of basket options, Quant. Finance, 12, 3, 63-72, (2005) · Zbl 1134.91425
[24] Chen, X.; Deelstra, G.; Dhaene, J.; Vanmaele, M., Static super-replicating strategies for a class of exotic options, Insurance Math. Econom., 42, 3, 1067-1085, (2008) · Zbl 1141.91427
[25] Linders, D.; Dhaene, J.; Hounnon, H.; Vanmaele, M., Index options: a model-free approach. technical report AFI-1265, (2012), KU Leuven
[26] Dhaene, J.; Denuit, M.; Goovaerts, M. J.; Kaas, R.; Vyncke, D., The concept of comonotonicity in actuarial science and finance: theory, Insurance Math. Econom., 31, 1, 3-33, (2002), 5th IME Conference (University Park, PA, 2001) · Zbl 1051.62107
[27] Dhaene, J.; Denuit, M.; Goovaerts, M. J.; Kaas, R.; Vyncke, D., The concept of comonotonicity in actuarial science and finance: applications, Insurance Math. Econom., 31, 2, 133-161, (2002) · Zbl 1037.62107
[28] Deelstra, G.; Dhaene, J.; Vanmaele, M., An overview of comonotonicity and its applications in finance and insurance, (Advanced Mathematical Methods for Finance, (2011), Springer Heidelberg), 155-179 · Zbl 1233.60006
[29] Vanduffel, S.; Hoedemakers, T.; Dhaene, J., Comparing approximations for risk measures of sums of non-independent lognormal random variables, N. Am. Actuar. J., 9, 4, 71-82, (2005) · Zbl 1215.91038
[30] Vanmaele, M.; Deelstra, G.; Liinev, J.; Dhaene, J.; Goovaerts, M., Bounds for the price of discrete arithmetic Asian options, J. Comput. Appl. Math., 185, 1, 51-90, (2006) · Zbl 1131.91027
[31] Denuit, M.; Dhaene, J.; Goovaerts, M.; Kaas, R., Actuarial theory for dependent risks: measures, orders and models, (2005), Wiley Chichester
[32] Rogers, L. C.G.; Shi, Z., The value of an Asian option, J. Appl. Probab., 32, 4, 1077-1088, (1995) · Zbl 0839.90013
[33] Vanduffel, S.; Shang, Z.; Henrard, L.; Dhaene, J.; Valdez, E. A., Analytic bounds and approximations for annuities and Asian options, Insurance Math. Econom., 42, 3, 1109-1117, (2008) · Zbl 1141.91550
[34] Vanduffel, S.; Chen, X.; Dhaene, J.; Goovaerts, M.; Henrard, L.; Kaas, R., Optimal approximations for risk measures of sums of lognormals based on conditional expectations, J. Comput. Appl. Math., 221, 1, 202-218, (2008) · Zbl 1154.91021
[35] Hainaut, D., Default probabilities of a holding company, with complete and partial information, J. Comput. Appl. Math., 271, 380-400, (2014) · Zbl 1319.91150
[36] Stochastic modeling: theory and reality from an actuarial perspective, (2010), International Actuarial Association
[37] Campbell, J.; Chan, M.; Li, K.; Lombardi, L.; Lombardi, L.; Purushotham, M., Modeling of policyholder behavior for life insurance and annuity products. technical report, (2014), Society of Actuaries
[38] Policyhoder behavior in the tail variable annuity guaranteed benefits survey 2012 results. technical report, (2013), Society of Actuaries
[39] Bernard, C.; Cui, Z.; Vanduffel, S., Impact of flexible periodic premiums on variable annuity guarantees, N. Am. Actuar. J., (2016), in press
[40] Delong, Ł., Pricing and hedging of variable annuities with state-dependent fees, Insurance Math. Econom., 58, 24-33, (2014) · Zbl 1304.91098
[41] C. Bernard, M.R. Hardy, A. MacKay, State-dependent fees for variable annuity guarantees, 2013.
[42] Vyncke, D.; Goovaerts, M. J.; Dhaene, J., An accurate analytical approximation for the price of a european-style arithmetic Asian option, Finance, 25, 121-139, (2004)
[43] Liu, X.; Jang, J.; Kim, S. M., An application of comonotonicity theory in a stochastic life annuity framework, Insurance Math. Econom., 48, 271-279, (2011) · Zbl 1218.91085
[44] Owen, D. B, Tables for computing bivariate normal probabilities, Ann. Math. Stat., 27, 1075-1090, (1956) · Zbl 0073.13405
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.