Forward mortality rates in discrete time. I: Calibration and securities pricing. (English) Zbl 1461.91246

Hunt, Andrew; Blake, David

N. Am. Actuar. J. 25, Suppl. 1, S482-S507 (2021).

Summary: Many users of mortality models are interested in using them to place values on longevity-linked liabilities and securities. Modern regulatory regimes require that the values of liabilities and reserves are consistent with market prices (if available), though the gradual emergence of a traded market in longevity risk needs methods for pricing new types of longevity-linked securities quickly and efficiently. In this study, we develop a new forward mortality framework to enable the efficient pricing of longevity-linked liabilities and securities in a market-consistent fashion. This approach starts from the historical data of the observed mortality rates, i.e., the force of mortality. Building on the dynamics of age/period/cohort models of the observed force of mortality, we develop models of forward mortality rates and then use a change of measure to incorporate whatever market information is available. The resulting forward mortality rates are then used to value a number of different longevity-linked securities, such as \(q\)-forwards, \(s\)-forwards, and longevity swaps.
For Part II, see [the authors, ibid. 25, Suppl. 1, S508–S533 (2021; Zbl 1461.91247)].

MSC:

91G05	Actuarial mathematics
91G20	Derivative securities (option pricing, hedging, etc.)

Citations:

Zbl 1461.91247

References:

[1]	Alai, D. H.; Ignatieva, K.; Sherris., M., Modelling longevity risk: Generalizations of the Olivier-Smith model, Tech. Rep (2013)
[2]	Ballotta, L.; Haberman., S., The fair valuation problem of guaranteed annuity options: The stochastic mortality environment case, Insurance: Mathematics and Economics, 38, 1, 195-214 (2006) · Zbl 1101.60045
[3]	Barbarin, J., Heath-Jarrow-Morton modelling of longevity bonds and the risk minimization of life insurance portfolios, Insurance: Mathematics and Economics, 43, 1, 41-55 (2008) · Zbl 1140.91377
[4]	Barrieu, P. M.; Bensusan, H.; Karoui, N. E.; Hillairet, C.; Loisel, S.; Ravanelli., C., Understanding, modelling and managing longevity risk: Key issues and main challenges, Scandinavian Actuarial Journal, 3, 203-231 (2012) · Zbl 1277.91073
[5]	Bauer, D.; Börger, M.; Ruß, J.; Zwiesler., H., The volatility of mortality, Asia-Pacific Journal of Risk and Insurance, 3, 10, 1-35 (2008)
[6]	Bauer, D.; Kramer, F., Risk and valuation of mortality contingent catastrophe bonds, Tech. Rep (2007)
[7]	Bayraktar, E.; Young., V. R., Hedging life insurance with pure endowments, Insurance: Mathematics and Economics, 40, 3, 435-44 (2007) · Zbl 1183.91067
[8]	Blake, D.; Cairns, A. J.; Coughlan, G. D.; Dowd, K.; MacMinn., R., The new life market, Journal of Risk and Insurance, 80, 3, 501-58 (2013)
[9]	Blake, D.; Cairns, A. J.; Dowd., K., Living with mortality: Longevity bonds and other mortality-linked securities, British Actuarial Journal, 12, 1, 153-97 (2006)
[10]	Börger, M.; Fleischer, D.; Kuksin., N., Modeling the mortality trend under modern solvency regimes, ASTIN Bulletin, 44, 1, 1-38 (2013)
[11]	Cairns, A. J., A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27, 3, 313-30 (2000) · Zbl 0971.62063
[12]	Cairns, A. J., A multifactor generalisation of the Olivier-Smith model for stochastic mortality, Tech. Rep (2007)
[13]	Cairns, A. J., Robust hedging of longevity risk, Journal of Risk and Insurance, 80, 3, 621-48 (2013)
[14]	Cairns, A. J.; Blake, D.; Dowd., K., Pricing death: Frameworks for the valuation and securitization of mortality risk, ASTIN Bulletin, 36, 1, 79-120 (2006) · Zbl 1162.91403
[15]	Cairns, A. J.; Blake, D.; Dowd., K., A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration, Journal of Risk and Insurance, 73, 4, 687-718 (2006)
[16]	Cairns, A. J.; Blake, D.; Dowd, K.; Coughlan, G. D.; Epstein, D.; Khalaf-Allah., M., Mortality density forecasts: An analysis of six stochastic mortality models, Insurance: Mathematics and Economics, 48, 3, 355-67 (2011)
[17]	Cairns, A. J.; Blake, D.; Dowd, K.; Coughlan, G. D.; Epstein, D.; Ong, A.; Balevich., I., A quantitative comparison of stochastic mortality models using data from England and Wales and the United States, North American Actuarial Journal, 13, 1, 1-35 (2009)
[18]	Continuous Mortality Investigation, Working Paper 38—A prototype mortality projection model: Part one—an outline of the proposed approach (2009)
[19]	Continuous Mortality Investigation, Working Paper 39—A prototype mortality projections model: Part two—detailed analysis (2009)
[20]	Continuous Mortality Investigation, Working Paper 69—The CMI mortality projections model, CMI_2013, and feedback on the consultation on the future of the CMI Library of Mortality Projections and the CMI mortality projections model (2013)
[21]	Coughlan, G. D.; Epstein, D.; Sinha, A.; Honig., P., q-Forwards: Derivatives for transferring longevity and mortality risks (2007), London: JPMorgan Pension Advisory Group
[22]	Cox, S. H.; Lin., Y., Natural hedging of life and annuity mortality risks, North American Actuarial Journal, 11, 3, 1-15 (2007)
[23]	Cox, S. H.; Lin, Y.; Wang., S., Multivariate exponential tilting and pricing implications for mortality securitization, Journal of Risk and Insurance, 73, 4, 719-36 (2006)
[24]	Denuit, M. M., An index for longevity risk transfer, Journal of Computational and Applied Mathematics, 230, 2, 411-17 (2009) · Zbl 1173.91443
[25]	Denuit, M. M.; Devolder, P.; Goderniaux., A.-M., Securitization of longevity risk: Pricing survivor bonds with Wang transform in the Lee-Carter framework, Journal of Risk and Insurance, 74, 1, 87-113 (2007)
[26]	Denuit, M. M.; Goderniaux, A.-M., Closing and projecting life tables using log-linear models, Bulletin of the Swiss Association of Actuaries, 1, 29-49 (2005) · Zbl 1333.62251
[27]	Dhaene, J.; Kukush, A.; Luciano, E.; Schoutens, W.; Stassen., B., On the (in-)dependence between financial and actuarial risks, Insurance: Mathematics and Economics, 52, 3, 522-31 (2013) · Zbl 1284.91226
[28]	Dowd, K., Survivor bonds: A comment on Blake and Burrows, Journal of Risk and Insurance, 70, 2, 339-48 (2003)
[29]	Dowd, K.; Cairns, A. J.; Blake., D., Mortality-dependent financial risk measures, Insurance: Mathematics and Economics, 38, 3, 427-40 (2006) · Zbl 1168.91411
[30]	European Insurance and Occupational Pensions Authority, Technical specification for the preparatory phase (Part I). Tech. Rep (2014), European Insurance and Occupational Pensions Authority: European Insurance and Occupational Pensions Authority, Frankfurt
[31]	Fetiveau, C.; Jia, C., Longevity risk hedging with population based index solution—A study of basis risk based on England & Wales population, Tech. Rep (2014)
[32]	Frederic, P.; Lad., F., Two moments of the logitnormal distribution, Communications in Statistics - Simulation and Computation, 37, 7, 1263-69 (2008) · Zbl 1148.60004
[33]	Gerber, H.; Shiu, E., Option pricing by Esscher transforms, Transactions of the Society of Actuaries, 46, 99-191 (1994)
[34]	Haberman, S.; Renshaw., A., On age-period-cohort parametric mortality rate projections, Insurance: Mathematics and Economics, 45, 2, 255-70 (2009) · Zbl 1231.91195
[35]	Haberman, S.; Renshaw., A., A comparative study of parametric mortality projection models, Insurance: Mathematics and Economics, 48, 1, 35-55 (2011)
[36]	Hobcraft, J.; Menken, J.; Preston., S. H., Age, period and cohort effects in demography: A review, Population Index, 48, 1, 4-43 (1982)
[37]	Holland, B.; Ahsanullah, M., Further results on a distribution of Meinhold and Singpurwalla, American Statistical Association, 43, 4, 216-19 (1989)
[38]	(2014)
[39]	Hunt, A.; Blake., D., A general procedure for constructing mortality models, North American Actuarial Journal, 18, 1, 116-38 (2014) · Zbl 1412.91045
[40]	Hunt, A.; Blake., D., Modelling longevity bonds: Analysing the Swiss Re Kortis Bond, Insurance: Mathematics and Economics, 63, 12-29 (2015) · Zbl 1348.91150
[41]	Hunt, A.; Blake., D., Modelling mortality for pension schemes, ASTIN Bulletin, 47, 2, 601-29 (2017) · Zbl 1390.91189
[42]	Hunt, A., and D. Blake. 2020a. On the structure and classification of mortality models. North American Actuarial Journal (forthcoming).
[43]	Hunt, A., and D. Blake. 2020b. A Bayesian approach to modelling and projecting cohort effects. North American Actuarial Journal (forthcoming).
[44]	Hunt, A., and D. Blake. 2020c. Forward mortality rates in discrete time II: Longevity risk and hedging strategies. North American Actuarial Journal (forthcoming).
[45]	Hunt, A., and D. Blake. 2020d. Identifiability in age/period mortality models. Annals of Actuarial Science (forthcoming).
[46]	Hunt, A., and D. Blake. 2020e. Identifiability in age/period/cohort mortality models. Annals of Actuarial Science (forthcoming).
[47]	Kijima, M., A multivariate extension of equilibrium pricing transforms: The multivariate Esscher and Wang transforms for pricing financial and insurance risks, Tech. Rep (2005) · Zbl 1162.91418
[48]	Lane, M., Longevity risk from the perspective of the ILS markets, The Geneva Papers on Risk and Insurance Issues and Practice, 36, 4, 501-15 (2011)
[49]	Lee, R. D.; Carter., L. R., Modeling and forecasting U.S. mortality, Journal of the American Statistical Association, 87, 419, 659-71 (1992) · Zbl 1351.62186
[50]	Lemoine, K., Mortality regimes and longevity risk in a life annuity portfolio, Scandinavian Actuarial Journal, 8, 689-724 (2015) · Zbl 1401.91163
[51]	Li, J., A quantitative comparison of simulation strategies for mortality projection, Annals of Actuarial Science, 8, 2, 281-97 (2014)
[52]	Li, J. S.-H.; Luo., A., Key q-duration: A framework for hedging longveity risk, ASTIN Bulletin, 42, 2, 413-52 (2012)
[53]	Li, N.; Lee, R. D.; Tuljapurkar., S., Using the Lee-Carter method to forecast mortality for populations with limited data, International Statistical Review, 72, 1, 19-36 (2004) · Zbl 1330.62349
[54]	Loeys, J.; Panigirtzoglou, N.; Ribeiro., R., Longevity: A market in the making (2007), London: JPMorgan Pension Advisory Group
[55]	Michaelson, A.; Mulholland., J., Strategy for increasing the global capacity for longevity risk transfer: Developing transactions that attract capital markets investors, Journal of Alternative Investments, 17, 1, 18-27 (2014)
[56]	Milevsky, M.; Promislow., S., Mortality derivatives and the option to annuitise, Insurance: Mathematics and Economics, 29, 299-318 (2001) · Zbl 1074.62530
[57]	Milevsky, M.; Promislow, S.; Young, V. R.; Bayraktar., E., Financial valuation of mortality risk via the instantaneous Sharpe ratio, Journal of Economic Dynamics and Control, 33, 3, 676-91 (2005) · Zbl 1170.91406
[58]	Milidonis, A.; Lin, Y.; Cox., S. H., Mortality regimes and pricing, North American Actuarial Journal, 15, 2, 266-89 (2011) · Zbl 1228.91043
[59]	Miltersen, K. R.; Persson, S.-A., Is mortality dead? Stochastic forward force of mortality rate determined by no arbitrage, Tech. Rep (2005)
[60]	Norberg, R., Forward mortality and other vital rates—Are they the way forward?, Insurance: Mathematics and Economics, 47, 2, 105-112 (2010) · Zbl 1231.91459
[61]	Oeppen, J.; Vaupel., J., Broken limits to life expectancy, Science, 296, 5570, 1029-31 (2002)
[62]	Olivier, P.; Jeffrey., T., Stochastic mortality models (2004)
[63]	Pelsser, A., Pricing and hedging guaranteed annuity options via static option replication, Insurance: Mathematics and Economics, 33, 2, 283-96 (2003) · Zbl 1103.91352
[64]	Plat, R., On stochastic mortality modeling, Insurance: Mathematics and Economics, 45, 3, 393-404 (2009) · Zbl 1231.91227
[65]	Renshaw, A.; Haberman., S., A cohort-based extension to the Lee-Carter model for mortality reduction factors, Insurance: Mathematics and Economics, 38, 3, 556-70 (2006) · Zbl 1168.91418
[66]	Schrager, D., Affine stochastic mortality, Insurance: Mathematics and Economics, 38, 1, 81-97 (2006) · Zbl 1103.60063
[67]	Smith, A., Stochastic mortality modelling (2005)
[68]	Sweeting, P. J., A trend-change extension of the Cairns-Blake-Dowd model, Annals of Actuarial Science, 5, 2, 143-62 (2011)
[69]	Tappe, S.; Weber., S., Stochastic mortality models: An infinite-dimensional approach, Finance and Stochastics, 18, 1, 209-48 (2013) · Zbl 1308.60065
[70]	Wang, C.-W.; Huang, H.; Liu., I.-C., A quantitative comparison of the Lee-Carter model under different types of non-Gaussian innovations, The Geneva Papers on Risk and Insurance Issues and Practice, 36, 4, 675-96 (2011)
[71]	Wang, S., A class of distortion operators for pricing financial and insurance risks, Journal of Risk and Insurance, 67, 1, 15-36 (2000)
[72]	Wang, S., A univeral framework for pricing financial and insurance risks, ASTIN Bulletin, 32, 2, 213-34 (2002) · Zbl 1090.91555
[73]	Zhou, R.; Li, J. S.-H.; Tan., K. S., Economic pricing of mortality-linked securities: A tatonnement approach, Journal of Risk and Insurance, 82, 1, 65-95 (2015)
[74]	Zhu, N.; Bauer., D., Applications of forward mortality factor models in life insurance practice, Geneva Papers on Risk and Insurance Issues and Practice, 36, 567-94 (2011)
[75]	Zhu, N.; Bauer., D., Coherent modeling of the risk in mortality projections: A semi parametric approach. Tech. Rep (2011), Georgia State University
[76]	Zhu, N.; Bauer., D., A cautionary note on natural hedging of longevity risk, North American Actuarial Journal, 18, 1, 104-115 (2014) · Zbl 1412.91061