Volterra mortality model: actuarial valuation and risk management with long-range dependence. (English) Zbl 1460.91240

The authors propose a novel class of Volterra mortality models that incorporate long-range dependence (LRD) into the actuarial valuation, retain tractability, and are consistent with the existing continuous-time affine mortality models.
To include the LRD, the authors suppose that the counting process \(N(t)\) of the number of deaths has an intensity \(\mu(t)=f(X(t))\) where \(X(t)\) follows a fractional Brownian motion which is equivalent to a Volterra process.
The survival probability is derived in closed form by taking into account the historical health records. The models can be used for valuing mortality-related products such as death benefits, annuities, longevity bonds, as well as for offering optimal mean-variance mortality hedging rules.
Numerical studies are conducted to examine the effect of incorporating LRD into mortality rates on various insurance products and hedging efficiency.


91G05 Actuarial mathematics
60G22 Fractional processes, including fractional Brownian motion
Full Text: DOI arXiv


[1] Abi Jaber, E.; Larsson, M.; Pulido, S., Affine Volterra processes, Ann. Appl. Probab., 29, 5, 3155-3200 (2019) · Zbl 1441.60052
[2] Antonio, K.; Bardoutsos, A.; Ouburg, W., Bayesian Poisson log-bilinear models for mortality projections with multiple populations, Eur. Actuar. J., 5, 2, 245-281 (2015) · Zbl 1329.91111
[3] Baudoin, F.; Nualart, D., Equivalence of Volterra processes, Stochastic Process. Appl., 107, 2, 327-350 (2003) · Zbl 1075.60519
[4] Biffis, E., Affine processes for dynamic mortality and actuarial valuations, Insurance Math. Econom., 37, 3, 443-468 (2005) · Zbl 1129.91024
[5] Biffis, E.; Millossovich, P., The fair value of guaranteed annuity options, Scand. Actuar. J., 2006, 1, 23-41 (2006) · Zbl 1142.91036
[6] Blackburn, C.; Sherris, M., Consistent dynamic affine mortality models for longevity risk applications, Insurance Math. Econom., 53, 1, 64-73 (2013) · Zbl 1284.91208
[7] Blake, D.; Cairns, A.; Dowd, K.; MacMinn, R., Longevity bonds: Financial engineering, valuation, and hedging, J. Risk Insurance, 73, 4, 647-672 (2006)
[8] Brouhns, N.; Denuit, M.; Vermunt, J. K., A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance Math. Econom., 31, 3, 373-393 (2002) · Zbl 1074.62524
[9] Chuang, S. L.; Brockett, P. L., Modeling and pricing longevity derivatives using stochastic mortality rates and the esscher transform, N. Am. Actuar. J., 18, 1, 22-37 (2014) · Zbl 1412.91040
[10] Danesi, I. L.; Haberman, S.; Millossovich, P., Forecasting mortality in subpopulations using Lee-Carter type models: A comparison, Insurance Math. Econom., 62, 151-161 (2015) · Zbl 1318.91109
[11] Delgado-Vences, F.; Ornelas, A., Modelling Italian mortality rates with a geometric-type fractional Ornstein-Uhlenbeck process (2019), arXiv preprint arXiv:1901.00795
[12] Duffie, D.; Filipović, D.; Schachermayer, W., Affine processes and applications in finance, Ann. Appl. Probab., 13, 3, 984-1053 (2003) · Zbl 1048.60059
[13] Duffie, D.; Pan, J.; Singleton, K., Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68, 6, 1343-1376 (2000) · Zbl 1055.91524
[14] Filipović, D., Time-inhomogeneous affine processes, Stochastic Process. Appl., 115, 4, 639-659 (2005) · Zbl 1079.60068
[15] Gompertz, B., On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Philos. Trans. R. Soc. Lond., 115, 513-583 (1825)
[16] Han, B.; Wong, H. Y., Mean-variance portfolio selection with Volterra Heston model, Appl. Math. Optim. (2020)
[17] Jevtić, P.; Luciano, E.; Vigna, E., Mortality surface by means of continuous time cohort models, Insurance Math. Econom., 53, 1, 122-133 (2013) · Zbl 1284.91240
[18] Jevtić, P.; Regis, L., A continuous-time stochastic model for the mortality surface of multiple populations, Insurance Math. Econom., 88, 181-195 (2019) · Zbl 1425.91226
[19] Lee, R. D.; Carter, L. R., Modeling and forecasting US mortality, J. Amer. Statist. Assoc., 87, 419, 659-671 (1992) · Zbl 1351.62186
[20] Leonenko, N.; Scalas, E.; Trinh, M., Limit theorems for the fractional non-homogeneous Poisson process, J. Appl. Probab., 56, 1, 246-264 (2019) · Zbl 1418.60025
[21] Li, N.; Lee, R., Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method, Demography, 42, 3, 575-594 (2005)
[22] Milevsky, M. A.; Promislow, S. D., Mortality derivatives and the option to annuitise, Insurance Math. Econom., 29, 3, 299-318 (2001) · Zbl 1074.62530
[23] Renshaw, A. E.; Haberman, S., Lee-Carter mortality forecasting with age-specific enhancement, Insurance Math. Econom., 33, 2, 255-272 (2003) · Zbl 1103.91371
[24] Schrager, D. F., Affine stochastic mortality, Insurance Math. Econom., 38, 1, 81-97 (2006) · Zbl 1103.60063
[25] Toczydlowska, D.; Peters, G.; Fung, M.; Shevchenko, P., Stochastic period and cohort effect state-space mortality models incorporating demographic factors via probabilistic robust principal components, Risks, 5, 3, 42 (2017)
[26] Villegas, A. M.; Haberman, S., On the modeling and forecasting of socioeconomic mortality differentials: An application to deprivation and mortality in England, N. Am. Actuar. J., 18, 1, 168-193 (2014) · Zbl 1412.91057
[27] Wang, Y.; Zhang, N.; Jin, Z.; Ho, T. L., Pricing longevity-linked derivatives using a stochastic mortality model, Comm. Statist. Theory Methods, 48, 24, 5923-5942 (2019)
[28] Wong, T. W.; Chiu, M. C.; Wong, H. Y., Managing mortality risk with longevity bonds when mortality rates are cointegrated, J. Risk Insurance, 84, 3, 987-1023 (2017)
[29] Yan, H.; Peters, G.; Chan, J., Mortality models incorporating long memory improves life table estimation: a comprehensive analysis, Ann. Actuar. Sci. (2018), (in press)
[30] Yan, H.; Peters, G.; Chan, J., Multivariate long memory cohort mortality models, Astin Bull., 50, 1, 223-263 (2020) · Zbl 1431.91346
[31] Yaya, O. S.; Gil-Alana, L. A.; Amoateng, A. Y., Under-5 mortality rates in G7 countries: Analysis of fractional persistence, structural breaks and nonlinear time trends, Eur. J. Popul., 35, 675-694 (2019)
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.