A subordinated Markov model for stochastic mortality. (English) Zbl 1273.91239

Summary: In this paper we propose a subordinated Markov model for modeling stochastic mortality. The aging process of a life is assumed to follow a finite-state Markov process with a single absorbing state and the stochasticity of mortality is governed by a subordinating gamma process. We focus on the theoretical development of the model and have shown that the model exhibits many desirable properties of a mortality model and meets many model selection criteria laid out in [A. J. G. Cairns et al., Astin Bull. 36, No. 1, 79–120 (2006; Zbl 1162.91403); Scand. Actuar. J. 2008, No. 2–3, 79–113 (2008; Zbl 1224.91048)]. The model is flexible and fits either historical mortality data or projected mortality data well. We also explore applications of the proposed model to the valuation of mortality-linked securities. A general valuation framework for valuing mortality-linked products is presented for this model. With a proposed risk loading mechanism, we can make an easy transition from the physical measure to a risk-neutral measure and hence is able to calibrate the model to market information. The phase-type structure of the model allows us to apply the matrix-analytic methods that have been extensively used in ruin theory in actuarial science and queuing theory in operations research (see [S. Asmussen, Applied probability and queues. 2nd revised and extended ed. New York, NY: Springer (2003; Zbl 1029.60001); S. Asmussen and H. Albrecher, Ruin probabilities. 2nd ed. Hackensack, NJ: World Scientific (2010; Zbl 1247.91080); M. F. Neuts, Matrix-geometric solutions in stochastic models. An algorithmic approach. London: The Johns Hopkins University Press (1981; Zbl 0469.60002)]. As a result, many quantities of interest such as the distribution of future survival rates, prediction intervals, the term structure of mortality as well as the value of caps and floors on the survival index can be obtained analytically.


91B30 Risk theory, insurance (MSC2010)
91B70 Stochastic models in economics
91G20 Derivative securities (option pricing, hedging, etc.)
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[1] Aalen OO (1995) Phase type distributions in survival analysis. Scand J Stat 22:447–463 · Zbl 0836.62095
[2] Asmussen S (1987) Applied probability and queues. Wiley, New York · Zbl 0624.60098
[3] Asmussen S, Albrecher H (2010) Ruin probabilities, 2nd edn. World Scientific Publishing, Singapore · Zbl 1247.91080
[4] Ballotta L, Haberman S (2006) The fair valuation problem of guaranteed annuity options: the stochastic mortality environment case. Insur Math Econ 38:195–214 · Zbl 1101.60045
[5] Biffis E (2005) Affine processes for dynamic mortality and actuarial valuation. Insur Math Econ 37:443–468 · Zbl 1129.91024
[6] Biffis E, Denuit M (2006) Lee-Carter goes risk-neutral: an application to the Italian annuity market. Giornale dell’Istituto Italiano degli Attuari LXIX:1–21
[7] Blake D, Dowd K, Cairns AJG (2008a) Longevity risk and the grim reaper’s toxic tail: the survivor fan charts. Insur Math Econ 42:1062–1066 · Zbl 1141.91485
[8] Blake DP, Cairns AJ, Dowd K (2008b) The birth of the life market. Asia Pa J Risk Insur 3:6–36
[9] Boyle P, Hardy M (2003) Guaranteed annuity options. ASTIN Bull 33:125–152 · Zbl 1098.91527
[10] Brouhns N, Denuit M, Vermunt J (2002) A Poisson log-bilinear regression approach to the construction of projected lifetables. Insur Math Econ 31:373–393 · Zbl 1074.62524
[11] Cairns AJG, Blake D, Dowd K (2006) Pricing death: frameworks for the valuation and securitization of mortality risk. ASTIN Bull 36:79–120 · Zbl 1162.91403
[12] Cairns AJG, Blake D, Dowd K (2006) A two-factor model for stochastic mortality with parameter uncertainty. J Risk Insur 1:687–718
[13] Cairns AJG, Blake D, Dowd K (2008) Modelling and management of mortality risk: a review. Scand Actuar J 2:79–113 · Zbl 1224.91048
[14] CMI (2004) Projecting future mortality: a discussion paper. Continuous mortality investigation working paper 3
[15] CMI (2005) Projecting future mortality: towards a proposal for a stochastic methodology. Continuous mortality investigation working paper 15
[16] CMI (2007) Stochastic projection methodologies: Lee-Carter model features, example results and implications. Continuous mortality investigation working paper 15
[17] Dahl M (2004) Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts. Insur Math Econ 35:113–136 · Zbl 1075.62095
[18] Dahl M, Melchior M, Moller T (2008) On systematic mortality risk and risk minimization with survivor swaps. Scandinavian Actuarial Journal 2:114–146 · Zbl 1224.91054
[19] Dahl M, Møller T (2006) Valuation and hedging of life insurance liabilities with systematic mortality risk. Insur Math Econ 39:193–217 · Zbl 1201.91089
[20] Dowd K, Cairns AJG, Blake D (2006) Mortality-dependent financial risk measures. Insur Math Econ 38:427–440 · Zbl 1168.91411
[21] Foo J, Kevin Leder K, Michor F (2011) Stochastic dynamics of cancer initiation. Phys Biol. doi: 10.1088/1478-3975/8/1/015002
[22] GAD (2001) National population projections: review of methodology for projecting mortality. National statistics quality review series, report no. 8. Government Actuary’s Department
[23] Hautphenne S, Latouche G (2011) The Markovian binary tree applied to demography. J Math Biol. doi: 10.1007/s00285-011-0437-1 · Zbl 1279.60113
[24] Hurd T, Kuznetsov A (2007) Affine Markov chain model of multifirm credit migration. J Credit Risk 3:3–29
[25] Jarrow RA, Lando D, Turnbull SM (1997) A Markov model for the term structure of credit risk spreads. Rev Financial Stud 10:481–523
[26] Kay R (1986) A Markov model for analysing cancer markers and disease states in survival studies. Biometrics 42:855–865 · Zbl 0622.62100
[27] Lee RD, Carter LR (1992) Modeling and forecasting U.S. mortality. J Am Stat Assoc 87:659–675 · Zbl 1351.62186
[28] Lee SCK, Lin XS (2011) Modeling dependent risks with multivariate Erlang mixtures. ASTIN Bulletin, to appear
[29] Lee SCK, Lin XS (2010) Modeling and evaluating insurance losses via mixtures of Erlang distributions. N Am Actuar J 14(1):107–130 · Zbl 05941820
[30] Lin XS, Liu X (2007) Markov aging process and phase-type law of mortality. N Am Actuar J 11:92–109
[31] Longini IM, Clark WS, Gardner L, Brundage JF (1991) The dynamics of CD4+ T-lymphocyte decline in HIV-infected individuals: a Markov modelling approach. J Acquir Immune Defic Syndr 4:1141–1147
[32] Madan DB, Carr P, Chang EC (1998) The Variance Gamma process and option pricing. Eur Finance Rev 2:79–105 · Zbl 0937.91052
[33] Madan DB, Milne F (1991) Option pricing with V.G. martingale components. Math Finance 1:39–55 · Zbl 0900.90105
[34] Milevsky MA, Promislow SD (2001) Mortality derivatives and the option to annuitise. Insur Math Econ 29:299–318 · Zbl 1074.62530
[35] Milevsky MA, Promislow SD, Young J (2006) Killing the law of large numbers: mortality risk premiums and the Sharpe ratio. J Risk Insur 73:673–686
[36] Neuts MF (1981) Matrix-geometrix solutions in stochastic models. Johns Hopkins University Press, Baltimore
[37] Pitacco E (2004) Survival models in a dynamic context: a survey. Insur Math Econ 35:279–298 · Zbl 1079.91050
[38] Renshaw A, Haberman S (2003) Lee-Carter mortality forecasting with age-specific enhancement. Insur Math Econ 33:255–272 · Zbl 1103.91371
[39] Renshaw A, Haberman S (2006) A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insur Math Econ 38:556–570 · Zbl 1168.91418
[40] Tuljapurkar S, Boe C (1998) Mortality changes and forecasting: how much and how little do we know? N Am Actuar J 2:13–47 · Zbl 1081.91603
[41] Willets R (1999) Mortality in the next millennium. Presented at the Staple Inn Actuarial Society on 7 December
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