##
**Affine stochastic mortality.**
*(English)*
Zbl 1103.60063

The author proposes a new model for mortality intensity. The approach is based on the observation that if the mortality intensity is an affine function of a number of latent factors, the survival and death probabilities are known in closed form. Most of the results are based on the literature on affine term structure models and the credit risk literature based on the subfiltration approach. The contribution consists of the application of these ideas to model the evolution of mortality rates over time. The author provides in the need for a model of mortality forces which can be combined consistently with continuous time models known from the derivative pricing literature. For some well known functional dependencies between age and mortality intensity (i.e. the Thiele and Makeham mortality laws) a new setup is introduced. The three main advantages of the model are a rich analytical structure (inherited from the affine setup), clear interpretation of the latent factors and the aforementioned consistency with derivative pricing models. In contrast to previous work, the mortality intensity is simultaneously considered for all ages. The author does not explicitly focus on the time series properties of mortality (although the model is extremely well suited for estimation to empirical data), rather he has a pricing and risk management application in mind. Four types of mortality risk are usually distinguished: trend (i.e. longevity), level (portfolio versus population), volatility (discrepancies between trend/level and observed mortality) and catastrophe. The model captures three of these types and the risks are directly quantified by parameter estimates. The author shows, using historical Dutch mortality rates, that the proposed Thiele and Makeham functional forms fit the data sufficiently well. Assuming independence of financial and mortality risk one can easily combine the proposed model with, for instance, a term structure model. One could then easily price several well studied options embedded in insurance contracts under stochastic mortality. To illustrate the effect of stochastic mortality on the pricing of a guaranteed annuity option the author calculates the value of an option on a life long annuity in a combined single factor Hull-White Makeham model.

Reviewer: Alexandr B. Vasil’ev (Odessa)

### MSC:

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

91G20 | Derivative securities (option pricing, hedging, etc.) |

### Keywords:

affine models; mortality laws; longevity risk; market price of mortality risk; mortality options; Gaussian Thiele model; Kalman filter estimation; valuation of endowments; annuities; guaranteed annuity options
PDF
BibTeX
XML
Cite

\textit{D. F. Schrager}, Insur. Math. Econ. 38, No. 1, 81--97 (2006; Zbl 1103.60063)

### References:

[1] | Bacinello, A.R.; Ortu, F., Pricing equity linked life insurance with endogenous minimum guarantee, Insurance mathematics and economics, 12, 245-257, (1993) · Zbl 0778.62093 |

[2] | Boyle, P.; Hardy, M., Guaranteed annuity options, Astin bulletin, 33, 2, 125-152, (2003) · Zbl 1098.91527 |

[3] | Brennan, M.J.; Schwartz, E.S., The pricing of equity linked life insurance policies with an asset value guarantee, Journal of financial economics, 3, 195-213, (1976) |

[4] | Dahl, M., Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance mathematics and economics, 35, 113-136, (2004) · Zbl 1075.62095 |

[5] | De Jong, F., Time series and cross section information in affine term structure models, Journal of business and economic statistics, 18, 300-314, (2000) |

[6] | Duffee, G.R., Estimating the price of default risk, Review of financial studies, 12, 197-226, (1999) |

[7] | Duffee, G.R., Stanton, R.H., 2004. Estimation of dynamic term structure models, working paper. |

[8] | Duffie, D.; Kan, R., A yield-factor model of interest rates, Mathematical finance, 6, 379-406, (1996) · Zbl 0915.90014 |

[9] | Duffie, D.; Pan, J.; Singleton, K., Transform analysis and asset pricing for affine jump diffusions, Econometrica, 68, 1343-1376, (2000) · Zbl 1055.91524 |

[10] | Elliott, R.; Jeanblanc, M.; Yor, M., On models of default risk, Mathematical finance, 10, 179-195, (2000) · Zbl 1042.91038 |

[11] | Filipovic, D., Time-inhomogeneous affine processes, Stochastic processes and their applications, 115, 639-659, (2005) · Zbl 1079.60068 |

[12] | Geman, H.; El Karoui, N.; Rochet, J.-C., Changes of numeraire, changes of probability measure and option pricing, Journal of applied probability, 32, 443-458, (1995) · Zbl 0829.90007 |

[13] | Green, W.H., Econometric analysis, (1997), Prentice Hall |

[14] | Jamshidian, F., Contingent claim evaluation in the Gaussian interest rate model, Research in finance, 9, 131-170, (1991) |

[15] | Jamshidian, F., LIBOR and swap market models and measures, Finance and stochastics, 1, 293-330, (1998) · Zbl 0888.60038 |

[16] | Jamshidian, F., Valuation of credit default swaps and swaptions, Finance and stochastics, 8, 343-371, (2004) · Zbl 1063.91034 |

[17] | Karoui, N.E., Rochet J., 1989. A pricing formula for options on coupon bonds, Cahier de Recherche du GREMAQ-CRES, no. 8925. |

[18] | Koopman, S.J.; Durbin, J., Fast filtering and smoothing for multivariate state space models, Journal of time series analysis, 21, 281-296, (2000) · Zbl 0959.62081 |

[19] | Lando, D., On Cox processes and credit risky securities, Review of derivatives research, 2, 99-120, (1998) · Zbl 1274.91459 |

[20] | Lee, R.D., The lee – carter method of forecasting mortality, with various extensions and applications, North American actuarial journal, 4, 80-93, (2000) · Zbl 1083.62535 |

[21] | Lee, R.D.; Carter, L.R., Modeling and forecasting US mortality, Journal of the American statistical association, 87, 659-671, (1992) · Zbl 1351.62186 |

[22] | Makeham, W.M., On the law of mortality and the construction of mortality tables, Journal of the institute of actuaries, 8, (1860) |

[23] | Milevsky, M.A.; Promislow, S.D., Mortality derivatives and the option to annuitize, Insurance mathematics and economics, 29, 299-318, (2001) · Zbl 1074.62530 |

[24] | Pelsser, A.A.J., Pricing and hedging guaranteed annuity options via static option replication, Insurance mathematics and economics, 33, 283-296, (2003) · Zbl 1103.91352 |

[25] | Schrager, D.F.; Pelsser, A.A.J., Pricing rate of return guarantees in regular premium unit linked insurance, Insurance mathematics and economics, 35, 369-398, (2004) · Zbl 1103.91049 |

[26] | Schrager D.F., Pelsser A.A.J., 2004b. Pricing swaptions in affine term structure models, working paper. · Zbl 1130.91028 |

[27] | Thiele, T.N., On a mathematical formula to express the rate of mortality throughout life, Journal of the institute of actuaries, 16, 313-329, (1872) |

[28] | Willemse W.J., 2004. Rational reconstruction of frailty-based mortality models by a generalisation of Gompertzâ€™ law of mortality, working paper. · Zbl 1183.62186 |

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.