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European Actuarial Journal

08-11-2023 | Letter

A new approximation of annuity prices for age–period–cohort models

Authors: Jean-François Bégin, Nikhil Kapoor, Barbara Sanders

Published in: European Actuarial Journal

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Abstract

This letter presents a new general formula for estimating annuity prices within a wide range of stochastic mortality models. The formula is constructed using two building blocks: an approximation technique based on the Wentzel–Kramers–Brillouin method for calculating the sum of correlated lognormal random variables, and an approximate expression for the moment generating function of the lognormal distribution. Notably, this formula is applicable to virtually all age–period–cohort models where period effects are represented by vector autoregressive models. This broad assumption encompasses the majority of existing stochastic mortality models in literature. Through a numerical illustration, we also demonstrate the reliability and precision of our new method in determining annuity prices.

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Appendix
Available only for authorised users
Footnotes
1
See Section SM.B of the Supplementary Material for an exhaustive literature review on the topic of annuity pricing for stochastic mortality models.
 
2
For simplicity, we only focus on longevity risk in this letter. One could also incorporate interest rate risk by replacing \(e^{-r s}\) in \(\ddot{a}_{x,t}\) by the price of an s-year zero-coupon bond if longevity and interest rate risks are independent.
 
3
In our setting, we use death and exposure counts available up to time t to compute annuity prices; note that the period effects at time \(t-1\) capture the mortality experience up to time t in our notation. It would be possible to consider future annuity prices at times after t by marginally modifying the proposed formula. To streamline our presentation, however, we focus on the annuity price based on the information up to time t.
 
4
Here, \(t-x\) can be interpreted as the year of birth—a proxy for the cohort. Moreover, all indices are integers.
 
5
We assume that \(\varvec{A}\) and \(\varvec{\Sigma }\) satisfy the usual conditions; see Kilian and Lütkepohl [16] for more details.
 
6
The approximate lognormality of sums of lognormals is a well-known rule of thumb since at least the Fenton–Wilkinson approximation (see [14]). See Boyle and Jiang as well as Vanduffel et al. [5, 23] and references therein for more details.
 
7
For more details on our implementation of Lo’s [19] approximation, see Wutzler  [25].
 
8
See Corless et al. [9] for more details on the Lambert W function.
 
9
The approximation does not apply directly to future cohorts, such as unborn generations; the method could nonetheless be adjusted to allow for such cohorts by including the first two moments of the future cohort’s term to Proposition 1.
 
10
The two approximations used in this letter are very accurate for reasonable APC model parameters. For more details, see Section SM.C of the Supplementary Material.
 
11
For simplicity, we use the parameters estimated based on 1970–2020 throughout our illustration.
 
12
The new approximation method has also been tested with the LC and CBD models (with a logarithmic link function) as robustness tests, and the results—similar to those presented in the letter—are available in Section SM.E of the Supplementary Material.
 
Literature
1.
Asmussen S, Jensen JL, Rojas-Nandayapa L (2016) On the Laplace transform of the lognormal distribution. Methodol Comput Appl Prob 18:441–458MathSciNetCrossRef
2.
Bacinello A, Millossovich P, Viviano F (2021) An efficient Monte Carlo based approach for the simulation of future annuity values. Working Paper
3.
Bégin J-F, Sanders B, Xu X (2023) Modeling and forecasting subnational mortality in the presence of aggregated data. North Am Actu J (Forthcoming)
4.
5.
Boyle P, Jiang R (2023) A note on portfolios of averages of lognormal variables. Insur Math Econ (Forthcoming)
6.
Cairns AJ (2011) Modelling and management of longevity risk: approximations to survivor functions and dynamic hedging. Insur Math Econ 49(3):438–453MathSciNetCrossRef
7.
Cairns AJ, Blake D, Dowd K (2006) A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. J Risk Insur 73(4):687–718CrossRef
8.
Cairns AJ, Kallestrup-Lamb M, Rosenskjold C, Blake D, Dowd K (2019) Modelling socio-economic differences in mortality using a new affluence index. ASTIN Bull 49(3):555–590MathSciNetCrossRef
9.
10.
Denuit M (2008) Comonotonic approximations to quantiles of life annuity conditional expected present value. Insur Math Econ 42(2):831–838MathSciNetCrossRef
11.
Dowd K, Blake D, Cairns AJ (2011) A computationally efficient algorithm for estimating the distribution of future annuity values under interest-rate and longevity risks. North Am Actu J 15(2):237–247MathSciNetCrossRef
12.
Dowd K, Cairns AJ, Blake D (2020) CBDX: a workhorse mortality model from the Cairns–Blake–Dowd family. Ann Actu Sci 14(2):445–460CrossRef
13.
Dowd K, Cairns AJ, Blake D, Coughlan GD, Khalaf-Allah M (2011) A gravity model of mortality rates for two related populations. North Am Actu J 15(2):334–356MathSciNetCrossRef
14.
Fenton L (1960) The sum of log-normal probability distributions in scatter transmission systems. IRE Trans Commun Syst 8(1):57–67CrossRef
15.
16.
Kilian L, Lütkepohl H (2017) Structural vector autoregressive analysis. Cambridge University Press, CambridgeCrossRef
17.
Lee RD, Carter LR (1992) Modeling and forecasting US mortality. J Am Stat Assoc 87(419):659–671
18.
19.
Lo C-F (2013) WKB approximation for the sum of two correlated lognormal random variables. Appl Math Sci 7(128):6355–6367MathSciNet
20.
Milevsky MA, Promislow SD (2001) Mortality derivatives and the option to annuitise. Insur Math Econ 29(3):299–318MathSciNetCrossRef
21.
22.
Renshaw AE, Haberman S (2006) A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insur Math Econ 38(3):556–570CrossRef
23.
Vanduffel S, Hoedemakers T, Dhaene J (2005) Comparing approximations for risk measures of sums of nonindependent lognormal random variables. North Am Actu J 9(4):71–82MathSciNetCrossRef
24.
Villegas AM, Kaishev VK, Millossovich P (2018) StMoMo: stochastic mortality modeling in R. J Stat Softw 84(3):1–38CrossRef
25.
Metadata
Title
A new approximation of annuity prices for age–period–cohort models
Authors
Jean-François Bégin
Nikhil Kapoor
Barbara Sanders
Publication date
08-11-2023
Publisher
Springer Berlin Heidelberg
Published in
European Actuarial Journal
Print ISSN: 2190-9733
Electronic ISSN: 2190-9741
DOI
https://doi.org/10.1007/s13385-023-00370-4