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European Actuarial Journal

08.11.2023 | Letter

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

verfasst von: Jean-François Bégin, Nikhil Kapoor, Barbara Sanders

Erschienen 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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Fußnoten
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.
 
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Metadaten
Titel
A new approximation of annuity prices for age–period–cohort models
verfasst von
Jean-François Bégin
Nikhil Kapoor
Barbara Sanders
Publikationsdatum
08.11.2023
Verlag
Springer Berlin Heidelberg
Erschienen in
European Actuarial Journal
Print ISSN: 2190-9733
Elektronische ISSN: 2190-9741
DOI
https://doi.org/10.1007/s13385-023-00370-4