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European Actuarial Journal
Published in: European Actuarial Journal 1/2022

13-05-2021 | Original Research Paper

Pricing participating longevity-linked life annuities: a Bayesian Model Ensemble approach

Author: Jorge Miguel Bravo

Published in: European Actuarial Journal | Issue 1/2022

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Abstract

Participating longevity-linked life annuities (PLLA) in which benefits are updated periodically based on the observed survival experience of a given underlying population and the performance of the investment portfolio are an alternative insurance product offering consumers individual longevity risk protection and the chance to profit from the upside potential of financial market developments. This paper builds on previous research on the design and pricing of PLLAs by considering a Bayesian Model Ensemble of single population generalised age-period-cohort stochastic mortality models in which individual forecasts are weighted by their posterior model probabilities. For the valuation, we adopt a longevity option decomposition approach with risk-neutral simulation and investigate the sensitivity of results to changes in the asset allocation by considering a more aggressive lifecycle strategy. We calibrate models using Taiwanese (mortality, yield curve and stock market) data from 1980 to 2019. The empirical results provide significant valuation and policy insights for the provision of a cost effective and efficient risk pooling mechanism that addresses the individual uncertainty of death, while providing appropriate retirement income and longevity protection.
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Footnotes
1
See, e.g., Bravo and Silva [10] and Simões et al. [64] for single and multiple ALM interest rate risk immunization strategies for pension funds and annuity providers.
 
2
Investment guarantees may be in the form of a “technical interest rate” implicit within the actuarial structure of the product or explicit as a minimum annual return [52]. Blake et al. [4] propose participating annuities which pay survivor credits to annuitants according to the mortality experience of a given pool of annuitants.
 
3
A similar but narrower approach can be found in Denuit et al. [26] in which only the systematic component of longevity risk is passed to annuitants and caps and floors can be introduced to limit the profit-loss share. A related approach is found in Bravo et al. [15] in which annuity payments are updated only if observed survivorship rates exceed a given threshold. Lüthy et al. [44] suggest updating benefits based on the ratio between the annuity factor computed at contract inception and the one based on the latest mortality forecast.
 
4
Alternative ways of sharing longevity risk have also been proposed in the context of the design and reform of public pension schemes, e.g., introducing periodically revised annuities in NDC pension schemes in which benefits are updated periodically based on the relationship between expected and observed life expectancy [1], conditional indexation in collective DC plans [5] or NDB/NDC schemes [14], the reform project of the first pillar in Belgium with the adoption of a points system with Musgrave rule [28], the automatic balancing mechanism in NDC schemes (Sweden), two-tier benefit schemes and group-specific annuities [2, 12].
 
5
To reduce basis risk, the mortality dynamics of the reference population should be closely linked to that of the target annuity portfolio or pension fund.
 
6
Note that in life insurance contracts including risk-sharing mechanisms (e.g., participating annuities), the determination of the actual mortality and investment surplus generated by the insurer and the percentage to be distributed to annuitants often depend on national insurance codes and the supervising authority. For instance, in Germany the Minimum Profit-Sharing Act (MindZV) establishes that the minimum amount that has to be shared with the annuitants is 90% of asset returns, 75% of mortality returns, and 50% of other return sources [46].
 
7
The risk (and profit)-sharing mechanism can further be limited by specifying at contract inception a maximum age to apply the benefit adjustment (1), \(x_{b}^{\max },\) eventually in combination with caps and floors. In this case, the annuity benefits evolve over time as follows:
$$\begin{aligned} b_{t_{0}+k}=\left\{ \begin{array}{ll} b_{t_{0}}\times {\mathcal {I}}_{t_{0}+k}\times {\mathcal {R}}_{t_{0}+k}, &{} k=1,\ldots ,x_{b}^{\max }-x_{0} \\ b_{t_{0}+x_{b}^{\max }-x_{0}}, &{} k=x_{b}^{\max }-x_{0}+1,\ldots ,\omega -x_{0} \end{array} \right. \end{aligned}$$
i.e., the contract structure combines a temporary PLLA with maturity \(x_{b}^{\max }-x_{0}\) with a deferred life annuity with unknown benefit at time 0.
 
8
The fair value of a deferred PLLA due with initial benefit \(b_{t_{0}}=1\) payable from time \(t_{0}+u\) to an individual then aged \(x_{0}+u\) is given by
$$_{u|}\ddot{a}_{x_{0}}^{PLLA}\left( t_{0}\right) =\mathop {\sum }\limits _{k=u}^{\omega -x_{0}-u}E^{{\mathbb {Q}}}\left[ \left. DF\left( t_{0},t_{0}+k\right) \cdot _{k}p_{x_{0}}^{\left[ {\mathcal {F}}_{t_{k}}\right] }(t_{k})\cdot {\mathcal {I}} _{t_{0}+k}\cdot {\mathcal {R}}_{t_{0}+k}\right| {\mathcal {F}}_{t_{0}}\right]$$
(7)
 
9
In a symmetrically designed contract in which annuity payments can also increase if observed longevity improvements are lower than predicted, Bravo and Freitas [11] show that the fair value of a PLLA can be decomposed into a long position in a classical fixed annuity, a long position in an embedded European-style longevity cap and a short position in an embedded European-style longevity floor with underlying \({\mathcal {I}}_{t_{0}+k}\), constant strike equal to one unit of currency and maturity \(\omega -x_{0}.\)
 
10
Similarly, for a deferred PLLA due with a deferral period of u years, the fair value of the annuity and embedded deferred longevity floor can be computed as follows
\(_{u|}\ddot{a}_{x_{0}}^{PLLA}\left( t_{0}\right) =_{u|}\ddot{a}_{x_{0}}^{ \left[ {\mathcal {F}}_{t_{0}}\right] }\left( t_{0}\right) -{\mathcal {L}} _{u}^{F}\left( t_{0}\right)\) (12)
with
\({\mathcal {L}}_{u}^{F}\left( t_{0}\right) =\mathop {\sum }\limits _{k=u}^{\omega -x_{0}-u}E^{ {\mathbb {Q}}}\left[ \left. DF\left( t_{0},t_{0}+k\right) \cdot _{k}p_{x_{0}}^{ \left[ {\mathcal {F}}_{t_{k}}\right] }(t_{0})\cdot \left( 1-{\mathcal {I}} _{t_{0}+k}\right) ^{+}\right| {\mathcal {F}}_{t_{0}}\right]\) (13)
For a capped PLLA, Bravo and El Mekkaoui [11] suggest three equivalent decompositions for the annuity payoff: (1) a protective longevity collar, i.e., a portfolio comprising the underlying, a long position in a longevity floorlet with strike \({\mathcal {I}}_{t_{0}+k}^{\min }\) and a short position in a longevity caplet with strike \({\mathcal {I}}_{t_{0}+k}^{\max };\) (2) A long position in a longevity caplet spread, i.e., a portfolio comprising the minimum benefit, a long position in a longevity caplet with strike \(\mathcal { I}_{t_{0}+k}^{\min }\) and a short position in a longevity caplet with strike \({\mathcal {I}}_{t_{0}+k}^{\max };\) (3) A short position in a longevity floorlet spread, i.e., a portfolio comprising the maximum benefit, a long position in a longevity floorlet with strike \({\mathcal {I}}_{t_{0}+k}^{\min }\) and a short position in a longevity floorlet with strike \({\mathcal {I}} _{t_{0}+k}^{\max }.\)
 
11
In a symmetrically designed contract in which annuity payments can decrease (increase) if observed longevity improvements are higher (lower) than predicted, the fair value of a PLLA can be decomposed into a long position in a classical fixed annuity, a long position in an embedded European-style longevity cap and a short position in an embedded European-style longevity floor with underlying \({\mathcal {I}}_{t_{0}+k}\), constant strike equal to one unit of currency and maturity \(\omega -x_{0}.\)
 
12
Alternative choices for the posterior probability allocation include, for example, the normalized C-probability, the natural odds-based probability, the extreme C-probability, the normalized extreme C-probability or the Sigmoid function.
 
13
The method is applied before pricing so that the resulting probabilities are risk adjusted using the Wang distortion operator for pricing purposes.
 
14
Alternative approaches have been proposed to price longevity-linked securities, including the arbitrage-free pricing framework of interest-rate derivatives, using the instantaneous Sharpe ratio, adopting the Equivalent Utility Pricing Principle, the CAPM- and CCAPM-based approaches or the cost of capital approach.
 
15
We conducted a sensitivity analysis on the impact of the GIR on longevity option prices and concluded that for non-participating LLAs, higher guaranteed interest rates reduce the fair value of the embedded options because of the discounting effect. Owing to space constraints, these results are not reported in the paper but can be obtained from the authors upon request.
 
16
An alternative formulation could be to pursue a more aggressive lifecycle strategy up to a fixed old age (e.g., 90 years old), resuming from that age on to a static conservative asset allocation strategy.
 
17
See, e.g., Chen and Suchanecki [21] and Chamboko and Bravo [19, 20].
 
18
In Chile, this guarantee is backed by budgetary resources, but there are other possibilities including the introduction of a small fund financed by the industry.
 
19
Similar results were obtained for females, with slightly lower volatility in expected longevity developments and embedded PLLA option prices.
 
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Metadata
Title
Pricing participating longevity-linked life annuities: a Bayesian Model Ensemble approach
Author
Jorge Miguel Bravo
Publication date
13-05-2021
Publisher
Springer Berlin Heidelberg
Published in
European Actuarial Journal / Issue 1/2022
Print ISSN: 2190-9733
Electronic ISSN: 2190-9741
DOI
https://doi.org/10.1007/s13385-021-00279-w

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