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The income stability of a closed pooled annuity fund is studied. The focus is on quantifying the impact of inhomogeneous initial savings amounts on idiosyncratic longevity risk. Besides wealth inhomogeneity, the members of the pool are independent and identical copies of each other. We ignore systematic investment risk or mortality risk and define income stability as keeping the income within a specified tolerance of each member's initial income payment in a fixed proportion of future scenarios. For a given group of people with different savings amounts, we find an analytical expression that closely approximates the time for which the fund provides a stable income. Our main result uses this expression to determine if everyone benefits from pooling their funds with the whole group or how to split a cohort into different pooled annuity funds.

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For insurance companies in Europe, the introduction of Solvency II leads to a tightening of rules for solvency capital provision. In life insurance, this especially affects retirement products that contain a significant portion of longevity risk (e.g., conventional annuities). Insurance companies might react by price increases for those products, and, at the same time, might think of alternatives that shift longevity risk (at least partially) to policyholders. In the extreme case, this leads to so-called tontine products where the insurance company’s role is merely administrative and longevity risk is shared within a pool of policyholders. From the policyholder’s viewpoint, such products are, however, not desirable as they lead to a high uncertainty of retirement income at old ages. In this article, we alternatively suggest a so-called tonuity that combines the appealing features of tontine and conventional annuity. Until some fixed age (the switching time), a tonuity’s payoff is tontine-like, afterwards the policyholder receives a secure payment of a (deferred) annuity. A tonuity is attractive for both the retiree (who benefits from a secure income at old ages) and the insurance company (whose capital requirements are reduced compared to conventional annuities). The tonuity is a possibility to offer tailor-made retirement products: using risk capital charges linked to Solvency II, we show that retirees with very low or very high risk aversion prefer a tontine or conventional annuity, respectively. Retirees with medium risk aversion, however, prefer a tonuity. In a utility-based framework, we therefore determine the optimal tonuity characterized by the critical switching time that maximizes the policyholder’s lifetime utility.

There is growing interest in the design of pension annuities that insure against
idiosyncratic
longevity risk while pooling and sharing
systematic
risk. This is partially motivated by the desire to reduce capital and reserve requirements while retaining the value of mortality credits; see for example, Piggott
et al.
(2005) or Donnelly
et al.
(2014). In this paper, we generalize the
natural retirement income tontine
introduced by Milevsky and Salisbury (2015) by combining heterogeneous cohorts into one pool. We engineer this scheme by allocating tontine shares at either a premium or a discount to par based on both the age of the investor and the amount they invest. For example, a 55-year old allocating $10,000 to the tontine might be told to pay $200 per share and receive 50 shares, while a 75-year old allocating $8,000 might pay $40 per share and receive 200 shares. They would all be mixed together into the same tontine pool and each tontine share would have equal income rights. The current paper addresses existence and uniqueness issues and discusses the conditions under which this scheme can be constructed
equitably
— which is distinct from
fairly
— even though it isn't
optimal
for any cohort. As such, this also gives us the opportunity to compare and contrast various pooling schemes that have been proposed in the literature and to differentiate between arrangements that are socially equitable, vs. actuarially fair vs. economically optimal.

This paper considers linear fair risk sharing rules and the conditional mean risk sharing rule for independent but heterogeneous losses that are gathered in an insurance pool. It studies the asymptotic behavior of individual contributions to total losses when the number of participants to the pool tends to infinity. It is shown that (i) insurance at pure premium is obtained for an infinitely large pool and (ii) the difference between the actual contribution and the pure premium becomes ultimately Normally distributed. The linear fair risk sharing rule approximating the conditional mean risk sharing rule is then identified, providing practitioners with a useful simplification applicable within large pools. Also, the approximate number of participants required to keep the volatility of individual contributions within an acceptable range is obtained from the established asymptotic Normality.

With the advancements of medical technology and the improvements in quality of life, the demand for innovative retirement products designed to address increasing longevity risks has been growing in recent decades. Tontines and tontine-like products, where the insurers and policyholder share longevity risks, are being explored as an alternative to annuities. As of now, homogeneous policyholders are often assumed in the design process of this mortality-pooling product type. Inspired by the work of Milevsky M. A. & Salisbury T. S. [(2016). Equitable retirement income tontines: mixing cohorts without discriminating. ASTIN Bulletin 46(3), 571–604] in which heterogeneous cohorts are considered, we also extend the tontine products to heterogeneous policyholders. Different from the method employed in Milevsky M. A. & Salisbury T. S. [(2016). Equitable retirement income tontines: mixing cohorts without discriminating. ASTIN Bulletin 46(3), 571–604], we establish the explicit expression of optimal withdrawal rates under the actuarial fairness budget constraint held for one single cohort. In addition, we propose a numerical procedure to achieve approximate fairness among the cohorts by choosing participation rates (or the share prices) properly.

Using risk-reducing properties of conditional expectations with respect to convex order, Denuit and Dhaene [Denuit, M. and Dhaene, J. (2012). Insurance: Mathematics and Economics 51, 265–270] proposed the conditional mean risk sharing rule to allocate the total risk among participants to an insurance pool. This paper relates the conditional mean risk sharing rule to the size-biased transform when pooled risks are independent. A representation formula is first derived for the conditional expectation of an individual risk given the aggregate loss. This formula is then exploited to obtain explicit expressions for the contributions to the pool when losses are modeled by compound Poisson sums, compound Negative Binomial sums, and compound Binomial sums, to which Panjer recursion applies. Simple formulas are obtained when claim severities are homogeneous. A couple of applications are considered: first, to a peer-to-peer insurance scheme where participants share the first layer of their respective risks while the higher layer is ceded to a (re)insurer; second, to survivor credits to be shared among surviving participants in tontine schemes.

A single-period tontine is an arrangement in which a group of members contribute to an investment pool, and after a fixed period of time, the pool is distributed to those members who are still alive. The distribution is made in unequal amounts based on member death probabilities and contribution amounts. In a companion article we apply the single-period tontine to propose products that we feel have commercial potential. Here, we focus on the analytics. The analysis applies not just to single-period tontines, but also to other arrangements that can be analyzed as a sequence of single-period tontines, such as pooled annuity funds.The core of the paper is the development of formulas for the mean and variance of the random amount that a surviving member receives in a single-period tontine. We use the formulas to resolve an open question about bias in pooled annuity funds and to show practical conditions under which it can be made negligible. We show how the provider can use the formulas to manage the subscription process, determining who is allowed to participate and how much they are allowed to contribute, so that the statistics of the tontine are favorably controlled. We use the formulas to compare mixing different cohorts within a single tontine versus creating a separate tontine for each cohort, finding that mixing cohorts is better because it reduces both idiosyncratic risk and systematic risk.

Stochastic processes occur everywhere in sciences and engineering, and need to be understood by applied mathematicians, engineers and scientists alike. This is a first course introducing the reader gently to the subject. Brownian motions are a stochastic process, central to many applications and easy to treat.

The financial industry has recently seen a push away from structured products and towards transparency. The trend is to decompose products, such that customers understand each component as well as its price. Yet the enormous annuity market combining investment and longevity has been almost untouched by this development.
We suggest a simple decomposed annuity structure that enables cost transparency and could be linked to any investment fund. It has several attractive features: (i) it works for any heterogeneous group; (ii) participants can leave before death without financial penalty; and (iii) participants have complete freedom over their own investment strategy.

Group self-annuitization (GSA) schemes are designed to share uncertain future mortality experience including systematic improvements. Challenges for designing group pooled schemes include decreasing average payments when mortality improves significantly, decreasing numbers in the pool at older ages, and the impact of dependence from systematic mortality improvements across different ages of members in the pool. This article uses a multiple-factor stochastic mortality model in a simulation study to show how pooling can be made more effective and to quantify the limitations of these pooling schemes arising from the impact of systematic longevity risk.

Various types of structures that enable a group of individuals to pool their mortality risk have been proposed in the literature. Collectively, the structures are called pooled annuity funds. Since the pooled annuity funds propose different methods of poolingmortality risk, we investigate the connections between them and find that they are genuinely different for a finite heterogeneous membership profile. We discuss the importance of actuarial fairness, defined as the expected benefits equalling the contributions for each member, in the context of pooling mortality risk and comment on whether actuarial unfairness can be seen as solidarity between members. We show that, with a finite number of members in the fund, the group self-annuitization scheme is not actuarially fair: some members subsidize the other members. The implication is that the members who are subsidizing the others may obtain a higher expected benefit by joining a fund with a more favorable membership profile. However, we find that the subsidies are financially significant only for very small or highly heterogeneous membership profiles.

An annuity is an arrangement in which an individual makes a one-time payment (the premium) to an insurer in exchange for a lifetime payment stream. Ideally, the expected present value of the payment stream matches the premium, making it a fair annuity. In practice, the payment stream is smaller than that, providing the insurer with a profit margin. This paper proposes a fair tontine annuity (FTA), an arrangement that provides a lifetime payment stream whose expected present value matches that of a fair annuity. The FTA is based on a fair tontine. In a fair tontine, a group of members contribute to a pool, and each time a member dies, her contribution is divided among surviving members. The distribution to surviving members is made in unequal portions, according to a plan that provides each member with a fair bet, meaning a bet whose expected gain is zero. We show that, under broad conditions, such a plan exists and is readily constructed. Members may be of any age and gender and may contribute any desired amount. New members may join at any time, allowing the fair tontine to operate in perpetuity.The FTA is formed by adding a few enhancements to the fair tontine. The result is something that closely resembles an annuity. The member makes a one-time contribution and receives payments on a fixed schedule (e.g., monthly) that last for his lifetime. The value of each payment is a random amount, with an expected value that is identical to the payment that would be made by a fair annuity. Thus, the FTA offers a higher expected payout than an insurer-provided annuity, since no profit is being extracted. Simulations show that the FTA outperforms a typical insurer-provided FTA not just on the average, but for virtually every member who lives more than just a few years, even with a pool as small as a few hundred members. Since the FTA imposes no risk on the provider, it can be offered by vendors other than insurers, such as mutual fund houses, retail brokers, etc.

Building on the analysis of Bernhardt and Donnelly (2021), we defined stability as keeping the income payments within two bounds for as long as possible with some certainty. The setup References Bernhardt

- Piggott

We have investigated the idiosyncratic risk of the pooled annuity fund by Piggott et al. (2005) in
the case of one cohort with different initial savings. Our focus has been on the time that the fund
can provide a stable income to its members. And how the composition of the cohort influences that
time. Building on the analysis of Bernhardt and Donnelly (2021), we defined stability as keeping
the income payments within two bounds for as long as possible with some certainty. The setup
References
Bernhardt, T. and Donnelly, C. (2021) Quantifying the trade-off between income stability and
the number of members in a pooled annuity fund. ASTIN Bulletin, 51(1):101-130. DOI:
10.1017/asb.2020.33.

- P Billingsley

Billingsley, P. (1999) Convergence of Probability Measures, Second Edition. New York, Chichester,
Weinheim Brisbane, Singapore, Toronto: John Wiley & Sons, Inc. DOI: 10.1002/9780470316962.

University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany)

Human Mortality Database (2019)
Data:
U.S.A, Life tables (period 1x1), Total (both sexes).
University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany).
Downloaded on the 30/04/2021.
URL:
https://www.mortality.org/hmd/USA/STATS/bltper 1x1.txt.

The simple analytics of a pooled annuity fund

- P Piggott
- E Valdez
- B Detzel

Piggott, P., Valdez, E. and Detzel, B. (2005) The simple analytics of a pooled annuity fund. The
Journal of Risk and Insurance, 72(3):497-520. DOI: 10.2307/3519963.