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The Simple Analytics of a Pooled Annuity Fund
Abstract
This article provides a formal analysis of payout adjustments from a longevity risk-pooling fund, an arrangement we refer to as group self-annuitization (GSA). The distinguishing risk diffusion characteristic of GSAs in the family of longevity insurance instruments is that the annuitants bear their systematic risk, but the pool shares idiosyncratic risk. This obviates the need for an insurance company, although such instruments could be sold through a corporate insurer. We begin by deriving the payout adjustment for a single entry group with a single annuity factor and constant expectations. We then show that under weak requirements a unique solution to payout paths exists when multiple cohorts combine into a single pool. This relies on the harmonic mean of the ratio of realized to expected survivorship rates across cohorts. The case of evolving expectations is also analyzed. In all cases, we demonstrate that the periodic-benefit payment in a pooled annuity fund is determined based on the previous payment adjusted for any deviations in mortality and interest from expectations. GSA may have considerable appeal in countries which have adopted national defined contribution schemes and/or in which the life insurance industry is noncompetitive or poorly developed. Copyright The Journal of Risk and Insurance.
... Ever since the resurrection of tontines by Piggott et al. (2005), Sabin (2010) and Milevsky and Salisbury (2015), modern tontines and tontine-like products have gained increasing popularity both in practice and among academic research. The word "tontine" is widely used throughout the academic literature to refer to any retirement plan with a mortality risk sharing component leaving pensioners with the largest part of the mortality risk. 1 In a tontine, a group of policyholders shares mortality risks and the benefit provider mostly serves as an administrator. ...
... There are two different sources of mortality risk: Unsystematic (or idiosyncratic) risk refers to the risk that individuals' future lifetimes 1 There exist many different ways to accomplish this feature, sometimes carrying names other than "tontine". Piggott et al. (2005) introduce a so-called group-self-annuitization scheme, sometimes also referred to as pooled annuity fund (see also Valdez et al. (2006) and Stamos (2008)). To the best of our knowledge, this is the first modern tontine design proposed in the literature. ...
... Typically, the optimal pattern for the tontine payoff d(t) displays a decreasing structure over time. This characteristic circumvents the drastic individual increases in retirement benefits as time progresses, as highlighted in works like Piggott et al. (2005), Sabin (2010), and Milevsky and Salisbury (2015). Sabin (2010) refers to this product design as a "noisy version of a constant annuity," as the benefits will fluctuate around those of a constant annuity, contingent on the precise mortality experience within the tontine pool. ...
Tontines, a form of financial arrangement in which a group of participants is pooled together and receives periodic payments that increase as other participants pass away, raise unique ethical considerations that warrant in-depth examination. This paper explores the ethical dimensions surrounding the practice of mortality risk sharing specifically in the context of tontines. Through a comprehensive analysis of existing literature and ethical frameworks, this study investigates the ethical implications of tontines as a mechanism for sharing mortality risks and provides policy recommendations to minimize ethical concerns.
... 3. Admissible strategies. There is a rich literature which explores different decumulation strategies (and products), with Pitacco (2016), Pitacco and Tabakova (2022) and Piggott et al. (2005) exploring innovative guarantee structures and participation mechanisms. Many of these strategies may help to protect retirement savings as improvement in healthcare continues to extend life expectancy. ...
... In Sect. 4, we find a set of admissible strategies in a generalised family of annuities which we construct with inspiration from Piggott et al. (2005). In Sect. ...
... We develop a family of pooling policies based on the original group self-annutisation (GSA) formulation due to Piggott et al. (2005) and discussed at depth by Zhou (2020) and Bär and Gatzert (2022). We denote d(t) = φg(t) where φ is the payment rate and g(t) is the payment vehicle. ...
A retiree’s appetite for risk is a common input into the lifetime utility models that are traditionally used to find optimal strategies for the decumulation of retirement savings. In this work, we consider a retiree with potentially differing appetites for the key financial risks of decumulation: liquidity risk and investment risk. We set out to determine whether these differing risk appetites have a significant impact on the retiree’s optimal choice of decumulation strategy. To do so, we design and implement a framework which selects the optimal decumulation strategy from a general set of admissible strategies in line with a retiree’s goals, and under differing appetites for the key risks of decumulation. Overall, we find significant evidence to suggest that a retiree’s differing appetites for different decumulation risks will impact their optimal choice of strategy at retirement. Through an illustrative example calibrated to the Australian context, we find results which are consistent with actual behaviours in this jurisdiction (in particular, a shallow market for annuities), which lends support to our framework and may provide some new insight into the so-called annuity puzzle.
... Various arrangements, products, and monikers fit the broad description of lifetime pension pools in the literature: group self-annuitization (GSA) plans (Piggott et al., 2005;Valdez et al., 2006;Qiao and Sherris, 2013;Hanewald et al., 2013), pooled annuity funds (Stamos, 2008;Donnelly et al., 2013), annuity overlay funds (Donnelly et al., 2014;Donnelly, 2015), retirement tontines Salisbury, 2015, 2016;Chen and Rach, 2019;Fullmer, 2019;Iwry et al., 2020;Gemmo et al., 2020;Weinert and Gründl, 2021;Chen et al., 2021), variable annuities (Balter and Werker, 2020;Dees et al., 2021), and variable payout annuities (Horneff et al., 2010;Boyle et al., 2015). Note that all these designs can be viewed as implicit or explicit tontines. ...
... We select a very simple structure for the dynamics of our lifetime pension pool. The pool operation is similar to that explained in Piggott et al. (2005), Qiao andOlivieri et al. (2022) in the context of GSA plans. It is also reminiscent of the benefit update rule used by the College Retirement Equities Funds in the US and the University of British Columbia Faculty Pension Plan in Canada. ...
... Using common actuarial notation, we know thatä (m)x−h = h + e −yh h p x−hä (m)x when mortality is deterministic (see, e.g.,Piggott et al., 2005). This relationship, however, is not satisfied for stochastic mortality models, meaning that C t will only equal one if there are no changes to future mortality expectations.17 ...
Lifetime pension pools—also known as group self-annuitization plans, pooled annuity funds, and retirement tontines in the literature—allow retirees to convert a lump sum into lifelong income, with payouts linked to investment performance and the collective mortality experience of the pool. Existing literature on these pools has predominantly examined basic investment strategies like constant allocations and investments solely in risk-free assets. Recent studies, however, proposed volatility targeting, aiming to enhance risk-adjusted returns and minimize downside risk. Yet they only considered investment risk in the volatility target, neglecting the impact of mortality risk on the strategy. This study thus aims to address this gap by investigating volatility-targeting strategies for both investment and mortality risks, offering a solution that keeps the risk associated with benefit variation as constant as possible through time. Specifically, we derive a new asset allocation strategy that targets both investment and mortality risks, and we provide insights about it. Practical investigations of the strategy demonstrate the effectiveness and robustness of the new dynamic volatility-targeting approach, ultimately leading to enhanced lifetime pension benefits.
... The term 'pooled annuities' generally refers to insurance company annuities offered without any insurance company guarantees (Piggott et al. 2005;Donnelly 2015). 7 Instead, the annuitants bear all of the risks. ...
... 1. Tontines are named after Lorenzo de Tonti, the 17th-century Italian who is credited with the idea (Milevsky 2015). Group self-annuitization (Piggott et al. 2005;Qiao and Sherris 2013) is a similar concept that could be used. Participating variable index-linked annuities (Maurer, Mitchell, et al. 2013;Maurer, Rogalla, and Siegelin 2013) could also be used-provided that the participation applies fully on both the upside and the downside. ...
... That study found that the comparison results depended significantly on the assumptions used. 7. Similar arrangements go by other names, including participating annuities and group self-annuitization schemes (Piggott et al. 2005;Qiao and Sherris 2013;Maurer, Mitchell, et al. 2013;Maurer, Rogalla, and Siegelin 2013). 8. ...
Notwithstanding the terrible price the world has paid in the Coronavirus pandemic, the fact remains that longevity at older ages is likely to continue to rise in the medium and longer term. This volume explores how the private and public sectors can collaborate via public-private partnerships (PPPs) to develop new mechanisms to reduce older people’s risk of outliving their assets in later life. As we show in this volume, PPPs typically involve shared government financing alongside private-sector partner expertise, management responsibility, and accountability. In addition to offering empirical evidence on examples where this is working well, our contributors provide case studies, discuss survey results, and examine a variety of different financial and insurance products to better meet the needs of the aging population. The volume will be informative to researchers, plan sponsors, students, and policymakers seeking to enhance retirement plan offerings.
... Extensions of this article include, but are not limited to, for example, Davidoff et al. (2005) and Peijnenburg et al. (2016). In addition to guaranteed retirement products like annuities, ever since the articles of Piggott et al. (2005) and Sabin (2010), there has been an increasing interest in innovative retirement plans which shift the mortality risk from insurers to policyholders, frequently referred to as modern tontines (see, e.g., Stamos (2008), Hanewald et al. (2013), Donnelly et al. (2014), Salisbury (2015, 2016), Chen et al. (2019Chen et al. ( , 2020Chen et al. ( , 2021a and Weinert and Gründl (2021)). While some of these articles propose a new tontine design (e.g. ...
... While some of these articles propose a new tontine design (e.g. Piggott et al. (2005), Sabin (2010) and Donnelly et al. (2014)), others focus on analyzing optimal consumption patterns and withdrawal plans in existing tontines (e.g. Stamos (2008), Chen et al. (2021a) and Chen and Rach (2023)), and some do both (Milevsky and Salisbury (2015) and Chen et al. (2019Chen et al. ( , 2020). ...
We study time consistency in optimal consumption problems of annuities and tontines. We find the annuity problem to be time-consistent, hence delivering the same optimal consumption at each time. The tontine problem, however, is found to be time-inconsistent, opening the possibility for individuals to increase or decrease their overall expected utility by changing the ex ante fixed consumption profile. However, such an increase in the utility of the tontine cannot lead to a larger level of expected utility than in the optimal annuity.
... Products that share longevity risk in a pool with differing payment profiles are referred to as pooled annuity products. Group Self-Annuitization (GSA) is a form of pooled annuity considered in Piggott et al. (2005) that aims to generate a payment profile similar to a traditional life annuity. Annuity payments depend on the mortality experience within the pool, as well as the investment performance of the pooled fund. ...
... We base our analysis on the GSA structure introduced by Piggott et al. (2005) whereby the pool shares investment risk as well as both idiosyncratic and systematic longevity risk. At time 0, there is a pool of l x homogeneous annuitants all aged x, each contributing the same amount to the pool. ...
Pooled annuity products, where the participants share systematic and idiosyncratic mortality risks as well as investment returns and risk, provide an attractive and effective alternative to traditional guaranteed life annuity products. While longevity risk sharing in pooled annuities has received recent attention, incorporating investment risk beyond fixed interest returns is relatively unexplored. Incorporating equity investments has the potential to increase expected annuity payments at the expense of higher variability. We propose and assess a strategy for incorporating equity investments along with managed-volatility for pooled annuity funds. We show how the managed volatility strategy improves investment performance, while reducing pooled annuity income volatility and downside risk, as well as an investment strategy that reduces exposure to investment risk over time. We quantify the impact of pool size when equity investments are included, showing how these products are viable with relatively small pool sizes.
... Such products are known as pooled annuity funds, group self-annuitization schemes and tontines (cf. Piggott et al. (2005), Sabin (2010), Donnelly et al. (2014), Milevsky and Salisbury (2015)). Although they carry different names, all these products have the similarity that a pool of pensioners shares mortality risks. ...
... However, the inclusion of safety loadings in annuity premiums opens a market for mortality risk-carrying products like tontines, which require no safety loadings because providers do not promise guaranteed benefits (cf. Piggott et al. (2005), Stamos (2008), Sabin (2010), Hanewald et al. (2013), Donnelly et al. (2014) and Milevsky and Salisbury (2015)). Thus, if tontines can be designed fairly for each participant, they could be offered at actuarially fair prices, which represents a huge advantage for any retiree who is not fond of full annuitization. 1 . ...
We study actuarial fairness in tontines with heterogeneous cohorts. For a given tontine, we show that both collective and individual actuarial fairness are achievable. While it is impossible to design a tontine scheme with mixed cohorts which is optimal (utility-maximizing) for each single cohort (Milevsky and Salisbury (2016); Chen et al. (2021d)), we design a socially optimal tontine maximizing the collective’s expected utility characterized through a weighted sum of individual utility functions. In particular, we compare the resulting collectively optimal tontine to existing schemes in the literature. We find that a tontine constructed by a social planner can outperform existing tontine schemes as it is able to better reflect individual risk preferences.
... In the academic literature, a lot of different tontine designs have been proposed in recent years, see e.g. Piggott et al. (2005), Sabin (2010), Donnelly et al. (2014), Milevsky and Salisbury (2015) and Bernhardt and Donnelly (2019). A brief overview of some tontine providers in practice can be found in If both annuities and tontines are priced actuarially fair and analytically convenient utility preferences are assumed, annuities are typically preferred to tontines (cf. ...
Retirement income tontines are retirement plans where benefits are obtained from the inheritance of non-surviving participants. This paper analyzes stochastic dominance relations between tontines and annuities. In the presence of risk loadings and/or subjective probabilities underestimating the insurer's pricing probabilities, we find the benefits of a properly designed tontine to dominate the benefits of a constant annuity in the almost first order stochastic dominance (AFSD) sense as defined by Leshno and Levy (2002). In particular, we show that this AFSD converges to first order stochastic dominance as the pool size in the tontine tends to infinity.
... (3.12) hence, Property 3.1(ii) is satisfied. In essence, the group gain G i in Equation (3.10) is a scaling factor to adjust for actual deaths compared to expected deaths, as suggested in Piggott et al. (2005) and Qiao and Sherris (2013). ...
We consider the holder of an individual tontine retirement account, with maximum and minimum withdrawal amounts (per year) specified. The tontine account holder initiates the account at age 65 and earns mortality credits while alive, but forfeits all wealth in the account upon death. The holder wants to maximize total withdrawals and minimize expected shortfall at the end of the retirement horizon of 30 years (i.e., it is assumed that the holder survives to age 95). The holder controls the amount withdrawn each year and the fraction of the retirement portfolio invested in stocks and bonds. The optimal controls are determined based on a parametric model fitted to almost a century of market data. The optimal control algorithm is based on dynamic programming and the solution of a partial integro differential equation (PIDE) using Fourier methods. The optimal strategy (based on the parametric model) is tested out of sample using stationary block bootstrap resampling of the historical data. In terms of an expected total withdrawal, expected shortfall (EW-ES) efficient frontier, the tontine overlay dramatically outperforms an optimal strategy (without the tontine overlay), which in turn outperforms a constant weight strategy with withdrawals based on the ubiquitous four per cent rule.
... Benefit payments are made to surviving members (Cfa & Fin, 2015). The calculation of the GSA method resembles the calculation of a whole-life annuity so that the pricing procedure includes calculating the annuity payment level (Chen & Rach, 2022) and (Piggott et al., 2005). ...
This study aims to calculate Indonesian pension funds using the Group Self Anuitization method and Makeham's death law. The calculation of the GSA method is almost the same as the calculation of an annuity for life, so the price determination procedure includes calculating the level of annuity payments. The death rate is projected by Makeham Mortality Law based on Indonesian Mortality Table IV. Based on an analysis with the same premium amount, it is known that the benefits of male pension funds are greater than women for each age at which benefits are paid, pension funds paid to policyholders increase from 2019-2021 and decrease in 2022, the higher the entry age payment of pension funds, the benefits obtained will be even greater Keywords: Group Self Annuitization; Pension Fund; Makeham Law; Premiums; Annuity Abstrak Penelitian ini bertujuan untuk menghitung dana pensiun Indonesia dengan menggunakan metode Group Self Anuitization dan hukum kematian Makeham. Perhitungan pada metode GSA hampir sama dengan perhitungan anuitas seumur hidup, sehingga prosedur penentuan harga meliputi perhitungan tingkat pembayaran anuitas. Angka kematian diproyeksikan oleh Hukum Kematian Makeham berdasarkan Tabel Mortalita Indonesia IV. Berdasarkan analisis dengan besaran premi yang sama, diketahui bahwa manfaat dana pensiun laki-laki lebih besar dari perempuan untuk setiap usia masuk pembayaran manfaat, dana pensiun yang dibayarkan kepada pemegang polis meningkat dari tahun 2019-2021 dan menurun pada tahun 2022, semakin tinggi usia masuk pembayaran dana pensiun maka manfaat yang diperoleh akan semakin besar. Kata Kunci: Annuitisasi Diri Kelompok; Dana Pensiun; Hukum Mortalita Makeham; Premil; Anuitas
... As an important application of our results, we discuss the case of group selfannuitization (Piggott et al. 2005). Here, a group of people join forces in that each of the group members agrees to invest some amount and to earn an investment return while sharing longevity risk. ...
Actuarial fairness pertains to the situation in which the price of an insurance contract is equal to its expected outcome. We show that actuarial fairness leads to “unfairness” in that annuitants with higher survival rates can choose a better payoff in the sense of second-order stochastic dominance than those with lower survival rates. To deal with this issue, we propose equal utility pricing, i.e., we determine prices such that all contracts have the same (nonlinear) utility from the viewpoint of a third party (e.g., a social planner). This approach is of particular relevance with respect to the design of group self-annuitization schemes.
... It is nevertheless possible to generate lifelong incomes by investing in survivor funds over successive time intervals. This offers an alternative to group self-annuitization schemes of Piggott et al. (2005) as well as to modern tontines or pooled annuity funds developed for example by Bernhardt and Donnelly (2021), Chen et al. (2021), and Hieber and Lucas (2022). ...
Survivor funds are financial arrangements where participants agree to share the proceeds of a collective investment pool in a predescribed way depending on their survival. This offers investors a way to benefit from mortality credits, boosting financial returns. Following Denuit (2019, ASTIN Bulletin , 49 , 591–617), participants are assumed to adopt the conditional mean risk sharing rule introduced in Denuit and Dhaene (2012, Insurance: Mathematics and Economics , 51 , 265–270) to assess their respective shares in mortality credits. This paper looks at pools of individuals that are heterogeneous in terms of their survival probability and their contributions. Imposing mild conditions, we show that individual risk can be fully diversified if the size of the group tends to infinity. For large groups, we derive simple, hierarchical approximations of the conditional mean risk sharing rule.
... Since tontines are much less influenced by longevity prospects than regular annuity products (Milevsky and Salisbury (2015), Chen et al. (2020b)), we expect adverse selection to play a much smaller role in the tontine market than in the annuity market. This argument is also highlighted by Valdez et al. (2006) for the pooled annuity introduced by Piggott et al. (2005): 3 If there is adverse selection in both types of annuities, it is usually smaller in the pooled annuity. As for the subjective survival probabilities, this is not an objective advantage of tontines over annuities for the individual investors and may even cause individuals to regret their decision to participate in a tontine after the first few years of retirement. ...
Under both normative and descriptive decision theory, tontines and tontine-related retirement income products have proven their superiority to classical annuity products. In the present paper, we show that benefits of a properly designed tontine dominate the benefits of an equally priced annuity as the pool size tends to infinity, leading individuals to prefer tontines with a sufficiently large pool size to annuities under utility preferences which are increasing and continuous in consumption. Such preferences include but are not limtied to cumulative prospect theory and generalized expected utility preferences, which we use as examples to illustrate our theoretical findings. Our results present an interesting puzzle which we call “tontine puzzle”, raising the question why the development of the tontine market is still in its infancy in practice.
... Today, modern versions of these original tontines exist, for example the TIAA-CREF retirement fund in the USA, the Lifetimeplus solution of Mercer in Australia, or "Le Conservateur" in France. In the literature, these modern versions are named pooled annuities, group self annuitization schemes (see, e.g., Piggott et al., 2005;Valdez et al., 2006;Stamos, 2008;Qiao and Sherris, 2013;Donnelly et al., 2013;Donnelly et al., 2014) or (modern) tontines (see, e.g., Sabin, 2010;Forman and Sabin, 2015;Milevsky and Salisbury, 2015;Forman and Sabin, 2016;Fullmer and Sabin, 2018;Li and Rothschild, 2019;. These articles follow a long tradition of mutual with profits products where mortality or investment surplus is shared through an appropriate bonus distribution (see, for example, the well-cited book by Fisher and Young, 1965). ...
The tendency of insurance providers to refrain from offering long-term guarantees on investment or mortality risk has shifted attention to mutual risk pooling schemes like (modern) tontines, pooled annuities or group self annuitization schemes. While the literature has focused on mortality risk pooling schemes, this paper builds on the advantage of pooling mortality and morbidity risks, and their inherent natural hedge. We introduce a modern “life-care tontine”, which in addition to retirement income targets the needs of long-term care (LTC) coverage for an ageing population. In contrast to a classical life-care annuity, both mortality and LTC risks are shared within the policyholder pool by mortality and morbidity credits, respectively. Technically, we rely on a backward iteration to deduce the smoothed cashflows pattern and the separation of cash-flows in a fixed withdrawal and a surplus from the two types of risks. We illustrate our results using real life data, demonstrating the adequacy of the proposed tontine scheme.
... A number of pooling structures, where a group of individuals create a fund which can be invested in the capital markets whilst periodically drawing down depending on survival, have been proposed in literature. Such products include group self-annuitization (GSA) schemes (Piggott et al., 2005;Valdez et al., 2006;Qiao and Sherris, 2013), pooled annuity funds (Stamos, 2008;Donnelly et al., 2013;Donnelly, 2015), tontines (Milevsky, 2014;Milevsky and Salisbury, 2015;Chen et al., 2019;Weinert and Gründl, 2020) among others. The design of these pooled products has mainly been analysed considering simple investment strategies. ...
While the current pandemic is causing mortality shocks globally, the management of longevity risk remains a major challenge for both individuals and institutions. It is high time there be private market solutions designed for efficient longevity risk transfer among various stakeholders such as individuals, pension funds and annuity providers. From individuals’ point of view, appealing features of post-retirement solutions include stable and satisfactory benefit levels, flexibility, meeting bequest preferences and low fees. This paper proposes a dynamic target volatility strategy for Group Self-Annuitization (GSA) schemes aimed at enhancing living benefits for pool participants. More specifically, we suggest investing GSA funds in a portfolio consisting of equity and cash, continuously rebalanced to maintain a target volatility level. The performance of a dynamic target volatility strategy is assessed against the static case which does not involve portfolio rebalancing. Benefit profiles are assessed by analysing quantiles and alternative strategies involving varying equity compositions. The case of death benefits is included, and the fund dynamics analysed by assessing resulting investment returns and the mortality credits. Overall, higher living benefit profiles are obtained under a dynamic target volatility strategy. From the analysis performed, a trade-off between the equity proportion and the impact on the lower quantile of the living benefit amount emerges, suggesting an optimal proportion of equity composition.
... As an important application of our results, we discuss the case of group self annuitization (Piggott et al. (2005)). Here, a group of people join forces in that each of the group members agrees to invest some amount and to earn an investment return while sharing longevity risk. ...
... As an important application of our results, we discuss the case of group self annuitization (Piggott et al. (2005)). Here, a group of people join forces in that each of the group members agrees to invest some amount and to earn an investment return while sharing longevity risk. ...
Actuarial fairness pertains to the situation in which the price of an insurance contract is equal to its expected outcome. This paradigm is at odds with financial pricing: If two financial contracts have the same expected value, but one is better than the other in the sense of second order stochastic dominance, then the better contract will be strictly more expensive. We show that actuarial fairness leads to unfairness in that annuitants with higher survival rates have a better payoff than those with lower survival rates. To deal with this issue we propose equal utility pricing, i.e., we determine prices such that all contracts have the same (nonlinear) utility from the viewpoint of a third party (e.g., a social planner). This approach is of particular relevance with respect to the fair design of group self annuitization.
... A number of pooling structures, where a group of individuals create a fund which can be invested in the capital markets whilst periodically drawing down depending on survival, have been proposed in literature. Such products include group self-annuitization (GSA) schemes (Piggott et al., 2005;Valdez et al., 2006;Qiao and Sherris, 2013), pooled annuity funds (Stamos, 2008;Donnelly et al., 2013;Donnelly, 2015), tontines (Milevsky, 2014;Milevsky and Salisbury, 2015;Chen et al., 2019;Weinert and Gründl, 2020) among others. The design of these pooled products has mainly been analysed considering simple investment strategies. ...
There is a significant potential demand around the world for a flexible product to manage individual longevity risk. However, annuity markets remain thin, driven by factors including lack of pricing transparency, high product loadings, bequest motives, loss aversion, and the difficulty to hedge the risk. This paper proposes an individual retirement product to allow flexible management of longevity risk. The product combines a lifetime income with a flexible death benefit to meet individual bequest needs. It also benefits the issuers by lower mortality risk due to the natural hedging, and thus lower capital cost as a risk margin. We apply actuarial models that provide transparent pricing for interest rate and mortality risk, the construction of immunized bond portfolios, and the determination of a loading and solvency margin for systematic longevity risk. We also quantify the natural hedging benefits arising from the flexible inclusion of both survival benefits and death benefits.
... Although models in this paper are single-period, they are also closely related to decentralized retirement planning tools, such as, fair transfer plan, fair tontine annuity, tontine pensions in Forman & Sabin (2015), annuity overlay fund in Donnelly et al. (2013), pooled annuity fund in Piggott et al. (2005), Stamos (2008), Qiao & Sherris (2013), and tontines in Milevsky & Salisbury (2015). ...
Peer-to-peer (P2P) insurance is a decentralized network in which participants pool their resources together to compensate those who suffer losses. The rise of P2P insurance in Western countries, such as Friendsurance and Lemonade, has been viewed as a disruption to the traditional insurance industry in the same way Uber is to the taxi industry. A similar business model of mutual aid, such as the model developed by Xianghubao, has become popular in the East. It is a model designed to provide financial support to those in need and to distribute the cost among all participants.
Despite the fast-changing landscape in this field, there has been scarce literature on the theoretical underpinning of P2P insurance and mutual aid. This paper presents the first effort to build a unified framework to quantify and assess the exchange of risks in a network of participants from different risk classes. Under this framework, the paper aims to address several essential research questions, including the fair exchange of heterogeneous risks and the optimality of algorithmic designs under various criteria. The modeling of multi-risk exchange is done with a P2P network structure. We show that these network structures not only can be used to explain the current practice in the industry, but also provide new tools to develop better designs. The paper concludes with a comparison of traditional insurance and P2P risk sharing from the standpoint of stability and cost reduction.
... Bommier and Schernberg (2021)). Concurrently, the contemporary actuarial literature, as exemplified by works such as Piggott et al. (2005), Sabin (2010), Milevsky and Salisbury (2015), and Chen et al. (2019), introduces a range of innovative retirement products characterized by their primary feature: directly linking the benefits of these products to realized survival probabilities. ...
We study the demand for retirement products given that individuals have access to innovative plans depending on realized survival probabilities, such as tontines, alongside traditional annuities. To model the preferences of these individuals, we employ a generalized life-cycle utility function that incorporates temporal risk aversion, indicating that individuals are risk averse about their lifetime. We identify conditions for pricing bounds under which individuals displaying temporal risk aversion are more inclined to opt for partial tontinization combined with partial annuitization than full annuitization. Our findings suggest that temporal risk aversion can account for the relatively low demand for constant annuities while boosting the demand for tontines. In an extended model with differential mortality and wealth, we analyze a utilitarian social planner’s retirement product demand, focusing on wealth transfers between the groups.
... We investigate three initial balance levels of $200,000 (low), $500,000 (average) and $800,000 (high). The levels are indicative, but were selected with reference to account balance statistics from the Association of Superannuation Funds of Australia (Clare, 2019) while allowing for the fact that balances are likely to grow as 2 A variety of annuity-like products such as variable annuities (Ledlie et al., 2008) and pooled annuities (Piggott et al., 2005) offer an alternative to annuities to transfer part of investment and/or longevity risk through smoothing and/or risk sharing structures. The more complex payoff structures of these products may limit their attractiveness, noting that one reason the literature provides for low annuitisation rates is complexity and lack of financial literacy. ...
This paper was previously titled "Optimal strategies for retirees in Australia with realistic risk transfer", but has been substantially revised.
ABSTRACT: Optimal investment and drawdown strategies vary significantly across different types of retiree • Welfare for loss aversion preferences is most dependent on hedging using annuities • Welfare for risk aversion preferences is most dependent on appropriate drawdown rates • Annuities add value for some retiree types, but not all Abstract We investigate optimal investment and drawdown decisions in retirement, and show that the asset mix and drawdown strategy vary significantly with financial circumstances and preferences. Loss aversion preferences lead to hedging strategies to secure the target consumption through use of immediate life and deferred life annuities, asset allocation and drawdown decisions. Risk aversion preferences lead to use of annuities to smooth and set a minimum level of consumption, with the strategies adopted varying with risk aversion. Welfare gains are more dependent on annuitisation decisions under loss aversion, and drawdown levels under risk aversion. A means-tested and government-provided Age Pension influences decisions of the risk averse but not the loss averse retiree.
... Pension gainsharing is typically the sharing of unanticipated investment gains with some or all of the pension plan population-usually current retirees. Favorable (higher) mortality experience has also been discussed as a possible gainsharing design (Piggott et al. 2005), although that article anticipated decreases in annuity payments for unfavorable (lower) mortality experience. ...
This paper examines the effect of gainsharing provisions on the selection of a discount rate for a defined benefit pension plan. The paper uses a traditional actuarial approach of discounting liabilities using the expected return of the associated pension fund. A stochastic Excel model was developed to simulate the effect of varying investment returns on a pension fund with four asset classes. Lognormal distributions were fitted to historical returns of two of the asset classes; large company stocks and long-term government bonds. A third lognormal distribution was designed to represent the investment returns of alternative investments, such as real estate and private equity. The fourth asset class represented short term cash investments and that return was held constant. The following variables were analyzed to determine their relative impact of gainsharing on the selection of a discount rate: hurdle rate, percentage of gainsharing, actuarial asset method smoothing period, and variations in asset allocation. A 50% gainsharing feature can reduce the discount rate for a defined benefit pension plan from 0.5% to more than 2.5%, depending on the gainsharing design and asset allocation.
... e.g. Piggott et al. (2005), Valdez et al. (2006), Stamos (2008), Donnelly et al. (2013), Hanewald et al. (2013), Milevsky and Salisbury (2015), Bernhardt and Donnelly (2019), Chen et al. (2019Chen et al. ( , 2020bChen et al. ( , 2021, Chen and Rach (2022)). Unlike annuities, where an insurance company promises guaranteed retirement benefits to the policyholders, tontines and pooled annuities largely shift mortality risk to the policyholders. ...
We analyze individuals’ demand for retirement plans under cumulative prospect theory (CPT), considering traditional and unit-linked annuities as well as innovative, unit-linked tontines, in which benefits depend on mortality and financial market risks. We find that the main parameters of CPT, subjective probabilities and loss aversion heavily affect individuals’ retirement income preferences, explaining why not sufficiently many individuals purchase traditional annuities. Subjective probabilities increase the attractiveness of risk-carrying retirement plans (like unit-linked annuities and tontines) and lead individuals to prefer such to traditional annuities, whereas loss aversion lowers the attractiveness of all retirement products and can lead individuals not to purchase any retirement plan.
... 4 The second category of products refers to non-insurance (closed or open) pooling arrangements, which merely pool risks among participants, in which no financial or longevity guarantees are explicitly provided by an insurer. This group includes group self-annuitisation (GSA) arrangements [7,38,53,56,66], in which the provider pools wealth from participants and invests it according to a given investment strategy, a regular income determined by the provider is expected to be paid for life but without any guarantee, and pool participants share both financial, and idiosyncratic and systematic longevity risk. This group includes also pooled annuity funds (PAF) [8,65] which differentiate from a GSA in that participants have freedom in consumption, instead of receiving a benefit determined at inception by the provider. ...
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.
This paper considers the optimal investment and consumption problem for the pooled annuity funds, in which both the financial market and the mortality hazard rate of participants in the pool are partially observable. We manage to achieve the explicit expressions for optimal consumption and investment strategies employing filtering techniques and Hamilton-Jacobi-Bellman (HJB) equation. What is more, we also discuss the models where both the instantaneous rate of return of financial market and mortality of plan members are observable and obtain the optimal investment strategies accordingly. In addition, we look into this optimization problem under different exit mechanism including infinite exit time for the plan members. Last, but not the least, we carry out numerical analysis demonstrating the impact of observability of information.
In this paper, we propose two financial-longevity risk-sharing methods based on an existing risk-linked annuity. Within a specific framework, we first propose the risk-sharing group self-annuitization (GSA), where different proportions of financial and longevity risks are shared between a group of annuitants and the insurer. Secondly, we propose the complete risk-sharing GSA. We implicitly define a set of annuity products which depend on the risk proportion borne by a group of annuitants. The defined set of annuities goes from the classical annuity to the GSA.
This chapter explains how state governments could create new low-cost lifetime assurance funds to help provide retirement income security for millions of private-sector workers who currently lack pension coverage. Basically, an assurance fund operates like a mutual fund held within a defined contribution plan, but with the added features of mortality pooling and fully funded lifetime payouts. As we envision them, assurance funds would be offered as annuity-like investment options on the new investment platforms being created by states like Oregon, California, and Maryland that offer their citizens the opportunity to participate in state-sponsored retirement savings plans. Adding an assurance fund could effectively turn these retirement savings plans into lifetime pensions. To ensure their sustainability, assurance funds would operate under a strict budget constraint and be organized as either tontines or pooled annuities.
The question how individuals should (optimally) annuitize their wealth remains of high relevance in light of longevity risk and volatile capital markets. In this article, we first present traditional and innovative products and strategies for the decumulation of wealth during retirement, based on a review of 72 selected academic articles in peer-reviewed journals. We further identify relevant factors that generally influence the conception of these products from the retirees’ perspectives, and derive implications for product developers, before concluding with avenues of future research. Our results indicate that innovative suggestions often comprise tontine-like structures, exploit actuarial and accounting smoothing in various ways, defer annuitization to higher ages, or combine it with long-term care options, for instance. Key areas of future research in this field include the consideration of both insurer and retiree perspectives in the analysis of products, using behavioral considerations when evaluating the retirees’ perspective, and taking into account the impact of costs or expenses. While recent articles increasingly consider these aspects, manifold opportunities for future research remain.
We investigate optimal investment and drawdown strategies in retirement and show that the asset mix and drawdown strategy vary significantly with preferences and other retiree attributes. Loss aversion preferences lead to hedging strategies to secure the target consumption through use of immediate life and deferred life annuities, asset allocation and drawdown decisions. Risk aversion preferences lead to use of annuities to smooth and set a minimum level of consumption, with the strategies adopted varying with risk aversion. Welfare gains are more dependent on annuitisation decisions under loss aversion, and drawdown levels under risk aversion. A means-tested and government-provided Age Pension influences decisions of the risk averse but not the loss averse retiree.
In this chapter I provide some background on the reasons (I think) a traditional fund company might want to introduce a modern tontine as it relates to the unique challenges that people face managing their financial affairs towards the end of the human lifecycle.
We consider annuity designs in which the benefit amount is allowed to fluctuate (up or down), based on a given mortality/longevity experience. This way, guarantees are relaxed in respect of traditional annuity arrangements. On the other hand, while the annuitant is exposed to the risk of a future reduction of the benefit amount because of higher longevity, he/she can immediately take advantage of a lower premium loading, as well as of a future increase of the benefit amount in the case of higher mortality. Flexibility in the annuity design could be welcomed by individuals, as the conservative features of traditional products partly explain their lack of attractiveness in most markets. To further contribute to the flexibility of the product, we suggest a pricing structure based on periodic fees applied to the policy fund, instead of the usual upfront loading at issue. Periodic fees are more suitable to support a revision of the arrangement after issue, which is currently not allowed in traditional annuity products. We show that periodic fees can be introduced by identifying a discount factor to be used for pricing and reserving. We assume stochastic mortality, and we compare alternative mortality/longevity linking solutions, by assessing the periodic fees and other quantities.
In the context of global aging population, improved longevity and ultra-low interest rates, the question of pension plan under-funding and adequate elderly financial planning is gaining awareness worldwide, both among experts, regulatory bodies, and popular media. Additional emergence of societal changes—Peer to Peer business model and Financial Disintermediation—have contributed to the resurgence of the concept of “Tontines” in various papers and the proposal of further models. These generalizations can offer efficient decumulation schemes with high longevity protection which is particularly well adapted for retirement needs—both for its members and carriers. In this paper, we revisit the mechanism proposed by Fullmer and Sabin (Journal of Accounting and Finance, 2019. https://doi.org/10.33423/jaf.v19i8.2615)—which allows the pooling of Modern Tontines through a self-insured community. This “Tontine” generalization retains the flexibility of an individual design: open contribution for a heterogeneous population, individualized asset allocation and predesigned annuitization plan. The actuarial fairness is achieved by allocating the deceased proceedings to survivors using a specific individual pool share which is a function of the prospective expected payouts for the period considered. After a brief introduction, this article provides a formalization of the mathematical framework with prospective analysis, characterizes the inherent bias, generalizes the mechanism to joint lives, and analyses simulated outcomes based on various assumptions. A reverse moral hazard limit is exposed and discussed (the “Term Dilemma”). Some solutions are then proposed to overcome scheme shortcomings and some requirements for practical implementation are discussed.
This paper proposes an innovative retirement product focusing on longevity risk sharing, a contract we refer to as tail index-linked annuity (TILA). Specifically, the proposed TILA pays out variable annual payments, which will be equal to a regular nominal amount when a reference survival index is lower than a predetermined threshold (i.e. normal evolution of longevity risk), and a reduced, index-dependent payment when the threshold is passed (i.e. highly unfavorable evolution of longevity risk). The proposed TILA aims at not only improving the benefits of the policyholders, which has been the focus in recent literature on innovative retirement products, but also reducing the longevity risk exposure of the insurer, particularly for advanced retirement ages. Using real-world mortality data and a stochastic multi-population mortality model, we find that the proposed TILA leads to higher expected lifetime utility than regular annuities for policyholders with different degrees of risk aversions. Meanwhile, numerical analysis shows that the proposed TILA could greatly mitigate the solvency risk of the insurer, leading to a substantially lower loss probability and expected (tail-) loss than regular annuities in the presence of a longevity shock, and therefore could reduce the insurer's required solvency capital under the latest solvency regulations.
In this paper I extend the work of Bernhardt and Donnelly (2019) dealing with modern explicit tontines, as a way of providing income under a specified bequest motive, from a defined contribution pension pot. A key feature of the present paper is that it relaxes the assumption of fixed proportions invested in tontine and bequest accounts. In making the bequest proportion an additional control function I obtain, hitherto unavailable, closed-form solutions for the fractional consumption rate, wealth, bequest amount, and bequest proportion under a constant relative risk averse utility. I show that the optimal bequest proportion is the product of the optimum fractional consumption rate and an exponentiated bequest parameter. I show that under certain circumstances, such as a very high bequest motive, a life-cycle utility maximisation strategy will necessitate negative mortality credits analogous to a member paying life insurance premiums. Typical scenarios are explored using UK Office of National Statistics life tables.
This paper studies the incorporation of an explicit fee in modern tontine schemes and investigates how it affects the attractiveness of these products from the point of view of both the insurer and tontine participants. We consider a single initial fee and a variable fee, where the latter can be designed to meet different liquidity needs and risk aversions. We find that the “indifference fee”, the fee for the tontine at which the participants are indifferent to choosing between the tontine or an annuity, comes extremely close to that of the annuity. This presents insurers with a large spectrum of fees on which to base their charge for managing the tontine. Tontine providers and participants can agree on a fee level which makes tontines an attractive alternative to annuities for tontine participants and at the same time, allows insurers to retain a large part of the fee as a profit, given the low risk contained in a tontine.
A new empirical study of the relation between money, nominal income, prices, and real output in postwar quarterly U.S. data rejects virtually all of the conclusions reached by Families provide individuals with risk sharing opportunities which may not otherwise be available. Within the family there is a degree of trust and a level of information which alleviates three key problems in the provision of insurance by markets open to the general public, namely, moral hazard, adverse selection, and deception. The informational advantages of pooling risk within families must be set against the inability of families to provide complete insurance because of the small size of the risk pooling group. This paper demonstrates how families can provide insurance against uncertain dates of death. Death risk sharing family arrangements effectively constitute an incomplete annuities market. Our analysis indicates that these arrangements even in small families can substitute by more than70% for complete annuities. Given the adverse selection problem and transactions costs in public annuity markets, risk pooling in families may well be preferred to purchasing market annuities. In the absence of organized public markets in annuities, these risk sharing arrangements provide powerful economic incentives for marriage and family formation. The paper suggests that inter-family transfers need have nothing to do with altruistic feelings; rather, they may simply reflect risk sharing behavior of completely selfish family members.
Longer lifespans are generally seen as a positive outcome of economic growth. Yet life extension also means that more people face the risk of living too long -- that is, outliving their assets and means of support. A range of financial products exists currently or can be envisioned for the future that would be useful in helping to protect people against having to dramatically curtail consumption in old age. This paper reviews the usefulness of life annuities in providing protection against longevity risk.
In this paper we derive and compare the value of life annuity products in an international context. Our specific goal is to assess the money's worth and adverse selection impact of annuities in two countries Singapore and Australia that have mandatory DC-type retirement plans. This similarity in plan type is offset by differences in the two countries' national retirement policies. Our comparison therefore exploits the natural experiment in annuity pricing and purchase behaviour under alternative retirement regimes. The results show that after controlling on administrative loadings, there appear to be important differences in measured adverse selection across countries. Specifically, selection appears to be far stronger in the presence of a generous public benefit scheme that provides a first line of defence against the risk of old-age poverty.
In this paper, we derive a formula for the optimal investment allocation (derived from a dynamic programming approach) in a defined contribution (DC) pension scheme whose fund is invested in n assets. We then analyse the particular case of n=2 (where we consider the presence in the market of a high-risk and a low-risk asset whose returns are correlated) and study the investment allocation and the downside risk faced by the retiring member of the DC scheme, where optimal investment strategies have been adopted. The behaviour of the optimal investment strategy is analysed when changing the disutility function and the correlation between the assets. Three different risk measures are considered in analysing the final net replacement ratios achieved by the member: the probability of failing the target, the mean shortfall and a value at risk (VaR) measure. The replacement ratios encompass the financial and annuitisation risks faced by the retiree. We consider the relationship between the risk aversion of the member and these different risk measures in order to understand better the choices confronting different categories of scheme member. We also consider the sensitivity of the results to the level of the correlation coefficient.
net income. It draws on the experience gained when the social security system in Italy was reformed in the early 1990s, which led to drastic reductions in the number of claims against the statutory pension scheme. The various sectors of the population (elderly people, young people, public employees as opposed to private sector employees, etc.) were affected to differing degrees. From Brugiavini's estimates, it becomes clear that a reduction in claims against the statutory pension system has led to a markedly increased willingness to save, particularly in the sectors of the population which were most affected. Reinhold Schnabel starts by discussing possible consequences of a pension cut from a German perspective. He discusses possible effects on saving and labour supply. In the second part of his comments he questions whether all of Brugiavini's interpretations of the "Italian experiment" are convincing. He doubts in particular, whether the reform was unexpected.
At retirement, most individuals face a choice between voluntary annuitization and discretionary management of assets with systematic withdrawals for consumption purposes. Annuitization ‘buying a life annuity from an insurance company’ assures a lifelong consumption stream that cannot be outlived, but it is at the expense of a complete loss of liquidity. On the other hand, discretionary management and consumption from assets ‘self-annuitization’ preserves flexibility but with the distinct risk that a constant standard of living will not be maintainable. We compute the lifetime and eventual probability of ruin (PoR) for an individual who wishes to consume a fixed periodic amount ‘a self-constructed annuity’ from an initial endowment invested in a portfolio earning a stochastic (lognormal) rate of return. The lifetime PoR is the probability that net wealth will hit zero prior to a stochastic date of death. The eventual PoR is the probability that net wealth will ever hit zero for an infinitely lived individual.
One of the key problems facing annuity providers is mortality risk, the risk of underestimating mortality improvements. The authors argue that the government could help the issuers of annuities to hedge aggregate mortality risk by introducing a new type of bond, which the authors call a survivor bond. The future coupons on this bond depend on the percentage of the population of retirement age on the issue date who are still alive on the future coupon payment dates. The coupons on the bond therefore decline over time but continue in payment until the last members of this population cohort have died. The government would therefore be assuming a risk that has hitherto been borne by the private sector. However, governments now issue inflation-indexed bonds, and the authors would argue that inflation risk is a much greater risk than mortality risk in aggregate. Furthermore, governments directly contribute to mortality risk: for example, they promote public health campaigns that, if successful, lead to mortality improvements that are difficult to predict many years ahead. Survivor bonds enable pension provision to be a shared responsibility between the public and private sectors.
The present paper considers a retiree of a certain age with an initial endowment of investable wealth facing the following alternative investment opportunities. One possibility is to buy a single premium immediate annuity-contract. This insurance contract pays a life-long constant pension payment of a certain amount, depending e.g. on the age of the retiree, the operating cost of the insurance company and the return the company is able to realize from its investments. The alternative possibility is to invest the single premium into a portfolio of mutual funds and to periodically withdraw a fixed amount, in the present paper chosen to be equivalent to the consumption stream generated by the annuity . The particular advantage of this self annuitization strategy compared to the life annuity is its greater liquidity. However, the risk of the second opportunity is to outlive the income stream generated by this investment. The risk in this sense is specified by considering the probability of running out of money before the uncertain date of death. The determination of this personal ruin probability with respect to German mortality and capital market conditions is the objective of the following paper.
Straightforward geometry can be used to explain the role for life annuities in smoothing individual consumption during retirement. In the aggregate, there is a surprising analogy with a standard diagram from trade theory. That diagram takes careful account of the government revenues foregone when there is a move from tariffs to free trade. Here the authors find a similar geometric interpretation of the proceeds from deceased estates which need to be accounted for when comparing a no-annuities economy with its annuitized counterpart. By the same token, there are considerable gains from opening up a market for life annuities, even after deceased estates are taken into account. Copyright 1999 by Blackwell Publishers Ltd and The Victoria University of Manchester
This paper uses the Panel Study of Income Dynamics to test whether risk-sharing is complete between or within American families. The tests accommodate wide variety in the configuration and availability of family data. The test results reject inter- as well as intra-family full risk-sharing even assuming that leisure is endogenous or that leisure and consumption are nonseparable. Copyright 1996 by The Econometric Society.
This paper deals with solvency requirements for life annuities portfolios and funded pension plans. Particular emphasis is devoted to longevity risk, i.e. the risk arising from uncertainty in future mortality trends. This risk must be faced by insurance companies and pension plans that have guaranteed lifelong payoffs.
Solvency is investigated referring to immediate annuities, and hence the so-called decumulation phase is addressed. To assess solvency, assets are compared with the random present value of liabilities. Several requirements are considered, each leading to a required asset level that must be financed both with premiums (or contributions) and capital allocation.
The present paper considers a retiree of a certain age who is endowed with a certain amount of wealth and is facing alternative investment opportunities. One possibility is to buy a single premium immediate (participating) annuity-contract. This insurance product pays a life-long pension payment of a certain amount, depending e.g. on the age of the retiree, the operating cost of the insurance company and the return the company is able to realize from its investments. The alternative possibility is to invest the single premium into a portfolio of mutual funds and to periodically withdraw a fixed amount that is assumed to be equivalent to the consumption stream generated by the annuity. The particular advantage of this self-annuitization strategy compared to the life annuity is its greater liquidity and the possibility of leaving money for heirs. However, the risk of self-annuitization is to outlive the assets before the uncertain date of death. The risk can thus be specified by considering the probability of running out of money before the uncertain date of death. The determination of this personal probability of consumption shortfall with respect to German insurance and capital market conditions is the objective of this paper.
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