This paper investigates time-varying risk sharing between annuity buyer and provider. It explores Pareto optimal (PO) and viable Pareto optimal (VPO) risk-sharing designs, in which the share of the reserve deviation transferred to the policyholder varies over time. The optimization problem, based on a weighted average of mean-variance preferences, results in a complex quartic objective function. Such optimization problems are difficult to solve, and checking their convexity is known to be NP-hard. A heuristic method is introduced to simplify the problem, providing a closed-form solution that closely approximates the numerical results. The paper also highlights factors influencing the existence of VPO designs, with age playing a critical role, thereby suggesting the suitability of these designs as retirement products.

This paper is a contribution to a symposium on Michael Otsuka’s book, How to Pool Risk Across Generations. Following Otsuka, one may distinguish three distinct systems of cooperation within a standard pension arrangement: the retirement system, the longevity risk pool and the investment risk pool. It is important to observe, however, that only the retirement system constitutes a genuine system of intergenerational cooperation, the other two are essentially intragenerational, in that they pool risks among members of a cohort. Otsuka is faulted for being occasionally less than clear on these distinctions.

Twenty years ago, the adjustment to monthly Social Security benefits for early or delayed claiming was, on average, roughly actuarially fair, although some subsets of individuals could gain from delay. Since then, delaying claiming has become much more attractive thanks to three factors: a more generous delayed retirement credit, improvements in mortality, and historically low real interest rates. In this article, I examine how these three factors influence optimal claiming behavior. I also discuss empirical patterns of claiming across individuals and over time, as well as explanations for these patterns. I argue that although many people appear to claim suboptimally early, this behavior may be changing as information spreads about the importance of the claiming decision. Finally, I discuss policy toward claiming and the impact that an increase in strategic claiming could have on Social Security's finances.

This study examines the interest in different pension payout schemes when full annuitization is the default. We focus on three possible pension payout schemes: a flat-rate annuity, a high/low annuity-based profile, and a partial lump sum at retirement with a lower flat-rate annuity after that. We make use of a vignette study and find substantial interest in each of the three payout schemes. Interest in the lump sum scheme increases when a higher percentage can be taken out as a lump sum or when interest rates or replacement rates are lower. Interest in a high/low annuity-based profile increases when the high annuity is valid for a shorter period.

This paper documents trends over the last two decades in retirement behavior and retirement income choices of participants in TIAA, a large and mature defined contribution plan. From 2000 and 2018, the average age at which TIAA participants stopped contributing to their accounts, which is a lower bound on their retirement age, rose by 1.2 years for female and 2.0 years for male participants. There is considerable variation in the elapsed time between the time of the last contribution to and the first income draw from plan accounts. Only 40% of participants take an initial income payment within 48 months of their last contribution. Later retirement and lags between retirement and the first retirement income payout led to a growing fraction of participants reaching the required minimum distribution (RMD) age before starting income draws. Between 2000 and 2018, the fraction of first-time income recipients who took no income until their RMD rose from 10% to 52%, while the fraction of these recipients who selected a life-contingent annuitized payout stream declined from 61% to 18%. Among those who began receiving income before age 70, annuitization rates were significantly higher than among those who did so at older ages. Aggregating across all income-receiving beneficiaries at TIAA, not just new income recipients, the proportion with a life annuity as part of their payout strategy fell from 52% in 2008 to 31% in 2018. By comparison, the proportion of all income recipients taking an RMD payment rose from 16% to 29%. About one-fifth of retirees received more than one type of income; the most common pairing was an RMD and a life annuity. In the later years of our sample, the RMD was becoming the de facto default distribution option for newly retired TIAA participants.

There is a lack of publicly available information covering the practices insurers employ to manage their exposure to reinsurance recapture risk. A working party was set-up to shed light on the different approaches insurers use to mitigate this complicated to manage risk. This report is intended to form part of a publicly available information repository that market practitioners can refer to and reflect on as best practice evolves and develops.

This paper proposes a shift in the valuation and production of long-term annuities, away from the classical risk-neutral methodology towards a methodology using the real-world probability measure. The proposed production method is applied to three examples of annuity products, one having annual payments linked to a mortality index and the savings account and the others having annual payments linked to a mortality index and an equity index with a guarantee that is linked to the same mortality index and the savings account. Out-of-sample hedge simulations demonstrate the effectiveness of the proposed less-expensive production method. In contrast to classical risk-neutral production, which revolves around the savings account as reference unit, the long-term best-performing portfolio, the numéraire portfolio of the equity market, is employed as the fundamental reference unit in the production of the annuity. The numéraire portfolio is the strictly positive, tradable portfolio that when used as denominator or benchmark makes all benchmarked non-negative portfolios supermartingales. Under real-world valuation, the initial benchmarked value of a benchmarked contingent claim equals its real-world conditional expectation. The proposed real-world valuation and production can lead to significantly lower values of long-term annuities and their less-expensive production than suggested by the risk-neutral approach.

One of the most important financial decisions that pension participants make concerns how they access their pension assets when they terminate employment with their plan sponsor. Their choices depend both on own preferences and the options offered by their retirement plan. This paper examines both past and future pension withdrawal choices for those with defined benefit (DB) and defined contribution (DC) pensions, separately. Our data are drawn from a set of pension distribution questions we fielded in the Understanding American Study. Results show significant differences in distribution choices based on the type of retirement plan, with individuals covered by DB plans significantly more likely to select annuities compared to similar employees covered by DC plans. We also find differences in how higher annual income affects annuity choices based on coverage by DB plans. Individuals with lower levels of financial literacy and lower annual income have less knowledge of basic pension characteristics.

Parametric mortality models permit detailed analysis of risk factors for actuarial work. However, finite data volumes lead to uncertainty over parameter estimates, which in turn gives rise to mis-estimation risk of financial liabilities. Mis-estimation risk can be assessed on a run-off basis by valuing the liabilities with alternative parameter vectors consistent with the covariance matrix. This run-off approach is especially suitable for tasks like pricing portfolio transactions, such as bulk annuities, longevity swaps or reinsurance treaties. However, a run-off approach does not fully meet the requirements of regulatory regimes that view capital requirements through the prism of a finite horizon, such as Solvency II’s one-year approach. This paper presents a methodology for viewing mis-estimation risk over a fixed time frame, and results are given for a specimen portfolio. As expected, we find that time-limited mis-estimation capital requirements increase as the horizon is lengthened or the discount rate is reduced. However, we find that much of the so-called mis-estimation risk in a one-year value-at-risk assessment can actually be driven by idiosyncratic variation, rather than parameter uncertainty. This counter-intuitive result stems from trying to view a long-term risk through a short-term window. As a result, value-at-risk mis-estimation reserves are strongly correlated with idiosyncratic risk. We also find that parsimonious models tend to produce lower mis-estimation risk than less-parsimonious ones.

Evidence shows that people have difficulty understanding complex aspects of retirement planning, which leads them to under-utilize annuities and claim Social Security benefits earlier than is optimal. To target this problem, we developed vignettes about the consequences of different annuitization and claiming decisions. We evaluated our vignettes using an experiment with a representative online panel of nearly 2,000 Americans. In our experiment, respondents were either assigned to a control group with no vignette, to a written vignette, or to a video vignette. They were then asked to give advice to hypothetical persons on annuitization or Social Security claiming, and were asked factual questions about these concepts. We found evidence that being exposed to vignettes led respondents to give better advice. For example, the gap between advised claim age for a relatively healthy person versus a relatively sick person was larger by nearly a year in the vignette treatments than in the control group. Furthermore, the vignettes increased financial literacy related to these concepts by 10–15 percentage points. Interestingly, the mode of communication did not have a significant impact – the video and written vignettes were equally effective.

This paper examines the impact of uncertainties in the future trends of mortality on annuity values in Singapore's compulsory purchase market. We document persistent population mortality improvement trends over the past few decades, which underscores the importance of longevity risk in this market. Using the money's worth framework, we find that the life annuities delivered expected payouts valued at 1.019–1.185 (0.973–1.170) per dollar of annuity premium for males (females). Even in a low mortality improvement scenario, the annuities provide an expected value exceeding 0.950. This suggests that participants in the national annuity pool have access to attractively priced annuities, regardless of sex, product, and premium invested.

This note derives analytic expressions for annuities based on a class of parametric mortality “laws” (the so-called Makeham–Beard family) that includes a logistic form that models a decelerating increase in mortality rates at the higher ages. Such models have been shown to provide a better fit to pensioner and annuitant mortality data than those that include an exponential increase. The expressions derived for evaluating single life and joint life annuities for the Makeham–Beard family of mortality laws use the Gauss hypergeometric function and Appell function of the first kind, respectively.

Compound interest was known to ancient civilisations, but as far as we know it was not until medieval times that mathematicians started to analyse it in order to show how invested sums could mount up and how much should be paid for annuities. Starting with Fibonacci in 1202 A.D., techniques were developed which could produce accurate solutions to practical problems but involved a great deal of laborious arithmetic. Compound interest tables could simplify the work but few have come down to us from that period. Soon after 1500, the availability of printed books enabled knowledge of the mathematical techniques to spread, and legal restrictions on charging interest were relaxed. Later that century, two mathematicians, Trenchant and Stevin, published compound interest tables for the first time. In 1613, Witt published more tables and demonstrated how they could be used to solve many practical problems quite easily. Towards the end of the 17th century, interest calculations were combined with age-dependent survival rates to evaluate life annuities, and actuarial science was created.

Who values life annuities more? Is it the healthy retiree who expects to live long and might become a centenarian, or is the unhealthy retiree with a short life expectancy more likely to appreciate the pooling of longevity risk? What if the unhealthy retiree is pooled with someone who is much healthier and forced to pay an implicit loading? To answer these and related questions this paper examines the empirical conditions under which retirees benefit (or may not) from longevity risk pooling by linking the economics of annuity equivalent wealth to actuarially models of aging. I focus attention on the Compensation Law of Mortality which implies that individuals with higher relative mortality (e.g., lower income) age more slowly and experience greater longevity uncertainty. Ergo, they place higher utility value on the annuity. The impetus for this research today is the increasing evidence on the growing disparity in longevity expectations between rich and poor.