Moshe Arye Milevsky

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.

This paper develops a computational framework for inverting Gompertz–Makeham mortality hazard rates, consistent with compensation laws of mortality for heterogeneous populations, to define a longevity-risk-adjusted global (L-RaG) age. To illustrate its salience and possible applications, the paper calibrates and presents L-RaG values using country data from the Human Mortality Database (HMD). Among other things, the author demonstrates that when properly benchmarked, the longevity-risk-adjusted global age of a 55-year-old Swedish male is 48, whereas a 55-year-old Russian male is closer in age to 67. The paper also discusses the connection between the proposed L-RaG age and the related concept of Biological age, from the medical and gerontology literature. Practically speaking, in a world of growing mortality heterogeneity, the L-RaG age could be used for pension and retirement policy. In the language of behavioral finance and economics, a salient metric that adjusts chronological age for longevity risk might help capture the public’s attention, educate them about lifetime uncertainty and induce many of them to take action — such as working longer and/or retiring later.

The consensus among researchers is that (some) longevity risk pooling is the optimal strategy for drawing down wealth in retirement, and a robust literature has developed around its measurement via annuity equivalent wealth. However, most of the published work is conducted numerically, and authors usually report only a handful of limited values. In this article we derive closed-form expressions for the value of longevity risk pooling with fixed life annuities under constant relative risk aversion preferences.

We solve a lifecycle model in which the consumer’s chronological age does not move in lockstep with calendar time. Instead, biological age increases at a stochastic non-linear rate in time like a broken clock that might occasionally move backwards. In other words, biological age could actually decline. Our paper is inspired by the growing body of medical literature that has identified biomarkers which indicate how people age at different rates. This offers better estimates of expected remaining lifetime and future mortality rates. It isn’t farfetched to argue that in the not-too-distant future personal age will be more closely associated with biological vs. calendar age. Thus, after introducing our stochastic mortality model we derive optimal consumption rates in a classic (Yaari, 1965) framework adjusted to our proper clock time. In addition to the normative implications of having access to biological age, our positive objective is to partially explain the cross-sectional heterogeneity in retirement spending rates at any given chronological age. In sum, we argue that neither biological nor chronological age alone is a sufficient statistic for making economic decisions. Rather, both ages are required to behave rationally.

We determine the optimal lifecycle purchasing strategy for deferred income annuities (DIAs)—which are distinct from single-premium income annuities (SPIAs)—for an individual who wishes to maximize the expected utility of his/her annuity income at a fixed time in the future. In contrast to the vast portfolio-choice literature for SPIAs, we focus on the stochasticity of the DIA’s payout yield and address concerns that rates are currently “too low” to justify irreversible annuitization. We assume a mean-reverting model for payout yields and show that a risk-neutral consumer who wishes to maximize his/her expected retirement income should wait until yields reach a threshold—which lies above historical averages—and then purchase the DIA in one lump sum. In contrast, a risk-averse consumer who is concerned the payout yield will remain below average for an extended period and worries about losing mortality credits while waiting, should employ a barrier purchasing strategy, as in the portfolio choice problem under transaction costs. We illustrate how this insight is applied in the context of annuitization. In fact, the optimal behavior of a risk-averse consumer resembles an asymmetric dollar-cost averaging strategy, with a portion of the DIA-budget spent even while payout rates are below historical averages. As part of our analysis we offer an easy-to-use asymptotic approximation for the optimal purchasing strategy (threshold) and provide some numerical examples to illustrate the concept.

Many assume that in the short run, annuity prices promptly and efficiently respond to changes in interest rates. Using a unique database of quotes, we show this is not the case. Prices are less sensitive to changes in rates than expected, and responses are asymmetric. Prices react more rapidly and with greater sensitivity to an increase than to a decrease in rates. The results are robust, but there is a small degree of heterogeneity in the responses of different insurance companies. When rates increase, larger firms are slightly quicker to improve prices. The opposite is true when rates decline. In sum, we show that the microstructure of annuity dynamics is more complicated than (simply) adding mortality credits to bond yields.

In this paper we model and solve a retirement consumption problem with differentially taxed accounts, parameterized by longevity risk aversion. The work is motivated by some observations on how Canadians de-accumulate financial wealth during retirement — which seem rather puzzling. While the Modigliani lifecycle model can justify a variety of (pre-tax) de-accumulation or draw down rates depending on risk preferences, the existence of asymmetric taxes implies that certain financial accounts should be depleted faster than others. Our analysis of data from the Survey of Financial Security indicates that Canadian retirees maintain approximately two-thirds of their financial wealth in tax-sheltered accounts and a third in taxable accounts regardless of age. The ratio of taxable to tax-sheltered wealth increases slightly or remains relatively constant depending on household income which is not what one would expect from the lifecycle model. Indeed, using our model we cannot locate a plausible tax function that justifies a constant “account ratio” regardless of age. For example under flat rates taxable accounts should be depleted well before tax-sheltered accounts are ever touched. The account ratio should go to zero quite rapidly in the absence of government mandated withdrawals. We also demonstrate that under progressive income taxes withdrawals are made from both accounts but at different rates depending on account size, pension income and longevity risk preferences. Again, the “account ratio” should eventually decline. We postulate that this sort of behavior is likely due to irrational considerations linked to mental accounting, etc. It remains to be seen whether this will persist over time and under a more careful analysis of Canadian cohorts or if retirees in other countries exhibit the same behavior.

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.

Tontines were once a popular type of mortality-linked investment pool. They promised enormous rewards to the last survivors at the expense of those died early. While this design appealed to the gambling instinct, it is a suboptimal way to generate retirement income. Indeed, actuarially-fair life annuities making constant payments–where the insurance company is exposed to longevity risk–induce greater lifetime utility. However, tontines do not have to be structured the historical way, i.e. with a constant cash flow shared amongst a shrinking group of survivors. Moreover, insurance companies do not sell actuarially-fair life annuities, in part due to aggregate longevity risk.

We describe a recursive algorithm that computes the timing and quantity of purchase of deferred income annuities (DIAs) within target-date funds (TDF) in defined contribution (DC) plans, although the algorithm could also be applied within any retirement account. We map a relatively small number of statistical parameters into a rule that conveys the dollar amount of DIAs to be purchased at any given age and time. Our model is of particular relevance given the recent announcement by the U.S. Treasury Department approving the inclusion of life annuities in 401(k) plans and in TDFs in particular. Note that to qualify as a TDF requires a methodology based on “generally accepted investment theories using a consistent investment strategy.” This article offers one possible such theory in the context of DIAs.

We analyze an insurance instrument called a ruin‐contingent life annuity (RCLA), which is a stand‐alone version of the option embedded inside a variable annuity (VA) but without the buyer having to transfer investments to the insurance company. The annuitant’s payoff from an RCLA is a dollar of income per year for life, deferred until a certain wealth process hits zero. We derive the partial differential equation (PDE) satisfied by the RCLA value assuming no arbitrage, describe efficient numerical techniques, and provide estimates for RCLA values. The practical motivation is twofold. First, numerous insurance companies are now offering similar contingent deferred annuities (CDAs). Second, the U.S. Treasury and Department of Labor have encouraged DC plans to offer longevity insurance to participants and the RCLA might be the ideal product.

This paper employs a micro-economic framework to examine the Registered Retirement Income Fund (RRIF) required withdrawal schedule in the context of current interest rates and longevity projections. It argues that today’s demographic and economic realities require that the schedule be revised to remain justifiable and fair. The methodology employed in this paper differs from other policy-based (or probabilistic) arguments. Namely, the paper compares the legislated withdrawal schedule with an optimal withdrawal schedule in a consumption-smoothing lifecycle model (LCM) for a longevity risk-averse retiree. This paper argues that while the LCM might be able to justify the RRIF withdrawal rates in place during the late 1980s – which was a period with higher interest rates and lower longevity – a quarter of a century later the schedule has become outdated.

Tontines and life annuities both insure against longevity risk by guaranteeing (pension) income for life. The optimal choice between these two mortality-contingent claims depends on personal preferences for consumption and risk. And, while pure tontines are unavailable in the twenty-first century, the first longevity-contingent claim (and debt) issued by the English government in the late seventeenth century offered an option to select between the two. This paper analyzes financial and economic aspects of King William’s 1693 tontine that have not received attention in the financial economic literature. In particular, I compare the stochastic present value (SPV) of the tontine vs. the life annuity and discuss characteristics of investors who selected one versus the other. Finally, I investigate whether the recorded 1693 tontine survival rates — which are abnormally high relative to population mortality rates in the late 17th century — should be attributed to anti-selection effects or perhaps to fraudulent behaviour. In sum, this paper is an empirical examination of annuitization decisions made by investors over three hundred years ago.

This paper offers a financial economic perspective on the optimal time (and age) at which the owner of a Variable Annuity (VA) policy with a Guaranteed Lifetime Withdrawal Benefit (GLWB) rider should initiate guaranteed lifetime income payments. We bypass issues related to utility, bequest and consumption preference by treating the VA as liquid and tradable. This allows us to use an American option pricing framework to derive a so-called optimal initiation region. Our main practical finding is that given current design parameters in which volatility (asset allocation) is restricted to less than 20%, while guaranteed payout rates (GPR) as well as bonus (roll-up) rates are less than 5%, GLWBs that are in-the-money should be turned on by the late 50s and certainly the early 60s. The exception to the rule is when a non-constant GPR is about to increase to a higher age band, in which case the optimal policy is to wait until the new GPR is hit and then initiate immediately. Also, to offer a different perspective, we invert the model and solve for the bonus (roll-up) rate that is required to justify delaying initiation at any age. We find that the required bonus is quite high and more than what is currently promised by existing products. Our methodology and results should be of interest to researchers as well as to the individuals that collectively have over $1 USD trillion in aggregate invested in these products. We conclude by suggesting that much of the non-initiation at older ages is irrational (which obviously benefits the insurance industry).

The article examines the implications for the pricing of longevity insurance and life annuities of a plateauing in the instantaneous force of mortality (IFM) at advanced ages. Given the increasing popularity of advanced life delayed annuities (ALDAs, or deferred-income annuities) and the current low-interest-rate environment, the present-value cost of misestimating the dynamics of late-life mortality can be substantial. The article also offers some comments about the possibility of using ALDA prices to imply market expectations of mortality dynamics and plateaus in a manner similar to implied volatility in the options market. All this has obvious implications for annuity buyouts, buy-ins, and other forms of longevity risk transfer as well the most pressing retirement problem for individuals—how to make their money last for the remainder of their lives.

Practitioners are well aware of the pernicious effect of the “sequence of investment returns” on retirement income sustainability. Poor markets early in the withdrawal phase increase the “lifetime ruin probability” and reduce the longevity of a portfolio. In this article, the authors investigate how and when traded equity options can be used to extend the life of a retiree’s investments. They label this class of strategies longevity extension overlays (LEOs) and use simulation techniques to analyze the strategy’s theoretical properties. They also provide evidence on the efficacy of simple LEOs during the 2007–2013 period. Our results are encouraging and offer a justifiable alternative for wealth managers who want to avoid using (more complex and opaque) insurance-product solutions.

We extend the lifecycle model (LCM) of consumption over a random horizon (also known as the Yaari model) to a world in which (i) the force of mortality obeys a diffusion process as opposed to being deterministic, and (ii) consumers can adapt their consumption strategy to new information about their mortality rate (also known as health status) as it becomes available. In particular, we derive the optimal consumption rate and focus on the impact of mortality rate uncertainty versus simple lifetime uncertainty — assuming that the actuarial survival curves are initially identical — in the retirement phase where this risk plays a greater role.