Encyclopedia of Gerontology and Population Aging

Living Edition
| Editors: Danan Gu, Matthew E. Dupre

Annuities

  • Ronald RichmanEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-69892-2_519-1

Synonyms

Definition of Annuities

An annuity is defined as a contract between two parties, the first of which, the annuity provider, is the party who undertakes to provide a series of payments to the second party, the annuitant, for as long as the annuitant is still alive, in exchange for a premium paid by the annuitant to the annuity provider. In general, annuity providers are life insurance companies and investment companies, and annuitants are older people who are eligible for the series of payments, or the annuity benefit, due to a premium that has been paid in respect of the annuity.

Note that this definition includes annuity benefits that result from the Guaranteed Lifetime Withdrawal Benefit (GLWB) benefits on so-called variable annuity products (sold mainly in the United States), as well as tontines, but excludes other financial products which do not provide a lifetime income benefit, such as so-called retirement annuities in South Africa. Furthermore, pensions benefits paid from a defined benefit pension scheme are excluded from the definition, since these benefits are not directly linked to a premium paid by the annuitant. For more discussion of what constitutes an annuity product, we refer the reader to Chapter 1 in OECD (2016).

Overview

Key Concepts of Annuities

Annuities, or similar products, have been used since ancient times; see, for example, Kopf (1926), who finds evidence of annuities purchased in Ancient Egypt, and Milevsky (2013), who notes that a life annuity is mentioned in the Book of Kings in the Old Testament. For more on the history of annuities, we refer the reader to these sources, as well as to Trenerry (1926) and Cannon and Tonks (2008).

Modern annuities are a financial product which enables older people to exchange a capital for a guaranteed lifetime income, thus allowing older people to manage their risk of living longer than their accumulated capital can support them. Annuities are generally purchased with retirement savings accumulated during a person’s working life, potentially within a tax-favored retirement product, such as a defined contribution pension fund. In some jurisdictions, the funds accumulated within a tax-favored vehicle must be used to buy an annuity, in other words, regulations specify the forced annuitization of specific funds, whereas in other jurisdictions, the funds may be taken as a lump sum. The empirical observation that many older people choose not to annuitize their retirement savings, leading to the “annuity puzzle,” is discussed in more detail later.

If funds for the purchase of annuity have been accumulated within a tax favored retirement product, such as a 401(k) fund in the United States or a defined contribution fund in the United Kingdom, then tax will not yet have been paid on these funds, and then the income received from an annuity will generally be taxed in the hands of the annuitant. Since the annuitant will be receiving a smaller income in retirement than while working, the marginal taxation rates will generally be lower than what would have been paid during the employed phase of an annuitant’s life. Thus, there are tax gains to be made through the purchase of an annuity. On the other hand, if the funds were not accumulated in this manner, and tax had already been paid on the funds used to purchase the annuity, then, in most jurisdictions, some tax relief will be granted to the annuitant; see, for example, Milevsky (2013) who discusses the complicated manner in which tax relief is granted in the United States.

Since annuities are usually purchased with a set amount of capital, the price of an annuity is often quoted using annuity factors, which is the amount of capital that must be paid to achieve a single unit of lifetime income. For example, suppose a potential annuitant has capital of $100,000, and is quoted an annuity factor of 14, then an annual income of $7143 ($100,000/14) can be achieved by paying the $100,000 as an annuity premium to the company providing the quote. The same logic applies if a potential annuitant wishes to calculate the amount of capital needed to achieve a specified annual income, for example, to achieve an income of $10,000 per annum, $140,000 of capital will be needed for the annuity premium.

What Determines Annuity Prices?

Annuity prices are determined mainly by two assumptions, mortality and investment returns. Since annuities pay an income for life, insurers need to estimate the expected total outgo over the course of the annuity contract, by projecting the probability that an annuitant is alive in each future period in which they will make an annuity payment and the size of the payments that will be made if this is the case.

Empirically, annuitants have been observed to survive longer than the general population, which is explained by the fact that those able to buy an annuity are usually wealthier than the general population and are able to afford medical care as well as by the fact that an older person is unlikely to purchase an annuity if they believe that they will survive for only a short period (Bissonnette et al. (2017) show that subjective assessments of life expectancy are relatively accurate, but may suffer from some optimism), thus annuitants are likely to be healthier than the general population. Furthermore, life insurers offering annuities may suffer from a unique form of moral hazard, whereby annuitants may use the income derived from an annuity to take steps to enhance their health, for example, by accessing high-quality medical care or purchasing healthier food (Ramsay and Oguledo 2018), leading to improved mortality relative to retirees without annuity income.

Thus, to price annuities, insurers will often use mortality tables containing mortality rates based on the recent mortality experience of a group of annuitants, which will likely be significantly lighter than population mortality. These annuitant mortality rates are then adjusted for likely changes in future mortality, generally by applying an assumed mortality improvement rate to the mortality table. Actuaries will generally rely on mortality projection models to derive the mortality improvement rate assumptions, such as the Lee and Carter (1992) or Cairns et al. (2006) mortality models, or standard industry projections, provided, for example, by the Continuous Mortality Investigation group in the United Kingdom. These models are usually calibrated on the basis of population mortality data, and the results may then be adjusted to reflect the faster rates of mortality improvement that annuitants exhibit compared to the general population (for an empirical demonstration of how mortality improvement rates vary by socioeconomic circumstances in the United Kingdom, see Lu et al. (2014)).

Therefore, a set of mortality assumptions that vary with the year of the projection will be produced for calculating the expected annuity payments, which are found by multiplying the contractual annuity payments with the probability that these payments will be made. In addition to the contractual annuity payments, insurers will also project other cash flows relating to the expected cost of servicing the annuities in each period, as well as cash flows which represent the profit margins charged by the insurer.

The expected annuity payments are then discounted using an investment return assumption, based on the returns that an insurer believes are achievable by using the annuity premium to buy a portfolio of assets, such as bonds and stocks, to support the annuity payments. The investment portfolios chosen by insurers will generally be relatively low risk, and, in some jurisdictions, the assets into which an insurer may invest are prescribed by regulation.

The value of the discounted expected payments is thus determined by the assumptions that the insurer makes regarding future mortality rates, investment returns, servicing costs, and profit margins, and therefore, these assumptions are the key determinant of annuity factors observed in the market.

Value of Annuities

Insurers calculate annuity premiums using expected values since they pool together the mortality risk of many lives, in other words, although the insurer does not know how long a particular annuitant will survive for, the survival rates of a group of annuitants can be predicted with relative certainty, using the law of large numbers. Thus, annuity premiums offer particularly good value for money for those annuitants who survive to enjoy the benefits of the contract, since part of their income stream has been financed by other, deceased annuitants. Milevsky (2013) notes that compared to other financial products, such as bonds, annuities offer an enhanced investment yield to annuitants that is difficult to replicate using other financial instrument. For a novel method of quantifying this benefit of annuities, the Implied Longevity Yield, see Milevsky (2005a).

Annuity Products

A substantial number of different annuity products have been developed, which offer variations on the basic annuity contract that can be categorized into several main types. Some variations relate to the manner in which the annuity benefits are paid, others extend the annuity contract from one life to several, and lastly, some products offer an enhanced benefit to annuitants whose health is different from the rest of the population.

Traditional annuities provide fixed payments at set intervals, usually monthly, to the annuitant, for as long as she is alive. The payments are determined at the outset of the contract, by dividing of the premium paid by the annuitant for the annuity, with the so-called annuity factor that has been calculated by the insurance company; see the section on actuarial management of annuities below. If the payments begin shortly after the inception of the contract, then the annuity is “immediate”; however, some annuity products allow for a lengthy period of time between the payment of the premium and the first annuity benefit, in which case, the annuity is “deferred.”

Advanced life-differed annuities, first described in an article by Milevsky (2005b), are a type of deferred annuity bought at retirement, but with payments only commencing at a more advanced age, such as at age 75. These products allow older people to insure their longevity risk at a much lower price than a traditional annuity, and are ideally combined with another form of retirement income until the age at which the annuity payments commence, such as managed withdrawals from an investment account. These products are relatively popular in Chile, but have not achieved popularity in developed markets (OECD 2016).

In real terms (in other words, after the consideration of inflation), traditional annuities offer a decreasing income stream, as the purchasing power of the fixed payment is eroded by inflation. Inflation-linked annuities offer annuity benefits that increase with a measure of inflation, often Consumer Price Inflation (CPI), thus providing benefits that remain constant in real terms and escalate in nominal terms. Other things being equal, inflation linked annuities are more expensive than traditional annuities. A similar product is a unit-linked annuity that pays out annuity benefits denominated in a set number of “units,” the value of which are linked to the investment return on an underlying pool of assets, which may include stocks and bonds.

An alternative product, at one stage very popular in the United Kingdom, is participating annuities, which entitle the annuitant to a share of the profits of the life insurance company, which, depending on the contract, may include investment returns, mortality profits, and expense savings. Upon the achievement of a profit, usually as determined by the actuarial valuation of the life insurer, an award of these profits, known as a bonus declaration, is made to the annuitant and the set payments are increased by the amount of the bonus.

From the perspective of the annuitant, there is a risk that death occurs soon after the annuity premium is paid, and thus the total annuity benefits received may be much less than the annuity premium paid. To offset this risk, some annuity contracts include a guarantee period in which the benefits are paid regardless of whether the annuitant is alive, or not, and in the latter case, the benefits are paid to a designated beneficiary. Other things being equal, annuities with a guarantee period are more expensive than those without. A similar product variation returns the original annuity premium net of payments made to the annuitant, if death occurs in a period specified within the contract.

Considering that the purpose of an annuity is generally to provide an income during retirement, the cessation of payments on the death of the annuitant is not practical if the annuitant has a surviving spouse who requires support. Thus, another variation on annuities are so-called “joint life” contracts, which provide a set benefit as long as both spouses are alive, and “last survivor” contracts which provide a benefit as long as one of the spouses are alive. Since last survivor contracts are expected to be paid for longer than single life contracts, other things being equal, these contracts are more expensive than single life contracts.

Tontines are an interesting variation on the concept of annuities contingent on multiple lives. Whereas last survivor annuities pay a fixed benefit as long as one of the specified annuitants are alive, tontines offer variable benefits that are determined by the number of survivors in a mortality pool. As members of the pool die, the survivors’ benefits are increased, until the last survivor of the pool is receiving a benefit equal to the sum of all of the benefits of the other members. Although tontines have effectively been superseded by modern annuity products, the potential advantages of tontines over annuities have recently been revisited by several authors, see for example, Weinert and Gründl (2016) and Milevsky and Salisbury (2015).

Enhanced annuity products, also known as substandard annuities in the United States, are priced based on multiple factors in addition to the traditional factors of age and sex (in the countries where pricing on the basis of sex is still allowed), and include a variety of products that seek to provide a greater income to those annuitants whose life expectancy is lower than the average annuitant population. These factors include the health status of the annuitant, such as whether the annuitant suffers from diseases such as cancer, diabetes, or stroke; lifestyle factors, such as smoking status and body mass index; and postcode, which is a popular pricing factor in the United Kingdom. Offering enhanced annuity products raises the problem of cannibalization for insurance providers (Gatzert and Klotzki 2016), which is the situation where a life insurer sells an enhanced annuity product to an annuitant who, in the absence of an enhanced product, would have bought a standard annuity from the same company. Since those buying standard annuities from the life insurer will now be expected to live longer, to maintain profitability in the standard annuity book, the life insurer will be forced to raise rates, leading to diminished sales or even to losses, if the rate increase would render the standard pricing uncompetitive. A similar issue, anti-selection, occurs when some life insurers offer enhanced annuities, attracting annuitants with lower life expectancy, and leading those annuitants with higher life expectancy to buy products from life insurers who only offer standard products, who are thus forced either to raise their rates or also to offer enhanced annuities.

Variable annuities, also known as unit-linked funds in the United Kingdom or segregated funds in Canada, are a retirement product allowing for the accumulation of funds for retirement and thus, intrinsically, do not fall within the definition of annuities in this entry. However, many of these products include the option to convert the fund balance into an income stream at a guaranteed rate, thus emulating a key feature of traditional annuities. There options and guarantees are called, in the context of variable annuity products, Guaranteed Lifetime Withdrawal Benefits (GLWBs). A potential advantage of this type of product over traditional annuities is that the annuitant maintains some control over the investment allocation of the fund, and can, for example, benefit from better than expected returns, as well as the option of surrendering the fund for a lump sum value. A further advantage is that, upon death of the annuitant, the annuitant’s estate inherits the balance of the fund value. The disadvantages relative to traditional annuities are the relatively higher costs of variable annuities and the lower annuity rate offered by GLWBs (Milevsky 2013). Notably, variable annuities are extremely popular in the United States and Canada (OECD 2016).

For a recent overview of the most popular types of annuity products in different jurisdictions, see OECD (2016).

Key Research Findings

Annuities in an Aging World

The world’s population is aging, meaning to say, that the proportion of the population in older age groups, such as those aged 60+, is increasing (United Nations 2017), driven primarily by decreasing fertility and increased longevity. This subsection firstly addresses the potential macroeconomic impact of population aging on financial markets, which could affect annuity pricing and sustainability, and secondly considers how population aging might affect annuity markets.

An interesting strand of the macroeconomic literature attempts to link the demography of populations to financial markets, by considering the question of the likely impact of population aging on asset prices and returns. Several studies, surveyed in Brooks (2006) and Mitchell (2018), rely on the lifecycle hypothesis of Modigliani and Brumberg (1954) to predict a depressive effect of the aging of the baby-boomer generation on asset prices, leading some economists in the early years of the new millennium to predict an “asset price meltdown” (Brooks 2006, p. 236). However, an empirical study of the impact of population aging on asset prices by Brooks (2006), who used an econometric model applied to data from 16 countries, finds little evidence for this, and indeed the evidence favors the opposite effect that in some countries asset prices will rise as the population ages, thus Mitchell (2018, p. 120) concludes that the evidence in favor of a market meltdown has been “undermined.” Perhaps most relevant to the discussion of annuities is the finding in Brooks (2006) that as the population ages, the relative price of stocks to treasury bills declines, perhaps because older investors prefer less risky investments. Since insurers often use treasury bills and other fixed income assets to back their annuity products, rising prices on these investments would lead to more expensive annuity products. However, with the benefit of hindsight, it appears that the persistent low interest rates in the developed world in the aftermath of the financial crisis have been more detrimental to annuity providers and the price of annuity products than these predicted effects of demographic shifts on financial markets.

In examining the effect of population aging on annuities, Mitchell and McCarthy (2002) consider the effect of demand-side and supply-side considerations on the annuity market. On the demand-side, they consider that decreasing mortality rates (and the need for individuals to pool their idiosyncratic longevity risk (Mitchell 2018)) and the move away from defined-benefit pension schemes (which are an annuity product substitute) will lead to increased demand for annuities, and similarly, on the supply-side, they predict that government pension systems reform will lead to the substitution of annuity products by older people. Unfortunately, the factors considered in the next section on the “annuity puzzle” seemingly indicate that older people do not find annuity products particularly attractive, and these predictions do not appear to have materialized. In a similar analysis, Mitchell et al. (2006) conclude that despite numerous ideas for product innovations spurred by population aging, nonetheless, the markets for these products remains underdeveloped, leading them to consider the potential benefit of a supranational organization overseeing annuity markets. This conclusion is unchanged in an updated study by Mitchell (2018, p. 122), who writes that the growth of private annuity markets has been “disappointing” and suggests several ways that insurers might deal with the uncertainty around future mortality improvements, such as by using mortality derivatives (such as survivor swaps or longevity bonds), or by selling variations on traditional annuities, such as with-profits annuities, products similar to tontines, or annuities, which embed an element of long-term care coverage.

The Annuity Puzzle

The annuity puzzle has its roots in the conclusion of Yaari (1965, p. 145) that the utility maximizing consumer should choose to hold all of her assets in annuities (referred to by Yaari as actuarial notes), who writes “the consumer’s assets (or liabilities) will always be held in actuarial notes rather than in regular notes.”

Although Yaari’s model depends on relatively strong assumptions, Davidoff et al. (2005) arrive at a similar conclusion in a much less restrictive setup. These conclusions stand in stark contrast to the reality that, in fact, annuities are not popular compared to other products or strategies (such as a drawdown approach, whereby a retiree does not buy an annuity but rather withdraws a percentage of funds each year for use as income)for producing an income in retirement, as noted by Modigliani (1986, p. 164) that it is a “well known fact that annuity contracts, other than in the form of group insurance through pension systems, are extremely rare.” Modigliani’s observation can be affirmed by recent events in the United Kingdom annuity market, which constitute an interesting natural experiment. Prior to pension reforms in 2015, holders of defined contribution pensions were strongly encouraged via tax incentives to annuitize their pensions (Thurley 2015). However, this incentive was removed in April 2015, leading to a decline in annuity sales in the United Kingdom from £11bn in 2014 to £4bn in 2017 and the withdrawal of several annuity providers from the market (Ramsay and Oguledo 2018). Thus, in the absence of strong tax incentives, demand for annuities has decreased markedly.

This contradiction between the prediction of Yaari’s normative economic model and the reality of the annuity market is called the annuity puzzle in the literature. Interestingly, Milevsky (2013) attributes the annuity puzzle to a much earlier source, Huebner (1927), who questions why more annuities are not purchased to avoid the “terror” of running out of funds during retirement.

In this section, some of the major approaches to answering the annuity puzzle are presented briefly, following closely the recent survey of Ramsay and Oguledo (2018); for an in-depth analysis, we refer the reader to that work, as well as to Section 3 of Milevsky (2013). Two major approaches, which can be described as the rational economic approach and the behavioral or psychological approach, have been proposed to resolve the annuity puzzle.

The rational approach seeks economic reasons why a consumer would choose not to buy an annuity, and the most prominent of these explanations are:
  • The insurers’ response to adverse selection, as mentioned above, is to use mortality tables that are substantially lighter than population mortality, leading to unattractive pricing for the majority of the population. Thus, enhanced annuities might play an important role in leading to more annuity purchases.

  • Since many older people appear to have bequest motives, they are deterred from purchasing traditional annuities, which involve the loss of capital upon death.

  • Older people feel confident that their self-annuitization strategies provide better value than traditional annuities, especially if they have family arrangement which allow them implicitly to pool their risk.

  • Many older people already have a substantial portion of their wealth annuitized via pensions or, in the case of the United States, Social Security (this approach is followed by Benartzi et al. (2011)).

  • Older people’s fear of the need to fund uninsured healthcare costs, which requires access to capital that is surrendered on the purchase of a traditional annuity.

In contrast to the rational approach, the behavioral approach seeks cognitive or psychological reasons to explain why older people might not act in accordance with economic theory (see Benartzi et al. (2011) for a discussion of the annuity puzzle along these lines), and prominent explanations in this vein are:
  • The manner in which the annuitization decision is framed, whether a as a consumption decision, where the annuity provides income to be spent for life, or an investment decision, which appears to be a risky gamble because the return depends on how long the older person will live, has a large impact on annuitization behavior.

  • Similar issues to framing, which are the endowment effect, which is an aversion to giving up an asset that is currently owned (the annuity premium) for an uncertain, and potentially risky asset (the annuity payments), and the effect of loss aversion (cumulative prospect theory predicts that losses decrease utility more than equivalent gains provide utility) on consumers who might view the annuitization decision as a costly gamble, as opposed to a consumption decision.

  • That consumers avoid considering annuities as a defense mechanism that protects them from thinking about death.

  • The availability heuristic, in which events which are easily imagined are given greater (subjective) likelihood than events which are hard to imagine, might lead older people to overemphasize the probability of dying shortly after purchasing an annuity.

Given the comprehensive explanations in the literature for the annuity puzzle, Milevsky (2013, p. 104) writes that “for those writing in 2012, the annuity puzzle is not as perplexing as it was 45 years ago.”

Summary and Future Directions of Research

With the extensive development of an array of annuity products, and the advances in understanding and modeling of mortality, it would be fair to assume that annuities are popular consumer products widely perceived to meet customer needs. However, the annuity puzzle, and recent events in the United Kingdom annuity market, seems to indicate otherwise, and perhaps for this reason, policy prescriptions to enhance the annuity market have appeared in most of the studies discussed in this entry. While a prominent feature of the literature is suggested changes to product design to make annuities more attractive, a seeming gap is a detailed survey of older people to understand why current annuity products might not be appealing. Closing this gap through future research would likely provide valuable insights into the design of new annuity products that would better meet consumer needs. Ultimately, resolving the annuity puzzle could lead to new types of annuity product and annuity markets that meet the financial and risk management needs of the increased number of older people in the aging world.

Cross-References

References

  1. Benartzi S, Previtero A, Thaler R (2011) Annuitization puzzles. J Econ Perspect 25(4):143–164CrossRefGoogle Scholar
  2. Bissonnette L, Hurd MD, Michaud PC (2017) Individual survival curves comparing subjective and observed mortality risks. Health Econ 26(12):285–303CrossRefGoogle Scholar
  3. Brooks R (2006) Demographic change and asset prices. In: Kent C, Park A, Rees D (eds) Demography and financial markets. Reserve Bank of Australia, Sydney, AustraliaGoogle Scholar
  4. Cairns AJG, Blake D, Dowd K (2006) A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. J Risk Insur 73(4):687–718CrossRefGoogle Scholar
  5. Cannon E, Tonks I (2008) Annuity markets. Oxford University Press, OxfordCrossRefGoogle Scholar
  6. Davidoff T, Brown J, Diamond P (2005) Annuities and individual welfare. Am Econ Rev 95(5):1573–1590CrossRefGoogle Scholar
  7. Gatzert N, Klotzki U (2016) Enhanced annuities: drivers of and barriers to supply and demand. Geneva Pap Risk Insur Issues Pract 41(1):53–77CrossRefGoogle Scholar
  8. Huebner S (1927) The economics of life insurance: human life values: their financial organization, management, and liquidation. D. Appleton-Century Company, New York CityGoogle Scholar
  9. Kopf E (1926) The early history of the annuity. Proceedings of the Casualty Actuary Society, 1926–1927, Volume XIIIGoogle Scholar
  10. Lee RD, Carter LR (1992) Modeling and forecasting US mortality. J Am Stat Assoc 87(419):659–671Google Scholar
  11. Lu J, Wong W, Bajekal M (2014) Mortality improvement by socio-economic circumstances in England (1982 to 2006). Br Actuar J 19(1):1–35CrossRefGoogle Scholar
  12. Milevsky M (2005a) The implied longevity yield: a note on developing an index for life annuities. J Risk Insur 72(2):302–320CrossRefGoogle Scholar
  13. Milevsky M (2005b) Real longevity insurance with a deductible: introduction to advanced-life delayed annuities (ALDA). North Am Actuar J 9(4):109–122CrossRefGoogle Scholar
  14. Milevsky M (2013) Life annuities: an optimal product for retirement income. CFA Institute, CharlottesvilleGoogle Scholar
  15. Milevsky M, Salisbury T (2015) Optimal retirement income tontines. Insur: Math Econom 64:91–105Google Scholar
  16. Mitchell OS (2018) Enhancing risk management for an aging world. Geneva Risk Insur Rev 43(2):115–136CrossRefGoogle Scholar
  17. Mitchell OS, McCarthy D (2002) Annuities for an ageing world. National Bureau of Economic Research, Cambridge, MACrossRefGoogle Scholar
  18. Mitchell OS, Piggott J, Sherris M, Yow S (2006) Financial innovation for an aging world. National Bureau of Economic Research, Cambridge, MACrossRefGoogle Scholar
  19. Modigliani F (1986) Life cycle, individual thrift, and the wealth of nations. Science 234(4777):704–712CrossRefGoogle Scholar
  20. Modigliani F, Brumberg R (1954) Utility analysis and the consumption function: an interpretation of cross-section data. In: Post-Keynesian economics. Rutgers University Press, New Brunswick, pp 388–436Google Scholar
  21. OECD (2016) Life annuity products and their guarantees. OECD Publishing, ParisCrossRefGoogle Scholar
  22. Ramsay C, Oguledo V (2018) The annuity puzzle and an outline of its solution. North Am Actuar J 22(4):623–645CrossRefGoogle Scholar
  23. Thurley D (2015) Pensions: annuities. UK House of Commons Library Note SN 65552Google Scholar
  24. Trenerry C (1926) The origin and early history of insurance: including the contract of bottomry. The Lawbook Exchange, London, United KingdomGoogle Scholar
  25. United Nations (2017) World population ageing 2017 – highlights. Department of Economic and Social Affairs, Population Division, United Nations, New YorkGoogle Scholar
  26. Weinert J, Gründl H (2016) The modern tontine: an innovative instrument for longevity risk management in an aging society. ICIR working paper series no 22/2016Google Scholar
  27. Yaari M (1965) Uncertain lifetime, life insurance, and the theory of the consumer. Rev Econ Stud 32(2):137–150CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Independent ResearcherJohannesburgSouth Africa

Section editors and affiliations

  • Bernardo L. Queiroz
    • 1
  1. 1.Department of DemographyUniversidade Federal de Minas GeraisBelo HorizonteBrazil