A coherent approach to annuitisation

Which of the following two financial instruments would you prefer if you were a 65-year old (or her fi nancial advisor, or insurance company):

(A) A regular risk-free bond with a current value of 100.000 EUR;
(B) An annuity which pays off nothing in the very unlikely case (say with probability of 1%) that you die this year; whose discounted pay-off s represent 100.000 EUR if you die between now and twenty years (89%); and if you happen to live longer the equivalent of a present value of 500.000 EUR?

Secondly, would you prefer

(C) An insurance policy which pays 100.000 EUR either in the very unlikely case that you die in the next year or in the rather unlikely event you live beyond 85, and nothing otherwise;
(D) A longevity annuity which pays off the equivalent of 500.000 EUR but only if you live beyond 85, and nothing otherwise?

The reader will, of course, have recognised the paradox of Allais (1953); the Allais choice A&D contradicts expected utility theory because if you prefer A to B , U(100) > 0, 1U(500) + 0,89U(100)  which is equivalent to 0,11U(100) > 0,1U(500) , and thus you should prefer C to D and vice versa. The paradox remarkably well captures consumer behaviour in retirement planning: people are averse to immediate-start annuities but take to the lotteryticket-like longevity annuity.

Coherent decision-making

This entry makes use of a unifying approach to rational decision-making in the social sciences, based on the concepts of imprecise probabilities and coherent previsions. A lower prevision P_(X) for a transaction X is the supremum price m for which it is asserted that the gamble X – m is desirable to you. The upper prevision P^(X) = -P_(-X); it is the minimum price m for which m - X is desirable. Only if the upper and lower prevision coincide, can a fair price be defined. The set of statements must be coherent in the sense that no combination of those statements can be undesirable to her; in particular, no arbitrage opportunity may arise implying a sure loss. Invoking coherence forms the basis of many fundamental principles of inference in economics, finance or game theory.

Crucially, coherence can extend beyond an individual decision-maker. When two or more agents form a market, the gambles they consider desirable, and dually, the convex constraints on their personal probability assessments interact. By communicating their beliefs and preferences and engaging in transactions, the agents will exploit arbitrage opportunities, exchange information, redistribute wealth, share risk, hedge their bets, exhibit herding, update their previsions. The market (or its representative agent) will typically be more complete than each of the individual agents participating.

We suggest that the intersubjective decision theory may aid in understanding and, possibly, resolving the annuity puzzle in retirement planning for the babyboomers. A rephrasing of the well-known Allais paradox remarkably well captures consumer behaviour in retirement planning: people are averse to immediate-start annuities but take to the lottery-ticket-like longevity annuity.

Annuities are perceived as risky gambles on the outcome that the retiree lives longer than expected rather than as a longevity insurance. As a result, demand in the market for actuarially fairly priced annuities is wanting. The framework here transcends the often contradictory theories that have been put forward to explain human behaviour. It focuses on the real agents using their limited knowledge to make decisions which are continually challenged by their environment. The decision whether to annuitise can consequently best be framed by confronting the consumer with the implications for her other assets and liabilities. In so doing, rationality emerges as a social concept.

The annuity puzzle

An actuarial note or annuity is a note which stays on the books until the consumer dies, at which time it is automatically cancelled. Selling an annuity corresponds then to obtaining a life-insured loan, and taking out life insurance is equivalent to purchasing regular bonds and simultaneously selling actuarial notes for the same amount. The di fference in interest between what one has to pay on the actuarial notes and what one receives on the bond is the insurance premium.

Menahem Yaari (1965) has shown that in the general case annuities and life insurance allow the consumer to optimally balance his marginal utility for consumption with the marginal utility of bequests, e ffectively separating the consumption decision from the bequest decision. The case that concerns us here is when there are no “loved dependents”.

The heart of the argument can be gleaned from the simple setting where only one future period is considered together with two securities: a bond returning B units of consumption whether the consumer is alive or not, and an annuity that returns A in the next period if the consumer is alive and nothing otherwise. The following result is readily derived from coherence:

If there is no a bequest motive, and the (lower) probability of dying is strictly positive, then:

(1) A is strictly larger than B;
(2) The consumer fully annuitises.

We refer to Davido , Brown, and Diamond (2005) for an extension of the argument to a multiperiod setting.

If markets are incomplete and some consumption patterns are impossible to implement, axiomatic decision theory can explain why full annuitisation is no longer appropriate, although an extensive simulation literature has established that the optimal proportion of annuitisation is still quite high. In reality most retirement portfolios show a near universal lack of annuitisation, a phenomenon that is dubbed the annuity puzzle. It is generally acknowledged that solving the puzzle calls for “psychological or behavioral considerations at play in the market for life-contingent products that have not yet been incorporated into standard economic models” according to Davido , Brown, and Diamond (2005).

A game of life and death

An entire zoo of anomalies has been identi ed to explain consumer behaviour: Hu & Scott (2007) mention mental accounting, cumulative prospect theory, the availability heuristic, money illusion and hyperbolic discounting, the conjunction fallacy, and ambiguity aversion. Each of these “heuristics and biases” may indeed be relevant, in fact: in the next paragraph we will add another to the list, which we feel most concisely captures the crux of the matter. But we should heed the admonition of de Finetti (1937, our translation) that “we are sometimes led to make a statement with a purely subjective meaning, and to do so is quite legitimate; but, when one subsequently seeks to replace it with something objective, one does not make progress but a mistake”. An intersubjective decision framework based on coherence transcends the often contradictory theories that have been put forward to explain human behaviour. It focuses on the mechanism with which these motivations are translated into individual actions kept in check by our social environment.

Annuities are perceived as risky gambles on the outcome that the retiree lives longer than expected rather than as a longevity insurance. As a result, demand in the market for actuarially fairly priced annuities is wanting. Such behaviour is all the more unexpected since these annuities are based on “objectively true” frequencies. People are not objective in gauging their own life expectancy: they tend to overestimate the probability of dying prematurely (and at the same time overestimate the probability of living to a very old age). The departure from fair price is further compounded by mis-discounting.

Suppose that you attach a non-negative weight 1-w close to 1 to the life expectancy P_obj(X) quoted in your social security mortality table and you form your own complementary estimate P(X) to come up with a weighted estimate (1-w)P_obj(X)+wP(X). If you feel completely ignorant, meaning that any prevision can take the place of P, the resulting coherent lower prevision will be the linear-vacuous mixture of Walley (1991, 2.9.2) P_(X) = (1-w)P_obj(X)+w inf X which will be smaller than P_obj(X). Of course, you can resort to a quantitative model of human behaviour to constrain the set M(P) in Walley; Hu & Scott (2007) for instance derive “maximum acceptable prices” for various annuities from cumulative prospect theory. Alternatively, M(P) can result from eliciting an estimate for your remaining life from all of your acquaintances…

Coherence by paradox

Which of the following two financial instruments would you prefer if you were a 65-year old (or her fi nancial advisor, or insurance company):

(A) A regular risk-free bond with a current value of 100.000 EUR;
(B) An annuity which pays off nothing in the very unlikely case (say with probability of 1%) that you die this year; whose discounted pay-off s represent 100.000 EUR if you die between now and twenty years (89%); and if you happen to live longer the equivalent of a present value of 500.000 EUR?

Secondly, would you prefer

(C) An insurance policy which pays 100.000 EUR either in the very unlikely case that you die in the next year or in the rather unlikely event you live beyond 85, and nothing otherwise;
(D) A longevity annuity which pays off the equivalent of 500.000 EUR but only if you live beyond 85, and nothing otherwise?

The reader will, of course, have recognised the paradox of Allais (1953); the Allais choice A&D contradicts expected utility theory because if you prefer A to B , U(100) > 0, 1U(500) + 0,89U(100) , 0,11U(100) > 0,1U(500) , you should prefer C to D and vice versa. The paradox remarkably well captures consumer behaviour in retirement planning: people are averse to immediate-start annuities but take to the lotteryticket-like longevity annuity. Probabilities and preferences are assessed quite diff erently “in the vicinity of certainty” (Allais) compared to in a world where winning is a long shot anyway, the case in point of mental accounting.
It is interesting that the overwhelming majority of (immediate-start) annuities actually sold are a combination of A&D: the life-with-period-certain annuity provides a number of guaranteed payments plus a longevity annuity commencing after the guarantee period. Irrespective of any bequest motive, the bond component appears to off er downside protection by eliminating the early-death “generalised scenarios” in M(P), eff ectively increasing the coherent lower prevision we constructed in the previous paragraph!

We suggest that the annuity puzzle can at least be mitigated, if not resolved by invoking coherence. After all, annuities are meant to relieve the retiree of having to solve her intertemporal consumption/investment decision over an unknown time horizon, i.e. annuities by defi nition must be “coherent” with the remainder of her portfolio. The decision whether to annuitise can consequently best be framed by confronting the consumer with the implications for her other assets and liabilities. More speculatively, insurers may induce demand for annuities by quoting offer prices away from actuarial fairness while balancing their book by quoting appropriate prices for products involving short positions in annuities, in particular life insurance. (We ignore administrative costs and pro fit margins.) Note that consumers cannot easily turn suppliers, severely obstructing exploitation of any arbitrage opportunity they may perceive for individual products.

We do not agree with Hu & Scott (2007) that “exploiting behavioral anomalies would be an ignoble way to induce annuity demand” if only because we fi nd it difficult to consider consistent behaviour on the scale witnessed in the life-changing context of retirement planning an anomaly. Is it more ignoble to adapt nancial instruments to the way in which humans actually make decisions and forecasts, rather than to submit to a questionable but convenient set of “normative” axioms? Coherence does not exclude “unfair prices” of individual products (or individual consumers). Natural extension only requires consistency among probability assessments to avoid sure loss; rationality is a social concept.

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