SURVIVAL PESSIMISM AND THE DEMAND FOR ANNUITIES
Cormac O’Dea and David Sturrock*
Abstract—The “annuity puzzle” refers to the fact that annuities are rarely
purchased despite the longevity insurance they provide. Most explanations
for this puzzle assume that individuals have accurate expectations about
their future survival. We provide evidence that individuals misperceive their
mortality risk and study the demand for annuities in a setting where annu-
ities are priced by insurers on the basis of objectively-measured survival
probabilities but in which individuals make purchasing decisions based on
their own subjective survival probabilities. Subjective expectations have the
capacity to explain significant rates of nonannuitization, yielding a quanti-
tatively important explanation for the annuity puzzle.
IO.
introduzione
ANNUITIES insure individuals against longevity risk by
allowing them to exchange wealth for an income stream
guaranteed until death. Theory predicts that under general
conditions, risk-averse individuals will purchase a fairly-
priced annuity (Yaari, 1965; Davidoff, Brown, & Diamond,
2005). Few households, Tuttavia, ever purchase an annuity.1
This divergence between theory and experience has become
known as the “annuity puzzle.”
Most of the explanations that have been proposed for
this puzzle2 attempt to rationalize nonpurchase by individ-
uals who are assumed to have accurate perceptions of their
survival probabilities. Individuals are not, Tuttavia, BENE-
Received for publication May 31, 2019. Revision accepted for publication
Febbraio 22, 2021. Editor: Shachar Kariv.
∗O’Dea (corresponding author): Yale University, NBER and Institute for
Fiscal Studies; Sturrock (corresponding author): University College London
and Institute for Fiscal Studies.
Funding from the IFS Retirement Savings Consortium (2016–2018) È
gratefully acknowledged. The group comprised Age UK, Association of
British Insurers, Chartered Insurance Institute, Department for Work and
Pensions, HM Revenue and Customs, HM Treasury, Investment Associa-
zione, Legal and General Investment Management, Money Advice Service,
and Tax Incentivised Savings Association. We are grateful to representa-
tives of these groups for comments through the funding period. Co-funding
from the Economic and Social Research Council (through a Knowledge Ex-
change Grant, through grants ES/N011872/1, ES/P001831/1 and through
the ESRC Centre for the Microeconomic Analysis of Public Policy (CPP)
(ES/M010147/1)) is gratefully acknowledged. We thank participants at a
seminar at the Oxford Institute for Population Ageing and to participants at
several conferences. We are also grateful to Joseph Altonji, James Banks,
Jochem de Bresser, Agar Brugiavini, Rowena Crawford, Carl Emmerson,
Eric French, Costas Meghir, John Eric Humphries, Gemma Tetlow, Lind-
sey Uniat, Guglielmo Weber, Ebonya Washington, Basit Zafar, e due
anonymous referees for helpful comments. Any errors are our own.
A supplemental appendix is available online at https://doi.org/10.1162/
rest_a_01048.
1Lockwood (2012) reports that less than 5% of a sample of single retirees
in the United States own an annuity; Inkmann, Lopes, and Michaelides
(2011) show that only 6% of older households in the UK voluntarily pur-
chase an annuity.
2These explanations include adverse selection (Brugiavini, 1993; Finkel-
stein & Poterba, 2004, 2014), bequest motives (Lockwood, 2012; Gan et al.,
2015), precautionary saving for medical and long-term care expenses (Re-
ichling & Smetters, 2015; Ameriks et al., 2011), existing annuity provision
from social security income (Pashchenko, 2013), cognitive limitations on
individuals’ abilities to value annuities (Brown et al., 2017), and costs of ad-
ministration (Mitchell, Poterba, & Warshawsky, 1999). See Brown (2007)
for a general review of this literature.
informed about their survival probabilities (Hurd & McGarry,
1995; Elder, 2013; Wu, Stevens, & Thorp, 2015).
We study the demand for annuities in a setting where those
annuities are priced by insurers on the basis of objectively-
assessed survival probabilities but in which individuals make
purchasing decisions on the basis of their own subjective sur-
vival probabilities. We estimate subjective survival curves
for a sample of older individuals using directly-measured
expectations. Consistent with an established literature (Vedere,
per esempio., Hurd & McGarry, 1995; Elder, 2013; Wu, Stevens, &
Thorp, 2015), our study finds that, on average, individu-
als underestimate their probability of survival through their
50S, 60S, and 70s and overestimate their chances of survival
through their late 80s and beyond. Overall, pessimism dom-
inates, and most respondents would perceive an annuity that
is priced fairly from an actuarial point of view as one which is
unfairly-priced.
As with all insurance products, individuals might, depend-
ing on their preferences, still purchase an annuity that is
unfairly-priced as the longevity insurance provided by the
annuity might be worth the low apparent (to them) “money’s
worth.” Whether survival pessimism is an important driver
of the demand for annuities by risk-averse individuals is an
open question. To assess the quantitative importance of sur-
vival pessimism for annuity purchases, we embed these sub-
jective survival curves in a lifecycle model of consumption,
saving, and annuitization. We estimate the proportion of their
wealth that individuals would choose to annuitize, given their
idiosyncratic subjective survival curves. Parameterizing our
model with plausible levels of risk aversion and patience,
we are able to explain high rates of nonannuitization. Nel
setting of Yaari (1965), patient individuals with “objective”
expectations about their own survival will choose to annu-
itize their entire stock of wealth when offered an actuarially
fair annuity. We find that, when behaving according to their
“subjective” expectations, the average rate of annuitization
for such individuals would be between 42% E 64%, for a
plausible range of levels of risk aversion. To benchmark the
quantitative importance of this channel, we compare these
results to those we obtain by introducing into our model ac-
tuarially unfair pricing caused by adverse selection (or trans-
action costs or other market imperfections), a leading ratio-
nalization of low annuity demand. In questo caso, the average
share of wealth annuitized would range from 34% A 69% for
the same range of levels of risk aversion. We therefore find
that survival pessimism is quantitatively as important as the
higher prices caused by adverse selection.
This result does not depend on other explanations that have
been given for nonannuitization; in our model, individuals
have only modest social security income, do not have be-
quest motives, face no medical cost risk, do not have access
to means-tested income floors and annuities are priced fairly
The Review of Economics and Statistics, Marzo 2023, 105(2): 442–457
© 2021 The President and Fellows of Harvard College and the Massachusetts Institute of Technology. Published under a Creative Commons Attribution 4.0
Internazionale (CC BY 4.0) licenza.
https://doi.org/10.1162/rest_a_01048
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SURVIVAL PESSIMISM AND THE DEMAND FOR ANNUITIES
443
given objectively-measured survival rates. The difference
between “objective” and individual-specific “subjective”
survival curves is large enough, for many individuals, A
outweigh the insurance value of annuitizing much of their
retirement wealth.
Subjective expectations of survival have been shown to be
empirically important in explaining a number of economic
decisions. Hurd, Smith, and Zissimopoulos (2004) find that
those with particularly low expectations of survival are more
likely to retire earlier and to claim Social Security bene-
fits earlier; de Bresser (2020) studies a similar phenomenon
and shows that bringing individual-level variation in survival
probabilities into a lifecycle model can help explain the tim-
ing of retirement and benefit claiming. Bloom et al. (2006)
find that a higher subjective probability of survival is as-
sociated with higher wealth levels. Gan et al. (2015) find
that a model of wealth decumulation and bequests includ-
ing subjective survival expectations better fits decumulation
and bequest behavior than does one with life table survival
probabilities. Heimer, Myrseth, and Schoenle (2019) solve a
lifecycle model with subjective mortality beliefs and show
that “pessimism” about survival to older age, combined with
“optimism” at the oldest ages can explain both undersaving
for retirement and slow decumulation of wealth at the end of
life. None of these papers considers an annuitization choice,
as we do.
There is evidence of a correlation between individual
longevity—realized as well as expected—and decisions
around annuitization. Examining the voluntary market for
annuities in the United Kingdom, Finkelstein and Poterba
(2004) find a positive association between ex-post survival
and features of annuities purchased (per esempio., those who buy back-
loaded annuities are longer-lived). Teppa and Lafourcade
(2013) find that stated optimism around survival is positively
correlated with stated demand for annuities while Inkmann
et al. (2011) find a similar link between subjective expec-
tations and annuity purchases.3 That paper also experiments
with subjective survival probabilities in a lifecycle model and
shows that if subjective survival probabilities are reduced by
10% at each age (a quantity that is not empirically grounded),
no households would demand an annuity. Wu et al. (2015)
embed estimated subjective survival curves within a lifecycle
model of consumption and savings. While they make calcu-
lations of the perceived money’s worth of annuities (finding
that annuities are perceived to offer less than actuarially fair
value, on average), their model does not contain an annuiti-
zation choice. The contribution of our paper is to study the
importance of the observed divergence between reported sur-
vival expectations and objective survival rates—the “survival
pessimism” discussed above—for the annuitization decision.
Given that sufficiently risk-averse agents may choose to buy
unfairly priced insurance products in preference to remaining
uninsured, we do this by combining subjective expectations
3Tuttavia, Brown (2001) finds no evidence of this phenomenon.
data with an economic model of consumption, savings and
annuitization.
We proceed as follows. Section II outlines the data. Sez-
tion III compares average reported survival expectations to
official life tables and sets out our method for constructing
‘subjective’ survival curves from stated beliefs. In section
IV, we outline the model of annuitization and the impact of
introducing subjective survival curves on predicted rates of
annuitization. Section V concludes.
II. Data
We draw on data from the English Longitudinal Study of
Ageing (ELSA) (Marmot et al., 2017), a biennial panel rep-
resentative of the English household population aged 50 E
above. ELSA is part of a network of longitudinal aging stud-
ies around the world, modeled on the U.S. Health and Retire-
ment Study (HRS). One module of the survey asks individuals
about their expectations that certain events will happen in the
future, including whether or not they will leave an inheri-
tance, whether they will still be in work at a certain age and
whether at some point in the future they will not have enough
resources to meet their financial needs. This battery of ques-
tions opens with the following statement: “Now I have some
questions about how likely you think various events might be.
When I ask a question I’d like you to give me a number from
0 A 100, Dove 0 means that you think there is absolutely no
chance an event will happen, E 100 means that you think
the event is absolutely certain to happen.”
As part of this module, individuals are asked a question of
the form “What are the chances that you will live to be age X
or more?", where the age X depends on the current age of the
respondent. All individuals aged 65 and under are asked about
survival to age 75. Those aged 66 or older are asked about the
age which is between 11 E 15 years ahead of their current
age and is a multiple of 5. Per esempio, those aged 75–79
are asked about survival to age 90. Additionally, from wave 3
onwards, all individuals aged under 70 were asked a second
question about survival to age 85. We denote individual i’s
reported probability of survival to age α as Ri(α).
Over the first seven waves of ELSA, 16,345 unique indi-
viduals are asked one or more survival questions in 67,201
separate interviews. In all of our analysis, unless otherwise
stated, we weight observations by the cross-sectional weights
available in the ELSA data.
UN. Evaluating the Content of Subjective Reports
Before using individual responses to survival probability
questions in analysis, we wish to assess whether individuals
appear to understand the meaning of these questions and to
be able to engage with the probabilistic concepts involved.
Prossimo, assuming that participants understand these questions,
we would like to establish, as far as possible, whether an-
swers constitute considered, reflective judgements that might
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444
THE REVIEW OF ECONOMICS AND STATISTICS
FIGURE 1.—DISTRIBUTION OF ANSWERS GIVEN TO FIRST SURVIVAL QUESTION BY AGE-GROUP
Fonte: ELSA waves 1–7. 66,210 answers from 16,345 unique individuals.
plausibly guide behavior, or are instead picked with little prior
thought.
In just 1.5% of interviews, individuals answer “do not
know” to one or more survival probability questions, sug-
gesting a willingness to answer in almost all cases. Figura 1
shows the distribution of reported survival probabilities for
the full sample of first questions asked, in bins of 10 per cento-
age points. We split the sample into those aged below 65 E
those aged 65 and above, with the younger group much more
likely to report high chances of survival.
Some individuals answer “0%” or “100%” to the survival
questions (5.3% E 6.8%, rispettivamente). Both of these an-
swers might be evidence of a lack of understanding of the
question. Tuttavia, neither is conclusively so; respondents
are asked to report probabilities on a discrete scale, and so
rounding, combined with a terminal diagnosis or extreme
optimism, rispettivamente, could rationalize these answers. Noi
include these individuals in the sample, but show in appendix
B.3 that our results change very little if we exclude either or
both of these groups.
When individuals are asked two survival questions, Essi
can report a higher chance of survival to the older age than to
the younger age. This happens in 8.3% of interviews. Given
that such responses indicate a fundamental misunderstanding
of the question, we remove these individuals from all of the
remaining analysis.
One may have reservations about the fact that a high
proportion—20.5%—of answers are “50%.” There could be
a concern that individuals pick this focal answer when want-
ing to give a response but not understanding the question. Noi
assess this by examining these individuals’ answers to other
probability questions. Those individuals who answer “50%”
almost always give a range of answers to other questions
and are no more likely to answer “50%” to other probability
questions than are the rest of the sample (appendix A.1 gives
further details). Of the 16,345 individuals who answered one
or more survival questions, only 41 individuals (0.2%) an-
swered “50%” to all survival questions in all waves. On the
basis of this evidence, we retain answers of “50%” in our main
sample but show in appendix B.3 the (minimal) sensitivity of
our results to their removal.
Given that the overwhelming majority of individuals give
answers that do not indicate a lack of understanding of prob-
abilities, we perform four further tests aimed at assessing the
informational content of responses and whether they relate to
economic behaviour. Firstly, we find that responses are corre-
lated with known mortality risk factors (per esempio., smoking, drink-
ing, and health conditions) in a way that is consistent with
existing evidence. Per esempio, current smokers report a 6–8
percentage points lower probability of survival over the 11–
15 year horizon, relative to current nonsmokers.4 Secondly,
using the panel nature of the survey we find that reports ‘up-
date’ over time in response to news relevant to mortality such
as diagnoses of new health conditions. Per esempio, a new
cancer diagnosis was associated with a 4 percentage point re-
duction in the stated probability of surviving to an age 11 A
15 years ahead.5 Thirdly, exploiting a link to administrative
death records from the English National Health Service, we
4This result is obtained using linear regression of reported probability on
smoker status, controlling for a range of other risk factors, demographic
variables, health conditions and self-reported health. Full details are given
in appendix A.2.
5This result is from a linear fixed effects regression of reported sur-
vival probability on a range of variables for whether individuals have been
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SURVIVAL PESSIMISM AND THE DEMAND FOR ANNUITIES
445
find that reported expectations are correlated with actual sub-
sequent mortality over a 10-year horizon.6 While it is possible
that individuals’ answers to survival expectations questions
could represent their “actual” expectations even if there was
no association with the above outcomes, these findings pro-
vide additional evidence that answers represent meaningful,
reflective judgements.7 Fourth, in line with expectations be-
ing drivers of economic behavior, we find that stated survival
expectations are negatively correlated with purchases of life
insurance, a product analogous to selling an annuity.8
III. Assessing the Accuracy of Subjective
Expectations of Survival
In this section, we describe the patterns in subjective re-
ports, compare them to actual mortality rates and projec-
tions and derive idiosyncratic survival curves that will be
used, together with our model, to evaluate the importance of
these curves for the annuitization decision. The results that
we find—that individuals are mostly pessimistic about sur-
vival to younger ages and optimistic regarding survival to
older ages—are consistent with a well-established literature
analyzing the accuracy of self-reported survival probabilities
and self-reported life expectancy across a number of coun-
tries and in a variety of survey settings. We give a brief review
of this literature before proceeding.
In the first studies comparing subjective expectations to an
objective benchmark, Hurd and McGarry (1995) and Hurd
and McGarry (2002) analysed the survival expectations data
available in the first wave and first two waves, rispettivamente,
of the HRS and compared it to the period life tables avail-
able at that time. These studies concluded that while there
was some evidence that men underestimated their chances of
survival to age 75 relative to life tables (with women approxi-
mately accurate) and that women overestimated their chances
of survival to age 85 relative to life tables (with men approxi-
mately accurate), mean expectations were broadly consistent
with the period life tables. Tuttavia, subsequent research us-
ing the HRS drew upon more expectations data and, due to
the passage of time, was able to compare stated expectations
to subsequent survival of the sample. Elder (2013) compared
reported survival probabilities from the first waves of data
in the HRS and AHEAD surveys to the respondents’ actual
diagnosed with particular health conditions, as well as other risk factors and
demographic variables. Full details are given in appendix A.2.
6Full details are given in appendix A.2.
7Hurd and McGarry (2002) make a similar point with respect to subjective
expectations in the Health and Retirement Survey.
8This result is from a linear regression in which the outcome is the per-
centage of individuals’ total wealth portfolio held as life insurance. Stated
expectations are negatively correlated with this outcome. Full details are
given in appendix A.2. A lifecycle model with some positive weight placed
on heirs will predict a ceteris paribus negative relationship between subjec-
tive expectations of own survival and life insurance demand. Quantifying
the full implications of survival pessimism for the level of this demand
depends on the form and strength of the motive for leaving bequests. IL
implications of survival pessimism for life insurance demand is an interest-
ing avenue for future research.
subsequent survival to the age they were asked about. Doing
so revealed a substantial (Sopra 10 percentage points) under-
estimation of survival probabilities, on average, for those in
their early 60s (who were asked about survival to age 75), UN
mild underestimation by those in their early 70s (who were
asked about survival to age 85) and growing optimism about
survival for those aged over 75 (who were asked about ages
between 11 E 15 years ahead of their current age). Ludwig
and Zimper (2013) found the same patterns in waves 5 A 7 Di
the HRS, when using Human Mortality Database and Social
Security Administration (SSA) life tables. Grevenbrock et al.
(2020) confirm these patterns in waves 8 A 12 of the HRS,
comparing subjective reports to estimated objective survival
probabilities. Heimer et al. (2019) use the 2014 wave of the
HRS, the Survey of Consumer Finances (SCF) and a survey
of their own to document further evidence that U.S. seniors
underestimate their near-term survival up until the age of
around 70 (by between 15 E 20 percentage points), after
which they become gradually more optimistic about survival.
The pattern of substantial pessimism about survival
through the 50s and 60s and early 70s, turning to relative
optimism about survival to the oldest ages, that is found in
NOI. surveys is also present in a variety of surveys across
a range of other countries. These include the Netherlands
(Teppa & Lafourcade, 2013), Australia (Wu et al., 2015), E
Germany (Bucher-Koenen & Kluth, 2013). Hurd, Rohwed-
der, and Winter (2005) document similar patterns across a
number of European countries using the Survey for Health,
Ageing and Retirement in Europe. Boyer et al. (2020), ask-
ing only about survival expectations to 85, find evidence of
optimism about survival to that age in a survey of Canadians
aged 55 A 75.
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UN. Comparing Reports to Actual Mortality Data
The UK Office for National Statistics (ONS) life tables
contain actual and projected mortality data for the England
and Wales population by sex and year of birth. These tables
would be a natural benchmark against which to assess sub-
jective expectations if the ELSA sample were representative
of the whole English population.9 As ELSA is representa-
tive only of the noninstitutionalized population (meaning that
those in residential care, per esempio., are excluded), mortality rates
for the ELSA population are slightly lower than given by the
ONS life tables.10 We use administrative death records linked
to ELSA to “rescale” the data in the ONS life tables by the
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9While ELSA includes only English residents, ONS cohort life tables
are available only for England and Wales combined and not England-only,
for the cohorts we analyze. Tuttavia, this will make little difference to our
analysis as Wales makes up around 6% of the England and Wales population
and has very similar mortality patterns.
10The ELSA data in linked to administrative death records giving the date
of death for any ELSA respondent who died on or before February 2013.
Amongst those aged 50 A 70 in ELSA wave 1, the actual mortality rate
over the period until February 2013 (approximately a 10-year period) era
10.5%. The average ONS life table implied death rate over that period for
the same sample was 13.0%.
446
THE REVIEW OF ECONOMICS AND STATISTICS
FIGURE 2.—COMPARISON OF ONS COHORT SURVIVAL CURVE AND “SCALED” SURVIVAL CURVE FOR MEN (LHS) AND WOMEN (RHS) AGED 60 AND BORN 1950
Fonte: ELSA waves 1–7 and ONS 2014-based cohort life tables for England and Wales.
observed difference in average mortality rates between the
ELSA sample and those implied by the ONS life tables. In
appendix B.1, we show that our main results are somewhat
attenuated, but qualitatively unchanged, if we use the ONS
life tables without rescaling them.
ELSA is linked to administrative death records such that we
know if any individual (including those who attrit) has died
up until February 2013. We use this information to “rescale”
the official life tables in the following way. We calculate, for
each year of age, the actual mortality hazard rate observed in
the ELSA sample and the expected mortality hazard rate if
each individual faced the hazard rate implied by the ONS life
table for their sex and year of birth. For each sex, we fit a cubic
in age to actual hazard rates using OLS and calculate the ratio
of the fitted hazard rate to the ONS hazard rate at each age. Noi
take the mean ratio across ages and use this to rescale the haz-
ard rates underlying the ONS survival curves for each sex and
year-of-birth, yielding a set of “scaled” ONS survival curves.
Our method yields hazard rates for men and women that are
71% E 69% of their original level, respectively—that is,
those in the ELSA sample have lower mortality probabilities
(and therefore higher life expectancies) than the population
at large. Figura 2 shows a comparison of the original ONS
survival curve with the “scaled” survival curve for 60-year
old men and women, born in 1950. We see that, for exam-
ple, men are estimated to have a 50% chance of survival to
age 90, in comparison to the 38% figure in the ONS life ta-
ble. The corresponding figures for women are 60% E 48%,
rispettivamente.
We use the “scaled” ONS survival curves life tables as
an “objective” benchmark to assess whether particular age-
sex-cohort groups have positively biased (“optimistic”) O
negatively biased (“pessimistic”) expectations of survival to
various “target ages” (cioè., the age about which the individual
is asked the question). We conduct this analysis at the most
granular level for which these comparisons are possible: we
calculate for each combination of age in years, year of birth,
sex, and target age, the average reported survival probabil-
ità, and compare this to the relevant scaled life table proba-
bility. A clear pattern emerges. Individuals are, on average,
“pessimistic” about their chances of survival to ages 75, 80,
85, E 90 and then become increasingly optimistic as they
get older and are asked about survival to older ages. While
the degrees of “optimism” and “pessimism” vary slightly be-
tween cohorts, these patterns are consistently found across
those born in the 1920s through to the 1950s. Comparing
men and women, we see that women tend to be slightly more
pessimistic on average, than men.
Figura 3 illustrates the comparison between mean subjec-
tive survival probabilities and scaled life tables for men and
women born in the 1930s. We see that individuals in their
early 60s underestimate survival to age 75 by around 25 A
30 percentage points while those in their late 60s and early
70s underestimate survival to age 85 by 15 A 20 percentage
points. Turning to those in their late 80s, we see that they
are close to accurate about their probability of survival, SU
average.11
These findings are in line with those in the existing litera-
ture, including Elder (2013) and Heimer et al. (2019), Quale
establish in a number of settings the pattern of overestima-
tion of mortality hazard rates at ages until around the mid-80s,
with underestimation of mortality rates at older ages.
B. Constructing Subjective Survival Curves
In this section, we describe how we use stated survival
expectations to estimate the individual-specific subjective
11Note that in figure 3, each “subjective” data point corresponds to an
average over respondents with different birth years. The “objective” data
points are constructed by weighting the corresponding scaled life table
survival probabilities according to the proportions of individuals with each
birth year in the sample.
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SURVIVAL PESSIMISM AND THE DEMAND FOR ANNUITIES
447
FIGURE 3.—COMPARISON OF MEAN “SUBJECTIVE” REPORTS AND SCALED ONS COHORT SURVIVAL RATES/PROJECTIONS FOR MEN (LHS) AND WOMEN (RHS)
BORN 1930–1939
Different colored series correspond to different ages about which respondents are asked questions.
Fonte: ELSA waves 1–7 and ONS 2014-based cohort life tables for England and Wales.
survival curves that will be used in our life-cycle model. There
are an infinite number of possible survival curves consistent
with the answers that individuals give to the survival ques-
zioni. We are therefore required to make further assumptions
if we are to infer individuals’ subjective survival curves from
their reports about their survival to specific ages. We make an
assumption on the functional form of individuals’ subjective
survival curves. Specifically, we assume that subjective sur-
vival probabilities follow a Weibull distribution. The Weibull
distribution is widely used in the epidemiological literature
and for modeling aging processes generally.12
The Weibull distribution is a two-parameter (λi, ki) distri-
bution defined in the following way. Person i with age z, ha
probability of surviving to at least age α:
(cid:3)
(cid:5)
(cid:2)
Si(α) = exp
−
: λi, ki > 0.
(1)
(cid:4)
ki
α − z
λi
We estimate subjective survival curves for all individuals
in our sample who answered two survival questions (cioè., Tutto
those aged under 70 who answered two questions and were
not removed from the sample due to giving “impossible”
answers). We make one additional weak assumption—that
individuals believe that they are almost certain not to live
beyond age 110—by including the relevant scaled life table
survival probability for each individual for target age 110 COME
a third “report” (we denote this third subjective “report” by
Ri(110)). We fit the individual’s Weibull-distributed subjec-
tive survival curve by estimating λi and ki using these three
reports and nonlinear least squares. Questo è, denoting the set
12See Bissonnette, Hurd, and Michaud (2017) for an example of use of the
Weibull distribution to construct objective and subjective survival curves
using data from the HRS. The authors report that their results are similar
when using the Weibull or the (also widely used) Gomertz distribution.
Di 3 ages for which we have subjective reports by Ai, we
choose the parameter vector (ˆλi, ˆki) that satisfies
(cid:7)
(cid:4)
ki
(cid:5)(cid:8)
(cid:6)
(cid:3)
(cid:2)
2
(ˆλi, ˆki) = arg min
Ri(α) − exp
−
.
(2)
α − z
λi
λi,ki
α∈Ai
Figura 4 illustrates the curve-fitting procedure applied to
the median responses from men and women born in the 1940s
and compares these subjective survival curves to the relevant
scaled life table survival curves. The subjective curve implies
that at age 60, this group of individuals is pessimistic about
survival to all ages up until around age 100 and optimistic
about survival to ages beyond this. The subjective life ex-
pectancy measures implied by these curves are 8.6 E 9.6
years lower than the life expectancies calculated using life
table survival curves for men and women, rispettivamente. Questo
is equivalent to life expectancies at the age of 60 that are 31%
(for men) E 33% (for women) lower than those implied by
life tables, on average.
Average overall pessimism of this magnitude is found
across sexes and cohorts when interviewed in their 50s and
60S. Using the full sample of individuals for whom we have
a subjective survival curve, we can compare “subjective” and
scaled life table life expectancy at the individual level. Sub-
jective life expectancy is lower than scaled life table life ex-
pectancy by 6.1 years, O 22%, amongst men, E 7.7 years,
O 25%, amongst women, on average.
IV.
Subjective Survival Expectations and Annuitization
Annuities provide insurance against longevity risk. IL
decision about how much of one’s wealth to annuitize at a
given price ought to depend on the individual’s assessment
of this longevity risk. Individuals who underestimate their
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448
THE REVIEW OF ECONOMICS AND STATISTICS
FIGURE 4.—COMPARISON OF MEDIAN “SUBJECTIVE” AND “OBJECTIVE” SURVIVAL CURVES FOR MEN (LHS) AND WOMEN (RHS) BORN 1940–49
Fonte: ELSA waves 1–7 and ONS 2014-based cohort life tables for England and Wales.
longevity may perceive an annuity as a worse deal than it
truly is. This is a potential explanation for the unpopularity
of annuities.
Primo, we can assess whether, given their subjective expec-
tations, individuals would perceive an annuity as offering at
least an actuarially “fair” deal. An annuity rate is defined as
actuarially fair with respect to a given discount rate and set of
survival probabilities, if it enables the purchase of a guaran-
teed income stream until death that has expected discounted
value equal to its price. For each individual for whom we
have fitted a subjective survival curve, we calculate the actu-
arially fair annuity rate given their subjective survival curve
and given their scaled life table survival curve. The actuar-
ially fair annuity rate for an individual of age z, given the
survival curve Si(α), is given by
θ =
(cid:2)
110(cid:6)
α=z
(cid:5)−1
,
Si(α)
(1 + R)α−z
(3)
where r is the interest rate.
Figura 5 compares these “subjective” and “objective” an-
nuity rates for our sample assuming a real interest rate of
0% in both cases. Variation in objective annuity rates comes
from variation in gender, age, and year of birth; variation in
the subjective annuity rates comes from the estimated sub-
jective survival curves. 88% of individuals would perceive
an annuity that is priced fairly for the average person of their
age, sex, and cohort as offering a less than fair annuity rate.
An individual who perceives an annuity as being unfairly
priced may, Ovviamente, choose to annuitize some of their
wealth if doing so offers sufficiently large insurance value.
To examine whether survival “pessimism,” and the implied
divergence between subjective and life table-based annuity
rates, could lead to low rates of annuitization of retirement
savings, we specify a model of consumption, saving and an-
nuitization. We compare results for our sample in the case
where individual survival expectations are consistent with
scaled life tables to those where they are consistent with their
subjective survival curve. We account for the fact that indi-
viduals have public pension entitlements, giving them some
already-annuitized income. We use data from ELSA on pri-
vate pension and financial wealth and model the choice of
how much (if any) of this wealth to annuitize at a rate that is
actuarially fair given the individual’s age, sex, and cohort.13
UN. Model
In this section, we outline the model of consumption, sav-
ing and annuitization. Just-retired agents (indexed by i) Avere
initial wealth, ai0, and receive public pension income (state
pension/social security), pi, in each period. In period 0, agents
choose to annuitize some fraction of their wealth. In each
of the following periods, the agents choose how much of
their resources (the sum of their public pension income, their
annuity income and their unannuitized wealth) to consume.
Borrowing is not allowed.
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Individuals make choices consistent with a survival curve –
which we can specify to be either their subjective survival
io (α)). When
io (α)) or their objective survival curve (So
curve (Ss
beliefs are consistent with the subjective survival curve, an
individual of age z believes that they have a probability of sur-
vival to each age α ≤ 110 that is described by their estimated
Weibull survival curve:
io (α) = ex p
Ss
−
.
(4)
(cid:2)
(cid:3)
(cid:5)
(cid:4)ˆki
α − z
ˆλi
13We do not include housing wealth in the measure of wealth which may
be annuitized: appendix B.2 presents a version of our model in which we add
a consumption flow coming from owner-occupied housing. The differences
in annuitization rates between the “objective” and “subjective” cases in this
version of the model are very similar to those in the baseline model.
SURVIVAL PESSIMISM AND THE DEMAND FOR ANNUITIES
449
FIGURE 5.—COMPARISON OF “OBJECTIVE” AND “SUBJECTIVE” BASED ANNUITY RATES
“Subjective” annuity rates are the actuarially fair rate implied by the subjective survival curve constructed from the individual’s responses to the survival expectations questions. “Objective” annuity rates are the
actuarially fair rate implied by the scaled ONS life table survival curve for the individual’s sex, age, and year of birth.
Fonte: ELSA waves 3–7 and ONS 2014-based cohort life tables for England and Wales.
We assume that all individuals believe that they will die at
the end of their 110th year at the latest. Individuals have a
constant relative risk aversion utility function and they dis-
count the future according to a geometric discount rate (β).
They perceive their expected lifetime utility to be
U =
110−z(cid:6)
t=0
βt Sx
io (z + T )
c1−γ
Esso
1 − γ
(5)
for x ∈ {S, o} depending on whether they are assumed to have
subjective or objective survival curves. Time is indexed by t,
where t = 0 is the year in which we observe a household in
the survey, and z is their age in that year.
At time zero, individuals can irreversibly annuitize any
fraction of their wealth, bi ∈ [0,1], at rate θi, which is the actu-
io (α), the objectively-measured
arially fair annuity rate given So
survival curve for someone of their sex, year of birth, and age:
(cid:2)
110(cid:6)
α=z
θi =
(cid:5)−1
.
io (α)
So
(1 + R)α−z
(6)
Their annuity income in each future period is therefore de-
fined as anni = θi · bi · ai0. An individual’s problem is there-
fore to choose their consumption in each remaining period
of life {cit } and the proportion of their initial wealth that they
annuitize (bi):
max
{cit },bi
(cid:9)
s.t.
110−z(cid:6)
βt Si(z + T )
c1−γ
Esso
1 − γ
t=0
ait+1 = (ait + pi + anni − cit )(1 + R)
ait+1 ≥ 0
(7)
(8)
.
(cid:10)
(cid:10)
z + T
= β(1 + R)si
(cid:11)
−γ
C
t+1 where si
Individuals’ optimal choice of consumption is character-
ized by the Euler equation which will bind with equal-
ity whenever the no-borrowing constraint does not hold:
−γ
≡ Si(z + T +
C
T
1)/Si(z + T ). Individuals will be inclined to annuitize more
of their wealth the higher is β (and so the fact that consump-
tion from annuitized wealth cannot be front-loaded does not
imply a substantial welfare cost) and the higher is γ (E
so the longevity insurance provided by annuities is more
valued).
z + T
(cid:11)
We solve the model for each individual in waves 3, 4, E 5
of ELSA who has begun drawing their public pension, holds
positive assets, and for whom we are able to construct a sub-
jective survival curve.14 We use the first observation for any
individuals observed multiple times, yielding 2,848 observa-
zioni. Each individual’s initial level of wealth (ai0) is taken as
the sum of their household private pension wealth and gross
financial wealth.15 We take public pension income (pi) to be
that level reported in the data. For each observation, we solve
the model twice. In one case, the individual’s expectations are
consistent with the scaled life table survival curve for their
sex, year of birth, and age, and in the other, expectations re-
flect their fitted subjective survival curve estimated from their
survey responses.
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14In this period, the male public pension age was 65 and the female state
pension age increased from age 60 to age 62. The public pension age is the
age at which individuals can first claim their state pension. Over 99% Di
individuals begin to claim at this age. We drop 231 individuals over their
public pension age who do not report deferring their state pension, but yet
report a public pension income of zero.
15Pension wealth information is available up until wave 5. Wealth mea-
surements are at the household level. For individuals in a couple, we there-
fore take half of this wealth level.
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THE REVIEW OF ECONOMICS AND STATISTICS
FIGURE 6.—PERCENTAGE OF INDIVIDUALS ANNUITIZING AT EACH PARAMETER COMBINATION
Fonte: Model predictions using ELSA waves 3–5 and ONS 2014-based cohort life tables for England and Wales. 2,848 observations.
B. Results
We illustrate the impact of subjective survival expectations
by comparing the mean rate of annuitization in the case where
individuals behave according to the scaled life table survival
curve for their age, sex, and cohort with the case where they
behave according to their own subjective survival curve. Noi
define the mean rate of annuitization as the simple mean of
N
individual rates, 1
i=1 bi. In appendix B, we show a qual-
N
itatively similar result holds when the outcome of interest is
the share of aggregate initial wealth annuitized.
(cid:12)
Figura 6 shows the mean rate of annuitization at various
parameter combinations of patience (β) and risk aversion (γ).
The real interest rate is set at 0%. Panel (UN) shows the rate
of annuitization when individuals have objectively-measured
expectations. The annuity we consider is one which pays a
constant (real) income. When households are fully patient
(β = 1), all households will fully annuitize—any risk-averse
fully-informed individual will prefer a constant stream of in-
come to self-insuring against longevity through a risk-free
legame. As patience is decreased, the rate of annuitization
falls.16 At a coefficient of relative risk aversion of 3, the an-
nuitization rate is 69% when the discount factor is set at 0.98,
E 51% with β = 0.96.
For a fixed pair of preferences parameters, variation across
individuals in the decision over how much of their wealth to
16The annuity puzzle is less stark at lower levels of patience as we assume
(realistically) that individuals cannot negotiate an actuarially-fair annuity
with a bespoke payment schedule that fits their patience and risk aversion.
If complete markets for annuities existed, the annuitization rate would be
100%, regardless of patience. Inkmann et al. (2011) show that this type
of market incompleteness, combined with estimated risk aversion and pa-
tience, can rationalize much of the lack of demand for annuities. To see
the effects of our results in a complete markets context, the reader should
focus only on the row with β = 1, the level of patience for which the offered
contract is the preferred one.
annuitize is driven by the relative level of public pension
entitlements (pi) and the level of wealth (ai0). Those who
annuitize a smaller proportion of wealth are those who have
little liquid wealth relative to the size of their accrued public
pension entitlements. For these households, the value of the
small amount of additional longevity insurance they would
receive by annuitizing is less than the additional value of
consuming that wealth sooner.
Panel (B) shows an equivalent set of results for the case
when individuals are making decisions based on their sub-
jective survival expectations. Comparing these rates to those
reported in panel (UN), it is clear that subjective expectations
have the capacity to substantially reduce annuity demand.
For fully patient individuals, the rate of annuitization falls
from 100% to between 42% (log utility) E 64% (coeffi-
cient of relative risk aversion of 5). With modest impatience
(β = 0.98), rates of annuitization fall from 47% A 20% COME-
suming log utility, and from 77% A 52% assuming a high
rate of risk aversion (γ = 5).
Appendix B shows that the difference between the pro-
portion of wealth annuitized under “objective” and “subjec-
tive” expectations is qualitatively insensitive to (1) including
a flow of consumption of housing services for homeowners,
(2) using ONS life tables without “rescaling,” and (3) various
alternative choices of sample selection and weighting.
To put the size of the falls in annuitization rates in con-
testo, figure 7 shows the average rate of annuitization in a
model where individuals have objectively-estimated survival
expectations but are faced with an annuity rate which, due to
adverse selection and other market imperfections, as well as
transactions costs, is offered at 17.5% below the actuarially-
fair rate.17 In panel (B) of figure 7, we show the model
17We use 17.5% based on the analysis of annuity rates available on the U.S.
market by Mitchell et al. (1999) who report that “the expected discounted
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SURVIVAL PESSIMISM AND THE DEMAND FOR ANNUITIES
451
FIGURE 7.—PERCENTAGE OF INDIVIDUALS ANNUITIZING AT EACH PARAMETER COMBINATION
Fonte: Model predictions using ELSA waves 3–5 and ONS 2014-based cohort life tables for England and Wales. 2,848 observations.
predictions in the case where individuals act according to
their “objective” survival curve, but face an annuity payout
equal to 82.5% of the actuarially fair rate. We reproduce panel
(UN), where individuals have “objective” expectations and face
the actuarially fair annuity rate, for comparison. This is in-
tended to illustrate the impact of the degree of actuarial un-
fairness observed in UK annuities markets which may be
attributable to adverse selection or administrative loading.
At all parameter combinations, the actuarially-unfair pric-
ing causes declines in annuitization rates comparable to those
caused by misperceived survival probabilities. With mod-
erate levels of impatience and risk aversion, the effects of
actuarially-unfair pricing are marginally greater than the
effects of misperceived survival probabilities: at (β, γ) =
(0.98, 3), the rate of annuitization with objectively-measured
expectations is 69% but falls to 38% under the reduced annu-
ity payouts and 43% when individuals make decisions based
on their subjective survival expectations. The effect of sub-
jective survival expectations has a slightly larger impact than
adverse selection at higher levels of risk aversion and higher
levels of patience – at (β, γ) = (1, 5), the average proportion
of wealth annuitized is 100% given objective expectations
and actuarial fairness, È 69% given objective expectations
and reduced payouts, and is 64% under subjective expecta-
tions and actuarial fairness.18
value of annuity payouts per dollar of annuity premium averages between
80 E 85 cents for an individual chosen at random from the population.”
18The reason for these patterns can be understood by comparing the rea-
sons in each case for annuities becoming less attractive products. Con
adverse selection raising annuity prices, all annuity payouts are discounted
relative to the actuarially fair benchmark. With subjective survival rates,
the annuity payouts late in life are not considered valuable—as individuals
wrongly perceive that they will likely be dead by then. When individuals
are less patient therefore, reducing annuity payments every period implies
a greater reduction in welfare than does treating payments in the distant fu-
Overall, we take these results as indicating that the effect
of individuals misperceiving their survival probabilities is as
large as the effect of adverse selection.
Finalmente, we note that, as shown by Heimer et al. (2019), In-
dividuals at younger ages than in our sample are likely to also
be pessimistic about their later-life survival and consequently
accumulate less retirement wealth under subjective expecta-
tions than they would do if their expectations were unbiased.
As the individual annuitization rate is increasing in wealth in
our model, this implies that the effect of subjective expecta-
tions on the rate of annuitization would be shown to be even
greater if this savings effect were taken into account.
V. Conclusione
Incorporating individual “subjective” survival curves into
a model of annuitization, consumption and saving has the
capacity to explain part of the “annuity puzzle.” While market
incompleteness and informational asymmetries play a role
in rationalizing low annuity demand, we take our results as
showing that misperceptions of survival probabilities are as
important for explaining behavior.
Our results are important for government policy in rela-
tion to annuities and retirement provision more generally.
While resources do exist to inform individuals about their
life expectancy,19 the divergence between self-assessed and
ture as “wasted.” When risk aversion is high, the welfare cost of the higher
price is low relative to the cost of insurance against the modest possibility
of living a long time. This insurance is less valuable as the likelihood of that
longevity appears to diminish. Di conseguenza, the higher is risk aversion, IL
greater is the effect of subjective survival probabilities relative to reduced
annuity rates.
19Per esempio, this online life expectancy calculator from the Social Se-
curity Administration: https://www.ssa.gov/planners/lifeexpectancy.html.
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THE REVIEW OF ECONOMICS AND STATISTICS
TABLE 1.—DISTRIBUTION OF NUMBER OF EXPECTATIONS QUESTIONS TO WHICH INDIVIDUALS ANSWER “50%”
All individuals
Answered “50%” to 1st survival Q
0
55.42%
52.09%
1
35.34%
35.78%
Number of “50%” answers given
2
8.07%
10.20%
3
1.05%
1.62%
4
0.12%
0.30%
5
0.00%
0.01%
Other probability questions include those related to the probability of moving out of one’s home in the future, of being in work in a number of years’ time, of having insufficient financial resources to meet needs at
some point in the future, of it raining tomorrow and of giving and receiving an inheritance. Fonte: ELSA waves 1–7. 66,210 interviews from 16,345 unique individuals.
objective life expectancies and the associated implications for
annuity purchases leave a role for larger policy interventions
to improve households’ understanding of the length of retire-
ment that they might have to fund. As individuals approach
retirement with increasingly large shares of their wealth in
nonannuitized form, ensuring that individuals adequately un-
derstand their longevity in this way will become only more
pressing.
Appendix A Details of Further Analysis and Tests
from Section II
A.1 Analysis of “50%” Answers
Tavolo 1 details the distribution of individuals by the number
of times they answered “50%” to questions in the expecta-
tions module other than the survival questions. The reporting
patterns in the 20.5% of interviews in which individuals an-
swered “50%” to the first survival question are very similar
to distribution of responses amongst the whole sample.
A.2 Correlation of Subjective Reports with Risk Factors,
New Information, Subsequent Mortality,
and Holdings of Life Insurance
Tavolo 2 details the results of a regression of an individual’s
answer to the first survival question they are asked on a range
of risk factors as well as a full set of wave by single-year-
of-age dummy interactions. Results are split by gender and
by whether or not self-reported health is controlled for. IL
coefficients reported are in percentage point deviations. For
esempio, a male current smoker reports an 8.3 percentage
points lower chance of survival to an age 11–15 years ahead
of their current age, on average, when compared to current
non-smokers (6.8 percentage points when controlling for self-
reported health).
Tavolo 3 reports the results of a fixed effects regression of
individuals’ answers to the survival expectations questions
on a range of dummies for whether or not they have received
a new diagnosis of a health condition since their last inter-
view. We control linearly for age. We find that individuals do
respond to new information by revising their survival expec-
tations. Per esempio, a new diagnosis of cancer or a case of a
stroke cause large and statistically significant downward re-
visions in survival expectations of 4 E 6 percentage points,
rispettivamente.
Figura 8 shows the 10 year mortality rates of individuals
according to their answer to the first survival question. Noi
use the linked death records which give us a 10-year horizon
for those interviewed in wave 1 of ELSA. We see clear dif-
ferences in mortality rates according to stated expectations.
These differences are statistically significant. The correlation
between expected and actual mortality remains even when
we control for age and sex-specific average mortality risk
and the range of health factors controlled for in the previous
regressions.
Finalmente, we examine the correlation of subjective expecta-
tions and purchases of life insurance, a product that is anal-
ogous to selling an annuity. In the ELSA questionnaire, In-
dividuals are asked whether they hold life insurance and the
amount that would be paid out to others in the event of their
death. We use this information, along with information about
financial wealth, pension wealth, and housing wealth, to gen-
erate a variable that puts the size of the (potential) insurance
payout in the context of the net worth of the respondents. Noi
call this variable the “share” of wealth held as life insurance
and define it as
Sharei ≡
Payouti × Die10
Payouti × Die10 + PWi + FWi + HWi
,
(9)
where Payouti is the amount that will be paid out in the event
of the individual’s death and PWi, FWi, and HWi are the indi-
vidual’s pension wealth, financial wealth and housing wealth,
rispettivamente. Die10 is the individual’s “objective” probabil-
ity of dying within the next 10 years. The idea of multiplying
the life insurance payout by this number is to capture the
expected life insurance payout. In the absence of informa-
tion about the term of individuals’ life insurance products,
we assume a term of 10 years.20 In the ELSA sample, 31%
of individuals hold some life insurance and the mean port-
folio share is 7% (23% amongst those who have some life
insurance). Life insurance is more prevalent at younger ages.
We use these constructed variables to run a set of OLS re-
gressions where the dependent variable is the share of wealth
held in life insurance and the independent variable is the sub-
jective report. We control for the the interaction of the respon-
dent’s sex with a full set of age dummies, dummy variables
20We obtain very similar results if we instead assume a remaining term of
5 O 15 years and if, instead of discounting the payout by a probability, we
allow it to enter in equation (9) undiscounted. This last measure would be
the share of wealth accounted for by a life insurance payout if the respondent
were to die immediately.
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SURVIVAL PESSIMISM AND THE DEMAND FOR ANNUITIES
453
TABLE 2.—RELATIONSHIP BETWEEN STATED SURVIVAL PROBABILITIES AND RISK FACTORS
Ex. self-reported health
Inc. self-reported health
Smoking (relative to nonsmoker)
Former occasional smoker
Former regular smoker
Former smoker, DK frequency
Current smoker
Alcohol consumption (relative to once or twice a month)
At least 3–4 days a week
Once or twice a week
A few times a year
Not at all
Age mother died (relative to 60–64)
Under 50
50–59
60–64
65–69
70–74
75–79
80–84
85+
Age father died (relative to 60–64)
Under 50
50–59
60–64
65–69
70–74
75–79
80–84
85+
Health conditions
Hypertension
Heart condition
Stroke
Cancer
Male
−3.1*
−1.6*
−2.8*
−8.3***
−0.1
0.0
−0.9
−2.2
2.4
1.2
0.0
3.4*
2.2
2.5
3.6*
6.5***
1.6
−0.1
0.0
0.7
2.6*
4.3***
4.4***
7.7***
−2.2***
−3.3***
−2.4
−6.5***
Female
−0.6
−0.2
−1.8
−7.3***
0.5
0.2
−1.1
−1.4
1.6
−2.5
0.0
−1.7
−0.5
0.7
2.7*
7.4***
−0.0
0.5
0.0
−1.6
1.7
3.0**
2.5*
4.6***
−2.7***
−2.7**
−0.3
−3.5***
Male
−2.8*
−1.1
−2.2
−6.8***
−0.9
−0.2
−0.3
−1.2
2.5
1.0
0.0
3.3*
2.1
2.1
3.2*
6.0***
1.9
0.3
0.0
0.7
2.6*
4.3***
4.2***
7.6***
−0.8
−1.7
−0.3
−4.4***
Female
−0.0
0.2
−1.3
−6.1***
0.1
−0.1
−0.7
−0.4
2.0
−2.4
0.0
−1.3
−0.4
0.8
2.9*
7.5***
0.1
0.7
0.0
−1.1
1.7
3.0**
2.5*
4.5***
−1.6***
−1.7*
1.5
−2.2**
Coefficients represent percentage point deviations in mean response. Statistical significance at the 5%/1%/0.1% level is denoted by */**/***. Standard errors are clustered at the individual level. Other control
variables, for which coefficients are not reported, are whether in a couple, income and wealth quintile, education level, whether working and dummy variables for whether diagnosed with Alzheimer’s, angina, arthritis,
diabetes, lung disease, osteoporosis, Parkinson’s and psychiatric disorders, whether the individual is white or nonwhite and a full set of dummy variables for each single year-of-age and wave interaction. Fonte: ELSA
waves 1–7. 43,146 observations of 13,739 unique individuals.
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TABLE 3.—REVISION TO SURVIVAL EXPECTATIONS FOLLOWING DIAGNOSIS WITH
MAJOR HEALTH CONDITIONS
Alzheimer’s disease
Cancer
Dementia
Heart attack
Lung disease
Parkinson’s disease
Psychiatric problems
Stroke
Observations
First survival
question
−4.4
−4.4***
3.1
−3.5*
−1.5
−2.5
−2.6*
−6.5***
48,917
Second survival
question
−18.5
−3.2*
5.4
−3.1
1.0
−6.8*
−1.3
−5.7**
22,926
vant. We select respondents aged between 50 E 69 (Anche se
we get very similar results if we further restrict to those aged
50 A 59).
Tavolo 4 presents the results. We see that a 1 percentage
point increase in the subjective belief about the probability
of survival is associated with a 0.043 percentage points lower
portfolio share held in life insurance (0.036 when controlling
for wealth), significant at the 1% level. Columns 3 E 4 show
an equivalent set of results but where the sample includes only
those who hold life insurance.
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Coefficients represent percentage point deviations in mean response. Statistical significance at the
5%/1%/0.1% level is denoted by */**/***. Standard errors are clustered at the individual level. Fonte:
ELSA waves 3–7. 48,917 observations of 13,811 unique individuals (22,926 observations of 8,135 unique
individuals for the second question).
for whether the respondent has a partner and whether the re-
spondent has children, and the interaction of their partner’s
sex and a full set of dummies for their partner’s age. Noi
show results with and without controls for the individual’s
total wealth (equal to the sum of their wealth and expected
life insurance payout). We focus on ages where life insurance,
which is ordinarily purchased to insure earnings, is most rele-
Appendix B Robustness of Results from Section IV.B
B.1 Robustness of Main Results to Using ONS Life Tables
Without Rescaling
We show here results of the model in the case where we use
the ONS life table survival curves without rescaling as our
measure of “objective” survival probabilities. Annuity rates
are calculated using the unscaled ONS survival curves. Tutto
other details of the model are as given in section IV.A. Figura
9 shows the model predictions.
454
THE REVIEW OF ECONOMICS AND STATISTICS
FIGURE 8.—10 YEAR MORTALITY RATES BY ANSWER TO SURVIVAL QUESTION
Reported probability of death is 100 minus the reported probability of survival in the first survival question the individual is asked.
Fonte: ELSA wave 1 and linked death records. 11,502 individuals.
TABLE 4.—ASSOCIATION OF SUBJECTIVE EXPECTATIONS AND LIFE
INSURANCE HOLDINGS
(1)
(2)
(3)
(4)
Subjective report
Controls for wealth
Has life insurance
Observations
−0.0434*** −0.0361*** −0.119*** −0.0984***
(0.00893)
No
No
14915
(0.00888)
Yes
No
14915
(0.0244)
Yes
Yes
5103
(0.0250)
No
Yes
5103
Statistical significance at the 10%/5%/1% level is denoted by */**/***. Standard errors are clustered at
the individual level. Fonte: ELSA waves 3–5.
B.2 Definition of Model Including Utility from Housing
Consumption
In table 5, we show the results from various further robust-
ness checks. We describe one of these checks in more detail
here by defining the model used. We run an alternative ver-
sion of the model where homeowners receive utility from the
consumption of housing services. We assume that individuals
receive a per-period flow of housing services, h equal to 4%
of the value of their primary house, as reported in the ELSA
dati. This value is fixed in real terms in future periods. Their
utility function is of the form
subjective expectations and objective expectations with a
17.5% rate reduction (i.e. the results in figures 6 E 7).
For brevity, we select one central parameter combination,
β = 0.98 and γ = 3. We show robustness to various sample
selection choices: (1) removal of individuals who respond
“100%” to one or more questions, (2) removal of individuals
who respond “0%” to one or more questions, (3) removal of
individuals who respond “50%” to one or more questions,
E (4) removal of individuals who respond “0%,” “100%,”
or “50%” to one or more questions. We show robustness to
(5) not weighting our results using the ELSA sample weights.
We show results from (6) an alternative model in which indi-
viduals can choose only whether to annuitize the entirety of
their wealth or none of it rather than being able to annuitize
any fraction of their wealth. We show results (7) weight-
ing the individual annuitization rates by individuals’ initial
wealth—that is showing the percentage of aggregate wealth
that is annuitized. Finalmente, we show (8) the results from a
model in which there is a utility flow from housing (as de-
scribed in section VII.B). These robustness checks are shown
in table 5, with the baseline results from figure 6 shown in
the first row for comparison.
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U =
110−z(cid:6)
t=0
βt Si(z + T )
(ct + H)1−γ
1 − γ
.
All other details of the model are as given in section IV.A.
B.3 Further Robustness of Main Results
We here show further robustness of our main results on
the average annuitization rate under objective expectations,
(10)
Appendix C Computational Appendix
We solve the model numerically. For each individual i, IL
model outlined in equation (8) can be expressed recursively.
We outline this below, first focusing on the periods after the
age at which individuals are observed in the data. We denote
this age as 0. In these periods the only decision they face is a
consumption and saving choice. We then outline the problem
at age 0 where individuals make an annuitization choice as
well as consumption and saving choice.
SURVIVAL PESSIMISM AND THE DEMAND FOR ANNUITIES
455
FIGURE 9.—PERCENTAGE OF INDIVIDUALS ANNUITIZING AT EACH PARAMETER COMBINATION (UNSCALED ONS LIFE TABLES)
Fonte: Model predictions using ELSA waves 3–5 and ONS 2014-based cohort life tables for England and Wales. 2,848 observations.
TABLE 5.—ANNUITIZATION RATES UNDER OBJECTIVELY-MEASURED
EXPECTATIONS, SUBJECTIVELY-ELICITED EXPECTATIONS, AND OBJECTIVE
EXPECTATIONS WITH A RATE REDUCTION, UNDER ALTERNATIVE SAMPLE
SELECTION, WEIGHTING AND MODELLING ASSUMPTIONS, WITH MODEL
PARAMETERS β = 0.98 AND γ = 3
Specification
Objective Subjective
Objective +
rate reduction
Baseline results
No “100%”s
No “0%”s
No “50%”s
No “0%”s, “50%”s or“100%”s
Unweighted
‘Discrete’ annuitization choice
Weighted by initial wealth
Utility flow from housing
69
70
70
70
72
70
76
90
54
43
45
45
45
51
44
44
68
26
38
38
39
39
40
39
36
67
17
Fonte: Model predictions using ELSA waves 3–5 and ONS 2014-based cohort life tables for England
and Wales. 2,848 observations.
C.1 Recursive Form of the Model
C.1.1 Periods After Annuitization Decision Has Been Made.
At all ages t > 0 the model can be expressed in recursive
form as:
subjective survival curve estimated from individual reports.
pi is the public pension income of individual i.
The fact that the public pension income stream and sur-
vival probabilities vary across individuals means that the
value function differs across individuals. This implies that
the value function must be solved for separately for each in-
dividual in our data. For notational convenience, we suppress
the i subscript for most of the rest of this appendix.
By assuming that there is an age (T ) beyond which there
is a zero probability of survival (110 in our application), we
get sT +1 = 0 and equation (11) for period T reduces to:
VT (aT , ann; P) = max
cT
tu(cT )
(12)
and the function VT can be obtained by maximizing the util-
ity function subject to the budget constraint. Knowledge of
VT allows the maximization in the recursion for T − 1 to be
undertaken and for VT −1 to be obtained. This recursive proce-
dure can be repeated for each period back to (and including) 1
which is the age immediately after the individual is observed
in the data.
Vit (ait , anni; pi)
= max
cit
tu(cit ) + βsi(T + 1)Vit+1(ait+1, anni; pi)
(11)
subject to the constraints given in equation (8) which we do
not repeat here. Consumption in period t (cit ) is a choice,
tu(.) is a utility function, V (.) is a value function, and assets
(ait ) and annuity income (anni) are state variables. si(T + 1)
is the probability of individual i surviving to period t + 1
conditional on having survived to period t and could be either
that calculated using life tables or could be that implied by
their objective survival curve based on life tables or their
C.1.2
Initial Period. The problem in period 0 differs
from that in future periods as agents need to make an an-
nuitization decision as well as a consumption decision:
V0(a0; P) = max
c0,b
tu(c0) + βs(1)V1(a1, ann; P)
(13)
where b, a choice variable, is the share of period 0 wealth
annuitized and where the maximization is subject to the con-
straints given in equation (8). With knowledge of V1(.) In
hand from the steps outlined in the previous subsection,
the maximization in (13) can be undertaken. This allows
us to obtain policy functions, in particular a function which
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456
THE REVIEW OF ECONOMICS AND STATISTICS
relates initial wealth holdings to annuitization decisions for
each individual. This is the function that yields the quantity
of initial wealth annuitized for each individual in the data,
which represent our central results. To be clear about what
this object depends on, we can write the policy function as
B(ai0; pi, si; β, γ) where we making explicit that the propor-
tion annuitized depends on (io) initial wealth ai0, (ii) individual
circumstances: public pension income (pi) and the individ-
ual’s entire survival curve (si), E (iii) the values of β and γ,
which we vary in our application.
C.2 Computational Implementation
In the absence of an analytical solution to the agents’ prob-
lem, we solve for value functions in equation (1) E (2) nu-
merically. We take a standard approach, by discretizing the
state variables and solving for the value function at those dis-
crete points. We define a grid of 50 points for assets from 0 A
ai0 (initial assets for individual i). Annuity income is placed
on a grid of 10 points which are equally spaced from 0 A
θiai0 where θi is the annuity rate faced by individual i. Noi
restrict the annuity choice to be be on this grid of 10 shares
(questo è, individuals can annuitize 0% of their wealth, 100%
of their wealth or can choose any of other 8 shares equally
spaced between the two extremes). Consumption is a contin-
uous choice, obtained at each point in the discretized state
space using golden section search.21
We have confirmed that our results are not sensitive to
the choices we have made over the manner of discretiza-
tion and the number of grid points used. As an example,
when expressed to two decimal places, our main result (from
figure 6) for the average share of wealth annuitized for
β = 0.98, γ = 3 È 69.33% under objectively-measured sur-
vival expectations and 42.51% under subjectively-measured
survival expectations. When we quintuple each of the num-
ber of asset grid points, annuity income grid points and the
grid of annuity share options to 250, 50, E 50, rispettivamente,
the shares would become 70.07% E 42.46%, rispettivamente.
21Vedere, Per esempio, Miranda and Fackler (2002).
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