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Longevity Risk, Retirement Savings, and Individual Welfare

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Title: Longevity Risk, Retirement Savings, and Individual Welfare


1
  • Longevity Risk, Retirement Savings, and
    Individual Welfare
  • Joao F Cocco and Francisco J. Gomes
  • London Business School and CEPR
  • June 2007

2
Introduction
  • Over the last few decades there has been an
    unprecedented increase in life expectancy.
  • In 1970 a 65 year old United States male
    individual had a life expectancy of 13.04 years.
  • In 2000 a 65 year old male had a life expectancy
    of 16.26 years.
  • This is an increase of 3.37 years in just three
    decades, or 1.12 years per decade.
  • To understand what such increase implies in terms
    of the savings needed to finance retirement
    consumption.
  • Consider a fairly-priced annuity that pays 1
    real per year, and assume that the real interest
    rate is 2 percent.
  • The price of such annuity for a 65 year old male
    would have been 10.52 in 1970, but it would have
    increased to 12.89 by 2000.
  • This is an increase of roughly 23 percent. A 65
    year old male in 2000 would have needed 23
    percent more wealth to finance a given stream of
    real retirement consumption than a 65 year old
    male in 1970.

3
Introduction
  • These large increases in life expectancy were, to
    a large extent, unexpected and as a result they
    have often been underestimated by actuaries and
    insurers.
  • This is hardly surprising given the historical
    evidence on life expectancy.
  • From 1970 to 2000 the average increase in the
    life expectancy of a 65 year old male was 1.12
    years/decade, but over the previous decade the
    corresponding increase had only been 0.15 years.
  • In the United Kingdom, the average increase in
    the life expectancy of a 65 year old male was
    1.23 years/decade from 1970 to 2000, but only
    0.17 years/decade from 1870 to 1970.
  • These unprecedented longevity increases are to a
    large extent responsible for the underfunding of
    pay as you go state pensions,\and of
    defined-benefit company sponsored pension plans.
  • In October 2006 British Airways reported that the
    deficit on its defined-benefit pension scheme had
    risen to almost 1.8 billion pounds, from a value
    of 928 million pounds in March 2003. The main
    reason for such an increase was the use of more
    realistic and prudent life expectancy assumptions.

4
Introduction
  • The response of governments has been to decrease
    the benefits of state pensions, and to give tax
    and other incentives for individuals to save
    privately, through pensions that tend to be
    defined contribution in nature.
  • Likewise, many companies have closed company
    sponsored defined benefit plans to new members.
  • For individuals who are not covered by
    defined-benefit schemes, and who have failed to
    anticipate the observed increases in life
    expectancy, a longer live span may also mean a
    lower average level of retirement consumption.
  • The purchase of annuities at retirement age
    provides insurance against longevity risk as of
    this age, but a young individual saving for
    retirement faces substantial uncertainty as to
    what aggregate life expectancy and annuity prices
    will be when he retires.
  • Our paper studies individual consumption and
    savings decisions in the presence of longevity
    risk.

5
Introduction
  • We first document the increases in life
    expectancy that have occurred over time, using
    long term data for a collection of 28 countries.
  • We focus our analysis on life expectancy at ages
    30 and 65.
  • Due to our focus on the relation between
    longevity and retirement saving.
  • We also consider the existing debate on how one
    should model mortality, and improvements in
    survival probabilities late in life.
  • We use the empirical evidence to parameterize a
    simple life-cycle model of consumption and saving
    choices in the presence of longevity risk.
  • We study how the individual's consumption and
    saving decisions, and welfare are affected by
    longevity risk.

6
Introduction
  • Model results When the agent is informed of the
    current survival probabilities, and correctly
    anticipates the probability of a future increase
    in life expectancy, longevity risk has a modest
    impact on individual welfare.
  • This is in spite of the fact that the agent in
    our model does not have available financial
    assets that allow him to insure against longevity
    risk.
  • When agents are uninformed of improvements in
    life expectancy, or are informed but make an
    incorrect assessment of the probability of future
    improvements in life expectancy, the effects of
    longevity risk on individual welfare can be
    substantial.

7
Outline of the Presentation
  • Empirical Evidence on Longevity
  • A Model of Longevity Risk
  • Model Parameterization
  • Model Results
  • Future Research and Concluding Remarks

8
Empirical Evidence on Longevity
  • Data from the Human Mortality Database, from the
    University of California at Berkeley.
  • Contains survival data for a collection of 28
    countries, obtained using a uniform method for
    calculating such data.
  • The database is limited to countries where death
    and census data are virtually complete, which
    means that the countries included are relatively
    developed.
  • We focus our analysis on period life
    expectancies.
  • Calculated using the age-specific mortality rates
    for a given year, with no allowance for future
    changes in mortality rates. For example, period
    life expectancy at age 65 in 2006 would be
    calculated using the mortality rate for age 65 in
    2006, for age 66 in 2006, for age 67 in 2006, and
    so on.
  • Period life expectancies are a useful measure of
    mortality rates actually experienced over a given
    period. Official life tables are generally
    period life tables for these reasons.
  • It is important to note that period life tables
    are sometimes mistakenly interpreted by users as
    allowing for subsequent mortality changes.

9
Empirical Evidence on Longevity
  • We focus our analysis on life expectancy at ages
    30 and 65.
  • Over the years there have been very significant
    increases in life expectancy at younger ages.
  • For example, in 1960 the probability that a male
    newborn would die before his first birthday was
    as high as 3 percent, whereas in 2000 that
    probability was only 0.8 percent.
  • In England, and in 1850, the life expectancy for
    a male newborn was 42 years, but by 1960 the life
    expectancy for the same individual had increased
    to 69 years.
  • Our focus on life expectancy at ages 30 and 65 is
    due to the fact that we are interested on the
    relation between longevity risk and saving for
    retirement.
  • The increases in life expectancy that have
    occurred during the last few decades have been
    due to increases in life expectancy in old age.
  • This is illustrated in Figure 1.

10
Figure 1 Life expectancy in the United States
and England for a male individual at selected ages
11
Table 1 Average annual increases in life
expectancy in number of years for a 65 year old
male
\endtable
12
Figure 2 Conditional probability of death for a
male US individual
13
Empirical Evidence on Longevity
  • A commonly used model for mortality data is the
    Gompertz model.
  • It was first proposed by Benjamin Gompertz in
    1825.
  • It has been extensively used by medical
    researchers and biologists modeling mortality
    data.
  • It is a proportional hazards model, for which the
    hazard function, or the probability that the
    individual dies at age t, conditional on being
    alive at that age, is given by
  • ht? exp(?t)
  • We estimate the parameters of the model using
    maximum likelihood. Figure 3 shows the fit of a
    Gompertz model to these conditional probabilities
    of death
  • The Gompertz model fits these probabilities well
    in the 30 to 80 years old range.
  • But not at later ages mortality rates observed
    in the data increase at a lower rate than those
    predicted by the model. This phenomenon is known
    in the demography literature as late life
    mortality deceleration.

14
Figure 3 Actual and fitted conditional
probability of death
15
Empirical Evidence on Longevity
  • In this version of the paper we use the Gompertz
    model to model survival probabilities
  • We plan to consider other possibilities in future
    versions of the paper.
  • But currently there is considerable discussion
    and uncertainty
  • As to how one should model mortality, and
    improvements in survival probabilities, in late
    life.
  • With respect to the magnitude of future increases
    in life expectancy.
  • Cohort life expectancies are calculated using
    age-specific mortality rates which allow for
    known or projected changes in mortality in later
    years.

16
Figure 4 Life expectancy for a 65 year old
United Kingdom male individual
17
A Model of Longevity Risk
  • Life cycle model of consumption and saving
    choices of an individual.
  • We let t denote age, and assume that the
    individual lives for a maximum of T periods.
    Obviously T can be made very large.
  • We use the Gompertz model to describe survival
    probabilities
  • ht? exp(?t)
  • When gamma is equal to zero the hazard function
    is equal to lambda for all ages so that the
    Gompertz model reduces to the exponential. When
    gamma is positive the hazard function, or the
    probability of death, increases with age.
  • The larger is gamma the larger is the increase in
    the probability of death with age.

18
The Model
  • We model longevity increases by assuming that in
    each period with probability pi that there is a
    permanent reduction in the value of gamma equal
    to Delta gamma. With probability (1-pi) the value
    of gamma remains unchanged.
  • Note
  • In this simplest version of our model we do not
    allow for decreases in life expectancy. The
    decreases that we observe in the data seem to be
    temporary, and the result of wars or pandemics.
  • More generally, one could allow for changes in
    both lambda and gamma.
  • pt denotes the probability that the individual
    is alive at date t1, conditional on being alive
    at date t, so that pt1-ht

19
The Model
  • Preferences time separable power utility.
  • Labor income
  • Deterministic component function of age and
    other individual characteristics.
  • Permanent income shocks.
  • Temporary income shocks
  • Financial assets
  • Single financial asset with riskless interest
    rate R

20
Solution Technique
  • The model was solved using backward induction.
  • In the last period the policy functions are
    trivial (the agent consumes all available wealth)
    and the value function corresponds to the
    indirect utility function.
  • We can use this value function to compute the
    policy rules for the previous period and given
    these, obtain the corresponding value function.
    This procedure is then iterated backwards.
  • The sets of admissible values for the decision
    variables were discretized using equally spaced
    grids. To avoid numerical convergence problems
    and in particular the danger of choosing local
    optima we optimized over the space of the
    decision variables using standard grid search.
  • Following Tauchen and Hussey (1991), approximate
    the density function for labor income shocks
    using Gaussian quadrature methods, to perform
    the necessary numerical integration.
  • In order to evaluate the value function
    corresponding to values of cash-on-hand that do
    not lie in the chosen grid we used a cubic spline
    interpolation in the log of the state variable.

21
Table 2 Model Parameterization
\endtable
22
Figure 8 Conditional Survival Probability (Model)
23
Table 3 Life Expectancy at Age 65 in the Model
24
Model Results
  • We use the optimal policy functions to simulate
    the consumption and savings profiles of thirty
    thousand agents over the life-cycle.
  • In Figure 9 we plot the average simulated income,
    wealth and consumption profiles.

25
Figure 9 Simulated Consumption, Income and
Wealth in the Baseline Model Average across
30,000 realizations
26
Welfare Results
  • In order to assess the impact of longevity risk
    on individual choices and welfare, we carry out
    the following exercise.
  • We solve our model assuming a deterministic
    improvement in life expectancy, which in each
    period is exactly equal to the average increase
    that occurs in our baseline model.
  • We then compare individual welfare in the
    baseline model with individual welfare in this
    alternative scenario in which there is no
    longevity risk.
  • This welfare comparison is carried out using
    standard consumption equivalent variations. More
    precisely, for each scenario (baseline and no
    risk), we compute the constant consumption stream
    that makes the individual as well-off in expected
    utility terms. Relative utility losses are then
    obtained by measuring the percentage difference
    in this equivalent consumption stream between the
    baseline case and the no risk scenario.

27
Table 4 Welfare Gains in The Form of Consumption
Equivalent Variations
Table 4 Welfare Gains in The Form of
Consumption Equivalent Variations
28
Figure 10 Simulated Consumption, Income and
Wealth in the Baseline Model for Two Different
Individuals Who Face the Same Labor Income
Realizations but Different Survival Probabilities
29
Comparative Statics
  • In the recent years there has been a trend away
    from defined benefit pensions, and towards
    pensions that are defined contribution in nature.
  • In the future, the level of benefits that
    individuals will derive from defined benefit
    schemes are likely to be smaller than the one
    that we have estimated using historical data.
  • This is important since defined benefit pension
    plans because of their nature provide insurance
    against longevity risk.
  • Consider as a scenario a lower replacement ratio.
  • Longevity risk is likely to affect more agents
    who are more averse to risk,
  • Consider a higher risk aversion scenario.

30
Table 4 Welfare Gains in The Form of Consumption
Equivalent Variations
Table 4 Welfare Gains in The Form of
Consumption Equivalent Variations
31
The Cost of Mistakes
  • Agents are uninformed about improvements in life
    expectancy or make mistakes in their assessment
    of the probability of an increase in life
    expectancy. Consider three possibilities
  • Uninformed agent an agent that at the initial
    age knows the current survival probabilities, but
    that in subsequent periods is unaware that these
    probabilities have changed.
  • Agent who in each period is informed about the
    current survival probabilities, or the current
    value of ?, but incorrectly think that the
    probability of a future increase in life
    expectancy, or the value of p, is only 0.10.
  • Agent who is informed about the current survival
    probabilities, or the current value of ?, that
    starts his life thinking that the probability of
    an increase in life expectancy is 0.10, but that
    updates this value based on what has happened
    during his life,

32
Table 4 Welfare Gains in The Form of Consumption
Equivalent Variations
Table 4 Welfare Gains in The Form of
Consumption Equivalent Variations
33
Conclusion
  • We have documented that existing evidence on life
    expectancy.
  • We have solved a life cycle model with longevity
    risk, and investigated how much such risk affects
    the consumption and saving decisions, and the
    welfare of an individual saving for retirement.
  • When the agent is informed of the current
    survival probabilities, and correctly anticipates
    the probability of a future increase in life
    expectancy, longevity risk has a modest impact on
    individual welfare.
  • However, when agents are uninformed about
    improvements in life expectancy, or are informed
    but make an incorrect assessment of the
    probability of future improvements in life
    expectancy, the effects of longevity risk on
    individual welfare can be substantial.
  • This is particularly so for more risk averse
    individuals, and in the context of declining
    payouts of defined benefit pensions.

34
Future Research
  • More realistic alternatives for longevity risk,
    other than the Gompertz model.
  • The agent may face uncertainty about the true
    model, and the parameters of the model. This
    could be done in a Bayesian setting.
  • Financial assets that allow agents to insure
    against longevity risk, and analyze the demand
    for these assets.
  • Alternative means to insure against longevity
    risk such as labor supply flexibility.
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