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.

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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.

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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.

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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

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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.

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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.