Longevity Risk, Retirement Savings, and
Individual Welfare
Joao F Cocco and Francisco J. Gomes
London Business School and CEPR
June 2007
Page 1
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.
Page 2
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 definedbenefit 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.
Page 3
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.
Page 4
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.
Page 5
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.
Page 6
Outline of the Presentation
 Empirical Evidence on Longevity
 A Model of Longevity Risk
 Model Parameterization
 Model Results
 Future Research and Concluding Remarks
Page 7
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.
Page 8
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.
Page 9
Figure 1: Life expectancy in the United States and
England for a male individual at selected ages
Page 10
Table 1: Average annual increases in life expectancy in number of years
for a 65 year old male
United
States
Canada
England
Sweeden
Germany
France
Italy
Japan
Sample
Period
1959 2002
1921 2003
1841 2003
1751 2004
1956 2002
1899 2003
1872 2003
19472004
Whole
Sample
0.08
0.05
0.03
0.03
0.09
0.06
0.05
0.15
Pre 1970
-0.01
0.01
0.01
0.01
-0.04
0.03
0.03
0.12
1970 -2000
0.11
0.09
0.12
0.09
0.13
0.13
0.11
0.15
1960 - 1969 -0.01
0.03
-0.01
-0.02
-0.08
-0.03
-0.08
0.07
1970 - 1979
0.13
0.09
0.07
0.04
0.12
0.14
0.08
0.22
1980 - 1989
0.08
0.08
0.12
0.12
0.13
0.15
0.14
0.16
1990 - 1999
0.11
0.11
0.15
0.11
0.14
0.11
0.12
0.08
Page 11
Figure 2: Conditional probability of death for a male
US individual
Page 12
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=λ exp(γt)

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.
Page 13
Figure 3: Actual and fitted conditional probability of
death
Page 14
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.
Page 15
Figure 4: Life expectancy for a 65 year old United
Kingdom male individual
Page 16
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=λ exp(γt)
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.
Page 17
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 t+1,
conditional on being alive at date t, so that pt=1-ht
Page 18
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
Page 19
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-onhand that do not lie in the chosen grid we used a cubic spline interpolation in
the log of the state variable.
Page 20
Table 2: Model Parameterization
Description
Parameter
Value
Survival probabilities
Initial parameters of the distribution
lambda
0.000142
gamma
0.081194
Prob. of an increase in life expectancy
0.5
Magnitude of the increase in life expectancy
0.00025088
Time Parameters
Initial age
30
Retirement age
65
Terminal age
110
Preference Parameters
Discount rate
0.98
Risk aversion
3
Bequest motive
0
Labor Income and Asset Returns
Variance of temporary income shocks
0.0738
Variance of permanent income shocks
0.01065
Replacement ratio
0.68212
Interest rate
2%
Page 21
Figure 8: Conditional Survival Probability (Model)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 94 98 102 106 110
Age
Initial distributon
After 10 increases
Page 22
Table 3: Life Expectancy at Age 65 in the Model
Number of
increases in
gamma
Life expectancy at age
65 in number of years
Life expectancy at age
65 – expected age of
death
0
12.15
77.15
1
12.28
77.28
5
12.82
77.82
10
13.53
78.53
20
15.04
80.04
30
16.66
81.66
40
18.41
83.41
50
20.18
85.18
60
21.75
86.75
Page 23
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.
Page 24
Figure 9: Simulated Consumption, Income and
Wealth in the Baseline Model – Average across
30,000 realizations
Page 25
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.
Page 26
Table 4: Welfare Gains in The Form of
Consumption Equivalent Variations
No risk
Uninformed
agent
Agent uses wrong
probability (10%)
Agent learns
the prob.
Welfare gain at age 30
Baseline
0.03%
-0.89%
-0.23%
-0.18%
Lower rep.
ratio
0.04%
-2.27%
-0.33%
-0.25%
Lower rep. and
higher risk
aversion
0.08%
-8.44%
-1.19%
-0.91%
Welfare gain at age 65
Baseline
0.10%
-6.34%
-1.76%
-1.31%
Lower rep.
ratio
0.12%
-10.39%
-2.26%
-1.61%
0.24%
-15.17%
-2.71%
-1.96%
Lower rep. and
higher risk
aversion
Page 27
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
140
Thousand US Dollars
120
100
80
60
40
20
0
30
35
40
45
50
55
60
65
70
75
80
85
90
95 100 105 110
Age
Cons
Cons Zero Increase
Income
Wealth
Wealth Zero Increase
Page 28
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.
Page 29
Table 4: Welfare Gains in The Form of
Consumption Equivalent Variations
No risk
Uninformed
agent
Agent uses wrong
probability (10%)
Agent learns
the prob.
Welfare gain at age 30
Baseline
0.03%
-0.89%
-0.23%
-0.18%
Lower rep.
ratio
0.04%
-2.27%
-0.33%
-0.25%
Lower rep. and
higher risk
aversion
0.08%
-8.44%
-1.19%
-0.91%
Welfare gain at age 65
Baseline
0.10%
-6.34%
-1.76%
-1.31%
Lower rep.
ratio
0.12%
-10.39%
-2.26%
-1.61%
Lower rep. and
higher risk
aversion
0.24%
-15.17%
-2.71%
-1.96%
Page 30
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:
1. 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.
2. Agent who in each period is informed about the current survival
probabilities, or the current value of γ, but incorrectly think that the
probability of a future increase in life expectancy, or the value of π,
is only 0.10.
3. Agent who is informed about the current survival probabilities, or
the current value of γ, 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,
Page 31
Table 4: Welfare Gains in The Form of
Consumption Equivalent Variations
No risk
Uninformed
agent
Agent uses wrong
probability (10%)
Agent learns
the prob.
Welfare gain at age 30
Baseline
0.03%
-0.89%
-0.23%
-0.18%
Lower rep.
ratio
0.04%
-2.27%
-0.33%
-0.25%
Lower rep. and
higher risk
aversion
0.08%
-8.44%
-1.19%
-0.91%
Welfare gain at age 65
Baseline
0.10%
-6.34%
-1.76%
-1.31%
Lower rep.
ratio
0.12%
-10.39%
-2.26%
-1.61%
0.24%
-15.17%
-2.71%
-1.96%
Lower rep. and
higher risk
aversion
Page 32
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.
Page 33
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.
Page 34