longevity risk, retirement savings, and individual welfare joao f cocco and francisco j. gomes

Longevity Risk, Retirement Savings, and Individual Welfare

Joao F Cocco and Francisco J. Gomes

London Business School and CEPR

June 2007

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.

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.

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.

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.

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.

Outline of the Presentation

Empirical Evidence on Longevity

A Model of Longevity Risk

Model Parameterization

Model Results

Future Research and Concluding Remarks


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.


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.

Figure 1: Life expectancy in the United States and England for a male individual at selected ages

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

1947-2004

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

Figure 2: Conditional probability of death for a male US individual

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.

Figure 3: Actual and fitted conditional probability of death


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.

Figure 4: Life expectancy for a 65 year old United Kingdom male individual

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.

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

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

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.

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%

Figure 8: Conditional Survival Probability (Model)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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

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

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.

Figure 9: Simulated Consumption, Income and Wealth in the Baseline Model – Average across 30,000 realizations

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.

No riskUninformed 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%


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%

Table 4: Welfare Gains in The Form of Consumption Equivalent Variations

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

0

20

40

60

80

100

120

140

30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110

Age

Tho

usan

d U

S D

olla

rs

Cons Cons Zero Increase Income Wealth Wealth Zero Increase

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.





Baseline 0.03% -0.89% -0.23% -0.18%

Lower rep. ratio 0.04% -2.27% -0.33% -0.25%


0.08% -8.44% -1.19% -0.91%


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%


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,





Baseline 0.03% -0.89% -0.23% -0.18%

Lower rep. ratio

0.04% -2.27% -0.33% -0.25%


0.08% -8.44% -1.19% -0.91%


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%


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.

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.

longevity risk, retirement savings, and individual welfare joao f cocco and francisco j. gomes

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