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- 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

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.

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.

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

\endtable

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? 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.

Figure 3 Actual and fitted conditional

probability of death

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.

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? 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.

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

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

\endtable

Figure 8 Conditional Survival Probability (Model)

Table 3 Life Expectancy at Age 65 in the Model

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.

Table 4 Welfare Gains in The Form of Consumption

Equivalent Variations

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

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.

Table 4 Welfare Gains in The Form of Consumption

Equivalent Variations

Table 4 Welfare Gains in The Form of

Consumption Equivalent Variations

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,

Table 4 Welfare Gains in The Form of Consumption

Equivalent Variations

Table 4 Welfare Gains in The Form of

Consumption Equivalent Variations

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.