Longevity Risk, Retirement Savings, and Individual Welfare Joao F Cocco and Francisco J. Gomes

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 \end{table}

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=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.

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=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.

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 t+1, conditional on being alive at date t, so that pt=1-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 \end{table}

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 γ, but incorrectly think that the probability of a future increase in life expectancy, or the value of π, is only 0.10. • 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,

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.