This paper examines how simulation modeling can be used to select a retirement age under defined benefit pension plans. This approach construes the variables affecting pension benefits as probabilistic variables. Simulations are then run to generate probabilistic values for the real value of pension benefits for alternative retirement ages. By construing variables affecting pension benefits as probability distributions, this approach reflects the uncertainty facing individuals contemplating retirement. By generating estimates of retirement benefits as probability distributions rather than as single deterministic values, the model provides individuals with a more realistic and complete frame of reference for making the retirement decision. (JEL D12, J26)
As the baby boom generation (i.e., the approximately 75 million Americans born between 1946 and 1964) continues to age and approaches the retirement years, there is an increasing interest in the various aspects of retirement planning. Important aspects of retirement planning include (a) determining retirement income needs, (b) assessing work-leisure preferences, and (c) developing saving and investment strategies for meeting retirement income needs.
At what age should the individual retire? In an abstract sense, this depends primarily on (a) the work-leisure preferences of the individual and (b) the individual's accumulated wealth relative to the resource needs during the retirement period.
For individuals covered by defined benefit pension plans, the benefits provided by such plans are usually an important source of income during the retirement years (McDonnell 2000). But how do individuals covered by such plans choose their retirement age? If they retire today, will the benefits provided by the plan be sufficient to meet their wants during the retirement years? What will be the effect on pension benefits of working an additional 1, 2, 5, or 10 years?
This paper examines how simulation modeling can be used as a tool to assist individuals who are attempting to select a retirement age under a defined benefit pension plan. The approach construes the critical variables affecting the real value (at the time of retirement) of average annual pension benefits received over the retirement period as probabilistic variables. Monte Carlo simulations are then run to generate probabilistic values for the real value of average annual pension benefits during the retirement years under alternative retirement ages. By construing input variables such as the wage growth rate, the inflation rate, and life expectancy as probability distributions, this approach properly reflects the uncertainty facing individuals contemplating retirement. By generating estimates of retirement benefits as probability distributions rather than as single deterministic values, the model provides the individual with a more realistic and complete frame of reference for making the retirement decision. For illustration purposes the simulations are constructed using the provisions of the State of Delaware Pension Plan. However, the approach could easily be adapted to any situation for any defined benefit plan.
Review of Literature
There has been little research that specifically examines the effect that choosing alternative retirement dates by individuals covered by defined benefit pension plans has on the real value of their pension benefits. A number of predictive models that attempted to explain the age at which individuals have chosen to retire have been developed and tested (Samwick 1998). Montalto, Yuh, and Hanna (2000) attempted to explain the planned retirement age rather than the actual retirement age. All of these models were developed within the framework of the traditional work-leisure and/or life cycle models. These studies found the following factors to affect the actual or planned retirement age: (a) pension and social security benefits received at the time of retirement, Co) earnings from employment immediately prior to retirement, (c) type of employment, (d) health, (e) accumulated wealth, and (f) a host of demographic variables. All of these studies attempted to identify predictors of the actual or planned retirement age. However, they do not provide specific insight for individuals who are attempting to determine the age at which to retire.
There has been some research that provides an analytical framework for individuals wishing to evaluate the benefits provided by different retirement plans under alternative retirement ages. Newmark and Walden (1995) provided an analytical framework for individuals to use in evaluating whether to opt to retire and receive Social Security benefits at age 62 or wait until they are eligible for full Social Security benefits. They provided the reader with a framework for computing the total present value of Social Security benefits for alternative retirement ages. However, the authors' present value approach leaves it up to the individual reader to determine the appropriate values to use for the rate of return that he could earn on assets and his life expectancy.
Woerheide (2000) provided an analytical framework for assessing the net salary of working one more year for individuals covered by defined benefit and defined contribution pension plans. He noted that if an individual chooses to work one more year, he would receive a salary for an additional year and higher pension benefits upon retirement. However, he would then receive pension benefits for one less year. Woerheide defined net salary as the individual's wage income for that additional year plus the present value of the increase in future retirement income less the retirement income he would have received in that year. Several conclusions follow from Woerheide's model: (a) the longer an individual expects to live, the higher the net salary from working one more year; (b) higher discount rates lower the net salary; and (c) the salary growth rate has a minimal effect on the net salary. While Woerheide provides an excellent analytical framework and includes the critical pension parameters in his analysis, the reader is left to his own devices in selecting values for the life expectancy, discount rate, and wage growth rate.
One approach that can be used to extend the analytical models proposed by Newmark and Walden and Woerheide is to construct a Monte Carlo simulation model of defined benefit pension plans. Monte Carlo simulation is a sampling experiment whose objective is to estimate the distribution of an outcome variable that depends on several probabilistic input variables. The term Monte Carlo was assigned to the technique by scientists who used it in the computer simulations of nuclear fission during the development of the atom bomb. The term Monte Carlo was assigned to the approach because of its similarity to the random sampling found in the games of chance played at the casinos of Monte Carlo (Evans and Olson 1998).
The Monte Carlo technique has been used for decades in the physical sciences. With the development of more powerful desktop computers beginning in the early 1990s, the technique became much more widely accessible. While researchers and practitioners in the areas of financial planning and investment management have been slow to utilize simulation modeling (McCarthy 2000), there have been some applications of the technique in these areas. Srikanth (2001) developed a Monte Carlo simulation model to evaluate the level of compensation risk for individuals who are employed by publicly listed companies and receive part of their compensation in the form of stock options. Wiess (2001) used Monte Carlo simulation to investigate whether dynamic rebalancing can add portfolio lifespan for a given initial withdrawal rate or increase the retirement portfolio's value for the target longevity. Abeysekera and Rosenbloom (2000) developed a Monte Carlo simulation model to compare the risk-reward trade-offs of lump sum and dollar-cost-averaging investment strategies.
A number of resources are now available to financial planning practitioners who wish to incorporate Monte Carlo simulation into the analysis phase of their practices. For example, Financial Engines (http://www.financialengines.com) provides a Website that provides a set of tools that the practitioner can access to make projections based on Monte Carlo simulations. T. Rowe Price (http://www.troweprice.com) provides a retirement income calculator that uses a Monte Carlo simulation model that simulates 500 potential market scenarios to estimate the probability of receiving a target level of income during the retirement period from a given portfolio. It also shows the spending rates that can be sustained based on the probabilities selected (Farrell, 2001).
In summary, there have been a number of applications of Monte Carlo simulation in some areas of financial analysis and retirement planning. However, the technique has not been used to examine the level of real average annual pension benefits that individuals covered by defined benefit pension plans would receive over their retirement period under alternative retirement ages.
Input Variables and Assumptions
The RVALB under alternative retirement ages is determined by (a) the wage growth rate for the additional years over which the individual chooses to work rather than spend in retirement, (b) the rate of inflation that is used to convert the stream of nominal benefits received over the retirement period into their real value at the time of retirement, and (c) the individual's life expectancy at the time of retirement. All of these variables are construed as probability distributions.
The probability distribution for the wage growth rate was constructed using the average hourly earnings data for private nonagricultural industries for the period 1947-99 as compiled by the Bureau of Labor Statistics and reported in The Economic Report of the President (various issues). This series was used because it is the only series available for the period that roughly encompasses the entire post World War II period. This time period was selected as the reference period because it encompasses the wide range of economic conditions that existed in the post World War II U.S. economy. By using such a long historical period to select the wage growth rate distribution, the simulation model provides a reasonably realistic range of wage growth rates that individuals contemplating continued employment could be expected to face. Over the reference period, the annual rate of change in the average hourly earnings of private nonagricultural industries ranged from a low of 2.2 percent to a high of 8.9 percent. Based upon inspection of the frequency distribution and the Kolmogorov-Smirnov goodness of fit test for the data points in the reference period, the wage growth rate variable is assumed to be approximately normally distributed. The mean and standard deviation of the wage growth rate for the period 1947-99 and the computed value for the Kolmogorov-Smirnov goodness of fit test for these data points are shown in Table 1.
The flow of nominal pension benefits over the retirement period was converted into the real value of benefits at the time of retirement using the U.S. inflation rate for the period 1947-99 as measured by the Consumer Price Index compiled by the Bureau of Labor Statistics and reported in The Economic Report of the President (various issues) as the inflation rate. The rate of inflation was used to convert the flow of nominal benefits into the real value of benefits because, under a defined benefit plan, annual nominal pension benefits are fixed, and their real value at the time of retirement is affected only by the rate of inflation. As was the case for the wage growth rate, the period 1947-99 provides a reasonably realistic range of inflation rates that those individuals contemplating retirement could be expected to face over their retirement period. Over the reference period, the annual rate of change in the price level as measured by the Consumer Price Index ranged from a low of -1.8 percent to a high of 13.3 percent. Based upon inspection of the frequency distribution and the Kolmogorov-Smirnov goodness of fit test of the data points for the reference period, the inflation rate variable is assumed to be approximately normally distributed. The mean and standard deviation for the inflation rate for the period 1947-99 and the computed value for the Kolmogorov-Smirnov goodness of fit test for these data points are shown in Table 1. The probability distributions for life expectancy are age, race, and gender specific. The probability distributions for life expectancy were obtained from the 1997 United States Life Tables (LJ.S. Department of Health and Human Services 1999). Based upon inspection of life expectancy frequency distributions and the Kolmogorov-Smirnov goodness of fit tests for different age, race, and gender categories, the life expectancy variable is assumed to approximate the lognormal distribution. The mean, standard deviation, and Kolmogorov-Smirnov goodness of fit test for the specific case used in the illustrative simulations are shown in Table 1.
Some Illustrative Simulations
Monte Carlo simulations were run using the risk analysis and forecasting program Crystal Ball (Crystal Ball 2000). Illustrative simulations are run for the case of a black male who is 55 years of age, currently earns $60,000 per year, has 25 years of employment, and is covered by the pension plan. Simulations are run for the following scenarios: (a) retirement at age 55 with 25 years of employment, (b) retirement at age 60 with 30 years of employment, and (c) retirement at age 65 with 35 years of employment.
The results of the simulations are shown in Figure 1 and Table 2. The real values of the average annual pension benefits for all three scenarios are computed for the time at which the individual is age 55. Figure 1 contains the frequency distributions of the forecasts generated by the 1,000 simulation replications for the alternative retirement ages. The mean forecast value of the real average annual benefits for retirement at age 55 with 25 years of employment is $15,083 and ranges from a low of $11,613 to a high of $18,667. The mean forecast value for retirement at age 60 with 30 years of employment is $22,073 and ranges from a low of $15,151 to a high of $31,027. The mean forecast value for retirement at age 65 with 35 years of employment is $22,742 and ranges from a low of $13,548 to a high of $36,460.
Table 2 provides further information generated by the simulations. As is reflected by the small difference between the mean and median values and the small sizes of the coefficients of skewness, the forecast values are distributed rather symmetrically about the mean forecast value. The values of the ranges, standard deviations, and coefficients of variation indicate that the variation in forecast values increases as the time that elapses before the retirement date is extended. This is partially attributable to the increased uncertainty in real pension benefits that are associated with variations in the wage growth rate as the individual extends his worklife. In addition, variations in the inflation rate have a greater impact on the real value of pension benefits that are received in the distant future than on benefits received in the near future.
Applications of the Simulations in Retirement Planning
There are a number of uses that can be made of the simulation model in the retirement planning process. For example, assume that an individual is covered by a defined benefit plan and has determined the amount of annual expenditures in today's dollars that will be necessary to sustain his desired lifestyle. The simulation results can then be used to determine the probability that the benefits from the defined benefit plan will provide for a given proportion of these expenditures during the retirement period. In the above illustration only the real value of the average annual pension benefits for the entire retirement period were simulated. However, the model could also be used to simulate the distribution of real benefits at any point during the retirement period. The individual could then determine the probability that the real benefits would fall below a prescribed level at different points in the retirement period. By assessing these probabilities for alternative retirement ages, the individual has more complete information with which to make the retirement decision.
The results of the pension simulations could also be combined with the results of simulations of the income stream generated by the individual's portfolio of assets that were accumulated to provide retirement income to assess the probability that the combined income from the two sources would provide the required level of income during the retirement period. Finally, if the individual has the simulation results of the real pension benefits for a target retirement age and finds that the probability of real benefits falling below the desired level is too high, he then will be able to better determine the additional years of employment and/or savings that will be necessary to provide the desired level of income with the desired level of certainty.
Summary and Implications
The purpose of this study was to examine how simulation modeling can be used as a tool to assist individuals who are attempting to select a retirement age under a defined benefit pension plan. The results indicate that ascertaining the real value of average annual pension benefits, even under a defined benefit pension plan in which the nominal annual benefits are fixed, involves uncertainty. This uncertainty is attributable to the uncertainty of the factors (i.e. the rate of inflation over the retirement period, the wage growth rate during the remaining years of employment and life expectancy at the time of retirement) that determine the real value of average annual pension benefits under alternative retirement ages.
Currently, a common approach to retirement planning by financial planning professionals is to simply assume single deterministic values for factors such as retirement income needs, the rate of inflation, life expectancy and the return on assets. The results of this study clearly show that projecting values for such factors is not quite that simple. Moreover, the conclusions drawn from such deterministic approaches may actually be misleading to clients and give them a false sense of control and security. By using a probabilistic approach to forecast factors such as real pension benefits, real income needs and real returns on portfolio assets, financial planning professionals will be able to provide their clients with a more realistic albeit less certain frame of reference for making their retirement decisions.
1 It is possible that the individual, in a three-year period prior to the year immediately preceding retirement, may have earned a higher average salary than the average salary in the three years immediately prior to retirement For example, he may have served on a temporary basis in a higher paying position and then returned to his prior position.
2 Different subgroups within the labor force have for some time periods in the past experienced different wage growth rates (Blackburn et al. 1990; Burtless 1990). In selecting wage growth rates for the simulation model, consideration was given to making the wage growth rate specific to the individual's demographic, occupational, and educational profile. However, the values for the wage growth rate used in the simulation model are based on annual data for a long historical period. Such data are not available for specific labor market subgroups. The lack of more subgroup-specific wage growth rates is mitigated by the fact that the variation in wage growth rates for a group of individuals covered by a single pension plan over a particular time period is likely to be significantly less than for the variation in wage growth rates for the overall labor force.
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Richard F. Bieker*
* Richard F. Bieker, School of Management, Delaware State University, Dover, DE 19901, email@example.com. The author is grateful to an anonymous referee and Joachim Zietz, JEF editor, for helpful comments.…