# «Chapter Two SIMULATION APPROACH AND ASSUMPTIONS In this chapter, we describe our simulation approach and the assumptions we make to implement it. We ...»

Chapter Two

## SIMULATION APPROACH AND ASSUMPTIONS

In this chapter, we describe our simulation approach and the assumptions we make

to implement it. We use this approach to address questions about the relative generosity of FERS compared with that of CSRS, about the retirement and separation incentives embedded in each system, and the incentives to switch to FERS by those

covered by CSRS.

The typical approach for comparing the relative generosity of benefits under different retirement systems is to compare their replacement rates at different retirement ages—the fraction of final pay that is covered by the retirement plan’s annuity (see, for example, GAO, 1997). A problem with this approach is that the replacement rate does not easily account for differences in contribution rates between retirement systems. For example, the annuity under one system may be more generous, but if employees contribute more of their earnings to the system, their expected lifetime wealth may not be greater.

Replacement rates also do not account for differences in cost-of-living-adjustment (COLA) provisions between systems. For example, an annuity may be larger under one system and yield a higher replacement rate, but if it is not inflation protected, the overall value of the benefit could be lower. The replacement rate approach also does not account for what the individual could earn in alternative employment. A retirement system may be more generous in terms of its replacement rate, but so may the replacement rate in alternative employment, so the individual may not be better off.

Other problems with the replacement rate approach are that it does not account for mortality risk or for the fact that the replacement rate may be higher at older retirement ages but the payout period shorter, resulting in potentially lower lifetime benefits.

Because of these flaws, we do not use replacement rates to compare benefits under FERS with those under CSRS. Instead we use a measure that accounts for such factors as mortality risk, contribution rates, payout length, COLA provisions, and the value of the alternative. Before developing our approach, we note that labor economists have developed several alternative models for analyzing retirement and separation decisions and for comparing the incentives embedded in alternative retirement systems. One class of models is based on stochastic dynamic programming (Gotz and McCall, 1980; Rust, 1989; and Daula and Moffitt, 1995). In stochastic dynamic programming models, the incentive to remain with an employer rather than 6 Separation and Retirement Incentives in the Federal Civil Service separating or retiring may be shown to be a weighted average of the incentive to remain one more period and then leave, two more periods and then leave, and so forth.

Weights are based on the individual’s probability of remaining to each future point and then separating. These probabilities depend, in turn, upon the individual’s preferences and upon random shocks to the stay-leave decision at each point in time.

Asch and Warner (1994) employ a version of the Gotz-McCall stochastic dynamic programming model to analyze the military compensation and personnel systems.

A simpler model is based upon deriving a future time horizon that is the focal point of current-period decisionmaking. This model has come to be known as the Annualized Cost of Leaving (ACOL) and is developed in detail in Warner and Goldberg (1984) and Black, Moffitt, and Warner (1990a and 1990b). Black, Moffitt, and Warner applied the model to the separation decisions of an entry cohort of DoD employees tracked for their first 10 years of employment. Lazear and Moore (1983) and Stock and Wise (1990) developed similar models of retirement decisions of workers in large firms.

ACOL is defined as the expected DPV of a person’s lifetime earnings, net of his or her wealth accumulation in an alternative job, annuitized over the length of the employment period. ACOL includes differences in future retirement pay and Social Security accumulations. The rest of the report will often refer to ACOL as the expected net lifetime earnings and retirement wealth of the individual. Since ACOL is annuitized DPV, it becomes the average annual pay differential between employment in the current job and the alternative.

In the economics literature, ACOL is also called the option value of staying in a given job. There has been much discussion about the relative strengths of the ACOL or option value approach and the stochastic dynamic programming approach (see the discussion in Stock and Wise, 1990; Lumsdaine, Stock, and Wise, 1992; and the exchange between Gotz, 1990, and Black, Moffitt, and Warner, 1990a and 1990b). Because we seek to describe only the retirement and separation incentives embedded in the two systems and do not attempt to estimate a structural model from actual data, we eschew the stochastic dynamic programming approach in favor of the simpler ACOL approach. This approach is well-suited for our purposes.

In both classes of models, nonmonetary factors also influence retirement and separation decisions. Models typically recognize two sources of nonmonetary disturbances. One source is “permanent preference factors.” Some jobs offer better working conditions and better amenities than do other jobs. Below, we let the symbol τ represent the value an individual places upon the nonpecuniary aspects of federal versus nonfederal employment. The other source arises from unexpected, or purely random, “shocks” to the retirement or separation decision. Poor health or an unexpectedly good job offer elsewhere are random factors that can cause even a person with strong preferences for the current employer (i.e., a high τ) to retire or separate.

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puter simulate the ACOL values for a “representative” individual. These computer simulations allow us to analyze the retirement and separation incentives embedded in FERS compared with those in CSRS.1

## DEFINING THE ACOL VARIABLE

To compute the ACOL, we subtract from the DPV of the employee’s future earnings from staying the DPV of his or her future earnings (i.e., earnings wealth) if he or she leaves the civil service immediately. The net earnings and retirement wealth if the employee leaves includes the DPV of pay in the alternative sector, the DPV of the civil service retirement benefit he or she would be eligible for upon leaving immediately as of the current period, and the DPV of the Social Security benefits that the employee would be eligible for at retirement.2 To account for differences in the length of time over which discounting is done when the career horizon changes, the net wealth measure is annuitized to create the ACOL variable. All dollars are discounted to the entry age.Formally, we denote the cost of leaving today, at time t, compared with that of staying until a future time period N as COL(N,t). If SN is the value of staying until period N, and Lt is the value of leaving today at time t, then COL(N,t) equals

The value of staying until a future period N is given by ______________

1 Our simulations focus on the provisions for immediate and deferred retirement under FERS and CSRS.

They ignore the provisions for “early” retirement. The early retirement benefit is available in certain involuntary separation cases and in cases of voluntary separations during a reduction-in-force. Since our focus is on normal voluntary separation incentives, we ignore this part of FERS and CSRS.

2 It should be noted that in computing the discounted present value of future retirement annuities we account for mortality risk in our calculations using a life table that gives the probability that an individual will survive to each age.

8 Separation and Retirement Incentives in the Federal Civil Service

The first right-side term in Eq. (2.3) is the difference between the DPV of earnings from a career path that includes staying N more years in the federal sector and then working T – N more years before withdrawal from the labor force (W NC + WN A) and the DPV of a T – t year career within alternative employment (WtA). The second term in Eq. (2.3) measures the increase in the DPV of retirement pay if the individual stays N – t more years rather than leaving immediately. Similarly, the third term measures the net change in the DPV of Social Security benefits as a result of N – t more years of employment in the federal sector.

Let β = 1/(1 + ρ) where ρ is the individual’s personal discount rate. Then, the annualized cost of leaving now rather than remaining N – t more periods is

Since ACOL(N,t) is the annuity equivalent of COL(N,t), ACOL(N,t) measures the average annual earnings differential between employment in the federal and nonfederal sectors, including not just pay differences while employed but also differences in expected future retirement benefits between sectors as well as differences due to Social Security accumulations.

**DECISION RULE FOR DETERMINING OPTIMAL RETIREMENT AGE**

Notice that there are T – t values of ACOL for a given individual: ACOL(t + 1,t), ACOL(t + 2,t),..., ACOL(T,t) or N – t values from ACOL(t + 1,t) through ACOL(N,t).

To determine the optimal retirement age, we assume that the individual stands at the entry age (i.e., t is assumed to equal 1) and looks at every possible future career horiSimulation Approach and Assumptions 9 zon N, calculates ACOL(N,1), and chooses the N or age where his or her expected ACOL(N,1) is maximized (denoted ACOL*(N,1)). At this point, the person will maximize his or her expected net earnings and retirement wealth over his or her lifetime relative to the entry age.

As will be discussed in the context of separation incentives, individuals may leave before the optimum career length once nonmonetary and random factors are considered. For example, if the individual receives an unexpectedly good outside opportunity that exceeds this maximum or if he or she finds that the disamenities of the civil service outweigh this maximum, then he or she will leave prior to the age when the ACOL is maximized. That is, the ACOL indicates the financial net gain to staying (or the financial cost of leaving) over the time horizon that maximizes wealth, but other factors can also influence the decision to leave.

As an example, Figure 2.1 graphs ACOL(N,1) for alternative N and shows the N at which we find ACOL*(N,1). For someone who enters at age 40, the maximized ACOL(N,1) is ACOL*(20,1), i.e., the ACOL is maximized at 20 YOS and age 60.

To compare retirement age incentives under FERS with those under CSRS, we simulate ACOL(N,1) for a representative individual under FERS and under CSRS, holding entry age constant. We then find the N where ACOL*(N,1) occurs for each system. If the maximized ACOL(N,1) occurs at the same N, we conclude that FERS and CSRS embed the same retirement age incentives, given our assumptions.

Figure 2.1—Annualized Cost of Leaving by Leaving Age 10 Separation and Retirement Incentives in the Federal Civil Service

## DECISION RULE FOR WHETHER TO SEPARATE AT A GIVEN AGE

Although the time horizon over which ACOL is maximized, ACOL*(N,1), indicates the optimal retirement point, it does not, by itself, indicate whether a person will remain until that point or separate. As noted above, preferences and such random factors as sudden ill health or an unexpectedly strong or weak economy will also affect the separation decision in period t.To examine separation incentives at each t, we no longer set t equal to 1 as we do in our examination of retirement incentives. Instead, we let t vary, and we find ACOL*(N,t) for every t, given entry age. We then compare ACOL*(N,t) to the value of nonmonetary and random factors. Formally, if τ is an individual’s net preference for federal employment and ε t denote random shocks to the current-period separation decision, an individual remains in federal employment at time t if ACOL*(N,t) + τ + εt 0 or ACOL*(N,t) –( τ + ε t). In other words, the individual stays in period t if the maximum expected future annualized pay differential (or expected net lifetime wealth) exceeds his or her net preference for nonfederal employment plus the (negative of the) value of new shocks to the decision.

As the individual progresses through his or her career, he or she is assumed to compare ACOL*(N,t) with τ + εt when deciding whether to separate at time t. As it turns out in our simulation analysis of FERS and CSRS, we find that the age or the N at which ACOL(N,t) is maximized does not generally vary with t. In other words, ACOL*(N,t) maximizes at the same N, for all t, holding entry age constant. For example, for someone who enters the civil service at age 20, ACOL(N,t) is maximized at age 55 when the individual would have 35 YOS (i.e., N equals 35 at the maximum).

We find that, if it was optimal to stay until age 55 at the beginning of the individual’s career, it is usually optimal for him or her to stay until 55 as the career progresses and the individual ages.

Although the N at which ACOL*(N,t) occurs does not vary with t, ACOL*(N,t) does vary with t. For example, for someone who enters the civil service at age 20, ACOL*(35,1) may equal $4,000 for someone contemplating leaving after the entry age.

If the leaving decision is contemplated at age 30, ACOL*(35,10) may equal $7,000. If it is contemplated at age 40, ACOL*(35,20) may equal $13,000. Figure 2.2 illustrates this example. The age or N at which ACOL*(35,1), ACOL*(35,10), and ACOL*(35,20) occurs is 55. Nonetheless, ACOL*(35,t) varies with t when t equals 1, 10, or 20. Since ACOL*(N,t) varies with t, there may be some t’s at which ACOL*(N,t) –( τ + εt) and therefore at which it is optimal for the individual to leave the civil service.