Civic
Report
No. 90 June 2014
Better Pay, Fairer Pensions II:
Modeling Preferences Between DefinedBenefit Teacher Compensation Plans
Josh McGee, Vice President of Public Accountability, Laura and John Arnold Foundation
Marcus A. Winters, Senior Fellow, Manhattan Institute
Executive Summary
Most public school teachers in the United States participate in traditional, finalaveragesalary definedbenefit (FASDB) pension plans. A common, but underappreciated, feature of FASDB plans is their heavy reliance on backloaded retirement benefits. Under these plans, teachers generally earn relatively meager benefits during their first several years in the classroom and then rapidly accrue far more valuable benefits late in their careers, as they near their plan’s retirement eligibility thresholds. At present, the majority of teachers leave—to take teaching jobs in other states or to work in the private sector—before earning substantial benefits.
In a recent paper (Manhattan Institute Civic Report no. 79, “Better Pay, Fairer Pensions: Reforming Teacher Compensation”), we compared the backloaded benefits offered under existing FASDB plans, in each of the ten largest U.S. school districts, with a less backloaded, alternative definedbenefit plan that we refer to as a smoothaccrual definedbenefit (SADB) plan. Using each pension plan’s own assumptions about teacher attrition, we showed that the majority (and sometimes the vast majority) of teachers would be better off under a less backloaded retirement plan.
Nonetheless, teachers’ preference between these two DB plans is not immediately clear from basic descriptive comparisons. On the one hand, SADB plans ensure that all teachers would leave with meaningful retirement benefits, even if they do so before reaching their plan’s retirement age. On the other hand, FASDB plans offer larger benefits to teachers who work their entire careers under a single plan.
This paper builds upon our previous work to answer the question: Which of these designs should the rational teacher prefer at entry? For each of the ten largest U.S. public school systems, we use a standard economic model—incorporating variables such as level of financial risk aversion and degree of uncertainty over future career plans—to determine entering teachers’ preferences between current backloaded FASDB pension plans and our less backloaded SADB alternative.
Key findings include:
 When offered a choice between two plans with the same expected retirement benefit, a rational, riskaverse teacher would consistently prefer an SADB plan because it offers a smaller difference between potential benefit payouts.
 The magnitude of preference for an SADB plan depends both upon the severity of an FASDB plan’s backloading and the amount of early and midcareer teacher turnover. Yet in nearly every case, the strength of teacher preference for the less backloaded system (SADB) is large, given reasonable, empirically grounded assumptions about risk aversion.
 Indeed, certain FASDB plans provide remarkably little retirement security to entering teachers. For instance, a new teacher in Hawaii’s school system with a typical level of risk aversion would be wiser to accept a lumpsum payment of merely $279, rather than participate in the system’s pension plan (some 77 percent of Hawaii’s entering teachers will leave the system before earning $279 in retirement compensation).
 As for teachers who expect to work under the same system for many years—who would, accordingly, experience a higher probability of benefiting from backloading—many would nevertheless benefit from a less backloaded SADB system.
 For example, teachers certain to work under the same retirement system for five years would still prefer the alternative SADB plan over current FASDB plans. Likewise, in six of ten school systems—New York City, Chicago, Philadelphia, Clark County (Nevada), Hawaii, and Houston—a teacher certain to remain for at least ten years would prefer the SADB plan. (Teachers certain to work under the same retirement system for 20 years would, however, universally prefer an FASDB plan.) In reality, moreover, few can forecast with absolute certainty that they will remain employed under the same retirement plan for the ensuing twenty, ten, or even five, years.
In short, this paper reinforces our earlier findings: reforms to teacher compensation in favor of SADB plans would help school districts offer significantly more attractive teacher compensation packages, without the need for higher taxes or reduced services.
About the Authors
Josh McGee is vice president of public accountability at the Laura and John Arnold Foundation. McGee also serves as an adjunct faculty member at Rice University where he has taught in the Rice education entrepreneurship program at the Jones Graduate School of Business. McGee has produced highquality, policyrelevant research spanning a number of important areas, including public pension structure, cost and labor market effects, K12 education policy, and economic development. His work has appeared in scholarly journals including the Journal of Development Economics, Education Finance and Policy, and Education Next. Throughout his career, McGee has worked to actively shape public policy. He has provided expert testimony, policy advice, and technical assistance on the topics of K12 education policy and public pension reform in a number of states, including Arizona, Arkansas, California, Florida, Kentucky, Illinois, Rhode Island, and Texas. During his tenure with the Foundation, McGee has focused primarily on addressing the problems with the nation’s public retirement systems by educating the public and policymakers about the nature and size of the problem, as well as potential structural reforms that would create a retirement system that is affordable, sustainable, and secure.
McGee holds a B.S. and M.S. in industrial engineering and a Ph.D. in economics from the University of Arkansas. McGee contributed to this report as an independent scholar and not in his official capacity at the Laura and John Arnold Foundation.
Marcus A. Winters is a senior fellow at the Manhattan Institute and an assistant professor at the University of Colorado Colorado Springs. He conducts research and writes extensively on education policy, including topics such as school choice, high school graduation rates, accountability, and special education. Winters has performed several studies on a variety of education policy issues including highstakes testing, performancepay for teachers, and the effects of vouchers on the public school system. His research has been published in the journals Journal of Policy Analysis and Management, Educational Researcher, Educational Evaluation and Policy Analysis, Education Finance and Policy, Educational Finance, Economics of Education Review, and Teachers College Record. His oped articles have appeared in numerous newspapers, including The Wall Street Journal, Washington Post, and USA Today, and he is often quoted on education issues.
Winters received his B.A. in political science from Ohio University in 2002 and Ph.D. in economics from the University of Arkansas in 2008.
Introduction
Since the 2008 financial crisis, public pension reform has gained national attention. Policy discussions around public pensions have, in turn, primarily focused on cost: unfunded liabilities and annual payments tied to such plans have increased substantially in recent years, resulting in difficult fiscal tradeoffs in several jurisdictions. Meanwhile, the benefits side of the pension equation has received relatively scant attention. As states and school systems across the nation alter their retirement benefits, now is an ideal time to also consider whether the benefits offered under current systems meet the needs of today’s workforce.
Too often, pension reform is characterized as a binary choice between finalaveragesalary definedbenefit (FASDB) plans—whereby workers receive an annuity based on years of service and final average salary—and definedcontribution (DC) plans similar to 401(k) plans common in the private sector. Indeed, numerous previous papers have considered workers’ preferences between DB and DC systems and have found each style of plan to be desirable under certain circumstances.[1]
Comparisons between FASDB and DC plans are, however, complicated by the large number of different plan features built in to such analyses and, generally, do not consider numerous other available pension structures. What’s more, many, if not all, of the assumed plan features are not inherent to one particular model and could instead be incorporated into any retirement savings plan.
One widely known attribute of definedbenefit plans is that they generally provide workers protection from investment risk. As long as the plan is well designed, well managed, and well funded, DB plan benefits do not vary with market returns. All else equal, riskaverse workers would prefer a plan providing investment protection, over the plan that does not.
What is less well understood is that retirement benefits under the dominant publicsector model, FASDB, are inherently backloaded—above and beyond accumulated contributions and interest. Backloading, in turn, subjects entering teachers to a different type of risk: attrition. In practice, teachers generally earn relatively meager pension benefits during their first several years in the classroom (often the first 20 years or so) and then rapidly earn much more valuable benefits as they near their plan’s retirement eligibility thresholds late in their careers. As a result, employees who exit a system before reaching retirement eligibility often are left with benefits much less valuable than those who work until their plan’s retirement eligibility threshold.[2]
In this paper, we isolate one important retirement plan feature—namely, how workers earn benefits across their careers—and consider reforms that would alter this feature strictly within the DB structure. The reforms we model simply shift benefits within teachers’ careers, are costneutral to taxpayers, and would continue to offer teachers investment protection and lifetime annuities.
In a 2013 study,[3] we compared the pattern of retirement benefit accrual for new teachers in existing FASDB plans, in each of the ten largest U.S. school districts, with an actuarially equivalent, but less backloaded, DB plan that we refer to as a smoothaccrual definedbenefit (SADB) plan. As its name suggests, under the SADB plan, teachers earn benefits equal to a constant percentage of cumulative wages, resulting in a smoothed benefit accrual pattern. Under an SADB, the value of teachers’ retirement benefits (payable as a lifetime annuity) equals annual contributions, plus guaranteed earnings on those contributions.
Importantly, teachers’ preference between these two DB plans is not immediately clear from basic descriptive comparisons. On the one hand, SADB plans ensure that all teachers exit with meaningful retirement benefits, even if they exit before the plan’s retirement age. On the other hand, current FASDB plans generally offer larger benefits to teachers who work a full career under the same plan than do costequivalent SADB plans. This paper—by estimating the magnitude of entering teachers’ preferences between these two plans, given various conditions—expands upon our earlier work.
I. Modeling Current Teacher Pension Systems
The majority of public school teachers in the United States earn retirement benefits under an FASDB structure. In such systems, teachers earn a lifetime annuity, which can be accessed once they reach their plan’s retirement eligibility thresholds. The starting annuity amount increases as teachers accrue more years of service in the same system and as their salaries increase. The dollar value of an employee’s starting annual annuity for a given age at separation a_{s} and age a_{t} retirement ar is given by equation (1) below.
In equation (1): B is the starting annual annuity beginning at age a_{r}, given age of separation a_{s}; M is the benefit multiplier; R is an indicator for retirement eligibility; E is the percent reduction for early retirement; YOS is the number of years worked for the sponsor; and FAS is final average salary.
The present value of a teacher’s retirement benefit, PVB, can be calculated at various ages of separation, a_{s}, using standard actuarial techniques (see Appendix 2).[4] In principle, PVB is the cash value of the annuity that a teacher has earned to date.[5] Teachers should, in theory, be indifferent as to whether they receive this lumpsum payment or the annuity.
We calculate the present value of teachers’ retirement benefits, net of their contributions (i.e., isolating the portion of the benefit funded by the employer). Teachers will be interested in the total benefit provided under the plan; but from a retirement income perspective, netting out the value of their own contributions provides a measurement of the employerfunded benefit, or retirement compensation. Looking at benefits, net of teacher contributions, allows us to better understand how retirement benefits fit into teachers’ total compensation package—and to analyze whether teachers might prefer a different pattern of compensation.
II. Modeling The SmoothAccrual DefinedBenefit (SADB) Plan
In this paper, our aim is to compare teachers’ preference between current retirement savings plans (described in Section I) and less backloaded, costequivalent definedbenefit plans. We model a plan in which benefits earned by teachers are a constant percentage of their cumulative earnings. In pension parlance, the constant accrual rate for our SADB plans is equal to the normal cost of benefits under the current FASDB plans, calculated using the “entry age normal” method.
The SADB percentage is calculated by dividing the expected value of future retirement benefits, at age of workforce entry, by expected cumulative wages at entry (see calculations in Appendix 3). The SADB system modeled below is equivalent to a cashbalance definedbenefit retirement plan, where employer contributions equal the SADB percentage, and annual interest earned on those contributions equals the interest rate used to discount liabilities (5 percent, in our case).
Table 1 shows the calculated SADB percentage for the plans currently offered by each of the ten largest U.S. school districts. To produce the SADB wealthaccrual patterns, we multiply this constant rate by teachers’ cumulative earnings, at each point in their careers.
There are meaningful differences in the generosity of pension plans across systems. And though, as we will see, the distribution of benefits across teachers’ careers produces substantial differences in the benefits actually earned by individual teachers within each system, most school systems offer a meaningful benefit valued at near 10 percent or more of salary per year. However, two school systems (Hawaii and Philadelphia) offer retirement compensation on average that amounts to only slightly more than 5 percent of salary. Meanwhile, Los Angeles teachers receive retirement compensation that is the equivalent of 18.51 percent of pay.
Importantly, these calculations leave out the value of Social Security, in which six of the ten districts participate. When Social Security contributions are included, teachers in these districts are generally offered average retirement compensation that is more generous than what is commonly offered in the private sector.[6] However, as we will see in the next section, most teachers receive retirement compensation that is far less valuable than the average, while a select few receive retirement packages far in excess of these averages.
III. The Pattern of Retirement Benefits Accrual
Under the dominant retirement plan offered to teachers today, retirement compensation is heavily backloaded—far more so than other elements of total compensation. Figure 1 provides an illustration of retirement compensation over the career of a 25yearold entrant into New York City’s public teaching workforce. The figure presents three lines, including: (i) the present value of the total benefit, net of employee contributions (retirement compensation); (ii) costequivalent smooth accrual; and (iii) cohort survival, according to the plan’s decrement table. (Appendix 1 presents similar figures for each of the nation’s ten largest school districts.)
We first consider pension benefits under the current system (black line). Figure 1 clearly illustrates the current backloaded nature of retirement compensation. Teachers earn relatively meager retirement benefits early in their careers. The rate at which they accrue benefits then increases rapidly later in their careers, illustrated by the steepening of the line around age 50. In New York, the actuarial maximum pension benefit is earned at age 65. After 65, teachers’ benefits actually lose value each additional year that they remain employed by the school system: each year in the classroom, after reaching retirement eligibility, represents one year fewer of pension payments.
The dark grey line represents pension benefits under the alternative SADB plan. Unlike the current FASDB, teachers earn benefits at a relatively constant rate throughout their careers. As a result, teachers earn significantly more valuable benefits early in their careers than they would under the existing plan. At the same time, teachers who remain in the system for a long period of time earn less under SADB than they would under FASDB.
Because we are modeling a costequivalent plan, the value of SADB benefits, relative to the current FASDB, is driven by the rate at which teachers leave the plan in the early and middle portions of their careers. The SADB line in Figure 1 can also be viewed as the current pension plan’s expected value of employer contributions and earnings on those contributions over time. Under the current FASDB system, for a portion of teachers’ careers, the contributions made by their employer are more valuable than the benefits they have earned. When workers leave the system during this period, contributions made on their behalf in excess of the value of their benefits are used by the plan to pay teachers whose benefits exceed the employer contributions made on the latter’s behalf, respectively. In other words, a costequivalent SADB plan only offers less valuable benefits relative to the FASDB plan late in workers’ careers—to the extent that the current system relies on teacher turnover to keep total employer cost low. Finally, the light grey line addresses the issue of teacher turnover: the pension plan’s own assumption for the percentage of entering teachers expected to remain employed under the retirement plan until a given age. Figure 1 demonstrates that by the time benefits under the two plans are equal—represented by the point at which the FASDB and SADB lines cross (age 56)—only about 42 percent of teachers who started within this cohort are expected to be still employed by the New York City system. That is, 58 percent of teachers are expected to leave before they would benefit from the backloaded structure of the current plan.
Even in New York City, which has very low teacher turnover for an urban district, the majority of 25yearold entering teachers are not expected to make it to the point where their retirement compensation is at least as valuable as the employer contributions made on their behalf. Indeed, the issue affects far more teachers in certain other systems that impose more severe backloading and have higher earlycareer attrition.
Philadelphia is one such example: the city’s plan heavily backloads pension benefits late in teachers’ careers (Figure 2). Meanwhile, teacher attrition in Philadelphia is very high for earlycareer teachers. As Figure 2 shows, teachers who would remain in the system until age 50 would benefit from the existing system. However, only about 18 percent of teachers who enter at age 25 are expected to remain employed under the plan until age 50.
IV. Uncertain Teacher Preferences Between Plans
Thus far, we have discussed how teachers earn retirement benefits under current plans and less backloaded, costequivalent plans. Relative to the current FASDB plan, the SADB plan would provide higher benefits to teachers who separate earlier in their careers—at the expense of lower benefits for teachers who work under a single plan until reaching retirement eligibility.
If a teacher were to know at the beginning of his career the exact number of years he would work under the same retirement plan, his preference between the systems would be obvious. Alas, there is little reason to believe that entering teachers are able to reliably forecast their tenure in a single retirement system.
We consider teacher preferences between plans by treating the issue as a choice under uncertainty. Each retirement plan yields a series of discrete payoffs across service that entering teachers can expect to achieve with probability (p). The series of payouts, combined with teachers’ uncertainty about their tenure, constitutes a “lottery” for retirement benefits. We examine which pension plan (i.e., which lottery) would be preferred by an entering teacher, given preferences about risk.
Our primary analysis considers the preferences of a hypothetical 25yearold female entrant to the teaching workforce, one naive about the number of years she will work under her current pension plan. That is, we assume that our entering teacher’s probability of exit matches the historical pattern of teacher attrition, within that particular pension system. We use the pension plans’ decrement tables—assumed hazard rates across age and service—to calculate the exit probabilities for our hypothetical teacher. Because our exit probabilities are tethered to the plans’ expected attrition rates, we argue that this analysis offers the best estimate for an average entering teacher. Later, we consider the preferences of teachers who enter with the knowledge that they are certain to remain in the plan for a minimum number of years (and thus may have different preferences from those of teachers lacking such certainty).[7]
In our model, teachers consider their expected retirement compensation when they enter employment at a school district. When offered a choice between two plans, a riskneutral entering teacher would prefer the lottery offering the highest expected value, at the point of entry. By design, the SADB plans we evaluate have the same expected value as the respective current system with which it is directly compared. Thus, a riskneutral entering teacher, naive about the number of years she will remain in the classroom, is indifferent as to which plan she wants.
Empirical research, however, suggests that workers are at least somewhat riskaverse.[8] In addition, at least one laboratory experiment finds that teachers are particularly riskaverse.[9] As such, we can meaningfully expand upon our analysis by considering teacher preferences under various assumptions accounting for their aversion to risk.
V. RiskAverse Individuals Prefer the SADB Plan to Current FASDB Plans
While the SADB plans that we model are designed to have the same expected value as current plans, they have smaller variance than do current backloaded plans (see Appendix 3). Table 2 reports the standard deviation for pension benefits accrual under each current, and alternative, SADB plan. Together, these facts imply that a riskaverse entering teacher would always prefer the SADB plan because less backloaded plans secondorder stochastic dominate current plans.[10] The proof derives from the argument regarding mean preserving spreads of distributions. In short, when offered choices between two distributions with the same expected value, a riskaverse teacher will prefer the distribution with smaller variance: the SADB plan offers the same expected return, but less difference between potential benefit payouts.
VI. Considering the Magnitude of Teacher Preferences
Though it is a potentially important finding that riskaverse teachers would prefer a less backloaded plan to their current plan, the analysis thus far does not indicate the magnitude of this preference. This is an essential factor to consider, if only because altering the structure of public pension systems is a difficult logistical and political undertaking—and therefore might only be worthwhile (from a policymaker’s perspective) if teacher preference for the SADB system is substantial.
To estimate the magnitude of teachers’ preference for the less backloaded plan, we must specify a utility function. The utility function is a tool commonly used by economists to map an individual’s wellbeing (often thought of as “happiness”), under different states of the world, by formalizing the relationship between changes in a given good (in our case, retirement benefits) on an individual’s rate of satisfaction.
Consistent with similar previous research, we assume that teacher utility follows a CRRA isoelastic utility function, taking the form in equation (2) below:
In equation (2): W represents the present value of retirement benefits accrued at time of separation (as considered by the entering teacher at time of hire) and η represents the coefficient of risk aversion. Where η = 0, individuals are riskneutral; as η increases, individuals become more riskaverse. We consider teacher preferences, under various assumptions, for aversion to risk. Previous empirical research in economics suggests that η = 0.71 offers a good approximation of workers’ risk aversion.[11] We might consider this finding, based on research evaluating all workers, to be a lower bound in the case of teachers.
For this analysis, we use previously determined values for the present value of pension benefits, as well as the probability of exiting the retirement plan, to calculate the teacher’s expected utility from each plan’s lottery (see Appendix 2).
We then convert these utility calculations into the certainty equivalents for each lottery and level of risk aversion. The certainty equivalent is a conventional economic calculation representing the dollar amount that an entering teacher would be willing to accept, in lieu of participating in a respective retirement plan.
Table 3 shows the certainty equivalents for each system, under a variety of assumed levels of risk aversion. The plan that produces the higher certainty equivalent at a given level of risk aversion is preferred by the average teacher entering the system. Intuitively, an entering teacher would require a larger upfront payment to be persuaded to forgo participation in the preferred plan, versus the alternative plan. The third column for each system reports the percentage increase in the certainty equivalent for the SADB plan, relative to that of the respective current system. Because the plans, by design, have the same expected value, the plans have the same certainty equivalent for a riskneutral teacher.[12] As the assumed level of risk aversion is increased, so too does the difference between the certainty equivalents for the plans.
As an example, consider the values for New York City. Under both plans, entering teachers are expected to earn average retirement compensation equal to $98,380—shown by the certainty equivalent for a riskneutral teacher (η = 0). An entering teacher, naive about her tenure, should be indifferent as to whether she accepts $98,380 when she is hired or participates in either of the retirement plans. (To counter the risk of exiting the system before earning the equivalent retirement benefits by participating in the plans, the teacher would, in theory, be willing to accept this upfront payment.)
A riskaverse teacher, on the other hand, would be willing to accept a lower amount up front, in lieu of participating in the pension plan, because she is more concerned about the risk that she might leave the system having earned less than the average amount. According to Table 3, a 25yearold teacher entering the New York City public school system who exhibits risk aversion—such that η = 0.3—would be indifferent as to whether to accept an upfront payment of $72,847 or participate in the current FASDB retirement plan. In comparison, we calculate that the same teacher would require an upfront payment of $87,077, in lieu of participating in the SADB plan. Thus, because she would require a higher upfront payment to forgo participating in the SADB plan, this suggests that riskaverse teachers would prefer the SADB plan over the current FASDB plan. Reasoned intuitively, this preference occurs because the SADB plan offers a larger benefit early in teachers’ careers, thereby reducing the penalty that teachers face for exiting the system prior to retirement eligibility.
Consistent with the previous example, once risk aversion is introduced (η > 0), the SADB plan is preferred over the current plan in each case. Indeed, the preference for the SADB plan increases as risk aversion increases.
The magnitude of the preference between a given current system and its SADB counterpart differs substantially across systems, depending on the severity of the current plan’s backloading and probability of remaining employed under the same plan for a prolonged period of time. For instance, in the heavily backloaded Philadelphia system, even at the relatively low level of risk aversion (where η = 0.3), we see that the teacher would require a 70.7 percent higher payoff at entry in lieu of participating in the SADB plan, than he would require in lieu of participating in the current FASDB plan. On the other hand, even at η = 0.7, the payoff required—for a teacher to choose not to participate in the SADB Chicago plan—is only about 15.6 percent higher than she would require not to participate in Chicago’s relatively less backloaded FASDB system.
In cases where benefits are heavily backloaded, with significant earlycareer turnover, certainty equivalents for the current system can appear so low, once risk aversion is introduced, that they seem implausible. In Hawaii, for instance, the certainty equivalent for an individual with η = 0.7 (a level consistent with empirical research) is only $279. In other words, an entering teacher should be willing to accept $279 at the time of hire, rather than participate in the system’s pension plan. But when one considers the fact that 77 percent of Hawaii’s entering teachers are expected to leave the system before earning $279 in retirement compensation (see Appendix 1), this finding no longer seems so odd.
VII. Preferences of Teachers With High Expectations For Longevity in System
Thus far, we have considered the preferences of teachers who enter a school system with no particular expectation of the amount of time that they will remain employed under the same plan. Since our “naive” teacher’s exit probabilities are based on real attrition rates, we argue that, on average, this analysis is the best way to consider the preferences of teachers.
However, we might be especially concerned with the preferences of teachers who expect to remain in a particular school system for a sustained period of time. Schools might wish to attract teachers who expect to remain in the system for their entire careers—or, at least, longer than the average teacher. Retaining, as well as rewarding, employees is a frequent justification for the backloaded nature of current pension plans.[13]
As Appendix 1 makes clear, current plans are quite beneficial for teachers working under the same plan long enough to benefit from the steep benefit accrual earned late in their careers. For this reason, entering teachers anticipating long careers under the same retirement system might prefer current plans to less backloaded ones.
Any given teacher, it is true, might enter a school system with any (among an infinite) number of expectations for his or her tenure. To limit our analysis to a manageable number of alternatives, we therefore consider teachers certain to remain in a particular school system for a fixed number of years: five, ten, or 20 (i.e., we consider the preferences of a hypothetical teacher with a 0 percent chance of exiting the retirement plan within the first five years, and so on). We utilize, as before, information from the respective pension plan’s decrement tables for the probability that a teacher exits after any given year of service—but we assign a probability of exit at earlier years of service (five, ten, or 20) to be 0.
The certainty equivalents for certaintoremain teachers with riskaversion values equal to 0.3 or 0.7 are reported in Table 4. We use boldface type to highlight the preferred plan in each comparison. We focus our discussion on the case where η = 0.7—the level suggested by previous empirical findings.
In all plans, teachers certain to remain in the same school system for 20 or more years prefer the current FASDB plan to the alternative SADB plan. At first blush, this might surprise, given that in each of the districts, the SADB system is more valuable than the FASDB at 20 years of service. However, at this point in a teacher’s career, she is relatively close to the maximum benefit offered under these plans, and the probability that she leaves the system sometime over the next ten years is quite low.
Meanwhile, the story is entirely different for teachers with lower levels of certainty about their tenure. For each of the plans in our analysis, teachers only certain to remain in the retirement system for at least five years prefer the SADB plan. In addition, in six systems—New York City, Chicago, Philadelphia, Clark County, Hawaii, and Houston—a teacher certain to remain for at least ten years would continue to prefer the SADB plan over the existing pension plan.
The results in Table 4 are particularly revealing, given the extreme nature of the test under consideration: few, if any individuals, at the time they are hired, can realistically say with absolute certainty that they will remain employed under the same retirement plan for the ensuing ten years. Our result suggests that even if such an individual did exist, he would prefer, under a conventional assumption for risk aversion, the SADB plan to current backloaded FASDB plans in the majority of the ten largest American school districts.
Conclusion
In this paper, we used information from pension plans in the ten largest U.S. school districts to demonstrate that—under a variety of conditions—teachers would prefer less backloaded retirement benefits. We have also demonstrated that—under reasonable assumed levels of risk aversion—teachers’ preference for a less backloaded plan is often quite large.
For policymakers, our findings offer several important insights. Most startling, perhaps, is our calculation of just how undesirable many current pension plans are from a teacher’s perspective, after accounting for even the slightest amount of risk aversion. Indeed, some current plans are so heavily backloaded—offering entering teachers such little probability of earning meaningful retirement benefits—that riskaverse teachers would be willing to accept relatively small certain amounts at the time of hire, rather than participate in the pension plan. All of this strongly suggests that future teachers might greatly benefit from less backloaded benefits.
In addition, our analysis makes clear some similarities, as well as meaningful differences, between plans offered in districts across the United States. Each of the current systems considered is backloaded and can, in fact, be improved (from the riskaverse teacher’s perspective) by adopting a less backloaded plan. However, the magnitude of the preference varies considerably across systems: some current pension plans are severely backloaded, and others less so.
To our knowledge, our paper represents the first formal consideration of teacher preferences between current FASDB systems and less backloaded alternative definedbenefit plans. Additional research is, admittedly, needed to ascertain all the effects of a policy change toward the latter. In particular, future research should more formally consider the likely impact of such a change in pension plans on teacher attrition patterns—and, consequently, on the distribution of teacher quality within American public schools.
Appendix 1. EmployerSponsored Retirement Wealth over Time, Under Alternative Systems
Appendix 2. Calculating Net Present Value of Retirement Wealth
The present value of an employee’s retirement benefit, PVB, can be calculated at various ages of separation, a_{s}, using standard actuarial techniques.[14] Retirement rules may allow employees to begin receiving an annuity immediately—or may require them to defer until meeting the retirement eligibility thresholds. The present value of the employee’s retirement benefit, at any given age, is given by equation (A1) below. The equation calculates the maximum pension benefits that an employee may achieve at each age, a_{s}.
In equation (1) (see Section I), B(a_{r}a_{s}) is the starting annuity that an employee would begin receiving at the age of a_{r}, given that the employee separated at age a_{s}; AF is the annuity factor, representing the value of a dollar of annuity, beginning at the age of retirement a_{r}; f(a_{r}a_{s}) is the conditional probability of survival from a_{s} to a_{r}; and r is the interest rate used to discount future cash flows.
In principle, PVBa_{s} represents the cash value of the annuity that an employee has earned at age a_{s}.[15] An employee should be indifferent as to whether to receive the lump sum PVBa_{s} or the annuity B(a_{r}a_{s}).
The present value of an employee’s retirement benefit can be calculated net of employee contributions (i.e., isolating the portion of the benefit funded by the employer), PVB_{net}, shown in equation (A2) below. TotCont represents cumulative employee contributions, up to a specified age. While workers will be interested in the total benefit provided under the plan (from a retirement income perspective), netting out the value of worker contributions provides a measurement of the employerfunded benefit, or retirement compensation. Looking at benefits, net of worker contributions, allows us to better understand how retirement benefits fit into workers’ total compensation package— and analyze whether workers might prefer a different pattern of compensation.
Appendix 3. Calculating the Constant Accrual Rate Under the SmoothAccrual DefinedBenefit Plan
In this paper, our aim is to compare the current retirement savings system, described in Section I, with a costequivalent system featuring a smoothaccrual pattern across a worker’s career. (By “smooth,” we mean that the benefits earned by teachers are a constant percentage of their cumulative earnings.) In pension parlance, the constant accrual rate in our SADB system is equal to the normal cost of benefits calculated using the Entry Age Normal method. The SADB percentage is calculated by dividing the expected value of future retirement benefits (at age of workforce entry) by expected cumulative wages at entry. The SADB system, modeled below, is equivalent to a cashbalance definedbenefit retirement system, where employer contributions are equal to the SADB percentage—and annual interest on those contributions is equal to the interest rate used to discount liabilities (5 percent, in our case).
Equation (A3)—the numerator of the SADB percentage formula—calculates the expected value of retirement benefits standing at entry age, a_{e}, where g(a_{s}) represents the separation probability distribution for a given entry age.[16] The summation extends to the last possible age at which an employee might separate from employment, a_{z}.
Equation (A4)—the denominator of the SADB percentage formula—calculates expected cumulative wages for an employee entering the pension plan at age a_{e}.
Equation (A5) represents the SADB percentage, the constant percentage of cumulative wages resulting in a smoothaccrual pattern that is costequivalent to the current backloaded FASDB accrual pattern. Equation (A5) is simply the quotient of equations (A3) and (A4). Importantly, the smoothaccrual percentage is specific to a particular entry age.
Appendix 4. Utility and Calculating the Certainty Equivalent
For this analysis, we use previously determined values—for the present value of pension benefits and probability of exiting the pension system at given years—to calculate the teacher’s expected utility from each lottery. The expected utility from a lottery equals the sum of the utility products deriving from the retirement benefits accrued from leaving the system at time period i (U_{i}(W_{i})); and the probability (p_{i}) of exiting after i years of service. Keeping with the decrement tables used by the pension plans, we consider teacher exits at year one through 50. Formally:
Each value necessary to calculate equation (A6) is known for each system considered. We can thus directly compare expected utility, resulting from each lottery for a teacher entering the profession, under various assumptions for η.
To better illustrate the magnitude of teacher preferences between systems, we calculate the certainty equivalent for each lottery—for each particular level of risk aversion. The certainty equivalent is the certain amount of benefits that an individual is indifferent about accepting at the time of hire, to avoid participating in a particular pension lottery. Combining equations (A1) and (A6), and performing some algebra, yields for the calculation of the certainty equivalent:
Endnotes
 See, e.g., D. McCarthy, A Lifecycle Analysis of Defined Benefit Pension Plans, University of Michigan Retirement Research Center, working paper 2003053 (2003); J. Poterba et al., “Defined Contribution Plans, Defined Benefit Plans, and the Accumulation of Retirement Wealth,” Journal of Public Economics 91, no. 10 (2007): 2062–86; and A. Schrager, “The Decline of Defined Benefit Plans and Job Tenure,” Journal of Pension Economics and Finance 8, no. 3 (2009): 259–90.
 Consider a teacher who began working in the New York City public school system at age 25. If she exited the system 38 years later, she would, we calculate, retire with an employerfunded lifetime annuity worth about $635,090—equivalent to about $16,713 in retirement compensation for each year in the classroom. Had that same teacher instead exited the system after 20 years of service—perhaps to take another teaching job in another state—New York City’s contribution to her retirement would be a lifetime annuity worth about $70,738—just $3,537 per year of service.
 Josh McGee and Marcus A. Winters, “Better Pay, Fairer Pensions: Reforming Teacher Compensation,” Manhattan Institute, Civic Report no. 79 (September 2013).
 The methods used here follow R. Costrell and M. Podgursky, “Peaks, Cliffs, and Valleys: The Peculiar Incentives in Teacher Retirement Systems and Their Consequences for School Staffing,” Education Finance and Policy 4, no 2 (2009): 175–211; and Robert M. Costrell and Josh B. McGee, “Teacher Pension Incentives, Retirement Behavior, and Potential for Reform in Arkansas,” Education Finance and Policy 5, no. 4 (2010): 492–518.
 For all present value calculations, we use a nominal interest rate of 5 percent and an inflation rate of 2.5 percent. We use mortality tables—dictated for use under ERISA—compiled and updated by the IRS. Specifically, we use the 2013 static mortality table based on the RP2000 Mortality Tables Report, adjusted for mortality improvement using Projection Scale AA, http://www.irs.gov/pub/irsdrop/n0885.pdf.
 According to the Bureau of Labor Statistics (its latest update of the National Compensation Survey), the average privatesector manager and professional earns 10.6 percent of total earnings in the form of deferred retirement savings. This time series is maintained by Robert Costrell,
http://www.uaedreform.org/downloads/2013/12/quarterlyemployercontributionchartupdate.pdf.
 In a related work, we also later consider the fact that adoption of an SADB plan might alter teacher incentives to remain in the system—and thereby change attrition patterns. We find that reasonable changes to teacher attrition, due to the adoption of the SADB plan, have no significant influence on our overall findings.
 See, e.g., R. Chetty, “A New Method of Estimating Risk Aversion,” American Economic Review 96, no. 5 (2006): 1821–34.
 See D. H. Bowen et al., “Risky Business: An Analysis of Teacher Risk Preferences,” University of Arkansas Department of Education Reform, working paper 201301 (2013).
 We do not claim here that the offered SADB plan would secondorder stochastic dominate all other accrual patterns. One could imagine plans with the same expected value but a smaller variance. As an extreme example, one could imagine a plan in which, after one year of service, all employees were given pension benefits equal to the expected value of the current system. Such a plan would have the same expected value as the offered SADB plan but a variance of 0. We argue, however, that the SA plans that we consider offer a plausible alternative to current systems and, thus, are worthy of consideration.
 Chetty, “A New Method of Estimating Risk Aversion.”
 Our calculations consider the net present value of pension wealth in a given year, net of the employee’s cumulative contributions. For some plans, there are at least some years early in a teacher’s career when the employee’s contribution is greater than the present value of pension wealth—in these cases, employees who exit in such a year would have negative net pension wealth. This poses a problem for our utility calculations because the CRRA utility function cannot calculate utility when wealth is negative. In such cases, we adjust the negative benefits value to “0” when calculating the individual’s utility. This change slightly alters the expected value under each plan. Nevertheless, the impact on our calculations is insubstantial and does not meaningfully influence our findings. For these cases, we replace the calculated certainty equivalent under the SADB plan, for a riskneutral individual, to be equal to the value under the respective FASDB plan. We make this change because equivalent expected value is the definition of riskneutrality.
 Though we do not directly consider the impact of such a change on teacher quality, it is notable that one rationale for backloaded pension plans is to encourage more productive behavior from employees, who otherwise might shirk their responsibilities (Edward P. Lazear, “Incentive Contracts,” NBER working paper no. 1917 (1986)). Still, C. Koedel, M. Podgursky, and S. Shi, “Teacher Pension Systems, the Composition of the Teaching Workforce, and Teacher Quality,” Journal of Policy Analysis and Management 32, no. 3 (2013): 574–96, provide sound empirical evidence suggesting that this argument likely does not hold true in the case of public school teachers.
 Methods used follow Costrell and Podgursky, “Peaks, Cliffs, and Valleys; and Costrell and McGee, “Teacher Pension Incentives, Retirement Behavior, and Potential for Reform in Arkansas.”
 For all presentvalue calculations, we use a nominal interest rate of 5 percent and an inflation rate of 2.5 percent. We use the mortality tables, dictated for use under ERISA and compiled and updated by the IRS. Specifically, we use the 2013 static mortality table—based on the RP2000 Mortality Tables Report, adjusted for mortality improvement using Projection Scale AA, http://www.irs.gov/pub/irsdrop/n0885.pdf.
 The separation probability function, g(), is estimated using the decrement tables reported by each plan in the CAFRs.
