1. Introduction and summary

The funding plans for traditional teacher-pension systems are built upon a highly uneven set of benefits, varying widely in value by age of entry and exit.Footnote ¹ This can be illustrated by the variation in the annual cost to fund each individual's benefits. These inequities are masked by a uniform contribution rate for pensions. For example, the annual joint contribution (employer and employee) for pension normal costs may be 15% of each teacher's salary. These contributions are designed to fund the future retirement benefits as they are earned,Footnote ² for the system as a whole. Such contributions are conceptually comparable with those made to account-based plans, such as 401(k)’s, 403(b)’s, and cash balance (CB) defined benefit (DB) plans. However, unlike account-based plans, the annual cost of benefits for individual teachers under traditional pension formulas may deviate widely from this overall average (Costrell and McGee, Reference Costrell and McGee2017a; Costrell, Reference Costrell2018c; Costrell and Fuchsman, Reference Costrell and Fuchsman2018). It is well-established that the benefits for early leavers are of much lower value than for those who retire at the ‘sweet spot;’ contributions by or for the former effectively include a cross-subsidy to the latter. The further step taken here is to examine the impact of the discount rate on this system of redistribution. As is well-known, a drop in the discount rate raises the overall normal cost rate − this is regularly reported in valuation reports when the assumed rate of return is reduced. However, the impact on the distribution of individual normal costs has not been previously explored, and this is the subject of this paper.

There are two possible interpretations and motivations for this inquiry. First, since the discount rate is the expected return on the investment portfolio, under current public sector practice, this analysis considers how the cross-subsidies vary as the expected return is reduced in funding plans to better align with evolving market conditions. The gap between assumed returns and actual returns in recent times has led to rising and persistent unfunded liabilities. Consequently, public plans are widely, if belatedly, gradually cutting their assumed return.Footnote ³ In doing so, the plans’ normal cost rates, previously held artificially low by high-assumed returns, are rising. In this paper, we examine whether the measured degree of redistribution of pension benefits has also been held artificially low by high-assumed returns.

A second motivation for examining the impact of the discount rate is to assess the distribution of the value of the pension guarantee to members of defined benefit plans. Since the expected return includes a risk premium, netting that out of the discount rate (i.e., using a risk-free rate) tacks on a measure of the market value of the guarantee (Brown and Wilcox (Reference Brown and Wilcox2009), Novy-Marx and Rauh (Reference Novy-Marx and Rauh2009), and, relatedly, Biggs (Reference Biggs2011)). We know that the guarantee raises the value of benefits, as measured by the overall normal cost rate (Richwine and Biggs, Reference Richwine and Biggs2011). In this paper, I examine whether it raises the individual values uniformly or whether – and how much – it tilts the distribution of benefits and increases the cross-subsidies.

Analytically, the impact of a lower-discount rate (for either reason given above) on the degree of redistribution is not immediately obvious. The effect on individual normal costs varies with the age of exit for multiple reasons, and the impact does not vary monotonically. Overall, however, using the illustration of the California State Teachers’ Retirement System (CalSTRS), I find that a lower-discount rate raises the degree of redistribution significantly.Footnote ⁴ Put differently, although the cross-subsidies embedded in the funding plan can be substantial, these understate the redistribution of benefits as the expected return is reduced. When evaluated with the risk-free rate, to assess the market value of the pension guarantee, I find the value of that guarantee is highly concentrated and very large indeed for career teachers, but non-existent for short-termers.

The structure of the paper is as follows. First, I will review the basic math of individual normal costs and the associated cross-subsidies embedded in the funding plan, under any given discount rate. I illustrate with the case of CalSTRS, using the plan's current expected return. I then compare the magnitude of the cross-subsidies with those under the higher expected return held until recently, and possibly lower-assumed returns in the future. I then examine the math behind the effect of the discount rate on individual normal cost rates and how that effect varies, to better understand the distributional impact in question. Next, I consider a risk-free rate; although it is unlikely to be used for funding purposes under current public sector practices, it does illustrate the extremely wide variation in individual values of the benefit guarantee. Finally, I will briefly discuss potential policy implications, limitations of this work, and extensions to it.

2. Individual normal cost rates and cross-subsidization

Pension plans calculate the normal cost rate at the aggregate level, to fund a cohort's benefits as they accrue. Individual cost rates, based on age of entry and exit are implicitly embedded within the calculation (Costrell and McGee, Reference Costrell and McGee2017a, Appendix), but they are not publicly reported. Specifically, consider an individual of type (e,s), where e is the age of entry and s (for separation) is the age of exit. For each type (e,s), one can identify an individual normal cost rate, n _es, as a constant percent of salary over one's career. This rate is calculated to generate a stream of contributions sufficient to fund the individual's future benefits. That is, the present value (PV) of contributions must equal the PV of benefits.

Formally, for an individual of type (e,s), we must have n _esW_es = B _es, where W _es is the PV of earnings (so n _esW_es is the PV of contributions) and B _es is the PV of benefits (both evaluated at entry). It immediately follows that the individual cost rate is the ratio of the two:

(1)$$n_{es} = {\rm} B_{es}/W_{es}{\rm.} $$

This is the rate that, applied to the individual's annual earnings over her career, would prefund her benefits. It represents the value of her benefits earned annually, as a percent of earnings – an individual fringe benefit rate for pensions, analogous to contributions to a retirement account.

Across individuals with different entry and exit ages, (e,s), cost rates, n _es, vary widely. In general, for any given e, n _es rises with s, from some point after vesting up through a peak value retirement age. This is a manifestation of the well-known back-loading of benefits that favors long-termers under traditional pension formulas (Costrell and Podgursky, Reference Costrell and Podgursky2009, Reference Costrell and Podgursky2010a). The variation in n _es with e, for any given s, is less obvious, and can go either way.Footnote ⁵

Traditional pension plans levy a joint (employee plus employer) contribution rate, n, that is uniform, applied to all members of the cohort (of varying entry ages) throughout their careers (of varying length), calculated to fund the benefits of the whole entering cohort. Formally, denote the joint frequency of ages of entry and exit, e and s, among entrants, as p _es. It can be readily shown that the uniform cost rate required to fund the cohort's projected benefits isFootnote ⁶:

(2)$$\; n = \displaystyle{{\sum\nolimits_e^{} {\sum\nolimits_s^{} {n_{es}\lpar {\,p_{es}W_{es}} \rpar }}} \over {\left( {\sum\nolimits_e^{} {\sum\nolimits_s^{} {\,p_{es}W_{es}}}} \right)}}.$$

This is the ratio of the PV of the cohort's benefitsFootnote ⁷ to the PV of the cohort's earnings: the same relationship we saw for the individual normal cost rate holds for the cohort as a whole. This expression also shows, importantly, that n is a weighted average of individual normal cost rates n _es across ages of entry and exit. The weights for n _es are (p _esW_es)/(∑_e∑_sp_esW_es), representing the share of type (e,s) in the cohort's PV of earnings.Footnote ⁸

The deviations (n _es − n) are positive and negative, as the cost of funding any individual's benefit exceeds or falls short of the cohort's uniform contribution rate, n. They effectively comprise a system of cross-subsidies. Moreover, by the nature of averages, these cross-subsidies must add up to zero, when properly weighted, by shares of the cohort's PV of earnings:

(3)$$\displaystyle{{\sum\nolimits_e^{} {\sum\nolimits_s^{} {\lpar {n_{es}-n} \rpar \lpar {\,p_{es}W_{es}} \rpar }}} \over {\left( {\sum\nolimits_e^{} {\sum\nolimits_s^{} {\,p_{es}W_{es}}}} \right)}}\; = 0.$$

The negative cross-subsidies provided by the losers fund the positive cross-subsidies enjoyed by winners. I will illustrate the system of cross-subsidies under CalSTRS’ current plan, followed by an examination of how this system of redistribution is affected by the discount rate.

2.2 Variation in normal cost rates by age of entry and exit

Consider the variation of normal cost rates under the current assumed return, r = 7.0%. Figure 1 depicts n _es for selected (but representative) ages of entry and all exit ages. The variation is wide, from 7.3% to 25.0% (the full range, for entry ages not shown, is 6.8% to 27.1%). The pattern for any given entry age (e.g., age 25) is depicted along each curve, as exit ages vary. Prior to vesting, and for some years beyond, the benefit is a refund of employee contributions. The normal cost rate, therefore, starts at the employee contribution of 10.2%Footnote ¹³: each curve begins at the dashed horizontal line representing that rate. The cost then declines, falling slowly below the employee contribution rate. That is because the interest credit of 3.0% is below the fund's assumed return of 7.0%, with the fund retaining the difference. The contribution rate needed to cover the refund falls as this difference accumulates.

Note: Estimated using current CalSTRS assumptions and benefit formula for new hires, slightly modified.

Figure 1. Normal Cost rate, by entry age and age of exit, r = 7.0%.

At a certain point, the pension becomes more attractive than the refund. A 25-year-old entrant reaches that point at age 44; at this age the pension would still be deferred, but exceeds in PV that of employee refunds. Beyond that point, the normal cost rate rises as the deferral becomes shorter, and then, beyond age 55, there is no deferral, but the normal cost rate continues to rise for two reasons. First, the age-specific multiplier grows.Footnote ¹⁴ In addition, as years of higher salary are tacked on to the end of one's career, FAS rises relative to career earnings, along with the annual pension payment, further raising n _es. Beyond age 55, each year of delayed retirement is a year of forgone pension payments, but prior to age 65, this effect is outweighed by the other factors. After age 65 the multiplier stops growing, and the normal cost declines, due to the decreasing number of years the pension will be paid. This pattern is reflected in Figure 1along each curve, corresponding to any given entry age, e: n _es rises with s up to age 65 and falls thereafter. I have explained in detail this pattern of n _es rising with s to a peak value and then declining because it will be pertinent when analyzing the impact of a lower-discount rate.

Finally, in addition to the variation within entry age cohorts (i.e., along the curves depicted), Figure 1 also shows the variation across entry ages (i.e., the spread between curves). This widens the overall spread in individual normal cost rates, and thereby adds to the redistributive structure of such pension plans.

3. The impact of the assumed return on cross-subsidies

The analysis above was based on r = 7.0%, the investment return assumed since the 2017 valuation. This represents a reduction from 7.5% in 2015 and 7.25% in 2016. In reducing its assumed return from 7.5% to 7.0%, CalSTRS has, of course, raised its normal cost rate for new hires, n (about 2.1 percentage points by my estimate). The point here, however, is that the degree of redistribution also rose, from 3.0% of total income to 3.4%.

To illustrate the distributional impact of further rate reduction, consider what the system would look like if r were reduced to r = 6.0% (comparable with recent 10-year rolling averages). The individual normal cost rates are depicted in Figure 2. All the normal cost rates are increased from those depicted in Figure 1: with lower assumed investment returns, contributions must be higher to fund the benefits. This much is well-known. What is perhaps less widely understood is that a drop in the assumed return will increase the degree of redistribution embedded in the funding plan. Stated alternatively, an over-optimistic assumed return not only underfunds the plan, but also understates the degree of redistribution.

Note: Estimated using current CalSTRS assumptions and benefit formula for new hires, slightly modified.

Figure 2. Normal cost rate, by entry age and age of exit, r = 6.0%.

For example, the extreme points depicted in Figure 2 now represent cross-subsidies of −14.4% to +7.5%, widening the previous range (−10.3 to +7.4), especially among the losers.Footnote ²⁰ On average, losers provide cross-subsidies that widen from −4.8% of their income to −6.9%, while winners receive cross-subsidies that rise from +2.7% of income to +3.4%. The zero-sum result on cross-subsidies still holds (with winners’ share of lifetime earnings now at 0.67), and, finally, taking the absolute values, we find that our measure of redistribution rises from 3.4% of total income to 4.5% (0.67 × 3.4% + 0.33 × 6.9%).

3.1 A graphical illustration of the distributional impact

Why does a cut in the discount rate increase the cross-subsidies and the degree of redistribution? On a purely mechanical level, the reason lies in the manner of variation of the impact of r on the individual normal cost rates n _es. If there were no variation, such that all n _es were to rise by the same amount with a drop in r, then n would rise by approximately that amountFootnote ²¹; the cross-subsidies (n _es − n) would remain nearly unchanged, and so would the degree of redistribution. This is not the case. The non-uniformity is seen in Figure 3, which depicts the rise in normal cost rates, by entry and exit age, as r drops from 7.0% to 6.0%. All the individual normal cost rates rise, but generally more so on the right side of the diagram than on the left side.Footnote ²² The uniform cost rate, n, is a weighted average of the individual rates, so it should rise by an amount that exceeds the (smaller) rise of the individual rates on the left side of Figure 3 and is generally less than the (greater) rise on the right side. And so it does: n rises by 5.2% (from 17.6% to 22.8%), while the individual normal costs rise by amounts close to zero for early departures, and up to 7% for departures at age 65.

Note: Estimated using current CalSTRS assumptions and benefit formula for new hires, slightly modified.

Figure 3. Rise in individual normal cost rates, r = 6.0% vs. 7.0%.

What does this mean for the cross-subsidies? These are the gaps (negative or positive) between individual normal cost rates and the uniform rate. Since the rise in the uniform rate exceeds the rise in individual rates on the left side of Figure 3, this widens the gap (by an average of 2.1% of pay, from −4.8% to −6.9%). Conversely, on the right side, since the individual rates generally rise by more than the uniform rate, this widens the gap here, too (by an average of 0.7% of pay, from +2.7% of pay to +3.4%). Thus, on both sides, we find an increase in the magnitude (absolute value) of the cross-subsidies provided and received: a drop in the assumed return thereby increases my measure of the degree of redistribution.Footnote ²³ Conversely, overstating the return most markedly understates the PV of long-termers’ benefits and, correspondingly, understates the cross-subsidies to support those benefits.

3.2 The analytics of individual normal cost rates and the discount rate

As we have seen empirically, a drop in the discount rate raises normal cost rates for individuals of early exit ages by less than those of late exit ages. More precisely, as illustrated in Figure 3, for each entry age, the rise in individual normal cost rates increases with the age of exit, up to the point of peak normal cost. What explains this pattern? The simple answer lies in the back-loading itself of FAS pensions.Footnote ²⁴ As I will argue, the main impact of a drop in the discount rate is to amplify that back-loading, thereby increasing the redistribution.

To elaborate a bit, before plunging into the math, as a first approximation, n _es rises proportionately with a drop in r. That is, a drop in r magnifies individual cost rates n _es more, the larger n _es is to begin with. Thus, since n _es rises with s up to a peak of 65 in CalSTRS, for reasons discussed earlier, Δn _es also rises with s up to 65 as r drops. Formally, we can write:

(4)$$\Delta n_{es} = n_{es}[\Delta n_{es}/n_{es}] \sim n_{es}[\Delta B_{es}/B_{es}-\Delta W_{es}/W_{es}].$$

My claim is that the variation across individuals of the impact of Δr on Δn _es primarily reflects variation in the initial level of n _es – the first term on the right-hand side (RHS) – rather than variation in the proportional change of n _es – the bracketed term on the RHS.

More precisely, to examine the distributional impact of a drop in r, we want to understand how −∂n _es/∂r varies across individuals. First, we back up and examine more closely why a drop in r raises the individual rates, n _es, to begin with. Let us write out n _es:

(5)$$n_{es} = B_{es}/W_{es} = \mathop \sum \limits_{a \gt s} \lpar {1 + r} \rpar ^{\lpar {e-a} \rpar }b\lpar {a \vert e,s} \rpar f\lpar {a \vert s} \rpar /\mathop \sum \limits_{a = e}^s \lpar {1 + r} \rpar ^{\lpar {e-a} \rpar }w_{a \vert e}.$$

The numerator of n _es − the PV of pension benefits − is the sum for all ages a, following the individual's separation s, of the PV (dated to entry e) of her annual pension benefit, b(a|e,s), weighted by her probability of survival to age a, conditional on survival to exit age s, f(a|s). Similarly, the denominator of n _es, is the sum across all ages from her entry, e, to exit, s, of the PV of her annual earnings, w _a|e. Using (5), we can then analyze the differential form of (4):

(6)$$\eqalign{-\partial n_{es}/\partial r = &-\!n_{es}\cdot \partial {\rm ln}\lpar {n_{es}} \rpar /\partial r = \; - n_{es}\lsqb {\partial {\rm ln}\lpar {B_{es}} \rpar /\partial r\; -\partial {\rm ln}\lpar {W_{es}} \rpar /\partial r\;} \!\!\rsqb \cr = &\; n_{es}\left[ {\mathop \sum \limits_{a \gt s} \lpar {a-e} \rpar \theta_{es}-\mathop \sum \limits_{a = e}^s \lpar {a-e} \rpar \gamma_{es}} \right]/\lpar {1 + r} \rpar ,} $$

where θ _es and γ _es are weights that sum to one. Thus, −∂n _es/∂r > 0 for all (e,s) since the first term in brackets is a weighted average of years since entry, for the period after separation, and the second term is an average of years since entry for the period before separation. The point here is simple: a fall in the discount rate raises all the individual cost rates, n _es = B _es/W_es, because the sequence of benefits follows in time (after separation) the sequence of earnings (before separation), so the PV of benefits, B _es, rises proportionally more than that of earnings, W _es. That is, n _es rises as r drops because of the non-overlapping time patterns in benefits and earnings.

We now turn to the paper's main point: how the impact of a drop in r varies across individuals, particularly by s, resulting in a greater degree of redistribution. More formally, having examined the first derivative of n _es with respect to r, we are now, in effect, interested in the cross-partial with respect to r and s.Footnote ²⁵ What is the impact of s on the final expression in (6)? Figure 3 suggests that the impact of s on the first term − n _es itself − is decisive; the back-loading of normal costs up to the peak age of 65 shapes the impact of s on the rise in n _es as r falls.

By comparison, the impact of s on the bracketed term of (6) is complex and appears not to be decisive – indeed its sign is ambiguous. The complexity is readily seen by noting that s appears at four points in the bracketed term: lower and upper limits, respectively, on the two summations, and the two sets of weights within those summations. Consider the impact of s through the sums’ limits. Raising s eliminates the lowest ages in the first sum and adds higher ages to the second sum. Consequently, raising s increases the average number of years since entry, both post-separation and pre-separation. Its impact on the difference between the two − the bracketed term in (6), representing the non-overlapping time patterns in benefits and earnings − is ambiguous. The impact of s on the weights θ _es and γ _es is also complex and ambiguous. Since the bracketed term governs the proportional change in n _es as r falls, these complex and ambiguous analytical effects of s are consistent with our empirical finding that they are dominated by the absolute impact on n _es, which is the back-loading itself.

To summarize, the back-loading of benefits under traditional FAS plans, which generates much of the redistributive cross-subsidies at issue, is magnified by reductions in the discount rate. As will be discussed further in the conclusion, this suggests that, by contrast, for account-based systems that are not heavily back-loaded, the distributional impact of a cut in r will be more muted. Put differently, while a high-assumed return understates the redistribution in FAS plans, this will be much less so of account-based systems: the redistribution in such systems will be minimal, independent of the discount rate.

4. The distribution of the value of the pension guarantee

Pension benefits are free of market risk, while public pension funds typically include a sizeable component of risky assets in their portfolio. Consequently, there has been animated debate between the field of finance economics (Brown and Wilcox, Reference Brown and Wilcox2009; Novy-Marx and Rauh, Reference Novy-Marx and Rauh2009; Biggs, Reference Biggs2011) and public pension fund actors (notably their actuaries, fund managers, and advocacy organizations) over the proper discount rate for pension liabilities. It is important to distinguish between the various uses of the discount rate, as the disagreements vary by use. These uses include: (i) financial reporting; (ii) determination of contribution rates; (iii) evaluation of plan costs, including the cost of market risk; and (iv) evaluation of benefits to the members. The critique of actuarial discount rate practice for public plans has focused most sharply on (i), financial reporting (GASB rules).Footnote ²⁶ The critique is more hedged for (ii), the determination of contribution rates, as this is recognized as more of a political decision,Footnote ²⁷ linked to the equally political decision regarding the risk profile of fund assets.

Finance economists and public plan advocates disagree sharply on (iii), the evaluation of the plan's full costs. Advocates argue that costs are fully accounted for by discounting with the expected return on risky assets because of time diversification for immortal public plans. In other words, it is claimed, there is no cost for market risk in public plans because the law of large numbers diversifies away the variance of the annual return over time. Finance economists, however, have shown long ago the fallacy of time diversification, as the variance in total return increases over time (Samuelson, Reference Samuelson1963). The cost of that risk may not be borne as a premium attached to the average annual contribution rate, but the variation in annual contributions (under finite amortization schedules) imposes costs on taxpayers, current or future, whether or not they are acknowledged by fund managers. Moreover, for mature public plans (where benefit outlays exceed contributions, as they now do nationally (Center for Retirement Research, 2018)), portfolios with variable annual return incur the risk of insolvency, even if the average annual return is certain (Boyd and Yin, Reference Boyd and Yin2018); if the period of losses precedes that of gains, the losses will be on a larger base than the gains, and the fund can deplete prematurely. This risk is also costly, but, again, it may not be factored into annual contribution rates; public plans and the taxpayers that stand behind them may, in effect, self-insure, but that does not eliminate these costs of investment in risky assets, even if they are unreported.Footnote ²⁸

The key point here is (iv), these costs of market risk are not borne by the plan members, the beneficiaries of the defined benefit guarantee. On this point, there is actually agreement between advocates and critics of discounting by the expected return (and of DB plans more generally). Even if they disagree on whether the costs of risky investment are borne by taxpayers or whether there is a ‘free lunch’ from time diversification, both sides agree that the pension guarantee is valuable to plan members (see, for example, the arguments in favor of DB plans by Rhee and Fornia (Reference Rhee and Fornia2017)). To evaluate this guarantee, finance economists argue that benefits should be discounted at something close to the risk-free rate, even if contributions continue to be determined based on expected returns. In doing so, Richwine and Biggs (Reference Richwine and Biggs2011) show how the standard normal cost rate for teachers dramatically understates the overall benefit.

Here, I consider how the value of the pension guarantee is distributed. To do so, I take the annualized value of the guarantee to be the difference between n _es evaluated at a risk-free rate and that evaluated at the expected return. The former represents what the market would charge an individual of known e and s, as a percent of annual earnings, to provide an annuity following separation that replicates the pension's annual payments. By contrast, the pension fund evaluates n _es at the expected return – including what the market considers a risk premium – because the fund either believes its immortality diversifies away the risk, or it is willing to bear the cost of that risk (or, more accurately, to impose the cost of that risk on taxpayers). Either way, the member receives a risk-free benefit that would cost more in the market than the fund is charging. The difference between the two is the market value of the pension guarantee.Footnote ²⁹

Figure 4 illustrates the total value of pension benefits, including the value of the guarantee, by age of entry and exit, with n _es evaluated at 4.0%. The average annualized value is nearly 40%, and the spread is wide, ranging from about 10% (the value of employee contributions, for those taking refunds) to 45% or 50% for those retiring at 65.Footnote ³⁰ Netting out the employee contribution, the value of employer-provided risk-free benefits ranges from 0% to 40%. One-third of entrants will receive employer-provided benefits worth at least 30% of pay. These are very high fringe benefit rates, but they are also highly concentrated. At the other end of the spectrum is a larger group (38% of entrants, whose benefit is the refund) for whom there are no employer-provided benefits.

Note: Estimated using current CalSTRS assumptions and benefit formula for new hires, slightly modified.

Figure 4. Annualized value of risk-free individual benefits, r = 4.0%.

My estimates for the individual market values of the pension guarantee alone are presented in Figure 5. This is the difference between the total value of the benefit, given in Figure 4 and the value of the benefit evaluated at the plan's assumed expected return, 7.0%, given in Figure 1. For those who stay to age 65, the market value of the guarantee provided by the employer is quite large, adding the equivalent of another 20–30% of pay to the 10–15% benefit covered by employer contributions and the 10% paid for by the employee herself. However, for those who receive refunds instead of a pension, the value of the guarantee is minimal – under 1 percentage point.Footnote ³¹ Of course, this makes sense: since the value of the guarantee is ascertained by stripping out the risk-premium from the discount rate, this will accumulate most significantly for those whose benefit is most far-removed from entry.

Note: Difference between n_es evaluated at 4.0% and 7.0% for CalSTRS new hires.

Figure 5. Annualized market value of pension guarantee.