2 Relationship with the literature

A recent overview by Munnell et al. (Reference Munnell, Aubry, Hurwitz and Medenica2013) illustrates the urgency of the financial condition of many US state pension funds. For a sample of 109 state plans and 17 locally-administered plans, the paper estimates the aggregate funding ratio (i.e., the ratio of assets over liabilities) for 2012 at 73%. Almost a quarter of the plans have a funding ratio below 60%, while only a small fraction of 6% have a funding ratio above 100%. The funding ratios are calculated on the basis of standards that prescribe that assets are reported on an actuarially-smoothed basis, while the discount rate for the liabilities is typically set at around 8%, reflecting the expected long-term investment return on the fund's assets.Footnote ² These standards have been criticized by economists (Bader and Gold, Reference Bader and Gold2003; Novy-Marx and Rauh, Reference Novy-Marx and Rauh2009), who claim that the future streams of benefit payments should be discounted at a rate reflecting their riskiness. As the state pension benefits are protected under most state laws, these payments can be seen as guaranteed and so this would plead for discounting them against the risk-free interest rate. In fact, Brown and Pennacchi (Reference Brown and Pennacchi2015) argue that for calculating a pension plan's funding ratio, liabilities should always be discounted against the default-free term structure, no matter what is the actual default risk of the plan. Doing so would lead to a severe fall in the already-low average funding ratios of the state plans. In fact, a precise assessment of the riskiness of state pension promises likely remains elusive for the foreseeable future. For example, the recent default of the city of Detroit on its obligations may be an instructive case of whether pensions should be seen as a contractual obligation that cannot be diminished or impaired, or whether the holders of pension rights should be treated like other creditors, so that benefit cuts cannot be ruled out. Indeed, a US bankruptcy judge declared in December 2013 that legally pensions can be cut (Bomey and Priddle, Reference Bomey and Priddle2013).

The value-based ALM approach that we apply in this paper to US state pension funds has its roots in the pioneering papers of Sharpe (Reference Sharpe1976) and Sharpe and Treynor (Reference Sharpe and Treynor1977) in utilizing derivative pricing to value contingent claims within pension funds. More recent applications of derivative pricing to pension plans are, among others, Blake (Reference Blake1998), Exley et al. (Reference Exley, Mehta and Smith1997), Chapman et al. (Reference Chapman, Gordon and Speed2001), Ponds (Reference Ponds2003), Bader and Gold (Reference Bader and Gold2007), Hoevenaars and Ponds (Reference Hoevenaars and Ponds2008) and Ponds and Lekniute (Reference Ponds and Lekniute2011). Our paper is closest in spirit to Biggs (Reference Biggs2010), which also employs an option-based approach to value the market price of pension liabilities of US state pension plans. However, it does not address reform plans and the associated redistribution effects. Value-based ALM has also been employed in the Dutch discussion on the redesign of the second-pillar pension funds offering DB plans. In particular, the CPB Netherlands Bureau for Economic Policy Analysis (2012) applies it to investigate the generational fairness of various pension reform plans, while Platanakis and Sutcliffe (Reference Platanakis and Sutcliffe2015) apply it to quantify the redistribution associated with the 2011 reform of the UK pension plan for universities (USS).

Our paper is complementary to related contributions studying the degree of underfunding of US state pension plans, the causes of the underfunding and the measures that could be taken to solve plan deficits. Novy-Marx and Rauh (Reference Novy-Marx and Rauh2011) conduct a careful analysis of the value of the pension promises made by US state pension funds under different assumptions about the riskiness of state pension plans and their seniority relative to state debt in the case of default. Several papers explore the extent of underfunding and the question what should be the appropriate target for the funding ratio (compare D'Arcy et al., Reference D'Arcy, Dulebohn and Oh1999; Lucas and Zeldes, Reference Lucas and Zeldes2009; Bohn, Reference Bohn2011). Brown et al. (Reference Brown, Clark and Rauh2011) argue that adequate funding requires 100% funding of the liabilities.

Regarding the causes of the underfunding of state pension plans some papers point at the implications of what may be called the ‘accounting game’. The accountability horizon of politicians and public sector union leaders is much shorter than the horizon over which pension promises have to be met by adequate funding. Hence, union leaders and politicians tend to downplay the long-term costs of pension promises in favor of higher short-term wages and benefits (Mitchel and Smith, Reference Mitchel and Smith1994; Schieber, Reference Schieber2011), as well as higher state spending in the short run. Shnitser (Reference Shnitser2013) stresses the role of the institutional design of pension plans in explaining the large variation in funding discipline. Institutional rules facilitating transparency and pre-commitment to expert-based financial and actuarial advice, in particular, appear to be conducive to mitigating underfunding and limiting the shift of costs to future tax payers. Kelley (Reference Kelley2014) and Elder and Wagner (Reference Elder and Wagner2014) apply a public choice perspective to explaining underfunding issues.

Essentially, the literature has explored three broad directions of solutions to the underfunding problem: higher contributions, lower benefits and higher investment returns. Novy-Marx and Rauh (Reference Novy-Marx and Rauh2014b) explore by how much contributions need to be raised to reach full funding in 30 years time. They compute a necessary increase of the order of 2.5 times the prevailing contribution level. The option of reducing benefits has long been seen as virtually impossible, as in many states public pensions are interpreted as hard contractual obligations, protected by civil law and state constitutions (Monahan, Reference Monahan2012). However, several states have already enacted benefit cuts and other measures to scale back the generosity of pensions. Further, as documented by Munnell et al. (Reference Munnell, Aubry and Cafarelli2014a), in response to the economic and financial crisis, many public plan sponsors have reduced or eliminated cost-of-living-adjustments (COLAs) in one way or the other. Munnell et al. (Reference Munnell, Aubry and Cafarelli2014a) report that between 2010 and 2013 this has been done by 17 states and conclude that benefits are no longer as secure as they were perceived to be before the financial crisis. Novy-Marx and Rauh (Reference Novy-Marx and Rauh2014a) explore performance-linked indexation rules comparable with the one in the Wisconsin Retirement System or the conditional indexation rule in the Netherlands (Ponds and van Riel, Reference Ponds and van Riel2009; Beetsma and Bucciol, Reference Beetsma and Bucciol2010, Reference Beetsma and Bucciol2011a, Reference Beetsma and Bucciolb). Shnitser (Reference Shnitser2013) stresses that simply scaling back generosity or imposing higher contribution rates will not be enough. She claims that foremost the institutional design has to be reframed to practices and rules that have proven to be successful in promoting funding discipline. A third route proposed to solve funding deficits is to raise expected investment returns. This option is already being exploited by US state pension funds, as many of them have high portfolio shares in equity, substantially higher than of pension funds outside the USA (OECD, 2011; Andonov et al., Reference Andonov, Bauer and Cremers2013). However, finance-based papers (Black, Reference Black1989; Peskin, Reference Peskin2001; Bader and Gold, Reference Bader and Gold2003; Lucas and Zeldes, Reference Lucas and Zeldes2009; Pennacchi and Rastad, Reference Pennacchi and Rastad2011) recommend that the strategic asset allocation should be set in line with the perspective of the tax payers as the party bearing the residual risk, which suggests to limit mismatch risk between the funds’ assets and the market value of their liabilities. This may imply no equity investment at all when one aims at full protection of accrued benefits in the short run. Lucas and Zeldes (Reference Lucas and Zeldes2009), employing a long-term perspective, suggest that the optimal portfolio should include some equity holdings as future pension liabilities are related to future wage growth and this growth is to some extent correlated with the equity returns.

3 The model

This section briefly sketches the model. The full details of the model are found in Appendix A. It first describes the demography, then the wage developments and, finally, the pension fund. The participant population of the pension fund consists of individuals of ages 25–99 years. Individuals enter the fund at the age of 25 and remain with the fund for the rest of their life. We distinguish between males and females and apply projections of survival probabilities to calculate the size of these cohorts in the future. The survival probabilities are deterministic, hence there is no longevity risk. Crucial for the calculation of the pension liabilities are the wage developments. We assume a uniform wage level within each cohort, while over one's life the wage level follows a certain career profile that is constant over time (i.e., the relative wage between two cohorts of specific ages is constant over time). Because we do not have gender-specific age-wage profiles, we set the wage levels of males and females equal. The economy-wide wage growth rate is stochastic and will be modelled as explained in Section 4 below.

State pension funds can differ in many respects from each other. Some of the differences are parametric, such as the value of the discount rate to be applied when calculating the liabilities, while other differences are more fundamental. We model a pension fund based on the most common features across the entire population of pension funds available to us from the Public Plans Database of the Center for Retirement Research at Boston College (2015). In doing so, we try to stay as close as possible to the actual practice of GASB accounting employed by pension funds, and use their most common actuarial assumptions to calculate their financial health in actuarial terms, which is the basis for their instrument settings (contributions, indexation and benefits). However, our simulations of the fund over time, and the valuation of the cash flows, will be based on an economic scenario set and a term structure constructed from actual and survey projections of economic and financial data.

3.1 The calculation of the actuarial assets and liabilities

As inputs for policy decisions, we need to calculate the actuarial assets $A_t^{act} $ and actuarial liabilities $L_t^{act} $ to the participants of the fund. These consist of the retired and the workers. Compared with the market value of the assets, the dynamics of the actuarial assets are smoothed, because each period they change only by expected investment income plus a fraction of the difference between realized and expected investment income. The actuarial liabilities are calculated as the present value of the future benefit payments to all current fund participants, taking into account the survival probabilities, and based on the actuarial assumptions typically used by pension funds. The actuarial liabilities should be distinguished from what we will term the economic liabilities, the projected benefits based on the economic developments expected by the market participants and discounted against a proper market interest rate. The economic liabilities will be discussed in more detail below.

To calculate the actuarial liabilities, see Munnell et al. (Reference Munnell, Haverstick, Sass and Aubry2008b), we need to calculate the projected future benefits of both the retirees and the workers. The projected future benefits equal the current pension rights adjusted for the actuarially projected future COLAs,Footnote ³ intended to protect the retirees’ living standards and calculated on the basis of the fund's own future inflation projection. The current pay-out to a retiree equals their current pension rights, which equals their pension rights at the moment of retirement compounded by the realized COLA since retirement. In turn, pension rights at retirement are the product of the annual accrual rate, the number of years in the workforce and the average over the latest (usually 1–5 years) wages earned by the individual (Munnell et al., Reference Munnell, Aubry, Hurwitz and Quinby2012).

Thus, we need to distinguish the actuarial projection of the COLA and the realized COLA. The latter covers a certain share of realized inflation and is often capped at a predetermined level (Center for Retirement Research at Boston College, 2015). Hence, under our baseline plan and most alternative plans we model the realized COLA as

(1) $$COLA_t = \max \left[ {\min \left( {\xi \cdot cpi_t,cap} \right),0} \right]$$

where cpi _t stands for actual consumer-price inflation in period t, ξ is the fraction that is compensated and cap is the maximum indexation rate. Here, both cpi and cap are expressed as a fraction of unity. The minimum indexation rate is bounded from below at zero, hence benefit payments are prevented from declining in nominal terms. In reality, ξ is often below one. However, in order to avoid an arbitrary parametrization, under our baseline pension plan we will set ξ = 1, while we will consider an alternative plan in which ξ is below one.

In contrast to the retirees, active participants are expected to continue working and, hence, accrue additional years of service. Therefore, there are various ways in which the liabilities associated with the worker's current pension rights can be recognized. Under the Accrued Benefit Obligation (ABO) method, only the pension rights accrued up to now are taken into account, while future wage rises are not taken into account. The Projected Benefit Obligation (PBO) method also takes into account the effect of actuarially projected salary increases on the rights accrued up to now. Finally, the Projected Value of Benefits (PVB) method in addition takes into account the rights that the employees will acquire in the future if they continue working in their current job until retirement. As a result, a cohort that has just entered the labor force already has a stake in the fund's liabilities, as opposed to the ABO or PBO method. For people of working age PVB liabilities would normally exceed PBO liabilities, which, in turn would exceed ABO liabilities. However, all three measures converge for a given individual at the moment of retirement.

The PVB method is of particular relevance here since state civil service jobs are relatively secure. We use the PVB method to calculate our liability measures and the associated pension contributions. The calculations are done under the assumption that individuals spend their full career as fund participants. We make this assumption, because the Public Pensions Database has no information on the entry and exit likelihoods at different ages at the individual fund level or at the aggregate level. Given the upward-sloping age-wage profiles, ignoring entry at later age biases all the liability measures downward, while ignoring job exits that take place before retirement produces an upward bias in our PBO and PVB measures, because the projected salary increases of the quitters would drop out of these measures. In view of the absence of the appropriate data, we have chosen to ignore entries later in working life and premature exits, rather than making arbitrary assumptions about the age distribution of the people entering and leaving the fund.

3.2 The funding ratio

The evaluation of a pension fund's financial health and its policies, including indexation decisions, is usually based on a funding ratio that is calculated on the basis of the fund's actuarial assumptions. We refer to this funding ratio as the policy funding ratio $(FR_t^P )$ , defined as the ratio of the fund's actuarial assets and actuarial liabilities:

(2) $$FR_t^P = \displaystyle{{A_t^{act}} \over {L_t^{act}}}. $$

However, the policy funding ratio often gives an overly optimistic picture of the financial health of the pension fund, in particular due to the use of an unrealistically high discount rate for the calculation of the actuarial liabilities. Therefore, we also calculate an economic funding ratio $(FR_t^E )$ defined as the ratio of the market value of the assets A _t and the economic liabilities L _t :

(3) $$FR_t^E = \displaystyle{{A_t} \over {L_t}}.$$

The economic liabilities are computed by discounting against the (default-free) nominal term structure the projected future benefits based on the expected price and nominal wage developments implied by our model of the dynamics of the state variables, conditional their values in period t (see Section 4 below).

Brown and Pennacchi (Reference Brown and Pennacchi2015) provide a number of arguments for using the default-free nominal term structure for DB pension plans if the funding ratio is intended to measure the fund's financial health. First, the funding ratio thus calculated is independent of the investment portfolio, in contrast to when the liabilities are calculated on the basis of the expected investment return. Second, the funding ratio calculated in this way immediately shows how much more a plan sponsor needs to contribute in the case of underfunding (or would receive back in the case of overfunding) for an insurance company to take over the pension obligations and guarantee the DB payments. Note that the floor and the cap on the indexation rate prevent accumulated entitlements from fully keeping up with inflation. Appropriate account of the floor would shift the term structure for the discounting down, while the opposite is the case when appropriate account is taken of the cap. However, incorporating the effects of the floor and the cap in a fully consistent way is so complicated that we decide to apply the nominal term structure without any further adjustments.

3.3 Benefits and contributions

The total amount of benefits B _t paid out by the fund is the sum over all retirement ages of the number of retired at each age times the current pension rights at that age. The total contribution rate c _t is total contributions C _t divided by the aggregate wage sum. We have that

(4) $$\eqalign{c_t = & c_t^{NC} + c_t^{Amort} + c_t^{SS} \cr = & c_t^{act} + c_t^{SS},} $$

where $c_t^{NC} $ , $c_t^{Amort} $ , and $c_t^{SS} $ are the so-called normal cost payment, the (actual) amortization payment and the sponsor support payments, respectively, all as fractions of the aggregate wage sum. Finally, we refer to $c_t^{act} = c_t^{NC} + c_t^{Amort} $ as the actuarial contribution rate.

The normal cost equals the value of the additional pension rights earned in a given year. The amortization payment is based on the so-called unfunded actuarial accrued liability (UAAL), which, in turn, is the difference between the actuarial accrued liability, $L_{accr,t}^{act} $ , and the actuarial value of the assets, i.e., $UAAL_t = L_{accr,t}^{act} - A_t^{act} $ . We assume a smoothing period of u years for the amortization of the UAAL _t . The amortization payment cannot be negative. If there is a funding surplus, i.e., UAAL _t < 0, the actuarial contribution is kept at the normal cost covering contribution $c_t^{NC} $ , hence $c_t^{Amort} = 0$ . Munnell et al. (Reference Munnell, Haverstick, Aubry and Golub-Sass2008a) show that in only about half of all the pension plans the required amortization contribution is paid. In our simulations we thus allow for only a fraction of the required amortization payment to be actually paid.

Forecasting the development of a pension fund over a long horizon implies that there exist scenarios in which the fund's assets get depleted.Footnote ⁴ We assume that, whenever the fund's assets A _t become insufficient to finance the net cash outflow B _t  − C _t , the employer, i.e., the tax payer,Footnote ⁵ provides sponsor support SS _t . Hence, $SS_t = (B_t - C_t) - A_t$ , if this is positive. Otherwise, SS _t  = 0. Once the sponsor support becomes positive, the fund effectively continues to be run on a pay-as-you-go basis. Both the policy and economic funding ratios remain at zero from that moment onwards.

The actuarial contribution rate can be split into a contribution rate $c_t^{PP} $ paid by the plan participant and a contribution rate $c_t^{TP} $ paid by the tax payer (as employer):

(5) $$c_t^{act} = c_t^{PP} + c_t^{TP}. $$

Typically, as we will also assume in our simulations, the contribution rate paid by the worker is fixed, i.e., $c_t^{PP} $ is constant, while the tax payer as employer pays the remainder.

4 The economic scenario generator

We estimate a quarterly vector autoregression (VAR) model on historical data for the USA and use this model to generate a set of economic scenarios.Footnote ⁶ It is generally not known which scenario will occur in reality. Hence, we evaluate the performance of an object of interest under all the scenarios that we generate. For our purposes, there is no need for a very refined model specification. Hence, we simply use a first-order VAR model linking the state vector in quarter q + 1 to that in quarter q:

(6) $$X_{q + 1} = \left[ {\matrix{ {y_{q + 1}} \cr {xs_{q + 1}} \cr {cpi_{q + 1}} \cr {w_{q + 1}} \cr}} \right] = \alpha + \Gamma X_q + \varepsilon _{q + 1},$$

where y _q+1 is the short-term quarterly interest rate, xs _q+1 is the excess return on stocks, cpi _q+1 is consumer price inflation and w _q+1 is real wage growth. The first two variables are calculated by taking the natural logarithm of the gross rate. The latter two variables are calculated as changes in the logarithm of the relevant index. The error terms ε _q+1 are independently and identically distributed and follow a multivariate normal distribution with mean zero and variance-covariance matrix Σ. Further, $\alpha = (I - \Gamma )\mu $ , where I denotes the identity matrix and μ is the unconditional mean of the vector X _q for all q. The estimates $\hat \alpha $ and $\hat \Gamma $ of α and Γ, respectively, are obtained by estimating equation (6) using OLS. We obtain $\hat \Sigma $ as the variance-covariance estimate of the error terms.Footnote ⁷ Given $\hat \Gamma $ and $\hat \alpha $ , the estimate of μ is:

(7) $$\hat \mu = (I - \hat \Gamma )^{ - 1}\hat \alpha. $$

The parameter estimates associated with (6) are used to price the cash flows implied by the pension arrangement.

5 Valuation

A pension plan can be seen as a financial contract. If we perceive this contract as a combination of contingent claims, we can value the pension deal using the derivative pricing techniques of risk-neutral valuation introduced by Black and Scholes (Reference Black and Scholes1973). Below, we explain how we calculate the value of the pension contract for all pension fund stakeholders, i.e., for the individual cohorts of participants and the individual cohorts of tax payers. Aggregating over the individual cohorts we obtain the contract value for all participants as a group and all the tax payers as a group.

To value the contract we use the scenarios for the underlying state vector produced by our scenario generator. The scenarios are generated at the quarterly frequency. We draw both real-world (‘P world’) scenarios for the classic ALM model and risk-neutral (‘Q world’) scenarios for the value-based ALM model. Under the classic ALM we generate paths for the state vector drawing shocks from a zero-mean normal density function, while under value-based ALM we generate these paths using risk-neutral sampling, which effectively amounts to a negative shift in the means of the components of the original shock vector.

The path for the state vector thus gives us a path for the stock returns. Also, for each quarter into the simulation, we determine the term structure using an affine model based on the state variables (e.g., see Dai and Singleton, Reference Dai and Singleton2000; Ang and Piazzesi, Reference Ang and Piazzesi2003; Ang et al., Reference Ang, Bekaert and Wei2008; Le et al., Reference Le, Singleton and Dai2010). This allows us to calculate the return on the fixed income part of the fund's portfolio, so that we now have the return on the entire fund portfolio. Using the path of the state vector generated under the risk-neutral sampling, as well as the returns on the fund's portfolio based on this same path of the state vector, allows us to generate the annual cash flows associated with the pension contract under the value-based ALM. These are then discounted against the risk-free rate of interest. Appendix B shows the details of the pricing of the cash flows and Appendix C of the transformation to the risk-neutral scenarios needed to calculate the value of the pension contract to the various stakeholders.

We aim at valuing the pension contract to the different stakeholders of the fund. Therefore, we will calculate the value of the net benefits to each cohort of fund participants and each cohort of tax payers. We will value the cash flows over a horizon of T years. Recall that the scenarios are simulated at the quarterly frequency, while the cash flows materialize at the annual frequency.

The value $V_0^{PP,a} $ of the pension contract to plan participants of the cohort aged a at t = 0 is:

(8) $$V_0^{PP,a} = {\rm E}_0^Q \left[ {\sum\limits_{t = 1}^T \left( {\prod\limits_{q = 1}^{4t - 2} {(Y_q)}^{ - 1}} \right)NB_t^a + \left( {\prod\limits_{q = 1}^{4T - 2} {(Y_q)}^{ - 1}} \right)\tilde L_T^a} \right],$$

where ${\rm E}_0^Q $ is the risk-neutral expectation under the Q measure of the cash flows discounted against the quarterly gross risk-free rate Y _q , obtained through the simulation of the state vector, and q denotes the quarter. $NB_t^a $ is the net benefit of the cohort aged a in period t, which for workers is their contribution and for pensioners is the pension benefit they receive. Hence, $NB_t^a $ is negative for workers and positive for retirees. Because $NB_t^a $ occurs in the middle of time period t, it is discounted only for half of year t. Further, $\tilde L_T^a $ is the final value of the economic liabilities to the cohort of age a. Since the working participants accrue pension entitlements in exchange for their contributions, we add the discounted final value of the economic liabilities to the discounted value of the net benefits to arrive at the value of the contract for each cohort. Unlike benefits and contributions, the outstanding pension entitlements at the end of the simulation period are not actual cash flows that materialize during the simulation. Rather, they form an estimate of the outstanding pension promises that have been given in exchange for contributions paid earlier. To sum up, the contract value to a cohort is defined as the present value of the benefits they receive, minus present value of their contributions, plus the present value of the pension entitlements accrued at the end of the simulation horizon in exchange for the contributions made earlier. The contract value to all participants is the sum of the contract values for each cohort participating in the fund during the evaluation horizon, i.e., $V_0^{PP} = \sum\nolimits_a V_0^{PP,a} $ .

The value of the contract to the tax payer cohort of age a in period 0 is:

(9) $$V_0^{TP,a} = - {\rm E}_0^Q \left[ {\sum\limits_{t = 1}^T \left( {\prod\limits_{q = 1}^{4t - 2} {(Y_q)}^{ - 1}} \right)\left( {C_t^{TP,a} + SS_t^a} \right) - V_0^{R,a}} \right],$$

where $C_t^{TP,a} $ is the tax-payers’ contribution in dollars, i.e., the normal cost plus the actual amortization payment minus the contribution by the employees, and $SS_t^a $ the sponsor support. The contract value to the total population of tax payers is $V_0^{TP} = \sum\nolimits_a V_0^{TP,a} $ . The final term in (9) is the part of the so-called residual value of the pension fund absorbed by the cohort of age a. The residual value $V_0^R = {\rm E}_0^Q \left[ {\prod\nolimits_{q = 1}^{4T - 2} {(Y_q)}^{ - 1}(A_T - {\tilde L}_T)} \right]$ is the difference between the present value of the assets remaining at the end of the evaluation horizon and the aggregate of the final values of the economic liabilities. Therefore, it is what is left over in present value terms for the tax payers at the end of the simulation horizon after redeeming their obligations to the fund participants still alive at that moment. Since the contributions associated with these accrued liabilities by the tax payers as employers have already been made, we can view the residual value as the amount of resources that the tax payers alive at the end of the horizon would still need to provide to (if the residual is negative) or receive from (if the residual is positive) the fund participants. We therefore distribute the residual value over all cohorts of tax payers alive at the end of the horizon.

The cohort-specific plan participant values are immediately available, as participants always contribute a certain part of their cohort-specific salary and receive cohort-specific benefit payments. Determining the cohort-specific tax payer values is not completely straightforward. The aggregate cash flows in a year from all the tax payers together are immediately available. However, calculating the contract values for the individual cohorts of tax payers aged a in period 0 requires an assumption about the allocation of the aggregate cash flows across the cohorts of tax payers. We assume that the demographic structure of the tax payer population and the age-wage profiles of the tax payers are identical to those of the participant population. The cash flows assigned to a specific tax payer cohort are then proportional to the share of this specific cohort in aggregate income. For tax payers of working age, the relative contribution of each cohort is fixed over time through the age-wage profile, because the latter is constant over time. The relative contribution of a cohort of retired tax payers is proportional to the average income of the participant cohort (i.e., the retirement benefit) relative to average aggregate income, where average income is the average over all scenarios in a specific year under the baseline plan. When allocating the residual value, we assume that it is absorbed by the different cohorts of tax payers alive at the end of the horizon in proportion to the present value of their projected remaining lifetime income.

6 The data

All our data are for the USA. Our dataset comprises macroeconomic data, data from financial markets, data on state pension funds and demographic data. The economic scenario generator described earlier is based on historical data spanning the first quarter of 1990 up to and including the second quarter of 2015.

We use historical time series of the short interest rate, stock returns, price inflation and wage inflation. We obtain the real wage growth rate by adjusting the nominal wage growth rate by the growth rate in the nominal price index. The short interest rate is the 3-month treasury rate series and is obtained from the Board of Governors of the Federal Reserve System (2015). For the stock returns we use the NYSE/Amex/Nasdaq value-weighted return with dividends, which is available from the Center for Research in Security Prices (2015). For price inflation we use the CPI index provided by the Bureau of Labor Statistics (2015). Finally, wage inflation is obtained from the compensation of employees, wages and salary accruals and is retrieved from the US. Bureau of Economic Analysis (2015). These data are all quarterly or of higher frequency and transformed into quarterly data. The US treasury nominal yield curve for maturities from 1 to 10 years is retrieved from the Federal Reserve Board (2015). We obtain the treasury real yield curve rates for maturities 5, 7 and 10 from the US Department of the Treasury (2015). However, they are only available starting from 2003. Finally, we obtain pension plan data from the Public Plans Database of the Center for Retirement Research at Boston College (2015).

As far as the demography is concerned, we use the National Population Estimates for 2010 by single year of age and sex provided by the United States Census Bureau (2014). Further, we use the survival rates derived from the Cohort Life Tables for the Social Security Area by Year of Birth and Sex provided by the US Social Security Administration (2014) These life tables are available for birth years with 10 years intervals. Therefore, we linearly interpolate the survival rates for the birth years in between the ones directly available.

7 Benchmark settings for our US state plan

We set the benchmark simulation horizon to 75 years, which is a commonly-used horizon for pension policy evaluation in the USA and which allows us to take into account a large number of cohorts that enter the fund after the start of the evaluation horizon. Hence, we evaluate the pension contract for cohorts born up to 50 years from now, in effect assuming that for cohorts born later contributions will be set such that their net contract value is zero. Extending the simulation horizon, we observe that the present values of both the end-of-the-horizon assets and economic liabilities fall and, hence, become less relevant relative to the present value of the benefits paid out over the horizon. In Subsection 9.3.3, we show that extending the horizon to 100 years has no qualitative and only limited quantitative effect on our results.

The demographic structure of our fund at the start t = 0 of the simulation horizon is assumed to be identical to that of the US population in 2010. Hence, the relative sizes of the cohorts are equal to those for the USA in 2010. We set the population of our pension fund at t = 0 equal to the aggregate number of participants in 2012 of all the state plans covered by the data from the Center for Retirement Research at Boston College (2015). As a result, our pension fund initially has approximately 26 million participants. While these data do not include all the US public pension plans, they do include a very substantial share of the US public sector workers at the sub-national level. Hence, the magnitude of the net burden on the tax payers that we will expose below provides a good indication of the seriousness of the public sector pension underfunding problem for the USA, although it may only be a lower bound to the problem. This is not only because a number of public plans have not been included in our dataset – and a priori there is little reason to assume that these plans are healthier than those that are part of our dataset –, but also because the lion's share of the cost to the tax payers will be in the most underfunded, riskiest plans, since tax payer guarantees are increasing and convex in risk.Footnote ⁸ Obviously, when financial pressure accumulates, the subnational authorities will start shifting new participants into DC plans. However, we aim at assessing the underfunding problem under current plans and plausible alternatives, so here we ignore other potential policy reactions to the problem.

The fund's demography over the simulation horizon evolves as follows. Throughout the simulation horizon new cohorts aged 25 enter the labor market and become participants of the pension fund. Hence, for the first 25 years of the simulation horizon we determine the sizes of the 25-year old cohort by taking the sizes of the cohorts younger than 25 in the US population initially and projecting the size of each such cohort when it reaches the age of 25 by using the officially projected (gender-specific) survival probabilities. The number of new entrants for the remaining 50 years of a simulation run is calculated by extending over time the trend in the size of the 25-year old cohort, i.e., by calculating the average annual growth rate of the 25-year old cohort over the first 25 years of the simulation horizon and taking this as the annual growth rate for the next 50 years. At the start of the simulation there are about 280 thousand 25-year old males and about 270 thousand 25-year old females entering the fund. During the first 25 years the number of new 25-year olds decreases on average by 0.34% per year. Extrapolating this trend over the next 50 years yields roughly 210 thousand new male entrants and 200 thousand new female entrants in the final simulation year.

Our simulations are based on the most common characteristics of the US state pension plans. Its liabilities are computed by using the entry age normal (EAN) actuarial method for liability recognition and are based on the following assumptions. Individuals retire at the age of 65. Hence, a full career means that they work for 40 years. The career (age-wage) profile is obtained from Bucciol (Reference Bucciol2012). Because the shape of this profile does not change over time, the wage earned at a given age increases each year with the common wage growth rate in the economy. The t = 0 average wage rate is set such that the fund's initial liabilities L ₀ are equal to 3900 billion dollars, which is the aggregate amount of liabilities in 2012 over all pension funds in the Public Plans Database of the Center for Retirement Research at Boston College (2015).Footnote ⁹ The benefit factor, or accrual rate, is 2% for each additional year of service, while the earnings base for the retirement benefit is the average of the final 3 years of pay during the career. During retirement there is full indexation for consumer price inflation when inflation is non-negative, while indexation is set at zero when inflation is negative.

The actuarial assumptions by our pension fund are based on the median assumptions used by the funds in the Public Plans Database of the Center for Retirement Research at Boston College (2015) in 2012 (the most recent year for which there is a full sample available). As a result, projected benefits are heightened up with a projected nominal salary increase of 3.75% for each working year and a projected annual rate of price inflation of 3.16% for each year in retirement, while the future retirement benefits are discounted at a fixed rate of 7.75% a year. We assume the initial accrued pension rights, $B_0^a $ , to have been fully indexed up to time t = 0.

Contributions to the fund are calculated as follows. The annually required contribution is set to the normal cost plus the required amortization payment, which we assumed to be fixed at zero in the case of a fund surplus (one-sided policy). The normal cost is calculated as a percentage of the projected career salary based on the EAN actuarial method. The amortization payment is determined by spreading the unfunded actuarial accrued liability UAAL in equal annual payments over the next 30 years, with a moving 30-years window. Hence, we use the so-called open amortization period. Employees pay a fixed 6% contribution of their salary, while the tax payer as employer pays the remainder of the required contribution.

Our sample data show that in 2012, assuming that the normal cost is paid in full, the actual amortization payment is on average just slightly more than half of the required amortization payment. Because of the lack of potentially more representative data we assume that the actual contribution is set to 100% of the normal cost, plus 50% of the required amortization payment. However, because our estimate of the share of required amortization paid varies substantially across plans and because of the importance of this parameter to the funding situation of the plan, we will also examine variations on the baseline plan in which less or more of the required amortization payment is paid in a year.

We set the initial actuarial funding ratio FR ₀ equal to 71%, which is the median funding ratio of the pension funds in our database in 2012. Based on FR ₀ and the initial liabilities, which are thus both matched to our data, we obtain the initial actuarial assets of the fund:

(10) $$A_0^{act} = FR_0 \cdot L_0.$$

The actuarial funding position of the pension fund in the subsequent periods is calculated as the ratio of the actuarial assets and liabilities.Footnote ¹⁰

We assume that under the baseline plan the fund invests 50% of its assets in fixed income (with an average annual return of 4.7%) and the other 50% in risky assets (with an average annual return of 5.5%). This implies an average annual return on the fund's portfolio of 5.1%. We thus abstract from investments in assets other than these two classes.Footnote ¹¹

8 The alternative plans

We consider a number of variations on the baseline plan, which we refer to as Plan 0, to explore what various policy changes imply for the contract values of the different stakeholders. This can give us leads about the effectiveness of different measures in reducing the likelihood that the pension fund's assets get depleted and, hence, in reducing reliance on the support from the tax payers.

We consider three groups of measures, which we summarize in Table 1. The first set of measures, Plans 1.1–1.4, addresses the contribution rate. Plans 1.1 and 1.2 vary the fraction of the required amortization payment actually paid, Plan 1.3 shortens the period over which the amortization payment is spread, while Plan 1.4 doubles the contribution payment by the participants from 6% to 12% of their salary, but leaves the total contribution rate and, hence, the fund's financial health unchanged.

Table 1. Summary of the alternative plans

The second set of alternatives addresses the degree of indexation. Under all plans, including the baseline plan, there is a 0% floor on indexation. Plan 2.1 halves indexation when CPI inflation is positive. Under Plan 2.2, if CPI inflation is positive, then indexation to CPI inflation is conditional on the level of the policy funding ratio FR ^P . Specifically, if cpi ≥ 0 and FR ^P <0.5, indexation is 0, while if cpi ≥ 0 and FR ^P  ≥ 0.5, it equals $(2FR^P - 1)cpi$ . Therefore, if CPI inflation is positive, indexation is zero for funding ratios below one half, while it increases linearly with the funding ratio for funding ratios of at least one half. Hence, a funding ratio above unity implies more than full indexation. This way of providing conditional indexation is closely related to the way most Dutch pension funds index their pension rights – see Ponds and van Riel (Reference Ponds and van Riel2009) and Beetsma and Bucciol (Reference Beetsma and Bucciol2011b).

We continue to set contributions on the basis of the normal cost that was calculated under the assumption of full indexation. We do this in order to see the isolated effect of a change in indexation. Otherwise, the comparison with the baseline plan would be contaminated by a fall of the normal cost when indexation is reduced, which through lower contributions would in turn dampen the beneficial effect of less indexation on the fund's financial health.

The third group of measures concerns the composition of the fund's asset portfolio, which we vary from zero to 100% stocks.

10 Concluding remarks

This paper has explored the financial and redistributive aspects of a typical US state DB pension fund under unchanged policies. The results confirm what is generally feared, namely that continuing current pension policies leads to the depletion of the fund's assets. Even a substantially more optimistic outlook for financial markets than that under our benchmark parameter setting does not solve the long-run financial sustainability issues of the US state pension sector. In fact, the magnitude of this problem exposed by the current paper may well form a lower bound to its true extent, because our dataset did not include all civil servants’ pension funds. A priori, there is little reason to assume that the funds that have not been included are financially healthier than the funds that were part of our dataset. Hence, a continuation of current pension policies is likely to confront the tax payers with an enormous bill, because in principle they have to guarantee the accumulated entitlements of the funds’ participants. Even rather substantial shifts in the structure of the pension contributions or substantial reductions in inflation indexation are unlikely to resolve the long-run financial issues of the US state pension sector, although they may slow down the deterioration of its financial health somewhat.

The looming financial burden on tax payers and the prospective deterioration of public services may discourage individuals of working age from working in states where the pension funding problems are worst. An outcome with falling spending on public goods and rising taxes merely paid to finance civil servants’ retirement schemes may become politically unsustainable and result in states defaulting on their pension obligations. This could well cause larger value losses to fund participants than those associated with orderly reforms backed by a majority of the population.

The analysis in this paper can be extended into various directions, for example by exploring the consequences of closing the fund to new participants or closing it even for contributions from current participants. Also, by applying our method of value-based ALM, we can study value shifts between tax payers and participants for individual state funds and explore which states face the most urgent need for pension reform.