Stochastic infinite horizon forecasts for us social security finances.

Lee, Ronald ; Anderson, Michael

Even over a 75-year horizon, forecasts of PAYGO pension finances are misleadingly optimistic. Infinite horizon forecasts are necessary, but are they possible? We build on earlier stochastic forecasts of the US Social Security trust fund which model key demographic and economic variables as historical time series, and use the fitted models to generate Monte Carlo simulations of future fund performance. Using a 500-year stochastic projection, effectively infinite with discounting, we find a fund balance of -5.15 per cent of payroll, compared to the -3.5 per cent of the 2004 Trustees' Report, probably reflecting different mortality projections. Our 95 per cent probability bounds are -10.5 and -1.3 per cent. Such forecasts, which reflect only 'routine' uncertainty, have many problems but nonetheless seem worthwhile.

Keywords: Social security; sustainable; infinite horizon; stochastic; forecast. JEL classification: H55; JII; C15

Introduction

Population ageing threatens the long-term finances of public pension programmes around the world. Many nations are discussing parametric adjustments or deeper reforms to address this problem. However, the design of effective pension policies requires a correct assessment of the magnitude of the problem. In the US, the recent debate about reform was accompanied by sharp controversy about the appropriate way to measure the imbalance. To measure the cost of keeping social security solvent, the Social Security Trustees and Office of the Actuary have traditionally relied primarily on the Summary Actuarial Balance calculated over a 75-year horizon. (1) Others, including the advocates of privatisation, argue for an assessment over the infinite horizon, since some of the advantages of privatisation accrue after 75 years, and are not reflected in the standard measure.

Whatever the merits of privatisation, we find that the 75-year horizon measure seriously understates the size of the true imbalance, and that some sort of infinite horizon measure or the equivalent is required. Many analysts agree that there are severe problems with the 75-year summary actuarial balance as a measure of the long-term fiscal soundness of the system, because it takes no account of what happens after the 75 year horizon, and therefore does not measure what it would cost to put the system on a sustainable footing. For closely related reasons, it is not time-consistent: given exactly the same underlying economic and demographic projections, it will nonetheless deteriorate from one year to the next due to the loss of a good year at the start of the evaluation period and the addition of a bad year at the end.

Lee and Yamagata (2003) developed analytic methods for making infinite horizon projections, and also discuss simpler measures which can be calculated from the standard 75-year projection under certain assumptions about stability, in which case they are equivalent to a certain kind of infinite horizon projection. One of these, which they call the Flat Fund Ratio Tax, is the immediate and permanent tax increase that would leave the ratio of the Trust Fund to Costs constant at the end of the projection horizon, a measure that has also been included in recent Trustees' Reports.

Before 1965 the Actuaries assessed solvency over an infinite horizon or "in perpetuity" (Myers, 1959). After 1965, they moved to a 75-year horizon on the recommendation of the Advisory Council. (2) This had a relatively small effect on the long-run cost projections at that time, because costs were projected to remain flat in any case, rather than rising exponentially as they do now (Goss, 1999). Starting in 1973, new legislation linked benefits to past earnings, so the projections began to assume a changing time path for earnings and benefits and consequently use of an infinite horizon might now make a considerable difference.

The 2003 Trustees' Report (TR) included an infinite horizon measure for the first time in many decades, which agreed with the Lee-Yamagata calculations in showing that the budget shortfall was about twice as great relative to payroll as for the 75-Year Summary Actuarial Balance. These estimates were repeated in the 2004 and 2005 Trustees' Reports. The method is described as follows: "The [infinite horizon] extension assumes that the current-law OASDI (3) programme and the demographic and economic trends used for the 75-year projection continue indefinitely." (TR 2004:58, text in brackets is added by us.)

It is well known that central forecasts by economists, demographers and actuaries often deviate seriously from actual outcomes even a few years ahead, which is in the nature of the undertaking. Lee and Tuljapurkar (1998a and b, and 2000) and Lee, Anderson and Tuljapurkar (2003) developed probabilistic or stochastic projections of the finances of the Social Security system. Details of the methods are described in the papers cited. Subsequently, the Congressional Budget Office (2001) developed a stochastic projection model, and then in 2003 the Office of the Actuary of the Social Security Administration also developed its own version. The projections of date of fund exhaustion were compared across these three models (Burdick and Manchester, 2003), and agreement was quite close among these three macro-stochastic forecasts.

These stochastic forecasts indicate a considerable range of uncertainty over a 75-year horizon. Given the substantial uncertainty inherent in long-term forecasts, it is questionable how seriously we should take infinite horizon forecasts. Can they really be trusted as the basis for serious policy decisions today? The very phrase 'infinite horizon forecast' makes many people snicker, and indeed many serious demographers believe it is pointless and misleading to forecast population beyond 25 years or so. (4) In principle, it would therefore be useful to have probability intervals for the infinite horizon forecasts, giving an idea of how far from the central forecast the actual outcomes might lie. This project will investigate the possibilities of developing probability intervals for infinite horizon forecasts, building on unpublished work by Lee and Yamagata and using the stochastic projection model described earlier. Of course, construction of probability intervals also requires assumptions about the regularity of history, and one need only count back 500 years from the present to a time before Shakespeare to get some idea of how difficult and potentially misleading the whole enterprise might be. Nonetheless, we believe it is worthwhile to make the attempt. However, it should be understood and kept in mind that we incorporate into our forecasts only uncertainty that arises within the context of assumed structural continuity and homogeneity. This means we assume that expected or long-term average rate of growth of covered wages remains the same over 500 years, as does the expected rate of decline of mortality, the expected level of fertility, and the expected level of real interest rates on government bonds. Random variations about these expected values do occur, but the expected values themselves are constant. Also, we assume that the current programme structure remains the same except for already legislated changes in the normal retirement age. Thus the forecasts are conditional on current programme structure, as they should be if they are intended to illuminate the extent of need for policy change. We call this kind of forecast uncertainty 'routine' or 'business as usual' to acknowledge that it excludes deep structural change, and kinds of shocks and trend breaks that were not observed in the past century. Thus the uncertainty we include understates the true amount of uncertainty.

Assumptions, procedures and measures

An overview of the basic stochastic model of social security finances

Over the past ten years, Lee, Tuljapurkar and Anderson have developed and extended a stochastic forecasting model for Social Security finances, which is described in Lee and Tuljapurkar (1998a and b) and Lee, Anderson and Tuljapurkar (2003), with recent extensions discussed in Lee, Miller and Anderson (2004). We will here sketch its structure, starting with the demography and then turning to the economy. In both cases, the approach is to extrapolate historical trends and variations without taking into account any potential feedback processes. We do not include capital stock or technology, nor savings rates.

The OASDI Trust Fund balance at the end of each year is equal to the balance at the end of the previous year, plus annual payroll tax income, plus taxes on benefits, plus interest earned on the Trust Fund, minus benefit payments, railroad retirement, and administrative expenses. The simulation starts with the Trust Fund balance from the 2003 Trustees' Report, which states total assets of $1.378 trillion at the end of calendar year 2002. It also starts with the population age distribution as of the beginning of 2003. Based on these accounting identities, it is straightforward to project the finances of the Social Security system for any given input sequence of fertility, mortality, interest rates and productivity growth rates. We produce sequences of this sort through Monte Carlo simulations of statistical time series models for the input variables, where the models are described below. Typical forecasts cover a 75-year horizon. Each 75-year sequence produces a single stochastic sample path. By repeating this procedure, we can generate a collection of sample paths for the system's finances. We can then calculate frequency distributions for outcomes of interest, and these form our primary forecasting output. For most purposes, between 1000 and 10,000 sample paths is enough to yield relatively stable distributions.

To project the population from its initial level and age structure, we need time series of age-specific fertility, mortality, and net immigration. Mortality is projected using the Lee-Carter method, which uses a combination of statistical time series methods and demographic concepts to extrapolate long-term age-specific trends (Lee and Carter, 1992). The level of mortality is summarised by a single time varying index, which is modelled as a random walk with drift. The random walk and uncertainty about the true level of the estimated drift term produce uncertainty in the mortality forecast. This method has performed well in the US in both prospective tests based on the 1992 forecasts, and in hypothetical projections made from various start dates throughout the 20th century (Lee and Miller, 2001).

The level of fertility is modelled as a time series process, with an unchanging proportional age distribution. We constrain the central value of the fertility process to be equal to the Intermediate assumption made by the Social Security Office of the Actuary (SSA), or 1.95 births per woman (the current Total Fertility Rate in the US has been between 2.0 and 2.1 in recent years). The level is modelled as a stationary time series process. This generates a variance and autocovariance structure for the forecast errors. Immigration is treated deterministically, in part because it is a policy instrument, and set at the level of the Intermediate assumptions of SSA (see Lee et al., 2004, for a stochastic treatment of immigration).

The economic variables we model are the productivity growth rate, (5) the real interest rate, and for some purposes, but not used in this paper, the rate of return on equities (S&P 500). These are modelled as vector autoregressive processes, but in fact only equities and the interest rate are found to be correlated. The central values of the interest rate and productivity growth processes are constrained to equal the Intermediate assumptions of SSA, as was done for fertility.

Either from cross-sectional survey data or from administrative records, we calculate a base year age profile of per capita payroll tax payments. At each iteration of the projection model, this is multiplied by the projected level of productivity or the covered real wage, divided by the base year level. There is an implicit assumption that the earnings threshold for payroll taxes is adjusted upwards as real wages rise. Per capita benefit levels by age are also estimated from either survey (6) or administrative data, and then shifted upwards for each generation according to the aggregate level of real wages when it turns 60. The determination of benefit levels is somewhat complicated in the model, because it must reflect patterns of continuing work by some after the early retirement age, and after the normal retirement age, leading to higher benefits after retirement, as well as patterns of divorce and widowhood as the generations age which also affect benefits. We use repeated cross-sections of benefits by age to estimate patterns of benefit change over time within cohorts. In addition, we model the effects of the scheduled increases in the normal retirement age on labour supply and retirement. The bottom line is that benefits, like tax payments, are driven by productivity growth and by base year per capita age profiles.

These stochastic models of the basic economic and demographic processes incorporate uncertainty in our forecasts to a degree that is consistent with the variability observed in the historical series. Other sources of uncertainty are not reflected in these forecasts, however. Some uncertainty is behavioural: will workers postpone retirement when the normal retirement age is raised? Will women continue to join the labour force at increasing rates? Other uncertainty arises from the possibility of deep structural shifts, through technological change, globalisation, climate change, dramatic biomedical advances, the emergence of new diseases, or radical changes in the family. Our estimates of forecast uncertainty do not attempt to incorporate such possibilities, to the degree that they have not been reflected in past history.

How long is forever?

We follow the same strategy described above in the Trustees' Report (2004). In the past, we have made probabilistic forecasts over horizons of 75 or 100 years. For the infinite horizon we just keep going, assuming that the same ultimate levels of fertility, rate of real wage growth, real interest rate, and rate of mortality decline continue,

while the structure of Social Security conforms to current law (including the legislated increase in the normal retirement age to 67). We are interested in present values and their ratios over the infinite horizon. Deterministic present values involve discounting at the real rate of 3.0 per cent annually.

Define the residual discount rate to be the 3 per cent real rate of interest that is assumed, less the 1.1 per cent growth rate in covered real wages that is assumed, less the approximate rate of population growth of 0.3 per cent, for a residual discount rate of 1.6 per cent per year. A proportional variation in the system's accounts S years in the future will count for only [e.sup.-0.016.S] relative to the present. (7)

Here are the relative weights that result.

Horizon S Weight

 0 1
 50 0.449329
 100 0.201897
 150 0.090718
 200 0.040762
 250 0.018316
 300 0.00823
 500 0.000335

A proportional variation in the first year is worth five times as much as one in the 100th year, more than 100 times as in the 250th year, and about 3000 times as much as one in the 500th year.

Because of this discounting, it is not necessary to go to the literal infinite horizon. Instead, we forecast over a 500-year horizon.

Stochastic infinite horizon methods Of the four variables we forecast probabilistically, only mortality is projected to have a continuing trend. Based on the Lee-Carter method, each age-specific death rate is assumed to decline at its own exponential rate for 500 years, where the rates of decline are based on historical estimates. Chart 1 plots the life expectancy forecast for females that results from this procedure, along with the 95 per cent probability interval that is generated by the method. Surprisingly (to us, at least), our fitted model predicts that life expectancy will not exceed 100 years until nearly 2400, and will only slightly exceed 100 by 2500. We view these forecasts as too low. There are several reasons why this might be so. First, we fit the model using a weighted Singular Value Decomposition (SVD) procedure (Wilmoth, 1993) instead of our usual unweighted SVD, and we believe that this leads to lower forecasts even over normal horizons. Second, our procedures constrain survivorship to become zero at age 120. Third, our projections assume that the age pattern of rates of decline remains unchanged. However, it is entirely possible that this age pattern of rate of decline will itself change dramatically in the coming centuries, which could permit much larger increases in life expectancy.

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We note that a recent paper by Oeppen and Vaupel (2002) found that recorded life expectancy among the world's populations with suitable data had been increasing linearly at nearly 2.5 years per decade or 25 years per century since 1840. If that trend were to continue, then life expectancy would be roughly 200 years by the year 2500, or twice our projected level. There are many biologists who believe that life expectancies of 150 or more could be attained in this century. There are also biologists who believe that life expectancy is unlikely to exceed 85 or so. Without necessarily subscribing to any of the views we just mentioned, we nonetheless conclude that our central projection is too low.

The 95 per cent probability intervals on our projected life expectancies are less than ten years wide, which also strikes us as unrealistically narrow for such a long horizon. Indeed, the span of the 95 per cent interval for 2500 is not much different than the span of the interval for 2075. One reason for the narrow uncertainty range is that the age schedule of mortality rises rapidly after age 30 or so, with death rates doubling every seven years of age, consistent with Gompertz's Law. This means that even large variations in the general levels of the age-specific death rates translate into small variations in life expectancy. The same degree of uncertainty about the level of death rates five centuries in the past would have translated into 95 per cent probability intervals wide enough to encompass current levels of life expectancy, because of the very different age structure of death rates at that time. (8) As noted above, the age pattern or age-specific mortality could change dramatically in the coming centuries, leading to very different mortality trends. This sort of uncertainty is not reflected in our probability intervals. One might say that our probability intervals reflect only a small portion of the true extent of the unknown.

All these projections or input series assume constant central tendencies in levels or (for mortality) in rates of change. An alternative approach would be to use structural time series models in which the central tendencies themselves can vary over time, for example as random walks or ARIMA processes, as done in Lee, Miller and Anderson (2004) (see also Holmer, 2003). Over a 500-year horizon, a random walk would most likely stray implausibly far from its starting value, leading to impossible levels of fertility or interest rate, for example. However, a structural process with a stochastic but equilibrating central tendency could perhaps reflect uncertainty more adequately. In any event, here we will proceed as described, although mindful of the problems with doing so.

Simplified infinite horizon measures

Lee and Yamagata (2003) developed two measures of long-run sustainability which do not require actually carrying out detailed projections beyond the standard 75-year horizon. One of these is the Flat Fund Ratio Tax, which was described above. They show that under an assumption of strong stability (revenues and costs grow at equal and constant exponential rates after the end of the projection period) this Flat Fund Ratio Tax is identical to the infinite horizon summary actuarial balance. Consequently, we will often call it the Stable measure. This simple measure has the virtue that it is easy to explain to policymakers and the public. The idea that the ratio of the Trust Fund to expenditures on benefits should be level at the end of the projection period is just common sense. In what follows, we will investigate the performance of this measure relative to the actual 500-year horizon calculations. We also try calculating a probability distribution of the Flat Fund Ratio Tax in a stochastic projection over the 75-year horizon, after smoothing the projected paths near the 75-year horizon so that changes in the Fund Ratio are not dominated by short-term noise.

If we calculate the actual rates of change of revenues and costs from the Trustees' Report, we find that costs are growing more rapidly than revenues, so that the stability assumption discussed above is not warranted. For this reason, the Flat Fund Ratio Tax will generally underestimate the infinite horizon imbalance. This leads us to a second measure that is slightly more complicated but can also be calculated directly from the 75-year projections regularly published by the Trustees. This measure assumes that costs and revenues continue to change after the 75-year horizon at the same exponential rate that they were changing at the end of the projection period, say between year 74 and year 75. We call this the Unstable measure. Under current conditions (rate of growth of costs exceeding that of revenues) it will indicate a larger infinite horizon imbalance than the Flat Fund Ratio Tax measure. As with the Stable measure, it is possible to calculate a probability distribution for the Unstable measure, based on stochastic projections over a 75-year horizon.

Below, we will assess the accuracy and usefulness of these simpler measures, and of their associated probability measures, taking the actual 500-year projection as the gold standard.

Results

For reference purposes, we note that the 2003 Trustees' Report projected an infinite horizon Actuarial Balance of -3.7 per cent of payroll, while the 2004 Trustees' Report projected -3.5 per cent, both consistent with those projections by Lee and Yamagata (2003) which used the Social Security Actuary's mortality projection.

The 75-year actuarial balance is not a sustainability measure

According to the 2003 Trustees' Report, the Summary Actuarial Balance (present value of annual balances divided by present value of payroll) was 1.92 per cent. In principle, raising the payroll tax by this amount should restore the system to solvency. The problem with this measure is illustrated in chart 2 which shows the probability distribution of the projected Trust Fund Ratio, assuming the payroll tax has been raised by 1.92 per cent. The median ratio crosses zero and turns negative in 2074, falling to -82 in 2200. The mean ratio crosses zero a bit later than the median, but falls much faster, ending below -150 by 2200. The upper 67 per cent probability bound (that is, at 83.3 per cent) also becomes increasingly negative soon after 2120. Clearly, a tax increase of 1.92 per cent will not put the system on a sustainable footing, even with quite a lot of luck.

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Simple measures of sustainability

We can calculate the Stable or Flat Fund Ratio measure for each stochastic sample path at a 75-year horizon. The probability distribution for the measure is then given by the frequency distribution of these estimates, which is plotted in chart 3. The median is 4.36 per cent, indicating that under the stable assumption, the infinite horizon imbalance would be 4.36 per cent of the present value of payroll. Alternatively, under this assumption an immediate and permanent tax increase of 4.36 per cent would put the system in sustainable balance. This compares to a similar calculation of 4.2 per cent reported in Lee and Yamagata (2003). 99 per cent of the probability distribution for this measure lies between 0 per cent and 10 per cent.

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Similarly, we can find the probability distribution for the Unstable measure of infinite horizon imbalance (not shown). As expected, the Unstable measure indicates a larger imbalance, with a median equal to 5.21 per cent of the present value of payroll.

Results of 500-year stochastic projections

We described above how the infinite horizon projections are carried out. Actuarial Balances can be calculated for different horizons, and the Trustees' Reports generally give them for horizons of 25, 50 and 75 years, as well as for the infinite horizon since 2003. Chart 4 plots percentiles of the probability distributions for Actuarial Balances calculated over horizons up to 500 years. Thus, for example, the line labelled '50 percentile' portrays the median Actuarial Balance by length of horizon. We see that it drops rapidly to increasingly negative values in the first century, reaching about -3 per cent in 2100, and around -4 per cent in 2200, while dropping more and more slowly thereafter, and approximately levelling off in the fifth century. This is the case for all the other probability percentile lines, which indicates that the probability distribution has stabilised by 500 years and extending the forecast horizon further would add little.

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Now consider one specific horizon, at 500 years. The percentiles in chart 4 at a horizon of 500 years provide five points on the full probability distribution for the year 2500. Chart 5 plots the probability distribution in more detail for Actuarial Balances at this 500-year horizon. Here we see that the median is -5.15 per cent of the present value of payroll (which corresponds to the 50 percentile line shown in chart 4 at this horizon), and the mean is -0.37 per cent. Much as with the probability distribution for the Flat Fund Ratio measure, the probability mass is almost entirely between 0 per cent and -10 per cent, but with a bit more out on the negative tail for the 500-year projection. These various measures are contrasted in the following summary table.

Infinite horizon measure Percent of PV of
of actuarial balance taxable payroll

2003 (2004) Trustees' Report -3.7 (3.5)
Stochastic Projection (median) -5.15
Flat Fund ratio (median of
 stochastic projection at 75 years) -4.36
Unstable measure (median of
 stochastic projection at 75 years) -5.21

We note that the unstable measure agrees very closely with the 500-year projection, which suggests, but does not establish, that it might be a simple alternative to carrying out the detailed projections over 75 years, at least to obtain a central value. The Flat Fund Ratio measure also does surprisingly well, given its simplicity, and the strong and false assumption on which it rests (rates of growth of revenues and costs are equal and constant after 75 years). We also note that all three of the measures based on our stochastic projections show substantially greater imbalances than does the Trustees' Report. This most likely reflects the difference in the mortality projection, which Lee and Yamagata (2003) found to make about twice as much difference over the infinite horizon (about 1 to 2 per cent) as over the 75-year horizon (about 0.5 per cent).

The probability interval for the Flat Fund Ratio estimate of the infinite horizon balance is very similar to the interval for the actual 500-year Actuarial Balance, although the medians are somewhat different. However, closer inspection suggests that this is just a lucky accident. The Flat Fund Ratio criterion is not identifying trajectories that are headed in particularly costly or particularly inexpensive directions. When one thinks about the quite different basis for the two distributions, that is not surprising.

We can conclude that it is best to carry out the full stochastic projection over a long horizon of around 500 years, but that a pretty good approximation of the central tendency may be obtained from the 75-year Unstable measure, at least in the present circumstances. Whether it will be a good approximation in other circumstances as well remains to be seen. The Flat Fund Ratio also provides an acceptable measure, although when costs are rising more rapidly than revenues it will underestimate the imbalance as it does here.

So far, we have expressed the imbalance relative to the present value of payroll. Chart 6 reports the distribution of the infinite horizon imbalance in 2003 dollars. The median present value of the imbalance is 18 trillion dollars, or about 1.6 times GNP in 2003. This is substantially larger than the 10.5 trillion reported by the 2003 Trustees' Report for the open group infinite horizon obligation, at 3.7 per cent of payroll, or the corresponding figures of 10.4 trillion and 3.5 per cent in the 2004 Report. Our greater figure is roughly half due to a higher projected present value of payroll, and half due to a greater proportional imbalance. Both differences reflect our somewhat higher projections of future life expectancy, but the fact that we are reporting the median of a stochastic outcome, rather than a deterministic projection, may have a role to play too.

The high cost of delay

Charts 4 and 5 showed the amount of uncertainty in the 500-year actuarial balance; the 95 per cent of actuarial balance interval ranged from 1 per cent to 11 per cent of the present value of payroll. It is tempting to view this dispersion, and to conclude that the difference between raising payroll taxes by 0 per cent or 1.92 per cent or 5.15 per cent is swamped by uncertainty in the amount needed, and that any needed adjustment could easily be made later when the amount needed became clearer. If the delay is relatively short, this might be so, but the information gained by waiting would be little. However, we must keep in mind that the actuarial imbalances shown in charts 4 and 5 are not simple averages of all the imbalances year by year from now over the next 500 years. Rather they are heavily weighted averages, where the weights are the discount factors given in an earlier table. A 1 per cent increase in the payroll tax today has an effect that is more than twice as big as an increase delayed for 50 years, and five times as powerful as an increase made in 100 years, relative to the scales of the economies. Put differently, it would take a percentage point increase twice as large if postponed for 50 years, and five times as large if postponed for a century.

Dates of insolvency

These measures are intended to indicate the size of the policy adjustment needed to put the system on a sustainable path. Nonetheless, a probability of insolvency would still remain, and it would be poor policy indeed to raise taxes or cut benefits to the point where the probability of future insolvency approaches zero. It is much more sensible to adjust taxes and benefits as the future reveals itself, so as to keep the programme on track. The sustainability measures we have discussed and estimated strive to indicate the likely centre of the range of policy adjustments that would prove necessary.

If payroll taxes were raised by 5.15 per cent, which is the median Actuarial Balance over the 500-year horizon, then we should expect that the system would nonetheless become insolvent (Trust Fund equal to zero) in less than 500 years about 50 per cent of the time. This is shown in chart 7, which gives the histogram of dates of insolvency under a 5.15 per cent tax increase (from 12.4 per cent to 17.55 per cent of payroll). We see that there is a 10 per cent chance of insolvency by 2100 despite this hefty tax increase, a 20 per cent chance by 2125, and a 28 per cent chance by 2150.

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Similar probability distributions for the Trust Fund Ratio, assuming the Flat Fund Ratio Tax increase of 4.36 per cent, show a 36 per cent chance that the system would stay solvent for 500 years, and nearly a 20 per cent chance that it would become insolvent by 2100, rising to 35 per cent by 2125, and to 43 per cent by 2150.

How big would the trust fund get with a sustainable tax increase?

The Actuarial Balance is conveniently interpreted as the size of the immediate and permanent increase in the payroll tax needed to balance the system, over the horizon used in the calculation, whether finite or infinite. However, it is important to keep in mind that this measure and interpretation do not amount to a policy prescription. Any long-term imbalance could be addressed by a wide range of policies, each with different implications for intergenerational distribution. The 'immediate and permanent' policy is just one of many, and would place a relatively heavy burden on early generations which are projected to have shorter life expectancies than later generations. Other policies would include reductions in benefits, gradual increases in tax rates, indexation of benefit levels to generational life expectancies, and so on. Whatever policy is chosen, it could be designed to respond flexibly and regularly to emerging economic and demographic trends.

In this section we will explore the implications of instituting the immediate and permanent tax increase corresponding to the measured imbalances. In particular, we will examine the trends in trust fund ratios and in size of trust fund implied by these tax increases, and the probability distributions of these quantities. Fixing a payroll tax level today and leaving it unchanged forever would make no sense at all as a policy. Nonetheless, exploring the consequences of doing so sheds light on a different dimension of infinite horizon uncertainty.

Chart 8 plots the median Trust Fund Balance in 2002 dollars under the Flat Fund Ratio Tax increase of 4.36 per cent. We see that it rises to 100 trillion dollars around 2150, before declining precipitously to -35 trillion by 2200. These are huge numbers, and difficult to interpret without considering the scale of the economy as a whole, which is greatly expanded by labour force growth and real wage growth. Expressing the Trust Fund Balance relative to the level of expenditures on benefits in each year gives us the Trust Fund Ratio, a standard measure of solvency. However, because the costs of benefits vary relative to the scale of the economy, it is also useful to express the size of the Trust Fund relative to GDP in each year. We will show both of these.

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It is useful to adjust for the scale of the economy, since only variations in total shortfall relative to the size of the economy are relevant. For example, the Trust Fund Ratio is formed by dividing the level of the Fund by the system's Costs. We begin by looking at median Trust Fund Ratio through 2200, for tax increases equal to the Trustees' 75-year Actuarial Balance measure, 1.92 per cent; equal to the Flat Fund Ratio Tax, 4.36 per cent; and equal to the Unstable measure, 5.21 per cent (very close to the infinite horizon Actuarial Balance measure of 5.15 per cent, which will therefore not be shown separately). These are shown in chart 9. We have already noted that the Trust Fund Ratio becomes negative in 2074 with the 1.92 per cent increase, as was shown in chart 3, having peaked in 2019 at 6.4 times Costs. The Trust Fund Ratio for the Flat Fund Ratio Tax is constant in the 2070s by construction, where it peaks at 13.3 times Costs, but it declines steadily thereafter and has become negative before 2200. We have already indicated that it is an underestimate of the adjustment needed for sustainability, but the rapidity of its decline is nonetheless surprising to us. With an increase equal to the Unstable amount (or nearly equivalently, to the infinite horizon Actuarial Balance), the Trust Fund Ratio reaches a high near 27 times Costs toward 2200, and appears to have stabilised.

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Selected probability quantiles for the Trust Fund Ratio under a 4.36 per cent tax increase are plotted in chart 10. We note that the distribution is very wide after 200 years, and we also note that while the median ratio decreases to 0 before 2200, the mean ratio is increasing monotonically, evidently driven by the fortunate upper tail of the distribution.

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The cost of benefits is rising relative to GDP as the population ages, so a constant Trust Fund Ratio would mask a rising proportion of GDP going to Social Security. For this reason, a clearer picture of the size of a surplus or deficit is gained by direct comparison to GDP. Chart 11 shows selected percentiles of the probability distribution for this ratio, expressed as a per cent, for a tax increase equal to the Flat Fund Ratio Tax increase of 4.36 per cent, through 2080. The median ratio reaches 105 per cent of GDP in the 2070s. While the median flattens out, and then declines (outside the range of this graph), the mean continues to rise, surpassing 170 per cent of GDP by the end of the plot range, with no end in sight. We also show the 2/3 probability range, with its upper bound (labelled 83.3 per cent) asymmetrically high and rising rapidly, reaching above 300 per cent of GDP by the late 2070s. The lower 16.7 per cent bound remains positive throughout, but has dropped below 30 per cent by the late 2070s. This great range of outcomes for a policy that is supposed to be sustainable demonstrates the necessity of maintaining policy flexibility to adjust to changing circumstances as they unfold.

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As we would expect, a similar plot for the infinite horizon Actuarial Balance tax increase of 5.15 per cent shows an even greater increase of the Trust Fund relative to GDP (not shown). The median reaches nearly 150 per cent of GDP by 2075, and the 83.3 per cent bound reaches 375 per cent by the late 2070s.

Balances of this size could not be held as government bonds, since that would exceed current levels of government debt relative to GDP, and defeat the purpose of this sort of partial prefunding. An alternative would be to hold the Trust Fund in equities, in which case it would account for a sizeable portion of the domestic capital stock, but not need to be so large. But these issues are beyond the scope of this paper. The main point to be taken from these stochastic simulations is the importance of formulating adaptable or self-correcting policies for addressing the infinite horizon imbalance. One simple policy of this sort would be to adjust the payroll tax rate every year, or every ten years, to aim for the Flat Fund Ratio at a 75-year horizon. If the Trust Fund began to accumulate unexpectedly rapidly, that would lead to an automatic reduction in the payroll tax rate. Similar policies could be designed based on adjustment of benefits. Implementing such a policy would not require projections beyond the 75-year horizon for which they are currently prepared.

Conclusions

Both the Flat Fund Ratio (Stable) measure and the Unstable measure are useful simple approximations to the deterministic or median infinite horizon open group imbalance measure. The Flat Fund Ratio is the immediate and permanent tax increase that would be needed to hold the ratio of the Trust Fund to Costs constant over the last few years of the 75-year projection. It is more intuitive and therefore easier to explain than the Unstable measure, but it underestimates the imbalance, whereas the Unstable measure gives a very good approximation to the infinite horizon measure, at least under current circumstances. The 2004 Trustees' Report indicates an infinite horizon open group imbalance equal to 3.5 per cent of payroll, consistent with Lee and Yamagata (2003) when the Actuary's mortality projection is used. Based on our 500-year projection, with somewhat more rapid mortality decline, we estimate it to be 5.15 per cent, substantially larger. The two simple methods, based on our 75-year projections, indicate imbalances of 4.36 per cent of the present value of payroll for the Flat Fund Ratio measure, and 5.21 per cent for the Unstable measure.

Many issues surround infinite horizon forecasts, and the whole enterprise can certainly be questioned. Nonetheless, we have found it useful simply to extend the range of the stochastic forecasting models to very distant horizons. We call these 'routine' or 'business as usual' stochastic forecasts, because their uncertainty does not reflect the possibility of structural shifts, so they understate full uncertainty. Good estimates of the uncertainty of the simple measures described above cannot be derived from stochastic forecasts over the 75-year horizon, at least using the methods we have attempted, so these measures are useful only for central tendency. The 'routine' uncertainty surrounding the infinite horizon estimates of the Summary Actuarial Balance gives a 95 per cent probability interval ranging from -1.3 per cent to -10.5 per cent of the present value of payroll. This 9.2 per cent range can be compared to the 6.5 per cent range over the 75-year horizon. It is bigger, to be sure, but not nearly as much wider as one would have expected. One reason is that uncertainty about the scale of the economy is removed by this ratio measure.

These measures are interpreted as the hypothetical immediate and permanent tax increase needed to achieve a reasonable probability of balancing the system over the very long run. But it is important to recall that they are measures, and not policy prescriptions. In practice, there is no need to impose a tax increase now that will be fixed for the indefinite future. Our simulations show that doing so would risk accumulating enormous surpluses or deficits, even if the median outcome were desirable. Policies can instead adjust taxes today to aim for a sustainable trajectory, and then readjust taxes periodically to reflect changing circumstances. For example, a public pension system could impose today a tax consistent with the Flat Fund Ratio criterion for a 75-year horizon. Then the tax level could be recalibrated every ten years based on the same criterion applied to the new 75-year horizon. We have not done a stochastic simulation of the performance of such policies, but it seems a natural next step in this research.

REFERENCES

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NOTES

(1) The Summary Actuarial Balance is the present value of future fund balances and imbalances over a given horizon (e.g. 75 years), expressed as a percentage of the present taxable payroll. If this amount was paid into the Social Security trust fund today, the fund would remain solvent for 75 years, assuming accurate forecasts.

(2) The Social Security Act originally required the appointment of an Advisory Council every four years to review the status of the Social Security trust fund and its long-term commitments. (This requirement has since been repealed, and the Advisory Councils have been replaced by the Social Security Advisory Board.) The last Advisory Council was appointed in late 1994 and released its final report in 1997.

(3) Old-Age, Survivors and Disability Insurance (OASDI) is the combined retirement, survivors and disability insurance programme; we refer to it as 'Social Security' for simplicity.

(4) So-called 'infinite horizon forecasts' actually stabilise at a horizon of several centuries, as discussed in a later section of this paper, 'How Long Is Forever?'. In practice, infinite horizon forecasts are extended for up to 500 years or so. Discounting accords geometrically declining weights to more distant periods.

(5) Actually, the growth rate of the covered real wage is needed for the Social Security forecasts, and this differs from the productivity growth rate due to changing hours supplied per worker, changes in the share of compensation taken as fringe benefits, and the difference between the GDP deflator and the Consumer Price Index.

(6) When survey data are used, both reported taxes and benefits must be proportionately adjusted to match administrative control totals, since both are significantly underreported.

(7) That is, in S years the scale of the system will be greater by a factor of e.014S, which is discounted by [e.sup.-0.035].

(8) We considered the age pattern of mortality when life expectancy at birth is 35. In this case, death rates in infancy and childhood are very high. When these death rates are proportionately reduced, many years of life are saved, and there is a big impact on life expectancy at birth. When life expectancy is at 77 as in the US today, variations in the level of mortality have their biggest effects at old ages, where mortality is still high. Saving the life of a 75 year old does not add much to life expectancy at birth.

Ronald Lee, Demography and Economics, University of California, Berkeley. e-mail: rlee@demog.berkeley.edu. Michael Anderson, Centre for the Economics and Demography of Aging, University of California at Berkeley, 2232 Piedmont Avenue, Berkeley, CA 94720-2120. e-mail: mikeandl@comcast.net. This research was supported by the US Social Security Administration through grant #10-P-98363-1 to the National Bureau of Economic Research as part of the SSA Retirement Research Consortium. The opinions and conclusions expressed are solely those of the authors and do not represent the opinions or policy of SSA, any agency of the Federal Government, or the NBER.

Chart 6. Probability distribution of the 500-year open
group actuarial deficit in trillions of 2003 dollars

2.5 percentile: -316
16.7 percentile: -60
Median: -18
83.3 percentile: -5.7
97.5 percentile: -1.8

Note: Left-most bin is open-ended.