Significant uncertainty surrounds the estimates under the intermediate assumptions, especially for a period as long as 75 years. This appendix presents a way to illustrate the uncertainty of these estimates. The stochastic projections supplement the traditional methods of examining such uncertainty.1. BackgroundThe Trustees have traditionally shown estimates using the low-cost and high-cost sets of specified assumptions to illustrate the presence of uncertainty. These alternative estimates provide a range of possible outcomes for the projections. However, they do not provide an indication of the probability that actual future experience will be inside or outside this range. This appendix presents the results of a model, based on stochastic modeling techniques, that estimates a probability distribution of future outcomes of the financial status of the combined OASI and DI Trust Funds. This model, which was first included in the 2003 report, is subject to further development.Other sections of this report provide estimates of the financial status of the combined OASI and DI Trust Funds using a scenario-based model. For the scenario-based model, the Trustees use three alternative scenarios (low-cost, intermediate, and high-cost) that make assumptions about levels of fertility, changes in mortality, legal and other immigration levels, legal and other emigration levels, changes in the Consumer Price Index, changes in average real wages, unemployment rates, trust fund real yield rates, and disability incidence and recovery rates. In general, the Trustees assume that each of these variables will reach an ultimate value at a specific point during the long-range period, and will maintain that value throughout the remainder of the period. The three alternative scenarios assume separate, specified values for each of these variables. Chapter V contains more details about each of these assumptions.This appendix presents estimates of the probability that key measures of OASDI solvency will fall in certain ranges, based on 5,000 independent stochastic simulations. Each simulation allows the above variables to vary throughout the long-range period. The fluctuation of each variable over time is simulated using historical data and standard time-series techniques. Generally, each variable is modeled using an equation that: (a) captures a relationship between current and prior years’ values of the variable; and (b) introduces year-by-year random variation as observed in the historical period. For some variables, the equations also reflect relationships with other variables. The equations contain parameters that are estimated using historical data for periods of at least 5 years and at most 110 years, depending on the nature and quality of the available data. Each time-series equation is designed so that, in the absence of random variation over time, the value of the variable for each year equals its value under the intermediate assumptions.1For each simulation, the stochastic method develops year-by-year random variation for each variable using Monte Carlo techniques. Each simulation produces an estimate of the financial status of the combined OASI and DI Trust Funds. This appendix shows the distribution of results from 5,000 simulations of the model.Readers should interpret the results from this model with caution and with an understanding of the model’s limitations. Results are very sensitive to equation specifications, degrees of interdependence among variables, and the historical periods used for the estimates. For some variables, recent historical variation may not provide a realistic representation of the potential variation for the future. Also, results would differ if additional variables (such as labor force participation rates, retirement rates, marriage rates, and divorce rates) were also allowed to vary randomly. Furthermore, more variability would result if statistical approaches were used to model uncertainty in the central tendencies of the variables. Time-series modeling reflects only what occurred in the historical period. Future uncertainty exists not only for the underlying central tendency but also for the frequency and size of occasional longer-term shifts in the central tendency. The future will bring with it the likelihood of substantial shifts, as predicted by many experts and as seen in prior centuries, that are not fully reflected in the current model. As a result, readers should understand that the true range of uncertainty is larger than indicated in this appendix.Figure VI.E1 displays the probability distribution of the year-by-year OASDI cost rates (that is, cost as a percentage of taxable payroll). The range of the cost rates widens as the projections move further into the future, which reflects increasing uncertainty. Because there is relatively little variation in income rates across the 5,000 stochastic simulations, the figure includes the income rate only under the intermediate assumptions. The two extreme lines in this figure illustrate the range within which future annual cost rates are projected by the current model to occur 95 percent of the time (i.e., a 95-percent confidence interval). In other words, the current model indicates that there is a 2.5 percent probability that the cost rate in a given year will exceed the upper bound and a 2.5 percent probability that it will fall below the lower bound. Other lines in the figure delineate additional confidence intervals (80‑percent, 60‑percent, 40‑percent, and 20‑percent) around future annual cost rates. The median (50th percentile) cost rate for each year is the rate for which half of the simulated outcomes are higher and half are lower for that year. These lines do not represent the results of individual stochastic simulations. Instead, for each given year, they represent the percentile distribution of cost rates based on all stochastic simulations for that year.
Figure VI.E1.—Long-Range OASDI Cost Rates From Stochastic Modeling Figure VI.E2 presents the simulated probability distribution of the annual trust fund ratios for the combined OASI and DI Trust Funds. The lines in this figure display the median set (50th percentile) of estimated annual trust fund ratios and delineate the 95‑percent, 80‑percent, 60‑percent, 40‑percent, and 20‑percent confidence intervals expected for future annual trust fund ratios. These lines are not the results of individual stochastic simulations. For each given year, they represent the percentile distribution of trust fund ratios based on all stochastic simulations for that year.Figure VI.E2 shows that the 95‑percent confidence interval for the trust fund depletion year ranges from 2028 to 2041, and there is a 50‑percent probability of trust fund depletion by the end of 2033 (the median depletion year). The median depletion year is the same as the Trustees project under the intermediate assumptions. The figure also shows confidence intervals for the trust fund ratio in each year. For example, the 95‑percent confidence interval for the trust fund ratio in 2025 ranges from 242 to 82 percent of annual cost.
This section compares results from two different approaches for determining ranges of uncertainty for trust fund actuarial status. One approach uses results from the low-cost, intermediate, and high-cost alternative scenarios. The other approach uses stochastic distributions of results. Each of these approaches provides insights into uncertainty. Comparison of the results requires an understanding of the differences in the approaches. Two fundamental differences exist between the approach using alternative scenarios and the stochastic approach.The first fundamental difference relates to the presentation of results. Figure VI.E3 shows projected OASDI annual cost rates for the low-cost, intermediate, and high-cost alternatives along with the annual cost rates at the 97.5th percentile, 50th percentile, and 2.5th percentile for the stochastic simulations. While all values on each line for the alternatives are results from a single specified scenario, the values on each stochastic line may be results from different simulations for different years. The one stochastic simulation (from the 5,000 simulations) that yields results closest to a particular percentile in one year may yield results that are distant from that percentile in another year. Thus, the stochastic presentation illustrates distributions of the range of potential results one year at a time, with no direct relationship of the results among years.Even with this fundamental difference in the presentation of results, figure VI.E3 shows similar results between the range of OASDI cost rates resulting from the alternatives and from the 95-percent confidence interval of stochastic results for years before 2030. After 2030, results for the alternatives show a wider range. The cost rates for the high-cost alternative are somewhat higher than the stochastic year-by-year results at the 97.5th percentile. The intermediate alternative results generally show somewhat lower cost rates than the stochastic year-by-year results at the 50th percentile. The cost rates are lower for the low-cost alternative than for the stochastic year-by-year results at the 2.5th percentile for years after 2030.Both the alternatives and the stochastic results suggest that the range of potential cost rates above the central levels (those for the intermediate alternative and for the median, respectively) is larger than the potential range below these central results. The difference between the central results and the higher cost levels (the high-cost alternative and the upper end of the 95-percent confidence range, respectively) is about 1.5 times as large as the difference between the central and lower cost levels for both models by the end of the projection period.
Figure VI.E3.—OASDI Cost Rates: Comparison of Stochastic to Low-Cost, Intermediate, and High-Cost Alternatives The second fundamental difference between the alternatives and the stochastic simulations is the method of assigning values for assumptions in the simulations. For the alternatives, the Trustees assign specific values for key demographic and economic variables. In comparison to the intermediate alternative, every value assigned to the high-cost alternative tends to raise estimated program cost and every value assigned to the low-cost alternative tends to reduce it throughout the projection period. In contrast, the stochastic method randomly assigns values for the key demographic and economic variables for each year in each of the 5,000 independent stochastic simulations. For each of the stochastic simulations, randomly assigned values for the various assumptions may have varying effects on projected cost, with some tending toward higher cost and some tending toward lower cost.Figure VI.E4 compares the ranges of trust fund (unfunded obligation) ratios for the alternative scenarios and the 95-percent confidence interval of the stochastic simulations. This figure extends figure VI.E2 to show unfunded obligation ratios, expressed as negative values below the zero percent line. Unfunded obligation ratios are the ratio of the unfunded obligation through the beginning of the year to the present value of annual cost for that year. Figure VI.E4 presents a more complete picture of the difference between the results from the three alternative scenarios and the stochastic simulations.
Figure VI.E4.—OASDI Trust Fund (Unfunded Obligation) Ratios: Comparison of Stochastic to Low-Cost, Intermediate, and High-Cost Alternatives1
An unfunded obligation, shown as a negative value in this figure, is equivalent to the amount the trust funds would need to have borrowed to date in order to pay all scheduled benefits (on a timely basis) after trust fund asset reserves are depleted. Note that current law does not permit the trust funds to borrow.
The range of stochastic results for trust fund (unfunded obligation) ratios in Figure VI.E4 appears to be consistent with the range seen for the stochastic cost rates. That is, the difference between the central result (the median) and the high-cost result (the lower level in the figure) is about 1.5 times as large as the distance between the central and low-cost result by the end of the projection period. However, the range for the alternatives is substantially different. For the alternatives, the distance between the central and high-cost projected trust fund (unfunded obligation) ratios is smaller than the difference between central and low-cost ratios, the opposite of the results for cost rates.The difference in the ranges of trust fund (unfunded obligation) ratios between the alternatives and the stochastic results may be a little surprising given the similarity of the ranges for cost rates. Several factors contribute to this finding, including the fact that cost rates are annual measures, while trust fund (unfunded obligation) ratios are cumulative measures of all financial activity up to that date. However, a clear difference between the ratios for the alternatives and the ratios for the stochastic model is the assignment of interest rates.For the stochastic model, real interest rates are assigned essentially randomly, and as a result, the range for trust fund (unfunded obligation) ratios is consistent with the range for cost rates. But for the alternatives, real interest rates are specified to be higher for the low-cost alternative and lower for the high-cost alternative. This assignment has the effect of shifting the trust fund (unfunded obligation) ratios up (higher or less negative) for both the high-cost and low-cost alternatives. High interest rates boost the level of the positive trust fund ratio in alternative 1, and low interest rates reduce the magnitude of the negative ratio (unfunded obligation) for alternative 3. This assignment of real interest rates contributes substantially to the upward shift in the range of the ratios for the alternatives.It is important to understand that the stochastic model’s 95‑percent confidence intervals for any summary measure of trust fund finances would tend to be narrower than the range produced for the low-cost and high-cost alternatives, even if the stochastic model’s 95-percent confidence interval for annual cost rates were identical to the range defined by the low-cost and high-cost projections. This is true because summary measures of trust fund finances depend on cost rates for many years, and the probability that annual cost rates, on average for individual stochastic simulations, will be at least as low (high) as the 2.5 (97.5) percentile line is significantly lower than 2.5 percent. As a result, the relationship between the ranges presented for annual cost rates and summary measures of trust fund finances is fundamentally different for the stochastic model than it is for the low-cost and high-cost alternatives.This contrast in results and methods does not mean that either approach to illustrating ranges of uncertainty, alternative scenarios or stochastic simulations, is superior to the other. The ranges are different and explainable.Table VI.E1 displays long-range actuarial estimates for the combined OASDI program using the two methods of illustrating uncertainty: (1) alternative scenarios and (2) stochastic simulations. The table shows stochastic estimates for the median (50th percentile) and for the 95‑percent and 80‑percent confidence intervals. For comparison, the table shows scenario-based estimates for the intermediate, low-cost, and high-cost assumptions. Each individual stochastic estimate in the table is the level at that percentile from the distribution of the 5,000 simulations. For each given percentile, the values in the table for each long-range actuarial measure are generally from different stochastic simulations.The median stochastic estimates displayed in table VI.E1 are, in general, slightly more pessimistic than the intermediate scenario-based estimates. The median estimate of the long-range actuarial balance is ‑2.99 percent of taxable payroll, about 0.11 percentage point lower than projected under the intermediate assumptions. The median first projected year that cost exceeds non-interest income (as it did in 2010 through 2013), and remains in excess of non-interest income throughout the remainder of the long-range period, is 2014. This is the same year as projected under the intermediate assumptions. The median year that asset reserves first become depleted is 2033, also the same as projected under the intermediate assumptions. The median estimates of the annual cost rate for the 75th year of the projection period are 18.62 percent of taxable payroll and 6.26 percent of gross domestic product (GDP). The comparable estimates under the intermediate assumptions are 18.19 percent of payroll and 6.12 percent of GDP.For four measures in table VI.E1 (the actuarial balance, the first year cost exceeds non-interest income and remains in excess through 2088, the first projected year asset reserves become depleted, and annual cost in the 75th year as a percent of taxable payroll), the 95‑percent stochastic confidence interval is narrower than the range defined by the low-cost and high-cost alternatives. In other words, for these measures, the range defined by the low-cost and high-cost alternatives contains the 95‑percent confidence interval of the stochastic modeling projections. For the remaining two measures (the open group unfunded obligation, and the annual cost in the 75th year as a percent of GDP), one or both of the bounds of the 95‑percent stochastic confidence interval fall outside the range defined by the low-cost and high-cost alternatives.
First projected year cost exceeds non-interest income and remains in excess through 2088b First year asset reserves become depletedd
The annual balance is projected to be negative for a temporary period, returning to positive levels before the end of the projection period.
For some stochastic simulations, the first year in which trust fund reserves become depleted does not indicate a permanent depletion of reserves.
More detail on this model, and stochastic modeling in general, is available at