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.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 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. As mentioned above, three 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. Each variable fluctuates using standard time-series modeling, a method designed to make inferences based on historical data. 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 between 25 years and 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.^{1}For each simulation, the stochastic method develops year-by-year random variation in most of the variables using Monte Carlo techniques. The one exception is that the model varies net other immigration directly rather than as the difference of its components (other immigration minus other emigration). 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 could result if statistical approaches were used to model shifts in the central tendencies of the variables. The historical period utilized for most variables does not reflect many substantial shifts, and time-series modeling reflects only what occurred in the historical period. As a result, readers should understand that the true range of uncertainty is likely to be larger than indicated in this appendix. Substantial shifts, as predicted by many experts and as seen in prior centuries, are not fully reflected in the current model.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. The figure includes the income rate under the intermediate assumptions to indicate the patterns of cash flow for the OASDI program. The figure includes only this income rate, because there is relatively little variation in income rates throughout the projection period. The two extreme lines in this figure illustrate the range within which future annual cost rates are projected to occur 95 percent of the time (i.e., a 95-percent confidence interval). In other words, the 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 display additional confidence intervals (80‑percent, 60‑percent, 40‑percent, and 20‑percent) around future annual cost rates. The median cost rate for each year is the rate that falls exactly in the middle of possible outcomes 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 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.The median estimate for each year indicates that the assets of the combined OASI and DI Trust Funds become exhausted by the end of 2033 with a probability of 50 percent. This exhaustion date is the same as the year of exhaustion the Trustees project under the intermediate assumptions. Figure VI.E2 shows that the 95‑percent confidence interval for the trust fund ratio in 2025 ranges from 249 to 94 percent of annual cost.

The difference in the ranges of the projected trust fund ratios between two of the methods for illustrating uncertainty (alternative scenarios and stochastic simulations) is substantially due to the different assignment of real interest rates in these two methods. The next section includes an explanation of the different treatments.4. Comparison of Results: Stochastic to Low-Cost, Intermediate, and High-Cost AlternativesThis 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 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 1 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 1 year at a time, with no direct relationship of the results among years.

Figure VI.E3.—OASDI Cost Rates: Comparison of Stochastic to Low-Cost, Intermediate, and High-Cost Alternatives 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. The cost rates for the high-cost alternative are similar to the stochastic year-by-year results at the 97.5th percentile. The intermediate alternative results show slightly higher cost rates than the stochastic year-by-year results at the 50th percentile. The largest differences are between the low-cost alternative and the stochastic year-by-year results at the 2.5th percentile. For this comparison, cost rates are higher for the low-cost alternative than for the stochastic year-by-year results at the 2.5th percentile for years before 2020 and after 2040.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, each value assigned to the high-cost alternative tends to raise estimated program cost and each value assigned to the low-cost alternative tends to reduce it. In contrast, the stochastic method randomly assigns values for the key demographic and economic variables in each of the 5,000 independent stochastic simulations. For each of the stochastic simulations, 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. Nonetheless, figure VI.E3 shows that the ranges in cost rates for the alternatives and the 95-percent confidence interval of stochastic simulations are similar. The principal difference is that the low-cost and intermediate scenarios generate cost rates after 2040 that are somewhat higher than the 2.5th-percentile and median stochastic results, respectively. Accordingly, the alternatives produce a narrower, less optimistic range of cost rates than do the stochastic simulations.In contrast, the alternatives produce a wider, more optimistic range of trust fund (unfunded obligation) ratios than do the stochastic simulations. 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 IV.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 at 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.As with cost rates, the trust fund (unfunded obligation) ratios differ most notably in the comparison of the results from the low-cost alternative to the 97.5th-percentile results from the stochastic simulations. However, the direction of the difference reverses. While cost rates are considerably less optimistic for the low-cost alternative than for the 2.5th-percentile results of the stochastic simulations, the trust fund (unfunded obligation) ratios for the low-cost scenario are more optimistic than the 97.5th-percentile results of the stochastic simulations. A similar relationship exists for the high-cost results, where the alternative scenario and the stochastic results have similar cost rates but the alternative scenario has higher (more favorable) trust fund (unfunded obligation) ratios toward the end of the period.This reversal is explainable. Projections of trust fund (unfunded obligation) ratios shown in figure VI.E4 require an additional variable not reflected in the cost rates shown in figure VI.E3. This additional variable is the real interest rate. For the alternatives, the Trustees assign higher real interest rates for the low-cost alternative and lower real interest rates for the high-cost alternative. Under the limitations imposed by the law, where the trust funds cannot borrow, a lower real interest rate is relatively pessimistic and thus consistent with the high-cost alternative. However, in order to show the size of the cumulative shortfall of non-interest income relative to scheduled cost, or the unfunded obligation, that would not be payable under current law, the Trustees project the cost of scheduled benefits, even after the point at which trust fund reserves become exhausted. In the case of the high-cost alternative, the relatively low assumed interest rates have the effect of making this unfunded obligation smaller than it otherwise would be. For the low-cost alternative, the relatively high assumed real interest rates help maintain trust fund reserves and account for the fact that the trust fund reserves remain positive throughout the 75-year projection period. This assignment of real interest rates elevates the level of the trust fund (unfunded obligation) ratio for both low-cost and high-cost alternatives compared to the expected result without variation in real interest rates across alternatives.The stochastic model, however, assigns real interest randomly, yielding rates with no correlation to the overall “optimism” or “pessimism” of the other variable assignments. The tendency for elevated trust fund (unfunded obligation) ratios resulting from the assignment of real interest rates in both the high-cost and low-cost alternatives is not present in the stochastic results. The relationship between cost rates for the alternatives and cost rates for the stochastic simulations, as shown in figure IV.E3, is therefore different from the relationship between the trust fund (unfunded obligation) ratios for the alternatives and the stochastic simulations as shown in figure IV.E4. Figure IV.E4 shows trust fund (unfunded obligation) ratios that tend to be higher (more optimistic) for the extreme alternatives than for the extreme stochastic results, which is contrary to the elevated cost rates (more pessimistic) for the extreme alternatives. This contrary effect is more evident for the low-cost alternative, which has substantially higher cost rates (more pessimistic) than the stochastic 2.5th percentile for most years, but has substantially higher trust fund reserves (more optimistic) throughout the projection period.

Figure VI.E4.—OASDI Trust Fund (Unfunded Obligation) Ratios: Comparison of Stochastic to Low-Cost, Intermediate, and High-Cost Alternatives^{}

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 assets are exhausted. Note that current law does not permit the trust funds to borrow.

This contrast in results and methods does not mean that either approach to illustrating ranges of uncertainty 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 optimistic than the intermediate-alternative scenario-based estimates. The median estimate of the long-range actuarial balance is ‑2.50 percent of taxable payroll, about 0.17 percentage point higher than projected under the intermediate assumptions. The median year that cost first exceeds non-interest income (and remains in excess of non-interest income throughout the remainder of the long-range period) is 2012, the same year as projected under the intermediate assumptions. The median year that assets first become exhausted 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 17.43 percent of taxable payroll and 5.77 percent of gross domestic product (GDP). The comparable estimates under the intermediate assumptions are 17.83 percent of payroll and 6.10 percent of GDP.A comparison of the 95‑percent confidence interval to the range of variation defined by the traditional low-cost and high-cost alternatives follows. For three measures in table VI.E1 (the actuarial balance, the open group unfunded obligation, and the first year assets become exhausted), 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 one measure (the first year cost exceeds non-interest income and remains in excess through 2086), the low-cost and high-cost estimates are consistent with the bounds of the 95-percent stochastic confidence interval. For the remaining two measures (the annual costs in the 75th year), 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.

Traditional

scenario-based model Intermediate Low-

cost High-

cost 80-percent

confidence interval 95-percent

confidence interval 10th

percentile 90th

percentile 2.5th

percentile 97.5th

percentile Actuarial balance Open group unfunded obligation (in trillions) First projected year cost exceeds non-interest income and remains in excess through 2086^{a} First year assets become exhausted^{d} Annual cost in 75th year (percent of taxable

payroll) Annual cost in 75th year (percent of GDP)

The annual balance is negative for a temporary period and returns to positive levels before the end of the projection period.

For some stochastic simulations, the first year in which trust fund assets become exhausted does not indicate a permanent exhaustion of assets.

More detail on this model, and stochastic modeling in general, is available at

www.socialsecurity.gov/OACT/stochastic/index.html.

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