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.
1Figure 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.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.
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.
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.
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.
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.
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.
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.
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.