A Stochastic Model
of the Long-Range Financial Status
of the OASDI Program—September 2004
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This chapter presents the OSM simulation results in three sections. The results presented in this chapter are based on sets of 5,000 simulations. The first section shows the results of the assumptions which include projected probability distributions for the equations presented in chapter II. In order to better illustrate how uncertainty is quantified, confidence interval bounds from the OSM results are often compared with the values assumed in the low cost (TR04I) and high cost (TR04III) alternatives from the 2004 Trustees Report. The second section contains selected actuarial estimates including annual trust fund ratios and balances, as well as summary actuarial measures (e.g., actuarial balances and summarized cost rates). The third section illustrates the sensitivity of the OSM to variations in the assumptions.
The tables in the following subsections display results of the OSM for the equations presented in chapter II. The three rows of values shown in the tables are the level in the 75th projection year (2078), the average level over the entire 75-year projection period (2004-78), and the average over the final 50 years of the projection period (2029-78).1 The corresponding columns for the three rows are the value from the TR04II, median value, values from the TR04I and TR04III, and bounds of the 95-, 90-, and 80-percent confidence intervals. For the mortality subsection, the increase in period life expectancy is shown rather than the 50-year and 75-year averages.
Unless otherwise noted, the average value is computed as an arithmetic average. When the geometric mean is specified, the values are transformed by adding one, averaged geometrically, and reverse transformed by subtracting one. This method is analogous to computing an average effective annual rate for compound interest (Kellison 1991).
The figures in the following subsections graph the values of the assumption variables throughout their entire historical and projection periods. For most variables, the expected future annual values from the equations presented in chapter II are shown along with the 95-percent confidence intervals for the annual values and the cumulative averages of each simulation. For the mortality subsection, the 95-percent confidence interval is shown for the annual values only. For the immigration subsection, values shown for legal emigration and net other immigration during the historical period are estimated since actual data are not available.
Table IV.1 lists the results of the total fertility rate equation presented in chapter II. For the final 50-year average, the limits of the 80-percent confidence interval closely match the TR04I and the TR04III values (2.24 vs. 2.20 and 1.65 vs. 1.70, respectively).
Figure IV.1 displays the total fertility rate throughout the entire historical and projection periods. The annual values indicate that the standard deviation of the projected total fertility rate is about 0.5 child per woman.2
In order to simplify the presentation, mortality results are given here as period life expectancies at birth and age 65 for males and females, separately. Thus, the number of age-sex groups is reduced from 42 to just four.
Tables IV.2 and IV.3 list the resulting period life expectancies at birth for males and females, respectively, from the mortality equations presented in chapter II. The increases shown are calculated as the difference between the values in 2078 and 2004 or the difference between the values in 2078 and 2029. The upper limit of the 90-percent confidence interval in 2078 is nearly equal to the TR04III value (85.8 vs. 85.9 for males, and 89.5 vs. 89.2 for females) while the lower limit is only slightly less than the TR04I value (77.5 vs. 78.0 for males, and 81.6 vs. 82.1 for females).
Figures IV.2 and IV.3 display the period life expectancies at birth for males and females, respectively, throughout the entire historical and projection periods.
Tables IV.4 and IV.5 list the resulting period life expectancies at age 65 for males and females, respectively, from the mortality equations presented in chapter II. The values shown are analogous to those shown in the previous subsection. The upper limit of the 80-percent confidence interval in 2078 is close to that of the TR04III value (23.2 vs. 23.6 for males, and 26.3 vs. 26.1 for females) while the lower limit is slightly larger than the TR04I value (18.4 vs. 17.7 for males, and 20.7 vs. 20.3 for females).
Figures IV.4 and IV.5 display the period life expectancies at age 65 for males and females, respectively, throughout the entire historical and projection periods.
Table IV.6 displays the results (in thousands) of the legal immigration equation presented in chapter II. For the final 50-year average, the bounds of the 80-percent confidence interval are close to the TR04I and the TR04III values (1,051 vs. 1,063 and 548 vs. 675, respectively).
Figure IV.6 shows a graph of legal immigration throughout the entire historical and projection periods. The annual values indicate that the standard deviation of the projected legal immigration level is about 300,000.3
Table IV.7 displays the results (in thousands) of the legal emigration equation presented in chapter II. The confidence intervals shown are not directly comparable with the values assumed for the TR04I and TR04III. This is because the Trustees' values are established by applying an assumed percentage to the level of legal immigration (20 percent for the TR04I and 30 percent for the TR04III). As a result, the level of emigration under the TR04I and the TR04III are both greater than that of the TR04II.
Figure IV.7 graphs the level of legal emigration throughout the entire historical and projection periods. The annual values indicate that the standard deviation of the projected legal emigration level is about 25,000.4
Table IV.8 displays the results (in thousands) of the net other immigration equation presented in chapter II. For the final 50-year average, the bounds of the 80-percent confidence interval are considerably wider than the interval defined by the TR04I and the TR04III values (505 vs. 450 and 86 vs. 200, respectively).
Figure IV.8 graphs the level of net other immigration throughout the entire historical and projection periods. Discontinuities are shown in the graph over the historical and the projected periods. The discontinuities shown over the historical years reflect that the estimated levels of net other immigration are determined as average annual levels for 10-year periods between census years. The discontinuities shown over the projected years are caused by the Trustees' assumption of a 50,000 decrease in the level of net other immigration in years 2014 and 2024. Ignoring the discontinuities, the graph of the projected values is quite typical of that of a random walk in that it produces a parabola. This is expected since the standard error of the forecast increases with the square root of the time variable.
Table IV.9 shows the results of the unemployment rate equation presented in chapter II. The values presented in the table are expressed in percents. The lower and upper bounds of the 95-percent confidence interval for the 75-year cumulative average (4.74 and 6.68 percent, respectively) are close to the values assumed under the TR04I and the TR04III (4.61 and 6.40 percent, respectively).
Figure IV.9 shows a graph of the unemployment rate over the historical and projection periods. Due to the log-odds transformation of the unemployment rate variable, the confidence intervals are not symmetrical around the median value.
Table IV.10 displays the results of the CPI inflation rate equation presented in chapter II. The values presented in the table are growth rates expressed in percents, and the averages shown are annual compound averages. The lower bound of the 95-percent confidence interval for the 75-year cumulative average, at 1.75 percent, closely matches the 1.77 percent assumed under the TR04I, while the upper bound of 4.64 percent is much higher than the 3.80 percent assumed under the TR04III.
Figure IV.10 shows a graph of the inflation rate over the historical and projection periods. The asymmetry in the confidence intervals around the median values is due to the logarithmic transformation that was applied to the inflation rate variable.
Table IV.11 shows the results of the real interest rate equation presented in chapter II. The values presented in the table are expressed in percents, and the averages shown are annual compound averages. For the 75-year cumulative average, the limits of the 90-percent confidence interval (3.73 percent and 2.26 percent) are close to the values assumed under the TR04I and the TR04III (3.69 percent and 2.24 percent, respectively).
Figure IV.11 shows a graph of the real interest rate on new special issue securities over the historical and projection periods.
Table IV.12 displays the results of the real average covered wage equation presented in chapter II. The values presented in the table are growth rates expressed in percents, and the averages shown are annual compound averages. For the 75-year cumulative average, the limits of the 95-percent confidence interval (1.69 percent and 0.59 percent) closely match the values assumed under the TR04I and the TR04III (1.63 and 0.62 percent, respectively).
Figure IV.12 shows a graph of the real average covered wage for the historical and projection periods.
Tables IV.13 and IV.14 show the results of the age-adjusted disability incidence rate equations for males and females, respectively. The disability incidence rates shown are per thousand. For the 75-year cumulative average, values for males under the TR04I and the TR04III (4.95 and 7.27 per thousand, respectively) fall outside the limits of the 95-percent confidence interval (5.58 and 6.56 per thousand, respectively). Comparable values for females, under the TR04I and the TR04III (4.24 and 6.22 per thousand, respectively) also fall outside the limits of the 95-percent confidence interval (4.71 and 5.69 per thousand, respectively).
Figures IV.13 and IV.14 show graphs of male and female age-adjusted disability incidence rates, respectively, over the historical and projection periods.
Tables IV.15 and IV.16 show the results of the disability recovery rate equations for males and females, respectively. The disability recovery rates presented are per thousand. For the 75-year cumulative average, values for males under the TR04I and the TR04III (13.64 and 8.42 per thousand, respectively) fall outside the limits of the 95-percent confidence interval (12.37 and 10.59 per thousand). Comparable values for females assumed under the TR04I and the TR04III (12.33 and 7.74 per thousand, respectively) also fall outside the limits of the 95-percent confidence interval (11.21 and 9.54 per thousand).
Figures IV.15 and IV.16 show graphs of the age-adjusted disability recovery rates for males and females, respectively, over the historical and projection periods.
1A final 50-year average is presented because most of the variables in this section reach an assumed ultimate value prior to the end of the 25th projection year. Typically, this assumed ultimate value is a constant for the variable of interest. A notable exception is the mortality assumption in which death rates do not reach a constant level since they are derived from a multiple decrement model (where causes of death compete). For mortality, increases in life expectancies over the 75-year and final 50 years of the projection period, rather than average levels, are presented.
2The width of the 95-percent confidence interval for the annual values is approximately two children per woman. For a normal distribution, this width represents about four standard deviations.
3The width of the 95-percent confidence interval for the annual values is approximately 1,200,000. For a normal distribution, this width represents about four standard deviations.
4The width of the 95-percent confidence interval for the annual values is approximately 100,000. For a normal distribution, this width represents about four standard deviations.
