A STOCHASTIC MODEL

OF THE LONG-RANGE FINANCIAL STATUS 

OF THE OASDI PROGRAM

ACTUARIAL STUDY NO. 128

K. Mark Bye, ASA

Michael L. Miller

Michael Morris, ASA

Sven H. Sinclair, FSA, MAAA

Social Security Administration

Office of the Chief Actuary

March 2024

SSA Pub. No. 11-11555
.
FOREWORD

The purpose of the OCACT Stochastic Model (OSM) is to provide probability distributions for 

the range of possible future experience of the OASDI program under current law. This probabilistic representation of uncertainty augments the presentations of low-cost and high-cost alternative scenarios and sensitivity analyses that have traditionally been included in the Annual Reports of the OASDI Board of Trustees (Trustees Reports). 

The OSM was introduced in the 2003 Trustees Report. Actuarial Study No. 117, published in 

September 2004, provided details about the model’s first stage of development and the projections produced using it. The OSM was enhanced in the 2021 Trustees Report to include parameter uncertainty. This study (Actuarial Study No. 128) provides details about the version of the 

OSM used for the 2023 Trustees Report and the projections produced using it. We expect to continue making improvements and refinements to the OSM in the future. 

Many of our colleagues, both inside and outside the Office of the Chief Actuary, provided input 

to the development of this study. We would like to acknowledge Anthony Cheng, David Pattison, Jason Schultz, and Robert Weathers for their technical expertise and support in developing 

the enhanced parameter uncertainty methodology. Kyle Burkhalter, Sharon Chu, Skylar Lowe, 

Johanna Maleh, Kent Morgan, Daniel Nickerson, Karen Rose, and Katie Sutton contributed to 

the ongoing development of the model and the writing and editing of this study. Erica Ciccotto 

assisted in the publication of this study. 

Karen P. Glenn, FSA, EA, MAAA 

Deputy Chief Actuary

Stephen C. Goss, ASA, MAAA

Chief Actuary
.
iii

TABLE OF CONTENTS

I. Introduction.............................................................................................................................1

II. Stochastic Model......................................................................................................................3

A. Basic Model .........................................................................................................................3

B. Incorporating Parameter Uncertainty Into the Basic Model................................................4

III. Equation Selection and Parameter Estimation.....................................................................7

A. Fertility ................................................................................................................................8

B. Mortality ............................................................................................................................10

C. Immigration .......................................................................................................................11

1. LPR New Arrival Immigration......................................................................................11

2. Adjustments of Status....................................................................................................12

3. Legal Emigration...........................................................................................................13

4. Other-than-LPR Immigration ........................................................................................14

D. Unemployment Rate, Inflation Rate, and Real Interest Rate.............................................16

1. Description of the Vector Autoregression Process........................................................16

2. Variables........................................................................................................................17

3. Equations and Fits .........................................................................................................18

E. Real Average Covered Wage ............................................................................................22

F. Disability Incidence Rate...................................................................................................24

G. Disability Recovery Rate...................................................................................................26

IV. Results....................................................................................................................................28

A. Equation Results................................................................................................................28

1. Fertility ..........................................................................................................................29

2. Mortality........................................................................................................................32

3. Immigration...................................................................................................................39

4. Unemployment Rate......................................................................................................50

5. Inflation Rate .................................................................................................................52

6. Real Interest Rate...........................................................................................................55

7. Real Average Covered Wage ........................................................................................57

8. Disability Incidence Rate...............................................................................................59

9. Disability Recovery Rate...............................................................................................63

B. Actuarial Estimates............................................................................................................67

1. Annual Measures...........................................................................................................67

2. Summary Measures.......................................................................................................70

C. Sensitivity Analysis...........................................................................................................73

1. Stochastic Analysis........................................................................................................73

2. Deterministic Analysis..................................................................................................74

V. Appendices.............................................................................................................................76

A. Documentation of Computer Program...............................................................................76
iv

1. Organization ..................................................................................................................76

2. Modules.........................................................................................................................77

B. Principal Measures of Financial Status..............................................................................92

1. Trust Fund Operations...................................................................................................92

2. Annual Income Rates, Cost Rates, and Balances..........................................................93

3. Comparison of Workers to Beneficiaries ......................................................................94

4. Trust Fund Ratios..........................................................................................................94

5. Actuarial Balance, Summarized Rates, Open Group Unfunded Obligation..................95

C. Time-Series Analysis.........................................................................................................97

D. Monte Carlo Simulation...................................................................................................102

E. Parameter Uncertainty Implementations..........................................................................103

F. Equation Regression Statistics, Parameter Estimates, and Bounding Statistics..............114

1. Regression Statistics....................................................................................................114

2. Assumed Annual Projected Values for Each Variable from the

2023 Trustees Report.................................................................................................. 130

3. Bounding Results.........................................................................................................141

G. Bibliography ....................................................................................................................143
v

LIST OF FIGURES

III. Equation Selection and Parameter Estimation

Figure III.1— Total Fertility Rate, Calendar Years 1917-2021...................................................9

Figure III.2— LPR New Arrival Immigration, Calendar Years 1991-2020..............................12

Figure III.3— Adjustments to LPR Status, Calendar Years 1978-2020....................................13

Figure III.4— Legal Emigration, Calendar Years 1941-2020 ...................................................14

Figure III.5— Other-than-LPR Immigration, Calendar Years 1999-2020 ................................15

Figure III.6— Unemployment Rate, Calendar Years 1961-2020 ..............................................19

Figure III.7— Inflation Rate (CPI-W), Calendar Years 1961-2020 ..........................................20

Figure III.8— Real Interest Rate, Calendar Years 1961-2020...................................................21

Figure III.9— Real Average Covered Wage (Percent Change),

Calendar Years 1968-2020 .................................................................................23

Figure III.10— Male Disability Incidence Rate, Calendar Years 1970-2022..............................25

Figure III.11— Female Disability Incidence Rate, Calendar Years 1970-2022 ..........................25

Figure III.12— Male Disability Recovery Rate, Calendar Years 1984 – 2022 ...........................27

Figure III.13— Female Disability Recovery Rate, Calendar Years 1984 – 2022........................27

IV. Results

Figure IV.1— Total Fertility Rate: Selected Values and Intervals Under the

Deterministic and Stochastic Models.................................................................30

Figure IV.2a— Total Fertility Rate Annual Values, Calendar Years 1917-2097........................31

Figure IV.2b— Total Fertility Rate Cumulative Average Values,

Calendar Years 1917-2097 .................................................................................31

Figure IV.3— Period Life Expectancies at Birth in 2097: Selected Values and 

Intervals Under the Deterministic and Stochastic Models..................................33

Figure IV.4— 75-year Increases in Period Life Expectancies at Birth in 2097: Selected 

Values and Intervals Under the Deterministic and Stochastic Models ..............34

Figure IV.5— Final 50-year Increases in Period Life Expectancies at Birth in 2097: Selected 

Values and Intervals Under the Deterministic and Stochastic Models..............34

Figure IV.6— Male Period Life Expectancies at Birth, Calendar Years 1900-2097.................35

Figure IV.7— Female Period Life Expectancies at Birth, Calendar Years 1900-2097 .............35

Figure IV.8— Period Life Expectancies at Age 65 in 2097: Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................37

Figure IV.9— 75-year Increases in Period Life Expectancies at Age 65 in 2097: Selected 

Values and Intervals Under the Deterministic and Stochastic Models ............. 37

Figure IV.10— Final 50-year Increases in Period Life Expectancies at Age 65 in 2097: Selected

Values and Intervals Under the Deterministic and Stochastic Models ..............38

Figure IV.11— Male Period Life Expectancies at Age 65, Calendar Years 1900-2097..............38
vi

Figure IV.12— Female Period Life Expectancies at Age 65, Calendar Years 1900-2097 ......... 39

Figure IV.13— LPR New Arrival Immigration: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................40

Figure IV.14a—LPR New Arrival Immigration Annual Values,

Calendar Years 1991-2097 .................................................................................41

Figure IV.14b—LPR New Arrival Immigration Cumulative Average Values,

Calendar Years 1991-2097 .................................................................................41

Figure IV.15— Adjustments of Status: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................43

Figure IV.16a—Adjustments of Status Annual Values, Calendar Years 1978-2097 ...................44

Figure IV.16b—Adjustments of Status Cumulative Average Values,

Calendar Years 1978-2097 .................................................................................44

Figure IV.17— Legal Emigration Rate: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................45

Figure IV.18a—Legal Emigration Annual Values, Calendar Years 1941-2097...........................46

Figure IV.18b—Legal Emigration Cumulative Average Values, Calendar Years 1941-2097.....47

Figure IV.19— Other-than-LPR Immigration: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................48

Figure IV.20a—Other-than-LPR Immigration Annual Values,

Calendar Years 1999-2097 .................................................................................49

Figure IV.20b—Other-than LPR Immigration Cumulative Average Values,

Calendar Years 1999-2097 .................................................................................49

Figure IV.21— Unemployment Rate: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................51

Figure IV.22a—Unemployment Rate Annual Values, Calendar Years 1961-2097......................52

Figure IV.22b—Unemployment Rate Cumulative Average Values,

Calendar Years 1961-2097 .................................................................................52

Figure IV.23— Inflation Rate: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................53

Figure IV.24a—Inflation Rate Annual Values, Calendar Years 1961-2097 ................................54

Figure IV.24b—Inflation Rate Cumulative Average Values, Calendar Years 1961-2097 ...........54

Figure IV.25— Real Interest Rate: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................55

Figure IV.26a—Real Interest Rate Annual Values, Calendar Years 1961-2097..........................56

Figure IV.26b—Real Interest Rate Cumulative Average Values,

Calendar Years 1961-2097 .................................................................................56

Figure IV.27— Real Average Covered Wage: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................57

Figure IV.28a—Real Average Covered Wage Annual Values, Calendar Years 1968-2097........58

Figure IV.28b—Real Average Covered Wage Cumulative Average Values,

Calendar Years 1968-2097 .................................................................................58
vii

LIST OF FIGURES

Figure IV.29— Male Disability Incidence Rate: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................60

Figure IV.30— Female Disability Incidence Rate: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................60

Figure IV.31a—Male Disability Incidence Annual Rate, Calendar Years 1970-2097 .................61

Figure IV.31b—Male Disability Incidence Cumulative Average Rate,

Calendar Years 1970-2097 .................................................................................62

Figure IV.32a—Female Disability Incidence Annual Rate, Calendar Years 1970-2097..............61

Figure IV.32b—Female Disability Incidence Cumulative Average Rate,

Calendar Years 1970-2097 .................................................................................62

Figure IV.33— Male Disability Recovery Rate: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................64

Figure IV.34— Female Disability Recovery Rate: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................64

Figure IV.35a—Male Disability Recovery Annual Rate, Calendar Years 1984-2097 .................65

Figure IV.35b—Male Disability Recovery Cumulative Average Rate,

Calendar Years 1984-2097 .................................................................................66

Figure IV.36a—Female Disability Recovery Annual Rate, Calendar Years 1984-2097..............65

Figure IV.36b—Female Disability Recovery Cumulative Average Rate,

Calendar Years 1984-2097 .................................................................................66

Figure IV.37a—Annual Trust Fund Ratios,

Calendar Years 2023-97, Without Parameter Uncertainty.................................68

Figure IV.37b—Annual Trust Fund Ratios,

Calendar Years 2023-97, With Parameter Uncertainty ......................................68

Figure IV.38a—Annual Cost and Income Rates,

Calendar Years 2023-97, Without Parameter Uncertainty .................................69

Figure IV.38b—Annual Cost and Income Rates,

Calendar Years 2023-97, With Parameter Uncertainty ......................................70

Figure IV.39— Actuarial Balance Percentage Frequency Distribution 

 of 5,000 Simulations........................................................................................... 72

Figure IV.40— Actuarial Balance Cumulative Percentage Frequency Distribution

of 5,000 Simulations ..........................................................................................72

V. Appendices

Figure V.1a— Other-than-LPR Immigration Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1999-2097 ..............105

Figure V.1b— Other-than-LPR Immigration Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1999-2097 ..............105

Figure V.2a— Total Fertility Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1917-2097 ..............106

Figure V.2b— Total Fertility Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1917-2097 ..............106
viii

Figure V.3a— Unemployment Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097 ..............107

Figure V.3b— Unemployment Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097 ..............107

Figure V.4a— Inflation Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097 ..............108

Figure V.4b— Inflation Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097 ..............108

Figure V.5a— Real Interest Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097 ..............109

Figure V.5b— Real Interest Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097 ..............109

Figure V.6a— Male Disability Incidence Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1970-2097 ..............110

Figure V.6b— Male Disability Incidence Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1970-2097 ..............110

Figure V.7— Annual Trust Fund (Unfunded Obligation) Ratios,

Calendar Years 2023-97...................................................................................111

Figure V.8— Annual Cost Rates, Calendar Years 2023-97 ...................................................111

Figure V.9— Actuarial Balance Cumulative Percentage Frequency of 5,000 Simulations

with Full Parameter Uncertainty Added...........................................................113
LIST OFTABLES

ix

IV. Results

Table IV.1— Total Fertility Rate: Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................29

Table IV.2— Male Period Life Expectancies and Increase in

Life Expectancies at Birth: Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................32

Table IV.3— Female Period Life Expectancies and Increase in 

Life Expectancies at Birth: Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................33

Table IV.4— Male Period Life Expectancies and Increase in 

Life Expectancies at Age 65: Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................36

Table IV.5— Female Period Life Expectancies and Increase

in Life Expectancies at Age 65: Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................36

Table IV.6— LPR New Arrival Immigration (in thousands): Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................40

Table IV.7— Adjustments of Status (in thousands): Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................42

Table IV.8— Legal Emigration Rate: Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................45

Table IV.9— Other-than-LPR Immigration (in thousands): Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................48

Table IV.10— Unemployment Rate (in percent): Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................50

Table IV.11— Inflation Rate (in percent): Selected Values and Intervals

Under the Deterministic and Stochastic Models ................................................53

Table IV.12— Real Interest Rate (in percent): Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................55

Table IV.13— Percentage Change in Real Average Covered Wage..........................................57

Table IV.14— Male Disability Incidence Rate (per thousand): Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................59

Table IV.15— Female Disability Incidence Rate (per thousand): Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................59

Table IV.16— Male Disability Recovery Rate (per thousand): Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................63

Table IV.17— Female Disability Recovery Rate (per thousand): Selected Values and

Intervals Under the Deterministic and Stochastic Models..................................63

Table IV.18— Estimates of the OASDI Program, Calendar Year 2097: Selected

Values and Intervals Under the Deterministic and Stochastic Models...............67

Table IV.19— Estimates of the OASDI Program: Selected Values and Intervals Under the

 Deterministic and Stochastic Models..................................................................71
x

Table IV.20— Actuarial Balance Sensitivity to Varying Selected Assumptions........................73

Table IV.21— Actuarial Balance Comparison of Deterministic and Stochastic Models...........75

V. Appendices

Table V.1— Module Dependencies ........................................................................................77

Table V.2— Nonstationary Draw Percentages Under Full Parameter Uncertainty ..............103

Table V.3— Actuarial Balance (as a percentage of taxable payroll)

in the Stochastic Model.....................................................................................112

Table V.4— Percentage of Runs Where a Bound Was Applied in at Least One Year .........141
xi
xii
A STOCHASTIC MODEL1 OF THE LONG-RANGE FINANCIAL

STATUS OF THE OASDI PROGRAM

I. INTRODUCTION

The Annual Report of the Board of Trustees of the Federal Old-Age and Survivors Insurance 

(OASI) and Disability Insurance (DI) Trust Funds includes three separate sets of long-range (75-

year) assumptions for key demographic, economic, and programmatic variables that affect the 

future financial status of the combined OASI and DI (OASDI) programs. The intermediate 

(alternative II) set of assumptions represents the Trustees’ best estimate for future experience, 

while the low-cost (alternative I) and high-cost (alternative III) sets of assumptions are more and 

less favorable, respectively, from the perspective of the trust funds’ future financial outlook. The 

Office of the Chief Actuary (OCACT) of the Social Security Administration (SSA) uses the three 

sets of assumptions to project the principal factors affecting the financial status of the OASDI 

program. Taken together, these three projections give policymakers a sense of the range of 

variation in the assumptions and in the financial status of the OASDI program. However, this 

deterministic approach makes no attempt to assign values to the likelihood of these sets of 

assumptions. 

Stochastic variation is introduced by developing equations based on standard time-series models. 

Generally, an equation may include the following: the variable’s prior-period values, prior-period 

error terms, and other variables. In addition, each equation includes a random error term. The data 

used for the regressions depend on the nature and quality of the historical data. Projected values 

for each variable in each year are computed using Monte Carlo techniques to assign the degree of 

stochastic variation around the Trustees’ intermediate assumptions. Each simulation projects 

annual values for each variable over the 75-year period, in addition to summary measures of the 

financial status of the combined OASI and DI Trust Funds (e.g., the long-range actuarial balance). 

The OCACT Stochastic Model (OSM) assigns random variation for some of the key demographic, 

economic, and programmatic assumptions. These include total fertility rates, changes in mortality 

rates, new arrival lawful permanent resident (LPR) and other-than-LPR immigration levels, rates 

of adjustment of status (from other-than-LPR to LPR), rates of legal emigration (from the 

population of citizens and LPRs), changes in the Consumer Price Index, changes in average real 

wages, unemployment rates, trust fund real interest rates, and disability incidence and recovery 

rates. The OSM is designed so that the projected values for each variable are centered on the 

intermediate assumptions of the Trustees Report—at the time of this study, the 2023 Trustees 

1 A stochastic model is used for projecting a probability distribution of potential outcomes. Such models allow for random 

variation in one or more variables through time. The random variation is generally based on fluctuations observed in historical 

data for a selected period. A large number of simulations, each of which reflects random variation in the variable(s), produce a 

distribution of potential outcomes. This contrasts with a deterministic model, which has specified assumptions for, and 

relationships among, variables. Under such a model, any specified set of assumptions fully determines a single outcome directly 

reflecting the specifications. For other definitions of terms that are used in this study, see the 2023 Trustees Report at

www.ssa.gov/oact/TR/2023/VI_I_glossary.html#1042403. 
Introduction

2 

Report. This updated study2 documents OSM version 2023.1. OSM version 2023.1 is the model 

referred to in The 2023 Annual Report of the Board of Trustees of the Federal Old-Age and 

Survivors Insurance and Federal Disability Insurance Trust Funds.

3 A stochastic model can 

incorporate parameter uncertainty for the expected means or for all parameters. Parameter 

uncertainty allows regression coefficients to vary between simulations. Parameter uncertainty is 

discussed in more detail in section II.B. Appendix E discusses why the OSM version 2023.1 uses 

parameter uncertainty for the expected mean rather than full parameter uncertainty.

The results presented here should be interpreted with caution and with a full understanding of the 

inherent limitations of the model. Results are sensitive to equation specifications, degrees of 

interdependence among variables, and the historical periods used for estimating model 

coefficients. For some variables, recent historical variation may not provide a realistic 

representation of the potential variation for the future. In addition, results would differ if variables 

other than those mentioned above (such as labor force participation rates, retirement rates, 

marriage rates, and divorce rates) were also allowed to vary randomly. Finally, if no parameter 

uncertainty or full parameter uncertainty were used instead of parameter uncertainty for the 

expected mean, then results would be different. 

The remaining chapters of this study provide detailed information about the OSM. Chapter II 

presents the general approach to stochastic modeling as well as how parameter uncertainty was 

introduced into the model. Chapter III presents the equations used to model random variation in 

the assumptions. Chapter IV presents projection results, including the projected probability 

distributions for the stochastic assumptions and the summary actuarial measures used to assess the 

long-range financial status of the OASDI program. Results with and without parameter uncertainty 

are shown. As expected, results with parameter uncertainty are distributed in a wider range.

Chapter IV also presents a sensitivity analysis for each stochastic assumption. 

Chapter V includes eight appendices to this study. Appendix A explains the overall structure of 

the OSM and its modules.4 Appendix B contains background material on various financial 

estimates of the OASDI program. Appendices C and D contain introductions to time-series 

modeling and Monte Carlo simulation, respectively. Appendix E explains why the Trustees only 

use parameter uncertainty for the expected mean rather than full parameter uncertainty. Appendix 

F provides additional details of the time-series equations used in the OSM as well as the projected

central tendency parameters and bounding statistics. Appendix G is a bibliography of references 

used in the development of this study. 

2 The first version of the study was based on OSM version 2004.1 and is located at 

www.ssa.gov/oact/NOTES/pdf_studies/study117.pdf. 

3 See www.ssa.gov/OACT/TR/2023/tr2023.pdf. 

4 A module is defined as a section of related program code.
Basic Model

3 

II. STOCHASTIC MODEL

This chapter describes the OSM’s general approach to introducing stochastic variation into some 

of the key demographic, economic, and programmatic assumptions. This variation is modeled 

using standard time-series techniques. Section A describes the basic stochastic model without 

parameter uncertainty for autoregressive moving average (ARMA) models, where the only source 

of random variation is from the error terms. Section B describes how parameter uncertainty is 

introduced into the basic model for ARMA models. Three of the economics variables use a vector 

autoregression model, which is described in detail in section III.D and appendix C. 

A. BASIC MODEL

The equations used to model the assumptions are mostly ARMA models with the additional 

requirement that the mean of the variable YYtt is always equal to its value under the intermediate 

assumptions of the 2023 Trustees Report. The value of the variable under the 2023 Trustees Report 

intermediate assumptions is written YYtt

TTTT. In general, YYttTTTT changes in the early years of the

projection period as it grades into an ultimate value that is attained in the later years. We call the 

resulting model a modified autoregressive moving average model. 

An ARMA(p,q) model with p autoregressive lags and q moving average lags takes the form 

YYtt = φφ0 + φφ1YYtt−1 + φφ2YYtt−2 + ⋯ + φφppYYtt−pp + εεtt + θθ1εεtt−1 + θθ2εεtt−2 + ⋯ + θθqqεεtt−qq

If this time series is stationary, then the mean μμφφ of this equation is computed to be 

μμφφ = φφ0

1 − φφ1 − φφ2−. . . − φφpp

 .

The above ARMA(p,q) equation can be written in “deviations form”, with yytt = YYtt − μμφφ, such that 

YYtt = μμφφ + φφ1yytt−1 + φφ2yytt−2 + ⋯ + φφppyytt−pp + εεtt + θθ1εεtt−1 + θθ2εεtt−2 + ⋯ + θθqqεεtt−qq

For the modified ARMA model, there is the additional requirement that the mean of the variable 

YYtt is always equal to its value under the 2023 Trustees Report intermediate alternative assumptions.

To satisfy this requirement, the above equation is re-written as

YYtt = YYtt

TTTT + φφ1yytt−1 + φφ2yytt−2 + ⋯ + φφppyytt−pp + εεtt + θθ1εεtt−1 + θθ2εεtt−2 + ⋯ + θθqqεεtt−qq

In this basic model, variation in YYtt over the projection period would come from the correlations of 

the variable with itself over prior periods and the random shocks from the error term εεtt. 

As the mean is fixed and equal to the value under the 2023 Trustees Report intermediate 

assumptions (i.e., YYtt

TTTT), the cumulative average of a time-series of simulated YYtt, over a sufficiently 

long enough time series, converges to its assumed mean, YYtt

TTTT. This result can be seen, to a certain 

degree, in the simulations without parameter uncertainty presented in Chapter IV. Forcing the 
Stochastic Model

4

mean equal to the values under the Trustees Report intermediate assumptions has the advantage of 

not allowing shocks, such as the COVID-19 pandemic, to affect stochastic projections from one 

Trustees Report to the next Trustees Report as much, other than increasing the variance of the 

projections.

B. INCORPORATING PARAMETER UNCERTAINTY INTO THE BASIC MODEL

In the previous section, the estimated parameters μμφφ,φφ1, ... ,φφpp, θθ1, θθ2,... , θθqq are assumed to be 

fixed for each equation and to not vary from simulation to simulation. However, these parameters 

are only estimates of the true parameters and are obtained from a sample of the true population. 

Thus, these estimates should not be treated as though they are certain. 

Econometric software packages (including EViews, which is used for the OSM) produce not only 

the parameter estimates of the equation, but also the parameter variance-covariance matrix 

associated with the estimated equation.5 When incorporating parameter uncertainty, this variancecovariance matrix V, which is (1+p+q) × (1+p+q) in dimension, is used to generate parameters 

for each simulation. This is done by multiplying the lower triangular matrix L, which is the 

Cholesky decomposition6 of the variance-covariance matrix V (such that VV = LLLL′), by a vector of 

independent standard normal random variables (ZZ⃗ ) to produce a vector of adjustments that are 

added to the estimated parameters. For an ARMA(p,q) equation, the process to obtain parameters 

adjusted for parameter uncertainty can be written in matrix notation in the following way:

⎣

⎢

⎢

⎢

⎢

⎢

⎡

μμψψ

ψψ1

...

ψψpp

φφ1

...

φφqq⎦

⎥

⎥

⎥

⎥

⎥

⎤

 = 

⎣

⎢

⎢

⎢

⎢

⎢

⎡

μμφφ

φφ1

...

φφpp

θθ1

...

θθqq ⎦

⎥

⎥

⎥

⎥

⎥

⎤

+ LLZZ⃗

The vectors above are (1+p+q) × 1 in dimension. μμψψ is the mean term adjusted for parameter 

uncertainty; ψψ1, ... , ψψpp are the autoregressive parameters adjusted for parameter uncertainty; 

φφ1, ... , φφqq are the moving average parameters adjusted for parameter uncertainty; 

μμφφ,φφ1, ... ,φφpp, θθ1, θθ2,... , θθqq are the parameters estimated originally from the ARMA(p,q) equation; 

and the product LLZZ⃗ is the vector of adjustments that is added to the vector of originally estimated 

parameters.

5 Variance-covariance matrices from these ARMA models are calculated using Newey-West estimators. Newey-West estimators 

extend Ordinary Least Square (OLS) estimators to overcome possible issues with serial correlation and heteroskedasticity.

6 See appendix C for details on this technique.
Incorporating Parameter Uncertainty into the Basic Model

5

Each simulation has its own vector of adjustments such that the same parameters adjusted for 

parameter uncertainty are used throughout the entire projection period, but the parameters vary 

from one simulation to the next.

Thus, for simulations with parameter uncertainty, there is a different mean associated with each 

simulation, and, assuming that the ARMA(p,q) process with parameters adjusted for parameter 

uncertainty is stationary, the mean implied by these parameters is μμ̂

ψψ. This mean is different from 

the process mean from the originally estimated equation, μμ̂

φφ. The mean to be used in the stochastic 

simulation with parameter uncertainty is:

μμtt

PPPP = μμttTTTT + ζζtt ∗ (μμ̂

ψψ − μμ̂φφ)

where μμtt

TTTT is the 2023 Trustees Report intermediate assumption value in year t, and ζζtt is the value 

of a 10-year phase-in factor in year t. The 10-year phase-in factor is applied to the difference 

between the μμ̂

ψψ and μμ̂φφ to avoid a sudden disconnect from the historical period to the beginning 

of the projection:

ζζtt = mmmmmm �1,

tt – (TTTT TTTTTT − 1)

10 �

where TRYEAR is the Trustees Report year (2023).

The equation for performing simulations with parameter uncertainty becomes:

YYtt = μμtt

PPPP + ψψ1(YYtt−1 − μμtt−1PPPP ) + ψψ2(YYtt−2 − μμtt−2PPPP ) +. . .+ ψψpp�YYtt−pp − μμtt−ppPPPP � + εεtt + φφ1εεtt−1

+ φφ2εεtt−2 +. . . + φφqqεεtt−qq

where the ψψ’s and the φφ’s are the autoregressive and moving average parameters with parameter 

uncertainty, respectively. The (YYtt−nn − μμtt−nn PPPP ) terms in the equation above are zero for all years 

(tt − ) prior to 2023. When running the model with parameter uncertainty for the estimated mean 

only, the autoregressive and moving average parameters calculated above are ignored and assumed 

to be the same values as without parameter uncertainty.

Some of the equations estimated under full parameter uncertainty can be nonstationary; that is, the 

variance in the projections grows exponentially over time. As a consequence, the range between 

the 2.5th and the 97.5th percentiles of the projections would grow over time. A non-stationary 

process is inconsistent with the assumptions under which the equations were estimated. Such 

nonstationary runs are eliminated from our simulations.

For an ARMA(p,q) process, the set of parameters will pass the stationarity test if the largest 

eigenvalue of the square matrix (p×p) lies inside the unit circle:
Stochastic Model

6

⎣

⎢

⎢

⎢

⎡

ψψ1 ψψ2 ⋯ ψψpp−1 ψψpp

1 0 ... 0 0

0 1 ⋯ 0 0

... ... ... ... ...

0 0 ... 1 0 ⎦

⎥

⎥

⎥

⎤

For parameter uncertainty for the estimated mean, a stationarity check is unnecessary since the 

original autoregressive parameters are unchanged from the original equations, which are known 

and required to be stationary. In other words, the μμψψ,φφ1,... ,φφpp, θθ1, θθ2,..., θθqq parameters from 

above are used (and not the μμφφ, ψψ1, ..., ψψpp, φφ1, ..., φφqq parameters). Since some equations have a 

high percentage of nonstationary draws and the mean parameter uncertainty is the most significant, 

the estimates used in the 2023 Trustees Report only allow for parameter uncertainty for the 

estimated mean term. For more details on the various forms of parameter uncertainty, see appendix 

E. In the rest of this study, unless otherwise specified, “parameter uncertainty” refers to the results 

of parameter uncertainty for the estimated mean.

A general approach to perform stochastic simulations with parameter uncertainty for an 

ARMA(p,q) model is described in this section. Pattison (2006) describes how stochastic 

simulations with and without parameter uncertainty could be implemented for the total fertility 

rate variable using an ARMA(4,1) equation, the same equation specification that is used in the 

OSM. However, Pattison takes a slightly different approach and does not estimate the means 

directly as we do for the OSM.
Fertility

7

III. EQUATION SELECTION AND PARAMETER ESTIMATION

Equations were selected for a set of assumption variables that include total fertility rates, changes 

in mortality rates, new arrival lawful permanent resident (LPR) and other-than-LPR immigration 

levels, rates of adjustment of status (from other-than-LPR to LPR), rates of legal emigration (from 

the population of citizens and LPRs), changes in the Consumer Price Index, changes in average 

real wages, unemployment rates, trust fund real interest rates, and disability incidence and recovery 

rates. The parameters of the equations were estimated using standard time-series modeling 

techniques, and then modified so that the projected variation was centered on the 2023 Trustees 

Report intermediate assumptions. Appendix C discusses the theory behind time-series modeling. 

This chapter briefly describes and presents each equation, while appendix F provides more detailed 

information and statistics.

For most variables, an ARMA(p,q) equation was selected and parameters were estimated using 

standard time-series analysis on the range of available data.7 The fit equation is of the form:

YYtt = μ + φ1(YYtt−1 − μ) + ... + φpp�YYtt−pp − μ� + εεtt + θ1εεtt−1 + ... + θqqεεtt−qq.

In this equation, YYtt represents the variable being modeled in year t, μ represents the estimated mean 

from the equation, and εεtt represents the random error in year t. In order to center the variable 

around the intermediate alternative, the modified equation is:

YYtt = YYtt

TTTT + φ1yytt−1 + ... + φppyytt−pp + εεtt + θ1εεtt−1 + ... + θqqεεtt−qq.

YYtt

TTTT represents the variable’s projected value from the intermediate assumptions of the 2023 

Trustees Report in year t, and yytt represents the deviation of the variable’s value from the 

intermediate assumptions of the 2023 Trustees Report value in year t (i.e., yytt = YYtt − YYtt

TTTT ).

The equation for performing simulations with parameter uncertainty is:

YYtt = YYtt

PPPP + φ1(YYtt−1 − YYtt−1

PPPP ) + ... + φpp(YYtt−pp − YYtt−pp

PPPP ) + εεtt + θ1εεtt−1 + ... + θqqεεtt−qq.

In this equation, YYtt

PPPP is the mean, which varies from one simulation to another, and around which 

each simulation is centered. This equation can be rewritten as:

YYtt = YYtt

PPPP + φ1yytt−1

PPPP + ... + φppyytt−pp

PPPP + εεtt + θ1εεtt−1 + ... + θqqεεtt−qq,

where yytt

PPPP = YYtt − YYttPPPP.

7 Variables that deviate from this pattern are three economic variables (the unemployment rate, inflation rate, and the real interest 

rate), which are jointly estimated in a vector autoregressive (VAR) model (section III.D), and the real average covered wage,

which is modeled as the dependent variable regressed on lagged values of the unemployment rate (section III.E).
Equation Selection and Parameter Estimation

8

A. FERTILITY

The total fertility rate is the sum of age-specific birth rates8 for girls and women aged 14 through 

49. Thus, the total fertility rate may be interpreted as the average number of children that would 

be born to a woman in her lifetime if she were to experience, at each age of her life, the birth rate 

observed in, or assumed for, a specified year, and if she were to survive the entire childbearing 

period.

Historical values for the total fertility rate in the U.S. for 1917 through 1979 are available from the 

National Center for Health Statistics (NCHS).9 Values for total fertility rate in the U.S. for 1980 

through 2021 are calculated from NCHS birth data and the U.S. Census Bureau10 population 

estimates. Over the historical period prior to 1990, the total fertility rate ranged from a minimum 

of 1.74 in 1976 to a maximum of 3.68 in 1957. From 1990 through 2007, it remained relatively 

stable, fluctuating between 1.97 and 2.12, but has dropped significantly since the start of the Great 

Recession. The rate was 1.66 in 2021.

For the total fertility rate, an ARMA(4,1) equation was selected and parameters were estimated 

using the entire range of data. Figure III.1 presents the actual and fitted values. The R-squared 

value is 0.98. The equation is:

FFtt = FFtt

PPPP + 1.96fftt−1

PPPP − 1.46fftt−2PPPP + 0.88fftt−3PPPP − 0.41fftt−4PPPP + εεtt − 0.60εεtt−1,

where FFtt is the total fertility rate in year t; FFtt

PPPP is the mean (which varies from one simulation to 

another, and around which each simulation is centered); fftt−nn PPPP = FFtt−nn − FFtt−nn PPPP ; and εt represents the 

random error in year t.

8 Age-specific birth rates are defined as the ratio of: (1) the number of live births to mothers of a specified age, to (2) the midyear 

female population of that age.

9 See www.cdc.gov/nchs. 

10 See www.census.gov. 
Fertility

9

Figure III.1 – Total Fertility Rate, Calendar Years 1917-2021
Equation Selection and Parameter Estimation

10

B. MORTALITY

The annual rate of decrease in the central death rate11 (which is sometimes referred to as the annual 

rate of improvement in mortality) is calculated as the negative of the percent change in the central 

death rate for a given year. Thus, a positive value represents a decrease in the central death rate 

from one year to the next.

Central death rates were calculated for 42 age-sex groups (under 1, 1-4, 5-9, 10-14, ..., 85-89, 90-

94, and 95+; male and female) for the period 1900 through 2020.12 Data for the annual numbers 

of deaths and the U.S. resident population for the population under age 65 are from the National 

Center for Health Statistics and the U.S. Census Bureau, respectively. For the population aged 65 

and older, annual deaths and enrollments are from the Centers for Medicare & Medicaid Services.

An AR(1) equation was selected for the annual rate of decrease in the central death rate for each 

age-sex group. The general form of the equation is:

MMkk,tt = MMkk,tt

PPPP + φφkk�mmkk,tt−1PPPP � + εεkk,tt,

where MMkk,tt is the rate of mortality improvement for age group k in year t; MMkk,tt

PPPP is the mean (which 

varies from one simulation to another, and around which each simulation is centered); mmkk,tt−nn PPPP =

MMkk,tt−nn − MMkk,tt−nn PPPP ; and εεkk,tt represents the random error for age group k in year t. Appendix F 

contains the estimates of the parameters φφkk for each age group k.

A Cholesky decomposition was performed using the residuals from the 42 fitted equations. 

Appendix C discusses this technique. The Cholesky matrix used was 42 xx 42 with the age groups 

in ascending order with alternating male and female groups.

11 The central death rate is defined as the annual number of deaths for a particular group divided by the midyear population of that 

group.

12 A detailed description of the methodology used in calculating death rates by age and sex can be found in Actuarial Study No. 120. 

See www.ssa.gov/oact/NOTES/pdf_studies/study120.pdf. 
Immigration

11

C. IMMIGRATION

The OSM models five categories of immigration flows:

1. Lawful permanent resident (LPR) new arrival immigration: Flow of persons who enter the 

Social Security area and are granted LPR status.

2. Adjustments of Status: Flow of persons who are already in the Social Security area and 

adjust their status to become LPRs.

3. Legal emigration: Flow of LPRs and citizens who leave the Social Security area

population.

4. Other-than-LPR immigration: Flow of persons who enter the Social Security area and stay 

to the end of the year without being granted LPR status, such as undocumented immigrants, 

and foreign workers and students entering with temporary visas.

5. Other-than-LPR emigration: Flow of other-than-LPR immigrants who leave the Social 

Security area population or who adjust their status to become LPRs.

The first four of these categories are discussed in the following four subsections. The last category, 

other-than-LPR emigration, does not have its own specific equation because the flow of those 

leaving the Social Security area is modeled using rates of departure, which do not vary 

stochastically.

1. LPR New Arrival Immigration

LPR immigration is defined as the flow of persons lawfully admitted for permanent residence into 

the United States, whether as a new arrival or as an adjustment of status.13 The level of LPR 

immigration largely depends on legislation that serves to define and establish limits for certain 

categories of immigrants. The Immigration Act of 1990, which is currently the legislation in force, 

establishes limits for three classes of immigrants: family-sponsored preferences, employmentbased preferences, and diversity immigrants. However, no numerical limits currently exist for 

immediate relatives of U.S. citizens.

Historical data for LPR immigration for years 1901 through 2020 are from the U.S. Citizenship 

and Immigration Services. LPR immigration averaged nearly one million per year from 1900 

through 1914, then decreased substantially to about 23,000 in 1933. Since the mid-1940s, LPR 

immigration has generally increased through about the year 2000. From 2000 – 2019, it remained 

generally steady, averaging a little over one million per year. Due to the COVID-19 pandemic, 

LPR immigration dropped to just a little over 700,000 persons in 2020.

LPR immigration, as defined above, includes both new arrivals and adjustments of status. The 

drop in total LPR immigration in 2020 is mostly due to the drop in new arrivals. These two 

categories are modeled with separate equations. For new arrival LPR immigration, an ARMA(1,1) 

equation was selected and parameters were estimated using historical data from 1991 – 2020. 

13 For more detailed information, refer to the Yearbook of Immigration Statistics at www.dhs.gov/immigrationstatistics/yearbook. 
Equation Selection and Parameter Estimation

12

Figure III.2 presents the actual and fitted values. The R-squared value is 0.69. The equation is:

LLtt = LLtt

PPPP + 0.72lltt−1

PPPP + εεtt + 0.59εεtt−1,

where LLtt is the new arrival LPR immigration in year t; LLtt

PPPP is the mean (which varies from one 

simulation to another, and around which each simulation is centered); lltt−nn PPPP = LLtt−nn − LLtt−nn PPPP ; and εt

represents the random error in year t.

Figure III.2 – LPR New Arrival Immigration, Calendar Years 1991-2020

2. Adjustments of Status

Adjustments of status is defined as the flow of persons within the United States who adjust from 

other-than-LPR status to LPR status.

As mentioned above, adjustments of status are a subset of LPR immigration, with the other subset 

being LPR new arrival immigration. The transfer rate is the percentage of the beginning-of-year 

other-than-LPR population that adjusts status to become LPRs. Having the regression based on the 

transfer rate, rather than adjustments of status values themselves, ensures that large numbers of 

adjustments of status will not cause the other-than-LPR population to become negative. To model 

the transfer rate, an AR(1) model was selected and parameters were estimated using historical data 

from 1978 – 2020. Figure III.3 presents the actual and fitted adjustments of status levels. The Rsquared value is 0.49. The equation is:

SStt = SStt

PPPP + 0.70sstt−1

PPPP + εεtt,
Immigration

13

where SStt is the LPR immigration transfer rate in year t; SStt

PPPP is the mean (which varies from one 

simulation to another, and around which each simulation is centered); sstt−nn PPPP = SStt−nn − SStt−nn PPPP ; and εt

represents the random error in year t.

Figure III.3 – Adjustments to LPR Status, Calendar Years 1978-2020

3. Legal Emigration

Legal emigration is defined as the flow of citizens and LPRs who leave the United States and are 

no longer considered to be a part of the Social Security area population. Although annual 

emigration data are not collected in the United States, we estimate, using U.S. Census Bureau 

estimates as a guide, that the level of legal emigration for the past century roughly totaled onefourth of the level of LPR immigration. The legal emigration rate is the percentage of the 

beginning-of-year LPR plus citizen population who emigrate during the year. Having the 

regression based on the emigration rate, rather than legal emigrant values themselves, ensures that 

large numbers of emigrants will not cause the total population to become negative. Using annual 

legal emigration values set at one-fourth of the LPR immigration values for 1941 – 2020, an AR(3) 

equation was selected based on these values divided by the LPR plus citizen population at the 

beginning of each year and parameters were estimated. Figure III.4 presents the actual and fitted 

emigration levels. The R-squared value is 0.95. The equation is:

tt = tt

PPPP + 1.27eett−1

PPPP − 0.63eett−2PPPP + 0.23eett−3PPPP + εεtt,

where tt is the legal emigration in year t; tt

PPPP is the mean (which varies from one simulation to 

another, and around which each simulation is centered); eett−nn PPPP = tt−nn − tt−nn PPPP ; and εt represents 

the random error in year t.
Equation Selection and Parameter Estimation

14

Figure III.4 – Legal Emigration, Calendar Years 1941-2020

4. Other-than-LPR Immigration

Other-than-LPR immigration is defined as the flow of persons into the United States who do not 

meet the above definition for LPR immigration. This includes unauthorized persons and those 

entering legally but without having LPR status, such as legal temporary workers and students.

Other-than-LPR immigration data is derived by subtracting LPR new arrivals from foreign-born 

new arrivals, based on the American Community Survey. An AR(1) model was selected and 

parameters were estimated using historical data from 1999 – 2020. Figure III.5 presents the actual 

and fitted values. The R-squared value is 0.42. The equation is:

OOtt = OOtt

PPPP + 0.66oott−1

PPPP + εεtt ,

where OOtt is the other-than-LPR immigration in year t; OOtt

PPPP is the mean (which varies from one 

simulation to another, and around which each simulation is centered); oott−nn PPPP = OOtt−nn − OOtt−nn PPPP ; and 

εt represents the random error in year t.
Immigration

15

Figure III.5 – Other-than-LPR Immigration, Calendar Years 1999-2020
Equation Selection and Parameter Estimation

16

D. UNEMPLOYMENT RATE, INFLATION RATE, AND REAL INTEREST RATE 

1. Description of the Vector Autoregression Process

The unemployment rate, inflation rate (growth rate in the CPI-W), and real interest rate are

simulated together using a vector autoregression (VAR), in order to capture the relationship among 

these three variables that economic theory suggests are related.14 In a vector autoregression, each 

variable is regressed on prior-period values of all variables. The general form of a VAR process is

YYtt = μμ + ΦΦ1(YYtt−1 − μμ) + ⋯ + ΦΦpp�YYtt−pp − μμ� + εεtt.

In this equation, YYtt is a vector of dimension n (the number of variables in the model), such that its 

i

th element is the value of the ith variable in year t, i.e., YYtt

ii

; μ is the vector of means, such that its i

th

element is the mean of the i

th variable, i.e., μμii = mean�YYii

�; p is the longest lag (highest number 

of prior periods) used; ΦΦkk, kk = 1, ... , pp is the matrix of autoregressive coefficients for lag k, such 

that its element φφkk

iiiiis the coefficient of k-years-lagged j

th variable in the equation for the ith variable;

and εεtt is the random error in year t. The vector equation is equivalent to a system of equations 

such that the equation for the i

th variable is

YYtt

ii = μμii + ∑ ∑ φφkk

iiii�YYtt−kk − μμ � nn

=1

pp

kk=1 + εεtt

ii

,

Where n is the number of variables, p is the number of lags (prior periods) used, YYtt

ii is the value of 

the i

th variable in year t, μi is the mean of the ith variable, φφkk

iiiiis the coefficient of k-years-lagged j

th

variable in the equation for the i

h variable, and εεtt

ii is the random error in year t associated with the

i

th variable.

For simulating projected values, just like in the single-variable equations in the preceding sections, 

the means are replaced by values from the intermediate alternative of the deterministic model in 

the case without parameter uncertainty, and with the simulated central tendencies, which vary from 

one simulation to another, in the case with parameter uncertainty.

In our case, there are three variables—the unemployment rate, the inflation rate, and the real 

interest rate. Vector autoregressions of different prior-period lengths were tested and it was 

determined that a vector autoregression including two prior years provided a good fit. Thus, using 

the notation above, n = 3 and p = 2. The historical period used for estimating the model is 1961 to 

2020.

14 Foster (1994) suggested that a multivariate approach might capture a more appropriate range of variability for these economic 

variables.
Unemployment Rate, Inflation Rate, and Real Interest Rate

17

2. Variables

a) Unemployment Rate

The unemployment rate is the number of unemployed persons seeking work as a percentage of the 

civilian labor force. Historical values are published by the Bureau of Labor Statistics (BLS).15 The 

annual levels are averages of the 12 corresponding monthly rates. 

Between 1961 and 2020, the unemployment rate averaged 6.3 percent, reaching a low of 3.5 

percent in 1969 and a high of 9.7 percent in 1982. In the VAR equation, the unemployment rates 

are transformed to log-odds ratios16 to bound the values of the unemployment rate between 0 and 

100 percent.

In the equations that follow, the transformed unemployment rate in year t is denoted UUtt, its central 

tendency for a given simulation in year t is denoted UUtt

PPPP, and its deviation from the central 

tendency is denoted uutt

PPPP (i.e., uuttPPPP = UUtt − UUttPPPP). In the case without parameter uncertainty, the 

value from the intermediate alternative of the deterministic model, UUtt

TTTT, is used for the central 

tendency in all simulations, and uutt = UUtt − UUtt

TTTT.

b) Inflation Rate

The inflation rate is defined as the annual growth rate in the Consumer Price Index for Urban 

Wage Earners and Clerical Workers (CPI-W). BLS publishes historical values for the CPI-W.17

BLS periodically introduces improvements to the CPI-W that affect its annual growth rate but does 

not revise earlier values. Consequently, OCACT has adjusted the CPI-W by back-casting the 

effects of the improvements on earlier values so that rates are comparable over the historical and 

projection periods. 

Over the historical period from 1961 to 2020, the adjusted inflation rate ranged from a low of -0.7 

percent in 2009 to a high of 11.1 percent in 1980. The estimation period includes a period of high 

and rising inflation rates in the 1970s, followed by a tightened monetary policy and a decrease in 

the inflation rate in the early 1980s, and eventually a period of explicit inflation targeting in the 

21st century. In the VAR equation, a logarithmic transformation18 is applied to the adjusted 

inflation rates so that the underlying rates have a lower bound. Instead of a traditional logtransformation of the inflation rate series, 3.0 percent is added to the inflation rate series prior to 

the log-transformation. This gives the inflation rates a lower bound of -3.0 percent.

In the equations that follow, the transformed inflation rate in year t is denoted IItt, its central 

tendency for a given simulation in year t is denoted IItt

PPPP, and its deviation from the central tendency 

15 See www.bls.gov/cps/home.htm. 

16 Ut = log[RUt / (1 – RUt)], where RUt is the unemployment rate in year t expressed as a decimal and Ut is the transformed 

unemployment rate.

17See www.bls.gov/cpi. 

18 lt = log(πt + 0.03) where πt is the percent change in the adjusted inflation rate in year t expressed as decimals.
Equation Selection and Parameter Estimation

18

is denoted tt

PPPP (i.e., ttPPPP = IItt − IIttPPPP). In the case without parameter uncertainty, the value from the 

intermediate alternative of the deterministic model, IItt

TTTT, is used for the central tendency in all 

simulations, and tt = IItt − IItt

TTTT.

c) Real Interest Rate

All securities held by the OASI and DI Trust Funds are special issues (i.e., securities issued only 

to the trust funds) that are issued by the Federal Government. Historical data on actual nominal 

interest rates of new purchases of these securities are published by OCACT.19 The nominal interest 

rate on new purchases of these securities for a given month is set equal to the average market yield 

on all marketable federal obligations that are not callable and do not mature within the next 4 

years.20 Annual nominal interest rates are the average of the 12 monthly rates, which, in practice, 

are compounded semiannually.21 The real interest rate earned on these obligations in year t is 

equal to the annual (compounded) nominal yield in year t-1 divided by the inflation rate in year t.

Looking at the period from 1961 to 2020, real interest rates on new purchases of special issues 

were at relatively low levels in the 1960s and 1970s. Rates rose to much higher levels in the 1980s, 

as investors demanded higher risk premiums for increased uncertainty surrounding the 

unexpectedly high rates of inflation. Since then, the rate of inflation and the real interest rate have 

declined to lower levels.

In the equations that follow, the real interest rate in year t is denoted tt , its central tendency for a 

given simulation in year t is denoted tt

PPPP, and its deviation from the central tendency is denoted 

rrtt

PPPP (i.e., rrttPPPP = tt − ttPPPP). In the case without parameter uncertainty, the value from the 

intermediate alternative of the deterministic model, tt

TTTT, is used for the central tendency in all 

simulations, and rrtt = tt − tt

TTTT.

3. Equations and Fits

The equation for the transformed unemployment rate with parameter uncertainty is:

UUtt = UUtt

PPPP + 1.10uutt−1

PPPP − 0.44uutt−2PPPP + 0.09 tt−1PPPP + 0.08 tt−2PPPP − 0.91rrtt−1PPPP + 0.11rrtt−2PPPP + εεtt .

In this equation, UUtt

PPPP is the central tendency of the transformed unemployment rate for a given 

simulation; uutt

PPPP is the deviation of the transformed unemployment rate from its central tendency 

in year t (i.e., uutt

PPPP = UUtt − UUttPPPP); ttPPPP is the deviation of the transformed inflation rate from its 

central tendency in year t (i.e., tt

PPPP = IItt − IIttPPPP); rrttPPPP is the deviation of the real interest rate from 

19 See www.ssa.gov/OACT/ProgData/newIssueRates.html. 

20 For more details on the history of trust fund investment policy, see Actuarial Note 142, Social Security Trust Fund Investment 

Policies and Practices, by Jeff Kunkel, at www.ssa.gov/OACT/NOTES/pdf_notes/note142.pdf.

21 For example, the annualized nominal yield on a special issue with a 5.0-percent nominal interest rate is equal to (1 + 0.05 / 2)2 -

1 = 5.06 percent.
Unemployment Rate, Inflation Rate, and Real Interest Rate

19

its central tendency in year t (i.e., rrtt

PPPP = tt − ttPPPP); εεtt is the random error associated with U in 

year t. 

For the unemployment rate equation, the R-squared value is 0.71. The actual and fitted values are 

shown in figure III.6.

Figure III.6 – Unemployment Rate, Calendar Years 1961-2020

The equation for the transformed inflation rate with parameter uncertainty is:

IItt = IItt

PPPP − 0.21uutt−1

PPPP + 0.16uutt−2PPPP + 0.30 tt−1PPPP + 0.53 tt−2PPPP − 6.82rrtt−1PPPP + 6.57rrtt−2PPPP + εεtt

II

.

In this equation, IItt

PPPP is the central tendency of the inflation rate for a given simulation; uuttPPPP is the 

deviation of the transformed unemployment rate from its central tendency in year t (i.e., uutt

PPPP =

UUtt − UUtt

PPPP); ttPPPP is the deviation of the transformed inflation rate from its central tendency in year 

t (i.e., tt

PPPP = IItt − IIttPPPP); rrttPPPP is the deviation of the real interest rate from its central tendency in 

year t (i.e., rrtt

PPPP = tt − ttPPPP); εεttII is the random error associated with I in year t. 

For the inflation rate equation, the R-squared value was 0.56. The actual and fitted values are 

shown in figure III.7.
Equation Selection and Parameter Estimation

20

Figure III.7 – Inflation Rate (CPI-W), Calendar Years 1961-2020

The equation for the real interest rate with parameter uncertainty is:

tt = tt

PPPP + 0.01uutt−1

PPPP − 0.01uutt−2PPPP + 0.04 tt−1PPPP − 0.02 tt−2PPPP + 1.28rrtt−1PPPP − 0.43rrtt−2PPPP + εεtt .

In this equation, tt

PPPP is the central tendency of the real interest rate for a given simulation; uuttPPPP is 

the deviation of the transformed unemployment rate from its central tendency in year t (i.e., uutt

PPPP =

UUtt − UUtt

PPPP); ttPPPP is the deviation of the transformed inflation rate from its central tendency in year 

t (i.e., tt

PPPP = IItt − IIttPPPP); rrttPPPP is the deviation of the real interest rate from its central tendency in 

year t (i.e., rrtt

PPPP = tt − ttPPPP); εεtt is the random error associated with R in year t. 

For the real interest rate equation, the R-squared value is 0.71. The actual and fitted values are 

shown in figure III.8. 
Unemployment Rate, Inflation Rate, and Real Interest Rate

21

Figure III.8 – Real Interest Rate, Calendar Years 1961-2020
Equation Selection and Parameter Estimation

22

E. REAL AVERAGE COVERED WAGE

The real average covered wage is defined as the ratio of the average nominal OASDI covered 

wage to the adjusted CPI. Because of the expansion of covered employment, the historical annual 

growth rate in the real average covered wage differs from the annual growth rate in the real average 

economy-wide wage. In the future, however, the annual growth rates in the two measures are 

expected to be approximately equal, since projected coverage changes are insignificant. Hence, 

the historical variation of the annual percent change in the real average economy-wide wage is 

used to model the future variation of the annual percent change in the real average covered wage.

The real average economy-wide wage is the ratio of the average nominal wage to the adjusted CPI. 

The nominal wage is the ratio of wages and salaries as published by the Bureau of Economic 

Analysis’ (BEA) National Income and Product Accounts (NIPA) to total wage employment. Total 

wage employment is the sum of civilian wage employment, as published by the BLS from its 

Household Survey, and total U.S. Armed Forces from the Census Bureau. The BLS periodically 

introduces improvements to its employment data but does not revise earlier data. However, the 

BLS has developed adjustment factors to improve the comparability of employment data with 

earlier years.22 OCACT uses these factors to adjust the wage employment data. 

The formula for calculating the annual percent change in the real average wage, given a nominal 

wage series, is:

WWtt =

� NNWWtt

NNWWtt−1

�

� CCCCIItt

CCCCIItt−1

�

− 1.

Wt is the annual percent change in the real average wage expressed in decimals in year t; NWt is 

the level of the nominal average wage in year t; and CPIt is the level of the CPI in year t.

The annual percent changes in the real economy-wide wage are modeled as a function of the 

current unemployment rate and the unemployment rate for the previous year, expressed as logodds ratios. The coefficients were estimated using data from 1968 to 2020. The R-squared value 

is 0.10. The actual and fitted values are shown in figure III.9. The equation is:

WWtt = WWtt

PPPP − 1.53uuttPPPP − 0.39uutt−1

PPPP + εεtt .

In this equation, Wt represents the percent change in the real average covered wage in year t; WWtt

PPPP

is the mean around which each simulation is centered, which varies from one simulation to another;

uutt

PPPP represents the deviation of the log-odds transformed unemployment rate from the mean logodds transformed unemployment rate under parameter uncertainty (which also varies from one 

22 For a detailed description of the methodology, refer to the article titled Creating Comparability in CPS Employment Series at

www.bls.gov/cps/cpscomp.pdf. 
Real Average Covered Wage 

23

simulation to another) in year t (i.e., uutt

PPPP = UUtt − UUttPPPP); and εt represents the random error in year 

t.

Figure III.9 – Real Average Covered Wage (Percent Change), Calendar Years 1968-2020
Equation Selection and Parameter Estimation

24

F. DISABILITY INCIDENCE RATE

The disability incidence rate for a given year is the proportion of the exposed population at the 

beginning of that year that becomes newly entitled to disability benefits during the year. The 

exposed population consists of workers who are disability insured but not entitled to disability 

benefits. The historical disability incidence rates used to fit the equations are age-adjusted to the 

exposed population in year 2000. The age-adjusted disability incidence rates (male and female) 

are the rates that would occur using the disability exposed population as of January 1, 2000, if that 

population were to experience the observed or assumed age-sex specific disability incidence rates 

in the selected year.

Data on disability incidence were obtained from SSA administrative records, and the age-adjusted 

disability incidence rates were computed by OCACT. Over the historical period from 1970 to 

2022, disability incidence rates varied widely due to changes in legislation and program 

administration as well as economic and demographic factors. 

The equations for disability incidence rates were selected separately for men and women. The 

disability incidence rates are expressed as log-odds ratios to bound the values between 0 and 100 

percent. For the remainder of this section, all references to the equations for the disability incidence 

rates refer to the transformed rates. 

The male and female disability incidence rates were modeled individually as AR(2) processes. The 

actual and fitted values for men and women are shown in figures III.10 and III.11, respectively. 

The R-squared values for the male and female disability incidence rate equations are both 0.86. 

The age-adjusted male disability incidence rate equation is:

DD tt = DD tt

PPPP + 1.29dd tt−1

PPPP − 0.34dd tt−2PPPP + εεtt ,

where DD tt is the age-adjusted male disability incidence rate in year t; DD tt

PPPP is the mean (which 

varies from one simulation to another, and around which each simulation is centered); dd tt−nn PPPP =

DD tt−nn − DD tt−nn PPPP ; and εt represents the random error in year t.
Disability Incidence Rate

25

Figure III.10 – Male Disability Incidence Rate, Calendar Years 1970-2022

The age-adjusted female disability incidence rate equation is:

DD tt = DD tt

PPPP + 1.35dd tt−1

PPPP − 0.44dd tt−2PPPP + εεtt ,

where DD tt is the age-adjusted female disability incidence rate in year t; DD tt

PPPP is the mean (which 

varies from one simulation to another, and around which each simulation is centered); dd tt−nn PPPP =

DD tt−nn − DD tt−nn PPPP ; and εt represents the random error in year t.

Figure III.11 – Female Disability Incidence Rate, Calendar Years 1970-2022
Equation Selection and Parameter Estimation

26

G. DISABILITY RECOVERY RATE

The disability recovery rate for a given year is the proportion of disabled-worker beneficiaries 

whose disability benefits terminate as a result of their recovery from disability. The age-adjusted 

disability recovery rates (male and female) are the rates that would occur in the in-current-payment 

population as of January 1, 2000, if the population were to experience the observed or assumed 

age-sex specific disability recovery rates in the selected year.

Data on disability recovery were obtained from SSA administrative records, and OCACT 

computed the age-adjusted disability recovery rates. Over the historical period from 1970 to 2022, 

there has been substantial variation in the age-adjusted disability recovery rates. The rate of 

recovery is affected by budget appropriations for continuing disability reviews. Changes in law 

have also caused variation. For example, the age-adjusted disability recovery rate for men 

increased from 10.3 per thousand in 1996 to 24.5 per thousand in 1997, largely as a result of the 

effects of a provision in Public Law 104-121 that prohibited benefit payments to individuals for 

whom drug addiction and/or alcoholism was material to the determination of disability.

The equations for disability recovery rates are modeled separately for men and women. The 

historical disability recovery rates used to fit the equations are age-adjusted to the 2000 in-currentpayment population. The disability recovery rates are expressed as log-odds ratios to bound the 

values between 0 and 100 percent. For the remainder of this section, all equations for the disability 

recovery rates refer to the transformed rates. Due to the frequent changes in the law, the period 

considered has been narrowed to 1984 through 2022. The value for 1997 is excluded in the 

development of the equation due to the change in the law described above.

The AR(1) model provides the best fit of the models that have been tested. The actual and fitted 

values for males and females are shown in figures III.12 and III.13, respectively. The R-squared 

values for the male and female disability recovery rate equations are 0.55 and 0.57, respectively.

The male age-adjusted disability recovery rate equation is:

DD tt = DD tt

PPPP + 0.62dd tt−1

PPPP + εεtt,

where DD tt is the age-adjusted male disability recovery rate in year t; DD tt

PPPP is the mean (which 

varies from one simulation to another, and around which each simulation is centered); dd tt−nn PPPP =

DD tt−nn − DD tt−nn PPPP ; and εt represents the random error in year t.
Disability Recovery Rate

27

Figure III.12 – Male Disability Recovery Rate, Calendar Years 1984 – 2022

The female age-adjusted disability recovery rate equation is:

DD tt = DD tt

PPPP + 0.71dd tt−1

PPPP + εεtt ,

where DD tt is the age-adjusted female disability recovery rate in year t; DD tt

PPPP is the mean 

(which varies from one simulation to another, and around which each simulation is centered); 

dd tt−nn PPPP = DD tt−nn − DD tt−nn PPPP ; and εt represents the random error in year t.

Figure III.13 – Female Disability Recovery Rate, Calendar Years 1984 – 2022
Results

28

IV. RESULTS

This chapter presents the OSM simulation results in three sections. The results presented in this 

chapter are based on two sets of 5,000 simulations; one set is with parameter uncertainty and the 

other set is without parameter uncertainty. The first section describes the OSM-simulated 

probability distributions for the equations presented in chapter III. In order to better illustrate the 

different approaches to assessing uncertainty, frequency interval bounds from the OSM results are 

compared with the values assumed in the low-cost and high-cost deterministic alternatives from 

the 2023 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 cost rates). The third section illustrates the sensitivity of the OSM to variations in the 

assumptions.

A. EQUATION RESULTS

The tables and figures in the following subsections display the OSM results for the equations 

presented in chapter III. The three rows of values shown in the tables are the level in the 75th

projection year (2097), the average level over the entire 75-year projection period (2023-97), and 

the average over the final 50 years of the projection period (2048-97).23 The corresponding 

columns for the three rows are the value from the 2023 Trustees Report intermediate alternative 

assumptions (alt II), the range from 2023 Trustees Report low-cost (alt I) and 2023 Trustees Report 

high-cost assumptions (alt III), the median and bounds of the 95-percent frequency interval for 

5,000 runs without parameter uncertainty, and the median and bounds of the 95-percent frequency 

interval for 5,000 runs with parameter uncertainty. 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 average 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 following subsections contain figures displaying values of the key variables throughout their 

entire historical and projection periods. For most variables, the expected future annual and 

cumulative average values from the equations presented in chapter III are shown along with the 

95-percent frequency intervals for 5,000 runs without parameter uncertainty and 5,000 runs with 

parameter uncertainty. The first figure displays the annual results and the second figure displays 

the cumulative average results. For the mortality subsection, the 95-percent frequency interval is 

shown for the annual values only. Note that the effects of the COVID-19 pandemic and the ensuing 

recession/recovery are evident through the first few years of the 75-year projection.

23 A 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 that variable. A notable exception is the 

mortality assumption in which death rates do not reach a constant level because 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.
Equation Results

29

1. Fertility

Table IV.1 lists the results of the total fertility rate equation presented in chapter III, and figure 

IV.1 displays a graphical representation of these results. The final 50-year average 95-percent

frequency interval without parameter uncertainty is approximately 0.61 children per woman wider 

than the range defined by alternatives I and III. With parameter uncertainty, the 95-percent

frequency interval is approximately 0.85 children per woman wider than the range defined by 

alternatives I and III. Note that stochastic results are approximately symmetrical around the median 

even though that is not true for alternative I and III relative to alternative II.

Table IV.1 – Total Fertility Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 2.00 1.97 2.00

Alt I and III Interval [1.70, 2.20] [1.68, 2.16] [1.70, 2.20]

Stochastic Model –

Without Parameter 

Uncertainty

Median 2.00 1.96 2.00

95% Frequency Interval [0.88, 3.08] [1.53, 2.37] [1.45, 2.56]

Stochastic Model –

With Parameter 

Uncertainty

Median 2.00 1.97 2.00

95% Frequency Interval [0.83, 3.15] [1.40, 2.52] [1.32, 2.67]
Results

30

Figure IV.1 – Total Fertility Rate: Selected Values and Intervals Under 

the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.2a displays the annual total fertility rate throughout the entire historical and projection 

periods. Figure IV.2b displays the annual total fertility rate throughout the historical period and 

the cumulative average total fertility rate in the projection period. The annual values indicate that 

the average standard deviation of the final 50 years of the projected total fertility rate is about 0.56 

children per woman without parameter uncertainty and 0.60 children per woman with parameter 

uncertainty.24

24 Without parameter uncertainty, the width of the 95-percent frequency interval for the annual values for the final 50 years averages 

approximately 2.20 children per woman. With parameter uncertainty, the width of the 95-percent frequency interval for the annual 

values for the final 50 years averages approximately 2.33 children per woman. For a normal distribution, this width represents 

about four standard deviations. 
Equation Results

31

Figure IV.2a – Total Fertility Rate Annual Values, 

Calendar Years 1917-2097

Figure IV.2b – Total Fertility Rate Cumulative Average Values, 

Calendar Years 1917-2097
Results

32

2. Mortality

In order to simplify the presentation, mortality results are given here as male and female period 

life expectancies at birth and at age 65, separately. Thus, the number of age-sex groups is reduced 

from 42 to just four.

a. Period Life Expectancies at Birth

Tables IV.2 and IV.3 list the male and female period life expectancies at birth, resulting from the 

mortality equations presented in chapter III. The increases shown are calculated as the difference 

between the values in 2097 and 2023 or the difference between the values in 2097 and 2048.

Without parameter uncertainty, the 95-percent frequency interval in 2097 is slightly lower than the

lower bound of the alternative I and III range, and higher than the upper bound of the alternative I 

and III range. With parameter uncertainty, the 95-percent frequency interval expands on both sides.

Table IV.2 – Male Period Life Expectancies and Increase in Life Expectancies at Birth:

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

increase

Final

50-year

increase

Deterministic Model

Alt II 83.41 7.35 4.52

Alt I and III Interval [78.68, 88.32] [2.93, 11.89] [1.83, 6.81]

Stochastic Model –

Without Parameter 

Uncertainty

Median 84.32 8.26 4.97

95% Frequency Interval [77.91, 89.96] [2.86, 12.84] [3.01, 7.14]

Stochastic Model –

With Parameter 

Uncertainty

Median 84.42 8.35 5.10

95% Frequency Interval [74.08, 91.67] [-0.98, 14.54] [-0.35, 8.51]
Equation Results

33

Table IV.3 – Female Period Life Expectancies and Increase in Life Expectancies at Birth:

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

increase

Final

50-year

increase

Deterministic Model

Alt II 87.25 6.07 3.66

Alt I and III Interval [83.42, 91.11] [2.49, 9.62] [1.52, 5.39]

Stochastic Model –

Without Parameter 

Uncertainty

Median 87.91 6.72 4.07

95% Frequency Interval [83.16, 93.40] [2.73, 11.43] [2.27, 6.59]

Stochastic Model –

With Parameter 

Uncertainty

Median 87.92 6.73 4.06

95% Frequency Interval [81.05, 95.13] [0.62, 13.15] [0.46, 8.05]

Graphical representations of the data in table IV.3 are shown in figure IV.3 through figure IV.5.

Figure IV.3 – Period Life Expectancies at Birth in 2097:

Selected Values and Intervals Under the Deterministic and Stochastic Models 

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]
Results

34

Figure IV.4 – 75-year Increases in Period Life Expectancies at Birth in 2097:

Selected Values and Intervals Under the Deterministic and Stochastic Models 

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.5 – Final 50-year Increases in Period Life Expectancies at Birth in 2097:

Selected Values and Intervals Under the Deterministic and Stochastic Models 

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]
Equation Results

35

Figures IV.6 and IV.7 display the male and female period life expectancies at birth, throughout the 

entire historical and projection periods. 

Figure IV.6 – Male Period Life Expectancies at Birth, 

Calendar Years 1900-2097

Figure IV.7 – Female Period Life Expectancies at Birth, 

Calendar Years 1900-2097
Results

36

b. Period Life Expectancies at Age 65

Tables IV.4 and IV.5 show the period life expectancies at age 65 for men and women, respectively, 

resulting from the mortality equations presented in chapter III. The values shown are analogous to 

those shown in the previous subsection. As with the results for life expectancy at birth, the bounds 

of the 95-percent frequency intervals are farther from the upper bound of the alternative I and III 

range than from the lower bound. With parameter uncertainty, these bounds of the 95-percent

frequency interval extend even farther, particularly at the upper end.

Table IV.4 – Male Period Life Expectancies and Increase in Life Expectancies at Age 65:

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

increase

Final

50-year

increase

Deterministic Model

Alt II 22.56 4.46 2.72

Alt I and III Interval [19.71, 25.74] [1.78, 7.42] [1.07, 4.33]

Stochastic Model –

Without Parameter 

Uncertainty

Median 22.95 4.84 2.96

95% Frequency Interval [19.24, 27.75] [1.67, 9.07] [1.31, 5.40]

Stochastic Model –

With Parameter 

Uncertainty

Median 22.97 4.86 3.01

95% Frequency Interval [18.37, 28.89] [0.80, 10.22] [0.56, 6.43]

Table IV.5 – Female Period Life Expectancies and Increase in Life Expectancies at Age 65:

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

increase

Final

50-year

increase

Deterministic Model

Alt II 24.68 3.98 2.41

Alt I and III Interval [22.14, 27.46] [1.61, 6.56] [0.95, 3.79]

Stochastic Model –

Without Parameter 

Uncertainty

Median 25.16 4.46 2.71

95% Frequency Interval [21.22, 30.23] [1.09, 8.92] [0.97, 5.20]

Stochastic Model –

With Parameter 

Uncertainty

Median 25.16 4.46 2.74

95% Frequency Interval [20.39, 31.75] [0.25, 10.43] [0.25, 6.60]
Equation Results

37

Graphical representations of the data in table IV.5 are shown in figure IV.8 through figure IV.10.

Figure IV.8 – Period Life Expectancies at Age 65 in 2097:

Selected Values and Intervals Under the Deterministic and Stochastic Models 

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.9 – 75-year Increases in Period Life Expectancies at Age 65 in 2097:

Selected Values and Intervals Under the Deterministic and Stochastic Models 

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]
Results

38

Figure IV.10 – Final 50-year Increases in Period Life Expectancies at Age 65 in 2097:

Selected Values and Intervals Under the Deterministic and Stochastic Models 

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figures IV.11 and IV.12 display the period life expectancies at age 65 for men and women, 

respectively, throughout the entire historical and projection periods.

Figure IV.11 – Male Period Life Expectancies at Age 65, 

Calendar Years 1900-2097
Equation Results

39

Figure IV.12 – Female Period Life Expectancies at Age 65, 

Calendar Years 1900-2097

3. Immigration

a. Lawful Permanent Resident (LPR) New Arrival Immigration

Table IV.6 displays the results (in thousands) of the LPR immigration equation presented in 

chapter III, and figure IV.13 displays a graphical representation of these results. For the final 50-

year average, the lower bound of the 95-percent frequency interval is about 11,000 persons higher 

than the alternative III level, and the upper bound is about 13,000 persons lower than the 

alternative I level without parameter uncertainty. With parameter uncertainty, the lower bound of 

the 95-percent frequency interval is about 54,000 persons lower than the alternative III level, and 

the upper bound is about 56,000 persons higher than the alternative I level.
Results

40

Table IV.6 – LPR New Arrival Immigration (in thousands):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 600 607 600

Alt I and III Interval [500, 700] [507, 707] [500, 700]

Stochastic Model –

Without Parameter 

Uncertainty

Median 601 606 600

95% Frequency Interval [356, 850] [533, 681] [511, 687]

Stochastic Model –

With Parameter 

Uncertainty

Median 602 608 603

95% Frequency Interval [329, 879] [466, 748] [446, 756]

Figure IV.13 – LPR New Arrival Immigration:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.14a displays the annual new LPR arrivals throughout the entire historical and projection 

periods. Figure IV.14b displays the annual new LPR arrivals throughout the historical period and 

the cumulative average new LPR arrivals in the projection period. The annual values indicate that 
Equation Results

41

the average standard deviation of the final 50 years of the projected annual new LPR arrivals is 

about 128,000 without parameter uncertainty and 143,000 with parameter uncertainty.25

Figure IV.14a – LPR New Arrival Immigration Annual Values, 

Calendar Years 1991-2097

Figure IV.14b – LPR New Arrival Immigration Cumulative Average Values, 

Calendar Years 1991-2097

25 Without parameter uncertainty, the width of the 95-percent frequency interval for the annual values for the final 50 years averages 

approximately 500,000. With parameter uncertainty, the width of the 95-percent frequency interval for the annual values for the 

final 50 years averages approximately 559,000. For a normal distribution, this width represents about four standard deviations. 
Results

42

b. Adjustments of Status

As described in chapter III, “adjustments of status” are those persons within the United States 

who adjust from other-than-LPR status to LPR status. The transfer rate is the percentage of the 

beginning-of-year other-than-LPR population that adjusts status to become LPRs. Table IV.7 

displays the results (in thousands) of the transfer rate equation (transformed back to actual 

numbers of adjustments of status) presented in chapter III, and figure IV.15 displays a graphical 

representation of these results. Without parameter uncertainty, the lower bound of the final 50-

year average 95-percent frequency interval is about 33,000 persons higher than the alternative 

III value, and the upper bound is about 1,000 persons higher than the alternative I value. With 

parameter uncertainty, the lower bound of the 95-percent frequency interval is about 18,000 

persons lower than the alternative III value, and the upper bound is about 63,000 persons higher 

than the alternative I value.

Table IV.7 – Adjustments of Status (in thousands):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 450 450 450

Alt I and III Interval [350,550] [350, 550] [350, 550]

Stochastic Model –

Without Parameter 

Uncertainty

Median 445 460 459

95% Frequency Interval [266, 749] [396, 532] [383, 551]

Stochastic Model –

With Parameter 

Uncertainty

Median 443 459 458

95% Frequency Interval [242, 794] [351, 586] [332, 613]
Equation Results

43

Figure IV.15 – Adjustments of Status:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.16a displays the annual transfers throughout the entire historical and projection 

periods. Figure IV.16b displays the annual transfers throughout the historical period and the 

cumulative average transfers in the projection period. Because transfer rates are applied to the 

other-than-LPR stock, the frequency intervals of final transfer values are not quite symmetrical 

around the median value. The annual values indicate that the average standard deviation of the 

final 50 years of the projected annual adjustments of status is about 124,000 without parameter 

uncertainty and 135,000 with parameter uncertainty.

26

26 Without parameter uncertainty, the width of the 95-percent frequency interval for the annual values for the final 50 years averages 

approximately 484,000. With parameter uncertainty, the width of the 95-percent frequency interval for the annual values for the 

final 50 years averages approximately 527,000. For a normal distribution, this width represents about four standard deviations. 
Results

44

Figure IV.16a – Adjustments of Status Annual Values, 

Calendar Years 1978-2097

Figure IV.16b – Adjustments of Status Cumulative Average Values, 

Calendar Years 1978-2097

c. Legal Emigration

Table IV.8 displays the results of the legal emigration rate of beginning of year LPR plus citizen 

population equation presented in chapter III, and figure IV.17 displays a graphical representation 
Equation Results

45

of these results. The final 50-year average 95-percent frequency interval is about 0.015 percent 

wider than the alternative I and III range without parameter uncertainty. With parameter 

uncertainty, the final 50-year average 95-percent frequency interval is about 0.025 percent wider

to the alternative I and III range.

Table IV.8 – Legal Emigration Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 0.060 0.070 0.066

Alt I and III Interval [0.047, 0.074] [0.061, 0.075] [0.056, 0.074]

Stochastic Model –

Without Parameter 

Uncertainty

Median 0.060 0.072 0.068

95% Frequency Interval [0.035, 0.100] [0.060, 0.088] [0.054, 0.086]

Stochastic Model –

With Parameter 

Uncertainty

Median 0.059 0.073 0.068

95% Frequency Interval [0.034, 0.103] [0.056, 0.094] [0.050, 0.093]

Figure IV.17 – Legal Emigration Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]
Results

46

Figure IV.18a displays the annual legal emigration level resulting throughout the entire historical 

and projection periods. Figure IV.18b displays the annual legal emigration level resulting 

throughout the historical period and the cumulative average legal emigration level in the projection 

period. Because the original equation is based on rates that are applied to the LPR plus citizen 

stock, the frequency intervals of final legal emigration values are not quite symmetrical around the 

median value. The annual values indicate that the average standard deviation of the final 50 years 

of the projected annual legal emigration is about 76,000 without parameter uncertainty and 84,000 

with parameter uncertainty.

27 Also noteworthy with figures IV.18a and IV.18b is the narrow range 

of the deterministic alternatives relative to the 95-percent frequency interval of the stochastic 

model.

Figure IV.18a – Legal Emigration Annual Values, 

Calendar Years 1941-2097

27 Without parameter uncertainty, the width of the 95-percent frequency interval for the annual values for the final 50 years averages 

approximately 297,000. With parameter uncertainty, the width of the 95-percent frequency interval for the annual values for the 

final 50 years averages approximately 326,000. For a normal distribution, this width represents about four standard deviations. 
Equation Results

47

Figure IV.18b – Legal Emigration Cumulative Average Values, 

Calendar Years 1941-2097

d. Other-than-LPR Immigration

Table IV.9 displays the results (in thousands) of the other-than-LPR immigration equation 

presented in chapter III, and figure IV.19 displays a graphical representation of these results. The 

final 50-year average 95-percent frequency interval is about 696,000 narrower than the alternative 

I and III range without parameter uncertainty. With parameter uncertainty, the final 50-year 

average 95-percent frequency interval is closer to the alternative I and III range but still somewhat 

narrower.
Results

48

Table IV.9 – Other-than-LPR Immigration (in thousands):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 1350 1361 1350

Alt I and III Interval [850, 1850] [861, 1861] [850, 1850]

Stochastic Model –

Without Parameter 

Uncertainty

Median 1363 1359 1349

95% Frequency Interval [738, 1958] [1206, 1510] [1168, 1537]

Stochastic Model –

With Parameter 

Uncertainty

Median 1359 1360 1348

95% Frequency Interval [652, 2019] [1020, 1698] [971, 1719]

Figure IV.19 – Other-than-LPR Immigration:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.20a displays the annual other-than-LPR immigration level throughout the entire 

historical and projection periods. Figure IV.20b displays the annual other-than-LPR immigration 

level throughout the historical period and the cumulative average other-than-LPR immigration 

level in the projection period. The annual values indicate that the average standard deviation of the 
Equation Results

49

final 50 years of the projected annual other-than-LPR immigration is about 310,000 without 

parameter uncertainty and 350,000 with parameter uncertainty.

28

Figure IV.20a – Other-than-LPR Immigration Annual Values, 

Calendar Years 1999-2097

Figure IV.20b – Other-than LPR Immigration Cumulative Average Values, 

Calendar Years 1999-2097

28 Without parameter uncertainty, the width of the 95-percent frequency interval for the annual values for the final 50 years averages 

approximately 1,212,000. With parameter uncertainty, the width of the 95-percent frequency interval averages approximately 

1,372,000. For a normal distribution, this width represents about four standard deviations.
Results

50

4. Unemployment Rate

Table IV.10 shows the results of the unemployment rate equation presented in chapter III, and 

figure IV.21 displays a graphical representation of these results. The final 50-year average 95-

percent frequency interval (3.96 percent to 5.27 percent) is a bit narrower than the alternative I and 

III range (3.51 percent to 5.26 percent) without parameter uncertainty but shifted up slightly due 

to the log-odds transformation of the unemployment rate. With parameter uncertainty, the 

frequency interval is close in size to the alternative I and III range but also shifted upwards (3.75

percent to 5.53 percent).

Table IV.10 – Unemployment Rate (in percent):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 4.43 4.43 4.43

Alt I and III Interval [3.51, 5.26] [3.50, 5.31] [3.51, 5.26]

Stochastic Model –

Without Parameter 

Uncertainty

Median 4.42 4.57 4.58

95% Frequency Interval [2.50, 7.65] [4.06, 5.13] [3.96, 5.27]

Stochastic Model –

With Parameter 

Uncertainty

Median 4.42 4.56 4.57

95% Frequency Interval [2.47, 7.70] [3.86, 5.37] [3.75, 5.53]
Equation Results

51

Figure IV.21 – Unemployment Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.22a displays the annual unemployment rate throughout the entire historical and 

projection periods. Figure IV.22b displays the annual unemployment rate throughout the historical 

period and the cumulative average unemployment rate in the projection period. Due to the logodds transformation of the unemployment rate variable, the frequency intervals are not 

symmetrical around the median value. This asymmetry is intended and desirable, as the typical 

values of the unemployment rate are much closer to its natural lower bound (zero) than to its 

natural upper bound (100 percent).
Results

52

Figure IV.22a – Unemployment Rate Annual Values, 

Calendar Years 1961-2097

Figure IV.22b – Unemployment Rate Cumulative Average Values, 

Calendar Years 1961-2097

5. Inflation Rate

Table IV.11 displays the results of the CPI-W inflation rate equation presented in chapter III, and 

figure IV.23 displays a graphical representation of these results. The values presented in the table 

are growth rates expressed in percent, and the averages shown are geometric averages. Without 

parameter uncertainty, the 50-year geometric average 95-percent frequency interval is much wider 
Equation Results

53

than the alternative I and III range but shifted up slightly due to the logarithmic transformation of 

the inflation rate. With parameter uncertainty, this frequency interval is even wider, with the 

upward shift remaining.

Table IV.11 – Inflation Rate (in percent):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 2.40 2.42 2.40

Alt I and III Interval [1.80, 3.00] [1.85, 3.01] [1.80, 3.00]

Stochastic Model –

Without Parameter 

Uncertainty

Median 2.39 2.69 2.67

95% Frequency Interval [-0.24, 7.70] [1.47, 4.27] [1.22, 4.60]

Stochastic Model –

With Parameter 

Uncertainty

Median 2.34 2.70 2.67

95% Frequency Interval [-0.44, 8.66] [0.89, 5.27] [0.71, 5.56]

Figure IV.23 – Inflation Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.24a displays the annual inflation rate throughout the entire historical and projection 

periods. Figure IV.24b displays the annual inflation rate throughout the historical period and the 
Results

54

cumulative average inflation rate in the projection period. The asymmetry in the frequency 

intervals around the median values is due to the logarithmic transformation that was applied to the 

inflation rate variable. This asymmetry is intended and desirable, as episodes of high inflation are 

empirically much more common than episodes of significant deflation.

Figure IV.24a – Inflation Rate Annual Values, 

Calendar Years 1961-2097

Figure IV.24b – Inflation Rate Cumulative Average Values, 

Calendar Years 1961-2097
Equation Results

55

6. Real Interest Rate

Table IV.12 shows the results of the real interest rate equation presented in chapter III, and figure 

IV.25 displays a graphical representation of these results. The values presented in the table are 

expressed in percent, and the averages shown are geometric averages. The 50-year geometric 

average 95-percent frequency interval is significantly wider than the alternative I and III range 

without parameter uncertainty. The frequency interval is even wider with parameter uncertainty.

Table IV.12 – Real Interest Rate (in percent):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 2.30 2.17 2.30

Alt I and III Interval [1.80, 2.80] [1.69, 2.67] [1.80, 2.80]

Stochastic Model –

Without Parameter 

Uncertainty

Median 2.37 2.22 2.36

95% Frequency Interval [-2.20, 7.71] [0.84, 3.82] [0.63, 4.43]

Stochastic Model –

With Parameter 

Uncertainty

Median 2.41 2.23 2.38

95% Frequency Interval [-2.19, 8.15] [0.38, 4.76] [0.28, 5.24]

Figure IV.25 – Real Interest Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]
Results

56

Figure IV.26a displays the real interest rate on new special issue securities throughout the entire 

historical and projection periods. Figure IV.26b displays the real interest rate on new special issue 

securities throughout the historical period and the cumulative average real interest rate on new 

special issue securities in the projection period.

Figure IV.26a – Real Interest Rate Annual Values, 

Calendar Years 1961-2097

Figure IV.26b – Real Interest Rate Cumulative Average Values, 

Calendar Years 1961-2097
Equation Results

57

7. Real Average Covered Wage

Table IV.13 displays the results of the real average covered wage equation presented in chapter 

III, and figure IV.27 displays a graphical representation of these results. The values presented in 

the table are growth rates expressed in percent, and the averages shown are geometric averages.

Without parameter uncertainty, the 50-year geometric average 95-percent frequency interval is 

slightly narrower than the alternative I and III range. With parameter uncertainty, the frequency 

interval is very close to the alternative I and III range.

Table IV.13 – Percentage Change in Real Average Covered Wage

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 1.12 1.17 1.12

Alt I and III Interval [0.53, 1.72] [0.57, 1.80] [0.52, 1.72]

Stochastic Model –

Without Parameter 

Uncertainty

Median 1.11 1.16 1.12

95% Frequency Interval [-2.18, 4.41] [0.74, 1.58] [0.60, 1.65]

Stochastic Model –

With Parameter 

Uncertainty

Median 1.09 1.17 1.12

95% Frequency Interval [-2.20, 4.42] [0.60, 1.74] [0.46, 1.80]

Figure IV.27 – Real Average Covered Wage:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]
Results

58

Figure IV.28a displays the real average covered wage throughout the entire historical and 

projection periods. Figure IV.28b displays the real average covered wage throughout the historical 

period and the cumulative average real average covered wage in the projection period.

Figure IV.28a – Real Average Covered Wage Annual Values, 

Calendar Years 1968-2097

Figure IV.28b – Real Average Covered Wage Cumulative Average Values, 

Calendar Years 1968-2097
Equation Results

59

8. Disability Incidence Rate

Tables IV.14 and IV.15 show the results of the age-adjusted disability incidence rate equations for 

men and women, respectively. Figures IV.29 and IV.30 are a graphical display of tables IV.14 and 

IV.15, respectively. The disability incidence rates shown are per thousand disability-exposed 

individuals. Without parameter uncertainty, the 50-year average 95-percent frequency interval is 

larger than the alternative I and III range for men and about the same size for women but shifted 

up slightly due to the log-odds transformation of the incidence rates. With parameter uncertainty, 

the 95-percent frequency interval is notably wider than the alternative I and III range for men and 

slightly wider for women, and shifted upwards for both sexes.

Table IV.14 – Male Disability Incidence Rate (per thousand):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 4.98 4.99 4.98

Alt I and III Interval [3.96, 5.98] [3.98, 5.98] [3.96, 5.98]

Stochastic Model –

Without Parameter 

Uncertainty

Median 4.96 5.11 5.13

95% Frequency Interval [2.84, 8.69] [3.88, 6.70] [3.61, 7.05]

Stochastic Model –

With Parameter 

Uncertainty

Median 4.99 5.11 5.12

95% Frequency Interval [2.03, 11.52] [2.52, 9.92] [2.34, 10.40]

Table IV.15 – Female Disability Incidence Rate (per thousand):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 5.07 5.08 5.07

Alt I and III Interval [4.03, 6.08] [4.05, 6.09] [4.03, 6.08]

Stochastic Model –

Without Parameter 

Uncertainty

Median 5.08 5.19 5.19

95% Frequency Interval [3.11, 8.18] [4.33, 6.16] [4.16, 6.40]

Stochastic Model –

With Parameter 

Uncertainty

Median 5.08 5.20 5.20

95% Frequency Interval [2.92, 8.78] [3.80, 7.00] [3.67, 7.24]
Results

60

Figure IV.29 – Male Disability Incidence Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.30 – Female Disability Incidence Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]
Equation Results

61

Figures IV.31a and IV.32a display the disability incidence rate for men and women, respectively, 

throughout the entire historical and projection periods. Figures IV.31b and IV.32b display the 

disability incidence rate for men and women, respectively, throughout the historical period and the 

cumulative average disability incidence rate in the projection period.

Figure IV.31a – Male Disability Incidence Annual Rate, 

Calendar Years 1970-2097

Figure IV.32a – Female Disability Incidence Annual Rate, 

Calendar Years 1970-2097
Results

62

Figure IV.31b – Male Disability Incidence Cumulative Average Rate, 

Calendar Years 1970-2097

Figure IV.32b – Female Disability Incidence Cumulative Average Rate, 

Calendar Years 1970-2097
Equation Results

63

9. Disability Recovery Rate

Tables IV.16 and 17 show the results of the disability recovery rate equations for men and women, 

respectively. Figures IV.33 and IV.34 display tables IV.16 and IV.17 graphically, respectively.

The disability recovery rates presented are per thousand disabled worker beneficiaries. Without 

parameter uncertainty, the 50-year average 95-percent frequency interval is slightly narrower for 

men and approximately the same for women when compared to each respective alternative I and 

III range. In addition, the median is slightly higher without parameter uncertainty due to the logodds transformation of the disability recovery rates. With parameter uncertainty, the 95-percent

frequency intervals are a bit wider than the alternative I and III ranges and still shifted upwards. 

Table IV.16 – Male Disability Recovery Rate (per thousand):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 9.62 9.86 9.64

Alt I and III Interval [7.61, 11.69] [7.95, 11.77] [7.62, 11.69]

Stochastic Model –

Without Parameter 

Uncertainty

Median 9.76 10.28 10.01

95% Frequency Interval [5.21, 17.05] [8.91, 11.71] [8.42, 11.70]

Stochastic Model –

With Parameter 

Uncertainty

Median 9.70 10.27 9.98

95% Frequency Interval [5.05, 17.76] [7.99, 13.13] [9.98, 13.13]

Table IV.17 – Female Disability Recovery Rate (per thousand):

Selected Values and Intervals Under the Deterministic and Stochastic Models

Value for

2097

75-year

average

Final

50-year

average

Deterministic Model

Alt II 8.73 8.92 8.73

Alt I and III Interval [6.92, 10.55] [7.20, 10.63] [6.92, 10.54]

Stochastic Model –

Without Parameter 

Uncertainty

Median 8.76 9.32 9.07

95% Frequency Interval [4.07, 15.82] [7.84, 10.91] [7.39, 11.03]

Stochastic Model –

With Parameter 

Uncertainty

Median 8.79 9.31 9.05

95% Frequency Interval [4.42, 16.87] [6.89, 12.52] [6.44, 12.63]
Results

64

Figure IV.33 – Male Disability Recovery Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]

Figure IV.34 – Female Disability Recovery Rate:

Selected Values and Intervals Under the Deterministic and Stochastic Models

[Deterministic interval is defined by alternatives I and III;

stochastic intervals are the 95-percent frequency intervals]
Equation Results

65

Figures IV.35a and IV.36a display the disability recovery rate for men and women, respectively, 

throughout the entire historical and projection periods. Figures IV.35b and IV.36b display the 

disability recovery rate for men and women, respectively, throughout the historical period and the 

cumulative average disability recovery rate in the projection period.

Figure IV.35a – Male Disability Recovery Annual Rate, 

Calendar Years 1984-2097

Figure IV.36a – Female Disability Recovery Annual Rate, 

Calendar Years 1984-2097
Results

66

Figure IV.35b – Male Disability Recovery Cumulative Average Rate, 

Calendar Years 1984-2097

Figure IV.36b – Female Disability Recovery Cumulative Average Rate, 

Calendar Years 1984-2097
Actuarial Estimates

67

B. ACTUARIAL ESTIMATES

The tables in the following subsections contain estimates for the OASDI program as a whole. Each 

column in the tables contains estimates of a particular measure. For each measure, the rows show 

results from the 2023 Trustees Report deterministic model, the OSM without parameter 

uncertainty, and the OSM with parameter uncertainty. The Trustees Report intermediate 

assumption (alt II) value is shown followed by the low-cost alternative (alt I) and high-cost 

alternative (alt III) range. Then, the median is shown followed by the 95-percent frequency interval 

without parameter uncertainty. Finally, the median is shown followed by the 95-percent frequency 

interval with parameter uncertainty.

1. Annual Measures

Table IV.18 shows selected measures of the OASDI program for the 75th projection year, 2097. 

The median estimates shown from the OSM are slightly more pessimistic (from the perspective of 

the OASDI program) for results with and without parameter uncertainty than those from the 

Trustees Report intermediate alternative. Without parameter uncertainty, the 95-percent frequency 

interval is narrower than the low-cost alternative and high-cost alternative range for cost rate as a 

percentage of payroll and slightly wider than the low-cost alternative and high-cost alternative 

range for cost rate as a percentage of gross domestic product (GDP) and beneficiaries per 100 

workers. With parameter uncertainty, the 95-percent frequency interval is slightly wider than the 

low-cost alternative and high-cost alternative range for cost rate as a percentage of payroll and 

wider than the low-cost alternative and high-cost alternative range for cost rate as a percentage of 

GDP and beneficiaries per 100 workers. 

Table IV.18 – Estimates of the OASDI Program, Calendar Year 2097:

Selected Values and Intervals Under the Deterministic and Stochastic Models

Cost Rate 

(% Payroll)

Cost Rate

(% GDP)

Beneficiaries per 

100 Workers

Deterministic Model

Alt II 17.75 5.98 47

Alt I and III Interval [12.34, 27.48] [4.52, 8.54] [36, 66]

Stochastic Model –

Without Parameter 

Uncertainty

Median 18.32 6.16 50

95% Frequency Interval [13.37, 25.99] [4.54, 8.62] [37, 69]

Stochastic Model –

With Parameter 

Uncertainty

Median 18.25 6.14 50

95% Frequency Interval [12.52, 27.93] [4.25, 9.21] [35, 74]

Figure IV.37a shows the estimated probability distribution of the annual trust fund ratio for the 

OASDI program from the OSM without parameter uncertainty, and figure IV.37b shows the 

estimated probability distribution of the annual trust fund ratio for the OASDI program from the 

OSM with parameter uncertainty. Note that the smooth lines that result do not represent the path 

of any particular simulation. Instead, for each given year, the curved lines represent the distribution 
Results

68

of trust fund ratios based on all stochastic simulation results for that year. The two extreme curves 

in this figure comprise a 95-percent frequency interval.

29 Additionally, the median is shown along 

with the curves that bound the 80-percent frequency interval.

Figure IV.37a – Annual Trust Fund Ratios, Calendar Years 2023-97, 

Without Parameter Uncertainty

Figure IV.37b – Annual Trust Fund Ratios, Calendar Years 2023-97, 

With Parameter Uncertainty

29 Note that bounds of the 95-percent frequency interval are the 2.5th and 97.5th percentiles.
Actuarial Estimates

69

An estimate of the depletion year of the combined OASI and DI Trust Fund reserves can be 

obtained from figures IV.37a and IV.37b by inspecting the x-intercept of a given curve. The xintercept of the median curve indicates that according to the OSM (both without and with 

parameter uncertainty), there is about the same chance of trust fund reserve depletion prior to 2033

as there is after 2033. This depletion year is one year less than the reserve depletion year indicated 

by the 2023 Trustees Report intermediate alternative. Furthermore, the lower limit of the 95-

percent frequency interval of the trust fund reserve depletion year indicated by the OSM with and 

without parameter uncertainty is also the same as the year in the 2023 Trustees Report high-cost 

alternative (2031). However, the upper limit as indicated by the OSM (2040 with parameter 

uncertainty and 2039 without parameter uncertainty) is less optimistic without parameter 

uncertainty than that of the 2023 Trustees Report low-cost alternative (2067).

Figures IV.38a and IV.38b show the estimated probability distribution of the annual cost rates for 

the OASDI program from the OSM without parameter uncertainty and with parameter uncertainty, 

respectively. The figures also include the income rates for the intermediate scenario. No 

distribution for the income rates is shown because there is relatively little variation in income rates 

across the 5,000 stochastic simulations. As before, the smooth curves which result do not represent 

the path of any particular simulation. Once again, the two extreme curves in this figure comprise 

a 95-percent frequency interval and the additional curves depict the median and the 80-percent 

frequency interval.

Figure IV.38a – Annual Cost and Income Rates, Calendar Years 2023-97, 

Without Parameter Uncertainty
Results

70

Figure IV.38b – Annual Cost and Income Rates, Calendar Years 2023-97, 

With Parameter Uncertainty

2. Summary Measures

Table IV.19 shows selected measures of the OASDI program for the 75-year projection period as 

a whole. The median estimates shown from the OSM are, in general, close to those from the 2023 

Trustees Report intermediate alternative. Estimates with parameter uncertainty have wider ranges 

than estimates without parameter uncertainty.
Actuarial Estimates

71

Table IV.19 – Estimates of the OASDI Program:

Selected Values and Intervals Under the Deterministic and Stochastic Models

Actuarial 

balance

Summarized 

cost rate

Summarized 

income rate

Open-group 

unfunded 

obligation (in 

trillions of 

dollars)

First projected 

year cost 

exceeds noninterest 

income and 

remains in 

excess 

through 2097

First year trust 

fund reserves 

become 

depleted

Deterministic 

Model

Alt II -3.61 17.38 13.78 22.42 2023 2034

Alt I and III 

Interval [-8.37, -0.10] [13.59, 22.52] [13.49, 14.16] [-0.43, 42.90] [2023, 1] [2031, 2067]

Stochastic 

Model –

Without 

Parameter 

Uncertainty

Median -3.72 17.52 13.79 23.22 2023 2033

95% Frequency 

Interval [-6.39, -1.68] [15.35, 20.31] [13.55, 14.07] [9.03, 51.37] [2023, 2096] [2031, 2039]

Stochastic 

Model –

With Parameter 

Uncertainty

Median -3.69 17.50 13.80 22.85 2023 2033

95% Frequency 

Interval [-7.17, -1.18] [14.85, 21.09] [13.49, 14.14] [5.83, 60.85] [2023, 1] [2031, 2040]

1 Cost is projected to exceed non-interest income for a temporary period, before falling below non-interest income by the end of the projection period.

Figure IV.39 shows a frequency distribution of the long-range actuarial balances estimated from 

the OSM with and without parameter uncertainty. The width of each interval on the figure is 0.2 

percent and the actuarial balance is expressed as a percentage of taxable payroll. It is interesting 

to note that none of the 5,000 simulations resulted in a positive long-range actuarial balance 

without parameter uncertainty, and only 5 of the 5,000 simulations resulted in a positive longrange actuarial balance with parameter uncertainty. In other words, according to the OSM, there 

is no measurable probability that the OASDI program has a positive actuarial balance for the period 

2023-97 without parameter uncertainty and just a 0.1 percent probability with parameter 

uncertainty.
Results

72

Figure IV.39 – Actuarial Balance Percentage Frequency Distribution of 5,000 Simulations

Figure IV.40 shows a cumulative frequency distribution of the long-range actuarial balances 

estimated from the OSM. The scale on the horizontal axis is the same as in figure IV.39. The 2023 

Trustees Report high-cost, intermediate, and low-cost alternatives are each identified on the figure 

and labeled with the probability (according to the OSM) that the actuarial balance is less than or 

equal to this level.

Figure IV.40 – Actuarial Balance Cumulative Percentage Frequency Distribution of 5,000 Simulations
Sensitivity Analysis

73

C. SENSITIVITY ANALYSIS

This section presents two types of sensitivity analysis for the OSM: stochastic and deterministic. 

A stochastic sensitivity analysis consists of performing a set of simulations in which a single 

equation (or group of related equations) is modeled stochastically while all other equations are 

modeled as for the 2023 Trustees Report intermediate alternative (alt II). A deterministic 

sensitivity analysis consists of a single simulation in which the mean of an equation (or group of 

related equations) is replaced with the 2023 Trustees Report low-cost (alt I) or high-cost (alt III) 

value, while the remaining equations are left with their means equal to the alt II values.

1. Stochastic Analysis

A stochastic sensitivity analysis is a process in which one, or some, of the demographic, economic, 

and programmatic variables are varied independently, while the other variables are set to their 

values under the 2023 Trustees Report intermediate alternative. This allows a measurement of how 

much of the variance of the final distributions can be attributed to each variable or set of variables.

Table IV.20 presents the median and average of the long-range actuarial balance, along with the 

standard deviation, for each of the stochastic sensitivity runs, both with parameter uncertainty and 

without. Each of these sensitivity runs is based on 5,000 simulations of the stochastic model. The 

first sensitivity run presented in table IV.20 allows only the total fertility rate to vary. The next 

sensitivity run varies only the 42 mortality improvement rates (21 age groups for boys/men, and 

21 age groups for girls/women). The third run, listed as immigration, allows only the four 

immigration assumptions (level of LPR new arrival immigration, rate of legal emigration, rate of 

transfer from other-than-LPR to LPR status, and level of other-than-LPR immigration) to vary. 

Grouping the four economic assumptions (unemployment, inflation, real interest, and real average 

wage growth rates) and allowing only these variables to vary produces the economic sensitivity 

run. The last two sensitivity runs vary only the disability incidence rates and disability recovery 

rates, respectively. 

Table IV.20 – Actuarial Balance Sensitivity to Varying Selected Assumptions

Actuarial Balance1

Without Parameter Uncertainty With Parameter Uncertainty

Median Average

Standard

Deviation Median Average

Standard

Deviation

Effect of Changing Individual 

Assumption or Set of Assumptions:

Fertility -3.62 -3.66 0.70 -3.61 -3.66 0.82

Mortality Improvement -3.71 -3.73 0.67 -3.72 -3.73 0.76

Immigration -3.62 -3.62 0.07 -3.62 -3.62 0.12

Economic -3.57 -3.58 0.61 -3.54 -3.55 0.84

Disability Incidence -3.64 -3.65 0.19 -3.64 -3.69 0.42

Disability Recovery -3.60 -3.60 0.01 -3.60 -3.60 0.02

Effect of Changing All 

Assumptions Together: -3.72 -3.82 1.20 -3.69 -3.82 1.51

1 The long-range actuarial balance under the 2023 Trustees Report intermediate alternative is -3.61 percent of taxable payroll.
Results

74

2. Deterministic Analysis

A deterministic sensitivity analysis is a process in which the mean of one, or some, of the 

demographic, economic, and programmatic assumptions are set to equal the alternative I or the 

alternative III value, while the remaining variables are set to equal the alternative II values. In this 

case, the variance of all key input variables is set to zero and only one simulation is performed, as 

the zero variance ensures that the input variables for any simulation match the alternative I, II, or 

III values for those variables. This type of sensitivity analysis is similar to that done in the Trustees 

Report and can therefore be used to compare the underlying nature of the OSM and the 

deterministic model presented in the Trustees Report.

Table IV.21 compares the long-range actuarial balance results from the deterministic sensitivity 

run of the OSM to the results in the Trustees Report. The first row of each sensitivity grouping is 

from the sensitivity analysis in the Trustees Report, while the second row contains the sensitivity 

result from the OSM. Technically speaking, the first grouping (“All”) is not really a sensitivity 

analysis, but instead shows the results of runs done using all of the assumptions based on 

alternatives I, II, and III. It is included as a point of reference. The next three results (fertility, 

mortality, and immigration) are the demographic deterministic sensitivity runs. These are followed 

by the three economic deterministic sensitivity runs (inflation, real interest, and real average 

wage). The final deterministic sensitivity run is programmatic (disability incidence). Note that 

unemployment and disability recovery sensitivity runs are not included because no comparable 

sensitivity analysis is performed for the Trustees Report. 
Sensitivity Analysis

75

Table IV.21 – Actuarial Balance Comparison of 

Deterministic and Stochastic Models

Sensitivity:

Actuarial Balance

Deterministic 

Low Cost

Deterministic 

Intermediate1

Deterministic 

High Cost

All

Trustees Report -0.10 -3.61 -8.37

OSM -0.13 -3.61 -8.49

Fertility

Trustees Report -3.14 -3.61 -4.32

OSM -3.13 -3.61 -4.34

Mortality

Trustees Report -2.91 -3.61 -4.39

OSM -2.71 -3.61 -4.67

Immigration

Trustees Report -3.21 -3.61 -4.02

OSM -3.22 -3.61 -4.02

Inflation

Trustees Report -3.47 -3.61 -3.74

OSM -3.48 -3.61 -3.78

Real Interest

Trustees Report -3.43 -3.61 -3.79

OSM -3.42 -3.61 -3.78

Real Average Wage

Trustees Report -2.44 -3.61 -4.81

OSM -2.32 -3.61 -4.90

Disability Incidence

Trustees Report -3.25 -3.61 -3.98

OSM -3.28 -3.61 -3.91

1 This column does not show a sensitivity analysis but is shown for comparison purposes.

Table IV.21 shows that the actuarial balances produced by the OSM are generally similar to those 

from the deterministic model.
Appendices

76

V. APPENDICES

A. DOCUMENTATION OF COMPUTER PROGRAM

This chapter describes the details of the computer program used to run the OSM.

1. Organization

The OSM contains nine modules. They are executed sequentially in the following order: 

Assumptions, Population, Economics, Insured, DI (Disability Insurance Beneficiaries), RSB (OldAge and Survivors Insurance Beneficiaries), Awards, Cost, and Summary Results. In a sequential 

model, the output from an earlier module may become input to a later module. The flow of data 

among the OSM modules is summarized in table V.1. The first column lists the nine modules in 

the order in which they are executed. For each module, the second column lists the modules from 

which it receives input, while the third column lists the modules to which it provides input. For 

example, the Population Module receives input from only one module (i.e., Assumptions) and 

provides output (that then becomes input) to six modules (i.e., Economics, Insured, DI, RSB, 

Awards, and Cost). The Assumptions Module does not receive data from any of the other modules, 

and the Summary Results Module does not send data to any of the other modules. The only instance 

in which a module sends data to an earlier module is that the DI Module sends data to the 

Economics Module. This is possible because the DI data sent there are from the prior (not current) 

year within the simulation.

The computer program is organized to go through three main phases: initialization, simulation, 

and wrap-up. In the initialization phase, the program prepares input and output files and variables 

needed by each module. In the simulation phase, the program solves the first eight modules using 

two nested loops. The first (outermost) is the run-number loop. It loops once for each simulation. 

The second is the year loop. It loops from the first year of the simulation (the current Trustees 

Report year) through the last year of the simulation. In the wrap-up phase, the final module, 

Summary Results, sorts and prints the final output results.
Documentation of Computer Program

77

Table V.1 – Module Dependencies

Module Input Modules Output Modules

Assumptions N/A Population

Economics

DI

Cost

Summary Results

Population Assumptions Economics

Insured

DI

RSB

Awards

Cost

Summary Results

Economics Assumptions

Population

DI

Insured

Awards

Cost

Insured Population

Economics

DI

RSB

Awards

DI Assumptions

Population

Insured

Economics

RSB

Cost

RSB Population

Insured

DI

Awards

Cost

Awards Population

Economics

Insured

RSB

Cost

Cost Assumptions

Population

Economics

DI

RSB

Awards

Summary Results

Summary Results Assumptions

Population

Cost

N/A

2. Modules

All of the modules, with the exception of the Assumptions and Summary Results Modules, are 

adapted from the deterministic computer model used to prepare the 2023 Trustees Report. The 

modules are written so that the set of non-stochastic inputs required to begin the projections is 

identical to the input assumptions used when running the deterministic model under the 2023 

Trustees Report intermediate alternative. Moreover, the mean value for each stochastic variable is 

set equal to the value assumed for the variable under the 2023 Trustees Report intermediate 

alternative. 
Appendices

78

a. Assumptions

The Assumptions Module contains 55 equations, one for each stochastic variable. The 55 

stochastic variables are the total fertility rate, changes in mortality rates (21 male age groups and 

21 female age groups), new arrival LPR and other-than-LPR immigration levels, rates of 

adjustments of status (from other-than-LPR to LPR), rates of legal emigration (from the population 

of citizens and LPRs), changes in the Consumer Price Index, changes in average real wages, 

unemployment rates, trust fund real interest rates, disability incidence rates (male and female), and 

disability recovery rates (male and female). These equations are described in detail in chapter III.

The equations are used to set the annual values for the stochastic variables. In any particular year, 

the value for a stochastic variable is determined, in part, by the equation’s error term. If the error 

term for an equation is not dependent on the error terms of other equations, then a random number 

is drawn for each year from a normal distribution with mean zero and standard deviation equal to 

the estimated standard error for the equation. If the error term for an equation is dependent on the 

error terms of other equations, then a Cholesky decomposition is used to assign the appropriate 

level of covariance. See appendix C for more details on this process.

The final step of the Assumptions Module is to use the error terms to calculate the results of each 

equation. Chapter III provides more details about these calculations for each equation.

For the mortality equations, there is an additional step to decompose the annual rates of decrease 

in the central death rates by age group into single years of age.

b. Population

The Population Module projects the Social Security area population by sex, single year of age, and 

marital status. The Population Module also projects the other-than-LPR population. The 

components of change—fertility, mortality, and immigration—are applied each year throughout 

the projection period based on levels and rates generated in the Assumptions Module. The 

population is grouped by marital status using the relative proportions for each age-sex group. 

The population is projected by starting with the beginning of the year population, adding births 

and immigration, and subtracting deaths and emigration. The total fertility rate is distributed 

among women of childbearing age using the relative proportions of age-specific birth rates for 

each year. The age-specific birth rates are then applied to the midyear population to calculate the 

number of births. For the mortality projection, central death rates are computed by applying the 

rates of decrease in the single year of age central death rates to the previous year’s central death 

rates. Death probabilities are derived from the central death rates by assuming a uniform 

distribution of deaths for each age (except age 0). The death probabilities are then applied to the 

beginning of the year population to calculate the number of deaths for each single year of age and 

sex group. For each type of immigration, the annual levels are distributed among the age-sex 

groups by using the relative proportions. The resulting population is then distributed by marital 

status using the relative marital proportions for each age-sex group.
Documentation of Computer Program

79

c. Economics

The Economics Module receives data from the Assumptions, Population, and DI Modules. The 

Assumptions Module passes civilian unemployment and inflation rates, along with the growth rate 

in the real average covered wage. The Population Module passes the age-sex levels of the Social 

Security area population and their life expectancies. The DI Module passes one-year-lagged agesex levels of disabled-worker beneficiaries in current-payment status. 

For employment-related variables, the module projects various measures for the total U.S. 

economy and then converts them to OASDI covered concepts. For the earnings variables, the 

module initially projects OASDI covered wages then converts them to a U.S. economy-wide 

concept. The module estimates levels for most key variables by projecting deviations from values 

produced for the intermediate assumptions.

Labor Force Participation

Future civilian labor force participation rates by age and sex are influenced by projected disability 

prevalence rates, current and lagged unemployment rates, and life expectancies. For a given year, 

the civilian labor force is summed from the products of the civilian labor force participation rates 

and civilian noninstitutional populations by age and sex. For each age-sex group, the civilian 

noninstitutional population is the product of the Social Security area population and the ratio of 

the noninstitutional population to the Social Security area population. 

Employment

The civilian unemployment rates by age and sex are projected by distributing the stochastically 

projected aggregate rate to its age-sex components. Projected economy-wide employment by age 

and sex is derived from the corresponding components of the civilian labor force and 

unemployment rates. The concept of economy-wide employment represents an “average” level of 

employment for a particular year. The projected economy-wide employment by age and sex is then 

used to estimate the number of workers with employment at any time during the year, a concept 

closer to OASDI covered employment. 

Covered Employment

Total OASDI covered employment by age and sex is projected by removing non-covered workers, 

such as certain categories of government employees, foreign workers in certain visa categories, 

and undocumented immigrants, from the total at-any-time employment. Total OASDI covered 

employment is then broken down into those with wages and those with self-employed income 

only.

Covered Wages

Total OASDI covered wages are the product of the number of covered wage workers and their 

average nominal covered wage. The average nominal covered wage is determined using the annual 

inflation rate and the real average OASDI covered wage from the Assumptions Module. Total U.S. 

economy-wide wages are projected as a ratio to OASDI covered wages, adjusted for relative 

differences in other-than-LPR immigration. Total compensation for wage workers, total and 
Appendices

80

covered self-employed income, taxable wages, and taxable self-employed net income are all 

derived from assumed relationships in the intermediate alternative. Multi-employer refund wages 

are projected as a ratio to OASDI covered wages, adjusted for relative differences in the 

unemployment rate. 

Taxable Payroll, Average Wage Index, and Cost of Living Adjustment (COLA)

The OASDI taxable payroll is the sum of taxable wages and self-employed income, less one-half 

of multi-employer refund wages. The average wage index is determined using the annual growth 

rate in the economy-wide average wage, defined as the ratio of total U.S. economy-wide wages to 

total at-any-time wage employment. The COLA is determined by the inflation rate.

d. Insured (Fully Insured and Disability Insured)

Fully insured status is required to receive worker benefits and is determined by a worker’s 

accumulation of quarters of coverage (QCs). Prior to 1978, one QC was credited for each calendar 

quarter in which at least $50 was earned. Quarterly reporting was replaced by annual reporting in 

1978. The minimum annual required amount, starting with $250 for each QC in 1978, is adjusted 

each year according to the average wage index. This value for 2023 is $1,640. Thus, if a worker 

earns at least $6,560 in covered employment any time during 2023, then the worker receives credit 

for four quarters of coverage.

Fully insured status is determined by the number of earned QCs and the worker’s age. To be fully 

insured, a worker must have a total number of QCs greater than or equal to the number of years 

elapsed after attaining age 21 (with a minimum of six QCs required). Once reaching 40 QCs, the 

worker remains permanently fully insured. Disability insured status is acquired by any fully 

insured worker over age 30 who has accumulated 20 QCs during the 40-quarter period ending with 

the quarter in which the disability began. A fully insured worker aged 24-30 needs to accumulate 

at least one-half of the quarters elapsed after attaining age 21. A fully insured worker under age 24 

needs to have accumulated six QCs during the 12-quarter period immediately before becoming 

disabled.

In the intermediate alternative, projections of the fully insured population, as a percentage of the 

Social Security area population, are made by age and sex for each birth cohort beginning with 

1900. Percentages for everyone except the other-than-LPR immigrant population are based on 

30,000 simulated work histories for each sex and birth cohort. The simulated work histories are 

constructed to reproduce fairly closely the historical insured percentages from 1990 to date, using 

the historical portions of the following data:

• Median earnings, by age and sex,

• Covered workers and Social Security area population, by age and sex, and

• Net LPR immigrants and other-than-LPR immigrants, by age and sex.

The projected portions of the above data are then used to extend the simulation of work histories 

throughout the projection period. Projected fully insured percentages for each sex and birth cohort 

are then determined by identifying all simulated work histories that meet the QC requirement for 
Documentation of Computer Program

81

fully insured status as a percentage of the 30,000 simulated cases, which represent everyone except 

the other-than-LPR immigrant population. For the other-than-LPR immigrant population, the 

model generates substantially lower percentages of individuals attaining fully insured status. A 

similar process is applied to produce the disability insured percentages.

In the OSM, the Insured Module projects the percentages of the population that will be fully 

insured and disability insured for each birth-sex cohort. For everyone except the other-than-LPR 

immigrant population, projections of fully insured percentages are based on the baseline projection 

in the intermediate alternative and an adjustment that accounts for the difference between the 10-

year moving averages of the covered worker rates30 from the Economics Module and the 

intermediate alternative. A small portion of the other-than-LPR immigrant population is then 

added to calculate the fully insured rate of the Social Security area population. Projections of the 

disability insured percentages are modeled in a similar manner. Finally, these percentages are 

multiplied by the Social Security area population from the Population Module to produce the 

numbers of insured individuals. 

e. Disability Insurance Beneficiaries (DI)

The DI Module begins with projections of the disabled-worker beneficiaries in current-payment 

status. The projections are based on the age-sex specific disability insured population passed from 

the Insured Module, the age-sex specific mortality rates passed from the Population Module, and 

the age-adjusted male and female incidence and recovery rates passed from the Assumptions 

Module. Additionally, the DI Module estimates31 the number of those currently entitled (as of the 

beginning of the projection period) to a disabled-worker benefit as a starting value.

Disabled Workers

The number of disabled-worker beneficiaries at the end of a year is calculated by adding those 

newly entitled to a disabled-worker benefit during the year to those currently entitled at the 

beginning of the year and subtracting those who recover, die, or convert to a retirement benefit 

upon reaching normal retirement age during the year. New entitlements are calculated by 

multiplying the incidence rate by the exposed population (disability insured less those currently 

entitled). For each sex, the future age-specific incidence rates are assumed to grow at the same rate 

as the growth in the age-adjusted incidence rate.32

Deaths and recoveries are calculated by applying the death and recovery rates to the number of 

people who are currently entitled at the beginning of the year and to the number of people who are 

newly entitled during the year. Death rates by age, sex, and duration since entitlement are projected 

to improve at the same rate as the general population aged 15 through 64. For each sex, the future 

30 Covered worker rates are defined as the number of covered workers, expressed as a percentage of the Social Security area 

population.

31 The current number of entitled disabled-worker beneficiaries is not completely known because of the time lag between 

entitlement to and receipt of benefits. 

32 Growth is determined relative to a 2005-14 base period for ages 15-60. Rates for ages 60 and over include adjustments for the 

increase in normal retirement age.
Appendices

82

age-specific recovery rates are assumed to grow at the same rate as growth in the age-adjusted 

recovery rate.

33 The number of disabled-worker beneficiaries in current-payment status is then 

estimated by reducing the number of those currently entitled by the number of those for whom 

payment has not yet begun.

Dependents of Disabled Workers

The projected number of auxiliary beneficiaries of disabled workers depends on the projections of 

disabled workers and the Social Security area population. Minor child beneficiaries of disabled 

workers are projected as the product of the child population and factors which represent the 

probabilities that a worker is under normal retirement age, is disability insured, and is disabled, 

and a statistical residual factor. Student and disabled-adult-child beneficiaries are calculated 

similarly. Married aged-spouse beneficiaries of disabled workers are projected as a percentage of 

disabled-worker beneficiaries. This percentage is set as in the intermediate alternative and a factor 

that adjusts for differences between the OSM and the intermediate alternative projected 

distributions of the age 62 or older married population. Young-spouse and divorced aged-spouse 

beneficiaries are calculated similarly, but with their respective populations. 

f. Old-Age and Survivors Insurance Beneficiaries (RSB)

The RSB module receives variables passed from the Population, Insured, and DI Modules. The 

Population Module passes the Social Security area population by age, sex, and marital status. The 

Insured Module passes the number of fully insured persons, by age, sex, and marital status, and 

also passes the LPR fully insured rates by age and sex. The DI Module passes disability prevalence 

rates34 and the numbers of disabled-worker and converted disabled-worker beneficiaries, by age 

and sex. Using these data, the RSB Module estimates the number of retired-worker beneficiaries, 

along with five categories of auxiliary and survivor beneficiaries who are eligible to receive 

benefits based on the earnings of a retired or deceased worker (also referred to as the primary 

account holder). These categories are aged-widow(er), aged-spouse, disabled-widow(er), children 

(minor, student, and disabled adult), and young-spouse beneficiaries.

Retired Workers

To calculate the number of retired-worker beneficiaries, the population aged 62 or older is 

multiplied by the probability that:

• The worker is fully insured,

• The worker is not receiving disability benefits, and

• The worker is not an insured aged widow(er).35

33 Growth is determined relative to a 2011-15 base period.

34 A disability prevalence rate is the ratio of the number of disabled workers to the number of disability insured workers.

35 In this case, the worker is fully insured and, therefore, eligible to receive his/her own retired-worker benefit. Instead, he/she has 

decided not to apply for a retired-worker benefit and is receiving only an aged-widow(er) benefit.
Documentation of Computer Program

83

Retirement prevalence rates36 used in the intermediate alternative are then applied to calculate the 

number of retired-worker beneficiaries. Due to the elimination of the file and suspend and deemed 

filing claiming strategies by the Bipartisan Budget Act of 2015, additional factors are included in 

the calculation of the prevalence rates. These factors increase the projected number of married and 

divorced retired workers to account for those who will no longer delay receiving a benefit as a part 

of the eliminated claiming strategies.

Aged Widow(er)s

Aged widow(er)s are divided into two subcategories: insured and uninsured. The number of 

insured aged-widow(er) beneficiaries is projected as the product of the widowed and divorced 

population aged 60 or older, and the probability that:

• The primary account holder is deceased,

• The primary account holder was fully insured at death, and

• The aged-widow(er) is fully insured but, if at least age 62, did not apply for a retiredworker benefit based on his/her own earnings (assuming that his/her own retiredworker benefit is less than his/her widow(er) benefit).

The number of uninsured aged-widow(er) beneficiaries is projected as the product of the widowed 

and divorced population aged 60 or older, and the probability that:

• The primary account holder is deceased,

• The primary account holder was fully insured at death,

• The aged widow(er) is not fully insured,

• The aged widow(er) is not receiving a young-spouse benefit for the care of a child, and

• The aged-widow(er)’s benefits are not withheld because of receipt of a significant 

government pension based on earnings in noncovered employment. 

For both the insured and uninsured categories, an additional probability is applied which accounts 

for other conditions not previously mentioned. For example, in the case of an aged widow(er), the 

additional factor includes the probability that the widow(er) did not remarry before age 60. In the 

case of a divorced widow(er), the factor includes the probability that the marriage to the primary 

account holder lasted at least 10 years. 

Aged Spouses

The number of aged spouses of retired workers is projected as the product of the married and 

divorced population aged 62 or older, and the probability that:

• The primary account holder is alive and fully insured,

36 A retirement prevalence rate is the ratio of the number of retired workers to the number of fully insured workers (not receiving 

disability or widow(er)’s benefits).
Appendices

84

• The primary account holder is receiving a retirement benefit (not required for divorced 

spouses),

• The aged spouse is not receiving a young-spouse benefit for the care of a child,37

• The aged spouse is not insured, 

• The aged spouse’s benefits are not withheld because of receipt of a significant 

government pension based on earnings in noncovered employment, and

• The couple is engaging in a filing strategy in which the primary account holder files 

for their benefit and the fully insured spouse files only for their aged spouse benefit.

In addition to the stated conditions, an adjustment is made for other requirements. One such 

requirement is that the aged spouse has been married to the primary account holder for at least one 

year. In the case of a divorced aged spouse, the requirement is that their marriage had lasted at 

least 10 years. As is the case with many of the listed requirements, there are exceptions to this 

requirement.

Disabled Widow(er)s

To calculate the number of disabled-widow(er) beneficiaries, the widowed and divorced 

population ages 50 through 69 is multiplied by the probability that:

• The primary account holder is deceased,

• The primary account holder was fully insured at time of death,

• The surviving spouse is disabled, and

• The disabled widow(er) is not receiving another type of benefit. 

Finally, an additional factor is applied to account for other eligibility requirements. For example, 

there is a 7-year deadline for surviving spouses to qualify for benefits on the basis of disability.

Children

The RSB Module calculates the number of child beneficiaries for three different child categories: 

minor, student, and disabled adult. Child beneficiaries are estimated separately for retired and 

deceased primary account holders. The population of potential beneficiaries for minor children 

includes children under age 18, while student status includes children of age 18 (and occasionally 

also age 19). Disabled adult status includes all disabled persons over age 17 who were disabled 

prior to age 22. 

To calculate the number of minor children of male retired workers, the population of children 

under age 18 with a parent aged 62 and over whose other parent is not deceased is multiplied by 

the ratio of the number of retired workers aged 62 to 71 to the number of members of the population 

aged 62 to 71. We maintain the total number of minor children of female retired workers at the 

37 The upper age limit to be eligible for a young-spouse benefit is 69, as long as there is a dependent child under 16 or disabled. 

Since the youngest age to receive an aged-spouse benefit is 62, there is a chance that some spouses between the ages of 62 and 

69 are still receiving young-spouse benefits.
Documentation of Computer Program

85

five-year historical average percentage of minor children of female retired workers to the total 

number of minor children of male and female retired workers. We then use the proportion of 

beneficiaries at each age in the last historical year to distribute the minor children of female retired 

workers among the ages 0 to 17. 

For minor children of deceased workers, the population of children under age 18 is multiplied by 

the probability that:

• The parent is deceased,

• The parent was fully or currently insured at time of death, and

• The child is not receiving a benefit based on his/her other parent’s earnings. 

Student and disabled adult children of retired and deceased workers are calculated similarly using 

their respective age-specific populations. The calculations for student and disabled children of 

retired workers have the additional condition that the parent is receiving a benefit. 

For each child category, an adjustment is made for other conditions, such as the marital status of 

the child (more common in the case of a disabled adult child) or the dependency status of the child. 

For example, if a child marries, he/she is no longer entitled to a benefit. Also, if it is determined 

that the child is not dependent upon the parent (or was not at the time of the parent’s death) then 

he/she is not entitled to receive benefits.

Young Spouses

Young-spouse beneficiaries are broken into two categories, young spouses of retired workers and 

young spouses of deceased workers (also referred to as mother-survivor and father-survivor 

beneficiaries). To estimate the number of young spouses of retired workers, the married population 

under age 70 is multiplied by the probability that:

• The primary account holder is age 62 or older,

• The young spouse has an entitled child (under age 16 or a disabled adult) in their care, 

and

• The young spouse is not already receiving benefits based on another child in their care.

To estimate the number of young spouses of deceased workers, the population of widowed and 

divorced spouses under age 70 is multiplied by the probability that:

• The primary account holder is deceased,

• The young spouse has an entitled child (under age 16 or a disabled adult) in their care,

• The young spouse is not already receiving benefits based on another child in their care, 

and

• The young spouse has not remarried.
Appendices

86

As with all categories, an additional factor is applied to account for other eligibility requirements, 

such as ensuring that the young spouse is not entitled to a widow(er) benefit and is not receiving a 

retired-worker benefit based on his/her own earnings.38

g. Awards

The Awards Module uses a stratified sample of newly entitled worker beneficiaries with their 

earnings histories. A one-percent sample is used for OASI beneficiaries and a five-percent sample 

is used for DI beneficiaries. In addition, the Awards Module receives the number of covered 

workers (historical and projected) from the Economics Module, the population (historical and 

projected) from the Population Module, and fully insured rates (historical and projected) from the 

Insured Model. The Awards Module also utilizes other historical and projected data from the 

Economics Module, such as average wage and average taxable earnings, to produce projected 

taxable earnings used in the calculation of future awards. Additionally, the Awards Module utilizes 

the distribution of the number of in-current-pay retired beneficiaries by age at retirement from the 

RSB Module to project the shifting of worker retirement to later ages relative to the initial 

entitlement sample. The Awards Module ultimately produces projected levels of benefits, in terms 

of average indexed monthly earnings (AIME), for those beneficiaries newly entitled by age, sex, 

and trust fund. These projected values are passed to the Cost Module.

Awards Sample

The sample of worker beneficiaries who are newly awarded in 2019 is the foundation of the 

Awards Module. The sample contains a total of 52,285 newly awarded beneficiaries: 25,807

retired-worker beneficiaries and 26,478 disabled-worker beneficiaries. This sample is a subset of 

the 10-percent sample of newly-entitled worker beneficiaries used for the intermediate alternative. 

Each record in the sample includes the worker’s history of covered earnings and taxable earnings 

under the OASDI program, as well as additional information such as date of birth, sex, type of 

benefit, month of entitlement, and eligibility year of the worker. This information allows us to 

compute the benefits and classify each beneficiary in the sample as either a retired-worker or a 

disabled-worker beneficiary. Some preliminary calculations made on the sample are utilized 

within the model. These include the sample’s covered worker rates, calculated separately for 

retired-worker and disabled-worker beneficiaries. These rates are determined using the sample’s 

earnings histories for 1951 through 2018. They are defined for each age group, sex, and trust fund 

as the ratio of (1) the number of beneficiaries with covered earnings to (2) the total number of 

beneficiaries. 

Components of the Awards Module

A goal of the Awards Module is to modify earnings histories and earnings levels in the sample to 

represent the earnings levels of worker beneficiaries who will be newly entitled in future years. 

38 Individuals must be age 69 or younger in order to receive a young-spouse benefit. Therefore, we exclude those who are in the 

age range to receive a young-spouse benefit but are receiving their own retired-worker benefit and are thus counted in the retiredworker calculation.
Documentation of Computer Program

87

These modified earnings levels are used to estimate future benefit levels. The modifications 

include:

• Covered Worker Rates: The earnings histories in the sample are modified to represent 

the projected covered worker rate of the sample in each projected entitlement year. To 

do this, the covered worker rates of current and future sample cohorts are calculated 

using data provided by both the Economics Module and the Population Module. Then 

the percentage changes from the historical to the projected covered worker rates39 are 

computed. The percentage changes are used to compute the number of years of earnings 

that need to be removed from, or added to, the sample for the projected sample earnings 

to meet the projected sample covered worker rates. 

• Contribution and Benefit Base: The earnings posted for beneficiaries in the sample 

are limited by the historical contribution and benefit base. Prior to 1975, the maximum 

annual amount of earnings on which OASDI taxes were paid was determined by ad hoc 

legislation. After 1974, the annual maximum level was legislated to be determined 

automatically, based on the increases in the national average wage index (AWI). Prior 

to these automatic contribution and benefit base increases, a relatively large portion of 

workers earned amounts above the contribution and benefit base. Additional legislation 

raising the annual maximum taxable level occurred in 1979, 1980, and 1981. In 

addition, the AWI used in the automatic calculation of the annual taxable maximum 

was modified in the early 1990s to include deferred compensation amounts. Hence, an 

adjustment must be made to incorporate earnings above the historical contribution and 

benefit base in the sample to reflect the taxable earnings levels for future samples.

• Average Taxable Earnings: Average taxable earnings (ATE) may grow at a different 

rate than the AWI. All projected earnings in the sample are adjusted to reflect the 

overall increase in the ATE of future cohorts as projected by the Economics Module.

The resulting adjusted earnings histories are used to compute AIMEs for future entitlements. The 

Awards Module calculates the distribution of the AIME values by entitlement age, sex, and 

projection year, and passes the results to the Cost Module. 

h. Cost

The Cost Module serves two broad purposes. The first is to compute the year-by-year progress of 

the combined OASI and DI Trust Funds for a 75-year projection period.40 The second is to produce 

the summary measures used to assess the long-range financial status of the OASDI program for 

the 75-year projection period. 

39 Changes in covered worker rates are determined as the ratio of the absolute difference in the rates to the potential difference in 

the rates.

40 Additionally, a projection for the 76th year is required to estimate the target fund, equal to the present value of the cost in the 76th 

year. See appendix B.5 for more information.
Appendices

88

Progress of Trust Funds

In order to determine the progress of the trust funds, the dollar amounts of income and cost are 

computed for each year. Income includes payroll tax revenue, taxation of benefits, and interest. 

Cost consists of scheduled benefits, administrative expenses, and the railroad retirement 

interchange. Once the values of income and cost have been determined for a given year, the amount 

of end-of-year asset reserves is determined by starting with beginning-of-year reserves, adding 

income and subtracting cost. Each simulation of the model projects the progress of the trust funds 

for 75 years. For any simulation, full scheduled benefits are assumed to be paid from the trust 

funds, even if the reserves of the trust funds become depleted at some point in the projection period. 

Payroll Tax Revenue

The OASDI payroll tax rate for the employer and employee, each, is currently specified 

as 6.2 percent, resulting in a combined employer/employee tax rate of 12.4 percent. 

Income to the trust funds from payroll taxes is first computed by multiplying the 

combined payroll tax rate by the effective taxable payroll from the Economics Module. 

Because there is a time lag between incurring and collecting payroll taxes, an 

adjustment is made when computing the annual revenue from payroll taxes.

Taxation of Benefits

Income from taxation of benefits in a given year is computed by applying a ratio to the 

projected benefit payments scheduled for that year. This ratio is the same as in the 

intermediate alternative. 

Interest

The nominal yield on new-issue special public-debt obligations held by the trust funds 

is computed by multiplying the increase in the Consumer Price Index by the increase 

in the real interest rate assumed for those special public-debt obligations. Both of these 

values are allowed to vary and are obtained from the Assumptions Module. The 

nominal yield on the combined OASI and DI Trust Funds is computed in the 

Assumptions Module as a 7-year moving average of the nominal yields on new-issue 

special public-debt obligations held by the trust funds.

The amount of interest on trust fund asset reserves at the beginning of the year is 

calculated by multiplying the nominal yield by the amount of these reserves. 

Additionally, interest is calculated for amounts that enter and leave the trust funds 

during the year. Dollar amounts of tax income, taxation of benefits, scheduled benefits, 

railroad retirement interchange, and administrative expenses are exposed to, or from, 

the point in the year at which, on average, they are received or paid out. The amount of 

interest on the trust funds is then obtained by multiplying the net exposed amounts by 

the nominal yield for the appropriate fund.

Scheduled Benefits

A key component in projecting the amount of benefit payments for a given year is the 

projection of average benefits of worker beneficiaries who are newly awarded and 
Documentation of Computer Program

89

those who are in current-payment status. The primary insurance amount (PIA) formula 

factors are as specified by law. The two bend points for 2023 are indexed by the 

increase in the average wage, as supplied by the Economics Module.

The Cost Module uses a starting average benefit matrix for worker beneficiaries in 

current-payment status based on a 100-percent sample of the Master Beneficiary 

Record (MBR)41 as of December 31, 2022. For each sex, the starting matrix of retiredworker average benefits is broken out by age of entitlement to benefits (62 through 

70+) and by current age (62 through 95+). The average benefits of retired workers who 

have converted from disability status are classified in a separate age of entitlement

category. Similarly, for each sex, the starting matrix of disabled-worker average 

benefits is broken out by benefit duration (0 years through 9+ years) and by current 

age (20 through 66).

The Cost Module updates this starting matrix each year of the projection. First, average 

benefits are computed for worker beneficiaries who are newly awarded during the year. 

To compute these amounts, the distributions of AIME levels by age and sex for OASI 

and DI are obtained from the Awards Module. The average benefit for each age and 

sex is computed by applying this distribution to the intervals of potential AIME dollars, 

and then converting to the average PIA. Benefit amounts are determined by applying 

adjustment factors necessitated by provisions in law (e.g., actuarial reduction factors 

and delayed retirement credits). To complete the update of the matrix, two adjustments 

are made to the benefit levels of those who were awarded benefits in a previous year. 

The first is increasing benefit levels due to the COLA, as received from the Economics 

Module. The other adjustment is to reflect earnings after entitlement and mortality

being higher for beneficiaries with lower benefits. Average benefits for worker 

beneficiaries are determined using the updated benefit matrix and the beneficiary 

matrix from the RSB and DI Modules.

Average benefits of auxiliary beneficiaries are computed by using the historical trend 

of the level of these benefits, as a percentage of either the average PIA or monthly 

benefit amount (MBA) of the primary worker beneficiary. The dollar amounts of these 

average benefits are computed by applying these factors to the appropriate average PIA 

or MBA projection. Computations are made by sex and type of auxiliary beneficiary.

Finally, the aggregate value of benefits scheduled to be paid is the sum of the products 

of average benefits and the numbers of beneficiaries by type of benefit with an 

adjustment to account for benefits paid on a retroactive basis. The numbers of 

beneficiaries by type of benefit are from the RSB and DI Modules.

Administrative Expenses and Railroad Retirement Interchange

The projection of administrative expenses and the railroad retirement interchange uses 

the same methodology as in the intermediate alternative.

41 The MBR is a complete record of beneficiaries which contains items such as the social security number, beneficiary 

identification code, date of birth, date of entitlement, etc.
Appendices

90

Year-by-Year Actuarial Measures

Once the various components of income and cost have been computed for a given year, 

the Cost Module computes the annual cash-flow measures and the trust fund ratio for 

that year. The annual cash-flow measures include income rates, cost rates, and annual 

balances (all as a percentage of taxable payroll and as a percentage of GDP).

Long-Range Actuarial Measures

The Cost Module computes various summary measures that are used to assess the financial status 

of the OASDI program during the projection period. These measures summarize, on a presentvalue basis, the projected annual income and cost components of the combined OASI and DI Trust 

Funds over the 75-year period. The assumed nominal yield on trust fund asset reserves is used to 

discount the annual income and cost components. The summary measures include the summarized 

income rate, summarized cost rate, and actuarial balance, all as percentages of taxable payroll and 

as percentages of GDP. An additional important summary measure is the 75-year open group 

unfunded obligation. All of these summary measures are stored for each simulation of the trust 

funds. Additional details concerning the calculations of these measures are given in appendix B.

The Cost Module also stores the years during the 75-year projection period that relate to certain 

trust fund events. For each simulation of the trust funds, the following special years are determined, 

if applicable: (1) the first year trust fund reserves are depleted, (2) the first year trust fund reserves 

are depleted and remain depleted throughout the projection period, (3) the first projected year that 

cost exceeds non-interest income, (4) the first projected year that cost exceeds non-interest income 

and remains in excess through the projection period, and (5) the first year that cost exceeds total 

income.

i. Summary Results

The Summary Results Module receives data from the Assumptions, Population, and Cost Modules 

for each of the stochastic simulations. It then computes the estimated probability distributions of 

both annual data and summarized data. 

Annual Data

The Summary Results Module produces the estimated distributions of projected data available on 

an annual, or year-by-year, basis. Such data include the annual variables that are projected in the 

Assumptions Module using standard time-series modeling. In the Population Module, life 

expectancy data are computed and sent to the Summary Results Module for analysis. In addition, 

annual data projected in the Cost Module and sent to the Summary Results Module for analysis 

include trust fund ratios, income and cost rates, and covered workers per beneficiary. For each of 

the stochastic simulations, the data for a given year are sorted using Heapsort (Press, et al., 2001). 

The probability distributions for each year are then computed. For each piece of data (e.g., the total 

fertility rate), the Summary Results Module computes the smoothed empirical estimates 

(Klugman, et al., 1998) of the 2.5th, 5th, 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, 90th, 95th, and 

97.5th percentiles. The Summary Results Module also computes various 75-year and final 50-year 

averages (arithmetic or geometric, as appropriate) for each piece of annual data.
Documentation of Computer Program

91

Summarized Data

Some data are only available as a summary measure for each of the stochastic simulations.

Examples are the actuarial balance, summarized income and cost rates, the open group unfunded 

obligation, and the depletion year of the combined OASI and DI Trust Fund reserves. The 

Summary Results Module processes these data, most of which are passed to the module at the end 

of each run. At the conclusion of all of the stochastic simulations, the module computes the 

estimated frequency distributions for each piece of data. It computes the mean, median, and bounds 

for 95-, 90-, 80-, 60-, 40-, and 20-percent frequency intervals.
Appendices

92

B. PRINCIPAL MEASURES OF FINANCIAL STATUS

Three types of financial measures are useful in assessing the actuarial status of the combined OASI 

and DI Trust Funds under the financing approach specified in current law: (1) annual cash-flow 

measures, including income rates, cost rates, and annual balances; (2) trust fund ratios; and (3) 

summary measures such as actuarial balances and open group unfunded obligations. In assessing 

the financial condition of the program, particular attention should be paid to the level of the annual 

balance at the end of the long-range period, and the time at which the annual balance may change 

from positive to negative. The ratio of beneficiaries to covered workers on a year-by-year basis (a 

program ratio rather than a financial ratio), provides a useful comparison to the cost of the OASDI 

program over the entire 75-year valuation period.

Also important is the pattern of projected trust fund ratios. The trust fund ratio is the proportion 

of a year’s projected cost that can be paid with the funds available at the beginning of the year. 

Particular attention should be paid to the amount and year of maximum trust fund ratio, to the trust 

fund reserve depletion year (which is the year prior to which the trust fund ratio becomes negative), 

and to the stability of the trust fund ratio in cases where the ratio remains positive at the end of the 

long-range period.

The final measures discussed in this appendix summarize the total income and cost over valuation 

periods that extend through 75 years. These measures indicate whether projected income will be 

adequate for the period as a whole. The first such measure, the actuarial balance, indicates the size 

of any shortfall (when negative) as a percentage of the taxable payroll over the period. The second, 

the open group unfunded obligation, indicates the size of any shortfall in present-value discounted 

dollars. 

If the 75-year actuarial balance is zero or positive, then the trust fund ratio at the end of the period, 

by definition, will be at least 100 percent, and financing for the program is considered to be 

adequate for the 75-year period as a whole. (Financial adequacy for each year is determined by 

whether the trust fund is non-negative throughout the year.) Whether or not financial adequacy is 

stable in the sense that it is likely to continue for subsequent 75-year periods in succeeding reports 

is also important when considering the actuarial status of the program. One indication of this 

stability is the behavior of the trust fund ratio at the end of the projection period. If projected trust 

fund ratios for the last several years of the long-range period are positive and are also stable, or 

rising, then it is likely that subsequent Trustees Reports will also show projections of financial 

adequacy (assuming no changes in demographic, economic, and programmatic assumptions). 

The remaining portion of this appendix gives an overview of the trust fund operations and 

discusses these basic measures of assessing the actuarial status of the combined OASI and DI Trust 

Funds. Intermediate values from the 2023 deterministic model are often given to aid in the 

discussion.

1. Trust Fund Operations

Starting with the dollar level of asset reserves in a given year, the combined OASI and DI Trust 

Funds receive income from payroll taxes, income from taxation of benefits, and interest income. 
Principal Measures of Financial Status

93

In turn, scheduled benefits, administrative expenses, and the net financial interchange to the 

Railroad Retirement Board are all paid from the combined OASI and DI Trust Funds. Hence, the 

dollar value of reserves at the end of a given year is equal to the dollar value of reserves at the 

beginning of the year, plus payroll taxes, plus income from taxation of benefits, plus interest 

income, less scheduled benefits, less administrative expenses, less net financial interchange to the 

Railroad Retirement Board. 

2. Annual Income Rates, Cost Rates, and Balances 

Basic to the consideration of the long-range actuarial status of the trust funds are the concepts of 

income rate and cost rate, each of which is expressed as a percentage of taxable payroll. The annual 

income rate is the ratio of income from revenues (payroll tax contributions and income from the 

taxation of benefits) to the OASDI taxable payroll for the year. The OASDI taxable payroll consists 

of the total earnings which are subject to OASDI taxes, with some relatively small adjustments. 

These adjustments include reflecting that individuals who work for more than one employer and 

have OASDI covered wages exceeding the contribution and benefit base during the year will be 

reimbursed for the payroll taxes they paid on wages in excess of the contribution and benefit base 

for that year. However, each employer pays, on behalf of each employee, payroll taxes on annual 

wages up to the contribution and benefit base, regardless of the amount of wages an employee 

receives from other employers. Because the taxable payroll reflects such adjustments as these, the 

annual income rate is defined as the sum of the OASDI combined employee-employer contribution 

rate (or the payroll tax rate) scheduled in the law and the rate of income from taxation of benefits 

(which is, in turn, expressed as a percentage of taxable payroll). As such, it excludes net investment 

income. The annual cost rate is the ratio of the cost (or expenditures) of the program to the taxable 

payroll for the year. In this context, the cost is defined to include benefit payments, administrative 

expenses, and net transfers from the trust funds to the Railroad Retirement program under the 

financial-interchange provisions. For any year, the income rate minus the cost rate is referred to as 

the annual balance for the year. Social Security’s cost as a percentage of the total U.S. economy 

(i.e., as a percentage of GDP) provides an additional perspective.

The year-by-year relationship of the income rates and cost rates illustrates the expected pattern of 

cash flow for the OASDI program over the full 75-year period. The pattern of the OASDI 

program’s estimated cost rate is generally much different than that of the income rate. The income 

rate generally increases only slightly during the next 75 years, from a little under 13.0 percent to a 

little under 13.5 percent, as income from taxation of benefits slowly increases. Under the 

intermediate alternative, the OASDI cost rate is estimated to exceed the income rate in every year 

of the projection period, as it has since 2010. The cost rate is projected to rise rapidly between 

2023 and 2035 as the baby-boom generation continues to reach retirement age. The projected 

continued reductions in death rates and relatively low birth rates will cause a significant upward 

shift in the average age of the population and will push the cost rate to about 18 percent of taxable 

payroll under the intermediate alternative. Annual deficits generally increase throughout the 

remainder of the 75-year projection period, reaching almost 5.1 percent of taxable payroll in 2078 

and decreasing slightly to 4.3 percent by 2097.
Appendices

94

3. Comparison of Workers to Beneficiaries

The primary reason that the estimated OASDI cost rate generally increases rapidly after 2023 is 

that the number of beneficiaries is projected to increase more rapidly than the number of covered 

workers. This occurs because the relatively large number of persons born during the baby-boom 

generation will reach retirement age, and begin to receive benefits, while the relatively small 

number of persons born during the subsequent period of low fertility rates will comprise the labor 

force. Based on the intermediate alternative, the number of covered workers per OASDI 

beneficiary declines from 2.7 in 2020 to 2.3 by 2040, and to 2.1 by 2097.

The impact of the demographic shifts on the OASDI cost rates can also be illustrated by 

considering the projected number of beneficiaries per 100 workers. As compared to the 2020 level 

of 37 beneficiaries per 100 covered workers, this ratio is estimated to be 47 in 2097 under the 

intermediate alternative. The trend of the OASDI cost rates is essentially the same as the trend of 

the number of beneficiaries per 100 covered workers over the 75-year projection period. This 

comparison of the trends emphasizes the extent to which the cost of the OASDI program as a 

percentage of taxable payroll is determined by the age distribution of the population. Because the 

cost rate is basically the product of the number of beneficiaries and their average benefit divided 

by the product of the number of covered workers and their average taxable earnings (and because 

average benefits rise at about the same rate as average earnings), it is to be expected that the pattern 

of the annual cost rates is similar to that of the annual ratios of beneficiaries to workers. 

4. Trust Fund Ratios 

Trust fund ratios are critical indicators of the adequacy of the financial resources of the Social 

Security program. The combined OASDI trust fund ratio for a given year is, by definition, the 

amount of asset reserves at the beginning of the year expressed as a percentage of cost for the year. 

In general, a yearly trust fund ratio of 100 percent or greater is desired. For any year in which the 

projected trust fund ratio is positive (i.e., the trust fund holds reserves at the beginning of the year) 

but is not positive for the following year, the trust fund is projected to become depleted during the 

year. Under current law, the OASI and DI Trust Funds do not have the authority to borrow, other 

than in the form of advance tax transfers. Therefore, depletion of the reserves in either fund during 

a year would mean that there are no longer sufficient funds to cover the full amount of benefits 

scheduled under current law. 

The trust fund ratio serves an additional important purpose in assessing the actuarial status of the 

program. If the projected trust fund ratio is positive throughout the long-range period and is either 

level or increasing at the end of the period, then it is likely that subsequent reports will continue to 

show projected financial adequacy for the long-range period. Under these conditions, the program 

has achieved sustainable solvency.

The OASDI program experienced negative cash flows in 2021 for the first time since 1982, and 

projected income, including interest, is projected to remain lower than cost. Under the intermediate 

alternative, the trust fund ratio for the combined OASI and DI Trust Funds declines from 204 

percent at the beginning of 2023 until the combined funds become depleted in 2034.
Principal Measures of Financial Status

95

5. Actuarial Balance, Summarized Rates, Open Group Unfunded Obligation

The deterministic model projects nominal dollar levels of trust fund reserves over a 75-year period. 

In order to assess the financial status of the OASDI program quantitatively, in particular, to view 

the program as a whole at a specified point in time, the notion of present value, or time-value of 

money, is required. An essential ingredient in computing present values is the interest rate. Under 

the intermediate alternative, the ultimate value of the annual nominal yield on the combined OASI 

and DI Trust Funds is 4.76 percent, achieved in 2046. In the short-range period, projected interest 

earned by new special issue securities, along with interest earned on the existing reserves of the 

trust funds, determine implicitly the nominal yield earned by the individual OASI and DI Trust 

Funds in years 2023-32. The annual nominal yield of the combined OASI and DI Trust Funds, 

calculated through a weighting process of the two individual funds, reaches the ultimate value in 

2046 under the intermediate alternative. The interest rate on the combined funds is used for present 

value purposes.

The 75-year long-range actuarial balance (henceforth referred to as the actuarial balance) is a 

measure, as a single number, of the program’s financial status for the 75-year valuation period as 

a whole. It is essentially the difference between the income and cost of the program expressed as 

a percentage of taxable payroll summarized over the valuation period. When the actuarial balance 

is negative, the magnitude of the negative actuarial balance, the actuarial deficit, can be interpreted

as the percentage that would have to be added to the current law income rate in each of the next 

75 years, or subtracted from the cost rate in each year, to bring the funds into actuarial balance. 

Under the intermediate alternative, there is an actuarial deficit of 3.61 percent of taxable payroll 

for the combined OASI and DI Trust Funds.

Summarized values for the full 75-year period are useful in analyzing the long-range adequacy of 

financing for the program over the period as a whole under current law. The summarized income 

rate for the 75-year valuation period is equal to the present value of tax income, plus the beginningof-year 2023 reserves, expressed as a percentage of the present value of taxable payroll. Similarly, 

the summarized cost rate for the 75-year valuation period is equal to the present value of the cost 

for the 75-year period, plus the present value of the 76th year’s cost (the so-called “target fund”), 

expressed as a percentage of the present value of taxable payroll. These summarized rates are 

useful for comparing the total cash flows of tax income and expenditures, as an indicator of the 

degree to which accumulated reserves and tax income during the period are sufficient to meet the 

cost estimated for the long-range period. The OASDI long-range actuarial balance is defined to be 

the difference between the summarized 75-year income rate and the summarized 75-year cost rate.

The summarized income rate and summarized cost rate, as described above, are expressed as a 

percentage of the present value of taxable payroll. However, it is also useful to provide summary 

measures expressed in terms of the total economy of the United States, or GDP. Specifically, 

summary values for the OASDI program can be expressed as a percentage of the present value of 

GDP (instead of payroll). The summarized long-range (75-year) balance as a percentage of GDP 

for the OASDI program is estimated to be -1.27 percent of GDP under the intermediate alternative. 

Conceptually, the open group unfunded obligation is the present value of the cash deficits observed 

over the 75-year period less the accumulated trust fund reserves at the beginning of the period. In 
Appendices

96

particular, it is the present value of the cost of the program, less the sum of the present value of tax 

income and the reserves on hand at the beginning of the 75-year valuation period. Under the 

intermediate alternative, the open group unfunded obligation was estimated to be $22.4 trillion.
Time-Series Analysis

97

C. TIME-SERIES ANALYSIS

Time-Series Modeling

Time-series analysis is a standard set of projection techniques in econometric modeling. The 

equations used in the OSM to project the assumptions are fitted using these techniques. This 

appendix provides details about the models in general, presents statistical methods employed to 

test these models, and underscores the nuances inherent in the method of determining the 

equations. The reader may wish to consult Box, Jenkins, and Reinsel (1994), or Hamilton (1994),

for a standard presentation of this material. A more elementary reference is Pindyck and Rubinfeld 

(1998). 

Stationary Time Series

A time series Yt is covariance stationary if neither the mean nor variance depends on time t:

• Yt has a constant mean μ.

• For fixed k and all t, the covariance of Yt and Yt-k is γγkk , a constant depending on k.

In particular, the variance of Yt is always equal to σσ2 = γγ0. A different concept is that of strict 

stationarity. This means that any k values of the time series have the same joint distribution as any 

other set of k values of the time series. If a time series Yt is strictly stationary, then it is stationary. 

Henceforth, when we say that a time series Yt is stationary we mean that it is strictly stationary. 

However, it is often more intuitively helpful to think of a time series as stationary in the simpler 

sense. 

Stationarity is an important assumption in many estimation methods and approaches, because it is 

typically desirable that the estimated parameters depend only on the data values and not on the 

time at which the sample begins or ends.

Gaussian White Noise Process

Suppose that εt, the error term at time t, is normally distributed with mean zero and constant 

variance σσεε

2 . If εr and εs are uncorrelated for rr ≠ ss, then the series εt is called a Gaussian white 

noise process. A Gaussian white noise process is stationary.

Random Walk

The simplest of all time-series models is the random walk. Here, if Yt is the series to be estimated, 

then the random walk process is given by YYtt = YYtt−1 + εεtt. The error term series, εt , is a white noise 

process. The forecast error variance increases as a linear function of the forecast lead time, l. In 

other words, the variance of the l-step-ahead forecast error is ll εε

2 . The uncertainty spread in these 

forecasts will therefore widen as l increases. In other words, a random walk is not stationary.

Moving Average (MA) Models

A time series is called a moving average model of order q, or simply an MA(q) process, if
Appendices

98

YYtt = μμ + εεtt + θθ1εεtt−1 + θθ2εεtt−2 + ⋯ + θθqqεεtt−qq.

In this equation, θθ1,... , θθqq are the moving average parameters and μ is the process mean. The error 

term series, εt, is a white noise process. An MA(q) process is always stationary for a finite q.

Autoregressive (AR) Models and Checking for Stationarity

A time series is called an autoregressive model of order p, or simply an AR(p) process, if

YYtt = φφ0 + φφ1YYtt−1 + φφ2YYtt−2 + ⋯ + φφppYYtt−pp + εεtt .

In this equation, φφ1, ... φφpp are the autoregressive parameters and φφ0 is a drift term. The naming of 

this model is apt because the coefficients of the time-series equation are obtained by regressing 

the equation on itself or, more precisely, on its own p lagged values. The error term series, εt, is a 

white noise process. If this time series is stationary, then the mean μ of this process is computed 

to be

μμ = φφ0

1 − φφ1 − φφ2−. . .−φφpp

.

There are two ways to determine whether an AR(p) process is stationary. One way involves finding 

the characteristic roots of the characteristic equation. This involves re-writing the AR(p) process 

as a lag polynomial:

�1 − φφ1LL − φφ2LL2 − ⋯ φφppLLpp�yytt = εεtt

and substituting the lag operator L with a complex variable z and setting the polynomial to be 

equal to zero: 

1 − φφ1zz − φφ2zz2 − ⋯ − φφppzzpp = 0.

The AR(p) process is stationary if all the values of z that solve the equation above lie outside the 

unit circle.

The alternative (and equivalent) way is to re-write the AR(p) process in matrix notation and 

determine whether all the eigenvalues of the p×p matrix lie inside the unit circle. For example, the 

matrix notation for an AR(2) process would be:

� YYtt

YYtt−1

� =�

φφ0

0 � + �

φφ1 φφ2

1 0 ��

YYtt−1

YYtt−2

� + �

εεtt

0�

Defining A as the 2×2 matrix in the equation above, λ is an eigenvalue of A and x is an eigenvector 

if:

= λλ => ( − λλII2)xx = 0
Time-Series Analysis

99

=> − λλIIpp is not invertible => det( − λλII2) = 0

where 

det( − λλII2) = det � �

φφ1 φφ2

1 0 � − �

λλ 0

0 λλ

� � = det ��

φφ1 − λλ φφ2

1 −λλ

��

= λλ2 − φφ1λλ − φφ2 = 0.

Solving the above equation for λ:

λλii = �φφ1 ± �(φφ1

2 + 4φφ2) �

2 = 1,2

If all the eigenvalues (λλ1 and λλ2 in this example) lie inside the unit circle, then the AR(2) process 

is stationary.

Autoregressive Moving Average (ARMA) Models

As its name indicates, an autoregressive moving average model of order (p,q), or simply an 

ARMA(p,q) model, is a natural combination of an autoregressive and moving average model. The 

equation which specifies an ARMA(p,q) model takes the form

YYtt = μμφφ + φφ1yytt−1 + φφ2yytt−2 + ⋯ + φφppyytt−pp + εεtt + θθ1εεtt−1 + θθ2εεtt−2 + ⋯ + θθqqεεtt−qq.

An ARMA(p,q)series is stationary if and only if its autoregressive part is stationary.

Deviations Form of ARMA Model Equations

It is often more convenient to transform an ARMA model equation into the deviation form using 

the equation

yytt = YYtt − μμ,

where μ is defined, as above, to be the mean of the process. The transformed model may be written 

as

yytt = φφ1yytt−1 + φφ2yytt−2 + ⋯ + φφppyytt−pp + εεtt + θθ1εεtt−1 + θθ2εεtt−2 + ⋯ + θθqqεεtt−qq

and has a mean of zero.

Adding the original process mean to both sides of the equation produces

YYtt = μμ + φφ1yytt−1 + φφ2yytt−2 + ⋯ + φφppyytt−pp + εεtt + θθ1εεtt−1 + θθ2εεtt−2 + ⋯ + θθqqεεtt−qq.

The lagged variables are left in deviations form, and the constant term, μ, on the right-hand side is 

the process mean.
Appendices

100

Cholesky Decomposition

Suppose that ZZ⃗ = (ZZ1,... , ZZnn)′ is a vector of independently and identically distributed standard 

normal random variables. (If A is a matrix, then ′ denotes its transpose.) Suppose we want to use 

these random variables to obtain a random vector XX⃗ = (XX1,... , XXnn)′ from a multivariate normal 

distribution with mean μμ⃗ = (μμ1, ... , μμnn)′ and an n×n variance-covariance matrix VV = (σσiiii) [with 

σσiiii = CC CC�XXii,XX � for ,jj = 1, ... , ]. Since V is positive definite and symmetric, a standard result 

in linear algebra yields a lower triangular matrix, L, such that VV = LLLL′. The random vector XX =

μμ⃗ + LLZZ⃗ is then a random vector with the desired properties. The decomposition VV = LLLL′ is called

a Cholesky decomposition; see Atkinson (1989) for more details. In what follows we call the 

matrix L a Cholesky matrix.

For our applications, a Cholesky decomposition is used to convert a random vector, εε⃗, of 

independent standard normal variates to a multivariate normal distribution with mean μμ⃗ = 0�⃗ (the 

zero vector) and a variance-covariance matrix V obtained by using historical data. If L is a lower 

triangular Cholesky matrix associated with V, then the vector LLεε⃗ has the required multivariate 

normal distribution.

Vector Autoregressive Models

Vector autoregressive models allow the joint modeling of time-series processes. For the sake of 

simplicity, suppose that three variables yy1,tt , yy2,tt , aa yy3,tt depend on time t. Data may indicate 

that these variables may be related to each other’s past values. The simplest such case is when the 

relationship is limited to a 1-year lag, i.e., when yy1,tt , yy2,tt aa yy3,tt may be modeled in terms of 

yy1,tt−1 , yy2,tt−1 , aa yy3,tt−1. In this case, a three-variable VAR(1) model takes the form 

�

yy1,tt

yy2,tt

yy3,tt

� = BB �

yy1,tt−1

yy2,tt−1

yy3,tt−1

� + �

εε1,tt

εε2,tt

εε3,tt

�

for some 3 x 3 matrix B.

Alternatively stated, if yy⃗tt = �yy1,tt, yy2,tt, yy3,tt�

′ and εε⃗tt = (εε1,tt, εε2,tt, εε3,tt)′ then the model takes the 

form yy⃗tt = BByy⃗tt−1 + εε⃗tt for some 3×3 matrix B.

The k-variable VAR(p) model, with p lags, naturally extends from this. If yy⃗tt =

�yy1,tt, yy2,tt,... , yykk,tt�

′

and εε⃗tt = (εε1,tt, εε2,tt,... , εεkk,tt)′ then the k-variable VAR(p) model takes the form 

yy⃗tt = BB1yy⃗tt−1 + BB2yy⃗tt−2+. . .+BBppyy⃗tt−pp + εε⃗tt for k×k matrices BB1, ... ,BBpp.

Adapting the Vector Autoregressive Model to Parameter Uncertainty

This section describes the procedure to obtain parameters adjusted for parameter uncertainty for 

the three-equation vector autoregressive model with two lags that was used to model the 

unemployment rate UUtt, the inflation rate IItt, and the real interest rate tt . 
Time-Series Analysis

101

The three-equation VAR with two lags can be written in matrix notation as:

�

UUtt

IItt

tt

� = �

μ

μII

μ

� + A1 �

UUtt−1 − μ

IItt−1 − μII

tt−1 − μ

� + A2 �

UUtt−2 − μ

IItt−2 − μII

tt−2 − μ

� + �

εε1tt

εε2tt

εε3tt

�

where

A1 = �

aa111 aa121 aa131

aa211 aa221 aa231

aa311 aa321 aa331

�

A2 = �

aa112 aa122 aa132

aa212 aa222 aa232

aa312 aa322 aa332

�

As there are 21 parameters to estimate (μ , μII, μ , and the parameters in matrices A1 and A2), there 

will be 21 parameters that need adjustments for full parameter uncertainty. When applying 

parameter uncertainty only around the expected means, only the adjustments on the three mean 

terms are used and the adjustments to the constants in the A1 and A2 matrices are ignored.

For full parameter uncertainty, the lower triangular Cholesky matrix of the 21 x 21 parameter 

variance-covariance matrix is multiplied by a 21 x 1 vector of standard normal random numbers 

to obtain a 21 x 1 vector of adjustments which are added to the estimated parameters to produce a 

final set of parameters. 

To verify that the parameters with full parameter uncertainty pass the stationarity test, the 

maximum eigenvalue of the following matrix is calculated:

⎣

⎢

⎢

⎢

⎢

⎡

aa�111 aa�121 aa�131 aa�112 aa�122 aa�132

aa�211 aa�221 aa�231 aa�212 aa�222 aa�232

aa�311 aa�321 aa�331 aa�312 aa�322 aa�332

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0 ⎦

⎥

⎥

⎥

⎥

⎤

In the matrix above, the aa�iiii terms denote the autoregressive parameters adjusted for parameter 

uncertainty. If the maximum eigenvalue of the matrix lies within the unit circle, then the 

parameters pass the stationarity test and are retained for stochastic simulation. However, when the 

model is run with parameter uncertainty only around the mean terms, these stationarity tests are 

unnecessary.
Appendices

102

D. MONTE CARLO SIMULATION

A standard reference for this appendix is Knuth (1997). In choosing a random number generator, 

it is essential to consider the following:

• the generator should have the correct statistical properties

• the stream should be easy to reproduce

• the cycle length should be long

• computational efficiency and storage needs should be met

• the generator should be portable across platforms

• it should be easy to generate separate streams

Given a seed I0 and a judicious choice of a and m, the linear congruence II +1 = aaII (mm mm) will 

generate a sequence of numbers. The values aa = 16,807 and mm = 231 − 1 have the property that, 

for any nonzero initial seed, a large number of variables may be generated before cycling occurs. 

Upon division by m, a sequence of variates Uj of a uniform distribution on the interval (0,1) is 

generated. This linear congruential generator with these values of a and m is the so-called 

“Minimal Standard” generator of Park and Miller and appears in Press et al. (2001). As explained 

there, the value of 0 must never be allowed as the initial seed. Also, since a and m are relatively 

prime, the value of 0 will never occur provided that the initial seed is nonzero. This choice of a 

and m gives a cycle of maximum possible length 231 − 2. The Minimal Standard generator can be 

modified so as to remove low-order serial correlations inherent in the random numbers produced 

by that method. The method is adapted from Press et al. (2001) and uses the Park-Miller algorithm 

along with a Bays-Durham shuffle. This is the process the OSM uses to generate uniform (0,1) 

variates. There are other methods which will further eliminate such serial correlations. We chose 

this particular method to balance the need to remove such serial correlations with the expense of 

run-time.

To generate the standard normal (mean 0 and variance 1) variates used to generate the random 

error terms for the equations, one may transform uniform variates using the Box-Muller method: 

generate two independent uniform (0,1) variates, U1 and U2 , and compute 

XX = �−2lnUU1 ⋅ cos (2ππUU2) and YY = �−2lnUU1 ⋅ sin (2ππUU2). In order to avoid the use of 

trigonometric functions, the Polar method is used. This method, as a preliminary step, transforms 

the two uniform (0,1) variates U1 and U2 to uniform (–1,1) variates V1 and V2 (using the 

transformation V = 2U – 1). If SS = VV1

2 + VV22, then XX = VV1�−2lnSS

SS

and YY = VV2�−2lnSS

SS

will be 

two independent random draws from a standard normal distribution, provided that S is less than 

one. (If not, then this process is repeated until it is.) Although this method produces two standard 

normal variates, the first one (X) is used and the second one (Y) is discarded. Each equation is 

separately seeded to have its own random number stream. 
Parameter Uncertainty Implementations

103

E. PARAMETER UNCERTAINTY IMPLEMENTATIONS

There are two ways to implement parameter uncertainty: (1) parameter uncertainty around the 

mean and (2) full parameter uncertainty. As mentioned in section II.B, under full parameter 

uncertainty, if a particular set of random numbers results in AR coefficients that are nonstationary

for a particular assumption, this draw is ignored and not used as one of the 5,000 simulations. For 

more discussion on nonstationarity, see section II.B and appendix C. However, as a quick 

summary, changing the coefficients requires a stationarity recheck, because stationarity is required 

for the equations in the OSM. If an equation is nonstationary, summary measures can increase in 

magnitude without bounds, resulting in implausible scenarios. Additionally, the equations are 

estimated assuming stationarity and thus, nonstationary series would be inconsistent with the 

estimation procedure. For these reasons, stationarity is a requirement for ARMA and VAR 

equations in the OSM. Nonstationarity is determined as having a maximum absolute value 

eigenvalue of greater than or equal to 1.0. Table V.2 shows the percentage of draws that result in 

nonstationarity when using full parameter uncertainty.

Table V.2 – Nonstationary Draw Percentages Under Full Parameter Uncertainty

Assumption1

Percentage 

Nonstationary

Total Fertility Rate 9.1

LPR New Arrival Immigration 0.6

Legal Emigration Rate 0.0

Other-than-LPR Immigration 0.6

Transfer Rate 0.0

Economic VAR Framework2 6.7

Male Disability Incidence Rate 21.1

Female Disability Incidence Rate 5.1

Male Disability Recovery Rate 5.5

Female Disability Recovery Rate 6.1

1 The 42 age-group/sex mortality assumptions are not listed. However, none of the draws in any of the simulations resulted in 

equations that were nonstationary. The real average covered wage is also not listed because it is based on unemployment only and 

therefore only the percentage of nonstationary draws for unemployment (i.e., the economic VAR framework) is relevant for this 

assumption.

2 The economic vector autoregression (VAR) framework determines the unemployment rate, inflation rate, and real interest rate 

simultaneously.

As shown in table V.2, several assumptions have a relatively high number of nonstationary draws 

under full parameter uncertainty. These are the total fertility rate, economic VAR framework, and 

both sets of disability rates (incidence and recovery). Particularly notable is the 21.1 percent of 

draws that are nonstationary for the male disability incidence rate assumption. For each of these 

assumptions, figures are presented below that correspond to those shown in section IV.A, but also 

contain additional lines for full parameter uncertainty. In addition, for comparison, a figure is 

presented for an assumption that does not have a large number of nonstationary draws (other-thanLPR immigration).
Appendices

104

Removing the nonstationary draws may lead to biased distributions of results. One would expect 

full parameter uncertainty to have a wider 95-percent frequency interval than parameter 

uncertainty for the expected mean in the short term, which can be seen in all of the figures below.

However, removing nonstationary draws can result in a skewed distribution or excess kurtosis 

(more weight of the distribution in the tails relative to a normal distribution). A skewed distribution 

or excess kurtosis can occur because draws that would result in values at the upper or lower portion 

of the distribution are removed, thus biasing the distribution in the opposite direction. One reason 

to use parameter uncertainty for the expected mean (rather than full parameter uncertainty) is that 

normally distributed frequency distributions (with skew and excess kurtosis both close to zero) are 

desired in simulations like the OSM.

Another reason to use parameter uncertainty for the expected mean is that most of the additional 

variation in the resulting distributions comes from parameter uncertainty around the mean and not 

from parameter uncertainty of the coefficients. In fact, one can show that in the long run, full 

parameter uncertainty and parameter uncertainty for the expected mean have cumulative averages 

that converge under certain assumptions (including certain conditions of normality). This 

convergence is evident in figure V.1b, which shows cumulative average values for other-than-LPR

immigration. Other-than-LPR immigration is an example of an assumption with a low number of 

nonstationary draws. However, convergence is also evident in figure V.2b, which shows 

cumulative average values for the total fertility rate, even with the obvious issues of skew and 

excess kurtosis in the annual values in figure V.2a. Convergence is not evident for the assumptions 

with a high number of nonstationary draws, as shown in figures V.3b, V.4b, V.5b, and V.6b.

However, we know that these full parameter uncertainty distributions are potentially distorted due 

to the dismissal of all the nonstationary draws. Note in particular figures V.6a and V.6b show a 

very high distortion under full parameter uncertainty with male disability incidence rates. The fact 

that male disability incidence rate shows the highest distortion should not be surprising given this 

assumption has the highest number of nonstationary draws of all variables in table V.2.
Parameter Uncertainty Implementations

105

Figure V.1a – Other-than-LPR Immigration Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1999-2097

Figure V.1b – Other-than-LPR Immigration Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1999-2097
Appendices

106

Figure V.2a – Total Fertility Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1917-2097

Figure V.2b – Total Fertility Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1917-2097
Parameter Uncertainty Implementations

107

Figure V.3a – Unemployment Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097

Figure V.3b – Unemployment Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097
Appendices

108

Figure V.4a – Inflation Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097

Figure V.4b – Inflation Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097
Parameter Uncertainty Implementations

109

Figure V.5a – Real Interest Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097

Figure V.5b – Real Interest Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1961-2097
Appendices

110

Figure V.6a – Male Disability Incidence Rate Annual Values,

with Full Parameter Uncertainty Added, Calendar Years 1970-2097

Figure V.6b – Male Disability Incidence Rate Cumulative Average Values,

with Full Parameter Uncertainty Added, Calendar Years 1970-2097

The summary values for parameter uncertainty for the expected mean and full parameter 

uncertainty are also of interest. Below are figures showing the trust fund ratios (figure V.7) and 

cost rates as a percentage of taxable payroll (figure V.8) for full parameter uncertainty, parameter 

uncertainty for the expected mean, and no parameter uncertainty.
Parameter Uncertainty Implementations

111

Figure V.7 – Annual Trust Fund (Unfunded Obligation) Ratios, Calendar Years 2023-97

Figure V.8 – Annual Cost Rates, Calendar Years 2023-97

As expected in figures V.7 and V.8, the ranges with parameter uncertainty for the expected mean 

are narrower than those with full parameter uncertainty.
Appendices

112

Finally, data about the actuarial balance is shown in table V.3.

Table V.3 – Actuarial Balance (as a Percentage of Taxable Payroll) in the Stochastic Model

Simulation

Metric No Parameter 

Uncertainty

Parameter Uncertainty 

for the Expected Mean

Full Parameter 

Uncertainty

Median -3.72 -3.69 -3.80

Average -3.82 -3.82 -3.95

2.5th percentile -6.39 -7.17 -7.59

97.5th percentile -1.68 -1.18 -1.19

Standard Deviation 1.20 1.51 1.63

Looking at table V.3, the difference in the actuarial balance between the median (or average) with 

full parameter uncertainty and the median (or average) with no parameter uncertainty is larger than 

the difference between the median (or average) with parameter uncertainty for the expected mean 

and the median (or average) with no parameter uncertainty. Without any parameter uncertainty, 

the 95-percent frequency interval ranges from -6.39 percent of taxable payroll to -1.68 percent of 

taxable payroll, for a total range of 4.71 percent of taxable payroll. With parameter uncertainty for 

the expected mean, the 95-percent frequency interval is notably larger, ranging from -7.17 percent 

of taxable payroll to -1.18 percent of taxable payroll, for a total range of 5.99 percent of taxable 

ratio. Under full parameter uncertainty, the 95-percent frequency interval ranges from -7.59 

percent of taxable payroll to -1.19 percent of taxable payroll, for a total range of 6.40 percent of 

taxable payroll. This range is not much larger than the 95-percent frequency interval with 

parameter uncertainty for the expected mean. Similarly, the standard deviation of the actuarial 

balance is 1.20 percent of taxable payroll without any parameter uncertainty but is 1.51 percent of 

taxable payroll with parameter uncertainty for the expected mean. Using full parameter uncertainty 

leads to a slightly larger standard deviation of 1.63 percent of taxable payroll. Figure V.9 shows 

the cumulative frequency distributions for the actuarial balance. Note that the line for full 

parameter uncertainty (yellow) is almost on top of the line for parameter uncertainty for the 

expected mean (purple).
Parameter Uncertainty Implementations

113

Figure V.9 – Actuarial Balance Cumulative Percentage Frequency of 5,000 Simulations

with Full Parameter Uncertainty Added

As explained above, there are complications inherent in implementing full parameter uncertainty. 

In particular, removing a large number of draws due to nonstationarity for certain assumptions can 

lead to skewed distributions and excess kurtosis. In addition, most of the effect of full parameter 

uncertainty is captured by having parameter uncertainty for the expected mean. Therefore, we 

decided to use parameter uncertainty for the expected mean for the current version of the OSM, 

beginning with the 2021 Trustees Report. We will continue to explore the feasibility of 

implementing full parameter uncertainty.
Appendices

114

F. EQUATION REGRESSION STATISTICS, PARAMETER ESTIMATES, AND 

BOUNDING STATISTICS

This section of the appendix contains the specification of the equations as they were estimated as 

well as the estimated parameter values, standard errors, and parameter variance-covariance 

matrices. The parameter values include, where applicable, the estimated autoregressive (AR) and 

moving average (MA) coefficients. In addition, the assumed annual projected values of each 

variable from the 2023 Trustees Report alternative II (e.g., FFtt

TTTT) are presented in a table in the 

second part of this appendix. For each equation presented here, there is a corresponding equation 

described in chapter II.

The standard errors of each regression were used to generate random variation in each of the 

variables. The random variation in one variable is usually generated independently of the random 

variation in another variable. However, the mortality improvement variables, economic VAR 

framework variables, disability incidence variables, and disability recovery variables were each 

generated in such a way that the random error terms were correlated. For each of these, a Cholesky 

decomposition was performed on a residual variance-covariance matrix. The correlated random 

error terms were then obtained by multiplying the lower triangular Cholesky matrix by a vector of 

standard normal random errors.

The parameter variance-covariance matrix from each estimated equation was used to introduce 

parameter uncertainty into the estimated parameters. For each parameter variance-covariance 

matrix, a Cholesky decomposition was performed, and adjustments to the estimated parameters 

were created by multiplying the lower triangular Cholesky matrix by a vector of standard normal 

random numbers. When applying parameter uncertainty for the estimated mean, as discussed in 

appendix E, only the adjustments to the mean term were added to the assumed annual projected 

values from the 2023 Trustees Report, and the other adjustments to the AR and MA parameters 

calculated by the multiplication above were set to zero. The adjusted means change from one 

simulation to the next but remain the same over the 75-year projection period for each simulation.

Some equations are bounded for reasonableness. The third part of this appendix displays the 

percentage of runs that are bounded. 

1. Regression Statistics

Total Fertility Rate

FFtt = μ + φφ1(FFtt−1 − μ) + φφ2(FFtt−2 − μ) + φφ3(FFtt−3 − μ) + φφ4(FFtt−4 − μ) + εεtt + θθ1εεtt−1,

Note: Because the total fertility rate must be non-negative, its lower bound was set at zero. An upper bound 

for the fertility rate was set at an effectively infinite value of 100,000.
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

115

Regression Mean Over

AR parameters MA parameters Historical Period Standard Error

φ1 = 1.961953 θ1 = -0.599494 μ = 2.302764 σσεε = 0.083837

φ2 = -1.456856

φ3 = 0.881344

φ4 = -0.405045

Cholesky decomposition of the parameter variance-covariance matrix from the estimated fertility 

equation, with the parameters in the following order: μ, φ1, φ2, φ3, φ4, θ1:

⎣

⎢

⎢

⎢

⎢

⎡

0.194726 0 0 0 0 0

0.068651 0.185479 0 0 0 0

−0.152632 −0.320726 0.152159 0 0 0

0.139752 0.149100 −0.286136 0.049555 0 0

−0.053947 −0.009164 0.137069 −0.048859 0.006084 0

−0.060572 −0.132401 −0.037497 −0.037215 −0.007022 0.077445⎦

⎥

⎥

⎥

⎥

⎤
Appendices

116

Rate of Decrease in Central Death Rate 

MMkk,tt = μkk + φφkk1(MMkk,tt−1 − μkk) + εεkk,tt,

Regression Mean Over

Group (k) Sex Age AR Parameter φφkk1 Historical Period μk

1 M Under 1 -0.321102 2.649174

2 F Under 1 -0.346321 2.614814

3 M 1 to 4 -0.415383 3.200755

4 F 1 to 4 -0.420574 3.258776

5 M 5 to 9 -0.230105 2.764799

6 F 5 to 9 -0.280705 2.767535

7 M 10 to 14 -0.231467 1.943431

8 F 10 to 14 -0.191579 2.100112

9 M 15 to 19 -0.180280 0.843267

10 F 15 to 19 -0.161958 1.716487

11 M 20 to 24 -0.165854 0.550723

12 F 20 to 24 -0.190482 1.429141

13 M 25 to 29 -0.205471 0.213847

14 F 25 to 29 -0.181090 1.030135

15 M 30 to 34 -0.202313 0.315454

16 F 30 to 34 -0.205214 1.144602

17 M 35 to 39 -0.181789 0.653697

18 F 35 to 39 -0.203269 1.268761

19 M 40 to 44 -0.082093 0.821361

20 F 40 to 44 -0.180428 1.296906

21 M 45 to 49 -0.129337 0.817384

22 F 45 to 49 -0.206066 1.164831

23 M 50 to 54 -0.071034 0.720460

24 F 50 to 54 -0.236083 1.110494

25 M 55 to 59 -0.023481 0.651642

26 F 55 to 59 -0.191735 1.006140

27 M 60 to 64 -0.076255 0.622407

28 F 60 to 64 -0.262722 1.001435

29 M 65 to 69 -0.171710 0.666208

30 F 65 to 69 -0.226507 0.990396

31 M 70 to 74 -0.255254 0.696362

32 F 70 to 74 -0.272665 0.970823

33 M 75 to 79 -0.244379 0.615691

34 F 75 to 79 -0.292959 0.856685

35 M 80 to 84 -0.327045 0.593276

36 F 80 to 84 -0.306926 0.758256

37 M 85 to 89 -0.244672 0.395173

38 F 85 to 89 -0.202911 0.508494

39 M 90 to 94 -0.258088 0.258191

40 F 90 to 94 -0.228274 0.385726

41 M 95 and Older -0.176282 0.076520

42 F 95 and Older -0.226741 0.247714
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

117

Cholesky decomposition of the residual variance-covariance matrix:

V = LL′ L = �

A1 B1 C1

A2 B2 C2

A3 B3 C3

� (L is a 42 x 42 matrix)

1 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

4.39 0 0 0 0 0 0 0 0 0 0 0 0 0

4.31 1.09 0 0 0 0 0 0 0 0 0 0 0 0

4.11 1.29 6.09 0 0 0 0 0 0 0 0 0 0 0

4.42 1.95 5.96 2.77 0 0 0 0 0 0 0 0 0 0

3.17 1.19 4.17 1.72 3.65 0 0 0 0 0 0 0 0 0

3.60 0.71 5.47 1.97 3.16 3.29 0 0 0 0 0 0 0 0

3.07 0.96 4.55 1.72 3.45 1.26 4.04 0 0 0 0 0 0 0

4.12 0.88 5.98 1.99 4.23 2.25 1.91 3.89 0 0 0 0 0 0

4.34 1.78 7.88 3.12 4.07 1.58 4.26 1.85 4.39 0 0 0 0 0

3.71 1.31 6.61 2.57 3.69 1.69 3.24 2.59 2.09 2.89 0 0 0 0

3.93 2.43 8.45 3.23 4.26 1.58 4.07 1.70 4.49 −0.70 3.39 0 0 0

4.62 1.79 8.12 3.68 4.22 2.31 4.25 2.42 3.21 2.07 0.41 3.39 0 0

5.10 2.65 10.16 4.42 5.17 2.49 4.88 2.77 5.40 0.49 1.81 2.01 2.99 0

5.08 2.45 9.92 4.46 4.86 2.78 4.76 3.31 4.08 2.29 0.18 3.96 1.63 2.24⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

2 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

4.92 2.57 8.85 3.47 4.32 2.02 5.12 2.57 4.24 0.72 1.16 1.86 2.25 0.53

4.15 2.05 7.54 3.22 3.85 2.20 3.93 2.29 2.88 1.40 0.40 2.41 0.95 1.42

3.52 1.50 5.55 1.92 2.54 0.78 3.74 1.41 2.32 0.71 0.81 0.20 1.32 0.33

2.71 1.08 4.42 1.76 2.12 1.01 2.85 1.19 1.39 1.10 0.60 0.94 0.15 0.82

2.43 0.95 2.90 0.41 1.07 0.09 2.52 0.57 1.22 0.72 0.81 −1.03 0.12 −0.02

1.95 0.59 2.47 0.52 1.01 0.48 1.97 0.64 0.86 1.07 0.76 −0.05 −0.34 0.30

1.87 0.40 1.68 −0.20 0.46 −0.20 1.94 0.32 0.80 0.53 0.53 −0.88 −0.06 0.10

1.53 0.25 1.81 −0.02 0.49 0.29 1.43 0.27 0.43 0.83 0.39 −0.12 −0.32 0.36

1.39 0.22 0.89 −0.42 0.03 −0.49 1.27 −0.07 0.51 0.65 0.66 −0.91 −0.23 0.24

1.08 0.44 1.49 −0.20 0.38 0.01 1.11 0.04 ]0.13 0.77 0.51 −0.24 −0.30 0.51

0.96 −0.01 0.59 −0.59 0.09 −0.29 1.07 −0.13 0.35 0.53 0.70 −1.12 −0.33 0.35

0.81 0.15 1.01 −0.24 0.04 0.04 1.05 −0.18 −0.18 0.76 0.57 −0.76 −0.44 0.43

0.74 0.18 0.45 −0.72 0.00 −0.14 1.29 0.07 0.03 0.41 0.33 −1.09 −0.27 0.38

0.58 0.13 0.69 −0.36 −0.16 0.10 1.19 −0.01 −0.28 0.65 0.22 −0.60 −0.45 0.52 ⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

3 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

0.66 −0.02 0.50 −0.47 −0.23 −0.21 1.10 −0.13 0.23 0.49 0.26 −1.10 −0.29 0.36

0.49 0.17 0.68 −0.52 −0.34 −0.01 1.07 0.05 −0.20 0.67 0.22 −0.77 −0.51 0.45

0.30 0.07 0.36 −0.78 −0.14 −0.07 1.02 −0.01 0.07 0.64 0.37 −1.18 −0.62 0.16

0.43 0.15 0.35 −0.58 −0.33 0.09 0.99 0.20 −0.36 0.85 0.48 −0.87 −0.84 0.45

0.48 −0.22 0.44 −0.72 −0.42 −0.28 0.98 −0.20 −0.22 0.53 0.23 −1.11 −0.63 0.44

0.54 −0.14 0.34 −0.68 −0.65 −0.09 0.94 −0.18 −0.52 0.56 0.34 −0.89 −0.71 0.59

0.37 −0.08 0.27 −0.81 −0.47 −0.22 1.00 −0.19 −0.30 0.59 0.37 −1.45 −0.92 0.32

0.49 0.06 0.22 −0.79 −0.65 −0.10 0.89 −0.29 −0.52 0.65 0.57 −1.16 −1.02 0.54

0.17 −0.32 0.07 −0.91 −0.55 0.14 1.04 0.03 −0.58 0.82 0.25 −0.94 −1.00 0.55

0.36 −027 −0.02 −0.97 −0.56 0.15 1.02 −0.05 −0.62 0.75 0.36 −0.99 −1.11 0.74

0.30 −0.23 0.03 −1.05 −0.58 0.36 0.95 0.17 −0.57 0.80 0.30 −0.94 −1.06 0.52

0.42 −0.16 0.02 −1.16 −0.61 0.30 1.08 −0.12 −0.67 0.88 0.42 −0.98 −1.17 0.65

0.40 −0.00 −0.04 −0.95 −0.74 0.78 0.83 0.34 −0.47 0.68 0.10 −1.04 −1.08 0.66

0.46 −0.10 0.13 −1.33 −0.75 0.62 1.05 0.08 −0.67 0.70 0.18 −1.13 −1.12 0.69⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤
Appendices

118

BB1 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

BB2 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

2.24 0 0 0 0 0 0 0 0 0 0 0 0 0

1.15 1.84 0 0 0 0 0 0 0 0 0 0 0 0

1.83 −0.02 2.49 0 0 0 0 0 0 0 0 0 0 0

0.77 0.74 1.62 1.72 0 0 0 0 0 0 0 0 0 0

1.74 −0.02 2.20 0.50 2.00 0 0 0 0 0 0 0 0 0

0.66 0.50 1.26 0.71 1.15 1.64 0 0 0 0 0 0 0 0

0.99 −0.08 1.83 0.62 1.31 0.42 1.53 0 0 0 0 0 0 0

0.34 0.38 1.09 0.54 0.73 0.56 0.82 1.25 0 0 0 0 0 0

0.75 −0.17 1.67 0.54 1.41 0.34 0.64 0.20 1.50 0 0 0 0 0

0.44 −0.00 1.01 0.67 0.58 0.71 0.28 0.49 0.95 1.22 0 0 0 0

0.90 0.07 1.27 0.58 1.26 0.24 0.86 0.23 0.75 0.16 1.28 0 0 0

0.59 0.27 0.97 0.96 0.54 0.42 0.56 0.30 0.31 0.37 1.11 1.08 0 0

0.88 −0.20 1.26 0.55 0.88 0.12 1.00 0.25 1.03 0.08 0.31 0.05 1.29 0

0.61 0.02 1.02 0.74 0.48 0.42 0.70 0.25 0.54 0.43 0.15 0.43 0.79 0.99⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

BB3 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡ 0.57 −0.05 1.25 0.64 1.15 0.21 0.66 0.03 0.81 0.04 0.66 0.25 0.66 0.11

0.31 0.36 0.89 0.73 0.72 0.42 0.56 0.10 0.51 0.21 0.54 0.58 0.26 0.47

0.71 −0.10 1.10 0.56 0.83 0.34 0.84 0.10 0.90 0.19 0.57 0.05 0.95 0.25

0.56 0.18 0.84 0.66 0.49 0.46 0.68 0.25 0.69 0.46 0.44 0.24 0.57 0.55

0.18 0.13 1.18 0.83 0.72 0.32 0.77 0.20 0.93 0.14 0.78 0.25 0.59 0.66

0.08 0.49 1.07 1.08 0.64 0.41 0.68 0.23 0.85 0.33 0.67 0.50 0.40 0.67

0.28 0.13 1.00 0.82 0.74 0.53 0.67 0.09 1.01 0.54 0.68 0.12 0.76 0.59

0.12 0.41 0.97 0.95 0.71 0.50 0.55 0.04 1.13 0.55 0.66 0.23 0.68 0.74

0.07 0.28 1.28 0.92 0.36 0.23 0.70 0.31 1.08 0.18 0.72 0.20 1.01 0.76

−0.21 0.52 1.37 1.19 0.42 0.10 0.91 0.16 1.18 0.41 0.61 0.11 0.96 0.90

0.02 0.38 1.39 0.93 0.40 0.22 0.56 0.30 1.00 0.25 0.64 0.28 0.96 0.80

−0.29 0.46 1.39 1.03 0.36 0.12 0.82 0.19 1.13 0.41 0.60 0.13 0.97 0.98

0.04 0.33 1.73 1.00 0.30 0.25 0.31 0.27 0.80 0.14 0.88 0.53 0.79 0.67

−0.08 0.49 1.57 1.02 0.27 0.16 0.61 −0.01 1.02 0.38 0.63 0.26 0.86 1.08⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

CC1 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

119

CC2 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

CC3 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

1.12 0 0 0 0 0 0 0 0 0 0 0 0 0

0.79 1.12 0 0 0 0 0 0 0 0 0 0 0 0

0.46 0.08 1.07 0 0 0 0 0 0 0 0 0 0 0

0.25 0.65 0.77 0.75 0 0 0 0 0 0 0 0 0 0

0.57 0.47 0.23 −0.05 1.09 0 0 0 0 0 0 0 0 0

0.44 0.91 0.10 0.50 0.73 0.78 0 0 0 0 0 0 0 0

0.72 0.30 0.55 0.26 0.35 0.12 1.17 0 0 0 0 0 0 0

0.39 0.48 0.63 0.67 0.25 0.42 0.87 0.78 0 0 0 0 0 0

0.70 0.56 0.35 0.31 0.71 0.36 0.63 0.40 1.37 0 0 0 0 0

0.63 1.00 0.22 0.63 0.65 0.48 0.72 0.73 0.90 0.87 0 0 0 0

0.62 0.57 0.29 0.36 0.70 0.40 0.72 0.52 1.45 0.00 0.58 0 0 0

0.62 1.04 0.14 0.63 0.63 0.52 0.80 0.73 0.91 0.84 0.24 0.39 0 0

0.48 0.67 0.12 0.64 0.60 0.53 0.96 0.75 1.58 −0.12 1.05 0.10 1.31 0

0.53 1.19 0.10 0.76 0.65 0.56 0.98 0.93 1.05 0.83 0.71 0.24 0.61 0.60⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

Cholesky decomposition of the parameter variance-covariance matrix from each of the estimated rate of 

decrease in central death rate equations, with the parameters in the following order: μκ, φφkk1:

Boys age 0

� 0.301443 0

−0.029782 0.095315�

Boys 1 to 4

�

0.475674 0

0.040191 0.077393�

Boys 5 to 9

�

0.481556 0

0.025627 0.092059�

Boys 10 to 14

�

0.578565 0

0.015952 0.085717�

Boys and Men 15 to 19

�

0.846049 0

0.028506 0.081654�
Appendices

120

Men 20 to 24

�

0.862519 0

0.037702 0.110139�

Men 25 to 29

�

1.046622 0

0.027024 0.062383�

Men 30 to 34

�

0.977386 0

0.028512 0.082742�

Men 35 to 39

�

0.675712 0

0.013943 0.146976�

Men 40 to 44

� 0.507667 0

−0.050636 0.149500�

Men 45 to 49

� 0.360972 0

−0.025556 0.108190�

Men 50 to 54

� 0.322319 0

−0.015266 0.070982�

Men 55 to 59

�

0.325910 0

0.016391 0.065924�

Men 60 to 64

� 0.300593 0

−0.008491 0.087176�

Men 65 to 69

�

0.273716 0

0.006429 0.106968�

Men 70 to 74

�

0.234159 0

0.021269 0.078799�

Men 75 to 79

�

0.246191 0

0.024471 0.080904�
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

121

Men 80 to 84

�

0.226873 0

0.020026 0.063280�

Men 85 to 89

� 0.268296 0

−0.006886 0.062842�

Men 90 to 94

� 0.268629 0

−0.004097 0.059072�

Men 95 and over

� 0.351433 0

−0.016281 0.064372�

Girls age 0

� 0.286609 0

−0.031943 0.083546�

Girls 1 to 4

�

0.512660 0

0.042046 0.071712�

Girls 5 to 9

�

0.555165 0

0.015179 0.065718�

Girls 10 to 14

�

0.694121 0

0.000339 0.081920�

Girls and Women 15 to 19

�

0.799380 0

0.018441 0.091412�

Women 20 to 24

�

0.989556 0

0.029893 0.068435�

Women 25 to 29

�

1.151190 0

0.028311 0.049099�

Women 30 to 34

�

0.912062 0

0.028093 0.073177�

Women 35 to 39

�

0.599478 0

0.021685 0.127704�
Appendices

122

Women 40 to 44

� 0.422401 0

−0.006536 0.148542�

Women 45 to 49

� 0.321234 0

−0.014699 0.161921�

Women 50 to 54

�

0.246731 0

0.017169 0.096181�

Women 55 to 59

� 0.269373 0

−0.013094 0.085901�

Women 60 to 64

� 0.239761 0

−0.014617 0.106417�

Women 65 to 69

�

0.226724 0

0.018589 0.101835�

Women 70 to 74

�

0.207535 0

0.003028 0.085607�

Women 75 to 79

�

0.235224 0

0.027749 0.074606�

Women 80 to 84

�

0.236155 0

0.016700 0.064967�

Women 85 to 89

� 0.313665 0

−0.005681 0.067138�

Women 90 to 94

� 0.310660 0

−0.006154 0.068316�

Women 95 and over

� 0.346683 0

−0.001743 0.064907�
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

123

LPR New Arrival Immigration

LLtt = μ + φφ1(LLtt−1 − μ) + εεtt + θθ1εεtt−1, 

Note: An upper bound for the LPR new arrival immigration was set at 30 million.

Regression Mean Over

AR parameters MA parameters Historical Period Standard Error

φ1 = 0.716520 θ1 = 0. 594647 μ = 535,440.7 σσεε = 59,777.54

Cholesky decomposition of the parameter variance-covariance matrix from the estimated legal immigration 

equation, with the parameters in the following order: μ, φ1, θ1:

�

63622.76 0 0

0.002365 0.116440 0

−0.065367 −0.063506 0.152604

�

Adjustments of Status (Transfers)

SStt = μ + φφ1(SStt−1 − μ) + εεtt, 

In the equation above, SStt represents the ratio of the number of transfers to the number of other-than-LPR 

immigrants and is expressed as a log-odds ratio.

Regression Mean Over

AR parameters Historical Period Standard Error

φ1 = 0.702792 μ = -3.390283 σσεε = 0.189973

Cholesky decomposition of the parameter variance-covariance matrix from the estimated transfers 

equation, with the parameters in the following order: μ, φ1:

� 0.085554 0

−0.023485 0.061967�

Legal Emigration

tt = μ + φφ1( tt−1 − μ) + φφ2( tt−2 − μ) + φφ3( tt−3 − μ) + εεtt , 

In the equation above, tt represents the ratio of the number of legal emigrants to the total number of LPR 

immigrants plus citizens and is expressed as a log-odds ratio.

Regression Mean Over

AR parameters Historical Period Standard Error

φ1 = 1.269066 μ = -7.375162 σσεε = 0.115638

φ2= -0.633007 

φ3 = 0.233052
Appendices

124

Cholesky decomposition of the parameter variance-covariance matrix from the estimated legal emigration 

equation, with the parameters in the following order: μ, φ1, φ2, φ3:

�

0.101544 0 0 0

−0.036127 0.133759 0 0

0.023605 −0.164998 0.072390 0

0.017126 0.044731 −0.065557 0.026498�

Other-than-LPR Immigration

OOtt = μ + φφ1(OOtt−1 − μ) + εεtt , 

Note: A lower bound for the number of other-than-LPR immigrants arriving was set at 100,000 and an 

upper bound was set at 15 million.

Regression Mean Over

AR parameters Historical Period Standard Error

φ1 = 0.656336 μ = 978,691.8 σσεε = 233,249.9

Cholesky decomposition of the parameter variance-covariance matrix from the estimated other-than-LPR

immigration equation, with the parameters in the following order: μ, φ1:

� 161512.8 0

−0.033522 0.130315�

Economic VAR Framework

Means from the VAR

μ = -2.753272

μII = -2.804968

μ = 0.024333

 Unemployment Rate (expressed as log-odds ratio)

UUtt = μ + aa111(UUtt−1 − μ ) + aa112(UUtt−2 − μ ) + aa121(IItt−1 − μII) + aa122(IItt−2 − μII) + aa131( tt−1

− μ ) + aa132( tt−2 − μ ) + εε1tt

Unemployment Mean and AR Parameters

aa111 = 1.103985 aa112 = -0.439009

aa121 = 0.086839 aa122 = 0.076847

aa131 = -0.908053 aa132 = 0.110108

 Inflation Rate (transformed into logs with lower bound of -3.0 percent)

IItt = μII + aa211(UUtt−1 − μ ) + aa212(UUtt−2 − μ ) + aa221(IItt−1 − μII) + aa222(IItt−2 − μII) + aa231( tt−1

− μ ) + aa232( tt−2 − μ ) + εε2tt
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

125

Note: In cases where the nominal interest rate is less than 0 when combining the inflation rate and real 

interest rate, the real interest rate is set such that the nominal interest rate is 0. 

Constant and AR Parameters

aa211 = -0.205316 aa212 = 0.156342

aa221 = 0.303731 aa222 = 0.525542

aa231 = -6.815745 aa232 = 6.568878

 Real Interest Rate

tt = μ + aa311(UUtt−1 − μ ) + aa312(UUtt−2 − μ ) + aa321(IItt−1 − μII) + aa322(IItt−2 − μII)

+ aa331( tt−1 − μ ) + aa332( tt−2 − μ ) + εε3tt

Note: Bounds of plus and minus 40 percent were put on the inflation rate and real interest rate. For the 

real interest rate, the plus/minus 40-percent bound is imposed before the bound requiring a nominal 

interest rate of at least 0 percent.

Constant and AR Parameters

aa311 = 0.005701 aa312 = -0.005288

aa321 = 0.041293 aa322 = -0.024433

aa331 = 1.281961 aa332 = -0.427983

 VAR Matrix Notation

�

UUtt

IItt

tt

� = �

μ

μII

μ

� + A1 �

UUtt−1 − μ

IItt−1 − μII

tt−1 − μ

� + A2 �

UUtt−2 − μ

IItt−2 − μII

tt−2 − μ

� + �

εε1tt

εε2tt

εε3tt

�

A1 = �

aa111 aa121 aa131

aa211 aa221 aa231

aa311 aa321 aa331

�

A2 = �

aa112 aa122 aa132

aa212 aa222 aa232

aa312 aa322 aa332

�

 Cholesky decomposition of the residual variance-covariance matrix:

�

0.152470 0 0

−0.061977 0.222491 0

0.002770 −0.010109 0.008994

�
Appendices

126

Cholesky decomposition of the parameter variance-covariance matrix from the vector autoregression, 

with the parameters in the following order:

1. Regression mean from the unemployment rate equation

2. Unemployment rate lagged one period from the unemployment rate equation

3. Unemployment rate lagged two periods from the unemployment rate equation

4. CPI lagged one period from the unemployment rate equation

5. CPI lagged two periods from the unemployment rate equation

6. Real interest rate lagged one period from the unemployment rate equation

7. Real interest rate lagged two periods from the unemployment rate equation

8. Regression mean from the CPI equation

9. Unemployment rate lagged one period from the CPI equation

10. Unemployment rate lagged two periods from the CPI equation

11. CPI lagged one period from the CPI equation

12. CPI lagged two periods from the CPI equation

13. Real interest rate lagged one period from the CPI equation

14. Real interest rate lagged two periods from the CPI equation

15. Regression mean from the real interest rate equation

16. Unemployment rate lagged one period from the real interest rate equation

17. Unemployment rate lagged two periods from the real interest rate equation

18. CPI lagged one period from the real interest rate equation

19. CPI lagged two periods from the real interest rate equation

20. Real interest rate lagged one period from the real interest rate equation

21. Real interest rate lagged two periods from the real interest rate equation

VV = LLLL′ LL = �

CC11 DD11

CC22 DD22

� (L is a 21 x 21 matrix)

CC11 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

0.067 0 0 0 0 0 0 0 0 0 0

0.042 0.189 0 0 0 0 0 0 0 0 0

−0.048 −0.123 0.071 0 0 0 0 0 0 0 0

0.008 0.024 0.033 0.058 0 0 0 0 0 0 0

0.072 −0.040 0.069 −0.022 0.106 0 0 0 0 0 0

−0.037 −0.045 −0.026 −0.058 0.004 0.040 0 0 0 0 0

−0.121 −0.589 0.096 0.826 −0.205 −0.301 1.036 0 0 0 0

0.001 −0.003 0.006 0.000 0.005 0.002 0.003 0.005 0 0 0

−0.063 0.562 −0.029 −0.776 0.358 0.373 −1.095 −0.125 0.502 0 0

0.027 −0.012 0.086 −0.032 −0.120 −0.063 0.066 0.028 −0.015 0.203 0

0.069 0.035 −0.118 0.041 0.142 0.050 −0.042 0.010 0.021 −0.185 0.068⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

127

CC22 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

−0.052 −0.043 0.037 0.029 −0.016 −0.017 0.013 0.003 −0.021 0.040 0.003

0.011 −0.022 −0.046 0.002 0.061 0.018 −0.041 −0.041 0.010 −0.091 −0.051

−0.526 1.044 −0.291 0.118 −0.753 −0.084 0.300 0.832 0.081 −0.021 0.164

0.003 −0.896 0.120 −0.186 0.215 −0.209 −0.355 −1.003 −0.536 0.413 −0.715

0.002 −0.001 −0.005 0.001 0.009 −0.000 −0.005 −0.003 −0.000 −0.008 −0.002

−0.006 −0.000 0.006 −0.000 −0.010 0.002 0.005 0.002 −0.000 0.007 −0.000

0.001 −0.002 0.000 0.002 0.002 0.001 0.000 0.000 −0.001 −0.003 −0.001

−0.000 0.004 0.001 −0.001 −0.006 0.000 0.002 0.003 0.002 0.006 0.003

0.032 −0.066 0.033 0.033 0.075 0.004 −0.007 −0.022 −0.055 −0.022 −0.024

−0.007 0.061 −0.030 −0.038 −0.053 −0.003 −0.001 0.018 0.075 0.009 0.050 ⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

DD11 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

DD22 =

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

0.123 0 0 0 0 0 0 0 0 0

−0.123 0.049 0 0 0 0 0 0 0 0

1.245 −0.595 0.823 0 0 0 0 0 0 0

−0.930 0.590 −0.609 0.553 0 0 0 0 0 0

0.000 0.002 −0.001 0.000 0.003 0 0 0 0 0

−0.001 −0.002 0.001 −0.000 −0.004 0.002 0 0 0 0

−0.004 0.001 0.001 0.001 0.001 −0.000 0.003 0 0 0

0.004 −0.003 0.000 −0.002 −0.001 0.001 −0.001 0.002 0 0

−0.035 0.063 −0.028 0.034 −0.009 −0.005 0.031 −0.021 0.053 0

0.035 −0.053 0.019 −0.071 0.016 −0.001 −0.043 0.021 −0.045 0.024⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

Real Average Wage (percent change)

WWtt = ββ0 + ββ1uutt + ββ2uutt−1 + εεtt

Note: The annual percent change in the real average wage is estimated as a function of the current and 

lagged unemployment rates (expressed as log-odds ratios) deviating from the historical mean.

Estimated Parameters Standard Error

ββ0 = 0.998129 σσεε = 1.569990

ββ1 = −1.530037

ββ2 = −0.393138
Appendices

128

Cholesky decomposition of the parameter variance-covariance matrix from the estimated real average 

wage equation, with the parameters in the following order: mean of the regression (ββ0), unemployment 

rate, lagged unemployment rate:

�

0.210740 0 0

0.729548 2.566081 0

−0.951809 −2.412212 0.514165

�

Disability Incidence – Male (expressed as log-odds ratio)

DD MMtt = μ + φφ1(DD MMtt−1 − μ) + φφ2(DD MMtt−2 − μ) + εεtt , 

Regression Mean Over

AR parameters Historical Period Standard Error

φ1 = 1.287210 μ = -3.065424 σσεε = 0.081089

φ2 = -0.344726

Cholesky decomposition of the parameter variance-covariance matrix from the estimated male disability 

incidence equation, with the parameters in the following order: μ, φ1, φ2:

�

0.360431 0 0

0.003622 0.210198 0

−0.065427 −0.203654 0.040104

�

Disability Incidence – Female (expressed as log-odds ratio)

DD FFtt = μ + φφ1(DD FFtt−1 − μ) + φφ2(DD FFtt−2 − μ) + εεtt , 

Regression Mean Over

AR parameters Historical Period Standard Error

φ1 = 1.345328 μ = -3.113388 σσεε = 0.081417

φ2 = -0.439248

Cholesky decomposition of the parameter variance-covariance matrix from the estimated female disability 

incidence equation, with the parameters in the following order: μ, φ1, φ2:

�

0.140141 0 0

0.000497 0.202243 0

−0.030797 −0.195458 0.045644

�

Cholesky decomposition of the residual variance-covariance matrix:

�

0.081089 0

0.077243 0.025735 �
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

129

Disability Recovery – Male (expressed as log-odds ratio)

DD MMtt = μ + φφ1(DD MMtt−1 − μ) + εεtt, 

Regression Mean Over

AR parameters Historical Period Standard Error

φ1 = 0.619281 μ = -2.167094 σσεε = 0.258440

Cholesky decomposition of the parameter variance-covariance matrix from the estimated male disability 

recovery equation, with the parameters in the following order: μ, φ1:

�

0.126645 0

0.100781 0.216991�

Disability Recovery – Female (expressed as log-odds ratio)

DD FFtt = μ + φφ1(DD FFtt−1 − μ) + εεtt , 

Regression Mean Over

AR parameters Historical Period Standard Error

φ1 = 0.711241 μ = -2.216565 σσεε = 0.238036

Cholesky decomposition of the parameter variance-covariance matrix from the estimated female disability 

recovery equation, with the parameters in the following order: μ, φ1:

�

0.151912 0

0.080653 0.171401�

Cholesky decomposition of the residual variance-covariance matrix:

� 0.259058 0

0.233211 0.050449 �
Appendices

130

2. Assumed Annual Projected Values for Each Variable from the 2023 Trustees Report

t Ft

TR M1,tTR M2,tTR M3,tTR M4,tTR M5,tTR M6,tTR M7,tTR M8,tTR

2023 1.7038 1.4913 1.5398 5.2051 5.0485 4.4623 4.6125 4.8200 4.3476

2024 1.7225 1.5218 1.5623 2.4098 2.2891 2.2322 1.9338 2.0970 1.6997

2025 1.7421 1.5464 1.5803 1.4205 1.3311 1.4456 1.0480 1.1721 0.8428

2026 1.7631 1.5661 1.5947 1.4097 1.3463 1.4409 1.1238 1.2154 0.9438

2027 1.7853 1.5818 1.6062 1.4008 1.3581 1.4370 1.1843 1.2498 1.0245

2028 1.8085 1.5944 1.6153 1.3934 1.3673 1.4336 1.2324 1.2772 1.0889

2029 1.8325 1.6044 1.6225 1.3873 1.3744 1.4307 1.2707 1.2988 1.1402

2030 1.8572 1.6124 1.6283 1.3820 1.3798 1.4281 1.3010 1.3158 1.1810

2031 1.8687 1.6186 1.6328 1.3775 1.3838 1.4257 1.3249 1.3291 1.2133

2032 1.8811 1.6236 1.6363 1.3736 1.3867 1.4236 1.3438 1.3394 1.2390

2033 1.8940 1.6274 1.6390 1.3702 1.3887 1.4216 1.3586 1.3474 1.2592

2034 1.9071 1.6304 1.6411 1.3671 1.3900 1.4198 1.3701 1.3535 1.2751

2035 1.9202 1.6327 1.6427 1.3644 1.3908 1.4180 1.3790 1.3581 1.2875

2036 1.9330 1.6344 1.6439 1.3619 1.3911 1.4164 1.3858 1.3614 1.2972

2037 1.9450 1.6357 1.6447 1.3596 1.3910 1.4148 1.3909 1.3638 1.3046

2038 1.9559 1.6366 1.6453 1.3575 1.3907 1.4133 1.3947 1.3654 1.3102

2039 1.9656 1.6373 1.6456 1.3555 1.3901 1.4118 1.3974 1.3664 1.3144

2040 1.9738 1.6377 1.6458 1.3536 1.3894 1.4104 1.3993 1.3669 1.3175

2041 1.9806 1.6379 1.6459 1.3517 1.3885 1.4090 1.4005 1.3669 1.3197

2042 1.9861 1.6379 1.6458 1.3500 1.3875 1.4076 1.4011 1.3667 1.3211

2043 1.9903 1.6378 1.6457 1.3483 1.3864 1.4062 1.4013 1.3662 1.3219

2044 1.9934 1.6377 1.6455 1.3467 1.3853 1.4049 1.4012 1.3655 1.3223

2045 1.9957 1.6374 1.6452 1.3451 1.3841 1.4036 1.4008 1.3647 1.3223

2046 1.9973 1.6371 1.6449 1.3435 1.3828 1.4022 1.4001 1.3637 1.3220

2047 1.9983 1.6376 1.6451 1.3415 1.3820 1.4007 1.4021 1.3642 1.3253

2048 1.9990 1.6370 1.6446 1.3401 1.3806 1.3995 1.4005 1.3627 1.3238

2049 1.9994 1.6365 1.6441 1.3387 1.3792 1.3982 1.3990 1.3612 1.3223

2050 1.9997 1.6359 1.6436 1.3372 1.3777 1.3970 1.3974 1.3598 1.3208

2051 1.9998 1.6353 1.6430 1.3358 1.3763 1.3957 1.3959 1.3583 1.3193

2052 1.9999 1.6348 1.6425 1.3344 1.3748 1.3944 1.3943 1.3568 1.3178

2053 1.9999 1.6342 1.6420 1.3329 1.3734 1.3932 1.3927 1.3553 1.3163

2054 2.0000 1.6336 1.6415 1.3315 1.3719 1.3919 1.3912 1.3538 1.3148

2055 2.0000 1.6330 1.6410 1.3301 1.3705 1.3907 1.3896 1.3523 1.3133

2056 2.0000 1.6324 1.6404 1.3286 1.3691 1.3894 1.3880 1.3508 1.3118

2057 2.0000 1.6319 1.6399 1.3272 1.3676 1.3881 1.3865 1.3493 1.3103

2058 2.0000 1.6313 1.6394 1.3258 1.3662 1.3869 1.3849 1.3478 1.3088

2059 2.0000 1.6307 1.6388 1.3244 1.3647 1.3856 1.3833 1.3463 1.3073

2060 2.0000 1.6301 1.6383 1.3229 1.3633 1.3844 1.3817 1.3448 1.3059

2061 2.0000 1.6295 1.6378 1.3215 1.3618 1.3831 1.3801 1.3433 1.3044

2062 2.0000 1.6289 1.6372 1.3201 1.3604 1.3819 1.3786 1.3418 1.3029

2063 2.0000 1.6283 1.6367 1.3187 1.3589 1.3806 1.3770 1.3403 1.3014

2064 2.0000 1.6277 1.6361 1.3173 1.3575 1.3793 1.3754 1.3387 1.2999

2065 2.0000 1.6271 1.6356 1.3158 1.3561 1.3781 1.3738 1.3372 1.2984

2066 2.0000 1.6265 1.6350 1.3144 1.3546 1.3768 1.3722 1.3357 1.2969

2067 2.0000 1.6259 1.6345 1.3130 1.3532 1.3756 1.3706 1.3342 1.2954

2068 2.0000 1.6253 1.6339 1.3116 1.3517 1.3743 1.3690 1.3327 1.2939

2069 2.0000 1.6247 1.6334 1.3102 1.3503 1.3731 1.3674 1.3312 1.2924

2070 2.0000 1.6241 1.6328 1.3088 1.3488 1.3718 1.3658 1.3297 1.2909

2071 2.0000 1.6235 1.6322 1.3074 1.3474 1.3706 1.3642 1.3282 1.2894

2072 2.0000 1.6229 1.6317 1.3060 1.3459 1.3693 1.3626 1.3267 1.2879

2073 2.0000 1.6223 1.6311 1.3045 1.3445 1.3681 1.3609 1.3252 1.2864

2074 2.0000 1.6216 1.6305 1.3031 1.3431 1.3668 1.3593 1.3237 1.2849

2075 2.0000 1.6210 1.6300 1.3017 1.3416 1.3656 1.3577 1.3222 1.2835

2076 2.0000 1.6204 1.6294 1.3003 1.3402 1.3643 1.3561 1.3207 1.2820

2077 2.0000 1.6198 1.6288 1.2989 1.3387 1.3631 1.3545 1.3191 1.2805

2078 2.0000 1.6191 1.6282 1.2975 1.3373 1.3619 1.3528 1.3176 1.2790

2079 2.0000 1.6185 1.6276 1.2961 1.3359 1.3606 1.3512 1.3161 1.2775

2080 2.0000 1.6179 1.6271 1.2947 1.3344 1.3594 1.3496 1.3146 1.2760

2081 2.0000 1.6172 1.6265 1.2934 1.3330 1.3581 1.3480 1.3131 1.2745

2082 2.0000 1.6166 1.6259 1.2920 1.3315 1.3569 1.3463 1.3116 1.2730

2083 2.0000 1.6159 1.6253 1.2906 1.3301 1.3556 1.3447 1.3101 1.2716

2084 2.0000 1.6153 1.6247 1.2892 1.3287 1.3544 1.3430 1.3086 1.2701

2085 2.0000 1.6146 1.6241 1.2878 1.3272 1.3532 1.3414 1.3070 1.2686

2086 2.0000 1.6140 1.6235 1.2864 1.3258 1.3519 1.3398 1.3055 1.2671

2087 2.0000 1.6133 1.6229 1.2850 1.3243 1.3507 1.3381 1.3040 1.2656
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

131

t Ft

TR M1,tTR M2,tTR M3,tTR M4,tTR M5,tTR M6,tTR M7,tTR M8,tTR

2088 2.0000 1.6127 1.6223 1.2836 1.3229 1.3495 1.3365 1.3025 1.2641

2089 2.0000 1.6120 1.6217 1.2823 1.3215 1.3482 1.3348 1.3010 1.2627

2090 2.0000 1.6113 1.6210 1.2809 1.3200 1.3470 1.3332 1.2995 1.2612

2091 2.0000 1.6107 1.6204 1.2795 1.3186 1.3457 1.3315 1.2980 1.2597

2092 2.0000 1.6100 1.6198 1.2781 1.3172 1.3445 1.3298 1.2964 1.2582

2093 2.0000 1.6093 1.6192 1.2768 1.3157 1.3433 1.3282 1.2949 1.2567

2094 2.0000 1.6087 1.6186 1.2754 1.3143 1.3420 1.3265 1.2934 1.2553

2095 2.0000 1.6080 1.6179 1.2740 1.3129 1.3408 1.3249 1.2919 1.2538

2096 2.0000 1.6073 1.6173 1.2727 1.3114 1.3396 1.3232 1.2904 1.2523

2097 2.0000 1.6066 1.6167 1.2713 1.3100 1.3384 1.3215 1.2889 1.2508
Appendices

132

t M9,t

TR M10,tTR M11,tTR M12,tTR M13,tTR M14,tTR M15,tTR M16,tTR M17,tTR

2023 4.3556 7.6346 7.5666 7.2803 7.4677 6.9534 6.9462 6.7615 6.6349

2024 1.6985 2.4645 2.3970 2.1522 2.3040 1.8805 1.8630 1.7275 1.6025

2025 0.8339 1.5759 1.5120 1.3121 1.4289 1.0979 1.0737 0.9838 0.8673

2026 0.9286 0.6384 0.5773 0.4143 0.5023 0.2466 0.2167 0.1639 0.0542

2027 1.0042 0.6759 0.6178 0.4861 0.5502 0.3575 0.3236 0.3008 0.1984

2028 1.0646 0.7058 0.6501 0.5437 0.5885 0.4464 0.4094 0.4105 0.3142

2029 1.1126 0.7296 0.6758 0.5899 0.6192 0.5177 0.4782 0.4983 0.4071

2030 1.1509 0.7485 0.6963 0.6269 0.6437 0.5749 0.5333 0.5686 0.4817

2031 1.1812 0.7635 0.7127 0.6565 0.6634 0.6205 0.5775 0.6248 0.5415

2032 1.2052 0.7753 0.7257 0.6801 0.6790 0.6571 0.6128 0.6696 0.5894

2033 1.2242 0.7847 0.7360 0.6990 0.6915 0.6862 0.6411 0.7054 0.6277

2034 1.2391 0.7920 0.7441 0.7141 0.7015 0.7095 0.6637 0.7340 0.6583

2035 1.2508 0.7977 0.7505 0.7261 0.7094 0.7280 0.6817 0.7567 0.6828

2036 1.2599 0.8022 0.7556 0.7356 0.7157 0.7427 0.6961 0.7747 0.7023

2037 1.2669 0.8056 0.7596 0.7431 0.7207 0.7544 0.7076 0.7890 0.7179

2038 1.2722 0.8082 0.7626 0.7491 0.7246 0.7637 0.7167 0.8003 0.7302

2039 1.2763 0.8101 0.7650 0.7538 0.7278 0.7710 0.7239 0.8092 0.7401

2040 1.2792 0.8115 0.7668 0.7574 0.7302 0.7767 0.7297 0.8162 0.7478

2041 1.2813 0.8125 0.7681 0.7603 0.7321 0.7812 0.7342 0.8216 0.7540

2042 1.2828 0.8132 0.7691 0.7625 0.7336 0.7847 0.7378 0.8258 0.7588

2043 1.2837 0.8136 0.7698 0.7642 0.7347 0.7874 0.7406 0.8290 0.7626

2044 1.2841 0.8138 0.7703 0.7655 0.7356 0.7894 0.7428 0.8315 0.7656

2045 1.2842 0.8138 0.7706 0.7664 0.7362 0.7909 0.7444 0.8333 0.7678

2046 1.2841 0.8136 0.7707 0.7671 0.7367 0.7921 0.7457 0.8346 0.7696

2047 1.2873 0.8152 0.7727 0.7710 0.7393 0.7982 0.7519 0.8420 0.7779

2048 1.2860 0.8145 0.7722 0.7706 0.7390 0.7977 0.7516 0.8413 0.7775

2049 1.2847 0.8138 0.7718 0.7702 0.7388 0.7971 0.7513 0.8406 0.7770

2050 1.2835 0.8132 0.7713 0.7698 0.7386 0.7966 0.7510 0.8399 0.7766

2051 1.2822 0.8125 0.7709 0.7694 0.7383 0.7961 0.7507 0.8392 0.7762

2052 1.2809 0.8118 0.7705 0.7690 0.7381 0.7956 0.7504 0.8384 0.7758

2053 1.2797 0.8112 0.7700 0.7686 0.7379 0.7950 0.7501 0.8377 0.7753

2054 1.2784 0.8105 0.7696 0.7682 0.7376 0.7945 0.7498 0.8370 0.7749

2055 1.2771 0.8099 0.7692 0.7678 0.7374 0.7940 0.7495 0.8363 0.7745

2056 1.2759 0.8092 0.7688 0.7674 0.7372 0.7935 0.7492 0.8356 0.7741

2057 1.2746 0.8086 0.7683 0.7671 0.7369 0.7930 0.7489 0.8349 0.7737

2058 1.2734 0.8080 0.7679 0.7667 0.7367 0.7925 0.7486 0.8342 0.7733

2059 1.2721 0.8073 0.7675 0.7663 0.7365 0.7920 0.7483 0.8335 0.7729

2060 1.2709 0.8067 0.7671 0.7659 0.7362 0.7915 0.7481 0.8328 0.7725

2061 1.2696 0.8061 0.7667 0.7655 0.7360 0.7910 0.7478 0.8321 0.7721

2062 1.2684 0.8054 0.7662 0.7652 0.7358 0.7905 0.7475 0.8315 0.7717

2063 1.2671 0.8048 0.7658 0.7648 0.7356 0.7900 0.7472 0.8308 0.7713

2064 1.2659 0.8042 0.7654 0.7644 0.7354 0.7895 0.7470 0.8301 0.7709

2065 1.2647 0.8036 0.7650 0.7641 0.7351 0.7890 0.7467 0.8294 0.7705

2066 1.2634 0.8030 0.7646 0.7637 0.7349 0.7885 0.7464 0.8287 0.7701

2067 1.2622 0.8024 0.7642 0.7633 0.7347 0.7880 0.7461 0.8281 0.7697

2068 1.2610 0.8018 0.7638 0.7630 0.7345 0.7876 0.7459 0.8274 0.7693

2069 1.2597 0.8011 0.7634 0.7626 0.7343 0.7871 0.7456 0.8267 0.7689

2070 1.2585 0.8005 0.7630 0.7622 0.7341 0.7866 0.7453 0.8261 0.7685

2071 1.2573 0.7999 0.7626 0.7619 0.7338 0.7861 0.7451 0.8254 0.7682

2072 1.2561 0.7994 0.7623 0.7615 0.7336 0.7857 0.7448 0.8248 0.7678

2073 1.2548 0.7988 0.7619 0.7612 0.7334 0.7852 0.7445 0.8241 0.7674

2074 1.2536 0.7982 0.7615 0.7608 0.7332 0.7847 0.7443 0.8235 0.7670

2075 1.2524 0.7976 0.7611 0.7605 0.7330 0.7843 0.7440 0.8228 0.7666

2076 1.2512 0.7970 0.7607 0.7601 0.7328 0.7838 0.7438 0.8222 0.7663

2077 1.2500 0.7964 0.7604 0.7598 0.7326 0.7833 0.7435 0.8215 0.7659

2078 1.2488 0.7958 0.7600 0.7594 0.7324 0.7829 0.7433 0.8209 0.7655

2079 1.2476 0.7953 0.7596 0.7591 0.7322 0.7824 0.7430 0.8203 0.7652

2080 1.2463 0.7947 0.7592 0.7587 0.7320 0.7820 0.7427 0.8196 0.7648

2081 1.2451 0.7941 0.7589 0.7584 0.7318 0.7815 0.7425 0.8190 0.7644

2082 1.2439 0.7936 0.7585 0.7581 0.7316 0.7811 0.7422 0.8184 0.7641

2083 1.2427 0.7930 0.7581 0.7577 0.7314 0.7806 0.7420 0.8178 0.7637

2084 1.2415 0.7925 0.7578 0.7574 0.7312 0.7802 0.7418 0.8171 0.7633

2085 1.2403 0.7919 0.7574 0.7571 0.7310 0.7797 0.7415 0.8165 0.7630

2086 1.2392 0.7913 0.7571 0.7567 0.7308 0.7793 0.7413 0.8159 0.7626

2087 1.2380 0.7908 0.7567 0.7564 0.7306 0.7789 0.7410 0.8153 0.7623

2088 1.2368 0.7902 0.7563 0.7561 0.7304 0.7784 0.7408 0.8147 0.7619

2089 1.2356 0.7897 0.7560 0.7558 0.7303 0.7780 0.7405 0.8141 0.7616
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

133

t M9,t

TR M10,tTR M11,tTR M12,tTR M13,tTR M14,tTR M15,tTR M16,tTR M17,tTR

2090 1.2344 0.7892 0.7556 0.7554 0.7301 0.7776 0.7403 0.8135 0.7612

2091 1.2332 0.7886 0.7553 0.7551 0.7299 0.7771 0.7401 0.8129 0.7609

2092 1.2320 0.7881 0.7549 0.7548 0.7297 0.7767 0.7398 0.8123 0.7605

2093 1.2309 0.7875 0.7546 0.7545 0.7295 0.7763 0.7396 0.8117 0.7602

2094 1.2297 0.7870 0.7543 0.7542 0.7293 0.7759 0.7394 0.8111 0.7598

2095 1.2285 0.7865 0.7539 0.7539 0.7291 0.7754 0.7391 0.8105 0.7595

2096 1.2273 0.7860 0.7536 0.7535 0.7290 0.7750 0.7389 0.8099 0.7592

2097 1.2262 0.7854 0.7533 0.7532 0.7288 0.7746 0.7387 0.8094 0.7588
Appendices

134

t M18,t

TR M19,tTR M20,tTR M21,tTR M22,tTR M23,tTR M24,tTR M25,tTR M26,tTR

2023 7.0045 6.6236 7.6097 7.2081 8.0440 7.8311 8.0001 7.9595 7.6933

2024 1.9453 1.6020 2.4700 2.1093 2.8445 2.6461 2.8043 2.7522 2.5489

2025 1.1727 0.8766 1.6099 1.3001 1.9210 1.7456 1.8878 1.8327 1.6892

2026 0.3293 0.0712 0.6959 0.4272 0.9558 0.7990 0.9280 0.8705 0.7751

2027 0.4458 0.2213 0.7524 0.5200 0.9690 0.8295 0.9463 0.8886 0.8322

2028 0.5392 0.3417 0.7976 0.5944 0.9795 0.8539 0.9607 0.9030 0.8777

2029 0.6139 0.4381 0.8336 0.6540 0.9877 0.8734 0.9720 0.9144 0.9138

2030 0.6735 0.5153 0.8622 0.7017 0.9941 0.8890 0.9806 0.9232 0.9423

2031 0.7211 0.5770 0.8850 0.7397 0.9990 0.9013 0.9872 0.9300 0.9647

2032 0.7591 0.6264 0.9030 0.7701 1.0027 0.9111 0.9921 0.9352 0.9823

2033 0.7893 0.6658 0.9172 0.7943 1.0055 0.9187 0.9957 0.9389 0.9960

2034 0.8134 0.6973 0.9284 0.8135 1.0074 0.9246 0.9981 0.9415 1.0066

2035 0.8324 0.7224 0.9372 0.8288 1.0087 0.9291 0.9996 0.9432 1.0146

2036 0.8475 0.7424 0.9439 0.8408 1.0095 0.9326 1.0005 0.9442 1.0206

2037 0.8593 0.7582 0.9491 0.8503 1.0099 0.9352 1.0007 0.9446 1.0250

2038 0.8687 0.7708 0.9531 0.8577 1.0100 0.9370 1.0005 0.9445 1.0281

2039 0.8759 0.7808 0.9560 0.8635 1.0098 0.9383 0.9999 0.9440 1.0301

2040 0.8815 0.7886 0.9581 0.8680 1.0094 0.9391 0.9991 0.9432 1.0313

2041 0.8859 0.7948 0.9596 0.8714 1.0088 0.9395 0.9980 0.9421 1.0319

2042 0.8891 0.7996 0.9605 0.8740 1.0081 0.9397 0.9966 0.9409 1.0319

2043 0.8915 0.8034 0.9611 0.8758 1.0072 0.9396 0.9952 0.9395 1.0315

2044 0.8933 0.8062 0.9613 0.8772 1.0063 0.9393 0.9936 0.9379 1.0307

2045 0.8945 0.8084 0.9612 0.8781 1.0053 0.9389 0.9919 0.9363 1.0297

2046 0.8953 0.8101 0.9610 0.8787 1.0042 0.9383 0.9901 0.9345 1.0285

2047 0.9013 0.8184 0.9633 0.8835 1.0039 0.9393 0.9894 0.9339 1.0299

2048 0.9004 0.8179 0.9622 0.8827 1.0027 0.9383 0.9873 0.9319 1.0278

2049 0.8994 0.8173 0.9611 0.8819 1.0014 0.9372 0.9852 0.9298 1.0256

2050 0.8985 0.8167 0.9599 0.8811 1.0001 0.9362 0.9831 0.9277 1.0235

2051 0.8975 0.8161 0.9588 0.8803 0.9988 0.9351 0.9810 0.9256 1.0214

2052 0.8966 0.8155 0.9577 0.8794 0.9975 0.9341 0.9789 0.9236 1.0192

2053 0.8956 0.8149 0.9566 0.8786 0.9962 0.9330 0.9768 0.9215 1.0171

2054 0.8947 0.8143 0.9555 0.8778 0.9949 0.9320 0.9747 0.9194 1.0149

2055 0.8938 0.8138 0.9544 0.8770 0.9936 0.9309 0.9727 0.9174 1.0128

2056 0.8928 0.8132 0.9532 0.8762 0.9924 0.9299 0.9706 0.9153 1.0107

2057 0.8919 0.8126 0.9521 0.8754 0.9911 0.9289 0.9685 0.9133 1.0085

2058 0.8910 0.8120 0.9510 0.8746 0.9898 0.9278 0.9664 0.9112 1.0064

2059 0.8901 0.8115 0.9499 0.8738 0.9885 0.9268 0.9643 0.9092 1.0042

2060 0.8891 0.8109 0.9488 0.8730 0.9873 0.9257 0.9623 0.9072 1.0021

2061 0.8882 0.8103 0.9477 0.8722 0.9860 0.9247 0.9602 0.9051 1.0000

2062 0.8873 0.8098 0.9466 0.8714 0.9847 0.9237 0.9581 0.9031 0.9978

2063 0.8864 0.8092 0.9455 0.8706 0.9834 0.9227 0.9560 0.9011 0.9957

2064 0.8855 0.8087 0.9445 0.8698 0.9822 0.9216 0.9540 0.8991 0.9936

2065 0.8846 0.8081 0.9434 0.8691 0.9809 0.9206 0.9519 0.8971 0.9914

2066 0.8837 0.8076 0.9423 0.8683 0.9796 0.9196 0.9499 0.8951 0.9893

2067 0.8828 0.8070 0.9412 0.8675 0.9784 0.9186 0.9478 0.8931 0.9872

2068 0.8819 0.8065 0.9401 0.8667 0.9771 0.9176 0.9458 0.8911 0.9851

2069 0.8810 0.8059 0.9390 0.8660 0.9759 0.9166 0.9437 0.8891 0.9829

2070 0.8801 0.8054 0.9380 0.8652 0.9746 0.9156 0.9417 0.8871 0.9808

2071 0.8792 0.8049 0.9369 0.8644 0.9734 0.9146 0.9396 0.8851 0.9787

2072 0.8784 0.8043 0.9358 0.8637 0.9721 0.9136 0.9376 0.8832 0.9766

2073 0.8775 0.8038 0.9348 0.8629 0.9709 0.9126 0.9356 0.8812 0.9745

2074 0.8766 0.8033 0.9337 0.8621 0.9696 0.9116 0.9335 0.8792 0.9723

2075 0.8757 0.8027 0.9326 0.8614 0.9684 0.9106 0.9315 0.8773 0.9702

2076 0.8749 0.8022 0.9316 0.8606 0.9671 0.9096 0.9295 0.8753 0.9681

2077 0.8740 0.8017 0.9305 0.8599 0.9659 0.9086 0.9275 0.8734 0.9660

2078 0.8731 0.8012 0.9295 0.8591 0.9646 0.9076 0.9255 0.8714 0.9639

2079 0.8723 0.8007 0.9284 0.8584 0.9634 0.9066 0.9234 0.8695 0.9618

2080 0.8714 0.8001 0.9274 0.8576 0.9622 0.9056 0.9214 0.8676 0.9597

2081 0.8706 0.7996 0.9263 0.8569 0.9609 0.9046 0.9194 0.8656 0.9576

2082 0.8697 0.7991 0.9253 0.8562 0.9597 0.9037 0.9174 0.8637 0.9555

2083 0.8689 0.7986 0.9242 0.8554 0.9585 0.9027 0.9154 0.8618 0.9534

2084 0.8680 0.7981 0.9232 0.8547 0.9572 0.9017 0.9135 0.8599 0.9513

2085 0.8672 0.7976 0.9222 0.8540 0.9560 0.9007 0.9115 0.8580 0.9492

2086 0.8664 0.7971 0.9211 0.8532 0.9548 0.8998 0.9095 0.8561 0.9472

2087 0.8655 0.7966 0.9201 0.8525 0.9536 0.8988 0.9075 0.8542 0.9451

2088 0.8647 0.7961 0.9191 0.8518 0.9524 0.8978 0.9055 0.8523 0.9430

2089 0.8639 0.7956 0.9181 0.8511 0.9511 0.8969 0.9036 0.8505 0.9409
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

135

t M18,t

TR M19,tTR M20,tTR M21,tTR M22,tTR M23,tTR M24,tTR M25,tTR M26,tTR

2090 0.8631 0.7951 0.9170 0.8504 0.9499 0.8959 0.9016 0.8486 0.9389

2091 0.8622 0.7946 0.9160 0.8497 0.9487 0.8950 0.8997 0.8467 0.9368

2092 0.8614 0.7941 0.9150 0.8489 0.9475 0.8940 0.8977 0.8449 0.9347

2093 0.8606 0.7937 0.9140 0.8482 0.9463 0.8931 0.8958 0.8430 0.9327

2094 0.8598 0.7932 0.9130 0.8475 0.9451 0.8921 0.8938 0.8412 0.9306

2095 0.8590 0.7927 0.9120 0.8468 0.9439 0.8912 0.8919 0.8394 0.9286

2096 0.8582 0.7922 0.9110 0.8461 0.9427 0.8902 0.8900 0.8375 0.9265

2097 0.8574 0.7917 0.9100 0.8454 0.9415 0.8893 0.8880 0.8357 0.9245
Appendices

136

t M27,t

TR M28,tTR M29,tTR M30,tTR M31,tTR M32,tTR M33,tTR M34,tTR M35,tTR

2023 7.8068 7.7698 7.6969 8.9568 8.7604 8.9596 9.0063 8.9639 9.0474

2024 2.6286 2.6104 2.5361 2.8715 2.6998 2.8521 2.8853 2.8227 2.8759

2025 1.7444 1.7401 1.6778 1.9114 1.7849 1.8891 1.9287 1.8592 1.9123

2026 0.8105 0.8171 0.7643 0.9151 0.8246 0.8896 0.9339 0.8579 0.9101

2027 0.8531 0.8677 0.8252 0.9066 0.8486 0.8812 0.9322 0.8529 0.9088

2028 0.8871 0.9080 0.8738 0.8991 0.8672 0.8737 0.9302 0.8480 0.9068

2029 0.9141 0.9398 0.9123 0.8924 0.8814 0.8670 0.9277 0.8433 0.9044

2030 0.9355 0.9649 0.9429 0.8864 0.8919 0.8608 0.9248 0.8386 0.9015

2031 0.9523 0.9847 0.9669 0.8808 0.8995 0.8551 0.9216 0.8340 0.8984

2032 0.9653 1.0000 0.9858 0.8756 0.9048 0.8497 0.9182 0.8294 0.8949

2033 0.9754 1.0119 1.0004 0.8707 0.9082 0.8446 0.9146 0.8248 0.8912

2034 0.9831 1.0210 1.0117 0.8660 0.9100 0.8396 0.9108 0.8203 0.8873

2035 0.9889 1.0278 1.0204 0.8615 0.9107 0.8349 0.9069 0.8158 0.8833

2036 0.9931 1.0329 1.0268 0.8572 0.9104 0.8303 0.9029 0.8114 0.8791

2037 0.9960 1.0364 1.0315 0.8530 0.9093 0.8258 0.8989 0.8069 0.8749

2038 0.9979 1.0389 1.0348 0.8489 0.9076 0.8214 0.8947 0.8025 0.8706

2039 0.9991 1.0404 1.0371 0.8449 0.9055 0.8171 0.8905 0.7981 0.8663

2040 0.9995 1.0411 1.0384 0.8410 0.9029 0.8129 0.8863 0.7937 0.8619

2041 0.9995 1.0413 1.0390 0.8371 0.9001 0.8087 0.8821 0.7894 0.8576

2042 0.9990 1.0409 1.0390 0.8333 0.8970 0.8046 0.8779 0.7850 0.8532

2043 0.9982 1.0402 1.0385 0.8295 0.8937 0.8005 0.8736 0.7807 0.8488

2044 0.9971 1.0392 1.0377 0.8258 0.8902 0.7964 0.8694 0.7764 0.8444

2045 0.9958 1.0379 1.0366 0.8221 0.8867 0.7924 0.8652 0.7722 0.8400

2046 0.9943 1.0365 1.0352 0.8184 0.8830 0.7885 0.8610 0.7679 0.8356

2047 0.9950 1.0374 1.0368 0.8146 0.8806 0.7844 0.8568 0.7637 0.8314

2048 0.9928 1.0352 1.0345 0.8110 0.8766 0.7805 0.8526 0.7595 0.8270

2049 0.9907 1.0329 1.0322 0.8075 0.8726 0.7767 0.8484 0.7554 0.8226

2050 0.9885 1.0306 1.0299 0.8040 0.8686 0.7728 0.8442 0.7512 0.8182

2051 0.9863 1.0284 1.0276 0.8005 0.8647 0.7690 0.8401 0.7471 0.8139

2052 0.9841 1.0261 1.0253 0.7970 0.8607 0.7653 0.8359 0.7430 0.8096

2053 0.9819 1.0238 1.0229 0.7935 0.8568 0.7615 0.8318 0.7389 0.8053

2054 0.9798 1.0216 1.0206 0.7901 0.8529 0.7578 0.8277 0.7349 0.8011

2055 0.9776 1.0193 1.0183 0.7866 0.8490 0.7540 0.8236 0.7308 0.7968

2056 0.9754 1.0170 1.0160 0.7832 0.8451 0.7504 0.8196 0.7268 0.7926

2057 0.9732 1.0148 1.0137 0.7798 0.8412 0.7467 0.8155 0.7229 0.7884

2058 0.9711 1.0125 1.0114 0.7765 0.8374 0.7430 0.8115 0.7189 0.7843

2059 0.9689 1.0102 1.0091 0.7731 0.8336 0.7394 0.8075 0.7150 0.7801

2060 0.9667 1.0079 1.0068 0.7698 0.8298 0.7358 0.8035 0.7111 0.7760

2061 0.9646 1.0057 1.0045 0.7664 0.8260 0.7322 0.7996 0.7072 0.7719

2062 0.9624 1.0034 1.0022 0.7631 0.8222 0.7287 0.7956 0.7033 0.7678

2063 0.9602 1.0011 0.9998 0.7599 0.8185 0.7251 0.7917 0.6995 0.7638

2064 0.9581 0.9989 0.9975 0.7566 0.8148 0.7216 0.7878 0.6957 0.7598

2065 0.9559 0.9966 0.9952 0.7534 0.8111 0.7181 0.7840 0.6919 0.7558

2066 0.9538 0.9944 0.9929 0.7501 0.8074 0.7146 0.7801 0.6882 0.7518

2067 0.9516 0.9921 0.9906 0.7469 0.8037 0.7112 0.7763 0.6845 0.7478

2068 0.9495 0.9898 0.9883 0.7437 0.8001 0.7078 0.7725 0.6808 0.7439

2069 0.9474 0.9876 0.9860 0.7406 0.7965 0.7044 0.7687 0.6771 0.7400

2070 0.9452 0.9853 0.9837 0.7374 0.7929 0.7010 0.7650 0.6734 0.7361

2071 0.9431 0.9831 0.9814 0.7343 0.7893 0.6976 0.7612 0.6698 0.7322

2072 0.9410 0.9808 0.9791 0.7312 0.7857 0.6943 0.7575 0.6662 0.7284

2073 0.9388 0.9785 0.9769 0.7281 0.7822 0.6910 0.7538 0.6626 0.7246

2074 0.9367 0.9763 0.9746 0.7250 0.7786 0.6877 0.7502 0.6591 0.7208

2075 0.9346 0.9740 0.9723 0.7220 0.7751 0.6844 0.7465 0.6556 0.7170

2076 0.9325 0.9718 0.9700 0.7189 0.7717 0.6812 0.7429 0.6521 0.7133

2077 0.9304 0.9695 0.9677 0.7159 0.7682 0.6779 0.7393 0.6486 0.7096

2078 0.9283 0.9673 0.9654 0.7129 0.7647 0.6747 0.7357 0.6451 0.7059

2079 0.9261 0.9650 0.9632 0.7099 0.7613 0.6716 0.7322 0.6417 0.7022

2080 0.9240 0.9628 0.9609 0.7070 0.7579 0.6684 0.7286 0.6383 0.6986

2081 0.9220 0.9606 0.9586 0.7040 0.7545 0.6653 0.7251 0.6349 0.6950

2082 0.9199 0.9583 0.9563 0.7011 0.7512 0.6621 0.7216 0.6316 0.6914

2083 0.9178 0.9561 0.9541 0.6982 0.7478 0.6591 0.7182 0.6283 0.6878

2084 0.9157 0.9539 0.9518 0.6953 0.7445 0.6560 0.7147 0.6250 0.6843

2085 0.9136 0.9516 0.9496 0.6924 0.7412 0.6529 0.7113 0.6217 0.6808

2086 0.9115 0.9494 0.9473 0.6896 0.7380 0.6499 0.7079 0.6185 0.6773

2087 0.9095 0.9472 0.9450 0.6868 0.7347 0.6469 0.7046 0.6152 0.6738

2088 0.9074 0.9450 0.9428 0.6840 0.7315 0.6439 0.7012 0.6120 0.6704

2089 0.9053 0.9427 0.9405 0.6812 0.7282 0.6410 0.6979 0.6089 0.6670
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

137

t M27,t

TR M28,tTR M29,tTR M30,tTR M31,tTR M32,tTR M33,tTR M34,tTR M35,tTR

2090 0.9033 0.9405 0.9383 0.6784 0.7251 0.6380 0.6946 0.6057 0.6636

2091 0.9012 0.9383 0.9361 0.6756 0.7219 0.6351 0.6913 0.6026 0.6602

2092 0.8992 0.9361 0.9338 0.6729 0.7187 0.6322 0.6880 0.5995 0.6569

2093 0.8971 0.9339 0.9316 0.6702 0.7156 0.6293 0.6848 0.5964 0.6535

2094 0.8951 0.9317 0.9294 0.6675 0.7125 0.6265 0.6816 0.5934 0.6502

2095 0.8931 0.9295 0.9271 0.6648 0.7094 0.6236 0.6784 0.5903 0.6470

2096 0.8910 0.9273 0.9249 0.6621 0.7063 0.6208 0.6752 0.5873 0.6437

2097 0.8890 0.9251 0.9227 0.6595 0.7033 0.6180 0.6721 0.5844 0.6405
Appendices

138

t M36,t

TR M37,tTR M38,tTR M39,tTR M40,tTR M41,tTR M42,tTR LtTR EtTR

2023 8.9506 9.1552 4.5715 4.7213 4.6880 5.0545 4.8686 858,000 -6.8924

2024 2.7504 2.8915 1.6461 1.7461 1.6616 1.7846 1.6881 858,000 -6.8976

2025 1.7919 1.9126 0.6263 0.7037 0.6123 0.6569 0.5934 600,000 -7.1229

2026 0.7923 0.8953 0.6348 0.7045 0.6277 0.6787 0.6189 600,000 -7.1278

2027 0.7981 0.8929 0.6409 0.7046 0.6393 0.6953 0.6386 600,000 -7.1328

2028 0.8018 0.8900 0.6451 0.7039 0.6480 0.7079 0.6536 600,000 -7.1377

2029 0.8038 0.8868 0.6479 0.7028 0.6541 0.7171 0.6647 600,000 -7.1426

2030 0.8045 0.8832 0.6493 0.7011 0.6583 0.7236 0.6729 600,000 -7.1476

2031 0.8040 0.8794 0.6498 0.6991 0.6609 0.7280 0.6785 600,000 -7.1526

2032 0.8026 0.8754 0.6495 0.6968 0.6622 0.7307 0.6822 600,000 -7.1574

2033 0.8006 0.8712 0.6486 0.6943 0.6626 0.7320 0.6843 600,000 -7.1622

2034 0.7979 0.8668 0.6471 0.6915 0.6620 0.7322 0.6852 600,000 -7.1669

2035 0.7949 0.8624 0.6452 0.6887 0.6609 0.7316 0.6851 600,000 -7.1716

2036 0.7915 0.8579 0.6430 0.6857 0.6592 0.7303 0.6842 600,000 -7.1761

2037 0.7878 0.8533 0.6406 0.6826 0.6571 0.7284 0.6826 600,000 -7.1806

2038 0.7838 0.8487 0.6379 0.6795 0.6547 0.7262 0.6805 600,000 -7.1850

2039 0.7798 0.8441 0.6351 0.6763 0.6520 0.7235 0.6780 600,000 -7.1893

2040 0.7756 0.8394 0.6321 0.6730 0.6491 0.7207 0.6752 600,000 -7.1935

2041 0.7712 0.8348 0.6291 0.6698 0.6460 0.7175 0.6722 600,000 -7.1977

2042 0.7669 0.8301 0.6259 0.6665 0.6429 0.7143 0.6689 600,000 -7.2016

2043 0.7624 0.8255 0.6227 0.6632 0.6396 0.7109 0.6655 600,000 -7.2055

2044 0.7579 0.8208 0.6195 0.6599 0.6362 0.7074 0.6620 600,000 -7.2093

2045 0.7534 0.8162 0.6162 0.6566 0.6328 0.7038 0.6584 600,000 -7.2129

2046 0.7489 0.8116 0.6130 0.6533 0.6294 0.7002 0.6547 600,000 -7.2164

2047 0.7449 0.8071 0.6102 0.6501 0.6266 0.6974 0.6521 600,000 -7.2199

2048 0.7403 0.8024 0.6068 0.6468 0.6230 0.6935 0.6481 600,000 -7.2232

2049 0.7357 0.7979 0.6034 0.6435 0.6193 0.6896 0.6441 600,000 -7.2265

2050 0.7311 0.7933 0.6000 0.6402 0.6157 0.6857 0.6401 600,000 -7.2298

2051 0.7266 0.7888 0.5967 0.6370 0.6121 0.6819 0.6362 600,000 -7.2330

2052 0.7221 0.7842 0.5933 0.6337 0.6085 0.6781 0.6322 600,000 -7.2363

2053 0.7176 0.7797 0.5900 0.6305 0.6049 0.6743 0.6283 600,000 -7.2396

2054 0.7131 0.7753 0.5867 0.6272 0.6013 0.6705 0.6244 600,000 -7.2429

2055 0.7087 0.7708 0.5834 0.6240 0.5978 0.6667 0.6205 600,000 -7.2462

2056 0.7043 0.7664 0.5802 0.6208 0.5943 0.6629 0.6167 600,000 -7.2497

2057 0.6999 0.7620 0.5769 0.6176 0.5908 0.6592 0.6128 600,000 -7.2532

2058 0.6955 0.7576 0.5737 0.6145 0.5873 0.6554 0.6090 600,000 -7.2569

2059 0.6912 0.7533 0.5705 0.6113 0.5838 0.6517 0.6052 600,000 -7.2607

2060 0.6869 0.7490 0.5673 0.6082 0.5803 0.6480 0.6014 600,000 -7.2645

2061 0.6827 0.7447 0.5641 0.6051 0.5769 0.6443 0.5976 600,000 -7.2685

2062 0.6784 0.7404 0.5610 0.6020 0.5735 0.6407 0.5939 600,000 -7.2726

2063 0.6742 0.7362 0.5578 0.5989 0.5700 0.6370 0.5902 600,000 -7.2768

2064 0.6700 0.7319 0.5547 0.5959 0.5667 0.6334 0.5864 600,000 -7.2811

2065 0.6659 0.7277 0.5516 0.5928 0.5633 0.6298 0.5827 600,000 -7.2855

2066 0.6618 0.7236 0.5485 0.5898 0.5599 0.6262 0.5791 600,000 -7.2899

2067 0.6577 0.7194 0.5455 0.5868 0.5566 0.6226 0.5754 600,000 -7.2944

2068 0.6536 0.7153 0.5424 0.5838 0.5533 0.6190 0.5718 600,000 -7.2989

2069 0.6496 0.7112 0.5394 0.5808 0.5500 0.6155 0.5682 600,000 -7.3034

2070 0.6456 0.7072 0.5364 0.5778 0.5467 0.6119 0.5646 600,000 -7.3079

2071 0.6416 0.7031 0.5334 0.5749 0.5434 0.6084 0.5610 600,000 -7.3124

2072 0.6377 0.6991 0.5305 0.5720 0.5402 0.6049 0.5575 600,000 -7.3168

2073 0.6337 0.6951 0.5275 0.5691 0.5370 0.6015 0.5539 600,000 -7.3213

2074 0.6299 0.6912 0.5246 0.5662 0.5338 0.5980 0.5504 600,000 -7.3257

2075 0.6260 0.6872 0.5217 0.5633 0.5306 0.5946 0.5469 600,000 -7.3300

2076 0.6222 0.6833 0.5188 0.5604 0.5274 0.5911 0.5435 600,000 -7.3343

2077 0.6184 0.6795 0.5159 0.5576 0.5243 0.5877 0.5400 600,000 -7.3385

2078 0.6146 0.6756 0.5131 0.5548 0.5211 0.5843 0.5366 600,000 -7.3427

2079 0.6109 0.6718 0.5102 0.5520 0.5180 0.5810 0.5332 600,000 -7.3468

2080 0.6071 0.6680 0.5074 0.5492 0.5149 0.5776 0.5298 600,000 -7.3509

2081 0.6035 0.6642 0.5046 0.5464 0.5119 0.5743 0.5264 600,000 -7.3550

2082 0.5998 0.6605 0.5018 0.5436 0.5088 0.5710 0.5231 600,000 -7.3590

2083 0.5962 0.6568 0.4991 0.5409 0.5058 0.5677 0.5198 600,000 -7.3631

2084 0.5926 0.6531 0.4963 0.5382 0.5028 0.5644 0.5165 600,000 -7.3671

2085 0.5890 0.6494 0.4936 0.5355 0.4998 0.5612 0.5132 600,000 -7.3711

2086 0.5855 0.6458 0.4909 0.5328 0.4968 0.5579 0.5099 600,000 -7.3751
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

139

t M36,t

TR M37,tTR M38,tTR M39,tTR M40,tTR M41,tTR M42,tTR LtTR EtTR

2087 0.5820 0.6421 0.4882 0.5301 0.4938 0.5547 0.5067 600,000 -7.3791

2088 0.5785 0.6386 0.4856 0.5275 0.4909 0.5515 0.5035 600,000 -7.3831

2089 0.5750 0.6350 0.4829 0.5249 0.4880 0.5484 0.5003 600,000 -7.3872

2090 0.5716 0.6315 0.4803 0.5222 0.4851 0.5452 0.4971 600,000 -7.3913

2091 0.5682 0.6280 0.4777 0.5196 0.4822 0.5421 0.4939 600,000 -7.3955

2092 0.5648 0.6245 0.4751 0.5171 0.4794 0.5389 0.4908 600,000 -7.3997

2093 0.5615 0.6210 0.4726 0.5145 0.4765 0.5358 0.4877 600,000 -7.4039

2094 0.5582 0.6176 0.4700 0.5120 0.4737 0.5328 0.4846 600,000 -7.4081

2095 0.5549 0.6142 0.4675 0.5094 0.4709 0.5297 0.4816 600,000 -7.4125

2096 0.5517 0.6108 0.4650 0.5069 0.4681 0.5267 0.4785 600,000 -7.4168

2097 0.5484 0.6075 0.4625 0.5044 0.4654 0.5236 0.4755 600,000 -7.4212
Appendices

140

tS

t

TR

O

t

TR

Ut

TR

It

TR

R

t

TR

Wt

TR DIMtTR DIFtTR DRMtTR DRFtTR

2023-3.4494 1,752,000 -3.1136-2.6592-0.0098 0.1484-3.2041-3.1350-1.8958-2.0009

2024-3.5169 1,752,000 -3.0154-2.8952 0.0102 1.2022-2.9845-2.9089-1.7666-1.8803

2025-3.5785 1,350,000 -3.0227-2.9188 0.0115 1.6219-2.9200-2.8786-1.8210-1.9432

2026-3.6110 1,350,000 -3.0433-2.9188 0.0116 1.6554-2.8920-2.8711-2.0423-2.1637

2027-3.6417 1,350,000 -3.0635-2.9188 0.0133 1.6384-2.8658-2.8482-2.0770-2.1951

2028-3.6708 1,350,000 -3.0666-2.9188 0.0151 1.5737-2.9350-2.9180-2.1140-2.2286

2029-3.6986 1,350,000 -3.0677-2.9188 0.0179 1.5951-2.9576-2.9431-2.1450-2.2559

2030-3.7252 1,350,000 -3.0689-2.9188 0.0203 1.5675-2.9730-2.9539-2.1760-2.2845

2031-3.7507 1,350,000 -3.0700-2.9188 0.0218 1.5395-2.9619-2.9413-2.2087-2.3148

2032-3.7751 1,350,000 -3.0714-2.9188 0.0223 1.3293-2.9479-2.9272-2.2432-2.3474

2033-3.7985 1,350,000 -3.0732-2.9188 0.0228 1.2430-2.9480-2.9274-2.2354-2.3421

2034-3.8209 1,350,000 -3.0749-2.9188 0.0230 1.2296-2.9480-2.9277-2.2337-2.3417

2035-3.8424 1,350,000 -3.0767-2.9188 0.0230 1.2277-2.9480-2.9280-2.2338-2.3427

2036-3.8630 1,350,000 -3.0785-2.9188 0.0230 1.2259-2.9480-2.9283-2.2335-2.3418

2037-3.8828 1,350,000 -3.0807-2.9188 0.0230 1.2271-2.9480-2.9286-2.2313-2.3412

2038-3.9018 1,350,000 -3.0828-2.9188 0.0230 1.2207-2.9480-2.9289-2.2294-2.3394

2039-3.9201 1,350,000 -3.0847-2.9188 0.0230 1.2226-2.9480-2.9291-2.2287-2.3394

2040-3.9376 1,350,000 -3.0862-2.9188 0.0230 1.2027-2.9480-2.9294-2.2267-2.3378

2041-3.9545 1,350,000 -3.0870-2.9188 0.0230 1.1817-2.9480-2.9297-2.2255-2.3376

2042-3.9707 1,350,000 -3.0875-2.9188 0.0230 1.1615-2.9480-2.9300-2.2243-2.3378

2043-3.9864 1,350,000 -3.0875-2.9188 0.0230 1.1487-2.9481-2.9300-2.2276-2.3391

2044-4.0015 1,350,000 -3.0871-2.9188 0.0230 1.1303-2.9480-2.9300-2.2296-2.3408

2045-4.0161 1,350,000 -3.0863-2.9188 0.0230 1.1132-2.9481-2.9300-2.2313-2.3430

2046-4.0301 1,350,000 -3.0849-2.9188 0.0230 1.0984-2.9481-2.9300-2.2326-2.3435

2047-4.0437 1,350,000 -3.0834-2.9188 0.0230 1.1042-2.9480-2.9300-2.2337-2.3448

2048-4.0569 1,350,000 -3.0819-2.9188 0.0230 1.1008-2.9481-2.9300-2.2341-2.3456

2049-4.0695 1,350,000 -3.0803-2.9188 0.0230 1.1034-2.9481-2.9300-2.2352-2.3467

2050-4.0818 1,350,000 -3.0789-2.9188 0.0230 1.0987-2.9480-2.9300-2.2361-2.3473

2051-4.0936 1,350,000 -3.0774-2.9188 0.0230 1.0989-2.9481-2.9300-2.2368-2.3481

2052-4.1049 1,350,000 -3.0760-2.9188 0.0230 1.0991-2.9481-2.9300-2.2376-2.3498

2053-4.1159 1,350,000 -3.0746-2.9188 0.0230 1.1012-2.9480-2.9300-2.2388-2.3499

2054-4.1265 1,350,000 -3.0732-2.9188 0.0230 1.0953-2.9481-2.9300-2.2398-2.3505

2055-4.1367 1,350,000 -3.0719-2.9188 0.0230 1.0939-2.9481-2.9300-2.2404-2.3508

2056-4.1465 1,350,000 -3.0707-2.9188 0.0230 1.0959-2.9480-2.9300-2.2401-2.3508

2057-4.1560 1,350,000 -3.0698-2.9188 0.0230 1.1025-2.9481-2.9300-2.2402-2.3511

2058-4.1651 1,350,000 -3.0690-2.9188 0.0230 1.1103-2.9481-2.9300-2.2396-2.3502

2059-4.1739 1,350,000 -3.0685-2.9188 0.0230 1.1175-2.9481-2.9300-2.2386-2.3496

2060-4.1823 1,350,000 -3.0682-2.9188 0.0230 1.1229-2.9481-2.9300-2.2384-2.3495

2061-4.1904 1,350,000 -3.0681-2.9188 0.0230 1.1289-2.9481-2.9300-2.2387-2.3498

2062-4.1982 1,350,000 -3.0682-2.9188 0.0230 1.1328-2.9480-2.9300-2.2381-2.3491

2063-4.2057 1,350,000 -3.0684-2.9188 0.0230 1.1347-2.9481-2.9300-2.2381-2.3489

2064-4.2129 1,350,000 -3.0688-2.9188 0.0230 1.1354-2.9480-2.9300-2.2374-2.3482

2065-4.2199 1,350,000 -3.0693-2.9188 0.0230 1.1373-2.9481-2.9300-2.2372-2.3481

2066-4.2266 1,350,000 -3.0699-2.9188 0.0230 1.1368-2.9481-2.9300-2.2362-2.3468

2067-4.2330 1,350,000 -3.0706-2.9188 0.0230 1.1402-2.9481-2.9300-2.2364-2.3474

2068-4.2392 1,350,000 -3.0713-2.9188 0.0230 1.1366-2.9481-2.9300-2.2365-2.3473

2069-4.2453 1,350,000 -3.0718-2.9188 0.0230 1.1382-2.9481-2.9300-2.2367-2.3468

2070-4.2511 1,350,000 -3.0723-2.9188 0.0230 1.1401-2.9481-2.9300-2.2371-2.3464

2071-4.2567 1,350,000 -3.0726-2.9188 0.0230 1.1376-2.9481-2.9300-2.2375-2.3467

2072-4.2622 1,350,000 -3.0728-2.9188 0.0230 1.1333-2.9481-2.9300-2.2373-2.3466

2073-4.2675 1,350,000 -3.0729-2.9188 0.0230 1.1312-2.9481-2.9300-2.2372-2.3472

2074-4.2727 1,350,000 -3.0730-2.9188 0.0230 1.1344-2.9481-2.9300-2.2368-2.3454

2075-4.2778 1,350,000 -3.0730-2.9188 0.0230 1.1356-2.9481-2.9300-2.2367-2.3462

2076-4.2827 1,350,000 -3.0729-2.9188 0.0230 1.1382-2.9480-2.9300-2.2367-2.3453

2077-4.2874 1,350,000 -3.0727-2.9188 0.0230 1.1394-2.9481-2.9300-2.2366-2.3456

2078-4.2921 1,350,000 -3.0724-2.9188 0.0230 1.1375-2.9481-2.9300-2.2370-2.3461

2079-4.2966 1,350,000 -3.0720-2.9188 0.0230 1.1348-2.9481-2.9300-2.2376-2.3459

2080-4.3009 1,350,000 -3.0716-2.9188 0.0230 1.1304-2.9481-2.9300-2.2377-2.3462

2081-4.3052 1,350,000 -3.0712-2.9188 0.0230 1.1326-2.9481-2.9300-2.2373-2.3462

2082-4.3093 1,350,000 -3.0708-2.9188 0.0230 1.1319-2.9481-2.9300-2.2374-2.3463

2083-4.3133 1,350,000 -3.0705-2.9188 0.0230 1.1322-2.9481-2.9300-2.2370-2.3459

2084-4.3172 1,350,000 -3.0702-2.9188 0.0230 1.1332-2.9480-2.9300-2.2373-2.3458

2085-4.3209 1,350,000 -3.0699-2.9188 0.0230 1.1311-2.9481-2.9300-2.2369-2.3461

2086-4.3244 1,350,000 -3.0698-2.9188 0.0230 1.1342-2.9481-2.9300-2.2368-2.3456

2087-4.3279 1,350,000 -3.0697-2.9188 0.0230 1.1295-2.9481-2.9300-2.2376-2.3457

2088-4.3312 1,350,000 -3.0696-2.9188 0.0230 1.1276-2.9481-2.9300-2.2379-2.3459

2089-4.3344 1,350,000 -3.0697-2.9188 0.0230 1.1250-2.9481-2.9300-2.2377-2.3467
Equation Regression Statistics, Parameter Estimates, and Bounding Statistics

141

t St

TR OtTR UtTR ItTR RtTR WtTR DIMtTR DIFtTR DRMtTR DRFtTR

2090 -4.3374 1,350,000 -3.0698 -2.9188 0.0230 1.1233 -2.9481 -2.9300 -2.2378 -2.3469

2091 -4.3403 1,350,000 -3.0700 -2.9188 0.0230 1.1214 -2.9481 -2.9300 -2.2382 -2.3470

2092 -4.3431 1,350,000 -3.0703 -2.9188 0.0230 1.1213 -2.9481 -2.9300 -2.2387 -2.3470

2093 -4.3457 1,350,000 -3.0706 -2.9188 0.0230 1.1212 -2.9481 -2.9300 -2.2385 -2.3473

2094 -4.3483 1,350,000 -3.0709 -2.9188 0.0230 1.1210 -2.9480 -2.9300 -2.2388 -2.3474

2095 -4.3507 1,350,000 -3.0712 -2.9188 0.0230 1.1205 -2.9481 -2.9300 -2.2389 -2.3472

2096 -4.3530 1,350,000 -3.0715 -2.9188 0.0230 1.1207 -2.9481 -2.9300 -2.2392 -2.3473

2097 -4.3552 1,350,000 -3.0719 -2.9188 0.0230 1.1213 -2.9481 -2.9300 -2.2395 -2.3475

3. Bounding Results

The program also bounds certain assumptions for reasonableness. If a bound is breached in a 

particular year, then the final value is set to that bound and the program continues running. In both 

the runs without parameter uncertainty and with parameter uncertainty for the estimated mean, the 

bounds are:

1. Total Fertility Rate: The total fertility rate is bounded by 0 on the low end. We set a high 

bound of 100,000, although the highest value reached is 4.62.

2. LPR New Arrival Immigration: 0 persons through 30 million persons.

3. Other-than-LPR Immigration: 100,000 persons through 15 million persons.

4. Economic VAR Framework (Unemployment Rate, Inflation Rate, and Real Interest Rate): 

In cases where the nominal interest rate is less than 0 when combining the inflation rate 

and real interest rate, the real interest rate is set such that the nominal interest rate is 0.

Bounds of plus and minus 40 percent were put on the inflation rate and real interest rate.

For the real interest rate, the plus/minus 40-percent bound is imposed before the bound 

requiring a nominal interest rate of at least 0 percent.

5. Percentage Change in Average Real Wage: The percentage change in average real wage is 

bounded by plus and minus 40 percent.

Table V.4 shows the percentage of runs where a bound was applied in at least one year. 

Table V.4 – Percentage of Runs Where a Bound Was Applied in at Least One Year

Assumption Without Parameter 

Uncertainty

With Parameter Uncertainty for 

the Estimated Mean

Total Fertility Rate 0.3 0.5

LPR New Arrival Immigration 0.0 0.1

Other-than-LPR Immigration 0.2 1.2

Inflation Rate 0.0 0.0

Real Interest Rate 53.7 53.2

Real Average Covered Wage 0.0 0.0

All instances of applying a bound to the real interest rate are caused by the resulting nominal 

interest rate being less than zero. Note that although many simulations have at least one year where 

the real interest rate is restricted by the bound, the bound is typically not breached in many years 

in the same simulation. Without parameter uncertainty, for each run with at least one year restricted 

by the bound, the average number of years that were restricted was about four per run. With 

parameter uncertainty for the expected mean, this value is about nine per run. These frequencies 
Appendices

142

are consistent with the observation that the zero lower bound on nominal interest rates occasionally 

constrains monetary policy.

The other assumptions (those not listed here) do not have any bounds imposed, because the 

transformations of the original data make bounding unnecessary. The regressions for the transfer 

rate, legal emigration rate, unemployment rate, male and female disability incidence rates, and 

male and female disability recovery rates are all performed on data transformed to be between 0 

and 100 percent.

Although it is not a bounding issue per se, three runs with parameter uncertainty for the estimated 

mean produced a negative number of LPR immigrants plus citizens for at least one year/sex/age 

combination. When this occurred, the number of LPR immigrants plus citizens for these particular

year/sex/age combinations was reset to one.
Bibliography

143

G. BIBLIOGRAPHY

Adams, Jeanne C., Brainerd, W. S., Martin, J. T., and Smith, B. T. (1994). Fortran Top 90. 

Albuquerque: Unicomp.

Atkinson, Kendall E. (1989). An Introduction to Numerical Analysis, 2nd edition. New York: John 

Wiley & Sons, Inc.

Bell, Felicitie C. and Miller, M. L. (2005). Life Tables for the United States Social Security Area 

1900-2100. Social Security Administration Actuarial Study No. 120. Baltimore, MD: Social 

Security Administration, Office of the Chief Actuary.

Barndorff-Nielson and Shou, G. (1973). “On the Parametrization of Autoregressive Models by 

Partial Autocorrelations,” Journal of Multivariate Analysis 3, pages 408-419.

Board of Trustees of the Federal Old-Age and Survivors Insurance and Disability Insurance Trust 

Funds (2003). The 2003 Annual Report of the Board of Trustees of the Federal Old-Age and 

Survivors Insurance and Disability Insurance Trust Funds. Washington, D.C.: U.S. Government 

Printing Office. 

Board of Trustees of the Federal Old-Age and Survivors Insurance and Disability Insurance Trust 

Funds (2023). The 2023 Annual Report of the Board of Trustees of the Federal Old-Age and 

Survivors Insurance and Disability Insurance Trust Funds. Washington, D.C.: U.S. Government 

Printing Office. 

Box, George E. P., Jenkins, G.M., and Reinsel, G.C. (1994). Times Series Analysis, Forecasting 

and Control, 3rd edition. New York: Prentice Hall.

Burdick, Clark and Manchester, J. (2004). Stochastic Models of the Social Security Trust Funds. 

Social Security Bulletin, Vol. 65, No. 1.

Cheng, Anthony W., Miller, Michael L., Morris, Michael, Schultz, Jason P., Skirvin, J. Patrick, 

and Walder, Danielle P. (2004) A Stochastic Model of the Long-Range Financial Status of the 

OASDI Program. Social Security Administration Actuarial Study No. 117, Baltimore, MD: Social 

Security Administration, Office of the Actuary.

Congressional Budget Office. (2001). Uncertainty in Social Security’s Long-Term Finances: A 

Stochastic Analysis. Technical Report.

Edwards, Ryan D., et al. (2003). Key Equations in the Tuljapurkar-Lee model of the Social Security 

System, Berkeley: Stanford University Press.

Feng, Runhuan, (November 2016). Nested Modeling for Insurance Compaines.
Appendices

144

Foster, Richard S. (1994). A Stochastic Evaluation of the Short-Range Economic Assumptions in 

the 1994 OASDI Trustees Report. Social Security Administration Actuarial Study No. 109. 

Baltimore, MD: Social Security Administration, Office of the Chief Actuary.

Hamilton, James D. (1994). Time-Series Analysis. Princeton: Princeton University Press.

Harris, Amy, Meyerson, N., and Smith, J. (2001). "Social Insecurity? The Effects of Equity 

Investments on Social Security Finances." National Tax Journal, September 2001, pp. 645-648.

Hartman, Brian, Richardson, Robert, and Bateman, Rylan (April 2017). Parameter Uncertainty.

Holmer, Martin R. (2003). Introductory Guide to SSASIM. Washington, D.C.: Policy Simulation 

Group, January 2003. 

Holmer, Martin R. and Bender, C. S. (1996). “Stochastic Simulation of Trust Fund AssetAllocation,” Report of the 1994–1995 Advisory Council on Social Security, Volume II, pages 

431-450.

Jones, M.C. (1987). “Randomly choosing Parameters from the Stationarity and Invertibility 

Region of Autoregressive-Moving Average Models,” Journal of the Royal Statistical Society. 

Series C (Applied Statistics), Volume 36, No. 2, pages 134-138.

Kellison, Stephen G. (1991). Theory of Interest, (2nd ed.). Boston: Irwin/McGraw-Hill.

Kleinow, Torsten and Richards, Stephen J. (February 2016). Parameter risk in time-series 

mortality forecasts. Edinburgh, U.K.

Klugman, Stuart A., Panjer, H. H., and Willmot, G. E. (1998). Loss Models: From Data to 

Decisions. New York: John Wiley & Sons, Inc.

Knuth, Donald E. (1997). Art of Computer Programming, Volume 2: Seminumerical Algorithms, 

3rd Edition. Boston: Addison Wesley.

Lee, Ronald D., Anderson, M. W., and Tuljapurkar, S. (2003). Stochastic Forecasts of the Social 

Security Trust Fund: Report for the Social Security Administration.

Marriott, John, Ravishanker, N. and Gelfand, Alan E. (June 24, 1993). Bayesian Analysis of ARMA 

Processes: Complete Sampling Based Inference Under Full Likelihoods. Stanford, CA.

Monahan, John F. (August 1984). “A Note on Enforcing Stationarity in Autoregressive-Moving 

Average Models,” Biometrika, Volume 71, No. 2, pages 403-404.

Park, Stephen K. and Miller, Keith W. (1988). “Random Number Generators: Good Ones are Hard 

to Find,” Communications of the ACM, Volume 31, No. 10, pages 1192-1201.
Bibliography

145

Pattison, David H. (2006). Parameter Uncertainty Simulations of the Total Fertility Rate. 

Unpublished manuscript.

Pattison, David H. (2013). A Protocol for ARMA(P,Q). Unpublished manuscript.

Pattison, David H. (2016). VAR(P) Protocol for Stochastic Simulation. Unpublished manuscript.

Pindyck, Robert S. and Rubinfeld, D. L. (1998). Econometric Models and Economic Forecast, 4th 

edition. Burr Ridge, IL: Irwin/McGraw-Hill.

Press, William H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2001). Numerical 

Recipes in Fortran 77, Volume 1, 2nd Edition, New York: Cambridge University Press.

Ross, Sheldon M. (1997). Simulation, 2nd edition, San Diego: Academic Press.

Stock, James H. and Watson, Mark W. (2001). “Vector Autoregressions,” Journal of Economic 

Perspectives, Volume 15, No. 4, pages 101-115.

Technical Panel on Assumptions and Methods (1999). Report to the Board: 1999 Technical Panel 

on Assumptions and Methods Washington, D.C.: U.S. Government Printing Office.

Wise, J. (June 1956). “Stationarity Conditions of Stochastic Process of the Autoregressive and 

Moving-Average Type,” Biometrika, Volume 43, No. 1/2, pages 215-219.

Wooldridge Jeffrey M. (2005). Introductory Econometrics, 3rd edition, Chapter 12: Serial 

correlation and heteroskedasticity in time series regressions.