A STOCHASTIC MODEL

OF THE LONG-RANGE FINANCIAL STATUS

OF THE OASDI PROGRAM

ACTUARIAL STUDY NO. 117

Primary Authors:

Anthony W. Cheng

Michael L. Miller 

Michael Morris 

Jason P. Schultz 

J. Patrick Skirvin 

Danielle P. Walder

Additional Authors:

Robert E. Baldwin

Sharon K. Chu

Josiah M. Lynch

William M. Piet

Lesley B. Reece

Social Security Administration

Office of the Chief Actuary

September 2004

SSA Pub. No. 11-11555
.
i

FOREWORD

The 2004 Annual Report of the OASDI Board of Trustees includes a brief description of the

OCACT Stochastic Model (OSM, Version 2004.1) and projections produced using it. Actuarial

Study No. 117 is intended to provide more extensive details of this model and its results.

The purpose of the OSM is to provide probability distributions for the range of possible future

experience of the OASDI program under present law. This probabilistic representation of uncertainty augments the presentations of low and high cost alternatives and sensitivity analyses that

have traditionally been included in the Trustees Report. It should be noted that this model is in its

first stage of development. Future improvements and refinements to the model are expected. In

particular, future revisions are expected to more fully reflect the range of uncertainty about future

experience.

Anthony W. Cheng and Michael L. Miller developed the equations for the OSM, Version 2004.1.

Programmers of the individual modules are identified below:

Seung An and Sue Kunkel of OCACT provided direction and support throughout the development process. Sheldon Baker and Lise Holler assisted in the publication of the study. In addition,

an acknowledgement must be made of the Long Term Modeling Group of the Congressional Budget Office and the Office of Policy of the Social Security Administration, both of which provided

consultation during the early development of the OSM.

 Alice H. Wade 

Deputy Chief Actuary

September 30, 2004

Stephen C. Goss 

Chief Actuary 

September 30, 2004

Assumptions Michael Morris

Population Michael L. Miller 

Economics William M. Piet 

Insured Sharon K. Chu 

DIB Robert E. Baldwin 

OASIB Danielle P. Walder 

Awards Michael D. Clingman and Lesley B. Reece

Cost Jason P. Schultz

Summary Results Josiah M. Lynch
.
TABLE OF CONTENTS

iii

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

II. Equation Selection and Parameter Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

A.Fertility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

B. Mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

C. Immigration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1. Legal Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. Legal Emigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3. Net Other Immigration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

D.Related Economic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1. Unemployment Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. Inflation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3. Real Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

E. Real Average Covered Wage (Percent Change) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

F. Disability Incidence Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

G.Disability Recovery Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

III. Documentation of the Computer Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

A.Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

B. Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1. Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2. Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3. Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4. Insured (Fully Insured and Disability Insured) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5. Disability Insurance Beneficiaries (DIB). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6. Old-Age and Survivors Insurance Beneficiaries (OASIB) . . . . . . . . . . . . . . . . . . . . 24

7. Awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8. Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9. Summary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

IV. Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A.Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1. Fertility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2. Mortality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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

4. Unemployment Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5. Inflation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6. Real Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7. Real Average Covered Wage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8. Disability Incidence Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

9. Disability Recovery Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
iv

B. Actuarial Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1. Annual Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2. Summary Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

C. Sensitivity Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1. Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2. Deterministic Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

V. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A.Principal Measures of Financial Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

1. Trust Fund Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2. Annual Income Rates, Cost Rates, and Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3. Comparison of Workers to Beneficiaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4. Trust Fund Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5. Actuarial Balance, Summarized Rates, Open Group Unfunded Obligation . . . . . . . 60

B. Time-Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

C. Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

D.Equation Parameter Estimates and Regression Statistics . . . . . . . . . . . . . . . . . . . . . . . . 67

E. Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

F. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
LIST OF TABLES

v

III. Documentation of the Computer Program

Table III.1— Module Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

IV. Results

Table IV.1— Total Fertility Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Table IV.2— Male Period Life Expectancies at Birth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Table IV.3— Female Period Life Expectancies at Birth. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Table IV.4— Male Period Life Expectancies at Age 65. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Table IV.5— Female Period Life Expectancies at Age 65 . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Table IV.6— Legal Immigration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Table IV.7— Legal Emigration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Table IV.8— Net Other Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Table IV.9— Unemployment Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Table IV.10— Inflation Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Table IV.11— Real Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Table IV.12— Real Average Covered Wage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Table IV.13— Male Disability Incidence Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Table IV.14— Female Disability Incidence Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Table IV.15— Male Disability Recovery Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Table IV.16— Female Disability Recovery Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Table IV.17— Estimates of the OASDI Program, Calendar Year 2078. . . . . . . . . . . . . . . . . 50

Table IV.18— Estimates of the OASDI Program, Calendar Years 2004-78 . . . . . . . . . . . . . 53

Table IV.19— Sensitivity to Varying Selected Assumptions, Calendar Years 2004-78 . . . . 55

Table IV.20— Comparison of Stochastic and Deterministic Models,

Calendar Years 2004-78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

V. Appendices

Equation Parameters from the 2004 Trustees Report Assumptions . . . . . . . . 74
.
LIST OF FIGURES

vii

II. Equation Selection and Parameter Estimation

Figure II.1— U.S. Total Fertility Rate, Calendar Years 1917-2002. . . . . . . . . . . . . . . . . . . 4

Figure II.2— U.S. Legal Immigration, Calendar Years 1901-2002 . . . . . . . . . . . . . . . . . . . 7

Figure II.3— Unemployment Rate, Calendar Years 1960-2002. . . . . . . . . . . . . . . . . . . . . . 10

Figure II.4— Inflation Rate (CPI), Calendar Years 1960-2002 . . . . . . . . . . . . . . . . . . . . . . 11

Figure II.5— Real Interest Rate, Calendar Years 1960-2002 . . . . . . . . . . . . . . . . . . . . . . . . 12

Figure II.6— Real Average Wage (Percent Change), Calendar Years 1968-2002. . . . . . . . 14

Figure II.7— Male Disability Incidence Rate, Calendar Years 1970-2003 . . . . . . . . . . . . . 16

Figure II.8— Female Disability Incidence Rate, Calendar Years 1970-2003 . . . . . . . . . . . 16

IV. Results

Figure IV.1— Total Fertility Rate, Calendar Years 1917-2078 . . . . . . . . . . . . . . . . . . . . . . . 35

Figure IV.2— Male Period Life Expectancy at Birth, Calendar Years 1900-2078 . . . . . . . . 37

Figure IV.3— Female Period Life Expectancy at Birth, Calendar Years 1900-2078 . . . . . . 37

Figure IV.4— Male Period Life Expectancy at Age 65, Calendar Years 1900-2078 . . . . . . 38

Figure IV.5— Female Period Life Expectancy at Age 65, Calendar Years 1900-2078. . . . . 39

Figure IV.6— Legal Immigration, Calendar Years 1901-2078 . . . . . . . . . . . . . . . . . . . . . . . 40

Figure IV.7— Legal Emigration, Calendar Years 1980-2078 . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure IV.8— Net Other Immigration, Calendar Years 1960-2078. . . . . . . . . . . . . . . . . . . . 42

Figure IV.9— Unemployment Rate, Calendar Years 1960-2078. . . . . . . . . . . . . . . . . . . . . . 43

Figure IV.10— Inflation Rate, Calendar Years 1960-2078 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Figure IV.11— Real Interest Rate, Calendar Years 1960-2078 . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure IV.12— Real Average Covered Wage, Calendar Years 1968-2078 . . . . . . . . . . . . . . . 46

Figure IV.13— Male Disability Incidence Rate, Calendar Years 1970-2078 . . . . . . . . . . . . . 47

Figure IV.14— Female Disability Incidence Rate, Calendar Years 1970-2078 . . . . . . . . . . . 47

Figure IV.15— Male Disability Recovery Rate, Calendar Years 1986-2078 . . . . . . . . . . . . . 49

Figure IV.16— Female Disability Recovery Rate, Calendar Years 1986-2078. . . . . . . . . . . . 49

Figure IV.17— Annual Trust Fund Ratio, Calendar Years 2004-78 . . . . . . . . . . . . . . . . . . . . 51

Figure IV.18— Annual Cost and Income Rates, Calendar Years 2004-78 . . . . . . . . . . . . . . . 52

Figure IV.19— Frequency Distribution of the Long-Range Actuarial Balance, 

Calendar Years 2004-78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure IV.20— Cumulative Distribution of the Long-Range Actuarial Balance, 

Calendar Years 2004-78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
.
1

A STOCHASTIC MODEL OF THE LONG-RANGE FINANCIAL

STATUS OF THE OASDI PROGRAM

I. INTRODUCTION

Each year the Board of Trustees of the Federal Old-Age and Survivors Insurance (OASI) and Disability

Insurance (DI) Trust Funds provides three separate sets of long-range (75-year) assumptions for key

demographic and economic 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 policy makers 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.

This Actuarial Study documents the OCACT Stochastic Model (OSM), Version 2004.1. The OSM

assigns random variation for some of the key demographic and economic assumptions. These include the

total fertility rate, rates of change in mortality, levels of immigration, emigration, and net other immigration, unemployment rate, inflation rate, real interest rate, growth rate in the real average wage, and disability incidence and recovery rates. The OSM is designed such that the projected values for each

variable are centered on the intermediate assumptions of the 2004 Trustees Report (TR04II).

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 ranges of 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 OASDI Trust Funds

(e.g., the long-range actuarial balance). 

It is important to note that the results presented here should be interpreted with caution and with a full

understanding of the inherent limitations of the process. If certain changes are made to the model specifications, then the projections could be significantly altered. For example, if any one equation is respecified, or if the degree of interdependency among variables is modified, or if the historical period used in

fitting any equation is changed, the results would be different. In addition, if variables other than those

mentioned above (such as labor force participation rates, retirement rates, marriage rates, and divorce

rates) were included in the stochastic modeling, the results would differ. Finally, additional variability

would be expected to result from incorporating statistical approaches that would allow for potential structural shifts in the long-range central tendencies (i.e., parameter uncertainty). In conclusion, the current

OSM’s projected variation is likely to be narrower than the true range of uncertainty for the future.
Introduction

2

The remaining chapters of this Actuarial Study provide detailed information from the OSM.

Chapter II presents the equations used to model random variation in the assumptions. Chapter III

explains the overall structure of the OSM and its modules. 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. Chapter

IV also presents a sensitivity analysis for each stochastic assumption.

Six appendices to this Actuarial Study are included. Appendix A contains background material on

various financial estimates of the OASDI program. Appendices B and C contain introductions to

time-series modeling and Monte Carlo simulation, respectively. Appendix D provides additional

details of the time-series equations used in the OSM. Appendix E provides a glossary of terms

used in this study, and appendix F is a bibliography of references cited in this study.
Fertility

3

II. EQUATION SELECTION AND PARAMETER ESTIMATION

Equations were selected for a set of assumption variables that include the total fertility rate,

changes in mortality, levels of immigration, emigration, and net other immigration, unemployment rate, inflation rate, real interest rate, growth rate in the real average wage, and disability incidence and recovery rates. The parameters of the equations were estimated using standard timeseries modeling techniques, and then modified so that the projected variation was centered on the

TR04II. Appendix B discusses the theory behind this procedure. This chapter briefly describes

and presents each equation, while appendix D provides more detailed information and statistics.

Historical time-series data can be obtained from OCACT.1

A. FERTILITY

The total fertility rate is the sum of age-specific birth rates2 for women aged 14 through 49. Thus,

the total fertility rate for a given year may be interpreted as the average number of children that

would be born to a woman throughout her lifetime if she were to survive the entire childbearing

period and experience the observed age-specific birth rate each year of her life.

Historical values for the total fertility rate in the U.S. for 1917 through 2002 are available from

the National Center for Health Statistics3 and the U.S. Census Bureau.4 The total fertility rate

ranged from a minimum of 1.74 in 1976 to a maximum of 3.68 in 1957, and has remained relatively stable, near 2.00, since 1990. The rate was 2.01 in 2002.

Using time-series analysis, an ARMA(4,1) equation was selected and parameters were estimated

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

value was 0.98. The modified equation is:

(1)

In this equation, Ft

 represents the total fertility rate in year t; represents the projected total

fertility rate from the TR04II in year t; ft represents the deviation of the total fertility rate from

the TR04II total fertility rate in year t (i.e. ) and εt

 represents the random error in

year t.

1 Contact OCACT (actuary@ssa.gov).

2 Age-specific birth rates are defined as the number of live births to women of a given age divided by the estimated female population of the given age at midyear.

3 www.cdc.gov/nchs

4 www.census.gov

12 3 4 1 1.99 1.51 0.91 0.42 0.67 . TR FF f f f f tt t t t t t t ε ε = + − + − +− −− − − −

TR Ft

TR

t tt f FF = −
Equation Selection and Parameter Estimation

4

Figure II.1—U.S. Total Fertility Rate, Calendar Years 1917-2002

1.50

2.00

2.50

3.00

3.50

4.00

1915 1925 1935 1945 1955 1965 1975 1985 1995 2005

Calendar Year

Rate

Actual

Fitted
Mortality

5

B. MORTALITY

The annual rate of decrease in the central death rate5 (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 2000.6 Data for the annual numbers of

deaths and the U.S. resident population are from the National Center for Health Statistics and the

U.S. Census Bureau, respectively. For the population aged 65 or older, annual deaths and enrollments are from the Centers for Medicare & Medicaid Services.

Using the approach of other researchers (Congressional Budget Office, 2001), 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 modified equation is:

(2)

In this equation, MRk,t represents the annual rate of decrease in the central death rate for group k

in year t; represents the projected annual rate of decrease from the TR04II for group k in

year t; mrk,t represents the deviation of the annual rate of decrease from the TR04II value for

group k in year t; and εk,t represents the random error for group k in year t. Appendix D contains

the estimates of the parameters, φk, in Equation (2).

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

Appendix B discusses this technique. The Cholesky matrix used was with the age groups

in ascending order with alternating male and female groups.

5 The central death rate is defined as the annual number of deaths for a particular group divided by the estimated population of

that group at midyear.

6 A detailed description of the methodology used in calculating death rates by age and sex can be found in Actuarial Study

No. 116. www.socialsecurity.gov/OACT/NOTES/as116/as116_Foreword.html

, , ,1 , . TR MR MR mr kt kt k kt kt φ ε =+ + −

,

TR MRk t

42 42 ×
Equation Selection and Parameter Estimation

6

C. IMMIGRATION

Total immigration is defined here as legal immigration minus legal emigration plus net other

immigration. Each component is modeled separately.

1. Legal Immigration

Legal immigration is defined as persons lawfully admitted for permanent residence into the

United States.7 The level of legal immigration largely depends on legislation which basically

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, employment-based preferences, and diversity immigrants. However, no numerical limits currently exist for immediate relatives of U.S. citizens.

Historical data for legal U.S. immigration for years 1901 through 2002 are from the U.S. Citizenship and Immigration Services.8 Legal immigration averaged nearly one million per year from

1900 through 1914, then decreased substantially to about 23,000 in 1933. Since the mid-1940s,

legal immigration increased steadily to over one million in 2002.

An ARMA(4,1) equation was selected and parameters were estimated using the entire range of

historical data. The R-squared value was 0.92. Figure II.2 presents the actual and fitted values.

The modified equation is:

(3)

In this equation, IMt

 represents the annual level of legal immigration in year t; represents

the projected level of legal immigration from the TR04II in year t; imt

 represents the deviation of

the annual level of legal immigration from the TR04II value in year t; and εt

 represents the random error in year t.

7 For more detailed information, refer to the Yearbook of Immigration Statistics - 

uscis.gov/graphics/shared/aboutus/statistics/ybpage.htm.

8 Formerly known as the Immigration and Naturalization Service (INS).

12 34 1 1.08 0.54 0.69 0.31 0.49 . TR

t t t t t tt t IM IM im im im im ε ε = + − + − ++ −− −− −

TR

t IM
Immigration

7

2. Legal Emigration

Legal emigration is defined as the number of persons who lawfully leave the United States, and

are no longer considered to be a part of the Social Security program. Although annual emigration

data are not collected in the United States, the U.S. Census Bureau estimates that the level of emigration for the past century roughly totaled one-fourth of the level of legal immigration.

Using the Census estimates as an approximate guide, the parameters of Equation (3) are multiplied by one-fourth.9 The modified equation is:

(4)

In this equation, EMt

 represents the annual level of legal emigration in year t; represents

the projected annual level of legal emigration from the TR04II in year t; emt

 represents the deviation of the annual level of legal emigration from the TR04II value in year t; and εt

 represents the

random error in year t.

3. Net Other Immigration

Net other immigration is defined as the annual flow of persons into the United States minus the

annual flow of persons out of the United States who do not meet the above definition of legal

immigration or legal emigration. Thus, net other immigration includes unauthorized persons and

those not seeking permanent residence.

Figure II.2—U.S. Legal Immigration, Calendar Years 1901-2002

9 It is important to note that legal emigration is simulated independently from legal immigration.

0

200

400

600

800

1,000

1,200

1,400

1,600

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Calendar Year

Level (thousands)

Actual

Fitted

12 34 1 0.27 0.13 0.17 0.08 0.12 . TR EM EM em em em em t t t t t tt t ε ε = + − + − ++ −− − − −

TR EMt
Equation Selection and Parameter Estimation

8

Since complete data does not exist for net other immigration, we rely on indirect measurements

from the U.S. Census Bureau for our estimate. The Census Bureau accomplishes this by comparing two consecutive decennial census populations, applying known components of change, and

assigning the residual to net other immigration. The annual level of net other immigration is

assumed to follow a random walk. The modified equation is of the form:

(5)

In this equation, ∆Ot

 represents the change in the annual level of net other immigration in year

t; represents the projected annual change in the level of net other immigration consistent

with the TR04II in year t; and εt

 represents the random error in year t. The equation is initialized

with the level of net other immigration in 2003 from the TR04II.

2003 2003 ∆ =∆ + = ε ; . TR TR OO O O t tt

TR ∆Ot

2003 , TR O
Related Economic Variables

9

D. RELATED ECONOMIC VARIABLES 

The unemployment rate, inflation rate and real interest rate are simulated together using a vector

autoregression, in order to capture the relationship among the three variables that economic theory suggests are related.10 In the vector autoregression, each variable is regressed on the

prior-period values of all three variables. Vector autoregressions of different prior-period lengths

were tested and it was determined that a vector autoregression including 2 prior years provided a

reasonable fit. The historical period considered was 1960 to 2002. 

For the vector autoregression, the unemployment rates (as defined in section II.D.1) were

expressed as log-odds ratios11 to bound the values between 0 and 100 percent. In addition, the

adjusted inflation rates (as defined in section II.D.2) had a logarithmic transformation12 applied to

give them a lower bound for the vector autoregression. Instead of simply log-transforming the

inflation rate series, 3.0 percent was added to the inflation rate series prior to the log-transformation. This gave the inflation rates a lower bound of -3.0 percent. For the remainder of this section

and for section II.E, references to the unemployment rates and inflation rates refer to the transformed rates.

1. 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).13

The annual levels are an average of the 12 monthly rates. 

Between 1960 and 1974, the unemployment rate was relatively stable, ranging from a low of 3.5

percent in 1969 to a high of 6.7 percent in 1961. Between 1975 and 1994, the unemployment rate

moved to higher levels, and peaked at 9.7 percent in 1982. From 1994 to 2000, a rapid economic

expansion resulted in unemployment rates falling to 4.0 percent.

For the unemployment rate equation, the R-squared value was 0.85. The actual and fitted values

are shown in figure II.3. The modified equation is:

(6)

In this equation, Ut

 represents the unemployment rate in year t; is the unemployment rate

from the TR04II in year t; ut

 represents the deviation of the unemployment rate from the TR04II

unemployment rate in year t; it

 is the deviation of the inflation rate from the TR04II inflation rate

in year t; rt

 is the deviation of the real interest rate from the TR04II real interest rate in year t; and

ε1t is the random error in year t.

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

variables. CBO implemented this approach in their stochastic model.

11 where RUt

 is the unemployment rate in year t expressed as a decimal.

12 where is the percent change in the adjusted inflation rate in year t expressed as decimals.

13 www.bls.gov/cps/home.htm

log[ /(1 )], U RU RU t tt = −

log( 0.03), t t I = + π π t

1 2 1 2 1 21 0.96 0.30 0.40 0.08 0.75 0.61 . TR UU u u i i r r tt t t t t t t t ε =+ − + − + + + − − −− −−

TR Ut
Equation Selection and Parameter Estimation

10

2. 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). The BLS publishes historical values for the CPI.14 The

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

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

of the improvements on earlier values to improve consistency. The inflation rate is important

because it determines the annual cost-of-living adjustment (COLA) for OASDI benefits.

Over the historical period from 1960 to 2002, the adjusted inflation rate ranged from a low of 0.8

percent in 1961 and 1962 to a high of 10.9 percent in 1980, and was 1.4 percent in 2002. 

For the inflation rate equation, the R-squared value was 0.83. The actual and fitted values are

shown in figure II.4. The modified equation is:

(7)

In this equation, It

 is the CPI inflation rate in year t; is the CPI inflation rate from the TR04II

in year t; ut

 represents the deviation of the unemployment rate from the TR04II unemployment

rate in year t; it

 is the deviation of the inflation rate from the TR04II inflation rate in year t; rt is

the deviation of the real interest rate from the TR04II real interest rate in year t; and ε2t

 is the random error in year t.

Figure II.3—Unemployment Rate, Calendar Years 1960-2002

14 www.bls.gov/cpi

2.0

4.0

6.0

8.0

10.0

1960 1970 1980 1990 2000

Calendar Year

Percent

Actual

Fitted

1 2 1 2 1 22 0.77 0.72 0.60 0.30 4.85 1.80 . TR

tt t t t t t t t II u u i i r r ε =− + + + − + + − − −− −−

TR

t I
Related Economic Variables

11

3. Real Interest Rate

All securities held by the OASI and DI Trust Funds are issued by the Federal Government.

Almost all of these securities are special issues (i.e., securities issued only to the trust funds). Historical data on actual nominal interest rates of new purchases of these securities are published by

OCACT.15 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.16 Annual nominal interest rates are the average of the 12

monthly rates, which in practice are compounded semiannually.17 The real interest rate earned on

these obligations is equal to the annual (compounded) nominal yield divided by the inflation rate.

Looking at the period from 1960 to 2002, real interest rates on new purchases of special issues

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.

For the real interest rate equation, the R-squared value was 0.81. The actual and fitted values are

shown in figure II.5. The modified equation is:

(8)

Figure II.4—Inflation Rate (CPI), Calendar Years 1960-2002

15 www.socialsecurity.gov/OACT/ProgData/newIssueRates.html

16 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, www.socialsecurity.gov/OACT/NOTES/note142.html.

17 For example, the annualized nominal yield on a special issue with a 6.0-percent nominal interest rate is equal to 

(1+.06/2)2 -1 = 6.09 percent.

0.00

2.00

4.00

6.00

8.00

10.00

12.00

1960 1970 1980 1990 2000

Calendar Year

Percent

Actual

Fitted

1 2 1 2 1 23 0.06 0.05 0.03 0.03 1.23 0.32 . TR RR u u i i r r tt t t t t t t t ε =+ − + − + − + − − −− − −
Equation Selection and Parameter Estimation

12

In this equation, Rt represents the real interest rate in year t; represents the real interest rate

from the TR04II in year t; ut represents the deviation of the unemployment rate from the TR04II

unemployment rate in year t; it

 represents the deviation of the inflation rate from the TR04II

inflation rate in year t; rt

 represents the deviation of the real interest rate from the TR04II real

interest rate in year t; and ε3t is the random error term in year t.

Figure II.5—Real Interest Rate, Calendar Years 1960-2002

TR Rt

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

10.00

1960 1970 1980 1990 2000

Calendar Year

Percent

Actual

Fitted
Real Average Covered Wage

13

E. REAL AVERAGE COVERED WAGE (PERCENT CHANGE)

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

wage to the adjusted inflation rate. Because of the expansion of covered employment, the annual

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

real average economy-wide wage series. In the future, however, the annual growth rates in the two

measures are expected to be approximately identical since projected coverage changes are insignificant. Hence, the historical variation of the annual percent change in the real average economywide 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 wage disbursement as published by the Bureau of Economic Analysis’ (BEA) National Income and Product Accounts (NIPA) to civilian employment.

Civilian employment is the sum of total 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.18 OCACT has used 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:

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 model estimates the annual percent changes in the real economy-wide wage as a function of

the current unemployment rate and the unemployment rate of the previous year, expressed as logodds ratios, over the period from 1968 to 2002. The value for 1974 was an outlier and therefore

was excluded in the development of the equation. The R-squared value was 0.53. The actual and

fitted values are shown in figure II.6.

The estimated coefficients and standard error of the regression are then used to simulate the percent change in the real average covered wage. The modified equation is: 

(9)

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

t; represents the percent change in the real average covered wage from the TR04II in year

t; ut represents the deviation of the (log-odds transformed) unemployment rate from the TR04II

unemployment rate in year t; and εt

 represents the random error in year t.

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

www.bls.gov/cps/cpscomp.pdf.

1 1 ( / ) /( / ) 1. W NW NW CPI CPI t tt t t = − − −

1 0.06 0.04 . TR WW u u tt t t t ε =− + + −

TR Wt
Equation Selection and Parameter Estimation

14

Figure II.6—Real Average Wage (Percent Change), Calendar Years 1968-2002

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

1965 1975 1985 1995 2005

Calendar Year

Percent

Actual

Fitted
Disability Incidence Rate

15

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 who become newly entitled to disability benefits during the year. The

exposed population is comprised 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 1996 exposed population. The age-adjusted disability incidence rates (male and female) are

the crude rates that would occur in the disability exposed population as of January 1, 1996, if that

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

in the selected year.

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

disability incidence rates are computed by OCACT. Over the historical period from 1970 to 2003,

disability incidence rates have varied widely due to changes in legislation and program administration as well as economic and demographic factors. 

The equations for disability incidence rates are selected separately for males and females. Over

the historical period of 1970 through 2003, both the male and the female series fail their tests for

stationarity. However, in this model it is assumed that the nonstationarity in these series is due to

the various changes in the law over the historical period. Therefore, both series were modeled

without correcting for nonstationarity. Using time-series analysis, both series were modeled individually as AR(2) processes. The R-squared values for the male and female disability incidence

rate equations were 0.89 and 0.87, respectively. The actual and fitted values for males and females

are shown in figures II.7 and II.8, respectively.

The modified male disability age-adjusted incidence rate equation is:

(10)

In this equation, DIMt

 represents the male disability incidence rate in year t; represents

the male disability incidence rate from the TR04II in year t; dimt

 represents the deviation of the

male disability incidence rate from the TR04II male disability incidence rate in year t; and εt is the

random error in year t.

1.47 0.63 . TR DIM DIM dim dim t t t-1 t-2 t =+ − + ε

TR DIMt
Equation Selection and Parameter Estimation

16

The modified female age-adjusted disability incidence rate equation is:

(11)

In this equation, DIFt

 represents the female disability incidence rate in year t; represents

the female disability incidence rate from the TR04II in year t; dift

 represents the deviation of the

female disability incidence rate from the TR04II female disability incidence rate in year t; and εt

is the random error in year t.

Figure II.7—Male Disability Incidence Rate, Calendar Years 1970-2003

Figure II.8—Female Disability Incidence Rate, Calendar Years 1970-2003

3.0

4.0

5.0

6.0

7.0

8.0

9.0

1970 1980 1990 2000

Calendar Year

Rate per thousand

Actual

Fitted

1 2 1.45 0.62 . TR DIF DIF dif dif t t t tt ε =+ − + − −

TR DIFt

2.0

3.0

4.0

5.0

6.0

7.0

8.0

1970 1980 1990 2000

Calendar Year

Rate per thousand

Actual

Fitted
Disability Recovery Rate

17

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 the individual’s recovery from disability. The

age-adjusted disability recovery rates (male and female) are the crude rates that would occur in

the in-current-payment population as of January 1, 1996, if the population were to experience the

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

Data on disability recovery are obtained from SSA administrative records and the age-adjusted

disability recovery rates are computed by OCACT. Over the historical period from 1970 to 2003,

there has been substantial variation in the age-adjusted disability recovery rates. This variation is

believed to be mostly due to changes in the law. For example, the age-adjusted disability recovery

rate for males jumped from 10.3 per thousand in 1996 to 24.4 per thousand in 1997, largely as a

result of the effects of a provision in Public Law 104-121. This provision prohibited benefit payments to individuals where drug addiction and/or alcoholism was material to the determination of

disability.

The equations for disability recovery rates were modeled separately for males and females. The

historical disability recovery rates used to fit the equations were age-adjusted to the 1996 in-current-payment population. Due to the frequent changes in the law, the time period considered was

narrowed to 1985 through 2003. The value for 1997 was excluded in the development of the

equation due to the change in the law described above. The AR(1) model provided the best fit of

the models that were tested. The R-squared values for the male and female disability recovery rate

equations were 0.83 and 0.50, respectively.

The modified male age-adjusted disability recovery rate equation is:

(12)

In this equation, DRMt

 represents the male disability recovery rate in year t; represents

the male disability recovery rate from the TR04II in year t; drmt

 represents the deviation of the

male disability recovery rate from the TR04II male disability recovery rate in year t; and εt is the

random error in year t.

The modified female age-adjusted disability recovery rate equation is: 

(13)

In this equation, DRFt

 represents the female disability recovery rate; represents the

female disability rate from the TR04II in year t; drft

 represents the deviation of the female disability recovery rate from the TR04II female disability recovery rate in year t; and εt

 is the random error in year t.

1 0.58 . TR DRM DRM drm t t tt ε =+ + −

TR DRMt

1 0.57 . TR DRF DRF drf t t tt ε =+ + −

TR DRFt
Documentation of the Computer Program

18

III. DOCUMENTATION OF THE COMPUTER PROGRAM

This chapter describes the details of the computer program used to run the OSM. The model was

written in Fortran 90/95 and compiled using the Compaq Visual Fortran compiler, version 6.1.A.

The program has about 26,000 lines of source code and was written in modular format with 20

source code files. It uses over 160 data files as input and about 425 MB of RAM. On a personal

computer with a 2.8 GHz Intel Pentium 4 processor, running 5,000 simulations takes about 34

hours.

A. ORGANIZATION

The OSM contains nine modules. They are executed sequentially in the following order: Assumptions, Population, Economics, Insured, DIB (Disability Insurance Beneficiaries), OASIB (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 III.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, DIB, OASIB,

Awards, and Cost). The Assumptions Module does not receive data from any of the other modules, while the Summary Results Module does not send data to any of the other modules. It is

important to note that in this table the only instance in which a module sends data to an earlier

solved module is the DIB Module sending data to the Economics Module. This is possible

because the DIB data sent there is from the prior (not current) year.

The computer program used to solve the modules 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 (the Trustees Report year

plus 75). Thus, for the 2004 Trustees Report, the year loop starts in 2004 and ends in 2079 (a total

of 76 years for each simulation). In the wrap-up phase, the program sorts and prints the final output results.
Organization

19

Table III.1—Module Dependencies

Module Input Modules Output Modules

Assumptions N/A Population

Economics

DIB

Cost

Summary Results

Population Assumptions Economics

Insured

DIB

OASIB

Awards

Cost

Economics Assumptions

Population

DIB

Insured

Awards

Cost

Insured Population

Economics

DIB

OASIB

DIB Assumptions

Population

Insured

Economics

OASIB

Cost

OASIB Population

Insured

DIB

Cost

Awards Population

Economics

Cost

Cost Assumptions

Population

Economics

DIB

OASIB

Awards

Summary Results

Summary Results Assumptions

Population

Cost

N/A
Documentation of the Computer Program

20

B. 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 2004 Trustees Report. The

modules are written so that the set of nonstochastic inputs required to begin the projections is

identical with the input assumptions used when running the deterministic model under the

TR04II. Moreover, the mean value for each stochastic variable is assumed to be the same as the

value assumed for the variable under the TR04II. 

1. Assumptions

The Assumptions Module contains 54 equations, one for each stochastic variable. These equations are described in detail in chapter II, and include ones for the total fertility rate, rates of mortality improvement (21 male age groups and 21 female age groups), immigration level,

emigration level, net other immigration level, unemployment rate, inflation rate, real interest rate,

percent change in real average covered wages, disability incidence rates (male and female), and

disability recovery rates (male and female).

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 B for more details on this process.

The final step of this module is to use the error terms to calculate the results of each equation.

Chapter II provides more details on specific equations.

For the mortality equations, an additional step decomposes the annual rates of decrease in the central death rates by age group into single years of age.

2. Population

The Population Module projects the Social Security area population by sex, single year of age,

and marital status. The components of change—fertility, mortality, and immigration—are applied

each year throughout the projection period based on levels generated in the Assumptions Module.

The population is grouped by marital status using the relative proportions for each age-sex group

projected under the TR04II. 

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 from the TR04II. 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
Modules

21

central death rates. Death probabilities are derived from the central death rates by assuming a uniform distribution of deaths for each age. 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 from the TR04II. The resulting population is then distributed by marital

status using the relative martial proportions for each age-sex group projected in the TR04II.

3. Economics

The Economics Module receives data from the Assumptions, Population, and DIB 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 DIB Module passes age-sex 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 TR04II.

Labor Force Participation

Future civilian labor force participation rates by age and sex are influenced by projected disability

prevalence ratios, business cycles, life expectancies, and Social Security area population. For a

given year, the civilian labor force is summed from the products of the age-sex civilian labor force

participation rates and civilian noninstitutionalized populations. For each age-sex group, the civilian noninstitutionalized population is the product of the Social Security area population and the

ratio of the noninstitutional to Social Security area population from the TR04II. 

Total Employment

The civilian unemployment rates by age and sex are projected by distributing the stochastically

projected aggregate rate to its age-sex components. Projected total economy-wide employment is

summed from age-sex components derived from the corresponding components of the civilian

labor force and unemployment rates. The concept of total economy-wide employment is consistent with the Bureau of Labor Statistics’ Current Population Survey, and thus represents an “average” level of employment for a particular year. The projected total 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. Total at-any-time employment by age

and sex is influenced by the relative number of illegal immigrants in the workforce and by the

proportion of the population employed. 
Documentation of the Computer Program

22

Covered Employment

Total OASDI covered employment by age and sex is projected by removing noncovered workers,

including those assumed illegal, from the total at-any-time employment. Total OASDI covered

employment is then distributed to those with wages and to those with self-employed net 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 illegal immigration. Total compensation for wage workers, total and covered self-employed income, taxable wages, and taxable self-employed net income are all derived

from assumed relationships from the TR04II. 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 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.

4. 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 2004 is $900. Thus, if a worker

earns at least $3,600 in covered employment anytime during 2004, 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.
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In the TR04II, 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. These percentages 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 legal immigrants and other immigrants, by age and sex.

The projected portions of the above data are then used in order to continue the simulation process

of work histories of the Social Security area population 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 fully insured status as a percentage

of the 30,000 simulated cases which represent the Social Security area population. 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. Projections of fully insured percentages

are based on the baseline projection in the TR04II and an adjustment that accounts for the difference between the 10-year moving averages of the covered worker rates19 from the Economics

Module and the TR04II. 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. 

5. Disability Insurance Beneficiaries (DIB)

The DIB 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 received

from the Insured Module and the age-adjusted male and female incidence and recovery rates

passed from the Assumptions Module. Additionally, the module uses an estimate20 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

19 Covered worker rates are defined as the number of covered workers, expressed as a percentage of the Social Security area population.

20 The current number of entitled disabled-worker beneficiaries is not completely known because of the time lag between entitlement to and receipt of benefits.
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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.21

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 rates22 by age, sex, and duration since entitlement are projected to improve at the same rate as the general population aged 19 through 64. For each sex, the

future age-specific recovery rates are assumed to grow at the same rate as growth in the ageadjusted recovery rate.22 The number of disabled-worker beneficiaries in current-payment status

is then estimated by reducing those currently entitled by those for whom payment has not yet

begun.

Dependents of Disabled Workers

The projected number of auxiliary beneficiaries of disabled workers basically 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 TR04II and a factor

that adjusts for differences between the OSM and the TR04II 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. 

6. Old-Age and Survivors Insurance Beneficiaries (OASIB)

The OASIB module receives variables passed from the Population, Insured, and DIB 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, also by age, sex, and marital status. The DIB Module passes disability prevalence rates,23 and the numbers of disabled-worker

and converted disabled-worker beneficiaries, by age and sex. Using the data received, the OASIB

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.

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:

21 Growth is determined from a 1994-96 base period.

22 Growth is determined from a 1991-95 base period.

23 A disability prevalence rate is the ratio of the number of disabled workers to the number of disability insured workers.
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• 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. 

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).24

Retirement prevalence rates25 used in the TR04II are then applied to calculate the number of

retired-worker beneficiaries. 

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: 

24 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.

25 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).
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• The primary account holder is alive and fully insured,

• 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,26

• The aged-spouse is not insured, and

• The aged-spouse’s benefits are not withheld because of receipt of a significant government pension based on earnings in noncovered employment.

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 1

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 64 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 OASIB 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 persons over age 17 who were disabled

prior to age 22. 

To calculate the number of minor children of retired workers, the population of children under age

18 is multiplied by the probability that:

26 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 minimum 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.
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• The parent is fully insured,

• The parent is indeed receiving a retired-worker benefit, and

• The child is not receiving a benefit based on his/her other parent’s earnings. 

For minor children of deceased workers, the same population 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. 

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). In order to estimate the number of young spouses of retired workers, the married population under age 65 is multiplied by the probability that:

• The primary account holder is age 62 or older,

• The young spouse has a child (under age 16 or a disabled adult) in his/her care, and

• The child is entitled to a child benefit.

To estimate the number of young spouses of deceased workers, the population of widowed and

divorced spouses under age 65 is multiplied by the probability that:

• The primary account holder is deceased,

• The young spouse has a child in his/her care (under age 16 or a disabled adult),

• The child is entitled to a child benefit, and

• The young spouse has not remarried.

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, or is not receiving a

retired-worker benefit based on his/her own earnings.27

27 The eligible age requirement to receive a young-spouse benefit is 69 or younger. Therefore it is necessary to exclude those who

are of age to receive a young-spouse benefit, but are receiving their own retired-worker benefit and are counted in the retired- worker calculation.
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7. Awards

The Awards Module uses a stratified sample of newly entitled worker beneficiaries. A one-percent sample is used for OASI accounts and a five-percent sample is used for DI accounts. In addition, the Awards Module is passed historical and projected covered workers and population from

the Economics and Population Modules, respectively. The module also utilizes historical and projected data from the Economics Module such as average wage, and average taxable earnings in

order to produce the modified earnings levels and the earnings history of the sample for the representation of future awards. The module ultimately produces projected levels of benefits, in terms

of Average Indexed Monthly Earnings (AIME), for those beneficiaries newly awarded by age,

sex, and trust fund. These projected levels are passed to the Cost Module.

Awards Sample

The sample of worker beneficiaries who are newly awarded in 2003 is the foundation of the

Awards Module. The sample contains a total of 29,002 newly awarded beneficiaries, 14,412

retired-worker beneficiaries and 14,590 disabled-worker beneficiaries. This sample is a subset of

the 10-percent sample used for the TR04II. 

Each record in the sample includes the worker’s history of taxable earnings under the OASDI program as well as additional information such as birth date, sex, type of benefit, and month of entitlement of the worker. This information allows us to compute the benefits and classify each

beneficiary in the sample. Some preliminary calculations made on the sample are utilized within

the model. These include the sample’s covered worker rates calculated separately for retiredworker and disabled-worker beneficiaries. These rates are determined using the sample’s earnings

histories for 1951 through 2002. The rates are defined for each age group and sex as the ratio of

(1) the number of beneficiaries with earnings to (2) the total number of beneficiaries. 

Breakdown of the Awards Module

A goal of the Awards Module is to adjust earnings histories and earnings levels in the sample to

represent those of future awards. The Awards Module is composed of three distinct components:

• The Coverage Loads component applies an adjustment to the earnings histories to

reflect the projected changes in covered worker rates.

• The Base Loads component adjusts the earnings levels to incorporate earnings above

the historical maximum taxable earnings, or wage base.

• The AIME component computes an AIME level for each beneficiary in the sample. A

distribution of these AIME levels is the basic input for determining average benefits

by the Cost Module.
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Coverage Loads

In order to estimate future benefit levels, the earnings histories in the sample are modified to represent those of the sample in a given projection year. First, the covered worker rates of current and

future sample cohorts are calculated based on sex and age group using data provided by both the

Economics Module and the Population Module. The percentage changes from these rates28 are

then applied to the corresponding sample’s covered worker rates by randomly removing or adding

earnings in such a way as to maintain the levels in the sample’s average taxable earnings for each

year by sex. 

Base Loads

The earnings posted for beneficiaries in the sample are limited by the historical wage base. Prior

to 1975, the maximum annual amount of earnings on which OASDI taxes were paid was determined by ad hoc legislation. After 1974, however, the annual maximum level was legislated to be

determined automatically, based on the increases in the Social Security Average Wage Index

(AWI). Prior to these automatic wage base increases, a larger portion of workers earned income at

or above the base. Additional ad hoc 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 wage

base in the sample to reflect the earnings levels for future samples. 

AIME

Average taxable earnings may grow at a different rate than the AWI. Therefore, in the AIME component, all earnings levels in the sample are adjusted to reflect the overall increase in the average

taxable earnings of future cohorts, relative to that in the AWI, as projected by the Economics

Module.

Based on the modified earnings levels and earnings histories, the new AIMEs can be obtained for

the future samples. The module then calculates the distribution of the AIME levels. 

8. Cost

The Cost Module serves two broad purposes. The first is to compute the year-by-year progress of

the combined OASDI Trust Funds for a 75-year projection period.29 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. 

28 Changes in rates are determined as the ratio of the absolute difference in the rates to the potential difference in the rates.

29 A projection for the 76th year is required to estimate the target fund, equal to the present value of the outgo in the 76th year. See

appendix A.5 for more information.
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Progress of Trust Funds

In order to determine the progress of the trust funds, the dollar amounts of income and outgo are

computed for each year. Income includes payroll taxes, the taxation of benefits, and interest.

Outgo consists of scheduled benefits, administrative expenses, and the railroad retirement interchange. Once the values of income and outgo have been determined for a given year, the amount

of end-of-year assets is determined by starting with beginning-of-year assets, adding income and

subtracting outgo. 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 assets of the trust funds become exhausted 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.

Benefit Outgo

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

those who are in current-payment status. The primary insurance amount (PIA) formula

factors are as specified in current law. The two bend points for 2003 are indexed by the

increase in the average wage supplied by the Economics Module.

The Cost Module uses a starting average benefit matrix for worker beneficiaries in

current-payment status based on a one-percent sample of the Master Beneficiary

Record (MBR) as of December 31, 2003. This starting matrix is read in once by the

Cost Module in its initialization subroutine. For each sex, the starting matrix of

retired-worker 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 through 9+) 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 current 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
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benefits in a previous year. The first is increasing benefit levels due to the COLA

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 OASIB and DIB 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. The numbers

of beneficiaries by type of benefit are from the OASIB and DIB Modules.

Taxation of Benefits, Administrative Expenses, and Railroad Retirement Financial

Interchange

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

TR04II. The projection of administrative expenses and railroad retirement interchange

uses the same methodology as in the TR04II.

Interest

The nominal yield on the combined OASDI Trust Funds is computed by multiplying

the increase in the Consumer Price Index by the increase in the real interest rate

assumed for the special public-debt obligations held by the trusts funds. Both of these

values are allowed to vary and are obtained from the Assumptions Module.30

The amount of interest on trust fund assets at the beginning of the year is calculated by

multiplying the nominal yield by the amount of these assets. 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.

30 The simulated nominal yield on new issues is converted to a nominal yield on the combined OASDI Trust Funds in the

Assumptions Module.
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32

Year-by-Year Actuarial Measures

Once the various components of income and outgo 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 rate, cost rate, and annual

balance (all as a percentage of payroll).

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 outgo components of the combined OASDI Trust

Funds over the 75-year period. The assumed nominal yield on trust fund assets is used to discount

the annual income and outgo components. The summary measures include the summarized

income rate, summarized cost rate, and actuarial balance, all as a percentage of taxable payroll

and as a percentage of gross domestic product (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 A.

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 possible: (1) the first year trust funds are exhausted, (2) the first year trust funds are

exhausted and remain exhausted, (3) the first year that outgo exceeds income excluding interest,

and (4) the first year that outgo exceeds total income.

9. 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 two different types of data: 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. These variables are the total fertility

rate, legal immigration, legal emigration, net other immigration, mortality changes by five-year

age groups, unemployment rate, inflation rate, real interest rate, change in average wages, and disability incidence and recovery rates. 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 ratio,

income and cost rates, and covered workers per beneficiary. The Summary Results Module processes the data, most of which is stored in arrays dimensioned by valuation year and stochastic

iteration. For each of the stochastic simulations, the data for a given year is sorted using Heapsort

(Press, et al., 2001). The probability distributions of the year may then be computed. For each
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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. These percentiles are stored in new arrays and are written

to disk, where they are accessible to post-processing steps used to create graphs and other printed

output. 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.

Summarized Data

Some data are only available as a summary measure for each of the stochastic simulations. Examples are actuarial balance, summarized income and cost rates, open group unfunded obligation,

and the exhaustion year of the combined OASDI Trust Funds. The Summary Results Module processes these data, most of which are passed to the module run by run. At the end 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 confidence

intervals.
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IV. RESULTS

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

chapter are based on sets of 5,000 simulations. The first section shows the results of the assumptions which include projected probability distributions for the equations presented in chapter II. In

order to better illustrate how uncertainty is quantified, confidence interval bounds from the OSM

results are often compared with the values assumed in the low cost (TR04I) and high cost

(TR04III) alternatives from the 2004 Trustees Report. The second section contains selected actuarial estimates including annual trust fund ratios and balances, as well as summary actuarial measures (e.g., actuarial balances and summarized cost rates). The third section illustrates the

sensitivity of the OSM to variations in the assumptions.

A. ASSUMPTIONS

The tables in the following subsections display results of the OSM for the equations presented in

chapter II. The three rows of values shown in the tables are the level in the 75th projection year

(2078), the average level over the entire 75-year projection period (2004-78), and the average

over the final 50 years of the projection period (2029-78).31 The corresponding columns for the

three rows are the value from the TR04II, median value, values from the TR04I and TR04III, and

bounds of the 95-, 90-, and 80-percent confidence intervals. For the mortality subsection, the

increase in period life expectancy is shown rather than the 50-year and 75-year averages.

Unless otherwise noted, the average value is computed as an arithmetic average. When the geometric mean is specified, the values are transformed by adding one, averaged geometrically, and

reverse transformed by subtracting one. This method is analogous to computing an average effective annual rate for compound interest (Kellison 1991).

The figures in the following subsections graph the values of the assumption variables throughout

their entire historical and projection periods. For most variables, the expected future annual values from the equations presented in chapter II are shown along with the 95-percent confidence

intervals for the annual values and the cumulative averages of each simulation. For the mortality

subsection, the 95-percent confidence interval is shown for the annual values only. For the immigration subsection, values shown for legal emigration and net other immigration during the historical period are estimated since actual data are not available.

1. Fertility

Table IV.1 lists the results of the total fertility rate equation presented in chapter II. For the final

50-year average, the limits of the 80-percent confidence interval closely match the TR04I and the

TR04III values (2.24 vs. 2.20 and 1.65 vs. 1.70, respectively).

31 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 the variable of interest. A notable exception is the mortality assumption in which death rates do not reach a constant level since they are derived from a multiple decrement model (where causes of death compete). For mortality, increases in life expectancies over the 75-year and final 50 years of

the projection period, rather than average levels, are presented.
Assumptions

35

Figure IV.1 displays the total fertility rate throughout the entire historical and projection periods.

The annual values indicate that the standard deviation of the projected total fertility rate is about

0.5 child per woman.32

2. Mortality

In order to simplify the presentation, mortality results are given here as period life expectancies at

birth and age 65 for males and females, separately. Thus, the number of age-sex groups is reduced

from 42 to just four.

Table IV.1—Total Fertility Rate

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 1.95 1.94 2.20 1.70 3.07 0.82 2.91 1.02 2.69 1.23

75-year

average 1.96 1.96 2.17 1.75 2.31 1.61 2.25 1.66 2.18 1.73

Final

50-year

average 1.95 1.95 2.20 1.70 2.39 1.50 2.32 1.57 2.24 1.65

32 The width of the 95-percent confidence interval for the annual values is approximately two children per woman. For a normal

distribution, this width represents about four standard deviations.

Figure IV.1—Total Fertility Rate, Calendar Years 1917-2078

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

1900 1920 1940 1960 1980 2000 2020 2040 2060 2080

Calendar Year

Rate

Actual Expected Annual Cum. Average

95% Confidence Intervals
Results

36

a. Period Life Expectancies at Birth

Tables IV.2 and IV.3 list the resulting period life expectancies at birth for males and females,

respectively, from the mortality equations presented in chapter II. The increases shown are calculated as the difference between the values in 2078 and 2004 or the difference between the values

in 2078 and 2029. The upper limit of the 90-percent confidence interval in 2078 is nearly equal to

the TR04III value (85.8 vs. 85.9 for males, and 89.5 vs. 89.2 for females) while the lower limit is

only slightly less than the TR04I value (77.5 vs. 78.0 for males, and 81.6 vs. 82.1 for females).

Figures IV.2 and IV.3 display the period life expectancies at birth for males and females, respectively, throughout the entire historical and projection periods. 

Table IV.2—Male Period Life Expectancies at Birth

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 81.4 81.8 78.0 85.9 76.5 86.6 77.5 85.8 78.6 84.9

75-year

increase 6.9 7.4 3.6 11.3 2.9 11.3 3.7 10.6 4.7 9.8

Final

50-year

increase 4.1 4.4 2.2 6.8 2.8 6.4 3.1 6.0 3.4 5.5

Table IV.3—Female Period Life Expectancies at Birth

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 85.2 85.4 82.1 89.2 80.9 90.4 81.6 89.5 82.4 88.6

75-year

increase 5.6 5.9 2.6 9.6 2.1 10.2 2.7 9.5 3.4 8.7

Final

50-year

increase 3.6 3.8 1.8 5.9 2.0 6.2 2.3 5.8 2.6 5.3
Assumptions

37

b. Period Life Expectancies at Age 65

Tables IV.4 and IV.5 list the resulting period life expectancies at age 65 for males and females,

respectively, from the mortality equations presented in chapter II. The values shown are analogous to those shown in the previous subsection. The upper limit of the 80-percent confidence

interval in 2078 is close to that of the TR04III value (23.2 vs. 23.6 for males, and 26.3 vs. 26.1 for

females) while the lower limit is slightly larger than the TR04I value (18.4 vs. 17.7 for males, and

20.7 vs. 20.3 for females).

Figure IV.2—Male Period Life Expectancies at Birth, Calendar Years 1900-2078

Figure IV.3—Female Period Life Expectancies at Birth, Calendar Years 1900-2078

45

50

55

60

65

70

75

80

85

90

95

1900 1920 1940 1960 1980 2000 2020 2040 2060 2080

Calendar Year

Years

Actual Expected Annual

95% Confidence Interval

45

50

55

60

65

70

75

80

85

90

95

1900 1920 1940 1960 1980 2000 2020 2040 2060 2080

Calendar Year

Years

Actual Expected Annual

95% Confidence Interval
Results

38

Figures IV.4 and IV.5 display the period life expectancies at age 65 for males and females, respectively, throughout the entire historical and projection periods. 

Table IV.4—Male Period Life Expectancies at Age 65

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 20.3 20.6 17.7 23.6 17.2 24.8 17.7 24.0 18.4 23.2

75-year

increase 4.2 4.5 1.7 7.5 1.6 8.3 2.0 7.6 2.6 6.8

Final

50-year

increase 2.7 2.9 1.1 4.8 1.4 5.1 1.6 4.6 1.9 4.2

Table IV.5—Female Period Life Expectancies at Age 65

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 22.8 23.3 20.3 26.1 19.4 28.1 19.9 27.3 20.7 26.3

75-year

increase 3.9 4.3 1.4 7.1 0.9 8.6 1.4 7.9 2.1 7.0

Final

50-year

increase 2.6 2.9 1.1 4.6 1.2 5.3 1.4 5.0 1.8 4.4

Figure IV.4—Male Period Life Expectancies at Age 65, Calendar Years 1900-2078

10

12

14

16

18

20

22

24

26

28

30

1900 1920 1940 1960 1980 2000 2020 2040 2060 2080

Calendar Year

Years

Actual Expected Annual

95% Confidence Interval
Assumptions

39

3. Immigration

a. Legal Immigration

Table IV.6 displays the results (in thousands) of the legal immigration equation presented in chapter II. For the final 50-year average, the bounds of the 80-percent confidence interval are close to

the TR04I and the TR04III values (1,051 vs. 1,063 and 548 vs. 675, respectively).

Figure IV.6 shows a graph of legal immigration throughout the entire historical and projection

periods. The annual values indicate that the standard deviation of the projected legal immigration

level is about 300,000.33

Figure IV.5—Female Period Life Expectancies at Age 65, Calendar Years 1900-2078

Table IV.6—Legal Immigration (thousands)

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 800 797 1,063 675 1,440 153 1,349 252 1,232 371

75-year

average 812 811 1,070 685 1,127 491 1,079 549 1,018 606

Final

50-year

average 800 797 1,063 675 1,193 409 1,125 473 1,051 548

33 The width of the 95-percent confidence interval for the annual values is approximately 1,200,000. For a normal distribution,

this width represents about four standard deviations.

10

12

14

16

18

20

22

24

26

28

30

1900 1920 1940 1960 1980 2000 2020 2040 2060 2080

Calendar Year

Years

Actual Expected Annual

95% Confidence Interval
Results

40

b. Legal Emigration

Table IV.7 displays the results (in thousands) of the legal emigration equation presented in chapter

II. The confidence intervals shown are not directly comparable with the values assumed for the

TR04I and TR04III. This is because the Trustees’ values are established by applying an assumed

percentage to the level of legal immigration (20 percent for the TR04I and 30 percent for the

TR04III). As a result, the level of emigration under the TR04I and the TR04III are both greater

than that of the TR04II.

Figure IV.7 graphs the level of legal emigration throughout the entire historical and projection

periods. The annual values indicate that the standard deviation of the projected legal emigration

level is about 25,000.34

Figure IV.6—Legal Immigration, Calendar Years 1901-2078

Table IV.7—Legal Emigration (thousands)

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 200 200 213 203 154 246 162 238 170 230

75-year

average 203 203 214 206 196 210 197 209 198 208

Final

50-year

average 200 200 213 203 191 208 193 207 194 206

34 The width of the 95-percent confidence interval for the annual values is approximately 100,000. For a normal distribution, this

width represents about four standard deviations.

0

200

400

600

800

1,000

1,200

1,400

1,600

1900 1920 1940 1960 1980 2000 2020 2040 2060 2080

Calendar Year

Level (thousands)

Actual Expected Annual Cum. Average

95% Confidence Intervals
Assumptions

41

c. Net Other Immigration

Table IV.8 displays the results (in thousands) of the net other immigration equation presented in

chapter II. For the final 50-year average, the bounds of the 80-percent confidence interval are considerably wider than the interval defined by the TR04I and the TR04III values (505 vs. 450 and 86

vs. 200, respectively).

Figure IV.8 graphs the level of net other immigration throughout the entire historical and projection periods. Discontinuities are shown in the graph over the historical and the projected periods.

The discontinuities shown over the historical years reflect that the estimated levels of net other

immigration are determined as average annual levels for 10-year periods between census years.

The discontinuities shown over the projected years are caused by the Trustees’ assumption of a

50,000 decrease in the level of net other immigration in years 2014 and 2024. Ignoring the discontinuities, the graph of the projected values is quite typical of that of a random walk in that it produces a parabola. This is expected since the standard error of the forecast increases with the

square root of the time variable.

Figure IV.7—Legal Emigration, Calendar Years 1980-2078

Table IV.8—Net Other Immigration (thousands)

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 300 299 450 200 733 -128 663 -64 577 14

75-year

average 320 320 470 207 570 75 526 110 481 152

Final

50-year

average 300 302 450 200 622 -16 565 28 505 86

0

50

100

150

200

250

300

350

400

1980 2000 2020 2040 2060 2080

Calendar Year

Level (thousands)

Estimated Expected Annual Cum. Average

95% Confidence Intervals
Results

42

4. Unemployment Rate

Table IV.9 shows the results of the unemployment rate equation presented in chapter II. The values presented in the table are expressed in percents. The lower and upper bounds of the 95-percent

confidence interval for the 75-year cumulative average (4.74 and 6.68 percent, respectively) are

close to the values assumed under the TR04I and the TR04III (4.61 and 6.40 percent, respectively).

Figure IV.9 shows a graph of the unemployment rate over the historical and projection periods.

Due to the log-odds transformation of the unemployment rate variable, the confidence intervals

are not symmetrical around the median value.

Figure IV.8—Net Other Immigration, Calendar Years 1960-2078

Table IV.9—Unemployment Rate

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 5.45 5.43 4.59 6.29 3.18 9.13 3.49 8.40 3.88 7.67

75-year

average 5.48 5.63 4.61 6.40 4.74 6.68 4.86 6.49 5.02 6.31

Final

50-year

average 5.47 5.59 4.58 6.34 4.53 6.92 4.67 6.70 4.87 6.45

-200

-100

0

100

200

300

400

500

600

700

800

1960 1980 2000 2020 2040 2060 2080

Calendar Year

Level (thousands)

Estimated Expected Annual Cum. Average

95% Confidence Intervals
Assumptions

43

5. Inflation Rate

Table IV.10 displays the results of the CPI inflation rate equation presented in chapter II. The values presented in the table are growth rates expressed in percents, and the averages shown are

annual compound averages. The lower bound of the 95-percent confidence interval for the 75-

year cumulative average, at 1.75 percent, closely matches the 1.77 percent assumed under the

TR04I, while the upper bound of 4.64 percent is much higher than the 3.80 percent assumed under

the TR04III.

Figure IV.10 shows a graph of the inflation rate over the historical and projection periods. The

asymmetry in the confidence intervals around the median values is due to the logarithmic transformation that was applied to the inflation rate variable.

Figure IV.9—Unemployment Rate, Calendar Years 1960-2078

Table IV.10—Inflation Rate

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 2.80 2.84 1.80 3.80 -0.13 8.73 0.23 7.66 0.65 6.37

75-year

geometric

average 2.75 3.03 1.77 3.80 1.75 4.64 1.93 4.36 2.14 4.05

Final 

50-year

geometric

average 2.80 3.06 1.80 3.80 1.53 5.07 1.76 4.73 2.03 4.34

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

1960 1980 2000 2020 2040 2060 2080

Calendar Year

Percent

Actual Expected Annual Cum. Average

95% Confidence Intervals
Results

44

6. Real Interest Rate

Table IV.11 shows the results of the real interest rate equation presented in chapter II. The values

presented in the table are expressed in percents, and the averages shown are annual compound

averages. For the 75-year cumulative average, the limits of the 90-percent confidence interval

(3.73 percent and 2.26 percent) are close to the values assumed under the TR04I and the TR04III

(3.69 percent and 2.24 percent, respectively).

Figure IV.11 shows a graph of the real interest rate on new special issue securities over the historical and projection periods.

Figure IV.10—Inflation Rate, Calendar Years 1960-2078

Table IV.11—Real Interest Rate

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 3.00 2.96 3.70 2.20 8.31 -2.28 7.40 -1.49 6.38 -0.60

75-year

geometric

average 3.01 2.98 3.69 2.24 3.86 2.12 3.73 2.26 3.57 2.43

Final

50-year

geometric

average 3.00 2.98 3.70 2.20 4.14 1.85 3.95 2.03 3.73 2.24

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

1960 1980 2000 2020 2040 2060 2080

Calendar Year

Percent

Actual Expected Annual Cum. Average

95% Confidence Intervals
Assumptions

45

7. Real Average Covered Wage

Table IV.12 displays the results of the real average covered wage equation presented in chapter II.

The values presented in the table are growth rates expressed in percents, and the averages shown

are annual compound averages. For the 75-year cumulative average, the limits of the 95-percent

confidence interval (1.69 percent and 0.59 percent) closely match the values assumed under the

TR04I and the TR04III (1.63 and 0.62 percent, respectively).

Figure IV.12 shows a graph of the real average covered wage for the historical and projection

periods.

Figure IV.11—Real Interest Rate, Calendar Years 1960-2078

Table IV.12—Real Average Covered Wage

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 1.07 1.05 1.57 0.57 4.31 -2.01 3.74 -1.55 3.12 -0.97

75-year

geometric

average 1.14 1.13 1.63 0.62 1.69 0.59 1.59 0.67 1.49 0.78

Final

50-year

geometric

average 1.07 1.07 1.57 0.57 1.75 0.38 1.62 0.48 1.51 0.61

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

1960 1980 2000 2020 2040 2060 2080

Calendar Year

Percent

Actual Expected Annual Cum. Average

95% Confidence Intervals
Results

46

8. Disability Incidence Rate

Tables IV.13 and IV.14 show the results of the age-adjusted disability incidence rate equations for

males and females, respectively. The disability incidence rates shown are per thousand. For the

75-year cumulative average, values for males under the TR04I and the TR04III (4.95 and 7.27 per

thousand, respectively) fall outside the limits of the 95-percent confidence interval (5.58 and 6.56

per thousand, respectively). Comparable values for females, under the TR04I and the TR04III

(4.24 and 6.22 per thousand, respectively) also fall outside the limits of the 95-percent confidence

interval (4.71 and 5.69 per thousand, respectively).

Figure IV.12—Real Average Covered Wage, Calendar Years 1968-2078

Table IV.13—Male Disability Incidence Rate

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 6.25 6.25 5.00 7.50 4.13 8.42 4.49 8.07 4.89 7.63

75-year

average 6.08 6.08 4.95 7.27 5.58 6.56 5.67 6.49 5.76 6.40

Final

50-year

average 6.25 6.25 5.00 7.50 5.64 6.86 5.74 6.76 5.86 6.64

Table IV.14—Female Disability Incidence Rate

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 5.27 5.29 4.22 6.33 3.24 7.34 3.58 7.01 3.96 6.64

75-year

average 5.21 5.21 4.24 6.22 4.71 5.69 4.80 5.61 4.89 5.52

Final

50-year

average 5.27 5.27 4.21 6.33 4.66 5.86 4.76 5.77 4.87 5.66

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

1960 1980 2000 2020 2040 2060 2080

Calendar Year

Percent

Actual Expected Annual Cum. Average

95% Confidence Intervals
Assumptions

47

Figures IV.13 and IV.14 show graphs of male and female age-adjusted disability incidence rates,

respectively, over the historical and projection periods.

Figure IV.13—Male Disability Incidence Rate, Calendar Years 1970-2078

Figure IV.14—Female Disability Incidence Rate, Calendar Years 1970-2078

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

1960 1980 2000 2020 2040 2060 2080

Calendar Year

Rate (per thousand)

Actual Expected Annual Cum. Average

95% Confidence Intervals

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

1960 1980 2000 2020 2040 2060 2080

Calendar Year

Rate (per thousand)

Actual Expected Annual Cum. Average

95% Confidence Intervals
Results

48

9. Disability Recovery Rate

Tables IV.15 and IV.16 show the results of the disability recovery rate equations for males and

females, respectively. The disability recovery rates presented are per thousand. For the 75-year

cumulative average, values for males under the TR04I and the TR04III (13.64 and 8.42 per thousand, respectively) fall outside the limits of the 95-percent confidence interval (12.37 and 10.59

per thousand). Comparable values for females assumed under the TR04I and the TR04III (12.33

and 7.74 per thousand, respectively) also fall outside the limits of the 95-percent confidence interval (11.21 and 9.54 per thousand). 

Figures IV.15 and IV.16 show graphs of the age-adjusted disability recovery rates for males and

females, respectively, over the historical and projection periods.

Table IV.15—Male Disability Recovery Rate

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 9.82 9.77 11.99 7.72 13.85 5.77 13.24 6.35 12.49 7.10

75-year

average 11.46 11.47 13.64 8.42 12.37 10.59 12.23 10.73 12.06 10.89

Final

50-year

average 9.89 9.89 12.03 7.77 10.98 8.82 10.82 8.99 10.62 9.19

Table IV.16—Female Disability Recovery Rate

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Value in

2078 9.28 11.80 11.26 7.34 13.06 5.46 12.47 6.03 11.80 6.70

75-year

average 10.37 10.38 12.33 7.74 11.21 9.54 11.08 9.69 10.92 9.84

Final

50-year

average 9.32 9.32 11.27 7.38 10.33 8.31 10.17 8.48 9.99 8.67
Assumptions

49

Figure IV.15—Male Disability Recovery Rate, Calendar Years 1986-2078

Figure IV.16—Female Disability Recovery Rate, Calendar Years 1986-2078

0.00

5.00

10.00

15.00

20.00

25.00

1980 2000 2020 2040 2060 2080

Calendar Year

Rate (per thousand)

Actual Expected Annual Cum. Average

95% Confidence Intervals

0.00

5.00

10.00

15.00

20.00

25.00

1980 2000 2020 2040 2060 2080

Calendar Year

Rate (per thousand)

Actual Expected Annual Cum. Average

95% Confidence Intervals
Results

50

B. ACTUARIAL ESTIMATES

The tables in the following subsections contain estimates for the OASDI program. A row in the

tables contains estimates of a particular measure. For each measure, the columns show results

from both the OSM and the 2004 Trustees Report. The value from the TR04II and the median

from the OSM are shown first since they both provide an estimate of the central case. The TR04I

and TR04III values are shown next followed by the 95-, 90-, and 80-percent confidence intervals

from the OSM. These pairs of values provide ranges of the uncertainty for each measure. Where

necessary, the limits of the confidence intervals have been inverted to simplify the visual comparison of these to the TR04I and TR04III values.

1. Annual Measures

Table IV.17 shows selected measures of the OASDI program for the 75th projection year, 2078.

The median estimates shown from the OSM are slightly more pessimistic (from the OASDI program perspective) than those from the TR04II. For each measure in the table, the upper bound of

the 95-percent confidence interval (the second value listed under the 95-percent confidence interval) is more pessimistic than the TR04III value. Thus, the 95-percent confidence interval includes

the values of these measures under the TR04III. In contrast (with the exception of the cost rate as

a percent of GDP), the 95-percent confidence interval does not include the values of these measures under the TR04I. 

Figure IV.17 shows the estimated probability distribution of the annual trust fund ratio for the

OASDI program from the OSM. Note that the smooth curves which result do not represent the

path of any particular simulation. Instead, for each given year, the curved lines represent the distribution of trust fund ratios based on all stochastic simulation results for that year. The two

extreme curves in this figure comprise a 95-percent confidence interval,35 and deciles are shown

for the range of the 10th through the 90th percentiles.

Table IV.17—Estimates of the OASDI Program, Calendar Year 2078

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Cost Rate

(% Payroll) 19.29 19.78 14.01 27.23 14.38 27.88 15.17 26.32 16.08 24.70

Cost Rate

(% GDP) 6.62 6.78 5.20 8.61 4.95 9.54 5.23 9.00 5.52 8.46

Beneficiaries

per 100

Workers 54 56 41 71 43 76 45 72 47 68

35 Note that bounds of the 95-percent confidence interval are the 2.5th and 97.5th percentiles.
Actuarial Estimates

51

An estimate of the exhaustion year of the combined OASDI Trust Funds can be obtained from figure IV.17 by inspecting the x-intercept of a given curve. The x-intercept of the median curve indicates that according to the OSM, there is about the same chance of trust fund exhaustion prior to

2042 as there is after 2042. This exhaustion year is exactly the same as that indicated by the

TR04II. Furthermore, the lower limit of the 95-percent confidence interval of the trust fund

exhaustion year indicated by the OSM is only 1 year greater than that of the TR04III (2032 vs.

2031). However, the upper limit, as indicated by the OSM is more pessimistic than that of the

TR04I (2071 vs. no exhaustion during the 75-year projection).

Figure IV.18 shows the estimated probability distribution of the annual cost and income rates for

the OASDI program from the OSM. 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 confidence interval, and deciles are shown for the range of the 10th through the 90th

percentiles. Note that for the annual income rate, the range of the 95-percent confidence interval

is too narrow to resolve the decile values.

Figure IV.17—Annual Trust Fund Ratio, Calendar Years 2004-78

0%

100%

200%

300%

400%

500%

600%

700%

2004 2019 2034 2049 2064 2078

Calendar Year

50%

97.5%

2.5%

90%

10%
Results

52

Notice that the upper limit of the 95-percent confidence interval for the income rate never exceeds

the lower limit of the 95-percent confidence interval for the cost rate after about 2025.

2. Summary Measures

Table IV.18 shows selected measures of the OASDI program for the 75-year projection period.

The median estimates shown from the OSM are equal to or slightly more pessimistic than those

from the TR04II. Estimates of the long-range actuarial balance, summarized cost rate, open group

unfunded obligation, and exhaustion year measures under the TR04I and TR04III fall outside the

bounds of the 95-percent confidence intervals from the OSM. Estimates of the summarized

income rate and the first year cost exceeds tax income under the TR04I and TR04III fall inside or

on the bounds of the 95-percent confidence intervals.

Figure IV.18—Annual Cost and Income Rates, Calendar Years 2004-78

5%

10%

15%

20%

25%

30%

2004 2019 2034 2049 2064 2078

Calendar Year

50%

97.5%

2.5%

90%

10%

Income Rate

Cost Rate
Actuarial Estimates

53

Figure IV.19 shows a frequency distribution of the long-range actuarial balances estimated from

the OSM. The width of each bar 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 only 34 of the 5,000 simulations

resulted in a positive long-range actuarial balance. In other words, according to the OSM, there is

a 99.3 percent probability that the OASDI program has a negative actuarial balance for the period

2004-78.

Table IV.18—Estimates of the OASDI Program, Calendar Years 2004-78

TR04II Median TR04I TR04III

Confidence Intervals

95-percent 90-percent 80-percent

Actuarial

Balance -1.89 -1.98 0.41 -4.96 -0.33 -4.02 -0.55 -3.63 -0.85 -3.25

Summarized

Cost Rate 15.73 15.83 13.30 18.94 14.08 17.96 14.35 17.57 14.65 17.15

Summarized

Income Rate 13.84 13.84 13.72 13.98 13.62 14.08 13.65 14.04 13.69 13.99

Open Group

Unfunded

Obligation

(in trillions) $3.7 $4.0 -$1.1 $10.3 $0.4 $9.2 $0.9 $8.1 $1.5 $7.1

First Year Cost

Exceeds Tax

Income 2018 2018 2022 2013 2023 2013 2022 2013 2021 2014

Exhaustion

Year 2042 2042 1/

1 There is no exhaustion year in the period 2004-78 for the TR04I.

2031 2071 2032 2063 2033 2056 2035

Figure IV.19—Frequency Distribution of the Long-Range Actuarial Balance, Calendar Years 2004-78

0

50

100

150

200

250

300

350

400

450

500

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0

Actuarial Balance

Frequency
Results

54

Figure IV.20 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.19. The TR04III,

TR04II, and TR04I have each been 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.20—Cumulative Distribution of the Long-Range Actuarial Balance, Calendar Years 2004-78

0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

4,500

5,000

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0

Actuarial Balance

Cumulative Frequency

TR04III

p=0.4%

TR04II

p=54.3%

TR04I

p=99.8%
Sensitivity Analysis

55

C. SENSITIVITY ANALYSIS

Two types of sensitivity analyses have been performed. 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 the TR04II. 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 TR04I or TR04III value, while the remaining equations are

left with their means equal to the TR04II values.

1. Stochastic Analysis

A stochastic sensitivity analysis is the process in which one, or some, of the demographic, economic, and disability variables are varied independently, while the other variables are set to their

values under the TR04II. 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.19 presents the expected long-range actuarial balance, along with the standard deviation,

for each of the stochastic sensitivity runs. Each of these sensitivity runs is based on 5,000 simulations of the stochastic model. The first sensitivity run presented in table IV.19 allows only the

total fertility rate to vary. The next sensitivity run varies only the 42 mortality improvement rates

(21 age groups for males and females). The third run, listed as immigration, allows only the three

immigration assumptions (legal immigration, emigration, net other 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 sensitivity run varies only the disability incidence rates (male and female together). Varying the disability recovery rates resulted in a negligible change.

The long-range actuarial balance under the TR04II is -1.89 percent of taxable payroll.

2. Deterministic Analysis

A deterministic sensitivity analysis is the process in which the mean of one, or some, of the demographic, economic, and disability equations are set to equal the TR04I or the TR04III value, while

the remaining variables are set to equal the TR04II value. In this case, the variance of all key input

variables is set to zero and only one simulation is performed, since the zero variance ensures that

all simulations produce the same results. This type of sensitivity analysis is similar to that done in

Table IV.19—Sensitivity to Varying Selected Assumptions, Calendar Years 2004-78

Effect of Changing Individual Assumption:

Actuarial Balance

Median Average

Standard

Deviation

Fertility -1.92 -1.93 0.49

Mortality Improvement -1.96 -1.97 0.56

Immigration -1.89 -1.89 0.10

Economic -1.90 -1.90 0.51

Disability Incidence -1.89 -1.88 0.08

Effect of Changing All Assumptions Together -1.98 -2.03 0.95
Results

56

the Trustees Report, and can therefore be used to compare the underlying nature of the OSM and

the Trustees’ model.

Table IV.20 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 OSM. The second row contains the actual sensitivity result from the Trustees Report.

Technically speaking, the first sensitivity is not really a sensitivity analysis at all, but is included

as a point of reference. The next three results (fertility, mortality, and immigration) are the demographic deterministic sensitivity runs. This is followed by three economic deterministic sensitivity

runs (inflation, real interest, and real average wage). The final deterministic sensitivity run is disability incidence. Note that unemployment and disability recovery sensitivity runs are not

included since no comparable sensitivity analysis is performed for the Trustees Report. 

Table IV.20—Comparison of Stochastic and Deterministic Models,

Calendar Years 2004-78

Sensitivity:

Actuarial Balance

Low Cost Intermediate 1

1 This column does not show a sensitivity analysis, but is shown for comparison purposes.

High Cost

None

OSM 0.40 -1.89 -4.99

TR 0.41 -1.89 -4.96

Fertility

OSM -1.57 -1.89 -2.21

TR -1.61 -1.89 -2.18

Mortality

OSM -1.17 -1.89 -2.76

TR -1.29 -1.89 -2.59

Immigration

OSM -1.68 -1.89 -2.03

TR -1.63 -1.89 -2.08

Inflation

OSM -2.10 -1.89 -1.63

TR -2.11 -1.89 -1.67

Real Interest

OSM -1.39 -1.89 -2.47

TR -1.38 -1.89 -2.48

Real Average Wage

OSM -1.32 -1.89 -2.50

TR -1.35 -1.89 -2.42

Disability Incidence

OSM -1.57 -1.89 -2.22

TR -1.61 -1.89 -2.17
Principal Measures of Financial Status

57

V. APPENDICES

A. PRINCIPAL MEASURES OF FINANCIAL STATUS

Three types of financial measures are useful in assessing the actuarial status of the combined

OASDI Trust Funds under the financing approach specified in current law: (1) annual balances,

income rates, and cost rates, (2) trust fund ratios, and (3) summary measures like actuarial balance

and open group unfunded obligation. 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 values. 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. Another important measure is the pattern of projected trust fund ratios. The trust

fund ratio is the proportion of a year’s projected outgo 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 year of exhaustion of the funds, and to 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, actuarial balance, indicates the size of any shortfall as a

percentage of the taxable payroll over the period. The second, 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 100 percent (or greater) 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 zero or positive 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 constant, or rising, then it is likely that subsequent Trustees Reports will also show projections of financial adequacy (assuming no changes in demographic and economic assumptions). 

The remaining portion of this appendix discusses these basic measures of assessing the actuarial

status of the combined OASDI Trust Funds. Values from the 2004 deterministic model are often

given in order to aid in the discussion.

1. Trust Fund Operations

Starting with the dollar level of assets in a given year, the combined OASDI Trust Funds receive

income from payroll taxes, income from taxation of benefits, and interest income. In turn, scheduled benefits, administrative expenses, and the net financial interchange to the Railroad Retirement Board, are all paid from the combined OASDI Trust Funds. Hence, the dollar value of assets

at the end of a given year is numerically equal to the dollar value of assets at the beginning of the

year, plus payroll taxes, plus income from taxation of benefits, plus interest income, less sched-
Appendices

58

uled 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 can be defined to be 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 outgo,

expenditures, or disbursements) of the program to the taxable payroll for the year. In this context,

the outgo 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. Considering Social Security’s cost as a percentage of the total U.S. economy, GDP provides an additional perspective.

The year-by-year relationship of the income 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 just under to just over 13 percent as

income from taxation of benefits increases. There is only a very small difference in the income

rate among the TR04I, TR04II, and TR04III. Under the TR04II, the OASDI cost rate is estimated

to remain fairly stable and well below the income rate for the next several years, until about 2010.

It then begins to increase rapidly and first exceeds the income rate for 2018, producing cash-flow

deficits thereafter. The cost rate continues rising through about 2030 as the baby-boom generation

reaches retirement age. By 2078, 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 over 19 percent of taxable payroll under the TR04II. Annual deficits generally increase throughout the remainder of the 75-year projection period, reaching nearly 6 percent

of taxable payroll by 2078. The OASDI cost rates for the TR04I and TR04III follow generally

similar patterns, but differ significantly in magnitude from those projected for the TR04II. 
Principal Measures of Financial Status

59

3. Comparison of Workers to Beneficiaries 

The primary reason that the estimated OASDI cost rate generally increases rapidly after 2010 is

that the number of beneficiaries is projected to increase more rapidly than the number of covered

workers. This generally 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 TR04II, the number of covered workers per OASDI beneficiary declines to 2.1 by

2031, and to 1.9 by 2062. Under the TR04I, for which high fertility rates and small reductions in

death rates are assumed, this ratio declines to 2.3 by 2032, and then rises back to a level of 2.4 by

2044. Under the TR04III, for which low fertility rates and large reductions in death rates are

assumed, the decline in this ratio is much greater, reaching 1.8 by 2037, and 1.4 by 2074. 

The impact of the demographic shifts under the TR04I, TR04II, and TR04III on the OASDI cost

rates is better understood by considering the projected number of beneficiaries per 100 workers.

As compared to the 2003 level of 30 beneficiaries per 100 covered workers, this ratio is estimated

to rise significantly by 2080 to 54, 41, and 71 under the TR04II, TR04I, and TR04III, respectively. The significance of these numbers can be seen by noting that 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 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 

The trust fund ratio is a useful indicator of the adequacy of the financial resources of the Social

Security program at any point in time. The combined OASDI Trust Fund ratio for a given year is,

by definition, the ratio of the assets at the beginning of the year in the combined OASDI Trust

Funds to the total outgo during that year, expressed as a percentage. 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 assets at the beginning of the year), but is not positive for the

following year, the trust fund is projected to become exhausted during the year. Under present

law, the combined OASDI Trust Funds do not currently have the authority to borrow. Therefore,

exhaustion of the assets in either fund during a year would mean that there are no longer sufficient

funds to cover the full amount of benefits scheduled under present law. 

The trust fund ratio also serves an additional important purpose in assessing the actuarial status of

the OASDI program. When the financing is adequate for the timely payment of full benefits

throughout the long-range period, the stability of the trust fund ratio toward the end of the period

indicates the likelihood that this projected adequacy will continue for subsequent Trustees
Appendices

60

Reports. If the trust fund ratio toward the end of the period is level or increasing, then projected

adequacy for the long-range period is likely to continue for subsequent reports. 

Under the TR04II, the trust fund ratio for the combined OASDI Trust Funds rises from 306 percent at the beginning of 2004 to a peak of 448 percent at the beginning of 2015. Thereafter, the

ratio declines, with the combined funds becoming exhausted in 2042. The trust fund ratio first

declines in 2016, about 2 years before annual expenditures begin to exceed noninterest income.

After 2015, the dollar amount of assets is projected to continue to rise through the beginning of

2028, because interest income more than offsets the shortfall in noninterest income. Beginning in

2018, the OASDI program is projected to experience increasingly large cash-flow shortfalls that

will require the trust funds to redeem special public-debt obligations of the General Fund of the

Treasury. The pattern of the trust fund ratios under the TR04III is similar to that under the TR04II,

with a lower maximum value and an earlier exhaustion year (2031). While the pattern of the trust

fund ratios under the TR04I seems similar to that under the TR04II, the trust fund ratios under the

TR04I stabilize generally and remain over 480 percent for most of the second half of the 75-year

long-range period. 

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

The deterministic model projects nominal dollar levels of trust fund assets 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 TR04II, the ultimate value of the annual nominal yield on the combined OASDI Trust Funds

is 5.88 percent, achieved in 2018. The ultimate value of the annual nominal yield rate is 5.57 and

6.08 under the TR04I and TR04III respectively. In the short-range period, interest earned by these

new issues, along with interest earned on the existing assets of the trust funds, determine implicitly the nominal yield earned by the combined OASDI Trust Funds in the short-range period

2004-13. In a five-year blending period, 2014-18, the annual yield rate on the combined OASDI

Trust Funds is linearly interpolated between the yield rate in 2013 and the ultimate value achieved

in 2018 to obtain the nominal yield on the combined funds in those years.

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 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 TR04II, there is an actuarial deficit of 1.89 percent of taxable payroll for the combined

OASDI Trust Funds. Under the TR04I, the actuarial balance is 0.41 percent of taxable payroll

while under the TR04III, the actuarial deficit is 4.96 percent of taxable payroll.

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 present law. The summarized income

rate for the 75-year valuation period is equal to the present value of tax income, plus the begin-
Principal Measures of Financial Status

61

ning-of-year 2004 assets, 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 outgo (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 assets and tax income during the period are sufficient to meet the

outgo 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. The income and cost of the

OASDI program may also be expressed in terms of GDP. In this case, summary values for the

OASDI program are 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 -0.70 percent of GDP under the TR04II. 

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 assets at the beginning of the

period. In particular, it is the present value of the cost of the program, less the sum of the present

value of tax income and the assets on hand at the beginning of the 75-year valuation period. Under

the TR04II, the open group unfunded obligation was estimated to be $3.7 trillion. It is -$1.1 trillion and $10.3 trillion under the TR04I and TR04III, respectively. A negative estimate of the open

group unfunded obligation, as observed for the TR04I, indicates that over the 75-year period the

OASDI program runs a net surplus (in present-value terms), i.e., the sum of accumulated trust

fund assets and the present value of tax income is strictly greater than the present value of outgo

over the 75-year period.
Appendices

62

B. TIME-SERIES ANALYSIS

1. Background

Time-Series Modeling

Time-series analysis is a standard projection technique in econometric modeling. The equations

used to project the assumptions are fit 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 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 γk, a constant depending on k.

In particular, the variance of Yt is always equal to 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 intend that it is strictly stationary.

However, it is more straightforward to think of a time series as stationary in the simpler sense. 

White Noise Process

Suppose that εt, the error term at time t, is normally distributed with mean zero and constant variance If εr and εs are uncorrelated for then the series εt

 is called a white noise process.

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 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 The confidence intervals for

these forecasts will therefore widen as l increases.

Moving Average (MA) Models

A time series is called a moving average model of order q, or simply an MA(q) process, if

2

0 σ γ = .

2

. σ ε r s ≠ ,

1 ε . Y Y tt t = + −

2

. σ ε l

11 2 2 . Yt t t t q tq μ ε θε θε θ ε =+− − −− −− − "
Time-Series Analysis

63

In this equation, are constant parameters and μ is the process mean. The error term

series, εt, is a white noise process. 

Autoregressive (AR) Models

A time series is called an autoregressive model of order p, or simply an AR(p) process, if

In this equation, are constant parameters and δ 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, more precisely, with 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

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

Deviations Form of ARMA Model Equations

It is often more convenient to transform an ARMA model equation into deviations form using the

equation

where μ is defined, as above, to be the mean of the process. The transformed model may be written as

and has a mean of zero.

Adding the original process mean to both sides of the equation produces

The lagged variables are left in deviations form, and the constant term, μ, on the right-hand side is

the process mean.

1, , θ θ ... q

11 2 2 . YY Y Y t t t ptp t φ φ φ δε = + + + ++ −− − "

1, , φ φ ... p

1 2

. 1

δ

μ φφ φ = −−−− " p

11 2 2 11 2 2 . Y YY Y t t t ptp t t t q tq δ φ φ φ ε θε θε θε =+ + + + + − − − − −− − −− − " "

, t t y Y = − μ

11 2 2 11 2 2 , t t t p tp t t t q tq yy y y φ φ φ ε θε θε θε = + ++ +− − −− −− − −− − " "

11 2 2 11 2 2 . Y yy y t t t p tp t t t q tq μ φ φ φ ε θε θε θε =+ + ++ +− − −− −− − −− − " "
Appendices

64

Cholesky Decomposition

Suppose that 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 from a multivariate normal distribution with mean and an variance-covariance matrix 

[with for ]. Since V is positive definite and symmetric, a standard

result in linear algebra yields a lower triangular matrix, L, such that The random vector

 is then a random vector with the desired properties. The decomposition 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, of independent standard normal variates to a multivariate normal distribution with mean (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 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 and 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 the time-1 lag, i.e., when and may be modeled in terms of

 and In this case, a three-variable VAR(1) model takes the form 

for some matrix B.

Alternatively stated, if and then the model takes the form

 for some matrix B.

The k-variable VAR(p) model, with p lags, naturally extends from this. If

and then the k-variable VAR(p) model takes the form

 for matrices 

2. Methods

Modified Autoregressive Moving Average (ARMA) Models

The equations used to model the assumptions are autoregressive moving average models with the

additional requirement that the mean of the variable Yt

 is always equal to its value under the

TR04II. The value of the variable under the TR04II is written Thus, the mean of Yt

 varies, in

1 (, , ) ZZ Zn = ′ G ...

A′

1 (,, ) XX Xn = ′ G ...

1 (, , ) μμ μn = ′ G ... n n × ( ) V = σ ij

(, ) σ ij i j = Cov X X ij n , 1, , = ...

V LL = ′.

X Z = + μ L G G V LL = ′

ε ,

G

μ = 0

G G

Lε

G

1, ,t y 2, ,t y 3,t y

1, ,t y 2, ,t y 3,t y

1, 1, t− y 2, 1, t y − 3, 1. t y −

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

+

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

= ⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

−

t

t

t

t

t

t

t

t

t

y

y

y

B

y

y

y

3,

2,

1,

3, 1

2, 1

1, 1

3,

2,

1,

ε

ε

ε

3 3 ×

1, 2, 3, (, , ) t ttt y yyy = ′ G 1, 2, 3, (, , ) t ttt ε εεε = ′ G

t tt 1 y By ε = + −

G G G 3 3 ×

1, 2, , ( , ,, ) t t t kt y yy y = ′ G ...

1, 2, , ( , , , ), t t t kt ε εε ε = ′ G ...

t t t p tp 11 2 2 y By By B y ε = + ++ + −− −

GG G G G " k k × 1,,. B B ... p

. TR Yt
Time-Series Analysis

65

general, in the early years of the long-range period. The resulting model is called a modified

autoregressive moving average model. In general, the equation in a modified autoregressive moving average model takes the form

Writing this equation in “deviations form,” with gives

Since we obtain the expression 

Equations in Chapter II are all presented in this form, with the letters Y and y replaced with a more

suggestive naming.

11 2 2 11 2 2 . Y YY Y t t t t ptp t t t q tq δ φ φ φ ε θε θε θ ε =+ + ++ +− − −− −− − −− − " "

= − μ , ttt y Y

11 2 2 11 2 2 . Y yy y t t t t p t p t t t q tq μ φ φ φ ε θε θε θε = + + ++ +− − −− −− − −− − " "

, TR μt t = Y

11 2 2 11 2 2 . TR YY y y y t t t t p t p t t t q tq φ φ φ ε θε θε θ ε = + + ++ +− − −− −− − −− − " "
Appendices

66

C. MONTE CARLO SIMULATION

A standard reference for this appendix is Knuth (1997). In choosing a random number generator,

it is essential to make the following considerations:

• 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 will

generate a sequence of numbers. The values and 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 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 we used 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 runtime.

To generate the standard normal (mean 0 and variance 1) variates used to generate the random

error terms for our equations, one may transform uniform variates using the Box-Muller method:

generate two independent uniform (0,1) variates, U1 and U2, and compute

 and In order to avoid the use of trigonometric functions, we instead used the Polar method. This, 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 then and will be two

independent random draws from a standard normal distribution, provided that S is less

than one. (If not, then we repeat this process until it is.) Although this method produces

two standard normal variates, we always used the first one (X) and discarded the second

one (Y). Each equation for each assumption is separately seeded to have its own random

number stream. 

1 (mod ) I aI m j j + =

a =16,807 31 m = − 2 1

31 2 2. −

1 2 X UU = −2ln cos(2 ) π 1 2 Y UU = −2ln sin(2 ). π

2 2

1 2 SV V = + 1

2ln S X V

S

− = 22ln S Y V

S

− =
Equation Parameter Estimates and Regression Statistics

67

D. EQUATION PARAMETER ESTIMATES AND REGRESSION STATISTICS

This appendix contains the parameter values and standard errors for each numbered equation

described in chapter II. 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 2004 Trustees Report alternative II (e.g., ) are presented in a

table at the end of this appendix.

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 variable

VAR, and disability 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 variancecovariance matrix. The correlated random error terms were then obtained by multiplying the

lower triangular Cholesky matrix by a vector of standard normal random errors.

Total Fertility Rate

(1)

Note: The total fertility rate was bounded on this interval because the computer model became unstable if the simulated total fertility rate was less than 0.5. The upper bound was set at 3.4 to preserve the symmetry around the ultimate TR04II value of 1.95.

AR Parameters MA Parameters

Process mean over

historical period

μ = 2.469526

Standard Error

TR Ft

11 2 2 3 3 4 4 11 φ φ φ φ ε θε , 0.5 3.4 = + + + + +− ≤ ≤ −−−− −

TR FF f f f f F tt t t t t t t t

1 φ = 1.991694 1 θ = 0.665162

2 φ = -1.508138

3 φ = 0.914735

4 φ = -0.419396

ˆ 0.090936 σ ε =
Appendices

68

Rate of Decrease in Central Death Rate

(2)

Cholesky Decomposition of Residual Variance-Covariance Matrix

Group (k) Sex Age AR Parameter (φk)

Process mean over

historical period

1 M Under 1 -0.351201 2.869243

2 F -0.372752 2.842529

3 M 1 to 4 -0.416565 3.501578

4 F -0.427767 3.565887

5 M 5 to 9 -0.227505 2.949994

6 F -0.271437 2.997825

7 M 10 to 14 -0.234871 2.155480

8 F -0.182700 2.320293

9 M 15 to 19 -0.193210 0.906328

10 F -0.168664 1.871810

11 M 20 to 24 -0.175331 0.822418

12 F -0.200078 1.914533

13 M 25 to 29 -0.213296 0.689647

14 F -0.185487 1.657118

15 M 30 to 34 -0.214036 0.882346

16 F -0.221462 1.788244

17 M 35 to 39 -0.213548 1.141628

18 F -0.237834 1.777003

19 M 40 to 44 -0.137733 1.173295

20 F -0.223119 1.681488

21 M 45 to 49 -0.153830 1.058397

22 F -0.253837 1.501890

23 M 50 to 54 -0.082579 0.992394

24 F -0.250153 1.418907

25 M 55 to 59 -0.025914 0.891541

26 F -0.221067 1.234432

27 M 60 to 64 -0.107071 0.783572

28 F -0.340946 1.133483

29 M 65 to 69 -0.217861 0.634373

30 F -0.219608 0.975814

31 M 70 to 74 -0.280665 0.610547

32 F -0.269514 0.978308

33 M 75 to 79 -0.256160 0.552180

34 F -0.280206 0.894627

35 M 80 to 84 -0.331191 0.540486

36 F -0.295512 0.816556

37 M 85 to 89 -0.241108 0.361083

38 F -0.185227 0.580038

39 M 90 to 94 -0.242183 0.274489

40 F -0.207037 0.467668

41 M 95 and Up -0.158566 0.139841

42 F -0.184255 0.322523

, , ,1 , φ ε =+ + −

TR MR MR mr kt kt k kt kt

V LL L A B C L == × ′ ( is a 42 42 matrix) [ ]
Equation Parameter Estimates and Regression Statistics

69

4.62 0 0 0 0 0 0 0 0 0 0 0 0 0

4.58 1.07 0 0 0 0 0 0 0 0 0 0 0 0

4.50 1.57 6.41 0 0 0 0 0 0 0 0 0 0 0

4.82 2.12 6.61 2.12 0 0 0 0 0 0 0 0 0 0

3.45 1.26 4.79 1.53 3.63 0 0 0 0 0 0 0 0 0

3.82 0.85 6.13 1.90 3.14 3.30 0 0 0 0 0 0 0 0

3.48 0.70 5.14 2.19 3.50 1.33 3.41 0 0 0 0 0 0 0

4.53 1.17 6.53 2.51 4.34 2.71 2.22 3.50 0 0 0 0 0 0

5.06 1.36 8.82 3.80 4.25 1.69 3.76 1.85 4.24 0 0 0 0 0

4.19 1.04 7.47 2.90 3.63 2.02 3.12 2.56 1.90 2.92 0 0 0 0

4.46 2.21 9.39 3.80 4.60 1.71 3.99 1.57 4.46 -0.84 3.49 0 0 0

5.07 1.73 9.02 4.46 4.51 2.66 4.05 2.27 3.36 2.56 0.08 3.12 0 0

5.75 2.49 11.22 5.42 5.59 2.70 4.82 2.51 5.68 0.76 1.61 1.75 3.13 0

5.73 2.47 10.87 5.51 5.31 3.18 4.58 3.06 4.38 2.75 -0.26 3.59 1.61 2.25

5.54 2.37 9.69 4.50 4.59 2.22 4.86 2.42 4.51 0.99 0.89 1.51 2.42 0.56

4.61 1.93 8.22 4.26 3.99 2.32 3.70 2.16 3.17 1.72 0.08 2.13 0.87 1.45

4.06 1.26 5.96 2.36 2.74 0.88 3.18 1.38 2.35 0.74 0.72 0.10 1.50 0.10

3.05 0.96 4.76 2.28 2.24 1.13 2.41 1.20 1.34 1.31 0.49 0.72 0.17 0.70

A =

2.80 0.73 3.13 0.53 1.06 0.13 1.78 0.82 1.01 0.59 0.88 -1.08 0.39 -0.19

2.21 0.31 2.67 0.79 1.03 0.61 1.19 0.97 0.63 0.96 0.79 0.02 -0.30 0.04

2.28 0.11 1.71 -0.11 0.53 -0.02 1.17 0.48 0.47 0.46 0.68 -0.91 0.15 -0.13

1.77 -0.06 1.91 0.04 0.57 0.46 0.93 0.38 0.15 0.70 0.30 -0.06 -0.25 0.16

1.62 -0.15 0.93 -0.37 0.00 -0.41 0.59 0.14 0.17 0.45 0.77 -0.88 -0.09 0.11

1.23 0.16 1.57 -0.09 0.41 0.11 0.70 0.16 -0.10 0.68 0.44 -0.40 -0.23 0.38

1.20 -0.41 0.54 -0.55 0.15 -0.16 0.44 0.01 0.09 0.48 0.87 -1.28 -0.23 0.15

1.02 -0.12 1.02 -0.27 0.01 0.20 0.55 -0.10 -0.40 0.69 0.68 -0.86 -0.42 0.24

0.97 -0.12 0.38 -0.61 -0.06 0.01 0.72 0.17 -0.26 0.26 0.46 -1.11 -0.13 0.27

0.77 -0.07 0.71 -0.38 -0.25 0.21 0.62 0.19 -0.67 0.54 0.37 -0.45 -0.29 0.45

0.99 -0.28 0.47 -0.41 -0.36 0.03 0.40 -0.02 -0.13 0.37 0.51 -1.04 -0.10 0.26

0.74 -0.11 0.66 -0.45 -0.48 0.18 0.51 0.19 -0.62 0.59 0.44 -0.59 -0.35 0.44

0.57 -0.24 0.31 -0.72 -0.22 0.10 0.39 0.11 -0.39 0.53 0.68 -1.01 -0.39 0.08

0.61 -0.12 0.31 -0.56 -0.41 0.28 0.59 0.28 -0.83 0.72 0.67 -0.70 -0.70 0.48

0.76 -0.57 0.41 -0.67 -0.54 -0.12 0.25 -0.09 -0.74 0.40 0.55 -0.86 -0.40 0.43

0.72 -0.46 0.27 -0.58 -0.75 0.07 0.41 -0.13 -0.98 0.42 0.53 -0.72 -0.56 0.62

0.61 -0.41 0.24 -0.76 -0.63 -0.04 0.39 -0.08 -0.75 0.46 0.68 -1.34 -0.79 0.29

0.66 -0.22 0.13 -0.67 -0.80 0.08 0.40 -0.26 -0.91 0.53 0.78 -1.11 -0.97 0.58

0.35 -0.63 -0.01 -0.76 -0.69 0.36 0.54 0.08 -1.00 0.77 0.50 -0.79 -0.90 0.64

0.50 -0.58 -0.19 -0.75 -0.63 0.37 0.56 -0.04 -1.04 0.68 0.52 -0.95 -1.04 0.85

0.42 -0.54 -0.07 -0.86 -0.71 0.59 0.50 0.27 -1.02 0.69 0.49 -0.81 -0.93 0.65

0.53 -0.51 -0.13 -1.00 -0.69 0.57 0.64 -0.07 -1.11 0.78 0.56 -0.92 -1.09 0.80

0.51 -0.29 -0.16 -0.72 -0.86 1.13 0.26 0.40 -0.79 0.52 0.25 -1.00 -1.04 0.76

0.60 -0.36 -0.13 -0.87 -0.84 0.91 0.50 0.14 -1.07 0.62 0.32 -1.14 -1.06 0.87

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

2.30 0 0 0 0 0 0 0 0 0 0 0 0 0

1.08 1.78 0 0 0 0 0 0 0 0 0 0 0 0

1.66 -0.16 2.39 0 0 0 0 0 0 0 0 0 0 0

0.48 0.72 1.46 1.69 0 0 0 0 0 0 0 0 0 0

B =

1.48 -0.11 1.96 0.44 1.96 0 0 0 0 0 0 0 0 0

0.45 0.56 0.99 0.59 1.04 1.48 0 0 0 0 0 0 0 0

0.76 -0.09 1.55 0.49 1.14 0.23 1.51 0 0 0 0 0 0 0

0.20 0.44 0.87 0.45 0.48 0.22 0.68 1.13 0 0 0 0 0 0

0.57 -0.33 1.53 0.51 1.29 0.10 0.63 -0.03 1.54 0 0 0 0 0

0.25 -0.05 0.86 0.61 0.43 0.64 0.17 0.39 0.98 1.18 0 0 0 0

0.67 0.05 0.99 0.47 1.25 0.10 0.84 0.26 0.90 -0.04 1.14 0 0 0

0.39 0.31 0.70 0.91 0.55 0.33 0.53 0.44 0.39 0.23 1.06 1.05 0 0

0.62 -0.27 1.01 0.43 0.75 -0.03 1.11 0.29 1.19 0.03 0.08 -0.11 1.21 0

0.36 0.04 0.76 0.62 0.25 0.29 0.74 0.35 0.67 0.52 0.15 0.24 0.64 0.92

0.33 -0.08 1.08 0.57 1.15 0.22 0.62 0.21 0.96 -0.02 0.55 0.16 0.60 -0.06

0.09 0.52 0.78 0.68 0.56 0.31 0.39 0.14 0.60 0.16 0.65 0.50 0.26 0.26

0.54 -0.06 0.89 0.48 0.75 0.28 0.79 0.22 1.11 0.15 0.50 -0.10 0.97 0.10

0.44 0.30 0.76 0.61 0.36 0.37 0.63 0.29 0.84 0.50 0.61 0.07 0.59 0.38

-0.05 0.25 1.00 0.77 0.51 0.15 0.67 0.23 1.09 0.12 0.89 0.06 0.59 0.41

-0.12 0.64 1.03 1.11 0.47 0.32 0.62 0.22 0.97 0.31 0.88 0.37 0.48 0.54

0.07 0.21 0.90 0.77 0.59 0.54 0.62 0.23 1.24 0.48 0.69 -0.07 0.76 0.52

-0.09 0.47 0.93 0.90 0.62 0.52 0.54 0.11 1.32 0.51 0.76 0.06 0.71 0.77

-0.15 0.38 1.34 0.87 0.16 0.18 0.64 0.41 1.28 0.14 0.87 0.00 1.10 0.64

-0.49 0.52 1.38 1.20 0.21 0.07 0.88 0.13 1.34 0.39 0.83 -0.12 1.02 0.83

-0.22 0.48 1.50 0.91 0.18 0.14 0.51 0.30 1.15 0.22 0.91 0.09 1.13 0.69

-0.57 0.52 1.45 1.01 0.14 0.06 0.77 0.11 1.30 0.39 0.83 -0.09 1.04 0.90

-0.12 0.41 1.98 1.11 0.05 0.12 0.24 0.20 0.82 0.12 1.29 0.30 1.13 0.68

-0.31 0.47 1.72 1.05 0.01 0.09 0.48 -0.08 1.11 0.41 1.07 0.06 1.04 1.08
Appendices

70

Legal Immigration

(3)

Note: Because legal immigration is a non-negative quantity, its lower bound was set at zero. The upper limit was chosen to preserve the symmetry around the TR04II value.

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

C =

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

00000000000000

1.06 0 0 0 0 0 0 0 0 0 0 0 0 0

0.69 1.13 0 0 0 0 0 0 0 0 0 0 0 0

0.25 -0.01 1.07 0 0 0 0 0 0 0 0 0 0 0

0.08 0.63 0.77 0.70 0 0 0 0 0 0 0 0 0 0

0.38 0.33 0.08 -0.12 1.09 0 0 0 0 0 0 0 0 0

0.30 0.89 0.02 0.43 0.78 0.75 0 0 0 0 0 0 0 0

0.67 0.23 0.58 0.22 0.41 0.05 1.18 0 0 0 0 0 0 0

0.38 0.47 0.71 0.63 0.36 0.35 0.83 0.81 0 0 0 0 0 0

0.62 0.44 0.29 0.21 0.72 0.30 0.55 0.45 1.42 0 0 0 0 0

0.69 1.05 0.33 0.46 0.80 0.34 0.61 0.75 0.96 0.83 0 0 0 0

0.69 0.37 0.20 0.28 0.66 0.31 0.63 0.61 1.48 -0.03 0.54 0 0 0

0.71 1.03 0.23 0.50 0.77 0.36 0.69 0.76 0.94 0.85 0.23 0.34 0 0

0.85 0.41 0.04 0.58 0.44 0.35 0.71 0.89 1.53 -0.24 0.94 0.23 1.23 0

0.84 1.08 0.10 0.63 0.68 0.35 0.77 0.95 1.03 0.80 0.67 0.31 0.57 0.41

AR Parameters MA Parameters

Process mean over

historical period

μ = 460,545

Standard Error

1 1 2 2 3 3 4 4 11 φ φ φ φ ε θε ,0 2 = + + + + + − ≤ ≤× −−−− −

TR TR

t t t t t ttt t t IM IM im im im im IM IM

1 φ = 1.079587 1 θ = -0.487496

2 φ = -0.538171

3 φ = 0.689674

4 φ = -0.309780

ˆ 84,770 σ ε =
Equation Parameter Estimates and Regression Statistics

71

Legal Emigration

(4)

Note: Because legal emigration is a non-negative quantity, its lower bound was set at zero. The upper limit was chosen to preserve the symmetry around the TR04II value.

Net Other Immigration

(5)

Economic Variable VAR

Unemployment Rate (expressed as log-odds ratio)

(6)

Inflation Rate (transformed into logs with lower bound of -3.0%)

(7)

AR Parameters MA Parameters

Standard Error

Standard Error

AR Parameters

Process mean over

historical period

μ = -2.796475

AR Parameters

Process mean over

historical period

μ = -2.719855

1 1 2 2 3 3 4 4 11 φ φ φ φ ε θε ,0 2 = + + + + + − ≤ ≤× −−−− −

TR TR EM EM em em em em EM EM t t t t t ttt t t

1 φ = 0.269897 1 θ = -0.121874

2 φ = -0.134543

3 φ = 0.172419

4 φ = -0.077445

ˆ 21,192 σ ε =

∆ =∆ + ε TR O O t tt 2003 2003 = = 400,000 TR O O

ˆ 25,371 σ ε =

111 1 112 2 121 1 122 2 131 1 132 2 1 ε =+ + + + + + + − −−− −−

TR U U au au ai ai ar ar tt t t t t t t t

111 a = 0.960146 112 a = -0.304386

121 a = 0.400633 122 a = -0.083907

131 a = 0.746191 132 a = 0.610372

211 1 212 2 221 1 222 2 231 1 232 2 2 ε =+ + + + + + + − −−− −−

TR

tt t t t t t t t I I au au ai ai ar ar

211 a = -0.769000 212 a = 0.721055

221 a = 0.595616 222 a = 0.301637

231 a = -4.846046 232 a = 1.803246
Appendices

72

Real Interest Rate

(8)

Note: The real interest rate was bounded below such that the nominal interest rate would be non-negative.

Matrix Notation

Cholesky Decomposition of Residual Variance-Covariance Matrix

Real Average Wage (percent change)

 (9)

AR Parameters

Process mean over

historical period

μ = 0.030517

AR Parameters

Process mean over

historical period

μ = 0.012822

Standard Error

311 1 312 2 321 1 322 2 331 1 332 2 3 ε =+ + + + + + + − −−− −−

TR R R au au ai ai ar ar tt t t t t t t t

311 a = 0.060540 312 a = -0.048520

321 a = 0.028107 322 a = -0.027562

331 a = 1.234137 332 a = -0.319613

1 21

1 22

1 23

A A 1 2

ε

ε

ε

− −

− −

− −

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ =+ + +

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

TR

tt t t t

TR

tt t t t

TR

tt t t t

UU u u

II i i

RR r r

111 121 131 112 122 132

211 221 231 212 222 232

311 321 331 312 322 332

 A A 1 2

⎡ ⎤⎡ ⎤

⎢ ⎥⎢ ⎥ = =

⎣ ⎦⎣ ⎦

aaa aaa

aaa aaa

aaa aaa

 0.113117 0 0

 -0.050445 0.139394 0 

0.000952 -0.007164 0.009837

V LL L

⎡ ⎤

⎢ ⎥ = = ′

⎣ ⎦

=++ + ββ ε 2 31−

TR WW u u tt t t t

2 β = -0.061391

3 β = 0.035378

ˆ 0.011806 σ ε =
Equation Parameter Estimates and Regression Statistics

73

Disability Incidence - Male

(10)

Disability Incidence - Female

(11)

Cholesky Decomposition of Residual Variance-Covariance Matrix

Disability Recovery - Male

(12)

Disability Recovery - Female

(13)

Cholesky Decomposition of Residual Variance-Covariance Matrix

AR Parameters

Process mean over

historical period

μ =5.559148

AR Parameters

Process mean over

historical period

μ =4.53556

AR Parameters

Process mean over

historical period

μ =10.51046

AR Parameters

Process mean over

historical period

μ =9.590448

=++ + φφ ε 1 12 2 − −

TR DIM DIM dim dim t t t tt

1 φ = 1.465965

2 φ = -0.634930

=++ + φφ ε 1 12 2 − −

TR DIF DIF dif dif t t t tt

1 φ = 1.447164

2 φ = -0.618748

0.369409 0 0.310160 0.198227

V LL L ⎡ ⎤ = = ′ ⎢ ⎥ ⎣ ⎦

=++ φ ε 1 1−

TR DRM DRM drm t t tt

1 φ = 0.577874

= ++ φ ε 1 1−

TR DRF DRF drf t t tt

1 φ = 0.569980

1.697850 0 1.538223 0.470810

V LL L ⎡ ⎤ = = ′ ⎢ ⎥ ⎣ ⎦
Appendices

74

Equation Parameters from the 2004 Trustees Report Assumptions

t Ft

TR MR1,tTR MR2,tTR MR3,tTR MR4,tTR MR5,tTR MR6,tTR MR7,tTR MR8,tTR

2004 2.0167 2.9097 2.7474 3.1458 2.9534 3.0533 2.5483 1.8772 1.4612

2005 2.0139 2.6733 2.5448 2.7958 2.6463 2.7069 2.3225 1.7531 1.4460

2006 2.0112 2.4845 2.3829 2.5161 2.4008 2.4350 2.1433 1.6580 1.4344

2007 2.0084 2.3338 2.2535 2.2923 2.2041 2.2203 2.0006 1.5840 1.4253

2008 2.0056 2.2133 2.1500 2.1128 2.0464 2.0501 1.8865 1.5260 1.4178

2009 2.0028 2.1170 2.0672 1.9688 1.9197 1.9145 1.7949 1.4800 1.4114

2010 2.0000 2.0399 2.0009 1.8530 1.8178 1.8062 1.7213 1.4432 1.4058

2011 1.9973 1.9781 1.9478 1.7598 1.7356 1.7194 1.6619 1.4134 1.4007

2012 1.9945 1.9287 1.9051 1.6846 1.6693 1.6495 1.6137 1.3892 1.3960

2013 1.9917 1.8889 1.8709 1.6239 1.6156 1.5931 1.5746 1.3692 1.3917

2014 1.9889 1.8571 1.8434 1.5746 1.5721 1.5474 1.5427 1.3528 1.3875

2015 1.9861 1.8314 1.8213 1.5347 1.5366 1.5103 1.5165 1.3390 1.3836

2016 1.9833 1.8107 1.8034 1.5021 1.5076 1.4800 1.4950 1.3273 1.3798

2017 1.9806 1.7940 1.7890 1.4754 1.4839 1.4552 1.4772 1.3174 1.3762

2018 1.9778 1.7805 1.7773 1.4535 1.4643 1.4347 1.4623 1.3089 1.3727

2019 1.9750 1.7695 1.7677 1.4354 1.4480 1.4177 1.4498 1.3015 1.3693

2020 1.9723 1.7605 1.7599 1.4203 1.4344 1.4035 1.4392 1.2950 1.3660

2021 1.9695 1.7532 1.7535 1.4077 1.4229 1.3916 1.4301 1.2893 1.3627

2022 1.9667 1.7471 1.7483 1.3970 1.4132 1.3815 1.4223 1.2841 1.3596

2023 1.9639 1.7421 1.7439 1.3879 1.4049 1.3728 1.4155 1.2794 1.3564

2024 1.9611 1.7379 1.7402 1.3801 1.3976 1.3653 1.4094 1.2751 1.3534

2025 1.9584 1.7344 1.7371 1.3733 1.3912 1.3587 1.4041 1.2712 1.3503

2026 1.9556 1.7314 1.7344 1.3673 1.3856 1.3529 1.3992 1.2675 1.3473

2027 1.9528 1.7288 1.7321 1.3619 1.3805 1.3477 1.3947 1.2639 1.3444

2028 1.9500 1.7212 1.7255 1.3486 1.3684 1.3348 1.3849 1.2576 1.3408

2029 1.9500 1.7203 1.7247 1.3458 1.3656 1.3321 1.3822 1.2549 1.3380

2030 1.9500 1.7193 1.7238 1.3431 1.3629 1.3294 1.3794 1.2523 1.3352

2031 1.9500 1.7184 1.7229 1.3404 1.3601 1.3266 1.3766 1.2497 1.3324

2032 1.9500 1.7175 1.7220 1.3377 1.3573 1.3239 1.3738 1.2471 1.3296

2033 1.9500 1.7165 1.7212 1.3350 1.3546 1.3212 1.3711 1.2445 1.3268

2034 1.9500 1.7156 1.7203 1.3323 1.3518 1.3185 1.3683 1.2419 1.3240

2035 1.9500 1.7146 1.7194 1.3296 1.3491 1.3158 1.3656 1.2394 1.3213

2036 1.9500 1.7137 1.7185 1.3270 1.3464 1.3131 1.3628 1.2368 1.3185

2037 1.9500 1.7127 1.7176 1.3243 1.3436 1.3105 1.3601 1.2343 1.3157

2038 1.9500 1.7117 1.7167 1.3216 1.3409 1.3078 1.3573 1.2317 1.3130

2039 1.9500 1.7108 1.7158 1.3190 1.3382 1.3051 1.3546 1.2292 1.3103

2040 1.9500 1.7098 1.7149 1.3163 1.3355 1.3025 1.3519 1.2267 1.3075

2041 1.9500 1.7088 1.7139 1.3137 1.3328 1.2998 1.3491 1.2242 1.3048

2042 1.9500 1.7078 1.7130 1.3110 1.3301 1.2972 1.3464 1.2217 1.3021

2043 1.9500 1.7068 1.7121 1.3084 1.3274 1.2946 1.3437 1.2193 1.2994

2044 1.9500 1.7058 1.7111 1.3058 1.3247 1.2920 1.3410 1.2168 1.2967

2045 1.9500 1.7048 1.7102 1.3032 1.3220 1.2893 1.3383 1.2144 1.2940

2046 1.9500 1.7038 1.7092 1.3006 1.3194 1.2868 1.3357 1.2120 1.2913

2047 1.9500 1.7027 1.7083 1.2980 1.3167 1.2841 1.3330 1.2095 1.2887

2048 1.9500 1.7017 1.7073 1.2954 1.3141 1.2816 1.3303 1.2071 1.2860

2049 1.9500 1.7007 1.7063 1.2929 1.3114 1.2790 1.3276 1.2047 1.2834

2050 1.9500 1.6996 1.7054 1.2903 1.3088 1.2764 1.3250 1.2024 1.2807

2051 1.9500 1.6986 1.7044 1.2877 1.3062 1.2739 1.3223 1.2000 1.2781

2052 1.9500 1.6975 1.7034 1.2852 1.3035 1.2713 1.3197 1.1977 1.2755

2053 1.9500 1.6965 1.7024 1.2826 1.3009 1.2688 1.3170 1.1953 1.2729

2054 1.9500 1.6954 1.7014 1.2801 1.2983 1.2663 1.3144 1.1930 1.2703

2055 1.9500 1.6943 1.7004 1.2776 1.2957 1.2637 1.3118 1.1907 1.2677

2056 1.9500 1.6932 1.6994 1.2751 1.2931 1.2612 1.3091 1.1884 1.2651

2057 1.9500 1.6921 1.6984 1.2726 1.2905 1.2587 1.3065 1.1861 1.2625

2058 1.9500 1.6910 1.6973 1.2701 1.2880 1.2563 1.3039 1.1838 1.2600

2059 1.9500 1.6899 1.6963 1.2676 1.2854 1.2538 1.3013 1.1816 1.2574

2060 1.9500 1.6888 1.6953 1.2651 1.2828 1.2513 1.2987 1.1793 1.2549

2061 1.9500 1.6877 1.6942 1.2627 1.2803 1.2488 1.2962 1.1771 1.2524

2062 1.9500 1.6866 1.6932 1.2602 1.2777 1.2464 1.2936 1.1749 1.2498

2063 1.9500 1.6855 1.6921 1.2578 1.2752 1.2440 1.2910 1.1727 1.2473

2064 1.9500 1.6843 1.6910 1.2553 1.2727 1.2415 1.2885 1.1704 1.2448

2065 1.9500 1.6832 1.6900 1.2529 1.2702 1.2391 1.2859 1.1683 1.2423

2066 1.9500 1.6820 1.6889 1.2505 1.2677 1.2367 1.2834 1.1661 1.2399

2067 1.9500 1.6809 1.6878 1.2481 1.2652 1.2343 1.2809 1.1639 1.2374

2068 1.9500 1.6797 1.6867 1.2457 1.2627 1.2319 1.2784 1.1618 1.2349

2069 1.9500 1.6785 1.6856 1.2433 1.2602 1.2295 1.2758 1.1597 1.2325

2070 1.9500 1.6773 1.6845 1.2409 1.2578 1.2272 1.2733 1.1575 1.2301

2071 1.9500 1.6762 1.6834 1.2385 1.2553 1.2248 1.2708 1.1554 1.2276

2072 1.9500 1.6750 1.6823 1.2362 1.2528 1.2225 1.2683 1.1533 1.2252

2073 1.9500 1.6738 1.6812 1.2338 1.2504 1.2201 1.2659 1.1513 1.2228

2074 1.9500 1.6726 1.6801 1.2315 1.2480 1.2178 1.2634 1.1492 1.2204

2075 1.9500 1.6713 1.6789 1.2291 1.2455 1.2155 1.2609 1.1471 1.2180

2076 1.9500 1.6701 1.6778 1.2268 1.2431 1.2132 1.2585 1.1451 1.2157

2077 1.9500 1.6689 1.6766 1.2245 1.2407 1.2109 1.2561 1.1431 1.2133

2078 1.9500 1.6676 1.6755 1.2222 1.2383 1.2086 1.2536 1.1411 1.2110

2079 1.9500 1.6664 1.6743 1.2199 1.2360 1.2063 1.2512 1.1390 1.2086
Equation Parameter Estimates and Regression Statistics

75

Equation Parameters from the 2004 Trustees Report Assumptions

t MR9,t

TR MR10,tTR MR11,tTR MR12,tTR MR13,tTR MR14,tTR MR15,tTR MR16,tTR MR17,tTR

2004 1.2537 1.0197 1.4051 1.1711 1.5643 0.7808 0.5897 0.0972 0.1171

2005 1.1661 0.9759 1.2872 1.0909 1.4114 0.7787 0.6290 0.2320 0.2625

2006 1.0973 0.9413 1.1944 1.0295 1.2938 0.7799 0.6686 0.3430 0.3855

2007 1.0429 0.9139 1.1212 0.9820 1.2027 0.7827 0.7054 0.4335 0.4883

2008 0.9999 0.8920 1.0631 0.9449 1.1316 0.7858 0.7380 0.5070 0.5731

2009 0.9656 0.8744 1.0169 0.9156 1.0757 0.7887 0.7660 0.5661 0.6425

2010 0.9382 0.8602 0.9800 0.8923 1.0316 0.7910 0.7895 0.6135 0.6989

2011 0.9162 0.8486 0.9504 0.8736 0.9965 0.7928 0.8088 0.6513 0.7443

2012 0.8986 0.8391 0.9267 0.8585 0.9685 0.7940 0.8244 0.6812 0.7807

2013 0.8843 0.8313 0.9076 0.8462 0.9460 0.7947 0.8369 0.7048 0.8097

2014 0.8728 0.8248 0.8922 0.8361 0.9278 0.7949 0.8468 0.7232 0.8326

2015 0.8634 0.8193 0.8797 0.8277 0.9131 0.7947 0.8545 0.7376 0.8507

2016 0.8557 0.8148 0.8695 0.8207 0.9011 0.7941 0.8603 0.7486 0.8647

2017 0.8495 0.8109 0.8612 0.8148 0.8912 0.7933 0.8647 0.7570 0.8755

2018 0.8443 0.8076 0.8544 0.8098 0.8831 0.7922 0.8679 0.7633 0.8837

2019 0.8400 0.8047 0.8489 0.8055 0.8764 0.7910 0.8700 0.7679 0.8898

2020 0.8365 0.8022 0.8442 0.8017 0.8708 0.7896 0.8715 0.7712 0.8943

2021 0.8335 0.8000 0.8404 0.7984 0.8660 0.7882 0.8723 0.7733 0.8974

2022 0.8309 0.7980 0.8371 0.7955 0.8620 0.7867 0.8726 0.7746 0.8994

2023 0.8288 0.7962 0.8344 0.7929 0.8585 0.7851 0.8725 0.7753 0.9006

2024 0.8269 0.7946 0.8320 0.7905 0.8555 0.7834 0.8720 0.7754 0.9011

2025 0.8253 0.7931 0.8300 0.7883 0.8528 0.7818 0.8714 0.7750 0.9010

2026 0.8239 0.7918 0.8283 0.7862 0.8505 0.7801 0.8705 0.7744 0.9005

2027 0.8226 0.7905 0.8267 0.7843 0.8484 0.7784 0.8695 0.7734 0.8997

2028 0.8195 0.7881 0.8227 0.7809 0.8435 0.7770 0.8705 0.7759 0.9034

2029 0.8188 0.7872 0.8220 0.7796 0.8423 0.7752 0.8688 0.7738 0.9011

2030 0.8182 0.7863 0.8213 0.7782 0.8412 0.7735 0.8672 0.7718 0.8989

2031 0.8176 0.7854 0.8206 0.7768 0.8401 0.7718 0.8655 0.7699 0.8967

2032 0.8170 0.7844 0.8199 0.7755 0.8390 0.7701 0.8639 0.7679 0.8945

2033 0.8164 0.7835 0.8192 0.7742 0.8379 0.7684 0.8623 0.7660 0.8924

2034 0.8158 0.7827 0.8185 0.7729 0.8368 0.7667 0.8608 0.7641 0.8903

2035 0.8151 0.7818 0.8179 0.7716 0.8358 0.7651 0.8592 0.7622 0.8882

2036 0.8146 0.7809 0.8172 0.7703 0.8347 0.7635 0.8577 0.7603 0.8861

2037 0.8140 0.7800 0.8166 0.7690 0.8337 0.7619 0.8562 0.7585 0.8840

2038 0.8134 0.7792 0.8160 0.7678 0.8326 0.7603 0.8547 0.7567 0.8820

2039 0.8129 0.7784 0.8153 0.7665 0.8316 0.7587 0.8532 0.7549 0.8800

2040 0.8123 0.7775 0.8147 0.7653 0.8306 0.7572 0.8517 0.7531 0.8780

2041 0.8118 0.7767 0.8141 0.7641 0.8296 0.7556 0.8503 0.7513 0.8761

2042 0.8112 0.7759 0.8135 0.7628 0.8286 0.7541 0.8488 0.7496 0.8741

2043 0.8107 0.7751 0.8129 0.7617 0.8277 0.7526 0.8474 0.7479 0.8722

2044 0.8101 0.7743 0.8123 0.7605 0.8267 0.7511 0.8460 0.7462 0.8703

2045 0.8096 0.7735 0.8117 0.7593 0.8258 0.7496 0.8446 0.7445 0.8684

2046 0.8091 0.7727 0.8111 0.7582 0.8248 0.7482 0.8432 0.7428 0.8665

2047 0.8086 0.7719 0.8106 0.7570 0.8239 0.7468 0.8419 0.7412 0.8647

2048 0.8081 0.7712 0.8100 0.7559 0.8230 0.7453 0.8406 0.7395 0.8629

2049 0.8076 0.7704 0.8094 0.7548 0.8221 0.7439 0.8392 0.7379 0.8611

2050 0.8071 0.7697 0.8089 0.7537 0.8212 0.7425 0.8379 0.7363 0.8593

2051 0.8066 0.7689 0.8083 0.7526 0.8204 0.7412 0.8366 0.7348 0.8576

2052 0.8061 0.7682 0.8078 0.7515 0.8195 0.7398 0.8354 0.7332 0.8558

2053 0.8056 0.7675 0.8073 0.7505 0.8186 0.7385 0.8341 0.7317 0.8541

2054 0.8052 0.7668 0.8068 0.7494 0.8178 0.7371 0.8329 0.7302 0.8524

2055 0.8047 0.7660 0.8062 0.7484 0.8170 0.7358 0.8316 0.7287 0.8507

2056 0.8043 0.7653 0.8057 0.7473 0.8161 0.7346 0.8304 0.7272 0.8491

2057 0.8038 0.7646 0.8052 0.7463 0.8153 0.7332 0.8292 0.7257 0.8474

2058 0.8034 0.7640 0.8047 0.7453 0.8145 0.7320 0.8280 0.7243 0.8458

2059 0.8029 0.7633 0.8042 0.7443 0.8137 0.7307 0.8268 0.7228 0.8442

2060 0.8025 0.7626 0.8037 0.7433 0.8129 0.7295 0.8257 0.7214 0.8426

2061 0.8020 0.7619 0.8032 0.7423 0.8121 0.7283 0.8245 0.7200 0.8410

2062 0.8016 0.7613 0.8028 0.7414 0.8114 0.7271 0.8234 0.7186 0.8395

2063 0.8012 0.7606 0.8023 0.7404 0.8106 0.7258 0.8223 0.7173 0.8379

2064 0.8008 0.7600 0.8018 0.7395 0.8098 0.7247 0.8211 0.7159 0.8364

2065 0.8004 0.7593 0.8014 0.7385 0.8091 0.7235 0.8201 0.7146 0.8349

2066 0.7999 0.7587 0.8009 0.7376 0.8084 0.7224 0.8190 0.7133 0.8334

2067 0.7995 0.7581 0.8005 0.7367 0.8076 0.7212 0.8179 0.7119 0.8320

2068 0.7991 0.7575 0.8000 0.7358 0.8069 0.7201 0.8168 0.7106 0.8305

2069 0.7987 0.7568 0.7996 0.7349 0.8062 0.7190 0.8158 0.7094 0.8291

2070 0.7984 0.7562 0.7991 0.7340 0.8055 0.7178 0.8147 0.7081 0.8277

2071 0.7980 0.7556 0.7987 0.7331 0.8048 0.7167 0.8137 0.7069 0.8263

2072 0.7976 0.7550 0.7983 0.7323 0.8041 0.7156 0.8127 0.7056 0.8249

2073 0.7972 0.7545 0.7979 0.7314 0.8034 0.7146 0.8117 0.7044 0.8235

2074 0.7968 0.7539 0.7974 0.7305 0.8028 0.7135 0.8107 0.7032 0.8222

2075 0.7965 0.7533 0.7970 0.7297 0.8021 0.7125 0.8097 0.7020 0.8208

2076 0.7961 0.7527 0.7966 0.7289 0.8014 0.7114 0.8088 0.7008 0.8195

2077 0.7958 0.7521 0.7962 0.7281 0.8008 0.7104 0.8078 0.6997 0.8182

2078 0.7954 0.7516 0.7958 0.7272 0.8001 0.7094 0.8069 0.6985 0.8169

2079 0.7950 0.7510 0.7954 0.7264 0.7995 0.7084 0.8059 0.6974 0.8156
Appendices

76

Equation Parameters from the 2004 Trustees Report Assumptions

t MR18,t

TR MR19,tTR MR20,tTR MR21,tTR MR22,tTR MR23,tTR MR24,tTR MR25,tTR MR26,tTR

2004 -0.0165 0.1965 0.5096 0.9120 1.1420 1.7302 1.3338 1.7585 0.9555

2005 0.1443 0.3353 0.5679 0.9229 1.0793 1.5762 1.2363 1.5985 0.9336

2006 0.2764 0.4528 0.6169 0.9365 1.0300 1.4559 1.1587 1.4721 0.9166

2007 0.3840 0.5509 0.6574 0.9503 0.9909 1.3614 1.0966 1.3718 0.9032

2008 0.4710 0.6318 0.6904 0.9631 0.9596 1.2867 1.0465 1.2918 0.8923

2009 0.5410 0.6979 0.7169 0.9740 0.9342 1.2272 1.0060 1.2278 0.8830

2010 0.5971 0.7515 0.7380 0.9831 0.9134 1.1797 0.9729 1.1762 0.8751

2011 0.6417 0.7946 0.7546 0.9902 0.8963 1.1413 0.9458 1.1345 0.8681

2012 0.6771 0.8290 0.7674 0.9955 0.8820 1.1102 0.9234 1.1006 0.8617

2013 0.7050 0.8563 0.7772 0.9993 0.8699 1.0848 0.9047 1.0728 0.8558

2014 0.7268 0.8778 0.7845 1.0017 0.8596 1.0639 0.8890 1.0498 0.8503

2015 0.7438 0.8945 0.7898 1.0029 0.8507 1.0465 0.8756 1.0307 0.8451

2016 0.7569 0.9074 0.7935 1.0032 0.8429 1.0318 0.8642 1.0146 0.8401

2017 0.7669 0.9171 0.7958 1.0027 0.8360 1.0195 0.8542 1.0010 0.8352

2018 0.7743 0.9243 0.7972 1.0015 0.8298 1.0088 0.8455 0.9893 0.8305

2019 0.7798 0.9295 0.7977 0.9997 0.8243 0.9996 0.8378 0.9791 0.8259

2020 0.7836 0.9331 0.7975 0.9976 0.8191 0.9915 0.8309 0.9702 0.8215

2021 0.7862 0.9354 0.7968 0.9951 0.8144 0.9843 0.8246 0.9622 0.8171

2022 0.7878 0.9366 0.7957 0.9923 0.8100 0.9778 0.8188 0.9551 0.8127

2023 0.7886 0.9370 0.7943 0.9893 0.8058 0.9719 0.8135 0.9486 0.8085

2024 0.7887 0.9368 0.7926 0.9861 0.8018 0.9664 0.8085 0.9425 0.8043

2025 0.7884 0.9360 0.7907 0.9828 0.7980 0.9613 0.8038 0.9369 0.8001

2026 0.7876 0.9347 0.7887 0.9794 0.7944 0.9565 0.7993 0.9316 0.7960

2027 0.7865 0.9332 0.7865 0.9759 0.7908 0.9520 0.7950 0.9266 0.7920

2028 0.7894 0.9362 0.7861 0.9740 0.7863 0.9449 0.7888 0.9189 0.7879

2029 0.7870 0.9332 0.7835 0.9701 0.7832 0.9412 0.7852 0.9148 0.7839

2030 0.7847 0.9303 0.7808 0.9663 0.7801 0.9376 0.7817 0.9108 0.7800

2031 0.7824 0.9275 0.7782 0.9625 0.7771 0.9340 0.7782 0.9069 0.7762

2032 0.7801 0.9246 0.7756 0.9587 0.7741 0.9304 0.7747 0.9029 0.7723

2033 0.7778 0.9218 0.7730 0.9550 0.7711 0.9269 0.7713 0.8990 0.7685

2034 0.7756 0.9190 0.7705 0.9513 0.7682 0.9233 0.7679 0.8951 0.7648

2035 0.7734 0.9162 0.7680 0.9476 0.7653 0.9198 0.7645 0.8912 0.7610

2036 0.7711 0.9135 0.7655 0.9440 0.7624 0.9163 0.7612 0.8873 0.7574

2037 0.7690 0.9108 0.7630 0.9404 0.7596 0.9128 0.7578 0.8835 0.7537

2038 0.7668 0.9081 0.7606 0.9368 0.7568 0.9093 0.7546 0.8797 0.7501

2039 0.7647 0.9054 0.7582 0.9333 0.7540 0.9059 0.7513 0.8759 0.7465

2040 0.7626 0.9028 0.7558 0.9298 0.7513 0.9024 0.7481 0.8721 0.7430

2041 0.7605 0.9002 0.7535 0.9263 0.7485 0.8990 0.7449 0.8684 0.7395

2042 0.7585 0.8976 0.7511 0.9229 0.7459 0.8956 0.7418 0.8646 0.7360

2043 0.7564 0.8951 0.7489 0.9195 0.7432 0.8923 0.7387 0.8610 0.7326

2044 0.7544 0.8925 0.7466 0.9162 0.7406 0.8889 0.7356 0.8573 0.7292

2045 0.7524 0.8900 0.7443 0.9128 0.7380 0.8856 0.7325 0.8536 0.7258

2046 0.7505 0.8876 0.7421 0.9095 0.7354 0.8823 0.7295 0.8500 0.7225

2047 0.7485 0.8851 0.7399 0.9063 0.7329 0.8790 0.7265 0.8464 0.7192

2048 0.7466 0.8827 0.7377 0.9030 0.7304 0.8757 0.7235 0.8428 0.7159

2049 0.7447 0.8803 0.7356 0.8998 0.7279 0.8725 0.7206 0.8392 0.7127

2050 0.7428 0.8779 0.7335 0.8967 0.7254 0.8692 0.7177 0.8357 0.7095

2051 0.7410 0.8756 0.7314 0.8935 0.7230 0.8660 0.7148 0.8322 0.7063

2052 0.7391 0.8733 0.7293 0.8904 0.7206 0.8628 0.7120 0.8287 0.7032

2053 0.7373 0.8710 0.7273 0.8874 0.7183 0.8596 0.7092 0.8253 0.7002

2054 0.7355 0.8687 0.7252 0.8843 0.7159 0.8565 0.7064 0.8218 0.6971

2055 0.7338 0.8665 0.7232 0.8813 0.7136 0.8534 0.7036 0.8184 0.6941

2056 0.7320 0.8643 0.7213 0.8783 0.7113 0.8503 0.7009 0.8150 0.6911

2057 0.7303 0.8621 0.7193 0.8754 0.7091 0.8472 0.6982 0.8116 0.6881

2058 0.7285 0.8599 0.7174 0.8724 0.7068 0.8441 0.6955 0.8083 0.6852

2059 0.7269 0.8578 0.7155 0.8696 0.7046 0.8410 0.6929 0.8050 0.6823

2060 0.7252 0.8556 0.7136 0.8667 0.7025 0.8380 0.6903 0.8017 0.6795

2061 0.7235 0.8535 0.7117 0.8639 0.7003 0.8350 0.6877 0.7984 0.6766

2062 0.7219 0.8515 0.7099 0.8611 0.6982 0.8320 0.6852 0.7952 0.6738

2063 0.7203 0.8494 0.7080 0.8583 0.6961 0.8291 0.6826 0.7920 0.6711

2064 0.7187 0.8474 0.7062 0.8556 0.6940 0.8261 0.6801 0.7887 0.6683

2065 0.7171 0.8453 0.7045 0.8528 0.6920 0.8232 0.6777 0.7856 0.6656

2066 0.7155 0.8434 0.7027 0.8502 0.6899 0.8203 0.6752 0.7824 0.6629

2067 0.7140 0.8414 0.7009 0.8475 0.6879 0.8174 0.6728 0.7793 0.6603

2068 0.7125 0.8395 0.6992 0.8449 0.6860 0.8145 0.6704 0.7762 0.6577

2069 0.7110 0.8375 0.6975 0.8423 0.6840 0.8116 0.6680 0.7731 0.6551

2070 0.7095 0.8356 0.6958 0.8397 0.6821 0.8088 0.6657 0.7700 0.6525

2071 0.7080 0.8337 0.6942 0.8372 0.6802 0.8060 0.6633 0.7670 0.6500

2072 0.7065 0.8319 0.6925 0.8347 0.6783 0.8032 0.6610 0.7640 0.6475

2073 0.7051 0.8300 0.6909 0.8322 0.6764 0.8004 0.6588 0.7610 0.6450

2074 0.7037 0.8282 0.6893 0.8297 0.6746 0.7977 0.6565 0.7580 0.6426

2075 0.7023 0.8264 0.6877 0.8273 0.6728 0.7950 0.6543 0.7551 0.6402

2076 0.7009 0.8246 0.6862 0.8249 0.6710 0.7922 0.6521 0.7522 0.6378

2077 0.6995 0.8229 0.6846 0.8225 0.6692 0.7895 0.6499 0.7493 0.6355

2078 0.6982 0.8211 0.6831 0.8201 0.6674 0.7869 0.6478 0.7464 0.6331

2079 0.6968 0.8194 0.6816 0.8178 0.6657 0.7842 0.6457 0.7436 0.6308
Equation Parameter Estimates and Regression Statistics

77

Equation Parameters from the 2004 Trustees Report Assumptions

t MR27,t

TR MR28,tTR MR29,tTR MR30,tTR MR31,tTR MR32,tTR MR33,tTR MR34,tTR MR35,tTR

2004 1.5425 0.6333 1.0988 0.1406 1.1434 0.1198 0.9292 0.0663 0.4925

2005 1.4227 0.6763 1.0477 0.2716 1.0847 0.2540 0.9204 0.2200 0.5784

2006 1.3283 0.7123 1.0088 0.3790 1.0397 0.3654 0.9150 0.3476 0.6485

2007 1.2536 0.7417 0.9789 0.4664 1.0047 0.4568 0.9115 0.4523 0.7050

2008 1.1940 0.7654 0.9553 0.5370 0.9772 0.5312 0.9088 0.5376 0.7502

2009 1.1463 0.7842 0.9366 0.5936 0.9551 0.5912 0.9065 0.6065 0.7861

2010 1.1077 0.7989 0.9212 0.6386 0.9371 0.6393 0.9042 0.6616 0.8142

2011 1.0764 0.8100 0.9085 0.6742 0.9222 0.6774 0.9017 0.7054 0.8360

2012 1.0508 0.8182 0.8978 0.7020 0.9097 0.7073 0.8990 0.7399 0.8526

2013 1.0296 0.8241 0.8885 0.7236 0.8990 0.7307 0.8960 0.7667 0.8650

2014 1.0119 0.8280 0.8803 0.7401 0.8896 0.7485 0.8927 0.7873 0.8740

2015 0.9970 0.8302 0.8730 0.7524 0.8812 0.7620 0.8892 0.8028 0.8802

2016 0.9843 0.8312 0.8663 0.7615 0.8738 0.7718 0.8854 0.8142 0.8840

2017 0.9733 0.8312 0.8602 0.7678 0.8669 0.7788 0.8815 0.8223 0.8862

2018 0.9637 0.8303 0.8544 0.7720 0.8605 0.7835 0.8773 0.8277 0.8868

2019 0.9552 0.8287 0.8490 0.7745 0.8545 0.7862 0.8731 0.8310 0.8863

2020 0.9476 0.8266 0.8438 0.7756 0.8488 0.7875 0.8686 0.8324 0.8848

2021 0.9407 0.8240 0.8388 0.7756 0.8433 0.7875 0.8642 0.8325 0.8825

2022 0.9343 0.8211 0.8340 0.7747 0.8381 0.7866 0.8596 0.8315 0.8797

2023 0.9284 0.8180 0.8293 0.7731 0.8330 0.7849 0.8550 0.8296 0.8763

2024 0.9229 0.8146 0.8247 0.7709 0.8281 0.7826 0.8503 0.8270 0.8726

2025 0.9176 0.8111 0.8202 0.7683 0.8232 0.7798 0.8456 0.8238 0.8686

2026 0.9126 0.8074 0.8157 0.7654 0.8185 0.7766 0.8409 0.8201 0.8644

2027 0.9078 0.8037 0.8114 0.7622 0.8138 0.7731 0.8361 0.8161 0.8599

2028 0.9012 0.8018 0.8069 0.7631 0.8089 0.7743 0.8323 0.8176 0.8585

2029 0.8970 0.7976 0.8027 0.7586 0.8045 0.7695 0.8274 0.8120 0.8532

2030 0.8929 0.7934 0.7985 0.7543 0.8001 0.7647 0.8226 0.8065 0.8480

2031 0.8888 0.7893 0.7944 0.7500 0.7957 0.7600 0.8178 0.8011 0.8428

2032 0.8848 0.7852 0.7903 0.7457 0.7913 0.7554 0.8130 0.7957 0.8376

2033 0.8807 0.7811 0.7862 0.7414 0.7870 0.7507 0.8083 0.7903 0.8325

2034 0.8767 0.7771 0.7822 0.7373 0.7827 0.7461 0.8035 0.7850 0.8273

2035 0.8727 0.7732 0.7782 0.7331 0.7785 0.7416 0.7989 0.7798 0.8223

2036 0.8687 0.7692 0.7743 0.7290 0.7743 0.7371 0.7943 0.7746 0.8173

2037 0.8648 0.7654 0.7703 0.7249 0.7701 0.7327 0.7897 0.7694 0.8123

2038 0.8609 0.7615 0.7664 0.7209 0.7659 0.7283 0.7851 0.7643 0.8073

2039 0.8569 0.7577 0.7625 0.7169 0.7618 0.7239 0.7806 0.7592 0.8024

2040 0.8531 0.7539 0.7587 0.7130 0.7577 0.7196 0.7761 0.7542 0.7975

2041 0.8492 0.7502 0.7549 0.7091 0.7537 0.7154 0.7717 0.7493 0.7927

2042 0.8454 0.7465 0.7511 0.7052 0.7497 0.7111 0.7672 0.7444 0.7878

2043 0.8416 0.7428 0.7473 0.7014 0.7457 0.7070 0.7629 0.7395 0.7831

2044 0.8378 0.7392 0.7436 0.6977 0.7417 0.7028 0.7585 0.7347 0.7783

2045 0.8341 0.7356 0.7399 0.6939 0.7378 0.6987 0.7542 0.7299 0.7737

2046 0.8304 0.7321 0.7362 0.6902 0.7339 0.6947 0.7499 0.7252 0.7690

2047 0.8267 0.7285 0.7326 0.6866 0.7301 0.6907 0.7457 0.7205 0.7644

2048 0.8230 0.7251 0.7290 0.6830 0.7263 0.6867 0.7415 0.7159 0.7598

2049 0.8194 0.7216 0.7255 0.6794 0.7225 0.6828 0.7373 0.7113 0.7552

2050 0.8158 0.7182 0.7219 0.6758 0.7187 0.6790 0.7332 0.7068 0.7507

2051 0.8122 0.7149 0.7184 0.6723 0.7150 0.6751 0.7291 0.7023 0.7462

2052 0.8086 0.7115 0.7149 0.6689 0.7113 0.6713 0.7250 0.6979 0.7418

2053 0.8051 0.7082 0.7114 0.6654 0.7076 0.6676 0.7210 0.6935 0.7374

2054 0.8015 0.7050 0.7080 0.6621 0.7040 0.6639 0.7170 0.6892 0.7330

2055 0.7981 0.7018 0.7046 0.6587 0.7004 0.6602 0.7131 0.6849 0.7287

2056 0.7946 0.6985 0.7013 0.6554 0.6969 0.6566 0.7091 0.6807 0.7244

2057 0.7911 0.6954 0.6979 0.6521 0.6933 0.6530 0.7053 0.6764 0.7201

2058 0.7877 0.6923 0.6946 0.6489 0.6898 0.6495 0.7014 0.6723 0.7159

2059 0.7844 0.6892 0.6914 0.6457 0.6864 0.6459 0.6976 0.6682 0.7117

2060 0.7810 0.6861 0.6881 0.6425 0.6830 0.6425 0.6938 0.6641 0.7075

2061 0.7777 0.6831 0.6849 0.6394 0.6796 0.6391 0.6900 0.6601 0.7034

2062 0.7744 0.6802 0.6817 0.6363 0.6762 0.6357 0.6863 0.6561 0.6993

2063 0.7711 0.6772 0.6785 0.6332 0.6728 0.6323 0.6827 0.6522 0.6953

2064 0.7678 0.6743 0.6754 0.6302 0.6695 0.6290 0.6790 0.6483 0.6913

2065 0.7646 0.6714 0.6723 0.6272 0.6662 0.6257 0.6754 0.6445 0.6873

2066 0.7614 0.6686 0.6692 0.6242 0.6630 0.6225 0.6718 0.6407 0.6834

2067 0.7582 0.6657 0.6662 0.6213 0.6598 0.6193 0.6683 0.6369 0.6795

2068 0.7550 0.6629 0.6632 0.6184 0.6566 0.6161 0.6647 0.6332 0.6756

2069 0.7519 0.6602 0.6602 0.6155 0.6534 0.6130 0.6613 0.6295 0.6718

2070 0.7488 0.6574 0.6572 0.6127 0.6503 0.6099 0.6578 0.6259 0.6680

2071 0.7457 0.6548 0.6543 0.6099 0.6472 0.6069 0.6544 0.6223 0.6642

2072 0.7427 0.6521 0.6513 0.6072 0.6441 0.6039 0.6510 0.6187 0.6605

2073 0.7396 0.6495 0.6485 0.6044 0.6411 0.6009 0.6477 0.6152 0.6568

2074 0.7366 0.6469 0.6456 0.6017 0.6381 0.5979 0.6443 0.6118 0.6531

2075 0.7336 0.6443 0.6428 0.5991 0.6351 0.5950 0.6410 0.6083 0.6495

2076 0.7307 0.6418 0.6400 0.5964 0.6322 0.5921 0.6378 0.6049 0.6459

2077 0.7278 0.6392 0.6372 0.5938 0.6292 0.5893 0.6346 0.6016 0.6424

2078 0.7249 0.6368 0.6345 0.5913 0.6263 0.5865 0.6314 0.5983 0.6389

2079 0.7220 0.6343 0.6317 0.5887 0.6235 0.5837 0.6282 0.5950 0.6354
Appendices

78

Equation Parameters from the 2004 Trustees Report Assumptions

t MR36,t

TR MR37,tTR MR38,tTR MR39,tTR MR40,tTR MR41,tTR MR42,tTR IMtTR EMtTR

2004 -0.0041 0.0426 -0.0753 -0.4091 -0.4196 -1.4868 -1.3316 1,033,333 258,333

2005 0.1766 0.1738 0.0691 -0.1827 -0.2000 -1.0437 -0.9210 1,000,000 250,000

2006 0.3255 0.2810 0.1901 0.0006 -0.0190 -0.6872 -0.5883 966,667 241,667

2007 0.4473 0.3681 0.2905 0.1485 0.1293 -0.4008 -0.3196 933,333 233,333

2008 0.5460 0.4385 0.3728 0.2675 0.2500 -0.1711 -0.1031 900,000 225,000

2009 0.6255 0.4952 0.4400 0.3629 0.3478 0.0130 0.0709 866,667 216,667

2010 0.6891 0.5405 0.4944 0.4393 0.4267 0.1602 0.2106 833,333 208,333

2011 0.7395 0.5767 0.5382 0.5002 0.4900 0.2778 0.3223 800,000 200,000

2012 0.7792 0.6053 0.5732 0.5486 0.5407 0.3715 0.4116 800,000 200,000

2013 0.8100 0.6278 0.6011 0.5869 0.5810 0.4461 0.4828 800,000 200,000

2014 0.8337 0.6453 0.6230 0.6170 0.6129 0.5052 0.5393 800,000 200,000

2015 0.8516 0.6589 0.6401 0.6406 0.6380 0.5521 0.5840 800,000 200,000

2016 0.8648 0.6692 0.6533 0.6590 0.6575 0.5890 0.6193 800,000 200,000

2017 0.8742 0.6770 0.6633 0.6731 0.6726 0.6179 0.6469 800,000 200,000

2018 0.8804 0.6826 0.6708 0.6838 0.6841 0.6405 0.6685 800,000 200,000

2019 0.8843 0.6865 0.6762 0.6918 0.6928 0.6579 0.6851 800,000 200,000

2020 0.8861 0.6891 0.6800 0.6976 0.6990 0.6712 0.6978 800,000 200,000

2021 0.8863 0.6906 0.6823 0.7016 0.7034 0.6813 0.7073 800,000 200,000

2022 0.8852 0.6912 0.6836 0.7042 0.7063 0.6887 0.7142 800,000 200,000

2023 0.8831 0.6911 0.6841 0.7056 0.7080 0.6940 0.7192 800,000 200,000

2024 0.8802 0.6904 0.6838 0.7062 0.7087 0.6976 0.7224 800,000 200,000

2025 0.8767 0.6893 0.6829 0.7060 0.7086 0.6999 0.7244 800,000 200,000

2026 0.8726 0.6878 0.6816 0.7053 0.7078 0.7010 0.7254 800,000 200,000

2027 0.8681 0.6861 0.6799 0.7040 0.7066 0.7014 0.7255 800,000 200,000

2028 0.8698 0.6882 0.6833 0.7089 0.7121 0.7126 0.7362 800,000 200,000

2029 0.8635 0.6853 0.6801 0.7057 0.7088 0.7093 0.7329 800,000 200,000

2030 0.8573 0.6823 0.6769 0.7026 0.7055 0.7062 0.7296 800,000 200,000

2031 0.8511 0.6793 0.6737 0.6995 0.7022 0.7030 0.7262 800,000 200,000

2032 0.8450 0.6764 0.6705 0.6964 0.6989 0.6998 0.7229 800,000 200,000

2033 0.8390 0.6735 0.6673 0.6933 0.6956 0.6966 0.7196 800,000 200,000

2034 0.8329 0.6706 0.6642 0.6903 0.6923 0.6935 0.7163 800,000 200,000

2035 0.8269 0.6677 0.6611 0.6872 0.6891 0.6903 0.7131 800,000 200,000

2036 0.8210 0.6648 0.6579 0.6842 0.6858 0.6872 0.7098 800,000 200,000

2037 0.8152 0.6619 0.6548 0.6811 0.6826 0.6841 0.7065 800,000 200,000

2038 0.8093 0.6591 0.6517 0.6781 0.6794 0.6810 0.7033 800,000 200,000

2039 0.8036 0.6562 0.6486 0.6751 0.6762 0.6779 0.7000 800,000 200,000

2040 0.7979 0.6534 0.6456 0.6721 0.6729 0.6748 0.6968 800,000 200,000

2041 0.7922 0.6506 0.6425 0.6691 0.6697 0.6718 0.6936 800,000 200,000

2042 0.7866 0.6477 0.6395 0.6661 0.6666 0.6687 0.6904 800,000 200,000

2043 0.7810 0.6450 0.6364 0.6632 0.6634 0.6657 0.6871 800,000 200,000

2044 0.7755 0.6422 0.6334 0.6602 0.6603 0.6627 0.6839 800,000 200,000

2045 0.7700 0.6394 0.6304 0.6573 0.6571 0.6596 0.6808 800,000 200,000

2046 0.7646 0.6367 0.6274 0.6544 0.6540 0.6566 0.6776 800,000 200,000

2047 0.7593 0.6339 0.6245 0.6515 0.6509 0.6536 0.6744 800,000 200,000

2048 0.7539 0.6312 0.6215 0.6486 0.6478 0.6507 0.6713 800,000 200,000

2049 0.7487 0.6285 0.6185 0.6457 0.6447 0.6477 0.6681 800,000 200,000

2050 0.7435 0.6258 0.6156 0.6428 0.6416 0.6447 0.6650 800,000 200,000

2051 0.7383 0.6231 0.6127 0.6400 0.6385 0.6418 0.6619 800,000 200,000

2052 0.7332 0.6204 0.6098 0.6371 0.6355 0.6389 0.6588 800,000 200,000

2053 0.7281 0.6178 0.6069 0.6343 0.6324 0.6360 0.6557 800,000 200,000

2054 0.7231 0.6151 0.6040 0.6315 0.6294 0.6331 0.6526 800,000 200,000

2055 0.7182 0.6125 0.6011 0.6287 0.6264 0.6302 0.6495 800,000 200,000

2056 0.7133 0.6099 0.5983 0.6259 0.6234 0.6273 0.6464 800,000 200,000

2057 0.7084 0.6073 0.5954 0.6231 0.6204 0.6244 0.6434 800,000 200,000

2058 0.7036 0.6047 0.5926 0.6204 0.6174 0.6216 0.6403 800,000 200,000

2059 0.6988 0.6021 0.5898 0.6176 0.6145 0.6187 0.6373 800,000 200,000

2060 0.6941 0.5996 0.5870 0.6149 0.6115 0.6159 0.6343 800,000 200,000

2061 0.6895 0.5970 0.5842 0.6122 0.6086 0.6131 0.6313 800,000 200,000

2062 0.6848 0.5945 0.5815 0.6095 0.6057 0.6103 0.6283 800,000 200,000

2063 0.6803 0.5920 0.5787 0.6068 0.6028 0.6075 0.6253 800,000 200,000

2064 0.6757 0.5895 0.5760 0.6041 0.5999 0.6048 0.6223 800,000 200,000

2065 0.6713 0.5870 0.5733 0.6015 0.5970 0.6020 0.6194 800,000 200,000

2066 0.6668 0.5845 0.5705 0.5988 0.5941 0.5993 0.6164 800,000 200,000

2067 0.6625 0.5820 0.5679 0.5962 0.5913 0.5965 0.6135 800,000 200,000

2068 0.6581 0.5796 0.5652 0.5935 0.5884 0.5938 0.6106 800,000 200,000

2069 0.6538 0.5772 0.5625 0.5909 0.5856 0.5911 0.6077 800,000 200,000

2070 0.6496 0.5747 0.5599 0.5884 0.5828 0.5884 0.6048 800,000 200,000

2071 0.6454 0.5723 0.5572 0.5858 0.5800 0.5858 0.6019 800,000 200,000

2072 0.6413 0.5699 0.5546 0.5832 0.5772 0.5831 0.5990 800,000 200,000

2073 0.6372 0.5675 0.5520 0.5806 0.5745 0.5805 0.5962 800,000 200,000

2074 0.6331 0.5652 0.5494 0.5781 0.5717 0.5778 0.5933 800,000 200,000

2075 0.6291 0.5628 0.5468 0.5756 0.5690 0.5752 0.5905 800,000 200,000

2076 0.6252 0.5605 0.5443 0.5731 0.5662 0.5726 0.5877 800,000 200,000

2077 0.6212 0.5581 0.5417 0.5706 0.5635 0.5700 0.5849 800,000 200,000

2078 0.6174 0.5558 0.5392 0.5681 0.5608 0.5674 0.5821 800,000 200,000

2079 0.6135 0.5535 0.5367 0.5656 0.5582 0.5649 0.5793 800,000 200,000
Equation Parameter Estimates and Regression Statistics

79

Equation Parameters from the 2004 Trustees Report Assumptions

t ∆Ot

TR UtTR ItTR RtTR WtTR DIMtTR DIFtTR DRMtTR DRFtTR

2004 0 -2.8137 -3.1669 0.0323 0.0238 5.2575 5.2156 14.0153 11.7905

2005 0 -2.8354 -3.1018 0.0329 0.0275 5.2389 5.1614 14.7945 12.2154

2006 0 -2.8334 -3.0007 0.0317 0.0189 5.2879 5.1323 15.9836 13.2971

2007 0 -2.8345 -2.9096 0.0314 0.0151 5.3314 5.1024 16.0449 13.2619

2008 0 -2.8361 -2.8531 0.0310 0.0137 5.3753 5.0688 17.5710 14.4340

2009 0 -2.8361 -2.8478 0.0307 0.0125 5.4026 5.0187 17.7771 14.4573

2010 0 -2.8369 -2.8454 0.0303 0.0122 5.4279 4.9677 18.0307 14.4862

2011 0 -2.8382 -2.8477 0.0302 0.0119 5.4606 4.9230 18.3203 15.8297

2012 0 -2.8391 -2.8468 0.0300 0.0121 5.4968 4.8817 18.6180 15.8735

2013 0 -2.8401 -2.8483 0.0301 0.0110 5.5336 4.8394 18.9424 15.9427

2014 -50,000 -2.8415 -2.8473 0.0300 0.0109 5.5953 4.8770 18.1460 15.2665

2015 0 -2.8430 -2.8473 0.0300 0.0103 5.6561 4.9157 17.3567 14.6057

2016 0 -2.8439 -2.8473 0.0300 0.0105 5.7185 4.9563 16.5604 13.9452

2017 0 -2.8436 -2.8473 0.0300 0.0104 5.7826 4.9957 15.7101 13.2610

2018 0 -2.8423 -2.8473 0.0300 0.0100 5.8435 5.0321 14.8327 12.5886

2019 0 -2.8431 -2.8473 0.0300 0.0105 5.9027 5.0689 13.9549 11.9400

2020 0 -2.8438 -2.8473 0.0300 0.0108 5.9631 5.1053 13.0818 11.3106

2021 0 -2.8449 -2.8473 0.0300 0.0098 6.0423 5.1523 12.1935 10.6925

2022 0 -2.8460 -2.8473 0.0300 0.0117 6.1223 5.2007 11.2658 10.0499

2023 0 -2.8460 -2.8473 0.0300 0.0111 6.2016 5.2480 10.3280 9.4046

2024 -50,000 -2.8461 -2.8473 0.0300 0.0105 6.2155 5.2538 10.3248 9.4176

2025 0 -2.8463 -2.8473 0.0300 0.0099 6.2282 5.2581 10.3208 9.4420

2026 0 -2.8469 -2.8473 0.0300 0.0097 6.2436 5.2644 10.3084 9.4631

2027 0 -2.8476 -2.8473 0.0300 0.0117 6.2436 5.2648 10.2754 9.4658

2028 0 -2.8472 -2.8473 0.0300 0.0112 6.2453 5.2674 10.2361 9.4581

2029 0 -2.8469 -2.8473 0.0300 0.0106 6.2473 5.2701 10.1971 9.4462

2030 0 -2.8468 -2.8473 0.0300 0.0100 6.2472 5.2707 10.1689 9.4495

2031 0 -2.8468 -2.8473 0.0300 0.0099 6.2462 5.2695 10.1362 9.4432

2032 0 -2.8470 -2.8473 0.0300 0.0117 6.2441 5.2682 10.0944 9.4271

2033 0 -2.8468 -2.8473 0.0300 0.0114 6.2411 5.2652 10.0497 9.4033

2034 0 -2.8467 -2.8473 0.0300 0.0108 6.2385 5.2637 10.0070 9.3877

2035 0 -2.8466 -2.8473 0.0300 0.0101 6.2379 5.2631 9.9843 9.3740

2036 0 -2.8466 -2.8473 0.0300 0.0100 6.2393 5.2639 9.9690 9.3749

2037 0 -2.8467 -2.8473 0.0300 0.0115 6.2435 5.2678 9.9495 9.3648

2038 0 -2.8468 -2.8473 0.0300 0.0114 6.2461 5.2711 9.9275 9.3471

2039 0 -2.8470 -2.8473 0.0300 0.0110 6.2479 5.2732 9.9120 9.3334

2040 0 -2.8473 -2.8473 0.0300 0.0102 6.2483 5.2729 9.9027 9.3281

2041 0 -2.8476 -2.8473 0.0300 0.0099 6.2477 5.2707 9.8982 9.3270

2042 0 -2.8480 -2.8473 0.0300 0.0112 6.2478 5.2703 9.8964 9.3294

2043 0 -2.8484 -2.8473 0.0300 0.0113 6.2469 5.2700 9.8798 9.3134

2044 0 -2.8488 -2.8473 0.0300 0.0111 6.2457 5.2700 9.8694 9.3020

2045 0 -2.8492 -2.8473 0.0300 0.0103 6.2444 5.2702 9.8600 9.2999

2046 0 -2.8497 -2.8473 0.0300 0.0100 6.2436 5.2692 9.8610 9.2958

2047 0 -2.8503 -2.8473 0.0300 0.0111 6.2464 5.2706 9.8674 9.3086

2048 0 -2.8507 -2.8473 0.0300 0.0111 6.2483 5.2723 9.8601 9.2960

2049 0 -2.8510 -2.8473 0.0300 0.0109 6.2500 5.2738 9.8542 9.2902

2050 0 -2.8514 -2.8473 0.0300 0.0101 6.2501 5.2734 9.8488 9.2901

2051 0 -2.8517 -2.8473 0.0300 0.0099 6.2485 5.2717 9.8531 9.2927

2052 0 -2.8520 -2.8473 0.0300 0.0109 6.2496 5.2727 9.8573 9.3020

2053 0 -2.8522 -2.8473 0.0300 0.0110 6.2487 5.2713 9.8520 9.2962

2054 0 -2.8524 -2.8473 0.0300 0.0109 6.2475 5.2705 9.8446 9.2910

2055 0 -2.8525 -2.8473 0.0300 0.0103 6.2455 5.2688 9.8361 9.2859

2056 0 -2.8524 -2.8473 0.0300 0.0100 6.2418 5.2663 9.8389 9.2848

2057 0 -2.8524 -2.8473 0.0300 0.0109 6.2440 5.2679 9.8431 9.2950

2058 0 -2.8524 -2.8473 0.0300 0.0109 6.2455 5.2689 9.8399 9.2920

2059 0 -2.8524 -2.8473 0.0300 0.0107 6.2475 5.2705 9.8363 9.2940

2060 0 -2.8524 -2.8473 0.0300 0.0104 6.2488 5.2722 9.8381 9.2888

2061 0 -2.8525 -2.8473 0.0300 0.0102 6.2479 5.2724 9.8345 9.2894

2062 0 -2.8525 -2.8473 0.0300 0.0109 6.2479 5.2707 9.8385 9.2932

2063 0 -2.8525 -2.8473 0.0300 0.0108 6.2458 5.2684 9.8339 9.2893

2064 0 -2.8525 -2.8473 0.0300 0.0107 6.2438 5.2674 9.8291 9.2872

2065 0 -2.8525 -2.8473 0.0300 0.0106 6.2432 5.2676 9.8275 9.2850

2066 0 -2.8524 -2.8473 0.0300 0.0106 6.2461 5.2696 9.8265 9.2831

2067 0 -2.8524 -2.8473 0.0300 0.0108 6.2458 5.2695 9.8294 9.2862

2068 0 -2.8524 -2.8473 0.0300 0.0107 6.2460 5.2701 9.8294 9.2854

2069 0 -2.8525 -2.8473 0.0300 0.0107 6.2462 5.2705 9.8278 9.2856

2070 0 -2.8526 -2.8473 0.0300 0.0107 6.2465 5.2706 9.8248 9.2845

2071 0 -2.8528 -2.8473 0.0300 0.0107 6.2467 5.2704 9.8257 9.2818

2072 0 -2.8529 -2.8473 0.0300 0.0107 6.2468 5.2705 9.8253 9.2813

2073 0 -2.8530 -2.8473 0.0300 0.0107 6.2469 5.2707 9.8241 9.2838

2074 0 -2.8531 -2.8473 0.0300 0.0107 6.2470 5.2706 9.8218 9.2816

2075 0 -2.8532 -2.8473 0.0300 0.0107 6.2471 5.2707 9.8246 9.2820

2076 0 -2.8534 -2.8473 0.0300 0.0107 6.2471 5.2705 9.8233 9.2802

2077 0 -2.8535 -2.8473 0.0300 0.0107 6.2472 5.2705 9.8214 9.2803

2078 0 -2.8536 -2.8473 0.0300 0.0107 6.2472 5.2704 9.8190 9.2842

2079 0 -2.8537 -2.8473 0.0300 0.0107 6.2474 5.2705 9.8203 9.2839
Appendices

80

E. GLOSSARY

Actuarial balance. The difference between the summarized income rate and the summarized

cost rate over a given time period.

Administrative expenses. Expenses incurred by the Social Security Administration and the

Department of the Treasury in administering the OASDI program and the provisions of the Internal Revenue Code relating to the collection of contributions. Such administrative expenses are

paid from the OASI and DI Trust Funds.

Age-adjusted rates. The crude rate that would occur in the standard population if it were to

experience the same rates by age as in the selected year.

Alternatives I, II, or III. See “Assumptions.”

Annual maximum taxable level. See “Contribution and benefit base.”

Assets. Treasury notes and bonds, other securities guaranteed by the Federal Government, certain Federally sponsored agency obligations, and cash, held by the trust funds for investment purposes. 

Assumptions. Values relating to future trends in certain key factors which affect the balance in

the trust funds. Demographic assumptions include fertility, mortality, net immigration, marriage,

and divorce. Economic assumptions include unemployment rates, average earnings, inflation,

interest rates, and productivity. Program-specific assumptions include retirement patterns, and

disability incidence and termination rates. Three sets of demographic, economic, and programspecific assumptions are presented in the 2004 Trustees Report:

• Alternative II (intermediate assumptions): represents the Trustees’ best estimates of likely

future demographic, economic, and program-specific conditions.

• Alternative I (low cost assumptions): assumes relatively rapid economic growth, low

inflation, and favorable (from the standpoint of program financing) demographic conditions.

• Alternative III (high cost assumptions): assumes relatively slow economic growth, high

inflation, and unfavorable (from the standpoint of program financing) demographic conditions.

Automatic cost-of-living benefit increase. The annual increase in benefits, effective for

December, reflecting the increase in the cost of living. The benefit increase equals the percentage

increase in the Consumer Price Index for Urban Wage Earners and Clerical Workers (CPI-W)

measured from the average over July, August, and September of the preceding year to the average

for the same 3 months in the current year. If the increase is less than one-tenth of 1 percent, when

rounded, there is no automatic increase for the current year; the increase for the next year would

reflect the net increase in the CPI over a 2-year period.
Glossary

81

Average indexed monthly earnings (AIME). The amount of earnings used in determining the

primary insurance amount (PIA) for most workers who attain age 62, become disabled, or die

after 1978. A worker’s actual past earnings are adjusted by changes in the average wage index, in

order to bring them up to their approximately equivalent value at the time of retirement or other

eligibility for benefits.

Average wage index. The average amount of total wages for each year after 1950, including

wages in noncovered employment and wages in covered employment in excess of the OASDI

contribution and benefit base. (See Title 20, chapter III, section 404.211(c) of the Code of Federal

Regulations for a more precise definition.) These average wage amounts are used to index the

taxable earnings of most workers first becoming eligible for benefits in 1979 or later, and for

automatic adjustments in the contribution and benefit base, bend points, earnings test exempt

amounts, and other wage-indexed amounts. 

Award. An administrative determination that an individual is entitled to receive a specified type

of OASDI benefit. Awards can represent not only new entrants to the benefit rolls but also persons already on the rolls who become entitled to a different type of benefit. Awards usually result

in the immediate payment of benefits, although payments may be deferred or withheld depending

on the individual’s particular circumstances.

Bend points. The dollar amounts defining the AIME or PIA brackets in the benefit formulas.

Each bracket is an AIME or PIA interval throughout which each dollar generates a uniform benefit increment.

Central death rate. The annual number of deaths for a particular group divided by the estimated

population of that group at midyear.

Consumer Price Index (CPI). An official measure of inflation in consumer prices. In this study,

all references to the CPI relate to the Consumer Price Index for Urban Wage Earners and Clerical

Workers (CPI-W). 

Contribution and benefit base. Annual dollar amount above which earnings in employment

covered under the OASDI program are neither taxable nor creditable for benefit computation purposes. (Also referred to as maximum contribution and benefit base, annual creditable maximum,

taxable maximum, and maximum taxable.) 

Contributions. The amount based on a percentage of earnings, up to an annual maximum, that

must be paid by:

• employers and employees on wages from employment under the Federal Insurance Contributions Act,

• the self-employed on net earnings from self-employment under the Self-Employment

Contributions Act, and

• States on the wages of State and local government employees covered under the Social

Security Act through voluntary agreements under section 218 of the Act.

Generally, employers withhold contributions from wages, add an equal amount of contributions,

and pay both on a current basis. Also referred to as taxes, and as tax contributions.
Appendices

82

Cost-of-living adjustment (COLA). See “Automatic cost-of-living benefit increase.”

Cost rate. The cost rate for a year is the ratio of the cost of the program to the taxable payroll for

the year. In this context, the cost is defined to include scheduled benefit payments, special

monthly payments to certain uninsured persons who have three or more quarters of coverage (and

whose payments are therefore not reimbursable from the General Fund of the Treasury), administrative expenses, net transfers from the trust funds to the Railroad Retirement program under the

financial-interchange provisions, and payments for vocational rehabilitation services for disabled

beneficiaries; it excludes special monthly payments to certain uninsured persons whose payments

are reimbursable from the General Fund of the Treasury (as described above), and transfers under

the interfund borrowing provisions.

Covered earnings. Earnings in employment or self-employment covered by the OASDI program.

Covered employment. All employment for which earnings are creditable for Social Security

purposes. Almost all employment is covered under the program. Some exceptions are:

• State and local government employees whose employer has not elected to be covered

under Social Security and who are participating in an employer-provided pension plan.

• Current Federal civilian workers hired before 1984 who have not elected to be covered.

• Self-employed workers earning less than $400 in a calendar year.

Covered worker. A person who has earnings creditable for Social Security purposes on the basis

of services for wages in covered employment and/or on the basis of income from covered selfemployment.

Covered worker rate. The ratio of OASDI covered workers to the Social Security area population.

Current-payment status. The status of beneficiaries presently scheduled to be paid regular

monthly benefits; includes those scheduled to be paid less than a full month’s benefits, but

excludes those who receive retroactive benefits that are not scheduled to be paid regularly.

Currently insured status. See “Insured status.”

Deterministic model. A model with specified assumptions for and relationships among variables. Under such model, any specified set of assumptions fully determines a single outcome

directly reflecting the specifications.

Disability incidence rate. The proportion of the exposed population at the beginning of that

year who become newly entitled to disability benefits during the year.

Disability Insurance (DI). See “Trust fund.”

Disability insured status. See “Insured status.”
Glossary

83

Disability recovery rate. The proportion of disabled-worker beneficiaries whose disability benefits terminate as a result of the individual’s recovery from disability during the year.

Disbursements. See “Outgo.”

Earnings. Unless otherwise qualified, all wages from employment and net earnings from selfemployment, whether or not taxable or covered. 

Emigration. See “Legal emigration.”

Entitlement. The conditions upon which an individual may collect a Social Security cash benefit. There are many definitions, depending on the type of benefit. 

Retired-worker benefit—An individual is entitled to a retirement insurance benefit if that

individual:

(a) is at least 62 for a full month and is fully insured; and

(b) has filed an application for retirement benefits, or is entitled to disabled-worker

benefits in the month before the month of attainment of the normal retirement age.

If the individual meets the requirements in (a) before the month the individual filed the

application, that individual may be entitled to retirement insurance benefits back to the

first month in which the individual met those requirements, subject to certain limitations. 

Disabled-worker benefit—An individual is entitled to a disability insurance benefit if that

individual:

(a) is disabled according to Social Security's definition of disability,

(b) has filed an application for disabled-worker’s benefits,

(c) has disability insured status,

(d) has completed a five-month waiting period (unless exempt), and

(e) has not attained the normal retirement age.

For other definitions, please refer to the Social Security Handbook, available for download at www.socialsecurity.gov/OP_Home/handbook/download.html.

Exhaustion year. See “Year of exhaustion.”

Financial interchange (Railroad). Provisions of the Railroad Retirement Act providing for

transfers between the trust funds and the Social Security Equivalent Benefit Account of the Railroad Retirement program in order to place each trust fund in the same position as it would have

been in if railroad employment had always been covered by Social Security.

Fully insured status. See “Insured status.”
Appendices

84

Gross Domestic Product (GDP). The total dollar value of all goods and services produced by

labor and property located in the United States, regardless of who supplies the labor or property.

High cost assumptions. See “Assumptions.”

Immigration. See “Legal immigration.” See “Total immigration.” See “Net other immigration.”

Income rate. Ratio of income from tax revenues on a liability basis (payroll-tax contributions

and income from the taxation of scheduled benefits) to the OASDI taxable payroll for the year.

Inflation rate. The annual growth rate in the Consumer Price Index for Urban Wage Earners and

Clerical Workers (CPI-W).

Insured status. There are three types of insured status which can be acquired by a worker under

the OASDI program. Each of these statuses is determined by the number and recency of quarters

of coverage (QCs) earned.

• Fully insured status is acquired by any worker whose total number of QCs is greater than

or equal to the number of years elapsed after the year of attainment of age 21 (with a minimum of six QCs required). Once a worker has accumulated 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 current quarter, any

fully insured worker aged 24-30 who has accumulated QCs during one-half of the quarters

elapsed after the quarter of attainment of age 21 up to and including the current quarter,

and any fully insured worker under age 24 who has accumulated six QCs during the

12-quarter period ending with the current quarter.

• Currently insured status is acquired by any worker who has accumulated six QCs during

the 13-quarter period ending with the current quarter.

Periods of disability are excluded from the above described QC requirements for insured status

(but do not reduce the minimum of six QCs).

Interest. A payment in exchange for the use of money during a specified period.

Interest rate. Amount of interest payment per unit of principal. Interest rates on new public-debt

obligations issuable to Federal trust funds are determined monthly. (See “Special public-debt

obligation.”) Such rates are set equal to the average market yield on all outstanding marketable

U.S. securities not due or not callable until after four years from the date of the determination.

The effective interest rate for a trust fund is the ratio of the interest earned by the fund over a

given period of time to the average level of assets held by the fund during the period. The effective rate of interest thus represents a measure of the overall average interest earnings on the

fund’s portfolio assets. 

Intermediate assumptions. See “Assumptions.”
Glossary

85

Labor force participation rate. The number of persons who are employed or actively seeking

work as a percent of the civilian noninstitutional population.

Legal emigration. The number of persons who lawfully leave the United States, and are no

longer considered to be a part of the Social Security program.

Legal immigration. The number of persons lawfully admitted for permanent residence into the

United States.

Life expectancy (period). The average number of years of life remaining if a group of people at

a given age were to experience the mortality rates for that year over the course of their remaining

lives. This quantity is computed for each year of each simulation, and is done so

• At age 65, and

• At birth.

Long range. The next 75 years. Long-range actuarial estimates are made for this period because

it is approximately the maximum remaining lifetime of current Social Security participants.

Low cost assumptions. See “Assumptions.”

Master Beneficiary Record (MBR). A complete record of beneficiaries which contains such

items as social security number, beneficiary identification code, date of birth, date of entitlement,

etc. The x% MBR represents a sampling of x% of these records from the collection of all records.

Module. In Fortran, a nonexecutable program unit that contains type definitions, object declarations, procedure definitions (module procedures), external procedure interfaces, user-defined

generic names, and user-defined operators and assignments. Any such definitions not specified to

be private to the module containing them are available to be shared with those programs that use

the module. Thus, modules provide a convenient sharing and encapsulation mechanism for data,

types, procedures, and procedure interfaces. The OSM contains the following modules:

• Assumptions: Projects some of the key demographic and economic assumptions using

time-series models.

• Population: Projects the Social Security area population for each year based on levels of

fertility, mortality, and immigration generated in the Assumptions Module.

• Economics: Projects U.S. economy-wide employment and wages, OASDI covered

employment and wages, taxable payroll, the average wage index and the COLA. 

• Insured: Projects the percentage of the population that will be fully insured and disability

insured.

• Disability Insurance Beneficiaries (DIB): Projects the annual values for the disabledworker beneficiaries in current-payment status.
Appendices

86

• Old-Age and Survivors Insurance Beneficiaries (OASIB): Projects the number of retiredworker beneficiaries and all aggregate dependent beneficiaries.

• Awards: Projects levels of benefits, in terms of Average Indexed Monthly Earnings

(AIME), for those beneficiaries newly awarded.

• Cost: Computes the year-by-year progress of the combined OASDI Trust Funds and produces summary measures used to assess the long-range financial status of the OASDI program.

• Summary Results: Computes the estimated probability distributions of annual and summarized data received from the Assumptions, Population, and Cost Modules.

Monthly benefit amount (MBA). The actual cash benefit scheduled to be paid to an entitled

individual.

Net other immigration. The annual flow of persons into the United States minus the annual

flow of persons out of the United States who do not meet the definition of legal immigration or

legal emigration. Thus, net other immigration includes unauthorized persons and those not seeking permanent residence.

Nominal yield (on special public-debt obligation). The effective annual interest rate, based on

the semiannual compounding of interest. See “Interest rate.”

Normal retirement age (NRA). The age at which a person may first become entitled to unreduced retirement benefits. For persons reaching age 62 before 2000, the normal retirement age is

65. It will increase gradually to 67 for persons reaching that age in 2027 or later, beginning with

an increase to 65 years and 2 months for persons reaching age 65 in 2003.

OCACT Stochastic Model (OSM). The particular stochastic model developed by the Office of

the Chief Actuary. See “Stochastic model.”

Old-Age and Survivors Insurance (OASI). See “Trust fund.”

Open group unfunded obligation. This measure is determined as of the valuation date over a

specified time period (such as over the long-range 75-year period). It is computed as the difference between: 

(a) The present value of the future cost of the program between the valuation date and

the end of the specified time period, and

(b) The sum of the assets in the trust fund as of the valuation date and the present

value of the future scheduled tax income of the program between the valuation

date and the end of the specified time period.

Outgo. Actual expenditures (outgo) made or expected to be made under current law, including

benefits paid or payable, special monthly payments to certain uninsured persons who have three

or more quarters of coverage (and whose payments are therefore not reimbursable from the Gen-
Glossary

87

eral Fund of the Treasury), administrative expenses, net transfers from the trust funds to the Railroad Retirement program under the financial-interchange provisions, and payments for vocational

rehabilitation services for disabled beneficiaries; it excludes special monthly payments to certain

uninsured persons whose payments are reimbursable from the General Fund of the Treasury (as

described above), and transfers under the interfund borrowing provisions.

Payroll taxes. A tax levied on the gross earned income of workers.

Period life expectancy. See “Life expectancy (period).”

Population in the Social Security area. The population comprised of (i) residents of the 50

States and the District of Columbia (adjusted for net census undercount); (ii) civilian residents of

Puerto Rico, the Virgin Islands, Guam, American Samoa and the Northern Mariana Islands; (iii)

Federal civilian employees and persons in the U.S. Armed Forces abroad and their dependents;

(iv) crew members of U.S. merchant vessels; and (v) all other U.S. citizens abroad.

Present value. The equivalent value, at a given time, of a future stream of payments (either

income or cost). The present value of a future stream of payments may be thought of as the lumpsum amount that, if invested today, together with interest earnings would be just enough to meet

each of the payments as they fell due. Present values are widely used in calculations involving

financial transactions over long periods of time to account for the time value of money (interest).

For the purpose of present-value calculations for this study, values are discounted by the effective

yield on trust fund assets.

Primary insurance amount (PIA). The monthly amount payable to a retired worker who

begins to receive benefits at normal retirement age or (generally) to a disabled worker. This

amount, which is related to the worker’s average monthly wage or average indexed monthly earnings, is also the amount used as a base for computing all types of benefits payable on the basis of

one individual’s earnings record.

Primary-insurance-amount formula. The mathematical formula relating the PIA to the AIME

for workers who attain age 62, become disabled, or die after 1978. The PIA is equal to the sum of

90 percent of AIME up to the first bend point plus 32 percent of AIME up to the second bend

point, plus 15 percent of AIME in excess of the second bend point. Automatic benefit increases

are applied beginning with the year of eligibility. 

Quarters of coverage. Basic unit of measurement for determining insured status. In 2004, a

worker receives one quarter of coverage (up to a total of four) for each $900 of annual covered

earnings. The amount of earnings required for a quarter of coverage is subject to annual automatic increases in proportion to increases in the Average Wage Index.

Railroad retirement. A Federal insurance program, somewhat similar to Social Security,

designed for workers in the railroad industry. The provisions of the Railroad Retirement Act provide for a system of coordination and financial interchange between the Railroad Retirement program and the Social Security program.
Appendices

88

Real average covered wage. The ratio of the average nominal OASDI covered wage to the

adjusted inflation rate.

Real interest rate. The annual (compounded) nominal yield divided by the inflation rate.

Self-employment. Operation of a trade or business by an individual or by a partnership in which

an individual is a member.

Social Security area population. See “Population in the Social Security area.”

Special public-debt obligation. Securities of the United States Government issued exclusively

to the OASI, DI, and other Federal trust funds. Section 201(d) of the Social Security Act provides

that the public-debt obligations issued for purchase by the OASI and DI Trust Funds shall have

maturities fixed with due regard for the needs of the funds. The usual practice has been to spread

the holdings of special issues, as of each June 30, so that the amounts maturing in each of the next

15 years are approximately equal. Special public-debt obligations are redeemable at par value at

any time and carry interest rates determined by law. (See “Interest rate.”) 

Stochastic model. A model 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. Distributions of potential outcomes are derived from a large number of simulations, each of which

reflects random variation in the variable(s).

Summarized cost rate. The ratio of the present value of cost to the present value of the taxable

payroll for the years in a given period, expressed as a percentage. This percentage can be used as

a measure of the relative level of cost during the period in question. For purposes of evaluating

the financial adequacy of the program, the summarized cost rate is adjusted to include the cost of

reaching and maintaining a target trust fund level. Because a trust fund level of about one year’s

cost is considered to be an adequate reserve for unforeseen contingencies, the targeted trust fund

ratio used in determining summarized cost rates is 100 percent of annual cost. Accordingly the

adjusted summarized cost rate is equal to the ratio of (a) the sum of the present value of the cost

during the period plus the present value of the increase needed to attain the targeted ending trust

fund level, to (b) the present value of the taxable payroll during the projection period.

Summarized income rate. The ratio of the present value of scheduled tax income to the present

value of taxable payroll for the years in a given period, expressed as a percentage. This percentage can be used as a measure of the relative level of income during the period in question. For the

purposes of evaluating the financial adequacy of the program, the summarized income rate is

adjusted to include assets on hand at the beginning of the period. Accordingly, the adjusted summarized income rate equals the ratio of (a) the sum of the trust fund balance at the beginning of

the period plus the present value of the total income from taxes during the period, to (b) the

present value of the taxable payroll for the years in the period.

Taxable earnings. Wages and/or self-employment income, in employment covered by the

OASDI and/or HI programs, that is under the applicable annual maximum taxable limit. For 1994

and later, no maximum taxable limit applies to the HI program.
Glossary

89

Taxable payroll. A weighted average of taxable wages and taxable self-employment income.

When multiplied by the combined employee-employer tax rate, it yields the total amount of taxes

incurred by employees, employers, and the self-employed for work during the period. 

Taxable self-employment income. The maximum amount of net earnings from self-employment by an earner which, when added to any taxable wages, does not exceed the contribution and

benefit base. 

Taxable wages. See “Taxable earnings.”

Taxation of benefits. During 1984-93, up to one-half of an individual’s or a couple’s OASDI

benefits was potentially subject to Federal income taxation under certain circumstances. The revenue derived from this provision was allocated to the OASI and DI Trust Funds on the basis of

the income taxes paid on the benefits from each fund. Beginning in 1994, the maximum portion

of OASDI benefits potentially subject to taxation was increased to 85 percent. The additional revenue derived from taxation of benefits in excess of one-half, up to 85 percent, is allocated to the

HI Trust Fund.

Taxes. See “Contributions.”

Total fertility rate. The average number of children who would be born to a hypothetical

woman in her lifetime if she were to experience the birth rates by age observed in, or assumed

for, a specified calendar year, and if she were to survive the entire childbearing period. 

Total immigration. Legal immigration minus legal emigration plus net other immigration.

TR04I, TR04II, TR04III. See “Assumptions.”

Trust fund. Separate accounts in the United States Treasury into which are deposited the taxes

received under the Federal Insurance Contributions Act and the Self-Employment Contributions

Act, contributions resulting from coverage of State and local government employees; any sums

received under the financial interchange with the railroad retirement account; voluntary hospital

and medical insurance premiums; and transfers of Federal general revenues. Funds not withdrawn for current monthly or service benefits, the financial interchange, and administrative

expenses are invested in interest-bearing Federal securities, as required by law; the interest

earned is also deposited in the trust funds. 

• Old-Age and Survivors Insurance (OASI). The trust fund used for paying monthly benefits to retired-worker (old-age) beneficiaries and their spouses and children, and to survivors of deceased insured workers.

• Disability Insurance (DI). The trust fund used for paying monthly benefits to disabledworker beneficiaries and their spouses and children and for providing rehabilitation services to the disabled.

Trust fund ratio. A measure of the adequacy of the trust fund level. Defined as the assets at the

beginning of the year, including advance tax transfers (if any), expressed as a percentage of the

cost during the year. The trust fund ratio represents the proportion of a year’s cost which could

be paid with the funds available at the beginning of the year.
Appendices

90

Unemployment rate. The number of unemployed persons seeking work as a percentage of the

civilian labor force.

Wage base. See “Contribution and benefit base.”

Year of exhaustion. The year in which a trust fund would become unable to pay benefits when

due because the fund has no assets.
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91

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