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by Tim Zayatz, A.S.A.

V. Technical Appendix on Table Construction Methodology

A. Overview

Select-and-ultimate tables 7A-9B present probabilities of decrement from the Social Security Disability Insurance (DI) beneficiary rolls. Table construction is based on 100 percent data from the Social Security Administration Master Beneficiary Record (MBR). [Click here for List of Tables] Over 6.6 million records of disabled worker beneficiaries were analyzed over a 5-year period--January 1, 1991 through December 31, 1995. The primary variables for the study include: the reason for decrement from the DI rolls, and duration since entitlement. Concomitant variables include the age at entitlement to benefits, and gender of the beneficiary. The analysis reflects a total of roughly 13.6 million life-years of exposure for males and 8.1 million life-years of exposure for females. A 10-year select period was chosen for this study, implying that decrement for participants 10 or more years beyond selection is a function of attained age only.

The reasons for termination of DI benefits can be categorized as follows:

Decrement due to "all other reasons" is treated as withdrawal from the study, and directly affects exposure. However, separate decrement tables were not developed for this category. The table below provides a breakdown of the termination data collected from the MBR. Note that there exist differences in classifying various disability actions among the various data sources consulted for this study. As a result, termination counts stated here may differ from those stated elsewhere.


DI Disabled Worker Terminations by Reason

(January 1991-December 1995)











































Source: MBR database as of January 1998.


As stated in the study, various exogenous variables have an impact on disability recovery rates without actually affecting the underlying rate of medical improvement. These include the level of continuing disability review activity, budget restrictions, and legislation. To some extent, the same may be true for death rates. In any event, the prevailing administrative policy can have a large impact on the nature of allowances, and the degree of overall impairment- severity of the DI rolls. This results in fluctuating recovery and mortality rates over time. Ideally, the observation period would exhibit limited legislation and minimal effects from exogenous variables. However, it should be noted that the DI program experienced significant growth in claims during 1991-1995, which limited the number of reviews performed over that period.

B. Data Considerations

The mortality experience reported in this study is affected by several unique circumstances. First, it is recognized that a claimant may die while waiting for a disability determination. Since observation of a participant is contingent upon entitlement, a disability which results in death prior to entitlement will not be an "observed" death. As a result, the probability of death during the first year of entitlement may be artificially low.

Second, non-death and non-recovery terminations are almost exclusively comprised of beneficiaries who have either been automatically converted to old-age benefits upon attainment of normal retirement age (a disability action termed conversion ), or have opted to receive old-age benefits in lieu of disability benefits prior to normal retirement age. In either case, observation stops with the last month in which DI benefits are paid. Since the decision to switch prior to full retirement age might be based on any number of unknown personal reasons (health, economic, etc.), it is not clear whether the mortality profile of the DI rolls changes significantly for attained ages 62 and older.

C. Data Collection

The model for this study utilizes a customized database that was constructed from 100 percent MBR data as of January 1998. The general record selection criteria are outlined below. The final data was subjected to numerous screenings and categorizations.

Included in the study are:

Excluded from the study are:

D. Underlying Methodology

The availability of complete data on each participant in the study (including date of birth, date of entitlement, and cause of decrement) allows us to directly estimate the multiple-decrement probabilities Formula , where i represents the cause of decrement. Alternatively, we could directly estimate the associated single-decrement (absolute) rates Formula , which could then be used to derive the multiple-decrement probabilities. We chose the former method, making use of scheduled exposure of the observed population. Under this approach, the ordered pair ( r, s ) is determined for each age interval Formula for which a participant is observed. The concept is that each participant enters the interval at age Formula , and is scheduled to exit the interval at age Formula . Numerically, s - r is the amount of time (measured in life-years) that the participant is potentially under observation and exposed to the risk of decrement. Summing over all participants, we can calculate the scheduled exposure contributed to an interval. 2

A participant may survive to the end of an interval, or may exit the study prior to the end of the interval in the event of:

Based on this criteria, a scheduled ending age , x + s , is established for an age interval in which the participant is expected to either survive to the end ( s = 1 ) , convert to old-age benefits prior to the end ( s < 1 ) , or become an observed ender ( s < 1 ) .3 Scheduled exposure is then credited to the appropriate interval (or duration since selection) using the following conditions: if the participant survives to the end of the interval, then exposure is credited from x + r to x + 1; if the participant dies, converts to old-age benefits, or is an observed ender within the interval, then exposure is credited from x + r to x + s ; if the participant withdraws from the study during the interval (for example, recovers or switches to old-age benefits prior to age 65), then exposure is credited from age x + r to x + s .

Note that in previous DI experience studies published by our office, the crediting of exposure for observed deaths was based on the actuarial method . Under this approach, exposure for a death is credited from x + r to x + 1 regardless of any scheduled ending age. For example, assume x + s is a participant's expected age at the close of the observation period. If death occurs during the interval that closes the observation period, the actuarial method would credit exposure up to age x + 1 , which would be beyond the end of the observation period whenever s < 1 . We follow Hoem's method in this regard and credit exposure only up to age x + s .

Multiple-decrement probabilities for an interval are found by dividing the observed number of deaths or recoveries by the aggregate scheduled exposure. Single-decrement rates are then derived from the probabilities using a constant force assumption for the distribution of decrement within a given interval. The method used in the construction of the select-and-ultimate multiple-decrement tables found in this study is detailed in chapter 9 of Actuarial Mathematics (Bowers et al. 1986).

For computation purposes, all dates are expressed as decimal-dates rounded to two decimal places (for example, 10-July-1993 is expressed as 1993.52). Coding detail is documented below for internal purposes. Variable acronyms are equivalent to those found in the description of the MBR database.

Entitlement dates

Various entitlement dates are converted to decimal-dates under the assumption that entitlement occurs on the first day of the month. Entitlement date variables include month and year of: initial entitlement (DOEI), disability entitlement (DOED), and current entitlement (DOEC).

Date of death

The decimal-date of death is calculated from the beneficiary date of death variable (BDOD), which states month, day, and year of death. Death on the 15th of the month is assumed for instances where the day of death is not recorded. Death on the 1st of the month is assumed for instances where the beneficiary dies in the month of the anniversary of entitlement. As an example, consider the following data for a participant who neither recovers nor switches to old-age benefits:


Date of entitlement: 1-July-1992

Date of death: 10-July-1993


Given death on 10-July-1993, the last month of entitlement to DI benefits is June-1993; hence the participant is only observed during the period July-1992 through June-1993. However, allowing that death occurs on 10-July-1993 is tantamount to allowing entry into the next observation interval--which would have begun at the entitlement anniversary on 1-July-1993--and Hoem's method would credit exposure from 1-July-1993 through the scheduled ending age for that interval. To avoid crediting exposure for the second interval (since the participant was not observed), the date of death is modified to be 1-July-1993, and exposure is credited only for the first interval. Also note that death is counted as having occurred exactly at the endpoint of the first interval, and is therefore included in that interval.

With respect to the MBR database, death carries a disability action code (termed LAF code ) of "T1".

Date of conversion

With regard to conversion to old-age benefits, the last month of entitlement to DI worker benefits is the month before the month in which the worker attains age 65 4. For programming purposes, it is assumed that conversion takes place on the first day of the month of attainment of age 65; therefore, the calculated age at conversion is fractionally less than 65. Also note that if death occurs in the month of attainment of age 65, the case is counted as a conversion rather than an observed death.

Date of withdrawal

All other decrements occur on the first day of the month as coded in the date of suspension or termination (DOST), or date of disability benefit cessation (DDBC) variables. DOST provides detail on the date of suspension or termination of benefits for the most recent period of disability. Note that DOST corresponds to the value of the LAF code and is therefore subjected to continuous updating. As such, DOST will not provide a "history" of benefit termination for individuals with multiple periods of disability.

DDBC refers to the first month for which disability benefits are not payable. This variable is referenced for participants having multiple periods of disability, and for times when the DOST variable cannot provide an historical account of disability activity. The following examples illustrate the coding conventions used for the disability action variables DOST, DDBC, and LAF:

Example 1

DI benefits are payable for the month in which a beneficiary medically recovers, as well as the next 2 months. Consider a beneficiary having a single period of disability with recovery in April. The beneficiary remains entitled through June. At that time, LAF is coded as "T8" (indicating recovery) and DOST is coded as July.

Example 2

Consider a DI beneficiary who, though not medically recovered, begins to participate in SGA, with the following benefit payment history:

In this case DDBC is coded as the first month in which DI benefits are not payable; and DOST would vary with the different LAF codes, ultimately reflecting the date at which Medicare benefits terminate.

Select age

Since most beneficiaries do not become entitled to benefits on their exact birthdays, entitlement would normally occur at some fractional age. To facilitate exposure calculations, a participant's insuring age and corresponding insuring date of birth are substituted for the actual age at entitlement and actual date of birth. In this study, the insuring age is calculated to be the beneficiary's age last birthday as of entitlement. For example, consider the following beneficiary data:


Date of entitlement: 1-February-1992

Date of birth: 10-July-1960

Actual age at entitlement: 31 years, 206 days

Insuring age: 31 years

Insuring date of birth: 1-February-1961


Use of insuring age results in an integral select age at entitlement ensuring that subsequent durations begin on the entitlement anniversary. This is true whether the participant enters the study during the observation period, or is already part of the entitlement group when the observation period opens. However the entry age into the study for those already part of the entitlement group when the observation period opens may not be integral.

Entry age into the study (YI)

For selection during the observation period, the entry age is the beneficiary's age as of the date of entitlement. For selection prior to the beginning of the observation period, the entry age is the beneficiary's age as of January 1, 1991. In either case, the beneficiary's age is measured from the insuring date of birth.

Scheduled exit age from the study (ZI)

Participants are scheduled to exit the study either as an old-age conversion or an observed ender. If the participant will attain normal retirement age prior to the end of the observation period then ZI is the conversion age--as previously noted, this usually will be fractionally less than age 65. If the beneficiary will not attain age 65 prior to the end of the observation period, then ZI is the beneficiary's age as of December 31, 1995.

Actual exit age from the study due to death (THI) or withdrawal (PSI)

If a participant dies within the observation period-- prior to conversion--then THI would reflect the date-of-death conventions mentioned earlier. If a participant withdraws from the study as a result of old-age conversion, then PSI would be the conversion age. If withdrawal occurs for any other reason, then PSI would be the age as of the withdrawal date. Both THI and PSI would be used in calculating exposure for absolute death and withdrawal rates Formula and Formula .


The intervals during which a participant is observed-- measured from the select age--are referred to as durations . For each select age [x] and duration n , we produce the ordered pair ( r , s ) representing each participant's scheduled exposure contribution to the observation interval Formula . For durations extending beyond the 10-year select period, exposure is credited to the appropriate attained age interval.

Fractional times

The fractional-time variables used to determine scheduled exposure contributions to an observation interval are defined below:





The amount of exposure contributed at each duration is summarized as follows:




Table construction

For each combination of select age and duration, the multiple-decrement probabilities Formula and Formula are calculated as the ratio of (a) the observed number of deaths or recoveries to (b) aggregate scheduled exposure. The probabilities are graduated and then used to obtain estimates for the absolute rates of decrement Formula and Formula as outlined below (note that d, r, and Formula superscripts refer to death, recovery, and total decrement, respectively):

Estimate the total decrement probability:


Under the assumption of constant force for each decrement over each age interval, derive estimates for the absolute rates:



The graduated, multiple-decrement probabilities are shown in tables 7A-9B; absolute rates appear in appendix tables A.3A, A.3B, A.4A and A.4B. The life functions, expected future lifetimes, and life annuities presented in this study are based on the probabilities.

Probabilities versus absolute rates

The data for this study was collected in a multiple-decrement environment, however, we explicitly consider only two major decrements--death and recovery. The quantity Formula represents the probability of death in the presence of the other decrements. Mathematically, this is represented by:


Formula = Formula

where Formula is the probability of neither dying nor recovering prior to death at age x + t; and Formula is the force of mortality.

For each of the causes of decrement in a multiple-decrement model, it is possible to define a single-decrement model that depends only on a particular cause of decrement. The quantity Formula represents the absolute rate of death in the associated single-decrement environment. Mathematically, this is represented by:


Formula = Formula

where Formula is the probability that death will not occur prior to age x + t.

By defining a new fractional-time variable k and withdrawal exposure (as shown below), absolute death rates could be calculated directly from the data as the ratio of deaths to aggregate exposure:






The select-and-ultimate multiple-decrement probabilities were graduated using the two-dimensional Whittaker-Henderson Type B method 5. The horizontal and vertical smoothing coefficients were chosen to obtain smoothness both across rows and down columns, while deviating as little as possible from the original estimates.

Life functions

Life tables 10A, 10B and 10C are constructed from the blended select-and-ultimate probabilities shown in appendix tables A.1A, A.1B and A.1C. The blended tables are constructed from disability mortality (Table 7A and Table 7B) and general population mortality for 1995. Results of the blending are highlighted in the appendix tables. The two-step blending process involves the following:

The life functions Formula are first constructed for select age [x] = 20, using a radix of 100,000. Life functions for select ages [x] > 20 are derived using the survival probabilities of the given select age. For these ages, Formula is calculated so that Formula is attained. Note that Formula represents the number alive at the beginning of duration t from those originally entitled at [x].

The blended death probabilities are added to the graduated recovery probabilities to produce the blended total termination probabilities shown in appendix tables A.2A and A.2B. These probabilities are used to construct life tables 11A and 11B, respectively.

Expected future lifetime and annuity tables

Expectation tables 12A-14C and annuity tables 15A-18D are produced from the life functions described above using basic actuarial principles found in any standard actuarial text on life contingencies. [Click here for List of Tables]

E. Programs

For internal purposes, this section outlines the series of programs used to process data from the 100 percent MBR as of January 1998.

EXPCLASS--Program which classifies each record in the sample, and writes out the record (as it appears) along with the record's classification code . The main purpose is to distinguish which of the three most recent periods of disability ("DIB lines") contain relevant data with respect to the established observation period. A total of 14, 9, and 6 possible classifications were established for the first-most, second-most, and third-most recent periods of disability, respectively.

EXFORT--An original record may contain up to three periods of disability that must be split out, and an individual record written for each period. This program reconstructs the MBR record as it would have originally appeared at the time of disability. Depending on the individual classifications of the DIB lines, certain information is written to create up to three separate records from the original record.

EXT5MAIN--Program which processes the records written out by EXFORT and defines all variables needed to tabulate exposure, and classify deaths, recoveries, and all other terminations by select age, gender, and duration.

EXT5EXP--Program which calculates exposure in aggregate life-years based on output from EXT5MAIN.

EXT5DTH--Program which tabulates DI deaths based on output from EXT5MAIN.

EXT5RCV--Program which tabulates DI recoveries based on output from EXT5MAIN.

EXT5OTH--Program which tabulates DI terminations for "all other reasons" based on output from EXT5MAIN.

wh2dgrad.f--Program which performs the Whittaker-Henderson Type B two-dimensional graduation. Required input includes the ungraduated values, graduation weights, and the choice of horizontal and vertical smoothing coefficients.

1 DI beneficiaries born on or before February 1, 1926 would have been automatically converted to old-age benefits prior to January 1, 1991.

2 For a complete discussion, refer to chapter 6 of Survival Models and Their Estimation (London 1988, second edition).

3 A participant who dies during the interval does so at age Formula ; a participant observed to withdraw from the study during the interval does so at age Formula .

4 Under present law, beneficiaries born on the 1st of the month revert to the previous month as the month of attaining the next integral age. In this case, conversion to old-age benefits would occur on the first day of the preceding month in which the DI beneficiary attains age 65.

5 For details, refer to chapter 8 of Graduation: The Revision of Estimates (London 1985).

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July 30, 1999