ACTIVE PARTICIPANTS IN ME-PIMS 

March 19, 2021 

Advanced Analytical Consulting Group, Inc. 

Constantijn W.A. Panis, PhD 

213-784-6400 

stanpanis@aacg.com 

Karthik Padmanabhan, MS, MBA 

312-551-9001 

karpad@aacg.com 
 i 

SUMMARY 

The Pension Benefit Guaranty Corporation (PBGC) insures participants in private pension 

plans against loss of benefits in case their plan is unable to pay benefits. The agency uses a 

stochastic modeling system, the Multiemployer Pension Insurance Modeling System (MEPIMS), to project future expected liabilities related to multiemployer plans. Future expected 

liabilities depend, in part, on annual employer contributions to multiemployer pension plans, 

which in turn depend on the number of active participants in such plans. This report reviews 

ME-PIMS’s current approach to projecting active participation in multiemployer plans. 

The current model assumes that the growth rate of active participants is distributed 

uniformly between -3.3% and 0.7%, with a long-term target of -1.3% per year. It assumes 

that all plans grow at the same rate. The projected rate can vary over time and scenarios, 

but it is invariant to industry, plan funding, macroeconomic indicators, or other factors. 

We analyze historical rates of change of active participation in multiemployer plans and find 

that the current model is at odds with different rates of change across plans. We develop a 

simple alternative model in which historical rates of change are explained by plan maturity, 

plan risk status, Construction industry, and the rate of return on equity. We recommend 

that PBGC consider that model, or a similar one, to maintain heterogeneity among 

multiemployer plans and to recognize factors that can affect plans’ financial outlook both 

directly and through their link with active participation and resulting cash flow. 
 ii 

CONTENTS 

Summary .................................................................................................................. i

1. Introduction ....................................................................................................... 1

2. Current Modeling Approach ................................................................................ 1

Overview.............................................................................................................. 1

Strengths and Potential Weaknesses of the Current Approach ..................................... 2

3. Analysis of Active Participation .......................................................................... 3

Data Source ......................................................................................................... 3

Descriptive Statistics ............................................................................................. 4

Regression Model Analysis .................................................................................... 10

Projections ......................................................................................................... 12

4. Recommendation ............................................................................................. 13

Model Specification .............................................................................................. 13

Likely Effects and Merits of an Alternative Active Participation Model ......................... 14

Sensitivity Analysis .............................................................................................. 14

Disclaimer ............................................................................................................. 15
 1 

1. INTRODUCTION 

The Pension Benefit Guaranty Corporation (PBGC) projects the solvency of its program to 

insure the benefits promised by multiemployer defined benefit (DB) pension plans based on 

a microsimulation model, the Multiemployer Pension Insurance Modeling System (ME-PIMS). 

PBGC retained Advanced Analytical Consulting Group, Inc. (AACG) to review ME-PIMS’s 

current modeling of the number of active participants in multiemployer plans. This 

document is the final report of our review. 

Employers that participate in a multiemployer plan make contributions on behalf of their 

employees who are active participants in the plan. Active participants may therefore be 

viewed as the contribution base of multiemployer plans. Their number directly correlates 

with the flow of cash to such plans and is therefore of vital importance to plan funding. By 

extension, accurate modeling of plans’ numbers of active participants is important to the 

validity of ME-PIMS. 

This report starts with a discussion of the current approach to modeling active participation 

in multiemployer plans, including its strengths and weaknesses (Section 2). Section 3 

presents a quantitative analysis of the number of active participants in recent years. Section 

4 concludes with recommendations for future modeling of active participation. 

2. CURRENT MODELING APPROACH 

Overview 

Our understanding of the current approach to modeling the number of active participants is 

based on a 2014 memorandum by Steven Boyce (“Boyce Memo”),1 which documented a 

prior version of the model, and private communication of recent changes. The number of 

active participants is equal to the previous year’s number plus a random rate of growth: 

N௜௧ = N௜௧ିଵ(1 + α + ε௧) (1) 

where N௜௧ represents the number of active participants in plan i at time t, α denotes the 

average targeted growth rate, and ε௧ is a random component. Equivalently, the annual rate 

of change is: 

(N௜௧ − N௜௧ିଵ)/N௜௧ିଵ = α + ε௧

 (2) 

It is our understanding that α=-0.013, i.e., active participation is assumed to contract by 

1.3% per year, on average. The random component is uniformly distributed from -0.02 to 

0.02 (-2% to 2%). It varies across projection years and scenarios, but is the same for all 

1

 PBGC provided us this memorandum, which was last modified on 7/21/2014. A nearidentical version appears at https://www.pbgc.gov/about/projections-report/pensioninsurance-modeling-system/projection-of-levels-of-future-active. 
 2 

plans.2 Equivalently, the assumption is that the rate of change is assumed to be uniformly 

distributed from -3.3% to 0.7%. 

The targeted average growth rate, -1.3%, was based on historical numbers of active 

participants as reported on Schedule B/MB of 1991–2011 Form 5500 filings. 

Strengths and Potential Weaknesses of the Current Approach 

Perhaps the main virtue of the current approach is the fact that it is straightforward to 

understand. The model relies on only an average growth target (-1.3%) and the range of 

the uniformly distributed random component. 

The model’s parsimony is, of course, also the source of a number of potential weaknesses. 

We identified the following potential issues. 

All plans grow at the same rate. The model’s random component, ε௧, varies over time and 

across scenarios, but is assumed to be the same for all plans in a certain year and scenario. 

In reality, even in a single year, growth rates may vary across plans because of differences 

in non-union competition, regional variation, unequal challenges facing various industries, 

and other factors. 

No Differences by Industry. The current model does not account for industry. However, 

long-term changes in the number of active participants may differ across industry sectors. 

(This is distinct from potential within-year variation across industries discussed in the 

previous paragraph.) 

No Variation by Plan Characteristics. The current model does not account for any plan 

characteristics. However, the inclination of employers to continue contributing to a plan may 

depend on its financial outlook or other factors. For example, an underfunded plan with 

mostly inactive participants will likely need to increase its contribution rates to meet its 

benefit obligations, and employers may be discouraged by such a prospect. 

No Interactions with the Macro Economy. The Boyce Memo demonstrates that active 

participation in multiemployer plans declined substantially in years with high unemployment 

and tended to increase in years with low unemployment. However, the current model does 

not account for any macroeconomic metric. 

Long-Term Growth May Miss the Target. The model targets -1.3% growth on average, with 

annual growth rates that are distributed symmetrically around -1.3%. However, this 

average breaks down when growth is compounded over time. For example, suppose a plan 

with 1,000 active participants grows 10% in the first year (to 1,100 participants), and 

contracts 10% in the next year (to 990 participants). The arithmetic average of 10% and 

-10% is 0%, but the plan contracted by 1% because 1.10 x 0.90 = 0.99. Similarly, 

symmetry around a target of -1.3% may, in the long run, cause faster contraction than 

1.3% annually. 

2

 There is one exception. ME-PIMS assumes that contributions to plans in the Critical and 

Declining zone remain constant, and this assumption is implemented through a 0% rate of 

change (FY 2019 PBGC Projections Report, page 32). 
 3 

Estimation Period. The current target rate of change model was estimated on data from 

1991 through 2011. We will evaluate active participant changes using newer data. 

The next section explores whether these potential issues are material to active participation 

and can be incorporated into a formal model. 

3. ANALYSIS OF ACTIVE PARTICIPATION 

Data Source 

The current model was estimated on participant counts reported on Form 5500 filings of 

multiemployer plans for 1991-2011. Form 5500 filings are available from the DOL website 

since plan year 1999.3 

The Form 5500 asks about active participation in multiemployer pension plans in two places. 

First, the main Form 5500 asks about numbers of plan participants at the end of the 

reporting period. (In addition, active participation at the beginning of the plan year needs to 

be reported since 2014.) See Figure 1. Second, Schedule MB asks about the number of 

participants at the beginning of the plan year; see Figure 2. Schedule MB replaced 

Schedule B, starting in 2008; the same participant questions were asked on Schedule B. 

Figure 1. Participation Questions on the Form 5500 

6 Number of participants as of the end of the plan year unless otherwise stated 

(welfare plans complete only lines 6a(1), 6a(2), 6b, 6c, and 6d). 

a(1) Total number of active participants at the beginning of the plan year ..........

a(2) Total number of active participants at the end of the plan year...................

b Retired or separated participants receiving benefits ..................................

c Other retired or separated participants entitled to future benefits ..............

d Subtotal. Add lines 6a(2), 6b, and 6c. .....................................................

e Deceased participants whose beneficiaries are receiving or are entitled to 

receive benefits. .................................................................................. 

f Total. Add lines 6d and 6e. ...................................................................

g Number of participants with account balances as of the end of the plan year 

(only defined contribution plans complete this item) ................................ 

h Number of participants who terminated employment during the plan year 

with accrued benefits that were less than 100% vested ............................. 

Source: 2014–2019 Form 5500. Line 6a(1) does not apply prior to 2014. 

3

 See https://www.dol.gov/agencies/ebsa/about-ebsa/our-activities/publicdisclosure/foia/form-5500-datasets. The analysis data were extracted on January 4, 2021. 
 4 

Figure 2. Participation Questions on Form 5500, Schedule MB 

2 Operational information as of beginning of this plan year:

a [...]

b “RPA ‘94” current liability/participant count breakdown: 

(1) Number of 

participants 

(2) Current 

liability 

(1)For retired participants and beneficiaries receiving payment.........

(2)For terminated vested participants ............................................

(3)For active participants:

(a)Non-vested benefits...........................................................

(b)Vested benefits .................................................................

(c) Total active ......................................................................

(4)Total .....................................................................................

c [...]

Source: 2008–2019 Form 5500 Schedule MB. 

Participant counts on the main Form 5500 and Schedule B/MB are generally consistent. It is 

our understanding that ME-PIMS uses Schedule B/MB participant counts, and our focus is 

likewise on Schedule B (through 2007) or Schedule MB (2008 and later). Where Schedule 

B/MB participant counts were missing, we used the main Form 5500 for imputation 

purposes. Schedule B/MB asks for the number of participants at the beginning of the 

reporting period. The calendar year in which a reporting period started is known as the plan 

year. 

Descriptive Statistics 

Figure 3 shows the number of active, terminated/vested, and retired participants in 

multiemployer plans, as reported on Schedule B/MB, for plan years 2001 through 2018. 

Years 1999-2000 and 2019 are not shown because the data appeared incomplete; no 

Schedule MB data are available electronically for 2008. While the total number of 

participants has generally increased, active participation generally decreased until 2012 and 

remained approximately level after 2012. 
 5 

Figure 3. Active, Terminated/Vested, and Retired Participants in Multiemployer 

Plans, By Plan Year 

The number of plans that reported nonzero active participants decreased from 1,530 in 

2001 to 1,228 in 2018, indicating that many multiemployer plans merged or terminated. 

The aggregate number of active participants may therefore not be a good basis for 

estimating and projecting organic changes in active participation of ongoing plans. 

Figure 4 shows the distribution of yearly changes in active participation based on pairs of 

consecutive filings. The distribution is mostly bell-shaped, except at the extremes: 0.9% 

experienced a change of -100% (from any active participants to none) and 0.5% reported a 

change of 100% or more. Figure 5 shows the same, weighted by active participants.4 The 

weighted distribution is generally tighter, suggesting that large changes are associated with 

smaller plans. Again 0.9% of the distribution is at -100%, which means that plans covering 

0.9% of active participants reduced to zero active participants in any single year. We 

manually reviewed the filing histories of the 20 largest plans that reported zero active 

participants for the first time; in all but one case, the issue appeared to be misreporting 

because active participation was restored to approximately the same level as before in the 

next filing, and active participation as reported on the main Form 5500 was uninterrupted. 

In other words, filings that report a change of -100% often appear erroneous. 

4

 All weighted analyses use the number of active participants in the preceding year as 

weight. For example, if a plan’s active participants rose from 1,000 to 1,050, the change 

was 5% and is weighted by 1,000. Weighting by the preceding year’s number ensures that 

the average change is equal to the change of the aggregate number. 

0

2m

4m

6m

8m

10m Number of participants

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

Source: Form 5500, Schedule B/MB.

Retired

Terminated

Active
 6 

Figure 4. Distribution of Annual Changes in Active Participation (1999–2019) 

Figure 5. Distribution of Annual Changes in Active Participation, Weighted by 

Active Participants (1999–2018) 

0%

3%

6%

9%

12%

15%

18%

-100% -80% -60% -40% -20% 0% 20% 40% 60% 80% 100%

or more

Annual change in Active Participants

Source: Form 5500, Schedule B/MB

0%

3%

6%

9%

12%

15%

18%

-100% -80% -60% -40% -20% 0% 20% 40% 60% 80% 100%

or more

Annual change in Active Participants

Source: Form 5500, Schedule B/MB
 7 

Apart from extreme changes, the distribution of annual change in active participation is bellshaped. This suggests that the distribution of the model’s random component should not be 

uniform but bell-shaped (e.g., normal). Our finding is consistent with the Boyce Memo, 

which likewise documented a bell-shaped distribution of rates of change. 

Figure 4 above shows the distribution of annual changes in active participation for pooled 

data from 1999 to 2019. Figure 6 shows the same for a single plan year, 2018. The 

distribution is again bell-shaped and there is substantial variation around the mean. This 

suggests that the current model’s assumption that all plans grow at the same rate in a 

certain year is at odds with reality. The solution is straightforward, namely to draw a 

random component not just for every year (ε௧), but for every plan in every year (ε௜௧). 

Figure 6. Distribution of Annual Changes in Active Participation (2018) 

An objective of ME-PIMS, and thus of our analysis, is to project the change in active 

participants of plans as they grow or contract organically. Following the analysis in the 

Boyce Memo, we excluded reported changes outside the range from -30% to +30%. This 

criterion should exclude plan mergers and terminations from the analysis. For the remaining 

multiemployer plans, Figure 7 shows the average yearly change in active participation over 

time. The blue bars are unweighted and the red bars are weighted by active participants. 

0%

3%

6%

9%

12%

15%

18%

-100% -80% -60% -40% -20% 0% 20% 40% 60% 80% 100%

or more

Annual change in Active Participants

Source: Form 5500, Schedule B/MB
 8 

Figure 7. Average Change in Active Participation, by Plan Year 

The participant-weighted average rate of change ranged from -6.2% in 2010 to 3.1% in 

2001. Recall that the current model assumes that active participation in all plans grows or 

contracts at the same rate in a year, and that it draws the change randomly from a uniform 

distribution between -3.3% and 0.7%. In light of the historical annual averages shown in 

Figure 7, that range appears too narrow. 

Taking the average over multiple years of annual average rates of change, the average 

participant-weighted change in 2000–2019 was -0.70% per year: -0.25% per year in 2000–

2009 and -1.15% per year in 2010–2019. For comparison, the Boyce Memo found an 

average change of -1.3% between 1992 and 2011. 

The annual average rates of change in Figure 7 are based on growth rates defined as r௜௧ =

(N௜௧ − N௜௧ିଵ)/N௜௧ିଵ, where r௜௧ is the growth rate and N௜௧ the number of active participants 

of plan i in year t. As argued on page 2, projections that use growth rates distributed 

symmetrically around a target growth rate may miss their target. For example, suppose the 

target is 0% growth and the projections are 10% in the first year and -10% in the next 

year. At the end of the second year, the active population is the starting population times 

1.10 x 0.90 = 0.99, i.e., a net contraction of 1% rather than 0%. Instead, we prefer 

measuring the growth rate as ln(1 + r௜௧). We refer to this metric as the “geometric growth 

rate” or “geometric rate of change.” At r௜௧=0%, the growth rate and the geometric growth 

rate are equal because ln(1 + r௜௧) = ln(1 + 0) = 0. Suppose again the target is 0% and the 

projections of the geometric growth rate are 10% in the first year and -10% in the next 

year. A 10% geometric growth rate means ln(1 + r௜௧)=0.1, i.e., r௜௧=exp(0.1)-1= 

10.517092% and -10% corresponds to r௜௧=exp(-0.1)-1= -9.516258%. At the end of the 

second year, the active population is equal to the starting population times 1.10517092 x 

0.90483742 = 1, i.e., a compounded change of 0%. Because the geometric growth rate 

-8%

-6%

-4%

-2%

0%

2%

4% Average Annual Change in Active Participants

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

Unweighted

Active participant-weighted

Source: Form 5500, Schedule B/MB.

Note: Excludes plans with changes in excess of 30%.
 9 

preserves the target in projections, much of the remainder of this report is based on the 

geometric growth rate. 

The average participant-weighted geometric rate of change in 2000–2019 was -0.93% per 

year: -0.52% per year in 2000–2009 and -1.35% per year in 2010–2019.5

Table 1 provides more detail about average participant-weighted geometric rates of change 

by industry sector.6 Annual growth rates for 2000–2019 were -0.11% for plans in the 

Construction sector, -1.96% for Retail plans, -1.91% for Transportation plans, and -1.20% 

in Other or Unknown industries. 

Table 1. Plans, Active Participants, and Average Annual Participant-Weighted 

Geometric Changes in Active Participants, by Sector 

Finally, Figure 8 shows annualized participant-weighted geometric rates of change in active 

participants by plans’ risk status. Risk status is taken from Schedule MB, line 4a; it was first 

asked in 2008 and has been available electronically since 2009. Plans at greater risk 

generally contract faster than relatively healthy plans, ranging from -1.0% per year among 

plans that are in neither endangered nor critical status to -5.1% per year for plans in critical 

and declining status. The average rate of contraction among seriously endangered plans is 

smaller than among endangered plans, but that rate is based on a small sample: only 1.7% 

of plan filings reported seriously endangerment. 

5

 To preserve comparability, the calculations of these rates excluded changes―not 

geometric changes―in excess of 30%, abs(r௜௧)>0.30. This exclusion also applies to all 

results discussed below. 

6

 Industry sectors are based on Form 5500 business codes, along with corrections and 

industry groupings as provided by Bill Marx on 2/13/2019. 

Plan year 2018: Average annual

Ac tive geometric rate of change

Plans participants 2000-2009 2010-2019 2000-2019

Construction 672 1.58m 0.79% -1.00% -0.11%

Retail 59 0.53m -2.95% -0.97% -1.96%

Transportation 120 0.44m -1.84% -1.98% -1.91%

Other/Unknown 320 1.11m -0.76% -1.64% -1.20%

All sec tors 1,171 3.66m -0.52% -1.35% -0.93%

Source: Schedule MB of 1999-2019 Form 5500 filings.

Based on plans with annual changes between -30% and +30% only.
 10 

Figure 8. Average Annual Participant-Weighted Geometric Changes in Active 

Participation, by Risk Status (2009–2019) 

Regression Model Analysis 

Table 2 shows parameter estimates from regression models. The outcome variable is the 

geometric rate of change of active participation in a plan between two years: 

ln(1 + r௜௧) = β′x௜௧ + ε௜௧

The data consists of filings from 1999 to 2019, i.e., the earliest year contributing to the 

regression is 2000. Our preferred specification is in the fourth column, labeled “Weighted.” 

The explanatory variables are as follows. 

 Inactive/total participants. The ratio of inactive to total participants may be 

interpreted as a measure of plan maturity. It ranges from 0 to almost 1 with a mean 

of 0.58. A coefficient of -0.14 means that a 10 percentage point difference in 

maturity translates into reduced annual growth (sharper contraction) by about 1.4 

percentage points (0.10 x -0.14 = -0.014 = -1.4%). 

 Plan is at risk and Risk status is missing. The former is an indicator variable for 

whether the plan reported that it was in endangered or critical status, including 

seriously endangered and declining; the latter is an indicator for whether risk status 

is missing. Risk status is available since 2009 only. In earlier years, or when the plan 

did not report its risk status, the risk status indicator was set equal to its mean over 

Not in

Endangered

or Critical

Status

Endangered

Status

Seriously

Endangered

Status

Critical

Status

Critical

and

Declining

Status

-1.0%

-2.5%

-1.9%

-3.0%

-5.1%

-6%

-5%

-4%

-3%

-2%

-1%

0% Annual Geometric Rate of Change

Source: Form 5500, Schedule B/MB.

Note: Excludes plans with changes in excess of 30%.
 11 

non-missing values (0.415) and the missing indicator was turned on.7 Lack of 

statistical significance suggests that no significant difference exists between plans 

with missing or non-missing risk status, and that it may be omitted from the 

specification. When included in the third and fourth specifications, the missing 

indicator is indeed statistically insignificant (not shown). For that reason, the 

specifications in the third and fourth columns omit the missing indicator. 

 Construction, Retail, and Transportation. These are indicator variables for plans in 

the corresponding industry sectors. They are statistically significant, but their signs 

and magnitude suggest that plans in those three sectors grow faster (contract less) 

than plans in other sectors, which is only partially consistent with the raw differences 

documented in Table 1. In other words, while rates of change differ across industry 

sectors, those differences appear to be driven by factors such as maturity and risk 

status. We therefore excluded industry indicators from our preferred specification, 

except for Construction because employers participating in Construction sector plans 

are subject to more lenient withdrawal liability provisions than those in other sectors 

(29 USC §1388). 

 Lagged 5-year equity return. This variable is the five-year trailing average 

annualized geometric rate of return (including dividends) on stocks in the S&P 500 

index, lagged by one year. For example, the value in plan year 2015 is the 

annualized geometric rate of total return on the S&P 500 index in 2010–2014. It 

ranges from -2.3% (in 2005) to 25.1% (in 2000) with a mean of 6.8%. Its estimated 

effect is positive, meaning that the rate of change in active participation relates 

positively to lagged 5-year returns on large cap stocks. We found its effect to be 

highly robust to alternative model specifications.8

 Lagged geometric change. This variable is the one-year lagged geometric rate of 

change in active participation, i.e., an autoregressive variable. Its effect is positive 

and statistically significant, but it is small: all else equal, only about 1.5% of last 

year’s rate of change persists in the next year. Given its lack of economic 

significance and the complications that autoregression introduces for calibration 

(discussed below), we omitted it from our preferred model specification. 

7

 Attempts to impute risk status based on funded level were abandoned because they did 

not improve model fit. 

8

 Alternative predictors of active participation include the unemployment rate or rate of 

growth of Gross Domestic Product, but their future values are not projected in ME-PIMS. In 

contrast, ME-PIMS projects the rate of return on equity. 
 12 

Table 2. Regression Estimates of the Geometric Rate of Change 

 Baseline AR(1) Simplified Weighted 

Inactive/total participants -0.1408 *** -0.1448 *** -0.1410 *** -0.1346 *** 

 (0.0035) (0.0038) (0.0034) (0.0033) 

Plan is at risk -0.0063 *** -0.0054 *** -0.0059 *** -0.0081 *** 

 (0.0017) (0.0017) (0.0017) (0.0012) 

Risk status is missing 0.0017 0.0031 ** 

 (0.0012) (0.0013) 

Construction 0.0133 *** 0.0105 *** 0.0100 *** 0.0052 *** 

 (0.0013) (0.0014) (0.0012) (0.0009) 

Retail 0.0057 ** 0.0058 ** 

 (0.0027) (0.0028) 

Transportation 0.0121 *** 0.0134 *** 

 (0.0021) (0.0022) 

Lagged 5-year equity return 0.2005 *** 0.2256 *** 0.1997 *** 0.1914 *** 

 (0.0077) (0.0091) (0.0077) (0.0059) 

Lagged geometric change 0.0146 *** 

 (0.0027) 

Intercept 0.0429 *** 0.0456 *** 0.0471 *** 0.0505 *** 

 (0.0027) (0.0028) (0.0023) (0.0019) 

Observations 24,072 22,082 24,072 24,072 

Weighted? No No No Yes 

R-Squared 0.1023 0.1032 0.1010 0.1134 

Root MSE 0.0880 0.0872 0.0880 0.0671 

Standard errors in parentheses. Significance: *=10%, **=5%, ***=1%. 

We explored model specifications that control for plan size, as measured by active or total 

participants (not shown). All else equal, larger plans appear to grow faster (contract less) 

than smaller plans. However, we excluded plan size from the model specification because it 

may cause ME-PIMS to run “off the rails” in the sense that large plans would be projected to 

grow ever faster. 

The third and fourth columns of Table 2 control for the same variables, but differ in that the 

fourth column was estimated using weights equal to the (previous year’s) number of active 

participants. The weighted regression reflects our preferred specification because, in the 

absence of controls for plan size, its estimates are influenced by plans in proportion to their 

active population. 

Projections 

The bottom row of Table 2 displays the square root of the mean squared error (MSE) of 

each regression, which is an estimate of the standard deviation of the residual, ε௜௧. It plays 

a role in stochastic projections based on regression models. 

Recall that the model referenced in Table 2 is ln(1 + r௜௧) = β′x௜௧ + ε௜௧. Denote the 

estimated parameters by βመ and the estimated standard deviation of the residual (root MSE) 

by σො. The projected geometric rate of change of plan i in year t is: 

ln(1 + r௜௧) = β′ ෡ x௜௧ + σොz௜௧,
 13 

where z௜௧ is a random draw from the standard normal distribution. The projected rate of 

change is r௜௧ = exp൫β′ ෡ x௜௧ + σොz௜௧൯ − 1. 

Table 2 shows that the R-Squared of the preferred specification is 0.1134, which means that 

about 11% of the variation in the growth rate is explained by variation in the model’s 

independent variables and that almost 89% stems from random variation in the residual. 

We therefore strongly suggest stochastic projections (projections that include a random 

draw of the residual) to maintain the distribution of growth rates within a single year (such 

as illustrated by Figure 6 on page 7). Should a compelling reason exist to exclude residual 

draws from the projections, then a variance correction needs to be applied.9

4. RECOMMENDATION 

Model Specification 

The current implementation of ME-PIMS assumes that, for a year within a scenario, the 

numbers of active participants at all multiemployer plans grow or contract at the same rate. 

The projected rate of change is drawn from a uniform distribution that ranges from -3.3% 

to 0.7%. The current model is parsimonious but it does not capture the diversity and range 

of the rates at which multiemployer plans have changed over the past two decades. It also 

does not take account of plan-level or macroeconomic variation. However, such 

relationships can be important. For example, we found that at-risk plans tend to lose active 

participants faster than not-at-risk plans. The contribution base of at-risk plans may 

therefore shrink faster than ME-PIMS currently projects, and ME-PIMS may underestimate 

the fraction of plans that will require assistance from PBGC. 

Even within a single year, plans have grown or contracted at widely varying rates, which is 

at odds with the current assumption of equal rates of change among all plans within a year. 

An important objective of a microsimulation model such as ME-PIMS is to preserve 

heterogeneity across plans, including heterogeneity in rates of change of active 

participation. We therefore recommend ME-PIMS draw a random rate of change for every 

plan in every year and every scenario. 

We estimated a simple single-equation model that describes rates of change in active 

participation with just four determinants: plan maturity (defined as the ratio of inactive to 

total plan participants), plan at-risk status, Construction sector, and the rate of return on 

equities. Each of these factors has a statistically significant and economically meaningful 

effect on the rate of change in active participation. They also matter directly for a plan’s 

financial outlook, and it may be important to capture their additional effect on a plan’s cash 

flow via active participation. We recommend that ME-PIMS adopt a similar model. 

9

 The variance correction depends on the distribution of ε௜௧. If ε௜௧ is distributed normally 

with mean zero and standard deviation σ, the expected value of exp(ε௜௧) is exp(σ

ଶ/2), so 

that the projection of r௜௧ is exp൫β′ ෡ x௜௧ + σො

ଶ/2൯ − 1. See, for example, NIST/SEMATECH eHandbook of Statistical Methods, section 1.3.6.6.9, available at 

https://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm. 
 14 

Likely Effects and Merits of an Alternative Active Participation Model 

Adopting an alternative active participation model may be expected to have two primary 

effects. The first is to improve the distribution of the projected financial position of PBGC’s 

multiemployer program. The projected rate of change in active participation would connect 

to plan characteristics, macroeconomic developments, and unobserved―but very 

real―determinants. As a result, the projected paths of individual plans may be expected to 

be more realistic and to better preserve the diversity of paths that is observed in historical 

data. Ultimately, the resulting stochastic projection of the financial position of PBGC’s 

multiemployer program will better capture its future distribution. The second effect is to 

brighten the outlook of PBGC’s multiemployer program. Subject to the observations on a 

sensitivity analysis in the next section, PBGC should consider raising the target average rate 

of change to around -0.9%, up from the current -1.3%. The effect of such a change is likely 

an improvement in the financial outlook of PBGC’s multiemployer program. 

An alternative active participation model that factors in plan characteristics may further 

enhance the ability of ME-PIMS to carry out hypothetical policy scenarios. For example, the 

indicator for the Construction industry may be interpreted as capturing withdrawal liability 

rules that differ from those in other sectors. Its estimate (to slow contraction by 0.52%) can 

help inform the likely effects of potential proposals to modify liability withdrawal rules in 

other industries. 

Sensitivity Analysis 

The average annual participant-weighted rate of change in active participation ranged from 

-6.2% to 3.1% in 2000–2019, with an overall geometric average over that period of 

-0.93%. However, historical rates have fluctuated, even over longer periods. At issue is 

whether the historical average continues to hold in future years, or whether any 

adjustments to the model are advisable. A relatively straightforward way to achieve a 

targeted rate is by calibrating the intercept of the model. 

We suggest that experts develop plausible bounds for the expected rate of change in active 

participation and apply those scenarios in ME-PIMS to gauge the sensitivity of PBGC’s 

financial outlook to alternative levels. These scenarios may be viewed as low-cost and highcost scenarios. Eventually, a baseline scenario can be adopted, and it would be of great 

interest to understand how sensitive the model is to unexpectedly high or low rates of 

change. If the sensitivity is high, then a periodic recalibration of active participation rate 

changes should be employed. 
 15 

This document is the Final Report on ME Active Participant Projections, Part 3 of Deliverable 

3 under Contract 16PBGC20C0015 (PIMS Peer Review — Single Employer Contribution 

Policy and Risk Transfer Analysis and Multi-Employer Active Population Analysis). 

DISCLAIMER 

The views, opinions, and/or findings contained in this report should not be construed as an 

official Government position, policy or decision, unless so designated by other 

documentation issued by the appropriate governmental authority. 

We call your attention to the possibility that other professionals may perform procedures 

concerning the same information or data and reach different findings than Advanced 

Analytical Consulting Group, Inc. (AACG) for a variety of reasons, including the possibilities 

that additional or different information or data might be provided to them that was not 

provided to AACG, that they might perform different procedures than did AACG, or that 

professional judgments concerning complex, unusual, or poorly documented matters may 

differ. 

This document contains general information only. AACG is not, by means of this document, 

rendering business, financial, investment, or other professional advice or services. This 

document is not a substitute for such professional advice or services, nor should it be used 

as a basis for any decision or action. Before making any decision or taking any action, a 

qualified professional adviser should be consulted. AACG, its affiliates, or related entities 

shall not be responsible for any loss sustained by any person who relies on this publication.