Deborah J. Lucas

English

Research Papers in Economics

Commentary
Deborah J. Lucas
THE PBGC’S RISK EXPOSURE
The PBGC assumes responsibility for a plan’s
defined-benefit pension obligations when two
conditions are simultaneously met: the sponsoring
firm is financially distressed and the pension plan
is sufficiently underfunded. As such, PBGC insur-
ance is a compound put option held by defined-
benefit plan sponsors, and PBGC liabilities can be
valued using options pricing methods. Recently,
Wendy Kiska and Marvin Phaup of the Congres-
sional Budget Office (CBO) and I have developed
an options-pricing model to quantify the PBGC’s
prospective net costs and to serve as a tool to
evaluate the effect of various policy alternatives.
The results described here are drawn from that
CBO (2005) analysis.
To briefly describe the model, it employs a
Monte Carlo simulation that takes into account
the evolution of firm assets, firm liabilities, pen-
sion assets, and pension liabilities and their inter-
action with program rules. For simplicity, firm
and pension assets are assumed to be stochastic,
whereas firm and pension liabilities are taken to
be deterministic. Both firm and pension assets are
affected by correlated market risk, and taking into
account this risk adds significantly to the esti-
mated value of the put option. The model is cali-
brated using 2004 data covering the top 1,179
companies with defined-benefit pension plans.
Although reported underfunding in 2004
totaled $450 billion, the forward-looking estimate
of the PBGC’s net cost is only a fraction of this.
Over a 10-year horizon, we project a net cost of
T
he recent failures of several very large
corporations with severely underfund-
ed pension plans (e.g., United Airlines,
U.S. Airways, and Bethlehem Steel)
have made the risk exposure of the Pension
Benefit Guaranty Corporation (PBGC), the 
government agency that insures defined-benefit
plans, front page news. Further, the prospect that
other large corporations are likely to follow has
motivated legislators to introduce several new
proposals aimed at limiting the PBGC’s risk expo-
sure. In his paper, Bodie (2006) reminds us of the
straightforward but often ignored fact that much
of the risk to the PBGC could be avoided if limits
were imposed on the share of pension assets
invested in stocks and other risky assets. Presum-
ably, the lack of interest by Congress in imposing
such restrictions is due to very strong resistance
from the business community. The fundamental
question, then, is why do managers believe that
it is imperative to invest pension assets predomi-
nantly in stocks, despite the volatility in funding
requirements that they have experienced follow-
ing this strategy?
In these comments I will focus on two broad
questions raised by Bodie’s analysis: First, what
are the main drivers of the PBGC’s risk exposure?
Second, why do pension managers choose to
invest pension assets the way they do and when
should the optimal hedge portfolio contain some
stocks? 
Deborah Lucas is a professor of finance at Northwestern University and a research associate at the National Bureau of Economic Research.
The author thanks Wendy Kiska, Marvin Phaup, and Steve Zeldes for many interesting conversations about these issues.
Federal Reserve Bank of St. Louis Review, July/August 2006, 88(4), pp. 331-35.
© 2006, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
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FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2006 331
           about $63 billion; and this increases to $119 bil-
lion over a 20-year horizon. The forward-looking
cost is much lower than the amount of contem-
poraneous underfunding because, for any com-
pany, the probability of the joint occurrence of
bankruptcy and underfunding is much lower
than the probability that the pension plan is
underfunded at a point in time. That is, as long
as a firm remains solvent, it gradually must close
any funding gap that arises, although shocks to
pension assets and liabilities will continue to
generate new episodes of underfunding. 
The model is useful for quantifying the savings
that might be realized from Bodie’s suggestion of
limiting the share of pension fund assets invested
in stocks. In the base-case analysis, stocks are set
to the typical 70 percent share of each company’s
asset portfolio. Limiting the stock share to 30 per-
cent saves $9.9 billion of the $63 billion in forward-
looking net costs. There are several reasons why
this cap on stock holdings reduces but does not
eliminate the forward-looking cost. 
One reason that the PBGC remains at risk in
this experiment is that stocks still comprise 30
percent of pension investments, so there is still
an assumed mismatch between the risk of assets
and liabilities. More importantly, however, when
the PBGC takes over a pension plan, there is typi-
cally a jump-up in pension liabilities that aver-
ages 20 percent. For example, U.S. Air reported
that it was 90 percent funded in the year prior to
termination but was found to be only 45 percent
funded at termination. “Termination liabilities”
are systematically higher than “current liabilities”
(the basis for funding requirements) because of
factors such as the triggering of early-retirement
benefits and the propensity of distressed compa-
nies to stop making contributions to their pen-
sion plans. This implies that even a fully funded
plan can generate substantial costs if the sponsor
becomes distressed. How regulation should deal
with this phenomenon is not obvious. Forcing all
firms to fund to the level of termination liabilities
instead of current liabilities would result in sys-
tematic overfunding and would further discour-
age healthy firms from staying in the system. 
Basing funding requirements on current lia-
bilities rather than on the systematically higher
termination liabilities is one example of how
underestimating liabilities increases costs to the
PBGC. Liabilities also will be systematically
underestimated if regulators allow firms to use
too high a discount rate. This puts the PBGC at
risk because it makes the target for full funding
too low and reduces average funding levels. For
example, the model implies that making perma-
nent the higher discount rate that has been in effect
for the past few years would add $8.1 billion to net
costs over a 10-year horizon, under the assump-
tion that the original, risk-free discount rate is
the correct rate. Again, though, forcing firms to
value liabilities too conservatively can place an
unfair burden on pension providers. 
As Bodie mentions, there is considerably more
interest in raising premiums or making them more
sensitive to risk than in limiting investment risk.
However, the premium increases being contem-
plated are far below the level the model suggests
are actuarially fair. Abstracting from behavioral
responses, the model generates a fair premium
multiple of 6.5 times the current premium rate,
whereas the leading proposal calls for an increase
that is only 2.1 times the current rate. Potentially,
plan managers’ incentives for controlling risk
could be improved by implementing risk-based
premiums; but again it is unlikely that it is politi-
cally feasible to create fair differentials between
low- and high-risk firms. For example, the model
implies that a fair premium for below-investment-
grade firms would have to be 18.5 times higher
than it is at present to match the current cost to
the PBGC for investment-grade firms. These kinds
of estimates, both because of their magnitude and
because they impose very high costs on firms that
are the most distressed, suggest that, although
incremental improvements could be made through
premium reform, controlling the risk of the system
through the premium structure is not feasible. 
Bodie draws an analogy to the Savings and
Loan (S&L) Crisis, where the mismatch in the
market sensitivity of assets and liabilities created
predictably huge losses when interest rates moved
sharply higher. Also, as for the S&Ls, it is likely
that regulatory forbearance will exacerbate the
problems of the PBGC, this time because of the
reluctance of Congress to force struggling com-
Lucas
332 JULY/AUGUST 2006 FEDERAL RESERVE BANK OF ST. LOUIS REVIEWpanies to devote scarce capital to funding their
pension plans. 
The propensity of S&Ls to take on excessive
risk is often attributed to the incentives created
by deposit insurance—the “down-side risk” is
partially absorbed by the government. An impor-
tant and unresolved question is whether PBGC
insurance is an important reason why plan spon-
sors prefer to invest in stocks. Although many
observers assume this to be the case, the CBO’s
analysis suggests that, at least for healthy com-
panies, the value of the insurance put option is
very small and that those companies’ plan man-
agers should largely internalize the investment
risk. Consistent with this view, Rauh (2006) finds
no evidence that higher-risk companies gamble
more with pension plan assets than do lower-risk
companies. 
Why, then, do even healthy companies choose
to invest such a high proportion of pension assets
in stocks? Bodie argues that managers are misled
by the fallacy that stocks always outperform bonds
in the long run and so mistakenly believe that
their strategy is less risky than it is. Although this
is quite possibly a contributing factor for many
managers, in the remainder of this discussion I
will consider a rational alternative—that there is
a role for stocks in hedging future pension liabil-
ities because some liabilities behave more like
stocks than like bonds. 
WHAT SHOULD THE HEDGE
PORTFOLIO LOOK LIKE?
Under Bodie’s assumption that future pension
obligations are fixed nominal quantities, it is clear
that nominal bonds provide a perfect hedge. In
the example of a pension liability of $1,000 that
comes due in 10 years, putting away the present
value of $1000/(1 + r)10 in risk-free bonds ensures
that the payoff will match the liability. 
In practice, defined-benefit pensions typically
link the level of the retirement annuity to a
worker’s years of service and to wage earnings in
the final year (or years) of service. This means that
the future benefit is a random variable and that
the best hedge portfolio is maximally correlated
with what wage earnings, and hence the benefit,
will turn out to be. For example, the benefit for a
worker expected to retire in 10 years might be 40
percent of wage earnings in 10 years. If the correct
object to hedge is this broadly defined liability
(i.e., the broadly defined PBO [pension benefit
obligation] rather than the ABO [accrued benefit
obligation]), then the best hedge is to invest an
amount equal to the present value of the liability
in a portfolio maximally correlated with earnings
in 10 years. 
This line of reasoning implies that if wage
earnings and stock returns are correlated and if
the broadly defined pension obligation is the right
measure of the liability to hedge, then the optimal
hedge portfolio will contain stocks as well as
bonds. Intuitively, it seems likely that wages and
stock returns should be correlated over long
horizons. Black (1989) suggested this as a reason
for pension funds to hold stocks, but he did not
quantify the effect. 
When wage earnings and stocks are correlated,
the value of the pension liability can be modeled
as a derivative on the stock market. Such a model
can also be solved for the time-varying share of
stock in the optimal hedge portfolio. An illustra-
tive example of this is given here (see Lucas and
Zeldes, 2006, for a much more complete analysis). 
The model proposed for the joint process for
stocks, human capital, and wage earnings is con-
sistent with the near-zero correlation between
wage earnings growth and stock returns observed
at a 1-year horizon and also with the hypothesis
that there is a higher correlation over longer
periods. The model, as parameterized in Table A1,
produces a correlation of 0.11 between wage earn-
ings growth and stock returns over three years
and 0.36 over five years.
Specifically, I assume that the aggregate value
of stock, S, evolves according to
where dzs is a draw from a standard normal dis-
tribution, the expected return on stocks is rs, the
dividend yield is div, and the standard deviation
of stock returns is σs. The time step is h, taken in
the calibrations to be one year.
S S r div h h dz th t s s s s + =− − + () exp ( . ) ( ) , 0 5 2 σσ
Lucas
FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2006 333The aggregate value of human capital, H,
evolves according to
where dzw is (idiosyncratic) risk and α is the
average drift. Human capital slowly adjusts
toward the long-run human capital to stock ratio,
T*, at an annual rate of γ. The stock of human
capital is reduced by earnings at time t, Wt, which
is analogous to a dividend. Finally, wage earnings
evolve according to
where next-period earnings equals current earn-
ings plus a term that pulls earnings toward a target
fraction of current human capital, rw, at an annual
rate of β. 
With regard to pension benefits, the benefit
is assumed to increase with service years at a
rate of 2 percent per year. The lifetime annuity at
retirement is based on wage earnings in the year
of separation or retirement, times service years,
times the 2 percent. Separation is stochastic, as is
mortality. All parameters are reported in Table A1. 
The results of the analysis are reported in
Table 1. The table shows the implied present value
of pension liabilities as a function of remaining
years to retirement, under various assumptions
about the discount rate for liabilities. Specifically,
liabilities are discounted at the risk-free rate, the
average assumed stock return, and using the deriv-
WWh r H W th t w t t + =+ − β() ,
HH h h d z
hT
H
S
t ht w w w
t
t
+ =− + ()
+−

exp ( . ) ( )
*
ασ σ
γ
05 2
  

  − SW tt ,
ative pricing implied by the model (and, hence,
at the correct rate). The assumption that the cor-
relation between wage earnings and stock returns
increases over time suggests that for young work-
ers the liability is more like a stock than a bond.
For instance, for a worker with 20 years to retire-
ment, the correct discount rate would have a 39
percent weighting on stocks and a 61 percent
weighting on bonds. Wrongly assuming that the
liability is like a bond and discounting at the risk-
free rate overstates its value by more than 30
percent. Lucas and Zeldes (2006) show that the
corresponding optimal hedge portfolio is more
heavily skewed toward stocks for active workers
than suggested by the discount-factor weights
because of the dynamic nature of the portfolio.   
CONCLUSIONS
The bottom line of this analysis is that there
is a role for stocks in the investment portfolios of
defined-benefit pension plans. The optimal share
of pension assets invested in stocks increases with
employment horizon and changes over time with
the demographics of a firm’s workforce. When
workers separate from the firm, their benefits
become bond-like and any stocks invested on their
behalf should be reallocated to bonds. Bonds also
are the natural hedge for firms whose obligations
are predominantly to retired workers and their
dependents. Firms with mostly young workers,
however, have a legitimate reason to prefer to
hold at least a portion of their investments in the
stock market.
Lucas
334 JULY/AUGUST 2006 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW
Table 1
Simulation Results
Initial wage replacement rate 0.6 0.5 0.3 0
Years to retirement 5 10 20 35
Present value using correct risk adjustment 0.218 0.179 0.109 0.040
Present value discounting at the risk-free rate 0.220 0.189 0.138 0.074
Present value discounting at the average stock return 0.189 0.140 0.076 0.026
Initial share in stocks to get the average discount rate 0.063 0.195 0.389 0.589
Share of cohort making it to 65 with the firm 0.766 0.5948 0.371 0.1694I conclude from this analysis that a blanket
prohibition on stock investments in defined-
benefit pension plans would be inappropriate
and for some firms it could actually increase risk.
Nevertheless, further analysis (Lucas and Zeldes,
2006) suggests that the typical firm holds far more
in stocks than can be justified by this hedging
demand. We argue there that Financial Accounting
Standards Board rules for how pensions are
accounted for in earnings may provide a strong
incentive to overinvest in stocks, a point Bodie
briefly alludes to also in his paper.  
Finally, any policy that seriously addresses the
PBGC funding gap will likely accelerate the switch
from defined-benefit to defined-contribution
pensions. Hence, the costs of PBGC insurance
must be considered in the broader context of the
goals of an employer-based retirement savings
system.
REFERENCES
Black, Fischer. “Should You Use Stocks to Hedge
Your Pension Liability?” Financial Analysts Journal,
January/February 1989, 45(1), pp. 10-12.
Bodie, Zvi. “On Asset-Liability Matching and Federal
Deposit and Pension Insurance.” Federal Reserve
Bank of St. Louis Review, July/August 2006, 88(5),
pp. 323-29.
Congressional Budget Office. “The Risk Exposure of
the Pension Benefit Guaranty Corporation.”
Washington, DC: September 2005; www.cbo.gov/
ftpdocs/66xx/doc6646/09-15-PBGC.pdf.
Lucas, Deborah and Zeldes, Stephen P. “Valuing and
Hedging Defined-Benefit Pension Obligations: The
Role of Stocks Revisited.” Working paper,
Northwestern University, 2006.
Rauh, Joshua. “Who Manages Pension Fund Risk?”
Working paper, University of Chicago, 2005.
Lucas
FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2006 335
APPENDIX
Table A1
Parameter Values for Simulations
Dividend yield 0.02
Standard deviation of stock return 0.18
Standard deviation of idiosyncratic wage return 0.02
Standard deviation of idiosyncratic own-firm return 0.2
Mean growth of human capital 0.02
Speed of reversion to target 0.1
Speed of reversion in wages 0.5
Target human to physical capital 2
Risk-free rate 0.02
Inputs to defined-benefit pension
Initial replacement rate Various
Years of earnings Various
Separation and mortality
Mortality rate # age 65 0.003
Mortality rate > age 65 0.05
Separation rate x < age 35 0.06
Separation rate age 34 < x < age 46 0.045
Separation rate age 45 < x < age 56 0.04336 JULY/AUGUST 2006 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW

Office. “The Risk Exposure of the Pension Benefit Guaranty Corporation.”

On Asset-Liability Matching and Federal Deposit and Pension Insurance.” Federal Reserve Bank of St. Louis Review,

Should You Use Stocks to Hedge Your Pension Liability?”

Valuing and Hedging Defined-Benefit Pension Obligations: The Role of Stocks Revisited.” Working paper,

Who Manages Pension Fund Risk?” Working paper,

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