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Economic Quarterly— Volume 98, Number 4— Fourth Quarter 2012— Pages 231–
254

Some Theoretical
Considerations Regarding
Net Asset Values for Money
Market Funds
Huberto M. Ennis

O

n Tuesday, September 16, 2008, the day after Lehman Brothers …led for bankruptcy, the Reserve Primary Fund, a large
prime money market fund, announced that it would not be
able to redeem investors’funds one for one. The fund had “broken the
buck” mainly due to losses on its holdings of Lehman’ debt instrus
ments. In the days that followed, out‡
ows from prime money funds
spiked, with investors withdrawing, in the space of a week, approximately $300 billion— roughly 15 percent of total assets invested in these
funds at the time (Financial Stability Oversight Council 2012). By Friday of that week, the U.S. Treasury and the Federal Reserve would
decide to implement several major interventions aimed at stabilizing
the money market funds industry. While out‡
ows did, in fact, slow
down in the following weeks, money funds continued divesting large
amounts of commercial paper and other assets for some time.
The interventions announced by the U.S. Treasury and the Federal
Reserve on September 19, 2008, were broad and unprecedented. The
Temporary Guarantee Program adopted by the Treasury Department
guaranteed that shareholders of those funds opting to participate would
receive the fund’ stable net asset value (NAV) per share were the fund
s
to suspend redemptions and fully liquidate. At the same time, the
I would like to thank Todd Keister, Je¤ Lacker, Jon Lecznar, Ned Prescott, Zhu
Wang, and Alex Wolman for comments on an earlier draft. All errors and imprecisions are of course my exclusive responsibility. The views expressed in this
article are those of the author and do not necessarily represent the views of
the Federal Reserve Bank of Richmond or the Federal Reserve System. E-mail:
huberto.ennis@rich.frb.org.

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Federal Reserve Bank of Richmond Economic Quarterly

Federal Reserve created the Asset-Backed Commercial Paper Money
Market Mutual Funds Liquidity Facility that was used to extend central bank credit to banks buying high-quality asset-backed commercial
paper from money market funds (see Duygan-Bump et al. [2013]).
Money market funds (or, money funds, for short) are open-end
mutual funds that invest in short-term high-credit-quality debt instruments such as commercial paper, large certi…cates of deposit, Treasury
bonds, and repurchase agreements. Most money funds maintain a stable redemption value of shares, usually set at a value equal to one, and
pay dividends that re‡ the prevailing short-term interest rates. As
ect
of September 2012, there were 632 money market funds in the United
States with total assets under management of approximately $2.9 trillion. In comparison, deposits at banking institutions amount to about
$11 trillion. So, the size of the U.S. money market fund industry is
signi…cant.
SEC rule 2a-7 pursuant to the Investment Company Act of 1940
provides the regulatory framework for these funds. The rule permits
funds to use the amortized cost method of valuation to compute net
asset values and allows the funds to round such value to the nearest
1 percent.1 The possibility of stable net asset values is a consequence
of these provisions. At the same time, the rule puts limitations on
the type of assets that the funds can hold: Funds must hold low-risk
investment instruments with remaining maturity no longer than a given
maximum date.
Within the broader category of money market funds, there are different sub-categories based on the main investments taken by the funds.
Prime money funds hold predominantly private debt instruments. Government funds, instead, are restricted to invest only in governmentissued securities. Prime money funds tend to be more exposed to credit
risk (Rosengren 2012) and they are the ones that experienced serious
…nancial distress during the second half of 2008.
In February 2010, partly as a response to the problems with prime
money funds during the crisis, the Securities and Exchange Commission
(SEC) adopted amendments to rule 2a-7 intended to make money funds
more resilient and less likely to break the buck. The changes tightened
restrictions on the amount of risk that money funds can assume and,
for the …rst time, required that money funds maintain liquidity bu¤ers
to help them withstand sudden demands for redemptions. The new
1
The amortized cost method allows the funds to value assets at their acquisition
cost rather than market value, and interest earned on the asset is accrued uniformly
over the maturity of the asset (adjusting for amortization of any premium or accretion
of any discount involved upon purchase).

H. M. Ennis: Net Asset Values for Money Market Funds

233

rules also enhanced information disclosure by funds and provided a
framework for the liquidation of funds that break the buck and suspend
redemptions.
Even after the wide-ranging revisions of rule 2a-7 in 2010, many
policymakers and interested parties believe that a more comprehensive
reform of the money funds industry is still necessary. In November
2012, the Financial Stability Oversight Council (FSOC) made public a set of proposed recommendations to the SEC for further reform
(Financial Stability Oversight Council 2012). The Council proposed
three di¤erent avenues for reform. The …rst alternative is to remove
the valuation and pricing provisions in rule 2a-7 and to require money
market funds to have a ‡
oating NAV that re‡
ects the market value of
their assets.
The second alternative is to require funds to maintain a bu¤er of
assets in excess of the value implied by a …xed (and stable) NAV on
outstanding shares. This bu¤er would be combined with a minimum
balance at risk— in certain circumstances a small percentage of each
investor’ shares would be made available for redemption only on a
s
delayed basis (see McCabe et al. [2012] for a detailed analysis of the
minimum balance at risk idea). Finally, the third proposal is to require
funds to hold a risk-based bu¤er and combine it with requirements on
portfolio diversi…cation, liquidity, and disclosure.2
To assess the Council’ proposals, or any other reform proposal,
s
it seems crucial …rst to be able to discern what is the ultimate function that money funds perform in the economy and how appropriate
regulations depend on that. There are (at least) two possible ways to
think about this issue. On one hand, some observers have argued that
money funds provide useful maturity transformation by issuing claims
(shares) that can be redeemed on demand while, at the same time,
investing in longer-term …nancial instruments. Even though the funds’
portfolios are concentrated in relatively short-term instruments, funds
stand ready to redeem shares on demand and, hence, are exposed to a
maturity mismatch and the threat of illiquidity.
On the other hand, it may be that the main role of money funds is
to manage the portion of investors’portfolios intended to be allocated
to relatively short-term money market instruments. In other words,
according to this view, money funds are expert “cash” managers and,
for this reason, it is e¢ cient for investors to delegate to them the administration of part of their short-term and liquid investment strategy.
2

See the FSOC document for a thorough description and evaluation of the reform
proposals (Financial Stability Oversight Council 2012). The document also provides a
good summary of the institutional details of the U.S. money market funds industry.

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Federal Reserve Bank of Richmond Economic Quarterly

Assessing which of the two alternative views best describes the economic value associated with money funds is important for choosing
the appropriate design of a regulatory framework. In particular, how
redemption values should be computed often depends on this assessment. The aim of this article is to illustrate this point by presenting
and comparing the implications of using di¤erent methods for computing NAVs in two very simple models that capture, in a stark way, the
two aforementioned views about the function of money funds.
The …rst model is a version of the canonical maturity transformation framework introduced by Diamond and Dybvig in 1983. We …nd
that, to the extent that NAV regulations are designed in a way that
still allow funds to ful…ll their basic function, then illiquidity and potential instability are likely to remain an integral feature of the money
fund business. Furthermore, from this standpoint, computing appropriate market-sensitive NAVs requires an estimation of the amount of
withdrawals that the fund can be expected to face. This process of anticipation is especially di¢ cult because it involves predicting economic
behavior that depends on agents’ expectations about the decisions of
others.3
The second model maintains many of the structural features of the
…rst model, but is modi…ed so that the motives investors have to deposit
money with the fund are di¤erent. In particular, investors no longer
derive value from maturity transformation but, instead, they rely on
the funds exclusively to manage their investments.4 In this case, we
…nd di¤erent implications relative to the …rst model. Computing NAVs
that accurately re‡ market valuations is perfectly compatible with
ect
the role played by the funds and can actually make the funds more
stable. The model also illustrates how a wave of withdrawals from a
poorly performing fund may just be the way that the system has to
implement the best possible allocation of resources. Trying to stop that
process would, in fact, be detrimental to economic e¢ ciency.
Obviously, it is hard to determine which is the main function that
money funds are performing in the economy, or even if they are essential organizations to pursue the highest attainable welfare of society. This article considers two candidate functions, one at a time.
However, it is certainly possible that money funds perform, at least
to a certain extent, these and potentially other functions simultaneously. Sorting these issues out is essentially an empirical undertaking,
3
Chen, Goldstein, and Jiang (2010, Appendix A) study a di¤erent, yet related
model of a mutual fund where the redemption strategies of agents are also interdependent in equilibrium and can generate the conditions for fund instability.
4
The recent article by Parlatore Siritto (2013) also studies a model where the main
function of money funds is to manage the assets of investors.

H. M. Ennis: Net Asset Values for Money Market Funds

235

beyond the scope of our study. The objective in this article is, instead,
rather theoretical. The point we want to illustrate is that once one has
taken a stand on the answer to the empirical question, some theoretical implications follow that can help guide the design of an appropriate
regulatory policy for money funds.
In principle, the models we present could be extended and modi…ed
to evaluate the other reform proposals currently being considered. For
example, to understand the implications of requiring a bu¤er of assets
one would need to take a stand on the way the bu¤er is being funded
and model the objectives of the agents providing such funding. While
this is potentially a productive activity, it would complicate the models
in a way that would reduce the clarity of the results related to NAV
policies. For this reason, we choose to limit our discussions to the NAV
proposals.5
Before turning to the models, we should mention here that there is,
in fact, a third commonly held perspective on the role of money funds
in the economy, which we will not discuss in this article. The money
funds industry developed and grew briskly in the 1970s, a period when
banks were subject to strict interest rate ceilings imposed by regulation.
These restrictions on the ability of banks to pay competitive rates did
not apply to money funds and allowed money funds to become a natural
alternative to banks (see Rosen and Katz [1983] for example). Even
though the restrictions have been mostly removed now, funds may still
be a vehicle for regulatory arbitrage to the extent that they are not
subject to strict capital requirements and other regulations faced by
banks.
The rest of the article is organized as follows. In the next two
sections, we study two alternative frameworks that can be used to
think about the problem of setting the appropriate redemption value
of shares in a mutual fund. The …rst model, presented in Section 1,
considers the case in which the role of the fund is to perform a maturity
transformation function. The second model, in which the fund is just an
investment vehicle that performs no essential maturity transformation,
is the subject of Section 2. We close the article in Section 3 with a brief
conclusion.
5
Another aspect left unexplored in this article is the possibility of contingent support from an institutional sponsor when the fund experiences …nancial distress. Sponsor
support has played a signi…cant role in the recent history of U.S. money market funds
(Rosengren 2012). For a theoretical analysis of the issue, see Parlatore Siritto (2013).

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1.

Federal Reserve Bank of Richmond Economic Quarterly

MATURITY TRANSFORMATION

The canonical framework for studying maturity transformation in …nancial economics is the Diamond and Dybvig (1983) model of banking.
A way to obtain desirable allocations in such an environment is to allow
for an institutional arrangement that resembles a mutual fund. In this
section, we analyze the implications of this theory for the determination
of the fund’ net asset value.6
s

The Model
There is a continuum of agents of mass 1. Agents are risk averse and
each owns one unit of resources at the beginning of time. Time is
denoted by t = 0; 1: Agents are homogeneous ex ante, but in period 0 a
proportion q of the agents gets a preference shock and needs to consume
at that time to be able to get any utility. We call these agents impatient
and the 1 q remaining agents, patient. Patient agents are indi¤erent
about consuming at time 0 or 1. There is a productive technology that
returns R > 1 units of resources in period 1 per unit of resources (not
consumed and) invested in period 0. Resources can be taken out of
the production technology during period 0 at a one-for-one basis (one
unit per unit invested); in other words, there are no liquidation “costs”
from interrupting the production process at an early stage.

A Benchmark Optimal Allocation
Since R > 1, there is a clear bene…t from delaying consumption in this
economy. For this reason, it is generally optimal to have patient agents
consume only in period 1. Impatient agents, however, must consume
in period 0.
Consider the solution (c0 ; c1 ) to the following planning problem:
max qu (c0 ) + (1
c0 ;c1

q) u (c1 )

(PP1)

subject to
(1

q) c1 = R (1

qc0 ) :

We take such a solution as a benchmark optimal allocation in this
environment. It is the allocation that maximizes the sum of the total
6
There is an extensive literature dedicated to the study of possible extensions of
the Diamond-Dybvig model (see, for example, Freixas and Rochet [2008]). We use the
simplest version of the model that allows us to illustrate the general points we are
trying to make. Studying the implications for money funds of extensions of the model
in various directions is a potentially fruitful activity. We consider this section a …rst
step in that direction.

H. M. Ennis: Net Asset Values for Money Market Funds

237

utility of both groups of agents, patient and impatient, subject to the
resource constraint. In this constraint, 1 qc0 is the amount of resources
left after making a payment of value c0 to each of the q impatient
agents. This amount remains invested in the productive technology
and is multiplied by the return R after waiting until period 1. In
period 1, the resulting resources are divided between the remaining
1 q patient agents and each of them gets an amount equal to c1 .
When investors’coe¢ cient of relative risk aversion is greater than
one it can be shown that
1 < c0 < c1 < R:
The thing to notice here is that patient and impatient agents share the
return from the productive investment in the optimal allocation. This
is a form of insurance. Impatient agents get more than their initial
resources even though the productive investment has not yielded any
returns at the time that these agents wish to consume. This insurance
is possible because only a proportion of the agents is expected to be
impatient.

Institutions: An Open-End Mutual Fund
There are two main categories of mutual funds: those that are openend and those that are closed-end. Open-end mutual funds stand ready
to redeem shares held by investors at an announced net asset value.
Closed-end mutual funds, instead, issue a …xed number of shares that
in principle trade in a securities market but do not redeem shares on
demand. Money market funds in the United States are predominantly
open-end funds. Given the focus of our study, we restrict attention to
this arrangement in the main body of the article. The reasons for the
prevalence of open-end funds is the subject of active academic research
(see, for example, Stein [2005]). We do not address the issue here but
we present a brief analysis in the Appendix of how a closed-end fund
would work in this environment.7
Suppose that at the beginning of time agents form an open-end
mutual fund and deposit their endowment with the fund. The fund
then invests the resources and sets dividend payments and a NAV that
determines how much an agent is entitled to withdraw from the fund
at each time.
7

There are many complex issues associated with the economics of closed-end mutual funds. For a survey of the subject see Lee, Shleifer, and Thaler (1990). Cherkes,
Sagi, and Stanton (2008) is an interesting recent contribution.

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Federal Reserve Bank of Richmond Economic Quarterly

One way for the fund to implement the optimal allocation (c0 ; c1 ) is
to set a NAV equal to 1 and assign c0 1 new shares to each investors in
period 0 in the form of a dividend payment. At that point, then, each
agent has in their account c0 shares of the fund. If only the proportion
q of agents that need to consume early decide to withdraw from the
fund, then total withdrawals from the fund equal qc0 and there will be
enough resources to pay the rest (a proportion 1 q) of the agents an
amount equal to c1 in period 1. Since c1 > c0 , an investor that expects
these payments and does not need to consume early will be willing to
wait to withdraw. For this reason, the optimal allocation is a possible
outcome associated with this mutual fund scheme.
As is well-known from the bank-run literature, when withdrawals
from the fund happen sequentially, there is another possible outcome
associated with this scheme (see Diamond [2007] for a simple exposition). Given that c0 is greater than unity, if all agents attempt to
withdraw at time 0 then the fund would not have enough resources
to cover all the required payments. As a result, if agents expect that
all other agents will attempt to withdraw from the fund, then they
also have incentives to try to withdraw, creating a situation that would
resemble a run on the fund.8
It is also well-known from the bank-run literature that a scheme
that allows the suspension of redemptions after q withdrawals will be
able to costlessly rule out the run equilibrium. In reality, money funds
can and have asked the SEC to authorize them to suspend redemptions
after experiencing a wave of withdrawals. However, the authorization is
usually granted under the assumption that the fund will fully liquidate
and terminate operations after that. To the extent that the requirement of full liquidation still imposes costs on the fund, the suspension
becomes less e¤ective in limiting the incidence of runs.
In the model, the possibility of runs arises because, after the fund
has distributed the new shares as dividends, if all agents are expected
to want to withdraw from the fund at time 0; then the current value of
fund assets is not su¢ cient to justify a NAV equal to 1. In particular,
at time 0 total assets in the fund have a current (liquidation) value of
1. Agents, however, own c0 > 1 shares which, with a NAV of 1, entitle
them to total time-0 payments that are greater than the current (liquidation) value of assets (one unit). An obvious solution to this problem
8
The fact that withdrawals take place sequentially during time 0 implies that the
fund initially makes payments without knowing the total number of time-0 withdrawals
that will ultimately happen. If the fund would be able to observe the total number of
withdrawal requests before making any actual payments, then it is easy to show that
the fund would adjust the value of those payments in such a way that runs could not
happen in equilibrium.

H. M. Ennis: Net Asset Values for Money Market Funds

239

is not allowing the fund to allocate new shares in the form of dividends
before the actual returns are realized. However, the “early” dividends
are essential for implementing the benchmark optimal allocation when
the NAV is set to equal 1.9
In general, however, the fund may not want to value assets at their
liquidation value (i.e., using a NAV equal to 1). Suppose, instead, that
the fund sets a NAV equal to the future discounted value of the cash
‡ from the assets (FDV for short). If the manager of the fund (or
ow
some regulator) looks at the assets currently in the fund and disregards
the withdrawal issue, following FDV would require setting a NAV equal
R
to 1+r , where r is an appropriate discount rate.
Since we are considering a situation without discounting, one possibility would be to take r = 0. In this case, the fund’ NAV will be set
s
to equal R. We know, however, that if agents withdrawing at time 0
get a payment equal to R; then the optimal allocation will not be implemented (since c0 < R). Furthermore, if q agents get R in period 0;
then there will not be enough resources to pay R or c1 to those agents
withdrawing (and consuming) at time 1. If withdrawals from the fund
happen sequentially, the only optimal withdrawal strategy for all investors under these payments is to try to withdraw early in a situation
resembling a run.
Given that the rate of return on investment between t = 0 and
t = 1 is equal to R, another possibility would be to use 1 + r = R as
the appropriate discounting to compute the FDV. In this case, then,
the fund’ NAV will be set to equal unity and again, without an early
s
distribution of shares in the form of dividends, the optimal allocation
would not be obtained. An attractive aspect of setting this value for
the NAV is that the unique equilibrium in this case is for only impatient
agents to withdraw at t = 0. While this conveys a sense of stability to
the fund, it is also the case that impatient agents consume only one unit
(not c0 ) in this situation and, hence, the fund no longer performs the
maturity transformation function that was the purpose of its creation.
It is unclear the extent to which money funds in reality are able to
make higher payments to investors in anticipation of future expected
returns. In the model, implementing a value of c0 greater than 1 requires such anticipation. Money funds may not be performing the type
of maturity transformation suggested by this model. We will consider
an alternative model in the next section.
9
Initially each agent owns one share with a NAV equal to 1. As impatient agents
need to consume c0 > 1 to conform with the benchmark optimal allocation, an entitlement of extra shares needs to be assigned to agents in period 0 so that impatient
agents can actually consume an amount greater than 1 (c0 ) at the appropriate time.

240

Federal Reserve Bank of Richmond Economic Quarterly

Even if the model in this section is the relevant one, it could be
that due to legal (or “best practice” restrictions, money funds do not
)
perform the function described here. For example, suppose that the
law requires that the fund pays dividends only after returns have been
realized and always sets the NAV at the current liquidation value of the
assets. In that case, the fund would set a NAV equal to 1 in period 0
and the payments would be given by c0 = 1 and c1 = R. This payment
scheme, again, makes the fund immune to runs even when withdrawals
are restricted to happen in a sequential manner.
The main insight thus far is that the maturity transformation function may involve a tradeo¤ between e¢ ciency and stability. Some
schemes result in a system that is immune to runs but does not provide bene…cial insurance to impatient agents. Other schemes transfer
resources appropriately among agents but make funds open to instability. The setting of the NAV plays a crucial role in the design of these
schemes.

Variable Liquidation Terms
Suppose the fund is not able to liquidate and recover the invested resources one for one at time 0. Instead, the fund can only get per
each unit initially invested and later liquidated during period 0. In
principle, the value of may depend on the amount x being liquidated
early. That is, is a function of x.
An optimal arrangement is one that delivers the consumption allocation (c0 ; c1 ) obtained by solving the following problem:
max qu (c0 ) + (1

c0 ;c1 ;x

q) u (c1 )

(PP2)

subject to

(1

qc0 = x (x) ;
q) c1 = R (1 x) :

Here, the …rst constraint indicates that to make a payment of c0 to
each of the q impatient agents, the fund needs to liquidate x units
of investment, which allows it to obtain x (x) units of resources at
time 0 when the payments to impatient agents need to occur. After
liquidating x units of resources, 1 x units are left in the productive
technology and, hence, result in R (1 x) available resources at time
1. The second constraint, then, says that these resources will be used
to pay an amount c1 to each of the 1 q patient agents.

H. M. Ennis: Net Asset Values for Money Market Funds

241

It is easy to see that if (x) = R for all x, then c0 = c1 = R. In this
case, a NAV equal to R per share implements the optimal allocation.10
However, if (x) < R for some x then it becomes less obvious how
to compute an appropriate NAV. For example, if (x) = e < R for
all x then c0 < c1 < R and a fund trying to implement the best
arrangement for its investors could need to set a NAV that would expose
it to instability. The benchmark situation we studied before is the
particular case when e = 1.
When funds liquidate, they usually sell assets in the market. It
is often argued that the price of the assets may depend on how much
is being liquidated. In our simple framework, liquidation at time 0
does not involve market prices but rather the direct technological costs
of liquidating productive investment. Still, using the ‡
exibility of the
function we can consider some cases that produce valuable insights
about the more complex situation in which market prices play a role
during liquidation. In particular, consider the case in which (x) = R
as long as x q and (x) = e < R if x is greater than q. Here, again,
the appropriate NAV would depend on the expected number of withdrawals. Suppose that the fund expects to have q withdrawals. Then,
using a NAV equal to R allows the fund to implement the allocation
c0 = c1 = R with only impatient agents withdrawing from the fund at
time 0.
However, if unexpected extra withdrawals were to happen (that is,
if more than q agents decide to withdraw at time 0), the NAV would
have to be drastically adjusted. Evidently, a crucial issue is how soon
in the withdrawal process would the fund realize that withdrawals will
be higher than q. If this realization comes after the …rst q withdrawals
have already happened, then the fund will have to adjust the NAV
at that point. The appropriate value of the NAV would depend on
how many more withdrawals are expected after the …rst q. Suppose
that after seeing that withdrawals continue beyond the …rst q the fund
expects q 0 > q withdrawals. Then, setting a NAV equal to e would
make the fund solvent but would destroy any insurance possibilities
that the fund could still try to exploit given that q 0 is expected to be
lower than 1.
This extension of the model captures in a stylized manner the technological (or market-based) costs that are often associated with the
10
Notice here that when (x) = R for all x; the fund has the ability to come
up with resources immediately at no cost. For each unit of resources that the fund
invests in the productive technology, it can get R units immediately, without waiting or
bearing any risk. For this reason, the case of (x) = R seems of limited applicability
for understanding actual real life investment situations.

242

Federal Reserve Bank of Richmond Economic Quarterly

early liquidation of an investment position. The analysis clearly illustrates that liquidation costs, in interaction with expectations about the
number of early withdrawals, signi…cantly complicate the setting of an
appropriate NAV.

Portfolio Choice: Adding a Liquid Asset
Suppose now that in the setup just studied the liquidation value of the
productive technology is (x) = e < 1 for all x. This situation may
seem peculiar since some costly liquidation is taking place even though
it is completely predictable. In other words, given that the fund is
expecting at least q redemptions, it would be better to invest some
resources in an asset that, while less productive, avoids any signi…cant
liquidation costs (i.e., a more liquid asset).
To address this issue, we extend the previous setup to include an
alternative technology that returns, per unit invested at the beginning
of time 0, one unit of resources at any time. Then, an optimal arrangement would produce the allocation that solves the following problem:
max qu (c0 ) + (1

c0 ;c1 ; ;x

q) u (c1 )

(PP3)

subject to
(1

qc0 =
+ xe;
q) c1 = R (1

x) ;

where is the portion invested in the liquid asset and x, again, is the
amount liquidated at time 0 of the fund’ investment in the productive
s
technology, 1
. As before, the two constraints are resource constraints on payments at time 0 and 1, respectively. The …rst constraint
shows that the investment in the liquid asset is fully used to make
payments to impatient agents. In the second constraint, total unliquidated productive investment is now equal to 1
x. Multiplying
this amount by R > 1, we obtain the total available resources at time
1 that can be used to make payments of value c1 to each of the 1 q
patient agents.
When e < 1 and the fund expects that exactly q agents will withdraw at t = 0, it is optimal to choose x = 0 and
= qc0 . Furthermore, the optimal values of c0 and c1 are given by the same c0 and
c1 obtained in the benchmark optimal allocation (problem PP1). The
perfect predictability of the number of withdrawals, combined with the
fund’ access to a liquid asset, implies that costly liquidation never
s
happens.
How should the fund compute its NAV at time 0? Here, again,
combining the payment of early dividends with a NAV equal to 1 would

H. M. Ennis: Net Asset Values for Money Market Funds

243

be consistent with obtaining the optimal allocation as an equilibrium
outcome. The alternative approach based on calculating a FDV with
a discount rate r = 0 would result in a value of the NAV equal to
1 + (1
) R: While the FDV method is often considered natural,
it is easy to show that in this case the implied NAV is greater than
c0 and, hence, it would provide too much consumption to those agents
withdrawing in period 0 (relative to the optimal allocation).11
The fact that the fund can perfectly predict the amount of withdrawals is important and may be considered unrealistic. Uncertainty
over q signi…cantly complicates the calculations. To gain some perspective on this issue, consider a situation where the fund was expecting q
withdrawals but instead q > q withdrawals happen. After making the
e
…rst q payments the fund would have to reassess the rest of its planned
payments. Suppose that after making the …rst q payments the fund
immediately discovers that the number of withdrawals will be q > q.
e
Then, the optimal continuation payments would solve the following
problem:
max (e
q

q) u c0 + (1
0

subject to
(e
q
(1

q) c0 = xe;
0

q ) c0 = R (1
e 1

q ) u c0
e
1

(PP4)

x) :

The …rst constraint indicates that for the fund to be able to make a
payment of value c0 to q q agents in period 0 it will have to liquidate an
e
0
amount x of productive investment that, given liquidation costs, results
in xe available resources. It is important to realize here that the fund
has already made q payments of size c0 , and since
= qc0 , there are
no more liquid assets available to make extra payments in period 0.
The second constraint (over payments in period 1) is similar to that
in the previous problem. Let us denote by c0 and c0 the solution to
0
1
problem PP4.12
Setting the appropriate continuation NAV in this case is again a
di¢ cult issue. Note that there are only (1
) units of the asset left at
the fund after the initial q withdrawals. These assets can be liquidated
at a rate of e < 1 and the fund has to still make 1 q payments. In
11

We know that c0 < c1 , c0 = =q, and c1 = R (1
) = (1 q). Then, we have
=q < R (1
) = (1 q) ; which can be rearranged to
+(1
) R > =q = c0 .
12
We do not discuss here whether the fund managers would have the incentives
at this point to redesign payments so as to maximize the remaining investors’ utility.
Perhaps reputational issues could be brought to bear in explaining a behavior of the
fund in line with that suggested by the optimal continuation payments studied here.
that

244

Federal Reserve Bank of Richmond Economic Quarterly

principle, using current values of the assets, the fund would set a NAV
equal to (1
) e=(1 q) and it can be shown that c0 is actually
0
greater than this number. The reason for the discrepancy between
the optimal continuation payment c0 and the NAV computed using
0
current valuations is essentially the same as we discussed before: The
fund does not expect to have to liquidate all assets (as long as q < 1)
e
and, as a consequence, it can still provide some insurance (maturity
transformation) to the agents requesting early redemptions. In the
optimal continuation, the fund’ payments to these agents are such
s
that they receive a portion of the returns coming from the productive
investment that will be held to maturity.
This last extension of the model shows that when the fund holds
a portfolio of investments, some more liquid than others (as it would
want to do, given that it expects some withdrawals to happen early and
some to happen late), the standard methods for computing NAVs again
may fail to deliver the most desirable allocations. In summary, then,
setting appropriate values for NAVs within the maturity transformation
paradigm often involves a tradeo¤ between e¢ ciency and stability. This
is the case in the simplest version of the model and it remains true even
when we consider liquidation costs and a non-trivial portfolio choice
available to the fund.

2.

INVESTMENT MANAGEMENT

In this section, we study a model in which the mutual fund performs
the function of investment management. The underlying justi…cation
is an assumption that the fund can administer the allocation of funds
to productive activities more e¢ ciently than individual investors. For
this reason, then, investors delegate management functions to the fund
by investing directly in it. The model is again very simple. We attempt
to stay as close as possible to the formal analysis of the previous section
but introduce some modi…cations that produce a di¤erent perspective
on the recent experiences with money funds.

The Model
There is a mass 1 of risk averse agents and each of them own one unit of
resources at the beginning of time. Time is again given by t = 0; 1. Different from the model in the previous section, here all agents are patient
(that is, they are indi¤erent between consuming at either time 0 or 1).
There is a risky productive technology that returns a random amount
R of resources in period 1 per unit of resources invested in period 0.
The value of R gets realized after investment in this risky technology

H. M. Ennis: Net Asset Values for Money Market Funds

245

has taken place. However, resources can be removed from the risky
productive technology at any time during period 0 on a one-for-one basis. Agents can also invest in an alternative riskless technology at any
time during period 0 that returns a …x gross return Rz > 1 in period 1
per unit of resources invested in period 0. Call z the amount invested
in this alternative riskless technology.

A Benchmark Optimal Allocation
Since z can be decided after observing the realization of R, it is optimal
to make z a function of R. The optimal allocation of resources solves
the following planning problem:
max E [u (c (R))] ;

c(R);z(R)

(PP5)

subject to
c (R) = R [1

z (R)] + Rz z (R)

and
0

z (R)

1

for all R:

The expectation in the objective function is taken with respect to the
random variable R. The …rst constraint is a resource constraint that
must hold pointwise, for each possible value of R. It says that consumption is equal to the return on the portfolio of investment implied
by z (R). The second constraint re‡
ects natural non-negativity requirements on the amount invested in each of the two technologies.
Let us denote by z (R) the optimal investment strategy implied
by the solution to this problem. We have that z (R) = 1 whenever
R < Rz and z (R) = 0 when R > Rz . If Rz = R; then the value of
z is not pinned down by this problem and it is irrelevant for payo¤s.
Just for concreteness assume that z (Rz ) = 0.

Institutions: An Investment Fund
Since all agents are equally exposed to the underlying uncertainty in
the environment, risk-sharing is no longer a reason for them to pool
resources in a fund. Assume, however, that only the fund has the necessary infrastructure (expertise) to be able to invest in the technology
with random return R. Agents have to decide whether to invest in the
fund before the value of R is realized. Let e be the amount of the initial
resources that each agent decides to keep outside the fund. Hence, the
amount 1 e of resources is invested in the fund.

246

Federal Reserve Bank of Richmond Economic Quarterly

Once the value of R is realized and observed, agents may want to
withdraw some of the resources initially invested in the fund. At that
time, the fund calculates a NAV and allows withdrawals according to
that value. Suppose R can take a …nite number of possible values. We
use the subindex j 2 J to indicate the di¤erent values of R, where J is
a …nite set. Let pj be the probability that R = Rj for each j 2 J and,
of course, j2J pj = 1. Denote by hj and zj the NAV set by the fund
and the amount that an agent withdraws from the fund, respectively,
when R = Rj . Then, the optimization problem faced by an investor is
the following:
X
max
pj u (cj )
(IP)
e;fcj ;zj gj2J

j2J

subject to
cj = Rj (1

e

zj ) + Rz (hj zj + e)

and 0
zj
1 e for all j 2 J, and 0
e
1. Agents initially
invest 1 e at the fund and then withdraw zj after they discover that
returns will be equal to Rj . The shares zj withdrawn from the fund are
valued at a NAV equal to hj and, hence, the total amount withdrawn
equals hj zj . Agents re-invest this amount in the alternative riskless
technology, together with the previously invested amount e. Hence,
total consumption equals the sum of resources obtained from the fund,
Rj (1 e zj ), and from the riskless technology, Rz (hj zj + e).

The Case of a Fixed NAV Equal to One
Since the fund can physically liquidate investment one for one, setting
hj = 1 for all j is feasible. When Rj > Rz for some j 2 J and the fund
sets hj = 1 for all j, agents will be willing to invest all their endowment
0
in the fund at the beginning of time. To see this, de…ne zj = zj + e for
0
0
all j 2 J and note that now we can write cj = Rj 1 zj +Rz zj since
hj = 1 for all j. Given that we still have the constraint zj 1 e as a
requirement, choosing e = 0 relaxes the domain constraints on zj and,
consequently, can only improve the solution to the agent’ problem.
s
In particular, note that when hj = 1 and e = 0 the problem of the
agent is the same as the planning problem for the benchmark optimal
allocation (PP5), but where now z (Rj ) = zj stands for withdrawals
from the fund in state j. Parallel to the solution of problem (PP5),
then, whenever Rj is less than Rz the optimal value of zj equals 1 and
agents withdraw all their investments from the fund. Even though this
event could look like a run on the fund, it is actually part of the process
involved in obtaining an optimal allocation of resources.

H. M. Ennis: Net Asset Values for Money Market Funds

247

This result provides an interesting perspective on some proposals to
reform the regulatory framework for money market funds. Speci…cally,
some reform proposals are designed to provide investors with a disincentive to withdraw from a troubled fund. The objective is to reduce
the incidence of runs. However, we see here that limiting the ability of
investors to reallocate resources at certain points in time could stand
in the way of economic e¢ ciency.
Note that we have considered only the case when investment in
the fund actually constitutes a risky alternative for the agents. It is
often the case, however, that money funds are considered a relatively
safe investment alternative. It would not be hard to modify the model
so that Rz is random and R is a …xed (safe) return. While the results
have a similar ‡
avor, some of the interpretations may not be as natural.
For example, investors would want to withdraw from the fund at those
times when Rz is relatively high. In other words, run-like episodes in
relatively safe funds would tend to be associated with “good times”
(high returns) for investors.

Variable Liquidation Terms
So far, we have studied a situation where the fund can liquidate investment one for one. More generally, suppose that the fund can obtain
resources equal to j per unit liquidated of the risky productive technology, with j 2 J. To simplify the calculations in what follows, assume
that J = fL; Hg with RH > RL and pL = p (so that 1 p is the
probability that R = RH ).
An optimal arrangement in this case produces an allocation that
solves the following problem:
max

e;fcj ;zj gj=L;H

pu (cL ) + (1

p) u (cH )

(PP6)

subject to
cj = Rj (1

e

zj ) + R z

j zj

+e

and 0 zj 1 e for j = L; H, and 0 e 1.
In principle, the liquidation values could be independent of the
observed value of R. When L = H = 1, problem (PP6) is equivalent
to problem (IP) with hj = 1 for all j. Then, when RH > Rz ; it is
optimal to set e equal to zero (recall that e must be chosen before
the realization of R can be observed). More generally, however, when
L = H = for some value of 2 (0; 1) and RL < Rz ; it is possible to
have an optimal value of e that is di¤erent from zero. There are two
cases to consider, depending on whether Rz is greater or less than RL .

248

Federal Reserve Bank of Richmond Economic Quarterly

When Rz < RL ; it is never optimal to liquidate investments in the
funds, and the expressions for consumption are given by:
cL = RL + (Rz RL ) e;
cH = RH (RH Rz ) e:

(NL)

It is clear here that there is a tradeo¤ involved in choosing the optimal
value of e: Investing more in the fund (lower e) increases consumption
when returns are high (when R = RH ) but decreases consumption when
returns are low (when R = RL < Rz ). For some parameter values the
optimal value of e is positive.
When Rz > RL ; it is optimal to liquidate investments when the
realization of R is known to be equal to RL . Given this, the expressions
for consumption are now given by:
cL = Rz + (Rz
cH = RH (RH

Rz ) e;
Rz ) e:

(FL)

Notice the similarities with respect to the previous expressions, (NL).
As a result, it is not hard to see that a similar logic applies and that
for certain parameter values the way to balance the tradeo¤ of returns
is to choose an interior (positive) value of e.
It is important to realize here that, given the information constraints implied by the environment, this situation re‡
ects ex ante ef…cient choices. However, when R = RL ; costly liquidation takes place.
This liquidation may be regarded as a regrettable outcome ex post but
it should be understood that trying to avoid it through regulation could
be detrimental to ex ante welfare.
Even though we do not model explicitly a market for assets we can
use the model, as in the previous section, to help us think about a
situation in which the fund is liquidating assets by selling them (potentially at a discount) in the market. To this end, let us consider
the case in which j is positively correlated with Rj . One particular,
simple version of this correlation is when j = Rj for j = L; H. This
assumption implies that the liquidation value of assets re‡
ects immediately the deterioration in prospective future returns, as one would
expect would happen in a market. We turn to the study of this case
next.
First, it is easy to see that if Rz > 1 then it is always optimal
to set zL = zH = 1 e and liquidate all investments from the fund
immediately after making them. This seems an implausible situation,
mainly due to the stark timing in the model. Hence, we will proceed
here under the assumption that Rz 1.
When Rz < 1 it is optimal to set zL = zH = 0 and the expressions
for consumption are the same as those labeled (NL) above. As before,

H. M. Ennis: Net Asset Values for Money Market Funds

249

then, the choice of e re‡
ects a tradeo¤ between lower returns in good
times and higher returns in bad times.13
Comparing the problem for the optimal arrangement, (PP6), with
the problem of the private investor, (IP), we can see that by setting
hj = Rj for j = L; H the fund would be able to provide the agents
with the optimal contract. Under this arrangement, agents do not
liquidate any of their investments in the fund, regardless of the state of
asset returns. That is, agents choose zL = zH = 0 and the fund never
experiences a wave of withdrawals.
The key to understanding this result is to note that when the return
R is expected to be low, the NAV set by the fund immediately adjusts
to re‡
ect the lower valuation of the fund’ assets. By the time the
s
investors get a chance to withdraw, the losses are already re‡
ected in
the withdrawal values. There is no way in which withdrawing from the
fund can be used by investors as a way to “escape”the expected losses
associated with the low returns from the fund’ assets.
s

Delays in Adjusting the NAV
Suppose, as before, that j = Rj for j = L; H: Now, however, assume
that the fund is not able to immediately adjust the NAV when the news
about the returns of the assets are …rst revealed. As an example, suppose that the fund initially sets an (unconditional) redemption value of
shares h equal to one (before any information about returns have been
revealed) and that the fund is only able to adjust h after q investors
have had an opportunity to withdraw from the fund.14
The payments to the …rst q investors are now given by:
cL = RL (1

e

zL ) + Rz zL + Rz e;

cH = RH (1 e zH ) + Rz zH + Rz e;
and it is optimal for these investors to set zL = 1 e and zH = 0. In
other words, those investors that are able to withdraw from the fund at
a NAV equal to 1 will withdraw all their investments when the return
on the assets is expected to be low and will leave all their investments
in the fund if the return on the assets is expected to be high.
When R = RL , after the …rst q agents have redeemed their shares,
the fund will be able to reset its NAV. At that point, the fund would
13
Under constant relative risk aversion, it is easy to show that the amount invested
in the fund 1 e is increasing in the average return R and decreasing on the (meanpreserving) variance of R.
14
This timing can perhaps be motivated by thinking of a gradual process of di¤usion of information, whereby only some agents …nd out that returns will be low before
the fund is able to (or willing to) adjust redemption values.

250

Federal Reserve Bank of Richmond Economic Quarterly

have already liquidated s = q (1 e) = RL units of the initial (1 e)
investments and the payo¤ to the remaining investors would have to be
recalculated. In particular, if the fund sets a NAV equal to RL , the
payo¤ to these agents from withdrawing from the fund equals
RL

1

e
1

s
q

Rz :

The payo¤ from not withdrawing equals
RL

1

e
1

s
q

:

Given that Rz < 1; these agents will prefer not to withdraw.
This example illustrates how delays in updating the NAV of an
investment fund may create the conditions for an initial rush of withdrawals resembling a run, which only stops after the NAV has been appropriately adjusted. Within the context of this interpretation about
the nature of money funds, ‡oating NAVs that adjust every time an
investor has an opportunity to withdraw could be helpful in reducing
fund instability.
At this point, it is natural to ask why delays in the adjustment
of NAVs would happen. Current regulation allows money funds not
to re‡ in their redemption value deviations from the market value
ect
of their assets as long as they are small (fewer than 50 basis points).
Furthermore, it seems possible that announcing changes in redemption
values that were otherwise expected to be relatively constant would
raise awareness and doubts among investors. If fund managers perceive
a threshold-like e¤ect from making these announcements they would
have incentives to delay them on the hope that new information arrives
and reverts the negative news previously received.

3.

CONCLUSION

Money market funds experienced considerable distress in 2008 during
the U.S. …nancial crisis. Their resiliency was questioned again in 2011
during the European sovereign crisis (see Chernenko and Sunderam
[2012] and Rosengren [2012]). Currently, a generalized concern exists
that the instability of money funds may have systemic consequences
(Financial Stability Oversight Council 2012). For these reasons, there is
a heated ongoing debate about the appropriate reform of the regulatory
framework that applies to these funds.
In this article, we have presented two models that represent, in a
stylized manner, two possible alternative interpretations of the
economic function ful…lled by money funds. In both models, money
funds may experience waves of withdrawals that resemble runs. The

H. M. Ennis: Net Asset Values for Money Market Funds

251

frameworks, however, are not ‡
exible enough to address systemic concerns such as contagion and economy-wide disruptions triggered by the
troubles in the money funds industry. Still, some important insights
about fund stability and regulation arise from the analysis. One of
the main lessons of the article is that the appropriate regulation of
money market funds depends on the stand taken with respect to the
fundamental economic function performed by the funds.
In particular, if money funds are mainly providers of maturity
transformation services, then the setting of the redemption value of
shares needs to take into account the optimal insurance component involved in this kind of arrangement. Extreme versions of ‡
oating net
asset values may undermine this function, just as narrow banking tends
to undermine the maturity transformation function of banks. Perhaps
some instability is inextricably associated with maturity transformation, and trying to completely rule out instability translates into ruling
out any degree of maturity transformation. Under this view, stable
money funds can, in e¤ect, be redundant institutions.
However, in the second model we presented in this article, we took
on the interpretation that money funds are instead investment managers that are able to access, select, and implement bene…cial assetallocation strategies for their investors. Under this view, money funds
do not perform any maturity transformation. We learned that in this
case a timely adjustment of the fund’ redemption value of shares (such
s
as a ‡
oating NAV) may be conducive to stability and is compatible
with the fund’ intended function. To a certain extent, then, alternas
tive reform-proposals involving NAVs indirectly re‡ di¤erent perect
spectives about the main function that money funds perform in the
economy.

APPENDIX
In this appendix we study an arrangement resembling a closed-end
fund in the environment presented in Section 1. We can interpret this
arrangement as a version of the …nancial intermediation system proposed by Jacklin (1987).
Suppose that at the beginning of time, investors form a fund that
issues shares in exchange for investors’ endowment. The fund, then,
invests in a productive technology with return R. The value of each
share is set to equal 1 and each share pays a dividend dt at t = 0; 1:
In other words, each share represents the right to a dividend stream.

252

Federal Reserve Bank of Richmond Economic Quarterly

At time t = 0 investors holding a share receive the dividend d0 and
a market for ex-dividend shares opens. Redemptions of shares are not
allowed at time t = 0 (i.e., it is a closed-end fund).
Clearly, the q impatient agents will want to sell their shares. If the
fund sets d0 = qc0 and d1 = R (1 qc0 ) ; we have that market clearing
in the shares market is given by
(1

q) d0 = vq;

where v is the price of a share and (1 q) d0 is the total amount of
resources in the hands of patient agents that can be used to buy the
q shares of the impatient agents. The equilibrium price is given by
v = (1 q) c0 . Note that, for each share, patient agents pay (1 q) c0
and receive in the following period R (1 qc0 ). Since R (1 qc0 ) =
(1 q) c1 > (1 q) c0 ; patient agents want to buy the shares at the
price v . Patient agents, as a group, then consume d1 since they own
all the shares in period t = 1 and each of them consume
d1
1

q

=

R
1

q

(1

qc0 ) = c1 :

Impatient agents consume d0 + v (the dividend plus the proceeds from
selling the shares) and we have that
d0 + v = qc0 +

(1

q) qc0
= c0 :
q

We see here, then, that a closed-end fund could also implement the
optimal allocation in this environment. In fact, this arrangement would
make the fund immune to runs. The reasons for why funds choose to
be open-end were left unmodeled in this article. See Stein (2005) for a
general discussion of this issue and for a possible explanation.

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Liquidity-Based Theory of Closed-End Funds.” Review of
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253

Chernenko, Sergey, and Adi Sunderam. 2012. “Frictions in Shadow
Banking: Evidence from the Lending Behavior of Money Market
Funds.” Fisher College of Business Working Paper 2012-4
(September).
Diamond, Douglas W. 2007. “Banks and Liquidity Creation: A Simple
Exposition of the Diamond-Dybvig Model.” Federal Reserve Bank
of Richmond Economic Quarterly 93 (Spring): 189–
200.
Diamond, Douglas W., and Philip H. Dybvig. 1983. “Bank Runs,
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91 (June): 401–
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Duygan-Bump, Burcu, Patrick M. Parkinson, Eric S. Rosengren,
Gustavo A. Suarez, and Paul S. Willen. 2013. “How E¤ective Were
the Federal Reserve Emergency Liquidity Facilities? Evidence
from the Asset-Backed Commercial Paper Money Market Mutual
Fund Liquidity Facility.” Journal of Finance 68 (April): 715–
37.
Financial Stability Oversight Council. 2012. “Proposed
Recommendations Regarding Money Market Mutual Fund
Reform.” Washington, D.C.: U.S. Department of the Treasury
(November).
Freixas, Xavier, and Jean-Charles Rochet. 2008. Microeconomics of
Banking. Cambridge, Mass.: The MIT Press.
Jacklin, Charles. 1987. “Demand Deposits, Trading Restrictions, and
Risk Sharing.” In Contractual Arrangements for Intertemporal
Trade, edited by E. Prescott and N. Wallace. Minneapolis:
University of Minnesota Press, 26–
47.
Lee, Charles M. C., Andrei Shleifer, and Richard H. Thaler. 1990.
“Anomalies. Closed-End Mutual Funds.” Journal of Economic
Perspectives 4 (Fall): 153–
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McCabe, Patrick E., Marco Cipriani, Michael Holscher, and Antoine
Martin. 2012. “The Minimum Balance at Risk: A Proposal to
Mitigate the Systemic Risks Posed by Money Market Funds.”
Federal Reserve Bank of New York Sta¤ Report No. 564 (July).
Parlatore Siritto, Cecilia. 2013. “The Regulation of Money Market
Funds: Adding Discipline to the Policy Debate.” Manuscipt, New
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Rosen, Kenneth T., and Larry Katz. 1983. “Money Market Mutual
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Rosengren, Eric S. 2012. “Money Market Mutual Funds and Financial
Stability.” Speech given at Federal Reserve Bank of Atlanta 2012
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U.S. Securities and Exchange Commission. 2010. “Money Market
Fund Reform: Final Rule.” Available at
www.sec.gov/rules/…nal/2010/ic-29132.pdf.
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Economic Quarterly— Volume 98, Number 4— Fourth Quarter 2012— Pages 255–
307

Debt Default and the
Insurance of Labor Income
Risk
Kartik B. Athreya, Xuan S. Tam, and Eric R. Young

R

ecent research (e.g., Chatterjee et al. 2007, Livshits, MacGee,
and Tertilt 2007) has found that allowing for debt default, such
as through the relatively lenient U.S. bankruptcy code, is likely
to improve ex ante welfare relative to more strict forms of debt forgiveness. The welfare gains come from improved consumption insurance
provided by the option to not repay debt in some circumstances. Thus
far, however, all instances where quantitative work …nds a bene…cial
role for default have been ones with large and transitory shocks directly
to household consumption expenditures. It is clear therefore that these
“expense shocks” that lead to involuntary reductions in net worth are
su¢ cient, given the speci…cation of non-expense-related income risk in
current models, to justify debt relief in forms resembling U.S. personal
bankruptcy provisions.
The availability of bankruptcy, and more generally, default, will be
re‡
ected in the pricing on consumer debt, and so will a¤ect households’
ability to smooth consumption across dates and states of nature. It is
therefore important to note that a signi…cant amount of the risk to
lifetime household resources may come from persistent shocks to labor
income (Huggett, Ventura, and Yaron 2010). As a result, to the extent
Athreya is an economist at the Richmond Fed; Tam is a¢ liated with the University
of Cambridge; Young is an economist at the University of Virginia. This article
previously circulated under the title “Are Harsh Punishments for Default Really
Better?” We would like to thank seminar and conference participants at UT-Austin,
the Board of Governors, the Cleveland Fed, Georgetown University, the Philadelphia
Fed, and Queen’ University for comments on the earlier versions. We thank the EQ
s
committee, especially Huberto Ennis, for detailed comments. Tam thanks the John
Olin Foundation for …nancial support. The opinions expressed here do not re‡ect
those of the Federal Reserve System or the Federal Reserve Bank of Richmond. All
errors are the responsibility of the authors. E-mail: kartik.athreya@rich.frb.org.

256

Federal Reserve Bank of Richmond Economic Quarterly

that one might be able to locate other, more targeted, ways of insuring
expense shocks, it is useful to better understand how e¤ective debt
forgiveness is for managing income risk in isolation.
In this article, we evaluate in detail the role of debt forgiveness in
altering the impact of income risk in the absence of expense shocks. The
experiments we present can be thought of as asking: “If we insure the
out-of-pocket expenses that constitute expenditure shocks, is there still
a role of debt relief as a form of insurance against ‘
pure labor income
risk’?” We address this question by studying a range of speci…cations
for households’attitudes toward the intra- and intertemporal properties
of income, when expense shocks are not present. Our main …nding is
that, absent expenditure shocks, the ability to default very generally
hinders the ability of households to protect themselves against labor
income risk.
Despite the nature of our results, we stress that our work is not to
be taken as a strong statement about the overall desirability of U.S. personal bankruptcy law, for two reasons. First, to the extent the expense
shocks are a feature of reality, our model is missing a feature known
to be capable of justifying bankruptcy protection. Second, informal
default or “delinquency” whereby a borrower simply ceases making
payments (and leaves themselves open to legally protected collections
e¤orts) may simply increase if formal bankruptcy is made stricter or
disallowed altogether. Indeed, in ongoing work (Athreya et al. 2013),
we …nd that this channel is quantitatively relevant. These related, and
coexisting, options to avoid debt repayment are not modeled here. Instead, our results apply more narrowly: They suggest that labor income
risk alone may not provide a strong rationale for allowing households to
default. In other words, our …ndings suggest that the scope of shocks
that debt forgiveness is providing insurance against is limited, perhaps
limited principally to relatively catastrophic outcomes.
It is interesting to note that similar results are now being located
in the literature on sovereign debt. Namely, it has proved very di¢ cult
to …nd plausible circumstances in which the bene…ts to being able to
repudiate debts (or perhaps more accurately, the costs of being unable
to commit to repayment of sovereign debt) are positive. The reasons
for the similarity of the results are natural. Most importantly, the
models themselves are largely isomorphic in the optimization problems
they lead to, and do not di¤er substantially enough in their quantitative speci…cation of either preferences or risk. Moreover, even though
sovereign debt models di¤er somewhat in the interpretation of the debt
itself (i.e., that is public debt, not private), the standard assumption
in that literature is that government is benevolent and seeks to borrow
on behalf of households who themselves wish to smooth consumption.

Athreya, Tam, and Young: Debt Default and Labor Income Risk 257
This blurs the distinction between the path of public debt the government chose and that which households would have chosen.
Our results come from comparing allocations arising from two underlying trading environments. First, we study allocations arising from
what we will refer to as the textbook, or “standard model” (SM), of
consumption and saving in which households face uninsurable earnings risks with persistent and transitory components. In this model,
households can only borrow using nondefaultable debt and also face
liquidity constraints. Canonical examples of SM include those laid out
in Deaton (1992, chapter 7) and Carroll (1997). To be consistent with
the view that borrowing limits should be endogenously determined by
repayment incentives, under SM, we investigate primarily the so-called
“natural borrowing limit” case.1
The second trading arrangement we consider is one where, as before, households face life-cycle consumption/savings problems in which
they encounter identical risks as in SM, but can issue defaultable debt.
We will refer to this as the “default model”(DM). Benchmarks in this
literature are Chatterjee et al. (2007) and Livshits, MacGee, and Tertilt
(2007). Following these articles, default in the DM will be represented
as a procedure whereby those with negative net worth can stop paying
obligations, subject to any costs that may be present. The two trading
arrangements we consider are thus clearly di¤erent. Nonetheless, they
are related in a simple way: SM is the limiting case of DM as default
becomes prohibitively costly.
To focus directly on the role of default in insuring labor income
risk relative to the SM, we take two steps. First, as already noted, we
deliberately set aside expenditure shocks. The presence of such shocks
rules out the comparison of models with default against the standard
model as budget sets would be empty for some dates and states were it
not for the possibility of default. Second, we will examine a wider array
of household preferences than has been done in the literature thus far.
Speci…cally, we (i) separate risk aversion from the intertemporal willingness of households to substitute consumption, and (ii) evaluate the
role of ambiguity aversion (or uncertainty aversion) when households
are unsure of the stochastic environment they populate.
Both the separation of risk aversion from intertemporal elasticities and the possibility of ambiguity have been previously identi…ed
with a bene…cial role for debt default. However, neither has been
studied formally. The logic for suspecting that they may be important in delivering a welfare-enhancing role for default is as follows.
1

See, e.g., Ljungqvist and Sargent (2004, p. 577).

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Federal Reserve Bank of Richmond Economic Quarterly

First, the tradeo¤ between intertemporal and intratemporal smoothing was …rst suggested in Livshits, MacGee, and Tertilt (2007) in a
life-cycle model of personal default. Assessing the relative importance
of these motives therefore requires allowing for preferences in which the
two attitudes can be distinct, irrespective of the uncertainty surrounding income. However, prior work has employed constant relative risk
aversion (CRRA) preferences that con‡
ate the two aspects of household preferences. In contrast, we employ Epstein-Zin recursive utility
(Epstein and Zin 1989), which we select because of its tractability and
demonstrated ability to improve the performance of asset pricing models, of which defaultable debt is a special case.
Second, with respect to the role of ambiguity in determining the
value of an option to default, the legal and political history of bankruptcy law suggests that allowing for the release of debtors subject
only to modest penalties is a policy that improves welfare if households
are not perfectly sure of the probabilistic structure of income risk (see
Jackson [2001] for one example).2 This view is not con…ned to legal
experts. As noted as early as Friedman (1957), agents will typically
be unsure about the process that generates their labor income shocks,
instead accepting that a family of potential distributions that may be
di¢ cult to distinguish are possible. Within this class of preferences, an
agent who displays ambiguity aversion (Epstein and Schneider 2003)
will solve a max-min problem— the agent will choose the member of
the class that makes utility lowest and then choose consumption and
savings in order to deliver the highest utility in this worst case.3 It is
precisely this feature of the problem that will allow for a more nuanced
understanding of how penalties can be “excessive”and thereby welfarereducing: Eliminating default through harsh penalties may leave the
agent unwilling to borrow at all. As a result, such a policy could
perversely inhibit both intertemporal and intratemporal consumption
smoothing, despite “mechanically”alleviating the limited commitment
problem that the young and poor face. U.S. bankruptcy law, for instance, appears directly predicated on the idea that penalties can indeed be excessive, in the sense that they may leave would-be borrowers
unwilling to do so (see Jackson [2001]).
The potential role for ambiguity in altering the welfare implications
of having defaultable debt is also suggested by the observation that, in
2
Miao and Wang (2009) study the decision to exercise an option under ambiguity. Due to the presence of …xed costs, bankruptcy has option value. We focus on a
related setting but are interested in the quantitative aspects associated with household
consumption smoothing.
3
These preferences are a special case of the more general ambiguity-averse preferences axiomatized by Klibano¤, Marinacci, and Mukerji (2009).

Athreya, Tam, and Young: Debt Default and Labor Income Risk 259
all extant work on consumer default, the relative gains seen in the
SM relative to DM strongly depend on the “worst case” for household
income. In particular, the large welfare losses in the DM relative to SM
stem from the ability of young agents to borrow out to the natural debt
limit. The natural debt limit is, however, extremely sensitive to small
changes in the value of the worst-possible labor income realization,
particularly for (i) young agents for whom the annuity value of future
labor income is particularly high, and (ii) all agents when the riskfree borrowing rate is low.4 This lower bound is di¢ cult to estimate
accurately (see Deaton [1992] or Pemberton [1998]) and the worst-case
outcomes are the primary focus of ambiguity-averse agents; thus, it
seems important to understand whether the superiority of SM hinges
entirely on the lowest value of income.
Our main …nding along these dimensions is that even in the presence of very high levels of uninsurable labor income risk, high risk
aversion, an unwillingness to substitute intertemporally, and the presence of ambiguity, the ability of households to default on debt leads
to allocations that all households prefer less than the outcome that
arises when they retain full commitment to repay. The intuition for
our welfare results involves the relationship between the current economic situation of the borrower and the price of debt. When shortterm debt is used in a setting with household labor income risk that
is persistent, limited commitment to debt repayment will make credit
expensive anytime the household experiences a negative shock; pricing
“moves against” the unlucky borrower. (In Athreya, Tam, and Young
[2009], we argue that unsecured credit markets are not insurance markets for precisely this reason.) As a result, agents who most “need”debt
to smooth consumption are exactly those that …nd themselves unable
to obtain it, because they also pose the highest risk of default. Tam
(2009) extends this result to longer-term arrangements; speci…cally, he
…nds that competitively priced longer-period debt (in which the pricing
function is held …xed over a number of periods) is welfare-dominated
by one-period debt.
In contrast, the possibility of welfare gains from lowering penalties
by enough to yield default in equilibrium was …rst suggested by Dubey,
4

Denoting by ymin > 0 the lowest realization of potential labor income and r the
risk-free interest rate on debt, the natural borrowing limit for an in…nitely lived agent is
ymin
given by bnat
, a function that asymptotes to 1 as interest rates go to zero.
r
Assuming a credit card interest rate of 14 percent (the modal interest rate in Survey
of Consumer Finances data in 1983 adjusted for a measure of realized in‡ation), the
natural debt limit moves roughly seven times as much as the minimum income level.
For good borrowers, for whom interest rate discounts have recently appeared (Furletti
2003; Livshits, MacGee, and Tertilt 2008), the natural debt limit will be even more
sensitive.

260

Federal Reserve Bank of Richmond Economic Quarterly

Geanakoplos, and Shubik (2005). Theirs was a setting where borrowers of di¤erential default risk were pooled together and thereby did
not pay the individually actuarially fair price for their debt issuance.
As a result of the stylized nature of their two-period model, it is not
suitable for determining whether defaultable debt is welfare-improving
in a more quantitatively oriented model economy. In some quantitative settings where pooling is imposed exogenously, Athreya (2002) and
Mateos-Planas and Seccia (2006) …nd that welfare is higher in SM than
DM. More recently, in a setting where private information allows for
equilibrium pooling, the …ndings of Athreya, Tam, and Young (2009)
suggest again that, as a quantitative matter, short-term defaultable
debt is unlikely to be able to function as a form of insurance. Viewing
these …ndings as a whole, they support the notion that the bene…ts of
slacker borrowing constraints outweigh the costs of having no default
option.
Lastly, with respect to political support for a policy allowing debt
default, in addition to the welfare gains from having defaultable debt
available in the presence of expense shocks, it seems possible that such
provisions would enjoy support even in their absence. One obvious possibility is that the current regime may simply re‡ objectives other
ect
than the maximization of the welfare of newborn agents. We therefore
ask if ex post welfare can account for the evident political support enjoyed by proponents of relatively lax rules on default. Speci…cally, we
ask whether model agents would choose to allow the option to default
on debt in an economy where it was not already present (taking into
account all changes resulting from the policy change). We …nd some
support for such a change, but it falls well short of a majority. Support
for the default option comes from relatively unlucky middle-aged college graduates: These are agents who borrowed a lot when young, in
(rational) anticipation of higher income in middle age. When realized
income did not materialize as expected, such households have signi…cant debt as they approach retirement, and so will bene…t from having
debt obligations removed. Young agents, by contrast, are almost uniformly opposed to allowing defaultable debt, and even less-educated
workers do not generally support it.

1.

MODEL

Households in the model economy live for a maximum of J < 1
periods. We assume that the economy is small and open, so that
the risk-free interest rate is exogenous, while the wage rate is still

Athreya, Tam, and Young: Debt Default and Labor Income Risk 261
determined by a factor price condition.5 As a result, our welfare calculations will be biased toward …nding a positive role for bankruptcy,
since any lost resources arising from the implementation of default procedures like bankruptcy courts and legal costs will be ignored.

Households
Each household of age j has a probability j < 1 of surviving to age
j + 1 and has a pure time discount factor < 1. Households value conc
sumption per household member nj and attach a negative value j;y
j
(in terms of a percentage of consumption) to all nonpecuniary costs of
defaulting, which depend on type y to be de…ned below. Their prefn oJ
cj
erences are represented by a recursive utility function U
nj
j=1

that we detail below. Households retire exogenously at age j < J.
We follow Chatterjee et al. (2007) in allowing for household-level
costs from default that are primarily nonpecuniary in nature. The
existence of nonpecuniary costs of default are also suggested by the
calculations and evidence in Fay, Hurst, and White (1998) and Gross
and Souleles (2002), respectively. The former article shows that a large
measure of households would have “…nancially bene…ted”from debt default via personal bankruptcy but did not …le for protection, while both
articles document signi…cant unexplained variability in the probability
of default across households even after controlling for a large number of
observables. These results suggest the presence of implicit unobserved
collateral that is heterogeneous across households, including (but not
limited to) any “stigma” associated with default along with any other
costs that are not explicitly pecuniary in nature (as in Athreya [2004]).
We will therefore sometimes refer to j;y as stigma in what follows,
although we intend it to be more encompassing.
The household budget constraint during working age is given by
cj + q (bj ; I) bj +

1 (dj = 1)

aj + (1

) W ! j;y ye ;

(1)

where q is an individual-speci…c bond price that depends on bond issuance bj and a vector of individual characteristics I: Net worth after
the current-period default decision is denoted aj , and therefore satis…es aj = bj 1 if the household does not default and aj = 0 otherwise;
is the pecuniary cost of …ling for default. The last term is after-tax
5
In our previous work we introduce a class of “special” agents who hold large
amounts of capital for the purpose of endogenously obtaining a low, risk-free rate in the
presence of low asset holdings for the median agent. Here we ignore the general equilibrium determination of returns and thus drop the special households from the model
because their presence is irrelevant to the question at hand.

262

Federal Reserve Bank of Richmond Economic Quarterly

current labor income ( is the tax rate). Log labor income is the sum of
…ve terms: the aggregate wage index W , a permanent shock y realized
prior to entry into the labor market, a deterministic age term ! j;y , a
persistent shock e that evolves as an AR(1):
log e0 = & log (e) + 0 ;

(2)

and a purely transitory shock log ( ). Both e and log ( ) are independent mean zero normal random variables with variances that are
y-dependent.6 The budget constraint during retirement is
cj + q (bj ; I) bj

aj + 1 (dj = 1) + W ! j

1;y yej

1 j

1+

W; (3)

where, for simplicity, we assume that pension bene…ts are composed of
a fraction 2 (0; 1) of income in the last period of working life plus a
fraction of average income (which is normalized to 1).
The survival probabilities j;y and the deterministic age-income
terms ! j;y di¤er according to the realization of the permanent shock.
We interpret y as di¤erentiating between non-high school, high school,
and college education levels, as in Hubbard, Skinner, and Zeldes (1994),
and the di¤erences in these life-cycle parameters will generate di¤erent
incentives to borrow across types. In particular, college workers will
have higher survival rates and a steeper hump in earnings; the second
is critically important as it generates a strong desire to borrow early in
the life cycle. Less importantly, they also face slightly smaller shocks
than the other two education groups. The life-cycle aspect of our model
is key— in the data, defaults are skewed toward young households (who
borrow at least in part for purely intertemporal reasons), particularly
those who do not report medical expenses as a main contributor to
their default.7
Nonpecuniary costs, , follow a two-state Markov chain with realizations f L;y ; H;y g that are independent across households, but serially dependent with transition matrix
=

1
1

:

Due to data limitations, we assume that the transition probability matrix is symmetric and type-invariant, so the only di¤erence across types
in terms of stigma costs are their realizations. Our parametrization is
more ‡
exible than we used in previous work (Athreya, Tam, and Young
2009, 2012) so that we can match the default rates across education
groups. As we show in a subsequent section, the process is still not
6
We approximate both e and with …nite-state Markov chains. This approximation
has the convenient property that income is bounded.
7
See Sullivan, Warren, and Westbrook (2000).

Athreya, Tam, and Young: Debt Default and Labor Income Risk 263
‡
exible enough to match all the targets of interest, although it does a
reasonable job. Households cannot borrow or save during the period
in which they declare default; however, they face no restriction in any
subsequent period.8

Loan Pricing
We focus throughout on competitive domestic lending. There exists
a competitive market of intermediaries who o¤er one-period debt contracts and utilize available information to o¤er individualized credit
pricing. Let I denote the information set for a lender and b : b I !
[0; 1] denote the function that assigns a probability of default to a loan
of size b given information I; b (b; I) is identically zero for positive levels of net worth and is equal to 1 for some su¢ ciently large debt level.
The break-even pricing function q( ) satis…es
( 1
if b 0
1+r
(4)
qj (b; I) =
(1 b(b;I)) j
if b < 0
1+r+

given b (b; I).
In terms of loan pricing, some remarks are in order. In earlier work,
Athreya (2002) speci…ed an exogenous credit limit and then limited the
sensitivity of loan pricing by forcing all loans to be priced identically.
This approach has the bene…t of plausibly capturing the “optionality”
of the typical unsecured debt contract, whereby households can count
on being able to borrow at a predetermined interest rate up to a predetermined credit limit, i.e., a credit “line.”A second bene…t from this
approach is that it might allow a shortcut to analyzing pooling outcomes that arise from private information on borrower characteristics.
However, there are clear drawbacks to this approach as well. First, for
the counterfactuals we are interested in, we desire a setting in which
both the supply side of the credit market and prices jointly respond
to changes in borrowing and repayment incentives. By contrast, in
Athreya (2002), only prices responded. For large changes in default
incentives, such as what we will examine, this is not a desirable limitation. More recently, Mateos-Planas and Seccia (2006) extended the
approach of Athreya (2002) to allow for changes in credit limits, but
both it and Athreya (2002) in the end employ a framework substantially di¤erent enough to make the comparison to the existing models
described at the outset di¢ cult. Second, from even a purely empirical
perspective, there are reasons to avoid the use of pooling contracts. As
8
That is, exclusion from credit markets beyond the initial period is not sustainable
as a punishment.

264

Federal Reserve Bank of Richmond Economic Quarterly

documented in Livshits et al. (2012), and Athreya, Tam, and Young
(2012), among others, the variation in unsecured credit terms is now
large and appears sensitive to household-level conditions. Lastly, while
not directly observable, it is plausible that while individual credit contracts are best characterized by a single interest rate and credit limit,
the proper interpretation of credit in the model is the sum of all credit
available to the household. In this case, then, the question is the extent to which the household would have to pay more, sooner or later, to
acquire additional credit. Our chosen approach features pricing that responds to default in a manner that yields supply-side e¤ects and makes
the marginal cost of credit an increasing function.
Returning to the model, r is the exogenous risk-free saving rate and
is a transaction cost for lending, so that r + is the risk-free borrowing rate; the pricing function takes into account the automatic default
by those households that die at the end of the period.9 We assume I
contains the entire state vector for the household: I = (a; y; e; ; ; j).
Zero pro…t for the intermediary requires that the probability of default used to price debt must be consistent with that observed in the
stationary equilibrium, implying that
X
0
0
0
b (b; I) =
j d b (a; y; e; ; ; j) ; e0 ; 0 ; 0 :
e e je
0 0 0
e; ;

(5)
Since d b; e0 ; 0 ; 0 is the probability that the agent will default in state
e0 ; 0 ; 0 tomorrow at debt level b, integrating over all such events tomorrow produces the relevant default risk. This expression also makes
clear that knowledge of the persistent component e is critical for predicting default probabilities; the more persistent e is, the more useful
it becomes in assessing default risk.

Government
The only purpose of government in this model is to fund pension payments to retirees. The government budget constraint is
Z
W y! j;y e (a; y; e; ; ; j < j ) =
Z
W ( ! j 1;y yej 1 j 1 + ) (a; y; e; ; ; j j ) :

The left-hand side is the total revenue obtained by levy of a ‡
at
tax rate on all working agents, where the distribution of working
9
We assume any savings of households who die is taxed at 100 percent and used
to fund wasteful government spending.

Athreya, Tam, and Young: Debt Default and Labor Income Risk 265
households (those for whom j < j ) over productivity levels and age is
given by ( ). The right-hand side is the total expenditure on retirees
(those for whom j
j ). Recall that to provide a tractable representation of social security and retirement bene…ts, we assume that
retirement income is composed of a fraction 2 (0; 1) of income in the
last period of working life plus a fraction
of average income (which
is normalized to 1).

Price Determination
We assume that the risk-free rate r is exogenous and determined by
the world market for credit. Given r, pro…t maximization by domestic
production …rms implies that
W = (1

)

r

1

;

where is capital’ share of income in a Cobb-Douglas aggregate pros
duction technology. Our assumption that the risk-free rate is exogenous
deserves discussion. It is certainly reasonable to assume that the U.S.
capital market is open, so empirically it is not implausible. Furthermore, if we close the economy we confront the high concentration of
wealth puzzle directly— the median-wealth agent in the United States
has little or no wealth and thus cares about default policy, since they
may borrow in the future if unlucky, while the mean agent holds substantial wealth and is unlikely to be concerned with the default policy
in place.10 There is a caveat, however. Li and Sarte (2006) is an early
article that establishes a role for general equilibrium feedback e¤ects
that overturn partial equilibrium implications. Though we suspect our
…ndings are robust to the determination of the risk-free rate via general equilibrium restrictions, it is not known for sure whether this is
the case.

Preferences
Here we present the recursive representations of the preferences we
study.
10

Chatterjee et al. (2007) calibrate their model to match the wealth distribution
in the United States in a dynastic setting. As we have argued, life-cycle considerations
are important for assessing the welfare e¤ects of bankruptcy.

266

Federal Reserve Bank of Richmond Economic Quarterly

Constant Relative Risk Aversion

The agent’ problem is standard under CRRA preferences, with the
s
Bellman equation for a household of age j given by

v (a; y; e; ; ; j) =

EU =
0

V b; y; e0 ; 0 ;

0

0

max

b;d(e0 ; 0 ; 0 )2f0;1g

X

0

nj

0

b; y; e0 ; 0 ;
1

d e0 ; 0 ;

cj
nj

+
j;y (EU )

)

e0 je

e0 ; 0 ;

V

j

;j + 1 = 1
d e0 ; 0 ;

e

(

0

0

; j+

v b; y; e0 ; 0 ;

v D 0; y; e0 ; 0 ;

0

0

;j + 1 +

;j + 1 ;

(6)

subject to the budget constraint given in (1) and (3), depending on
their age.
The value function for a household that defaulted in the current
period is given by

EU

=

X

cj
nj

nj

v D (0; y; e; ; ; j) = max

e0 ; 0 ;
0

0

+

j;y

(EU )

e0 je

e
0

j

v

0; y; e0 ; 0 ;
1

0

; j+

: (7)

1
0 is the coe¢ cient of relative risk aversion and also the inverse
of the elasticity of intertemporal substitution. Given our assumptions,
the budget constraints remain the same as for all other agent types,
aside from current net worth being zero as a result of the default.

Athreya, Tam, and Young: Debt Default and Labor Income Risk 267
Epstein-Zin
Under Epstein-Zin preferences, a household of age j solves the dynamic
programming problem
)1
(
c
nj nj +
j
v (a; y; e; ; ; j) =
max
b;d(e0 ; 0 ; 0 )2f0;1g
(EU ) 1
j;y
X
0
EU =
e e je
0 0 0
e; ;
0

V b; y; e0 ; 0 ;
v

b; y; e0 ; 0 ;
1

0

0

;j + 1

; j+

=

0

d e0 ; 0 ;

1

V b; y; e0 ; 0 ;

j

0

;j + 1

0

1

d e0 ; 0 ;

+

0

v D 0; y; e0 ; 0 ;

0

;j + 1

1

;

(8)

subject to the usual budget constraints, and where
cj
nj

D

v (0; y; e; ; ; j) = max nj
EU

=

X

e0 ; 0 ;
0

j

e

0

v

e0 je

1

+

j;y

(EU ) 1

0

0; y; e0 ; 0 ;
1

0

; j+

(9)

is the value of default.
0 governs the household’ aversion to
s
‡
uctuations in utility across states of nature while
1 controls the
substitutability between current and future utility; speci…cally, is the
coe¢ cient of relative risk aversion with respect to gambles over future
consumption and 1 1 is the elasticity of intertemporal substitution in
consumption. When = 1
; these preferences generate the same
ordering over stochastic streams of consumption as expected utility
does.

2.

RESULTS

The results are organized into two subsections. First, we study the
roles played by aversion to ‡
uctuations in consumption over time and
across states-of-nature. We begin with expected utility preferences.
We then relax this by employing Epstein-Zin preferences. Throughout
this subsection, we consider parameter values that lie near the values
implied by the benchmark calibration; these values ensure that model

268

Federal Reserve Bank of Richmond Economic Quarterly

outcomes remain in congruence with cross-sectional facts on consumption and income inequality. We show that welfare under the default
option is lower, at least ex ante. Second, based on this result, we
ask the “inverse” question: Are there economies in which welfare in
the standard model is worse? In this subsection, we no longer restrict
ourselves to parameters dictated by U.S. data; rather, our goal is to understand whether any parameterizations within the parametric classes
we study are capable of generating lax default as a welfare-improving
policy. Speci…cally, we consider shocks with counterfactually large persistent and transitory components and preferences that display ambiguity aversion.
As noted at the outset, our approach throughout will shut down
expense shocks in an otherwise standard consumption smoothing problem. A caveat is in order. While we have argued that this is informative
about a case in which insurance is introduced where it was previously
missing, it should be recognized that this is not necessarily identical to
that case. In particular, the most direct route to addressing the question of whether default would remain useful if society located a way to
insure what are presently uninsurable expenses is to explicitly model
such an option. We opt for a simpler approach here in part because the
form of such insurance, were it to become available, is not obvious a
priori. This is primarily because it is unlikely to be provided privately,
given that it has not emerged to this point. As a result, the form it
takes will likely be as part of a tax-transfer scheme. Our model lacks
the detail needed to address the associated incentive-related costs. Our
approach is therefore similar to the thought-experiment of Lucas (1987)
in which the costless removal of risks was employed as a benchmark for
the gains from business cycle stabilization. Still, the reader should keep
in mind the indirect nature of our approach and the limits it places on
the interpretation of our results. In particular, our approach leads us
to calibrate more than once, sometimes with only partial success, depending on the case under study, as opposed to calibrating once at
the outset. We acknowledge this limitation and leave the alternate
approach for future work.

Does Default Help Insure Labor
Income Risk?
In this subsection, we evaluate the implications of default relative to the
standard model for a variety of empirically plausible values for agent
attitudes toward intra- and intertemporal consumption smoothing. Before evaluating these alternatives, we present our argument for why
default regimes must be a matter of policy rather than an endogenous

Athreya, Tam, and Young: Debt Default and Labor Income Risk 269
outcome of decentralized trading arrangements. The most prevalent
form of explicitly unsecured credit is that arising from the open-ended
revolving debt plan o¤ered by credit card lenders. Credit card lending,
in turn, has been (certainly since the mid-1990s) extremely competitive.11 The relevance of the competitiveness of the U.S. unsecured
lending industry is that the credit market cannot be punitive in its
treatment of those who default. That is, no single …rm would be willing
to treat an individual borrower any worse than the current assessment
of their state would justify. As a result, a household contemplating
default in such a setting can safely rule out being “punished”for it. In
the case where default conveys no additional information to a lender
than what it was able to observe ex ante, there is literally no change
in terms that are “caused” by the act of reneging on a payment obligation. Conversely, when default does reveal information, the change in
terms is again not “punitive”in nature, but instead re‡
ects an updated
assessment of default risk. As a result, “high” ex post interest rates
following default are implausibly ascribed to deadweight loss-inducing
penalties. In the symmetric-information and competitive setting we
study, punishments that are ex post ine¢ cient will not be sustainable. Even if any single lender could withhold credit after default, the
presence of other lenders would undermine the possibility of anything
purely punitive. As a result, default costs capable of sustaining unsecured credit markets are likely to require intervention by policymaking
authorities.12 Thus, in the market for unsecured consumer debt, it is
likely that any costs of default …ling that are in any way punitive have
to be policies.13
At the outset, we noted that for plausible parameterizations of preferences that admit an expected utility representation, the
11
The average interest rate on credit card balances is high— currently 14 percent—
relative to more secured forms of debt. As Evans and Schmalensee (2005) have pointed
out, however, it is straightforward to account for the interest rate after funding costs,
transactions costs, and, most crucially, default costs are taken into account, without
relying on market power distortions.
12
Most dynamic contracting models of limited borrower commitment, for example,
currently use implicit or explicit appeals to public institutions with commitment to punish, in order to motivate penalties for the value of autarky. In recent work, Krueger and
Uhlig (2006) show that the inability of the supply side of the credit market to commit
to punishments can have severe implications for the existence of the market itself. In
the “normal” case, Krueger and Uhlig (2006) show that competition in fact collapses
credit and insurance markets completely even without informational frictions.
13
We want to be clear that what we call “penalties” di¤ers from the usage in
Ausubel and Dawsey (2008), where rates imposed after late or missed payments are labeled punitive. They attribute the high values of such rates to a common agency problem. Modeling the bilateral contracting problem that would arise in the presence of
noncompetitive intermediation is well beyond the goals for this article. We are pursuing
the endogenous determination of interest rate hikes for delinquent borrowers in other
work.

270

Federal Reserve Bank of Richmond Economic Quarterly

standard model typically maximizes welfare. Our …rst step is to understand whether this argument against default obtains only because of
the restriction to expected utility or is a more fundamental property of
models of life-cycle consumption smoothing. To collapse the model to
the standard model, the speci…c quantitative experiment we consider is
the imposition of a cost of default
that is large enough to eliminate
all default on the equilibrium path.14 Before proceeding, we note the
following property of our model.
Proposition 1 For each (a; y; e; ; j) there exists
b (b) = 0.

large enough that

This result relies on the nonnegativity condition for consumption—
if
exceeds the labor income of the household in the current period,
default cannot occur since consumption would have to be negative.
Given that total labor income is bounded (by assumption) and borrowing is proscribed in the period of default, we can always impose a
cost of …ling su¢ cient to generate zero default along the equilibrium
path. We then compute the change in lifetime utility for each individual given a
that exceeds the maximum required; in the absence
of general equilibrium e¤ects, we can compute these changes for each
individual, rather than simply for newborns, without the need to track
transitional dynamics. We will focus in general on ex ante welfare of
newborns.
Calibration
We consider a benchmark case of expected utility, where = 1
=
1. We choose ( ; L;y ; H;y ; ) to match the default rates of each type
y, the measure of negative net worth as a fraction of gross domestic
product for each type y, the fraction of borrowers, and the discharge
ratio (mean debt removed via default divided by mean income at time
of …ling). Table 1 contains the constellation of parameters that …ts best
(when viewed as exactly identi…ed generalized method of moments with
an identity weighting matrix). Other parameters are identical to those
in Athreya, Tam, and Young (2009)— these include the resource cost
of default , the income processes faced by each type, the measure of
each type, and the parameters of the retirement system ( ; ).15
14

Similar results would obtain if the government could impose “shame” on households by choosing values for , provided it could make large enough to guarantee zero
default on the equilibrium path. In our model, the Inada condition on consumption implies that such a
always exists.
15
Speci…cally, we set = 0:35,
= 0:2, = 0:03,
= 0:03, & = 0:95, 2 = 0:033,
n;
2
= 0:04, 2 = 0:025, 2 = 0:021, 2 = 0:016, and 2 = 0:014.
n;
c;
c;
h;
h;

Case
Parameter,Target
h
nhs ; nhs = 1:03%
l
b
nhs ; E( y jb < 0)nhs = 0:1552
h
hs ; hs = 1:11%
l
b
hs ; E( y jb < 0)hs = 0:5801
h
coll ; coll = 0:57%
l
b
coll ; E( y jb < 0)coll = 0:7251
; Pr(b < 0) = 12:5%
E(bjd=1)
; E(yjd=1) = 0:56

= 1;
Parameter
0.8972
0.7624
0.8832
0.7135
0.7067
0.5698
0.9765
0.8597

=2
Outcome
0.31%
0.2071
0.97%
0.1835
0.63%
0.1504
17.5%
0.3986

= 0:5; = 2
Parameter
Outcome
0.8668
1.24%
0.6929
0.2104
0.8064
1.29%
0.6933
0.1825
0.7136
0.79%
0.6352
0.1506
0.9895
13.3%
0.6655
0.4073

= 1;
Parameter
0.9376
0.7538
0.8872
0.6236
0.7055
0.4205
0.9532
0.7658

=5
Outcome
0.51%
0.1561
1.31%
0.2553
0.76%
0.2194
12.5%
0.4630

Athreya, Tam, and Young: Debt Default and Labor Income Risk 271

Table 1 Calibration

272

Federal Reserve Bank of Richmond Economic Quarterly

Our model is not capable of exactly matching the entire set of
moments— for example, we underpredict default rates and discharge,
generally underpredict debt-to-income ratios, and overpredict the measure of borrowers. This inability arises because the model actually
places very tight links between some variables, restricting the minimization routine’ ability to independently vary them.16 In the end,
s
one either accepts that expense shocks do indeed play a very dominant
role in default data, or one is left with a puzzle relative to standard
consumption-savings models. Still, we note that the qualitative …ndings from our analysis do not appear to depend on our speci…cation of
the stochastic process for .17
Expected Utility and Ex Ante Welfare
We consider two environments— one with the calibrated value for and
one with a cost su¢ cient to eliminate default on the equilibrium path.
Table 2 contains the welfare gain from the standard model in which it
is infeasible for any household to declare default. Consistent with our
previous work, we …nd that welfare is higher in the standard model ex
ante for every newborn (independent of type). College types bene…t
the most from the change, and their welfare gain is substantial (1:2
percent of lifetime consumption). To aid the discussion in subsequent
sections where we alter preference parameters, we quickly summarize
the reasons for the welfare gains here.
In the standard model, the loss of resources generated by the …ling
cost is not present. Since we do not impose an economy-wide resource
constraint, these lost resources are not important. Instead, the welfare
gain is driven by an improved allocation of consumption. By the law
of total variance, the variance of consumption over the life cycle can be
decomposed into two components:
V ar (log (c)) = V ar (E [log (c) jage]) + E [V ar (log (c) jage)] :
16
Consider an attempt to improve the model’ prediction for the measure of bors
rowers by increasing . Holding all other parameters constant, this reduces default rates
and debt-to-income ratios for all types (and these variables are generally already too
small). To counteract this e¤ect, one might then move
for each type and each state.
Consider …rst increasing both H and L for one type i. While this change would ini
i
crease the default rate— default becomes less costly— it would via a supply-side e¤ect
tend to reduce debt levels (see Athreya [2004]). By contrast, suppose we increase H
i
and decrease L ; this change has countervailing e¤ects on both default rates and debt
i
levels and default rates could rise because it becomes cheaper for H types, but fall
as it becomes more expensive for L types. A similar tension exists for debt-to-income
ratios— driving it up for one type tends to drive it down for the other.
17
In the real world, “stigma” may also be a function of aggregate default rates (an
agent cares less about default if everyone else is defaulting), in which case this invariance
may break. To analyze this case would be of interest, but it poses some challenges with
respect to calibration. We therefore defer it to future work.

Athreya, Tam, and Young: Debt Default and Labor Income Risk 273

Table 2 Welfare Gains (without Recalibration)
= 2 & EIS = 0:5
DM ! SM
= 2 & EIS = 0:67
DM ! SM
= 5 & EIS = 0:5
DM ! SM

Coll
1:21%
Coll
0:58%
Coll
0:47%

HS
0:54%
HS
0:21%
HS
0:16%

NHS
0:52%
NHS
0:13%
NHS
0:13%

We label the …rst term the “intertemporal”component of consumption smoothing; it represents how expected consumption di¤ers across
time periods. The second term is the “intratemporal” component; it
measures how much consumption varies across agents of a given age.
Roughly speaking, how costly the …rst component is in terms of welfare depends on the elasticity of intertemporal substitution, because it
measures the deterministic variance of consumption over time, whereas
the welfare cost of the second part is governed by static risk aversion.
In Figure 1 we see that the standard model, or “no-default”case (SM),
improves intertemporal smoothing (the curve gets ‡
atter) because all
lending becomes risk-free. Thus, as we noted in the introduction, the
only debt limit that is relevant is the natural debt limit, which is very
large in our model for newborn agents. Turning to the intratemporal component, in Figure 2 we see that the SM improves this as well,
restating the analysis in Athreya, Tam, and Young (2009) that unsecured credit markets do not provide insurance. Here, bad shocks trigger
tightening of credit constraints, making consumption smoothing across
states of nature more di¢ cult. As a result, young agents are unable to
respond e¤ectively to bad income realizations when they can default,
causing their consumption to be highly volatile. Under the SM, the
natural debt limit is su¢ cient to protect them against adverse shocks;
by middle age, default has ceased to be relevant and thus the two cases
largely coincide.18
The di¤erences in outcomes across the DM and SM cases are given
in Figures 3, 4, and 5, and are driven by changes in the pricing functions
agents face. In Figure 3 we show the pricing functions in the low costs
of default environment facing a young college agent across realizations
of the persistent shock e. The initial ‡ segment is driven by and is
at
increasing in the current realization of the persistent shock e. As debt
18

The …gures are drawn for the aggregate, since the results are the same for each
type qualitatively. Figures decomposed by type are available from the authors upon
request.

274

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Intertemporal Consumption Smoothing, Expected
Utility

increases, more realizations of e0 would trigger default, causing q to
decline until it reaches zero; looking across e values we see that higher
e realizations permit more borrowing. Of course, higher e realizations
in our model are typically associated with less, not more, borrowing, so
these increased debt limits are not particularly valuable; instead, the
tightening of credit limits when e is low generates substantial costs for
poor agents. In contrast, under SM pricing is ‡ out to the natural
at
debt limit. Crucially, transitory shocks do not impact pricing; because
0 cannot be predicted using , the current transitory shock has no
e¤ect on the default decision tomorrow conditional on b (b is changed
by the transitory shock, however).
The potential tradeo¤ between the two components of smoothing
motivated the life-cycle analysis of Livshits, MacGee, and Tertilt (2007)
and Athreya (2008), so why doesn’ default generate this tradeo¤? As
t
discussed in Athreya, Tam, and Young (2009), default can either help
or hinder intratemporal smoothing, depending on which agent you ask.
An agent facing an income process with low intertemporal variance

Athreya, Tam, and Young: Debt Default and Labor Income Risk 275

Figure 2 Intratemporal Consumption Smoothing, Expected
Utility

but high intratemporal variance— that is, tomorrow’ expected income
s
is close to current income but tomorrow’ income has substantial risk—
s
may bene…t from default; the intertemporal distortion is minimal while
the potential to truncate the consumption distribution at the low end
conveys signi…cant bene…ts (even once pricing is taken into account).
In contrast, an agent facing the opposite process— income that grows
over time and is relatively safe— generally does not bene…t; default is
not used because pricing prevents it and the intertemporal distortion is
substantial, leading to signi…cant welfare losses. In our model, a young
agent is of the second type, especially a college-educated one, while
older households are members of the …rst type.
Ex Post Welfare— Voting over Default Policy
Because we study a small open-economy model in which the risk-free
rate is …xed, but also allow all pricing to be individualized, there are

276

Federal Reserve Bank of Richmond Economic Quarterly

Figure 3 Pricing, Expected Utility

no “pecuniary” externalities. We can therefore compute the welfare
consequences of policy changes for any agent at any point in the state
space; since the distribution plays no role in pricing (and therefore no
role in welfare), we do not need to calculate the transitional dynamics
of the model to get the welfare changes. We ask agents of a given age
and type whether, conditional on their current state, they would be
in favor of eliminating the option to declare default. Figure 6 displays
the measure of each type, conditional on age, that would support retaining default with the calibrated . A substantial portion of college
types oppose elimination, but they are all middle-aged and have experienced histories of bad shocks; the peak in opposition occurs earlier for
high-school types and later for non-high-school types, with correspondingly fewer such households opposing overall. For the convenience of
the reader, Table 3 presents the aggregate measures of each type that
oppose eliminating default (the column labeled “DM Regime” they
);
are small for each education group. Furthermore, as is clear from the
…gures, almost no newborns oppose eliminating the option.

Athreya, Tam, and Young: Debt Default and Labor Income Risk 277

Figure 4 Pricing, Expected Utility

We now consider the inverse of the preceding experiment: Agents
of di¤erent ages and types are asked if they would prefer to introduce
default (again, with
set to its calibrated value) into a setting in
which it is currently prohibited. As seen in Figure 7, a nontrivial fraction of agents would like to introduce default. The intuition here is
that the no-default case allows signi…cant borrowing at the risk-free
rate. As a result, many households, especially the college-educated,
borrow when young in anticipation of higher earnings. The relatively
unlucky among them then …nd themselves indebted by middle age and
thereby will bene…t from the discharge of debts. Moreover, by virtue of
being middle-aged, these households place relatively low value on being able to access the cheap unsecured debt later in life. This e¤ect is
especially strong for the college-educated, for whom purely intertemporal consumption smoothing motives dictate a strong e¤ort to save for
retirement beyond middle age. As a result, a substantial proportion
of high-school- and college-educated household groups would support
the introduction of default when they reach middle age. In contrast,

278

Federal Reserve Bank of Richmond Economic Quarterly

Figure 5 Pricing, Expected Utility

those who have not completed high school support the introduction
of default only late in working life, when the subsequent increase in
borrowing costs is not long-lasting. However, as Table 3 shows (the
column “SM Regime” the aggregate number of agents who vote in
),
favor of introduction falls well short of majority status.
Separating Risk Aversion from
Intertemporal Substitution
As discussed above, the two pieces of the variance decomposition have
welfare costs that depend (mainly) on di¤erent aspects of preferences.
Our benchmark case using CRRA expected utility restricted these two
aspects of preferences to be reciprocals of each other. Here, we relax
that requirement by using the Epstein-Zin preference structure, and
consider two particular deviations. First, we make households more
tolerant of intertemporal variance than in the expected utility benchmark by employing a high value for . Second, the default option

Athreya, Tam, and Young: Debt Default and Labor Income Risk 279

Figure 6 Fraction Supporting Bankruptcy, BK Regime

may shrink the volatility of intratemporal consumption, at least for
some ages. Given this, making intratemporal variance more painful
to households may help us explain the presence of low default costs.
We therefore select a relatively high value for . It is important to
note that this particular combination of insensitivity to the timing of
consumption and sensitivity to the income state in which it occurs is
the arrangement that gives default its best chance of improving ex ante
welfare and does not lie within the class of expected utility preferences.
The speci…c experiments we investigate involve changing and
without recalibrating the entire model. This type of change generates
two e¤ects— an e¤ect conditional on borrowing (which we call the price
e¤ect) and an e¤ect caused by changes in the number of borrowers (the
extensive e¤ect). We then compare the results with cases where the
model is recalibrated (to the extent that is possible) in an attempt to
isolate the two e¤ects.
We …rst consider changes in . To understand how this change affects welfare, it is helpful to …rst consider the extreme case of = 1,

280

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 Measure of Agents in Favor of Bankruptcy
Education
College
High School
Non-High School
Total

DM Regime
6:45%
4:05%
0:16%
4:05%

SM Regime
4:09%
3:26%
0:24%
2:98%

making the household in…nitely willing to move consumption deterministically through time. As ! 1, the Bellman equation converges to
the form
P
0
( 0)
e0 ; 0 ; 0 e (e je)
v (a; y; e; ; ; j) = max cj +
0
j;y
0; 0; 0; j + 1
j V b; y; e
Here, the household will either completely frontload or backload
consumption, depending on the relationship between the discount factor and the interest rate. For the parametrization we use, the e¤ective discount factor ( times the survival probability) lies between the
risk-free saving and borrowing rates for almost every age, meaning that
households don’ wish to borrow and, critically, do not value the default
t
option no matter how risk averse they are. For some older households,
whose survival probabilities are relatively low, the e¤ective discount
factor is su¢ ciently low that they want to borrow and “frontload”their
consumption; the option to default makes borrowing expensive enough
to render complete frontloading impossible. This, in turn, reduces the
welfare of these households— since they face no uncertainty, default is
either probability zero or one and pricing therefore eliminates it. Obviously such extreme consumption behavior is inconsistent with U.S.
cross-sectional facts; in particular, the model with = 1 would miss
very badly on the life-cycle pattern of consumption inequality, which
in the data is substantially smaller than income inequality.
Returning to less extreme values for , Figure 8 displays the pricing
function across several di¤erent values of and demonstrates the e¤ect
on loan prices. As increases, the pricing function shifts downward
because at any given level of debt an agent with a higher is more
willing to default. The intuition for this result is not straightforward.
When increases, the household is more willing to accept variability
in consumption across time. If a household enters the current period
with some debt and wishes not to lower debt, they have two options:
(i) borrow more if possible or (ii) default and void those obligations.
Borrowing more is only feasible if there is a reasonable commitment to
repay. But since a bad shock would lead to low mean consumption,

1
1

!

:

Athreya, Tam, and Young: Debt Default and Labor Income Risk 281

Figure 7 Fraction Supporting Bankruptcy, NBK Regime

default becomes attractive, and households lack strong commitment to
repay debt. As a result, they cannot borrow easily. For the cases with
“intermediate” values for , the creation of strong default incentives
makes intertemporal smoothing more costly, but the latter is relatively
unimportant.
Consider next an experiment where , the risk aversion with respect
to gambles over future utility, is increased. Again, turning …rst to the
polar case, let ! 1, so that the household becomes in…nitely risk
averse. In this case, the limiting household Bellman equation takes the
form
)1
(
cj
nj nj +
;
v (a; y; e; ; ; j) = max
0 0 0
j;y mine0 ; 0 ; 0 V b; y; e ; ; ; j + 1
subject to the usual budget constraints seen earlier in equations (1)
and (3).
When households are in…nitely risk averse, they choose not to borrow for the reasons outlined in Athreya, Tam, and Young (2009)—

282

Federal Reserve Bank of Richmond Economic Quarterly

Figure 8 Pricing, Epstein-Zin with Di erent EIS

unsecured credit markets do not provide insurance and thus agents
will be unwilling to pay the transaction cost to borrow. As a result,
there is a welfare gain to living in the standard model, as no household
has negative net worth. Again, extreme preferences render the model
grossly inconsistent with cross-sectional facts; here, consumption inequality would be essentially zero over all ages.
Returning again to more intermediate cases, we see that changes
in risk aversion generates two e¤ects. The extensive margin e¤ect is
similar to increasing , but for di¤erent reasons. When
is large,
households have a strong demand for precautionary savings; for = 5,
for example, we see a clear decline in the measure of total borrowers,
again making default overall less damaging. The pricing e¤ect is also
similar; by increasing risk aversion, we make the household less willing
to have consumption di¤er across states of the world tomorrow. Conditional on borrowing, the pricing functions reveal a stronger desire
to default— for any given b, the price of debt is decreasing in (see
Figure 9). As above, there are only two options for a household with

Athreya, Tam, and Young: Debt Default and Labor Income Risk 283

Figure 9 Pricing, Epstein-Zin with Di erent Risk Aversion

debt; since even a moderately bad outcome will cause a highly riskaverse agent to default, commitment is not possible, leaving default as
the only option for smoothing consumption across states.19 Combining
these results into one statement, we see that no combination of ( ; )
leads to default being a welfare-improving policy, although for extreme
cases it will be nearly innocuous.
Table 2 shows that welfare is higher (for newborns) in the standard
model (SM), but that the gains from (imposing the high ) decline
with risk aversion and elasticity of intertemporal substitution (EIS).
>1
, which is satis…ed when either parameter increases, implies
the household has a preference for early resolution of uncertainty; thus,
default appears to be least damaging when households prefer to resolve
their risk early rather than late.
19

Our model satis…es the conditions noted in Chatterjee et al. (2007) that imply
default occurs only if current debt cannot be rolled over: If d ( 0 ; 0 ; 0 ) > 0 for some
0 ; 0 ; 0 ; then there does not exist b such that a + y
q (b; Y ) b > 0 for total income Y .

284

Federal Reserve Bank of Richmond Economic Quarterly

Table 4 Welfare Gains (with Recalibration)
= 2 & EIS = 0:5
DM ! SM
= 2 & EIS = 0:67
DM ! SM
= 5 & EIS = 0:5
DM ! SM

College
1:21%
College
0:28%
College
1:28%

High School
0:54%
High School
0:05%
High School
0:57%

Non-High School
0:52%
Non-High School
0:04%
Non-High School
0:56%

All of these results are obtained without recalibrating the model. To
ensure that our …ndings are not particularly sensitive to this strategy,
we also recalibrate the model for di¤erent values of and , to the
extent that this recalibration is possible; Table 1 contains the new
parameter values that best …t the targets under alternative settings.
By doing so, we attempt to shut o¤ the extensive margin, although we
are not completely successful. When we recalibrate, we …nd that with
high EIS all welfare gains from eliminating default are substantially
reduced, with both noncollege types now barely bene…ting at all (see
Table 4), while for high risk aversion the welfare gains increase slightly.
As noted above, this welfare change is entirely due to the shifts in the
pricing function that higher EIS and/or higher risk aversion engender.
Thus, for no parameter combination that we consider do we observe
welfare gains from retaining the default option.
A summary of …ndings thus far is that default signi…cantly worsens
allocations for income risk and preference parameters that are empirically plausible for U.S. data, as well as for more extreme values of preference parameters within the class of Epstein-Zin non-expected utility
preferences. We turn now to the question of whether such policies continue to remain desirable under two additional (and more substantial)
departures from the settings studied so far.

Is the Standard Model Ever Worse?
We begin this section by allowing for the underlying volatility of income
to be driven by relatively more and less persistent income shocks. For
this experiment, we hold the unconditional variance of labor income
…xed and vary the relative contributions of the persistent component
e and the transitory component . We then ask whether a relaxation
in the household’ understanding of the probabilistic structure of earns
ings risk can open the door for welfare-improving default. For this

Athreya, Tam, and Young: Debt Default and Labor Income Risk 285
experiment, we allow for households to display ambiguity aversion in
the sense of Klibano¤, Marinacci, and Mukerji (2009).20
The Roles of Persistent and Transitory
Income Risk
It has long been known that self-insurance, and therefore also the bene…t of insurance markets, hinges critically on the persistence of the
risks facing households. As a general rule, the more persistent are
shocks, the more di¢ cult they are to deal with via the accumulation
of assets in good times and decumulation and borrowing in bad times.
In contrast, purely transitory income shocks can typically be smoothed
e¤ectively. In a pure life-cycle model, however, there are additional impediments to self-insurance: Young households are born with no wealth
and often face incentives to borrow arising from purely intertemporal
considerations. In particular, those with relatively high levels of human capital, especially the college-educated, can expect age-earnings
pro…les with a signi…cant upward slope into late middle age. As a result, such households would like to borrow even in the absence of any
shocks to income, often substantially, against their growing expected
future income. In contrast, those households with low human capital
face a far less income-rich future, and as a result borrow primarily to
deal with transitory income risk.
In order to understand the role that the persistence of income risk
plays in the welfare gains or losses arising from U.S.-style bankruptcy
and delinquency, we now evaluate the e¤ects of changes in the persistent
component of household income risk for all three classes of households.
However, in order to avoid con‡
ating persistence and overall income
volatility, we adjust the variance of transitory income volatility such
that the overall variance of log labor income remains constant.21 Figure
10 and Tables 5 and 6 present the welfare and consumption smoothing
implications of the standard model under varying income shock persistence. The …rst column of each table documents the fraction of total
variance contributed by the persistent component.
Normatively, three …ndings are noteworthy. First, and perhaps
most importantly, the standard model displays higher welfare irrespective of the nature of shocks accounting for observed income
20

There are connections between ambiguity aversion and the concept of Knightian
uncertainty from Bewley (2002), although the latter concept does not permit preferences
to be represented by a utility function and is therefore hard to analyze quantitatively.
There are also connections between ambiguity aversion and robust decisionmaking as
de…ned by Hansen and Sargent (2007).
21
Athreya, Tam, and Young (2009) are primarily concerned with the role of income
variance in models of default.

286

Federal Reserve Bank of Richmond Economic Quarterly

Figure 10 Welfare Gains from Eliminating Bankruptcy

volatility. This result strengthens our …ndings thus far, and it further
suggests that defaultable debt is simply unlikely to be useful to households. It is also a particularly important form of robustness, given both
the general importance of persistence for the e¢ cacy of self-insurance
and borrowing and because estimates of income shock persistence vary
dramatically— see Guvenen (2007), Hryshko (2008), or Guvenen and
Smith (2009) for discussions of the debate between so-called “restricted
income pro…les” (RIP), in which all households draw earnings from a
single stochastic process, and “heterogeneous income pro…les” (HIP),
in which households vary in the processes from which they derive earnings. This debate has implications for models like ours because these
two models di¤er, sometimes strongly, in the persistence of earnings
shocks their structure implies. Most recent work now suggests that
income-process parameters vary over the life cycle as well (Karahan
and Ozkan 2009).

1:0%
10:0%
20:0%
30:0%
40:0%
50:0%
60:0%
70:0%
80:0%
90:0%
99:0%

Coll
0:0306
0:0377
0:0459
0:0538
0:0619
0:0700
0:0779
0:0859
0:0946
0:1053
0:1248

Intra
HS
0:0462
0:0561
0:0807
0:0884
0:1013
0:1146
0:1280
0:1413
0:1543
0:1681
0:1863

NHS
0:0575
0:0872
0:1092
0:1367
0:1472
0:1613
0:1797
0:1992
0:2182
0:2368
0:2566

Coll
0:0359
0:0343
0:0336
0:0327
0:0316
0:0305
0:0294
0:0283
0:0272
0:0258
0:0235

Inter
HS
0:0364
0:0367
0:0347
0:0327
0:0301
0:0284
0:0264
0:0247
0:0231
0:0212
0:0187

NHS
0:0386
0:0357
0:0325
0:0297
0:0280
0:0263
0:0241
0:0224
0:0211
0:0199
0:0180

Coll
0:0665
0:0720
0:0795
0:0865
0:0925
0:1005
0:1065
0:1141
0:1218
0:1311
0:1483

Total
HS
0:0826
0:0938
0:1154
0:1211
0:1314
0:1430
0:1544
0:1660
0:1774
0:1893
0:2050

NHS
0:0961
01229
0:1417
0:1664
0:1752
0:1876
0:2038
0:2216
0:2393
0:2567
0:2680

Athreya, Tam, and Young: Debt Default and Labor Income Risk 287

Table 5 Consumption Smoothing (DM)

288

1:0%
10:0%
20:0%
30:0%
40:0%
50:0%
60:0%
70:0%
80:0%
90:0%
99:0%

Coll
0:0196
0:0271
0:0360
0:0444
0:0524
0:0600
0:0673
0:0743
0:0811
0:0878
0:0935

Intra
HS
0:0307
0:0397
0:0541
0:0683
0:0820
0:0951
0:1076
0:1197
0:1314
0:1428
0:1528

NHS
0:0474
0:0577
0:0771
0:0971
0:1173
0:1364
0:1550
0:1729
0:1903
0:2072
0:2218

Coll
0:0318
0:0315
0:0311
0:0306
0:0300
0:0295
0:0291
0:0288
0:0285
0:0282
0:0280

Inter
HS
0:0314
0:0298
0:0290
0:0284
0:0277
0:0271
0:0267
0:0262
0:0258
0:0255
0:0253

NHS
0:0120
0:0124
0:0131
0:0137
0:0144
0:0151
0:0158
0:0164
0:0171
0:0178
0:0182

Coll
0:0514
0:0586
0:0671
0:0750
0:0824
0:0895
0:0964
0:1031
0:1096
0:1160
0:1215

Total
HS
0:0621
0:0695
0:0831
0:0967
0:1097
0:1222
0:1343
0:1495
0:1627
0:1638
0:1781

NHS
0:0594
0:0801
0:0902
0:1108
0:1317
0:1515
0:1708
0:1893
0:2075
0:2250
0:2400

Federal Reserve Bank of Richmond Economic Quarterly

Table 6 Consumption Smoothing (SM)

Athreya, Tam, and Young: Debt Default and Labor Income Risk 289
Second, the e¤ect of the contribution of persistent shocks to income
volatility depends on the education level of households. In particular,
when volatility is driven primarily by persistent shocks, the relatively
well-educated bene…t from the elimination of default substantially more
than their less-educated counterparts. Conversely, when most income
variability is driven by large but transitory shocks, it is the relatively
less-educated who bene…t most from the elimination of the default option. The intuition for this result comes from the nature of borrowing:
College types borrow primarily to use future expected income today
while less-educated types borrow to smooth shocks.
Third, within each educational class, the welfare losses from default
decline monotonically as the relative contribution of the persistence of
the shock grows; default on debt is least (most) useful when income
volatility is driven primarily by shocks that are transitory (persistent).
What is surprising, but in keeping with the main theme of our results, is
that in no case is it true that U.S.-style default is ex ante more desirable
than allocations obtaining under the standard model. Moreover, even
in the case where essentially all income risk is delivered in the form
of persistent shocks where credit markets are least useful in dealing
with income risk, outcomes that allow for default are worse for agents
than those arising in the standard model. The welfare in the standard
model is non-trivially higher, at up to 1:24 percent of consumption for
college-educated households (as seen in Figure 10).
In Figures 11 and 12 we display the measure of borrowers at each
age and the conditional mean of debt among those who borrow for two
levels of the importance of persistent income risk.22 The fact that the
losses from allowing default rise for all agent types with the importance
of transitory shocks is a consequence of the increased usefulness of
credit in dealing with transitory income risk. Conversely, when shocks
are primarily persistent, a negative realization requires more frequent
borrowing and leads, all else equal, to more debt in middle age; the
combination is ultimately unable to stem the transfer of income risk to
consumption volatility. In Tables 5 and 6, we see that, irrespective of
default policy, persistence translates into higher consumption volatility,
and that the presence of lax default policy seen in Table 6 does little
to stem the ‡ of income risk into consumption risk (echoing our
ow
previous result in Athreya, Tam, and Young [2009]).
We turn next to the relationship between shock persistence and
equilibrium default rates, displayed in Figure 13. Default is “U-shaped,”
with high default rates at both ends. To understand this shape, con22
From the perspective of a newborn, the measure of borrowers of a given age
equals the probability of the newborn borrowing at that age.

290

Federal Reserve Bank of Richmond Economic Quarterly

Figure 11 Fraction of Borrowers

sider …rst the case where the labor income shocks are nearly all transitory (the left side of the graph). Here, agents can generally manage
their risk e¤ectively via saving and dissaving, but they choose to augment the self-insurance mechanism with default at higher rates than
they do in the benchmark setting. The reason they do so is that riskbased pricing is not e¤ective here, because there is no useful information contained in the current labor income of the borrower that would
identify future bad risks. In contrast, in the case where labor income
is driven entirely by the persistent component (the right side of the
graph), high default is the result of agents being generally unable to
smooth consumption; persistent shocks are hard to smooth using assets
alone (and if permanent are in fact impossible). As a result, despite
the pricing e¤ects, borrowers will use default relatively often (and pay
the costs to do so). The middle parts of the graph, where default is
lowest, balance these two e¤ects.
Intuitively, in the standard model, borrowers realize that debt must
be repaid, and under high persistence, heavy borrowing in response

Athreya, Tam, and Young: Debt Default and Labor Income Risk 291

Figure 12 Mean Debt of Borrowers

to a negative shock makes low future consumption relatively likely.
Nonetheless, credit markets are willing to lend to such households at
the risk-free rate (adjusted for any transactions costs of intermediation), making total debt rise. When default is available, borrowing
today to deal with persistent income risk does not expose the borrower to severe consumption risk in the long term as default o¤ers an
“escape valve,” but it does expose lenders to severe credit risk in the
near term. Creditors then price debt accordingly; as seen in Figure 14,
when shocks are primarily persistent, as the current shock deteriorates
so do the terms at which borrowers can access credit. Moreover, under
a bad current realization of income, households facing persistent risk
see a disproportionate decline in the price of any debt they may issue,
while the reverse occurs in the event of a good current realization of income; the pricing functions essentially “switch places.”Yet, despite the
increased sensitivity of loan pricing to the borrower’ current income
s
state under relatively high persistence, the welfare gains under the SM,
though still positive, fall. This result obtains because of the reduction

292

Federal Reserve Bank of Richmond Economic Quarterly

Figure 13 Default Rates

in the ability of self-insurance, inclusive of borrowing, to prevent income ‡
uctuations from a¤ecting consumption. To sum up, income risk
is quantitatively relevant in governing the losses conferred by default,
but irrelevant for altering the qualitative welfare property that, in the
absence of expense shocks, the default option lowers welfare.
Ambiguity Aversion
We turn next to the question of whether default can improve outcomes when households are not perfectly certain about the probabilistic structure of income risk. Households that face ambiguity are
uncertain about the probability process for their incomes; if ambiguityaverse, these households behave pessimistically and therefore adopt
views about their income that would, for example, imply that it would
mean-revert more slowly from low realizations. In such a situation, borrowing to smooth away temporary falls may not be optimal, since asset

Athreya, Tam, and Young: Debt Default and Labor Income Risk 293

Figure 14 Pricing Functions

decumulation is not e¤ective against permanent shocks, and therefore
in the absence of a default option households may be unwilling to do
so. In contrast, if default is an option, the household may be willing
to borrow since, even if their pessimism is validated, consumption can
be protected via discharge. We formalize this idea, as in Klibano¤,
Marinacci, and Mukerji (2009), by assuming agents are averse to ambiguity. In this formulation, a household of age j solves the dynamic

294

Federal Reserve Bank of Richmond Economic Quarterly

programming problem

v (a; y; e; ; ; j) =

EU =

X

e0 ; 0 ; 0
0 0 0

V b; y; e ; ;

e

8
>
<

max
0

b;d(e0 ; 0 ; )2f0;1g >
:

e0 je

0

nj
1

j;y

0

P

e0 ;

1

cj
nj
0

+

p (e0 ; 0 je;

)

(EU )

j

;j + 1 ;

9
>
=
>
;
(10)

subject to budget constraints, (1) and (3), where ( ) is given as follows:
(
1 exp( x)
if > 0
1 exp( )
(x) =
x
if = 0
determines preferences over ambiguity.
0 controls the attitude
toward ambiguity; as increases, the household becomes more averse
to ambiguous stochastic processes. The restrictions on the choices of
p (e0 ; 0 je; ) are that they must sum to 1 for each (e; ) and every element must lie in some set P
[0; 1]; we nest the standard model by
setting the P to be an arbitrarily small interval around the objective
probabilities.23 We use to denote objective probabilities and p to denote subjective ones; note that households are assumed to be uncertain
only about the distribution of income shocks, not the process for .
Because we are interested in these preferences only to the extent
that they may provide an environment in which relatively low-cost default and debt discharge are welfare-enhancing, we will deliberately take
the most extreme case of = 1, yielding the max-min speci…cation
from Epstein and Schneider (2003):
)
(
cj 1
nj
+
1
nj
v (a; y; e; ; ; j) =
max
b;d(e0 ; 0 ; 0 )2f0;1g
minp(e0 ; 0 je; ) EU
j;y
X
EU =
p e0 ; 0 je;
0 0 0
e; ;
0

j

V b; y; e0 ; 0 ;

0

;j + 1

=

1

V b; y; e0 ; 0 ;

d e0 ; 0 ;

d e0 ; 0 ;

0

0

0

;j + 1

v b; y; e0 ; 0 ;

v D 0; y; e0 ; 0 ;

0

0

;j + 1 +

;j + 1 ;

(11)

23
We do not require that the household assume that the probabilities of the independent events are independent in every distribution that is considered. That is, the
household may be concerned that the independence property is misspeci…ed and therefore select a worst-case distribution in which the events are correlated.

Athreya, Tam, and Young: Debt Default and Labor Income Risk 295
where
v D (0; y; e; ; ; j + 1) = max
EU

=

X

(

e0 ; 0 ;
0

j

nj
1
j;y
0

cj
nj

minp(e0 ;

1
0 je;

+
) EU

)

p e0 ; 0 je;
v 0; y; e0 ; 0 ;

0

;j + 1

(12)

is the value of default.
The min operator that appears in front of the summation re‡
ects
the agent’ aversion to uncertainty; as shown by Epstein and Schneider
s
(2003), a household who is in…nitely uncertainty-averse chooses the
subjective distribution of future events that is least favorable and then
makes their decisions based on that subjective distribution. The size
of the set of possible processes P measures the amount ofi ambiguity
h
[0; 1].24
agents face; a typical pij element lies in the interval pij ; pij
2
1
Standard ambiguity aversion models imply that households will
learn over time and reject stochastic processes that are inconsistent
with observed data (for example, a household who initially entertains
the possibility of permanently receiving the worst possible income level
forever will dismiss this process as soon as one non-worst realization
occurs). For simplicity, we will focus our attention on a special case
of extreme ambiguity aversion in which this learning does not occur; if
default is not useful in this environment, it is likely of less use to households than when they face less uncertainty over time. The intuition is
that the income process we bu¤et agents with is a non-unit process.
To the extent that households would realize by a certain age that the
data they’ received makes unit-root earnings unlikely, they would be
ve
able to rule out such a persistent process and thereby smooth more
e¤ectively, and as a result, may not value default as much as someone
viewing shocks as permanent.
Given the quali…cations and considerations discussed above, we now
evaluate outcomes in the standard model in the case where P = [0; 1],
the most extreme case possible (households behave as if the minimum
income draw will be realized with probability 1 next period). The
intuition is that such a case o¤ers the possibility, discussed at the outset, that lax penalties for default might actually encourage the use of
credit for consumption in a setting where the agent’ aversion to ams
biguity would otherwise preclude becoming indebted. And in fact, we
do …nd that this case delivers default as welfare-improving for some
24
Hansen and Sargent (2007) provide an interpretation of P in terms of detection
probabilities.

296

Federal Reserve Bank of Richmond Economic Quarterly

Table 7 Welfare E ects Under Ambiguity Aversion
P = [0; 1]
DM ! SM
P = min(1; + 0:5)
DM ! SM

Non-High School
0:215%
Non-High School
0:296%

High School
0:189%
High School
0:219%

College
0:185%
College
0:044%

agents (see Table 7). However, this …nding is very limited: Benchmark
default costs improve welfare for only the college type and the welfare
gain is tiny (under 0:2 percent of consumption). As a result, unconditional ex ante welfare is negative since college types are not a large
enough group to overcome the losses to the remainder of the population. It is interesting to see, however, that the welfare changes from
allowing default are now reversed— the largest gains are experienced
by the most educated, while the least educated su¤er more. Part of
the intuition for this result is that it is the best educated who face
the steepest mean age-earnings pro…les. Therefore, these agents would
have the strongest purely intertemporal motives to borrow, absent any
ambiguity. Low default costs mitigate the e¤ect of ambiguity and allow
for states in which a temporarily unlucky college-educated agent would
…nd borrowing desirable.
Pricing is presented in Figures 15 and 16. Notice that for the low
realization of e, the pricing function under ambiguity aversion is everywhere below the baseline expected utility case, but for the higher realization they switch places; ambiguity-averse agents with high income
actually pose less of a default risk. The di¤erence in pricing stems only
from a di¤erence in the households’willingness to default next period
for a given b. Since default has a …xed cost component ( ), households
want to time their usage of default; in particular, households must
balance the gains from defaulting tomorrow from those arising from
waiting until additional shocks have been realized. This fact places
the expectations of income in periods after tomorrow at the heart of
the timing of default decisions, and here households who face ambiguity about the income process act quite di¤erently from those in the
benchmark economy.25
25
The exposition is simpler if we refer to the expectations of the households facing
ambiguity as coinciding with the choice of p, because the ambiguity-averse agents act as
if those probabilities were the objective ones. Of course, if one were to ask ambiguityaverse agents about their forecasts of future income, they would use the true objective
probabilities; they just do not use these probabilities for decisions. The proper phrasing of our statement “ambiguity-averse agents expect low future income” would be the
more cumbersome “ambiguity-averse agents act as if they expect low future income.” We

Athreya, Tam, and Young: Debt Default and Labor Income Risk 297

Figure 15 Pricing, Ambiguity Aversion, Low e

Take …rst the household with low e. For a “rational expectations”
household, income in the distant future is expected to be better than
whatever is realized tomorrow, as e is persistent but mean-reverting; for
the household facing ambiguity, however, income is actually expected
to be no better, or even worse, than tomorrow’ realization. Since
s
ambiguity-averse households do not think the future will be better,
they may as well default next period if the realization of income is
bad; lenders must therefore o¤er them higher interest rates to break
even. In contrast, the ambiguity-averse household with higher e views
a realization near the mean for next period as unexpectedly good, but
does not expect better times in the more-distant future. Default in the
next period is therefore not as valuable as waiting for a future period
when those bad states are expected to occur. In contrast, without
ambiguity a bad realization will induce the household to substantially

abuse the notion of expectation slightly as a result, and beg for the reader’ indulgence
s
on this matter.

298

Federal Reserve Bank of Richmond Economic Quarterly

Figure 16 Pricing, Ambiguity Aversion, High e

revise their future expectations downward, making default today more
attractive (the decline in future income makes the …xed cost of default
worth paying).26 The result is that ambiguity-averse households with
high current income obtain better terms.
Is such extreme ambiguity aversion “reasonable?” It seems highly
unlikely that households entertain a stochastic process in which they
receive the worst possible outcome forever with probability one as reasonable, at least not for long— after all, they need only observe the
fact that their income is occasionally higher than the lower bound
to discard this process empirically. As we noted above, we could introduce this learning into the model— since the households are simply learning about an exogenous process, it can be done “o- ine”
—
but it is computationally quite burdensome to condition the set of
26
The median e has the pricing functions crossing, so that agents who face ambiguity are more likely to default on small debts but less likely to default on large ones.

Athreya, Tam, and Young: Debt Default and Labor Income Risk 299
permissive stochastic processes on the history of observations.27 It is
also the case that this extreme ambiguity leads to a discrepancy between model and data in terms of borrowing patterns; there is far too
little debt, which lessens our interest in making this economy “more
realistic.”If we consider smaller limits for P, such as 10 percent above
or below the objective value, we …nd that default is welfare-reducing
for all education levels. Thus, while ambiguity aversion provides a theoretical foundation for default options, it does not appear to provide
an empirically tenable one.

3.

CONCLUDING REMARKS

We have studied the e¢ cacy of default in helping households better insure labor income risk in a large range of settings in which risk aversion,
intertemporal smoothing motives, income risk, and uncertainty— and
attitudes to uncertainty— over income risk itself were all varied. Our
…ndings here suggest that within the broad class of models used thus
far to develop quantitative theory for unsecured consumer credit and
default, relatively generous U.S.-style default does not appear to be
capable of providing protection against labor income risk.
Despite the fact that we …nd that labor income risk is not well
hedged from the ex ante perspective, we also show that there are ex
post bene…ciaries from allowing default as it currently is; speci…cally, we
show that the standard model generates a positive measure of agents
ex post who would vote to introduce default. Our calibrated model
predicts that these agents do not constitute a majority, though, since
they are primarily college-educated middle-aged households who have
been unlucky enough to still have signi…cant debt. This result warrants
further investigation since it may help explain why default penalties are
becoming less stringent over time (with the exception of some aspects
of the most recent reform).
Our results also suggest that “expense” shocks or catastrophic
movements in net worth are likely to be essential to justify the view
of default as a welfare-improving social institution. To the extent that
uninsured, catastrophically large, and “involuntary” expenditures are
indeed a feature of the data, a natural question is whether consumer
default is the best way to deal with such events. Given the nature of resource transfers created by default and the constraints that it imposes
27
Since this learning is not Bayesian, it can be quite di¢ cult to write recursively,
and, in any case, learning about discrete processes generally involves a large number of
states. Campanale (2008) investigates non-Bayesian learning in a two-state model where
the approach taken introduces only one additional state.

300

Federal Reserve Bank of Richmond Economic Quarterly

on the young, who disproportionately account for both the income-poor
and uninsured, this statement seems unlikely.
With respect to future work, it is worth stressing that since expense shocks and their absence seem so important to the implications
of the class of models considered here, the value of purely empirical
work better documenting the nature of expense shocks, and their (a
priori plausible) connection to income shocks (for example, job loss
leading to insurance loss, which in turn exposes households to out of
pocket expenditures), is high. Relatedly, the pivotal role played by
borrowing costs “moving against” unlucky borrowers seems important
to independently substantiate. In the absence of such work, it remains
a possibility that the welfare …ndings of this article (and essentially all
others) hinges too much on an institutional arrangement for borrowing
that is inaccurate. Use of detailed household level credit card pricing
and income information seems productive.
In addition to the preceding, in light of the …ndings of this article
and the larger quantitative theory of consumer default, two directions
seem particularly useful. First, a more “normative”approach that asks
if observed default procedures can arise an optimal arrangement under
plausible frictions, may yield di¤erent conclusions. One interesting
example of the latter approach is the theoretical work of Grochulski
(2010), where default is shown to be one method for decentralizing
a constrained Pareto optimum in the presence of private information.
Quantifying models with default and endogenously derived asset market structures may lead to better understanding of policy choices in
this area (such as why Europe has chosen to make default available
under very strict conditions, and social insurance generous, while the
United States has chosen the opposite).
Second, with respect to the experiments we studied, we were led
to allow for two speci…c preference extensions beyond CRRA expected
utility in order to accurately assess the particular tradeo¤s created by
default. While we emphatically did not attempt to turn the article
into a survey of any larger variety of non-expected utility preferences,
some further extensions seem potentially important: disappointment
aversion (Gul [1991] or Routledge and Zin [2010]), deviations from geometric discounting (Laibson 1997), habit formation (Constantinides
1990), and loss aversion (Barberis, Huang, and Santos 2001). Why
these preferences speci…cally? In each case, the more general preference structure breaks the link between risk aversion and intertemporal
substitution (and generally makes risk aversion state-dependent), and
some (such as nongeometric discounting and loss aversion) provide arguments for government intervention; there is also extensive empirical
work supporting many of them. A recent contribution to this literature

Athreya, Tam, and Young: Debt Default and Labor Income Risk 301
is Nakajima (2012), who investigates whether the temptation preferences of Gul and Pesendorfer (2001) alter the consequences of default
reform.28 We suspect other work will follow.

APPENDIX:

COMPUTATIONAL CONSIDERATIONS

We make some brief points here regarding the computation of the
model. The model is burdensome to calibrate, and all programs are
implemented using Fortran95 with OpenMP messaging.
In all the models we study, the objective function (the right-hand
side of the Bellman equation) is not globally concave, since the discrete
nature of the bankruptcy decision introduces convex segments around
the point where the default option is exercised (we …nd that, as in
Chatterjee et al. [2007], the default decision encompasses an interval
and in our case it extends to b = 1 as
is smaller than even the
worst income realization). The nonconcavity poses a problem for local
optimization routines, so we approach it using a global strategy. We
use linear splines to extend the value function to the real line and a
golden section search to …nd the optimum, with some adjustments to
guarantee that we bracket the global solution rather than the local
one. It is straightforward to detect whether we have converged to the
local maximum at any point in the state space, as the resulting price
function will typically have an upward jump.
For the ambiguity aversion case we have a saddlepoint problem to
solve. By the saddlepoint theorem we can do the maximization and
minimization in any order; the minimization (conditional on b and d)
is a linear program that we solve using a standard simplex method conditional on some b (as in Routledge and Zin [2009]). We then nest this
minimization within our golden section search, again with adjustments
to deal with the presence of the local maximum. For our model, this
linear program turns out to be extremely simple to solve— the household puts as much weight as allowed on the worst possible outcome,
then as much weight as allowed on the next worst, and so on.
To impose boundedness on the realizations of income, we approximate both e and by Markov chains using the approach in Flodén
(2008). Having income be bounded above is convenient since it implies
28
Nakajima (2009) …nds that increasing borrowing constraints in a model with
quasi-geometric discounting is not always welfare-improving, similar to Obiols-Homs
(2011).

302

Federal Reserve Bank of Richmond Economic Quarterly

Figure 17 Optimal Choice of b given q

that there always exists a cost of default
such that bankruptcy is
completely eliminated because it becomes infeasible. Quite naturally,
bankruptcy is also likely not to occur when
is high enough even if
…ling is feasible for some types; in general, households with high income
are not interested in the default option in our model.29
Figure 17 shows a typical objective function for a household in our
1 = 2). The objective
benchmark case (expected utility with =
function has three distinct segments. The …rst segment is at the far
right, where the values for both the low- and high-cost types coincide.
In this region, default is suboptimal because borrowing either does not
or barely exceeds . The second segment is at the other end, where
q (b) = 0; although impossible to see in the picture, the low-cost de29
Households with high income realizations do not want to pay the stigma cost
(which is proportionally higher for them) even if they are currently carrying a large
amount of debt (which is very rare due to persistence). Thus, our model does not predict
any “strategic” default, which can arise in models that rely on exclusion as a punishment
for bankruptcy.

Athreya, Tam, and Young: Debt Default and Labor Income Risk 303
fault experiences slightly more utility in this region since default is
less painful. The action is all in the middle segment. For this particular individual, the high-cost type ( L ) borrows signi…cantly more
than the low-cost type; this extra borrowing re‡
ects primarily the pricing function (as seen in the lower panel) and not any particular desire
to borrow. High-cost types have more implicit collateral and are less
likely to default at any given debt level, so they face lower interest
rates. As a result, high-type borrowers today who become low-type
borrowers tomorrow are a main source of default in our model— they
both have debts and are not particularly averse to disposing of those
debts through the legal system. Since type is persistent, low-type borrowers today will not generally make the same choice— the supply side
of their credit market will contract.

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Economic Quarterly— Volume 98, Number 4— Fourth Quarter 2012— Pages 309–
348

Regulation and the
Composition of CEO Pay
Arantxa Jarque and Brian Gaines

I

t is well known that the use of stock options for compensating
executives in large U.S. companies was widespread during the last
15 years. But were all …rms using them with equal intensity? We
are interested in the answer to this question because option grants are
di¤erent from other compensation instruments in the type of incentives
they provide, how transparent they are to investors, and the level of
insider trading that they allow. In this article, we provide an empirical
examination of the trends in the last two decades of the use of di¤erent
compensation instruments, mainly focusing on restricted stock grants
and option grants. We …nd that there have been important changes,
and that they coincide in time with two changes in regulation: the
modi…cations to reporting requirements for option grants introduced by
the passage of the Sarbanes-Oxley Act in 2002, and the 2006 adoption of
revised accounting standards from the Financial Accounting Standards
Board (FASB) included in statement no. 123R (FAS 123R), which
mandated the expensing of option grants.
Today, companies pay their top executives through some or all of
the following instruments: a salary, a bonus program, stock grants
(usually with restrictions on the ability to sell them), grants of options
on the stock of the …rm, and perks and long-term incentive plans that
specify retirement and severance payments, as well as pension plans
and deferred bene…ts. The most accepted explanation for the inclusion
of compensation instruments that are contingent on the performance of
the …rm is the existence of a moral hazard problem: The separation of
ownership and control of the …rm implies the need to provide incentives
The views expressed here do not necessarily re‡ect those of the
Reserve Bank of Richmond or the Federal Reserve System.
arantxa.jarque@rich.frb.org.

Federal
E-mail:

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Federal Reserve Bank of Richmond Economic Quarterly

to the chief executive o¢ cer (CEO) that align his interests with those
of the …rm owners.
In the presence of moral hazard, the optimal contract prescribes
that the pay of the executive should vary with the results of the …rm.
However, in spite of the need for incentives, limited funds on the part
of CEOs or risk aversion considerations imply that exposing the CEO
to the same risk as shareholders is typically either an unfeasible or
an ine¢ cient arrangement. The optimal contract should balance incentives and insurance. Some part of the compensation should not
be subject to risk, like the annual salary, providing some insurance to
the CEO against bad performance of the …rm over which he does not
have control. However, some part of the compensation should be variable and tied to some measure of performance of the …rm. The main
variable pay instruments can be classi…ed in three categories. First,
bonus plans, which make annual pay dependent on yearly accounting
results. Second, grants of stock of the …rm (often referred to as “restricted stock,”since the executive cannot sell them for some time after
they are granted, typically about three or four years); these make pay
in the longer term dependent on the results of the …rm over a longer
time horizon. Third, grants of stock options, which allow the executive
to purchase stock of the …rm at a pre-established price (the “exercise
price” and also typically are granted with restrictions as to how soon
)
they can be exercised; these also provide incentives for longer-term performance, but they only pay o¤ for the executive if the stock price of
the …rm is above the exercise price.1
These di¤erent compensation instruments di¤er in how transparent they make compensation to shareholders or outside investors. For
example, bonus schemes that are based on both objective and subjective performance targets may be more di¢ cult for an outside investor
to evaluate than a plain restricted stock grant. These instruments also
di¤er on how robust they are to insider trading and other opportunistic
behavior; the exercise of stock options or the sales of vested stock can
potentially be timed by the CEO to the disclosure of particularly good
or bad news on the prospects of the …rm, for example, and bonuses
may be sensitive to creative accounting practices where some annual
results are made to look better by using the degree of freedom present
in accounting standards, or by fraudulent misrepresentation of …nancial
results.
1
When options are granted with the exercise price equal to the stock market price
at the date of grants, they are called “at the money;” this is the most popular practice,
although some options are occasionally also granted “in the money” (with exercise price
below market price) or “out of the money” (with exercise price above market price).

A. Jarque and B. Gaines: CEO Pay

311

Another important factor is that various compensation instruments
are treated di¤erently for taxation purposes and are subject to di¤erent
disclosure requirements and accounting standards. As an example of
heterogeneity in tax treatment, non-quali…ed option grants, which have
been the most popular type of option grant in the last two decades,
trigger a tax deduction for the company when they are exercised by an
employee; salaries or any other compensation that is not performancebased (like plain restricted stock awards) in excess of one million dollars,
instead, do not qualify for a deduction.2
As an example of heterogeneity in disclosure requirements, compensation that is given to executives in the form of perks does not need to
be detailed in the compensation disclosure tables of proxy statements
if its value is less than $10,000; when the value exceeds that sum, the
disclosure is only in a footnote. Salary, bonus, stock, and option grants
are disclosed in the mandatory compensation table instead.3
These di¤erences have historical origins, and are likely subject to
political pressures. One cannot ignore, however, the distortion that the
tax, disclosure, or accounting treatment may potentially have on the
choice of instruments, and through that— as we just argued— on the
e¢ ciency of incentives and the transparency of compensation practices
to shareholders. Hence, in this article we ask the following questions:
Are …rms in certain industries or of larger size more likely to use option
grants? Are …rms that use options more likely to pay higher compensation to their CEOs? Have restricted stock grants replaced option
grants after expensing and reporting rules were changed in 2002 with
Sarbanes-Oxley, and then again in 2006 with the adoption of Statement
of Financial Accounting Standards 123R (SFAS 123R) accounting standards? Has the relative importance of salary and bonuses decreased in
recent years in favor of option or stock grants?

Regulatory Changes
In this article we consider two major changes in the U.S. regulation of
compensation practices. The …rst change is the Sarbanes-Oxley Act
of 2002, which aimed to improve corporate governance after several
earnings management scandals surfaced in the early 2000s. The second
change is a revision of accounting rules introduced by the SFAS 123R in
2006, which for the …rst time mandated a positive expense for options
2

This di¤erential tax treatment was introduced in 1993, IRC section 162(m). See
Hall and Liebman (2000) for an analysis of the taxation of executive compensation.
See Meyers (2012) for a recent explanation of requirements on stock awards that are
considered performance-based and qualify for a tax deduction.
3
See Securities and Exchange Commission (2006).

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Federal Reserve Bank of Richmond Economic Quarterly

awarded “at the money”(with exercise price equal to the stock market
price at the date of the grant).
We use available data on executive compensation from 1993 to 2010
to evaluate the e¤ect of these two regulation changes in the choice of
compensation instruments of large public U.S. …rms. Note that the
Dodd-Frank Act, which was motivated by the …nancial crises of 2008,
was passed in 2010 and it a¤ected …nancial …rms only. It would certainly be interesting to know how the increased scrutiny of incentive
schemes that the Act mandates (both for executives and lower level employees) is a¤ecting pay practices at large …nancial institutions. However, we do not have enough observations in our sample to deal with
that regulatory change in this article.
The Sarbanes-Oxley Act, as part of its e¤ort to improve transparency to shareholders, decreased the time window allowed for the
disclosure of insider trades to two business days.4 Before the Act,
…rms had until the end of the …scal year to report any type of insider
trading, including the grants of options to employees of the …rm. As
it became apparent after some investigations, a number of …rms were
able to exploit the lax reporting requirements to engage in “backdating,” the (illegal) practice of arti…cially changing the grant date of
options to the day with the lowest stock price in the time window allowed for reporting.5;6 The bene…ts of this practice were twofold, and
hinged on both the accounting standards and the tax treatment of options. First, it allowed the …rm to report higher earnings. At the time,
accounting standards under SFAS 123 allowed …rms to expense grant
options according to their “intrinsic value,” which is zero for options
granted at the money. Instead, the intrinsic value of an option in the
money (which is what was being e¤ectively granted without the backdating) would have been positive, and hence a compensation expense
would have been deducted from the …rm’ income, resulting in lower res
ported earnings. Second, it allowed a larger tax deduction for the …rm
at the time that the option was exercised. Under Internal Revenue
Code (IRC) section 162(m), …rms can deduct from their tax liability
any compensation costs that originate in incentive pay. In contrast,
4
“Ownership Reports and Trading by O¢ cers, Directors and Principal Security
Holders,” Release No. 34-46421 (Aug. 27, 2002) [56 FR 56461] at Section II.B.
5
Investigations pointing to the existence of backdating became well-known only in
2005. Since Sarbanes-Oxley was passed in 2002, it may be the case that the change in
reporting requirements was not directly aimed at preventing backdating. In other words,
in trying to improve corporate governance in general, the Act inadvertedly limited the
possibility of backdating.
6
For a discussion of the issues and anecdotal evidence, see the Wall Street Journal article “The Perfect Payday” (March 18, 2006). For an academic evaluation of the
backdating practice, see Heron and Lie (2007) and references therein.

A. Jarque and B. Gaines: CEO Pay

313

there is a limit of one million dollars for deducting compensation that
is not tied to incentives. Hence (provided the employee was already
receiving one million dollars in non-incentive compensation), the tax
deduction would have been lower for an option in the money, since the
di¤erence between the stock price at the time of grant and the exercise price would not have been considered incentive pay.7 Backdating
options without proper disclosure, then, implied both misreporting to
investors the amount of incentive pay given to employees, and engaging in fraudulent accounting to save on taxes.8 The Act, by decreasing
the time window allowed to report the granting of options to two business days after the trade takes place, constrained the …rms’ability to
misreport the actual date of the grant, and hence made options a less
attractive compensation instrument for …rms that were backdating, or
for those that were considering the possibility of doing it at some point.
The second piece of regulation that we consider is SFAS 123R, a
revision to accounting standards SFAS 123, which was adopted by the
Security and Exchange Commission (SEC) in 2006. The main change
introduced by the revision was a homogenized method of valuation of
options to “fair value” calculations, such as Black and Scholes. Previously, the “intrinsic valuation” method was allowed, which attributed
a zero value to options granted at the money. Because option grants
are accounted for as expenses in the income statement of the …rm,
this change in valuation method e¤ectively eliminated the possibility
of not charging any expense of compensation for options granted at the
money.9 The general view on this piece of regulation is that, after its
adoption, companies were no longer able to “hide” the dent of option
grants on their accounting pro…t. This view is supported by the numerous complaints by large U.S. corporations when the measure was
…rst proposed, arguing that lower earnings per share would hurt, for
example, their ability to borrow and grow, hindering innovation and
job creation. However, under the disclosure requirements in SFAS 123
before 2006, …rms were already required to report (in a footnote in
7
This tax treatment applies to “non-quali…ed” option grants, which are the most
common in executive compensation packages during the time period that we study.
Firms are also allowed to grant “quali…ed” options, or “incentive stock options,” to
their employees, which are limited to a maximum value of $100,000, and hence are not
usually granted to executives. See Bickley (2012) for details on the taxation of employee
stock grants.
8
Because backdating implies a violation of the SEC’ disclosure rules, a violation of
s
accounting rules, and a violation of tax laws, the SEC has sued a number of companies
suspected to have engaged in this practice. See, for example, the testimony of Christopher Cox as Chairman of the SEC on September 6, 2006 (available at www.sec.gov),
where he states that charges related to this matter were made as early as 2003.
9
Accounting standards and a detailed description of accepted “intrinsic value” calculations can be found in APB 25, from the FASB.

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Federal Reserve Bank of Richmond Economic Quarterly

their proxy statement) enough information about their grants of employee options for any interested shareholder to compute the cost of
these (using, for example, the Black and Scholes valuation). Hence,
the economic impact of this change in regulation remains unclear, and
it somehow hinges on the assumption that the information disclosed in
the footnotes was somewhat less available to the public than after it
was o¢ cially included as an expense in the income statement.10
It is important to note that the …rst proposal for the expensing of
options was drafted as far back as 1993. Due to strong opposition from
the corporate sector and other political forces, the …nal recommendations in FASB 123 issued in 1995 merely recommended the expense, but
did not mandate it. The public debate about the pros and cons of expensing, which involved senators, congressmen, the SEC, and lobbyists
from the corporate sector, was ongoing for more than a decade. Finally,
in 2006, the SEC endorsed the revision SFAS 123R, which mandates
expensing. It is worth noting that many large public …rms started the
expensing on a volunteer basis as early as 2002; some commentators
have noted that this voluntary adhesion, and the …nal political push
that lead to the mandatory requirement, were rooted in the Enron and
other accounting scandals in 2002.11 Hence, the e¤ect of SFAS 123R is
potentially present as early as the passage of the Sarbanes-Oxley Act,
preventing the separate identi…cation in the data of the e¤ect of the two
regulations. Nevertheless, in our analysis we …nd signi…cant changes in
the patterns of usage of stock and option grants coinciding with both
changes in regulation.

Outline
In this article, we start by describing the data. We provide a motivating example that illustrates the primary di¢ culties in using the
currently available data on CEO compensation to answer the main
questions of interest to us. In Section 2 we brie‡ review some previy
ous attempts in the academic literature to shed light on similar issues,
and the di¤erences with the approach we take here. We proceed with
our main analysis in two parts: First, in Section 3, we document facts
related to the extensive margin (i.e., when and by which …rms are stock
and option grants used), and second, in Section 4, we discuss facts related to the intensive margin (i.e., what is the relative importance of
10

See Guay, Kothari, and Sloan (2003) and Guay, Larcker, and Core (2005) for a
clear exposition of these issues.
11
See, for example, Brown and Lee (2011), or “Reporting Employee
Stock Option Expenses:
Is the Debate Over?” by Paulette A. Ratli¤
(www.nysscpa.org/cpajournal/2005/1105/essentials/p38.htm).

A. Jarque and B. Gaines: CEO Pay

315

stock, options, and other forms of pay for the …rms that use them).
We document the change in compensation practices across the di¤erent regulatory regimes. We also explore the correlation of other …rm
characteristics, like size, industry classi…cation, and executive characteristics, like age, tenure, and gender, with the choice and importance
of the di¤erent available compensation instruments. We also examine
the relationship of usage of stock and option grants with the level of
pay. We conclude in Section 5.

1.

SAMPLE DESCRIPTION AND DATA
INTERPRETATION ISSUES

Thanks to disclosure requirements by the SEC, we have data available
on pay to the top executives of public U.S. companies starting in 1992.
This data is collected systematically by Compustat into a database
called Execucomp. Many academic studies have used Execucomp and
other available data to document the regularities in the level of pay
and its sensitivity to …rm performance, across time and also for …rm
characteristics like size and industry.12
The Execucomp data set is published by Compustat four times per
year. Each release includes the new information for companies that
…led their proxy statements with the SEC in that period (companies
can decide when their …scal years start, and hence there is variation in
when annual proxies are …led). Execucomp tries to collect data on the
…rms that are listed in the S&P 1500 index, which roughly corresponds
to the 1,500 largest U.S. …rms by market capitalization. This article
uses the information on CEO pay of the October 2011 edition of the
Execucomp data, which covers 1992 to 2010, for a total of 19 complete
…scal years. We exclude observations in year 1992, since there are very
few and they may not be representative. We exclude CEOs who own
a large fraction of the …rm’ stock, since presumably pay is not set to
s
provide incentives for these owner-CEOs. Next, we elaborate on the
issues in choosing the threshold value for this selection.
12
For the analysis of sensitivity of pay to performance, see the seminal contributions of Jensen and Murphy (1990), Rosen (1992), and Hall and Liebman (1998).
For the relationship of pay level and sensitivity to …rm size in the cross section, see
Schaefer (1998) and Baker and Hall (2004). A more recent study of the variation of the
level of pay over time and its potential relationship to …rm size is Gabaix and Landier
(2008). Frydman and Saks (2010) provides a comprehensive historical overview of both
level and sensitivity of pay facts using a small sample of …rms over an unusually long
period, from 1936 to 2005.

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Federal Reserve Bank of Richmond Economic Quarterly

Ownership, Incentives, and Steve Jobs
In this article we are interested in the decisions of …rms to use or not
use a given compensation instrument. One potential concern with this
analysis is that the choice of a …rm of not using stock or options may
be explained by the fact that its CEO is a founder of the company, or
that he or she is very vested in the …rm already. An example of this
would be Steve Jobs, who is, in our sample from 1997 to 2010, listed
as the CEO of Apple, Inc.
Jobs’ history of compensation over 12 years is easily summarized.
s
In 1997, the year he took the CEO position, Jobs received, as a director
of the company, 30,000 stock options with an exercise price of $23, to
be vested proportionally over a three-year period.13 The salary of Jobs
was $1 for all the years we observe him in the sample. He received
sporadic bonus and “other compensation” payments, stock in 2003,
and options in 1997, 2000, and 2002.14;15 In the company’ own words:
s
“In 2010, Mr. Jobs’ compensation consisted of a $1 annual salary.
s
Mr. Jobs owns approximately 5.5 million shares of the Company’
s
common stock. Since rejoining the Company in 1997, Mr. Jobs
has not sold any of his shares of the Company’ stock. Mr. Jobs
s
holds no unvested equity awards. The Company recognizes that
Mr. Jobs’ level of stock ownership signi…cantly aligns his interests
s
with shareholders’ interests. From time to time, the Compensation
Committee may consider additional compensation arrangements for
16
Mr. Jobs given his continuing contributions and leadership.”

Jobs’ ownership shares are only reported in Execucomp, combined
s
with option holdings, for four of the years, and they never exceed 1.35
percent of the total shares outstanding, which is about the 67th percentile ownership in the original sample of CEOs. Even if one may be
tempted to think that Jobs was not an “agent” for the shareholders
of Apple due to the great value of the stock that he owned (especially
after the 2003 grant, valued at more than $80 billion at the grant date),
a closer look at the evolution of his ownership shows that he went from
owning one share in 1997 to owning 5.5 million shares mainly as a result of his compensation packages. Moreover, Jobs had a considerable
amount of wealth from his investment in Pixar, and one could argue
13

See Apple’ De…nitive Proxy statement on March 16, 1998.
s
According to Execucomp, Jobs received a bonus payment in 2001 and 2002, and
two big sums as “other compensation” in 2001 and 2002.
15
Given the compensation pattern of Jobs, it is interesting to note that Apple
stated in April 2003 its decision to voluntarily expense option grants to its employees
according to FASB recommendations
16
See Apple’ De…nitive Proxy statement on January 7, 2011.
s
14

A. Jarque and B. Gaines: CEO Pay

317

that stakes had to be necessarily high in order to provide him with adequate incentives. Finally, when Jobs’ illness was made public, markets
s
reacted, providing proof that the value that Jobs was bringing to the
company was real.
The case of Steve Jobs is easy to check and understand, but in general the data on ownership in Execucomp shows some inconsistencies,
and there are many missing values, since ownership is recorded only if
it is over 1 percent. Hence, a back-of-the-envelope calculation of the
value of the stock held by the CEO is not always available, and even
if it were it would be hard to determine when the CEO is subject to a
moral hazard problem based on those numbers.17 From our analysis of
the Jobs case, however, we conclude that we cannot rule out that ownership, in our sample, is a result of dynamic incentives provided by the
…rm. Hence, we are most comfortable adopting a conservative criteria
of only dropping CEOs from our sample if their ownership reaches 50
percent in any of the years that they worked for a given …rm, as opposed to more restrictive selection criteria in the literature.18 Our …nal
sample includes information on 6,146 di¤erent executives, and 3,248
…rms, which amounts to 6,416 unique executive-…rm pairs. In the year
1993, we observe 1,147 …rms, and every year after that the number is
at least 1,500, with a maximum of 2,010 …rms in the year 2007.

Compensation Measures
Our focus in this article is on the choice of compensation instruments
by the …rm, and we use the information readily available in Execucomp
about each of the components of total compensation: salary, bonus and
incentive compensation, stock and option grants, and “other compensation”such as pension plans, life insurance premiums, or perks. Note
that to avoid discontinuity issues with the “bonus”and “incentive compensation”variables due to changes in reporting requirements in 2006,
we sum these two to construct a single series, which we refer to as
BIC throughout the article. Also, in spite of the di¤erent accounting
standards during the sample period, Execucomp contains the Black
and Scholes valuation of option grants for the whole period: Companies that used alternative valuations prior to SFAS 123R were required
17
In spite of the sparse availability, we did construct a value of shares owned for
the CEOs for which we had data: The average value for those that we classi…ed as
non-owners was $1,928,000, compared to a mean total compensation of $2,507,000.
18
Clementi and Cooley (2010) used the more restrictive threshold of 1 percent ownership. We conducted a robustness check of our main analysis by dropping all CEOs
who owned 3 percent or more shares on average over their tenure and results did not
change qualitatively.

318

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Average CEO Compensation

Notes: “BIC” stands for bonus and incentive compensation.

to provide the parameters necessary to calculate the Black and Scholes
value. Whenever we need a measure of total compensation, we use the
sum of these components (the variable T DC1 in Execucomp).19
Figure 1 presents the evolution of the mean total compensation in
our sample over time, and its components. All the amounts here and
in the rest of the article are normalized to thousands of 2010 dollars
using the consumer price index. The year 2000 stands out as the peak
in our measure of compensation, with an average of $8,553,690 and
a median of $3,107,580. The year 2009 seems to be the last one of
a decreasing compensation trend coinciding with the …nancial crisis:
Mean compensation for this year was $4,637,950, while the median was
$3,030,940.
The most salient fact about the composition of pay in Figure 1 is
that the variation of pay with the business cycle is implemented through
19

For recent studies that use this same measure of total compensation, see Gabaix
and Landier (2008); Frydman and Saks (2010); and Cheng, Hong, and Scheinkman
(2012).

A. Jarque and B. Gaines: CEO Pay

319

the grants of stock and options, rather than through salary, bonus, or
other compensation. For example, the graph shows that the decline
in average total compensation between 2000 and 2003 is driven by a
decline in the value of stock options. However, after 2002, the category
BIC becomes somewhat cyclical as well. It is important to keep in
mind that, of these components of total compensation, only bonus and
incentive payments are mechanically related to the results of the …rm.
For example, the amount used to construct Figure 1 is the expected
value of the grant at the time when it was awarded. Hence, the fact
that compensation was the highest in the year 2000 is not due to a high
value of past grants driven by a stock market boom, but rather to a
conscious decision by the …rms to increase the value of compensation
for their CEOs.20

2.

PREVIOUS LITERATURE

Before we start our analysis of the data, we review the relevant literature and explain our contribution.
In an in‡
uential chapter of the Handbook of Labor Economics,
Murphy (1999) provides some suggestive evidence for a sample of …rms
between 1992 and 1996 that the importance of the di¤erent compensation instruments in pay packages (salary, bonus, stock, and option
grants) varies across …rms according to their size and the industry to
which they belong.21
In his graphical analysis for the e¤ect of size, Murphy compares
S&P 500 industrials, mid-cap industrials, and small-cap industrials.
We replicate and extend his analysis (including data up to 2010) in
Figure 2, where we classify …rms in our sample, year by year, into four
quantiles according to their volume of sales.
The most striking fact that emerges from Figure 2 is that …rms
with larger sales …gures have higher levels of pay. The variation in
the relative importance of the di¤erent compensation instruments is
di¢ cult to evaluate in a systematic manner, although it is clear that
larger …rms have a larger portion of their pay given in stock and options.
Also, the increase in the relative importance of options in the late 1990s
that has been frequently commented on both in the academic and the
popular press seems to have been disproportionately concentrated in
the quantile of the largest …rms.
20
Note that …rms amortize the expense from these grants over their vesting period,
and hence compensation expenses are actually smoothed out over time by the …rms.
21
See Figures 2 and 3 in Murphy (1999).

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Average Compensation and its Components, by
Quartiles of Sales Volume

Murphy’ graphical analysis for the regularities across di¤erent ins
dustries is limited to S&P 500 …rms, and it uses a classi…cation of
SIC industries in four groups: mining and manufacturing …rms, …nancial services …rms, utilities …rms, and other industries. In Figure 3
we replicate this evidence, again extending the sample to include data
from 1992 to 2010, as well as all …rms in the S&P 1500. Figure 3
does not allow us to draw any clear conclusions. If anything, it seems
to suggest that …rms in utilities seem to rely more on restricted stock
than option grants. In a related study, and for the period 1992–
2001,
Murphy (2003) classi…es …rms into “new economy” versus “old economy”according to the industry sector they belong to, and he …nds that
new economy …rms (those competing in the computer, software, internet, telecommunications, or networking …elds) use stock-based compensation (both restricted stock and options) more often and to a larger
extent.
One important shortcoming of the simple facts reported in
Murphy (1999, 2003) is that they do not inform us about the relationship between combinations of individual characteristics (industry and
size together, for example) and usage of instruments. Also, the information about the variation in the cross-section is lost in the graphs. Our

A. Jarque and B. Gaines: CEO Pay

321

Figure 3 Average Compensation and its Components, by
Industry Group

contribution in this article consists of analyzing the data according to
…rm characteristics by running some simple regressions. Our analysis is still partial, since we are not exploiting the panel component
in the data, but we are able to provide a more accurate description
of the facts by controlling for several individual …rm characteristics.
We also split our analysis into the extensive margin (which compensation instruments are used) and the intensive margin (given a set of
instruments that is being used, what is their individual share of total
compensation).
In addition to answering the questions posed above about the trends
in the usage of di¤erent compensation instruments, we explore whether
factors other than …rm size or industry classi…cation may be associated
with the usage of certain instruments. For example, given the limits
on tax deductions imposed on salaries, …rms that— for reasons other
than their industry and size— choose to compensate their CEO with a
larger sum of money may bene…t more from issuing non-quali…ed option grants or restricted stock grants. As another example, executives
who have longer tenures may need fewer restricted stock grants if they
already hold a large number of shares of the …rm from previous grants.

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Federal Reserve Bank of Richmond Economic Quarterly

This last point, which is an interesting one, refers to the dynamic
nature of incentives for CEOs. There have been important e¤orts in
the literature of CEO compensation that track the evolution of the
portfolio of grants of the executives, so that at each point in time
we have a better understanding of how the executive’ wealth would
s
vary with a particular realization of the …rm’ results. Some impors
tant examples are Hall and Liebman (1998), Core and Guay (1999),
and, more recently, Clementi and Cooley (2010). These measures of
incentives are a way of controlling for outstanding past issues of stock
and option grants. The focus of these studies, however, has not generally been the trends in the usage of compensation instruments. An
important exception is Core and Guay (1999), who study this in detail
for a shorter time period than the one we are analyzing here. They
construct a model of the optimal level of stock holdings of the CEO,
for incentives purposes. They …nd evidence that new grants (combining
stock and options) are aimed at maintaining that level of incentives, as
old grants expire or go out of the money. However, as far as we know,
none of the studies that construct the portfolio measures address the
potential e¤ects of regulation on the trends in the usage of individual
compensation instruments.
One important shortcoming of our data set is that it starts in 1993.
Regulations on tax deductibility of CEO pay had just changed at the
time (see IRC section 162(m)). Data on compensation practices prior
to 1993 would be useful to the understanding of the distortions that
162(m), and other tax advantages introduced earlier, may have induced
on pay practices.22 Detailed compensation data for a broad representative set of …rms going further back in time is not available; however,
Frydman and Saks (2010) provide a historical analysis of a limited set
of …rms.23
As part of their analysis, Frydman and Saks (2010) plot the median
of the partial sums of salary and bonus payments, successively adding
the value of stock and option grants. They …nd that, even though
the usage of options picks up considerably after taxation advantages
are introduced in 1950, their relative importance in total compensation, as well as that of stock grants, does not become signi…cant until
the 1980s.24 Since their sample of …rms is necessarily limited (because
of the long historical scope), and for comparison purposes, we replicate their graphical analysis for our sample in Figure 4.25 For the
22
23
24
25

See
See
See
See

Jarque (2008) for a review.
also Lewellen (1968).
Frydman and Saks (2010, Figure 2, p. 2,108).
Frydman and Saks (2010, Figure 1, p. 2,107).

A. Jarque and B. Gaines: CEO Pay

323

Figure 4 Median Total Compensation and its Main
Components

overlapping period from 1992 to 2005, and again with the caveat of not
controlling for individual characteristics in this simple graphical analysis, we con…rm their …ndings: Option grants have been an increasingly
important component of the median pay of CEOs for the whole period,
while the importance of stock awards started to pick up around 2002.
With respect to the mean compensation that we plotted in Figure 1,
we see that the importance of options was not as marked for median
pay in the 1999–
2001 period as it was for mean pay. Other than that,
the main patterns seem to align between the two …gures.

3.

THE COMPOSITION OF PAY PACKAGES: THE
EXTENSIVE MARGIN

We start this section by documenting the usage of the di¤erent
compensation instruments over time. Then we proceed to analyze more
formally which …rm characteristics may be relevant for the choice of
instruments of compensation. We …nd that variables like size and industry classi…cation have some explanatory power over whether …rms
decide to include options or stock in their compensation packages.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 5 Evolution of the Percentage of Firms that Use Each
Instrument

Changes in regulation during the period we are studying exhibit the
highest correlation with changes in usage patterns.

The Use of Di erent Compensation
Instruments: A First Look
For all the …rms in our sample, we check year by year which ones use
each instrument (for example, a …rm “uses”stock if it reports a positive
stock grant to their CEO, regardless of the amount of the grant). This
is plotted in Figure 5.
As is apparent from the graph, the use of both salary and other
compensation is fairly universal and fairly constant over time (with a
slight trend up for other compensation in the last …ve years). The
use of bonus and incentive compensation is volatile around 85 percent,
with no obvious trends. But the most striking feature in Figure 5 is
the run-up in the use of restricted stock grants starting around 2003,
which coincides with an important decrease in the use of option grants.
Given the strong variation over time in the usage of stock and
options, it is worth thinking about the factors that could potentially
be determining the decision of a …rm to include either type of grant

A. Jarque and B. Gaines: CEO Pay

325

in the compensation package to its CEO. Here we point to three main
factors: (1) di¤erences in tax advantages and accounting standards, (2)
di¤erences in sensitivity to …rm performance, and (3) …xed costs of
adoption of each instrument.
First we turn to tax and expensing di¤erences. As we discuss at
length in the introduction, both the passage of the Sarbanes-Oxley
Act in 2002, and, especially, the change in expensing requirements and
valuation of options in SFAS 123R approved in 2006 (and expected
and voluntarily adopted by many …rms as early as 2002), seem to have
decreased the relative attractiveness of option grants over stock grants.
We can summarize the comparison between the two instruments as
follows. Restricted stock grants do not qualify for a tax deduction,
they have to be accounted for as compensation expenses, and, before
2002, they had to be reported as insider transactions within 10 days
of the grant. Options were more advantageous than stock before 2002
because they only had to be reported as insider transactions by the end
of the …scal year of the company; after Sarbanes-Oxley, both types of
grants have to be reported within two business days of the transaction.
Options were more advantageous than stock before 2006 because (i)
they could be deducted for tax purposes, and (ii) they did not need
to be expensed; after 2006, advantage (i) is still present, but (ii) is no
longer there.
Second, stock and options may implement di¤erent incentives for
the CEO. That is, in principle, without any accounting or tax di¤erential treatment, stock and options could be substitutes in a compensation package: One could transfer a given amount of resources to the
CEO either with a stock grant or with an option grant of equal expected
value. However, the value of each of these two grants could change differently with changes in the value of the …rm, i.e., the sensitivity of the
compensation may be di¤erent depending on whether it includes only
options or only stock (or both). Hence, idiosyncratic characteristics
of the …rm, like industry, size, or …nancial health, may determine the
optimal sensitivity of pay to performance, and hence instrument choice.
Third, there may be a …xed cost of including an extra instrument in
a compensation package (perhaps related to communication of new or
more complex compensation practices to shareholders and creditors);
this would imply that larger …rms decide to include a di¤erent set of
compensation instruments than their smaller counterparts.
To shed some light on these and other potential hypotheses, we will
formally analyze the correlation of di¤erent …rm characteristics on the
choice of compensation instruments. We start our analysis of the data
by classifying …rms into four mutually exclusive groups, I; according

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 6 Evolution Over Time of the Percentage of Firms in
Each I Group

to which set of compensation instruments they use:
I = fS; O; B; N g ;
with typical element I: That is,
a …rm with I = S includes restricted stock grants (but no options) in its compensation package to the CEO,
a …rm with I = O includes options (but no stock),
a …rm with I = B includes both restricted stock and options,
and
a …rm with I = N includes none of the two.

A. Jarque and B. Gaines: CEO Pay

Table 1 Descriptive Statistics of the Sample

327

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 6 presents the evolution of the proportion of …rms in each of
these four groups over our sample period. The evidence is consistent
with the changes in regulation prompting …rms to switch from using
options only (O) to using stock only (S); but it is also apparent that
a higher portion of …rms use both instruments (B), suggesting that
some …rms may have chosen to add stock to the use of options, rather
than completely substituting options with stock. Next, we formally
evaluate the role of changes in regulation in these variations in usage
patterns, after also considering other potential determinant factors for
these patterns, such as the size of the …rm and the industry to which
it belongs.

The Determinants of the Composition of
Pay Packages
Table 1 presents the breakup of …rms in the compensation groups in
I according to the regulatory periods, the industry group, and several
…rm and CEO characteristics available in Execucomp. It also includes
statistics that describe the cross relations between these variables. We
have established the existence of three di¤erent subperiods in our sample determined by important changes in regulation. Table 1 reports, in
its …rst three rows, the fraction of …rms that choose each instrument
in the sample, and the changes in these fractions after the two changes
in regulation. We denote as period I observations those from 1993 to
2001, as period II those from 2002 to 2005, and as period III those
from 2006 to 2010. We see in the table that the fraction of …rms in S
increases in both subperiods, but especially in the later one. The fraction of …rms in O options decreases, again more sharply in the later
subperiod. The fraction of …rms in N remains fairly constant over time
around its overall mean of 22 percent. As we explained in the previous
subsection, both regulations had the e¤ect of decreasing the relative
attractiveness of options over stock grants. Given this, it is useful to
trace the changes in the fraction of …rms that use options at all, i.e.,
O [ B. In period I, we see that 73 percent of …rms had options in
their compensation packages. In period II this fraction remains fairly
constant, at 71 percent: The decrease in O is almost exactly o¤set by
the increase in B. That is, in the second subperiod …rms were more
likely to use options together with stock, rather than alone, but still as
likely as before to use options at all. However, in the last subperiod the
fraction drastically decreases to 54 percent: Although the fraction of
…rms in B increases, the decrease in O is three times as large. This is
consistent with the annual evidence presented in Figures 5 and 6, which

A. Jarque and B. Gaines: CEO Pay

329

show that the main adjustment in the usage of options was gradual and
took place mainly over the course of period II.
For industry classi…cation, we use the simple four groups of …rms
proposed by Murphy (1999). Firms are classi…ed into: 1) mining and
manufacturing, 2) …nance and real estate (FIRE), 3) utilities, and 4) a
mixed group containing any other …rm. Table 1 reports that the choice
of S is relatively more likely in FIRE and utilities, while that of O is
more likely in mining and manufacturing and other. The choice of B is
relatively more likely in FIRE, and less in other. Finally, the proportion
of …rms choosing N is much lower in mining and manufacturing and
FIRE.
Next we report the breakout into compensation groups according
to size. The literature has established that size is an important factor
in the determination of pay levels. We use total assets as a measure of
size.26 Year by year, we classify the …rms in our sample according to
which of the four quantiles of the distribution of asset value they belong.
Table 1 reports the fraction of …rms that choose each instrument in the
sample, and the di¤erences from the fractions for the control group,
which is the quantile of smallest …rms. The patterns of usage of S
seem to be independent of size, while O and N are relatively more
popular in smaller …rms. On the other hand, using stock and options
together (B) is more frequent in larger …rms.
Other potentially important characteristics are the tenure, age, and
gender of the CEO. We brie‡ discuss each of these in turn.
y
Younger executives may have di¤erent career concerns than older
ones, less experience, or di¤erent attitudes toward risk. More tenured
executives may be more vested in the …rm by means of historical grants,
or …rm-speci…c human capital. In Table 1 we see that the choice of S
seems to be fairly independent of both age and tenure. The choice of O,
instead, is more frequent for younger executives, while, interestingly,
given the natural correlation of these two variables, it is less frequent
for shorter tenured ones. The frequencies of choice of B are humpshaped with respect to age, and decreasing for tenure. Firms seem
more likely to use none of the instruments more frequently for longtenured executives, and less frequently for middle-aged ones.
Some have argued that women are more risk averse than men (see
Schubert et al. [1999] for a discussion of the evidence); this could
in‡
uence the choice of compensation instrument. We only have 548
26
For recent estimates, see Gabaix and Landier (2008, Table I, p. 66). Other size
measures used in the literature are the number of employees and sales value. We con…rm
that in our data set the size measure with the highest R 2 for the level of pay is asset
value. Details are available upon request.

330

Federal Reserve Bank of Richmond Economic Quarterly

…rm-executive-year observations that correspond to a female CEO,
versus 30,032 for males. Despite this, we report the average use of
instruments by gender in Table 1 to point to one apparently signi…cant
di¤erence: Female CEOs are about 10 percent less likely than male
CEOs to receive options exclusively.
We now proceed to validate these raw statistics by performing a
formal check on the e¤ect of …rm characteristics on the choice of instrument. We model the value to a given …rm i of choosing a set of
instruments I at time t as
V (I)it =

+

3
X

k

industryi +

k=1

+

7

ageit +

2
X

3+k

periodt +

6

tenureit

k=1

8 ln (assets)it

+

9

f emaleit + "it :

(1)

In words, the value V (I)it is assumed to depend linearly on a constant,
; dummy variables for the three distinct regulatory periods in the
sample, an indicator variable for the industry group to which that …rm
i belongs, and the characteristics of …rm i in year t that we selected
based on our sample analysis. We do not observe directly the value
V (I)it ; but rather the discrete choice of …rms for I 2 fS; O; B; N g :
Hence, our statistical model is
Pr V (I)it > V I 0

it

8I 0 6= I;

where the probability of observing the choice of a given I depends on
whether V (I)it ; the value derived by a …rm i from using instrument I
at time t; is higher than the value of the other instruments. We assume
the noise term "t has a type I extreme value distribution, so our discrete
choice regression is a multinomial logit.
One concern with the interpretation of the results of the regression
is the potential for colinearity. Table 1 reports, starting in the column
labeled “Period,”the averages of each variable in the subgroups de…ned
in the di¤erent rows of the table. When analyzing those, the most
salient fact is the uneven average size across periods (average size is
increasing) and across industry groups (FIRE contains …rms that are,
on average, 10 times the size of …rms in mining and manufacturing).
However, when we plot the actual size distribution across periods it
is not signi…cantly di¤erent, thanks to the high variation in the size
of …rms within periods that we get by using the cross section. The
di¤erence in the distribution of size across industry groups is more

A. Jarque and B. Gaines: CEO Pay

Table 2 Regression Results

331

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Federal Reserve Bank of Richmond Economic Quarterly

apparent. However, we perform a robustness check of the main qualitative features of our regression results by running our regression in four
di¤erent samples according to industry group, and we con…rm them
all.27
The results of the regression using the benchmark speci…cation for
V (I) in (1) are reported in Table 2, under the column labeled as regression speci…cation (1). In the …rst row of Table 2, we report the regression sample averages, or, equivalently, the average predicted probability
in the model. The numbers di¤er slightly from those reported in Table
1 because some of the observations have missing values for some of the
regressors, and hence they are dropped from the sample.
In order to provide an intuitive sense of the estimated relative importance of each regressor, the rest of the rows in the table report the
average of the partial derivatives of the probability of usage, or the
average marginal e¤ect of each explanatory variable xij in the vector
of all explanatory variables Xi , de…ned as
AM E (j)

M eani

@ Pr (IjXi )
xij

:

That is, using the estimated coe¢ cients, we calculate how much the
probability of using each instrument changes for each of the …rms in the
sample when we marginally increase the value of a given explanatory
variable xij ; evaluated at the true value of the vector of regressors Xi
for …rm i; then we take the average of those marginal changes over
i. Note that the marginal e¤ects are calculated in a slightly di¤erent
way for discrete variables. The marginal e¤ects with respect to the
variable “Period,” for example, represent the average change induced
by hypothetically switching a …rm from the base period, I, to each of
the remaining periods.28 Formally,
h
i
x
x
AM E (j) M eani Pr IjXi ij ; xij = n Pr IjXi ij ; xij = base ;

for all n di¤erent than base; where base denotes the value of the regressor xj ; in this case period I, and n 6= base represents period II
x
and period III. The notation Xi ij represents the vector of regressors
Xi excluding regressor xj : In order to provide a benchmark to evaluate these discrete changes in probability, we also report, for discrete
27

Details are available upon request.
To calculate the marginal e¤ect for “2002–
2005,” Stata calculates the predicted
probabilities by setting to 1 the dummy for the baseline period of “1993–
2001” into
each observation while leaving all other regressors at their true sample values. Then
it calculates this predicted probability again by substituting “2002–
2005” instead. The
average of this di¤erence is the reported marginal e¤ect.
28

A. Jarque and B. Gaines: CEO Pay

333

regressors, the level of the average predicted probability in the sample
when setting xij = base (we denote this by ALE (j) in the table).
Some of the strongest economic e¤ects are associated with the three
regulatory subperiods. Size and industry are also statistically significant for all groups. Despite the di¤erences reported in Table 1, and
possibly due to high standard deviations, our controls for gender, age,
and tenure of the CEO often do not have a statistically signi…cant effect on the choice of compensation instruments, so we choose to not
report the AM Es for these variables.29 We now summarize the …ndings
regarding period, industry, and size.
Regulatory periods Firms move away from compensation packages
that include only options after both regulatory changes, either to
use only stock or to use options together with stock. They mainly
add stock to their compensation packages during period II, and
they mainly substitute stock for options in period III.
We …nd that if the same …rm went from living in period I, before
Sarbanes-Oxley, to period II, the probability of it choosing O would decrease by a substantial 12 percentage points (pp), while that of choosing
S would increase 5 pp and that of choosing B would increase by almost
10 pp. In period III, after FAS 123R went into e¤ect, the probability
of using options would be 39 pp lower than in the initial period, leaving it at about 20 percent. The most favored category in that switch
would be stock, with a 23 pp increase, followed by both, with a 19 pp
increase. The use of none decreases at a modest 2– pp in each of the
3
two periods.
Industry classi…cation Firms in FIRE and utilities favor packages
that include stock exclusively, or no grants at all, more frequently
than the average …rm. Both these industries make less use of
packages that include options exclusively, or stock and options
together. Firms in mining and manufacturing, in contrast, use
options exclusively, or together with stock, slightly more than the
average …rm, and they are less likely to compensate without using
any grants at all.
The control industry, “other,” aligns with the average usage probabilities in the overall sample. We see that switching from “other” to
“mining and manufacturing” is associated with a shift away from using N into O, or B, in comparable magnitude. Switching to FIRE is
associated with an important shift away from O and B (by 2 and 5 pp,
29

Details are available upon request.

334

Federal Reserve Bank of Richmond Economic Quarterly

respectively), into S and N (by 4 and 3 pp, respectively). Switching to
utilities, which includes transportation, communications, electric, gas,
and sanitary services, presents, perhaps surprisingly, a similar pattern
than FIRE. For utilities, however, the e¤ects are even stronger: The
decrease in B and O is by 4 and 8 pp, respectively, and the increase in
N and S is by 4 and 8 pp, respectively.
Size Larger …rms are less likely to use compensation packages that
include options exclusively, and more likely to use those that include both stock and options. They are also less likely to compensate without using any type of grants at all.
Firm size is a continuous variable (the log of the value of assets
measured in thousands of 2010$). An increase in …rm size signi…cantly
increases the probability of choosing B; to the detriment of choosing
O or N: It is di¢ cult to compare the economic importance of size
with respect to the discrete variables that we just commented on, since
the numbers in the table represent the e¤ect of in…nitessimal increases
in size, not changes in industry or period as above. For comparison,
we can calculate the implied average change in the probabilities for
an increase in log size equal to one standard deviation. This backof-the-envelope calculation implies that the probability of choosing B
increases by approximately 9 pp, while that of O decreases by 1 and
that of N by 7. This suggests that the magnitude of changes associated
with size is similar to that of changes in the industry classi…cation, but
smaller than that of the regulatory subperiods.
The Role of Individual Firm Characteristics
In Figure 7 we provide a simple graphical evaluation of the …t of the
model in equation (1). We have plotted the sample percentage of users
of each instrument by year in each of the subplots (blue line), the
predicted probability of usage by the full model in (1) (red line), and
a limited model that includes as regressors only the period dummy
indicators (green line).30 The main result that emerges from this comparison is that the explanatory power of the variables that indicate the
di¤erent subperiods is high. Although the econometric model is not
able to …t the smooth decline in the use of options, we have already
pointed out earlier in this article that the new standards in FAS 123R
30
The SEC adopted FAS 123R reporting rules for …rms …ling their proxy statements after December 15, 2006. Hence, we classify …rms as being in period III if the
month in which their …scal year ends falls after November 2006. This means that in
the year 2006, the …rms in the sample are split across two subperiods. This explains
the extra kink in the predictions using the restricted model.

A. Jarque and B. Gaines: CEO Pay

335

Figure 7 Fit of the Model Over Time

were already recommended by the FASB in 1993, and some companies
started adopting them earlier than 2006. This may explain most of the
discrepancies between the green line and the true data.
In contrast with the good …t of the model over time, a low pseudoR2 seems to suggest that, even including the individual characteristics,
our model does not do a good job in explaining individual
cross-sectional variation in the use of instruments. As is apparent from
the …gure, the individual characteristics of the …rm that we include in
the full regression do not add much to the explanatory power of the
model over time. In fact, the pseudo-R2 of the regression using only
the regulatory subperiods as explanatory variables is already 7 percent,
compared to the 11 percent of the full model in speci…cation (1). More
work is needed to understand which individual characteristics of …rms
determine their choice of compensation instruments.
The Role of the Level of Pay
One potential explanatory variable of the choice of instrument that we
left out of our analysis in regression speci…cation (1) is the level of
pay itself. It may be the case that tax advantages, or transparency
concerns of the …rm, make it more convenient for the …rm to pay large
sums to its CEO in the form of stock or options, rather than through

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Federal Reserve Bank of Richmond Economic Quarterly

Table 3 Average Total Compensation and Average Mean
Compensation by Firm

S

Period
I

T DC1it
mean(T DC1)i

II

T DC1it
mean(T DC1)i

III

T DC1it
mean(T DC1)i

Across
Periods

T DC1it
mean(T DC1)i

O

B

N

Across
Groups

5,683
(37,035)
5,752
(7,804)
5,672
(8,023)
5,422
(6,447)
5,539
(7,861)
5,236
(6,163)
5,585
(16,417)
5,350
(6,499)

6,211
(16,821)
5,707
(7,707)
5,677
(7,399)
5,380
(5,876)
4,754
(6,645)
4,616
(5,196)
5,893
(14,020)
5,486
(6,409)

8,809
(15,811)
7,457
(7,707)
8,854
(10,304)
7,206
(7,510)
7,621
(7,827)
6,688
(6,421)
8,299
(11,448)
7,056
(7,125)

1,668
(3,360)
4,227
(6,211)
2,185
(3,627)
4,380
(7,398)
1,762
(2,866)
3,625
(5,605)
1,804
(3,299)
4,095
(6,335)

5,519
(16,176)
5,630
(7,002)
5,808
(8,106)
5,660
(6,771)
5,372
(7,218)
5,310
(6,090)
5,542
(12,412)
5,542
(6,691)

Notes: Standard deviations in parenthesis.

a salary or a bonus program. To explore this possibility, we present in
Table 3 the average level of total compensation (…rst row of all periods,
labeled as T DC1) by group and subperiod. We can see that …rms in N
have a remarkably lower level of compensation, across all subperiods.
Moreover, …rms choosing B have the highest average compensation in
all subperiods. The statistics for the group using only stock or only
options are interesting: The level of pay is higher for O in period I,
when options were more likely to be used on their own than stock (see
Table 1). During period II, average pay is equal across the two groups
of users. As we discussed in Section 3, this is a period when SarbanesOxley had just been passed, making the choice of options more costly
— at least in terms of opportunities for backdating and maybe in terms
of public image. In period III, after the new accounting standards that
made the valuation of options less arbitrary became compulsory, the
ranking of average pay reverses: CEOs of …rms that are users of stock
are paid, on average, more than those that are users of options.
In order to explore formally the explanatory power of the level of
pay after controlling for …rm characteristics, we replicate the regression
in equation (1), but add the level of total compensation (the log of the
variable T DC1 in Execucomp) as a regressor. The results are reported
in Table 2, under the column labeled as regression speci…cation (2).

A. Jarque and B. Gaines: CEO Pay

337

The meaningfulness of these estimated e¤ects needs to be evaluated in
the context of the mechanics of compensation, since there is an obvious
relation between the level of compensation and the use of grants. This
mechanical relationship exists unless we think that sometimes …rms issue grants that are very small in value. In the sample, the minimum
value for stock and option grants is in the order of $3; the 1st percentile
value is $18,667 for stock grants and $38,355 for options; the 10th percentiles values are about $200,000 and $250,000, respectively. Given
that the 10th percentile of salary payments in the sample is $364,000,
and that of total compensation is in the order of $750,000, these statistics suggest that fairly low values of the grants are possible and not
that uncommon.
Another concern with the results of regression speci…cation (2) is
endogeneity: Since stock and options are risky assets, CEOs receiving
their compensation in the form of grants (as opposed to salary or other
less risky instruments, such as bonuses) may need to be compensated
for their risk aversion with higher levels of pay. See Hall and Murphy
(2002) for a formal explanation and quanti…cation of the e¤ect of risk
aversion on the value of grants to executives, and the comparison of
that value to the cost for the …rm.
Total compensation is indeed a signi…cant variable according to
the results of the multinomial logit. Also, the pseudo-R2 doubles with
respect to speci…cation (1). We …nd that a marginal increase in the
level of pay leaves the probability of using stock almost unchanged,
while it increases the one for choosing O by 10 percent and of B by 12
percent; it decreases the probability of using N by 23 percent.
The e¤ects of size change signi…cantly in speci…cation (2). An increase in size now has a small but signi…cant positive e¤ect on the
probability of S: The e¤ects on O remain negative but increase signi…cantly in magnitude. Moreover, the positive relation between size and
choosing B becomes negligible (and insigni…cant) when controlling for
the level of pay. Finally, the negative e¤ect of size on the probability
of choosing N changes to positive when controlling for the level of pay,
suggesting that if a given …rm is granting a relatively high level of total
compensation, the fact that it is a larger …rm actually makes it less
likely to include stock or options in its compensation package.
We can also consider the changes in the marginal e¤ects of the rest
of the regressors with respect to those reported in speci…cation (1). As
for the period variable, the new model has similar implications both
for period II and III when it regards the choice of S: However, the
negative e¤ect on the probability of choosing O is even stronger, while
that of choosing B is still positive but weaker. The e¤ect on choosing N
changes sign in both periods with respect to speci…cation (1): Although

338

Federal Reserve Bank of Richmond Economic Quarterly

magnitudes are small, …rms are more likely to choose N in later periods
than in the initial one. Note in Table 3 that there is no clear trend of
average compensation over time across groups; however, compensation
is lower in later periods for …rms that include options in their packages.
As for the industry dummy, while the results for mining and manufacturing are very robust, for FIRE and utilities we observe some
important changes. While …rms in FIRE were more likely to choose
S or N in speci…cation (1), when including the level of total compensation as a control they are more likely to choose S; and O or B are
now favored (in similar magnitudes), while N is now less likely to be
chosen. Utilities is also more likely to choose S or N in speci…cation
(1); in speci…cation (2) it becomes a likely user of B and a less likely
user of N . A possible explanation of these reversals in the sign of the
coe¢ cients is that, given their size, …rms in FIRE and utilities tend to
have lower levels of pay, which are associated with a lower probability
of choosing B and a higher probability of choosing N ; when the level of
pay is not a control, that e¤ect is assigned by the model to the industry
dummy.
One may suspect that the covariance of the regressors with the level
of pay is a potential cause of these changes in the estimated coe¢ cients.
However, the covariance is not perfect, and both size and pay remain
signi…cant in the robustness check, suggesting that speci…cation (1)
may have an omitted variable problem. Numbers need to be taken
with caution.
As a …nal robustness check, we replicate the regression in equation
(1) but add as a regressor the average level of total compensation (the
log of the average of the variable T DC1 in Execucomp) of a …rm across
the years that it stays in the sample, rather than the actual level of
T DC1 in each year. The results are reported in Table 2, under the column labeled as regression speci…cation (3). Table 3 reports the average
and standard deviation of this measure of pay in the sample (labeled
mean(T DC1)i ). The most striking feature is the much higher pay for
…rms choosing N when compared to the average of contemporary level
of pay. This re‡
ects the fact that many of the …rms in N are in one of
the other compensation groups in some of the years.
The hope in including average pay as a regressor is that this may
break slightly the mechanical link between the level of pay and the
presence of grants, and rather pick up some …rm characteristics that
are correlated with, for example, the outside opportunity of the CEO,
or any other characteristic that determines his average pay across the
years but not necessarily the timing of the grants. We see in Table
3 that the pseudo-R2 is higher than in speci…cation (1), but much
lower than in speci…cation (2). The average level of pay is a signi…cant

A. Jarque and B. Gaines: CEO Pay

339

explanatory variable for O; B; and N; and the sign of the coe¢ cients is
aligned with that of the contemporaneous level of pay, but its economic
importance is much smaller, con…rming that some of the e¤ects of the
level of pay on the choice of instruments are purely mechanical.

Choosing N and the Timing of Grants
There is some anecdotal evidence that companies tend to have …xed
timing rules when it comes to giving stock or option grants to their
executives. Hence, when we observe a …rm choosing N in our sample
it may just mean that the …rm is in a “non-granting”year, but that it
will grant again the following year, or in a couple of years, depending
on its timing rule. For example, taking the compensation of Steve
Jobs over his tenure as Apple CEO (see Section 1), according to our
classi…cation, the company chose N in 10 of the years, O in 3, and
S in 1. Why …rms may not want to smooth out grants is, to our
knowledge, an open question, and beyond the scope of this article.
However, the common practice of having the selling restrictions of both
stock and option grants vest progressively over time does provide some
smoothing. Unfortunately, there is no good data readily available on
these vesting periods. Nonetheless, we should keep in mind that if
the practice of timing grants on a regular basis is really prevalent,
then the statistics about usage presented here should be understood as
informative about the timing of grants, and changes in usage patterns
would be informative about changes in this timing.
A thorough analysis of the reincidence patterns in the usage of
instruments is beyond the scope of this article, and is left for future
research. However, in order to provide a sense of how much of the
variation in instrument choice in the data is not coming from timing
of grants, we now report on a measure of the frequency of instrument
use at the individual …rm level: We calculate the fraction of years that
a given …rm is in each of the groups, or the “…rm’ time share of I:”
s
Denoting by tiI the number of years that a …rm i is in compensation
group I; and by Ti the total number of years that …rm i is in the sample,
…rm i’ time share of I is de…ned
s
tiI
iI
Ti
for each I in I. To give more meaning to the extreme values iI = 0 and
iI = 1, we construct a balanced subset of the sample that includes only
…rms that we observe for at least six years (Ti 6; a total of 489 …rms
out of the original full sample of 3,248 …rms). From the fact that there
are mass points at 0 (and, to a lesser extent, at 1) in the frequencies
for this subsample, we conclude that, provided compensation cycles

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Federal Reserve Bank of Richmond Economic Quarterly

Table 4 Percentage of Firms that are Never in a Given
Compensation Group
Percent of Firms
with iI = 0
Period
I
II
III

S
Bal.
.82
.80
.59

O
Full
.86
.80
.55

Bal.
.08
.26
.65

B
Full
.12
.29
.64

Bal.
.49
.39
.29

N
Full
.61
.53
.42

Bal.
.40
.65
.72

Full
.42
.60
.59

are shorter than six years, not all the …rms are following alternating
times for the inclusion of options or stock grants in their compensation
packages. This means that at least some of the variation that we see in
the data comes from meaningful choices about the usage of the di¤erent
compensation instruments.
Because the timing choices themselves may be in‡
uenced by the
regulation period, in Tables 4 and 5 we report statistics of iI by regulatory subperiod. We report this for the balanced subsample (denoted
“Bal.” as well as for our original full sample (denoted “Full”
),
).
Table 4 reports the fraction of …rms with iI = 0; i.e., they are
never in compensation group I: It shows a pattern consistent with the
evidence in our previous regression results: The fraction of …rms that
never were in group S or B decreases over time, while that of …rms
never choosing O increases. Interestingly, the increase in the fraction
of …rms with iN = 0 over time suggests that, if anything, timing
decisions have changed toward using grants more frequently.
Table 5 reports the average value of iI contingent on it being positive; that is, the average time share iI for …rms that are in compensation group I for at least one year. To report both the averages and
their signi…cances, we run an ordinary least squares regression of iI
on period dummies, for each I: The …rst column under each I reports,
for the balanced sample, the coe¢ cients for the constant (the level in
the control period, I) and the included dummies (the change in the
average iI in each subsequent period with respect to period I), while
the second column reports the same coe¢ cients for the larger sample
of …rms that are in the data for at least six years. The patterns and
signi…cances are remarkably similar across the two samples of …rms.
There is no evidence of a signi…cant decrease in the fraction of years
that …rms choose to grant options only ( iO ) in period II, while it is
signi…cant both statistically and economically in period III. There is an
important upward trend for both iS and iB ; and a lot less markedly

A. Jarque and B. Gaines: CEO Pay

341

Table 5 Mean Time Shares
Mean (

iI j iI

> 0)

S

Period

Bal.

I: Level

.26
(.03)
.15
(.04)
.31
(.04)

II: Change from
Period I
III: Change from
Period I
Adjusted R2
N

.17
387

O
Full

N

Full

Bal.

Full

Bal.

.28
.62
.65
(.02)
(.01)
(.01)
.17 [ :02] [ :02]
(.02)
(.02)
(.01)
.27
.14
.14
(.02)
(.03)
(.02)

.36
(.02)
.19
(.02)
.33
(.02)

.40
(.01)
.15
(.01)
.24
(.01)

.34
(.01)
.07
(.02)
[:05]
(.03)

.18
894

.10
2,531

.13
1,345

Bal.

B

.03
979

.03
3,532

Full
.41
(.01)
.10
(.01)
.08
(.01)

.01
.03
602 2,489

Notes: Time share iI represents the fraction of years that …rm i belongs to group
I; out of the total number of years that …rm i is in the sample. This table reports
mean time shares for each I, for …rms with positive iI . Square brackets indicate
insigni…cance at the 5 percent con…dence level.

for iN :31 That is, (1) …rms that choose O do so less frequently in period III, (2) …rms that choose S, or B, do so more frequently in the
later periods than in the initial one, and (3) …rms that choose N do so
only slightly more often in the last two periods than in the …rst one.
Since these changes in grant timing patterns align with the trends in
the usage of instruments that we have reported in Table 2, we conclude that our results could be due, at least partly, to a change in the
frequency of usage of stock and options, rather than a change in the
number of di¤erent …rms that use them.
It is important to keep in mind that the evidence on the timing
of grants that we have provided in this section is partial, since it does
not control for the amount of past grants and it only exploits the panel
aspect of the data in a limited way. It would be interesting to perform
the analysis of usage that we do here with a comprehensive measure of
the wealth of the CEO vested in the …rm at each point in time (as in
Clementi and Cooley [2010]), as a way of controlling for outstanding
incentives. This is left for future research.
31
Note that Table 5 is providing evidence for …rms that have iI > 0; and these
…rms di¤er across Is; hence, the percentages across rows do not typically sum up to 1.

342

Federal Reserve Bank of Richmond Economic Quarterly

Table 6 Shares of Total Compensation, by Instrument
Salary
(%)
All
S
O
B
N

4.

BIC
(%)

Stock
(%)

Option
(%)

Other
(%)

.32
.27
.27
.19
.59

.23
.23
.20
.20
.30

.11
.45
0
.26
0

.28
0
.49
.31
0

.06
.05
.04
.04
.10

THE IMPORTANCE OF DIFFERENT
COMPENSATION INSTRUMENTS: THE
INTENSIVE MARGIN

In the previous section we asked what determines the choice of compensation instruments. A natural complementary question to that is what
is the relative importance of each instrument in the total compensation
of the CEO. In this section, we provide some simple statistics about the
share of total compensation that salary, bonus and incentive compensation (BIC), stock grants, option grants, and “other compensation”
represent.
Table 6 documents the average of these shares in our sample, disaggregated by groups of users. The most salient feature of those statistics
is the di¤erence in the shares of grants across …rms in S, O, and B:
Firms in B have a combined share of grants of 57 percent, higher than
the shares of grants for …rms using stock exclusively (45 percent) or
options exclusively (49 percent). The share of BIC is similar for …rms
in S; O; and B; around 20 percent. In contrast, …rms in N , who do not
use stock or options, use both BIC and “other compensation” more
intensely than the rest of …rms, but the share of the only incentive
instrument, BIC, is 30 percent, well below the combined shares of incentive instruments (BIC + stock + option) of the rest of the …rms. In
other words, BIC, stock, and options do not appear to be perfect substitutes for each other. This evidence complements what we presented
in Table 3 about the relationship between the level of compensation
and the usage choices, suggesting that the relative importance of different instruments may be related to the choice of instruments through
the level of pay. We saw in Table 3 that …rms in N have levels of
total compensation between one-third and one-fourth of the rest of
…rms. In spite of this, the relative importance of the salary is much
higher for them. Hence, there seems to be a …xed component in the determinant of the salary, or a “cap,” which is somewhat independent of
whether the …rms choose to also award grants or not. The most obvious

A. Jarque and B. Gaines: CEO Pay

343

Table 7 Shares of Total Compensation, by Instrument
Salary
(%)

Freq.

BIC
(%)

Stock
(%)

Option
(%)

Other
(%)

.04
.57
.16
.24

Period I
S
O
B
N

.35
.34
.28
.22
.62

.22
.26
.20
.18
.28

.04
.33
0
.20
0

.33
0
.48
.36
0

.05
.07
.04
.04
.09

.09
.45
.26
.20

Period II
S
O
B
N

.29
.27
.25
.18
.56

.23
.26
.20
.20
.32

.10
.42
0
.25
0

.32
0
.51
.33
0

.05
.06
.04
.04
.11

.26
.19
.35
.20

Period III
S
O
B
N

.29
.25
.27
.18
.55

.23
.22
.21
.21
.32

.24
.49
0
.31
0

.18
0
.48
.27
0

.06
.05
.04
.04
.12

explanation is the limits to tax deductions for salaries above a certain
level.32 However, other factors may be important, like the need to provide incentives through variable pay. This also possibly plays a role in
explaining the di¤erence in the shares of salary across the …rms in S,
O, and B. The share of salary is the lowest (19 percent) for …rms in B;
which are the ones that have the highest total compensation according
to Table 3. However, the share of salary is equal for …rms in S than
for …rms in O; in spite of the average total compensation in S being 90
percent of that in O.
Our previous analysis has shown that the use of instruments differs importantly across subperiods, and to some extent across industry
groups. Hence, we now look at the average shares controlling for these
two variables.
Table 7 presents evidence on the changes in the relative importance of the instruments over the three di¤erent regulation subsamples.
For convenience, we replicate the sample frequencies of each group of
compensation, within a period, that we already discussed following
Table 1.
32
The Omnibus Budget Reconciliation Act Resolution 162(m) of 1992 imposed a
$1 million cap on the amount of the CEO’ non-performance-based compensation that
s
quali…es for a tax deduction. See Jarque (2008) for a review of the academic literature
that studied the e¤ects of that change of regulation on pay practices.

344

Federal Reserve Bank of Richmond Economic Quarterly

When we look at the shares for all the users together, we see that
while the shares of BIC and “other compensation” remained fairly
constant at about 23 percent and 5 percent, respectively, the share
of salary was higher in period I (35 percent as opposed to 29 percent
post-2002). The share granted in the form of options also experienced a
sharp decline, but only in period III, when it went from 32 percent to 18
percent. The share of compensation that is no longer granted through
salary after 2002 and no longer granted through options after 2006 is
granted through stock: There is an increase in the share of stock of 6 pp
in period II, and then of 14 extra pp in period III. These changes in the
share of stock over time (intensive margin) are in line with the changes
in the choice of S reported in Table 1 (extensive margin), where we
saw that …rms tended to “add”stock to their compensation package in
period II, rather than completely substitute options for stock. Note,
however, that these numbers for the share of total compensation that
are given in the form of stock are representative both of …rms in S and
B in Table 1. We discuss the data in each compensation group next.
When we look at the statistics disaggregated by user groups, we see
slightly di¤erent changes over the regulatory periods for each of them.
The most striking fact may be the increase in the share of stock, which
happens both for …rms that are in S and in B. For …rms in S, the
share of stock increases by 9 pp in period II (compensated mainly by a
decrease in the share of salary of 7 pp), and then by 7 pp in period III
(compensated mainly by a decrease in the share of BIC by 4 pp). For
…rms in B; the share of stock increases by about 5 pp each period, while
the share of options decreases (3 pp in period II, 6 extra pp in period
III). In addition, for …rms in O the share of options stays constant
overall (and it even increases by 3 pp in period II). In other words, for
the …rms that continue to rely exclusively on option grants in spite of
the regulatory hurdles, the relative importance of options with respect
to salary, BIC, and “other compensation” does not decrease. That is,
if what we observe is a response to the regulatory changes, it seems to
have taken place through the extensive margin (with …rms in O going
from 57 percent of the sample to 19 percent), rather than the intensive
one. This suggests that there might be some …xed cost to adopting a
new instrument of compensation, maybe related to accounting costs or
perhaps to communication to shareholders.
Table 8 presents the shares of each compensation instrument by
industry group. The variation in the shares across instrument users,
within a given industry group, is fairly in line with the patterns by users
that we described in Table 6, so we do not report the

A. Jarque and B. Gaines: CEO Pay

345

Table 8 Shares of Total Compensation, by Industry Group
Salary
(%)
Min/Man
FIRE
Utilities
Other

BIC
(%)

Stock
(%)

Option
(%)

Other
(%)

.32
.29
.34
.33

.22
.27
.25
.20

.11
.15
.14
.11

.31
.23
.21
.30

.05
.06
.06
.06

disaggregated numbers here.33 One main conclusion stands out from
Table 8— mining and manufacturing and other use options and salary
more intensely than do FIRE and utilities, which rely more on BIC and
stock. FIRE, which includes …nancial …rms, has in fact the lowest share
for salary. It is important to keep in mind that, as reported in Table
1, the proportion of …rms in each user group is not constant across
industry groups; this, together with the (omitted) evidence that shares
for user groups within industry align with those reported in Table 6,
implies that most of the variation across industries is due to composition e¤ects, without important industry-speci…c patterns for the shares
of each compensation instrument.

5.

CONCLUSION

In the last decade several regulatory changes took place in the United
States regarding the reporting and expensing of stock option grants.
This article provides an empirical analysis of the impact of these changes
in the composition of pay packages for CEOs at the largest U.S. …rms
from 1993 to 2010. Both the passage of the Sarbanes-Oxley Act in 2002
and the changes in accounting standards in SFAS 123R mandated by
the SEC in 2006 erased some advantage of granting options versus stock
as part of the compensation of CEOs. We …nd evidence indicating that
…rms may have responded to this by shifting away from options and
into stock. Even though, after the two regulatory changes, there is still
a signi…cant portion of …rms in the sample that choose to grant options to their CEO (about 55 percent of …rms in the 2006–
2010 period,
compared to 67 percent before 2002), alone or combined with stock,
the fraction of …rms that are awarding options but not stock in a given
year decreases (from 57 percent before 2002 to 19 percent after 2006).
33
A more detailed table with shares across industries and compensation groups is
available upon request.

346

Federal Reserve Bank of Richmond Economic Quarterly

However, while only 4 percent of …rms used exclusively stock grants
before 2002, this percentage increases over the period we analyze to
reach 26 percent after 2006.
How …rms decide whether to include options, stock, both, or none
of the two types of grants in their pay packages remains to be understood, but we …nd some regularities. Firms in …nance and in utilities
are more likely to use stock or neither, while …rms in mining and manufacturing are more likely to use options, or stock and options together.
Larger …rms tend to use stock and options together, although this e¤ect
disappears if we control for the level of pay, which is higher at larger
…rms. A higher level of pay is associated with a higher probability of
using stock and options together, or only options.
We also …nd that di¤erent compensation instruments do not appear
to be perfect substitutes within compensation packages. The relative
importance of bonuses in overall compensation has not decreased over
time, while that of the salary has, in favor of stock and option grants.
Perhaps surprisingly given the decrease in the popularity of option
grants starting in the early 2000s, the relative importance of options
in relation to the total amount of compensation has not decreased over
time for …rms that still include options in their compensation packages.

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