DERIVATIVES AND EARNINGS MANAGEMENT
By
LIN NAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF
PHILOSOPHY
UNIVERSITY OF FLORIDA
2004
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DERIVATIVES AND EARNINGS MANAGEMENT
By
Lin Nan
August 2004
Chair: Joel S. Demski
Major Department: Fisher School of Accounting
Derivative instruments are popular hedging tools for firms to reduce risks.
However, the complexity of derivatives brings earnings management concern and there
are recent accounting rules trying to regulate the use of derivatives. This dissertation
studies the joint effect of derivatives' hedging function and earnings management
function, and explores how the recent rules influence firms' hedging behavior.
The two-edged feature of derivatives is modeled by bundling a hedging
option and a misreporting option. A mean preserving spread structure is employed
to capture the risk-reduction theme of hedging. This dissertation shows a trade-off
between the benefit from hedging and the dead weight loss from misreporting. It is
shown that when the manager's misreporting cost declines with the effectiveness of
hedging, the principal's preference for the hedge-misreport bundle does not change
monotonically with the effectiveness of hedging. Specifically, when hedging is highly
effective, the principal's preference for the bundle increases in the effectiveness, while
when hedging's effectiveness is moderate, the principal's preference decreases in the
effectiveness. When hedging is only slightly effective, whether the principal prefers the
bundle is not influenced by the effectiveness. In addition, this dissertation shows that
sometimes it is not efficient to take any measure to restrain earnings management.
Recent regulations require firms to recognize the ineffective portion of hedges
into earnings. This dissertation indicates that this early recognition may change the
firms' hedging behavior. Since the early recognition increases the interim earnings'
riskiness, hedging may become inefficient even though it still reduces the total risk.
In this sense, the new regulations may not benefit investors, though their intention is
to provide more information about the risk and value of derivatives to the investors.
Copyright 2004
by
Lin Nan
To my parents Manping Wang
and Weihan Nan,
and to Laurence
ACKNOWLEDGEMENTS
I am very grateful to Joel S. Demski, my Chair, for his guidance and
encouragement. I also thank David Sappington, Karl Hackenbrack, Froystein Gjesdal,
and Doug Snowball for their helpful comments.
TABLE OF CONTENT
ACKNOWLEDGMENTS....................................................... .......... iv
A B ST R A C T ........................................................................................ v
CHAPTER
1 BACKGROUND AND LITERATURE REVIEW.............................. ..............
B ackground............................................................................ ........................1. .
Literature Review on Hedging................................................... ..................... 3
Review of Earnings Management and Information Content...............................9
Review of LEN Framework.......................................................................11
Summ ary. .................................................... ................. ... .......... ....... .14
2 BASIC MODEL........................................................... ........ 15
The M odel............................................................................. ...............15
Basic Setup..................................... .. ...... ............................ ..... 16
B enchm ark ........................................................................................................ 17
Hedging and Earnings Management Options..................................................... 19
Sum m ary................................................. ... .............................. ...................30
3 HEDGE-MISREPORT MODEL..................................................32
Bundled Hedging and Misreporting Options...................................................32
W whether to Take the Bundle................................................................... 34
Sum m ary.................................................. ........................................... ...37
4 MANIPULATION RESTRAINED BY HEDGE POSITION............................. 41
Cost of Earnings Management...................................................................41
"Strong Bundle" M odel..................................................................................41
Whether to Take the Strong Bundle.......................................................... ..... 43
Sum m ary................................................................. .... ......... 47
5 EARLY RECOGNITION MODEL ............ ... ............................ .... ..... ...... 51
Early Recognition of Hedging ...... .............................................................. 51
Centralized Case......................................... ....... ........ 53
Delegated Case...................................................................................54
Sum m ary.................... ...................... .........................................................59
6 OTHER RELATED TOPICS.......................................................................61
Riskiness and Agency....................................................................................... 61
Informative Earnings Management: Forecast Model........................................ 62
Sum m ary......................... ................................................................................ 64
7 CONCLUDING REMARKS...................................................................... 66
A P P E N D IX ............................................................................................................. 68
REFERENCE LIST........................................................................................................78
BIOGRAPHICAL SKETCH...................................................................................82
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DERIVATIVES AND EARNINGS MANAGEMENT
By
Lin Nan
August 2004
Chair: Joel S. Demski
Major Department: Fisher School of Accounting
Derivative instruments are popular hedging tools for firms to reduce risks.
However, the complexity of derivatives brings earnings management concern and there
are recent accounting rules trying to regulate the use of derivatives. This dissertation
studies the joint effect of derivatives' hedging function and earnings management
function, and explores how the recent rules influence firms' hedging behavior.
The two-edged feature of derivatives is modeled by bundling a hedging
option and a misreporting option. A mean preserving spread structure is employed
to capture the risk-reduction theme of hedging. This dissertation shows a trade-off
between the benefit from hedging and the dead weight loss from misreporting. It is
shown that when the manager's misreporting cost declines with the effectiveness of
hedging, the principal's preference for the hedge-misreport bundle does not change
monotonically with the effectiveness of hedging. Specifically, when hedging is highly
effective, the principal's preference for the bundle increases in the effectiveness, while
when hedging's effectiveness is moderate, the principal's preference decreases in the
effectiveness. When hedging is only slightly effective, whether the principal prefers the
bundle is not influenced by the effectiveness. In addition, this dissertation shows that
sometimes it is not efficient to take any measure to restrain earnings management.
Recent regulations require firms to recognize the ineffective portion of hedges
into earnings. This dissertation indicates that this early recognition may change the
firms' hedging behavior. Since the early recognition increases the interim earnings'
riskiness, hedging may become inefficient even though it still reduces the total risk.
In this sense, the new regulations may not benefit investors, though their intention is
to provide more information about the risk and value of derivatives to the investors.
CHAPTER 1
BACKGROUND AND LITERATURE REVIEW
Background
Derivatives include "a wide variety of financial and commodity instruments
whose value depends on or is derived from the value of an underlying asset/liability,
reference rate, or index" (Linsmeier 2003). Financial derivatives have been developed
rapidly and employed widely since the early 1990s. Alan Greenspan, the Chairman of
the Federal Reserve Board, said in a speech before the Futures Industry Association
in 1999 that "by far the most significant event in finance during the past decade has
been the extraordinary development and expansion of financial derivatives."
Financial derivatives are very popular tools for risk-reduction in many firms
and the global derivatives market has grown rapidly. At the end of June 2003,
the Bank for International Settlements estimated that the total estimated notional
amount of over-the-counter (OTC) derivatives contracts stood at US$169.7 trillion
with a gross market value of $7.9 trillion.1 The derivatives have played an important
role in the firms' risk management activities.
On the other hand, the complexity of financial derivatives raises investors'
concern about how the derivatives change the firms' performance, and suspicion about
whether the derivatives are abused in the interest of the management. During the
past ten years, there are numerous scandals involving the use of derivatives. For
example, in March 2001, a Japanese court fined Credit Suisse First Boston 40 million
Yen for using complex derivatives transactions to conceal losses. Also in 2001, Enron,
the seventh largest company in the United States and the largest energy trader in
'Notional amount is the market value of the hedged items whose risk is hedged by the use of
derivatives. It is different from the market value of the derivatives.
the world, collapsed. Investigations revealed that it had made extensive use of energy
and credit derivatives to bolster revenues.
Responding to the investors' concern, the Financial Accounting Standards
Board (FASB) recently issued several new statements on the measurement and disclosure
of derivatives, including SFAS 133, 137, 138 and 149. There are also tons of detailed
implementation guidance from the Emerging Issues Task Force (EITF). As of the
November 12-13, 2003, EITF meeting, there are at least 38 issues directly addressing
the accounting for derivative instruments.
Although the FASB makes all these new regulations, it is unlikely that
the new regulations will eliminate earnings management through derivatives. In
addition, some new rules may even provide new avenues for earnings management.
For example, SFAS 133 requires firms recognize both derivatives and hedged items
at their fair value, even before the settlement of the derivative contracts. When the
increase/decrease in a derivative's fair value cannot offset the decrease/increase in its
hedged item's fair value, the uncovered part is regarded as the ineffective portion of the
hedge and is recorded immediately into earnings. However, evaluating the "fair value"
of unsettled derivatives is often subjective. Managers can either estimate the fair value
based on the current market price of other derivatives, or invoke "mark-to-model"
techniques. With a subjective estimation of the fair value, the estimation of the
ineffective portion is also discretionary.
Derivative instruments, therefore, have a two-edged feature. They can be
used as tools to reduce firms' risks (that is, tools of hedging), but they can also be
used as tools of earnings management. This two-edged feature provides us an ideal
setting to study the joint effect of hedging and earnings management, which is one of
the focuses of this dissertation.
Literature Review on Hedging
Among the research on hedging activities, the incentives for hedging, how
hedging activities influence contracting between the principal and the manager, and
how the FASB regulations affect hedging behavior are most explored.
Hedging Incentives
First, for the research on hedging incentives, there are some finance studies
on why firms hedge. Smith and Stulz (1985) analyze the determinants of firms'
hedging policies from the perspective of firm value. They examine taxes, bankruptcy
cost, and managers' wealth. Regarding hedging activities as a kind of insurance,
Mayers and Smith (1982) use the theory of finance to analyze the corporate demand
for insurance from the perspectives of taxes, contracting costs and the impact of
financing policy on firms' investment decisions.
Smith and Stulz (1985) conclude that hedging reduces a firm's tax liability if
the post-tax firm value is a concave function of the pre-tax firm value. Nance, Smith,
and Smithson (1993) provide the empirical evidence that firms with more of the range
of their pretax income in the statutory progressive region of the tax schedule, or have
more tax preference items, are more likely to hedge. This supports Smith and Stulz's
statement on the corporate hedging incentive from tax liability. However, Graham
and Rogers (2002) quantify the convexity-based benefits of hedging by calculating
the tax savings that result from hedging and do not find evidence that firms hedge
in response to tax convexity.
Smith and Stulz (1985) also study hedging from the perspective of the
manager's wealth. They indicate that the manager would like to hedge if his end-of-period
wealth is a concave function of the end-of-period firm value, while the manager would
not hedge if his end-of-period wealth is a convex function of firm value. Rogers
(2002) considers both the manager's stock holdings and option holdings, and finds
the risk-taking incentives from options are negatively associated with the use of
derivatives. This evidence is consistent with Smith and Stulz's suggestion that the
manager's wealth plays a role in hedging decision, and is also consistent with the
notion that derivatives are to reduce firms' risks. Mayers and Smith (1982) argue that
the corporation does not "need" insurance to reduce the stockholders' risk since the
stockholders can eliminate insurable risk through diversification. Instead, insurance is
optimally designed to shift the risk of managers and employees to stockholders, since
the managers and other employees do not have enough ability to diversify claims on
human capital. It then follows that the higher the employees' fraction of the claims
to the firm's output, the higher the probability that the firm will purchase insurance.
Smith and Stulz (1985) point out that hedging lowers the probability of
incurring bankruptcy costs. Graham and Rogers (2002) show empirical evidence that
firms with more expected financial distress hedge more, which is consistent with the
idea that firms hedge to reduce risks. Nance, Smith and Smithson (1993) find that
firms who have less coverage of fixed claims (a proxy for financial distress) hedge more,
which is another piece of empirical evidence to support the risk reduction incentive.
Contrary to works such as Smith and Stulz (1985) and Nance, Smith and
Smithson (1993), Schrand and Unal (1998) emphasize that hedging is a means of
allocating a firm's risk exposure among multiple sources of risk rather than reducing
total risk. However, in some sense, their allocation theory is still consistent with the
risk-reduction theory. In Schrand and Unal's paper, the total risk is reflected in the
variability of a loan portfolio's cash flow. They further classify the risk into two types:
core-business risk, from which the firm can earn economic rent for bearing since it
has an information advantage in the activities related to this risk; and homogenous
risk, where the firm does not have any information advantage and cannot earn rent.
Schrand and Unal claim that the firms use hedging to increase core-business risk and
reduce homogenous risk. In other words, the firms confidently play with core-business
risk in which they have expertise and more "control," but reduce the uncontrollable
homogenous risk by hedging. The uncontrollable homogenous risk is the real risk for
the firms.
Though there are a number of incentives for hedging activities, here in this
paper I focus on the risk-reduction incentive. Risk reduction is the initial intention
of the invention of various financial derivatives, and it is the main purpose of hedging
activities.
Hedging and Contracting
Secondly, for the research on contracting, there are some, but not many,
studies on how the hedging activities affect contracting. Campbell and Kracaw
(1987) focus on optimal insurance through hedging by the manager. They show that
under certain incentive contracts, shareholders will be hurt by the manager's hedging
behavior since the manager will deviate from the optimal managerial effort level with
the acquisition of insurance. However, if the contract anticipates the hedging, then
the shareholders will benefit from that hedging. The reason is that the shareholders
can reduce fixed payments to the manager to acquire the direct gain from hedging
reaped by the manager and they can induce the manager to exert more effort by
raising the manager's share of risky returns. Their result is consistent with this
paper's conclusion that hedging reduces compensating wage differentials in a LEN
framework. However, unlike the model in this paper, they assume the outputs are
always public and their work does not consider earnings management. In addition,
their analysis focuses on how hedging influences firm value, while in the current paper
my focus is how hedging influences the information content of performance measures.
Another related paper is DeMarzo and Duffie (1995), who also analyze
hedging behavior from the perspective of information content. They show that
financial hedging improves the informativeness of corporate earnings as a signal of
managerial ability and project quality. However, in their model, the manager's action
is given, so there is no need to motivate the manager to work diligently. Moreover,
unlike the model in my work, their model does not consider manipulation of earnings
by the managers.
Recent Hedging Regulations
Thirdly, recent new FASB regulations on hedging have triggered some research
on how the new rules affect firms' hedging behavior. The recent rapid development
of various financial derivatives and the complexity of derivatives raise the concern
that the explosion in their use may endanger investors' interest, since it is difficult for
investors to understand from the financial reports the magnitude, value and potential
risk of the derivatives. Since 1994, the FASB has issued several new regulations on
the disclosure of derivatives. Among these regulations, SFAS 119 (which was later
superseded by SFAS 133) requires more disclosure on the use of derivatives, and
SFAS 133 requires that an entity recognize all derivatives as either assets or liabilities
in the statement of financial position and measure those instruments at fair value.
These regulations are complex, to say the least, and have led to considerable debate.
Responding to the controversy, the FASB issued SFAS 137 to defer the effective date
for SFAS 133 from June 1999 to June 2000, and later also issued SFAS 138 and 149
as amendments of SFAS 133.
Recent research provides both pros and cons of the hedging disclosure policy.
Barth et al. (1995) respond to the FASB Discussion Document about hedge accounting
and the FASB Exposure Draft. They argue that a mark-to-market accounting is the
best approach to accounting for hedging activities and they support the disclosure
policy. They claim that disclosures about management objectives in using derivatives
for risk management are useful to financial statement users and are value relevant.
Kanodia, Mukherji, Sapra, and Venugopalan (2000) examine the social benefits
and costs under different hedge accounting methods from a macroeconomic view.
They claim that hedge accounting provides information about firms' risk exposure to
the market, thus helps make the futures price more efficient.
Melumad, Weyns, and Ziv (1999) also compare different hedge accounting
rules, but they focus on the managers' decision making process. They show that
under comprehensive fair-value hedge accounting, investors can infer the necessary
information from the reported earnings for the valuation of the firm, though the
market does not directly observe intermediate output and hedge position. In this
case the managers choose the same hedge position as in the public information case.
In contrast, under the no-hedge accounting rules, the managers choose a lower hedge
position than in the public information case. This is because the investors cannot
infer new information from the earnings report to revise their evaluation and the
market price at the interim date has less variance. Melumad, Weyns and Ziv support
hedge accounting against no-hedge accounting, but they believe only comprehensive
fair-value hedge accounting is efficient while recent accounting regulations, e.g., SFAS
No. 133, only require limited fair-value hedge accounting or cash-flow hedge accounting.
On the other side, DeMarzo and Duffie (1995) argue that it is optimal for
the shareholders to request only the aggregate output instead of disclosure of hedging
positions. The reason is that with nondisclosure of hedging positions, managers choose
risk minimization (full hedging) since reduced output variability leads to a more
stable wage. However, when the hedging positions are required to be disclosed, the
managers deviate from optimal full hedging since disclosed hedging eliminates noise
from the outputs and makes the outputs more sensitive signals of the managers'
ability. Holding the variability of output fixed, this disclosure makes the managerial
wage more variable.
A related concern is that the recorded value for derivatives can be manipulated
by managers. A derivative's "fair value" refers to the current market price of similar
derivatives. However, it is often difficult to find "similar" derivatives. Managers then
must estimate the "fair value" of the derivatives based on the current market price of
other derivatives or invoke "mark-to-model" techniques. This leaves the "fair value"
of the derivatives up to the managers' discretion. In this sense, the new accounting
rules for derivatives provide a new field for earnings management, and the complexity
of many derivatives is itself a choice variable. Therefore, derivatives can function
both as tools for hedging and for earnings management. Even in Barth, Elliott et al.
(1995) and Barth, Collins et al. (1995), they admit that the mark-to-market approach
is arbitrary and creates opportunities for earnings management, though they insist
that the "fair value" approach to report hedging activities is still the best option.
Among empirical studies, Barton (2001) is probably the first to study the
interaction between hedging and earnings management. He measures derivatives using
notional amounts and discretionary accruals using the modified Jones model. Barton
finds that firms with larger derivatives portfolios have lower levels of discretionary
accruals. He suggests managers use discretionary accruals and derivatives as partial
substitutes to smooth earnings so as to reduce agency costs, income taxes, and
information asymmetry, and to increase personal wealth and utility. Since he believes
that the earnings management through accruals and derivatives use are some kind of
substitutes, his paper suggests that SFAS No. 133 may cause fewer hedging activities
but more accrual management. Unlike the analysis of the current paper, Barton's
study only considers the hedging function of derivatives and does not consider the
earnings management function of derivatives.
Focusing on the oil and gas industry, Pincus and Rajgopal (2002) also explore
the interaction between accrual management and hedging. Their empirical result is
partially consistent with Barton's result. However, they emphasize that their result
suggests a sequential process whereby the managers first decide how much to hedge
and manage the accruals only during the fourth quarter, which weakens Barton's
substitution hypothesis.
Review of Earnings Management and Information Content
Literature on earnings management is as vast as the ocean. There are
numerous studies attempting to document the existence of earnings management,
and a lot of research on when and why the managers manage the earnings. I am
not so ambitious as to review all related literature, not to say there are already some
good reviews of the earnings management literature, such as Schipper (1989), Healy
and Wahlen (1999), Beneish (2001), and Lev (2003). Here I focus on the literature
on "efficient" earnings management.
For empirical studies, researchers usually use the market reaction to an
earnings announcement to measure the information content in the earnings. Abnormal
discretionary accruals are usually regarded as proxies for managers' manipulation
and many studies focus on whether the market can "see through" the managed
earnings. The hypothesis is that if the market is efficient, then the investors will not
be fooled by the manipulation. There are further studies on who is more easily fooled
(unsophisticated investors). The unspoken words here are that the manipulation
of earnings is "bad," and it is just a garbling of the information. This opinion
about earnings manipulation is even stronger among investors and regulators after
the scandals of Enron and WorldCom. The SEC has been taking measures to fight
earnings management. Walter P. Schuetze, a former chief accountant in the SEC,
even proposes mark-to-market accounting to eliminate the manipulation of earnings.
However, recently researchers began to question whether it is worth eliminating
managers' manipulation. First, the elimination of discretion will shut the doors of
communicating managers' private information to investors. Taken to the extreme, if
we totally eliminate any manipulation and return to a mark-to-market accounting,
then the existence of accounting is meaningless. Financial statements should be
sources of information and communication tunnels between management and investors.
If financial statements were reduced to be just records of the investors' estimates
about the firms' operating activities based on other information, then we did not
"need" financial statements or accounting. Accrual accounting provides a tunnel for
the managers to use their judgement and discretion to better communicate with the
financial statements users. Beneish (2001) states that there are two perspectives
on earnings management. One is the opportunistic perspective, which claims that
managers seek to mislead investors. The other is the information perspective, which
regards the managerial discretion as a means for managers to reveal to investors their
private expectations about the firm's future cash flows. Similarly, Dechow and Skinner
(2000) call for attention on how to distinguish misleading earnings management
from appropriate discretion. They further indicate that it is hard to distinguish
certain forms of earnings management from appropriate accrual accounting choices.
Demski (1998) demonstrates that in a setting with blocked communication, when
the manager's manipulation requires his high effort level, it may be efficient for the
principal to motivate earnings management. If we eliminate discretion in order to put
a stop to the "detrimental" earnings management, we may close the communication
channel between the managers and the investors.
Secondly, eliminating earnings management, even if earnings management
merely garbles information, may be too costly. Liang (2003) analyzes equilibrium
earnings management in a restricted contract setting and shows that the principal
may reduce agency cost by tolerating some earnings management when the contract
helps allocate the compensation risk efficiently. Arya, Glover and Sunder (2003) tell
a story of "posturing," that when the commitment is limited and the information
system is not transparent, allowing manipulation is more efficient than forbidding it.
Demski, Frimor and Sappington (2004) show that assisting the manager to manipulate
in an early period may help ease the incentive problem of a later period. In their
model the principal's assistance reduces the manager's incentive to devote effort to
further manipulation and induces the manager to devote more effort to production.
As stated in Arya, Glover and Sunder (1998), when the Revelation Principle
is broken down (that is, when communication is limited, the contract is restricted,
or commitment is missing), earnings management may be efficient. In this paper,
in most cases the existence of earnings management in equilibrium comes from the
violation of the unrestricted contract assumption, since I assume a linear contract. I
also suppose an uneven productivity setting to induce earnings management, following
a design in Liang (2003). This paper shows that, even if earnings management merely
destroys information, sometimes it is not only inefficient to motivate truth-telling but
also inefficient to take any measure to restrain earnings management, since restraining
manipulation may be too costly. In the last chapter of this paper, I also include an
analysis of informative earnings management (the forecast model), in which earnings
management conveys the manager's private information.
Review of LEN Frameworks
Most of my analysis in this paper is in a LEN framework. LEN (linear
contract, negative exponential utility function, and normal distribution) is a helpful
technology for research in agency and has been employed in more and more analytical
research. Among the three assumptions of the LEN framework, exponential utility
and normal distribution have been widely used and accepted, while the linear contract
assumption is more controversial.
Lambert (2001) gives a good review on LEN. He summarizes three common
justifications for the linear contracting setting. The first is that according to Holmstrom
and Milgrom (1987), a linear contract is optimal in a continuous time model where
the agent's action affects a Brownian motion process. However, it is not easy to fit
their model into multi-action settings. The second justification is that the contracts
in practice are usually simple, instead of in the complicated form of the optimal
contracts from agency models. The argument against this justification, however, is
that even in practice, contracts are not strictly linear. The third justification is
tractability. Linear contracts are usually not the optimal contracts, but they provide
great tractability and help researchers explore some agency questions that were hard
to analyze in the conventional agency models. For example, LEN is especially helpful
in multi-action or multi-period agency problems. This is the most important reason
that LEN has become more popular. However, as mentioned by Lambert, we achieve
this tractability with a cost of restricting the generality. Nevertheless, except for
some questions that cannot be addressed by linear assumptions such as the study on
contract shapes, the LEN setting still provides plentiful insight.
Feltham and Xie (1994) are among the first to employ LEN in analytical
analysis. They focus on the congruity and precision of performance measures in a
multidimensional effort setting, where the agent allocates his effort on more than one
task. They show that with single measure, noncongruity of the measure causes a
deviation from the optimal effort allocation among tasks, and noise in the measure
makes the first-best action more costly. In their setting, the use of additional measures
may reduce risk and noncongruity.
There are some works on selecting performance measures for contracting
purposes using LEN frameworks. Indjejikian and Nanda (1999) use a LEN framework
to study the ratchet effect in a multi-period, multi-task contract. They show that in a
two-period setting, when the performance measures are positively correlated through
time, and when the principal cannot commit not to use the first period performance
in the second period contracting, the agent is inclined to supply less effort in the
first period, since a better first period performance increases the "standard" for the
second period performance. To motivate the agent to work hard in the first period,
the principal has to pay more for the first period. Indjejikian and Nanda also suggest
that an aggregate performance measure may be better than a set of disaggregate
measures, and consolidating multi-task responsibility to one agent may be better
than specializing the responsibilities to avoid the ratchet effect. Autrey, Dikolli and
Newman (2003) model career concerns in a multi-period LEN framework. In their
model, there are both public performance measures and private measures that are
only available to the principal. Their work shows that the public measures create
career concerns and help the principal reduce the compensation to the agent, while
the private measures enable the principal to reward the agent more efficiently. They
suggest that it is better to use a combination of both public and private measures in
contracting.
LEN technology is also employed in the research on valuation and accrual
accounting. Dutta and Reichelstein (1999) adopt a multi-period LEN framework in
part of their analysis on asset valuation rules. In a setting where the agent's effort
affects the cash flows from operating activities, they show that incentive schemes
based only on realized cash flows are usually not optimal since it is difficult for the
principal to eliminate the variability in cash flow from financing activities.
Discretionary reporting and earnings management are also among the topics
explored using LEN frameworks. Christensen and Demski (2003) use a covariance
structure to model reporting schemes (either conservative or aggressive) under a
linear contract in a two-period setting. They discuss when reporting discretion is
preferred to inflexible reporting and when it is not, and further explore the role
of an auditor in this setting. They use an asymmetric piece-rate to model the
incentive for the exercise of reporting discretion. Similarly, Liang (2003) uses a
time-varying production technology and therefore uneven bonus schemes through time
to explore earnings management. He studies the equilibrium earnings management
in a two-period, multi-player (managers, shareholders, and regulators) setting. His
work shows that a zero-tolerance policy to forbid earnings management may not be
economically desirable.
In this paper, LEN provides great tractability, and also induces earnings
management.
Summary
Prior research on derivatives explores the incentives for hedging, how hedging
influences contracting, and the pros and cons of the recent derivatives disclosure
regulations. Derivatives are popular instruments for hedging. However, the complexity
of derivatives also makes them handy tools for managers to manipulate earnings. Up
to now there is little theoretical research on the joint effect of the hedging function
and the earnings management function of derivatives, though there is a lot of research
on either hedging or earnings management.
In addition, this chapter also provides a review of the LEN framework,
which is a component of my following models to study the joint effect of hedging and
manipulation.
CHAPTER 2
BASIC MODEL
To study the joint effect of hedging and earnings management through the
use of derivatives, I use a two-period LEN model. A no-hedging, no-misreporting
case is employed as the benchmark. The hedging option and the misreporting option
then will be introduced into the benchmark to study the influence of hedging and
manipulation.
The Model
The main model in this paper is a two-period model in a LEN framework.
There is a risk neutral principal and a risk averse agent (manager). The principal tries
to minimize her expected payment to the manager while motivating the manager to
choose high as opposed to low actions in both periods. The manager's preference for
total (net) compensation is characterized by constant absolute risk aversion, implying
a utility function of u(S c) = -e-r(-), where S is the payment to the agent, c is
the manager's cost for his actions, and r is the Arrow-Pratt measure of risk aversion.
Without loss of generality, the manager's reservation payment is set at 0. In other
words, his reservation utility is -e-r().
Performance signals (outputs) are stochastic, and their probability is affected
by two factors: the manager's action and some exogenous factor. The manager's
action is binary. In each period, the manager either supplies low action, L, or high
action, H, H > L. Without loss of generality, L is normalized to zero. The manager's
personal cost for low action is zero. His personal cost for high action is C > 0 in each
period. The principal cannot observe the manager's actions. An exogenous factor
also affects realized output. The effect of this exogenous factor on the output can be
hedged at least partially by using derivatives. Neither the principal nor the manager
can foresee the realization of the exogenous factor. Here, "output" represents a noisy
performance measurement of the manager's action levels (e.g., earnings); "output"
does not narrowly refer to production and can be negative. I use x1 to represent the
output for the first period, and x2 to represent the output for the second period.
Basic Setup
Assume x1 = klal + ei and x2 = k2a2 + E2, with ai E {H, 0}, i E {1, 2}. ai
represents the action level for period i. k1, k2 are positive constants and represent
the productivity in the first and the second periods, respectively. Suppose k, > k2.
The uneven productivity follows a design in Liang (2003). The different productivity
induces different bonus rates through time and is important for the ensuing of earnings
management. (The assumption ki > k2 is relaxed in Chapter 5.) The vector [e, e2]
follows a joint normal distribution with a mean of (0, 0]. There is no carryover effect
of action, and the outputs of each period are independent of each other.'
If the outputs are not hedged, the covariance matrix of [E, 2] is E =
o2 0
S If the second period output is hedged (as I discuss in a later section,
0 22 o
any such hedge is confined to the second period), the matrix is Ed =
0 O
a2 < a2. The hedging process is stylized with a mean preserving spread structure:
assuming the same action level, the hedged production plan has a lower variance,
aO, than that of the unhedged one, a2, though they share the same mean. Thus the
unhedged production plan is a mean preserving spread of its hedged counterpart. In
this way, hedging lowers the variance of output due to the uncontrollable exogenous
factor and reduces the noisy output risk. This structure captures the risk reduction
theme of Rothschild and Stiglitz (1970), and also offers tractability.
SThe conclusions in this paper persist when the outputs have a non-zero covariance.
The manager's contract or compensation function is assumed to be "linear"
in the noted output statistics. Specifically, S = S(xl, X2) = W + axl + /32, where
W is a fixed wage, and a and 3 are the bonus rates respectively assigned to the first
period output, xl, and the second period output, x2.
Benchmark
The benchmark is a public-output, no-hedge-option model. There is no
option to hedge in this benchmark, and earnings management (misreporting actual
outputs) is impossible since the output for each period is observed publicly.
To solve the principal's design program in this benchmark, I start from
the second period. To motivate the manager's high action in the second period, the
principal sets the contract so that the manager's certainty equivalent, when he chooses
high action, is at least as high as that when he chooses low action, for each realization
of xi. Denote the manager's certainty equivalent when he chooses a2 given x, at the
beginning of the second period as CE2(a2; xl), the incentive compatibility constraint
for the second period is
CE2(H; xl) CE2(0; x), V x (IC2)
With x1 known and x2 a normal random variable with mean k2a2 and
variance a2, it is well known that
CE2(a2; l) = W + ax + Ok2a2 c(al, a2) 23a2.
(c(al, a2) represents the manager's cost for his actions.) Thus, (IC2) can be expressed
as
W + ax + k2H C 32a2 > W + axI j2a2,
which reduces to > j, regardless of xl.
Denote the manager's certainty equivalent at the beginning of the first period
when he chooses al followed by a2 regardless of xl as CE (al; a2). To motivate al = H
given high action in the second period, the incentive compatibility constraint for the
first period is
CEI(H; H) 2 CE(0; H) (IC1)
If a2 = H, regardless of x1, (IC2) is satisfied, then S(zx, X2) is a normal
random variable with mean W + akcial +,3k2H and variance a2c2 + /322. Thus (IC1)
implies W+akiH+ 3k2H- 2C- !(a 2 +2 +22) > V W+ k2H -C- (2a22 + 2a2),
which reduces to a > C
kiH
The individual rationality constraint requires the manager's certainty equivalent
when he chooses high actions in both periods is not lower than his reservation wage,
normalized to 0. The individual rationality constraint therefore is
CE (H; H) > 0 (IR)
Expanding (IR), we get W + aklH + fk2H 2C --(cT2a2 + /2U2) > 0.
The principal minimizes her expected payment to the manager, E[W +
a(kiH + e1) + f(k2H + E2)] = W + oakH + 3k2H. Her design program in this
benchmark model is
min W +akiH+3k2H Program[A]
s. t. W + akH + k2H 2C -(a2 2q2) > 0 (IR)
a> (IC1)
i kiH (IC2)
The individual rationality constraint must be binding, as otherwise the
principal can always lower W. Thus, the optimal fixed wage must be -akiH -
Ok2H+2C+ (a2a2+/ 32u2), and the principal's expected cost is 2C+((2a2 a2+f2a2).
Therefore, the principal's design program reduces to the minimization of (a (2a2 +
02a2) subject to the two incentive constraints.2 The optimal fixed wage is chosen to
ensure that the individual rationality constraint binds. I therefore focus on the bonus
rates in the optimal contracts in our analysis. Denote a* as the optimal first period
bonus rate and /3 the optimal second period bonus rate, we now have
2This result has been shown in, for example, Feltham and Xie (1994).
Lemma 2.1: The optimal contract in the benchmark model exhibits a* = -- and
P* C
A k2H
Proof: See the Appendix.
In a full-information setting the principal only needs to pay for the reservation
wage and the personal cost of high actions, 0 + 2C. In the present benchmark setting,
the principal needs to pay 0 + 2C + (aoU2 + f32U2). The principal pays more since
the manager bears compensation risk with a risk premium or compensating wage
differential of (a*2a2 + 0322).
Next I introduce the hedging and earnings management options.
Hedging and Earnings Management Options
Hedging Option
Initially suppose the second period output can be hedged, but no possibility
of managing earnings is present. In practice, a hedging decision is usually made to
reduce the risk in the future output. To capture this feature, assume that the hedging
decision is made at the beginning of the first period, but the hedge is for the second
period output only and doesn't influence the first period output.
Recent FASB regulations on derivatives, e.g. SFAS No. 133, require that
firms recognize the ineffective portion of hedges into earnings even before the settlement
of the derivatives." In this chapter I do not consider the recognition in earnings
from unsettled derivatives. (That is, I assume hedging only influences the output
of the second period, when the hedge is settled.) The estimation of the hedge's
ineffectiveness involves earnings management, and I will address the manipulation
associated with the use of derivatives later.
3Consider a fair value hedge as an example. At the date of financial reporting, if the
increase/decrease in the fair value of the derivative doesn't completely offset the decrease/increase
in the fair value of the hedged item, the uncovered portion is regarded as the ineffective portion of
the hedge, and is recognized into earnings immediately. However, this gain or loss from unsettled
derivatives is not actually realized, and the estimation of the hedge's ineffectiveness is usually
subjective (for the evaluation of derivatives' fair value is usually subjective).
Centralized-hedge case
First, consider a centralized-hedge case, where the principal has unilateral
hedging authority. (Later in this chapter I will delegate the hedging option to the
manager.) Notice that the benchmark is identical to the case here if the principal
decides not to hedge.
If the principal hedges, the principal's design program changes slightly from
the one in the benchmark. The expected payment is still IV + aklH + f3k2H. The
incentive constraints for the manager remain the same, since the hedging decision is
not made by the manager and the action choice incentives are unaffected by hedging
activities. However, the individual rationality constraint changes to be W + akH +
/3k2H 2C -!(a2.2 + 32 o) > 0.
The principal's design program in the centralized-hedge model when she
hedges is
min W + aklH + /k2H Program [B]
W,a,3
s. t. W + akH +3k2H- 2C --a(L22 + /32) > 0 (IR)
a > c (IC1)
kjH
S> !H( (IC2)
I use a*, 0* to denote respectively the optimal bonus rates in the first and
the second periods in Program [B]. Paralleling Lemma 2.1, I immediately conclude
Lemma 2.2: The optimal contract in the centralized-hedge model exhibits a* = kiH
and /B = k2H
Proof: See the Appendix.
The optimal contract shares the same bonus rates with that in the benchmark,
because the manager's action affects the output mean, while hedging only affects the
output risk. As implied by the (IR) constraint, when there is no hedging option or
when the principal does not hedge, the principal's expected payment is 2C+ (Oa*2a2+
/2a2), while its counterpart with hedging is 2C + (~o2 + Af dj). With hedging,
her expected cost is reduced by ~2 (a2 -oU). Obviously, the principal prefers hedging
to no hedging. Using d = 0 to represent the strategy of no hedging, and d = 1 to
represent the strategy of hedging, we have
Lemma 2.3: The principal prefers d = 1 in the centralized-hedge model.
Proof: See the above analysis.
The principal's expected cost is lower when she hedges, because hedging
reduces the noise in using output to infer the manager's input, and thus provides a
more efficient information source for contracting. Therefore, the compensating wage
differential is reduced.
Delegated-hedge case
Next, I change the setting into a delegated one. In practice, managers, not
shareholders, typically decide on the use of derivative instruments, since the managers
usually have expertise in financial engineering. To capture this fact, I change the
model so the manager, rather than the principal, makes the hedging decision. This
decision is made by the manager at the beginning of the first period, but hedges
the second period's output. The time line of this delegated-hedge model is shown in
Figure 2-1.
It has been shown that when the principal makes the hedging decision,
she prefers hedging since hedging reduces the compensating wage differential. The
question now, is whether hedging is still preferred when the hedging decision is
delegated to the manager.
With the hedging decision delegated to the manager, although the manager
has the option not to hedge, the manager will always choose to hedge. This is
because hedging reduces the output variance and therefore reduces the manager's
compensation risk derived from noisy output signals.
To illustrate this conclusion, I use CE (al, d; a2) to denote the manager's
certainty equivalent at the beginning of the first period when he chooses al and d in
the first period and chooses a2 in the second period. By hedging,
CE,(al, 1; a2) = W + akla, + 3k2a2 c(ai, a2) r(a2 2 + 020 2).
If the manager doesn't hedge.
CE (ai, 0; a2) = W + aklal + 3k2a2 c(al, a2) (au2 + 02a2).
Since a, 3 > 0, CE, (al, 1; a2) > CE1(al, 0; a2) for o2 < (2. Therefore d = 1
is always preferred.
Lemma 2.4: For any action choice, the manager always prefers d = 1.
Proof: See the above analysis.
As with the centralized model, allowing the manager to hedge the output is
efficient. The manager gladly exercises this option and in equilibrium the compensating
wage differential is reduced.
Proposition 2.1: Hedging is efficient regardless of whether the manager or the
principal is endowed with unilateral hedging authority.
Proof: See the analysis in this section.
In the LEN framework, hedging lowers the output variance but has no effect
on the output mean, while the manager's action affects the output mean but not the
variance. Therefore, there is separability between the action choices and the hedging
choice. The optimal bonus rates are not affected by the hedging choice.
Misreporting Option
To this point, I have focused on settings where the realized output is observed
publicly. Here I introduce the option to manipulate performance signals. I presume
hedging is not possible in this subsection, but will combine both hedging and earnings
management options later in the next chapter.
Suppose the output for each period is only observable to the manager. The
manager chooses the first period action level, al e {H, 0}, at the beginning of the first
period. At the end of the first period the manager observes privately the first-period
output xl. He reports x 6E {xl, xz A} to the principal and chooses his action level
for the second period, a2 E {H, 0}. At the end of the second period, again the manager
observes privately the second-period output, 2, and reports x2 = Xz 21 + x2. The
principal observes the aggregate output of the two periods at the end of the second
period and pays the manager according to the contract. The linear contract here
becomes S = W + ail + T2.
The manager may have an option to misreport the output by moving A
from the first to the second period. (A can be negative. Negative A implies that
the manager moves some output from the second to the first period.) Assume the
manager manipulates at a personal cost of -A2, which is quadratic in the amount of
manipulation.4 The manager faces the misreporting option with probability q, and
he doesn't know whether he can misreport until the end of the first period.5 I use
m = 1 to represent the event that the misreporting option is available, and m = 0 to
represent its counterpart when the misreporting option is unavailable. The time line
is shown in Figure 2-2.
Notice in the benchmark case of Lemma 2.1 where there is no option to
misreport, the bonus rates for the two periods are not equal, and a* = < 3* =
__. If the outputs were observed privately by the manager, the manager would have
a natural incentive to move some output from the first to the second period, since he
receives greater compensation for each unit of output produced in the second period.
If m = 1, the manager's certainty equivalent at the beginning of the second
period becomes
W + axi + 3k2a2 + ( a)A A2 c(a, a2) 322.
Notice that with a linear contract, the manager's manipulation choice is separable
from his action choices and the output risk. This separability implies that the optimal
4The quadratic personal cost follows a similar design in Liang (2003). It reflects the fact that
earnings management becomes increasingly harder when the manager wants to manipulate more.
5The manager, even though determined to manipulate, may not know whether he can manipulate
at the beginning, but has to wait for the chance to manipulate.
"shifting" occurs where '[(l a)A _A2] = 0, or A*= 1 a. The only way to
deter manipulation in this setting is to set a = 3.
I again solve the principal's design program starting from the second period.
Since the manager always chooses A* = / a as long as he gets the misreporting
option, we use CE2(a2, A*; x1, m = 1) to denote the manager's certainty equivalent at
the beginning of the second period when he gets the misreporting option and chooses
a2 after privately observing the first period output xl. We use CE2(a2; X, m = 0)
to denote the manager's certainty equivalent when he doesn't get the misreporting
option. The incentive constraints for the second period are CE2(H, A*; xI, 1) >
CE2(0, A*; xi, 1), and CE2(H; x1, 0) > CE2(0; x1, 0), Vxl. With the noted separability,
it is readily apparent that both constraints collapse to 3 > C-, just as in the
k2H'
benchmark.
I use CEl(al;a2) (to distinguish from CEl(al;a2) in the benchmark) to
denote the manager's certainty equivalent at the beginning of the first period when
he chooses al followed by a2 in the second period. At the beginning of the first period,
since m is random, the manager's expected utility at the beginning of the first period
is: (1 q)Eu[W +a(kial + fe) + 3(k2a2 + 62) c(a, a2)] +qEu[W + a(klal + e- A)
+P(k2a2 + 2 A*) c(al, a2) 1A*2]. Therefore, CE1(al; a2) is the solution to
u(CEl(a; a2)) = (1-q)Eu[W +a(klal +l)+3(k2a2+ 2) -c(al, a2)] +qEu[W +
a(kial + eI A*) +f(k2a2 + E2 + A*) -c(a, a2) 2]
= -(1 q)e-r[lW+aklal +k2a2-c(al,a2)- 1 (a2 +O2U2)J
_qe-r[W+aklal+Ok2a2-c(a1,a2)- ~(a 2a+/02a2) -A*2+( -4 )A*j
= _e-r[W+ak1ia1+k2a2-c(ai,a2)- (a2o+ 2a2)] [(1 -rq[( -)A* -A*2]
= -e-r(CEl(al;a2))
Thus, -r(CE1(a(;a2)) = -r[W + akial + 3k2a2 c(a,) -(a2 2 +
032a2)] + ln[(l q) + qe-(-)2], and CE (al; a2) = W + akial + Ok2a2 c(al, a2) -
2(a2"2 + 32a2) 1ln[(1 q) + q- ].
Comparing CE (al; a2) with CE (aa; a2) in the benchmark, we have CE, (a,; a2)
= CE (a,; a2) i ln[(l q) + qe- -)]
Importantly, now, the agent's risk premium reflects the summation of the
earlier risk premium, due to the variance terms, and an additional component due
to the shifting mean effects introduced by earnings management. The additional
component comes from the extra bonus from manipulation and the higher uncertainty
of the compensation.
Given a2 = H, the incentive compatibility constraint for the first period
is CE1(H; H) > CE1(0; H). Again, thanks to the separability between the action
choice and the manipulation amount choice, this constraint reduces to a > c just
as in the benchmark case.
The individual rationality constraint in this model is CEI(H; H) > 0, or
W + aklH + fk2H 2C --(a22 + 32) ln[(l q) + qe-i(-)2]'] 2 0. The
principal's expected payment to the manager, upon substituting the manager's A
choice, is
(1 q)E[W + a(k1H + Eq) + i(k2H + E2)] +qE[W + a(k1H + e1 (/ a)) +
O3(k2H + Q2 + (3 a))]
= (1 q)[W + akiH + Sk2H] +q[W + a(kiH (3 a)) + 1(k2H + (/ a))]
= W + akH + pk2H + q(P a)2.
Now the principal's design program in this misreporting model is:
min W + ak1H + /k2H + q(3 a)2 Program [C]
W,a, 3
s. t.
W + akiH + k2H 2C -(22 + 2a2) ln[(l q) + qe- )2] 0
(IR)
a> C (IC1)
> k2H (IC2)
Similar to the previous models, here the individual rationality constraint
must bind, and the principal's expected cost can be expressed as 2C +!(a2ao2 +,32a2)
+1 ln[(1 q) + qe- -)2] + -a)2.
For later reference, the reduced program is written below.
min (a22 + 12) + 1 ln[(l q) + qe-(-o)2] + q(_ a)2 Program [C]
s. t. a > c (IC1)
S> 2H (IC2)
Define a*. 3* as the optimal bonus rates in Program [C'], we have the
following results.
Proposition 2.2: The optimal contract in the misreporting model exhibits
C
k1H -- ac k-- c*
Corollary 2.1: When q is sufficiently low, the optimal contract in the misreporting
model exhibits ac = and P H = -
Corollary 2.2: When q is sufficiently high and ki is sufficiently large, the optimal
contract in the misreporting model exhibits a* > C- and *c = cj.
Proof: See the Appendix.
Compare Program [C'] with the benchmark: when q = 0. we revert to our
benchmark case; however, when q > 0, the misreporting option introduces a strict loss
in efficiency. The principal must compensate for the manager's risk from the uncertain
misreporting option. There is also a bonus payment effect for the manipulated amount
of output. In addition, the principal may choose to raise the first period bonus rate,
which increases the riskiness of the unmanaged compensation scheme.
Note in this model we always have a* < 0*. Although the misreporting
option merely garbles the information and does not benefit the principal, it is never
efficient for the principal to motivate truth-telling and completely eliminate the
manager's incentive to misreport by setting a = 3. Instead, it is efficient to tolerate
some misreporting. This surprising fact is also shown in Liang (2003). Liang documents
that the optimal contract exhibits a* $ P, while the analysis in the present paper
provides more details on the optimal contact.
More surprisingly, the principal not only tolerates some misreporting by
setting ac < 3*, but sometimes she even maintains the bonus rates at the levels
in the benchmark case where there is no misreporting option. Although an uneven
bonus scheme leads to manipulation, it may not be efficient for the principal to
adjust the bonus scheme to restrain manipulation. The reason for this conclusion is
the following.
Since the induced misreporting is given by A* = a a, the dead weight
loss of misreporting can be reduced by lowering 3 a. To lower 3 a, the principal
either lowers 8 or raises a. However, / has a binding lower bound at c, and the
principal cannot reduce 3 below that bound. Thus the optimal 3 remains at its
bound. By raising a, the principal reduces the dead weight loss of misreporting,
but simultaneously increases the riskiness of the unmanaged compensation scheme
( (a2a2 +/2a2) goes up). Hence there is a trade-off. When the chance of misreporting
is small (q is sufficiently low), the principal finds it inefficient to raise the bonus
rate, since the corresponding reduction in the misreporting dead weight loss does not
outweigh the increase in the riskiness of the unmanaged compensation scheme. On
the other hand, when the probability of misreporting is sufficiently high, the losses
from misreporting constitute a first order effect. In this case, the principal may find
it optimal to raise the first period bonus rate. In addition, when the first period
productivity k1 is high, the lower bound for the first period bonus is low, and the
principal is more willing to raise the first period bonus above the lower bound to
reduce the misreporting dead weight loss.
Table 2-1 shows a numerical example to illustrate Proposition 2.2, and
Corollaries 2.1 and 2.2. In this numerical example, I fix the values of the cost of
high action C = 25, high action level H = 10, output variance a2 = 0.5, risk aversion
r = 0.5, and the second period productivity k2 = 15. I focus on how the optimal
contract changes with the misreporting probability q and the first period productivity
kl. When q is very small (q = 0.01), the optimal contract has both a* and 3* at
their lower bounds. However, when q is high (q = 0.9) and ki is large (ki = 200),
the first period bonus rate a* in the optimal contract deviates from its lower bound
c
kiH'
For simplicity, I use the case q = 1 to explore more details on the misreporting
option.
Ubiquitous misreporting opportunities (q = 1)
When q = 1, Program [C] becomes
min W + aklH +pk2H + (3 a)2 Program [C(q = 1)]
W,a,B
s. t. W + ak1H +/k2H -2C + I(/ a)2 (a22 + /2a2) > 0 (IR)
a 2 (IC1)
S> k1H (IC2)
The situation when q = 1 is special, because there is no uncertainty about
the misreporting option. The principal knows the manager will always shift 3 a to
the second period to get additional bonus income of (/3-a)A* = (0-a)2. Responding
to this, she can cut the fixed wage by (3 a)2 to remove the bonus payment effect.
However, although the principal removes fully the certain bonus payment from her
expected payment to the manager, she must compensate for the manager's dead
weight personal cost of misreporting.
Define a 1, I/3 as the optimal bonus rates for the misreporting model when
q = 1. We have the following result.
Proposition 2.3: When q = 1, the optimal contract in the misreporting model
exhibits
Ci = T and -c =k- if ki > k2(1 + ro2);
a1 = and 3e = otherwise.
kIH C k2H
Proof: See the Appendix.
If the productivity of the two periods is very different (that is, kl is much
higher than k2), the naive bonus rates for the two periods are very different too,
and the manager prefers to move a great amount of output between periods to take
advantage of the uneven bonus scheme. In this case, the principal raises the first
period bonus rate to make the bonus scheme more even to reduce the dead weight
loss of misreporting. However, keep in mind that this brings a cost of higher risk
premium for unmanaged noisy output.
Unexpectedly, even when q = 1, in some cases the principal still maintains
the bonus rates at the levels as in the setting where there is no misreporting option.
The optimal contract may still exhibit (a*1, 0Z*) = (a*, ,3). In other words, even
when the misreporting opportunities are ubiquitous, it may still be optimal not to
restrain misreporting. With an attempt to restrain misreporting by raising a from
its lower bound, the increase in the riskiness of the unmanaged compensation scheme
may outweigh the reduction in the dead weight loss of misreporting.
It is a general belief of investors and regulators that we must take measures
to address detrimental earnings management. In September 1998, Arthur Levitt,
Chairman of SEC, warned that earnings management is tarnishing investors' faith
in the reliability of the financial system, and kicked off a major initiative against
earnings management. From then on, the SEC has taken a variety of new and renewed
measures to fight earnings management.
However, according to our results in Propositions 2.2 and 2.3, in some
cases it is optimal not to take any measure to restrain earnings management. Even
when misreporting opportunities are ubiquitous, it may be better just to live with
misreporting instead of taking any action to fight it. This conclusion may sound
counter-intuitive and cowardly, but is in the best interest of investors.
30
Summary
This chapter shows that in a LEN framework, hedging option reduces the
compensating wage differential for the principal and reduces the compensation risk
for the manager. On the other hand, the introduction of a misreporting option
complicates the agency problem. Surprisingly, although manipulation is detrimental
to the principal, sometimes it is not efficient to take any action to restrain earnings
management.
2nd Period
Manager chooses xl observed publicly. X2 observed publicly.
al E {H, 0}, and Manager chooses Manager gets paid.
dE {0,1}. a2 {H,0}.
Figure 2-1: Time line for delegated-hedge model
1st Period
2nd Period
Manager chooses Manager observes
al E {H,0}. privately Xl and m.
i1 reported.
Manager chooses a2 E {H, 0}.
Figure 2-2: Time line for misreporting model
Manager observes
privately X2. 22
reported. (Principal
sees aggregate output.)
Table 2-1: Numerical Example for Proposition 2.2, Corollaries 2.1 and 2.2
kC C a
kl q kiH k2H C
20 0.01 0.125 0.1667 0.1250 0.1667 C = ji, C = H
200 0.9 0.125 0.1667 0.1304 0.1667 aC > C ,B = C=
...... .__ ...._ I_ __ II IIC___ C k2H
1st Period
CHAPTER 3
HEDGE-MISREPORT BUNDLE MODEL
The last chapter analyzes the hedging and misreporting options respectively.
Now I bundle the delegated hedging option and the misreporting option. This
bundling allows us to study the joint effect of the hedging function and manipulation
function of derivatives.
Bundled Hedging and Misreporting Options
Suppose at the beginning of the first period, the manager chooses his action
level al E {H, 0}, and has the option to hedge. Hedging again only affects the output
in the second period but the hedging decision is made at the beginning of the first
period. Further, suppose the hedging option is bundled with the misreporting option.
If and only if the manager chooses hedging, with probability q can he later misreport
the output by moving some amount of output between periods. This "hedge-misreport
bundle" setting reflects the current concern that managers use derivatives to reduce
risks but can also use the derivatives as tools of earnings management.
At the end of the first period the manager observes privately the first-period
output x1, and at this point he observes whether he can misreport (if he chose
hedging). Similar to the misreporting model, if the manager gets the misreporting
option, he shifts 3 a from the first to the second period. The time line of the
hedge-misreport bundle model is shown in Figure 3-1.
Similar to Lemma 2.4 in the last chapter, here, the manager always prefers
to hedge. To illustrate this, I use CE1 (a, d; a2) to represent the manager's beginning
certainty equivalent when he chooses al and d followed by a2. When the manager
chooses to hedge, CE1 (a,, 1; a2) = W + aklat + /k2a2 c(al, a2)- (a2a + 2a) -
Sln[(l q) + qe-i(-o)21]. When the manager does not hedge, CEi(al, 0; a2) = W +
32
akcia + /k2a2 c(al, a2) ~a2a2 + 222). It is easy to verify that CEi(al, 1; a2)>
CE (al, 0; a2). That is, the manager always hedges in this bundled model. Therefore
the bundled model is identical to the misreporting model with the second period
output hedged (in other words, with the second period output variance reduced to
O2). The principal's design program in this bundled model is
min W + ak1H + fk2H + q(3 a)2 Program [D]
w,a,3
s. t.
W + aklH + Pk2H 2C -(a2 + 2) ln[(l q) + qe-(-2] >2 0
(IR)
a > (IC1)
k2 (IC2)
Again, Program [D] can be reduced to the minimization of !(a(2 2+/32o"2)+
In[(1 q) + qe- (-~)2J+ q(0 a)2 subject to the incentive constraints. For the
convenience of later analysis, I show the reduced program below.
min (a2 r2 + 032),2 1 ln[(l q) + qe-(3-a)2]+ q(/ a)2 Program [D']
s. t. a > c (IC1)
k2H (IC2)
It is readily verified that the optimal bonus rates in the bundled model are
identical to those in the misreporting model, since the manager's action choices do
not depend on his hedging choice. However, compared with the misreporting model,
the principal's expected cost is reduced by 2(,2 2- 2), thanks to the hedging option.
The principal does not need to motivate the manager to hedge, since the manager
always exercises the hedging option.
Although the hedging option and the misreporting option both affect output
signals, they affect the signals in different ways. The hedging option influences
output signals through variances. The greater is the reduction in the noisy output's
variance (a2 oa), the more beneficial is the hedging option. On the other hand, the
misreporting option influences output signals through means. With the misreporting
option, the manager shifts 3* OD from the mean of the first period output to the
mean of the second period output. The amount of manipulation / aD and the
increased compensating wage differential due to misreporting depend on a host of
factors, including the misreporting probability q, the action productivity, and ra2.
In addition, note that although we target earnings management associated with the
use of derivatives, the analysis of the influence on performance signals from earnings
management holds for general earnings management activities.
Whether to Take the Bundle
The hedge-misreport bundle is a mixture of "good" and "bad." As analyzed
in the prior chapter, the hedging option alone, no matter whether centralized or
delegated, is always preferred, since it lowers the output variance and reduces the
compensating wage differential. However, the misreporting option complicates the
agency problem. It is just a garbling of information and reduces the reliability of
performance signals. In their classic article, Ijiri and Jaedicke (1966) define the
reliability of accounting measurements as the degree of objectivity (which uses the
variance of the given measurement as an indicator) plus a bias factor (the degree of
"closeness to being right"). In their terms, we say that the hedging option improves
the objectivity of performance signals, and therefore improves the reliability of the
signals. On the other hand, the misreporting option increases the bias by shifting
output between periods, and reduces the reliability of performance signals. If the
principal faces a take-it-or-leave-it choice on the hedge-misreport bundle, she needs
to see whether the increase in the reliability of performance signals from the hedging
option exceeds the decrease in the reliability from the misreporting option.
Lemma 3.1: When q is sufficiently low, the hedge-misreport bundle is preferred to
the benchmark.
Lemma 3.2: When q is sufficiently high and k1 is sufficiently large, the hedge-misreport
bundle is preferred to the benchmark only if a2 o2 is sufficiently large.
Proof: See the Appendix.
Table 3-1 illustrates numerical examples for the above lemmas. I suppose
the cost of high action C = 25, high action H = 10, risk aversion degree r = 0.5,
and the second period productivity k2 = 1.5. From the numerical examples, we see
when q is sufficiently low (q = .1), the principal prefers the bundle. While when q is
sufficiently high (q = .8) and ki is sufficiently large (ki = 5), the principal does not
prefer the bundle if a2 o2 is small (a2 a_ = .25). She prefers the bundle when
o2 o2 is sufficiently large (a2 = 6).
When the probability that the manager can misreport, q, is sufficiently
low, misreporting is of limited concern. The increase in agency cost due to the
misreporting option is outweighed by the reduction in agency cost due to hedging,
and it is optimal for the principal to take the hedge-misreport bundle. With high
probability of misreporting option and high first period productivity ki, however, the
misreporting option in the bundle can impose severe damage and greatly increases the
compensating wage differential. In this case, the principal admits the hedge-misreport
bundle only when the benefit from hedging is sufficiently large. That is, only when the
hedging option greatly reduces the noise in the output signals (a2 a2 is sufficiently
large) does the principal take the bundle here.
Now focus on the q = 1 case. From Program [D'], when q = 1, the program
becomes
min (a 2a2 + 3202) +(( a)2 Program [D'(q = 1)]
s. t. a > (IC1)
k2 (IC2)
Define aD1, 3l as the optimal bonus rates in the bundled model when q = 1,
we have
Lemma 3.3: When q = 1, the optimal contract in the hedge-misreport bundle model
exhibits
aD1 2 = and D1i = -2 if k12 k2(1 rr2);
aD =C and Xi" = otherwise.
Proof: See the Appendix.
As mentioned earlier, due to the separability between the manager's hedging
choice and his action choices, the optimal bonus scheme in the bundled model is
identical to that in the misreporting model.
Proposition 3.1: When q = 1,
(1) if ki > k2(1 + ro2), the hedge-misreport bundle is preferred to the benchmark
when
k2 _2-(i+ra2)(a2) 2 )
fc( > 1(I+ra2) (a)
and
(2) if kI < k2(1 + ra2), the bundle is preferred to the benchmark when
(k-k2 (b)
(kl -k2)2 > r7_,)
Proof: See the Appendix.
Proposition 3.1 provides a detailed analysis on the trade-off between the
benefit from the hedging option and the cost from the misreporting option when
q=1.
First, for both the cases of kl > k2(1 + ra2) and ki < k2(1 + ra2), the
hedge-misreport bundle is more likely to be preferred (in other words, condition (a)
or (b) is more likely to be satisfied) when a2 is high.' Intuitively, when the unhedged
output signals are very noisy, the principal has a strong preference for hedging to
reduce the noise and is more likely to take the bundle, regardless of the accompanied
cost of misreporting.
af2-(l+ra2)(a2--a, -(1+ra2)2a2-ra2)2%2
'Define Q --~r(i+) in condition (a), we have = -0(l+r a)- < 0.
It is easy to verify that in condition (b) also decreases in o2.
Second, for both cases, it is readily verified that conditions (a) and (b) are
more likely to be satisfied when the Arrow-Pratt degree of risk aversion, r, is high.
When the manager is very risk averse, the principal must pay a high compensating
wage differential to the manager for the noisy output signals. Thus, the principal
would like to reduce the noise in output signals and so has a strong preference for
hedging. Hence she is more likely to take the hedge-misreport bundle regardless of
the accompanying cost of misreporting.
Third, for both cases, the harm from misreporting behavior is smaller when
ki and k2 are similar in magnitude. (ki and k2 are similar in magnitude implies the
bonus rates can be set closer, and the potential damage from misreporting is small.)
The more similar are ki and k2, the more likely is condition (a) or (b) to be satisfied,
hence the more likely is the hedge-misreport bundle to be preferred.
Fourth, for both cases, the benefit from hedging is higher when hedging
greatly reduces the noise in output signals (that is, when a2 ao2 is large). The more
the hedge can lower the second period output variance, the more likely is condition
(a) or (b) to be satisfied, hence the more likely is the principal to prefer the bundle.
The numerical examples in Table 3-2 illustrate these comparative static
observations. We see from the examples that the principal is more likely to take
the bundle when o2 od2 is large, a2 is high, r is high, or ki is close to k2.
Summary
A hedge-misreport bundle is used to model the two-edged feature of derivative
instruments in this chapter. I analyze the trade-off between the improvement of
performance signals' objectivity brought by hedging and the increase in the performance
signals' bias due to earnings management. With a LEN framework, hedging makes
it easier for the principal to infer the manager's action from output signals, and
thus helps lower the compensating wage differential. In addition, hedging is efficient
regardless of whether the manager or the principal is endowed with unilateral hedging
38
authority. On the other hand, earnings management merely garbles information. It
is also shown that hedging and earnings management influence performance signals
from different angles. and their net influence depends on various factors.
2nd Period
Manager chooses
a E {H, 0},
dE {0,1}
Figure 3-1: Time
Manager observes priv,
xz and m (if chose d =
x1 reported. Manager
chooses a2 E {H, 0}.
line for hedge-misreport
lately x2 observed privately.
1). x2 reported.
(Principal sees
aggregate output.)
bundle mode
Table 3-1: Numerical Examples for Lemma 3.1, 3.2
parameters principal's expected cost
benchmark ? bundle
q=.1
ki = 3 50.4340 -< 50.2977
a2 a = .25
q =.8
ki = 5 50.3785 >- 50.4454
a2 a = .25
q=.8
k = 5 50.6056 -< 50.5239
a2 -- o=.6
1st Period
Table 3-2: Numerical Examples for Proposition 3.1
parameters key parameters principal's expected cost
benchmark ? bundle
a2 = .75
r = .5
(U22 high; a2 a' high) 50.8138 50.5534
ki =2
k2I = 1.5
a2 = .30 50.3255 50.3548
9 = .25
(a2 low; a2 a low)
kl =2
r = .9 (r high) 50.9766 -< 50.7509
k2 = 1.5
a2 =.5
r = .01 (r low) 50.0109 < 50.0104
ad = .25
ki = 1.6, k2 = 1.5
r = .5 (kl, k2 close) 50.6524 < 50.4842
a2 = .5
a] = .25 kl = 5, k2 = 1.5 50.3785 50.4514
(kl, k2 not close)
CHAPTER 4
MANIPULATION RESTRAINED BY HEDGE POSITION
In the hedge-misreport bundle model in Chapter 3, a hedging option is
bundled with a misreporting option, but the influence on the outputs from misreporting
is separable from the effect of hedging. In this chapter, a stronger bond is tied between
hedging and misreporting. This stronger bond reflects the fact that greater use of
derivatives can provide expanded opportunities for earnings management.
Cost of Earnings Management
Firms' risks involve many uncontrollable factors, such as interest rate changes,
foreign exchange rate changes, credit defaults, and price changes. Reducing firms'
risks more effectively requires more hedging, while the increase in the use of derivatives,
in turn, makes it easier to manipulate earnings. When the manager reduces risks more
effectively with more derivatives, he also obtains additional ways to manipulate. In
other words, the cost of earnings management is lowered. To reflect the association
between the cost of earnings management and the extent of hedging, I suppose the
manager's personal cost of earnings management is ( -) A2. To simplify the
notation, define D =- a2 a2, and express the manipulation cost as 2A2. The
personal cost of manipulation is still quadratic in the amount of manipulation. In
addition, now the marginal cost of manipulation is associated with the hedge position.
I call this new setting the "strong bundle" model. As in the previous model, I suppose
the manager can manipulate only if he hedges. Without hedging, there is no way to
manipulate earnings, or it is extremely costly to manipulate.
"Strong Bundle" Model
With personal cost of manipulation A2, the manager's optimal manipulation
amount will again occur when the marginal cost of manipulation equals the marginal
benefit, that is, when = 2(0-). This implies A* = D(/ a). As in the
previous hedge-misreport bundle model, suppose the manager gets the misreporting
option with probability q given that he hedges, and he does not know whether he can
misreport until the end of the first period. If he obtains the misreporting option, he
will shift D(P a) from the first to the second period to capture additional bonus.
Let CE (al; a2; d = 1) denote the manager's certainty equivalent at the beginning of
the first period, given that he hedges. We readily find
CE'(al; a2; d = 1) = W + akla + fk2a2 c(al, a2) Oa 22 +2) ln[(l q) +
qe-D(-0)2.
Now the manager's benefit from misreporting, -' ln[(l q) + qe-D(-)2]1,
is associated with the hedging position, D. In addition, the larger the D, the more
the benefit from misreporting for the manager, regardless of the additional increased
riskiness from the misreporting option2.
Compared to the previous model in Chapter 3, the manager benefits not
only from hedging through reduced riskiness of outputs, but also from an increased
marginal gain from manipulation. Therefore, it is easy to verify that the manager
still always prefers hedging (d = 1), regardless of his action choices.
Similar to the analysis in Chapter 3, the principal's design program to
encourage the manager's high actions is:
min W + ak1H + Ok2H + qD(f a)2 Program [E]
W,a,1
s. t. CE(H; H; d= 1) > 0 (IR)
a > C. (IC1)
k1H (IC)
C H (IC2)
'Notice that -In[(1 q) + qe-gD(e-a' ] is positive. It is the result of joint effects from
increased riskiness of compensation and additional bonus from manipulated output net of personal
manipulation cost.
2Define Q ln[(1 q) + qe-zD(-)2] We have a = 2 qe- -'2("- ) > 0.
o 2 l--q(l--e-f D("-U'))
As usual, the (IR) constraint must bind, so W* = -aklal /Sk2a2 + 2C +
r(a22 + /2o.) + I ln[(l q) + qe-1D(-a)2], and the principal's objective function
can be rewritten as (a2a2 + 2ao) + 1 ln[(l q) + qe-1D(,-~)2] + qD(3 a)2.
Now the principal's design program reduces to
min (a(22 + /32) + 1 ln[(1 q) + qe-D(-a")2] + qD(3 a)2
s. t. Program [E']
a > c (IC1)
Sk2H (IC2)
Define a), 0* as the optimal bonus rates in Program [E']. We have
Proposition 4.1: The optimal contract in the strong bundle model has
kiH E < k~H P
Corollary 4.1: When q is sufficiently low, the optimal contract in the strong bundle
model has a* = and 3* -.
kH E k2H
Corollary 4.2: When q is sufficiently high, D is sufficiently large, and the difference
between ki and k2 is sufficiently large, the optimal contract in the strong bundle model
has aE > and *E =
Proof: See the Appendix.
In contrast to the model in Chapter 3, the hedge position now plays a role
in deciding whether to linit earnings management. When D is sufficiently large, the
cost of misreporting (-A2) is low. If the probability of misreporting is not trivial,
the potential misreporting damage is severe. With a severe misreporting threat, it is
efficient for the principal to raise a to limit earnings management.
Whether to Take the Strong Bundle
In the hedge-misreport bundle model in Chapter 3, although hedging is a
prerequisite for the manager to misreport, the benefit from hedging and the cost
from misreporting are separable. Whether it is efficient to take the bundle depends
on a clear-cut trade-off between the reduced output variance and the garbled output
means. In the current strong bundle model, hedging is not only a prerequisite for
earnings management, but also affects the manager's manipulation amount through
its effect on the personal cost of earnings management. The trade-off between hedging
and earnings management becomes more complex.
Lemma 4.1: When q is sufficiently low, the strong bundle is preferred to the benchmark.
Proof: See the Appendix.
Obviously, if the probability of misreporting option is very low, misreporting
is all second order effect, even though hedging reduces the misreporting cost.
However, when q is high and k1 is significantly larger than k2, D plays
a double role in the principal's view of the strong bundle. A highly effective hedge
greatly reduces the noise in the output signals, but also greatly reduces the manager's
personal cost of manipulation. This is most evident when q = 1.
Ubiquitous Misreporting Opportunities
When q = 1, that is, when the manager can always misreport, the principal's
design program becomes
mm (2(a2 + 2) + D( a)2
s. t. Program [El]
a > (IC1)
k2> H (IC2)
Define a*, 31, as the optimal bonus rates for the strong bundle model when
q = 1. We have the following result.
Proposition 4.2: When q = 1, the optimal contract in the strong bundle model has
E1 D n C ,r2 <,2
o and 371 = C if D > (or 2, < ( ));
k2H kl/k2-1 r kl/k2-1
EI = Hand = -i otherwise.
Proof: See the Appendix.
We use Figure 4-1 to show how the first period optimal bonus rate, a*l,
changes with the hedging position, D.
Notice that ac, = 2C > C when D > r2. In other words, the
+ra2k2H k-H ki/k2-1
principal raises a from its lower bound when D > 2 Intuitively, if D is large,
earnings management is going to be a first order effect, and the optimal contract has to
reduce the manipulation incentive by increasing a. More precisely, when D < ,r2
the optimal first period bonus rate is fixed at -, while when D > the optimal
first period bonus rate increases gradually at a decreasing rate.3 Right after D exceeds
ra2
/k2-1, as the hedge becomes more effective (D gets larger), the principal has to
increase the first period bonus rate sharply to cope with the increased manipulation.
However, when the hedge gets even more effective (D approaches oa2), the principal
does not need to improve the first period bonus so much to deal with manipulation.
Turning to the question at hand, we have the following characterization.
Proposition 4.3: When q = 1 :
Case (1): If -< 2(1_ k/k
the strong bundle is preferred to the benchmark when
k2 (2-2)(a2-ru2) (
k c 2(2_ +,ra2)2 (A')
Case (2): If a > a2(1 2-1)
the strong bundle is preferred to the benchmark when
1> 1- -Vr (B')
Otherwise, it is not efficient for the principal to take the strong bundle.
Proof: See the Appendix.
Corollary 4.3: In Case (1) or (2) of Proposition 4.3,
condition (A') or (B') is less stringent with higher risk aversion degree r;and
condition (A') or (B') is less stringent when kl and k2 are similar in magnitude.
Proof: See the Appendix.
Condition (A') or (B') is less stringent means the condition is more likely to
be satisfied. With a less stringent condition (A') or (B'), the principal is more likely
3Here -' r ra c (- ) ,= r-2 <
3Here -- TT= ^ k > 0, and o72 = 2 kT < O.
,H';fi7( T-D),k
to prefer the strong bundle. Intuitively, a higher r indicates that the manager is more
risk averse. When the manager is very risk averse, the principal must pay a high risk
premium to the manager for the noisy output signals. Thus, the principal would like
to reduce the noise and has a strong preference for hedging. Hence she is more likely
to take the strong bundle.
In addition, when productivity does not change much through time (kl and
k2 are similar in magnitude), the bonus rates will not change much through time
either. With little additional bonus gain from misreporting, even when the cost of
misreporting is low, the manager's manipulation will not bring great damage. Thus
the principal is more likely to take the strong bundle. In other words, the conditions
for the strong bundle to be preferred are less stringent.
Corollary 4.4: When a o- 2(1 k/k21)
(a) if a(T2 < 2( 1 ), then the smaller the a2 the less stringent is condition (A');
(b) if a2( 1+) < ,2 < a2 k/-l), t then the larger the o2 the less stringent is
condition (A').
Corollary 4.5: When o, > a2(1 )k,)), o does not influence the principal's
decision on whether to take the strong bundle.
Proof: See the Appendix.
In the strong bundle, the "good" from hedging and the "bad" from manipulation
are colinear, so a low ao in and of itself is not a cause for joy. The principal's preference
for the bundle is not monotonic in a2. To explore the intuition behind these results,
recall Figure 4-1:
In Figure 4-1, when D is small (o- is large), the principal does not raise
the first period bonus rate a from its lower bound, since it is not worth raising the
bonus rate to restrain earnings management. Correspondingly, here in Corollary 4.5,
when acr is sufficiently large, the effectiveness of hedging does not affect the principal's
decision on whether to take the strong bundle. Since the hedge is poorly effective
and the manipulation cost is high, neither hedging nor manipulation has significant
influence. In Figure 4-1, when D gets larger and exceeds "'2 (od gets smaller
than a2(1 r_)), the principal has to raise a greatly to deal with the increased
manipulation, while when D approaches a2 (oa approaches zero), the principal raises
the first period bonus rate only at a decreasing rate. In Corollary 4.4, we see when
ad is intermediate, the marginal benefit from hedging's effectiveness cannot beat
the marginal loss from manipulation, and the principal is more likely to take the
bundle with a less effective hedge. When ad approaches zero, however, the marginal
benefit from hedging's effectiveness beats the marginal loss from manipulation, and
the principal is more likely to take the bundle with a more effective hedge.
Figures 4-2A and 4-2B show how the principal's preference for the strong
bundle changes with the effectiveness of the hedge (represented by ad).
To further illustrate the tension between the benefit from the hedge's effectiveness
and the loss from the more manipulation around a21r2~) in Figure 4-2A, we use a
numerical example to show how the principal's preference for the effectiveness of the
hedge changes. In this example we suppose r = .5, ki = 5, k2 = 1, a2 = .5 so that
or2( r2) a2(1 l/r-1) is satisfied. The value of a2(`2j) here is .4165. The
numerical example is presented in Table 4-1.
Summary
From the analysis in this chapter, we see when the manager's personal cost of
manipulation decreases with the effectiveness of the hedge, there is more tension when
the principal decides whether to take the hedge-misreport bundle. When the hedge
is highly effective, the more effective the hedge, the more likely the principal prefers
the bundle since the marginal benefit from hedging's effectiveness beats the marginal
loss from lower manipulation cost. When the hedge is moderately effective, however,
the principal is more likely to take the bundle with a less effective hedge, since the
marginal loss from misreporting is more considerable compared with the marginal
48
benefit from the hedge's effectiveness. But when the hedge is poorly effective, the
effectiveness plays no role in the principal's decision on whether to take the bundle,
since both the hedging benefit and the loss from manipulation are insignificant.
I C
Ir1 H T -
C
k, 4-1: D and
0
k.
Figure 4-1: D and a*
________ &
1k1 -I
the smaller the ad,
the more likely
the bundle is preferred
the larger the ad,
the more likely
the bundle is preferred
ad has no influence
2(1 k/k2-1
the smaller the ad,
the more likely
the bundle is preferred
Ca has no influence
Sk1/k2-1 )+r
Figure 4-2: oa and preference. A)When +2(- ) < k 2(1 /2- ). B)When a2( ) >
o2(1 kl/k2-1)
Table 4-1: Numerical Example for principal's preference and a2
principal's expected cost
ad <- 2( +-) = .4165 benchmark strong bundle
a = .25 50.8125 -< 50.7813
2 = .35 50.8125 > 50.8398
a > 2( 1) = .4165
o2 = .42 50.8125 50.8456
T = .4999 50.8125 ~ 50.8125
CHAPTER 5
EARLY RECOGNITION MODEL
The FASB has recently issued several new regulations on the measurement
and disclosure of derivatives. There is also substantial detailed implementation guidance
from the Emerging Issues Task Force (EITF). The new regulations are intended to
recognize the effect of a hedge on earnings and use "mark-to-market" techniques to
evaluate the unsettled derivatives, presumably so investors understand the potential
risk and value of the derivative contracts held by firms. However, the "mark-to-market"
technique may be problematic. The prior chapters have discussed the case in which
this technique is abused to manipulate earnings. In this chapter, I further explore its
impact on the firms' risk management behavior.
Early Recognition of Hedging
In practice, over-the-counter derivative contracts are difficult to evaluate,
for they are not traded on exchange markets. For OTC derivatives, their "fair value"
is more easily manipulated. However, for derivative instruments that are frequently
traded on exchange markets, such as futures and options, their market value is readily
available. Manipulation may not be a major concern for these derivatives. However,
the new regulation of evaluating unsettled derivatives may still influence the firms'
risk management behavior.
According to FASB Statement 133, the ineffectiveness of a hedge would
result from "a difference between the basis of the hedging instrument and the hedged
item or hedged transaction, to the extent that those bases do not move in tandem,"
or "differences in critical terms of the hedging instrument and hedged item or hedging
transaction, such as differences in notional amounts, maturities, quantity, location, or
delivery dates." In practice, the effectiveness of hedging refers to the degree to which
the fair value changes in the derivatives offset the corresponding fair value changes in
the hedged item.
For this chapter, I assume the manager cannot manipulate the fair value
of unsettled derivatives. There is no other option of earnings management either.
Without hedging, xz = klal + cl and x2 = k2a2 + 62, where [1, 62] follow a joint
a 2 0
normal distribution with zero means, and the covariance matrix is As
0 a2
in the previous models, suppose any hedge is for the second period only. If the firm
does not recognize the hedging influence earlier, the second period output will be
x2 = k2a2 + E2, where e2 ~ N(0, a2) with oa < a2. In this way, again we capture the
risk reduction theme of hedging by a mean preserving spread structure. Since in this
chapter I am not interested in earnings management, I relax the earlier assumption
kI > k2, and allow kl and k2 to be any positive value. With hedging (but without
early recognition), the second period output variance is reduced from a2 to oa. If firms
have to recognize the ineffective portion of hedge before the settlement of derivative
contracts, I suppose x1 = kial + e1 + pe' and x2 = k2a2 + (1 p)E, p E (0, 1). Due
to the early recognition, part of the reduced variance is recognized in the first period
and the remaining variance is recognized in the second period, when the derivative
contract is settled.
Centralized Case
First consider the case in which the principal is endowed with the hedging
authority and decides to hedge. Suppose the outputs are publicly observed.
Paralleling earlier work, it is routine to verify that the incentive constraints
collapse to the following two constraints:
a > (IC1)
SI3k (IC2)
Moreover, the individual rationality constraint is W + aklH + 3k2H 2C
-r2 (a2 + p2a2) z 2(1 P)2a2 > 0.
Therefore, the principal's design program is
min W + aklH + 3k2H Program[F]
W,a43
s. t.
W + akH + 3k2H 2C -:a2(a 2 + p2o ) /2(1 p)2o2 > 0 (IR)
SkiH (IC1)
Sk2H (IC2)
Since, as usual, (IR) must bind, the reduced version of Program [F] is
min !a2(a2 + p2ao) + !/32(1 p)2o02 Program[F']
s. t. a > C (IC1)
2 (IC2)
Notice the only difference from the prior centralized case in Chapter 2 is
that part of the second period output variance is moved to the first period. Denote
the optimal bonus rates for Program [F] as a* and P.. It is readily verified that
the optimal solution is a] = C and /3. = k. The optimal bonus rates here are
identical to those in the benchmark case where either the principal does not hedge
or there is no hedging option at all. This follows because hedging only influences the
output variance and does not affect the incentive constraints.
In this case, the principal's expected cost is B 2C + ac(2 + p2d) +
02(1 p)2a.d Compared to her expected cost if she does not hedge, A = 2C +
ra2 + 2o2, hedging is efficient when A B > 0, or when < ~2-() Since
S= and = --, this implies hedging is efficient when
k2 ^20.2 2 [&2
kI > 2-(1-=)2Cr = 2^ I[Q]
2 a7 (' d -(1.p)2
Proposition 5.1: In the centralized case, hedging is efficient when condition [Q] is
satisfied.
Proof: See the above analysis.
From condition [Q]. we see when hedging greatly reduces the output noise
(when o2 is small), the principal prefers to hedge. In addition, when ki >> k2, the
principal prefers to hedge. With ki >> k2, a* << *. When the first period bonus
rate is low, the compensating wage differential for the increased risk in the first period
output is not significant, and the principal is willing to hedge to obtain the benefit
from hedging regardless of the increased risk in the first period output. Moreover,
when p is small, there is only a small increase in the riskiness of the first period
output since only a small portion of the ineffectiveness is recognized earlier, and the
insignificant early recognition will not overturn the preference for hedging.
Delegated Case
Now consider the case in which the manager makes the hedging decision.
As in the centralized case, suppose the outputs are publicly observed. When [Q] is
satisfied, will hedging still be efficient in the delegated case?
Since the manager makes the hedging decision, to motivate him to hedge,
there are two more incentive constraints besides the constraints in Program [F]. Using
CE1 (al, d; a2) to denote the manager's certainty equivalent at the beginning of the
first period, these two constraints are:
CE (H, d = 1; H) > CE (H, d = 0; H)
CE(H, d = 1; H) > CE1(0, d = 0; H)
Expanding the constraints, we get
W + aklH +lk2H -2C- a2(2 + p2 2) :2(1 p)2oa W + aklH + k2H -
2C -a2 a2
W + akH + 3k2H 2C (a22 p2) 2 p)2 > W +k2H C -
hr2 227 2
They reduce to
2[ (1 p)2oa] > a2p2aO (ICa)
a -> C- { i "[-(-)_hp)2V(-C2p2a)
a kH (IClb)
The second constraint is redundant, for 32[u2 (1 p)2a2] > a2p2C2 and
a> C
kiH
Now the new design program is:
min Ia2(2 + p2o2) + 2(1 p)2 Program[G'j
s.. t. > C (IC1)
Sk2 (IC2)
2[a2 (1 p)201] 2 ap2D (IC1a)
Plug the optimal bonus rates ac = and = into the IC constraints.
For (ICla), we have 2[(2-(1- p)2 2]-)2 p2a = ( )2 2-(1 ) ] Condition
u 2-(l-p)2,2 p 02 2
[Q] implies ( 2 2 0, thus (IC,) is satisfied. Therefore, the optimal
bonus rates in Program [F] are also the optimal bonus rates in Program [G'J. In
addition, (ICla) satisfied implies the manager will always hedge as long as [Q] holds,
thus it is free for the principal to motivate hedging. No matter whether it is the
principal or the manager who has the authority to hedge, hedging is efficient when
condition [Q] is satisfied.
Lemma 5.1: When [Q] holds and hedging is motivated, the optimal contract has
a= and /* =
Proof: See the above analysis.
The intuition behind this proposition is the following: When [Q] is satisfied,
under the benchmark bonus scheme a* = C and 3" = although the first
period output is more risky with the hedge, the total risk premium for the two periods
outputs, *2(02 +p22) +1./2(1 -p)2o2, is still lower than the risk premium without
the hedge, a*2a2 + *2 2.
However, when [Q] is not satisfied, a* = c and = are not the
optimal rates for Program [G'].
Define < ~2-- as condition [Q]. We have the following lemma.
'2 7-(l-p)2
d
Lemma 5.2: When [Q] holds and hedging is motivated, the optimal contract has
a* = and 3* = /- p)2
kH V a 2-(1-p)2 kH'
Proof: See the Appendix.
From the above lemma, when [Q] holds, the principal has to set a higher
second period bonus rate to motivate hedging. [Q] implies that ki is low. When ki is
low, the lower bound of a, --, is high. With a high bonus rate in the first period, the
induced increased risk in the first period will be high. Thus the manager is reluctant
to hedge. To motivate hedging, the principal has to reduce the weight of the first
period bonus in the manager's compensation, either by reducing a or by increasing
p. Since a has its lower bound at --, the principal cannot reduce the first period
rate below that bound. The only way is to increase the second period bonus rate, /.
Here hedging is not so attractive as in the case when [Q]. The question now
is what if the principal does not encourage hedging when [Q].
If the principal does not encourage hedging, the incentive constraints for
the second period are the same as when the principal encourages hedging, since
the hedging choice is already made at the beginning of the first period. Using
CE (al, d; a2) to denote the manager's certainty equivalent at the beginning of the
first period, the incentive constraints for the first period given a2 = H become
CE (H,d = O;H) 2 CE'(H, d = 1;H) (ICl')
CE'(H, d = 0; H) > CE(O, d = 1; H)
CE (H, d = O; H) CE (O, d = 0; H)
Rewrite the (IC1') constraints, we have
2[o2 (1 p)2_,] < a2p2a2 (IC12)
>C+-51 {2-(1-p)2]-2p2o} (IC
> kH (IC1,)
a > C (ICl1)
Here (IClb) is redundant, since 2[a2 (1 p)2a2] < a2p a2. The individual
rationality constraint in this case is CE (H, d = 0; H) > 0, and it must bind.
Therefore, the program can be rewritten as
min 'aa2r2 + 22a2 Program[H']
s. t. a > (IC1:)
k> H c
3 k2H (IC2')
a2p2a > [a (1 p)22] (ICI:)
Lemma 5.3: When [Q] holds, the optimal contract that precludes hedging has a* =
k6H and 3* -= c
Proof: See the Appendix.
Now, when [Q] holds and hedging is discouraged, the principal's expected
cost is
2C+6 ( C )222_ri C 2H22 (A')
while when hedging is encouraged, her expected cost is
2C + ( )2(o2 + p2a2) + ( )2(1 p)22 (B')
Comparing A' with B', we have
B'-A'=r( C)2(2 + p2 )+ ( C)2 2 2 (1 22 C )22 ( r )202
2k- H d 2 k t2-(l-p)2a21 P) -d 2k -- -H)
+r( c 2 _p_)2a2
=r- c 2 +2 .2 2 2
r(C)2,2[ 1 P2a 1
_r(@C202 I 1[Z /,2r
-2-() O fjo,_-(i- _1 i] > 0
Therefore, with condition [Q] satisfied, B'-A'>0. Thus hedging is not efficient.
In other words, the principal prefers not to hedge when < ( a =<2
k2a:P --(l--p)2 '
ad
no matter whether the principal or the manager has the hedging authority.
Proposition 5.2: If [Q] holds, encouraging hedging is efficient, and the optimal
contract has a* = H and /* = ; If [Q] holds, discouraging hedging is efficient,
and the optimal contact has a* = and 3* = k.H
Proof: See the above analysis.
Table 5-1 has a numerical example to illustrates the conclusion in Proposition
5.2. I assume r = 0.5, C = 25, and H = 10. I use d = 1 to denote the case that
the principal encourages hedge, and use d = 0 to denote the case that hedge is
discouraged.
From the above proposition and the numerical example in Table 5-2, it is
shown that the delegated case is similar to the centralized case in that hedging is
k2 2
efficient only when 5 > -2 _P- In the delegated case, there is no moral hazard
2 -ay-(1-p)
d
problem on hedging. The principal still follows the rules in the centralized case to
decide whether to encourage hedging. The principal need not motivate hedging when
[Q], and need not forbid hedging when [Q]. This is because the principal and the
manager share the same interest. The lower the induced output risk for the manager,
the less the compensating wage differential the principal has to pay.
More importantly, recall that in the previous chapters when there is no
early recognition, hedging is always efficient since it reduces the compensating wage
differential. However, with the early recognition of hedging's ineffective portion,
in some cases hedging is not efficient any more, since it adds more risk to the
first period output, though the risk in the second period output is reduced. When
the first period output has a sufficiently great weight in deciding the manager's
compensation, increased riskiness in the first period output greatly increases the
manager's compensation risk. The principal has to pay a relatively large compensating
wage differential, thus hedging becomes unattractive. In addition, the less effective is
the hedge (that is, 2 is closer to 1), the less likely is hedging to be efficient. Moreover,
d
the larger the portion of the ineffectiveness is recognized earlier (that is, the higher
the p), the less likely hedging is efficient.
Summary
In Chapters 2 and 3, hedging is efficient since it reduces the second period
output variance, and therefore reduces both the manager's compensation risk and
the principal's compensating wage differential. However, when I introduce the early
recognition of a portion of the hedge's ineffectiveness, it is shown that sometimes
hedging becomes undesirable. The reason is that the early recognition increases the
riskiness of the first period output. When the first period output carries a great weight
in the manager's compensation, or when a large percentage of the ineffectiveness has
to be recognized early, the early recognition policy makes hedging unattractive. In
addition, without the early recognition, as long as a2 > 4a, hedging is efficient; while
with the early recognition, only when the effectiveness is sufficiently high will hedging
be efficient.
The analysis of this chapter sheds light on how some recent accounting
regulations may influence the firms' risk management behavior. Recent accounting
regulations require that firms recognize the ineffectiveness of hedge into earnings even
before the settlement of derivatives contracts. Although the intention of the new rules
is to provide investors with more information on the firms' use of derivatives, they
may have a side effect of discouraging the firms' risk management activities.
Table 5-1: Numerical Example for Proposition 5.2.
a* kf" k _C principal's expected cost
d = 1 1.250 1.667 50.2631
Q1 1.25 1.667
d = 0 4.410 1.667 52.7778
d = 1 2.500 .095 50.8929
2 22.50 .833
d = 0 2.500 .833 50.8681
'For the case with condition [Q], I assume kl = 2, k2 = 1.5,a2 = .5, a = .25, and p = .5.
Therefore = 1.7 > -- = .142, which satisfies [Q].
2For the case with condition [Q], I assume ki = 1, k2 = 3,u2 = .5, a' = .25, and p = .5. Therefore
k2 2 satisfy
- = 0.111 < = .142, which satisfies [Q].
,4 -W
CHAPTER 6
OTHER RELATED TOPICS
In this chapter I briefly explore the relationship between riskiness and agency
problems. In addition, I also analyze a model with "informative" earnings management,
where manipulation is desirable.
Riskiness and Agency
For the main model in this paper I assume output follows a normal distribution,
and show that hedging reduces the firms' risks and helps reduce the compensating
wage differential. However, we need to be cautious not to take this result casually and
conclude that "as long as hedging reduces the risk in output, it improves the agency
problem." The normal distribution assumption may play an important role here. In
a continuous setting, Kim and Suh (1991) illustrate that if there are two information
systems whose distributions belong to the normal family, the system with the higher
likelihood ratio variance is more efficient (costs less for the principal to induce the
manager's certain action level). In a binary action setting, it is easy to verify that the
hedged plan also has a higher likelihood ratio variance. A likelihood ratio distribution
with a higher variance makes it easier for the principal to infer the manager's action
from the output, and therefore helps reduce the compensating wage differential.
Without the normal distribution assumption, reduction in riskiness may
not improve an agency problem. Consider a finite support numerical example in
which hedging drives up the compensating wage differential. For simplicity, assume
a one-period, centralized-hedge case with three possible outputs, {1,2,3}. When
the principal does not hedge (or when there is no hedging option), the probability
distribution of {1,2,3} given the manager's high action is PH = (1, -, 1), and when
the manager chooses low action the distribution is PL = (, 3, -). But when the
principal hedges, the probability distribution given high action is PHd = (, i, ),
and the distribution given low action is PLd = (, ). Also assume C = 25 and
r = 0.01. Given the action, it is readily verified that PH is a mean preserving spread
of PHd and PL is a mean preserving spread of PLd. In other words, the unhedged plan
is more risky, according to Rothschild and Stiglitz (1970). However, in this example,
the principal pays 54.1149 to encourage high action when she hedges, while she only
pays 37.7216 when she does not hedge or when there is no hedging option.
Risk reduction is usually believed to be beneficial to investors. However, as
illustrated here there is no necessary connection between risk reduction and improvement
in the agency problem. Counter-intuitively, risk-reducing activities may increase the
compensating wage differential. In other words, even though hedging activities reduce
firms' risks, in some cases they are detrimental to investors.
Informative Earnings Management: Forecast Model
To this point, the misreporting behavior is just garbling, and it merely
destroys information. That is, the misreporting behavior is bad for the principal,
although sometimes the principal tolerates some misreporting behavior because the
elimination is too costly. However, when misreporting carries some private information,
in some cases it is good for the principal to encourage earnings management.
Consider an extreme case in which earnings management is not only encouraged
but enforced by the principal for her interest. I will show that with some engineering,
encouraging manipulation may lead to first best solution.
As in the previous models, the agent chooses a first period action level and
decides whether to hedge the second period's output at the beginning of the first
period. Here I further assume the agent can hedge only when he chooses high action
in the first period. That is, I assume the hedging activities need effort, and a slack
manager will not be able to hedge. In addition, if the agent decides to hedge and
chooses high action in the second period, he can also perfectly forecast the output
of the second period at the beginning of the second period. The principal cannot
observe the agent's choices or the outputs for each period, but can observe the actual
aggregate output at the end of the second period. For simplicity, assume the cost
of misreporting is zero and the agent can misreport freely, as long as the aggregate
reported output, i1 + x2, is equal to the actual aggregate output, xz + x2. The time
line for the forecast model is shown in Figure 6-1.
In this model, although the principal cannot observe the output of each
period and cannot know the agent's forecast, she can design a contract that achieves
first best to encourage high actions and hedge. Think about the contract that pays
the agent the first-best compensation if the agent reports equal outputs for period
1 and period 2, but pays the agent a penalty if the reported outputs for the two
periods are not equal. Under this contract, the agent can forecast the second period's
output only when he works hard and hedges, and only when he forecasts the second
period's output is he able to manage the earnings so that the two periods' outputs are
equivalent. With any other choice of actions, he cannot perfectly smooth the earnings
to avoid the penalty, and the chance to get two equivalent outputs by accident is small.
Therefore, the only choice for the agent to avoid the penalty is to supply high effort
in both periods, hedge, and smooth the reported earnings. In this case, earnings
management is not only encouraged but enforced. It helps the principal to reap the
rent from the agent. Income smoothing here is desirable to the principal.
The program for the principal in the forecast model is
min J S1 fH(x )fHd(X2)dxldx2 Program [I]
So,Si
s.t. ff u(Si 2C)fH(xl)fHd(x2)dxldx2 > -U
f fu(SI 2C) fH(Xl)fHd(x2)dxldx2 > E(u) for any choice other than e1,e2
H, hedge, and equal 21, 2,
where S1 is the payment to equivalent reported outputs, and So is a penalty.
Proposition 6.1: In the forecast model, first best can be achieved by S, = 2C -
r In(-7) for equivalent reported outputs for the two periods and penalty So << 0
otherwise.
Proof: See the Appendix.
Summary
In this chapter it is shown that there is no necessary connection between
the reduction in riskiness and the improvement in agency problems. It is a general
belief that risk reduction activities are beneficial to the investors, while this chapter
illustrates that this may not be true in some cases.
A model with informative earnings management is also included in this
chapter. When earnings management conveys the manager's private information,
manipulation may be efficient and desirable. It is shown that in a well-constructed
model, encouraging manipulation can achieve the first best.
1st Period
2nd Period
Agent chooses
al E {H, L}. If al = H,
can hedge for 2nd
period output.
Agent observes xl,
report il. Choose
a2 E {H, L}.Can forecast
x2 if a2 = H and hedged
Agent observes
x2, reports x2.
Principal observes
x1 + 2.
Figure 6-1: Time line for forecast model
CHAPTER 7
CONCLUDING REMARKS
Derivative instruments arouse mixed feelings. They are inexpensive hedging
instruments that cost much less than real option hedging, while their complexity
makes them harbors for earnings management. Investors and regulators are concerned
and nervous about the potential damage from abusing derivatives, but cannot forgo
the convenience and benefit from hedging through derivatives. Derivative instruments
are like nuclear power stations, when they work well, they provide users with clean and
cheap energy, while when anything goes wrong, their destructive power is dreadful.
This has led to great effort aimed at restraining the abuse of derivatives. To
help investors get more information about and more control of firms' use of derivatives,
the FASB has issued various rules recently on the recognition and disclosure of
derivatives, such as SFAS 133, 137, 138 and 149. There are also numerous detailed
guidance from Emerging Issues Task Force (EITF) on how to implement these
complicated new rules. The main strategy of the regulators to fight the abuse of
derivatives is to require firms to disclose the fair value of both the derivatives and the
hedged items.
The regulators believe investors can understand better the value of the
derivative contracts through the managers' estimates of the derivatives' fair value.
However, to do this, discretionary evaluation of the fair value is necessary, since many
unsettled derivatives' fair value is not available from the market. The more discretion
for the managers may offer more earnings management opportunities, contrary to the
initial intention of the new accounting rules.
Moreover, another intention of the new regulations is to help the investors
understand better the potential risk of derivatives through the managers' early disclosure
of the ineffectiveness of hedging. However, the early recognition of the hedging's
ineffectiveness raises the riskiness of interim earnings. With a higher risk in the
interim earnings, the firms may be discouraged from risk reduction activities, which
may not be a desirable consequence for investors and regulators. In addition, the
discouragement of hedging may force the managers to look for other ways to secure
their wealth. Unfortunately, more earnings management is a promising candidate.
As shown in this dissertation, the current accounting regulations on derivative
instruments may be inefficient. But shall we give up the effort to restrain the abuse of
derivatives? Or shall we just discard derivative instruments? I would say no. It is not
the purpose of this paper to criticize the current rules and claim the effort is totally
in vain. Instead, the intention is to explore the complicated feature of derivatives so
we get better ideas on how to keep the benefit of cheaper hedging while minimizing
the potential destruction from derivatives abuse. An ancient Chinese saying says,
"a thorough understanding of both yourself and your enemy guarantees a victory." I
hope this research may shed some light on the feature of our enemy, the dark-side of
derivatives, and help us find more efficient ways to regulate the use of derivatives.
APPENDIX
Chapter 2:
Proof for Lemma 2.1:
Proof. We use /t1, /2 to denote the Lagrangian multipliers for (IC1) and (IC2)
respectively. With the reduced program, the first order conditions are -ra2a+pt1 = 0
and -ra23 + A2 = 0. Since a > > > 0 and 3 C > 0, we get p = ra22a > 0 and
'2 = ra2 > 0. This implies both (IC1) and (IC2) bind, or a* = -, *1 = C
Proof for Lemma 2.2:
Proof. The principal's design program can be expressed as the minimization of
2(a2o2 + /2o2) subject to the incentive constraints. Again let D1, 2 denote the
Lagrangian multipliers of (IC1) and (IC2) respectively, with the reduced program,
we have the first order conditions -ra2a + ~ = 0 and -rad + p2 = 0. Hence
/1P = ra2a > 0 and /2 = ra2d/ > 0. This implies both (IC1) and (IC2) are binding,
and thus a* = Ci and 03 = c. n
Proof for Proposition 2.2:
Proof. Define it as the Lagrangian multiplier for (IC1) and p2 for (IC2). With the
reduced Program [C'], the first order conditions are
-ra2a q,-f)2(-2 + 2q(3 a) + Pi = 0 (FOC1)
l-q+qe (8-)
and -ra213 + q_- ) -2q( a) + p2 = 0. (FOC2)
l-q+qe -5(0-)2
In the optimal solution, if neither constraint is binding, P1i = #2 = 0. Substitute
Pi = 112 = 0 into the first order conditions and add the two conditions together, we
get -ra2, ra2a = 0, which implies a = 0 = 0. This contradicts a > > 0 and
S> ~2 > 0. Therefore, MP = /2 = 0 is not true in the optimal solution, and at least
one of the constraints is binding.
If /1 > 0, 12 = 0, then a = C and3 2 (FOC2) implies 3 = M Q,
where M _= 2- 1-2
where M = 2- q+qe-7(- Rewriting M, we have M = 1M_( {(1- )(1-
e-i(-)2) +[1-q(1-e-i(Pa))]} > 0. With M > 0, we get / = 2aa < a = -
However, ki > k2 implies / does not satisfy the constraint /3 > Therefore,
(a = iH, /3 ) -) cannot be true. Hence, regardless of IL, (IC2) always binds,
implying/ k-T
Thus, p2 > 0 and p, > 0. Moreover, if (IC1) binds, then a = -; if (IC1) is slack,
then from (FOC1) a = SM 3 < 3. Hence we always have a* < /3. *
Proof for Corollary 2.1:
Proof. Using the first order conditions displayed in the proof of Proposition 2.2, we
see when q is sufficiently near zero, (FOC1) reduces to -ra2a + e1 + p, = 0 and
(FOC2) reduces to -rr2/3 + E2 + A2 = 0, where E1 and E2 are small. This implies
Ii > 0 and 12 > 0. That is, when q is sufficiently small, both incentive constraints
bind and aC = X = -7.
Proof for Corollary 2.2:
Proof. Using the first order condition (FOC1) in the proof of Proposition 2.2 again,
if 1 > 0 and a = then
A1 = ro72 -C (2q )ge ( O57) > 0. (i)
k1H I-q +g e r )2 Clk
(i) can be re-expressed as
r (2 ()- 2 ( ) > 0. (ii)
k ,1-q+qe 2 _-" k' 1l
Now suppose q is sufficiently high and ki is sufficiently large. This implies the
inequality in (ii) is reversed and a* > C
Proof for Proposition 2.3:
Proof. Define A1, A2 as the Lagrangian multipliers of the two constraints respectively.
We get the following first order constraints:
-ro2a + (/3 -a) + 1 =0 fociI')
-ra2/ (3 a) + 12 = 0 (FOC2')
From Proposition 2.2, we know p2 > 0, and z1 > 0.
If /l = 0, [2 > 0, then 3= -, and from fociI') we get -r2a + ( a)
0, which implies a = 1/3 = 1 If 1 > then both incentive
-ra 1+r-TH -2IH'
= lfr2 k2H and are the optimal bonus
constraints are satisfied, and ac = 12 and = are the optimal bonus
rates. The condition 12 c > c reduces to ki > k2(1 + r2).
l+ra2 k2H k-H
Ifi1 C < C then a = C does not satisfy the constraint a > C,
I1+r" k2 H kh 1+ra2 k2H -- k1H
and the optimal contract must have abc = 7 and /i = ', as both incentive
constraints bind. *
Chapter 3:
Proof for Lemma 3.1:
Proof. From Chapter 2, when q is sufficiently low, the optimal contract in the
misreporting model is (ab = C-, 3c = ). The misreporting model, Program
[C'], is identical to the bundled model, Program [D'], except that the second period
output variance decreases to ad in the bundled model. It is easy to verify that
(a* = H, 0C* = rc) remains optimal in the bundled model when q is sufficiently
near zero.
The principal's expected cost in the bundled model therefore gets close to 2C +
S(aga2 +/ or 3 ") when q is near zero. In the benchmark model where there is neither
hedging nor misreporting option, her expected cost is 2C + -(oc2 + / 2), which
is higher than 2C + j (o022 + 3*22d). The hedge-misreport bundle is preferred. m
Proof for Lemma 3.2:
Proof. From Corollary 2.2, when q is sufficiently high and k: is sufficiently large, the
optimal contract exhibits a* > -H, 0 = C. We rewrite a' as ( + e, e > 0. In
H C C
addition, from Proposition 2.2, we know ab = C + e < /3 = .
In the benchmark model where there is neither a hedging nor a misreporting
option, the principal's expected cost is 2C +(a*o*22 + n*o2,2), while in the bundled
model it is 2C + j(au72 + ,C ) + ln[(l e- -c2] q( a)2. If
r(a*<2 2) {*22 ,+ R2 + ln[(1 q) + qe-(-V)~2+ q(o a)2}
A A 2 C C 2r 2 2
=r(.C)2(2 2- u2) {(2[( +E)2 -( )2]
k2Hc c 2 2 q c
+- ln[(1 q) + qe-z2 2-H kH 2] + q H- k )2} > 0 (iii)
then hedge-misreport bundle is preferred to no hedging, no misreporting.
But (iii) is positive only when a2 car is sufficiently large. *
Proof for Lemma 3.3:
Proof. Refer to the proof for Proposition 2.3. Program [C'(q = 1)] is identical to
Program [D'(q = 1)] except that in the bundled model program's objective function,
the second period variance is a2 instead of a2. It is easy to verify these two programs
share the same optimal bonus coefficients. (a*\D) Dl) = (a*l *c1)" u
Proof for Proposition 3.1:
Proof. In the benchmark model where there is neither a hedging nor a misreporting
option, the principal's expected cost is 2C +-(a2a*2 + /2Ua2), while in the bundled
model it is 2C +V(aa*2 2 + 2 + (,3 ca1)2. As long as (af*2a2 + *a2)-
[j(a*2 a2 + *2 ,) + (31i aD1)2] > 0, the hedge-misreport bundle is preferred.
(1). When kl > k2(l + ro2), the optimal contract exhibits (a*I 1+ k2H
D = gC) Substitute ac, O and a*D(, 3 into rU 2 + 2a2) [r(a*22
DI3 2) + 2(/ D D1)2]. We have
r(a*2,2 + Ia2) [(a.2 2 + / 2) + 1 1)2]
1 (2ra2 ra2 r(a2-a2) (ra2)2
T2TH) k"2 (l+rcr2)2k2jJ
= =kl2 -L" -?. -(r --
SG)2 r2(1+r2)2k 2r 2k 2+r(2 -)(1+ 22 -(22)2k
-"2H ]lr2 2 (l+ra2)2k k'
-21H (1+ro2)2k;12
= )2(+r )2 r (1 ra2)2k2 k2[r2 r( a2)(1 + ra2)2 + (ra2)2]}
Thus, we need ra2(1 + ra2)2k2 k2[ra2 r(a2 a2)( + ra2)2 + (ra2)2] > 0. This
fes k2 -(1+r2)(a2-2)
implies > r2(1+r:2)
(2). When ki < k2(l+ra2), the optimal contract exhibits (a*o = =
(aOD1, lD1) is identical to (aA, 0*). Substitute a, A* and a* into 1( a 0 2 +r
-u2) [ D(l1"2 ) + 2d D 2 -1 D 1)2], we have
S*22 2 0*2 2 \ *2 .*2 2 1 2 1 C2r 2) 1 1)2]
r(a or (a a+a- a -aC*1)2] = 1(C)2 2 (1)2]
Thus, we need r2 ( )2 > 0, which implies > 3
2 '(k-k2)d >"
Chapter 4:
Proof for Proposition 4.1:
Proof. Define 1 as the Lagrangian multiplier for (ICl) and /2 for (IC2). The first
order conditions are
-ra2a -q+qe3-D(3_)2 + 2qD( a a) + 1 = 0 (FOC1)
1-q+qe- JD(3-.a 2
-ra2 + qe--D(-) ) -2qD( a) + /2 = 0. (FOC2)
l-q+qe7D(3-)2
Suppose neither constraint is binding, implying /,, p2 = 0. Substitute l, /12 = 0
into the first order conditions and add the two conditions together, we get -ro-a -
ra2a = 0, which implies a = 0 = 0. This contradicts a >_ c > 0 and > c > 0.
Therefore, /1, A2 = 0 is not true and at least one of the constraints is binding.
If #1 > 0, A2 = 0, then a = and/ > C (FOC2) implies a = Pa, where
T- D(3-a)2 -
T (2 +e-qi(3-)2 )D. Rewriting T, we have T = D_1-e-{(1 )(1 -
1 {(1 -
e-D(- a)2) q( -D(2-a)2)]} > 0. With T > 0, we get 0 = -,a
C However, kl > k2 implies 0 doesn't satisfy the constraint 3 > Therefore,
(a = j, > C-) cannot be true.
k1H k2H
C
Hence, regardless of [I, (IC2) always binds, implying *E -- k2i*
Thus, A2 > 0 and Pli 2 0. Moreover, if (IC1) binds, then a = ; if (IC1) is slack,
then from (FOC1) a = 2+Tq 3 < 3. Hence we always have aE < E. *
Proof for Corollary 4.1:
Proof. Using the first order conditions displayed in the proof of Proposition 4.1, we
see when q is sufficiently near zero, (FOC1) reduces to -ra2c0 + E1 + 1 = 0 and
(FOC2) reduces to -roa2$ + 62 + [2 = 0, where el and E2 are small. This implies
A, > 0 and A2 > 0. 0
Proof for Corollary 4.2:
Proof. Using the first order condition (FOC1) in the proof of Proposition 4.1 again,
if 1i > 0 and a = ', then
-JD(OE_ -F 2
02 C -k-7)
A1 = 2 (2q qe- E )2 )D(E k7) > 0. (i)
1-q+qe 1 kl"
(i) can be re-expressed as
rr21 q(2 e )D( )> 0. (ii)
T1q- -_JD(j )2(. __))D(k2 kT2 T .1
1-q+ge 2 2l
Now suppose q is sufficiently high, kl and k2 are sufficiently different, and D is
large. This implies the inequality in (ii) is reversed and ac > k-.
e 2 1
In addition, define G = (2 e )D( ). We have
1-q+qe D( 2 Weh
G 1 1 )2- e-D(-a2 e- JD(8-)2 ~ )2 )2(1-)
D k2 ki 1-q+qe- D(-)2 (1-q+qe- D(0-)2 2
and
a D[(2 e-(-a)2 e-~ ( D-a)2 D ()22( )2(1-q)
a =+ 2 H k )2(l
J2 JD ) -q+qe- D(O-)2 (I-) q+e- -D(-)- 2
D(O-a)2
From the proof for Proposition 4.1, we know T = (2- _e- a-o )D >
1-q+qe D(-a) )
0.Therefore (2- e"-iD(BO .)2) > 0, and both 0 > 0 and G > 0.This implies
1-q+qe k2 k1
that the larger a2 da, the more likely (ii) is reversed and ac > k- ; Also, the larger
(k -) (in other words, the greater the difference between kl and k2), the more
likely (ii) is reversed and a* > C.
Proof for Lemma 4.1:
Proof. From Corollary 4.1, when q is sufficiently low, the optimal contract in the
strong bundle model is a* = and /3 = ,2 as in the benchmark, where there is
neither a hedging nor a misreporting option.
The principal's expected cost in the bundled model therefore gets close to 2C +
j[( T)2"2 + (kCT)2d2] when q is near zero. In the benchmark model where there is
neither a hedging nor a misreporting option, her expected cost is 2C + '[('/)202 +
(f)2 a2], which is higher than 2C + [(k )22 + C )2U]. The strong bundle is
preferred. *
Proof for Proposition 4.2:
Proof. Define 1, /2 as the Lagrangian multipliers of the two constraints respectively.
We get the following first order constraints:
-ru2a + D(3 a) + p = 0 fociI')
-rad D(3 a) + 12 = 0 (FOC2')
From Proposition 4.1, we know 12 > 0, and y1 > 0.
If p, = 0, P2 > 0, then / = c, and from fociI') we get -ra2a + D(/ a) = 0,
which implies a = = D+- If > C then both incentive
,,7kD+ra2 k2H-Dr2 k1
= D C and X*i C are the optimal bonus
constraints are satisfied, and a* = and E\ = are the optimal bonus
rates. The condition D- kC- > C can be rewritten into D > r2 (or a <
D+rorTH-- -- kl/k2-1 d
a2(1 kl/I )).
If D < r2 /k- then a D+= 2 does not satisfy IC1 and the optimal contract
must have a 1 = and 3E = as both incentive constraints bind. n
Proof for Proposition 4.3:
Proof. In the benchmark model where there is neither a hedging nor a misreporting
option, the principal's expected cost is A 2C +w[(-C-)2a2 + () i)202], while in the
strong bundle model it is B 2C +(r (12 ,2 2 + *D(2 a )2. As long as
El E1Vd) -- D(3E I As long as
A B > 0, the strong bundle is preferred.
(1). When a2 < ,2(1 k/2-), the optimal contract has ao1 = D- k- and
E1 = Then B = (D ) ( 22 +)2 k)22-D + D ( )2( )2 We have
A B = l 2f + I2 Iro2 rra22
A-B ){ 2 1k2 2 -D+ro2 2 r-D( D+r)2
= ( )2f I 2 ftl )22 + D( 2 rD)]}
A B > 0 requires ra- r [( ) D 2r + D(D )2 D > 0, which implies
k2 > ( D )2 ra2 D
> (D+- ) + D 2
k Da2( D+ra2) (D+ra2)2
k > a2(D+ra2)2
kc2 (.2- u)(,2-ru2)
2 r the(optimal2)2r
(2). When oa > a2(1 k/ ), the optimal contract exhibits a*l = and
3 = .2 Then A B ( )2D 1 )2( )2 C )2D[ 1 -- 1)2].
Al 2 > r 2( -) 2 > kic iml > H1 H U k2k
A-B > requires -1 2 )2 > 0, which implies k' > 1 Vr.
Proof for Corollary 4.3:
Proof. In condition (A'), case (1), define Z ,2("2_ )2 = 2D(-2). (A') is
rewritten as > Z.
1
Z D [(D+ra2)2(-a2)-2(D+ra2)2 (,2 -r22)]
=r a2 (D+ro2)4
D[-(D+ro2)a-2a2 _-ra-2)
(D+roa2)3
D(-(D+ra2)-2(or-ra2)
(D+ra2)3
= (D+ro2)3
(02_-a)[-(1-r)ja2_ag
(D+ra2)3 < 0
Since < 0 the higher the risk aversion degree r, the smaller the Z, and the
more likely the condition (A') is satisfied. In other words, (A') is less stringent with
higher r.
In condition (B'), case (2), it is easy to verify that the higher the r, the smaller
is 1 Vr, and more likely is condition (B') satisfied. In other words, (B') is less
stringent with higher r.
It is also easy to verify that in either case (1) or (2), the closer are ki and k2, the
less stringent condition (A') or (B') is. *
Proof for Corollary 4.4 and 4.5:
Proof. In case (1), ad < a2(1 k7k-1), and the strong bundle is preferred when
k2 (,22)( 2 ro) (u -2)(u2-r2)
ff> 2(Lr2)2 Define Z = ,2,22 we have
az 2 r(a2-& +r2)2{-a2d+r +(a 2- )}+2 (2-(a 2 + _)(a2 -)(a- 2)
a? 2((2-oo2+r-2)2
2t 2 o2 12 ( 2 2 2 a 2 2 22a 2 d2
a 2(,2-a2+r` 2){((t 2-oZA+p`,)((g -2o2+ ` )+2(g2-%)(%-rg2)}
,a4(a-_,2+ra2)4
f 4-2 2,2+2ra 4--2, +2a --ra2a2+r `,+2a4 --2r4 --24d+2ra2a'2
-- 02(,2 _-_+r0`2)3
r o4-+22r-ra2a+ 224
-- 02(u2--.2+ru2)3
(l+r2)u2-(l+r),2
(o2-a'+o2)3
When (1+r2)2 (1+r)r2 > 0, that is, when a2 < a2( 1+), z > 0, the smaller
the o2, the less constraining is > Z.
When (1 + r2)U2 (1 + r)a < 0, that is, when a2 > o r(1r ), < 0, the larger
~1dd d =,9a
2k2
the a2, the less constraining is > Z.
In case (2), from (B') we see cr does not play a role in the principal's decision on
whether to take the bundle. *
Chapter 5:
Proof for Lemma 5.2:
Proof. Denote the Lagrangian coefficients for (ICl), (IC2) and (ICla) in Program
[G'] as A1, /2 and /3. The first order conditions are:
-ra(a2 + p2a2) + p 2ap2a2 13 = 0 (FOC1)
-rp(1l- p)2a+ + 2[a2 (1 p)2 2]3 = 0 (FOC2)
From (FOC1), if p, = 0, then -ra(a2 + p22) 2ap2p3 = 0, which implies
/13 < 0. Therefore we must have M1 > 0, that is, a* = C.
From (FOC2), if /2 = 0, P3 = 0, then -r3(l p)2a2 = 0, which is not true.
Therefore, we cannot have both /2 and p3 = 0.
If A2 = 0,13 > 0, then 2[2 (1 p)2] a= 2p2o2, which implies =
___ c a* c th en P
P2- O)_ 2 If this > then a* = C and 3* =- -, are
S02-(j-p)~ kkH Va2-(1-p)2d kl
o l *p2___ c_2_2 k2
optimal. = 2_ )2~2 > implies .(p) > which is condition
2--(1-- p)2 kIH a,-(' 'd
[Q]. In other words, if condition [Q] is satisfied, the optimal contract has a* -=
and 3* = p= $ c
V2 -(1-p)2au ki H
Proof for Lemma 5.3:
Proof. Define the Lagrangian coefficients for (IC1C), (IC2') and (ICl') in Program
[H'] as /i, /4 and p3. The first order conditions are:
-raa2 + p + 2ap2a2,/, = 0 (FOCI')
-r3a2 + pI 23p'[[a2d (1 p)2a] = 0 (FOC2')
From (FOC2'), it is easy to verify we cannot have /4 = 0, since in that way /j < 0.
Therefore, /Z > 0, and 0* = .
If p/ > 0, = 0, then a* = and a*2p2 2 > 3*2[a2 (1 p)2a2], which
k2 2 2
implies condition [Q, 4 < 2 __ 2 In other words, with condition [Q] satisfied,
the optimal contract shows a* = C, and 3* = c
Chapter 6:
Proof for Proposition 6.1:
Proof. The program for the principal is
min f f S1 fH(l)fHd(X2)dxldX2
So,S1
s.t. ff u(S1 2C)fH(xl)fHd(X2)dXldx2 > U (IR)
ff u(S1-2C)fH(xl)fHd(x2)dxIdx2 f ff u(So-C)fH(zl)fL(x2)dxldX2+ f f u(Si-
X2Xl 52=Xl
C)fH(l)fL(x2)dxldx2 (IC1)
f f u(S-2C)fH(xl)fHd(Z2)dxldx2 f f u(So-C)fL(xl)fH(x2)dxldx2+ ff u(S1-
X2*xl X2=XI
C)fL(l)fH (x2)dxldx2 (IC2)
f fu(S1 2C)fH(xl)fHd(x2)dxldx2 >2 f u(So)fL()fL(x2)dxldx2 +
f f u(Sl)fL(xl)fL(X2)dxldx2 (IC3)
ff u(Si 2C)fH(xl)fHd(x2)dxdx2 > f f u(So 2C)fH(xZ)fHd(x2)dxldx2 +
f u(SI 2C)fH(x,)fHd(x2)dxldx2 (IC4)
X2=X1
With a sufficiently low So, it is obvious that the IC constraints are not binding.
(Think about So = -oo All the right hand sides of the IC constraints are equivalent
to -oo then. Therefore none of the IC constraints is binding.)
With none of the IC constraints binding, we reduce the program into a first-best
one. Therefore we have, f f u(Si 2C) fH(x)fHd(Z2)dxidzx = U or u(S 2C) = U.
That is, the optimal S1 = 2C In(-U). *
r-
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BIOGRAPHICAL SKETCH
Lin Nan was born in Beijing, China, in spring 1973. In June 1995, she
received a Bachelor of Engineering in industrial economics from Tianjin University in
Tianjin, China. She then worked at the Industrial and Commercial Bank of China
(ICBC) for two years as Presidential Assistant. In 1997, Lin came to the United States
and started her graduate education at the West Virginia University in Morgantown,
West Virginia. She received her Master of Arts in economics in August 1999 and then
joined the accounting doctoral program at the University of Florida in Gainesville,
Florida. She is expected to graduate with a Ph.D. degree in August 2004.
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Joel S. Demski, Chair
Frederick E. Fisher Eminent Scholar of
Accounting
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adeqaIte, in cope and quality, as a--
dissertation for the degree of Doctor of Philosop
Dv dE. M. Sappington
Lanzillotti-McKethan Eminent Scholar of
Economics
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequa ,,in scope ad quality as a
dissertation for the degree of Doctor of Philosophy. /
Karl tEHackenbrack
Associate Professor of Accounting
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosoph
professor of Accounting
This dissertation was submitted to the Graduate Faculty of the Fisher School of
Accounting in the Warrington College of Business Administration and to the Graduate
School and was accepted as partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
August 2004
Dean, Graduate School
LD
1780
202,.m
UNIVERSITY OF FLORIDA
III 1IIII III IIIIi II I 2 II 11111IIIIII IIIIl
3 1262 08554 2073
