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