Jan 23, 2004 - Hedging and earnings management influence performance signals from ... earnings management option, the manager shifts output between ...
Derivatives and Earnings Management Lin Nan∗ Fisher School of Accounting University of Florida January 23, 2004
Abstract Hedging and earnings management influence performance signals from different angles, and their influence is decided by various factors. When bundled together with the use of derivatives, there is a trade-off between the higher objectivity of performance signals brought by hedging and the larger bias in signals due to earnings management. Surprisingly, sometimes it is optimal not to take any measure to restrain earnings management, even though earnings management is detrimental. In addition, risk reduction from hedging does not always lead to improvement in the informativeness of performance signals.
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Introduction
Alan Greenspan, 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." Derivative instruments are popular hedging instruments that help firms reduce various risks such as interest rate risk, foreign exchange risk and price fluctuation risk. However, the complexity of derivatives also offers opportunities for earnings management. In 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 the world, collapsed. Investigations revealed that it had made extensive use of energy and credit derivatives to bolster revenues. Although the FASB has recently issued several new regulations on the measurement and disclosure of derivatives, it is unlikely that the new regulations will eliminate the earnings management through deriv∗ 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.
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atives. In addition, some new rules may even provide new avenues for earnings management.1 Derivative instruments, therefore, have a two-edged feature. They can be tools for both hedging and earnings management. This two-edged feature of derivative instruments provides us with an ideal scene to study the interaction between hedging and earnings management. The present research analyzes the joint effect of hedging and earnings management in a two-period model. The risk reduction theme of hedging is captured by a mean preserving spread structure, and earnings management in this paper is induced by uneven bonus rates through time. 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 authority. On the other hand, earnings management merely garbles information. Surprisingly, despite the detrimental earnings management, in some cases it is not optimal for the principal to take any measure to restrain earnings management, even when misreporting opportunities are ubiquitous. Moreover, it is never efficient to motivate truth-telling. The reason behind these results is that fighting earnings management may be too costly. When a hedging option and an earnings management option are bundled together with the use of derivatives, whether derivative instruments are worth encouraging depends on a trade-off between the higher objectivity of performance signals brought by hedging and the larger bias of performance signals due to misreporting. Hedging influences performance signals through variances, and its influence depends on how much it can reduce the noise in the signals. Earnings management influences performance signals through means. With the earnings management option, the manager shifts output between periods to take advantage of the uneven bonus rates. The effect of earnings management depends on a host of factors including the probability of obtaining the chance to misreport and the uneven productivity through time. In addition, this paper also documents "undesirable" hedging. Hedging reduces firms’ risks, and risk reduction is widely believed to be beneficial to investors. Counter-intuitively, this paper indicates that risk reduction does not 1 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.
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always lead to improved informativeness. Hedging may weaken information systems and drive up agency costs, even though it reduces firms’ risks. Although the literature contains no theoretical research that explicitly studies the joint effect of hedging and earnings management, there are numerous studies on either earnings management or hedging. Arya, Glover and Sunder (1998) conclude that economic explanations for earnings management require that key assumptions of the Revelation Principle be violated.2 In the present paper, the existence of earnings management comes from the violation of the unrestricted contract assumption. The present paper also adopts an uneven productivity setting similar to Liang (2003) to induce earnings management. Among the studies on hedging, the most comparable research to the present paper is that of DeMarzo and Duffie (1995), which also analyzes how hedging behavior affects the information content of earnings signals. DeMarzo and Duffie explore the corporate incentives for hedging and show that financial hedging improves the informativeness of corporate earnings as signals of managerial ability and project quality. However, unlike in the present paper, in their model the manager’s action is given, so there is no need to motivate the manager to work diligently. In addition, their model doesn’t consider the manipulation of earnings by the manager. Among empirical works, Barton (2001) is probably the first to examine the relationship between hedging activities and earnings management. He measures derivatives using notional amounts and discretionary accruals using the modified Jones model, and finds that “firms with larger derivatives portfolios have lower levels of discretionary accruals.” Barton thus suggests that hedging and earnings management are partial substitutes. However, he doesn’t consider the earnings management function of derivatives and regards derivatives merely as hedging instruments. In addition, the substitution hypothesis is weakened by Pincus and Rajgopal (2002), who show that managers do not make hedging and earnings management decisions simultaneously. The rest of this paper proceeds as follows: Section 2 presents the basic setup of the model and a benchmark. Section 3 studies the hedging and earnings management options respectively. Section 4 analyzes the trade-off between the hedging and earnings management options when they are bundled together. Section 5 explores "undesirable" hedging. Section 6 concludes the paper and discusses future research. 2 The Revelation Principle states that truth-telling is in equilibrium when (1) communication is not blocked, (2) contract is not restricted, and (3) commitment is not restricted.
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2
The Model
The main model in this paper is a two-period model in a LEN framework.3 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(S−c) , 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(0) . Performance signals (outputs) are stochastic, and their probability is affected by two factors: the manager’s action and some exogenous factor. For the manager’s action, assume a binary setting where 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. In this paper, "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. We use x1 to represent the output for the first period, and x2 to represent the output for the second period.
2.1
Basic Setup
Assume x1 = k1 a1 + 1 and x2 = k2 a2 + 2 , with ai ∈ {H, 0}, i ∈ {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 k1 > k2 . The uneven productivity follows a design in Liang (2003). As in Liang’s model, the different productivity induces different bonus rates through time and is important for our later analysis on earnings management. The vector [ 1 , 2 ] 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.4 If the outputs are not hedged, the covariance matrix 3 LEN refers to linear contract, exponential utility function, and normal distribution. The LEN framework is a technology that provides tractability for research in many agency problems, especially those with rich settings such as multi-action and multi-period models. Recent works using LEN include Dutta and Reichelstein (1999), Indjejikian and Nanda (1999), Christensen and Demski (2003), Liang (2003), and Autrey, Dikolli and Newman (2003). 4 The conclusions in this paper persist when the outputs have a non-zero covariance.
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of
1,
2
is Σ =
·
σ2 0
0 σ2
¸
. If the second period output is hedged (as we
discuss in a later·section, any ¸ such hedge is confined to the second period), the σ2 0 matrix is Σd = , σ 2d < σ2 . The hedging process is stylized with a 0 σ2d mean preserving spread structure: assuming the same action level, the hedged production plan has a lower variance, σ 2d , than that of the unhedged one, σ2 , 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. The manager’s contract or compensation function is restricted to be "linear" in the noted output statistics. Specifically, S = S(x1 , x2 ) = W + αx1 + βx2 , where W is a fixed wage, and α and β are the bonus rates respectively assigned to the first period output, x1 , and the second period output, x2 .
2.2
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, we 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 higher than that when he chooses low action, no matter what x1 is obtained. Denote the manager’s certainty equivalent when he chooses a2 given x1 at the beginning of the second period as CE2 (a2 ; x1 ), the incentive compatibility constraint for the second period is CE2 (H; x1 ) ≥ CE2 (0; x1 ), ∀ x1
(IC2)
With x1 known and x2 a normal random variable with mean k2 a2 and variance σ 2 , it is well known that CE2 (a2 ; x1 ) = W +αx1 +βk2 a2 −c(a1 , a2 )− 2r β 2 σ2 . Thus, (IC2) can be expressed as W +αx1 +βk2 H−C− 2r β 2 σ 2 ≥ W +αx1 − 2r β 2 σ2 , which reduces to β ≥ k2CH , regardless of x1 . Denote the manager’s certainty equivalent at the beginning of the first period when he chooses a1 followed by a2 regardless of x1 as CE1 (a1 ; a2 ). To motivate a1 = H given high action in the second period, the incentive compatibility constraint for the first period is
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CE1 (H; H) ≥ CE1 (0; H)
(IC1)
If a2 = H, regardless of x1 , (IC2) is satisfied, then S(x1 , x2 ) is a normal random variable with mean W + αk1 a1 + βk2 H and variance α2 σ 2 + β 2 σ 2 . Thus (IC1) implies W + αk1 H + βk2 H − 2C − 2r (α2 σ 2 + β 2 σ 2 ) ≥ W + βk2 H − C − 2 2 r C 2 2 2 (α σ + β σ ), which reduces to α ≥ k1 H . 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 CE1 (H; H) ≥ 0
(IR)
Expanding (IR), we get W + αk1 H + βk2 H − 2C − 2r (α2 σ 2 + β 2 σ 2 ) ≥ 0. The principal minimizes her expected payment to the manager, E[W + α(k1 H + 1 ) + β(k2 H + 2 )] = W + αk1 H + βk2 H. Her design program in this benchmark model is
min W +αk1 H +βk2 H
Program[A]
W,α,β
s. t. W +αk1 H +βk2 H −2C − r2 (α2 σ 2 +β 2 σ 2 ) ≥ 0 α ≥ k1CH β≥
C k2 H
(IR) (IC1) (IC2)
The individual rationality constraint must be binding, as otherwise the principal can always lower W . Thus, the optimal fixed wage must be −αk1 H − βk2 H +2C + 2r (α2 σ 2 +β 2 σ 2 ), and the principal’s expected cost is 2C + r2 (α2 σ 2 + β 2 σ2 ). Therefore, the principal’s design program reduces to the minimization of r2 (α2 σ 2 + β 2 σ 2 ) subject to the two incentive constraints.5 The optimal fixed wage is chosen to ensure that the individual rationality constraint binds. We therefore focus on the bonus rates in the optimal contracts in our analysis. Denote α∗A as the optimal first period bonus rate and β ∗A the optimal second period bonus rate, we now have Lemma 1 The optimal contract in the benchmark model exhibits α∗A = and β ∗A = k2CH .
C k1 H
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 5 This
result has been shown in, for example, Feltham and Xie (1994).
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∗2 2 2 setting, the principal needs to pay 0 + 2C + 2r (α∗2 A σ + β A σ ). The principal pays more since the manager bears compensation risk with a risk premium or ∗2 2 2 compensating wage differential of r2 (α∗2 A σ + β A σ ).
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Hedging and Earnings Management Options
Next we introduce the hedging and earnings management options. In this section, the hedging and earnings management options will be studied respectively.
3.1
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 output of a future period. 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. Consider 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). In this subsection we do not consider the recognition in earnings from unsettled derivatives. (That is, we 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 we will address the manipulation associated with the use of derivatives later.
3.1.1
Centralized-Hedge Model
Consider a centralized-hedge case, where the principal has unilateral hedging authority. (Later in this paper we 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 W + αk1 H + βk2 H.
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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 + αk1 H + βk2 H − 2C − r2 (α2 σ2 + β 2 σ 2d ) ≥ 0. The principal’s design program in the centralized-hedge model when she hedges is min W + αk1 H + βk2 H
Program [B]
W,α,β
s. t. W +αk1 H +βk2 H −2C − r2 (α2 σ 2 +β 2 σ 2d ) ≥ 0 α ≥ k1CH β ≥ k2CH
(IR) (IC1) (IC2)
We use α∗B , β ∗B to denote respectively the optimal bonus rates in the first and the second periods in Program [B]. Paralleling Lemma 1, we immediately conclude Lemma 2 The optimal contract in the centralized-hedge model exhibits α∗B = ∗ C C k1 H and β B = k2 H . 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 doesn’t hedge, the principal’s ex∗2 2 2 pected payment is 2C + 2r (α∗2 A σ + β A σ ), while its counterpart with hedging ∗2 2 r ∗2 2 is 2C + 2 (αA σ + β A σd ). With hedging, her expected cost is reduced by r ∗2 2 2 2 β A (σ − σ d ). 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 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.6 6 We also know that hedging modeled in this fashion is efficient in a single period model where an optimal contract without linear restriction, subject to limited liability, is in place. (Proof available upon request.)
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3.1.2
Delegated-Hedge Model
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, we change the model so the manager instead of 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 below. 1st Period
Manager chooses a1 ∈ {H, 0}, and d ∈ {0, 1}.
2nd Period
x1 observed publicly. Manager chooses a2 ∈ {H, 0}.
x2 observed publicly. Manager gets paid.
Time Line 1: Delegated-hedge model 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 hedging. This is because hedging reduces the output variance and therefore reduces the manager’s compensation risk brought by noisy output signals. Lemma 4 For any action choice, the manager always prefers d = 1. Proof. See the Appendix. 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 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.
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3.2
Misreporting Option
Up to now we have focused on settings where the realized output is observed publicly. Here we introduce the option to manipulate performance signals. We presume hedging is not possible in this subsection, but will combine both hedging and earnings management options later in Section 4. Suppose the output for each period is only observable to the manager. The manager chooses the first period action level, a1 ∈ {H, 0}, at the beginning of the first period. At the end of the first period the manager observes privately the first-period output x1 . He reports x b1 ∈ {x1 , x1 − ∆} to the principal and chooses his action level for the second period, a2 ∈ {H, 0}. At the end of the second period, again the manager observes privately the second-period output, x2 , and reports x b2 = x1 −b x1 +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 + αb x1 + βb x2 . The manager may have an option to misreport the output by moving ∆ from the first to the second period. (∆ can be negative. Negative ∆ implies that the manager moves some output from the second to the first period.) Assume the manager manipulates at a personal cost of 12 ∆2 , which is quadratic in the amount of manipulation.7 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. We 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 below. 1st Period
Manager chooses a1 ∈ {H, 0}.
2nd Period
Manager observes privately x1 and m. x b1 reported. Manager chooses a2 ∈ {H, 0}.
Manager observes privately x2 . x b2 reported. (Principal sees aggregate output.)
Time Line 2: Misreporting model
Notice in the benchmark case of Lemma 1 where there is no option to misreport, the bonus rates for the two periods are not equal, and α∗A < β ∗A . 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. If m = 1, the manager’s certainty equivalent at the beginning of the second period becomes W + αx1 + βk2 a2 + (β − α)∆ − 12 ∆2 − c(a1 , a2 ) − r2 β 2 σ 2 . Notice that with a linear contract, the manager’s manipulation choice is separable from 7 The
quadratic personal cost follows a similar design in Liang (2003).
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his action choices and the output risk. This separability implies that the optimal ∂ "shifting" occurs where ∂∆ [(β − α)∆ − 12 ∆2 ] = 0, or ∆∗ = β − α. The only way to deter manipulation in this setting is to set α = β. We again solve the principal’s design program starting from the second period. Since the manager always chooses ∆∗ = β − α as long as he gets the misreporting option, we use CE2 (a2 , ∆∗ ; 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 x1 . We use CE2 (a2 ; x1 , 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, ∆∗ ; x1 , 1) ≥ CE2 (0, ∆∗ ; x1 , 1), and CE2 (H; x1 , 0) ≥ CE2 (0; x1 , 0), ∀x1 . With the noted separability, it is readily apparent that both constraints collapse to β ≥ k2CH , just as in the benchmark. d 1 (a1 ; a2 ) (to distinguish from CE1 (a1 ; a2 ) in the benchmark) to We use CE denote the manager’s certainty equivalent at the beginning of the first period when he chooses a1 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 + α(k1 a1 + 1 ) + β(k2 a2 + 2 ) − c(a1 , a2 )] +qEu[W + α(k1 a1 + 1 − ∆∗ ) +β(k2 a2 + 2 + ∆∗ ) − c(a1 , a2 ) − 12 ∆∗2 ]. Therefore, d 1 (a1 ; a2 ) is the solution to CE d
d 1 (a1 ; a2 )) = −e−r(CE 1 (a1 ;a2 )) = (1 − q)Eu[W + α(k1 a1 + 1 ) + β(k2 a2 + u(CE 1 ∗2 ∗ ∗ 2 )−c(a1 , a2 )] +qEu[W +α(k1 a1 + 1 −∆ ) +β(k2 a2 + 2 +∆ ) −c(a1 , a2 )− 2 ∆ ] 2 2 2 2 r = −(1 − q)e−r[W +αk1 a1 +βk2 a2 −c(a1 ,a2 )− 2 (α σ +β σ )] 2 2 2 2 ∗2 ∗ r 1 −qe−r[W +αk1 a1 +βk2 a2 −c(a1 ,a2 )− 2 (α σ +β σ )− 2 ∆ +(β−α)∆ ] 2 2 2 2 ∗ ∗2 r 1 = −e−r[W +αk1 a1 +βk2 a2 −c(a1 ,a2 )− 2 (α σ +β σ )] [(1 − q) + qe−r[(β−α)∆ − 2 ∆ ] ].
d 1 (a1 ; a2 ) = W + αk1 a1 + βk2 a2 − c(a1 , a2 ) − r (α2 σ2 + β 2 σ 2 ) − Thus, CE 2 1 − r2 (β−α)2 ] ]. r ln[(1 − q) + qe d 1 (a1 ; a2 ) with CE1 (a1 ; a2 ) in the benchmark, we have Comparing CE d 1 (a1 ; a2 ) = CE1 (a1 ; a2 ) − CE
1 r
r
2
ln[(1 − q) + qe− 2 (β−α) ] ].
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. Given a2 = H, the incentive compatibility constraint for the first period is d 1 (H; H) ≥ CE d 1 (0; H). Again, thanks to the separability between the action CE choice and the manipulation amount choice, this constraint reduces to α ≥ k1CH , just as in the benchmark case. 11
d 1 (H; H) ≥ 0, or The individual rationality constraint in this model is CE 2 r 2 2 r 1 2 2 W + αk1 H + βk2 H − 2C − 2 (α σ + β σ ) − r ln[(1 − q) + qe− 2 (β−α) ] ] ≥ 0. The principal’s expected payment to the manager, upon substituting the manager’s ∆ choice, is (1 − q)E[W + α(k1 H + 1 ) + β(k2 H + 2 )] +qE[W + α(k1 H + 1 − (β − α)) + β(k2 H + 2 + (β − α))] = (1 − q)[W + αk1 H + βk2 H] +q[W + α(k1 H − (β − α)) + β(k2 H + (β − α))] = W + αk1 H + βk2 H + q(β − α)2 . Now we have the principal’s design program in this misreporting model: min W + αk1 H + βk2 H + q(β − α)2
Program [C]
W,α,β
s. t. 2 r W + αk1 H + βk2 H − 2C − r2 (α2 σ 2 + β 2 σ 2 ) − 1r ln[(1 − q) + qe− 2 (β−α) ] ] ≥ 0 (IR) α ≥ k1CH (IC1) (IC2) β ≥ k2CH Similar to the previous models, here the individual rationality constraint must bind, and the principal’s expected cost can be expressed as 2C + r2 (α2 σ2 + 2 r β 2 σ2 ) + 1r ln[(1 − q) + qe− 2 (β−α) ] + q(β − α)2 . For later reference, the reduced program is written below. r (α2 σ 2 + β 2 σ 2 ) + 1r α,β 2 s. t. α ≥ k1CH β ≥ k2CH
min
r
0
2
ln[(1 − q) + qe− 2 (β−α) ] + q(β − α)2 Program [C ] (IC1) (IC2) 0
Define α∗C , β ∗C as the optimal bonus rates in Program [C ], we have the following results. Proposition 2 The optimal contract in the misreporting model exhibits ∗ C C ∗ k1 H ≤ αC < k2 H = β C . Corollary 1 When q is sufficiently low, the optimal contract in the misreporting model exhibits α∗C = k1CH and β ∗C = k2CH . Corollary 2 When q is sufficiently high and k1 is sufficiently large, the optimal contract in the misreporting model exhibits α∗C > k1CH and β ∗C = k2CH . Proof. See the Appendix. 0
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 12
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 α∗C < β ∗C . 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 α = β. Instead, it is efficient to tolerate some misreporting. This surprising fact is also shown in Liang (2003). Liang documents that the optimal contract exhibits α∗C 6= β ∗C , 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 α∗C < β ∗C , but sometimes she even maintains the bonus rates at the levels in the benchmark case where there is no misreporting option. Although manipulation is induced by an uneven bonus scheme, it may not be efficient for the principal to adjust the bonus scheme to restrain manipulation. Since the induced misreporting is given by ∆∗ = β − α, the dead weight loss of misreporting can be reduced by lowering β − α. To lower β − α, the principal either lowers β or raises α. However, β has a binding lower bound at C k2 H , and the principal cannot reduce β below that bound. Thus the optimal β remains at its bound. By raising α, the principal reduces the dead weight loss of misreporting, but simultaneously increases the riskiness of the unmanaged compensation scheme ( r2 (α2 σ 2 + β 2 σ 2 ) 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 1 shows a numerical example to illustrate Proposition 2, and Corollaries 1 and 2. C C k1 q k1 H k2 H 20 0.01 0.125 0.1667 200 0.9 0.125 0.1667 Table 1. Numerical example for
α∗C β ∗C 0.1250 0.1667 α∗C = k1CH , β ∗C = k2CH 0.1304 0.1667 α∗C > k1CH , β ∗C = k2CH Proposition 2, and Corollaries 1 and 2.
In this numerical example, we fix the values of the cost of high action C = 25, high action level H = 10, output variance σ2 = 0.5, risk aversion r = 0.5, and 13
the second period productivity k2 = 15. We focus on how the optimal contract changes with the misreporting probability q and the first period productivity k1 . When q is very small (q = 0.01), the optimal contract has both α∗C and β ∗C at their lower bounds. However, when q is high (q = 0.9) and k1 is large (k1 = 200), the first period bonus rate α∗C in the optimal contract deviates from its lower bound k1CH . For simplicity, we use the case q = 1 to explore more details on the misreporting option.
3.2.1
Ubiquitous Misreporting Opportunities (q = 1)
When q = 1, Program [C] becomes min W + αk1 H +βk2 H + (β − α)2
Program [C(q = 1)]
W,α,β
s. t. W +αk1 H +βk2 H −2C − 12 (β −α)2 − r2 (α2 σ 2 +β 2 σ 2 ) ≥ 0 α ≥ k1CH β ≥ k2CH
(IR) (IC1) (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 β−α to the second period to get additional bonus income of (β − α)∆∗ = (β − α)2 . Responding to this, she can cut the fixed wage by (β − α)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 α∗C1 , β ∗C1 as the optimal bonus rates for the misreporting model when q = 1. We have the following result. Proposition 3 When q = 1, the optimal contract in the misreporting model exhibits ∗ 1 C C 2 α∗C1 = 1+rσ 2 k H and β C1 = k H if k1 ≥ k2 (1 + rσ ); 2 2 ∗ α∗C1 = k1CH and β C1 = k2CH otherwise. Proof. See the Appendix. If the productivity of the two periods is very different (that is, k1 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.
14
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 (α∗C1 , β ∗C1 ) = (α∗A , β ∗A ). 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 α 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 measures to fight earnings management. However, according to our results in Propositions 2 and 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.
4
Bundled Hedging and Misreporting Options
Now we mix the delegated hedging option and the misreporting option. Suppose at the beginning of the first period, the manager chooses his action level a1 ∈ {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 β − α from the first to the second period. The time line of the hedge-misreport bundle model is shown below.
15
1st Period
Manager chooses a1 ∈ {H, 0}, d ∈ {0, 1}
2nd Period
Manager observes privately x1 and m (if chose d = 1). x b1 reported. Manager chooses a2 ∈ {H, 0}.
x2 observed privately. x b2 reported. (Principal sees aggregate output.)
Time Line 3: Hedge-misreport bundle model
According to a variation on Lemma 4, the manager always prefers to hedge.8 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 σ 2d ). It’s straightforward that the principal’s design program in this bundled model is min W + αk1 H + βk2 H + q(β − α)2
Program [D]
W,α,β
s. t. 2 r W + αk1 H + βk2 H − 2C − 2r (α2 σ 2 + β 2 σ 2d ) − 1r ln[(1 − q) + qe− 2 (β−α) ] ] ≥ 0 (IR) α ≥ k1CH (IC1) (IC2) β ≥ k2CH Again, Program [D] can be reduced to the minimization of r2 (α2 σ 2 +β 2 σ 2d )+ 1 − r2 (β−α)2 ]+ q(β − α)2 subject to the incentive constraints. For r ln[(1 − q) + qe the convenience for later analysis, we show the reduced program below. r (α2 σ 2 + β 2 σ 2d ) + 1r α,β 2 s. t. α ≥ k1CH β ≥ k2CH
min
r
2
0
ln[(1 − q) + qe− 2 (β−α) ]+ q(β − α)2 Program [D ] (IC1) (IC2)
In addition, it is easy to verify 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 2r β 2 (σ 2 − σ 2d ), thanks to the hedging option. The principal does not need to motivate the manager to hedge, since the manager always exercises the hedging option. 8 We use CE d 1 (a1 , d; a2 ) to represent the manager’s beginning certainty equivalent when he chooses a1 and d followed by a2 . Similar to the argument in Lemma 4, it can be shown that d 1 (a1 , 0; a2 ). That is, the manager always hedges in this bundled model. d 1 (a1 , 1; a2 ) ≥ CE CE
16
4.1
Whether to Take the Bundle
The hedge-misreport bundle is a mixture of "good" and "bad". As analyzed in previous sections, 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 5 When q is sufficiently low, the hedge-misreport bundle is preferred to the benchmark. Lemma 6 When q is sufficiently high and k1 is sufficiently large, the hedgemisreport bundle is preferred to the benchmark only if σ 2 − σ2d is sufficiently large. Proof. See the Appendix. 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 k1 , 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 (σ 2 − σ 2d is sufficiently large). 0
Now focus on the q = 1 case. From Program [D ], when q = 1, the program becomes r (α2 σ 2 + α,β 2 s. t. α ≥ k1CH β ≥ k2CH
min
0
β 2 σ2d ) + 12 (β − α)2
Program [D (q = 1)] (IC1) (IC2)
17
Define α∗D1 , β ∗D1 as the optimal bonus rates in the bundled model when q = 1, we have Lemma 7 When q = 1, the optimal contract in the hedge-misreport bundle model exhibits ∗ 1 C C 2 α∗D1 = 1+rσ 2 k H and β D1 = k H if k1 ≥ k2 (1 + rσ ); 2 2 ∗ C C ∗ αD1 = k1 H and β D1 = k2 H otherwise. Proof. See the Appendix. As mentioned, 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 4 When q = 1, (1) if k1 ≥ k2 (1 + rσ 2 ), the hedge-misreport bundle is preferred to the benchmark2 when2 k2 σ −(1+rσ 2 )(σ 2 −σ 2d ) > ; (a) σ 2 (1+rσ 2 ) k12 and (2) if k1 < k2 (1 + rσ 2 ), the bundle is preferred to the benchmark when k12 1 (b) (k1 −k2 )2 > r(σ 2 −σ 2 ) . d
Proof. See the Appendix. Proposition 4 provides detailed analysis on the trade-off between the benefit from the hedging option and the cost from the misreporting option when q = 1. Specifically, (i). For both the cases of k1 ≥ k2 (1 + rσ2 ) and k1 < k2 (1 + rσ 2 ), the hedgemisreport bundle is more likely to be preferred (in other words, condition (a) or (b) is more likely to be satisfied) when σ 2 is high.9 Intuitively, when the unhedged output signals are very noisy, the principal has strong preference for hedging to reduce the noise and is more likely to take the bundle, regardless of the accompanied cost of misreporting. (ii). For both cases, it is easy to verify that condition (a) or (b) is 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 high risk premium to the manager for the noisy output signals. Thus, the principal would like to reduce the noise in output signals and has strong preference for hedging. Hence she is more likely to take the hedge-misreport bundle regardless of the accompanied cost of misreporting. 9 Define
σ 2 −(1+rσ 2 )(σ 2 −σ 2 d) σ 2 (1+rσ 2 ) verify that r(σ21−σ2 ) d
Q≡
It is easy to
in condition (a), we have
∂Q ∂σ 2
=
in condition (b) also decreases
18
2 2 −(1+rσ 2 )2 σ 2 d −r(σ ) [σ 2 (1+rσ 2 )]2 in σ2 .
< 0.
(iii). For both cases, the harm from misreporting behavior is smaller when k1 , k2 are close. (k1 , k2 are close implies that the bonus rates can be set closer, and the potential damage from misreporting is small.) The closer are k1 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. (iv). For both cases, the benefit from hedging is higher when hedging greatly reduces the noise in output signals (that is, when σ2 −σ 2d 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. 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 (σ 2 − σ 2d ), 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 β ∗D − α∗D from the mean of the first period output to the mean of the second period output. The amount of manipulation β ∗D − α∗D and the increased compensating wage differential due to misreporting depend on a host of factors, including the misreporting probability q, the action productivity, and rσ2 . 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.
5
"Undesirable" Hedging
For the main model in this paper we assume output follows normal distributions, and it is shown that when the unhedged production plan is a mean preserving spread of its hedged counterpart, hedging improves the informativeness of the output measure and helps reduce agency cost. However, we need to be cautious not to take this result casually and announce that "as long as hedging reduces the risk in output, it improves the output signals’ informativeness". The normal distribution assumption is not idle here. It provides a nice structure that a hedged plan leads to a more dispersed likelihood ratio distribution (a likelihood ratio distribution with larger variance).10 A more dispersed likelihood ratio distribution makes it easier for the principal to infer the manager’s action from the output, and is the key to the improvement in informativeness of output signals. If hedging cannot provide a more informative likelihood ratio distribution, it does not help reduce the compensating wage differential, and it is not desirable for a risk neutral principal. 1 0 Detailed
analysis available upon request.
19
Consider a finite support numerical example in which hedging doesn’t help improve the informativeness of outputs, and 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 doesn’t hedge (or when there is no hedging option), the probability distribution of {1, 2, 3} given the manager’s high action is PH = ( 13 , 16 , 12 ), and when the manager chooses low action the distribution is PL = ( 13 , 13 , 13 ). But when the principal hedges, the 5 probability distribution given high action is PHd = ( 14 , 13 , 12 ), and the distribu1 1 1 tion given low action is PLd = ( 4 , 2 , 4 ). Also assume C = 25 and r = 0.01. It’s easy to verify that PH is a mean preserving spread of PHd and PL is a mean preserving spread of PLd . However, in this example, the principal pays 54.1149 to encourage high action when she hedges, while she only pays 37.7216 when she doesn’t hedge or when there is no hedging option. The reversed result in this example is because after hedging, the likelihood ratios are less dispersed than before, and therefore the information system is worsened. Without hedging, the likelihood ratio distribution for outputs {1, 2, 3} is PPHL = (1, 2, 23 ), while Ld after hedging, the likelihood ratio distribution becomes PPHd = (1, 32 , 35 ). With a narrower spread of likelihood ratios, it becomes more difficult for the principal to infer the agent’s action level from the output, and the compensating wage differential goes up. 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 informativeness of performance signals. Counter-intuitively, risk-reducing activities may worsen the information system and increase the compensating wage differential. In other words, even though hedging activities reduce firms’ risks, in some cases they are detrimental to investors. Normal distributions, however, are a natural assumption. It is also adopted in prior analytical research on hedging and risk management, such as Campbell and Kracaw (1987). In addition, the normal distribution is not the only distribution that provides a connection between risk reduction and informativeness. Hedging improves the informativeness of performance signals as long as the hedged production plan has a more dispersed likelihood ratio distribution.
6
Conclusions and Future Research
This paper studies the interaction between hedging and earnings management. We analyze how contracting is influenced by earnings management and hedging activities. For the hedging function of derivatives, we use a mean preserving spread structure to capture the risk reduction theme. It is shown in a two-period LEN framework that hedging is efficient regardless of whether the principal or the manager gets the hedging authority. 20
For the earnings management case, it is shown that although earnings management garbles information and is detrimental to the principal, sometimes it is optimal not to take any measure to fight earnings management. Even when earnings management opportunities are ubiquitous, it may still be optimal to live with earnings management instead of taking any action to restrain it. In addition, it is never efficient to motivate truth-telling. The two-edged feature of derivative instruments is modeled with a hedgemisreport bundle. We analyze the trade-off between the improvement of performance signals’ objectivity brought by hedging and the increase in performance signals’ bias due to earnings management. It is shown that hedging and earnings management influence performance signals from different angles, and their net influence depends on various factors. We also document "undesirable" hedging as a supplement to the main discussion. It is shown that there is no necessary connection between risk reduction and improvement in performance signals’ informativeness. Although hedging reduces firms’ risks, in some cases it lowers the information content of the contracting variables and drives up compensating wage differentials. Many directions for future research remain. First, exploring "desirable" earnings management may be fruitful. In this paper we focus on detrimental earnings management that merely garbles information. However, researchers have realized that earnings management can be beneficial. When earnings management conveys the manager’s private information, in some cases it is efficient to encourage manipulation.11 We currently have a forecast model (not included in this paper) in which the manipulated earnings reveal information about the manager’s action levels and his forecast of the future period output. In that model, under tightly controlled conditions, we can even achieve the first-best by encouraging a hedge-misreport bundle. More exploration on the joint effect of "desirable" earnings management and hedging is under way. Secondly, more work will be done on "undesirable" hedging. As we mentioned in Section 5, there is no necessary connection between risk reduction and informativeness improvement. The relationship between risk reduction and informativeness improvement is interesting and important by itself, and it is worth more exploration. Moreover, if both "undesirable" hedging and earnings management garble information, we may find a substitute relationship, just as Barton (2001) suggests. Thirdly, the variety of financial derivatives offers many avenues for future research. For example, this paper focuses on the financial derivatives designated as hedging instruments. However, in practice there are also derivatives for trading purposes. Modeling the derivatives held and issued for trading purposes 1 1 Demski (1998), Arya, Glover and Sunder (1998), and Demski and Frimor (1999), among others, document "desirable" earnings management.
21
will be different, for they are traded based on asymmetric information about a future market, and possibly on investors’ heterogenous beliefs. In addition, there are many different types of financial instruments such as swaps, futures, forwards, options, etc.. How managers choose derivative instruments and how different instruments influence the output are topics for future research.
22
Appendix Proof for Lemma 1: Proof. We use µ1 , µ2 to denote the Lagrangian multipliers for (IC1) and (IC2) respectively. With the reduced program, the first order conditions are −rσ 2 α + µ1 = 0 and −rσ2 β + µ2 = 0. Since α ≥ k1CH > 0 and β ≥ k2CH > 0, we get µ1 = rσ2 α > 0 and µ2 = rσ2 β > 0. This implies both (IC1) and (IC2) bind, or α∗A = k1CH , β ∗A = k2CH . Proof for Lemma 2: Proof. The principal’s design program can be expressed as the minimization of r2 (α2 σ 2 + β 2 σ 2d ) subject to the incentive constraints. Again let µ1 , µ2 denote the Lagrangian multipliers of (IC1) and (IC2) respectively, with the reduced program, we have the first order conditions −rσ 2 α+µ1 = 0 and −rσ 2d β +µ2 = 0. Hence µ1 = rσ 2 α > 0 and µ2 = rσ 2d β > 0. This implies both (IC1) and (IC2) are binding, and thus α∗A = k1CH and β ∗A = k2CH . Proof for Lemma 4: Proof. We use CE1 (a1 , d; a2 ) to denote the manager’s certainty equivalent at the beginning of the first period when he chooses a1 and d in the first period and chooses a2 in the second period. By hedging, CE1 (a1 , 1; a2 ) = W + αk1 a1 + βk2 a2 − c(a1 , a2 ) − r2 (α2 σ 2 + β 2 σ 2d ). If the manager doesn’t hedge, CE1 (a1 , 0; a2 ) = W +αk1 a1 +βk2 a2 −c(a1 , a2 )− 2r (α2 σ 2 +β 2 σ2 ). (c(a1 , a2 ) represents the manager’s cost for his actions.) We have CE1 (a1 , 1; a2 ) > CE1 (a1 , 0; a2 ) for σ2d < σ 2 . Therefore d = 1 is always preferred. Proof for Proposition 2: Proof. Define µ1 as the Lagrangian multiplier for (IC1) and µ2 for (IC2). With 0 the reduced Program [C ], the first order conditions are − r (β−α)2 2 (β−α) r 2 1−q+qe− 2 (β−α) − r (β−α)2 qe 2 (β−α) 2 r 2 1−q+qe− 2 (β−α)
−rσ 2 α − qe
+ 2q(β − α) + µ1 = 0
and −rσ β +
−2q(β − α) + µ2 = 0.
(FOC1) (FOC2)
In the optimal solution, if neither constraint is binding, µ1 , µ2 = 0. Substitute µ1 , µ2 = 0 into the first order conditions and add the two conditions together, we get −rσ2 β − rσ 2 α = 0, which implies α = β = 0. This contradicts α ≥ k1CH > 0 and β ≥ k2CH > 0. Therefore, µ1 , µ2 = 0 is not true in the optimal solution, and at least one of the constraints is binding. q If µ1 > 0, µ2 = 0, then α = k1CH and β ≥ k2CH . (FOC2) implies β = rσ2M+Mq α, r
2
e− 2 (β−α) r 2 1−q+qe− 2 (β−α) 2 r − 2 (β−α)
where M ≡ 2−
. Rewrite M, we have M = 2
r
1 {(1− r 2 1−(1−e− 2 (β−α) )q
) +[1 − q(1 − e− 2 (β−α) )]} > 0. With M > 0, we get β = q)(1 − e Mq C α < α = rσ 2 +M q k1 H . However, k1 > k2 implies β doesn’t satisfy the constraint β ≥ k2CH . Therefore, (α = k1CH , β ≥ k2CH ) cannot be true. Hence, regardless of µ1 , (IC2) always binds, implying β ∗C = k2CH .
23
Thus, µ2 > 0 and µ1 ≥ 0. Moreover, if (IC1) binds, then α = k1CH ; if (IC1) ∗ ∗ is slack, then from (FOC1) α = rσ2Mq +Mq β < β. Hence we always have αC < β C . Proof for Corollary 1: Proof. Using the first order conditions displayed in the proof of Proposition 2, we see when q is sufficiently near zero, (FOC1) reduces to −rσ 2 α + ε1 + µ1 = 0 and (FOC2) reduces to −rσ 2 β + ε2 + µ2 = 0, where ε1 and ε2 are small. This implies µ1 > 0 and µ2 > 0. That is, when q is sufficiently small, both incentive constraints bind and α∗C = k1CH , β ∗C = k2CH . Proof for Corollary 2: Proof. Using the first order condition (FOC1) in the proof of Proposition 2 again, if µ1 > 0 and α = k1CH , then µ1 = rσ 2 k1CH − (2q −
2 C − r (β ∗ C − k1 H ) 2
qe
− r (β ∗ − C )2 1−q+qe 2 C k1 H
)(β ∗C − k1CH ) > 0.
(i) can be re-expressed as 2 − r ( C ) ( 1 − 1 )2 2 H k2 k1 rσ 2 k11 −(2q− qe )( k12 − k11 ) > 0. 2 − r ( C ) ( 1 − 1 )2 k2 k1 1−q+qe 2 H
(i)
(ii)
Now suppose q is sufficiently high and k1 is sufficiently large. This implies the inequality in (ii) is reversed and α∗C > k1CH . Proof for Proposition 3: Proof. Define µ1 , µ2 as the Lagrangian multipliers of the two constraints respectively. We get the following first order constraints: 0 −rσ 2 α + (β − α) + µ1 = 0 (FOC1 ) 0 −rσ 2 β − (β − α) + µ2 = 0 (FOC2 ) From Proposition 2, we know µ2 > 0, and µ1 ≥ 0. 0 If µ1 = 0, µ2 > 0, then β = k2CH , and from (FOC1 ) we get −rσ 2 α+(β −α) = 1 1 C 1 C C 0, which implies α = 1+rσ 2 β = 1+rσ 2 k H . If 1+rσ 2 k H ≥ k H , then both 2 2 1 ∗ 1 C C incentive constraints are satisfied, and α∗C1 = 1+rσ 2 k H and β C1 = k H are 2 2 1 C C the optimal bonus rates. The condition 1+rσ2 k2 H ≥ k1 H reduces to k1 ≥ k2 (1 + rσ 2 ). 1 C C 1 C If 1+rσ 2 k H < k H , then α = 1+rσ 2 k H doesn’t satisfy the constraint α ≥ 2 1 2 ∗ C C C ∗ k1 H , and the optimal contract must have αC1 = k1 H and β C1 = k2 H , as both incentive constraints bind. Proof for Lemma 5: Proof. From Corollary 1, when q is sufficiently low, the optimal contract in the misreporting model is (α∗C = k1CH , β ∗C = k2CH ). The misreporting model, 0 0 Program [C ], is identical to the bundled model, Program [D ], except that the second period output variance decreases to σ 2d in the bundled model. It is easy to verify that (α∗C = k1CH , β ∗C = k2CH ) remains optimal in the bundled model when q is sufficiently near zero. 24
The principal’s expected cost in the bundled model therefore gets close to ∗2 2 2 2C + r2 (α∗2 C σ +β C σ d ) when q is near zero. In the benchmark model where there 2 is neither hedging nor misreporting option, her expected cost is 2C + 2r (α∗2 Cσ + ∗2 2 ∗2 2 2 β C σ ), which is higher than 2C + 2r (α∗2 C σ +β C σ d ). The hedge-misreport bundle is preferred. Proof for Lemma 6: Proof. From Corollary 2, when q is sufficiently high and k1 is sufficiently large, the optimal contract exhibits α∗C > k1CH , β ∗C = k2CH . We rewrite α∗C as k1CH + ε, ε > 0. In addition, from Proposition 2, we know α∗C = k1CH + ε < β ∗C = k2CH . In the benchmark model where there is neither a hedging nor a misreporting ∗2 2 2 option, the principal’s expected cost is 2C + r2 (α∗2 A σ + β A σ ), while in the ∗ 2 ∗2 2 ∗ 1 2 − r2 (β ∗ C −αC ) ]+ q(β bundled model it is 2C + 2r (α∗2 C− C σ +β C σ d )+ r ln[(1−q)+qe ∗ ∗ 2 r ∗2 2 ∗2 2 r r 1 ∗ 2 ∗2 2 ∗2 2 − 2 (β C −αC ) ]+ αC ) . If 2 (αA σ + β A σ ) − { 2 (αC σ + β C σd ) + r ln[(1 − q) + qe q(β ∗C − α∗C )2 } > 0, the hedge-misreport bundle is preferred to no hedging, no misreporting. Now, ∗ 2 ∗2 2 ∗2 2 ∗ r r 1 ∗2 2 ∗2 2 − r2 (β ∗ C −αC ) ]+ q(β C− 2 (αA σ +β A σ )−{ 2 (αC σ +β C σ d )+ r ln[(1−q)+qe ∗ 2 αC ) } = r2 ( k2CH )2 (σ 2 − σ 2d ) − { r2 σ 2 [( k1CH + ε)2 − ( k1CH )2 ] r
C
C
2
+ 1r ln[(1 − q) + qe− 2 ( k2 H − k1 H −ε) ] + q( k2CH − k1CH − ε)2 } But (iii) is positive only when σ 2 − σ 2d is sufficiently large.
(iii)
Proof for Lemma 7: 0 Proof. Refer to the proof for Proposition 3. Program [C (q = 1)] is identical 0 to Program [D (q = 1)] except that in the bundled model program’s objective function, the second period variance is σ 2d instead of σ2 . It is easy to verify that these two programs share the same optimal bonus coefficients. (α∗D1 , β ∗D1 ) = (α∗C1 , β ∗C1 ). Proof for Proposition 4: Proof. In the benchmark model where there is neither a hedging nor a mis∗2 2 2 reporting option, the principal’s expected cost is 2C + 2r (α∗2 A σ + β A σ ), while ∗2 2 ∗ 1 2 ∗ 2 in the bundled model it is 2C + 2r (α∗2 D1 σ + β D1 σ d ) + 2 (β D1 − αD1 ) . As long ∗2 2 ∗2 2 ∗ r r 1 ∗2 2 ∗2 2 ∗ 2 as 2 (αA σ + β A σ ) − [ 2 (αD1 σ + β D1 σ d ) + 2 (β D1 − αD1 ) ] > 0, the hedgemisreport bundle is preferred. 1 C (1). When k1 ≥ k2 (1+rσ 2 ), the optimal contract exhibits (α∗D1 = 1+rσ 2 k H, 2 ∗2 2 r 2 ∗2 2 β ∗D1 = k2CH ). Substitute α∗A , β ∗A and α∗D1 , β ∗D1 into r2 (α∗2 A σ +β A σ )−[ 2 (αD1 σ + ∗2 2 ∗ 1 ∗ 2 β D1 σ d ) + 2 (β D1 − αD1 ) ]. We have ∗2 2 ∗2 2 ∗ r r 1 ∗2 2 ∗2 2 ∗ 2 2 (αA σ + β A σ ) − [ 2 (αD1 σ + β D1 σ d ) + 2 (β D1 − αD1 ) ] r(σ 2 −σ 2d ) (rσ 2 )2 rσ 2 − (1+rσ 2 )2 k2 + 2 )2 k2 ] (1+rσ k22 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 C 2 rσ (1+rσ ) k2 −rσ k1 +r(σ −σ d )(1+rσ ) k1 −(rσ ) k1 = 2(H ) 2 2 2 2 (1+rσ ) k1 k2 C 2 = 12 ( H ) (1+rσ21)2 k2 k2 {rσ 2 (1 + rσ2 )2 k22 − k12 [rσ 2 − r(σ 2 1 2 2 2 2
C 2 rσ ) [ k2 − = 12 ( H
(rσ ) ]}
25
− σ2d )(1 + rσ2 )2 +
Thus, we need rσ 2 (1 + rσ 2 )2 k22 − k12 [rσ 2 − r(σ 2 − σ 2d )(1 + rσ 2 )2 + (rσ 2 )2 ] > 0.
σ 2 −(1+rσ 2 )(σ 2 −σ 2d ) . σ 2 (1+rσ 2 ) 2 (2). When k1 < k2 (1+rσ ), the optimal contract exhibits (α∗D1 = k1CH , β ∗D1 = ∗ ∗ ∗ ∗ C ∗ ∗ ∗ ∗ k2 H ). (αD1 , β D1 ) is identical to (αA , β A ). Substitute αA , β A and αD1 , β D1 into ∗2 2 ∗ r r 1 ∗2 2 ∗2 2 ∗2 2 ∗ 2 2 (αA σ + β A σ ) − [ 2 (αD1 σ + αD1 σ d ) + 2 (β D1 − αD1 ) ], we have 2 2 ∗2 2 ∗ r r 1 1 C 2 r(σ −σ d ) ∗2 2 ∗2 2 ∗2 2 ∗ 2 − 2 (αA σ +β A σ )−[ 2 (αD1 σ +αD1 σ d )+ 2 (β D1 −αD1 ) ] = 2 ( H ) [ k22 r(σ 2 −σ 2d ) k12 1 1 2 1 1 2 − ( k2 − k1 ) > 0, which implies (k1 −k2 )2 > ( k2 − k1 ) ]. Thus, we need k22 1 . r(σ 2 −σ 2d )
This implies
k22 k12
>
26
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