High-powered incentives and fraudulent behavior: Stock-based versus stock option-based compensation

Economics Letters 101 (2008) 122–125

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Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o n b a s e

High-powered incentives and fraudulent behavior: Stock-based versus stock option-based compensation Rainer Andergassen ⁎ Department of Economics, University of Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy

a r t i c l e

i n f o

Article history: Received 2 February 2006 Received in revised form 3 July 2008 Accepted 7 July 2008 Available online 12 July 2008

a b s t r a c t This paper examines the trade-off shareholders face between providing managers with incentives to exert beneficial effort and to engage in costly fraudulent activity. We provide a solution to the optimal compensation problem, given that shareholders can either grant (restricted) stock or stock options and given fixed average compensation costs. © 2008 Elsevier B.V. All rights reserved.

Keywords: Executive compensation Executive stock options Restricted stock Fraud Incentives JEL classification: G30 M52

1. Introduction The 1980s and 1990s witnessed an explosion in stock option-based compensation packages in the U.S. (Murphy, 1999; Hall and Murphy, 2003). Stock options are granted in order to align the interests of managers with those of shareholders, thereby reducing agency costs. Over the past few years, the proportion of pay through stock options has declined. A recent survey by Pearl Meyer and Partners for the New York Times shows a decrease in the percentage of total remuneration in options from 66% to 31%, while restricted stock and long-term incentive plans (LTIP) over the same period increased from 14% to 30%. Thus, the current trend in corporate compensation is to switch from stock options to alternative compensation schemes such as restricted stocks. The most famous example is Microsoft, which previously was one of the largest users of employee stock options. One explanation for this change in corporate compensation is the relative cost of employee stock options. If a manager is risk averse, then stock options are not a very effective instrument for aligning shareholder goals with those of managers (Hall and Murphy, 2003). The recent wave of accounting scandals, such as Enron, Tyco, Worldcom and Xerox, hints at a strong tie between executive compensation packages and fraudulent behavior. Compensation schemes are designed to induce effort, but they may also induce

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costly fraudulent activity (see Goldman and Slezak, 2006). A selfinterested manager may misrepresent and manipulate earnings or disseminate fraudulent information to inflate the stock price and, as a result, his compensation and wealth. Recent empirical literature finds a positive link between option-based compensation packages and earnings management, while finding no significant impact of restricted stock and LTIP on earnings management (see Burns and Kedia, 2006; Gao and Shrieves, 2002). This paper considers the case where a manager can boost the short-term cash flow either by increasing his effort or manipulating earnings. By increasing his effort, the manager also boosts the company's long-term value, whereas earnings manipulation is assumed here to reduce the company's long-term value.1 The manager and shareholder face different time horizons. While the shareholder is interested in the company's long-term value, the manager's compensation is a function of its short-term value. Shareholders face a tradeoff between providing managers with incentives to exert beneficial effort and to engage in costly fraudulent activity. We provide a solution to the optimal compensation problem, given that shareholders can either grant (restricted) stock or stock options and given fixed average compensation costs. We also show that the fraud to effort ratio is an increasing function of the strike price. Furthermore, we find a threshold level for the long-term negative effects of fraud 1 For empirical evidence on the negative relationship between earnings manipulation and the future stock price, see Chan et al. (2006), Teoh et al. (1998a,b) and Jensen (2003).

R. Andergassen / Economics Letters 101 (2008) 122–125

according to which stock-based compensation is optimal for larger values, whereas for lower values stock option-based compensation is optimal. Last, we show that the optimal strike price is decreasing in the size of long-term negative effects of fraud on the company's value.

― costs at W. The shareholder's problem can formally be stated as 4 follows ~  max E S3

~ S2 ¼




S þ αe þ βf þ ~e2 S þ αe þ ~e2

with probability 1−p with probability p

ð1Þ

where˜ ε2 ∈U[−h,h] is a uniformly distributed random variable compounding project- and market-specific risk, its value being realized at time 2. Effort increases the company's short-term and long-term value, while earnings management boosts only its short-term value and decreases its long-term value. Thus, the company's expected long-term value is ~  E S3 ¼ S þ αe−δf

ð2Þ

where δf indicates the long-term loss in profitability due to the manager's fraudulent behavior. The shareholder defines the compensation package (λ, K), consisting of λ ≥ 0 stock options, vesting at time 2, with strike price K ≥ 0. For K = 0 the shareholder grants (restricted) stock, while for K N 0 he grants stock options. The shareholder and the manager face different time horizons. We assume that the stock options vest at time 2, and thus the executive is interested in the firm's short-term value. Probability p is therefore the probability of detecting fraudulent behavior before the executive's stock options are vested. The executive's compensation is ˜ W2 = λ˜ V2, where ˜ V2 = {˜ S 2 − K}+. On the other hand, shareholders are interested in the company's long-term value (Eq. (2)). Effort (e) and fraud (f) are not observable. Fraudulent behavior does not directly produce disutility like effort does; however, if the manager is caught, then a penalty is imposed, its size increasing in f. Let P ð f Þ ¼ 2
f2 be the penalty, then at time 1 the executive's expected utility is  ~  ~  f 2 e2 E U W 2 uE W 2 −p
− : 2 2 The shareholder defines the strike price K and the number of options granted λ, maximizing the company's expected long-term present value (Eq. (2)) subject to the incentive compatibility constraint and to a budget constraint which fixes the expected compensation

2

See also Kadan and Yang (2005). This assumption is supported by the finding in Erickson et al. (2004) that the value of equity holdings of top managers accused of fraud was substantially overstated prior to the revelation of fraudulent behavior. 3

ð3Þ

K

2. The model We consider an economy consisting of three time periods: 1, 2 and 3. Times 1 and 2 are the short-term, while time 3 represents the long-term. We consider two agents, one being an effort-averse manager and the other being a representative shareholder. Both the manager and the representative shareholder are risk-neutral. In assuming risk neutrality, we eliminate the trade-off shareholders face between providing the manager with incentives and insurance, allowing us to focus on the trade-off between effort and fraud2. The risk-neutral interest rate is normalized to zero. Throughout the paper, random variables are indicated with a tilde (~) and realized variables without a tilde. At time 1, the manager undertakes a risky project, whose fundamental value depends on the effort e ≥ 0 exerted during the same time period. Furthermore, at time 1, the manager may inflate the company's short-term value by engaging in earnings management f ≥ 0. Let p be the exogenous probability that fraud is detected at time period 2. Assuming that instantaneous price correction takes place if fraudulent behavior is discovered3 and that the detection event is independent of ˜ ε2, the company's value at time 2 is

123

s:t:

 ~  e; f ¼ arg max E U W 2

ð4Þ

e;f

~  λE V 2 ¼ W

ð5Þ

where Eq. (4) is the incentive compatibility constraint and Eq. (5) is the project's budget constraint. Let us define X = min{S − K + αe, h} and X′ = min{S − K + αe + βf, h}, then   ~  1 h2 −X′2 E V2 ¼ þ ð1−pÞ ðS−K þ αe þ βf Þðh þ X′Þ þ 2h 2   h2 −X 2 þp ðS−K þ αeÞðh þ X Þ þ : 2

f

g

ð6Þ

Three cases arise. Case (1): X = X′ = h, i.e. S − K + αe + βf N S − K + αe N h; Case (2) X′ = h and X = S − K + αe, i.e. S − K + αe + βf N h N S − K + αe; and Case (3) X′ = S−K + αe + βf and X = S − K + αe, i.e. h N S − K + αe + βf N S − K + αe, with (3a) S + αe + h N K and (3b) S + αe + h ≤ K b S + αe + βf + h.5 We define ðiÞ the fraud to effort ratio Φ = {Φ(1), Φ(2), Φ(3,a), Φ(3,b)} where ΦðiÞ ¼ ef ðiÞ is the fraud to effort ratio for Case (i). Lemma 1. Given the budget constraint (5), the fraud to effort ratio as from Eq. (4) is a continuous and increasing function of the strike price. Proof. Case (1). From the first-order conditions (FOC) of the manager's problem (4), we obtain eð1Þ ¼ αλ and f ð1Þ ¼ βλ

1−p : p


ð7Þ ð1Þ

The fraud to effort ratio Φð1Þ u ef ð1Þ ¼ αβ 1−p p
is independent of K. Case (2). From the FOC of the manager's problem (4), we obtain eð2Þ ¼ αλ

pðS−K Þ þ ð2−pÞh 1−p : and f ð2Þ ¼ βλ p
2h−pα 2 λ

ð8Þ

The fraud to effort ratio is Φð2Þ u

f ð2Þ β 1−p 2h−pα 2 λ : ¼ eð2Þ α p
pðS−K Þ þ ð2−pÞh

ð9Þ

To prove continuity of the function Φ in K, we observe that Φ(2) is continuous in K. Furthermore, for e(1) = e(2), S − K + αe(1) = S − K + αe(2) = h and, since f (1) = f (2), Φ(1) = Φ(2). In order to prove that Φ(2) is increasing in K, we apply the implicit function theorem to the budget constraint (5) and obtain pðS−K Þþhð2−pÞ

dλ 2h−α 2 λp ¼λ   :  2 ~ dK ð Þþð2−pÞh p S−K 1 E V 2 þ λα 2 þλð1−pÞ2 β2 p
2h−pα 2 λ

ð10Þ

4 In what follows, we assume that parameters are such that the manager's ― participation constraint is always satisfied. In particular, we require W to be such that E(U(˜ W 2)) ≥ U0, where U0 represents the executive's utility for a given outside option. 5 We neglect the case where the strike price K is such that the option is always outof-the-money also in the state of non-detection (i.e. K ≥ S + αe + βf + h) since, in this case, the compensation has no incentive effect.

124

R. Andergassen / Economics Letters 101 (2008) 122–125

ð12Þ

where the option is out-of-the-money for some realizations of ε˜2 if fraud is detected or not. Compared with the previous case, we observe a reduced incentive per option to engage in fraud and effort, but the effect is asymmetric since the incentives to engage in fraud depend just on the option value in the non-detection state, while the incentives to exert effort depend on both the option value in the detection and non-detection states and both values are decreasing in K. In Case (3b), the option is always out-of-the-money in the detection state, while it remains in-the-money in the non-detection state for some realizations of ε˜2; therefore, fraud and effort are again affected in the same way by K, leading to a constant fraud to effort ratio. Given the budget constraint (5), by increasing K the shareholder induces relatively more fraudulent behavior than productive activity. Thus, in choosing the optimal incentive structure (K), he trades off long-term gains from an increased effort against long-term losses due to fraudulent behavior. The following proposition describes the optimal compensation structure.

ð13Þ

Proposition 1. For δ N δ̄ stock-based compensation is optimal (K = 0), while for δ b δ ̄ stock option-based compensation is optimal (K N 0), where 2 p
δu αβ 1−p .

Taking the derivative of Eq. (9) with respect to K we obtain that the fraud to effort ratio increases if dλ 2h−pα 2 λ b : dK α 2 ð pðS−K Þ þ ð2−pÞhÞ

ð11Þ

By substituting Eq. (10) into the left-hand side of inequality (11) ~  1 and rearranging the terms, we obtain λE V 2 þ λ2 ð1−pÞ2 β2 p
N0; which is always satisfied, establishing the result. Case (3). We divide this case into two parts: the option in the case of detection is (a) for low values of K, in-the-money for some realizations of ε~2, and (b) for large values of K, always out-of-the-money. Part (a). From the FOC of the manager's problem (4) we obtain, after rearranging the terms, ð3;aÞ

e

¼ αλp

  ðS−K þ hÞ 2h
−β2 λð1−pÞ 2hp
ð2h−α 2 λÞ−β2 λð1−pÞð2h−pα 2 λÞ

f ð3;aÞ ¼ βλð1−pÞ

2hðS−K þ hÞ 2hp
ð2h−α 2 λÞ−β2 λð1−pÞð2h−pα 2 λÞ

:

Proof. First we show that for α 2 bβδ 1−p p
, K = 0 is optimal. For K = 0 we are always in Case (1). Using Eq. (7) we obtain

The fraud to effort ratio is ð3;aÞ

Φ

f ð3;aÞ β 1−p 2h : u ð3;aÞ ¼ α p 2h
−β2 λð1−pÞ e

ð14Þ

To prove continuity, we show that if S − K + αe(3,a) + β f (3,a) = h, then Φ = Φ(3,a). Using Eqs. (12) and (13), condition S − K + αe(3,a) + β f (3,a) = h reads (2)

ðS−K þ hÞ

2hp
2hp
ð2h−α 2 λÞ−β2 λð1−pÞð2h−pα 2 λÞ

¼ 1:

ð15Þ

By substituting Eqs. (12) and (13) into equality Φ(2) = Φ(3,a) and rearranging the terms, Eq. (15) can be obtained. ~  AλE V2 It is easy to see that AK b0. Furthermore, simple algebra shows ~  ð3;aÞ ð3;aÞ AλE V2 that AeAλ N0 and AfAλ N0 and consequently Aλ N0. Applying the implicit function theorem to the budget constraint (5) we obtain dλ N0, dK and thus we can observe from Eq. (14) that Φ(3,a) is increasing in K. Part (b). Part (b) arises for a strike price K such that S − K + h + αe(3,a) = 0 and larger values. Using Eq. (12), this equality reads 2hpϕ = β2λ(1 − p). If the option has a positive value only in the non-detection state, then from the FOC we obtain straightforwardly Φð3;bÞ u

f ð3;bÞ β ¼ eð3;bÞ α
p

which is independent of K. Furthermore, note that for S − K + h + αe(3,a) = □ 0 (i.e. for 2hpϕ = β2λ(1 − p)), Φ(3,a) = Φ(3,b). The reason for this result is that the incentives per option to engage in fraud are affected by the strike price only in the non-detection of fraud state, while the incentives to exert effort are affected by the strike price in both the detection and the non-detection of fraud states. An increase in the strike price K leads to a reduction of the options' value, and thus the budget constraint requires an increase in λ which in turn affects e and f. In Case (1), the option is always in-themoney and consequently effort and fraud are affected in the same way by an increase in λ; the fraud to effort ratio, therefore, remains constant as K changes. In Case (2), the option is always in-the-money if fraud is not detected, while, in the case of fraud detection, it is outof-the-money for some realizations of ε˜2. Thus, compared with Case (1), the incentives per option granted to engage in fraud are the same, while the incentives to exert effort are reduced and further are decreasing in K since now the option in the fraud detection state is worth less and further its value is decreasing in K. Consequently, the fraud to effort ratio is increasing in K. The same occurs in Case (3a),

  ~  1−p E S3 ¼ S þ λ α 2 −βδ p


ð16Þ

" # ~  ð1−pÞ2 E V 2 ¼ S−K þ λ α 2 þ β2 : p


ð17Þ

Using Eq. (17), the budget constraint (5) and condition λ N 0, we obtain

λ¼

−ðS−K Þ þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS−K Þ2 þ4ηW

ð18Þ

2η 2

Þ where η ¼ α 2 þ β2 ð1−p p
. The expected firm's value at time 3 is decreasing in λ. Thus, since λ as in Eq. (18) is strictly increasing in K, the optimal strike price is K = 0. To complete the proof, we have ~  dE S to show that dK is also negative in Cases (2) and (3). Consider first ˜3) is decreasing in K if f z α, where fK(2) and e(2) Case (2). Note that E(S K δ e 3

ð2Þ K ð2Þ K

(1) are derivatives taken with respect to K. If α 2 Vβδ 1−p ≤ δf (1). p
, then αe Since the fraud to effort ratio is increasing in K, we have that ef z ef z αδ and that ef z ef . Putting these results together, we obtain ef z ef z α δ establishing the result. Case (3): the proof is the same as in Case (2). Consider now the case where α 2 Nβδ 1−p p
. The expected firm's value at time 3 (Eq. (16)) is now increasing in λ. Thus, since λ as in Eq. (18) is strictly increasing in K, there does not exist a solution for S − K + αe(1) N h. Thus, K = 0 is never optimal. Note that for S − K + αe(1) = S − K + αe(2) = h, ð2Þ K ð2Þ K

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð1Þ

ð2Þ ð2Þ K ð2Þ K

ð1Þ ð2Þ

ð2Þ

ð2Þ

and since Φ(1) is independent of K, f Kð2Þ ¼ ef ð2Þ b αδ , from which eK follows that E(S˜3) is increasing in K. Two cases may arise: (i) Φð3;bÞ z α f ð2Þ eð2Þ

ð1Þ

¼ ef ð1Þ b αδ

δ

and (ii) Φð3;bÞ b αδ . In case (i), since Φ(3,b) is independent of K,~it follows dE S that, for S − K + h + αe(3,a) = 0, ef N ef ¼ ef ¼ ef z αδ and thus dK b0 from f f α which follows that a maximum exists with either ¼ or ¼ αδ. In   δ e e dE ~S f f α case (ii) e ¼ e b δ and hence dK N0, from which follows that K→S+αe(3,b) + □ βf (3,b) +h. Increased incentives lead to more fraudulent behavior and more effort. If the long-term negative effects of fraud are sufficiently large, then the long-term stock price decreases as the incentives increase (that is, as K increases); therefore, stock-based compensation is optimal. On the other hand, if the long-term effects of fraud are sufficiently low, then it is optimal to increase incentives (increasing the strike price K), and, as a result, stock option-based compensation is optimal. The threshold level δ ̄ is increasing in the detection ð3;aÞ K ð3;aÞ K

ð3;bÞ K ð3;bÞ K

ð3;bÞ

ð3;bÞ

3

ð3;bÞ K ð3;bÞ K

ð3;bÞ

ð3;aÞ

ð3;bÞ

ð3;aÞ

3

ð2Þ K ð2Þ K

ð3;aÞ K ð3;aÞ K

R. Andergassen / Economics Letters 101 (2008) 122–125

probability (p), the punishment in the case of detection (ϕ) and in the effectiveness of effort (α), since these encourage the manager's good behavior, while it is decreasing in the effectiveness of fraud (β). The following comparative statics result for the optimal strike price can be proven. Proposition 2. Consider the case where δ b δ ̄. The optimal strike price (K) is decreasing in δ. Proof. Consider first the case ef z αδ. We show that, for the optimal K, fK fK α eK is increasing in K and thus, as δ increases, FOC eK ¼ δ requires K to fK f f decrease. eK is increasing in~K if e N e . The second-order condition for a maximum requires that d dEKS b 0 or equivalently efKK N αδ, consequently, in KK ð3;bÞ fK α the maximum efKK N ¼ , establishing the result. In the case ef ð3;bÞ b αδ, eK δ KK the optimal strike price is independent of δ, but the condition itself depends on it; thus, as δ increases it is more likely that it is being □ violated which leads to a reduction in the optimal K. In solving the optimal compensation problem, the shareholder trades off gains from effort against costs arising from fraudulent behavior. The larger the long-term loss in profitability due to fraudulent behavior (δ) is, the larger the costs associated with incentives and thus the lower the optimal strike price K (i.e. the lower the incentives). Conflicts between shareholders and the manager are eliminated in this framework if both face the same time horizon. If stock options vest at time 3, fraudulent behavior decreases the manager's wealth as well as the shareholder's. In a similar vein, a longer vesting period, i.e. a higher detection probability p, increases the range of parameter values where stock option-based compensation is optimal. This result is in accordance with recent suggestions to increase stock option vesting periods as a way to reduce conflicts between shareholders and CEOs.

125

holder, in providing incentives, trades off gains from beneficial effort against costs arising from fraudulent behavior. We find a threshold level for the cost of fraud, above which stock-based compensation is optimal. Furthermore, we find that the optimal strike price is decreasing in the size of the long-term negative effects of fraud on the company's value. Our analysis also suggests that increasing the stock option vesting period mitigates conflicts between shareholders and managers.

ð3;bÞ

ð3;bÞ

2

KK

K

KK

K

3

2

3. Conclusion We studied the optimal managerial compensation package if shareholders can either grant stock or stock options. The share-

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