Using Executive Stock Options to Pay Top Management

Transcription

1 Using Executive Stock Options to Pay Top Management Douglas W. Blackburn Fordham University Andrey D. Ukhov Indiana University 17 October 2007 Abstract Research on executive compensation has been unable to explain the vast use of executive stock options. We address this problem by proposing a simple principal-agent model where the principal must choose a compensation contract, consisting of stock and options, that will induce the agent to exert e ort on the principal s project. The agent will accept the contract only if his expected utility under the contract is greater than his expected participation constraint. Including uncertainty in the participation constraint allows the agent s risk aversion to play a greater role. If the agent declines the principal s contract, there is no gaurantee that he will nd an equal or better contract elsewhere. Hence, the agent chooses between the principal s contract or an uncertain employment market. We nd that the agent will accept stock options in exchange for not having to face the risk of the uncertain market. As the agent s risk aversion increases, total compensation decreases, the proportion of compensation paid in stock decreases, and the proportion of compensation paid in options increases. Further, we show that rm characteristics, agent quality and the characteristics of the outside market all a ect the total compensation and the optimal mix of stock and options. Skilled agent s are paid more and receive more options than less skilled agents. Agent s managing high quality projects (projects that have a high probability of success) are paid less overall but receive a larger proportion of options than agent s managing low quality projects (projects with low probability of success). Finally, the riskiness of the rm s project a ects the composition but not the expected level of total compensation. The agent facing a low volatility project receives more options than the agent facing a highly volatile project. JEL Classi cation: G3, M52, J33 Keywords: Executive compensation; Manager shareholder incentive con icts; Risk preferences We would like to thank Michael Jensen for helpful discussions and John Wald for many thoughtful comments on an earlier draft. Address correspondence to Andrey D. Ukhov at Kelley School of Business, Indiana University, 1309 East Tenth Street, Bloomington, IN 47401, or aukhov@indiana.edu.

2 Using Executive Stock Options to Pay Top Management. Abstract Research on executive compensation has been unable to explain the vast use of executive stock options. We address this problem by proposing a simple principal-agent model where the principal must choose a compensation contract, consisting of stock and options, that will induce the agent to exert e ort on the principal s project. The agent will accept the contract only if his expected utility under the contract is greater than his expected participation constraint. Including uncertainty in the participation constraint allows the agent s risk aversion to play a greater role. If the agent declines the principal s contract, there is no gaurantee that he will nd an equal or better contract elsewhere. Hence, the agent chooses between the principal s contract or an uncertain employment market. We nd that the agent will accept stock options in exchange for not having to face the risk of the uncertain market. As the agent s risk aversion increases, total compensation decreases, the proportion of compensation paid in stock decreases, and the proportion of compensation paid in options increases. Further, we show that rm characteristics, agent quality and the characteristics of the outside market all a ect the total compensation and the optimal mix of stock and options. Skilled agent s are paid more and receive more options than less skilled agents. Agent s managing high quality projects (projects that have a high probability of success) are paid less overall but receive a larger proportion of options than agent s managing low quality projects (projects with low probability of success). Finally, the riskiness of the rm s project a ects the composition but not the expected level of total compensation. The agent facing a low volatility project receives more options than the agent facing a highly volatile project. JEL Classi cation: G3, M52, J33 Keywords: Executive compensation; Manager shareholder incentive con icts; Risk preferences

3 There is only one way to get anybody to do anything. And that is by making the other person want to do it. Dale Carnegie The separation of rm ownership and control creates the opportunity for moral hazard problems. How do the owners persuade those in control to make decisions that will maximize shareholder wealth? One suggested solution is to link manager s pay to the performance of the rm by using equity-based compensation (Jensen and Meckling (1976)). In reality, however, this has been frequently translated into giving more instead of carefully choosing a package that will align the manager s interests with those of the shareholders. Hall and Liebman (1998) nd that the mean direct CEO compensation has increased 209% (in real terms) over the period of 1980 to They further document that the percentage of CEOs receiving option grants has increased from 30% in 1980 to 70% in Jensen and Murphy (1990) comment that, On average, corporate America pays its most important leaders like bureaucrats. Is it any wonder then that so many CEOs act like bureaucrats rather than the value-maximizing entrepreneurs companies need to enhance their standing in world markets? They claim that it is not how much the CEO is paid but how he is paid. The theoretical literature has had di culty explaining the use of stock options. This was the punch-line of Yermack (1995). Yermack nds little evidence that stock options decrease agency 1

4 problems. He also concludes that theories relating the use of stock options to liquidity, taxes and earnings management are not able to explain the use of stock options in the data. Following Holmstr½om (1979), Dittman and Maug (2007) develop a principal-agent model where the principal o ers a contract to an e ort and risk averse agent. Unlike Holmstr½om, Dittman and Maug allow the agent to be paid with base salary, stock and options. They nd that incentive compensation is provided through stock and that base salaries are low. However, their model is unable to explain the use of stock options. They conclude that conventional principal-agent models are unable to explain observed executive compensation in general and the use of executive stock options in particular. We explain the use of stock options by proposing a simple principal-agent model where the principal must choose a compensation contract that will entice the agent to exert all e ort toward the principal s project. At the same time, the agent faces an uncertain outside employment market (alternative project). The alternative project acts as a participation constraint that incorporates both the risk and the expected return of an uncertain market for employment. Agents in traditional principal-agent models are willing to participate as long as their utility is above some minimum level of utility (a constant). By allowing uncertainty to enter into the participation constraint we allow risk aversion to play a greater role. When the risk averse agent believes, either rationally or irrationally, that he faces the possibility of failing in the outside employment market, he will be willing to take the risky options in order to avoid facing the uncertainty of the market. We nd that, depending on agent characteristics, rm characteristics and outside employment opportunities, the optimal contract can consist of all stock, all options and a combination of stock and options. Additionally, we nd that the optimal compensation contract is a ected by the riskiness of obtaining alternate employment, the talent of the agent, the quality of the rm s project, and the volatility of the rm s project. Skilled agents are paid more and receive more 2

5 options than less skilled agents. Agents managing high quality projects (projects that have a high probability of success) are paid less overall but receive a large proportion of options than agent s managing low quality projects (projects with low probability of success). Finally, the riskiness of the rm s project (volatility of rm value) a ects the composition but not the expected level of total compensation. The agent facing a low volatility project receives more options than the agent facing a highly volatile project. This paper contributes to the literature in three ways. First, we show that modeling risky outside employment opportunities is important to linking theory to observed CEO compensation. Recent research has suggested the importance of the market for CEOs. Oyer (2004) considers a model where the market updates its information concerning the ability of a CEO after observing the performance of his rm - which depends on both the ability of the CEO and the general state of the economy. Hence, the CEO s participation constraint is time varying. Firms must either increase the xed pay or link compensation to rm value in order to keep their good CEO. This theory was investigated empirically by Rajgopal, Shevlin and Zamora (2006) who nd that the demand for skilled CEOs increases with industry and market-wide shocks. Compensation for highly skilled CEOs increases more than the compensation for less skilled CEOs. Himmelberg and Hubbard (2000) nd that CEOs of small, easy to manage rms are paid as agency theory suggest since the supply of CEOs for such rms is large. However, the compensation of CEOs managing large, complex rms is more sensitive to aggregate shocks suggesting that the supply of CEOs with the skill to manage these complex rms is inelastic. Whereas Oyer models time varying employment opportunities, we model a risky expected outside opportunity. The mean of the outside employment opportunity describes the pay the agent believes he can obtain elsewhere, while the variation allows for the possibility of the agent getting more or less than he expects and even the possibility of not nding any immediate outside employment. Modeling the participation 3

6 constraint in such a way allows risk aversion to play a more important role. 1 Fear of an unknown employment market provides the avenue for a variety of optimal compensation contracts. Second, we show that both stock and options provide important sources of incentives. Though the empirical literature describing the use and incentive a ects of various forms of compensation is vast 2, the theoretical literature describing the optimal composition of compensation is small. As mentioned, Dittman and Maug (2007) model does not support the use of options. We, on the other hand, nd many instances where options play an important role in the agent s compensation - in some cases the optimal contract consists only of options and no stock. Hence, our model increases our understanding of the role that stock options play in the optimal compensation contract. Third, we show how the optimal compensation structure depends on agent characteristics, rm characteristics and employment opportunities. Depending on these characteristics, we nd that the optimal composition of compensation may consist of only stock, only options and a combination of stock and options. This nding is consistent with Zhou (2001). From a panel of 619 rms from the ExecuComp database, 30% of CEOs hold an insigni cant quantity of options, 40% have similar quantities of both stock and options and the remaining 30% hold mostly options 3. able to produce the wide variation in compensation contracts observed in the data. Our model is We further describe how the optimal contract changes with rm characteristics. Palia (2001), controlling for 1 The Holmstrom (1979) model uses a constant participation constraint. It is true that any risky payo can be modeled using its certainty equivalent, however, the certainty equivalent depends on the agent s risk aversion. Changing risk aversion without changing the participation constraint implies that the participation constraint can be obtained with certainty since it does not depend on risk aversion. 2 See, for example, Core and Guay (1999), Demsetz and Lehn (1985), Hall and Liebman (1998), Hall and Murphy (2003), Lippert and Moore (1994), McConnell and Servaes (1990), Morck, Shleifer and Vishny (1988), and Yermack (1995). 3 A related literature investigates when executives nd their options to be valuable. For an executive stock option valuation model see Cai and Vijh (2004) and references therein. 4

7 various characteristics that describe the CEO s quality and the riskiness of the rm, nds that rms are in equilibrium when determining compensation. Core and Guay (1999) and Demsetz and Lehn (1985) nd that such features as rm size, growth opportunities, volatility, monitoring di culty and the managers risk aversion are related to equity ownership. All these results together illustrate the Jensen and Murphy (1990) claim that the composition of the agent s contract does matter. This paper continues as follows. We present our model and describe the equilibrium in Section 2. In Section 3, we describe the optimal compensation contracts and how they are related to rm characteristics, manager characteristics, and the opportunities of outside employment. Concluding remarks are given in Section 4. I The Model We consider a one period principal-agent model where all decisions are made at time 0 and all uncertainty is resolved at time 1. Figure 1 illustrates the model in a tree diagram. The risk averse agent is initially endowed with a xed amount of e ort,, that the agent must allocate between two projects. For simplicity, we normalize to unity. The risk neutral principal hires the agent to manage a risky project that will yield a rm value of either S H ( high ) or S L ( low ), at time 1, where 2 [0; 1] is the amount of e ort allocated to the rm s project and S H > S L > 0 are the rm values when the agent exerts all e ort to the rm s project ( = 1). Notice that the rm value is a function of the agent s e ort. As the agent exerts more time and energy to advancing the project, the value of the rm in both the high an low states increase. A Compensation and E ort As compensation, the risk neutral principal o ers a package consisting of some combination of stock and call options. The principal writes a compensation contract (the total value of the expected 5

8 compensation and the mix of stock and options) in order to encourage the agent to exert e ort so as to maximize rm value. The number of shares is denoted in terms of the fraction of the total rm value. Since S H > S L > 0, the stock will provide compensation in all states as long as the agent exerts some e ort ( > 0) to the rm s project. The option, with strike price X such that S L < X < S H, has a payo function of Max[S i X; 0] for i = H; L. The options are in the money only in the high state and only when the agent exerts enough e ort so that S H > X: In addition to the rm s project, the agent can receive payo from an uncertain alternative outside project that is unrelated to the principal s project. This outside (alternative) project pays either (1 )A H or (1 )A L where A H A L 0. The payo of the outside project also depends on the allocation of e ort. As moves closer to zero, the agent exerts more e ort to the alternative causing the payo in both the high and low state to increase. Hence, if the agent allocates all e ort to the outside project ( = 0), the agent will not receive any wealth from the rm s project. Also, when A H = A L, the outside project pays with certainty in proportion to the e ort allocated to the alternative project. This case is similar to Holmstr½om s e ort averse agent. Since the agent is e ort averse, he gains certain utility by not working. B Probabilities and E ort The probabilities of realizing the high and low rm values as well as the high and low outside project payo s depend on both nature ( for the rm s project and for the outside project) and e ort. Hence, e ort a ects the payo of both projects and the probabilities of realizing the high value of both project. The more e ort the agent allocates to a particular project increases the likelihood of the high payo. This causes the agent to have direct control over the means and variances of each project. Nature ( and ) describes the exogenous factors that a ect the success of the projects. How- 6

9 ever, nature can also be interpreted as agent talent or project quality. The agent may have some speci c skill that increases the likelihood of success beyond that of simply working hard. In addition, some projects inherently have higher probability of success than others. Our model captures these di erences. C Timing of Decisions At time 0, the agent selects the level of e ort, 2 [0; 1], to be applied to the rm s project in order to maximize his own expected utility of wealth. The choice of e ort,, a ects the realized rm value by increasing the payo and by a ecting the probabilities of each state. If all e ort is applied within the rm, = 1, then the probabilities of the high and low rm value are governed solely by nature. With probability, the rm s project will yield the high share price, S H. In this case the payo on the outside project is zero with certainty and either state 2 or 4 will occur (see Figure 1). If all e ort is applied outside the rm, = 0, then the payo on the stock is zero since the payo of the stock depends on e ort, and state 3 occurs with probability and state 4 with probability (1 ). Also at time 0, the principal decides on the value and composition of the compensation contract. The principal chooses the value of the compensation as well as the composition in order to encourage the agent to exert the e ort required to maximize the expected value of the rm - the expected payo less the expected compensation owed to the agent. All uncertainty is resolved at time 1 when payo s on the two projects are realized. To illustrate how the payo s S H, S L, A H ; A L ; and the parameters and describe the economy, consider the following special cases. The parameter 2 [0; 1] a ects the probability (nature) of receiving the high outside payo for a given level of e ort. If = 0, then the probability of receiving A H is zero, irrespective of the level of e ort,. Nature is against him. If = 1, then the probability 7

10 of obtaining the high payo depends only on e ort: (1 ) + (1 ) (1 ) = (1 ) : In this case, the manager can receive A H with certainty by diverting all e ort away from the rm, = 0. Decreasing the e ort exerted on the outside project both decreases the probability of the high payo and decreases the outside project s payo in each state. Similarly, for a given level of e ort, the parameter 2 [0; 1] a ects the probability (nature) of achieving the high or low rm value. If = 0, then the probability of the high value is zero, irrespective of the level of e ort. The low rm value is realized with certainty. If = 1, the probability of obtaining S H is equal to the amount of e ort allocated to the rm s project,. In this case, if the agent places all e ort with the rm, then the rm s project will yield the high rm value, S H, with certainty. The rm value at time 0, S 0, is the expected payo on the project and depends on the probabilities of the four states and on e ort. Thus, the expected rm payo is S 0 (; ; ) = () (S H ) + (1 ) (S L ) = 2 (S H S L ) + S L : Similarly, the expected option price at time 0, O 0, is given by O 0 (; ; ) = () Max (S H X; 0) : Again, the options only payo in the high rm payo state and only when the agent exerts a su cient amount of e ort so that S H > X. D Equilibrium Several quantities are jointly determined in equilibrium. Knowing the agent s utility function, risk preferences and his outside employment opportunities, the principal chooses the compensation 8

11 package that will maximize rm value. The agent is awarded N S 0 shares of stock and N O 0 option contracts. The rm has only one share of stock yielding the constraint N O + N S 1. At the same time, knowing the details of the compensation contract, the agent decides on the level of e ort. The two components of the equilibrium problem the agent s problem, and the principal s problem are considered next. Both problems are solved simultaneously. Agent s problem. Given the compensation package fn S ; N O g, the manager selects e ort to maximize expected utility of wealth h i max EU fw : 01 For a power utility of wealth function with risk aversion parameter a, U (W ) = W 1 the probabilities of the four states, the utility maximization problem becomes a 1 a, and given h i EU fw = h i max EU fw ; 01 (1 ) [N 1 a SS H + N0 Max (S H X) ; 0 + (1 )A H ] 1 a [1 (1 )] + [N 1 a SS H + N0 Max (S H X; 0) + (1 ) A L ] 1 a (1) [1 ] (1 ) + [N 1 a SS L + (1 ) A H ] 1 a [1 ] [1 (1 )] + [N 1 a SS L + (1 ) A L ] 1 a Principal s Problem. The principal chooses the total value of the compensation package, the number of shares, and the number of options in order to encourage the agent to exert the e ort necessary to maximize the rm s expected value. Hence, the principal chooses N S and N O that will maximize the rms objective function de ned as the expected rm payo less the expected agent 9

12 compensation: Max N S ; N o h ( ) 2 (S H N S h ( ) 2 (S H S L ) + S L i S L ) + S L i N O [( ) Max ( S H X; 0)] : Shareholders have the objective of maximizing the equilibrium rm value. De nition 1 (Equilibrium) The equilibrium consists of the level of e ort, 2 [0; 1], the number of shares N S and the number of option contracts N O such that 1. Given the number of shares NS and the number of option contracts N O, the agent decides on the level of e ort to maximize his expected utility of wealth, max EU 0 " # fw 1 a : 1 a 2. Given the agent s level of e ort,, the principal chooses the composition of the compensation package, fns ; N O g, to maximize the rm value h max ( ) 2 (S H N S ;N o N S h ( ) 2 (S H S L ) + S L i S L ) + S L i N O [( ) Max ( S H X; 0)] : II The Principal s Problem The principal desires to maximize the value of the rm. To do this, he must entice the agent to exert the desired amount of e ort by choosing the optimal compensation package. decreases rm value and therefore, the rm will want to keep compensation low. Compensation The principal can choose to pay the agent a fraction of the rm by giving stock and he can pay the agent options that payo only if the agent exerts enough e ort and the high state is realized. In this section, we will show that the principal desires the agent to exert all e ort on the rm s project. 10

13 The principal seeks to maximize the objective function F V = h () 2 (S H S L ) + S L i N S h () 2 (S H S L ) + S L i N O [() Max (S H X; 0)] = [(1 N S ) ( (S H S L ) + S L ) N O Max (S H X; 0)] : As can be seen, the objective function is non-linear in e ort,, and the function is not di erentiable when the options are at the money, = X=S H. We rst consider the cases when options are not part of the compensation (N O = 0). In these cases, N O Max(S H X; 0) = 0. The rst derivative of FV with respect to is then df V d = (1 N S) [2 (S H S L ) + S L ] : The derivative is positive and increasing in e ort when N S < 1: (note that when N S = 1 the rm pays to the agent the entire value of the rm in stock). When options are either not part of the compensation or e ort is su ciently low so that options are out of the money, then more e ort increases rm value. Let be the level of e ort that maximizes rm value. Proposition 1 When options are not o ered and stock is the only form of compensation then 1) = 1 when N S < 1, 2) = 0 when N S 1: Proof. The rst derivative of rm value with respect to e ort is df V d = (1 N S) [2 (S H S L ) + S L ] : Since S H > S L, the df V d > 0 only when N S < 1, and df V d 0 only when N S 1. Since the derivative is linear in and 2 [0; 1], then the extrema occur on the boundary. Thus, when 11

14 N S < 1, rm value is increasing in and the maximum rm value occurs when = 1. When N S 1, the rm value is not increasing and the maximum rm value occurs when = 0: This proposition shows that whenever stock is the only form of compensation, the principal either wants the agent to exert all e ort on the rm or the principal does not want the agent to work at all. The principal will want the agent to work elsewhere only when the required compensation is greater than the total value of the rm. A similar result is obtained when options are added to the compensation. Consider the case when options are o ered to the agent and e ort is large enough to make the options in the money, X=S H < 1 and N O Max(S H X; 0) = N O (S H X). The rst derivative of rm value is now df V d = (1 N S) [2 (S H S L ) + S L ] N O [2S H X] : By observation, the rst term is increasing in as long as N S < 1. The following proposition will show that rm value is maximized when the agent exerts all e ort toward the rm. The key to the proof is that since the rm has one share of stock that is in nitely divisible, the number of shares and options is restricted by N S + N O 1. Proposition 2 When N O > 0, then the rm maximizing e ort is = 1. Proof. The rst derivative of rm value with respect to e ort is df V d = (1 N S) [2 (S H S L ) + S L ] N O [2S H X] : It su ces to show that for all possible contracts fn S ; N O g that df V d > 0. Since the rm has one in nitely divisible share of stock, then N S + N O 1. We start with the realization that rm value decreases as compensation increases. Thus, setting N S + N O = 1 gives 12

15 the rst inequality: df V d = (1 N S ) [2 (S H S L ) + S L ] N O [2S H X] > N O [2 (S H S L ) + S L ] N O [2S H X] = N O [S L (1 2) + X] > N O [S L (1 2) + X] > N O [S L (1 2) + (S L + )] : The last two inequalities come from evaluating the inequality at = 1 since the inequality is decreasing in e ort and the fact that S L < X. In the last line, > 0 is some value such that S L + = X. To nish the proof, df V d > N O [S L (1 2) + (S L + )] = N O [ + S L (1 )] > 0: Thus, rm value is increasing in e ort for all admissable contracts and all permissible parameter values. The rst proposition shows that when options are not available then the principal will seek all the agent s e ort as long as the principle can a ord it. The rm cannot give more than one share of stock. The second proposition shows that when options are used, then the rm will still seek all of the agent s e ort. Hence, for all feasible contracts fn S ; N O g such that N S + N O 1, rm value is largest when the agent s e ort is = 1. This is an important result because it is the solution to the rst part of the optimization problem. The rm wants all e ort and must choose the number of stock and options that will persuade the agent to give all e ort. III Optimal Compensation Contracts Now that we know the optimal level of e ort desired by the principal, we can focus on solving the compensation problem. Equation (1) gives the expected utility of the agent as a function of e ort 13

16 and compensation. As can be seen, the function is highly non-linear. Due to the complexity of the model, analytic solutions are di cult to nd and, therefore, we use numerical techniques to characterize the optimal compensation structure. The technique used essentially searches for the set of all compensation contracts fn S ; N O g that encourages the agent to exert all e ort for the rm. The rm-maximizing contract is then found from the set of feasible solutions. In this section, we investigate the optimal compensation structure implied by the model. We rst show that agent characteristics risk aversion, and the value of outside alternatives are an important determinant of the compensation structure. We then investigate the impact of rm characteristics on the optimal compensation structure. For the sake of our discussion, we consider the base-case set of parameters describing the rm s project: S H = $1; 598:68M, S L = $1; 007:30M, X = $1; 445:36M, and = 0:74. These values come from market data. The average annual CRSP value weighted return from 1926 to the present is 12.04% with a variance of 4.04%. Over this time period, the market experienced an annual positive return 74% of the time. The average rm value over this same time period is $1; 290:07M. S H and S L were chosen to match the 12.04% annual return and the 4.04% variance where the probability of the high state is The exercise price is the expected value of the rm. A The Alternative We begin by investigating the importance of the agent s alternative source of utility. We begin by considering = 0:5 with various choices of A H and A L. The choice of = 0:5 indicates that nature has equal a ects on success and failure. Even if the agent exerts all e ort on the outside opportunity, he still only has a 0.5 probability of realizing the high payo. We choose A H and A L so as to preserve the expected payo of the alternative at 5M (assuming the agent exerts all e ort toward the outside project) while only changing the variance. We study values (A H, A L )={(10,0) 14

17 , (7,3), (5,5)}. Two cases are worth noting. When (A H, A L )=(10,0), the alternative pays like an option. There is a positive probability that the agent will not receive compensation from the outside opportunity. The agent will only receive compensation if the high state occurs. This models the case where the agent expects he can nd outside compensation of $5M, but there is a positive probability that he will not be successful in the outside job or that he will not be able to quickly nd outside employment. When (A H, A L )=(5,5), exerting all e ort to the outside project guarantees a payo of $5M regardless of nature. This case is similar to the commonly used participation constraint. The principal must o er enough compensation to the agent so that his utility is greater than some minimum value that is supposed to describe opportunity costs. Though any risky gamble can be described by its certainty equivalent, the certainty equivalent is a function of risk aversion. Past studies have often investigated the role of risk aversion without allowing the participation constraint (certainty equivalent) to change. By doing so, they either assume that the agent can obtain a certain level of utility with certainty or they allow the risk and return characteristics that describe the participation constraint to change so as to maintain a constant participation constraint for any level of risk aversion. Table I and Figures 2-A, B and C show the optimal number of shares and options as well as the total compensation for the three cases described as a function of risk aversion (ranging from 0.05 to 0.95). We rst notice the role of risk aversion and variance of the outside project in determining the optimal total compensation. As risk aversion increases, the total compensation required for the agent to exert all e ort to the rm decreases for all cases except (A H, A L )=(5, 5). Further, compensation is lowest when the outside project has the highest variance. The risk averse agent wants to escape the uncertainty of the market for managers. Since the probability of the high rm payo is 0.74, the agent will give up wealth in order to obtain a more certain high payo. 15

18 When (A H, A L )=(5,5), the agent is guaranteed a job paying $5M. Therefore, the principal must pay a premium to tempt the agent toward the rm s risky project. The premium increases with risk aversion. This is a common result for principal-agent problems but it is misleading. Any opportunity (participation constraint) that competes with the compensation earned by managers will have some uncertainty. Therefore, as in the cases (A H, A L )=(10,0) and (7,3), the risk averse manager will give up compensation in order to avoid the risky outside opportunity. Our result highlights the importance of uncertainty in the participation constraint. What is even more interesting is the composition of the optimal compensation. In all cases except (A H, A L )=(10,0), the rm uses only stock and no options. To see why this is the case, we must understand how the agent s utility function works. Figure 3 shows two curves describing the expected utility of an agent who is paid with only options and only stock. The utility has two local maximums. One at zero e ort ( = 0) and one at full e ort ( = 1). This implies that the agent will choose to specialize in one project and will not diversify. The goal of the rm is to give just enough compensation to cause the utility at = 1 to be higher than the utility at = 0. Hence, the utility at = 0 acts as a participation constraint. For any level of e ort greater than zero, the agent will bene t from owning stock since the stock value is a function of the amount of e ort exerted. Therefore, stock will a ect the shape of the entire utility curve. Options, however, only pay o if the agent exerts enough e ort so that the stock price exceeds the strike price. The option-only utility curve increases only when the options are in the money, but a large number of options are required to force the utility to exceed the participation constraint. It is much more e cient to use stock. This argument is di erent when (A H, A L )=(10,0). We nd that the agent is willing to be compensated with options when he has the risk of not getting any compensation from choosing the alternative project. The number of options in this case is signi cant - with value of options 16

19 exceeding $1M for risk aversion near 0.7. The option-like payo of the outside project is the key to this result. Figure 4 shows the expected utility as a function of e ort for this case. Unlike the previously seen utility curves, the two local maximums do not occur on the boundary. Instead, there is a local maximum near e ort of The participation constraint is in the interior. The agent facing the possibility of receiving nothing from the outside project will want to hedge the risk by working both projects. As long as the agent exerts a little e ort toward the rm, any stock compensation will pay o - even in the bad state. Hence, stock compensation a ects the participation constraint, but options do not a ect the reservation constraint since options only a ect the utility curve when they are in the money - when e ort is su ciently large. Therefore, it is most e cient for the principal to use a combination of stock and options. Using stock will keep the principal s cost low and options will discourage the agent from choosing the low level of e ort 4. Our nal observation is that the optimal compensation can be all stock, a combination of stock and options, or all options depending on the parameter values. Consider the (A H, A L )=(10,0) case in Table I. When risk aversion is between 0.0 and 0.35, only stock is used. For risk aversion between 0.35 and 0.8, it is most e cient to use a combination of both stock and options, and when risk aversion is greater than 0.8, only options are used. This results di ers from Dittman and Maug (2007) who nd that the principal-agent models are not able to produce compensation plans the include options. A.1 The Risky Market for Managers We have found that the use of options is an optimal form of compensation when (A H, A L )=(10,0). Interpreting the alternative project as the market for outside employment, the CEO is willing to 4 This results is also obtained for su ciently low values of A L. Hence, A L = 0 is not a necessary assumption. 17

20 be compensated with options when he faces the risk of not nding alternate employment (as in the above case where (A H, A L )=(10,0)), but it is not optimal to use options when the CEO faces no risk of nding alternate employment (as in the case where (A H, A L )=(7,3) and (5,5)). However, in the above case we allow the probability,, of not nding outside employment to be 0.5. This is likely to be an unreasonably high value. In this section we show that options are an important part of the compensation contract even when the likelihood of not nding outside employment is small. Figures 5-A, B and C show the total compensation, the proportion of compensation in options and the proportion of compensation in stock as a function of risk aversion. We consider four cases, = 0:5; 0:8; 0:9; 1:0. The parameter indicates the probability with which the agent believes he will obtain outside employment. We adjust the parameters A H and A L so that the expected compensation from the outside market is set at $5M for all cases of. That is, the agent always believes he can obtain $5M, but we allow the probability of not nding a job to vary. When = 1, the agent is certain to gain compensation from the outside market. As found previously, total compensation (Figure 5-A) increases with risk aversion. He must be compensated for taking the principal s risky project since he is certain to receive $5M (the high payo ) from the outside market. For all other values of < 1, the agent s total compensation is a decreasing function of risk aversion. As the risk of not nding a job increases, the agent is willing to accept less and less compensation from the principal. The structure of the total compensation also changes with. When = 1, the agent is paid only with stock. Options are an ine cient form of incentive when the agent is facing the certain market for agents. It will take a large quantity of options to overshadow the added risk that options bring to the compensation contract. It is a di erent story for the cases when < 1. Options become an important part of the compensation contract in these cases. 18

21 Figure 5-B shows the proportion of compensation from options. When is small ( nding an outside job is highly uncertain), options are used for agents with risk aversion between 0.35 and 0.95 and the total option value reaches a peak of $1.10M. As increases, the stock options are used for agents with higher levels of risk aversion and the maximum dollar amount of options also increases. Interestingly, the range of risk aversion narrows as increases, and when reaches one, the range of risk aversion vanishes. Hence, when the agent faces an outside market that has an option-like payo structure, options become an important component of the optimal compensation package. A.2 The Talented Agent One of the arguments against the use of stock-based compensation is that managers are rewarded for luck rather than skill. CEOs are highly compensated for positive market movements. One possible explanation for the use of stock-based compensation in light of this problem is the market for talented managers. Himmelberg and Hummel (2000) nd that the demand for talented CEOs increases during boom markets. As a result, talented CEOs must earn more during these times to be convinced to stay at their current job. One way of doing this is by linking pay to the market, through stock or options, so that compensation will naturally increase with booming markets. Oyer (2004) considers a model where the participation constraint varies with the market. Since adjusting the contract is costly, the principal has incentive to link compensation to the market so that the manager is paid more during those times he is in greatest demand. We investigate the optimal mix of stock and options for managers facing a positive employment market. In the previous section we model the risk of the outside opportunity by the range of the high and low outcome and the probability of realizing the high outcome, - probability due to nature. We now reinterpret as the probability due to nature plus an adjustment that accounts for the 19

22 CEOs talents that are highly desired by the outside market. A high indicates that either the agent has desired characteristics that will make alternate employment more likely or that the agent is facing a good market for CEOs. Combining skill with nature is consistent with the above mentioned literature since it is unknown whether the success of a project is due to luck (nature) or skill. Unlike the previous case, we allow the mean of the distribution to change. Therefore, as a result of the talented CEO having a high, the skilled manager will also face a higher expected payo from the outside opportunities since the combination of skill and luck will make him more desirable than an agent with a lower. As in the previous section, we choose values of the outside project (A H, A L )={(10,0), (7,3), (5,5)} but this time we assume that the combination of the CEOs skill and luck causes to be 0.7 (compared to 0.5 used in the previously - see Figure 2). Figures 6-A, B and C and Table II show these results. As before, the higher the risk aversion, the lower the compensation in all cases but (A H, A L )=(5,5), and as expected, the total compensation levels are much higher for the skilled CEO than the unskilled CEO. A skilled agent with risk aversion of 0.5 and facing an outside alternative of (A H, A L )=(10,0) receives an expected $5.06M (Table II) compared to the unskilled agent who receives an expected $2.70M (Table I). The increase in total compensation decreases with the variability of the payo of the outside opportunity. Both the skilled and unskilled CEO, for all levels of risk aversion and facing a certain outside payo (A H, A L )=(5,5), receive nearly the same compensation. This again shows the importance of modeling the risk and return characteristics of agent s opportunity costs (participation constraint). The a ects of risk aversion on the total compensation is dramatic. Total compensation decreases as a function of risk aversion and the downward slope is greatest when the volatility of the alternative is highest. The (A H, A L )=(10,0) curve shows a compensation of $6.88M for risk aversion of 0.05 and a compensation of $1.54M for risk aversion of The low volatility outside alternative (A H, A L )=(7,3), on the other hand, shows a compensation of $5.78M and $5.55M for risk aversions 20

23 0.05 and 0.95, respectively. The highly risk averse agent facing a highly unknown outside market for CEOs is willing to accept low compensation in order to avoid this alternative. This result is true in all cases except when the alternative o ers a certain payo - when (A H, A L )=(5,5). In this case, the rm must pay a risk premium. The premium increases with risk aversion. When risk aversion is 0.05, the optimal total compensation is just above $5.00M (nearly risk neutral), but when risk aversion is 0.95, the rm must pay $5.09M. The composition of stock and options result is similar to the previous case. Options become an e ective part of the compensation only when the volatility of the alternative payout is very high - when (A H, A L )=(10,0). This agent faces the choice between two possible allocations of e ort. He will exert all e ort to the rm if the compensation is high enough or he will choose to divide his e ort between the principal s project and the outside opportunity (Figure 4). Facing the possibility of getting nothing, the risk averse agent will not choose to put all of his e ort in the alternative project but he will instead want to allocate a small fraction of his e ort to the rm s project. The stock from the rm s compensation protects the agent from the bad state when the alternative pays nothing. The rm, then, will choose to allocate fewer number of shares to prevent the agent from using their stock as a hedge and will instead use options. Options only a ect the agent s utility when e ort is su ciently large. Hence, options prevent the agent from allocating e ort between the two projects, and instead cause the agent to exert e ort to the rm. As seen in Figure 6-B, options become a signi cant part of the agent s total compensation when the his risk aversion is above When risk aversion is lower than 0.55, the compensation consists solely of stock. For risk aversion between 0.55 and 0.9, the compensation consist of both stock and options with options make up an increasing fraction of total compensation as risk aversion increases. When risk aversion is 0.95, the agent is paid completely with options. Again, the model is able to produce a wide variety of compensation contracts. 21

24 Large compensation packages are obtained when a talented CEO faces an uncertain outside employment market. Further, we show that the total compensation can consist of high dollar values of options. This signi cantly adds to the ndings of Dittman and Maug (2007). The main requirement for this result is a positive probability of not receiving any payout from the alternative opportunity. This is a reasonable assumption. A zero payout can result from not succeeding at the alternative project, from an unexpected downturn in the market for CEOs, or from simply being unlucky in the employment market. Hence, it is the possibility of receiving a no compensation from the alternative (even if it is remote) that makes stock compensation valuable to the agent. As long as the agent exerts at least some e ort on the rm s project, he will always gain a positive payo from stock - even in the bad state. Options become an incentive in that they only pay o in the good state and only when the agent exerts su cient e ort. A large number of options e ectively and e ciently tempt the agent to work fully for the principal. B Firm Characteristics Past studies, such as Demsetz and Lehn (1985) and Himmelberg, Hubbard and Palia (1999), have shown that the rm characteristics a ect compensation. In this section we investigate how changes in the risk characteristics of the rm a ect the composition of the manager s compensation. B.1 Probability of Success In our model, the probability,, describes the likelihood of obtaining the high rm payo that is not a ected by managerial e ort - nature. Hence, this parameter captures the quality of the project and exogenous market conditions. A good project developed at the right time will have a high probability of success whereas a bad project implemented at the wrong time will have a low probability of success. We adjust the probability to determine how the quality of the project 22

25 a ects the manager s compensation. Figures 7-A, B and C and Table III show the total compensation for three cases: = 0:74 (high quality), = 0:65 and = 0:5 (low quality). The di erence in total compensation between = 0:74 and = 0:65 is negligible. When drops to 0.5, the total compensation for all levels of risk aversion is higher. As the quality of the project decreases, the manager demands more compensation to cover the risk of the bad project failing. Figures 7-B and C show the amount of compensation in options and stocks. As the quality of the project decreases, the manager demands more compensation in the form of stock. This is true for all levels of risk aversion. Stock provide some wealth even when the project fails while options only pay when the project succeeds. The risk averse agent managing a low quality project with low probability of success, therefore, will prefer stock compensation. Options, however, yield an interesting result (Figure 7-B). For risk aversion between 0.4 and 0.8, the manager facing the good project ( = 0:74) is willing to take a higher fraction of his wealth in options. The number of options decreases with the quality of the project. This is as expected. Since options only pay o in the high state, then a manager only wants options if there is a su ciently high probability of success. At higher levels of risk aversion, the order switches. The manager facing the low quality project has a high proportion of his wealth in options while the manager facing the high quality project has nearly no options. the low quality project. This result is caused by the large compensation required by the CEO with This CEO of the high quality project receives both little stock and few options. The manager facing the low quality project demands the highest compensation. He is therefore given both the largest amount of stock and the most options. 23

26 B.2 Volatility of the Firm s Project While maintaining the mean of the rm s project, we allow the variance of the payo to increase. The base case is S H = $1; 598:68M and S L = $1; 007:30M. We consider two additional cases: the high volatility case with S H = $1; 798:68M and S L = $807:30M, and the low volatility case with S H = $1; 498:68M and S L = $1; 107:30M: The strike price remains at X = $1; 445:36M. We nd a surprising result regarding the total compensation. As seen in Figure 8-A, total compensation does not depend on the variance of the project s payo. The manager is paid the exact same amount in all three scenarios. Firms that are equal in every way except in terms of volatility should provide their top management the same expected compensation. However, the forms of the compensation are di erent. Figures 8-B and 8-C and Table IV show total compensation decomposed into options and stock. A higher number of options are used when the variance is low and for a wider range of risk aversion. In the low variance case, options are used when risk aversion is as low as 0.35 while options are not used in the high volatility case until risk aversion reaches 0.5. At the same time, stock plays the opposite role. Stock is used in greater quantities in the high volatility case than in the low volatility case. Again, this intuition is clear. Risk averse managers facing a risky project prefer safe forms of compensation. When the project has low variance, stock becomes a very safe choice since stock also has low variance. Options still have the risk of not paying in the event of the bad state. With a nearly certain stock payo, the agent is willing to deal with some risk associated with holding options. When the project has high variance, the manager s stock payo also has high variance. To control the risk of his compensation, the manager will want fewer options. 24