“Stock Options, Restricted Stock, and Incentives”

STOCK OPTIONS, RESTRICTED STOCK, AND INCENTIVES Richard A. Lambert David F. Larcker The Wharton School University of Pennsylvania Philadelphia, PA 19104-6365 Revised, April 2004 We would like to thank workshop participants at the University of Pennsylvania, the University of Rochester, University of Toronto, and Stanford University for their helpful comments. 2 Stock Options, Restricted Stock, and Incentives Abstract Prior work has suggested that options represent an inefficient form of compensation because the value placed on an option by a risk-averse employee is much less than the cost of the option from the perspective of the firm. However, much of this work ignores or fails to properly incorporate the incentive effect of option-based contracts into their analysis. We use agency theory to model the optimal mix of options and stock in the compensation contract. In contrast to prior work, we show that restricted stock is generally not the optimal contract form. We present comparative static results to show how the mix between options and stock and the optimal exercise price of the options varies as a function of the exogenous parameters. Stock Options, Restricted Stock, and Incentives 1. Introduction Over the past decade, we have witnessed an explosion in the use of equity-based compensation (stock options and restricted stock) for top executives of companies. We have seen the use of options extend further down into lower levels of organizations, especially in socalled "new economy" firms (e.g., see Murphy [1999] and Ittner, Lambert, and Larcker [2003]). Despite the growing popularity of stock options and restricted stock, there is considerable academic and professional controversy regarding the relative costs and benefits of equity-based compensation. On the one hand, these plans are viewed as providing “high-powered incentives” that help align the interests of employees with shareholders and help attract and retain scarce managerial and technical talent. However, critics in the popular press claim that options give away too much value by diluting the interests of shareholders. Recently, a number of academic papers have criticized option contracts as being an "inefficient" mechanism for providing compensation and incentives. Meulbroek [2001] argues that risk averse and undiversified executives do not place enough value on the risky payout they will receive from an option to justify the cost given up by shareholders (and implicitly the incentives the option will provide). Unfortunately, Meulbroek [2001] does not model the incentive effect of the stock option and this makes it problematic to assess the net benefit to shareholders from using a stock option for compensating managers. Hall and Murphy [2002] and Jenter [2000] argue that restricted stock (which is an option with an exercise price of zero) dominates options with non-zero exercise prices. However, their analysis is "partial equilibrium" that does not formally incorporate the cost of the option, the value to the employee, or the incentives provided by the options into an optimization program. In contrast, Feltham and Wu 2 [2001] develop a fully-specified optimization model that includes stock options or restricted stock. When the agent affects only the mean of the outcome, they show that restricted stock contracts dominate option-based contracts. However, when the agent's actions affect both the mean and the variance of the outcome, they show that restricted stock contracts are no longer necessarily optimal. In this paper, we develop and analyze an agency theory model to examine the optimal structure of option-based (which includes restricted stock as a special case) contracts. In our model, the role of options and stock in the contract is to provide incentives. We do not consider screening issues in which the contract is used to attract agents with the right "type." The principal chooses the parameters of the compensation package to maximize his net profits, taking into consideration the incentive effects of the contract and the amount of compensation that is paid to the agent.1 Our model differs from that of Feltham and Wu [2001] in four important respects. First, we assume that contracts possess limited liability (i.e., the compensation to the agent cannot be negative). Among other things, this precludes contracts that "sell the firm" to the agent. Second, we allow the contract to include both options and restricted stock, whereas they restrict their analysis to contracts with include only options or only restricted stock, but not both. Third, we allow for a more general specification of the agent’s utility function. Finally, we do not represent the agent's action choice using the first-order-condition approach. As we show, the convexity of an option's payoff can make the agent's expected utility a non-concave function of his effort, in which case the first-order-condition approach fails. 1 When the agent is risk neutral, we provide conditions under which option-based contracts are optimal. This work extends the results in Innes [1990]. However, when the agent is risk averse, we do not claim that option-based contracts are optimal. To our knowledge there are no theoretical papers that have derived an option-based structure to be optimal when the agent is risk averse. Several agency papers have derived results in which the optimal contract is convex (see Holmstrom [1979], Banker and Datar [1989], Hemmer, Kim, and Verrecchia [2000], Lambert [1986, 2001], and Core and Qian [2002]). 3 Our results indicate that option contracts with positive exercise prices generally dominate restricted stock based contracts. In fact, the exercise price in the optimal contract is frequently far "out of the money," particularly in situations where the agent is not very risk averse or where there is little risk in the economic setting. A major advantage of an option with a nonzero exercise price is that it reduces the amount of the firm the principal "gives away" to the agent. By shifting some of the incentives in the contract away from the extreme lower tail of the distribution of outcomes and concentrating the incentives on the rest of the distribution, the principal can reduce his compensation cost while still providing the agent with incentive to work hard. However, this benefit from option contracts must be traded off against the extra risk that an option contract imposes upon the agent. We conduct comparative statics analyses to show how the structure of the optimal option based compensation contract varies with characteristics of the agent (e.g., risk aversion), characteristics of the production function (e.g., marginal productivity of effort and risk-return tradeoffs), and characteristics of the environment (variance in the outcome distribution). The remainder of the paper is organized as follows. In Section 2, we describe the basic model structure. Section 3 analyzes the case where the agent is risk neutral and compensation contracts are not required to satisfy any limited liability constraints. In section 4, we examine the contracting problem for a risk neutral agent where limited liability exists. We introduce risk aversion on the part of the agent in Section 5. An analysis of how the contract changes when the agent’s actions affect both the mean and the variance of the outcome is presented in Section 6. Finally, concluding remarks and directions for future work are discussed in Section 7. 4 2. Model We model a simple agency setting in which a principal hires an agent to perform tasks on his behalf. The risk-neutral principal’s utility is defined over the end-of-period “gross” value of the firm, x, minus the agent’s compensation, s. The agent’s utility is defined over his monetary compensation, s, his other wealth, W, and the action he selects, a. Consistent with most of the agency literature, the agent’s utility function is assumed to be additively separable into monetary and non-monetary components, H(W+s, a) = U(W + s) - D(a). His monetary utility function is modeled as being risk averse or risk neutral: U ' > 0, U′′ ≤ 0 . We assume the agent's other wealth is non-stochastic. We interpret the agent’s action as representing the level of effort he supplies. As is customary in agency models, the agent’s contract is written over the end-of-period outcome, gross of the agent’s compensation. While it is common to express the contract in terms of the total gross value of the firm (x), it will facilitate comparisons with commonly observed stock-based compensation to express the contract in terms of stock price, P. We assume that there are N shares of stock, all initially held by the principal (or shareholders). Therefore, if the end-of-period stock price is P, the principal receives the gross value of the firm (x = NP) minus whatever is paid to the agent. 2 The end of period stock price, P, is assumed to take on only non-negative values. Higher levels of effort will increase the expected value of the end-of-period gross value of the firm. Let f(P|a) denote the probability density of the outcome for a given level of effort. In subsequent sections, we will be more specific about the functional form of f(P|a). 2 It is also possible to do the analysis with the contract expressed in terms of the price net of the compensation paid to the agent. This simply involves transforming the variables. However such a transformation considerably complicates the analysis. Therefore, we present our results using the gross end-of-period outcome, as is customary in the agency literature. 5 If increasing the expected outcome could be done “costlessly,” the principal and agent would both agree to increase firm value to the maximum amount possible, and there would be no incentive problem to address. To eliminate this uninteresting case, we assume there is a cost to the agent’s action. In particular, we model the agent as bearing a nonpecuniary cost or disutility of effort, D(a), were D′ > 0 and D′′ > 0. With this structure, we can write the principal’s problem as maximize ∫ [ NP − s( P)] f ( P | a)dP subject to (1) (1a) (1b) s(P),a ∫ U [W + s( P)] f ( P | a)dP − D(a) > H a maximizes ∫ U [W + s( P)] f ( P | a)dP − D(a) The principal’s problem is to choose the (contract, action) combination to maximize his expected net profits subject to two constraints. First, the contract must offer the agent an acceptable level of expected utility (modeled in equation (1a). The agent’s reservation level of utility is exogenously assumed to be H.3 Second, the (contract, action) pair must ensure that the agent’s action is incentive compatible. That is, the action chosen must be the one that maximizes the agent’s expected utility given the contract offered by the principal. Note that we do not replace the incentive compatibility constraint by the agent’s first-order-condition: ∫ U [W + s ( P)] f a ( P | a )dP − D′(a) = 0. Specifically, we show in Section 4.1 that the convex shape of option-based contracts can render invalid the commonly used first-order-condition approach to modeling the agent’s choice of effort. The reservation utility will depend on how (or whether) the agent's other wealth changes if he leaves the firm for another employment opportunity. We assume the agent's other wealth is non-stochastic and is not affected by what firm that employs the agent. 3 6 Finally, we need to specify the set of contracts that are available to the principal. In subsequent sections, we will vary this set of contracts. Initially, the contract shape is unrestricted. We then impose limited liability, on the part of the agent, or a lower bound on the allowable payment, s(P) > s for all x. Finally, we will impose a piecewise linear shape for the contract form. 3. Risk Neutral Agent, Unlimited Liability Contracts As a benchmark for our subsequent analysis, we begin by examining the special case where the agent is risk neutral and there are no limited liability restrictions on the contract. In this case, it is well known that the first-best solution can be obtained by "selling" the firm to the agent. That is, the optimal contract can be written as s(P) = α + βP where β is set equal to the total number of shares of the firm, N. Assuming P takes on only non-negative values, the contract can also be interpreted as an option-based contract in which the exercise price of the contract is zero (i.e., restricted stock). Intuitively, this contract works because giving the agent 100 percent of any incremental profit his actions generate causes him to completely internalize the incentive problem. Of course, this contract also results in the agent bearing all the risk associated with variability in the firm's outcome. However, since the agent is risk neutral, he does not mind bearing this risk. The salary portion of the contract (α) then allocates the expected outcome between the principal and the agent in the desired fashion. It is less well discussed in the agency literature that the "selling the firm" makes extremely strong assumptions about agent's ability to fulfill the terms of this contract. To see this, note that the intercept of the contract is the (negative of the) payment the agent makes to the principal to buy the firm, and since the principal will only be willing to sell the firm to the agent for a positive price, the intercept of the contract must be negative. In particular, the intercept of the contract of the contract is the negative of the net value of the firm in the first best solution. 7 That is, the principal receives a fixed payment from the agent of -α = NE(P|afb) - D(afb) - H, which is the expected gross value of the firm under the first best actions (afb) minus an amount sufficient to compensate the agent for this actions and meet his reservation level of utility. The agent's overall compensation is then α + βP = H + D(afb) + N[P - E(P|afb) ]. Note that the agent must be sufficiently wealthy to be able to absorb any deviations of the realized outcome (no matter how bad) relative to the expected outcome of the firm. For large publicly traded firms, this outcome would easily be in excess of typical managerial wealth levels. It is also less well documented that the first-best solution can also be achieved using other contract forms. In particular, the first-best solution can be achieved using "option-like" contracts with positive exercise prices as long as the number of options can be chosen to motivate the firstbest level of effort. Not surprisingly, the greater the exercise prices of the options, the greater the number of options that must be granted to achieve the same level of incentives.4 Since an option contract provides a flat payment for all realizations of the outcome below the exercise price, the agent does not care where the outcome falls when it is within the flat part of the range. Because there are no incentives over the flat part of the range, the incentives must be increased over the “in the money” range to provide the desired level of ex-ante overall incentives. As long as the first-best action can be motivated, the first-best gross value of the firm will be achieved. Because there are no restrictions on the size of base salary, it is always possible to choose a base salary that divides the value of the firm between the principal and the agent in whatever fashion is desirable. Since an option-based contract with an exercise price of zero, which corresponds to a restricted stock contract, requires the number of options in the contract to equal the total number of shares of the firm in order to achieve the first best solution, this means than an option-based contract with a positive exercise must offer more options than there are shares of stock in the firm. While this could not literally be implemented using options, it could be done with a cash-based contract that replicates the payoff of an option. 4 8 4. Risk Neutral Agent, Limited Liability Contracts In this section, we examine the impact of limited liability (or limited wealth) constraints on the contracting problem. Limited liability implies that the agent’s compensation must satisfy a lower bound, i.e., s(P) > s for all P. The magnitude of the lower bound, s, is exogenous to our model, but more generally it would a function of institutional features of the legal system and the agent's wealth.5 We assume that the end-of-period value P is bounded below by zero. Therefore, limited liability in a stock-based contract implies the contract can be written as, s(P) = α + β0P, with the restriction that α > s. Similarly, limited liability in a contract that contains restricted stock and multiple layers of options with m different exercise prices, s ( P) = α + β 0 x + ∑ β i max( P − K i ,0) i =1 m where βi is the number of options with exercise price Ki, also implies that the agent’s base salary must be meet the lower bound, α > s. Clearly, the principal could still choose a restricted stock based contract with β0 = N and βi = 0 for i = 1 through n, and the agent would respond with the first best level of effort. However, since the principal's expected net profits would then be ( N − β 0 ) P − α , this "solution" gives the entire value of the firm to the agent (or more, since α must be non-negative). Therefore, the principal must reduce the incentives offered to the agent in order to retain some of the value of the firm. At the other extreme, the principal could offer just enough shares to satisfy the agent’s reservation utility. Unfortunately, if the agent’s reservation utility is small compared to the market value of the firm, the number of shares offered would be quite small, and the incentives provided by the contract would not be very large. Therefore, in choosing the slope We assume the lower bound is such that s > -First best Net Profits. This restriction ensures that the agent is not wealthy enough to be able to buy the firm outright, so that the first best solution cannot be obtained trivially by simply "selling the firm" to the agent. 5 9 coefficient in the contract (which is the number of shares of the firm he gives to the agent), the principal must trade-off: (a) the value of the marginal share of the firm he gives away and (b) the increase in the size of the firm that results from increasing the agent’s incentives to work hard.6 In contrast, in the “unlimited liability” case in the previous section, the sole function of the slope coefficient was to provide incentives. We begin the comparison of restricted stock and option contracts with an example in which we vary the exercise price of the option, and calculate the number of options which are required to motivate the agent to select a given amount of effort. After fixing the incentives of the contract, we then compare the expected cost of these alternative compensation packages options. This example will also highlight some technical issues that are important in solving the complete agency problem we will discuss later. Specifically, assume the end-of-period gross stock price is normally distributed with an expected value equal to 100.0 + a, where a is the agent’s effort. 7 The standard deviation is 10.0. There are initially N = 10,000 shares of stock. The agent’s disutility of effort is D(a) = 100a2, and the agent’s reservation level of utility is 1000.0. The agent has zero other wealth, and the 6 More generally, suppose E(x|a) = L + Ma, and the agent’s disutility of effort satisfies D(a) = .5da2. The parameter L represents the “base” value of the firm (i.e., the value of the firm if the agent provides zero effort). The parameter M is the marginal productivity of the agent’s effort. In this case, the agent’s choice of effort satisfies a = β0M/d, and the optimal slope coefficient (assuming it is positive) is β0 = .5 − Ld . 2M 2 As long as the "base value" of the firm (i.e., the expected end-of-period value of the firm if the agent puts in zero effort) is non-negative (L > 0), we will have β0 < .5. That is, the principal would never give more than half the firm to the agent. The optimal share of the firm to give to the agent is a decreasing function of the “base” level of outcome (L), the marginal disutility of effort (d), and it is an increasing function of the marginal productivity of effort (M). Moreover, note that if the base level of outcome is high enough, the optimal slope coefficient specified in the above equation is negative, which is clearly infeasible. That is, under these circumstances, the principal would prefer to not hire the agent, and to simply pocket the base value of the firm, L. If the principal was forced to retain the agent, he would increase the number of shares to the point where the agent’s reservation level of utility was met. The slope coefficient which satisfies the incentive compatibility constraint (a = β0M/d) and exactly meets the reservation utility constraint β0[L+Ma] - .5da2 = H is the value of β0 which satisfies .5(M2/d)β02 + Lβ0 – H = 0. 7 Since stock price cannot go below zero, we truncate the distribution at zero. 10 contract must provide the agent with non-negative compensation (s = 0). Given these parameters, the first best level of effort is 50.0, and the principal’s expected utility is $1,249,000 in the first best solution. Using two examples, Table 1 shows that the number of options required to motivate a given level of effort is a nondecreasing function of the exercise price of the option. More interestingly, Table 1 shows that the cost of the option package is a decreasing function of the exercise price of the option. In fact, among all exercise prices for which it is feasible to motivate the desired level of effort, restricted stock (i.e., the exercise price equals zero) is the most expensive contract. That is, restricted stock gives up the most value to the agent relative to the effort level it can be used to induce. This presents no problems when the principal does not have to worry about limited liability constraints because he can simply reduce the agent’s fixed salary to offset the value given up by the restricted stock contract. However, with a limited liability contract, the principal is now constrained in his ability to do this. Intuitively, this result occurs because a restricted stock contract “wastes” value in regions of the outcome where there are low incentives. That is, the slope coefficient in a restricted stock contract is, by definition, constant over all ranges of the outcome. However, the impact of the agent’s actions on the probability of an outcome occurring is not constant over all ranges of the outcome. Specifically, in equilibrium a marginal change in the agent’s effort has virtually no effect on the probability of very small outcomes (i.e., stock prices near zero). In the example in Panel A, the restricted stock contract must offer the agent 5,000 shares to motivate the desired level of effort. These shares will pay the agent a large amount of money even if the stock price falls far below expectation. For example, even if the stock price falls as low as $50 (which is 11 virtually impossible if the agent is supplying the desired level of effort), the agent still earns 5,000 x $50 = $250,000. By offering a contract with a nonzero exercise price (and more options), the principal is able to shift the slope of the contract to the regions of the outcome where the probabilities are more sensitive to the agent’s actions. That is, even though an option-based contract offers lower incentives than a restricted stock contract in the lower range of outcomes, it offers much more incentives in higher regions. By properly choosing the exercise price of the options (and number of options), the principal can tailor the contract to optimally trade off the cost of the options with their incentives. In fact, with a little more structure, we can not only show that option-based contracts dominate restricted stock contracts, but that they are the optimal contact structure. To do this, we adopt the following additional assumptions: (A1) The agent’s payoff, s(P), and the principal payoff , NP – s(P), are non-decreasing in the price, P. (A2) The distribution of price satisfies the Monotone Likelihood Ratio Property (MLRP). Formally, the monotone likelihood ratio property implies that P. Assumption (A1) implies, for example, that the number of options plus shares of restricted stock in the contract cannot exceed the number of shares in the firm. Assumption (A2) is common in many agency models, but is fairly restrictive.8 The MLRP is a stronger condition f a ( P | a) is increasing in the price f ( P | a) This condition is commonly assumed in the agency literature. For example, in the classic Holmstrom [1979] paper, the monotone likelihood ratio is necessary to ensure the optimal compensation scheme is a increasing function of the firm’s outcome. When the outcome distribution is normally distributed and the agent’s effort affects 8 12 than simply assuming the expected value of the outcome increases with the agent’s effort; it also implies that higher outcomes are stronger signals that the agent worked hard (see Milgrom [1981] for a discussion).9 With this structure, we can extend results in Innes (1990) to show the optimality of an option-based contract (see Appendix A for proof) Proposition 1: Assume the agent is risk neutral, the outcome distribution satisfies the MLRP, and that the contract satisfies a limited liability constraint, s(P) > s for all P, Further assume that both the principal’s and agent’s monetary payoffs are nondecreasing functions of the outcome. Then the optimal contract consists solely of salary and stock options with single positive exercise price. Moreover, in the optimal contract, the number of options granted is equal to the total number of shares in the firm. In the lower range of outcomes, this contract gives the least amount of incentives allowable (the slope of the contract is required to be non-negative), and in the upper range of outcomes it gives and the highest amount of incentives allowable (attempting to further increase the slope coefficient would make the principal’s marginal share of the outcome become negative). When the outcome distribution satisfies the MLR property, incentives are most effective in the upper tail of the distribution, and this is exactly where the option contract provides them.10 ∂E ( P | a ) ∂a only the expected value of the outcome, m(a), the likelihood ratio is proportional to [ P − E ( P | a )], Var ( P) which is increasing (and linear) in price. The monotone likelihood ratio also holds with the analogous log-normal distribution. Finally, for a large class of common distributions, Banker and Datar [1989] show that the likelihood ratio is increasing and linear. The MLRP is frequently violated when the agent's actions can affect both the mean and the variance of the outcome distributions. Since incentives to affect the variance are viewed to be a particularly important dimension of the incentive effects of option-based contracts, we will relax the MLRP property later. In fact, we will show that relaxing the MLRP property provides an additional reason to prefer options over restricted stock. We will also relax (A1). However, in relaxing these assumptions, we will be unable to prove our results analytically and we will use numerical techniques for solving the model. 10 9 This result is the “reverse” of the Mirrlees [1976] result. Mirrlees shows that if the outcome is normally distributed and penalties can be unbounded below, the first best solution can be approximated to an arbitrarily close degree through a contract that imposes gigantic penalties on the agent when the outcome is in the extreme lower tail 13 Proposition 1 does not specify how the exercise price is chosen or how the contract changes as a function of the underlying parameters of the model. We are unable to provide closed-form analysis of these issues. Therefore, we use numerical optimization techniques to solve for the optimal contract. Even using numerical methods, solving the principal’s problem is difficult because the agent’s response to the contract parameters is not mathematically well behaved. We discuss these problems in the next section. 4.1 Technical Issues in Solving the Model The results in Table 1 indicate one “problem” the optimization solution technique must overcome. In particular, as the exercise price increases beyond a certain range, it becomes infeasible to motivate a given level of effort.11 This occurs because the agent’s expected utility is not a nicely behaved concave function of his effort. To see why this is, note that the increase in the agent’s expected compensation as a function of his effort is analogous to an increase in the Black-Scholes value of an option as a function of the spot price of the firm’s stock. It is wellknown that value of a call option is an increasing convex function of the spot price. Similarly, with an option-based contract, the agent’s expected compensation in our model is an increasing convex function of his choice of effort. When the exercise price of the option is high, the number of options that must be granted to attempt to implement a given level of effort becomes of the distribution. Mirrlees shows that the expected penalties can be made arbitrarily small while still providing the agent with incentive to select the first best solution. Holmstrom [1979] shows conditions where the same result can be achieved using extremely large rewards that are paid only in the extreme upper tail of the outcome distribution. Unlike the Mirrlees and Holmstrom papers, the first best solution cannot be achieved in our model for three reasons. First, the limited liability feature of the model restricts the penalties that can be imposed in the lower tail of the distribution. This eliminates the Mirrlees result as a feasible contract in our model. Second, the magnitude of any reward offered in the upper tail is limited by constraining the number of options granted to the agent to be less than or equal to the number of shares in the firm. This eliminates the Holmstrom result as a feasible contract. Finally, even if the constraint on the number of options is relaxed, we show in the next section that the first best solution cannot necessarily be achieved due to convexities in the agent’s expected utility function that arise when the number of options he is granted gets large. 11 See Appendix B for a proof that there is an upper bound on the exercise price that can be used to motivate a given level of effort. 14 so high that the convexity of the option’s payoff can overwhelm the convexity of the agent’s disutility.12 As a result, the agent’s expected utility is not a concave function of his effort. We illustrate this technical issue in Figure 1, which graphs the agent’s expected utility as a function of his effort level for two contracts that are “close” to one of the optimal contracts from Table 1. Note that in each case, the agent’s expected utility is not a nicely behaved concave function of his effort level. As Figure 1 illustrates, if the principal offers an option-based contract with an exercise price of 124.9 (the upper curve), the agent’s expected utility as a function of his effort has two local maxima. The right-most one is slightly better for the agent, and this one also yields the higher expected net profits for the principal. The principal can do slightly better than this contract by increasing the exercise price to 125.0 (the middle curve). Now the agent is indifferent between the two levels of effort that yield the two peaks. Assuming the agent selects the one that is best for the principal given his own indifference, the agent selects the higher amount of effort. If the principal selects a slightly higher exercise price (125.1 which is the lower curve), the agent makes a very different decision. Now the local peak associated with the lower level of effort is strictly better for the agent. However, the effort level that yields the peak to the right gives a higher level of expected profits to the principal. If the agency model was formulated using the first-order-condition approach, this combination of (exercise price = 125.0, effort = 44.16) would appear to be a solution to the principal’s problem. It satisfies the agent’s first-order condition on effort and yields a higher level of expected profits than what the principal can achieve by offering contacts with either of the two lower exercise prices. However, 12 Since the partial derivative of the agent’s expected compensation with respect to his effort is bounded above, but his marginal disutility of effort is unbounded above as his effort level gets arbitrarily large, the convexity of the option’s payoff will eventually be dominated by the convexity of the disutility function. Therefore, for a given option contract, the agent’s expected utility eventually becomes concave for extremely high levels of effort. It is the intermediate levels of effort where the convexity can occur. 15 this effort level is not incentive compatible for the agent given this contract; the agent would select an effort level of 5.54, not 44.16. Therefore, the first-order-condition approach (e.g., Jewitt, 1988 and Rogerson, 1985) to modeling the agent’s effort is not valid for our setting. Figure 1 also illustrates that the agent’s optimal response is not always a continuous function of the contract parameters offered by the principal, and therefore the principal’s expected net profits are not always a continuous function of the contract parameters. For example, in Panel B of Figure 1, as the principal increases the number of shares in the contract from 9999.0 to 10,000, the agent’s optimal effort jumps discretely from 5.68 to 44.32 (as he moves from one local peak to another), and the principal’s expected net profits jump from $1,047,971 to $1,241,158. Rather than relying on a brute-force exhaustive search or a programming procedure that assumes the objective function is concave, we use a hybrid solution approach consisting of a genetic algorithm (GA) to find the approximate region of the maximum, followed by a nonlinear programming algorithm (NLP) to refine the final result. A genetic algorithm (GA) is a computational search technique that is based on the biological models of evolution and natural selection (see Holland, 1975; Goldberg, 1989; Mitchell, 1996). The estimation results from the GA are known to be able to identify the approximate “neighborhood” of the optimal solution. We then generate a large grid of starting values for the principal’s problem by separately adjusting each dimension of the solution of the principal's problem (i.e., the contract parameters) by small increments around the GA solution. 4.2 Comparative Statics In this section, we develop comparative statics to examine how the solution to problem (1) depends on the sensitivity of the outcome to the agent’s effort and the variance of the 16 outcome distribution. Analogous to the previous example, the end-of-period outcome (e.g. the stock price gross of the agent’s compensation) is normally distributed.13 The expected value of the outcome depends on the agent’s action in a linear fashion: E(P|a) = 100 + Ma, where M is the marginal productivity of effort, or the sensitivity of the expected outcome to the agent’s effort. Since the scale of the agent’s effort is arbitrary, the linear specification is without loss of generality. The variance of the end-of-period outcome distribution does not depend on the agent’s action. The principal initially owns all N shares of stock, where N = 10,000. Therefore, the principal receives NP minus the compensation paid to the agent at the end of the period. In our simulation, we allow the contract to include salary, shares of restricted stock, and stock options. In this way, we can verify whether our optimization approach reproduces those features of the contract that we can prove analytically (e.g., that the optimal contract will consist of zero salary, zero shares, and 10,000 options for the agent). The results in Table 2 confirm the theoretical results, and also show how the exercise price and the agent’s effort vary with the exogenous parameters of the problem. The results in Table 2 indicate that as the sensitivity of the expected outcome to the agent’s effort increases, both the agent's effort and the net profits to the principal increase. As the variance of the outcome distribution increases, the optimal contract motivates a lower amount of effort and a lower level of expected net profits for the principal. That is, as the variance of the outcome We truncate the outcome distribution from below at zero to reflect the fact that stock prices must be non-negative. As long as the expected value of the outcome is large relative to variance, the truncation feature has no significant effect on our numerical results. We also conducted our numerical analysis using a log-normal specification for the outcome distribution. The advantages of the log-normal distribution are that this distribution is naturally bounded below by zero and it is consistent with the stock price return process assumed in conventional option pricing models (e.g., Black-Scholes). The disadvantage of the log-normal process is that it does not allow for as clean a separation of a mean effect and a variance effect. Since these two separate effects were found to be critical in the work of Feltham and Wu [2001], we adopted the normal distribution framework. 13 17 distribution increase, the value of an option increases to the risk neutral agent. Ceteris paribus, this transfers wealth from the principal to the agent. To reduce this wealth transfer, the principal increases the exercise price of the options. Unfortunately, this has the effect of reducing the agent’s incentives to work hard, so his effort level falls. An interesting aspect of this result is that in the absence of a limited liability constraint on the contract, the variance of the outcome would have no impact on the solution to an agency problem between a risk neutral principal and agent. For a given sensitivity of the outcome to the agent’s effort, the optimal exercise price of the option is not monotonic in the variance of the outcome distribution.14 However, Table 2 indicates that the “moneyness” of the option in the optimal contract declines monotonically as the variance increases. We calculate the moneyness of the option to be the number of standard deviations that the expected end-of period outcome given the optimal effort level of the agent exceeds the exercise price of the option. That is, the moneyness of the option is an endogenous characteristic of the option, and it takes into consideration the equilibrium level of incentives induced by the option contract. As the variance of the outcome distribution increases, the option moves from being far “in the money” to less “in the money” to “out of the money.” In particular, note that the optimal contract can result in an option that has an exercise price that is far out of the money. If the moneyness of the option was evaluated relative to the current share 14 In fact, the exercise price is an S-shaped function of the standard deviation, with an increasing, then decreasing, then increasing range. The initial increasing range occurs where the reservation level of utility is met before the “nonconcave problem” with the agent’s effort is reached. The decreasing range during the transition phase between the values for which the reservation level of utility is met before the nonconcave problem is reached and the values for which the opposite occurs. As the standard deviation continues to increase, the exercise price also continues to increase. 18 prices, as it is more conventionally expressed, the optimal exercise price in Table 2 would appear to be even more “out of the money.” 15 4.3 Comparative Statics - No Upper Bound on the Number of Options In this section, we examine the effect of relaxing the constraint that the number of options is bounded above by the number of shares in the firm. As discussed earlier, this could be implemented with a cash based compensation scheme that had the same structure as a stock option plan.16 When this constraint is removed, the results in Proposition 1 do not apply, and an option-based contract is in general, no longer optimal (see Innes [1990]). We will solve problem (1) for the optimal contract within the class of limited liability contracts that consist of salary, restricted stock, and stock options. That is, we examine contracts of the form: s(x) = α + β0 X + β1 max(X − K,0), where β0 is the number of restricted shares, β1 is the number of options, and K is the exercise price of the options. We require α > s = 0, β0 and β1 > 0, but allow β0 and/or β1 to be greater than the number of shares (which is N = 10,000 in our simulation). In Table 3 we document the results for the same set of parameters used to generate the solutions in Table 2. We find once again that the number of shares is equal to zero in each case, and the optimal contract consists solely of options. Although in some instances the first best solution can be achieved, this is not always the case. Using the intuition from the previous section, by simultaneously increasing the exercise price of the options and the number of options, 15 The current stock price, of course, depends on the information the market has regarding the outcome distribution in general, including the incentive effects of the contract offered to the agent. If the market has the same information as the principal and agent do (or if the principal is “the market”) the current stock price is simply the discounted expected future stock price. 16 In addition, this constraint is generally not binding in our results when the agent is risk averse. Therefore, to make the comparative statics more comparable between the risk neutral and risk averse cases, it is of interest to examine both sets of results when there is no constraint on the number of options. 19 the principal is able to motivate the same (or higher) level of effort at a lower cost.17 However, there is a limit as to how high the exercise price (and corresponding number of options) can be and still motivate a given level of effort.18 If the number of options gets too high, the convexity of the options' payoff overwhelms the convexity of the agent's disutility of effort. When this happens, the desired level of effort moves from being the global maximum to a local maximum or a point in the convex portion of the agent's expected utility (see Figure 1). If the principal can raise the exercise price and number of options so that he can motivate the first best level of effort and also lower the agent's expected utility to his reservation level of utility before the "non-convex" feature occurs, the first best solution is attained. However, if the agent's induced utility function turns convex before the contract can lower the agent to his reservation utility, the first best solution is not obtained. The results in Table 3 demonstrate that the induced level of effort increases with increases in the sensitivity of the outcome to the agent's effort. The number of options needed to motivate the desired level of effort falls, because the extra incentive comes directly through the "effort sensitivity" parameter. Moreover, as the effort sensitivity of the outcome distribution increases, the moneyness of the option increases. When the variance of the outcome distribution increases, we observe decreases in both the induced level of effort and the principal's expected net profits. The number of options used in the contract increases, as does the expected utility of 17 A similar result is noted in Hall and Murphy [2002]. However, when they attempt to do a more complete optimization program, this result disappears and their optimal option-based contract sets the exercise price at the other extreme (e.g., zero). We discuss their results more fully in Section 5. Feltham and Wu (2001) also derive an upper bound on the exercise price of an option contract that can be used to implement a give level of effort. However, the upper bound in their analysis is driven by the riskiness of the option contract. For this reason, their analysis applies only to risk averse agents. They claim there is no upper bound for risk neutral agents. However, they assume the agent’s effort choice can be represented using his first-order condition. As we have demonstrated, this condition does not generally hold for large exercise prices. 18 20 the agent. Finally, the moneyness of the options granted falls as the variance of the outcome distribution increases. As in the previous section, the optimal exercise price can result in the options being granted in the money or out of the money. 5. Risk Averse Agent, Limited Liability Contract When the agent is risk averse, the principal must take into consideration the cost of imposing risk on the agent. Since higher exercise prices impose more risk on the agent to motivate the same level of effort (see Table 1), intuitively we might expect the principal to use option contracts with lower exercise prices as the agent becomes more risk averse. We use agency theory to provide some theoretical structure for how risk aversion interacts with other factors in determining the functional form of an optimal contract. In particular a straightforward modification of Holmstrom [1979] to incorporate limited liability in the contract and other wealth of W for the agent implies that the optimal contract satisfies the following condition  f ( P | a)  1 = max λ , λ R +λa a U ′[W + s ( P)] f ( P | a)    (2) where λR is the Lagrange multiplier on the agent's reservation constraint and λa is the Lagrange multiplier on the action incentive compatibility constraint.19 The parameter λ determines the payment to the agent for those values of the outcome P where the minimum payment constraint is binding. For those values of P where the minimum payment constraint is not binding, the f ( P | a) . payment is determined, in part, by the expression λR + λa a f ( P | a) 19 The Holmstrom characterization of the contract assumes that the agent's action choice can be represented using the first-order-condition approach. As demonstrated earlier, this can be problematic when the contract has an option-like structure. Our intent in this section is not to use equation (2) to represent the solution to the principal's problem, but to use it to help provide intuition for how different contract structures will perform. 21 Equation (2) shows that the shape of the optimal contract will depend on of three factors: (a) whether the minimum compensation constraint is ever binding, (b) the form of the agent’s utility function; (c) the function that determines how the agent’s action affects the probability f ( P | a) ). Equation (2) also suggests distribution of the outcome (through the term λR + λa a f ( P | a) that a single contract shape is unlikely to be optimal across a broad class of situations. When the limited liability feature of the contact is binding for a range of outcomes (i.e., f ( P | a) < λ), this will introduce a convex feature there is a range of outcomes for which λR + λa a f ( P | a) into the structure of the contract, ceteris paribus. To illustrate the other two features affecting the shape of the contract, assume the agent’s monetary utility is a member of the power class of utility functions: U(W+s) = n(δ ) (W + s )1−δ for δ > 0.20 Higher values of δ correspond to more 1−δ risk averse utility functions. The power class of utility functions provides additional motivation for the requirement of limited liability in the contract (i.e., the power utility function is not defined for negative values of compensation). With this additional structure, the characterization of the optimal contract in equation (2) becomes:   f ( P | a )  δ  W + s ( P) = n(δ ) max λ , λR + λa a  f ( P | a )     1 (3) Equation (3) demonstrates that the degree of the agent's risk aversion will affect the shape of the optimal contract. In particular, if 0 < δ < 1, the optimal contract is a convex function of the term in the interior the brackets in equation (3). As δ approaches zero (the agent’s risk aversion goes 20 Lambert, Larcker and Verrecchia [1991] and Hall and Murphy [2002] use the power class of utility functions in their analyses of the cost of options to a firm versus the value of options to executives. 22 to zero), the contract becomes “extremely” convex, ceteris paribus. As δ approaches 1.0, the utility function approaches the logarithmic utility function, and equation (3) is linear in the term in brackets. On the other hand, if δ > 1, the optimal contract is a concave function of this term. The structure of how the agent's action affects the shape of the probability distribution will also affect the shape of the contract. For example, when the outcome is normally distributed and the agent's actions affect only the mean of the distribution, the likelihood ratio is ∂E ( P | a ) f a ( P | a) M [ P − (100 + Ma)] ∂a [ P − E ( P | a )] = , which is linear in the outcome P.21 = f ( P | a) Var ( P) Var ( P) However, in other situations, such as when the agent's actions affect both the mean and the variance of a normal distribution, the likelihood ratio is not linear, and its shape will have an impact on the shape of the optimal contract. We analyze such a case in Section 6. There are some settings in which the combination of these factors results in the optimal contract being linear (which corresponds to a restricted stock contract) or piece-wise linear (which corresponds to a stock option contract). In particular, if the probability distribution is a member of the exponential family such that the likelihood ratio is linear in x, and the agent's utility function is logarithmic, the optimal contract in equation (2) reduces to: W + s( P) = β max(γ , γ 0 + γ 1P ) . If the minimum payment constraint is not binding for any range of outcomes, the optimal contract will be linear. However, if there is a range of outcomes for [ ] Banker and Datar [1989] show that the likelihood ratio is linear in x for many common distributions such as the truncated normal, exponential, gamma, and chi-squared, which are referred to as the exponential class of distributions. 21 23 which the minimum payment constraint is binding, the optimal contract is piece-wise linear. The "kink" in the piece-wise linear contract can be thought of as the exercise price of the option. 22 The analysis above suggests that, in general, a combination of restricted stock and options is unlikely to be optimal relative to a broader set of functional forms for the contract. For these situations, we can view the principal's "optimization" problem in our setting as one where he is attempting to choose the parameters within a restricted set of contracts to try to get as close as possible to the contract that would be optimal from an unconstrained set. We assume the principal chooses a contract that can consist of salary, restricted stock, and stock options. That is, we examine contracts of the form: s ( P) = α + β 0 P + β1 max(P − K ,0), where β0 is the number of restricted shares, β1 is the number of options, and K is the exercise price of the options. We require α > s = 0, β0 and β1 > 0, and we require the sum of β0 and β1 to be no greater than the total number of shares owned by the principal (N = 10,000). In Table 4, we present the optimal contract, and demonstrate how it varies as a function of exogenous parameters to the model. We present the results in four panels, corresponding to two choices of risk averse (i.e., a "low" risk aversion of δ = 0.50 and a “high” risk aversion of δ = 2.00) crossed with two choices for sensitivity of the outcome to the agent's action (i.e., a “low” sensitivity of 0.6 and a “high” sensitivity of 1.0). Within each panel, we also vary the standard deviation of the outcome (which the agent's actions are assumed to not affect). While there is no problem comparing results within a given utility function for the agent (i.e., a given level of risk aversion, δ), care must be taken in making comparisons across risk aversion parameters. In particular, as we change the risk aversion parameter δ, not only is the 22 Unfortunately, it is difficult to determine whether this constraint will be binding because the Lagrange multipliers, λ. λR, and λa are all endogenous variables, as is the agent's effort. 24 agent's aversion to risk changing, but his marginal utility of income is also changing. Of course, the marginal utility of income relative to his disutility of effort greatly affects the agent's effort choice. This can impact the agent's incentives to work hard even when there is no risk imposed on the agent. To allow for sensible comparisons across utility functions, we scale the utility functions such that the optimal effort and optimal level of compensation in the first best solution does not depend on the agent’s risk aversion (i.e., in the first best solution where no risk is imposed on the agent, the agent’s risk aversion should be irrelevant). Specifically, we normalize the utility functions, so that when the sensitivity of expected outcome is equal to 1.0, the firstbest action is 50.0, and the agent receives $251,000 to achieve his reservation level of utility.23 In each panel of Table 4, as the standard deviation of the outcome increases, we observe corresponding decreases in the optimal level of effort and the principal’s expected utility. The intuition for this is identical to other agency models where motivating the agent to increase his effort by increasing the sensitivity of his compensation to the outcome is more expensive when the outcome has a higher variance. For each degree of risk aversion, as the sensitivity of the outcome to the agent's effort increases, the principal designs the contract to motivate a higher level of effort, and the principal's expected net profits increase. For the “low” degree of risk aversion (δ = .5), equation (3) suggests that the optimal contract will be a convex function of the performance measure P.24 Therefore, we expect options to be a part of this contract. The observation that the contract consists of a nontrivial number of options in Panels A and B is consistent with this intuition. As the standard deviation of outcome increases, the exercise price of the options declines. However, after adjusting for the decline in 23 This means that that the results are not comparable across utility functions when the sensitivity of the expected outcome to the agent’s effort is 0.6 See Hemmer, Kim, and Verrecchia [2000] for further analysis of this idea. 24 25 the effort level induced, the options become decreasingly “in the money.” As the variability increases, the number of options and number of shares eventually decline, consistent with the principal’s desire to motivate lower levels of effort in this range. In fact, for high levels of volatility, virtually no shares are used. However, at lower levels of volatility, the number of options and number of shares do not vary monotonically. In particular, in Panel B (higher sensitivity of the outcome to the agent’s effort), as the volatility of the outcome increases, the number of options used falls, then rises, then falls again, while the number of shares increases, then decreases.25 For the higher level of risk aversion (δ = 2.0), the theory does not provide an unambiguous result regarding the optimal contract shape because there are countervailing forces pushing for convexity and for concavity. Our results in Panels C and D show that for low levels of volatility, the optimal contract consists solely of options. As the volatility increases, the number of options and the exercise price of the options decline. However, as before, after adjusting for the lower level of effort being motivated, these lower priced options are less in the money than the options for contracts at lower levels of volatility. As the volatility increases, the contract eventually reaches the point where no options are used and the contract consists primarily of shares of stock. At this point, the contract also begins to include salary. The intuition for the use of salary is that it helps to offset the agent’s risk aversion by providing him protection in the lower range of outcomes.26 As the volatility further increases, the salary and 25 A finer presentation of results confirms that these functions are smooth. The number of options is “double humped;” it falls, then rises, then falls again as the volatility increases. The number of shares increases, then eventually falls. This protection is important because for the power = 2.0 utility function (or any power greater than 1.0), the agent’s utility approaches negative infinity as the agent’s payment approaches the lower bound of zero. 26 26 number of shares increase. Moreover, options re-appear in the contract. Interestingly, these options are far out of the money. Our results stand in stark contrast to the results in Hall and Murphy [2002] and Feltham and Wu [2001], who both find that restricted stock contracts dominate option-based contracts. However, Hall and Murphy [2002] do not analyze a fully specified optimization problem.27 Moreover, they restrict their analysis to risk aversion parameters of 2.0 and 3.0.28 29 As discussed above, for less risk averse agents, the contract is unambiguously convex. However, we do find that for risk aversion coefficients in the range considered by Hall and Murphy, there are parameter values for which the optimal contract is “substantively” a restricted stock-based contract. However, for most parameter values, the optimal contract contains a mixture of stock and options, or consists entirely of options. Moreover, for the restricted stock contract to be optimal, we require a combination of a high degree of risk aversion and a high level of exogenous variability. We might expect that self selection and sorting would make such a combination unlikely. As we show in the next section, even for these parameter values, options are part of the optimal contract if the agent’s actions also affect the variance of the outcome. Specifically, Hall and Murphy [2002] evaluate the incentives from an option-based contract as the partial derivative of the agent's expected monetary utility with respect to the stock price. However, the stock price they use in evaluating the partial derivative is exogenously specified. Therefore, this stock price does not incorporate the anticipated incentive effects of the agent's actions. As a result, they do not model the relation between the agent's actions and the distribution of future stock price (and therefore the current stock price). The agent's monetary incentives depend not only on how much his expected monetary utility is affected by increasing the stock price, but also on how his actions affect stock price. Moreover, by not incorporating a non-monetary dimension to the agent's utility, there is no trade-off the agent must make between increasing his expected monetary utility versus incurring additional disutility of effort. Finally, Hall and Murphy allow the agent's salary to adjust when they compare optionbased contracts, but they do not (explicitly) place bounds on the magnitude of the salary adjustments, which could potentially violate a limited liability constraint. 28 27 We repeated the analysis for utility functions with a risk aversion parameter of δ = 3.0, with qualitatively similar results. We are not aware of any conclusive empirical evidence on the degree of risk aversion for executives. There is evidence on the risk aversion of “representative” investors, but there is little consensus even here. Intuitively, we would expect that executives in entrepreneurial firms, where options are the dominant form of compensation, to be considerably less risk averse than the “average” investor. 29 27 Feltham and Wu [2001] allow the payments in the contract to be unbounded below (i.e., they do not assume the contract has a limited liability feature). Moreover, their specification of the agent’s utility function is most consistent with a negative exponential utility function, or U(S) = 1 exp(−ρS). Substituting this utility function into equation (2), along with the normally ρ distributed outcome in which the agent’s effort affects only the mean of the distribution, yields: 1 M s(x) = log[λ R + λ a 2 (x − E(x | a))] . Note that the expression inside the log is linear in the ρ σ outcome x. Therefore, the optimal contract in their setting using the Holmstrom [1979] characterization is a concave function of the outcome. Since they restrict the principal to choosing between a restricted stock contract (which is linear in the outcome) and stock option contract (which is convex in the outcome), it is not surprising to see that the principal prefers the linear contract. This is especially true because the contracts are also assumed to have unlimited liability (i.e., the intercept of the contract can be made negative to extract any "excessive" value the restricted stock contract provides to the agent).30 It is important to recognize that agent’s personal valuation of the model is explicitly a part of the optimization program. Since the agent is risk averse, undiversified, and cannot undo the incentives and risk imposed upon him through the compensation contract, he does not value the option using the Black-Scholes formula. To illustrate, consider the optimal contract for the situation where the agent’s risk aversion coefficient is 0.5, the sensitivity of effort is 0.6, and the standard deviation is 24 (Table 4, Panel A, row 6). For this contract, given the equilibrium level of effort induced by the contract, the Black-Scholes value of the options is $22.63 per option. We do not examine negative exponential utility functions because the precision limits of computers do not allow us to perform numerical integrations using the negative exponential function. For example, computers have considerable difficulty distinguishing between –exp(-100) and –exp(-1000). 30 28 However, the agent only places $16.31 of value per option. That is, given the effort level (22.73) induced by the contract, the agent receives the same expected utility from the options and stock in the contract as he would if each of the options were replaced by an extra $16.31.31 While this may appear to be “inefficient” (e.g., Meulbroek [2001] and Hall and Murphy [2002]), it is important to remember is that the agent’s effort would not be the same with the contact that includes options and the contract that replaces it with salary. The principal’s optimization program explicitly trades off the agent’s valuation of the contract, the principal’s assessment of the contract’s cost, and the incentive effects of the contract. Moreover, the agent’s valuation and principal’s assessment of the cost are calculated based on the equilibrium level of effort induced by the compensation contract. 6. Agent’s Actions Affect the Variance In this section, we examine the impact of allowing the agent’s actions to affect the variance of the stock price. An important part of many executives’ responsibilities include project and investment choice. These decisions generally affect both the expected value and the variability of the outcome. While we do not explicitly model a project (or portfolio of projects) selection choice by the agent, we model the risk-return tradeoff by assuming the agent’s action’s affect both the mean and the variance of the outcome. As before, we assume the gross outcome is normally distributed with mean equal to m(a), and standard deviation equal to σ(a). The functions m(a) and σ(a) are assumed to be increasing functions of the agent’s actions. Their relative slopes determine the steepness of the risk-return frontier. Similarly, when the agent’s risk aversion coefficient is 0.5, the sensitivity of the expected outcome to the agent’s effort is 1.0, and the standard deviation is 32, the Black-Scholes value of the options is $30.88 per option, while the agent only values each option at $25.05. 31 29 Under these, conditions, the likelihood ratio in equations (2) and (3) becomes: f a ( P | a) 1 ∂σ (a ) [ P − m(a )] ∂m(a ) [ P − m(a )]2 ∂σ (a ) =− + + f ( P | a) ∂a ∂a σ ∂a σ2 σ3 This expression is quadratic in the outcome x, and therefore represents an additional reason for why the optimal contract would be convex. The intuition for this result is that the convexity of the contract is used to offset the agent’s risk aversion in order to motivate the risk-averse agent to make risk-return trade-offs more compatible with those desired by the risk-neutral principle.32 We illustrate the impact of having the agent’s actions affect both the mean and the variance of the outcome distribution in Table 5. The mean of the outcome distribution is m(a) =100 + a, and the standard deviation is σ(a) = L + Ja. We focus attention on those parameter values in Table 4 where the contract contained a substantial amount of stock.33 The results in Table 5 indicate that, holding the “base volatility” (i.e., the parameter L) fixed, increasing the sensitivity (J) of the standard deviation to the agent’s action has three primary effects. First, the optimal level of effort decreases. This occurs because the added variability of the outcome that accompanies an increase in the agent’s effort increases the cost of motivating that effort. That is, increasing the agent’s effort requires an increase in the sensitivity of the agent’s compensation to the outcome, which is then magnified by the higher standard deviation of the outcome that accompanies the higher effort. Accordingly, the principal’s expected net profits decline as the sensitivity of the standard deviation to the agent’s actions increases. 32 33 See Meth [1996] for additional analysis. As described in the previous section, by focusing attention on the situation where the sensitivity of the expected outcome to the agent’s effort = 1.0, the scaling in the utility functions makes the results comparable across the degrees of risk aversion. While not reported in the Tables, we also conducted the analysis when the agent is risk neutral. Similar to the results in Tables 2 and 3, the optimal contract consists solely of options, with no stock. We also conducted the analysis for the parameter values in Table 4 for which options were already the primary component of the contract, and found that this continued to be true when the agent could affect the variance. 30 Second, we find that as sensitivity of the standard deviation to the agent’s action increases, the number of options used in the contract increases and the exercise price of the options increases (both in absolute terms, and in terms how far the options are “out of the money”). 34 Therefore, contract offers very high rewards for high outcomes in order to motivate the agent to be willing to take on risk. The optimal number of shares goes to zero with increases in the sensitivity of the outcome standard deviation to agent effort. Finally, the principal begins to use increasing levels of salary to provide the agent with protection at the lower end of the outcome distribution. That is, a high salary with a large number of “out of the money” options is a more efficient way to motivate the desired agent action than the use of restricted stock. The results in Table 5 demonstrate that in order to evaluate the “efficiency” of observed compensation contracts it necessary know more than the equilibrium stock price parameters. It is also critical to know how the agent’s actions affect the parameters of the stock price distribution and the nature of the mean-variance tradeoff. 7. Concluding Remarks Although stock options are an important component of typical managerial compensation contracts, there is considerable debate regarding the “efficiency” of these remuneration arrangements. Recently, Meulbroek [2001], Hall and Murphy [2002], Feltham and Wu [2001], Jenter [2000], and others have concluded that restricted stock (an option with an exercise price of zero) dominates options with non-zero exercise prices for contracting purposes. The basic conclusion of these authors is that stock options are “inefficient” compensation mechanisms. Unfortunately, with the exception of Feltham and Wu [2001], these conclusions are based on a 34 In Table 6 (Panel B), the number of options for a base sigma of 32 or 36 and a sensitivity of sigma to effort of 0.0 have a very large exercise price. For this level of exercise price, the stock option has such a low value that the compensation contract is essentially all shares and zero options. 31 very “partial equilibrium” analysis which ignores the incentive effects associated with alternative compensation arrangements. Using a simple agency model and using a fully specified optimization approach, we find that stock options with positive exercise prices generally dominate restricted stock contracts. Moreover, it is frequently the case that the optimal contract involves stock options that are far “out of the money.” Thus, our results are in stark contrast to the existing numerical analyses comparing the efficiency of stock options to that of restricted stock. There are a number of interesting ways to extend our basic model. For example, rather than restrict the principal to granting all the options with a single exercise price, we could allow multiple layers of options with different exercise prices. This could allow the principal to construct a more convex compensation scheme than can be done solely with salary, restricted shares, and single grant of options. A second direction is to consider the impact of "market-adjusting" the option's exercise price. That is, the option's exercise price would depend on what happened to a market or industry or peer group index. Holmstrom [1982] shows that basing contracts on an agent's relative performance has benefits because it allows the principal to maintain incentive for the agent to work hard yet reduce the agent's exposure to "market" risk. In our analysis, the principal is generally better off when the variability of the outcome is smaller. This suggests that decreasing the variance of the performance measure by extracting the “market component” that the agent cannot affect will have benefits. Finally, it would be of interest to allow the agent not only to have other wealth, but to allow him to adjust his portfolio of wealth in response to the incentive contract offered by the principal. For example, the agent might have choices about effort level and then re-allocate his 32 existing wealth between a risk-free asset, a risky market portfolio, and perhaps existing stock and options. 33 Appendix A Proof of Proposition 1 Let the end-of-period stock price gross of the agent’s compensation be P. Let the agent’s other wealth be W, the compensation to the agent be s(P), and the lower bound on the contract payment be s. Let the agent’s disutlity of effort be D(a). Assuming the principal has N shares of stock, the principal’s net payoff is NP – s(P). We will show that the optimal contract has the form s(P) = s + N max (0,P-K), where K > 0 is the exercise price of the option. The principal’s payoff is then NP – Nmax(0,P-k) = N min(P,K). To prove this proposition, suppose that the optimal contract is sR(P), and that this contract does not have this property. Let aR be the agent’s effort under this contract.. Let sR be the lowest possible payment under this contract. Then we can find an exercise price K > 0, such that the corresponding option based contract, s*(P) = sR + N max(0,P-K) has the following property: if the agent selects aR, his expected compensation under the new contract s* is the same as under contract sR. That is, the agent's overall expected payoff, W + E(s|aR) –D(aR), is the same under contract s* as it is under contract sR. This implies that the principal’s net profits are also identical under the two contracts as long as the agent selects aR under both. Moreover, if contract sR(x) satisfies the limited liability constraint then so does s*(x). Similarly, if the contract sR(P) satisfies the agent’s reservation utility constraint, then so does s*(P). Note that the slope of the option-based contract when P < K is the lowest allowable slope. Similarly, the slope of this contract when P > K is the highest allowable slope. This means there ˆ will be a threshold level of outcome, P , such that the agent’s compensation under s* is weakly ˆ lower than under sR for all P < P , and the agent’s compensation under s* is weakly greater than ˆ under sR for all P > P (with strict inequality for some ranges of P both above and below the 34 ˆ threshold value P . Therefore, the contract s* starts below the contract sR, and crosses it once. This means that the agent’s overall wealth W + s(P) –D(a) satisfies the single-crossing property when s(P) moves from sR to s*. Using the results in Athey (2002), when the distribution of f(P|a) satisfies the monotone likelihood ratio property, the agent’s effort under contract s* will be greater than under contract sR because s* is “steeper” than sR. Let a* denote the agent’s optimal action under contract s*. It is important to note that this result does not depend on the agent’s expected utility being strictly concave or on the assumptions that the optimal effort levels under contracts s* or sR are unique. Every optimal effort under contract s* will be higher than the greatest optimal effort level under contract sR. Since the agent could have chosen aR under contract S*, but chooses not to, his expected utility under (s*,a*) must be greater than under (s*,aR) which is by construction the same as his expected utility under (sR,aR). Moreover, since the principal receives a non-decreasing share of the outcome, the principal’s net profits under (s*,a*) must be greater than under (s*,aR), which is equal by construction to his net profits under (sR,aR). Therefore the principal prefers contract s* to sR. This contradicts that sR is the optimal contract. Q.E.D. 35 Appendix B Proof There is an Upper Bound on the Exercise Price that can be used to Implement a Given Level of Effort Let the mean of the outcome, m(a), be a linear function of the agent's action and assume the standard deviation σ does not depend on the agent’s action. Then the agent’s expected monetary utility from receiving n options with an exercise price of K is nE[max(P − K ,0) | a] = nσ [φ ( z ) − (1 − Φ( z )) z ] , where z is the number of standard deviations the exercise price of the option exceeds the mean K − m(a) of the distribution of x, z = , σ φ(z) is the density function of a standard normal random variable evaluated at z, 1 1 φ(z) = exp[− z 2 ], and 2 2π Φ(z) is the cumulative distribution function of a standard normal random variable z 1 1 evaluated at z, Φ (z) = ∫ exp[− t 2 ]dt. 2 2π −∞ Assuming the agent’s disutility of effort is of the form D(a) = .5da2, the agent’s firstorder condition on effort when given n options with an exercise price of K is 1 ∂m(a) − da. Therefore, the number of options necessary to motivate a given σ ∂a nσ[1 − Φ (z)] level of effort is, assuming the first-order-condition is sufficient to characterize the agent’s optimum, n = da ∂m(a) [1 − Φ (z)] ∂a . As the exercise price of the options increases, Φ(z) increases, and the number of options necessary to satisfy the agent’s first-order-condition on effort increases. Similarly, the agent's second-order condition on effort is −1[n ∂Φ (z) ∂m(a) + d] . ∂a ∂a 36 Substituting for n above, the agent’s second-order-condition on effort is  a ∂m(a)   φ(z)   d    − 1 .  σ ∂a   1 − Φ (z)   Holding the desired level of effort fixed, as the exercise price of the options increases, it  φ(z)  can be shown that the ratio,   , is monotonically increasing and approaches infinity.  1 − Φ (z)  This implies there will be a threshold level for the exercise price such that the number of options that causes the first-order-condition to equal zero at the given level of effort will result in the agent’s second-order-condition on effort being positive. Therefore, the pre-specified level of effort cannot be optimal for the agent. In conclusion, the desired level of effort cannot be implemented with an option contract with an exercise price above some threshold. Q.E.D. 37 Table 1 Number of Options Needed to Induce a Given Level of Effort, Expected Cost, and Riskiness of Compensation as a Function of the Exercise Price of the Option A. Desired Level of Effort = a = 25, Standard Deviation of Outcome = 12 Exercise Price Probability No. of Options Expected Value Options Finish Needed to Motivate of Agent's Out of the Desired Effort Compensation Money Level 0.000 0.000 0.000 0.000 0.000 0.046 0.183 0.282 0.301 0.319 0.366 0.399 5,000.0 5,000.0 5,000.0 5,000.0 5,000.1 5,094.8 5,590.7 6,268.2 6,465.2 N/A N/A N/A 625,000 575,000 525,000 375,000 250,004 127,784 87,254 71,205 68,363 N/A N/A N/A Standard Deviation of Compensation 60,000 60,000 60,000 60,000 60,000 60,128 61,067 62,500 62,953 N/A N/A N/A 0 10 20 50 75 100 110 115 116 117 120 125 B. Desired Effort Level = a = 50, Standard Deviation of Outcome = 36 Exercise Price 0 25 50 75 100 125 140 145 146 150 175 Probability No. of Options Expected Value Standard Options Needed to of Agent's Deviation of Finishes Out Motivate Desired Compensation Compensation of the Money Effort Level 0.000 10,000.2 1,500,024 360,000 0.001 10,002.6 1,250,346 360,006 0.008 10,027.4 1,003,040 360,082 0.046 10,189.6 766,707 360,767 0.152 10,898.4 559,662 364,681 0.313 13,222.3 399,211 379,413 0.384 16,409.4 326,751 399,396 0.395 18,010.5 306,183 408,951 0.396 N/A N/A N/A 0.399 N/A N/A N/A 0.313 N/A N/A N/A Both the principal and agent are risk neutral. The outcome (stock price gross of the agent's compensation) is normally distributed with an expected value equal to 100.0 + a. The agent's disutility for effort is D(a) = 100a2. The first best level of effort is 50.0. N/A means the desired level of effort cannot be motivated at this exercise price. 38 Figure 1 Illustration of the Non-Concavity of the Agent’s Expected Utility as a Function of His Effort 7,000 6,000 Agent's Expected Utility 5,000 4,000 3,000 2,000 1,000 0 0 10 20 30 40 50 60 Agent's Effort Ex Price = 124.9 Ex Price = 125.0 Ex Price = 125.1 A. Agent's Response as a Function of the Exercise Price (Number of Options =10,000) Low Effort Peak Exercise Price 124.90 125.00 125.10 Agent's Expected Effort Utility 5.84 5,728.11 5.68 5,612.93 5.54 5,500.76 Principal's Expected Utility 1,049,236.16 1,047,983.09 1,046,786.07 High Effort Peak Agent's Expected Principal's Expected Utility Utility 6,500.76 1,240,435.56 5,612.93 1,241,157.98 4,728.11 1,241,864.65 Effort 44.46 44.32 44.16 B. Agent's Response as a Function of the Number of Options (Exercise Price = 125.0) Low Effort Peak Number of Options 9,998.00 9,999.00 10,000.00 Agent's Expected Effort Utility 5.68 5,611.16 5.68 5,612.04 5.68 5,612.93 Principal's Expected Utility 1,047,959.59 1,047,971.34 1,047,983.09 High Effort Peak Agent's Expected Principal's Expected Utility Utility 5,572.54 1,241,173.00 5,592.73 1,241,165.52 5,612.93 1,241,157.98 Effort 44.30 44.31 44.32 39 Table 2 Optimal Contract (salary, stock, and options) for a Risk Neutral Agent A. Sensitivity of Expected Outcome to Agent’s Effort = 0.6 Number of Options 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 Exercise Price of Options 108.92 108.20 107.96 110.22 114.98 122.19 131.76 143.65 157.83 Agent’s Effort 29.59 22.08 17.55 13.36 9.70 6.75 4.49 2.85 1.72 Expected Outcome 117.76 113.25 110.53 108.01 105.82 104.05 102.69 101.71 101.03 Std Deviations "In the money" 2.21 0.63 0.21 (0.14) (0.46) (0.76) (1.04) (1.31) (1.58) Agent's Expected Utility 1,000 14,553 31,044 35,581 32,796 26,611 19,668 13,434 8,523 Principal's Expected Net Profits 1,088,984 1,069,169 1,043,445 1,026,724 1,016,012 1,009,313 1,005,254 1,002,919 1,001,768 Standard Deviation of Outcome 4 8 12 16 20 24 28 32 36 B. Sensitivity of Expected Outcome to Agent’s Effort = 1.0 Number of Options 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 Exercise Price of Options 124.90 124.90 124.99 125.00 123.78 122.18 121.53 121.80 123.01 Agent’s Effort 50.00 49.96 48.82 44.32 38.33 35.59 32.85 30.14 27.46 Expected Outcome 150.00 149.96 148.82 144.32 138.33 135.59 132.85 130.14 127.46 Std Deviations "In the money" 6.28 3.13 1.99 1.21 0.73 0.56 0.40 0.26 0.12 Agent's Expected Utility 1,000 1,000 1,000 5,612 25,873 50,679 69,392 82,831 91,549 Principal's Expected Net Profits 1,249,000 1,248,999 1,248,861 1,241,157 1,210,502 1,178,552 1,151,189 1,127,734 1,107,647 Standard Deviation of Outcome 4 8 12 16 20 24 28 32 36 In all cases, the optimal salary = 0, and the optimal number of shares = 0. Both the principal and agent are risk neutral. The outcome (stock price gross of the agent's compensation) is normally distributed with an expected value equal to 100.0 + Ma, where M is the marginal productivity of the agent, or the sensitivity of the outcome to the agent's effort. The principal's ownership consists of 10,000 shares of stock before any compensation to the agent. The agent's outside wealth is W = 0. The limited liability constraint restricts the payments to be at least s = 0. The agent's disutility for effort is D(a) = 100a2. The agent's reservation level of utility is 1000.0. The following table shows the first best level of effort and first best level of net profits for the principal for the cases analyzed above. M = Sensitivity of Output to Agent's Effort 0.6 1.0 First Best Effort Level 30.0 50.0 First Best Expected Net Profits for Principal $1,089,000 $1,249,000 40 Table 3 Optimal Contract for a Risk Neutral Agent when there is No Upper Bound on the Number of Options A. Sensitivity of Expected Outcome to Agent’s Effort = 0.6 Standard Deviation of Outcome 4 8 12 16 20 24 28 32 36 Number of Options 10,126 12,697 16,902 22,372 29,156 37,385 48,550 70,589 115,401 Expected End of Pd Outcome 118.00 118.71 118.38 117.63 115.66 113.50 110.81 108.46 106.88 Standard deviations "In the money" 2.24 0.91 0.26 (0.16) (0.53) (0.84) (1.16) (1.50) (1.84) Agent's Expected Utility 1,000 5,238 16,661 30,191 42,081 49,933 50,846 46,076 40,861 Principal's Expected Net Profits 1,089,000 1,084,621 1,073,299 1,059,771 1,046,397 1,034,439 1,024,813 1,018,694 1,014,915 Exercise Price 109.03 111.43 115.21 120.14 126.24 133.65 143.19 156.51 173.01 Agent's Effort 30.00 31.19 30.63 29.38 26.10 22.50 18.02 14.11 11.46 B. Sensitivity of Expected Outcome to Agent’s Effort = 1.0 Standard Deviation of Outcome 4 8 12 16 20 24 28 32 36 Number of Options 10,001 10,008 10,210 11,008 12,026 13,279 14,734 16,351 18,120 Expected End of Pd Outcome 150.00 150.00 150.00 151.24 151.80 152.04 152.09 151.73 150.89 Standard deviations "In the money" 6.28 3.13 2.04 1.48 1.09 0.79 0.54 0.34 0.16 Agent's Expected Utility 1,000 1,000 1,000 3,927 10,002 18,721 29,469 41,757 55,033 Principal's Expected Net Profits 1,249,000 1,249,000 1,249,000 1,245,921 1,239,676 1,230,863 1,220,095 1,207,943 1,194,889 Exercise Price 124.90 124.92 125.51 127.52 130.07 133.20 136.84 140.88 145.30 Agent's Effort 50.00 50.00 50.00 51.24 51.80 52.04 52.09 51.73 50.89 In all cases, the optimal salary = 0, and the optimal number of shares = 0. Both the principal and agent are risk neutral. The outcome (stock price gross of the agent's compensation) is normally distributed with an expected value equal to 100.0 + Ma, where M is the marginal productivity of the agent, or the sensitivity of the outcome to the agent's effort. The principal's ownership consists of 10,000 shares of stock before any compensation to the agent. The agent’s disutility for effort is D(a) = 100a2. The agent's outside wealth is W = 0. The limited liability constraint restricts the payments to be at least s = 0. The agent's reservation level of utility is 1000.0. 41 Table 4 Optimal Contract (salary, stock, and stock options) for a Risk Averse Agent whose Action does not Affect the Variance A. Risk Aversion Coefficient = 0.5, Sensitivity of Expected Outcome to Agent’s Effort = 0.6 Standard Deviation of Outcome 4 8 12 16 20 24 28 32 36 Number of Options 9,168.19 9,149.92 8,616.75 7,827.01 6,751.83 5,597.04 4,644.90 3,834.07 3,182.68 Exercise Price of Options 113.44 109.45 105.11 101.66 97.87 93.75 89.62 85.32 80.97 Agent’s Effort 37.49 35.91 33.58 30.37 26.67 22.73 19.35 16.35 13.88 Expected Outcome 122.50 131.55 120.15 118.22 116.00 113.64 111.61 109.81 108.33 Standard Deviations “In the money” 2.27 2.76 1.25 1.04 0.91 0.83 0.79 0.77 0.76 Principal’s Expected Net Profits 1,068,969 1,061,846 1,050,308 1,036,664 1,022,447 1,009,153 997,624 988,080 980,432 Salary 0 0 0 0 0 0 0 0 0 Shares 594.23 338.83 135.84 52.29 15.12 5.06 2.64 4.45 10.46 B. Risk Aversion Coefficient = 0.5, Sensitivity of Expected Outcome to Agent’s Effort = 1.0 Standard Deviation of Outcome 4 8 12 16 20 24 28 32 36 Number of Options 9,266.11 9,067.96 9,227.86 9,247.43 9,109.09 8,851.63 8,412.38 8,012.22 7,628.45 Exercise Price of Options 132.50 131.85 129.22 125.69 122.39 119.45 116.40 114.15 112.42 Agent’s Effort 49.84 49.35 48.54 47.40 45.92 44.13 41.99 39.63 37.08 Expected Outcome 149.84 149.35 148.54 147.40 145.92 144.13 141.99 139.63 137.08 Standard Deviations “In the money” 4.33 2.19 1.61 1.36 1.18 1.03 0.91 0.80 0.68 Principal’s Expected Net Profits 1,247,406 1,242,638 1,234,798. 1,224,077 1,210,807 1,195,431 1,178,484 1,160,573 1,142,444 Salary 0 0 0 0 0 0 0 0 5.91 Shares 602.61 614.66 470.02 293.26 159.82 73.87 21.72 3.88 0.04 42 C. Risk Aversion Coefficient = 2.0. Sensitivity of Expected Outcome to Agent’s Effort = 0.6 Standard Deviation of Outcome 4 8 12 16 20 24 28 32 36 Number of Options 8,403.72 4,645.56 3,162.39 2,400.47 1,935.24 571.15 1,094.76 1,713.34 3,428.60 Exercise Price of Options 101.36 80.73 59.27 37.99 16.85 262.35 316.65 364.18 398.56 Agent’s Effort 42.93 41.94 40.72 39.55 38.39 37.20 35.93 32.81 31.05 Expected Outcome 125.76 125.16 124.43 123.73 123.03 122.32 121.56 119.69 118.63 Standard Deviations in the money 6.10 5.55 5.43 5.36 5.31 (5.83) (6.97) (7.64) (7.78) Principal’s Expected Net Profits 1,052,582 1,045,238 1,038,273 1,031,472 1,024,837 1,017,347 1,006,482 989,308 976,053 Salary 0.10 0.10 0.10 0.10 0.10 0.25 8.79 99.55 531.08 Shares 0.00 0.00 0.00 0.00 0.00 1,682.90 1,720.24 1,733.39 1,768.14 D. Risk Aversion Coefficient = 2.0, Sensitivity of Expected Outcome to Agent’s Effort = 1.0 Standard Deviation of Outcome 4 8 12 16 20 24 28 32 36 Number of Options 8,891.83 5,531.16 3,777.13 2,885.09 2,327.44 1,955.28 0.00 3,448.42 3,696.45 Exercise Price of Options 121.24 103.29 81.42 60.27 38.89 17.70 N/A 314.86 361.32 Agent’s Effort 49.08 48.56 47.78 47.17 46.42 45.74 45.31 44.04 43.86 Expected Outcome 149.08 148.56 147.79 147.17 146.42 145.74 145.31 144.04 143.86 Standard Deviations in the money 6.96 5.66 5.53 5.43 5.38 5.33 N/A (5.34) (6.04) Principal’s Expected Net Profits 1,243,239 1,235,203 1,228,040 1,220,972 1,213,964 1,207,023 1,199,172 1,185,590 1,172,002 Salary 0.10 0.10 0.10 0.10 0.10 0.10 0.18 4.74 38.71 Shares 0.00 0.00 0.00 0.00 0.00 0.00 1,747.49 1,768.70 1,852.64 The principal is risk neutral. The agent is risk averse with a power utility function: U(s) = s1-δ. Therefore, higher values of δ imply greater degrees of risk aversion. The outcome (stock price gross of the agent's compensation) is normally distributed with an expected value equal to 100.0 + Ma, where M is the marginal productivity of the agent, or the sensitivity of the outcome to the agent's effort. The principal's ownership consists of 10,000 shares of stock before any compensation to the agent. The agent’s disutility for effort is D(a) = 100a2. The agent's outside wealth is W = 0. The limited liability constraint restricts the payments to be at least s = 0. In all cases, the agent’s reservation level of utility constraint was exactly met. Table 5 Optimal Contract (salary, stock, and stock options) for a Risk Aversion Agent whose Actions Affect the Variance of the Outcome A. Agent’s Risk Aversion Coefficient = 0.5 Base Sigma 24 24 24 28 28 28 32 32 32 36 36 36 Sensitivity of Sigma to Effort 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2 Salary 0 0 33,129 0 9 33,084 0 242 33,391 0 836 33,838 Number of Options 8,851.63 9,764.92 9,999.90 8,412.38 9,773.47 9,999.90 7,794.90 9,774.18 9,999.90 7,590.44 9,773.05 9,999.90 Exercise Price of Options 119.45 128.26 134.80 116.40 130.19 139.10 113.00 132.90 144.04 112.23 136.24 149.45 Number of Shares 73.87 235.08 0.10 21.72 226.53 0.00 0.24 225.82 0.00 0.01 226.95 0.00 Agent’s Effort 44.13 36.91 33.19 41.99 34.30 31.23 39.67 31.98 29.67 37.04 29.94 28.43 Standard Std Expected Deviation Deviations Principal’s Gross of In the Expected Outcome Outcome Money Net Profits 144.13 136.91 133.19 141.99 134.30 131.23 139.67 131.98 129.67 137.04 129.94 128.43 24.00 27.7 30.6 28.00 31.4 34.2 32.00 35.2 37.9 36.00 39.0 41.7 1.03 0.31 (0.05) 0.91 0.13 (0.23) 0.83 (0.03) (0.38) 0.69 (0.16) (0.50) 1,195,431 1,162,257 1,147,419 1,178,484 1,144,997 1,134,669 1,160,546 1,129,603 1,124,626 1,142,297 1,116,193 1,117,098 44 B. Agent’s Risk Aversion Coefficient = 2.0 Sensitivity of Sigma to Effort 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2 Base Sigma 24 24 24 28 28 28 32 32 32 36 36 36 Salary 0. 47,542 93,465 0. 60,009 96,573 5 68,019 98,897 39 74,089 100,689 Number of Options 1,955.28 3,024.63 3,718.18 0.00 2,873.02 3,541.86 3,448.42 2,733.19 3,375.60 3,696.45 2,607.96 3,221.95 Exercise Price of Options 17.70 78.97 108.19 N/A 81.10 110.41 314.86 82.40 112.96 361.32 84.14 115.74 Number of Shares 0.00 3.86 0.00 1,747.49 3.86 0.00 1768.70 0.00 0.00 1,852.64 0.00 0.00 Agent’s Effort 45.74 31.17 25.63 45.31 28.05 23.30 44.04 25.38 21.45 43.86 23.08 19.96 Standard Std Expected Deviation Deviations Principal’s Gross of In the Expected Outcome Outcome Money Net Profits 145.74 131.17 125.63 145.31 128.05 123.30 144.04 125.38 121.45 143.86 123.08 119.96 24.0 27.1 29.1 28.0 30.8 32.7 32.0 34.5 36.3 36.0 38.3 40.0 5.33 1.93 0.60 N/A 1.52 0.39 (5.34) 1.24 0.23 (6.04) 1.02 0.11 1,207,023 1,103,840 1,070,495 1,199,172 1,080,534 1,052,932 1,185,590 1,060,458 1,039,185 1,172,002 1,043,487 1,028,606 The principal is risk neutral. The agent is risk averse with a power utility function: U(s) = s1-δ. Therefore, higher values of δ imply greater degrees of risk aversion. The outcome (stock price gross of the agent's compensation) is normally distributed with an expected value equal to 100.0 + Ma, where M =1 is the marginal productivity of the agent, or the sensitivity of the outcome to the agent's effort. The standard deviation of the outcome is equal to (Base Sigma + Ja), where J is the sensitivity of the standard deviation of the outcome to the agent’s effort. The principal's ownership consists of 10,000 shares of stock before any compensation to the agent. The agent’s disutility for effort is D(a) = 100a2. The agent's outside wealth is W = 0. The limited liability constraint restricts the payments to be at least s = 0. In all cases, agent’s reservation utility constraint was exactly met. References Athey, S., 2002. 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