# Maturity and exercise price of executive stock options.

Chongwoo Choe (*)Received 19 December 2000; received in revised form 4 May 2001; accepted 31 August 2001

Abstract

Using a simple three-period model in which a manager can gather information before making an investment decision, this paper studies optimal contracts with various stock options. In particular, we show how the exercise price of executive stock options is related to a base salary, the size of the option grant, leverage, and the riskiness of a desired investment policy. The optimal exercise price increases in the size of grant and the base salary and decreases in leverage and the riskiness of a desired investment policy. Other things equal, the optimal exercise price of European options with a longer maturity should increase more for an increase in the base salary and the size of grant and decrease more for an increase in leverage than the one with a shorter maturity. The optimal exercise price of American options is determined by the optimal exercise prices of European options with different maturities. Given the fixed exercise price, the size of the option grant does not decrease in the face value of debt. [C ] 2002 Elsevier Science Inc. All rights reserved.

Keywords: Executive stock options; Exercise price; Maturity

1. Introduction

Stock options are the fastest growing instrument in compensating top management of corporations. While the use of stock options is gaining popularity across industrialized economies, the increase has been most prominent in the US (Murphy, 1999). The amount of wealth represented by executive stock options, as well as the way in which options are treated for tax and accounting purposes, have attracted heated debate both in public and in academic arenas (Economist, 1999).

Most existing studies of executive stock options are empirical in nature, mainly focused on estimating the incentive effects of options (Core & Guay, 1998; Hall, 1998; Hall & Liebman, 1998; Jensen & Murphy, 1990; Yermack, 1995). The principal findings are that the most of incentives to CEOs are provided though stock options and that, when options are properly valued, the pay-performance sensitivity of executive compensation is larger than that originally estimated by Jensen and Murphy (1990). However, there is a lack of rigorous theoretical studies incorporating stock options as part of an optimal CEO compensation package.

Haugen and Senbet (1981) is the earliest study that explains how the mix of put and call options can overcome the agency problems of external capital. However, put options are rarely used in practice. John and John (1993) analyze optimal CEO compensation for mitigating the problem of asset substitution when the firm has debt in its capital structure. While their focus is not on executive stock options per se, they derive a conclusion that more levered firms should sever the link between managerial interests and shareholder interests by making smaller the pay-performance sensitivity parameter of managerial contract. (1) Garvey and Mawani (1999) explain how the asset substitution problem can be overcome though adjusting the exercise price of executive stock options without having to dilute the effort incentives of executives. More importantly, they find from Canadian data that more levered firms tend to set the exercise prices lower so that the effective exercise prices faced by executives remain more or less s table within an industry.

A number of recent studies focus exclusively on how stock options motivate managers. Hall and Murphy (2000) present a numerical example showing how at-the-money options can be justified for risk-averse managers. Carpenter (2000) argues that executive stock options may not necessarily encourage excessive risk-taking if managers are risk averse. However, her focus is on the manager's trading strategy, taking the stock option contract as given. Acharya, John, and Sundaram (2000), Brenner, Sundaram, and Yermack (2000), Chance, Kumar, and Todd (2000), and Johnson and Tian (2000a, 2000b) address the optimality of resetting the exercise price of executive stock options or the valuation of various executive stock options, again without deriving stock options as part of the optimal contract.

The purpose of this paper is to address several additional aspects of executive stock options. Compared to conventional bonus or stock ownership, stock options provide additional choice variables that firms can use in designing managerial contracts. In particular, the exercise price and the maturity of stock options need to be part of the contracting problem. This paper incorporates these variables explicitly in an optimal contracting framework. The specific questions asked are the following: what is the relationship among the size of the option grant, the exercise price, and other components of executive compensation; how do they depend on the riskiness of an investment policy desired by shareholders; how do they depend on financial leverage; and what is the relationship between executive stock options with different maturities? Answers to these questions are provided in a simple model similar to John and John (1993), in which the manager can gather private information before making an investment decision. T he owner or the representative shareholder designs managerial contracts that consist of a base salary and various types of stock options. Managerial contracts, as well as the manager's investment decision, are observed by the market, which then prices the firm in a way consistent with these observations.

This paper draws a number of conclusions. First, other things equal, the optimal exercise price increases in the size of the option grant and the base salary and decreases in leverage and the riskiness of a desired investment policy. The intuition behind this becomes clear when the exercise value of options is viewed as an incentive component of a contract. When the size of the option grant increases, the exercise price needs to be increased as well to leave the exercise value of options unchanged. An increase in leverage reduces the exercise value of options by increasing the effective exercise price, which is the sum of the exercise price and the face value of debt. Thus, the exercise price must be adjusted downwards. When shareholders desire a more risky investment policy, they can induce the manager to do so by making risk-taking more attractive to the manager. Reducing the exercise price in this case makes the exercise value of options larger, making options more valuable when risk-taking pays off. When the base salary increases, the exercise value of options can be reduced to leave the manager's expected utility the same, and so there is a negative relation between the two.

Second, other things equal, the optimal exercise price of European options with a longer maturity can be increased more for an increase in the base salary and the size of the option grant and decreased more for an increase in leverage than the one with a shorter maturity. This is a direct consequence of the fact that shareholders have more flexibility in providing incentives to the manager by using options with a longer maturity. The longer the maturity is, the more of the manager's private information can be incorporated into the contract with stock options. Consequently, for the same amount of increase in the option grant (and base salary), the exercise price can be increased more (to the benefit of shareholders) for options with a longer maturity.

Third, the optimal exercise price of American options is determined by the optimal exercise prices of European options with different maturities if there were only a discrete number of dates when options could be exercised. In reality, when American options can be exercised at any point in time at the manager's discretion, the optimal exercise price could be approximated using the optimal exercise prices of various European options.

Fourth, for a fixed exercise price, the size of the option grant does not decrease in the face value of debt. This last conclusion holds when the debt is priced optimally so that, unlike in John and John (1993), asset substitution is not an issue. In that case, higher leverage requires a lower exercise price. Therefore, if the exercise price is fixed, either the size of the option grant or the base salary must be increased to keep the manager's expected utility the same.

An additional conclusion from this study is that financial leverage should not necessarily distort the investment decision made by managers who are motivated by self-interest. Numerous studies have provided various solutions to this asset substitution problem. The first of those is changing the financial structure of a firm, either by reducing debt, by using complex securities such as warrants or convertibles, or by providing covenants to debt holders. The second is concerned with redesigning managerial contracts. This paper is in line with the second strand of literature. In particular, we show how the exercise price of managerial stock options should be adjusted to make contracts incentive compatible when the firm has debt in its capital structure.

The rest of the paper is organized as follows. Section 2 presents the model. Section 3 analyzes optimal contracts with European stock options in unlevered firms. The analysis of American stock options is provided in Section 4. The results are extended in Section 5 for levered firms. Section 6 summarizes main findings and discusses their empirical implications. All proofs are relegated to Appendix.

2. The model

We adopt a slightly modified version of John and John (1993). The firm consists of two agents: the owner (or a representative shareholder) and the manager. There are two investment projects. A safe project, denoted S, returns I>0 for sure. A risky project, denoted R, returns [pi]>I with probability q and 0 with probability 1 - q. (2) If necessary, the owner issues debt to partially finance a project. The debt is a zero-coupon bond, repayable at the terminal date, whose face value is denoted by D. (3) The net interest rate is assumed to be zero, and the manager's reservation utility is denoted by W>0. Both agents are assumed to be risk neutral, interested only in maximizing expected payoffs. Assuming risk neutrality allows us to separate the risk-sharing aspect from the incentive aspect of optimal contracts. (4) Modeling the behavior of risk-averse managers will not change the qualitative results of the paper, in particular, as regards the comparative static predictions of the optimal contract and the relation ship among optimal contracts with different types of stock options. (5)

Prior to making a project choice decision, the manager alone can gather information, which leads to a forecast of probability q. For simplicity, it is assumed that the manager directly observes q. (6) Without information gathering, the owner and the manager share the same information about the distribution of q, which is given by a positive, differentiable density function f(q) and a distribution function F(q). Since both agents are risk neutral, the first-best investment policy is the one that maximizes the expected return from projects. Define [q.sup.*] by [q.sup.*] [pi]=I or [q.sup.*] = I/[pi]. That is, [q.sup.*] is a cutoff probability above which the risky project has a larger expected return than the safe project. Therefore, the first-best investment policy is to choose R if and only if q[member of][[q.sup.*],1] and S otherwise. An investment policy, which selects R for [q,1], will be simply denoted by q. Thus, the first-best investment policy is [q.sup.*]. By its definition, [q.sup.*] is the policy that maximizes the expected return, which is denoted by [V.sub.0] [equivalent to] [[integral].sup.1.sub.q.sup.*] q[pi]f(q)dq + [[integral].sup.q.sup.*.sub.0] If (q)dq = [alpha]([q.sup.*])[pi] + F([q.sup.*])[pi]I where [alpha](q) [equivalent to] [[integral].sup.1.sub.q] rf(r)dr is defined as the ex ante cumulative probability of [pi] under the investment policy q. The timing of the model is as follows. At date 1, the owner offers the manager a contract and, if necessary, issues debt. When debt is issued to finance a project, we restrict attention to the case where the face value of debt is less than I, i.e., D<I. Moreover, debt is priced based on the expectation of an investment policy. If the investment policy q is expected, then the price of debt is equal to the expected payment at the terminal date, given by F(q)D + [alpha](q)D. Given that the net interest rate is zero, this exactly cancels with D if the policy q is indeed implemented. At date 2, the manager observes q and chooses an investment proj ect. At date 3, the return from the chosen project is realized and the terms of initial contract are executed, including the repayment of debt. We assume that the choice of project at date 2 and the returns from projects at date 3 are publicly observable and will be incorporated in the value of the firm. In solving for the optimal contract, we look at the owner's problem of maximizing her expected payoff subject to the constraints that the manager cannot be penalized with negative compensation (limited liability), that the manager should voluntarily accept the contract (individual rationality), and that the manager should choose an investment policy that is also in the interest of the owner (incentive compatibility).

3. Optimal contracts with European stock options in unlevered firms

In the rather simple contracting environment described above, many different forms of contracts can be considered. This has been studied elsewhere and the purpose of this paper is to have a closer look at contracts with various stock options. (7) In discussing contracts with stock options, we first look at the case where D = 0. A contract with stock options is denoted by [SIGMA] [equivalent to] (B, [sigma], X) where B is a base salary payable when the return is [pi] or I (8) and [sigma] is the fraction of the value of the firm, which the manager can buy either at date 2 or 3 at an exercise price given by X [greater than or equal to] 0. (9) In other words, [sigma] represents the size of call options on stocks awarded to the manager at date 1. Limited liability requires B to be nonnegative, and individual rationality and incentive compatibility will impose restrictions on [sigma] and X. To clarify these restrictions, it is necessary to study how the value of the firm is determined in the market. Since the marke t is assumed rational, it will value the firm based on the expectation of an investment policy inferred from the managerial contract, which is confirmed at the equilibrium.

The value of the firm at different dates can be determined recursively. At date 3, after the realization of return, the value of the firm is [{r - B}.sup.+] if the return is r where [{Y}.sup.+] = max {O,Y}. Needless to say, these values are independent of any investment policy chosen. Denote them by Eq. (1):

[V.sub.s] [equivalent to] I - B, [V.sub.3] [equivalent to] [pi] - B. (1)

Suppose now the market expects the investment policy q. If S is chosen, then the value of the firm at date 2 is again [V.sub.s]. When the market observes the choice of R at date 2, the probability of [pi] is updated. Using Bayes' rule, this conditional probability can be written as Pr(q\R) = q/[[integral].sup.1.sub.q]f(q)dq = q/1-F(q) if q [greater than or equal to] q and Pr (q\R) = 0 otherwise. Since the date 2 value of the firm when R is chosen should be equal to the date 3 value of the firm from R expected at date 2, we have

[V.sub.2](q) [equivalent to] [[integral].sup.1.sub.q]q/1-F(q) [V.sub.3]f(q)dq = [alpha](q)/1-F(q) [V.sub.3] (2)

where [alpha](q) [equivalent to] [[integral].sup.1.sub.q]rf(r)dr was defined earlier as the date 1 cumulative probability of [pi] under the policy q (Eq. (2)). Finally, the date 1 value of the firm can be determined. Since the announcement of choosing S leads the value of the firm equal to [V.sub.S], and since the probability of choosing S under the policy q is F(q), the date 1 value of the firm is (Eq. (3))

[V.sub.1](q) [equivalent to] F(q)[V.sub.S] + [1 - F(q)][V.sub.2](q) = F(q)[V.sub.S] + [alpha](q)[V.sub.3]. (3)

For future reference, we note here how the above values change as the investment policy expected by the market changes.

Lemma 1: Let q be the investment policy the market expects from the firm. Then, (1) [V.sub.1](0) = E(q) [V.sub.3], [V.sub.1](1) = [V.sub.S], [V'.sub.1](q)>/<0 if and only if [V.sub.S]</>q[V.sub.3], and (2) [V.sub.2](0)=E(q)[V.sub.3], lim q[right arrow]1 [V.sub.2](q) = [V.sub.3], [V'.sub.2](q)[greater than or equal to]0, where E(q), the expectation of q, is with respect to f(q) and the prime indicates a derivative.

The date 1 value of the firm changes from the expected value from the risky project to the value from the safe project as the investment policy changes from the least to the most conservative one. It reaches a maximum when [V.sub.S]=q[V.sub.3], which can be restated as q=[q.sup.*]-((1-[q.sup.*])B)/([V.sub.3]). If B=0, then the date 1 value of the firm is maximized when the first-best investment policy is expected. As we shall see later, however, B[greater than or equal to]0 is necessary to implement the policy [q.sup.*], implying that the date 1 value of the firm is maximized if the market expects an investment policy more aggressive than the first-best one. After the choice of project is announced at date 2, the market knows that the risky project has been chosen simply because it is deemed profitable. As a higher value of q then implies a higher probability of [pi], [V.sub.2](q) increases in q. Fig. 1 describes the changes in the value of the firm in response to the market's expectation of investment policy . The point labeled A corresponds to the investment policy q = [q.sup.*] - (1-[q.sup.*])B/([V.sub.3]).

We are now ready to incorporate stock options as part of managerial contracts. Our model admits three types of stock options. European options, which expire at dates 2 and 3, are the focus of this section. (10) American options are studied in Section 4. In each of the three cases, an equilibrium is a collection of an investment policy, the manager's contract, and the pricing rule for the value of the firm, such that (1) given the pricing rule, the investment policy and the manager's contract constitute a solution to the owner's optimization problem, and (2) the pricing rule is consistent with the investment policy given by the solution to the owner's optimization problem.

3.1. European options expiring at date 2

Suppose the owner wants the investment policy q [member of](0,1) to be implemented and consider a contract [SIGMA] [equivalent to] (B,[sigma],X) for which the exercise of options is restricted to date 2 after the announcement of project choice. (11) For this contract to be incentive compatible, the manager who observed q<q should not have incentives to select R or B + [sigma][{[V.sub.s] - X}.sup.+] [greater than or equal to] [sigma][{[V.sub.2](q - X}.sup.+] + qB. Similarly, the manager who observed q[greater than or equal to]q should not have incentives to select S or B + [sigma][{[V.sub.s] - X}.sup.+] [less than or equal to] [sigma][{[V.sub.2](q) - X}.sup.+] + qB. From these follow the following incentive compatibility constraints.

Lemma 2: Let (B,[sigma],X) be a contract with European options, which expire at date 2. Then, the owner can almost surely implement the investment policy q if and only if (1) X[less than or equal to][V.sub.s] and (1-q)B = [sigma][[V.sub.2](q) - [V.sub.s]] or (2) [V.sub.s]<X[less than or equal to][V.sub.2](q) and (1 - q)B = [sigma][[V.sub.2](q) - X].

In Lemma 2, (1 - q)B measures the loss in the base salary from choosing R instead of S at the cutoff probability q. The other sides of the equalities are the gains from exercising options after choosing R instead of S at the cutoff probability q. Thus, Lemma 2 implies that the loss and the gains have to be balanced at the cutoff probability q. Given this and noting that [V.sub.2](q) is increasing in q, the manager with q<q will choose the safe project since the loss outweighs the gains. Similarly, the manager with q[less than or equal to]q will choose the risky project.

If [SIGMA] = (B,[sigma],X) is incentive compatible for the policy q, then the manager's expected utility can be expressed as U(q,[SIGMA]) = [sigma][[V.sub.2](q) - X] + B[qF(q) + [alpha](q)]. It is easy to check that, at the solution to the owner's optimization problem, the manager's individual rationality constraint is binding: U(q,[SIGMA]) = W. Since the owner has a residual claim on the return from the project, which is maximized with the first-best investment policy, it follows that the owner will implement the first-best investment policy. This establishes the following result.

Proposition 1: An optimal contract with European options, which expire at date 2, is given by [SIGMA] = (B,[sigma],X) such that satisfies the conditions in Lemma 2 for [q.sup.*] and of [[V.sub.2]([q.sup.*]) - X] + B[[q.sup.*]F(q*) + [alpha]([q.sup.*])] = W.

Some remarks on Proposition 1 are in order. First, in order to implement [q.sup.*], it must be that B>0. The reason is as follows. The manager can choose S to secure B as well as possible gains from exercising options or R to pursue larger gains from exercising options at the cost of possible loss in B. If B = 0 and if the market expects the policy [q.sup.*], then the manager would always choose R since the gains from exercising options after announcing R outweigh those after announcing S without incurring the loss in B. Second, there is a continuum of optimal contracts. For a given base salary B, the owner can choose from a menu of ([sigma],X) that satisfies the equality in Proposition 1.

Albeit obvious, we can make several observations. First, the exercise price and the size of the option grant are positively related. For an increase in the exercise price, the necessary adjustment in the size of grants depends on the manager's reservation utility, the base salary, and the distribution of return from the risky project. Second, for a given size of grant, a higher base salary can be accompanied by a higher exercise price and vice versa. Third, there is no a priori ground for selecting a particular exercise price: options could be granted out of the money, at the money, or in the money as long as [sigma] and B are adjusted accordingly.

To formalize the above observations, use Proposition 1 to express the optimal exercise price in terms of other variables and parameters. Denoting this by [X.sub.2], we have (Eq. (4))

[X.sub.2] = [V.sub.2]([q.sup.*] - W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[sigma] (4)

from which follows:

Proposition 2: Let [X.sub.2] be the optimal exercise price of European options, which expire at date 2. Then,

(1) [partial][X.sub.2]/[partial][sigma] = W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[[sigma].sup.2] [greater than or equal to] 0,

(2) [partial][X.sub.2]/[partial]B = [q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])/[sigma] [greater than or equal to] 0,

(3) [partial][X.sub.2]/[partial]W = -1/[sigma] [less than or equal to] 0,

(4) [partial][X.sub.2]/[partial][q.sup.*] = [V'.sub.2]([q.sup.*]) + BF([q.sup.*])/[sigma] [greater than or equal to] 0. (12)

The first two results in Proposition 2 show how the optimal exercise price needs to be adjusted when other contractual variables are changed. The next two are comparative static results showing how the optimal exercise price should change in response to changes in exogenous parameters. In particular, the last part of the above proposition suggests a relationship between the exercise price and investment policy. First, to induce a more conservative policy (higher [q.sup.*]), the exercise price needs to be increased. The reason is the lower the exercise price, the more inclined the manager is to take a chance by choosing the risky project. While the safe project rewards the manager with B and sure gains from exercising options, the risky project entails the risk of losing B, which has to be compensated through expected gains from options. Thus, for a lower exercise price, the manager is more likely to go for the risky project. Second, holding W, B, and [sigma] fixed, the exercise price can be higher or lower th an the value of the firm at the grant date, depending again on what type of policy the owner wants. It is easy to see that [X.sub.2] = E(q)[V.sub.3] - (W - BE(q)))/([sigma]) when [q.sup.*] = 0, which is smaller than [V.sub.1](0) = E(q)[V.sub.3]. The implication is that options are more likely to be granted in the money if the owner wants a more aggressive policy. When [q.sup.*] increases, [X.sub.2] increases and [V.sub.1]([q.sup.*]) decreases, implying that more conservative policies are more likely to be accompanied by options granted out of the money. However, since the optimal exercise price depends on W, B, and [sigma], one cannot derive a systematic relationship between the exercise price and the value of the firm at the grant date. More on this will follow as we go on.

3.2. European options expiring at date 3

As before, suppose the owner wants the investment policy q[epsilon](0,1) to be implemented and consider a contract (B,[sigma],X) for which the exercise of options is restricted to date 3. Essentially, the same argument used for Lemma 2 leads to the following.

Lemma 3: Let (B,[sigma],X) be a contract with European options, which expire at date 3. Then, the owner can almost surely implement the investment policy q if and only if (1) X [less than or equal to] [V.sub.s] and (l - q)B = q[sigma]([V.sub.3] - X) - [sigma]([V.sub.s] - X) or (2) [V.sub.s]<X[less than or equal to][V.sub.3] and (1 - q) B = q[sigma]([V.sub.3] - X).

For a contract [SIGMA] = (B,[sigma],X), which is incentive compatible for the policy q, the manager's expected utility can be written as U(q,[SIGMA]) = [q,F(q) + [alpha](q)][[sigma]([V.sub.3] - X) + B]. Again, it is easy to show that, at the solution to the owner's optimization problem, the manager's individual rationality constraint is binding and the first-best investment policy is implemented.

Proposition 3: An optimal contract with European options, which expire at date 3, is given by [SIGMA] = (B, [sigma], X) such that satisfies the conditions in Lemma 3 for [q.sup.*] and [[q.sup.*] F([q.sup.*])+[alpha]([q.sup.*])][[sigma]([V.sub.3]-X)+B] = W.

Denote the optimal exercise price of European options, which expire at date 3, by [X.sub.3]. From Proposition 3, it is given by Eq. (5)

[X.sub.3] = [V.sub.3] - W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[sigma][[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]. (5)

The following result is immediate.

Proposition 4: Let [X.sub.3] be the optimal exercise price of European options, which expire at date 3. Then,

(1) [partial][X.sub.2]/[partial][sigma] = W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[[sigma].sup.2] [greater than or equal to] 0,

(2) [partial][X.sub.3]/[partial]B = 1/[sigma] [greater than or equal to] 0,

(3) [partial][X.sub.3]/[partial]W = -1/[sigma][[q.sup.*]F([q.sup.]) + [alpha]([q.sup.*])] [less than or equal to] 0,

(4) [partial][X.sub.3]/[partial][q.sup.*] = WF([q.sup.*])/[sigma][[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])].sup.2] [greater than or equal to] 0.

As before, the exercise price is positively related to the size of grants and the base salary and negatively related to the manager's reservation utility. As the optimal investment policy becomes more conservative (larger [q.sup.*], the exercise price needs to be increased.

3.3. Optimal exercise prices and maturity

The comparison of the two exercise prices provides a better understanding of the nature of incentives provided through stock options. Intuitively, the owner has more latitude in providing incentives to the manager by using options that expire at date 3 than those that expire at date 2. It is because the uncertainty about the return from the risky project is still held privately by the manager at date 2. If possible, the manager can disguise this private information and gain from exercising options at date 2. With options expiring at date 3, however, such a possibility no longer exists.

Lemma 4: (1) ([partial][X.sub.3])/([partial][sigma]) [greater than or equal to] ([partial][X.sub.2])/([partial][sigma]), (2) ([partial][X.sub.3])/([partial]B) [greater than or equal to] ([partial][X.sub.2]/([partial]B), (3) ([partial][X.sub.3])/([partial]W) [greater than or equal to] ([partial][X.sub.2]/([partial]W).

For a given increase in the size of a grant, the necessary increase in the exercise price is higher for options, which expire at date 3. Conversely, for a given increase in the exercise price, the owner needs to increase the grant size more for options, which expire at date 2. Similarly, a given increase in the base salary can be accompanied by a larger increase in the exercise price for options, which expire at date 3. Other things being equal, an increase in the manager's reservation utility needs to be accompanied by a decrease in the exercise price but by less for options, which expire at date 3. However, a direct comparison between [X.sub.2] and [X.sub.3] does not render a systematic relationship possible: depending on the underlying distribution of q and other variables, [X.sub.3] may or may not be larger than [X.sub.2]. This will be illustrated by examples after we study contracts with American options.

4. Optimal contracts with American options in unlevered firms

Suppose the owner wants the investment policy q [member of] (0,1) to be implemented and let [SIGMA] = (B,[sigma],X) be a contract for which options can be exercised either at date 2 or 3. Because American options can be exercised at either date, the incentive compatibility constraints are more complicated than with European options. On the other hand, as there are only two possible dates for exercise in the current model, the exercise price can be chosen to control the manager's exercise decision. (13) Using a similar argument as for European options, we first characterize the conditions for incentive compatibility.

Lemma 5: Let (B,[sigma],X) be a contract with American options. Then, the owner can almost surely implement the investment policy q if and only if

(1) X [less than or equal to] min {[V.sub.s], [V.sub.2](q) - q[V.sub.3]/1 - q} and (1 - q)B = [sigma][[V.sub.2](q) - [V.sub.s] or

(2) [V.sub.2](q) - q[V.sub.3] < X [less than or equal to] [V.sub.s] and (1 - q)B = q[sigma]([V.sub.3] - X) - [sigma]([V.sub.s] - X) or

(3) [V.sub.s] < X [less than or equal to] [V.sub.2](q) - q[V.sub.3]/1 - q and (1 - q)B = [sigma][[V.sub.2](q) - X] or

(4) max{[V.sub.s], [V.sub.2](q) - q[V.sub.3]/1 - q} < X [less than or equal to] [V.sub.3](q) and (1 - q)B = q[sigma]([V.sub.3] - X).

In cases (1) and (3) of Lemma 5, the manager who observed q[greater than or equal to]q selects R and exercises options at date 2. The manager's expected utility is then U(q,[SIGMA])= [sigma][[V.sub.2](q - X] +B [qF(q)+[alpha](q)], which the same as for European options that expire at date 2. In the other two cases, options are exercised at date 3, resulting in the manager's expected utility U(q,[SIGMA])=[qF(q)+[alpha](q)][[sigma]([V.sub.3]-X)+B], as in the contract with European options, which expire at date 3. As with European options, it is easy to show that, at the solution to the owner's optimization problem, the manager's individual rationality constraint is binding and the first-best investment policy is implemented. Thus, we have

Proposition 5: An optimal contract with American options is given by [SIGMA]= (B,[sigma], X) such that the conditions in Lemma 5 for [q.sup.*] are satisfied, and [sigma][[V.sub.2] ([q.sup.*]) -X]+B[[q.sup.*]F([q.sup.*])+ [alpha]([q.sup.*])]= W if X [less than or equal to] ([V.sub.2]([q.sup.*]) - [q.sup.*] [V.sub.3])/(l - [q.sup.*]) and [[q.sup.*]F([q.sup.*])+[alpha]([q.sup.*])][[sigma]([V.sub.3] - X) + B] = W if X [greater than or equal to] ([V.sub.2]([q.sup.*]) - [q.sup.*][V.sub.3])/(1 - [q.sup.*]).

From Proposition 5, the optimal exercise price of American options can be completely characterized. Denote C([q.sup.*])[equivalent to] ([V.sub.2]([q.sup.*]) - [q.sup.*] [V.sub.3])/(l - [q.sup.*]), the critical value on which the exercise decision of American options is based, and recall [X.sub.i] is the optimal exercise price of European options, which expire at date i, i=2, 3.

Lemma 6: The optimal exercise price of American options denoted by [X.sub.A] is given by (1) [X.sub.A]= [X.sub.2] if C([q.sup.*]) [greater than or equal to] max {[X.sub.2], [X.sub.3]}, (2)[X.sub.A]= [X.sub.3] if C([q.sup.*]) [less than or equal to] min{[X.sub.2],[X.sub.3]}, and (3) [X.sub.A]= [X.sub.2] or [X.sub.A] = [X.sub.3] if [X.sub.3] [greater than or equal to] C([q.sup.*]) [greater than or equal to] [X.sub.2].

Since the optimal exercise price of American options is equal to either of the optimal exercise prices of European options, American options can replicate the incentives provided through European options except for one important difference. With European options, the exercise price has to be smaller than the expected value of the firm at the expiration date of the options, which constrains the choice of other variables such as B and [sigma]. With American options, the exercise price can be set to induce the manager's exercise decision. Accordingly, there is more flexibility in choosing B and [sigma] as well as setting the exercise price of American options, compared to when only one type of European options is available.

As mentioned before, the exact relationships among [V.sub.1]([q.sup.*]), [X.sub.2], and [X.sub.3] depend on other variables (W B,[sigma]) and the distribution of q. Generally, both exercise prices can be made larger (smaller) than [V.sub.1] ([q.sup.*]) if B or [sigma] is increased (decreased). Even holding these variables fixed, the relationship between [X.sub.2] and [X.sub.3] depends on the distribution of q. We illustrate this with examples. Suppose first q has a uniform density, f(q)= 1, and fix [sigma]=0.5, B=5, W = 10, [pi]= 100, and vary I from 5 to 50. Then, an optimal policy changes from [q.sup.*]0.05 to 0.5. In this case, European options, which expire at date 2, are always granted in the money, while European options, which expire at date 3, are always granted out of the money. If American options are available and if a higher exercise price is desirable for some reasons, then the owner will set the exercise price along the path of [X.sub.3]. (14) This is shown in example 1 of Fig. 2. In example 1, the exercise price of American options is equal to [X.sub.3] to the right of the point labeled A. To the left of A, the exercise price of American options could be either [X.sub.2] or [X.sub.3]. In example 2, we hold all other variables fixed but change [sigma] to 0.2. Then, both European options are granted in the money, and the exercise price of American options is equal to [X.sub.2]. A similar effect can be seen from decreasing B or increasing W while holding other variables fixed.

In the next examples, q has a reverse unimodal density, f(q) = 2-4q if q [less than or equal to]0.5 = 4q - 2 if 0.5[less than or equal to]q[less than or equal to]1. In example 3, the values for other variables are the same as before: [sigma] = 0.5, B = 5, W = 10, [pi] = 100, and I varies from 5 to 50. This gives us a picture similar to example 1. In example 4, we change [sigma] to 0.2 and I from 35 to 80 while holding other variables the same, so that [q.sup.*] changes from 0.35 to 0.8. Then, both European options are granted in the money, and the one with shorter maturity can have a higher exercise price to the left of the point labeled [A.sub.1]. The exercise price of American options is equal to [A.sub.2] to the left of [A.sub.2] and [X.sub.3] to the right of [A.sub.3]. Between these two points, [X.sub.A] could be either [X.sub.2] or [X.sub.3]. Needless to say, American options are also granted in the money.

The above examples also illustrate possible problems with the near-universal practice of granting at-the-money options. If, for some reason, options are restricted to being granted only at the money, then the owner has only two contractual variables left to control the manager's incentives. While there are still many combinations of base salary and size of the option grant that can implement the desired investment policy, the set of optimal contracts shrinks dramatically. This can be seen from examples 1 and 3. Suppose the exercise price of European options with short maturity is exogenously set equal to the grant date value of the firm ([X.sub.2] = [V.sub.1]([q.sup.*])). Continue to assume that B is fixed at 5. Then, as [q.sup.*] changes from 0.05 to 0.5, [sigma] decreases from 3.01 to 0.52 in example 1. Since the grant size above 1 is not feasible in practice, at-the-money options are feasible only when 0.17 [less than or equal to] [q.sup.*] [less than or equal to]0.5. In example 3, at-the-money options are feasible only when 0.12 [less than or equal to] [q.sup.*] [less than or equal to]0.45. This is shown in Fig. 3. While changing the base salary in this case can admit more values of grant size, there is also a limit to this since the base salary has a lower bound due to limited liability and an upper bound due to firm's financial constraints. (15) The point is that exogenously fixing the option exercise price at the grant date stock price (or at any other levels for that matter) reduces the flexibility with which the manager's incentives can be controlled. While Hall and Murphy (2000) provide an example with risk-averse managers in which the optimal exercise price falls within a wide range that includes the stock price at the grant date, this is not a definitive justification for granting at-the-money options.

5. Optimal contracts with stock options in levered firms

The analysis in this section is more or less the same as before. The key difference is that the effective exercise price of options is the sum of the exercise price and the face value of debt. These two are substitutes in the contracting problem since the value of options now depends on the value of equity rather than the value of the firm. Therefore, we will mainly highlight differences and additional implications. Continue to denote [SIGMA] = (B,[sigma],X) to be a contract with stock options, and let 0<D<I be the face value of debt. The value of the firm still depends only on investment policies, although the value of equity now decreases by the face value of debt. However, debt will not appear in the owner's objective function insofar as it is optimally priced. It thus follows that it is still in the owner's interest to implement the first-best investment policy [q.sup.*]. In other words, debt does not distort investment decisions. (16)

Given D and [q.sup.*], the value of stocks at different dates is given by Eqs. (6) and (7)

[V.sub.S](D) [equivalent to] I - B - D, [V.sub.3](D) [equivalent to] [pi] - B - D, (6)

[V.sub.2]([q.sup.*],D) [equivalent to] [alpha]([q.sup.*])/1-F([q.sup.*]) [V.sub.3](D), [V.sub.1]([q.sup.*],D) [equivalent to] F([q.sup.*])[V.sub.S](D) + [alpha]([q.sup.*])[V.sub.3](D). (7)

Following similar steps as before, we can show that the optimal exercise prices of European options are

[X.sub.2](D) = [V.sub.2]([q.sup.*],D) - W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[sigma] (8)

[X.sub.3](D) = [V.sub.3](D) - W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[sigma][[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])] (9)

and the optimal exercise price of American options is as in Lemma 6 with C([q.sup.*]) replaced by C([q.sup.*],D)[equivalent to]([V.sub.2]([q.sup.*],D)-[q.sup.*] [V.sub.3](D))/(l - [q.sup.*]). This leads us to the relationship between the exercise prices and leverage.

Proposition 6. ([partial][X.sub.3](D))/([partial]D)=- 1 [less than or equal to] ([partial][X.sub.2](D))/([partial]D)=-([alpha]([q.sup.*]))/(l - F([q.sup.*])) [less than or equal to] 0.

Since the optimal exercise prices of European options are decreasing in leverage, so is the optimal exercise price of American options. The implication is that, other things equal, the exercise price of executive stock options should be lower for firms with higher leverage. The idea is simple. Higher leverage implies a lower exercise value of options, which requires downward adjustments of exercise prices relative to the value of equity if other components of executive compensation are held fixed. Put differently, higher leverage encourages the manager to take more risks, which needs to be mitigated through a lower exercise price of options. This last point becomes more evident when we see that the necessary adjustment of the optimal exercise price is larger for European options with longer maturity. For European options expiring at date 3, the face value of debt and the exercise price of options are perfect substitutes: since there is no more public information to be impounded in the stock price at date 3, a dollar increase in the face value of debt should be accompanied by a dollar decrease in the exercise price. For European options expiring at date 2, such an adjustment is smaller, discounted by the (date 2) probability (given by ([alpha]([q.sup.*]))/(1 - F([q.sup.*]))) that the firm will be solvent with the risky project. The fact that more levered firms grant options with lower exercise prices is consistent with the empirical findings of Garvey and Mawani (1999). However, there is no a priori reason why options should be granted at the money. As a matter of fact, any combinations of B, [sigma], and X satisfying Eq. (8) or Eq. (9) are optimal. As shown previously in Fig. 2, options could be granted in, at, or out of the money as long as necessary adjustments in B and [sigma] follow.

Finally, we examine the relationship between the size of the option grant and leverage. John and John (1993) show that the pay-to-shareholder-wealth sensitivity parameter of executive compensation should decrease in the face value of debt. (17) The main reason, it is argued, is that larger debt increases the manager's risk-shifting incentives if her interests are aligned with those of shareholders. This is true if shareholders want to implement the asset substitution policy [q.sup.~][equivalent to](I-D)/([pi]-D). However, as long as debt is priced optimally given the managerial contract and the expected investment policy, shareholders optimally implement the first-best investment policy. (18) In other words, shareholders are interested in maximizing the value of the firm rather than the value of equity. In this case, the argument that managerial contracts should be designed to sever the link between shareholder interests and managerial interests does not have bite. Moreover, for a fixed base salary and the ex ercise price of options, decreasing the size of the option grant in response to higher debt will decrease the manager's compensation and may violate the manager's individual rationality constraint. Of course, these other components will change as well if the amount of debt changes. To see these changes and to compare our results with John and John's, we restrict analysis to European options, which expire at date 3, while fixing the exercise price of options. As there are three variables (B, [sigma], and X) to be determined from two constraints (incentive compatibility and individual rationality), fixing one variable is necessary for comparative static purpose.

Proposition 7: Let (B(D), [sigma](D), [X.sub.3](D)) be an optimal contract with European options, which expire at date 3. When [X.sub.3](D) is held fixed, (1) ([partial][alpha](D))/([partial]D)[greater than or equal to]0 and ([partial]B(D))/([partial]D)=0 if [V.sub.s](D)<[X.sub.3](D)[less than or equal to][V.sub.3](D), (2) ([partial][sigma](D))/([partial]D)=0 and ([partial]B(D))/([partial]D)[greater than or equal to]0 if [X.sub.3](D)[less than or equal to] [V.sub.s](D).

The above proposition is in contrast to John and John (Proposition 5) where pay-to-shareholder-wealth sensitivity parameter declines in the face value of debt. The difference between the above result and John and John's stems mainly from the way compensation contracts can be structured. In John and John, the base salary is paid regardless of the financial situation of the firm, while insolvency incurs a penalty [PHI] to the manager, which is effectively taken as exogenous. Therefore, the only endogenous variable that enters the incentive constraint is the size of the option grant. Thus, it follows that a higher face value of debt should be accompanied by a smaller option grant, which has to be compensated through an increase in the base salary to keep the manager's individual rationality constraint satisfied. While it is debatable which of the two scenarios better describes actual practice of executive compensation, it seems reasonable to think that firms have various instruments with which they can affect th e incentives of their chief executive officers. Depending on how these instruments can be flexibly set, the pay-to-shareholder wealth sensitivity may or may not decrease in leverage. (19)

6. Empirical implications and discussions

This paper is primarily concerned with stock options as a means of compensating top management of corporations. Three types of contracts are studied: contracts with European options with short maturity, contracts with European options with long maturity, and contracts with American options. Particular attention is paid to the optimal exercise price of options. The main findings are (1) other things equal, the optimal exercise price increases in the size of the option grant and the base salary and decreases in leverage and the riskiness of a desired investment policy; (2) other things equal, the optimal exercise price of European options with a longer maturity should increase more for an increase in the base salary and the size of grant and decrease more for an increase in leverage than the one with a shorter maturity; (3) the optimal exercise price of American options is determined by the optimal exercise prices of European options with different maturities; (4) for a fixed exercise price, the size of the opt ion grant does not decrease in the face value of debt.

In drawing empirical implications from these findings, note first that the size of the option grant in this paper represents the fraction of the stock market value of the firm on which options are awarded. As such, it could be interpreted as the pay-to-shareholder-wealth sensitivity parameter as in John and John (1993). Given this interpretation, one can derive several testable hypotheses. The first is regarding the relationship between executive stock options and leverage. Across firms with a similar pay-to-shareholder-wealth sensitivity, the exercise price and leverage should be inversely related. On the other hand, given the fixed exercise price, the pay-to-shareholder-wealth sensitivity parameter should not decrease in leverage. The second is about the relationship between the optimal exercise price and the riskiness of a firm's investment policy. Firms in growth industries tend to have more investment opportunities where risk taking is an essential part of their success. Therefore, we could predict that, other things equal, the exercise price of executive stock options relative to the value of equity should be lower for firms in growth industries than those in mature industries. The third implication relates to the change in the level of exercise price for options with different maturities. When other contracting variables such as the base salary or the pay-to-shareholder-wealth sensitivity parameter change, the change in the exercise price should be larger for options with a longer maturity. The same is true in relation to the change in leverage.

Among the various issues left out in this paper, two points deserve special attention. The first is the usual practice of granting options at the money. (20) As was shown in the paper, there is no a priori reason why this has to be the case. In particular, fixing the exercise price equal to the stock price at the grant date reduces the number of available choice variables and hence the flexibility in using stock options relative to, for example, restricted stocks. While Hall and Murphy (2000) provide an example in which granting at-the-money options to risk-averse managers may be justified for 'reasonable' parameter values, we do not yet have a convincing explanation why this has to be the case. Some attribute the practice of granting options at the money to its favorable tax and accounting treatment (Murphy, 1999). However, this does not explain why options are not granted out of the money or why we have such a favorable treatment. The second is why options are gaining increasing popularity. While additional elements such as managers' risk aversion, information content of accounting earnings, or the degree of market efficiency may all be important for possible explanations of this trend, tax and accounting considerations are often deemed some of the more important factors. A better understanding of executive stock options therefore may elucidate whether or how tax and accounting rules should be modified. (21)

Acknowledgments

This paper was written during my visit to the Bank of Portugal and the University of Bristol. I am grateful to Bernardino Adao, Isabel Correia, Gerry Garvey, Ian Jewitt, InUck Park, and Pedro Teles for stimulating discussions, to the Bank of Portugal for financial support, to an anonymous referee, and to Diane Denis (guest editor) for constructive comments.

(*.) Australian Graduate School of Management, University of New South Wales, Sydney, NSW 2052, Australia. Tel.: +61-2-9931-9528; fax: +61-2-9313-7279.

E-mail address: c.choe@agsm.edu.au (C. Choe).

(1.) Their optimal contract includes stock ownership rather than stock options or, interpreted differently, stock options with zero exercise price.

(2.) Assuming that the return from the risky project is zero in a bad state simplifies notation. It allows us to ignore managerial compensation in that state. Alternatively, one could assume a positive return in that state but reduce managerial compensation by an exogenous penalty as in John and John (1993). This will complicate notation at no additional benefit.

(3.) In our paper, there is no a priori reason why debt has to be issued. Like John and John (1993), we do not combine the issues of optimal capital structure and optimal executive compensation. While this is an obvious avenue for future research, our main interest in this paper is how debt affects the nature of an optimal contract. For an example of research in which capital structure and managerial compensation are simultaneously determined, see Berkovitch, Israel, and Spiegel (1999) or Dybvig and Zender (1991).

(4.) In John and John (1993), the manager incurs a fixed private cost when the firm issues debt and subsequently becomes insolvent. This makes the manager partially risk averse, which is the potential source of the conflict of interests between the manager and the owner. In our model, a potential conflict of interests exists because the manager has to incur the loss of base salary when the firm defaults on its debt, whereas the base salary is always paid in John and John. While the current model can be easily modified in line with John and John's model, the motivation for the current model is provided in footnotes 2 and 7.

(5.) An obvious avenue for future research is to derive an optimal stock options contract incorporating the manager's risk aversion and the volatility of stock price. This will be especially important for the valuation of executive stock options. To our knowledge, Carpenter (2000) comes closest to this approach. As mentioned before, however, her focus is not on the derivation of an optimal contract but rather on the trading strategy of a risk-averse manager given a stock options contract.

(6.) For additional simplicity, we assume that information gathering is costless. In Choe (1999), the manager has to pay some private cost to observe q. This makes the incentive constraints more complicated but does not change the general conclusion of the paper.

(7.) For example, Choe (1999) shows the equivalence of three widely observed forms of contracts: standard bonus contracts in which bonus depends on the final return from the project, restricted stock ownership contracts where the manager is awarded stocks for which trading date is properly restricted, and various forms of stock option contracts.

(8.) When the return is zero, the base salary is reduced to zero. Salary reduction for CEOs in financially distressed firms is often observed in practice (Gilson & Vetsuypens, 1993). Reducing the base salary to some exogenously given positive amount instead of zero does not make any qualitative changes to our results.

(9.) We assume that the manager can costlessly borrow funds to exercise options and then immediately resell the stocks. Thus, options are a pure pecuniary incentive device in this paper. Issues such as control reallocation through option exercise are beyond the scope of this paper.

(10.) One can also consider European options that can be exercised before the manager announces the investment policy. Such a contract cannot be incentive compatible simply because the market does not have any additional information (other than the manager's contract) based on which to update the value of the firm. The same is true even when the manager has to pay the cost of information gathering. As long as information gathering is a private activity, the market again lacks additional information if options can be exercised before the announcement of the project choice. The point is that, for stock options contracts to be incentive compatible, the manager's private information needs to be communicated to the market through some public signals before the expiration of options. Otherwise, the market cannot serve the informational role in motivating the manager. This is also the reason why executive stock options in practice have waiting periods and vesting schedules.

(11.) Hereafter, we will ignore two polar investment policies of q=0 and q=1.

(12.) Since [q.sup.*] = I/[pi], a change in [q.sup.*] is due to either a change in [pi] or a change in I. For simplicity, suppose [pi] is fixed and a change in [q.sup.*] is entirely due to a change in I. Then, a change in [q.sup.*] leads to a change in [V.sub.s] but not [V.sub.3].

(13.) In practice, it will be virtually impossible to use the exercise price to time the manager's exercise decision as American options can be exercised at any point in time once they are vested. The point of this paper is that the optimal exercise price of American options can be approximated by using a discrete number of optimal exercise prices for European options.

(14.) For example, tax and accounting considerations could lead to a preference for a higher exercise price.

(15.) For example, B cannot be larger than I.

(16.) If debt is given exogenously instead of priced optimally, then there is the usual problem of asset substitution. That is, for any given D [greater than or equal to] 0, the owner would want to implement the asset substitution policy, q [equivalent to] I-D/[pi]-D [less than or equal to] [q.sup.*]. When D is optimally priced at the time of issue, however, such a problem does not exist. We do not mean to say that the asset substitution problem is not important. At any point in time, the firm may have some level of debt, which has been carried over from planning periods long before. As long as there is time inconsistency in pricing debt and covenants are not perfect, the asset substitution problem could be real.

(17.) Their model is a special case of ours when options are of European type with zero exercise price, which expire at date 3, and when there is an exogenous penalty to the manager for insolvency. In our model, the manager is penalized for insolvency through the loss of the base salary rather than a penalty. Their pay-to-shareholder-wealth sensitivity parameter is essentially the size of the option grant in this case.

(18.) This is also the case in John and John (see their Lemma 2, p. 963).

(19.) Yermack (1995) finds no significant association between financial leverage and incentives from stock option awards.

(20.) See Kole (1997) or Murphy (1999) for the case of the US and Garvey and Mawani (1999) for the case of Canada.

(21.) Empirical evidence on the relationship between the use of stock options and tax and accounting rules is mixed. Hall and Liebman (2000) and Yermack (1995) find that the effect of tax and accounting considerations on the use of executive stock options is not significant in the US, while Klassen and Mawani (1999) report a positive correlation between the two in Canada. This discrepancy may be due to the difference in tax treatment of executive stock options in the two countries.

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Appendix

Proof of Lemma 1: (1) follows from the definition of [V.sub.1](q). For (2), [V.sub.2](0)=[q.sup.*][V.sub.3] is straightforward. Applying L'Hopital's rule gives us [lim.sub.q[right arrow]1] [V.sub.2](q)=[V.sub.3]. Finally, [V'.sub.2](q)=([alpha]'(q[1 - F(q)] + [alpha](q)f(q))/[((1 - F(q)).sup.2])[V.sub.3] of which the numerator is f(q)([alpha](q) - q (1 - F(q))) = f(q) [[integral].sup.1.sub.q](q-q) f(q)dq [greater than or equal to] 0, [V'.sub.2](q) [greater than or equal to] 0.

Proof of Lemma 2: Incentive compatibility requires B+[sigma][{[V.sub.s]-X}.sup.+][greater than or equal to][sigma][{[V.sub.2](q)-X}.sup.+]+qB for all q<q and the reverse inequality for all q[greater than or equal to]q. Thus, we must have B+[sigma][{[V.sub.s]-X}.sup.+] = [sigma][{[V.sub.2](q)-X}.sup.+]+qB. Consider first X[less than or equal to][V.sub.s]. Then, options are in the money whether S or R are chosen at date 2. Given the condition (1), it is straightforward to show that the manager who observed q<p is strictly better off by choosing S, and the manager who observed q>q is strictly better off by choosing R. Given that q has a continuous support so that q=q is an event of measure zero, the claim follows. The same argument applies to the case [V.sub.s]<X[less than or equal to][V.sub.2](q). Finally, if X>[V.sub.2](q), then options are out of the money at date 2 regardless of the chosen project. Then, incentive compatibility holds only in the trivial case of B=O. However, this violates individual rational ity for the manager.

Proof of Lemma 3: The result follows from Propositions 2 and 4 if we could show that [q.sup.*]F([q.sup.*])+[alpha]([q.sup.*])[less than or equal to]1. However, 1-[q.sup.*]F([q.sup.*] - [alpha]([q.sup.*])= 1 - [[integral].sup.q.sup.*.sub.o] [q.sup.*]f(q)dq- [[integral].sup.1.sub.q.sup.*]qf(q)dq = [[integral].sup.q.sup.*.sub.o] (1- [q.sup.*])f(q)dq + [[integral].sup.1.sub.q.sup.*](1-q)f(q)dq[greater than or equal to]0.

Proof of Lemma 4: Consider first that the case q<q is observed. By choosing S, the manager can secure B+[sigma][{[V.sub.s]-X}.sup.+] whether options are exercised at date 2 or 3. Alternatively, the manager can choose R and exercise options at date 2 for the expected utility qB+[sigma][{[V.sub.2](q)-X}.sup.+] or exercise options at date 3 for the expected utility qB+q[sigma][{[V.sub.3]-X}.sup.+]. For a contract (B,[sigma],X) to be incentive compatible for the policy q, the manager who observed q<q should not have incentives to select R. As this has to be true for all q<q, we must have

B+[sigma][{[V.sub.s] - X}.sup.+] [greater than or equal to] sup {qB + [sigma][{[V.sub.2](q) - X}.sup.+], qB + q[sigma][{[V.sub.3] - X}.sup.+] : q < q}

= max {qB + [sigma][{[V.sub.2](q) - X}.sup.+], qB + q[sigma][{[V.sub.3] - X}.sup.+]}.

Suppose now q [greater than or equal to] q is observed. Then, the manager will select R as long as the expected utility from exercising options at either date is larger than B+[sigma][{[V.sub.s]-X}.sup.+]. This has to be true for all q[greater than or equal to]q. Thus, we have

B + [sigma][{[V.sub.s] - X}.sup.+] [less than or equal to] min {max {qB + [sigma][{[V.sub.2](q) - X}.sup.+], qB + q[sigma][{[V.sub.3] - X}.sup.+]} : q [greater than or equal to] q

= max {qB + [sigma][{[V.sub.2](q) - X}.sup.+], qB + q[sigma][{[V.sub.3] - X}.sup.+]}.

Combining the two leads us to

B + [sigma][{[V.sub.s] - X}.sup.+] = max {qB + [sigma][{[V.sub.2](q) - X}.sup.+], qB + q[sigma][{[V.sub.3] - X}.sup.+]}.

We will simplify the above conditions in several cases. First, suppose X [less than or equal to] Vs. Then, options are always in the money, and qB+[sigma][[V.sub.2](q)-X][greater than or equal to]qB+q[sigma]([V.sub.3]-X) if and only X[less than or equal to]([V.sub.2](q)-q[V.sub.3]/(1-q)[equivalent to]C(q). For future reference, it is necessary to compare C(q) with [V.sub.2](q). Noting

[V.sub.2](q)=([sigma](q))/(1-F(q))[V.sub.3] and C(q)=([sigma](q)-q[1-F(q)])/((1-q)[1-F(q)])[V.sub.3], it is easy to see that C(q)[less than or equal to][V.sub.2](q). Note also that [V.sub.s][less than or equal to][V.sub.2](q)[less than or equal to][V.sub.3]. If X[less than or equal to]min{[V.sub.s],C(q)}, then options are always exercised at date 2 once R is chosen. Thus, the above conditions become B+[sigma]([V.sub.s]-X)=qB+[sigma][[V.sub.2](q)-X] or (1-q)B=[sigma][[V.sub.2](q)-[V.sub.s]]. This give us (1). Suppose next C(q)<X[less than or equal to][V.sub.s], then options are exercised at date 3 when R is chosen, which leads us to (2). If [V.sub.s]<X[less than or equal to]C(q), then options are in the money only when R is chosen and will be exercised at date. 2. This give us (3). If max {[V.sub.s],C(q)}<X[less than or equal to][V.sub.3](q), then options are in the money only when R is chosen and will be exercised at date 3 regardless of whether or not [V.sub.2](q)[greater than or equal to]X. Thus, we have th e conditions in (4). Finally, if X>[V.sub.3], then options are always out of the money, in which case incentive compatibility holds only in the trivial case of B=0. However, this violates individual rationality for the manager.

Conversely, given these conditions, it is straightforward to check that the manager is strictly better off by implementing the policy q with the only exception of when q=q, in which case the manager is indifferent between S and R. However, this is an event of measure zero.

Proof of Lemma 5: From Proposition 5, the exercise decision of American options when R is chosen depends on the critical value C([q.sup.*]). Options are exercised at date 2 [date 3] if [X.sub.A][less than or equal to]C([q.sup.*]) [[X.sub.A][greater than or equal to]C([q.sup.*])] in which case the exercise price is equal to [X.sub.2] [[X.sub.3]], the optimal exercise price of European options, which expire at date 2 [date 3]. We need to consider four cases. First, if C([q.sup.*])[greater than or equal to]max{[X.sub.2],[X.sub.3]}, then options are exercised at date 2 and so [X.sub.A]=[X.sub.2]. Second, if C([q.sup.*])[less than or equal to]min{[X.sub.2],[X.sub.3]}, then options are exercised a date 3. Hence, [X.sub.A]=[X.sub.3]. Next, if [X.sub.3][greater than or equal to]C([q.sup.*])[greater than or equal to][X.sub.2], then the exercise price can be either [X.sub.2] in which case options are exercised at date 2 or [X.sub.3] when options are exercised at date 3. Finally, we need to show that the remaining case [X.sub.2][greater than or equal to]C([q.sup.*])[greater than or equal to][X.sub.3] with at least one strict inequality cannot happen. Suppose [X.sub.2]>[X.sub.3]. Then, using the expressions in (2) and (3), we are led to

[V.sub.3] - [V.sub.2]([q.sup.*]) < W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[sigma] (1/[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*]) - 1).

Then, we must have

[X.sub.2] - C([q.sup.*]) = [V.sub.2]([q.sup.*]) - W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[sigma] - [V.sub.2]([q.sup.*]) - [q.sup.*][V.sub.3]/1 - [q.sup.*]

= [q.sup.*]/1 - [q.sup.*][[V.sub.3] - [V.sub.2]([q.sup.*])] - W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[sigma]

< W - B[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])]/[sigma] ([q.sup.*][1 - F([q.sup.*])] - [alpha]([q.sup.*])/(1 - [q.sup.*])[[q.sup.*]F([q.sup.*]) + [alpha]([q.sup.*])])

[less than or equal to] 0

where the penultimate inequality is from the supposition [X.sub.2]>[X.sub.3] and the last inequality follows from [q.sup.*][1 - F([q.sup.*])][less than or equal to][alpha]([q.sup.*]). Thus if we have [X.sub.2]>[X.sub.3], then it must be that C([q.sup.*])>[X.sub.2].

Proof of Lemma 6: Suppose first [V.sub.s](D)<[X.sub.3][less than or equal to][V.sub.3](D). Then, the incentive compatibility constraint for [q.sup.*] can be written as (1-[q.sup.*])B=[q.sup.*][sigma][[V.sub.3](D)-[X.sub.3]] and the individual rationality constraint becomes [F([q.sup.*])+[alpha]([q.sup.*])]B+ [alpha]([q.sup.*])[sigma][[V.sub.3](D)-[X.sub.3]]= W. Totally differentiating these two and using Cramer's rule, we obtain ([partial][sigma])/([partial]D)=([sigma])/([V.sub.3](D)-[X.sub.3])[gr eater than or equal to]0 and ([partial]B)/([partial]D)=0. For the case [X.sub.3](D)[less than or equal to][V.sub.s](D), the incentive compatibility constraint is (1-[q.sup.*])B=[q.sup.*][sigma][[V.sub.3](D)-[X.sub.3]]+[sigma][[V.su b.s](D)-[X.sub.3]] and the individual rationality constraint is [F([q.sup.*])+[alpha]([q.sup.*])B+ [alpha]([q.sup.*])[sigma][[V.sub.3](D)-[X.sub.3]]+F([q.sup.*])[sigma] [[V.sub.s](D)-[X.sub.3]]=W. The same procedure leads to ([partial][sigma])/([partial]D)=0 and ([partial]B)/([partial]D)[gre ater than or equal to]0.

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Author: | Choe, Chongwoo |
---|---|

Publication: | Review of Financial Economics |

Article Type: | Statistical Data Included |

Geographic Code: | 1USA |

Date: | Jun 22, 2001 |

Words: | 12220 |

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