the design of executive stock options

MANAGERIAL AND DECISION ECONOMICS, VOL. 16,129-143 (1995)

The Design of Executive Stock Options

Leslie Young The University of Texas at Austin, USA and The Chinese University of Hong Kong

and Socorro M. Quintero Oklahoma City University, USA

A contract between manager and shareholder comprising salary plus options which are sometimes out of the money implies less risky managerial income but weaker incentives than a contract comprising salary plus stocks (or options which are always in the money) which leaves the manager as well off. Increasing the exercise price and the salary so that (1) there are some states where the options cannot be exercised but (2) the manager is as well off as before always leads initially to a reduction in effort which outweighs any gains from improved risk sharing, leaving the shareholder worse off.

effects of a high option exercise price, it is disap- pointing to some observers that most firms use INTRODUCTION

Executive contracts typically give managers some stake in their firm by relating their compensation to firm performance, most commonly by grants of stocks or stock options.’ The recent trend toward granting options may be explained, in part, by tax considerations: options generally do not create tax obligations until they are exercised, whereas stocks are taxed in the year that they are granted. It is also a common view that options provide stronger incentives for managerial performance than the underlying stocks since the manager must drive the stock price up beyond the exercise price before his or her options have value-although the possibility of the options having a zero value would seem to leave the manager more exposed to risk. Thus, at a recent roundtable on corporate structure and managerial incentives published in the Journal of Applied Corporate Fi- nance, Stewart argued against rewarding managers with options which were deep in the money because just in-the-money options leverage up the incentive effects of their investment, while Jensen suggested that raising the exercise price at a rate equal to the cost of capital would lead to a contract with a very high pay-for-performance sensitivity.’

Given these presumptions about the incentive

options in a manner which seems geared toward insulating managers from risk, at the cost of un- dermining the incentives that the options provide? Most executive options expire only after ten years, yet are issued at the money. Some firms also issue ‘discount options’ with exercise prices well below the market price of the stock at the date of issue.4 In an environment where stock prices are expected to trend upward, even executives making minimal efforts are unlikely to perceive any risk of their options expiring unexercised. Moreover, if the options become out of the money to any extent in the interim, then they are often re-issued at a lower exercise price. These corporate practices essentially eliminate the possibility of the options expiring unexercised, so the managers might as well be given the stock directly, were it not for the unfavorable tax consequence^.^ Such practices, apparently subverting the intended incentive effects of options, seem particularly per- verse in view of the tax advantages to issuing options which are out of the money.6

Thus, an important issue for compensation committees charged with designing incentive contracts is the choice of the exercise price of the options included in the contract. This paper ad- dresses this issue in a static principal-agent model

CCC 0143-6570/95 /020129- 15 0 1995 by John Wiley & Sons, Ltd.

130 L. YOUNG AND S. M. QUINTERO

in which shareholders use some combination of salary plus options to motivate a manager whose unobservable efforts shift the probability distribution of profits. We contrast the incentive and the risk-sharing properties of contracts which differ in terms of the option exercise price and the salary but leave the manager equally well off. The contrast between the use of stocks and of options which can expire out of the money is a special case of this analysis, since in a static model, stocks are equivalent to options with exercise prices low enough that they will always be exercised, plus a salary increase sufficient to pay for exercising the options.

In this framework we first consider the impact of the option exercise price on the riskiness of managerial income and on the manager’s effort. This leads to surprising conclusions: options with exercise prices which are high enough that they have some chance of being out of the money lead to less risky income but weaker incentives for the manager than the underlying stocks or options which are always in the money, when his salary is adjusted to leave him equally well off. Thus, although the way in which options are commonly used might appear to reduce the risk exposure of managers at the cost of reducing their performance incentives, closer analysis reveals that it has the opposite efsects. We then consider the tradeoff between these two aspects of the choice of option exercise price. Incentive considerations always dominate when the exercise price is low: if the exercise price is initially at the level which just leaves the options always in the money, then a small increase in this price, accompanied by a salary increase leaving the risk-averse manager as well off, will reduce the risk-neutral shareholder’s expected utility because he or she loses more from the reduction in managerial effort than he gains from the more efficient sharing of risk. Thus, the observed corporate policies of setting the exercise price so low that the options are always in the money cannot be improved upon, at least with small increases in the exercise price. Strikingly, this is true whatever the relative strengths of the manager’s aversion to effort and to income risk (apart from the extreme cases where his effort is completely inelastic or where he applies a maxmin criterion to income). Finally, we show that these conclusions remain valid when the number of options can be varied: even allowing for this possibility, a contract comprising salary plus options

which are always in the money remains locally Pareto optimal among the set of contracts comprising salary plus options. The intuition for these conclusions can be found in the third section of this paper.

THE LITERATURE

Discussions of managerial compensation in the press tend to focus on compensation packages of sensational magnitude, while offering anecdotal evidence that these sums are not warranted by corporate per f~rmance .~ Contrasting evidence is provided by Coughlan and Schmidt (19851, Mur- phy (1985) and Jensen and Murphy (1990) who find that executive compensation is generally positively related to stock market performance. There is also evidence on the positive impact of performance-related contracts on stock returns. Favorable stock market reaction to announcements of such contracts is reported by researchers such as Larcker (1983), Tehranian and Waegelein (19851, Brickley et al. (1985) and DeFusco et al. (1990). Tehranian and Waegelein also find positive abnormal returns ten months after the an- nouncement. Earlier, Masson (1971) found that firms whose managers’ financial rewards were more closely aligned with stockholder interests performed better in the stock market.

Despite this evidence that compensation committees reward executive performance and that performance contracts do improve the align- ment of the interests of managers and stockholders, empirical studies provide little guidance for choosing among the wide variety of performance contracts in use. Brickley et al. (1985) are unable to differentiate between stock market re- actions to the announcements of different types of plans, a result consistent with Smith and Watt’s (1983) view that firms face a variety of conflicts of interest between managers and other claim holders so that the compensation contracts which minimize agency costs can vary from firm to firm. Hite and Long (1982) argue that the switch from tax-qualified to non-qualified options in the 1970s was driven largely by tax changes, while Miller and Scholes (1982) cannot reject the hypothesis that executive compensation plans were enacted for tax purposes, although they recognize that incentive and tax considerations could jointly in- fluence the adoption of these plans. Lewellen et

THE DESIGN OF EXECUTIVE STOCK OPTIONS 131

al. (1987) find that the components of executive compensation vary across firms in a manner which is not readily explained by tax effects, although they do appear to be aimed at controlling for limited horizon and risk exposure problems.

The theoretical agency literature also provides limited guidance for the design of performance contracts. One branch, growing out of Jensen and Meckling (1976), considers how the problems arising from the separation of ownership from control-such as excessive perquisite consumption and risk taking by managers-can be alleviated by changes in the ownership structure or creative use of existing financial instruments.8 A second branch, pioneered by Mirrlees (1975, 1976), Har- ris and Raviv (19791, Holmstrom (1979) and Shave11 (19791, considers the problems arising because the manager’s effort is difficult to observe, necessitating performance contracts based on output, whose relationship to effort is subject to random disturbances. By modelling explicitly the manager’s choice of effort, this literature seeks to characterize the contract giving the Pareto-opti- ma1 trade-off between incentive and risk-sharing considerations. While this yields theoretical in- sights, the technical difficulty of the problem limits the practical guidance that emerges for the design of performance contracts. Explicit solutions are unavailable for cases of any generality and, in- deed, do not even exist in some natural cases. The solutions available for special cases are typically nonlinear and rather distant from the performance contracts actually used.g This paper uses the same framework as this second branch of the agency literature but, like the first branch, focuses on performance contracts which are close to those actually used.

RESTRICTED STOCK VERSUS STOCK OPTIONS

We use the standard principal-agent framework where a principal (the representative shareholder of a corporation) uses a performance contract to motivate effort by his agent (the manager) to increase corporate cash flows, which depend both on effort and on an unobservable random variable. Contracts which assure the manager an expected utility M * , reflecting his alternative em- ployment opportunities, are compared in terms of

the shareholder’s expected utility from the cash flows left after the manager has been paid. While it is possible, in principle, to calculate the managerial effort which maximizes the shareholder’s expected utility subject to the manager attaining his reservation utility, implementation of this first-best solution requires that the manager’s effort be accurately observed by the shareholder and directly rewarded using a ‘forcing contract’. We follow the bulk of the principal-agent literature in assuming that this is impossible” and restrict attention to contracts relating the manager’s compensation to the firm’s cash flows rather than to managerial effort itself; in particular, to contracts comprising salary plus options or stocks. As a result, both the risk sharing and the incentive aspects of the contract form an integral part of the analysis. After stating the model we treat these two aspects separately then compare their relative importance. We then discuss the determi- nation of contracts which are Pareto optimal in the set of contracts comprising salary plus options. Formal proofs are relegated to the Ap- pendix.

The Model

The firm’s time 1 cash flows e + 0 are assumed to depend additively on the manager’s effort e between time 0 and time 1 and on a non-negative state variable 0 with a probability density function 4(0) and cumulative probability function W e ) , embodying the time 0 beliefs about 0 held jointly by shareholder and manager. We maintain the following assumption on 4, which holds for all commonly used densities:

Assumption A 4 is positive over a connected interval I and is differentiable over this interval.

We shall assume that the shareholder has sufficient opportunities to diversify away firm-specific risks that he is concerned only with the expected value of his shares, i.e. he adopts a risk-neutral attitude. However, we shall generally assume that the manager cannot fully diversify away the risks inherent in having his human capital largely tied up in the firm and thus exhibits risk-averse behavior with respect to his income from the firm.” Let u(e, i) be the manager’s utility function over his time 1 income i and his effort e between time 0


and time 1. u is assumed to have continuous second derivatives with:

ui > 0, uii < 0, u, < 0, u,, < 0 and u,; = 0

where subscripts indicate partial differentiation with respect to the corresponding variable. Thus, the manager is non-satiated and risk averse with respect to income, effort is increasingly disliked by the manager and his utility function is additively separable in income and effort. Under these assumpt ions , which a r e s t a n d a r d in principal-agent models,’* it would be Pareto-optimal to let the shareholder bear all the risk in the firm’s cash flows, were it not necessary to motivate the manager by relating his reward to these cash flows.

At time 0, a contract is negotiated which relates the manager’s time 1 compensation to the firm’s time 1 cash flows. This compensation comprises a state-independent salary w plus call options on a fraction c (0 < c < 1) of the firm’s one share. This share constitutes a claim on its time 1 cash flows, net of the manager’s salary, so its price is 8 + e - w. Let x be the time 1 exercise price of the executive options. H (L) indexes the set of ‘high’ (‘low’) states 8 > ( < ) x + w - e where the options are in (out of) the money. The probabilities of low and high states are:

w L = a( x + w - e) and wH = 1 - @ ( x + w - e)

A: Manager

Thus, increased effort increases the probability of the options being in the money. The expectations operators over low and high states are:

and

E[..] =EL[..] +EH[..] denotes the unconditional expectations operator. When this is applied to the utility functions of the manager and shareholder or their derivatives it will be notationally convenient to omit their arguments, since the incomes appearing in those arguments have different functional forms in low and high states.

Figure 1 (A) depicts the manager’s time 1 income i ( 8 ) in various states 8. For low states, this is represented by the horizontal schedule:

iL(8) = w

For high states, it is represented by the upward- sloping schedule:

giving his salary w plus the value c(8 + e - w) - CT of his options when exercised.

B: Shareholder

t

x+w-e Figure 1. Incomes as a function of 8 when the contract comprises salary plus options.


Figure 1 (B) depicts the shareholder’s time 1 income I(0) in various states 8. For low states, this is represented by the schedule:

which gives the firm’s residual cash f l 0 ~ s . l ~ For high states, it is represented by the schedule:

giving his share (1 - cXe + e - w ) of the residual cash flows plus the manager’s payments cx for exercising his options.

In comparing (a) the contract comprising salary wa plus a fraction c of the firm’s one share with (b) the contract comprising salary w,, plus c options with exercise price x which are sometimes in the money, it is convenient to re-interpret contract (b) as comprising salary plus options which are always in the money, as follows. Let 5 be the lowest realization of the stock price under the effort e_ resulting from contract (a). Then (a) is equivalent to (a’) the contract comprising salary w = wa + cx/(l - c ) plus c options with exercise price x.14 Since (a’) and (b) both comprise a salary plus c options, it is simple to compare them when there are only marginal differences in their salaries and in the exercise prices of their options. However, when we re-interpret our conclusions to arrive at a comparison of contracts (a) and (b), we should bear in mind that, even if the salaries in contracts (a’) and (b) differ only marginally, the salaries in contracts (a) and (b) can differ substantially, since the salaries in contracts (a) and (a’) differ by cx/(l - c) . For example, in Proposi- tion 4 below, which shows that the contract (a) comprising salary plus c shares Pareto dominates the contract (b) comprising salary plus c options with a small probability of being out of the money, the salaries in (a) and (b) differ substantially-by almost cx/(l - c) .

Using Options Rather than Stocks Stabilizes the Manager‘s Income

Replacing the stocks in performance contract by options on those stocks might seem to expose the manager to more risk if the options can have zero value. However, to compensate for this possibility and keep the manager at his reservation utility,

his salary would have to increase by more than the amount required simply to offset the exercise price of the options. As a result, the manager’s income in low states would be higher than under the original, contract. However, the salary increase must leave his income lower in high states, else his expected utility would exceed its reservation level. Thus, the switch from stocks to options reduces the probabilities both of extremely high and of extremely low payoffs to the manager, i.e. his income becomes less risky. More formally:

Proposition 1: Consider:

(a) A contract comprising c shares plus a salary sufficient to bring the manager’s expected utility up to his reservation level M* and

(b) A contract comprising c options on the firm’s shares with an exercise price high enough that they are out of the money in some states, plus a salary sufficient to bring the manager’s expected utility up to his reservation level M*.

For a fixed effort by the manager, we can identify incomes i* and I* for the manager and shareholder such that the switch from (a) to (b): (i) Decreases the probability that the manager’s

income will be less than any specified I < i*, as well as the probability that it will exceed any specified I > i*.

(ii) Increases the probability that the shareholder’s income will be less than any specified I < I * , as well as the probability that it will exceed any specified I > I*.

Thus, the income of the manager is ‘less risky’ under contract (b) in that its probability distribution has less probability mass in its tails than under contract (a), while the income of the shareholder is ‘more risky‘ under contract (b) in that its probability distribution has more probability mass in its tails than under contract (a).” In this sense, a switch from stocks to options in the performance contract would stabilize the manager’s income and de-stabilize the shareholder’s income.

The above argument is readily extended to show that, given the manager’s effort, an increase in the exercise price of the options in a performance contract, accompanied by the increase in salary required to keep the manager at his reservation


utility, would stabilize the manager’s income and de-stabilize the shareholder’s income in the sense of Proposition 1. This suggests that if the shareholder is risk neutral while the manager is risk averse, then these changes would lead to a Pareto superior contract in the absence of incentive effects. This is confirmed by:

Proposition 2: Suppose that the shareholder is risk neutral with respect to income while the manager is risk averse and makes a fixed effort, so that risk-sharing considerations predominate. Given any contract comprising salary plus options, it is possible to increase the shareholder’s expected utility by increasing the exercise price of the options while increasing the manager’s salary so as to keep him at his reservation utility. Thus, the contract (a) comprising a salary wa plus a fraction c of the firm’s shares is Pareto dominated by some contract (b) comprising a salary wb > wa +cx/(l - c ) plus c options on those shares with an exercise price high enough that the options are out of the money in some states.

Before providing the intuition for this result, we explain our analytical approach, which will be used throughout the paper. Given the contract comprising salary w plus c options with exercise price x , we could consider the direction of Pareto improvements by seeing how the shareholder’s expected utility is affected by an increase A x in the exercise price, keeping the manager at his reservation utility by a ‘compensating’ increase Aw in his salary above the level w. However, the effects of the latter on the welfare of the con- tracting parties are complex, precluding clean results. Proposition 2 is proved by showing that the shareholder can benefit from an increase A x in x when the manager is kept at his reservation utility by an increase Aw in w plus an additional increase Aw(1 - c ) / c in x . The latter offsets the impact of Aw on the manager’s high-state income (1 - c ) w - cx + c( 8 + e) , so this ‘compensating’ combination of contract changes increases the manager’s income in low states by Aw but has no direct effect on his income in high states. Its effects on the expected utilities of shareholder and manager are therefore easily analyzed. We shall term this combination of contract changes ‘an increase A y in low-state salary’. The new contract thus comprises a fixed salary w + Aw plus c options with exercise price 5 + A x + Aw(1 - c ) / c .

The intuition for Proposition 2 is as follows. An increase Ax in x decreases the manager’s income and increases the shareholder’s income in all high states. As a result of the decrease in his income in high states, the risk-averse manager loses utility at an expected rate which is less than his marginal utility of income in low states times the probability of high states; the risk-neutral shareholder gains utility at an average rate equal to his marginal utility of income in low states times the probability of high states. It follows that the increase A y in the low-state salary required to keep the manager at this reservation utility leaves the shareholder with higher expected utility. Risk has been shifted to the party more inclined to bear it.

Recall that the contract comprising salary wa plus c shares is equivalent to a salary w = wa + cx/(l - c ) plus c options with an exercise price 5. The above argument implies that a Pareto superior contract can be obtained by an increase A x in the exercise price of the options (from the base 5) accompanied by the increase A y in the low- state salary (from the base w ) required to keep the manager at his reservation utility. These changes result in an exercise price higher than 5. Moreover, the increase in the salary w shifts the distribution of the stock price 8 + e - w to the left. Hence, the exercise price certainly ends up higher than the lowest realization of the stock price under the new contract. This completes the argument underlying Proposition 2.

Incentive Implications of Options Versus Stocks

The view that options provide stronger incentives for managerial effort than stocks would be valid if the effects of effort on the firm’s cash flows were known with certainty. Thus, if the manager is currently exerting the effort required to bring the stock price up to some level, then his effort could be increased by replacing his stocks by options with an exercise price higher than that level. The resulting contract is just one of the ‘forcing contracts’ which could be used in these circumstances to target the manager’s effort by making payment closely dependent upon the known outcome of that effort. A different conclusion would emerge, however, if the impact of effort on cash flows is not certain but depends on a random variable. The switch from stocks to options would then mean that the manager would not enjoy the bene-


fits of increased effort for some realizations of the random variable-namely, those leading to a stock price below the exercise price. This could attenuate his incentives.

We can formalize the above argument within our model, where the manager must choose his effort e while still uncertain about the random variable 8 determining the time 1 cash flows 8 + e resulting from this effort. Under a contract (c, w, x ) he chooses e to maximize his expected u ti1 i ty :

M ( e , c, w , x ) = E , [ u ( e , w)l + E , [ u ( e , w + c ( e + e - w - x ) ) l

(1)

The first- and second-order conditions for his choice of e = e(c , w, x ) are:16

Applying the Implicit Function Theorem to Eqn (2) and invoking the Fundamental Theorem of Calculus:

(1 - C ) E H [ C U , , ] - cu,(e, w ) c $ ( x + w - e ) e , =

-Met (4)

(5 1 ex =

The second-order condition (3) implies that these expressions have the signs of their numera- tors. Thus, an increase in the exercise price will increase (decrease) the manager’s effort if the manager is strongly (weakly) risk averse. The intuition is as follows. As displayed in Eqn (2), the marginal impact of effort on the manager’s expected utility comprises the disutility of the effort:

- C 2 E H [ U , , ] - cu, (e , w)c$(x + w - e ) - M e e

plus his expected marginal utility from the increase in the exercise value of his options:

An increase in the exercise price does not affect Eqn (6) since his utility is additively separable in income and effort. However, by lowering his income in high states, it raises his marginal utility of income and increases the marginal benefits (7) of the increased option income in high states that results from increased effort. This tends to increase effort, as indicated by the first term in the numerator of Eqn (5) . Moreover, for a given effort, the increase in x shifts rightward the borderline state 8 = x + w - e where the options are at the money. Hence, the options are in the money in fewer states so increased effort by the manager increases the value of his options in fewer states. This decreases effort in proportion to the marginal probability of the borderline state, as indicated by the second term in the numerator of Eqn (5) . If the manager were risk neutral, then only the second effect would operate and an increase in the exercise price would certainly decrease his effort. In this case, the polar opposite to that addressed in Proposition 2, we have:

Proposition 3: Suppose that both shareholder and manager are risk neutral with respect to income, so that incentive considerations predominate. Any performance contract comprising a salary plus options which are in the money in some realized states and out of the money in others is Pareto dominated by a contract comprising a lower salary plus options with a lower exercise price.

The Pareto improvement in contract design in Proposition 3 arises from a decrease - A x in the exercise price plus a decrease - A y in the low- state salary, as defined in the discussion of Propo- sition 2. The new contract thus comprises a fixed salary w - Aw plus c options with exercise price x - A x - Aw(1 - c)/c. The intuition for Proposi- tion 3 is as follows. A decrease - A x in exercise price accompanied by the decrease - A y in low- state salary which keeps the manager at his reservation utility will redistribute income between shareholder and manager in various states, but if both parties are risk neutral, then this redistribu- tion by itself has no net effect on the expected utility of either: any effect comes from changes in


the manager’s effort. Since the manager always chooses effort to maximize his expected utility (i.e. the first-order condition for effort always holds), any effort increase from marginal contract changes will not affect his expected utility, although it benefits the shareholder.

If the manager is risk neutral, then his marginal utility ui of income is affected neither by a reduction in exercise price nor by a reduction in low- state salary. However, for a given effort, both the initial decrease in exercise price x and the compensating decrease in low-state salary-comprising decreases in both w and x-shift leftward the borderline state 8 = x + w - e where the options are at the money. Hence, the options are in the money in more states and increased effort by the manager would increase the value of his options in more states. As a result, these contract changes increase effort in proportion to the marginal probability of the original borderline state. This marginal probability is positive, given the hypothesis that the original options are in the money in some realized states and out of the money in others, plus Assumption A that the density function of the state variable is positive over a connected interval. Thus, effort is certainly increased and the shareholder made better off.

Comparing the Risk Sharing and Incentive Aspects of Stocks and Options

Propositions 2 and 3 drive home the implications of risk sharing and incentive considerations for option design by treating the polar cases where just one of these considerations is operative. If effort is fixed, as in Proposition 2, then risk-sharing considerations call for the exercise price to be set at the highest realized stock price; if the manager is risk neutral, as in Proposition 3, then incentive considerations call for the exercise price to be set at the lowest realized stock price.” It is thus natural to conjecture that if both considerations are operative, because the manager both responds to incentives and is risk averse, then the Pareto-optimal exercise price would leave the options in the money in some states but out of the money in others. However, such a conjecture would be valid only if the negative incentive effects from increasing the exercise price become small relative to the gains from more efficient risk sharing when the exercise price is low. The following argument shows that, on the contrary, the

negative incentive effects become relatively large when the exercise price is low, so that it is not possible to Pareto improve upon a contract where the options are always in the money, at least with small increases in their exercise price. As shown in the Appendix, an increase in the

exercise price, accompanied by the increase in the low-state salary keeping the manager at his reservation utility, has the following marginal effect on the shareholder’s expected income:

(8)

The first term is positive when the manager is risk averse, reflecting the shareholder’s expected gain from the more efficient risk sharing identified in Proposition 2. The second-order condition (3) for effort requires that Me, < 0, so the second term is also positive, reflecting the shareholder’s expected gain from the increase in managerial effort that arises because an increase in exercise price lowers the manager’s income in high states, increasing his expected marginal utility (7) from the increase in option income from increased effort. The third term is negative, reflecting the shareholder’s expected loss from the decrease in the manager’s effort that emerged in Proposition 3 because both the increase in x and the compensating increase in low-state salary shift the borderline state rightward, so that his options now pay off in fewer states. This reduction in effort is proportional to the marginal probability &x+w-ee) of the borderline state. Since the impact of an increase in low-state salary on the expected utility of the manager is proportional to the probability rL = @ ( x + w - e) of a low state, the decrease in effort from the shifts in the borderline state under the postulated ‘compensated’ increase in the exercise price, captured in the third term in Eqn (81, is proportional to CP(X + w + e) /@(x + w - e), the ratio of the marginal probability of the borderline state to its cumulative probability. For an exercise price near the lowest realization g of the stock price under a


contract comprising salary plus stocks, the borderline state is close to the lowest realized state; for probability densities + ( d ) which are positive and differentiable over a connected interval (Assump- tion A), the ratio # J ( O ) / @ ( d ) becomes very large for 0 near the lowest realized state (see Eqn (Al2) in the Appendix). Thus, the negative third term in Eqn (8) dominates the first two terms. This leads a very general conclusion, holding whatever the relative strengths of the manager’s aversion to effort and to income risk:

Proposition 4 If the manager is risk averse but responds to incentives, then any contract comprising salary plus options with only a small chance of being out of the money is Pareto dominated by some contract comprising salary plus the underlying stocks (or options which are always in the money): the greater effort that the latter calls forth outweighs its less efficient sharing of risk.

Varying the Number of Options

Given a contract comprising salary w1 plus c1 options which are always in the money (i.e., with exercise price x), we might seek a Pareto superior contract by raising the exercise price of the options, keeping the manager at his reservation utility not by increasing his salary but by increasing the number of options. However, even if we found a Pareto superior contract comprising a salary w 2 plus c2 options with exercise price x 2 > 5, where c2 > cl, Proposition 4-which holds whatever the value of c-would indicate that we could do better still by fixing the number of options at c2 but reducing their exercise price below x 2 again, while decreasing the salary to keep the manager at his reservation utility. These two changes in the contract can be viewed as steps in the search for a Pareto-optimal contract, so it is illuminating to reformulate Proposition 4 as a characterization of a local Pareto optimum in the set of contracts comprising salary plus options:

Proposition 4’: If the manager is risk averse but responds to incentives, then there is an x1 > 5 such that any contract (c, w, x ) comprising salary w plus c 5 1 options with exercise price x E [s xl] which satisfies the first-order necessary conditions for a Pareto-optimal contract must, in fact, set x = x. Thus, any such contract is equiva-

lent to one comprising a salary w’= w -cx/ (1 - c) plus c shares.

Proposition 4’ confirms that a contract comprising salary w plus c options which are always in the money is locally Pareto optimal, even when the number of options can be varied. The Pareto-optimal combination of salary plus options of this type can be determined as follows. In the first-order conditions for w and c in the maximization of expected shareholder income S subject to the constraint on the manager’s expected utility M, elimination of the Kuhn-Tucker multi- plier associated with this constraint yields:

By taking a second-order approximation to the manager’s utility function in this equation, as in Stiglitz (1975, pp. 559-60), we can relate the Pareto-optimal level of c to the riskiness of output and the manager’s relative risk aversion and supply elasticity of effort. w is then determined by the constraint on the manager’s expected utility. This procedure characterizes a contract which is locally Pareto optimal in the set of contracts comprising salary plus options.

By contrast, the globally Pareto optimal contract comprising salary plus options is difficult to characterize since the objective and constraint functions are complex, non-concaue functions of the contract parameters w, x and c. Even if a solution were found for particular functional forms, implementation would require detailed information about the manager’s preference parameters which the shareholder is unlikely to possess. In the absence of such information, a given compensated increase in x above 5 might leave the shareholder worse off. This is certainly true for some increases in x , since the exercise price x = is locally Pareto optimal. Thus, it is no surprise to find, as noted in the Introduction, that in practice most managerial contracts comprise salary plus stocks or options which are always in the money.

CONCLUSIONS

Jensen and Murphy (1990) have suggested that the ‘implicit regulation’ of stockholders and pub- lic opinion imposes a significant constraint on


compensation committees. Furthermore, the tax code encourages the issue of options which are out of the money since some restrictions on the number that can be issued to an executive without a tax penalty are then waived. In face of these political and tax considerations, the use of options with high exercise prices and performance targets would appear to be an attractive way to tie managerial rewards visibly to the shareholder benefits from their efforts. Therefore, the observed reluctance of compensation committees to adopt such policies might, at first, suggest that they are more interested in insulating managers from risk than in providing performance incentives. Our analysis indicates that opposite is true: in a risky environment, the use of options which have low exercise prices can strengthen performance incentives, at the cost of increasing the risk exposure of managers.

Our model assumes that effort by the manager shifts the entire distribution of cash flows, without affecting their riskiness. This assumption seems appropriate for situations where the primary objective of the performance contract is to encour- age effective management of the firm’s routine operations. However, as emphasized in the first branch of the agency literature cited in the second section of this paper, managers also make strategic choices among investment projects which affect the riskiness of the firm’s cash flows, with implications for both stock and bond holders. It should therefore be interesting to make Pareto comparisons of performance contracts when executive decisions can modify the riskiness of the firm’s cash flows, as well as their level. Our model might also be extended to many periods to incor- porate the arrival of information about the impact of effort on cash flows between the time that the contract is signed and the times that the manager chooses his effort. The trade-offs between risk bearing and incentive considerations in such a model could help explain the way in which firms adjust the terms of performance contracts as information arrives, for example by reis- suing executive options at lower exercise prices when the stock price falls.

APPENDIX

Proof of Proposition 1

Consider the contract (a) comprising a salary w,

plus c shares. Let g be the lowest realization of the stock price under the effort g resulting from this contract, which is equivalent to one comprising salary w = w, + cg/(l - c) plus c options with exercise price g. In state 8, the incomes of the manager and shareholder under this contract are:

and

Now suppose that the options with exercise price 5 are replaced by options with exercise price x b > g. Since the latter options have zero value in some states, this reduces the manager’s expected utility so, to keep him at his reservation utility M*, his salary must rise above w. However, this salary increase must be less than c(xb - x)/ (1 - c), else in all states his income would be higher than under the original contract, implying an expected utility higher than his reservation level. Thus, the salary wb under the new contract (b) satisfies:

At the manager’s original effort g, the incomes of manager and shareholder under the new contract are:

Figure 2 depicts the incomes of manager and, shareholder under the two contracts. For 8 close to 0, 8 + g - w is close to g, so i , ( 8 ) is close to w. For such 8, ib (w)=wb. Thus, Eqn (Al) implies that i b ( 8 ) > i , ( 8 ) for 8 close to 0. Eqn (Al) also implies that ib(8) < i , (8 ) for 8 >xb + wb - g. Hence, if 8* is the unique state where the manager’s income is the same under the two contracts, i.e.

THE DESIGN OF EXECUTIVE STOCK OFTIONS 139

A: Manager

A B: Shareholder

i,(e)

I (e l a

e* Figure 2. Incomes when the salary comprises (a) salary plus stock and (b) salary plus options.

then pensated’ increase in the exercise price x , i.e. a marginal increase in x , accompanied by the increase in low-state salary keeping the manager at his reservation utility. The direct marginal effect of an increase in low-state salary on a variable is given by the partial differential operator:

e $ e* implies i ,(e) $ i , (e)

and

i,,(e) $ I , ( e )

For i > i , ( e* ) = i , (e*) , let e b ( i ) be the unique a(.) 1 - c a( . ) +-- c d x (.)Y = dw value of 0 such that i b ( e ) = i and let O,( i ) be the unique value of 0 such that i , (8) = i. As can be seen in Fig. 2, e,(i) < eb(i ) . Hence: Changes in the exercise price or the low-state

salary can affect a variable not only directly but also indirectly via the induced change in effort, as indicated by the total differential operators:

prob(i,(o) < i } = prob(O< e , ( i ) }

> prob(8 < e,(i)) = prob{i,( 8 < i )

d(.) - a(.> + a(.> dx - d x de ex --- -

Applying similar reasoning to the other cases, we conclude that:

i 2 < a i ( e * ) = i@*) implies

prob tib( 8) < i )

$ prob ( i , ( e ) < i }

I $ I,< e* = I,( e* implies Applying these operators to the expected utility of the manager and the expected income of the shareholder: prob ( I , ( e ) < I }

$ prOb(Ib(8) < I } M e , c , w , x ) = E , [ u ( e , w) l Setting i* = i , ( O * ) = i , ( B * ) and I* =I,(e*)=

I,( e * ) yields Proposition 1. + ~ , [ ~ ( e , w + c ( B + e - w - x ) ) l

Proof of Proposition 2 S(e , c, w , x ) =EL[ e + e - w ]

We calculate the marginal effects of a ‘com- + E , [ ( l - c > ( e + e - w ) +a]


- .rrLui(e, w ) dM -- dY

- C7rH + ITL + (1 - c ) T H l e x (A5) dS dx _ -

- (.rrL + (1 - c).rr,le, - .rrL (A61 dS _ - dY

In these calculations, the effects of changes in x or in y via changes in the limits of integration of the operators EL[.] and EH[. ] cancel in the manner explained in note 14. Moreover, in Eqns (A3) and (M), the first-order condition (2) for the manager’s effort has been used to eliminate the effects of the induced changes in effort on his expected utility, i.e. the Envelope Theorem has been applied.

If effort e is fixed, then Eqns (A3)-(A6) imply that:

If the manager is risk neutral with respect to income, so that uii = 0, then:

dS d S dM dM + ( x + w - e ) cE[ui l ---- .- dx dy dx = L Me,

(A10)

The hypothesis that the options are in the money in some realized states and out of the money in others, plus Assumption A implies that + ( x + w - e ) > 0. The second-order condition (3) for effort then implies that Eqn (A101 is negative. Thus, a compensated decrease in the exercise price increases the shareholder’s expected income.

Since the manager is risk averse, u is strictly Proof of Proposition 4 concave and ui(e, w + c(8 + e - w -x)) < ui(e , w ) for 8 > x + w - e. Taking expectations over these high states, we conclude that E H [ U , ] < .rrHui(e, w ) so Eqn (A7) is positive. Thus, an increase in the exercise price, accompanied by the increase (dM/dx)/(dM/dy) in low-state salary required to keep the manager at his reservation utility leaves the shareholder better off. These

5 is the lowest stock price realized when the manager’s effort g is that chosen under a contract comprising a salary w’ plus a fraction c of the firm’s stock. Without loss of generality, we can suppose that 0 is the left end of the interval over which the density 4(0) is positive, i.e. 0 is the lowest realization of 8. Then:

contract changes raise both the salary and the exercise price of the option.

5 = g - w forw = w‘ + cx/(l- c ) 0

Our assumption that u(e, i) is twice differentiable

Proofs of Eqn (8) and Proposition 3 ensures that e(c, w , x) is a continuous function of x for x 2 x. Hence: -

If the manager’s effort responds to incentives, then by Eqns (4) and (5): e ( c , w , x) -, g

and

(A81 x + w - e ( c , w , x ) + O as x + x c2EH[u,i] + cui(e , w)+(x + w - e )

Me, e, =

1 - c u i ( e , w)c$(x + w - e ) (A9) Therefore e y = e e , + - ex = C

Substituting Eqns (A8) and (AS) into (Mi) and


To show that this limit is infinite, note that if +(8) has a positive limit as 8 approaches 0, then for a sufficiently small 8, @(8) will be smaller than any specified number and 4(f3)/@(8) will exceed any specified number. If +(8) has a limit of 0 as 8 approaches 0, then Assumption A implies that + is positively sloped in an interval just to the right of 0. For any 8 in this interval:

Therefore +(8)/@(8) > l /8 . Again, for a sufficiently small 8, +( 8 )/@( 8 will exceed any specified number. We conclude that for densities satis- fying Assumption A:

(A121

Since the first and second derivatives of u are continuous, the first two terms in Eqn (8) have a finite positive limit as x + x, while, in the third term, cE[ui]/Me, has a strictly negative limit. Therefore Eqns (81, (Al l ) and (A12) imply that:

Thus, given any contract comprising salary plus options with exercise price x > 5, a small compensated reduction in the exercise price increases the shareholder’s expected income. 0

Proof of Proposition 4

The Pareto-optimal contract solves the con- strained maximization problem:

max S(e(c, w, x ) , c, w, x ) subject to (c, w , x )

M ( e ( c , w, x ) , c, w , x ) 2 M* and x 2 x (A131

The Kuhn-Tucker Theorem implies that, at a Pareto optimum, we can associate multipliers A and p with the two constraints which satisfy the complementary slackness conditions:

M2M*[A>O] and x 2 5[p10] (A141

such that the following first-order conditions hold:

(A19

(A161

(A17)

d S d M d x - A d x + p d S d M d w - A d w d S d M dc - ’dc

--

--

_ -

Equations (A151, (A161 and definition (A2) of changes in the low-state salary imply that:

(A18) _ - d S d M 1 - c -A- - - dy dy c

Equations (A151 and (A181 imply that:

by Eqns (A31 and (A4) (A19)

Equation (A121 implies that (dS /dx) - (dS/dyXdM/dx)/(dM/dy) is negative for all x in some interval [x, x l ] where x1 > 5. The term in braces on the right side of Eqn (A19) is positive for c 2 1, so Eqn (A191 implies that p < 0 for any contract (c, w, x ) with x E [x, x , ] and c I 1 which satisfies the first-order conditions (A15HA17). The complementary slackness condition (A14) on p then implies that the constraint x 2 5 is binding for any such contract.

This argument does not invoke the first-order condition (A171 for c, but this does not jeopardize its validity, since the first-order conditions (A151 and (A161 which are invoked are necessary for a Pareto optimum: they must hold at whatever value of c obtains there. W

Acknowledgements We are grateful to Glenn Boyle, Stu Gilson, Jeff Kanter, Richard Kimball, Ken Wiles and two referees of this journal for helpful comments.

NOTES

1. According to Cook‘s (1989a, b) most recent surveys of the 200 largest industrial companies and the 200 largest service companies, 91% use stock options in


2.

3.

4.

5.

6.

7. 8. 9.

10.

11.

12.

their executive performance contracts, 49% use stock appreciation rights and 42% use restricted stock. Stock appreciation rights entitle the executive to a sum proportional to the appreciation in the stock from the date that he signed the contract and thus resemble options with an exercise price equal to the market price at that date. Restricted stock cannot be sold before a specified date. See comments by Stewart and Jensen in Apple- baum er al. (1990, pp. 22-3) ‘Only after executives exercise their options do they become truly owners. Until then they have no capital at risk, and if the stock sags, they can expect a new grant the next year at a lower price. Some companies even cancel “underwater” options and replace them with new ones ...’ (Stewart, 1990, p. 94). Cook (1989, p. 8), Lambert et al. (1989, p. 412), Steward (1990, p. 93). The CEO of Walt Disney received 500000 stock options (out of 2 million) with an exercise price $10 over the price ($71.06) of Disney stock at the date that he signed the contract. However, few other executives are issued stock options which are out of the money (Byme et al., 1990, p. 61). A grant of stock would save the manager from paying the exercise price of the options but if these savings accrue in all states, then they can be offset by a lower salary. Under Section 422(a) of the Economic Recovery Act 1981, the tax advantages accorded Incentive Stock Options are available only if the executives owns less than 10% of his firm’s common stock, but this condition is waived for options with an exercise price more than 10% above the market price when issued. e.g. Stewart (1990) and Byrne et al. (1990). e.g. Haugen and Senbet (1981). See the survey by Hart and Holmstrom (1987). See Jensen and Murphy (1990, p. 2261, Holmstrom (1979, p. 553) and Shavell (1979, p. 56). Harris and Raviv (1979) and Holmstrom (1979) treat the case where effort can be observed imperfectly. Managers might also hedge the stocks and options in their compensation packages with boutique derivatives written by brokerage houses. The extreme case where derivative security markets are sufficiently complete that the manager also be- haves in a risk-neutral fashion is analytically straightforward. If his effort level is fixed, as in Proposition 2 below, then it becomes a matter of indifference which combination of salary and stocks or options is included in the compensation pack- age: all that is required is that the contract bring the manager up to his reservation utility. The case where both manager and shareholder are risk neutral and the manager’s effort responds to incentives is addressed directly in Proposition 3. Harris and Raviv (19791, Holmstrom (19791, Shavell (1979), Grossman and Hart (1983). Proposition 3 assumes that the manager is risk neutral to isolate

the implications of incentive considerations for option design.

13. To keep notation simple, we implicitly assume that 8 + e - w is always positive. Recognition of limited liability would result in slightly more complicated formulae, but would not affect our conclusions.

14. Under both contracts, the manager’s income is (1 - c)w - cx + c(8+ e) for any 8 and e, so he chooses the same e under both contracts. Hence, the incomes of manager and shareholder are the same under both.

15. In the terminology used by Rothschild and Stiglitz (1970) in developing their concept of ‘increasing risk‘, the cumulative probability functions of the incomes of the manager under contracts (a) and (b) have the ‘single crossing property’, as do the incomes of the manager under these two contracts. We cannot directly apply their concept of ‘increasing risk’ as a mean-preserving spread of the underlying probability distribution since we are comparing pairs of distributions with different means. However, we could apply this concept indirectly by first normalizing all incomes to have a unit mean. A simple extension of the proof of Proposition 1 shows that the normalized income of the manager under contract (a) is a mean-preserving spread of his normalized income under contract (b), the re- verse being true for the shareholder.

16. A change in e affects the borderline value, x + w - e, of 8 which forms the upper limit of integration of the expectations operator over low states, EL[.. .] , and the lower limit of integration of the expectations operator over high states, EH[ ...I. By the Fundamental Theorem of Calculus, the marginal effects of e on M via changes in these two limits are -u(e , w)C$(x + w - e) and u(e, w)C$(x + w - e) , respectively. These terms cancel, leading to the expression for Me in Eqn (2). Similar cancellations occur in the calculations leading to Eqns (3)-(5).

17. These conclusions are foreshadowed in the well- known result that if the manager is risk averse and exerts a fixed effort while the shareholder is risk neutral, then the globally Pareto efficient contract assigns the shareholder the firm’s cash flows, less a fixed payment to the manager which keeps him at his reservation utility. However, if both shareholder and manager are risk neutral, but the manager responds to incentives, then the globally Pareto efficient contract assigns the manager the firm’s cash flows, less a fixed payment to the shareholder which keeps the manager at his reservation utility (Shavell, 1979, Proposition 1).

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