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Article Excerpt
Summary. In this paper we are concerned with the performance of stock option contracts in the provision of managerial incentives. In our simple framework, we restrict the space of contracts available to the principal to those conformed by a fixed payment and a call option on the firm's stock. As compared to the fixed payment and the option grant, we find that the strike price plays an intermediate role in the provision of insurance and incentives. We also develop some methods for the calibration of a standard principal-agent model based upon observed CEO earnings schedules and the volatility of the firm's value in the stock market. These methods are useful to address some important issues such as the performance of stock option contracts, the degree of risk aversion compatible with current earnings profiles and the sensitivity of compensation to changes in firm's characteristics.
Keywords and Phrases: Stock option contract, Optimal contract, CEO earnings schedule, Stock price.
JEL Classification Numbers: C6, D83.
1 Introduction
Most principal-agent models yield rather complex payment schedules, and it seems quite challenging to test the predictions of these models. (3) In practice there is a widespread use of simple compensation schemes such as linear or piecewise linear (stock option) contracts. As is well understood (e.g., Stiglitz, 1991) these simple contracts can only be generated under some specialized assumptions that seem unsuitable for the construction of economic theories. Hence, a question that follows naturally is if simpler contracts can be optimal or nearly optimal. We address this question in a standard principal agent framework in which we evaluate the cost of restricting compensation to a family of contracts composed of a fixed salary and a call option on the company's stock. Our analysis relies on numerical techniques, since the optimal contract cannot be calculated analytically. We also develop some methods for the calibration of a principal agent model. These methods should be of independent interest.
In our simple setting, the principal represents the shareholders of a firm, and the agent plays the role of the CEO. The random outcome, realized after the agent's action choice, is interpreted as the firm's stock price. One major task is to understand the incentives provided by each component of the stock option compensation scheme. Clearly, the fixed payment discourages high effort. The strike price and the option grant (4) perform better as instruments to induce high effort, although the option grant turns out to be the most powerful instrument. Thus, a marginal increment in cost devoted to lower the strike price has a greater impact on the provision of incentives but it yields less utility than a marginal increment devoted to increase the fixed component, and it has a lesser impact on the provision of incentives but it yields more utility than a marginal increment invested in the option grant.
At a later stage in our analysis we address the issue of optimality of stock option contracts. Our main purpose is to compare the cost of the stock option contract with the cost of the optimal contract. First, some environments where the welfare loss approaches zero are discussed. As one may anticipate, if the agent is risk neutral, there exists a continuum of stock option contracts which are all first-best optimal. Moreover, in some rather limited environments the stock option contract may be the outcome of a general maximization program even if the agent is risk averse. Second, we develop a methodology for the calibration of a general principal-agent model, and we carry out several quantitative experiments to evaluate the performance of the stock option contract. The most critical issue in the calibration of a principal-agent model is the specification of the manager's effort technology, that is, the quantification of the effects on the firm's stock value of the effort exerted by the manager (cf. Haubrich, 1994; Haubrich and Popova, 1998; Margiotta and Miller, 2000). We assess these effects using a new approach based upon observed CEO earnings schedules and the volatility of the firm's stock value. This calibration exercise should be of interest for further quantitative experiments in models of agency.
In an influential paper, Jensen and Murphy (1990) claimed that observed contracts are fairly insensitive to changes in shareholders' wealth. Hall and Liebman (1998) report a higher pay-performance sensitivity in more recent years. We analyze this empirical issue using our quantitative framework. We find that observed CEO earnings schedules can be generated by plausible risk aversion coefficients, which may range between and 7. These numerical experiments confirm that observed pay schedules can be generated by principal-agent models. We then analyze the performance of the stock option contract. In all numerical exercises, our stock option contract fares well in cost with the optimal contract, even though both payoff profiles may take on different shapes, and may be quite distant from each other. The small extra cost associated with restricting the choice space to three-parameter contracts may account for the widespread use of stock option contracts.
Finally, we perform some other comparative numerical exercises of certain general interest. Thus, from some controlled numerical exercises it appears that fixing arbitrarily the strike price within a reasonable range does not result in a substantial additional cost of the contract. As discussed above, in the optimal provision of insurance and incentives the strike price p plays an intermediate role between the option grant q and the fixed wage T. Hence, deviations from the optimal p may be partially offset by a suitable reshuffling of both q and T. This seems key to explain why in these contracts most options are granted at the money (i.e., at the grant-date price), since these options receive a better fiscal treatment and the cost of departing from the optimal p may be minor. Also, we find that the optimal wage schedule is steeper if the gains to the agent of exerting low effort are higher and if low effort is less detrimental to the firm. These results will be useful to understand how the stock option contract varies with shifts in the mean and variance of the firm's returns, and with changes in the rank and productivity of the agent.
The rest of the present article is organized as follows. In the next section we describe our environment along with the basic assumptions about the primitives. Section 3 studies qualitative properties of the stock option contract. Section 4 presents several numerical experiments related to the calibration of the model and the aforementioned comparative analysis exercises. In Section 5 we conclude and discuss some ideas for future research. All proofs are gathered in a separate appendix.
2 The framework of analysis
The present section begins with a standard principal-agent model with fully-contingent payment schemes. This model will serve as a useful benchmark to evaluate some dimensions of the stock option contract in the provision of insurance and incentives and in a further quantitative study of its welfare properties.
2.1 A standard principal-agent model
We portray a stylized model of agency (e.g., see Mas-Collel et al., 1995, Ch. 14; Hart and Holmstrom, 1987) for the managing of a corporation or a production activity. The observable outcome y is assumed to be the market price of the firm's stock, and often called the value of the firm. The principal, representing the shareholders, offers a contract specifying a compensation scheme, w(dot), and a recommended action, a. If the agent accepts the contract and chooses action a, the value of the firm is realized according to the cumulative distribution F([dot]; a). Then, the payoff of the principal is given by y - w(y), and the payoff of the agent is v(w(y)) - a, where v(dot) - a is the agent's utility function representing preferences over monetary payments and effort.
The details concerning the fundamentals of the contracting problem in this simple static framework are specified by the following assumptions:
Assumption 2.1 The agent selects an action from a finite set A = {[a.sub.1],..., [a.sub.N]}, where [a.sub.i] [member of] R for all i = 1,..., N and [a.sub.1] > ... > [a.sub.N].
Assumption 2.2 The value of the firm, y, is a continuous random variable, which takes on values in the interval Y = [[y.bar], [bar.y]] [??] [R.sub.+]. For each fixed a [member of] A, function F([dot]; a) is the probability distribution of y with continuous density f([dot]; a) such that f(y; a) > for all y [member of] Y.
Assumption 2.3 Strong monotone likelihood ratio property (SMLRP): For all pairs of actions [a.sub.i] and [a.sub.j], if [a.sub.i] < [a.sub.j] then the ratio f(y; [a.sub.i])/f(y; [a.sub.j]) is non-increasing in y.
Assumption 2.4 Concavity of distribution function condition (CDFC): For any triple of actions a*, a', and a", if a* = [lambda]a' + (1 - [lambda])a", for [lambda] [member of] (0, 1), then it must hold that F(y; a*) < [lambda]F(y; a') + (1 - [lambda])F(y; a") for all y [member of] int(Y).
Assumption 2.5 The agent's utility of consumption, c, and effort, a, is given by v(c) - a, where v : R [right arrow] R is bounded, strictly increasing, strictly concave, and continuously differentiable.
These assumptions are commonly used in the literature of principal-agent models (e.g., see Grossman and Hart, 1983), and ensure that the optimal contract is monotone in the observable outcome. Regarding Assumption 2.1, our analysis is easily extended to a countable number of actions, and it seems plausible to consider a continuum convex domain of actions. Note that in Assumption 2.2 the observable outcome y is a continuous variable. This condition will be convenient for the computation and calibration of the model. Assumptions 2.3 and 2.4 are crucial to guarantee the monotonicity of the optimal contract, and to build the analysis on the comparison of two carefully chosen actions. Finally, the separable form of the utility function in Assumption 2.5 will simplify the analysis considerably but it is not needed for most of our results.
The expected utility of the agent under compensation scheme w(dot) and action a is defined as
V (w; a) = [[integral].sub.Y][v(w(y)) - a]f(y; a)dy.
The agent will accept the contract offer and take action a if two conditions are satisfied. The...
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