Optimal CEO Compensation with Search: Theory and Empirical Evidence

The following post comes to us from Melanie Cao of the Finance Area at York University and Rong Wang of the Finance Group at Singapore Management University.

Two issues concerning executive compensation deserve particular attention. The first is how a firm’s risk affects the executive’s pay-to-performance sensitivity (hereafter PPS), i.e., the ratio of incentive pay to firm performance. Standard agency models predict that the PPS does not change with the firm’s risk if the agent is risk neutral and decreases with the firm’s risk if the agent is risk averse. Notable examples are Bolton and Dewatripont (2005), Holmstrom (1982), and Murphy (1999). In contrast to this theoretical prediction, the empirical evidence on the effect of the firm’s risk on the PPS is ambiguous. For example, Core and Guay (1999) and Oyer and Shaefer (2005) find a positive relationship while Aggarwal and Samwick (1999) document a negative relationship.

The second issue is the large increase in CEO compensation along with the increase in firm size in the past three decades. This large increase has generated an intense debate in the public and the academia on whether CEOs are over-compensated. Although the increase in firm value contributed partly to the increase in CEO pay, a closer look at the data reveals two notable features (see section IV for a detailed description of the data). First, incentive pay, which is the predominant component of CEO pay, has increased more rapidly than the increase in firm value. From 1994 to 2009, median incentive pay increased by 244% in real terms, compared with a 40% increase in median firm value, and its share in total pay increased from 41% to 78.8%. Second, and related to the first feature, total CEO pay outpaced firm value. The ratio between CEO pay and firm value increased from $1.59 in 1994 to $1.73 in 2009 per a thousand dollars. These features suggest that the key to understanding the increase in CEO compensation is to understand what factors determine the PPS.

We believe that two factors are intuitively important for the PPS, both arising from the notion that executive contracts should be designed to maximize firm value in a market economy. One is job mobility of CEOs. When different firms compete for CEOs, each firm has incentive to design contracts to increase the retention probability. Thus, changes in the market conditions can affect the PPS by affecting the severity of competition for CEOs. Another factor is the composition of risks faced by a firm. By switching from one firm to another, a CEO can change the amount of idiosyncratic risk to which he is exposed, but not the aggregate systematic risk since all firms face the similar systematic risk. Thus, the PPS should depend on the two types of risks differently. To incorporate these factors, in our paper, Optimal CEO Compensation with Search: Theory and Empirical Evidence, forthcoming in the Journal of Finance, we integrate an agency model into search theory to determine incentive contracts in a market equilibrium, and then empirically evaluate the model. Search theory endogenizes CEOs’ and firms’ outside options and enables us to distinguish idiosyncratic risks from systematic risks. The integrated model captures the intuitive mechanism that competition among firms for CEOs affects incentive contracts in the equilibrium by affecting a CEO’s incentive to participate in a firm. To isolate the effect of competition on the incentive contract from the effect of risk aversion, we focus on risk-neutral and effort-averse CEOs.

In our model, there are many firms and CEOs. In each period, a firm’s output depends on an aggregate shock, an idiosyncratic shock, and the CEO’s effort. The aggregate shock is publicly observed while the idiosyncratic shock, measuring the match quality between the firm and the CEO at a particular time, is the CEO’s private information. The firm offers an incentive contract, which can be contingent on its output and the aggregate shock, but not directly on the idiosyncratic shock and the CEO’s effort. The CEO decides whether to accept the offer after observing the idiosyncratic shock. If he quits, he can search for a new job. Due to the competition among firms, a CEO’s outside option depends on the probability of getting a new job and the compensation at the new job. This link between a CEO’s outside option and other firms’ contracts implies that a market equilibrium must determine all firms’ contracts and agents’ outside options simultaneously. We focus on a stationary and symmetric equilibrium where all firms offer the same type of contracts.

To determine the equilibrium, we first analyze an individual firm’s optimal contract under arbitrarily fixed outside options for CEOs and firms. The value of the outside option to a participant of a contract is defined as the difference between the value of quitting and searching and the future value of staying in the contractual relationship. We prove that the optimal PPS is less than one, in spite of a risk neutral CEO. This result arises because a CEO can choose whether or not to quit after privately observing the idiosyncratic shock. If the idiosyncratic shock is contractible or the CEO is forbidden to quit, the optimal contract would set the PPS to one, as is well known in agency models with a risk-neutral agent. Such a contract would align the CEO’s effort perfectly with the objective of maximizing the joint surplus of the match, and the firm would vary the base wage with the idiosyncratic shock to obtain the maximum share of the joint surplus. However, because the idiosyncratic shock is the CEO’s private information, it is not feasible to make the base wage contingent on such a shock. The CEO will choose to stay to obtain the high payoff when the idiosyncratic shock is high, and will quit to insulate himself from the low payoff when the shock is low. In this setting, it is optimal for the firm to set the PPS below one in order to get part of the high surplus when the idiosyncratic shock is high and compensate for the low payoff when the CEO quits. In fact, the firm chooses the PPS and the base wage to obtain the optimal trade-off between the retention probability and the expected profit conditional on retention.

Once the optimal PPS is below one, it can be affected by the aggregate and idiosyncratic risks. When the outside options are arbitrarily fixed, the two risks have the same qualitative effect on the PPS. Specifically, they affect the PPS negatively if and only if the joint value of the CEO’s and the firm’s outside options for the current period is positive. This effect of the risks on the PPS arises from a new mechanism in our model whereby a firm makes a trade-off between retention and profit conditional on retention, not from risk aversion as in standard agency models cited above. To see this, let us examine the case where the risk (either aggregate risk or idiosyncratic risk) increases. An increase in the risk increases a firm’s expected profit conditional on retention, which increases a firm’s incentive to retain the CEO. However, when the value of CEO’s outside option is high, the probability of retaining the CEO is low. Therefore, it is optimal for the firm to increase retention probability by increasing the base wage and reducing the PPS. The opposite holds when the risk decreases. Thus, overall, the PPS is negatively related to aggregate and idiosyncratic risks under positive outside options.

Next, we endogenize the outside options, determine the market equilibrium and explore new predictions of the equilibrium. In contrast to the partial equilibrium with fixed outside options, the two risks now have opposite effects on the PPS in the market equilibrium. This difference stems from two externalities due to endogenizing outside options: One is the interactions between firms’ contracts and the other is the dependence of the matching probability on the contracts through competitive entry of vacancies. Under these externalities, the joint value of a firm’s and a CEO’s outside options increases with the idiosyncratic risk and decreases with the aggregate risk. The intuition is as follows. An increase in the idiosyncratic risk increases the dispersion in match value, which induces both the CEO and the firm to search for a new match with a higher expected profit. In contrast, an increase in the aggregate risk increases the profit of all firms uniformly, and thus reduces the motivation for the CEO and the firm to search. This link between the risks and the outside options serves as a bridge between the risks and the PPS. Specifically, an increase in the idiosyncratic risk increases the CEO’s outside option, which intensifies the competition among firms for CEOs. As a result, firms must increase the equilibrium PPS so that the CEO can capture more of the surplus and is less likely to quit. In contrast, an increase in the aggregate risk reduces the CEO’s incentive to search, which weakens the competition and triggers firms to lower the equilibrium PPS. It should be noted that the opposite effect of two risks on the optimal PPS is unique to the market equilibrium with search. When the outside options are exogenous as commonly assumed in the agency literature, the PPS responds to the two risks in the same direction, as discussed above.

Finally, we empirically test two new predictions of our model. First, the equilibrium PPS depends negatively on the systematic risk and positively on the idiosyncratic risk. Second, because the PPS responds to the two risks differently, so does the ratio of a CEO’s total compensation to firm value in the equilibrium. This ratio depends negatively on the systematic risk and positively on the idiosyncratic risks. The empirical tests find robust support for these predictions. We contribute to the labor search literature (e.g., Mortensen and Pissarides (1994)) by being the first to integrate incentive contracts into a search model to examine CEO compensation and empirically test its implications. To the principal-agent literature (e.g., Bolton and Dewatripont (2005) and references therein), our paper contributes in three dimensions. First, we explicitly model CEOs’ quitting decisions and study incentive contracts that induce both optimal effort and optimal retention. Second, we endogenously determine the effects of market conditions on a CEO’s outside option. Third, we analyze the optimal contract in a dynamic equilibrium in which firms interact in the CEO job market. This dynamic equilibrium structure contrasts with typical agency models, such as Jin (2002) and Garvey and Milbourn (2003). They analyze the optimal contract in a single agent-firm pair static setting where the joint outside option value is assumed to be positive since the CEO has a positive reservation utility while the principal has zero reservation value. The common conclusion of these studies is that the PPS decreases with the idiosyncratic risk. The negative effect of the idiosyncratic risk on the PPS is consistent with our partial equilibrium analysis when the joint outside option value is positive. However, in a dynamic equilibrium setting like our model, the joint outside option value for the current period can be either positive or negative. Specifically, when the future value of continuing the match is higher than the value of breaking up the match, the joint outside option value for the current period is negative, in which case the PPS increases with the idiosyncratic risk. More importantly, our market equilibrium analysis shows that the equilibrium PPS increases with the idiosyncratic risk due to contracting externalities. This new result offers a possible explanation for the mixed evidence on the empirical relationship between a firm’s total risk and the PPS.

Our paper is also related to Oyer (2004) and Edmans, Gabaix and Landier (2009). Similar to our model, Oyer (2004) recognizes that an agent may choose not to participate in a contract in certain states of the world. However, he assumes that the outside option is exogenous and does not study a market equilibrium. Moreover, he studies broad-based stock option plans for lower-ranked workers and abstracts from the effort-inducing mechanism on the ground that such plans have limited incentive effects on workers. Edmans, Gabaix and Landier (2009) use the same assumption as ours that the shocks and the agent’s effort are multiplicative in a firm’s profit function. However, they study a different mechanism (i.e., positive assortative matching) and their objective is to explain the negative relationship between the CEO’s effective equity stake and firm size. They do not analyze the effects of risks on the PPS.

The full paper is available for download here.

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