CONVEXITY, MAGNIFICATION, AND TRANSLATION: THE EFFECT OF MANAGERIAL OPTION‐BASED COMPENSATION ON CORPORATE CASH HOLDINGS

• Author: Ephraim Clark,Yacine Belghitar
• Published date: 01 June 2014
• DOI: http://doi.org/10.1111/jfir.12034
• Date: 01 June 2014

I

The growing interest in executive compensation and agency problems has spawned a large and increasing literature that looks at the option‐based incentives embedded in managerial compensation and examines their effect on corporate financial decisions. This literature is motivated by the assumption that the convex compensation schedule of an option can offset the concavity of the utility function of an underdiversified and overly risk averse manager and make him or her less risk averse (see, e.g., Haugen and Senbet ; Smith and Watts ; Smith and Stulz ; DeFusco, Johnson, and Zorn ; Billett, Mauer, and Zhang ). Ross (, p. 207), however, shows that “without further conditions on utility functions beyond monotonicity and risk aversion, this is not correct” and that, in fact, the opposite can be true. He shows that when examining the relation between option‐based incentives and corporate decision making three distinct effects should be considered. The first, called the convexity effect, is how the individual option‐based incentives affect managerial risk aversion at a given level of wealth. The second, called the magnification effect, reflects how different levels of exposure to the firm's share price affect managerial risk aversion. The third, called the translation effect, is how different levels of wealth affect managerial risk aversion. Failure to make these distinctions in empirical applications can lead to problems of model misspecification and erroneous inference. No empirical study (that we know of) in the extant literature on the relation between option‐based compensation and corporate financial decisions has formally made these distinctions.

Cash holdings are particularly adapted to the study of managerial incentives and corporate decision making because the decision to deploy or accumulate cash in excess of what is necessary to meet the needs of normal business transactions and any contractual obligations such as liquidity covenants is to a large extent at the discretion of managers with little scope for external scrutiny. Thus, given the propensity for accumulated cash to lower firm risk (Kim, Mauer, and Sherman ; Opler et al. ; Ozkan and Ozkan ), it is an excellent instrument for a manager seeking to implement personally advantageous corporate policies that are inconsistent with the risk preferences of shareholders. This is reflected in the fact that it has figured prominently in the recent compensation literature (e.g., Chava and Purnanandam ; Liu and Mauer ; Tong ).

As in several recent studies that use the concepts of vega and delta from option pricing theory to examine the relation between managerial financial decisions and managerial risk incentives (e.g., Guay ; Coles, Daniel, and Naveen ), there is no consistent evidence in the cash holdings literature of how and why vega and delta affect managerial behaviour. For instance, Chava and Purnanandam () document a significant positive relation between chief executive officer (CEO) delta and cash holdings, and an inverse relation between CEO vega and cash holdings. Tong () documents a significant negative relation between CEO delta and vega and cash holdings, whereas Liu and Mauer () provide evidence of a weak negative relation between delta and cash holdings but a strong positive relation between vega and cash holdings. The authors argue that their results support the argument of increased precautionary cash holdings to avoid costly external funding as well as a higher cost of debt to satisfy debt holders.

These conflicting arguments, and mixed results warrant further investigation and analysis, especially with respect to the apparent contradictory interpretations of vega and delta. For example, in the studies presented above, both vega and delta can either be risk‐inducing incentives or proxies for increased risk aversion. In Chava and Purnanandam () vega is the option incentive that reduces cash holdings and delta is the risk aversion proxy that increases them. In Liu and Mauer () vega is risk reducing and delta is risk inducing. In Tong () both vega and delta are risk‐inducing incentives.

We resolve some of the foregoing contradictions and extend the literature by developing an innovative empirical model based on the Ross () results that distinguish among the convexity effect, the magnification effect, and the translation effect. This makes it possible to identify the relevant option‐based incentives and their effect on managerial risk aversion and specify how each affects cash holdings. Our findings provide explanations for some of the discrepancies in the outstanding literature and have important implications for corporate policies and legislative regulations on executive compensation.

Using the Black–Scholes option pricer to estimate option values, deltas, and vegas, we show that the convexity and magnification effects reduce managerial risk aversion. We also identify the change in delta as a key variable and show that it is negatively related to the reduction in risk aversion. The implication is that the alignment of managerial and shareholder interests that comes about through lower risk aversion, which induces managers to lower cash holdings in favor of riskier assets, is reduced by increases in delta. As in Ross (), the translation effect depends on the managerial utility function. For utility functions with decreasing absolute risk aversion (DARA), the most common assumption, its effect is negative. Its effect is positive with increasing absolute risk aversion (IARA) and neutral with constant absolute risk aversion (CARA), and there is no reason why absolute risk aversion could not be constant or increasing. For example, IARA due to low diversification would explain the undiversified manager's decision to increase cash holdings at the expense of more profitable risky assets. We show that the effect of delta and vega on risk aversion depends on the managerial utility function. The implication is that the effect of delta and vega on cash holdings depends on whether the manager has DARA, IARA, or CARA.

In the main contribution of this article, we use UK data to provide strong empirical evidence that the change in delta is a key option‐based compensation incentive that is positively related to cash holdings. The implication is that this reflects the positive relation between the change in delta and risk aversion due to its effect on magnification and convexity. We also provide evidence that because of the significant negative relation between delta and vega and cash holdings, UK managers generally exhibit DARA. These results are robust with respect to alternative specifications, when incentives are extended to include all executive board members and when the sample is broken down according to different risk characteristics. We show that omitting the change in delta as an explanatory variable renders delta insignificant as an explanatory variable, an indication of potential misspecification in models where the change in delta is not included.

II

The Ross () analysis of the effect of convex compensation structures and managerial attitudes to risk are based on the Pratt () measure of absolute risk aversion and the assumption that managers are risk averse. Risk aversion means that each manager has a utility function u(w) satisfying the following conditions: where primes denote first and second derivatives with respect to wealth, denoted as w. Utility functions such as these are strictly concave. Pratt shows that maximizing the expected utility of a risk‐averse economic agent is approximately equal to: where A represents the degree of absolute risk aversion and measures how much the economic agent (in the case of Ross, the manager) dislikes the uncertainty he faces. A can be increasing in w (the first derivative with respect to w is positive), decreasing in w (the first derivative with respect to w is negative), or constant (the first derivative with respect to w is zero). This gives rise to the terminology increasing, decreasing, and constant absolute risk aversion (IARA, DARA, and CARA respectively).

Starting from here, Ross defines the derived utility function as u(f(x)) = v(x), where wealth (f) is a derivative security whose value depends on the value of an underlying asset denoted as x. The derived coefficient of absolute risk aversion is given as:

Ross shows that whether the derived utility function is more or less risk averse than the original depends on three effects—the convexity effect, the magnification effect, and the translation effect—where the net effect on risk aversion can be measured as:

The convexity effect measures how the payoff schedule of the wealth function affects managerial risk aversion. For example, the convexity of a payoff schedule like a call option clearly makes risky bets more desirable. It is defined as:

The magnification effect reflects how different levels of exposure to the firm's share price affect managerial risk aversion. It is defined as:

The translation effect measures how different levels of wealth affect managerial risk aversion as the fee schedule shifts or translates the evaluation of any bet to a different portion of the domain of the agent's utility function. Thus, it depends on whether the agent has DARA, IARA, or CARA. It is defined as:

As Ross () emphasizes, one merit of this decomposition is that these three effects are locally independent, in the sense that for a given utility function, each can vary without affecting the others. We now use this insight and these three effects in the context of the Black–Scholes option pricer to develop an empirical model for testing the effect of firm‐based compensation on managerial risk aversion.

Consider...