Leverage decision and manager compensation with choice of effort and volatility Abel Cadenillasa, Jakˇsa Cvitani´cb, Fernando Zapateroc aUniversity of Alberta, Department of Mathematical and Statistical Sciences, Edmonton, Alberta, Canada bUSC, Departments of Mathematics and Economics, Los Angeles, CA 90089, USA cUSC, Marshall School of Business, FBE, Los Angeles, CA 90089, USA (Received 10 October 2002; accepted: 16 June 2003) ———————————————————————————————————— Abstract We study the incentive effects of granting levered or unlevered stock to a risk-averse manager. The stock is granted by risk-neutral shareholders who choose leverage and compensation level. The manager applies costly effort and selects the level of volatility, both of which affect expected return. The results are driven by the attempt of the risk-neutral shareholders to maximize the value of their claims net of the compensation package. We consider a dynamic setting and find that levered stock is optimal for high-type managers, firms with high momentum, large firms, and firms for which additional volatility only implies a modest increase in expected return. JEL Classification: C61, G39 Keywords: Capital structure, principal-agent, stochastic control ———————————————————————————————————— The research of A. Cadenillas was supported by the Social Sciences and Humanities Research Council of Canada grant 410-2000-0631. The research of J. Cvitani´c was supported in part by the National Science Foundation, under Grant NSF-DMS-00-99549. This paper is based on a previous paper titled “Executive Stock Options with Effort Disutility and Choice of Volatility.” We are especially grateful to Kevin Murphy for many detailed comments on that paper. We also thank H´ector Chade, Li Jin, Fulvio Ortu, Rafael Repullo, Manuel Santos, Luigi Zingales, and seminar participants at HEC (Montreal), NYU, McGill, USC, Princeton, Wharton, BI (Oslo), Copenhagen School of Business, California Institute of Technology, Florida International University, CEMFI, ASU, Humboldt Universit¨at zu Berlin, the Winter 2000 Meeting of the Canadian Mathematical Society, the 2001 Canada-China Math Congress, and the Second World Congress of the Bachelier Finance Society (2002) for comments. Last, but not least, many comments and suggestions of an anonymous referee (including a key note with regard to the focus of the paper) are gratefully acknowledged. Existing errors are our sole responsibility. 1. Introduction Modigliani and Miller (1958) establish that if the firm’s investment policy is fixed and if there are no taxes and contracting costs, then the firm’s value is independent of itscapitalstructure. Sincethatseminalresult, financialeconomistshaverelaxedsome of these assumptions to analyze optimal capital structure. For example, Modigliani and Miller (1963) show that because debt provides the firm a tax shield, it may be of benefit for firms to issue debt. However, debt may also induce some costs, for example bankruptcy costs (Kraus and Litzenberger, 1973) or agency costs (Jensen and Meckling, 1976). By balancing such costs and benefits, one should then be able to determine optimal capital structure. WhileModiglianiandMillerapplytheirargumentstofinancingpolicy, theirresult generallyappliestoallfirmpolicies. Inthispaper, weexaminethejointdetermination of compensation and leverage decisions in a dynamic framework, notably, we analyze the impact of the choice of effort and volatility on the value of the firm. We ignore taxes and bankruptcy costs to keep the analysis simple. We emphasize that because weuseadynamicsettingourconclusionsdependdirectlyontheabilityofthemanager to adjust the level of effort in a state contingent way. The specific problem we consider is as follows. A risk-averse manager whose com- pensation consists exclusively of (possibly) levered stock has to choose optimal levels of effort and volatility. Effort is costly, unlike volatility. The manager’s compensation is determined by existing risk-neutral shareholders, where existing shareholders are outside investors who decide how many shares of stock to give to the manager and how much debt to issue. Both of these shareholder decisions are interlaced: the ob- jective of the risk-neutral shareholders is to maximize the expected value of the firm (net of the value of the compensation package) by selecting the levels of compensation and leverage that will provide optimal incentives to the manager. Our setting sheds light on the factors that determine some of the leverage ratios observed in practice. Our main conclusions are the following. The optimal policy is to grant stock with high leverage to “good” managers (those who are more efficient in affecting firm value througheffort)andstockwithlowleverageto“bad”managers. Themomentumofthe firm’svaluehasthesameeffect, highmomentumisequivalenttoahigh-typemanager. Also, low leverage seems to be preferable for firms for which extra volatility translates into very high expected return. The former result is obviously difficult to test (a distinction between good and bad managers has to be established). The latter result seemstobeconsistentwiththeempiricalfindingsofSmithandWatts(1992)whofind that leverage tends to be lower in firms with good growth prospects. Finally, optimal leverage will be lower for smaller companies than large companies. Overall, our model is in the spirit of the results of Lewellen (2002), which finds empirical evidence of the interaction between risk-aversion of the managers and financing decisions. Our model isalsocomplementarytotheresultsofMorellec(2003), whopresentsacomprehensive model with taxes, possible default, and agency problems, and concludes that optimal 1 leverage tends to be low, as observed in practice. We also mention here the work of Bertrand and Schoar (2002), who document that manager characteristics help explain the financing decision, the premise of this paper. Obviously, our paper is related to the executive compensation literature (see Mur- phy, 1999, for an extensive review) and the incentive effects of different types of com- pensation packages, as studied in Jensen and Murphy (1990). Given the option-like features of levered stock, our paper complements the literature on executive stock options and their incentive effects. In particular, the choice of an optimal strike price in Hall and Murphy (2000, 2002) is a problem similar to the choice of optimal leverage that we consider in this paper. On the technical side, our paper is related to the growing literature on dynamic principal-agent models. This literature started with Holmstrom and Milgrom (1987) and includes Scha¨ttler and Sung (1993), Sung (1995), Bolton and Harris (2001), and Ou-Yang (2003). Carpenter (2000) also considers a dynamic setting. The paper is structured as follows. In Section 2 we describe the dynamics of firm valueandtheeffectsofthemanager’sactionsonthosedynamics. Wealsocharacterize the objectives of the two players in our model: the manager and the firm. In Section 3 we derive the optimal effort and volatility choices of the manager, as well as the optimal number of shares and leverage to be chosen by the firm. In Section 4 we present some numerical examples and derive the main economic results of the paper. We close this work with some conclusions. 2. The setting We consider the problem of a risk-neutral shareholder who wants to maximize the value of its claims net of the compensation package of the firm’s manager. As in Stulz (1990) and Morellec and Smith (2003), we assume that shareholders have decision rights over financing and compensation policy. We consider a dynamic setting. As we will see below, the economic results we derive are essentially dynamic and do not obtain in the equivalent static setting. We discuss the static setting and its solution at the end of the next section. 2.1. Firm value and stock Following Black and Scholes (1973) and Merton (1974), the literature on com- pensation policy and capital structure decisions typically assumes that the value of unlevered assets is governed by a geometric Brownian motion dV t = µdt+σdW , (1) t V t with starting value V . The process (W ) is a standard Brownian motion process 0 t t≥0 and µ and σ are constant parameters. In our framework, the manager can affect both 2 the growth rate and volatility of this value process. In particular, we assume that (V ) satisfies t t≥0 dV = µV dt+δu dt+αv V dt+v V dW , (2) t t t t t t t t where u and v are adapted stochastic processes chosen by the manager, δ ∈ [0,∞) (cid:82) (cid:82) and α ∈ (0,∞). We assume that E[ T |u |2dt] < ∞ and E[ T |v V |2dt] < ∞, 0 t 0 t t where T is the horizon of the manager, as described later. The constant parameter µ is exogenous, and we interpret it as the momentum of the firm’s value, which the manager of the firm cannot affect. In an equilibrium setting, this parameter would have to be related to the interest rate prevailing in the economy. The control u is the level of effort the manager expends in running the company. The higher the effort u, the higher the expected value of the firm. We will assume later that the manager’s effort is costly. On the other hand, the choice of v is equivalent to the choice of the firm’s volatility, although it also has an impact on the expected value. We interpret the choice of v as a choice of projects, such that the manager selects a project or strategy from a set characterized by different levels of risk. The parameter α is a measure of the benefits associated with taking more risk, and is a characteristic of the firm. One possible interpretation of α (and, potentially, a way to estimate it empirically) is that it is the slope of the equivalent of the Capital Market Line which results from all the projects available to the firm (more about the distinction between firm-specific and market risk can be found below). Arguably, α will be high for those companies with relatively better growth prospects, such as new companies. This interpretation will be important when we compare our results with the empirical observations of Smith and Watts (1992). The parameter δ is a measure of the impact of the manager’s effort on the value of the firm, and can be interpreted as an indicator of the type (quality) of the manager. Carpenter (2000) studies the optimal choice of v for the case in which δ = 0. We emphasize that this is a partial equilibrium setting. We do not compare the dynamics of the value of our firm with the dynamics of other firms. If all the investors had perfect information, the value of our firm would be decided in equilibrium and the manager would not be able to influence the drift independent of the volatility; i.e., δ would be equal to zero. As discussed earlier, shareholders can issue debt. We denote by B the face value of the debt. At maturity of the debt, the value of the stock, i.e., the residual claim, is S = (V −B)+. (3) In this paper we only consider one source of uncertainty, a single Brownian mo- tion process W. It would be more realistic to distinguish between idiosyncratic and systematic risk, and in our setting, we could do that by including in the dynamics of V a second Brownian motion process that is independent of the first one. The diffu- sion term of the dynamics of the firm’s value would be of the type vVdW +σVdW , 1 2 where the process W would describe the idiosyncratic risk, while W would represent 1 2 3 the systematic or market risk. The manager would choose v as above, while σ would be exogenous. However, to make this a complete-markets setting (as is needed for technical reasons), we should allow the manager to trade in a security that depends on W . This is realistic because managers cannot sell the stock of the company until 2 it is vested, but they are allowed to trade in other securities. In particular, as shown by Jin (2002), managers can diversify the market risk component of the underlying stock by trading (a portfolio that replicates) the market index. As Jin (2002) shows, the market risk becomes irrelevant. Hence, the model we study in this paper can be considered a reduced form of the model described here. Our specific setting is tractable and considers the relevant risk component. 2.2. The manager The manager is risk averse and exerts costly effort. We assume that the manager receives stock as the only source of compensation. In our model, the manager chooses u and v to maximize expected utility. The objective of the manager is (cid:183) (cid:90) (cid:184) (cid:169) (cid:170) 1 T maxE log n(V −B)+ − u2dt . (4) u,v T 2 t 0 In (4), n is the number of shares of stock the manager receives as compensation, and B = 0 corresponds to an unlevered firm. The second term of the objective function represents the cost of effort, which might be, for example, the result of spending more time working for the firm. We assume that the cost of effort is an increasing, convex function, with quadratic cost being a convenient approximation. The horizon of the manager is T, and is determined by the vesting period, the maturity of the debt, and the time the manager plans to stay with the company. In order to simplify the problem, we collapse all those variables into the single constant parameter T. The manager will optimally choose a non-negative level of effort u. The other control, v, involves the choice of projects the firm will undertake and has no effect on the cost of the manager’s effort since it does not require any effort. On the other hand, v does affect the expected value of the compensation package, as the manager has a menu of projects and decides the level of risk to undertake. The projects are, in principle, comparable in quality since projects with higher risk also offer a higher expected return. By allowing the manager to choose the level of volatility we intend to address the problem of the potential incentives to increase volatility that the levered firm’s options-like features might trigger. As discussed in Jensen and Meckling (1976), the manager that owns equity of the firm might prefer a higher level of volatility when the firm is levered than when it is unlevered. Similarly, John and John (1993) show that a risk-neutral manager that receives levered stock will have incentives to choose riskier projects. As we will see later, our results are different because in our model the manager is risk-averse and does not enjoy any perks from holding a management position. 4 We do not introduce a weighting parameter that would measure the relative im- portance of the utility from the stock compensation versus the cost of effort. The reason is that this constant would be equivalent to the parameter δ that represents the type of the manager: a manager with high δ is equivalent to a manager that does not care much about the cost of effort. Given logarithmic utility, the number of shares of stock n becomes irrelevant for incentive purposes, although it is important to determine the total compensation of the manager and whether the compensation satisfies the manager’s participation constraint (discussed later). In the next section we discuss the solution to this problem. We also consider the equivalent static problem, in which the agent chooses ex ante a constant value for both u and v, rather than treating them as functionals of V. 2.3. The shareholders (outside investors) Shareholders(outsideinvestors)compensatethemanagerwithstockthatbecomes vested at time T. Shareholders choose both the number of shares of stock, n, to grant to the manager and the level of debt of the company, B. A debt level of zero indicates an unlevered firm. We assume that shareholders are risk-neutral and care about the final value of the company net of the cost of compensation. On the other hand, when choosing the compensation package and debt level, shareholders must guarantee that the utility of the manager is at least as great as a reservation utility R. This can be interpreted as the utility that the manager would achieve in the best alternative job offer from another firm. This restriction amounts to a participation constraint, which is standard in the principal-agent literature. Werequirethelevelofdebt, B, isnolargerthantheinitialvalueofthefirm, V .As 0 we will argue later, in our setting, even if the nominal value of debt exceeds the value of the firm, at maturity the stock will have a positive value given the optimal effort of the manager. Since we are only interested in qualitative results, our conclusions do not depend on this. Let h(B,n) = λE[V ]−nE[(V −B)+] (5) T T and (cid:189) (cid:183) (cid:90) (cid:184) (cid:190) (cid:169) (cid:170) 1 T A(R) = (B,n) ∈ [0,V ]×[0,∞) : maxE log n(V −B)+ − u2dt ≥ R . 0 u,v T 2 t 0 (6) The objective of the firm is then max h(B,n), (7) (B,n)∈A(R) where λ is an exogenous constant that represents the relative importance for the firm of the expected value of the stock with respect to the compensation package. For a 5 given debt level B, the ratio n/λ is an indicator of the proportion of the firm granted to the manager as compensation. The time horizon of the shareholders matches the time horizon of the manager, and the time at which the compensation becomes vested. The value R represents the minimum utility the manager must be able to achieve through the optimal choice of effort and volatility in order to work for the firm.1 3. Optimal strategies In this section we derive the solution to the problems of the manager and share- holders as described in the previous section. We will show that the solution to the problem of the manager is essentially dynamic and thus different from the solution to the static version of the problem that we consider in this paper. At the end of this section, we discuss the solution to the static problem. In order to derive analytic solutions, we impose some restrictions. Nevertheless, it appears that the model can provide some robust insights to the main problem addressed in this paper, namely, the optimal leverage of the firm when the principal is less risk-averse than the agent and the agent can choose both the effort and the volatility of the firm. In particular, there are two assumptions of our model that determine a subset of our quantitative results, but upon which the intuition of our general conclusions does not seem to depend. First, we constrain the shareholders to grant stock (with some optimal degree of leverage) as the single source of compensation to the manager; i.e., we do not allow other types of payments to the manager, like cash, for example. Second, the utility of the manager is logarithmic (any utility of the CRRA class would yield similar qualitative results). With these two assumptions, it will always be optimal for the manager to apply a level of effort that is at least sufficient to guarantee that the stock will have a positive value and the firm will avoid bankruptcy at the end of the horizon considered, as discussed in footnote 1. However, the utility of the manager depends heavily on the level of effort, so, given the participation constraint, high leverage will be suboptimal for many parameter values. In this setting, the optimal contract with no restriction in the type of compensation would consist of a fixed amount of cash that would match the reservation utility R plus an infinitesimal amount of stock; in this case, the manager would choose v = ∞, which would guarantee an infinite expected value of the firm. Since shareholders are risk- neutral, they do not care about the volatility of the value of the firm. This unrealistic 1We assume that in case of bankruptcy the payoff to the shareholders is zero and, as we explain later, for δ > 0, default never occurs. However, in practice, deviations from absolute priority take place in bankruptcy (see e.g. Franks and Torous, 1989). If shareholders expect to have a positive payoff in default, it might be optimal for the manager to allow the firm to default. The payoff (minus possible bankruptcy costs) would have to be an argument of equations (5) and (6). Unfortunately, that problem does not have a closed-form solution, and there does not exist a clear tractable numerical algorithm to solve it. 6 result arises because of the combination of risk-neutral shareholders and unbounded volatility. We point out that this result would also obtain in a static setting. Ideally, we would have risk-averse shareholders (same type of utility as the manager, but less risk-aversion) and some type of boundary for volatility. Unfortunately, an explicit solution to that problem does not appear feasible, neither in a static nor dynamic setting. While the restricted setting we consider in this paper prevents the optimality of infinite volatility, shareholders can provide the manager with incentives to choose a sufficiently high level of volatility. We now derive the optimal strategies. Note that the dynamic setting is essential for our results, as we explain later. Finally, we stress the importance of the fact that the manager cannot diversify the risk resulting from the firm’s value. The choice of optimal effort and volatility depends directly on that detail. If the manager did have the possibility to diversify some of this risk, then the optimal effort and volatility would be less sensitive to the level of leverage B chosen by the shareholder, and the manager would be able to undo the effect of leverage by trading other securities. 3.1. Optimal effort and volatility We will present the optimal effort uˆ and the optimal choice of volatility vˆ of the manager. First we introduce the auxiliary exponential martingale Z, (cid:189) (cid:190) 1 Z = exp − α2t−αW , (8) t t 2 where α is the parameter in (2) that represents the tradeoff between the volatility and expected return of the projects available to the manager. Also, consider the following ¯ function of time T, (cid:40) e−2µt[e(α2−2µ)(T−t)−1] if α2 (cid:54)= 2µ ¯ T = α2−2µ . (9) t e−α2t(T −t) if α2 = 2µ Using the previous notation and given the following quadratic equation in z, δ2T¯ z2 +(V −Be−µT)z −1 = 0, (10) 0 0 where δ is the parameter that measures the type of the manager and B is the level of debt of the company, we denote by zˇ the constant positive solution (the Lagrange multiplier) of (10): (cid:189) (cid:113) (cid:190) 1 zˇ = (Be−µT −V )+ (Be−µT −V )2 +4δ2T¯ . (11) 2δ2T¯ 0 0 0 0 We now solve for the optimal controls of the manager. 7 Theorem 1 Consider the problem of the manager described in Section 2.2. Consider ¯ also the exponential martingale of (8), the time function T of (9), and the positive number zˇ of (11). Assume δ > 0. Then, the optimal effort uˆ of the manager is uˆ = δzˇe−µtZ (12) t t and the optimal choice of volatility vˆ is given by αeµt vˆV = +αzˇδ2eµtZ T¯. (13) t t t t zˇZ t The optimal effort and volatility together determine that the value of the firm be given by eµt V = +Be−µ(T−t) −zˇδ2eµtZ T¯. (14) t t t zˇZ t If V > Be−µT, the above formulas remain valid even for δ = 0, but with 0 1 zˇ= . (15) V −Be−µT 0 If V < Be−µT and δ = 0, then the utility of the manager is negative infinity since it 0 is impossible in this case to guarantee V > B with probability one. T Proof. See the Appendix. (cid:50) Remark 1 When δ > 0, equation (14) with t = T indicates that the manager applies enough effort to guarantee that V > B with probability one, otherwise expected utility T is negative infinity. Therefore, bankruptcy never occurs. When δ = 0 and V > 0 Be−µT, the manager still can (and will) avoid bankruptcy by choosing the appropriate volatility. However, when δ = 0 and V ≤ Be−µT, the manager cannot guarantee 0 V > B with probability one, and the problem does not have a solution. T We observe that the optimal levels of effort and volatility can also be written as functions of the value of the firm. That is, when δ > 0, (cid:189) (cid:113) (cid:190) e−2µt uˆ = (Be−µ(T−t) −V )+ (Be−µ(T−t) −V )2 +4δ2e2µtT¯ (16) t ¯ t t t 2δT t and αδ vˆV = +αδe2µtuˆ T¯ t t t t uˆ t 2αδ2T¯e2µt = (cid:112) t (Be−µ(T−t) −V )+ (Be−µ(T−t) −V )2 +4δ2e2µtT¯ (cid:183) t (cid:113) t t (cid:184) α + (Be−µ(T−t) −V )+ (Be−µ(T−t) −V )2 +4δ2e2µtT¯ . (17) t t t 2 8 In Table 1 we study the effect of different parameter values on optimal effort. First, we observe that uˆ is increasing in the level of debt B. This effect is different from the asset substitution effect of Jensen and Meckling (1976); here the manager is risk-averse and the stock of a levered firm is the single source of compensation and utility to the manager. The higher the leverage, the higher the impact of the effort on the value of the compensation, since the manager receives more shares of a levered firm than an unlevered firm. Leverage increases the incentives of the manager, which is consistent with the findings of Hall and Murphy (2000, 2002). In Section 4 we elaborate further on this point. Also, the longer the time horizon, the lower the effort of the manager. The intuition is clear: a larger T has a similar effect on the manager as a lower level of debt. The effect of δ, the “type” of manager, depends on the relationship between V and B. When V = B, the optimal effort is independent t t of δ, as we can see from (16). We can also verify that when V > B, the effort is t increasing in δ, and when V < B, the optimal effort decreases with δ. The fact that t the relationship changes for V = B is due to the logarithmic utility function and the t absence of cash compensation. CRRA utility forces the manager to drive the value of the firm above the debt level. When V ≥ B, the main risk (default) is greatly t lowered, and thus the higher the marginal productivity of effort, the higher the effort exercised by the manager. The effect of µ, the exogenous component of the drift that does not depend on the effort of the manager, is straightforward: the higher this component, the lower the effort of the manager. In a favorable economic environment the manager has to apply less effort than in difficult circumstances. Finally, the effect of α is also unambiguous: the higher α, the lower the optimal effort, or, the better the menu of projects from which the manager can choose, the lower the effort. We study the dependence of volatility on the different parameter values in Table 2. For δ > 0, as in Carpenter (2000), high volatility might not be optimal (it is always optimal in the risk-neutral setting of John and John, 1993). The relationship between volatility v V and leverage B is not monotonic: v V is a decreasing function of B o 0 0 0 when B ∈ [0,V ], and is an increasing function when B ∈ (V ,∞). The intuition of 0 0 this result is as before. When the value of the firm is higher than the nominal level of debt, then the higher the leverage, the higher the volatility chosen by the manager. When the level of debt is higher than the valueof the firm and, therefore, the payoffof the manager would be zero if debt expired, the manager chooses lower volatility and relies more on effort to ensure there is no default. Similarly, the relationship between volatility and the exogenous component of the drift µ is also nonlinear. Rather, this relation depends on the leverage ratio and the horizon. For a long horizon, optimal volatility is always increasing in µ. In the short horizon, it increases with µ if the level of debt is lower than the value of the company, but decreases with µ when B > V. We also see that the volatility is increasing in the type of manager δ. Finally, as expected, the level of volatility increases with α. It is also interesting to study the correlation between optimal effort and optimal volatility. By Ito’s Lemma and equation (13), the dynamics of the optimal volatility 9