On managerial risk-taking incentives when compensation may be hedged against

Abstract

We consider a continuous time principal-agent model where the principal/firm compensates an agent/manager who controls the output’s exposure to risk and its expected return. Both the firm and the manager have exponential utility and can trade in a frictionless market. When the firm observes the manager’s choice of effort and volatility, there is an optimal contract that induces the manager to not hedge. In a two factor specification of the model where an index and a bond are traded, the optimal contract is linear in output and the log return of the index. We also consider a manager who receives exogenous share or option compensation and illustrate how risk taking depends on the relative size of the systematic and firm-specific risk premia of the output and index. Whilst in most cases, options induce greater risk taking than shares, we find that there are also situations under which the hedging manager may take less risk than the non-hedging manager.

Appendix

Proof of Proposition 2

The manager wants to maximize $E[U_A(C_T\,-\,G_T\,+\,H_T)]$, which, under our assumptions, is the same as maximizing $E[U_A(C_T\,-\,G_T\,+\,H_T\,-\,H_0B_T)]$. The standard martingale/duality approach to portfolio selection (see, e.g., [19] for diffusion models, or [20] for general semimartingale models) says that she will hedge so that

$$\begin{aligned} C_T-G_T+H_T-H_0B_T=I_A(yY_T) \end{aligned}$$

(6.1)

where $y,Y_T$ solves, over constant numbers $y$ and SDFs $Y_T\in \mathcal{Z}$, the dual problem

$$\begin{aligned} \bar{V}(H_0):=\min _{y,Y}E[U_A(I_A(yY_T))-yY_TI_A(yY_T)+yY_T(C_T-G_T) +yY_T(H_T-H_0B_T)] \end{aligned}$$

(6.2)

if a solution exists, and if we have $V(H_0)=\bar{V}(H_0)$. The last term in (6.2) disappears, because $E[Y_T(H_T-H_0B_T)]=0$. Then, if (3.5) is satisfied, it is easily seen that the above is minimized for $yY_T=zZ_T$. From (6.1) this implies $H_T=H_0B_T$. Conversely, if the optimal hedging strategy results in $H_T=H_0B_T$, we see from (6.1) that $C_T$ has to be of the form (3.5). $\square$

Computations for Sect. 3.2: Consider the more general model where output is given by

$$\begin{aligned} dX_t=[\delta u_t + \alpha _x x_t +\alpha _y y_t]dt+ x_tdW_t+y_tdM_t \end{aligned}$$

and where the manager pays the cost $G_T= \int _0^T g(u_t) dt$ with $g(u) = \frac{u^2}{2k}$ for $k>0$ when exerting the effort $u$. Under optimal behavior of the firm, the manager is given a contract of the form $C_T=aX_T+b\log (S_T) +c$. Denote by $\tilde{c}$ the certainty equivalent (CE) of $c$ and with $\tilde{R}$ the CE of the manager’s reservation utility. Also denote by $\pi _A$ the amount of capital manager invest in $S$, and by $\pi _P$ the analogous hedging amount for the firm. Given that at the optimum $\pi _A,\pi _P$ and effort $u$ are constant in our framework, the manager’s CE is then

$$\begin{aligned}&\tilde{c}+a[X_0+\alpha _y yT +\delta uT]+[\alpha _xax+ \lambda \sigma \pi _A-g(u)]T\nonumber \\&\quad +\,\,b\left[ \log S_0+\left( \mu -\frac{\sigma ^2}{2}\right) T\right] -\frac{\gamma _A T}{2}\left[ (ax+\pi _A\sigma +b\sigma )^2+a^2y^2\right] \end{aligned}$$

(6.3)

Maximizing over $u,\pi _A$ and $y$ we get

$$\begin{aligned}&\displaystyle ax+\sigma \pi _A+b\sigma =\frac{\lambda }{\gamma _A}\end{aligned}$$

(6.4)

$$\begin{aligned}&\displaystyle y= \frac{\alpha _y}{a\gamma _A}\end{aligned}$$

(6.5)

$$\begin{aligned}&\displaystyle g'(u)=a\delta \end{aligned}$$

(6.6)

This means that the manager’s CE is

$$\begin{aligned}&\tilde{R}=\tilde{c}+a\left[ X_0+\left( \frac{\alpha _y^2}{a\gamma _A} +\delta u\right) T\right] -g(u)T\nonumber \\&\quad +\,\,(\alpha _x-\lambda )ax+\lambda \left[ \frac{\lambda }{\gamma _A}-b\sigma \right] T+b[\log S_0+(\mu -\sigma ^2/2)T]\end{aligned}$$

(6.7)

$$\begin{aligned}&\quad -\,\,\frac{T}{2}\gamma _A\left[ \left( \frac{\lambda }{\gamma _A}\right) ^2 +\left( \frac{\alpha _y}{\gamma _A}\right) ^2\right] \end{aligned}$$

(6.8)

The firm’s CE is

$$\begin{aligned}&-\tilde{c}+(1-a)\left[ X_0+ \left( \frac{\alpha _y^2}{a\gamma _A} +\delta u\right) T\right] +[\alpha _x(1-a)x+\lambda \sigma \pi _P]T -b[\log S_0\nonumber \\&\quad +(\mu -\sigma ^2/2)T]\end{aligned}$$

(6.9)

$$\begin{aligned}&-\frac{T}{2}\gamma _P\left[ ((1-a)x+\pi _P\sigma -b\sigma )^2+(1-a)^2 \left( \frac{\alpha _y}{a\gamma _A}\right) ^2\right] \end{aligned}$$

(6.10)

Fixing $\tilde{R}$ and computing $\tilde{c}$ from the previous expression we get the firm’s CE as

$$\begin{aligned}&-\tilde{R}-g(u)T + (\alpha _x-\lambda )ax +\lambda \left[ \frac{\lambda }{\gamma _A}-b\sigma \right] T-\frac{T}{2}\gamma _A \left[ \left( \frac{\lambda }{\gamma _A}\right) ^2 +\left( \frac{\alpha _y}{\gamma _A}\right) ^2\right] \nonumber \\&\quad +\left[ X_0+ \left( \frac{\alpha _y^2}{a\gamma _A} +\delta u\right) T\right] \end{aligned}$$

(6.11)

$$\begin{aligned}&+[\alpha _x(1-a)x+\lambda \sigma \pi _P]T -\frac{T}{2}\gamma _P\left[ ((1-a)x+\pi _P\sigma -b\sigma )^2+(1-a)^2 \left( \frac{\alpha _y}{a\gamma _A}\right) ^2\right] \nonumber \\ \end{aligned}$$

(6.12)

This means that the hedging portfolio $\pi _P$ is chosen so that

$$\begin{aligned} \gamma _P[(1-a)x+\pi _P\sigma -b\sigma ]=\lambda \end{aligned}$$

(6.13)

and hence the firm needs to maximize

$$\begin{aligned} \delta u-g(u)-\lambda b\sigma + \frac{\alpha _y^2}{a\gamma _A} +(\alpha _x-\lambda )x+\lambda \sigma b -\frac{\gamma _P}{2}(1-a)^2 (\frac{\alpha _y}{a\gamma _A})^2 \end{aligned}$$

(6.14)

We see that the principal is indifferent with respect to the choice of $b$. In case there is no effort $u$, $u=g(u)=0$, we get

$$\begin{aligned} a=\frac{\gamma _P}{\gamma _A+\gamma _P} \end{aligned}$$

(6.15)

In case $g(u)=u^2/(2k)$ is quadratic, taking into account that $g'(u)=\delta a$, we can check that the derivative of the firm’s CE is

$$\begin{aligned} \frac{1}{a^3}[a^3(1-a)k\delta ^2-\frac{1}{\gamma _A}\alpha _y^2[a(1+\gamma _P/\gamma _A)-\gamma _P/\gamma _A] \end{aligned}$$

This is positive for $a$ positive and close to zero and negative for $a$ close to one, thus there is exactly one maximizer $\hat{a}\in [0,1]$. Moreover, it is seen that increasing $\alpha _y^2$ also increases the derivative in absolute value. Note also that the firm’s CE converges to $-\infty$ at $a=0$, and has a higher value at $a=1$ for higher $\alpha _y^2$. A combination of all these properties is possible only if the maximizer $\hat{a}$ moves to higher values when $\alpha _y^2$ is increased.