Clawback enforcement, executive pay, and accounting manipulation

Abstract

Clawback provisions entitle shareholders to recover previously-awarded compensation from managers involved in accounting manipulation or misconduct. In a principal-agent model, we show that strong clawback enforcement tilts managerial compensation towards the short-term but may increase manipulation. In contrast, weak enforcement alleviates the shareholders’ incentives to tilt compensation towards the short-term and reduces manipulation. While weak enforcement and lack of commitment may generate a time inconsistency problem, the clawback adoption decision may foster further governance changes that elicit ex post enforcement. We discuss the regulatory implications of the theory and its consistency with results in empirical studies.

Appendix A: Proofs

1.1 Proof of Lemma 1

Assume first that a contract with $w_{LL}>0$ is optimal. Consider a new contract such that $w_{LL}^\prime =w_{LL}-\epsilon$ and $w_{L}^\prime =w_{L}+\beta \epsilon$, with $\epsilon >0$ arbitrarily small. Notice that the new contract still satisfies the incentive compatibility constraint. Moreover, the decision m is unchanged with respect to the original contract from problem (2). The new contract reduces the expected compensation costs by $(1-{\overline{e}})(1-m)(1-\beta )\epsilon >0$ and is feasible. Thus, the original contract cannot be optimal. The optimal contract reduces then to solve:

$$\begin{aligned} \underset{\begin{array}{c} w_H,w_{HH},w_{HL} \\ w_L, c \end{array}}{\min }&{\overline{e}}(w_H + w_{HH}) + (1-{\overline{e}}){\widehat{m}}\big [w_H + w_{HL} + \kappa c (w_{H}-w_L) \big ] + (1-{\overline{e}})(1-{\widehat{m}})w_L \\ s.t. \hspace{0.1cm}&{\overline{e}}\big [(1-{\widehat{m}})\big (w_H-w_L\big ) + \beta w_{HH} - {\widehat{m}}\beta w_{HL} + G(m)\big ] \ge B \\&{\widehat{m}} = \frac{w_H-w_L+\beta w_{HL}}{\gamma } \\&(w_H,w_L,w_{HH})\in \mathbb {R}_+^3 \\&w_{HL}\ge - c\ell \big (w_{H}-w_{L}\big ),\ c\in \{0,1\} \end{aligned}$$

Taking the first order condition with respect to $w_{HL}$, we obtain:

$$\begin{aligned} w_{HL}:\ (1-{\overline{e}}){\widehat{m}} + (1-{\overline{e}})\frac{\partial {\widehat{m}}}{\partial w_{HL}}\big [w_H+w_{HL}-w_L + \kappa c (w_{H}-w_L) \big ] + \mu _p\beta {\widehat{m}} - \mu _{HL}^+ = 0 \end{aligned}$$

where $\mu _p$ is the Lagrange multiplier associated with the incentive compatibility constraint and $\mu _{HL}^+$ is the multiplier associated with the clawback constraint for $w_{HL}$. If $\mu _{HL}^+=0$ in the optimal contract then we would have that $w_{HL}>-c\ell (w_H-w_L)$, but at the same time, in order to satisfy the first order condition, the contract must also satisfy that $w_H+w_{HL}-w_L<0$, or $w_{HL}<-(w_H-w_L)$, reaching a contradiction.

From the first order conditions with respect to $w_H$ and $w_L$ we obtain:

$$\begin{aligned}&w_{H}:\ {\overline{e}} + 2{\widehat{m}}(1-{\overline{e}})(1-\ell c + \kappa c) -\mu _p[1-{\widehat{m}}(1-\beta \ell c)] - \mu _{H}^+ = 0 \\&w_{L}:\ (1-{\overline{e}})[1 - 2{\widehat{m}}(1-\ell c + \kappa c)] + \mu _p[1 - {\widehat{m}}(1-\beta \ell c)] - \mu _{L}^+ = 0 \end{aligned}$$

where $\mu _{H}^+$ and $\mu _{L}^+$ represent, respectively, the Lagrange multipliers associated with the non-negativity constraints of $w_H$ and $w_L$. Assume that in the optimal contract $w_L>0$, so that $\mu _{L}^+=0$. This means that:

$$\begin{aligned} -(1-{\overline{e}})[1 - 2{\widehat{m}}(1-\ell c+ \kappa c)] = \mu _p[1 - {\widehat{m}}(1-\beta \ell c)] \end{aligned}$$

Notice that we can rewrite the first order condition for $w_H$ as

$$\begin{aligned} 1 -(1-{\overline{e}})[1 - 2{\widehat{m}}(1-\ell c + \kappa c)] - \mu _p[1-{\widehat{m}}(1-\beta \ell c)] - \mu _{H}^+ = 0 \end{aligned}$$

which implies that $\mu _{H}^+=1$, $w_H=0$ and ${\widehat{m}}=0$. This would also imply that $\mu _{L}^+>0$, which contradicts the assumption that $w_L>0$.

Finally, the incentive compatibility constraint binds. Otherwise, nodes with positive compensation, $(w_H,w_{HH})$, can be readjusted until the incentive compatibility condition binds, reducing the expected compensation costs. Moreover, a reduction in $w_{H}$ reduces m which also reduces the expected compensation costs. $\square$

1.2 Proof of Proposition 1

Solving for $w_{HH}$ in the effort incentive compatibility constraint, we obtain:

$$\begin{aligned} w_{HH} = \frac{1}{\beta }\left[ \frac{B}{{\overline{e}}} - (1-{\widehat{m}})w_H - G({\widehat{m}})\right] \end{aligned}$$

Plugging the previous expression into the objective function, the optimization problem boils down to:

$$\begin{aligned} \underset{w_H\ge 0}{\min }\ \ \ {}&{\overline{e}}w_H + \frac{{\overline{e}}}{\beta }\left[ \frac{B}{{\overline{e}}} - (1-{\widehat{m}})w_H - G({\widehat{m}}) \right] + (1-{\overline{e}}){\widehat{m}}w_H\\ s.t. \hspace{0.1cm}&{\widehat{m}} =\min \left\{ 1,\frac{w_H}{\gamma }\right\} \end{aligned}$$

The first order condition with respect to $w_H$, after invoking the envelope theorem and assuming ${\widehat{m}}<1$, is given by:

$$\begin{aligned} {\overline{e}} - \frac{{\overline{e}}}{\beta }(1-{\widehat{m}})+ (1-{\overline{e}}){\widehat{m}} + \frac{(1-{\overline{e}})}{\gamma }{\widehat{w}}_H= 0 \end{aligned}$$

The second-order optimality conditions confirm that the objective function is convex in $w_{H}$. Inserting the optimal choice of manipulation as a function of $w_{H}$, we reach the optimal choice of $w_H$:

$$\begin{aligned} {\widehat{w}}_{H,0} = \frac{{\overline{e}}(1-\beta )\gamma }{{\overline{e}}+2\beta (1-{\overline{e}})} \end{aligned}$$

The optimal intensity of manipulation reads as:

$$\begin{aligned} {\widehat{m}}_0 = \frac{{\overline{e}}(1-\beta )}{{\overline{e}}+2\beta (1-{\overline{e}})} \le 1 \end{aligned}$$

The optimal level of long-term pay follows from the incentive-compatibility constraint:

$$\begin{aligned}&{\widehat{w}}_{HH,0} = \frac{1}{\beta }\left[ \frac{B}{{\overline{e}}} - (1-{\widehat{m}}_0){\widehat{w}}_{H,0} - G({\widehat{m}}_0)\right] = \frac{1}{\beta }\left[ \frac{B}{{\overline{e}}} -{\widehat{w}}_{H,0} + \frac{{\widehat{w}}_{H,0}^2}{2\gamma }\right] \end{aligned}$$

$\square$

1.3 Proof of Proposition 2

Setting $c=1$, the optimal contracting problem with clawbacks reads as follows:

$$\begin{aligned} \underset{\begin{array}{c} w_H,w_{HH} \end{array}}{\min }&{\overline{e}}(w_H + w_{HH}) + (1-{\overline{e}}){\widehat{m}}w_H(1-\ell + \kappa )\\ s.t. \hspace{0.1cm}&\big [1-{\widehat{m}}(1-\beta \ell )\big ]w_H + \beta w_{HH} + G({\widehat{m}}) \ge \frac{B}{{\overline{e}}} \\&{\widehat{m}} = \frac{(1-\beta \ell )w_H}{\gamma } \ \ (w_H,w_{HH})\in \mathbb {R}_+^2 \end{aligned}$$

Using the incentive compatibility constraint to solve for $w_{HH}$ in the objective function, and taking the first order condition with respect to $w_H$, we reach the optimal choice of $w_H$:

$$\begin{aligned} {\overline{e}} - \frac{{\overline{e}}}{\beta }\big [1-{\widehat{m}}(1-\beta \ell )\big ] + (1-{\overline{e}}){\widehat{m}}(1-\ell +\kappa ) + \frac{(1-{\overline{e}})}{\gamma }(1-\beta \ell )(1-\ell + \kappa ){\widehat{w}}_H= 0 \end{aligned}$$

Using the optimal choice of manipulation by the manager, the optimal levels of short-term compensation and manipulation are given by:

$$\begin{aligned} {\widehat{w}}_{H,1}&= \frac{{\overline{e}}(1-\beta )\gamma }{(1-\beta \ell )\left[ {\overline{e}}(1-\beta \ell )+2\beta (1-{\overline{e}})(1-\ell +\kappa )\right] } \\ {\widehat{m}}_1&= \frac{{\overline{e}}(1-\beta )}{{\overline{e}}(1-\beta \ell )+2\beta (1-{\overline{e}})(1-\ell + \kappa )} \le 1 \end{aligned}$$

The optimal level of long-term pay follows from the incentive-compatibility constraint:

$$\begin{aligned}&{\widehat{w}}_{HH,1} = \frac{1}{\beta }\left\{ \frac{B}{{\overline{e}}} - [1-{\widehat{m}}_1(1-\beta \ell c)]{\widehat{w}}_{H,1} - G({\widehat{m}}_1)\right\} = \frac{1}{\beta }\left[ \frac{B}{{\overline{e}}} -{\widehat{w}}_{H,1} + \frac{(1-\beta \ell c)^2{\widehat{w}}_{H,1}^2}{2\gamma }\right] \end{aligned}$$

$\square$

1.4 Proof of Proposition 3

Given a choice $c\in \{0,1\}$, the manager’s manipulation intensity in the optimal contract arises from:

$$\begin{aligned} {\overline{e}} - \frac{{\overline{e}}}{\beta }\big [1-{\widehat{m}}_c(1-\beta \ell c)\big ] + (1-{\overline{e}}){\widehat{m}}_c(1-\ell c+\kappa c) + (1-{\overline{e}})(1-\ell c + \kappa c){\widehat{m}}_{c}= 0 \end{aligned}$$

Collecting terms, we obtain that:

$$\begin{aligned} {\overline{e}}\left( 1-\frac{1}{\beta }\right) + {\widehat{m}}_c\left\{ \frac{{\overline{e}}}{\beta } + 2(1-{\overline{e}}) - c\left[ {\overline{e}}\ell + 2(1-{\overline{e}})(\ell -\kappa )\right] \right\} = 0 \end{aligned}$$

(9)

Notice that clawback adoption decreases manipulation, ${\widehat{m}}_1 - {\widehat{m}}_0<0$ if the last term multiplying c has a negative sign. This is the case if:

$$\begin{aligned} \kappa \ge {\underline{\kappa }} = \ell \frac{2 - {\overline{e}}}{2(1-{\overline{e}})} > \ell \end{aligned}$$

where ${\underline{\kappa }}$ is increasing in $\ell$ and ${\overline{e}}$. The change in the level of short-term compensation from the clawback adoption decision satisfies:

$$\begin{aligned} {\widehat{w}}_{H,1} - {\widehat{w}}_{H,0}&\propto {\overline{e}}+2\beta (1-{\overline{e}}) - (1-\beta \ell )\left\{ {\overline{e}}(1-\beta \ell )+2\beta (1-{\overline{e}})(1-\ell +\kappa )\right\} \end{aligned}$$

Short-term compensation decreases if:

$$\begin{aligned}&\kappa \ge {\widehat{\kappa }} = \ell \left[ \frac{\beta + (1-\beta \ell )}{1-\beta \ell } + \frac{(2-\beta ){\overline{e}}}{2(1-{\overline{e}})(1-\beta \ell )}\right] \end{aligned}$$

which is greater than ${\underline{\kappa }}$ and is also increasing in $\ell$. To study the effects on long-term compensation, first notice that

$$\begin{aligned} {\widehat{w}}_{HH,1} - {\widehat{w}}_{HH,0}&\propto ({\widehat{w}}_{H,0}-{\widehat{w}}_{H,1}) + \frac{\gamma }{2}({\widehat{m}}_1^2 - {\widehat{m}}_0^2) \end{aligned}$$

At ${\widehat{\kappa }}$, we have that ${\widehat{w}}_{H,0}-{\widehat{w}}_{H,1}=0$ and ${\widehat{m}}_1^2 - {\widehat{m}}_0^2<0$, so that ${\widehat{w}}_{HH,1} - {\widehat{w}}_{HH,0}<0$. Moreover, define the function

$$\begin{aligned}&J(\kappa ) = \frac{\gamma }{2}\left[ {\widehat{m}}_1(\kappa )\right] ^2 - {\widehat{w}}_{H,1}(\kappa ) - \frac{\gamma }{2}{\widehat{m}}_0^2 + {\widehat{w}}_{H,0} = -\gamma {\widehat{m}}_1(\kappa )\left( \frac{1}{1-\beta \ell }-\frac{{\widehat{m}}_1(\kappa )}{2} \right) + constant, \end{aligned}$$

where $J({\widehat{\kappa }})<0$, and $\underset{\kappa \rightarrow \infty }{\lim }J(\kappa )=constant={\widehat{w}}_{H,0}-\gamma {\widehat{m}}_0^2/2$ is positive in an interior solution for $w_{HH,0}$. Notice also that $J(\kappa )$ is monotonically increasing in $\kappa$:

$$\begin{aligned}&J'(\kappa ) = -\gamma \left( \frac{1}{1-\beta \ell } - {\widehat{m}}_{1}\right) \frac{\partial {\widehat{m}}_{1}}{\partial \kappa } > 0, \end{aligned}$$

where the sign comes from $\frac{\partial {\widehat{m}}_{1}}{\partial \kappa }<0$. Thus, there must exist ${\overline{\kappa }}$ that solves $J({\overline{\kappa }})=0$, with ${\widehat{\kappa }}<{\overline{\kappa }}$. Implicit differentiation yields that ${\overline{\kappa }}$ is increasing in $\ell$ since $\frac{\partial {\widehat{m}}_{1}}{\partial \ell }>0$. $\square$

1.5 Proof of Proposition 4

Let $c\in \{0,1\}$ denote the clawback adoption decision. The cost function for shareholders can be written as:

$$\begin{aligned}&\omega _c = {\overline{e}}{\widehat{w}}_{H,c} + \frac{{\overline{e}}}{\beta }\left[ \frac{B}{{\overline{e}}} - {\widehat{w}}_{H,c} + \frac{(1-\beta \ell c)^2{\widehat{w}}_{H,c}^2}{2\gamma } \right] + (1-{\overline{e}})\frac{(1-\beta \ell c){\widehat{w}}_{H,c}^2}{\gamma }(1-\ell c + \kappa c) \end{aligned}$$

Define $\Delta \omega =\omega _1 - \omega _0$. Notice that $\Delta \omega$ is monotonically increasing in $\kappa$, by the envelope theorem:

$$\begin{aligned} \frac{\partial (\Delta \omega )}{\partial \kappa } = \frac{\partial \omega _1}{\partial \kappa } = (1-{\overline{e}})\frac{(1-\beta \ell ){\widehat{w}}_{H,1}^2}{\gamma } > 0 \end{aligned}$$

At $\kappa ={\widehat{\kappa }}$, we have that ${\widehat{w}}_{H,1} = {\widehat{w}}_{H,0} = {\widehat{w}}$ and

$$\begin{aligned}&\Delta \omega = \frac{{\overline{e}}\gamma }{2\beta }({\widehat{m}}_{1}^2-{\widehat{m}}_{0}^2) + (1-{\overline{e}}){\widehat{w}}[{\widehat{m}}_{1}(1-\ell +{\widehat{\kappa }})-{\widehat{m}}_{0}] \\&{\widehat{m}}_{1}-{\widehat{m}}_{0} = -\beta \ell {\widehat{m}}_0 = -\beta \ell \frac{{\widehat{w}}}{\gamma } \\&{\widehat{m}}_{1}^2-{\widehat{m}}_{0}^2 = \left[ (1-\beta \ell )^2-1\right] {\widehat{m}}_0^2 = \left[ (1-\beta \ell )^2-1\right] \frac{{\widehat{w}}^2}{\gamma ^2} \end{aligned}$$

We obtain, then:

$$\begin{aligned}&\Delta \omega \propto {\overline{e}}\left[ (1-\beta \ell )^2-1\right] - 2\beta (1-{\overline{e}}) + 2\beta (1-\beta \ell )(1-{\overline{e}})(1-\ell +{\widehat{\kappa }}) \end{aligned}$$

which, using the definition of ${\widehat{\kappa }}$, yields $\Delta \omega ({\widehat{\kappa }})=0$. By the monotonicity of $\Delta \omega$ in $\kappa$ then ${\widehat{\kappa }}$ is the only value that sets $\Delta \omega (\kappa )=0$.

Notice that the adoption decision will be more likely, i.e. takes place for a broader space of parameters, after a change in a parameter $\theta$ as long as $\partial {\widehat{\kappa }}/\partial \theta >0$. Using the definition of ${\widehat{\kappa }}$ from Proposition 3, we reach the following comparative statics:

$$\begin{aligned} \frac{\partial {\widehat{\kappa }}}{\partial \ell }&=\frac{{\widehat{\kappa }}}{\ell }+\ell \left\{ \frac{\beta ^2}{(1-\beta \ell )^2}+\frac{\beta {\overline{e}}(2-\beta )}{2(1-{\overline{e}})(1-\beta \ell )^2}\right\}> 0 \\ \frac{\partial {\widehat{\kappa }}}{\partial {\overline{e}}}&=\frac{(2-\beta ) \ell }{2 (1-{\overline{e}})^2 (1-\beta \ell )} > 0 \\ \frac{\partial {\widehat{\kappa }}}{\partial \gamma }&=0 \end{aligned}$$

The comparative statics with respect to $\beta$ arise from:

$$\begin{aligned} \frac{\partial {\widehat{\kappa }}}{\partial \beta }&=-\ell \frac{{\overline{e}} (3-2 \ell )-2}{2 (1-{\overline{e}}) (\beta \ell -1)^2} \end{aligned}$$

Notice that the expression above is positive when:

$$\begin{aligned} \ell >\ell _b=\frac{3{\overline{e}}-2}{2{\overline{e}}}\ . \end{aligned}$$

Such an $\ell _b\in (0,1)$ may exist as in the last expression the numerator is smaller than the denominator.

Although ${\widehat{\kappa }}$ does not change with $\gamma$, we can show the derivative of $\Delta \omega$ with respect to $\gamma$ satisfies:

$$\begin{aligned} \frac{\partial \Delta \omega }{\partial \gamma }&\propto {\widehat{w}}_{H,1}^2(1-\beta \ell )\left[ {\overline{e}}(1-\beta \ell )+2\beta (1-{\overline{e}})(1-\ell +\kappa )\right] - {\widehat{w}}_{H,0}^2\left[ {\overline{e}}+2\beta (1-{\overline{e}})\right] \end{aligned}$$

The expression above is negative for $\kappa \le {\widehat{\kappa }}$ and positive otherwise. $\square$

1.6 Proof of Proposition 5

If shareholders cannot commit to the enforcement of a clawback, ex post enforcement only can happen when ${\widehat{w}}_{H,1}(\kappa -\ell )\le 0$, i.e., $\kappa <\ell$. For $\kappa >\ell$ ex post enforcement is suboptimal, clawbacks generate no incentive effect on managers and adoption generates no changes in optimal compensation. From Proposition 3, we have that $\ell <{\underline{\kappa }}$. Thus, in the region where clawback adoption induces changes in compensation, the adoption decision only involves an increase in short-term compensation and an increase in manipulation. $\square$

1.7 Proof of Proposition 6

An equilibrium is represented by the same choice of enforcement $d^\prime$ at $t=2$ as the one anticipated by shareholders as of $t=0$. Trivially, a sufficient condition for enforcement is that $\kappa \le \ell$, since $\pi \ge 0$. Notice that if directors anticipate that the clawback is not enforced at $t=2$, $d^\prime =0$, then the optimal choice of short-term compensation with and without clawbacks is the same, ${\widehat{w}}_{H,1}(d=0)={\widehat{w}}_{H,0}$.

We must analyze two different cases. First, suppose that ${\widehat{w}}_{H,1}(d=1)<{\widehat{w}}_{H,0}$. This means that $\kappa >{\widehat{\kappa }}$ from Proposition 3. The threshold ${\widehat{\pi }} = (\kappa -\ell ){\widehat{w}}_{H,0}$ determines the lower bound of the penalties for directors such that a contract that is designed under the assumption that the clawback is not enforced will actually lead to enforcement. Hence, enforcement is an equilibrium, $d=1$, for $\pi \ge {\widehat{\pi }}$. Notice that there exists the possibility that

$$\begin{aligned}&(\kappa -\ell ){\widehat{w}}_{H,1}(d=1) \le \pi < (\kappa -\ell ){\widehat{w}}_{H,0}\ . \end{aligned}$$

The expressions above suggest that enforcement and no enforcement represent, respectively, candidate equilibrium decisions. However, as Proposition 3 shows, the costs of a clawback contract exceed the costs of the no clawback contract when $\kappa >{\widehat{\kappa }}$. Hence, by eliminating strictly dominated strategies, no enforcement, $d=0$, is the optimal decision from a cost-minimization perspective for $\pi <{\widehat{\pi }}$.

Second, if ${\widehat{w}}_{H,1}(d=1)\ge {\widehat{w}}_{H,0}$ this means that $\kappa <{\widehat{\kappa }}$ from Proposition 3. Therefore, $\pi \ge (\kappa -\ell ){\widehat{w}}_{H,1}(d=1)$ is a sufficient condition for full enforcement, $d=1$, and $\pi \le (\kappa -\ell ){\widehat{w}}_{H,0}$ is a sufficient condition for no enforcement, $d=0$. If $\pi \in \big [{\widehat{\pi }},(\kappa -\ell ){\widehat{w}}_{H,1}(d=1)\big )$, there is no equilibrium with corner solutions, but we can find an equilibrium intensity of enforcement ${\overline{d}}\in (0,1)$ such that $\pi = (\kappa -\ell ){\widehat{w}}_{H,1}\left( d={\overline{d}}\right)$, where ${\widehat{w}}_{H,1}\left( d={\overline{d}}\right) >{\widehat{w}}_{H,0}$. $\square$