Hedging and accounting-based RPE contracts for powerful CEOs

Abstract

Several firms prohibit their CEOs from trading in the stock of peer firms. This is puzzling since hedging by the CEO through private trading in the capital market can reduce the CEO’s exposure to systematic compensation risk. When the CEO’s incentive contract comprises relative performance evaluation, we find that the firm might want to disallow private hedging even though there are no technological interdependencies or strategic interactions to peer firms. In the analysis, we highlight two frequently observed characteristics of incentive contracts. First, the use of accounting benchmarks is widespread in compensation contracts for CEOs. Second, empirical and anecdotal evidence suggests that powerful CEOs have influence on the process of designing their own compensation. We find that in the presence of a powerful CEO, the firm can benefit from disallowing private hedging. In particular, the firm’s decision to allow or to disallow private hedging depends on the characteristics of the accounting benchmarks and the characteristics of the peer firms.

Appendices

Appendix A: Proofs

Proof of Observation 1

(i) If the BoD does not implement RPE and the CEO cannot privately hedge industry-specific risk, the “no RPE” constraint (5d) and the “no hedging” constraint (5e), $h = 0$, are binding. Substituting the “no RPE” constraint (5d), the “no hedging” constraint (5e), and the CEO’s action choice (3) into the BoD’s optimization problem (6) yields the BoD’s unconstrained optimization problem:

$$ \mathop {\hbox{max} }\limits_{{v_{f} }} \;\varPi = v_{f} - \frac{{v_{f}^{2} }}{2} - \frac{r}{2}v_{f}^{2} (\sigma_{f}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} ). $$

(13)

Differentiating the BoD’s unconstrained optimization problem with respect to $v_{f}$ and solving the first-order condition gives the incentive weight on the firm performance, $v_{f}$ in (7).

(ii) Substituting the “no hedging” constraint (5e) and the CEO’s action choice (3) into the BoD’s optimization problem (6) yields the BoD’s unconstrained optimization problem:

$$ \mathop {\hbox{max} }\limits_{{v_{f} ,v_{p} }} \;\varPi = v_{f} - \frac{{v_{f}^{2} }}{2} - \frac{r}{2}(v_{f}^{2} (\sigma_{f}^{2} + \sigma_{\eta }^{2} ) + v_{p}^{2} (\sigma_{p}^{2} + \sigma_{\eta }^{2} ) + (v_{f} + v_{p} )^{2} \sigma_{\theta }^{2} ). $$

(14)

Differentiating the BoD’s unconstrained optimization problem with respect to $v_{f}$ and $v_{p}$ and solving the first-order conditions gives the incentive weight on the firm performance, $v_{f}^{R}$ in (8a), and the incentive weight on the peer-firm performance, $v_{p}^{R}$ in (8b). ${\square }$

Proof of Observation 2

(i) If the BoD does not implement RPE and the CEO privately hedges industry-specific risk, the CEO’s certainty equivalent is given by:

$$ \begin{aligned} {\text{CE}}(a,h) = c_{0} + v_{f} a + h(\bar{x}_{p} + \mu_{p} - c_{p} ) \hfill \\ & & - \frac{{a^{2} }}{2} - \frac{1}{2}r\left( {v_{f}^{2} \left( {\sigma_{f}^{2} + \sigma_{\eta }^{2} } \right) + h^{2} \left( {\sigma_{p}^{2} + \sigma_{m}^{2} } \right) + (v_{f} + h)^{2} \sigma_{\theta }^{2} } \right). \hfill \\ \end{aligned} $$

(15)

The CEO’s action choice follows as in (3).

Note that in perfect and competitive markets, the price the CEO pays for acquiring one share of the peer firm equals the expected value of the stock, $\bar{x}_{p} + \mu_{p}$. The CEO chooses his hedging strategy to maximize the certainty equivalent in (15) considering the share price. The first-order condition with respect to $h$ is given by:

$$ \frac{{\partial {\text{CE}}(a,h)}}{\partial h} = - r\left( {h\left( {\sigma_{p}^{2} + \sigma_{m}^{2} } \right) + (v_{f} + h)\sigma_{\theta }^{2} } \right) = 0. $$

(16)

Rearranging the first-order condition yields the CEO’s hedging strategy, $h^{H}$ in (9b).Substituting the “no RPE” constraint (5d), the CEO’s action choice (3) and the CEO’s hedging strategy (9b) into the BoD’s optimization problem (6) yields the BoD’s unconstrained optimization problem:

$$ \mathop {\hbox{max} }\limits_{{v_{f} }} \;\varPi = v_{f} - \frac{{v_{f}^{2} }}{2} - \frac{r}{2}v_{f}^{2} \left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} }}] + \sigma_{\eta }^{2} } \right). $$

(17)

Differentiating the BoD’s unconstrained optimization problem with respect to $v_{f}$ and solving the first-order condition gives the incentive weight on the firm performance, $v_{f}^{H}$ in (9a).

(ii) If the BoD implements RPE and the CEO privately hedges industry-specific risk, the CEO’s certainty equivalent is given by:

$$ {\text{CE}}(a,h) = c_{0} + v_{f} a + v_{p} \mu_{p} + h(\bar{x}_{p} + \mu_{p} - c_{p} ) - \frac{{a^{2} }}{2} $$

$$ - \frac{1}{2}r\left( {v_{f}^{2} (\sigma_{f}^{2} + \sigma_{\eta }^{2} ) + v_{p}^{2} \sigma_{\eta }^{2} + h^{2} \sigma_{m}^{2} + (v_{f} + v_{p} + h)^{2} \sigma_{\theta }^{2} + (v_{p} + h)^{2} \sigma_{p}^{2} } \right). $$

(18)

Analogous to part (i), differentiating the CEO’s certainty equivalent with respect to $h$ and solving the first-order condition gives the CEO’s hedging strategy, $h^{RH}$ in (10c).

Substituting the CEO’s action choice (3) and the CEO’s hedging strategy (10c) into the BoD’s optimization problem (6), differentiating with respect to $v_{f}$ and $v_{p}$ and solving the first-order conditions gives the incentive weight on the firm performance, $v_{f}^{RH}$ in (10a), and the incentive weight on the peer-firm performance, $v_{p}^{RH}$ in (10b). ${\square }$

Proof of Observation 3

The first part of Observation 3 follows from considering the first derivative of the CEO’s hedging strategy, $h^{RH}$ in (10c), with respect to the incentive weight on the peer-firm performance, $v_{p}^{RH}$ in (10b):

$$ \left| {\frac{{\partial h^{RH} }}{{\partial v_{p}^{RH} }}} \right| = \frac{{\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} } \right)}}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} }} < 1. $$

(19)

The second part of Observation 3 follows from comparing the firm value when RPE as well as private hedging by the CEO are used to reduce industry-specific risk in CEO compensation with firm value when the BoD implements RPE and firm value when the CEO privately hedges industry-specific risk. The difference of the firm value when RPE as well as private hedging by the CEO are used and firm value when the BoD solely implements RPE follows from:

$$ \begin{aligned} &\varPi^{RH} - \varPi^{R} = \frac{1}{2}\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} (\sigma_{m}^{2} + \sigma_{\eta }^{2} )}}{{(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} )(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} ) - (\sigma_{p}^{2} + \sigma_{\theta }^{2} )^{2} }}] + \sigma_{\eta }^{2} } \right)}} \hfill \\ & \quad - \frac{1}{2}\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} }}] + \sigma_{\eta }^{2} } \right)}}. \hfill \\ \end{aligned} $$

(20)

This term is larger than zero if $r > 0$. The difference of the firm value when RPE as well as private hedging by the CEO are used and firm value when the CEO privately hedges industry-specific risk follows from:

$$ \begin{aligned} & \varPi^{RH} - \varPi^{H} = \frac{1}{2}\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} (\sigma_{m}^{2} + \sigma_{\eta }^{2} )}}{{(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} )(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} ) - (\sigma_{p}^{2} + \sigma_{\theta }^{2} )^{2} }}] + \sigma_{\eta }^{2} } \right)}} \hfill \\ & \quad - \frac{1}{2}\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} }}] + \sigma_{\eta }^{2} } \right)}}. \hfill \\ \end{aligned} $$

(21)

This term is larger than zero if $r > 0$. ${\square }$

Proof of Observation 4

If the BoD implements RPE and the CEO cannot privately hedge industry-specific risk, the CEO’s certainty equivalent is given by:

$$ {\text{CE}}(a,h = 0) = c_{0} + v_{f} a + v_{p} \mu_{p} - \frac{{a^{2} }}{2} - \frac{r}{2}\left( {v_{f}^{2} (\sigma_{f}^{2} + \sigma_{\eta }^{2} ) + v_{p}^{2} (\sigma_{p}^{2} + \sigma_{\eta }^{2} ) + (v_{f} + v_{p} )^{2} \sigma_{\theta }^{2} } \right). $$

(22)

The CEO’s action choice follows as in (3). Further, differentiating the CEO’s certainty equivalent with respect to the incentive weight on the peer-firm performance, $v_{p}$, and solving the first-order condition yields the incentive weight on the peer-firm performance, $v_{p}^{{{{\ddag }}R}}$ in (11b). Substituting the “no hedging” constraint in (5e), $h = 0$, the CEO’s action choice (3) and the CEO’s choice of the incentive weight on the peer-firm performance into the BoD’s optimization problem (6) yields the BoD’s unconstrained optimization problem:

$$ \mathop {\hbox{max} }\limits_{{v_{f} }} \;\varPi = v_{f} - \frac{{v_{f}^{2} }}{2} - \frac{r}{2}v_{f}^{2} \left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} }}] + \sigma_{\eta }^{2} } \right) - \frac{{\mu_{p}^{2} }}{{2r(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} )}}. $$

(23)

Differentiating the BoD’s unconstrained optimization problem with respect to $v_{f}$ and solving the first-order condition gives the incentive weight on firm performance, $v_{f}^{{{{\ddag }}R}}$ in (11a). ${\square }$

Proof of Proposition 1

If the BoD implements RPE and the CEO privately hedges industry-specific risk, the CEO’s certainty equivalent is given in (18). The CEO’s action choice follows as in (3). Differentiating the CEO’s certainty equivalent in (18) with respect to the incentive weight on the peer-firm performance, $v_{p}$, and the CEO’s hedging strategy, $h$, and solving the first-order conditions yields the incentive weight on the peer-firm performance, $v_{p}^{{{{\ddag }}RH}}$ in (12b), and the CEO’s hedging strategy, $h^{{{{\ddag }}RH}}$ in (12c).

Substituting the CEO’s action choice (3), the CEO’s hedging strategy (12c) and the CEO’s choice of the incentive weight on the peer-firm performance (12b) into the BoD’s optimization problem (6), differentiating with respect to $v_{f}$ and solving the first-order condition gives the incentive weight on firm performance, $v_{f}^{{{{\ddag }}RH}}$ in (12a). ${\square }$

Proof of Proposition 2

When the CEO cannot privately hedge industry-specific risk, the BoD implements RPE if:

$$ \begin{aligned} & \varPi^{{{{\ddag }}R}} - \varPi^{{{\ddag }}} = \frac{1}{2}\left( {\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} }}] + \sigma_{\eta }^{2} } \right)}} - \frac{{\mu_{p}^{2} }}{{r\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} } \right)}}} \right) \hfill \\ & \quad - \frac{1}{2}\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} } \right)}} > 0. \hfill \\ \end{aligned} $$

(24)

The condition is satisfied if

$$ \mu_{p} < \mu_{p}^{\Delta R} = r\sigma_{\theta }^{2} \sqrt {\frac{1}{{\left( {1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} }}] + \sigma_{\eta }^{2} } \right)} \right)}}} \cdot \sqrt {\frac{1}{{\left( {1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} } \right)} \right)}}} . $$

(25)

When the CEO privately hedges industry-specific risk, the BoD implements RPE if:

$$ \begin{aligned} & \varPi^{{{{\ddag }}RH}} - \varPi^{{{{\ddag }}H}} = \frac{1}{2}\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} (\sigma_{m}^{2} + \sigma_{\eta }^{2} )}}{{(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} )(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} ) - (\sigma_{p}^{2} + \sigma_{\theta }^{2} )^{2} }}] + \sigma_{\eta }^{2} } \right)}} \hfill \\ & \quad - \frac{1}{2}\frac{{\mu_{p}^{2} }}{{\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} } \right) - \frac{{\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} } \right)^{2} }}{{\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} } \right)}}}} \hfill \\ & \quad - \frac{1}{2}\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} }}] + \sigma_{\eta }^{2} } \right)}} > 0. \hfill \\ \end{aligned} $$

(26)

The condition is satisfied if:

$$ \begin{aligned} & \mu_{p} < \mu_{p}^{\Delta R|H} = \frac{{r\sigma_{\theta }^{2} \sigma_{m}^{2} }}{{(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} )}} \cdot \sqrt {\frac{1}{{\left( {1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} (\sigma_{m}^{2} + \sigma_{\eta }^{2} )}}{{(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} )(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} ) - (\sigma_{p}^{2} + \sigma_{\theta }^{2} )^{2} }}] + \sigma_{\eta }^{2} } \right)} \right)}}} \hfill \\ & \quad \cdot \sqrt {\frac{1}{{\left( {1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} }}] + \sigma_{\eta }^{2} } \right)} \right)}}} . \hfill \\ \end{aligned} $$

(27)

With $r > 0$, $\mu_{p}^{\Delta R}$ is larger than $\mu_{p}^{\Delta R|H}$. ${\square }$

Proof of Proposition 3

When the BoD implements RPE in the presence of a powerful CEO, the BoD benefits from private hedging by the CEO if:

$$ \begin{aligned} & \varPi^{{{{\ddag }}RH}} - \varPi^{{{{\ddag }}R}} = \frac{1}{2}\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} (\sigma_{m}^{2} + \sigma_{\eta }^{2} )}}{{(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} )(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} ) - (\sigma_{p}^{2} + \sigma_{\theta }^{2} )^{2} }}] + \sigma_{\eta }^{2} } \right)}} \hfill \\ & \quad - \frac{1}{2}\frac{{\mu_{p}^{2} }}{{r\left( {\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} } \right) - \frac{{\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} } \right)^{2} }}{{\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} } \right)}}} \right)}} \hfill \\ & \quad - \frac{1}{2}\left( {\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} }}] + \sigma_{\eta }^{2} } \right)}} - \frac{{\mu_{p}^{2} }}{{r\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} } \right)}}} \right). \hfill \\ \end{aligned} $$

(28)

With $r > 0$, this term is larger than zero if, and only if, $\mu_{p} < \mu_{p}^{\Delta H|R}$, with

$$ \begin{aligned} \mu_{p}^{\Delta H|R} = \frac{{r\sigma_{\theta }^{2} \sigma_{\eta }^{2} }}{{\left( {\sigma_{p}^{2} + \sigma_{\theta }^{2} } \right)}} \cdot \sqrt {\frac{1}{{\left( {1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} (\sigma_{m}^{2} + \sigma_{\eta }^{2} )}}{{(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} )(\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} ) - (\sigma_{p}^{2} + \sigma_{\theta }^{2} )^{2} }}] + \sigma_{\eta }^{2} } \right)} \right)}}} \hfill \\ \cdot \sqrt {\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} }}] + \sigma_{\eta }^{2} } \right)}}} \hfill \\ \end{aligned} $$

(29)

The threshold $\mu_{p}^{\Delta H|R}$ is increasing in the accrual noise of earnings, i.e., $\frac{{\partial \mu_{p}^{\Delta H|R} }}{{\partial \left( {\sigma_{\eta }^{2} } \right)}} > 0$. ${\square }$

Proof of Proposition 4

The BoD prefers to implement RPE only when (i) $\varPi^{{{{\ddag }}R}} > \varPi^{{{{\ddag }}H}}$ and (ii) $\varPi^{{{{\ddag }}R}} > \varPi^{{{{\ddag }}RH}}$. Rearranging the two conditions yields a range for the expected performance of the peer-firm, $\mu_{p}$, $\mu_{p}^{\Delta H|R} < \mu_{p} < \mu_{p}^{R - H}$, for which implementation of RPE only maximizes firm value. The thresholds are given in (29) and:

$$ \begin{aligned} \mu_{p}^{R - H} = \frac{{r\sigma_{\theta }^{2} \sqrt {\sigma_{m}^{2} - \sigma_{\eta }^{2} } }}{{\sqrt {\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} } }} \cdot \sqrt {\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{\eta }^{2} }}] + \sigma_{\eta }^{2} } \right)}}} \hfill \\ \cdot \sqrt {\frac{1}{{1 + r\left( {\sigma_{f}^{2} + \sigma_{\theta }^{2} [1 - \frac{{\sigma_{\theta }^{2} }}{{\sigma_{p}^{2} + \sigma_{\theta }^{2} + \sigma_{m}^{2} }}] + \sigma_{\eta }^{2} } \right)}}} . \hfill \\ \end{aligned} $$

(30)

From (30), $\varPi^{{{{\ddag }}H}} > \varPi^{{{{\ddag }}R}}$ for all $\mu_{p}$ if $\sigma_{\eta }^{2} > \sigma_{m}^{2}$.

The BoD prefers to implement RPE and private hedging by the CEO when (i) $\varPi^{{{{\ddag }}RH}} > \varPi^{{{{\ddag }}H}}$ and (ii) $\varPi^{{{{\ddag }}RH}} > \varPi^{{{{\ddag }}R}}$. Rearranging yields two conditions, $\mu_{p} < \mu_{p}^{\Delta H|R}$ and $\mu_{p} < \mu_{p}^{\Delta R|H}$, where $\mu_{p}^{\Delta R|H}$ is given by (27).

The BoD prefers private hedging by the CEO when (i) $\varPi^{{{{\ddag }}H}} > \varPi^{{{{\ddag }}RH}}$ and (ii) $\varPi^{{{{\ddag }}H}} > \varPi^{{{{\ddag }}R}}$. Rearranging yields conditions $\mu_{p} > \mu_{p}^{\Delta R|H}$ and $\mu_{p} > \mu_{p}^{R - H}$. ${\square }$

Appendix B: Additional analysis

In this appendix, we consider a more general setting where shares of both firms are traded in the capital market and the stock prices of the two firms are differentially affected by market risk. We show that our main results hold qualitatively if the BoD uses the stock price of the own firm in addition to accounting earnings for contracting with the CEO.

Assume in the following that the stock price of the firm $f$ and the stock price of the peer firm $p$ are given by

$$ P_{f} = x_{f} + \beta_{f} \varepsilon_{m} = \bar{x}_{f} + a + \varepsilon_{f} + \theta + \beta_{f} \varepsilon_{m} , $$

(31)

$$ P_{p} = x_{p} + \beta_{p} \varepsilon_{m} = \bar{x}_{p} + \mu_{p} + \varepsilon_{p} + \theta + \beta_{p} \varepsilon_{m} , $$

(32)

where $\beta_{f}$ ($\beta_{p}$) denotes the impact of market risk on the performance of firm $f$ (peer firm $p$).

To provide the CEO with incentives to exert effort, the BoD uses the accounting earnings and the stock price of the own firm. Further, in order to reduce the CEO’s exposure to compensation risk, the BoD can implement an RPE contract using accounting earnings of the peer firm as a benchmark. The contract offered by the BoD specifies the fixed wage, $c_{0}$, the incentive weight on the firm’s accounting earnings, $v_{f}$, the incentive weight on the firm’s stock price, $v_{m}$, and the incentive weight on the accounting earnings of the peer firm, $v_{p}$. Thus, the CEO’s compensation is given by

$$ c(y_{f} ,P_{f} ,y_{p} ) = c_{0} + v_{f} y_{f} + v_{m} P_{f} + v_{p} y_{p} . $$

If the CEO can trade in the stock of the peer firm, his wealth reflects the compensation and the proceedings from his hedging strategy, $c(y_{f} ,P_{f} ,y_{p} ) + h(P_{p} - (\bar{x}_{p} + \mu_{p} ))$.

If the BoD determines all parameters of the compensation contract, Fig. 5 shows the firm value depending on the accrual noise in the accounting benchmark for three cases: when (1) the BoD filters industry-specific risk using RPE, (2) the CEO privately hedges industry-specific risk, and (3) when the BoD implements RPE and allows private hedging by the CEO.

Fig. 5

Firm value in the benchmark setting where the contract is based on accounting earnings and stock price (with $\sigma_{f}^{2} = \sigma_{p}^{2} = \sigma_{m}^{2} = 1, \, \sigma_{\theta }^{2} = 4, \, \beta_{f} = 0.1, \, \beta_{p} = 1, \, r = 1$)

The result illustrated by Fig. 5 is similar to the setting where the BoD solely uses accounting earnings to compensate the CEO. In particular, the BoD prefers to implement RPE and to allow private hedging by the CEO through trading in the stock of the peer firm. This result holds even if the firm’s and the peer firm’s exposure to market risk differs. In the extreme case where the stock prices of the firm and the peer firm are identically affected by market risk, market risk can be completely filtered out by the CEO through trading in the stock of the peer firm. In this case, the BoD does not implement RPE and the CEO privately manages compensation risk.

If the CEO has the power to influence the design of his RPE contract and chooses the incentive weight on the performance of the peer firm, the BoD is less likely to use RPE. Figure 6 shows the threshold for the use of RPE if a powerful CEO can or cannot hedge market risk. Figure 6 corresponds to Fig. 2 and shows that the principal implements RPE when the expected performance of the peer firm is small.

Fig. 6

Threshold for the use of RPE when the powerful CEO does or does not privately hedge industry-specific risk and the contract is based on accounting earnings and stock price (with $\sigma_{f}^{2} = \sigma_{p}^{2} = \sigma_{\theta }^{2} = \sigma_{m}^{2} = 1, \, \beta_{f} = 0.1, \, \beta_{p} = 1, \, r = 1$)

Figure 7 illustrates the finding that in the presence of a powerful CEO, the BoD does not necessarily prefer the CEO to privately hedge market risk. Similar to the main setting, the BoD restricts private hedging by the CEO if the expected performance of the peer firm is above a threshold, where the threshold is increasing in the accounting earnings’ noise. Figure 7 corresponds to Fig. 3 and shows that the principal allows private hedging by the CEO when the expected performance of the peer firm is small.

Fig. 7

Threshold for allowing hedging by a powerful CEO when the BoD uses RPE and the contract is based on accounting earnings and stock price (with $\sigma_{f}^{2} = \sigma_{p}^{2} = \sigma_{\theta }^{2} = \sigma_{m}^{2} = 1, \, \beta_{f} = 0.1, \, \beta_{p} = 1, \, r = 1$)

Finally, Fig. 8 illustrates the trade-offs faced by the BoD in the presence of a powerful CEO who is able to trade in the stock of the peer firm. Similar to the main setting presented in Fig. 4, depending on $\mu_{p}$ and $\sigma_{\eta }^{2}$, the BoD implements RPE, allows the CEO to privately hedge market risk, or uses both instruments to reduce the CEO’s compensation risk.

Fig. 8

Thresholds for implementing RPE and allowing hedging by a powerful CEO and the contract is based on accounting earnings and stock price (with $\sigma_{f}^{2} = \sigma_{p}^{2} = \sigma_{\theta }^{2} = \sigma_{m}^{2} = 1, \, \beta_{f} = 0.1, \, \beta_{p} = 1, \, r = 1$)