Balancing difficulty of performance targets: theory and evidence

Abstract

We examine how firms balance difficulty of performance targets in their annual bonus plans. We present an analytical model showing that managerial allocation of effort is a function of not only relative incentive weights but also the difficulty of performance targets. We find that relative incentive weights and target difficulty can either be complements or substitutes in motivating effort depending on the extent to which managers have alternative employment opportunities. To test the predictions of our model, we use survey data on performance targets in annual bonus plans. Our sample of 877 survey respondents consists primarily of financial executives in small- and medium-size private companies where annual bonuses are important both for motivation and retention. Consistent with our model, we find that relative incentive weights are negatively (positively) associated with perceived target difficulty when concerns about managerial retention are high (low). It follows that performance measures included in annual bonus plans have sometimes easy and other times challenging targets depending on their relative incentive weights and retention concerns.

Appendices

Appendix 1—Analytical framework and proofs

Proof of Lemma 1. The manager solves the problem:

$$ \underset{e_i}{\max }s+{w}_1{bP}_1+{w}_2{bP}_2-\frac{1}{2}{c}_1{e}_1^2-\frac{1}{2}{c}_2{e}_2^2 $$

(3)

Differentiating with respect to e _i gives the incentive constraint (IC):

$$ {e}_i=\frac{w_ib}{c_i}g\left({e}_i-{t}_i\right) $$

(IC)

and second order sufficient condition (SOSC):

$$ 1-\frac{w_ib}{c_i}{g}^{\prime}\left({e}_i-{t}_i\right)>0. $$

(4)

Assume SOSC holds. Using (4) and differentiating IC with respect to each of the contract choices gives:

$$ \frac{\partial {e}_i}{\partial b}=\frac{g\left({e}_i-{t}_i\right)\frac{w_i}{c_i}}{1-\frac{w_ib}{c_i}{g}^{\prime}\left({e}_i-{t}_i\right)}>0, $$

(5)

$$ \frac{\partial {e}_i}{\partial {w}_i}=\frac{\frac{b}{c_i}g\left({e}_i-{t}_i\right)}{1-\frac{w_ib}{c_i}{g}^{\prime}\left({e}_i-{t}_i\right)}>0, $$

(6)

$$ \frac{\partial {e}_i}{\partial {t}_i}=\frac{-\frac{w_ib}{c_i}{g}^{\prime}\left({e}_i-{t}_i\right)}{1-\frac{w_ib}{c_i}{g}^{\prime}\left({e}_i-{t}_i\right)}>0\kern0.72em \mathrm{iff}\kern0.30em {e}_i>{t}_i.\mathrm{QED}. $$

(7)

Proof of Lemma 2. Rearranging IC and using the fact that g is symmetric around zero yields

$$ {t}_i={e}_i\pm {g}^{-1}\left(\frac{c_i{e}_i}{w_ib}\right). $$

(8)

Thus there exists δ > 0, low target ${t}_i^L={e}_i-\delta$, and a high target ${t}_i^H={e}_i+\delta,$ such that g(e _i  − t _i ^L) = g(δ) = g(−δ) = g(e _i  − t _i ^H) and IC is satisfied both for ${t}_i^L$ and ${t}_i^H$. QED.

Proof of Lemma 3. Fix an effort level e ₀ = e(t, w) from the manager’s problem. Using (6) and (7), we obtain by the implicit function theorem:

$$ \frac{\partial {t}_i}{\partial {w}_i}=-\frac{\partial e/\partial {w}_i}{\partial e/\partial {t}_i}=\frac{g\left({e}_i-{t}_i\right)}{w_i{g}^{\prime}\left({e}_i-{t}_i\right)}. $$

(9)

Given that g is increasing only over its negative domain, ${g}^{\prime}\left({e}_i-{t}_i^L\right)={g}^{\prime}\left(\delta \right)<0$ and so by (9), $\frac{\partial {t}_i^L}{\partial {w}_i}<0$. Similarly, ${g}^{\prime}\left({e}_i-{t}_i^H\right)={g}^{\prime}\left(-\delta \right)>0$, and therefore $\frac{\partial {t}_i^H}{\partial {w}_i}>0$. QED.

Proof of Proposition 1. The firm selects a contract ω = (s, b, t _i , w _i ) that maximizes gross profits, subject to the participation (PC), incentive (IC), limited liability (LL) and compensation cap constraints (CC). For ease of exposition, we first describe the firm’s optimization problem with the former two constraints only and subsequently discuss the effect of adding the latter two constraints.

Suppose that only the PC and IC constraints are binding. We know that the firm can implement first-best effort ${e}_i^{\ast }$ because both contracting parties are risk neutral.¹⁹ From Lemma 2, we also know that each effort level ${e}_i^{\ast }$ can be implemented with either a high (${t}_i^H$) or a low (${t}_i^L$) target, for a fixed bonus b and relative incentive weights w _i . Let s _jk be the salary when measure 1 is j = L , H and measure 2 is k = L , H. It follows that any first-best effort ${e}_i^{\ast }$ can be implemented with four different contracts ${\omega}_{jk}=\left({s}_{jk},b,{t}_1^j,{t}_2^k,{w}_1,{w}_2\right)$ for jk = LL , LH , HL , HH. The expected utility of the manager under each of the four contracts is ${EU}_{jk}={s}_{jk}+{E}_1^j+{E}_2^k-C\left({e}_1,{e}_2\right)$, where ${E}_i^L$ and ${E}_i^H$denote the expected bonus contingent on performance measure i under a low and high target, respectively. Low targets increase the probability of success and the expected bonus so that ${E}_i^L>{E}_i^H$. Given a binding participation constraint PC, ${EU}_{jk}=\overline{u}$, the salary under high targets must exceed the salary under low targets (s _LL  < s _HH ).

An increase in the reservation utility $\overline{u}$ does not affect the choice of first-best effort level ${e}_i^{\ast }$, but the firm has to adjust the contract to increase the manager’s expected utility. Specifically, the firm can take one of the following four actions.

• 1. Increase salary. The firm can raise salary s _jk , which has no effect on incentives and implements the same ${e}_i^{\ast }$.

• 2. Increase target bonus. The firm can raise target bonus b but that would also lead to an increase in effort(∂e _i /∂b > 0). To keep effort fixed at ${e}_i^{\ast }$, the firm can either increase the high target or reduce the low target, because $\partial {e}_i/\partial {t}_i^L>0$ and $\partial {e}_i/\partial {t}_i^H<0$. As shown below, both of these changes also increase the manager’s expected utility.

Specifically, the IC constraint implies a functional relationship between target bonus b and targets ${t}_i^j$ if effort is to remain unchanged at ${e}_i^{\ast }$:

$$ b=\frac{e_i^{\ast }c}{w_ig\left({e}_i-{t}_i^j\right)}. $$

The expected bonus is ${E}_i^j={w}_i bG\left({e}_i-{t}_i^j\right)={e}_i cG\left({e}_i-{t}_i^j\right)/g\left({e}_i-{t}_i^j\right).$ Differentiating this expected bonus with respect to target ${t}_i^j$ yields the following:

$$ \frac{\partial {E}_i^j}{\partial {t}_i^j}={e}_ic\left[\frac{-g{\left({e}_i-{t}_i^j\right)}^2+G\left({e}_i-{t}_i^j\right){g}^{\prime}\left({e}_i-{t}_i^j\right)}{g{\left({e}_i-{t}_i^j\right)}^2}\right]. $$

For low targets ${g}^{\prime}\left({e}_i-{t}_i^L\right)<0$, and therefore $\partial {E}_i^j/\partial {t}_i^L<0$, so decreasing the low target will increase expected bonus. For high targets ${g}^{\prime}\left({e}_i-{t}_i^j\right)>0$, but log-concave G implies that Gg ^' > g ², which assures that $\partial {E}_i^j/\partial {t}_i^H>0$, so increasing the high targets also increases expected bonus.

• 3. Change relative incentive weights. The firm can also change relative incentive weightsw _i . WLOG, suppose the performance measures are sorted in the sense that i = 1 denotes the performance measure that accounts for a majority of the target bonus. The manager’s expected utility can be increased without changing ${e}_i^{\ast }$ as follows. If the target accounting for the majority of the target bonus is low (${t}_1^L$), the firm can simultaneously increase w ₁ (which increases effort) and reduce ${t}_1^L$ (which decreases effort), holding effort unchanged. If the target accounting for the majority of the target bonus is high (${t}_1^H$), the firm can simultaneously increase w ₁ (which increases effort) and increase ${t}_1^H$ (which decreases effort), also holding effort unchanged. Given that w ₁ > 0.5, marginal changes in total expected bonus will have the same sign as marginal changes in ${E}_1^j$. As shown above, $\partial {E}_i^j/\partial {t}_i^j<0$ for low and $\partial {E}_i^j/\partial {t}_i^j>0$ for high targets, and the IC constraint implies that higher w ₁ has to be accompanied by decreasing the low target and increasing the high targets.

• 4. Switch targets. The firm can leave target bonus b and relative incentive weights w _i unchanged and select between the four different contracts ω _jk by switching from high targets to low targets. This will increase the expected bonus since the probability of success is greater under a low target $\left({P}_i^L=G\left({e}_i-{t}_i^L\right)>G\left({e}_i-{t}_i^H\right)={P}_i^H\right)$ but implement the same effort ${e}_i^{\ast }$ by Lemma 2.

Without additional assumptions, the firm can choose any (combination) of the above four actions to increase the agent’s expected utility until PC binds. However, the third action becomes infeasible at some point, as the reservation utility changes, because w _i has to be between zero and one by definition. Similarly, the LL and CC constraints impose bounds on the salary ($\underline{s}\le s\le \overline{s}$) and target bonus ($0\le b\le \overline{b}$), and when these constraints are binding, switching targets (the fourth action) becomes the only feasible way to further increase or reduce the manager’s expected compensation.

For example, the highest possible expected utility for a given choice of targets jk is ${\overline{u}}_{jk}^{\ast }=\overline{s}+{w}_1\overline{b}G\left({e}_1^{\ast }-{t}_1^j\right)+{w}_2\overline{b}G\left({e}_2^{\ast }-{t}_2^k\right)-C\left({e}_1^{\ast },{e}_2^{\ast}\right)$.

For any reservation utility above this threshold, the firm must switch some high targets to low targets. For sufficiently high reservation utility, $\overline{u}>\max \left\{{\overline{u}}_{LH}^{\ast },{\overline{u}}_{HL}^{\ast}\right\}$, the firms will choose low targets on both measures, jk = LL. A symmetric argument applies for decreases in reservation utility. QED.

Appendix 2—Survey questions

SALARY: Your annual base salary in [year_t-1]²⁰ was approximately

TBONUS: If [year]²¹ performance meets all targets, the [year] annual bonus will be approximately

If your [year_t-1] bonus plan included a nonfinancial performance target fitting one or more of the broad categories below, please check the box next to the categories. You can also describe your nonfinancial performance targets in the text boxes.

Customers, market, and strategy

(e.g., market share, customer satisfaction, strategic milestones)

Operations

(e.g., efficiency, safety, quality, process improvement, cost control)

Sustainability

(e.g., energy use, emissions, social reporting, stakeholder satisfaction)

Financing and investment

(e.g., working capital management, capex planning, M&A deals, divestitures, investor relations)

Accounting, reporting, and IT systems

(e.g., timeliness and efficiency of reporting, management satisfaction, IT projects)

Teamwork and human resource management

(e.g., employee turnover, leadership, collaboration, and communication)

If [year] performance meets all targets, what percentage of this bonus will you earn based on

WEIGHT: Financial performance targets

WEIGHT_t Nonfinancial performance targets(e.g., market share, strategy milestones, customer satisfaction)

WEIGHT_t [Alternatively] Nonfinancial performance targets related to [category label]²²

Achievements evaluated subjectively (i.e., without objective targets)

WEIGHT_t Other

Given the current business environment, how likely is it that you will meet your [year] bonus targets?

Bonus target refers to the performance level that earns you the full targeted bonus (as opposed to some minimum performance level below which no bonuses are paid or some maximum performance level at which bonuses may be capped).

PROB: Earnings targetPROB: Other financial performance targetsPROB_t: Nonfinancial performance targetsPROB_t [Alternatively] Nonfinancial performance targets related to [category label]

To what extent do you agree with the following statements?

RETAIN: Retention of executives is the key objective of our [year] bonus plan

CAPITAL: Our [entity] has adequate (access to) capital for the near term

Scales: Strongly agree / Somewhat agree / Neither agree nor disagree / Somewhat disagree / Strongly disagree / N/A

SALES: Sales of your company in [year] were approximately (in $ millions):

SIZE: Number of [entity] employees in [year_t-1]?

ROS and FAIL: Profitability of your company in [year_t-1] was approximately (in $ millions)?

Actual profit/loss

Budgeted profit/loss

GROWTH: How would you characterize the long-term (5–10 years) business prospects of your company?

Expected annual growth in sales

Scale: Negative / 0–5% / 6–12% / 13–20% / More than 20% / N/A

NOISE: To what extent do financial performance measures reflect management’s overall performance?

Scale: Not at all / Low / Medium / High / Very high / Don’t know

CEO, CFO: Which of the following best describes your job?

CEO (the top executive)

CFO (or similar title referring to the top financial executive)

Other financial executive (reporting to the top financial executive)

Other, please specify:

PUBLIC: Is the company you are a part of:

Publicly traded

Privately owned

BU: Are you answering for:

Corporate level

Division level

Other, please specify

INDUSTRY: Please describe your industry. Select from the list below

Manufacturing / Finance and Insurance / Wholesale Trade / Retail Trade / Transportation and Warehousing / Construction / Real Estate / Professional, Scientific and Technical Services / Hospitality and Food Services / Healthcare / Information and Media / Education / Arts, Entertainment and Recreation / Utilities / Mining and Oil & Gas / Agriculture, Forestry, Fishing and Hunting / Holding Company or Conglomerate / Other