Relative Performance Evaluation in Executive Compensation Contracts

J. Carr Bettisa

Arizona State University, Incentive Lab

John Bizjakb

Texas Christian University

Jeffrey Colesc

Arizona State University

Brian Youngd

Mississippi State University

First Attempt: April 28, 2012

This Draft: January 30, 2014

Preliminary and Incomplete

Please do not quote without permission.

a W.P. Carey School of Business, Arizona State University, Tempe, AZ, 85287, USA; carr@asu.edu b Neeley School of Business, Texas Christian University, Fort Worth, TX, 76129, USA; j.bizjak@tcu.edu c W.P. Carey School of Business, Arizona State University, Tempe, AZ, 85287, USA; jeffrey.coles@asu.edu d Mississippi State University, Mississippi State, MS 39762, USA; brian.young@msstate.edu

The authors are grateful to Incentive Lab, a Scottsdale, Arizona compensation data and science firm, for generously

providing the data. The analysis and conclusions in this paper are those of the authors and have been formulated

independently of the views of Incentive Lab. For financial support, Bizjak thanks TCU, Coles is grateful to ASU, and

Young thanks MSU. The authors thank seminar participants at the University of Utah for helpful comments.

Relative Performance Evaluation in Executive Compensation Contracts

Abstract

Using data that includes specific contractual details of Relative Performance Evaluation

(RPE) contracts granted to executives for 1,833 firms for the period 1998 to 2012, we develop new

methods to characterize RPE awards and measure their value and incentive properties. The

frequency in the use of these awards has grown over time with 37% of the firms in our sample

granting an RPE award in 2012. When RPE awards are used they are typically granted to the five

named executive officers and they represent about 32% of total recipient compensation. Stock is

most frequently the instrument conveyed, followed by cash, and options are almost never granted.

RPE awards are more likely to be used at firms with diversified business lines, less concentrated

industries, greater exposure to systematic risk, larger size, lower M/B, higher dividend yield, fewer

insiders on the board, greater institutional ownership, and that engage a compensation consultant.

The typical award is a rank-order tournament based on three year stock returns compared

to a select group of 13 peers (median) and is paid out with stock. Payout functions typically include

regions of concavity, convexity, explicit inelasticity, and implicit inelasticity. The median firm

achieves a threshold for at least some payout of stock or cash about 70% of the time and target

payout about 50% of the time. In general, RPE grant value differs significantly from the fair

market value reported by firms. We find that RPE awards convey to executives the incentive to

increase shareholder wealth. RPE awards of stock contingent on either stock or accounting

performance and RPE awards of cash contingent on accounting performance convey the incentive

to increase firm risk, while RPE cash awards do not. These incentives can be significant in

comparison to those conveyed by APE grants with similar attributes.

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I. Introduction

In the context of the standard agency problem (Holmstrom, 1979; Shavell, 1979), the

intuition for using relative performance evaluation (RPE) in the structure of incentive

compensation is compelling. When agents face common shocks then their compensation should

depend not only on the agent’s own performance but also on the performance of the others

(Holmstrom, 1982). For example, if firms face a common exogenous random shock, because of

macroeconomic movements or shifts in industry conditions, for instance, then an optimal

compensation contract for a CEO or other top executive would be contingent on performance of

the firm relative to other firms. RPE is consistent with removing forces that the CEO can’t control,

thereby increasing the principal’s power of inference from observables about unobservable

managerial actions. Insulating the executive from such risk permits better risk sharing and more

powerful incentive alignment than would be possible otherwise.

Further impetus for using RPE in executive pay arises from shareholders, institutional

investors, and the media. A common shared criticism of stock and option grants is that corporate

executives can and frequently do benefit from broad stock market gains for which the executives

are not responsible (e.g., see the analysis in Garvey and Milbourn, 2006). On the other side, one

suspects that executives themselves resist the notion that their equity-based awards should be

structured so as to expose award value to broad downward movements in the stock market. Such

“pay for luck, whether it is good or bad, can be diminished or perhaps eliminated by paying

executives based on own-firm performance relative to performance of other firms.

RPE could be implemented by various contracts, including indexed options or formulaic

payouts based on return compared to a broad index of performance, industry performance, or

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performance of a peer group. Likewise, Lazear and Rosen (1981) suggest that in some

circumstances a rank-order tournament among competitors is optimal. Without knowing what

specific forms might characterize contracts, researchers have searched for implicit evidence of

RPE. The general approach has been to regress CEO pay against the firm’s return and some

benchmark performance measure, such as industry or market return. A positive coefficient on firm

return and a negative coefficient on the benchmark return would be evidence in favor of RPE. As

Albuquerque (2009, Table 1) indicates, however, numerous attempts to detect RPE have been

largely unsuccessful. Among others, Antle and Smith (1986), Barro and Barro (1990),

Janakiraman et al (1992), Jensen and Murphy (1990), Aggarwal and Samwick (1999a, b),

Himmelberg and Hubbard (2000), Garvey and Milbourn (2003, 2006), and Rajgopal et al. (2006)

find little support for the existence of RPE. Gibbons and Murphy (1990) find some evidence of

RPE, but the mechanism appears to be through board discretion over salary and bonus rather than

a formulaic contract specifying payout of cash, stock, or options.

Misspecification of the regression model is an obvious potential explanation for the paucity

of supporting evidence. For example, it is likely that firms face varying economic circumstances,

so that it is appropriate for some firms to use RPE while other firms should not and do not. Even

among firms that do use RPE, the functional dependence of CEO compensation on firm

performance is unlikely to be the same as specified by the regression models. Furthermore, a

regression model must specify the elements of pay covered by RPE (e.g., cash bonuses versus

stock or options), the timing of pay, any performance measure(s) (e.g., accounting versus market

performance) and measurement period(s), and the component contributors to the performance

benchmark (such as the industry or members of a peer group), and control variables. Finally,

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endogeneity concerns arise when performance affects pay and causation likely runs in the other

direction as well. While some studies attempt to address misspecification,1 for RPE Lewellen

(2013, p. 1) asserts that “thirty years of empirical research has found little evidence consistent with

the agency hypothesis.”

Perhaps a more promising line of attack would pursue explicit evidence from surveys or

financial disclosures. Based on data from a 1997 Towers Perrin survey of 177 large U.S. firms,

Murphy (1999) reports claimed RPE usage by 51 of those firms in the annual cash bonus plan.

These data are quite limited but serve as early evidence of RPE usage. Carter et al (2009) examine

the 2002 financial reports for UK firms in the FTSE 350 index and find 129 of 252 firms report

usage of RPE as a component of total compensation. Gong et al (2011) examine the proxy

statements of the firms in the S&P 1500 for the year 2006, just following the implementation by

the SEC of enhanced compensation disclosure requirements, and find 361 of 1,419 firms report

using RPE in compensating executives and that implicit tests fail to detect RPE in the 2006 data.

These three papers combined provide some initial indication that at least some firms use RPE

contracts. Nonetheless, due in part to data limitations, these studies rely on small samples, provide

little detail on the specific characteristics of grants to executives, and provide no evidence on usage

patterns through time.

From Incentive Lab we obtain hand-collected data from proxy statements for 1,833 large

US firms on all long-term grants of stock, options, and cash to named executive officers (NEOs)

1 Albuquerque (2009) and Lewellen (2013) attack misspecification of the peer group. Antle and Smith (1986) in

part address the notion that perhaps only some firms use RPE.

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over the period 1998-2012.2 The data facilitate a comprehensive characterization of the incidence

and form of both performance-vesting (p-v) and traditional time-vesting (t-v) provisions in large

US listed companies. As Figure 1 indicates, traditional time-vested grants, which are not

contingent on a performance metric, are being displaced by awards that use a p-v provision that

specifies the size of the award contingent on some measure of firm performance. Approximately

86 percent of large US companies use a p-v provision in one or more awards to one or more NEOs

during 2012. While Bettis, Bizjak, Coles, and Kalpathy (2013) use these data to focus on p-v

awards that use absolute performance evaluation (APE), we focus attention on RPE. Of the firms

using conditional p-v awards in 2012, almost half specify that performance be measured relative

to performance of other firms.

RPE has been and continues to be a common feature of executive compensation contracts.

Beginning with about 13 percent usage in 1998, RPE is present over the entire 1998-2012 period.

In 2012 three in eight large U.S. companies will have issued an RPE grant to one or more NEOs.

Conditional on any RPE usage in 2012, on average 4.6 of the (five) NEOs receive an RPE award,

and RPE represents about 32 percent of total compensation among those recipients, based on the

grant date value reported in firm financial disclosures. The first contribution of this study is to

document frequent, longstanding usage of RPE in compensation contracts of corporate executives.

Second, the propensity to use RPE increases when the industry is less concentrated, firm

exposure to systematic risk is greater, and when the firm is larger and has lower M/B, higher

2 Incentive Lab initiated the data collection project in 2002. Because of the wide span of data items collected, wide

variation in how firms report compensation data, and extensive efforts for quality control, the Incentive Lab data have

become publicly available only in 2013. Both the data collection time interval and dataset contain 2006, the data year

used in Gong, Li, and Shin (2011),

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dividend yield, fewer insiders on the board, greater institutional ownership, and the firm uses a

compensation consultant. Consistent with the theoretical predictions from the agency model, RPE

usage is less likely when aggressive price or quantity competition is destructive and more likely

when the stock return of the firm is a noisier signal of managerial actions, the firm is more exposed

to systemic (common) shocks, and the firm operates across diverse business segments.

Third, when disclosure is sufficiently comprehensive, careful examination of the proxy

statements allows us to characterize fully the RPE award as a function of performance. We find

that 79% to 89% of the time that a firm makes a RPE grant the firm uses a rank-order tournament.

Under this schema performance is measured for the target firm and a group of peers for a defined

period of time. After the performance period ends, the granting firm is ranked by performance

among the peer firms. The percentile rank is then mapped grant schedule to the payout of shares,

options, or cash to the executive. Figure 2, Panel A depicts a roughly-representative p-v RPE

percentile stock grant to the CEO of Allete, Inc. The number of shares to be conveyed to the CEO,

specified as a percentage of a target number of shares, depends on the ranking of Allete total stock

return (TSR) over a three-year performance period relative to TSR of 16 peer firms. This

tournament in TSR yields a lumpy grant schedule as a function of finishing position (Panel A).

Recasting the number of shares granted based on Allete TSR as the domain, the grant schedule

resembles a saw blade, with the location of steps depending on the ex post realization of

performance for the 16 peers. Panels B and C depict the schedule for two possible realizations of

performance of the 16 peers.

The other primary form of RPE grant schedule specifies the number of units of the back-

end instrument conveyed as a piecewise-linear, mostly-continuous function of firm performance

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net of a benchmark. Panel A of Figure 3 depicts a grant schedule, with cash as the back-end

instrument, for the CEO of American Express in 2008, that depends on three-year annualized TSR

of the company net of a benchmark of TSR for the S&P 500.

While p-v RPE grant schedules vary widely in structure, some rough general statements

are possible. Stock is the most common back-end instrument, cash is second in frequency, and

RPE grants of options are rare. The most common performance measure is stock return, though

some grants use accounting or other metrics. Most grants use a single performance metric, though

others use two or more. The most common measurement period for performance is three years.

Often awards are described by a target number of back-end units and then the grant schedule is

defined by percent of target. Both percentile (saw-blade) and “smooth” (or benchmark-adjusted)

grants can have a jump in payout at a threshold, an increasing number of shares granted as

performance or percentile increases over an incentive zone (which often contains a point

designated as target), and a ceiling on units granted. Our data indicate that, when the grant

schedule contains such milestones (threshold, target, ceiling), the corresponding performance

requirements pose meaningful hurdles. Finally, many grant schedules have convexity in the

incentive zone and at threshold, concavity at the ceiling and threshold, discernible performance

sensitivity in the incentive zone around target, and no additional performance sensitivity (beyond

the value of the shares or options) below threshold and above the maximum. On the other hand,

some grant schedules display significant concavity around target in the incentive zone.

Finally, we develop and implement new methods to measure the value and incentive

properties of p-v RPE grants. Even relatively basic RPE grant schedules, such as the Allete and

American Express examples, can be complex or “messy,” as are the implied mappings from firm

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and peer (and benchmark) performance to value of the grant at the end of the performance period.

For this reason, and because the grant is contingent on one or more accounting or other non-price

measure, the underlying arguments for using risk-neutral valuation methods often are not

applicable. Simulations based on our risk-adjusted approach suggest that the economic value of

the typical RPE award departs significantly from the value reported in the firm’s financial

disclosure. This departure has broad implications. All parties to the formation of pay in practice,

including shareholders, institutional investors, and regulators, and perhaps even the media, have a

fundamental need for the facts. Moreover, solving the measurement problem is a necessary

condition for assessing whether the level and incentive properties of executive pay are appropriate,

vary in the cross-section according to hypothesized economic factors, or affect firm value and risk.

For a given grant the typical measure of executive incentive alignment, delta, compares the

value of the grant before and after perturbation of stock price upward. The typical measure of the

incentive to take risk, known as vega, gauges convexity though a comparison of grant value after

and before a change in volatility of stock return. For a p-v APE grant based on stock performance,

these measures are intuitive and not too difficult to apply (see Bettis, Bizjak, Coles, and Kalpathy,

2010, 2013). For a p-v grant based fully or in part on accounting performance or for a p-v RPE

grant contingent on peer or benchmark performance, some adjustments to delta and vega are

required. Because accounting performance and peer performance often are correlated with stock

returns, we develop new measures of delta and vega that aggregate the direct, traditional effects

on executive incentives that come through stock performance and stock return volatility and the

indirect effects that arise through accounting and peer-firm performance and volatility. We find

that p-v RPE grants convey significant managerial incentives. Our analysis indicates that RPE

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provisions shape the incentives of executives to advance shareholder interests and take risk, effects

that are likely to imply substantial consequences for firm performance, risk, investment policy,

and financial policy.

The remainder of this paper is as follows. Section II describes the data. Section III

provides descriptive statistics for RPE usage, hypotheses for RPE usage, tests regarding the

determinants of usage, and descriptive statistics for the role of RPE in overall compensation.

Section IV presents statistics on contractual details. Section V develops a new framework for

characterizing RPE grants and measuring their value. Section VI develops new measures of the

incentive properties of RPE awards. Section VII empirically implements these new methods to

analyze award outcomes and measure grant date expected value and incentives conveyed by RPE

contracts. Section VIII concludes.

II. Data

We obtain from Incentive Lab detailed data from proxy statements (DEF 14A) on the

various aspects of long- and short-term stock, option, and cash awards to named executive officers

(NEOs) over the period 1998-2012. The sample of firms is based on the largest 750 firms,

measured by market capitalization, in each of those years. The set of 750 largest firms changes

from year to year. Back- and forward-filling yields 1,833 firms during the period between 1998

and 2012, though data will not be available for some firms in a given year for the usual reasons

(e.g., merger, not listed). In each year, the sample fully contains the S&P 500 and encompasses

about 80% of the S&P 400 (mid-cap) and 5% of the S&P 600 (small-cap). We have 19,435 firm-

years in sample. In 3,620 of those firm-years the company made one or more awards using RPE

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to one or more NEOs. The vehicle for RPE always is a performance-vesting provision. Some p-

v provisions do not use RPE so we also record the presence and characteristics of p-v awards that

use absolute performance evaluation (APE).

Several items are noteworthy. Since multiple individuals can receive an award, there can

be many firm-year-person observations per firm-year. Some individuals receive at the same time

multiple RPE and APE awards. In some cases the grant “components” are side-by-side and do not

explicitly interact (though value can be correlated). In other cases one grant is contingent on

another. Thus, there can be multiple firm-year-person-component observations per firm-year-

person. As a further complication, each component may have its own performance period and

payout table in the DEF 14A. Herein, each table specifies the level of reporting. For some analysis

it is appropriate to roll data up to the firm-year level.

At the award level the data contain recipient name and title, award date, target payout

amount, ex-post vesting conditions, the number of components, and type of each component

(relative or absolute). The interaction between components, if any, also can be available. At the

component level the data contain the performance measure (i.e. stock returns or accounting data),

the specific details on construction of the performance measure, and the characteristics of the

performance benchmark(s) (including peer firms or performance index). For each associated

performance period and payout table (typically just one), the data contain the method of

comparison of performance measures, the range of dates of the performance period, and the details

of the grant function. More recent data contain the name of the compensation consultant used -

we have these data for 3,711 firm-years from 2006 to 2011. We supplement our data from CRSP

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and Compustat. We obtain board data from Risk Metrics. Data on institutional ownership come

from 13F filings made available by Thomson Reuters.

III. RPE Usage and Role in Overall Compensation

An advantage of the data used in this study is that it is the only longitudinal data set for

RPE awards to executives. In this section we present statistics on the role of RPE in compensation

and test for factors associated with RPE usage, first-time adoption, and discontinuation.

III.A. RPE Usage

Panel A of Table 1 reports grant patterns through time to named executive officers (NEOs).

Firms using long-term (performance period exceeds one year) RPE awards constitute 18.6% of the

firm-years in our total sample, with firm-year usage rising through time to 37% of all firms in

2012. Including only long-term awards, in 2012 69.9% of the firms in our sample grant stock,

options, or cash with a p-v provision that extends beyond one year.3 Of these, roughly half use an

absolute p-v provision, while the other half (34.4% of firms in 2012) employ long-term RPE in the

grant schedule.4 Three in eight large U.S. firms issue one or more short- and/or long-term RPE

awards in 2012. RPE usage is consistent after adoption, with the median firm using an RPE award

in 87.5% of the years subsequent to initial adoption. Even though reporting prior to the 2006

enhancement in standards is likely to be incomplete, Panel A reports direct evidence of RPE usage

for large U.S. firms over the full sample period. When used, RPE awards constitute a significant

3 This is consistent with the data in BBCK (2013), which indicate that over the past 15 years performance vesting is

displacing time vesting in stock and option grants. 4 Compare the RPE usage figures in Table 1 to 25%, 29%, 51%, and 28% for Gong et al (2011), Murphy (1999),

Carter et al (2009), and Bannister and Newman (2003) respectively – from small samples that cover one year only.

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portion of compensation. Based on reported fair market value at the date of grant (or the grant-

date value of the target award when FMV is not available), RPE awards represent approximately

32% of overall compensation from 2006 to 2012. The reported proportion of value from RPE is

even higher in earlier periods.

Panel B shows that, conditional on an RPE award, by 2012 the back-end instrument has

shifted primarily to stock (82.4% of grants), with options rarely used (1.9%), and cash used with

intermediate frequency (27.5%). Panel C describes who receives RPE awards. If the firm uses

RPE, CEOs are included 91% to 96% of the time and on average 4.5 of the top five executives

receive an RPE award. In contrast, non-employee directors rarely receive RPE grants.

III.B. Determinants of Usage

We now explore firm and industry characteristics that are associated with RPE usage. We

generate several hypotheses and apply standard statistical tests. Nonetheless, without stronger

identification strategies we are unwilling to make strong claims about causation. We view the

results in this section as largely descriptive.

The Informativeness Principle (Holmstrom, 1982) states that an optimal contract should

depend on any variable that is additionally informative about the agent’s actions. A corollary is

that RPE, based on one or more variables that reflect common shocks, should be a component of

the optimal contract. Restated in the negative, if there is no common shock to remove from

performance, so there is little association between industry, peer-firm, or systematic returns and

own-firm performance, then RPE as a contractual feature adds little value. Other circumstances

as well likely reduce the desirability of an RPE contract. First, Lazear and Rosen (1981) point out

that RPE can be destructive in concentrated industries where cooperation among firms, in terms

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of price or quantity decisions, for example, may prove beneficial to all. Second, it is likely to be

more difficult for firms operating across multiple segments to assemble a set of peers that reflect

similar business lines and remove common shocks. Third, a manager with a lower skill level or

lesser desire (higher cost) to work may prefer to be paid based on a noisier signal of output.

Accordingly, firms with a governance structure that imposes low monitoring would be less likely

to use RPE. Finally, it is likely that RPE contracts are sufficiently complex that the advice of a

compensation consultant is required. It is also possible that the compensation consultants justify

their fees by introducing innovative and complex compensation contracts, so that the use of RPE

is driven by the presence of a compensation consultant. Regardless of the direction of causation,

in the data we observe many similarities in RPE contractual structures and wording that lead us to

believe the compensation consultant is central in the adoption and form of these awards. We

propose the following hypotheses:

H1a The propensity to use RPE increases as firm return is associated more strongly

with systematic return.

H1b The propensity to use RPE increases with lower industry concentration.

H1c The propensity to use RPE increases in firm focus.

H1d

The propensity to use RPE increases with governance structures that imply

stronger monitoring (a smaller board, fewer insiders on the board, and higher

institutional ownership).

H1e The propensity to use RPE increases if the firm utilizes a compensation

consultant.

To test these hypotheses, we estimate the following logistic regression for RPE versus no

RPE and, conditional on the presence of a p-v provision, for p-v RPE versus p-v APE.

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logit[Pr(𝑅𝑃𝐸 = 1)]

= 𝛽0 + 𝛽1𝑆𝐸𝐺_𝐻𝐸𝑅𝐹 + 𝛽2 𝐼𝑁𝐷_𝐻𝐸𝑅𝐹 + 𝛽3𝑀𝐾𝑇_𝑅𝐼𝑆𝐾

+ 𝛽4 𝐿𝑁_𝐴𝑆𝑆𝐸𝑇𝑆 + 𝛽5𝑀/𝐵 + 𝛽6𝑅𝑂𝐴 + 𝛽7 𝐼𝑁𝐷𝐴𝐷𝐽_𝑅𝑂𝐴 + 𝛽8𝑅𝐸𝑇

+ 𝛽9𝐼𝑁𝐷𝐴𝐷𝐽_𝑅𝐸𝑇 + 𝛽10 𝑅𝐸𝑇_𝑉𝑂𝐿 + 𝛽11𝐶𝐹_𝑉𝑂𝐿 + 𝛽12 𝑆𝐴𝐿𝐸𝑆_𝐺𝑅

+ 𝛽13 𝐼𝑁𝑉 + 𝛽14𝐷𝐼𝑉_𝑌𝐼𝐸𝐿𝐷 + 𝛽15 𝑃𝐶𝑇_𝐼𝑁𝑆𝐼𝐷𝐸

+ 𝛽16𝐵𝑂𝐴𝑅𝐷_𝑆𝐼𝑍𝐸 + 𝛽17 𝐼𝑁𝑆𝑇𝑂𝑊𝑁 + 𝛽18𝐶𝑂𝑁𝑆𝑈𝐿𝑇 + ��

(1)

All variables are defined in Appendix A. We lag all dependent variables by one year.

MKT_RISK is constructed per Gong et al (2011). This variable describes the portion of firm

returns that can be explained by market returns. Since higher values of this variable mean

systematic risk has a greater effect on the firm’s stock returns, a positive coefficient for this

variable supports Hypothesis H1a. IND_HERF measures industry concentration using the

Herfindahl Index. The maximum value of one signifies a pure monopoly while a lower value

approaching zero signifies perfect competition. A negative coefficient provides support for H1b.

SEG_HERF serves as a proxy for the ability to define a peer group that has similar expected

returns. The maximum value of one means the firm’s sales are fully concentrated in one industry

and a value approaching the minimum of zero is firm diversified into many lines of business. A

positive coefficient supports H1c. PCT_INSIDE and BOARD_SIZE proxy for strength of

governance.5 Larger values for each conventionally are viewed as signifying weaker governance,

so a negative coefficient supports H1d. INSTOWN proxies for the level of monitoring of the board

by large shareholders, so a positive coefficient supports H1d. CONSULT is a dichotomous

5 While this broad generalization is often accepted by researchers, Coles et al (2008) demonstrate that some firms have

characteristics that require larger board size and a less independent board.

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variable equal to one if the firm uses a compensation consultant. A positive coefficient supports

Hypothesis 1e. All other variables serve as controls.

Panel A of Table 2 presents our results. In models (1) and (2), the dependent variable takes

the value of 1 if a firm made an RPE award to one or more NEOs in that year, and 0 otherwise (no

p-v or p-v APE). Model (2) differs from model (1) only by inclusion of a variable indicating use

of a compensation consultant.

In model (1), the coefficient for MKT_RISK supports the hypotheses (H1a) that firms are

more likely to use RPE if the firm has greater exposure to systematic risk. The negative coefficient

on IND_HERF suggests that a less competitive product market leads to a lower likelihood of RPE

usage, which is consistent with H1b and similar in direction to the results in Gong et al. (2011) in

the 2006 cross section. The estimated coefficient, however, is statistically insignificant. Our

approach differs from that in Gong et al. (2011) insofar as we measure industry concentration using

the three-digit SIC and cluster our errors at the firm level. Clustering errors reduces the t-statistic

from -3.32 to -1.90. The coefficient on SEG_HERF is inconsistent with H1c. Firms operating

across more business lines are more likely to use RPE. This result warrants further investigation

to identify, for example, the characteristics of peers specified by conglomerate firms.

Of the governance variables, only the positive coefficient on PCT_INSIDE supports the

hypotheses (H1d) that “better” governance increases the propensity to use RPE. In contrast, the

estimated coefficients on BOARD_SIZE, INSTOWN, and DUALITY are insignificant in both

specifications.

Model (2) differs from model (1) by inclusion of the consultant dummy, CONSULT, and

by the lack of data on compensation consultants prior to 2006. The estimated coefficient on

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CONSULT is positive and highly significant, which is consistent with the hypothesis (H1e) that

the presence of compensation consultant is associated with an increased propensity to use RPE.

The empirical support for the effect of common risk and managerial ownership, however, is

diminished. In both models, RPE firms tend to be larger and have higher sales growth and dividend

yield. Though we use different models, data, explanatory variables, and controls than Gong, Li,

and Shin (2011), our results are roughly supportive of their estimates, which are based only on

2006 data.

Models (3) and (4) repeat the analysis for the subsample of firms that use p-v RPE or no

p-v RPE (meaning APE) in a given year. Because we exclude firms in years that the firm makes

no p-v award, as compared to models (1) and (2) the samples are smaller and the results

(potentially) differ. Indeed, the signs of the coefficients are similar to those in models (1) and (2),

though statistical significance in some cases is lessened.

RPE usage is persistent once adopted, so it is possible that using our panel data essentially

duplicates the same cross-section. Moreover, we have an interest in identifying the reasons for

switching from RPE to APE or the reverse. Models (5) and (6) perform logit analysis of adoption

of RPE and elimination of RPE, respectively. Exposure to common shocks is a primary factor

positively associated with adoption of RPE (model (5)) and negatively related to dropping RPE

(model (6)).

Panel B provide results for the propensity to use a stock performance metric (MEASRET

= 1, 0 otherwise), stock (versus cash or options) as the back-end instrument (STOCKPAY = 1, 0

otherwise), or both (STOCKRET = MEASRET x STOCKPAY). Higher exposure to systematic

risk reduces the likelihood that the RPE grant uses a performance metric based on stock price

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(model (8)) and increases the probability that stock is the back-end instrument (model (7)). Higher

stock return volatility reduces both, while use of a compensation consultant increases both.

IV. RPE Contractual Details

This study provides explicit evidence of frequent and persistent usage of RPE in

compensation contracts for executives. Based on output of the Incentive Lab data project, we now

describe in detail the characteristics of observed RPE contracts.

Performance-vesting awards can consist of a single relative or absolute award. Other

awards have multiple components. In our sample, 45.8% of RPE awards consist of a single relative

component. Of awards with multiple components, 25% of the awards have interacting

components.6 For example, one award in our sample to the CEO of Macy’s consists of an initial

hurdle of $8 billion in EBITDA. If that hurdle were to be surmounted, the CEO would receive

company shares in three components, with 50% of the award continuously contingent on EBITDA

scaled by sales, 30% contingent on ROIC, and 20% contingent on TSR relative to a peer group.

A common element in RPE design is selection of a peer group. As shown in Table 3, the

proportion of RPE awards based on selection of individual firms to comprise a peer group varies

from 62% to 76% across years. Otherwise, 16% to 30% of the time the firm uses the constituents

of an industry-specific index as the definition of the peer group, and 12% to 21% use the

constituents of a broad index, such as the S&P 500. Once again, the percentages add to more than

100% because of simultaneous usage by a firm in a grant.

6 Various types of interactions include the achievement of one award to trigger the use of another award, the

substitution of one award if the other isn’t met, and modification of the payout plan contingent on performance with

the other award.

17

An RPE award measures a signal that proxies for the executive’s actions. Panel A of Table

4 reports that 70% to 83% of RPE awards use a performance metric based on stock price, while

28% to 39% specify an accounting metric. Some awards use both, so the figures in a given year

sum to more than 100%. These proportions are very different for absolute p-v awards. Per BBCK

(2013), as of 2012 approximately 75% of conditional p-v APE grants use an accounting

performance metric, while 48% use a stock performance metric. Panel B describes the accounting

metric that is measured when the conditional p-v RPE grant is based on accounting performance.

The two largest categories in our sample are Earnings Growth (38% to 60%, depending on year)

and Return on X (35% to 71%, across years), where X could stand for Equity, Assets, or Invested

Capital, to name a few.

Panel C of Table 4 indicates that the manner in which the conditional RPE grant schedule

compares the performance metric against the peer or industry benchmark(s) comes in two main

forms. 79% to 89% of all RPE awards evaluate firm performance based on a percentile ranking

within the peer group. This mode is consistent with standard tournament theory. The other

significant mode of comparison compares firm performance to a benchmark, such as an industry

index or broader index or to the mean or median of performance of a smaller peer group. Recall

the example of Allete, Inc., which illustrates the former mode of comparison, while the example

from American Express illustrates the latter mode. In both instances, there is a formula or table

that maps a quantitative comparison to an actual payout for the award of stock, options, or cash at

the end of the performance period. We describe these examples in more detail in Section V.

Panel D provides statistics on the length in years of the period over which firm performance

is calculated and compared to peer or benchmark performance. In the latter years of our sample,

18

approximately 80% of firms using RPE have at least one award with a three year performance

period and around 25% have at least one award with a one year performance period. Between 7%

and 16% of RPE awards, depending on year, are broken down into multiple components with

different performance periods. These multiple performance periods may or may not overlap each

other and can also include provisions to roll one performance period into the next if some level of

performance is not achieved.

Long-term performance periods essentially behave the same as vesting periods for stock

and options, requiring the executive to remain with the firm to receive the award. Nonetheless,

8% to 12% of RPE grantors attach additional time-based vesting to the back-end instrument, with

the typical vesting period ranging from 21 to 30 months.

RPE awards typically have a payout table that maps the performance measurement statistic

into a payout. The Grants of Plan Based Awards typically states a target award, often in amount

of cash or number of shares, and the payout table or grant schedule converts relative performance

into a percent of target for the actual payout. The following sections provide detailed examples.

For the remainder of this section and in Table 5, we limit our discussion to the largest class of

awards, those based on percentile rank of firm performance relative to peers. In general, the grant

schedule comes in two forms – either as a single-step function or as a more complex multi-step

function. The former conveys no units of the back-end instrument up to a certain percentile rank

and jumps fully to a target number of units at that percentile. More complex, multi-step schedules

typically have an initial threshold for any units, then an incentive zone over which the number of

shares, options, or cash increases step-wise as each additional peer is passed, and then a ceiling

beyond which increased relative performance earns no more back-end units. The incentive zone

19

often contains a kink point in step size at a target level of relative performance and units. Single-

step function grant schedules account for 11.8% of our sample, and 55.0% of those awards interact

with another PV award component – serving perhaps as a trigger for another function. Multi-step

percentile grant schedules represent 88.2% of our subsample of percentile grants.

Table 5 provides descriptive statistics for various elements of the domain and range of the

grant function. For single-step function grants, the majority pay out for median performance or

above, while other awards require stronger performance. For multi-step awards, the median

payout levels for threshold, target, and maximum performance are 27%, 100%, and 200% of target,

respectively. The median percentile rank required to achieve threshold, target, and maximum

payouts are 30th percentile, 50th percentile, and 80th percentile, respectively.

V. A Framework for Characterizing and Measuring the Value of RPE Grants

V.A. The RPE Grant Schedule

For purposes of describing our methodology for characterizing RPE awards and for

measuring their value and incentive properties, suppose that there are I performance metrics, �� =

(𝑋1, 𝑋2, … , 𝑋𝐼) and, when there is relative performance evaluation, the performance metrics are

evaluated relative to a vector of J benchmarks, �� = (𝑌1, 𝑌2, … , 𝑌𝐽). A benchmark for a given

performance metric can be single-valued, such as median performance of a group of peer firms or

an industry or market index defined in terms of the associated performance metric. A benchmark

vector can contain multiple elements for each performance metric. One common case specifies

that the number of shares or options to be granted depends on a performance ranking among a

20

group of peer firms, in which case the benchmarks would comprise the vector of performance

values, one element for each of the members of the peer group.

For concreteness, suppose there are two performance metrics (I = 2), 𝑋1 = 𝐴 and 𝑋2 = 𝑃,

where 𝑋𝑡1 = 𝐴𝑡 is meant to represent the level of an accounting metric (e.g., EPS, earnings, ROA,

ROE, ROI, etc.) or other performance measure (such as sales, market share, FDA approval, etc.)

and 𝑋𝑡2 = 𝑃𝑡 is a metric based solely on stock price of the firm. Likewise, further suppose that

there is a single benchmark for each performance metric, given by 𝑌𝑡1 = 𝐵𝑡 for accounting

performance and 𝑌𝑡2 = 𝑄𝑡 for stock performance.7 The subscript indicates time t, for t ≥ 0, the

initial grant date (t = 0), and thereafter.

The performance and benchmark vectors have initial values at the grant date, t = 0, denoted

𝑋0 = (𝑋0

1, 𝑋02, … , 𝑋0

𝐼) and 𝑌0 = (𝑌0

1, 𝑌02, … , 𝑌0

𝐽). For purposes of determining the number of

units of the back-end instrument to be conveyed to the executive, performance is evaluated over

some time horizon, the performance period, from t = 0 to t = τ.

The means by which RPE impinges on executive wealth is always through a conditional

performance-vesting provision. Let the number of units of the back-end instrument, stock, options

or even cash, that are conveyed to the executive at time τ, the end of the performance period, be

given by the grant function or grant schedule, 𝑁(𝑋𝜏|𝑌𝜏

). In the two-by-two example, this grant

schedule is written 𝑁(𝑋𝜏|𝑌𝜏

) = 𝑁((𝐴𝜏, 𝑃𝜏)|(𝐵𝜏, 𝑄𝜏)).

7 For example, measurement of A versus B and P versus Q can be A – B and P – Q. Obviously, more complex

specifications are accommodated in this framework. While this example and associated notation are meant to focus

on the mix of one accounting metric and one stock-price metric, more generally both A and P can be accounting

performance metrics or both can be stock-price metrics.

21

Grant schedules in practice vary widely in design. In a simple example, for 𝑁(𝑋𝜏|𝑌𝜏

) =

𝑁((𝐴𝜏, 𝑃𝜏)|(𝐵𝜏, 𝑄𝜏)), the schedule could specify a step function, with the number of shares or

options granted equal to zero if one or both of an accounting hurdle, 𝐵𝜏, and stock price hurdle,

𝑄𝜏, are not reached, but some N > 0 for any (𝐴𝜏, 𝑃𝜏) with 𝐴𝜏 ≥ 𝐵𝜏 and 𝑃𝜏 ≥ 𝑄𝜏 (performance

surmounts both hurdles). Some schedules specify relative hurdle rates at some threshold, target,

and ceiling, for both 𝐴𝜏 and 𝑃𝜏, how the number of units conveyed depends on the performance

metrics, and whether and how the hurdles are defined relative to peer, industry, or market

benchmarks (𝐵𝜏, 𝑄𝜏). In many instances the performance comparison is implemented using a

function of stock price or an accounting number. For example, RPE schedules often rely on a

comparison among firms of annualized total stock return (TSR).

Again, measuring performance relative to accounting and/or stock price benchmarks is a

form of RPE. The expression 𝑁(𝑋𝜏|𝑌𝜏

) represents a performance-vesting provision contingent

on performance of 𝑋𝜏 relative to benchmarks 𝑌𝜏

.8

V.B. Value of the Back-end Instrument after the Performance Period

Define the value of the stock, options, or cash conveyed at any time t ≥ τ (at or after the

end of the performance period) as 𝑉(𝑋𝑡), or 𝑉(𝐴𝑡, 𝑃𝑡) in the illustrative case. Note that the award

itself could be a number of shares or options that vest fully at time τ. Otherwise, shares or options

could be restricted by a subsequent time-based or p-v schedule, with the relevant performance

period beginning at the end of the first performance period.

8If the grant schedule reflects absolute performance evaluation (APE), rather than RPE, 𝑌𝜏

is absent and 𝑁(𝑋𝜏|𝑌𝜏

) =

𝑁(𝑋𝜏).

22

We are particularly interested in t = τ, the point at which the number of units of the back-

end instrument is known. Setting aside for the moment any question of dilution, if the back-end

instrument is shares unrestricted by additional time vesting, then 𝑉(𝑋𝜏) = 𝑉(𝐴𝜏, 𝑃𝜏) = 𝑃𝜏, the

stock price per share at τ. If the back-end instrument is options or stock appreciation rights

unrestricted by additional time vesting, and remaining time to maturity is positive, the Black-

Scholes option value adjusted for dividends may be suitable. We denote the Black-Scholes value

(or simulated value, if that approach is employed) adjusted for dividends of the single call option

as 𝑐(𝑃𝑡, 𝐾, 𝜎𝑟 , 𝑑(Ŧ𝑡), 𝑇 – 𝑡, 𝑟𝑓), where Pt is stock price at t ≥ τ, K is exercise price (set at the initial

grant date t = 0), σr is the standard deviation of return on the stock, d(Ŧt) is the schedule of N

remaining (at time t) dividend payments as a proportion of stock price to be paid on dates Ŧt = {t1,

t2, ... ,tN} (with t1 ≥ t and tN ≤ T), remaining time to maturity is T – t, and rf is the risk free rate. In

the data, when cash is the back-end instrument, the value depends on neither Pτ nor Aτ, so

𝑉(𝐴𝜏, 𝑃𝜏) = $1.

V.C. Value of the RPE Grant at the End of the Performance Period (Ex Post)

Ex post value of the conditional p-v grant, at the end of the performance period, is the

product of the number of units earned through the performance-vesting provision and the value at

time t = τ per unit of the back-end instrument, 𝑁(𝑋𝜏|𝑌𝜏

)𝑉(𝑋𝜏). In the simple two-metric case with

no dependence of the back-end instrument on 𝐴𝜏, ex post value is 𝑁((𝐴𝜏, 𝑃𝜏)|(𝐵𝜏, 𝑄𝜏))𝑉(𝑃𝜏).

V.D. Two Examples: RPE Grant Function and Ex Post Award Value

V.D.1. Rank Order Tournament Example: Allete, Inc.

In our sample 88% of the RPE firm-years use a rank-order tournament. Under this schema,

the firm grants an RPE award to the executive, whereby performance is measured for the target

23

firm and a group of peers for a defined period of time. After the measurement period ends, the

target firm is pooled with the peers and they are ranked by performance. The percentile rank is

then mapped by a payout function from percentile or rank to the payout to the executive.

For example, the 2008 Proxy Statement for Allete (ALE)9 states that Donald J. Shippar,

the CEO, was granted an RPE award on February 1, 2008. The award is based on a rank-order

tournament among Allete and 16 named peer firms. Total Shareholder Return (TSR), meaning

total stock return including dividends, is the measure of performance by which the firms were to

be compared after a three-year period. Thus, 𝑋𝑡 = 𝑃𝑡 and 𝑌𝑡 is the 16-vector of stock prices for

the members of the peer group. The TSR calculation is embedded in the grant schedule using the

functional definition of TSR, as represented by 𝑇𝑆𝑅𝑡 = 𝑅(𝑃𝑡) . The award has a target payout of

8,282 shares of stock. After the three-year measurement period, Allete’s percentile rank among

the peers will be determined and the payout function shown in Panel A of Figure 2 will determine

the percent of the target shares that is actually paid out to recipient, 𝑝𝑐𝑡 𝑝𝑎𝑦𝑜𝑢𝑡 (𝑋𝜏|𝑌𝜏 ), for τ =

3. For example, if Allete ranks 5th (73.5th percentile) for TSR from February 2008 to January 2011,

then the recipient receives 150% of the target, which is 12,423 shares. There is a severe drop-off

for performance below the 44.1st percentile, the payout step size increases at the 61.8th percentile,

and the award is capped at 200% payout of target (16,564 shares) if Allete TSR is in the top three.

9 Allete (http://www.allete.com/our_businesses/) offers the following description. “ALLETE (NYSE: ALE) is well-

positioned as a reliable provider of competitively-priced energy in the upper Midwest, and has a strategic investment

in the American Transmission Company. ALLETE's Minnesota Power electric utility serves 144,000 residents, 16

municipalities and some of the nation's largest industrial customers. Other businesses include BNI Coal in North

Dakota; ALLETE Clean Energy, a developer of energy projects with limited environmental impact; ALLETE

Properties, which owns 10,000 acres of real estate in northeast Florida; Superior Water, Light & Power in Superior,

Wisconsin; and ALLETE Renewable Resources, which operates and maintains wind generation facilities in North

Dakota for ALLETE's utility and nonutility companies.”

24

While the above description is a complete account of how the Allete RPE award works, it

is useful to depict the grant schedule in terms of three-year TSR rather than as a percentile of TSR

in the peer group. Ex ante, in 2008, one will not know the ex post (2011) distribution of peer

performance. One possibility is that peer returns will be uniformly distributed three years hence.

Suppose peer group performance realized in 2011 is distributed evenly in 5% increments from a

minimum at -40%% to a maximum of 35%. Under this assumption, the payout schedule for the

Allete CEO as a function of scaled three-year TSR is depicted in Panel B of Figure 2. Note that

in a rank-order tournament the percentage payout only changes as Allete displaces with one of its

peers, thus creating ranges of return with the same percentage payout. The ex post grant schedule

is lumpy with steps. The example in Panel B also suggests that for some realizations of returns by

Allete’s peers, it is possible for Allete TSR to be negative and for the CEO still to receive a positive

payout of shares. Of course, it is highly unlikely that the distribution of peer performance three

years from the initial grant date will be so well-behaved. Panel C of Figure 2 depicts an example

in which ex post peer returns are distributed less evenly.

Allete’s stock price on the grant date was $39.10. The market value of this payout, at τ =

3, is 𝑁(𝑋𝜏|𝑌𝜏

)𝑉(𝑋𝜏) = 8,282 × 𝑝𝑐𝑡 𝑝𝑎𝑦𝑜𝑢𝑡 (𝑋𝜏|𝑌𝜏

) × $39.10 × (1 + 𝑇𝑆𝑅𝜏)𝜏, assuming for

the present no dividends (or the award is dividend-protected), where the percent of target share

payout is given by the grant schedule and initial share value is multiplied by one plus three-year

scaled TSR. Restated, at τ = 3, 𝑁(𝑋𝜏|𝑌𝜏 ) = 8,282 × 𝑝𝑐𝑡 𝑝𝑎𝑦𝑜𝑢𝑡 (𝑋𝜏|𝑌𝜏

) and 𝑉(𝑋𝜏) = $39.10 ×

(1 + 𝑇𝑆𝑅𝜏)𝜏. If ex post returns for peers are distributed as in Panel B, then Panel D of Figure 2

depicts the ex post value of the award. In Panel B, each step is associated with more shares.

Holding shares constant on each step, until Allete TSR surmounts TSR of the adjacent peer leading

25

Allete, the value of the number of shares granted rises linearly in ex post stock price. In terms of

the ex post value of the award, each step in Panel D has positive slope, with the slope increasing

for higher steps because the number of shares granted will have been larger for higher steps.

Panels B through D are based on a specific set of assumed returns for the peers. Holding

constant peer relative performance (as in Panel B), Panel E depicts ex post award value depending

on whether the peer group performs well or not, with the ex post spread in TSR maintained across

all peer group members. Better peer group performance means that Allete TSR must be higher,

all else equal, for the CEO to receive the same ex post award value. Of course, this is only one of

an infinity of possible ex post award value functions. An infinite number of curves and surfaces,

with the location and size of steps depending on the realizations of TSR for the set of peer firms,

are possible.

V.D.2. Example of Performance Net of a Benchmark: American Express

In contrast to the rank-order tournament, about 12% of our sample compares firm

performance to a single peer group statistic, such as the weighted average return. For example,

the 2008 Proxy Statement for American Express (AMEX) states that Kenneth I. Chenault, the

CEO, was granted an RPE award on January 31, 2008. The award pays out cash (as compared to

stock in the previous example) based on adjusted three-year return, defined by the AMEX total

shareholder return net of the S&P 500 TSR. The target payout is $1,636,593. At the end of the

performance period, the CEO is paid 25%, 100%, or 350% of target for an adjusted return of -9%,

0%, or 13%, respectively, with interpolation between points. There is no payout for a benchmark-

adjusted return below -9%. This grant function is depicted in Panel A of Figure 3. Panel B restates

the award in more familiar terms of award value versus absolute return for an assumed index

26

return. Finally, Panel C provides a three-dimensional representation of the award value versus

absolute return for all possible index returns.

For all three panels, because the back-end instrument is cash, the ex post value of the award

and the grant schedule are the same. If, instead, the back-end instrument were stock or options,

the flat portions of ex post value above threshold would be increasing in stock price (or

performance) and, for all parts of the schedule below the ceiling except at steps, would be convex

in stock price.

V.D.3. More Complex Grant Schedules and Back-End Instruments

The RPE awards for Allete and American Express are representative of the most typical

awards, but awards do vary along many dimensions. Of course, the functional form 𝑁(𝑋𝜏|𝑌𝜏)

accommodates more complex comparisons of 𝑋𝜏 versus 𝑌𝜏 and more complex grant schedules than

the piecewise linear schedules in Figures 2 and 3. Some awards use multiple performance

measures and benchmarks that interact. For example, an award can specify that both an industry-

adjusted accounting hurdle and a peer-group-adjusted TSR hurdle be satisfied for threshold

performance to be surmounted and the threshold number of back-end units to be conveyed to the

executive. Other multiple-metric grants use “or” conditions. Some awards mix RPE and APE

conditions. Some RPE awards are granted alongside single or multiple APE p-v awards or

alongside other RPE awards.

The framework accommodates more complex incarnations of cash, stock, and options as

back-end instruments. Aspects of a time-based vesting schedule that are relevant would be

discounting, the probability of dismissal and forfeiture of the unvested portion, and the likelihood

that the board would accelerate some of the unvested units even conditional on dismissal (e.g.,

27

without cause) or departure. These features require estimated parameters on departure

probabilities from the company in question. For options, we can model the interaction of the

effects of early exercise on value with the above-mentioned effects on value of time-vesting

subsequent to the performance period. See Bettis, Bizjak, and Lemmon (2005) on early exercise.

In many cases, the executive receives warrants rather than call options. Standard methods allow

us to value warrants and stock grants that are dilutive. This does not seem to make a large

difference (Anderson and Core, 2013). It also is possible to approximate value for a back-end

instrument that has no or diminished voting rights. While we see no evidence of such in the data,

it is possible to imagine cash payments or stock or option grants for which the value of each unit

depends on At. For example, the back-end unit at the end of the performance period could be

another p-v grant with the performance-vesting provision contingent on both At and Pt. In addition

to setting aside the above complexities, we value the back-end instrument from the point of view

of the diversified investor. See Hall and Murphy (2002) and Ingersoll (2006) on the subjective

value perceived by a risk-averse, undiversified executive.

V.E. Grant Date (Ex Ante) Discounted Expected Value of Ex Post Realized Value

Let ρ be the discount rate applied to 𝑁(𝑋𝜏|𝑌𝜏

)𝑉(𝑋𝜏), the product of the number of units

of the back-end security conveyed times value per unit, both at the end of the performance period.

If 𝑁(𝑋𝜏|𝑌𝜏

) is not exposed to systematic risk and risk of the back-end instrument can be hedged

away, a risk-neutral pricing framework is applicable. Or if investors can fully hedge away the risk

of 𝑁(𝑋𝜏|𝑌𝜏

)𝑉(𝑋𝜏), then it is appropriate to use a risk-neutral pricing framework.

28

In general it is unlikely that a p-v RPE provision joined with the back-end instrument,

𝑁(𝑋𝜏|𝑌𝜏

)𝑉(𝑋𝜏), can be perfectly hedged. First, in many cases, there are no instruments and

strategies one might use to hedge the relevant state variables, such as accounting performance,

market share, FDA approval, or the benchmarks for such metrics. These state variables are

imperfectly correlated with stock price and any hope of an approximation hedge with stock-price

instruments is unlikely to be realized. Second, even if the p-v RPE provision is based solely on

stock performance, the ex post payoff function is often nonlinear and complex and the hedging

strategy complex and correspondingly expensive to execute.

Accordingly, we construct and implement an approach that uses drift rates predicted from

an asset pricing model (APM) and risk-adjusted discounting using that APM. Our approach will

be familiar. Alongside the risk-neutral approach, Black and Scholes (1973) allude to a basic

version of our approach. The discount rate is based on exposure of 𝑁(𝑋𝜏|𝑌𝜏

)𝑉(𝑋𝜏), not just

𝑉(𝑋𝜏), to priced risk, per the logic in standard APMs.

Because the applicable discount rate potentially varies across realizations of the state

variables, we write 𝜌(𝑋𝜏, 𝑌𝜏

). Ex ante expected value at the grant date is 𝐸[𝑁(𝑋𝜏|𝑌𝜏

)𝑉(𝑋𝜏)/(1 +

𝜌(𝑋𝜏, 𝑌𝜏

))𝜏], or in continuous time 𝐸[𝑁 (𝑋𝜏 |𝑌𝜏

) 𝑉(𝑋𝜏)/𝑒𝜌(𝑋𝜏

,𝑌𝜏 )𝜏], where the expectation is

taken across the joint distribution function of (𝑟𝑋 , 𝑟𝑌 )~𝐹(��, Σ), estimated parameters (��, Σ), and

initial values (𝑋0, 𝑌0

).10

10 Once the performance period is complete and uncertainty embedded in 𝑁(𝑋𝜏

|𝑌𝜏 ) is resolved, the grant is worth

𝑁𝑉(𝑃𝜏) and risk-neutral valuation methods can be applied.

29

For illustration, consider the simple two-by-two case with state variables

((𝐴𝑡, 𝑃𝑡), (𝐵𝑡, 𝑄𝑡)) drifting according to 𝐹(��, Σ), with �� = (𝜇𝐴, 𝜇𝑃, 𝜇𝐵, 𝜇𝑄) and Σ =

{𝑐𝑜𝑣(𝑟𝐶 , 𝑟𝐷)}𝐶,𝐷=𝐴,𝑃,𝐵,𝑄. Define 휀𝐶𝐷 = (𝐷 𝐶⁄ )(𝜕𝐶 𝜕⁄ 𝐷) as the elasticity of C in D, where C, D =

A, B, P, Q, M, V, or N and C ≠ D. Performance metrics, A and P, and benchmarks, B and Q, are

defined as above, while M is value of the market portfolio. V and N are functions given by 𝑉(𝑃𝜏)

and 𝑁((𝐴𝜏, 𝑅𝜏)|(𝐵𝜏, 𝑄𝜏)). Then the sensitivity of return on the p-v grant to return on the market

is

( ( , ) | , ) ( ))(1)

( , | , ) ( )

P d N A P B Q V P M P

N A P B Q V P dP P M

where Mτ is value of the market at time t = τ and (Mτ/Pτ)(∂Pτ/∂Mτ) = βPM, which is the traditional

CAPM beta of the firm’s stock return with stock market return.

Turn now to providing an expression for the two bracketed terms of equation (1), which

together represent the full elasticity of 𝑁((𝐴𝜏, 𝑃𝜏)|(𝐵𝜏, 𝑄𝜏))𝑉(𝑃𝜏) in Pτ through all channels. By

all channels, we mean directly through 𝑃𝜏 in the grant schedule and value of the back-end

instrument and also by way of the statistical association between Pτ and the accounting metric, Aτ,

and the benchmarks for stock and accounting performance, Bτ, Qτ. Time subscripts are suppressed

when the meaning is clear and it is convenient. In general, the derivatives are taken at the end of

the performance period, t = τ.

Differentiating fully, the elasticity of 𝑁((𝐴𝜏, 𝑃𝜏)|(𝐵𝜏, 𝑄𝜏))𝑉(𝑃𝜏) in Pτ is

30

( )P d N V P V N N A N B N QV

N V dP N V P P A P B P Q PN

This expression can be rewritten as

( )(2)VP NP NB BPNA AP NQ QP

P d N V

N V dP

The sensitivity of the return on the p-v grant to return on the market is given by

( )(3)PMVP NP NB BPNA AP NQ QP

P d N V M P

N V dP P M

Restated, the beta of return on 𝑁((𝐴𝜏, 𝑃𝜏)|(𝐵𝜏, 𝑄𝜏))𝑉(𝑃𝜏) relative to return on the market is

, (4)NV M PMVP NP NB BPNA AP NQ QP

The elasticities 휀𝐴𝑃, 휀𝐵𝑃, and 휀𝑄𝑃 can be measured using historical data on ((𝐴, 𝑃), (𝐵, 𝑄)). 휀𝑁𝑃,

휀𝑁𝐴, 휀𝑁𝐵, and 휀𝑁𝑄 are based on the functional form of the grant schedule, 𝑁((𝐴𝜏, 𝑃𝜏)|(𝐵𝜏, 𝑄𝜏)),

and 휀𝑉𝑃 is based on the ex post value of the back-end instrument, 𝑉(𝑃𝜏). These elasticities

potentially can vary across realizations of (𝐴𝜏, 𝑃𝜏, 𝐵𝜏, 𝑄𝜏).

For example, if the back-end security is a simple vested call option and adjusted Black-

Scholes is appropriate, 휀𝑉𝑃 would be based on the derivative of 𝑉(𝑃𝜏) =

𝑐(𝑃𝑡, 𝐾, 𝜎𝑟 , 𝑑(Ŧ𝑡), 𝑇 – 𝑡, 𝑟𝑓) in 𝑃𝜏, a derivative that varies in 𝑃𝜏. On the other hand, 휀𝑉𝑃 = 1 if the

back-end instrument is stock and there are no complications, such as an additional time-vesting

restriction or dilution. Note that if there is no RPE then 휀𝑁𝐵 = 휀𝑁𝑄 = 0, and if there is no

31

performance-vesting at all then 휀𝑁𝑃 = 휀𝑁𝐴 = 0, and (4) reduces to the expression, 𝛽𝑉𝑁,𝑀 =

휀𝑉𝑃𝛽𝑃𝑀, which is equation (15) in Black and Scholes (1973).

The application of this discounting approach seems simple for the sort of grant schedule

depicted in Figure 3. The only problem arises at the threshold where the grant function has infinite

slope, though this occurs only for a subset of realizations in the state space of firm and

peer/benchmark performance that has measure zero. This problem is more apparent for percentile-

based RPE grant schedules. For example, the Allete, Inc. grant schedule exposes the recipient to

systematic risk because it depends on Allete stock performance and also peer stock performance.

But, per Figure 2 (Panels A, B, C), if the ex post schedule were treated as certain, the measured

discount rate would reflect zero additional systematic risk on the flat part of each step and infinite

exposure to systematic risk for the few realizations of Allete TSR that exactly match a realization

for one of the peer firms (the small number of vertical parts of the steps). On the other hand, given

that own performance and peer performance are uncertain (or trembling) just prior to the end of

the performance period, so is the ex post location of the steps in the ex post grant function. To

accommodate this aspect of percentile grant schedules, for purposes of discounting only we

approximate exposure to systematic risk by smoothing the grant function, with smoothing to

depend on each ex post realization. For each step above threshold in each possible ex post grant

function we linearize using the corner of each step. We also smooth at the first step, threshold, by

anchoring the lower end of the smoothing line at performance of the adjacent trailing peer. If there

is no trailing peer, we create a synthetic trailing peer that trails the threshold peer by the same

amount as the adjacent peer above threshold exceed threshold performance.

32

To illustrate, for steps above threshold suppose there is a percentile grant function using

stock price of the firm and peers as the metric. For a given step in the grant schedule there are two

immediately adjacent peers, one leading in performance (designated L) and the other following

(F). Assume the follower and leader firms have final stock prices of $1.25 and $1.50 respectively.

If trading places with the follower firm would induce a payout of 40% of target, and trading places

with the leader firm would induce a payout of 50%, then we assume that the firm in question is

paid linearly between 40% and 50% of target for final prices ranging between $1.25 and $1.50.

Similarly, if the leader (follower) receives 𝑁𝐿 (𝑁𝐹) units for performance of 𝑃𝐿 (𝑃𝐹), then, under

the linearization, for stock price 𝑃 the executive receives 𝑁 = [(𝑃 − 𝑃𝐹)/(𝑃𝐿 − 𝑃𝐹)](𝑁𝐿 − 𝑁𝐹).

The linearized grant schedule captures systematic risk through a positive derivative in own

performance (𝑃) and negative derivatives in performance of each peer (𝑃𝐿 , 𝑃𝐹). Then for each step

and associated adjacent peers, exposure to the market return is given by equation (4) above, with

𝑁 representing the smoothed grant function, absence of an accounting metric, 𝐵 = 𝑃𝐿, and 𝑄 =

𝑃𝐹, which is written as, L L F FNV M PMVP NP NP P P NP P P .

The form of 𝜌(𝑋𝜏, 𝑌𝜏

) (or 𝜌(𝐴𝜏, 𝑃𝜏, 𝐵𝜏, 𝑄𝜏)) depends on the asset pricing model. The above

calculations of beta, sensitivity of the value of the grant at the end of the performance period to

return on the market, are oriented towards a CAPM or CAPM-like market model.11 We employ

the CAPM, 𝑟 = 𝑟𝑓 + 𝛽𝑁𝑉,𝑀[𝐸(𝑟𝑀) − 𝑟𝑓] + ��, where �� is the error term. Then

( , , , ) [ ( ) ] (5)f VP NP NA AP NB BP NQ QP PM M fA P B Q r E r r

11 We can extend our approach to a Fama-French 3-factor model or Carhart 4-factor model.

33

Then, again, ex ante expected value at the grant date is

0 0 0 0[ ( , | , ) ( , ) / (1 ( , , , )) | ( , , , )] (6)E N A P B Q V A P A P B Q A P B Q

To empirically generate the expectation, for each grant schedule in each firm we simulate 10,000

paths in the vector of state variables.

V.F. Evolution of the State Variables, (𝑿𝝉 , 𝒀𝝉

)

A probabilistic model of how the state variables (𝑋𝑡, 𝑌𝑡

) (or ((𝐴𝑡, 𝑃𝑡), (𝐵𝑡, 𝑄𝑡))) evolve

from initial values (𝑋0, 𝑌0

) (or ((𝐴0, 𝑃0), (𝐵0, 𝑄0))) over the performance period, t = 0 to t = τ,

and beyond to any other t > τ of interest, is required. We assume that the rate of change in (𝑋𝑡, 𝑌𝑡

)

has a stationary multivariate cumulative distribution, (𝑟𝑋 , 𝑟𝑌 )~ F(��, Σ), with (I+J) x 1 vector of

expected values given by �� = (𝜇𝑋 , 𝜇𝑌 ) and the (I+J) x (I+J) covariance matrix by Σ. Think of the

parameters as determining the drift and volatility of (𝑋𝑡, 𝑌𝑡

) per unit time. In the simple two-by-

two case, state variables ((𝐴𝑡, 𝑃𝑡), (𝐵𝑡, 𝑄𝑡)) drift according to F(��, Σ), with �� = (𝜇𝐴, 𝜇𝑃, 𝜇𝐵, 𝜇𝑄)

and Σ = {𝑐𝑜𝑣(𝑟𝐶 , 𝑟𝐷)}𝐶,𝐷=𝐴,𝑃,𝐵,𝑄.

V.G. Estimating the Parameters

The expectation in (6) requires 𝛽𝑃𝑀, the joint distribution function of (𝑟𝑋 , 𝑟𝑌 ), ~ F(𝜇, Σ),

initial values (𝐴0, 𝑃0, 𝐵0, 𝑄0), estimated parameters (��, Σ), and length of the performance period

𝜏. We assume that drift rates for most accounting metrics are normally distributed and drift rates

for stock price and other truncated performance measures, such as sales, costs, and dividends, are

log-normally distributed. For any specific vector of state variables (𝑋𝑡, 𝑌𝑡

), for the granting firm

we estimate the parameters (��, Σ) using five years of quarterly data on (𝑟𝑋 , 𝑟𝑌 ) over t = -5 to t = -

34

1, with the last observation from the quarter just prior to the quarter containing the grant date. We

do the same for each other firm in the same Fama French 48 (FF48) industry over the same 20

quarters. Because the data are noisy and the time series is short, for the granting firm we estimate

(��, Σ) as the average across all firms in the same FF48 industry.12 13 For the granting firm we

estimate 𝛽𝑃𝑀 = 𝑐𝑜𝑣(𝑟𝑝, 𝑟𝑀)/𝑐𝑜𝑣(𝑟𝑀, 𝑟𝑀) using three years of weekly data on firm stock return

(including dividends) and return on the market. We use the S&P 1500 as the market index. The

risk-free rate is the Treasury bill rate with the maturity date closest to the end of the performance

period.

VI. The Incentive Properties of RPE Awards

For simplicity, the following discussion is based primarily on the simple two-by-two

model. The approach to measuring the incentive properties of RPE grants can be extended easily

to the case of I performance metrics and J benchmarks.

12 A technical problem arises because earnings drift is not well defined for companies with either zero or negative

lagged earnings. If lagged earnings is positive, then we define drift in earnings as change in earnings divided by

lagged earnings. If lagged earnings is less than zero then we use the absolute value in the denominator. We do not

have this difficulty with sales, which also is frequently used in practice, as it is strictly positive. These assumptions

are discussed in Bens, Nagar, Skinner and Wong (2003). 13 We use the industry-level approach for the large-sample analysis in this paper. [Since this study involves large

firms, we use the Incentive Lab universe of firms (which encompasses the Execucomp universe) to compute industry

averages.] The benefit of the industry-level approach, absent large differences across companies, is additional

precision in the estimates. The cost is that this assumes that all firms in the same industry have the same process

generating the state variables. Accordingly, in analyzing awards for some individual firms, one can implement an

approach that customizes the parameter estimates to better reflect the circumstances of the specific company. Among

other things, the method can use a mix of industry and firm parameters, characteristics of prior grants (including

threshold, target, ceiling, and type of metric), and whether the firm achieved critical performance levels specified in

prior grants, so as to enhance the precision of the forecast of (𝜇, Σ).

35

VI.A. Marginal Sensitivity of Grant Value to Stock and Net Stock Performance

One conventional way to calculate the pay-performance sensitivity of a stock or option

grant (delta) is to calculate the change in value of that grant arising from a 1% change in stock

price.14 Likewise, for an RPE grant of stock, options, or cash we perturb initial 𝑃0 by 1%, simulate

ex ante value (per equation (6)) based on the different initial condition (𝐴0, (1.01)𝑃0, 𝐵0, 𝑄0), and

take the difference between the simulated values based on different initial starting points,

(𝐴0, (1.01)𝑃0, 𝐵0, 𝑄0) versus (𝐴0, 𝑃0, 𝐵0, 𝑄0). This difference indicates the effect of changing the

stock price by 1% on the ex ante expected discounted value of the RPE grant, holding (𝐴0, 𝐵0, 𝑄0)

constant. We call this the marginal delta in own stock performance and denote it as 𝛿𝑃.

The intuition for RPE implies a focus on stock performance net of any benchmark(s). As

a first step, note that expected discounted value of the award depends on the benchmark 𝑄, with

the dependence likely to be negative. Perturb initial 𝑄0 by 1%, simulate ex ante value based on

the different initial condition (𝐴0, 𝑃0, 𝐵0, (1.01)𝑄0), and take the difference between the simulated

values based on different initial starting points, (𝐴0, 𝑃0, 𝐵0, (1.01)𝑄0) versus (𝐴0, 𝑃0, 𝐵0, 𝑄0). This

difference indicates the effect of changing the stock price by 1% on the ex ante expected discounted

value of the p-v grant, holding (𝐴0, 𝑃0, 𝐵0) constant. We call this the marginal delta in stock

performance benchmark and denote it as 𝛿𝑄. In the simple two-by-two case, there is a single stock

performance benchmark, but when there are multiple benchmarks, such as for a percentile grant,

the approach applies in the same manner to the benchmarks for the group of peers together. For

example, if there are 16 peers, with initial benchmark values (𝑄01, 𝑄0

2, … , 𝑄016), then we generate

14 For example, see Coles, Daniel, and Naveen (2006) for usage of the semi-elasticity form of pay-performance

sensitivity (delta) in the absence of vesting provisions.

36

𝛿𝑄 based on the change in ex ante expected value of the grant for

(𝐴0, 𝑃0, 𝐵0, (1.01)𝑄01, (1.01)𝑄0

2, … , (1.01)𝑄016) versus (𝐴0, 𝑃0, 𝐵0, 𝑄0

1, 𝑄02, … , 𝑄0

16).

For the second step, note that 𝑃 and 𝑄 covary. Indeed, it is precisely this covariation that

the inclusion of the benchmark 𝑄 can be intended to remove, so that the inference problem yields

a cleaner signal and the incentive contract can be written to be more effective by focusing on

performance the executive can affect or control. The statistical relation between 𝑃 and 𝑄 is given

by 휀𝑄𝑃 = 𝑐𝑜𝑣(𝑄, 𝑃)/𝑐𝑜𝑣(𝑃, 𝑃). Thus, for a 1% change in 𝑃 one would on average expect an

휀𝑄𝑃% change in 𝑄. Perturb initial 𝑃0 by 1% and initial 𝑄0 by 휀𝑄𝑃%, simulate ex ante value based

on the different initial condition, (𝐴0, (1.01)𝑃0, 𝐵0, (1 + .01휀𝑄𝑃)𝑄0), and take the difference

between the simulated values based on the different initial starting points, (𝐴0, (1.01)𝑃0, 𝐵0, (1 +

.01휀𝑄𝑃)𝑄0) versus (𝐴0, 𝑃0, 𝐵0, 𝑄0). We call this difference the marginal RPE delta in stock

performance and denote it as 𝛿𝑃𝑄𝑅𝑃𝐸.

In this simple case, with a single stock performance benchmark, a linear approximation of

this incentive effect is 𝛿𝑃𝑄𝑅𝑃𝐸 ≅ 𝛿𝑃 + 휀𝑄𝑃𝛿𝑄. Based on the logic for RPE in contract design, in

general we expect 휀𝑄𝑃 > 0 and 𝛿𝑄 < 0, insofar as 𝑄 is meant to remove the variation in 𝑃 over

which the executive has no control. Thus, we have 𝛿𝑃𝑄𝑅𝑃𝐸 < 𝛿𝑃.

For percentile grants with multiple peers, the analogous calculation is to perturb initial 𝑃0

by 1% and initial 𝑄0𝑘 by 휀𝑄𝑘𝑃%, for each peer k = 1, …,K. Then to generate 𝛿𝑃𝑄

𝑅𝑃𝐸 we simulate ex

ante value based on the different initial condition, (𝐴0, (1.01)𝑃0, 𝐵0, (1 + .01휀𝑄1𝑃)𝑄01, … , (1 +

.01휀𝑄𝐾𝑃)𝑄0𝐾) and take the difference between the simulated values based on the different initial

starting points, (𝐴0, (1.01)𝑃0, 𝐵0, (1 + .01휀𝑄1𝑃)𝑄01, … , (1 + .01휀𝑄𝐾𝑃)𝑄0

𝐾) versus

37

(𝐴0, 𝑃0, 𝐵0, 𝑄01, 𝑄0

2, … , 𝑄0𝐾). To the extent that the effect on value of increasing the benchmark for

each peer individually is negative, and supposing 휀𝑄𝑘𝑃 > 0 for k = 1, …,K, then again 𝛿𝑃𝑄𝑅𝑃𝐸 < 𝛿𝑃.

VI.B. Marginal Sensitivity of Grant Value to Gross and Net Accounting Performance

It is possible to extend the above approach to accounting performance or any other

performance metric. As above, one could perturb initial 𝐴0 by 1%, simulate ex ante value based

on the different initial condition ((1.01)𝐴0, 𝑃0, 𝐵0, 𝑄0) versus (𝐴0, 𝑃0, 𝐵0, 𝑄0), and take the

difference between the simulated values at the different starting points. The difference indicates

the effect of changing accounting performance (or some other performance metric) by 1% on the

ex ante expected discounted value of the p-v grant, holding (𝑃0, 𝐵0, 𝑄0) constant. We call this the

marginal delta in own accounting (or other) performance, denoted as 𝛿𝐴. Likewise, perturbing

the accounting performance benchmark by 1% gives the marginal delta in accounting benchmark,

which we denote 𝛿𝐵. For accounting-based percentile grants with multiple peers, as for stock-

based grants, marginal delta in the accounting benchmark is based on a 1% perturbation of all

benchmarks.

To construct the RPE delta in accounting performance, perturb initial 𝐴0 by 1% and initial

𝑄0 by 휀𝐵𝐴%, simulate ex ante value based on the different initial condition, ((1.01)𝐴0, 𝑃0, (1 +

.01휀𝐵𝐴)𝐵0, 𝑄0), and take the difference between the simulated values based on the different initial

starting points, ((1.01)𝐴0, 𝑃0, (1 + .01휀𝐵𝐴)𝐵0, 𝑄0) versus (𝐴0, 𝑃0, 𝐵0, 𝑄0). We call this difference

the marginal RPE delta in accounting (or other) performance and denote it as 𝛿𝐴𝐵𝑅𝑃𝐸. A linear

approximation of this incentive effect is 𝛿𝐴𝐵𝑅𝑃𝐸 ≅ 𝛿𝐴 + 휀𝐵𝐴𝛿𝐵. For 휀𝐵𝐴 > 0 and 𝛿𝐵 < 0, we have

𝛿𝐴𝐵𝑅𝑃𝐸 < 𝛿𝐴. For percentile grants with multiple peers, the construction is analogous to that for

38

stock-price-based percentile grants, with the perturbation of each peer benchmark determined by

the elasticity of that benchmark in the accounting metric of the granting firm.

VI.C. Aggregate Sensitivity of RPE Grant Value to Stock Performance

Presumably stock performance and other performance metrics, such as accounting

performance, are related. For example, it can be precisely through observable improvements in

accounting performance that price discovery and formation in stock markets leads to increased

stock price. Moreover, the benchmarks can be correlated with stock performance. Empirically,

the parameters, estimated using historical data, used to construct F(��, Σ), represent the relation.

For each 1% change in stock price, on average there will have been some associated change in the

accounting metric, 휀𝐴𝑃%, where 휀𝐴𝑃 is the historical elasticity of 𝐴 in 𝑃. Similarly, when the award

contains RPE, 휀𝐵𝑃 and 휀𝑄𝑃, which are likely to be positive, represent the association between the

benchmarks and the stock price metric. Thus, the marginal RPE delta in stock price is an

incomplete measure of the sensitivity of the ex ante value of the p-v RPE grant to stock price.

Accordingly, when we perturb initial 𝑃0 by 1%, the expected associated perturbations in

accounting performance and the benchmarks are: 𝐴0 by 휀𝐴𝑃%, 𝐵0 by 휀𝐵𝑃%, and 𝑄0 by 휀𝑄𝑃%.

Simulate ex ante value based on the initial condition ((1 + .01휀𝐴𝑃)𝐴0, (1.01)𝑃0, (1 +

.01휀𝐵𝑃)𝐵0, (1 + .01휀𝑄𝑃)𝑄0), and take the difference between the simulated values based on the

different initial starting points, ((1 + .01휀𝐴𝑃)𝐴0, (1.01)𝑃0, (1 + .01휀𝐵𝑃)𝐵0, (1 + .01휀𝑄𝑃)𝑄0)

versus (𝐴0, 𝑃0, 𝐵0, 𝑄0), to obtain the aggregate RPE delta in stock performance, denoted 𝛿𝐴𝑔𝑔𝑅𝑃𝐸.

The obvious linear approximation is 𝛿𝐴𝑔𝑔𝑅𝑃𝐸 ≅ 𝛿𝑃 + 휀𝐴𝑃𝛿𝐴 + 휀𝐵𝑃𝛿𝐵 + 휀𝑄𝑃𝛿𝑄, which is the

approximately the same as 𝛿𝐴𝑔𝑔𝑅𝑃𝐸 ≅ 𝛿𝑃𝑄

𝑅𝑃𝐸 + 휀𝐴𝑃𝛿𝐴𝐵𝑅𝑃𝐸when 휀𝐵𝑃 ≅ 휀𝐵𝐴휀𝐴𝑃

39

For percentile grants with multiple peers, we apply the same procedures as above. For

example, consider a percentile grant based on an accounting metric and a stock-price metric, J peer

benchmarks in accounting performance, and K peer benchmarks in stock performance. To

generate 𝛿𝐴𝑔𝑔𝑅𝑃𝐸 we take the difference in value based on ((1 + .01휀𝐴𝑃)𝐴0, (1.01)𝑃0, (1 +

.01휀𝐵1𝑃)𝐵01, … , (1 + .01휀𝐵𝐽𝑃)𝐵0

𝐽, (1 + .01휀𝑄1𝑃)𝑄01, … , (1 + .01휀𝑄𝐾𝑃)𝑄0

𝐾) versus

(𝐴0, 𝑃0, 𝐵01, 𝐵0

2, … , 𝐵0𝐽, 𝑄0

1, 𝑄02, … , 𝑄0

𝐾).

To our knowledge, all but the marginal delta in stock price, 𝛿𝑃, are new constructs for

measuring incentive alignment. In implementation, we focus for now on percentile awards based

on a single stock or accounting performance metric. J and K are the number of peers. For ex post

value given by 𝑁(𝑃𝜏| 𝑄1,𝜏, … , 𝑄𝐾,𝜏)𝑉(𝑃𝜏), we calculate 𝛿𝑃, 𝛿𝑄, and 𝛿𝐴𝑔𝑔𝑅𝑃𝐸 = 𝛿𝑃𝑄

𝑅𝑃𝐸, while noting

that 𝛿𝐴 = 𝛿𝐵 = 𝛿𝐴𝐵𝑅𝑃𝐸 = 0. For ex post value given by 𝑁(𝐴𝜏| 𝐵1,𝜏, … , 𝐵𝐽,𝜏)𝑉(𝑃𝜏), we calculate

𝛿𝑃 = 𝛿𝑃𝑄𝑅𝑃𝐸, 𝛿𝐴, 𝛿𝐵, 𝛿𝐴𝐵

𝑅𝑃𝐸, and 𝛿𝐴𝑔𝑔𝑅𝑃𝐸, while noting that 𝛿𝑄 = 0. Also note that 𝛿𝑃 = 𝛿𝑃𝑄

𝑅𝑃𝐸 = 0 if

cash is the back end instrument for a percentile grant based on a (single) accounting metric. We

set aside for now, to be analyzed in a forthcoming draft, the more complex case of percentile grants

with multiple performance metrics and the simpler case of benchmark-adjusted awards.

VI.D. Marginal Sensitivity of RPE Grant Value to Performance and Benchmark Volatility

The sensitivity of expected value of an award to the volatility of stock performance is seen

as a measure of the incentive conveyed by the award to the executive to take risk on behalf of

shareholders (e.g., Core and Guay. 2002; Guay, 1999). One measure of vega is the change in

expected value of the award associated with a 1% proportional change in the annualized standard

deviation of stock return (see Anderson and Core, 2013, for example). In the absence of vesting

40

provisions, in general the convexity in payoff arising from options is large relative to any convexity

arising from shares (Core and Guay, 2002), so historically almost all of vega in an executive’s

portfolio arose from the accumulation of options net of dispositions. More recently, as illustrated

in Figures 2 and 3 for RPE awards and per BBCK (2010, 2013) for APE awards, there appears to

be significant local convexity in vesting provisions, though often there are other portions of the

grant schedule with significant concavity.

To measure vega incentives, the increased usage of p-v poses two problems. First, even

for p-v provisions using APE, the usage of one or more accounting performance metrics, either on

their own or in conjunction with stock performance metrics, requires an adjustment for variation

in the accounting measure(s) and covariation in the accounting measure(s) with stock return.

BBCK (2013) have addressed this problem for p-v APE awards based on accounting performance.

Second, using RPE in a p-v award introduces the question of volatility of the benchmarks and

covariation in the benchmarks with the accounting and stock performance metrics. We adopt a

modified version of the BBCK (2013) solution to covariation in accounting and stock performance

and develop a new approach to address the benchmark question. In both instances, unlike BBCK

(2013), we suppose that the correlation matrix, rather than the covariance matrix, represents the

underlying primitive relation among performance metrics and benchmarks.

Consider the two-by-two case. Recall that (𝑟𝐴, 𝑟𝑃, 𝑟𝐵, 𝑟𝑄)~F(��, Σ), based on estimated

parameters. For 𝛲 ≡ {𝑐𝑜𝑟𝑟(𝑟𝐶 , 𝑟𝐷)}𝐶,𝐷=𝐴,𝑃,𝐵,𝑄, we can write Σ =

{𝑐𝑜𝑟𝑟(𝑟𝐶 , 𝑟𝐷)√𝑐𝑜𝑣(𝑟𝐶 , 𝑟𝐶)√𝑐𝑜𝑣(𝑟𝐷, 𝑟𝐷)}𝐶,𝐷=𝐴,𝑃,𝐵,𝑄. For a given p-v RPE grant of stock or

options, we proportionally increase √𝑐𝑜𝑣(𝑟𝑃, 𝑟𝑃), by 1%. Note that this means that, in addition to

41

the variance of stock return, we also perturb some off-diagonal elements of the covariance matrix.

We then simulate ex ante value based on the different initial condition based on

(1.01)√𝑐𝑜𝑣(𝑟𝑃, 𝑟𝑃) and take the difference between the simulated expected values based on

different initial starting points, (1.01)√𝑐𝑜𝑣(𝑟𝑃, 𝑟𝑃) versus √𝑐𝑜𝑣(𝑟𝑃, 𝑟𝑃), holding all other

parameters constant). For the covariance matrix, note that this change multiples 𝑐𝑜𝑣(𝑟𝑃, 𝑟𝑃) by

(1.01)2 and 𝑐𝑜𝑣(𝑟𝑃, 𝑟𝐶) by (1.01) for C = Q, A, B. This difference indicates the effect of changing

the stock return volatility proportionally by 1% on the ex ante expected discounted value of the p-

v grant. We call this the marginal vega in own stock performance and denote it as νP.

Likewise, following BBCK (2013), we calculate the effect of a change in volatility of the

accounting (or other) performance metric. We proportionally increase √𝑐𝑜𝑣(𝑟𝐴, 𝑟𝐴) by 1%. This

change multiples 𝑐𝑜𝑣(𝑟𝐴, 𝑟𝐴) by (1.01)2 and 𝑐𝑜𝑣(𝑟𝐴, 𝑟𝐶) by (1.01) for C = P, Q, B. The difference

between the simulated expected values based on different initial starting points

((1.01)√𝑐𝑜𝑣(𝑟𝐴, 𝑟𝐴) versus √𝑐𝑜𝑣(𝑟𝐴, 𝑟𝐴), holding all other parameters constant, indicates the

effect of changing the volatility of drift in the accounting metric proportionally by 1% on the ex

ante expected discounted value of the p-v grant. We call this the marginal vega in own accounting

performance and denote it as νA.

In similar manner, to calculate the effects of an increase in riskiness of the performance

benchmarks, we proportionally increase the standard deviation of the accounting or stock

performance benchmarks. If there are multiple peers in a percentile grant, then we increase by 1%

either √𝑐𝑜𝑣(𝑟𝐵𝑗 , 𝑟𝐵𝑗), for all j = 1, …,J, or √𝑐𝑜𝑣(𝑟𝑄𝑘 , 𝑟𝑄𝑘), for all k = 1,…,K. This scales up the

corresponding diagonal and cross elements of the covariance matrix, some by 1.01 and others by

42

(1.01)2. We then compare the value of the RPE grant based on the different starting conditions.

We denote the marginal vega in the accounting performance benchmark(s) as 𝜈𝐵 and the marginal

vega in the stock performance benchmark(s) as 𝜈𝑄.

If convexities in the grant schedule and back-end instrument are sizable relative to

concavities in the grant schedule (e.g., at threshold or ceiling), then 𝜈𝐴, 𝜈𝑃, 𝜈𝐵, and 𝜈𝑄are likely to

be positive. If concavities dominate, then these measures can be negative.

One caveat is required. Anderson and Core (2013) assert, contrary to conventional wisdom

(Core and Guay, 2002), that managerial holdings of debt and levered equity also provide

significant vega. If this analysis survives further inspection, then our model can be adjusted to

include these sources of vega.

VI.E. Aggregated Sensitivity of RPE Grant Value to Performance and Benchmark Volatility

To aggregate all sources of volatility and, we apply the analogous procedure to all elements

of the covariance matrix. That is, we scale up all elements of the covariance matrix per (1.01)2Σ,

which is the same as multiplying every √𝑐𝑜𝑣(𝑟𝐶 , 𝑟𝐷), C, D = A, P, B, Q, by 1.01. This procedure

both acknowledges the covariation of the accounting and stock performance metrics and

benchmarks and also maintains the elements in the covariance matrix in the same proportions. We

then simulate ex ante grant value based (1.01)2Σ versus Σ and take the difference. We call this

aggregate RPE vega and denote it as 𝜈Σ.

While 𝜈𝑃 is well-known, to our knowledge the marginal vega in accounting (or other)

metric, 𝜈𝐴, marginal vegas in performance benchmarks, 𝜈𝐵 and 𝜈𝑄, and aggregate 𝜈Σ are new

constructs for measuring executive incentives to take risk. Again, in implementation we focus for

43

now on percentile awards based on a single stock or accounting performance metric. For ex post

value given by 𝑁(𝑃𝜏| 𝑄𝜏1, … , 𝑄𝜏

𝐾)𝑉(𝑃𝜏), we calculate 𝜈𝑃, 𝜈𝑄, and 𝜈Σ, while noting that 𝜈𝐴 = 𝜈𝐵 =

0. For ex post value given by 𝑁(𝐴𝜏| 𝐵𝜏1, … , 𝐵𝜏

𝐽)𝑉(𝑃𝜏), we calculate 𝜈𝑃, 𝜈𝐴, 𝜈𝐵, and 𝜈Σ, while

noting that 𝜈𝑄 = 0. Also note that 𝜈𝑃 = 0 if cash is the back end instrument for a percentile grant

based on a (single) accounting metric. Again, we set aside for now the more complex case of

percentile grants with multiple performance metrics and the simpler case of benchmark-adjusted

awards.

VII. Outcomes, Value, and Incentives for Percentile P-V RPE Grants to Executives

In this section we apply the methods developed in the prior two sections to characterize the

outcomes, value, and incentive properties of RPE awards. For now we focus on the largest part of

the sample, percentile awards of stock and/or cash based on accounting or stock performance. For

simplicity, in this draft we restrict the analysis to awards based on a single performance metric and

a single measurement period.

To derive payouts and estimate grant date value, delta, and vega of RPE awards, we

simulate stock price (and accounting) drift over the specified performance period for the granting

firm and its stated RPE peer group. Motion is based on a vector of drift rates and a covariance

matrix for drift rates.15 For any grant, we simulate 10,000 paths for the state (performance)

15 Some additional details follow. We determine the drift rates using the CAPM model. We estimate firm-specific

beta for all firms in the CRSP universe by regressing weekly returns in excess of the 10-year government bond against

weekly returns of the CRSP value-weighted market portfolio in excess of the same risk-free rate. The estimation

period is a three year window and is repeated at the end of the every calendar year available from 1973 to 2012. To

utilize the most stable value, we take the median beta for each firm as the firm beta. We winsorize the CRSP universe

of betas at the 5th and 95th percentiles. Each firm is classified by two-digit SIC and size quintile as of mid-date of

our sample. If data are not available on that date, we take the closest available date. We then assume every firm in

44

variables. For each simulation, we find the final percentile rank of the granting firm, apply the

payout multiplier as defined in the proxy, and determine the value of the award at the end of the

performance period (𝜏).16 For cash awards the value of the award at 𝑡 = 𝜏 is the product of the

target amount and the multiplier. For stock awards it is the product of the target number of shares,

the multiplier, and the ending stock price as determined by each individual simulation. For some

comparisons it is useful to equalize all initial stock prices to a value of $1 at the beginning of the

performance period.

VII.A. Distribution of Grant Schedule Performance Rank and Payout Rate at Threshold,

Target, and Ceiling

We first examine the magnitude of payout (as a percentage of target payout) at threshold,

target, and ceiling. One possibility is that these milestones are “cream-puff” hurdles and payout

at any performance milestone is a high proportion of target. An alternative is that the milestones

provide meaningful objectives. We consider single-step or “cliff” schedules on their own because

in essence threshold and ceiling (and target) are all located at the same step. Multi-step percentile

grant schedules all have a different threshold and ceiling.

each industry/size group share the same median value for that group. If the value is missing for any firm we repeat

the process based on industry only. The CAPM estimation is completed by using the contemporaneous 10-year

government bond rate and a market premium of 4.1%. To estimate the volatility of each firm, for each industry/size

group we calculate the average 5-year annualized standard deviation of monthly returns for each firm in the CRSP

database. The minimum number of observations is 36 months per firm and five firms per industry/size group. The

estimation window spans the 5-year period ending the month before the awards grant date. As with the drift rate, we

match the estimated volatility to each firm in the simulation by industry and size. If the value is missing, we repeat

the process based on industry only. To estimate the correlation matrix for the firms in the simulation, we rely again

on historical returns. For each possible combination of firms in an RPE simulation, we develop a list of firms for each

of the two firms. The lists consist of all CRSP firms in the same industry/size group as the simulation firm. We then

find the correlation of returns among all combinations of firms between the two lists, discarding any observation with

less than 36 observations. When five or more correlations are present, the average correlation among the remaining

observations is the estimated correlation between the two firms. The correlation of any firm to itself is set to one. If

the value is missing we repeat the process based on industry only. 16 The payout percentages are normalized so that the average payout percentage, equally weighted across all percentile

ranks, is 100%. This facilitates comparability.

45

Table 5 reports the distribution of payout as a proportion of target across percentile grant

schedules. For the median multi-step percentile RPE grant schedule, threshold payout is 27% of

target shares or cash and ceiling payout is 200% of target. For the median multi-step percentile

RPE grant, the percentile performance required to earn threshold, target (100% of target), and the

maximum are 30th percentile, 50th percentile, and 90th percentile, respectively. Less onerous grant

schedules, such as the 10th percentile among schedules) hit the maximum (75th percentile

performance) and threshold (20th percentile performance) payouts at lower performance

percentiles.

Table 6 tabulates the distribution across the sample of single-step and multi-step percentile

grant schedules of the simulated likelihood of achieving threshold, target, and ceiling performance.

For each grant, based on 10,000 simulated paths for the state variables based on historical

parameters, we calculate the percentage of those paths for which the realization of performance

implied payout at threshold or better, target or better, and ceiling. By way of illustration, among

multi-step saw-blade RPE grant schedules that pay shares as the back-end instrument the median

proportion of the 10,000 simulated paths for which such grants cleared threshold is 0.73. The

median (across such grant schedules) likelihood time that ceiling (target) payout is attained is 0.12

(0.51).

VII.B. Simulated Ex Post Value of Percentile RPE Awards of Stock and Cash

Table 7 describes the distribution of ex-post payout for the sample of percentile RPE

awards of cash and stock by whether the p-v RPE award is contingent on a (single) stock price or

accounting metric. Consider awards based on stock price. For cash awards, the median RPE

award has an average payout of $1.01 per $1.00 of target value. For stock awards the multiplier

46

is the fraction of target shares that are granted to the award recipient ex-post. Similar to cash

awards, on average the median firm receives 100.8% of the target, but the median normalized stock

price after the measurement period has increased from $1.00 to $1.23. The average payout for the

median firm is $1.62 per $1.00 of grant target value. The reason this exceeds the product of the

average target payout rate and average ex post value of the back-end instrument (stock) is that

often there is significant convexity in the product of a grant schedule that depends on stock price

and the stock price.

This effect does not appear for grant schedules dependent on accounting performance. For

the median across simulations of percentile p-v RPE grants based on accounting performance that

pays shares, the average simulated payout is 105.9% of target, ex post average stock price is $1.25

per dollar of initial stock price, and average payout per dollar of initial stock price is $1.31. Though

such grant schedules often have significant convexity, at least locally, induced convexity is small

because accounting and stock performance are not highly correlated. Of course, if cash is the

back-end instrument there is no convexity induced beyond what is embedded in the grant schedule

as a function of performance.

VII.C. Ex Ante Value of Percentile RPE Awards of Stock and Cash versus Disclosed FMV

For percentile RPE awards of stock and cash, we now report ex ante value for the instances

in which we have a full characterization of the ex post grant schedule and compare that estimate

of ex ante economic award fair market value to the FMV disclosed by the company in the DEF

14A. Table 8 presents the results.17

17 We lose some observations because of incomplete reporting. Prior to the enhanced reporting requirements

implemented in 2006 reporting on FMV and the target number of back-end units was often incomplete. In the latter

case, for example, this means that calculations per $1 of target value are possible but aggregate value can’t be

47

Consider percentile RPE awards of shares based on a single stock price performance

metric. The present value at the grant date of the median stock award is $1.17 per $1 of initial

stock value. Unnormalized, the ex ante value of the median percentile RPE grant is $1,125,378.

The median difference between ex ante value and fair market value reported by the company is a

modest $70,948. But for awards with value below the median, reported FMV exceeds the grant

date present value by much more, such as by about $1.2 million at the 10th percentile in grant date

present value. In contrast, RPE stock awards with higher grant date present value exceed reported

FMV by significant amounts, such as by more than $400,000 at the 90th percentile in grant date

present value. There is a significant disconnect in the economic value of the awards versus FMV

reported by companies. This disconnect extends all four subsets of awards depicted in Table 8.

One possible reason is that for awards contingent on stock performance firms are required

to report an FMV using standard, risk-neutral simulation methods. We make the same calculation

for purposes of comparison. The median difference of grant date value versus risk-neutral value

is -$0.14, which is a significant difference per dollar of initial stock price. Our valuation

framework delivers estimates of economic value that differ substantially from value estimated

using standard risk-neutral methods. While these differences are largest for stock grants

contingent on stock performance, they are significant as well for grants of cash and grants

contingent on accounting performance.

To isolate the effect of RPE, we calculate the average number of back-end units, as

determined by the simulation, multiplied by the present value of the back-end award. The present

calculated. Attrition due to missing disclosed FMV is even worse for grants based on accounting performance and

grants of cash, because firms likely are not required to report figures for these classes of grants.

48

value of the back-end instrument is the grant date stock price for stock awards and $1 discounted

by the risk-free rate over the performance period for cash awards. Table 8 indicates that, for stock

awards, RPE adds value to the grant, while the reverse is true for cash awards.

VII.D. Incentive Properties of Percentile RPE Awards of Stock and Cash

Tables 9 and 10 report, for single-metric percentile RPE awards of stock and cash, our

results on the incentives of executives to increase shareholder value and incentives to take risk.

Table 9 confirms that the RPE delta in stock performance is less than the delta in own stock

performance. The reason is that stock performance is measured in comparison to peer stock

performance, higher peer stock performance is negatively related to the number of back-end units

of stock or cash (and ex post value) conveyed to the executive (𝛿𝑄 < 0), and own-firm and peer-

firm stock performance are positively correlated (휀𝑄𝑃 > 0). Of course, the benefit of this effect is

to allow the firm to use higher-powered contracts with larger 𝛿𝑃 than would otherwise be optimal.

RPE strips out variation in performance over which the executive has no control. Note that this

effect is much smaller for stock and cash awards based on a single accounting performance

measure. The reason is that the correlation between own-firm and peer-firm accounting

performance tends to be quite small (휀𝐵𝐴 ≅ 0). Accounting RPE does little to remove common

shocks.

For all four classes of awards, there is large variation across RPE grants in delta incentives.

For example, for a p-v RPE grant of stock contingent on stock performance, the aggregate RPE

incentive to increase shareholder value varies from $3,449 at the 10th percentile to $71,773 at the

90th percentile, approximately a factor of 20. The other three categories of awards in Table 9

reflect large differences of the same order of magnitude for the 90th versus 10th percentiles.

49

The literature provides very little opportunity to assess the economic importance of RPE

award incentives. One reason is that most prior empirical studies measure delta and vega without

including the effects of any p-v provision that might be employed (e.g., Jensen and Murphy, 1990,

and Coles, Daniel, and Naveen, 2006). One exception is BBCK (2013), which measures marginal

and aggregate delta and vega incentives for p-v absolute performance evaluation grants. For stock

grants based on a single stock performance metric, median aggregate RPE delta calculated herein

($19,947) exceeds slightly the median APE delta ($16,335, per BBCK, 2013, Table 6). For stock

grants contingent on accounting performance only, the median aggregate RPE delta ($7,223) is

more than twice that of the median APE award ($3,268, per BBCK, 2013, Table 6). RPE grants

convey to executives as much or more incentive to increase shareholder wealth, through the direct

stock price channel, indirectly through accounting performance of the firm, and indirectly through

competing with peers to affect peer performance, than APE awards.

In terms of risk-taking incentives, it is not clear as to whether increased variability should

increase or decrease the value of a p-v RPE grant. Grant schedules and ex post value, as functions

of the state variables, meaning the performance metrics and the associated benchmarks, tend to

contain both concavities and convexities. Table 10 indicates that RPE grants tend to increase the

appetite of executives for volatile firm and peer returns. Particularly for RPE grants of stock, for

only a small proportion of awards is aggregate RPE vega negative and, when it is, the vega is close

to zero. Otherwise, 𝜈Σ tends to be positive and large. Moreover, for awards of stock contingent

on stock performance, the median RPE award conveys 𝜈Σ = $5,081, as compared to the median

aggregate vega from an APE grant of $891 (BBCK, 2013, Table 7). For stock grants based on a

single accounting metric, median aggregate vegas are almost the same ($429 for RPE versus $417

50

for APE (BBCK, 2013)). Also note that aggregate vega tends to be discernibly larger than the

marginal vega in stock price performance. This comparison reinforces the notion that including

RPE is likely to increase the appetite of executives for risk. Note that even RPE awards of cash

contingent on stock performance convey positive aggregate vega. The median is $498. In contrast,

median vega for cash awards contingent on accounting performance appears to be negligible.

In terms of executive incentives, however, it is important to focus on what executives can

control. In this respect, because executives are unlikely to be able to affect the risk characteristics

of peer performance, perhaps the most relevant incentive measures are the marginal vegas in stock

and accounting performance. From BBCK (2013, Table 7), for APE awards of stock based on a

stock performance metric, median stock performance vega is $891, as compared to the median of

$3,808 in Table 10. For stock awards based on accounting performance, median APE marginal

accounting vega is $211, as compared to the RPE median of $2,305, while APE aggregate vega in

stock performance is $417 versus $971 for RPE awards. Based on median grants, it appears that

percentile RPE grants of stock based on either accounting or stock performance, relative to

analogous APE awards, convey significant marginal incentives to executives to take risk. This

conclusion, however, is tentative, insofar as a significant proportion of RPE grants have negative

marginal accounting and stock performance vegas.

VIII. Conclusion

The intuition for the use of RPE contracts is compelling. Nonetheless, empirical tests

rarely detect usage of RPE in contract design. Likely reasons include various forms of

misspecification. Specification problems likely include: assuming that all firms use RPE when

51

some do not; incorrect performance metric(s); incorrect benchmark(s); incorrect peer groups or

industry; and usage of an incorrect functional form. In this context, our data indicate persistent,

common usage of RPE in executive compensation contracts in large U.S. companies. Consistent

with the paucity of prior evidence supporting the use of RPE, the level of detail in our data

demonstrates that implicit tests for RPE suffer from all of the above-mentioned forms of

misspecification.

We examine a full spectrum of issues pertaining to the use of RPE in compensation contract

design. In particular, we address: the frequency and persistence of RPE usage by firms; the

functional form of RPE grant schedules in firm and peer/benchmark performance; the performance

metrics and benchmarks employed; who receives RPE awards; the economic determinants of RPE

usage; the significance in value of RPE awards; and the incentive properties, for value creation

and risk-taking, of RPE awards. These last two contributions require that we develop and

implement new models for valuation and new measures of executive incentives.

In short, RPE awards tend to be used more often when the product market is more

competitive and when it is possible to remove common shocks by using RPE. In terms of the form

of the grant schedule, the milestones in RPE grants, including thresholds and ceilings based on

percentile or relative stock or accounting performance, represent significant hurdles that are

achievable by executives with at best modest likelihood. Using our valuation approach, our

simulations indicate that in general the grant date value of an RPE grant differs significantly from

the fair market value of the grant reported by the firm in public disclosures. The accuracy and

utility of reported FMV, as an indicator of the economic value of p-v RPE awards of stock, cash,

and options to executives, appear to be low. Many RPE awards convey significant incentives to

52

executives to increase shareholder value and at least some incentives to increase the volatility of

firm stock and accounting returns. Some of these incentives appear to be significant compared to

p-v APE awards with otherwise-similar attributes.

53

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Appendix A: Variable Definitions

Definitions of RPE Usage (all variables dichotomous at the firm-year level)

ADOPT

This variable equals one the first year a firm grants RPE to any individual.

We exclude all occurrences where the first year is before 2000 to reduce the

likelihood of incorrectly identifying a repeat RPE observation as the initial

observation. The value is set to zero for all years for firms with no RPE

observations or for observations at least three years before initial adoption.

For each firm we keep only one randomly selected firm-year where ADOPT

equals zero.

DROP

This variable equals one if the firm granted RPE to any individual the prior

year and granted no RPE for the current year. The value is set to zero if the

firm granted RPE in both the prior and the current year.

RPE This variable equals one if the firm grants RPE to any individual and zero

otherwise.

PV-RPE Equals one if the firm grants RPE to any individual and zero if the uses a p-v

grant to any individual but uses APE and no RPE.

MEASRET For an RPE contract, the variable equals 1 if the contract relies on one or

more performance metrics based on stock price, equals 0 otherwise.

STOCKPAY For an RPE contract, equals 1 if the back-end instrument uses only stock as

the back-end instrument, equals 0 otherwise.

STOCKRET The product of MEASRET and STOCKPAY.

Other Explanatory Variables

BOARD_SIZE The number of members on the board of directors

CF_VOL The standard deviation of the quarterly operating cash flows divided by sales

for the previous five years

CONSULT

Dichotomous variable equal to one if the firm used a compensation

consultant and equal to zero otherwise (for the limited years of data

available).

DIV_YIELD Dividends paid over the fiscal year divided by the market value of equity at

the end of the fiscal year

57

IND_HERF The sum of the squared values of firm sales for the fiscal year divided by

total industry sales, where the industry is defined by three-digit SIC code

INDADJ_RET

Stock return of the firm for the fiscal year less the median industry stock

return for the same time period, where the industry is defined by two-digit

SIC code

INDADJ_RO

A

Firm ROA for the fiscal year less the median ROA for the industry, where

ROA is operating income divided by assets and industry is defined by two-

digit SIC code

INSTOWN The aggregate percentage ownership of all shareholders with 5% or more

ownership average over the fiscal year

INV The sum of R&D, capital expenditures, and advertising for the fiscal year

divided by total assets

LN_ASSETS The natural log of total assets

M/B Market value of equity divided by book value of equity

MKT_RISK The R2 from regressing monthly stock returns for the firm against monthly

stock returns for the value-weighted market returns for the CRSP universe

OPTGRANT Options granted during the fiscal year divided by total shares outstanding as

of the end of the fiscal year

OPTOUT Options outstanding at the end of the fiscal year divided by total shares

outstanding as of the end of the fiscal year

OPTPRICE The value of an option grant at the end of the fiscal year divided by the stock

price at the end of the fiscal year

PCT_INSIDE Number of board of director members who are firm employees divided by

the total number of members

RET Total stock return for the fiscal year

RET_VOL Standard deviation of daily stock returns for the firm for the fiscal year

ROA Operating income divided by total assets

SALES_GR The percentage growth in sales for the fiscal over the prior fiscal year

58

SEG_HERF The sum of the squared values of sales per segment divided by total firm

sales

59

Figure 1: Yearly Usage Rates for Large US Firms of Time-vesting,

Performance-vesting (P-V), and P-V RPE Grants of Stock, Options,

and Cash to Executives

0

0.2

0.4

0.6

0.8

1

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Time Trend in Incentive PayFirm-Year Usage for NEOs

Time Vest Options Time Vest Restricted Stock All P-V Awards RPE Awards

60

Figure 2. Example of RPE Award - Allete, Inc.

This figure depicts possible payouts for a stock award based on an RPE performance vesting provision.

The performance measure is three-year annualized total stock return (TSR). The number of shares granted,

defined as a proportion of a target number, depends on the relative percentile rank of Allete TSR as

compared to a group of 16 peer firms. The target number of Allete shares is 8, 282. The grant was made

to the CEO of Allete, Donald J. Shippar, on February 1, 2008.

Panel A: Payout as a Function of Relative Percentile Rank

Panel B: Payout as a Function of Return Assuming an Even Ex Post Distribution of Peer Returns

0%

50%

100%

150%

200%

250%

0.00 20.00 40.00 60.00 80.00 100.00

Per

cen

t P

ay

ou

t of

Targ

et

Percentile Rank

0%

50%

100%

150%

200%

250%

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40%

Per

cen

t P

ay

ou

t of

Targ

et

Measurement Period Return

Payout

Rank Percentile Target

Payout

11 - 17 <44.12 0%

10 44.12 50%

7 61.76 100%

1 - 3 ≥85.29 200%

61

Panel C: Payout as a Function of Return Based on a Less Even Ex Post Distribution of Peer

Returns

Panel D: Ex Post Market Value of Payout (Based on an Even Ex Post Distribution of Peer

Returns)

0%

50%

100%

150%

200%

250%

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40%

Per

cen

t P

ay

ou

t of

Targ

et

Measurement Period Return

0

200,000

400,000

600,000

800,000

1,000,000

-50% -30% -10% 10% 30%

Ex

-Post

Mark

et V

alu

e of

Pay

ou

t ($

)

Measurement Period Return

62

Panel E: Market Value of Payout for Various Returns of Peers

(Based on an Even but Movable Ex Post Distribution of Peer Returns)

Allete

Measurement

Period Return

Median Peer

Return

Ex-Post Value

of Payout ($)

63

Figure 3. Example of Benchmark-Adjusted-Return RPE Award – American

Express

This figure depicts the possible payouts for the RPE award of cash granted to the CEO of American Express on January

31, 2008.

Panel A: Payout as a Function of Benchmark-Adjusted Return

Panel B: Market Value of Payout Assuming 12% Return for S&P 500

0%

50%

100%

150%

200%

250%

300%

350%

400%

-25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25%

Per

cen

t P

ay

ou

t of

Targ

et

Measurement Period Return – RetS&P500

$0

$1,000,000

$2,000,000

$3,000,000

$4,000,000

$5,000,000

$6,000,000

$7,000,000

-10% -5% 0% 5% 10% 15% 20% 25% 30% 35%

Ex

-Post

Mark

et V

alu

e of

Pay

ou

t ($

)

Measurement Period Return

Payout

TSR –

RetS&P500

Target

Payout

< -9% 0%

-9% 25%

0% 100%

>= 13% 350%

64

Panel C: Market Value of Payout as a Function of Index Return

AMEX

Measurement

Period Return

S&P 500 Wtd

Avg Return

Ex-Post Value

of Payout ($)

65

Table 1: RPE Usage Statistics

The following tables provide descriptive statistics of RPE usage in our sample. Panel A reports the portion

of firms using at least some options, stock, and performance-vested awards. In addition Panel A provides

the fraction of total compensation that RPE represents for RPE-granting firms. To determine the portion

of total compensation, RPE ex-ante value is the reported Fair Market Value or the value of the target at the

time of grant. Panel B reports the back-end instrument of at least some of the RPE awards for each firm-

year when reported. Panel C describes the depth of RPE awards among the named executive officers and

directors. Year represents the calendar year of the end of the fiscal year. Note that for all tables it is possible

for a firm to have multiple types of compensation and even multiple RPE awards with varying

characteristics. Thus rows do not necessarily add up to 100%. Appendix A provides all variable definitions.

Panel A: Usage of Plan-based Awards

All P-V Awards RPE P-V Award

Year N T-V

Options

T-V

Restricted

Stock

Annual

Incentive

Plan

Long-

term

Incentive

Plan

Annual

Incentive

Plan

Long-

term

Incentive

Plan

N

(RPE

Details

Reported)

RPE

as Pct

of Tot

Comp

1998 1,134 84.0% 18.1% 22.8% 23.3% 4.5% 9.9% 62 44.1%

1999 1,417 84.6% 17.9% 22.3% 19.8% 3.5% 7.7% 63 33.4%

2000 1,406 84.3% 21.2% 24.6% 19.4% 4.3% 8.0% 52 33.6%

2001 1,393 86.5% 22.0% 23.0% 19.8% 3.7% 7.8% 58 36.5%

2002 1,390 84.5% 26.1% 25.8% 20.1% 4.0% 8.9% 57 30.5%

2003 1,374 81.4% 29.9% 27.3% 23.4% 3.7% 10.4% 63 45.7%

2004 1,368 79.2% 38.0% 27.6% 28.8% 3.7% 12.9% 71 34.3%

2005 1,348 73.2% 42.3% 26.3% 33.8% 3.9% 14.6% 107 37.3%

2006 1,306 71.1% 51.8% 61.1% 46.5% 4.8% 17.1% 220 31.9%

2007 1,273 70.5% 63.8% 73.2% 53.3% 5.7% 18.9% 256 31.6%

2008 1,253 70.3% 63.8% 75.9% 55.9% 5.4% 20.2% 266 31.7%

2009 1,232 67.1% 65.9% 75.1% 55.8% 6.1% 21.5% 284 32.8%

2010 1,210 65.1% 68.3% 77.8% 60.3% 6.3% 25.2% 320 33.6%

2011 1,182 63.2% 69.1% 77.0% 66.6% 6.4% 28.2% 349 32.1%

2012 1,149 56.7% 64.8% 76.8% 69.9% 6.2% 34.0% 386 31.8%

(Continued)

66

Table 1-Continued Panel B: Back-end Instrument for RPE Awards

Year Stock Options Cash

1998 52.8% 4.2% 53.5%

1999 48.6% 2.9% 56.4%

2000 50.3% 2.0% 58.3%

2001 47.3% 0.0% 60.3%

2002 52.8% 0.0% 55.8%

2003 61.3% 1.7% 49.7%

2004 63.1% 2.0% 48.3%

2005 64.8% 0.9% 43.2%

2006 68.7% 3.5% 42.1%

2007 69.9% 2.9% 43.7%

2008 73.9% 2.1% 39.1%

2009 73.4% 3.6% 39.3%

2010 76.1% 2.0% 34.6%

2011 79.1% 1.6% 30.3%

2012 82.4% 1.9% 27.5%

Panel C: Individuals Receiving RPE

Year CEOs

Granted RPE

Directors

Granted

RPE

Avg Number of

NEOs Granted

RPE

1998 95.1% 7.7% 4.5

1999 95.7% 5.7% 4.6

2000 94.7% 7.9% 4.5

2001 96.6% 10.3% 4.6

2002 94.5% 10.4% 4.6

2003 94.2% 7.5% 4.6

2004 93.1% 7.9% 4.6

2005 94.3% 9.3% 4.5

2006 91.5% 6.2% 4.6

2007 92.8% 6.1% 4.5

2008 91.2% 6.3% 4.5

2009 92.5% 5.8% 4.5

2010 93.1% 6.9% 4.6

2011 95.4% 9.1% 4.6

2012 95.1% 6.4% 4.6

(Continued)

67

Table 2: Determinants of RPE Usage

Panel A provides maximum likelihood estimates from a logistic regression for various factors associated

with the propensity to use, adopt, or discontinue RPE. Panel B does the same for the propensity to pay

stock (versus cash) as the back-end instrument (STOCKPAY), the propensity to use stock (versus

accounting) performance in the grant schedule (STOCKRET), and the length of the measurement period

(MEASRET). All variable definitions are provided in Appendix A. All continuous variables are

Winsorized at the 2nd and 98th percentiles. The standard errors are calculated after adjusting for firm-level

clustering in all models with the exception of model three (firms do not have repeating observations). We

report absolute values of Z-statistics in parentheses. Significance is denoted by ***, **, and * at less than

1%, 5%, and 10% levels, two-tailed tests, respectively. The coefficient for the constant is not shown.

Panel A: Logit Estimates for RPE Usage

(1) (2) (3) (4) (5) (6)

RPE/no RPE RPE/no RPE RPE/APE RPE/APE ADOPT DROP

SEG_HERF -0.607** -0.710** -0.529** -0.630* -0.592 -0.295

(-2.42) (-2.16) (-2.07) (-1.88) (-1.64) (-0.72)

IND_HERF -1.041* -1.270* -1.006* -1.050 -1.076 -0.181

(-1.90) (-1.83) (-1.85) (-1.44) (-1.56) (-0.25)

MKT_RISK 0.959*** 0.885 -0.0714 0.740 2.537*** -1.380**

(2.92) (1.46) (-0.21) (1.18) (4.06) (-2.43)

LN_ASSETS 0.329*** 0.297*** 0.337*** 0.379*** 0.130 -0.135

(5.35) (3.49) (5.24) (4.20) (1.38) (-1.38)

M/B -0.0823*** -0.0158 -0.0790*** -0.0396 -0.124*** 0.0409

(-2.81) (-0.38) (-2.64) (-0.83) (-2.59) (1.09)

ROA 2.530** 1.449 1.601 1.616 3.045 -0.514

(2.21) (0.99) (1.40) (1.10) (1.60) (-0.30)

INDADJ_ROA -1.583* -0.671 -1.380 -0.324 -0.926 1.199

(-1.77) (-0.64) (-1.59) (-0.30) (-0.74) (1.03)

RET 0.111 0.408** 0.162 0.545*** 0.254 -0.185

(1.08) (2.18) (1.44) (2.70) (1.02) (-0.78)

INDADJ_RET -0.0523 -0.198 -0.0610 -0.325 -0.510* -0.293

(-0.46) (-0.95) (-0.50) (-1.44) (-1.88) (-1.19)

RET_VOL -0.874*** -0.131 -0.173 0.168 -1.159* 0.582

(-3.17) (-0.30) (-0.60) (0.37) (-1.91) (1.14)

CF_VOL 0.0957 0.236 0.539*** 1.189*** -0.0921 -0.187

(0.60) (1.00) (3.03) (2.91) (-0.39) (-0.56)

SALES_GR -0.449*** -0.663** -0.482*** -0.847** -0.375 -0.309

(-2.83) (-2.03) (-2.95) (-2.47) (-0.70) (-0.81)

68

INV -3.616 -9.042* -5.002 -9.995* 6.032 1.714

(-0.86) (-1.74) (-1.18) (-1.87) (1.21) (0.31)

DIV_YIELD 22.09*** 19.14*** 19.57*** 19.76*** 4.689 -13.60**

(6.37) (4.63) (5.26) (4.39) (0.72) (-2.28)

PCT_INSIDE -3.806*** -2.152* -2.555*** -1.936 -4.120*** -1.412

(-5.38) (-1.87) (-3.50) (-1.59) (-4.36) (-1.16)

BOARD_SIZE -0.0147 -0.0648 -0.0212 -0.0850* 0.0283 0.0473

(-0.48) (-1.36) (-0.63) (-1.74) (0.55) (0.88)

DUALITY -0.0932 0.190 0.102 0.204 -0.677*** 0.114

(-0.78) (1.05) (0.81) (1.10) (-3.39) (0.59)

INSTOWN 0.433 0.0785 0.223 0.273 1.274 0.430

(0.76) (0.10) (0.38) (0.34) (1.42) (0.44)

CONSULT 0.667** 0.523*

(2.33) (1.87)

N 7517 1858 4720 1677 701 1355

Pseudo-R2 0.160 0.118 0.117 0.133 0.150 0.033

Panel B: Characteristics of the RPE Contract

(7) (8) (9)

STOCKPAY

(Stock is paid at

back end)

MEASRET

(Stock price

metric)

STOCKRET

(Both)

SEG_HERF -0.249 -0.943* -0.310

(-0.55) (-1.92) (-0.70)

IND_HERF -0.309 -0.146 -0.926

(-0.31) (-0.15) (-0.98)

MKT_RISK 1.590*** -1.214** 0.158

(2.61) (-2.12) (0.28)

LN_ASSETS -0.00849 -0.110 -0.0363

(-0.07) (-0.79) (-0.29)

M/B -0.0616 -0.0339 -0.0930*

(-1.19) (-0.74) (-1.81)

69

ROA 0.207 -2.356 -0.426

(0.09) (-1.07) (-0.19)

INDADJ_ROA 0.714 1.682 1.911

(0.48) (1.19) (1.34)

RET 0.0905 -0.139 -0.151

(0.43) (-0.69) (-0.72)

INDADJ_RET -0.0372 0.143 0.132

(-0.16) (0.64) (0.59)

RET_VOL -1.083** -1.335** -1.266**

(-2.02) (-2.34) (-2.47)

CF_VOL -0.0812 1.204 0.0942

(-0.33) (1.39) (0.37)

SALES_GR -0.0154 0.0667 -0.104

(-0.06) (0.23) (-0.42)

INV 9.511 3.582 8.908

(1.15) (0.53) (1.26)

DIV_YIELD 18.39*** 11.34 17.58***

(2.86) (1.49) (2.90)

PCT_INSIDE -4.629*** -2.130 -3.182**

(-3.39) (-1.47) (-2.31)

BOARD_SIZE -0.0970 0.0775 -0.0373

(-1.62) (1.17) (-0.63)

DUALITY -0.117 -0.301 -0.0768

(-0.56) (-1.26) (-0.37)

INSTOWN 0.842 -0.175 -0.375

(0.84) (-0.16) (-0.37)

N 1314 1312 1290

Pseudo-R2 0.074 0.062 0.060

70

Table 3

RPE Peer Groups

The following table describes the type of peer group used for measuring relative performance. Select Peers means the firm has

identified the constituents of its peer group by name. Broad Index means the firm is using the constituents of a pre-defined broad

index such as the S&P 500. Industry Index means the firm is using the constituents of a pre-defined industry-specific index.

Panel A: Peer Group Type

Year Select Peers Broad Index or

Peer Group

Industry Index or

Peer Group

1998 65.4% 17.6% 25.0%

1999 62.3% 21.0% 26.1%

2000 64.0% 16.7% 27.3%

2001 63.4% 18.3% 29.6%

2002 64.8% 12.3% 29.6%

2003 66.9% 16.0% 28.4%

2004 71.0% 15.0% 24.5%

2005 67.6% 17.8% 22.7%

2006 71.2% 19.1% 21.4%

2007 73.3% 18.4% 19.1%

2008 70.5% 19.9% 19.9%

2009 72.5% 17.7% 19.0%

2010 75.9% 16.6% 17.4%

2011 74.7% 19.0% 15.8%

2012 72.3% 19.9% 17.1%

71

Table 4

Performance Evaluation The following table describes how firm performance is measured for RPE awards. Panel A reports the firm performance metric

used. Panel B provides further details for awards based on accounting results. Panel C describes representation of RPE by

functional form. Panel D describes the performance period length, the existence of any ex-post vesting after the performance

period, and the average time to vest.

Panel A: Performance Metric Year N Stock Return Accounting

1998 139 79.1% 36.0%

1999 140 75.0% 34.3%

2000 150 76.7% 36.0%

2001 145 70.3% 43.4%

2002 163 72.4% 39.3%

2003 173 74.6% 41.6%

2004 202 75.7% 35.6%

2005 226 76.1% 35.4%

2006 257 73.9% 38.1%

2007 275 75.6% 36.4%

2008 278 74.8% 34.2%

2009 301 75.4% 35.2%

2010 338 78.1% 34.3%

2011 365 80.0% 32.1%

2012 417 83.2% 28.5%

Panel B: Accounting Metric

Year N Earnings

Growth

Return on

X

Cash

Flow

Growth

EVA

Growth Other

Profit

Margin

Sales

Growth

1998 50 38.0% 70.0% 4.0% 0.0% 14.0% 4.0% 16.0%

1999 48 43.8% 70.8% 2.1% 2.1% 20.8% 2.1% 14.6%

2000 52 46.2% 67.3% 7.7% 1.9% 13.5% 1.9% 13.5%

2001 61 52.5% 65.6% 6.6% 4.9% 11.5% 3.3% 16.4%

2002 59 50.8% 61.0% 6.8% 5.1% 11.9% 0.0% 10.2%

2003 69 55.1% 50.7% 5.8% 1.4% 13.0% 1.4% 14.5%

2004 70 60.0% 54.3% 2.9% 0.0% 7.1% 1.4% 17.1%

2005 78 60.3% 50.0% 2.6% 1.3% 10.3% 2.6% 15.4%

2006 98 55.1% 45.9% 6.1% 1.0% 17.3% 3.1% 15.3%

2007 101 48.5% 43.6% 6.9% 1.0% 19.8% 2.0% 22.8%

2008 96 43.8% 43.8% 4.2% 1.0% 24.0% 4.2% 20.8%

2009 107 44.9% 35.5% 4.7% 1.9% 22.4% 4.7% 24.3%

2010 117 42.7% 38.5% 0.0% 1.7% 26.5% 7.7% 18.8%

2011 117 38.5% 40.2% 1.7% 1.7% 28.2% 7.7% 23.1%

2012 122 37.7% 44.3% 0.8% 1.6% 24.6% 6.6% 20.5%

(Continued)

72

Table 4-Continued Panel C: Functional Form of RPE Grant

Year Percentile

Rank

Benchmark-

Adjusted

1998 79.1% 26.7%

1999 78.7% 26.6%

2000 76.8% 25.3%

2001 85.5% 18.1%

2002 84.9% 18.3%

2003 86.5% 15.4%

2004 89.6% 12.8%

2005 88.3% 13.1%

2006 86.3% 15.1%

2007 88.9% 12.3%

2008 86.9% 15.5%

2009 85.8% 15.3%

2010 89.3% 13.3%

2011 87.4% 14.4%

2012 87.3% 15.8%

Panel D: Performance Period and Ex-Post Vesting

Year 1-Year 2-Year 3-Year 4-Year 5+ Years

Multiple

Perf

Periods

Use of

Ex-Post

Vesting

Avg Ex-Post

Vest

(Months)

1998 36.9% 5.0% 58.9% 10.6% 8.5% 9.8% 9.4% 25.3

1999 36.4% 2.9% 60.0% 10.7% 5.0% 9.3% 9.4% 24.2

2000 39.1% 6.0% 63.6% 6.6% 6.0% 10.6% 9.3% 21.2

2001 34.2% 2.1% 67.1% 6.8% 4.1% 8.2% 4.1% 26.3

2002 33.7% 6.1% 66.9% 4.3% 3.7% 7.4% 7.5% 21.0

2003 34.9% 5.2% 66.9% 6.4% 4.7% 6.9% 11.1% 29.8

2004 28.2% 6.4% 74.8% 4.0% 4.5% 7.4% 10.5% 29.4

2005 27.1% 5.8% 74.2% 3.6% 4.4% 7.5% 9.4% 27.9

2006 28.3% 4.3% 74.4% 6.2% 1.9% 11.6% 12.5% 26.5

2007 29.9% 2.9% 74.8% 4.7% 2.5% 15.1% 9.7% 30.4

2008 27.1% 2.8% 75.7% 6.3% 2.1% 14.8% 9.3% 29.9

2009 26.0% 4.5% 75.6% 4.2% 1.9% 16.6% 7.5% 28.0

2010 26.0% 4.9% 78.9% 3.2% 0.9% 13.8% 8.8% 27.6

2011 24.7% 4.6% 78.0% 3.8% 1.6% 14.5% 8.4% 25.5

2012 22.7% 5.0% 79.0% 3.8% 1.7% 13.7% 12.1% 29.1

73

Table 5

Payout Function for RPE Percentile Grants of Stock, Cash, or Options

This table describes how relative performance translates to the ex-post payout. We limit this table to

percentile grant schedules.

Percent of Target Payout

Grant Schedule Percentile

N 10th 25th 50th 75th 90th

Single-Step Schedule

Percentile Performance

Rank for Threshold 1,689 50% 50% 50% 75% 75%

Multi-Step Schedule

Threshold Payout as %

of Target 12,671 0% 23% 27% 50% 50%

Maximum Payout as %

of Target 12,671 100% 150% 200% 200% 200%

Percentile Performance

Rank for Threshold

12,671 20% 25% 30% 39% 50%

Percentile Performance

Rank for Target 12,671 50% 50% 50% 60% 74%

Percentile Performance

Rank for Maximum 12,671 75% 75% 80% 90% 100%

74

Table 6: Simulation Results – Hit Rates Distribution of Simulated Percentage Hit Rates for Threshold, Target, and Ceiling across Percentile RPE

Grant Schedules by Single- versus Multi-step Schedules and by Shares versus Cash as the Back-end

Instrument

Grant Schedule Percentile

N 10th 25th 50th 75th 90th

Stock Price Metric; Paid in Stock

Simulated Ex-Ante Avg Pctl/Rank 1158 0.45 0.49 0.50 0.52 0.54

Single Step Awards - Simulated % times hurdle met 55 28.8% 44.5% 50.9% 58.2% 62.8%

Multi-Step Awards - Simulated % time threshold met 732 52.3% 62.9% 73.0% 80.2% 89.6%

Multi-Step Awards - Simulated % time target met 707 40.7% 45.2% 50.5% 55.1% 60.0%

Multi-Step Awards - Simulated % time max met 732 0.0% 0.0% 12.0% 22.0% 25.3%

Stock Price Metric; Paid in Cash

Simulated Ex-Ante Avg Pctl/Rank 295 0.46 0.49 0.50 0.52 0.54

Single Step Awards - Simulated % times hurdle met 11 26.5% 27.6% 40.7% 45.4% 46.0%

Multi-Step Awards - Simulated % time threshold met 162 45.2% 54.8% 68.7% 77.4% 87.3%

Multi-Step Awards - Simulated % time target met 142 37.9% 44.3% 48.2% 52.9% 57.5%

Multi-Step Awards - Simulated % time max met 162 0.0% 0.0% 15.2% 21.4% 24.3%

Accounting Metric; Paid in Stock

Simulated Ex-Ante Avg Pctl/Rank 361 0.40 0.48 0.51 0.56 0.61

Single Step Awards - Simulated % times hurdle met 19 28.0% 45.5% 55.3% 67.5% 74.1%

Multi-Step Awards - Simulated % time threshold met 214 50.6% 61.6% 76.0% 85.7% 100.0%

Multi-Step Awards - Simulated % time target met 199 31.7% 40.6% 52.9% 61.3% 69.1%

Multi-Step Awards - Simulated % time max met 214 0.0% 0.0% 8.4% 20.9% 27.7%

Accounting Metric; Paid in Cash

Simulated Ex-Ante Avg Pctl/Rank 278 0.46 0.50 0.51 0.54 0.58

Single Step Awards - Simulated % times hurdle met 17 14.5% 20.3% 34.6% 53.5% 78.5%

Multi-Step Awards - Simulated % time threshold met 145 49.4% 58.6% 75.9% 84.0% 100.0%

Multi-Step Awards - Simulated % time target met 139 34.9% 42.8% 50.5% 55.7% 62.5%

Multi-Step Awards - Simulated % time max met 145 0.0% 0.0% 14.5% 22.2% 25.9%

75

Table 7: Simulation Results – Payout Rates and Value Distribution of Payout Rates and Values as a Proportion of Target Payout across Percentile RPE Grant

Schedules per Dollar of Initial Stock Price (Stock Grant) or per Dollar of Cash Target (Cash Grant)

Grant Schedule Percentile

N 10th 25th 50th 75th 90th

Stock Price Metric; Paid in Stock

Avg target multiplier 787 87.3% 95.0% 101.0% 105.1% 112.5%

Avg price per $1 of original stock value 1158 $1.15 $1.19 $1.23 $1.27 $1.30

Avg payout per $1 of original stock value 787 $1.39 $1.49 $1.64 $1.76 $1.87

Std Dev of payout per $1 of original stock value 787 $1.25 $1.44 $1.83 $2.36 $3.10

Stock Price Metric; Paid in Cash

Avg payout per $1 target 173 $0.87 $0.95 $1.02 $1.05 $1.09

Std Dev of payout per $1 target 173 $0.62 $0.74 $0.83 $1.00 $1.18

Accounting Metric; Paid in Stock

Avg target multiplier 233 69.3% 85.0% 105.9% 119.5% 134.2%

Avg price per $1 of original stock value 361 $1.08 $1.19 $1.25 $1.28 $1.31

Avg payout per $1 of original stock value 233 $0.84 $1.06 $1.31 $1.53 $1.78

Std Dev of payout per $1 of original stock value 233 $1.02 $1.20 $1.46 $1.82 $2.27

Accounting Metric; Paid in Cash

Avg payout per $1 target 162 $0.82 $0.95 $1.03 $1.11 $1.22

Std Dev of payout per $1 target 162 $0.59 $0.67 $0.79 $0.98 $1.14

76

Table 8: Simulated Ex Ante Value of Percentile P-V RPE Grants of Stock and

Cash versus Fair Market Value Disclosed by the Company Grant date value is the simulated ex ante present discounted value at the grant date of the percentile p-v

RPE award of stock or cash. The table also reports the present value of the award if risk-neutral methods

are applied and the value of the award disclosed by the company in the proxy statement. The table reports

an estimate of the value of the award if there is no RPE, which is calculated as the average number of back-

end units granted, as determined by the simulation, multiplied by the present value of the back-end award,

which is grant date stock price for stock awards and $1 discounted by the risk-free rate over the performance

period for cash awards. Grant Value Percentile

N 10th 25th 50th 75th 90th

Stock Price Metric; Paid in Stock

Elasticity of Grant Date Value to Stock Price 707 2.70 3.33 4.24 5.54 7.42

Grant Date Value (per $1 of target) 707 $1.02 $1.07 $1.17 $1.25 $1.35

Grant Date Value - Risk-Neutral Value (per

$1 of target)

707 -$0.22 -$0.18 -$0.14 -$0.11 -$0.08

Grant Date Value - Value of Award w/o RPE

Grant Schedule (per $1 of target)

707 $0.00 $0.05 $0.13 $0.27 $0.44

Total Grant Date Value 578 $202,220 $529,993 $1,125,738 $2,237,212 $4,224,330

Total Grant Date Value - Disclosed FMV 344 -$1,187,007 -$443,024 -$70,948 $74,611 $408,458

Stock Price Metric; Paid in Cash

Elasticity of Grant Date Value to Stock Price 142 1.99 2.71 3.85 4.97 9.85

Grant Date Value (per $1 of target) 142 $0.67 $0.73 $0.77 $0.82 $0.90

Grant Date Value - Risk-Neutral Value (per

$1 of target)

142 -$0.18 -$0.15 -$0.12 -$0.09 -$0.07

Grant Date Value - Value of Award w/o RPE

Grant Schedule (per $1 of target)

142 -$0.17 -$0.16 -$0.13 -$0.09 -$0.08

Total Grant Date Value 113 $36,788 $196,639 $538,934 $1,411,060 $2,437,508

Total Grant Date Value - Disclosed FMV 9 -$3,261,163 -$1,167,305 -$699,614 -$674,540 -$137,918

Accounting Metric; Paid in Stock

Elasticity of Grant Date Value to Stock Price 199 0.94 1.04 1.23 1.59 2.25

Grant Date Value (per $1 of target) 199 $0.71 $0.84 $1.05 $1.18 $1.33

Grant Date Value - Risk-Neutral Value (per

$1 of target)

199 -$0.14 -$0.07 -$0.02 $0.01 $0.06

Grant Date Value - Value of Award w/o RPE

Grant Schedule (per $1 of target)

199 -$0.02 -$0.01 $0.00 $0.01 $0.02

Total Grant Date Value 139 $97,476 $222,062 $649,662 $1,328,209 $3,983,840

Total Grant Date Value - Disclosed FMV 24 -$1,609,960 -$850,151 $9,490 $240,408 $629,998

Accounting Metric; Paid in Cash

Elasticity of Grant Date Value to Stock Price 139 0.00 0.04 0.35 0.79 1.52

Grant Date Value (per $1 of target) 139 $0.76 $0.87 $0.97 $1.04 $1.12

Grant Date Value - Risk-Neutral Value (per

$1 of target)

139 -$0.07 -$0.04 -$0.01 $0.00 $0.03

Grant Date Value - Value of Award w/o RPE

Grant Schedule (per $1 of target)

139 -$0.04 -$0.02 -$0.01 $0.00 $0.00

Total Grant Date Value 65 $105,847 $161,915 $328,312 $468,685 $759,008

Total Grant Date Value - Disclosed FMV 0 . . . . .

77

Table 9: Ex Ante Delta Incentive Properties of Percentile P-V RPE Grants of

Stock and Cash Based on a Single Performance Metric

Grant Incentive Percentile

N 10th 25th 50th 75th 90th

Stock Price Metric; Paid in Stock

Marginal Delta in Own Stock Performance (𝛿𝑃) 620 $3,814 $10,552 $23,976 $49,864 $94,458

Marginal RPE Delta in Stock Price Benchmark(s) (𝛿𝑄) 620 -$56,014 -$28,475 -$12,498 -$5,234 -$1,873

Marginal RPE Delta in Stock Performance = Aggregate RPE

Delta in Stock Performance (𝛿𝑃𝑄𝑅𝑃𝐸=𝛿𝐴𝑔𝑔

𝑅𝑃𝐸) 619 $3,449 $9,046 $19,947 $40,004 $71,773

Stock Price Metric; Paid in Cash

Marginal Delta in Own Stock Performance (𝛿𝑃) 139 $342 $2,289 $7,319 $20,138 $48,149

Marginal RPE Delta in Stock Price Benchmark(s) (𝛿𝑄) 139 -$43,316 -$19,001 -$6,933 -$2,553 -$335

Marginal RPE Delta in Stock Performance = Aggregate RPE

Delta in Stock Performance (𝛿𝑃𝑄𝑅𝑃𝐸=𝛿𝐴𝑔𝑔

𝑅𝑃𝐸) 139 $278 $1,682 $5,163 $12,324 $26,638

Accounting Metric; Paid in Stock

Marginal Delta in Own Accounting Performance (𝛿𝐴) 151 $262 $719 $2,305 $6,049 $12,721

Marginal Delta in Accounting Performance Benchmark(s) (𝛿𝐵) 151 -$10,922 -$6,152 -$2,500 -$812 -$252

Marginal RPE Delta in Accounting Performance 𝛿𝐴𝐵𝑅𝑃𝐸 151 $266 $720 $2,268 $6,264 $12,709

Marginal Delta in Own Stock Performance (𝛿𝑃) 151 $971 $2,386 $6,843 $13,662 $33,622

Aggregate RPE Delta in Stock Performance (𝛿𝐴𝑔𝑔𝑅𝑃𝐸) 151 $992 $2,688 $7,223 $14,495 $34,219

Accounting Metric; Paid in Cash

Marginal Delta in Own Accounting Performance (𝛿𝐴) 69 $461 $896 $1,591 $2,163 $4,075

Marginal Delta in Accounting Performance Benchmark(s) (𝛿𝐵) 69 -$3,735 -$2,349 -$1,650 -$726 -$485

Marginal RPE Delta in Accounting Performance 𝛿𝐴𝐵𝑅𝑃𝐸 69 $461 $883 $1,589 $2,159 $3,999

Marginal Delta in Own Stock Performance (𝛿𝑃) 69 $0 $0 $0 $0 $0

Aggregate RPE Delta in Stock Performance (𝛿𝐴𝑔𝑔𝑅𝑃𝐸) 69 -$153 $87 $377 $618 $963

78

Table 10: Ex Ante Vega Incentive Properties of Percentile P-V RPE Grants of

Stock and Cash Based on a Single Performance Metric

Grant Incentive Percentile

N 10th 25th 50th 75th 90th

Stock Price Metric; Paid in Stock

Marginal Vega in Own Stock Performance (νP) 620 $481 $1,524 $3,808 $8,641 $18,102

Marginal Vega in the Stock Performance Benchmark(s) (νQ) 620 -$712 $4 $848 $2,551 $6,001

Aggregate RPE Vega (𝜈Σ) 620 $723 $2,121 $5,081 $10,424 $22,567

Stock Price Metric; Paid in Cash

Marginal Vega in Own Stock Performance (νP) 139 -$4,686 -$1,997 -$393 $5 $436

Marginal Vega in the Stock Performance Benchmark(s) (νQ) 139 -$1 $163 $1,401 $3,730 $9,175

Aggregate RPE Vega (𝜈Σ) 139 -$74 $30 $498 $1,891 $6,205

Accounting Metric; Paid in Stock

Marginal Vega in Own Accounting Performance (νA) 151 -$38,692 -$15,878 -$6,768 -$2,788 -$726

Marginal Vega in Accounting Performance Benchmark(s) (νB) 151

-

$515,284 -$96,485 $7,262 $243,014 $549,979

Marginal Vega in Own Stock Performance (νP) 151 $104 $316 $971 $1,733 $4,380

Aggregate RPE Vega (𝜈Σ) 151 -$625 -$36 $429 $2,825 $8,329

Accounting Metric; Paid in Cash

Marginal Vega in Own Accounting Performance (νA) 69 -$8,485 -$6,194 -$2,898 -$1,020 -$345

Marginal Vega in Accounting Performance Benchmark(s) (νB) 69

-

$156,114 -$78,613 -$2,616 $38,083 $170,765

Marginal Vega in Own Stock Performance (νP) 69 $0 $0 $0 $0 $0

Aggregate RPE Vega (𝜈Σ) 69 -$421 -$196 -$6 $339 $1,050