Voting Efficiency and the Even-Odd Effects of Corporate Board:
Theory and Evidence
Xin Deng, Huasheng Gao, Wei-Lin Liu*
This Version: July 2012
Abstract: We analyze a simple model of board voting and find that in comparison to boards with
an even number of directors (even boards), those with an odd number of directors (odd boards)
improve voting efficiency by better aggregating directors’ information. Consistent with the
model’s implications, our empirical analysis reveals that firms with an odd board derive higher
Tobin’s Q, deliver better operating performance, exhibit stronger Chief Executive Officer (CEO)
turnover-performance sensitivity, and have lower CEO compensation but higher CEO payperformance sensitivity, than do firms with an even board. Furthermore, these even-odd effects
diminish as board size increases. Overall, our findings are consistent with the even-odd
characteristic of board playing an important role in influencing board voting efficiency and thus
the quality of board decisions.
Keywords: Voting Efficiency, Odd Boards, Even Boards, Firm Performance, Corporate
Governance, CEO Turnover, CEO Compensation
JEL Classification: G32; G34; K22
*
Xin Deng (deng0021@e.ntu.edu.sg), Huasheng Gao (hsgao@ntu.edu.sg), and Wei-Lin Liu (wlliu@ntu.edu.sg) are
from the Nanyang Technological University. We thank Chishen Wei, Suman Banerjee, Charlie Hadlock, Chuan
Yang Hwang, and seminar participants at the Nanyang Technological University, National University of Singapore,
University of Hong Kong, Shanghai University of Finance and Economics (SHUFE), and 24th Australasian Banking
and Finance conference, for helpful comments. We also thank Zheng Qiao for excellent research assistance. All
remaining errors are our own.
Understanding the relation between board characteristics and the efficacy of board decisions is an
important issue that has attracted considerable research interests. Extant finance literature has
put under scrutiny the effects of key board characteristics, including board size, fraction of
independent directors, and Chief Executive Officer (CEO)-chairman duality, etc., on boards’
roles in advising CEOs on major corporate strategy, monitoring CEO conduct, and, when
circumstances necessitate, disciplining CEOs. 1 This paper extends the existing literature by
examining the link between board characteristics and board voting efficiency – the extent to
which voting outcomes aggregate directors’ information.
Since major board decisions are
generally preceded by a voting process, voting efficiency is of critical importance to the quality
of such decisions.
The board characteristic we focus on relates to the even-odd nature of the number of
directors. This focus is motivated in part by anecdotal evidence suggesting that the even-odd
characteristic of a board may influence the board’s voting process significantly. For example,
the 2003 proxy filing of Del Global Technologies Corp., a manufacturer of medical devices,
described how the company, in an effort to improve board voting outcomes, switched to an odd
board by expanding its board members from four to five.2 As another example, the corporate
governance guidelines of Enliven Corporation (Nasdaq: ENLV), in addition to prescribing the
range of board size, states explicitly that an odd number of directors is desirable, though not
1
A partial list of the previous studies include Weisbach (1988), Byrd and Hickman (1992), Brickley et al. (1994),
Yermack (1996, 2006), Baliga et al. (1996), Brickley et al. (1997), Eisenberg et al. (1998), Adams and Ferreira
(2008a, 2008b), and Linck et al. (2008). Also, see Adams et al. (2010) for an excellent survey of the recent research
on corporate boards. 2
http://findarticles.com/p/articles/mi_m0EIN/is_2003_May_13/ai_101653816/
1 required.3 A similar statement appears in the corporate governance guidelines of Gleacher &
Company, Inc. (Nasdaq: GLCH).4
Our focus on the even-odd characteristic of board is also motivated by Yermack’s (1996)
classical study on corporate board. While Yermack (1996) focuses on the effects of board size,
his findings also reveal a possible relation between the even-odd characteristic of boards and firm
values. In particular, Figure 1, reproduced from Yermack (1996), shows that odd boards tend to
be associated with a higher Tobin’s Q relative to even boards, especially among small boards.
Reading from the figure, for instance the average Tobin’s Q is 2.1 for five-member boards,
whereas for four-member and six-member boards, the averages of Tobin’s Q assume much
smaller values of 1.6 and 1.55, respectively. Surprisingly, this empirical pattern appears to have
eluded the attention of finance researchers so far.
As a result, there remains a lack of
understanding of the potentially distinctive role of the even-odd characteristic of boards in
determining firm values.
[Insert Figure 1 Here]
To explore the relation between the even-odd characteristic of board and voting
efficiency, we first analyze a simple model of board voting inspired by the previous studies on
group voting in economics and political science. The key element of our model is that each
director has both a performance preference and a conformity preference. The performance
preference aligns a director’s incentive with ensuring high quality board decisions, and thus
motivates the director to vote based on his own information. The conformity preference, on the
3
http://www.enliven.com/downloads/CorporateGovornanceGuidlines.pdf
http://www.gleacher.com/investorrelations/corporategovernance/Documents/Gleacher%20%20Company%20%20Amended%20and%20Restated%20Corporate%20Goverance%20Guidelines%20_5-9-11_.pdf
4
2 other hand, induces an incentive for the director to vote for the decision favored by a majority of
other directors.5
The analysis of our model reveals that because of the conformity preference, voting in
odd boards may aggregate directors’ information better than that in even boards. The intuition
behind this finding is as follows. Because of the conformity preference, in deciding what
decision to vote for, each director considers not only his own information, but also how the other
directors vote. In an odd board, each director faces an even number of other directors. Since, on
average, opposing votes among an even number of directors tend to balance out one another, in
an odd board, the conformity preference becomes moot, and the performance preference causes
each director to vote based on his own information. In contrast, in an even board, each director
faces an odd number of other directors, among whom opposing votes generally do not balance
out one another. Consequently, a strong conformity preference can cause a director to vote in
accordance to the anticipated net vote by the other directors, even if his own information
suggests otherwise.
To examine the implications of our model empirically, we analyze a large sample of
corporate boards over the period 1999–2009. Testing our model directly, however, requires an
accurate measurement of voting efficiency, which appears to be quite challenging.
To
circumvent this problem, we follow most of the previous studies on corporate boards by
assuming that a higher quality of board decisions should manifest in improved firm performance
and strengthened corporate governance. Our tests, therefore, focus on the implications of the
even-odd characteristic of boards on firm performance and corporate governance.
5
Director conformity preference has previously been employed by several theoretical studies to provide insights to a
multitude of empirical regularities of corporate boards (Gillette et al. (2003), Chemmanur and Fedaseyeu (2010), and
Malenko (2011)).
3 Our empirical analysis yields several important findings. First, we find that firms with an
odd board are associated with significantly higher firm values, as measured by Tobin’s Q, than
do firms with an even board. In terms of economic significance, the firm fixed-effect regression
shows that odd boards on average derive a 2.7% higher Tobin’s Q relative to even boards. To
put this number in perspective, Yermack (1996) finds that in the firm fixed-effect regression,
expanding an eight-person board by one member results in a reduction in Tobin’s Q by 4%.
Thus, in terms of firm value, the even-odd effect we document represents close to 70% of the
above size effect, suggesting that the former effect provides an important modification to the
latter effect. Moreover, we show that firms with an odd board are associated with significantly
better operating performance, as measured by return-on-asset (ROA), than do firms with an even
board. In sum, consistent with our model implications, odd boards benefit firms by improving
firm values and operating performance significantly relative to even boards.
Second, we find that the benefits of odd boards exhibit patterns of cross-sectional
variations that are consistent with improved voting efficiency being the source of these benefits.
In particular, we find that the differences in firm value and operating performance between firms
with even and odd boards are especially evident when directors have a strong conformity
preference but weak performance preference. Furthermore, since the extent of information
aggregation in board decisions is likely to be of critical importance to firms that actively make
informationally intensive investments (e.g., R&D expenditure), the difference in firm value and
operating performance between even and odd boards should be particularly pronounced among
firms with high R&D expenditures. Our result also confirms this observation.
Third, we find significant differences in the likelihood of CEO turnover and in CEO
compensation between firms with an odd board and those with an even board. In particular, firms
4 with an odd board show higher CEO turnover-performance sensitivity and have lower CEO
compensation but higher pay-performance sensitivity than do firms with an even board.
Moreover, both of these differences are generally more pronounced when directors have a strong
conformity preference but a weak performance preference and when firms actively make R&D
investments.
Prior studies suggest that CEO turnover policy and CEO pay are important
governance mechanisms that align a CEO’s incentive with that of the shareholders’ (e.g.,
Weisbach (1988), Core et al. (1999), and Hartzell and Starks (2003)). Our findings, therefore,
indicate that odd boards enhance the boards’ effectiveness in corporate governance relative to
even boards and that the enhanced effectiveness is likely to arise from improved board voting
efficiency.
Finally, as an extension of our model, we consider the possibility that directors may face
varying costs in acquiring information critical to board decisions or in participating in board
voting. We argue that because of these costs, the even-odd effects tend to diminish as board size
increases. The results from additional analyses on Tobin’s Q, operating performance, CEO
turnover, and CEO compensation strongly support this argument.
There may be two concerns with our findings. First, as with any study on board structure,
our results may suffer from endogeneity problems. To alleviate these problems, we control for
an extensive list of firm characteristics in our regression analysis. In addition, we control
explicitly for other board characteristics, including board size, proportion of independent
directors, and CEO-chairman duality, to mitigate the concern that the differences between even
and odd boards are a mere reflection of the effects associated with these other characteristics. To
further mitigate possible endogeneity problems, we conduct two additional robustness tests. In
the first test, we use firm fixed effects to control for firm-specific and time-invariant factors that
5 potentially correlate with firms’ propensities of having an odd board and that affect firm
performance and corporate governance. Next, we perform a two-stage treatment regression,
using as instrumental variables the prevalence of odd boards in the firm’s industry and the state
in which the firm is located. We find that our key results on the differences between firms with
even and odd boards survive both tests.
Second, if our findings suggest genuine benefits of odd boards, it might appear that firms
should invariably opt for odd boards. Yet, many firms in our sample have even boards. Our
responses to the seemingly conflicting evidence are as follows. First, our model reveals that a
CEO whose incentive is not aligned with attaining optimal board decisions prefers an even than
an odd board. An even board enables the self-interested CEO to exploit the directors’ conformity
preference and the consequential voting inefficiency so as to ensure that the board adopts the
CEO’s preferred decisions, even if the decisions are detrimental to the firm.
Our second
response follows from the views in Adams et al. (2010) and Coles et al. (2010). Specifically,
since each firm operates within the confines of its exogenously given environment, if changing
the environment is costly or takes time a firm may optimally adopt an even board as the best
solution to the constrained optimization problem relating to the design of the board.6 These
responses also reinforce the importance of the fixed-effect and two-stage treatment regressions,
which ensure that our findings capture the effects of the difference in voting efficiency between
even and odd boards on firm performance and corporate governance beyond those associated
with possible managerial incentive problems or exogenous constraints.
6
For instance, a firm wishing to switch from an even board to an odd board may be constrained from doing so when
the supply of competent directors is limited. Since the firm cannot freely change the availability of competent
directors, staying with an even board may be the firm’s solution to the constrained optimization problem. 6 This paper makes two important contributions to the literature. First, while there is
voluminous literature on group voting in economics and political science, the finance literature
on board voting remains small and mostly theoretical in nature (e.g., Warther (1998), Gillette et
al. (2003), Raheja (2005), Harris and Raviv (2008), Baranchuck and Dybvig (2009), Chemmanur
and Fedaseyeu (2010), and Malenko (2011)). Lack of sufficiently detailed data on the process
and outcome of board voting renders challenging direct tests of these theories. Perhaps for this
reason, empirical analysis on board voting remains scarce. Our model and empirical analysis, on
the other hand, focus on an easily measurable board characteristic: the even-odd nature of the
number of directors. Our findings help establish an empirical link between board characteristics
and board voting efficiency.
Second, we complement the existing literature on corporate board by identifying the
even-odd characteristic of boards as a new measure of the boards’ effectiveness in improving
firm performance and strengthening corporate governance. Our analysis suggests that this new
measure represents an economically significant yet under-explored aspect of boards distinct from
those captured by the conventional measures. As such, our results complement the existing
studies by Yermack (1996) and Coles et al. (2008) in providing a precise characterization of the
relation between firm value and the number of directors.
The rest of the paper is organized as follows. Section I presents a simple model of board
voting and develops the empirical implications of the model. Section II describes the data and
the summary statistics of the key variables. Section III presents the main empirical findings. We
describe the results of additional tests in Section IV, and Section V concludes the paper.
7 I. A Simple Model of Board Voting and Model Implications
A. A Simple Model of Board Voting
A firm can undertake one of two actions: sticking to the status quo, denoted as a = 0, or
adopting a new strategy, denoted as a = 1. The new strategy can be a decision to change CEO
compensation scheme, replace the current CEO, adapt key firm policies, and so on. The firm’s
performance improvement
which can be either high
,
depends on the suitability of the new strategy
or low
, and the action undertaken:
,
,
to the firm,
1
0
0;
1,
,
(1)
1
1.
(2)
From Eq. (1), if the status quo is maintained, the performance improvement is invariably
0. On the other hand, from Eq. (2), adopting the new strategy enhances the firm’s performance if
it is highly suitable to the firm, but leads to a decline in performance if its suitability is low. In
the example of CEO replacement decision (a = 1 if the current CEO is replaced, and a = 0 if the
current CEO is retained), suitability θ is determined by the difference in the overall abilities
between a new CEO candidate and the current CEO. If the new CEO candidate is more (less)
competent than the current CEO, replacing the current CEO is a highly suitable (unsuitable)
strategy, and it improves (reduces) firm performance.
The firm’s board determines the firm’s choice of action through voting. The board has n >
2 directors. 7 Prior to casting their votes, the directors learn aspects of the new strategy.
Specifically, director i, i = 1, 2, …, n, can privately learn the ith aspect of the new strategy
which can be either good,
1, or bad,
1. Before the directors learn about the
7
The smallest board in our sample has three directors. 8 ,
s,
they share a common prior belief that each
is equally likely to be good or bad. Since
s
represent distinct aspects of the new strategy, they are independently distributed.
Collectively, the various aspects of the new strategy stochastically determine its
suitability.
Γ Φ
Specifically, let Φ = (
∑
,…,
) be the set of the directors’ information, and
be the n directors’ collective information about the new strategy.
The
probabilities of high and low suitability conditional on Γ Φ are:
|Γ Φ
Γ Φ ,
(3)
|Γ Φ
Γ Φ .
(4)
Eqs. (3) and (4) suggest that positive (negative) collective information, i.e., Γ Φ
(Γ Φ
0
0), is indicative of high (low) suitability. Moreover, as Γ Φ increases, high suitability
becomes increasingly more likely. In the limit when all the directors receive positive (negative)
information, i.e.,
1(
1 for all i, suitability
(
with certainty. Note
that before the directors obtain their information, according to the common prior belief,
and
are equally likely.
In the running example of CEO replacement, suitability
is likely to reflect the aggregate
of the differences between the two individuals in multiple aspects. These aspects may include
knowledge and understanding about the firm’s business, creative ideas about how to grow the
firm, the abilities to work with the firm’s other senior executives and provide leadership,
personal charisma, and so on. Collectively, the differences in these aspects determine which one
of the two individuals is likely to be the more competent CEO.
9 1 or
Upon obtaining information about , each director votes for either
0.8 In
casting his vote, a director does not know the information that other directors have or how each
of the other directors votes, but holds rational expectations about others’ information and voting
strategies. After the directors cast their votes, each vote is revealed, and the board chooses the
action to implement based on the pre-specified voting rule
1, if the number of directors voting for
i.e.,
, so that the new strategy is adopted,
1 is greater or equal to
.
The directors have the same utility function, consisting of two parts. The first part,
,
, produces a performance preference perfectly aligned with maximizing the firm’s
expected performance improvement. Specifically,
improvement,
,
, so that
,
,
,
is proportional to the performance
, where constant
0 and measures the
strength of the directors’ preference for performance maximization. The directors’ performance
preference may arise directly from their share ownership, as greater ownership benefits directors
more when firm performance improves. Performance preference can also arise from directors’
reputational concerns: the director labor market may see poor firm performance as a sign of the
directors’ inabilities in providing quality monitoring and advising services, thereby jeopardizing
the directors’ pursuits of retaining their incumbent directorships or obtaining new directorships
(Fama and Jensen (1983)).
The basic setup outlined so far parallels that widely used in the studies on common value
voting (e.g., Austen-Smith and Banks (1996), and Feddersen and Pesendorfer (1996)). We
enrich the basic setup by adding a second part to the directors’ utility function that gives rise to a
8
Implicitly, we are assuming that the directors cannot choose to abstain. In our model, this is without loss of
generality, as directors will either vote based on own information when performance preference dominates
conformity preference, or vote to conform to the majority opinion. Thus, even if abstention is allowed, directors will
not invoke that option.
10 conformity preference – to vote for the same action as the one that the board ends up adopting
(Gillette et al. (2003), Chemmanur and Fedaseyeu (2010), and Malenko (2011)). Specifically,
0 if the action he votes for turns out to
we assume that each director faces a personal cost
disagree with the action that the board chooses. Thus, the total utility for director i
, ,
where
,
,
(5)
is an indicator function that takes the value of one if director i’s vote disagrees with
the board’s decision and zero otherwise.
Cost
measures the strength of the directors’
conformity preference.
B. Analysis of the Model
Before examining the voting equilibrium, consider the optimal decision that utilizes the
directors’ collective information to maximize the expected performance improvement. Since
based on the prior belief the two actions yield the same expected performance improvement, one
good (bad) aspect of the new strategy tips the balance toward favoring action a = 1 (a = 0). Thus,
the optimal decision is determined by the difference between the numbers of the good and bad
aspects that the directors observe.
,...,
Given a set of the directors’ observations, Φ
aspects is
let
Φ
∑
, while that of the bad aspects is
1 /2. If
Φ
( Φ
, the total number of the good
Φ . When n is an odd number,
, the directors observe a strictly greater (smaller)
number of good aspects than bad aspects, so the optimal decision chooses
1. If
n is an even number, we set
Φ
( Φ
1(
0). When
1 , the directors observe
a strictly greater (smaller) number of good aspects than bad aspects, so the optimal decision
chooses
1(
0). If
Φ
1, there is an equal number of good and bad aspects. In
11 this case, the two alternative actions provide the same expected performance improvement, and
without loss of generality, we assume that the optimal decision chooses
0.
To achieve the optimal decision through board voting, it is essential that in equilibrium,
each director follows an informative voting strategy, according to which a director votes for
action
1(
0) upon observing a good (bad) aspect. It follows from the above discussion
on the optimal decision that the appropriate voting rule is a simple majority rule with
(the proof of Proposition 1 shows that this is indeed the optimal rule). The following proposition
describes the respective voting equilibrium for an odd board (n is odd) and an even board (n is
even). Appendix B provides the proof of the proposition.
Proposition 1: In an odd board, informative voting is an equilibrium strategy for the directors.
Thus, voting fully aggregates the directors’ information, and board decision coincides with the
optimal decision. In an even board, if
However, when
, informative voting is also an equilibrium strategy.
, informative voting is not an equilibrium strategy, so board voting fails
to aggregate directors’ information. In this case, the board’s decision may not coincide with the
optimal decision.
Since the performance preference is aligned with maximizing the firm’s expected
performance improvement, this preference produces an incentive for the directors to vote
informatively. However, as Proposition 1 indicates, for an even board, the conformity preference
can conflict with the performance preference, and when the former is stronger than the latter, the
directors no longer vote informatively. In particular, even after observing a good (bad) aspect of
the new strategy, a director may vote for
0(
1) if he expects that other directors’ votes
are likely to lead the board to stick to the status quo (adopt the new strategy). In other words, in
an even board when the directors’ conformity preference dominates the performance preference,
12 they tend to vote based on their conjectures about how the other directors will vote instead of on
their own information.
To illustrate the differential effects of the conformity preference on voting in even and
odd boards, consider an odd board with n = 5, and the voting decision by one of the directors,
say director 1, when all the other directors vote informatively. Because director 1 does not know
the other directors’ information, he thinks that the other directors’ possible voting profiles can be
(4, 0), (3, 1), (2, 2), (1, 3), or (0, 4), where the first (second) component in each binary is the
3, profile
number of other directors who vote for a = 0 (a = 1). Given voting rule
(2, 2) represents a pivotal case in which director 1’s vote determines the board’s choice of action.
In this case, director 1 can always ensure his vote to be in line with the board’s decision, so the
conformity preference does not bias director 1’s choice between the two alternative actions.
For the four (even number of) remaining non-pivotal cases, the board’s choice of action is
independent of director 1’s vote. However, because these non-pivotal cases are paired, on net
conformity preference does not bias director 1’s choice between the two actions either.
Specifically, when the other directors’ voting profile is (1, 3), the board chooses action a = 1
regardless of director 1’s vote, so the conformity preference causes director 1 to bias toward
voting for a = 1. But, in the case of voting profile being (3, 1), the board chooses a = 0
regardless of director 1’s vote, so director 1 is biased toward voting for a = 0. Since the directors’
information is statistically independent, director 1 views (1, 3) and (3, 1) as equally likely. 9
Thus, for the paired equal probable profiles (1, 3) and (3, 1), the conformity preference creates
exactly offsetting biases between the two actions. The same logic applies to the pair (4, 0) and (0,
9
The probability for profile (3, 1) is
!
!
3
4
!
!
, while the probability for profile (1, 3) is
.
13 1
4
4).
Consequently, in an odd board, conformity preference does not bias directors’ voting
decisions, which will be based on their own information, and board voting fully aggregates
directors’ information.
Consider next director 1’s voting decision in an even board with n = 4. To director 1, the
other directors’ possible voting profiles include (3, 0), (2, 1), (1, 2), and (0, 3). Given voting rule
1
3, profile (1, 2) is the pivotal case, for which the conformity preference does not
bias director 1’s choice between the two actions. However, among the three (odd number of)
remaining non-pivotal profiles, one profile must be unpaired, thereby creating a net bias in
director 1’s preference between the two actions. Specifically, it is clear that (3, 0) and (0, 3) are
paired equal probable profiles for which the conformity preference creates exactly offsetting
biases between the two actions. In the case of the unpaired profile (2, 1), the board adopts action
a = 0 regardless of director 1’s vote, so the conformity preference biases director 1’s decision
towards voting for a = 0. When this bias is stronger than the incentive that performance
preference creates to vote informatively, director 1 votes for a = 0 even after observing a good
aspect of the new strategy, rendering an equilibrium with informative voting infeasible. Thus, in
an even board, conformity preference creates systematical biases in directors’ voting decisions,
and may prevent the board voting from fully aggregating directors’ information.
The above discussion provides an interesting implication about a CEO’s preference
between an even and odd board, when his incentive is not aligned with ensuring the optimality of
board decisions. Specifically, suppose that the CEO invariably desires to implement a = 0 even
if the action does not maximize the firm’s expected performance improvement. From the above
discussion, when directors have strong conformity preference, i.e.
14 , in an even board the
directors vote for a = 0 irrespective of their respective information about the new strategy, but in
an odd board the directors vote based on their information. Consequently, to ensure that the
board always ends up adopting a = 0, the CEO prefers an even than an odd board. We
summarize this observation as a corollary to the proposition.
Corollary: Suppose that a CEO invariably desires to implement a = 0. If
, the CEO
prefers an even board than an odd board.
C. Discussion
The main intent of our model is to provide a simple and focused illustration of the
difference in information aggregation between voting in odd and even boards. Our model can be
extended in a variety of ways. For example, in our model, directors can both costlessly acquire
private information about the relative merits of the two actions and costlessly participate in
voting.
It is conceivable that information acquisition may be costly as in Fedderson and
Pensendorfer (1997) and Persico (2004), and directors may have to incur private costs in
attending board meetings and participating in voting as in Borgers (2004). Furthermore, in our
model, directors proceed directly to the formal voting after they obtain information about the
performance consequences of the actions. In practice, directors may take part in pre-voting
communication that can be modeled as directors taking a straw poll (e.g., Coughlan (2000)) or
engaging in cheap talk (e.g., Gerardi and Yariv (2007), and Lizzeri and Yariv (2011)).
Other than briefly considering the impact of directors’ costs of information acquisition
and participating in voting in Section IV, we do not attempt to formally pursue the above and
other extensions to our model. Undoubtedly, the extensions will enrich the characterization of
board voting. However, these enrichments will necessarily bring into the model additional key
variables that are likely to be hard to measure empirically. The main implications from our
15 simple model revolve around the even-odd characteristic of boards, which can be measured
easily and unambiguously. Ultimately, whether our simple model provides a useful abstraction
of board voting process and the revealed difference between odd and even boards bears a firstorder effect on the quality of board decisions are empirical issues. Consequently, rather than
seeking to provide a comprehensive model of board voting, we believe that it is more fruitful to
take our model to the data.
D. Model Implications
Our model indicates that odd boards enhance the quality of board decisions by better
aggregating directors’ information than do even boards.
Previous studies show that better
decision making by the board generally leads to increased firm value and operating performance.
For example, supporting the arguments by Lipton and Lorsch (1992) and Jensen (1993) that
small boards improve board decisions by affording efficient communication, Yermack (1996)
finds that small boards are associated with greater firm value and better operating performance.
Following the previous studies, we measure firm value by Tobin’s Q and operating performance
by ROA. Our model, thus, implies that firms with an odd board, on average, derive higher
Tobin’s Q and ROA than do firms with an even board.
Furthermore, our model suggests that directors’ conformity preference is the culprit of the
low voting efficiency of even boards.
In our empirical analysis, we measure conformity
preference by CEO tenure. The idea here is that a CEO with a longer tenure tends to have
greater influence over the board (e.g., Hermalin and Weisbach (1998), and Coles et al. (2010)).
With a more influential CEO, the directors are likely to try harder to anticipate the board’s final
decision and vote in support of that decision for two reasons. First, in instances when the board
16 sides with the CEO in its final decision, a dissident director can be denied future nomination for
reelection, as the influential CEO can exercise significant control over the selection of directors
(Mace (1971), Lorsch and MacIver (1989), Shivdasani and Yermack (1999), and Tejada (1997)).
Second, when the board decides against an influential CEO, a director may also suffer a
significant personal cost from dissenting from the majority. For example, Farrell and Whidbee
(2000) examine forced CEO succession, a process that can get rather contentious, especially if
the CEO has normally been quite influential. Farrell and Whidbee find that outside directors that
are closely aligned with the outgoing CEO face increased likelihood of leaving the board
subsequent to the departure of the CEO. In sum, longer CEO tenure is likely to be associated
with strengthened director conformity preference.
Directors’ performance preference, on the other hand, provides a countervailing force that
mitigates the effect of conformity preference. In our empirical analysis, we use the average
director ownership as the proxy for directors’ performance preference.
This is reasonable
because directors receive more benefits from improvements in firm performance when they hold
larger financial stakes in the firms.10
Taken together, our model implies that the differences in Tobin’s Q and ROA between
firms with an even board and those with an odd board increase as CEO tenure increases and as
average director ownership decreases. In stating the above implication, we fully recognize that
CEO tenure and director ownership may influence Tobin’s Q and ROA through other effects.
For example, the larger CEO influence over the board that comes with longer CEO tenure may
lead to greater CEO entrenchment and thus managerial agency problems, which can negatively
10
Directors’ reputational concerns can also provide strong performance preference (Fama and Jensen (1983) and
Yermack (2004)). However, it is not clear how the average reputational concern for directors can be measured
empirically in a meaningful way.
17 affect Tobin’s Q and ROA. On the other hand, by better aligning directors’ interests with those
of shareholders, larger director ownership can have a positive effect on Tobin’s Q and ROA.
However, there are no obvious reasons why these other effects should operate differently
between even and odd boards. The unique aspect of our model implication is that because of the
disparate influences on board voting in even and odd boards, CEO tenure and director ownership
affect Tobin’s Q and ROA differently between the two types of boards.
Finally, board voting efficiency is likely to have varying benefits to different firms. In
particular, board decision making that better aggregates directors’ information should be
especially beneficial to firms that more actively make investments whose payoffs are highly
uncertain and informationally sensitive. A primary example of such type of investments is R&D
investment.
Consequently, our model implies that the differences in Tobin’s Q and ROA
between firms with an even board and an odd board are particularly pronounced when firms
make large amounts of R&D investments. These arguments lead to the following implication.
Implication 1: All else equal, firms with an odd board are associated with higher Tobin’s Q and
ROA than are firms with an even board. These differences tend to be larger among firms with
longer CEO tenure and lower average director ownership, and among those that more heavily
engage in R&D investments.
Better board decision making should also improve boards’ effectiveness in corporate
governance. A key governance function of boards involves properly evaluating CEOs and acting
promptly to replace those who are performing poorly.
Prior studies show that poor CEO
performance is associated with high likelihood of CEO turnover (Coughlan and Schmidt (1985),
Warner et al. (1988), and Huson et al. (2001)). Moreover, evidence shows that boards that are
more effective tend to be timelier in taking actions against underperforming CEOs, elevating the
18 turnover-performance sensitivity. For example, Weisbach (1988) finds that boards with more
independent directors are more likely to remove poorly performing CEOs promptly. On the
other hand, Goyal and Park (2002) find that captured boards are slower in replacing poorly
performing CEOs. Thus, in parallel with Implication 1, our model provides the following
implication regarding CEO turnover decision.
Implication 2: All else equal, firms with an odd board show higher sensitivity of CEO turnover to
performance than do firms with an even board. This difference tends to be larger among firms
with longer CEO tenure and lower average director ownership, and among those that more
heavily engage in R&D investments.
Another important governance function of boards is to set appropriate managerial
incentives through well-designed CEO compensation. A large number of previous studies show
a close link between the quality of corporate governance and CEO pay. For example, Core et al.
(1999) and Faleye (2007) find that in firms where boards are less capable of providing effective
corporate governance, CEOs tend to receive higher compensation and their pay tends to be less
sensitive to firm performance. Thus, in parallel with the previous implications, our model
provides the following implication.
Implication 3: All else equal, firms with an odd board provide lower CEO compensation but
higher pay-performance sensitivity than do firms with an even board. This difference tends to be
larger among firms with longer CEO tenure and lower average director ownership, and among
those that more heavily engage in R&D investments.
II. Data and Summary Statistics
Our starting point is the RiskMetrics database, which covers directors of S&P 1500
companies. We further require that sample firms have available CEO compensation data from
19 Execucomp, accounting data from Compustat, and stock price data from CRSP. CEO turnover
events are also obtained from Execucomp.
Our final sample consists of 12,075 firm-year
observations from 1999 to 2009.11
[Insert Table I Here]
Table I presents descriptive statistics of sample firms. All dollar values are in 2009
dollars, and all continuous variables are winsorized at the 1st and 99th percentiles. On average,
the board has 9.3 directors, 68.7% of them are independent directors, and the dollar-value
director ownership is 27.6 million. Turning to firm characteristics, on average, the sample firms
have a Tobin’s Q of 1.9, ROA of 8.8%, and annual stock return of 8.2%. Moreover, they make
considerable investment with the average Capex (R&D) of 7.2% (4.1%) of the total sales. The
sample firms are quite large with average market value of equity of $7,381 million and a
moderate leverage ratio of 55.1%. Consistent with Kaplan and Minton (2010), we find that
around 10% of sample firms experience CEO changes in a given year.
The mean CEO
compensation is around 5.6 million. About 80% of the CEOs are also the chairman of the board,
and their average tenure is 6.8 years.
We also split the sample into subsamples of firms that respectively have an even board
and an odd board. There are 6,462 (54%) odd boards and 5,613 (46%) even boards. These
numbers indicate a greater but not significantly larger likelihood of odd boards, and thus, might
appear to be inconsistent with the hypothesized benefits of odd boards. Caution needs be
exercised in jumping to this conclusion. As we pointed out earlier, a firm’s choice between an
even and odd board may be influenced by both managerial incentive problems and the exogenous
11
We start from 1999 because the director ownership information is available in RiskMetrics from 1998 and we use
lagged ownership data in our regression analysis.
20 constraints the firm faces in attempting to solve the optimization problem relating to the design
of board structure. In addition, the comparison based on the pooled sample masks variations in
the even-odd nature of boards over time. We examine such variations in Section IV and show
that they exhibit significant dependence on firm characteristics in ways that are consistent with
our model implications.
Comparisons between the subsamples of firms with even and odd boards show several
differences between the two types of firms. Specifically, in comparison to firms with an even
board, those with an odd board have smaller boards and fewer independent directors, have
simpler corporate structure with a smaller number of business segments, are younger, and have
slightly longer CEO tenure. Comparisons based on median show that firms with an odd board
are smaller in size, as measured by the market value of equity, while comparisons based on the
mean show firms with an odd board use less debt. Furthermore, consistent with our model
implications, firms with an odd board have higher mean Tobin’s Q and median ROA, experience
more CEO turnover, and pay less to their CEOs, than do firms with an even board.
II. Empirical Results
A. Firm Value
We begin our investigation of the ramifications of odd boards by examining firm value,
as measured by Tobin’s Q. Table II reports the results. In all regression models, we control for
an extensive set of board, firm, and CEO characteristics. We also control for year fixed effects
and, except for firm fixed effects regression, industry fixed effects. Here and throughout our
analysis, all standard errors are adjusted for heteroscedasticity and firm clustering.
[Insert Table II Here]
21 In the baseline regression model, we focus on Odd, a dummy variable that takes the value
of one if the firm has an odd board and zero otherwise. We calculate the Odd dummy based on
the board structure at the end of the previous fiscal year, because the impact of board decisions
are likely to take some time to show up in firm performance. Likewise, we use lagged values for
all the other independent variables. In unreported tests, we have also experimented with using
the contemporaneous variables and find that results remain the same qualitatively.
Column (1) of Table II shows that the coefficient of the Odd dummy is positive at 0.060
and significant at the 1% level. Thus, consistent with Implication 1 of our model, firms with an
odd board have, on average, a significantly higher Tobin’s Q than do firms with an even board.
Column (1) also shows that the coefficient of big board dummy, which takes the value of one if
board size is above sample median size and zero otherwise, is negative at –0.17 and significant at
the 1% level. Thus, consistent with Yermack (1996), board size bears a negative effect on firm
value. Similar to Hermalin and Weisbach (1991), Column (1) shows that the proportion of
independent directors does not have a significant effect on Tobin’s Q.
Our theoretical arguments suggest that the difference in Tobin’s Q between firms with an
odd board and those with an even board increases as CEO tenure (the proxy for directors’
conformity preference) increases. To test this prediction, in the second regression model, we
include the interaction term between the Odd dummy and CEO tenure.
Consistent with
Implication 1, Column (2) shows that the coefficient of the interaction term is positive and
significant at the 1% level.
Furthermore, our theoretical arguments suggest that the difference in Tobin’s Q between
firms with an odd board and those with an even board narrows when the directors’ average
22 ownership (the proxy for directors’ performance preference) increases. In Column (3), we
include the interaction between the Odd dummy and average director ownership. The coefficient
of the interaction term is negative and significant at the 5% level.
Thus, consistent with
Implication 1 of our model, this result indicates that the contrast in Tobin’s Q between firms with
even and odd boards is more evident when directors have lower ownerships.
Finally, our theoretical arguments suggest that improved board voting efficiency is likely
to be especially beneficial to firms that heavily engage in R&D investments. To test for this
prediction, we interact the Odd dummy with R&D expenditure in Column (4).12 Consistent with
Implication 1, the interaction term is positive and significant at the 1% level.
The last two regressions in Table II provide additional checks of the findings. First, to
account for possible biases due to omitted variables associated with firm-specific and timeinvariant characteristics, we perform a firm fixed-effect regression by including fixed-effect
dummies in the baseline regression. Column (5) shows that the coefficient on the Odd dummy is
0.027 and statistically significant, indicating that on average, odd boards derive a 2.7% higher
Tobin’s Q relative to even boards. In comparison to the size effect in Yermack (1996), which
produces a 4% increase in Tobin’s Q when the number of directors in an eight-person board is
reduced by one, the even-odd effect on Tobin’s Q is smaller but comparable.
Second, the univariate comparison in Table I shows that odd boards are generally smaller
than are even boards. While we have tried to control for board size using the big board dummy,
this control may nevertheless be imperfect. In light of the findings in Yermack (1996), it is
important to further verify that the even-odd effect we document is not simply a reflection of the
12
To address the concern that firms in financial and utility industries tend to have very small R&D intensities, we
exclude all the financial and utility firms from our sample and redo all the regressions. The results are largely the
same.
23 board-size effect. To control for board size more precisely, we select boards with 6, 7, and 8
directors, with 10, 11, and 12 directors, with 14, 15, and 16, and so on. The idea here is that in
the group of boards with 6, 7, and 8 directors, the even boards (with 6 and 8 directors) have an
average board size close to 7, so the comparison between even and odd boards within the group
is conducted with closely matched board size. Similar logic applies to the comparison in the
group of boards with 10, 11, and 12 directors, and to those in the other groups.
We then rerun the baseline regression in the sample of the selected firms. To account for
the differences in the average board size across the groups, we also include a set of group
dummies in the regression. The Odd dummy in this size-matched regression, therefore, measures
the average difference between odd and even boards across the groups. Column (6) of Table II
reports the regression result. As Column (6) shows, the Odd dummy remains positive and
significant.
In an unreported test, we repeat the above size-matched regression by selecting boards
with 4, 5, and 6 directors, with 8, 9, and 10 directors, and so on. Again, we find that the Odd
dummy remains positive and significant. Note that we cannot combine the two size-matched
regressions as boards with 6 directors will belong to both the group with 4, 5, and 6 directors and
the group with 6, 7, and 8 directors, confounding the interpretation of the Odd dummy. The
same problem applies to boards with 8 directors, and so on. In the rest of the paper, when
referring to size-matched regression, we report the one where the boards are selected as in the
previous regression. However, each time in the unreported test, we also verify that the result
remains similar when the boards are selected as in the second regression.
24 Taken together, our findings show a pronounced and robust difference in Tobin’s Q
between firms with an even board and those with an odd board, and this difference exhibits
cross-sectional variations that are consistent with Implication 1 of our model.
B. Firm Operating Performance
We compare ROA between firms with an even board and those with an odd board. Table
III presents the regression results. In the regressions, we include the same set of control variables
as in Table II.
[Insert Table III Here]
Column (1) of Table III shows that the coefficient of the Odd dummy is 0.251 and
significant at the 1% level. Thus, in comparison to firms with an even board, those with an odd
board deliver significantly better operating performance.
Next, we respectively include the interaction term between the Odd dummy with CEO
tenure and with director ownership in Columns (2) and (3). We find that the coefficient of Odd
× (CEO tenure) is significantly positive in Column (2), while the coefficient of Odd × (Director
ownership) is significantly negative in Column (3). When we include the interaction term
between the Odd dummy with R&D expenditure in Column (4), we find that the interaction term
is positive and significant at the 1% level. Finally, we perform the fixed-effect and size-matched
regressions respectively in Columns (5) and (6). We find that in each of the regressions, the
coefficient on the Odd dummy remains positive and significant.
In sum, the analysis of firm operating performance provides results consistent with
Implication 1 of our model.
25 C. CEO Turnover
In this subsection, we compare CEO turnover decision and, in particular, CEO turnoverperformance sensitivity between firms with an even board and those with an odd board. We
estimate the probability of CEO turnover using logit regression, where the dependent variable is
the CEO turnover indicator, which equals one if the CEO is in his last year in office, and zero
otherwise. Based on the recent findings by Kaplan and Minton (2010) and Jenter and Lewellen
(2010), we do not separate turnover events into forced and unforced ones. Kaplan and Minton
(2010) show that the determinants of forced turnovers are similar to those of voluntary turnovers,
because turnovers labeled as unforced using the algorithms in, for example, Parrino (1997), may
not be de facto voluntary. Furthermore, Jenter and Lewellen (2010) suggest that treating all
turnovers equally can avoid the bias caused by misclassifying forced ones as voluntary ones.13
[Insert Table IV Here]
Table IV presents the results of the logit estimation. The key explanatory variables are
the Odd dummy and firm stock return over the previous three years. We use past three-year
performance because using short-term performance (e.g., performance in the previous 12 or 24
months) tends to underestimate turnover-performance sensitivity (Jenter and Lewellen (2010)).
Table IV contains the results of the logit analysis. The reported coefficient on each
variable is the estimate of the marginal effect of the variable on the probability of turnover when
all the other independent variables are held at their respective mean values. Looking at Panel A
of Table IV, Column (1) shows that the marginal effect of Odd dummy is 0.015 and significant at
the 1% level, indicating that firms with an odd board are 1.5% more likely to experience CEO
turnover relative to firms with an even board. Moreover, consistent with the findings in the
13
As a robustness check, we delete events where CEO departures are likely to be due to retirements, namely CEOs
who are either over 60 or over 65. We find similar results as in the full sample.
26 previous studies, the marginal effect of the firm’s past stock performance is negative and
significant at the 10% level, indicating that CEO turnover becomes more likely subsequent to
poor firm performance.
To examine the difference in the performance sensitivity of CEO turnover between firms
with an even board and those with an odd board, in the second regression we include the
interaction between the Odd dummy with past stock performance. To compute the marginal
effect of the interaction term, we follow the approach developed by Ai and Norton (2003).
Column (2) shows that the marginal effect on the interaction term is negative and significant.
This result indicates that firms with an odd board are more likely to fire the CEO in response to
poor firm performance than are firms with an even board, consistent with Implication 2 of our
model.
We control for firm fixed effects in Column (3)14 and run a size-matched regression in
Column (4), and find that in both instances, the marginal effect of Odd × 3-year return remains
negative and significant.15
In Panel B of Table IV, we conduct a sub-sample analysis on CEO turnover-performance
sensitivity. In the first two columns of Panel B, we divide the sample based on the sample
median CEO tenure. We find that the marginal effect of Odd × 3-year return is –0.006 (–0.017)
for the subsample with short (long) CEO tenure, and is insignificant (significant at the 1% level).
Thus, in terms of both economic magnitude and statistical significance, the effect of odd boards
14
Controlling for firm fixed effects greatly reduces the number of observations in the regression, because firms
experiencing no CEO turnover are dropped.
15
Given that we are using three-year past stock performance, we conduct robustness checks by focusing on the
subsample of CEOs who stay in office for at least three years, and our results are largely the same.
27 in strengthening turnover-performance sensitivity is greater for firms with longer CEO tenure,
consistent with Implication 2 of our model.
In Columns (3) and (4) of Panel B, we split the sample into two subsamples based on the
sample median director ownership. The marginal effect of the interaction term, Odd × 3-year
return, is –0.014 in the low director ownership subsample and –0.007 in the high director
ownership subsample. While the two marginal effects are of similar statistical significance, the
former is two times in magnitude relative to that of the latter. Thus, in terms of the economic
significance, the difference in turnover-performance sensitivity between firms with an odd board
and those with an even board is substantially more pronounced when director ownership is lower,
consistent with Implication 2 of our model.
In the last two columns of Panel B, we conduct subsample analysis based on R&D expense.
The marginal effect on Odd × 3-year return is –0.015 (–0.012) for the high (low) R&D
subsample, and is significant at the 5% (10%) level. Thus, in terms of both economic magnitude
and statistical significance, the effect of an odd board in enhancing CEO turnover-performance
sensitivity is more pronounced for the high R&D firms, consistent with Implication 2 of our
model.
D. CEO Compensation
In this subsection, we compare both the level of CEO pay and the pay-performance
sensitivities between firms with an odd board and those with an even board. Table V contains
28 the results of this analysis. To alleviate the influence of extreme observations, we use the natural
logarithm of total compensation as the dependent variable in the regressions.16
[Insert Table V Here]
The first regression model examines the total CEO compensation (Execucomp Item
TDC1). Column (1) of Panel A in Table V shows that the coefficient on the Odd dummy is –
0.042 and is significant at the 5% level, indicating that CEOs of firms with an odd board tend to
receive around 4% less total compensation. On the other hand, the coefficient on past three-year
stock return is positive and significant, indicating that good past performance leads to high
compensation to CEO.
A possible reason behind the lower total compensation might be that CEO compensation
for firms with an odd board has lower performance sensitivity so that less pay is needed to
compensate CEOs for bearing the compensation risk. To examine this possibility, in Column (2)
we include the interaction term between the Odd dummy and past three-year stock return.
Column (2) shows that the coefficient on the interaction term is positive and significant. Thus, in
comparison to firms with an even board, CEO compensation of firms with an odd board is tied
more closely to firm performance. In sum, in consistency with Implication 3 of our model firms
with an odd board have lower total CEO compensation but higher pay-performance sensitivity.
We control for firm fixed effects in Column (3) and run a size-matched regression in
Column (4). In both instances, we continue to find that firms with an odd board pay less to their
CEOs (though the Odd dummy loses significance in the firm fixed-effect regression) and have
stronger CEO pay-performance sensitivity than do firms with an even board.
16
We also conduct a robustness check by focusing on the subsample of CEOs who stay in office for at least three
years; our results are the same. 29 In Panel B of Table V, we conduct a subsample analysis on CEO compensation, and the
results are generally consistent with Implication 3 of our model. In Columns (1) and (2), we find
that the coefficient on Odd × 3-year return is –0.004 (0.045) and is insignificant (significant) for
the subsample with short (long) CEO tenure. In Columns (3) and (4), we divide the sample
based on the median director ownership. The coefficient on the interaction term Odd × 3-year
return is significantly positive in the low director ownership subsample, but is negative and
insignificant in the high director ownership subsample. Finally, in Columns (5) and (6), we
divide the sample based on the median R&D expenditure. The coefficient on Odd × 3-year
return is significantly positive in the high R&D subsample, but is close to zero and insignificant
in the low R&D subsample.
Taken together, the results in this subsection show that consistent with Implication 3, odd
boards are associated with lower CEO compensation but higher pay-performance sensitivity.
Furthermore, the higher pay-performance sensitivity of firms with an odd board is especially
evident when CEOs have long tenures, directors have low ownerships, and firms actively engage
in R&D investments.
IV. Additional Tests
A. The Dependence of Even-Odd Effects on Board Size
In the regression analysis so far, we have focused on the Odd dummy variable as the
measure of the average even-odd effects and used the big board dummy to control for board size.
Intuitions suggest that the significance of the even-odd effects is likely to diminish as board size
increases. Specifically, extending our model in Section I, suppose that directors face varying
costs to obtain information about the suitability of the new strategy or to participate in board
30 voting. Since a director’s chance of being pivotal in voting declines as board size increases, in
large boards, directors facing high costs of information acquisition or participating in voting have
reduced incentives to acquire information (e.g., Persico (2004)), or to show up for voting (Adams
and Ferreira (2008a, 2008b)). On the other hand, as the analysis in Section I makes clear, the
higher voting efficiency of odd boards requires that all directors are endowed with valuable
information about the suitability of new strategy. Thus, as directors face reduced incentives to
engage in information acquisition or participate in voting, voting in odd boards may also fail to
capture sufficient information. As a result, the difference in the voting efficiency between even
and odd boards narrows when board size increases.
[Insert Table VI Here]
We formally test this prediction in Table VI. In Columns (1) and (2), the dependent
variables are Tobin’s Q and ROA, respectively.
In both columns, the coefficient on the
interaction term, Odd × Big board dummy, is negative and significant, indicating that the
differences in Tobin’s Q and ROA between firms with an odd board and those with an even
board abate when board size increases. The result on Tobin’s Q is also consistent with Figure 1
in Yermack (1996).
Next, we form subsample based on the sample median board size and examine CEO
turnover and CEO compensation in the respective subsamples. The results on CEO turnover are
in Columns (3) and (4). We find that the interaction between Odd dummy and firm’s past threeyear stock performance is significantly negative in the subsample of firms with small boards but
is not significant in the subsample of firms with large boards. Similarly, from Columns (5) and
(6), the coefficient on Odd dummy is significantly negative and that on the interaction between
31 Odd dummy and past three-year stock performance is significantly positive, only in the
subsample of firms with below median board sizes. These results reveal that the differences in
CEO turnover and CEO compensation between firms with even and odd boards decline as board
size increases.
Overall, the results in Table VI show that the even-odd effects are most pronounced in
small boards. In addition, these results appear to suggest that costs of information acquisition
and participating in voting present a significantly negative effect on the voting efficiency of large
boards.
B. The Treatment Regression
In the previous regression models, we have tried to mitigate potential endogeneity
problems by including a comprehensive list of control variables in the regression models and
running firm fixed-effect regressions. To further substantiate our results, we conduct a treatment
regression analysis that rectifies the potential selection problem – the odd-even characteristic of a
board represents the firm’s optimal choice.
The selection problem causes the usual OLS
estimators to be inconsistent (Heckman (1979)).
The underlying empirical model for the treatment analysis can be specified as follows:
,
,
1,
0;
0,
.
In the equation above, the dependent variable represents Tobin’s Q, ROA, CEO
compensation, or the probability of CEO turnover. Variable X is a list of control variables. The
32 coefficient of key interest is
.
indicates the latent propensity of a firm having
an odd board. For the purpose of identification, we include instrument variables
that affect a
firm’s propensity of having an odd board, but do not directly affect the dependent variable. The
Odd dummy is allowed to be endogenous in the sense that the correlation between the noises
corr( ,
0. A positive (negative) correlation biases upward (downward) the coefficient
estimate on odd dummy or on the interaction of odd dummy × 3-year return in OLS or logit
regression when the self-selection problem is not properly corrected for. To allow for timevarying unobserved heterogeneity across firms, we estimate the above model using the maximum
likelihood estimator developed by Maddala (1983, Chapter 5).
We use two instrumental variables. The first is the industry odd board, which is computed
as the ratio of the number of companies with an odd board to the total number of companies in
the firm’s industry. The second is the state odd board, which is measured as the ratio of the
number of companies with an odd board to the total number of companies in the state in which
the firm is located. Since peer firms in the same industry or geographic region tend to face
similar product market, factor market, and legal environment, an individual firm is likely to share
a similar propensity to have an odd board as the peer firms are. Supporting this view, Knyazeva
et al. (2009) provide evidence that a firm’s board structure is significantly influenced by the
same-industry firms in the same state. Thus, the instruments are likely to satisfy the relevance
condition. Furthermore, due to the exogeneity of industry(state)-level variables, there are no
clear reasons to believe that the instruments directly affect firm performance and corporate
governance practice after controlling for various firm characteristics. Thus, the instruments are
also likely to satisfy the exogeneity condition.
[Insert Table VII Here]
33 Table VII reports the results.
We find that in the first-stage probit regression, the
coefficient estimates on industry and state odd board are 2.549 and 2.695, respectively and are
significant at the 1% level. Test of joint significance of the two instruments results in an Fstatistic of 62.72, confirming the validity of our instruments (Staiger and Stock (1997)). In the
second-stage OLS regression of Tobin’s Q, we find that the coefficient on the Odd dummy is
0.351 and is significant at the 5% level, indicating that after controlling for self-selection bias,
the effect of odd board on firm value is still positive. In Column (3), we report the second-stage
OLS regression of ROA and find that after controlling for self-selection bias, the coefficient on
Odd dummy continues to be positive and significant at the 1% level. In Columns (4) and (5), we
report the results of the second-stage logit regression of CEO turnover and OLS regression of
CEO compensation, respectively. Confirming our previous findings, the coefficients on the
interaction Odd × 3-year return are significantly negative in the turnover regression and
significantly positive in the compensation analysis, respectively.
In conclusion, the results of treatment regression reveal that our key findings are robust
for controlling for the endogeneity problem due to self-selection.
C. Switches from Even to Odd Boards
In evaluating the even-odd effects of board, we have so far used pooled regressions. To
gain further insights into these effects, we explore the time-series aspect of our sample, focusing
in particular on instances when firms with an even board switch to an odd board. Given that the
advantage of odd boards over even boards is more pronounced when CEO tenure is longer,
director ownership is lower, and R&D expenditure is higher, we expect that a firm with an even
board is more likely to switch to an odd board in response to long CEO tenure, low director
ownership, or large R&D expenditure.
34 [Insert Table VIII Here]
In our sample, there are 877 firm-year observations where an even board changes to an
odd board. In Table VIII, we apply a propensity score matching method to investigate what
drive these changes. The matching procedure that we employ is a one-to-one nearest neighbor
matching with replacement (Heckman et al. (1997)).
Specifically, we start with a probit
regression where the dependent indicator variable takes a value of one if a firm switches from an
even to an odd board and zero otherwise. The independent variables include the number of
directors, the proportion of independent directors, prior-year stock return, Ln(MV), leverage,
Ln(firm age), and year and industry fixed effects. Using the predicted probability from the
estimated probit regression, i.e., the propensity score, we match each firm that switches from an
even to an odd board to a firm that switches from an odd to an even board and provides the
minimum absolute difference in propensity scores. Table VIII provides the mean values of CEO
tenure, director ownership, and R&D expenditure, in the respective samples. As the last column
of Table VIII shows, compared to the matched sample, the firms that switch from even to odd
boards are associated with longer CEO tenure, lower director ownership, and larger R&D
expenditure.
In sum, Table VIII shows that firms with even boards have a tendency to switch to odd
boards when the benefits associated with the latter type of boards are especially large, further
substantiating the implications of our model.
V. Conclusions
This paper examines the relation between boards’ even-odd characteristic and board
voting efficiency. We develop a simple model of board voting that predicts that odd boards
enhance voting efficiency by enabling better aggregation of directors’ information relative to
35 even boards. Our empirical analysis provides evidence supporting the implications of the model.
In particular, we find that in comparison to firms with an even board, those with an odd board
have higher firm values and better operating performance. Moreover, our analysis reveals that
odd boards strengthen corporate governance by increasing both CEO turnover-performance
sensitivity and CEO pay-performance sensitivity. Finally, the cross-sectional variations in the
benefits of odd boards display patterns of cross-sectional variations that are consistent with
improved voting efficiency as the source of these benefits. Taken together, our findings help
establish a link between board composition and board voting efficiency.
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40 Appendix A: Variable Definitions
Variable
Definitions
Big board
An indicator variable that takes the value of one if the number of directors
is greater than the sample median and zero otherwise.
Board independence
The number of independent directors normalized by the total number of
directors.
Capex
Capital expenditure normalized by sale.
CEO duality
An indicator variable that takes the value of one if the CEO is the chairman
of the board, and zero otherwise.
CEO tenure
The number of years since the person became CEO.
CEO total pay
The variable TDC1 in Execucomp, which consists of salary, bonus, value
of restricted stock granted, value of options granted (using Black-Scholes),
long-term incentive payouts, and other compensation.
Director ownership
The average number of shares owned by directors times the stock price at
the fiscal year end.
Firm age
The number of years since the firm first appears in CRSP.
Industry odd board
The number of firms with an odd board in an industry normalized by the
total number of firms in that industry.
Leverage
The book value of total asset minus book value of equity normalized by the
book value of total assets.
MV
The number of shares outstanding times the stock price at the fiscal year
end.
Number of segments
The number of segments the firm has.
Odd
An indicator variable that takes the value of one if an odd number of
directors are on the board, and zero otherwise.
R&D
Research and development expense normalized by sale.
ROA
Return on total assets, calculated as (Operating income before depreciation
– Net interest expense – Cash taxes – Change in net working capital) /
Total assets.
Sale growth
The ratio of sales over previous year sales.
State odd board
The number of firms with an odd board in a state normalized by the total
number of firms in that state.
Stock return
The buy-and-hold return on the firm’s stock for the prior 12 months.
Stock volatility
The standard deviation of monthly stock return for the prior 60 months.
Tobin’s Q
Market value of assets (total book value of assets minus book value of
equity plus market value of equity) over book value of assets.
Turnover
An indicator variable that takes the value of one if the CEO is in his last
year in office, and zero otherwise
3-year return
The buy-and-hold return on the firm’s stock for the prior three years.
41 Appendix B: Proof of Proposition 1
Proof of Proposition 1: We generalize the examples in the main text to the case where the board
3. To proceed, consider director 1’s voting decision when all the other directors follow
has
the informative voting strategy. As in the main text, we denote the other directors’ voting profile
,
as
, when
of them vote for a = 0 and
of them vote for a = 1.
1 /2, profile ((n-1)/2,
Suppose first that n is an odd number. Given voting rule
(n-1)/2) is the pivotal case, where director 1’s vote determines the board’s choice of action.
Since the other directors follow the informative voting strategy, in the pivotal case their
collective information ∑
0. Thus, when
1(
1 , director 1 views the new
strategy as providing a strictly positive (negative) expected performance improvement. On the
other hand, since the board always ends up adopting the action director 1 votes for, the
conformity preference does not bias director 1 voting decision. Consequently, the pivotal case
motivates director 1 to follow the informative voting strategy.
,
Consider next a non-pivotal case
, where
.
In this case, the board
chooses a = 1 independent of director 1’s vote. Thus, by voting for a = 1, director 1 avoids the
disconformity cost. By doing so, however, director 1 incurs the disconformity cost if the other
directors’ voting profile turns out to be
,
, in which case, the board chooses action a = 0.
Conversely, by voting for a = 0, director 1 avoids the disconformity cost in the case
incurs the cost in the case
,
and
,
,
. Since the
but
s are uncorrelated, director 1 views profiles
as equally likely. Thus, these two paring non-pivotal cases leave director 1
indifferent between voting for a = 1 and for a = 0.
42 ,
When n is an odd number, the other n-1 directors produce a total of n possible voting
profile, namely, (n-1, 0),…, (0, n-1). Excluding the pivotal case, there is an even number ((n-1))
of non-pivotal cases. Thus, each non-pivotal case
paired with an equally likely non-pivotal case
,
,
, can be uniquely
, where
. Thus, in aggregate, the non-pivotal cases
do not produce a strict preference for director 1 between the two actions.
The above discussions suggest that when other directors follow the informative voting
strategy, director 1 also prefers to adopt the same strategy. Thus, in an odd board, informative
voting strategy is an equilibrium strategy for the directors.
Suppose next that n is an even number. Given voting rule
/2
1, director 1 is
pivotal when the other directors’ voting profile is ((n/2)-1, n/2). Since the other directors follow
the informative voting strategy, in this pivotal case the other directors’ collective information
∑
1. Thus, if
1(
1), director 1 views the new strategy as providing a
strictly positive (zero) expected performance improvement, and based on performance preference,
director 1 strictly (weakly) prefers to vote for a = 1 (a = 0). On the other hand, since in the
pivotal case, the board’s choice of action is always aligned with director 1’s choice, the
conformity preference leaves director 1 indifferent between the two actions.
Among the (n-1) (odd number) non-pivotal cases, one case does not have an equally
probable paring non-pivotal case. It is easy to see that this unpaired non-pivotal case is (n/2,
(n/2)-1). In this case, the board adopts action a = 0, so director 1 strictly prefers to vote for a = 0.
Because of the conformity preference, therefore, the non-pivotal cases collectively produce a
strict preference for director 1 to vote for a = 0.
43 The above discussions indicate that that upon observing
1 , the conformity
preference reinforces the performance preference, so director 1 strictly prefers to vote based on
the negative information, i.e., vote for a = 0. But, when director 1 observes
1, conformity
preference conflicts with the performance preference. In this case, by voting for a = 1 and
ensuring that the board adopts a = 1 in the pivotal case ((n/2)-1, n/2), which occurs with
probability
1
, director 1 derives an expected gain of
. However, voting
for a = 1 imposes an expected disconformity cost due to non-pivotal case ((n/2)-1, n/2), which
occurs with probability
1
. The expected disconformity cost is
, director 1 is better off voting for a = 1 after getting
equilibrium strategy. In contrast, if
. If
1, so informative voting is an
, director 1 strictly prefers to vote for a=0, and
informative voting fails to be an equilibrium strategy.
Finally, it follows from the discussions on both the cases where n is odd and when n is
even that the simple majority rule is optimal. Specifically, it is clear that in the case when n is
odd, any other voting rule will create unpaired non-pivotal case(s), which, because of conformity
preference, will bias director 1’s voting decision. In the case when n is even, voting rules other
than the simple majority rule will create more unpaired non-pivotal cases that exacerbate the bias
that conformity preference creates in director 1’s decision.
44 Table I. Summary Statistics
The sample consists of 6,462 firm-year observations with an odd number of directors and 5,613 firm-year observations with an
even number of directors from 1999 to 2009. We collect the board of director information from RiskMetrics, accounting
information from Compustat, stock price data from CRSP, and CEO compensation and turnover information from ExecuComp.
Definitions of all variables are provided in Appendix A. All dollar values are in 2009 dollars. All continuous variables are
winsorized at the 1st and 99th percentiles. Superscripts ***, **, and * denote statistical significance at the 1%, 5%, and 10%
levels, respectively.
Full sample
Number of director
Board independence
Director ownership ($M)
Tobin’s Q
ROA
Stock return
Sale growth
Capex
R&D
MV($M)
Leverage
Number of segments
Firm age
Stock volatility
CEO turnover
CEO total pay ($K)
CEO duality
CEO tenure
Odd board
subsample
Test of differences
Mean
(1)
Median
(2)
Mean
(3)
Median
(4)
Mean
(5)
Median
(6)
t-test
(3) – (5)
9.3
68.7%
27.6
1.9
8.8%
8.2%
111.3%
7.2%
4.1%
7,381.6
55.1%
2.5
24.3
12.1%
0.112
5,636
0.8
9.0
71.4%
6.4
1.5
8.9%
5.1%
108.4%
3.9%
0.0%
1,685.3
56.1%
2.0
18.0
10.7%
0.000
3,284
1.0
9.2
68.4%
26.9
1.9
8.9%
8.4%
111.1%
7.2%
4.1%
7,265.3
54.8%
2.5
23.9
12.1%
0.122
5,537
0.8
9.0
71.4%
6.3
1.5
9.0%
5.1%
108.4%
3.9%
0.0%
1,612.0
56.0%
2.0
17.0
10.7%
0.000
3,175
1.0
9.4
69.0%
28.4
1.8
8.7%
8.1%
111.4%
7.3%
4.1%
7,515.6
55.5%
2.5
24.8
12.1%
0.101
5,752
0.8
10.0
71.4%
6.4
1.5
8.8%
5.2%
108.3%
4.0%
0.0%
1,773.8
56.1%
2.0
19.0
10.7%
0.000
3,372
1.0
–0.2***
–0.6%**
–1.5
0.1**
0.2%
0.3%
–0.3%
–0.1%
–0.0%
–250.3
–0.7%*
–0.0*
–0.9**
–0.0%
0.021***
–215
–0.0
Wilcoxon
test
(4) – (6)
–1.0***
0.0%**
–0.1
0.0
0.2%*
–0.1%
0.10%
–0.1%
0.0%
–161.8***
–0.1%
0.0*
–2.0**
0.0%
0.000***
–197***
0.0
6.8
5.0
6.9
5.0
6.7
5.0
0.2*
0.0**
45 Even board
subsample
Table II. Firm Value and Odd Board
The sample consists of 6,462 firm-year observations with an odd number of directors and 5,613 firm-year observations
with an even number of directors from 1999 to 2009. The dependent variable is Tobin’s Q, calculated as market value
of assets (total book value of assets minus book value of equity plus market value of equity) over book value of assets.
Odd dummy takes the value of one if an odd number of directors are on the board and zero otherwise. All other
controls are defined in Appendix A. Industry fixed effects are based on the two-digit SIC code. In Column (5), we
control for the firm fixed effects. In Column (6), we control for the board size fixed effects, which include a group of
dummy variables to flag the boards with 6, 7, and 8 directors, with 10, 11, and 12 directors, with 14, 15, and 16
directors, and so on. The p-values in parentheses are based on standard errors adjusted for heteroscedasticity and firm
clustering. Superscripts ***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.
46 Oddt-1
(1)
(2)
(3)
(4)
(5)
(6)
0.060***
(0.001)
0.010
(0.667)
0.007***
(0.006)
0.074***
(0.000)
–0.030
(0.155)
0.027*
(0.088)
0.046**
(0.048)
–0.084**
(0.016)
–0.001
(0.548)
1.548***
(0.000)
0.268***
(0.000)
1.123***
(0.005)
0.026
(0.575)
–0.079
(0.113)
–0.417
(0.439)
0.012
(0.682)
–0.173
(0.232)
–0.027**
(0.011)
–0.305***
(0.000)
1.075**
(0.029)
0.017
(0.570)
–0.001
(0.687)
2.652***
(0.000)
–0.001
(0.260)
1.904***
(0.000)
0.254***
(0.000)
3.863***
(0.000)
0.107
(0.191)
–0.194
(0.368)
3.814***
(0.000)
0.160***
(0.000)
–0.154
(0.296)
–0.055***
(0.000)
–0.024
(0.335)
0.549
(0.221)
–0.074**
(0.028)
0.002
(0.413)
1.095***
(0.000)
Odd t-1×CEO tenure t-1
Odd t-1× Director ownership t-1
–0.110**
(0.034)
Odd t-1×R&D t-1
–0.170***
(0.000)
–0.000
(0.629)
1.819***
(0.000)
0.269***
(0.000)
6.125***
(0.000)
0.035
(0.549)
–0.186
(0.332)
3.923***
(0.000)
0.133***
(0.000)
–0.042
(0.743)
–0.049***
(0.000)
–0.029
(0.177)
1.080**
(0.010)
–0.055*
(0.065)
0.003
(0.151)
0.530***
(0.000)
–0.169***
(0.000)
–0.000
(0.631)
1.802***
(0.000)
0.270***
(0.000)
6.118***
(0.000)
0.036
(0.539)
–0.181
(0.345)
3.921***
(0.000)
0.132***
(0.000)
–0.044
(0.736)
–0.049***
(0.000)
–0.029
(0.178)
1.075**
(0.011)
–0.055*
(0.063)
–0.000
(0.848)
0.559***
(0.000)
–0.217***
(0.000)
–0.001
(0.434)
0.203***
(0.002)
0.258***
(0.000)
3.363***
(0.000)
0.154**
(0.031)
–0.064
(0.752)
0.367***
(0.000)
0.216***
(0.000)
–0.435***
(0.001)
–0.067***
(0.000)
–0.037
(0.127)
2.270***
(0.000)
–0.096***
(0.003)
0.005**
(0.046)
0.453***
(0.007)
2.233***
(0.000)
–0.194***
(0.000)
0.000
(0.901)
1.676***
(0.000)
0.250***
(0.000)
5.409***
(0.000)
0.038
(0.520)
–0.141
(0.469)
0.214***
(0.000)
0.165***
(0.000)
–0.255**
(0.043)
–0.058***
(0.000)
–0.033
(0.139)
2.243***
(0.000)
–0.073**
(0.019)
0.004*
(0.097)
0.519***
(0.001)
Year FE
Industry FE
Firm FE
Board size FE
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Observations
Adj R2
12075
45%
Big board t-1
Board independence t-1
Director ownership t-1
Return t-1
ROA t-1
Sale growth t-1
Capex t-1
R&D t-1
Ln(MV) t-1
Leverage t-1
Number of segments t-1
Ln(firm age) t-1
Stock volatility t-1
CEO duality t-1
CEO tenure t-1
Constant
Yes
Yes
Yes
Yes
12075
45%
12075
44%
47 Yes
12075
43%
12075
76%
8882
41%
Table III. Operating Performance and Odd Board
The sample consists of 6,462 firm-year observations with an odd number of directors and 5,613 firm-year
observations with an even number of directors from 1999 to 2009. The dependent variable is ROA, where ROA is
calculated as (Operating income before depreciation – Net interest expense – Cash taxes – Change in net working
capital) normalized by total asset. Odd dummy takes the value of one if an odd number of directors are on the board
and zero otherwise. All other controls are defined in Appendix A. Industry fixed effects are based on the two-digit
SIC code. In Column (5), we control for the firm fixed effects. In Column (6), we control for the board size fixed
effects, which include a group of dummy variables to flag the boards with 6, 7, and 8 directors, with 10, 11, and 12
directors, with 14, 15, and 16 directors, and so on. The p-values in parentheses are based on standard errors adjusted
for heteroscedasticity and firm clustering. Superscripts ***, **, and * denote statistical significance at the 1%, 5%,
and 10% levels, respectively.
48 Odd t-1
(1)
(2)
(3)
(4)
(5)
(6)
0.251***
(0.007)
–0.102
(0.729)
0.049*
(0.098)
0.459
(0.164)
0.294
(0.261)
0.240*
(0.086)
0.179*
(0.090)
–0.592
(0.332)
–0.015
(0.249)
–0.177
(0.231)
2.319***
(0.000)
0.300***
(0.000)
–5.692
(0.354)
–0.780
(0.769)
–11.075*
(0.053)
0.913
(0.285)
–3.141*
(0.061)
–0.014
(0.884)
–0.815
(0.295)
–0.511
(0.928)
–0.017
(0.941)
–0.016
(0.415)
10.590**
(0.027)
–0.002
(0.678)
0.936
(0.213)
1.894***
(0.000)
0.530***
(0.000)
–0.370
(0.358)
2.843*
(0.057)
–3.744*
(0.056)
0.416***
(0.000)
–0.126
(0.815)
–0.147***
(0.001)
0.005
(0.960)
–7.691***
(0.000)
–0.398**
(0.010)
0.009
(0.381)
1.973
(0.154)
Odd t-1×CEO tenure t-1
Odd t-1× Director ownership t-1
–2.698**
(0.049)
Odd t-1×R&D t-1
–0.323**
(0.025)
–0.003
(0.427)
0.209
(0.797)
1.821***
(0.000)
0.520***
(0.000)
–0.605*
(0.094)
3.264**
(0.011)
–4.013**
(0.021)
0.395***
(0.000)
–0.300
(0.513)
–0.140***
(0.000)
–0.018
(0.832)
–9.007***
(0.000)
–0.430***
(0.002)
0.016*
(0.076)
3.689***
(0.000)
–0.793***
(0.002)
0.010
(0.171)
–0.469
(0.771)
1.837***
(0.000)
0.862***
(0.000)
–4.581***
(0.000)
1.300
(0.352)
–10.625***
(0.000)
0.435***
(0.000)
–0.129
(0.887)
0.058
(0.405)
–0.243
(0.102)
–1.457
(0.577)
–0.520**
(0.047)
–0.019
(0.375)
3.689***
(0.001)
–0.806
(0.199)
0.016
(0.377)
0.026
(0.765)
1.669***
(0.000)
0.840***
(0.000)
–4.548*
(0.089)
–2.755
(0.330)
–24.354**
(0.032)
0.710**
(0.021)
–2.440
(0.368)
0.032
(0.633)
–0.375*
(0.079)
–2.355
(0.266)
–0.647***
(0.009)
0.016
(0.493)
3.648*
(0.064)
1.915***
(0.000)
–0.575
(0.230)
0.012
(0.341)
–0.578
(0.303)
2.106***
(0.000)
0.852***
(0.000)
–4.686*
(0.096)
0.785
(0.773)
–6.473***
(0.000)
0.445***
(0.004)
–0.672
(0.664)
0.068
(0.461)
–0.287
(0.120)
–9.066**
(0.039)
–0.519***
(0.001)
0.012
(0.579)
4.344*
(0.077)
Year FE
Industry FE
Firm FE
Board size FE
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Observations
Adj R2
11968
60%
Big board t-1
Board independence t-1
Director ownership t-1
Return t-1
ROA t-1
Sale growth t-1
Capex t-1
R&D t-1
Ln(MV) t-1
Leverage t-1
Number of segments t-1
Ln(firm age) t-1
Stock volatilityt-1
CEO dualityt-1
CEO tenuret-1
Constant
Yes
Yes
Yes
Yes
11968
63%
11968
64%
49 Yes
11968
63%
11968
74%
8806
60%
Table IV. CEO Turnover and Odd Board
The sample consists of 6,462 firm-year observations with an odd number of directors and 5,613 firm-year observations
with an even number of directors from 1999 to 2009. The dependent variable, Turnover, is a dummy variable that
takes the value of one if the CEO is in his last year in office and zero otherwise. Odd dummy takes the value of one if
an odd number of directors are on the board and zero otherwise. The coefficients reported are estimates of the
marginal effect on the probability when all of the independent variables are at their mean value. All other controls are
defined in Appendix A. Industry fixed effects are based on the two-digit SIC code. Board size fixed effects include a
group of dummy variables to flag the boards with 6, 7, and 8 directors, with 10, 11, and 12 directors, with 14, 15, and
16 directors, and so on. Panel A presents the full sample analysis and Panel B presents the subsample analysis. The pvalues in parentheses are based on standard errors adjusted for heteroscedasticity and firm clustering. Superscripts ***,
**, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.
50 Panel A: Full Sample Analysis
(1)
(2)
(3)
(4)
0.015***
(0.004)
–0.006*
(0.080)
–0.065
(0.163)
0.023***
(0.000)
0.000
(0.209)
–0.013
(0.811)
–0.020
(0.118)
0.027
(0.407)
0.000
(0.999)
0.002
(0.336)
–0.008
(0.601)
0.002
(0.247)
0.002
(0.646)
0.124*
(0.063)
0.019***
(0.003)
0.001**
(0.033)
0.016***
(0.000)
–0.012***
(0.002)
0.002
(0.175)
–0.052
(0.147)
0.018***
(0.000)
0.000**
(0.024)
–0.008
(0.848)
–0.024**
(0.018)
0.014
(0.586)
–0.003
(0.924)
0.002
(0.279)
–0.011
(0.365)
0.001
(0.281)
0.001
(0.634)
0.098*
(0.061)
0.012**
(0.014)
0.001**
(0.014)
0.025***
(0.010)
–0.020***
(0.004)
0.003
(0.442)
–0.051
(0.638)
0.043**
(0.010)
0.001***
(0.008)
0.159
(0.260)
–0.012
(0.561)
–0.141
(0.138)
0.177
(0.189)
0.013
(0.236)
–0.039
(0.438)
–0.000
(0.985)
–0.023
(0.427)
–0.362*
(0.055)
0.041***
(0.007)
0.020***
(0.000)
0.021***
(0.000)
–0.015***
(0.004)
0.002
(0.211)
–0.057
(0.147)
Year FE
Industry FE
Firm FE
Board size FE
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Observations
Pseudo R2
11252
2.5%
Odd t-1
Odd t-1×3-year return
3-year return
ROA t-1
Big board t-1
Board independence t-1
Director ownership t-1
Sale growth t-1
Capex t-1
R&D t-1
Ln(MV) t-1
Leverage t-1
Number of segments t-1
Ln(firm age) t-1
Stock volatility t-1
CEO duality t-1
CEO tenure t-1
Yes
Yes
11252
2.6%
51 0.000
(0.105)
0.003
(0.953)
–0.027**
(0.020)
0.031
(0.319)
–0.036
(0.372)
0.001
(0.497)
–0.013
(0.378)
0.001
(0.518)
0.003
(0.406)
0.094
(0.119)
0.010*
(0.069)
0.001*
(0.077)
6890
10.9%
8208
3.0%
Panel B: Subsample Analysis
Short CEO
tenure
subsample
(1)
Long CEO
tenure
subsample
(2)
0.005
(0.422)
–0.006
(0.305)
–0.005
(0.231)
–0.049
(0.311)
0.009
(0.160)
0.000
(0.460)
0.053
(0.302)
–0.007
(0.616)
0.010***
(0.002)
–0.023
(0.596)
–0.002
(0.508)
–0.035**
(0.022)
–0.002
(0.283)
0.001
(0.844)
0.108
(0.113)
0.012**
(0.040)
0.030***
(0.000)
–0.017***
(0.006)
0.005**
(0.025)
–0.038
(0.572)
0.028***
(0.001)
0.001**
(0.027)
–0.095
(0.249)
–0.052***
(0.003)
0.029
(0.331)
0.014
(0.796)
0.005*
(0.079)
0.023
(0.303)
0.006***
(0.006)
0.001
(0.790)
0.013
(0.847)
0.014
(0.152)
Year FE
Industry FE
Yes
Yes
Observations
Pseudo R2
5407
3.9%
Odd t-1
Odd t-1×3-year return
3-year return
ROA t-1
Big board t-1
Board independence t-1
Director ownership t-1
Sale growth t-1
Capex t-1
R&D t-1
Ln(MV) t-1
Leverage t-1
Number of segments t-1
Ln(firm age) t-1
Stock volatility t-1
CEO duality t-1
Low director High director
ownership
ownership
subsample
subsample
(3)
(4)
High R&D
subsample
(6)
0.016**
(0.018)
–0.012*
(0.091)
0.001
(0.901)
–0.057
(0.261)
0.023***
(0.002)
0.000
(0.778)
0.048
(0.366)
–0.017
(0.288)
0.041
(0.200)
0.020**
(0.012)
–0.015**
(0.019)
0.002
(0.239)
–0.038
(0.311)
0.018**
(0.041)
0.001*
(0.074)
–0.119
(0.219)
–0.026**
(0.046)
0.016***
(0.000)
0.017**
(0.019)
–0.014*
(0.095)
–0.001
(0.847)
0.028
(0.528)
0.029***
(0.001)
0.000
(0.189)
0.018**
(0.011)
–0.007*
(0.086)
0.003
(0.158)
–0.038
(0.440)
0.014*
(0.097)
0.000
(0.501)
–0.029*
(0.084)
0.023
(0.158)
0.074
(0.140)
–0.006*
(0.084)
–0.013
(0.500)
–0.000
(0.971)
0.001
(0.850)
0.044
(0.531)
0.019**
(0.028)
0.002***
(0.005)
–0.023
(0.160)
–0.038
(0.321)
–0.008
(0.888)
0.006**
(0.024)
–0.002
(0.906)
0.004*
(0.096)
0.000
(0.940)
0.103
(0.105)
0.017**
(0.037)
0.000
(0.471)
0.003
(0.361)
–0.010
(0.597)
0.001
(0.650)
0.001
(0.860)
0.285***
(0.000)
0.022***
(0.004)
0.001***
(0.010)
0.001
(0.677)
–0.017
(0.422)
0.003
(0.248)
0.003
(0.524)
0.025
(0.760)
0.010
(0.276)
0.000
(0.486)
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
5845
3.8%
5626
3.5%
5626
3.5%
6202
3.8%
5050
3.0%
CEO tenure t-1
52 Low R&D
subsample
(5)
Table V. CEO Compensation and Odd Board
The sample consists of 6,462 firm-year observations with an odd number of directors and 5,613 firm-year observations
with an even number of directors from 1999 to 2009. The dependent variable is Ln(CEO total pay), where total pay is Item
TDC1 in Execucomp, which consists of salary, bonus, value of restricted stock granted, value of options granted (using
Black-Scholes), long-term incentive payouts, and other compensation. Odd dummy takes the value of one if an odd
number of directors are on the board and zero otherwise. All other controls are defined in Appendix A. Industry fixed
effects are based on the two-digit SIC code. Board size fixed effects include a group of dummy variables to flag the boards
with 6, 7, and 8 directors, with 10, 11, and 12 directors, with 14, 15, and 16 directors, and so on. Panel A presents the full
sample analysis, and Panel B presents the subsample analysis. The p-values in parentheses are based on standard errors
adjusted for heteroscedasticity and firm clustering. Superscripts ***, **, and * denote statistical significance at the 1%, 5%,
and 10% levels, respectively.
53 Panel A: Full Sample Analysis
(1)
(2)
(3)
(4)
–0.042**
(0.011)
0.129***
(0.000)
–0.575***
(0.008)
0.051*
(0.061)
0.005***
(0.000)
–1.521***
(0.000)
–0.108***
(0.010)
–0.227
(0.147)
0.051
(0.797)
0.465***
(0.000)
0.641***
(0.000)
0.007
(0.386)
–0.032*
(0.055)
1.870***
(0.000)
0.103***
(0.000)
–0.005**
(0.025)
3.984***
(0.000)
–0.066***
(0.000)
0.075**
(0.010)
0.027
(0.321)
–0.446**
(0.046)
0.046*
(0.091)
0.005***
(0.000)
–1.549***
(0.000)
–0.070*
(0.093)
–0.237
(0.134)
0.036
(0.853)
0.469***
(0.000)
0.639***
(0.000)
0.007
(0.393)
–0.033*
(0.053)
1.868***
(0.000)
0.097***
(0.000)
–0.005**
(0.032)
3.923***
(0.000)
–0.025
(0.125)
0.072***
(0.002)
0.042**
(0.050)
–0.085
(0.711)
–0.005
(0.847)
0.002*
(0.095)
0.078
(0.762)
–0.069
(0.106)
–0.012
(0.942)
-0.495
(0.139)
0.291***
(0.000)
0.050
(0.623)
0.019*
(0.077)
–0.011
(0.838)
0.659
(0.130)
0.008
(0.751)
–0.001
(0.798)
5.885***
(0.000)
–0.039*
(0.061)
0.047*
(0.078)
0.061***
(0.005)
–0.323
(0.151)
Year FE
Industry FE
Firm FE
Board size FE
Yes
Yes
Yes
Yes
Yes
Observations
Adj R2
10520
53%
Odd t-1
Odd t-1×3-year return
3-year return
ROA t-1
Big board t-1
Board independence t-1
Director ownership t-1
Sale growth t-1
Capex t-1
R&D t-1
Ln(MV) t-1
Leverage t-1
Number of segments t-1
Ln(firm age) t-1
Stock volatility t-1
CEO duality t-1
CEO tenure t-1
Constant
Yes
Yes
Yes
Yes
10520
52%
54 0.006***
(0.000)
–1.381***
(0.001)
–0.110**
(0.019)
–0.308*
(0.090)
0.063
(0.774)
0.462***
(0.000)
0.600***
(0.000)
0.011
(0.208)
–0.052***
(0.005)
2.006***
(0.000)
0.094***
(0.000)
–0.004
(0.137)
3.529***
(0.000)
10520
76%
7731
53%
Panel B: Subsample Analysis
Short CEO
tenure
subsample
(1)
Long CEO
tenure
subsample
(2)
–0.033
(0.109)
–0.004
(0.901)
0.149***
(0.000)
–0.691***
(0.005)
0.059*
(0.054)
0.004***
(0.000)
–0.606
(0.137)
–0.162***
(0.004)
–0.340**
(0.039)
–0.108
(0.663)
0.454***
(0.000)
0.551***
(0.000)
0.021**
(0.019)
–0.050***
(0.005)
2.223***
(0.000)
0.088***
(0.001)
–0.062**
(0.015)
0.045*
(0.071)
0.037**
(0.037)
–0.327
(0.313)
0.055
(0.135)
0.008***
(0.000)
–0.688***
(0.000)
–0.048
(0.411)
–0.178
(0.405)
0.179
(0.509)
0.466***
(0.000)
0.742***
(0.000)
0.001
(0.909)
–0.020
(0.408)
1.407***
(0.001)
0.085**
(0.030)
4.214***
(0.000)
Year FE
Industry FE
Observations
Adj R2
Odd t-1
Odd t-1×3-year return
3-year return
ROA t-1
Big board t-1
Board independence t-1
Director ownership t-1
Sale growth t-1
Capex t-1
R&D t-1
Ln(MV) t-1
Leverage t-1
Number of segments t-1
Ln(firm age) t-1
Stock volatility t-1
CEO duality t-1
High R&D
subsample
(6)
–0.039
(0.121)
–0.000
(0.998)
0.109***
(0.000)
–0.370
(0.210)
0.071*
(0.061)
0.005***
(0.000)
–1.366***
(0.004)
–0.093
(0.103)
–0.152
(0.435)
–0.058**
(0.019)
0.068*
(0.069)
0.035
(0.248)
–0.728***
(0.004)
0.007
(0.856)
0.005***
(0.000)
–1.614***
(0.010)
–0.069
(0.242)
–0.486**
(0.047)
0.461***
(0.000)
0.953***
(0.000)
0.018
(0.110)
–0.031
(0.232)
3.079***
(0.000)
0.129***
(0.000)
–0.003
(0.375)
3.548***
(0.000)
0.469***
(0.000)
0.349***
(0.000)
–0.003
(0.779)
–0.011
(0.625)
0.925**
(0.033)
0.067*
(0.059)
–0.006**
(0.038)
4.247***
(0.000)
–0.048*
(0.060)
–0.030
(0.220)
0.144***
(0.000)
–0.648***
(0.001)
0.076***
(0.007)
0.007***
(0.000)
3.645***
(0.000)
–0.141***
(0.000)
–0.038
(0.732)
–0.023**
(0.048)
0.468***
(0.000)
0.512***
(0.000)
0.031***
(0.000)
–0.045***
(0.000)
2.013***
(0.000)
0.108***
(0.000)
0.000
(0.973)
3.758***
(0.000)
–0.040
(0.346)
–0.135
(0.380)
–0.055
(0.413)
0.432***
(0.000)
0.767***
(0.000)
–0.023***
(0.002)
0.004
(0.814)
1.639***
(0.000)
0.091***
(0.001)
–0.006***
(0.000)
3.140***
(0.000)
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
5005
58%
5515
51%
5260
57%
5260
47%
5608
52%
4912
55%
55 Low R&D
subsample
(5)
–0.025
(0.158)
0.030*
(0.091)
0.079***
(0.000)
–0.444***
(0.002)
0.038*
(0.067)
0.004***
(0.000)
CEO tenure t-1
Constant
Low director High director
ownership
ownership
subsample
subsample
(3)
(4)
Table VI. Board Size and the Even-Odd Effects
The sample consists of 6,462 firm-year observations with an odd number of directors and 5,613 firm-year observations
with an even number of directors from 1999 to 2009. Odd dummy takes the value of one if an odd number of directors are
on the board and zero otherwise. The dependent variable is Tobin’s Q in Column (1), ROA in Column (2), Turnover
dummy in Columns (3) and (4), and Ln(CEO total pay) in Columns (5) and (6). The coefficients reported in Columns (3)
and (4) are estimates of the marginal effect on the probability when all of the independent variables are at their mean value. All other controls are defined in Appendix A. Industry fixed effects are based on the two-digit SIC code. The p-values in
parentheses are based on standard errors adjusted for heteroscedasticity and firm clustering. Superscripts ***, **, and *
denote statistical significance at the 1%, 5% and, 10% levels, respectively.
56 Tobin’s Q
(1)
Odd t-1
Odd t-1×Big boardt-1
0.093***
(0.007)
–0.124***
(0.001)
ROA
(2)
0.353***
(0.002)
–0.314**
(0.015)
Odd t-1×3-year return
3-year return
Big board t-1
Board independence t-1
Director ownership t-1
Return t-1
ROA t-1
Sale growth t-1
Capex t-1
R&D t-1
Ln(MV) t-1
Leverage t-1
Number of segments t-1
Ln(firm age) t-1
Stock volatilityt-1
CEO dualityt-1
CEO tenuret-1
Constant
Year FE
Industry FE
Observations
Adj R2 /Pseudo R2
Turnover
Small board
subsample
(3)
Big board
subsample
(4)
0.022***
(0.001)
0.025***
(0.002)
–0.045*
(0.054)
-0.018
(0.472)
–0.010*
(0.098)
0.004
(0.418)
–0.016
(0.101)
–0.007
(0.295)
0.088***
(0.004)
0.010
(0.708)
–0.016
(0.618)
0.133***
(0.000)
Small board
subsample
(5)
Big board
subsample
(6)
–0.005
(0.914)
0.000*
(0.092)
–0.142*
(0.078)
0.000
(0.214)
0.005***
(0.000)
–2.719***
(0.000)
0.006***
(0.000)
–0.755**
(0.031)
0.026
(0.712)
–0.037***
(0.007)
–0.014
(0.708)
0.053
(0.208)
–0.002
(0.363)
0.000
(0.995)
0.002
(0.344)
0.001
(0.781)
0.052
(0.435)
0.013*
(0.058)
0.000
(0.661)
–0.026
(0.688)
0.008
(0.671)
0.061
(0.164)
–0.120
(0.115)
0.007***
(0.009)
–0.034
(0.162)
0.002
(0.401)
0.007
(0.198)
0.183
(0.123)
0.020*
(0.056)
0.002***
(0.001)
–0.387
(0.131)
–0.081
(0.137)
0.006
(0.971)
0.008
(0.972)
0.492***
(0.000)
0.575***
(0.000)
0.003
(0.766)
–0.025
(0.222)
1.456***
(0.000)
0.104***
(0.001)
–0.008***
(0.002)
3.896***
(0.000)
–0.488*
(0.097)
–0.107*
(0.089)
–0.870***
(0.002)
0.162
(0.628)
0.449***
(0.000)
0.748***
(0.000)
0.015
(0.158)
–0.023
(0.331)
2.970***
(0.000)
0.076**
(0.038)
0.002
(0.397)
3.868***
(0.000)
–0.140***
(0.000)
–0.000
(0.639)
1.802***
(0.000)
0.265***
(0.000)
6.103***
(0.000)
0.031
(0.592)
–0.180
(0.354)
3.899***
(0.000)
0.138***
(0.000)
–0.038
(0.770)
–0.049***
(0.000)
–0.026
(0.226)
1.086***
(0.010)
–0.055*
(0.061)
0.004
(0.142)
0.469***
(0.002)
–0.279*
(0.052)
–0.003
(0.429)
0.163
(0.840)
1.810***
(0.000)
51.967***
(0.000)
–0.617*
(0.088)
3.275**
(0.011)
–4.077**
(0.019)
0.410***
(0.000)
–0.277
(0.546)
–0.138***
(0.000)
–0.008
(0.926)
–9.041***
(0.000)
–0.432***
(0.001)
0.016*
(0.076)
3.524***
(0.000)
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
12075
45%
11968
60%
6541
3.8%
4711
4.3%
6307
45%
4213
58%
57 Ln(CEO total pay)
Table VII. Controlling for Self-Selection Bias
The sample consists of 6,462 firm-year observations with an odd number of directors and 5,613 firm-year
observations with an even number of directors from 1999 to 2009. Column (1) reports the first-stage probit
regression of the likelihood of a firm having an odd board, where we use Industry odd board and State odd board
as two instrumental variables. Industry odd board is the number of firms with an odd board in an industry
normalized by the total number of firms in that industry. State odd board is the number of firms with an odd board
in a state normalized by the total number of firms in that state. The dependent variable is Tobin’s Q in Column (2),
ROA in Column (3), Turnover dummy in Column (4), and Ln(CEO total pay) in Column (5). Column (4) reports
the marginal effect on the probability when all of the independent variables are at their mean value. All other
controls are defined in Appendix A. Industry fixed effects are based on the two-digit SIC code. The p-values in
parentheses are based on standard errors adjusted for heteroscedasticity and firm clustering. Superscripts ***, **,
and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.
58 1st stage
Odd
2nd stage
Tobin’s Q
2nd stage
ROA
2nd stage
Turnover
2nd stage
Ln(CEO total pay)
(2)
(3)
(4)
(5)
0.351**
(0.012)
5.054***
(0.000)
–0.056
(0.358)
–0.011***
(0.001)
0.003
(0.105)
0.087
(0.998)
0.075***
(0.000)
0.025***
(0.000)
–0.628***
(0.000)
–0.002**
(0.028)
–0.181
(0.305)
–0.013
(0.664)
–0.040
(0.828)
–0.086
(0.122)
0.099
(0.388)
0.058**
(0.021)
0.038***
(0.000)
0.243***
(0.000)
0.009
(0.226)
0.027*
(0.094)
–1.652***
(0.000)
0.022
(0.454)
–0.001
(0.705)
–2.604***
(0.000)
–0.076*
(0.053)
–0.000
(0.673)
1.846***
(0.000)
0.268***
(0.000)
6.128***
(0.000)
0.047
(0.223)
–0.193*
(0.070)
3.883***
(0.000)
0.131***
(0.000)
–0.085*
(0.099)
–0.051***
(0.000)
–0.034***
(0.003)
1.366***
(0.000)
–0.058***
(0.004)
0.004***
(0.001)
–0.644
(0.307)
–1.475***
(0.000)
–0.001
(0.746)
0.300
(0.689)
1.756***
(0.000)
49.006***
(0.000)
–0.508**
(0.029)
2.768***
(0.000)
–5.612***
(0.000)
0.553***
(0.000)
–0.374
(0.207)
–0.130***
(0.000)
0.155**
(0.028)
–9.804***
(0.000)
–0.426***
(0.001)
0.009
(0.220)
–0.667
(0.855)
0.053
(0.391)
0.000**
(0.011)
0.023*
(0.056)
0.010
(0.567)
0.005***
(0.000)
–1.549***
(0.000)
–0.015
(0.745)
–0.024**
(0.024)
0.026*
(0.095)
0.021
(0.806)
–0.003
(0.896)
–0.018
(0.333)
0.001
(0.374)
–0.004
(0.779)
0.100*
(0.091)
0.012**
(0.025)
0.001**
(0.010)
–0.469***
(0.000)
–0.070**
(0.035)
–0.243***
(0.008)
0.016
(0.893)
0.473***
(0.000)
0.650***
(0.000)
0.009*
(0.085)
–0.028***
(0.005)
1.857***
(0.000)
0.096***
(0.000)
–0.005***
(0.000)
3.722***
(0.000)
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
12,075
12,075
–0.373
6.19**
11,968
–0.568
64.28***
11,252
0.018
0.078
10,520
–0.133
0.250
(1)
Odd t-1
Odd t-1×3-year return
3-year return
Instrument variables:
Industry odd boardt-1
State odd board t-1
Control variables:
Big board t-1
Board independence t-1
Director ownership t-1
Stock return t-1
ROA t-1
Sale growth t-1
Capex t-1
R&D t-1
Ln(MV) t-1
Leverage t-1
Number of segments t-1
Ln(firm age) t-1
Stock volatility t-1
CEO duality t-1
CEO tenure t-1
Constant
Year FE
Industry FE
Observations
Rho
Chi-2 statistic
2.549***
(0.000)
2.695***
(0.000)
59 Table VIII. Firms that Change from an Even Board to an Odd board, Propensity Score Matching
The sample consists of 877 firm-year observations where an even board changes to an odd board from 1999 to 2009. We
match each observation to a firm-year observation where an odd board changes to an even board using the nearest
neighborhood matching approach. The variables used in the matching are the number of directors, the proportion of
independent directors, prior-year stock return, Ln(MV), leverage, Ln(firm age), and year and industry fixed effects. All
continuous variables are winsorized at the 1st and 99th percentiles. P-values based on bootstrapped standard errors of 50
replications with replacement are reported in parentheses. Superscripts ***, **, and * denote statistical significance at the
1%, 5%, and 10% levels, respectively.
CEO tenure Firms that change from an even
board to an odd board
Mean
(1)
6.28
Matched firms
Test of differences
Mean
(2)
5.46
(1) - (2)
0.82***
(0.003)
Director ownership ($M) 22.48
28.21
-5.73**
(0.021)
R&D 6.11%
4.20%
1.91%*
(0.091)
60 Figure 1. Board Size and Tobin’s Q
This graph is extracted from Figure 1 of Yermack (1996) and illustrates sample means and medians of Tobin’s Q for different
sizes of boards of directors. Yermack’s sample consists of 3,438 annual observations from 452 firms between 1984 and 1991.
Figure 1
61