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Working Paper 94 16
EXECUTIVE COMPENSATION:
A CALIBRATION APPROACH
by Joseph G. Haubrich and Ivilina Popova
Joseph G. Haubrich is an economic advisor at the Federal
Reserve Bank of Cleveland, and Ivilina Popova is a graduate
student in the Department of Operations Research, Case
Western Reserve University, Cleveland. The authors thank
Ben Craig for helpful comments.
Working papers of the Federal Reserve Bank of Cleveland
are preliminary materials circulated to stimulate discussion
and critical comment. The views stated herein are those of
the authors and not necessarily those of the Federal Reserve
Bank of Cleveland or of the Board of Governors of the
Federal Reserve System.
December 1994
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Abstract
We use a version of the Grossman and Hart (1983) principal-agent model with 10
actions and 10 states to produce quantitative predictions for executive compensation.
Performance incentives derived from the model are compared with the performance
incentives of 350 firms from a survey by Michael Jensen and Kevin Murphy.
The
results suggest both that the model does a reasonable job of explaining the data and that
actual incentives are close to the optimal incentives predicted by theory.
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1. Introduction
Economists as far back as Adam Smith and Alfred Marshall have wondered about
the incentives of top executives. 'The principal-agent model provides an elegant theory
of incentives, but very little practical advice on how large those incentives should be.
Because of this, much recent work aimed at reconciling theory with the data on
executive pay has focused on the qualitative predictions of the theory. Papers such as
Murdoch (1993), Habib (1993), and Kole (1993) document, for example, that highgrowth firms provide more compensation via stock-option plans. Such work, though
important, avoids key quantitative issues, such as whether compensation provides
sufficient incentives to maximize firm value (see Jensen and Murphy [1990a,b] and
Cowan [1992]). We directly address the quantitative issues by comparing actual CEO
incentives with the predictions of a finite-state principal-agent model.
Haubrich (1994) took a preliminary step in this direction and showed how a simple
parameterization of the Grossman and Hart (1983) principal-agent model produced
performance incentives broadly in line with those documented by Jensen and Murphy.
We generalize those results along two dimensions here. First, instead of a two-state
model (with a closed-form solution), we consider a 10-state, 10-action model. This
allows incentive pay to be nonlinear. Wang (1994) generalizes in a different direction,
developing a multiperiod model with two states and two actions. Second, where
Haubrich (and Wang) made a simple comparison between the model and the mean of
Jensen and Murphy's performancelpay ratio, we take a calibration approach.
Specifically, we choose parameters that minimize the distance between the data and the
model's output for 350 firms, explicitly comparing the distributions. This calibration
approach has an added benefit: It provides an estimate of CEO productivity, a central
but difficult aspect of executive pay.
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The calibration begins with 350 firms chosen from Jensen and Murphy's "New
Survey of Executive Compensation" (1990b). These firms appeared in the dataset of the
Center for Research in Security Prices (CRSP) long enough for us to calculate the
standard deviation of shareholder value.
This variance, along with some global
parameters, pins down the principal-agent problem for each firm. The program then
solves the problem for many values of risk aversion and for another parameter that
measures CEO productivity. It conducts a grid search for the values that minimize the
distance between the 350 predictions and the actual values calculated by Jensen and
Murphy. We use several metrics, including the sum of squared errors and the difference
of sample means.
Determining if the principal-agent model correctly describes executive incentives
does matter. Competing models have very different implications. Jensen (1 989) argues
that political constraints keep firms from tying compensation closely enough to firm
performance, and that as a consequence, leveraged buyouts will replace corporations.
The underinvestment model of Myers and Majluf (1984), by contrast, argues that
compensation is tied too closely to firm performance. The desirability of proposals
pending before Congress -- and shareholders -- depends on the resolution of this issue.
2. The Model and Solution Technique
The key question in the modeling of executive compensation was aptly put by
Marglin (1974): "What do bosses do?" Grossman and Hart's (1983) answer is that
bosses raise the likelihood of good outcomes. In their model, increased effort by the
agent increases the probability of good states occurring. The boss adds value to the
firm, but observing output does not let you infer his actions. A good outcome may
reflect luck as well as hard work.
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2.1. A Discrete Principal-Agent Model
More formally, we assume that the firm has 10 profit levels: q, < q2 <...< q,, .
The action set A consists of 10 possible actions: a,,a2,..., a,,. The restrictions come
from limitations of GAMS, the software we use to solve the nonlinear programming
problem (see Brooke, Kendrick, and Meeraus [1992]). If we used, for example, the
industrial version of GAMS, the number of profit levels and actions could be increased
significantly. xi(a) denotes the probability of state (i.e., profit level) i given action a. To
forestall some technical problems, xi(a)>O for all a and i.
The agent's utility depends on actions and income, expressed as U(a,I). Grossman
and Hart consider a fairly general form, but for calibration purposes, we use constant
absolute risk aversion (CARA),
(a, I) = -e-'('-"), in which effort appears as negative
income.' Choosing the correct hnctional form has its difficulties, but Grossman and
Hart find this utility hnction particularly useful in principal-agent theory, in part because
it has a mutiplicatively separable representation, U(a,I)=K(a)V(I). In addition, since
compensation depends on the disutility of effort, treating effort as negative income
makes the resulting contract easier to interpret. For a more extensive discussion of the
choice, see Haubrich (1994). The agent also has a reservation utility
0,derived from
alternative employment or a leisure-time activity.
Grossman and Hart concentrate on the cost of getting the agent to choose a
particular action. When the principal observes the action, the cost is simply the agent's
reservation price for action a, denoted CFB(a)=h[U/K(a)], where h=V
-'.
The point of the principal-agent problem is that the principal cannot observe the
action taken by the agent. She can only make payment dependent on the observed
1
Wang (1 994) uses the slightly more general specification -e7""'
.
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output state. This incentive scheme {Il...Il~),a set of payments contingent on the state,
gives the agent utility levels VI=V(II)...vlo=V(Ilo).
Although the principal cannot observe the action, she can design the incentive
scheme so that the agent chooses a particular action. The expected value of payments to
the agent defines the second-best cost of an action a, C(a). For a given action, the
incentive scheme minimizes the principal's cost (the expected value of the incentive
scheme) subject to three constraints.
The first is the incentive compatibility constraint, which states that the agent takes
action a only if it gives a higher payoff than any other action. The second and third
constraints are the participation constraints, which state that the agent must get a certain
minimum utility and that some income level produces that utility.
Several incentive schemes (I or v sets) may induce the agent to choose action a
4
(that is, to implement a). Define C(a) as the least costly of these (technically, the
greatest lower bound [infinum or infl of x n , [ a ] @ v , ]on
) the constraint set. If the
principal cannot induce action a (an empty constraint set), set C(a) to infinity.
A little terminology about the principal completes the basic notation. The risk-
neutral principal receives the gross profits q;, so her expected gross benefit from the
agent's action is B(a)=C ~i(a)q;.Her expected net benefit, B(a)-C(a), subtracts the cost
of the action, and the (second-best) optimal action maximizes her expected net benefit.
Grossman and Hart take a simple approach to solving the principal-agent problem.
First, they compute the cost C(a) for each action a. Then, they optimize the net benefit,
B(a)-C(a), over all actions a.
Of central concern here is the proper specification of the n;(a) function: Measuring
the CEO's contribution to the firm is the most problematic aspect of calibrating the
principal-agent problem.
Perhaps the best evidence comes from studies of CEO
turnover, where the effects, though at times substantial, are generally small but
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significant (Weisbach [1988]). At one extreme, when James Crosby, the controlling
shareholder and chief executive of Resorts International, died, Resorts' stock increased
by 37 percent in a single day (Holderness and Sheehan [1991]).
With 10 states and 10 actions, the specification problem becomes even more
difficult, since there are many ways that an action can make good states more likely.
Finding an intuitively appealing specification proved difficult. Even after restricting the
search to probability structures that satisfjl the "spanning condition" (SC), where better
performance means higher pay, many structures had only degenerate feasible solutions
or implied implausible CEO productivity.
This point -- how CEO effort benefits the firm
-- is clearly the major difficulty in
using the direct quantitative approach. Squarely confronting that problem gives us a
better idea of what we lack, both in terms of the data we would like to have and in
regard to the theoretical concepts that need clarification.
We generated the probabilities x;(a) that satisfjl the SC in Grossman and Hart,
A
-
namely, that there exist vectors n , n such that for each action ~ E A ,
A
(1) ~ ( a=) R(a)n+[l -R(a)];
for some 0 5 h (a)
1
and
(2)
gi
is nonincreasing in I.
xi
This precise form is a technical condition to ensure that the incentive scheme
increases in effort. The finction h(a) measures the effects of CEO effort and describes
how much better the probability distribution gets as the CEO expends more effort and
takes increasingly difficult actions. We use a h(a) finction that is decreasing in a. This
means that increasing a moves the probability distribution away fiom the "bad vector
2 and closer to the "good" vector G , making good states more likely.
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Clearly, a major parameter in the calibration exercise is the fbnction h(a). We did
an extensive search in this direction, initially starting with the linear fbnctions h(a)=a and
h(a)=l-a. Unfortunately, in both cases the optimal solution was action 1 or action 10.
For this fbnction, the problem reduces to the two-state case.2 Therefore, we use a
nonlinear fbnction to avoid the problem.
2.2. Solution Procedures
All the pieces are now in place to delineate the nonlinear programming problem. Given
specific risk aversion y and specific U , we have:
10
min
i=l
Ti
~n(-v,)
(a*)(-I)
10
e ~C
' Z, (a*)):> e-p
For every action a* from the action set A, GAMS produced the optimal solution whenever the
problem was feasible. It did this by using one of its solvers, MINOS, which implements some of
We do not yet have analytical proof of this, but it holds true for every set of parameters we
have checked. Let's take a closer look at the linear case:
A(a)=l-a,~(a)=(l-a)%+a?l.
The hnction B(a) has the following form:
Here, K1 and K2 are constants not depending on action.
Hence, max {B(a) - C(a))= max {K1+ K2a - C(a)). In every case we checked, all of the
a
a
problems except those for actions 1 and 10 are infeasible, and so the values of their cost fbnctions
are infinite. M here fore, the maximum will be obtained at either action 1 or action 10.
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the most popular algorithms for solving nonlinear programming problems, i.e., problems in
which at least the objective function or the constraint set is a nonlinear function. In our case, it
is obvious that the objective function is nonlinear. GAMS specifies the above general model for
every particular action from the action set. It then tries to find the optimum. As usual, the first
step is to find a feasible solution. If successful, the next step is to find the optimum. In this
specific example, because the objective function is convex
[-
ln(iVi)), the Kuhn-Tucker-
Karush theorem guarantees that the optimal solution exists, and this is the result that GAMS
[-y)
produces. From this, GAMS generates the cost-function value C(a), where
10
C(a) = m i n x ni(a)
i= 1
for every action from the action set, together with the vector
V1,"2,...,"10.
For some configurations, however, there are no points v, ,v, ,...,vlo that simultaneously
satisfy the constraints. Some actions "a" cannot be implemented by the principal at any cost.
For those cases, we assign an infinitely large number to be the value of the objective function,
The next step is to choose which action to implement, that is, to choose a E A so as to
x
10
maximize B(a) - C(a), where ~ ( a=)
ir, (a)qi
.
i=I
As mentioned before, in the case of linear h(a) functions, only actions 1 and 10 are
-&
feasible. A nonlinear function avoids the reduction to the two-state case. We use A(a) = e
where 6 is a parameter measuring how additional effort affects output,
aeA={O. 1,O.Z...0.9,1.0). Figure 1 plots h(a) for three representative 6's.
In our empirical study, we chose the following values for the vectors
ir
and
k:
,
and
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The above briefly describes one cycle of our procedure. The actual steps are
Step 1
Choose some initial starting values for risk aversion y and CEO
productivity 6.
Ster, 2
Generate the probability distribution from equation (1) for every action a.
Step 3
Use GAMS to solve the nonlinear problem (NP) for every action a.
Produce as an output C(a) and v, ,v,, ...,v,,.
Step 4
Find m a x { ~ ( a-) ~ ( a ) .}Obtain the second-best optimal action a^
Step 5
Compute SSE, BAR, and DSSQ statistics (defined below) using data for
350 companies.
Step 6
Increment y and 6.
acA
Detailed description of Step 5:
The data comprised those 350 companies in Jensen and Murphy's (1990b) "New Survey of
Executive Compensation" for which we could extract shareholder value from the CRSP
database. We extracted the stock price and the number of shares outstanding for the last
trading day of each quarter for the years 1982 - 1990, then used this information to generate the
profit levels q, based on the standard deviations computed for every company from the set.
This meant rescaling the qi's given i3. We next compared the profit shares produced by our
procedure with the real profit shares obtained from Jensen and Murphy. Following their
approach, we define profit share as the fraction of increased shareholder wealth that the CEO
receives in total compensation. In our model, that translates to
l o
- . We arrive at this by
410-91
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using the following finctions:
If we denote by x,,x2,...,x3,, the profit shares fiom
JensenlMurphy and by y, ,y2,...,y3,, the profit shares from our procedure, then
(I)
SSE =
Z(X,-yi)
2
These three statistics are actually finctions of y and 6 . The procedure was to minimize them
with respect to 6 and y.
The three metrics all have a natural interpretation. The first, the sum of squared errors, is
the standard quadratic loss finction. The others attempt to match. specific moments. BAR
matches the means, and DSSQ matches second moments.
3. Results
In looking at the results, three questions stand out: 1) What parameters does the
calibration choose, 2) How closely do we match the data, and 3) What does the optimal
compensation contract look like? Answering these questions resolves the deeper issue
-- What have we learned about principal-agent theory and executive compensation?
We
see where the theory falters and what missing factors hold promise of better fits.
3.1. Basic Results
As described in section 2, the calibration approach searches across parameter
combinations for the values that best match the observed profit shares. Figures 2, 3, and
4 illustrate the procedure by plotting the three different loss finctions against risk
aversion. Figure 2 plots the sum of squared errors,
figure 3 plots the absolute
difference in sample means, aiming at matching the first moment, and figure 4 plots the
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absolute difference in sample standard deviations, aiming to match the second moment.
Table 1 reports the underlying numbers.
The hnctions differ, of course, but a common pattern emerges. The global
minimum occurs at a low level of risk aversion. For the sum of squared errors, there is a
global minimum at 0.125. The two moment-matching cases show lower risk aversion.
The mean case selects 0.025, and the variance case selects 0.025, a boundary value,
suggesting that the actual minimum may occur at even lower values. The results for 6
show greater variability. The SSE and BAR metrics produce values of 10 and 13.5, but
matching variance produces a lower value of 3.5.
What do these parameter values tell us? The risk aversion parameters may initially
seem rather low, but they represent absolute risk aversion, not the relative risk aversion
calculated in most consumption and asset-pricing studies. To convert absolute to
relative risk aversion, we multiply by wealth. One measure of wealth, the median value
of CEO stockholdings, is $3.5 million in the Jensen and Murphy sample. Since our
paper works in million-dollar units, this suggests adjusting risk aversion by a factor
between one and ten. With this in mind, the numbers look reasonable but still low.
The meaning of the parameter 6, labeled CEO productivity, is less obvious. It
describes how increased effort heightens the probability of good states, moving away
from probability vector 7; toward ?I. For the SSE optimal value of 6=10, for example,
with the lowest level of effort al=O.1, the probability of the best state is 0.11; for a2=0.2,
the probability is 0.13; and for a p l , the probability is 0.14. For the DSSQ optimal
value of 6=3.5, the corresponding good-state probabilities are 0.08, 0.10, and 0.14.
How well does this calibration match the data? Table 1 and figures 2, 3, and 4
provide one set of answers (since they are explicit metrics), but these are hard to
interpret. Another way to look at the match is as follows. The average actual
performancelpay ratio for the 350 firms in the sample is 0.01003 (the CEO gets $10.03
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for every $1,000 increase in shareholder value), and the sample standard deviation is
0.032. For the mean (BAR) case, the corresponding figures are 0.01004 and 0.008.
The calibration matches the mean quite well, to within 1 cent per $1,000 of shareholder
wealth. It seriously understates the standard deviation, however, a topic we pursue in
the next section. The calibration designed to match standard deviations did better, of
course.
Figure 5 plots the optimal incentive scheme (compensation contract) for each
metric and lists the optimal action chosen under each scheme. The incentive schemes
are monotonic, meaning that the agent gets paid more in good states. (This must
happen because the probabilities satisfjl the SC.) However, they are also nonlinear: A
given increase in firm profits (a constant difference in gross profits from one state to the
next) corresponds to a different change in the agent's income.
Figure 5 also indicates that the linear compensation scheme, implicitly assumed in
Jensen and Murphy's empirical work and explicitly assumed in Rosen (1990), Haubrich
(1994), and Wang (1994), is not the filly optimal contract. The CEO receives greater
rewards for improving a bad state than for improving a good state. Kaplan (1994) finds
evidence that incentives may differ across states in this manner. Figure 5 also suggests
that the model takes this too far, overemphasizing the negative payments in bad states.
Wang (1 994) argues that a dynamic approach avoids this problem.
3.2. Comparing Distributions
Formal metrics have the advantage of being explicit, but they can also hide
information about the distributions being compared. The problem boils down to
comparing two distributions. We use a series of graphs developed by statisticians to
examine the total distributions in more detail. Figure 6 shows a percentile plot of the
actual profit shares and the profit shares generated by the model (SSE case).
A
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percentile plot graphs the value against its percentile, allowing easy comparison between
percentiles. Examining two together provides a picture of how the distributions differ,
even if single numbers such as means match up closely. For the actual values, notice the
small number of extreme values at the top. For the model, notice the absence of both
negative and very high values. No predicted profit share exceeds 0.05, while 30 actual
values do so, reaching as high as 0.43. In general, figure 6 shows that the model slightly
overpredicts profit shares for most companies, but never produces the large profit shares
found at the high end of the data. Our judgment is that in these extreme cases, such as
An Wang, where the CEO is also a substantial stockholder of the firm,the distinction
between principal and agent breaks down, making our model inappropriate. These
major errors also explain why matching the standard deviation and the sum of squared
errors is difficult.
Figures 7 and 8 take the comparison one step further. Figure 7 shows apercentile
comparison graph (see Cleveland [I 985]), which plots the ordered values of one dataset
against the ordered values of another. Identical distributions result in a perfect x = y
line, while a small amount of noise results in random deviations around that line. One
defect of the graph is that the human eye is a poor judge of distance from a slanted line.
The Tukey Sum-Difference graph (figure 8) resolves the problem, plotting Yi-xi against
yi+xi and in effect rotating the 45" line to the horizontal. Notice that for most values,
the model predicts a profit share that is a little too high. For larger values, the model
underpredicts profit share. This problem gets worse for larger values.
3.3. Truncated Sample Results
The comparisons in section 3.2 indicate that the model fails in cases of high
performance pay. In these cases, the CEO is also a (often the) major stockholder in the
firm, a point emphasized by looking at the names: Barron Hilton of Hilton Hotels, An
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Wang of Wang Laboratories, and Richard Timken of Timken Industries. It no longer
seems clear that the CEO is the agent of the stockholders, and it is not surprising that
the model breaks down.
To account for this, we truncated our sample by removing all executives (12) with
a performancelpay ratio above 0.05 ($50/$1,000). After checking each to make certain
that the high ratio was due to large shareholdings, we recalibrated the model. The
results are presented in table 2 and figures 9 and 10.
Table 2 shows that in addition to matching the mean, the model can also match the
standard deviation of the truncated sample very closely. The percentile plots show an
even closer match, but a similar pattern to before: overprediction of profit shares for
most firms, underprediction for the highest. Figures 9 and 10 compare the actual and
predicted distributions for the truncated sample. The extreme outliers are gone, though
the model again does worse at high levels. Note the relative paucity of profit shares
above 0.0 1.
The model can clearly generate a distribution of profit shares that closely matches
the actual distribution. This is not the same as accurately predicting each firm's profit
share, however. Figure 1 1 illustrates this, plotting the predicted profit share for each
firm against its actual profit share. While the model produces a distribution similar to
that found in the data, a firm with a high predicted profit share may or may not have a
high actual profit share.
3.4. Parameter Uncertainty
Calibration chooses parameters, but some degree of uncertainty necessarily
surrounds the parameters chosen. More important, because we care little about the
uncertainty in y and &perse, there is uncertainty in the predicted profit shares.
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Cecchetti, Lam, and Mark (1993) correctly emphasize that this uncertainty has two
parts. One part arises because the input data, the variance of shareholder value, is only
an estimate of the true variance and thus has its own uncertainty, i.e., its own finite
sample distribution. The other part exists because of the error in the final estimate; we
estimate y and 6 with efror, so the predicted distribution of profit shares, which depends
crucially on these parameters, also has an associated error.3
Simulations can address the first source of uncertainty. For a normal population,
(n - l)s2
the sample variance has a
distribution, or more precisely,
with (n-1)
o2
-
degrees of freedom. We took 100 draws from a
distribution and rescaled the
~ 3 4 9 ~
sample variances to produce new input data. This new data, in conjunction with the old
optimal contract, yielded new predictions of profit shares and a corresponding value for
the distance between those predictions and the actual profit shares. Table 3 reports the
results -- how the distribution varies when the underlying variance changes, given
particular values for y and 6 . The first panel reports the findings for the y and 6 that
minimize the SSE in the actual data, while the second and third panels report the
combinations that minimize differences in means and variances.
We find that the uncertainty does matter: The variation around the optimal is
nontrivial. This is particularly noticeable in the mean and variance case, which matched
the original data most closely. For example, originally, mean-predicted pay matched the
actual mean to within 1 cent in $1,000; changing the variances dropped the match to
between $3 and $5 per $1,000.
procedure such as the Generalized Method of Moments would explicitly introduce these two
types of uncertainty. Unfortunately, our model, both because of its particular form and because it
has no closed-form solution, makes it difficult (if not impossible) to apply the required
orthogonality conditions.
3~
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The other source of uncertainty, the estimated values of y and 6, already showed
up in table 1, which revealed how the difference between predicted and actual values
changes with shifts in y and 6
-- and
how the distribution depends on the estimated
parameters, which is exactly what we wish to know.
4. Conclusion
No one would deny the insights gained from looking at the qualitative
correspondence between economic theory and executive compensation.
Still, as
recognized by Jensen and Murphy (1990a), exercises such as correlating CEO
compensation with firm risk can run afoul of the biblical injunction about straining gnats
and swallowing camels. The finer nuances may not matter if the CEO has inadequate
incentives. Our results show the feasibility of using calibration to undertake a direct,
quantitative approach.
Beyond demonstrating feasibility, calibration produces some usehl information by
forcing us to look at questions that would not come up in most purely econometric
settings. In so doing, we get an estimate of CEO productivity: By taking the best
action rather than the worst, the CEO increases the probability of the most profitable
outcome from 0.08 to 0.14. We also find that theory predicts a decidedly nonlinear pay
schedule for top executives, one that rewards improvements from bad outcomes more
than improvements from good outcomes.
Our results suggest that standard principal-agent theory predicts low profit shares
for CEOs. Results such as those of Jensen and Murphy should not be taken as strong
evidence that CEO compensation schemes are seriously out of line with proper
incentives.
Quantitatively, the theory can be said to match the data success&lly by two
criteria. First, by matching moments, the mean of the predicted values differs from the
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mean of the actual values by only 1 cent in $1,000. Next, on a more subjective level, the
percentile comparison plots show broad coherence between actual and predicted
distributions. This occurs despite ignoring differences known to affect CEO pay, such
as company size (Rosen [1990]) and CEO tenure (Gibbons and Murphy [1992]).
Calibration has contributed substantially to our understanding of asset pricing and
business cycles. We believe that taking the quantitative predictions of theoretical models
seriously can also contribute to the study of executive compensation and, more broadly,
to corporate finance as well.
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Marglin, Stephen A., "What Do Bosses Do?: The Origins and Functions of Hierarchy in
Capitalist Production," Review of Radical Political Economy, vol. 6, Summer 1974, pp. 60112.
Murdoch, Jane, "Factors Explaining the Variation in the Use of Executive Incentive Contracts,"
Charles River Associates, Boston, unpublished manuscript, June 1993.
Myers, Stuart C., and Nicholas S. Majluf, "Corporate Financing and Investment Decisions
When Firms Have Information That Investors Do Not Have," Journal of Financial Economics,
vol. 13, 1984, pp. 187-221.
Rosen, Sherwin, "Contracts and the Market for Executives," NBER Working Paper No. 3542,
December 1990.
Wang, Cheng, "Incentives, CEO Compensation, and Shareholder Wealth in a Dynamic Agency
Model," University of Iowa, Department of Economics, unpublished manuscript, April 1994.
Weisbach, Michael S., "Outside Directors and CEO Turnover," Journal of Financial
Economics, vol. 20, 1988, pp. 43 1-460.
clevelandfed.org/research/workpaper/index.cfm
clevelandfed.org/research/workpaper/index.cfm
Figure 2 3D Plot SSE
Source: Authors' cal~ulati~ns.
clevelandfed.org/research/workpaper/index.cfm
clevelandfed.org/research/workpaper/index.cfm
clevelandfed.org/research/workpaper/index.cfm
Figure 5: Incentive Pay in Each State for the Different Optimal Values of Gamma and Delta
Pay
15 T
IS
S
E - Action 4 ----BAR
- Action 4
-DSSQ - Action 10 1
Source: Authors' calculations.
clevelandfed.org/research/workpaper/index.cfm
FIGURE 6: PERCENTILE PLOT, ACTUAL AND
PREDICTED PROFIT SHARES, 350 FIRMS
Profit share
Predicted
I
I
I
I
60
Percent
SOURCES: Michael C. Jensen and Kevin J. Murphy, "A New Survey of Executive Compensation" (1990); and authors' calculations.
I
clevelandfed.org/research/workpaper/index.cfm
FIGURE 7: PERCENTILE COMPARISON PLOT, ACTUAL
AND PREDICTED PROFIT SHARES, 350 FIRMS
Predicted profit share
0.250
0.200
0.150
0.100
0.050
0.000
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Actual profit share
SOURCES: Michael C. Jensen and Kevin J. Murphy, "A New Survey of Executive Compensation" (1990); and authors' calculations.
0.45
clevelandfed.org/research/workpaper/index.cfm
FIGURE 8: TUKEY SUM-DIFFERENCE GRAPH
OF PERCENTILES, 350 FIRMS
Predicted - Actual profit share
-0.008
0.042
0.092
0.142
0.192
0.242
0.292
0.342
0.392
0.442
Predicted + Actual profit share
SOURCES: Michael C. Jensen and Kevin J. Murphy, "A New Survey of Executive Compensation" (1990); and authors' calculations.
0.492
clevelandfed.org/research/workpaper/index.cfm
FIGURE 9: PERCENTILE PLOT, ACTUAL AND
PREDICTED PROFIT SHARES, 338 FIRMS
Profit share
0.050
0.040
0.030
0.020
0.010
0.000
I
-0.010
0
20
40
60
80
100
Percent
SOURCES: Michael C. Jensen and Kevin J. Murphy, "A New Survey of Executive Compensation" (1990); and authors' calculations.
clevelandfed.org/research/workpaper/index.cfm
clevelandfed.org/research/workpaper/index.cfm
cn
z
[I:
U.
[I:
a
ui
[I:
a
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cn
clevelandfed.org/research/workpaper/index.cfm
Table 1.: MINIMIZINGTHE LOSS FUNCTIONS
SSE
Y~S
0.001
0.005
0.025
0.125
0.225
0.325
0.425
0.525
0.625
0.725
0.825
0.925
14
0.55111
0.43298
0.38553
0.37591
0.38400
0.38516
0.38594
0.38628
0.38580
0.39535
0.39547
0.39559
13.5
0.68391
0.40798
0.38417
0.37225
0.38470
0.38580
0.38661
0.38707
0.38703
0.38531
0.39559
0.39571
13
0.80316
0.41208
0.37865
0.37209
0.38092
0.38644
0.38723
0.38775
0.38793
0.38749
0.39571
0.39581
12.5
0.70816
0.40503
0.38635
0.37545
0.37635
0.38705
0.38782
0.38837
0.38866
0.38858
0.38761
0.39591
12
0.65400
0.44856
0.37702
0.37659
0.37672
0.38765
0.33838
0.38893
0.38928
0.38938
0.38906
0.38719
11.5
0.65583
0.47376
0.37645
0.37517
0.37861
0.37910
0.38892
0.38946
0.38984
0.39003
0.38996
0.38935
11
0.68357
0.44765
0.37248
0.37583
0.38013
0.38003
0.38943
0.38994
0.39033
0.39058
0.39064
0.39042
10.5
0.76505
0.44017
0.38336
0.37484
0.37918
0.38193
0.37903
0.39040
0.39079
0.39106
0.39119
0.39114
14
0.01880
0.00453
0.00078
0.00524
0.00746
0.00771
0.00786
0.00793
0.00784
0.00934
0.00936
0.00937
13.5
0.02211
0.00469
0.0049
0.00761
0.00784
0.00799
0.00807
0.00807
0.00784
0.00937
0.00939
13
0.02387
0.00586
0.00048
0.00455
0.00671
0.00796
0.00810
0.00820
0.00823
0.00815
0.00939
0.00940
12.5
0.01999
0.00773
0.00054
0.00443
0.00617
0.00807
0.00821
0.00830
0.00835
0.00834
0.00817
0.00941
12
0.01796
0.00981
0.00038
0.00477
0.00568
0.00818
0.00831
0.00840
0.00846
0.00847
0.00842
0.00810
11.5
0.01889
0.00979
0.00030
0.00511
0.00594
0.00656
0.00840
0.00848
0.00854
0.00858
0.00857
0.00847
11
0.02230
0.00792
0.00088
0.00522
0.00624
0.00641
0.00848
0.00856
0.00862
0.00866
0.00867
0.00864
6
5.5
5
4.5
4
3.5
0.00678
0.00690
0.008540.00846
0.00954
0.00957
0.00955
0.00944
0.00606
0.00748
0.00878
0.00881
0.00858
0.00965
0.00965
0.00960
0.00628
0.00788
0.00784
0.00903
0.00899
0.00859
0.00972
0.00969
0.00581
0.00693
0.00832
0.00797
0.00919
0.00907
0.00976
0.00975
0.00351
0.00627
0.00762
0.00704
0.00857
0.00932
0.00928
0.00895
0.00977
0.00371
0.00566
0.00703
0.00805
0.00886
0.00865
0.00938
0.00921
0.00976
10
0.79702
0.45097
0.38728
0.38095
0.38134
0.38271
0.39083
0.39120
0.39148
0.39166
0.39170
9.5
0.70971
0.46733
0.37681
0.37185
0.38185
0.38296
0.38288
0.38262
0.39159
0.39187
0.39206
0.39215
9
0.77705
0.49516
0.38660
0.37505
0.37464
0.38393
0.38454
0.38387
0.39195
0.39221
0.39242
0.39254
10.5
0.02494
0.00834
0.00170
0.00494
0.00642
0.00679
0.00662
0.00863
0.00869
0.00873
0.00876
0.00875
10
0.02397
0.00945
0.00142
0.00439
0.00666
0.00700
0.00705
0.00870
0.00876
0.00880
0.00882
0.00883
9.5
0.02262
0.01109
0.00123
0.00408
0.00693
0.00722
0.00734
0.00703
0.00881
0.00886
0.00888
0.00890
9
0.02592
0.01145
0.00190
0.00415
0.00580
0.00744
0.00758
0.00755
0.00887
0.00891
0.00894
0.00895
3
2.5
2
0.00487
0.00543
0.00628
0.00725
0.00819
0.00893
0.00941
0.00926
0.00969
0.00592
0.00656
0.00610
0.00805
0.00857
0.00901
0.00936
0.00911
0.00951
0.00675
0.00722
0.00726
0.00756
0.00864
0.00890
0.00912
0.00930
0.00946
1.5
0.00696
0.00728
0.00757
0.00764
0.00777
0.00832
0.00877
0.00911
0.00903
0.01101
1
0.00724
0.00735
0.00746
0.00732
0.00699
0.00781
0.01101
0.01101
0.01101
0.01101
BAR
~16
0.001
0.005
0.025
0.125
0.225
0.325
0.425
0.525
0.625
0.725
0.825
0.925
1
1
DSSQ
Y~S
0.025
0.125
0.225
0.325
0.425
0.525
0.625
0.725
0.825
0.925
00.00433
. 0 00.00429
289
wl
0.00409 0.00382
Source: Authors' calculations.
clevelandfed.org/research/workpaper/index.cfm
Table 2: RESULTS FOR 338 FIRMS
SSE
Yfi
0.025
0.125
0.225
0.325
0.425
0.525
0.625
0.725
0.825
0.925
BAR
Yfi
0.025
0.125
0.225
0.325
0.425
0.525
0.625
0.725
0.825
0.925
DSSQ
Yfi
0.025
0.125
0.225
0.325
0.425
0.525
0.625
0.725
0.825
0.925
14.5
0.04331
0.01787
0.01888
0.01905
0.01920
0.01923
0.01901
0.02294
0.02300
0.02306
14
0.04694
0.01796
0.01897
0.01919
0.01938
0.01947
0.01934
0.02300
0.02306
0.02312
13.5
0.04186
0.01734
0,01909
0.01934
0.01956
0.01969
0.01968
0.01901
0.02312
0.02317
13
0.03799
0.01869
0.01776
0.0195 1
0.01974
0.01990
0.01996
0.01982
0.02317
0.02322
14.5
0.00561
0.001 14
0.00244
0.00270
0.00286
0.00288
0.00266
0.00444
0.00445
0.00447
14
0.00565
0,00039
0.00260
0.00284
0.00300
0.00306
0.00297
0.00445
0.00447
0.00449
13.5
0.00490
o.ooo1o
0.00275
0.00297
0.00312
0.00321
0.00320
0.00298
0.00449
0.00450
14.5
0.00555
0.00117
0.00247
0.00273
0.00289
0.00291
0.00270
0.00447
0.00449
0.00451
14
0.00561
0.00042
0.00263
0.00287
0.00303
0.00309
0.00300
0.00449
0.00451
0.00452
13.5
0.00487
0.00014
0.00278
0.00300
0.00315
0.00324
0.00323
0.00301
0.00452
0.00454
12.5
0.04235
0.02026
0.01969
0.01993
0.02011
0.02021
0.02018
0.01986
0.02327
12
0.03700
0.01978
0.01798
0.01987
0.02011
0.02030
0.02043
0.02047
0.02035
0.01973
11.5
0.03772
0.01957
0.01807
0.01744
0.02030
0.02049
0.02063
0.02071
0.02068
0.02045
11
0.03901
0.01905
0.01836
0.01803
0.02048
0.02068
0.02082
0.02092
0.02095
0.02086
10.5
0.05058
0.01944
0.01824
0.01830
0.01776
0.02085
0.02100
0.02111
0.02117
0.02115
10
0.04985
0.01863
0.01891
0.01823
0.01823
0.02102
0.02 117
0.02129
0.02136
0.02138
13
0.00428
o.ooo3o
0.00184
0.00309
0.00323
0.00333
0.00336
0.00328
0.00450
0.00451
12.5
0.00427
0.00044
0.00130
0.00320
0.00334
0.00343
0.00348
0.00347
0.00330
0.00453
12
0.00441
o.ooo1o
0.00081
0.00331
0.00343
0.00352
0.00358
0.00360
0.00354
0.00323
11.5
0.00509
0,00029
0.00106
0.00169
0.00352
0.00361
0.00367
0.00370
0.00369
0.00359
11
0.00571
0.00039
0.00136
0.00153
0.00360
0.00369
0.00375
0.00378
0.00379
0.00376
10.5
0.00652
0.00158
0.00191
0.00176
0.00376
0.00382
0.00386
0.00388
0.00387
10
0.00629
0.00044
0.00181
0.00214
0.00216
0.00382
0.00388
0.00392
0.00395
0.00395
13
0.00426
0.00027
0.00 187
0.00312
0.00327
0.00336
0.00339
0.00331
0.00454
0.00455
12.5
0.00426
0.00041
0.00133
0.00323
0.00337
0.00346
0.00351
0.00350
0.00333
0.00456
12
0.00438
11.5
0.00503
0 00031
0.00 109
0.00172
0.00356
0.00364
0.00370
0.00373
0.00372
0.00363
11
0.00564
0.00041
0.00138
0.00156
0.00364
0.00372
0.00378
0.00382
0.00383
0.00379
10.5
0.00647
0.00013
0.00160
0.00194
0.00179
0.00379
0.00385
0.00389
0.00391
0.00390
10
0.00624
0.00040
0.00183
0.00217
0.00220
0.00386
0.00391
0.00395
0.00398
0.00399
3..
:.. . . . .,
0.00084
0.00334
0.00347
0.00356
0.00362
0.00363
0.00358
0.00326
.
Source: Authors' calculations.
!..!,..
4:.i::.: ........... .,,;;!,!;
clevelandfed.org/research/workpaper/index.cfm
Table 3 : EFFECT OF INPUT VALLIE UNCERTAINTY ON LOSS FUNC'IIONS
Optimal results from the model
Simulation results
100 draws
Average
StDev
High
Low
Optimal results from the model
Simulation results
100 draws
Average
StDev
High
Low
Optimal results from the model
Simulation results
100 draws
Average
StDev
High
Low
SSE
0.370024
0.382051
0.003645
0.389342
0.371578
BAR
DSSQ
0.004395
0.006860
0.000149
0.007248
0.006441
0.005366
0.0078%
0.000151
0.008215
0.007396
0.384171
0.381940
0.006583
0.395234
0.364810
0.000013
0.004497
0.000261
0.005066
0.003571
0.000962
0.005443
0.000265
0.006026
0.004498
0.398710
0.390155
0.007941
0.41 1602
0.367236
0.000810
0.004549
0.000302
0.005087
0.003725
0.000110
0.005464
0.000312
0.006011
0.004620
Source: Authors' calculations.
@FedFRASER