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Incentive efficiency of compensation based on accounting and market performance
Journal of Accounting and Economics 16 (1993) 25-53. North-Holland
Incentive efficiency of compensation based on accounting and market performance* Oliver Kim and Yoon Suh ~niUersi1.Vof California aI Los Angeles, Los Angeles. CA 90024-1481. USA
Received May 1991, final version received February 1992
This paper analyzes how earnings and price are used in executive compensation contracts. Riskaverse shareholders collectively design a contract and individually trade in the stock market. In the optimal linear contract the use of earnings and price depends critically on incentive efficiency, i.e., how precisely the measures convey the true outcome. The relative weight of price to earnings exaggerates the true relative importance of price because price impounds traders’ overall information while its informational value lies in the incremental information it provides. The use of price allows shareholders to share trading risks with managers.
1. Introduction
Recent empirical studies find that executive compensation is closely related to accounting and market measures of firm performance.’ We analyze how riskaverse shareholders’ individual investment decisions are interrelated with the design of a top manager’s compensation contract. Compensation contracts are assumed to be based on earnings and price. Price is endogenously determined in a securities market and aggregates (with noise) the information held by traders. We derive the optimal linear contract in equilibrium and investigate its properties.2 Among theoretical papers that examine the use of price in compensation contracts, our study is closely related to Bushman and Indjejikian
Correspondence to: Oliver Kim, Anderson Graduate School of Management, University of California at Los Angeles, Los Angeles, CA 90024-1481, USA. l We gratefully acknowledge the comments and suggestions of David Hirshleifer, Bruce Miller, and the participants of a workshop at the University of Washington and the 1991 KPMG Peat Marwick/Olin Conference. We also thank Ray Ball (the editor) and Robert Verrecchia (the referee) for many helpful suggestions. ‘See, for example, Jensen and Murphy (1990). ‘Holmstrom and Milgrom (1987) demonstrate that linear contracts are viable.
0165-4101/93/505.00 Q 1993-Elsevier
Science Publishers B.V. All rights reserved
26
0. Kim and Y. Sub. Compensation based on accounring and market performance
(1993).3 They analyze how the use of earnings in addition to price in contracts alleviates the inefficiency of price in allocating the manager’s efforts between two activities. Our model is similar to their special case with only one activity. The major difference is that in their model shareholders are assumed risk-neutral as the principals (but not as investors), while in our model shareholders are always risk-averse. Our economic setting is one in which the principals are risk-averse shareholders and the agent is the top manager. Following Verrecchia (1986), Dye (1988), and Bushman and Indjejikian (1993), we assume that the true performance (economic earnings) of a firm cannot be used as a contracting variable. Both earnings and price are noisy measures of economic earnings: accounting earnings is subject to measurement error and share price is subject to various demand and supply noise. When shareholders design a compensation contract (through a board of directors), they must anticipate how share price is determined in the market and how the use of price in the contract affects their investment decisions. In order to analyze these explicitly, we use a noisy rational expectations model in the fashion of Diamond and Verrecchia (1981) and Hellwig (1980). In this way we link the noisy rational expectations literature, which captures the behavior of price as a (less than perfect) aggregator of public and private information, with the principal-agent paradigm, which captures the second-best contract as a (less than perfect) incentive device for managers. We analyze how price and its informativeness are determined in equilibrium and affect compensation. In particular, we show that the use of price in a compensation contract enables shareholders to share trading risks with managers. We identify a problem in measuring how managers’ compensation is related to earnings and price. This measurement problem arises from the fact that price impounds the earnings report as well as other common and private information. The informational value of price in addition to earnings for both portfolio and contracting decisions depends on the incremental information conveyed by price. We extract the price signal that represents this incremental information by filtering other information (earnings and the prior information) from (the raw) price.4
sAmong other related theoretical papers, Diamond and Verrecchia (1982) consider the use of security prices in incentive contracts in relation to the availability of other information and to systematic and nonsystematic risks associated with the firm’s output. Paul (1991) shows that price aggregates information about the manager’s different activities in such a way that the manager is not motivated to allocate his efforts efficiently among dimerent activities. This is because price reflects the value of the firm while an optimal aggregator of information would be the value-added of the manager. 4Likewise, the informational value of earnings in addition to price is due to the incremental information it conveys. In our model earnings and the filtered price each represents the incremental information.
0. Kim and Y. Sub. Compensation based on accounting and market performance
27
The importance of earnings and the filtered price as contracting variables is represented by their respective weights (coefficients) in the optimal linear contract. We first examine the overall efficiency of the two measures by analyzing their combined weight (the sum of the weights). We show that the combined weight is determined so as to achieve the optimal trade-off between risk-sharing and incentives. The weight of earnings relative to the filtered price is shown to depend mostly on their respective precisions.5 However, shareholders place extra weight on price because the use of price in contracts allows shareholders to share some of their trading risks with the manager. Although the importance of earnings and price can be better measured in terms of the weights on earnings and the filtered price, the latter is often unobservable to the researcher. This is because information that has to be filtered out is unobservable. Thus, most empirical studies estimate weights on performance measures based on raw prices (returns). In order to link our results to empirical studies, we convert the contract based on the filtered price to one based on the raw price. Both the absolute and relative weights change as a result of this conversion. It is shown that the weight on price is greater than the weight on filtered price, whereas the weight on earnings is less when based on earnings and the raw price. This is because price has a component that reflects earnings. As a result, the relative weight of price to earnings is greater than the relative weight of the filtered price to earnings. Our results are relevant to empirical studies that focus on the absolute [e.g., Jensen and Murphy (1990)] and the relative weights [e.g., Lambert and Larcker (1987) and Sloan (1993)] on accounting and market measures of firms’ performance in executive compensation. Empirical estimates of the weights on these measures should be interpreted with care.6 The plan of this paper is as follows. The model is explained in section 2, and a market equilibrium for a given contract is obtained in section 3. The expected utility of shareholders and the manager are derived in section 4. A benchmark case in which effort is observed is considered in section 5. The optimal contract under moral hazard is derived and its characteristics are examined in section 6. The pay-performance-measure sensitivities in the optimal contract are discussed in relation to some empirical findings in section 7, and some concluding remarks are given in section 8. 2. The model We consider a firm whose shares are traded in a competitive securities market. There are a countably infinite number of identical shareholders, and for every sThis is similar to the results of Banker and Datar (1989) on a linear aggregation compensation contracts and to those of Bushman and Indjejikian (1993).
of signals
6For example, our results are particularly relevant to those in which compensation is regressed earnings and price (returns) and their slope coefficients are used as measures of the importance earnings and price.
J.A.E.-
B
in on of
28
0. Kim and Y. Sub. Compensarion based on accounring and market performance
N < x shareholders there is one manager who works for the shareholders. We assume that all managers are identical and, without loss of generality, concentrate on one representative subset which consists of N shareholders (the principal) and one manager (the agent).’ The securities market structure of this model is a standard noisy rational expectations framework with three time periods 0, 1, and 2. There is one risky asset, namely the firm’s shares, and riskless bonds. One riskless bond pays off one unit of consumption good in period 2. The final payoff to one share of the firm (before compensation) is assumed to be the sum of the level of effort exerted by the manager, denoted by e, and a random environmental factor, denoted by r18 It is assumed that ii is normally distributed with mean ti and precision (inverse of variance) h. The per capita supply of the risky asset is 2 > 0 and is known to everyone. Shareholders are risk-averse and their preferences can be represented by a negative exponential utility function with risk tolerance r > 0, i.e., G(W) = -exp( - W/r), where W is the representative shareholder’s wealth. In period 0 each shareholder is endowed with zero riskless bonds and .Ushares of the firm, while the manager is endowed with zero riskless bonds.’ The manager and the shareholders then enter a compensation contract for the work to be provided by the manager to the principal. Given the contract, the manager decides how much effort to provide. The manager’s utility function is also a negative exponential with risk tolerance ra > 0, i.e., H( W,) = - exp( - [ W, - C(e)],+,), where W, is the manager’s compensation. The function C(e) measures the disutility of providing effort in terms of dollars.” It is assumed that C(e) = (c/2)e*.” Four events occur in period 1. First, the firm announces earnings before compensation represented by a public signal j = e + ii + 4, where fi is normally
‘The economic setting we try to model here is one in which there are N shareholders and a single top manager (e.g., a CEO). However, analyses within a noisy rational expectations framework become clean only with an infinite number of investors. On the other hand, if there is an infinite number of shareholders and if the manager’s compensation is finite, each shareholder’s share of compensation is negligible. As a result, the moral hazard problem faced by individual shareholders becomes trivial. Our model avoids these mathematical problems and is a reasonable approximation of a model with a single manager and N < cc shareholders. *This additivity assumption independent of the environment.
implies
that
the increment
of the final payotT due
to effort
is
‘Under the assumption of constant risk aversion of both the principal and the agent, zero riskless endowments are assumed for convenience and without loss of generality. Shareholders’ risky endowments are assumed identical because such an allocation would be the equilibrium allocation if they were allowed to trade in period 0. “‘Measuring disutility of effort in units of wealth is consistent Holmstrom and Milgrom (1987) for their linearity result. “This assumption cost function would
with the restriction
allows for a closed form solution to our problem. not qualitatively change our results.
Assuming
imposed
by
a more general
0. Kim and Y. Sub, Compensation based on accounting and marker performance
29
distributed with mean zero and precision m.” The precision m can be interpreted as either the quality or the amount of the earnings information. Second, shareholder i observes a private signal z’i= e + l;i+ Ei, where Ziis independently and normally distributed with mean zero and precision s. In other words, investors have diverse private information of the same quality. Third, shareholders trade in a competitive market based on new information, J and their 4’s. There is also a separate group of liquidity traders whose aggregate demand (per share) for the risky asset is a random variable, denoted by d’,which is normally distributed with mean zero and precision t. Last, the manager is compensated according to the contract. In this model we consider contracts which are linear (affine) in two publicly observed measures of the performance of the trm, namely, earnings (before compensation) jj and the share price, denoted by P. In period 2 the firm liquidates and shareholders consume their final wealth from their portfolios. Shareholders in this model make two types of decisions. They collectively design a compensation contract and individually select the number of shares to buy or sell. In a rational expectations equilibrium each choice is made correctly taking into account all other decisions made in the economy. For convenience, we separate the two decisions. In the next section we analyze shareholders’ investment decisions, and derive the equilibrium share price for a exogenously given compensation contract. The optimal contract is analyzed in section 6.
3. Market
equilibrium
In this section a market equilibrium is derived for a given compensation contract. Let a linear compensation contract be expressed as w, = Ni(cr* + p*jj + r*P).
(1)
Then, the final payoff per share, denoted by v’,is o’= e + ii -(x*
+ /I*p + y*P).
(2)
In a rational expectations equilibrium traders conjecture how the market price of a share reflects the information held by market participants and the conjecture is self-fulfilling. Let such a conjecture be expressed as
= bO + b,jj + b2(e + u’) + b,d-
bri,
“Using earnings after compensation instead would not mathematically change our model.
(3)
0. Kim and Y. Sub. Compensarion based on accounring and market performance
30
where the second equation follows from the law of large numbers and the fact that the 2i’s are independent. Define the normalized price signal 4 as +(P-ho-b,j+b,r) 2
= e + ii + BJ,
(4)
where b2 # 0 and B z b3/b2. I3 The information set {J, ii, p} is equivalent to (~,ii, ci}. The signal 4 represents the pure addition of knowledge due to observing the price beyond what is available from other pieces of information.r4 Using (1) and (4) we can rewrite the manager’s compensation per share as
r* + /?*j + y*p = r* + p*j + i’*(bo
+ b,j + b2ij - b4R)
= (x* + y*bo - -,-*bd.u)+ (fl* + y*b,)j =
cl +
pj
+
+ y*bzij
74.
(5)
(5) shows how a linear function of j and P’ can be converted to a linear function of j and (I, and vice versa. For convenience we use the normalized price signal 4 for an initial calculation of the optimal linear contract. The final payoff is now rewritten as
= --a+(1
+I--y)(e+i+j?+-yBd:
(6)
Given normalized signals, j, z’i, and q, it is straightforward E[filp,z’i, 41 =
$
h(e + ii) +
m_f+ Sz’i+
to calculate
sij -
(x + pL: + ;,g,
where
K is the sum of the precisions of the prior and the three normalized signals, and can be interpreted as the amount of total information about the final payoff
“It
can be shown
that there is no equilibrium
if bz = 0. See section
7 for a discussion
of b2.
‘The signal 4 represents the pure addition of knowledge in the sense that the increase in the var-‘(e+ lilj.T,),istheprecision posterior precision due to observing price, var -‘(e + liIJ,5i,4)of the error of 4.
0. Kim and Y. Suh. Compensarion based on accounring and marker performance
31
possessed by each shareholder in period 1. The conditional expectation of the final payoff is simply the weighted average of the four sources of information. The shareholder’s expected utility conditional on signals j, Zi, and P’ is EIGIJ,Zi,F]
= E
= -exp
i
- “5 - [E(Clj. Zi, -
by using the moment-generating function of a normal random variable. The utility-maximizing individual (gross) demand for shares, di, can be found as
= rK
Aggregating over i and applying the zero excess demand condition (in per capita terms), we get X = rK
h(e + ii) + mj + s(e + ii) + +ij
which can be rewritten as
P-=; h(e+fi)+my+s(e+fi)++j
I
i
-
(ct + pj + yij) +
$ - -$.
(8)
Since the conjecture (3) must be self-fulfilling in equilibrium, (3) and (8) are equivalent. Therefore,
32
0. Kim and y. 5th. Compensation bused on accounting and market per/ormance
and B is solved as B = Ilrs. Also b
=
0
b2 =
0 +4 7
-
%
s + r’s”t K -‘r’,
(9)
where K is rewritten as K = h + m + s + r*s*t. Eqs. (7) and (8) in conjunction with B = l/rs give a complete characterization of the unique market equilibrium given any u, b, and 7 such that y # (s + r’s*t)/K.
4. The expected utility In this section we derive the expressions for the expected utility of the representative shareholders and the manager and provide an interpretation of the terms obtained. Since a contract has to be offered in period 0 before observing any information signals, 2, /?, and 7 (or r*, /?*, and y*) have to be determined based on expectations at the time of contracting. The ex ante expected utility of a shareholder and that of the manager are first derived in the following lemma. All proofs not shown in the text are provided in an appendix. Lemma 1. For any given SL,/?, and y the ex ante utility of the shareholder, denoted by EG, and the manager, denoted by EH, are s+r
+g
.f* I
-Zt-
1
K
(1 -/I--r)*h-‘+/?*m-‘+
-f{-1+(1-_B-;‘)(e+Q)]
YZW
+ 4
r2s2t(K + s + r-*t-l)
II
0. Kim and Y. Suh. Compensation
based on accounting
and marker
performance
33
and EH = - exp
- F
(@!+ (/I + r)(e + ri,> + $
(B+.,)2h-‘+BW+-&
* .
The exponent in the manager’s expected utility in Lemma 1 is simply a combination of the mean and the variance of the uncertain compensation and the private cost (disutility) of exerting effort. The expression for the shareholder’s expected utility is similar but contains some additional terms. For a better understanding of this expression, we rewrite it as l+
EC=
( x
exp
x2
K )( Zr2
rVt(K
- Y2r-2f-1
+ s + r-2t-1)
>I
EG
’
(10)
where
EGE
-
exp
-f{-%+(I
-/I-y)(e+ii)f
- SE(C) + $ -
var(L’)
[
=
-exp [
1
is the expected utility if the shareholder did not trade. Note that EG is not entirely independent of the market because it depends on the share price used in the contract. The two multiplicative terms in (lo), (1 + (s + r-2t-1)/K)-1’2 and exp[ - y2f2/2r6s2t2(K + s + r-‘t-l)], capture the gains from trade because both are less than one and EG is negative. In order to understand the two multiplicative terms, it is convenient to consider the added return and risk from trading separately. To see the returnrisk trade-off when trading, consider two cases: one in which the liquidity demand is very high or low and, as a result, the price tends to be very high or low; the other in which the liquidity demand and thus the price are moderate. In
34
0. Kim and Y. Sub. Compensation based on accounting and marker performance
the first case, the return from trading tends to be large because the extreme price is due to demand noise that is irrelevant to the final payoff. As a consequence, the shareholder can profit by using her private information, either by buying at a low price or by selling at a high price. On the other hand, she faces a higher risk when disposing of her endowed shares. In the second case, her return from informed trading is small but the risk in disposing of her endowments is also small. When the contract does not depend on share price (7 = 0). the combination of added return and risk result in a net gain represented by the first multiplicative term in ( 10).15 When share price is used in the contract, the shareholder can shift some trading risk to the manager through the contract. These additional opportunities to share risks between the shareholders and the manager give rise to the second multiplicative term in (10). As the contract depends more heavily on share price (larger y2), the risk-sharing opportunities increase and this term decreases. Also, as the demand noise increases (t decreases) for a given ;‘, it also decreases. (Note that a decrease in the term implies that the gain it represents increases.) Therefore, the role of sharing trading risk with the manager through compensation contracts becomes more important when the demand noise is greater. This beneficial role of price increases the importance of price as an incentive device relative to earnings as analyzed in section 6. Taking a monotonic transformation of the shareholder’s utility we get G E - rln(-
EC)
(
s+r
=iln
l+
-;
i
-zt-I
K
+ x{ - X + (1 - p - r)(e + Li,) )
(1 -B-
‘i)2h-’ + D2m-l + r2s2,,;2ys;!2r_I)
I
. (11)
Similarly, the manager’s utility can be expressed as
EH)
H E - r,ln(= Nl{r
+ (p + y)(e + ti)} -
N2,f2 -y&--
e
i
$
(/?+~)2h-‘+p?l-‘+~ . r2s2t
IsThis term is obtained in many other including Admati and Pfleiderer (1987).
I
studies
that use noisy
rational
(12) expectations
models,
0. Kim and Y. Sub, Compensarion bused on accounting and marker performance
35
By (11) and (12) we represent the expected utility of the shareholder and the manager in terms of certainty equivalents. For the manager it simply becomes a mean-variance representation due to the normality assumption. We allow the manager to be risk-neutral, which can be considered as a subcase of (12) by allowing r. to reach infinity, but the shareholder’s risk tolerance is restricted to be less than infinity in order to guarantee the existence of a market equilibrium.
5. The observable effort case
We first analyze a benchmark case in which the manager’s effort choice is observed by the principal and can be contracted upon. The principal’s problem is then to choose %,/I, and 7 to achieve the first-best optimal risk-sharing among N shareholders and the manager. Formally, the principal’s problem is to maximize (11) with respect to x, /?, and ;’ subject to the minimum utility condition
H
2 ii.
(13)
fi measures the minimum utility in terms of dollars and can be interpreted as the manager’s opportunity wage. The optimal linear contract in this situation is referred to here as the first-best contract. Since any desired effort level can be enforced by a forcing contract, the Lagrangean form of this problem is to
(
max Liln .,P,?.l.P
l+
(1
-g
s+r
-zt-1
K
+.u{-a+(1
-
-/I-y)(e+U)}
>
_/3_y)2h-l
+jZm-l
i
+
‘i2(K + s) r2s2t(K + s + r-2t-‘)
+ 1
[ -y-
I
Nf{r + (j? + ;I)(e + ii))
N2Z2
*
I
(jl+y)‘h-’
+fi2m-’
The solution is shown in the following proposition.
+&
}+q.
36
0. Kim and Y. Suh. Compensarion
Proposition
1.
The first-best
hasett on uccounring mti murkrr per~bnancr
contract
is characteri:ed
br
m + (1 + 6)‘r’s’t :rn + (I + G)r’s’tj,’
’
I?1
/3/ =
Qf,
171+ (I + 6)r’s’t (1 + G)r’s’t
*, ” -
Qf,
m + (1 + d)r2s2t
1 )-f =N’ where *++J(
m+(l+@r_2s2t ) h + m + (1 + b)r2s2r
and
6
r. Nr + r’t(K
Each shareholder G,=G
+ s)(r, + Nr)’
achieves utility (in dollars) of
0
+E+?Q 2c
2rh
”
where
Go+
(
l+
s+re2t-’
-
K
-2
-_x+xrT-h
>
is the utility when the ejiort leoel is zero and the lump sum fi is paid to the manager.
0. Kim und Y. .%/I. Compensation bused on uccounting and market performance
37
The result L, = 1,‘s is obtained because N shareholders contribute to one manager. The choice of the effort level N.$‘c maximizes the total expected wealth of the N shareholders. where the marginal cost of effort, ce, equals the marginal product of effort, NY. The optimal risk-sharing is attained when /?, + ;‘/ = L?,, which is the product of two ratios shown in the proposition. The first ratio is the relative magnitude of the manager’s risk tolerance to the sum of those of all parties (one manager and N shareholders). Besides the risk tolerances of the contracting parties, the optimal risk-sharing also depends on the precision of the signals used in the contract. which is captured by the second ratio. This is because the risk concerning t? is shared via two noisy signals, y and 4. For example, if earnings is a perfect signal (m = CL)or if the prior is extremely diffuse (h = 0), then the second ratio becomes one and the optimal risk-sharing is such that risk is shared in proportion to risk tolerance. Note that letting m = x8 and N = 1 in our setting will yield fl, = re/(ro + r) and ;‘, = 0, which corresponds to the first-best solution in standard agency models with one principal and perfect outcome observability. As the signals become noisier, the additional risk from the noise in the signals, namely, the variances of fi and d’,increases and the optimal contract depends less on those signals. Shareholders in our model are subject to two types of risk. The first type is project risk, due to the inherently random nature of the outcome of firm’s activities, represented by uncertain u’. This is the risk the shareholders desire to share with the manager. However, the only available sharing devices are noisy signals of ii. The use of these devices exposes the shareholders to the second type of risk, noise risk, due to measurement error and,‘or noise in the signals. The optimal contract is devised so as to give the best trade-off between project risk and noise risk. This is reflected in the choice of $2,. Also, the relative weights of the two noisy signals naturally depend on their respective precisions. A more detailed discussion of the relative weights is provided in the next section.
6. The optimal contract with unobservable effort We now turn to our main problem in which the effort choice is unobservable and the manager has to be induced to exert effort by means of an incentive contract. The manager’s problem of choosing an effort level given I, /I, and 7 is to maximize (12) with respect to e. The first-order condition is N,F(/3 + 7) - ce = 0.
(14)
It is easy to check that (14) describes a unique optimal e for any given /? and 7.
The principal’s problem in the Lagrangean form is to l+s+rK
-;
-1
-2
(
max L-iln ../?.7.i.p.u
’
+ .U( - 3 + (1 - /I - 7)(e + 17)) > +/pm-l
(1 _p_#h-l
i
+
y’(K + s) r’s’t(K
+ p[N.f(p
+ s + r-2t-‘)
-t y) - ce].
The difference of this problem from the benchmark problem is that p and y simultaneously play both the role of inducing effort and the role of risksharing. The contract that solves this problem is referred to here as the secondbest contract, or simply the (optimal) contract, and is characterized by the following proposition. Proposition 2.
The second-best contract is characterized by
a+17R_32+32’ 0
D= Y=
( (
m
m + (1 + S)r2s2t 1
(1 + 6)r2s2t
m+(l
+b)r2s2t >
1 A=----, iv P=
0f
(1 - Q),
Q, R ’
(
‘+ h
m + (1 + 8)2r2s2t {m + (1 + B)r2s2t)’ > ’
0. Kim and Y. Sub. Compensation based on accounting and murket performance
39
where r,(Nr + ch-‘)
RG
r,Nr+c(r,+
Nr)
i+
1 m + (1 + 8)rzszt
and 6 is as defined in Proposition I. Each shareholder achieves utility (in dollars) of 2
G z G,, + z
=
x2
R’ + 2r
II
R2 (r, + Nr) r,f’(.Vr
Go + 2cr
r,Nr + c(r, + A;r)
‘+ h
1 m + (1 + 6)r2s2t
+ ch-‘)2
1 m + (1 + B)r2s2t
Note that the agent’s optimal action choice does not depend on the minimum certainty equivalent, A, in both the first-best and the second-best cases. Under the assumption of constant risk aversion both the manager’s action choice and risk-sharing are independent of the constant term, a, of a given contract. Accordingly, /I, 7, and the action choice, e, are determined independently of fi and a change in fi only affects B.‘~ We first verify that a welfare loss exists due to the unobservability of the manager’s effort choice. Corollary 1. There is a positiue welfare loss due to the unobsercability of the manager’s eflort choice. That is, G, - G > 0.
Corollary 1 shows that a moral hazard problem exists in our main problem. We now analyze the weights, a and 7, on the two performance measures, j and 4, in the second-best contract. For a better understanding of /? and 7, it is useful to consider a situation in which only one signal, say F = ii + G, is used in the contract, where c is normally distributed with mean zero and precision M. In this case the weight on yin the first-best and the second-best contract, denoted by PI and $, respectively, can be calculated as
and p^=
r,(Nr + ch-‘) r,Nr + c(r, + Nr)(h-’
+ M-l)’
16This property of exponential utility function Fellingham, Newman. and Suh (1985).
has be-en used to derive a no-memory
contract
in
40
0.
Kim and Y. Suit. Compensation
bused on accounrtng
and math!
performance
Comparing these to the expressions for R, and R in Propositions 1 and 2, it is clear that the combined weight, R, of the two performance measures, 4: and 4, depends directly on their combined precision, m + (1 + 6)rZs2r. Therefore, it is useful to analyze first the combined weight, R, and then to see how the combined weight is decomposed into individual weights, j? and 7. The most important difference between the second-best contract and the first-best contract in Proposition 1 lies in the combined weights, B + 7 G R and P, + ‘J/ = Q,. The difference is summarized in the following corollary. Corollary 2. The combined weight, Q, of two performance measures in the second-best contract lies between one that achieves the best incentive (one) and one that procides the best risk-sharing (Q,). That is, R, < R < 1.
Corollary 2 shows how the existence of moral hazard affects the combined weight. The best incentive is achieved when R = 1 because from (14) the manager would then choose e such that N% = ce, which is the condition that maximizes the principal’s utility when the principal provides effort herself. The differences in the effort level, e, and the constant term in the contract, a, can also be understood through this difference in R. The marginal cost of inducing effort, ,u, is positive because R < 1. Corollary 3. The combined weight, Q, of performance measures in the contract is decreasing in the manager’s work aversion, and is increasing in the combined precision of the performance measures. That is, iW/& < 0 and S2/i3{tn + (1 + 6)r2sZt} > 0.
The magnitude of R depends, conceptually, on two aspects of incentives, namely, incentive effectiveness and incentive efficiency. Incentive effectiveness concerns the end result, i.e., the increase in the shareholder’s utility, given incentive devices of fixed quality, i.e., given fixed m + (1 + B)r2s2t. Incentive effectiveness depends on how sensitively the manager responds to given incentives, how productive he is for given effort exerted, and other additional related benefits and costs, if any. One variable that captures effectiveness in our model is the degree of the manager’s work aversion, c. This is because c is proportional to the manager’s private cost of exerting effort that has to be compensated in order to keep the manager. Therefore, a smaller c corresponds to more effective incentives. R is decreasing in c because as it becomes more costly to induce effort, it is optimal to give less incentives, holding the manager’s productivity constant. As c approaches infinity, inducing any positive level of effort becomes too costly. As a result, R approaches R, in which the role of using the measures in the contract is pure risk-sharing. At the other extreme, when c approaches zero, a large amount of effort can be induced costlessly. As a result, Q approaches one and the maximum level of effort is
0. Kim and Y. Sub. Compensation bused on accounting and market performance
41
induced. Incentive ineffectiveness can arise for many other reasons that are not modeled here. A manager may not be responsive to incentives because he is too lazy, or because social or other monitoring mechanisms work well and he is already exerting enough effort. In this case any additional effort is too costly to him as well as to the agency. Incentive efficiency concerns how precisely the incentive devices (jj and 4’)convey the true outcome (e + ii). The second part of Corollary 3 says that the combined weight is increasing in incentive efficiency measured by the overall quality of the signals. Given our understanding of the combined weight, the following corollary shows how the weight on each of the two measures is determined in the contract. The same result applies to both the first-best and the second-best contracts and directly follows from Propositions 1 and 2. Corollary 4. The relatice importance of earnings, Jt, and the normalized price, q, in the contract is largely determined in proportion to their precisions. That is, pkj = m/(1 + 8)r2s2t.
The use of noisy signals in the compensation contract subjects both the principal and the agent to noise risk. The optima1 combination of the two noisy signals thus naturally depends on their precisions, m and r2s2t. In addition, the use of price in the contract has a beneficial effect of sharing trading risks between shareholders and the manager as discussed in section 4. Therefore, the relative weight on the market measure, 4, is its precision. r2s2t, combined with this effect. This is expressed in a precision-equivalent term as br2s2t. Given fixed precision, this makes q more desirable, which is reflected by the fact that S is positive. Most of the comparative statics with respect to Proposition 2 are intuitive. We single out one that is related to the literature on information disclosure. Corollary 5. An increase in the precision of earnings reduces shareholders’ welfare through its adcerse risk-sharing efect. but alleviates the moral haxrd problem. That is, aGO jam< 0 and a(G - G,), Sm > 0.
Recall that GO is the shareholder’s utility when the manager’s effort level is zero and the lump sum fl is paid to the manager. The first part of Corollary 5 implies that, in the absence of a moral hazard problem, a more precise disclosure makes shareholders worse off. This effect of disclosure, called the adverse risk-sharing effect, has been extensively discussed in the literature.” The importance of Corollary 5 is that it identifies the second effect. G - Go is the portion of the shareholder’s utility that corresponds to moral hazard considerations. The second part of Corollary 5 says that a more precise disclosure “See, for example, discussions.
Hirshleifer
(1971). Verrecchia
(19SZ), and
Diamond
(1985) for detailed
42
0. Kim and Y. Suh, Compensation based on accounting and marker performance
increases this portion of utility. That is, it alleviates the moral hazard problem. This is because as the accounting measure, 9, conveys a more precise information about the performance of the firm, the variances of the shareholders’ and the manager’s wealth both decrease. As a result, both are made better off.
7. Price and incremental information The informational value of price both for investment and contracting purposes mainly arises from the fact that price carries incremental information over and above what is known from other sources of information. This incremental information is represented by the normalized price signal, 4. However, in most empirical studies [e.g., Jensen and Murphy (1990), Lambert and Larcker (1987), and Sloan (1993)J the estimation of the sensitivity of compensation to market and/or accounting measures is based on the ‘raw’ price. From (8) the relation between price P’ and the normalized price signal 4 can be written as F
=
(----$a)
+(;-p)j+(~+~~2~-y)@(15)
Me+ 4 K
Eq. (15) shows that price impounds both its incremental information, 4, and other information (i.e., the prior and 4: in our model). In order to link our results to the empirical literature, we now convert the optimal contract based on normalized signals, j and 4, to one based on earnings, B, and price, p’. BY O), a = r* + 7*bo - y*b4i&
y = -j*b2.
Solving for CX*, p*. and y* and using (9), we obtain the following result. Proposition 3.
a* =
The optimal linear contract x* + p*j + y*P’ is such that s + r2s2t h(e + U) Q -77 K s -I-r2s2t K
-T r’
3
ap!Aold 6aq1 asn~~aq pw suo!s!3aJd JO SUIW u! Quo passaJdxa aJe 6aql asnesaq ]InsaJ s!ql u! pasn a,e saylenbau! aql ‘,v pue O.tSE qms sJa]aruemd Jaqlo uo puadap L pue ‘61‘*A ‘J ‘s,q%!am aqler
pasvq asyd 01 sbu~uma~o amvllodur! aaflvlar ayl my1 ra]pxus s! (.J/,$) a3!rd MVI ayl uo pasvq lm.~uo~ ayl u! a+d 01 s&ma/o l@a.+t aoflvlar ayl ‘L Lrv110~03 ~~pxus%u!aq c, uo puadap lou op ~1nsaJlxau aql pue s!q~ yx~%!s aD!Jd pazyeruJou aql uo paseq asoql ql!M aDpd MBJ aql UO paseq SaJnS~aUllayJWJlpUE%l!lUnO33~aqlJOSlq%!aMaA~l -Olafaql saJedmo3 ICJr?lloJo3 lxau aq~ *ax.wJodm! ~eluauIaJxI! anJ1 J!aqiu10q luaJa&p aq 01 Llag are (suJnlaJ)ayJd MBJ ayi put! %u!uJea uo pawq sa!pnls uo!ssaJ?iaJ ald!llnuJu! sal~uysa luaggao3
aql leql slsa%%ns 9 AJ~IIoJo~
ayl uo pasvq am Sly&ah4 ayl uayM uvy (C,+ d < *L + .d) dalvaA6 s! lyB!ar\\ pawqLuo3 ayl puv ‘(X < .&) ralval6 s! az+!d uo iy6!aM ayl ‘(d > *d) ssal s! s6uwva uo lyB!a,n ayl ‘d ‘as!rd MVI ayl uo pasvq am samsvaw ammrojIIad uo sly6iaM ayl uayd *y/(izszd + s) 01 anyvlad ~pxus haa s! h lvyt asoddns ‘9 rC1v110.10~ .(swo
u~!yi
JaieaJ% y3nw aq
01 AJt2110J03 %U!MOIIOJ aql u! pawwe lnq ‘alqeAJasqouns! q3y~) UO!~EUJJOJU! IlEJaAO J!aql 01 aD!Jd UIOJJpUI2Sa3JnOS alt?A!Jd UIOJJ UO!ll?UIJOJU! ,SJOlSaAU! JO ax.wlJoduJ!aA!lelaJ aql s! Jaw] aq_L.Ly/(lZSZ1 + s)ol aA!ie[aJ(g)o'()uI?qlSSa~ say -!A!l!suas axxemJoJJad-h?d 1JOdaJ dlp_?cyddl sa!pnls[~j!J!drua)~~erus hJaA s! XWql %h.Iynssoapw.us!uos!Jedwo3 aqL *~CJI?~~OJO~ %~!~0~[0~aqlu!‘I(~aA!l3adsaJ'~ + d aql aJt?duJo3 lsry aM pue ‘x‘9'10aSoq1 ql!M z& + *d pUe ‘.& ‘*d JO S~pIIl$hII .aD!Jduo paseq salwuysa pD!J!dLLIa %J!laJdJalU!6lpaJJO3 JOJ 1uelJodLy 6llegualod ale saxIaJa~!p asaqJ_ *ayJd pazyxuJou aJe a+d
aql pue s%u!uJt?a uo ‘I. pue d ‘slq%!aMaql UJOJJluaJaa!pdlleJaua%
pue S%I!UJt?auo ‘*X pur! *d ‘Slq%!aMaql 1eqlsMoqs
‘z uoy!sodo~d
E UO!l!SOdOJd
u! uaa!B am C.puv ‘d ‘xi rojsuoynjos
c.‘
ayl araym
>I
l,S,J + s = *.i’ L
44
0. Kim and Y. Sub. Compensation
based on accounting and marker performance
on the normaked price (/IF;). The discrepancy is more severe for a firm for which shareholders’ private information constitutes a smaller portion of their overall information. More precisely, S
K+s
p*pj* <---iK
s
PI?
The true relative importance of earnings to price is /Q considering only the incremental information carried by price. For a better understanding of Corollary 7, suppose that the estimate of p*/-i* is 0.1. In this case K /I E
K-IS
10s
For example, if shareholders’ private information comprises only 10% of their total information, i.e., if s/K = 0.1, then /?h is between 1.0 and 1.1. If s/K is 0.33, /?pl is between 0.3 and 0.4. The above examples illustrate that empirical estimates of the weights on accounting and market measures [such as those of Lambert and Larcker (1987) and Sloan (1993)] may seriously exaggerate the relative importance of price to the accounting measure. The problem is more severe for firms about which investors have a small amount of private information relative to their overall information. l9 Finally, the following result summarizes Corollaries 4 and 7. Corollary 8. The relative weight of earnings to price in the contract is roughly the product of the ratio of their precisions and the fraction of private information in investors’ overall information. More precisely, ms p* r*s*tK + s(s + r*s*t) < -y* <
ms r2s2tK’
Both the upper and the lower bounds of /3*k;* are increasing in m and are decreasing in h, s, and t. This implies that an increase in the precision of earnings increases the relative weight of earnings as expected, whereas an increase in the precision of any other information (either private information or other public information) decreases the relative weight of earnings.
“Since precisions are not directly observable, it is difficult to determine for what types of firms investors’ private information constitutes a larger fraction of their overall information. See Im (1989) for an indirect method of grouping firms based on the fraction of private information based on analysts’ forecasts and a simple Bayesian model.
0. Kim and Y. Sub. Compensarion bused on accounting and marker performance
45
8. Conclusion
We have analyzed how accounting and market performance measures are used in compensating top executives. In particular, we have demonstrated that empirical estimates of the extent to which accounting and market performance measures are related to compensation should be interpreted with care. This is because price impounds the earnings report, as well as other common and private information, while the informational value of price in addition to earnings for contracting purposes depends on the incremental information it conveys in addition to earnings.
Appendix
Proof of Lemma 1
For arbitrary individual risky endowments, Xi, the expected utility is
=E[Er[-exp{
where E[v’lj, z’i,P] E pi, var[v’jj, z’i, P] E l/K, and
= -exp
=-
ev
1-
K(fii -
i
-
P)(fii- P) +
K’(,Ci 2K
P)’ I
1
f(fii- F)2
7
which is obtained by using the moment-generating
function.
46
0. Kim and Y. Sub. Compensarion bused on accounring and marker performance To
simplify notation define
-f
brfii-p--=-
rK
1
.
K
The expected utility can be rewritten as -p -
EG=E[-exp{-[d-1+(*
y)(e
+
c)
_
_$ : 1
-
T(b+-&>‘}I.
(6,6) has mean (0,O) and let V, s var(a’), V, 3 var(6), C,,* E cov(& g), and F = V, Vb - C,“,. EC above is now expressed as an integral
Since the exponent of the integrand is a quadratic function, the integration is a straightforward exercise using the normal density. Integrating with respect to ii and then to b’, it becomes
-&&exP[- {2
--+pjp 2r2K
F xf b
1
z + (1 - p - v)(e + W)- $
-}xi r
0. Kim and Y. Suk. Compensation based on accounting and market performance
-&iv
=
-
.f 0 -
r
[- i
2
1
-r+(l
-8-;)(e+E)-2
3 -1
Xi
K+V;‘-
47
C’ob
’
07
V;(K
+ I’;‘)
d6
-
u +
2
xxi
c:b Vb(KVb
(1 - p - y)(e + Is) -
+
Cab
l)+FKV,
1 '
V, =(-K + s + r-2t-‘)/K2 + (1 - /? - ?)‘/I-’ Xi = .~, substituting _p2m-’ - y2/r2s2t, vb = (s + r-‘t-‘)/K2, and C,,b= (YK - s - r2s2t)/r2stK2, the above is simpified as BY
EC=-
(
l+
s+r
-zt-1
-l/2
K
-:{-r+ll-/?-y)(e+E)}
exp >
(1 -j?-;1)2h-‘+/?2m-‘+
[
y2(K + s) r2s2f(K + s + rm2t-‘)
The expression for EH is calculated by using the moment-generating the normal random variable.
* 11
function of
Proof of Proposition 1 I. = l/N is obtained from the first-order condition with respect to s( and is substituted in the first-order conditions with respect to fi, 7, A,and e to generate,
48
0. Kim and Y. Sub. Compensu~ion bused on accounting
and murkrt performuncr
respectively,
)(I - p- ;,)/I_’ - irm-‘)-~{(ll+7)h-L+Bm-l!=o, (16) ;
Y(K + 4
(1 _/j_yp-‘_
r2sZt(K + s +
i
-; (P+y)P+-& I (I
Nx(r
r-2t-1)
I
=o,
(17)
i
(p + y)(e + ii,} -
+
ce2 --_=
N2.f2 2r,
..2
(p + i’)2h-1 + B2n-’ +
(18)
A,
2
9-;=o. By subtracting
(19)
(17) from (16) and multiplying
-4 Y
by r,r we get
K+s
K+s+r-2t-l+Nr TO
r2s2t
or /I
‘i a=-
(r, + Nr)(K
+ s + re2t-‘)
m (r, + Nr)(K + s) + Nrrq2t-’
Nr + r2r(K + s)(r, + Nr)
3
$1+6).
Let R, E /I + 7. Then
mQs
p=
m + (1 + b)r2s2t’ y=
-&
(1 + B)r2s2tQf m+(l
+6)r2s2t’
49
0. Kim and Y. Sub. Compensation based on accounring and marker performance
By substituting these back into (16) we get !+ h
1 m + (1 + b)r2s2t
or r,h-l
1 m + (1 + B)r2s2t ).
Now the solution for z is obtained by substituting the above into (18), and e is solved directly from (19). G, is obtained by using all the solutions in the expression for the shareholder’s utility. Proof of Proposition
2
i. = l/N from the first-order condition with respect to a is used in the first-order conditions with respect to 8, 7, i., p, and e to generate , respectively,
$((I - p - y)h-’
-pm-‘l-F{(p+y)h-‘tpm-l)+Np=O, (20)
4
-P__.Jh-‘-
(1
i
-F
Nf{r
{
i’(K + s) r2s2t(K + s + r-2t-‘)
@+y)h-I+-&
+ (p + y)(e + ~7)) - F
ce2 --_= 2
I
+N,u=O, (P+V)Zh-l+fi2m-l+;.2
II
(21)
r2sZt
H,
N_f(P + ;,) - ce = 0,
(23)
f-+=0.
(24)
Let R E b + 7. Then from (20) and (21) P= y=
mQ m + (1 + d)r2s2t’
(1 + 8)r2s2tQ m + (1 + b)r2s2t’
50
0. Kim and Y. WI,
Compvnsurion based on uccounrin~ rrnd market performonce
as in the proof of Proposition
1. By substituting these back into (20) we get
1 m + (1 + S)rzszt
+ N,u = 0.
(25)
From (23) and (24) we get p =
;(I -
Q).
By substituting this into (25) R is solved, and using (22) x is solved. Now e is solved by using (23), and G is obtained by using all the solutions in the expression for the shareholder’s utility. Proof of Corollary I
The difference in the utility can be easily calculated as G,-G=
1 m + (1 + S)r*s’t > 0. Proof of Corollary 2 Q is clearly less than one by inspection. Also, Q is greater than R, because it can be generated by adding the same positive number r. Nr/c to the numerator and the denominator of r,h-’
1 m + (1 + d)r2s2t
which is less than one.
0. Kim and Y. WI. Compensation based on uccounring ad marker performance
51
Proof of Corol1ar.v 3
The first part is true because r,Nr[(r,
af2 -=?.c
i
+ Nr){m + (1 + b)r’s2tj-’
+ Nrh-‘1
2 1 k+ m + (1 + d)r’?t )I (
r,Nr+c(r,+Nr)
< 0.
The second part is obvious by inspection. Proof of Corollary
5
For the first part, by inspection G,, is decreasing in K and thus in m. For the second part, it is clear from the expression for G that m affects G - GO only through m + (1 + 8)r’s’t. Since G - GO is increasing in m + (1 + d)r2s2t, it suffices to show that m + (1 + b)r2s2t is increasing in m, rOr2t(rs + Nr)r2s2t
a{m + (1 + B)r2s2t} = 1 _
{Nr + r2t(K + s)(r, + Nr)).’
am
rOr2t(r, + Nr)r’s’t ’ ’ - {r’t(K
>l-F
+ s)(r, + Nr)j2
S2
(K + s)-
> 0. Proof of Corollary
6
For the first part, we have from Proposition s +
p*, P
r2s2t
---
K s + r2s2t K
3
ym BK
_+ J
If ‘J is negligible relative to (s + r2s2t)/K, this is approximately s + r2s2t -
(1 + B)r2s2t
s + r2s2t
=
s - br2s2t s + r2s2t ’
equal to
52
0. Kim und Y. Suh. Compensation
bused on accounrmg
By the definition of 6 in Proposition s - 6r2s2t = s -
1,
r2s2tr Nr + r’t(K
+ s:(r. + Nr)
S2
>s--
=-
and murket per/ormunce
K+s SK
K+s
The claim follows from the above and the fact that 6 is positve. The second part follows from Proposition 3 if y is negligible relative to (s + r2s2t)/K. For the third part, we have from Proposition 3 ps +;‘s’t /j* + ‘J* =
%
+ yh + s;
r2s2t
s + r2s2t _. i’ K (p + ~){m(s + r2s2t) + (1 + 6)rZs2t(h + s + r2s2t)] (s + r2s2t){m + (1 + 6)r2s2t).
= (P + 7)
1 + (s +
(1 + b)hr2s2t r2szt){m + (1 + 6)r2s2t)
1’
The claim is true from the facts that (1 + S)r2s2t/{m + (1 + 8)r2s2t} is increasing in 6 and that 0 -c 6r2s2t c
Proof of Corollary
From Proposition
s= K +s’
7
3, -1
= >
m(s - Sr’s’t)
K(l + 6)r2s2t’
Our claim follows from this and the fact that 0 c 6r2s2t < s2/(K + s).
0. Kim and Y. Sub. Compensation based on accounring and market performance
53
Proof of Corollary 8
From the proof of Corollary 7,
P* -_= i’*
m(s - 8r2s2t)
K(l + 6)r2szt’
which is decreasing in 6. P*lr* is less than m,IrZs2t K because S is positive. Also, since 8r2s2t < s2/(K + s), we have
=
r2s2rK + s(s + r’s’t)’
References Admati, A.R. and P. Pfleiderer, 1987, Viable allocations of information in financial markets, Journal of Economic Theory 43, 76-l 15. Banker, R.D. and SM. Datar, 1989, Sensitivity, precision, and linear aggregation of signals for performance evaluation, Journal of Accounting Research 27, 21-39. Bushman, R.M. and R.J. Indjejikian, 1993, Accounting income, stock price, and managerial compensation, Journal of Accounting and Economics 16. this issue. Diamond, D.W. and R.E. Verrecchia, 1981, Information aggregation in a noisy rational expectations economy, Journal of Financial Economics 9. 221-235. Diamond, D.W. and R.E. Verrecchia, 1982, Optimal managerial contracts and security prices, Journal of Finance 41, 275-287. Dye, R.. 1988, Earnings management in an overlapping generations model, Journal of Accounting Research 26, 195-235. Fellingham, J., P. Newman, and Y. Sub, 1985, Contracts without memory in multiperiod agency models, Journal of Economic Theory 41, 340-355. Hellwig, M., 1980, On the aggregation of information in competitive markets, Journal of Economic Theory 22, 477-498. Hirshleifer, J., 1971, The private and social value of information and the reward to inventive activity, American Economic Review 61, 561-574. Holmstrom, B., 1979, Moral hazard and observability, Bell Journal of Economics 10. 74-91. Holmstrom, B. and P. Milgrom, 1987. Aggregation and linearity in the provision of intertemporal incentives, Econometrica 55, 303-328. Im, C., 1988, Differential predisclosure information and the magnitude of stock price responses to earnings announcements, Ph.D. dissertation (Univ-ersity of Pennsylvania, Philadelphia, PA). Jensen, M.C. and K.J. Murphy, 1990, Performance pay and top-management incentives. Journal of Political Economy 98, 225-264. Lambert, R.A. and D.F. Larcker, 1987, An analysis of the use of accounting and market measures of performance in executive compensation contracts. Journal of Accounting Research 25.85-125. Paul, J.M., 1992, On the efficiency of stock-based compensation, Review of Financial Studies 5, 471-502. Sloan, R.G., 1993, Accounting earnings and top executive compensation, Journal of Accounting and Economics 16, this issue. Verrecchia, R.E., 1982, The use of mathematical models in financial accounting, Journal of Accounting Research 20, Supplement, l-42. Verrecchia, R.E., 1986, Managerial discretion in the choice among financial alternatives, Journal of Accounting and Economics 8, 175-196.
Incentive efficiency of compensation based on accounting and market performance* Oliver Kim and Yoon Suh ~niUersi1.Vof California aI Los Angeles, Los Angeles. CA 90024-1481. USA
Received May 1991, final version received February 1992
This paper analyzes how earnings and price are used in executive compensation contracts. Riskaverse shareholders collectively design a contract and individually trade in the stock market. In the optimal linear contract the use of earnings and price depends critically on incentive efficiency, i.e., how precisely the measures convey the true outcome. The relative weight of price to earnings exaggerates the true relative importance of price because price impounds traders’ overall information while its informational value lies in the incremental information it provides. The use of price allows shareholders to share trading risks with managers.
1. Introduction
Recent empirical studies find that executive compensation is closely related to accounting and market measures of firm performance.’ We analyze how riskaverse shareholders’ individual investment decisions are interrelated with the design of a top manager’s compensation contract. Compensation contracts are assumed to be based on earnings and price. Price is endogenously determined in a securities market and aggregates (with noise) the information held by traders. We derive the optimal linear contract in equilibrium and investigate its properties.2 Among theoretical papers that examine the use of price in compensation contracts, our study is closely related to Bushman and Indjejikian
Correspondence to: Oliver Kim, Anderson Graduate School of Management, University of California at Los Angeles, Los Angeles, CA 90024-1481, USA. l We gratefully acknowledge the comments and suggestions of David Hirshleifer, Bruce Miller, and the participants of a workshop at the University of Washington and the 1991 KPMG Peat Marwick/Olin Conference. We also thank Ray Ball (the editor) and Robert Verrecchia (the referee) for many helpful suggestions. ‘See, for example, Jensen and Murphy (1990). ‘Holmstrom and Milgrom (1987) demonstrate that linear contracts are viable.
0165-4101/93/505.00 Q 1993-Elsevier
Science Publishers B.V. All rights reserved
26
0. Kim and Y. Sub. Compensation based on accounring and market performance
(1993).3 They analyze how the use of earnings in addition to price in contracts alleviates the inefficiency of price in allocating the manager’s efforts between two activities. Our model is similar to their special case with only one activity. The major difference is that in their model shareholders are assumed risk-neutral as the principals (but not as investors), while in our model shareholders are always risk-averse. Our economic setting is one in which the principals are risk-averse shareholders and the agent is the top manager. Following Verrecchia (1986), Dye (1988), and Bushman and Indjejikian (1993), we assume that the true performance (economic earnings) of a firm cannot be used as a contracting variable. Both earnings and price are noisy measures of economic earnings: accounting earnings is subject to measurement error and share price is subject to various demand and supply noise. When shareholders design a compensation contract (through a board of directors), they must anticipate how share price is determined in the market and how the use of price in the contract affects their investment decisions. In order to analyze these explicitly, we use a noisy rational expectations model in the fashion of Diamond and Verrecchia (1981) and Hellwig (1980). In this way we link the noisy rational expectations literature, which captures the behavior of price as a (less than perfect) aggregator of public and private information, with the principal-agent paradigm, which captures the second-best contract as a (less than perfect) incentive device for managers. We analyze how price and its informativeness are determined in equilibrium and affect compensation. In particular, we show that the use of price in a compensation contract enables shareholders to share trading risks with managers. We identify a problem in measuring how managers’ compensation is related to earnings and price. This measurement problem arises from the fact that price impounds the earnings report as well as other common and private information. The informational value of price in addition to earnings for both portfolio and contracting decisions depends on the incremental information conveyed by price. We extract the price signal that represents this incremental information by filtering other information (earnings and the prior information) from (the raw) price.4
sAmong other related theoretical papers, Diamond and Verrecchia (1982) consider the use of security prices in incentive contracts in relation to the availability of other information and to systematic and nonsystematic risks associated with the firm’s output. Paul (1991) shows that price aggregates information about the manager’s different activities in such a way that the manager is not motivated to allocate his efforts efficiently among dimerent activities. This is because price reflects the value of the firm while an optimal aggregator of information would be the value-added of the manager. 4Likewise, the informational value of earnings in addition to price is due to the incremental information it conveys. In our model earnings and the filtered price each represents the incremental information.
0. Kim and Y. Sub. Compensation based on accounting and market performance
27
The importance of earnings and the filtered price as contracting variables is represented by their respective weights (coefficients) in the optimal linear contract. We first examine the overall efficiency of the two measures by analyzing their combined weight (the sum of the weights). We show that the combined weight is determined so as to achieve the optimal trade-off between risk-sharing and incentives. The weight of earnings relative to the filtered price is shown to depend mostly on their respective precisions.5 However, shareholders place extra weight on price because the use of price in contracts allows shareholders to share some of their trading risks with the manager. Although the importance of earnings and price can be better measured in terms of the weights on earnings and the filtered price, the latter is often unobservable to the researcher. This is because information that has to be filtered out is unobservable. Thus, most empirical studies estimate weights on performance measures based on raw prices (returns). In order to link our results to empirical studies, we convert the contract based on the filtered price to one based on the raw price. Both the absolute and relative weights change as a result of this conversion. It is shown that the weight on price is greater than the weight on filtered price, whereas the weight on earnings is less when based on earnings and the raw price. This is because price has a component that reflects earnings. As a result, the relative weight of price to earnings is greater than the relative weight of the filtered price to earnings. Our results are relevant to empirical studies that focus on the absolute [e.g., Jensen and Murphy (1990)] and the relative weights [e.g., Lambert and Larcker (1987) and Sloan (1993)] on accounting and market measures of firms’ performance in executive compensation. Empirical estimates of the weights on these measures should be interpreted with care.6 The plan of this paper is as follows. The model is explained in section 2, and a market equilibrium for a given contract is obtained in section 3. The expected utility of shareholders and the manager are derived in section 4. A benchmark case in which effort is observed is considered in section 5. The optimal contract under moral hazard is derived and its characteristics are examined in section 6. The pay-performance-measure sensitivities in the optimal contract are discussed in relation to some empirical findings in section 7, and some concluding remarks are given in section 8. 2. The model We consider a firm whose shares are traded in a competitive securities market. There are a countably infinite number of identical shareholders, and for every sThis is similar to the results of Banker and Datar (1989) on a linear aggregation compensation contracts and to those of Bushman and Indjejikian (1993).
of signals
6For example, our results are particularly relevant to those in which compensation is regressed earnings and price (returns) and their slope coefficients are used as measures of the importance earnings and price.
J.A.E.-
B
in on of
28
0. Kim and Y. Sub. Compensarion based on accounring and market performance
N < x shareholders there is one manager who works for the shareholders. We assume that all managers are identical and, without loss of generality, concentrate on one representative subset which consists of N shareholders (the principal) and one manager (the agent).’ The securities market structure of this model is a standard noisy rational expectations framework with three time periods 0, 1, and 2. There is one risky asset, namely the firm’s shares, and riskless bonds. One riskless bond pays off one unit of consumption good in period 2. The final payoff to one share of the firm (before compensation) is assumed to be the sum of the level of effort exerted by the manager, denoted by e, and a random environmental factor, denoted by r18 It is assumed that ii is normally distributed with mean ti and precision (inverse of variance) h. The per capita supply of the risky asset is 2 > 0 and is known to everyone. Shareholders are risk-averse and their preferences can be represented by a negative exponential utility function with risk tolerance r > 0, i.e., G(W) = -exp( - W/r), where W is the representative shareholder’s wealth. In period 0 each shareholder is endowed with zero riskless bonds and .Ushares of the firm, while the manager is endowed with zero riskless bonds.’ The manager and the shareholders then enter a compensation contract for the work to be provided by the manager to the principal. Given the contract, the manager decides how much effort to provide. The manager’s utility function is also a negative exponential with risk tolerance ra > 0, i.e., H( W,) = - exp( - [ W, - C(e)],+,), where W, is the manager’s compensation. The function C(e) measures the disutility of providing effort in terms of dollars.” It is assumed that C(e) = (c/2)e*.” Four events occur in period 1. First, the firm announces earnings before compensation represented by a public signal j = e + ii + 4, where fi is normally
‘The economic setting we try to model here is one in which there are N shareholders and a single top manager (e.g., a CEO). However, analyses within a noisy rational expectations framework become clean only with an infinite number of investors. On the other hand, if there is an infinite number of shareholders and if the manager’s compensation is finite, each shareholder’s share of compensation is negligible. As a result, the moral hazard problem faced by individual shareholders becomes trivial. Our model avoids these mathematical problems and is a reasonable approximation of a model with a single manager and N < cc shareholders. *This additivity assumption independent of the environment.
implies
that
the increment
of the final payotT due
to effort
is
‘Under the assumption of constant risk aversion of both the principal and the agent, zero riskless endowments are assumed for convenience and without loss of generality. Shareholders’ risky endowments are assumed identical because such an allocation would be the equilibrium allocation if they were allowed to trade in period 0. “‘Measuring disutility of effort in units of wealth is consistent Holmstrom and Milgrom (1987) for their linearity result. “This assumption cost function would
with the restriction
allows for a closed form solution to our problem. not qualitatively change our results.
Assuming
imposed
by
a more general
0. Kim and Y. Sub, Compensation based on accounting and marker performance
29
distributed with mean zero and precision m.” The precision m can be interpreted as either the quality or the amount of the earnings information. Second, shareholder i observes a private signal z’i= e + l;i+ Ei, where Ziis independently and normally distributed with mean zero and precision s. In other words, investors have diverse private information of the same quality. Third, shareholders trade in a competitive market based on new information, J and their 4’s. There is also a separate group of liquidity traders whose aggregate demand (per share) for the risky asset is a random variable, denoted by d’,which is normally distributed with mean zero and precision t. Last, the manager is compensated according to the contract. In this model we consider contracts which are linear (affine) in two publicly observed measures of the performance of the trm, namely, earnings (before compensation) jj and the share price, denoted by P. In period 2 the firm liquidates and shareholders consume their final wealth from their portfolios. Shareholders in this model make two types of decisions. They collectively design a compensation contract and individually select the number of shares to buy or sell. In a rational expectations equilibrium each choice is made correctly taking into account all other decisions made in the economy. For convenience, we separate the two decisions. In the next section we analyze shareholders’ investment decisions, and derive the equilibrium share price for a exogenously given compensation contract. The optimal contract is analyzed in section 6.
3. Market
equilibrium
In this section a market equilibrium is derived for a given compensation contract. Let a linear compensation contract be expressed as w, = Ni(cr* + p*jj + r*P).
(1)
Then, the final payoff per share, denoted by v’,is o’= e + ii -(x*
+ /I*p + y*P).
(2)
In a rational expectations equilibrium traders conjecture how the market price of a share reflects the information held by market participants and the conjecture is self-fulfilling. Let such a conjecture be expressed as
= bO + b,jj + b2(e + u’) + b,d-
bri,
“Using earnings after compensation instead would not mathematically change our model.
(3)
0. Kim and Y. Sub. Compensarion based on accounring and market performance
30
where the second equation follows from the law of large numbers and the fact that the 2i’s are independent. Define the normalized price signal 4 as +(P-ho-b,j+b,r) 2
= e + ii + BJ,
(4)
where b2 # 0 and B z b3/b2. I3 The information set {J, ii, p} is equivalent to (~,ii, ci}. The signal 4 represents the pure addition of knowledge due to observing the price beyond what is available from other pieces of information.r4 Using (1) and (4) we can rewrite the manager’s compensation per share as
r* + /?*j + y*p = r* + p*j + i’*(bo
+ b,j + b2ij - b4R)
= (x* + y*bo - -,-*bd.u)+ (fl* + y*b,)j =
cl +
pj
+
+ y*bzij
74.
(5)
(5) shows how a linear function of j and P’ can be converted to a linear function of j and (I, and vice versa. For convenience we use the normalized price signal 4 for an initial calculation of the optimal linear contract. The final payoff is now rewritten as
= --a+(1
+I--y)(e+i+j?+-yBd:
(6)
Given normalized signals, j, z’i, and q, it is straightforward E[filp,z’i, 41 =
$
h(e + ii) +
m_f+ Sz’i+
to calculate
sij -
(x + pL: + ;,g,
where
K is the sum of the precisions of the prior and the three normalized signals, and can be interpreted as the amount of total information about the final payoff
“It
can be shown
that there is no equilibrium
if bz = 0. See section
7 for a discussion
of b2.
‘The signal 4 represents the pure addition of knowledge in the sense that the increase in the var-‘(e+ lilj.T,),istheprecision posterior precision due to observing price, var -‘(e + liIJ,5i,4)of the error of 4.
0. Kim and Y. Suh. Compensarion based on accounring and marker performance
31
possessed by each shareholder in period 1. The conditional expectation of the final payoff is simply the weighted average of the four sources of information. The shareholder’s expected utility conditional on signals j, Zi, and P’ is EIGIJ,Zi,F]
= E
= -exp
i
- “5 - [E(Clj. Zi, -
by using the moment-generating function of a normal random variable. The utility-maximizing individual (gross) demand for shares, di, can be found as
= rK
Aggregating over i and applying the zero excess demand condition (in per capita terms), we get X = rK
h(e + ii) + mj + s(e + ii) + +ij
which can be rewritten as
P-=; h(e+fi)+my+s(e+fi)++j
I
i
-
(ct + pj + yij) +
$ - -$.
(8)
Since the conjecture (3) must be self-fulfilling in equilibrium, (3) and (8) are equivalent. Therefore,
32
0. Kim and y. 5th. Compensation bused on accounting and market per/ormance
and B is solved as B = Ilrs. Also b
=
0
b2 =
0 +4 7
-
%
s + r’s”t K -‘r’,
(9)
where K is rewritten as K = h + m + s + r*s*t. Eqs. (7) and (8) in conjunction with B = l/rs give a complete characterization of the unique market equilibrium given any u, b, and 7 such that y # (s + r’s*t)/K.
4. The expected utility In this section we derive the expressions for the expected utility of the representative shareholders and the manager and provide an interpretation of the terms obtained. Since a contract has to be offered in period 0 before observing any information signals, 2, /?, and 7 (or r*, /?*, and y*) have to be determined based on expectations at the time of contracting. The ex ante expected utility of a shareholder and that of the manager are first derived in the following lemma. All proofs not shown in the text are provided in an appendix. Lemma 1. For any given SL,/?, and y the ex ante utility of the shareholder, denoted by EG, and the manager, denoted by EH, are s+r
+g
.f* I
-Zt-
1
K
(1 -/I--r)*h-‘+/?*m-‘+
-f{-1+(1-_B-;‘)(e+Q)]
YZW
+ 4
r2s2t(K + s + r-*t-l)
II
0. Kim and Y. Suh. Compensation
based on accounting
and marker
performance
33
and EH = - exp
- F
(@!+ (/I + r)(e + ri,> + $
(B+.,)2h-‘+BW+-&
* .
The exponent in the manager’s expected utility in Lemma 1 is simply a combination of the mean and the variance of the uncertain compensation and the private cost (disutility) of exerting effort. The expression for the shareholder’s expected utility is similar but contains some additional terms. For a better understanding of this expression, we rewrite it as l+
EC=
( x
exp
x2
K )( Zr2
rVt(K
- Y2r-2f-1
+ s + r-2t-1)
>I
EG
’
(10)
where
EGE
-
exp
-f{-%+(I
-/I-y)(e+ii)f
- SE(C) + $ -
var(L’)
[
=
-exp [
1
is the expected utility if the shareholder did not trade. Note that EG is not entirely independent of the market because it depends on the share price used in the contract. The two multiplicative terms in (lo), (1 + (s + r-2t-1)/K)-1’2 and exp[ - y2f2/2r6s2t2(K + s + r-‘t-l)], capture the gains from trade because both are less than one and EG is negative. In order to understand the two multiplicative terms, it is convenient to consider the added return and risk from trading separately. To see the returnrisk trade-off when trading, consider two cases: one in which the liquidity demand is very high or low and, as a result, the price tends to be very high or low; the other in which the liquidity demand and thus the price are moderate. In
34
0. Kim and Y. Sub. Compensation based on accounting and marker performance
the first case, the return from trading tends to be large because the extreme price is due to demand noise that is irrelevant to the final payoff. As a consequence, the shareholder can profit by using her private information, either by buying at a low price or by selling at a high price. On the other hand, she faces a higher risk when disposing of her endowed shares. In the second case, her return from informed trading is small but the risk in disposing of her endowments is also small. When the contract does not depend on share price (7 = 0). the combination of added return and risk result in a net gain represented by the first multiplicative term in ( 10).15 When share price is used in the contract, the shareholder can shift some trading risk to the manager through the contract. These additional opportunities to share risks between the shareholders and the manager give rise to the second multiplicative term in (10). As the contract depends more heavily on share price (larger y2), the risk-sharing opportunities increase and this term decreases. Also, as the demand noise increases (t decreases) for a given ;‘, it also decreases. (Note that a decrease in the term implies that the gain it represents increases.) Therefore, the role of sharing trading risk with the manager through compensation contracts becomes more important when the demand noise is greater. This beneficial role of price increases the importance of price as an incentive device relative to earnings as analyzed in section 6. Taking a monotonic transformation of the shareholder’s utility we get G E - rln(-
EC)
(
s+r
=iln
l+
-;
i
-zt-I
K
+ x{ - X + (1 - p - r)(e + Li,) )
(1 -B-
‘i)2h-’ + D2m-l + r2s2,,;2ys;!2r_I)
I
. (11)
Similarly, the manager’s utility can be expressed as
EH)
H E - r,ln(= Nl{r
+ (p + y)(e + ti)} -
N2,f2 -y&--
e
i
$
(/?+~)2h-‘+p?l-‘+~ . r2s2t
IsThis term is obtained in many other including Admati and Pfleiderer (1987).
I
studies
that use noisy
rational
(12) expectations
models,
0. Kim and Y. Sub, Compensarion bused on accounting and marker performance
35
By (11) and (12) we represent the expected utility of the shareholder and the manager in terms of certainty equivalents. For the manager it simply becomes a mean-variance representation due to the normality assumption. We allow the manager to be risk-neutral, which can be considered as a subcase of (12) by allowing r. to reach infinity, but the shareholder’s risk tolerance is restricted to be less than infinity in order to guarantee the existence of a market equilibrium.
5. The observable effort case
We first analyze a benchmark case in which the manager’s effort choice is observed by the principal and can be contracted upon. The principal’s problem is then to choose %,/I, and 7 to achieve the first-best optimal risk-sharing among N shareholders and the manager. Formally, the principal’s problem is to maximize (11) with respect to x, /?, and ;’ subject to the minimum utility condition
H
2 ii.
(13)
fi measures the minimum utility in terms of dollars and can be interpreted as the manager’s opportunity wage. The optimal linear contract in this situation is referred to here as the first-best contract. Since any desired effort level can be enforced by a forcing contract, the Lagrangean form of this problem is to
(
max Liln .,P,?.l.P
l+
(1
-g
s+r
-zt-1
K
+.u{-a+(1
-
-/I-y)(e+U)}
>
_/3_y)2h-l
+jZm-l
i
+
‘i2(K + s) r2s2t(K + s + r-2t-‘)
+ 1
[ -y-
I
Nf{r + (j? + ;I)(e + ii))
N2Z2
*
I
(jl+y)‘h-’
+fi2m-’
The solution is shown in the following proposition.
+&
}+q.
36
0. Kim and Y. Suh. Compensarion
Proposition
1.
The first-best
hasett on uccounring mti murkrr per~bnancr
contract
is characteri:ed
br
m + (1 + 6)‘r’s’t :rn + (I + G)r’s’tj,’
’
I?1
/3/ =
Qf,
171+ (I + 6)r’s’t (1 + G)r’s’t
*, ” -
Qf,
m + (1 + d)r2s2t
1 )-f =N’ where *++J(
m+(l+@r_2s2t ) h + m + (1 + b)r2s2r
and
6
r. Nr + r’t(K
Each shareholder G,=G
+ s)(r, + Nr)’
achieves utility (in dollars) of
0
+E+?Q 2c
2rh
”
where
Go+
(
l+
s+re2t-’
-
K
-2
-_x+xrT-h
>
is the utility when the ejiort leoel is zero and the lump sum fi is paid to the manager.
0. Kim und Y. .%/I. Compensation bused on uccounting and market performance
37
The result L, = 1,‘s is obtained because N shareholders contribute to one manager. The choice of the effort level N.$‘c maximizes the total expected wealth of the N shareholders. where the marginal cost of effort, ce, equals the marginal product of effort, NY. The optimal risk-sharing is attained when /?, + ;‘/ = L?,, which is the product of two ratios shown in the proposition. The first ratio is the relative magnitude of the manager’s risk tolerance to the sum of those of all parties (one manager and N shareholders). Besides the risk tolerances of the contracting parties, the optimal risk-sharing also depends on the precision of the signals used in the contract. which is captured by the second ratio. This is because the risk concerning t? is shared via two noisy signals, y and 4. For example, if earnings is a perfect signal (m = CL)or if the prior is extremely diffuse (h = 0), then the second ratio becomes one and the optimal risk-sharing is such that risk is shared in proportion to risk tolerance. Note that letting m = x8 and N = 1 in our setting will yield fl, = re/(ro + r) and ;‘, = 0, which corresponds to the first-best solution in standard agency models with one principal and perfect outcome observability. As the signals become noisier, the additional risk from the noise in the signals, namely, the variances of fi and d’,increases and the optimal contract depends less on those signals. Shareholders in our model are subject to two types of risk. The first type is project risk, due to the inherently random nature of the outcome of firm’s activities, represented by uncertain u’. This is the risk the shareholders desire to share with the manager. However, the only available sharing devices are noisy signals of ii. The use of these devices exposes the shareholders to the second type of risk, noise risk, due to measurement error and,‘or noise in the signals. The optimal contract is devised so as to give the best trade-off between project risk and noise risk. This is reflected in the choice of $2,. Also, the relative weights of the two noisy signals naturally depend on their respective precisions. A more detailed discussion of the relative weights is provided in the next section.
6. The optimal contract with unobservable effort We now turn to our main problem in which the effort choice is unobservable and the manager has to be induced to exert effort by means of an incentive contract. The manager’s problem of choosing an effort level given I, /I, and 7 is to maximize (12) with respect to e. The first-order condition is N,F(/3 + 7) - ce = 0.
(14)
It is easy to check that (14) describes a unique optimal e for any given /? and 7.
The principal’s problem in the Lagrangean form is to l+s+rK
-;
-1
-2
(
max L-iln ../?.7.i.p.u
’
+ .U( - 3 + (1 - /I - 7)(e + 17)) > +/pm-l
(1 _p_#h-l
i
+
y’(K + s) r’s’t(K
+ p[N.f(p
+ s + r-2t-‘)
-t y) - ce].
The difference of this problem from the benchmark problem is that p and y simultaneously play both the role of inducing effort and the role of risksharing. The contract that solves this problem is referred to here as the secondbest contract, or simply the (optimal) contract, and is characterized by the following proposition. Proposition 2.
The second-best contract is characterized by
a+17R_32+32’ 0
D= Y=
( (
m
m + (1 + S)r2s2t 1
(1 + 6)r2s2t
m+(l
+b)r2s2t >
1 A=----, iv P=
0f
(1 - Q),
Q, R ’
(
‘+ h
m + (1 + 8)2r2s2t {m + (1 + B)r2s2t)’ > ’
0. Kim and Y. Sub. Compensation based on accounting and murket performance
39
where r,(Nr + ch-‘)
RG
r,Nr+c(r,+
Nr)
i+
1 m + (1 + 8)rzszt
and 6 is as defined in Proposition I. Each shareholder achieves utility (in dollars) of 2
G z G,, + z
=
x2
R’ + 2r
II
R2 (r, + Nr) r,f’(.Vr
Go + 2cr
r,Nr + c(r, + A;r)
‘+ h
1 m + (1 + 6)r2s2t
+ ch-‘)2
1 m + (1 + B)r2s2t
Note that the agent’s optimal action choice does not depend on the minimum certainty equivalent, A, in both the first-best and the second-best cases. Under the assumption of constant risk aversion both the manager’s action choice and risk-sharing are independent of the constant term, a, of a given contract. Accordingly, /I, 7, and the action choice, e, are determined independently of fi and a change in fi only affects B.‘~ We first verify that a welfare loss exists due to the unobservability of the manager’s effort choice. Corollary 1. There is a positiue welfare loss due to the unobsercability of the manager’s eflort choice. That is, G, - G > 0.
Corollary 1 shows that a moral hazard problem exists in our main problem. We now analyze the weights, a and 7, on the two performance measures, j and 4, in the second-best contract. For a better understanding of /? and 7, it is useful to consider a situation in which only one signal, say F = ii + G, is used in the contract, where c is normally distributed with mean zero and precision M. In this case the weight on yin the first-best and the second-best contract, denoted by PI and $, respectively, can be calculated as
and p^=
r,(Nr + ch-‘) r,Nr + c(r, + Nr)(h-’
+ M-l)’
16This property of exponential utility function Fellingham, Newman. and Suh (1985).
has be-en used to derive a no-memory
contract
in
40
0.
Kim and Y. Suit. Compensation
bused on accounrtng
and math!
performance
Comparing these to the expressions for R, and R in Propositions 1 and 2, it is clear that the combined weight, R, of the two performance measures, 4: and 4, depends directly on their combined precision, m + (1 + 6)rZs2r. Therefore, it is useful to analyze first the combined weight, R, and then to see how the combined weight is decomposed into individual weights, j? and 7. The most important difference between the second-best contract and the first-best contract in Proposition 1 lies in the combined weights, B + 7 G R and P, + ‘J/ = Q,. The difference is summarized in the following corollary. Corollary 2. The combined weight, Q, of two performance measures in the second-best contract lies between one that achieves the best incentive (one) and one that procides the best risk-sharing (Q,). That is, R, < R < 1.
Corollary 2 shows how the existence of moral hazard affects the combined weight. The best incentive is achieved when R = 1 because from (14) the manager would then choose e such that N% = ce, which is the condition that maximizes the principal’s utility when the principal provides effort herself. The differences in the effort level, e, and the constant term in the contract, a, can also be understood through this difference in R. The marginal cost of inducing effort, ,u, is positive because R < 1. Corollary 3. The combined weight, Q, of performance measures in the contract is decreasing in the manager’s work aversion, and is increasing in the combined precision of the performance measures. That is, iW/& < 0 and S2/i3{tn + (1 + 6)r2sZt} > 0.
The magnitude of R depends, conceptually, on two aspects of incentives, namely, incentive effectiveness and incentive efficiency. Incentive effectiveness concerns the end result, i.e., the increase in the shareholder’s utility, given incentive devices of fixed quality, i.e., given fixed m + (1 + B)r2s2t. Incentive effectiveness depends on how sensitively the manager responds to given incentives, how productive he is for given effort exerted, and other additional related benefits and costs, if any. One variable that captures effectiveness in our model is the degree of the manager’s work aversion, c. This is because c is proportional to the manager’s private cost of exerting effort that has to be compensated in order to keep the manager. Therefore, a smaller c corresponds to more effective incentives. R is decreasing in c because as it becomes more costly to induce effort, it is optimal to give less incentives, holding the manager’s productivity constant. As c approaches infinity, inducing any positive level of effort becomes too costly. As a result, R approaches R, in which the role of using the measures in the contract is pure risk-sharing. At the other extreme, when c approaches zero, a large amount of effort can be induced costlessly. As a result, Q approaches one and the maximum level of effort is
0. Kim and Y. Sub. Compensation bused on accounting and market performance
41
induced. Incentive ineffectiveness can arise for many other reasons that are not modeled here. A manager may not be responsive to incentives because he is too lazy, or because social or other monitoring mechanisms work well and he is already exerting enough effort. In this case any additional effort is too costly to him as well as to the agency. Incentive efficiency concerns how precisely the incentive devices (jj and 4’)convey the true outcome (e + ii). The second part of Corollary 3 says that the combined weight is increasing in incentive efficiency measured by the overall quality of the signals. Given our understanding of the combined weight, the following corollary shows how the weight on each of the two measures is determined in the contract. The same result applies to both the first-best and the second-best contracts and directly follows from Propositions 1 and 2. Corollary 4. The relatice importance of earnings, Jt, and the normalized price, q, in the contract is largely determined in proportion to their precisions. That is, pkj = m/(1 + 8)r2s2t.
The use of noisy signals in the compensation contract subjects both the principal and the agent to noise risk. The optima1 combination of the two noisy signals thus naturally depends on their precisions, m and r2s2t. In addition, the use of price in the contract has a beneficial effect of sharing trading risks between shareholders and the manager as discussed in section 4. Therefore, the relative weight on the market measure, 4, is its precision. r2s2t, combined with this effect. This is expressed in a precision-equivalent term as br2s2t. Given fixed precision, this makes q more desirable, which is reflected by the fact that S is positive. Most of the comparative statics with respect to Proposition 2 are intuitive. We single out one that is related to the literature on information disclosure. Corollary 5. An increase in the precision of earnings reduces shareholders’ welfare through its adcerse risk-sharing efect. but alleviates the moral haxrd problem. That is, aGO jam< 0 and a(G - G,), Sm > 0.
Recall that GO is the shareholder’s utility when the manager’s effort level is zero and the lump sum fl is paid to the manager. The first part of Corollary 5 implies that, in the absence of a moral hazard problem, a more precise disclosure makes shareholders worse off. This effect of disclosure, called the adverse risk-sharing effect, has been extensively discussed in the literature.” The importance of Corollary 5 is that it identifies the second effect. G - Go is the portion of the shareholder’s utility that corresponds to moral hazard considerations. The second part of Corollary 5 says that a more precise disclosure “See, for example, discussions.
Hirshleifer
(1971). Verrecchia
(19SZ), and
Diamond
(1985) for detailed
42
0. Kim and Y. Suh, Compensation based on accounting and marker performance
increases this portion of utility. That is, it alleviates the moral hazard problem. This is because as the accounting measure, 9, conveys a more precise information about the performance of the firm, the variances of the shareholders’ and the manager’s wealth both decrease. As a result, both are made better off.
7. Price and incremental information The informational value of price both for investment and contracting purposes mainly arises from the fact that price carries incremental information over and above what is known from other sources of information. This incremental information is represented by the normalized price signal, 4. However, in most empirical studies [e.g., Jensen and Murphy (1990), Lambert and Larcker (1987), and Sloan (1993)J the estimation of the sensitivity of compensation to market and/or accounting measures is based on the ‘raw’ price. From (8) the relation between price P’ and the normalized price signal 4 can be written as F
=
(----$a)
+(;-p)j+(~+~~2~-y)@(15)
Me+ 4 K
Eq. (15) shows that price impounds both its incremental information, 4, and other information (i.e., the prior and 4: in our model). In order to link our results to the empirical literature, we now convert the optimal contract based on normalized signals, j and 4, to one based on earnings, B, and price, p’. BY O), a = r* + 7*bo - y*b4i&
y = -j*b2.
Solving for CX*, p*. and y* and using (9), we obtain the following result. Proposition 3.
a* =
The optimal linear contract x* + p*j + y*P’ is such that s + r2s2t h(e + U) Q -77 K s -I-r2s2t K
-T r’
3
ap!Aold 6aq1 asn~~aq pw suo!s!3aJd JO SUIW u! Quo passaJdxa aJe 6aql asnesaq ]InsaJ s!ql u! pasn a,e saylenbau! aql ‘,v pue O.tSE qms sJa]aruemd Jaqlo uo puadap L pue ‘61‘*A ‘J ‘s,q%!am aqler
pasvq asyd 01 sbu~uma~o amvllodur! aaflvlar ayl my1 ra]pxus s! (.J/,$) a3!rd MVI ayl uo pasvq lm.~uo~ ayl u! a+d 01 s&ma/o l@a.+t aoflvlar ayl ‘L Lrv110~03 ~~pxus%u!aq c, uo puadap lou op ~1nsaJlxau aql pue s!q~ yx~%!s aD!Jd pazyeruJou aql uo paseq asoql ql!M aDpd MBJ aql UO paseq SaJnS~aUllayJWJlpUE%l!lUnO33~aqlJOSlq%!aMaA~l -Olafaql saJedmo3 ICJr?lloJo3 lxau aq~ *ax.wJodm! ~eluauIaJxI! anJ1 J!aqiu10q luaJa&p aq 01 Llag are (suJnlaJ)ayJd MBJ ayi put! %u!uJea uo pawq sa!pnls uo!ssaJ?iaJ ald!llnuJu! sal~uysa luaggao3
aql leql slsa%%ns 9 AJ~IIoJo~
ayl uo pasvq am Sly&ah4 ayl uayM uvy (C,+ d < *L + .d) dalvaA6 s! lyB!ar\\ pawqLuo3 ayl puv ‘(X < .&) ralval6 s! az+!d uo iy6!aM ayl ‘(d > *d) ssal s! s6uwva uo lyB!a,n ayl ‘d ‘as!rd MVI ayl uo pasvq am samsvaw ammrojIIad uo sly6iaM ayl uayd *y/(izszd + s) 01 anyvlad ~pxus haa s! h lvyt asoddns ‘9 rC1v110.10~ .(swo
u~!yi
JaieaJ% y3nw aq
01 AJt2110J03 %U!MOIIOJ aql u! pawwe lnq ‘alqeAJasqouns! q3y~) UO!~EUJJOJU! IlEJaAO J!aql 01 aD!Jd UIOJJpUI2Sa3JnOS alt?A!Jd UIOJJ UO!ll?UIJOJU! ,SJOlSaAU! JO ax.wlJoduJ!aA!lelaJ aql s! Jaw] aq_L.Ly/(lZSZ1 + s)ol aA!ie[aJ(g)o'()uI?qlSSa~ say -!A!l!suas axxemJoJJad-h?d 1JOdaJ dlp_?cyddl sa!pnls[~j!J!drua)~~erus hJaA s! XWql %h.Iynssoapw.us!uos!Jedwo3 aqL *~CJI?~~OJO~ %~!~0~[0~aqlu!‘I(~aA!l3adsaJ'~ + d aql aJt?duJo3 lsry aM pue ‘x‘9'10aSoq1 ql!M z& + *d pUe ‘.& ‘*d JO S~pIIl$hII .aD!Jduo paseq salwuysa pD!J!dLLIa %J!laJdJalU!6lpaJJO3 JOJ 1uelJodLy 6llegualod ale saxIaJa~!p asaqJ_ *ayJd pazyxuJou aJe a+d
aql pue s%u!uJt?a uo ‘I. pue d ‘slq%!aMaql UJOJJluaJaa!pdlleJaua%
pue S%I!UJt?auo ‘*X pur! *d ‘Slq%!aMaql 1eqlsMoqs
‘z uoy!sodo~d
E UO!l!SOdOJd
u! uaa!B am C.puv ‘d ‘xi rojsuoynjos
c.‘
ayl araym
>I
l,S,J + s = *.i’ L
44
0. Kim and Y. Sub. Compensation
based on accounting and marker performance
on the normaked price (/IF;). The discrepancy is more severe for a firm for which shareholders’ private information constitutes a smaller portion of their overall information. More precisely, S
K+s
p*pj* <---iK
s
PI?
The true relative importance of earnings to price is /Q considering only the incremental information carried by price. For a better understanding of Corollary 7, suppose that the estimate of p*/-i* is 0.1. In this case K /I E
K-IS
10s
For example, if shareholders’ private information comprises only 10% of their total information, i.e., if s/K = 0.1, then /?h is between 1.0 and 1.1. If s/K is 0.33, /?pl is between 0.3 and 0.4. The above examples illustrate that empirical estimates of the weights on accounting and market measures [such as those of Lambert and Larcker (1987) and Sloan (1993)] may seriously exaggerate the relative importance of price to the accounting measure. The problem is more severe for firms about which investors have a small amount of private information relative to their overall information. l9 Finally, the following result summarizes Corollaries 4 and 7. Corollary 8. The relative weight of earnings to price in the contract is roughly the product of the ratio of their precisions and the fraction of private information in investors’ overall information. More precisely, ms p* r*s*tK + s(s + r*s*t) < -y* <
ms r2s2tK’
Both the upper and the lower bounds of /3*k;* are increasing in m and are decreasing in h, s, and t. This implies that an increase in the precision of earnings increases the relative weight of earnings as expected, whereas an increase in the precision of any other information (either private information or other public information) decreases the relative weight of earnings.
“Since precisions are not directly observable, it is difficult to determine for what types of firms investors’ private information constitutes a larger fraction of their overall information. See Im (1989) for an indirect method of grouping firms based on the fraction of private information based on analysts’ forecasts and a simple Bayesian model.
0. Kim and Y. Sub. Compensarion bused on accounting and marker performance
45
8. Conclusion
We have analyzed how accounting and market performance measures are used in compensating top executives. In particular, we have demonstrated that empirical estimates of the extent to which accounting and market performance measures are related to compensation should be interpreted with care. This is because price impounds the earnings report, as well as other common and private information, while the informational value of price in addition to earnings for contracting purposes depends on the incremental information it conveys in addition to earnings.
Appendix
Proof of Lemma 1
For arbitrary individual risky endowments, Xi, the expected utility is
=E[Er[-exp{
where E[v’lj, z’i,P] E pi, var[v’jj, z’i, P] E l/K, and
= -exp
=-
ev
1-
K(fii -
i
-
P)(fii- P) +
K’(,Ci 2K
P)’ I
1
f(fii- F)2
7
which is obtained by using the moment-generating
function.
46
0. Kim and Y. Sub. Compensarion bused on accounring and marker performance To
simplify notation define
-f
brfii-p--=-
rK
1
.
K
The expected utility can be rewritten as -p -
EG=E[-exp{-[d-1+(*
y)(e
+
c)
_
_$ : 1
-
T(b+-&>‘}I.
(6,6) has mean (0,O) and let V, s var(a’), V, 3 var(6), C,,* E cov(& g), and F = V, Vb - C,“,. EC above is now expressed as an integral
Since the exponent of the integrand is a quadratic function, the integration is a straightforward exercise using the normal density. Integrating with respect to ii and then to b’, it becomes
-&&exP[- {2
--+pjp 2r2K
F xf b
1
z + (1 - p - v)(e + W)- $
-}xi r
0. Kim and Y. Suk. Compensation based on accounting and market performance
-&iv
=
-
.f 0 -
r
[- i
2
1
-r+(l
-8-;)(e+E)-2
3 -1
Xi
K+V;‘-
47
C’ob
’
07
V;(K
+ I’;‘)
d6
-
u +
2
xxi
c:b Vb(KVb
(1 - p - y)(e + Is) -
+
Cab
l)+FKV,
1 '
V, =(-K + s + r-2t-‘)/K2 + (1 - /? - ?)‘/I-’ Xi = .~, substituting _p2m-’ - y2/r2s2t, vb = (s + r-‘t-‘)/K2, and C,,b= (YK - s - r2s2t)/r2stK2, the above is simpified as BY
EC=-
(
l+
s+r
-zt-1
-l/2
K
-:{-r+ll-/?-y)(e+E)}
exp >
(1 -j?-;1)2h-‘+/?2m-‘+
[
y2(K + s) r2s2f(K + s + rm2t-‘)
The expression for EH is calculated by using the moment-generating the normal random variable.
* 11
function of
Proof of Proposition 1 I. = l/N is obtained from the first-order condition with respect to s( and is substituted in the first-order conditions with respect to fi, 7, A,and e to generate,
48
0. Kim and Y. Sub. Compensu~ion bused on accounting
and murkrt performuncr
respectively,
)(I - p- ;,)/I_’ - irm-‘)-~{(ll+7)h-L+Bm-l!=o, (16) ;
Y(K + 4
(1 _/j_yp-‘_
r2sZt(K + s +
i
-; (P+y)P+-& I (I
Nx(r
r-2t-1)
I
=o,
(17)
i
(p + y)(e + ii,} -
+
ce2 --_=
N2.f2 2r,
..2
(p + i’)2h-1 + B2n-’ +
(18)
A,
2
9-;=o. By subtracting
(19)
(17) from (16) and multiplying
-4 Y
by r,r we get
K+s
K+s+r-2t-l+Nr TO
r2s2t
or /I
‘i a=-
(r, + Nr)(K
+ s + re2t-‘)
m (r, + Nr)(K + s) + Nrrq2t-’
Nr + r2r(K + s)(r, + Nr)
3
$1+6).
Let R, E /I + 7. Then
mQs
p=
m + (1 + b)r2s2t’ y=
-&
(1 + B)r2s2tQf m+(l
+6)r2s2t’
49
0. Kim and Y. Sub. Compensation based on accounring and marker performance
By substituting these back into (16) we get !+ h
1 m + (1 + b)r2s2t
or r,h-l
1 m + (1 + B)r2s2t ).
Now the solution for z is obtained by substituting the above into (18), and e is solved directly from (19). G, is obtained by using all the solutions in the expression for the shareholder’s utility. Proof of Proposition
2
i. = l/N from the first-order condition with respect to a is used in the first-order conditions with respect to 8, 7, i., p, and e to generate , respectively,
$((I - p - y)h-’
-pm-‘l-F{(p+y)h-‘tpm-l)+Np=O, (20)
4
-P__.Jh-‘-
(1
i
-F
Nf{r
{
i’(K + s) r2s2t(K + s + r-2t-‘)
@+y)h-I+-&
+ (p + y)(e + ~7)) - F
ce2 --_= 2
I
+N,u=O, (P+V)Zh-l+fi2m-l+;.2
II
(21)
r2sZt
H,
N_f(P + ;,) - ce = 0,
(23)
f-+=0.
(24)
Let R E b + 7. Then from (20) and (21) P= y=
mQ m + (1 + d)r2s2t’
(1 + 8)r2s2tQ m + (1 + b)r2s2t’
50
0. Kim and Y. WI,
Compvnsurion based on uccounrin~ rrnd market performonce
as in the proof of Proposition
1. By substituting these back into (20) we get
1 m + (1 + S)rzszt
+ N,u = 0.
(25)
From (23) and (24) we get p =
;(I -
Q).
By substituting this into (25) R is solved, and using (22) x is solved. Now e is solved by using (23), and G is obtained by using all the solutions in the expression for the shareholder’s utility. Proof of Corollary I
The difference in the utility can be easily calculated as G,-G=
1 m + (1 + S)r*s’t > 0. Proof of Corollary 2 Q is clearly less than one by inspection. Also, Q is greater than R, because it can be generated by adding the same positive number r. Nr/c to the numerator and the denominator of r,h-’
1 m + (1 + d)r2s2t
which is less than one.
0. Kim and Y. WI. Compensation based on uccounring ad marker performance
51
Proof of Corol1ar.v 3
The first part is true because r,Nr[(r,
af2 -=?.c
i
+ Nr){m + (1 + b)r’s2tj-’
+ Nrh-‘1
2 1 k+ m + (1 + d)r’?t )I (
r,Nr+c(r,+Nr)
< 0.
The second part is obvious by inspection. Proof of Corollary
5
For the first part, by inspection G,, is decreasing in K and thus in m. For the second part, it is clear from the expression for G that m affects G - GO only through m + (1 + 8)r’s’t. Since G - GO is increasing in m + (1 + d)r2s2t, it suffices to show that m + (1 + b)r2s2t is increasing in m, rOr2t(rs + Nr)r2s2t
a{m + (1 + B)r2s2t} = 1 _
{Nr + r2t(K + s)(r, + Nr)).’
am
rOr2t(r, + Nr)r’s’t ’ ’ - {r’t(K
>l-F
+ s)(r, + Nr)j2
S2
(K + s)-
> 0. Proof of Corollary
6
For the first part, we have from Proposition s +
p*, P
r2s2t
---
K s + r2s2t K
3
ym BK
_+ J
If ‘J is negligible relative to (s + r2s2t)/K, this is approximately s + r2s2t -
(1 + B)r2s2t
s + r2s2t
=
s - br2s2t s + r2s2t ’
equal to
52
0. Kim und Y. Suh. Compensation
bused on accounrmg
By the definition of 6 in Proposition s - 6r2s2t = s -
1,
r2s2tr Nr + r’t(K
+ s:(r. + Nr)
S2
>s--
=-
and murket per/ormunce
K+s SK
K+s
The claim follows from the above and the fact that 6 is positve. The second part follows from Proposition 3 if y is negligible relative to (s + r2s2t)/K. For the third part, we have from Proposition 3 ps +;‘s’t /j* + ‘J* =
%
+ yh + s;
r2s2t
s + r2s2t _. i’ K (p + ~){m(s + r2s2t) + (1 + 6)rZs2t(h + s + r2s2t)] (s + r2s2t){m + (1 + 6)r2s2t).
= (P + 7)
1 + (s +
(1 + b)hr2s2t r2szt){m + (1 + 6)r2s2t)
1’
The claim is true from the facts that (1 + S)r2s2t/{m + (1 + 8)r2s2t} is increasing in 6 and that 0 -c 6r2s2t c
Proof of Corollary
From Proposition
s= K +s’
7
3, -1
= >
m(s - Sr’s’t)
K(l + 6)r2s2t’
Our claim follows from this and the fact that 0 c 6r2s2t < s2/(K + s).
0. Kim and Y. Sub. Compensation based on accounring and market performance
53
Proof of Corollary 8
From the proof of Corollary 7,
P* -_= i’*
m(s - 8r2s2t)
K(l + 6)r2szt’
which is decreasing in 6. P*lr* is less than m,IrZs2t K because S is positive. Also, since 8r2s2t < s2/(K + s), we have
=
r2s2rK + s(s + r’s’t)’
References Admati, A.R. and P. Pfleiderer, 1987, Viable allocations of information in financial markets, Journal of Economic Theory 43, 76-l 15. Banker, R.D. and SM. Datar, 1989, Sensitivity, precision, and linear aggregation of signals for performance evaluation, Journal of Accounting Research 27, 21-39. Bushman, R.M. and R.J. Indjejikian, 1993, Accounting income, stock price, and managerial compensation, Journal of Accounting and Economics 16. this issue. Diamond, D.W. and R.E. Verrecchia, 1981, Information aggregation in a noisy rational expectations economy, Journal of Financial Economics 9. 221-235. Diamond, D.W. and R.E. Verrecchia, 1982, Optimal managerial contracts and security prices, Journal of Finance 41, 275-287. Dye, R.. 1988, Earnings management in an overlapping generations model, Journal of Accounting Research 26, 195-235. Fellingham, J., P. Newman, and Y. Sub, 1985, Contracts without memory in multiperiod agency models, Journal of Economic Theory 41, 340-355. Hellwig, M., 1980, On the aggregation of information in competitive markets, Journal of Economic Theory 22, 477-498. Hirshleifer, J., 1971, The private and social value of information and the reward to inventive activity, American Economic Review 61, 561-574. Holmstrom, B., 1979, Moral hazard and observability, Bell Journal of Economics 10. 74-91. Holmstrom, B. and P. Milgrom, 1987. Aggregation and linearity in the provision of intertemporal incentives, Econometrica 55, 303-328. Im, C., 1988, Differential predisclosure information and the magnitude of stock price responses to earnings announcements, Ph.D. dissertation (Univ-ersity of Pennsylvania, Philadelphia, PA). Jensen, M.C. and K.J. Murphy, 1990, Performance pay and top-management incentives. Journal of Political Economy 98, 225-264. Lambert, R.A. and D.F. Larcker, 1987, An analysis of the use of accounting and market measures of performance in executive compensation contracts. Journal of Accounting Research 25.85-125. Paul, J.M., 1992, On the efficiency of stock-based compensation, Review of Financial Studies 5, 471-502. Sloan, R.G., 1993, Accounting earnings and top executive compensation, Journal of Accounting and Economics 16, this issue. Verrecchia, R.E., 1982, The use of mathematical models in financial accounting, Journal of Accounting Research 20, Supplement, l-42. Verrecchia, R.E., 1986, Managerial discretion in the choice among financial alternatives, Journal of Accounting and Economics 8, 175-196.