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The Dynamics of Capital Allocation
Neal M. Stoughton
UC Irvine
Josef Zechner
University of Vienna
December 2000
Abstract
Financial institutions are facing increased pressure to enhance shareholder value. This has lead
to the popularity of practical techniques such as EVA r© and RAROCTM. The major purpose of this
study is to illustrate the interaction between incentive-based compensation and performance evalua-
tion in a multiperiod setting. We demonstrate that while EVA can justifiably be used to incentivize
managers to make better current investment decisions, performance measurement techniques such
as RAROC help the firm to better assess abilities for the future. The model is applied to under-
stand why hard position limits are employed as well as softer incentive contracts and what sort of
termination standard should be used for the investment manager.
1 Introduction
Risk management in financial institutions is a major priority due to modern developments such as
globalization and the information technology revolution. Influential figures such as Alan Greenspan
have called for increased attention to be paid to designing capital allocation mechanisms. Recently he
remarked that ”The uncertainties inherent in valuations of assets and the potential for abrupt changes
in perceptions of those uncertainties clearly must be adjudged by risk managers at banks and other
Future versions of the paper will be available on the web site of one of the authors athttp://www.gsm.uci.edu/stoughton. The authors thank Helmut Elsinger for valuable assistance with one of thelemmas. The paper has benefited from comments at seminars at the Catholic University of Louvain, Claremont-McKenna,Fullerton, Groningen, HEC, INSEAD, Norwegian School of Economics and Business Administration, London School ofBusiness, University of Pampeu Fabra, Toulouse and USC. In addition it has been presented at the European FinanceAssociation and the Mitsui Life Symposium at the University of Michigan. We thank the discussants, Yrjo Koskinen andS. Viswanathan for valuable comments.
1
financial intermediaries. At a minimum, risk managers need to stress test the assumptions underlying
their models and set aside somewhat higher contingency resources–reserves or capital–to cover the losses
that will inevitably emerge from time to time when investors suffer a loss of confidence. These reserves
will appear almost all the time to be a suboptimal use of capital. So do fire insurance premiums.”1
At the same time, financial institutions, as well as many other corporate entities, are facing increased
pressure to enhance shareholder value. This has lead to the popularity of practical techniques such as
Economic Value Added (EVA r©) and Risk Adjusted Return on Capital (RAROCTM).2 These methods
of achieving shareholder value maximization use residual income after a capital charge is applied. Not
surprisingly, therefore, the method of deriving the appropriate capital charge becomes critical. For
financial institutions, the amount of actual capital required for investment decisions is of less importance
than the amount of economic capital that is related to the riskiness of the activities within the firm.
Capital allocation therefore represents the problem of deriving an appropriate risk-based capital so that
EVA and standard capital budgeting procedures such as net present value can be adapted for financial
firms (Stulz, 1998).
This paper considers the question of how modern financial institutions can design incentives for their
internal divisional managers in a dynamic environment. The major purpose of this study is to illustrate
the interaction between incentive-based compensation and performance evaluation in a multiperiod
setting. In our previous paper, Stoughton and Zechner (1999), we demonstrated the applicability of
EVA and RAROC in a multidivisonal firm in which the externality of diversification benefits needed to
be internalized at the level of each individual division. While that paper showed that a form of internal
capital markets could be used, there essentially was no distinction between incentive compensation
based on EVA and performance evaluation using RAROC. In the present paper, we demonstrate that
while EVA can justifiably be used to incentivize managers to make better current investment decisions,
performance measurement techniques such as RAROC help the firm to better assess abilities for the
future. These two separate and fundamental functions can only be understood in a truly dynamic
environment. As Boquist, Milbourn and Thakor (1998) state, “a benefit of a dynamic capital budgeting1In a speech entitled “Measuring Financial Risk in the 21st Century,” before a conference sponsored by the Office of
the Comptroller of the Currency, October 14, 1999.2EVA is a registered trademark of Stern Stewart and Co. and RAROC is a trademark of Deutsche Bank/Bankers Trust.
2
system is the opportunity for learning.”
Risk management has been the subject of considerable academic and practitioner interest in recent
years. For instance, Froot, Scharfstein and Stein (1993), Merton and Perold (1993) and Froot and Stein
(1998) illustrate that due to imperfections in capital markets banks have a preference for debt finance.
The costs of financial distress are considerable and must be accounted for in determining the optimal
capital structure. Efficient risk management procedures can therefore free up costly equity capital and
thereby enhance shareholder value. Some of the practical difficulties present in the capital allocation
problem are discussed by Kimball (1997). Uyemura, Kantor and Pettit (1996) and Kimball (1998)
discuss the use of EVA in measuring performance in the financial industry. Zaik, Walter, Kelling and
James (1996) describe the use of RAROC at Bank of America.
Risk-based capital allocation procedures are extensively documented in Matten (1996). Milbourn
and Thakor (1996) develop an agency-based model of the capital allocation and managerial compen-
sation process. The theoretical justification for EVA and the use of other residual income based per-
formance measures goes back quite some time (Preinreich, 1937). More recent papers include Feltham
and Ohlson (1995), Rogerson (1997) and Reichelstein (1997).
Capital allocation can be regarded as an application of some of the principles of capital budgeting.
One of the earliest papers to look at the consequences of asymmetric information was Harris, Kriebel
and Raviv (1982). More recently impacts of delegation have been studied in Harris and Raviv (1996),
Harris and Raviv (1998) and Bernardo, Cai and Luo (2000).
In this paper we develop a two period model in which an investment manager observes a signal
of investment productivity and then must communicate to the firm what level of capital should be
allocated to his investment activities. Because the manager can undertake multiple investment projects
required capital is an increasing function of his activity choice. However, while risk increases with the
scale of investment activities, the amount of learning also increases. As a result there is a tradeoff
between the value of learning and the amount of risk undertaken. Learning benefits the firm in the
second period because it can utilize this information in the subsequent capital allocation process. An
agency problem arises in the delegation process because the manager has a tendency to increase risk
beyond the optimal point while attempting to maximize the amount of learning. As a result it becomes
3
1 2ManagerobservesState S
Managerselects
projects N
Firmraises
capital C
Cash flowsobserved yi
Manager paid
Updating ofmanagerability θp
New capitalraised C2
Cash flowsobserved
Figure 1: Timing of Events
costly to the firm to counter this risk-taking incentive of the manager.
The model is applied in a number of ways to indicate potential theoretical justification of procedures
often used in practice. For instance why are hard position limits often employed in addition to softer
incentive procedures? Furthermore, what sort of performance measurement standard should be used in
deciding whether to retain or terminate the manager after some initial experience?
The paper is divided into five subsequent sections. Section 2 sets forth the basic story. Section
3 solves for the optimal capital allocation schedule and section 4 shows how this is implemented in
a delegated managerial environment. Section 5 considers the case of optimal termination and the
implications of the paper are discussed in the concluding section 6.
2 The Story
In this section we set forth the basic model with the financial institution and a single divisional manager,
who is responsible for an investment portfolio. The essential features are similar to those in Holmstrom
and Ricart i Costa (1986).
2.1 Timing and Events
Initially the firm contracts with the manager over a two-period time horizon. We discuss the exact
nature of the contractual relationship with the manager in section 4. For the present purpose, we can
simply assume in the absence of any contracting imperfections that the manager is paid a fixed amount
in each period.
4
The sequencing of events is depicted in Figure 1. Initially, at time zero, a signal, S, is learned by the
division manager. This signal represents a market-wide macroeconomic factor that will be observable to
the firm at time one and can be used for compensation at that time. At the initial time, it is only seen
by the manager and therefore represents the necessary reason for employing the manager in the form
of various investment activities. In addition to this market index, the expected return to investment
activities is influenced by the ability of the manager. At the initial time we assume that the manager
and the firm have symmetric information about managerial ability.
On the basis of the privately observed signal, the manager reports the level of investment activities
that she would like to engage in over the initial period. This level, denoted by N , can be thought of
as representing the number of individual investments that are advocated by the manager. For instance
if the division represents bank lending (credit) activity, then N might represent simply the number of
loans made in a given geographical area for a specific purpose. Or if the division represents the bank’s
trading book, then N is related to the size of its trading portfolio. As a final example, if the division is
engaged in yield curve asset transformation, N would represent the magnitude of duration mis-matching
between the asset and liability side.
Another interpretation of the variable N is relevant. If we think of a time interval over which the
manager is allowed to operate before any form of performance assessment is made, then N could be
interpreted as the number of periods of cash flow observations that are made.
In many cases these investment activities will require actual capital to conduct. In other cases such
as in financial derivatives, very little up front capital is required. Nevertheless, since these activities
are risky they impose risk on the financial claimholders of the institution in relation to its financial
structure. The institution, either because of regulatory constraints, or its own internal optimal capital
structure considerations then raises equity capital, C, in order to support the desired level of economic
activity.
The first period is long enough so that further information about the project outcomes becomes
available at time 1. We refer to these outcome-related informational variables as cash flows, yi. In
general, though, these should be thought of as being sufficiently general to encompass other valuation-
related measures such as changes in the value of marketable securities. Cash flows will be stochastic,
5
although they will be related to a number of factors: the return on the market, S, the number of projects
engaged in, N and the ability of the manager at forecasting cash flows.
On the basis of the cash flow outcomes and the decisions of the manager, the firm then reassesses
the managers ability in order to decide on the appropriate investment strategy for the second period.
We denote the posterior ability level of the manager as θp as updated from the prior after observing
first-period cash flows. Once again the manager observes a second period signal, S2 and then reports
on a desired level of activities for the second period. The original amount of capital is adjusted to C2.
As before, at the end of the second period, cash flows from the second period are observed and payouts
to the manager and the firm are made in connection with the agreed-upon compensation contract.
2.2 Investment Opportunities and Activity Selection
The most significant assumption we make is that there is a negative relation between the marginal net
present value of an investment activity and the number of activities that the division engages in. We
represent this is a very simple way; the incremental expected cash flow from engaging in a marginal
project is a linearly decreasing function of the number of projects. This embodies the notion that better
projects will be adopted first. Specifically, we assume that the first-period cash flow for project, i, yi
satisfies the following linear relationship:
yi = S + εi − bi, (1)
where εi represents both the effect of managerial ability as well as idiosyncratic risk inherent in the
project. Project rankings are accounted for by the coefficient b. Equation (1) indicates that the cash
flow has three components: a market factor, an idiosyncratic factor that is related to managerial ability
and an index of profitability.
We assume that S is distributed over [0, S] on the positive real line with distribution function,
F . On the other hand, εi are all distributed normally with common mean θ and standard deviation,
σε. We incorporate learning about managerial ability into the model by assuming that at time zero,
the institution and manager believe that θ is normally distributed with prior mean θ0 and standard
6
deviation, σθ. After cash flows are observed the mean estimate will be updated. We assume that the
standard deviation is measured precisely ex ante, so that there is no updating of the precision of εi.
The combination of projects yields an overall portfolio risk. Here we denote this risk as σy, the
standard deviation of total cash flow∑
yi. The institution takes this cash flow risk into account in its
optimal capital structure. Because of a preference for debt as opposed to equity for given cash flow risk,
we assume that the binding constraint on equity capital is given by a value at risk (VaR) constraint.3
Equity capital, C, is set equal to the overall institution’s portfolio VaR. When the capital structure
constraint is represented in this way, we are assuming that yi is expressed as a cash flow to equity (i.e.,
net of any interest-related costs associated with the actual capital that must be invested up front). The
equity capital is then assumed to be invested risklessly over the period (so that it does not distort the
original VaR calculation).
If cash flows are assumed to be distributed normally, then
C = VaR = ασy, (2)
for some constant factor, α. The firm is assumed to face a constant cost of capital, r.
2.3 The objective of the firm
We are now ready to define the economic value added, EVA, of the sequence of projects selected by the
manager. EVA is given by
EV A =∑
yi − rασy. (3)
The question of how σy is determined depends on the information available to the firm at the time
capital is chosen. As we shall subsequently argue, the firm will be able to infer from the manager’s
project selection decisions what signal was observed. In this sense, the model is payoff equivalent to a
setting in which the manager reports the signal to the firm and the firm dictates the level of investment
activity. Hence, if capital is flexible so that it can respond to different signal values, S, the remaining3The value at risk of a cash flow distribution is equal to the maximum unexpected loss that can occur over a specified
time interval with a specified probability. In the case of recent Basle committee recommendations, the VaR governingequity capital for trading activities of banks is set equal to some multiplier (between three and four) times the value atrisk using a 99 percent confidence level over a 10 day period.
7
uncertainty is embodied in εi.
Even though the εi values are independent, for the purposes of choosing equity capital, the bank
is worried about “worst-case” outcomes. Therefore, we assume that rather than taking account of the
diversification benefits of independent projects, the bank is concerned about the possibility of systemic
risk and therefore allocates capital as though the idiosyncratic risks are perfectly correlated. In this
case σy = Nσε.
Consider now the single-period problem of maximizing expected EVA when capital is flexible (can
respond truthfully to reports of signals by the manager). Since the single-period problem is applicable
to either the first or second period, we denote the signal as S. Also, denote the current belief about
managerial ability as θ (whether updated or not). Expected cash flows are computed as
∫ N
0(S + θ − bn)dn = (S + θ)N − bN2
2.
(Note here that to avoid needless technicalities dealing with integral numbers of projects we simplify
matters by assuming that the last project can be scaled down proportionately in terms of expected value
as well as standard deviation.) Then the single-period optimal capital is determined by the solution to
the following problem:
max{N |N≥0}
(S + θ)N − bN2
2− rαNσε (4)
The first-order condition for the optimal capital yields the following result (for N ≥ 0; otherwise
N = 0).
N =S + θ − rασε
b. (5)
This solution is illustrated in Figure 2.
To determine the expected single-period benefit from optimal capital allocation, we substitute for
the optimal single-period number of projects from (5) above into the objective function (4). We then
obtain the following proposition, in which the expected EVA is evaluated conditional on signal S and
beliefs about managerial ability, θ:
Proposition 1. Suppose that capital is flexible and optimized over one period with respect to the market
8
S + θ
Number of projectsN
Slope = −b
Capital charge
EVA
Total EVA
Figure 2: Investment Opportunities
signal, S, and current beliefs about managerial ability, θ. Then expected EVA is equal to
E(EVA) = (1/2b){max[(S + θ)− rασε, 0]}2. (6)
Proof. See the appendix.
This proposition shows that expected EVA is a convex (quadratic) function of managerial ability,
θ. The important implication of Proposition 1 occurs in the multiperiod setting. Since equation (6)
represents the expected EVA in the second period, any first-period policy that increases the variance of
the posterior estimate of θ has value to the firm.4 Figure 3 illustrates the results of proposition 1.
3 Learning and Dynamics
We now consider the previous analysis in the context of multiperiod learning about managerial abil-
ity. In the course of this development, we will explore the implications of the initial choice of capital4By iterated expectations the posterior estimate of managerial ability satisfies the conditions for a second-degree stochas-
tic dominance ordering.
9
θ
ExpectedEVA
−S + rασε
Figure 3: Expected EVA for a single period
on performance measurement for the second period. We also consider the functional form of manage-
rial compensation that is capable of implementing the optimal solution when investment choices are
delegated.
3.1 The Value of Learning
Consider the second period. The firm wants to maximize the sum of first and second period expected
EVAs. Proposition 1 gives the expected EVA for the second period conditional on the value of the
signal in the second period as well as the updated belief of the expected managerial ability.
We simplify matters for the second period by assuming that there exists only a single value for the
market signal in the second period, S2. If the manager’s updated ability, θp, at the end of the first
period is high, then clearly (6) indicates that the second period expected EVA is great. On the other
hand, if the manager’s updated ability, θp < −S2 + rασε, then expected second period EVA is equal
to zero, because even taking on a single project would have a negative EVA. Thus, for low levels of
managerial ability, the firm should essentially “shut down” the division.
One alternative to shutting down the investment activities of the division would be to fire the
10
incumbent manager and hire another whose ability level is unknown. However, this will not necessary
change the results. If there is competition in the sense that investment activities using the average
manager type have zero EVA in the subsequent period, i.e., if a new manager has expected ability θ0
and S2 + θ0 − rασε = 0, then this is not better than closing the divisional activities. In this section we
maintain this assumption. It is relaxed in section 5.
The main benefit of the zero EVA assumption upon replacement is that the evaluation of the second-
period expected EVA from the perspective of the prior period is very simple. As is demonstrated below,
the updated ability estimate of the manager, θp after first-period cash flows are observed is normally
distributed with expected value θ0. Moreover, because of symmetry, the evaluation of expected EVA is
particularly simple:
E(EV A2) =12b
∫ ∞
θ0
(θp − θ0)21√2πσp
e−(θp−θ0)2/(2σ2p)dθp = (1/4b)σ2
p . (7)
That is expected EVA is proportional to the prior variance on the estimate of managerial ability. This
means that any first-period policy that increases the uncertainty in the estimate will have greater
second-period value.
3.2 Ability Updating
In order to consider the consequences of learning on expected EVA for the second period, we need to
compute the Bayesian posterior distribution of the estimate of managerial ability after the first period,
θp. Since we assume that the signal is observable to the firm at time one, and the firm knows the number
of projects that were taken by the manager, N , the idiosyncratic project risks are exactly identifiable
from equation (1) as
εi = yi − S + bi. (8)
Denote the sample mean as ε =∑
εi/N .
The problem now is to determine the distribution of the estimate, θp from the prior and the obser-
vations of εi. It is well-known that as the data are normally distributed and the prior also normal, the
11
estimate will also be normally distributed. In fact,
θp =θ0(1/σ2
θ ) + ε(N/σ2ε )
(1/σ2θ ) + (N/σ2
ε ). (9)
Lemma 1 derives the resulting formula for the variance of the estimate.
Lemma 1. The expected value of the estimate of managerial ability is equal to E(θp) = θ0 and the
variance of the estimate is equal to σ2p, where
σ2p =
(N/σ2
ε
(1/σ2θ) + (N/σ2
ε )
)2(σ2
ε
N+ σ2
θ
). (10)
Further, σ2p as a function of N is increasing, with upper bound equal to σ2
θ and has slope equal to
dσ2p
dN=
1/σ2ε(
(1/σ2θ ) + (N/σ2
ε ))2 . (11)
Proof. The details of the calculations appear in the appendix.
Lemma 1 shows that the value of learning for the second period is an increasing function of the
number of projects accepted in the first-period. For N = 0 the marginal value to learning exhibited in
equation (11) is greatest while the marginal value declines as the number of projects selected increases.
As a result, σ2p approaches σ2
θ as N → ∞. These are very intuitive featues. Figure 4 illustrates theresults of lemma 1.
3.3 RAROC Interpretation
Originally pioneered by Bankers Trust and now widely used by many US Banks such as Bank of America,
the concept of RAROC has been a major development in the financial industry. RAROC is usually
represented as a ratio with net income divided by the capital allocation and the cost of capital then
subtracted as follows:
RAROC =net income
capital allocation− r. (12)
12
σ2p
σ2θ
0 N
Figure 4: The variance of the updated estimate, σ2p
Performance is judged to be superior to that obtainable through zero EVA investments when RAROC
is greater than zero.
The ability updating procedure in the model can be readily interpreted in light of this RAROC
definition. Compare equation (9):
θp =θ0(1/σ2
θ ) +∑
εi(1/σ2ε )
(1/σ2θ ) + (N/σ2
ε ). (13)
For a diffuse prior variance, σ2θ → ∞,
θp →∑
εi
N=∑(yi + bi − S)
N.
Optimal capital raised in the second period, from (5), is given by
C2 = αN2σε =[S2 + θp − rασε
b
]ασε, (14)
as long as C2 ≥ 0. Since next periods capital is a linear function of θp, the following proposition shows
that capital allocated for the second period is a linear function of RAROC in the first period.
13
Proposition 2. For a diffuse prior variance of managerial ability, and for θp ≥ θ0, optimal second
period capital allocation, C2 is given by
C2 =[(RAROC)ασε + S2 + (b/2)N − S
b
]ασε. (15)
Proof. Rewriting (13) gives
θp =∑
yi
N+ (b/2)N − S.
We can interpret RAROC in our model as
RAROC =∑
yi
C− r;
therefore since C = αNσε,
θp = ασε(RAROC + r) + (b/2)N − S.
Substituting into (14) then yields
C2 =[S2 + (RAROC + r)ασε + (b/2)N − S − rασε
b
]ασε.
Cancelling the terms in the numerator involving r yields equation (15).
Proposition (2) shows that conditional on the first-period signal, second period capital is linear
in RAROC. The proposition also shows that in order to determine exact capital, RAROC should be
adjusted (indexed) by the ex ante signal, S, as well as the depreciation schedule of project rankings. Of
course second period investment opportunities, as embodied in S2 are relevant as well.
3.4 First-best Optimum
The multiperiod problem with dynamics and learning is therefore equivalent to:
max{N |N≥0}
(S + θ0)N − bN2
2− rαNσε + (1/4b)σ2
p(N). (16)
14
S0
First-best N∗
Single-period N
Optimal N
Figure 5: Optimal project selection in the multiperiod model
The first three terms above represent a concave function of N ; Lemma 1 shows that the remaining
term is also concave. Therefore the first-order condition characterizes the optimum. Using (11) and
differentiating in (16) we then can derive the following proposition for the optimum scale of project
activity in the multiperiod problem.
Proposition 3. The optimal scale of investment activity in the multiperiod problem is characterized
by the function N∗(S), which is the solution to the following equation.
S + θ0 − bN − rασε +(1/4b)(1/σ2
ε )((1/σ2
θ ) + (N/σ2ε ))2
= 0. (17)
Examining the terms in (17) reveals that the only term involving σθ is the final term, i.e., the benefit
of learning for the second period. In general it is easy to see the the benefit of learning is increasing
in the ratio σθ/σε. That is, the greater is the amount of uncertainty in the manager’s true ability as
compared to the amount of noise in the cash flows conditional on the signal observation.
Figure 5 illustrates the solution to problem (17), which we denote as N∗(S). For comparison, the
single period solution from (5) is depicted as N . Notice that in the case of very negative signals the
15
optimal single period investment decision would be to invest not at all. This occurs because the EVA
is negative for all projects. On the other hand, the multiperiod investment solution is positive because
of the value of learning about managerial ability. In the example depicted in figure 5 the negative EVA
is completely overcome by the ability to utilize specific managerial ability information to optimal select
the amount of capital for investment in the second period. As the first period signal improves, then it
becomes optimal to invest in a subset of the feasible projects even in a single period situation. However
even then learning still adds a set of incremental projects that, by themselves would have negative EVAs.
It is also true that the value of learning decreases marginally with the number of projects selected in
the first period. This causes the difference between the first and multiperiod optima to converge as the
initial signal improves.
4 Compensation and Agency
We now describe the nature of compensation contracting between the manager and firm in a setting
where decisions on the level of risk-taking are delegated to the manager.
4.1 Contracting Imperfections
As mentioned before, in the absence of any contracting imperfections between the institution and its
manager the firm could simply offer a fixed wage payment in both periods to the manager independent
of the signal and the manager would then voluntarily report the correct signal to the firm. The firm
then raises the optimal amount of capital and the manager undertakes the first-best level of risk in the
initial period. After the first period, the firm revises the amount of capital in the balance sheet and a
new investment decision is made.
The use of fixed wage payments to induce optimal behavior on the part of the manager is, however,
not consistent with real-world imperfections on the contracting environment. Therefore we consider two
main limitations to the firm’s ability to enforce fixed wage payments to the manager ex ante.
The first impediment considered deals with the feasible set of second-period contracts offered to
the manager. We assume that the manager will not commit to a second-period contract that does not
reflect the updated information about his ability. Specifically, from proposition 1, the manager’s second
16
period wage payment is proportional to the EVA in the second period,
w2(S2, θp) = (γ/4b){max[(S2 + θp)− rασε, 0]}2. (18)
This embodies the assumption that the manager must receive some fraction, γ, of the EVA benefits
that he can generate in the subsequent period.
The second contracting impediment considered deals with the nature of first-period contingent con-
tracts. We assume here that even though the signal is observable ex post the firm cannot impose
contracts contingent on the signal to the manager without his consent. This assumption is motivated
by the intertemporal nature of signal observations. If this assumption were not adopted, the firm could
essentially employ forcing contracts against the manager to costlessly coerce him into revealing the
signal ex ante. In the real world signals are generated by possibly complex and proprietary information
technologies. Managers would not be willing to enter commitments unless they are compensated by
receiving some rent for the use of such technologies. The fact that this assumption is made does not
preclude the use of contingent contracts. As we show below, the manager may agree to them as long as
he receives compensation equal to what he would if non-contingent contracts were utilized.
In addition to the above two major assumptions, we also assume that the manager must be paid at
least some minimum retention level at the end of the first period, independent of the signal realization.
We denote this minimum wage as w. To determine the reservation utility that the manager must be
given in order to participate, we consider the manager’s delegation problem given that non-contingent
contracts are used in the first period. In this case, the manager selects the desired risk level. Let w1(N)
denote the manager’s first period compensation depending on the number of projects selected. Then
the manager’s delegation problem is as follows:
maxN
w + w1(N) + E((γ/4b){max[(S2 + θp)− rασε, 0]}2)
= w + w1(N) + (γ/4b)σ2p(N), (19)
using equation (7).
17
4.2 The Impact of Delegation
First consider what would happen if a constant wage in the first period were paid to the manager.
In this case with w1(N) = 0 the manager would maximize σ2p(N), i.e., would choose N = ∞. This
occurs because of the second-period lack of contract commitment. Convexity causes the manager to
always benefit from the highest possible variance in his updated estimate, which always occurs when
the estimate is as close as possible to the true value.
Looking at the objective function of the manager (19) it is apparent that since the signal, S, does
not enter, the only way for the firm to induce the manager to select a signal-contingent risk level, such
as the first-best N∗ of the previous section, is if the manager’s two-period utility is independent of the
state, in which case the manager is free to do what is in the interest of the firm.
Since first-period compensation to the manager is bounded below by w, w1(N) ≥ 0, and the σ2p(N) <
σ2θ as long as the manager’s choice of risk level is unrestricted, the firm must provide the manager with
compensation equal to
w1(N) = (1/4b)γσ2θ − (1/4b)γσ2
p(N). (20)
Substituting (20) into (19) gives the manager’s two period utility as:
U = w + (1/4b)γσ2θ − (1/4b)γσ2
p(N) + (1/4b)γσ2p(N) = w + (1/4b)γσ2
θ . (21)
Equation (21) says that because the firm cannot force the manager into contingent contracts, the
manager’s reservation utility must be equal to the minimum wage level in the first period plus the
maximum potential value to him of learning about his ability in the second period. Therefore the firm
essentially must give up the manager’s share, γ of learning rent in the delegation problem.
Notice however, that because the manager’s utility is independent of the signal, the firm’s optimal
investment decision, N(S) will be equal to the first-best level. That is, the firm solves the following
problem for every signal realization, S:
max{N |N≥0}
(S + θ0)N − bN2
2− rαNσε + (1/4b)σ2
p(N)− U. (22)
18
Since U is independent of S, this has the same solution as equation (17), which is the first-best optimum
with learning.
Therefore we have shown that there is no impact on the optimal capital allocation decision due to
delegation. Naturally this follows from the fact that in this delegation problem there is always symmetric
information and no moral hazard problems. However even though no distortion is introduced in the
investment decision, delegation is costly to the firm, as the manager’s reservation utility level is higher
than the absolute minimum.
4.3 Contingent Contracts
The previous subsection showed that in order to implement a signal-contingent investment policy the
firm could neutralize the incentive to take excessive risk by introducing first-period compensation that
is a negative function of risk. The question we next investigate is whether the firm can duplicate the
optimum level of risk-taking by utilizing a contingent contract.
Even though a contingent contract that meets the manager’s reservation utility constraint cannot
do strictly better than a non-contingent wage, it may be interesting if it is able to do just as well,
nevertheless. The reason is that the firm may wish to encourage the manager to improve the signal
distribution ex ante, for instance through expenditure of unobservable effort. Although this is not
formally considered in the model, it is clear that the use of a non-contingent contract can never provide
any incentives in this respect.
Hence we now derive a first-period wage contract contingent on the state that implements the
first-best risk policy in a delegation problem. Consider the following contingent contract:
w1(S,N) = w1(S) + γ[(S + θ0)N − bN2
2− rαNσε], (23)
where
w1(S) = w + (1/4b)γσ2θ − (1/4b)γσ2
p(N∗S)− γ[(S + θ0)N∗(S)− bN∗(S)2
2− rαN∗(S)σε]. (24)
19
Using (23) then the manager solves the following delegation problem using the contingent contract:
max{N |N≥0}
w1(S) + γ[(S + θ0)N − bN2
2− rαNσε] + (1/4b)γσ2
p(N), (25)
which clearly yields the first-best optimum, N∗(S). The additional wage paid to the manager in the
first-period, w1(S), from (24) is chosen so that the manager’s two-period utility equals the reservation
utility constraint of (21) at the first-best optimum.
These results lead immediately to the following proposition.
Proposition 4. The optimal dynamic capital allocation scheme can be implemented by a contingent
contract based on a constant share of first-period EVA plus an additional signal-contingent wage payment.
4.4 Position Limits
There is considerable interest in whether risk should be controlled at the decentralized level using
“flexible” incentive contracts such as EVA or through “hard” position limits that specify the maximum
utilization of risk. We demonstrate below that a combination may be optimal from the standpoint of
the institution.
We noted above that the optimal non-contingent or contingent contract allows the manager to
extract rent whenever it is important to dissuade him from excessive risk-taking. This means that the
firm not only gives up the manager’s share of learning ability at the optimal investment choice, but also
a share of the maximum potential value. As this is costly to the firm, we now investigate the use of
position limits in addition to contingent contracting based on EVA.
Going back to equation (21) we see that if the firm decides ex ante to limit access to capital to some
maximum amount, N ≤ N , then it has the potential to eliminate some of this rent extraction by the
manager, at the cost of distorting investment policy for some signals. The question is whether the use
of position limits can be optimal.
The use of position limits can be modeled as an upper bound constraint, N ≤ N added to the
delegation problem (25). Notice that if the firm combines position limits with first period EVA-based
contingent compensation, then if position limits are binding they will be so only for higher signals.
20
If position limits are employed, it is easy to see that the manager’s reservation utility condition (21)
is replaced by the following
U = w + (1/4b)γσ2p(N). (26)
and therefore the firm’s optimal capital allocation problem becomes
max{N,N |0≤N≤N}
E[(S + θ0)N − bN2
2− rαNσε + (1/4b)σ2
p(N)]− w − (1/4b)γσ2p(N ), (27)
where the expectation is taken over all values of the signal S ∈ [0, S].Looking at equation (27) it is apparent that the benefit of imposing a strict position limit is that
managerial rent consumption due to excessive risk-taking is curtailed for all signals, S. The cost of
imposing limits is that there is underinvestment in projects that may either provide benefits in terms
of learning about managerial ability or in terms of foregone EVA. But because the costs are expended
only over high signal realizations, but the benefits are enjoyed over all signal realizations, it will always
be optimal to employ some form of position limits in optimal delegation scheme. This intuition is
formulated in the following proposition.
Proposition 5. The optimal linear contingent compensation scheme for the manager employs a fixed
fraction of first period EVA and a strict position limit that binds for a non-empty set of signals, S ∈[S, S].
Proof. We have already seen that whenever the upper bound contstraint in (27) is neglected, the optimal
number of projects is a strictly increasing function, N∗(S). Therefore if it is optimal to introduce a
constraint, the constraint will bind only for S ∈ [S, S], where S ∈ (0, S). Thus it suffices to prove thatit will always be optimal to introduce the constraint. Suppose that the constraint were not introduced.
This implies that N = N∗(S). Consider the first order condition for N in (27) evaluated at S. This
first order condition is
(S + θ0)− bN − rασε + σp(N)dσp(N)
dN− (1/4b)γσp(N)
dσp(N )dN
= 0− (1/4b)γσp(N)dσp(N )
dN< 0,
21
since the first four terms are zero at N = N∗(S). Since the first order condition evaluated at S is
negative, this implies that it is optimal to reduce N below N∗(S), in which case the constraint becomes
binding for some signals.
5 Termination and Competition
This section of the paper considers the implications of replacing the manager after the end of the first
period based on the updated ability estimate. This sets the stage for considering the capital allocation
problem with multiple divisions.
5.1 Threat of Firing
Previously we have assumed that if the manager were fired after the end of the first period, he would
be replaced by a manager with ability θ0 such that the value of investing in a single project in the next
period is zero, i.e., S2 + θ0 − rασε = 0. Therefore the firm may just as well not replace the manager
and eliminate the investment activity in the second period. This event occurs whenever the manager’s
updated ability θp < θ0.
Now we assume that the second period continuation value of investing is positive, in the sense that
competition between firms does not deplete the value of recruiting a new manager. Henceforth, we
assume that S2 + θ0 − rασε > 0. In this case the manager is replaced whenever θp < θ0.5 When the
manager is terminated, it is assumed that the manager is unable to find substitute employment with
another firm and therefore earns nothing in the second period. Otherwise, he earns, as before a fixed
fraction, γ, of the EVA generated. Figure 6 illustrates the second period EVA to the firm and to the
original manager. Notice that in the firing states (θp < θ0) there is a difference in the proportional
sharing between the firm and the manager as the difference is paid to a new manager.5The question of whether this is an optimal termination policy with commitment by the firm is considered later in this
section. It is clear that this policy is optimal without commitment.
22
θp
ExpectedEVA
Manager’s lossdue to firing
−S2 + rασε θ0
Figure 6: Second period EVA when termination is feasible
5.2 Optimal Investment Activity
The analysis proceeds in essentially the same was as in the previous subsection; we first characterize the
second-period expected EVA benefits as a function of the updated ability distribution and then solve
for the reservation utility of the manager and the optimal decision of the firm.
In the previous analysis on learning the benefit of learning was proportional to the variance of the
updated ability distribution, σ2p. This limits, to some degree, the upside potential of learning. With
optimal termination, the potential for learning is increased and will be a positive function of the EVA
generated in the future, S2 + θ0 − rασε. Lemma 2 evaluates the expected EVA under the optimal
termination and is the analogue to equation (7).
Lemma 2. Expected second-period EVA using the optimal termination policy is equal to
E(EV A2) =12b
[12σ2
p +2√2π(S2 + θ0 − rασε)σp + (S2 + θ0 − rασε)2
]. (28)
Proof. See the appendix.
23
Notice that equation (28) shows that the positive second-period EVA from marginal projects asso-
ciated with the new manager becomes important now in two ways. First, it raises the level of EVA
for all choices of first period project activity levels, N . Second, it multiplies the standard deviation of
the ability estimate, σp, which we have seen is an increasing function of N . Comparing (7) to (28) it
is apparent that the benefit from increased risk taking in the first period is greater due to enhanced
learning potential when the manager can be replaced with a better manager.
Although the expression for second period EVA has been derived above, when it comes to analyzing
the optimal decision of the firm in the first period, we must consider only the remaining EVA left after
the original manager has been compensated as well as the new manager.
Focus first on the utility of the original manager. As before, we assume that the firm cannot
enforce contingent contracts without the agreement of the manager. Using contracts contingent only
on first-period investment activity, we obtain
U = w + w1(N) +γ
2b
[12σ2
p +2√2π(S2 + θ0 − rασε)σp +
12(S2 + θ0 − rασε)2
]. (29)
Note importantly in the last term above that because the manager is fired and does not earn anything
in certain states in the second period, he does not share in the EVA earned by the firm in those states.
It then follows that the firm must provide disincentives to the manager for excessive risk-taking in the
form of a first-period compensation that is decreasing in the amount of risk taken. In fact if position
limits are not employed,
w1(N) =γ
2b
[12(σ2
θ − σ2p) +
2√2π(S2 + θ0 − rασε)(σθ − σp)
]. (30)
To evaluate the objective function of the firm, we must subtract the utility of the original manager
as well as the expected payments made to the new manager in the second period. Therefore the firm’s
24
objective function becomes
V (N,S) = (S + θ0)N − bN2
2− rαNσε + E(EV A2)− w
− γ
2b
[12σ2
θ +2√2π(S2 + θ0 − rασε)σθ +
12(S2 + θ0 − rασε)2
]. (31)
Hence this leads directly to the following proposition showing that the firm maximizes once again the
sum of first and second period EVA as before.
Proposition 6. When the firm can terminate the manager at the end of the first period, the optimal
investment policy satisfies the following first-order condition:
S + θ0 − bN − rασε +12b
(σp(N) +
2√2π(S2 + θ0 − rασε)
)dσp
dN, (32)
where σp(N) and dσp/dN are given by Lemma 1.
Proof. Differentiate equation (31) using Lemma 2.
Comparing propositions 3 and 6 it is easy to see that with managerial termination there will be
greater levels of initial investment activity according to the optimal schedule. This is because the
benefit of learning is greater. Importantly the value of learning is an increasing function of the extent
of value creation in the second period, S2 + θ0 − rασε.
The firm can implement the solution to proposition 6 in the same way as when replacement had
zero value. Using contracts contingent on the signal, the signal-contingent part of the manager’s first-
period compensation should be a fixed fraction of first-period EVA; he will then identify the optimum
of proposition 6 because his second-period compensation involves an equal share of second-period EVA
minus some non-contingent amount that will not distort his incentives.
Also, as before, if the firm implements strict position limits, the rent to the manager from excessive
risk-taking can be reduced. This causes underinvestment relative to optimal solution of proposition 6
only for certain high signal realizations.
25
5.3 Optimality with Commitment
The analysis with termination was conducted under the presumption that the manager would be fired
if and only if his updated ability level was less than that of a newly recruited manager which in turn
is equal to the prior estimate for the original manager. It is of interest to explore whether some sort of
commitment on the part of the firm might improve the welfare of its shareholders. The rent loss to the
manager occurs as result of the need to neutralize the excessive risk-taking potential.
The most direct form of commitment would be for the firm to adopt a policy of termination based
on the choice of N in the first period. Such a policy would not be immune to renegotiation, since the
firm knows that whenever θp ≥ θ0 it is better off by sticking with the incumbent. Thus this is something
that cannot reasonably be committed to. A better choice of termination policy would be to adopt a
threshold θ specified ex ante so that the manager is terminated iff θp < θ. This could be interpreted
as a threshold level of performance and could be built into the performance evaluation contract of the
manager. This form of commitment is never something that the manager would regret ex post.
Note that under the policy of termination with θp < θ0, the manager knows that regardless of the
choice of investment activity, his probability of termination is always 1/2. Thus, he wants to maximize
his upside payoff by choosing a high risk level. If θ > θ0 the probability of termination is greater than
1/2. However this causes an even greater propensity to take risk. This is easy to see, since if the variance
of the estimate, σ2p is small, the probability of termination is almost 1 and so the manager loses nothing
by gambling on retention.
The result is different when θ < θ0, i.e., when the threshold for retention means that the probability
of termination is less than 1/2. Now it is possible that this type of firing threshold limits the degree
of risk-taking by the manager. The reason is as follows. Here for small increases in σ2p near zero,
utility increases because the manager is unlikely to be fired and is operating on the “convex” part of
the expected payment schedule. However then for a range of risk increases, the loss of salary associated
with firing becomes important and the manager is worse off. For very high σ2p levels, this result reverses
and utility again increases with higher levels of risk. However since σ2p is bounded by σ2
θ , which is finite,
it is possible to design an appropriate firing threshold level so that the manager has the potential to take
on only a limited amount of risk. This is illustrated in figure 7, where the 95% confidence interval, 2σθ
26
θp
ExpectedEVA
2σθ
θ0θ
Figure 7: Optimal Firing Threshold
indicates the maximum risk that can be attained. In this example, it is possible for the firing threshold
to be set so that rent to the manager from excessive risk-taking is limited.
Therefore we have shown that the firm should give the manager a limited amount of “downside”
protection against termination. Interestingly this result contrasts with Heinkel and Stoughton (1994)
where it was shown that the manager would be provided with a hurdle to overcome. The reason for the
difference is due to the asymmetric information problem considered in the aforementioned work.
6 Implications
The paper has a number of significant implications for the design of incentives for modern financial
institutions. These are enumerated below.
6.1 Learning increases the optimal extent of risk taking
We showed that with the possibility of learning about managerial ability in a dynamic environment,
the firm would optimally induce the manager to take on a greater number of projects than in a single
27
period environment. Alternatively this would imply that the time interval over which the manager is
evaluated should be extended to provide the opportunity for more cash flow observations. Hence, at
the margin, the last few projects may have negative EVAs as viewed from a traditional perspective.
However this is overcome by their benefits in terms of providing more information about managerial
abilities.
6.2 RAROC is related to the estimate of managerial ability
We interpreted RAROC—a comparison of the ratio of net income to capital allocation to the cost of
capital—as measuring the posterior estimate of managerial ability. The reason for this interpretation is
that the magnitude of cash flow realizations should be weighted by the number of observations, which
is equivalent to the amount of risk capital employed.
6.3 RAROC should be indexed
In the model the signal of the manager is observable ex post. It is clear that cash flows net of this signal
should be used as the means to evaluate the manager’s ability. This provides justification for the widely
accepted practice of indexing project outcomes in order to better assess managers.
6.4 Learning creates a risk-incentive problem
We indicated that when the firm and division manager are in an imperfect contracting environment
where there are limitations to commitment, learning creates a convexity in the payoff function to the
manager. As a result, he will have a tendency to adopt extremely risky positions unless counteracted
through the incentive contract. Therefore we showed that if non-contingent contracts are used, the
initial period payment is decreasing in the extent of risk undertaken.
6.5 EVA-based compensation is optimal
We demonstrated that the firm will be successful in implementing the optimal risk investment level
through a contingent contract that is based on first period EVA of the division. Hence the current
interest on using EVA in delegation environments is well-founded.
28
6.6 Position limits improve shareholder value
Position limits may be combined with EVA compensation in order to reduce the amount of rent that is
appropriated by the manager through the risk incentive problem. This justifies the widespread use of
such hard constraints in practice, coupled with softer incentive programs.
6.7 Termination
We showed that lack of commitment prevents the firm from using the threat of termination to limit
the agency losses in the delegation environment. However with limited commitment, such as with the
ability to enforce a comparison standard, the firm can benefit. Such a comparison standard gives the
manager a small amount of “downside” protection.
6.8 Conclusion
The major conclusion of our paper is that the use of outright EVA compensation related to shareholder
value creation must be combined with performance measurement based on RAROC in order to provide
the right incentive system in a dynamic environment.
29
A Appendix
A.1 Proof of Proposition 1
For N ≥ 0, substitute the first-order condition (5) into the objective function (4) to get the following
sequence of equalities:
E(EVA) = (S + θ)[S + θ − rασε
b
]− b
2(S + θ − rασε)2
b2− rασε
(S + θ − rασε)b
= (1/2)(S + θ)2
b− (1/b)(S + θ)(rασε) + (1/2b)(rασε)2
=12b[(S + θ)− rασε]2.
A.2 Proof of Lemma 1
We want to calculate the variance of θp, which is denoted by σ2p. Notice from equation (9) that
σ2p =
(N/σ2
ε
(1/σ2θ ) + (N/σ2
ε )
)2
σ2ε , (33)
where σ2ε denotes the variance of ε. The following formula for the conditional variance is well-known.
σ2ε = E[σ2
ε |θ] + var[E(ε|θ)]
= E[σ2ε /N ] + var[θ]
= (σ2ε /N) + σ2
θ . (34)
Substituting (34) into (33) yields equation (10) in the text.
It is straightforward to see that σ2p = 0 for N = 0. We now consider what happens as N → ∞. For
large N ,
σ2p → (σ2
ε /N) + σ2θ → σ2
θ .
30
Finally consider the derivative of σ2p with respect to N :
dσ2p
dN= 2
(Nσ2
θ
σ2ε +Nσ2
θ
)1
σ2ε +Nσ2
θ
(σ2θ(σ
2ε +Nσ2
θ)− Nσ2θσ
2θ)((σ
2ε /N) + σ2
θ)−(
Nσ2θ
σ2ε +Nσ2
θ
)2σ2
ε
N2
=(
Nσ2θ
σ2ε +Nσ2
θ
)[2σ2
θσ2ε
(σ2ε +Nσ2
θ)2((σ2
ε /N) + σ2θ)−
Nσ2θ
σ2ε +Nσ2
θ
(σ2ε /N
2)]
=(
Nσ2θ
σ2ε +Nσ2
θ
)[2σ2
θσ2ε
N
(σ2
ε +Nσ2θ
σ2ε +Nσ2
θ
)− σ2
θσ2ε
N
]
=σ4
θσ2ε
(σ2ε +Nσ2
θ)2.
Equation (11) in the text is equivalent to the latter equation.
A.3 Proof of Lemma 2
The expression for expected EVA in this case is
E(EV A2) =12b
(∫ ∞
θ0
(S2 + θp − rασε)21√2πσp
e−(θp−θ0)2/(2σ2p)dθp + (1/2)(S2 + θ0 − rασε)2
). (35)
Let x = −S2 + rασε and note that the integral above can be expressed as follows:
∫ ∞
θ0
(θp − x)21√2πσp
e−(θp−θ0)2/(2σ2p)dθp
=∫ ∞
θ0
(θp − θ0)21√2πσp
e−(θp−θ0)2/(2σ2p)dθp
+∫ ∞
θ0
2(θp − θ0)(θ0 − x)1√2πσp
e−(θp−θ0)2/(2σ2p)dθp
+∫ ∞
θ0
(θ0 − x)21√2πσp
e−(θp−θ0)2/(2σ2p)dθp
= (1/2)σ2p + 2(θ0 − x)
∫ ∞
θ0
(θp − θ0)1√2πσp
e−(θp−θ0)2/(2σ2p)dθp + (1/2)(θ0 − x)2.
31
To evaluate the remaining integral above, notice that
∫ ∞
θ0
(θp − θ0)1√2πσp
e−(θp−θ0)2/(2σ2p)dθp
= − σp√2π
∫ ∞
θ0
d(e−(θp−θ0)2/(2σ2
p))
= − σp√2π(0− 1)
=σp√2π
.
Using these identities, we can determine expected EVA as
E(EV A2) =12b
[(1/2)σ2
p + 2(θ0 − x)σp√2π+ (1/2)(θ0 − x)2 + (1/2)(S2 + θ0 − rασε)2
].
Equation (28) is obtained from this expression after substituting for x.
32
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