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Working Paper Series
Banker Compensation and Bank Risk
Taking: The Organizational Economics
View
WP 13-03
Arantxa Jarque
Federal Reserve Bank of Richmond
Edward Simpson Prescott
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Banker Compensation and Bank Risk Taking:
The Organizational Economics View
Arantxa Jarque and Edward Simpson Prescott∗
Federal Reserve Bank of Richmond
March 4, 2013
Working Paper No. 13-03
Abstract
Models of banks operating under limited liability with deposit insurance and employee incentive problems are used to analyze how banker compensation contracts can
contribute to bank risk shifting. The first model is a multi-agent, moral-hazard model,
where each agent (e.g. a loan officer) operates a risky lending technology. Results differ
from the single-agent model; pay for performance contracts do not necessarily indicate
risk at the bank level. Correlation of returns is the most important factor. If loan
officer returns are uncorrelated, the form of pay is irrelevant for risk. If returns are
correlated, a low wage causes risk. If correlation is endogenous, relative performance
contracts that encourage correlation of returns can create bank risk. A sufficient condition for a contract to induce risk at the bank level is provided. The second model
adds a loan review and risk management function that affects risk characteristics of
loan officers’ loans. Counter to common perception, paying loan reviewers and risk
managers for performance does not necessarily create risk. The model also identifies
the importance of evaluating the quality of bank controls as a means for limiting bank
risk.
Keywords: incentive compensation, bank regulation, risk shifting
JEL Codes: D82, G21, G28, J33
∗
Earlier versions of this paper were presented at the Bank of Canada, the Federal Reserve Bank of St.
Louis, the 2010 Econometric Society World Congress, and the 2010 SAET meetings. The views expressed
here are those of the authors and not necessarily those of the Federal Reserve Bank of Richmond or the
Federal Reserve System.
1
1
Introduction
This paper uses organizational economics to analyze the connection between banker compensation and bank risk. In a framework where a bank operates under limited liability with
deposit insurance, a sequence of multi-agent, moral-hazard models is developed. Contracts
that implement both safe and risky lending decisions by a bank’s employees are provided.
These contracts are then used to address the question of what characteristics of compensation
arrangements encourage risk-taking behavior.
Controlling bank risk via regulation of compensation arrangements is a new focus of bank
regulation. The Federal Reserve Board in 2010 issued supervisory guidance to banks that
their compensation arrangements “Provide employees incentives that appropriately balance
risk and reward” (Federal Register, 2010). Similarly, the Dodd-Frank law requires that regulations be written that prohibit incentive-based compensation that encourages inappropriate
risks.1 These regulations are motivated by the belief that bank compensation practices were
a significant contributory factor to the recent financial crisis (e.g., Financial Stability Forum
(2009)).
Conceptually, there are two classes of people in a bank who could materially contribute
to the risk of a bank. The first is an individual, like a CEO or some traders, whose individual
decisions can materially affect the bank’s performance. The second is a group of individuals like loan officers whose decisions together can have a significant impact on the bank’s
performance.
This paper analyzes the second class of people. We do this for two reasons. First, a
CEO is limited in his ability to directly control the actions of his subordinates. Instead, he
has to rely on indirect methods, such as delegation of authority, usage of internal controls,
and compensation to direct the actions of subordinates. In the end, a bank’s risk profile is
determined by the actions of its lending officers and other employees. Second, despite the
high level of CEO pay, most labor compensation paid out by a bank goes to its employees,
so compensation regulations have the largest effect on them.
Our approach is to develop static, multi-agent, moral-hazard models where employees’
1
There is a precedent in that the Federal Deposit Insurance Corporation Improvement Act of 1991 gives
bank supervisors authority to prevent banks from paying “excessive” compensation, but this feature of the
Act is not directly connected with bank risk-taking.
2
interests are not necessarily aligned with those of bank equity or of society. The multi-agent
feature allows us to analyze the impact of compensation on bank risk when each individual
has a minuscule effect on the performance of the entire organization. As we show, this
factor alone implies that the degree to which pay depends on an individual’s performance
is not a primary factor in determining bank risk. The reason is that the idiosyncratic risk
of loan-officer loans averages out, regardless of how risky their individual loans are or how
they are paid. Instead, the main way in which loan-officer compensation affects bank risk is
through the degree to which it makes loan returns correlated. We then show how relative
performance contracts and monitoring mechanisms, like loan review, can increase or decrease
the correlation of loan returns.
One feature of compensation that we do not consider is how the timing of payments to
bankers affects bank risk. The timing of compensation is important, partly because regulations are pushing banks to use more deferred compensation that can be “clawed back,” that
is, reduced if the loan or project performs badly in the long run. We leave this feature out,
however, to focus on the connection between compensation and correlation of returns. For
work addressing the timing question using dynamic moral-hazard models with persistence,
see Jarque and Prescott (2010).
Section 2 reviews the literature on banker compensation. Section 3 develops a multiagent moral-hazard model where each agent has a small effect on the bank’s total return.
Section 4 analyzes the importance of correlation in returns. Section 5 studies monitoring
and bank controls. Section 6 concludes.
2
Literature
There are two theoretical literatures relevant for this paper. The first one is the literature
about the incentives of a bank to make inefficient investments. Both Merton (1977) and
Kareken and Wallace (1978) showed that, due to deposit insurance, banks have incentives to
take risky investment decisions in order to receive implicit transfers from the deposit insurer.
The large number of thrift failures in the U.S. S&L crisis of the 1980s is often considered to
be strong evidence of these incentives at work (e.g., White (1991)).
Much of the resulting theoretical banking literature on risk shifting incentives has focused
3
on the use of capital requirements or franchise value to mitigate this distortion. Examples
of papers along this line include Flannery (1989), Furlong and Keeley (1990), and Kim and
Santomero (1988).2 In this theoretical literature, management is implicitly assumed to act
in the interests of bank equity owners.
The second relevant literature assumes that management does not necessarily act in the
interests of bank equity owners. Jensen and Murphy (1990) is an early example of this
problem. However, only a few papers such as John, Saunders, and Senbet (2000), Phelan
(2009), and Bolton, Mehran, and Shapiro (2010) explicitly study this problem in the context
of banking.
In contrast, the empirical literature on compensation and risk shifting in banking is
bigger, but almost all of it looks at CEO compensation, mainly because of data availability.
One portion of this literature uses as its sample period the banking problems of the 1980s and
early 1990s. These papers often looked at proportions of pay that are variable, e.g., equity
based, and examined whether this is correlated with bank risk. Their findings were mixed.
For example, Houston and James (1995) studied commercial bank CEO pay from 1980-1990
and found that CEO compensation policies did not encourage excessive risk-taking in the
1980s. In contrast, Benston and Evans (2006) looked at a sample of highly levered banks
over the period 1988-1994 and found evidence of more use of short-term compensation, like
bonuses, at banks that failed. Using bank CEO data from 1976-1988, Crawford, Ezzell, and
Miles (1995) found some mixed evidence that highly levered banks increased incentives for
their CEOs, which is consistent with risk shifting.
Another portion of this literature looks at a sample period that includes the recent
financial crisis. Fahlenbrach and Stulz (2011) investigate whether the performance of banks
during the crises is related to incentives provided for their CEOs in the period leading up
to it. They find some evidence that banks with CEOs whose objectives were better aligned
with the interests of shareholders performed worse than other banks. Cheng, Hong and
Scheinkman (2010) find important and persistent differences across firms in the level of pay
and risk, with firms that took more risk also having the highest pay. Moreover, they find
that the firms that take more risks performed worse during the crisis period. The authors
2
Marshall and Prescott (2001, 2006) extend this work to allow more complicated capital structures that
include warrants and convertible debt.
4
also interpret their results as evidence that firms have different risk cultures. In a related
study, Balachandran, Kogut, and Harnal (2010) perform a similar analysis of the relation
between pay and risk and find a positive and significant correlation.
2.1
Empirical literature on non-CEO banker pay
There are very few studies of compensation of lower-level bank employees because this data
is proprietary. One exception is Agarwal and Wang (2009) who studied the results of an
experiment that was run at a bank, which for a period of time paid half of its small business
loan officers a wage and paid the other half with a wage plus an incentive. They found that
the incentive plan increased the loan approval rate by 47 percent and the size of loans by 45
percent. Unfortunately for the bank, the plan also increased the default rate by 24 percent,
so the plan was dropped.3
Berg, Puri, and Rocholl (2012) studied the data input behavior by loan officers who are
paid based on volume. These loan officers entered hard information, that is, non-judgmental
information, into a bank’s loan scoring system that determined approval. They find evidence of selective entering of hard information into the scoring system to improve a borrower’s chance of approval. Cole, Kanz, and Klapper (2011) ran laboratory experiments on
commercial bank loan officers where they varied the connection between compensation and
incentives. They found that the compensation structure had a large effect on lending and
the quality of the loans.
Finally, Hertzberg, Liberti, and Paravisini (2011) examined the connection between pay,
organizational structure, and reporting of information. They examined the use of loan officer
rotation at a large international bank and argued that it alleviates incentives to hide the
quality of poorly performing loans. In their analysis, the bank’s policies effectively tie pay to
loans under management, and they argue that the loan rotation along with career concerns
mitigate the incentive loan officers have to under report the risk in their portfolio of loans.
3
Interestingly, the incentive bonuses in the experiment were relatively small, only 15 percent of total pay,
and in the data most loan officers seemed to hit a volume amount that was labeled as 100 percent of the
bonus. The loan officers could earn more if they exceeded this volume, but earned proportionally less above
this threshold. Given the relatively small size of the bonuses, it may be that loan officers interpreted the
volume incentive as a target they had to reach rather than a pure commission type arrangement.
5
3
Loan Officer Compensation and Bank Risk
There is a bank that consists of depositors, equity holders, and a large number of loan officers.
There is a continuum of loan officers each of whom has an infinitesimally small effect on the
performance of the bank. Together, the size of the loan officers are measure one. Each loan
officer takes an action a ∈ A that produces a return r as a function of an idiosyncratic shock
and a common shock θ. The probability of a loan officer’s return is written f (r|θ, a). Both
shocks occur after the action is taken. There is a finite number of realizations of the common
shock, and the probability of shock θ is h(θ). There is also a finite number of possible returns
for each loan officer. Also, for most of the analysis there is a finite number of actions, though
in one subsection we allow for a continuum of actions.
A loan officer’s action and idiosyncratic shock are private information, while the common
shock is observed by the bank.4 A loan officer receives utility from consumption, c ≥ 0, and
action, a, of U (c) − V (a), where U is concave and increasing, U (0) ≥ 0, and V is increasing
and weakly convex. Each loan officer has an ex ante reservation utility level of Ū .
The bank finances the loan officers’ investment projects with an investment of size one.
The investment is financed by government insured deposits, 0 ≤ D ≤ 1, and equity 1 − D.
Because of deposit insurance, depositors receive the face value of deposits at the end of the
period no matter how the bank performs. For simplicity, we take the level of deposits as
given.5
The bank operates in the best interest of the equity holders, so we will often refer to the
bank and the equity holders interchangeably. The equity holders are treated as a single riskneutral principal with limited liability. The bank receives a total output of r̄(θ), which is the
sum of the loan officer’s returns, and pays out funds to depositors and compensation to loan
4
We could assume that θ is not observed by anyone, but as long as the mapping from a to the total return
is an invertible function, then θ could be identified from the contract. For that reason, we simply assume
that θ is public information.
5
The model can be extended to include franchise value, the value of a bank being a continuing concern. A
positive franchise value reduces risk-taking incentives because it is lost in the event of failure. The empirical
banking literature finds that franchise value has a significant impact on risk-taking. Keeley (1990) argued
that the low rate of bank failure pre-1980, before deregulation, was due to banks’ incentive to preserve the
positive franchise value that came with monopoly profits. The second half of the savings and loan crises is
often attributed to savings and loans institutions gambling for resurrection when they had negative franchise
value (e.g., White (1991)). Demsetz, Saidenberg, and Strahan (1996) find that franchise value is negatively
correlated with risk-taking in bank data from the 1990s. We leave franchise value out to keep the problem
simpler.
6
officers. The total compensation bill is c̄(θ). The principal’s utility is
P
θ
h(θ) max{r̄(θ) −
c̄(θ) − D, 0}.
The total return to the bank is the sum of the individual loan officers’ returns, which is
∀θ, r̄(θ) =
X
f (r|θ, a)r.
(1)
r
The bank gives each loan officer the same compensation schedule, c(r, θ), where r is the
return produced by a loan officer. The total compensation bill is then
∀θ, c̄(θ) =
X
f (r|θ, a)c(r, θ).
(2)
r
Finally, we assume that in the event of bankruptcy, depositors are paid before loan officers,
so if r̄(θ) < D then c(r, θ) = 0.
The problem for the bank is:
Bank Program
X
max
a,c(r,θ)≥0,c̄(θ)≥0,r̄(θ)
h(θ) max{r̄(θ) − c̄(θ) − D, 0}
(3)
θ
subject to (1), (2),
∀θ, c̄(θ) ≤ max{r̄(θ) − D, 0},
X
h(θ)
θ
X
θ
h(θ)
X
X
f (r|θ, a)U (c(r, θ)) − V (a) ≥ Ū ,
(5)
r
f (r|θ, a)U (c(r, θ)) − V (a) ≥
r
(4)
X
h(θ)
X
f (r|θ, â)U (c(r, θ)) − V (â), ∀â.
(6)
r
θ
Equation (4) limits total compensation to be less than bank revenue, net of payments to
depositors. Equation (5) is the participation constraint for a loan officer, and equation (6)
is the incentive constraint.
The piecewise linear objective function and the piecewise linear constraint, (4), make
this optimization problem non-differentiable. In order to derive results about compensation
from first-order conditions, we consider the subproblem of implementing a given action. For
each a, there is a set of bankruptcy states, Θ1 (a), where r̄(θ) ≤ D. Note that in these states
7
c(r, θ) = c̄(θ) = 0. There is also a set of solvency states, Θ2 (a), where r̄(θ) > D. These
latter states are those in which limited liability does not bind. Now consider the subproblem
of implementing action a and choosing c(r, θ) for θ ∈ Θ2 . This subproblem is
Bank Subprogram
X
max
∀θ∈Θ2 (a),c(r,θ)≥0,c̄(θ)≥0,r̄(θ)
h(θ)(r̄(θ) − c̄(θ) − D)
(7)
X
f (r|θ, a)r.
(8)
f (r|θ, a)c(r, θ).
(9)
θ∈Θ2 (a)
subject to
∀θ ∈ Θ2 (a), r̄(θ) =
r
∀θ ∈ Θ2 (a), c̄(θ) =
X
r
∀θ ∈ Θ2 (a), c̄(θ) ≤ r̄(θ) − D,
X
θ∈Θ1
h(θ)
X
h(θ)U (0) +
θ∈Θ1 (a)
X
h(θ)
X
θ∈Θ1 (a)
h(θ)U (0) +
(11)
f (r|θ, a)U (c(r, θ)) − V (a)
r
θ∈Θ2 (a)
X
f (r|θ, a)U (c(r, θ)) − V (a) ≥ Ū ,
r
θ∈Θ2 (a)
X
≥
X
h(θ)U (0) +
(10)
X
h(θ)
X
f (r|θ, â)U (c(r, θ)) − V (â), ∀â.
(12)
r
θ∈Θ2 (a)
Note that in the incentive constraint, the bankruptcy states on the right-hand side of (12) are
a function of a and not the deviating action â. The bankruptcy states are not defined by a
loan officer’s deviating action because in equilibrium all loan officers choose the recommended
action a and that determines the aggregate return and thus whether there is bankruptcy in
state θ.
The objective function and constraints in the subproblem are differentiable, so we can
use the Lagrangian multipliers to characterize an optimal compensation contract. Let ν(θ)
be the multiplier on (10), λ on (11), and µ(â) on (12). The first-order condition on c(r, θ)
gives
!
X
h(θ) + ν(θ)
f (r|θ, â)
=
λ
+
µ(â) 1 −
,
0
h(θ)U (c(r, θ))
f (r|θ, a)
â6=a
8
(13)
where λ ≥ 0 and µ(â) ≥ 0, and when c(r, θ) > 0.
There are two cases to consider. First, when 0 < c̄(θ) < r̄(θ) − D, ν(θ) = 0 and the
first-order condition is the same as in the standard moral hazard problem where consumption
decreases as the likelihood ratio increases. Second, when c̄(θ) = r̄(θ) − D ≥ 0, ν(θ) > 0, so
the upper bound on the total compensation bill reduces what the bank would pay out if this
constraint did not bind.
The subsequent analysis will consider the connection between compensation, c(r, θ), and
the risk profile of the bank. The analysis will also make frequent use of the likelihood ratio
in (13). Let LR(r, θ, â; a) be the likelihood ratio corresponding to the incentive constraint
where a is recommended and â is taken, that is,
LR(r, θ, â; a) =
4
f (r|θ, â)
.
f (r|θ, a)
The Importance of Correlation
In models with deposit insurance and limited liability, the bank’s preferences are not aligned
with those of society’s. The bank’s objective function is convex over profits and that can
make it willing to undertake negative net present value projects if enough of the downside
falls in the range where limited liability is binding. The more leveraged the bank is, the
stronger is this effect. In our problem, the bank must use compensation to induce riskaverse loan officers to take actions that generate risk to the bank, if that is indeed, what the
bank wants to do.
As we will see, there is not a direct mapping from the form of loan officer compensation to
bank risk. Compensation is important for risk at the individual level, but it is not necessarily
important for risk at the bank level. Instead, other factors, like correlation of returns, are
far more important. For example, as we will show, if loan officer risk is idiosyncratic, then
bank and social objectives are perfectly aligned.
When there is a common shock in addition to loan officer idiosyncratic risk, then the
limited liability distortions can affect bank decisions. In the following analysis, we work
through a sequence of functional forms of the production technology, f (r|θ, a), to study
connections between compensation and bank risk-taking. For each technology, we solve for
optimal contracts that implement actions that generate excessive risk and those that do not.
9
We then analyze what information an outsider would need to know to determine whether
the compensation arrangement generated excessive risk.
4.1
Uncorrelated Returns
Consider the extreme case where there is no correlation in loan officer returns. All risk is
idiosyncratic, so the gross return of the bank is a constant
∀θ, r̄ = r̄(θ) =
X
f (r|a, θ)r.
r
The bank’s return does not vary with the common shock θ. Consequently, the bank has
no risk. All that the form of compensation does is to determine the bank’s profits. Because
there is no variation in bank profits, the bank cannot gain anything from failing, since if it
fails it does so with probability one.
More formally, let r̄(a) be the bank’s gross return and let c̄(a) be the bank’s wage bill,
both as a function of a. The objective function for the bank is
max
max{r̄(a) − c̄(a) − D, 0}
a
while the objective function for society is
max
r̄(a) − c̄(a) − D.
a
As long as there exists an a such that bank profits are positive, a solution to the bank’s
problem is a solution to the society’s problem.
Proposition 1 When loan officer returns are uncorrelated, there is no connection between
the form of loan officer compensation and bank risk.
4.2
Perfectly Correlated Returns
Now consider the other extreme case, where loan officer returns are perfectly correlated. In
this case, the bank’s gross return does vary with θ and the bank may want to encourage its
loan officers to take on risk. Interestingly, loan officer compensation matters for risk, but in
a surprising way.
10
When returns are perfectly correlated, there is no idiosyncratic risk, so the bank can infer
a loan officer’s action from the common shock, θ, and the loan officer’s return r. Since the
bank essentially knows the action, it can pay each loan officer a wage if his return is what it
is supposed to be and zero otherwise. We assume that the zero payment penalty is enough
to induce the loan officer to take the recommended action. An alternative way of viewing
this contract – and the way we view it – is as a relative performance contract. Each loan
officer’s return is compared with that of everyone else’s. If his return is the same, he is paid
a wage. If it differs, he is paid zero.
The contract has strong incentives in it, but the incentives are not directly tied to his own
performance, but instead to how his performance compares with others. In equilibrium, loan
officers do not deviate, so what is observed is a compensation contract that is a wage that
does not vary with his return, though it may vary with the aggregate return if constraint
(10) binds. In particular, from the participation constraint, (11),
a = V −1 Ū −
X
h(θ)U (0) −
θ∈Θ1 (a)
X
h(θ)U (c̄(θ)) .
θ∈Θ2 (a)
The higher the compensation, the harder the loan officer works. Which effort level gives
the bank the best opportunity to exploit the safety net depends on the tradeoff between the
aggregate return and the aggregate wage bill. Indeed, it is possible that a bank pays a low
wage to increase its probability of failure as Figure 1 illustrates. The idea in that figure is
that the savings in wage payments increase the bank’s profits when it is successful and this
benefit outweighs the higher probability of failure, the cost of which, in any case, is borne
by the deposit insurer.
To see this more formally, bank profits given action a are
X
h(θ)(r̄(θ) − c̄(θ) − D)
θ∈Θ2 (a)
=
X
h(θ)(r̄(θ) − c̄(θ) − D) −
θ
=
X
X
h(θ)(r̄(θ) − c̄(θ) − D)
θ∈Θ1 (a)
h(θ)(r̄(θ) − c̄(θ) − D) −
θ
X
h(θ)(r̄(θ) − D),
θ∈Θ1 (a)
where the last equation holds because for θ ∈ Θ1 (a), c̄(θ) = 0. Substituting in for r̄(θ) and
11
c̄(θ), means that profits can also be written
X
θ
X
h(θ)(
f (r|θ, a)(r − c(r, θ)) − D) +
r
X
h(θ)(D −
X
f (r|θ, a)r).
r
θ∈Θ1
To simplify the notation, let E(r̄|a) be the expected return produced by the bank; let
E(c̄|a) be the expected compensation paid out by the bank; and let z(a) be the expected
value of the implicit transfers from the deposit insurer to the bank. Formally,
E(r̄|a) =
X
h(θ)
X
h(θ)
θ
z(a) =
f (r|θ, a)r,
r
θ
E(c̄|a) =
X
X
X
f (r|θ, a)c(r, θ),
r
h(θ)(D −
X
f (r|θ, a)r).
r
θ∈Θ1 (a)
This last term in bank profits, z(a), is sometimes referred to as the value of the deposit
insurance put option because the bank gets to put its losses onto the deposit insurer.
In the rest of this subsection, we make the following assumptions. First, we assume
that a is chosen from a continuum. This assumption is not essential, but simplifies the
analysis. Second, we assume that E(r̄|a) is differentiable and concave. This assumption is
standard and reflects diminishing returns in expected production. Third, we assume that
z 0 (a) < 0. This latter assumption will hold in many reasonable situations. For example, if
the probability distribution of the aggregate return is single peaked, the bankruptcy state
is to the left of the peak, and a shifts the distribution to the right, then the value of the
deposit insurance put option decreases with the action.
In terms of this new notation, the bank’s problem is
max
E(r̄|a) − E(c̄|a) − D + z(a),
a
and the optimal a satisfies
∂E(c̄|a)
∂E(r̄|a)
+ z 0 (a) =
.
∂a
∂a
At a social optimum, society takes into account that z(a) is a transfer. The social
optimum is the solution to
max
E(r̄|a) − E(c̄|a) − D,
a
12
so
∂E(r̄|a)
∂E(c̄|a)
=
.
∂a
∂a
Proposition 2 When loan officer returns are perfectly correlated, if E(c̄|a) is increasing
and convex, then the bank chooses an a that is less than the social optimum.
Proof: Follows directly from z 0 (a) < 0.
Normally, the monotonicity and convexity of expected pay would not be assumptions
requiring any justification. In this model, however, bankruptcy has an effect on the expected
wage bill that could violate these assumptions in certain extreme situations. In the absence
of bankruptcy, where loan officers receive a constant wage, the level of the wage is a convex
function of the effort level. However, with bankruptcy, it is possible that as a increases and
the number of bankruptcy states declines, the wage bill will drop (despite the higher effort)
because there are fewer states where the loan officers receive zero. This possibility would
seem mainly to be an issue when there is a high level of failure and increases in a lead to a
substantial marginal decrease in the probability of failure.
The inability to pay employees when there is failure is not in the traditional corporate
finance model of risk shifting and it illustrates an important point. As long as bank employees do not like risk, they have to be compensated to bear it, and that can make it more
expensive for a bank to take risk, which in turn reduces its incentive to exploit risk shifting.
Nevertheless, despite these costs, the bank will still not take the socially optimal a because of
the deposit insurance safety net factor z(a). In the case that we think is relevant most of the
time, namely, that the assumptions on expected compensation in Proposition 2 hold, loan
officers work less than is socially optimal and a bank fails more frequently than is socially
optimal. In this case, the compensation arrangement that encourages excessive risk is a low
wage!
4.3
Partial Correlation
When loan officer returns are partially correlated, there will be a trade off between the
identified effects. The regulator likes diversification because it reduces risk. The bank likes
correlation for two reasons, one bad and one good. The bad reason is that correlation allows
13
r()‐c()
D
r()
D+wage(al)
D+wage(ah)
Figure 1: Example of a bank that pays a low wage to increase bank risk. The variable wage(a)
is the wage paid to loan officers if a is taken and if the bank has positive profits. The solid
line that intercepts the x-axis is profits for the bank if al is taken and if r̄(θ) > D + wage(al ).
(For lower values of r̄(θ), either all the return net of deposits goes to workers or limited
liability binds.) The dashed line that intercepts the x-axis is profits if ah is taken and
r̄(θ) > D + wage(ah ). The solid curve is the density function of al and the dashed curve
is the density function of ah . For each density, the area under the curve to the left of D is
the probability of failure; it is much higher for al . Furthermore, wages are so large if ah is
taken that the bank gets little of the return in excess of D because so much of it goes to
loan officers. Consequently, the bank prefers to take al .
14
it to exploit the safety net. The good reason is that correlation relaxes the loan officer’s
incentive constraint, which reduces the cost to the bank of compensating the loan officers.
4.4
Identifying Risk-Creating Contracts
What these results imply for identifying contracts that generate bank risk depends on what
information is known about the bank. If a bank supervisor does not know the correlation
of bank returns, then a low wage may signal that the bank’s returns are highly correlated
and it is trying to shift risk to the deposit insurer. In contrast, if this supervisor knew the
correlation, then that information would be used in conjunction with the contract to assess
risk. For example, if correlation is low, then the supervisor need not worry about compensation because the bank has limited ability to do risk shifting. In contrast, if correlation
is high, then the supervisor needs to worry about risk shifting and should consider the details of the compensation contract. The supervisor will need to develop an understanding
of the technology, f (r|a, θ), and how the compensation contract may generate correlation of
returns.
In general, the relationship between compensation and risk mainly depends on the production technology, f (r|a, θ). However, a general simple test for risky compensation contracts
can be created directly from the incentive constraints if we assume that there are only two
actions for a loan officer to take, one that generates bank risk, al , and one that does not, ah ,
where V (al ) < V (ah ). The incentive constraint that induces a loan officer to take the risky
action is
X
θ
h(θ)
X
f (r, θ|al )U (c(r, θ)) >
r
X
h(θ)
X
θ
f (r, θ|ah )U (c(r, θ)) + V (al ) − V (ah ).
(14)
r
Because V (al ) − V (ah ) < 0, a sufficient condition to implement the risky action is
X
θ
h(θ)
X
r
f (r, θ|al )U (c(r, θ)) >
X
θ
h(θ)
X
f (r, θ|ah )U (c(r, θ)).
(15)
r
Equation (15) says that if the expected value of compensation – weighted by utility – is
higher for the risky action than for the safe action, then the loan officer will take the risky
action. It is not a necessary condition; there are compensation arrangements that do not
satisfy (18), but still induce the risky action. Nevertheless, equation (15) provides a simple
test that identifies a subset of the compensation arrangements that implement risk-taking.
15
This sufficient condition is useful because it does not use knowledge of effort disutility.
However, it still requires knowledge of the production function f (r|a, θ). The next section
works out the implications of two different specification of the production function.
4.5
Example Where Effort Affects Mean of Returns
In this specification, loan officer effort affects the mean of the return. Each loan officer can
take either al or ah , with 0 < al < ah < 1. There are also only two possible returns, failure
(r = 0) and success (r = 1). As before, θ is the common shock, though now it is restricted
to take on values between 0 and 1. Its mean is θ̄. The probability of success for a loan officer
is
f (r = 1|θ, a) = a(αθ̄ + (1 − α)θ).
(16)
The first term is the idiosyncratic component of the probability of success and the second
term is the common shock component. Notice that a loan officer’s expected return is aθ̄,
which does not depend on α.
Compensation is determined by the likelihood ratios. These are
â
,
a
1 − â(αθ̄ + (1 − α)θ)
.
LR(r = 0, θ, â; a) =
1 − a(αθ̄ + (1 − α)θ)
LR(r = 1, θ, â; a) =
Proposition 3 For the technology specified in (17), at an interior solution, consumption
for r = 1 does not vary with θ and consumption for r = 0 increases with θ.
Proof: Likelihood ratios comove with θ such that
∂LR(r = 1, θ, â, a)
∂c(r = 1, θ)
=0⇒
=0
∂θ
∂θ
∂LR(r = 0, θ, â, a)
∂c(r = 0, θ)
>0⇒
< 0.
∂θ
∂θ
Figure 2 illustrates the comovement of consumption with θ. Successful loan officers
receive a constant level of pay, while unsuccessful ones see their pay drop with the success of
the bank. The reason their pay drops is that failure is less likely when there is a high value
of θ.
16
c(r=1,θ)
c(r=0 θ)
c(r=0,θ)
θ
Figure 2: Optimal compensation in example where effort affects the mean of returns and
under the assumption of an interior solution. Note that θ̄ moves one for one with θ. Thus,
the x-axis also measure the total gross return of the bank.
17
Since output increases with θ, the comovement of compensation with θ directly corresponds to the comovement of total compensation with bank performance. This feature is
interesting because in investment banking a substantial portion of pay is often tied to performance of the bank or the line of business. For example, investment banks often decide
on and report on total compensation as a percentage of revenue.6 The share of revenue
distributed to loan officers – the only employees in this problem – is
W S(θ) =
r(θ)c(r = 1, θ) + (1 − r(θ))c(r = 0, θ)
r(θ)
for the range of consumption that is interior.
Proposition 4 describes the relationship for this technology.
Proposition 4 For the technology specified in this example and where consumption is interior
∂W S(θ)
< 0.
∂r(θ)
Proof: See the appendix.
In this example, at an optimum the employees’ share of income decreases with bank
performance.
4.5.1
Implication for regulation
In this example, the goal is to prevent the bank from taking the low action, which lowers the
mean of output and increases the probability of failure. For that technology, a pay structure
like that in Figure 2 would indicate a prudent compensation scheme. There are, however,
alternative technologies where actions could affect risk in different ways. For example, loan
officer actions could directly affect the correlation of the returns. And given our earlier
finding that correlation in returns is the big risk to the deposit insurer, we consider an
example where a loan officer’s action directly affects this variable.
6
There are other proposed explanations for this behavior, like sorting and retention of workers. See Oyer
and Schaefer (2005).
18
4.6
Example Where Effort Affects Correlation of Returns
The production technology is similar to the previous one, but where the loan officer chooses
the degree to which his loan is correlated with the bank’s performance. The loan officer
chooses the correlation of his loan by choosing either a low correlation loan, αl , or a high
correlation one, αh . The agent is not allowed to choose a, so for simplicity we drop it from
the technology. Formally,
f (r = 1|θ, α) = (αθ̄ + (1 − α)θ).
(17)
Recall that θ̄ = E(θ), so in this example the loan officer’s action does not affect the mean of
his return, but just its correlation with the performance of the firm. Finally, to reflect the
new choice variable, we write the utility function as
U (c) − V (α).
If the bank tries to implement the safe action, αh , then the likelihood ratios are
αl θ̄ + (1 − αl )θ
,
αh θ̄ + (1 − αh )θ
1 − (αl θ̄ + (1 − αl )θ)
LR(r = 0, θ, αl ; αh ) =
.
1 − (αh θ̄ + (1 − αh )θ)
LR(r = 1, θ, αl ; αh ) =
Proposition 5 For the technology specified in (17), at an interior solution consumption for
r = 1 decreases with θ and consumption for r = 0 increases with θ.
Proof: Likelihood ratios comove with θ such that
∂LR(r = 1, θ)
∂c(r = 1, θ)
>0⇒
< 0.
∂θ
∂θ
Similarly,
∂LR(r = 0, θ)
∂c(r = 1, θ)
<0⇒
> 0.
∂θ
∂θ
Figure 3 illustrates the comovement of consumption with θ. The consumption spread
between success and failure declines with bank performance. When a loan officer’s return is
high and the bank does poorly, the bank rewards the loan officer because his performance
is a signal that the loan officer’s return was not correlated with everyone else. In contrast,
19
c(r=0,)
c(r=1,)
Figure 3: Optimal contract that induces a loan officer to make a low correlation loan. Note
that θ̄) moves one for one with θ. Thus, the x-axis also measure the total gross return of the
bank.
if the loan officer does poorly and the bank does poorly then that is a signal that the loan
officer’s return is correlated, so the bank punishes him. In general, the bank wants to reward
the loan reviewer when the signal indicates no correlation and punish him when the signal
indicates correlation.
The effect of relative performance is so strong in this example that for θ ≥ θ̄ the optimal
contract is characterized by the loan officer being paid more for r = 0 than r = 1! This
result is specific to this example and need not be the case for technologies that also allow for
lower levels of effort that reduce the probability of success. We used this example to starkly
illustrate how correlation pushes contracts towards the use of relative performance.7
7
A guaranteed way to eliminate this result is to allow the loan officer to secretly destroy his own output
at no cost as in Innes (1990). This additional source of private information adds an incentive constraint that
takes the form ∀θ, c(r = 0, θ) ≤ c(r = 1, θ). The constraint imply that compensation is weakly monotonically
increasing in r, which is very appealing on empirical grounds. Optimal compensation would look similar to
that in Figure 3 for θ ≤ θ̄, but compensation would be a constant for θ > θ̄.
20
4.7
Relative Performance and Risk
The previous example provided a relative performance contract that reduced bank risk. Do
all relative performance contracts have this feature? The answer is no. The previous analysis
showed that an optimal contract that reduced risk is one that decreases pay for performance
– the spread between failure and success – as the bank’s overall performance improves. But
what kind of contracts would induce risk-taking? For the production function analyzed in
the previous example, the incentive constraint that implements the risky action simplifies to
X
h(θ) (θ̄ − θ)(U (c(r = 1, θ)) − U (c(r = 0, θ))) ≤
θ
V (αh ) − V (αl )
.
αh − αl
(18)
Equation (18) provides two characteristics of contracts that encourage risk
1. Pay for performance is small when the bank performs poorly, that is, for a low θ;
2. Pay for performance is high when the bank performs well, that is, for a high θ.
Relative performance that increases pay for performance when the bank does well, and decreases it when the bank does poorly, is a type of compensation arrangement that encourage
risk taking.
5
Loan Review and Team Production
In all but the smallest banks, a bank will typically employ people who monitor lending
and risk. For example, in trading, there are risk managers who attempt to monitor and
limit trader risk exposure. In mortgage lending, a loan that is originated by a loan officer is
evaluated by a loan underwriter who is a different person. In commercial lending, individuals
or groups other than the loan officer are required to approve a loan, particularly if it is a
larger loan. Loan review is not a new function. For example, Udell (1989) surveyed a large
number of Midwestern banks and found extensive use of loan review in the 1980s.
There are several classes of models that are applicable to aspects of loan review. These
include models where the loan reviewer audits the loan officer or where the loan reviewer
creates a signal that is used to help determine whether the loan is made. To keep the notation
and analysis simple, we consider a less general production function; we also assume that for
each loan officer there is one loan reviewer.
21
The loan reviewer takes an action, b, that is simply an input into the production function
g(r|θ, b, α). The loan reviewer’s utility function is U (c2 ) − V (b), where c2 denote the loan
reviewer’s compensation. The loan reviewer’s reservation utility is Ū2 .
Compensation for both the loan officer and the loan reviewer can depend on the return
and θ, so compensation for the loan officer is c(r, θ) and compensation for the loan reviewer
is c2 (r, θ). The bank’s problem with loan review is only slightly different than the earlier
problem. It is
Bank’s Problem with a Loan Reviewer
X
max
c(r,θ)≥0,c2 (r,θ)≥0,α,b,c̄(θ)≥0,r̄(θ)
h(θ) max{r̄(θ) − c̄(θ) − D, 0}
(19)
θ
subject to resource constraints
∀θ, r̄(θ) =
X
g(r|θ, b, α)r,
(20)
g(r|θ, b, α) (c(r, θ) + c2 (r, θ)) ,
(21)
r
∀θ, c̄(θ) =
X
r
c̄(θ) ≤ max{r̄(θ) − D, 0},
(22)
the participation constraints
X
h(θ)
X
g(r|θ, b, α)U (c(r, θ)) − V (α) ≥ Ū ,
(23)
g(r|θ, b, α)U (c2 (r, θ)) − V (b) ≥ Ū2 ,
(24)
r
θ
X
h(θ)
X
r
θ
and the incentive constraints
X
X
h(θ)
g(r|θ, b, α)U (c(r, θ)) − V (α)
(25)
r
θ
≥
X
h(θ)
X
X
g(r|θ, b, α̂)U (c(r, θ)) − V (α̂)), ∀α̂.
r
θ
h(θ)
X
g(r|θ, b, α)U (c2 (r, θ)) − V (b)
r
θ
≥
X
θ
h(θ)
X
g(r|θ, b̂, α)U (c2 (r, θ)) − V (b̂), ∀b̂.
r
22
(26)
Notice that in incentive constraint (25) the loan officer takes the loan reviewer’s action as
given and that in incentive constraint (26) the loan reviewer takes the loan officer’s action
as given.8
We do not write out the bank subproblem, but instead write out the first-order conditions
for an interior solution. For loan officers, they are
!
X
1
g(r|θ, b, α̂)
=λ+
µ(α̂) 1 −
,
0
U (c(r, θ))
g(r|θ, b, α)
α̂
(27)
where λ is the multiplier on the bank subproblem constraint that corresponds to (23) and
µ(α̂) on the incentive constraints that correspond to (25). For loan reviewers, they are
!
X
1
g(r|θ, b̂, α)
=
λ
+
,
µ(b̂) 1 −
2
0
U (c2 (r, θ))
g(r|θ, b, α)
(28)
b̂
where λ2 is the multiplier on the bank subproblem constraint that corresponds to (24) and
µ(b̂) on the incentive constraints that correspond to (26).
For both loan officers and loan reviewers, the first-order conditions show that optimal
compensation depends on the return r as a function of the likelihood ratios. The team
aspect can be seen in that the action of the other agent affects the likelihood ratio and thus
compensation. For loan reviewers, there is the interesting finding that loan reviewers and risk
managers should be paid based on the performance of the projects that they monitor. While
not surprising from standard incentive theory, it does go against the common perception
among bank regulators that risk management and other similar positions should not have
their pay tied to the performance of the line of business that they monitor.
5.1
A Loan Review and Risk Control Example
This subsection works through a specific functional form to derive some implications of loan
reviewer pay for risk control. We build on the same basic production function used earlier,
namely, that
f (r = 1|θ, α) = αθ̄ + (1 − α)θ,
(29)
and where the loan officer can make a risky loan, αl , or a safe one, αh .
8
An interesting variation to consider is to allow the loan reviewer and loan originator to collude. That
would generate a different set of incentive constraints.
23
The loan reviewer’s role is to review a loan originated by the loan officer and accept or
reject it. If accepted the probability of success of the loan is described by (29). If rejected,
the bank receives 0 < r̃ < 1 with certainty, which reflects the value of the funds being
allocated to some safe activity.
Acceptance or rejection is determined probabilistically as a function of the loan type and
the loan reviewer’s action b. If the loan officer originates an αl loan opportunity then it is
rejected with probability b. Loan review is not perfect, so if the loan officer generates an
αh loan opportunity, then the loan reviewer rejects it with probability z(b), where z(b) < b.
We also assume that z 0 (b) < 0, which means that if the loan reviewer works more, he rejects
risky loans more frequently and safe loans less frequently.
There are three possible outcomes:
1. The loan is rejected, so r = r̃;
2. The loan is not rejected and r = 0;
3. The loan is not rejected and r = 1.
The production function is
g(r = 0|θ, b, αl ) = (1 − b)f (r = 0|θ, αl )
g(r = 1|θ, b, αl ) = (1 − b)f (r = 1|θ, αl )
g(r = r̃|θ, b, αl ) = b.
g(r = 0|θ, b, αh ) = (1 − z(b))f (r = 0|θ, αh )
g(r = 1|θ, b, αh ) = (1 − z(b))f (r = 1|θ, αh )
g(r = r̃|θ, b, αh ) = z(b).
For loan officers, the likelihood ratios are
(1 − b)f (r = 1|θ, αl )
,
(1 − z(b))f (r = 1|θ, αh )
(1 − b)f (r = 0|θ, αl )
LR(r = 0, θ, b, αl ; αh ) =
,
(1 − z(b))f (r = 0|θ, αh )
b
LR(r = r̃, θ, b, αl ; αh ) =
.
z(b)
LR(r = 1, θ, b, αl ; αh ) =
24
The comparative statics for r = 0 and r = 1 are exactly the same as in the earlier analysis
of Section 4.6. For r = r̃, the loan officer receives a fixed level of compensation that does
not depend on bank performance, at least for interior solutions. This amount could be
interpreted as a fixed retainer for the loan officer, which is common in sales jobs.
For loan reviewers, the likelihood ratios are
1 − z(bl )
,
1 − z(bh )
1 − z(bl )
LR(r = 0, θ, bl , αh ; bh ) =
,
1 − z(bh )
z(bl )
.
LR(r = r̃, θ, bl , αh ; bh ) =
z(bh )
LR(r = 1, θ, bl , αh ; bh ) =
Loan reviewer compensation depends on whether a loan is made, but, if it is made, not on
how it performs.9 Furthermore, because
z(bl )
z(bh )
>
1−z(bl )
,
1−z(bh )
pay is lower if the loan reviewer
rejects the loan.
If the bank was trying to implement the risky strategy, by encouraging loan officers to
take αl , and then encouraging loan reviewers to supply less effort screening, then likelihood
ratios would be
1 − z(bh )
,
1 − z(bl )
1 − z(bh )
,
LR(r = 0, θ, bl , αl ; bl ) =
1 − z(bl )
z(bh )
LR(r = r̃, θ, bl , αl ; bl ) =
.
z(bl )
LR(r = 1, θ, bh , αl ; bl ) =
Here a pay structure that rewarded loan reviewers for rejecting loans would suggest a risk
problem.
More generally, however, the analysis in this section points to an alternative strategy to
regulating compensation for controlling risk. The loan review process could be evaluated to
determine how well it identifies and manages risk. Indeed, bank supervisors spend a lot of
resources evaluating the “controls,” that is, the quality of the processes used by banks to
approve loans, and in this model that would be reflected by evaluating rejection rates as well
as the quality of the loan reviewer’s assessment.
9
As noted in the previous analysis, this is not true in general. It is true in this example because loan
review probabilistically determines acceptance or rejection, so that decision is what contains information on
the loan reviewer’s action.
25
6
Conclusion
This paper worked through several organizational models of banker incentive compensation.
We found that the contribution of banker pay to bank risk depends on the amount of correlation in loan officer returns. If returns are uncorrelated, the form of banker pay is irrelevant.
If they are correlated, then low wages could be risk-creating. When the correlation of returns is endogenous, the structure of relative performance contracts greatly influences risk.
Pay for individual performance that increases with total bank performance was a source of
bank risk. A sufficient condition for a compensation arrangement to create bank risk was
provided.
A model of an organization with monitoring and controls in the form of loan review
was also introduced. The analysis indicated that monitoring and evaluating the loan review
process was an effective alternative to regulating compensation for controlling bank risk
and that paying risk managers for the performance of the employees they monitor does not
necessarily create risk.
More generally, the analysis illustrated an important principle of organizational economics: Compensation and its effect on the performance of a bank depend on the way that
the bank organizes its production. Looking at pay in isolation without considering the effects
of the bank’s monitoring and controls could not only lead to bans on compensation arrangements that do not necessarily create risk, but could also permit forms of compensation that
in isolation look harmless, but instead, generate excessive risk.
A
Proofs
Proof of Proposition 4
Let W S(θ) be the workers’ share of total revenue. Also, recall that r̄(θ) is not only the
total revenue, but also the fraction of workers who produce the high return of one. First,
W S(θ) =
c̄(θ)
r̄(θ)c(r = 1, θ) + (1 − r̄(θ))c(r = 0, θ)
=
r̄(θ)
r̄(θ)
c(r = 0, θ)
= c(r = 1, θ) +
− c(r = 0, θ).
r̄(θ)
26
Differentiating with respect to π gives
∂c(r=0,θ)
∂c(r = 1, θ) r̄(θ) ∂ r̄(θ) − c(r = 0, θ) ∂c(r = 0, θ)
dW S(θ)
=
+
−
dr̄(θ)
∂r̄(θ)
r̄(θ)2
∂r̄(θ)
Note:
∂c(r=1,θ)
∂ r̄(θ)
= 0 because
∂c(r=1,θ)
∂θ
= 0. Therefore,
dW S(θ)
∂c(r = 0, θ) 1
c(r = 0, θ)
=
(
− 1) −
< 0.
dr̄(θ)
∂r̄(θ)
r̄(θ)
r̄(θ)2
The term
∂c(r=0,θ)
∂ r̄(θ)
=
∂c(r=0,θ) ∂θ
∂θ
∂ r̄(θ)
< 0 because
∂c(r=0,θ)
∂θ
< 0 and
∂θ
∂ r̄(θ)
> 0. Furthermore,
r̄(θ) < 1, so the first term is negative. The second term is also negative.
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