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Financial Study Reports on Cost of Capital, Hurdle Rate, and Default Probability: A Robust Model of Internal Capital Markets:- A company's financial needs or goals for the future. Corporate financial planning involves identifying these financial objectives and determining how to achieve them.
Financial Study Reports on Cost of Capital, Hurdle Rate,
and Default Probability: A Robust Model of Internal
Capital Markets
Abstract. This paper adds to the understanding of the cost of capital (COC) by providing a micro-
level analysis of the endogenous relation between a firm's investment activities and COC. Building upon
recent results on multi-unit auctions in the mechanism design literature, I formulate a multi-division firm
model endogenously connecting a firm's COC to the hurdle rate for project selection, through the default
probability of the firm. Other models typically take the COC as given and examine how the hurdle rate or
other variables of interest are linked to the COC. In the model here, there is a transaction cost of financing,
namely, the legal cost of collection in case of default. Banks set the loan interest rate taking into account the
default risk factor, which can be affected by the number of projects financed by a firm. Anticipating the
impact on the COC (i.e., loan interest rate), the firm sets a hurdle rate accordingly to determine the number
of projects to invest and finance. As this affects the firm's default probability, the COC and the hurdle rate
are jointly determined. The model demonstrates how a firm may operate an internal capital market (ICM) to
elicit private information from division managers and allocate capital to their projects efficiently. It is robust
in the sense that the results are not sensitive to any prior belief about division managers' private information.
Under conditions labeled as "small bonuses, a few divisions," the default probability takes a simple
geometric form, which induces a downward sloping (inverse) capital supply curve. Interestingly, this leads
to "capital sponsoring" with a subsidized hurdle rate, rather than capital rationing. The model is also used to
understand why operating an ICM may be better than giving division managers complete autonomy in
investing and financing decisions. Such reasons include a lower COC and a smaller expected loss in
shareholder value in case of default, both due to reducing the default probability. (JEL G31, G33, D23, L22)
Keywords: Cost of capital, hurdle rate, default risk, project selection, project financing, capital
rationing.
Cost of Capital, Hurdle Rate, and Default Probability: A Robust Model of Internal Capital Markets
1. Introduction
It has been repeatedly mentioned in the literature that understanding the cost of capital (COC) is
important to management accountants and corporate finance specialists, as well as to academic researchers
and even government agencies regulating utility companies (see e.g. Modigliani and Miller's [1958] classic
paper). Empirical researchers pursuing studies on COC certainly are interested in knowing more about its
theoretical properties, in particular, how it is determined by firm attributes and factors of the surrounding
environments.
1
Regulatory agencies such as SEC also emphasize their decisions' impact on COC and
recognize it as an important consideration in setting regulations.
2
Despite its importance, COC remains a mystery. It is ofttimes an unexplained, exogenous constant in
theoretical models, and in practice a hard-to-estimate unknown. Management accounting textbooks instruct
students to use COC as the discount rate/required rate of return for selecting investment projects, yet with
little discussion of how to determine the COC (e.g., Garrison et al [2008]). Finance textbooks teach the use
of weighted average cost of capital (WACC) as an estimate for COC in spite of the recognized limitations of
this approximation (Brealey et al [2006]). Finance scholars and professionals have long relied on the capital
asset pricing model to estimate COC from stock returns, however, without being entirely satisfied with the
method (Ferson and Locke [1998]). A fast growing literature in accounting instead uses the abnormal
earnings model to infer COC from stock prices (e.g., Gebhardt et al [2001]). How useful this method is
remains to be ascertained (Easton and Monahan [2005]). What clear to us is that our understanding of COC
is still limited.
1
Empirical research on COC is growing quickly. For example, Hail and Leuz [2006] study whether cross-country
differences in COC are related to the legal institutions and securities regulations of different countries. Francis et al [2004]
study the relation between earnings attributes and COC. Gebhardt et al [2001] advocate their approach to estimating COC to
researchers studying the effects of disclosure policies on COC.
2
For example, in his speech on September 13, 1999, entitled "Small Business: The Lifeblood of Our Nation's
Economy," former SEC commissioner Norman S. Johnson said, "I look forward to working with you to find ways to
increase the availability and lower the cost of capital to small businesses." Similarly, on April 21, 1997, former SEC chairman
Arthur Levitt in his remarks entitled "Small Business Makes a Large Contribution" said, "[T]he SEC adopted new rules that I believe
will reduce the cost of private capital formation and especially benefit small businesses." Speaking even more directly, former SEC
chairman Richard C. Breeden in his remarks on September 21, 1990 said, "At the SEC, ... we are trying to take steps available to us
to reduce the cost of raising capital. ... [W]e will seek to promote the competitiveness and
vitality of U.S. business . by trying to reduce as much as possible the cost of capital to U.S. firms."
1
This paper adds to the limited understanding of COC by providing a micro-level analysis of the
endogenous relation between a firm's investment activities and COC. Specifically, I formulate a model of a
firm with multiple divisions, each with a project. The firm operates an internal capital market (ICM) to elicit
private information from division managers and allocate capital to their projects. With the model, I derive
results linking the COC endogenously to the hurdle rate for project selection, through the default probability
of the firm.
Recognizing the endogenous determination of COC has important implications to empirical work
and the valuation of firms. Two firms estimated to have similar COC and appearing to have similar growth
opportunities might actually face quite different capital supply curves. Consequently, they will react
unequally to similar investment opportunities available in the future. The valuation of these firms today
should thus be different. Correct valuation of these firms requires, among other things, estimating their
capital supply curves that constrain the project selections and ultimately determine their actual investments.
Related empirical work therefore should focus on the whole capital supply curve, rather than merely a given
COC determined by the curve for a particular moment in time.
There have been many models of firms with multiple divisions. For example, in accounting, models
have been proposed to examine how a transfer pricing system can be designed to allocate costs to divisions
optimally (e.g., Melumad, Mookherjee, and Reichelstein [1992]). These models typically focus on operating
activities of a firm and say little about their relation with the COC. There are also multi-division models of
capital budgeting that tie the hurdle rate for project selection to the COC (e.g., Baldenius, Dutta, and
Reichelstein [2007]). But the COC typically is exogenously given in the models.
3
So a firm's COC is not
affected by the firm's "growth opportunities", e.g., the number of good projects it has in a year.
In finance, models of multiple divisions have been used to study ICM. A partial list includes
Bernardo et al [2006, 2004], Inderst and Laux [2005], Ozbas [2005], de Motta [2003], Inderst and Muller
[2003], Stein [2002], Scharfstein and Stein [2000], Stein [1997], and Gertner, Scharfstein, and Stein [1994].
These models either assume only one "winning" division can obtain financing from the firm, or every
3
For multi-firm (more precisely, multi-asset) models, rather than single-firm, multi-division models like mine, an
exception is Lambert, Leuz, and Verrecchia's [2007] study. They, however, use a different modeling approach based on the
capital asset pricing model. Another exception is Hughes, Liu, and Liu's [2007] study that uses a multi-asset, factor-structure model
with informed and uninformed investors. Both studies focus on the cost of equity capital and say little about firms relying mainly
on debt capital, e.g., like business groups studied by Dewaelheyns and Van Hulle [2008].
2
division will receive some fraction of the available capital. In practice, it is likely that divisions compete for
financial resources in a way that is somewhere in between: not as tough as only one can win, nor as widely
spread as every surely gets a bite.
4
In contrast to others, the multi-division model of this paper endogenously links a firm's COC to the
number of projects invested by the firm; in other words, one or more divisions will "win" but usually not all.
With such an analysis, the paper provides a micro-foundation to the relation between financing and investing
activities. On top of this basic objective, it is hoped that the model can be robust and tractable. Robustness
gives some guarantee to the reliability of the results so that they are not too sensitive to details of the model
specification, such as a correctly specified prior belief about the division managers' private information.
Tractability allows the model to serve as a simple baseline construct for further refinements or extensions so
that a single family of models sharing the same structure can be used to understand a variety of corporate
finance and accounting issues.
I view the model of this paper a normative one, in the sense that it illustrates how a firm can elicit
private information from managers so as to allocate capital efficiently. Notwithstanding this, it can also have
testable implications. For example, for firms following practices suggested by the model, one should expect
to see more efficient investment across divisions than in others that allocate capital quite differently, of
course assuming other factors are constant.
Figure 1 is an overview of the model's important elements. While I draw upon recent results on
multi-unit auctions to build the model, I make no innovation in the multi-unit auction used as a mechanism
to elicit private information and allocate capital. The use of the mechanism in the model is twofold. First, it
converts an incomplete-information setting that is difficult to analyze into a more tractable complete-
information setting, upon which more complex models can be built. Second, the conversion cannot be done
in an arbitrary manner. To fulfill incentive compatibility and individual rationality constraints to be detailed
in Section 3, the multi-unit auction has to take the form of an ex post mechanism of which a component can
be naturally interpreted as a hurdle rate, which is widely used in practice. This and other structures of an ex
post mechanism provide the basis for deriving other results in the paper, which are my contributions.
4
In summarizing some survey evidence, Mukherjee and Henderson [1987] note that the reported project
acceptance rate ranged from over 75% to over 90%. Segelod's [1995] field study provides some institutional details on the
resource allocation process of a divisionalized firm.
3
Through operating an ICM (i.e., using a multi-unit auction to elicit private information and allocate
capital), the firm will become informed of the valuation profile · that represents managers' private
information about the expected returns of their projects. I assume that at no cost the firm can communicate
· to a bank. While unrealistic, I believe this is the best assumption to make at this stage of developing the
model. Components of the model have existed in the literature for some time but assembling them to form
this model is new. Given the complexity of the model, a more realistic assumption about how · can be
communicated will only confuse the readers more, instead of helping them to understand how the model
functions. This does not mean future extensions or refinements of the model should continue to adopt this
unrealistic assumption. Indeed, as discussed in Section 4, the roles of auditor and analyst come into play
when this restrictive assumption is relaxed, which is one of the reasons why the model is relevant to
accounting besides its relevance to corporate finance.
In the model, internally generated cash flow is supposed to be insufficient to fund project
investment. A second important simplifying assumption is that borrowing from a bank is the firm's only
financing source. While restrictive, it is not unrealistic; there are big firms and numerous smaller firms in the
world that are unlisted and rely heavily on bank loans to meet financing needs.
5
Discussions in Section 4
touch on how relaxing this restrictive assumption may lead to stronger relevance of the model to accounting.
For simplicity, I suppose that banks have perfect access to capital at the market interest rate r, and
there is perfect competition among banks. Given that · can be communicated to a bank at no cost, perfect
bank competition forces the loan interest rate c
0
(i.e., the firm's COC for financing projects) to be set by
adding on top of r a premium for default risk. The friction that drives my results is a transaction cost of debt
financing, namely the legal cost L of collection in case of default. For the shareholders, they also dislike
5
Understanding the overall COC of a firm requires knowing not only its cost of equity capital but also debt capital.
Debt financing is a source of capital second to none. Among the 30 countries studied by Demirguc-Kunt and Maksimovic
[1996], only four have a total debt to total equity ratio less than 1, with the US' equal to 1.791. In their 1999 study using the same
sample, they provide data of the total debt to total assets ratio for four firm classes ranging from small to very large; the ratios are
all within 50% to 60%. In another study based on World Bank firm-level survey data, Beck, Demirguc-Kunt, and Maksimovic
[2008] find that bank is without exception the largest source of external finance, with the responding US firms on average
reporting a 21.47% for bank versus a 3.24% for equity as two of the external finance sources. This survey, however, under-
represents large firms, which constitute only 20% of the sample (with the rest equally split between medium and small firms).
Large firms are more likely to use underwriting services in external financing. The Thomson Financial League Tables for 2003
and 2004 show that underwriting leaders in the loans and bonds categories raise proceeds that are each nearly 10 times of the
stocks category (http://en.wikipedia.org/wiki/Thomson_ Financial_league_tables). In short, it is important to understand the cost
of debt capital, and bank apparently is the most important source of debt financing.
4
default because there is a loss in shareholder value due to forced liquidation of assets. For simplicity,
regularity conditions on parameters of the model (to be detailed in Section 3) are assumed such that the
lending bank can always get full recovery of the loan amount plus interest in case of default. Therefore, the
main driver of the default risk is the firm's default probability q
0
.
In general, the default probability can depend on ·, as suggested by the characterization provided in
Section 3. However, in a later part of my analysis, I identify conditions, labeled as "small bonuses, a few
divisions," under which the default probability takes a simple geometric form. Restricting only to geometric
default probability highlights a sufficient condition for capital rationing not to occur in the model and
clarifies the important role of the capital supply curve. Other models typically assume the COC is given,
which is equivalent to assuming a perfectly elastic (inverse) capital supply curve. The model here provides a
micro-foundation demonstrating that default risk consideration can lead to a downward slopping capital
supply curve. Thereby it uncovers the rarely discussed incompatibility between a downward slopping capital
supply curve and capital rationing.
More concretely, geometric default probability implies that within the relevant range of k units of
capital to borrow (as restricted by the "small bonuses, a few divisions" conditions), the capital supply curve
takes the functional form: C(k)/k = (1+r) + Lq
k
/k, where q is the default probability of a division were it to
borrow like a legally separate entity. This capital supply curve is associated with a "marginal cost" curve,
C?(k), everywhere below the supply curve such that C?(k) < (1+r) < C(k)/k for any k> 0. My analysis shows
that if confining to a simple class of ex post mechanisms, called posted-price mechanisms, the hurdle rate h
will be set at a level such that (1+h) = C?(A), where A is the number of projects to be financed.
6
Note that not
only is h a "subsidized" rate with respect to the COC c
0
= C(A)/A - 1 but the rate is also "subsidized" even
when compared to the market interest rate r. This "capital sponsoring" result is illustrated in Figure 2.
6
Throughout the paper, I confine myself to a robust analysis of the model without specifying a prior belief about
the profile · that represents managers' private information. As such, I cannot discuss "the optimal" mechanism to operate an
ICM. Nonetheless, under perfect bank competition, I show that it is optimal for banks to use a posted-price mechanism to
allocate capital to the divisions, should the firm let them operate as separate legal entities to borrow from banks directly. The
rest of the analysis assumes that when operating an ICM, the firm also uses a posted-price mechanism to allocate capital to the
divisions. A posted-price mechanism need not be optimal had a prior been specified. However, it will be clear that with a properly
set hurdle rate, a posted-price mechanism will allocate capital efficiently, regardless of managers' private information. It is
therefore immune to ex post renegotiation. In contrast, the possibility of renegotiation can prevent an ex ante efficient mechanism
that optimizes with respect to a given prior from working as intended.
5
This paper contributes to the corporate finance and accounting literatures in several ways. First, it
provides a model connecting the COC to the hurdle rate endogenously. Other models typically take the COC
as given and examine how the hurdle rate or other variables of interest are linked to the COC. Second, the
model fills a gap between two extreme types of models in the ICM literature, namely "only one can win" and
"everyone gets a bite." The allowing for financing genuinely multiple projects, ranging from one to all, is the
reason why the COC can be endogenously linked to the hurdle rate through the default probability. Third,
the model provides a baseline construct with which extensions and refinements may be developed to
examine other issues not analyzed in this paper (see discussions in Section 4). It offers the potential to
organize the understanding of various related corporate finance and accounting issues using a single family
of models all developed from the one here as a baseline construct.
Fourth, the analysis of the model adds to the understanding of why operating an ICM may be better
than giving division managers complete autonomy in investing and financing decisions. Such reasons
include a lower COC and a smaller expected loss in shareholder value in case of default, both due to
reducing the default probability. Fifth, the analysis also shows that with geometric default probability, the
capital supply curve is downward sloping. This leads to the interesting phenomenon of a subsidized hurdle
rate and "capital sponsoring," which to my knowledge has not been examined in the literature before.
Finally, the model demonstrates to practitioners how a firm may operate an ICM to elicit private information
from division managers and allocate capital to their projects efficiently. Because the model is robust in the
sense that it does not depend on detailed specifications of managers' private information, the class of
mechanisms suggested here is easier to implement than suggestions derived from a Bayesian framework. A
Bayesian analysis usually requires correctly specifying a common prior belief, which is often difficult in
practice.
The rest of the paper is organized as follows. Next, I will give an overview of the model features and
main results. The model is formally introduced and analyzed in Section 3. There I use the model to
understand why operating an ICM can be better than giving division managers complete autonomy in
investing and financing decisions. In addition, I give a characterization of the default probability that affects
a firm's COC and is affected by the number of projects financed. Section 4 discusses the model's relevance
to corporate finance and to accounting; some unanswered questions on modeling are also posted there.
Concluding remarks are given in Section 5. Technical derivations and proofs are relegated to the Appendix.
6
2. Overview of Model Features and Main Results
2.1 Model Features
Many multi-division models have explored incentive issues related to getting managers to work hard
(e.g., Stein [2002] and Bernardo et al [2004, 2006]). In contrast, I focus on the capital allocation problem
and maintain tractability of the model by abstracting away from moral hazard due to hidden action. The only
incentive issue in the model is to get division managers to reveal truthfully their private information about
expected project returns.
In practice, a firm might find it hard to have some reliable prior beliefs about expected project
returns known only to managers. The model does not require a firm to form such beliefs before it can
implement the capital allocation scheme suggested. It is a robust model in the sense that the results derived
do not critically depend on details of such prior beliefs. Because of the robustness and accordingly the
comparatively easier implementation of the suggested scheme, practitioners might find the model more
relevant than others recommending "optimal" schemes derived from a Bayesian framework.
Consistent with the trend of emphasizing robustness in modeling (e.g., Bergemann and Morris
[2005]), I focus on direct-revelation mechanisms ensuring truth-telling is dominant-strategy incentive
compatible (DIC) and that participation is ex post individually rational (EIR). Such mechanisms are called ex
post mechanisms. To draw on Segal's [2003] characterization of ex post mechanisms, I assume each division
needs exactly one unit of capital to finance its project. While stylized, this is consistent with often found
non-scalability of investment projects.
Like many others in the literature, the model assumes risk neutrality. This is important because the
model is built upon results on multi-unit auctions in the mechanism design literature. These results rely
heavily on a quasi-linear functional form of agents' preferences (i.e., managers' here). The recent
characterization by Saitoh and Serizawa [2008] however opens up potential for relaxing this assumption to
allow preferences exhibiting income effects. This could provide a venue for future research, widening the
applicability of the model.
Residual income is a divisional performance evaluation measure widely taught in textbooks. In
particular, specific versions of it, such as EVA
®
, are highly advocated by consulting companies like Stern
Stewart & Co. (Stewart, Ellis, and Budington [2002]). The model assumes managers' objectives are to
maximize residual incomes of their projects, with the capital charges endogenously determined in the model,
7
rather than computed from an exogenously given required rate of return. Similarly, the firm's CEO is
assumed to maximize the value created from project investment, taking into account the cost of financing.
This can be viewed as the firm's "residual income," a performance measure analogous to the divisional
residual income of project investment.
Incentive compensations are determined by the bonus rates for the CEO and managers exogenously
specified in the model. While this differs from the widely accepted optimal contracting approach, the
constant bonuses rates will facilitate empirical work built around the model. By contrast, it is not easy for
researchers to test implications of the optimal contracting literature. Real-life compensation contracts often
have simple structures. Rarely will one find such contracts nicely align with fine-tuned optimal contracts
suggested by the literature. As my focus is on the capital allocation problem, optimal compensation design
issues are left aside to maintain tractability of the model.
In the model, borrowing from banks is supposed to be the only source of external financing. For
tractability, perfect bank competition is assumed so that any surplus from improved default risk is fully
captured by the firm through a lower cost of (debt) capital.
The model provides a framework for examining a number of interesting questions. For example,
within a firm, the capital charge is only a nominal accounting charge that serves the purpose of performance
evaluation and involves no actual payment. As such, there is no issue about whether a division is able to
make the "payment" or not. Suppose instead divisions are organized as separate legal entities with direct
access to external financing in the debt market. The "capital charge" becomes an actual payment for the loan
amount plus interest, and complications like default risk will kick in. The intuition is that an ICM can add
value by pooling together divisions' business risks and thereby relaxing the payment constraint that gives
rise to the default risk.
7
In analyzing the model, I contrast two polar cases of firm with different degrees of decentralization:
• ICM Firm: Divisions have autonomy in operating decisions but investing and financing
decisions are centralized.
• "Autonomous" Firm: Divisions have autonomy in investing and financing decisions, in addition
7
It should be emphasized that this is distinct from the usual diversification argument for settings with risk-averse
agents. All players here are risk-neutral. The efficiency gain from an ICM is due to the reduction in the default probability
that dictates the expected transaction cost of default to a bank, namely LPr{default}.
8
to operating decisions.
An ICM firm has an "active" CEO operating an ICM. The project selection and financing policy
used by the CEO is modeled as a direct-revelation mechanism. With the ICM, information asymmetry is
resolved before approaching a bank for debt financing. I suppose that in discussing with a bank, the CEO
can credibly communicate what she knows about the expected project returns at no cost.
For an "autonomous" firm without an "active" CEO, a bank must resolve the asymmetric
information problem itself. To give direct external financing the best chance to dominate an ICM, I assume
the bank can also use a direct-revelation mechanism to elicit private information from managers
approaching it for project financing. The only exception is that "capital charges" to divisions now involve
actual payments, rather than merely nominal accounting charges.
2.2 Main Results
Below are main results of the paper:
Ex post mechanism as a project selection and financing policy: Simply requiring truth-telling to be
dominant-strategy incentive compatible and participation to be ex post individually rational imposes an
interesting structure on a firm's or a bank's policy for project selection and financing. Specifically, the
policy is characterized by a hurdle rate above which all projects with at least that level of expected returns
will be selected and financed.
Posted-price mechanism as the policy used by banks under perfect bank competition: In the case of
an "autonomous" firm, perfect bank competition forces the policy used by a bank to take the form of a
posted-price mechanism with a particular hurdle rate. Specifically, the hurdle rate is set at the same level as
the "total cost" (i.e., 1 plus loan interest rate c) charged to a division, which is given by
(1+c) ÷ (1+r) + qL,
where r is the market interest rate at which banks have perfect access to capital, q is the default probability
of a division borrowing as a standalone legal entity, and L is the lending bank's legal cost of collection in
case of default.
Benefits of ICM: Because a firm needs to hire an "active" CEO to operating an ICM, whether it is
better to organize as an ICM firm depends on how large this hiring cost is, which is characterized by the
bonus rate for the CEO. Organizing as an ICM firm provides two potential benefits. By pooling together
divisions' business risks, it might reduce the default probability and thereby lower the COC. Additionally,
9
when the lending bank invokes legal procedures of collection, there is a loss in shareholder value due to
forced liquidation of assets in place. So the shareholders also want to avoid default for their own sake. These
benefits are independent of the collateral-based argument suggested by Stein [1997]. Taken together the
benefits will outweigh the cost if the CEO's bonus rate is sufficiently small.
Characterization of default probability: The default probability plays an important role in the model
to link hurdle rate and COC together endogenously. A characterization of the default probability adds
understanding to what are driving the probability and other results of the paper. Such a characterization is
provided.
Geometric default probability: Conditions are given under which the default probability takes the
simple geometric form, i.e., q
k
, where k is the loan amount borrowed by an ICM firm. Loosely speaking, it
requires the bonus rates for the CEO and managers to be "small" and that there are only "a few" divisions in
the firm. The simple structure of geometric default probability relies on the cash flow of only one
"successful" project to make up for the shortfall of all other "unsuccessful" projects with zero realized
returns. It would not be possible if there are too many divisions in the firm. Because the bonus to the CEO is
based on the value created from project investment, when all projects are "unsuccessful," a larger bonus rate
essentially means a bigger "rebate" by the CEO to help shouldering the borrowing cost of the firm. If the
bonus rate is too large, the "rebate" would be enough to ensure no default even when all projects are
"unsuccessful." For a similar reason, the bonus rate for managers must not be too large either.
Capital sponsoring and subsidized hurdle rate: In prevalent models in the literature, typically an
upward sloping capital supply curve is assumed, or otherwise an exogenous COC is assumed, which is
equivalent to a perfectly elastic capital supply curve. Both assumptions are consistent with standard settings
in economic theory. The model of this paper provides a micro-foundation for assuming possibly a downward
sloping capital supply curve. When the default probability is geometric, the "total cost" of borrowing C(k) ÷
(1+r)k + Lq
k
is strictly convex, with the "average cost" of borrowing C(k)/k = (1+r) + Lq
k
/k decreasing in the
loan amount k (i.e., a downward sloping capital supply curve). Moreover, the "marginal cost" of borrowing
C?(k) = (1+r) + Lq
k
ln(q) is below the "average cost" of borrowing. Consequently, the CEO would prefer
"capital sponsoring" with a subsidized hurdle rate set at the level such that (1+h) = C?(A), where A is the
number of projects approved for financing. This hurdle rate h is a "subsidized" rate in the sense that h =
C?(A) - 1 < C(A)/A - 1, which is the endogenously determined COC in the model. This is in stark contrast to
10
the typically assumed setting with a hurdle rate above a firm's COC and therefore not every positive NPV
project is financed, a phenomenon referred to as capital rationing.
While unexpected, the "capital sponsoring" result driven by geometric default probability is not
unreasonable. Given perfect bank competition, the geometric default probability is decreasing in the loan
amount borrowed. With a subsidized hurdle rate, more projects would be financed than otherwise would
have. Given a downward sloping capital supply curve induced by geometric default probability, the
financing of more projects means a more favorable term in borrowing. The positive externality that results
makes the subsidized hurdle rate sustainable.
Considering geometric default probability and the resulting downward sloping capital supply curve
is not to say that they are likely cases to see in reality. Indeed, even in the model, they occur under the
restrictive "small bonuses, a few divisions" conditions. These conditions, however, are not inconceivable. So
although they might not be met often, they may indeed exist. Yet the assumption of a downward sloping
capital supply curve is hardly seen in the literature. This is not difficult to understand. If the capital supply
curve is to be specified exogenously, as is often the case in most studies, one would naturally assume an
upward sloping capital supply curve to adhere to traditional economic theory. It is under the micro-level
analysis of this paper that a downward sloping capital supply curve is derived endogenously from some
admittedly restrictive, yet not inconceivable conditions. Identifying such circumstances in the real world
would provide a way to test the empirical validity of the model.
To sum up, the importance of conducting a micro-level analysis and considering the restrictive case
of geometric default probability is to provide a new angle in rethinking issues that might have been
overlooked when capital rationing is taken for granted as the necessary case.
3. Model and Analysis
3.1 Model Specifications
I consider the setting of a firm with n divisions (n> 2), each headed by a manager, competing for
capital allocated from the corporate headquarters, headed by the CEO. Each division has a project requiring
exactly 1 unit of capital to invest. The project of division i, if executed, will bring in some uncertain future
cash flow R
i
, with its mean ·
i
known only by the manager. This expected project return is referred to as
manager i's valuation.
It is common knowledge that managers' valuations follow some joint distribution with the support of
11
each marginal distribution equal to ( µ , µ ), where 0 s µ< µ<·. This is all the CEO knows about the
valuations. On the other hand, the CEO and managers commonly know that conditional on the valuation
profile · = (·
1
, ·
2
,., ·
n
), the project returns R
i
's are independently and identically distributed as follows:
R
i
= 0 with probability q e (0, 1), and
Pr{R
i
< R +z | ·} = q + (1-q)G(z | ·
i
) for all z e [0, z ),
where 0 < zs·, with R> 0 and G(0| ·
i
) = 0. The "probability density function" g(z | ·
i
) = G?(z | ·
i
) is
positive and continuous for z> 0, and as defined, } zg(z | ·
i
)dz = ·
i
/(1-q) so that E[R
i
| ·
i
] = ·
i
.
Each division has some assets in place required for its regular operations, which will bring in a
certain (net) cash flow regardless of what happens to the division's project. For simplicity, assume each
division has an identical cash flow from regular operations to be generated during the year, M> 0, and
beginning-of-the-year book value of assets, B> 0.
8
Besides what might be later arranged for project financing, the firm has no liabilities. For simplicity,
bank loan is assumed to be the only source available to the firm for financing projects. The loan amount plus
interest, denoted by (1+c
0
)K, is referred to as the "total cost" of borrowing K units of capital, where K> 1.
The interest rate c
0
at which the firm can borrow the loan amount K is referred to as its cost of (debt) capital.
Although the model may be viewed as representing a typical round of a multiple-period model, here
I assume it is one-shot. The sequence of events happening in the model is as follows:
8
In this paper, modeling choices are made with an attempt to balance among tractability, empirical relevance, and
simplicity. Tractability always comes first, for few results can an analytical model give without tractability. Most theorists
take the view that models should be "leanest," i.e., every model element should play an indispensable role in the analysis. I also
take this view but with the room to allow for model elements that are included for empirical relevance considerations.
Oftentimes empiricists interested in testing the implications of a model need to "hold other things constant" by including
a variety of control variables, or confine themselves to conditions under which the model's results are valid. This legitimate concern
is usually ignored by theorists who care most about formulating the "leanest" models, with little interest in giving empiricists an
easier life and more guidance. Lacking any suggestions, empiricists typically put controls into regression equations in an
additively separable fashion. This ignores the possibilities that the controls should enter the equations nonlinearly or interact with
other variables, had they been included in the original model. Similarly, parameters inessential to the main insights behind the
analysis are often normalized to 0 or 1 to give cleanest statements of the conditions in concern. When in reality such exogenous
variables have different values, empiricists are left alone to make their own judgmental adjustments.
In contrast, the model of this paper includes elements (e.g., cash flows from regular operations) that are not absolutely
necessary but I believe empiricists interested in using the model as a theoretical framework for their research would consider in
formulating their research designs. These elements stay in the model because simplicity is not seriously
jeopardized. Such elements, however, could be considered "clutters" by theorists taking the extremist "leanest" perspective.
12
At the beginning of a year, it is common knowledge that the project selection policy adopted by the
firm will continue as usual, namely, given any profile · of expected project returns reported by
managers, the policy ?x(?), t(?)? specifies an outcome constituted of a capital allocation x = (x
1
, x
2
,.,
x
n
) e {0, 1}
n
, with x
i
= 1 indicating the approval of division i's project, and a "payment" scheme t=
(t
1
, t
2
,., t
n
) e R
n
, with t
i
standing for an accounting charge to division i.
9
Though not confined to be
so at this point of the model specification, it will be clear shortly that t
i
's are zero for divisions with
"losing" projects, i.e., those not selected.
It is also common knowledge that the firm's compensation policy will continue as usual, namely:
o "Losing" managers, whose projects are not selected, will receive a basic salary, which is
normalized to zero for simplicity;
o "Winning" manager i will receive a basic salary, also normalized to zero, plus a bonus, at the
rate of |> 0, based on the approved project's residual income R
i
- t
i
.
10
o The CEO will receive a basic salary, also normalized to zero, plus a bonus, at the rate of |
0
> 0,
based on the value E
i
R
i
x
i
- (1+c
0
)K created from projects selected and financed at the cost of
capital c
0
.
11
Managers privately learn about the expected returns of their projects and submit reports to convey
the information to the CEO.
Everyone learns about the market interest rate r> 0 at which banks can have perfect access to
capital.
12
9
Recall that 1 is the unit of capital invested in a project. So (t
i
- 1)/1 is the required (rate of) return often discussed
in accounting textbooks under the topic of capital budgeting.
10
At the expense of more complicated notations, the bonus rate can be specified based on post-bonus, rather than
pre-bonus, residual income. See related discussions on bonus compensation computation in accounting textbooks such as
Kieso, Weygandt, and Warfield [2006] and Stice, Stice, and Skousen [2007].}
11
With more complicated notations, the bonus rate for the CEO can be specified based on shareholder value,
namely the expected equity value at the year end after closing income to retained earnings. This alternative specification is
consistent with a stock bonus. Some analyses of the paper however become intractable with this alternative specification.
12
The following dummy event may be included at this point to justify non-zero bonus rates exogenously set in the
model, which could be an endogenous outcome of expanded modeling: A manager can "take the project with him" to pursue
an outside option (e.g., seek venture capital to start his own company), which will give an expected payoff of no more than
13
The CEO contacts a banker to share the information about projects available. The banker makes an
offer on the loan terms, namely the schedule of interest rate at each loan amount requested. The offer
is competitive, and the CEO accepts it with the loan amount finalized.
The CEO allocates the capital to the divisions and imposes accounting charges accordingly.
Each manager continues his division's regular operations from which a cash flow is generated.
Alongside with the regular operations, "winning" managers execute the approved projects, with their
returns realized some time before the year end. Compensations to the CEO and managers are paid at
the year end accordingly.
Following the year end, if the cash available is enough to cover the "total cost" of borrowing, the
payment is made to the lending bank. Otherwise the firm defaults, and the bank seeks recovery by
forced liquidation, which results in a legal cost of L> 0 to the bank. The liquidation value of the
firm's assets is a fraction ì e (0, 1) of their beginning-of-the-year book value.
Managers are expected payoff maximizers. Conditional on the valuation profile ·, a manager's
expected payoff is equivalent to
E[R
i
x
i
(·) - t
i
(·) | ·] = ·
i
x
i
(·) - t
i
(·).
This quasi-linear functional form of managers' expected payoffs allows utilizing directly some results on
multi-unit auctions in the mechanism design literature. In particular, the following definition and lemma are
essentially due to Segal [2003]:
DEFINITION 1 (Segal [2003]): A mechanism ?x(?), t(?)? is an ex post mechanism if it satisfies
dominant-strategy incentive compatibility (DIC) and ex post individual rationality (EIR):
For any manager i, any valuation profile · e ( µ , µ )
n
, and any µ
i
e ( µ , µ ), ˆ
[DIC]: ·
i
x
i
(·) - t
i
(·) > ·
i
x
i
( µ
i
, ·
-
i
) - t
i
( µ
i
, ·
-
i
),
ˆ ˆ
[EIR]: ·
i
x
i
(·) - t
i
(·) > 0.
where ( µ
i
, ·
-
i
) means the valuation profile constructed from · by replacing ·
i
with µ
i
.
ˆ ˆ
|(·
i
- c
0
). So he stays. Similarly, the CEO can choose to start her own company and invite "winning" managers to join. This
outside option will give her an expected payoff of no more than |
0
[E
i
·
i
x
i
- (1+c
0
)K]. So she also stays.
14
LEMMA 1 (Segal [2003]): A deterministic mechanism ?x(?), t(?)? is an ex post mechanism if and
only if for each manager there exist functions p
i
, s
i
: ( µ , µ )
n
-1
÷ R
+
such that for every valuation profile · e
( µ , µ )
n
,
x
i
(·) = 1 if ·
i
> p
i
(·
-
i
), x
i
(·) = 0 otherwise, and
t
i
(·) = p
i
(·
-
i
) x
i
(·) - s
i
(·
-
i
).
In structuring a policy to allocate capital and charge for the use, the CEO is assumed to restrict
attention to ex post mechanisms only, which do not require any knowledge about the distribution of ·. The
CEO is also an expected payoff maximizer. Conditional on the valuation profile ·, her expected payoff is
equivalent to
E
i
·
i
x
i
(·) - (1+c
0
)K.
3.2 "Autonomous" versus ICM Firm
In an ICM firm that has an "active" CEO operating an ICM, information asymmetry is resolved
before approaching a bank. Suppose it is prohibitively costly for the CEO to commit criminal acts like
falsifying information in project proposals submitted by division managers. Then by demanding the CEO to
provide the proposals to support her claim about ·, the information can be credibly communicated to a bank.
I suppose that in discussing with a bank, the CEO can credibly communicate · at no cost. With
complete information and perfect competition from other banks, the bank will set c
0
to equate its cost of
providing the loan. Suppose further that even in the case of default the firm has enough assets in place for
the bank to get full recovery of the loan amount plus interest. Then c
0
will be set by adding to r a risk
premium due to the expected loss as a result of the legal cost and the default probability:
c
0
= r + q
0
(K, ·)L/K,
where q
0
(K, ·) denotes the default probability of the firm when K units of capital are borrowed to finance
projects from a pool characterized by ·.
Instead of operating an ICM, the firm could have organized divisions in a fully autonomous fashion,
namely, separately incorporating the divisions and delegating project selection and financing responsibilities
to them. In such an "autonomous" firm, the CEO plays merely a "passive" role of ordinary administration
and receives only a basic salary normalized to zero.
15
For simplicity, assume each division in the firm has enough assets in place for a lending bank to get
full recovery of the loan amount plus interest. Then the expected cost of providing a loan to division i is
given by
(1+c
i
) ÷ (1+r) + L Pr{default by division i | ·}.
Without an "active" CEO operating an ICM, a bank must resolve the asymmetric information
problem itself. To give direct external financing the best chance to dominate an ICM, I assume the bank can
also use an ex post mechanism to elicit private information about ·. The only exception is that the t
i
(·)'s
charged to divisions now involve actual payments, rather than merely accounting charges.
A bank's objective is to choose an ex post mechanism ?x(?), t(?)? to maximize its expected profit
from lending,
E
i
[t
i
(·) - (1+c
i
)x
i
(·)],
subject to the zero expected profit condition due to perfect bank competition:
E
i
[t
i
(·) - (1+c
i
)x
i
(·)] = 0.
When an optimum is attained, no change in ?x(?), t(?)? can lead to a positive expected profit to a bank.
Although a bank can guarantee a zero expected profit by refusing to issue any loan, this is not
optimal if issuing some loans can create efficiency surplus. When such unexploited surplus exists, there
would be a way for a competing bank to offer loan terms to realize the surplus and make a positive expected
profit, which however cannot happen under perfect competition. That is to say, optimality together with
competition requires a bank to issue loans efficiently, given the constraints imposed by an ex post
mechanism and the legal cost of collection in case of default.
If ignoring the DIC and EIR constraints of an ex post mechanism, efficient loan issuance would
imply
x
i
(·) = 1 if ·
i
> (1+c
i
), x
i
(·) = 0 otherwise.
This, however, cannot constitute an ex post mechanism unless (1+c
i
) ÷ (1+r) + LPr{default by division i | ·}
is unrelated to ·
i
. The regularity condition below specifies circumstances under which (1+c
i
) is indeed
unrelated to ·
i
. When the condition is met, a simple posted-price mechanism is optimal to a bank. This result
is given in Lemma 2.
REGULARITY CONDITION:
16
(a) G((1+r) + L - M/(1-|) | ·
i
) < 1 for any ·
i
;
(b) (1+r) s (ìB + M)/(1-|) - L;
(c) (1+r) s R + M/(1-|) - L;
(d) (M+L)/(1-|) s (1+r).
Part a of the regularity condition ensures that the default probability is not too close to 1. Part b of
the condition says the liquidation value ìB of a division's assets is not too small to prevent full recovery of
the loan amount plus interest when legal procedures are invoked. These two parts together with part c imply
the default probability of a division is at most q. Finally, part d guarantees that the legal cost L is not too
large to make invoking the legal procedures unwise.
LEMMA 2: Suppose the regularity condition holds. Then for a "winning" division (i.e., one provided
with a loan), the default probability is q and hence its cost of capital is c÷ r + qL; a posted-price mechanism
?x(?), t(?)? with
x
i
(·) = 1 and t
i
(·) = (1+c) if ·
i
> (1+c),
x
i
(·) = t
i
(·) = 0 otherwise,
is optimal to a bank.
Suppose the regularity condition is met and thus the default probability of any division provided
with a loan is q. If posted-price mechanisms are used by banks,
I(c) ÷ {i | ·
i
> (1+c) for ·
i
of ·}
would be the index set of the "winning" divisions provided with loans. Let K denote the number of
"winning" projects, i.e., K÷ |I(c)|. Conditional on ·, the expected equity value of an "autonomous" firm
(without an ICM) is given by
[AF]: nM - Kq(1-ì)B + (1-|)[E
i
eI(c
)
·
i
- (1+c)K].
The first term of this expression is the cash flow generated from regular operations of all n divisions.
The second term is the expected loss in shareholder value due to forced liquidation of assets in case of
default. The last term is the part of value created from "winning" projects that is retained by shareholders. A
fraction | of the value created is paid to "winning" managers as their bonuses.
For ease of exposition, suppose for the moment that a CEO operating an ICM also uses a posted-
17
price mechanism with t
i
(·) = p
i
(·
-
i
) = (1+c) and the loan amount she borrows is also K. Bank competition
forces the interest rate offered to the CEO to be set at a level given by the zero expected profit condition, i.e.,
(1+c
0
)K = (1+r)K + q
0
(K, ·)L,
where q
0
(K, ·) is the default probability of the firm given that K units of capital are borrowed to finance
projects from a pool characterized by ·. The way q
0
(K, ·) is defined assumes that successful voluntary
liquidation of some assets to avoid default cannot be completed before the lending bank invokes the legal
procedures to liquidate the firm.
Given the suppositions above, the following would be a "lower bound" of such an ICM firm's
expected equity value:
[IF]: nM - q
0
(K, ·)(1-ì)nB + (1-|)[E
i
eI(c
)
·
i
- (1+c)K] + (c - c
0
)K - |
0
[E
i
eI(c
)
·
i
- (1+c
0
)K].
This is a "lower bound" because the possibility of reorganization might result in a higher value to the
shareholders. If liquidating some of the assets is sufficient to meet the payment obligation, the second term
in the expression above would exaggerate the expected loss in shareholder value in case of default. The third
and fourth terms in the expression add up to
[E
i
eI(c
)
·
i
- (1+c
0
)K] - |[E
i
eI(c
)
·
i
- (1+c)K].
This is the value created by the "winning" projects minus the bonuses paid to managers given that "winning"
divisions are charged at (1+c), rather than the "average cost" of borrowing (1+c
0
). Finally, the last term in
the lower bound has no counterpart in the expected equity value of an "autonomous" firm. The "passive"
CEO of such a firm is presumed to receive only a basic salary normalized to zero. By contrast, the CEO of
an ICM firm receives also a bonus based on the value she creates by operating an "active" ICM. This bonus
is the last term in the lower bound.
3.3 Default Probability of ICM Firm
In general, an ICM firm might attain an expected equity value higher than the lower bound IF by
charging "winning" divisions (possibly asymmetrically) more or less than (1+c) and by borrowing a loan
amount different from K, as long as such adjustments constitute an ex post mechanism. Raising the capital
charges t
i
(·)'s can save bonus expenses but the project selection cutoffs p
i
(·
-
i
)'s might need to be tightened
accordingly, which can reduce the total value created by "winning" divisions. Depending on the
characteristics of the default probability q
0
(K, ·), borrowing less might reduce the cost of capital c
0
so
18
substantially that it compensates for the loss in value due to a smaller number of "winning" projects. Given
these flexibilities in improving upon a posted-price mechanism with given c and K, there is room for an ICM
firm to achieve a higher equity value than the lower bound IF.
In contrast, an "autonomous" firm's equity value AF can be lower than an ICM firm's lower bound
IF. This may be best seen by examining their difference with the former deducted from the latter:
?÷ [Kq - nq
0
(K, ·)](1-ì)B + [Kq - q
0
(K, ·)]L - |
0
[E
i
eI(c
)
·
i
- (1+c
0
)K].
Suppose the bonus rate |
0
for the CEO is sufficiently small and therefore the cost of hiring her to operate an
ICM is arbitrarily negligible. Then as long as q
0
(K, ·) < qK/n, the difference ? can be made positive. In
words, the decrease in expected loss in shareholder value due to default risk, i.e., [Kq - nq
0
(K, ·)](1-ì)B,
together with the saving in cost of capital due to co-insurance among divisions, i.e., [Kq - q
0
(K, ·)]L, will be
large enough to cover the cost of operating an ICM, i.e., |
0
[E
i
eI(c
)
·
i
- (1+c
0
)K]. This result is stated below as
Proposition 1.
PROPOSITION 1 (Benefits of internal capital markets): Suppose q
0
(K, ·) < qK/n and hence c
0
< c.
For sufficiently small |
0
> 0, an ICM firm's expected equity value can exceed an "autonomous" firm's.
It should be emphasized that Proposition 1 is a "can-be" result. Since the CEO's and the
shareholders' interests do not align with each other completely, the equity value of an ICM firm might fail to
exceed an "autonomous" firm's if the task of operating an ICM is fully delegated to the CEO. However, it is
conceivable that the shareholders might impose restrictions on the ex post mechanism used, e.g., require
capital rationing while leaving the project selection decision to the CEO. Such measures might prevent the
CEO's opportunistic behavior from jeopardizing the firm's equity value.
The behavior of the default probability q
0
(K, ·) of an ICM firm in general can be quite complex. To
get some sense of how it might behave, let's continue to suppose that the firm uses a posted-price
mechanism with t
i
(·) = p
i
(·
-
i
) = (1+c) and the loan amount borrowed is K÷ |I(c)|, where I(c) ÷ {i | ·
i
> (1+c)
for ·
i
of ·}.
13
Given these suppositions, the firm's year-end cash position would be
nM + E
i
eI(c
)
R
i
- |[E
i
eI(c
)
R
i
- (1+c)K] - |
0
[E
i
eI(c
)
R
i
- (1+c
0
)K]
13
This could be the case if the provision of capital follows the traditional price-based transfer pricing framework,
where the transfer price of internally provided goods is set at the outside market price.
19
= nM + (1-|-|
0
)[E
i
eI(c
)
R
i
- (1+c)K] + (c - c
0
)(1-|
0
)K + (1+c
0
)K.
Default occurs when the cash position is insufficient to meet the firm's payment obligation, i.e.,
E
i
eI(c
)
R
i
< K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o,
where o÷ (1-|-|
0
). Let S denote the number of "successful" projects, i.e. those with positive realized
returns. Given S, let J(S) denote an index set of the S "successful" projects. There are altogether K!/S!(K-S)!
such J(S)'s each indicating a specific combination of the projects that constitute the S "successful" projects.
Let O(S) denote the set of all such J(S)'s. Finally, define Z
i
= (R
i
- R )1
{
Ri
>0}
; in words, Z
i
is the part of R
i
exceeding R . Conditional on R
i
> 0, Z
i
follows the distribution G(z
i
| ·
i
).
With these notations, the default probability can be expressed as follows:
Pr{default | ·}
= E
s
Pr{default | S = s, ·}Pr{S = s | ·}
= E
s
E
J
(s)eO(s
)
Pr{default | J(s), S = s, ·}(1-q)
s
q
K
-s
where
Pr{default | J(s), S = s, ·}
= Pr{Z
J
(s
)
+ s R< K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o | ·}
and Z
J
(s
)
= E
i
eJ(s
)
Z
i
is the sum of the independent random variables Z
i
's for those i e J(s).
The distribution function of Z
J
(s
)
, which is referred to in probability theory as the convolution of the
distribution functions G(z
i
| ·
i
)'s, can depend on the ·
i
's in a non-trivial way. For tractability, I assume the
probability distribution function of Z
J
(s
)
simply takes the form G(z | ·
J
(s
)
), where ·
J
(s
)
= E
i
eJ(s
)
·
i
.
14
Given this
assumption, the conditional probability Pr{default | J(s), S = s, ·} can be expressed as follows:
Pr{default | J(s), S = s, ·}
= Pr{Z
J
(s
)
< K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - s R | ·}
= G(K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - s R | ·
J
(s
)
)
= G(f(q
0
, s, K, c) | ·
J
(s
)
),
where
14
One example is that Z
i
follows the gamma distribution Gam(·
i
,1-q), which has mean ·
i
/(1-q) and variance ·
i
/(1-
q) . Then Z
J
(s
)
follows the distribution Gam(E
i
eJ(s
)
·
i
,1-q). 2
20
f(q
0
, s, K, c) = K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - s R
and c
0
= r + q
0
(K, ·)L/K.
Below is the next main result of the paper, which provides an equation characterizing the default
probability of an ICM firm.
PROPOSITION 2 (Characterization of default probability): Suppose that an ICM firm uses a
posted-price mechanism with t
i
(·) = p
i
(·
-
i
) = (1+c) and the loan amount borrowed is K÷ |I(c)|, where I(c) ÷
{i | ·
i
> (1+c) for ·
i
of ·}. Moreover, for s > 1, suppose that the sum of the parts of positive realized returns
over R , namely Z
J
(s
)
÷ E
i
eJ(s
)
Z
i
, where Z
i
÷ (R
i
- R )1
{
Ri
>0}
, follows the distribution G(z | ·
J
(s
)
), where ·
J
(s
)
÷
E
i
eJ(s
)
·
i
; for s = 0, define instead G(z | ·
?
) = 0 for z s 0 and G(z | ·
?
) = 1 for z> 0. Then for any given · and
c and the K so determined, the default probability q
0
(K, ·) of the ICM firm is given by the equation below,
provided it admits an interior solution in [0, 1]:
q
0
= E
s
E
J
(s)eO(s
)
G(f(q
0
, s, K, c) | ·
J
(s
)
)(1-q)
s
q
K
-
s
,
where
f(q
0
, s, K, c) = K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - s R ,
c
0
= r + q
0
L/K, and o = (1-|-|
0
).
To understand more about the properties of q
0
(K, ·), let's consider the special case of K = n. Given
the complex "combinatorial" structure of the equation defining q
0
(K, ·), little can be said even for this
special case. However, if there are "not too many" divisions in the firm and the bonus rates |
0
and | for the
CEO and managers are "not too large" (both made specific shortly), then q
0
(n, ·) will take the simple form
of q
n
. This follows from the fact that f(q
0
, s, K, c) is decreasing in s. As a result, f(q
0
, 1, n, c) s 0 implies f(q
0
,
s, n, c) s 0 for all s > 1 and hence G(f(q
0
, s, n, c) | ·
J
(s
)
) = 0 for all s > 1. It can be shown that f(q
0
, 1, n, c) s 0
and f(q
0
, 0, n, c) > 0. As G(z | ·
?
) = 1 for z> 0, it follows that
q
0
= E
s
E
J
(s)eO(s
)
G(f(q
0
, s, n, c) | ·
J
(s
)
)(1-q)
s
q
n
-
s
,
= G(f(q
0
, 0, n, c) | ·
?
)q
n
= q
n
.
This result is stated as the lemma below.
21
LEMMA 3: Suppose the regularity condition holds. In addition, ns [ R - L/(1-|-|
0
)]/[(1+r) -
M/(1-|)] and |
0
< (1-|)(1- q|)L/(M+L). Then given · and c, if K = n, q
0
(n, ·) = q
n
.
This simple structure of the default probability relies on the cash flow of only one positive realized
return project to make up for the shortfall of all other zero realized return projects. It would not be possible if
there are too many divisions in the firm. Recall that the bonus to the CEO is based on the value created from
project investment. When all projects have zero realized returns, a larger bonus rate essentially means a
bigger "rebate" by the CEO to help shouldering the borrowing cost of the firm. If the bonus rate is too large,
the "rebate" would be enough to ensure no default even when all projects have zero realized returns. For a
similar reason, the bonus rate for managers must not be too large either.
Suppose further that |
0
, |, and n are sufficiently small such that (nM + q|L)/(1-|-|
0
) s (1+r). Then
the simple structure of default probability derived above carries over to any K e {1, 2, ., n}. To see why,
first note that given the additional assumption on n and |
0
,
f(q
0
, s, K, c) = {K[(1+r)o - q|L] - nM + q
0
(1-|
0
)L - s R }/o
is increasing in K. It has been shown in the proof of Lemma 3 that f(q
0
, 1, n, c) s 0. It follows that f(q
0
, 1, K,
c) s 0 for all Ks n. The additional assumption also implies f(q
0
, 0, 1, c) > 0. As f(q
0
, s, K, c) is increasing in
K, f(q
0
, 0, K, c) > 0 for all Ks n. Together they imply q
0
(K, ·) = q
K
for any K e {1, 2, ., n}. This result is
stated as the next proposition.
ASSUMPTION SBAFD1 ("Small bonuses, a few divisions" - first version): The bonus rates for the
CEO and managers are "small", and there are only "a few" divisions in the firm, i.e., |
0
, |, and n satisfy the
following conditions:
ns [ R - L/(1-|-|
0
)]/[(1+r) - M/(1-|)];
|
0
< (1-|)(1- q|)L/(M+L);
(nM + q|L)/(1-|-|
0
) s (1+r).
PROPOSITION 3 (Geometric default probability): Suppose the regularity condition holds and |
0
, |,
and n satisfy the assumption SBAFD1. If an ICM firm uses a posted-price mechanism with c = r + qL as the
(rate of) "required return" and borrows K÷ |I(c)| as the loan amount, where I(c) ÷ {i | ·
i
> (1+c) for ·
i
of ·},
then the firm's default probability is q
0
(K, ·) = q
K
. Consequently, its cost of capital is c
0
= r + Lq
K
/K, and
22
"total cost" of borrowing is (1+c
0
)K = (1+r)K + Lq
K
.
Given this simple structure of an ICM firm's default probability, the result in Proposition 1 can be
strengthened. A specific threshold for the CEO's bonus rate can now be provided. The stronger result is
stated as the proposition below.
PROPOSITION 4: Suppose the regularity condition and assumption SBAFD1 hold. Given · and c,
and the K so determined, if 2 s K s n s [2((1-ì)B + L)/q - L]/(1-ì)B and |
0
s |
0*
, where
|
0*
÷ [(q - nq
K
/K)(1-ì)B + (q - q
K
/K)L]/( µ - (1+r) - Lq
K
/K),
then the equity value of an ICM firm can exceed that of an "autonomous" firm.
This proposition relies on the geometric default probability derived earlier, which hinges on the
possibility of using the cash flow of only one "successful" project to make up for the shortfall of all other
"unsuccessful" projects. This however is impossible if K = 1. Because the geometric default probability also
requires the firm to have at most "a few" divisions, its total cash flow from regular operations alone is
insufficient to prevent default either. Given the presumption that all assets will be liquidated in case of
default, there would be too much loss in shareholder value if n is large. Hence, the proposition also requires
n s [2((1-ì)B + L)/q - L]/(1-ì)B.
3.4 Capital Sponsoring and Subsidized Hurdle Rate
So far, I have assumed an ICM firm uses a posted-price mechanism with t
i
(·) = p
i
(·
-
i
) = (1+c) and
the loan amount borrowed is K÷ |I(c)|, where c = r + qL is the interest rate divisions of an "autonomous"
firm can obtain from banks. The characterization of the default probability in Proposition 2 remains valid if
c and K are replaced by h and K(h) respectively, where h is a hurdle rate (of required return) used by the
firm and K(h) ÷ |I(h)| is the loan amount corresponding to the rate.
15
That is, for any given · and h and the
K(h) so determined, the default probability q
0
(K(h), ·) of the ICM firm is given by the equation below,
provided it admits an interior solution in [0, 1]:
q
0
= E
s
E
J
(s)eO(s
)
G(f(q
0
, s, K(h), h) | ·
J
(s
)
)(1-q)
s
q
K
(h)-
s
,
where
15
Other results have specifically used c = r + qL in the proofs and cannot be generalized trivially as this.
23
f(q
0
, s, K(h), h) = K(h)[(1+h) - (h - c
0
)(1-|
0
)/o] - nM/o - s R ,
c
0
= r + q
0
L/K(h), and o = (1-|-|
0
).
In addition, the result of a geometric default probability can be generalized to this more flexible posted-price
mechanism, provided that the bonus rates for the CEO and managers are "small" and there are only "a few"
divisions in the firm. This time, "small" and "a few" are in the following sense:
ASSUMPTION SBAFD2 ("Small bonuses, a few divisions" - second version): The bonus rates |
0
and | for the CEO and managers and the number n of divisions in the firm satisfy the following conditions:
ns [ R - L/(1-|-|
0
)]/[(1+r) - M/(1-|)];
(nM + µ |)/(1-|-|
0
) s (1+r).
PROPOSITION 5 (Geometric default probability for general posted-price mechanism): Suppose the
regularity condition and assumption SBAFD2 hold. If an ICM firm uses a posted-price mechanism with h as
the hurdle rate for project selection and K(h) ÷ |I(h)| as the loan amount for project financing, where I(h) ÷ {i
| ·
i
> (1+h) for ·
i
of ·}, then for any h > 0 with K(h) > 1, the firm's default probability is q
K
(h
)
. Consequently,
its "total cost" of borrowing is C(K(h)), where C(k) ÷ (1+r)k + Lq
k
.
Finally, it is interesting to examine whether the concern for default discussed here can naturally lead
to capital rationing. With perfect bank competition, surplus from reduced default risk is completely captured
by the firm. If it faces an upward sloping capital supply curve, the firm would be analogous to the buyer in
the classic monopsony case. In such circumstances, it would be optimal for the buyer to restrain the quantity
purchased (i.e., capital rationing in this context) to obtain a more favorable purchase price (i.e., a lower cost
of capital).
There are four caveats to this conclusion. First, in the model here the capital needed for investment
is a whole number. If an upward sloping capital supply curve is sufficiently flat, the gain from reducing the
loan amount by one unit can be too little to justify the loss in expected project returns. Second, the capital
supply curve need not be upward sloping and consequently capital rationing could reduce the value created
from project investment. Third, from the shareholders' perspective, they would like to balance the value
created from project investment with the expected loss in shareholder value due to default. The latter
depends on the default probability, which could decrease with the loan amount even when the capital supply
24
curve is upward sloping. So shareholders might dislike capital rationing even when the CEO prefers. Finally,
the whole issue is further complicated by the saving in bonuses paid to managers when a higher hurdle rate
is set for capital rationing. This alone benefits the shareholders but the CEO is indifferent.
When the default probability is geometric, the capital supply curve is downward sloping. Therefore,
capital rationing is not preferred by the CEO. A seemingly unexpected finding is that she would set the
hurdle rate below the cost of capital to "subsidize" divisions for project investment, resulting in some
"capital sponsoring." This however is reasonable given perfect bank competition and geometric default
probability, which is decreasing in the loan amount borrowed. With a subsidized hurdle rate, more projects
would be financed than otherwise would have, which leads to a more favorable term in borrowing. The
positive externality that results makes the subsidized hurdle rate sustainable. This last result is stated below.
PROPOSITION 6 (Capital sponsoring and subsidized hurdle rate): Suppose an ICM firm uses a
posted-price mechanism with h as the hurdle rate for project selection and K(h) ÷ |I(h)| as the loan amount
for project financing, where I(h) ÷ {i | ·
i
> (1+h) for ·
i
of ·}. If the default probability of the firm is
geometric, i.e., q
0
(k, ·) = q
k
, then the "total cost" of borrowing C(k) ÷ (1+r)k + Lq
k
is strictly convex, with
the "average cost" of borrowing C(k)/k = (1+r) + Lq
k
/k decreasing in the loan amount k. Moreover, the
"marginal cost" of borrowing C?(k) = (1+r) + Lq
k
ln(q) is everywhere below the "average cost" of borrowing.
Consequently, the CEO prefers capital sponsoring with a subsidized hurdle rate set at h = C?(A) - 1, where A
÷ max{ a | ·
e
(a
)
> C?(a) for a s n } is the number of projects approved for financing, and e(a) denotes the
index of the ath highest ·
i
of ·.
To illustrate the potential empirical relevance of this result of the model, let me casually point out
how one can obtain a lower-bound estimate of the expected return of the marginal project accepted by a
firm. Note that by definition the expected return of the marginal project, denoted by ·
e
(A
)
, is at least C?(A) =
(1+r) + Lq
A
ln(q). Since C(A)/A = (1+r) + Lq
A
/A and hence Lq
A
= C(A) - (1+r)A, it follows that
·
e
(A
)
> C?(A) = (1+r) + ln(q)[C(A)/A - 1 - r]A.
Empirically, A may be proxied by the borrowing amount of a firm in a period, and C(A)/A - 1 - r by the
interest rate premium. Together with a proxy for the market interest rate r and some estimate of the default
probability q for a division borrowing like a separate legal entity, one can compute an estimate of C?(A),
which serves as a lower-bound estimate of ·
e
(A
)
. Suppose firms are similarly efficient in operating their ICM.
25
Then such an estimate of ·
e
(A
)
could be useful for comparing the investment opportunities of different firms.
It may thus contribute to the valuation of the firms.
4. Discussions
4.1 Relevance to Corporate Finance
The model has potential to shed light on a variety of corporate finance issues, like capital rationing
as a response to managerial overconfidence, value of corporate diversification, boundaries of the firm,
spinoff and acquisition decisions, diversification discount, and "socialistic" capital allocation.
Capital rationing as a response to managerial overconfidence: With the model, I am able to
highlight the relation between capital supply curve and project financing. The analysis on geometric default
probability demonstrates that a downward sloping capital supply curve is incompatible with capital
rationing. In reality, a downward sloping capital supply curve might not occur very often and therefore
capital rationing is much often seen than "capital sponsoring." When capital rationing occurs in a firm with
a downward sloping capital supply curve, does this mean the model developed here is wrong?
Not necessary. Stein [2003] points out that "[a] . potentially very promising agency theory of
investment builds on the premise that managers are likely to be overly optimistic about the prospects of
those assets that are under their control." (p. 123) It has been well recognized that overconfidence as a
cognitive bias exists even in competitive business environments (e.g., Russo and Schoemaker [1992] and
Zacharakis and Shepherd [2001]). What Stein has not pointed out is "inflated" hurdle rate could be a simple
way to curb managerial overconfidence on expected project returns. This "overconfidence" explanation
implies seemingly inefficient capital rationing could be an efficient way to correct for otherwise inefficient
capital allocation. Taking this into account, capital rationing can arise even when a firm faces a downward
sloping capital supply curve. The model thus suggests the following testable hypothesis: Managerial
overconfidence on expected project returns is more likely to be found in firms imposing capital rationing
even with a downward sloping capital supply curve than in those with an upward sloping curve.
Value of corporate diversification: Thought not analyzed in this paper, a firm might be able to better
reduce its default risk and the cost of debt financing by diversifying into different business lines. In contrast,
the business risks of divisions of a focused firm tend to be highly correlated (e.g., due to systematic industry-
level risks), which limits the extent of diversifying the risks. An analysis along this line would need an
extension of the model to allow for correlated project returns; distributions of the expected project returns
26
however can still be independent.
Boundaries of the firm: When more and more business lines of different sectors are pooled together
under a single roof, chances are there would be insufficient talents to manage each of them efficiently. This
would pose a limit on the extent default risk can be reduced through bringing in more business lines of
different sectors. The limit could be a factor determining the boundaries of the firm. An analysis along this
line would need an extension of the model to allow for correlated project returns, as well as asymmetric
expected project returns.
Spinoff and acquisition decisions: A diversified firm might be very careful in keeping optimal the
combination of divisions in different sectors for the purpose of reducing default risk. But unanticipated
external shocks to industry sectors can affect divisions' expected project returns in such a way that some
divisions are no longer worth being included in the conglomerate, or new targets should be acquired to form
a better "portfolio" of divisions in different sectors. Because diversification is less costly when divisions in
different sectors have similar expected project returns, it is conceivable that sometimes a "strong" division
would be spun off, sometimes a "weak" one. What can be sure is that those retained under a single roof are
more similar to each other in terms of expected project returns. Analogously, acquisitions are more likely
when the acquirer's divisions and the acquiree are more similar in their expected project returns. An
extension of the model could also be used to examine such issues.
Diversification discount: Suppose a firm has specialty in its focused sector, which was why it
focused on the sector in the first place. Then it is likely that it does not initially have comparative advantages
in other sectors later brought in for the default risk reduction purpose; otherwise, it would have focused on
those sectors. If divisions of the firm have high expected project returns to begin with, its cost of equity
capital should be relatively low, and therefore it is not urgent to reduce the cost of debt capital either. So
diversification for default risk reduction is not likely to occur at that moment. Later if there are adverse
shocks to the firm's focused sector, expected project returns of its divisions go down and its cost of equity
capital goes up. Then it is both more fruitful and more urgent to consider reducing default risk by
diversification. If diversification takes place at this point, it might seem that diversification leads to a lower
equity value when in fact it is a lower equity value making diversification a more sensible option to
consider. As laid out here, an extension of the model might be useful for understanding the diversification
discount.
27
"Socialistic" capital allocation: The model captures the very essence of potential lack of relation
between current investment opportunities and past investment performance. Imagine the model is repeated
twice to form a two-period model. Suppose project returns are independently distributed among divisions
and across periods, and so are the expected project returns. Moreover, suppose the return of a project in
period 1 is spread out to both periods. Given the lack of relation between expected project returns in the two
periods, a division with a high expected return project in period 1 is not particularly likely to have it again in
period 2 and get the project funded. However, the high expected return project in period 1 is likely to result
in a high realized return spilling over to period 2. Consequently, it might appear that a division with
seemingly high investment efficiency in period 2 fails to prevent capital from flowing to another division
with seemingly low investment efficiency - a phenomenon referred to as "socialistic" capital allocation.
Empirical studies on ICM typically compute investment efficiency of a division on an annual basis.
Hypotheses on the direction of the capital flow are tested accordingly. This approach ignores the fact that (i)
cash flows from past investments affect the investment efficiency computed for the current period; (ii) the
capital flow should be related to current investment opportunities, not past investment performance. An
overlapping generation model using this paper's model as a building block can give guidance to empirical
work on ICM and shed light on the true reason behind the phenomenon of "socialistic" capital allocation.
4.2 Relevance to Accounting
Besides its relevance to a variety of corporate finance issues, the model is also relevant to several
issues in accounting.
Roles of auditor and analyst: Can auditors and analysts play some roles in the model? It has been
assumed that at no cost the CEO can communicate to a bank the expected project returns (i.e., valuation
profile ·) she has learnt through operating an ICM. This is unlikely in reality. Even if the CEO indeed would
be able to provide the managers' project proposals to support her claim on ·, a bank might find it more
convenient to use some "summary statistics" in making the loan term decision. Analyst earnings forecasts
may thus serve as a third-party source of evidence for assessing the truthfulness of the CEO's claim on ·.
By contrast, auditors' job might be literally interpreted as only verifying historical information and
therefore is unrelated to the forward-looking information ·. However, suppose that the model is extended to
an overlapping generation model where the realized return of a project is to arrive at two time points, one at
the end of current year and the other the end of the next year. Then the auditor's verification of the firm's
28
current-year earnings would be informative about the realized project returns of the next year. Consequently,
it would affect the firm's default probability of the next year and hence its COC (i.e., the loan interest rate it
is able to get) at the beginning of the next year for the upcoming year's project financing. Analogously, the
auditor's verification of last year's earnings would be relevant to determining the COC for current year's
project financing.
Even without extending the model this way, the one-shot model can provide a role for auditor if
there is imperfect information about B and M, i.e., the assets in place and the cash flow from regular
operations, respectively. These parameters of the model in general can affect the default probability. When
they are not known with certainty, a financial audit on the balance sheet and income statement of the firm
can reduce the uncertainty about B and M and potentially affect the COC through affecting the default risk
assessment.
Experimental studies on ICM: In recent years, there is a growing interest in accounting and
economics investigating whether behavioral factors like honesty, trust, fairness, etc can play a role in
competitive business environments. For example, Bruggen and Luft [2008] experimentally examine how
competition and honesty interact in the capital budgeting process of an ICM setting. Some of the predictions
are derived from informal theorizing because they could not find any existing model that considers a varying
degree of competition in ICM (i.e., whether only one, two, or all three of the divisions in their setting can
"win" the capital for project investment). The model of this paper fills a gap between two extreme types of
models in the literature, namely "only one can win" and "everyone gets a bite". It provides a theoretical
benchmark for comparing to behaviors of non-(purely-)economic agents in an ICM setting and could be
useful to experimental studies looking at similar settings.
Performance evaluation measures: Although not analyzed in this paper, the model may be used to
examine the relative merits of different accounting performance measures. For example, return on
investment (ROI) is widely discussed in accounting textbooks. Can it be a useful alternative to the residual
income (RI) presumed in the model? RI is often argued to be superior to ROI, yet the latter is believed to be
useful for comparing divisions with very different sizes. If the model is extended to allow for divisions with
asymmetric sizes, could ROI become superior to RI under particular circumstances?
Cost of capital and required rate of return: RI has been advocated as the correct approach to
comparing projects for investment decisions. Yet the specification of the required rate of return remains
29
problematic in practice. Some have argued it should be set at the WACC, which is a widely used estimate of
a firm's COC. The model here provides a micro-foundation for the relation between the required rate of
return (i.e., the hurdle rate h) and the "marginal cost" of borrowing (i.e., C?(k)) to which the COC (i.e.,
C(k)/k - 1) is closely tied. Further improvement on the model to incorporate stock price for shares of the
firm would clear up the role of WACC in this setting.
Accounting ratios and cost of capital: The model offers potential to build a micro-foundation for the
relations between widely used accounting ratios and COC. For example, the equity value of an ICM firm is
affected by the book value of assets in place and the default probability, with the latter tied to the COC. This
might shed light on the linkage between book-to-market ratio and COC. If the model can be generalized to
let the book value of assets and cash flow from regular operations affect the default probability, it might
shed light on the linkage between leverage ratio (or current ratio, etc) and COC as well. But these are
unrelated when restricted to geometric default probability.
4.3 Unanswered Questions
The analysis in this paper is only the first step in exploring the potential of the model and applying it
to understand some of the corporate finance and accounting issues that may be examined with the model. A
number of open questions remain.
Default probability: The analysis of the model is substantially simplified by the assumption of a
specific distributional structure of the project returns. The probability mass at the lower end of the
distribution's support and its disconnection from the smooth part of the distribution is critical to establishing
a default probability unrelated to the expected returns of projects in divisions of an "autonomous" firm. This
in turn allows a simple characterization of the COC charged to the divisions. Can alternative assumptions on
the distributional structure also lead to tractable analyses?
CEO's preference: Can the quasi-linear objective function of the CEO be formally justified by a
career-concern or other types of models? Ignoring the cost-of-borrowing component of the CEO's objective
function will lead to an often assumed preference for empire building (based on gross output). How would
this alternative assumption change the results here?
Division managers' preferences: That a manager cares about the residual income of his division's
project is important to making the model work. Otherwise, the accounting charges would have no impact on
managers' behaviors, and the model would fall apart. Can alternative preferences be assumed to make the
30
model work for other performance evaluation measures?
Compensation contracts observed in practice: The model as it stands excludes incentive
compensations that are found in practice, e.g., budget-based bonus schemes (Sprinkle, Williamson, and
Upton [2008] and Murphy [2001]). Generalizing the model to allow for such possibilities would provide
interesting venues for future research.
Other roles of the headquarters: The model assumes a highly decentralized firm environment, where
the only role of the corporate headquarters (HQ), headed by the CEO, is to allocate capital raised from
outside. It ignores the monitoring role the HQ can play and related control right issues that have been
extensively studied in the literature (e.g., Gertner, Scharfstein, and Stein [1994], Scharfstein and Stein
[2000], and Stein [1997, 2002]). It is certainly interesting to generalize the model to include monitoring
activities and control rights of the HQ.
Non-geometric default probability: The characterization provided in the paper admits a wide range
of default probability besides geometric. Can other assumptions be made to restrict the structure of default
probability to other tractable forms? Advances in this direction would widen applications of the model.
5. Concluding Remarks
This paper outlines a theoretical framework for studying the endogenous determination of the COC.
Recognizing the endogenous relationship between a firm's investment activities and COC has important
implications to empirical work and the valuation of firms. While the theoretical framework proposed here
remains in its infancy stage, its potential is worth further exploration, as suggested by the various corporate
finance and accounting issues that may be examined with this framework.
Using the model of this paper, I have derived results of which some are restricted to the case of
geometric default probability valid under certain conditions. The importance of conducting a micro-level
analysis and considering this restrictive case is to provide a new angle in rethinking issues that might have
been overlooked under the usually presumed case of an upward sloping capital supply curve. Further
analysis to go beyond geometric default probability is certainly interesting. This is left to future research.
A number of potentially interesting extensions of the model have been discussed in an earlier
section. Yet another one is to allow for asymmetric divisions and an endogenous mix of the two sources of
capital, i.e., internal financing via the ICM and external bank debt. As a first step in exploring the potential
of the model, this paper has restricted attention to the polar cases of "autonomous" or ICM firm.
31
Dewaelheyns and Van Hulle [2008] point out that for some business groups, "divisions" are legally separate
member firms with direct access to the external capital markets, alongside with internal debts provided by
parent companies. Allowing for an endogenous mix of internal and external debts might yield testable
implications suggesting interesting empirical work.
32
REFERENCES
Baldenius, T., Dutta, S., and Reichelstein, S. [2007]: "Cost Allocation for Capital Budgeting Decisions,"
Accounting Review, 82, 837-867.
Beck, T., Demirguc-Kunt, A., and Maksimovic, V. [2008]: "Financing patterns around the world: Are small
firms different?" Journal of Financial Economics, 89, 467-487.
Bergemann, D. and Morris, S. [2005]: "Robust Mechanism Design," Econometrica, 73, 1771-1813.
Bernardo, A. E., Cai, H., and Luo, J. [2004]: "Capital Budgeting in Multidivision Firms: Information,
Agency, and Incentives," Review of Financial Studies, 17, 739-767.
Bernardo, A. E., Luo, J., and Wang, J. [2006]: "A theory of socialistic internal capital markets," Journal of
Financial Economics, 61, 311-344.
Brealey, R. A., Myers, S. C., and Allen, F. [2006]: Corporate Finance, 8
th
Edition. McGraw-Hill/Irwin.
Bruggen, A. and Luft, J. [2008]: "Capital Rationing, Competition, And Misrepresentation in Budget
Forecasts," Working paper.
de Motta, A. [2003]: "Managerial Incentives and Internal Capital Markets," Journal of Finance, 58, 1193-
1120.
Demirguc-Kunt, A. and Maksimovic, V. [1996]: "Stock Market Development and Financing Choices of
Firms," World Bank Economic Review, 10, 341-369.
Demirguc-Kunt, A. and Maksimovic, V. [1999]: "Institutions, financial markets and firm debt maturity,"
Journal of Financial Economics, 54, 295-336.
Dewaelheyns, N. and Van Hulle, C. [2008]: "Internal Capital Markets and Capital Structure: Bank Versus
Internal Debt," European Financial Management, doi: 10.1111/j.1468-036X.2008.00457.x .
Easton, P. D. and Monahan, S. J. [2005]: "An Evaluation of Accounting-Based Measures of Expected
Returns," Accounting Review, 80, 501-538.
Ferson, W. E. and Locke, D. H. [1998]: "Estimating the Cost of Capital Through Time: An Analysis of the
Sources of Error," Management Science, 44, 485-500.
Francis, J., LaFond, R., Olsson, P. M., and Schipper, K. [2004]: "Costs of Equity and Earnings Attributes,"
Accounting Review, 79, 967-1010.
Garrison, R. H., Noreen, E. W., and Brewer, P. C. [2008]: Managerial Accounting, 12
th
Edition. McGraw-
Hill/Irwin.
Gebhardt, W. R., Lee, C. M. C., and Swaminathan, B. [2001]: "Toward an Implied Cost of Capital," Journal
of Accounting Research, 39, 135-176.
Gertner, R., Powers, E., and Scharfstein, D. [2002]: "Learning about Internal Capital Markets from
Corporate Spin-offs," Journal of Finance, 57, 2479-2506.
Gertner, R., Scharfstein, D., and Stein, J. [1994]: "Internal Versus External Capital Markets," Quarterly
33
Journal of Economics, 109, 1211-1230.
Hadlock, C., Ryngaert, M., and Thomas, S. [2001]: "Corporate Structure and Equity Offerings: Are There
Benefits to Diversification," Journal of Business, 74, 613-635.
Hail, L. and Leuz, C. [2006]: "International Differences in the Cost of Equity Capital: Do Legal Institutions
and Securities Regulation Matter?" Journal of Accounting Research, 44, 485-531.
Inderst, R. and Laux, C. [2005]: "Incentives in internal capital markets: capital constraints, competition, and
investment opportunities," Rand Journal of Economics, 36, 215-228.
Inderst, R. and Muller, H. M. [2003]: "Internal versus External Financing: An Optimal Contracting
Approach," Journal of Finance, 58, 1033-1062.
Khanna, N. and Tice, S. [2001]: "The Bright Side of Internal Capital Markets," Journal of Finance, 56,
1489-1528.
Kieso, D. E., Weygandt, J. J., and Warfield, T. D. [2006]: Intermediate Accounting, 12th Edition. Wiley.
Lambert, R., Leuz, C., and Verrecchia, R. E. [2007]: "Accounting Information, Disclosure, and the Cost of
Capital," Journal of Accounting Research, 45, 385-426.
Lamont, O. [1997]: "Cash flow and investment: Evidence from internal capital markets," Journal of
Finance, 52, 83-109.
Melumad, N., Mookherjee, D., and Reichelstein, S. [1992]: "A theory of responsibility centers," Journal of
Accounting and Economics, 15, 445-484.
Modigliani, F. and Miller, M. H. [1958]: "The Cost of Capital, Corporation Finance and the Theory of
Investment," American Economic Review, 48, 261-297.
Mukherjee, T. K. and Henderson, G. V. [1987]: "The Capital Budgeting Process: Theory and Practice,"
Interfaces, 17, 78-90.
Murphy, K. J. [2001]: "Performance Standards in Incentive Contracts," Journal of Accounting and
Economics, 30, 245-278.
Rajan, R., Servaes, H., and Zingales, L. [2000]: "Cost of Diversity: The Diversification Discount and
Inefficient Investment," Journal of Finance, 55, 35-80.
Russo, J. E. and Schoemaker, P. J. H. [1992]: "Managing Overconfidence," Sloan Management Review, 33,
7-17.
Saitoh, H. and Serizawa, S. [2008]: "Vickrey allocation rule with income effect," Economic Theory, 35, 391-
401.
Scharfstein, D. S. [1998]: "The Dark Side of Internal Capital Markets II: Evidence from Diversified
Conglomerates," NBER-w6352.
Scharfstein, D. S. and Stein, J. C. [2000]: "The dark side of internal capital markets: Divisional rent-seeking
and inefficient investment," Journal of Finance, 55, 2537-2564.
Segal, I. [2003]: "Optimal Pricing Mechanisms with Unknown Demand," American Economic Review, 93,
34
509-529.
Segelod, E. [1996]: "Resource allocation in divisionalized groups: a survey of major Swedish groups,"
unpublished manuscript. (http://publications.uu.se/abstract.xsql?dbid=2349)
Segelod, E. [1995]: Resource allocation in divisionalized groups: a survey of major Swedish groups.
(http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-2349).
Shin, H. and Stulz, R. [1998]: "Are internal capital markets efficient," Quarterly Journal of Economics, 113,
531-552.
Sprinkle, and Williamson, and Upton [2008]: "The effort and risk-taking effects of budget-based contracts,"
Accounting, Organizations, and Society, 33, 436-452.
Stein, J. C. [1997]: "Internal Capital Markets and the Competition for Corporate Resources,"Journal of
Finance, 52, 111-133.
Stein, J. C. [2002]: "Information Production and Capital Allocation: Decentralized versus Hierarchical
Firms," Journal of Finance, 57, 1891-1921.
Stein, J. C. [2003]: "Agency, Information and Corporate Investment," in Handbook of the Economics of
Finance, Constantinides, Harris, Stulz (ed.).
Stewart, III, G. B., Ellis, Mark, and Budington, D. [2002]: "Stern Stewart's EVA
®
Clients Outperform the
Market and Their Peers," EVAluation, October 2002. Stern Stewart Research, the Americas.
Stice, J. D., Stice, E. K., and Skousen, F. [2007]: Intermediate Accounting, 16th Edition. Thomson South-
Western.
Wulf, J. [2002]: "Internal capital markets and firm-level compensation incentives for division managers,"
Journal of Labor Economics, 20, 5219-5262.
Wulf, J. [2006]: "Influence and Inefficiency in the Internal Capital Market," SSRN-id145908.
Zacharakis, A. L. and Shepherd, D. A. [2001]: "The nature of information and overconfidence on venture
capitalists' decision making," Journal of Business Venturing, 16, 311-332.
35
APPENDIX
PROOF OF LEMMA 2: I will first show that under the regularity condition, the default probability
of a "winning" division (i.e., one provided with a loan) is q, regardless of ·. Consequently, t
i
(·) must be set
at (1+c) ÷ (1+r) + qL for a "winning" division. Requirements of an ex post mechanism then imply it is
weakly better to set p
i
(·
-
i
) = (1+c). As a result, the requirements also imply t
i
(·) = 0 for a "losing" division
(i.e., one declined for a loan), which would complete the proof.
First, note that given any ex post mechanism used by a bank, managers of the divisions will report
their ·
i
's truthfully. Moreover, bank competition ensures that t
i
(·) s (1+r) + L for a "winning" division.
Otherwise, a competing bank could have offered an ex post mechanism with t
i
(·) = (1+r) + L for a "winning
division." Any division willing to accept t
i
(·) > (1+r) + L will find this alternative offer more attractive. Since
division i's year-end cash position, with bonus to the manager already paid, is M + R
i
- |(R
i
- t
i
(·)),
parts a and b of the regularity condition imply that (i) the probability of default is less than 1, and (ii) if legal
procedures of collection are invoked, the competing bank will surely get back t
i
(·) = (1+r) + L for any
realized R
i
leading to default. Consequently, charging t
i
(·) = (1+r) + L will lead to a positive expected profit for
the competing bank, which violates the condition for perfect competition. Hence, the charge t
i
(·) to a
"winning" division must not exceed (1+r) + L.
Now recall that the year-end cash position of a division is M + R
i
- |(R
i
- t
i
(·)). This will be
sufficient to meet the payment obligation to the bank if M + R
i
- |(R
i
- t
i
(·)) > t
i
(·), which is equivalent to
t
i
(·) s R
i
+ M/(1-|). With probability 1 - q, the right hand side of this inequality is at least R + M/(1-|).
Under part c of the regularity condition, this is no less than (1+r) + L, which in turn is no less than t
i
(·).
Thus, the default probability of a division must be at most q, regardless of ·. Bank competition thus implies
t
i
(·) s (1+r) + qL for a "winning" division.
With probability q, R
i
= 0, and division i's year-end cash position is only M + |t
i
(·). For the division to meet
its payment obligation, it requires that t
i
(·) s M/(1-|). Note that t
i
(·) > (1+r), which is a floor of the cost of lending, even
without default risk. Under part d of the regularity condition, M/(1-|) < (1+r) s t
i
(·).
Thus, the division will default when R
i
= 0. In other words,
Pr{default by division i | ·} = q.
Now suppose t
i
(·) < (1+r) + qL for a "winning" division. This will result in a negative expected
profit to a bank unless invoking legal procedures in case of default is unwise sometimes. However, this
cannot be under part d of the regularity condition. To see this, recall that default occurs only when R
i
= 0,
which means the cash position of the division is only M + |t
i
(·). For this to be as great as the bank's payoff from
invoking the legal procedures, it requires t
i
(·) s (M+L)/(1-|). Since t
i
(·) > (1+r), the inequality cannot
hold under part d of the regularity condition. Therefore, it is always wise to invoke legal procedures in case
i
of default. As a result, setting t
i
(·) < (1+r) + qL for a "winning" division will lead to a negative expected
profit to a bank. This means only ex post mechanisms with t
i
(·) = (1+c) ÷ (1+r) + qL for a "winning"
division will be used.
Recall that for ?x(?), t(?)? to be an ex post mechanism, there must exist p
i
, s
i
: ( µ , µ )
n
-1
÷ R
+
such
that for every · e ( µ , µ )
n
,
x
i
(·) = 1 if ·
i
> p
i
(·
-
i
), x
i
(·) = 0 otherwise, and
t
i
(·) = p
i
(·
-
i
) x
i
(·) - s
i
(·
-
i
).
For a "winning" division, t
i
(·) = (1+c). Since s
i
(·
-
i
) > 0, it follows that p
i
(·
-
i
) > (1+c).
Suppose the loan approval cutoff p
i
(·
-
i
) is chosen to be above (1+c). Then potential surplus from a
division with ·
i
such that (1+c) < ·
i
s p
i
(·
-
i
) would not be realized under such a mechanism. A competing bank could
have used an ex post mechanism with t
i
(·) set strictly in between (1+c) and p
i
(·
-
i
) and an approval cutoff
identical to this t
i
(·). This competing bank would then do at least as good as the bank setting p
i
(·
-
i
) > (1+c) and
possibly better when indeed some divisions have ·
i
's falling in between this competing bank's t
i
(·) and the
other bank's p
i
(·
-
i
). In short, setting p
i
(·
-
i
) > (1+c) is weakly dominated. Consequently, an ex post mechanism with
p
i
(·
-
i
) = (1+c) is as good as any other ex post mechanisms.
Given that p
i
(·
-
i
) = t
i
(·) = (1+c) for a "winning" division, it must be that s
i
(·
-
i
) = 0 for ·
i
> (1+c).
However, s
i
(·
-
i
) cannot depend on ·
i
. It follows that s
i
(·
-
i
) = 0 even for ·
i
s (1+c).
In conclusion, a posted-price mechanism with
x
i
(·) = 1 and t
i
(·) = (1+c) if ·
i
> (1+c), and
x
i
(·) = t
i
(·) = 0 otherwise
is at least as good as any other ex post mechanisms to a bank. ?
PROOF OF LEMMA 3: As explained in the main text, it suffices to show that f(q
0
, 1, n, c) s 0 but
f(q
0
, 0, n, c) > 0. First, note that c = r + qL and when K = n, c
0
= r + q
0
L/n. Thus,
f(q
0
, 0, n, c) = n[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o
= [(1+c)o - (c - c
0
)(1-|
0
) - M]n/o
= {(1+r)o + [qo - (q - q
0
/n)(1-|
0
)]L - M}n/o
= {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M}n/o,
which is positive if
(1+r) > [M + (q| - (q
0
/n)(1-|
0
))L]/o.
To show that this inequality indeed holds, note that
o(M+L) - (1-|)[M + (q| - (q
0
/n)(1-|
0
))L]
= (1-|)(M+L) - |
0
(M+L) - (1-|)M - (1-|)(q| - (q
0
/n)(1-|
0
))L
= (1-|)[1- q| + (q
0
/n)(1-|
0
)]L - |
0
(M+L)
ii
> (1-|)(1- q|)L - |
0
(M+L)
> 0,
where the last inequality is due to the given assumption that |
0
< (1-|)(1- q|)L/(M+L). Hence,
(M+L)/(1-|) > [M + (q| - (q
0
/n)(1-|
0
))L]/o.
By part d of the regularity condition,
(1+r) > (M+L)/(1-|)
> [M + (q| - (q
0
/n)(1-|
0
))L]/o,
which implies f(q
0
, 0, n, c) > 0.
Now consider
f(q
0
, 1, n, c) = n[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - R
= f(q
0
, 0, n, c) - R ,
which is non-positive if f(q
0
, 0, n, c) s R . Note that the assumption ns ( R - L/o)/[(1+r) - M/(1-|)] implies
ns R /[(1+r) - M/(1-|) + L/no].
Thus,
f(q
0
, 0, n, c)
= {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M}n/o
s {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M} R /[(1+r) - M/(1-|) + L/no]o
= R {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M}/[(1+r) - M/(1-|) + L/no]o.
To see that {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M}/[(1+r) - M/(1-|) + L/no]o< 1, consider
(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M - [(1+r) - M/(1-|) + L/no]o
= - [1 - q
0
(1-|
0
)]L/n - q|L - |
0
M/(1-|)
< 0.
Since {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M} > 0, it follows that f(q
0
, 0, n, c) < R .
As f(q
0
, 1, n, c) s 0 and f(q
0
, 0, n, c) > 0, q
0
(n, ·) = q
n
. ?
PROOF OF PROPOSITION 3: As explained in the main text, it suffices to show that f(q
0
, 0, 1, c) >
0. Note that
f(q
0
, 0, 1, c) = [(1+r)o - q|L - nM + q
0
(1-|
0
)L]/o,
which is clearly positive given the additional assumption on n and |
0
. ?
PROOF OF PROPOSITION 4: Given the regularity condition and assumption SBAFD1, q
0
(K, ·) =
q
K
and (1+c
0
)K = (1+r)K + q
K
L. Therefore, the difference obtained by deducting an "autonomous" firm's
equity value AF from an ICM firm's equity value lower bound IF is as follows:
?÷ [Kq - nq
K
](1-ì)B + [Kq - q
K
]L - |
0
[E
i
eI(c
)
·
i
- (1+r)K - Lq
K
].
iii
First, recognize that µ K> E
i
eI(c
)
·
i
> (1+c)K = [(1+r) + qL]K > (1+r)K + q
K
L. Since
|
0*
÷ [(q - nq
K
/K)(1-ì)B + (q - q
K
/K)L]/( µ - (1+r) - Lq
K
/K),
it follows that |
0*
( µ K - (1+r)K - q
K
L) = (Kq - nq
K
)(1-ì)B + (Kq - q
K
)L, which is positive under the
condition that 2 s K s n s [2((1-ì)B + L)/q - L]/(1-ì)B. For |
0
s |
0*
,
|
0
[E
i
eI(c
)
·
i
- (1+r)K - q
K
L]
s |
0*
[E
i
eI(c
)
·
i
- (1+r)K - q
K
L]
< |
0*
( µ K - (1+r)K - q
K
L)
= (Kq - nq
K
)(1-ì)B + (Kq - q
K
)L,
which means ?> 0. ?
PROOF OF PROPOSITION 5: Note that
F(q
0
, s, h) ÷ f(q
0
, s, K(h), h)
= K(h) [(1+c
0
)(1-|
0
) - (1+h)|]/o - nM/o - s R
= K(h)[(1+ r)(1-|
0
) - (1+h)|]/o + q
0
L(1-|
0
)/o - nM/o - s R .
It suffices to show that for any h > 0 with K(h) > 1, F(q
0
, 0, h) > 0 and F(q
0
, 1, h) s 0.
To show that F(q
0
, 0, h) > 0 for any h > 0 with K(h) > 1, it suffices to consider only h< µ - 1;
otherwise, I(h) = ? and K(h) = 0. For h with K(h) > 1,
F(q
0
, 0, h)
= K(h)[(1+ r)(1-|
0
) - (1+h)|]/o + q
0
L(1-|
0
)/o - nM/o
> [(1+ r)(1-|
0
) - µ |]/o + q
0
L(1-|
0
)/o - nM/o
> [(1+ r)(1-|
0
) - ( µ | + nM)]/o,
which is non-negative under assumption SBAFD2. Thus, F(q
0
, 0, h) > 0 for any h with K(h) > 1.
Now consider
F(q
0
, 1, h)
= K(h)[(1+r)(1-|
0
) - (1+h)|]/o + q
0
L(1-|
0
)/o - nM/o - R
< K(h)(1+r)(1-|
0
)/o + q
0
L(1-|
0
)/o - nM/o - R
s n(1+r)(1-|
0
)/o + L(1-|
0
)/o - nM/o - R
= n[(1+r) - M/o] + L(1-|
0
)/o - R .
To see that this is negative, note that o = (1-|-|
0
) < (1-|) implies (1+r) - M/(1-|) > (1+ r) - M/o, which is
positive under assumption SBAFD2. Additionally, by the assumption,
ns ( R - L/o)/[(1+r) - M/(1-|)]
iv
< ( R - L/o)/[(1+ r) - M/o]
< [ R - L(1-|
0
)/o]/[(1+ r) - M/o].
Therefore, F(q
0
, 1, h) < 0 for any h > 0. ?
PROOF OF PROPOSITION 6: Recall that the CEO's objective is to maximize her bonus, which is
maximized when the value created from project investment is maximized, i.e.,
Max
h
E
i
eI(h
)
·
i
- C(K(h)),
where K(h) ÷ |I(h)| and I(h) ÷ {i | ·
i
> (1+h) for ·
i
of ·}.
Let e(a) denote the index of the ath highest ·
i
of ·, and hence ·
e
(a
)
is the ath highest ·
i
.
16
Define A÷ max{ a | ·
e
(a
)
>
C?(a) for a s n }.
It is straightforward to show that C(k) ÷ (1+r)k + Lq
k
is strictly convex with C?(k) = (1+r) + Lq
k
ln(q) > 0 for
k> k
*
÷ ln[-Lln(q)/(1+r)]/[-ln(q)]. The strict convexity of C(k) means C?(k) is increasing in k. So borrowing
a loan amount beyond A to finance some project with ·
e
(a
)
< ·
e
(A
)
can only reduce the value created from
project investment. Because K(h) = A for h = C?(A) - 1, setting the hurdle rate at this level is optimal to the
CEO if any loan amount a< A is suboptimal.
Note that the capital supply curve C(k)/k = (1+r) + Lq
k
/k is downward sloping. If the loan amount is
reduced to a< A, the "average cost" of borrowing (1+c
0
) = (1+r) + Lq
a
/a is higher for the projects financed
while the number of projects financed is smaller. With the same · given, the value created from project
investment must be lower if a loan amount a< A is borrowed. ?
16
For simplicity, ties are ignored; they are zero-probability events if · is drawn from a continuous probability
distribution.
v
Incomplete info. : · known only to managers
privately
ICM modeled as a multi-unit hurdle rate h set according to
auction: ¢x(?), t(?)² anticipated c
0
as a function of
A , the number of projects approved
Complete info : · learnt by the CEO thru'
operating the ICM
· communicated to
banks at no cost
Default probability: q 0
Transaction cost(s) of debt financing:
L, Legal cost of collection in case of default
(also, for shareholders, there is a loss in shareholder
value due to forced liquidation of assets)
Cost of capital: (1+c
0
)A
Perfect bank competition
(1+r )A + q
0
L
Figure 1. Important Elements of the Model
Figure 2. Geometric Default Probability Implies Capital
Sponsoring and Subsidized Hurdle Rate
(1+c)
(1+c)
(1+r)
mw(a)
·
e
(k)
C(k)/k
C(k) = (1+r)k + Lq
k
C(k)/k = (1+r) + Lq
k
/k
C?(k) = (1+r) + Lq
k
ln(q)
C(k)+
k
r)
(1+h) (1
/
C'(k)
C?(k)
A
1 2 3 4 5 6 7 8 9 10
k
doc_198676996.docx
Financial Study Reports on Cost of Capital, Hurdle Rate, and Default Probability: A Robust Model of Internal Capital Markets:- A company's financial needs or goals for the future. Corporate financial planning involves identifying these financial objectives and determining how to achieve them.
Financial Study Reports on Cost of Capital, Hurdle Rate,
and Default Probability: A Robust Model of Internal
Capital Markets
Abstract. This paper adds to the understanding of the cost of capital (COC) by providing a micro-
level analysis of the endogenous relation between a firm's investment activities and COC. Building upon
recent results on multi-unit auctions in the mechanism design literature, I formulate a multi-division firm
model endogenously connecting a firm's COC to the hurdle rate for project selection, through the default
probability of the firm. Other models typically take the COC as given and examine how the hurdle rate or
other variables of interest are linked to the COC. In the model here, there is a transaction cost of financing,
namely, the legal cost of collection in case of default. Banks set the loan interest rate taking into account the
default risk factor, which can be affected by the number of projects financed by a firm. Anticipating the
impact on the COC (i.e., loan interest rate), the firm sets a hurdle rate accordingly to determine the number
of projects to invest and finance. As this affects the firm's default probability, the COC and the hurdle rate
are jointly determined. The model demonstrates how a firm may operate an internal capital market (ICM) to
elicit private information from division managers and allocate capital to their projects efficiently. It is robust
in the sense that the results are not sensitive to any prior belief about division managers' private information.
Under conditions labeled as "small bonuses, a few divisions," the default probability takes a simple
geometric form, which induces a downward sloping (inverse) capital supply curve. Interestingly, this leads
to "capital sponsoring" with a subsidized hurdle rate, rather than capital rationing. The model is also used to
understand why operating an ICM may be better than giving division managers complete autonomy in
investing and financing decisions. Such reasons include a lower COC and a smaller expected loss in
shareholder value in case of default, both due to reducing the default probability. (JEL G31, G33, D23, L22)
Keywords: Cost of capital, hurdle rate, default risk, project selection, project financing, capital
rationing.
Cost of Capital, Hurdle Rate, and Default Probability: A Robust Model of Internal Capital Markets
1. Introduction
It has been repeatedly mentioned in the literature that understanding the cost of capital (COC) is
important to management accountants and corporate finance specialists, as well as to academic researchers
and even government agencies regulating utility companies (see e.g. Modigliani and Miller's [1958] classic
paper). Empirical researchers pursuing studies on COC certainly are interested in knowing more about its
theoretical properties, in particular, how it is determined by firm attributes and factors of the surrounding
environments.
1
Regulatory agencies such as SEC also emphasize their decisions' impact on COC and
recognize it as an important consideration in setting regulations.
2
Despite its importance, COC remains a mystery. It is ofttimes an unexplained, exogenous constant in
theoretical models, and in practice a hard-to-estimate unknown. Management accounting textbooks instruct
students to use COC as the discount rate/required rate of return for selecting investment projects, yet with
little discussion of how to determine the COC (e.g., Garrison et al [2008]). Finance textbooks teach the use
of weighted average cost of capital (WACC) as an estimate for COC in spite of the recognized limitations of
this approximation (Brealey et al [2006]). Finance scholars and professionals have long relied on the capital
asset pricing model to estimate COC from stock returns, however, without being entirely satisfied with the
method (Ferson and Locke [1998]). A fast growing literature in accounting instead uses the abnormal
earnings model to infer COC from stock prices (e.g., Gebhardt et al [2001]). How useful this method is
remains to be ascertained (Easton and Monahan [2005]). What clear to us is that our understanding of COC
is still limited.
1
Empirical research on COC is growing quickly. For example, Hail and Leuz [2006] study whether cross-country
differences in COC are related to the legal institutions and securities regulations of different countries. Francis et al [2004]
study the relation between earnings attributes and COC. Gebhardt et al [2001] advocate their approach to estimating COC to
researchers studying the effects of disclosure policies on COC.
2
For example, in his speech on September 13, 1999, entitled "Small Business: The Lifeblood of Our Nation's
Economy," former SEC commissioner Norman S. Johnson said, "I look forward to working with you to find ways to
increase the availability and lower the cost of capital to small businesses." Similarly, on April 21, 1997, former SEC chairman
Arthur Levitt in his remarks entitled "Small Business Makes a Large Contribution" said, "[T]he SEC adopted new rules that I believe
will reduce the cost of private capital formation and especially benefit small businesses." Speaking even more directly, former SEC
chairman Richard C. Breeden in his remarks on September 21, 1990 said, "At the SEC, ... we are trying to take steps available to us
to reduce the cost of raising capital. ... [W]e will seek to promote the competitiveness and
vitality of U.S. business . by trying to reduce as much as possible the cost of capital to U.S. firms."
1
This paper adds to the limited understanding of COC by providing a micro-level analysis of the
endogenous relation between a firm's investment activities and COC. Specifically, I formulate a model of a
firm with multiple divisions, each with a project. The firm operates an internal capital market (ICM) to elicit
private information from division managers and allocate capital to their projects. With the model, I derive
results linking the COC endogenously to the hurdle rate for project selection, through the default probability
of the firm.
Recognizing the endogenous determination of COC has important implications to empirical work
and the valuation of firms. Two firms estimated to have similar COC and appearing to have similar growth
opportunities might actually face quite different capital supply curves. Consequently, they will react
unequally to similar investment opportunities available in the future. The valuation of these firms today
should thus be different. Correct valuation of these firms requires, among other things, estimating their
capital supply curves that constrain the project selections and ultimately determine their actual investments.
Related empirical work therefore should focus on the whole capital supply curve, rather than merely a given
COC determined by the curve for a particular moment in time.
There have been many models of firms with multiple divisions. For example, in accounting, models
have been proposed to examine how a transfer pricing system can be designed to allocate costs to divisions
optimally (e.g., Melumad, Mookherjee, and Reichelstein [1992]). These models typically focus on operating
activities of a firm and say little about their relation with the COC. There are also multi-division models of
capital budgeting that tie the hurdle rate for project selection to the COC (e.g., Baldenius, Dutta, and
Reichelstein [2007]). But the COC typically is exogenously given in the models.
3
So a firm's COC is not
affected by the firm's "growth opportunities", e.g., the number of good projects it has in a year.
In finance, models of multiple divisions have been used to study ICM. A partial list includes
Bernardo et al [2006, 2004], Inderst and Laux [2005], Ozbas [2005], de Motta [2003], Inderst and Muller
[2003], Stein [2002], Scharfstein and Stein [2000], Stein [1997], and Gertner, Scharfstein, and Stein [1994].
These models either assume only one "winning" division can obtain financing from the firm, or every
3
For multi-firm (more precisely, multi-asset) models, rather than single-firm, multi-division models like mine, an
exception is Lambert, Leuz, and Verrecchia's [2007] study. They, however, use a different modeling approach based on the
capital asset pricing model. Another exception is Hughes, Liu, and Liu's [2007] study that uses a multi-asset, factor-structure model
with informed and uninformed investors. Both studies focus on the cost of equity capital and say little about firms relying mainly
on debt capital, e.g., like business groups studied by Dewaelheyns and Van Hulle [2008].
2
division will receive some fraction of the available capital. In practice, it is likely that divisions compete for
financial resources in a way that is somewhere in between: not as tough as only one can win, nor as widely
spread as every surely gets a bite.
4
In contrast to others, the multi-division model of this paper endogenously links a firm's COC to the
number of projects invested by the firm; in other words, one or more divisions will "win" but usually not all.
With such an analysis, the paper provides a micro-foundation to the relation between financing and investing
activities. On top of this basic objective, it is hoped that the model can be robust and tractable. Robustness
gives some guarantee to the reliability of the results so that they are not too sensitive to details of the model
specification, such as a correctly specified prior belief about the division managers' private information.
Tractability allows the model to serve as a simple baseline construct for further refinements or extensions so
that a single family of models sharing the same structure can be used to understand a variety of corporate
finance and accounting issues.
I view the model of this paper a normative one, in the sense that it illustrates how a firm can elicit
private information from managers so as to allocate capital efficiently. Notwithstanding this, it can also have
testable implications. For example, for firms following practices suggested by the model, one should expect
to see more efficient investment across divisions than in others that allocate capital quite differently, of
course assuming other factors are constant.
Figure 1 is an overview of the model's important elements. While I draw upon recent results on
multi-unit auctions to build the model, I make no innovation in the multi-unit auction used as a mechanism
to elicit private information and allocate capital. The use of the mechanism in the model is twofold. First, it
converts an incomplete-information setting that is difficult to analyze into a more tractable complete-
information setting, upon which more complex models can be built. Second, the conversion cannot be done
in an arbitrary manner. To fulfill incentive compatibility and individual rationality constraints to be detailed
in Section 3, the multi-unit auction has to take the form of an ex post mechanism of which a component can
be naturally interpreted as a hurdle rate, which is widely used in practice. This and other structures of an ex
post mechanism provide the basis for deriving other results in the paper, which are my contributions.
4
In summarizing some survey evidence, Mukherjee and Henderson [1987] note that the reported project
acceptance rate ranged from over 75% to over 90%. Segelod's [1995] field study provides some institutional details on the
resource allocation process of a divisionalized firm.
3
Through operating an ICM (i.e., using a multi-unit auction to elicit private information and allocate
capital), the firm will become informed of the valuation profile · that represents managers' private
information about the expected returns of their projects. I assume that at no cost the firm can communicate
· to a bank. While unrealistic, I believe this is the best assumption to make at this stage of developing the
model. Components of the model have existed in the literature for some time but assembling them to form
this model is new. Given the complexity of the model, a more realistic assumption about how · can be
communicated will only confuse the readers more, instead of helping them to understand how the model
functions. This does not mean future extensions or refinements of the model should continue to adopt this
unrealistic assumption. Indeed, as discussed in Section 4, the roles of auditor and analyst come into play
when this restrictive assumption is relaxed, which is one of the reasons why the model is relevant to
accounting besides its relevance to corporate finance.
In the model, internally generated cash flow is supposed to be insufficient to fund project
investment. A second important simplifying assumption is that borrowing from a bank is the firm's only
financing source. While restrictive, it is not unrealistic; there are big firms and numerous smaller firms in the
world that are unlisted and rely heavily on bank loans to meet financing needs.
5
Discussions in Section 4
touch on how relaxing this restrictive assumption may lead to stronger relevance of the model to accounting.
For simplicity, I suppose that banks have perfect access to capital at the market interest rate r, and
there is perfect competition among banks. Given that · can be communicated to a bank at no cost, perfect
bank competition forces the loan interest rate c
0
(i.e., the firm's COC for financing projects) to be set by
adding on top of r a premium for default risk. The friction that drives my results is a transaction cost of debt
financing, namely the legal cost L of collection in case of default. For the shareholders, they also dislike
5
Understanding the overall COC of a firm requires knowing not only its cost of equity capital but also debt capital.
Debt financing is a source of capital second to none. Among the 30 countries studied by Demirguc-Kunt and Maksimovic
[1996], only four have a total debt to total equity ratio less than 1, with the US' equal to 1.791. In their 1999 study using the same
sample, they provide data of the total debt to total assets ratio for four firm classes ranging from small to very large; the ratios are
all within 50% to 60%. In another study based on World Bank firm-level survey data, Beck, Demirguc-Kunt, and Maksimovic
[2008] find that bank is without exception the largest source of external finance, with the responding US firms on average
reporting a 21.47% for bank versus a 3.24% for equity as two of the external finance sources. This survey, however, under-
represents large firms, which constitute only 20% of the sample (with the rest equally split between medium and small firms).
Large firms are more likely to use underwriting services in external financing. The Thomson Financial League Tables for 2003
and 2004 show that underwriting leaders in the loans and bonds categories raise proceeds that are each nearly 10 times of the
stocks category (http://en.wikipedia.org/wiki/Thomson_ Financial_league_tables). In short, it is important to understand the cost
of debt capital, and bank apparently is the most important source of debt financing.
4
default because there is a loss in shareholder value due to forced liquidation of assets. For simplicity,
regularity conditions on parameters of the model (to be detailed in Section 3) are assumed such that the
lending bank can always get full recovery of the loan amount plus interest in case of default. Therefore, the
main driver of the default risk is the firm's default probability q
0
.
In general, the default probability can depend on ·, as suggested by the characterization provided in
Section 3. However, in a later part of my analysis, I identify conditions, labeled as "small bonuses, a few
divisions," under which the default probability takes a simple geometric form. Restricting only to geometric
default probability highlights a sufficient condition for capital rationing not to occur in the model and
clarifies the important role of the capital supply curve. Other models typically assume the COC is given,
which is equivalent to assuming a perfectly elastic (inverse) capital supply curve. The model here provides a
micro-foundation demonstrating that default risk consideration can lead to a downward slopping capital
supply curve. Thereby it uncovers the rarely discussed incompatibility between a downward slopping capital
supply curve and capital rationing.
More concretely, geometric default probability implies that within the relevant range of k units of
capital to borrow (as restricted by the "small bonuses, a few divisions" conditions), the capital supply curve
takes the functional form: C(k)/k = (1+r) + Lq
k
/k, where q is the default probability of a division were it to
borrow like a legally separate entity. This capital supply curve is associated with a "marginal cost" curve,
C?(k), everywhere below the supply curve such that C?(k) < (1+r) < C(k)/k for any k> 0. My analysis shows
that if confining to a simple class of ex post mechanisms, called posted-price mechanisms, the hurdle rate h
will be set at a level such that (1+h) = C?(A), where A is the number of projects to be financed.
6
Note that not
only is h a "subsidized" rate with respect to the COC c
0
= C(A)/A - 1 but the rate is also "subsidized" even
when compared to the market interest rate r. This "capital sponsoring" result is illustrated in Figure 2.
6
Throughout the paper, I confine myself to a robust analysis of the model without specifying a prior belief about
the profile · that represents managers' private information. As such, I cannot discuss "the optimal" mechanism to operate an
ICM. Nonetheless, under perfect bank competition, I show that it is optimal for banks to use a posted-price mechanism to
allocate capital to the divisions, should the firm let them operate as separate legal entities to borrow from banks directly. The
rest of the analysis assumes that when operating an ICM, the firm also uses a posted-price mechanism to allocate capital to the
divisions. A posted-price mechanism need not be optimal had a prior been specified. However, it will be clear that with a properly
set hurdle rate, a posted-price mechanism will allocate capital efficiently, regardless of managers' private information. It is
therefore immune to ex post renegotiation. In contrast, the possibility of renegotiation can prevent an ex ante efficient mechanism
that optimizes with respect to a given prior from working as intended.
5
This paper contributes to the corporate finance and accounting literatures in several ways. First, it
provides a model connecting the COC to the hurdle rate endogenously. Other models typically take the COC
as given and examine how the hurdle rate or other variables of interest are linked to the COC. Second, the
model fills a gap between two extreme types of models in the ICM literature, namely "only one can win" and
"everyone gets a bite." The allowing for financing genuinely multiple projects, ranging from one to all, is the
reason why the COC can be endogenously linked to the hurdle rate through the default probability. Third,
the model provides a baseline construct with which extensions and refinements may be developed to
examine other issues not analyzed in this paper (see discussions in Section 4). It offers the potential to
organize the understanding of various related corporate finance and accounting issues using a single family
of models all developed from the one here as a baseline construct.
Fourth, the analysis of the model adds to the understanding of why operating an ICM may be better
than giving division managers complete autonomy in investing and financing decisions. Such reasons
include a lower COC and a smaller expected loss in shareholder value in case of default, both due to
reducing the default probability. Fifth, the analysis also shows that with geometric default probability, the
capital supply curve is downward sloping. This leads to the interesting phenomenon of a subsidized hurdle
rate and "capital sponsoring," which to my knowledge has not been examined in the literature before.
Finally, the model demonstrates to practitioners how a firm may operate an ICM to elicit private information
from division managers and allocate capital to their projects efficiently. Because the model is robust in the
sense that it does not depend on detailed specifications of managers' private information, the class of
mechanisms suggested here is easier to implement than suggestions derived from a Bayesian framework. A
Bayesian analysis usually requires correctly specifying a common prior belief, which is often difficult in
practice.
The rest of the paper is organized as follows. Next, I will give an overview of the model features and
main results. The model is formally introduced and analyzed in Section 3. There I use the model to
understand why operating an ICM can be better than giving division managers complete autonomy in
investing and financing decisions. In addition, I give a characterization of the default probability that affects
a firm's COC and is affected by the number of projects financed. Section 4 discusses the model's relevance
to corporate finance and to accounting; some unanswered questions on modeling are also posted there.
Concluding remarks are given in Section 5. Technical derivations and proofs are relegated to the Appendix.
6
2. Overview of Model Features and Main Results
2.1 Model Features
Many multi-division models have explored incentive issues related to getting managers to work hard
(e.g., Stein [2002] and Bernardo et al [2004, 2006]). In contrast, I focus on the capital allocation problem
and maintain tractability of the model by abstracting away from moral hazard due to hidden action. The only
incentive issue in the model is to get division managers to reveal truthfully their private information about
expected project returns.
In practice, a firm might find it hard to have some reliable prior beliefs about expected project
returns known only to managers. The model does not require a firm to form such beliefs before it can
implement the capital allocation scheme suggested. It is a robust model in the sense that the results derived
do not critically depend on details of such prior beliefs. Because of the robustness and accordingly the
comparatively easier implementation of the suggested scheme, practitioners might find the model more
relevant than others recommending "optimal" schemes derived from a Bayesian framework.
Consistent with the trend of emphasizing robustness in modeling (e.g., Bergemann and Morris
[2005]), I focus on direct-revelation mechanisms ensuring truth-telling is dominant-strategy incentive
compatible (DIC) and that participation is ex post individually rational (EIR). Such mechanisms are called ex
post mechanisms. To draw on Segal's [2003] characterization of ex post mechanisms, I assume each division
needs exactly one unit of capital to finance its project. While stylized, this is consistent with often found
non-scalability of investment projects.
Like many others in the literature, the model assumes risk neutrality. This is important because the
model is built upon results on multi-unit auctions in the mechanism design literature. These results rely
heavily on a quasi-linear functional form of agents' preferences (i.e., managers' here). The recent
characterization by Saitoh and Serizawa [2008] however opens up potential for relaxing this assumption to
allow preferences exhibiting income effects. This could provide a venue for future research, widening the
applicability of the model.
Residual income is a divisional performance evaluation measure widely taught in textbooks. In
particular, specific versions of it, such as EVA
®
, are highly advocated by consulting companies like Stern
Stewart & Co. (Stewart, Ellis, and Budington [2002]). The model assumes managers' objectives are to
maximize residual incomes of their projects, with the capital charges endogenously determined in the model,
7
rather than computed from an exogenously given required rate of return. Similarly, the firm's CEO is
assumed to maximize the value created from project investment, taking into account the cost of financing.
This can be viewed as the firm's "residual income," a performance measure analogous to the divisional
residual income of project investment.
Incentive compensations are determined by the bonus rates for the CEO and managers exogenously
specified in the model. While this differs from the widely accepted optimal contracting approach, the
constant bonuses rates will facilitate empirical work built around the model. By contrast, it is not easy for
researchers to test implications of the optimal contracting literature. Real-life compensation contracts often
have simple structures. Rarely will one find such contracts nicely align with fine-tuned optimal contracts
suggested by the literature. As my focus is on the capital allocation problem, optimal compensation design
issues are left aside to maintain tractability of the model.
In the model, borrowing from banks is supposed to be the only source of external financing. For
tractability, perfect bank competition is assumed so that any surplus from improved default risk is fully
captured by the firm through a lower cost of (debt) capital.
The model provides a framework for examining a number of interesting questions. For example,
within a firm, the capital charge is only a nominal accounting charge that serves the purpose of performance
evaluation and involves no actual payment. As such, there is no issue about whether a division is able to
make the "payment" or not. Suppose instead divisions are organized as separate legal entities with direct
access to external financing in the debt market. The "capital charge" becomes an actual payment for the loan
amount plus interest, and complications like default risk will kick in. The intuition is that an ICM can add
value by pooling together divisions' business risks and thereby relaxing the payment constraint that gives
rise to the default risk.
7
In analyzing the model, I contrast two polar cases of firm with different degrees of decentralization:
• ICM Firm: Divisions have autonomy in operating decisions but investing and financing
decisions are centralized.
• "Autonomous" Firm: Divisions have autonomy in investing and financing decisions, in addition
7
It should be emphasized that this is distinct from the usual diversification argument for settings with risk-averse
agents. All players here are risk-neutral. The efficiency gain from an ICM is due to the reduction in the default probability
that dictates the expected transaction cost of default to a bank, namely LPr{default}.
8
to operating decisions.
An ICM firm has an "active" CEO operating an ICM. The project selection and financing policy
used by the CEO is modeled as a direct-revelation mechanism. With the ICM, information asymmetry is
resolved before approaching a bank for debt financing. I suppose that in discussing with a bank, the CEO
can credibly communicate what she knows about the expected project returns at no cost.
For an "autonomous" firm without an "active" CEO, a bank must resolve the asymmetric
information problem itself. To give direct external financing the best chance to dominate an ICM, I assume
the bank can also use a direct-revelation mechanism to elicit private information from managers
approaching it for project financing. The only exception is that "capital charges" to divisions now involve
actual payments, rather than merely nominal accounting charges.
2.2 Main Results
Below are main results of the paper:
Ex post mechanism as a project selection and financing policy: Simply requiring truth-telling to be
dominant-strategy incentive compatible and participation to be ex post individually rational imposes an
interesting structure on a firm's or a bank's policy for project selection and financing. Specifically, the
policy is characterized by a hurdle rate above which all projects with at least that level of expected returns
will be selected and financed.
Posted-price mechanism as the policy used by banks under perfect bank competition: In the case of
an "autonomous" firm, perfect bank competition forces the policy used by a bank to take the form of a
posted-price mechanism with a particular hurdle rate. Specifically, the hurdle rate is set at the same level as
the "total cost" (i.e., 1 plus loan interest rate c) charged to a division, which is given by
(1+c) ÷ (1+r) + qL,
where r is the market interest rate at which banks have perfect access to capital, q is the default probability
of a division borrowing as a standalone legal entity, and L is the lending bank's legal cost of collection in
case of default.
Benefits of ICM: Because a firm needs to hire an "active" CEO to operating an ICM, whether it is
better to organize as an ICM firm depends on how large this hiring cost is, which is characterized by the
bonus rate for the CEO. Organizing as an ICM firm provides two potential benefits. By pooling together
divisions' business risks, it might reduce the default probability and thereby lower the COC. Additionally,
9
when the lending bank invokes legal procedures of collection, there is a loss in shareholder value due to
forced liquidation of assets in place. So the shareholders also want to avoid default for their own sake. These
benefits are independent of the collateral-based argument suggested by Stein [1997]. Taken together the
benefits will outweigh the cost if the CEO's bonus rate is sufficiently small.
Characterization of default probability: The default probability plays an important role in the model
to link hurdle rate and COC together endogenously. A characterization of the default probability adds
understanding to what are driving the probability and other results of the paper. Such a characterization is
provided.
Geometric default probability: Conditions are given under which the default probability takes the
simple geometric form, i.e., q
k
, where k is the loan amount borrowed by an ICM firm. Loosely speaking, it
requires the bonus rates for the CEO and managers to be "small" and that there are only "a few" divisions in
the firm. The simple structure of geometric default probability relies on the cash flow of only one
"successful" project to make up for the shortfall of all other "unsuccessful" projects with zero realized
returns. It would not be possible if there are too many divisions in the firm. Because the bonus to the CEO is
based on the value created from project investment, when all projects are "unsuccessful," a larger bonus rate
essentially means a bigger "rebate" by the CEO to help shouldering the borrowing cost of the firm. If the
bonus rate is too large, the "rebate" would be enough to ensure no default even when all projects are
"unsuccessful." For a similar reason, the bonus rate for managers must not be too large either.
Capital sponsoring and subsidized hurdle rate: In prevalent models in the literature, typically an
upward sloping capital supply curve is assumed, or otherwise an exogenous COC is assumed, which is
equivalent to a perfectly elastic capital supply curve. Both assumptions are consistent with standard settings
in economic theory. The model of this paper provides a micro-foundation for assuming possibly a downward
sloping capital supply curve. When the default probability is geometric, the "total cost" of borrowing C(k) ÷
(1+r)k + Lq
k
is strictly convex, with the "average cost" of borrowing C(k)/k = (1+r) + Lq
k
/k decreasing in the
loan amount k (i.e., a downward sloping capital supply curve). Moreover, the "marginal cost" of borrowing
C?(k) = (1+r) + Lq
k
ln(q) is below the "average cost" of borrowing. Consequently, the CEO would prefer
"capital sponsoring" with a subsidized hurdle rate set at the level such that (1+h) = C?(A), where A is the
number of projects approved for financing. This hurdle rate h is a "subsidized" rate in the sense that h =
C?(A) - 1 < C(A)/A - 1, which is the endogenously determined COC in the model. This is in stark contrast to
10
the typically assumed setting with a hurdle rate above a firm's COC and therefore not every positive NPV
project is financed, a phenomenon referred to as capital rationing.
While unexpected, the "capital sponsoring" result driven by geometric default probability is not
unreasonable. Given perfect bank competition, the geometric default probability is decreasing in the loan
amount borrowed. With a subsidized hurdle rate, more projects would be financed than otherwise would
have. Given a downward sloping capital supply curve induced by geometric default probability, the
financing of more projects means a more favorable term in borrowing. The positive externality that results
makes the subsidized hurdle rate sustainable.
Considering geometric default probability and the resulting downward sloping capital supply curve
is not to say that they are likely cases to see in reality. Indeed, even in the model, they occur under the
restrictive "small bonuses, a few divisions" conditions. These conditions, however, are not inconceivable. So
although they might not be met often, they may indeed exist. Yet the assumption of a downward sloping
capital supply curve is hardly seen in the literature. This is not difficult to understand. If the capital supply
curve is to be specified exogenously, as is often the case in most studies, one would naturally assume an
upward sloping capital supply curve to adhere to traditional economic theory. It is under the micro-level
analysis of this paper that a downward sloping capital supply curve is derived endogenously from some
admittedly restrictive, yet not inconceivable conditions. Identifying such circumstances in the real world
would provide a way to test the empirical validity of the model.
To sum up, the importance of conducting a micro-level analysis and considering the restrictive case
of geometric default probability is to provide a new angle in rethinking issues that might have been
overlooked when capital rationing is taken for granted as the necessary case.
3. Model and Analysis
3.1 Model Specifications
I consider the setting of a firm with n divisions (n> 2), each headed by a manager, competing for
capital allocated from the corporate headquarters, headed by the CEO. Each division has a project requiring
exactly 1 unit of capital to invest. The project of division i, if executed, will bring in some uncertain future
cash flow R
i
, with its mean ·
i
known only by the manager. This expected project return is referred to as
manager i's valuation.
It is common knowledge that managers' valuations follow some joint distribution with the support of
11
each marginal distribution equal to ( µ , µ ), where 0 s µ< µ<·. This is all the CEO knows about the
valuations. On the other hand, the CEO and managers commonly know that conditional on the valuation
profile · = (·
1
, ·
2
,., ·
n
), the project returns R
i
's are independently and identically distributed as follows:
R
i
= 0 with probability q e (0, 1), and
Pr{R
i
< R +z | ·} = q + (1-q)G(z | ·
i
) for all z e [0, z ),
where 0 < zs·, with R> 0 and G(0| ·
i
) = 0. The "probability density function" g(z | ·
i
) = G?(z | ·
i
) is
positive and continuous for z> 0, and as defined, } zg(z | ·
i
)dz = ·
i
/(1-q) so that E[R
i
| ·
i
] = ·
i
.
Each division has some assets in place required for its regular operations, which will bring in a
certain (net) cash flow regardless of what happens to the division's project. For simplicity, assume each
division has an identical cash flow from regular operations to be generated during the year, M> 0, and
beginning-of-the-year book value of assets, B> 0.
8
Besides what might be later arranged for project financing, the firm has no liabilities. For simplicity,
bank loan is assumed to be the only source available to the firm for financing projects. The loan amount plus
interest, denoted by (1+c
0
)K, is referred to as the "total cost" of borrowing K units of capital, where K> 1.
The interest rate c
0
at which the firm can borrow the loan amount K is referred to as its cost of (debt) capital.
Although the model may be viewed as representing a typical round of a multiple-period model, here
I assume it is one-shot. The sequence of events happening in the model is as follows:
8
In this paper, modeling choices are made with an attempt to balance among tractability, empirical relevance, and
simplicity. Tractability always comes first, for few results can an analytical model give without tractability. Most theorists
take the view that models should be "leanest," i.e., every model element should play an indispensable role in the analysis. I also
take this view but with the room to allow for model elements that are included for empirical relevance considerations.
Oftentimes empiricists interested in testing the implications of a model need to "hold other things constant" by including
a variety of control variables, or confine themselves to conditions under which the model's results are valid. This legitimate concern
is usually ignored by theorists who care most about formulating the "leanest" models, with little interest in giving empiricists an
easier life and more guidance. Lacking any suggestions, empiricists typically put controls into regression equations in an
additively separable fashion. This ignores the possibilities that the controls should enter the equations nonlinearly or interact with
other variables, had they been included in the original model. Similarly, parameters inessential to the main insights behind the
analysis are often normalized to 0 or 1 to give cleanest statements of the conditions in concern. When in reality such exogenous
variables have different values, empiricists are left alone to make their own judgmental adjustments.
In contrast, the model of this paper includes elements (e.g., cash flows from regular operations) that are not absolutely
necessary but I believe empiricists interested in using the model as a theoretical framework for their research would consider in
formulating their research designs. These elements stay in the model because simplicity is not seriously
jeopardized. Such elements, however, could be considered "clutters" by theorists taking the extremist "leanest" perspective.
12
At the beginning of a year, it is common knowledge that the project selection policy adopted by the
firm will continue as usual, namely, given any profile · of expected project returns reported by
managers, the policy ?x(?), t(?)? specifies an outcome constituted of a capital allocation x = (x
1
, x
2
,.,
x
n
) e {0, 1}
n
, with x
i
= 1 indicating the approval of division i's project, and a "payment" scheme t=
(t
1
, t
2
,., t
n
) e R
n
, with t
i
standing for an accounting charge to division i.
9
Though not confined to be
so at this point of the model specification, it will be clear shortly that t
i
's are zero for divisions with
"losing" projects, i.e., those not selected.
It is also common knowledge that the firm's compensation policy will continue as usual, namely:
o "Losing" managers, whose projects are not selected, will receive a basic salary, which is
normalized to zero for simplicity;
o "Winning" manager i will receive a basic salary, also normalized to zero, plus a bonus, at the
rate of |> 0, based on the approved project's residual income R
i
- t
i
.
10
o The CEO will receive a basic salary, also normalized to zero, plus a bonus, at the rate of |
0
> 0,
based on the value E
i
R
i
x
i
- (1+c
0
)K created from projects selected and financed at the cost of
capital c
0
.
11
Managers privately learn about the expected returns of their projects and submit reports to convey
the information to the CEO.
Everyone learns about the market interest rate r> 0 at which banks can have perfect access to
capital.
12
9
Recall that 1 is the unit of capital invested in a project. So (t
i
- 1)/1 is the required (rate of) return often discussed
in accounting textbooks under the topic of capital budgeting.
10
At the expense of more complicated notations, the bonus rate can be specified based on post-bonus, rather than
pre-bonus, residual income. See related discussions on bonus compensation computation in accounting textbooks such as
Kieso, Weygandt, and Warfield [2006] and Stice, Stice, and Skousen [2007].}
11
With more complicated notations, the bonus rate for the CEO can be specified based on shareholder value,
namely the expected equity value at the year end after closing income to retained earnings. This alternative specification is
consistent with a stock bonus. Some analyses of the paper however become intractable with this alternative specification.
12
The following dummy event may be included at this point to justify non-zero bonus rates exogenously set in the
model, which could be an endogenous outcome of expanded modeling: A manager can "take the project with him" to pursue
an outside option (e.g., seek venture capital to start his own company), which will give an expected payoff of no more than
13
The CEO contacts a banker to share the information about projects available. The banker makes an
offer on the loan terms, namely the schedule of interest rate at each loan amount requested. The offer
is competitive, and the CEO accepts it with the loan amount finalized.
The CEO allocates the capital to the divisions and imposes accounting charges accordingly.
Each manager continues his division's regular operations from which a cash flow is generated.
Alongside with the regular operations, "winning" managers execute the approved projects, with their
returns realized some time before the year end. Compensations to the CEO and managers are paid at
the year end accordingly.
Following the year end, if the cash available is enough to cover the "total cost" of borrowing, the
payment is made to the lending bank. Otherwise the firm defaults, and the bank seeks recovery by
forced liquidation, which results in a legal cost of L> 0 to the bank. The liquidation value of the
firm's assets is a fraction ì e (0, 1) of their beginning-of-the-year book value.
Managers are expected payoff maximizers. Conditional on the valuation profile ·, a manager's
expected payoff is equivalent to
E[R
i
x
i
(·) - t
i
(·) | ·] = ·
i
x
i
(·) - t
i
(·).
This quasi-linear functional form of managers' expected payoffs allows utilizing directly some results on
multi-unit auctions in the mechanism design literature. In particular, the following definition and lemma are
essentially due to Segal [2003]:
DEFINITION 1 (Segal [2003]): A mechanism ?x(?), t(?)? is an ex post mechanism if it satisfies
dominant-strategy incentive compatibility (DIC) and ex post individual rationality (EIR):
For any manager i, any valuation profile · e ( µ , µ )
n
, and any µ
i
e ( µ , µ ), ˆ
[DIC]: ·
i
x
i
(·) - t
i
(·) > ·
i
x
i
( µ
i
, ·
-
i
) - t
i
( µ
i
, ·
-
i
),
ˆ ˆ
[EIR]: ·
i
x
i
(·) - t
i
(·) > 0.
where ( µ
i
, ·
-
i
) means the valuation profile constructed from · by replacing ·
i
with µ
i
.
ˆ ˆ
|(·
i
- c
0
). So he stays. Similarly, the CEO can choose to start her own company and invite "winning" managers to join. This
outside option will give her an expected payoff of no more than |
0
[E
i
·
i
x
i
- (1+c
0
)K]. So she also stays.
14
LEMMA 1 (Segal [2003]): A deterministic mechanism ?x(?), t(?)? is an ex post mechanism if and
only if for each manager there exist functions p
i
, s
i
: ( µ , µ )
n
-1
÷ R
+
such that for every valuation profile · e
( µ , µ )
n
,
x
i
(·) = 1 if ·
i
> p
i
(·
-
i
), x
i
(·) = 0 otherwise, and
t
i
(·) = p
i
(·
-
i
) x
i
(·) - s
i
(·
-
i
).
In structuring a policy to allocate capital and charge for the use, the CEO is assumed to restrict
attention to ex post mechanisms only, which do not require any knowledge about the distribution of ·. The
CEO is also an expected payoff maximizer. Conditional on the valuation profile ·, her expected payoff is
equivalent to
E
i
·
i
x
i
(·) - (1+c
0
)K.
3.2 "Autonomous" versus ICM Firm
In an ICM firm that has an "active" CEO operating an ICM, information asymmetry is resolved
before approaching a bank. Suppose it is prohibitively costly for the CEO to commit criminal acts like
falsifying information in project proposals submitted by division managers. Then by demanding the CEO to
provide the proposals to support her claim about ·, the information can be credibly communicated to a bank.
I suppose that in discussing with a bank, the CEO can credibly communicate · at no cost. With
complete information and perfect competition from other banks, the bank will set c
0
to equate its cost of
providing the loan. Suppose further that even in the case of default the firm has enough assets in place for
the bank to get full recovery of the loan amount plus interest. Then c
0
will be set by adding to r a risk
premium due to the expected loss as a result of the legal cost and the default probability:
c
0
= r + q
0
(K, ·)L/K,
where q
0
(K, ·) denotes the default probability of the firm when K units of capital are borrowed to finance
projects from a pool characterized by ·.
Instead of operating an ICM, the firm could have organized divisions in a fully autonomous fashion,
namely, separately incorporating the divisions and delegating project selection and financing responsibilities
to them. In such an "autonomous" firm, the CEO plays merely a "passive" role of ordinary administration
and receives only a basic salary normalized to zero.
15
For simplicity, assume each division in the firm has enough assets in place for a lending bank to get
full recovery of the loan amount plus interest. Then the expected cost of providing a loan to division i is
given by
(1+c
i
) ÷ (1+r) + L Pr{default by division i | ·}.
Without an "active" CEO operating an ICM, a bank must resolve the asymmetric information
problem itself. To give direct external financing the best chance to dominate an ICM, I assume the bank can
also use an ex post mechanism to elicit private information about ·. The only exception is that the t
i
(·)'s
charged to divisions now involve actual payments, rather than merely accounting charges.
A bank's objective is to choose an ex post mechanism ?x(?), t(?)? to maximize its expected profit
from lending,
E
i
[t
i
(·) - (1+c
i
)x
i
(·)],
subject to the zero expected profit condition due to perfect bank competition:
E
i
[t
i
(·) - (1+c
i
)x
i
(·)] = 0.
When an optimum is attained, no change in ?x(?), t(?)? can lead to a positive expected profit to a bank.
Although a bank can guarantee a zero expected profit by refusing to issue any loan, this is not
optimal if issuing some loans can create efficiency surplus. When such unexploited surplus exists, there
would be a way for a competing bank to offer loan terms to realize the surplus and make a positive expected
profit, which however cannot happen under perfect competition. That is to say, optimality together with
competition requires a bank to issue loans efficiently, given the constraints imposed by an ex post
mechanism and the legal cost of collection in case of default.
If ignoring the DIC and EIR constraints of an ex post mechanism, efficient loan issuance would
imply
x
i
(·) = 1 if ·
i
> (1+c
i
), x
i
(·) = 0 otherwise.
This, however, cannot constitute an ex post mechanism unless (1+c
i
) ÷ (1+r) + LPr{default by division i | ·}
is unrelated to ·
i
. The regularity condition below specifies circumstances under which (1+c
i
) is indeed
unrelated to ·
i
. When the condition is met, a simple posted-price mechanism is optimal to a bank. This result
is given in Lemma 2.
REGULARITY CONDITION:
16
(a) G((1+r) + L - M/(1-|) | ·
i
) < 1 for any ·
i
;
(b) (1+r) s (ìB + M)/(1-|) - L;
(c) (1+r) s R + M/(1-|) - L;
(d) (M+L)/(1-|) s (1+r).
Part a of the regularity condition ensures that the default probability is not too close to 1. Part b of
the condition says the liquidation value ìB of a division's assets is not too small to prevent full recovery of
the loan amount plus interest when legal procedures are invoked. These two parts together with part c imply
the default probability of a division is at most q. Finally, part d guarantees that the legal cost L is not too
large to make invoking the legal procedures unwise.
LEMMA 2: Suppose the regularity condition holds. Then for a "winning" division (i.e., one provided
with a loan), the default probability is q and hence its cost of capital is c÷ r + qL; a posted-price mechanism
?x(?), t(?)? with
x
i
(·) = 1 and t
i
(·) = (1+c) if ·
i
> (1+c),
x
i
(·) = t
i
(·) = 0 otherwise,
is optimal to a bank.
Suppose the regularity condition is met and thus the default probability of any division provided
with a loan is q. If posted-price mechanisms are used by banks,
I(c) ÷ {i | ·
i
> (1+c) for ·
i
of ·}
would be the index set of the "winning" divisions provided with loans. Let K denote the number of
"winning" projects, i.e., K÷ |I(c)|. Conditional on ·, the expected equity value of an "autonomous" firm
(without an ICM) is given by
[AF]: nM - Kq(1-ì)B + (1-|)[E
i
eI(c
)
·
i
- (1+c)K].
The first term of this expression is the cash flow generated from regular operations of all n divisions.
The second term is the expected loss in shareholder value due to forced liquidation of assets in case of
default. The last term is the part of value created from "winning" projects that is retained by shareholders. A
fraction | of the value created is paid to "winning" managers as their bonuses.
For ease of exposition, suppose for the moment that a CEO operating an ICM also uses a posted-
17
price mechanism with t
i
(·) = p
i
(·
-
i
) = (1+c) and the loan amount she borrows is also K. Bank competition
forces the interest rate offered to the CEO to be set at a level given by the zero expected profit condition, i.e.,
(1+c
0
)K = (1+r)K + q
0
(K, ·)L,
where q
0
(K, ·) is the default probability of the firm given that K units of capital are borrowed to finance
projects from a pool characterized by ·. The way q
0
(K, ·) is defined assumes that successful voluntary
liquidation of some assets to avoid default cannot be completed before the lending bank invokes the legal
procedures to liquidate the firm.
Given the suppositions above, the following would be a "lower bound" of such an ICM firm's
expected equity value:
[IF]: nM - q
0
(K, ·)(1-ì)nB + (1-|)[E
i
eI(c
)
·
i
- (1+c)K] + (c - c
0
)K - |
0
[E
i
eI(c
)
·
i
- (1+c
0
)K].
This is a "lower bound" because the possibility of reorganization might result in a higher value to the
shareholders. If liquidating some of the assets is sufficient to meet the payment obligation, the second term
in the expression above would exaggerate the expected loss in shareholder value in case of default. The third
and fourth terms in the expression add up to
[E
i
eI(c
)
·
i
- (1+c
0
)K] - |[E
i
eI(c
)
·
i
- (1+c)K].
This is the value created by the "winning" projects minus the bonuses paid to managers given that "winning"
divisions are charged at (1+c), rather than the "average cost" of borrowing (1+c
0
). Finally, the last term in
the lower bound has no counterpart in the expected equity value of an "autonomous" firm. The "passive"
CEO of such a firm is presumed to receive only a basic salary normalized to zero. By contrast, the CEO of
an ICM firm receives also a bonus based on the value she creates by operating an "active" ICM. This bonus
is the last term in the lower bound.
3.3 Default Probability of ICM Firm
In general, an ICM firm might attain an expected equity value higher than the lower bound IF by
charging "winning" divisions (possibly asymmetrically) more or less than (1+c) and by borrowing a loan
amount different from K, as long as such adjustments constitute an ex post mechanism. Raising the capital
charges t
i
(·)'s can save bonus expenses but the project selection cutoffs p
i
(·
-
i
)'s might need to be tightened
accordingly, which can reduce the total value created by "winning" divisions. Depending on the
characteristics of the default probability q
0
(K, ·), borrowing less might reduce the cost of capital c
0
so
18
substantially that it compensates for the loss in value due to a smaller number of "winning" projects. Given
these flexibilities in improving upon a posted-price mechanism with given c and K, there is room for an ICM
firm to achieve a higher equity value than the lower bound IF.
In contrast, an "autonomous" firm's equity value AF can be lower than an ICM firm's lower bound
IF. This may be best seen by examining their difference with the former deducted from the latter:
?÷ [Kq - nq
0
(K, ·)](1-ì)B + [Kq - q
0
(K, ·)]L - |
0
[E
i
eI(c
)
·
i
- (1+c
0
)K].
Suppose the bonus rate |
0
for the CEO is sufficiently small and therefore the cost of hiring her to operate an
ICM is arbitrarily negligible. Then as long as q
0
(K, ·) < qK/n, the difference ? can be made positive. In
words, the decrease in expected loss in shareholder value due to default risk, i.e., [Kq - nq
0
(K, ·)](1-ì)B,
together with the saving in cost of capital due to co-insurance among divisions, i.e., [Kq - q
0
(K, ·)]L, will be
large enough to cover the cost of operating an ICM, i.e., |
0
[E
i
eI(c
)
·
i
- (1+c
0
)K]. This result is stated below as
Proposition 1.
PROPOSITION 1 (Benefits of internal capital markets): Suppose q
0
(K, ·) < qK/n and hence c
0
< c.
For sufficiently small |
0
> 0, an ICM firm's expected equity value can exceed an "autonomous" firm's.
It should be emphasized that Proposition 1 is a "can-be" result. Since the CEO's and the
shareholders' interests do not align with each other completely, the equity value of an ICM firm might fail to
exceed an "autonomous" firm's if the task of operating an ICM is fully delegated to the CEO. However, it is
conceivable that the shareholders might impose restrictions on the ex post mechanism used, e.g., require
capital rationing while leaving the project selection decision to the CEO. Such measures might prevent the
CEO's opportunistic behavior from jeopardizing the firm's equity value.
The behavior of the default probability q
0
(K, ·) of an ICM firm in general can be quite complex. To
get some sense of how it might behave, let's continue to suppose that the firm uses a posted-price
mechanism with t
i
(·) = p
i
(·
-
i
) = (1+c) and the loan amount borrowed is K÷ |I(c)|, where I(c) ÷ {i | ·
i
> (1+c)
for ·
i
of ·}.
13
Given these suppositions, the firm's year-end cash position would be
nM + E
i
eI(c
)
R
i
- |[E
i
eI(c
)
R
i
- (1+c)K] - |
0
[E
i
eI(c
)
R
i
- (1+c
0
)K]
13
This could be the case if the provision of capital follows the traditional price-based transfer pricing framework,
where the transfer price of internally provided goods is set at the outside market price.
19
= nM + (1-|-|
0
)[E
i
eI(c
)
R
i
- (1+c)K] + (c - c
0
)(1-|
0
)K + (1+c
0
)K.
Default occurs when the cash position is insufficient to meet the firm's payment obligation, i.e.,
E
i
eI(c
)
R
i
< K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o,
where o÷ (1-|-|
0
). Let S denote the number of "successful" projects, i.e. those with positive realized
returns. Given S, let J(S) denote an index set of the S "successful" projects. There are altogether K!/S!(K-S)!
such J(S)'s each indicating a specific combination of the projects that constitute the S "successful" projects.
Let O(S) denote the set of all such J(S)'s. Finally, define Z
i
= (R
i
- R )1
{
Ri
>0}
; in words, Z
i
is the part of R
i
exceeding R . Conditional on R
i
> 0, Z
i
follows the distribution G(z
i
| ·
i
).
With these notations, the default probability can be expressed as follows:
Pr{default | ·}
= E
s
Pr{default | S = s, ·}Pr{S = s | ·}
= E
s
E
J
(s)eO(s
)
Pr{default | J(s), S = s, ·}(1-q)
s
q
K
-s
where
Pr{default | J(s), S = s, ·}
= Pr{Z
J
(s
)
+ s R< K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o | ·}
and Z
J
(s
)
= E
i
eJ(s
)
Z
i
is the sum of the independent random variables Z
i
's for those i e J(s).
The distribution function of Z
J
(s
)
, which is referred to in probability theory as the convolution of the
distribution functions G(z
i
| ·
i
)'s, can depend on the ·
i
's in a non-trivial way. For tractability, I assume the
probability distribution function of Z
J
(s
)
simply takes the form G(z | ·
J
(s
)
), where ·
J
(s
)
= E
i
eJ(s
)
·
i
.
14
Given this
assumption, the conditional probability Pr{default | J(s), S = s, ·} can be expressed as follows:
Pr{default | J(s), S = s, ·}
= Pr{Z
J
(s
)
< K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - s R | ·}
= G(K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - s R | ·
J
(s
)
)
= G(f(q
0
, s, K, c) | ·
J
(s
)
),
where
14
One example is that Z
i
follows the gamma distribution Gam(·
i
,1-q), which has mean ·
i
/(1-q) and variance ·
i
/(1-
q) . Then Z
J
(s
)
follows the distribution Gam(E
i
eJ(s
)
·
i
,1-q). 2
20
f(q
0
, s, K, c) = K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - s R
and c
0
= r + q
0
(K, ·)L/K.
Below is the next main result of the paper, which provides an equation characterizing the default
probability of an ICM firm.
PROPOSITION 2 (Characterization of default probability): Suppose that an ICM firm uses a
posted-price mechanism with t
i
(·) = p
i
(·
-
i
) = (1+c) and the loan amount borrowed is K÷ |I(c)|, where I(c) ÷
{i | ·
i
> (1+c) for ·
i
of ·}. Moreover, for s > 1, suppose that the sum of the parts of positive realized returns
over R , namely Z
J
(s
)
÷ E
i
eJ(s
)
Z
i
, where Z
i
÷ (R
i
- R )1
{
Ri
>0}
, follows the distribution G(z | ·
J
(s
)
), where ·
J
(s
)
÷
E
i
eJ(s
)
·
i
; for s = 0, define instead G(z | ·
?
) = 0 for z s 0 and G(z | ·
?
) = 1 for z> 0. Then for any given · and
c and the K so determined, the default probability q
0
(K, ·) of the ICM firm is given by the equation below,
provided it admits an interior solution in [0, 1]:
q
0
= E
s
E
J
(s)eO(s
)
G(f(q
0
, s, K, c) | ·
J
(s
)
)(1-q)
s
q
K
-
s
,
where
f(q
0
, s, K, c) = K[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - s R ,
c
0
= r + q
0
L/K, and o = (1-|-|
0
).
To understand more about the properties of q
0
(K, ·), let's consider the special case of K = n. Given
the complex "combinatorial" structure of the equation defining q
0
(K, ·), little can be said even for this
special case. However, if there are "not too many" divisions in the firm and the bonus rates |
0
and | for the
CEO and managers are "not too large" (both made specific shortly), then q
0
(n, ·) will take the simple form
of q
n
. This follows from the fact that f(q
0
, s, K, c) is decreasing in s. As a result, f(q
0
, 1, n, c) s 0 implies f(q
0
,
s, n, c) s 0 for all s > 1 and hence G(f(q
0
, s, n, c) | ·
J
(s
)
) = 0 for all s > 1. It can be shown that f(q
0
, 1, n, c) s 0
and f(q
0
, 0, n, c) > 0. As G(z | ·
?
) = 1 for z> 0, it follows that
q
0
= E
s
E
J
(s)eO(s
)
G(f(q
0
, s, n, c) | ·
J
(s
)
)(1-q)
s
q
n
-
s
,
= G(f(q
0
, 0, n, c) | ·
?
)q
n
= q
n
.
This result is stated as the lemma below.
21
LEMMA 3: Suppose the regularity condition holds. In addition, ns [ R - L/(1-|-|
0
)]/[(1+r) -
M/(1-|)] and |
0
< (1-|)(1- q|)L/(M+L). Then given · and c, if K = n, q
0
(n, ·) = q
n
.
This simple structure of the default probability relies on the cash flow of only one positive realized
return project to make up for the shortfall of all other zero realized return projects. It would not be possible if
there are too many divisions in the firm. Recall that the bonus to the CEO is based on the value created from
project investment. When all projects have zero realized returns, a larger bonus rate essentially means a
bigger "rebate" by the CEO to help shouldering the borrowing cost of the firm. If the bonus rate is too large,
the "rebate" would be enough to ensure no default even when all projects have zero realized returns. For a
similar reason, the bonus rate for managers must not be too large either.
Suppose further that |
0
, |, and n are sufficiently small such that (nM + q|L)/(1-|-|
0
) s (1+r). Then
the simple structure of default probability derived above carries over to any K e {1, 2, ., n}. To see why,
first note that given the additional assumption on n and |
0
,
f(q
0
, s, K, c) = {K[(1+r)o - q|L] - nM + q
0
(1-|
0
)L - s R }/o
is increasing in K. It has been shown in the proof of Lemma 3 that f(q
0
, 1, n, c) s 0. It follows that f(q
0
, 1, K,
c) s 0 for all Ks n. The additional assumption also implies f(q
0
, 0, 1, c) > 0. As f(q
0
, s, K, c) is increasing in
K, f(q
0
, 0, K, c) > 0 for all Ks n. Together they imply q
0
(K, ·) = q
K
for any K e {1, 2, ., n}. This result is
stated as the next proposition.
ASSUMPTION SBAFD1 ("Small bonuses, a few divisions" - first version): The bonus rates for the
CEO and managers are "small", and there are only "a few" divisions in the firm, i.e., |
0
, |, and n satisfy the
following conditions:
ns [ R - L/(1-|-|
0
)]/[(1+r) - M/(1-|)];
|
0
< (1-|)(1- q|)L/(M+L);
(nM + q|L)/(1-|-|
0
) s (1+r).
PROPOSITION 3 (Geometric default probability): Suppose the regularity condition holds and |
0
, |,
and n satisfy the assumption SBAFD1. If an ICM firm uses a posted-price mechanism with c = r + qL as the
(rate of) "required return" and borrows K÷ |I(c)| as the loan amount, where I(c) ÷ {i | ·
i
> (1+c) for ·
i
of ·},
then the firm's default probability is q
0
(K, ·) = q
K
. Consequently, its cost of capital is c
0
= r + Lq
K
/K, and
22
"total cost" of borrowing is (1+c
0
)K = (1+r)K + Lq
K
.
Given this simple structure of an ICM firm's default probability, the result in Proposition 1 can be
strengthened. A specific threshold for the CEO's bonus rate can now be provided. The stronger result is
stated as the proposition below.
PROPOSITION 4: Suppose the regularity condition and assumption SBAFD1 hold. Given · and c,
and the K so determined, if 2 s K s n s [2((1-ì)B + L)/q - L]/(1-ì)B and |
0
s |
0*
, where
|
0*
÷ [(q - nq
K
/K)(1-ì)B + (q - q
K
/K)L]/( µ - (1+r) - Lq
K
/K),
then the equity value of an ICM firm can exceed that of an "autonomous" firm.
This proposition relies on the geometric default probability derived earlier, which hinges on the
possibility of using the cash flow of only one "successful" project to make up for the shortfall of all other
"unsuccessful" projects. This however is impossible if K = 1. Because the geometric default probability also
requires the firm to have at most "a few" divisions, its total cash flow from regular operations alone is
insufficient to prevent default either. Given the presumption that all assets will be liquidated in case of
default, there would be too much loss in shareholder value if n is large. Hence, the proposition also requires
n s [2((1-ì)B + L)/q - L]/(1-ì)B.
3.4 Capital Sponsoring and Subsidized Hurdle Rate
So far, I have assumed an ICM firm uses a posted-price mechanism with t
i
(·) = p
i
(·
-
i
) = (1+c) and
the loan amount borrowed is K÷ |I(c)|, where c = r + qL is the interest rate divisions of an "autonomous"
firm can obtain from banks. The characterization of the default probability in Proposition 2 remains valid if
c and K are replaced by h and K(h) respectively, where h is a hurdle rate (of required return) used by the
firm and K(h) ÷ |I(h)| is the loan amount corresponding to the rate.
15
That is, for any given · and h and the
K(h) so determined, the default probability q
0
(K(h), ·) of the ICM firm is given by the equation below,
provided it admits an interior solution in [0, 1]:
q
0
= E
s
E
J
(s)eO(s
)
G(f(q
0
, s, K(h), h) | ·
J
(s
)
)(1-q)
s
q
K
(h)-
s
,
where
15
Other results have specifically used c = r + qL in the proofs and cannot be generalized trivially as this.
23
f(q
0
, s, K(h), h) = K(h)[(1+h) - (h - c
0
)(1-|
0
)/o] - nM/o - s R ,
c
0
= r + q
0
L/K(h), and o = (1-|-|
0
).
In addition, the result of a geometric default probability can be generalized to this more flexible posted-price
mechanism, provided that the bonus rates for the CEO and managers are "small" and there are only "a few"
divisions in the firm. This time, "small" and "a few" are in the following sense:
ASSUMPTION SBAFD2 ("Small bonuses, a few divisions" - second version): The bonus rates |
0
and | for the CEO and managers and the number n of divisions in the firm satisfy the following conditions:
ns [ R - L/(1-|-|
0
)]/[(1+r) - M/(1-|)];
(nM + µ |)/(1-|-|
0
) s (1+r).
PROPOSITION 5 (Geometric default probability for general posted-price mechanism): Suppose the
regularity condition and assumption SBAFD2 hold. If an ICM firm uses a posted-price mechanism with h as
the hurdle rate for project selection and K(h) ÷ |I(h)| as the loan amount for project financing, where I(h) ÷ {i
| ·
i
> (1+h) for ·
i
of ·}, then for any h > 0 with K(h) > 1, the firm's default probability is q
K
(h
)
. Consequently,
its "total cost" of borrowing is C(K(h)), where C(k) ÷ (1+r)k + Lq
k
.
Finally, it is interesting to examine whether the concern for default discussed here can naturally lead
to capital rationing. With perfect bank competition, surplus from reduced default risk is completely captured
by the firm. If it faces an upward sloping capital supply curve, the firm would be analogous to the buyer in
the classic monopsony case. In such circumstances, it would be optimal for the buyer to restrain the quantity
purchased (i.e., capital rationing in this context) to obtain a more favorable purchase price (i.e., a lower cost
of capital).
There are four caveats to this conclusion. First, in the model here the capital needed for investment
is a whole number. If an upward sloping capital supply curve is sufficiently flat, the gain from reducing the
loan amount by one unit can be too little to justify the loss in expected project returns. Second, the capital
supply curve need not be upward sloping and consequently capital rationing could reduce the value created
from project investment. Third, from the shareholders' perspective, they would like to balance the value
created from project investment with the expected loss in shareholder value due to default. The latter
depends on the default probability, which could decrease with the loan amount even when the capital supply
24
curve is upward sloping. So shareholders might dislike capital rationing even when the CEO prefers. Finally,
the whole issue is further complicated by the saving in bonuses paid to managers when a higher hurdle rate
is set for capital rationing. This alone benefits the shareholders but the CEO is indifferent.
When the default probability is geometric, the capital supply curve is downward sloping. Therefore,
capital rationing is not preferred by the CEO. A seemingly unexpected finding is that she would set the
hurdle rate below the cost of capital to "subsidize" divisions for project investment, resulting in some
"capital sponsoring." This however is reasonable given perfect bank competition and geometric default
probability, which is decreasing in the loan amount borrowed. With a subsidized hurdle rate, more projects
would be financed than otherwise would have, which leads to a more favorable term in borrowing. The
positive externality that results makes the subsidized hurdle rate sustainable. This last result is stated below.
PROPOSITION 6 (Capital sponsoring and subsidized hurdle rate): Suppose an ICM firm uses a
posted-price mechanism with h as the hurdle rate for project selection and K(h) ÷ |I(h)| as the loan amount
for project financing, where I(h) ÷ {i | ·
i
> (1+h) for ·
i
of ·}. If the default probability of the firm is
geometric, i.e., q
0
(k, ·) = q
k
, then the "total cost" of borrowing C(k) ÷ (1+r)k + Lq
k
is strictly convex, with
the "average cost" of borrowing C(k)/k = (1+r) + Lq
k
/k decreasing in the loan amount k. Moreover, the
"marginal cost" of borrowing C?(k) = (1+r) + Lq
k
ln(q) is everywhere below the "average cost" of borrowing.
Consequently, the CEO prefers capital sponsoring with a subsidized hurdle rate set at h = C?(A) - 1, where A
÷ max{ a | ·
e
(a
)
> C?(a) for a s n } is the number of projects approved for financing, and e(a) denotes the
index of the ath highest ·
i
of ·.
To illustrate the potential empirical relevance of this result of the model, let me casually point out
how one can obtain a lower-bound estimate of the expected return of the marginal project accepted by a
firm. Note that by definition the expected return of the marginal project, denoted by ·
e
(A
)
, is at least C?(A) =
(1+r) + Lq
A
ln(q). Since C(A)/A = (1+r) + Lq
A
/A and hence Lq
A
= C(A) - (1+r)A, it follows that
·
e
(A
)
> C?(A) = (1+r) + ln(q)[C(A)/A - 1 - r]A.
Empirically, A may be proxied by the borrowing amount of a firm in a period, and C(A)/A - 1 - r by the
interest rate premium. Together with a proxy for the market interest rate r and some estimate of the default
probability q for a division borrowing like a separate legal entity, one can compute an estimate of C?(A),
which serves as a lower-bound estimate of ·
e
(A
)
. Suppose firms are similarly efficient in operating their ICM.
25
Then such an estimate of ·
e
(A
)
could be useful for comparing the investment opportunities of different firms.
It may thus contribute to the valuation of the firms.
4. Discussions
4.1 Relevance to Corporate Finance
The model has potential to shed light on a variety of corporate finance issues, like capital rationing
as a response to managerial overconfidence, value of corporate diversification, boundaries of the firm,
spinoff and acquisition decisions, diversification discount, and "socialistic" capital allocation.
Capital rationing as a response to managerial overconfidence: With the model, I am able to
highlight the relation between capital supply curve and project financing. The analysis on geometric default
probability demonstrates that a downward sloping capital supply curve is incompatible with capital
rationing. In reality, a downward sloping capital supply curve might not occur very often and therefore
capital rationing is much often seen than "capital sponsoring." When capital rationing occurs in a firm with
a downward sloping capital supply curve, does this mean the model developed here is wrong?
Not necessary. Stein [2003] points out that "[a] . potentially very promising agency theory of
investment builds on the premise that managers are likely to be overly optimistic about the prospects of
those assets that are under their control." (p. 123) It has been well recognized that overconfidence as a
cognitive bias exists even in competitive business environments (e.g., Russo and Schoemaker [1992] and
Zacharakis and Shepherd [2001]). What Stein has not pointed out is "inflated" hurdle rate could be a simple
way to curb managerial overconfidence on expected project returns. This "overconfidence" explanation
implies seemingly inefficient capital rationing could be an efficient way to correct for otherwise inefficient
capital allocation. Taking this into account, capital rationing can arise even when a firm faces a downward
sloping capital supply curve. The model thus suggests the following testable hypothesis: Managerial
overconfidence on expected project returns is more likely to be found in firms imposing capital rationing
even with a downward sloping capital supply curve than in those with an upward sloping curve.
Value of corporate diversification: Thought not analyzed in this paper, a firm might be able to better
reduce its default risk and the cost of debt financing by diversifying into different business lines. In contrast,
the business risks of divisions of a focused firm tend to be highly correlated (e.g., due to systematic industry-
level risks), which limits the extent of diversifying the risks. An analysis along this line would need an
extension of the model to allow for correlated project returns; distributions of the expected project returns
26
however can still be independent.
Boundaries of the firm: When more and more business lines of different sectors are pooled together
under a single roof, chances are there would be insufficient talents to manage each of them efficiently. This
would pose a limit on the extent default risk can be reduced through bringing in more business lines of
different sectors. The limit could be a factor determining the boundaries of the firm. An analysis along this
line would need an extension of the model to allow for correlated project returns, as well as asymmetric
expected project returns.
Spinoff and acquisition decisions: A diversified firm might be very careful in keeping optimal the
combination of divisions in different sectors for the purpose of reducing default risk. But unanticipated
external shocks to industry sectors can affect divisions' expected project returns in such a way that some
divisions are no longer worth being included in the conglomerate, or new targets should be acquired to form
a better "portfolio" of divisions in different sectors. Because diversification is less costly when divisions in
different sectors have similar expected project returns, it is conceivable that sometimes a "strong" division
would be spun off, sometimes a "weak" one. What can be sure is that those retained under a single roof are
more similar to each other in terms of expected project returns. Analogously, acquisitions are more likely
when the acquirer's divisions and the acquiree are more similar in their expected project returns. An
extension of the model could also be used to examine such issues.
Diversification discount: Suppose a firm has specialty in its focused sector, which was why it
focused on the sector in the first place. Then it is likely that it does not initially have comparative advantages
in other sectors later brought in for the default risk reduction purpose; otherwise, it would have focused on
those sectors. If divisions of the firm have high expected project returns to begin with, its cost of equity
capital should be relatively low, and therefore it is not urgent to reduce the cost of debt capital either. So
diversification for default risk reduction is not likely to occur at that moment. Later if there are adverse
shocks to the firm's focused sector, expected project returns of its divisions go down and its cost of equity
capital goes up. Then it is both more fruitful and more urgent to consider reducing default risk by
diversification. If diversification takes place at this point, it might seem that diversification leads to a lower
equity value when in fact it is a lower equity value making diversification a more sensible option to
consider. As laid out here, an extension of the model might be useful for understanding the diversification
discount.
27
"Socialistic" capital allocation: The model captures the very essence of potential lack of relation
between current investment opportunities and past investment performance. Imagine the model is repeated
twice to form a two-period model. Suppose project returns are independently distributed among divisions
and across periods, and so are the expected project returns. Moreover, suppose the return of a project in
period 1 is spread out to both periods. Given the lack of relation between expected project returns in the two
periods, a division with a high expected return project in period 1 is not particularly likely to have it again in
period 2 and get the project funded. However, the high expected return project in period 1 is likely to result
in a high realized return spilling over to period 2. Consequently, it might appear that a division with
seemingly high investment efficiency in period 2 fails to prevent capital from flowing to another division
with seemingly low investment efficiency - a phenomenon referred to as "socialistic" capital allocation.
Empirical studies on ICM typically compute investment efficiency of a division on an annual basis.
Hypotheses on the direction of the capital flow are tested accordingly. This approach ignores the fact that (i)
cash flows from past investments affect the investment efficiency computed for the current period; (ii) the
capital flow should be related to current investment opportunities, not past investment performance. An
overlapping generation model using this paper's model as a building block can give guidance to empirical
work on ICM and shed light on the true reason behind the phenomenon of "socialistic" capital allocation.
4.2 Relevance to Accounting
Besides its relevance to a variety of corporate finance issues, the model is also relevant to several
issues in accounting.
Roles of auditor and analyst: Can auditors and analysts play some roles in the model? It has been
assumed that at no cost the CEO can communicate to a bank the expected project returns (i.e., valuation
profile ·) she has learnt through operating an ICM. This is unlikely in reality. Even if the CEO indeed would
be able to provide the managers' project proposals to support her claim on ·, a bank might find it more
convenient to use some "summary statistics" in making the loan term decision. Analyst earnings forecasts
may thus serve as a third-party source of evidence for assessing the truthfulness of the CEO's claim on ·.
By contrast, auditors' job might be literally interpreted as only verifying historical information and
therefore is unrelated to the forward-looking information ·. However, suppose that the model is extended to
an overlapping generation model where the realized return of a project is to arrive at two time points, one at
the end of current year and the other the end of the next year. Then the auditor's verification of the firm's
28
current-year earnings would be informative about the realized project returns of the next year. Consequently,
it would affect the firm's default probability of the next year and hence its COC (i.e., the loan interest rate it
is able to get) at the beginning of the next year for the upcoming year's project financing. Analogously, the
auditor's verification of last year's earnings would be relevant to determining the COC for current year's
project financing.
Even without extending the model this way, the one-shot model can provide a role for auditor if
there is imperfect information about B and M, i.e., the assets in place and the cash flow from regular
operations, respectively. These parameters of the model in general can affect the default probability. When
they are not known with certainty, a financial audit on the balance sheet and income statement of the firm
can reduce the uncertainty about B and M and potentially affect the COC through affecting the default risk
assessment.
Experimental studies on ICM: In recent years, there is a growing interest in accounting and
economics investigating whether behavioral factors like honesty, trust, fairness, etc can play a role in
competitive business environments. For example, Bruggen and Luft [2008] experimentally examine how
competition and honesty interact in the capital budgeting process of an ICM setting. Some of the predictions
are derived from informal theorizing because they could not find any existing model that considers a varying
degree of competition in ICM (i.e., whether only one, two, or all three of the divisions in their setting can
"win" the capital for project investment). The model of this paper fills a gap between two extreme types of
models in the literature, namely "only one can win" and "everyone gets a bite". It provides a theoretical
benchmark for comparing to behaviors of non-(purely-)economic agents in an ICM setting and could be
useful to experimental studies looking at similar settings.
Performance evaluation measures: Although not analyzed in this paper, the model may be used to
examine the relative merits of different accounting performance measures. For example, return on
investment (ROI) is widely discussed in accounting textbooks. Can it be a useful alternative to the residual
income (RI) presumed in the model? RI is often argued to be superior to ROI, yet the latter is believed to be
useful for comparing divisions with very different sizes. If the model is extended to allow for divisions with
asymmetric sizes, could ROI become superior to RI under particular circumstances?
Cost of capital and required rate of return: RI has been advocated as the correct approach to
comparing projects for investment decisions. Yet the specification of the required rate of return remains
29
problematic in practice. Some have argued it should be set at the WACC, which is a widely used estimate of
a firm's COC. The model here provides a micro-foundation for the relation between the required rate of
return (i.e., the hurdle rate h) and the "marginal cost" of borrowing (i.e., C?(k)) to which the COC (i.e.,
C(k)/k - 1) is closely tied. Further improvement on the model to incorporate stock price for shares of the
firm would clear up the role of WACC in this setting.
Accounting ratios and cost of capital: The model offers potential to build a micro-foundation for the
relations between widely used accounting ratios and COC. For example, the equity value of an ICM firm is
affected by the book value of assets in place and the default probability, with the latter tied to the COC. This
might shed light on the linkage between book-to-market ratio and COC. If the model can be generalized to
let the book value of assets and cash flow from regular operations affect the default probability, it might
shed light on the linkage between leverage ratio (or current ratio, etc) and COC as well. But these are
unrelated when restricted to geometric default probability.
4.3 Unanswered Questions
The analysis in this paper is only the first step in exploring the potential of the model and applying it
to understand some of the corporate finance and accounting issues that may be examined with the model. A
number of open questions remain.
Default probability: The analysis of the model is substantially simplified by the assumption of a
specific distributional structure of the project returns. The probability mass at the lower end of the
distribution's support and its disconnection from the smooth part of the distribution is critical to establishing
a default probability unrelated to the expected returns of projects in divisions of an "autonomous" firm. This
in turn allows a simple characterization of the COC charged to the divisions. Can alternative assumptions on
the distributional structure also lead to tractable analyses?
CEO's preference: Can the quasi-linear objective function of the CEO be formally justified by a
career-concern or other types of models? Ignoring the cost-of-borrowing component of the CEO's objective
function will lead to an often assumed preference for empire building (based on gross output). How would
this alternative assumption change the results here?
Division managers' preferences: That a manager cares about the residual income of his division's
project is important to making the model work. Otherwise, the accounting charges would have no impact on
managers' behaviors, and the model would fall apart. Can alternative preferences be assumed to make the
30
model work for other performance evaluation measures?
Compensation contracts observed in practice: The model as it stands excludes incentive
compensations that are found in practice, e.g., budget-based bonus schemes (Sprinkle, Williamson, and
Upton [2008] and Murphy [2001]). Generalizing the model to allow for such possibilities would provide
interesting venues for future research.
Other roles of the headquarters: The model assumes a highly decentralized firm environment, where
the only role of the corporate headquarters (HQ), headed by the CEO, is to allocate capital raised from
outside. It ignores the monitoring role the HQ can play and related control right issues that have been
extensively studied in the literature (e.g., Gertner, Scharfstein, and Stein [1994], Scharfstein and Stein
[2000], and Stein [1997, 2002]). It is certainly interesting to generalize the model to include monitoring
activities and control rights of the HQ.
Non-geometric default probability: The characterization provided in the paper admits a wide range
of default probability besides geometric. Can other assumptions be made to restrict the structure of default
probability to other tractable forms? Advances in this direction would widen applications of the model.
5. Concluding Remarks
This paper outlines a theoretical framework for studying the endogenous determination of the COC.
Recognizing the endogenous relationship between a firm's investment activities and COC has important
implications to empirical work and the valuation of firms. While the theoretical framework proposed here
remains in its infancy stage, its potential is worth further exploration, as suggested by the various corporate
finance and accounting issues that may be examined with this framework.
Using the model of this paper, I have derived results of which some are restricted to the case of
geometric default probability valid under certain conditions. The importance of conducting a micro-level
analysis and considering this restrictive case is to provide a new angle in rethinking issues that might have
been overlooked under the usually presumed case of an upward sloping capital supply curve. Further
analysis to go beyond geometric default probability is certainly interesting. This is left to future research.
A number of potentially interesting extensions of the model have been discussed in an earlier
section. Yet another one is to allow for asymmetric divisions and an endogenous mix of the two sources of
capital, i.e., internal financing via the ICM and external bank debt. As a first step in exploring the potential
of the model, this paper has restricted attention to the polar cases of "autonomous" or ICM firm.
31
Dewaelheyns and Van Hulle [2008] point out that for some business groups, "divisions" are legally separate
member firms with direct access to the external capital markets, alongside with internal debts provided by
parent companies. Allowing for an endogenous mix of internal and external debts might yield testable
implications suggesting interesting empirical work.
32
REFERENCES
Baldenius, T., Dutta, S., and Reichelstein, S. [2007]: "Cost Allocation for Capital Budgeting Decisions,"
Accounting Review, 82, 837-867.
Beck, T., Demirguc-Kunt, A., and Maksimovic, V. [2008]: "Financing patterns around the world: Are small
firms different?" Journal of Financial Economics, 89, 467-487.
Bergemann, D. and Morris, S. [2005]: "Robust Mechanism Design," Econometrica, 73, 1771-1813.
Bernardo, A. E., Cai, H., and Luo, J. [2004]: "Capital Budgeting in Multidivision Firms: Information,
Agency, and Incentives," Review of Financial Studies, 17, 739-767.
Bernardo, A. E., Luo, J., and Wang, J. [2006]: "A theory of socialistic internal capital markets," Journal of
Financial Economics, 61, 311-344.
Brealey, R. A., Myers, S. C., and Allen, F. [2006]: Corporate Finance, 8
th
Edition. McGraw-Hill/Irwin.
Bruggen, A. and Luft, J. [2008]: "Capital Rationing, Competition, And Misrepresentation in Budget
Forecasts," Working paper.
de Motta, A. [2003]: "Managerial Incentives and Internal Capital Markets," Journal of Finance, 58, 1193-
1120.
Demirguc-Kunt, A. and Maksimovic, V. [1996]: "Stock Market Development and Financing Choices of
Firms," World Bank Economic Review, 10, 341-369.
Demirguc-Kunt, A. and Maksimovic, V. [1999]: "Institutions, financial markets and firm debt maturity,"
Journal of Financial Economics, 54, 295-336.
Dewaelheyns, N. and Van Hulle, C. [2008]: "Internal Capital Markets and Capital Structure: Bank Versus
Internal Debt," European Financial Management, doi: 10.1111/j.1468-036X.2008.00457.x .
Easton, P. D. and Monahan, S. J. [2005]: "An Evaluation of Accounting-Based Measures of Expected
Returns," Accounting Review, 80, 501-538.
Ferson, W. E. and Locke, D. H. [1998]: "Estimating the Cost of Capital Through Time: An Analysis of the
Sources of Error," Management Science, 44, 485-500.
Francis, J., LaFond, R., Olsson, P. M., and Schipper, K. [2004]: "Costs of Equity and Earnings Attributes,"
Accounting Review, 79, 967-1010.
Garrison, R. H., Noreen, E. W., and Brewer, P. C. [2008]: Managerial Accounting, 12
th
Edition. McGraw-
Hill/Irwin.
Gebhardt, W. R., Lee, C. M. C., and Swaminathan, B. [2001]: "Toward an Implied Cost of Capital," Journal
of Accounting Research, 39, 135-176.
Gertner, R., Powers, E., and Scharfstein, D. [2002]: "Learning about Internal Capital Markets from
Corporate Spin-offs," Journal of Finance, 57, 2479-2506.
Gertner, R., Scharfstein, D., and Stein, J. [1994]: "Internal Versus External Capital Markets," Quarterly
33
Journal of Economics, 109, 1211-1230.
Hadlock, C., Ryngaert, M., and Thomas, S. [2001]: "Corporate Structure and Equity Offerings: Are There
Benefits to Diversification," Journal of Business, 74, 613-635.
Hail, L. and Leuz, C. [2006]: "International Differences in the Cost of Equity Capital: Do Legal Institutions
and Securities Regulation Matter?" Journal of Accounting Research, 44, 485-531.
Inderst, R. and Laux, C. [2005]: "Incentives in internal capital markets: capital constraints, competition, and
investment opportunities," Rand Journal of Economics, 36, 215-228.
Inderst, R. and Muller, H. M. [2003]: "Internal versus External Financing: An Optimal Contracting
Approach," Journal of Finance, 58, 1033-1062.
Khanna, N. and Tice, S. [2001]: "The Bright Side of Internal Capital Markets," Journal of Finance, 56,
1489-1528.
Kieso, D. E., Weygandt, J. J., and Warfield, T. D. [2006]: Intermediate Accounting, 12th Edition. Wiley.
Lambert, R., Leuz, C., and Verrecchia, R. E. [2007]: "Accounting Information, Disclosure, and the Cost of
Capital," Journal of Accounting Research, 45, 385-426.
Lamont, O. [1997]: "Cash flow and investment: Evidence from internal capital markets," Journal of
Finance, 52, 83-109.
Melumad, N., Mookherjee, D., and Reichelstein, S. [1992]: "A theory of responsibility centers," Journal of
Accounting and Economics, 15, 445-484.
Modigliani, F. and Miller, M. H. [1958]: "The Cost of Capital, Corporation Finance and the Theory of
Investment," American Economic Review, 48, 261-297.
Mukherjee, T. K. and Henderson, G. V. [1987]: "The Capital Budgeting Process: Theory and Practice,"
Interfaces, 17, 78-90.
Murphy, K. J. [2001]: "Performance Standards in Incentive Contracts," Journal of Accounting and
Economics, 30, 245-278.
Rajan, R., Servaes, H., and Zingales, L. [2000]: "Cost of Diversity: The Diversification Discount and
Inefficient Investment," Journal of Finance, 55, 35-80.
Russo, J. E. and Schoemaker, P. J. H. [1992]: "Managing Overconfidence," Sloan Management Review, 33,
7-17.
Saitoh, H. and Serizawa, S. [2008]: "Vickrey allocation rule with income effect," Economic Theory, 35, 391-
401.
Scharfstein, D. S. [1998]: "The Dark Side of Internal Capital Markets II: Evidence from Diversified
Conglomerates," NBER-w6352.
Scharfstein, D. S. and Stein, J. C. [2000]: "The dark side of internal capital markets: Divisional rent-seeking
and inefficient investment," Journal of Finance, 55, 2537-2564.
Segal, I. [2003]: "Optimal Pricing Mechanisms with Unknown Demand," American Economic Review, 93,
34
509-529.
Segelod, E. [1996]: "Resource allocation in divisionalized groups: a survey of major Swedish groups,"
unpublished manuscript. (http://publications.uu.se/abstract.xsql?dbid=2349)
Segelod, E. [1995]: Resource allocation in divisionalized groups: a survey of major Swedish groups.
(http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-2349).
Shin, H. and Stulz, R. [1998]: "Are internal capital markets efficient," Quarterly Journal of Economics, 113,
531-552.
Sprinkle, and Williamson, and Upton [2008]: "The effort and risk-taking effects of budget-based contracts,"
Accounting, Organizations, and Society, 33, 436-452.
Stein, J. C. [1997]: "Internal Capital Markets and the Competition for Corporate Resources,"Journal of
Finance, 52, 111-133.
Stein, J. C. [2002]: "Information Production and Capital Allocation: Decentralized versus Hierarchical
Firms," Journal of Finance, 57, 1891-1921.
Stein, J. C. [2003]: "Agency, Information and Corporate Investment," in Handbook of the Economics of
Finance, Constantinides, Harris, Stulz (ed.).
Stewart, III, G. B., Ellis, Mark, and Budington, D. [2002]: "Stern Stewart's EVA
®
Clients Outperform the
Market and Their Peers," EVAluation, October 2002. Stern Stewart Research, the Americas.
Stice, J. D., Stice, E. K., and Skousen, F. [2007]: Intermediate Accounting, 16th Edition. Thomson South-
Western.
Wulf, J. [2002]: "Internal capital markets and firm-level compensation incentives for division managers,"
Journal of Labor Economics, 20, 5219-5262.
Wulf, J. [2006]: "Influence and Inefficiency in the Internal Capital Market," SSRN-id145908.
Zacharakis, A. L. and Shepherd, D. A. [2001]: "The nature of information and overconfidence on venture
capitalists' decision making," Journal of Business Venturing, 16, 311-332.
35
APPENDIX
PROOF OF LEMMA 2: I will first show that under the regularity condition, the default probability
of a "winning" division (i.e., one provided with a loan) is q, regardless of ·. Consequently, t
i
(·) must be set
at (1+c) ÷ (1+r) + qL for a "winning" division. Requirements of an ex post mechanism then imply it is
weakly better to set p
i
(·
-
i
) = (1+c). As a result, the requirements also imply t
i
(·) = 0 for a "losing" division
(i.e., one declined for a loan), which would complete the proof.
First, note that given any ex post mechanism used by a bank, managers of the divisions will report
their ·
i
's truthfully. Moreover, bank competition ensures that t
i
(·) s (1+r) + L for a "winning" division.
Otherwise, a competing bank could have offered an ex post mechanism with t
i
(·) = (1+r) + L for a "winning
division." Any division willing to accept t
i
(·) > (1+r) + L will find this alternative offer more attractive. Since
division i's year-end cash position, with bonus to the manager already paid, is M + R
i
- |(R
i
- t
i
(·)),
parts a and b of the regularity condition imply that (i) the probability of default is less than 1, and (ii) if legal
procedures of collection are invoked, the competing bank will surely get back t
i
(·) = (1+r) + L for any
realized R
i
leading to default. Consequently, charging t
i
(·) = (1+r) + L will lead to a positive expected profit for
the competing bank, which violates the condition for perfect competition. Hence, the charge t
i
(·) to a
"winning" division must not exceed (1+r) + L.
Now recall that the year-end cash position of a division is M + R
i
- |(R
i
- t
i
(·)). This will be
sufficient to meet the payment obligation to the bank if M + R
i
- |(R
i
- t
i
(·)) > t
i
(·), which is equivalent to
t
i
(·) s R
i
+ M/(1-|). With probability 1 - q, the right hand side of this inequality is at least R + M/(1-|).
Under part c of the regularity condition, this is no less than (1+r) + L, which in turn is no less than t
i
(·).
Thus, the default probability of a division must be at most q, regardless of ·. Bank competition thus implies
t
i
(·) s (1+r) + qL for a "winning" division.
With probability q, R
i
= 0, and division i's year-end cash position is only M + |t
i
(·). For the division to meet
its payment obligation, it requires that t
i
(·) s M/(1-|). Note that t
i
(·) > (1+r), which is a floor of the cost of lending, even
without default risk. Under part d of the regularity condition, M/(1-|) < (1+r) s t
i
(·).
Thus, the division will default when R
i
= 0. In other words,
Pr{default by division i | ·} = q.
Now suppose t
i
(·) < (1+r) + qL for a "winning" division. This will result in a negative expected
profit to a bank unless invoking legal procedures in case of default is unwise sometimes. However, this
cannot be under part d of the regularity condition. To see this, recall that default occurs only when R
i
= 0,
which means the cash position of the division is only M + |t
i
(·). For this to be as great as the bank's payoff from
invoking the legal procedures, it requires t
i
(·) s (M+L)/(1-|). Since t
i
(·) > (1+r), the inequality cannot
hold under part d of the regularity condition. Therefore, it is always wise to invoke legal procedures in case
i
of default. As a result, setting t
i
(·) < (1+r) + qL for a "winning" division will lead to a negative expected
profit to a bank. This means only ex post mechanisms with t
i
(·) = (1+c) ÷ (1+r) + qL for a "winning"
division will be used.
Recall that for ?x(?), t(?)? to be an ex post mechanism, there must exist p
i
, s
i
: ( µ , µ )
n
-1
÷ R
+
such
that for every · e ( µ , µ )
n
,
x
i
(·) = 1 if ·
i
> p
i
(·
-
i
), x
i
(·) = 0 otherwise, and
t
i
(·) = p
i
(·
-
i
) x
i
(·) - s
i
(·
-
i
).
For a "winning" division, t
i
(·) = (1+c). Since s
i
(·
-
i
) > 0, it follows that p
i
(·
-
i
) > (1+c).
Suppose the loan approval cutoff p
i
(·
-
i
) is chosen to be above (1+c). Then potential surplus from a
division with ·
i
such that (1+c) < ·
i
s p
i
(·
-
i
) would not be realized under such a mechanism. A competing bank could
have used an ex post mechanism with t
i
(·) set strictly in between (1+c) and p
i
(·
-
i
) and an approval cutoff
identical to this t
i
(·). This competing bank would then do at least as good as the bank setting p
i
(·
-
i
) > (1+c) and
possibly better when indeed some divisions have ·
i
's falling in between this competing bank's t
i
(·) and the
other bank's p
i
(·
-
i
). In short, setting p
i
(·
-
i
) > (1+c) is weakly dominated. Consequently, an ex post mechanism with
p
i
(·
-
i
) = (1+c) is as good as any other ex post mechanisms.
Given that p
i
(·
-
i
) = t
i
(·) = (1+c) for a "winning" division, it must be that s
i
(·
-
i
) = 0 for ·
i
> (1+c).
However, s
i
(·
-
i
) cannot depend on ·
i
. It follows that s
i
(·
-
i
) = 0 even for ·
i
s (1+c).
In conclusion, a posted-price mechanism with
x
i
(·) = 1 and t
i
(·) = (1+c) if ·
i
> (1+c), and
x
i
(·) = t
i
(·) = 0 otherwise
is at least as good as any other ex post mechanisms to a bank. ?
PROOF OF LEMMA 3: As explained in the main text, it suffices to show that f(q
0
, 1, n, c) s 0 but
f(q
0
, 0, n, c) > 0. First, note that c = r + qL and when K = n, c
0
= r + q
0
L/n. Thus,
f(q
0
, 0, n, c) = n[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o
= [(1+c)o - (c - c
0
)(1-|
0
) - M]n/o
= {(1+r)o + [qo - (q - q
0
/n)(1-|
0
)]L - M}n/o
= {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M}n/o,
which is positive if
(1+r) > [M + (q| - (q
0
/n)(1-|
0
))L]/o.
To show that this inequality indeed holds, note that
o(M+L) - (1-|)[M + (q| - (q
0
/n)(1-|
0
))L]
= (1-|)(M+L) - |
0
(M+L) - (1-|)M - (1-|)(q| - (q
0
/n)(1-|
0
))L
= (1-|)[1- q| + (q
0
/n)(1-|
0
)]L - |
0
(M+L)
ii
> (1-|)(1- q|)L - |
0
(M+L)
> 0,
where the last inequality is due to the given assumption that |
0
< (1-|)(1- q|)L/(M+L). Hence,
(M+L)/(1-|) > [M + (q| - (q
0
/n)(1-|
0
))L]/o.
By part d of the regularity condition,
(1+r) > (M+L)/(1-|)
> [M + (q| - (q
0
/n)(1-|
0
))L]/o,
which implies f(q
0
, 0, n, c) > 0.
Now consider
f(q
0
, 1, n, c) = n[(1+c) - (c - c
0
)(1-|
0
)/o] - nM/o - R
= f(q
0
, 0, n, c) - R ,
which is non-positive if f(q
0
, 0, n, c) s R . Note that the assumption ns ( R - L/o)/[(1+r) - M/(1-|)] implies
ns R /[(1+r) - M/(1-|) + L/no].
Thus,
f(q
0
, 0, n, c)
= {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M}n/o
s {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M} R /[(1+r) - M/(1-|) + L/no]o
= R {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M}/[(1+r) - M/(1-|) + L/no]o.
To see that {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M}/[(1+r) - M/(1-|) + L/no]o< 1, consider
(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M - [(1+r) - M/(1-|) + L/no]o
= - [1 - q
0
(1-|
0
)]L/n - q|L - |
0
M/(1-|)
< 0.
Since {(1+r)o + [(q
0
/n)(1-|
0
) - q|]L - M} > 0, it follows that f(q
0
, 0, n, c) < R .
As f(q
0
, 1, n, c) s 0 and f(q
0
, 0, n, c) > 0, q
0
(n, ·) = q
n
. ?
PROOF OF PROPOSITION 3: As explained in the main text, it suffices to show that f(q
0
, 0, 1, c) >
0. Note that
f(q
0
, 0, 1, c) = [(1+r)o - q|L - nM + q
0
(1-|
0
)L]/o,
which is clearly positive given the additional assumption on n and |
0
. ?
PROOF OF PROPOSITION 4: Given the regularity condition and assumption SBAFD1, q
0
(K, ·) =
q
K
and (1+c
0
)K = (1+r)K + q
K
L. Therefore, the difference obtained by deducting an "autonomous" firm's
equity value AF from an ICM firm's equity value lower bound IF is as follows:
?÷ [Kq - nq
K
](1-ì)B + [Kq - q
K
]L - |
0
[E
i
eI(c
)
·
i
- (1+r)K - Lq
K
].
iii
First, recognize that µ K> E
i
eI(c
)
·
i
> (1+c)K = [(1+r) + qL]K > (1+r)K + q
K
L. Since
|
0*
÷ [(q - nq
K
/K)(1-ì)B + (q - q
K
/K)L]/( µ - (1+r) - Lq
K
/K),
it follows that |
0*
( µ K - (1+r)K - q
K
L) = (Kq - nq
K
)(1-ì)B + (Kq - q
K
)L, which is positive under the
condition that 2 s K s n s [2((1-ì)B + L)/q - L]/(1-ì)B. For |
0
s |
0*
,
|
0
[E
i
eI(c
)
·
i
- (1+r)K - q
K
L]
s |
0*
[E
i
eI(c
)
·
i
- (1+r)K - q
K
L]
< |
0*
( µ K - (1+r)K - q
K
L)
= (Kq - nq
K
)(1-ì)B + (Kq - q
K
)L,
which means ?> 0. ?
PROOF OF PROPOSITION 5: Note that
F(q
0
, s, h) ÷ f(q
0
, s, K(h), h)
= K(h) [(1+c
0
)(1-|
0
) - (1+h)|]/o - nM/o - s R
= K(h)[(1+ r)(1-|
0
) - (1+h)|]/o + q
0
L(1-|
0
)/o - nM/o - s R .
It suffices to show that for any h > 0 with K(h) > 1, F(q
0
, 0, h) > 0 and F(q
0
, 1, h) s 0.
To show that F(q
0
, 0, h) > 0 for any h > 0 with K(h) > 1, it suffices to consider only h< µ - 1;
otherwise, I(h) = ? and K(h) = 0. For h with K(h) > 1,
F(q
0
, 0, h)
= K(h)[(1+ r)(1-|
0
) - (1+h)|]/o + q
0
L(1-|
0
)/o - nM/o
> [(1+ r)(1-|
0
) - µ |]/o + q
0
L(1-|
0
)/o - nM/o
> [(1+ r)(1-|
0
) - ( µ | + nM)]/o,
which is non-negative under assumption SBAFD2. Thus, F(q
0
, 0, h) > 0 for any h with K(h) > 1.
Now consider
F(q
0
, 1, h)
= K(h)[(1+r)(1-|
0
) - (1+h)|]/o + q
0
L(1-|
0
)/o - nM/o - R
< K(h)(1+r)(1-|
0
)/o + q
0
L(1-|
0
)/o - nM/o - R
s n(1+r)(1-|
0
)/o + L(1-|
0
)/o - nM/o - R
= n[(1+r) - M/o] + L(1-|
0
)/o - R .
To see that this is negative, note that o = (1-|-|
0
) < (1-|) implies (1+r) - M/(1-|) > (1+ r) - M/o, which is
positive under assumption SBAFD2. Additionally, by the assumption,
ns ( R - L/o)/[(1+r) - M/(1-|)]
iv
< ( R - L/o)/[(1+ r) - M/o]
< [ R - L(1-|
0
)/o]/[(1+ r) - M/o].
Therefore, F(q
0
, 1, h) < 0 for any h > 0. ?
PROOF OF PROPOSITION 6: Recall that the CEO's objective is to maximize her bonus, which is
maximized when the value created from project investment is maximized, i.e.,
Max
h
E
i
eI(h
)
·
i
- C(K(h)),
where K(h) ÷ |I(h)| and I(h) ÷ {i | ·
i
> (1+h) for ·
i
of ·}.
Let e(a) denote the index of the ath highest ·
i
of ·, and hence ·
e
(a
)
is the ath highest ·
i
.
16
Define A÷ max{ a | ·
e
(a
)
>
C?(a) for a s n }.
It is straightforward to show that C(k) ÷ (1+r)k + Lq
k
is strictly convex with C?(k) = (1+r) + Lq
k
ln(q) > 0 for
k> k
*
÷ ln[-Lln(q)/(1+r)]/[-ln(q)]. The strict convexity of C(k) means C?(k) is increasing in k. So borrowing
a loan amount beyond A to finance some project with ·
e
(a
)
< ·
e
(A
)
can only reduce the value created from
project investment. Because K(h) = A for h = C?(A) - 1, setting the hurdle rate at this level is optimal to the
CEO if any loan amount a< A is suboptimal.
Note that the capital supply curve C(k)/k = (1+r) + Lq
k
/k is downward sloping. If the loan amount is
reduced to a< A, the "average cost" of borrowing (1+c
0
) = (1+r) + Lq
a
/a is higher for the projects financed
while the number of projects financed is smaller. With the same · given, the value created from project
investment must be lower if a loan amount a< A is borrowed. ?
16
For simplicity, ties are ignored; they are zero-probability events if · is drawn from a continuous probability
distribution.
v
Incomplete info. : · known only to managers
privately
ICM modeled as a multi-unit hurdle rate h set according to
auction: ¢x(?), t(?)² anticipated c
0
as a function of
A , the number of projects approved
Complete info : · learnt by the CEO thru'
operating the ICM
· communicated to
banks at no cost
Default probability: q 0
Transaction cost(s) of debt financing:
L, Legal cost of collection in case of default
(also, for shareholders, there is a loss in shareholder
value due to forced liquidation of assets)
Cost of capital: (1+c
0
)A
Perfect bank competition
(1+r )A + q
0
L
Figure 1. Important Elements of the Model
Figure 2. Geometric Default Probability Implies Capital
Sponsoring and Subsidized Hurdle Rate
(1+c)
(1+c)
(1+r)
mw(a)
·
e
(k)
C(k)/k
C(k) = (1+r)k + Lq
k
C(k)/k = (1+r) + Lq
k
/k
C?(k) = (1+r) + Lq
k
ln(q)
C(k)+
k
r)
(1+h) (1
/
C'(k)
C?(k)
A
1 2 3 4 5 6 7 8 9 10
k
doc_198676996.docx