The Impact of Credit Rating Agencies on Capital Markets
Spyros Terovitis
∗
†
April 13, 2016
Abstract
In this paper I argue that the reason for potential problems emerging from Credit Rating Agencies (CRAs) is more pathological than the literature recognizes; even in the absence of conflicts
of interest or other distortions resulting from players’ behavior, the market structure itself can
be a source of inefficiency. I develop a setting, where, in the benchmark case, the CRA is capable
of perfect monitoring and reveals its private information truthfully and costlessly. I explore the
impact of such a “sincere” CRA on the interest rate and the probabilities of project financing
and default. I find that even under these- near-utopian-, conditions the introduction of a CRA
may not be beneficial. Specifically, a CRA may lead to under-financing of projects with positive
net present value that would otherwise be financed; a higher expected interest rate; and a higher
expected probability of default. These findings arise from the feedback effect inherent in capital
markets, and the asymmetric impact the effect has on firms of different quality. I argue that
restricting CRAs to provide hard evidence with their ratings might have a negative effect on
the probabilities of project financing and default.
JEL Classification: D82, D83, and G24
Keywords: Credit Rating Agencies, Feedback Effect, Financial Regulation.
∗
I am grateful to Motty Perry and Ilan Kremer for their support and their very useful comments. Moreover, I
would like to thank Herakles Polemarchakis, Dan Bernhardt, Vasiliki Skreta, Giulio Trigilia and the participants at
the European Winter Meeting of the Econometric Society, the RES Annual Conference 2015, the Game Theory and
Applications UECE Meeting and the Warwick Workshop. I also wish to thank the Royal Economic Society as well
as the State Scholarships Foundation of Greece for the financial support.
†
University of Warwick, Department of Economics, Coventry, CV4 7AL, United Kingdom, email: .terovitis@warwick.ac.uk
1
Introduction
The 2007-2008 financial crisis raised unprecedented scrutiny on Credit Rating Agencies (CRAs).
CRAs have been accused of failing to predict the financial crisis (Mason and Rosner, 2007), of
following investor opinions rather than leading them (Richard and Steward, 2003) and most importantly of facing conflicts of interest which leads to rating inflation (Mathis, McAndrews, and
Rochet 2009, Bolton, Freixas, and Shapiro 2012). In this paper I argue that the reason for potential problems emerging from CRAs is more pathological than the literature recognizes; potential
sources of inefficiencies lie not only in the conflicts of interest, the way the players act, and the
possibility that they are not rational, but in the market structure itself. To show this, I develop a
model where: there are no conflicts of interest; ratings are accurate and truthful; and there is no
price (rating fee) distortion. Even in this ideal environment, the introduction of a CRA may lead
to higher expected probability of default and under-financing of projects with positive NPV that
would be financed in the absence of a CRA.
CRAs play a significant role in the market by affecting major economic variables, such as the
interest rate and the probability of default. The main mechanism through which credit ratings
affect such variables is by shaping of investors’ beliefs, and hence investment decisions. Unlike
meteorologists, who predict the weather without affecting it, CRAs produce ratings that do affect
the quality of the asset they rate. Kuhner (2001) points out that a credit rating can take the form
of a self-fulfilling prophecy. Similarly, Manso (2013), captures the self-fulfilling prophecy nature of
ratings by showing that a rating deterioration may trigger a “death spiral” . The feedback effect
inherent in capital markets lies in the heart of my model.
I develop a model of project’s financing where the entrepreneur is privately informed about his
project’s efficiency level, and he cannot commit to exerting effort. The information asymmetry
about the entrepreneur’s type is partly resolved by a CRA. Following CRAs’ common practice, i.e.,
issuing ratings that correspond to a predetermined level of creditworthiness, I cluster entrepreneurs
into two categories: entrepreneurs with high expected cost of effort (HEC) and entrepreneurs with
low expected cost of effort (LEC). This distinction captures the different levels of efficiency in project
implementation. Thus, HEC entrepreneurs are of low credit-worthiness and LEC entrepreneurs are
of high credit-worthiness. The inherent information asymmetry about the entrepreneur’s efficiency
1
level, together with the absence of commitment to exerting effort, allows me to endogenously
determine the probability of default. In such an environment, a feedback effect emerges: the
interest rate is determined by the expected probability of default, which in turn determines the
realized probability of default.
I argue that once one takes the market structure into consideration, better information by introducing a CRA, may not result in a better outcome from a social perspective. The reason is that
in a setting characterized by information asymmetry and lack of commitment, introducing better
information bears an inherent trade-off: it alleviates the information asymmetry problem but it
exacerbates the adverse effect of non-commitment. The main findings suggest that the impact of
introducing an intermediary depends on the features of the environment. In particular, the introduction of a CRA: (i) leads to higher expected default probability when the entrepreneur is efficient
enough to raise funds independently of the presence of a CRA (mild information asymmetry), (ii)
leads to under-financing, when financing of an HEC entrepreneur is feasible only if he is pooled with
an LEC entrepreneur (moderate information asymmetry), and (iii) alleviates under-financing, when
financing of an LEC entrepreneur is feasible only if he can be differentiated by an HEC entrepreneur
(severe information asymmetry). I characterize the level of accuracy of the CRA’s private signal
which maximizes welfare, and I show that some degree of inaccuracy or rating inflation might be
optimal. Also I consider the case where the CRA charges the profit-maximizing fee. I find that
a positive fee adversely affects the probability of default, but it does not affect the probability of
market breakdown.
The intuition behind the findings presented in the previous paragraph relates to the feedback
effect in capital markets and its asymmetric impact on firms of different quality. In order to
understand this asymmetry, one should first understand how the feedback effect works in practice.
Lack of commitment implies that the entrepreneur exerts effort only if the cost of doing so is below
a certain threshold, ĉ. This cost is known to the entrepreneur but unknown to the investors. In
this setup the probability of default coincides with the probability of the cost of effort being above
ĉ. Consider the case where an investor wishes to finance an entrepreneur. First, he forms his
beliefs about the probability of default, and then, based on these beliefs he demands an interest
rate which allows him to break even. The raised funds adjusted by the interest rate (capital cost)
need to be paid back by the entrepreneur. For an entrepreneur whose cost of effort is such that he
2
is indifferent between exerting and not exerting effort, adding the capital cost makes the project
unprofitable, thus, the implementation threshold decreases. The lower implementation threshold
coincides with a higher probability of default, which results in investors demanding a higher interest
rate. This cycle continues until the interest rate converges to its equilibrium value. Because the
impact of the feedback effect is smaller for a more efficient entrepreneur, the negative effect of a
downgrade dominates the positive effect of an upgrade. Along these lines, Kliger and Sarig (2000),
use a natural experiment to show that credit ratings affect the cost of capital and Kisgen (2006)
shows that a firm’s structural decision is directly affected by credit ratings.
The finding that the introduction of a CRA can lead to under-financing of a positive NPV
project that would otherwise be financed is another consequence of the non-commitment assumption. The underlined intuition is captured in the following example: Consider a case where an
HEC entrepreneur has a bad rating and investors require an interest rate which makes exerting
effort not profitable. Investors anticipate that if the entrepreneur is financed, then he will not exert
effort, and, as a result, they are not willing to finance him. Now consider the case where there
is no CRA and an HEC entrepreneur is mixed with an LEC entrepreneur. The investors are now
more optimistic, and they require a lower interest rate. If the lower interest rate turns the choice of
exerting effort from unprofitable to profitable, investors anticipate that effect, and they are willing
to finance the entrepreneur.
Finally, I address whether restricting a CRA to provide hard evidence leads to better outcomes.
Notice that the hard-evidence assumption effectively restricts the rating policy to truthful disclosure, which, in turn, diminishes the role of the CRA. Relaxing the hard-evidence assumption gives
rise to a model similar to the Bayesian Persuasion model developed by Kamenica and Gentzkow
(2009). In this environment I explore the optimal disclosure policy when the CRA can commit in
advance to it. I show that when the CRA can pre-commit to a disclosure rule, rating inflation can
be part of the equilibrium. I differ from the related literature (Mathis et al., 2009) by showing
that rating inflation can emerge even if the fee is not rating contingent. Specifically, I find that
the optimal disclosure policy is either a truthful revelation of the type, or a babbling equilibrium
where no information is revealed. The optimal disclosure policy implies that the presence of a CRA
alone can alleviate market break-down, and increase financing opportunities compared to a regime
where the CRA needs to provide hard evidence. Hence, restricting a CRA to provide hard evidence
3
might worsen financing opportunities. Another implication of the optimal disclosure policy is that,
compared to the regime without CRA, a CRA can improve financing opportunities of an LEC type
but it can not improve financing opportunities of an HEC type.
Because the introduction of a CRA in capital markets can be also interpreted as a revelation
of a signal about the creditworthiness of a party, this paper has implications about the optimal
information disclosure policy imposed by a government. For instance, a key question after the recent
crisis is whether the results of stress tests for banks should be revealed (Goldstein and Leitner, 2013).
The answer that my analysis suggests is that no disclosure could be welfare enhancing, unless the
market breaks down in the absence of additional information. In addition, this work produces
implications related to the sovereign debt in eurozone countries.
Finally, my work suggests that future research should take into account capital markets’ characteristic feedback effect in order to analyze whether a conflict of interest exists, and, if so, whether
this affects the quality of the reported ratings. This finding opens the doors to policy considerations
and raises concerns regarding CRAs regulation.
This paper is organized as follows: Section 2 reviews the related literature. Section 3 introduces the model. Section 4 presents the three different regimes. Section 5 explores the impact of
introducing a CRA. Section 6 relaxes the assumption of hard evidence and explores the optimal
rating policy when the CRA can commit ex ante to it. Section 7 allows the agent to choose a
profit-maximizing fee. Section 8 discusses and concludes.
2
Literature Review
My paper relates primarily to two strands of the literature: the literature dealing with the effect of
CRAs on real economic variables, and the literature that recognizes the adverse effect of information.
Much of the literature on CRAs focuses on the quality of the reported ratings. There are three
main approaches to analyzing the quality of reported ratings: (i) by testing if there is a conflict of
interest, (ii) by examining the way the CRA chooses to disclose its private information, and (iii)
by modeling behavioral biases of investors, and ratings shopping by issuers.
The most popular way the literature addresses the question of whether the CRA has an incentive
to be truthful is by testing the validity of the main argument used by CRAs in response to the
4
criticism of rating inflation, i.e., the reputation-concerns argument. According to this, inflating the
ratings would harm a CRA’s reputation, and ultimately force it out of the market; hence, such
conflict of interests does not exist. Mathis et al. (2009) develop a model with rating-contingent
fees and they demonstrate that the reputation-concerns argument only works when enough of the
CRA’s income comes from sources other than rating complex projects. Bouvard and Levy (2012)
also test the reputation concerns argument. They show that a higher reputation for transparency
is not always desirable because while it results in a higher sale price conditional on the certification,
it comes at the cost of a lower probability of being certified. Another strand of literature links the
quality of ratings with information disclosure. Lizzeri (1999) adopts a mechanism design approach.
He shows that a monopolist CRA only reveals the minimum level of information, but if there are
multiple CRAs, information is disclosed fully. In contrast, Faure-Grimaud, Peyrache, and Quesada
(2007) adopt contingent fees and ex post efficient contracts, and they show that competition reduces
information revelation. Several papers attribute rating inflation to behavioral biases of investors
and to ratings shopping by issuers: if investors are not sophisticated enough to understand that a
rating is not necessarily accurate, then an issuer has an incentive to buy good ratings, and a CRA
wants to inflate its ratings to attract customers. Bolton et al. (2012) develop a model with the
interaction of sophisticated and naive investors, and the results show that a duopoly is generally
less efficient than a monopoly, because the entrepreneur has the opportunity to shop for a good
rating to exploit naive investors. Skreta and Veldkamp (2009) obtain similar findings, where the
direct implication of rating shopping is the systemic bias in disclosed ratings, even if each CRA
produces unbiased ratings. Opp, Opp, and Harris (2013) show that ratings inflation can emerge
if the face value - not the informational value - of the rating that matters. This is the case when
rating-contingent regulation is required for institutional investors.
My central approach deviates from this literature; I focus on a seemingly best case scenario
where the CRA always reports its private information truthfully, and hard evidence supports each
rating. This environment does not allow for a conflict of interest. As I recognize the ability of ratings
to affect the creditworthiness of a project, I focus on the impact of CRAs on capital markets. Save
for Boot, Milbourn, and Schmeits (2006), Kuhner (2001), Manso (2013) and to some extent Mathis
et al. (2009), the literature has ignored this effect. Boot et al. (2006) propose that credit ratings
can serve as a coordination mechanism in situations where multiple equilibria exist. Mathis et al.
5
(2009) show that the behavior of CRAs can lead to reputation cycles, with implications for credit
spreads. Kuhner (2001) shows that when CRAs care about reputation, they are more likely to
reveal their private information if their ratings cannot become self-fulfilling ex post.
Manso (2013) deals with the feedback effect of credit ratings. He describes an environment
where a single CRA interacts repeatedly with a firm that holds performance-sensitive debt, and
whose payout flows are linked to its rating. This framework enables him to incorporate the feedback
effect of credit ratings in a dynamic credit-rating model. He finds that when forming its rating
policy the CRA should focus not only on ratings accuracy, but also on the effect of the ratings on
the borrower’s probability of survival.
This paper differs from Manso (2013) in three important dimensions: (i) the nature of the
feedback effect, (ii) the information available to the CRA and (iii) the paper’s main focus. This
paper examines the feedback effect inherent in capital markets rather than the feedback effect of
ratings. In my setup the credit rating affects the project’s credit quality but, as opposed to Manso
(2013), the project’s quality does not affect the CRA’s rating behavior. Thus, this paper explores
the feedback loop between the project’s quality and the investors’ decisions rather than the feedback
loop between project’s quality and credit rating. The second departure from Manso (2013) is that
the CRA has an information advantage over potential investors. In Manso (2013) the cash flow
process, which is the only parameter which determines firm’s creditworthiness, is observed by all
market participants. In contrast, in this paper the CRA observes a private signal about the firm’s
creditworthiness. That characteristic enables me to provide micro-foundations for the impact of
CRAs on the cost of capital. Another departure from Manso (2013) concerns the methodology
underlying the determing of capital cost. In Manso (2013) the capital cost depends on ratings
exogenously, whereas in this paper, the capital cost is endogenously determined, capturing all the
parameters of the model. Finally, regarding the main focus of the paper, Manso (2013) is interested
in exploring the effect of the rating policy, i.e., the function which maps the cash flow into a rating,
in the economy. This work focuses instead on the impact of providing information via a CRA on
capital markets.
Contrary to this model which highlights the feedback effect inherent in capital markets, Goldstein and Guembel (2008) develop a model which allows for a feedback effect in financial markets.
They build a model where investors might be more informed than the firm about parameters which
6
determined profitability of a project. As a result, price reaction is informative, and it could be used
by the firm in determining its investment strategy, something which allows uninformed investors
to manipulate prices and affect real economic decisions. This papers differ from Goldstein and
Guembel (2008) by assuming that the firm and not the investors has information advantage and
by focusing on the capital-raising game rather the allocational role of prices.
In addition, my work pertains to the literature on the adverse welfare consequence of information
disclosure. Hirshleifer (1971) argues that an increase in the amount of information leads to welfare
reduction because it destroys hedging opportunities. More recently, Amador and Weill (2010) show
that the effect of releasing partial information about a monetary or productivity shock is twofold:
on the one hand, providing more information benefits the economy; but on the other hand, it forces
households to value the newly released public information more and their private information less.
As a result, the situation leads to reduction of the endogenous informational content of prices.
Kondor (2013) shows that when the correlation across the private information of different groups is
low, the release of public information increases disagreement among short-horizon traders about the
expected selling price. Kurlat and Veldkamp (2012) argue that disclosure of information reduces
an asset’s risk and hence its return. As a result, high-risk, high-return investments disappear and
investor welfare falls.
The mechanism through which better information can be harmful differs from the existing
literature. I suggest that the welfare reduction is a consequence of the coexistence of asymmetric
information and lack of commitment which generates an asymmetric impact of the feedback effect on
downward and upward deviations in beliefs of investors after the introduction of more information
(CRA).
This paper also relates to the literature on Bayesian Persuasion. In Section 8 I show that if the
CRA is not obliged to provide supporting evidence for its ratings, and if it can commit to a rating
policy, then the emerging setup is similar to Kamenica and Gentzkow (2009).
Finally, my paper contributes to the literature on optimal security design. It shows that when
the probability of implementing a project is endogenous, and when entrepreneurs know at the time
of contracting their realized return, the main finding of Nachman and Noe (1994) holds.
7
3
Benchmark Model
I consider a setting with three types of risk neutral players: a cashless entrepreneur, a CRA, and
investors of mass 1. The entrepreneur holds a project whose implementation requires investment
of size 1 which is raised by selling securities.1 The project returns R if it is successful and 0 if it
fails2 , where R is known to all parties3 . The project is successful with probability 1 as long as the
entrepreneur exerts effort. If no effort is exerted, then project fails with certainty.
4
The project’s
final outcome is observed by all parties. Exerting effort implies a cost, c, and the entrepreneur
cannot commit to exerting effort.
There is asymmetric information between the entrepreneur and investors; only the entrepreneur
observes the cost of effort. The entrepreneur’s cost of effort is a priori unknown to potential
investors. The information investors have is that the entrepreneur is of low expected cost (LEC)
with probability λ and of high expected cost (HEC) with probability 1−λ. Investors also know that
the effort cost of an LEC (HEC) entrepreneur is drawn by a distribution with fL (fH ) and FL (FH )
standing for the probability density and cumulative distribution function respectively. Throughout
this paper I assume that an HEC entrepreneur First Order Stochastically Dominates (FOSD) an
LEC entrepreneur, i.e., FL (c0 ) ≤ FH (c0 ). Thus, an entrepreneur of LEC can be interpreted as the
(in expectation) efficient entrepreneur and an entrepreneur of HEC as the inefficient entrepreneur.5
I restrict the analysis to the case where both LEC and HEC entrepreneur have positive net expected
return, i.e., R > 1 + E[ci ] for each i ∈ {H, L}.
Unlike potential investors, the CRA can monitor whether the entrepreneur is of LEC or HEC.
The intuition behind this assumption is the following: First, it captures the empirical observation
that CRAs have better information compared to outside investors about the creditworthiness of
firm because CRAs have more experience in monitoring, because they have a more complete picture
of the market conditions. Second, it reflects the fact that regardless of the experience that CRAs
1
Raising funds in exchange of a security is the reduced form of a setup where the entrepreneur sells a security at a
price P in order to finance a project of size 1 ≥ P . The reason is that a direct consequence of a pooling equilibrium
in the contracting stage is that there is a loss for the efficient entrepreneur (LEC), thus, he does not have incentive
to raise more than the capital required for the investment, i.e. P = 1.
2
This is without loss of generality as long as the return in case of failure, RF is lower than R
3
The main findings are robust to the case where R is drawn from a distribution which is known to the entrepreneur
and investors
4
The main findings are robust if we instead assume that exerting effort increases the probability of success.
5
The main findings are robust for any number of types of the entrepreneur.
8
may have, they are not able to know more about the creditworthiness of a firm than what the firm
itself does.
The reason the CRA only knows the type of the firm, i.e., the distribution from which the actual
cost is derived and not the realized return, relates to two particular characteristics of the rating
industry: the monitoring method and the clustering of ratings. Regarding the monitoring method,
CRAs usually base their ratings on a mix of indexes. Hence, two firms with similar index values
are likely to receive similar rating, even though the indexes do not reflect other parameters that
are vital for a firm’s profitability. Regarding the clustering of ratings, one of the characteristics
of the industry is that CRAs issue ratings following a specific rating scale and refrain from giving
predictions about the exact economic outcome.
The rest of this subsection is devoted to the timing of the events, the problems that each party
faces, and the equilibrium concept. A more detailed analysis of each player’s problem follows in
Section 4.
Timing
The timing of the events is as follows:
1) Nature determines the effort cost c̃ and the type of the entrepreneur.
2) The CRA announces its rating fee, H (zero in the benchmark model).
3) The entrepreneur decides:
(i) whether to ask for a rating from the CRA, and
(ii) the contract he offers.
4) Investors observe the rating (if any) and the contract, and then decide whether to invest.
5) Conditional on financing, the entrepreneur chooses whether to implement the project.
6) The contract is executed.
The CRA’s Problem
I assume that the CRA truthfully reveals its private information. This assumption could be justified by the observation that CRAs usually provide evidence supplementary to their ratings. In the
benchmark model I consider a CRA that can costlessly monitor the type of the entrepreneur, and
does not charge a rating fee. These assumptions suppress any conflict of interest and allow me: i)
9
to focus on the effect that a CRA has on capital markets by providing better information, and ii)
to explore the welfare implications of introducing a CRA. In Section 6, I relax the hard evidence
assumption whereas in Section 7, I relax the zero-fee assumption.
Entrepreneur’s Problem
First, the entrepreneur decides whether to ask for a rating. In doing so, he takes into consideration:
i) the expected rating, ii) what would be the contract that he would issue with and without rating,
and iii) whether he would exert effort in each case. In the benchmark model where the rating fee
is zero, an LEC entrepreneur can costlessly differentiate himself, which enables him to promise a
lower return to investors6 . In contrast, an HEC entrepreneur has no incentive to ask for a rating,
because by doing so, he would reveal his type. Thus, the problem of choosing whether to ask for a
rating becomes trivial; only an LEC entrepreneur ask for a rating.
Second, the entrepreneur decides the contract which maximizes his expected profit. It is shown
that the profit-maximizing contract promises a payment W if the the return is R and zero otherwise.
This simple form of the optimal contract is a consequence of non-verifiability (or only partial
verifiability) of the cost of effort. I analyze the optimality of such contract in following subsection.
Third, given the rating and the optimal contract, the entrepreneur decides whether to exert
effort. Recall that for a project to be successful: i) the investment needs to take place, and ii) the
entrepreneur has to exert effort. Thus, the entrepreneur’s utility in each case would be:
U (ef f ort) = R − W − c̃
U (not ef f ort) = 0
Hence, the entrepreneur exerts effort as long as:
c̃ ≤ R − W ≡ ĉ
(1)
A critical variable of the model is the threshold level ĉ. The intuition behind this is straightforward,
and it captures the fact that the entrepreneur will exert effort only if the benefit from doing so,
R − W , does not exceed the cost, c̃.
6
The costless differentiation is not possible when the CRA can charge a profit maximizing fee.
10
It is crucial to mention here that the main findings would be unaffected if the entrepreneur
observes a signal about the realized cost rather than the realized cost itself. The reason I present
the case where the entrepreneur observes the realized cost is mathematical convenience. In the
Appendix, I prove the coincidence of the two models.
I define as default the case where the entrepreneur fails to pay back his loan. In this model,
a default occurs when the project returns zero, which coincides with the case where no effort is
exerted. This leads to the following definition of the probability of default.
Definition 1 (Probability of default): The probability of default, d, coincides with the probability
of financing an entrepreneur whose realized cost of effort exceeds ĉ:
d ≡ P rob(def ault) = P rob(c > ĉ)
(2)
Similarly, the survival probability, s, coincides with the probability of financing an entrepreneur
whose realized cost of effort is below ĉ:
s ≡ P rob(not def ault) = P rob(c ≤ ĉ)
(3)
Because ĉ is negatively related to the payment W , the probability of default is positively related to
the endogenously determined W .7 To avoid the trivial case in which the firm can finance through
the issuance of risk-free debt, I assume that there are values of c such that the entrepreneur chooses
to default even if the implied interest rate is zero, i.e. W = 1.
The Investors’ Problem
Investors form their beliefs about the probability that the entrepreneur exerts effort, and they
require an interest rate that satisfies their participation constraint. I assume perfectly competitive
capital markets. Consequently, the participation constraint of investors binds and enables me to
7
What matters for the main findings is that the effort exertion threshold, which determines the probability of
default, is negatively related to the payment W . Alternatively, this could be achieved by assuming that the project
succeeds with certainty when the entrepreneur exerts effort with cost c and succeeds with probability q if there is no
effort. In such a setup the entrepreneur chooses to exert effort if an only if c̃ ≤ (R − W )(1 − q) which is associated
to a probability of success equal to P rob(c ≤ ĉ) + qP rob(c > ĉ) = q + (1 − q) E[P rob(c ≤ ĉ)|Ω].
11
determine the equilibrium interest rate by the investor’s zero profit condition:
W E[s|Ω] = 1
(4)
where the LHS is the expected benefit and the RHS is the borrowed capital. More specifically,
W stands for the payment the entrepreneur promises, and s̃ for the investors’ beliefs about the
probability of not defaulting given their information set Ω. Following Definition 1:
E[s|Ω] ≡ E[P rob(c ≤ ĉ)|Ω]
(5)
The investors are willing to finance the project as long as E[P rob(c ≤ ĉ)|Ω] ≥ W −1 . Thus, it
is the investors’ beliefs about the probability of default - not the probability of default itself that determine the interest rate. By substituting (1) into (5), given the information set, and by
substituting of (5) into (4), we obtain:
E[P rob(c ≤ R − W )|Ω = W −1
(6)
which determines the equilibrium payment. The feedback effect can be seen in the previous fixedpoint equation; positive probability of default implies a positive interest rate (W > 1) which
increases the probability of default, which in turn increases the interest rate.
The Optimal Contract
I now return to the entrepreneur’s problem, and, more specifically, to optimality of the suggested
contract. A contract, w, defines the payment in each state of the world, i.e., the realized value of c,
c̃, and the final output (R or 0). Recall that the cost of effort is not verifiable. Lack of verifiability of
the cost gives rise to two main implications: i) the contract pins down to two relevant-in-equilibrium
payments, and ii) the equilibrium in the capital raising problem is pooling. We explore these two
implications sequentially.
Regarding the equilibrium payments, note that the entrepreneur’s limited liability implies that
the payment in case where the output is zero is also zero. Thus, the only case where the payment
can be positive is when the projects succeeds. Suppose now that the contract w implies different
12
payment for different values of the realized cost. The entrepreneur would never find it optimal to
report a realized value of cost which does not correspond to the minimum payment, wmin . Hence,
in equilibrium the only relevant payments would be 0 if the project’s return is 0, and wmin if the
return is R.
To explore the lack of a separating equilibrium, we follow the intuition of Nachman and Noe
(1994). Suppose that a separating equilibrium exists where wc̃0 is a contract offered by type c = c̃0
and wc̃00 is a contract offered by type c = c̃00 with c̃00 > c̃0 . In case type c = c̃00 mimics type c = c̃0 the
gains for type c̃00 (if effort is exerted) is the difference between the expected payment under contract
wc̃0 , Ec̃00 wc̃0 and the expected payment under contract wc̃00 , Ec̃00 wc̃00 . Notice that this difference is
always non-negative since it can never be the case that type c = c̃00 chooses to exert effort (thus, it
has a positive return) and type c = c̃0 chooses not to exert effort. This is true because the utility of
type c = c̃00 is always lower than the utility of type c = c̃0 . Thus, we can focus on pooling equilibria
where the all types of entrepreneur offer the same contract w. Note also that there might be more
than one pooling equilibrium, which leads us to the following definition.
Definition 2 (Equivalent Contracts): Contracts with payments contingent to the reported value of
effort cost have the same expected return, and, hence, they are equivalent, if and only if they have
the same minimum positive payment.
The previous analysis allows me to assume without loss of generality that the contract is contingent
on the project’s return instead of the reported value because the two contracts imply the same
payment in equilibrium. Hence, the contract is associated with the following return for potential
investors:
Investor’s return


 W
if return R

 0
otherwise
(7)
I now deal with the uniqueness of the optimal contract. An entrepreneur who is not planning
to exert effort mimics an entrepreneur who exerts effort by offering the same contract. Thus, the
expected revenue of potential investors is W times the probability of financing an entrepreneur who
exerts effort, which coincides with the probability that W < R − c. This last point implies that
an efficient entrepreneur who plans to exert effort in equilibrium, should choose the value of W
13
taking into consideration the probability of default, such that the investors’ zero-profit condition
is satisfied. Note that depending on the distribution of effort cost c there could be multiple combinations of payment-default probabilities that satisfy the zero-profit condition. However, the only
combination/equilibrium which survives is the one that corresponds to the lowest value of W , W ∗ ,
because this equilibrium dominates all other equilibria from the entrepreneur’s perspective.
The question that emerges is whether such contracts are observed in capital markets. Offering
such contracts is a method of financing similar to debt financing; the entrepreneur issues bonds that
return min{(1 + r), V }, where V is the firm’s value and r the interest rate. Since the entrepreneur
has no wealth other than what the project returns, the value of the firm will be either R if effort is
exerted, or 0 otherwise. Note also that if (1 + r) ≥ V the entrepreneur would choose to not exert
effort because this would lead to negative profit. Hence, the return can only be (1 + r) if the project
is implemented (V = R), or 0 if the project is not implemented (V = 0). Direct consequence of
this is that the suggested contract is quantitative the same with debt financing where 1 + r = W .
For the rest of the analysis I adopt the debt-financing interpretation. Given the form of optimal
contract, equations (1), (4) ,and (6) are replaced by (9), (10), and (11).
c̃ ≤ R − (1 + r) ≡ ĉ
(8)
(1 + r) E[s|Ω] = 1
(9)
E[P rob(c ≤ R − (1 + r))|Ω] = (1 + r)−1
(10)
For notational convenience I denote E[P rob(c ≤ R − (1 + r))|Ω] with s̃. Thus, s̃ captures investors
beliefs about the probability of effort exertion given the information set Ω.
The previous analysis highlights the importance of investors’ beliefs about the default probability on the determination of the equilibrium interest rate. The introduction of a CRA affects
investors’ beliefs as follows: if the entrepreneur receives a bad rating then investor believe that he is
an HEC type, which implies that investors downgrade their beliefs about the implementation cost
compared to the case where there is no CRA. Similarly, if the entrepreneur receives a good rating,
then investors believe that he is an LEC type which implies that investors upgrade their beliefs
compared to the case with no CRA. Since the ĉ is related to survival probability, investors require
a higher (lower) interest rate to finance such an HEC (LEC) entrepreneur compared to the status
14
quo. In addition, after the initial expectations about the implementation cost have been formed,
a feedback mechanism arises as follows: a lower (higher) ĉ coincides with higher (lower) expected
default probability, which implies a higher (lower) required interest rate by the investors, which, in
turn, leads to an even higher (lower) default probability, triggering another feedback loop. Every
loop leads to a smaller change in the interest rate, until it converges to the equilibrium interest
rate. The main reason for convergence is that for an HEC (LEC) entrepreneur the conditional on
survival probability c decreases (increases) after each loop.
Equilibrium Concept
The equilibrium concept is Perfect Bayesian Equilibrium, under which: (i) the CRA chooses the
rating fee (H) that maximizes its expected revenue (forced to be zero in the basic model), (ii) the
entrepreneur chooses whether to ask for a rating, the contract (W̄ ), and whether to implement
the project to maximize his expected profits, and (iii) investors make financing decisions which
maximize their utility. Each player’s beliefs about the others player’s strategies in equilibrium are
correct.
4
Regime with and without CRA
In this section I characterize the equilibrium in two different regimes: (i) an Imperfect Information
without CRA regime and (ii) an Imperfect Information with CRA regime. In the Appendix, I
present the regime of Perfect Information. Note that the following analysis holds for any probability
density function of the cost as long as the FOSD assumption is satisfied. In the Appendix, I present
the case there the cost of effort is uniformly distributed.
4.1
Imperfect & Asymmetric Information without CRA
As opposed to the entrepreneur who observes the actual cost of effort, investors only know the
entrepreneur is of LEC type with probability λ and of HEC type with probability (1 − λ). Because
investors cannot distinguish between the different types of an entrepreneur, there is only one interest rate, rII .
15
The Investors’ Problem: As investors have no additional information, they use their prior beliefs
to form their beliefs about the probability of default:
s̃II = P rob(c̃ ≤ ĉ|LEC)P rob(LEC) + P rob(c̃ ≤ ĉ|HEC)P rob(HEC) =⇒
s̃II = λs̃L + (1 − λ)s̃H
s̃L ≡ P rob(c̃ ≤ ĉ|LEC) = FL (ĉ)
and
(11)
s̃H ≡ P rob(c̃ ≤ ĉ|HEC) = FH (ĉ)
where FL (ĉ) is the c.d.f of the cost of an LEC entrepreneur and FH (ĉ) is the c.d.f of the cost of an
HEC entrepreneur. Thus,
s̃II = λFL (ĉ) + (1 − λ)FH (ĉ) = ΨII (ĉ)
(12)
Equilibrium Interest rate: The equilibrium existence condition is given by the Equilibrium
Existence Theorem if I replace Ψ(ĉ) with Φ(ĉ|Ω) . The equilibrium interest rate is given by:
∗
(1 + rII
) = s̃−1
II
4.2
(13)
Imperfect & Asymmetric Information with CRA
In this section I introduce a CRA that monitors the true distribution of the entrepreneur’s cost of
effort and, by assumption, reveals this distribution truthfully and costlessly.
The analysis in this regime accounts for CRA’s problem in the determination of the equilibrium
interest rate. The CRA affects the equilibrium interest rate by shaping investors’ beliefs by affecting their information set. As a result, the equilibrium interest rates are conditional on the rating.
The Entrepreneur’s Problem: In this regime, the entrepreneur’s problem is augmented by the
choice of whether to ask for a rating. The decision to ask for a rating depends on what the CRA
can monitor (-i.e., the type of the entrepreneur (LEC or HEC)), and not on the actual cost. An
LEC entrepreneur asks for a rating since he expect to receive a good rating and differentiate himself
from an HEC type which will in turn enable him to pay a lower interest rate. An HEC entrepreneur
is indifferent between asking and not asking for a rating. The intuition is that if he asks for a rating
his type will be disclosed. Similarly, if he does not ask for a rating, investors will infer his type
16
because not asking for a rating is not consistent with an LEC type. Without loss of generality I
assume that an HEC entrepreneur asks for a rating and receives a bad rating.
Recall that the threshold ĉ depends on the interest rate which depends investors beliefs which
depend on the rating. Thus, there are two critical values for the effort cost, ĉGR if the rating is
good and ĉBR if the rating is bad.
The Investors Problem: Investors form their beliefs conditional on the rating, with s̃GR denoting
the beliefs following a good rating and s̃BR denoting the beliefs following a bad rating.
s̃ ≡



s̃GR = P rob(c̃ ≤ ĉGR |LEC)P rob(LEC|GR) + P rob(c̃ ≤ ĉGR |HEC)P rob(HEC|GR)







 s̃BR = P rob(c̃ ≤ ĉBR |LEC)P rob(LEC|BR) + P rob(c̃ ≤ ĉBR |HEC)P rob(HEC|BR)
Note that P rob(LEC|GR) = 1 and P rob(HEC|BR) = 1, as the CRA truthfully and accurately
reports its private information. Hence, I can re-write investor beliefs as follows:
s̃ ≡






s̃GR = P rob(c̃ ≤ ĉL |LEC) = FL (ĉGR )
if Good Rating
(14)




 s̃BR = P rob(c̃ ≤ ĉH |HEC) = FH (ĉBR )
if Bad Rating
Investors would be willing to finance an entrepreneur with a rating as long as s̃GR ≥ (1 + rGR )−1
and an entrepreneur a bad rating as long as s̃BR ≥ (1 + rBR )−1 .
Equilibrium Interest rates: The solution to the problem of the CRA, entrepreneur and investor
∗
∗
combined determines the equilibrium interest rates rGR
and rBR
for an entrepreneur with good or
bad (or no) rating respectively.
(1 + r∗ ) =


∗ ) = (F (ĉ
−1

(1 + rGR

L GR ))


if Good Rating




 (1 + r∗ ) = (FH (ĉBR ))−1
BR
if Bad Rating
17
(15)
5
Impact of a CRA
In the previous sections I derived the equilibrium conditions under three different regimes. In
addition I provided the conditions under which the market does not collapse. This section compares
those regimes in terms of three vital market variables: the expected interest rate, the expected
default probability, and the probability of project financing.
5.1
Important Variables
Once the equilibrium interest rate is derived, I can easily find the expected/market interest rate
and the expected/market default probability, rM and dM respectively. The expected interest rate
(default probability) consists of the interest rate (default probability) for each type of entrepreneur,
i, weighted by the probability the entrepreneur is of type i, where i ∈ {H, L} with H standing for
HEC and L standing for LEC. As ratings are perfectly accurate, a good rating is associated with
an LEC entrepreneur and a bad rating is associated with an HEC entrepreneur. Thus,
∗
∗
∗
(1 + rM
) = λ(1 + rH
) + (1 − λ)(1 + rL
)
(16)
d∗M = λd∗L + (1 − λ)d∗H = λ(1 − s∗L ) + (1 − λ)(1 − s∗H ) = 1 − s∗M
(17)
where d∗L = P rob(c ≥ ĉ∗GR |LEC) = 1 − FL (ĉ∗GR ) and d∗H = P rob(c ≥ ĉ∗BR |HEC) = 1 − FH (ĉ∗BR )
5.2
Comparison
I present the effect of introducing a CRA on these variables for both an LEC and an HEC entrepreneur; however, my main goal is to provide a welfare analysis by focusing on the effect at the
market level. Recall that a social planner whose objective is to maximize net surplus will finance
both types of entrepreneurs, since they are of positive NPV. In what follows, I define as “market
survival for an LEC (HEC) entrepreneur” when FL (r)(FH (r)) satisfies the sufficient condition of the
Equilibrium Existence Theorem, i.e., financing takes place if the type is known to be LEC (HEC).
Similarly, I define as “market survival independently of whether a CRA exists” when ΨII (r) satisfies that condition, i.e. financing takes place for investors’ prior beliefs about the entrepreneur’s
type.
18
5.2.1
In terms of equilibrium interest rates
Proposition 1 (Equilibrium Interest Rates)
If information asymmetry is mild such as the market survives independently of whether a CRA
exists, then the relation among the equilibrium interest rates is given by:
∗
∗
∗
(1 + rGR
) < (1 + rII
) < (1 + rBR
)
(18)
∗ , is always lower
In addition, the expected equilibrium interest rate in the absence of a CRA, rM
∗
than the expected equilibrium interest rate when a CRA exists, r̄CRA
:
∗
∗
∗
∗
∗
)
(1 + rM
) ≡ (1 + rII
) < λ(1 + rGR
) + (1 − λ)(1 + rBR
) ≡ (1 + r̄CRA
(19)
∗ − r ∗ > r ∗ − r ∗ ; the introduction of a CRA
A direct implication of Proposition 1 is that rBR
GR
II
II
results in an increase in the interest rate for an HEC entrepreneur that exceeds the reduction in
the interest rate that an LEC entrepreneur receives.
FL (r)
FH (r)
0.5FL (r) + 0.5FH (r)
(1 + r)−1
1
G
0.8
N
A
0.6
B
0.4
r
0.2
0.4
0.6
0.8
1
1.2
1.4
Graph 1: Interest rate and default probability with and without CRA
Here I present a graphical illustration of the proof and the underlined intuition. A more detailed
proof is provided in the Appendix. Graph 1, where the interest rate is on horizontal axis, illustrates
the equilibrium interest rates for the case where the entrepreneur is of an LEC or of an HEC type
with equal probability, and the cost of effort is uniformly distributed. The two implications of First
19
Order Stochastic Dominance assumption are that: i) the graphical illustration of FL (r) lies above
the graphical illustration of FH (r), and ii), the difference FL (r) − FH (r) is increasing in r. Points
G and B indicate the equilibrium interest rate of an LEC and an HEC entrepreneur when a CRA
exists. Point N indicates the equilibrium interest rate when there is no CRA.8
Part one follows directly from the FOSD assumption. Part two relates to the convexity of
(1 + r)−1 . Note that (1 + r)−1 is a downward convex function of r, so the point denoted by N
always lies to the left of the point A that determines the expected interest rate when a CRA exists.
Hence, the interest rate when there is no CRA always exceed the expected interest rate when a
CRA exists.
Introducing a CRA has a negative effect on an HEC and a positive effect on an LEC entrepreneur. I show that the former always offsets the latter in the case where asymmetric information is not severe enough to lead to market collapse. These findings arise from the feedback effect
in capital markets and its asymmetric impact on firms of different quality. In order to understand
this asymmetry, it is crucial to first understand how the feedback effect works in practice.
After investors observe the rating and form their beliefs about the expected effort cost and
the probability of default, they demand an interest rate which allows them to break even. The
raised funds adjusted by the interest rate (capital cost) need to be paid back by the entrepreneur.
For an entrepreneur who was indifferent between exerting effort or not, adding the capital cost
makes exerting effort unprofitable, and thus the implementation threshold ,ĉ, drops. The lower
implementation threshold coincides with a higher probability of default, which results in investors
demanding a higher interest rate. This cycle continues until the interest rate converges to its
equilibrium value.
Every loop leads to continuous updates in beliefs about the probability of default, and the
expected interest rates, but the effects decrease with each loop. This is a consequence of the always
decreasing expected cost conditional on exerting effort, which capturing the fact that the least
efficient entrepreneurs have already defaulted.
The feedback effect implies that the change in the interest rate is smaller for more-efficient
entrepreneurs, i.e., the equilibrium interest rate is a convex function of the unconditional expected
cost (the expected cost in the case the entrepreneur finances his project with his own funds). A
8
∗
∗
∗
∗
∗
∗
∗
In G: (1 + rGR
)−1 = FL (rGR
), in B: (1 + rBR
)−1 = FH (rBR
) and in N : (1 + rGR
)−1 = 0.5FL (rGR
) + 0.5FH (rBR
)
20
direct consequence of this is that the negative effect on an HEC entrepreneur will dominate the
positive effect on an LEC entrepreneur because the former has a lower unconditional expected cost.
5.2.2
In terms of Default Probability
Proposition 2 (Probability of Default)
If information asymmetry is mild such as the market survives independently of whether a CRA
exists, the relation among the equilibrium probabilities of default is:
∗
∗
∗
∗
dL (rGR
) < dL (rII
) < dH (rII
) < dH (rBR
)
(20)
In addition, the expected probability of default in absence of a CRA, d∗M , is lower than the expected
probability of default when a CRA exists, d¯∗CRA :
∗
∗
∗
∗
d∗M ≡ λdL (rII
) + (1 − λ)dH (rII
) < λdL (rGR
) + (1 − λ)dH (rBR
) ≡ d¯∗CRA
(21)
The first part of the proposition is a consequence of the FOSD assumption. The second part is
directly connected to part two of Proposition 1: In Graph 1, point N is always above point A. See
Appendix for a more detailed proof.
Another fundamental aspect of capital markets is the financing opportunities. The following
Proposition explores the impact of introducing a CRA on financing opportunities given the creditworthiness of the entrepreneur.
Proposition 3 (Probability of Financing)
In a regime with a CRA, the probability of financing an LEC entrepreneur is weakly higher than
the probability of financing an HEC entrepreneur. The probability of financing of an LEC (HEC)
entrepreneur in a regime without CRA is weakly lower (higher) than in a regime with CRA. There
is a non-empty set of values of the parameters for which the introduction of CRA alleviates underfinancing, and a non-empty set of values for which it leads to under-financing of entrepreneur with
positive rate of return.
21
FL (r)
λFL (r) + (1 − λ)FH (r)
FH (r)
(1 + r)−1
1
FL (r)
λFL (r) + (1 − λ)FH (r)
FH (r)
(1 + r)−1
1
0.5
0.5
r
1
2
3
4
r
5
Graph 2: CRA alleviates under-financing
1
2
3
4
5
Graph 3: CRA leads to under-financing
The intuition underlying Proposition 3 can be seen graphically. For any value of r, r̄, FL (r̄)
lies above λFL (r̄) + (1 − λ)FH (r̄) which lies above FH (r̄). Hence, for any productivity level, it is
more likely that an LEC entrepreneur will be financed if he can distinguish himself from an HEC
entrepreneur, and similarly, it is more likely that an HEC entrepreneur is financed if he is pooled
with an LEC entrepreneur. The last part of the proposition is depicted in Graph 2 and 3. Recall
that a benevolent planner would finance both types, given that they are of positive NPV. The
left panel illustrates the case where asymmetric information is severe and an LEC entrepreneur
is able to raise capital only if he can differentiate himself. Hence, introducing a CRA alleviates
under-financing.
The right panel illustrates the case where an HEC entrepreneur is able to raise capital only
if he is pooled with an LEC entrepreneur. Thus, introducing a CRA restricts this pooling from
happening, and leads to under-financing of an HEC entrepreneur even though it is socially optimal
to finance a positive NPV type.
The intuition behind the last observation is related to the fact that an entrepreneur cannot
commit to exerting effort. The distortion stemming from the lack of commitment is greater the
lower ex-ante efficiency of the entrepreneur. Hence, pooling an HEC with an LEC entrepreneur
alleviates the resulting inefficiency. This is evident in the case where an HEC entrepreneur can
not be funded in isolation, though he can be funded by being pooled with an LEC entrepreneur.
The reason is that pooling leads investors to attribute lower probability of default, which, in turn,
reduces the interest rate they require. If the lower interest rate alters the choice of exerting effort
22
from unprofitable to profitable, then, investors anticipate this, and they are willing to finance the
entrepreneur.
5.2.3
Optimal Level of Rating Precision
A direct consequence of Propositions 1-3 is that better monitoring does not necessarily corresponds
to welfare improvement. For example, consider an environment where the CRA receives a signal
about the entrepreneur’s type, and the social planner is able to affect the precision of this signal
through, for instance, regulating the evidence that an entrepreneur should provide to the CRA. A
natural question is what would be the precision level that would maximize financing opportunities.
To explore this I allow for the following modifications: I assume that the CRA receives a signal
which can be either good or bad, σ = {GS, BS} which reveals the true state with probability α,
i.e.,
P rob(LEC|GS) = P rob(HEC|BS) = α
where α ∈ (0.5, 1] is known to all parties involved.9 To avoid any unnecessary complications I
assume without loss of generality that the CRA reveals his private signal truthfully, i.e., the CRA
gives a good rating if the signal is good, and a bad rating if the signal is bad. The level of optimal
precision is given in Proposition 4.
Proposition 4: (Optimal Level of Precision)
The level of precision α that maximizes an entrepreneur’s financing opportunities does not necessarily coincide with 1 (perfectly precise signal). If an HEC entrepreneur can not be financed if his
type is perfectly revealed, then, the financing opportunities are maximized for an α which is weakly
smaller than one. The optimal level of precision, α∗ , solves:
FL (r̃BR )(1 − α∗ ) + FH (r̃BR )α∗ = (1 + r̃BR )−1
Proof. See Appendix.
9
For α = 1 the signal is perfectly informative.
23
(22)
The following Corollary summarizes the impact of introducing a CRA and Figure 1a and 1b illustrate the impact of introducing a CRA when the cost is uniformly distributed.
Corollary 1 (Impact of better Information)
The introduction of a CRA:
(i) leads to a higher expected default probability and interest rate when both an LEC and an HEC
entrepreneur can be finance independently of whether a CRA exists. (Area A)
(ii) leads to under-financing, when an HEC entrepreneur’a financing can be sustained only if he
is pooled with an LEC entrepreneur. (Area B)
(iii) alleviates under-financing, when the LEC entrepreneur’s financing can be sustained only if an
LEC entrepreneur can be differentiated by an HEC entrepreneur. (Area C)
Note that as the probability of the entrepreneur being LEC, λ, increases, area B spreads over
area C. The intuition is straightforward, as the information asymmetry becomes less severe, the
adverse effect of non-commitment which is amplified by the introduction of a CRA dominates.
I have developed a framework characterized by asymmetric information and a lack of commitment to exerting effort. In this environment, resolving part of the information asymmetry amplifies
the impact of non-commitment. Thus, there is an inherent trade-off of introducing a CRA; on
one hand, the CRA’s introduction alleviates the information-asymmetry problem, but on the other
hand, it exacerbates the adverse effect of non-commitment. It can been shown that if the entrepreneur can commit to exerting effort, then better information is always welfare enhancing- an
outcome, that may not be true if the entrepreneur is unable to commit. The net welfare effect of a
CRA depends on the relative extent of each problem. For instance, if the information asymmetry
is severe, -such as in case of a market collapse in the absence of a CRA, then introducing a CRA is
welfare enhancing. Antithetically, if the information asymmetry is mild, then introducing a CRA
amplifies the adverse impact of non-commitment of the less efficient entrepreneurs- a situation
results in a welfare drop.
24
Figure 1a: Equilibrium existence regions for:
cL ∼ U [0, c̄], cL ∼ U [γc̄, c̄], R = 5 & λ = 0.5
Figure 1b: Equilibrium existence regions for:
cL ∼ U [0, c̄], cL ∼ U [γc̄, c̄], R = 5 & λ = 0.8
25
6
Optimal Disclosure Policy
In the previous analysis I assumed that the ratings should be accompanied by hard evidence.
This assumption restricts the rating policy to truthful disclosure. Even though this assumption is
convenient when it comes to the determination of the impact of better information, it diminishes the
role of a CRA. In this section I relax the hard-evidence assumption and I explore the optimal rating
policy when the CRA can ex-ante commit to it. The rationale behind relaxing the assumption of
hard evidence is that although it is true that CRAs provide supporting evidence with the ratings,
there could be cases where the evidence does not perfectly reveal the true type.
The CRA designs its rating policy with the objective of maximizing its expected profits,
E[Π|ΩCRA ] = E[(p − d)D|ΩCRA ]
where p, c and D stand for the rating fee, the cost of acquiring a signal, and the demand for ratings
respectively.
In order to disentangle the rating policy from the decision of acquiring a private signal and the
rating fee determination, I assume that: i) the CRA can receive a signal about the entrepreneur’s
type at cost d = 0, and that ii) the CRA is price(fee)-taker and the fee, p, is not rating-contingent.10
Price-taker CRA implies that the objective of maximizing expected revenues coincides with the
objective of maximizing the expected demand,
E[D|ΩCRA ] = E[λILEC + (1 − λ)IHEC |ΩCRA ]
where ILEC (IHEC ) equals 1 if an LEC (HEC) type ask for a rating and zero otherwise. Note that
the entrepreneur will only ask for a rating if he expects that having a rating will enable him to
raise capital at a lower cost. Thus, the objective of maximizing demand pins down to designing a
disclosure rule such that the entrepreneur will ask for a rating independently of his type. For this
set of assumptions, the objective of the CRA is effectively the same with the planner’s objective,
i.e. financing of both types.
This setup is similar to the persuasion model developed in Kamenica and Gentzkow (2009)
10
The fee is substantially small, such that the entrepreneur may buy a rating as long as he is able to raise capital.
26
where a sender (CRA) observes a private signal (type of entrepreneur), and then sends a message
(rating) to a receiver (investor) ,who, in turn, takes an action (decides whether to finance and the
interest rate in case of financing), which affects the pay-off of both the sender and receiver.
The persuasion game is relevant when the message is pivotal, i.e., it can affect the action/financing
choice. This is the case in an environment where if the type of the entrepreneur is unknown only
an LEC entrepreneur would be financed, i.e. for values of r such that FL (r) > (1 + r)−1 and
FH (r) ≤ (1 + r)−1 .
To avoid unnecessary complications I assume that the signal can be of two values, good and bad
(σ̃ = {GS, BS}) and that it perfectly reveals the true type of the entrepreneur, i.e. P r(LEC|GS) =
1 and P r(HEC|BS) = 1. The disclosure policy is a mapping from a signal realization to a rating,
where the rating can similarly take two values, good and bad (R̃ = {GR, BR}). Thus, the disclosure policy is denoted by a quadruple DP : {P r(GR|GS), P r(BR|GS), P r(GR|BS), P r(BR|BS)}.
Since P r(GR|GS) is compliment to P r(BR|GS) and P r(GR|BS) is compliment to P r(BR|BS)
the disclosure policy is simplified to DP : {αG , αB } where αG = P rob(GR|GS) and αB =
P rob(BR|BS).11
Recall that as long as both ratings are issued in equilibrium, the market survives if there is at
least a value of r such that the following conditions hold:
P r(LEC|GR)FL (r) + P r(HEC|GR)FH (r) ≥ (1 + r)−1
P r(LEC|BR)FL (r) + P r(HEC|BR)FH (r) ≥ (1 + r)−1
where the first (second) condition refers to the case where a good (bad) rating is issued. By
implementing the Bayes rule I can re-write the previous conditions as follows:
(1 − αB )(1 − λ)
αG λ
FL (r) +
FH (r) ≥ (1 + r)−1
αG λ + (1 − αB )(1 − λ)
αG λ + (1 − αB )(1 − λ)
(1 − αG )λ
αB (1 − λ)
FL (r) +
FH (r) ≥ (1 + r)−1
(1 − αG )λ + αB (1 − λ)
(1 − αG )λ + αB (1 − λ)
From the first order stochastic dominance assumption it holds that FL (r) > FH (r) for any r, which
11
Truthful disclosure is implemented when αG = αB = 1.
27
implies that the market survives and financing takes place as long as the probability the investors
allocate to the event that the entrepreneur is of LEC type is sufficiently large.
Let me define as λ̂ the minimum probability that investors should allocate to the entrepreneur
being of LEC type such as financing takes place. λ̂ solves:
λ̂FL (r) + (1 − λ̂)FH (r) = (1 + r)−1
Thus, a CRA can achieve project financing for both types (ILEC = IHEC = 1) as long as αG and
αB are chosen in a way that investors’ beliefs about the probability the entrepreneur is of LEC
type exceed λ̂ for both good and bad rating. This translates into the CRA choosing a combination
of αG and αB such that:
M in{
(1 − αG )λ
αG λ
,
} = λ̂
αG λ + (1 − αB )(1 − λ) (1 − αG )λ + αB (1 − λ)
An emerging question is to what extent the CRA can affect investor’s beliefs and more specifically,
what is the maximum λ̂, λ̂max , that could be achieved by a CRA when it can pre-commit to a
disclosure policy. λ̂max solves the following problem:
Maximize M in{P r(LEC|GR), P r(LEC|BR)}
αG ,αB
subject to 0 ≤ αG ≤ 1 and 0 ≤ αB ≤ 1
The solution satisfy the following equation:
∗
∗
αG
= 1 − αB
(23)
and the updated beliefs are given by:
∗
∗
∗
∗
λ̂max ≡ P r(LEC|αG
, αB
, GR) = P r(LEC|αG
, αB
, BR) = λ
Note that the equilibrium that maximizes financing opportunities is a babbling equilibrium. In
a babbling equilibrium the updated beliefs equal the prior beliefs which coincide with investors’
28
beliefs in a regime without CRA.
Proposition 5:(Optimal Disclosure Policy)
The optimal disclosure policy when the CRA’s objective is to maximize its profit is:
∗ = 1 − α∗ (babbling equilibrium) if there is an r such that λF (r) + (1 − λ)F (r) ≥ (1 + r)−1
i) αG
L
H
B
∗ = α∗ = 1 (truthful disclosure) otherwise.
ii) αG
B
One implication of Proposition 5 is that compared to the regime without CRA, a CRA can not
improve financing opportunities of an HEC type, even in a setup where it is not obliged to provide
supporting evidence,and it is able to commit to a disclosure rule.
Rating Inflation and its Impact
∗ and α∗ which satisfy (23), and for α∗ high enough such as an LEC entrepreneur has
For αG
B
G
∗ > 0. A strictly positive
stronger incentive than an HEC entrepreneur to ask for a rating, αB
∗ 0 implies that the CRA inflates its rating, i.e., an HEC entrepreneur receives a rating which
αB
corresponds to an entrepreneur of higher level of creditworthiness.
Rating inflation has been documented in the literature on Credit Rating Agencies. The underlined mechanism behind the rating inflation differs from the related literature, where rating
inflation is a consequence of a rating-contingent fee. In this paper, we show that rating inflation
arises due to the CRA’s incentive to attract HEC entrepreneurs, which takes place by distorting
the ratings in order to make sure that an HEC will be able to raise capital after he asks for a rating.
Also, in terms of welfare, we show that once we allow for the feedback effect inherent in capital
markets, rating inflation might lead to financing of entrepreneur with positive NPV, that would not
otherwise be financed. A direct consequence of this finding is that restricting CRAs to provide hard
evidence with their ratings might have a negative effect on the probabilities of project financing
and default.
Corollary 2: Impact of Rating Inflation
When the CRA can pre-commit to a disclosure rule, rating inflation can be part of the equilibrium
even if the fee is not rating-contingent. In addition, rating inflation might lead to financing of
positive NPV projects that would not otherwise be financed.
29
7
Profit-Maximizing Fee
In this section I allow the CRA to choose a profit-maximizing fee. A direct consequence of the
weakly positive rating fee is that it modifies the implementation threshold, ĉ. The comparison with
the case where the rating fee is restricted to zero allows me to isolate the distortion that a rating
fee imposes.
The Entrepreneur’s Problem: The entrepreneur’s problem is more complex than before: now,
before facing the problem of the security choice or whether to exert effort , he has to choose whether
to ask for a rating. An HEC entrepreneur has no incentive to ask for a rating, because by doing
this he would reveal his type.
I first explore the incentives of an LEC entrepreneur in the case where the market does not
collapse in the absence of a good rating. Whether an LEC entrepreneur has incentive to ask for
a rating depends on which of the following two forces dominates. On one hand, asking for a
rating differentiates him from an HEC entrepreneur, and allows him to promise a lower interest
rate, r̂R instead of rII . On the other hand, asking for a rating implies that he needs to borrow a
higher amount in order to cover the rating fee, H, apart from the investment cost. Thus, an LEC
entrepreneur faces a trade-off between repaying a smaller loan (of 1, instead of 1 + H) with higher
interest rate (rII instead of r̂R ), or a larger loan with lower interest rate.12 The choice of asking
for a rating does not affect the payoff in the situation in which the entrepreneur chooses to not
exert effort. Thus, in an environment where a rating is unnecessary for financing, the entrepreneur
chooses the maximum of:
U (without rating) = max{R − (1 + rII ) − c̃, 0}
U (with rating) = max{R − (1 + r̂R )(1 + H) − c̃, 0}
(24)
(25)
where the max function reflects the fact that the entrepreneur always has the choice to not exert
effort. An LEC entrepreneur asks for a rating as long as:
(1 + r̂R )(1 + H) ≤ (1 + rII )
(26)
12
Here I implicitly assume that there is a lag between the time that the CRA gives the rating and the time CRA
is paid. Due to rational expectations of the CRA, such setup is feasible if the entrepreneur can commit to pay the
fee after the loan is taken.
30
and he exerts effort as long as:
c̃ ≤ ĉ ≡ max{R − (1 + rII ), R − (1 + r̂R )(1 + H)}
(27)
I now proceed to the case where the market collapses in the absence of a rating. It is shown
the CRA charges H̄ max , hence, an LEC entrepreneur exerts effort as long as his utility is greater
than or equal to zero, which is his outside option. Taking into consideration the CRA’s problem, I
re-write the implementation threshold of an LEC entrepreneur , ĉL , as:
ĉL ≡



∗
ĉL = R − 1 + rII

 c̄ˆL = R − (1 + r̄R )(1 + H̄ max )
If rating & rating not necessary
(28)
If rating & rating necessary
The related threshold for an HEC entrepreneur is:
ĉH ≡ R − (1 + rN R )
(29)
The CRA’s Problem: The CRA understands that only an LEC entrepreneur has incentive to
ask for a rating; thus, it chooses the rating fee H, such that an LEC entrepreneur is indifferent
between having a rating or not. The functional form of H max is related to the entrepreneur’s
outside option, which is determined by whether financing is feasible without a rating. If having a
rating is not necessary for financing, the CRA will charge a fee that makes an LEC entrepreneur
indifferent between asking for a rating and differentiating himself, or not asking for a rating and
pooling with an HEC entrepreneur. In contrast, when the absence of a good rating leads to no
financing, the entrepreneur’s outside option is zero. Hence the profit-maximizing CRA extracts all
the surplus, i.e., it charges the maximum fee for which an equilibrium exists. Hence, the following
conditions characterize the profit maximizing rating fee:13
H max ≡






H max =
(rII −r̂R )
(1+r̂R )
If rating is not necessary for financing
(30)




 H̄ max : F (ĉ|H = H̄ max ) = 0.5 If rating is necessary for financing
13
H̄ max = arg max H s.t.(1 + r̂R )−1 = F (ĉ|H = H̄ max )
31
The Investors’ Problem: Investors know whether project financing is feasible without the rating
and they update their beliefs after they observe whether the entrepreneur has a rating. Because the
CRA reports its private information truthfully and charges a fee that an LEC entrepreneur buys a
rating in equilibrium: P rob(LEC|GR) = 1 and P rob(HEC|N R) = 1. Hence, investors’ beliefs are
as follows:
s̃ =






s̃R = FL (ĉL )
If GR & rating not necessary for financing
s̃R̄ = FL (c̄ˆL )




 s̃N R = FH (ĉH )
If GR & rating necessary for financing
(31)
If No Rating
Investors would be willing to finance an entrepreneur with a rating as long as s̃R ≥ (1 + rR )−1 , if
rating is necessary, and as long as s̃R̄ ≥ (1 + rR̄ )−1 , if rating is not necessary. Similarly, investors
would be willing to finance an entrepreneur with no rating as long as s̃N R ≥ (1 + rN R )−1 .
7.1
Equilibrium Interest rates
The combined problems of the CRA, entrepreneur and the investors determine the equilibrium
∗ and r ∗
interest rates rR
N R for an entrepreneur with or without rating respectively.
(1 +
∗
rR
)
≡



(1 + r∗R ) = FL (ĉL )−1

 (1 + r̄∗ ) = FL (c̄ˆL )−1 = 2
R
If GR & rating not necessary
(32)
If GR & rating necessary
∗
−1
(1 + rN
R ) = FH (ĉH )
(33)
∗ of an LEC entrepreneur is affected by the probability of the
Observe that the interest rate rR
entrepreneur being LEC: a higher λ, or a lower the cost of an HEC entrepreneur, reduces the
∗ , improves the outside option of an LEC entrepreneur, and results in a lower rating
interest rate rII
fee and interest rate. Moreover, the higher the fee, the higher the probability of default. As an
HEC entrepreneur does not ask for a rating, the probability of default and his expected surplus are
unaffected.
Proposition 6: (Effect of Profit Maximizer CRA)
Allowing the CRA to be profit-maximizing fee leads to a (weakly) positive fee that is negatively
related to the probability λ and the cost of effort. This fee: (i) (weakly) increases the interest rate
32
and the probability of default of an LEC entrepreneur, (ii) (weakly) decreases expected surplus of
an LEC entrepreneur and the market expected surplus, and (iii) does not affect the probability of
project financing.
Proof. See Appendix.
8
Concluding Remarks and Future Research
In this paper I argue that the reason for potential problems emerging from Credit Rating Agencies
is more pathological than the literature recognizes; even in the absence of conflicts of interest or
other distortions resulting from players’ behavior, the market structure itself can be a source of
inefficiency.
I argue that if the economy is characterized by entrepreneurs with low expected efficiency
(LEC), introducing a CRA enables entrepreneurs with relatively high expected efficiency (LEC)
to distinguish themselves and raise the necessary funds for undertaking the investment. On the
antipode, if the economy is dominated by entrepreneurs with high expected efficiency, introducing
a CRA could raise the expected interest rate and the expected probability of default, and lead
to under-financing of HEC entrepreneurs who have positive expected productivity. Moreover, I
evaluate the suggested regulation policy of restricting CRAs to provide hard evidence with their
ratings, and I argue this policy might have an adverse effect on the probabilities of project financing
and default.
My findings have implications on the optimal information disclosure policy imposed by a government or a Central Bank. For instance, a key question after the recent crisis is whether the
results of stress tests for banks should be revealed. My analysis suggests that no disclosure of the
stress tests results could be welfare enhancing, unless the market breaks down in the absence of
additional information. This is a result of the amplification in the probability of bad banks; default,
which dominates the beneficial effect on good banks. Goldstein and Leitner (2013)
14
arrive at a
similar conclusion by adopting a different setting.
Corollary 1 could also relate to the recent debate on the borrowing interests rates that countries
14
Disclosure of some information may be necessary to prevent a market breakdown, but disclosing too much destroys
risk-sharing opportunities
33
in the eurozone face. Interest rates differ across countries due to differences in credit risk; this has
resulted in some peripheral countries borrowing with high spreads, which kept rising over time and
eventually led to some countries being close to default. This paper suggests that a policy that
requires countries with lower credit risk to guarantee for countries with high credit risks could be
welfare enhancing.
My analysis also suggests that future research examining whether conflicts of interest could
affect rating quality should account for the feedback effect that characterizes capital markets. Note
that the feedback effect in capital markets implies a self-fulfilling effect of ratings: two projects
with the same efficiencies, but different ratings, have different default probabilities. The literature
has overlooked the self-fulfilling effect of ratings- a concept that could be applied in future work in
the context of testing arguments on reputation concerns; CRAs claim that in practice there is no
conflict of interests because by issuing inaccurate ratings a CRA would harm its reputation which,
in turn, would lead to a loss of market share.
However, one could imagine a simple model where the project can be of a good or bad type,
with the bad type always failing and the good type succeeding with a positive probability if it
receives a good rating, and failing otherwise. In that regime, it is not obvious that the CRA will
always give a good rating to a good project because if it does so, and the project fails, the CRA’s
reputation will drop substantially.
The mechanism explained in the previous paragraph opens the door to policy considerations,
and it raises concerns regarding CRAs’ regulation. A message from this paper is that regulation of
CRAs has yet to consider their ability to affect crucial variables, such as the probability of default.
34
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36
Appendix A:
A.1
Equilibrium Existence Theorem
In order to find the conditions for the existence of an equilibrium I use the Bolzano Theorem: for
two real numbers rα and rβ , rα < rβ , if G(r) is (i) a continuous function and (ii) G(rα ) , G(rβ ) are
of opposite signs, then there exists an r∗ ∈ [rα , rβ ] such that G(r∗ ) = 0.
To find the equilibrium conditions I set E[P rob(c ≤ ĉ)|Ω] ≡ Φ(ĉ|Ω) where ĉ = R − (1 + r) and
I create a new function, G(r) = (1 + r)−1 − Φ(ĉ|Ω), which satisfies (i) as Φ(ĉ|Ω) and (1 + r)−1 are
continuous functions. To show that (ii) is satisfied I set rα = 0 for which the function (1 + r)−1
reaches its maximum value, 1. For such value of r, G(0) is non-negative as Φ(ĉ|Ω, r = 0) ≤ 1.
Hence, for an equilibrium to exist it suffices to show that there is an r̃ such that G(r̃) < 0. Graph
4 present the case where at least one equilibrium exists, and Graph 5 the case where there is no
equilibrium.
Φ(r)
G(r)
(1 + r)−1
1
Φ(r)
G(r)
(1 + r)−1
1
0.5
0.5
r
1
2
3
4
r
5
Graph 4: Equilibrium exists
A.2
1
2
3
4
5
Graph 5: Equilibrium does not exist
Feedback effect in practice
Before I proceed in exploring the effect of introducing a CRA, I provide a numerical example which
illustrates how the feedback works. Assume that c ∼ U [0, 8] and R = 7.5. Under these assumptions
and based on the previous analysis, investors’ belief about the probability of exerting effort is given
by s =
7.5−(1+r)
.
8
The entrepreneur’s objective is to minimize the interest rate paid to investors.
37
If the entrepreneur promises an interest rate equal to zero, then the probability of exerting effort,
which coincides with the probability the project producing a positive return (R) and investor being
paid, is 0.812. These values of r and s imply an expected return of 0.812 × (1 + 0%) = 0.812. As the
expected return is below the cost of financing, the entrepreneur will have to increase the interest
rate to 23% such as investors break even (0.812 × (1 + 23%) = 1). The increase in the interest
rate triggers a decrease in the probability of exerting effort to 0.783, since for a set of values of c
the entrepreneur does not find exerting effort profitable any more. Consequently, the entrepreneur
should promise an even higher interest rate in order to satisfy investors’ participation constraint,
which will in turn result in a decrease in the probability of exerting effort. Note that each feedback
loop produces an increase of the interest rate and the probability of not implementing of always
decreasing magnitude. This is a consequence of the fact that after each loop, the conditional value
of c decreases. This process continuous until the equilibrium (s∗ = 0.77 and r∗ = 28%) is reached.
Table 1 illustrates the described reasoning for different distributions of c.
Round
s
(1 + r)
s(1 + r)
s
(1 + r)
s(1 + r)
s
(1 + r)
s(1 + r)
1st
0.812
1
0.65
1
0.65
0.812
0.541
1
0.541
2nd
0.812
1.230
1
0.65
1.538
1
0.541
1.846
1
3rd
0.783
1.230
0.963
0.582
1.538
0.895
0.471
1.846
0.869
4th
0.783
1.276
1
0.869
1.677
1
0.471
2.122
1
5th
0.777
1.276
0.990
0.578
1.677
0.969
0.448
2.122
0.950
6th
0.768
1.287
1
0.578
1.71
1
0.448
2.231
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Nth
0.7765
1.287
1
0.576
1.734
1
0.432
2.313
1
Table 1: Feedback loops and equilibrium values of r and s for different distributions of c.
Columns 2-4: c ∼ U ni[0, 8], Columns 5-7: c ∼ U ni[0, 10], Columns 8-10: c ∼ U ni[0, 12]
The process described in the previous paragraph is only a mental exercise; the equilibrium is reached
immediately, i.e. the entrepreneur promises an interest rate which is consistent with the probability
of exerting effort and investors break even.
A crucial remark illustrated in Table 1, which is going to be useful for the analysis that follows,
is that the more pessimistic the investors about the efficiency of the entrepreneur are (captured by
38
the increase in the expected cost), the higher the impact of the feedback effect in the probability
of exerting effort. One can see this by comparing the probability of exerting effort before and after
the feedback effect takes place for different efficiency levels (different distributions). For c ∼ U [0, 8]
the feedback effect leads to a decrease from 0.812 to 0.775. For c ∼ U [0, 10] the feedback effect
resulted in a decrease from 0.65 to 0.576 and for c ∼ U [0, 12] in a decrease from 0.541 to 0.432.
A.3
Perfect Information - First Best
In order to find the first best, I analyze the regime where both the entrepreneur and the observe
the actual productivity, c̃, before the financing decision.
The entrepreneur’s problem is identical to the one explained in Section 3: the entrepreneur
implements the project as long as condition (1) is satisfied.
The Investors’ Problem: In this regime investors know with certainty whether the entrepreneur
will choose to exert effort for a given interest rate. Hence, investors beliefs are:
s̃P I =


 1
if c̃ < ĉ

 0
otherwise
(34)
Equilibrium Interest Rate: Substituting (11) into (10) yields the equilibrium interest rate rP∗ I .
The equilibrium interest rate depends on the effort cost - which is now observable - , as follows:
rP∗ I
=



if c̃ < R − (1 + rP∗ I )
0

 not defined
(35)
otherwise
Hence, there are two cases. If c̃ > R − 1, the entrepreneur does not exert effort (s̃ = 0), independently of the interest rate, and such an entrepreneur is never funded. If c̃ < R − (1 + rP∗ I ), the
entrepreneur always exerts effort, the project is funded and, due to perfectly competitive capital
markets, rP∗ I = 0.
A.4
Proof of Main Propositions
Firstly I provide the proof of Proposition 2 as it will be used in the proof of Proposition 1.
39
A.4.1
Proof of Proposition 2
Proof. Recall that when the type of the entrepreneur is known then the probability of default is
P rob(c̃ > ĉ) ≡ 1 − F (ĉ), where ĉ is the value of c below which the entrepreneur chooses to not exert
effort. In this setup is ĉ = R − (1 + r). For part one of the proposition I use the fact that F (ĉ) is
a non-decreasing function of ĉ. I only present the case for an LEC entrepreneur. Similarly I can
prove part one for an HEC entrepreneur.
∗ (1 + r ∗ ) if there is (there is not) an intermediary. FOSD
An LEC entrepreneur promises 1 + rGR
II
∗
∗ and by plugging this into relation (1) I derive the relation between
implies that 1 + rGR
< 1 + rII
the effort exertion thresholds, ĉGR > ĉII . The last part of the proof stems from the fact that F (ĉ)
is a non-decreasing function as it stands for the c.d.f. of c. Hence, F (ĉG ) > F (ĉII ) which implies a
lower probability of default in the case where a CRA exists.
FL (r)
FH (r)
0.5FL (r) + 0.5FH (r)
(1 + r)−1
1
G
Ḡ
0.8
0.6
B̄
0.4
B0
B
r
0.2
0.4
0.6
0.8
1
1.2
1.4
Graph 6: r and s̃ with and without CRA
In order to show part b of the proposition I use the assumption of FOSD. The distance of point G
(B) from the x-axis denotes the survival probability of an LEC (HEC) entrepreneur when a CRA
exists. Similarly, the distance of point Ḡ (B̄) from the x-axis denotes the survival probability of
an LEC (HEC) entrepreneur when there is no CRA. From FOSD, the difference of the vertical
distance of G and Ḡ from the x-axis is smaller than the the difference of the distance of B and B̄
from the x-axis. Hence, any linear combination of Ḡ and B̄, λḠ + (1 − λ)B̄, will corresponds to a
point which is above the point which corresponds to the linear combination of Ḡ and B̄ (for same
40
λ), λG + (1 − λ)B 0 . This is translated into:
∗
∗
∗
∗
λFL (rII
) + (1 − λ)FH (rII
) ≥ λFL (rGR
) + (1 − λ)FH (rBR
)
(36)
which implies part b of the Proposition:
∗
∗
∗
∗
λ[1 − FL (rGR
)] + (1 − λ)[1 − FH (rBR
)] ≥ λ[1 − FL (rII
)] + (1 − λ)[1 − FH (rII
)]
A.4.2
(37)
Proof of Proposition 1
Proof. Recall that the entrepreneur exerts effort as long as c̃ < R−(1+rk ) and that the equilibrium
interest rate condition for each case is given by:
(1 + rk∗ )−1 =






∗ )
FL (rGR
∗ )
FH (rBR




 λFL (r∗ ) + (1 − λ)FH (r∗ )
II
II
if k = GR
if k = BR
(38)
if k = II (no CRA)
For part one I use the assumption of the FOSD assumption. The proof is rather intuitive and is
based on the fact that for a given interest rate r, the probability of default is weakly higher for
an HEC project. As F is an non-increasing function of ĉ and ĉ is a decreasing function of r, the
curve of FL (r) crosses the curve of (1 + r)−1 before the curve of FH (r) does, which implies that
∗ ) < (1 + r ∗ ). Similarly, the curve λF (r) + (1 − λ)F (r) crosses the curve of (1 + r)−1
(1 + rGR
L
H
BR
on the right of the curve of FL (r) and on the left of the curve of FH (r) does, which implies that
∗
∗ < r∗ .
rGR
< rII
BR
In order to show part two I will use Proposition 2. From part two of Proposition 2 I know:
∗
∗
∗
∗
λFL (rII
) + (1 − λ)FH (rII
) ≥ λFL (rGR
) + (1 − λ)FH (rBR
)
41
(39)
After I substitute the zero profit conditions into equation (38) I get:
∗ −1
∗
∗
∗
(1 + rII
) ≥ λ(1 + rGR
)−1 + (1 − λ)(1 + rBR
)−1 =⇒ (1 + rII
)≤
∗ )(1 + r ∗ )
(1 + rGR
BR
∗ ) + (1 − λ)(1 + r ∗ )
λ(1 + rBR
GR
(40)
If I show that:
∗
∗
λ(1 + rGR
) + (1 − λ)(1 + rBR
)≥
∗ )(1 + r ∗ )
(1 + rGR
BR
∗ ) + (1 − λ)(1 + r ∗ )
λ(1 + rBR
GR
(41)
then, given (38) it should also hold that:
∗
∗
∗
λ(1 + rGR
) + (1 − λ)(1 + rBR
) ≥ (1 + rII
)
(42)
Hence, in order to show part b of Proposition 2, it suffices to show that (40) holds. (40) implies:
∗ )(1 + r ∗ ) + (1 − λ)λ[(1 + r ∗ )2 + (1 + r ∗ )2 ] + (1 − λ)2 (1 + r ∗ )(1 + r ∗ ) ≥
λ2 (1 + rGR
BR
BR
GR
GR
BR
∗ )(1 + r ∗ ) =⇒
(1 + rGR
BR
∗ )(1 + r ∗ ) + (1 − λ)λ(1 + r ∗ )2 + (1 − λ)λ(1 + r ∗ )2 ≥ 0 =⇒
2(λ2 − λ)(1 + rGR
BR
GR
BR
∗ )2 + (1 + r ∗ )2 − 2(1 + r ∗ )(1 + r ∗ )] ≥ 0 =⇒ (1 − λ)λ[(1 + r ∗ ) − (1 + r ∗ )]2 ≥ 0
(1 − λ)λ[(1 + rGR
BR
GR
BR
GR
BR
which is true since λ ∈ (0, 1).
A.4.3
Proof of Proposition 3
Proof. Part one of Proposition 3 implies as c is a random variable then, for any given r, r̄:
FL (r̄) ≥ (1 + r̄)−1 =⇒ FH (r̄) ≥ (1 + r̄)−1
which is satisfied as FL (r̄) ≥ FH (r̄) from the FOSD assumption.
Part two of Proposition 3 for an LEC entrepreneur implies that for any given r, r̄:
∗ ) ≥ (1 + r̄)−1 =⇒ F (r ∗ ) ≥ (1 + r̄)−1
FL (rGR
L II
which is satisfied as FL is non-increasing. The proof for the HEC entrepreneur and the nonimplementation cost is identical.
42
The case where the introduction of CRA alleviates under-financing of project with positive
rate of return refers to a state of the world where an LEC entrepreneur is financed only if it can
differentiate itself from a HEC entrepreneur. This is true when there exists r# such that the
following conditions are satisfied:
λFL (r# ) + (1 − λ)FH (r# ) < (1 + r# )−1
(43)
FL (1 + r# ) ≥ (1 + r# )−1
(44)
FL (r# ) > (1 + r# )−1 > λFL (r# ) + (1 − λ)FH (x + r# )
(45)
The two conditions combined:
which is a non-empty set as as FL (r# ) ≥ FH (r# ) from the FOSD assumption.
The case where the introduction of a CRA leads to under-financing of project with positive rate
of return refers to a state of the world where an HEC entrepreneur is financed only if they are
mixed with an LEC entrepreneur. This case emerges when there exists r## such that the following
conditions are satisfied:
λFL (r## ) + (1 − λ)FH (r## )) ≥ (1 + r## )−1
FL (r## ) < (1 + r## )−1
(46)
(47)
The two conditions combined:
λFL (r## ) + (1 − λ)FH (r## )) > (1 + r## )−1 > FL (r## )
which is a non-empty set as FL (r## ) ≥ FH (r## ) from the FOSD assumption.
43
(48)
A.4.4
Proof of Proposition 4
Following the assumption that the CRA costlessly reveals its private signal and given that the
CRA’s precision level is represented by α the investors beliefs are formed as follows:
s̃ ≡


 s̃GS = FL (ĉ)α + FH (ĉ)(1 − α)
if Good Signal

 s̃BS = FL (ĉ)(1 − α) + FH (ĉ)α
if Bad Signal
(49)
Given the level of precision α the equilibrium condition for an entrepreneur with a good or a bad
signal respectively, are:
(1 + r̃∗ ) =


 (1 + r̃∗ ) = [FL (ĉ)α + FH (ĉ)(1 − α)]−1
GS
if Good Signal

 (1 + r̃∗ ) = [FL (ĉ)(1 − α) + FH (ĉ)α]−1
BS
if Bad Signal
(50)
Note that since α > 0.5, if the market survives after a bad signal, this must be also true after
a good signal. For a given α = α̃ there is at least an interest rate r, r̃ for which the the following
two conditions can be satisfied simultaneously
FL (r̃BR )(1 − α̃) + FH (r̃BR )α̃ ≥ (1 + r̃BR )−1
(1 + r̃BR|α=1 )−1 ≥ FH (r̃BR|α=1 )
(51)
(52)
where the first relation implies that for α = α̃ an HEC entrepreneur raises funds and the second
relation implies that for α = 1 (benchmark case) an HEC entrepreneur can not raise funds. The
two relations can hold simultaneously since FH (r̃) < FL (r̃) from the FOSD assumption.
A.4.5
Proof of Proposition 5
Proof. I start by proving part three which explains the use of the word “weakly” in the Proposition.
This finding arises from the value of the optimal rating fee; the fee is strictly positive as long as
there is at least one value of r such that FL (r|H = 0) ≥ (1 + r)−1 . As a positive fee that would
shift the FL (r) curve downwards, this condition suffices that an equilibrium exists. The fee is zero
if for any r but r∗ FL (r|H = 0) < (1 + r)−1 and FL (r∗ |H = 0) ≥ (1 + r∗ )−1 hold simultaneously.
Observe that these two conditions reflect the minimum value of FL for which the market does not
44
collapse, which is unaffected by the profit maximizer assumption due to the non-positive fee. Hence
market survival is insensitive to the profit-maximizing CRA assumption. I put the word “weakly”
in a parenthesis, as this case is an almost zero-probability event.
Now it is easy to show that part one hold. For part one I simply use the fact that the ĉL is an
decreasing function of H max and FL is a non-decreasing function of ĉL . Hence, FL (r|H = H max ) ≤
FL (r|H = 0), and the equilibrium interest rate given by (1 + r∗ )−1 = FL (r∗ ), will be higher if the
CRA charges the profit maximizing fee.
For part two it is trivial to show that the HEC entrepreneur’s expected surplus is unaffected.
Regarding the expected surplus of an LEC entrepreneur, there is a decrease as a consequence the
definition of the rating fee which implies that for any given implementation threshold, ĉ, part of
the entrepreneur’s surplus will be extracted by the CRA.
45
A.5
Case where the Entrepreneur observes a signal about the realised cost
The aim of this section is to show the coincidence of a model where the entrepreneur observes
the realized cost with the model where the entrepreneur observes an informative signal about the
realized cost.
To do so, I explore the case where the entrepreneur receives a signal, σc̃ about the realized cost
c̃. For mathematical convenience and without loss of generality assume that this signal is unbiased:
σc̃ = c̃ + where ∼ N (0, σ2 ). This interpretation implies that if the realized signal is σc̃0 then bayesian
updating results in E[c̃|σc̃ = σc̃0 ] = σc̃0 .
I first deal with the implications on the Entrepreneur’s problem. Recall that exerting effort is
profitable as long as c̃ ≤ R − W . Since the entrepreneur observes σc̃ and not c̃, his critical condition
for exerting effort will depend on σc̃ (rather than c̃). Thus, the entrepreneur exerts effort as long
as E[c̃|σc̃ ] ≤ R − W which is true if σc̃ ≤ R − W . Hence the effort exertion threshold ĉ ≡ R − W
switches to σ̂c̃ ≡ R − W .
I now deal with the implications on the Investor’s problem. Recall that investors’ beliefs about
the probability of the entrepreneur exerting effort determine W . Following the analysis of the
previous paragraph, the investors’ beliefs about that probability equals E[P rob(σc̃ ≤ σ̂c̃ )|Ω] instead
of E[P rob(c ≤ ĉ)|Ω] where Ω is investor’s information set. Observe that even though ĉ = σ̂c̃ , in
general E[P rob(σc̃ ≤ σ̂c̃ )|Ω] differs from E[P rob(c ≤ ĉ)|Ω]. In order to show that the two models
ˆc̃ such that E[P rob(σc̃ ≤ σ̂
ˆc̃ )|Ω] =
are identical, it is enough to show that there exists a unique σ̂
ˆc̃ produces the same model. Hence, the
E[P rob(c ≤ ĉ)|Ω]. If this is the case, then replacing ĉ with σ̂
ˆc̃ which satisfies the aforementioned property exists.
whole exercise pins down to showing that a σ̂
The proof of that is straightforward. Since the cumulative distribution function of σc̃ and c̃ is
a continuous and strictly increasing function, then for each W there is a unique ĉ associated to
ˆ
ˆc̃
the value E[P rob(c ≤ ĉ)|Ω]. In addition, for each value of E[P rob(s ≤ ŝ)|Ω],
there is a unique σ̂
ˆc̃ for which E[P rob(σc̃ ≤ σ̂
ˆc̃ )|Ω] = E[P rob(c ≤ ĉ)Ω].
associated with it. Hence, there exists a unique σ̂
46
A.6
Uniform Distribution
In this section I assume that the cost of effort is uniformly distributed with cL ∼ U [αc̄, βc̄] and
cH ∼ U [γc̄, δc̄]. The analysis is meaningful only if the probability of default is strictly positive.
The probability of default is zero if effort is exerted for any c̃. From the entrepreneur’s problem
this is true when c̃ ≤ R − 1. Additionally, I keep the assumption that the implied expected rate of
i]
return of both types is non-negative ( R−1−E[c|c∼c
1+E[c|c∼ci ] ≥ 0). In order to satisfy these two conditions
and simplify calculations I set α = 0 and β = δ = 1. The parameter γ captures the efficiency level
of the HEC type compare to the efficiency level of the LEC type; as γ approaches 1, the efficiency
level of the HEC type approaches the efficiency level of the LEC type.
A.6.1
Imperfect Information Case without CRA
After I incorporate the entrepreneur’s problem, investors’ beliefs are given by:
s̃II ≡ λs̃L + (1 − λ)s̃H = λ
R − (1 + rII )
R − (1 + rII ) − γc̄
+ (1 − λ)
c̄
c̄(1 − γ)
(53)
∗ , is given by:
Substituting s̃II into the ZPC, the equilibrium interest rate, r̄II
∗
1 + rII
=
[(1 − λγ)(R) − γc̄(1 − λ)] −
p
[(1 − λγ)(R) − γc̄(1 − λ)]2 − 4(1 − λγ)(1 − γ)c̄
2(1 − λγ)
(54)
It can be shown that the probability of being LEC (λ) and the return of the project (R) is positively
related to the probability of market survival. In contrast, the cost of effort is negatively related
to the probability of market survival. For this set of distributions, the equilibrium condition is
quadratic in r. In the case where the equation has two distinct roots, the smallest one is associated
is with lower interest rates and lower default probability and it is the the only equilibrium since it is
the combination of interest rate and probability of default which maximizes entrepreneur’s profits.
A.6.2
Imperfect Information Case with non-profit maximizer CRA
The investor’s beliefs depend on the rating as follows:
s̃ =



s̃GR =

 s̃BR =
R−(1+rGR )
c̄
R−(1+rBR )−γc̄
c̄(1−γ)
47
if Good Rating
if Bad Rating
(55)
The equilibrium interest rates are given by:
∗
1+r =




 1 + r∗
BR
√
2
−4c̄
∗
1 + rGR
= R− R
2
√
(R−γc̄)− (R−γc̄)2 −4c̄(1−γ)
=
2
if Good Rating
(56)
if Bad Rating
The market existence condition for an HEC entrepreneur is (R − γc̄)2 ≥ 4c̄(1 − γ) and for an LEC
entrepreneur it is R2 ≥ 4c̄. When these conditions are satisfied, the introduction of CRA leads to
inferior results, as it does not improve financing opportunities and it increases the expected default
probability.
An interesting case is when an HEC entrepreneur is not efficient enough to restrict market
breakdown, but he can raise capital if he pools with an LEC entrepreneur because the emerging
2
≥ 4c̄(1 − γ) > (R − γc̄)2 ).
interest rate is smaller and easier to be sustained ( [(1−λγ)R−γc̄(1−λ)]
(1−λγ)
Without a CRA both types are financed, in contrast to the case where a CRA exists and restricts
this pooling from happening, and only an LEC entrepreneur is financed. Hence, the introduction
of a CRA leads to under-financing of an entrepreneur with positive expected net return.
Lastly, when an LEC entrepreneur can be financed only if they can be distinguished from an
2
HEC entrepreneur ( [(1−λγ)R−γc̄(1−λ)]
< 4c̄ ≤ R2 ), then, the absence of a CRA leads to market
(1−λγ)(1−γ)
breakdown and neither an LEC nor an HEC entrepreneur is financed. Thus, the introduction of a
CRA restricts the market of an LEC entrepreneur from collapsing, alleviating the problem of underfinancing. Figures 1a and 1b illustrate the effect of introducing a CRA for different combinations
of c̄, γ and λ. More specifically,
• Area A: Both an LEC and an HEC entrepreneur are financed, independently on whether a
CRA exists; CRA increase the probability of default.
• Area B: An HEC entrepreneur is financed only in the absence of a CRA. An LEC entrepreneur
is financed independently on whether a CRA exists; CRA leads to under-financing.
• Area C: An HEC entrepreneur is not financed independently on whether a CRA exists. An
LEC entrepreneur is finance only if a CRA exists; CRA alleviates under-financing.
• Area D: Neither an LEC nor an HEC entrepreneur is financed, independently on whether a
CRA exists; CRA plays no role.
48
A.6.3
Imperfect Information Case with profit maximizer CRA
Investors observe the rating and whether a rating is necessary for financing ([(1−λγ)R−γc̄(1−λ)]2 ≥
4(1 − λγ)(1 − γ)c̄), and they form their beliefs about the probability of default as follows:

R−(1+rR )(1+H max )


s̃GR =

c̄


R−(1+r̄
)(1+
H̄ max )
R
s̃ ≡
s̃GR
¯ =
c̄




NR)

s̃N R = R−(1+r
c̄
If GR & rating not necessary
If GR & rating necessary
(57)
If No Rating
The CRA’s, the entrepreneurs and the investors’ problems combined determine the equilibrium
∗ and r ∗
interest rates rR
N R for an entrepreneur with or without rating respectively.
(1 +
∗
rR
)
≡


 (1 + r∗ ) =
R


(1 +
c̄
∗ )
R−(1+rII
∗)=2
(1 + r̄R
∗
rN
R)
=
if rating not necessary
(58)
if rating necessary
(R − γc̄) −
p
(R − γc̄)2 − 4c̄(1 − γ)
2
(59)
As Proposition 6 states, the market existence conditions are unaffected by the profit maximizing
assumption.
49