Repeated Interaction and Rating In ation: A Model of Double Reputation∗ Sivan Frenkel† February 2014
Abstract public reputation for creddevelop a second, private rep-
Credit-rating agencies have an incentive to maintain a ibility among investors but also have an incentive to
utation for leniency among issuers. We show that in markets with few issuers, such as markets for structured assets, these incentives may lead rating agencies to in ate ratings as a strategic tool to form a
double reputation.
The model extends the
existing literature on cheap-talk reputation to the case of two audiences. Our results can explain why rating in ation occurred speci cally in markets for MBSs and CDOs during the recent nancial crisis. Policy implications are discussed.
JEL Classi cation: Keywords: ∗
G24, D82, L15, C73
credit-rating agencies, reputation, double reputation, two audiences.
This paper is based on part of my Ph.D. dissertation at the Eitan Berglas School of Economics, Tel
Aviv University. I am grateful to Zvika Neeman for his useful advice and huge support during my Ph.D. period. I thank Eddie Dekel, Alessandro Lizzeri, Marco Ottaviani, David Rahman and Asher Wolinsky for helpful comments and suggestions. I thank the three referees for their constructive comments, which signi cantly improved the paper. Part of the research for this paper was done during a visit to the Center for Mathematical Studies in Economics and Management Science at the Northwestern University Kellogg School of Management. Financial support by the Kellogg School of Management and the Berglas School of Economics is gratefully acknowledged.
†
Center
for
the
Study
of
Rationality,
Hebrew
University
of
Jerusalem.
[email protected]; Web page: https://sites.google.com/site/sivanfrenkel/.
1
E-mail:
1
Introduction
The nancial crisis of 2008 exposed a dramatic failure in the rating of structured assets such as mortgage-backed securities (MBSs) and collateralized debt obligations (CDOs). Prior to the crisis, a large proportion of these assets received top ratings; for example, 80-95% of a typical subprime MBS deal was assigned the highest possible AAA rating (Ashcraft, Goldsmith-Pinkham, and Vickery, 2010). When the crisis erupted, however,
1
these assets were severely downgraded, often below investment grade.
Critics have claimed that rating agencies knowingly ignored risks when rating MBSs and CDOs. These claims receive some support from recent empirical literature. Apparently, rating agencies ignored available data on risks when rating mortgage deals. In many cases out-of-the-model adjustments were made to ensure higher ratings.
2
The failure
has drawn attention to a potential con ict of interest in the rating agencies' issuer pays business model, and raised the possibility that ratings were in ated to attract more deals
3
and increase market shares.
One of the main dierences between mortgage-related securities, whose ratings plummeted, and plain corporate bonds, which did not incur such severe downgrades and defaults,
4
is the structure of the markets in which these assets are issued.
Corporate
1 For example, 90 percent of the CDOs that were rated AAA by S&P during the years 2005-07 were downgraded as of June 30, 2009, with 80 percent downgraded below investment grade. For AAA-rated MBSs, the percentages were 63 and 52 respectively (White, 2010, p. 221). Benmelech and Dlugosz (2009) oer additional data on the rating collapse of CDOs and MBSs.
2 Ashcraft, Goldsmith-Pinkham, and Vickery (2010) examine a sample of nearly 90 percent of the
MBS deals issued in the period of 2001-07, and report that during 2005-07 the fraction of highly rated MBSs in each deal remained constant, despite a signi cant increase in the average risk of new MBS deals. In addition, MBS deals backed by loans with observably risky characteristics did not get lower ratings. Gri n and Tang (2012) examine a sample of 916 CDOs and report that the formal rating model accounted for only half of the criteria for determination of credit rating. They report that 84 percent of the out-of-the-model adjustments are positive and account for an additional 12.1 percent of AAA tranches. They estimate that without such adjustments the average ratings of the AAA-rated tranches in their sample would have been rated BBB, resulting in a 20.1 percent lower valuation.
3 For example, the SEC report of an e-mail correspondence from a rating agency o cial that asserted
that We are meeting with your group this week to discuss adjusting criteria for rating CDOs of real estate assets this week because of the ongoing threat of losing deals (U.S. Securities and Exchange Commission, 2008, p. 26).
4 Obviously, during a nancial crisis there are more defaults, and therefore more credit downgrades,
than at other times.
A comparison by Standard and Poor's (2010) shows that ratings of assets other
than MBSs and CDOs were not downgraded in 2008 more than in previous stress years such as 1991 and 2001. S&P therefore claims that the ratings of theses assets served as predictors of the relative likelihood of default even during the 2008 crisis.
2
bonds are issued by many dierent commercial rms, and therefore their market is composed of thousands of issuers who approach the market infrequently. In contrast, MBSs and CDOs are issued by a relatively small number of specialized rms and big investment banks. During the years 2005-07, the peak of the securitization era, such rms repeatedly originated mortgages to securitize them and issue MBSs, an activity known as originate to distribute. In 2006, for example, the top ten subprime MBS issuers were responsible for almost 65 percent of market volume, and the top 25 were responsible for 95 percent
5
of market volume (Ashcraft and Schuermann, 2008, Table 2.3).
In this paper, we analyze how market concentration can in uence the reputational concerns of credit-rating agencies. Our results suggest that a dierence in market concentration can help explain the dierence between the observed rating in ation of MBSs
6
and CDOs and the informative rating of plain corporate bonds.
We develop a reputa-
tion model with the following results. In markets with a large number of issuers, most of whom issue only once, reputational concerns lead rating agencies to give truthful ratings in order to build a credible reputation. In such markets, truthful ratings mitigate the adverse selection problem between issuers and investors and create a surplus that
7
the CRA can extract.
In contrast, in a market with few issuers who repeatedly require
ratings of new deals, rating in ation may occur. These issuers are better informed about the truthfulness of the ratings because they can compare the published rating to the real quality of the rated deal. These issuers can therefore reward rating in ation with high fees, while investors realize that a rating is not truthful only in the case of a default. As a result, rating agencies may have an incentive to provide favorable ratings to create a
5 In a similar examination of a sample of 642 CDOs and residential MBSs deals, the SEC reports that 12 arrangers accounted for 80 percent of the deals, in both number and dollar volume (U.S. Securities and Exchange Commission, 2008, p. 32). He, Qian, and Strahan (2012) present additional similar evidence.
6 Other dierences between the market structure of corporate bonds and structured assets may have
also aected the quality of ratings. For example, in the corporate debt market unsolicited ratings are much more common compared to the markets for structured assets.
Structured assets are also more
complex and opaque. In this paper we focus on concentration as we believe it is an important aspect of the market that is overlooked by previous literature.
7 Rating agencies often claim that this reputation concern is enough to ensure their credibility.
See, for example, S&P's statement in the SEC public hearing on November 15, 2002:
[T]he on-
going value of Standard & Poor's credit ratings business is wholly dependent on continued market
con dence
in
the
credibility
and
reliability
of
its
credit
ratings.
No
single
issuer
fee
or
group of fees is important enough to risk jeopardizing the agency's reputation and its future. (http://www.sec.gov/news/extra/credrate/standardpoors.htm)
3
double reputation: the issuers recognize that the CRA is lenient and in ates ratings, while investors still believe that the CRA is credible. Due to the double reputation, the CRA is rehired for a high fee. We develop our results in a simple two-period communication model with perfectly rational players. A risk-neutral issuer attempts to sell an asset to a risk-neutral investor. The asset is risky, and can vary in quality, which is de ned as the probability of default. Due to adverse selection, there is no trade without some indication that the asset is of high quality. An intermediary, or credit-rating agency (CRA), can be hired by the issuer to rate the asset. The rating agency's fee depends on the issuer's expected revenue, which depends on the price the investor is willing to pay, which in turn depends on the rating. We assume that the fee for the rating agency is paid in advance in every period and is not contingent on the rating; thus there is no rating shopping in our model. We allow the rating agency to develop reputation, and we model that using two commitment types (Kreps and Wilson, 1982; Milgrom and Roberts, 1982): one truthful and one who publishes only good ratings. Players form expectations about the informativeness of the ratings based on their beliefs about the type of the CRA and the strategy of the third, non committed type. An investor will not pay a premium for a highly rated asset if he believes the rating is not truthful. However, for given prices, an issuer with a bad asset will pay more to a rating agency if he expects to get a good rating. We analyze two market structures.
In the rst, there is a dierent issuer in every
period. The second issuer has the same information as the investor: both observe only the previous rating and return. In the second structure, the same issuer sells two assets repeatedly. This issuer is better informed compared to the investor in the second period, as he knows whether the rating in the previous period was truthful or in ated.
We
show that in the former market, a strategic rating agency always bene ts from building reputation, and thus is always truthful in equilibrium.
In the latter case, however, a
strategic CRA in ates ratings with a positive probability if its initial reputation is high and the bad asset's default probability is low. We next extend our analysis to the case where an incumbent CRA knows that it will
4
have to compete with an entrant in the next period.
We show that in such a model,
in the absence of rating shopping, possible competition plays a disciplining role in the rating industry due to hiring considerations. This is because the issuer always hires the rating agency with the highest public reputation, and therefore the CRA has an additional incentive to practice in reputation building by giving conservative ratings. Nevertheless, rating in ation is still maintained in equilibrium if the reputation of the entrant is not too high, though it is lower than in the monopolistic case. In the remaining sections we describe additional empirical predictions of the model, that larger issuers will receive more favorable ratings and that the prices of their AAA assets will re ect higher yields.
We show that these predictions are supported by the
empirical literature. We also discuss the current changes to CRA regulation in light of the model. The remainder of the paper is organized as follows. Section 2 is a survey of the related literature. Section 3 describes the basic model. Section 4 describes the equilibrium of this model with and without repeated interaction between issuers and the CRA. Section 5 extends our results to the case where the CRA faces a threat of entry. Section 6 discusses additional predictions and compares our results to the empirical literature.
Section 7
discusses the policy implications of our analysis and Section 8 concludes. Proofs appear in the main text or the Appendix.
2
Related Literature
The economic literature on credit-rating agencies has signi cantly increased as a response to the nancial crisis of 2008 and its exposure of systematic rating in ation in MBSs and CDOs.
The most common explanation of the rating in ation is the issuer-pays
business model combined with rating shopping of issuers among several rating agencies (Faure-Grimaud, Peyrache, and Quesada, 2009; Mathis, McAndrews, and Rochet, 2009; Skreta and Veldkamp, 2009; Bolton, Freixas, and Shapiro, 2012).
8
Reputational concerns
8 Sangiorgi, Sokobin, and Spatt (2009) use an argument similar to one made by Skreta and Veldkamp (2009) to explain the process of notching in the rating of structured assets that are backed by rated assets, as well as the fact that unsolicited ratings tend to be lower than solicited ones.
5
of rating agencies appear in Mathis, McAndrews, and Rochet (2009) and Bolton, Freixas, and Shapiro (2012), but both papers consider such concerns as an incentive for the rating agency to provide truthful ratings. The analysis in these papers suggests that if fees are not contingent on ratings, truthful reporting will be established because reputation then becomes the prominent concern of rating agencies. In contrast, our model assumes that fees are not contingent on ratings and shows how reputational concerns may actually encourage rating in ation. These dierences in analysis have signi cant regulation implications, as discussed in Section 7 below. In addition, neither of the two above-mentioned papers focuses on repeated interaction between issuers and CRAs and on the degree of market concentration in order to explain the dierent behavior of rating agencies in the corporate bonds versus structured assets markets. This issue lies at the heart of our paper. In addition, unlike the models presented in Skreta and Veldkamp (2009) and Bolton, Freixas, and Shapiro (2012), our model does not require the assumption that investors are naive and accept published ratings at face value. Rating in ation can be pro table to the CRA in our model even when investors are rational and realize that the CRA may not be completely truthful.
9
Our paper is also related to the literature on communication (cheap talk) in dynamic settings, starting with Sobel (1985) and extended by Benabou and Laroque (1992). Morris (2001) and Ely and Välimäki (2003) have shown that even if the sender's preferences are aligned with those of the receiver, he or she may lie in an attempt to improve his or her reputation. In all these papers, the sender have preferences that may coincide or con ict with those of the receiver. Few papers have discussed reputation in cases where a sender faces two receivers with dierent preferences, as in our case.
10
Recently Bar-Isaac and Deb
(2013) presented such a setup with costly actions, where dierent receivers may receive dierent information about the actions of the sender. Their focus is dierent from ours, however, as they consider dierent communication environments, while we assume that
9 This is also a feature of Opp, Opp, and Harris (2013). However, in their paper investors want in ated ratings to be published due to regulatory frictions, while in the present paper investors are concerned with the informative value of the rating.
10 Gertner, Gibbons, and Scharfstein (1988) and Austen-Smith and Fryer (2005) present models of static
signaling with two audiences. They show that this may lead to more pooling in equilibrium.
6
11
the rating is publicly observed by all players.
The closest paper to ours is Bouvard and Levy (2012). As in our paper, they present a model where a rating agency has two con icting reputational concerns, and show that rating in ation may occur even if fees are not contingent on ratings. However, Bouvard and Levy (2012) present a model in which rating agencies are always truthful, and rating in ation is a result of underinvestment in a costly auditing process that detects bad assets.
In their model, the rating agency's payo is non monotone in reputation, and
this is true for any market structure and without repeated interaction. In contrast, we present a cheap-talk model where an informed rating agency chooses whether to report or misreport its information.
In such a setup, if issuers have the same information as
investors then the rating agency's payo monotonically increase in truthfulness, and therefore no rating in ation takes place. Manipulating information is only pro table in our model if the rating agency has the ability to create a double reputation. Indeed, the focus in our paper is on repeated interaction between a long-lived issuer and a rating agency, a setup that plays no role in Bouvard and Levy (2012).
3
The Basic Model
The game consists of three players:
a buyer/investor (b), a seller/issuer (s), and an
intermediary/credit-rating agency (CRA). All players are risk neutral.
In each period,
the issuer has a risky asset he wishes to sell to the investor. The asset's quality is known
12
to the issuer,
unknown to the investor, and the issuer can hire a rating agency to rate
the asset.
3.1
Assets
The investor can invest in a risk-free asset with a known return normalized to zero. Alternatively, he can buy the asset oered by the issuer, with a return equal to
R.
This
11 Farrell and Gibbons (1989) and Goltsman and Pavlov (2011) analyze similar questions of communicating with more than one receiver in a static cheap-talk model with two audiences.
12 The qualitative results continue to hold even if the issuer receives only a noisy signal about this
quality. The key issue is that the issuer is more informed than the investor.
7
asset is one of two equally likely qualities, periods are uncorrelated. A good asset (a normalized to one. A bad asset (a
= B)
and a negative return with probability
a ∈ {G, B}. = G)
The qualities of assets in dierent
always gives a positive return, which is
gives a positive return
R=1
with probability
π,
1 − π.
The issuer cannot credibly communicate the asset's quality to the investor. We restrict our attention to the case where the expected return of the bad asset equals
−` < −1.
In this case, the expected return of the risky asset is
E(R|a = B) =
E(R) = 0.5(1 − `) < 0,
and therefore there is no trade without a rating agency (due to adverse selection). We assume that
`
is close to one. Formally:
Assumption 3.1. ` = 1 + for some in nitesimal > 0. Assumption 3.1 is made only for ease of exposition: the results hold in full for higher levels if other parameters are constrained. For such levels, however, we have to assume that the CRA's initial reputation is above some minimum to assure that it is hired in both periods, and this makes the analysis more complicated.
3.2
13
Credit-Rating Agency
A credit-rating agency can be hired by the issuer for an endogenous fee of fee is not public and therefore not observed by the investor. its payo is normalized to zero. We assume that
w
14
w > 0.
The
If the CRA is not hired
is paid to the CRA in the beginning
of every period, at the time of hiring, and is therefore not contingent in any way on the rating. One the other hand, the CRA does care about its future fees. In addition, we assume that the issuer cannot prevent the publication of a rating or condition the fee on the rating published, and the CRA publishes a rating every time it is hired (no shadow ratings), and hence even bad ratings become public. Since the issuer observes the quality of the asset before he decides whether to hire the CRA, the fee can be conditional on
13 Given the plausible assumption of limited liability, all the investor can lose is an alternative (positive) return. In the model we normalize the risk-free return to zero, and therefore the return of the risky asset is negative following default. High
`
is therefore equivalent to assuming a high risk-free return.
14 We motivate this assumption by the empirical fact that rating agencies do not publish the fees that
they charge clients, and that rating fees of structured assets may vary signi cantly (Partnoy, 2006). For example, Moody's has reported that its fees may range from $1,500 to $2,400,000 (Moody's Investors Service, 2006).
8
the asset's quality. We denote by of quality
a ∈ {B, G}.15
wa
the fee paid to the CRA by an issuer with an asset
If the CRA is hired it learns the asset's quality after a costless
rating process. It then publishes a rating
r=b
r ∈ {b, g},
where
r=g
is the high rating and
is the low rating.
The CRA is one of three types, where
θ ∈ {S, H, C}.
The strategic type (θ
= S)
maximizes its pro t and may in ate ratings to increase its expected fee in the future. Thus we assume that:
Assumption 3.2.
A rating agency can in ate ratings, but it cannot de ate ratings by
giving a bad rating to a good asset.
The motivation for this assumption is that a bad rating demands some justi cation in the form of hard evidence; i.e., the rating agency has to explain why the asset is risky.
If the CRA gives a bad rating without such a justi cation or with a false one,
then the issuer can always prove that this rating is fallacious. Obviously, an issuer has no incentive to do so when rating in ation occurs, and therefore no evidence is needed in this case.
16
The other two types are commitment types who have a predetermined
strategy. The Honest type (θ and
r=b
if
a = B.
= H)
always gives truthful ratings, i.e.,
r=g
if
a=G
The honest type can be thought of as an ideal rating agency where
analysts have the proper incentives to remain truthful. The second commitment type is Corrupt (θ
= C)
and always publishes a good rating. This type may represent a CRA
or an analyst who always caters to the issuer and therefore intentionally in ates ratings. A second interpretation is to think of this type as a rating agency that, unlike the other two types, has a high cost for auditing.
Such a rating agency therefore prefers not to
15 An alternative assumption, where the quality of the asset is observed by the issuer only after he hires the CRA and therefore the fee does not depend on the quality, will not change the results because the expected fee in every period, which is maximized by the CRA, remains the same.
16 In this model rating de ation is undesirable from the point of view of issuer and investor alike. This is
because prices are endogenous and so the investor wants an informative rating, not a low one. However, in a model where a rating agency can de ate ratings, a strategic type may opt to do so in order to prove that it is not an uninformative type that gives only good ratings. This is a so-called bad reputation eect explored previously by Morris (2001) and Ely and Välimäki (2003). We wanted to abstract from bad reputation in order to focus on a standard reputation eect with two audiences. We conjecture that the addition of a third conservative commitment type that always gives bad ratings would, if the prior probability of this type is high enough, prevent rating de ation in equilibrium even without Assumption 3.2, since the CRA would prefer not to be identi ed as this type, but analysis of such a model is beyond the scope of this paper.
9
perform an auditing process, and since it does not collect any hard evidence that the asset is bad, it publishes a good rating. Under this interpretation, the
C
represents cost,
and rating in ation is the result of a moral hazard problem. The reputation of the CRA is a pair
µ + ρ ≤ 1.
(µ, ρ)
where
µ ≡ Pr(θ = H), ρ ≡ Pr(θ = C)
To maintain the association between reputation and good qualities such as
µ
honesty, we would say that reputation improves or is higher when
ρ
and
is increasing and
is decreasing. We denote the strategy of the strategic CRA by
x ∈ [0, 1], where x ≡ Pr(r = g|a = B).
Notice that the two pure strategies of the strategic type, actions of the truthful and corrupt type, respectively.
x=0
and
x = 1,
match the
We assume that all types are
17
possible in the beginning of the game:
Assumption 3.3.
All types have a positive probability:
µ > 0, ρ > 0, µ + ρ < 1.
It is useful to de ne the informativeness, or truthfulness, of the rating as the expected probability that ratings are not in ated, i.e., the probability that a bad asset receives a bad rating. We denote the informativeness of a CRA with reputation expected strategy of the strategic type is
x,
(µ, ρ),
by
T (µ, ρ, x) ≡ Pr (r = b|a = B; µ, ρ, x) = µ + (1 − µ − ρ)(1 − x).
Notice the obvious fact that
x,
when the
(1)
T (·) is (weakly) increasing in µ and decreasing in ρ for every
and, given Assumption 3.3, is strictly decreasing in
x.
17 Our results generally hold also for the case where Assumption 3.3 does not hold and only one of the commitment types is present. In such cases there are some equilibria where there are no reputational concerns, and the exposition is therefore more complicated (see footnote 20).
This fact is actually a
virtue of the model, since it is known in the reputation literature that the choice of commitment types may in uence the results (it is especially relevant whether the strategic type has an incentive to separate from a bad commitment type or to imitate a good commitment type; see Bar-Isaac and Tadelis, 2008, Section 5). This is not the case in our model, because each commitment type is preferred by one audience and disliked by the other. Since the payo of the CRA depends on the beliefs of both the investor and the issuer, it does not have a clear incentive to imitate or separate.
10
3.3
Timing θ ∈ {S, H, C}
is drawn by nature, where
Pr(θ = S) = 1 − µ − ρ.
The game consists of two
At the beginning of the game, the CRA's type
Pr(θ = H) = µ, Pr(θ = C) = ρ,
and
periods. In each period of the game, the following steps occur:
1. Nature determines the asset's quality
a ∈ {B, G}.
Both qualities are equally likely.
2. The issuer observes the asset's quality and decides whether to hire the CRA (for an endogenous fee
w).
3. If hired, the CRA observes the asset's quality and publishes a rating
4. The investor buys the asset for an endogenous price
p > 0
r ∈ {b, g}.
or refuses to buy the
asset.
5. The return of the asset is materialized and is observed by all players.
6. The issuer and the investor update their beliefs regarding the CRA's type.
3.4
Preferences
The investor's payo in case he buys the issued asset is of the asset, and
R
R − p,
where
p≥0
is the price
is the materialized return. The alternative payo from the safe asset
is zero. In what follows we assume that trade occurs only if not willing to buy the risky asset then
p = 0.
can extract all the investor's surplus. Let
pr
and if the investor is
For simplicity, we assume that the issuer
be price of an asset that is rated
pr = max {0, E(R|r)} .
Any alternative assumption where
p > 0,
r;
we have
(2)
p is an increasing function of the expected return would
not change the qualitative results. The issuer's payo is
p − w.18
We assume that the CRA can extract all the issuer's
18 We assume that the issuer cannot hold an asset until maturity, and therefore its payo does not depend on the quality of the asset. This assumption gives the issuer an incentive to sell. This assumption may arise due to liquidity constraints, and is also in line with the originate to distribute business model
11
expected surplus (given his asset's quality):
wa = E(p|a).
As before, any alternative assumption where
w
is an increasing function of the expected
price does not change the results qualitatively. The CRA's payo is simply its fee is hired, and zero otherwise. The CRA agrees to work for any positive fee
3.5
w
if it
w > 0.
Repeated Interaction and Beliefs
We assume that all players observe the CRA's past record, which includes its past ratings as well as past realized returns of rated assets. These returns do not necessarily reveal the quality of those assets, as even bad assets need not default. However, when an issuer hires the CRA more than once, he also knows the quality of his previously rated assets, and thus has an informational advantage over the investor. We analyze two opposing cases.
In the rst case we assume one-time issuers, each
active in a dierent period. The second issuer has the same information as the investor. In the second case there is a single issuer who issues two assets, one after the other. He has an informational advantage over the investor, as described above.
The former
case represents a market structure with many infrequent issuers, while the latter case represents a concentrated market with only a few issuers (and in the extreme case, one
19
issuer), who issue repeatedly.
In the beginning of the game the CRA has a commonly known reputation
(µ, ρ).
In
the second period players may have dierent beliefs about the type of the CRA. We denote the belief of the investor (buyer) and the issuer (seller) in the beginning of the second in the markets for MBSs and CDOs. Alternatively, we could retain the assumption that the issuer cannot enjoy the high return of a good asset, but may suer the loss from a bad asset if it is not sold. In this case, the expected payo of an issuer with a bad asset is issuer is willing to pay more than
E(p|a = B)
−`
if the asset is not sold, and therefore the
to the CRA. In this model the qualitative results are just
slightly dierent.
19 A general way to model the level of repeated interaction in a market would be by a parameter q , which
represents the probability that the CRA will meet the same issuer in the next period. Therefore, with probability
1−q
q
the issuer in the second period is better informed than the investor, while with probability
this issuer has the same information as the investor.
A higher
q
can be interpreted as a more
concentrated market. We focus our attention on the two extreme cases, where
12
q=0
and
q = 1.
period by
(µˆb , ρˆb )
and
(µˆs , ρˆs )
respectively. When the two posteriors dier, they can be
(µˆb , ρˆb ),
interpreted as a double reputation: the posterior of the investor,
is commonly
known to all players and represents the CRA's public reputation ; the posterior of the issuer,(µ ˆs , ρˆs ), is known only to the issuer and the CRA and represents a hidden or private
reputation.
4
Equilibrium
In this section we describe the perfect Bayesian equilibria in mixed strategies of the game, with a minor restriction that is described below. We focus on the rating decision of the CRA in the rst period of the game, which takes into account the expected fees and prices in the second period. In what follows, we rst present the prices of the assets and the fee of the rating agency in the second period. Then we nd the equilibrium strategy of the CRA for the two cases described in the previous section.
4.1
Rating Agency's Behavior in the Second Period
In every period, the CRA is paid before the rating process, and therefore its only concern when deciding on its rating strategy is future fees. This implies that in the second period the strategic CRA is indierent to all rating strategies. Therefore, the model has multiple equilibria by construction. We denote
x
and
xˆ
as the probability that the strategic type
in ates ratings in the rst period and second period, respectively. In what follows, we describe the set of possible equilibria parameters and
xˆ.
(x, xˆ),
where
xˆ ∈ [0, 1]
and
x
is a function of the
Notice that we limit ourselves to equilibria where
xˆ
is not contingent
on the history of the game. This is done to simplify the exposition: more complicated
20
second period strategies result in similar results.
20 Assumption 3.3 allows us to analyze reputational concerns for all
x ˆ ∈ [0, 1].
When only one commit-
ment type is present, there is always an equilibrium where reputation plays no role because both types act the same in the second period (when
ρ=0
it is the equilibrium where
µ=0
it is the equilibrium where
x ˆ = 0).
13
x ˆ = 1,
and in the case where
4.2
Prices of Rated Assets
Prices depend on the belief of the investor about the CRA's type
(µb , ρb )
and the rating
strategy (x). We can write Equation (2) explicitly as
pr = max {Pr (a = G|r; x, µb , ρb ) − ` Pr (a = B|r; x, µb , ρb ) , 0} .
Using Bayes Law and (1) we get
pg = 1T >0 ·
1 − ` [1 − T (µb , ρb , x)] , 2−T
(3)
pb = 0,
where
1X
denotes the index function that receives one in case condition
pb = 0
and zero otherwise.
21
4.3
`
is close to 1,
pg > 0
if and only if
the investor is willing to buy a highly rated asset if he believes that the
rating is not completely uninformative. period because
is satis ed
is immediate because bad ratings are only given to bad
assets (Assumption 3.2). Given Assumption 3.1 that
T (µb , ρb , x) > 0:
X
Given Assumption 3.3
pg
is positive in the rst
µ > 0.
Payo of the CRA
Because
g|a) · pg .
pb = 0,
the CRA's payo (and expected issuer's surplus) is simply
wa = Pr(r =
Given Assumption 3.3 the probability of a good rating is always positive in the
rst period because
Lemma 4.1.
ρ > 0.
We therefore conclude that
The CRA is always hired in the rst period.
Lemma 4.1 ensures that the decision of the issuer to hire the CRA in the rst period does not signal anything on the quality of the asset to the investor (since it is assumed that the fee is not public).
21 If Assumption 3.1 does not hold then
pg > 0
only if
such a case Lemma 4.1 below does not always hold.
14
T
is above some strictly positive threshold. In
Since both qualities are equally likely, the expected fee of the CRA in the second period is
wˆ = wB + wG /2,
which, using (3), can be rewritten as
ˆb 1 − ` 1 − T ˆ 2 − Ts , wˆ = 1Tˆb >0 · · 2 2 − Tˆb where the perceived informativeness
Tˆi = T (µˆi , ρˆi , xˆ) for i ∈ {b, s} is calculated using (1).
Remark 4.2 (Con icting Reputational Concerns). Notice that using (1) it is easy to verify that
public reputation (high
µˆb
∂w ˆ ∂ µˆb
and low
(4)
≥ 0,
ρˆb )
∂w ˆ ∂ ρˆb
≤ 0,
increases
fee. However, a poor private reputation (low
µˆs
pg
∂w ˆ ∂ µˆs
≤ 0,
∂w ˆ ∂ Tˆb
and
≥0
∂w ˆ ∂ ρˆs
and
≥ 0.
∂w ˆ ∂ Tˆs
≤ 0,
and
Thus, a better
and therefore increases the expected
and high
ρˆs )
also increases the fee as the
issuer believes that a good rating is more likely. These two opposing eects of reputation are the driving force behind our results.
Notation. In what follows, we sometimes abuse notation and write the beliefs and strategies in the second period not as functions of the prior and the strategies, but indirectly as a function of the history. Thus, the posteriors are sometimes written as and
ρˆi (a, r, R; x),
where:
a
µˆi (a, r, R; x)
is the quality of the rst asset (relevant only if known to the
issuer in the second period);
r
is the rating published in the rst period;
set's return (whether it defaulted or not); and CRA in the rst period. Similarly, we use
R
is the rst as-
x is the (believed) strategy of the strategic
w(a, ˆ r, R; x, xˆ) and Tˆ(a, r, R; x, xˆ) to denote the
indirect expected fee and informativeness of the CRA in the second period as a function of the game history in the rst period and the expected strategy of the strategic type in the second period
4.4
xˆ.
Equilibrium with a One-time Issuer
In this case, the issuer and the investor have the same information on what happened in the rst period, and therefore
Tˆb = Tˆs = Tˆ.
From (4) we can see that in such a case
the expected fee is increasing in the rating's informativeness (
∂w ˆ ∂ Tˆ
> 0).
This leads almost
immediately to the result that the CRA does not have an incentive to give a good rating to a bad asset.
15
Proposition 4.3.
In the case where issuers sell only one asset, and therefore the second
issuer has the same information as the investor in the second period, a strategic CRA never in ates ratings in the rst period (x
∗
= 0)
.
When reputation is common knowledge, a more informative rating means less adverse selection, and therefore more gains from trade. Since the CRA's fee equals the expected surplus from trade, the strategic CRA has an incentive to build a truthful reputation. The economic implication of this result is that in markets where issuers and investors have the same information, reputational concerns lead to market discipline. Notice that this result may still hold even if the CRA has some additional short-term incentives to in ate ratings, in the spirit of Bolton, Freixas, and Shapiro (2012) and Mathis, McAndrews, and Rochet (2009), as long as these incentives are not too strong.
4.5
Equilibrium with Repeated Interaction
When the issuer in the second period is the same one as in the rst period, he is potentially better informed than the investor, as he also knows the quality of the rst period's asset. This may result in
Tˆb 6= Tˆs .
Our results suggest rating in ation (x
∗
> 0)
under certain
conditions.
Proposition 4.4.
In the case where the issuer sells two assets,
1. if
π>
1 2
and
ρ<
2π − 1 1 < 4π + 1 5
then the game admits equilibria where the strategic CRA in ates ratings with positive probability in the rst period of the game.
2. If, in addition,
µ > 2(1 − π) + 4ρπ
16
then in all equilibria (i.e., for all
xˆ ∈ [0, 1])
the strategic CRA in ates ratings with
positive probability in the rst period of the game (x
µ>
∗
> 0).
If
2(1 + π) 4 > 1 + 4π 5
then in all equilibria the strategic CRA gives only good ratings (x
∗
= 1).
The intuition behind Proposition 4.4 is as follows. Following a bad rating the CRA is publicly identi ed as not corrupt, and its reputation improves (ρˆb
= ρˆs = 0, µˆs = µˆb > µ).
In contrast, if a bad asset receives a good rating, there are two possible future payos: with probability
1−π
the asset defaults and the rating in ation becomes commonly
known, leading to a poor common reputation (ρˆb fee. With probability where
π
ρ < ρˆb < ρˆs < 1
= ρˆs > ρ
and
µˆs = µˆb = 0)
and a low
the asset does not default, which results in a double reputation,
0 = µˆs < µˆb < µ.
and
A double reputation may lead to a fee
greater than the fee following a bad rating. If the conditions described in the proposition hold, then rating in ation is pro table. Our results provide us with the following comparative statics across the (unique) equilibrium that takes the strategic rm's behavior in the second period (x ˆ) as xed.
Proposition 4.5.
∂x∗ following properties: (1) ∂ρ ∂x∗ (5) ∂ x ˆ
≤0
xˆ ∈ [0, 1],
For a speci c
≤ 0;
∂x∗ (2) ∂µ
the rst-period optimal strategy
≥ 0;
∂x∗ (3) ∂π
≥ 0;
∂x∗ and (4) ∂`
≤ 0.
x∗
has the
In addition,
.
The initial reputation of the CRA has a signi cant impact on the results. If the initial
ρ
is high, then truthful rating is relatively more rewarding, because it identi es the CRA
as not corrupt. Therefore, if
ρ
is above some threshold, then a strategic CRA remains
honest (builds its reputation).
If
µ
is high, then rating in ation is relatively more
rewarding, because, following a good rating and a high return, the dierence between the two reputations is large
µˆs = 0
milks its reputation). If
µ is high enough then the probability that the CRA is strategic
(1
− µ − ρ)
any
is low and therefore
while
xˆ
µˆb
is still high and thus we get
xˆ > 0
(the CRA
is less important, and thus we get rating in ation for
xˆ. 17
Higher
π
`
and lower
represent safer bad assets, which makes rating in ation is less
risky and therefore more probable. Finally, if the strategic type is expected to in ate with high probability in the second period (high
xˆ),
then it has a greater incentive to imitate
the honest type and remain truthful. In contrast, if the strategic type is expected to be relatively truthful (low
xˆ),
then fees following a rating in ation and default will not drop
much, and so rating in ation becomes more aordable. Thus, even in our simple model we can see a trend where a period of building reputation is followed by a period of milking reputation, and vice versa.
5
Threat of Entry
In this section, we explore the possibility of creating a double reputation when there is possible competition.
We present a model where an incumbent CRA faces a threat
from potential entrant CRA with a known reputation in the second period. This setup is tractable and captures the strategic considerations of a CRA that expects future competition. As in the monopolistic case, we analyze two dierent cases: in the former issuers act only once, while in the latter they issue assets repeatedly. The analysis has two main results.
First, double reputation can be maintained in
equilibrium even under the threat of competition. We view this as a robustness result. The second result is that competition has a disciplining eect on the incumbent in some cases: there are conditions where a strategic CRA that faces potential competition in ates ratings with lower probability than a monopolistic CRA and even remains truthful while the monopolist in ates ratings. It is worth noting that there is a debate in the literature on whether competition leads to more or less rating in ation. While competition leads to more rating in ation when rating shopping is possible (Bolton, Freixas, and Shapiro, 2012; Skreta and Veldkamp, 2009), its impact is less clear when fees are not contingent on ratings.
22
In Bar-Isaac
22 Becker and Milbourn (2011) nd that competition may in fact decrease the informativeness of ratings even when rating shopping is not present. However, the authors do not consider structured assets, which lie at the heart of our analysis, and it is not clear what aspect of the corporate market delivers their results. This suggests that more empirical and theoretical work is needed.
18
and Shapiro (2013) competition may not lead to more rating in ation: if one CRA gives in ated ratings, it may get caught and exit in the next period, and so another CRA has an incentive to remain truthful in order to stay in business and receive monopolistic rents in the future. Jeon and Lovo (2011) focus on the behavior of an entrant CRA when the incumbent is assumed to report truthfully. They show that the entrant has an incentive to in ate ratings, and therefore is not hired. This creates a natural entry barrier in the credit-rating industry.
5.1
A Model with a Threat of Entry
We modify the game described in Section 3 by adding a possible entrant CRA in the second period, with a commonly known reputation
(µE , ρE ).
Thus, the rst period of
the game is similar to the benchmark model, but in the beginning of the second period the issuer chooses whether to rehire the incumbent CRA for another period, or to hire the entrant CRA. We assume without loss of generality that entrant's rating informativeness is that
TE ≡ µE .23
µE = 1 − ρ E ,
and so the
We ignore the extreme cases and assume
0 < TE < 1.
As in the basic model, we assume that the issuer can extract all the surplus of the investor. However, we do not continue to assume that the CRA can extract all the surplus of the issuer, as in such a case the issuer receives a zero payo and is indierent between hiring the incumbent and the entrant. To simplify the analysis, we assume that the CRA and the issuer split the expected surplus from each transaction.
Assumption 5.1. payo, where
The CRA's fee equals a known fraction
α
of the issuer's total expected
α ∈ (0, 1).
The issuer therefore hires the CRA that generates the highest expected payo. If the issuer is indierent between the two CRAs, then he rehires the incumbent.
23 Any reputation where equivalent to a reputation
µE + ρE < 1, and the strategic type in ates the rating with probability xE , (˜ µE , ρ˜E ) where µ ˜E = µE + (1 − µE − ρE )(1 − xE ) and ρ˜E = 1 − µ ˜E .
19
is
5.2
Hiring Decision and Signaling of Quality
In the second period the issuer privately knows the quality of his asset, and may also have privileged information about the type of the incumbent CRA. His hiring decision may signal part or all of this information to the investor. We rst establish the fact that in equilibrium the issuer's hiring decision does not signal whether he has a good asset (G type issuer) or a bad asset (B type issuer).
Lemma 5.2.
The game admits no equilibrium where the issuer's hiring decision in the
second period depends on the quality of its asset, if both CRAs are believed to in ate ratings with positive probability. Else, if
a=G
and does not hire any CRA if
Tˆs = 1,
then the issuer hires the incumbent if
a = B.
The intuition behind Lemma 5.2 is that an issuer with a bad asset always has an incentive to pool with the
G
type issuer.
24
In the next subsections we use Lemma 5.2 and analyze pooling equilibria where issuers with assets of varying quality hire the same CRA. To avoid unreasonable equilibria, we focus in this section solely on perfect Bayesian equilibria that satisfy the D1 criterion (Cho and Kreps, 1987). In the context of this paper, this means that the investor realizes that only an issuer with a bad asset has an incentive to hire a CRA that in ates ratings. A formal de nition of the D1 criterion in the context of the model appears in Section A.4 of the Appendix.
5.3
Competitive Equilibrium with One-time Issuers
In this subsection we describe the Bayesian equilibrium of the model, if each issuer issues only one asset. The results show that in such case, as in the monopolistic case, there is no rating in ation.
Proposition 5.3. In a model where an incumbent CRA faces a threat of entry and issuers sell only one asset, the equilibrium admits the following properties: 24 If the incumbent is truthful he will be hired by the
G
type while the
B
type cannot expect a positive
payo and thus does not hire. Such a separating equilibrium also exists in the monopolistic model when the CRA is known to be truthful in the beginning of the second period.
20
1. The second issuer hires the CRA that is more truthful (i.e., with the maximal
2. A strategic incumbent CRA never in ates ratings in the rst period (x
∗
= 0)
T ). if it
has a positive probability of being rehired. The fact that the issuer's only private information is his asset's quality, together with Lemma 5.2 and the D1 criterion, leads to the rst part of the proposition. Given that rst part, the analysis is similar to the monopolistic case, and so the second part of the proposition is similar to Proposition 4.3.
5.4
Competitive Equilibrium with Repeated Interaction
In such a case, the issuer may have private information both on the quality of his asset and on the type of the incumbent. As it turns out (and is proven in Appendix A.7), we can extend Lemma 5.2 and show that in this case all types of the issuer also pool and hire the CRA with the highest public informativeness. This, in turn, in uences the conditions for rating in ation in equilibrium, as stated in the next proposition.
Proposition 5.4.
In a model where an incumbent CRA faces a threat of entry and the
issuer sells two assets, if the following conditions are satis ed: 1.
π>
1 ; 2
2.
ρ<
2π−1 4π+1
3.
TE ≤ TE ,
<
1 ; and 5
where
TE
is de ned in Appendix A.7;
then the game admits equilibria where a strategic incumbent in ates ratings in the rst period (x
∗
> 0).
While the rst two conditions are similar to those in Proposition 4.4, there is an additional condition on the reputation of the entrant: the incumbent may in ate ratings only if the potential competitor is also likely to in ate ratings. Higher
TE
can be interpreted
as stronger competition faced by the incumbent, since the issuer hires the CRA with the highest public informativeness. If
TE
is above some threshold, the strategic incumbent
prefers to build his public reputation by giving a bad rating.
21
De ne
xm
as the rst period's equilibrium strategy of a monopolistic CRA (given by
Proposition 4.4) for a given second-period strategy
xˆ.
The next proposition describes
several properties of the equilibria.
Proposition 5.5.
If the incumbent has a strictly positive probability of being rehired, and
xing a strategic incumbent's second period behavior at
xˆ ∈ [0, 1]
(across both monopoly
and competition), then the unique equilibrium rst-period behavior properties: (1)
x∗ ≤ xm ;
∂x∗ (2) ∂T E
≤ 0.
∂x∗ In addition, (3) ∂ x ˆ
x∗
has the following
≤ 0.
Part (1) of the proposition simply shows that competition has a disciplining eect. Part (2) shows that there is strategic complementarity in the market for ratings.
The
intuition behind these results is that a good rating, even when not followed by a default, harms the rating agency's public reputation.
This (weakly) decreases the incumbent's
chances to be rehired. Therefore, rating in ation is always less pro table under competition, and its pro tability decreases as the competitor's reputation improves. A complete characterization of the equilibria with repeated interaction can be found in Section A.7 of the Appendix.
6
Predictions and Empirical Implications
In this section we describe testable predictions of the model, all derived from the core result that CRAs will cater to experienced issuers who obtain better information than investors about the quality of the ratings.
We examine these predictions using recent
empirical papers on the credit-rating industry.
Greater rating in ation and higher fees for ratings will be observed in markets with a small number of issuers. of our model.
As mentioned above, this is the main prediction
In the nancial crisis of 2008, ratings of MBSs and CDOs were found
to be much less informative than ratings of corporate bonds, which suered only a few downgrades (for more details see the Introduction). We view this as an empirical support for our claim, because markets for MBSs and CDOs had repeated interaction between
22
issuers and rating agencies. A caveat is that there are additional dierences between the markets, and so further empirical research is needed to show that repeated interaction played an important role in rating policies. Furthermore, the model predicts that CRAs' fees are expected to be higher in markets with few issuers. This prediction is in line with the high fees that were charged in the markets for MBSs and CDOs compared to plain corporate bonds (Partnoy, 2006). Other papers predict that rating in ation is a result of high rating-dependent fees (see, e.g, Bolton, Freixas, and Shapiro, 2012). Our model predicts the converse: issuers pay higher fees for ratings in these markets because they expect to get in ated ratings.
Repeated interaction with issuers will result in favorable and in ated ratings. Several papers present evidence that issuers seek long-term relationships, and that such relationships result in rating in ation. Faltin-Traeger (2009) examines the hiring decisions of issuers in a large sample of asset-backed securities. He reports that an issuer is more likely to choose the CRA that provided the most favorable rating in its previous deals and that the CRA it chooses is less likely to rate its subsequent deals lower than other CRAs. Kronlund (2013) nds similar results for a large sample of corporate bonds. Mählmann (2011) examines a large sample of U.S. publicly traded rms rated by S&P for at least three consecutive years for the years 1986-2005.
He nds that the average
rating of a bond is increasing in the length of the rm-agency relationship duration, even when controlled for a wide range of rm-, bond-, and industry-speci c characteristics. Firms with longer-term relationships, while having higher average ratings, do not possess lower default rates and their bonds have a signi cantly higher downgrade hazard. Moreover, relationship bene ts are higher for rms that face a high default risk. This suggests that the high ratings that follow long-term relationships are due to rating in ation and not high quality. An additional proxy for the impact of repeated interaction is comparing large and small issuers, because rms that issue more assets have more opportunities to obtain private information about the quality and truthfulness of their ratings.
He, Qian, and
Strahan (2011) examine 85,000 MBS tranches originated and issued between 2000 and
23
2006, and compare tranches sold by large issuers versus small issuers (where size is based on the market share). Controlling for diversi cation and collateral risk, they nd that the fraction of tranches that received AAA ratings in every deal was signi cantly higher for large issuers. Moreover, they show that AAA tranches of large issuers performed worse during the 2008 nancial crisis.
The market will undervalue assets with good ratings if those assets are issued by repeated issuers.
An important feature of our model is that all agents, including
25
the investors, are rational.
Therefore, our results suggest lower prices of assets with a
good rating in the case where there is repeated interaction between issuers and rating agencies.
In accordance with this prediction, Mählmann (2011) nds that initial yield
spreads on bonds of a rm in a long-term relationship with the same CRA is higher than similar bonds sold by a rm in a short-term or non relationship, conditional on the rating. He, Qian, and Strahan (2012), in a companion paper to the one described above (that uses the same data), nd that the initial yield of MBS tranches is about 10 percent higher on tranches sold by large issuers than on similarly rated tranches issued by small issuers during market boom years.
Rating in ation is more likely in boom periods.
Our model suggests that CRAs
in ate ratings only if the probability that bad assets default is below some threshold. In addition, the lower the default probability of a bad asset, the higher the probability that the CRA will in ate ratings. While the default probability is constant in our static model, in reality it changes with the business cycle.
For example, during the housing
boom years, the probability that mortgage-takers would default was low even if they had no collateral other than their home.
26
Thus, our model predicts more rating in ation in
boom periods. This is also a prediction of the model oered by Bar-Isaac and Shapiro (2013). While there is no direct evidence for this prediction, the empirical evidence of the
25 This is in contrast to other papers that assume naive investors, such as Bolton, Freixas, and Shapiro (2012) and Skreta and Veldkamp (2009). For details see Section 2.
26 For
years
evidence
2008-11
see,
that for
default-risk example,
for
S&P's
structured Global
assets
was
Structured
low
Finance
2005-07
compared
Default
Study,
to
the
1978-2011
(http://www.standardandpoors.com/ratings/articles/en/us/?articleType=HTML&assetID=1245331158575). Fitch publishes a similar trend for its rated assets.
24
2008 nancial crisis found that the rating standards deteriorated during the years of the housing bubble; see Ashcraft, Goldsmith-Pinkham, and Vickery (2010), Gri n and Tang (2012), and He, Qian, and Strahan (2012).
7
Policy Discussion
Previous theoretical literature has focused on the short-term con ict of interest that is inherent in the rating agencies' business model (as explained in Section 2 above), and therefore has mainly recommended the prevention of rating shopping and rating-contingent fees.
27
Our results suggest that rating in ation may also be a result of long-term repu-
tational concerns in the case of repeated interaction. Our results thus provide additional policy recommendations that are relevant in markets where there are few issuers. In what follows, we brie y discuss three possible regulatory remedies that are supported by the analysis of our model: severing the connection between the issuer and the CRA, public fee disclosure, and increasing the potential cost of rating in ation by holding rating agencies liable to investors' losses in such a case.
Prevent Continuous Relationships
Given our results on the detrimental eect of
continuous relationships between issuers and rating agencies, an immediate regulatory recommendation is to prevent such relationships by limiting the issuer's ability to select a rater for his assets (see, for example, Mathis, McAndrews, and Rochet, 2009). A legal amendment following this line was proposed by U.S. Senator Al Franken in 2010, but it did not pass the legislation process. Franken's amendment required that issuers pay the rating fees to a government authority, and that the o cers of that authority would then choose the rating agency. Similarly, the new EC regulation of CRAs (known as CRA 3) introduces mandatory rotation for CRAs when rating re-securitizations (i.e., structured
27 These recommendations are in line with the agreement made by Andrew Cuomo, then New York State Attorney General, with the three major credit-rating agencies in 2008. Under this agreement the rating agencies, in rating Residential MBSs, agreed to demand upfront fees and to disclose every rating deal, even in the case where the rating was not published. For more details see the Attorney General's o ce press release at http://www.ag.ny.gov/press-release/attorney-general-cuomo-announces-landmarkreform-agreements-nations-three-principal.
25
assets such as MBSs and CDOs).
Mandatory Fee Disclosure
28
A key factor of the model presented above is that investors
cannot observe the fees that are paid to rating agencies, and thus issuers can pay high fees for agencies they believe are lenient without signaling those beliefs to the market. In reality, rating agencies do not publish the fees that they charge from clients. Moreover, during the boom years of 2005-07, rating agencies were frequently hired by issuers to give consulting services regarding the design of structured assets. This can be seen as a way for issuers to pay higher fees to rating agencies.
29
As part of the U.S. Dodd-Frank nancial
reform act of July 2010, rating agencies must disclose fees they receive from non rating services. This is done explicitly to address the CRAs' potential con icts of interest.
Increasing Civil Liability of CRAs
Our model suggests that in some markets CRAs
may in ate ratings despite the fact that they suer a reputational cost (in the form of lower fees) when they are caught misreporting. One can, however, decrease the possibility of a rating in ation by increasing the cost that the rating agency suers in case of a publicly disclosed rating in ation.
One way to achieve this is by increasing the civil
liability of CRAs. The Dodd-Frank Wall Street Reform and Consumer Protection Act, which passed in the U.S. congress in the summer of 2010, grants investors the right to sue a CRA for damages if they prove that the CRA was grossly negligent in determining
30
a rating and as a result contributed substantially to the investor's economic loss.
Sim-
ilar legislation that increases the civil liability of rating agencies is also part of the new European Commission's regulation of CRAs (CRA 3).
28 The Regulation introduces a mandatory rotation rule forcing issuers of structured nance products with underlying re-securitised assets, who pay CRAs for their ratings ("issuer pays model"), to switch to a dierent agency every four years.
An outgoing CRA would not be allowed to rate re-securitised
products of the same issuer for a period equal to the duration of the expired contract, though not exceeding four years...[a] review clause provides the possibility for mandatory rotation to be extended to other instruments in the future.
For more information see the press release of June 18, 2013, at
http://europa.eu/rapid/press-release_MEMO-13-571_en.htm?locale=en.
29 Several authors have discussed the connection between ancillary services fees and rating in ation.
See, for example, White (2010), p. 220, and Partnoy (2006), p. 70.
30 Before the Dodd-Frank act, ratings were considered to be opinions by American law, thus aording
CRAs the protection of the First Amendment of the U.S. Constitution. As a result, CRAs never lost when sued by investors or issuers who claimed that they had been injured by the actions of the agencies (Partnoy, 2006, pp. 83-89).
26
8
Concluding Remarks
This paper analyzes the reputational concerns of credit-rating agencies using a dynamic model of communication with two audiences. Our results suggest that reputational concerns encourage truthful ratings in some markets, but discourage them in others.
In
markets with many sellers who act infrequently, credit-rating agencies have the proper incentives to be truthful. In such markets, CRAs help remedy adverse selection problems. In concentrated markets where few frequent sellers face many buyers, sellers have privileged information both on the quality of the assets they oer and on the quality of the
rating s they receive. This second level of information asymmetry creates an incentive for CRAs to distort information in favor of the better informed. Given the relative simplicity of the model, one may wonder about the robustness of the results.
The basic model allows for the two commitment types that stand for the
two pure strategies of the CRA (results hold even when only one of them has a positive probability), and for any strategy of the strategic type. Thus, our results do not depend on a speci c choice of strategy in the last period or a speci c choice of the commitment type. The model can be extended to include more than two periods, and we believe that the results will qualitatively hold. In such a model, a CRA who chooses to in ate ratings today knows that, in case it is caught misreporting, it can give a bad rating the next time it rates a bad asset, thus building its reputation again. Therefore, the reputational cost associated with revealed rating in ation is temporal, and the incentive to in ate remains. In Section 5 we have shown that the results hold even when the CRA takes into account hiring considerations. While an oligopolistic model where several CRAs are active in each period is not tractable, we believe that rating in ation will still be present in such a case. The key for rating in ation is that each CRA believes there is a signi cant probability that rating in ation will not be revealed publicly, and that public reputation following a good rating and no-default will remain high enough for the CRA to be rehired. There are many parameter values where these conditions apply and rating in ation takes place. One possible explanation of the fact that all three major CRAs gave high ratings to toxic assets in the period before the nancial crisis of 2008 is that those agencies were
27
involved in tacit collusion. Developing a model that predicts tacit collusion between CRAs is a challenge left for future research. Another challenge is to examine the case where there are several possible qualities and ratings. In such a model a CRA can in ate ratings of some assets and de ate ratings for others (for example, by giving an intermediate rating to all assets). Exploring such a model is beyond the scope of the paper, but it is needed to explore exactly the dynamics of misreporting due to reputational concerns. The model developed here presents an expert sender who decides whether or not to disclose his private information. or receivers:
The sender is faced with two opposing audiences
the rst is an uninformed audience that bene ts from full disclosure of
information, and the second is an informed audience that bene ts if the sender chooses a speci c message. While we use this model to analyze rating agencies, we believe that this general model ts many other institutional circumstances. One example that comes to mind comes from political economy: voters may elect an informed politician in the hope that she will choose the best policy according to her knowledge. However, a lobby with substantial political power may prefer a speci c policy to be implemented. It is reasonable to assume that the lobby is more informed about the impact of dierent policies than the voters (for example, oil companies are more knowledgeable than the average voter about the environmental risks of drilling in certain areas).
The politician obviously aims to
be elected by the voters. But if at the same time she relies on the lobby's support (for example, as a source of campaign funding), then she faces the same problem as does the rating agency in our model.
28
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32
A A.1
Technical Appendix Posteriors
Below is a table with the posteriors of the issuer and the investor at the beginning of the second period. We use these posteriors in the proofs below. The middle section describes the posteriors of the investor (µ ˆb , ρˆb ) and an issuer who sells only one asset
((µˆs , ρˆs )
of
Section 4.4) both only observe the rating and return. The right section describes the posteriors of an issuer who sells two assets ((µ ˆs , ρˆs ) of Section 4.5), and therefore also knows the quality of the asset.
History Quality
Investor / One-time Issuer
µ ˆ
ρˆ
µ ˆ
ρˆ
µ µ+(1−ρ−µ)(1−x)
0
µ µ+(1−ρ−µ)(1−x)
0
0
ρ ρ+(1−ρ−µ)x
0
ρ ρ+(1−ρ−µ)x
Rating & Return
a=B
r=b
a=B
r = g,
a=B
r = g,
a=G
r=g
default no-default
µ 1+π[ρ+(1−ρ−µ)x]
Comments: (1) common prior; (2)
ρ(1+π) 1+π[ρ+(1−ρ−µ)x]
µˆs < µˆb , ρˆs > ρˆb
µˆs > µˆb , ρˆs < ρˆb
A.2
Repeat Issuer
µ
ρ
in the repeated case; (3)
in the repeated case.
Proof of Proposition 4.3
Proof. The issuer in the second period has the same information as the investor regarding the incumbent's type. Substituting
(µˆs , ρˆs ) = (µˆb , ρˆb ) ≡ (ˆ µ, ρˆ).
into (4), the expected fee of the CRA in the second period is
1 − ` 1 − Tˆ wˆ = 1Tˆ>0 ·
2
33
,
(1) (1) (2) (3)
Tˆ = T (ˆ µ, ρˆ, xˆ) = µ ˆ + (1 − ρˆ − µ ˆ)(1 − xˆ). wˆ
where
is (weakly) increasing in
Tˆ.
It can be
easily veri ed using the posteriors in Section A.1 that
0=µ ˆ(r = g, default; x) < µ ˆ(r = g, no-default; x) < µ < µ ˆ(r = b; x) and
0 = ρˆ(r = b; x) < ρ < ρˆ(r = g, no-default; x) < ρˆ(r = g, default; x) x,
for all
and
Tˆ(B, b; x, xˆ) > Tˆ(·, g, no-default; x, xˆ) > Tˆ(B, g, default; x, xˆ)
for all
x
and so
x∗ = 0.
A.3
and
xˆ.
The CRA therefore always prefers to publish
r=b
(5)
in case that
a = B,
Proof of Proposition 4.4
Proof. We prove rst the conditions for rating in ation, and then the equilibrium strategy.
Conditions for rating in ation (x∗ > 0). because if
x=0
x = 0
Notice rst that
then, following a good rating and default
w(g, ˆ default; 0, xˆ) = 0
(ˆ µ, ρˆ) = (0, 1).
Therefore,
is not an equilibrium strategy if and only if
E w(B, ˆ g, ·; 0, xˆ) = π · w(B, ˆ g, no-default; 0, xˆ) > w(B, ˆ b; 0, xˆ).
Substituting (4) (with the proper posteriors from Section A.1 and
x = 0),
(6)
condition 6
becomes
`ˆ x(1 − µ − ρ) 2(1 + ρπ)(1 + `)π + − 2`π − 1 > 0. 1 + (1 + 2π)ρ + (1 − µ − ρ)ˆ x 1−ρ
(7)
One can verify that the following four conditions are su cient for (7) to hold for all
xˆ ∈ [0, 1]: 1 π> , 2
34
(8)
ρ<
2π − 1 1 < , 4π + 1 5
(9)
µ > 2(1 − π) + 4ρπ,
(10)
(1 − ρ) [1 + ρ − 2π 2 ρ − 2π(1 − ρ) + xˆ(1 − µ − ρ)] . xˆ(1 − µ − ρ) [ρ(4π + 1) − (2π − 1)] + xˆ2 (1 − µ − ρ)2 − 2π(1 + π)(1 − ρ)ρ
(11)
and
`<
Given that conditions (8)-(10) are satis ed, the RHS of (11) is always greater than one, and therefore condition (11) is satis ed by Assumption 3.1. It is also possible to verify that, while conditions (8), (9), and (11) are necessary for (7), condition (14) may be violated as long as
xˆ ≤
∗
= 1)
is low enough, and speci cally
2π − 1 − ρ(1 + 4π) . 1−ρ−µ
Equilibrium with full in ation (x∗ = 1). equilibrium (x
xˆ
A strategic CRA fully in ates ratings in
if
π · w(B, ˆ g, no-default; 1, xˆ) + (1 − π) · w(B, ˆ g, default; 1, xˆ) > w(B, ˆ b; 1, xˆ).
Substituting (4) (with the proper posteriors from Section A.1 and
x = 1),
(12)
condition (12)
can be rewritten as
[ˆ x(1 − µ − ρ) + ρ] {l [−(1 + π)ˆ x(1 − µ − ρ) + 2µπ − πρ + 1 − π − ρ − 2] + µπ} > 0. 2(1 − µ) [(1 + π)ˆ x(1 − µ − ρ) − µπ + πρ + π + ρ + 1] (13) This condition is satis ed for every
xˆ < 1
if and only if conditions (8) and (9) hold (they
are necessary conditions) together with
2(1 + π) 1 + 4π
(14)
µπ . (1 + π)(1 − µ − ρ)ˆ x + (1 + π)(1 + ρ) − 2µπ
(15)
µ>
and
l<
35
If conditions (8), (9), and (14) are satis ed, then the RHS of (15) is always strictly greater than 1, so (15) is always satis ed given Assumption 3.1. In addition, (13) is also satis ed if (8) and (9) hold and If
xˆ = 1,
then
1+π 3π
<µ≤
2(1+π) as long as 1+4π
w(B, ˆ g, default; ·, 1) = 0,
0 < xˆ <
3µπ−1−π . (1−µ)(1+π)
because following a public default all players
expect the rating in the second period to be uninformative. Substituting that into (12), we get an inequality similar to (13). It is easy to show that this inequality is satis ed if and only if conditions (8), (9), and (14) hold together with a dierent condition on
`,
which is satis ed given Assumption 3.1.
Equilibrium in mixed strategies (0 < x∗ < 1).
If (7) holds but (12) does not, then
the equilibrium strategy is mixed and calculated using the following indierence condition:
π · w(B, ˆ g, no-default; x∗ , xˆ∗ ) + (1 − π) · w(B, ˆ g, default; x∗ , xˆ∗ ) = w(B, ˆ b; x∗ , xˆ∗ ).
(16)
Using the relevant posteriors in Section A.1 and (4) we can rewrite (16) as
x∗ =
π(1 − ρ)(` + 1) − ` [2ρπ + xˆ(1 − µ − ρ) + 1 + ρ] , π(1 − µ − ρ) [1 + `(2 + xˆ)]
which is strictly greater than zero given (8)-(10).
A.4
Technical Description of the D1 Criterion Used in Section 5
Assume a pooling equilibrium where CRA
i
is hired by all types in the second period. In
the beginning of this period the issuer is one of several types, which re ect private information about the quality of his asset and possibly also about the type of the incumbent. Denote the set of types by
Θ.
For each type
in the Bayesian equilibrium and de ne where
pg
θ ∈ Θ let u∗ (θ) be the expected payo of this type
P(θ) = {pg : (1 − α) Pr(r = g|θ, −i) · pg ≥ u∗ (θ)},
is a maximal price that the investor is willing to pay for a good rated asset
following the hiring of CRA hiring CRA
−i)
−i. P(θ)
pro table for type
θ
is the set of prices
pg
that make deviation (i.e.,
(this set may be empty). Notice that
pg
is a direct
result of the investor's belief (about the type of the issuer) after he observes the deviation.
36
Let
β(θ| − i)
denote this belief.
De nition A.1.
An equilibrium satis es the D1 criterion if
those types for which
P(θ)
is supported by
such that
P(θ0 ) ⊆ P(θ00 ).
is maximal.
To understand the intuition assume two types, The interpretation is that
β(θ| − i)
θ00
θ0
and
θ00 ,
is more likely to deviate than
investor that makes deviation pro table for
θ0 ,
θ0 :
for every response of the
the deviation is also pro table for
there are prices where deviation is pro table for
θ00
and not for
θ0 .
A.5
and
Therefore, following a
deviation, it is not reasonable to believe that the type of the issuer is demands that
θ00 ,
θ0 ,
and the criterion
β(θ0 | − i) = 0.
Proof of Lemma 5.2
Proof. Assume to the contrary an equilibrium where, following some history, the hiring decision of a
G
type issuer diers from the hiring decision of a
B
type issuer.
history the investor learns the quality of the asset and the payo of a zero. If the CRA that is hired by the then
B 's
G
B
In this
type issuer is
type in ates ratings with a positive probability
decision is not optimal contradiction.
If the CRA is believed by the issuer to be completely truthful in the second period (by construction, this can only happen for the incumbent), then a separating equilibrium can be sustained by the following out-of-equilibrium belief: the investor believes that after observing that the incumbent is hired, unless this incumbent publishes case he believes that then a
a = B.
Given such beliefs, If a
G
r = b:
B
in that
type issuer hires the incumbent,
B type issuer expects a zero payo in every strategy and does not hire.
out-of-equilibrium beliefs, the
a=G
In any other
type issuer is believed to have a good asset with some
positive probability if he hires the same CRA as the
G
type, and therefore a separating
equilibrium does not exist.
A.6
Proof of Proposition 5.3
Proof. We solve the game backwards: rst we nd the equilibrium in the sub-game of the second period for general reputations, and then solve for the rating strategy of the
37
incumbent in the rst period.
Second period.
In the second period, both CRAs have commonly known reputation:
(µˆs , ρˆs ) = (µˆb , ρˆb ) ≡ (ˆ µ, ρˆ) Tˆ ≡ T (ˆ µ, ρˆ, xˆ)
and
TE
for the incumbent, and
second asset's quality, so Let
equilibrium. those where payo of a
Let
The only private information of the issuer is the
θ ∈ {B, G}.
Lemma 5.2 precludes a separating equilibrium.
be the informativeness of the CRA that is hired by both types in
We now show that the only equilibria that satisfy the D1 criterion are
n o T ∗ = max Tˆ, TE .
B
type issuer and a
u∗ (G) ≡ (1 − α)pg (T ∗ )
G
Notice that in a pooling equilibrium the expected type issuer are
respectively (p
g
g
g
g
(T ∗ )
u∗ (B) ≡ (1 − α)(1 − T ∗ )pg (T ∗ )
∗
P (B) = pg : pg ≥
(T ) . ∗
n o ˆ T = min T , TE , ∗
then
so the investor believes that the asset is good after he observes that the
more informative CRA is hired. In the latter case, the a contradiction. Finally, if
First period.
1−T ∗ pg 1−{Tˆ,TE }\T ∗
deviation and hiring the CRA with the
minimal informativeness results in zero payo. In contrast, if
P (B) ⊂ P (G),
and
is calculated using (3)). Using the de nition
P (G) = {p : p ≥ p (T )} and n o ∗ ˆ T = max T , TE then P (G) ⊂ P (B), so
in Section A.4, If
for the entrant.
be the expected informativeness of the incumbent and entrant
respectively as calculated from (1).
n o T ∗ ∈ Tˆ, TE
(µE , 1 − µE )
TE = Tˆ
G type has a pro table deviation
then both types of equilibria survive the D1 criterion.
Notice that since the hiring strategy does not signal anything about the
quality of the asset, then the prices in the second period are similar to the monopolistic prices calculated in (3). Given the common posterior in that period, the expected fee of the incumbent can be calculated in a similar way to (4) and is equal to
1 − ` 1 − Tˆ wˆ = 1Tˆ>TE · α
This fee is increasing in
2
.
Tˆ, and using the same argument as in the proof to Proposition 4.3
(see equation (5)) we argue that
Tˆ
is maximal following
r=b
for all
x
and
xˆ.
Using the
posteriors in Section A.1 we calculate that the maximal informativeness of the incumbent is when
xˆ. x = 0, Tˆ (B, b; 0, xˆ) = 1− 1−µ−ρ 1−ρ
If
Tˆ (B, b; 0, xˆ) ≥ TE
38
then
x∗ = 0 is a strict best
response for the incumbent and an equilibrium strategy. Notice that there are always equilibria where
Tˆ = 1 ⇐⇒ xˆ = 0, so
xˆ is low enough and the incumbent never in ates ratings
and is hired in the second period. However, if
Tˆ (B, b; 0, xˆ) < TE
then the incumbent is
never hired in the second period, and therefore all rating strategies are a best response (such equilibria never exist if
A.7
µ 1−ρ
> µE ).
Threat of Entry with Repeated Interaction Characterization of Equilibria and Proof of Propositions 5.4 and 5.5
In what follows we fully characterize equilibria where
xˆ
does not depend on
x.
We
rst describe which CRA is hired in the second period for every sub-game with general reputations, and then solve for the rating strategy of the incumbent in the rst period, nding
x
for every
Second period.
xˆ ∈ [0, 1].
In the second period, the incumbent has a double reputation only
following a history of good rating (r
= g)
and no-default. In all other cases (good rating
followed by default and bad rating), both CRAs have a commonly known reputation. In the latter cases we can use the same analysis as in Proposition 5.3 to conclude that the issuer hires the more informative CRA. For a history of has one of four types,
θ ∈ {BG, GG, BB, GB},
r=g
and no default the issuer
where the rst and second letters denote
the qualities of the issuer's rst and second assets respectively. First, we can use the same argument as in the proof of Lemma 5.2 to explain why there cannot be a separating equilibrium where the hiring decision signals the quality of
31
the second asset.
Next we show that there is no equilibrium where types
hire dierently than types
{GB, GG}.
{BB, BG}
Assume, on the contrary, that such an equilibrium
does exist. In such an equilibrium the investor discovers the quality of the rst asset, so the incumbent has a commonly known reputation and informativeness (denote it by Since
pg
is increasing in
T
(3), it is optimal for types
Tˆ).
BG and GG to hire the (same) CRA
31 The only case where the issuer's hiring decision discloses the quality of its second asset is if the incumbent is expected to be completely truthful (Tˆs
= 1).
can only happen following a bad rating, and in such a case hire any CRA.
39
As discussed in the proof of Lemma 5.2, this
BG
hires the incumbent while
BB
does not
with the maximal
T
a contradiction. Thus, in equilibrium all types of the issuer hire
the same CRA. The nal step is to prove that the only equilibria are those where the CRA with the maximal public informativeness is hired following a good rating and no-default. given prior
(µ, ρ)
and rating strategy
(x, xˆ),
For a
let
Tˆb ≡ T (µˆb (g, no − default; x), ρˆb (g, no − default; x), xˆ) be the public informativeness (i.e. expected by the investor ) of the incumbent following
r=g
and no-default; let
Tˆ ≡ T (µˆs (a = B, r = g; x), ρˆs (a = B, r = g; x), x)
be the private informativeness expected by an issuer of types
BG
and
BB
(that is,
following rating in ation); and let
Tˆ ≡ T (µˆs (a = G, r = g; x), ρˆs (a = G, r = g; x), x)
be the private informativeness expected by an issuer of types lowing a truthful good rating). in Appendix A.1 and satisfy
Tˆb , Tˆ
and
Tˆ < Tˆb < Tˆ
Tˆ
GG
and
GB
(that is, fol-
can be calculated using 1 and the posteriors
(given assumption 3.3).
Consider a pooling equilibrium where the same CRA is hired by all types, and let and
Ts∗ (θ)
Tb∗
be its expected public and private informativeness following a good rating and
no default respectively (if the entrant is hired then is hired then
Tb∗ = Tˆb
while
n o Ts∗ ∈ Tˆ, Tˆ ).
Ts∗ (θ) = Tb∗ = TE ;
if the incumbent
For each type, the set of prices that result in
40
deviation and hiring the alternative CRA (see Section A.4) is
P (BG) = P (GG) = {pg : pg ≥ pg (Tb∗ )} ; ∗ 1 − Ts o P (BB) = p g : pg ≥ n pg (Tb∗ ) ; Tˆ, TE \Ts∗ ∗ 1 − Ts o pg (Tb∗ ) . P (GB) = p g : pg ≥ n Tˆ, TE \T ∗
and
s
If
n o Tb∗ = max TE , Tˆb ,
then
P(θ)
is largest for either type
GB
or type
BB
(depending
whether the incumbent or the entrant are hired). Thus, by the D1 criterion, deviation
∗ results in zero payo. If, in contrast, Tb
BG
and
GG,
n o ˆ = min TE , Tb ,
then
P(θ)
is maximal for types
which gives an issuer of such types an incentive to deviate and receive a
payo of one. Thus, the only equilibria that survive the D1 criterion are those where all types of the issuer hire the CRA with the highest public informativeness.
First Period.
First, let
xm
denote the strategy of a monopolistic CRA, described in
Proposition 4.4. Second, using (1) and the posteriors in Appendix A.1, we can calculate the public informativeness (i.e., expected by the investor) in each of the three histories,
Tˆb (x, xˆ, r = b), Tˆb (x, xˆ, r = g, default), sions are weakly decreasing in
and
Tˆb (x, xˆ, r = g, no − default).
All three expres-
xˆ, while the rst two are also weakly increasing in x.
Given
Assumption 3.3,
Tˆb (x, xˆ, r = g, default) < Tˆb (x, xˆ, r = g, no − default) < Tˆb (x, xˆ, r = b).
Using the above as well as the results we obtained on the hiring strategy of the issuer in the second period, we identify four hiring cases: (1) the incumbent is never rehired, (2) the incumbent is rehired only following a bad rating, (3) the incumbent is rehired as long as rating in ation is not publicly exposed (i.e., except in case of good rating and a default), and (4) the incumbent is always rehired.
In what follows we describe the
conditions on the parameters that result in each case being part of an equilibrium, and the equilibrium strategy
x∗
of the incumbent in each case for a given
41
xˆ.
The set of all
equilibria we characterize is
Case (1)
If
(x∗ (ˆ x), xˆ)
for
TE ≤ Tˆb (xm , xˆ, r = g, default)
strategic CRA chooses
xm .
Since
xm
always hired (Proposition 4.4) then
Case (3)
If
x ∈ [0, 1]. then the incumbent is always rehired if the is by de nition optimal if the incumbent is
x∗ = xm .
Tˆb (0, xˆ, r = g, no − default) < TE ≤ Tˆb (0, xˆ, r = b)
rehired only following a bad rating. In this case,
x∗ = 0
then the incumbent is
is a dominant strategy for
the incumbent.
Case (4) and
If
Tˆb (0, xˆ, r = b) < TE
x∗ ∈ [0, x)
where
x
then in equilibrium the incumbent is never rehired
Tˆb (x, xˆ, r = b) = TE .
is de ned using the inequality
The
indierence is due to the fact that if the incumbent is never rehired following some strategy
y,
then it also not rehired following any strategy
x ≤ y.
In this case, the incumbent has no reputation incentives in period 1, and its payo is always zero.
Notice that
equilibria always exist if low levels of
Case (2)
xˆ)
µE
Tˆb (x, 0, r = b) = 1 > TE
is high enough, the game always admits equilibria (with
where the incumbent CRA is rehired.
Assume that the incumbent's strategy is some
TE ≤ Tˆb (xc , xˆ, r = g, no − default) r=g
and therefore, while such
and default.
xc > 0
xc .
If
Tˆb (xc , xˆ, r = g, default) <
then the incumbent is not rehired only after
if and only if
π · w(B, ˆ g, no-default; 0, xˆ) + (1 − π) · 0 > w(B, ˆ b; 0, xˆ).
Observe that this condition is identical to 6, the condition for in ation in the monopolistic case. Thus,
xc = 0 if any of the conditions (8)-(10) that appear in Proposition
4.4 do not hold. If conditions (8)-(10) hold then
xc = min {1, x}
where
x
is de ned
using the indierence condition
π · w(B, ˆ g, no-default; x, xˆ) = w(B, ˆ b; x, xˆ).
Comparing the equality above to (16), the indierence condition in the monopolistic
42
Equilibrium under competition, as a function of the entrant's reputation for a given x ˆ. If TE is low (region (1) ), the entrant is never hired, and therefore the incumbent's strategy equals the monopolistic case; if TE is in intermediate levels (region (2) ), the entrant is hired only Figure 1:
if a good rating is followed by a default, and the incumbent may in ate ratings, but less than
TE (region (3) ), the incumbent is rehired only r = b, and so remains truthful; nally, there is always a region where, due to high x ˆ, the
in the monopolistic case; in even higher levels of following
incumbent is never rehired ( shrinks as
x ˆ
region (4) ), and therefore is indierent to all strategies this region
decreases and disappears when
x ˆ = 0.
The gray areas represent areas of where two
types of equilibria exist: while regions (1) and (2) always overlap, this is not necessarily so for regions (2) and (3), and if some levels of
µE
x ˆ
is high then a gap is possible, meaning there are no equilibria for
with these levels of
case, we can see that Notice and
x ˆ.
x < xm
for every
w(B, ˆ g, default; x, 1) = 0
x = xm .
Thus,
so if
xˆ < 1
xˆ = 1
(unless, of course,
x = xm = 1).
then the two conditions are identical
xc ≤ xm .
This concludes all the cases. Figure 1 presents a sketch of an equilibrium for a speci c a function of if
xˆ <
Notice that
(1+π)ρ , and (1+π)ρ+µ
are levels of
(xc , xˆ)
TE .
TE
Tˆb (0, xˆ, r = g, no − default) ≤ Tˆb (xc , xˆ, r = g, no − default)
Tˆb (xc , xˆ, r = g, default) ≤ Tˆb (xm , xˆ, r = g, default)
are both equilibria for some
xˆ).
An equilibrium does not exist if
Tˆb (xc , xˆ, r = g, no − default) < TE ≤ Tˆb (0, xˆ, r = g, no − default). (µ, ρ)
µE , there are h i (1+π)ρ xˆ ∈ 0, (1+π)ρ+µ .
and
strategy in the second period is
Proofs
for all
where equilibria of type (1) and (2) are possible (that is,
some initial reputations
xˆ as
so there
(xm , xˆ)
and
(1+π)ρ and (1+π)ρ+µ
This means that for
only equilibria where the incumbent's
We can use the above to prove Proposition 5.4: de ne
TE ≡ Tˆb (xc , 0, r = g, no − default; µ, ρ).
43
xˆ >
xˆ,
If conditions (8)-(10) are satis ed, then rating in ation appears in some equilibria if and only if
TE ≤ TE .
As for Proposition 5.5, parts (1)-(3) are immediate from the description above, and the fact that
x
decreases as we move up the cases. For part (4), since all the thresholds
are decreasing in
xˆ,
then
x
is also decreasing in
44
xˆ.
