Feedback effects between credit ratings and financial markets

Accepted Manuscript Feedback effects between credit ratings and financial markets José Jorge PII:

S0264-9993(18)30601-1

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https://doi.org/10.1016/j.econmod.2018.11.019

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Economic Modelling

Received Date: 27 April 2018 Revised Date:

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Please cite this article as: Jorge, José., Feedback effects between credit ratings and financial markets, Economic Modelling (2018), doi: https://doi.org/10.1016/j.econmod.2018.11.019. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Markets∗ Jos´e Jorge†

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Feedback Effects Between Credit Ratings and Financial

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Faculdade de Economia, Universidade do Porto, CEF.UP ‡ November 2018

Abstract

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Credit rating agencies often make sharp adjustments in their pronouncements during times of stress in financial markets. These adjustments typically happen with a delay relative to shocks in market prices. Since prices convey information ∗

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I am grateful to Xavier Freixas, Christina Bannier, Nuno Cassola, Qinglei Dai, Lucia Gibilaro, Joseph Guerra, Charles Kahn, Walter Novaes, Joana Rocha, and Ana S´ a for helpful comments as well as participants in the ECB Workshop on ”The post-crisis design of the operational framework for the implementation of monetary policy”, the Reserve Bank of New Zealand Conference on ”Macro policies after the crisis”, the VI Lubrafin workshop, the European Financial Management 21 st Annual Meeting, the XXI International Conference on Money, Banking and Finance, the Department of Finance Seminars at the University of Illinois at Urbana-Champaign, the 47 th Annual Conference of the Canadian Economics Association, the 19 th International Conference on Computing in Economics and Finance, the 28th Annual Congress of the European Economic Association, and the 2015 Midwest Finance Association Annual Meeting. This paper supersedes earlier working papers that circulated under the titles ”Endogenous Credit Ratings”, ”Ratings and Sentiments”, and “Sovereign Ratings and Investor Behavior”. † Address: Faculdade Economia, Rua Dr. Roberto Frias, Porto, Portugal. Tel: +351 225 571 100; fax: +351 225 505 050. E-mail address: [email protected] ‡ Centre for Economics and Finance at the University of Porto. Assignment co-financed by the European Regional Development Fund (ERDF) through the COMPETE 2020 - Operational Programme Competitiveness and Internationalization (POCI) and national funds by the FCT under the POCI-010145-FEDER-006890 project. Jos´e Jorge gratefully acknowledges financial support from European Regional Development Fund (ERDF) through the Programme COMPETE and by the Portuguese Government through FCT - Foundation for Science and Technology, project FCOMP-01-0124-FEDER029223.

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about what market participants are doing and thinking, it is likely that rating agencies take into account market prices when issuing their pronouncements.

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In order to understand the relationship between credit ratings and financial prices, we develop a model of debt roll-over in which rating agencies incorporate information publicly available in financial markets. We find that (1) rating agencies respond to market prices, i.e. nonfundamental price volatility can shift

financing conditions from a low risk spread and high credit rating equilibrium to

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an equilibrium with high spread and low rating, and (2) rating agencies can anchor expectations about the equilibrium in financial markets, thus serving as an

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antidote to nonfundamental price volatility.

Keywords: Credit ratings, roll-over risk, creditor coordination.

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JEL Classification Codes: D82, G18, G24.

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Introduction

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A credit rating measures the likelihood that an entity will meet its financial commit-

ments on a timely basis. In many areas, there are no alternative measures of creditworthiness that are as transparent and simple to use, that allow for consistent implementation across markets, and that effectively differentiate credit risk as credit ratings.

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They are supposed to provide a stable measure of creditworthiness.

The determination of credit ratings in times of stress is, however, ill understood. On

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the one hand, credit rating agencies (CRAs) claim their pronouncements are based on a range of quantitative and qualitative factors which mirror borrowers’ fundamentals. On the other hand, CRAs insist that they do not base their decisions on market prices. But consider the example of European sovereign debt markets during the period 20092012, when some European states were suddenly put under severe stress. Countries

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whose sovereign ratings had remained remarkably stable for an extended period of time, faced a strong rise in credit default swap (CDS) spreads as a result of investor concerns about debt sustainability. With anxious investors demanding ever-higher CDS spreads, agencies adjusted their pronouncements through a series of sharp downgrades. Were

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these downgrades related to the spikes in CDS spreads observed during the crisis? This article argues that CRAs take into account the information in financial market prices

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(such as in CDS market prices) when producing their ratings. As described more in detail below, this mechanism is more important in cases with multiple equilibria, when financial market prices signal a shift in the market equilibrium (which often happens in times of stress and in crises). Crises are volatile periods when individuals monitor intensely endogenous sources of information such as financial prices, as they convey information about what others are doing and thinking—a critical feature when borrowers are exposed to roll-over risk. It is not palatable that CRAs deliberately ignore the information content of market prices,

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since the credibility of CRAs is at stake when they sustain opinions which diverge

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substantially from the assessment of financial markets for an extended period of time. Our premise in this paper is that CRAs incorporate information publicly available in financial markets, and multiplicity in equilibrium market prices then feeds into multiplicity in credit ratings. We consider:

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1. A market for short-term debt, with a borrower whose debt is about to mature and needs to roll over the debt.

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2. A market which prices a derivative security whose value depends on the likelihood of default of the borrower. This market represents those financial markets which trade securities whose price conveys information to short-term creditors, such as the market for long-term debt or the stock market in the case of a corporate

on a bond.

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borrower. The derivative security could be broadly interpreted as a CDS written

Our framework has two key ingredients:

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1. The borrower’s capacity to repay their debt depends on the ability to roll over their maturing bonds, and the price of the derivative security depends on roll-over

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risk.

2. The price of the derivative security signals investors about the quality of the maturing bonds—the standard information-aggregation role discussed in the rational expectations literature—and thus contributes to lenders’ willingness to roll over their debt claims. In our setup a high price for the derivative security is akin to a low CDS spread, and signals the good quality of debt. Feedback effects between the two markets arise when the return on the derivative security depends on roll-over risk. A high price of the derivative security signals the 4

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good quality of debt, thus inducing debt roll-over among short-term creditors. Reduced roll-over risk, in turn, raises the aggregate demand of the derivative security. As long as

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this feedback effect between market prices and roll-over decisions is sufficiently strong,

the aggregate derivative security demand is increasing with its price over some region. With a backward-bending derivative security demand curve, there are multiple equi-

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librium prices when the aggregate asset supply is inelastic. In this case, there is ( i) a good equilibrium with high derivative security prices where the borrower easily rolls over their maturing debt, and (ii) a bad equilibrium with depressed derivative security

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prices where the borrower may be unable to roll over their debt.

With multiple equilibrium prices, relying exclusively on information about the economic fundamentals is insufficient to predict default rigorously, since prices also convey useful information about which equilibrium is actually being played in financial markets. CRAs aiming at producing accurate ratings take into account not only the fundamen-

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tals of the borrower but also the financial prices. The best response of the CRA is to issue a favorable credit rating when the price of the derivative security is high (thus revealing the good equilibrium) and an unfavorable rating when the price is low (thus

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revealing the bad equilibrium). Hence, the CRA may issue either a favorable or an unfavorable rating for the same value of the fundamentals of the borrower.

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Another key insight in our paper is that rating agencies can make a strategic move which restores the uniqueness of equilibrium when (i) the amount of noise in financial markets is small, and (ii) many short-term creditors react mechanically to credit ratings. The strategic behavior of CRAs diverts the participants in the derivative market from the price multiplicity area, thus preventing financial markets from entering the area where nonfundamental price volatility (arising from noise or from shifts in ”market sentiment”) would lead to unstable ratings. Credit ratings anchor the expectations of market participants about the equilibrium, thus serving as an antidote to excess volatility in financial markets. 5

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The underlying assumptions of our model are described in Section 2. We start with the analysis of the debt roll-over problem of Morris and Shin (2004) to which

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we add a credit rating as a source of exogenous public information. In Section 3, we

extend the analysis to introduce a financial market a ` la Grossman and Stiglitz (1976) where individuals trade based on their private information and the equilibrium asset

price becomes an endogenous public signal. In Section 4, we formalize the idea that

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financial markets influence CRAs and vice versa. We follow Angeletos and Werning (2006) and let the asset return depend on the outcome of the roll-over problem thus

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creating a feedback loop between investors in financial markets and short-term creditors. This feedback loop creates multiple equilibria, where noise in financial markets and shifts in ”market sentiment” are responsible for unstable ratings. Up to this point, we consider that CRAs merely function as a pass-through entity of new information, thus establishing that sharp downgrades do not depend on any type of strategic behavior by CRAs. In Section 5, we formalize the idea that rating agencies are in a tricky position,

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as they become part of the risk they are trying to assess. We consider the case in which some debt roll-over depends mechanically on credit ratings, show how the CRAs help to sort out the multiple equilibria problem, and examine the procedures used by the

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agencies to lend credibility to their rating strategies. Section 6 concludes.

The failure of credit rating agencies during the 2007 financial

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Literature review

crisis has triggered considerable research on the shortcomings of credit ratings, but the role of financial markets in the determination of credit ratings has been overlooked in the literature. The theoretical literature naturally divides into two camps, dealing respectively with the misaligned incentives of CRAs and with the interaction between credit ratings and investor behavior. In the first camp, models which examine how incentives reduce the quality of the information disclosed to investors are useful (see, for example, Mathis, McAndrews and

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Rochet 2009, Skreta and Veldkamp 2009, Bar-Isaac and Shapiro 2011, Bolton, Freixas and Shapiro 2012, Mariano 2012, Pagano and Volpin 2012, Bar-Isaac and Shapiro

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2013, or Chiu and Koeppl 2015) but it is unlikely that these models fully explain the behavior of credit ratings during financial crises. First, the conditions under which

incentives become misaligned are often ex ante observable and identifiable, so that rational individuals should take corrective measures to offset the distortions in credit

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ratings. Second, the literature focusses on financial crises after 2007, but does not explain why did all CRAs end up with misaligned incentives at the same time. Since

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few papers addressed the incentive problems of CRAs before 2007, the literature did not consider this type of problems as being critical (and it is not obvious why these problems became so important after 2007).1 Third, many incentive problems discussed in the literature are not regarded as central to the failure of sovereign ratings, as most of the information used in the construction of sovereign ratings is public, there is competition among rating agencies, and ”ratings shopping” is not seen as a major

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problem in sovereign debt markets.

Our model falls into the second camp, concerned with the interaction between credit ratings and investment decisions. Existing models in this camp suggest that rating

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agencies are relevant because they help to sort out the multiple equilibria problem which arises from the existence of strategic complementarities among creditor decisions.

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Models in this camp posit that individuals react mechanically to credit ratings. Manso (2013) assumes the borrower uses ratings-based performance sensitive debt (which is a very specific case since, for example, sovereign debt is seldom contingent on ratings). Boot, Milbourn and Schmeits (2006) posit the existence of a set of institutional investors who blindly condition their decisions on the credit rating. We add to the literature by combining endogenous information produced in financial 1 Deregulation and the growing importance of complex securities in the structured finance segment in the early 2000s may have aggravated the incentive problems of CRAs, which may have later been responsible for the ratings failure after 2007. But it is less palatable that such an explanation applies to ratings on sovereign and corporate bonds.

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markets with the business of issuing credit ratings. More specifically, there are four key

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differences with the literature in the second camp:

1. We add a full-fledged financial market to a standard model with rating agencies and strategic complementarities among the decisions of creditors, something that

has not been tried before. For example, Manso (2013) considers an exogenous

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diffusion process for the cash flow of the borrower and gives no explicit role to those financial markets which provide information to market participants. In our

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case, the derivative market produces information to rating agencies and shortterm creditors.

2. On the one hand, the price of the derivative security signals the quality of debt and affects the roll-over decisions of short-term creditors, thus influencing the likelihood of default and the credit rating. On the other hand, the credit rating

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affects the roll-over decisions of short-term creditors, thus influencing the likelihood of default and the price of the derivative security. To the best of our knowledge, this feedback effect from rating agencies to the derivative market is

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new in the literature on credit ratings.

3. Multiplicity arises in the derivative market, and not just as a result of strategic complementarities among creditor decisions as in Boot, Milbourn and Schmeits

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(2006) or in Manso (2013).

4. CRAs play a coordination role new in the literature since credit ratings anchor the expectations about the equilibrium in the derivative market (whereas in Boot, Milbourn and Schmeits 2006, or in Manso 2013, CRAs fix the equilibrium in the coordination game), thus serving as an antidote to excess volatility in this market caused by the existence of multiple equilibria.

Explicitly adding financial markets to a model of credit ratings allows us to trace 8

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the links between market prices and credit ratings, thus providing a number of testable implications about the relationship between prices and ratings which have been absent

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in the theoretical literature. Yet, the empirical literature has examined some of these implications.

A number of empirical papers document features present in our model and which

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reveal the significance of the relationship between market prices and credit ratings, such

as the excessive volatility of sovereign ratings during financial crises (see, for example, Ferri, Liu and Stiglitz 1999, and Mora 2006), and the response of credit ratings to CDS

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markets: Afonso, Furceri and Gomes (2012) find that CDS spreads anticipate sovereign rating announcements in a 1 − 2 week window, and Norden and Weber (2004), Hull, Predescu and White (2004), and Finnerty, Miller and Chen (2013) confirm that CDS markets anticipate corporate rating downgrades. 2

With panel data for the Eurozone countries, Boumparisa, Milas, Panagiotidis (2015)

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suggest that government debt and the cumulative current account exert a stronger positive impact on credit ratings post-2008 compared to the period before, and Kinateder and Wagner (2017) document that changes in credit ratings negatively affect sovereign

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yield spreads.

The basic model with exogenous information

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The model builds on Morris and Shin (2004) and Carlson and Hale (2006). A sovereign government has an outstanding amount of one period debt that equals 1 and is about to mature.3 The government can and is willing to repay an exogenous share θ of this 2

It is hard to prove that changes in market prices unambiguously cause shifts in credit ratings. Since smoothing practices actually make ratings more prone to delayed responses, ratings are naturally slower to signal default risk than financial markets. 3 Although we consider the case of sovereign ratings, the issues we address are common to corporate, municipal, or structured product ratings. Sovereign debt markets are a natural setting for our model because these markets are less opaque, and sovereign ratings are less vulnerable to incentive problems.

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debt—θ may be interpreted as available cash—while the remaining amount of debt (1 − θ) needs to be rolled over. Government debt is held by a continuum of short-term

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creditors indexed by i and uniformly distributed over the [0, 1] interval. Each shortterm creditor individually decides whether or not to roll over his unit of debt, and we R1 define ai ∈ {0, 1} as individual investment. Let A = 0 ai di denote the aggregate level

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of debt roll-over.

The individual return to investment depends on the aggregate level of debt roll-over.

u (ai , A, θ, ψ) =

     

R

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Accordingly, short-term creditors have utility

when ai = 1 and ψ + A + θ ≥ 1

R − Δ when ai = 1 and ψ + A + θ < 1      1 when ai = 0

where R and Δ are constants with 0 < R − 1 < Δ ≤ R. Parameter ψ measures the

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amount of exogenous funding that the government can guarantee. We give the following interpretation to the payoff function. Provided the mass of creditors A is large enough, the government is able to fulfil its promise and repays R; otherwise, the country is forced to default on its debt and repays R − Δ (where Δ measures Loss Given Default,

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LGD). Alternative investment opportunities yield zero interest.

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For simplicity, we assume the borrower’s finances are viable in the long run as long as θ + ψ ≥ 0. If θ + ψ < 0, the country is not solvent. When θ + ψ ≥ 1, the debtor country faces no liquidity problems because the sovereign has enough funds to meet the maturing debt. In the intermediate range 0 ≤ θ + ψ < 1, the fate of the country lies in the hands of its short-term creditors, and it will default if it is unable to convince creditors to roll over their claims.

Information structure

The θ stands for the underlying economic fundamental. The

fundamental θ ∈ R is not known at the time the lending decisions are made, and short10

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term creditors have heterogeneous beliefs about θ. The fundamental θ is drawn from

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an improper uniform prior over the real line. The private information of short-term creditor i is summarized by a sufficient statis-

tic xi = θ + σ x ξ i , where σ x > 0 and ξ i is standard normal, independent of θ and ερ , and independent and identically distributed across agents. All short-term creditors

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have access to the credit rating—the source of public information in this section.

Rating agencies base their analysis on confidential information which borrowers

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agree to share with them. We assume there is one single representative rating agency which privately observes a signal ρ = θ + σ ρ ερ , where σ ρ > 0 and ερ is standard normal and independent of θ. The random variable ερ is associated with errors in the agency signal, and may be interpreted as mistakes resulting from the process of collecting information.

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The information structure is parametrized by the standard deviations σ x and σ ρ −2 or, equivalently, by precision of the signals αx = σ −2 x and αρ = σ ρ . The information

structure and the values of parameters αx and αρ are common knowledge. Private signals introduce heterogeneity in market expectations about the economic fundamental,

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information.

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and may be understood as heterogeneity in the reading and interpretation of public

The objective of the CRA

We abstract away from any conflicts of interest, and

the objective of the CRA is to set an accurate rating that informs investors about the credit quality of the sovereign borrower. Let pCRA be the probability that the sovereign is able to repay its debt, conditional on the information available to the CRA. The CRA ρ on sovereign debt is accurate maps the probability pCRA into rating grades. A rating b

if

b ρ (ρ) ∈ arg min |b ρ − pCRA | b ρ∈[0,1]

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which is equivalent to setting b ρ = pCRA .4 Accordingly, the CRA assigns a ”high” (”low”) rating when it believes that there is a high (low) probability that the sovereign

Equilibrium

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is able to repay its debt.5

There is a one-to-one mapping between the signal ρ and the credit

rating b ρ, so that these variables have identical informational content—it is indifferent

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which one is announced in equilibrium. The individual roll-over decision is a function

of x and ρ, and aggregate investment is a function of θ and ρ. An equilibrium is a

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rating function b ρ (ρ) and a set of individual strategies for investment in public debt

a (x, ρ) which satisfy the following conditions:

• The credit rating is accurate, that is b ρ = pCRA .

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• The decision of an individual short-term creditor maximizes their expected gain a (x, ρ) ∈ arg max E [u (a, A (θ, ρ) , θ, ψ) |x, ρ] . a∈{0,1}

• The aggregate level of investment equals A (θ, ρ) = E [a (x, ρ) |θ, ρ] .

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• The sovereign government defaults if and only if ψ + A (θ, ρ) + θ < 1.

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The above conditions define a Bayesian Nash equilibrium. Let Φ and φ denote, respectively, the standard normal cumulative distribution function and the standard normal density distribution function. 4 Ratings are often used as though they map into specific credit-risk metrics, including the Basel Accord standardized approach. The European Central Bank’s collateral framework has explicitly mapped credit ratings into one-year default probabilities (ECB, 2008). 5 Holden, Natvik and Vigier (2012) use the same equilibrium refinements which we apply to explain investor behavior. Additionally, they posit that reputational concerns penalize rating agencies for making inaccurate predictions and reward them for making accurate ones. Since the final result of the game is either ”default” or ”no default”, agencies have incentives to announce extreme ratings. Their contribution illustrates how the choice of the payoff function can distort the behavior of rating agencies and have a profound impact on the final results. Instead, we consider milder assumptions on the payoff function of rating agencies and assume that agencies want their rating grades to mirror ex post default probabilities.

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Proposition 1 Provided

√

2π ≥ αρ σ x , there exists a unique equilibrium. In this equi-

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librium the country defaults if and only if θ < θ ∗ (ρ), where θ∗ (ρ) is implicitly deter√
mined by θ∗ = Φ σ x αx + αρ Φ−1 1 − R−1 +αρ (θ∗ − ρ)]) − ψ. Δ The inequality condition suffices for the equilibrium to be unique and amounts to saying that public information cannot be too precise, otherwise there are multiple

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equilibria. Equilibrium is described by a critical value θ∗ (ρ) for the fundamental θ, below which the debtor country defaults. The default threshold θ∗ is decreasing in the

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signal ρ, since a favorable rating has a positive effect on roll-over decisions. In the next section we consider the case with endogenous information, and provide the necessary and sufficient conditions for the existence of a unique equilibrium. We thus prove that multiple equilibria does not hinge on the type of information (exogenous

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versus endogenous) revealed to short-term creditors.

Financial markets and endogenous information

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To investigate the role of prices, we extend the analysis of Angeletos and Werning (2006) thus introducing a financial market where agents trade a derivative security and where the rating is issued before playing the coordination game. Naturally, knowing the

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credit rating is useful for the roll-over decisions. Because the return on the derivative depends on the underlying fundamental, the equilibrium price conveys information that is valuable to short-term creditors in the coordination game.

Setup

As before, the fundamental θ is drawn from an improper uniform distribution

over the real line, and each agent i receives the exogenous private signal xi = θ + σ x ξ i . Agents base their analysis on their private signal and on public information (the credit rating and the price). For tractability purposes we separate the investment coordination 13

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game from the derivative market. Agents can be seen as interacting in segmented

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markets and in two consecutive stages. The first stage is a static game in which the CRA and agents in the derivative

market move simultaneously : The CRA discloses the credit rating, and agents trade a

risky security with gross return r at a price p. We adopt the CARA-normal framework

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introduced by Grossman and Stiglitz (1976). 6 The utility of agent i is v (wi ) = −e−γwi

for γ > 0, where wi = w0 + (r − p) ki is the final wealth, w0 is the initial endowment, and ki is investment in the derivative security. The aggregate supply of the security is

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uncertain and not observed, given by KS (εs ) = σ s εs , with σ s > 0, and εs is standard normal, independent of θ and ξ i . This formulation means that the derivative security exists in zero net supply plus some noise—parametrized by σ s —thus preventing a fully revealing equilibrium. 7

The second stage is essentially the same as the model with credit rating agency in the

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previous section: short-term creditors decide whether to invest or not; the sovereign government defaults if and only if ψ + A + θ < 1 and the payoff from this stage is u (ai , A, θ, ψ). The eventual default by the sovereign, the derivative’s return and the

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payoffs from both stages are realized at the end of stage 2.

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6 CDS are traded in over-the-counter (OTC) markets. These markets rely on networks of trading relationships, which imply search-and-bargain features which are not fully captured in the GrossmanStiglitz framework. Although significant for the formation of daily prices, these features are less important in the process of transmission of information to longer term equilibrium prices for the euro area sovereign CDS markets because arbitrage conditions establish a strong link between CDS and sovereign bond prices. Moreover, (i) many dealers quote each CDS every day and these markets are populated by institutional (i.e. well-informed) investors, (ii) advances in electronic trading platforms have improved the trading process in CDS markets (often blurring the distinction between traditional OTC markets and exchanges), and (iii) daily data about the CDS market is readily available from a number of sources and conveys useful information about how credit risk evolves through time. The stylized approach proposed by Grossman and Stiglitz highlights the informational role of financial prices, and is thus a useful starting point for describing and analyzing the feedback effects between the informational roles of prices and credit ratings. 7 Indeed, Stanton and Wallace (2011) suggest that volatility in CDS spreads is often driven by the demand and supply of credit protection, and not by the underlying risk itself.

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Equilibrium

Conditional on the market price p, there is a one-to-one mapping be-

tween the signal ρ and the credit rating b ρ, so that these variables have identical infor-

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mational content. Individual demand of the derivative security is a function of x, p and ρ, the realizations of the private and public signals, and the corresponding aggregate is

a function of θ, p and ρ. The individual roll-over decision is a function of x, p, and ρ and aggregate investment in sovereign debt is a function of θ, p, and ρ. An equilibrium

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is a rating function b ρ (ρ, p), a price function P (θ, εs , ερ ) and individual strategies for

the following conditions:8

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investment in the derivative and in public debt k (x, p, ρ) and a (x, p, ρ) which satisfy

• The credit rating is accurate, b ρ = pCRA .

• The individual demand for the derivative security maximizes the utility of the final wealth of the individual

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k (x, p, ρ) ∈ arg max E [v (w0 + (r − p) k) |x, p, ρ] . k∈R

• The aggregate demand for the derivative security equals K (θ, p, ρ) = E [k (x, p, ρ) |θ, p, ρ] .

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• Market clearing in the derivative market, KS (εs ) = K (θ, P (θ, εs , ερ )) .

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• The decision of an individual short-term creditor maximizes their expected gain a (x, p, ρ) ∈ arg max E [u (a, A (θ, p, ρ) , θ, ψ) |x, p, ρ] . a∈{0,1}

• The aggregate level of investment equals A (θ, p, ρ) = E [a (x, p, ρ) |θ, p, ρ] . • The sovereign government defaults if and only if ψ + A (θ, P (θ, εs , ερ ) , ρ) + θ < 1. 8

We posit that the CRA can condition the rating on the financial price. This should not be taken literally, as it is not fully consistent from a game theoretical perspective; rather, we think this assumption captures in a simple way the idea that rating agencies issue and revise ratings based on information about prices. Still, our specification extends to a simple repeated game with the CRA moving before the agents in the derivatives market, with a perfect Bayesian equilibrium in which the CRA does not have information about contemporaneous prices.

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The above conditions define a rational expectations equilibrium for the first stage— where the equilibrium price and the credit rating are determined—and a perfect Bayesian

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equilibrium for the coordination game in the second stage—where individual roll-over strategies are determined. In this section we consider the more simple case where the

return r depends only on the fundamental θ. In this case there is a one-way causality from the derivative market to the sovereign debt market, as the price of the derivative

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security informs about the quality of sovereign debt. In the next section we consider

effect in the opposite direction.

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that the return depends on the amount of debt roll-over A, thereby adding a feedback

For the sake of simplicity, we assume r = θ.  2 σ and C = σxρ B.

Equilibrium with exogenous return Define B =

γ 2 σ 6x σ 2s γ 2 σ 4x σ 2s σ 2ρ +γ 2 σ 6x σ 2s +σ 2x σ 2ρ

√ Proposition 2 Provided γ 2 σ 2s σ 3x σ 2ρ 2π ≥ γ 2 σ 2s σ 4x + σ 2ρ , there exists a unique equilib-

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rium. In this equilibrium the country defaults if and only if θ < θ ∗ (p, ρ), where θ∗ (p, ρ) is implicitly determined by

    αx 2 4 2 CB R−1 1−C −B + (1 − C) θ∗ − p− ρ −ψ. γ σ x σ s (1 − C − B) Φ−1 1 − Δ 1−B 1−B C

√

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θ∗ = Φ

The model behaves in a similar way to the model with exogenous information pre-

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sented in the previous section. The default threshold θ∗ is decreasing in the price p, since high prices signal a strong fundamental and favor debt roll-over. As in the previous section, we provide the conditions such that multiplicity does not arise in the individual roll-over strategies in the coordination game. This is in sharp contrast with other models of credit ratings where strategic complementarities among creditor decisions are the source of multiple equilibria (such as Boot, Milbourn and Schmeits 2006, or Manso 2013). In the next section we consider the existence of endogenous returns for the derivative security, and show that this is the key assumption 16

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for the existence of multiple equilibria in our framework.

Financial markets and endogenous returns

Since crises are likely to affect short-term returns in derivative securities, we now con-

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sider the case where the return on the derivative security is endogenously determined by

the coordination game (as in Angeletos and Werning, 2006). This framework captures

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in a stylized fashion the return on bond derivatives when a borrower is under stress. The model is the same as in the previous section, except for the endogeneity of the return of the derivative security. More specifically, we define its return as a function of the aggregate level of investment r = r (A), where r (A) = Φ−1 (A) so as to maintain the normality of the information structure. Albeit stylized, this function captures in a convenient way the idea that the value of the derivative security is higher when the

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borrower is clearly able to roll over its debt. The function illustrates the short-term rate of return of holding a CDS written on a sovereign bond. 9

The conditions for equilibrium are as defined in the previous section,

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Equilibrium

and describe a rational expectations equilibrium for the first stage and a perfect Bayesian

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equilibrium for the coordination game in the second stage. Proposition 3 In the economy with endogenous return, the individual roll-over strate√ gies are uniquely determined for a given price realization if 2π (1 − B − CB) ≥ √ αx CB. There are multiple equilibrium prices if σ x σ s σ ρ is sufficiently small, thus creating multiplicity in the coordination game. The endogenous return creates multiple equilibrium prices in the first stage, which 9 Strictly speaking, the derivative security is not a CDS but the results would be equivalent if we had considered this type of insurance contract. We could also have considered a secondary market which trades long-term bonds issued by the sovereign and provides information to market participants.

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then feed into multiplicity in the coordination game in the second stage. Multiplicity arises because a backward-bending demand curve is possible in the economy with

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endogenous return.

The two-way feedback between the return on the derivative security and the roll-

over decisions creates an inverted−S shaped demand curve of the derivative security.

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The intuition is as follows. A higher price of the derivative security signals shortterm creditors that the sovereign is unlikely to face liquidity problems, thus raising individual debt roll-over in the coordination game. Increased aggregate roll-over raises

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the derivative security’s return, thereby raising its demand. Provided either source of noise (i.e. σ x σ s σ ρ ) is small, this effect is strong enough for the aggregate security demand to increase with the price (see details in the Appendix).

The market price performs three distinct roles in the economy with endogenous return. First, it reflects the cost of purchasing the derivative security. Second, it

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aggregates dispersed private information, thus conveying information about the future return of the derivative security. Third, it affects roll-over decisions in the coordination game, thus changing the return of the derivative security in the second stage. This

return.

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third effect is responsible for the price multiplicity in the economy with endogenous

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Figure 1 depicts the case where an inverted−S shaped demand curve meets the supply of the derivative security KS (εs ). The solid vertical supply schedule meets the demand three times, thus generating three equilibrium prices. Equilibrium E1 features a high derivative security price which induces high debt roll-over among shortterm creditors, thus reducing the likelihood of default of the sovereign government and raising the return of the derivative security. Equilibrium E2 features a low asset price and high probability of default. Importantly, multiplicity does not arise in the individual strategies in the coordina-

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Figure 1: Inverted−S shaped demand curve of the derivative security, price multiplicity, and noise.

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tion game (for given price realization). The source of multiplicity lies in the derivative market, which then feeds into the coordination game. In this sense, price multiplicity is essential for equilibrium multiplicity. Multiple equilibrium prices then have obvious

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consequences for the strategy of the CRA.

Corollary 1 Multiplicity in the price function creates multiplicity in credit ratings.

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Given the ratings function b ρ (ρ, p), multiplicity in the equilibrium market price p

feeds into multiplicity in credit ratings. The demand curve of the derivative security

depends on the fundamental θ, so that the solid backward-bending line in Figure 1 depicts the demand curve for a given value of θ. For the same fundamental θ, there are three equilibrium prices which yield different levels of credit quality. The credit rating in equilibrium E1 is higher than the credit rating in equilibrium E2 , although the borrower shares the same fundamental θ and the CRA received the same signal ρ. Despite multiplicity, the ratings function continues to be invertible. Given the 19

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market price p, there is a one-to-one mapping between the signal ρ and the credit

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rating b ρ, so that these variables have the same informational content.

Our result on rating multiplicity is new. Boot, Milbourn and Schmeits (2006)

and Manso (2013) find multiplicity in credit ratings, but the source of multiplicity there is completely different. In these papers, there is no financial market producing

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a price which conveys endogenous information, and there is no endogenous return. Instead, multiple equilibria arises exclusively from strategic complementarities in the

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coordination game.

A consequence of multiplicity is that nonfundamental price volatility arises for two reasons. First, there is the usual instability associated with the existence of multiple equilibria, in which mere sunspots and arbitrary shifts in ”market sentiment” can switch the equilibrium to new credit conditions (for example from equilibrium E1 to

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equilibrium E2 in Figure 1).

Second, a small amount of noise in the supply of the derivative security εs can move the borrower from the favorable equilibrium E1 with high ratings to the depressed equilibrium E3 with low ratings, thus triggering sharp downgrades. The dashed line

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in Figure 1 shows a parallel shift in the asset supply as a result of a shock to εs . Only the depressed asset price equilibrium remains after a high shock ε0s —the positive

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shock to the supply of the derivative security will drive its price downwards to the new equilibrium E3 , which will be interpreted as reduced creditworthiness and will decrease debt roll-over.

The next result presents sufficient conditions for the existence of a unique equilibrium.

Corollary 2 There are critical thresholds ψ (θ, εs , ερ ) and ψ (θ, εs , ερ ) such that there is
a unique equilibrium if the amount of exogenous debt roll-over ψ ∈ / ψ (θ, εs , ερ ) , ψ (θ, εs , ερ ) . 20

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The intuition for this result is as follows: Consider the upper threshold ψ (θ, εs , ερ ) as the reasoning is similar for the lower threshold. As the amount of exogenous funding

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ψ increases, investors in the derivative security conjecture that the sovereign is fac-

ing reduced roll-over risk, thus shifting the aggregate demand rightwards. When the

amount of exogenous funding ψ is sufficiently large, the demand and supply curves cross

Equilibrium selection by credit rating agencies

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5

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only once and there is a unique equilibrium in the market for the derivative security.

We suggest that CRAs set credit ratings which keep the borrowers away from the multiplicity region, thus offsetting the undesirable effects of nonfundamental price volatility. We assume that some short-term creditors make a mechanical use of ratings in their roll-over decisions. We follow Boot, Milbourn and Schmeits (2006) and Manso (2013),

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who suggest that the strategic behavior of CRAs can restore uniqueness of equilibria when creditors react mechanically to credit ratings. Considering the mechanical use of credit ratings is palatable because ratings are

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embedded in many regulations and private contracts. Prudential regulations typically allow for less capital or liquidity to be held against highly rated securities. Central banks use ratings to determine which assets can serve as collateral in their money mar-

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ket operations. Suitability standards, which discipline fund managers by restricting investments to assets with certain risk characteristics, are often based on rating thresholds. Credit ratings are used as triggers for collateral calls in margin agreements in financing transactions. In these ways, ratings drive institutional demand and market liquidity.

Setup and equilibrium

The preceding sections contain descriptions drawn from

individuals’ optimal behavior. In this section we consider an additional set of creditors 21

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who use credit ratings as buy-sell triggers, and we reinterpret ψ as the amount of debt

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roll-over by those investors who use this type of triggers. The model is identical to the model presented in Section 4, except that the exoge-

nous funding ψ is replaced with ψ = ψ (b ρ). We assume ψ 0 > 0, thus capturing in a stylized fashion the mechanical use of ratings by short-term creditors, since part of the

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roll-over decisions in the coordination game depend on the credit rating b ρ.

We use the same definition of equilibrium as in Section 4, except for a modification

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of the condition for sovereign default. With the mechanical use of credit ratings, the sovereign government defaults if and only if ψ (b ρ) + A (θ, P (θ, εs , ερ ) , ρ) + θ < 1. When creditors using buy-sell triggers react strongly to credit ratings, the CRA can restore the uniqueness in equilibrium with an appropriate rating strategy.

Proposition 4 In the economy with endogenous return, unique individual roll-over

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strategies, and mechanical use of credit ratings, the credit rating agency can restore the   uniqueness in the equilibrium price p when ψ (b ρ (p, ρ)) ∈ / ψ (θ, εs , ερ ) , ψ (θ, εs , ερ ) .

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There is a unique equilibrium when the amount of debt roll-over based on triggers is above the critical threshold ψ (θ, εs , ερ ) or below the critical threshold ψ (θ, εs , ερ ).10 The intuition for obtaining uniqueness lies in the ability of the CRA to shift the aggre-

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gate demand of the derivative security when issuing a credit rating. The CRA can ease (or make more difficult) the debt roll-over by ”manipulating” those creditors who use ratings as buy-sell triggers, thus steering the expected return and shifting the demand of the derivative security. In this way, the CRA can achieve a unique equilibrium in the security’s market, thereby stabilizing the credit rating. To fix ideas, consider the case when ψ (b ρ (p, ρ)) > ψ (θ, εs , ερ ) as the intuition for 10

The uniqueness region becomes smaller when some short-term creditors use credit ratings as buysell triggers, that is ψ (θ, εs , ερ ) ≤ ψ (θ, εs , ερ ) and ψ (θ, εs , ερ ) ≤ ψ (θ, εs , ερ ). Details available in the Appendix.

22

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Figure 2: Equilibrium in the market for the derivative security with an intermediate credit rating b ρI (solid inverted−S shaped line) and with a high credit rating b ρH (dashed line). the lower threshold is similar. The CRA issues a credit rating b ρ based on its private

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signal ρ and the equilibrium price p. The solid backward-bending line in Figure 2 plots

the aggregate derivative demand when the CRA issues a intermediate credit rating b ρI

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compatible with the intermediate default probability associated with equilibrium EI .

Consider, instead, that the CRA strategically issues a high rating. Announcing the ρI will entice high debt roll-over ψ (b ρH ) > ψ (b ρI ) by those shorthigh credit rating b ρH > b

term creditors who use buy-sell triggers. Investors in the derivative security conjecture

that the sovereign is facing reduced roll-over risk, and anticipate a higher return from the derivative security. Their individual demand for the derivative increases, thus shifting aggregate demand rightwards. The dashed line in Figure 2 depicts the demand for the derivative when the CRA issues the high credit rating b ρH , thereby leading to a 23

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new equilibrium EH . The high credit rating is accurate as long as b ρH = b ρ (pH , ρ) with pH being the price associated with the low likelihood of default and high credit rating.

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In summary, the reaction of creditors in sovereign debt and investors in derivatives

leads to the new equilibrium EH with reduced likelihood of default, thus vindicating

the initial opinion of the CRA. The condition ψ (b ρ (pH , ρ)) > ψ (θ, εs , ερ ) is key to the strategic move of the CRA as it guarantees that the shift in demand is sufficiently large

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so as to guarantee uniqueness of equilibrium, as portrayed in Figure 2, thus suggesting that the mechanical response to credit ratings must be sufficiently strong.

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As with Boot, Milbourn and Schmeits (2006) and Manso (2013), credit ratings help fix the desired equilibrium when there are multiple equilibria. Agencies can pick the equilibrium if the mechanical reaction to credit ratings is sufficiently strong. Yet, we present a different mechanism since there is no moral hazard nor performance sensitive debt, and multiplicity does not arise in the coordination game. In our case, the CRA

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solves the multiplicity problem which arises in the derivative market.

Implementation of the equilibrium

To implement a specific equilibrium, the CRA

can make an unconditional strategic move (in the sense of Schelling, 1960) so as to alter

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the beliefs and actions of investors in the derivative market and of short term creditors.

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This strategic move must satisfy two conditions. First, the choice of the rating strategy must be observable—financial markets must know in advance the strategy chosen by the CRA—and, second, this choice must be irreversible—the agency must commit not to revoke the credit rating. Irreversibility poses the problem of the credibility, since it is hard to sustain a credit rating which diverges substantially from those appraisals implicit in market prices. To make the rating strategy credible, the CRA must take ancillary actions which lend credibility to the credit rating.

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Although discussing such ancillary actions in detail is beyond the purpose of this contribution, we point out two broad approaches to credibility undertaken by CRAs.

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These consist of devices which deliberately reduce the freedom of action of agencies by creating procedures which slow down the process of adjustment of credit ratings.

First, agencies use a variety of smoothing practices which ensure rating stability

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(Cantor and Mann, 2007) and automatically mute the impact of market volatility on their assessments. Second, credit ratings are determined by a rating committee which takes into account the information presented by a relevant analyst and then forms a

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judgment of the borrower. This judgment introduces stickiness in credit ratings as the size of the adjustments tends to be small, and agencies are generally careful to keep the number of rating changes to a minimum.

As these procedures slow down the process of adjustment of the credit rating, investors in the derivative market know in advance that those creditors who follow me-

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chanic rules will reallocate their funds gradually. Hence, investors do not anticipate sharp changes in debt roll-over, which lends credibility to the credit rating in the short

Conclusion

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6

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term.

Motivated by the correlation between prices and ratings, we developed a framework which combines endogenous information produced in financial markets with the business of issuing credit ratings. To the best of our knowledge, this approach is new in the literature on credit ratings. We focus on the case in which multiple equilibria in the derivative market feeds into multiplicity in the short-term debt market, with each equilibrium associated with a specific likelihood of default. Each equilibrium is associated with a market price, so that

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rating agencies use this information so as to predict default rigorously. With multiple equilibria, nonfundamental price volatility can shift the equilibrium from a favorable

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equilibrium with low CDS spread and high credit rating to a depressed equilibrium with high spread and low rating.

When roll-over decisions from short-term creditors depend mechanically on and

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react strongly to credit ratings, CRAs can use their pronouncements to induce debt roll-

over and ease the borrower’s liquidity problem, and thus fix the equilibrium when there are multiple equilibria. Although we made no normative prescriptions, the message of

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the debt roll-over model of Morris and Shin (2004) is clear. Conditional on positive fundamentals, the equilibrium without default is preferable from the social welfare point of view.

We finish by highlighting two policy concerns raised by our work: First, volatility in financial markets has an indirect effect on regulatory measures when regulation relies on

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credit ratings and, second, there is a policy dilemma when deciding whether regulation should rely on credit ratings or not. Regarding the first concern, government bodies worldwide have resisted using market-based indicators like financial prices as references

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in supervisory practices and regulation. For example, recent proposals to use CDS information to estimate the risk taken by banks have been criticized on the basis that market discipline has limitations when directly applied to regulation as it may unduly

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amplify the effect of noise. Utilizing credit ratings as references has an advantage over utilizing financial prices, as ratings are supposed to provide a stable measure of creditworthiness.11

But our work points out an indirect channel through which instability in financial markets gets transmitted to regulation and supervisory practices (via credit ratings). Since financial prices influence agencies’ pronouncements, references to credit ratings 11

Regulators have worked to eliminate regulatory reliance on credit ratings for financial safety. But it is one thing to identify the weaknesses of credit ratings, quite another to find solutions and alternative standards that are clearly better.

26

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in regulation and rules will then pass the volatility in financial prices on to regulation and rules. In times of stress, this indirect channel may have unintended systemic

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consequences.

Regarding the second concern, our work also raises a key policy dilemma since references to credit ratings in regulation and supervisory practices have opposing effects.

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On the one hand, these references have destabilizing effects when there is a shift from

a good equilibrium to a bad equilibrium and the concomitant downgrades discourage creditors to hold debt, thereby aggravating liquidity problems. On the other hand,

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these references raise the mechanical reaction to credit ratings, thereby raising the ability of CRAs to stabilize financial markets and select virtuous equilibria. 12

References

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[1] Afonso, A., D. Furceri, P. Gomes, 2012. Sovereign Credit Ratings and Financial Market Linkages: Applications to European Data. Journal of International Money and Finance 31, 606-638.

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[2] Angeletos, G., I. Werning, 2006. Crises and Prices: Information Aggregation, Multiplicity, and Volatility. American Economic Review 96, 1720-1736.

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[3] Bar-Isaac, H., J. Shapiro, 2011. Credit Rating Accuracy and Analyst Incentives. American Economic Review 101, 120-124.

[4] Bar-Isaac, H., J. Shapiro, 2013. Ratings Quality over the Business Cycle. Journal of Financial Economics 108, 62-78.

[5] Bolton, P., X. Freixas, J. Shapiro, 2012. The Credit Ratings Game. Journal of Finance 67, 85-111. 12 Replacing credit ratings by market prices is likely to reduce the mechanical response to credit ratings, thereby mitigating their stabilizing effect.

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[6] Boot, A., T. Milbourn, A. Schmeits, 2006. Credit Ratings as Coordination Mech-

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anisms. Review of Financial Studies 19, 81-118. [7] Boumparisa, P., C. Milas, T. Panagiotidis, 2015. Has the Crisis Affected the Be-

havior of the Rating Agencies? Panel Evidence from the Eurozone. Economics Letters 136, 118-124.

Stability. Journal of Fixed Income 16, 60–68.

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[8] Cantor, R., C. Mann, 2007. Analyzing the Tradeoff between Ratings Accuracy and

in Macroeconomics 6, article 8.

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[9] Carlson, M., G. Hale, 2006. Rating Agencies and Sovereign Debt Rollover. Topics

[10] Chiu, J., T. Koeppl, 2015. Livin’ on the Edge with Ratings: Liquidity, Efficiency and Stability. Queen’s Economics Department Working Paper 1335. [11] European Central Bank (ECB), 2008. Technical Specifications for the Temporary

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Expansion of the Central Framework. ECB Press Release, 17 October 2008. [12] Ferri, G., L. Liu, J. Stiglitz, 1999. The Procyclical Role of Rating Agencies: Evidence from the East Asian Crisis. Economic Notes 28(3), 335–55.

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[13] Finnerty, J., C. Miller, R. Chen, 2013. The Impact of Credit Rating Announcements on Credit Default Swap Spreads. Journal of Banking and Finance 37, 2011-

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[14] Grossman, S., J. Stiglitz, 1976. Information and Competitive Price Systems. American Economic Review 66, 246-53.

[15] Holden, S., G. J. Natvik, A. Vigier, 2012. An Equilibrium Model of Credit Rating Agencies. Norges Bank Research Working Paper 2012|23. [16] Holinski, N., C. Kool, J. Muysken, 2012. Persistent Macroeconomic Imbalances in the Euro Area: Causes and Consequences. Federal Reserve Bank of St. Louis Review 94, 1-20. 28

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[17] Hull, J., M. Predescu, A. White, 2004. The Relationship Between Credit Default Swap Spreads, Bond Yields, and Credit Rating Announcements. Journal of Bank-

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ing and Finance 28, 2789-2811.

[18] International Monetary Fund, 2010. Sovereigns, Funding, and Systemic Liquidity. Global Financial Stability Report, October 2010.

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[19] Jorge, J. and J. Rocha, 2015. A Primer on Global Games Applied to Macroeconomics and Finance. Journal of Economic Surveys 29, 869-886.

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[20] Kinateder, H., N. Wagner, 2017. Quantitative Easing and the Pricing of EMU Sovereign Debt. Quarterly Review of Economics and Finance 66, 1-12. [21] Manso, G., 2013. Feedback Effects of Credit Ratings. Journal of Financial Economics 109, 535-548.

[22] Mariano, B., 2012. Market Power and Reputational Concerns in the Ratings In-

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dustry. Journal of Banking and Finance 36, 1616-1626.

[23] Mathis, J., J. McAndrews, J.-C. Rochet, 2009. Rating the Raters: Are Reputation Concerns Powerful Enough to Discipline Rating Agencies? Journal of Monetary

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Economics 56, 657-674.

[24] Mora, N., 2006. Sovereign Credit Ratings: Guilty Beyond Reasonable Doubt?

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Journal of Banking & Finance 30, 2041-2062. [25] Morris, S., H. Shin, 2004. Coordination Risk and The Price of Debt. European Economic Review 48, 133-153.

[26] Norden, L. and M. Weber, 2004. Informational Efficiency of Credit Default Swap and Stock Markets: The Impact of Credit Rating Announcements. Journal of Banking and Finance 28, 2813-2843. [27] Pagano, M., P. Volpin, 2012. Securitization, Transparency and Liquidity. Review of Financial Studies 25, 2417-2453. 29

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[28] Pisani-Ferry, J., G. Wolff, A. Sapir, 2013. EU-IMF assistance to euro area coun-

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tries: an early assessment. Bruegel Blueprint 779. [29] Schelling, T., 1960. The strategy of conflict. Harvard University Press, Cambridge MA.

[30] Skreta, V., L. Veldkamp. 2009. Ratings Shopping and Asset Complexity: A Theory

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of Ratings Inflation. Journal of Monetary Economics 56, 678-695.

[31] Stanton, R., N. Wallace, 2011. The Bear’s Lair: Index Credit Default Swaps and

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the Subprime Mortgage Crisis. Review of Financial Studies 24, 3250-3280.

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A

Appendix

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Throughout the paper we extend the proofs in Jorge and Rocha (2015) so as to include the existence of a credit rating.

Proof of Proposition 1

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A.1

A strategy for short-term creditor i is a decision rule which defines an action conditional

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on the observed signal xi . Throughout the paper, we restrict attention to equilibria in which short-term creditors use switching strategies, in which case creditor i rolls over the loan whenever the private signal xi is higher than some given threshold level x∗ (ρ). Since xi = θ + σ x ξ i , aggregate investment is increasing in the fundamental θ. As a result, there is a threshold for the economic fundamental θ∗ (ρ) below which there is

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default because an insufficient number of short-term creditors chooses to roll over their loans. When θ ≥ θ∗ (ρ) there is no default, because a sufficient mass of short-term creditors receives private signals above x∗ (ρ).

Let probi be the probability that agent i attributes to θ ≥ θ∗ (ρ). The Indiffer-

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ence Threshold Condition states that an agent finds it optimal to invest when the expected return from roll-over is larger than the payoff from the alternative, that is,

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when probi R + (1 − probi ) (R − Δ) ≥ 1. An agent invests if and only if x ≥ x∗ , where x∗ solves probi R + (1 − probi ) (R − Δ) = 1 with probi = 1 − Φ





αρ αx θ − x∗ + ρ αx + αρ αx + αρ ∗

The mass of creditors equals A = 1 − Φ

√



p



αx + αρ .


αx (x∗ (ρ) − θ) . According to the State

Threshold Condition, the sovereign defaults if and only if θ < θ ∗ where θ∗ solves
√ ψ + A + θ∗ = 1, or equivalently θ∗ = Φ αx (x∗ − θ∗ ) − ψ. Solving for θ∗ yields θ∗ (ρ). 31

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√ We have considered a solution in switching strategies. Condition σ 2ρ 2π ≥ σ x is sufficient for there to be a unique dominance-solvable equilibrium and therefore is

A.2

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sufficient for uniqueness of equilibrium in any class of strategies.

Proof of Proposition 2

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Without loss of generality, we assume that the CRA discloses the signal ρ before the asset market pins down the equilibrium price p. In the first stage, we look for a linear

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price function p = (1 − B) θ + Bρ − (1 − B) σ p εs . Observing the price realization is equivalent to observing pe = θ − σ p εs since p = (1 − B) pe + Bρ.

Lemma 1 The posterior of θ conditional on x, ρ and pe is normally distributed with

mean (1 − C − B) pe + Cxi + Bρ and variance σ 2p (1 − C − B).

γ2 2 k V 2 i

ar[θ|xi ,e p,ρ]

. The first order conditions yield an individual

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−e−γw0 −γ(E[θ|xi ,ep,ρ]−p)ki +

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Individuals choose the amount of the risky asset which maximizes their expected   utility E [v (wi ) |xi , p, ρ] = E [−e−γwi |xi , pe, ρ] = E −e−γw0 −γ(θ−p)ki |xi , pe, ρ . Using the   moment-generating function of a normal random variable, one obtains E −e−γw0 −γ(θ−p)ki |xi , pe, ρ = asset demand equal to

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k (xi , pe, ρ) =

E [θ|xi , pe, ρ] − p C (xi − pe) = 2 γV ar [θ|xi , pe, ρ] γσ p (1 − C − B)

and aggregate demand becomes KD (θ, pe) =

Pe (θ, εs ) = θ −

γσ 2p (1−C−B)σ s εs , C

C(θ−e p) . γσ 2p (1−C−B)

Market clearing implies

which verifies the initial guess with σ p =

γσ 2p (1−C−B)σ s , C

and solving for σ p yields σ p = γσ 2x σ s . The second stage is equivalent to the benchmark model in the previous section.

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Compute

−

B 1−B ρ.

According to the Indifference Threshold Condition probi R +

(1 − probi ) (R − Δ) = 1, that is θ =

σ 2p (1

− C − B) Φ

−1



R−1 Δ



+ Cx∗ +

CB 1−C −B p+ ρ. 1−B 1−B

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∗



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p 1−B

∗

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probi

since pe =



θ∗ − [(1 − C − B) pe + Cx∗ + Bρ] = prob [θ ≥ θ |x , pe, ρ] = 1 − Φ σ 2p (1 − C − B) ! CB θ∗ − Cx∗ − 1−C−B 1−B p − 1−B ρ = 1−Φ σ 2p (1 − C − B) ∗

The State Threshold Condition remains the same as in the previous section, and replacing x∗ = θ∗ +

+ ψ) in the above expression yields

    αx 2 R−1 1−C −B CB σ p (1 − C − B) Φ−1 1 − + (1 − C) θ∗ − p− ρ −ψ. C Δ 1−B 1−B

√

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θ∗ = Φ

√1 Φ−1 (θ ∗ αx

The equation has a unique solution as long as the expression on the right-hand side has

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slope lower than one everywhere. Since the slope of the cumulative standard normal √ distribution has a maximum value equal to 1/ 2π, the uniqueness condition becomes 1 1≥ √ 2π

√

√
αx (1 − C) √ ⇔ σ 2p σ 2ρ 2π ≥ αx σ 2p σ 2x + σ 2x σ 2ρ C

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√ and substituting σ p = γσ 2x σ s yields the uniqueness condition γ 2 σ 3x σ 2s σ 2ρ 2π ≥ γ 2 σ 4x σ 2s + σ 2ρ .

A.3

Proof of Proposition 3

As in the previous section, the market clearing price p incorporates information about the signal ρ. We consider equilibria in which creditors roll over their claims if and only if their private signal is above the threshold x∗ (p, ρ), and aggregate investment 33

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to r = Φ−1 (A) =

√ √



√ αx (x∗ (p, ρ) − θ) = Φ αx (θ − x∗ (p, ρ)) with the return equal

αx (θ − x∗ (p, ρ)).

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equals A = 1 − Φ

The optimal individual investment decision in the first stage is k (xi , p, ρ) =

E[r|xi ,p,ρ]−p γV ar[r|xi ,p,ρ] .

It is equivalent to observe a price p of an asset which pays r, or a price pb on and asset which pays θ =

√r αx

+ x∗ (p, ρ). By arbitrage, pb =

√p αx

+ x∗ (p, ρ). Hence, the optimal

√

αx (E [θ|xi , p, ρ] − pb) . γαx V ar [θ|xi , p, ρ]

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E [r|xi , p, ρ] − p k (xi , p, ρ) = = γV ar [r|xi , p, ρ]

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investment decision is

It is equivalent to observe a price pb or a price pe, with pb = (1 − B) pe+ Bρ. We look for a

√ p − x∗ (p, ρ)] = linear price function pe = θ − σ P εs , which is equivalent to p = αx [b √ √ αx [(1 − B) pe + Bρ − x∗ (p, ρ)] = αx [(1 − B) (θ − σ P εs ) + Bρ − x∗ (p, ρ)]. As in σ 2p (1 − C − B) so that

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the previous section, E [θ|xi , p, ρ] = (1 − C − B) pe + Cxi + Bρ and V ar [θ|xi , p, ρ] = k (xi , p, ρ) =

√

√ αx (E [θ|xi , p, ρ] − pb) αx C (xi − pe) = . γαx V ar [θ|xi , p, ρ] γαx σ 2p (1 − C − B) C(θ−e p) √ , γ αx σ 2p (1−C−B) √ γ αx σ 2p (1−C−B)σ s . C

pe = θ −

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Aggregate asset demand is KD (θ, pe) = √ γ αx σ 2p (1−C−B)σ s εs C

with σ p =

and market clearing implies Hence, σ p = γσ x σ s which

AC C

verifies the initial guess.

The second stage is as before, with probi = prob [θ ≥ θ∗ |xi , pe, ρ] = 1−Φ

Replacing pe by pe = probi

√

p αx (1−B)

−

B 1−B ρ

+

x∗ (p,ρ) 1−B



yields



θ ∗ −[(1−C−B)e p+Cx∗ +Bρ] σ 2p (1−C−B)

 θ∗ − [(1 − C − B) pe + Cx∗ + Bρ] = 1−Φ σ 2p (1 − C − B)   (1−C−B)+C(1−B) ∗ CB θ∗ − √1−C−B p − ρ − x 1−B 1−B αx (1−B) . = 1 − Φ σ 2p (1 − C − B) 34



.

ACCEPTED MANUSCRIPT

The Indifference Threshold Condition becomes θ =

σ 2p (1

− C − B) Φ

−1



 1−C −B (1 − C − B) + C (1 − B) ∗ R−1 CB +√ ρ+ x p+ αx (1 − B) Δ 1−B 1−B

and substituting the State Threshold Condition x∗ = θ∗ +

√1 Φ−1 (θ ∗ αx

+ ψ) we obtain

    αx (1 − B) 2 CB R−1 1−C −B + (θ∗ − ρ) − √ σ p (1 − C − B) Φ−1 1 − p −ψ. αx (1 − B) Δ 1−B 1 − B − CB (1)

√

SC

θ∗ = Φ

RI PT

∗

M AN U

There is one single solution to the above equation if and only if

√ √ αx (1 − B) CB √ 1 ⇔ 2π (1 − B − CB) ≥ αx CB. 1≥ √ 2π 1 − B − CB 1 − B Aggregate asset demand becomes

TE D

C (θ − pe) = KD (θ, p) = √ γ αx σ 2p (1 − C − B)

 C θ−

x∗ (p,ρ) p B αx (1−B) + 1−B ρ − 1−B √ γ αx σ 2p (1 − C − B) √



and the slope of the demand equals

EP

   ∗ C 1 1 ∂KD (θ, p) ∂θ =− √ √ + 1+ √ ∗ 2 αx αx φ (θ + ψ) ∂p ∂p γ αx σ p (1 − C − B) (1 − B)

(1),

AC C

by the State Threshold Condition. By the implicit function theorem applied to equation

∂KD (θ, p) C = √ ∂p γ αx σ 2p (1 − C − B) (1 − B)

"

1 1+ √ αx φ (θ∗ + ψ)



1−C−B φ (θ∗ + ψ) 1−B−CB √

α CB

x 1 − φ (θ∗ + ψ) 1−B−CB

Aggregate asset demand is upward sloping when

sign



∂KD (θ, p) ∂p



= sign

"

1 1+ √ αx φ (θ∗ + ψ)

35



1−C−B φ (θ∗ + ψ) 1−B−CB √

α CB

x 1 − φ (θ∗ + ψ) 1−B−CB

1 −√ αx

#

>0

# 1 −√ . αx

ACCEPTED MANUSCRIPT

that is C √ (1 − C) αx

RI PT

φ (θ∗ + ψ) >

which holds when C is small, that is when σ 2p σ 2ρ = γ 2 σ 2x σ 2s σ 2ρ is small.

Proof of Proposition 4

SC

A.4

We proceed by steps.

M AN U

Lemma 2 The individual roll-over strategies are uniquely determined for a given price  √  √
√ realization if 2π (1 − B − CB) ≥ αx CB and σ 2ρ 1 + √α2πx ≥ σ 2p + σ 2ρ (1 − B). Proof. The rating on sovereign debt equals the probability that the actual value of the fundamental θ lies above the threshold θ∗ conditional on p and ρ (the information

TE D

available to the CRA in the first stage),



 b ρ (p, ρ) = prob [θ > θ ∗ |p, ρ] = prob [θ > θ ∗ |e p, ρ] = 1 − Φ  p αx (1−B)



B 1−B ρ

+

x∗ 1−B

EP

√

2  σp ρ

−

+

σ 2ρ

AC C

Recall pe =

b ρ (p, ρ) = Φ  



√

p αx (1−B)

with x∗ = θ∗ +

−

B 1−B ρ

+

θ∗ −



√1 Φ−1 (θ ∗ αx

θ ∗ + √1α Φ−1 (θ ∗ +ψ)

σ 2ρ σ 2p

x

1−B

σ 2p ρ σ 2p +σ 2ρ

+

σ 2ρ σ 2p σ 2p +σ 2ρ



σ 2ρ pe σ 2p +σ 2ρ

 .

+ ψ) so that 

−

σ 2p

+

σ 2ρ




∗



θ  . 

Modify equation (1) so as to incorporate the mechanical response to credit ratings. The CRA chooses the strategy b ρ (p, ρ), so that θ

∗

= Φ

√

    αx (1 − B) 2 R−1 1−C −B CB −1 ∗ 1− p + (θ − ρ) − √ σ p (1 − C − B) Φ αx (1 − B) Δ 1−B 1 − B − CB

−ψ (Φ (E))

(2)

36

ACCEPTED MANUSCRIPT

with +

σ 2ρ



√ p αx (1−B)

−

B 1−B ρ

+

θ ∗ + √1α Φ−1 (θ ∗ +ψ) x

1−B

σ 2ρ σ 2p




− σ 2p + σ 2ρ θ∗

RI PT

E=

σ 2p ρ

.

When compared with the model without mechanical reaction to credit ratings, the uniqueness condition in the coordination problem may become stricter, since the deriva-

"

σ 2ρ 1−B

Condition σ 2ρ

√

2π 1+ √ αx

!

≥

σ 2p

+

σ 2ρ






1 1+ √ αx φ (θ∗ + ψ)



−

σ 2p

M AN U

1 F = −ψ φ (E) 2 2 σρσp 0

SC

tive of the right-hand side now includes

σ 2ρ (1 − B) ⇔ 1−B

TE D

guarantees F ≤ 0 and together with condition

√

√

2π 1+ √ αx

!

+

σ 2ρ




#


− σ 2p + σ 2ρ ≥ 0

2π (1 − B − CB) ≥

√

αx CB they are

sufficient for unique roll-over strategies. Assume

√

2π (1 − B − CB) ≥

√

 αx CB and σ 2ρ 1 +

√  √ 2π αx


≥ σ 2p + σ 2ρ (1 − B), and

EP

consider that fundamentals (and shocks) put the sovereign in the area where the aggregate demand of the derivative is increasing with the price. This equilibrium has an ”intermediate” equilibrium price, which corresponds to ”intermediate” probability of

AC C

default and credit rating.

The issuance of the credit rating affects the behavior of those short-term creditors who react mechanically in the second stage, as ψ = ψ (b ρ). If the CRA announces the ”high” rating b ρH (instead of the ”intermediate” rating), the favorable pronouncement

raises debt roll-over in the second stage, thus increasing the return of the derivative security, thereby shifting the aggregate security demand rightwards.

ρH ) > ψ (θ, εs , ερ ), the shift in There is a threshold ψ (θ, εs , ερ ) such that when ψ (b

37

ACCEPTED MANUSCRIPT

the aggregate demand schedule will be sufficient for the demand schedule to intersect the supply curve only once. When the CRA strategy is explicit about choosing the

RI PT

high rating b ρH , there can be no other equilibrium. A similar reasoning applies to the threshold ψ (θ, εs , ερ ) .

When compared with the impact of the price in the economy with endogenous

SC

return in the previous section, the price of the derivative has a larger impact on roll-

over decisions in the coordination game, thus changing the return of the derivative security in the second stage by more than in the previous section. This effect increases

is positively sloped increases.

M AN U

the area of multiplicity of asset prices, since the size of the area where aggregate demand

AC C

EP

TE D

Corollary 3 ψ (θ, εs , ερ ) ≤ ψ (θ, εs , ερ ) and ψ (θ, εs , ερ ) ≤ ψ (θ, εs , ερ ).

38

ACCEPTED MANUSCRIPT Highlights Model of debt roll-over, with a credit rating agency and a derivative security. Multiplicity in the price of the derivative feeds into multiplicity in ratings. There is an equilibrium with low risk spread and high credit rating. There is an equilibrium with high risk spread and low credit rating. Rating agencies anchor expectations and restore the uniqueness of equilibrium.

AC C

EP

TE D

M AN U

SC

RI PT

    