Financ3

THE JOURNAL OF FINANCE • VOL. LXX, NO. 4 • AUGUST 2015

Transparency in the Financial System: Rollover Risk and Crises

MATTHIEU BOUVARD, PIERRE CHAIGNEAU, and ADOLFO DE MOTTA∗

ABSTRACT

We present a theory of optimal transparency when banks are exposed to rollover risk. Disclosing bank-specific information enhances the stability of the financial system during crises, but has a destabilizing effect in normal economic times. Thus, the reg- ulator optimally increases transparency during crises. Under this policy, however, information disclosure signals a deterioration of economic fundamentals, which gives the regulator ex post incentives to withhold information. This commitment problem precludes a disclosure policy that provides ex ante optimal insurance against aggre- gate shocks, and can result in excess opacity that increases the likelihood of a systemic crisis.

FINANCIAL CRISES ARE OFTEN associated with demands for an increase in the transparency of the financial system. Following the 2008 financial crisis, reg- ulators have periodically performed and disclosed stress tests on the largest financial institutions. This practice has been the subject of much debate, how- ever. For instance, Fed Chairman Ben Bernanke cautioned that “when the stress assessment was getting started, some observers had warned that the assessment and, in particular, the public disclosure of the results might back- fire.”1 Moreover, the accuracy and informativeness of these tests have been questioned, emphasizing the credibility issues that regulators face when ad- justing their disclosure policy. In this paper we develop a model in which the optimal level of transparency is contingent on the state of the financial sys- tem, and show that the regulator faces a commitment problem when trying to implement the optimal transparency policy.

∗Bouvard and de Motta are at McGill University, Desautels Faculty of Management, and Chaigneau is at HEC Montreal. We thank the Editor, Kenneth Singleton, the Associate Editor, and two anonymous referees. We are grateful to Sudipto Bhattacharya, Valentin Haddad, Au- gustin Landier, Frédéric Malherbe, Guillaume Plantin, Viktors Stebunovs, as well as audiences at the University of Geneva, Toulouse School of Economics, HEC Paris, Chicago Booth, McGill Uni- versity, Banque de France, the 2012 European Meeting of the Econometric Society in Malaga, the 1st Oxford Financial Intermediation Theory Conference, the McGill-Todai Market Frictions Con- ference in Tokyo, and the 2012 AFFI international finance meeting in Paris. Matthieu Bouvard acknowledges financial support from the French National Research Agency (Project “Regulation and information sharing”); Adolfo de Motta acknowledges financial support from Institut de Fi- nance Mathématique de Montreal. All errors remain our own.

1 Speech at the Federal Reserve Bank of Chicago, Illinois, on May 6, 2010.

DOI: 10.1111/jofi.12270

1805

1806 The Journal of Finance R©

We consider a stylized model of financial intermediation with rollover risk, in which financial institutions—banks—have exclusive access to a long-term in- vestment technology that is illiquid. Banks are ex ante identical but they differ ex post in the quality of their investment technology and hence in the quality of their balance sheets. Furthermore, the quality of banks in the financial system is affected by aggregate shocks. While investors may have information about these shocks, and therefore about the ex post average quality of banks, they do not observe the idiosyncratic component of each bank’s balance sheet. By contrast, the regulator has access to this bank-specific information and can choose to communicate it to the public.

In the baseline case in which investors observe aggregate shocks and there- fore know the average quality of banks in the financial system, the optimal disclosure policy depends on the realization of these shocks. When the average quality is high enough that investors are willing to roll over their credit, it is optimal not to disclose bank-specific information as transparency may expose lower-quality banks to a run. In contrast, when a negative shock pushes the average quality below a threshold, the regulator switches to transparency, that is, discloses the idiosyncratic component of each bank’s balance sheet. Oth- erwise, if investors knew that the average quality was low but could not tell which banks were of higher relative quality, there would be a run on the entire system.2 This result relies on the properties of the bank run equilibrium, in which the mass of withdrawals is a nonlinear function of a bank’s expected re- turn. Specifically, under good economic conditions, pooling the most profitable banks with a few lower-quality banks has little effect on the rollover risk of the former, while it significantly reduces the probability of runs on the latter.

We turn next to the leading case in which the regulator also has private infor- mation about aggregate shocks to the financial system. In this case, provided that investors receive sufficiently precise yet imperfect information about ag- gregate shocks, the equilibrium disclosure policy is still such that the regulator discloses bank-specific information when the average bank quality falls below a threshold. There is an important economic difference, however: when the regulator has private information about aggregate shocks, investors perceive the absence of information disclosure as a positive signal about the state of the financial system, that is, no news—opacity—conveys good news. This signaling aspect of the disclosure policy, which gives the regulator incentives to withhold information, generates a commitment problem. Specifically, the regulator’s ex post disclosure decision does not internalize that extending the opacity region to worse realizations of the aggregate shock makes investors more likely to run under opacity. As a result, the regulator keeps the system opaque in more states

2 Recent empirical evidence supports the view that stress tests provided useful information to market participants during the recent crises, despite the credibility issues that we discuss below. See, for instance, Peristiani, Morgan, and Savino (2010), Bayazitova and Shivdanasi (2012), Ellahie (2012), and Greenlaw et al. (2012).

Transparency in the Financial System 1807

than is ex ante optimal, which creates instability by increasing the likelihood of a systemic run.3

We also consider the case in which the precision of investors’ information about aggregate shocks is arbitrarily low, which widens the informational ad- vantage of the regulator. The presence of a large informational asymmetry re- inforces the signaling role of the regulator’s disclosure choice and exacerbates the commitment problem. In particular, the regulator may lose the ability to implement a state-contingent disclosure policy that provides insurance against shocks to the financial system.

Finally, we study the case in which the regulator can credibly disclose aggre- gate information without disclosing bank-specific information. The regulator’s private information about aggregate shocks then tends to unravel, which high- lights the fundamentally different economic principles that lead to the disclo- sure of aggregate and bank-specific information. In equilibrium, transparency still increases as fundamentals deteriorate, but there is now a gradation in the release of information: the regulator discloses aggregate information without bank-specific information after a moderate aggregate shock, and discloses both aggregate and bank-specific information after a large negative shock.4

This paper builds on seminal models by Bryant (1980) and Diamond and Dybvig (1983) in which strategic complementarities between depositors may trigger runs and lead to the early liquidation of solvent but illiquid banks. Be- cause these models typically have several equilibria, which makes the impact of public policies difficult to assess, we use the global games approach (Carlsson and van Damme (1993), Morris and Shin (1998), Morris and Shin (2003)) to ob- tain equilibrium uniqueness. Our paper is therefore related to Morris and Shin (2000) and Goldstein and Pauzner (2005), who use global games techniques to refine models of bank runs. Within this literature, we introduce heterogene- ity among banks, which makes the release of bank-specific information by the regulator a relevant issue.5

Our paper belongs to the literature on transparency in the banking sys- tem. (See Landier and Thesmar (2011) and Goldstein and Sapra (2014) for a review of the trade-offs related to transparency in financial systems.) A com- mon theme in this literature is that transparency allows investors to better

3 This commitment problem is consistent with the concerns raised about the leniency of some of the stress tests. For instance, while all banks had passed the 2010 stress test performed by the Committee of European Banking Supervisors, the 2011 stress tests conducted on Irish banks (largely by outside independent advisors) revealed a total capital need of 24 billion. See Schuer- mann (2013).

4 This prediction is consistent with the evolution of stress test disclosures in the United States between 2009 and 2011 (Schuermann (2013)).

5 See also Morris and Shin (2004), Morris and Shin (2009) for models of rollover risk using global games. Eisenbach (2013) also introduces heterogeneity among financial institutions and studies the optimality of short-term debt in the presence of aggregate risk. Plantin (2009) studies bond pricing when investors’ concerns about secondary market liquidity create strategic complementarities among them. Chen, Goldstein, and Jiang (2010) and Hertzberg, Liberti, and Paravisini (2011) provide empirical evidence consistent with the existence of strategic complementarities between investors in financial institutions.

1808 The Journal of Finance R©

monitor financial institutions, thereby enhancing market discipline and im- proving allocation efficiency. This literature points out, however, that informa- tion disclosure can also be associated with important costs. For instance, the release of public information reinforces coordination concerns, which can gen- erate multiple self-fulfilling equilibria (Rochet and Vives (2004)) and may lead market participants to put insufficient weight on private information (Morris and Shin (2002), Angeletos and Pavan (2007)). Moreover, public information crowds out private incentives to acquire information, which may prevent inef- ficient runs (He and Manela (2014)) but can also adversely affect the ability of the government to learn from market prices (Bond, Goldstein, and Prescott (2010), Bond and Goldstein (2014)). We contribute to this literature by showing that the contingent release of bank-specific information can improve the sta- bility of the banking sector by insuring banks against idiosyncratic as well as aggregate shocks, and by studying the regulator’s commitment problem when attempting to implement such an ex ante optimal contingent disclosure policy.6

The paper proceeds as follows. Section I presents the model and analyzes a benchmark case. Section II studies the regulator’s commitment problem. Sec- tion III studies the case in which the regulator can credibly disclose aggregate information without disclosing bank-specific information. Section IV concludes. All proofs are in the Appendices.

I. The Model

A. Description

Consider a risk neutral economy with one consumption good, three periods, t = 0, 1, 2, and no discounting. There is a continuum [0, 1] × [0, 1] of investors, each endowed with one unit of the consumption good. At t = 0 investors can invest their unit in a financial institution or store it. Financial institutions, which we call banks hereafter, have exclusive access to a long-term investment technology. Specifically, bank i’s long-term investment technology generates a gross return of 1 + ri per unit of consumption good at t = 2. Each active bank invests a mass one of the consumption good at t = 0, and there is free entry in the banking industry. Thus, if all investors were to deposit their goods in banks, there would be a continuum [0, 1] of banks each with a continuum [0, 1] of investors. 7

The net return of the long-term technology, ri , is stochastic. Specifically, ri = μ + ηi , where μ is a common parameter that captures the expected return

6 This central role of idiosyncratic information is reminiscent of Hirshleifer (1971), who provides a model in which the early knowledge of future realizations of uncertainty prevents individuals from sharing risk efficiently through transactions. The Hirshleifer effect underlies the analysis of Goldstein and Leitner (2013), who study the impact of information disclosure on the ability of banks to share risk in financial markets. The signaling role of the regulator’s policy in a banking system is related to Shapiro and Skeie (2014).

7 While we refer to the continuum [0, 1] of banks as the financial or banking system, it can be interpreted more generally as a subset of banks within the banking system that investors perceive as homogeneous.

Transparency in the Financial System 1809

of the banking sector, and ηi is a bank-specific component that captures the relative quality of bank i. The bank-specific component, ηi , can take values �η > 0 (for high-quality banks) and −�η (for low-quality banks) with probability p and 1 − p, respectively. The proportion p of high-quality banks is drawn from a uniform distribution on (0, 1), so that, at t = 0, E0(ηi ) = 0 for all i. The realization of p at t = 1 is interpreted as an aggregate shock to the expected return of the banking system, μ.

Investors who invest in banks at t = 0 can either leave their good in the bank or withdraw it at t = 1. Thus, banks face rollover risk and the possibility of early liquidation. Liquidation is costly because the technology is illiquid: if a proportion li of the resources invested in the long-term technology is withdrawn at t = 1, the per-unit return at t = 2 is reduced to ri − cli . For instance, if half of the bank’s investors withdraw, each of them gets one unit of the consumption good back at t = 1 and the other half gets 1 + ri − c2 at t = 2. This investment technology is similar to that in Morris and Shin (2000) and, in essence, models rollover risk as a coordination problem among investors.

We assume that early liquidation is inefficient, that is, the net expected return of the long-term technology is greater than zero, and that, when banks’ types are disclosed, low-quality banks face early liquidation while high-quality banks do not. As we show in the next subsection, this boils down to the following assumption on the parameters of the model:8

0 < −�η + μ < c 2

< �η + μ. (1)

The financial system is supervised by a regulator who has access to bank- specific information. That is, at t = 1, before investors make their withdrawal decisions, the regulator learns {ηi}i∈[0,1]. Investors cannot directly observe these idiosyncratic shocks, however. Hence, at t = 1, their belief E1(ηi ) about the re- turn of each bank in the system depends on the regulator’s disclosure policy. If the regulator chooses transparency, investors perfectly distinguish high-quality banks from low-quality ones: E1(ηi ) = ηi for all i. If the regulator chooses opac- ity, that is, if he does not disclose bank-specific information, E1(ηi ) depends on the information investors have at t = 1 about the aggregate shock p. Regarding p, we assume that, at t = 1, after the regulator has made a disclosure decision but before making their rollover decisions, investors observe

z = p + u, (2)

where u is uniformly distributed on [− δ2 , + δ2 ]. The parameter δ ≥ 0 measures the degree of information asymmetry about p between the regulator and

8 This assumption allows us to focus our analysis on the more interesting cases. In the Internet Appendix, available at The Journal of Finance website, we consider a case with both efficient and inefficient runs and show that the optimal disclosure policy is similar to that in Proposition 1, under reasonable assumptions about the impact of negative shocks on the distribution of returns across banks (i.e., a monotone likelihood ratio property).

1810 The Journal of Finance R©

t = 0

Investors invest in a

bank or in a storage

technology.

t = 1

1. {ηi}i∈[0,1] is realized.

2. The regulator makes disclosure decision.

3. Investors observe z and make rollover de-

cisions.

t = 2

Banks’ returns are

realized and distributed

to investors who rolled

over their investment.

Figure 1. Timeline of the model.

investors. 9 Finally, the objective of the regulator is to maximize total out- put, that is, the sum of what investors who withdraw early receive at t = 1 and what investors who roll over their investment receive at t = 2. Notice that this objective is equivalent to maximizing the aggregate expected return of the banking system or maximizing total consumption.10 Figure 1 summarizes the timeline of the model.

We finish the presentation of the model by discussing two assumptions that we maintain throughout the analysis. First, we assume that investors have the right to withdraw at t = 1, that is, financial institutions borrow short-term and face rollover risk. A feature of the 2008 financial crisis was the credit market freeze that led to the collapse of financial institutions that relied on the rollover of short-term debt in the asset-backed commercial paper and overnight secured repo markets (Acharya, Gale, and Yorulmazer (2011), Gorton and Metrick (2012)). Thus, our paper studies the optimal level of transparency in the financial system given the presence of rollover risk, that is, under the implicit assumption that, while banks may try to ameliorate rollover risk, they will not be able to eliminate it. From a theoretical point of view, rollover risk can be micro-founded through depositors’ demands for insurance against id- iosyncratic liquidity shocks as in Diamond and Dybvig (1983).11

Second, the analysis assumes that banks cannot credibly disclose their in- formation while the regulator can. That banks cannot credibly disclose soft information is straightforward: low-quality banks do not have incentives to disclose information that would lead them to cease operations due to a credit run. However, the regulator is concerned with the long-term viability of the

9 Note that the regulator learns p—the proportion of high-quality banks—perfectly at t = 1, since he observes the quality of each individual bank, {ηi }i∈[0,1].

10 In the model, the only alternative to investing with banks is the storage technology that delivers a net return of zero. Therefore, investors who withdraw at t = 1 consume one unit of the consumption good, regardless of whether they consume at t = 1 or store their unit and consume at t = 2.

11 Goldstein and Pauzner (2005) show that short-term debt, which provides risk-sharing bene- fits, is desirable even when its destabilizing effect is taken into account. See also Barnea, Haugen, and Senbet (1980), Calomiris and Kahn (1991), and Stein (2011) for micro-foundations of short- term debt.

Transparency in the Financial System 1811

entire banking system and not just a single bank, and thus has stronger incen- tives to disclose information truthfully when he chooses a transparent regime. Indeed, when disclosing {ηi}i∈[0,1], the regulator communicates relative perfor- mance, whereas individual banks can only communicate absolute performance. Chakraborty and Harbaugh (2007, 2010) show that an expert with information on multiple variables may be able to credibly communicate a ranking of these variables in cases in which communication about a single variable is impossible. Intuitively, comparative statements have the property of being simultaneously positive along one dimension and negative along another dimension.

B. Rollover Equilibrium

Deriving the regulator’s equilibrium disclosure policy requires first charac- terizing investors’ rollover decisions at t = 1. Consider an investor who invests in bank i at t = 0. At t = 1, he can either withdraw his unit of consumption good or roll over and receive a random payoff 1 + ri − cli at t = 2. Hence, his willingness to withdraw depends on li , that is, on the withdrawal decisions of all other investors in bank i. As in Diamond and Dybvig (1983), these strategic complementarities lead to multiple equilibria. In one equilibrium, investors roll over and banks’ assets pay off at t = 2. In another equilibrium, investors demand early withdrawal (i.e., run) causing fundamentally solvent banks to terminate productive investments and eventually to fail.

We resolve this equilibrium indeterminacy using standard global games tech- niques (Carlsson and van Damme (1993), Morris and Shin (1998)). Specifically, we perturb the model by transforming the parameter that governs the ex ante expected return of the financial system μ into a normally distributed ran- dom variable μ̃ with mean μ and precision hμ. We then introduce dispersed information among investors by assuming that each investor j receives a sig- nal sj = μ̃ + ε j between t = 0 and t = 1, where ε j is normally distributed with mean zero and precision hε, and is independent across investors. In this setup, if the precision of the signal, h� , is small relative to the precision of the ex ante distribution of μ̃, hμ, then there exists a unique threshold equilibrium. More- over, a well-known property of global games is that uniqueness carries over to the limit in which signals become infinitely precise. In that case, the model with dispersed private information converges to the original model in which μ is common knowledge. However, the rollover equilibrium remains unique and has investors in bank i running when μ + E1(ηi ) is below c2 . 12 That is, as uncertainty about μ vanishes, the global game refinement selects the bank run equilibrium when μ + E1(ηi ) is smaller than c2 , and the rollover equilib- rium when μ + E1(ηi ) is greater than c2 . Note that the properties of this equi- librium are economically sensible: bank runs are more likely when a bank’s expected return, μ + E1(ηi ), is lower, and when the strength of the negative

12 To simplify the exposition we assume that investors do not run at the margin. See Appendix B for a detailed analysis of the rollover game.

1812 The Journal of Finance R©

externality that early withdrawals exert on investors who roll over, c, is higher. The following lemma summarizes the above discussion.

LEMMA 1: At t = 1, every investor in bank i runs if and only if μ + E1(ηi ) < c2 . Equipped with this result, consider investors’ decisions to roll over under a

transparent regime. In that case, investors perfectly learn the quality of each bank: E1(ηi ) = ηi . Hence, from Lemma 1 and the assumption in (1), investors in bank i run if their bank is of low quality, ηi = −�η, and roll over if it is of high quality, ηi = �η. Turn now to the case in which the system is opaque. Investors then perceive all banks as of average quality: E1(ηi ) = [2E1( p) − 1]�η, where E1( p) is investors’ expectation of p given their information at t = 1. Hence, from Lemma 1, investors run under opacity if and only if E1( p) is smaller than a threshold p� defined as

p� ≡ 1 2�η

( c 2

− μ )

+ 1 2

. (3)

Notice that the assumption in (1) implies that p� ∈ (0, 1). Next, we use this equilibrium characterization of the rollover subgame to study the regulator’s disclosure decision.

C. Benchmark: Common Knowledge about Aggregate Shocks

Before analyzing the general case in which the regulator has private infor- mation about the aggregate shock p, it is useful to build intuition with the simpler case in which p is common knowledge, that is, δ = 0. Since in that case E1( p) = p, it follows from Lemma 1 and the discussion thereafter that, if p < p�, transparency dominates opacity as opacity would generate a run on every bank. Conversely, if p ≥ p�, opacity prevents any runs and dominates transparency. The next proposition states this result.

PROPOSITION 1: If δ = 0, the regulator follows a policy of transparency if and only if p is below a threshold p�.

The threshold disclosure policy in Proposition 1 relies on the properties of the bank run equilibrium, in which the mass of withdrawals is a nonlinear function of a bank’s expected return. Specifically, under good economic condi- tions, pooling the most profitable banks with a few lower-quality banks has little effect on the rollover risk of the former, while it significantly reduces the probability of runs on the latter. This pooling equilibrium without bank runs is sustainable as long as the proportion of high-quality banks in the financial system, p, is large enough. Otherwise, pooling leads to runs on all banks (both high- and low-quality).

A contingent disclosure policy where transparency increases in bad times is consistent with the heightened disclosure of information via stress tests during the recent financial crisis. Governor D. Tarullo from the Federal Reserve Board of Governors has argued that the publication of stress tests performed on U.S. banks helped stabilize the financial system during the crisis in 2008

Transparency in the Financial System 1813

and 2009.13 Recent empirical evidence also supports the view that stress tests provided useful information to market participants, despite some credibility issues (particularly in Europe, as discussed in Section II below). For instance, Ellahie (2012) finds that information asymmetry declined for tested banks following the disclosure of the 2011 stress test results in Europe, and Peristiani, Morgan, and Savino (2010) document that, although the market had correctly anticipated which banks would have a capital gap, the U.S. Supervisory Capital Assessment Program (SCAP) was informative about the size of the gap.14

We conclude this first analysis by briefly discussing the assumption that dis- closure policy involves either full transparency or full opacity. If the proportion of high-quality banks is below p�, then in principle the regulator could disclose information (and cause a run) on a subset of low-quality banks only, rather than disclose information on all banks and cause a run on every low-quality bank. Notice that, under this alternative disclosure rule, there would still be an increase in disclosure during financial crises and the purpose of disclosure would again be to prevent a run on the whole banking system. However, this policy of selective disclosure would give additional discretion to the regulator, a regulator that, as we show in the next section, faces a commitment problem.15

II. Private Information about Aggregate Shocks: The Regulator’s Commitment Problem

We now turn to the general case in which the regulator has superior infor- mation about the aggregate shock p. Our main result shows that, while the regulator may still be able to implement a p-contingent transparency policy (as in Proposition 1), the presence of asymmetric information about p creates a commitment problem. This commitment problem can generate excess opacity and increase the likelihood of a systemic run.

A. Excess Opacity and Systemic Runs

While it is natural to assume that a regulator who gathers bank-specific information also acquires superior information about aggregate shocks to the financial system, investors can also learn about aggregate shocks from a wide

13 “This departure from the standard practice of keeping examination information confidential was based on the belief that greater transparency of the process and findings would help restore confidence in U.S. banks at a time of great uncertainty.” (Keynote speech at the Federal Reserve Board International Research Forum on Monetary Policy, Washington, D.C., March 26, 2010.)

14 Bayazitova and Shivdanasi (2012) also find evidence of a certification effect from the U.S. SCAP in 2009.

15 We also consider an extension of the model in which investors observe an imperfect public signal about the quality ηi of each bank, and the regulator can make his disclosure policy contingent on this public information. In that case, information disclosure still increases as fundamentals deteriorate, but for moderate negative aggregate shocks disclosure is restricted to a subset of banks that investors perceive as vulnerable, in an attempt to prevent runs on some of these banks. See the Internet Appendix.

1814 The Journal of Finance R©

variety of sources: the stock market, consumption and investment data, unem- ployment figures, etc. Hence, we assume that the regulator has an informa- tional advantage over investors regarding p, that is, δ > 0, but we focus on the case in which investors still learn about p with a certain degree of precision:

δ < min{ p�, 1 − p�}. (4)

Condition (4) has the following economic interpretation: δ < 1 − p� guarantees the existence of a p-region at the top of which runs do not occur under opacity. Indeed, if p > p� + δ, even the smallest possible signal that investors can re- ceive, z = p − δ2 , reveals that p is larger than p�, that is, p − δ2 > p� + δ2 . Notice that this upper-dominance region ( p� + δ, 1) can be arbitrarily small provided it exists. That is, condition (4) only requires the existence of states of the econ- omy, however unlikely, in which the fundamentals are strong enough that the financial system does not face the risk of a systemic run. Symmetrically, δ < p�

guarantees that, for small enough realizations of p, investors can infer from z that p is smaller than p�. Again, this lower-dominance region (0, p� − δ) can be arbitrarily small. While we consider the case in which condition (4) is verified as the most economically relevant, in Section II.B, we also analyze the case in which information asymmetries about p can be arbitrarily large.

The following proposition characterizes the optimal disclosure policy and constitutes the main result of the paper.

PROPOSITION 2: If 0 < δ < min{ p�, 1 − p�}, there is a unique equilibrium. In this equilibrium, the regulator follows a policy of transparency if and only if p is below a threshold pNC , where 0 < pNC < p�. This equilibrium exhibits excess opacity: if the regulator could commit at t = 0, he would follow a policy of transparency if and only if p is below a threshold pC , where pNC < pC < p�.

The equilibrium characterization in Proposition 2 has three components: (i) the equilibrium is unique, (ii) it has a threshold form, and (iii) it is ex ante inefficient. We postpone the discussion of equilibrium uniqueness, and start with the threshold nature of the equilibrium.

The form of equilibrium disclosure policy in Proposition 2 is similar to the one in Proposition 1, in that disclosure takes places when p is below a threshold. There is an important economic difference, however: the regulator now has in- centives to use opacity as a positive signal on p, and therefore keeps the system more opaque than when p is common knowledge ( pNC < p�), which implies that systemic runs can now occur under opacity in equilibrium. Indeed, if the regu- lator chose the same disclosure threshold p� as in Proposition 1, then opacity would be a perfect signal that p ≥ p� and therefore investors would never run under opacity. However, the regulator would then have an incentive to deviate to opacity for some realizations of p just below p�. This deviation would be de- tected only with a small probability and, if undetected, would prevent runs on low-quality banks. It follows that pNC is below p�, which in turn implies that, if z is small enough, investors’ expectation of p given z and opacity can be smaller than p�. In equilibrium, this happens when z is smaller than an endogenous

Transparency in the Financial System 1815

Figure 2. Disclosure policy and rollover decisions in equilibrium.

threshold z�NC , and generates a systemic run. Hence, the regulator’s equilib- rium strategy strikes a trade-off between a transparent regime in which runs always happen but are confined to low-quality banks, and an opaque regime in which runs are less frequent but affect the entire banking system follow- ing a bad realization of z. When p is high, the probability of a systemic run under opacity, Pr[z < z�NC | p], is small, which makes opacity more attractive than transparency. As p decreases, systemic runs become more likely under opacity, until a threshold pNC below which the regulator prefers switching to transparency.

We now turn to the question of ex ante efficiency. From Proposition 2, if the regulator could commit to a disclosure policy at t = 0, disclosure would still take place when p is below a threshold, but that threshold pC would be strictly higher than pNC . That is, the lack of commitment ability results in opacity being chosen ex post in regions where it is ex ante optimal to be transparent. This tendency toward excess opacity is due to the signaling role of the equilibrium disclosure policy: because opacity conveys that p is above a threshold, that is, because “no news is good news,” the regulator has an ex post incentive to lower the disclosure threshold from pC to pNC , and use opacity to protect low-quality banks. However, when lowering the disclosure down to pNC , the regulator does not internalize that expanding the opacity region toward lower realizations of p weakens the positive news associated with opacity and makes investors more likely to run under opacity. This commitment problem is illustrated in Figure 2: ex post, the regulator does not factor in that increasing the disclosure threshold from pNC to pC would reduce the threshold below which investors run under opacity from z�NC to z

� C , which would strictly reduce the probability

1816 The Journal of Finance R©

of a systemic run for every p between pC and z�NC + δ2 . Hence, the regulator’s excessive tendency to withhold information creates instability in the financial system by increasing the probability of a systemic run.16

Proposition 2 also shows that the ex ante optimal disclosure policy, while less opaque than the equilibrium disclosure policy, is still more opaque than in the symmetric information benchmark: pC < p�. Intuitively, because investors do not perfectly observe p, there can now be pooling not only between high- and low-quality banks, but also across different realizations of p. That is, by setting pC below p�, the regulator is able to provide insurance against idiosyncratic shocks as well as against moderate aggregate shocks.

Next, we turn to the question of equilibrium uniqueness. Note that equi- librium uniqueness is not straightforward a priori, as signaling games are prone to equilibrium multiplicity. In fact, we show in the next subsection that when the information asymmetry about p becomes arbitrarily large, that is, when condition (4) is not verified, multiple equilibria can arise. Note also that equilibrium uniqueness in Proposition 2 does not rely on any constraint on out- of-equilibrium beliefs other that the requirement that an investor’s posterior about p be consistent with the signal z.17 In the next three paragraphs, we provide a sketch of the derivation of the unique equilibrium in Proposition 2.

The proof is in three steps. Let O denote the set of realizations of p for which the regulator chooses opacity. (Hence, O characterizes the regulator’s strat- egy.) The first step shows that we can characterize the form of investors’ equi- librium strategy under opacity without specifying the form of the regulator’s equilibrium strategy O. Specifically, if O is an equilibrium strategy, investors withdraw under opacity if and only if the signal z is below a threshold z�(O). Intuitively, in equilibrium, since a higher z is more likely to come from higher realizations of p, investors’ posterior belief about p is increasing in z for any given O. Therefore, in an equilibrium where investors anticipate a strategy O, there exists a unique threshold z�(O) such that E( p|O, z) < p� (and investors run) if and only if z < z�(O).

The second step shows that, if investors follow an equilibrium threshold strategy z�, then the regulator’s equilibrium strategy is also a threshold strat- egy, where the regulator chooses opacity if and only if p ≥ pT (z�) for some pT (z�) ∈ (0, 1), that is, O = [ pT (z∗), 1). As previously discussed, for every p, the regulator chooses between an opaque regime that generates a systemic run with probability Pr[z < z�| p], and a transparent regime that always generates a run on low-quality banks. The existence of upper- and lower-dominance regions (i.e., δ < min{ p�, 1 − p�}) ensures that, for p sufficiently high, Pr[z < z�| p] = 0, and for p sufficiently low, Pr[z < z�| p] = 1. Therefore, since Pr[z < z�| p] strictly

16 There is some evidence pointing to the European regulatory authorities withholding infor- mation during the crisis. For instance, the 2011 stress tests on Irish banks conducted largely by outside independent advisors (BlackRock) revealed a total capital need of 24 billion while all banks had previously passed the 2010 stress test performed by the Committee of European Banking Su- pervisors. See Schuermann (2013).

17 The support of investors’ posterior belief is a subset of [z − δ2 , z + δ2 ], as in Angeletos, Hellwig, and Pavan (2006).

Transparency in the Financial System 1817

increases when p decreases, there is a threshold pT (z�), strictly between zero and one, at which the regulator is indifferent between opacity and transparency.

The third step imposes rational expectations, and shows that there is a unique threshold disclosure policy O = [ pT (z�), 1) that is consistent with in- vestors’ strategy z�(O). Note that z�([ pT , 1)) is decreasing in pT : when pT is high, opacity sends a strong positive signal about p, which makes investors unlikely to run under opacity (i.e., when pT is high, z∗([ pT , 1)) is low). As a result, if pT is too high, the regulator has an incentive to deviate to opacity for p just below pT . Therefore, opacity expands from the top ( pT decreases), causing z∗([ pT , 1)) to rise and increasing the probability of a systemic run un- der opacity. When pT reaches pNC , the regulator is exactly indifferent between opacity and transparency, that is, pNC is the only disclosure threshold such that pT (z�([ pNC , 1))) = pNC .18,19

Note that underlying our main result in Proposition 2 is the assumption that the information asymmetry between the regulator and investors regarding the aggregate shock p is not too large: δ < min{ p�, 1 − p�}. Next, we show that relaxing this constraint can exacerbate the regulator’s commitment problem.

B. Large Information Asymmetry about Aggregate Shocks

As we discuss in Section II.B, investors can learn about aggregate shocks from a wide variety of sources, which makes the case in which the informa- tion asymmetry about p is not too large (Proposition 2) the most economically relevant. However, studying the case in which δ is unbounded helps a further understanding of the regulator’s commitment problem.

To build intuition, consider the limit case in which investors’ signal is com- pletely uninformative: δ → +∞. Recalling that μ is investors’ unconditional expectation of banks’ return, and c captures the strength of the negative exter- nality that early withdrawals exert on investors who roll over, the equilibrium disclosure policy is then as follows:

CLAIM 1: If δ → +∞, there is no equilibrium with a p-contingent transparency policy. There always exists a fully transparent equilibrium, and, if μ ≥ c2 , there also exists a more efficient equilibrium that is fully opaque.

From the above claim, a fully transparent equilibrium always exists: it is supported by investors’ out-of-equilibrium belief that the expectation of p con- ditional on observing opacity is strictly below p�. Under this out-of-equilibrium belief, the regulator does not deviate to opacity as opacity would trigger a

18 Specifically, z�([ pT , 1)) is decreasing in pT and pT (z�) is increasing z�. Hence, pT (z�([ p, 1))) is decreasing in p, which, together with continuity and the fact that pT (z�([ p, 1))) ∈ (0, 1) (from step 2), guarantees the existence of a unique fixed point pNC (i.e., pT (z�([ p, 1))) crosses the 45-degree line only once).

19 Here we only discuss the derivation of the equilibrium disclosure policy. Since we have already provided the intuition for the commitment problem, we refer the reader to Appendix A for the derivation of the ex ante optimal disclosure policy.

1818 The Journal of Finance R©

systemic run.20 Further, suppose that, in equilibrium, there is a p-region in which the regulator chooses opacity. Then, investors’ expectation of p con- ditional on observing opacity must be higher than p�. Otherwise, investors would run on every bank under opacity and the regulator would be better off switching to transparency. However, if investors do not run under opacity, then the regulator has an ex post incentive to choose opacity for any realization of p. Therefore, the only possible equilibrium featuring some opacity is, in fact, a fully opaque equilibrium. Since full opacity does not convey any information about p, this equilibrium exists if and only if investors’ unconditional expecta- tion of banks’ return, μ, is high enough.

The case in which δ → +∞ illustrates an extreme manifestation of the regu- lator’s commitment problem. When investors do not receive direct information on p, their behavior under opacity (in and out of equilibrium) is driven solely by their belief about the equilibrium disclosure policy. As a result, if investors believe that opacity is a negative signal about p, the regulator is “trapped” into a fully transparent regime. Conversely, in a candidate equilibrium in which opacity is a positive signal about p, the regulator has ex post incentives to withhold information for every p, which may eventually destroy equilibria with opacity. In other words, the regulator may need the discipline of a more precise investor signal z to curb his natural incentive to overexploit opacity and therefore to provide some insurance against aggregate and idiosyncratic shocks.21

Finally, the line of reasoning developed above is not specific to the extreme case in which δ → +∞. For δ large enough, that is, when investors receive information on p but information is very imprecise, fully opaque and fully transparent equilibria may coexist, as formally stated in the next proposition.

PROPOSITION 3: If δ > 1 − p�, there always exists a fully transparent equilib- rium. In addition, when μ ≥ c2 , there also exists a fully opaque equilibrium if δ >

μ+�η μ−�η .

C. Discussion

Our analysis in this section is related to Angeletos, Hellwig, and Pavan (2006), who show that a policy maker’s action can signal aggregate information in a coordination game between agents, in a similar way that the disclosure policy conveys aggregate information in the current paper. The equilibrium multiplicity we obtain in Proposition 3 is reminiscent of the self-fulfilling “policy

20 This out-of-equilibrium belief survives the intuitive criterion since there is no type p for which transparency is equilibrium dominated. It also survives D1: the set of investors’ response for which the regulator benefits from a deviation is the same for every type p.

21 Note that a fully transparent equilibrium is obviously ex ante inefficient: if the regulator could commit to the disclosure policy in Proposition 1 at t = 0, all banks would survive when p ≥ p� even under asymmetric information, as opacity would signal that p is higher than p�. One can show that the optimal ex ante policy in this case actually involves opacity above a threshold that is strictly lower than p�.

Transparency in the Financial System 1819

traps” that constitute the focus of their paper. However, Angeletos, Hellwig, and Pavan (2006) feature a signaling game in which the policy maker can take a costly action—a policy—that directly affects the agents’ payoff from taking an action that is detrimental to the policy maker, such as a currency attack or a bank run. By contrast, the regulator’s action in the current paper, disclosing information, is costless and only affects investors’ expected payoffs through an information channel. Moreover, because our game is one of information disclosure, it allows us to study the different economic principles that lead to the release of aggregate and bank-specific information.

The central role of idiosyncratic information also distinguishes our paper from earlier contributions on the role of public information in coordination games. As in Angeletos and Pavan (2007), in our model agents (investors) ex- ert payoff externalities on each other, and therefore information can have pos- itive or negative social value. However, in Angeletos and Pavan (2007) agents learn about a common aggregate component and hence public information is socially valuable to the extent that it brings agents closer to the socially opti- mal coordination level (see also Morris and Shin (2002) and Metz (2002)). By contrast, in our model agents learn about a continuum of idiosyncratic compo- nents and hence information is simultaneously positive in one dimension and negative in another. As a result, information disclosure leads some investors to coordinate on the socially efficient action (roll over) and other investors to coordinate on the socially inefficient one (withdraw). The overall social value of information depends on the relative importance of these two effects. This cen- tral role of idiosyncratic information is consistent with the design of the 2009 U.S. stress tests (SCAP). Indeed, as Hirtle, Schuermann, and Stiroh (2009) and Peristiani, Morgan, and Savino (2010) point out, SCAP was especially suited to determine banks’ relative value due to its horizontalism: banks were subject to simultaneous examination using the same underlying assumptions about economic conditions and loan losses, and the same quantitative techniques. In the next section, we further examine differences in the disclosure of aggregate and idiosyncratic information.

III. Disclosure of Aggregate Information

Our main result in Proposition 2 assumes that the regulator cannot credibly communicate aggregate information p without also disclosing bank-specific in- formation {ηi}i∈[0,1]. This section considers the case in which the disclosure of aggregate information can be decoupled from the disclosure of bank-specific in- formation. Specifically, we assume that the regulator (who still lacks the ability to commit to a disclosure policy) can credibly disclose p—the proportion of high- quality banks—without disclosing {ηi}i∈[0,1]. While credibility issues can make selective disclosure difficult to implement in practice, this case highlights dif- ferences in the incentives to disclose aggregate and idiosyncratic information. The next proposition characterizes the equilibrium disclosure policy.

1820 The Journal of Finance R©

PROPOSITION 4: If δ > 0, there is a unique threshold pCr > p� such that, in any equilibrium, the regulator discloses {ηi}i∈[0,1] if and only if p < p�, discloses p if p� ≤ p < pCr , and discloses p or follows opacity if p ≥ pCr .

Proposition 4 indicates that, when the regulator can credibly disclose p, the optimal disclosure policy produces the same economic outcome, namely, runs on low-quality banks if p < p�, as when p is common knowledge, even if δ is arbitrarily large. Indeed, when p ∈ [ p�, pCr ), the regulator has an ex post incentive to disclose p to avoid being pooled with lower p types. From an ex ante point of view, however, this means that there cannot be any pooling across realizations of p. This is another manifestation of the commitment problem that the regulator faces when he possesses private information about p: while the regulator still has the ability to retain bank-specific information and provide insurance across banks for p ≥ p�, the unraveling of aggregate information now prevents him from providing any insurance across different states of the economy (i.e., realizations of p).22

While the optimal disclosure policy in Proposition 4 is still such that trans- parency increases as fundamentals deteriorate, notice that there is now a gra- dation in the release of information: the regulator discloses aggregate informa- tion without bank-specific information after a moderate aggregate shock, and discloses both aggregate and bank-specific information after a large negative shock. This gradation in information release is consistent with the fact that the U.S. stress test of 2009 (SCAP), which was undertaken in the middle of the fi- nancial crisis, disclosed more information than the U.S. stress test of 2011, the Comprehensive Capital Analysis and Review (CCAR). More importantly, while SCAP disclosed bank-specific information, CCAR disclosed the macro scenario but no bank-level results (see Schuermann (2013)).

To gain intuition for the proof of Proposition 4, note first that, if p < p�, transparency strictly dominates disclosing p only (it preserves high-quality banks), and, if p ≥ p�, disclosing p only strictly dominates transparency (it preserves low-quality banks). Consider now a counterfactual equilibrium in which the regulator chooses opacity (i.e., discloses neither bank types nor p) in a region below p�. Within this region, the probability that investors roll over must be strictly positive as otherwise transparency (i.e., disclosing bank types) would be preferable. Specifically, for any p in the opaque region below p�, there must be (high enough) realizations of z such that investors believe the signal is likely to come from an opaque region above p�. This belief requires that the set of signals that can be generated from the opaque region below p� overlaps with the set of signals that can be generated from an opaque region above p�. This in turn implies that, for the smallest p in the opaque region above p�, there can be (low enough) realizations of z that investors believe the signal is likely to come from the opaque region below p� and consequently run. But this constitutes a contradiction since, for any p above p�, the regulator can avoid

22 Note that the disclosure policy in Proposition 4 (i.e., transparency below p�) is strictly domi- nated by the ex ante optimal policy in Proposition 2 (i.e., transparency below pC ).

Transparency in the Financial System 1821

runs by disclosing p. Therefore, in any equilibrium the system is transparent below p�.

Next, equilibrium transparency below p� requires the existence of an in- termediate region [ p�, pCr ) where the regulator discloses p without disclosing bank-specific information. Specifically, pCr is such that, if the system were fully opaque for some p ∈ [ p�, pCr ), then a deviation to opacity would be profitable for some p below p�: it would be detected with a small enough probability, and when undetected would preserve the entire system. By contrast, if the system is opaque only above pCr , then, for any p below p�, the probability that the sig- nal z could be interpreted as coming from an opaque region above pCr is small enough to deter a deviation to opacity. As a result, if p ≥ pCr , the regulator can follow opacity or disclose p, both of which generate the same economic outcome of no runs.

In Proposition 4 the regulator can credibly communicate aggregate informa- tion without disclosing bank-specific information, while in Proposition 2 it can- not. In practice, regulators also differ in their ability to credibly disclose aggre- gate information. For instance, Freixas and Laux (2012), Greenlaw et al. (2012), and Schuermann (2013) indicate that European regulatory authorities faced greater credibility issues than U.S. authorities during the recent financial cri- sis. When such credibility issues arise, it might be necessary to expand the scope of information release and disclose bank-specific details (as in Proposition 2) to convince investors. For example, the stress tests conducted exclusively on Irish banks in 2011 had a much higher degree of disclosure than the previ- ous 2009 and 2010 tests conducted at the European level. Similarly, the 2011 Europe-wide stress tests conducted by the European Banking Authority (EBA) also disclosed more information than the earlier European tests. Schuermann (2013) argues that this level of detail was necessary to make information re- lease credible.23

Overall, Propositions 2 and 4 together suggest that the general pattern in which transparency increases after negative aggregate shocks is robust to dif- ferent assumptions on the ability of the regulator to credibly disclose aggregate information without disclosing bank-specific information. Moreover, in both cases regulators’ commitment problem prevents the implementation of an ex ante optimal disclosure policy.

IV. Conclusion

The 2008 financial crisis has triggered demand for an increase in the trans- parency of the financial system. Regulatory authorities in Europe and the United States have attempted to enhance transparency by performing stress

23 “This is precisely what was done in the Irish bank stress test of 2011, an acute case of loss of confidence (and subsequent regaining), as well as the 2011 EBA stress test. Because credibility of European supervisors was rather low by that point, only with very detailed disclosure, bank by bank, of their exposures by asset class, by country, by maturity bucket, could the market do its own math and arrive at its own conclusions.” (Schuermann (2013)).

1822 The Journal of Finance R©

tests on banks and releasing their results to investors. One stated objective of these measures, to prevent a contagion of investor distrust to the entire system by providing information on the specific risk exposure of each financial insti- tution, is consistent with the idea that the banking crisis was due in part to a run on the liability side of banks’ balance sheets. In line with this view, in this paper we study the risk faced by solvent but potentially illiquid financial insti- tutions. We show that, when banks are exposed to rollover risk, the disclosure of bank-specific information following a negative aggregate shock does indeed increase the stability of the financial system. However, while transparency should ideally be contingent on the state of the financial system, the analy- sis emphasizes that regulatory authorities face a commitment problem when implementing such a state-contingent policy. In particular, under this policy, an increase in transparency signals a deterioration of economic fundamentals, which gives the regulator ex post incentives to manipulate investors’ beliefs by retaining information. As a result, the regulator lacks the ability to fully exploit his private information through an ex ante optimal disclosure policy.

While this paper focuses on a regulatory measure that has been central in the recent debate on the reform of the financial system, namely, transparency, reg- ulators have several instruments at their disposal to cope with liquidity crises. Among them, the provision of liquidity by central banks or governments, acting as lenders of last resort, has been an emergency recourse for financial insti- tutions during the recent credit crisis. There can be interesting interactions between public provision of liquidity to the banking system and transparency. In particular, to the extent that the regulator faces a trade-off between the size and frequency of bank runs when choosing a transparency regime, disclosure policy is likely to have an effect on the magnitude of the liquidity shock that a government or central bank would have to withstand in times of crisis in order to keep the financial system afloat. Moreover, disclosing bank-specific information implies putting pressure on the financial system’s weakest insti- tutions in an attempt to release pressure from the fittest ones. This suggests that, following information disclosure, the regulator should be ready to inter- vene. For instance, Greenlaw et al. (2012) argue that “in order for a stress test to alleviate concerns that deleveraging is imminent, the test must either produce a credible private capital-raising program or be accompanied with an adequate government backstop to make sure that wholesale creditors see no risk of suffering losses.” These are interesting avenues for future research on how to build a more stable financial system.

Initial submission: December 3, 2012; Final version received: February 10, 2015 Editor: Kenneth Singleton

Appendix A: Proof of Proposition 2

For clarity, we devote Appendix A to the proof of Proposition 2, which is divided into two parts: we first study the case in which the regulator cannot

Transparency in the Financial System 1823

commit to a disclosure policy, and then turn to the optimal ex ante policy. Appendix B contains the other proofs.

Part 1: Equilibrium Transparency Policy without Commitment

Suppose the regulator chooses a disclosure policy at t = 1, after observing the realization of p. We start by introducing notation. Let O denote the set of p’s for which the regulator chooses opacity (O therefore defines the regulator’s strategy). We can split O into two distinct sets:

O− ≡ O ∩ (0, p�) and O+ ≡ O ∩ [ p�, 1), (A1)

that is, O+ (O−) is the set of p’s above (below) the run threshold p� such that the system is opaque. We denote by Z the set of signals under opacity that are consistent with the regulator playing strategy O:

Z ≡ {

z : O ∩ [

z − δ 2

, z + δ 2

] �= {∅}

} . (A2)

In words, for any z ∈ Z, there exists a p ∈ O that can generate z. Hence, when observing z ∈ Z under opacity, investors can infer that p ∈ O. The proof of the proposition builds on the following lemma.

LEMMA A1: If δ < 1 − p�, then O+ and O− are nonempty in equilibrium, and there exists a threshold z� such that investors’ strategy under opacity is to with- draw if and only if z < z�.

PROOF OF LEMMA A1: We prove the three components of this lemma in three steps. Q.E.D.

1. O+ is nonempty. For any p in [ p� + δ, 1), even under the lowest possible signal, p − δ2 , investors assign probability one to p ≥ p�. Hence, under opacity the probability of a run is zero and choosing opacity is optimal.

2. O− is nonempty. Suppose that O− is empty. Then, for any p ∈ O+, the prob- ability of a run is zero under opacity (i.e., if O− is empty, then a policy of opacity reveals that p ≥ p�) and the net benefit of opacity (i.e., avoiding runs on low-quality banks) is strictly positive, that is, (1 − p)(μ − �η) > 0. At p = inf O+ − ε, for some small ε > 0, the benefit of deviating to opac- ity is weakly greater than (1 − ε

δ )(1 − inf {O+} + ε)(μ − �η) − εδ (inf {O+} −

ε)(μ + �η). (Intuitively, if the regulator deviates to opacity, either the de- viation is not detected (z ∈ Z ), which happens with probability 1 − ε

δ ,

and the deviation has the net benefit of saving low-quality banks, or the deviation is detected with probability ε

δ and then in the worst case there

is a run on all banks, so that the net cost of the deviation is the liquida- tion of high-quality banks.) This expression is strictly positive for ε small enough, and hence at p = inf {O+} − ε opacity is a profitable deviation, a contradiction.

1824 The Journal of Finance R©

3. There exists a z� such that all investors withdraw under opacity if and only if z < z�. Let p(z) : Z → (0, 1) be investors’ expectation of p given a signal z and opacity

p(z) ≡ E {

p| p ∈ O ∩ [

z − δ 2

, z + δ 2

]} . (A3)

The expectation p(z) is weakly increasing in Z. To see this, consider z′ < z. If z′ + δ < z, then obviously p(z′) < p(z). (If z′ + δ2 < z − δ2 , any p that can generate a signal z is higher than any p′ that can generate z′.) If z′ + δ ≥ z, then

p(z′) = E {

p| p ∈ O ∩ [

z′ − δ 2

, z′ + δ 2

]} ≤ E

{ p| p ∈ O ∩

[ z − δ

2 , z′ + δ

2

]} ≤ E

{ p| p ∈ O ∩

[ z − δ

2 , z + δ

2

]} = p(z).

(A4)

Let z� ≡ inf {z ∈ Z : p(z) ≥ p�}, that is, z� is the lower bound of the set of signals such that there is no run under opacity. Note that, since O+ is nonempty, this set is nonempty as well. Therefore, z� is well defined. �

The previous lemma implies that in any equilibrium there is a z� such that the probability of a run on all banks when the regulator chooses opacity for a given p is Pr(z < z�| p). Next, we show that in any equilibrium the regulator chooses opacity iff p ≥ pT for some pT . Consider an equilibrium strategy O. From Lemma A1, there exists a z� such that a run occurs under opacity iff z < z�. Since z ≥ p� + δ2 perfectly reveals p ≥ p�, it must be the case that z� ≤ p� + δ2 . Similarly, since z < p� − δ2 perfectly reveals p < p�, it must be the case that z� ≥ p� − δ2 .

Define the following function for p ∈ (0, 1): B( p) = Pr(z ≥ z�| p)[μ + (2 p − 1)�η] − p(μ + �η). (A5)

The function B( p) represents the net benefit of opacity if the strategy of in- vestors is to run under opacity iff z < z�. Hence, B( p) ≥ 0 is a necessary condi- tion for p ∈ O.

The function B( p) has the following properties:

1. For p ≥ z� + δ2 , Pr(z ≥ z�| p) = 1 and B( p) = (1 − p)(μ − �η), which is pos- itive and linearly decreasing in p. Since z� ≤ p� + δ2 and p� < 1 − δ, the interval [z� + δ2 , 1) is not empty.

2. For p ≤ z� − δ2 , Pr(z ≥ z�| p) = 0 and B( p) = − p(μ + �η), which is nega- tive and linearly decreasing in p. Since z� ≥ p� − δ2 and δ < p�, the inter- val (0, z� − δ2 ] is not empty.

Transparency in the Financial System 1825

3. For z� − δ2 < p < z� + δ2 , Pr(z ≥ z�| p) = 12 + p−z�

δ ∈ (0, 1), and hence B( p)

is continuous and strictly convex,

B′′( p) = 1 δ

4�η. (A6)

These observations together imply that there exists a unique pT ∈ (0, 1) such that B( pT ) = 0, and ∀ p ∈ (0, 1), B( p) > 0 iff p > pT . Since B( p) ≥ 0 is a neces- sary condition for p ∈ O, then for p < pT the regulator chooses transparency in equilibrium. (Note: B( p) ≥ 0 is a necessary condition for p ∈ O because B( p) < 0 implies that transparency is a profitable deviation if p ∈ O, a contradiction.)

Next we show that ( pT ,1) ⊂ O. Suppose otherwise, that is, that there exists a subset D ⊂ ( pT , 1) such that D ∩ O = {∅}. Then, since [ p� + δ, 1) ⊂ O (i.e., opacity is strictly optimal in [ p� + δ, 1)), there exists a p and an ε > 0 arbitrarily small such that ( p, 1) ⊂ O and p − ε ∈ D. Consider the incentive of type p − ε to deviate to opacity. Since ( p, 1) ⊂ O, a sufficient condition for such a deviation not to be detected is z > p − δ2 (i.e., for investors, any signal in [ p − δ2 , p − ε + δ2 ] could originate from p). In addition, in the event of detection, the worst possible outcome is a run on the entire system, in which case the net cost of opacity is ( p − ε)(μ + �η) (the liquidation of high-quality banks). Therefore, the net benefit of deviating to opacity at p − ε is bounded below by

Pr (

z ≥ max {

z�, p − δ 2

} | p − ε

) [μ + (2 p − ε − 1)�η] − ( p − ε)(μ + �η). (A7)

The expression in (A7) tends to B( p) > 0 as ε → 0, and hence it is strictly posi- tive for ε small. (Note that B( p) > 0 for all p ∈ ( pT ,1).) Therefore, there exists a type p − ε ∈ D for which the regulator has a profitable deviation to opacity, a contradiction. Hence, the system must be opaque for p > pT . Note finally that, at pT , the regulator is indifferent between opacity and transparency. Indeed, B( pT ) = 0 is the net benefit of opacity if pT ∈ O, but also if pT /∈ O because, since ( pT , 1) ⊂ O, the probability that a deviation to transparency is detected is zero at pT . Since p = pT is a zero-probability event, and the regulator is indifferent between policies, we simply assume that the regulator follows a policy of opacity at pT.

So far we have shown that in any equilibrium there is a z� and a correspond- ing pT such that the regulator follows a policy of transparency iff p < pT . That is, we have shown that any equilibrium is a threshold equilibrium. Next, we show that the pair (z�, pT ) is uniquely defined.

Note first that in principle, from the definition of z� in Lemma A1., it could be the case that

p(z�) = E {

p| p ∈ O ∩ [

z� − δ 2

, z� + δ 2

]} > p�. (A8)

1826 The Journal of Finance R©

However, O = [ pT , 1) and B( pT ) = 0 imply that p(z) is continuous on Z = [ pT − δ2 , 1 + δ2 ], strictly increasing, and such that p(.) changes sign on Z. There- fore, in equilibrium,

p(z�) = E {

p| p ∈ [

pT , z� + δ 2

]} = p�, (A9)

which is equivalent to

pT + ( z� + δ2

) 2

= p� ⇔ z� = 2 p� − pT − δ 2

. (A10)

Therefore,

Pr ( z > z�| pT

) = 1

2 + p

T − z� δ

= 1 − 2 δ

( p� − pT

) . (A11)

Plugging this expression into (A5) yields the following equilibrium require- ment, which pins down pT :

f ( pT ) ≡ [ 1 − 2

δ

( p� − pT

)] [μ + (2 pT − 1)�η] − pT (μ + �η) = 0. (A12)

From Lemma A1, O− is not empty, which by definition implies that pT < p�. Moreover, f ( pT ) is strictly convex, f ( p�) > 0 , and f ( p) < 0 ∀ p ∈ (0, p� − δ2 ]. Therefore, f ( pT ) has a unique root pNC (δ) in (0, p�) and p� > pNC > p� − δ2 .

Part 2: Optimal Ex Ante Transparency Policy

We turn now to the case in which the regulator can commit to a p- transparency policy at t = 0. From Lemma A1, for any strategy O such that O+ is nonempty, there exists a (signal) threshold z∗(O) such that investors withdraw if and only if their signal z is strictly below z∗(O).

Assume that strategy O is ex ante optimal. We first show that O+ is nonempty. If O+ were empty, investors would always

withdraw from low-quality banks so that O would be strictly dominated by O′ = [ p�, 1), where investors never withdraw from high-quality banks and withdraw from low-quality banks if and only if p < p�.

Since O+ is nonempty, the threshold z∗(O) is well defined and for any p ∈ (0, 1), the net benefit of opacity is

B( p, O) = Pr(z ≥ z∗(O)| p)[μ + (2 p − 1)�η] − p(μ + �η). (A13) Since O is ex ante optimal,

O ∈ argmax Õ

∫ p∈Õ

B( p, Õ)dp. (A14)

Transparency in the Financial System 1827

Using the same reasoning as in Part 1 of the proof, there exists a unique pT (O) in (0, 1) such that B[ pT (O), O] = 0, B( p, O) > 0 if p > pT (O), and B( p, O) < 0 if p < pT (O) . This has a series of implications:

1. The system is transparent almost everywhere in (0, pT (O)). Suppose that S ≡ (0, pT (O)) ∩ O has positive measure. Switching to transparency for p ∈ S both strictly increases welfare for every p ∈ S (since B( p, O) < 0) and (weakly) decreases the probability of a run for every p ∈ O ∩ [ pT (O), 1), a contradiction.

2. pT (O) ≤ p�. Suppose that pT (O) > p�. Then, given that the system is transparent almost everywhere below pT (O), O is strictly dominated by O′ ≡ [ p�, 1). Therefore, O is not ex ante optimal, a contradiction.

3. The system is opaque almost everywhere in [ p�, 1). Suppose that the system is transparent for a set S ⊂ [ p�, 1) of positive measure. Consider the alternative strategy O′ ≡ O ∪ S. By the law of iterated expectations,

E

[ p| p ∈

[ z∗(O) − δ

2 , z∗(O) + δ

2

] ∩ O′

] (A15)

= qE [

p| p ∈ [

z∗(O) − δ 2

, z∗(O) + δ 2

] ∩ O

] + (1 − q)E

[ p| p ∈

[ z∗(O) − δ

2 , z∗(O) + δ

2

] ∩ S

]

where q = ∫

p∈[z∗(O)− δ2 ,z∗(O)+ δ2 ]∩O dp∫

p∈[z∗(O)− δ2 ,z∗(O)+ δ2 ]∩O′ dp

.

Since by the definition of z∗(O), E [ p| p ∈ [z∗(O) − δ2 , z∗(O) + δ2 ] ∩ O

] ≥ p�,

and S ⊂ [ p�, 1),

E

[ p| p ∈

[ z∗(O) − δ

2 , z∗(O) + δ

2

] ∩ O′

] ≥ p�, (A16)

and therefore z∗(O′) ≤ z∗(O). This implies in turn that, for any p ∈ (0, 1), B( p, O′) ≥ B( p, O). Finally, since pT (O) ≤ p� and S ⊂ [ p�, 1), for any p ∈ S, B( p, O) ≥ 0 and hence B( p, O′) ≥ 0, with strict inequalities (i.e., B( p, O) > 0 and B( p, O′) > 0) for S \ p�. Therefore, since S has positive measure, it follows that

∫ p∈S B( p, O

′)dp > 0 and∫ p∈O′

B( p, O′)dp = ∫

p∈O B( p, O′)dp +

∫ p∈S

B( p, O′)dp > ∫

p∈O B( p, O)dp,

(A17)

which implies that O is not ex ante optimal, a contradiction.

1828 The Journal of Finance R©

4. There exists pC such that the system is opaque almost everywhere on [ pC , 1) and transparent almost everywhere on (0, pC ). Suppose such pC does not exist. Then there exist two intervals ( pa, pb) and ( pc, pd) such that pa < pb ≤ pc < pd, ( pa, pb) ⊂ O, ( pc, pd) ∩ O is empty, and pb − pa = pd − pc. Note that

- From implications 1 and 3 above, pT (O) ≤ pa < pd ≤ p�. - z∗(O) + δ2 ≥ p� by the definition of z∗(O), and pT (O) + δ2 ≥ z∗(O) by

the definition of pT (O). - Hence, z∗(O) + δ2 ≥ pd and pa ≥ z∗(O) − δ2 , and therefore ( pc, pd) ⊂

[z∗(O) − δ2 , z∗(O) + δ2 ] and ( pa, pb) ⊂ [z∗(O) − δ2 , z∗(O) + δ2 ]. - Finally, z∗(O) + δ2 ≥ p� implies that B( p, O) is strictly increasing on

[ pT (O), p�]. Consider the strategy O′ ≡ O ∪ ( pc, pd) \ ( pa, pb). Using conditional ex- pectations, and noticing that O ∪ O′ can be written as the union of the two disjoint sets O and ( pc, pd) or as the union of the two disjoint sets O′

and ( pa, pb), we have

(1 − q)E [

p| p ∈ [

z∗(O) − δ 2

, z∗(O) + δ 2

] ∩ O′

] + qE

[ p| p ∈ ( pa, pb)

] = (A18)

(1 − q)E [

p| p ∈ [

z∗(O) − δ 2

, z∗(O) + δ 2

] ∩ O

] + qE

[ p| p ∈ ( pc, pd)

] ,

where

q = pb − pa∫ p∈[z∗(O)− δ2 ,z∗(O)+ δ2 ]∩(O∪O′)

dp = pd − pc∫

p∈[z∗(O)− δ2 ,z∗(O)+ δ2 ]∩(O∪O′) dp

. (A19)

Since E [ p| p ∈ ( pa, pb)

] < E

[ p| p ∈ ( pc, pd)

] ,

E

[ p| p ∈

[ z∗(O) − δ

2 , z∗(O) + δ

2

] ∩ O′

] > E

[ p| p ∈

[ z∗(O) − δ

2 , z∗(O) + δ

2

] ∩ O

] ≥ p�, (A20)

which in turn implies z∗(O′) ≤ z∗(O). Hence, for any p ∈ (0, 1), B( p, O′) ≥ B( p, O). Furthermore, the strict monotonicity of B( p, O) on [ pT (O), p�] implies

∫ p∈( pa, pb)

B( p, O)dp < ∫

p∈( pc , pd) B( p, O)dp. (A21)

Transparency in the Financial System 1829

Therefore,∫ p∈O

B( p, O)dp = ∫

p∈O′ B( p, O)dp +

∫ p∈( pa, pb)

B( p, O)dp − ∫

p∈( pc , pd) B( p, O)dp

<

∫ p∈O′

B( p, O)dp (A22)

≤ ∫

p∈O′ B( p, O′)dp,

which implies that O is not optimal, a contradiction. 5. pNC < pC < p�.

Notice that pT (O) ≥ z∗(O) − δ2 and (from implication 1 above) pC ≥ pT (O) imply pC ≥ z∗(O) − δ2 . Notice also that z∗(O) − δ2 ≤ p� and p� < 1 − δ im- ply z∗(O) + δ2 < 1. Therefore,

E

{ p| p ∈

[ z∗(O) − δ

2 , z∗(O) + δ

2

] ∩ [ pC , 1)

} = p

C + z∗(O) + δ2 2

. (A23)

Equating this expression to p� yields

z∗(O) = 2 p� − δ 2

− pC . (A24)

If p ≥ 2 p� − pC , then z ≥ z∗(O), and depositors never run. If p ∈ [ pC , 2 p� − pC ] (which is nonempty since pC ≤ p�), then the probability of a run conditional on p is

Pr ( z < z∗(O)| p

) = Pr

( u > z∗(O) − p| p

) =

∫ δ 2

z∗(O)− p

1 δ

du = 1 − 2 p � − pC − p

δ . (A25)

The regulator’s objective function is then∫ pC 0

p(μ + �η)dp + ∫ 2 p�− pC

pC

( 1 − 2 p

� − pC − p δ

) [ μ + (2 p − 1)�η

] dp

+ ∫ 1

2 p�− pC [ μ + (2 p − 1)�η

] dp. (A26)

Differentiating with respect to pC yields

pC (μ + �η) − (

1 − 2 δ

( p� − pC ) ) [

μ + (2 pC − 1)�η ]

+ ∫ 2 p�− pC

pC

1 δ

[ μ + (2 p − 1)�η

] dp. (A27)

1830 The Journal of Finance R©

The second-order derivative with respect to pC is

(μ − �η) − 2 δ

[ μ + (2 pC − 1)�η

] < −(μ − �η) < 0, (A28)

where the last inequality uses δ < 1. From (A28) the problem is convex, and therefore the optimal pC equalizes (A27) to zero. From (A12) pNC satisfies(

1 − 2 δ

( p� − pNC ) ) [

μ + (2 pNC − 1)�η ] − pNC (μ + �) = 0, (A29)

and hence evaluating (A27) at pC = pNC yields∫ 2 p�− pNC pNC

1 δ

[ μ + (2 p − 1)�η

] dp > 0, (A30)

where the last inequality uses pNC < p�. Evaluating (A27) at pC = p� yields

p�(μ + �η) − [ μ + (2 p� − 1)�η

] < 0. (A31)

It follows that pNC < pC < p�. �

Appendix B: Other Proofs

PROOF OF LEMMA 1: Let ρ j denote investor j’s expectation of μ̃ conditional on sj , Q.E.D.

ρ j ≡ hμμ + hεsj

hμ + hε . (B1)

Conjecture an equilibrium in which investor j in bank i rolls over if and only if ρ j ≥ ρ�i . At the threshold ρ�i , an investor must be indifferent between rolling over and withdrawing:

ρ � i + E1(ηi − cli |ρ�i ) = 0, (B2)

where E1(.|ρ j ) denotes investor j’s expectation given his private information about μ̃, as captured by the posterior belief ρ j . Equation (B2) is equivalent to

ρ � i + E1(ηi ) = c

[√ γ (ρ�i − μ)

] , where γ ≡

h2μ(hμ + h� ) h� (hμ + 2h� )

. (B3)

If c2γ < 2π , there is only one solution to (B3), and hence a unique equilibrium in threshold strategies. This threshold equilibrium is in fact the unique equi- librium. The proof by iterated deletion of dominated strategies is standard and omitted (see Morris and Shin (2000)).

Transparency in the Financial System 1831

Finally, if hε → +∞, then, for any j, ρ j → μ̃ and since γ → 0, ρ�i → c2 − E1(ηi ). Hence, the probability of a run becomes Pr[μ̃ < c2 − E1(ηi )]. If, in addition, hμ → +∞ and γ still tends to zero, which is guaranteed by h

2 μ

hε → 0, then the

probability of a run tends to one if μ < c2 − E1(ηi ) and to zero if μ > c2 − E1(ηi ) (see Morris and Shin (2004)). �

PROOF OF CLAIM 1: We prove first that there cannot be a p-contingent disclo- sure policy. Suppose there exists a nonempty subset O ⊂ [0, 1], in which the regulator chooses opacity. Subset O must be such that

E[ p| p ∈ O] ≥ p�. (B4) Indeed, if E[ p| p ∈ O] < p�, Bayesian updating would lead investors to run on the entire system, and the regulator would then choose transparency for any p ∈ O. Notice next that, for any p /∈ O, switching from transparency to opacity increases total expected surplus from p(μ + �η) to p(μ + �η) + (1 − p)(μ − �η). Therefore, if O �= {∅}, O = [0, 1].

Full disclosure is always an equilibrium. It is sustained by (a continuum of) out-of-equilibrium beliefs that p < p� conditional on opacity.

Full opacity can be an equilibrium if and only if

μ + E[2 p − 1]�η ≥ c 2

⇔ μ ≥ c 2

. (B5)

�

PROOF OF PROPOSITION 3: We first show that, if δ ≥ 1 − p�, there exists an equi- librium with full transparency. Assume that δ ≥ 1 − p�. Consider the following off-path beliefs in an equilibrium with full transparency (F T ):

E( p|z, F T ) = min {

z − δ 2

, 0 }

, (B6)

where E( p|z, F T ) is defined as the posterior of p given both signal z and a policy of opacity. (These off-path beliefs assign the worse feasible expectation of p for a given realization of z.) Then the probability of a run if the regulator deviates to opacity is:

Pr (

z − δ 2

< p� )

= {

1 − p− p� δ

if p > p�

1 if p ≤ p�. (B7)

(Note: For p > p�, we have p� + δ2 − p ≥ − δ2 [i.e., even if p = 1 , we get δ ≥ 1 − p�, which is satisfied by assumption].) Given these off-path beliefs, there is no incentive to deviate if p ≤ p� as there would be a run with probability one. Consider the incentive to deviate to opacity if p > p�:

D( p) ≡ (

p − p� δ

) ( μ + (2 p − 1) �η

) − p

( μ + �η

) , (B8)

1832 The Journal of Finance R©

where p− p �

δ is the probability of avoiding a run on all banks under opacity, μ +

(2 p − 1) �η is the net output under opacity if there are no runs, and p ( μ + �η

) is the net output under transparency. Notice that

D( p�) < 0, (B9)

D (1) = (

1 − p� δ

) ( μ + �η

) −

( μ + �η

) ≤ 0, (B10)

∂ 2 D( p) ∂ p2

= 4�η δ

> 0. (B11)

This implies that D ( p) ≤ 0 for all p > p�, that is, there are no incentives to deviate for any p > p�. Hence, we conclude that, for δ ≥ 1 − p�, there is an equilibrium with full transparency that is supported by the aforementioned off-path beliefs.

We show next that, if μ ≥ c2 (equivalently, p� ≤ 12 ) and δ ≥ μ+�η μ−�η , there exists

an equilibrium with full opacity. We proceed in two steps.

1. If δ ≥ μ+�η μ−�η and the equilibrium is fully opaque, investors run if their

signal z is strictly lower than z� = 2 p� − δ2 . Notice first that, if z > δ2 , then

E( p|z) = z − δ 2 + min

{ 1, z + δ2

} 2

= min { 1 + z − δ2 , 2z

} 2

≥ min {

1 2

, δ

2

} ≥ p�, (B12)

where the last inequality uses δ ≥ μ+�η μ−�η > 1 and p

� ≤ 12 , which implies δ 2 ≥ p�. Hence, if z > δ2 , there is no run. Now suppose that z < δ2 . Then

E( p|z) = 0 + min { 1, z + δ2

} 2

= min {

1 2

, z + δ2

2

} , (B13)

and there is a run if and only if E( p|z) < p�. Therefore, given that p� ≤ 12 , there is a run if and only if

z + δ2 2

< p� ⇔ z < 2 p� − δ 2

. (B14)

2. If δ ≥ μ+�η μ−�η , then there exists an equilibrium with full opacity. Let B( p, p

�) denote the net benefit of opacity at p for a given p�. If p ≥ 2 p�, then, from step 1 above, z ≥ z� = 2 p� − δ2 . This in turn implies that the probability of a run is zero and the net benefit of opacity (i.e., avoiding runs on low-quality banks) is positive, that is, B( p, p�) = (1 − p)(μ − �η) > 0. If

Transparency in the Financial System 1833

p < 2 p�, the probability of a run is strictly positive and

B( p, p�) = p + δ 2 − z� δ

[μ + (2 p − 1)�η] − p(μ + �η) =

= (

1 − 2 p � − p δ

) [μ + (2 p − 1)�η] − p(μ + �η). (B15)

(Note that δ ≥ μ+�η μ−�η > 1 implies δ ≥ 2 p

�, and hence p+ δ2 −z�

δ = 1 − 2 p�− p

δ >

0.) There exists a fully opaque equilibrium if and only if

b( p�) ≡ min p∈(0,2 p�]

B( p, p�) ≥ 0. (B16)

Notice that, from (B15), B( p, p�) is linearly decreasing in p�. Hence, if B

( p, 12

) ≥ 0 ∀ p ∈ (0, 1), then a fortiori, when p� < 12 , B ( p, p�) ≥ 0 ∀ p ∈

(0, 1). This in turn implies b( p�) ≥ 0. Hence, a sufficient condition for the existence of an equilibrium is b( 12 ) ≥ 0. Note that B( p, 12 ) is convex in p and B(1,

1 2 ) = 0. Hence, a necessary

and sufficient condition for b( 12 ) ≥ 0 is ∂ B( p, 12 )

∂ p evaluated at p = 1 to be negative. This condition is equivalent to

δ ≥ μ + �η μ − �η

. (B17)

�

PROOF OF PROPOSITION 4: First, for p ≥ p�, disclosing p only prevents runs and hence strictly dominates disclosing {ηi}i∈[0,1] (which would result in runs on low- quality banks). Conversely, for p < p�, disclosing {ηi}i∈[0,1] strictly dominates disclosing p only (which would result in runs on all banks). The proof of the proposition builds on the following two lemmas. (Note: we refer to opacity as a policy in which the regulator does not disclose any information, neither p nor {ηi}i∈[0,1].) LEMMA B1: In any equilibrium, the set of p’s below p� for which opacity is the equilibrium strategy has measure zero.

PROOF OF LEMMA B1: Assume otherwise and let pO−below be the smallest p below p� for which a policy of opacity is followed in equilibrium. Let pO−above be the smallest p greater than or equal to p� for which a policy of opacity is followed in equilibrium. (Note that, if a policy of opacity is followed in equilibrium in a nonzero measure of p’s below p�, it must be the case that a policy of opacity is followed for some p ≥ p�. Otherwise, the probability of a run for those p’s below p� would be one under opacity, and hence transparency would be optimal ∀ p < p�.) Then it must be the case that

pO−below + δ 2

≥ pO−above − δ 2

, (B18)

1834 The Journal of Finance R©

as otherwise the probability of a run at pO−below would be one under opacity, and hence the regulator would have an incentive to deviate to transparency. However, if there is a nonzero measure subset of p’s between pO−below and p�, then pO−below + δ2 ≥ pO−above − δ2 implies that the expectation of p given both z = pO−above − δ2 and opacity is smaller than p�. (Intuitively, a signal z = pO−above − δ2 under opacity could come from pO−above or from all the p’s below p� for which opacity is the equilibrium strategy. If there is a nonzero measure of p’s between pO−below and p�, then the posterior of p would be below p�.) However, this implies that the probability of a run under opacity at pO−above

would be greater than zero, which in turn implies that opacity could not be an equilibrium for pO−above as it would be strictly dominated by disclosing p, a contradiction. � LEMMA B2: In any equilibrium, there is an interval of p’s above p� such that the regulator follows a policy of disclosing p only.

PROOF OF LEMMA B2: Suppose the regulator follows opacity in equilibrium from some realizations of p ≥ p�, and let pO−above be the smallest p ∈ [ p�, 1) for which opacity is followed. The probability of a run on all banks at pO−above is zero (as otherwise the regulator would disclose p ). Therefore, in equilibrium there is a run under opacity iff z is smaller than the lowest signal that can be generated from pO−above, that is, iff z < pO−above − δ2 . Then, for p ∈ (0, p�) and pO−above ∈ [ p�, 1), the net benefit of following opacity rather than transparency (i.e., rather than disclosing {ηi}i∈[0,1]) is

B( p, pO−above) = Pr [ u + p ≥ pO−above − δ

2

] ( μ + (2 p − 1) �η

) − p

( μ + �η

) = max

[ 1 − p

O−above − p δ

, 0 ] (

μ + (2 p − 1) �η ) − p

( μ + �η

) .

(B19)

If the regulator does not follow opacity for any p ∈ [ p�, 1), we adopt the conven- tion pO−above = 1, and, for any p ∈ (0, p�), the net benefit of following opacity rather than transparency is simply

B( p, 1) = − p(μ + �η), (B20) as opacity would cause a run in all banks.

If we set pO−above = p�, then B( p�, p�) > 0. This implies that, if pO−above = p�, then there is an interval of p’s just below p� (that is, [ p� − ε, p�) for some ε > 0) in which the regulator would have an incentive to follow a policy of opacity (i.e., B( p, p�) > 0 for all p ∈ [ p� − ε, p�)). Since the interval has measure ε > 0, this contradicts Lemma B1, which implies that pO−above > p�. �

Next, we show that in equilibrium the regulator follows transparency every- where below p�. (From Lemma B1, this holds almost everywhere below p�.) That is, we show that, for all p ∈ (0, p�), B( p, pO−above) < 0. From (B20), this is

Transparency in the Financial System 1835

immediate if pO−above = 1. Turn now to pO−above ∈ [ p�, 1). Suppose there exists a p̂ ∈ (0, p�) such that B( p̂, pO−above) ≥ 0. Then, from (B19), 1 − pO−above − p̂

δ > 0,

which implies that B( p, pO−above) is strictly convex in p around p̂. Strict convex- ity and B( p̂, pO−above) ≥ 0 imply the existence of an interval in a neighborhood of p̂ in which B( p, pO−above) > 0. This contradicts Lemma B1.

We next define pCr as the smallest p such that the regulator can follow opacity without causing p ∈ (0, p�) to deviate from transparency. Formally, pCr ≡ inf { p ∈ ( p�, 1] : ∀ p′ ∈ (0, p�), B( p′, p) < 0}. (Note: the set we consider is bounded and nonempty since it contains one, and thus it admits an infimum.)

Finally, note that the disclosure choice is uniquely determined for p ∈ (0, p�) (i.e., disclosing {ηi}i∈[0,1]) and for p ∈ [ p�, pCr ) (i.e., disclosing p only). For p ≥ pCr , the regulator can either disclose p only or follow opacity (both of which generate the same economic outcome of no runs). For instance, an equilibrium in which the regulator discloses p only for all p ∈ [ pCr , 1) can be sustained by the off-path belief of the minimum p consistent with the observed signal z (i.e., z − δ2 ) following a deviation to opacity. Further, an equilibrium in which the regulator follows opacity for all p ∈ [ pCr , 1) can be sustained by the off- path belief of the minimum p consistent with the observed signal z (i.e., z − δ2 ) following a deviation to opacity if z < pCr − δ2 . (Note: if z ≥ pCr − δ2 , then Bayes’s rule applies.) �

REFERENCES

Acharya, Viral V., Douglas Gale, and Tanju Yorulmazer, 2011, Rollover risk and market freezes, Journal of Finance 66, 1175–1207.

Angeletos, George-Marios, Christian Hellwig, and Alessandro Pavan, 2006, Signaling in a global game: Coordination and policy traps, Journal of Political Economy 114, 452–484.

Angeletos, George-Marios, and Alessandro Pavan, 2007, Efficient use of information and social value of information, Econometrica 75, 1103–1142.

Barnea, Amir, Robert A. Haugen, and Lemma W. Senbet, 1980, A rationale for debt maturity structure and call provisions in the agency theoretic framework, Journal of Finance 35, 1223– 1234.

Bayazitova, Dinara, and Anil Shivdasani, 2012, Assessing TARP, Review of Financial Studies 25, 377–407.

Bond, Philip, and Itay Goldstein, 2014, Government intervention and information aggregation by prices, Working paper, University of Pennsylvania.

Bond, Philip, Itay Goldstein, and Edward C. Prescott, 2010, Market-based corrective actions, Review of Financial Studies 23, 781–820.

Bryant, John, 1980, A model of reserves, bank runs and deposit insurance, Journal of Banking and Finance 4, 335–344.

Calomiris, Charles W., and Charles M. Kahn, 1991, The role of demandable debt in structuring optimal banking arrangements, American Economic Review 81, 497–513.

Carlsson, Hans, and Eric van Damme, 1993, Global games and equilibrium selection, Econometrica 61, 989–1018.

Chakraborty, Archishman, and Rick Harbaugh, 2007, Comparative cheap talk, Journal of Eco- nomic Theory 132, 70–94.

Chakraborty, Archishman, and Rick Harbaugh, 2010, Persuasion by cheap talk, 2010, American Economic Review 100, 2361–2382.

Chen, Qi, Itay Goldstein, and Wei Jiang, 2010, Payoff complementarities and financial fragility: Evidence from mutual fund outflows, Journal of Financial Economics 97, 239–262.

1836 The Journal of Finance R©

Diamond, Douglas W., and Philip H. Dybvig, 1983, Bank runs, deposit insurance, and liquidity, Journal of Political Economy 91, 401–419.

Eisenbach, Thomas M., 2013, Rollover risk as market discipline: A two-sided inefficiency, Federal Reserve Bank of New York Staff Report No. 597.

Ellahie, Atif, 2012, Capital market consequences of EU bank stress tests, Working paper, London Business School.

Freixas, Xavier, and Christian Laux, 2012, Disclosure, transparency and market discipline, in M. Dewatripont and X. Freixas, eds.: The Crisis Aftermath: New Regulatory Paradigms (CEPR, London, UK).

Goldstein, Itay, and Yaron Leitner, 2013, Stress tests and information disclosure, Working paper, Federal Bank of Philadelphia.

Goldstein, Itay, and Ady Pauzner, 2005, Demand–deposit contracts and the probability of bank runs, Journal of Finance 60, 1293–1327.

Goldstein, Itay, and Haresh Sapra, 2014, Should banks’ stress test results be disclosed? An analysis of the costs and benefits, Foundations and Trends in Finance 8, 1–54.

Gorton, Gary, and Andrew Metrick, 2012, Securitized banking and the run on repo, Journal of Financial Economics 104, 425–451.

Greenlaw, David, Anil K. Kashyap, Kermit Schoenholtz, and Hyun Song Shin, 2012, Stressed out: Macroprudential principles for stress testing, Chicago Booth Working Paper 12-08.

He, Zhiguo, and Asaf Manela, 2014, Information acquisition in rumor-based bank runs, Journal of Finance, forthcoming.

Hertzberg, Andrew, José Marı́a Liberti, and Daniel Paravisini, 2011, Public information and coor- dination: Evidence from a credit registry expansion, Journal of Finance 66, 379–412.

Hirshleifer, Jack, 1971, The private and social value of information and the reward to inventive activity, American Economic Review 61, 561–574.

Hirtle, Beverly, Til Schuermann, and Kevin J. Stiroh, 2009, Macroprudential supervision of finan- cial institutions: Lessons from the SCAP, Federal Reserve Bank of New York Staff Report 409.

Landier, Augustin, and David Thesmar, 2011, Regulating systemic risk through transparency: Tradeoffs in making data public, in Markus K. Brunnermeier and Arvind Krishnamurthy, eds: Risk Topography: Systemic Risk and Macro-Modeling (University of Chicago Press, Chicago, IL).

Metz, Christina E., 2002, Private and public information in self-fullfilling currency crises, Journal of Economics 76, 65–85.

Morris, Stephen, and Hyun Song Shin, 1998, Unique equilibrium in a model of self-fulfilling currency attacks, American Economic Review 88, 587–597.

Morris, Stephen, and Hyun Song Shin, 2000, Rethinking multiple equilibria in macroeconomic modeling, NBER Macroeconomics Annual 15, 139–161.

Morris, Stephen, and Hyun Song Shin, 2002, Social value of public information, American Economic Review 92, 1521–1534.

Morris, Stephen, and Hyun Song Shin, 2003, Global games: Theory and applications, in Mathias Dewatripont, Lars Peter Hansen, and Stephen J. Turnousky, eds.: Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress, Volume I (Cambridge University Press, Cambridge, UK).

Morris, Stephen, and Hyun Song Shin, 2004, Coordination risk and the price of debt, European Economic Review 48, 133–153.

Morris, Stephen, and Hyun Song Shin, 2009, Illiquidity component of credit risk, Working paper, Princeton University.

Peristiani, Stavros, Donald P. Morgan, and Vanessa Savino, 2010, The information value of the stress test and bank opacity, Federal Reserve Bank of New York Staff Report 460.

Plantin, Guillaume, 2009, Learning by holding and liquidity, Review of Economic Studies 76, 395–412.

Rochet, Jean-Charles, and Xavier Vives, 2004, Coordination failures and the lender of last resort: Was Bagehot right after all? Journal of the European Economic Association 2, 1116–1147.

Schuermann, Til, 2013, Stress testing banks, International Journal of Forecasting 30, 717–728.

Transparency in the Financial System 1837

Shapiro, Joel, and David Skeie, 2014, Information management in banking crises, Working paper, University of Oxford.

Stein, Jeremy, 2011, Monetary policy as financial-stability regulation, Quarterly Journal of Eco- nomics 127, 57–95.

Supporting Information

Additional Supporting Information may be found in the online version of this article at the publisher’s website:

Appendix S1: Internet Appendix