Predatory Short-Selling and Self-ful lling Crises: When Morris-Shin Meets Diamond-Dybvig

Transcription

1 Predatory Short-Selling and Self-ful lling Crises: When Morris-Shin Meets Diamond-Dybvig Xuewen Liu The Hong Kong University of Science and Technology Abstract This paper investigates the mechanism through which short-selling of a bank s stocks can cause the failure of a bank. In the model, creditors, who learn information from stock prices, will grow increasingly unsure about the true bank fundamentals in facing more noisy stock prices and thus a run on the bank is more likely because of creditors concave payo. Understanding this, speculators conduct short selling beforehand to amplify market (il)liquidity and add noise to stock prices, triggering a bank run, and subsequently pro t from the bank failure. We show that predatory short-selling of bank stocks essentially involves two runs: the aggressive run among speculators and the conservative run among creditors; the two runs interact with and reinforce each other through compound feedback spirals, drastically increasing the probability of the collapse of a bank. The model provides a theoretical justi cation for short-selling regulations in times of nancial (banking) crisis. The paper also makes methodological contributions to the global game literature. I am grateful to Enrico Bi s, Sudipto Dasgupta, Itay Goldstein, Cristian Huse, Sheng Li, Yang Lu, Filippos Papakonstantinou, Guillaume Plantin, Tianxi Wang, Xiaohui Wu and seminar participants at numerous institutions and conferences for comments. I thank Hyun Song Shin for his continued support and encouragement. All errors remain my own. Contact: Xuewenliuust.hk. Electronic copy available at:

2 Regulators imposed a temporary ban on short-selling of nancial stocks in four European nations (France, Italy, Spain, and Belgium), e ective Friday, in order to calm volatility. Friday, Aug 2, 20, CNNmoney Introduction There have been tough regulations on short selling recently, in particular short-selling of nancial stocks and naked credit default swaps (CDSs). These speculative activities have been blamed for playing an important and direct role in destabilizing the nancial system. Short selling of stocks has been accused of constituting a direct cause of the collapses of Bear Stearns and Lehman Brothers. In late 2008, several countries, including the US and UK, imposed an emergency order to ban the short-selling of nancial stocks. In executing the short sale-ban order (Securities Exchange Act NO / September 8, 2008), 2 the SEC concluded: Short selling in the securities of a wider range of nancial institutions may be causing sudden and excessive uctuations of the prices of such securities in such a manner so as to threaten fair and orderly markets. Similarly, naked CDSs have been accused of worsening the Greece sovereign debt crisis that has shaken the eurozone. On May 9th, 200, Germany temporarily banned naked short-selling of nancial shares and naked CDSs of euro-area government bonds. 3 On Aug 2, 20, in a new wave of the European debt crisis, four EU nations banned short-selling on bank stocks. The regulation of a ban on short-selling, however, is extremely controversial in the nancial industry as well as with academics. For example, several empirical studies have found that bans were actually harmful to nancial markets. On the theory side, there are no compelling explanations Naked CDS trading means buying swaps without holding a direct investment in the underlying bonds. Buying CDSs is equivalent to shorting the underlying bonds. Naked CDSs make unlimited shorting of bonds possible Naked short selling of stocks is the practice of short-selling stocks without rst borrowing the stocks. Thus, naked shorting is not limited by the actual number of shares outstanding. An in nite number of shares can be created to sell. 2 Electronic copy available at:

3 so far to the basic question of why the short selling can cause the failure of nancial rms and state nancing. This paper investigates the speci c mechanism through which short selling of stocks might cause failures of banks and in particular address the question of why banks (as opposed to standard corporations) are unique in su ering short-selling speculations. We show that the mechanism of speculative attacks on sovereign debt is essentially the same. Received wisdom usually attributes the danger of short-selling of a bank s stocks to that shortselling can arti cially bring down the bank s stock price. The low stock price then sparks concern about the bank s fundamental value and may even erode counterparties con dence. After observing a very low stock price, the counterparties may be unwilling to roll over their credit to the bank. This in turn causes the failure of the bank. 4 A critical question about the above argument, however, is why counterparties, as debtholders, should care about the stock price. First, the regulatory capital requirements of banks are not calculated based on stock prices. 5 Second, in a rational-expectations framework, counterparties of a bank would rationally anticipate predatory nature of trading by speculators and understand that the low stock price is due to speculators manipulation. So it is not obvious why counterparties should care about the arti cial stock price. In fact, Standard & Poor s, a rating agency, also con rms that a company s stock price, by itself, is never the primary factor in evaluating a company s debt rating. 6 This paper o ers a di erent explanation for why short-selling of stocks can cause the failure of a bank. In our model, the fundamental problem of short-selling is that it can amplify market (il)liquidity, leading to an increase in uncertainty and information asymmetry on the bank s fundamental value. In the model, speculators face random market liquidity in short-selling attacks. 4 See, for example, Soros (2009). 5 Regulatory capital is instead based on the mark-to-market value of net assets. The mark-to-market value of net assets is di erent from the stock market capitalization. 6 In an o cial document titled How stock prices can a ect an issuer s credit rating, Standard & Poor s states clearly: The price of a company s stock is one of the many factors that Standard & Poor s Ratings Services may consider in rating a company s debt. However, a company s stock price, by itself, is never the primary factor in how we analyze the creditworthiness of a company. Instead, we emphasize fundamental analysis in our approach (Standard & Poor s RatingsDirect, September 26, 2008). 3

4 When market liquidity is low, short-sellers can in uence and depress the stock price. When market liquidity is high, short-sellers cannot in uence the stock price very much. In the presence of imperfect information about fundamentals, creditors use the stock price to learn (in a Bayesian fashion) about the fundamental value. They will grow increasingly unsure about the true fundamental value in facing a low stock price. The low price may be due to short-selling or to bad fundamentals. In a rational-expectations equilibrium, creditors can still assess the fundamental value correctly, on average, by observing the stock price. However, uncertainty increases. That is, for a given stock price, both the downside and the upside risk of the fundamental value increase. Considering that creditors (debt) payo is concave in the bank s fundamental value, creditors mainly care about the downside risk. As a result, uncertainty leads to a reduction in debt value, which may cause creditors to run and eventually lead to the bank s failure. We believe that the mechanism presented above is in line with what happened in reality. In fact, in both the SEC s o cial document and the European Securities and Market Authority s statement, key words uctuation and volatility appear. 7 These terms, to some degree, con rm that short-selling ampli ed the market liquidity. The economic translation of those terms in our model is information asymmetry. We build a three-date model. At the initial date, a continuum of speculators receive imperfect private information about the bank s fundamentals (of the nal date). They decide whether to short the bank s stock. They face random market liquidity and thus can only partially in uence the stock price. At the intermediate date, a continuum of creditors of the bank, observing the stock price and also receiving imperfect private information about the bank s fundamentals, decide whether to roll over their short-term credit to the bank. If enough creditors decline to roll over, a creditor run occurs and the bank fails. Essentially, there are two runs in our model: the speculator run at the initial date (in the spirit of Morris and Shin (998)) and the creditor run at the intermediate date (in the spirit of Diamond and Dybvig (983)). Crucially, the two runs in our model interact with each other. The speculator run provides public information (stock prices) to the creditor run while the creditor run determines the payo s in the speculator run. We start solving the equilibrium by abstracting away the speculator run and focus on the second 7 4

5 stage of the game - the creditor run. We show an intuitive result: creditor runs are more easily triggered when stock prices are more noisy. In fact, when the stock price is very noisy and volatile, creditors have no clue about the bank s true fundamentals. In facing a low stock price, the creditors understand that it may be for some exogenous reasons and the stock is underpriced. But it can also be due to true bad fundamentals. Because of the asymmetry in payo s for the upside and the downside risk, the creditors react to the more noisy stock price by running on the bank at a higher threshold. Based on the above analysis, we move to the rst stage of the game and show that an endogenous short-selling attack by speculators arises. Essentially, we show that speculators have incentives to coordinate automatically to create noise in the stock price (by short-selling). 8 Indeed, we show that there exist endogenous strategic complementarities among speculators in the short-selling attack. Concretely, for a particular speculator, the more aggressive the other speculators are in short selling, the more noisy the stock price is; a more noisy stock price leads to a higher likelihood of a creditor run, which translates into a higher probability of success for the short-selling attack. Therefore, the expected gain for a particular speculator to participate in the short-selling attack has an increased probability to be higher than the expected cost, and thus she participates with a greater probability. That is, if other speculators are aggressive in short selling, in equilibrium it is optimal for an individual speculator to be aggressive as well. The endogenous strategic complementarities result in aggressive short selling by every speculator. Crucially, we show that the two runs not only interact with but also reinforce each other through compound feedback spirals, with the result of drastically increasing the probability of the collapse of the bank. We have already discussed one way of the feedback the speculator run impacts the creditor run (through the stock price informativeness channel). We discuss the other way of the feedback. A numerical example helps to illustrate the intuition. Consider two banks, A and B, with the same fundamentals but with di erent degrees of maturity mismatch. Bank A has a higher degree of maturity mismatch and hence has a more severe coordination problem among its 8 There seems some implicit or explicit coordination among speculators in short-selling attacks. On April 20, 200, Telegraph reported: Goldman was named in the court ling along with four other rms, including hedge funds SAC Capital and Citadel, as lawyers demanded to know whether it was partly responsible for triggering Lehman Brothers downfall by shorting its rival s shares. See Goldman Sachs implicated in shorting Lehman shares, Telegraph, April

6 creditors. Say that, the creditors of Bank A decide to decline to roll over when they observe the stock price to be below a threshold of $50 while those of Bank B use a threshold of $00. 9 We suppose that a collective force of speculators can bring down the stock price by $20 for either stock. Thus, speculators start to short Bank A when its stock price is $70 (= $50+20) while they start to attack bank B when B s stock price is $20. From here, we can see clearly that if creditors decline to roll over at a higher threshold, it means that speculators can bring down the bank more easily, in turn inducing them to attack at a higher level. However, we show that there is another reinforcing mechanism. If Bank A indeed fails and ends up with a zero stock price, speculators gain from attacking A is $70 (=$70-0), while the gain for a successful attack on B is $20. Therefore, ex ante, speculators have incentives to attack A at an even higher level, resulting in an upward spiral in the attacking level. 0 In equilibrium, the gap in the attacking levels between A and B can be much higher than $50. In short, there are two joint forces through which the creditor run feeds back to the speculator run. The model o ers two cross-sectional predictions. First, banks with lower fundamentals are more likely to be subject to short-selling attacks, and are more likely to fail. Second, for a given fundamental value, rms with a higher degree of maturity mismatch in their balance sheets are more likely to incur short-selling attacks. Indeed, we show that due to the compound reinforcing spirals, a slight di erence in maturity mismatch can make a large di erence in the probability of incurring a short-selling attack. This implies that predatory short-selling should be much more likely on nancial rms rather than standard corporations. Our research suggests that short-selling can fuel a crisis. In times of nancial (banking) crisis, rational speculators could coordinate (through market mechanisms) to attack and trigger a creditor run on a weak but viable bank. Their distinctive capital structure makes banks unique in both the likelihood of incurring short-selling attacks and the vulnerability to failure under attack. Interestingly, our paper shows the asymmetry between short-side and long-side speculation. In 9 The creditors of Bank A are more conservative because each of them understands that there is a more severe coordination problem among them and thus is more concerned about others run. 0 We will show that the cost in an unsuccessful attack does not vary much with the attacking level. Also, the probability of a successful attack decreases in the attacking level, so the attacking level spirals upward but converges to a limited number. 6

7 our model, both long-side and short-side attacks increase uncertainty and thus make a creditor run more likely. However, speculators need to gain from their positions and only shorting of stocks can allow them to bene t from the decline in the stock price. Therefore, speculators optimally choose short-side attacks. This is the opposite of the attacks through CDS. When a default happens, the price of CDSs increases rather than decreases so only the attacks by long positions in CDSs can gain from the default. In sum, our paper contributes to an understanding of why short-selling of stocks can trigger creditor runs and thereby bank failures. Our central argument is that creditor runs are more likely to occur in an environment of more noisy stock prices whereas speculators can at times use short sales to create noise in stock prices. On the methodology side, we use global game methods to model a double-run game, through which we are able to explicitly characterize compound feedback spirals that explain why banks have always been the central focus of short-selling regulations. Related Literature. Although short selling stabilizes prices most of the time, we study the destabilizing e ects of short selling at times of market turbulence. Brunnermeier and Pedersen (2005). 2 Our paper relates to They show that predator traders may deliberately dump the asset that a distressed trader holds, depressing the asset s price and triggering the trader s constraints to bind (e.g., margin calls). Their model captures well situations of predatory trading in asset markets in which traders like hedge funds typically face some external constraints, such as mark-to-market based margin calls from their brokers. However, for our research on short-selling of bank stocks, the stock price does not represent a relevant constraint for banks, as there are almost no explicit constraints for banks that are contingent on stock prices. 3 In our paper, we instead o er the insight that it is the change in stock price informativeness, rather than the stock price itself, that explains why short-selling can cause a failure of a bank. While several empirical studies (e.g., Boehmer, Jones, and Zhang (2009), Beber and Pagano (20)) nd that bans on speculation (short-selling) activities in a nancial crisis are not bene cial Friedman (953), Hart and Kreps (986), and DeLong et al. (990) contribute to the old debate of destabilizing speculation, where speculation is not speci ed as short-selling. 2 Carlin, Lobo, Viswanathan (2007) and La O (200) study predatory trading and address di erent questions from our paper. 3 Regulatory capital requirements of banks are not based on the stock price. Of course, if a bank requires recapitalization, a low stock price may be a problem. However, recapitalizations do not occur very often. 7

8 to nancial markets, the di erence between these studies and our paper lies in that these studies do not take into account the impact of speculation on the (real) business operation of attacked banks and the feedback, and lack a consideration of externalities in the welfare losses of these banks. 4 The model in our paper presents Morris-Shin meeting Diamond-Dybvig. In the model, two groups of investors move sequentially. The rst group of investors coordinates to conduct shortselling attacks, with the objective of triggering the conservative run of the second group. We show that there exists a two-way feedback between the actions of these two groups, and that the feedback generates a hurricane-like self-reinforcing spiral. We endogenize the strategic complementarities, the information structure and the payo structure of the global games in our model. Our work therefore contributes to the theoretical literature on global games by simultaneously endogenizing these three elements. 5 Our paper relates to Angeletos and Werning (2006) and Hellwig, Mukherji and Tsyvinski (2006) in that all papers endogenize public information of global games. 6 Crucially, there are two runs instead of one run in our model, which is di erent from previous work. The rst run generates the public information for the second run. Angeletos and Werning also study feedback between asset price, coordination actions and dividends. The feedback in their paper in part depends on the assumption that the payo of the rst-move group of investors is a function of the coordination actions of the later-move group of investors. Our paper complements their work and completely endogenizes the feedback. Hellwig, Mukherji and Tsyvinski (2006) endogenize the payo structure of the global games simultaneously with the information structure. In this respect, our paper is close to theirs. Our paper relates to the literature that studies feedback between nancial markets and the real economy (Hirshleifer, Subrahmanyam and Titman (2006) and Goldstein and Guembel (2008)). Hirshleifer, Subrahmanyam and Titman (2006) study the role of feedback in a model with irrational traders and show that such traders may survive in nancial markets when their trades a ect 4 In addition, the failure of one bank has the potential to impose systematic risk on the entire nancial market and the real economy (see also Brunnermeier (2009)). 5 Applications of global games include currency crises (Morris and Shin (998), Corsetti, Dasgupta, Morris, and Shin (2004)), bank runs (Rochet and Vives (2004), Goldstein and Pauzner (2005)), contagion of nancial crises (Dasgupta (2004), Goldstein and Pauzner (2004)), and market liquidity (Morris and Shin (2004), Plantin (2009)). 6 The information structure is endogenous also in Angeletos, Hellwig and Pavan (2007) and Dasgupta (2007). 8

9 rm cash ows. In Goldstein and Guembel s (2008) model, rm managers learn from the price about real investment opportunities and speculators trading can thus distort managers investment decisions. 7 The authors show that the presence of such feedback creates an incentive for an uninformed trader to sell the rm s stock. 8 Our paper di ers from Goldstein and Guembel (2008) in important ways. In our model, the manipulation happens because of the impact on creditors nancing (rollover) decisions rather than on managers real-investment decisions. In particular, the mechanism of speculation in our paper is di erent, which lies in that short-selling ampli es market liquidity and increases uncertainty and creditors are averse to uncertainty. Our model also explains in cross-section why predatory short-selling should be more frequently observed in nancial rms rather than standard corporations. The rest of the paper is organized as follows. Section 2 presents a model of short-selling of bank stocks. Section 3 discusses predictions and policy implications of the model. Section 4 analyzes speculative attacks on sovereign debt through naked CDSs. Section 5 concludes. 2 Model 2. The model setup The model has three dates: T 0, T and T 2. There is no discount factor between T and T 2. We discuss the three types of agents in our model in order: bank, speculators and creditors. The bank Consider a bank (investment or commercial bank) which holds one unit of asset. The asset, denoted as A, realizes a random cash ow e = + e at time T 2, where e is normally distributed as e s N(0; 2 ). The term represents the fundamentals of the bank. As in the literature of global 7 Goldstein, Ozdenoren and Yuan (2009) show strategic complementarities among speculators as a result of the feedback. Angeletos, Lorenzoni and Pavan (200) and Goldstein, Ozdenoren and Yuan (200) also derive endogenous complementarities as a result of learning from the aggregate actions of agents. The focus of these papers is overweighting of public information, with the consequence of excessive non-fundamental volatility. 8 Allen and Gale (992) provide a model in which a non-informed trader can make a pro t if investors think the manipulator may be an informed trader. 9

10 games, we assume that has an improper uniform prior over the real line. The value of is realized at T 0, while the uncertainty of e is resolved at T 2. The bank nances its asset with short-term debt (D S ) and equity (E). The short term debt is the borrowing from a continuum of lenders with unit mass. The debt is short-term in the sense that the creditors have the right to decide at T whether to roll over their lending or not. If a creditor declines to roll over, her claim is the face value of debt at T, denoted by F. If a creditor rolls over, her claim is the face value of debt plus interest, amounting to a total value of K at T 2, where K > F. For the equity, E, we assume that the bank has a single unit of divisible share outstanding. Speculators There is a continuum of risk-neutral speculators with unit mass. At T 0, these speculators receive private information regarding. The information is imperfect. Speci cally, speculator i observes a noisy signal i = +! i, where! i is normally distributed! i N(0; 2 ) and the noises are independent across speculators. Based on the private information received, a speculator decides whether to short-sell the bank s stock at T 0 or not: a speculator can only short-sell one unit of stock or decide not to take any position. If a bank run occurs at T and consequently the bank fails, it means that the short-selling attack is successful. The speculators payo (gain) in a successful short-selling attack is t. In contrast, if the bank does not fail at T, it means the short-selling attack is unsuccessful, in which case the speculators payo (loss) is r. At this stage, we assume that both t and r are xed numbers. We will endogenize t and r later. The xed cost of conducting short-selling (e.g., margin or opportunity cost), whether the short-selling attack is successful or not, is c. Speculator i s decision rule at T 0 is a map: i 7! (Short-sell, Not), where i is speculator i s information and (Short-sell, Not) is her decision set. Stock market We follow Brunnermeier and Pedersen s (2005) assumption regarding the predatory trading market. In addition to the strategic traders (i.e., speculators) who can pro t from stock price 0

11 swings, the stock market is populated by long-term investors, such as individual investors and other unsophisticated investors. Those investors are passive and price-takers. They do not have su cient information, skills, or time to predict short-term events such as the probability of a bank run at T. Their demand is only based on the long-term fundamental information. They are buy and hold investors. The demand from an individual investor among them is y i (p) = vi p d, where vi is the individual s private information regarding the long-term fundamentals of the stock and p is the stock price. 9 Hence, the aggregate net demand from those investors is Y (p) = R y i (p) = v p d, which is a function of the true long-term fundamentals of the stock, v, as noises of private information cancel out in aggregating. The long-term investors as market makers provide the residual demand curve facing the speculators. Let s represent the aggregate short-selling by the speculators at T In equilibrium, net supply equals demand, s = Y, and thus the stock price at T is given by p = v ds, where d measures the market liquidity (depth) of the stock. The downward-sloping demand curve of stocks is used in a large literature (see Grossman and Miller (988), Bernardo and Welch (2004), Morris and Shin (2004), Plantin, Sapra, and Shin (2008), et al.). 2 As v is the stock s long-term fundamentals, which in our model are E[max(0; e K)j], we have v e() = E[max(0; e K)j]. Therefore, the stock price at T is given by p = e() ds, a function of. As in some previous work, the stock price above is not forward-looking in incorporating the probability of the short-term bank s failure arising from a bank run at T. This is because the longterm investors the market markers are unsophisticated and do not understand short-term events. In fact, until the collapse of Lehman Brothers and of Bear Stearns, the failure of big banks had been beyond comprehension or expectation for most non-professional investors. Many individuals and other unsophisticated investors had seen the plunges of bank stocks as golden buying opportunities and did not jump away. Some industry reports clearly con rm that at times of crisis the market 9 Those investors are passive and price-taking, and do not learn information from the stock price. 20 The speculators borrow stocks from brokers. Brokers in turn use their clients stocks to lend. The nal lenders of stocks are therefore the existing shareholders, who are perhaps also long-term unsophisticated investors. They buy and hold and do not understand the predatory trading. 2 Empirically, Shleifer (986), Chan and Lakonishok (995),Wurgler and Zhuravskaya (2002), and others document downward sloping demand curves of stocks.

12 makers are quite often individual unsophisticated investors who intend to take advantage of market turbulence. A lot of other factors, such as the high possibility of government rescue, also contribute to many unsophisticated investors overcon dence. From a theoretical perspective, a richer model at the cost of complexity that explicitly models heterogeneous (potentially overcon dent) beliefs or preferences among stock traders regarding the likelihood of government bailouts, foreign large shareholders recapitalization, deep pockets takeover, etc., might endogenize this assumption. In order to have a clean analysis and to highlight the core mechanism of the model, we use reduced-form equity value: v e() = E[ e Kj]. So e() = K. 22 Hence, p = K ds. The mechanism of the model does not depend on this simpli cation. In the appendix, we give a formal proof that as long as e() is increasing and weakly convex in, the robustness of the model does not change. We denote P p + K, where P is interpreted as the stock price-based asset value of the bank. Therefore, we have P = ds. The equation P = ds is the central result in this subsection. Basically, as the stock price contains information about the asset value, investors can use the stock price to infer the asset value. The stock price-based asset value of the bank re ects two elements: the true fundamentals of the asset,, and the short-selling impact, ds. Further, we assume that the market liquidity d is random. Pastor and Stambaugh (2002) and others document time-varying random market liquidity. Randomness in market liquidity can be due to time-varying risk aversion of market makers. It can also be due to the fact that the size of the market-making sector is random. That is, there are times when a large number of long-term investors are able to provide liquidity to the market and d is low, whereas there are times when the opposite is true and d is high. 23 We will specify the probability distribution of d later. Basically, randomness of d implies that speculators face market liquidity risk in short-selling attacks. Creditors and bank run 22 The equity contract in Diamond (984) gives an intuitive interpretation for this simpli cation. When a rm defaults, equityholders incur some deadweight loss equal to the defaulting amount. So the equity payo is linear rather than convex. 23 Formally, suppose that the size of the market-making sector is rather than. Then, we have v p = s. Thus, d p = v d s. In order to save notations, we write d as d. Therefore, if is random, d is random. 2

13 Morris and Shin (2009) and Rochet and Vives (2004). 24 Roll over Withdraw Creditors are risk-neutral. At T, all creditors observe the stock price p, equivalent to knowing P. Further, they receive private information regarding. Creditor j s private signal is j = + & j, where & j is normally distributed as & j N(0; 2 ). Based on both stock price information and the private signal, each creditor needs to decide whether to roll over her lending. That is, creditor j s decision at T is a map: ( j ; P ) 7! (Roll over, Not). We assume that the asset of the bank is illiquid. The liquidation value of asset A at T is L, where L < F. Therefore, if more than L F proportion of creditors decline to roll over at T, the liquidation value is not su cient to cover the creditors claims, and consequently the bank fails. Alternatively, one may think of L as the collateral value of the bank s asset. This means that the bank can raise at most L amount of cash at T by using its asset as collateral. If the demand of cash exceeds L at T, the bank fails. This is the classic bank run problem. We use the term L (for a given F ) to measure the maturity mismatch of the balance sheet of the bank: the lower the L, the more severe the maturity mismatch. To fully describe the payo structure of creditors in a bank run, we need to write the payo of an individual creditor to roll over (vs. not roll over) as a continuous function of the total number of creditors who roll over. For the payo structure of a bank run with continuous states, we refer the reader to Diamond and Dybvig (983), Goldstein and Pauzner (2005) and Liu and Mello (200, 20). For our purposes, however, it is su cient to use the discrete-state setup, which follows More than L F proportion withdrawing (i.e., the bank fails) 0 (at T 2) F (at T ) Less than L F proportion withdrawing (i.e., the bank survives) K (at T 2) F (at T ) Table The above matrix shows the payo s for the creditors in a bank run. If less than L F proportion of creditors withdraw, the bank does not fail. The bank regains its initial status after the failed run. In this case, the repayments to creditors are not a ected: the payo is the notional value K in the case of rolling over, and F in the case of withdrawing. In contrast, if more than L F 24 Eisenbach (200) uses a similar discrete-state setup. proportion of 3

14 creditors withdraw, the bank fails. The creditors who roll over are left with nothing, whereas the creditors who withdraw have a payo of F. 25 This simpli ed setup avoids technical complications while su cing to catch the key feature of the bank-run game the strategic-complementary payo s. That is, if more than L F proportion of creditors withdraw, the dominant strategy for an individual creditor is withdrawing. If less than L F proportion of creditors withdraw, the dominant strategy is rolling over. Figure summarizes the main setup of the model. 2.2 A simple example of the mechanism of the model In order to facilitate the analysis in the next subsection, we use a simple example to illustrate the key mechanism of the model. Speci cally, we assume that d follows the simple two-state Bernoulli distribution: d can be either 0 or with 50% probability each. The economic interpretation is as follows. The long-term investors as market makers are sometimes very risk-averse and the demand curve is downwardsloping, i.e., d =, while sometimes they are much less risk-averse and almost risk-neutral, i.e., 25 If more than L proportion of creditors withdraw and thus the bank fails, the payo for a creditor who withdraws F should be slightly less than the face value F as the total liquidation value is L, less than F. However, as long as the payo is higher than 0, the dominant strategy is withdrawing. Thus, when the payo is written as F, the nature of strategic complementarities does not change. 4

15 d = 0. Intuitively, this is equivalent to the market depth being deterministic (i.e., not random), but speculators face noise-trading risk. In one state, noise traders are few, d is a positive number (normalized to be ) re ecting the true market depth, and hence speculators can in uence the stock price. In the other state, noise traders are abundant. They absorb the selling of speculators, and d = 0. Further, as will be shown later, in equilibrium, s is deterministic. At this stage, we assume that s is public information and s = 2. We further assume the following parameter values: = 7, K = 6, and = 0. We compute the debt value in a rational-expectations equilibrium. Note that P = ds. If there is no short-selling, then P = = 7. Hence, creditors can perfectly infer the debt value from the stock price: D = min(; K), which is 6. Suppose there is short-selling. Given the fundamental value = 7, the price P can be 7 or 5, depending on whether the market is liquid enough. In the case of P = 7, the creditors rationally expect (with Bayesian inference) that can be either 9 or 7 with equal probabilities, that is, E(jP = 7) = In the case of P = 5, the creditors rationally expect that can be either 7 or 5 with equal probabilities, that is, E(jP = 5) = 6. Therefore, the estimation of is unbiased, i.e., E[E(jP )j] =. For the debt value, however, we can easily obtain E(DjP = 7) = 6 and E(DjP = 5) = 5:5. Hence, E[E(DjP )j] < K. That is, short-selling creates uncertainty and increases information asymmetry, causing a reduction in the expected debt value. Figure 2 demonstrates the example. We have three observations from the above example. First, the more aggressive the short-selling 26 Note that has an improper prior over the real line. 5

16 (i.e., the higher the s), the higher the uncertainty creditors face in inferring the true fundamental value. This is because short-selling ampli es the market liquidity shocks. Second, short-selling does decrease the (expected) stock price. However, the lower stock price itself is not the underlying cause for the reduction in debt value. In fact, creditors can rationally expect that a low stock price may be due to short-selling. Thus, when estimating the fundamental value, they would take short-selling pressure into consideration and o set the possible price derivation. That is, creditors can correctly estimate the fundamental, on average. Instead, the key reason for the reduction in the (expected) debt value is the increase in uncertainty. Third, it might seem that long-side attacks can also increase uncertainty and cause the failure of the bank. However, speculators need to gain from their positions. Because only short positions can bene t from the fall in the stock price (i.e., the bank failure), speculators choose to attack by shorting. This is the opposite of the attacks on sovereign debt through CDS. When a sovereign defaults, the price of CDSs increases rather than decreases. So, only the attacks by long positions in CDSs can gain from the default. 2.3 Equilibrium of the model In the formal analysis of the model, we assume that d has the normal distribution d N(0; h 2 ). Note that d is typically positive. The assumption of normal distribution, however, is to obtain a closed-form solution of the model (for global game reasons). The mechanism of the model does not depend on this assumption. In fact, we can assume d N(a; h 2 ), where a > 0. In this case, the price P at T follows the distribution P N( as; h 2 s 2 ). So short selling not only depresses the mean of stock prices but also increases the variance. However, as illustrated in the previous example, the danger of shortselling does not lie in the change in the mean rather than in the variance. That is, in equilibrium, creditors can rationally anticipate the (expected) magnitude of short-selling and hence o set the term as. Using the language of statistics, from P there is always an unbiased estimation of, but the e ciency of the estimation decreases. In order to highlight the fact that it is the change in variance rather than in mean that causes problems, and to have a clean analysis, we assume a = 0. With the above setup, the price P at T has the distribution P N(; h 2 s 2 ). We also assume that 0 < h < + and 0 < < +, that is, both private and public information has value. 6

17 We consider the threshold (monotone) equilibrium of the model. That is, both the specula- 8tors and the creditors use threshold strategies. Speci cally, the speculators strategy is i 7! < Not short i, where is the threshold. The creditors use the strategy ( j ; P )7! : Short i < 8 < Roll over j (P ), where (P ) is the threshold, which itself is a function of : Not Roll over j < (P ) P. Solving the equilibrium of the model means working out the two thresholds and (P ), as shown in Figure. We solve the equilibrium in the following three cases to gradually expose the double-run game. First, we study one run the creditor run and abstract away the speculator run by assuming s as being exogenously given. Then, we move to double runs and endogenize s. Finally, we further endogenize the payo s to short-selling, t and r One run the creditor run under noisy stock prices with exogenous s In this subsection, we study the second stage of the double-run game the creditor run. Our aim in this subsection is to show one key insight of the paper: creditor runs are more easily triggered when stock prices are more noisy. To focus on this analysis, we abstract away the speculative attack and assume that s is exogenously given. As P has the distribution P N(; h 2 s 2 ), it is clear that the higher the s, the more noisy the price. Thus, s is a measure of the degree of noise in the price. In this subsection, we abstract away the economic source of s (i.e., the reason for noise in the price). Instead, we focus on studying the e ect of the noise on the trigger level for bank runs, that is, how s impacts (P ). Global game methods enable us to explicitly express the relation. We work out the creditors optimal strategy at T for a given s. If a creditor declines to roll over at T, her payo is F. If the creditor rolls over, her payo is the debt value D() = E[min( e ; K)j] We can write D() = E[max(0; min( e ; K))j]. In this case, D() is concave with respect to when is su ciently positive. This alternative expression does not a ect the results of our model. In fact, in our model, the parameter condition requires that F is close to. This condition guarantees that all relevant s are su ciently positive. So the K probability that D() is non-concave is very small. We can neglect the region of in which D() is non-concave. 7

18 Clearly, D() is globally concave with respect to. A creditor infers the debt value based on the stock price information and her private signal. At T, the price P is distributed as P N(; h 2 s 2 ) while the private signal has the distribution j N(; 2 ). Thus, the posterior expectation value of the debt for a creditor is E[D()jP; j ; s]. We have already de ned the threshold (P ), which is the threshold at which creditors decline to roll over at T. Here we de ne another threshold b (P ), which is the threshold of fundamental value below which the bank fails (for a given P ) at T. Intuitively, is the running point (of the creditors) while b is the failure point (of the bank). Solving the creditor run problem means working out these two thresholds. Fixing P, we solve and b simultaneously. Given the threshold, the proportion of creditors declining to roll over at T conditional on the realized fundamentals is pr( j < j) = ( than L F ). Recall that the bank fails at T if more proportion of creditors decline to roll over. We have the following equation: ( b ) = L F. (2) Next, given b, we consider the position for an individual creditor. In equilibrium, the creditor at margin who just receives the signal should be indi erent about rolling over or not. That is, Z b Z + 0 d + b D() f(jp; j = ; s)d = F, (3) where f(jp; j ; s) is the posterior pdf for P N(; h 2 s 2 ) and j N(; 2 ). In equation (3), the left-hand side expresses the payo for the margin creditor when she decides to roll over: If < b, the bank cannot survive to T 2 and she gets nothing, which is the rst term; conditional on the bank surviving to T 2, her expected payo is E[D()jP; j ; s], the second term. In equation (3), the creditor s threshold depends on s. This is because s determines the informativeness of the stock price. Also, the creditor s threshold depends on L, which measures the maturity mismatch. Considering that has an improper prior, it is easy to obtain the posterior density as f(jp; j ; s) = f(p j;s)z(j j ) Z + f(p j;s)z(j j )d density function of N( j ; 2 )., where f(p j; s) is the density function of P N(; h 2 s 2 ) and z(j j ) is the 8

19 Intuitively, when s increases, the posterior distribution jp; j gets fatter tails, meaning that both downside risk and upside risk of increase. But the creditors mainly care about the downside risk because D() is concave. So they may set a higher threshold (P ). Formally, by solving the system of equations (2)-(3), we can obtain and b, both of which are functions of P. The two thresholds (P ) and b (P ) fully characterize the equilibrium of the creditor-run game. b = We need to examine the existence and uniqueness of the threshold equilibrium. By (2), we have Z + ( L F ). Thus, we can combine (2) and (3) as ( L F ) D() f(jp; j = ; s)d = F. (4) If the private signal is in nitely more precise than the public signal, it is easy to prove that (4) admits a unique solution of (P ). In fact, in the limit! 0 for given h, (4) can be rewritten as L F D( ) = F. The equation becomes very intuitive: the term L F measures the coordination risk while D( ) corresponds to the fundamental risk. The uniqueness of the solution of is straightforward. Also, it clearly shows that is decreasing in L. In our setting, we stipulate that the public information is not in nitely small (i.e., h 0). We prove that the creditor-run game has a unique robust equilibrium in the sense that an equilibrium has to survive iterated deletion of strictly dominated strategies (Frankel, Morris and Pauzner (2003), Morris and Shin (2003)). Proposition summarizes the equilibrium and the comparative static result. Proposition The creditor-run game has a unique (robust) equilibrium, characterized by equation (4). The equilibrium threshold, (P ), increases in s if h that is, (P ) s > 0 and (P ) L < 0. is su ciently high, and decreases in L; Proof: see the Appendix. The intuition for Proposition is the following. The credit risk of a nancial institution can be decomposed into two parts: insolvency risk and illiquidity risk. 28 In our model, an increase 28 See, e.g., Morris and Shin (2009). 9

20 in s leads to a greater uncertainty on the bank asset value, which implies higher insolvency risk. Also, a lower L means a more severe coordination problem among creditors, which causes higher illiquidity risk. Both sources of risk lead to creditors running at a higher threshold (P ). Proposition explicitly expresses the relation between the degree of noise in stock prices and the trigger level for bank runs. The higher the degree of noise, the higher the trigger level. In the next subsection, we analyze the role of speculative attacks in creating the noise Double runs with endogenous s In this subsection, we endogenize s. In doing so we are able to know the exact degree of noise in the stock price in equilibrium. We show that the atomistic speculators automatically coordinate to attack, increasing noise in stock prices and thereby triggering a creditor run at a higher level. Essentially, we show that there exist endogenous strategic complementarities among speculators. We have two global games played subsequently and interacting in equilibrium. We work out the equilibrium by backward induction, from T to T 0. We nd the equilibrium in three steps. T. Step Given the speculators strategy at T 0, we work out the creditors optimal strategy at Conditional on the speculators using the threshold and the fundamental value being, the proportion of speculators short-selling at T 0 is s(; ) = Pr( i < j) = ( ), where () is the cumulative distribution function (cdf) of the standard normal distribution. So s(; ) is increasing in and decreasing in. We have solved the equilibrium for the creditor-run game in the last subsection. Here we only need to substitute the endogenous s(; ) for the exogenous s in equation (4). That is, Z + ( L F ) D() f(jp; j = ; )d = F, (5) where f(jp; j ; ) is the posterior pdf for P N(; (h s(; )) 2 ) and j N(; 2 ). As in Proposition, we have the comparative static result on the creditor-run game. This is Lemma. 20

21 Lemma The creditor-run game has a unique (robust) equilibrium, characterized by equation (5). The equilibrium threshold, (P ), increases in if h is su ciently high, and decreases in L; that is, (P ) > 0 and (P ) L < 0. Proof: See the Appendix. Step 2 Given the strategy used by the creditors in Step, we calculate the bank failure probability at T. Because P N(; (h s(; )) 2 ), we have the probability of bank failure Pr( < b (P )) = Pr(P < b ()) = ( b () hs(; ) ).29 So the probability depends on, b (P ), and. Considering that there is a one-to-one mapping between b (P ) and (P ) and (P ) is a function of and L, therefore, the probability ultimately depends on, and L. Thus, we can denote the bank failure probability as Pr(; ; L); where Pr(; ; L) = ( b () hs(; ) ). We have the following property regarding Pr(; ; L). Lemma 2 The bank failure probability Pr(; ; L) decreases in. For a given, Pr(; ; L) increases in when is high enough, and decreases in L. Proof: see the Appendix. The intuition of Lemma 2 is simple. The higher the fundamental value, the lower the bank failure probability. The more aggressive the short-selling attack (i.e., the higher and thus the higher threshold b (P )), the higher the bank failure probability. Finally, the higher the degree of maturity mismatch (i.e., the lower L and thus the higher b (P )), the higher the bank failure probability. Step 3 We go back to T 0 and work out the equilibrium strategy of speculators. A speculator s action depends on her belief about other speculators actions. Conditional on all other speculators using the threshold strategy with the threshold as and creditors using the 29 The complete results of the probability of bank failure are provided in the Appendix. 2

22 strategy in Step as the response to speculators, we compute the optimal strategy of speculator i. The threshold of speculator i, denoted as i, satis es: Z + f[pr(; ; L) t + ( Pr(; ; L)) r] cg dg(j i ) = 0, (6) where G(j i ) is the posterior cdf for i N(; 2 ). In equation (6), the left hand side is the expected net payo if speculator i decides to attack while the right hand side is her payo if she not attack. By the de nition of i, she is indi erent about attacking or not when she receives the signal i. We prove that, given, there exists a unique solution of i in (6). Further, if c enough (e.g., c r t r 2 ) or F K is high enough, i is increasing in. r t r is small Lemma 3 i is increasing in. That is, there exist strategic complementarities among speculators in short-selling. Proof: See the Appendix. Lemma 3 is an important result. It formally shows that there exist strategic complementarities among short-sellers: If other speculators are aggressive in short selling, it is optimal for a particular speculator to be aggressive as well. Importantly, the strategic complementarities among shortsellers in our paper are endogenous: this di ers from Morris and Shin (998). It is through the impact on creditors rollover decisions that the speculators create strategic complementarities among themselves. After showing the strategic complementarities, we are able to prove that there exists a unique threshold equilibrium among speculators. By symmetric equilibrium, we have i =. (7) We can combine equations (6) and (7) to obtain one equation, which is (8). We have one property regarding the equilibrium. That is, is an increasing function of t: The higher the gain of short-selling, the more aggressive the short-sellers are. 22