Predatory trading, Stigma and the Fed׳s Term Auction Facility

Journal of Monetary Economics 65 (2014) 57–75

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Journal of Monetary Economics journal homepage: www.elsevier.com/locate/jme

Predatory trading, Stigma and the Fed's Term Auction Facility$ Jennifer La'O a,b a b

Columbia University, Department of Economics, International Affairs Building, 420 West 118th St., New York, NY 10027, United States NBER, United States

a r t i c l e i n f o

abstract

Article history: Received 7 February 2014 Received in revised form 30 May 2014 Accepted 3 June 2014 Available online 12 June 2014

Predatory trading may affect the incentives for banks to raise liquidity in times of financial distress. In these periods, borrowing becomes a signal of illiquidity, exposing borrowers to predatory trading and possible insolvency. A stigma of borrowing thus arises, leading distressed banks to take on more illiquid positions than they would otherwise. The Fed's Term Auction Facility (TAF) can alleviate this problem. The TAF's competitive auction format allows auction winners to signal that they are illiquid but relatively strong. The TAF may therefore be an effective policy tool during financial crises: by altering the signal value of borrowing, this facility supports the injection of liquidity into distressed banks. In normal times, however, this auction facility becomes counterproductive: by creamskimming the relatively strong banks, the weakest banks are left as potential prey for predators. This suggests that the TAF is a policy best reserved for times of crisis. & 2014 Elsevier B.V. All rights reserved.

Keywords: Predatory trading Interbank lending Central bank Discount window Term auction facility Liquidity Asymmetric information Signalling games

1. Introduction The interbank lending market and the discount window of the Fed are two facilities which allow banks to borrow shortterm in order to meet temporary liquidity needs. However, these opportunities are not always availed by financial institutions. Banks at times appear reluctant to borrow, even when liquidity is needed the most. This paper studies how strategic interactions among banks may deter financial institutions from raising money in times of temporary financial distress and how the Fed's Term Auction facility can alleviate this problem. Financial markets are often modeled as interactions between small traders in perfectly competitive markets taking prices as given. However, in reality these markets are not devoid of large players with market impact. A recent literature has begun to emphasize strategic behavior among large financial institutions, and predatory trading is one such strategic interaction. Brunnermeier and Pedersen (2005) define predatory trading as “trading that induces and/or exploits the need of other investors to reduce their positions” (Brunnermeier and Pedersen, 2005, p. 1825). That is, predatory trading is a strategy in which one trader trades against another's position, driving an otherwise solvent but distressed trader into insolvency. The forced liquidation of the distressed trader leads to price swings from which the predator can then profit. Brunnermeier and

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This paper was prepared for the November 2013 Carnegie-Rochester-NYU Conference on Public Policy. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.jmoneco.2014.06.001 0304-3932/& 2014 Elsevier B.V. All rights reserved.

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J. La'O / Journal of Monetary Economics 65 (2014) 57–75

Pedersen (2005) provide a framework to study this type of interaction; they demonstrate how predatory trading can in fact induce a distressed trader's need to liquidate assets. Predatory trading and the stigma of borrowing: This paper explores how predatory trading may discourage banks from borrowing in times of financial distress. In general, a distressed bank may wish to raise money, either via the discount window or on the interbank market, in order to bridge temporary financial illiquidity. However, in an environment in which banks have private information about their own finances, seeking out extra liquidity from lenders may become a signal of financial weakness. Third-party traders can then exploit this information and predatorily trade against banks that they infer to be sufficiently weak. Therefore, the mere act of seeking out loans from other banks or at the Fed's discount window may expose a distressed bank to predatory trading and possible insolvency. The key assumption behind this result is that there exists asymmetric information among traders–that is, ex ante, traders have private information about their own balance sheet that is not available to other traders. Within this asymmetric information environment, actions undertaken by banks to relieve financial distress convey information about their underlying financial state. Hence, in deciding whether or not to seek out a loan, a distressed bank faces a trade-off between the financial cushion provided by extra liquidity and the information this action reveals. In equilibrium, a stigma associated with borrowing arises, leading some distressed banks to take on more illiquid positions than they would take otherwise. Predatory trading may therefore deter banks and financial institutions from raising liquidity in times when they need it the most. Anecdotal and (some) empirical evidence: The findings of this paper support certain historical and anecdotal evidence about strategic trading and the reluctance of financial institutions to seek out loans in times of distress. One of the most oft-cited examples of predatory trading is the alleged front-running against the infamous hedge fund Long-Term Capital Management (LTCM) in the fall of 1998. After realizing losses in a number of markets, LTCM began inquiring about loans at a number of Wall Street banks, most notably Goldman Sachs & Co. LTCM alleges that with this information Goldman then traded heavily against LTCM's positions in credit-default swaps, front-running LTCM's eventual unwinding. Business Week writes1 …if lenders know that a hedge fund needs to sell something quickly, they will sell the same asset—driving the price down even faster. Goldman Sachs & Co. and other counterparties to LTCM did exactly that in 1998. Goldman admits it was a seller but says it acted honorably and had no confidential information (Business Week, 2001). Similarly, in When Genius Failed: The rise and fall of Long-Term Capital Management, Lowenstein writes2 As it scavenged for capital, Long-Term had been forced to reveal bits and pieces and even the general outline of its portfolio… Meriwether bitterly complained to the Fed's Peter Fisher that Goldman, among others, was “frontrunning”, meaning trading against it on the basis of inside knowledge. Goldman, indeed, was an extremely active trader in mid-September, and rumors that Goldman was selling Long-Term's positions in swaps and junk bonds were all over Wall Street (Lowenstein, 2001). Empirical evidence of predatory trading is more limited. Cai (2003) uses a unique data set of audit transactions to examine the trading behavior of market makers in the Treasury bond futures market during LTCM's collapse. Cai finds strong evidence of predatory front running behavior by market makers, based on their informational advantages.3 Second, this paper provides an explanation for why financial firms may be reluctant to borrow during times of financial distress. During the 2008 financial crisis, sources report that Lehman Brothers was reluctant to publicly raise liquidity, a month or so before its collapse. According to an October 2008 Wall Street Journal article,4 As the credit crunch deepened, the Fed had set up a new lending facility for investment banks. Although the central bank doesn't reveal who borrows from it, the market generally figures it out, and there's a stigma associated with it. Lehman didn't do so over the summer, because it didn't want to be seen as needing Fed money, says one person familiar with the matter (Mollenkamp et al., 2008). In the same article, the Wall Street Journal further reported that Lehman eventually attempted to secretly raise funds from the European Central Bank: In the weeks before it collapsed, Lehman Brothers Holdings Inc. went to great lengths to conceal how fast it was careening toward the financial precipice. The ailing securities firm quietly tapped the European Central Bank as a financial lifeline (Mollenkamp et al., 2008). 1

“The Wrong Way to Regulate Hedge Funds,” Business Week, February 26, 2001, p. 90. See Lowenstein (2001). When Genius Failed: The Rise and Fall of Long-Term Capital Management: New York: Random House. According to the Lowenstein, his book is based on interviews with former employees and partners of the firm, as well as interviews conducted at the major Wall street investment banks. 3 Although identities are concealed in the transactions dataset, Cai finds one large clearing firm (coded “PI7” ) with large customer orders during the crisis period which closely match various features of LTCM's trades executed through Bear Stearns, including trade size, pattern and timing. More importantly, Cai finds that market makers traded on their own accounts in the same direction just 1 or 2 min before PI7 customer orders were executed. 4 See Mollenkamp et al. (2008). 2

J. La'O / Journal of Monetary Economics 65 (2014) 57–75

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Eventually, any loans Lehman could acquire were apparently not enough, and the investment bank infamously declared bankruptcy on September 15, 2008. The Associated Press writes5 If the mortgage meltdown is like a financial hurricane, then think of Lehman Brothers as a casualty that waited too long to cry for help (The Associated Press, 2008). Finally, there exists both anecdotal and suggestive empirical evidence of a stigma effect discouraging borrowing from the discount window. During the early stages of the financial crisis, the Federal Reserve encouraged banks to request loans at the discount window if they were in need of cash. Despite reductions in the discount rate, only but a few banks availed this resource. On April 3rd, 2009, in a speech at the Federal Reserve Bank of Richmond's Credit Markets Symposium, then Federal Reserve Board chairman Ben Bernanke stated6 In August 2007, banks were reluctant to rely on discount window credit to address their funding needs. The banks' concern was that their recourse to the discount window, if it became known, might lead market participants to infer weakness—the so-called stigma problem (Bernanke, 2009). In response to the low level of discount window lending, the Fed made credit available through special lending facilities. In December 2007, the Fed created the Term Auction Facility, or the TAF as it was called. Under this program, the Federal Reserve made short-term loans available to banks through competitive single-price auctions conducted every 2 weeks. Each auction featured a predetermined fixed supply of loans ($20–75 billion), allocated to the banks with the highest interest rate bids. Bank participation in TAF was significant–at its peak, the TAF amounted to more than $500 billion. Benmelech (2012) finds that the demand for funds exceeded the supply from the first auction in December 2007 until the auction of September 22, 2008. In the auction of September 22, 2008, following Lehman's collapse, “the facility was increased to $75 billion, but 85 banks submitted bids totaling $133.6 billion” (Benmelech, 2012, p. 75). Moreover, TAF loans were often auctioned at a premium over the discount rate, despite the same term length and collateral requirements. Armantier et al. (2011) find that from March 2008 until the bankruptcy of Lehman, a significant fraction of banks participating in the TAF submitted bids above the primary discount rate. The authors furthermore find that the equilibrium TAF stop-out rate (the equilibrium rate at which banks obtained TAF funds) was in the majority of these auctions well above the discount rate. In fact, the highest TAF bids were on average 37 basis points above the primary discount rate, and 150 basis points above after Lehman's bankruptcy. Cassola et al. (2013) similarly find that in European Central Bank auctions, financial firms were willing to borrow at interest rate premia exceeding 30 basis points over EONIA by the end of 2007. Both sets of authors contend that this evidence is suggestive of a stigma associated with borrowing at the discount window. An analysis of the Fed's Term Auction Facility: The second part of this paper examines how the Fed's Term Auction Facility may alleviate the stigma associated with borrowing. In addition to their original borrowing opportunities, distressed banks are given the option to participate in a competitive auction in which they bid for loans offered by the Fed. The supply of loans is restricted so that only the highest bidders win the TAF loans. The same informational asymmetry is assumed: that is, third party traders do not have any information about individual bank balance sheets, but they may infer information by observing which banks borrow via the TAF and which banks borrow through other means (such as at the discount window or on the interbank market). The only difference, then, between obtaining loans via TAF or outside of this facility is that the TAF loans are allocated through a competitive auction format. The analysis produces two main results. First, in equilibrium, winning the TAF auction becomes a positive, rather than negative, signal of financial strength. This signal thereby deters any predatory trading against banks who are observed to win the TAF auction. Furthermore, in equilibrium banks pay a premium for TAF loans over the discount rate. Therefore, when the stigma effect discouraging borrowing is strong, the TAF may support the injection of liquidity into distressed banks who would otherwise be reluctant to seek out loans. The analysis suggests that the primary reason as to why bank participation in TAF was so large relative to the discount window was due to its competitive auction format and auction-determined interest rate. While the Fed lent freely to any bank at the discount window satisfying collateral requirements, the TAF's auction format ensured that those who obtained TAF loans were only the highest bidders. In equilibrium, this selection may have meant that winning a TAF loan became a signal of financial strength rather than weakness, thereby attenuating any stigma effect. As a result, illiquid but relatively strong banks were willing to pay a premium to participate in the TAF auctions during the crisis, consistent with what is observed in the data. The second main result is that when the stigma effect against borrowing is sufficiently weak, such that no predatory trading occurs in equilibrium, introducing the Term Auction Facility can be detrimental. In this case the auction facility acts to separate the weakest banks from the stronger ones. As the TAF auction cream-skims the relatively stronger, more liquid, banks from the pool, it leaves the financially weakest banks as potential prey for predatory traders. 5

“Financial hurricane victim Lehman waited too long.” The Associated Press, September 14, 2008. “The Federal Reserve's Balance Sheet.” Speech given by Ben Bernanke at the Federal Reserve Bank of Richmond's 2009 Credit Markets Symposium, in Charlotte, North Carolina, April 3, 2009. 6

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This paper thus provides a unique theoretical interpretation as to why the TAF was an effective policy instrument of the Fed during the 2008 financial crises. At the time when the Fed suspected that a strong stigma effect was discouraging banks from borrowing at the discount window, TAF auctions became a novel way to alleviate the stigma problem and inject much needed liquidity into distressed banks. However, the effectiveness of the TAF during financial crises comes with an important caveat. In times when average bank liquidity is high and the stigma associated with borrowing is weak, introducing the TAF can be counterproductive. This suggests that the Term-Auction facility should be a policy instrument best reserved for crises situations only, when the stigma of borrowing is a substantial threat to bank liquidity. Related literature: Brunnermeier and Pedersen (2005) provide the basic framework for predatory trading used in this paper. Brunnermeier and Pedersen show that if a distressed trader is likely to liquidate a large position, other traders have the incentive to trade in the same direction in order to profit from large price swings. In their analysis, the predator is perfectly informed of the distressed trader's balance sheet, whereas in the present paper this assumption is relaxed–traders have private information about their own finances. This is motivated by the concern that banks often know more about their own balance sheets and portfolios than other financial institutions.7 A few recent papers address the adverse effects of information revelation through bank borrowing. Kondo and Papanikolaou (2012) provide a model in which information attained by lenders from borrowers can endogenously generate limits to arbitrage. In their model, financial firms are reluctant to borrow from one another in fear of being front-run on their assets by their creditor. Ennis and Weinberg (2013) consider a model in which banks sell assets of heterogeneous quality to outside investors. A bank may send a negative signal about the quality of its assets to financial market participants when it accesses the Fed discount window, but not when participating in the interbank market. Bank participation in the interbank lending market is assumed to be unobservable by third parties, while borrowing at the discount window is partially observable. As a result, discount window stigma arises: banks are willing to borrow in the interbank market at rates higher than the one offered at the discount window. Finally, Philippon and Skreta (2012) consider an alternative financial model with adverse selection in asset markets. In their framework, they solve a mechanism design problem for optimal government intervention in which the government takes into account that participation in its program may signal information to third parties. Layout: This paper is organized as follows. Section 2 describes a benchmark model without any predator and highlights the incentive for the trader to obtain a loan. Section 3 provides the full model, with the potential predator. Section 4 provides a complete characterization of the equilibria in the full model with predatory trading and compares these allocations to the benchmark case without predatory trading. Section 5 examines the policy implications of introducing the Term Auction Facility. Finally, Section 6 concludes. All proofs are provided in the online Appendix.

2. The benchmark model without a predator This section outlines the benchmark model with only one player—the distressed trader. This benchmark highlights the incentives for the distressed trader to obtain a loan when it faces liquidity risk. The predatory trader is added later in Section 3. There are 4 periods: t A f0; 1; 2; 3g. There is one trader, the distressed trader, labelled d. This trader is risk neutral and seeks to maximize his expected wealth in the final period. The interest rate is normalized to zero. Initial balance sheet of the trader: Before this game begins, we assume that the distressed trader has invested 1 unit of the numeraire in an illiquid investment which will yield payoff u 4 1 in period 3.8 This investment is illiquid in the following sense: it cannot be sold by the trader at any point in time before the investment has materialized in period 3 without a large penalty. The trader has financed his illiquid investment using short-term debt but also has enough cash holdings to cover this debt. One may understand that these cash holdings to be reserve balances at the Fed as well as short-term T-bills. In addition, the trader owns x shares of a tradeable asset which can be bought or sold at a price st. For now this tradeable asset plays no role; hence we simply assume that xdt ¼ x, st ¼ s, 8 t. This assumption will be relaxed in Section 3 when the game with the predator is considered. The balance sheet of this trader is thus given by Assets

Liabilities

u

1

1

ð1Þ

xs

7 Carlin et al. (2007) offer a complementary theory of predatory trading: they show how predation is a manifestation of a breakdown in cooperation between market participants. 8 Alternatively, this payoff could be a random variable with an expected value of Eu ¼ u 4 1.

J. La'O / Journal of Monetary Economics 65 (2014) 57–75

Stage 0 Distressed faces rollover risk. Observes type.

Stage 1

Stage 2

Distressed decides whether or not to seek out a loan. Loan value is realized.

61

Stage 3 Distressed trader faces liquidity shock. Final payoffs are realized.

Fig. 1. Timeline without predator.

On the assets side, there is the expected payoff of the illiquid investment, 1 unit of cash, and the mark-to-market value of the bank's position in the trade-able asset. On the liabilities side, the trader has the unit of debt incurred for investing in the illiquid asset. The illiquid investment cannot be traded. Therefore, the total available liquid assets of this trader are given by wd ¼ xs þ 1 The liquid assets wd must stay above some threshold w for all t. If at any time before the last period a trader's liquidity drops below the threshold level w, then the trader must liquidate all assets and face a penalty.9 This assumption of forced liquidation may be justified by margin constraints, risk management, or other considerations in connection with low wealth. Timeline: There are three pertinent stages. In stage 0 the distressed trader faces roll-over risk on his debt, in stage 1 he may seek out a loan, and in stage 3 there is a liquidity shock which affects the traders balance sheet, and then all payoffs are realized. Because the predator has not yet been introduced, stage 2 plays no role and is hence omitted. Thus, when there is only the distressed trader, the timeline for this game is presented in Fig. 1. Stage 0. Debt Roll-over stage: In stage 0, the distressed trader faces roll-over risk on his debt. The trader's creditor rollsover only vd 0 of debt, where vd 0 can take three values, each with 1/3 probability 8 > < vℓ with probability 1=3 vd0 ¼ vm with probability 1=3 ð2Þ > :v with probability 1=3 h Let vℓ ovm ovh ¼ 1. This last equality means that if vd0 ¼ vh , the trader is able to roll-over all of his debt. The realizations of vd 0 are independent of the trader's final payoff from investment, which remains at u. The trader must thus repay his creditor 1  vd0 , and the trader's new balance sheet is given by Assets

Liabilities

u

vd0

vd0

ð3Þ

xs After facing this roll-over risk, the agent's liquid assets become wd0 ¼ xs þ vd0 . One may think of vd 0 as the initial “type” of the distressed trader; that is, the distressed is a low-type if he has cash holdings vℓ , medium-type if he has cash holdings vm, and high-type if he has cash holdings vh. Stage 1. Loan seeking stage: Next, given this trader's initial type vd0 A fvℓ ; vm ; vh g, the distressed trader decides in stage 1 whether or not to seek a loan. Let ad denote the action taken by the distressed, ad A Ad  fS; NSg where S denotes the decision to seek out a loan and NS denotes the decision to not seek out a loan. If the distressed decides to not seek out a loan, ad ¼ NS, then nothing happens and the trader's balance sheet remains the same. If on the other hand the distressed decides to seek out a loan, ad ¼ S, then the trader receives a loan of size v^ vd0 , where vℓ o vm o v^ ovh . The agent's balance sheet is then given by Assets u v^

Liabilities v^

ð4Þ

xs ^ Thus, obtaining a loan increases the cash holdings of all types except and the agent's liquid assets are given by wd1 ¼ xsþ v. the high-type. Stage 2: In stage 2, nothing occurs. This changes in Section 3 when the predatory trader is introduced. Stage 3, Liquidity shock and final payoffs: Finally in stage 3, before the payoff of the illiquid investment is realized, the distressed trader faces an exogenous negative liquidity shock. This income shock has two outcomes, either it results in liquidity below the threshold w, forcing the distressed trader to liquidate, or it results in liquidity above the threshold in which case the trader survives. 9 One may think of this penalty as both the penalty for early liquidation of the illiquid investment, as well as the cost it bears by inducing a negative price impact when selling the tradeable asset.

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Let p denote the probability that liquid assets after this shock are above the threshold w. The higher the liquid assets ^ vh g there is a of the trader, the higher the probability of surviving the liquidity shock. That is, for each vd A fvℓ ; vm ; v; ^ ph g where pl o pm o p^ o ph . The high-type thus has the highest probability of surviving the liquidity corresponding p A fpℓ ; pm ; p; shock, and the low-type has the lowest probability of surviving this shock. In particular, let pℓ ¼ 0 and ph ¼ 1. That is, the hightype never faces the risk of hitting the lower bound on wealth while the low-type always hits the lower bound on wealth. Finally, if the distressed trader survives the liquidity shock, it gets the final payoff from his portfolio of w ¼ xz þu, where z 4 0 is the return on the risky, tradeable asset. On the other hand, if the distressed trader does not survive, he must liquidate his portfolio and receive liquidation value L ow, the fire-sale value of the entire portfolio. 2.1. Equilibrium without any predator The following proposition characterizes the optimal strategy of the distressed trader in the benchmark without any predators. Proposition 1. When there are no predators, the distressed follows a strategy in which the low and medium-types seek out a loan and the high-type does not. In the environment without any predators, there is a clear motive for the low and medium-type distressed traders to seek out a loan in stage 1. The liquidity shock in stage 3 puts the low and medium-types in the danger of hitting the lower threshold w if the trader does not have enough liquidity. Hitting the lower bound forces the trader to prematurely liquidate his portfolio, which leads to a loss of value and lower profits. Obtaining a loan in stage 1 increases the low and mediumtype's chances of avoiding this outcome by increasing his liquidity before the shock. The high-type, on the other hand, has no such incentive, as this trader knows he will survive the final liquidity shock. Thus, the low and medium-type distressed traders prefer to seek out a loan in stage 1, while the high-type trader does not. 3. The model with the predator This section introduces the predatory trader and examines how this trader affects the incentives of the distressed trader to seek out liquidity. Again, there are 4 periods: t A f0; 1; 2; 3g. There are now two strategic traders, the distressed trader and the (potential) predator, which are denoted by iA fd; pg. Both traders are risk neutral and seek to maximize their expected wealth in the final period. The interest rate is normalized to zero. Initial balance sheets: As in the previous section, each trader starts off with an investment in an illiquid asset with payoff u 4 1 in the final period. The balance sheet of the distressed trader is again given by (1), with total liquid assets wd ¼ xs þ1. The predator similarly has the following balance sheet: Assets u

Liabilities vp

vp xs with total liquid assets wp ¼ xs þ vp . The predator's liquid holdings vp is constant over time and is common knowledge. Traders may choose to buy or sell the risky tradeable asset. Trade in this asset is what opens the door for predatory trading. For a moment, we discuss the market for this tradeable asset. The tradeable asset: Let xit denote trader i's position in the risky asset at time t, and let st denote its price. Each strategic trader has a fixed initial endowment of x in the risky asset and is restricted to hold any position xit A ½ x; x.10 Thus, for simplicity, the trader's initial position is equal to its maximum long position. The risky asset has an aggregate supply of Q and a final payoff z in the last period, where z is a random variable with an expected value of Ez ¼ z 4 0. Each strategic trader is large, and hence, his trading impacts the equilibrium price. In this way, traders can be likened to large investment banks or financial institutions that have both a trading desk as well as a lending unit. The bank's trading desk trades in the risky tradeable asset, while the lending unit manages loans like the illiquid investment. In addition to the two large strategic traders, the market is populated by long-term investors. The long-term investors are price-takers and have at each point in time an aggregate demand curve given by 1 Y ðst Þ ¼ ðz st Þ:

λ

ð5Þ

This demand schedule has two important attributes. First, it is downward sloping: in order for long-term traders to hold more of the risky asset, they must be compensated in terms of lower prices.11 Second, the long-term traders' demand depends only on the current price st, that is, they do not attempt to profit from future price swings.12 10

This position limit can be interpreted more broadly as a risk limit or a capital constraint. This could be due to risk aversion or due to institutional frictions that make the risky asset less attractive for long-term traders. For instance, longterm traders may be reluctant to buy complicated derivatives such as asset-backed securities. 12 Long-term investors should be interpreted as pension funds, mutual funds, or individual investors. Under this interpretation, long-term investors may not have sufficient information, skills, or time to predict future price changes. 11

J. La'O / Journal of Monetary Economics 65 (2014) 57–75

Stage 0 Distressed faces rollover risk. Observes type.

Stage 1 Distressed decides whether or not to seek out a loan. Loan value is realized.

Stage 2 Predator decides whether or not to predatorily trade. If so, predation war occurs.

63

Stage 3 Survivingg distressed traders face liquidity shock. Final payoffs are realized.

Fig. 2. Timeline with predatory phase.

The market clearing price solves Q ¼ Yðst Þ þ xpt þ xdt . Market clearing implies that the equilibrium stock price is given by st ¼ z  λ½Q  ðxpt þ xdt Þ. Due to the constraint on asset holdings, strategic traders cannot take unlimited positions. Assuming the case of limited capital, i.e. 2x oQ , the equilibrium price is always lower than the fundamental value: st o z, 8 t. Therefore, the presence of long-term investors and limited capital implies positive expected profits for the strategic traders from holding the asset until the final period.13 As before, the liquid assets wit for either trader must stay above some threshold w for all t. Otherwise, the trader must liquidate their entire portfolio and face a penalty. Timeline: The sequence of events is similar to the benchmark presented in Section 2, with the exception that stage 2 is now active. In stage 2 the predator decides whether or not to predatorily trade against the distressed, and if so, a predation war occurs. These events are summarized in Fig. 2. Stage 0. Debt Roll-over stage: Stage 0 is exactly the same as in the previous section for the distressed trader; this trader faces roll-over risk on his debt. After facing this roll-over risk, the distressed's balance sheet is given by (3), and its liquid assets are wd0 ¼ xs þvd0 where the type vd0 A fvℓ ; vm ; vh g is drawn with equal probability (2). We now assume that the distressed's initial type is private information. That is, the distressed observes his initial type vd 0, however the distressed's type is not observed by the predator. This can be interpreted as each creditor conducting a valuation of the bank's projects, but this value is not released publicly. Finally, we assume that vℓ o vm o vp o vh , so that the predator has more liquidity than the low and medium-type distressed traders, but not as much as the high-type trader. Stage 1. Loan seeking stage: As in the previous section, given the distressed trader's initial type vd 0, the trader decides whether or not to seek out a loan, ad A fS; NSg. Again, if the distressed decides to not seek out a loan, ad ¼ NS, then nothing happens and the trader's balance sheet remains the same. If on the other hand the distressed decides to seek out a ^ Again, his balance sheet loan, ad ¼ S, then the trader receives a loan of size v^  vd0 , so that his cash holdings increase to v. becomes (4). A slight modification relative to the previous section is now introduced. Before deciding to seek a loan, the distressed does not know the precise value of this loan.14 In particular, the loan size is stochastic and takes on two values: v^ A fv^ w ; v^ s g where v^ w ovp o v^ s . Thus v^ w can be thought of as the weak-type since the distressed will have less liquidity than the predator in this case, even after obtaining the loan. Similarly, v^ s may be considered the strong-type as the distressed will now have more liquidity than the predator after obtaining the loan. The loan thereby gives the trader more liquidity, but it may not always be enough to make the trader stronger than the predator. In summary, vℓ o vm o v^ w o vp o v^ s o vh Furthermore, the probability of becoming stronger than the predator is increasing in the trader's initial type. For example, if the distressed trader is initially a medium-type vm and decides to seek out a loan, with probability πm the trader receives a loan which makes him the strong type v^ s , and with probability 1  π m he becomes the weak type v^ w . ( v^ w with probability: 1  π m ^vjvm ¼ v^ s with probability: π m Similarly, if the distressed initially is a low-type vℓ and decides to seek out a loan, with probability π ℓ he receives a loan which makes him the strong type v^ s , and with probability 1  π ℓ he receives a loan which makes him the weak type v^ w . ( v^ w with probability: 1  π ℓ ^ ℓ¼ vjv v^ s with probability: π ℓ We assume that 0 o π ℓ o π m . Conditional on seeking out a loan, the probability of becoming stronger than the predator is increasing the distressed's type. Being a medium-type thereby not only affects the relative probability of surviving the liquidity shock in stage 3, but it also affects the chances that the trader will be stronger than the predator after receiving a loan. Moreover, the assumption that π ℓ is strictly greater than zero means that the low-type has some chance of gaining enough liquidity to be stronger than the predator. 13 Note that is optimal for each trader to always hold x of the risky asset, unless engaged in a predation war. This corner solution is due to the longterm investor's demand curve and to the fact that traders have limited capital, so that the equilibrium price is always lower than the fundamental value. 14 The uncertainty over the value of the loan is why the terminology of “seeking out loans” is used for the distressed trader's action, rather than simply using the term “borrowing”. Specifically, the distressed must make the decision to seek out and inquire about a loan before knowing exactly how much liquidity the loan will eventually bring in.

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Finally, we assume that if the distressed trader receives a loan, then his new type v^ A fv^ w ; v^ s g is again private information —the value of v^ is observed by the distressed, but not by the predator. Stage 2, Predatory Phase: Although the predator never directly observes the distressed's type directly, the predator does observe whether or not the distressed sought out a loan in stage 1. Formally, the predator observes ad A fS; NSg. Thus, only the distressed's action of seeking out the loan is observable, not the loan size itself. After observing ad, the predator then decides whether or not to predatorily trade against the distressed. Let ap denote the action taken by the predator, ap A Ap  fP; NPg where P denotes the decision to “predatorily trade”, and NP denotes the action to “not predatorily trade”. If the predator decides to not predatorily trade, ap ¼ NP, then nothing happens. Both traders move on to the next stage with no change in their balance sheets. If, on the other hand, the predator decides to predatorily trade, ap ¼ P, then the two traders engage in a “predation war” which occurs via trading in the risky asset. The results of this predation war are derived from Brunnermeier and Pedersen (2005). The mechanics of this predation war are not the primary focus of this paper. For this reason, only the main reduced-form results that are pertinent for the understanding of the model are presented here. A more detailed description of the predation war is given in the Appendix. In the “predation war” both traders begin selling the risky asset as rapidly as possible until one trader hits the lower bound on liquidity w and is forced to exit the market. The trader who is forced to exit the market is the trader whose liquid assets fall below the threshold first—that is, the trader with the lower amount of liquidity. The losing trader must then liquidate his entire portfolio, including the illiquid asset. The predator therefore wins the predation war and the distressed loses if and only if vp 4vd . In this case, the predator buys back up to his optimal long position x, receives strictly positive profits m 40 and moves on to stage 3, while the distressed trader is forced into liquidation and receives final payoff L. Note that this occurs whenever the distressed's type is either vℓ , vm, or v^ w .15 On the other hand, if the predator loses and the distressed wins the predation war, the predator is forced into liquidation and receives final payoff L, while the distressed buys back up to his optimal position x and continues on to stage 3. Note that this occurs whenever the distressed's type is either v^ s or vh. There are two caveats to the predation war analysis. First, while m is assumed to be an exogenous constant, it is in fact an endogenous equilibrium quantity that arises in the predation war.16 Here it is simply treated as a reduced-form quantity, representing the predator's gains found in Brunnermeier and Pedersen (2005). Although this gain may in general depend on the relative liquidity of the traders, for tractability reasons, we abstract from this issue. One may justify this assumption by requiring that the liquid holdings of the distressed and the predator are close enough in levels so that the value of m is similar across different predation war scenarios to a first-order approximation. In any case, abstracting from this issue greatly simplifies the analysis. Second, it is assumed that the distressed trader does not make any profits or gains from winning the predation war. This assumption may be interpreted as the distressed trader not having the skills nor foresight to profit from the exit of the predator. Specifically, if the distressed trader survives the predation war, he buys back the asset at the same rate as he sold, thereby yielding no net gain. This assumption is again made for tractability: it allows the analysis to focus solely on the distressed trader's desire to avoid a predation war, without an additional desire to start one. Stage 3. Liquidity shock and final payoffs: Conditional on survival through stage 2 (by either not engaging or winning a predation war), the predator receives his final realized wealth from his portfolio, w ¼ x z þu. For the distressed trader, this stage is similar to that in the benchmark model. Conditional on survival to stage 3, the distressed trader still faces a negative liquidity shock. Let p denote the probability that the distressed's liquid assets after this shock are above the threshold w. The higher liquidity the trader has, the higher the probability of surviving the liquidity shock. That is, for each vd A fvℓ ; vm ; v^ w ; v^ s ; vh g there is a corresponding p A fpℓ ; pm ; p^ w ; p^ s ; ph g where pl o pm o p^ w ¼ p^ s o ph . For simplicity, we let p^  p^ w ¼ p^ s , so that the probability of surviving the liquidity shock is the same for either the strong or ^ As before, we assume that pℓ ¼ 0 and ph ¼ 1. weak type, and given by p. Finally, if the distressed trader survives the liquidity shock, he receives the final expected payoff from his portfolio of w ¼ xz þ u. On the other hand, if the distressed trader does not survive, he must liquidate his portfolio and receive liquidation value L o w. Let Δ  w  L denote the difference between the expected payoff from the portfolio and its fire sale value. This difference Δ 40 represents the penalty the trader incurs for liquidating prematurely. We assume that m o Δ so that the profit a predator obtains from successfully predatorily trading is less than the loss due to a forced liquidation. Remarks. The key decision for the distressed trader occurs in stage 1, the loan seeking stage. In this stage, after observing his initial type, the distressed trader decides whether or not to seek out a loan. In making this decision, there are two future risks that the distressed trader faces: predatory trading risk in stage 2 and the exogenous liquidity shock in stage 3. 15 Even after receiving a loan, the distressed firm may not necessarily be able to beat the predator, v^ w o vp . That is, a loan will not always bring the trader into the range where it is not subject to predation risk. Bolton and Scharfstein (1990) for example show that an optimal financial contract may leave a firm cash constrained and unable to fully counter predation risk. They consider product market predation, not financial market predation. 16 The derivation of m is provided in the online Predation War Appendix.

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Obtaining a loan is the distressed trader's only way of potentially protecting himself against the liquidity risk. For this reason a loan would be desirable, as emphasized in Section 2. However, the main disadvantage of seeking out a loan is its possible signal value–that is, the potential predator observes whether the distressed sought out a loan, and hence infers some information from this action. Therefore, in deciding whether or not to seek out a loan, the distressed bank faces a trade-off between the financial cushion provided by a loan and the information that borrowing may convey. The key decision for the predator occurs in stage 2. While the predator does not observe the distressed's type, he does observe whether or not the distressed sought out a loan. This is motivated by the fact that banks and financial institutions must contact outside lenders, counterparties, or central banks when they are in need of liquidity. Although the loan size received may not be observed by the market, the mere act of seeking out liquidity and inquiring about loans is likely to become public. Thus, predators may be able to infer information from the actions of other banks. 4. Equilibrium characterization This section provides a characterization of the equilibria of this game. There are two stages in this game in which the traders make choices, and the choices each trader makes may affect his final payoff. We formalize this as follows. The type space of the distressed in period 1 is given by V d ¼ ðvℓ ; vm ; vh Þ, the action space of the distressed in stage 1 is given by Ad ¼ ðS; NSÞ, and the action space of the predator in stage 2 is given by Ap ¼ ðP; NPÞ. With this in mind, we define the strategies of the traders as follows. Definition 1. A strategy f of the distressed trader is a mapping from the distressed's type space to an action, f : V d -Ad . A strategy g of the predator is a mapping from its information set to an action, g: Ad -Ap . Note that in this game, each agent—the distressed and the predator—may face the risk of liquidating prematurely and receiving final payoff wi ¼ L. For the predator, this is the outcome of losing a predation war. For the distressed, this could either be the outcome of losing a predation war, or experiencing a large enough liquidity shock in stage 3. In terms of final outcomes of the game, we say that a particular trader “survives” if he is never forced to liquidate. That is, “survival” refers to the event that the trader makes it through the entire game without liquidating and receives the final value his portfolio, w. In stage 0, the distressed trader, after observing his initial type vd0, decides whether or not to seek out a loan. In order to make this decision, the distressed trader must form beliefs about his survival probability that depend not only on his chosen action and his initial type, but also on the strategy of the predator. Let αðad jvd0 ; gÞ denote the distressed's belief he survives if he chooses action ad, conditional on being type vd0 and conditional on the predator following strategy g. Thus, given an initial type vd0, the distressed's expected payoff conditional on not seeking out a loan is given by Ed1 ½wd jNS; vd0 ; g ¼ w αðNSjvd0 ; gÞ þ ðw  ΔÞð1  αðNSjvd0 ; gÞÞ;

ð6Þ

since the distressed receives expected payoff w if he survives, and liquidation value L ¼ w  Δ otherwise. On the other hand, given initial type vd0, the distressed's expected payoff conditional on seeking out a loan is given by Ed1 ½wd jS; vd0 ; g ¼ w αðSjvd0 ; gÞ þ ðw  ΔÞð1  αðSjvd0 ; gÞÞ

ð7Þ

since he gets expected payoff w if he survives, minus the fixed cost of the loan, and liquidation value L ¼ w  Δ otherwise. Likewise, in stage 1, the predator, after observing the action of the distressed, ad, decides whether or not to predatorily trade. To make this decision the predator must form beliefs about his survival probability that depend not only on his chosen action, but also on the observed action of the distressed and the distressed's strategy. Let β ðap jad ; f Þ denote the predator's belief he survives if he chooses action ap, conditional on observing an action ad, and conditional on the distressed following strategy f. If the distressed chooses to predatorily trade, then the survival probability is simply the posterior probability that vd o vp , i.e. β ðPjad ; f Þ ¼ Pr½vd ovp jad ; f . On the other hand, if the predator does not predatorily trade, then β ðNPjad f Þ ¼ 1, for any ad ; f . Therefore, given the observed action ad, the predator's expected payoff conditional on predatorily trading is given by Ep2 ½wp jP; ad ; f  ¼ ðw þmÞβðPjad ; f Þ þðw  ΔÞð1  β ðPjad ; f ÞÞ

ð8Þ

since the predator gets expected payoff w plus profits m if it survives, and liquidation value L ¼ w  Δ otherwise. On the other hand, given an observed action ad, the predator's expected payoff conditional on not predatorily trading is given by Ep2 ½wp jNP; ad ; f  ¼ w

ð9Þ

The equilibrium of this game is then defined as follows. Definition 2. An equilibrium is a strategy for the distressed f : V d -Ad , a strategy for the predator g: Ad -Ap , a survival belief of the distressed α: Ad  V d -½0; 1, and a survival belief of the predator β: Ap  Ad -½0; 1, such that

(i) For each vd0 A V d , the distressed of initial type vd 0 seeks out a loan if and only if his expected payoff from doing so is greater than his expected payoff from not seeking out a loan, conditional on the predator following strategy g. f ðvd0 Þ ¼ S if and only if Ed1 ½wd jS; vd0 ; g 4Ed1 ½wd jNS; vd0 ; g;

ð10Þ

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(ii) For each ad A Ad , the predator who observes ad predatorily trades if and only if his expected payoff from doing so is greater than his expected payoff from not predatorily trading, conditional on the distressed following strategy f. gðad Þ ¼ P if and only if Ep2 ½wp jP; ad ; f  4 Ep2 ½wp jNP; ad ; f  ð11Þ (iii) The survival belief of the distressed, α, is based on the predator following strategy g. (iv) The survival belief of the predator β is formed using Bayes rule and based on the distressed following strategy f.

Conditions (i) and (ii) of Definition 2 require that the strategies of the distressed and the predator are sequentially rational given their beliefs. Conditions (iii) and (iv) state that the belief system must be consistent given the strategy profile of the players. Thus, the equilibrium definition is that of a standard perfect-Bayesian equilibrium, in which the distressed is the sender and the predator is the receiver. Furthermore, we prove shortly that in this game there are no out-of-equilibrium beliefs. Decision rule of the distressed trader: Consider first the decision of the distressed trader in stage 1. The expected payoffs for the distressed from seeking out loans and from not seeking out loans are given in (7) and (6), respectively. Combining these with the distressed's decision rule stated in (10), it follows that the optimal action for the distressed trader may be expressed as follows. Lemma 1. Given initial type vd 0 and conditional on the predator following strategy g, the distressed trader seeks out a loan if and only if

αðNSjvd0 ; gÞ o αðSjvd0 ; gÞ

ð12Þ

The above lemma gives a simple rule in terms of the distressed's beliefs for when it is optimal for the distressed to seek out a loan. Lemma 1 states that the distressed trader will seek out a loan if and only if the probability of survival from seeking out a loan is greater than the probability of survival from not seeking out a loan. This seems fairly intuitive–seeking out a loan is optimal only if it increases the distressed's chances of survival. Lemma 1 gives a simple decision rule for the distressed trader, as a function of his initial type. Using this decision rule, it is now clear that for the two extreme types—the low-type and the high-type—seeking out a loan and not seeking out a loan, respectively, are strictly dominant strategies. This is stated in the following lemma. Lemma 2. For any strategy of the predator, the low-type always finds it optimal to seek out a loan. Likewise, for any strategy of the predator, the high-type always finds it optimal to not seek out a loan. The proof of Lemma 2 is very simple. Consider first the low-type's decision. For any strategy of the predator, if the lowtype decides not to seek out a loan, his probability of survival is zero (pℓ ¼ 0), since no matter what happens in stage 2, the exogenous income shock in stage 3 will force the firm to liquidate. On the other hand, for any strategy of the predator, if the low-type decides to seek out a loan, his probability of survival is strictly positive. (12) is hence satisfied for all g. Therefore, no matter the strategy of the predator, the low-type always finds it optimal to seek out a loan. Similarly, consider the decision of the high-type in stage 1. For any strategy of the predator, the high-type's probability of survival, whether he seeks out a loan or not, is always equal to 1. This is due to the fact that neither the predator in stage 2 nor the income shock in stage 3 can force the high-type to liquidate (vh 4 vp and ph ¼ 1). Therefore, according to (12), the high-type always finds it optimal to not seek out a loan. Lemma 2 clarifies the type of equilibria that may exist in this game. The property that the low-type always seeks out a loan and the high-type never seeks out a loan, i.e. that there are dominance regions in the type space, implies that any possible equilibrium in this game must be a separating (or semi-separating) equilibrium. Any action observed by the predator is consistent with an equilibrium path, and hence no off-the-equilibrium beliefs need be specified. Finally, note that Lemma 2 also contributes to understanding the signalling nature in this game. Because the predator does not observe the type of the distressed, it can only infer information from the distressed's action. From Lemma 2, we see that regardless of the predator's strategy, it is strictly dominant for the low-type to seek out a loan, and it is strictly dominant for the high-type to not seek out a loan. Therefore, when the medium-type trader makes his decision whether or not to seek out a loan, he in effect chooses whether to be pooled with the low-types or to be pooled with the high-types. In this way, his action conveys information to the predator. The medium-type is thus the “marginal trader”. Decision rule of the predator: Consider next the decision of the predator in stage 2. The expected payoffs for the predator from trading and from not trading are given in (8) and (9), respectively. Combining these with the predator's decision rule stated in (11), it follows that the optimal action for the predator may be expressed as follows. Lemma 3. Conditional on observing action ad and on the distressed following strategy f, the predator predatorily trades if and only if 1  βðPjad ; f Þ m o o1 Δ βðPjad ; f Þ

ð13Þ

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Much like Lemma 1, Lemma 3 gives a simple threshold rule for the predator, in terms of his beliefs, for when it is optimal to predatorily trade. To see this, note that the left-hand side of Eq. (12) is simply the ratio of the probability of failing to the probability of succeeding, if the trader decides to predatorily trade. Lemma 3 states that the predator will predatorily trade if and only if this ratio is sufficiently low. The cut-off for this ratio, i.e. the right-hand side of Eq. (12), is a constant which is increasing in the gain from winning a predation war, m, but decreasing in the liquidation penalty Δ. Therefore, the predator finds it optimal to predatorily trade if the probability of survival conditional on predatorily trading is sufficiently high. Similarly, a lower gain from predatorily trading or a higher liquidation penalty makes it less likely that the predator will find it optimal to predatorily trade. The decision rules stated in Lemmas 1 and 3 greatly simplify the equilibrium analysis. In particular, they lead directly to the following non-existence results. Lemma 4. (i) No equilibrium exists in which the predator follows a strategy in which he always predatorily trades. (ii) No equilibrium exists in which the predator follows a strategy in which he predatorily trades if and only if he observes that the distressed did not seek out a loan. This lemma implies that in any equilibrium, the predator is either playing a strategy in which he never predatorily trades, or one in which he predatorily trades if and only if he observes that the distressed sought out a loan. In the former case, the equilibrium of the game will be similar to that in the benchmark without the predator. In the latter case, the predator's strategy creates an incentive for the distressed trader to refrain from seeking loans. This analysis leads to the main result of this section. Proposition 2. If pm 4 π m p^

and

m=Δ 4 κ ℓ ; where κ ℓ 

πℓ

1  πℓ

then there exists an equilibrium in which the low-type distressed trader seeks out a loan, the medium and high-type distressed traders do not seek out a loan, and the predator predatorily trades if and only if the distressed seeks out a loan. This is the central proposition of this section. The potential of predatory trading in stage 2 clearly affects the incentives of distressed banks to seek out loans. Recall that in the benchmark without any predator, the medium-type seeks out a loan. This loan helps relieve temporary financial distress, giving the distressed trader some liquidity to cushion the shock in period 3. On the other hand, with a predator, Proposition 2 demonstrates that under certain conditions seeking out a loan carries a strong stigma effect—it sends a negative signal about the distressed's liquidity, opening the door for predatory trading. In this case, the medium-type trader chooses not to seek a loan, thereby taking a more illiquid position than it would otherwise. The intuition for how this equilibrium is sustained is straight-forward. Consider the predator's decision. When the predator observes that the distressed trader seeks out a loan, he knows that this trader must be a low-type trader. Using Bayes rule, the predator forms posterior beliefs over the probability of winning in a predation war against this trader. Given that the predator knows that only the low-types seek out a loan, the predator knows it has a high chance of winning in a predation war. Thus, as long as the expected return to predatorily trading is high relative to the expected loss, the predator finds it optimal to predatorily trade when the distressed trader seeks out a loan. This condition is met when m=Δ 4 κ ℓ . The left-hand side is the ratio of the gain from winning to the loss from losing, while the right-hand side is the ratio of the probability of losing to the probability of winning the predation war. If only low-types seek out a loan, the ratio κ ℓ is rather low, and hence this condition is easily satisfied. Thus, even though the predator cannot directly observe types, seeking out a loan is a strong signal that the distressed is very illiquid. For this reason the predator finds it optimal to predatorily trade against any trader it observes seeking out a loan. The predator's equilibrium strategy provides a strong incentive for the distressed medium-type trader to refrain from seeking out loans. Consider the decision of the medium-type distressed trader. If the medium-type chooses to not seek out a loan, then he will not face any predation risk; the only risk he faces is the exogenous liquidity shock in stage 3. On the other hand, if the medium-type chooses to seek out a loan, he then engages in a predation war in stage 2, which he can win only if he receives a loan large enough to make him a strong-type trader. Therefore, when the probability of surviving the liquidity shock as a medium-type is high relative to the probability of becoming a strong type (that is, when the ratio pm =π mh is sufficiently high), then the medium-type finds it optimal to not seek out a loan. In other words, the medium-type trader prefers not to borrow and consequently face greater liquidity risk. In summary, the potential of predatory trading clearly affects the incentives of banks and financial institutions to seek out liquidity in times of distress. In deciding whether or not to seek out a loan, a distressed bank faces a trade-off between the financial cushion provided by a loan and the information that obtaining a loan reveals. In this equilibrium, the mediumtype distressed trader who would otherwise wish to obtain more liquidity is reluctant to do so in the presence of predators. Instead, in order to not signal any weakness, the medium-type pools himself with the high-type bank and refrains from seeking outside liquidity. This trader thus takes on a more illiquid position than he would otherwise, subjecting himself to greater liquidity risk.

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pm

m /

m /

m

pˆ

pm

m

pˆ

m

(

,

m

)

m /

Fig. 3. Equilibria regions of the parameter space.

The set of all possible equilibria. Finally, we may in fact characterize the set of all possible pure-strategy equilibria in this game. Proposition 3. Let κ ℓ  π ℓ =ð1  π ℓ Þ and κ ℓm  ðπ l þ π m Þ=ðð1  π l Þ þ ð1  π m ÞÞ. In this game there are three types of pure-strategy equilibria, each of which exist in particular regions of the parameter space. Type I: An equilibrium in which the low-type distressed trader seeks out a loan, the medium and high-type distressed trader do not seek out a loan, and the predator predatorily trades if and only if the distressed seeks out a loan exists whenever m=Δ 4 κ ℓ ^ and pm 4 π m p. Type II: An equilibrium in which the low and medium-type distressed trader seeks out a loan, the high-type distressed trader does not seek out a loan, and the predator predatorily trades if and only if the distressed seeks out a loan exists whenever ^ m=Δ 4 κ ℓm and pm o π m p. Type III: An equilibrium in which the low and medium-type distressed trader seeks out a loan, the high-type distressed trader does not seek out a loan, and the predator never predatorily trades exists whenever m=Δ o κ ℓm . These regions of equilibria are presented in a diagram in Fig. 3. As seen in the figure, in most regions of the parameter ^ space the equilibrium is unique. The only non-generic region in which there is not a unique equilibrium is when pm 4 π m p, and κ ℓ o m=Δ o κ ℓm . Here, two types of pure-strategy equilibria exist: type I and type III.17 Note that the existence of this region of multiple equilibria depends on κ ℓm being strictly greater than κ ℓ , which occurs when πm is strictly greater than πl. If instead π m ¼ π l , then κ ℓm ¼ κ ℓ , and in this case, there would be no region of multiplicity. In all parts of the parameter space, the equilibrium would be generically unique. The Type I equilibrium was discussed above in Proposition 2. We now discuss the intuition for the two other types of equilibria. ^ If the ratio pm =π m p^ is sufficiently low, that Type II : Starting from the type I equilibrium, suppose instead that pm o π m p. is, if the medium-type's probability of surviving the exogenous liquidity shock is low relative to the probability of becoming a strong type after receiving a loan, then the medium-type distressed trader finds it optimal to seek out a loan. In this case, the medium-type knows it's unlikely to survive the income shock without borrowing, and thus must seek out a loan despite the fact that this means it will be subject to a predation war. That is, when the ratio pm =π m p^ is sufficiently low, the increase in survival probability the medium-type gains from seeking out a loan is high enough to compensate for the increased predation risk. Therefore, despite the danger of predatory trading, the medium-type finds it optimal to mimic the low-type trader and seek out a loan. When the predator observes that the distressed trader seeks out a loan, he knows that this trader must be either a low or a medium-type. The predator forms posterior beliefs over the probability of winning in a predation war against this trader. Again, as long as the expected return to predatorily trading is higher than the expected loss, the predator finds it optimal to predatorily trade when the distressed trader seeks out loans. This condition is met when m=Δ 4 κ ℓm . The left-hand side is still the ratio of the gain from winning to the loss from losing, while the right-hand side is the ratio of the probability of losing to the probability of winning the predation war, conditional on both low and medium-types seeking out loans. Note that this condition is stricter than in the type I equilibria, because κ ℓm 4 κ ℓ . This is due to the fact that in this equilibrium, both medium and low-types seek out a loan, whereas in the type I equilibrium, only the low-types seek out a loan. Thus, the probability of losing the predation war is higher when medium-types also seek out a loan, and hence the ratio of gains to losses, m=Δ, must be higher in order for the predator to find it optimal to predatorily trade. 17 Of course, there also exists a third equilibrium in mixed strategies. In this equilibrium, the distressed trader's strategy is one in which the low-type seeks out a loan, the high-type does not seek out a loan, and the medium-type randomizes between the two actions. Furthermore, the predator's strategy is one in which it randomizes between predatorily trading and not predatorily trading after observing either action.

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When these conditions are met, there exists an equilibrium in which both the low and the medium-type distressed traders seek out a loan, the high-type does not seek out a loan, and the predator predatorily trades if and only if he observes that the distressed trader seeks out a loan. In fact, in this region, this equilibrium is unique. Type III : Finally, starting from the type II equilibrium, suppose instead that m=Δ o κ ℓm . In this case, the predator finds it optimal to never predatorily trade—as long as both the low and medium-types seek out a loan, the ratio of probabilities of losing the predation war to winning the predation war is too high relative to the relative gain from winning. As for the distressed trader, it is as if the predator did not exist. Hence, the distressed trader behaves in the same way as it does in the benchmark case without the predator; that is, the low and medium-types seek out a loan while the high-type does not.18 5. The Fed's Term Auction Facility The problem of discount window stigma is real and serious. The intense caution that banks displayed in managing their liquidity beginning in early August 2007 was partly a result of their extreme reluctance to rely on standard discount mechanisms… Central banks eventually were able to partially circumvent this stigma by designing additional lending facilities for depository institutions. —Donald Kohn, then Vice Chairman of the Federal Reserve Board of Governors, in a May 2010 speech at Carleton University, Ottawa, Canada19 In response to the low use of the Fed's discount window at the onset of the crisis, the Federal Reserve introduced the Term Auction Facility, or the TAF as it was called, in December 2007. This facility, with its competitive auction-determined interest rate and allocation of funds, was viewed as an attempt by the Fed to limit the stigma associated with accessing central bank liquidity.20 This section examines within the context of this model whether a policy such as the TAF may in fact alleviate the stigma effect on borrowing. The model with TAF: In order to incorporate a TAF-like facility, the model is augmented in the following way. First, we assume that there is a unit mass of distressed traders. The distressed trader's types are i.i.d. and drawn from the same distribution given in (2). By the law of large numbers, 1/3 of distressed traders are high-type, 1/3 are medium-type, and 1/3 are low-type. We further assume a unit mass of predators, one for each distressed trader.21 Next, suppose that after facing the debt roll-over risk, but before the loan seeking stage, there is another stage. This stage is called the TAF stage. After learning their initial types, the distressed traders now have the opportunity to participate in the TAF, in which the Fed auctions off loans from a fixed total loan supply of S. We assume that the loan supply is restricted.22 Each distressed trader that wants to participate submits a bid specifying the rate and the quantity at which it is willing to transact. Each bank is allowed only one rate-quantity offer.23 That is, each trader submits a step-like demand function as follows: ( 0 if r 4r i δi ðrÞ ¼ ^ v  vi if r rr i where r is the interest paid on the loan of size v^  vi . Aggregate demand for TAF loans is then given by the sum of all bids Z DðrÞ ¼ δi ðrÞ di I

where I is the set of all bidders. Once bids are submitted, the Fed then collects all bids and determines the maximum rate at which demand weakly exceeds supply. Specifically, market clearing occurs at the maximum rate r n such that S r Dðr n Þ. The market clearing rate r n is called the TAF “stop-out” rate. As in the real TAF auction, loans are allocated via a uniform-price auction. This means that all bidders pay the stopout price for all units they request at prices exceeding the stop-out-price.24 For simplicity, we assume that there is no rationing of leftover funds. Finally, a winning bidder does not pay the interest on the TAF loan until the very end of the game in period 3, when final payoffs are realized. ^ This is true for all values of pm =π m p. “The Federal Reserve's Policy Actions during the Financial Crisis and Lessons for the Future.” Speech given by Donald Kohn at Carleton University, Ottawa, Canada, May 13, 2010. 20 “By lending term funds at market-determined rates using an auction mechanism, one of the Fed's objectives in designing the TAF was to eliminate the stigma concerns that were believed to have affected the discount window” (Armantier et al., 2011, p. 3). 21 The set-up for the predatory war is thus exactly the same as in the preceding section: for each match, there is one predator versus one distressed trader. 22 We restrict our attention to fixed loan supplies low enough so that no more than one type of trader can win the TAF loan. Otherwise the TAF would be no different from the discount window. This is consistent with the fact that demand for loans exceeded supply from the first TAF auction in December 2007 up until the auction immediately following Lehman's bankruptcy in September 2008 (Benmelech, 2012). 23 The actual TAF allowed each bidder to make two rate-quantity offers. For simplicity we allow for only one bid. 24 “The Federal Reserve also chose a uniform-price (or single-price) auction rather than a discriminatory (pay-your-bid) auction in part to spur participation further. By using the uniform-price structure common in Treasury auctions, the Fed reasoned that banks would be more comfortable with bidding” (Armantier et al., 2008, p. 6). 18

19

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While the bids in the auction are sealed and not observable to outsiders, the identities of the TAF winners are observable. Thus, the predator, by observing which banks win the TAF, may be able to infer some information about the winners' types. After the TAF auction takes place and funds are distributed, the distressed traders who did not win a loan may still seek out loans in the non-TAF market. The rest of the game remains the same as before. Specifically, the game transitions to the loanseeking stage in the non-TAF market, and then to stages 2 and 3. As at the discount window, TAF borrowing is fully collateralized and the loan rules are the same across the two facilities. In fact, according to the rules posted on the Federal Reserve bank's website, assets used as collateral in TAF auctions are simply “any collateral eligible to secure discount window loans,” and that, “the Federal Reserve Bank's standard valuation and haircut procedures apply.”25 Hence, to make the loan rules across the two facilities identical, we assume that the v^ for TAF loans are drawn from the same distribution as that for non-TAF loans. The only difference, then, between borrowing at the discount window or via the Term Auction Facility is that TAF loans are dispersed through a competitive auction. Remarks. First, relative to the previous analysis, there are now two ways a bank can borrow: either via the TAF auction or through the original non-TAF market. In terms of interpretation, borrowing through the original non-TAF market should be understood as borrowing either at the Fed's discount window or through the interbank market; in this model, the distinction between the two is not made.26 This lack of distinction is made for simplicity and it presumes that within the model the discount rate is similar or equal to the interbank rate; we combine the two into one rate, simply referred to as the “non-TAF interest rate”. This assumption may be justified on the basis of the similarity of these two rates in the months leading up to the creation of the TAF; in August 2007, “the one-month term interbank rate was almost equal, on average, to the expected discount rate” (Armantier et al., 2008, p. 4). This analysis furthermore presumes that funds lent at the non-TAF rate, i.e. the primary discount rate, are rationed. That is, the Fed sets the discount rate, and then lends to any bank at that rate with the proper collateral. Thus, the non-TAF interest rate is not a market-determined interest rate like the TAF stop-out rate; it is a rate controlled by the Fed and rationed accordingly.27 For simplicity the non-TAF interest rate, as in the previous section, is normalized to zero. Moreover, this normalization implies that the market clearing TAF stop-out rate should be interpreted as the interest rate differential paid on TAF loans over the non-TAF interest rate. If this difference is positive, then the model implies that TAF loans sell at a premium over the primary discount rate; if it is negative, then TAF loans sell at a discount. As mentioned in the introduction, TAF loans during the crisis were often auctioned at a premium over the discount rate, despite the same term length and collateral requirements. Armantier et al. (2011) find that in 11 out of the 13 consecutive TAF auctions leading up to the Lehman bankruptcy, the TAF stop-out rate was well above the discount window primary credit rate. These authors moreover find that the highest bids in these auctions were on average 37 basis points over the primary discount rate, jumping to 150 basis points in the auction immediately following Lehman's bankruptcy.28 One of the primary goals of this section is to examine whether this model can deliver similar behavior in the corresponding interest rate differential. 5.1. TAF with unobservable winners In this paper the preferred specification for the predators' information regarding TAF is that the predators observe which banks win TAF loans, but do not observe the TAF loan sizes. This ensures that the only difference between obtaining a loan at the discount window and one through TAF is TAF's competitive auction format. However, in this subsection we first analyze a simple benchmark in which TAF lending is completely anonymous and unobservable by predators. In this case, introducing the TAF necessarily eliminates the stigma problem by allowing distressed banks to obtain liquidity without any risk of predation. To demonstrate this, we first restrict attention to the parameter region in which a type I equilibrium is in effect without the TAF. In the type I equilibrium presented in Section 4, the medium-type distressed traders are adversely affected by the stigma of borrowing. In this equilibrium, high-type traders are not affected by predatory trading at all, while the low-types, although predatorily traded against, still seek out loans on the interbank market or at the discount window, and are thereby not deterred by the stigma of borrowing. The medium-type traders, on 25 For institutional details on the TAF implementation, see Armantier et al. (2008) as well as the FAQ on the Federal Reserve System Board of Governors website, 〈http://www.federalreserve.gov/monetarypolicy/taffaq.htm〉. 26 See Ennis and Weinberg (2013) for a model which distinguishes between the two borrowing channels. 27 According to Marvin Goodfriend, editor and former Senior Vice President and Policy Advisor at the Federal Reserve Bank of Richmond: “Before 2003 the Fed kept its primary discount rate below the federal funds rate. The Fed ran a relatively extensive program to ration primary discount borrowing, to control the subsidy available to any bank at the discount window. Discount window officers at the Fed made sure that no single bank borrowed for too long or too often. The primary discount rate was moved permanently above the federal funds rate in 2003, i.e., to a penalty rate, in part to reduce the need to ration discount window borrowing. Only troubled banks would find it in their interest to borrow at the primary discount rate after 2003. A bank would borrow at the window after 2003 only when forced by the market to pay a ‘premium’ above the federal funds rate at which most private interbank borrowing occurred. The Fed might ration banks that came to the window, however, because such banks would still benefit from a subsidy—a primary discount rate lower than the rate they would have to pay to borrow in the interbank market.” 28 These auctions cover the period of April to September 2008; see Figure 8 of Armantier et al. (2011). The authors also find that the fraction of banks bidding above the discount rate during this period was large, more than 55 percent, with a generally increasing trend; see Figure 4 of Armantier et al. (2011).

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the other hand, are unwilling to borrow at the discount window (or from other banks) for fear of predatory trading. Therefore, in this case there is a strong stigma effect associated with borrowing as it deters these banks from seeking out loans. Starting from this equilibrium, the introduction of the Term Auction Facility with unobservable winners can eliminate stigma by providing the medium-type traders with a strong incentive to borrow from the Fed through this facility. This insight is formalized in the following proposition. Proposition 4. Suppose pm 4 π m p^ and κ ℓ o m=Δ and there is a type I equilibrium without the TAF. Suppose TAF winners are unobservable. In this case, the high-type distressed traders do not seek out a loan nor participate in the TAF market. Both the medium and low-type traders submit bids to TAF. If in equilibrium the medium-type traders win the TAF, then these traders are not predatorily traded against. If in equilibrium the low-type traders win the TAF, then as long as m=Δ is sufficiently low these traders are not predatorily traded against. Thus, when TAF winners are unobservable to predators, both the low-type and medium-type banks submit bids to the TAF auction. Bidding in the TAF becomes a dominant strategy for these traders, as winning TAF loans puts them in no risk of predatory trading. Comparing this equilibrium to the type I equilibrium before the introduction of the TAF, the medium-type banks that were originally reluctant to borrow at the discount window (or on the interbank market) are now willing and eager to borrow via the TAF. This is essentially due to the fact that the predator is unable to identify which banks obtain liquidity through TAF and which banks do not. Hence, the stigma problem is trivially eliminated. While this analysis demonstrates that the introduction of TAF can alleviate the stigma problem, this result is solely due to the assumption that TAF winners are unobservable to outsiders. However, one might consider this assumption quite strict. In particular it seems difficult to justify how the Fed can keep the identities of TAF winners a secret, but not the identities of banks borrowing through the discount window. Thus, one would expect that obtaining a loan through the Term Auction Facility, as at the discount window, may be partially observable to third parties. Furthermore, note that with unobservability of TAF winners, the competitive auction feature of the TAF plays no role in eliminating stigma. That is, the auction feature only determines which particular banks win the loans ex post, but it does not in any way affect the incentives for distressed banks to participate in the TAF ex ante. Participating in the TAF is a strictly dominant strategy for both low and medium-type banks, regardless of whether or not the lending facility is an auction.29 This underscores that the key factor for reducing stigma in this analysis is simply the assumption of unobservable TAF winners. 5.2. TAF with observable winners In this subsection the assumption of complete unobservability of TAF borrowing is relaxed. Specifically, we assume that bids submitted to the auction are sealed and unobservable by third parties, however the identities of the TAF winners are observable. This is a weaker and arguably more plausible assumption on the informational asymmetry between distressed and predatory traders. In this case, a distressed bank cannot necessarily avoid predatory trading by simply obtaining a TAF loan. Under these weaker conditions, the Term Auction Facility may still manage to alleviate the stigma effect associated with borrowing. To demonstrate this, we again restrict attention to the parameter region in which a type I equilibrium is in effect without the TAF. Starting from this equilibrium, the introduction of the Term Auction Facility even with observable auction winners can alter the equilibrium outcomes so that the medium-type traders become willing to borrow. This is formalized in the following Proposition. Proposition 5. Suppose pm 4 π m p^ and κ ℓ o m=Δ and there is a type I equilibrium without the TAF. If m=Δ A ðκ ℓ ; κ m Þ and vm is sufficiently greater than vℓ , then there exists bounds S 1 ; S such that for all loan supplies S A ðS 1 ; SÞ, there exists an equilibrium characterized as follows. The high-type distressed traders do not seek out loans nor participate in the TAF market. Medium and low-type traders submit bids to TAF. The medium-types win the TAF loans and pay a positive premium over the non-TAF interest rate. The low-types seek out loans in the non-TAF market. Finally, predators predatorily trade only against traders who seek out loans in the non-TAF market, but not against traders who win the TAF loans. Proposition 5 demonstrates that the Term Auction Facility can alleviate the stigma effect associated with borrowing. In this equilibrium, the medium-type banks require a smaller loan quantity than the low-type banks, and hence are willing to pay a higher interest rate for loans. The medium-type traders thus submit strictly higher interest-rate bids than the lowtype traders, and thereby win the TAF loans. In this equilibrium the signal from winning a TAF loan thus becomes a positive signal of liquidity. Because medium-type banks are financially stronger than the low-type banks, in equilibrium predators optimally choose not to predatorily trade against these banks.30 Furthermore, because the predators in equilibrium do not trade against the TAF winners, the medium-type traders are not deterred from bidding in TAF and winning these loans. 29 In other words, the equilibrium in Proposition 4 would be essentially the same if the Fed had instead opened another “discount window” facility, devoid of any auction mechanism for allocating funds, but ensured that borrower identities at this new facility were unobserved. 30 The condition that κ ℓ o m=Δ o κ m simply implies that the predator is not willing to trade against the medium-type traders who win the TAF, but are interested in trading against the low-types who receive loans outside of TAF.

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Interpretation and remarks: The equilibrium in Proposition 5 coincides with how the Term Auction Facility was understood and promoted by the Fed. In the fall of 2007, the Fed was concerned that a significant fraction of banks were in need of liquidity yet were unwilling to borrow at reasonably low rates at the discount window. This reluctance was interpreted as evidence of a possible stigma effect associated with borrowing. In response to the low level of discount window borrowing, the TAF was created and promoted by the Fed as another avenue through which distressed banks may obtain loans. It may be considered a puzzle as to why the TAF loans, with exactly the same term length and collateral requirements as the discount window, were sold at auction at considerable premiums over the primary discount rate. This paper provides a unique rational explanation for this behavior. In this model, the banks adversely affected by the stigma of borrowing are the medium-type banks, who, unlike the hightype banks, suffer from illiquidity concerns but, unlike the low-type banks, are reluctant to seek out loans. Without the TAF, medium-type banks are unwilling to borrow, thereby taking on illiquid positions. With the introduction of the TAF, however, medium-type banks are eager to use this facility to obtain liquidity even at a premium, as it allows them to signal that they are financially stronger than the weakest banks.31 In contrast to the preceding subsection in which the TAF necessarily eliminates stigma when TAF borrowing is unobservable to third parties, the result in Proposition 5 provides an explanation under much weaker informational assumptions. It demonstrates that even with allowing for the observability of TAF winners, the introduction of this facility can still reduce stigma. As borrowing though the TAF auction becomes a positive, rather than a negative, signal of financial liquidity, medium-type banks are willing to borrow at a premium for these loans. Note that it is primarily by winning the TAF loan that changes the signal value. This highlights that the competitive auction feature is now an essential component of the overall mechanism: it is precisely the competitive auction which makes the TAF effective. “What sets the TAF apart from the discount window, however, is the competitive auction format and use of a market-determined interest rate” (Armantier et al., 2008, p. 2).32 Note further that through the lens of this model, TAF is not a policy designed to help the low-type banks. The low-type banks are always willing to borrow, either through the interbank market, at the discount window, or via the TAF auction. Low-type banks are thus not deterred from borrowing by the stigma effect—even in the presence of predators they seek out loans otherwise they would fail. This distinction is important: as the TAF was a policy designed to alleviate stigma, it should thus be interpreted as an avenue for channeling liquidity to banks that were previously unwilling to borrow at the discount window or on the interbank market, i.e. the medium-type banks. It is only through separating the medium-types from the low-types via the auction that the Fed is able to inject liquidity into the former.33 Finally, one might wonder whether there exists an equilibrium in which the low-type banks win the TAF auction. Could it be possible that, in equilibrium, the financially weakest banks submit the highest TAF bids as these banks are the most desperate for liquidity? The answer to this question is considered next. Equilibria in which the low-types win the TAF auction: In Proposition 5, the Term Auction Facility reduces the stigma effect by channeling liquidity to the medium-type banks. However, this may not always be the case. The following proposition gives conditions under which the low-type banks win the TAF auction in equilibrium. We again start from the type I equilibrium without the TAF. In any equilibrium with the TAF, it is a dominant strategy for the high-type distressed traders to not seek out loans nor participate in the TAF market. Thus to avoid repetition, equilibria from now on will be stated only in terms of the behavior of the low-types, medium-types, and the predators. Proposition 6. Suppose pm 4 π m p^ and κ ℓ om=Δ and there is a type I equilibrium without the TAF. (i) If m=Δ A ðκ ℓ ; κ m Þ, then there cannot exist an equilibrium in which the low-types win the TAF auction and the medium-types lose the auction. (ii) If m=Δ 4 κ m , then there exists bounds S 2 ; S such that for all loan supplies S A ðS 2 ; SÞ there exists an equilibrium in which the low-types win the TAF auction. In this equilibrium, the medium-types lose the TAF auction and do not seek out a loan in the nonTAF market. Predators predatorily trade against all traders who win a TAF loan and all traders who seek out a loan in the non-TAF market. Finally, there is a zero interest rate spread between the TAF stop-out rate and the non-TAF interest rate. The first part of the Proposition 6 states that under certain parameter restrictions, there cannot exist an equilibrium in which the low-types win the TAF auction and the medium-types do not win. An equilibrium such as this cannot exist for the following reason. If the low-types win the TAF auction and the medium-types do not, predators find it optimal to predatorily trade only against the winners of the TAF auction and no one else. This is optimal for predators as long as m=Δ o κ m . However, if this were the case, then the low-type traders would have no incentive to win the TAF auction as they would be better off losing the TAF and mimicking the medium-type traders by borrowing outside of TAF. This immediately proves that the proposed equilibrium is not incentive compatible and hence cannot be sustained.

31

This intuition here is similar to the original signalling motive provided in Spence's (1973) job-market signalling model. In addition, Gilchrist (2012) writes, “[the TAF] also potentially differed from discount window lending in that the primary credit rate was the minimum bid rate but not necessarily the effective borrowing rate. Thus, the TAF gave the Federal Reserve the potential to ‘price-discriminate’ between the two forms of borrowing” (Gilchrist, 2012, p. 94). 33 If the Fed could somehow eliminate predatory trading entirely (or more generally any stigma-inducing behavior), this action would benefit the lowtype traders as well as the medium-type traders. However, under the constraint that the Fed cannot generally abolish predatory trading or any other stigma-inducing behavior, Proposition 5 establishes that the TAF may be effective in mitigating the stigma problem. 32

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The second part of the proposition states that if m=Δ is sufficiently large, it is possible to construct an equilibrium in which the low-type distressed traders win the TAF loans. In order for this equilibrium to exist, the predator must find it optimal to predatorily trade against all traders that obtain a loan, either through the TAF or outside of TAF. This is optimal for the predator as long as m=Δ 4 κ m . That is, the predator's expected benefit outweighs the expected cost of trading against either the low-types or the medium-types who have borrowed. In this equilibrium, the medium-type traders do not seek out a loan even outside of TAF. This implies that along the equilibrium path, no trader seeks out a loan in the non-TAF market. Thus, in order to ensure that the predator trades against any trader seeking out a loan, we must specify off-the-equilibrium path beliefs. Here the intuitive criterion is applied (Cho and Kreps, 1987). In particular, if off-the-equilibrium a trader is observed to seek out a loan in the non-TAF market, the predator infers this to be a medium-type. The reason for this is that the medium-type trader has a higher incentive to seek out a loan than the high-type trader, and therefore is more likely to deviate from the equilibrium action of not seeking out a loan. Therefore, by applying the intuitive criterion, the predator has the incentive to trade against this bank. Finally, if the predator predatorily trades both against the traders that win the TAF auction and the traders that seek loans in the non-TAF market, then the low-type is indifferent between either avenue for borrowing. For the low-type trader there is no difference between a loan obtained via the TAF and a loan obtained through the non-TAF market; given the predator's strategy, in either case the low-type is predatorily traded against. Therefore, in this equilibrium the low-type submits a TAF bid equal to the interest rate paid in the non-TAF loan market. Furthermore, given the predator's strategy, winning the TAF auction is worse for the medium-type trader than losing. The medium-type traders are thus only willing participate in the TAF at a negative interest rate (an interest rate lower than the non-TAF rate). In this equilibrium, the medium-types thereby lose the TAF auction to the low-types. Therefore, when m=Δ is sufficiently large, an equilibrium in which the low-types win the TAF auction can be constructed. This equilibrium works in the opposite way as one might expect: the medium-type chooses to lose the auction due to the underlying stigma effect. Furthermore, this equilibrium features no interest rate spread between TAF loans and non-TAF loans. As long as the low-type is indifferent between the TAF auction and the non-TAF market, these rates must be equal. Interpretation and remarks: Proposition 6 clarifies whether it is possible for an equilibrium to exist in which the financially weakest banks win the TAF. This proposition demonstrates that such an equilibrium may be possible, although its characteristics may not be compatible with what was observed during the 2008 financial crisis. First, if the predator is sufficiently weak, starting from a type I equilibrium without the TAF, there cannot exist an equilibrium in which the low-type banks win the TAF auction, as any such equilibrium cannot be sustained. On the other hand, if the predator is sufficiently strong, an equilibrium may be constructed in which the low-type banks win the auction. This equilibrium, however, can only be sustained at a zero interest rate spread between the TAF stop-out rate and the nonTAF interest rate, i.e. the primary discount rate. This lack of spread is a direct result of the low-type banks' indifference between winning TAF loans and borrowing at the discount window. In either case, the predator trades against the bank, and hence the two options yield the same probability of survival. The equilibrium's characteristics thus run counter to the possible intuition that the financially weakest banks submit the highest TAF bids out of desperation. Furthermore, although an equilibrium in which the low-types win the TAF auction can be constructed, this equilibrium seems unlikely to have materialized during the height of the 2008 financial crisis. The zero interest-rate spread featured in this equilibrium is clearly incompatible with the positive and significant premia on TAF loans over the discount rate observed in the majority these auctions (Armantier et al., 2011). Finally, note that in this equilibrium the TAF has no effect on equilibrium outcomes relative to the pre-TAF outcome. In terms of allocations, the TAF is a wash—the low-types obtain loans and have the same probability of survival as before, while the medium-types are still unwilling to borrow due to the stigma effect. As such, the allocation is identical to the allocation in the type I equilibrium without the TAF. Therefore, in this equilibrium, the TAF makes no progress towards alleviating stigma. The effects of TAF when the stigma effect is weak: Finally, we consider conditions under which the TAF may become a potentially harmful policy tool. Unlike the previous analyses, we restrict our attention to the region in which there is a type III equilibrium without the TAF. Recall that in this equilibrium the low-type and medium-type traders both seek out loans, and the predators never predatorily trade. The next proposition demonstrates that a lending facility such as the TAF may be problematic under such circumstances. Proposition 7. Suppose m=Δ o κ ℓm and there is a type III equilibrium without the TAF. That is, the low and medium-types both seek out a loan, the high-types do not seek out a loan, and the predators never predatorily trade. ^ and vm is sufficiently greater than vℓ , then there exists bounds S 1 ; S such that for all loan supplies If m=Δ A ðκ ℓ ; κ ℓm Þ, pm 4 π m p, S A ðS 1 ; SÞ there exists an equilibrium characterized as follows. Medium and low-types traders submit bids to TAF, the mediumtypes win the TAF loans, and the low-types seek out loans in the non-TAF market. Predators only predatorily trade against traders who seek out loans in the non-TAF market Proposition 7 suggests that the TAF can be counterproductive when the stigma effect on borrowing is particularly weak. Under these conditions, without TAF there is an equilibrium in which the low-types benefit from pooling with the mediumtype banks. This ensures that both medium and low-type banks obtain extra liquidity by borrowing. As long as m=Δ o κ ℓm , the predator's chances of winning in a predation war is too low to warrant predatory trading, as the predator is unsure as to

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whether he is facing a low-type or a medium-type bank. On average the pool of traders he faces is too strong, thereby deterring him from predatorily trading. However, the introduction of the TAF can lead to an equilibrium in which the medium-types win the TAF loans, separating themselves from the low-types, while the low-types are then left to seek out loans in the non-TAF market. The TAF auction thereby cream-skims the medium-type banks: medium-types win the TAF loans and consequently do not face any predatory trading. This nevertheless implies that the predators can now infer which traders have the lowest liquidity, as these are the only traders seen seeking out loans outside of TAF. The predators thus find it optimal to predatorily trade against these banks, making the low-types worse off in this environment. Interpretation and remarks: While Proposition 5 demonstrates that the TAF may be an effective policy tool when the stigma effect is strong, Proposition 7 warns that the TAF may be detrimental if employed during normal times. If the stigma effect on borrowing is weak, it is advantageous for the financially weakest banks to pool themselves with the relatively stronger banks. But in this case, a policy such as the TAF may separate the relatively strong banks from the general pool, thereby hurting the weakest banks and leaving them susceptible to predatory trading. In conclusion, this model suggests that the Term Auction Facility may be an effective policy tool, but should be used only in times of crisis, when a stigma effect strongly discourages banks from borrowing. Finally, as an end note, the model presented in this paper completely abstracts from issues of insolvency. All distressed traders are solvent, but some are more illiquid than others. This means that it is beneficial to protect all types of traders from predatory trading, including even the low-types. In reality, however, it may not be the case that all banks during financial crises remain solvent, and as such it may be efficient to allow some banks to fail. Future welfare analysis of the stigma effect on borrowing would potentially take both illiquidity and solvency into account, making the environment and policy analysis richer. 6. Conclusion This paper analyzes how predatory trading may affect the incentives of banks and financial institutions to borrow in times of financial distress. When a distressed trader is more informed than other traders about its own balance sheet, seeking out extra liquidity from lenders can become a signal of financial need, thereby opening the door for predatory trading and possible insolvency. Under certain conditions, an equilibrium may arise in which some distressed traders who would like to borrow short-term in order to meet temporary liquidity needs, may be reluctant to do so in the presence of potential predators. These banks thus take on more illiquid positions than they would otherwise. Predatory trading may therefore deter banks and financial institutions from raising liquidity in times when they need it the most. This paper furthermore provides a unique theoretical interpretation for why the Term Auction Facility may have been an effective policy instrument of the Fed during the 2008 financial crisis. At a time when the Fed suspected that a strong stigma effect was discouraging banks from borrowing at the discount window, TAF auctions became an effective way of injecting liquidity into distressed institutions. This model suggests that the primary reason as to why the TAF may have been effective is simply due to its competitive auction format and auction-determined interest rate. While the Fed lent freely to any bank at the discount window satisfying the Fed's collateral requirements, the auction format of the TAF ensured that the winners of these loans were only the highest bidders. Winning the TAF auction may have thus become a signal of a bank's financial strength rather than weakness, thereby attenuating any stigma associated with borrowing. As a result, banks that were originally reluctant to borrow at the Fed's discount window or on the interbank market were now willing to pay a premium for TAF loans. An intermediate result demonstrates that if predators are sufficiently strong, an equilibrium in which the financially weakest banks win the TAF auction can be constructed. In this case, the TAF has no effect on outcomes and the equilibrium features a zero interest-rate spread between the TAF stop-out rate and the primary discount rate. Thus, while this equilibrium may be possible under certain parameter restrictions, it is inconsistent with the significant and positive TAF rate premia observed during the crisis. Finally, this paper further suggests that the TAF should not be used without caution. Although the TAF may have been effective in injecting liquidity into distressed banks during the financial crisis, the facility was employed at a time when the stigma effect on borrowing was suspected to be quite strong. If instead the stigma effect had been much weaker, the TAF would have potentially been counterproductive—by separating the relatively stronger banks from the weaker ones, the latter could become easy prey for predators. Thus, in normal times an auction facility for distributing loans may not be desirable. For this reason, this paper suggests that the TAF is a policy best suited for crisis times only.

Acknowledgments I am extremely grateful for the detailed feedback of my discussant Richard Lowery and of the editor Marvin Goodfriend. For helpful comments and suggestions, I thank George-Marios Angeletos, Saki Bigio, Nina Boyarchenko, Markus Brunnermeier, Ricardo Caballero, V.V. Chari, Fernando Duarte, Thomas Eisenbach, Mark Gertler, Mike Golosov, Luigi Iovino, Guido Lorenzoni, Debbie Lucas, Rick Mishkin, Jaromir Nosal, Ricardo Reis, Jon Steinsson, Todd Walker, Venky

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Venkateswaran, Ariel Zetlin-Jones, and seminar participants at MIT, Chicago Booth, Columbia, the NYC Junior Macro Finance group, and the 2010 LACEA Annual Meeting in Medellín, Colombia. Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jmoneco. 2014.06.001.

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