Fund Runs and Market Frictions∗ Dong-Hyun Ahn†

Soohun Kim‡

Kyoungwon Seo§

November 14, 2017

Abstract We show that a financial crisis can arise even when funding liquidity is sufficiently provided in the market and there is no exogenous shock. The fully endogenous crisis is triggered by the interaction between asset prices and financial market frictions. We identify the conditions for such a crisis in the context of financial sector practices: (i) risk management of investors, (ii) performance-based fee for fund managers and (iii) margin requirements in funding markets. Our model provides novel insights into the unintended consequences of government intervention in financial markets. JEL classification codes: G01, G23, G28 Keywords: fund fee structure, leverage, risk management, systemic risk

∗

We are thankful to Andras Denis, Jiang Luo and seminar participants at KAIST, HKUST, the University of Indiana, Hitotsubashi University and the 2017 CICF for helpful feedback. Seo gratefully acknowledges the Institute of Finance and Banking and the Management Research Center at Seoul National University for providing financial support. Any remaining errors are solely ours. The usual disclaimer applies. † Department of Economics, Seoul National University, Seoul, South Korea. E-mail: [email protected]. ‡ Scheller College of Business, Georgia Institute of Technology, Atlanta, GA. E-mail: [email protected]. § Business School (Finance), Seoul National University, Seoul, South Korea. E-mail: [email protected].

1

Introduction

We show that a financial crisis can occur even when funding liquidity is sufficiently provided in the market and there is no exogenous shock. Because of sufficient funding liquidity, an equilibrium with no mispricing, referred to as an ideal equilibrium, exists in our model. However, another equilibrium with a financial crisis, referred to as a fund run equilibrium, can arise purely from a coordination failure among market participants without any exogenous shock. We show that such coordination failure can materialize in a financial market as a result of its systemic fragility induced by key features of the modern financial market, such as risk management, fee structure and margin requirements. So far, the literature on financial market crises has focused on illuminating how an exogenous shock can be amplified within a financial system through various forms of particular frictions. For examples among many, Bernanke and Gertler (1989) and Kiyotaki and Moore (1997) show that adverse shocks can depress economic activity further through the collateral channel in macroeconomics. In a context of financial markets, Gromb and Vayanos (2002) and Brunnermeier and Pedersen (2009) study the amplification of shocks through margin constraints imposed on arbitrageurs. He and Krishnamurthy (2013) endogenize the equity constraints on financial intermediary through moral hazard and show that the capital scarcity plays a critical role for price dynamics. In the literature on bank runs, many studies analyze how bad news on the fundamentals can be escalated into panics (e.g., Postlewaite and Vives (1987), Jacklin and Bhattacharya (1988), Chari and Jagannathan (1988)). Moreover, global games have been widely used in modeling the propagation of shocks through correlated signals (e.g., Morris and Shin (1998, 2001, 2004), Goldstein and Pauzner (2005)). All of these papers begin with an exogenous shock and study amplification mechanisms in which a small shock can lead to a severe crisis because of various market frictions. By contrast, our model shows that the assumption of an exogenous shock is not needed in explaining financial crises and identifies the market frictions under which a financial crisis can arise without an exogenous shock. External fundamental or preference shocks are not crucial for a fund run in our model. We only assume that a rare event occurs in the interim period. This event can be caused by any public signal, which may or may not be related to fundamentals of the economy. The only role of the event is to allow market participants to use the event as a public signal device for a coordination failure. The Flash Crash in 2010 is a good example where the price drop is not due to fundamental shocks. 2

We identify the conditions under which such a fund run can arise: (i) investors delegate their wealth management to professional fund managers but request redemption as a way to manage risk if the performance is not satisfactory, (ii) the fee schedule of fund managers may misalign the interests of investors and fund managers and (iii) in exploiting arbitrage profits, fund managers frequently employ leverage, subject to margin constraints. The risk management of investors in our model plays a critical role in transforming the best economy (ideal equilibrium) to one of the worst (fund run equilibrium), which is the opposite of the intended results from risk management. While the specific manifestations may change, a variety of risk management rules, either regulatory rules or internal rules, have long been present. For example, the BIS (Bank for International Settlements) and other regulatory bodies mandate diverse capital requirements on banks (BIS ratio), insurance firms (Risk-Based Capital Requirement) and security firms (Net Capital Rule) and also risk management guidelines on value at risk, short fall risk and many others. Many hedge funds, which are less bound by the external regulation, introduce pre-specified risk management rules on trades, such as loss-cut criteria on specific trades or various unwinding rules on all accounts under certain circumstances. Despite the popularity of external and internal risk management rules, criticism has mounted on their ‘procyclicality’, and many papers have extensively discussed the amplification of crisis through the feedback effect of risk management.1 Moreover, as various risk management strategies are employed over complex networks among sophisticated portfolio managers, regulatory agencies and households, incorporating the interaction of risk management with other market frictions, such as incentive fees of fund managers and margin requirements in the funding market, becomes crucial to assess the systemic risk of financial markets. Our model explicitly introduces the compensation structure for fund managers in practice which can misalign the incentive of fund managers with that of their investors. Specifically, the fee schedule for fund managers in our model can incorporate a typical compensation plan for hedge fund managers, such as the 2% management fee for the total size of the fund and the 20% performance fee for the profit over the benchmark. This performance-based fee schedule implicitly protects fund managers from some of the downside risk and may induce aggressive risk taking, which can lead to fund liquidations 1

See Brennan and Schwartz (1989), Gennotte and Leland (1990), Donaldson and Uhlig (1993), Basak (1995, 2002), Grossman and Zhou (1996), Basak and Shapiro (2001), Gromb and Vayanos (2002), Danielsson, Shin and Zigrand (2004, 2012), Brunnermeier and Pedersen (2009), Garleanu and Pedersen (2011), Oehmke (2014) among others.

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during a financial crisis. In fact, at the beginning of 1998, the debt-to-equity ratio of LTCM was more than 25 to 1.2 Still, while 968 new hedge funds were launched in 2015, an equivalently large number (979) of hedge funds were also liquidated the same year.3 Prior studies have examined the effect of the fee structure on the behavior of individual fund managers (see Goetzmann, Ingersoll and Ross (2003), Lan, Wang and Yang (2013), Buraschi, Kosowski and Sritrakul (2014)). In contrast, our interest lies in the equilibrium consequences of such a fee structure, i.e., whether a crisis can arise as a result of fee schedules. Furthermore, the liquidity provision through a funding market makes the aggressive strategy of fund managers viable. In fact, enhancing profits through leverage is what institutional investors do in practice. Ang, Gorovyy and Inwegen (2011) show that over the three years of 2005-2007, the overall hedge fund industry had the leverage around two and the relative value hedge funds spiked up the leverage upto five. Jiang (2017) finds that the average leverage of hedge funds is still close to two in the recent years of 2011-2015. The use of leverage comes with its cost. Aragon and Strahan (2012) find the empirical evidence that hedge funds using Lehman as prime broker suffer from the funding liquidity after the Lehman bankruptcy. To reflect this aspect of funding market, we put an upper bound on leverage, or equivalently require a minimum margin. Existing studies focus on the possibility that the margin constraints in funding market can amplify the exogenous shock (Gromb and Vayanos (2002) and Brunnermeier and Pedersen (2009)). By contrast, we assume a sufficient funding liquidity a priori and explore the possibility of the collective defaults of fund managers without any exogenous shock. A fund run in our economy can take place in the following way. Although a fund manager has specialized skills in identifying arbitrage opportunities, an investor does not know the ability or the intention of him (fund manager). Hence, she (investor) requests redemption as a way to manage risk when her fund value hits a certain threshold level. Our key observation is that the response of the fund manager to the potential redemption can go either way. On the one hand, the redemption risk can incentivize the fund manager to take excessive risk if the downside risk due to the fund liquidation is limited by the fund fee schedule. On the other hand, he may need to be defensive to avoid the fund liquidation. The fund run equilibrium of our model lies in the balance of the trade-off between whether the fund manager should take excessive leverage since 2

See Lowenstein (2000). See the report from Hedge Fund Research at https://www.hedgefundresearch.com/sites/ default/files/articles/pr_20160317.pdf. 3

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he is protected from the downside risk, or take defensive leverage so as to avoid a fund liquidation. We find that an equilibrium in which a significant proportion of funds are unnecessarily liquidated may also exist even in an economy in which a sufficient amount of funding is available, so that fund managers can eliminate any mispricing in another equilibrium. Another unique feature of the fund run equilibrium in our model is that it is supported by heterogeneous strategies across ex-ante homogeneous fund managers. Before the CDO and CDS market collapsed in 2007-2008, a group of people not only recognized that a crisis is near but also made large bearish bets. Steve Eisman, who got famous by betting against subprime mortgages at FrontPoint Partners, a unit of Morgan Stanley, and also by being portrayed by Steve Carell in the movie The Big Short, is an example of such people. As mentioned previously, a fund run in our economy exists at the balance of fund managers’ incentives between being aggressive and being defensive. Hence, a fund run equilibrium in our model explains the coexistence of extreme leverage and big short. Furthermore, our model provides novel insights into the unintended consequences of government intervention in financial markets. Specifically, our model predicts that a direct regulation on the leverage of the selected financial institutions, such as CCAR (Comprehensive Capital Analysis and Review), on top of individual investors’ risk management, intensifies systemic risk in financial markets by encouraging unregulated financial institutions (not subject to CCAR) to lever up before a crisis. Our model also presents a unique perspective that the government’s liquidity injection in the middle of a crisis, such as TARP (Troubled Asset Relief Program), can worsen the crisis because surviving in the crisis becomes less attractive and more funds go aggressive ahead of the crisis. In summary, this paper shows that coordination failure among delegated fund managers can arise fully endogenously in an economy in which sufficient funds are provided and no exogenous shock exists. Financial market frictions, such as risk management, fee structure and margin constraints, lie at the root of the coordination failure. Hence, our model is consistent with many financial crises, such as Black Monday, the savings and loan crisis, the LTCM scandal, the subprime mortgage crisis, the Flash Crash and others, the causes of which are hard to be found with the application of fundamentals only (Goldstein (2013)). Our study is related to several strands of literature. This paper puts a new spin on coordination failure, which has been dormant in the bank run literature, from the

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viewpoint of wide literature on other market frictions, such as the limits of arbitrage, risk management, margin constraints and fee schedule of fund managers. The literature on bank runs was pioneered by Diamond and Dybvig and extended by others to incorporate the uncertainty on the fundamentals (Chari and Jagannathan (1988), Jacklin and Bhattacharya (1988), Allen and Gale (1998) and Goldstein and Pauzner (2005)) and financial networks (Bhattacharya and Gale (1987) and Allen and Gale (2000)). In contrast with the above-mentioned models in which the amplification is caused by direct contractual linkages such as deposit contracts or interbank lendings, our model is distinctive in that the amplification in our fund run works indirectly through a price mechanism, and therefore the resulting crisis becomes market-wide, much beyond institution-level crises in the bank run literature. For example, when two funds hold a common financial asset, the liquidation of one fund can lower the price of the asset, triggering the liquidation of the other fund, without any direct contractual relationship between the two funds. Prior research has examined the indirect spillover through price mechanism in many other contexts, such as margin requirements (Brunnermeier and Pedersen (2009)) or performance-based fund flow (Shleifer and Vishny (1997)). Brunnermeier and Pedersen show that the firesale of collaterals can arise through the channel of increasing margin requirements by financiers, igniting the associated amplification between funding illiquidity and market illiquidity. In their model, the linkage between funding illiquidity and market illiquidity stems from the binding nature of the margin requirements of all investors. However, in a fund run of our economy, some defensive fund managers have leverage levels that are not limited by margin requirements. Furthermore, because we assume that sufficient funds can be provided in our economy, we can interpret illiquidity in a fund run crisis as an equilibrium outcome rather than a reason for amplified shocks as in their paper. In particular, Shleifer and Vishny (1997) provide the key step that enables our model to connect coordination failure with a market-wide crisis. However, while their main message is that the potential withdrawal (risk management of investors) will induce defensive behavior of fund managers, our model reveals the possibility that fund managers can become aggressive due to the protection from the downside risk in the fee structure for fund managers. In addition, Shleifer and Vishny’s key assumption is that funding in the market is not sufficient to bring prices to fundamental values while a mispricing in our fund run does not require such an assumption. Kondor (2009) shows that arbitrageurs can suffer from a loss without demand shocks when arbitrage opportunities randomly disappear for exogenous causes. Our paper proceeds further 6

and fully endogenizes the creation and destruction cycle of arbitrage opportunities through interactions among financial market frictions. Interestingly, the evolution of arbitrage opportunities in our fund run equilibrium without any exogenous demand shocks looks identical to that in Shleifer and Vishny (1997). Also closely related are Allen and Gale (1994) and Gennotte and Leland (1990). While Allen and Gale (1994) examine the effect of the limited market participation, induced by heterogeneous investor types, on the price behavior, we show how heterogeneous strategies can result from a prospective crisis among a priori homogeneous investors. Gennote and Leland (1990) also note the possibility of multiple equilibria when traders use portfolio insurance strategy. To the best of our knowledge, our paper is the first attempt to incorporate the interaction of risk management and fund managers’ incentives for crisis analysis. Lastly, our paper is related to the literature on global games. Morris and Shin (1998) model the currency attack. Morris and Shin (2001) and Goldstein and Pauzner (2005) apply global games to bank runs. In a more related setup, Morris and Shin (2004) use the global game analysis in selecting an equilibrium of an economy in which traders use loss-cut strategies. Liu and Mello (2011) connect global games to the literature on the limits to arbitrage through the uncertain nature of the equity capital in hedge funds. Cong, Grenadier and Hu (2017) study the dynamic effect of government intervention in a series of potential crises. Our approach to a crisis is contrasted with those in global games in two ways. First, we require only a public signal that fund managers may coordinate on while in a standard global game, players receive private information that induces a unique equilibrium.4 Second, in our model, coordination among fund managers can generate a crisis without fundamental shocks but global games rely on bad fundamentals as well as coordination. The reasons of mispricing in our model economy are found inside a fragile financial system with the described frictions, rather than from the asymmetric information that reflects the economic fundamentals. The rest of this paper proceeds as follows. Section 2 describes our model. Section 3 establishes the main result. Section 4 examines the policy implications. Section 5 concludes. All proofs are in the Appendix. 4

Even if our model is extended to a global game by adding private information, there is no guarantee that the extended game will have a unique equilibrium, because the endogenously determined asset price is publicly observed (see Angeletos and Werning (2006)).

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2

Setup

There are three time periods (t = 0, 1, 2) and two assets in the economy. One asset is a one-period risk-free asset that agents can borrow and lend at the normalized rate of zero. The other is a two-period risky asset that pays 1 at t = 2 and has no interim cash flow so that the fundamental value is 1. The total supply of the risky asset is assumed to be 1. There are two types of market participants: fund managers who manage arbitrage funds and investors who invest in the arbitrage funds. The fund managers have specialized knowledge about the risky asset, and only fund managers can engage in trading the assets on behalf of the investors. Specifically, at t = 0, only fund managers know the fundamental value of the risky asset and trade the risky asset. At t = 2, the fundamental value is known to the public, and the price becomes the fundamental value of 1. That is, there is no long-run fundamental risk like in the literature on the limits of arbitrage (Shleifer and Vishny (1997) and Gromb and Vayanos (2002)). The interim uncertainty of the economy stems from the following randomness at t = 1: With probability q (the b(bad) state), for some exogenous reasons, such as the risk transmission from another market, the sudden pessimism of investors, or internal control failures within funds, ε (≥ 0) proportion of funds are forced to be liquidated. We assume that the ε proportion of liquidated funds is randomly selected across all the funds. Therefore, we call this liquidation of ε proportion of funds as “random liquidations”. In the g(good) state, which occur with probability 1 − q, no funds are randomly liquidated. Note that the only difference between the two states is the existence of random liquidations. In our later analysis in Section 3, we will restrict our attention to an economy where the price in the state g is larger than or equal to the price in the state b. This underlying structure of the uncertainty is similar to that in Shleifer and Vishny (1997). However, we depart from typical models in the literature on the limits of arbitrage in which exogenous demand shocks are critical for mispricing. In our model, the mispricing of the risky asset arises not from the reason of noise trader shocks but as the result of various market frictions, which we clarify subsequently. We have a continuum of investors and fund managers, each with unit mass. At t = 0, the i-th investor, i ∈ [0, 1], deposits her wealth of Wi,0 = W0 to the fund operated by the i-th fund manager. That is, each fund has the same size of initial capital.5 Following Shleifer and Vishny (1997), we assume that the i-th investor (she) 5

The assumption of Wi,0 = W0 for all i ∈ [0, 1] can be relaxed. The required assumption is that

8

puts her wealth of W0 to the i-th fund manager (he) with the highest expected return to the best of her belief and that the i-th fund manager is associated with the i-th investor as a result of this matching process. Fund managers have specialized skills in detecting the arbitrage opportunities in the risky asset market, but investors do not have access to such information. The only observable information to the i-th investor t = 1 is the evolution of the net asset value of her fund over time. The i-th investor observes Wi,1 in at t = 1 and evaluates the skill of the i-th fund manager. Specifically, the i-th investor cuts losses of her fund by requesting redemption at t = 1 if Wi,1 − Wi,0 < −s · Wi,0 , where s ≥ 0. To highlight that the liquidation due to an investor’s redemption can be avoided by a fund manager’s leverage decision, we call this liquidation as “deliberate liquidation”, in contrast with “random liquidation” of the ε proportion of funds. We assume that all investors share the common s, the maximum loss ratio with which she can bear.6 As the smaller (larger) level of s means the tighter (easier) risk management, we interpret s as a measure of the sensitivity of investors to the fund performance, which can stem from the risk aversion of investors and/or the information asymmetry between investors and fund managers. For example, as an investor becomes more risk averse or the degree of information asymmetry becomes more severe, s moves closer to zero. There are a couple of important aspects of the risk management rule in our model. First, although the price will certainly become the fundamental value at t = 2, the fund withdrawal at t = 1 can be a completely rational decision from the perspective of investors. One of the key features in our model is the separation of brains and resources. As arbitrage markets are highly specialized, an investor may not understand what her fund manager is trying to achieve even when she observes the holdings of the fund, as Shleifer and Vishny (1997) argue. Furthermore, it is a common practice for a hedge fund manager not to reveal his strategy to protect his comparative advantage from being copied by competitors. Hence, as shown in Berk and Green (2004), it can be fully rational for an investor to request redemption simply based on the past performance.7 As another rationale for the fund withdrawal, the investor may have borrowed from another financier and needs to pledge additional margin when her position has lost value (Gromb and Vayanos (2002), Brunnermeier and Pedersen (2009)). Besides, in R1 W0 = 0 Wi,0 di is well-defined and bounded. In this case, W0 is interpreted as the ex-ante aggregate capital in the economy. 6 The assumption of constant s across investors is not necessary. However, the homogeneous s is assumed to emphasize that the heterogeneous behavior of agents in an equilibrium is not due to the ex-ante heterogeneity of agent types. 7 See Chen, Goldstein and Jiang (2010), Shek, Shim and Shin (2015), Goldstein, Jiang and Ng (2016) and Zeng (2017) for more studies on the flow-to-performance relationship.

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an extremely fast trading environment such as the modern financial market, a simple trading error can accumulate enormous losses before human traders can identify and analyze the reason. To prevent the occurrence of such an event, investors need to take a preemptive action. Second, the loss-cut rule is employed in our model as a representative of a wide variety of risk management in practice with the “pro-cyclical” feature (see Danielsson, Shin and Zigrand (2004, 2012) and Adrian and Shin (2010, 2014) for VaR (Valueat-Risk) and Brunnermeier and Pedersen (2009) for margin constraints). Along with capital requirements (BIS ratio on banks or Net Capital Rule for brokers or dealers) or VaR, the loss-cut rule induces the upward-sloping demand curve with higher selling pressure at a lower price. This property of our risk management rule captures the empirical feature of the fund industry. As mentioned in Lan et al. (2013), the poor performance of hedge funds frequently leads to redemption or liquidation of the funds. Similar patterns have been examined also in the context of mutual funds (Berk and Green (2004) and Chen, Goldstein and Jiang (2010)). The loss-cut rule also has an advantage of its simplicity over other risk management tools in managing rare but extreme events. Confronting infrequently observed crashes or because of a lack of understanding of those rare events due to information asymmetry, an individual investor may not be able to afford more complicated risk management methods that require a sufficient amount of timely data to make themselves operational. Next, we introduce the compensation and strategy of a fund manager. If the i-th fund survives till t = 2, the i-th investor compensates the i-th fund manager and gives (i) α > 0 proportion of the gain, max (Wi,2 − W0 , 0) , (performance fee) and (ii) β > 0 proportion of the total fund size Wi,2 at t = 2 (management fee). If the i-th fund is liquidated in the interim period from either random or deliberate liquidations, the i-th fund manager is penalized by −C, where C > 0. Liquidation cost, C, captures a reputation damage or a job search cost that a fund manager may face if his fund is liquidated. Even though we do not explicitly model an infinitely repeated entryexit game of fund managers, C implicitly reflects the difference in the continuation payoff between survival and exit. Note that this kinked linear compensation structure with a lump-sum penalty is frequently used to address general issues in the principalagent problem (See, for example, Stiglitz and Weiss (1981) and Lewis (1980)). Hence, our model can be useful in examining the incentives for various forms of financial institutions, which we will discuss in Section 4. The strategy of the i-th fund manager is a triplet of (li,0 , li,1 (g) , li,1 (b)) , where h i li,0 ∈ −1, l0 is the leverage (borrowing divided by the fund capital) of the i-th fund at 10

h

i

t = 0, and li,1 (g) and li,1 (b) ∈ −1, l1 are the leverages of the i-th fund in state g and state b at t = 1, respectively.8 The dollar demand from fund manager i is computed as follows. The fund manager i takes a position of W0 (1 + li,0 ) dollars in the risky asset at t = 0. In the following period of t = 1, fund i’s position is Wi,1 (g) (1 + li,1 (g)) in state g and Wi,1 (b) (1 + li,1 (b)) in state b if the fund survives. A fund manager is  constrained to use leverage at most l0 l1 at t = 0 (1) and is not allowed to take a short position on the risky asset.9 The maximum leverage constraint can be interpreted as the margin requirements from the lender (e.g., a prime broker) or a direct regulation by the government entity.10 We can introduce heterogeneity in the degree of funding constraints to study the interaction between different kinds of financial institutions. For example, while most hedge funds are minimally regulated in their leverage decision, some banks are severely regulated. The effect of heterogeneous funding constraints will be discussed further in Section 4. We now specify the market clearing prices of our economy. Given the strategy profile (li,0 , li,1 (g) , li,1 (b))i∈[0,1] of the fund managers, the loss-cut rule of the investors and the random liquidation of ε proportion of funds in state b, the market clearing price of p0 is determined by Z 1

p0 = W0

0

(1 + li,0 ) di

(2.1)

and the market clearing prices, p1 (g) and p1 (b), satisfy p1 (g) = where Wi,1 (g) = W0

Z 1



0

Wi,1 (g) (1 + li,1 (g)) · 1 (Wi,1 (g) ≥ W0 (1 − s)) di,

p1 (g) p0

p1 (b) = (1 − ε) 

(2.2)



(1 + li,0 ) − li,0 , and

Z 1 0

Wi,1 (b) (1 + li,1 (b)) · 1 (Wi,1 (b) ≥ W0 (1 − s)) di,

(2.3)



(1 + li,0 ) − li,0 . For (2.3), we resort to the assumption that where Wi,1 (b) = W0 p1p(b) 0 funds are randomly selected for the ε size of random liquidation. The risky asset market is said to be mispriced if any of p0 , p1 (g) and p1 (b) is different from the fundamental value of 1. Finally, we define an equilibrium of this economy as follows. 8

We assume l0 > −1 and l1 > −1. The lower bound of leverage, -1, means that fund managers are not allowed to short-sell the asset. Our theoretical results hold as long as the fund managers are not allowed to take unbounded short positions. We assume a specific level of the lower bound for convenience. 10 We discuss the implication of our model on the government regulation in greater detail in Section 4. 9

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Definition 2.1. An equilibrium of the economy is the leverage profile (li,0 , li,1 (g) , li,1 (b))i∈[0,1] and the price (p0 , p1 (g) , p1 (b)), such that 1) given the price (p0 , p1 (g) , p1 (b)) and the leverage of li,0 at t = 0, li,1 (g) solves the maximization problem, Ui,1 (g) = max [α max (Wi,2 (g) − W0 , 0) + βWi,2 (g)] · 1 (Wi,1 (g) ≥ W0 (1 − s)) li,1 (g)

− C·1 (Wi,1 (g) < W0 (1 − s))

(2.4)

where !

1 Wi,2 (g) =Wi,1 (g) (1 + li,1 (g)) − li,1 (g) , p1 (g) ! p1 (g) Wi,1 (g) =W0 (1 + li,0 ) − li,0 , p0

(2.5) (2.6)

and li,1 (b) solves the maximization problem, 0 Ui,1 (b) = max [α max (Wi,2 (b) − W0 , 0) + βWi,2 (b)] · 1 (Wi,1 (b) ≥ W0 (1 − s)) li,1 (b)

− C·1 (Wi,1 (b) < W0 (1 − s))

(2.7)

where !

1 (1 + li,1 (b)) − li,1 (b) , Wi,2 (b) =Wi,1 (b) p1 (b) ! p1 (b) Wi,1 (b) =W0 (1 + li,0 ) − li,0 ; p0

(2.8) (2.9)

2) given the price (p0 , p1 (g) , p1 (b)), li,0 maximizes U0 (li,0 ) = (1 − q) Ui,1 (g) + qUi,1 (b) ,

(2.10)

0 Ui,1 (b) = (1 − ε) Ui,1 (b) − εC;

(2.11)

where

3) given the strategy profile (li,0 , li,1 (g) , li,1 (b))i∈[0,1] , the price (p0 , p1 (g) , p1 (b)) satisfies the market clearing conditions, (2.1)-(2.3). Condition 1) gives the optimization problem of the i-th fund manager at t = 1. The ith fund manager solves (2.4) in state g and (2.7) in state b. In particular, the expression in (2.7) describes the utility of the i-th fund manager conditional on the event that the i-th fund is not randomly liquidated in state b. Given the price (p0 , p1 (g) , p1 (b)) and 12

the leverage of li,0 at t = 0, Wi,1 , the value of the i-th fund at t = 1, is evaluated by (2.6) in state g and (2.9) in state b. If Wi,1 < W0 (1 − s), the i-th fund manager is penalized by the liquidation cost of C. When Wi,1 ≥ W0 (1 − s), the i-th fund manager maximizes his compensation of α max (Wi,2 − W0 , 0) + βWi,2 where Wi,2 , the final value of the i-th fund at t = 2, is evaluated by (2.5) following state g and (2.8) following state b. Thus, the i-th fund manager finds the optimal leverage of li,1 at t = 1 by solving (2.4) in state g and (2.7) in state b. Condition 2) specifies the optimization problem of the i-th fund manager at t = 0. 0 Note that from the expression (2.11), Ui,1 (b) = (1 − ε) Ui,1 (b) − εC is the expected utility of fund manager i in state b before the random liquidation realizes. Hence, the optimal leverage of the i-th fund manager at t = 0 is the maximizer of (2.10). In the next section, we provide detailed procedures for finding equilibria described in Definition 2.1 for an economy with parameters of our interest. Then, our main theoretical results follow.

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Model Prediction

We show that there are two equilibria under certain conditions. In one equilibrium, a financial crisis is generated without any exogenous shock, and in the other, the fundamental value of the two-period asset is fully reflected in the market price, suggesting that funding liquidity is a priori sufficient in the market. The following lemma shows that overpricing is not possible, and therefore we focus only on the underpricing. Lemma 3.1. In any equilibrium, p1 (g) ≤ 1, p1 (b) ≤ 1 and p0 ≤ 1. The intuition behind this lemma is as follows. The compensation of a fund manager who has survived at t = 1 is strictly increasing in the final wealth of his fund as shown in (2.4) for state g and (2.7) for state b. Assume that p1 > 1, where p1 is either p1 (g) or p1 (b) . Then, because the fund manager knows that the price will become the fundamental value of 1 at t = 2, he will not buy any of the risky asset at t = 1 to maximize his fund size at t = 2. This optimal decision is applied to every fund manager who has survived, resulting in zero demand for the asset. The zero demand contradicts the assumption that p1 > 1. Similarly, we can show that p0 > 1 cannot hold because any rational fund manager will not hold any of the risky asset at t = 0 if p0 > 1. We argue that the underpricing is coordination failure in the sense that there is another equilibrium in which the fundamental value of the asset is fully reflected in 13

the market price. An important implication of this multiplicity of equilibria is that risk management may play a role of transforming the best economy into the worst economy, which is the complete opposite to the intended results from risk management. We elaborate on this striking outcome in the following two subsections. Section 3.1 provides the condition of fair pricing p0 = p1 (g) = p1 (b) = 1 in an equilibrium. Section 3.2 analyzes another equilibrium in which some funds are deliberately liquidated and the market price is strictly smaller than the fair value.

3.1

Ideal equilibrium

We define an ideal equilibrium as an equilibrium in which p0 = p1 (g) = p1 (b) = 1. Assumption 1 and Theorem 3.1 describe the sufficient and necessary conditions for an ideal equilibrium to exist. 







Assumption 1. W0 1 + l0 ≥ 1 and W0 1 + l1 (1 − ε) ≥ 1. Assumption 1 posits that funding liquidity is a priori enough to support the fair prices. At t = 0, every fund manager can take the leverage li,0 = W10 − 1 ≤ l0 , the inequality of which is from the first condition of Assumption 1. Then, themarket clearing  R price of p0 in (2.1) becomes the fundamental value because p0 = W0 1 + 01 li,0 di = 







W0 1 + W10 − 1 = 1. Given the price dynamics of p0 = p1 (g) = p1 (b) = 1, which needs to be verified, the aggregate wealth in the economy becomes W0 in state g and W0 (1 − ε) in state b because the ε proportion of fund managers are randomly liquiof which is dated only in state b. Then, with li,1 (g) = W10 − 1 ≤ l1 , the inequality  R from the second condition of Assumption 1, it holds that p1 (g) = W0 1 + 01 li,0 di = W0 1 + W10 − 1 = 1. Also, the second condition of Assumption 1 implies that every 1 fund manager who has survived in state b can take the leverage li,1 (b) = W0 (1−ε) −1 ≤ l1 . Then, the market hclearing price (2.3) becomes the fundamental value of 1 because  i 1 p1 (b) = W0 (1 − ε) 1 + W0 (1−ε) − 1 = 1. We want to show that a financial crisis can arise even when there is enough funding liquidity in the market, so we assume Assumption 1 hereinafter. The following theorem states that Assumption 1 is sufficient and necessary for fair pricing. Theorem 3.1. Assumption 1 holds if and only if an ideal equilibrium exists. This equilibrium is ideal in the following respects. The private information of the fund managers is fully reflected in the market price at t = 0, before the fundamental value of the two-period asset is known to the public. In addition, the price does not 14

fluctuate with the interim uncertainty in the random liquidation at t = 1. Theorem 3.1 states that if fund managers are allowed to raise sufficient funds through either capital financing of W0 from investors or debt financing by taking leverages up to l0 or l1 from the funding market, the equilibrium price can be equal to the fundamental value of the two-period risky asset. Previously, we suggested that li,0 = li,1 (g) = W10 − 1 and 1 − 1 for all i ∈ [0, 1] as one of the strategy profiles that support the li,1 (b) = W0 (1−ε) fundamental value as the market clearing price. Given the fair pricing of p0 = p1 (g) = p1 (b) = 1, the utility in (2.10) becomes βW0 − qεC, not depending on the leverage decision of fund managers. Thus, the suggested strategy profile is optimal.

3.2

Fund run equilibrium

This subsection provides our main theoretical results. We define a fund run equilibrium as an equilibrium in which some funds are deliberately liquidated. In this equilibrium, there exists an investor who runs to her fund manager to request redemption because the value of her fund hits the threshold level of W0 (1 − s). For convenience of further analysis, we assume that the price in state g is larger than or equal to the price in state b. Assumption 2. p1 (g) ≥ p1 (b). The main purpose of Assumption 2 is to differentiate between states g and b, and not to restrict the price dynamics. This aspect of Assumption 2 will become clear in our later discussion. The following theorem shows that there are only two possibilities of equilibrium prices. Theorem 3.2. Under Assumptions 1 and 2, i) there is no deliberate liquidation if and only if p0 = p1 (g) = p1 (b) = 1, and ii) there are some deliberate liquidations if and only if p1 (b) < p0 < p1 (g) = 1. Although we define a fund run equilibrium in terms of the behavior of investors, the theorem implies that this equilibrium can be characterized equivalently in terms of prices. An ideal equilibrium can also be characterized by a lack of deliberate liquidations. If both ideal and fund run equilibria exist at the same time, a fund run equilibrium may be considered a coordination failure. Because we assume Assumption 1, sufficient funds could have been provided to the market and a crisis need not have happened. In this sense, the fund run in our model is distinctive from the crisis in Brunnermeier and 15

Pedersen (2009), in that funding liquidity or interaction between funding liquidity and market liquidity is not the reason for the fund run in our model. A fund run materializes in our economy because frictions in the financial market, such as risk management, leverage limit or management fee structure, can make agents in the economy fail to coordinate their strategies to achieve an ideal equilibrium. In the rest of this section, we solve backwards for a fund run equilibrium and discuss why such a fund run is possible under market frictions. Relying on Theorem 3.2, we restrict our attention to the case of p1 (b) < p0 < p1 (g) . 3.2.1

Time 1

In this subsection, we solve for the equilibrium at t = 1, where p0 and (li,0 )i∈[0,1] are given. Then, we express the equilibrium outcomes of Ui,1 (g), Ui,1 (b) and p1 (g) , p1 (b) as functions of p0 and (li,0 )i∈[0,1] . We start with state g. First, we confirm that no fund is liquidated in state g. Lemma 3.2. In any equilibrium with Assumption 2, no fund is liquidated in state g. From Assumption 2, it holds that Wi,1 (g) ≥ Wi,1 (b) . Hence, if fund i is deliberately liquidated in state g, it will be deliberately liquidated in state b, too. For this case, the ith fund manager gets the utility of −C. Because he can always deviate to the minimum leverage -1 and avoid deliberate liquidations, this cannot occur in any equilibrium. Furthermore, the following lemma pins down the equilibrium price in state g. Lemma 3.3. In any equilibrium with Assumptions 1 and 2, it holds that p1 (g) = 1. The lemma above is from the following logic. Because no fund is liquidated in state g from Lemma 3.2, the aggregate wealth among funds is preserved from t = 0 to state g. Furthermore, with Assumption 1, funding can be sufficiently provided. With the common knowledge of p2 = 1, a fund manager will increase the position in the asset as long as p1 (g) < 1. Hence, from Lemma 3.1, it should hold that p1 (g) = 1. The equilibrium leverage in state g can be found as follows. Because p1 (g) = p2 = 1, the i-th fund manager knows that the fund size will not change at t = 2, Wi,2 (g) =  1 Wi,1 (g) = W0 p0 (1 + li,0 ) − li,0 , and therefore his leverage decision is not relevant for maximizing utility. Hence, the equilibrium leverage of li,1(g) i∈[0,1] just needs to satisfy  R1 1 the market clearing condition of 0 W0 p0 (1 + li,0 ) − li,0 (1 + li,1 (g)) di = 1.

16

From the above analysis, we can express the i-th fund manager’s equilibrium utility in state g as Ui,1 (g) = α max (Wi,2 (g) − W0 , 0) + βWi,2 (g) = max (Wi,1 (g) − W0 , 0) + βWi,1 (g) !

= α max W0

!

!

1 1 (1 + li,0 ) − li,0 − W0 , 0 + βW0 (1 + li,0 ) − li,0 . p0 p0 (3.1)

Resorting to Lemmas 3.2 and 3.3, we do not consider the deliberate liquidation in state g and take the fair pricing p1 (g) = 1 in state g. Hence, we can summarize the equilibrium utility in state g as (3.1) and the equilibrium in Definition 2.1 as the optimal leverage (li,0 , li,1 (b))i∈[0.1] and the market clearing price (p0 , p1 (b)) . Hereinafter, we restrict our attention to (li,0 , li,1 (b))i∈[0.1] and (p0 , p1 (b)). Now, we examine state b. For state b, those who are liquidated exit the market, and only the remaining fund managers demand the risky asset. Recall that the i-th fund can be liquidated by either (i) random liquidation with probability ε or (ii) deliberate liquidation due to an investor’s loss-cut. Because the i-th fund is deliberately liquidated if Wi,1 (b) < W0 (1 − s) , we can define the critical leverage l∗ at t = 0 as !

Wi,1 (b) = W0

!

p1 (b) p1 (b) + − 1 l∗ = W0 (1 − s) . p0 p0

This gives ∗

l =

p1 (b) p0

− (1 − s)

1−

p1 (b) p0

,

(3.2)

and it follows that li,0 ≤ l∗ ⇔ Wi,1 (b) ≥ W0 (1 − s) .

(3.3)

Then, we can express the i-th fund manager’s compensation at t = 2 as −C if li,0 > l∗ or fund i is randomly liquidated, and α max (Wi,2 (b) − W0 , 0) + βWi,2 (b) if li,0 ≤ l∗ and fund i is not randomly liquidated. (3.4) As C is constant, the leverage decision in state b is relevant only for the fund managers who have survived in state b. The following Lemma describes the optimal leverage of li,1 (b) for fund manager i who has survived in state b. 17

Lemma 3.4. If p1 (b) < 1, the optimal li,1 (b) for the i-th fund that survives in state b is l1 . The intuition of the lemma is as follows. Because the price will be the fundamental value of 1 at t = 2 and p1 (b) < 1, the payoff for the non-liquidation case in (3.4) is a  1 strictly increasing function of Wi,2 (b) = Wi,1 (b) p1 (b) (1 + li,1 (b)) − li,1 (b) , which is also a strictly increasing function of li,1 (b). Thus, a fund manager who has survived in state b will take the maximum leverage of l1 . 0 (b), the utility of fund manager i who has survived in state b, is given Therefore, Ui,1 by (2.7), where the final wealth in (2.8) is updated as follows: !

 1  Wi,2 (b) = Wi,1 (b) 1 + l1 − l1 p1 (b) ! !  p1 (b) 1  = W0 (1 + li,0 ) − li,0 1 + l1 − l1 . p0 p1 (b)

Next, we describe the equilibrium price p1 (b) of (2.3) in state b. From Lemma 3.4, we restrict p1 (b) of (2.3) by imposing li,1 (b) = l1 as follows: p1 (b) = (1 − ε)

Z 1 0





Wi,1 (b) 1 + l1 · 1 (Wi,1 (b) ≥ W0 (1 − s)) di.

(3.5)

This expression of p1 (b) reveals the key mechanism of a fund run. When the initial random liquidation of size ε occurs in the economy, it directly reduces the price by the ε ratio as shown in the first term of the RHS in (3.5). Although the size of ε may be minuscule, the ultimate effect is amplified because the value of fund,   p1 (b) Wi,1 (b) = W0 p0 (1 + li,0 ) − li,0 , is quoted by the market price of the risky asset. 



Note that Wi,1 (b) determines the size of the dollar leverage, Wi,1 (b) 1 + l1 , and more importantly the liquidation decision by investors, 1 (Wi,1 (b) ≥ W0 (1 − s)) . Not only is the direct impact of the decrease in p1 (b) on Wi,1 (b) negative, but the decrease in Wi,1 (b) also ignites the deliberate liquidations such that Wi,1 (b) < W0 (1 − s), forcing fund managers to sell more, which in turn causes afurther price decline and so on.  p1 (b) By plugging Wi,1 (b) = W0 p0 (1 + li,0 ) − li,0 into (3.5), we obtain p1 (b) as an implicit function of p0 and (li,0 )i∈[0,1] :11 p1 (b) = (1 − ε)

(3.6) Z 1 0

!

W0

!

  p1 (b) p1 (b) (1 + li,0 ) − li,0 1 + l1 · 1 (1 + li,0 ) − li,0 > 1 − s di. p0 p0

11

We show the existence of p1 (b) in (3.6) given p0 and (li,0 )i∈[0,1] with parameters of our interests in the proof of Theorems 3.1 and 3.3.

18

Then, l∗ defined in (3.2) is also an implicit function of p0 and (li,0 )i∈[0,1] . Finally, we can express the i-th fund manager’s expected utility in state b as a function of p0 and (li,0 )i∈[0,1] by Ui,1 (b) = (1 − ε) (α max (Wi,2 (b) − W0 , 0) + βWi,2 (b)) · 1 (l∗ ≥ li,0 ) − C · ((1 − ε) 1 (l∗ < li,0 ) + ε) ,

(3.7)

where !

 1  Wi,2 (b) = Wi,1 (b) 1 + l1 − l1 p1 (b) ! ! !  p1 (b) p1 (b) 1  = W0 + − 1 li,0 1 + l1 − l1 . p0 p0 p1 (b)

3.2.2

Time 0

We find a fund run equilibrium of our model economy in this subsection. Theorem 3.2 says that p0 < 1 in a fund run equilibrium. The following lemma describes the optimal leverage li,0 at time 0 for the i-th fund, which is deliberately liquidated in state b. Lemma 3.5. If p0 < p1 (g) = 1, the optimal li,0 for the i-th fund which is deliberately liquidated in state b, is l0 . The intuition is as follows. For the i-th fund to be deliberately liquidated, it holds that Wi,1 (b) < W0 (1 − s), implying that Ui,1 (b) = −C from (3.7). Then, after some algebras, we can express the i-th fund manger’s expected utility (2.10) as follows: !

U0 (li,0 ) = (1 − q) (α + β) W0

!

1 − 1 (1 + li,0 ) + βW0 − qC, p0

which is a strictly increasing function of li,0 when p0 < 1. Thus, the i-th fund manager will choose to lever up his position as much as possible (i.e., li,0 = l0 ). Resorting to Lemmas 3.4 and 3.5, we restrict our attention to the following strategy profile in finding a fund run equilibrium.12 We do not fix the proportion h and l∗ now but let them be determined in equilibrium. Definition 3.1. The bang-bang strategy profile refers to a strategy profile in which h ∈ (0, 1) proportion of fund managers take li,0 = l0 > l∗ and the other fund managers   take (lj,0 , lj,1 (b)) = l∗ , l1 , where l∗ = −1 or l∗ . 12

Recall that from Lemmas 3.2 and 3.3, we reduce the utility in state g into (3.1) and do not explicitly consider the strategy in state g.

19

Figure 3.1: Optimal Leverage Decision

(b)

(a)

(c)

This figure shows how the i-th fund manager maximizes U (l ) at t = 0 by choosing i 0 i,0 h the leverage at t = 0 over the admissible range of −1, l0 using the piecewise linearity of U0 (li,0 ) as expressed in (3.8). The i-th fund manager takes −1, l∗ , or l0 as the optimal leverage, as shown in panels (a), (b) and (c), respectively. In the bang-bang strategy profile, the h proportion of the total fund managers use the aggressive strategy that takes the maximum leverage l0 (> l∗ ) to accrue high profits from t = 1 to state g. The remaining (1 − h) proportion use the defensive leverage l∗ (≤ l∗ ) so that they can survive in state b with probability 1 − ε and buy the risky asset at a heavily discounted price with the maximum leverage of l1 . The objective of fund managers at t = 0 is to maximize U0 (·) in (2.10), U0 (li,0 ) = (1 − q) Ui,1 (g) + qUi,1 (b) ,

(3.8)

where Ui,1 (g) and Ui,1 (b) are now restricted by (3.1) and (3.7), the equilibrium utilities at t = 1, respectively. o n In the bang-bang strategy profile, we consider only −1, l∗ , l0 as possible leverages at t = 0. To show why this is justifiable intuitively, consider the case of α = 0 and β = 1.13 Then, U0 (li,0 ) in (3.8) becomes a piecewise linear function of li,0 with a discontinuity at l∗ . The piecewise linearityh of U0i(li,0 ) makes the optimal leverage decision at t = 0 over the admissible range of −1, l0 quite simple; the optimal leverage should be found n

o

within −1, l∗ , l0 as illustrated in Figure 3.1. The bang-bang strategy profile with h ∈ (0, 1) can be an equilibrium if the aggressive strategy is indifferent to the defensive strategy. Otherwise, the fund managers playing 13

Lemmas A.2 and A.6 provide the general case α, β > 0.

20

the inferior strategy have an incentive to deviate. This implies  

U0 (l∗ ) = U0 l0 ,

(3.9)

where U0 (·) is from (3.8). In addition, given the strategy profile, the market clearing prices of p0 and p1 (b) in (2.1) need to satisfy 







p1 (b) = (1 − ε) (1 − h) Wj,1 (b) 1 + l1 < p0 = W0 hl0 + (1 − h) l∗ < p1 (g) = 1, (3.10) where Wj,1 (b) is the wealth level of the j-th fund which survives in state b. If (3.9) and (3.10) hold for some h ∈ (0, 1), a fund run equilibrium exists. A remarkable prediction of our model is that some funds are deliberately liquidated, following randomly liquidated funds, even when liquidity can be sufficiently provided in the market. Theorem 3.3. Assume that ε > 0. Under Assumptions 1 and 2, if q, s and W0 are sufficiently small, a fund run equilibrium exists. From Theorem 3.1, Assumption 1 guarantees the existence of an ideal equilibrium, and thus Theorem 3.3 provides conditions under which both ideal and fund run equilibria coexist. As long as some funds of size ε > 0 are randomly liquidated, other funds also follow to be liquidated deliberately when some conditions of market frictions are met. The conditions for a fund run equilibrium are that the probability for state b is small, the initial capital size of an arbitrage fund is not too large (but large enough so that the fair pricing equilibrium exists) and the risk management rule is tight enough. The required conditions are economically justifiable in the following context. First, with small q, the risk management rule in our model is intended to protect the funds from disastrous events that happen infrequently. Second, for small s, after such a rare event occurs, investors in panic will compete to exit the market before others and they will end up using a tight risk management rule. Many existing models describe how such an amplification is triggered once a crisis is initiated (see, e.g., Diamond and Dybvig (1983), Morris and Shin (2004), Goldstein and Pauzner (2005) and Brunnermeier and Pedersen (2009)). Third, with small W0 , we stress that the risky-asset market of our model is an example of a highly specialized market. Hence, although there is a large amount of outside capital, many potential investors are reluctant to participate, limiting the initial capital of funds. While there is a potentially large amount of capital in the economy (Assumption 1), the arbitrage trades in the risky asset market are heavily dependent on the short-term funding. Theorem 3.3 shows that, under these circumstances, mispricing may arise as a result of coordination failure. 21

We provide an intuition on why a fund run can be supported as an equilibrium outcome. In the bang-bang strategy profile, the group of fund mangers who take the maximum leverage l0 at t = 0 bet on the gain of the risky asset position from t = 0 to state g, p10 − 1, and the rest of the fund managers, who choose l∗ at t = 0, put their stake on the gain of the risky asset position from state b to t = 1, p11(b) − 1. Hence, the utility of the first group increases as p0 decreases, and that of the second group increases as p1 (b) decreases. Next, we examine the effect of h, the size of fund managers with the aggressive strategy, on prices of p0 and p1 (b) and resulting optimal strategy choices. On the one hand, if a sufficiently large number of fund managers play the aggressive strategy, the defensive strategy becomes more attractive than the aggressive strategy. Note that   from (2.1), p0 = W0 hl0 + (1 − h) l∗ with the bang-bang strategy profile. If h is large enough such that p0 is close to 1, the gain of the risky asset position from t = 0 to state  g, p10 −1, shrinks to zero. Thus, with a sufficiently large h, it holds that U0 (l∗ ) > U0 l0 . On the other hand, if a sufficiently small number of fund managers play the aggressive strategy, the defensive strategy becomes less attractive than the aggressive strat  egy. From (3.5), the bang-bang strategy results in p1 (b) = (1 − ε) W0 1 + l1 (1 − h) 



if l∗ = −1 or p1 (b) = (1 − ε) W0 (1 − s) 1 + l1 (1 − h) if l∗ = l∗ . In either case, when h is small enough, p1 (b) can be close to p0 > p1 (b) . Then, the return from t = 0 to state g becomes similar to the return from state b to t = 2. If q is small enough, it is unlikely that a fund manager obtains the return from state b to t = 2. Therefore, fund managers find the aggressive strategy, exploiting the return from t = 0 to state g, more   attractive, or U0 (l∗ ) < U0 l0 with a sufficiently small h. Then, resorting to the intermediate value theorem and from the previous results,   we can find h such that U0 (l∗ ) = U0 l0 . The other conditions, small W0 and small s in Theorem 3.3, guarantee that the inequalities in (3.10) hold. For example, a small s implies a small l∗ , which in conjunction with a small W0 reduces p0 in (3.10) to below 1. Strikingly, the random liquidation is not essential for the described trade-off, and a fund run can arise without any random liquidation (i.e., completely endogenously). Theorem 3.4. Assume that ε = 0. Theorem 3.3 still holds. Note that when ε = 0, state b is essentially identical to state g.14 It is noteworthy that state b is not drawn from the outside force but endogenously generated within the fragile financial system. In particular, Theorem 3.4 casts a doubt on the conventional approach 14

Hence, Assumption 2 can be made without any loss of generality.

22

in defining states based on fundamentals. While there is no fundamental difference between state g and state b, we only assume that there exists a public signal device through which a significant mass of fund managers can exit the market simultaneously. The Flash Crash in 2010 is a good example of such an event in which some arbitrageurs or computerized traders decided to exit the market suddenly. The comparison of our fund run equilibrium to Shleifer and Vishny (1997, SV) follows. First, note that the price in SV’s model economy behaves like the price dynamics of p1 (b) < p0 < p1 (g) = 1 in our fund run equilibrium. However, they need to assume that exogenous demand shock dynamics to generate such price pattern. In their model, arbitrageurs do not fully exploit the given mispricing (p0 < 1) due to the possibility of the deeper mispricing (p1 (b) < p0 ) driven by accentuated demand shocks. In contrast, our model fully endogenizes the creation and destruction cycle of an arbitrage opportunity. We find that the financial market frictions can explain how an arbitrage opportunity arises (p0 < 1) as well as why it becomes more severe (p1 (b) < p0 ) or disappears (p1 (g) = 1) . It is worth emphasizing that our approach to a crisis is clearly different from crises in global games in two ways. First, our model needs only a public signal through which fund managers can coordinate. In contrast, a standard global game requires correlated private signals through which a unique equilibrium is derived. Second, a crisis in our model can arise without any fundamental shocks while a crisis in a global game is ignited by bad fundamentals. In our fund run equilibrium, it is a fragile financial system, not the economic fundamentals reflected in the correlated private information, that results in a mispricing. Another distinctive feature of our model is that the payoff of players does not have strategic complementarity. Typically, in a bank run model, an aggressive action becomes reinforcing among players through strategic complementarity. In contrast, a fund run in our economy exists at the balance of incentives between being aggressive or defensive. In addition, as we previously explained, more (less) players with the aggressive leverage make the payoff from the aggressive strategy less (more) attractive. The next lemma highlights this difference of our economy that emerges from the distinctive incentive structure. Lemma 3.6. In any equilibrium, there are some funds which survive in state b. In particular, a bank run in Diamond and Dybvig (1983) is triggered by the strategic complementarity of early withdrawals among depositors – if a depositor believes that other depositors will withdraw, it would be better for him or her to withdraw early. 23

In contrast, the key amplifier in our fund run is the price feedback mechanism, i.e., the mutual reinforcement between the downward pressure on the price and the selling pressure on the asset. The introduction of this price feedback in an economy with frictions enables us to model a market-wide crisis, distinctive from an institution-level crisis of Diamond and Dybvig. Lastly, we close this section by confirming that the market frictions in our economy are necessary for a fund run equilibrium. Specifically, we find that when market frictions of our model economy - the risk management rule of the investors, downside risk protection to fund managers and funding constraints - become sufficiently loose, a fund run cannot be sustained. That is, our model economy finds reasons for fund runs entirely within the fragile system of the financial market rather than from exogenous shocks. The following theorem, in conjunction with Theorem 3.4, establishes that a fund run equilibrium is a result of market frictions within the economy. Theorem 3.5. Under Assumption 1, only an ideal equilibrium exists if any of the following four conditions is met: i) s is sufficiently large, ii) C is sufficiently large, iii) l1 is sufficiently large, or iv) W0 is sufficiently large. The intuitions behind Theorem 3.5 are as follows. As s increases, the critical leverage l also increases to the point at which the risk management rule does not bind anymore, l∗ > l0 . Thus, the deliberate liquidation disappears. Also when C, the liquidation cost, is large enough, the strategy of being aggressive at t = 0 cannot be optimal, and therefore every fund manager decides not to be deliberately liquidated. Finally, because p0 < 1 in a fund run equilibrium, the size of funds that are being aggressive at t = 0 is limited, and therefore a non-negligible size of funds will survive in state b. Under this situation, large l1 or W0 increases p1 (b), resulting in a violation of p1 (b) < p0 . ∗

4

Policy Implications

Our model provides novel insights into the unintended consequences of government intervention in financial markets. Specifically, we examine three episodes; i) direct regulation on the leverage of systemically important financial institutions (SIFI), ii) the formation of optimistic beliefs and iii) liquidity injection during a crisis. The financial crisis of 2007-2009 and its aftermath revealed the fragility of the current financial system as well as the necessity of new regulations to prevent crises in the 24

future. One of the most prominent post-crisis changes in regulations is the adoption of Comprehensive Capital Analysis and Review (CCAR) in the United States or EU-wide Stress Test in the EU. Although the details of the two tests differ, those regulations share substantial commonalities. First, the subjects of those tests are not the most of the financial firms but selected large firms. For example, in 2016, CCAR was imposed on only 33 bank holding companies with $50 billion or more of total consolidated assets. And, only the top 39 Euro area banks, which covered 70% of aggregate assets in the banking sector of EU, participated in EU-wide Stress Test. Second, both tests are intended to make sure that SIFIs to hold sufficient capital so that they can absorb unexpected losses and remain to be solvent to consumers even in an adverse scenario. As a result, to meet regulatory capital requirements, those financial institutions need to have limited exposure to exotic asset classes. Contrary to the expectation of regulatory authorities, our model predicts that such a regulation, on top of individual investors’ risk management, intensifies a systemic risk in financial markets. Recall that we intentionally assume homogeneous fund managers in Section 2 to highlight that the heterogeneous strategies in a fund run is the result of an equilibrium, not from assumptions. We can extend some of these assumptions such as constant l0 to incorporate various forms of financial institutions and to examine the effect of various regulations across different financial institutions. To show this, we set the parameters of our model to satisfy the conditions in Theorem 3.3. Then, there exist both an ideal equilibrium and a fund run equilibrium. In particular, the fund run equilibrium is supported by the strategy profile that the h proportion of the total fund managers take the maximum leverage of l0 and the remaining (1 − h) proportion use the defensive leverage of l∗ . As an analog of the CCAR or EU-wide Stress Test, if l∗ is imposed as the upper bound of li,0 for the (1 − h) proportion of financial firms, Theorem 3.2 implies that the fair-pricing equilibrium is no longer feasible and only the fund run equilibrium survives. Corollary 4.1. When both an ideal equilibrium and a fund run equilibrium are feasible, after a government puts sufficiently strong capital requirements (e.g., the maximum leverage constraint of l∗ ) on a certain proportion (e.g., (1 − h) proportion) of financial institutions, only a fund run equilibrium may survive. Owing to the government’s restriction, regulated financial institutions cannot invest sufficiently in the risky asset. Hence, the risky asset is underpriced at t = 0. In turn, the mispricing of the risky asset justifies the aggressive leverage of unregulated financial institutions. As we have either an ideal equilibrium or a fund run equilibrium in our 25

economy under the condition of Theorem 3.3, as shown in Theorem 3.2, the regulation of the government effectively eliminates an ideal equilibrium. Corollary 4.1 is highly relevant at a time when a number of virtually unregulated hedge funds coexist with the heavily regulated large financial institutions. In addition, recall that the 2007-2009 crisis started by Bear Stearns’ hedge funds, which held highly leveraged positions in the mortgage-backed security market. Comparative statics in a fund run equilibrium provides intuitions on what external factors can drive the crisis more severely. Our model treats q, the probability of state b, as a given parameter. However, in reality, those with political and informational power have an incentive to shape or distort the beliefs of people for their personal benefits such as voting outcomes. An important lesson from our model is that an optimistic prospect can make a crisis even worse. Theorem 4.1. In a fund run equilibrium, for sufficiently small q, s and W0 , as q decreases (increases), more (less) funds are liquidated in state b. Intuitively, when fund managers expect that the market price is more likely to be recovered to the fundamental value, the aggressive leverage becomes more attractive than the defensive leverage. Therefore, more fund managers take the maximum leverage before a crisis, and accordingly, more funds are liquidated during a crisis. Last, our model presents a unique perspective on the effect of a government’s liquidity injection during a crisis. In the middle of the subprime mortgage crisis on 2008, the US Treasury started Troubled Asset Relief Program(TARP), which allows it to purchase or insure troubled assets. Since TARP, the US Federal Reserve has enacted multiple rounds of quantitative easing to stimulate the economy. Theorem 3.5 shows that a sufficiently large l1 can prevent the occurrence of an unnecessary crisis, justifying these drastic government interventions. However, our model also suggests that a crisis may become more severe by these stimulus programs. We elaborate on this issue by examining how the mispricing in the fund run equilibrium responds to the changes in l1 . Panel (a) of Figure 4.1 shows a numerical example in which p1 (b) decreases as l1 , the upper bound of leverage during the crisis state, increases in a fund run equilibrium. Here the (1 − h) proportion of funds, leveraged by l∗ at t = 0, survive in state b. Since p1 (b) = W0 (1 − s) (1 − h) (1 − ε) l1 , decreasing p1 (b) implies that (1 − h) (1 − ε), the size of funds who have survived in state b, decreases faster than the rate of l1 increases, as shown in Panel (b) of Figure 4.1. The intuition behind the negative relationship between (1 − h) (1 − ε) and l1 is as follows. Although an individual fund can increase profits by taking a higher leverage 26

Figure 4.1: The Effect of Liquidity Injection 0.68 0.66

p1 (b)

0.64 0.62 0.6 0.58 0.56 0.54 3

3.1

3.2

3.3

3.4

3.5

3.4

3.5

l1 (a) Price during crisis 0.48 0.46 0.44

1−h

0.42 0.4 0.38 0.36 0.34 0.32 3

3.1

3.2

3.3

l1 (b) Proportion of defensive fund managers Panel (a) shows a numerical example in which p1 (b) decreases as l1 , the upper bound of leverage during the crisis state, increases in a fund run equilibrium. Here, the (1 − h) proportion of funds, leveraged by l∗ at t = 0, can survive in state b with probability of 1 − ε. Panel (b) shows how (1 − h) (1 − ε), the size of funds who have survived in state b, decreases as l1 increases. Other parameters are set as W0 = 0.4, l0 = 3, q = 0.1, C = ε = 0 and s = 0.1. 27

as long as p1 (b) < 1, the relaxed funding constraint during the crisis decreases the aggregate return from state b to t = 2. Note that the return from state b to t = 2 is a decreasing function of l1 , given h function, because 1 + l1 1 1 + l1   − l1 = − l1 . − l1 = p1 (b) W0 (1 − s) (1 − h) (1 − ε) W0 (1 − s) (1 − h) 1 + l1 (1 − ε) Hence, if liquidity injection artificially supports the market price during the crisis, a strategy of surviving in the crisis becomes less attractive and more funds choose to be liquidated during the crisis as an equilibrium phenomenon. As a result, the market price during a crisis can react to the relaxed funding constraint in a counter-intuitive way. Remark 4.1. The mispricing of the risky asset in state b may become severe with a relaxed funding constraint. This observation provides a distinct perspective on the relationship between funding constraints and mispricing during a financial crisis. Existing models have focused on the amplification mechanism between the loss spiral and the margin (or collateral) spiral after a crisis is initiated, predicting a concurrent positive relationship between funding and market liquidity (see, e.g., Brunnermeier and Pedersen (2009) and Acharya and Viswanathan (2011)). In contrast, our model uncovers an intertemporal channel through which a loose funding constraint during a crisis can distort the incentive of financial institutions and crowd out the moderately leveraged financial institutions exante crisis, potentially resulting in more severe mispricing in a crisis.

5

Conclusion

The main objective of this paper is to show that a market-wide financial crisis can arise fully endogenously even in an economy in which funds can be sufficiently provided so that the fair value can be supported as an equilibrium price. Though provocative, we find that such a crisis exists in an economy with three characteristics of the modern financial market: (i) although investors delegate their wealth management to professional fund managers, they try to protect themselves by requesting redemption when the performances of fund managers are not satisfactory; (ii) fund managers are protected from the downside risk of the fund performance through fee schedule and (iii) fund managers can maximize their monetary rewards by taking an excessive leverage under margin constraints. Furthermore, we find that two of the three conditions are 28

not enough to induce a fund run. Exploiting the above market frictions, we succeed in completing the endogenous evolution of arbitrage opportunities without any external disturbances. Furthermore, our model offers novel perspectives on the adverse effects of government intervention on the systemic risk in financial markets. Specifically, we show that (i) when a tight regulation on the leverage is imposed on systemically important financial institutions (SIFI) as in CCAR, the regulation may eliminate a preferred equilibrium with fair pricing, and only a crisis may remain feasible; (ii) optimistic beliefs on the future economy can make the crisis more severe and (iii) the expected liquidity injection during a crisis can crowd out defensive investors from the market before a crisis arises, ironically making a crisis more severe than would be the case without the liquidity injection.

29

A

Proofs

Proof of Lemma 3.1 Assume that p1 (b) > 1. Since p1 (b) is positive, some fund managers have survived in state b. (Otherwise, p1 (b) = 0 by (2.3)) Suppose the i-th fund manager is one of them. He knows that for sure the price will decrease from state b in t = 1 to t = 2. Since the i-th fund manager’s compensation, α max (Wi,2 (b) − W0 , 0)+ βWi,2 (b) , is strictly increasing in Wi,2 (b) and, from (2.8), Wi,2 (b) is strictly decreasing in li,1 (b) by the assumption that p1 (b) > 1, he will take li,1 (b) = −1, the lowest leverage. This optimal leverage is applied to every manager who has survived, implying p1 (b) = 0 by (2.3). This contradicts p1 (b) > 1. Hence, p1 (b) ≤ 1. Assume that p1 (g) > 1. From the identical logic above, it leads to a contradiction. Hence, p1 (g) ≤ 1. Assume that p0 > 1. Since p1 (g) ≤ 1 and p1 (b) ≤ 1, the i-th fund manager knows that the price will decrease from t = 0 to both states in t = 1. Note that if li,0 > −1, by lowering li,0 , he can strictly increase both Wi,1 (g) and Wi,2 (b) , in turn strictly increasing his expected utility in (2.10). Hence, he will take the lowest leverage, -1. This leverage choice is applied to all fund managers at t = 0, implying p0 = 0 by (2.1). This contradicts p0 > 1. Hence, p0 ≤ 1. 

Proof of Theorem 3.1 ⇒: Consider a strategy profile that all fund managers take   1 1 − 1 ≤ l1 . Given the leverage of li,0 = li,1 (g) = W0 − 1 ≤ min l0 , l1 and li,1 (b) = W0 (1−ε) this strategy profile, the market clearing conditions (2.1)-(2.3) imply p0 = p1 (g) = p1 (b) = 1. Note that the i-th fund manager’s investment decision does not affect the prices. Plugging p0 = p1 (g) = p1 (b) = 1 into (2.10) gives U0 (li,0 ) = (1 − q) βW0 + q ((1 − ε) βW0 − εC) which does not depend on the i-th fund manager’s decision. Thus, he does not have an incentive to deviate from his strategy. Therefore, we have an ideal equilibrium. ⇐: Because li,0 ≤ l0 for each i ∈ [0, 1], the market clearing condition at time 0 implies   Z 

1



W0 1 + l0 ≥ W0 1 +

0

li,0 di = 1,

which is the first inequality of Assumption 1. Since p0 = p1 (b) = 1, Wi,1 (b) = W0 ≥ W0 (1 − s) , implying that there does not exist any deliberate liquidation in state b. Hence, the aggregate size of funds in state

30

b decrease only by the random liquidation of ε proportion of funds. It follows that the market clearing condition in state b is given by 

W0 (1 − ε) 1 +

Z 1 0



li,1 (b) di = p1 (b) = 1,

implying 





W0 1 + l1 (1 − ε) ≥ W0 1 +

Z 1 0



li,1 (b) di (1 − ε) = 1,

which is the second inequality of Assumption 1. This completes the proof.



Proof of Lemma 3.2 Assume that fund i is deliberately liquidated in state g, W0 (1 − s) > Wi,1 (g) . From Assumption 2, !

Wi,1 (g) = W0

!

p1 (g) p1 (b) (1 + li,0 ) − li,0 ≥ W0 (1 + li,0 ) − li,0 = Wi,1 (b) . p0 p0

Hence, the i-th fund is deliberately liquidated in both states, resulting in U0 (li,0 ) = −C. This contradicts Lemma A.1. Hence, the i-th fund is not liquidated in state b.  Proof of Lemma 3.3 Assume p1 (g) 6= 1 by contradiction. Lemma 3.1 implies p1 (g) < 1. Pick an arbitrary fund i. Recall that fund i is not liquidated in state g by Lemma 3.2, i.e., Wi,1 (g) ≥ W0 (1 − s) . Then, Ui,1 (g) in (2.4) is a strictly increasing function of Wi,2 (g) . Furthermore, with p1 (g) < 1, Wi,2 (g), given by (2.5), strictly increases with li,1 (g) , which in turn leads to li,1 (g) = l1 for all i ∈ [0, 1]. Because no fund is liquidated in state g by Lemma 3.2 and the fair pricing is not supported, we have that ! Z 1   1 W0 (1 + li,0 ) − li,0 1 + l1 di < 1. p0 0 With the relationship of as

R1

0 li,0 di

=

p0 W0

− 1 by (2.1), the inequality above can be written 



(1 − p0 + W0 ) 1 + l1 < 1.

(A.1)

Then, 







1 < W0 1 + l1 ≤ (W0 + 1 − p0 ) 1 + l1 < 1. The first inequality holds by Assumption 1, the second inequality is due to Lemma 3.1 and the last inequality is from (A.1). This is a contradiction and we have proved p1 (g) = 1.  31

0 (b) = α max (Wi,2 (b) − W0 , 0) + βW (b) from  Proof of Lemma 3.4 Note that Ui,1  i,2 1 (2.7) is strictly increasing in Wi,2 (b) and that Wi,2 (b) = Wi,1 (b) p1 (b) − 1 li,0 (b) + p11(b) is strictly increasing in li,0 (b) when p1 (b) < 1. Hence, the i-th fund which survives in state b will take the optimal leverage of l1 , proving the lemma. 

Proof of Lemma 3.5 For the i-th fund to be deliberately liquidated in state b, it holds that Wi,1 (b) < W0 (1 − s) , implying that Ui,1 (b) = −C. Note that p1 (g) = 1 from Lemma 3.3. Then, some algebras show that the i-th fund manger’s expected utility (2.10) can be obtained as follows: !

U0 (li,0 ) = (1 − q) (α + β) W0

!

1 − 1 (1 + li,0 ) + βW0 − qC. p0

Note that the above is strictly increasing in li,0 with p0 < 1, proving the lemma.



Proof of Lemma 3.6 Assume that no fund survives in state b in some equilibria. Consider two cases of (i) p1 (g) = 0 and (ii) p1 (g) > 0. Step 1. p0 > 0: From (2.1), p0 = 0 implies almost every fund takes li,0 = −1, no position in the risky asset at t = 0. This contradicts the assumption that no fund survives in state b. Hence, p0 > 0. Step 2. p1 (b) = 0: Since no fund survives in state b, this is implied by (2.3). 0 (b) = ∞ if the i-th fund manager survives: Note that for any li,1 (b) > Step 3. Ui,1 −1, (2.8) implies Wi,2 (b) = ∞ by Step 2 and hence Ui,1 (b)0 = ∞ from (2.7). Step 4. If p1 (g) = 0, every fund is liquidated in state g : Assume that some funds survive in state g. That is, there exists a fund i such that Wi,1 (g) ≥ W0 (1 − s) . Fund i maximizes the utility of (2.4), which is a strictly increasing in Wi,2 given by (2.5). Because Wi,2 is infinite with any li,0 (g) > −1, there should be a strictly positive demand. This contradicts p1 (g) = 0. Hence, no fund survives in state g. Step 5. The lemma holds: If p1 (g) = 0, U0 (li,0 ) = −C for all i ∈ [0, 1] because every fund is liquidated in state g (Step 4) and in state b (assumption). If p1 (g) > 0, U0 (li,0 ) is finite for all i ∈ [0, 1] in the equilibrium because p1 (g) > 0 (assumption), p0 > 0 (Step 1) and every fund is liquidated in state b. Hence, for any nonnegative p1 (g), U0 (li,0 ) is finite for all i ∈ [0, 1]. However, if the i-th fund manager deviates and takes li,0 = −1, he will survive in state b with probability of 1 − ε, which gives U0 (−1) =    0 0 (1 − q) Ui,1 (g) + q (1 − ε) Ui,1 (b) − εC ≥ − (1 − q) C + q (1 − ε) Ui,1 (b) − εC = ∞ by Step 3. This is a contradiction to the assumption that no fund survives in state b. Hence, the lemma holds.  32

We need the following lemma to prove Theorem 3.2. Lemma A.1. In any equilibrium, U0 (li,0 ) in (2.10) satisfies that U0 (li,0 ) ≥ ((1 − q) + q (1 − ε)) βW0 − qεC for any i ∈ [0, 1] . Proof Note that any fund manager can choose the strategy of the minimum leverages, li,0 = li,1 (b) = −1, yielding Wi,2 (g) = Wi,2 (b) = W0 , which in turn implies that 0 Ui,1 (g) = Ui,1 (b) = βW0 . Hence, for U0 (li,0 ) with the optimal leverage li,0 , it holds that U0 (li,0 ) ≥ (1 − q) βW0 + q ((1 − ε) βW0 − εC) , proving the lemma.



Proof of Theorem 3.2 i) ⇒: Consider an equilibrium without deliberate liquidation. It will be verified that p0 = p1 (b) = p1 (g) = 1. From Lemma 3.3, p1 (g) = 1. We show that p1 (b) = 1. Assume p1 (b) 6= 1 by contradiction. Lemma 3.1 implies p1 (b) < 1, which in turn leads to li,0 = l1 from Lemma 3.4. Hence, we have that (1 − ε)

Z 1 0

Because

R1

0 li,0 di

=

p0 W0

W0

!   1 (1 + li,0 ) − li,0 1 + l1 di < 1. p0

− 1 by (2.1), the inequality can be written as 



(1 − p0 + W0 ) 1 + l1 (1 − ε) < 1.

(A.2)

Then, 







1 < W0 1 + l1 (1 − ε) ≤ (W0 + 1 − p0 ) 1 + l1 (1 − ε) < 1. The first inequality holds by Assumption 1, the second inequality is due to Lemma 3.1 and the last inequality is from (A.2). This is a contradiction and we have proved p1 (b) = 1. Lastly, we show p0 = 1. Using Lemma 3.1, assume p0< 1 by contradiction. Then, R p0 < W10 ≤ 1 + l0 . Because li,0 ≤ l0 for by (2.1) and Assumption 1, 01 (1 + li,0 ) di = W 0 all i ∈ [0, 1], it should hold that li,0 < l0 for some i ∈ [0, 1]. Fix this i. Recalling that p1 (b) = p1 (g) = 1, implying no deliberate liquidation at t = 1, we compute (2.10), the expected utility of fund manager i. Some algebras give !

U0 (li,0 ) = (1 − q + q (1 − ε)) (α + β) W0 33

!

1 (1 + li,0 ) − li,0 − αW0 − qεC. p0

Because p0 < 1, from Lemma 3.5, li,0 = l0 is optimal. This is a contradiction, which proves p0 = 1. i) ⇐: Because p0 = p1 (g) = p1 (b) , it holds that Wi,1 (b) = Wi,1 (g) = W0 ≥ W0 (1 − s) , implying that there is no deliberate liquidation. ii) ⇒: Consider an equilibrium where some funds are deliberately liquidated. Let the i-th fund be one of the deliberately liquidated funds. We will show that p1 (b) < p0 < p1 (g) = 1. From Lemma 3.3, p1 (g) = 1. Hence, it suffices to show that p1 (b) < p0 < 1. We compute his expected utility at time 0 defined in (2.10). From Lemma 3.2, fund i is liquidated only in state b. Because Ui,1 (b) = −C, the expected utility of the i-th fund manager is expressed as: U0 (li,0 ) = (1 − q) (α max (Wi,2 (g) − W0 , 0) + βWi,2 (g)) − qC ≥ ((1 − q) + q (1 − ε)) βW0 − qεC, where the last inequality is from Lemma A.1. The inequality implies Wi,2 (g) > W0 because otherwise the inequality does not hold. Furthermore, because p1 (g) = 1 implies Wi,2 (g) = Wi,1 (g) , we have that !

1 (1 + li,0 ) − li,0 , p0

W0 < Wi,2 (g) = Wi,1 (g) = W0 implying that

!

1 − 1 > 0. (1 + li,0 ) p0 Because li,0 ≥ −1, p0 < 1.

(A.3)

Furthermore, if p1 (b) ≥ p0 , no fund will face a loss in state b, contradicting the assumption that the i-th fund is deliberately liquidated. Hence, it holds that p1 (b) < p0 .

(A.4)

From (A.3) and (A.4), we have that p1 (b) < p0 < 1. With Lemma 3.3, this completes the proof. ii) ⇐: This is implied by the contrapositive of i) ⇒ .  We need following lemmas to prove Theorems 3.3 and 3.4. Recall that l∗ = and fund manager i is not deliberately liquidated if li,0 ≤ l∗

34

p1 (b) −(1−s) p0 p (b) 1− 1p 0

Lemma A.2. Under Assumptions 1-2 and for a sufficiently small s or s = 0, any equilibrium satisfies h

i

max (Wi,2 (g) − W0 , 0) = Wi,2 (g) − W0 for any li,0 ∈ −1, l0 and max (Wi,2 (b) − W0 , 0) = Wi,2 (b) − W0 for any li,0 ∈ [−1, l∗ ] . Proof We prove the lemma in two parts by classifying equilibria into those without any deliberate liquidation and those with some deliberate liquidation. For the rest of the proof, we assume that 0≤s< 

1 α+β C β+ W



1−q (1−ε)q



0

1+l0 1+l1



,

(A.5)

+1

implying that 1 + l0 

C β+ W 0

α+β



(1−ε)q 1−q

< + 1 + l0

1 + l1 . + 1 + l1

s 1−s

(A.6)

Clearly, the RHS of (A.5) is positive. First, consider an equilibrium where no fund is deliberately liquidated. Theorem 3.2 shows that p0 = p1 (g) = p1 (b) = 1, implying Wi,2 (g) = Wi,2 (b) = W0 for any h i li,0 ∈ −1, l0 . Hence, the two equations in the lemma are valid. Second, consider an equilibrium where some funds are deliberately liquidated. Due to Lemma 3.2, those funds are liquidated only in state b. From Theorem 3.2, p0 < h i p1 (g) = 1. Hence, for any i ∈ [0, 1] and li,0 ∈ −1, l0 , !

Wi,1 (g) = W0

!

1 1 (1 + li,0 ) − li,0 = W0 − 1 (1 + li,0 ) + W0 ≥ W0 , p0 p0

proving the first equation in Lemma A.2. Turn to the second equation in the lemma. By assumption, some funds are deliberately liquidated, and by Lemma 3.6, some funds survive in state b. Also, from Theorem 3.2, it holds that p1 (b) < p0 < p1 (g) = 1. Take any i, j ∈ [0, 1] such that the i-th fund is deliberately liquidated and the j-th fund survives in state b. Then, lj,0 ∈ [−1, l∗ ] and it suffices to show that Wj,2 (b) ≥ W0 . Because fund manager j survives in state b, Wj,1 (b) ≥ (1 − s) W0 . Also from Lemma 3.4, lj,1 (b) = l1 is the optimal choice and hence we need to prove that !

 1  (1 − s) 1 + l1 − l1 ≥ 1, p1 (b)

35

(A.7)









which implies that Wj,2 (b) = Wj,1 (b) p11(b) 1 + l1 − l1 ≥ W0 . For the i-th fund manager which is deliberately liquidated, li,0 = l0 from Lemma 3.5. Furthermore, Lemma A.1 implies !

!

  1 − 1 1 + l0 + βW0 − qC p0

U0 (li,0 ) = (1 − q) (α + β) W0

≥ ((1 − q) + q (1 − ε)) βW0 − qεC. Rearranging the terms in the inequality, we have p0 ≤  β+

1 + l0 C W0



α+β

(A.8)

.

(1−ε)q (α+β)(1−q)

+ 1 + l0

Then, combining (A.6), (A.8) and Theorem 3.2 yields p1 (b) <

1 + l1 . + 1 + l1

s 1−s

This implies (A.7) and the proof is completed.



Lemma A.3. If 



2 + l0 + l1 1  ,  , W0 < W 0 ≡ min   1 + l0 1 + l1 2

(A.9)

it holds that 







(1 − h) W0 1 + l1 < hW0 1 + l0 < 1 h

i

for any h ∈ h, h where h and h are given by h≡

1 + l1 >0 2 + l0 + l1

and h≡

1 

W0 1 + l0

.

Moreover, h < h. 







Proof For (1 − h) W0 1 + l1 < hW0 1 + l0 , h>

1 + l1 =h 2 + l0 + l1 36

(A.10)

(A.11)





Also, for hW0 1 + l0 < 1, h<

1 

W0 1 + l 0

= h.



(A.12)

Lastly, from the the assumption, W0 < 

2 + l0 + l1 1 + l0



1 + l1

,

implying h= 

1 1 + l1   = h. < 2 + l0 + l1 W0 1 + l0



Hence, the interval of h, h is well-defined, which completes the proof.



Lemma A.4. Assume that W0 < W 0 where W 0 is defined in (A.9). Then, it holds that 

q≡

2+l0 +l1 W0 (1+l0 )(1+l1 )

2+l0 +l1 W0 (1+l0 )(1+l1 )



−1



−1



l0 + 1

2 + l0 + l1 + (1 − ε)





C −α W0

α+β







> 0. (A.13)

+ 1 + ε l1 − 1

In addition, if q < q, the following holds: 



C    −α 1  1 W0   − (1 − q) 1 + l1 − l1 + − 1 l0 + 1 < 0, q (1 − ε) f (h) α+β g (h)



!





where f (h) = (1 − h) W0 1 + l1 (1 − ε), g (h) = hW0 1 + l0 (A.10).



and h is given in

Proof First, we prove that q in (A.13) is strictly positive. Note that W 0 in (A.9) is 2+l0 +l1 strictly positive. Since W 0 < W 0 ≤ 1+l , we have that ( 0 )(1+l1 ) 2 + l0 + l1 

W0 1 + l0 Furthermore, from W0 < W 0 ≤ l1 > 1 and that 

(1 − ε)

C  W0

1 2



1 + l1





> 1.

(A.14)



and W0 1 + l1 > 1 (Assumption 1), it follows that



  −α C β + 1 + ε l1 − 1 = (1 − ε) + α+β W0 (α + β) α + β

!





+ ε l1 − 1 > 0. (A.15)

37

Using (A.14) and (A.15), we prove that q > 0. Next, turn to the latter inequality. Some algebras give 



C    −α 1  1 W0   − (1 − q) − 1 l0 + 1 q (1 − ε) 1 + l1 − l1 + f (h) α+β g (h)

 







=q (λ − 1) 2 + l0 + l1 + (1 − ε) 


C

2+l0 +l1 . W0 (1+l0 )(1+l1 )



−α

!



C  W0

    −α + 1 + ε l1 − 1  + (λ − 1) l0 + 1 α+β







    −α + 1 + ε l1 − 1  + (λ − 1) l0 + 1 = 0, α+β





C  W0





Note that (A.14) and (A.15) imply (λ − 1) 2 + l0 + l1 +





0 + 1 + ε l1 − 1 > 0, justifying the inequality above. This completes (1 − ε) Wα+β the proof of the second claim in the lemma. 

Lemma A.5. Suppose f , g and v be functions on (0, 1) × [0, ∞) that satisfy 

f (h, s) = W0 1 + hl0 + (1 − h) v (h, s)







g (h, s) = W0 (1 − h) (1 − s) 1 + l1 (1 − ε) and v (h, s) =

g(h,s) f (h,s)

− (1 − s)

1−

g(h,s) f (h,s)

.

If 0 < g (h0 , 0) < f (h0 , 0) < 1, then f (h, s) and g (h, s) are continuously differentiable and 0 < g (h, s) < f (h, s) < 1 in a neighborhood of (h0 , 0) . Proof It is obvious that g (h, s) is continuously differentiable. Plugging v (h, s) to f (h, s) yields the equation, !

f (h, s) = W0

f (h, s) (1 − s) − g (h, s) . 1 + hl0 + (1 − h) g (h, s) − f (h, s)

Under the assumption that f (h, s) > g (h, s), f (h, s) has a unique solution, r

B+



f (h, s) = 



where B = W0 (1 − h) s + h 1 + l0 



B 2 − 4W0 h 1 + l0 g (h, s) 2 

,

(A.16)

+ g (h, s). (In fact, some algebras show that



B 2 − 4W0 h 1 + l0 g (h, s) ≥ 0 for any s ≥ 0.) Hence, the expression of (A.16) along with the continuously differentiability of g (h, s) confirms that f (h, s) is continuously 38

differentiable in a neighborhood of (h0 , 0). Also, the inequalities of 0 < g (h, s) < f (h, s) < 1 follow from the continuity of f (h, s) and g (h, s) and the inequalities of 0 < g (h0 , 0) < f (h0 , 0) < 1.  Lemma A.6. Assume that s is sufficiently small. Given p0 and p1 (b), the expected utility of the i-th fund manger at t = 0, can be evaluated by U (li,0 ) = (δ0 + δ1 li,0 ) · 1 (li,0 ≤ l∗ ) + (γ0 + γ1 li,0 ) · 1 (li,0 > l∗ ) ,

(A.17)

where l∗ is given in (3.2), !

"

!

#

1 1 p1 (b) 1 + − 1 l1 , (A.18) δ0 = W0 (1 − q) + q (1 − ε) p0 p1 (b) p1 (b) p0 " ! !# ! ! 1 1 1 p1 (b) δ1 = W0 (1 − q) − 1 + q (1 − ε) −1 , + − 1 l1 p0 p1 (b) p1 (b) p0 (A.19) 



C −α 1 W0  and  γ0 = W0 (1 − q) − q (1 − ε) p0 α+β

(A.20)

!

1 −1 . γ1 = W0 (1 − q) p0

(A.21)

Proof Assume that s is sufficiently small so that Lemma A.2 applies. Note that (1 − q) Ui,1 (g) + qUi,1 (b) is equivalent to U (li,0 ) , where U (li,0 ) ≡

1 ((1 − q) Ui,1 (g) + qUi,1 (b) + αW0 (1 − q + q (1 − ε)) + qεC) . α+β

In addition, from that Ui,1 (g) = (α + β) Wi,2 (g) − αW0 , !

!

1 1 − 1 li,0 + , p0 p0

Wi,2 (g) = Wi,1 (g) = W0

Ui,1 (b) = (1 − ε) (((α + β) Wi,2 (b) − αW0 ) · 1 (li,0 ≤ l∗ ) − C · 1 (li,0 > l∗ )) − qεC, !

Wi,2 (b) = Wi,1 (b)

1 1 − 1 l1 + p1 (b) p1 (b)

!

and !

Wi,1 (b)=W0

!

p1 (b) p1 (b) − 1 l1 + , p0 p0

some algebras show that (A.17) holds.



39

Proof of Theorems 3.3 and 3.4 The proof strategy is as follows. First, the existence of a fund run equilibrium will be proved when s = 0. Then, it will be shown that the claim holds in a neighborhood of s = 0, or s ∈ [0, s) for a sufficiently small s > 0 which is to be determined below. For the rest of the proof, we assume that 0 ≤ ε < 1, 0 < W0 ≤ W 0 and 0 < q < q where W 0 and q are given in (A.9) and (A.13), respectively. Lemma A.4 shows q > 0. Note that the range of 0 ≤ ε < 1 includes the possible cases of ε in both Theorems 3.3 and 3.4. Resorting to Lemma 3.2, we do not consider the liquidation in state g. Also, from Lemma 3.3, we take the fair pricing p1 (g) = 1 in state g as given by using (3.1), the equilibrium utility in state g. For each case of s = 0 and s > 0, we will check the feasibility of leverages in state g which support the fair pricing before the end. Set s = 0. Note that from Lemma A.6, we consider the expected payoff of (A.17) as the maximizing objective of fund managers. We show that a fund run equilibrium can be supported by the bang-bang strategy profile in Definition 3.1 with l∗ = −1 and a properly chosen h. If the bang-bang strategy profile is implemented, the market clearing conditions (2.1) and (2.3) allow us to view the prices as functions of h: 

p0 (h) = hW0 1 + l0



(A.22)

and 



p1b (h) = (1 − h) W0 1 + l1 (1 − ε) .

(A.23)

We will show that d (h) = 0 for some h ∈ (0, 1), where d (h) ≡ U (−1; p0 (h) , p1b (h))−  U l0 ; p0 (h) , p1b (h) and U (·) is defined in (A.17) and here we indicate dependence on p0 (h) and p1b (h) explicitly. Recall that h and h are defined in Lemma A.3.   Step 1. 0 < p1b (h) < p0 (h) < 1 if h ∈ h, h : Since ε ≥ 0, Lemma A.3 implies that 





p1b (h) < p0 (h) < 1 for h ∈ h, h where h and h are defined in (A.10) and (A.11), respectively. Since ε is the proportion of random liquidation, ε < 1, implying that 0 < p1b (h) < p0 (h) < 1. Step 2. We have 







C    −α 1  1 W0  − (1 − q)   d (h) = W0 q (1 − ε) 1 + l1 − l1 + − 1 l0 + 1  p1b (h) α+β p0 (h)

for h ∈ [0, 1]: Clear by (A.18)-(A.21).

40

!

 

Step 3. d (h) < 0 < d h : In Lemma A.4, f (h) = p1b (h) and g (h) = p0 (h). Thus, the same lemma implies that d (h) < 0. On the other hand, setting h = h and utilizing Lemma A.3 give  





















 

p0 h = hW0 1 + l0 = 1 > 1 − h W0 1 + l1 > 1 − h W0 1 + l1 (1 − ε) = p1b h . Then,  

d h





=W0 q (1 − ε)  

=W0 q (1 − ε) 

1  

p1b h 1

 

p1b h



1 + l1





C W0







  −α 1  − (1 − q)    − 1 l0 + 1  − l1 + α+β p0 h 

 

− 1 1 + l1



β  C > 0. + + W0 (α + β) α + β

 









(A.24)

The second equality holds because p0 h = hW0 1 + l0 = 1. To see the inequality, note that 











 

1 > 1 − h W0 1 + l1 > 1 − h W0 1 + l1 (1 − ε) = p1b h

by Lemma A.3.   Step 4. d (h0 ) = 0 for some h0 ∈ h, h and such h0 is unique: By Step 3 and the 



continuity of d (h), there exists h0 ∈ h, h such that d (h0 ) = 0. Furthermore, since p0 (h) is strictly increasing in h and p1b (h) is strictly decreasing in h, Step 2 implies h i ∂d (h) > 0 where h ∈ h, h . ∂h

(A.25)

Thus, uniqueness is proved. Step 5. The strategy profile is a fund run equilibrium: Set h = h0 . Since d (h0 ) = 0,   U (−1; p0 (h0 ) , p1b (h0 )) = U l0 ; p0 (h0 ) , p1b (h0 ) . Note that p0 (h0 ) < 1 from Step 1 implies that γ1 > 0 from (A.21). Because l∗ = −1 for si = 0, U (li,0 ) = (γ0 + γ1 li,0 ) ·  ∗ 1 (li,0 > l ) is strictly increasing in li,0 for li,0 ∈ −1, l0 . Thus, U (li,0 ) ≤ U (−1) =  

h

i

U l0 for any li,0 ∈ −1, l0 , showing that fund manager i does not have an incentive to deviate. Finally, we show that the fair pricing can hold in state g with the bang-bang strategy profile with h = h0 . If the fair pricing holds in state g, the aggregate wealth in state g is Z 1 0

!

Wi,1 (g) di = h0 W0

 1  1 + l0 − l0 + (1 − h0 ) W0 ≥ W0 , p0 (h0 )

41

where the inequality is from p0 (h0 ) < 1 (Step 1). Hence, the fair price can be supported − 1 < W10 − 1 ≤ l1 across all fund by taking the uniform leverage l = R 1 1 0

Wi,1 (g)di

managers. Therefore, the strategy profile is an equilibrium and Step 1 implies that 0 < p1b (h) < p0 (h) < 1, which in turn showing that it is a fund run equilibrium by Theorem 3.2. Next, we examine the existence of a fund run equilibrium when s > 0. We analyze a fund run equilibrium around the neighborhood of s = 0 such that Lemma A.2 still holds. Hence, we consider the payoff of (A.17). For the ease of analysis, we classify the equilibria at s = 0 into two cases of δ1 > 0 and δ1 ≤ 0 where δ 1 is given by (A.19). First, we consider the case where δ1 ≤ 0 at (h, s) = (h0 , 0). We will show that the equilibrium strategy profile and prices for s = 0 are also an equilibrium for s > 0. For s = 0, we have shown that the bang-bang strategy profile with h = h0 is an equilibrium. Let p0 and p1 (b) denote the equilibrium prices for s = 0. We show that p0 and p1 (b) clear the markets for a sufficiently small s > 0. Because (2.1) does not involve s, p0 satisfies (2.1). Take s > 0 such that l∗ = −1 +

s 1−

p1 (b) p0

p1 (b) −(1−s) p0 p (b) 1− 1p 0

=

< l0 for any s < s. Then, the h0 proportion liquidate in state b, which is

the same as in s = 0. Thus, p1 (b) satisfies (2.3). h i   We show that U (−1) = U l0 ≥ U (li,0 ) for li,0 ∈ −1, l0 to claim the fund managers do not have incentives to deviate. We use (A.17). The assumption δ1 ≤ 0 implies that U (−1) ≥ U (li,0 ) for li,0i ≤ l∗ . Since p0 < 1, we have γ > 0, implying   1 ∗ that U l0 ≥ U (li,0 ) for li,0 ∈ l , l0 . The property U (−1) = U l0 comes from the construction of h0 . Therefore, if δ1 ≤ 0 at (h, s) = (h0 , 0), there is a fund run equilibrium for s > 0. The second case is δ1 > 0 at (h, s) = (h0 , 0). The proof idea is as follows. We restrict our attention to the bang-bang strategy profile in Definition 3.1 with l∗ = l∗ : the h proportion of fund managers (indexed by i) take li,0 = l0 and liquidate deliberately in   state b, and the remaining 1 − h proportion (indexed by j) take (lj,0 , lj,1 (b)) = l∗ , l1 . Like what we did in the case s = 0, we define 



d (h, s) = U (l∗ ; p0 , p1b ) − U l0 ; p0 , p1b . Then, we show hs satisfying d (hs , s) = 0 exists and the suggested strategy profile with h = hs yields a fund run equilibrium.

42

We express dependence of l∗ , p0 and p1b on (h, s) explicitly. Denoting l∗ in (3.2) as a function of prices gives ∗

l (h, s) =

p1b (h,s) p0 (h,s)

− (1 − s)

1−

p1b (h,s) p0 (h,s)

(A.26)

.

The prices satisfy the market clearing conditions (2.1) and (2.3): 



p0 (h, s) = W0 1 + hl0 + (1 − h) l∗ (h, s) and 

(A.27)



p1b (h, s) = [(1 − ε) (1 − h)] [(1 − s) W0 ] 1 + l1 .

(A.28)

The equality in (A.28) follows because, in state b, the (1 − ε) (1 − h) fund managers survive the random and deliberate liquidations and their time 0 leverage l∗ satisfies   Wi,1 (b) = W0 p1p(b) (1 + l∗ ) − l∗ = (1 − s) W0 . 0 Then, we can write d (h, s) as a function of (h, s) explicitly. Use (A.17)-(A.21) along with the above expressions to write d (h, s) as follows: d (h, s)  



=U (l∗ (h, s)) − U l0 = (δ0 − δ1 l∗ (h, s)) − γ0 + γ1 l0 



(A.29)



= (δ0 − γ0 ) − (δ1 − γ1 ) l∗ (h, s) − γ1 l0 − l∗ (h, s) !

!

!

!

1 1 p1b (h, s) p1b (h, s) =W0 q (1 − ε) + − 1 l1 − − 1 l∗ (h, s) p1b (h, s) p1b (h, s) p0 (h, s) p0 (h, s) !     C 1 + W0 q (1 − ε) − 1 l0 − l∗ (h, s) . (A.30) − α − W0 (1 − q) W0 p0 (h, s) The following steps show that there exists a fund run equilibrium. Step 1. d (h, s) and l∗ (h, s) are continuously differentiable and 0 < p1b (h, s) < p0 (h, s) < 1 in a neighborhood of (h0 , 0): We have shown that 0 < g (h0 , 0) < f (h0 , 0) < 1 from a fund run equilibrium with s = 0. Lemma A.5 implies p0 (h, s) and p1b (h, s) are continuously differentiable with respect to (h, s) and 0 < p1b (h, s) < p0 (h, s) < 1 in a neighborhood of (h0 , 0). Thus, l∗ (h, s) is a continuously differentiable function of (h, s) and so is d (h, s). h i Step 2. l∗ (h, s) ∈ −1, l0 in a neighborhood of (h0 , 0): Note that the continuously ∗

(h,s) |s=0 = differentiability of p1b (h, s) and p0 (h, s) from Lemma A.5 implies that ∂l ∂s 1 ∗ p1b (h,s) > 0 when h is close to h0 . Also, since l 0 > −1, the continuity of l (h, s) (Step 1−

p0 (h,s)

1) implies that −1 ≤ l∗ (h, s) < l0 in a neighborhood of (h0 , 0). Step 3. U (l∗ ) ≥ U (li,0 ) for li,0 ∈ [−1, l∗ ] in a neighborhood of (h0 , 0): Note that [−1, l∗ ] is non-empty by Step 2. We utilize the property that δ1 > 0 at (h0 , 0). From 43

the expression of δ1 in (A.19), we find that δ1 is continuous in p0 and p1 (b) if p0 > 0 and p1 (b) > 0. By the continuity of p0 (h, s) and p1b (h, s) in (h, s) around (h0 , 0) , verified in Lemma A.5, we can always find a neighborhood of (h0 , 0) such that δ1 > 0. This implies U (l∗ ) ≥ U (li,0 ) for li,0 ∈ [−1, l∗ ] . Step 4. There is s > 0 such that there exists a continuously differentiable function hs of s such that d (hs , s) = 0 for all s ∈ [0, s): By invoking the implicit function theorem, we conclude that there exist a continuously differentiable function hs and s > 0 such that d (hs , s) = 0 for any s ∈ [0, s) . Step 5. The suggested strategy profile constitutes a fund-run equilibrium: Consider the bang-bang strategy profile in Definition 3.1 with h = hs and l∗ = l∗ . From Step 4, we   know that U (l∗ ) = U l0 . Furthermore, Step 3 shows that U (l∗ ) ≥ U (li,0 ) for li,0 ≤ l∗ . Also, since p0 (hs , s) < 1 from p0 (h0 , 0) < 1 and the continuity ofp0 (h, s) in Lemma    ∗ A.5, we have γ1 > 0, implying that U l0 > U (li,0 ) for li,0 ∈ l , l0 . Combining these  

h

i

findings, we have U (l∗ ) = U l0 ≥ U (li,0 ) for li,0 ∈ −1, l0 , verifying that no fund manager has an incentive to deviate from the bang-bang strategy profile. Finally, we show that the fair pricing can hold with the bang-bang strategy profile with h = hs . If the fair pricing holds in state g, the aggregate wealth in state g is Z 1 0

!

Wi,1 (g) di = h0 W0

!

  1 1 1 + l0 − l0 +(1 − h0 ) W0 (1 + l∗ ) − l∗ ≥ W0 , p0 (hs , s) p0 (hs , s)

where the inequality is from p0 (hs , s) < 1 (Step 1). Hence, the fair price can be supported by taking the uniform leverage l = R 1 1 − 1 < W10 − 1 ≤ l1 across all 0

Wi,1 (g)di

fund managers. From Step 1, we have 0 < p1b (hs , s) < p0 (hs , s) < 1, implying that there exists deliberate liquidations by Theorem 3.2. Hence, the bang-bang strategy profile constitutes a fund-run equilibrium.  Proof of Theorem 3.5 Theorem 3.1 guarantees the existence of an ideal equilibrium. We show that a fund run equilibrium does not exist under the given conditions. i): Assume that s > l0 + 1 and there exists a fund run equilibrium. Then, ∗

l =

p1 (b) p0

− (1 − s)

1−

p1 (b) p0

= −1 +

44

s 1−

p1 (b) p0

> −1 + s > l0 ,

< 1 in a fund run by Theorem 3.2, where the first inequality follows from 0 < p1p(b) 0 and the second by the assumption. Hence, there is no deliberate liquidation. This is a contradiction. ii): Assume that 







    1 + l0 (1 − q)    − W0 1 + l0 − 1  − βW0 C> q (1 − ε) (1 − s) (1 − ε) 1 + l1

(A.31)

and there exists a fund run equilibrium. Then, there exists some funds which are deliberately liquidated in state b. Let the i-th fund be one of such funds. From Theorem 3.2, p0 < 1, yielding li,0 = l0 by Lemma 3.5. Exploiting Lemma A.1, we have U0 (li,0 ) = max (1 − q) (α max (Wi,2 (g) − W0 , 0) + βWi,2 (g)) − qC li,0

!

= (1 − q) (α + β) W0

!

 1  1 + l0 − l0 − αW0 − qC p0

≥ ((1 − q) + q (1 − ε)) βW0 − qεC. This inequality implies that 

p0 <

(1 − q) W0 1 + l0

 

q (1 − ε) (βW0 + C) + (1 − q) W0 1 + l0

(A.32)

.

Recalling that there exist h proportion of fund managers who take li,0 = l0 from Lemma 3.5, we have 



hW0 1 + l0 ≤ p0 = W0

Z 1 0

(1 + li,0 ) di.

(A.33)

From (A.32) and (A.33), we have 1−h>1−



p0 

W0 1 + l 0



>







q (1 − ε) (βW0 + C) + (1 − q) W0 1 + l0 − 1 

q (1 − ε) (βW0 + C) + (1 − q) W0 1 + l0



. (A.34)

Next, recall that there exists a fund who has survived in state b by Lemma 3.6. Suppose the j-th fund manager survives in state b. Since p1 (b) < 1 from Theorem 3.2, the optimal leverage at time 1, lj,1 (b), is l1 from Lemma 3.4. Noting that Wj,1 (b) is at least W0 (1 − s) and that (1 − h) (1 − ε) proportion of fund managers survive, 



p1 (b) ≥ W0 (1 − s) (1 − h) (1 − ε) 1 + l1 .

45

(A.35)

From (A.34) and (A.35), we have 



p1 (b) ≥ W0 (1 − s) (1 − ε) 1 + l1









q (1 − ε) (βW0 + C) + (1 − q) W0 1 + l0 − 1 

q (1 − ε) (βW0 + C) + (1 − q) W0 1 + l0



.

(A.36) Lastly, some algebras show that the assumption of (A.31) and two inequalities of (A.32) and (A.36) yield p0 < p1 (b) , contradicting the existence of a fund run equilibrium from Theorem 3.2. iii) and iv): Assume there exists a fund run equilibrium. Suppose h > 0 proportion of fund managers take li,0 ≥ l∗ . Since p0 < 1 from Theorem 3.2, li,0 = l0 from Lemma 3.5. Hence, it should hold that 



W0 h 1 + l 0 ≤ W0

Z 1 0

(1 + li,0 ) di = p0 < 1,

where the equality follows from (2.1). This implies W0 (1 − h) > W0 −

1 . 1 + l0

Note that (1 − h) (1 − ε) proportion of funds survive in state b and that Wi,1 (b) of each such fund is at least W0 (1 − s). Furthermore, since p1 (b) < 1 from Theorem 3.2, lj,1 (b) = l1 from Lemma 3.4. Hence, 



p1 (b) ≥ W0 (1 − s) (1 − h) (1 − ε) 1 + l1 . Combining the above two inequalities yields !

  1 (1 − s) (1 − ε) 1 + l1 , p1 (b) ≥ W0 − 1 + l0

implying p1 (b) ≥ 1 with a sufficiently large l1 or W0 . This contradicts to p1 (b) < 1 in a fund run equilibrium by Theorem 3.2.  Proof of Theorem 4.1 From (A.25), it holds that in a fund run equilibrium around (h0 , 0) ∂d (hs , s) > 0 when d (hs , s) = 0. (A.37) ∂h

46

Since p0 (h, s), p1b (h, s) and l∗ (h, s) in (A.27)-(A.26) are not affected by q, it follows from (A.30) that !

!

!

!

  ∂d (hs , s) 1 1 1 = W0 (1 − ε) + − 1 l1 (1 − s) + − 1 l0 − l∗ > 0. ∂q p1 (b) p1 (b) p0 (A.38) ∂d(hs ,s) ∂d(hs ,s) dhs Applying the implicit function theorem, we obtain dq = − ∂q / ∂h < 0, completing the proof. 

47

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