### Discussion Paper

## Abstract

A hedge fund’s capital structure is fragile because uninformed fund investors are highly loss sensitive and easily withdraw capital in response to bad news. Hedge fund managers, sharing common investors and interacting with each other through market price, sensitively react to other funds’ investment decisions. In this environment, panic-based market runs can arise not because of systematic risk but because of the fear of runs. The authors find that when the market regime changes from a normal state to a “bad” state (in which runs are possible), hedge funds reduce investment prior to runs. In addition, the market runs are more likely to occur in a market where hedge funds hold greater market exposure and uninformed traders have greater sensitivity to past price movement.

## Comments and Questions

This study theoretically shows that the irrational behavior of uninformed investors indeed hinder the investment decision of hedge fund in equilibrium. This study is meaningful for many researchers and practitioners in the sense that the problem may distort the investment decision. Based on this study, many follow-up studies can analyze ...[more]

... features of uninformed investors and their impact on the hedge fund.

See attached file

Author Responses and Revision Plans

Discussion Paper 2019-31

Title: Market runs of hedge funds during financial crises

Authors: Sangwook Sung, Hoon Cho, and Doojin Ryu

Thank you very much for giving us the two referee reports. The two referees highly evaluate this paper as follows.

1) The ...[more]

... goal of the paper, i.e. to shed light on the detailed interaction of agents in financial markets that gives rise to strategic fragility, is interesting and definitely deserves scholarly attention.The paper tells a reasonable story that indeed seems to match stylized facts from the financial crisis.

2) I think the paper investigates a relevant and interesting issue. Understanding the determinants behind fund managers’ decision to cut their arbitrage positions and how their initial asset allocation affects the exposure to runs is very important also in light of the recent crisis. In particular, I praise the authors for trying to endogenize the short-term return (also referred to as market price). This is not only relevant and more realistic than having a fixed liquidation value, but it is also, especially in the context of global games model, quite challenging. In my opinion, the endogenization of the price at which risky assets are sold in the market to raise liquidity at short notice would represent a key contribution of this paper to the existing literature.

As the referees suggests, we believe that this paper has a potential and contributes to the existing literature. After the editor makes a formal revision decision, we will rebuild our model setup and totally reconstruct our model and framework based on the referee comments. It will be a long journey. However, we will do our best to enhance the quality and readability of this paper considering all of referee comments.

Warm regards,

Doojin

Doojin Ryu (corresponding author) is a Tenured Professor at College of Economics, Sungkyunkwan University (SKKU). Ryu is the Editor of Investment Analysts Journal (SSCI) and the subject editors of Emerging Markets Review (SSCI).

https://sites.google.com/site/doojinryu/home/papers

see attached file

Author Responses and Revision Plans

Discussion Paper 2019-31

Title: Market runs of hedge funds during financial crises

Authors: Sangwook Sung, Hoon Cho, and Doojin Ryu

Thank you very much for giving us the two referee reports. The two referees highly evaluate this paper as follows.

1) The ...[more]

... goal of the paper, i.e. to shed light on the detailed interaction of agents in financial markets that gives rise to strategic fragility, is interesting and definitely deserves scholarly attention. The paper tells a reasonable story that indeed seems to match stylized facts from the financial crisis.

2) I think the paper investigates a relevant and interesting issue. Understanding the determinants behind fund managers’ decision to cut their arbitrage positions and how their initial asset allocation affects the exposure to runs is very important also in light of the recent crisis. In particular, I praise the authors for trying to endogenize the short-term return (also referred to as market price). This is not only relevant and more realistic than having a fixed liquidation value, but it is also, especially in the context of global games model, quite challenging. In my opinion, the endogenization of the price at which risky assets are sold in the market to raise liquidity at short notice would represent a key contribution of this paper to the existing literature.

As the referees suggest, we believe that this paper has potential and contributes to the existing literature. After the editor makes a formal revision decision, we will rebuild our model setup and totally reconstruct our model and framework based on the referee comments. It will be a long journey. However, we will do our best to enhance the quality and readability of this paper considering all of the referee comments.

Warm regards,

Doojin

Doojin Ryu (corresponding author) is a Tenured Professor at College of Economics, Sungkyunkwan University (SKKU). Ryu is the Editor of Investment Analysts Journal (SSCI) and the subject editors of Emerging Markets Review (SSCI).

https://sites.google.com/site/doojinryu/home/papers

Revised Paper

Authors’ Responses for the Referee Comments

Title: Market runs of hedge funds during financial crisis

Reviewer #1

Summary

The paper undergoes a theoretical analysis of runs by hedge funds. In the paper, these refer to the decision of hedge fund managers to cut their arbitrage positions, i.e., ...[more]

... to reduce their exposure to risky (profitable) investments and so increase their cash holdings.

The paper highlights the role of strategic uncertainty in triggering these runs: a fund manager chooses to disinvest because she/ he fears that others would do the same. This behaviour hinges on the existence of strategic complementarities. Strategic complementarities emerge in the paper because the proportion of fund manager disinvesting influences the short-term return (i.e., the market price) of the risky technology: The more fund managers run, the lower the short-term return. The endogenous short-term return represents the central and novel element of the paper.

Normally, a setting exhibiting strategic complementarities would feature multiple equilibria. By using global games the authors overcome this and uniquely pin down the run probability. This, in turn, allows them to characterize fund managers’ initial asset allocation (i.e., how much to invest in the risky asset and how much to hold as cash) taking into account how this decision impacts the run probability.

The basic setup of the paper is as follows. There are four dates and three groups of agents: fund managers, fund investors and trend followers. At the initial date, arbitrage opportunities arise as the price of the risky asset falls below its fundamental value. In the subsequent date, fund managers raise 1 unit of resources from investors and allocate it between a risky asset and cash. At t1, θ funds are hit by a funding shock in that they go bankrupt and exit the market. The proportion of funds hit by the shock is a uniformly distributed random variable, whose realization is not publicly observable. Fund managers only observe a private imperfect signal about it and based on this signal they decide whether to exit the market. If they do not exit the market, they need to meet the demand for liquidity by investors, which is assumed to correspond to the overall proportion of funds exiting the market λ. At the final date, all returns are produced and payoffs of remaining agents realize.

Major comments

Comment 1-1. Relevance and contribution

I think the paper investigates a relevant and interesting issue. Understanding the determinants behind fund managers’ decision to cut their arbitrage positions and how their initial asset allocation affects the exposure to runs is very important also in light of the recent crisis.

In particular, I praise the authors for trying to endogenize the short-term return (also referred to as market price). This is not only relevant and more realistic than having a fixed liquidation value, but it is also, especially in the context of global games model, quite challenging. In my opinion, the endogenization of the price at which risky assets are sold in the market to raise liquidity at short notice would represent a key contribution of this paper to the existing literature.

Despite finding the paper interesting and relevant, I have a number of reservations on the analysis. As I already mentioned, in my opinion, the endogenization of the market price represents a significant contribution. However, the way it is achieved in the current version of the paper raises a number of important issues regarding the way strategic complementarity is modelled and the determination of r (λ). Furthermore, the exposition in the current version of the paper is often obscure, making difficult to follow the derivations and to grasp the intuition behind the results.

Below some detailed comments about these issues.

Answer 1-1

We are grateful for your interest in our analysis of the fund manager’s decision process during crisis. We agree that although r(λ) is important in our model, explanation is insufficient to understand. To enhance readability, we revised the paper following referee’s suggestions.

Comment 1-2. Fund investors’ withdrawal decision, fund managers’ action and strategic complementarity

Unlike other papers in the literature (e.g., Liu and Mello (2011)), the paper focuses on the strategic complementarity between fund managers rather than fund investors. This allows the authors to consider the strategic interactions between different funds and so endogenize the market price. However, I find that the way this is modelled very ad hoc.

In the paper, the proportion of funds exiting the market is stochastic and each fund manager only receives an imperfect signal about it. Based on this signal they decide what to do- i.e., exiting the market or not. The proportion of fund managers exiting the market is λ. The variable λ also represent the proportion of resources that investors withdraw from the exiting funds. This is crucial as it affects the amount that remaining funds may need to raise at short notice in the market and so the market price.

Maybe I am missing something, but it seems to me that the proportion of funds exiting the market λ affects the market price only because it translates (in an exogenous and ad hoc way) in the amount of withdrawals by fund investors. I found this approach problematic as it highlights that, although stated differently in the paper, what really matters is the fund investors’ withdrawal decisions. If so, then, it is not clear why the authors do not simply consider the strategic complementarity between fund investors as in Liu and Mello (2011). They mention something in the introduction about this, but I do not find their argument particularly convincing.

Additionally, in the current model it is more complicated to appreciate and separate the two stages of the analysis. In current version, the fund managers’ decision is essentially analogous at t1 and t2. At t1, they choose the how much to invest in the risky asset and how much to hold in cash. At t2, the decision is similar in that it boils down to choose whether to hold just cash or also the risky asset. Focusing on the strategic complementarity between fund investors would allow overcoming this.

While reading the paper, I was wondering whether it would be worth to try an alternative approach and so model directly the strategic complementarity between fund investors within a single fund and those between investors in different funds, as it is done in Goldstein (2005). In other words, one could think to a situation where an investor in fund i is concerned not only about withdrawals in his own fund but also in other funds, since the more investors withdraw in other funds, the more these funds would need to liquidate and so the lower will be the market price. This approach would allow the authors to retain the interaction between different funds via the market price and so the its endogenous determination.

Answer 1-2

We thank the referee for their comment. λ is an important variable in our paper as referee has mentioned. However, the referee doubts that our model does not consider the strategic complementarity between the investor and simply assume that homogeneous investor determines withdrawal mechanically depending on λ. Our answer is as follows.

As the referee mentioned, there are key differences between our study and existing studies (e.g. Liu and Mello (2011)) in terms of model background. Liu and Mello assume informed and sophisticated fund investors and model game of fund investors. However, some studies argue that the fund investors are usually not the rational investors, especially in the distressed market. Accordingly, we focus more on the fund managers’ behavior, who are known as more informed and sophisticated, and model game of fund managers. To focus more on the fund mangers’ behavior, we simplify the fund investors’ role. We assume that fund investors observe λ after the surviving fund managers’ decision, and withdrawal portion g(λ) of their capital. As we assume that fund investors treat λ as the proxy for the economic state; that is, they conjecture that a higher λ means a riskier market situation, function g(x) should be increasing function of x. For the simplicity, we assume that g(x)=x. Nevertheless, we agree with the referee's opinion that it is unrealistic to assume that the withdrawal one-to-one corresponds with the aggregate portion of default and exit funds. We also think that to adopt a complex function or consider heterogeneous investors’ decision making would be a good further question. We added and revised following statement in Chapter 2.1.

“We denote λ as the aggregate portion of default and exit funds. Then, λ-θ is the portion of surviving and exit funds. After the surviving fund managers’ decision, fund investors observe λ, but they cannot distinguish which funds default or exit. We assume that they treat λ as the proxy of economic state and withdrawal portion g(λ) of their capital. Function g(x) should be increasing function of x, because they conjecture that a higher λ means a riskier market situation. For the simplicity, we assume that g(x)=x. That is, fund investors withdraw exact portion λ of their capital from remaining funds.”

Also, we added following footnotes in Chapter 2.1 and 2.2, respectively.

“It may be unrealistic to assume that the withdrawal one-to-one corresponds with the aggregate portion of default and exit funds. To adopt a complex function or consider heterogeneous investors’ decision making would be a good further question.”

“To solve the model analytically, we assume that new liquidity inflow into the risky asset market at t2 canceled out liquidity outflow from default fund; however, this assumption does not change the implications of the model because the negative relation between liquidity outflow and market return is maintained.”

We separate t1 and t2 to enhance readability, however, lack of explanation made it more confusing. In revised version, we insert two subsections provide more detailed explanation in Chapter 2. In Chapter 2.1 Main players, we introduce two main players: fund managers and fund investors and their decision making. In Chapter 2.2 Market price determination process, we describe changes in market price over time in detail.

Comment 1-3. Asset market

In general, I found the description of the functioning of the asset market and so the determination of the market price (equation 3) not very accurate. I think the current version of the paper could be improved significantly by adding a few more details. In particular, I did not really understand who is buying the assets sold by fund managers Other funds? Outside investors? How is the market price determined? Is it a cash-in-the-market price? All these elements are important to understand how the market price is determined.

Relatedly, while equation 3 is a key equation in the paper, the authors do not provide many details about how it is determined. I think much more details and explanations should be added. For example, where does the function LI(λ) come from? The authors refer to this as the new liquidity inflow in the risky asset market. Who is providing this liquidity? The exact expression assigned to LI(λ) by the authors does not seem so “neutral” as they state as it cancel out with another term in the expression for the short-market return. What am I missing?

Finally, still regarding equation 3, it seems to me that the derivations could be simplified if the authors were to assume right from the beginning that all funds are the same in terms of initial asset allocation. Why is this not the case? Which aspects of the analysis would be lost assuming that there is a continuum of mass one of funds and they have chosen the same asset allocation?

Answer 1-3

We thank to the referee to point out that although the market price determination in Equation (3) is a one of the core part of this study, it may not be delivered clearly to the readers. We added more detail to enhance readability. Our answer is as follows.

Due to the limitation of modeling, it is difficult to include all stock market participants in the model. We assume that liquidity can flow in from other market participants (e.g. a private and institutional investors). Nevertheless, assumption that the new liquidity inflow at t2 exactly cancel out liquidity outflow from the default fund may seem artificial. We agree with the comment, but we think this assumption is mathematically permissible since it helps to solve the following equations analytically. Furthermore, this assumption does not change the implications of the model because of the negative relation between liquidity outflow and the market return is maintained.

We agree to the referee’s comments that derivations could be simplified if the initial asset allocation strategy is assumed to be identical for all fund managers. This assumption is reasonable because the optimal strategy of risk-neutral fund managers is to maximize their expected final payoff and, since homogeneous fund managers, who know the optimal strategies of their peers, compete with each other in the market, all managers select the identical asset allocation strategy at equilibrium. So, without loss of generality, we set x≔x_1=x_2=⋯=x¬-_n.

Based on modifications, we revised and added following statement in Chapter 2.1. Main players.

“The optimal strategy of risk-neutral fund managers is to maximize their expected final payoff and, since homogeneous fund managers, who know the optimal strategies of their peers, compete with each other in the market, all managers select the identical asset allocation strategy at equilibrium. Using this identity condition, it is possible to assume that x≔x_1=x_2=⋯=x¬_n.”

We also added more detailed explanation for P2(λ) (which was r(λ) in the previous version) in Chapter 2.2 Market price determination process.

“The market price at t2, denoted by P2, is determined as follows. At t2, fund investors withdraw portion λ of their capital from surviving funds. If funds have enough cash, they do not have to sell any of risky asset in response to withdrawal. Therefore, if 0≤λ≤c, only market fear affects market price; that is, P2 = 1/Pτ. However, if λ is greater than c, capital outflows are greater than funds’ cash holdings. Accordingly, the funds are forced to liquidate some of their risky assets and due to their selling order, P2 decreases. Notably, P2 is decreasing function of λ and as λ increase, the funds are more likely to default. Let λ¬_d be the largest value of λ while the funds do not default; that is, if λ≥λ¬_d, funds default and otherwise, not. Then, if c≤λ<λ¬_d , funds have to sell (λ-c)/(P¬_2 ) of their risky asset to prepare for shortage of cash. For the sake of convenience, we index n funds such that the m highest are default and exit funds, that is, λ=m/n, and the remaining n-m lower-indexed funds are the surviving funds. Also, we assume that the buying or selling order affects market price proportional to current price. Then, the market price is determined as P¬_2= 1/Pτ - P_2*f/n*(n-m)*(λ-c)/(P¬_2 )= 1/Pτ - (n-m)/n{f(λ-c)}. Finally, if λ¬_d≤λ≤1, all funds exit the market and then the market price is P¬_2= 1/Pτ -(n-m)/n P¬_2 x. Equation (2) summarizes the market price at t2.”

P_2 (λ)={■(1/Pτ,&0≤λ<c@1/Pτ-(n-m)/n {f(λ-c)},&c≤λ<λ¬_d@1/Pτ-(n-m)/n P¬-_2 x,&λ¬_d≤λ≤1)┤

Comment 1-4. Market structure

At page 6, the authors state that fund managers compete with each other in the market and maximize their final asset value. In the same page, in the footnote, they state that the market is perfectly competitive. Which market are the authors referring to? Is it the market for resources or the asset market? I think a more detailed explanation is needed.

Answer 1-4

We thank to the referee’s comment. In the model, we assume a perfectly competitive market, that is, one with infinitely many funds. We agree that perfectly competitive market is ideal, but we think this assumption is mathematically and economically permissible since it helps to solve the following equations analytically. Furthermore, some existing studies argue that the matured stock market (e.g. U.S. stock market) can be somewhat treated as competitive market. In this sense, existing studies often model in the perfectly competitive market. For example, Grossman (1976) examines information efficiency in perfect market.

Comment 1-5. Trend followers

I do not find particularly clear the role that trend follower plays in the model. If I understand correctly, their presence allows the price of the risky asset to fluctuate around the fundamental value. I see the importance to have the price falling below the fundamental and then converging back to it in the final date. However, I am not sure having a third group of agents is really needed to do so. Would not be better to simply assume some shocks? The trend followers are really passive in the paper and their behaviour is simply assumed rather than modelled.

Answer 1-5

We thank the referee for their comment. Referee mentions that trend follower does not the play an important role in modeling and can be replace to some shock. In our model, the trend followers have unlimited capital but have no information on the fundamental values of risky assets, so they just follow market trends and their aggregate demand is therefore positively related to past market returns. The key role of trend followers is to diverge market price from fundamental price between t1 and t2. The reason that we introduce concept of trend followers is to illustrate market fear and analyze the effect of investor sensitivity to negative shock (i.e. τ) in distressed market. Allen and Gale (2004), Bernardo and Welch (2004), and Pedersen (2009) show that stock market investors overreact to the negative price shock and worsen market condition during financial crisis. Based on this idea, we illustrate additional price drop at t2 followed by negative price shock s. Therefore, the concept of trend followers is indeed important. However, we agree with the referee's opinion that the concept of a trend follower is ambiguous and confusion. Refer to the referee’s comment, we remove a concept of trend followers and describe model background more precisely using the concept of market fear. For example, we revised and added following statements in Chapter 2.2.

“In our model, we illustrate distressed market where market price is below the fundamental price. The fundamental and market price of risky asset over time is as follows. For the simplicity, we assume that fundamental price P is time-invariant, but market price diverges. At t_0, the market price P0 is equal to fundamental price P. Between t0 and t1, negative price shock occurs and drops the market price by s.”

“Between t_1 and t_2, market price decreases because of market fear. Allen and Gale (2004), Bernardo and Welch (2004), and Pedersen (2009) show that stock market investors overreact to the negative price shock and worsen market condition during financial crisis. We symbolize investor sensitivity to the negative shock by τ; that is, the greater τ implies that the market participants overreact to the negative shock and vice versa. We assume that market price is reduced in response to past market returns; that is, since the market return of the risky asset changes from P to 1 between t_0 and t_1, the market price between t_1 and t_2 again declines to 1/Pτ, where τ∈[1/P,1).”

Accordingly, the definition of τ is also changed from “trend followers’ sensitivity” to “stock market investors’ sensitivity to the negative shock”. As the definition has been changed, interpretation has also changed. τ become more important because it illustrates exogenous market characteristics; the greater τ implies that the market participants overreact to negative price shock and vice versa. Therefore, τ can be a cross-sectional or time-series asset market property. By altering τ, we can analyze the fund managers’ behavior in different period and market. For example, it is well known that developed market participants are more informed. We can reflect it with smaller τ. Also, even in the same market, investor sentiment is different over time. We can reflect it by altering τ. We describe it in the newly added Chapter 3.3.

Comment 1-6. Exposition and clarity

As a general comment, I think that the paper falls short in clarity. Little intuition is provided for the results and in some cases explanations are quite obscure. Several (important) details of the model are hidden in the text or missing, thus making reading the paper a bit cumbersome. Furthermore, I found the paper language sometimes inappropriate for being an applied theory paper. For example, expressions like “In the meantime, in the meanwhile” used in the model section instead of specifying the exact timing, make really difficult to understand the framework and the interactions among the various agents and their actions.

I think that the exposition of the paper should be significantly improved regarding the description of the setup (i.e., timing, agents’ actions, shocks etc.). For example, the authors talk about fund managers runs often together with fund investors’ withdrawals (which are usually what people have in mind when thinking of a run, especially given the existing literature). I think it would be useful to clearly separate the two things. In principle, fund investors’ withdrawal decisions could also be strategic complements, thus runs could also emerge as large withdrawals of fund investors.

As a suggestion, given the presence of several agents in the economy, the model setup could be organized in small sections, each describing actions and payoffs of the different agents. Finally, I think it would be useful to include a figure illustrating the timing of the model. The description at page 6 is not very clear.

Answer 1-6

We agree to referee’s comment that some details of the model are hidden and making reading the paper a bit cumbersome. Following referee’s comment, we added definitions and explanations to make paper more readable. We also removed unclear statements and correct typos. In particular, to improve description of the setup, we insert subsections in Chapter 2. In Chapter 2.1 Main players, we introduce two main players: fund managers and fund investors and their decision making. We revised and added following statement in Chapter 2.1.

“The brief time schedule of fund managers and investors is as follows. At t_1, fund investors provide aggregate capital f distributed equally among all funds. Fund i then allocates a portion x_i of capital to the risky asset and the remaining portion c_i to cash, where x_i∈[0,1]. Cash pays a return of one unit, but the payoff of the risky asset is uncertain, in that fund managers’ decision affects market price. The optimal strategy of risk-neutral fund managers is to maximize their expected final payoff and, since homogeneous fund managers, who know the optimal strategies of their peers, compete with each other in the market, all managers select the identical asset allocation strategy at equilibrium. Using this identity condition, it is possible to assume that x≔x_1=x_2=⋯=x¬_n. At t2, each fund manager receive funding liquidity shock on probability φ where φ is uniformly distributed on [0,1]. Fund managers who receive this shock should go bankrupt and make no profits. Let θ be a portion of default fund. Then, θ = φ and θ is also uniformly distributed on [0,1]. We name remaining non-default fund as surviving fund, which accounts for 1-θ. Surviving fund managers, however, do not know the exact value of θ because fund managers cannot observe the status of other funds but, instead, receive noisy signals, such that the signal of a surviving manager of fund i is θ_i=θ+ε_i, where ε_i is an independent and identically uniform distribution on the interval [-ε,ε] and ε is an arbitrarily small real number. Based on this signal, the manager of surviving fund i forecasts the strategies of other managers and decides whether to stay in or exit from the market. We denote λ as the aggregate portion of default and exit funds. Then, λ-θ is the portion of surviving and exit funds. After the surviving fund managers’ decision, fund investors observe λ, but they cannot distinguish which funds default or exit. We assume that they treat λ as the proxy of economic state and withdrawal portion g(λ) of their capital. Function g(x) should be increasing function of x, because they conjecture that a higher λ means a riskier market situation. For the simplicity, we assume that g(x)=x. That is, fund investors withdraw exact portion λ of their capital from remaining funds.”

We also think that readers may be confused by using price and return interchangeably. For better understand, we now use “fundamental and market price” and “investment payoff” instead of “long- and short-term market return” and “investment return”. We added and changed some notations as follows.

Before After

Long-term return R Fundamental price P

Market price of risky asset at t0, t1, t2, t3 P0, P1, P2, P3

Short-term return r(λ) Market price of risky asset at t2, P2(λ)

Investment return Investment payoff

In addition, we newly included following two figures which help the explanation.

Figure 2. Time schedule

This graph summarizes the time schedule of players and change in market price over time.

Figure 3. The graph of V(x,λ)

This graph illustrates decreasing function V(x,λ) and the relationship between π and λ_d. Refer to the definition, λ_d should satisfy λ¬_d=V(x,λ_d ).

Minor Comment

• Page 4, last paragraph, I found the sentence “equilibrium problems of panic-based crises” a bit misleading. I think I understand what the authors mean, but do not find the expression used precise enough. Multiplicity of equilibria is a feature of games characterized by strategic complementarity not a problem…I would suggest to rephrase it

We remove the sentence because formal sentence is similar but clearer.

• Page 10 “As in the benchmark case”, which benchmark are the authors referring to? Probably, the case where runs are not possible is the benchmark…

Thanks to referee’s comment, we can correct a typo. The original sentence was ambiguous because of a typo. Correct sentence is as follows.

“As the benchmark case, we first consider an equilibrium when surviving fund managers cannot exit the market.”

• Derivations at page 11 could be moved to the Appendix. This comment generally applies also to other parts of the paper. Derivations in the text could be moved to the appendix so to make the reading easier

We move derivation of Theorem 1 (which was Proposition 1 in the previous version) to appendix as referee suggested.

• Proof of theorem 1 is missing. I understand that it is standard and many steps of the proof are already in the text, but it would be useful, especially for the readers that are not very familiar with the global game literature to include the proof in the appendix.

We present the proof of Theorem 2 (which was Theorem 1 in the previous version) before Theorem 2, but it is not clear. We added following sentence before proof of Theorem 2 started.

“Using Equation (22), we can show that unique threshold equilibrium exist for a common threshold θ^*.”

Reviewer #2

Summary

In this paper the authors use a global games approach to show how interaction (a) among hedge funds and (b) between hedge funds and their investors can lead to market runs with fire sales of risky assets. The key idea is that any hedge fund’s investment decision has an impact on other funds through its impact on market prices. Moreover, withdrawals by loss-sensitive uninformed investors can trigger fire sales that further depress the market price.

In the model, an exogenous negative shock to the price of a risky asset motivates hedge funds (who know the fundamental value) to invest in the asset. To do so, they collect funds from investors and decide how much to invest in the risky asset vs. how much to hold in risk-free cash. Hedge funds’ purchases of the risky asset mitigate the drop in the asset price, but since uninformed trend followers amplify the initial drop in the price, it does not immediately return to its fundamental value. What is more, an exogenous "funding liquidity shock" wipes out a fraction θ of hedge funds and the managers of the surviving funds decide whether to keep their positions or whether to liquidate and exit the market as well.

This is where the global game component comes in: Individual hedge fund managers do not observe the actual fraction θ, but only a noisy signal. Since exits trigger proportional fund withdrawals by investors, there is a complementarity in fund managers’ actions: If more surviving funds exit, the remaining funds face higher withdrawals and may have to sell the risky asset, thereby depressing its price and making it less attractive to stay in the market. Fund manager i’s fear that other managers might exit can thus make it optimal to exit as well, even though the price drop associated with θ fundamental exits would not make fund i insolvent per se.

The authors derive a Nash equilibrium strategy according to which fund managers exit iff their private signal is above an endogenous threshold. They show that this threshold decreases (and so the probability of market runs increases) with investment in the risky asset. Hence, compared to a situation where runs are not allowed, fund managers optimally choose smaller market exposure when runs are possible. The authors claim that their model explains some stylized facts regarding hedge fund behavior before and during the financial crisis of 2007-2009.

Major comments

Comment 2-1

The goal of the paper, i.e. to shed light on the detailed interaction of agents in financial markets that gives rise to strategic fragility, is interesting and definitely deserves scholarly attention.

Comment 2-2

The paper tells a reasonable story that indeed seems to match stylized facts from the financial crisis. However, many parts of this story are, unfortunately, assumed exogenously to make the narrative coherent. Therefore, the actual contribution of the model in terms of economic mechanisms is rather marginal so far. What is more, as far as I can tell, the technical analysis is flawed. Details follow below.

Comment 2-3

In general, the paper uses appealing language and vocabulary. Sometimes, however, it lacks clarity because it uses terms without properly defining them.

Answer 2-3

We thank the referee for their comment. We agree that some expressions are not clear and confuse the readers. Following referee’s comment, we added definitions and explanations to make paper more readable. We also removed unclear statements and correct typos.

Comment 2-4

It is not clear why the authors included "trend followers" in the model. First of all, they have neither an objective function nor a choice to be made; they only serve as a narrative for why the price of the risky asset drops between t1 and t2. Second, I am not even convinced that this additional drop (on top of the exogenous price shock s) is even necessary for the results. In addition, the sensitivity analysis in section 4.2 is interesting, but in my opinion the authors oversell their interpretation of τ. Increasing τ simply means that the (exogenous!) price drop between t¬1 and t2 becomes more severe. It is not a deep, structural parameter of, say, risk aversion. Suggesting otherwise is misleading.

Answer 2-4

We thank the referee for their comment. Referee mentions that even though a trend follower does not the play an important role in modeling, the existence of trend followers confuse the readers. In our model, the trend followers have unlimited capital but have no information on the fundamental values of risky assets, so they just follow market trends and their aggregate demand is therefore positively related to past market returns. The reason for the confusion is as follows. First, trend followers do not make decisions through their objective function, but they exist only to explain the price change between t1 and t2. Second, there is no rational reason why additional price drops should occur following exogenous price shock s. Third, the interpretation of τ (sensitivity to the market) in section 4.2 is overselling. The key role of trend followers is to diverge market price from fundamental price between t1 and t2. The reason that we introduce concept of trend followers is to illustrate market fear and analyze the effect of investor sensitivity to negative shock (i.e. τ) in distressed market. Allen and Gale (2004), Bernardo and Welch (2004), and Pedersen (2009) show that stock market investors overreact to the negative price shock and worsen market condition during financial crisis. Based on this idea, we illustrate additional price drop at t2 followed by negative price shock s. Therefore, the concept of trend followers is indeed important. However, we agree with the referee's opinion that the concept of a trend follower is ambiguous and confusion. To make clear, we remove a concept of trend followers and describe model background more precisely. For example, we revised and added following statements in Chapter 2.2.

“In our model, we illustrate distressed market where market price is below the fundamental price. The fundamental and market price of risky asset over time is as follows. For the simplicity, we assume that fundamental price P is time-invariant, but market price diverges. At t_0, the market price P0 is equal to fundamental price P. Between t0 and t1, negative price shock occurs and drops the market price by s.”

“Between t_1 and t_2, market price decreases because of market fear. Allen and Gale (2004), Bernardo and Welch (2004), and Pedersen (2009) show that stock market investors overreact to the negative price shock and worsen market condition during financial crisis. We symbolize investor sensitivity to the negative shock by τ; that is, the greater τ implies that the market participants overreact to the negative shock and vice versa. We assume that market price is reduced in response to past market returns; that is, since the market return of the risky asset changes from P to 1 between t_0 and t_1, the market price between t_1 and t_2 again declines to 1/Pτ, where τ∈[1/P,1).”

Accordingly, the definition of τ is also changed from “trend followers’ sensitivity” to “stock market investors’ sensitivity to the negative shock”. As the definition has been changed, interpretation has also changed. τ become more important because it illustrates exogenous market characteristics; the greater τ implies that the market participants overreact to negative price shock and vice versa. Therefore, τ can be a cross-sectional or time-series asset market property. By altering τ, we can analyze the fund managers’ behavior in different period and market. For example, it is well known that developed market participants are more informed. We can reflect it with smaller τ. Also, even in the same market, investor sentiment is different over time. We can reflect it by altering τ. Nevertheless, we agree that the interpretation of τ is exaggerated. Therefore, we condensed the content in Chapter 4 and insert it into Chapter 3.3 as follows.

“So far, we investigate the effect of investor sensitivity τ on probability of market runs p_run. A change in τ affects P2, as well as funds’ investment payoff. Therefore, if an unexpected change in τ occurs, for a given x, fund managers would adjust their existing threshold strategy to reflect the new τ. In turn, this adjustment shifts the probability of market runs. Figure 5 illustrates this tendency for different levels of market exposure x. In this figure, the parameter values are f=4 and s=8. The horizontal axis represents τ from 0.2 to 0.9 and the vertical axis denotes p_run. Since the boundary condition restricts τ to the range [1/P ̅,1), we exclude both upper and lower extreme values of τ. Each curve indicates p_run as a function of τ for different x values, which increase from 0.2 to 0.9 by increments of 0.1.”

“In Figure 5, all the curves are monotonically increasing, which indicates the fragility of a highly sensitive market. In a financial market where investors respond sensitively to negative shock, even moderate levels of market shock can have a large impact on the market price, leading to a huge price drop. Price deterioration is more detrimental to funds that stay than to exiting funds, so a highly sensitive market is more prone to suffer from market runs. To put it concretely, an increase in sensitivity τ decreases the P2 and, accordingly, (Π^S ) ̅ falls more sharply than (Π^E ) ̅ does. Since it decreases the net investment return function, ∆Π ̅(λ)=(Π^S ) ̅-(Π^E ) ̅, the threshold θ^* shifts to a lower value because θ^* satisfies ∫_(θ^*)^1▒〖∆Π ̅(λ)dλ〗=0 (see Figure 4). Therefore, in response to an unexpectedly high trend sensitivity τ, fund managers lower the threshold θ^* and the probability of market runs, p_run, increases. Goldstein and Pauzner (2005) and Liu and Mello (2011) predict the similar result that short-term price deterioration raises the possibility of synchronized runs. However, their results are derived from the simple assumption of a constant short-term return, because their models do not consider a price determination mechanism. In contrast, we develop a market model in which short-term market return is endogenously determined by several factors and, among them, we note price sensitivity is an important source that affects the likelihood of market runs. Since price sensitivity is positively related to the probability of market runs, we suggest that lowering the information cost mitigates the irrational market fear and could help to resolve the synchronization problem in a distressed market, by enhancing market stability.”

Comment 2-5

A general problem in the exposition of the paper is that the authors often use the terms "price" and "return" interchangeably and thus confuse them. Most of the time they are talking about the effect of managers’ actions on the market return of the risky asset when in fact the price is affected. Even though the purchase price is normalized to 1, confusion remains and makes reading parts of the paper a bit tedious.

Answer 2-5

As referee mentioned, price and return can be used interchangeably because the purchase price is normalized to 1. Nevertheless, we agree that readers may be confused by using price and return interchangeably. For better understand, we now use “fundamental and market price” and “investment payoff” instead of “long- and short-term market return” and “investment return”. We added and changed some notations as follows.

Before After

Long-term return R Fundamental price P

Market price of risky asset at t0, t1, t2, t3 P0, P1, P2, P3

Short-term return r(λ) Market price of risky asset at t2, P2(λ)

Investment return Investment payoff

In addition, we clearly define fundamental and market price in Chapter 2.2 as follows.

“In our model, we illustrate distressed market where market price is below the fundamental price. The fundamental and market price of risky asset over time is as follows. For the simplicity, we assume that fundamental price P is time-invariant, but market price diverges. At t_0, the market price P0 is equal to fundamental price P.”

“At t_1, fund managers receive capital from investors and determine their investment strategy based on market conditions. After fund managers execute their investment strategies, the market price P1 rises by the demand for the risky asset.”

“The market price at t2, denoted by P2, is determined as follows.”

“Eventually, at t3, market price P3 converges to fundamental value P at t3.”

Comment 2-6

The contribution of the paper to the literature, especially compared to Liu & Mello (2011), is not clear. If the main contribution is the interaction between hedge funds through market prices and the withdrawals of investors, then unfortunately, I do not think the analysis is sufficiently rigorous. Investors are assumed to mechanically withdraw a fraction proportional to the share of exiting funds, so the actual economic incentives of uninformed investors remain obscure. Similarly, the market price of the asset is assumed to increase one-to-one with the amount of investment from hedge funds. Without any account of market structure, this is a very ad hoc way of generating the desired payoff structure for the global game. In my view, the additional insights with respect to Liu & Mello (2011) are not significant enough to justify the loss of rigor in modelling.

Answer 2-6

We thank the referee for their comment about contribution of our study. We are also afraid that readers do not recognize the contribution of our study compare to the previous studies, especially Liu and Mello (2011). It is true that their and our study seem similar in two reasons. First, we share same motivation; that is, we both analyze why the hedge funds hold more cash during financial crisis. Second, we share same methodology; the global game method. Nevertheless, there are key difference in terms of model background. Liu and Mello assume informed and sophisticated fund investors and model game of fund investors. However, some studies argue that the fund investors are usually not the rational investors, especially in the distressed market. Accordingly, we focus more on the fund managers’ behavior, who are known as more informed and sophisticated, and model game of fund managers. To model the game of fund managers, we develop a market model in which fund managers interact with other funds through the market price. In particular, if some fund managers exit the market, their selling order decrease the market price of risky asset and deteriorate the remaining fund value. Therefore, if a fund manager thinks that other funds are likely to exit, he or she should also exit, even though the risk itself is not that serious. This is what we called panic-based market run of fund managers. We added and revised statement in Chapter 1 to clarify the contribution of our study as follows.

“The study of Liu and Mello (2011) highlights synchronized withdrawal of fund investors as the major driver of the massive asset liquidation during the global financial crisis. They consider an isolated fund that cannot interact with the market or other funds and focus on fund-specific rather than economy-wide characteristics. Therefore, the linkage between fund runs and price deterioration is weak and the influence of fund managers’ decisions on the financial market is inferred indirectly. Although their and our study share same motivation (i.e. to reveal why the hedge funds reduce market exposure during financial crisis) and methodology (i.e. global game method), there are key differences in the model background. Liu and Mello assume informed and sophisticated fund investors and the game of fund investors. However, some studies argue that the fund investors are usually not the rational investors, especially in the distressed market. For example, according to the empirical evidence of Ben-David et al. (2011), during the global financial crisis, hedge fund investors actively withdraw their capital from poorly performing funds, which implies that investors determine their investment decisions depending on the past information of funds rather than information regarding their future. Accordingly, we focus more on the fund managers, who are known as informed and sophisticated, and model game of fund managers. To model the game, we develop a market model in which fund managers interact with other funds through the market price. Our approach allows us to identify a direct linkage between price deterioration and its impact on the decisions of fund managers.”

To focus more on the fund mangers’ behavior, we simplify the fund investors’ role. We assume that fund investors observe λ after the surviving fund managers’ decision, and withdrawal portion g(λ) of their capital. As we assume that fund investors treat λ as the proxy for the economic state; that is, they conjecture that a higher λ means a riskier market situation, function g(x) should be increasing function of x. For the simplicity, we assume that g(x)=x. Nevertheless, we agree with the referee's opinion that it is unrealistic to assume that the withdrawal corresponds one-to-one with the portion number of default and exit funds. We also think that to adopt a complex function or heterogeneous investors’ decision making would be a good further question. We added and revised following statement in Chapter 2.1.

“We denote λ as the aggregate portion of default and exit funds. Then, λ-θ is the portion of surviving and exit funds. After the surviving fund managers’ decision, fund investors observe λ, but they cannot distinguish which funds default or exit. We assume that they treat λ as the proxy of economic state and withdrawal portion g(λ) of their capital. Function g(x) should be increasing function of x, because they conjecture that a higher λ means a riskier market situation. For the simplicity, we assume that g(x)=x. That is, fund investors withdraw exact portion λ of their capital from remaining funds.”

Also, we added following footnote in Chapter 2.1.

“It may be unrealistic to assume that the withdrawal one-to-one corresponds with the aggregate portion of default and exit funds. To adopt a complex function or consider heterogeneous investors’ decision making would be a good further question.”

Comment 2-7

The execution of the global games exercise (starting from section 3.2) looks solid and here the authors explain very well what they are doing and why. Up to potential algebra mistakes it is nice that the authors find analytical expressions for some key equilibrium objects. The problems rather lie in the first part of the paper, i.e. in the attempt to micro-found the payoff structure.

Answer 2-7

We glad to point out that although the micro-found payoff structure is a one of the core parts of this study, it may not be delivered clearly to the readers. We agree with that there is not enough explanation for the definitions, concepts, or expressions. We insert two subsections in Chapter 2 as follows. In Chapter 2.1 Main players, we introduce two main players: fund managers and fund investors and their decision making. We revised and added following statement in Chapter 2.1.

“The brief time schedule of fund managers and investors is as follows. At t_1, fund investors provide aggregate capital f distributed equally among all funds. Fund i then allocates a portion x_i of capital to the risky asset and the remaining portion c_i to cash, where x_i∈[0,1]. Cash pays a return of one unit, but the payoff of the risky asset is uncertain, in that fund managers’ decision affects market price. The optimal strategy of risk-neutral fund managers is to maximize their expected final payoff and, since homogeneous fund managers, who know the optimal strategies of their peers, compete with each other in the market, all managers select the identical asset allocation strategy at equilibrium. Using this identity condition, it is possible to assume that x≔x_1=x_2=⋯=x¬_n. At t2, each fund manager receive funding liquidity shock on probability φ where φ is uniformly distributed on [0,1]. Fund managers who receive this shock should go bankrupt and make no profits. Let θ be a portion of default fund. Then, θ = φ and θ is also uniformly distributed on [0,1]. We name remaining non-default fund as surviving fund, which accounts for 1-θ. Surviving fund managers, however, do not know the exact value of θ because fund managers cannot observe the status of other funds but, instead, receive noisy signals, such that the signal of a surviving manager of fund i is θ_i=θ+ε_i, where ε_i is an independent and identically uniform distribution on the interval [-ε,ε] and ε is an arbitrarily small real number. Based on this signal, the manager of surviving fund i forecasts the strategies of other managers and decides whether to stay in or exit from the market. We denote λ as the aggregate portion of default and exit funds. Then, λ-θ is the portion of surviving and exit funds. After the surviving fund managers’ decision, fund investors observe λ, but they cannot distinguish which funds default or exit. We assume that they treat λ as the proxy of economic state and withdrawal portion g(λ) of their capital. Function g(x) should be increasing function of x, because they conjecture that a higher λ means a riskier market situation. For the simplicity, we assume that g(x)=x. That is, fund investors withdraw exact portion λ of their capital from remaining funds.”

In Chapter 2.2 Market price determination process, we describe changes in market price over time in detail. In particular, we modified some part related to market price at t2 as follows. First, without loss of generality, we set x≔x_1=x_2=⋯=x¬_n. This is because the optimal strategy of risk-neutral fund managers is to maximize their expected final payoff and, since homogeneous fund managers, who know the optimal strategies of their peers, compete with each other in the market, all managers select the identical asset allocation strategy at equilibrium. Second, we offer more detailed explanation depending on size of λ. Finally, we introduce λ¬_d; the largest value of λ while the funds do not default. In the previous version, function π_i (λ) and value π_i may confuse the readers. By using λ¬_d, we clarify the concept of V(x,λ) and π∶=V(x,λ_d ). We revised and added following statements in revised version.

“In our model, we illustrate distressed market where market price is below the fundamental price. The fundamental and market price of risky asset over time is as follows. For the simplicity, we assume that fundamental price P is time-invariant, but market price diverges. At t_0, the market price P0 is equal to fundamental price P. Between t0 and t1, negative price shock occurs and drops the market price by s. At t_1, fund managers receive capital from investors and determine their investment strategy based on market conditions. After fund managers execute their investment strategies, the market price P1 rises by the demand for the risky asset. The demand for risky asset is calculated as f/n ∑_(i=1)^n▒x_i =fx. For convenience, we normalize the market supply of t_1 to unity and express exogenous market factors such as market value, capital amounts, and price shocks relative to the price at t_1. In addition, without loss of generality, we assume that the market price at t_1 is 1. Then, the market price of risky asset at t¬1 is

P_1=1=P-s+fx

P=1+s-fx

where s is the impact of the negative market shock on the market price.

Between t_1 and t_2, market price decreases because of market fear. Allen and Gale (2004), Bernardo and Welch (2004), and Pedersen (2009) show that stock market investors overreact to the negative price shock and worsen market condition during financial crisis. We symbolize investor sensitivity to the negative shock by τ; that is, the greater τ implies that the market participants overreact to the negative shock and vice versa. We assume that market price is reduced in response to past market returns; that is, since the market return of the risky asset changes from P to 1 between t_0 and t_1, the market price between t_1 and t_2 again declines to 1/Pτ, where τ∈[1/P,1). The boundaries on τ restrict the impact of market fear such that 1/P<1/Pτ≤1. The first inequality, 1/P<1/Pτ, means that the effect of market fear does not dominate the market so that the market price at t2 does not fall below 1/P. The second inequality, 1/Pτ≤1, limits the market price at t2 as one, to focus on only a distressed market.

The market price at t2, denoted by P2, is determined as follows. At t2, fund investors withdraw portion λ of their capital from surviving funds. If funds have enough cash, they do not have to sell any of risky asset in response to withdrawal. Therefore, if 0≤λ≤c, only market fear affects market price; that is, P2 = 1/Pτ. However, if λ is greater than c, capital outflows are greater than funds’ cash holdings. Accordingly, the funds are forced to liquidate some of their risky assets and due to their selling order, P2 decreases. Notably, P2 is decreasing function of λ and as λ increase, the funds are more likely to default. Let λ¬_d be the largest value of λ while the funds do not default; that is, if λ≥λ¬_d, funds default and otherwise, not. Then, if c≤λ<λ¬_d , funds have to sell (λ-c)/(P¬_2 ) of their risky asset to prepare for shortage of cash. For the sake of convenience, we index n funds such that the m highest are default and exit funds, that is, λ=m/n, and the remaining n-m lower-indexed funds are the surviving funds. Also, we assume that the selling order decrease market price proportional to current price. Then, the market price is determined as P¬_2= 1/Pτ - P_2*f/n*(n-m)*(λ-c)/(P¬_2 )= 1/Pτ - (n-m)/n{f(λ-c)}. Finally, if λ¬_d≤λ≤1, all funds exit the market and then the market price is P¬_2= 1/Pτ -(n-m)/n P¬_2 x. Equation (2) summarizes the market price at t2.

P_2 (λ)={■(1/Pτ,&0≤λ<c@1/Pτ-(n-m)/n {f(λ-c)},&c≤λ<λ¬_d@1/Pτ-(n-m)/n P¬-_2 x,&λ¬_d≤λ≤1)┤

Note that fund investors’ cautious reaction to fund exits due to information asymmetry between fund managers and investors can stimulate runs of the remaining funds. Even a fund manager who knows that the funding liquidity shock is at a safe level can choose not to stay if he or she thinks that other managers will leave the market, because the exit of other funds not only induces withdrawals but also lowers market prices, deteriorating the investment payoff of the remaining funds. If the payoff deterioration seems severe enough, fund managers will decide to leave the market. After all, regardless of market fundamentals, market runs can be triggered by the fear of runs, which we call panic-based market runs. Eventually, at t3, market price P3 converges to fundamental value P at t3. Figure 2 briefly summarizes time schedule.

In our model, λ¬_d plays an important role in terms of fund maintenance. We now derive λ¬-_d and reveal its economic meaning. Since fund managers sell their risky asset for price P2, the liquidation value at t2 is calculated as V(x,λ)=xP¬_2 (λ)+c. Then, funds default when λ>V and do not default when λ≤V. Formerly, we show that P¬_2 (λ) is decreasing function of λ and so as V(x,λ). The graph of V(x,λ) in Figure 3 presents that in this case, λ¬-_d should satisfies λ¬_d=V(x,λ_d ). Also, we can define the minimum value of non-default funds’ investment payoff before capital outflow π as in Equation (3). By solving Equation (3), we can obtain π=x/(Pτ(1+fx))+c.”

Furthermore, we newly included following two figures which help the explanation.

Figure 2. Time schedule

This graph summarizes the time schedule of players and change in market price over time.

Figure 3. The graph of V(x,λ)

This graph illustrates decreasing function V(x,λ) and the relationship between π and λ_d. Refer to the definition, λ_d should satisfy λ¬_d=V(x,λ_d ).

Minor comments

• Please explain the purpose of the paper and your contribution before the literature review. Some references seemed quite far-fetched from the content of the paper.

In Chapter 1. Introduction, we change the order of parts. We move the part that describe our model and contribution forward and the literature review part back. Also, we remove some literatures which is far-fetched from the content of the paper. For example, we remove some literature about limit to arbitrage (e.g. Gromb and Vanyanos (2002), Liu and Longstaff (2004), and Brunnermeier and Pedersen (2009)).

• I do not think that "(funding) liquidity shock" is a good description of what θ does, because it suggests an endogenous reaction of the concerned hedge funds. In fact, it is just an exogenous solvency shock that kills a fraction of hedge funds.

As referee suggest, we changed “funding liquidity shock” to “exogenous funding shock”. We also distinguish probability of exogenous funding shock φ and portion of default fund θ. Following statements is inserted in the revised version.

“At t2, each fund manager receive funding liquidity shock on probability φ where φ is uniformly distributed on [0,1]. Fund managers who receive this shock should go bankrupt and make no profits. Let θ be a portion of default fund. Then, θ = φ and θ is also uniformly distributed on [0,1]. We name remaining non-default fund as surviving fund, which accounts for 1-θ.”

• Equation (1) is referred to as "market clearing condition", but I cannot see how the terms relate to demand, supply or general equilibrium. All I see is an ad hoc pricing equation where the price (normalized to unity) is equal to the fundamental value minus the exogenous shock plus the compensating fund purchases.

For better understand, we remove term “market clearing condition” and revise it to pricing equation as follows.

“In addition, without loss of generality, we assume that the market price at t_1 is 1. Then, the market price of risky asset at t¬1 is

P_1=1=P-s+fx

P=1+s-fx

where s is the impact of the negative market shock on the market price.”

• I think there is an error in equation (3). In the last three lines there should be no r multiplying the x’s. The point is that the market price r decreases by the amount f/n x_k for every defaulting hedge fund. This error leads to some follow-up errors later on, e.g. in (7) or the definition of πi on p. 8.

We assume that the buying or selling order affects market price proportional to current price. It means if someone sell z amount of risky assets, prices will fall by P2 * z. It is because we think that if investors trade the same amount of risky asset, the return should also be the same. We present it by P2 multiplying x’s.

• I think the expression for expected payoffs ex ante in (12) and its successors are wrong. The expectation operator in E[Π_i^S] is al with respect to θ, so how can it be taken out of the integral? Should it not read ∫_0^1▒〖(1-θ)∏_i^S▒(x,θ)dθ〗?

If the expectation value calculated at t2, referee’s comment is right. However, since the fund manager has not yet received the signal for θ at t1, although the default probability is determined, fund managers have to consider whole range. To clarify the expression, we introduce φ and rewrite the equation as follows.

E[Π^I ]=∫_0^1▒〖[(1-φ)E[Π^S ]] □(24&dφ)〗=E[Π^S ] ∫_0^1▒〖(1-φ) □(24&dφ)〗=1/2 E[Π^S ]=1/2 ∫_0^π▒〖Π^S □(24&dθ)〗

• The authors refer to "liquidity inflows" into the risky asset market at t2, but do not explain where these come from. It seems like all exiting funds sell their risky assets to new market participants (hence there is an inflow), but then why is there no corresponding inflow for the (λ − ck) portion of liquidated assets? Is this asymmetry really without loss of generality?

Due to the limitation of modeling, it is difficult to include all stock market participants in the model. We assume that liquidity can flow in from other market participants (e.g. a private and institutional investors). Nevertheless, assumption that the new liquidity inflow at t2 exactly cancel out liquidity outflow from the default fund may seem artificial. We agree with the comment, but we think this assumption is mathematically permissible since it helps to solve the following equations analytically. Furthermore, this assumption does not change the implications of the model because of the negative relation between liquidity outflow and the market return is maintained.

• The effect of ε → 0 should become clearer. Which results survive for general ε?

Based on the assumption that ε → 0, it is possible to solve Equation (25) through the variable transportation, which allows to derive the following results. Morris and Shin (1998) mention that if ε → 0, fundamental uncertainty disappears while maintaining strategic uncertainty.

References

Allen, F., and Gale, D. (2004) Financial fragility, liquidity, and asset price, Journal of the European Economic Association 2, 1015-1048.

Grossman, S. (1976) On the efficiency of competitive stock market where trades have diverse information, Journal of Finance 31, 573-585.

Morris, S. and Shin, H.S. (1998) Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks, American Economic Review 88, 587–597.

Pedersen, L. H. (2009) When Everyone Runs for the Exit, International Journal of Central Banking 5, 177-199.

Author Answers