Comments and Questions

Revised PaperAuthors’ Responses for the Referee CommentsTitle: Market runs of hedge funds during financial crisis Reviewer #1SummaryThe 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., 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 commentsComment 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 1-1We 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 complementarityUnlike 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 1-2We 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 1-3We 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