Abstract

We present a model of equilibrium allocation of capital for arbitrage. If asset prices may fall low enough, it is profitable to carry liquid capital to acquire assets in such states. Set against this, keeping capital in liquid form entails costs in terms of foregone profitable investments. This trade-off generates occasional fire sales and limited arbitrage capital as robust phenomena. With learning-by-doing effects, arbitrage capital moves in to acquire assets only if fire sales are steep. However, once arbitrage capital finds it profitable to acquire assets, it requires similar returns elsewhere, inducing contagious fire-sale prices even for unrelated assets. (JEL G21, G28, G38, E58, D62)

Our understanding of financial crises has been enhanced by a large and rapidly growing empirical literature that has documented the incidence and severity of fire sales by distressed parties in a wide range of asset classes. 1 Indeed, it would not be too much of an exaggeration to say that fire sales have been a defining feature of most financial crises, including the most recent crisis of 2007–2009.

The term “fire sale” carries the connotation that assets are being sold at prices that are below some benchmark, fair fundamental price that would prevail in the absence of a crisis. However, the notion that assets are being sold at prices below their fundamental value begs an important question. How can fire sales take place in a world in which arbitrage capital waits on the sidelines to take advantage of artificially low prices? If there were such arbitrageurs who wait on the sidelines, would they not compete with each other as soon as the crisis erupts, providing a cushion for prices? As well as these “positive” questions on the nature of the equilibrium outcome, there are also important normative questions on the social value of arbitrage capital. Is the equilibrium provision of arbitrage capital at the efficient level? If not, is the efficient level higher or lower than the equilibrium level? Our paper tries to answer these questions in a setting that is simple and transparent enough to help uncover the underlying economic mechanisms at work.

We show that the answers to both the positive and normative sets of questions rely on the interplay between two underlying allocative mechanisms in the economy. One operates at the ex post stage, and has to do with the efficient allocation of assets to those economic agents that can generate most value from them. The second operates at the ex ante stage, and has to do with how much of the economy’s resources are set aside in the form of “idle” arbitrage capital that waits on the sidelines.

In our baseline setup, engaging in production entails greater expertise to insiders through learning-by-doing effects, and these insiders are the natural holders of the assets in the sense of being able to generate greater value from them compared with outsiders who take over distressed assets of illiquid insiders.

The greater is the discount in the value realized by outsiders, the more severe must be the fire sale before outsiders enter to cushion the distress. Thus, for a finite pool of arbitrage capital at the ex post stage, there are states of the world with possibly steep price discounts, which then create the incentive to hold unproductive “idle” arbitrage capital at the ex ante stage. In equilibrium, the two choices–whether to invest in profitable activities or to set aside funds for arbitrage in the future–earn the same rate of return when viewed ex ante. As a consequence, limited provision of arbitrage capital and fire sales emerge as robust features of the equilibrium. However, there are two important normative consequences of fire sales. First, there is ex post inefficiency due to assets being held by outsiders in some states of the world. Second, there is also ex ante inefficiency due to the ex ante allocation of resources to arbitrage capital.

There are also implications for the depth of fire sales following long periods of low liquidity risk (good times). During periods when liquidity risk is low, illiquid projects are more attractive relative to holding cash balances. Hence, a higher fraction of agents choose to become insiders and there is less liquid capital put aside. As a result, when liquidity shocks do materialize fire-sale effects in asset prices are more severe as there was less liquid capital put aside for arbitrage. This can potentially explain why crises that erupt after good times are associated with sharper, more severe fire sales. 2

The picture that emerges from our analysis is that private incentives lead to the over-provision of arbitrage capital relative to its socially optimal level. Intuitively, in states in which there are few liquid insiders, asset prices must fall so that the market clears. Nevertheless, there is allocative efficiency as assets remain in the hands of insiders. The presence of arbitrage capital interferes with the efficient allocation. Ex post, there is inefficiency if arbitrageurs are less efficient than insiders in deploying assets. Importantly, even if arbitrageurs are as efficient as insiders, setting aside arbitrage capital ex ante implies passing over profitable investment opportunities. Somewhat counterintuitively perhaps, relative to the competitive outcome, the social optimum features lower asset prices ex post and greater profitable investments ex ante.

Because our welfare results arise from the interplay between the ex post asset allocation and the ex ante provision of arbitrage capital, one important message from our paper is that the opening of capital markets in the ex post period where liquid insiders can raise new funding from outsiders does not eliminate the inefficiency, although there is some mitigation of the inefficiency. We demonstrate this feature in an extension of our framework, where we allow insiders to raise additional funding from outsiders by selling financial claims. Because outsiders have the choice between acquiring physical assets or buying financial claims sold by insiders, arbitrage capital is allocated in such a way that the returns are equalized. The result is that fire-sale discounts in prices for acquisition of assets must equal that for provision of external finance, giving rise to a spillover or contagion from illiquidity in the market for real assets to that for financing of these assets. From the welfare standpoint, ex post efficiency of allocation is restored as arbitrageurs simply fund asset purchases of insiders, but they make same profits ex ante because of discounts they charge in funding insiders. As a result, it remains profitable for there to be some arbitrage capital in equilibrium and there continues to be some ex ante inefficiency in investment decisions.

In another extension of our benchmark model, we show that contagion can result across different asset classes whenever the provision of arbitrage capital in these asset markets is from a common pool; returns on different investments in the portfolio of an arbitrageur must be the same. The fact that the quantity of arbitrage capital is limited implies that these returns are positive in all markets where arbitrageurs allocate capital. 3 This channel of contagion operates even though the fundamentals of two assets are independent and the existence of fire sales in one asset can give rise to fire sales in the other.

Fire sales are not, of course, a new idea. The idea that asset prices may contain liquidity discounts when potential buyers are financially constrained and assets are not easily redeployable were discussed by Williamson (1988) and Shleifer and Vishny (1992). This early literature suggests that firms, whose assets tend to be specific (i.e. whose assets cannot be readily redeployed by firms outside of the industry) are likely to experience lower liquidation values because they may suffer from fire-sale discounts in cash auctions for asset sales, especially when firms within an industry become simultaneously financially or economically distressed. Since then, fire sales have often figured in models of crises (Allen and Gale 1994. 1998 ; among others). Intimately tied to the notion of fire sales is the idea that arbitrageurs wanting to buy assets at steep discounts may also face financing frictions due to principal-agent problems. The resulting “limits of arbitrage” (Shleifer and Vishny 1997 ) can entrench fire-sale prices for a period of time once they materialize. 4

Our contribution relative to this earlier literature is to focus on the ex ante decisions of investors, which much of the literature takes as given, and thereby to explain the origins of the limited nature of arbitrage capital as an equilibrium phenomenon. In this sense, our work is closest to the analysis by Allen and Gale (2004) of the portfolio choice of banks between holding safe versus risky assets. Gorton and Huang (2004). another closely related paper, also consider the equilibrium portfolio choice of firms, deriving that it is socially inefficient to hold large quantities of safe assets required to avoid fire sales, and study in this context the role of government bailouts during crises. Our framework is tractable and facilitates crisp conclusions on key comparative statics and welfare questions. In particular, our result that arbitrage capital is endogenously lower in good times and therefore that crises arising directly after good times feature deeper fire-sale discounts, is noteworthy and is one which (to our knowledge) has not been discussed so far in the literature.

Another advantage of our tractable framework is to open up for scrutiny the arbitrageurs’ access to different real and financial markets, and thereby identify a channel of contagion that relies purely on the limited nature of arbitrage capital. Our results on this front are closest to Gromb and Vayanos (2007) and Duffie and Strulovici (2012). Gromb and Vayanos (2007) consider arbitrageurs exploiting fire-sale opportunities across markets, and this equilibrates returns they can earn in different markets. Duffie and Strulovici (2012) study a similar setting, taking the financing friction of arbitrageurs as given. In both of these papers, the quantity of equilibrium arbitrage capital is exogenous, whereas our central theoretical concern is to endogenize the quantity of arbitrage capital and illustrate that its limited quantity as well as its limited expertise make fire sales a robust equilibrium phenomenon. 5

Rampini and Viswanathan (2010) consider a dynamic contracting setup for exploring which firms make investments and which preserve debt capacities for the future. This question is related to our analysis. Although we model fire sales as an outcome from market clearing when buyers are financially constrained, Rampini and Viswanathan model asset prices as being temporarily low because of low cash-flow realizations. Acharya, Shin, and Yorulmazer (2011) also consider gains from acquiring assets at fire-sale prices that make it attractive for firms (in their model, banks) to hold liquid assets ex ante but that when such fire-sale states are not too likely, banks hold too little liquidity because of the asset-substitution problem. Their focus is on analyzing how government or central bank interventions to resolve banking crises (that may be desirable ex post) affect ex ante liquidity in potentially adverse ways. 6. Bolton, Santos, and Scheinkman (2011) and Diamond and Rajan (2011) present models wherein once an adverse state of the world arises, decisions of individual banks affect when assets are sold–immediately or with delay. Immediate sales can increase returns to holding cash and lead to potentially excessive cash hoarding ex ante. In contrast to these models, our paper is more in the spirit of Allen and Gale (1994. 1998 ) and Gorton and Huang (2004) in that once the adverse state arises, there is an immediate (fire) sale of assets.

Our study is also related to the seminal work of Kiyotaki and Moore (1997) on credit cycles. In Kiyotaki and Moore (1997) and Krishnamurthy (2003). the underlying asset cannot be pledged because of inalienable human capital. Krishnamurthy (2003) differs from Kiyotaki and Moore (1997) in that all contingent claims on aggregate variables are subject to collateral constraints. However, land can be pledged and has value both as a productive asset and as collateral. Caballero and Krishnamurthy (2001) employ a Holmstrom and Tirole (1998) approach with exogenous liquidity shocks and allow firms to post collateral in a manner similar to Kiyotaki and Moore (1997).

Finally, there are models in which there are pecuniary externalities from fire sales of assets. Lorenzoni (2008). for example, considers a competitive model of intermediaries in which ignoring these externalities leads to excessive borrowing ex ante and excessive volatility ex post. In our model, there are no externalities from fire sales and the result is that the competitive equilibrium features too much setting aside of idle capital ex ante for the purpose of undertaking arbitrage ex post.

1. Model

The time line of the model is provided in Figure 1. There are three dates indexed by . There is a unit measure of risk-neutral agents. Each agent is endowed with one unit of the consumption good at .