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Liquidity Market and Funding Liquidity Markus K. Brunnermeier Princeton University Lasse Heje Pedersen New York University This version: June 2007 Abstract We provide a model that links an asset s market liquidity i.e., the ease with which it is traded and traders funding liquidity i.e., the ease with which they can obtain funding. Traders provide market liquidity, and their ability to do so depends on their availability of funding. Conversely, traders funding, i.e., their capital and the margins they are charged, depend on the assets market liquidity. We show that, under certain conditions, margins are destabilizing and market liquidity and funding liquidity are mutually reinforcing, leading to liquidity spirals. The model explains the empirically documented features that market liquidity (i) can suddenly dry up, (ii) has commonality across securities, (iii) is related to volatility, (iv) is subject to ight to quality , and (v) co-moves with the market, and it provides new testable predictions. Keywords: Liquidity Risk Management, Liquidity, Liquidation, Systemic Risk, Leverage, Margins, Haircuts, Value-at-Risk, Credit Crunch, Counterparty Credit Risk, Fat Tails We are grateful for helpful comments from Franklin Allen, Yakov Amihud, David Blair, Bernard Dumas, Denis Gromb, Charles Johns, Christian Julliard, John Kambhu, Markus Konz, Martin Oehmke, Ketan Patel, Guillaume Plantin, Felipe Schwartzman, Matt Spiegel (the editor), Jeremy Stein, Dimitri Vayanos, Jiang Wang, Pierre-Olivier Weill, and from an anonymous referee and Filippos Papakonstantinou and Felipe Schwartzman for outstanding research assistantship. We also thank seminar participants at the New York Federal Reserve Bank and the New York Stock Exchange, Citigroup, Bank for International Settlement, University of Z rich, INSEAD, Northwestern University, Stockholm Institute for Financial Research, Goldman u Sachs, IMF, the World Bank, UCLA, LSE, Warwick University, Bank of England, University of Chicago, Texas A&M, University of Notre Dame, HEC, University of Maryland, University of Michigan, Virginia Tech, Ohio State University, University of Mannheim, ECB-Bundesbank, and conference participants at the American Economic Association Meeting, FMRC conference in honor of Hans Stoll at Vanderbilt, NBER Market Microstructure Meetings, NBER Asset Pricing Meetings, NBER Risks of Financial Institutions conference, the Five Star conference, and American Finance Association Meeting. Princeton University, NBER and CEPR, Department of Economics, Bendheim Center for Finance, Princeton University, 26 Prospect Avenue, Princeton, NJ 08540-5296, e-mail: markus@princeton.edu, http://www.princeton.edu/ markus New York University, NBER and CEPR, 44 West Fourth Street, NY 10012-1126, e-mail: lpederse@stern.nyu.edu, http://www.stern.nyu.edu/ lpederse/ Trading requires capital. When a trader e.g. a dealer, hedge fund, or investment bank buys a security, he can use the security as collateral and borrow against it, but he cannot borrow the entire price. The di erence between the security s price and collateral value, denoted as the margin, must be nanced with the trader s own capital. Similarly, shortselling requires capital in the form of a margin; it does not free up capital. Therefore, the total margin on all positions cannot exceed a trader s capital at any time. Our model shows that the funding of traders a ects and is a ected by market liquidity in a profound way. When funding liquidity is tight, traders become reluctant to take on positions, especially capital intensive positions in high-margin securities. This lowers market liquidity, leading to higher volatility. Further, under certain conditions, low future market liquidity increases the risk of nancing a trade, thus increasing margins. Based on the links between funding and market liquidity, we provide a uni ed explanation for the main empirical features of market liquidity. In particular, our model implies that market liquidity (i) can suddenly dry up, (ii) has commonality across securities, (iii) is related to volatility, (iv) is subject to ight to quality or ight to liquidity, and (v) co-moves with the market. The model has several new testable implications that link margins and dealer funding to market liquidity: We predict that (i) a shock to speculators capital is a state variable a ecting market liquidity and risk premia, (ii) a reduction in capital reduces market liquidity, especially if capital is already low (a non-linear e ect) and for high-margin securities, (iii) margins increase in illiquidity if the fundamental value is di cult to determine, and (iv) speculators returns are negatively skewed (even if they trade securities without skewness in the fundamentals). Our model is similar in spirit to Grossman and Miller (1988) with the added feature that speculators face the real-world funding constraint discussed above. In our model, di erent customers have o setting demand shocks, but arrive sequentially to the market. This creates a temporary order imbalance. Speculators smooth price uctuations, thus providing market liquidity. Speculators nance their trades through collateralized borrowing from nanciers 1 who set the margins to control their value-at-risk (VaR). We derive the competitive equilibrium of the model and explore its liquidity implications. We de ne market liquidity as the di erence between the transaction price and the fundamental value, and funding liquidity as a speculator s scarcity (or shadow cost) of capital. 14% 12% Black Monday 10% 10/19/87 US/Iraq war LTCM 8% 6% 4% 2% 1989 mini crash Asian crisis 0% Jan-82 Jan-84 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04 Jan-06 Figure 1: Margins for S&P500 Futures. The gure shows margin requirements on S&P500 futures for members of the Chicago Mercantile Exchange as a fraction of the value of the underlying S&P500 index multiplied by the size of the contract. (Initial or maintenance margins are the same for members.) Each dot represents a change in the dollar margin. We rst analyze the properties of margins. We show that margins can increase in illiquidity when margin-setting nanciers are unsure whether price changes are due to fundamental news or to liquidity shocks and fundamentals have time-varying volatility. This happens when a liquidity shock leads to price volatility, which raises the nancier s expectation about future volatility. Figure 1 shows that margins did increase empirically for S&P 500 futures during the liquidity crises of 1987, 1990, and 1998. We denote margins as destabilizing if they can increase in illiquidity, and note that anecdotal evidence from prime brokers suggests that margins often behave in this way. The model also shows that margins can, in contrast, decrease in illiquidity and thus be 2 stabilizing. This happens when nanciers know that prices diverge due to temporary market illiquidity and know that liquidity will be improved shortly as complementary customers arrive. This is because a current price divergence from fundamentals provides a cushion against future adverse price moves, making the speculator s position less risky in this case. In summary, our model predicts that margins depend on market conditions and are more destabilizing in specialized markets in which nanciers cannot easily distinguish fundamental shocks from liquidity shocks or predict when a trade converges. Turning to the implications for market liquidity, we rst show that, as long as speculator capital is so abundant that there is no risk of hitting the funding constraint, market liquidity is naturally at its highest level and is insensitive to marginal changes in capital and margins. However, when speculators hit their capital constraints or risk hitting their capital constraints over the life of a trade then they reduce their positions and market liquidity declines. When margins are destabilizing or speculators have large existing positions, there can be multiple equilibria and liquidity can be fragile. In one equilibrium, markets are liquid, leading to favorable margin requirements for speculators, which in turn helps speculators make markets liquid. In another equilibrium, markets are illiquid, resulting in larger margin requirements (or speculator losses), thus restricting speculators from providing market liquidity. Importantly, any equilibrium selection has the property that small speculator losses can lead to a discontinuous drop of market liquidity. This sudden dry-up or fragility of market liquidity is due to the fact that with high levels of speculator capital, markets must be in a liquid equilibrium, and, if speculator capital is reduced enough, the market must eventually switch to a low-liquidity/high-margin equilibrium.1 The events following the Russian default and LTCM collapse in 1998 are a vivid example of fragility of liquidity since a relatively small shock had a large impact. Compared to the total market capitalization of the US stock and bond markets, the losses due to the Russian default were minuscule but, as Figure 1 shows, Fragility can also be caused by asymmetric information on the amount of trading by portfolio insurance traders (Gennotte and Leland (1990)), and by losses on existing positions (Chowdhry and Nanda (1998)). 1 3 caused a shiver in world nancial markets. Further, when markets are illiquid, market liquidity is highly sensitive to further changes in funding conditions. This is due to two liquidity spirals: rst, a margin spiral emerges if margins are increasing in market illiquidity because a reduction in speculator wealth lowers market liquidity, leading to higher margins, tightening speculators funding constraint further, and so on. For instance, Figure 1 shows how margins gradually escalated within a few days after Black Monday in 1987. Second, a loss spiral arises if speculators hold a large initial position that is negatively correlated with customers demand shock. In this case, a funding shock increases market illiquidity, leading to speculator losses on their initial position, forcing speculators to sell more, causing a further price drop, and so on.2 These liquidity spirals reinforce each other, implying a larger total e ect than the sum of their separate e ects. Paradoxically, liquidity spirals imply that a larger shock to the customers demand for immediacy leads to a reduction in the provision of immediacy during such times of stress. Consistent with our predictions, Mitchell, Pulvino, and Pedersen (2007) nd significant liquidity-driven divergence of prices from fundamentals in the convertible bond markets after capital shocks to the main liquidity providers, namely convertible arbitrage hedge funds. In the cross-section we show that the ratio of illiquidity to margin is the same across all assets for which speculators provide market liquidity. This is the case since speculators optimally invest in securities that have the greatest expected pro t (i.e. illiquidity) per capital use (determined by the asset s dollar margin). This common ratio is determined in equilibrium by the speculators funding liquidity i.e. capital scarcity. Our model thus provides a natural explanation for the commonality of liquidity across assets since shocks to speculators funding constraint a ect all securities. This may help explain why market liquidity is correlated across stocks (Chordia, Roll, and Subrahmanyam (2000), Hasbrouck and Seppi (2001) and Huberman and Halka (2001)), and across stocks and bonds The loss spiral is related to the multipliers that arise in Grossman (1988), Kiyotaki and Moore (1997), Shleifer and Vishny (1997), Chowdhry and Nanda (1998), Xiong (2001), Kyle and Xiong (2001), Gromb and Vayanos (2002), Morris and Shin (2004), Plantin, Sapra, and Shin (2005) and others. To our knowledge, our paper is the rst to model the margin spiral and the interaction between the two multipliers. 2 4 (Chordia, Sarkar, and Subrahmanyam (2005)). In support of the idea that commonality is driven at least in part by our funding-liquidity mechanism, Chordia, Sarkar, and Subrahmanyam (2005) nd that money ows ... account for part of the commonality in stock and bond market liquidity. Moreover, their nding that during crisis periods, monetary expansions are associated with increased liquidity is consistent with our model s prediction that the e ects are largest when traders are near their constraint. Coughenour and Saad (2004) provide further evidence of the funding-liquidity mechanism by showing that the comovement in liquidity among stocks handled by the same NYSE specialist rm is higher than for other stocks, commonality is higher for specialists with less capital, and decreases after a merger of specialists. Next, our model predicts that market liquidity declines as fundamental volatility increases, which is consistent with the empirical ndings of Benston and Hagerman (1974) and Amihud and Mendelson (1989).3 Further, the model can shed new light on ight to quality, referring to episodes in which risky securities become especially illiquid. In our model, this happens when speculator capital deteriorates. Indeed, a reduction in speculators capital induces them to provide liquidity mostly in securities that do not use much capital (low volatility stocks with lower margins), implying that the liquidity di erential between high-volatility and low-volatility securities increases. This capital e ect means that illiquid securities are predicted to have more liquidity risk.4 Recently, Hendershott, Moulton, and Seasholes (2006) test these predictions using inventory positions of NYSE specialists as a proxy for funding liquidity. Their ndings support our hypotheses that market liquidity of high volatility stocks is more sensitive to changes in inventory shocks and that this is more pronounced at times of low funding liquidity. Moreover, Pastor and Stambaugh (2003) and Acharya and Pedersen The link between volatility and liquidity is shared by the models of Stoll (1978), Grossman and Miller (1988), and others. What sets our theory apart is that this link is connected with margin constraints. This leads to testable di erences since, according to our model, the link is stronger when speculators are poorly nanced, and high-volatility securities are more a ected by speculator wealth shocks our explanation of ight to quality. 4 In Vayanos (2004) liquidity premia increase in volatile times. Fund managers become e ectively more risk averse because higher fundamental volatility increases the likelihood that their performance falls short of a threshold, leading to costly performance-based withdrawal of funds. 3 5 (2005) document empirical evidence consistent with ight to liquidity and the pricing of this liquidity risk. Market-making rms are often net long in the market. For instance, Ibbotson (1999) reports that security brokers and speculators have median market betas in excess of one. Therefore, capital constraints are more likely to be hit during market downturns, and this, together with the mechanism outlined in our model, helps to explain why sudden liquidity dryups occur more often when markets decline. Further, capital constraints a ect the liquidity of all securities, leading to co-movement as explained above. The fact that this e ect is stronger in down markets could explain that co-movement in liquidity is higher during large negative market moves as documented empirically by Hameed, Kang, and Viswanathan (2005). Finally, the very risk that the funding constraint becomes binding limits speculators provision of market liquidity. Our analysis shows that speculators optimal (funding) risk management policy is to maintain a safety bu er. This a ects initial prices, which increase in the covariance of future prices with future shadow costs of capital (i.e., with future funding illiquidity). Our paper is related to several literatures.5 Most directly related are the models with margin-constrained traders: Grossman and Vila (1992) and Liu and Longsta (2004) derive optimal strategies in a partial equilibrium with a single security; Chowdhry and Nanda (1998) focus on fragility due to dealer losses; and Gromb and Vayanos (2002) derive a general equilibrium with one security and study welfare and liquidity provision. Our paper contributes to the literature by considering the simultaneous e ect of margin constraints on multiple securities and by examining the nature of those margin constraints. Stated simply, the existing theoretical literature uses a xed or decreasing margin constraint say $5,000 per contract Market liquidity is the focus of market microstructure (Stoll (1978), Ho and Stoll (1981, 1983), Kyle (1985), Glosten and Milgrom (1985), Grossman and Miller (1988)), and is related to the limits of arbitrage (DeLong, Shleifer, Summers, and Waldmann (1990), Shleifer and Vishny (1997), Abreu and Brunnermeier (2002)). Funding liquidity is examined in corporate nance (Shleifer and Vishny (1992), Holmstr m and Tirole o (1998,2001)) and banking (Diamond and Dybvig (1983), Allen and Gale (1998, 2004, 2005)). Funding and collateral constraints are also studied in macroeconomics (Bernanke and Gertler (1989), Kiyotaki and Moore (1997), Lustig (2004)), and general equilibrium with incomplete markets (Geanakoplos (1997, 2003)). Finally recent papers consider illiquidity with constrained traders (Attari, Mello, and Ruckes (2005), Bernardo and Welch (2004), Brunnermeier and Pedersen (2005), Eisfeldt (2004), Morris and Shin (2004), and Weill (2004)). 5 6 and studies what happens when trading losses cause agents to hit this constraint, whereas we study how market conditions lead to changes in the margin requirement itself e.g., an increase from $5,000 to $15,000 per futures contract as happened in October 1987 and the resulting feedback e ects between margins and market conditions. We proceed as follows. We describe the model (Section 1) and derive our four main new results: (i) margins increase with market illiquidity when nanciers cannot distinguish fundamental shocks from liquidity shocks and fundamentals have time-varying volatility (Section 2); (ii) this makes margins destabilizing, leading to sudden liquidity dry ups and margin spirals (Section 3); (iii) liquidity crises simultaneously a ect many securities, mostly risky high-margin securities, resulting in commonality of liquidity and ight to quality (Section 4); and (iv) liquidity risk matters even before speculators hit their capital constraints (Section 5). Then we outline how our model s new testable predictions may be helpful for a novel line of empirical work that links measures of speculators funding conditions to measures of market liquidity (Section 6). Finally, we describe the real-world funding constraints for the main liquidity providers, namely market makers, banks, and hedge funds (Appendix A), and provide proofs (Appendix B). 1 Model Setup. The economy has J risky assets, traded at times t = 0, 1, 2, 3. At time t = 3, each security j pays o v j , a random variable de ned on a probability space ( , F, P). There is no aggregate risk and the risk-free interest rate is normalized to zero, so the fundamental value j of each stock is its conditional expected value of the nal payo vt = Et v j . Fundamental volatility has an autoregressive conditional heteroscedasticity (ARCH) structure. Speci cally, j vt evolves according to j j j j j vt+1 = vt + vt+1 = vt + t+1 j , t+1 (1) 7 where all j are i.i.d. across time and assets with a standard normal cumulative distribution t j function with zero mean and unit variance, and the volatility t has dynamics j j t+1 = j + j | vt |, (2) where j , j 0. A positive j implies that shocks to fundamentals increase future volatility. There are three groups of market participants: customers and speculators trade assets while nanciers nance speculators positions. The group of customers consists of three k risk-averse agents. At time 0, customer k = 0, 1, 2 has a cash holding of W0 bonds and zero shares, but nds out that he will experience an endowment shock of zk = {z 1,k , ..., z J,k } shares at time t = 3, where z are jointly normally distributed random variables such that the aggregate endowment shock is zero, 2 j,k k=0 z = 0. With probability (1 a), all customers arrive at the market at time 0 and can trade securities in each time period 0, 1, 2. Since their aggregate shock is zero, they can share risks and have no need for intermediation. The basic liquidity problem arises because customers arrive sequentially with probability a, which gives rise to order imbalance. Speci cally, in this case customer 0 arrives at time 0, customer 1 arrives at time 1, and customer 2 arrives at time 2. Hence, at time 2 all customers are present, at time 1 only customers 0 and 1 can trade, and at time 0 only customer 0 is in the market. Before a customer arrives in the marketplace, his demand is yk = 0, and after he arrives t he chooses his security position each period to maximize his exponential utility function k k U (W3 ) = exp{ W3 } over nal wealth. Wealth Wtk , including the value of the anticipated endowment shock of zk shares, evolves according to k Wt+1 = Wtk + pt+1 pt (yk + zk ) . t (3) The total demand shock of customers who have arrived in the market at time t is denoted 8 by Zt := t k k=0 z . The early customers trading need is accommodated by speculators who provide liquidity/immediacy. Speculators are risk-neutral and maximize expected nal wealth W3 . Speculators face the constraint that the total margin on their position xt cannot exceed their capital Wt : xj+ mj+ + xj mj Wt t t t t j (4) where xj+ 0 and xj 0 are the positive and negative parts of xj = xj+ xj , respectively, t t t t t and mj+ 0 and mj 0 are the dollar margin on long and short positions, respectively. t t The institutional features related to this key constraint for di erent types of speculators like hedge funds, banks, and market makers are discussed in detail in Appendix A. Speculators start out with a cash position of W0 and zero shares, and their wealth evolves according to Wt = Wt 1 + pt pt 1 xt 1 + t , (5) where t is an independent wealth shock arising from other activities, e.g., a speculator s investment banking arm. If a speculator loses all his capital, Wt 0, he can no longer invest because of the margin constraint (4), i.e. he must choose xt = 0. We let his utility in this case be t Wt , where t 0. Limited liability corresponds to t = 0, and a proportional bankruptcy cost (e.g., monetary, reputational, or opportunity costs) corresponds to t > 0. We focus on the case in which 2 = 1, that is, negative consumption equal to time-2 the dollar loss, and we discuss 1 in Section 5. Our results would be qualitatively the same with other bankruptcy assumptions.6 The nancier sets the margins to limit his counterparty credit risk. Speci cally, the nancier ensures that the margin is large enough to cover the position s -value-at-risk (where We could allow the speculators to raise new capital as long as this takes time. Indeed, the model would be the same if the speculators could raise capital only at time 2 (and in this case we need not assume that the customers endowment shocks z j aggregate to zero). 6 9 is a non-negative number close to zero, e.g., 1%): = P r( pj > mj+ | Ft ) t t+1 = P r( pj > mj | Ft ) . t t+1 (6) (7) Equation (6) means that the margin on a long position m+ is set such that price drops larger than the margin only happen with a small probability . Similarly, (7) means that price increases larger than the margin on a short position only happen with small probability. Clearly, the margin is larger for more volatile assets. The margin depends on nanciers information set Ft . We consider two important benchmarks: informed nanciers who know the fundamental value and the liquidity shocks z, Ft = {z, v0 , . . . , vt , p0 , . . . , pt , 1 , . . . , t }, and uninformed nanciers who only observe prices, Ft = {p0 , . . . , pt }. This margin speci cation is motivated by the real-world institutional features described in Appendix A. Theoretically, Stiglitz and Weiss (1981) show how credit rationing can be due to adverse selection and moral hazard in the lending market, and Geanakoplos (2003) considers endogenous contracts in a general-equilibrium framework of imperfect commitment. We let j be the (signed) deviation of the price from fundamental value t j j = pj vt , t t (8) and we de ne our measure of market illiquidity as the absolute amount of this deviation, | j |. We consider competitive equilibria of the economy: t De nition 1 An equilibrium is a price process pt such that (i) xt maximizes the speculators expected nal pro t subject to the margin constraint (4), (ii) each yk maximizes k-customers t expected utility after their arrival at the marketplace and is zero beforehand, (iii) margins are set according to the VaR speci cation (6), and (iv) markets clear, xt + 2 k k=0 yt = 0. 10 Outline of Equilibrium. We derive the optimal strategies for customers and speculators using dynamic programming, starting from time 2, and working backwards. A customer s value function is denoted and a speculator s value function is denoted J. At time 2, customer k s problem is k 2 (W2 , p2 , v2 ) = max E2 [e W3 ] k y2 k (9) k = max e (E2 [W3 ] 2 V ar2 [W3 ]) k y2 k (10) which has the solution j,k y2 = j v 2 pj 2 j ( 3 )2 z j,k (11) Clearly, since all customers are present in the market at time 2, the unique equilibrium is p2 = v2 . Indeed, when the prices are equal to fundamentals, the aggregate customer demand is zero, j,k k y2 = 0, and the speculator also has a zero demand. We get the k k customer s value function 2 (W2 , p2 = v2 , v2 ) = e W2 , and the speculator s value function J2 (W2 , p2 = v2 , v2 ) = W2 . The equilibrium before time 2 depends on whether the customers arrive sequentially or all at time 0. If all customers arrive at time 0, then the simple arguments above show that pt = vt at any time t = 0, 1, 2. We are interested in the case with sequential arrival of the customers such that the speculators liquidity provision is needed. At time 1, customers 0 and 1 are present in the market, but customer 2 has not arrived yet. As above, customer k = 0, 1 has a demand and value function of j,k y1 j v1 pj 1 = k k 1 (W1 , p1 , v1 ) = exp W1 + j ( 2 )2 z j,k j (v1 pj )2 1 j2 2 ( 2 ) (12) (13) j 11 k At time 0, customer k = 0 arrives in the market and maximizes E0 [ 1 (W1 , p1 , v1 )]. j At time t = 1, if the market is perfectly liquid so that pj = v1 for all j, then the 1 speculator is indi erent among all possible positions x1 . If some securities have p1 = v1 , then the risk-neutral speculator invests all his capital such that his margin constraint binds. The speculator optimally trades only in securities with the highest expected pro t per dollar j j used. The pro t per dollar used is (v1 pj )/mj+ on a long position and (v1 pj )/mj on 1 1 1 1 a short position. A speculator s shadow cost of capital, denoted 1 , is 1 plus the maximum pro t per dollar used as long as he is not bankrupt: j v 1 pj 1 j (v1 pj ) 1 1 = 1 + max max{ j mj+ 1 , mj 1 } , (14) where the margins for long and short positions are set by the nancier as described in the next section. If the speculator is bankrupt W1 < 0 then 1 = 1 . Each speculator s value function is therefore J1 (W1 , p1 , v1 , p0 , v0 ) = W1 1 . At time t = 0, the speculator maximizes E0 [W1 1 ] subject to his capital constraint (4). The equilibrium prices at times 1 and 0 do not have simple expressions. However, after having derived the margin conditions, we characterize several important properties of these prices, which illuminates the connection between market liquidity, | |, and speculators funding situation. (15) 2 Margin Setting and Liquidity (Time 1) A key determinant of speculators funding liquidity is their margin requirement for collateralized nancing. Hence, it is important to determine the margin function, m1 , set by, respectively, informed and uninformed nanciers. The margin at time 1 is set to cover a position s value-at-risk, knowing that prices equal the fundamental values in the next period 2, p2 = v2 . 12 We consider rst informed nanciers who know the fundamental values v1 and, hence, price divergence from fundamentals 1 . Since 2 = 0, they set margins on long positions at t = 1 according to = P r( pj > mj+ | F1 ) 2 1 j = P r( v2 + j > mj+ | F1 ) 1 1 (16) = 1 mj+ 1 j 2 j 1 , which implies that j mj+ = 1 (1 ) 2 + j 1 1 j = j + | v1 | + j 1 (17) where we de ne j = j 1 (1 ) (18) (19) = 1 (1 ) . The margin on a short position can be derived similarly and we arrive at the following surprising result: Proposition 1 (Stabilizing Margins and the Cushioning E ect) When the nancier is informed about the fundamental value and knows that prices will equal fundamentals in the next period t = 2, then the margins on long and short positions are, respectively, j mj+ = max{ j + | v1 | + j , 0} 1 1 j mj = max{ j + | v1 | j , 0} 1 1 (20) (21) The more prices are below fundamentals j < 0, the lower is the margin on a long position 1 13 mj+ , and the more prices are above fundamentals j > 0, the lower is the margin on a short 1 1 position mj . Hence, in this case illiquidity reduces margins for speculators who buy low and 1 sell high. The margins are reduced by illiquidity because the speculator is expected to pro t when prices return to fundamentals at time 2, and this pro t cushions the speculators from losses due to fundamental volatility. Thus, we denote the margins set by informed nanciers at t = 1 as stabilizing margins. Stabilizing margins are an interesting benchmark, and they are hard to escape in a theoretical model. However, real-world liquidity crises are often associated with increases in margins, not decreases. To capture this, we turn to the case of a nancier who is uninformed about the current fundamental so that he must set his margin based on the observed prices p0 and p1 . This is in general a complicated problem since the nancier needs to lter out the probability that customers arrive sequentially, and the values of z0 and z1 . The expression becomes simple however, if the nancier s prior probability of a liquidity shock is small so j that he nds it likely that pj = vt , implying a common margin mj = mj+ = mj for long t 1 1 1 and short positions in the limit: Proposition 2 (Destabilizing Margins) When the nancier is uninformed about the fundamental value, then, as a 0, the margins on long and short positions approach j mj = j + | pj | = j + | v1 + j | . 1 1 1 (22) Margins are increasing in price volatility and market illiquidity can increase margins. Intuitively, since liquidity risk tends to increase price volatility, and since an uninformed nancier may interpret price volatility as fundamental volatility, this increases margins.7 Equation (22) corresponds closely to real-world margin setting, which is primarily based on volatility estimates from past price movements. Equation (22) shows that illiquidity increases In the analysis of time 0, we shall see that margins can also be destabilizing when price volatility signals future liquidity risk (not necessarily fundamental risk). 7 14 j margins when the liquidity shock j has the same sign as the fundamental shock v1 or 1 is greater in magnitude, but margins are reduced if the liquidity shock counterbalances a fundamental move. We denote the phenomenon that margins can increase as illiquidity rises by destabilizing margins. As we will see next, the information available to the nancier i.e., whether margins are stabilizing or destabilizing has important implications for the equilibrium. 3 Fragility and Liquidity Spirals (Time 1) We next show how speculators funding problems can lead to liquidity spirals and fragility the property that a small change in fundamentals can lead to a large jump in illiquidity. We show that funding problems are especially escalating with uninformed nanciers, i.e. destabilizing margins. For simplicity we illustrate this with a single security J = 1. Fragility. To set the stage for the main fragility proposition below, we make a few brief de nitions. Liquidity is said to be fragile if the equilibrium price pt ( t , vt ) cannot be chosen to be continuous in the exogenous shocks, namely t and vt . Fragility arises when the excess demand for shares xt + 1 k=0 y1 can be non-monotonic in the price. While under normal circumstances, a high price leads to a low total demand (i.e., excess demand is decreasing), binding funding constraints along with destabilizing margins (margin e ect) or speculators losses (loss e ect) can lead to an increasing demand curve. Further, it is natural to focus on stable equilibria in which a small negative (positive) price perturbation leads to excess demand (supply), which, intuitively, pushes the price back up (down) to its equilibrium level. Proposition 3 (Fragility) There exist x, , a > 0 such that (i) With informed nanciers, the market is fragile at time 1 if speculators position |x0 | is larger than x and of the same sign as the demand shock Z1 . (ii) With uninformed nanciers the market is fragile as in (i) and additionally if the ARCH 15 parameter is larger than and the probability, a, of sequential arrival of customers is smaller than a. Numerical Example. We illustrate how fragility arises due to destabilizing margins or dealer losses by way of a numerical example. We consider the more interesting (and arguably more realistic) case in which the nanciers are uninformed, and we choose parameters as follows. The fundamental value has ARCH volatility parameters = 10 and = 0.3, which implies clustering of volatility. The initial price is p0 = 130, the aggregate demand shock of the customers who have arrived at time 1 is Z1 = z0 + z1 = 30, and the customers risk aversion coe cient is = 0.05. The speculators have an initial position of x0 = 0 and a cash wealth of W1 = 900. Finally, the nancier uses a VaR with = 1% and customers learn their endowment shocks sequentially with probability a = 1%. Panel A of Figure 2 illustrates how the speculators demand x1 and the customers supply (i.e., the negative of the customers demand as per Equation (12)) depend on the price p1 when the fundamental value is v1 = 120 and the speculators wealth shock is 1 = 0. Customers supply is given by the upward sloping dashed line since, naturally, their supply is greater when the price is higher. Customers supply Z1 = 30 shares, namely the shares that they anticipate receiving at time t = 3, when the market is perfectly liquid, p1 = v1 = 120 (i.e. illiquidity is | 1 | = 0). For lower prices, they supply fewer shares. The speculators demand, x1 , must satisfy the margin constraints. It is instructive to consider rst the simpler limiting case a 0 for which the margin requirement is simply m = + | p1 | = 2.326(10 + 0.3| p1 |). This implies that speculators demand |x1 | W1 /( + | p1 |). Graphically, this means that their demand must be inside the hyperbolic star de ned by the four (dotted) hyperbolas (that are partially overlaid by a solid demand curve in Figure 2). At the price p1 = p0 = 130, the margin is smallest and hence the constraint is most relaxed. As p1 departs from p0 = 130, margins increase and speculators become more constrained the horizontal distance between two hyperbolas shrinks. 16 170 170 160 160 150 150 140 p1 1 140 130 130 120 p 40 30 20 10 0 x1 10 20 30 40 50 120 110 110 100 100 90 50 90 50 40 30 20 10 0 x1 10 20 30 40 50 Figure 2: Speculator Demand and Customer Supply. This gure illustrates how margins can be destabilizing when nanciers are uninformed and the fundamentals have volatility clustering. The solid curve is the speculators optimal demand for a = 1%. The upward sloping dashed line is the customers supply, that is, the negative of their demand. In Panel A the speculators experience a zero wealth shock, 1 = 0, while in Panel B they face a negative wealth shock of 1 = 150, otherwise everything is the same. In Panel A, perfect liquidity p1 = v1 = 120 is one of two stable equilibria, while in Panel B the unique equilibrium is illiquid. After establishing the hyperbolic star, it is easy to derive the demand curve for a 0: For p1 = v1 = 120, the security s expected return is zero and the speculator is indi erent between all his possible positions on the horizontal line. For price levels p1 > v1 above this line, the risk neutral speculators want to short-sell the asset, x1 < 0, and their demand is constrained by the upper left side of the star. Similarly, for prices below v1 , speculators buy the asset, x1 > 0, and their demand is limited by the margin constraint. Interestingly, the speculators demand is upward sloping for prices below 120. As the price declines, the nanciers estimate of fundamental volatility and consequently of margins increase. We now generalize the analysis to the case where a > 0. The margin setting becomes more complicated since uninformed nanciers must lter out to what extent the equilibrium price change is caused by a movement in fundamentals v1 and/or an occurrence of a liquidity event with an order imbalance caused by the presence of customers 0 and 1, but not customer 2. Since customers 0 and 1 want to sell (Z1 = 30), a price increase or modest price decline is 17 most likely due to a change in fundamentals, and hence the margin setting is similar to the case of a = 0. This is why speculators demand curve for prices above 100 almost perfectly overlays the relevant part of the hyperbolic star in Figure 2. However, for a large price drop, say below 100, nanciers assign a larger conditional probability that a liquidity event has occurred. Hence, they are willing to set a lower margin (relative to the one implying the hyperbolic star) because they expect the speculator to pro t as the price rebounds in period 2 hence, the cushioning e ect discussed above reappears in the extreme here. This explains why the speculators demand curve is backward bending only in a limited price range and becomes downward sloping for p1 below roughly 100.8 Panel A shows that there are two stable equilibria: a perfect liquidity equilibrium with price p1 = v1 = 120 and an illiquid equilibrium with a price of about 91 (and an uninteresting unstable equilibrium with p1 just below 120). Panel B of Figure 2 shows the same plot as Panel A, but with a negative wealth shock to speculators of 1 = 150 instead of 1 = 0. In this case, perfect liquidity with p1 = v1 is no longer an equilibrium since the speculators cannot fund a large enough position. The unique equilibrium is highly illiquid because of the speculators lower wealth and, importantly, because of endogenously higher margins. This disconnect between the perfect-liquidity equilibrium and the illiquid equilibrium and the resulting fragility is illustrated more directly in Figure 3. Panel A plots the equilibrium price correspondence for di erent exogenous funding shocks 1 (with xed v1 = 10) and shows that a marginal reduction in funding cannot always lead to a smooth reduction in market liquidity. Rather, there must be a level of funding such that an in nitesimal drop in funding leads to a discontinuous drop in market liquidity. The dark (blue) line in Figure 3 shows the equilibrium with the highest market liquidity and the light (red) line shows the equilibrium with the lowest market liquidity. We note that We note that the cushioning e ect relies on the nanciers knowledge that the market will become liquid in period t = 2. This is not the case in the earlier period 0, though, and, in an earlier version of the paper, we showed that the cushioning e ect disappears in a stationary in nite horizon setting in which the complementary customers arrive in each period with a constant arrival probability. 8 18 150 150 140 140 130 130 p1 1 120 120 p 110 110 100 100 90 100 50 0 50 100 150 1 200 250 300 350 400 90 30 20 10 0 v1 10 20 30 Figure 3: Fragility due to Destabilizing Margins. The gure shows the equilibrium price as a function of the speculators wealth shock 1 (Panel A) and of fundamental shocks v1 (Panel B). This is drawn for the equilibrium with the highest market liquidity (light/ red line) and the equilibrium with the lowest market liquidity (dark/ blue line). The margins are destabilizing since nanciers are uninformed and fundamentals exhibit volatility clustering. The equilibrium prices are discontinuous, which re ects fragility in liquidity since a small shock can lead to a disproportionately large price e ect. the nanciers ltering problem and, hence, the margin function depend on the equilibrium selection. Since the margin a ects the speculators trades, the equilibrium selection a ects the equilibrium outcome everywhere prices are slightly a ected even outside the region (v1 region) with fragility. Panel B of Figure 3 plots the equilibrium price correspondence for di erent realizations of the fundamental shock v1 (with xed 1 = 0) and shows the same form of discontinuity for adverse fundamental shocks to v1 . The discontinuity with respect to v1 is most easily understood in conjunction with Panel A of Figure 2. As v1 falls, the horizontal line of speculator demand shifts downward, and the customer supply line moves downward. As a result, the perfect liquidity equilibrium vanishes. Panel B also reveals the interesting asymmetry that negative fundamental shocks lead to larger price movements than corresponding positive shocks (for Z1 := z0 + z1 > 0). This asymmetry arises even without a loss e ect since x0 = 0. Fragility can also arise because of shocks to customer demand or volatility. Indeed, the market can also be suddenly pushed into an illiquid equilibrium with high margins due to an 19 increase in demand and an increase in volatility. Paradoxically, a marginally larger demand for liquidity by customers can lead to a drastic reduction of liquidity supply by the speculators when it pushes the equilibrium over the edge. While the example above has speculators with zero initial positions, x0 = 0, it is also interesting to consider x0 > 0. In this case, lower prices lead to losses for the speculators, and graphically this means that the constraints in the hyperbolic star tighten (i.e., the gap between the hyperbolas narrows) at low prices. Because of this loss e ect, the discontinuous price drop associated with the illiquid equilibrium is even larger. In summary, this example shows how destabilizing margins and dealer losses give rise to a discontinuity in prices which can help to explain the sudden market liquidity dry-ups observed in many markets. For example, Russia s default in 1998 was in itself only a trivial wealth shock relative to global arbitrage capital. Nevertheless, it had a large e ect on liquidity in global nancial markets, consistent with our fragility result that a small wealth shock can push the equilibrium over the edge. Liquidity Spirals. To further emphasize the importance of speculators funding liquidity, we now show how it can make market liquidity highly sensitive to shocks. We identify two ampli cation mechanisms: a margin spiral due to increasing margins as speculator nancing worsens, and a loss spiral due to escalating speculator losses. Figure 4 illustrates these liquidity spirals : A shock to speculator capital ( 1 < 0) forces speculators to provide less market liquidity, which increases the price impact of the customer demand pressure. With uninformed nanciers and ARCH e ects the resulting price swing increases nanciers estimate of the fundamental volatility and, hence, increases the margin, thereby worsening speculator funding problems even further, and so on, leading to a margin spiral. Similarly, increased market illiquidity can lead to losses on speculators existing positions, worsening their funding problem and so on, leading to a loss spiral. Mathematically, the spirals can be expressed as follows: Proposition 4 20 reduced positions initial losses funding problems for speculators prices move away from fundamentals higher margins losses on existing positions Figure 4: Liquidity Spirals (i) If speculators capital constraint is slack then the price p1 is equal to v1 and insensitive to local changes in speculator wealth. (ii) (Liquidity Spirals) In a stable illiquid equilibrium with selling pressure from customers, Z1 , x1 > 0, the price sensitivity to speculator wealth shocks 1 is p1 1 = 1 2 m+ ( 2 )2 1 + m+ 1 p1 x1 (23) x0 and with buying pressure from customers, Z1 , x1 < 0, p1 1 m+ 1 p1 = 1 2 m ( 2 )2 1 m 1 p1 + m 1 p1 x1 . + x0 (24) A margin spiral arises if < 0 or > 0, which happens with positive probability if speculators are uninformed and a is small enough. A loss spiral arises if speculators previous position is in the opposite direction as the demand pressure x0 Z1 > 0. 21 This proposition is intuitive. Imagine rst what happens if speculators face a wealth shock of $1, margins are constant, and speculators have no inventory x0 = 0. In this case, the speculator must reduce his position by 1/m1 . Since the slope of each of the two customer 2 demand curves is9 1/( ( 2 )2 ), we get a total price e ect of 1/( ( )2 m1 ). 2 The two additional terms in the denominator imply ampli cation or dampening e ects due to changes in the margin requirement and to pro t/losses on the speculators existing positions. To see that, recall that for any k > 0 and l with |l| < k, it holds that 1 k 1 k l = + l k2 + l2 k3 + ...; so with k = 2 m ( 2 )2 1 and l = m 1 p1 x1 x0 , each term in this in nite series corresponds to one loop around the circle in Figure 4. The total e ect of the changing margin and speculators positions ampli es the e ect if l > 0. Intuitively, if e.g. Z1 > 0, then customer selling pressure is pushing down the price, and m 1 p1 m+ 1 p1 < 0 means that as prices go down, margins increase, making speculators funding tighter and thus destabilizing the system. Similarly, when customers are buying, > 0 implies that increasing prices leads to increased margins, making it harder for speculators to short-sell, thus destabilizing the system. The system is also destabilized if speculators lose money on their previous position as prices move away from fundamentals similar to e.g., Shleifer and Vishny (1997). Interestingly, the total e ect of a margin spiral together with a loss spiral is greater than the sum of their separate e ects. This can be seen mathematically by using simple convexity arguments, and it can be seen intuitively from the ow diagram of Figure 4. Note that spirals can also be started by shocks to liquidity demand Z1 , fundamentals v1 , or volatility. It is straightforward to compute the price sensitivity with respect to such shocks. They are just multiples of p1 1 . For instance, a fundamental shock a ects the price both because of its direct e ect on the nal payo and because of its e ect on customers estimate of future volatility and both of these e ects are ampli ed by the liquidity spirals. Our analysis sheds some new light on the 1987 stock market crash, complementing the standard culprit, portfolio insurance trading. In the 1987 stock market crash, numerous market makers hit (or violated) their funding constraint: 9 See Equation (12). 22 By the end of trading on October 19, [1987] thirteen [NYSE specialist] units had no buying power SEC (1988), page 4-58 Several of these rms managed to reduce their positions and continue their operations. Others did not. For instance, Tompane was so illiquid that it was taken over by Merrill Lynch Specialists and Beauchamp was taken over by Spear, Leeds & Kellogg (Beauchamp s clearing broker). Also, market makers outside the NYSE experienced funding troubles: the Amex market makers Damm Frank and Santangelo were taken over; at least 12 OTC market makers ceased operations; and several trading rms went bankrupt. These funding problems were due to (i) reductions in capital arising from trading losses and defaults on unsecured customer debt, (ii) an increased funding need stemming from increased inventory and, (iii) increased margins. One New York City bank, for instance, increased margins/haircuts from 20% to 25% for certain borrowers, and another bank increased margins from 25% to 30% for all specialists (SEC (1988) page 5-27 and 5-28). Other banks reduced the funding period by making intra-day margin calls, and at least two banks made intra-day margin calls based on assumed 15% and 25% losses, thus e ectively increasing the haircut by 15% and 25%. Also, some broker-dealers experienced a reduction in their line of credit and as Figure 1 shows margins at the futures exchanges also drastically increased (SEC (1988) and Wigmore (1998)). In summary, our results on fragility and liquidity spirals imply that during bad times, small changes in underlying funding conditions (or liquidity demand) can lead to sharp reductions in liquidity. The 1987 crash exhibited several of the predicted features, namely capital constrained dealers, increased margins, and increased illiquidity. 4 Commonality and Flight to Quality (Time 1) We now turn to the cross-sectional implications of illiquidity. Since speculators are riskneutral, they optimally invest all their capital in securities that have the greatest expected 23 pro t | j | per capital use, i.e., per dollar margin mj , as expressed in Equation (14). That equation also introduces the shadow cost of capital 1 as the marginal value of an extra dollar. The speculators shadow cost of capital 1 captures well the notion of funding liquidity: a high means that the available funding from capital W1 and from collateralized nancing with margins mj is low relative to the needed funding, which depends on the investment 1 opportunities deriving from demand shocks z j . The market liquidity of assets all depend on the speculators funding liquidity, especially for high-margin assets, and this has several interesting implications: Proposition 5 There exists c > 0 such that, for j < c for all j and either informed nanciers or uninformed with a < c, we have: (i) (Commonality of Market Liquidity) The market illiquidities | | of any two securities, k and l, co-move, Cov0 | k |, | l | 0 , 1 1 (25) and market illiquidity co-moves with funding illiquidity as measured by speculators shadow cost of capital 1 Cov0 | k |, 1 0 . 1 (26) (ii) (Commonality of Fragility) Jumps in market liquidity occur simultaneously for all assets for which speculators are marginal investors. (iii) (Quality=Liquidity) If asset l has lower fundamental volatility than asset k, l < k , then l also has lower market illiquidity, | l | | k | 1 1 k l if xk = 0 or |Z1 | |Z1 |. 1 (27) (iv) (Flight to Quality) The market liquidity di erential between high- and low-fundamentalvolatility securities is bigger when speculator funding is tight, that is, l < k implies that 24 | k | increases more with a negative wealth shock to the speculator, 1 | k | | l | 1 1 , ( 1 ) ( 1 ) k l k l if xk = 0 or |Z1 | |Z1 |. Hence, if xk = 0 or |Z1 | |Z1 | a.s., then 1 1 (28) Cov0 (| l |, 1 ) Cov0 (| k |, 1 ) . 1 1 (29) Numerical Example, continued. To illustrate these cross-sectional predictions, we extend the numerical example of Section 3 to two securities. The two securities only di er in their long-run fundamental volatility: 1 = 7.5 and 2 = 10. The other parameters are as before, except that we double W1 to 1800 since the speculators now trade two securities, the nanciers remain uninformed, and we focus on the simpler limited case with a 0. 160 140 120 100 p1, p2 11 80 60 40 20 0 1000 500 0 1 500 1000 Figure 5: Flight to Quality and Commonality in Liquidity. The gure plots the prices pj of assets 1 and 2 as functions of speculators funding shocks 1 . Asset 1 (darker curve) 1 has lower long-run fundamental risk than asset 2 (lighter curve), 1 = 7.5 < 10 = 2 . Figure 5 depicts the assets equilibrium prices for di erent values of the funding shock 1 . 25 First note that as speculator funding tightens and our funding illiquidity measure 1 rises, the market illiquidity measure | j | rises for both assets. Hence, for random 1 , we see our 1 commonality in liquidity result Cov0 | k |, | l | > 0. 1 1 The commonality in fragility cannot directly be seen from Figure 5, but it is suggestive that both assets have the same range of 1 with two equilibrium prices pj . The intuition 1 for this result is the following. Whenever funding is unconstrained, there is perfect market liquidity provision for all assets. If funding is constrained, then it cannot be the case that speculators provide perfect liquidity for one asset but not for the other, since they would always have an incentive to shift funds towards the asset with non-perfect market liquidity. Hence, market illiquidity jumps for both assets at exactly the same funding level. Our result relating fundamental volatility to market liquidity ( Quality=Liquidity ) is re ected in p2 being below p1 for any given funding level. Hence, the high-fundamental1 1 volatility asset 2 is always less liquid than the low-fundamental-volatility asset 1. The graph also illustrates our result on ight to quality. To see this, consider the two securities relative price sensitivity with respect to 1 . For large wealth shocks, market 1 2 liquidity is perfect for both assets, i.e. p1 = p2 = v1 = v1 = 120, so in this high range 1 1 of funding, market liquidity is insensitive to marginal changes in funding. For su ciently small funding levels, 1 < 157, market illiquidity of both assets increases as 1 drops since speculators must take smaller stakes in both assets. Importantly, as funding decreases, p2 1 decreases more steeply than p1 , that is, asset 2 is more sensitive to funding declines than 1 asset 1. This is because speculators cut back more on the funding-intensive asset 2 with its high margin requirement. The speculators want to maximize their pro t per dollar margin, | j |/mj and therefore | 2 | must be higher than | 1 | to compensate speculators for using more capital for margin. Both price functions exhibit a kink around = 1210, because, for su ciently low funding levels, speculators put all their capital into asset 2. This is because the customers are more eager to sell the more volatile asset 2, leading to more attractive prices for the speculators. 26 5 Liquidity Risk (Time 0) We now turn attention to the initial time period t = 0 and demonstrate that (i) funding liquidity risk matters even before margin requirements actually bind, (ii) the pricing kernel depends on future funding liquidity, t+1 , (iii) the conditional distribution of prices p1 is skewed due to the funding constraint (inducing fat tails ex-ante), and (iv) margins m0 and illiquidity 0 can be positively related due to liquidity risk even if nanciers are informed. The speculators trading activity at time 0 naturally depends on their expectations about the next period and, in particular, the time 1 illiquidity described in detail above. Further, speculators risk having negative wealth W1 at time 1, in which case they have utility t Wt . If speculators have no dis-utility associated with negative wealth levels ( t = 0), then they go to their limit already at time 0 and the analysis is similar to time 1 (see appendix). We focus on the more realistic case in which the speculators have dis-utility in connection with W1 < 0 and, therefore, choose not trade to their constraint at time t = 0 when their wealth is large enough. To understand this, note that while most rms legally have limited liability, the capital Wt in our model refers to pledgable capital allocated to trading. For instance, Lehman Brothers 2001 Annual Report (page 46) states: The following must be funded with cash capital: Secured funding haircuts, to re ect the estimated value of cash that would be advanced to the Company by counterparties against available inventory, Fixed assets and goodwill, [and] Operational cash ... Hence, if Lehman su ers a large loss on its pledgable capital such that Wt < 0, then it incurs monetary costs that must be covered with its unpledgable capital like operational cash (which could also hurt Lehman s other businesses). In addition the rm incurs non-monetary cost, like loss in reputation and in goodwill, that reduces its ability to exploit future pro table investment opportunities. To capture these e ects, we let a speculator s utility be 1 W1 , where 1 is given by the right-hand side of (14) both for positive and negative values of W1 . With this assumption, equilibrium prices at time t = 0 are such that the speculators do not 27 trade to their constraint at time t = 0 when their wealth is large enough. In fact, this is the weakest assumption that curbs the speculators risk taking since it makes their objective function linear. Higher bankruptcy costs would lead to more cautious trading at time 0 and qualitatively similar results.10 If the speculator is not constrained at time t = 0, then the rst-order condition for his position in security j is E0 [ 1 (pj pj )] = 0. (We leave the case of a constrained time-0 1 0 speculator for the appendix.) Consequently, the funding liquidity 1 determines the pricing kernel 1 /E0 [ 1 ] for the cross section of securities: pj = 0 E0 [ 1 pj ] Cov0 [ 1 , pj ] 1 1 = E0 [pj ] + . 1 E0 [ 1 ] E0 [ 1 ] (30) Equation (30) shows that the price at time 0 is the expected time-1 price which already depends on the liquidity shortage at time 1 further adjusted for liquidity risk in the form of a covariance term. The liquidity risk term is intuitive: The time-0 price is lower if the covariance is negative, that is, if the security has a low payo during future funding liquidity crises when 1 is high. An illustration of the importance of funding-liquidity management is the LTCM crisis. The hedge fund Long Term Capital Management (LTCM) had been aware of funding liquidity risk. Indeed, they estimated that in times of severe stress, haircuts on AAA-rated commercial mortgages would increase from 2% to 10%, and similarly for other securities (HBS Case N9-200-007(A)). In response to this, LTCM had negotiated long-term nancing with margins xed for several weeks on many of their collateralized loans. Other rms with similar strategies, however, experienced increased margins. Due to an escalating liquidity spiral, LTCM We note that risk aversion also limits speculators trading in the real world. Our model based on margin constraints di ers from one driven purely by risk aversion in several ways. E.g. an adverse shock that lowers speculator wealth at t = 1 creates a pro table investment opportunity which one might think partially o sets the loss a natural dynamic hedge. Because of this dynamic hedge, in a model driven by risk-aversion speculators (with a relative risk aversion coe cient larger than one) increase their t = 0 hedging demand, which in turn, lowers illiquidity in t = 0. However, exactly the opposite occurs in a setting with capital constraints. Capital constraints prevent speculators from taking advantage of investment opportunities in t = 1 so they cannot exploit this dynamic hedge. Hence, speculators are reluctant to trade away the illiquidity in t = 0. 10 28 could ultimately not fund its positions in spite of its numerous measures to control funding risk, and it was taken over by 14 banks in September 1998. Another recent example is the funding problems of the hedge fund Amaranth in September 2006, which reportedly ended with losses in excess of $6 billion. Numerical Example, continued. To better understand funding liquidity risk, we return to our numerical example with one security, 1 = 0 and a 0. We rst consider the setting with uninformed nanciers and later turn to the case with informed nanciers. Figure 6 depicts the price p0 and expected time-1 price E0 [p1 ] for di erent initial wealth levels, W0 , for which the speculators funding constraint is not binding in t = 0. The gure shows that even though the speculators are unconstrained at time 0, market liquidity provision is limited with prices below the fundamental value of E0 [v] = 130. The price is below the fundamental for two reasons: First, the expected time-1 price is below the fundamental value because of the risk that speculators cannot accommodate the customer selling pressure at that time. Second, p0 is even below E0 [p1 ], since speculators face liquidity risk: Holding the security leads to losses in the states of nature when speculators are constrained and investment opportunities are good, implying that Cov[ 1 , p1 ] < 0. The additional compensation for liquidity risk is Cov0 [ 1 ,pj ] 1 E0 [ 1 ] as seen in Equation (30), which is the di erence between the solid line p1 and the dashed E0 [p1 ].11 The funding constraint not only a ects the price level, it also introduces skewness in the p1 -distribution conditional on the sign of the demand pressure. For Z1 > 0, speculators 11 Our numerical analysis shows that Cov[ 1 , p1 ] is non-positive over the whole range of (non-binding) funding levels W0 . The correlation is non-positive since low v1 realizations typically lead to large price drops that p widen the gap v1 p1 . This increases 1 = 1 + v1m1 1 , despite the mitigating e ect that an increase in m1 = + | p1 | causes. As a function of W0 the correlation coe cient is U-shaped and below .8 for W0 [1000, 2500]. The U-shaped pattern in speculators wealth is best understood by considering extreme funding levels: for extremely large speculator wealth, the price approaches the fundamental value, and for zero speculator wealth the speculators become unimportant because they don t have the means to move the price. For su ciently low wealth levels, a more subtle mechanism is at work that explains why the correlation becomes less negative as the speculator wealth declines. High realization of v1 lead to a less sharp increase in price ( v1 > p1 ), since the price increase is held back by higher margins. This, in turn, increases 1 when p1 increases - a positive relationship. While this second mechanism is is at work for high realizations of v1 , ex-ante it is still dominated by the e ect driven by low realizations of v1 , and hence Cov[ 1 , p1 ] is non-positive for all W0 . 29 130 128 126 124 p0, E[p1] 122 120 118 116 114 1000 1500 W0 2000 2500 3000 Figure 6: Illiquidity at Time 0. This graph shows the price p0 at time 0 (solid line), the expected time-1 price E0 [p1 ] (dashed line), and the fundamental value E0 [v] = 130 (dotted line) for di erent levels of speculator funding W0 . The price p0 is below the fundamental value due to illiquidity, in particular, because of customer selling pressure and the risk that speculators will hit their capital constraints at time 1, even though speculators are not constrained at time 0 for the depicted wealth levels. take long positions and, consequently, negative v1 -shocks lead to capital losses with resulting liquidity spirals. This ampli cation triggers a sharper price drop than the corresponding price increase for positive v1 -shocks. Figure 7 shows this negative skewness for di erent funding levels W0 . The e ect is not monotone zero dealer wealth implies no skewness, for instance. For negative realizations of Z1 , customers want to buy (not sell as above), and funding constraints induce a positive skewness in the p1 -distribution. The speculator s return is negatively skewed, as above, since it is still its losses that are ampli ed. This is consistent with the casual evidence that hedge fund return indexes are negatively skewed. It also suggests that from an ex-ante point of view, i.e., prior to the realization of Z1 , funding constraints lead to higher kurtosis of the price distribution (fat tails). Finally, we can also show numerically that unlike at time t = 1, margins can be positively related to illiquidity at time 0, even when nanciers are fully informed.12 This is because of the liquidity risk between time 0 and time 1. To see this, note that if we reduce the 12 The simulation results are available upon request from the authors. 30 0 2 4 skewness of p1 6 8 10 12 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 W0 Figure 7: Conditional Price Skewness. The gure shows the conditional skewness of p1 for di erent funding levels W0 . While the funding constraint is not binding at time 0, it can become binding at time 1, leading to large price drops due to liquidity spirals. Price increases are not ampli ed, and this asymmetry results in skewness. speculators initial wealth W0 , then the market becomes less liquid in the sense that the price is further from the fundamental value. At the same time, the equilibrium price in t = 1 is more volatile and thus equilibrium margins at time 0 can actually increase. 6 New Testable Predictions Our analysis provides a theoretical framework that delivers a uni ed explanation for a host of stylized empirical facts. Our analysis further suggests a novel line of empirical work that tests the model at a deeper level namely its prediction that speculator funding is a driving force underlying these market liquidity e ects. First, it would be of interest to empirically study the determinants of margin requirements, e.g. using data from futures markets or from prime brokers. Our model suggests that both fundamental volatility and liquidity-driven volatility a ect margins (Propositions 1 and 2). Empirically, fundamental volatility can be captured using price changes over a longer time period, while the sum of fundamental and liquidity-based volatility can be captured by 31 short-term price changes as in the literature on variance ratios (see, e.g. Campbell, Lo, and MacKinlay (1997)). Our model predicts that, in markets where it is harder for nanciers to be informed, margins depend on the total fundamental and liquidity-based volatility. In particular, in times of liquidity crises, margins increase in such markets, and, more generally, margins should co-move with illiquidity in the time series and in the cross section.13 Second, our model suggests that an exogenous shock to speculator capital should lead to a reduction in market liquidity (Proposition 4). Hence, a clean test of the model would be to identify exogenous capital shocks such as an unconnected decision to close down a trading desk, a merger leading to reduced total trading capital, or a loss in one market unrelated to the fundamentals of another market, and then study the market liquidity and margin around such events. Third, the model implies that the e ect of speculator capital on market liquidity is highly non-linear: a marginal change in capital has a small e ect when speculators are far from their constraints, but a large e ect when speculators are close to their constraints illiquidity can suddenly jump (Propositions 3 and 4). Fourth, the model suggests that a cause of the commonality in liquidity is that the speculators shadow cost of capital is a driving state variable. Hence, a measure of speculator capital tightness should help explain the empirical co-movement of market liquidity. Further our result commonality of fragility suggests that especially sharp liquidity reductions occur simultaneously across several assets (Proposition 5(i) (ii)). Fifth, the model predicts that the sensitivity of margins and market liquidity to speculator capital is larger for securities that are risky and illiquid on average. Hence, the model suggests that a shock to speculator capital would lead to a reduction in market liquidity through a spiral e ect that is stronger for illiquid securities (Proposition 5(iv)). Sixth, speculators are predicted to have negatively skewed returns since, when they hit their constraints, they make signi cant losses because of the endogenous liquidity spirals, One must be cautious with the interpretation of the empirical results related to changes in Regulation T since this regulation may not a ect speculators but a ects the demanders of liquidity, namely the customers. 13 32 and, in contrast, their gains are not ampli ed when prices return to fundamentals. This leads to conditional skewness and unconditional kurtosis of security prices (Section 5). 7 Conclusion By linking funding and market liquidity this paper provides a uni ed framework that explains the following stylized facts: (i) Liquidity suddenly dries up; we argue that fragility in liquidity is in part due to destabilizing margins which arise when nanciers are imperfectly informed the fundamentals follow an ARCH process. (ii) Market liquidity and fragility co-moves across assets since changes in funding conditions a ects speculators market liquidity provision of all assets. (iii) Market liquidity is correlated with volatility, since trading more volatile assets requires higher margin payments and speculators provide market liquidity across assets such that illiquidity per capital use, i.e. illiquidity per dollar margin, is constant. (iv) Flight to quality phenomena arise in our framework since when funding becomes scarce speculators cut back on the market liquidity provision especially for capital intensive, i.e. high margin, assets. (v) Market liquidity moves with the market since funding conditions do. In addition to explaining these stylized facts, the model also makes a number of more speci c, testable predictions that could inspire further empirical research on margins. Our analysis also suggests policy implications for central banks. Central banks can help mitigate market liquidity problems by controlling funding liquidity. If a central bank is better than the typical nanciers of speculators at distinguishing liquidity shocks from fundamental shocks, then the central bank can convey this information and urge nanciers to relax their funding requirements as the Federal Reserve Bank of New York did during the 1987 stock market crash. Central banks can also improve market liquidity by boosting speculator funding conditions during a liquidity crisis, or by simply stating the intention to provide extra funding during times of crisis, which would loosen margin requirements immediately as nanciers worst-case scenarios improve. 33 A Appendix: Real World Margin Constraints A central element of our paper is the capital constraints that the main providers of market liquidity face. To set the stage for our model, we review these institutional features for securities rms such as hedge funds, banks proprietary trading desks, and market makers. (Readers mainly interested in our theory can skip to Section 1.) A.1 Funding Requirements for Hedge Funds We rst consider the funding issues faced by hedge funds since they have relatively simple balance sheets and face little regulation. A hedge fund s capital consists of its equity capital supplied by the partners, and possible long-term debt nancing that can be relied upon during a potential funding crisis. Since a hedge fund is a partnership, the equity is not locked into the rm inde nitely, as in a corporation. The investors (that is, the partners) can withdraw their capital at certain times, but to ensure funding the withdrawal is subject to initial lock-up periods and general redemption notice periods before speci c redemption dates (typically at least a month, often several months or even years). A hedge fund usually does not issue long-term unsecured bonds, but some (large) hedge funds manage to obtain debt nancing in the form of medium-term bank loans or in the form of a guaranteed line of credit.14 Recently, some hedge funds have even raised capital by issuing bonds or permanent equity (e.g., see The Economist 1/27/2007, page 75). The main source of leverage for hedge funds is collateralized borrowing nanced by the hedge fund s prime broker(s). The prime brokerage business is opaque since the terms of the nancing are subject to negotiation and are hidden to outsiders. We describe stylized nancing terms and, later, we discuss caveats. If a hedge fund buys at time t a long position of xj > 0 shares of a security j at price pj , t t it has to come up with xj pj dollars. The security can, however, be used as collateral for a tt A line of credit may have a material adverse change clause or other covenants subject to discretionary interpretation of the lender. Such covenants imply that the line of credit may not be a reliable source of funding during a crisis. 14 34 j new loan of, say, lt dollars. The di erence between the price of the security and the collateral j value is denoted as the margin requirement mj+ = pj lt . Hence, this position uses xj mj+ t t tt dollars of the fund s capital. The collateralized funding implies that the capital use depends on margins, not notional amounts. The margins on xed income securities and over-the-counter (OTC) derivatives are set through a negotiation between the hedge fund and the prime broker that nances the trade. The margins are typically set so as to make the loan almost risk free for the broker, that is, such that it covers the largest possible adverse price move with a certain degree of con dence (i.e., it covers the Value-at-Risk).15 If the hedge fund wants to sell short a security, xj < 0, then the fund asks one of its brokers to locate a security that can be borrowed, and then the fund sells the borrowed security. Du e, G rleanu, and Pedersen (2002) describe in detail the institutional arrangements of a shorting. The broker keeps the proceeds of the short sale to be able to repurchase the security if the hedge fund fails and, additionally, requires that the hedge fund posts a margin mj that covers the largest possible adverse price move with a certain degree of con dence. t In the US, margins on equities are subject to Regulation T, which stipulates that nonbrokers/dealers must have an initial margin (downpayment) of 50% of the market value of the underlying stock, both for new long and short positions. Hedge funds can, however, circumvent Regulation T by, for instance, organizing the transaction as a total return swap, which is a derivative that is functionally equivalent to buying the stock. The margin on exchange traded futures (or options) is set by the exchange. The principle for setting the margin for futures or options is the same as that described above. The margin is set so as to make the exchange almost immune to the default risk of the counterparty, and hence riskier contracts have larger margins. A hedge fund must nance all of its positions, that is, the sum of all the margin requireAn explicit equation for the margin is given by (6) in Section 1. Often brokers also take into account the delay between the time a failure by the hedge fund is noticed, and the time the security is actually sold. Hence, the margin of a one-day collateralized loan depends on the estimated risk of holding the asset over a time period that is often set as ve to ten days. 15 35 ments on long and short positions cannot exceed the hedge fund s capital. In our model, this is captured by the key Equation (4) in Section 1. At the end of the nancing period, time t + 1, the position is marked-to-market, which means that the hedge fund receives any gains (or pays any losses) that have occurred between t and t + 1, that is, the fund receives xj (pj pj ) and pays interest on the loan at t t+1 t the funding rate. If the trade is kept on, the broker keeps the margin to protect against losses going forward from time t + 1. The margin can be adjusted if the risk of the collateral has changed, unless the counterparties have contractually xed the margin for a certain period. Stock exchanges and self-regulatory organizations (e.g., NASD) also impose maintenance/continuation margins for existing stock positions. For example, the NYSE and the NASD require that investors maintain a minimum margin of 25% for long stock positions and 30% for short stock positions. Instead of posting risk-free assets (cash), a hedge fund can also post other risky assets, say asset k, to cover its margin on position, say xj . However, in this case a haircut, hk , t is subtracted from asset k s market value to account for the riskiness of the collateral. The funding constraint becomes xj mj Wt xk hk . Moving the haircut term to the left-hand side tt tt reveals that the haircut is equivalent to a margin, since the hedge fund could alternatively have used the risky security to raise cash and then used this cash to cover the margins for asset j. We therefore use the terms margins and haircuts interchangeably. We have described how funding constraints work when margins and haircuts are set separately for each security position. It is, however, sometimes possible to cross-margin , i.e. to jointly nance several trades that are part of the same strategy. This leads to a lower total margin if the risks of the various positions are partially o setting. For instance, much of the interest rate risk is eliminated in a spread trade with a long position in one bond and a short position in a similar bond. Hence, the margin/haircut of a jointly nanced spread trade is smaller than the sum of the margins of the long and short bonds. For a strategy that is nanced jointly, we can reinterpret security j as such a strategy. Prime brokers compete by, 36 among other things, o ering low margins and haircuts a key consideration for hedge funds which means that it is becoming increasingly easy to nance more and more strategies jointly. In the extreme, one can imagine a joint nancing of a hedge fund s total position such that the portfolio margin would be equal to the maximum portfolio loss with a certain con dence level. Currently, it is often not practical to jointly nance a large portfolio. This is because a large hedge fund nances its trades using several brokers; both a hedge fund and a broker can consist of several legal entities (possibly located in di erent jurisdictions); certain trades need separate margins paid to exchanges (e.g., futures and options) or to other counterparties of the prime broker (e.g., securities lenders); prime brokers may not have su ciently sophisticated models to evaluate the diversi cation bene ts (e.g., because they do not have enough data on the historical performance of newer products such as CDOs); and because of other practical di culties in providing joint nancing. Further, if the margin requirement relies on assumed stress scenarios in which the securities are perfectly correlated (e.g., due to predatory trading, as in Brunnermeier and Pedersen (2005)), then the portfolio margin constraint coincides with position-by-position margins. A.2 Funding Requirements for Banks A bank s capital consists of equity capital plus its long-term borrowings (including credit lines secured from commercial banks, alone or in syndicates), reduced by assets that cannot be readily employed (e.g., goodwill, intangible assets, property, equipment, and capital needed for daily operations), and further reduced by uncollateralized loans extended by the bank to others (see e.g., Goldman Sachs 2003 Annual Report). Banks also raise money using short-term uncollateralized loans such as commercial paper and promissory notes, and, in the case of commercial banks, demand deposits. These sources of nancing cannot, however, be relied on in times of funding crisis since lenders may be unwilling to continue lending, and therefore this short-term funding is often not included in measures of capital. The nancing of a bank s trading activity is largely based on collateralized borrowing. 37 Banks can nance long positions using collateralized borrowing from corporations, other banks, insurance companies, and the Federal Reserve Bank, and can borrow securities to shortsell from, for instance, mutual funds and pension funds. These transactions typically require margins which must be nanced by the bank s capital as captured by the funding constraint (4). The nancing of a bank s proprietary trading is more complicated than that of a hedge fund, however. For instance, banks may negotiate zero margins with certain counterparties, and banks can often sell short shares held in-house, that is, held in a customer s margin account (in street name ) such that the bank does not need to use capital to borrow the shares externally. Further, a bank receives margins when nancing hedge funds (i.e., the margin is negative from the point of view of the bank). However, often the bank wants to pass on the trade to an exchange or another counterparty and hence has to pay a margin to the exchange. In spite of these caveats, we believe that in times of stress, banks face margin requirements and are ultimately subject to a funding constraint in the spirit of (4). For instance, Goldman Sachs 2003 Annual Report (page 62) states that it seeks to maintain net capital in excess of total margins and haircuts that it would face in periods of market stress plus the total draws on unfunded commitments at such times. In addition, Goldman Sachs recognizes that it may not have access to short-term borrowing during a crisis, that margins and haircuts may increase during such a crisis, and that counterparties may withdraw funds at such times. Banks must also satisfy certain regulatory requirements. Commercial banks are subject to the Basel accord, supervised by the Federal Reserve system for US banks. In short, the Basel accord of 1988 requires that a bank s eligible capital exceeds 8% of the risk-weighted asset holdings, which is the sum of each asset holding multiplied by its risk weight. The risk weight is 0% for cash and government securities, 50% for mortgage-backed loans, and 100% for all other assets. The requirement posed by the 1988 Basel accord corresponds to Equation (4) with margins of 0%, 4%, and 8%, respectively. In 1996, the accord was 38 amended, allowing banks to measure market risk using an internal model based on portfolio VaRs rather than using standardized risk weights. US broker-speculators, including banks acting as such, are subject to the Securities and Exchange Commission s (SEC) s net capital rule (SEC Rule 15c3-1). This rule stipulates, among other things, that a broker must have a minimum net capital, which is de ned as equity capital plus approved subordinate liabilities minus securities haircuts and operational charges. The haircuts are set as security-dependent percentages of the market value. The standard rule requires that the net capital exceeds at least 6 2 % (15:1 leverage) of ag3 gregate indebtedness (broker s total money liabilities) or alternatively 2% of aggregate debit items arising from customer transactions. This constraint is similar in spirit to (4).16 As of August 20, 2004, SEC amended the net capital rule for Consolidated Supervised Entities (CSE) s such that CSE s may, under certain circumstances, use their internal risk models to determine whether they ful ll their capital requirement (SEC Release No. 34-49830). A.3 Funding Requirements for Market Makers There are various types of market-making rms. Some are small partnerships, whereas others are parts of large investment banks. The small rms are nanced in a similar way to hedge funds in that they rely primarily on collateralized nancing; the funding of banks was described in Section A.2. Certain market makers, such as NYSE specialists, have an obligation to make a market and a binding funding constraint means that they cannot ful ll this requirement. Hence, avoiding the funding constraint is especially crucial for such market makers. Market makers are in principle subject to the SEC s net capital rule (described in Section A.2), but this rule has special exceptions for market makers. Hence, market makers main regulatory requirements are those imposed by the exchange on which they operate. These constraints are often similar in spirit to (4). Let L be the lower of 6 2 % of total indebtedness or 2% of debit items and hj the haircut for security j; 3 P P then the rule requires that L W j hj xj , that is, j hj xj W L. 16 39 B Appendix: Proofs Proof of Propositions 1 and 2. These results follow from the calculations in the text. Proof of Proposition 3. We prove the proposition for Z1 > 0, implying p1 v1 and x1 0. The complementary case is analogous. To see how the equilibrium depends on the exogenous shocks, we rst combine the equilibrium condition x1 = speculator funding constraint to get m+ Z1 1 that is, G(p1 ) := m+ Z1 1 2 (v1 p1 ) p1 x0 b0 1 ( 2 )2 (32) 2 (v1 p1 ) ( 2 )2 1 k k=0 y1 with the b0 + p1 x0 + 1 (31) For 1 large enough, this inequality is satis ed for p1 = v1 , that is, it is a stable equilibrium that the market is perfectly liquid. For 1 low enough, the inequality is violated for p1 = 2v1 ( 2 )2 Z1 , that is, it is an equilibrium that the speculator is in default. We are interested in intermediate values of 1 . If the left-hand side G of (32) is increasing in p1 then p1 is a continuously increasing function of 1 , implying no fragility with respect to 1 . Fragility arises if G can be decreasing in p1 . Intuitively, this expression measures speculator funding needs at the equilibrium position, and fragility arises if the funding need is greater when prices are lower, that is, further from fundamentals. (This can be shown to be equivalent to a non-monotonic excess demand function.) When the nancier is informed, the left-hand side G of (32) is ( + | v1 | + p1 v1 ) Z1 + 2 (p1 v1 ) p1 x0 b0 ( 2 )2 (33) The rst product is a product of two positive increasing functions of p1 , but the second term, 40 p0 x0 , is decreasing in p1 if x0 > 0. Since the rst term does not depend on x0 , there exists x such that, for x0 > x, the whole expression is decreasing. When the nancier is uninformed, we rst show that there is fragility for a = 0. In this case, the left-hand side of (32) is G0 (p1 ) := ( + | p1 |) Z1 + 2 (p1 v1 ) p1 x0 b0 ( 2 )2 (34) When p1 < p0 , | p1 | = (p0 p1 ) decreases in p1 and, if is large enough, this can make the entire expression decreasing. (Since is proportional to , this clearly translates directly to .) Also, the expression is decreasing if x0 is large enough. Finally, on any compact set of prices, the margin function converges uniformly to (22) as a approaches 0. Hence, G converges uniformly to G0 . Since the limit function G0 has a decreasing part, choose pa < pb such that := G0 (pa ) G0 (pb ) > 0. By uniform convergence, 1 1 1 1 choose a > 0 such that for a < a, G di ers from G0 by at most /3. Then we have G(pa ) G(pb ) = G0 (pa ) G0 (pb ) + [G(pa ) G0 (pa )] [G(pb ) G0 (pb )] 1 1 1 1 1 1 1 1 = >0 33 3 (35) (36) which proves that G has a decreasing part. It can be shown that the price cannot be chosen continuously in 1 when the left-hand side of (32) can be decreasing. Proof of Proposition 4. When the funding constraint binds, we use the implicit function theorem to compute the derivatives. With y1 is given by (12), the equilibrium condition x1 = y1 , the fact that the speculator funding constraint binds in an illiquid equilibrium, and that v1 p1 > 0 when Z1 > 0, we have m+ Z 1 1 2 (v1 p1 ) ( 2 )2 41 = b0 + p1 x0 + 1 . (37) We di erentiate this expression to get m+ p1 1 p1 1 2 2 p1 p1 (v p1 ) + m+ = x0 + 1 , 1 21 2 ( 2 ) ( 2 ) 1 1 Z1 (38) which leads to Equation (23) after rearranging. The case of Z1 < 0 (i.e., Equation (24)) is analogous. Finally, spiral e ects happen if one of the last two terms in the denominator of the righthand side of Equations (23) and (24) is negative. (The total value of the denominator is positive by de nition of a stable equilibrium.) When the speculator is informed, and m 1 p1 m+ 1 p1 =1 = 1 using Proposition 1. Hence, in this case margins are stabilizing. m+ 1 1 m 1 p1 m 1 1 If the speculators are uninformed and a approaches 0, then using Proposition 2 we nd that m+ 1 p1 = approaches < 0 for v1 v0 + 1 0 < 0 and = approaches > 0 for v1 v0 + 1 0 > 0. This means that there is a margin spiral with positive probability. The case of a loss spiral is immediately seen to depend on the sign on x0 . Proof of Proposition 5. We rst consider the equation that characterizes a constrained j equilibrium. When there is selling pressure from customers, Z1 > 0, it holds that j ( 2 )2 j Z1 } , 2 j | j | = j = v1 pj = min{ 1 mj+ , 1 1 1 1 (39) j and if customers are buying, Z1 < 0, we have j ( 2 )2 j ( Z1 ) } . 2 j | j | = j = pj v1 = min{ 1 mj , 1 1 1 1 (40) We insert the equilibrium condition xj = 1 j,k k y1 j,k and Equation (12) for y1 into the 42 speculators funding condition to get mj+ 1 j Z1 > j+ 2 1 m1 j ( 2 )2 j Z1 2 1 mj+ 1 j ( 2 )2 + j Z1 > j 2 1 m1 j ( 2 )2 mj 1 j Z1 2 1 mj 1 j ( 2 )2 = j xj pj + b0 + 1 01 (41) where the margins are evaluated at the prices solving (39) (40). When 1 approaches in nity, the left-hand side of (41) becomes zero, and when 1 approaches zero, the left-hand side approaches the capital needed to make the market perfectly liquid. As in the case of one security, there can be multiple equilibria and fragility (Proposition 3). On a stable equilibrium branch, 1 increases as 1 decreases. 1 J Of course, the equilibrium shadow cost of capital 1 is random since 1 , v1 , . . . , v1 are random. To see the commonality in liquidity, we note that | j | is increasing in 1 for each j j = k, l. To see this, consider rst the case Z1 > 0. When the nanciers are uninformed, a = 0, and j = 0, then mj+ = k , and, therefore, Equation (39) shows directly that | j | 1 1 increases in 1 (since the minimum of increasing functions is increasing). When nanciers are informed and j = 0 then mj+ = k + j , and, therefore, Equation (39) can be solved to 1 1 1 be | j | = min{ 1+ 1 j , 1 k ( 2 )2 k 2 Z1 }, which increases in 1 . Similarly, Equation (40) shows that j | j | is increasing in 1 when Z1 < 0. Now, since | j | is increasing in 1 and does not depend on other state variables under these conditions, Cov | k ( )|, | l ( )| > 0 because any two functions which are both increasing in the same random variable are positively correlated (the proof of this is similar to that of Lemma 1 below). Since | j | is bounded, we can use dominated convergence to establish the existence of c > 0 such that part (i) of the proposition applies for any j , a < c. To see part (ii) of the proposition, note that, for all j, | j | is a continuous function of 1 , which is locally insensitive to 1 if and only if the speculator is not marginal on security j (i.e., if the second term in Equation (39) or (40) attains the minimum). Hence, | j | jumps if and only if 1 jumps. 43 To see part (iii), we write illiquidity using Equations (39) (40) as | j | = min{ 1 m1 1 j j,sign(Z1 ) , j ( 2 )2 j |Z1 | } . 2 (42) Hence, using the expression for the margin, if the nancier is uninformed and j = a = 0, then | j | = min{ 1 1 , j 1 and, if the nancier is informed and j = 0, then | j | = min{ 1 j 1 ( 2 )2 j 1 , j |Z1 | } . 1 + 1 2 j ( 2 )2 j |Z1 | } 2 (43) (44) In the case of an uninformed nancier as in (43), we see that, if xk = 0, 1 | k | = 1 1 > 1 1 | l | k l 1 1 k l and, if |Z1 | |Z1 |, k l ( 2 )2 k ( 2 )2 l |Z1 | } > min{ 1 1 , l |Z1 | } = | l | . 1 2 2 (45) | k | = min{ 1 1 , k 1 (46) Since k and l converge to these values as j , a approach zero, we can choose c so that inequality holds for j , a below c. With an informed nancier, it is seen that | k | | l | 1 1 using similar arguments. For part (iv) of the proposition, we use that | j | 1 | j | 1 1 = ( 1 ) 1 ( 1 ) Further, 1 ( 1 ) (47) | k | 1 1 | l | 1 1 . 0 and, from Equations (43) (44), we see that The result that Cov( k , ) Cov( l , ) now follows from Lemma 1 below. Lemma 1 Let X be a random variable and gi , i = 1, 2, be weakly increasing functions X, 44 where g1 has a larger derivative than g2 , that is, g1 (x) g2 (x) for all x and g1 (x) > g2 (x) on a set with non-zero measure. Then, Cov[X, g1 (X)] > Cov[X, g2 (X)] Proof. For i = 1, 2 we have (48) Cov[X, gi (X)] = E [(X E[X])gi (X)] X (49) gi (y)dy) . (50) = E (X E[X])( E[X] The latter expression is a product of two terms that always have the same sign. Hence, this is higher if gi is larger. Liquidity Risk (Time 0). Section 5 focuses on the case of speculators who are uncon- strained at t = 0. 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