Equilibrium theory of stock market crashes

Journal of Economic Dynamics & Control 60 (2015) 73–94

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Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Equilibrium theory of stock market crashes$ Sergey Isaenko n The John Molson School of Business, Concordia University, 1455 de Maisonneuve Blvd. W., Montreal, Quebec, Canada H3G 1M8

a r t i c l e i n f o

abstract

Article history: Received 29 August 2014 Received in revised form 10 August 2015 Accepted 13 August 2015 Available online 28 August 2015

We consider an equilibrium in illiquid stock market in which liquidity suppliers trade with investors and face significant trading costs. A similar situation was observed during the recent financial crisis. We find that the expected risk premium on the stock and its Sharpe ratio are positive and very large, while the expected stock return volatility is a few times bigger than in the liquid market. Investors sell stock shares due to their excessive leverage, whereas market makers try to compensate their trading costs with the profits expected from buying the stock shares with very high Sharpe ratio. Moreover, the short-term stock returns exhibit either a strong overreaction or a momentum effect depending on the state of the economy. & 2015 Elsevier B.V. All rights reserved.

JEL classification: G11 G12 Keywords: General equilibrium Financial crisis Liquidity Transaction costs

1. Introduction It has been well documented that selling pressure from investors in the stock market rises during financial crises, while the expected volatility of the stock market returns undergoes a substantial increase.1 Moreover, despite the realized decrease of the stock market price during financial crises, its expected return and the conditional Sharpe ratio rise dramatically during these times.2 The goal of this paper is to show that these stylized facts can be explained in a general equilibrium framework without assuming irrationality of investors. Our explanation is based on the assumption that liquidity providers suffer significant trading costs during financial distresses. To be general we assume that a liquidity provider could be a specialist, a dealer, a financial institution, or an individual investor, all of which we call market makers. A market maker trades with investors and supplies liquidity by either closing all transactions or placing limit orders. Consequently, market makers face losses due to opportunity costs, as well as inventory and search costs. While these costs are not significant during normal times, they become dramatically aggravated during financial crises.3 Moreover, market

☆ Earlier versions of this paper were circulated under the titles “Dynamic Equilibrium in Economy with Illiquid Stock Market” (2007) and “Illiquidity and Equilibrium Stock Returns” (2010). n Tel.: þ 1 514 848 2424x2797; fax: þ1 514 848 4500. 1 For example, the CBOE S&P 500 implied volatility index (VIX), which is used to measure short-term expected volatility of the stock market returns increased by about three times during the financial crisis of 2008. 2 See, for example, Graham and Harvey (2010). 3 The losses of market makers during a recent financial crisis were especially high in the OTC markets. Many market makers bought an excessive number of illiquid securities, which they had hard time selling. Some dealers continued buying risky assets and borrowing, while being close to their

http://dx.doi.org/10.1016/j.jedc.2015.08.004 0165-1889/& 2015 Elsevier B.V. All rights reserved.

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makers often finance their liquidity supply by borrowing, which can become problematic during crises and cause further deterioration of their solvency.4 On the opposite trading side, investors post market orders and also face increased transaction costs in terms of higher bid–ask spreads. Given the numerous stories of dramatic trading costs of market makers during financial turmoil, we assume that these costs are bigger than those of the investors that we neglect for the purpose of tractability.5 Facing significant transaction costs makes market makers less willing to trade. On the other hand, investors, who face smaller trading costs, have strong incentives to trade. Therefore, the prices have to be substantially adjusted to make market makers willing to trade more. We find that the risk premium, the expected Sharpe ratio, and the return volatility of the stock market behave in the same way in which they are observed during financial crises. For tractability, we ignore a microstructure of trading and devise losses of market makers on providing liquidity in the stock market by using exogenous convex transaction costs. Convexity of the costs is intuitive in illiquid market: as the size of the transaction increases, the market maker closes a deal with more difficulty, and, therefore his losses per share of providing liquidity increases with the size of a trade. Moreover, because we treat economy in continuous time, the convexity in the number of stock shares is converted into the convexity in the rate of share trading. Consequently, a market maker endogenously chooses to delay his investments and is not able to adjust his allocations quickly enough in response to systematic shocks. It follows that convex transaction costs result in a slow capital movement and could be used to model the opportunity costs of not transacting, investment delay costs and inventory and search costs. Moreover, assuming selffinancing portfolios, the convex costs could be related to the losses due to the borrowing limits faced by market makers.6 Furthermore, Duffie (2010) argues that a slow capital movement could be the reason for price distortions observed during the recent financial crisis. Our model confirms this expectation by linking capital inertia with mispricings of securities. Note that convex costs have been used to model illiquidity and trading delays by both practitioners7 and academia researchers.8 Investors and market makers behave competitively. Moreover, investors could be of the two types: those who have logarithmic preferences or those who have exponential preferences. The exponential investors can sustain large negative shocks to the stock market price and allow logarithmic investors to avoid default through risk sharing. Most investors have logarithmic preferences while investors with exponential preferences are introduced for the technical reasons so that equilibrium will exist at all times. Overall, logarithmic investors comprise a marginal agent in the economy who defines the behavior of stock prices in most states of the economy. The stock market is considered to be less liquid during financial turmoils than during regular times. Therefore, we call our economy illiquid, where illiquidity is defined by the presence of trading costs. As a benchmark case, we also consider an economy with a perfectly liquid stock market (with no trading costs). Investors and market makers in our economy trade for the purpose of risk sharing. Market makers face significant transaction costs and are not able to quickly unwind the positions of investors. We find that the illiquid economy could be in states in which investors have very long positions in the stock market. Investors are willing to hold these positions only if the stock market has a large risk premium and Sharpe ratio. In these states investors sell stock shares to market makers and buy liquid T-bills. On the other side of trades, market makers have small positions in the stock market. They are willing to be underexposed to securities with abnormally high expected Sharpe ratios, since buying them quickly is too costly. Market makers partly compensate their risk underexposure by buying the stock market that is expected to be underpriced. Note that investors do not become excessively leveraged in a perfectly liquid stock market since they can immediately adjust their allocations by trading with market makers. If the aggregate investor is sufficiently wealthy, and has significantly long or short positions in the stock, then stock return volatility becomes very large. This happens because of the wealth effect on this investor. If his position is long (short) and becomes longer (shorter), then the stock price decreases (increases), causing the aggregate investor to lose money on his current position in the stock market. The losses make the aggregate investor sell stock shares when stock price is falling and buy them when the stock price is rising. Therefore, he further decreases the stock price when it falls or increases it when it rises, causing an increase in stock return volatility. These results explain the increase in stock return volatility observed during the financial crises when volatility jumped to at least twice the level it was before these distresses. Notice that in a liquid market the wealth effect from the aggregate investor is offset by a substitution effect from market makers that makes the stock return volatility remain approximately the same across all states of the economy. The wealth effect on the

(footnote continued) capacity, and some even withdrew from the markets. See, for example, “Markets: Exchange or Over-the-Counter” by Randall Dodd at http://www.imf.org see also Choi and Shachar (2013) and Han and Wang (2014). 4 In 2008, there was the equivalent of a bank run on the money market funds, which frequently invest in commercial paper issued by corporations to fund their operations. The TED spread, an indicator of perceived credit risk in the general economy, spiked in July 2007, remained volatile for a year, before spiking even higher in September 2008, reaching a record 4.65% on October 10, 2008. Nearly one-third of the U.S. lending mechanism was frozen and continued to be frozen until at least June 2009. 5 Even though the median quoted bid–ask spread of S&P 500 increased during financial crisis of 2007–2009, it did so by less than two times and the total quoted spread remained quite low. See, for example, Angel et al. (2011). The quoted bid–ask spread in the OTC markets increases more significantly during the crisis but was still reasonably low. See, for example, “Got Liquidity?” at https://www.blackrock.com. 6 For a self-financed portfolio, the value of traded stocks must be equal to the negative value of traded T-bills. Therefore, if an investor cannot borrow fast, she will not be able to buy enough stock shares. 7 For example, see Grinold and Kahn (2000). 8 See Heaton and Lucas (1996), Isaenko (2010), Rogers and Singh (2010), Garleanu and Pedersen (2013), and Vayanos and Woolley (2013).

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aggregate investor becomes dominant in an illiquid market because trading by market makers is substantially limited by the presence of the transaction costs. It follows that our model predicts the states of the economy that are observed during financial crises: if investors happen to have excessively long positions in the stock market, then they aggressively sell their shares to market makers, the stock market is expected to be very volatile, while the expected stock returns and Sharpe ratio significantly increase. Market makers face significant trading costs which they plan to compensate by buying the stock that is expected to grow. The leverage of market makers can increase or decrease during a crisis depending on their precrisis leverage. Besides states in which the conditional Sharpe ratio and the expected returns are very high, our model also predicts the possibility of states in which the stock market has significantly negative conditional returns and the Sharpe ratio. The conditional returns on the stock market are negative when investors are short in this market. A situation such as this cannot exist in a liquid economy in which risk premium is positive in every state of the economy and an investor is always long in a stock market. It follows that the introduction of stock market illiquidity in the economy does not have to result in higher conditional premiums for holding the stock market. Market makers are willing to keep long positions in the stock market that is expected to fall since selling their shares is too expensive. A possibility of a negative liquidity premium is also manifested in Vayanos (1998) and Longstaff (2009). These premiums are, however, too small in magnitude to result in a negative risk premium. For a better intuition on stock returns in illiquid market, we analyze the short-term predictability of stock market returns. We find that the autocorrelation coefficient of short-term stock returns in a liquid economy is negative, and its magnitude does not exceed a small fraction of a percent. On the other hand, its value in an illiquid economy could be a few hundred times larger than it is in a liquid economy. It is negative for the most states and is highly significant where the wealth effect is strong. However, if the risk of the stock market allocation by the aggregate investor is excessive, then the autocorrelation coefficient becomes positive. The stock returns exhibit the momentum effect because the marginal stock premium with respect to the wealth of the aggregate investor is positive if he is very long in the stock market and negative if he is very short. We provide comparative statics by changing the calibration that affects stock market liquidity. We find that decreasing the transaction costs results in a weakening of the wealth effect and therefore in a smaller deviation of the stock price from the one when the stock market is perfectly liquid (we call this deviation “mispricing”). Conversely, a longer horizon for market illiquidity makes mispricing more pronounced. Furthermore, a smaller percentage of investors in the economy make the stock market less liquid, resulting in a significantly stronger mispricing in all states of the economy. Our study is related to recent papers by He and Krishnamurthy (2012, 2013). The authors model a financial crisis in a general equilibrium in which specialists trade risky assets for households and face funding illiquidity rising from contracts imposed by households. In the presence of a negative shock to his wealth, a specialist becomes excessively leveraged and requires a significant increase in the expected stock returns and Sharpe ratio of risky assets. He and Krishnamurthy (2012) show that stock return volatility increases during crises. However, the biggest increase in volatility they find is less than 10% of its value in a normal state of the stock market. Our model combines the leverage effect and the wealth effect. Both effects contribute to a dramatic increase of the risk premium and the Sharpe ratio while the wealth effect causes the volatility to spike. Moreover, in the models of He and Krishnamurthy, a specialist is a marginal agent in the economy, while in our model the marginal agent is an aggregate investor that trades with an aggregate liquidity provider. Our model complements that of He and Krishnamurthy. It suggests that besides the funding illiquidity faced by specialists, the behavior of the stock market observed during financial crises can also follow from trading illiquidity faced by liquidity providers. Funding liquidity as a source for significant mispricing was also studied by Garleanu and Pedersen (2011), who model funding limits by means of margin constraints. The authors report a strong impact of margin constraint on the risk premia and Sharpe ratios. Furthermore, in a related paper, Brunnermeier and Sannikov (2014) study equilibrium in the production economy with financial frictions and link intermediaries' financing position to risk premia and volatility of stock returns. Another important paper related to ours is that of Xiong (2001).9 Xiong considers dynamic equilibrium in the presence of long-term investors, convergence traders, and noise traders where a risky investment is a spread between two mispriced securities. The author shows that convergence traders can increase the volatility of mispricing in states in which their wealth is significant and the supply of the spread substantially fluctuates from its mean value. Xiong (2001) explains the increased volatility of stock mispricing and its conditional Sharpe ratio using wealth and substitution effects. Our paper complements Xiong (2001) along a few dimensions. First, we construct an equilibrium in which all agents are rational. Second, we provide a new economic intuition about the nature of trading strategies and mispricing of stock returns. Finally, we avoid the critique of Loewestein and Willard (2006), who show that if unlimited asset liability and storage technology in a perfectly elastic supply used by Xiong (2001) are replaced by a locally riskless bond market in which the external supply of bonds is zero, then mispricing vanishes. Mispricing is present in our model even though the bond market clears at zero external supply.10

9

See also Kyle and Xiong (2001). The impact of the wealth effect on prices is also studied by Kondor and Vayanos (2014), who develop the model in which hedgers trade multiple risky assets with arbitrageurs. 10

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Modeling stock illiquidity using transaction costs is common. Moreover, most existing studies of the effects of transactions costs on asset prices assume that these costs are linear in the number of traded shares or in their dollar value.11 As argued by Duffie (2010), proportional transaction costs cannot result in slow movement of capital. Therefore, the contribution of our study does not overlap with these papers. Overall, our paper adds to the growing theoretical literature analyzing illiquid markets in a general equilibrium framework. In addition to the above-cited references, important contributions include but are not limited to Amihud and Mendelson (1986), Huang (2003), Eisfeldt (2004), Acharya and Pedersen (2005), Brunnermeier and Pedersen (2005), Duffie et al. (2005, 2007), Vayanos and Wang (2007), Weill (2008), Brunnermeier and Pedersen (2009), Huang and Wang (2009), Lagos et al. (2011), and Cespa and Foucault (2014). The remainder of this paper is organized as follows. Section 2 describes the model. Section 3 presents equilibrium in an economy with a liquid stock market. Section 4 analyzes an equilibrium in the economy with an illiquid stock market. Section 5 concludes. Appendix A derives partial differential equations (PDEs) for value functions of two investors and a market maker; Appendix B finds the PDE for the stock price in an illiquid market; Appendix C describes the numerical algorithm for finding an equilibrium; Appendix D outlines the solution for an equilibrium without transaction costs; and Appendix E provides details for finding the autocorrelation coefficient. 2. Economic setting We consider a Markovian economy with a finite horizon T. We assume a filtered probability space (Ω; F ; fF t g; Q ), with uncertainty in the model being generated by a standard one-dimensional Brownian motion Wt, which is also adapted. 2.1. Financial securities We consider an economy in which trading takes place continuously between time zero and time T. The traded securities are claims to a risky asset (a stock market), which is in positive net supply of one, and a locally risk-free security (a bond), in zero supply, whose terminal payoff at T is always one unit of the consumption good. There is no consumption until time T, and there are no consumption goods until time T. The risky asset will not pay any dividend before time T, and at time T, each unit of a risky asset will pay out a lump sum dividend at an amount of DT (in units of consumption good). The lump sum dividend provides a consumption good at time T. The view of DT at time t is given by Dt, following the process dDt ¼ μD dt þ σ D dW t ;

ð1Þ

where μD and σD are constants and σD is positive. By definition, the price of the locally safe asset is always equal to one. The price of the risky asset is expressed in units of the price of the safe asset and follows dSt ¼ μSt dt þ σ St dW t ;

ð2Þ

where σ S 4 0, μS is a conditional dollar risk premium for holding the stock market per time-interval dt, which we will simply call conditional risk premium, and σS is a conditional standard deviation of a dollar return per time period dt, which we will call conditional standard deviation of the stock return or stock return volatility.12 2.2. Economic agents We assume that there are many individual economic agents of two types: market makers and investors. Investors trade with market makers that provide liquidity in the market.13 A market maker could be a specialist, a dealer, an individual investor, or a financial institution. Any individual economic agent behaves competitively. If the stock market is illiquid, then a market maker needs significant time and effort to find a buyer or seller of shares. Moreover, the more shares an investor is willing to trade within a given time interval, the bigger the efforts and costs per share a market maker must face. Consequently, a market maker is not able to trade shares fast and becomes suboptimal in his allocations. In a rational expectation equilibrium, the limited ability of market makers to provide liquidity is compensated by the adjustment in prices, making investors willing to trade less. Moreover, as a compensation for the time and risks, a market maker earns a spread between prices paid and those received by an investor. This spread is not nearly enough to compensate a market maker for providing liquidity during financial turmoils and will be neglected for the purpose of tractability. Investors trade self-financing portfolios and could be of the two types: those who have logarithmic utility function and those who have exponential utility function. Assuming CRRA preferences for investors of the first type introduces the wealth effect into economy. The wealth effect makes investors sell (buy) stock market shares when this market is falling (rising) and 11

See, for example, Vayanos (1998), Acharya and Pedersen (2005), and Buss and Dumas (2013). The dividend view, Dt, may take negative values, implying that the stock price also may be negative. This inconsistency with practice is not economically important since the intuition behind the strategies are based on returns. 13 An investor requires liquidity by placing market orders. A market maker provides liquidity by placing a limit order, finding another investor who is willing to close the transaction, or closing the transaction himself. 12

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plays an important role in a real economy. Investors of the second type can absorb negative shocks to the stock market price and therefore provide a channel for logarithmic investors to maintain their solvency through risk sharing. Note that the exponential investors' role in the model is purely technical. They allow the mathematical solution of the equilibrium to exists when logarithmic investors are poor. It will be confirmed that investors are marginal agents in the economy and the preferences of market makers have a rather small influence on the equilibrium prices.14 Therefore, we assume an exponential utility function for market makers, allowing us to reduce the number of state variables in the economy. Furthermore, the proportion of logarithmic investors in the economy is denoted by λ1, the proportion of exponential investors is λ2, and the proportion of market makers is 1  λ1  λ2 . For a better understanding of losses faced by market makers, let us first assume that trades take place at fixed discrete times 0; …; t k ; t k þ 1 ; …; T. If a market maker has to trade jΔN k j shares at time tk, he will face liquidity costs per share, which increase with the size of a trade and are given by α^ k jΔN k jεk þ 1 ; where α^ k ; and εk are positive numbers. Now let us assume that a market maker trades continuously in time at a finite rate of trading, so that the number of shares that he trades from time tk to t k þ 1 is jΔNk j  juk jΔt k , where uk is an average rate of trading during this time-interval and Δt k ¼ t k þ 1 t k . Therefore, the cost for a market maker from time tk to t k þ 1 is αk juk j1 þ εk Δt k , where αk ¼ α^ k ðΔt k Þεk . Furthermore, we assume that αk and εk are independent from Δt k and from the state of economy k. The former assumption establishes the connection between continuous- and discrete-time frameworks.15 It follows that a market maker faces the trading costs αjuj1 þ ε Δt within time-interval Δt. For simplicity of exposition, we also require that coefficient α be the same for buying and selling shares. The convexity of the transaction costs does not allow the rate of a market maker's trading to be infinite. Naturally, his share holding N becomes absolutely continuous and is given by dN t ¼ ut dt:

ð3Þ

We conclude that market makers change their allocations slowly (at a finite rate). The convex trading costs are appealing since they endogenize a finite rate of trading. The given type of trading may result not only from opportunity costs, inventory, or search costs but also from other market frictions. For example, market makers may trade stock shares slowly due to the presence of borrowing limits. Indeed, the self-financing condition suggests that dBt ¼ St dN t , where B is the number of locally risky securities. The last equality implies that if borrowing can be done only slowly so that dBt ¼ uBt dt, where uB is finite, then trading of stock shares also can be done only at a finite rate. Finally, we notice that it has been argued by Duffie (2010) that slow movement of capital could be the source of price disturbances observed during financial crises. The predictions of our model presented below agree with this expectation. 2.3. Formulations of portfolio choice problems In this subsection we describe portfolio choice problems faced by investors of the two types and by market makers. An investor of the first type maximizes his expected logarithmic utility function of the terminal wealth subject to the dynamic budget constraint: max

Ni1 A R

E0 lnðX i1T Þ;

ð4Þ

dX 1t ¼ X i1t π i1t ðμSt dt þ σ St dW t Þ; i

ð5Þ

X i1t ¼ Ni1t St þ Bi1t ; i X1

ð6Þ

where denotes an investor's wealth, is the number of shares that he holds, π and represents his investment in the locally risk-free security. An investor of the second type maximizes his expected exponential utility function of the terminal wealth subject to the dynamic budget constraint: 
 1 ð7Þ max E0  exp  γ 2 X i2T ; Ni2 A R

N i1t

i 1

¼ Ni1 =X i1 ,

Bi1t

γ2

dX 2t ¼ N i2t μSt dt þ Ni2t σ St dW t ; i

ð8Þ

X i2t ¼ Ni2t St þ Bi2t ;

ð9Þ

where γ2 is a coefficient of absolute risk aversion, is the wealth of an investor of the second type, of shares that he holds, and Bi2t represents his investment in the locally risk-free security. i X2

14

Ni2t

denotes the number

See Section 4.3 in which we find that the coefficient of risk aversion of market makers has a very small impact on prices in the illiquid economy. Once the dynamic portfolio decisions by a market maker are found in the continuous time, one can easily adopt them to discrete-time trading by setting ΔN k ¼ uk Δt k and α^ k ¼ α=ðΔt k Þε . 15

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A market maker maximizes the expected CARA utility function of his terminal wealth subject to his dynamic budget constraint:     1 m E  exp  γ X max ; ð10Þ 0 3 3T m

γ3

u3 A R

m 1þε Þ dt þ Nm dX 3t ¼ ðN m 3t μSt  αju3t j 3t σ St dW t ; m

ð11Þ

m

ð12Þ

dN 3t ¼ um 3t dt; m m Xm 3t ¼ N 3t St þ B3t ;

ð13Þ

where γ3 is a coefficient of absolute risk aversion and and stand for a market maker's rate of trading, wealth, number of stock shares, and bonds, respectively. For tractability, we assume that each market maker has the same initial allocation to the stock market. It will follow that each market maker has identical allocation to the stock market given by m Nm 3t ¼ N 3t ; 8 t A ½0; T and the same rate of trading given by u3t ¼ u3t ; 8 t A ½0; T. The above portfolio selection problems can be solved only in the dynamic programming approach, which relies on specification of state variables for each agent. The list of state variables for an agent includes his wealth and the state variables describing the aggregate state of the economy. Moreover, we can remove the stock allocation of an individual m market maker, N3 , from the list of his state variables since allocations across market makers are all the same. It will be shown in the next subsection that allocations of a logarithmic investor depend on his wealth. The stock market clearing condition implies that the list of macro-state variables should depend on an average wealth of logarithmic investors, X1.16 In addition, this list should also include variables D; N3 , and Y, where Y is the cumulative transaction costs of the aggregate market maker.17,18 However, because the conditional moments of the stock returns do not depend on Y explicitly and the indirect utility function of a market maker allows separation of his wealth (see below), Y becomes a redundant state variable in the economy. For the same reason, D is a redundant state variable for a market maker and for investors. um 3;

Xm 3;

Nm 3;

m B3

2.4. Portfolio choices Given the definitions of state variables in the economy, one can proceed with the solution for portfolio choice problems. We consider portfolio choice problems faced by investors in Appendix A. We conjecture that the utility function of logarithmic investor can be written as V i1 ðt; X i1 ; X 1 ; N3 Þ ¼ ln X i1 þ g 1 ðt; X 1 ; N3 Þ; where function g1 solves PDE (A.3). This conjecture implies that the weight of the stock market, π logarithmic investors and is given by

π1 ¼

ð14Þ i 1,

is the same for all

μS : σ 2S

ð15Þ

We also conjecture that the indirect utility function of investor 2 can be written as
 i 1 V i2 t; X i2 ; X 1 ; N3 ¼  e  γ 2 X 2 g 2 ðt; X 1 ; N3 Þ;

ð16Þ

γ2

i

where function g2 solves PDE (A.7), We show in Appendix A that the optimal allocation of this investor, N2, is the same for all investors of a given type and is given by   X 1 g 2X 1 μ N2 ¼ S 2 1 þ : ð17Þ g2 γ2σS The subscripts X 1 ; N3 ; D or t denote partial derivatives of a function with respect to these variables. We also use the subscript t with the state variables to show their dependence on time. The dynamic programming problem for a market maker is formulated in Appendix A. Solving this problem will provide m us with his indirect utility function V m 3 ðt; X 3 ; X 1 ; N 3 Þ. We conjecture that     1 m Vm exp  γ 3 X m 3 t; X 3 ; X 1 ; N 3 ¼  3 f ðt; X 1 ; N 3 Þ;

γ3

ð18Þ

We admit that the terminology “aggregate” is not precise here since an aggregate wealth of investors of the first type is λ1 X 1 rather than X1. We note that Y follows dY ¼ ð1  λ1  λ2 Þαju3 j1 þ ε dt: Moreover, the average wealth of exponential investors, X2, and the average wealth of market makers, X3, are not state variables in the economy since the controls of exponential investors and market makers are not affected by the values of their portfolios. 18 Because transaction costs faced by market makers dissipate from the economy, the latter is incomplete and cannot be fully described by one state variable D. Indeed, state variables X 1t ; N 3t ; and Y t depend on the historical realizations of state variable Dt and cannot be expressed as deterministic functions of t and D. 16 17

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m

where function f solves PDE (A.11). We find that the trading rate, u3 , is the same for all market makers and is given by  1=ε   jf N3 j u3 ¼  sgn f N3 : ð19Þ αð1 þ εÞγ 3 f The presence of trading costs forces a market maker into a state in which his expected utility is not maximal. Therefore, a market maker would like to change his allocation to the stock market even in the absence of idiosyncratic shocks. He trades in the direction of a state with a maximal utility and switches the sign of his trading rate in this state. This behavior of market makers is a distinct feature of our model that is intuitive for an illiquid market (see also Longstaff, 2001). 2.5. Equilibrium and its calculation Allocations and portfolio values of logarithmic investors do not have to be the same but must satisfy relation (15). Let N1 be an average allocation across these investors. We describe logarithmic investors using an agent with wealth X1 and allocation N1 that we call aggregate investor 1. Definition 1. An equilibrium is a price system (μS ; σ S ) and a set of trading strategies ðπ 1 ; N2 ; u3 ÞT of investors and market makers such that (i) individual agents choose their optimal portfolio strategies and (ii) the stock market clears, that is, 8 t A ½0; T

λ1 N1t þ λ2 N2t þð1  λ1  λ2 ÞN 3t ¼ 1:

ð20Þ

Note that the bond market clears as soon as the stock market does.19 The dissipation of costs does not allow the existence of a representative agent in our economy. Nonetheless, the economy has a pricing kernel due to the lack of arbitrage opportunities. The latter are absent because of investors who do not face transaction costs. In Appendix B we prove the following: Proposition 1. Assume that PDEs (A.4), (A.8) and (A.12) for functions g 1 ; g 2 ; and f have solutions. Moreover, assume that function hðt; X 1 ; N3 Þ solves PDE:  2 μ 1 μS 0 ¼ ht þ μD  S σ D þu3 hN3 þ X 1 hX 1 X 1 ; hðT; X 1 ; N3 Þ ¼ 0; ð21Þ 2 σS σS where

μS ¼ σ 2S Ω; σS ¼

σD

1  ΩhX 1 X 1

ð22Þ ;

1 ð1  λ1  λ2 ÞN 3   : Ω¼ g λ λ1 X 1 þ 2 1 þ X 1 2X 1 γ2 g2

ð23Þ

ð24Þ

Then the equilibrium stock price, Sðt; D; X 1 ; N 3 Þ, is equal to D þ hðt; X 1 ; N3 Þ. Moreover, equilibrium strategies of investors and market makers are given by  1=ε !T     X 1 g 2X 1 jf N3 j μS μS T ; 1þ : ; sgn f N3 ðπ 1 ; N 2 ; u3 Þ ¼ g2 αð1 þ εÞγ 3 f σ 2S γ 2 σ 2S

We will show that the stock market price in the economy with no trading costs is very close to the expected dividend D. Therefore, function h is a deviation of the stock price from fundamentals or simply the stock market's mispricing. Note that the presence of only logarithmic investors and market makers in the economy makes the former hold stock shares at time T. An unfavorable realization of the dividend payment may cause the wealth of a logarithmic investor to become negative. Therefore, the presence of exponential investors is essential for the existence of equilibrium. The latter allows the logarithmic investors to quickly liquidate their stock positions when their wealth becomes very small and to avoid default. The presence of the two types of investors would be redundant if preferences of investors could support negative wealth and exhibit the desired wealth effect. The equilibrium can be found only numerically by simultaneously solving the PDEs for the functions g 2 ; f , and h.20 The numerical approach for solving these PDEs is described in Appendix C. 19 The bond market clearing condition is λ1 X 1t þ λ2 X 2t þ ð1  λ1  λ2 ÞX 3t ¼ St  Y t , where X2 and X3 are the average values of investors' and market makers' portfolios, respectively. 20 It is assumed without a proof that the numerical results for g 2 ; f , and h converge to the true solutions in the limit of a very small time increment.

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3. Equilibrium without trading costs Let us consider the economy described in the previous section but with no transaction costs for market makers. For the purpose of tractability, we assume that investors of the second type and market makers have the same coefficients of risk aversion. Therefore, only two agents are present in the liquid economy: agent 1 with logarithmic utility function and the weight in the economy of λ1 and agent 2 with exponential utility function and the weight of 1  λ1 . The following proposition describes an equilibrium in this economy and is proven in Appendix D. Proposition 2. Assume that PDEs (D.3) and (D.5) have solutions given by functions νðt; DÞ and X 1 ðt; DÞ, respectively. Assume also that function Sðt; DÞ solves   μ 1 0 ¼ St þ μD  S σ D SD þ σ 2D SDD ; SðT; DÞ ¼ D; ð25Þ 2 σS where μσSS ¼  νDνσ D . Then the stock price in equilibrium is given by function Sðt; DÞ, while allocations of agent 1 and agent 2 are λ1 X 1D N 1 ¼ XS1D and N2 ¼ Sð1D  , respectively.  λ ÞS D 1

D

A state in this economy is completely defined by the current view on the dividend, D, and an assumed value of this view at some time in the past that is hidden in coefficient z0.21 Fig. 1, Panel A, shows the stock price (the thin dashed line), the conditional risk premium and the volatility of the stock returns (the thick solid and dashed lines, respectively), the conditional Sharpe ratio (the thin solid line), and the stock allocations by agent 1, N1 (the thin dotted line), versus the expected dividend, D, for the calibration to be used in the illiquid economy.22 As seen from the panel, the stock price follows the expected dividend very closely, and the volatility of the stock market returns is barely different from σD at any state of the economy. The existence of an equilibrium in this economy hinges on the presence of agent 2, who can sustain large negative shocks to his consumption. If agents expect the dividend to be low and this expectation worsens, then agent 1 reduces his stock holding, N1, to zero, while his wealth approaches zero as well. In these states, the stock market is dominated by agent 2, so that the conditional Sharpe ratio, stock return volatility, and the risk premium converge to γ 2 σ D =ð1  λ1 Þ; σ D , and γ 2 σ 2D =ð1  λ1 Þ, respectively (see Appendix D). The impact on the stock prices from agent 1 becomes significant in states in which the expected dividend is significant. Since the coefficient of absolute risk aversion of this agent decreases with the growing stock price, his share of the stock market increases with a higher expected dividend. In the limit of very large values of the expected dividend, the economy is dominated by agent 1, whose coefficient of absolute risk aversion becomes zero. He takes a very long position in the stock market and becomes much more wealthy than agent 2. Consequently, the conditional risk premium and the conditional Sharpe ratio converge to zero, while the stock return volatility converges to σD, and the allocation of agent 1 approaches 1=λ1 . Fig. 1, Panel A, shows that in any state of the economy both agents have long positions in the stock. Moreover, the stock allocation of agent 1 essentially follows the stock market price. If the dividend rate decreases then agent 1 can buy stock shares since the Sharpe ratio rises. This is referred to as the substitution effect. On the other hand, agent 1 loses money on his current position in the stock market since he has purchased shares at a price higher than today's price. The resulting losses cause the wealth effect that makes agent 1 sell his stock shares. Because agent 1 sells stock shares, we conclude that the wealth effect plays a dominant role for his trading strategy while the substitution effect defines trading of agent 2. It follows that if the stock price experiences a positive shock, then agent 1 buys more stock shares because of the wealth effect. This trading makes the volatility of the stock market returns increase. In spite of this, volatility barely changes, due to the substitution effect from agent 2. Overall, heterogeneity between agents' preferences results in only small variations of the conditional volatility of the stock market returns across different states of the economy. On the other hand, the conditional risk premium undergoes more significant variations, while always remaining positive. Furthermore, our analysis shows that the risk premium and stock holdings of each agent remain positive in all states under various calibrations of the economy, including those with a negative return of the dividend view, μD. We conclude that the risk premium in a liquid economy cannot be negative at any calibration. Moreover, the short initial allocations of market makers (investors) will not sustain in equilibrium and will be immediately adjusted to long ones by means of instant trading and price adjustments. Since the stock holdings of agent 1 and agent 2 are X 1 σμ2S and γ μσS 2 , respectively, the stock clearing condition suggests that 2 S S the allocation of agent 2 is N2 ¼ λ γ X 1þ 1  λ . If the volatility of stock holdings by agent 2, σ N2 t , is defined from 1 2 1 1 dN 2t ¼ μN2 t dt þ σ N2 t dW t then from Itô's lemma we find

σ N2 ¼

γ 2 λ1 X 1 μS =σ S

ðγ 2 λ1 X 1 þ 1  λ1 Þ2

:

ð26Þ

We confirm that the stock holdings of both agents have infinitely large first-order variations. This conclusion is expected for the liquid market since an investor can trade an arbitrarily large number of stock shares very quickly. 21 To be more precise, z0 is uniquely defined by setting D to some value D0 for a given time in the past. We assume that this time occurred before time zero. 22 We set z0 ¼ 0:5 for concreteness. We considered the liquid economy at various values of z0 and confirm that all the conclusions of this section hold qualitatively at any feasible choice of z0.

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Panel A

81

Panel B

2.0 −0.00005 1.5

CorrΔS

1.0

−0.00015

0.5 −0.00025 0.0

−0.5

−0.00035 −1

0

1 D

2

−2

0

2

4

D

Fig. 1. Panel A shows the conditional risk premium, μS (thick solid line), the volatility of the stock market returns, σS (thick dashed line), the stock market price (thin dashed line), the conditional Sharpe ratio (thin solid line), and the stock allocation by agent 1, N1 (thin dotted line) as functions of the expected dividend, D. Panel B depicts the autocorrelation coefficient of stock returns, Corr ΔS , when Δt ¼ 0:02. The stock market is liquid. We assume that γ 2 ¼ γ 3 ¼ 1; μD ¼ 0:07; σ D ¼ 0:2; T ¼ 0:5; λ1 ¼ 0:4; z0 ¼ 0:5, and t ¼ 0.

For the purpose of comparison with an illiquid economy, we also find the instantaneous autocorrelation coefficient between stock returns for the two consecutive time intervals Δt. The coefficient is defined as Covt ðΔSt ; ΔSt þ Δt Þ Corr ΔS ðt Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Var t ðΔSt ÞVar t ðΔSt þ Δt Þ

ð27Þ

where ΔSt ¼ Sðt þ ΔtÞ  SðtÞ and the expectations are conditioned on the information at time t. Lemma 1. If Δt is very small, then Corr ΔS ðt Þ ¼

1 StD þ μD SDD þ SDDD σ 2D 2 Δt: SD

ð28Þ

Eq. (28) is proved in Appendix E. Panel B of Fig. 1 depicts the autocorrelation coefficient of the stock market returns for a time period of 0.02 years. The autocorrelation coefficient of the stock returns is negative and does not exceed 0.05% in magnitude. The short-term predictability of stock returns is negative and very small due to a negative and very weak sensitivity of the conditional risk premium to idiosyncratic shocks: a random increase in the stock price is likely to result in a slightly smaller risk premium, causing a small decrease in the future stock price.

4. Equilibrium with trading costs In this section we show the results of the numerical analysis of an equilibrium in the economy with an illiquid stock market and provide their interpretation. We choose the following calibration of the model: γ 2 ¼ γ 3 ¼ 1; ε ¼ 1; T ¼ 0:5; μD ¼ 0:07; σ D ¼ 0:2; λ1 ¼ 0:4, and λ2 ¼ 0:1. We set T to be rather small because we are interested in an illiquid stock market. Moreover, by setting α to 0.5, we study an economy in which the stock market is significantly illiquid. We will discuss the changes in our results if α decreases. Furthermore, we find the equilibrium in N 3 X 1 -space and show the graphs for the equilibrium processes when time is zero and one of the state variables is fixed. For convenience, the graphical results will be shown versus total holdings of investors, N i ¼ 1  ð1  λ1  λ2 ÞN3 , rather than versus N3. For the same reason, we show the graphs for the rate of trading of investors, ui ¼  ð1  λ1  λ2 Þu3 , rather than for u3. A small weight of investors of the second type results in their impact on the stock prices being relatively small unless investors of the first type have vary high risk exposure, jN 1 j=X 1 . This observation often allows us to neglect the difference between aggregate investor 1 and the aggregate investor representing all investors.

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Panel A

Panel B

0.6

1.0

0.4

0.8

0.2

0.6 0.4

0.0

0.2

−0.2

0.0

−0.4

−0.2 −0.6 −2

−1

0 Ni

1

2

0.0

0.5

1.0

1.5

X1

Fig. 2. This figure shows the stock market premium, μS (thick solid line), the volatility of the stock market returns, σS (thick dashed line), the conditional Sharpe ratio (thin solid line), and investors’ rate of trading, ui (thin dashed line), against stock holdings by investors, Ni (Panel A), and against the wealth of the aggregate investor 1, X1 (Panel B). Panel A assumes that X 1 ¼ 0:5 and Panel B assumes that N i ¼ 1:5. All other parameters are set to α ¼ 0:5; ε ¼ 1; γ 2 ¼ γ 3 ¼ 1; μD ¼ 0:07; σ D ¼ 0:2; T ¼ 0:5; λ1 ¼ 0:4; λ2 ¼ 0:1; and t ¼ 0.

4.1. Strategies and returns Panel A of Fig. 2 depicts the conditional stock market premium and the volatility of the stock market returns (thick solid and dashed lines, respectively), the conditional Sharpe ratio (thin solid line), and investors' rate of trading, ui (thin dashed line), versus investors' allocation, Ni, when the wealth of the aggregate investor 1 is fixed. If investors are excessively long (Ni is large), they are willing to bear the risk only if the conditional risk premium and the conditional Sharpe ratio are high. On the other end of trading, market makers have insufficient positions in the stock market in these states and would like to buy shares. Consequently, the overall direction of the stock trading is to buy for market makers and sell for the aggregate investor. Moreover, because market makers face significant transaction costs, they are willing to have small positions in the stock market when the conditional risk premium and the conditional Sharpe ratio are high. Similarly, if investors are very short, they have excessively risky liabilities and are willing to keep them only if the conditional risk premium and the conditional Sharpe ratio are negative. On the other hand, market makers have excessively long positions in the stock market and want to sell their stock shares. Therefore, market makers sell stocks, while investors buy. Furthermore, market makers tolerate their long positions in the stock with a negative risk premium since adjustments of their stock allocations are very costly to them. Given the above behavior of the conditional stock returns, let us consider in more detail the aggregate investor's trading. Panel A of Fig. 2 shows that if investors are long and state variable Ni increases (or short and Ni decreases), then the risk premium rises (falls). Therefore, investors have an incentive to buy cheap stock shares when they are long or take a shorter position when they are short due to the substitution effect. On the other hand, in both cases the aggregate investor loses money on his current position in the stock market. The resulting losses cause the wealth effect that makes the aggregate investor sell his stock shares when he is long in the stock market or buy them when he is short. Because investors follow this strategy, we conclude that, like in the economy with no trading costs, the wealth effect plays a dominant role for them in the illiquid economy. If investors sell shares when they are excessively long in the stock market, they put pressure on the conditional Sharpe ratio to increase. Similarly, if investors are short or their allocations are insufficient, they buy stock shares and exert pressure on the conditional Sharpe ratio to decrease. Because the described behavior of returns is seen in Panel A of Fig. 2, we conclude that the wealth effect of investors has a dominant influence on the stock returns when they are significantly long or short. As in the liquid market, the wealth effect plays a central role in the trading strategies of the aggregate investor. However, while in the liquid market the resultant impact on the stock prices is offset by the substitution effect from market makers, in the illiquid market the role of market makers in price making is substantially weakened by the transaction costs, and the wealth effect from investors dominates price formation in the stock market.

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It is interesting that the illiquid economy allows states in which the conditional premium for holding stock shares is negative. These states are possible in economy where one of the representative agents is excessively long in the stock market and has difficulties in selling shares. As was shown in Section 3, this situation is not possible in the liquid economy: no matter what the calibration of an economy is, the stock market premium is always positive. That is, even though the fundamentals suggest that the conditional premium for holding stock shares should be positive, it could be negative in the illiquid economy. These conclusions could be surprising since one would normally expect that in the illiquid economy in which market makers have to face trading costs, the conditional premium for holding the stock market should be above that in the liquid economy. The possibility of a negative liquidity premium is also manifested in Vayanos (1998) and Longstaff (2009). However, the nature of the negative risk premium is different across the three cases. In Vayanos (1998) the risk premium is negative because of competition between an investor's incentive to hold fewer shares and the incentive to hold them for a longer time in the presence of transaction costs. In Longstaff (2009), however, the negative the risk premium arises from the willingness of the less patient investor to hold liquid assets and a strong demand for intertemporal risk sharing. Panel A of Fig. 2 shows that the stock market becomes very volatile in states in which investors have very long or very short positions in the stock. Volatility arises because of the wealth effect on investors: if the economy is in a state with a substantially positive (negative) stock market premium, and this premium rises (falls) so that the stock price decreases (increases), then investors sell (buy) more stock shares to make the stock price decrease (increase) further. This strategy causes stock return volatility to increase. This behavior of stock return volatility is quite different from that in the liquid economy, where upward pressure on stock return volatility from logarithmic investors is fully offset by the downward pressure from market makers. A related mechanism for an excessive stock return volatility as suggested in Xiong (2001).23 Panel B of Fig. 2 shows the stock market premium and the volatility of the stock market returns (thick solid and dashed lines, respectively), the conditional Sharpe ratio (thin solid line), and the rate of trading, ui (thin dashed line), versus the wealth of the aggregate investor 1, X1, at fixed positive position of investors, Ni. If aggregate investor 1 becomes poor, his absolute risk aversion increases, making him reduce his risk exposure. Therefore, the stock market is cleared by exponential investors that are willing to take a long position in the stock market only if the conditional Sharpe ratio is large. On the other end of the trading, market makers have low-risk exposure when the conditional Sharpe ratio is high, making them buy shares quickly. The volatility of the stock market returns is relatively small in these states because the impact of the wealth effect from aggregate investor 1 is negligible. Moreover, the stock market premium subsides with smaller wealth of loginvestors due to a diminishing wealth effect but still remains very high due to excessively leveraged exponential investors. The impact of the wealth effect rises as the wealth of log-investors increases from small values and becomes maximal when aggregate investor 1 is neither too rich to dominate the stock market nor too poor to be insignificant in this market. The wealth of aggregate investor 1, at which the wealth effect is maximal, increases with the growing stock position of investors so that they maintain their solvency. Moreover, the maximal strength of this effect increases with the size of this position as well since investors demand a higher compensation for a greater risk exposure.24 Based on the above analysis, we present a scenario related to a financial crisis when the stock market becomes significantly illiquid. As the stock market becomes illiquid, investors want to reduce their stock holdings due to illiquidity (e.g., see Isaenko, 2010) and more pessimistic expectations on stock returns. Therefore, in the illiquid economy investors start from excessively long allocations in the stock market. The extreme risk exposure of investors is compensated by very high expected risk premium and Sharpe ratio of the stock market. Nonetheless, they continuously sell their stock shares due to the wealth effect. Moreover, a strong wealth effect leads to a significant increase in the stock return volatility.25 At the other end of trading, market makers face substantial transaction costs in trying to accommodate a selling pressure from investors. They expect to be compensated from acquiring stock shares that have a very high expected Sharpe ratio. These strategies by investors and market makers, as well as the expected stock returns, are historically observed during dramatic financial distresses when the stock market liquidity plunges. We show that these strategies and stock returns are feasible in a general equilibrium and do not follow from irrational behavior. The cross-section in Ni-dimension is important for understanding the economical intuition behind stock market mispricings, while the cross-section in X1-dimension provides insight into the dynamics of stock returns. Indeed, if stock market illiquidity is high, then time variation in the economy is propelled by the dividend shocks channeled toward the wealth of the aggregate investor. 4.2. Predictability For a better understanding of the price dynamics during financial distress, we consider the short-term predictability of the stock market returns. This predictability is driven by the variations of the risk premium due to changes in the wealth of 23

See also Kondor and Vayanos (2014). The analysis of strategies and returns versus the wealth of aggregate investor 1 when investors are short is a straightforward extension of the analysis we carry for a fixed long position of investors and is not presented. 25 In an alternative scenario one may assume that the economy is illiquid at time zero and investors are long in the stock market but not necessarily excessively (for example, see Panel B of Fig. 2, where N i ¼ 1:5). If the stock price undergoes a negative shock, investors's average wealth drops to the values for which the wealth effect is strong, resulting in a sharp increase of volatility and expected returns. 24

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log-investors. Autocorrelation in returns of an illiquid stock has been considered in a few asset pricing models.26 These models predict a negative sign for the coefficient of autocorrelation implying a small price overreaction. We will show that price overreaction could be very strong in illiquid economy. Moreover, in the states of the economy in which investors' risk exposure is very high, the autocorrelation in stock returns is significantly positive, indicating a strong momentum effect. The latter behavior of the autocorrelation coefficient is very different from that in the liquid economy, where it is always negative and negligibly small. We consider only a short-term autocorrelation coefficient, defined in Eq. (27). The following lemma presents this coefficient and is proven in Appendix E. Lemma 2. If time between two consecutive returns, Δt, is very small then sgnðμS ÞΨ Δt ffi; Corr ΔS ðt Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ½σ D σ S =ðX 1 μS Þ þ hX 1 þ ΦΔt2 þ Υ Δt þ Ψ ðΔtÞ2

ð29Þ

where functions Υ ðt; X 1 ; N3 Þ; Ψ ðt; X 1 ; N 3 Þ, and Φðt; X 1 ; N3 Þ are given by Eq. (E.5)–(E.7) in Appendix E. Panel A of Fig. 3 shows the instantaneous autocorrelation coefficient of the stock returns versus allocations of investors, Ni, assuming that Δt ¼ 0:02 and the wealth of aggregate investor 1, X1, is fixed. The autocorrelation coefficient of the stock returns in the illiquid market is negative unless the aggregate investor has excessively leveraged or short positions. The coefficient is small in magnitude if the aggregate investor's risk exposure is small. However, the coefficient becomes highly significant with an increasing stock position and reaches the minimal value at allocations at which the wealth effect is strong. Since the marginal risk premium with respect to the wealth of aggregate investor is negative (positive) when the aggregate investor is sufficiently long (short), the rise (fall) of the wealth causes the stock drift to fall. A smaller drift today results in a smaller stock price tomorrow. Furthermore, the price overreaction is replaced by a strong momentum effect when the aggregate investor becomes excessively leveraged or excessively short in the stock market. Since the marginal risk premium with respect to the wealth of aggregate investor is positive (negative) when the aggregate investor is very long (short), the rise (fall) of the wealth makes the stock market premium increase. A higher premium today results in a higher stock price tomorrow. Panel B of Fig. 3 shows the instantaneous autocorrelation coefficient of the stock returns versus the wealth of aggregate investor 1 when investors' allocations are fixed. In agreement with our previous discussion, the autocorrelation coefficient is small and negative when aggregate investor is wealthy and the wealth effect is small. This coefficient trails the wealth effect with decreasing aggregate wealth of investors and approaches the minimal value when this effect is maximal. As the wealth of the aggregate investor decreases further, and the wealth effect weakens, the autocorrelation coefficient falls in magnitude and then becomes significantly positive when aggregate investor 1 is poor and has high leverage. In latter states, a positive shock to the stock price causes the wealth of investor 1 to fall, in turn making the premium increase. A higher premium implies a higher stock price tomorrow and an increase in aggregate investor 1' risk exposure. This investor avoids an escalation of the risk in his portfolio by means of risk sharing with exponential investors. 4.3. Comparative statics In this subsection we consider changes in equilibrium resulting from variations in the main parameters of the economy that define its liquidity: α; the weight of investors in economy, λ1 þ λ2 , the time horizon of the illiquid economy, T, and the convexity of transaction costs, ε. We analyze the behavior of the conditional moments of the stock returns and the rate of trading between agents. Extrapolation of our conclusions to the predictability of the stock returns is intuitive and will not be discussed. Finally, we evaluate the effect of risk aversion of market makers on the stock market returns. Fig. 4, Panels A and B, repeats the conditional moments of the stock market returns shown in Fig. 2 for a stock market with smaller coefficient α, while assuming all other parameters remain unchanged. For clarity of exposition, we do not show the rate of trading. Smaller α implies faster trading by market makers. As market makers trade faster, the time that the economy stays in the states with a strong wealth effect decreases, making the impact of this effect on the prices fall at the given allocation of investors, Ni. Hence, the stock market premium and the volatility of stock returns decrease in magnitudes if α subsides. In the limit of a very small α, where the stock market is almost liquid, the wealth effect becomes negligible for a very wide range of Ni, so that the prices are very close to those in the liquid market, except for the states in which the aggregate investor has a very significant position in the stock market and the mispricing cannot be neglected. Fig. 5, Panels A and B, repeats the conditional moments of the stock market returns shown in Fig. 2 for a smaller weight of investors, while keeping all other parameters the same. For a given value of the wealth of aggregate investor 1, the stock returns in the two economies should be compared at the same aggregate allocations of investors, Ni ¼ 1  ð1  λ1  λ2 ÞN 3 . Fewer investors translate to a decline in stock market liquidity. Consequently, the wealth effect and mispricing increases. Fig. 5 shows that even a small decrease in the proportion of investors in the economy results in a significant increase in stock 26

For example, see Ho and Stoll (1981), Grossman and Miller (1988), Huang and Wang (2009), and others.

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Panel A

85

Panel B 0.4

0.5

0.2 0.0 CorrΔS

CorrΔS

0.0

−0.2 −0.5 −0.4

−0.6

−1.0 −2

−1

0

1

2

0.0

Ni

0.5

1.0

1.5

2.0

X1

Fig. 3. This figure shows the autocorrelation coefficient, Corr ΔS , against stock holdings by market makers, Ni (Panel A), and against the wealth of the aggregate market maker, X1 (Panel B). Panel A assumes that X 1 ¼ 0:5 and Panel B assumes that N i ¼ 1:5. All other parameters are set to α ¼ 0:5; ε ¼ 1; γ 2 ¼ γ 3 ¼ 1; μD ¼ 0:07; σ D ¼ 0:2; T ¼ 0:5; λ1 ¼ 0:4; λ2 ¼ 0:1; Δt ¼ 0:02, and t ¼ 0.

Panel A

Panel B

0.6

0.7

0.6

0.4

0.5

0.2

0.4 0.0

0.3 −0.2

0.2

−0.4

0.1

−0.6

0.0 −2

−1

0 Ni

1

2

0.0 0.2 0.4 0.6 0.8 1.0 X1

Fig. 4. This figure shows the stock market premium, μS (solid lines) and the volatility of the market stock returns, σS (dashed lines) against stock holdings of market makers, Ni (Panel A) and against the wealth of the aggregate investor 1, X1 (Panel B). The thin lines correspond to α ¼ 0:5 and the thick lines correspond to α ¼ 0:1. Panel A assumes that X 1 ¼ 0:5 and Panel B assumes that N i ¼ 1:5. All other parameters are set to ε ¼ 1; γ 2 ¼ γ 3 ¼ 1; μD ¼ 0:07; σ D ¼ 0:2; T ¼ 0:5; λ1 ¼ 0:4; λ2 ¼ 0:1; and t ¼ 0.

market mispricing. Furthermore, the decreasing proportion of investors makes them trade much faster in a given state of the economy X 1 ; Ni . Panels A and B of Fig. 6 repeat the conditional moments of the stock market returns shown in Fig 2 for a longer illiquidity horizon T, while keeping all other parameters unchanged. Clearly, the longer the stock market remains illiquid, the stronger it will be mispriced today. If the investment horizon of investors increases, then returns from their stock holdings become

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Panel A

Panel B

0.6

1.0

0.4 0.8 0.2 0.6 0.0 0.4

−0.2

0.2

−0.4

0.0

−0.6 −2

−1

0

1

2

0.0

0.4

Ni

0.8

1.2

X1

Fig. 5. This figure shows the stock market premium, μS (solid lines) and the volatility of the stock market returns, σS (dashed lines) against stock holdings by investors, Ni (Panel A) and against the wealth of the aggregate investor 1, X1 (Panel B). The thin lines correspond to λ1 ¼ 0:4; λ2 ¼ 0:1 and the thick lines correspond to λ1 ¼ 0:3; λ2 ¼ 0:075. Panel A assumes that X 1 ¼ 0:5 and Panel B assumes that N i ¼ 1:5. All other parameters are set to ε ¼ 1; γ 2 ¼ γ 3 ¼ 1; μD ¼ 0:07; σ D ¼ 0:2; T ¼ 0:5; and α ¼ 0:5.

Panel A

Panel B

0.6

0.8

0.4 0.6 0.2

0.0

0.4

−0.2 0.2 −0.4

0.0

−0.6 −2

−1

0 Ni

1

2

0.0

0.4

0.8

1.2

X1

Fig. 6. This figure shows the stock market premium, μS (solid lines), and the volatility of the stock market returns, σS (dashed lines), against stock holdings by investors, Ni (Panel A), and against the wealth of the aggregate market maker, X1 (Panel B). The thin lines correspond to T ¼ 0:5 and the thick lines correspond to T ¼ 1:0. Panel A assumes that X 1 ¼ 0:5 and Panel B assumes that N i ¼ 1:5. All other parameters are set to α ¼ 0:5; ε ¼ 1; γ 2 ¼ γ 3 ¼ 1; μD ¼ 0:07; σ D ¼ 0:2; λ1 ¼ 0:4; λ2 ¼ 0:1; and t ¼ 0.

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87

0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −2

−1

0

1

2

Ni Fig. 7. This figure shows the stock market risk premium, μS, and the volatility of the stock market returns, σS, against stock holdings by investors, Ni, assuming that X 1 ¼ 0:5. The solid lines correspond to ε ¼ 2, the dashed lines correspond to ε ¼ 1 and the dotted lines correspond to ε ¼ 0:5. All other parameters are set to α ¼ 0:5; γ 2 ¼ γ 3 ¼ 1; μD ¼ 0:07; σ D ¼ 0:2; λ1 ¼ 0:4; λ2 ¼ 0:1; T ¼ 0:5; and t ¼ 0.

0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −2

−1

0

1

2

Ni Fig. 8. This figure shows the stock market risk premium, μS, and the volatility of the stock market returns, σS, against stock holdings by investors, N i , assuming that X 1 ¼ 0:5. The solid lines correspond to γ 3 ¼ 3, the dashed lines correspond to γ 3 ¼ 1 and the dotted lines correspond to γ 3 ¼ 0:3. All other parameters are set to α ¼ 0:5; ε ¼ 1; γ 2 ¼ 1; μD ¼ 0:07; σ D ¼ 0:2; λ1 ¼ 0:4; λ2 ¼ 0:1; T ¼ 0:5, and t ¼ 0.

more risky. Therefore, the stock market clears only if the risk premium increases where the aggregate investor is long and decreases where he is short. Furthermore, potential losses of market makers increase in the states with investors having high risk exposure, making them trade faster with a longer investment horizon. Based on Fig. 6 we conclude that the wealth effect and stock mispricing subsides relatively quickly over time. Fig. 7 depicts the conditional risk premium and the volatility of the stock returns when ε ¼ 1 (dashed line), ε ¼ 2 (solid line), and ε ¼ 0:5 (dotted line), while keeping all other parameters unchanged. Apparently, the changes in the convexity of trading costs have only a small effect on the stock market prices when the market is illiquid. The last conclusion holds in spite of a noticeable impact of the convexity on the rate of trading between agents. Lower convexity increases transaction costs when the trading rate is less than one and decreases it when this rate is greater than one. It follows that if the convexity decreases then a market maker will trade slower when he is close to his utility-maximizing allocation and faster when he is away from it. Nonetheless, the adjustments in the trading rate due to the changes in the convexity of transaction costs are not substantial enough to result in noticeable changes of prices in the illiquid economy. Finally, we confirm our statement made in Section 2.2 that the preferences of market makers have minor effect on stock prices in the illiquid market. Fig. 8 shows the conditional risk premium and the volatility of the stock market returns for the cases when γ3 is equal to 0.3, 1, and 3, while keeping the rest of the model calibration as in Section 4.1. Apparently, the tenfold increase in the coefficient of risk aversion of market makers have a negligible impact on stock prices as long as the market remains significantly illiquid.

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5. Conclusion In this paper we find dynamic equilibrium in the economy with an illiquid stock market, where investors trade with market makers that face significant transaction costs that are convex in the number of traded shares. Our model is able to explain the increased volatility, very large positive expected stock returns, and the conditional Sharpe ratios observed during financial crises. We find that market makers suffer losses during financial turmoils which they try to cover by buying stock shares with a high expected Sharpe ratio. We also find that the short-term stock returns exhibit a very strong overreaction or a momentum effect, depending on the state of the illiquid market, while the autocorrelation coefficient in the liquid market is always negative.

Acknowledgments I gratefully acknowledge financial support from the Social Sciences and Humanities Research Council of Canada (Grant number 410-2008-196). I am thankful to Harjoat Bhamra, Bernard Dumas, Anna Pavlova, Stylianos Perrakis, Martin Schneider, Dimitri Vayanos, the anonymous referees for helpful comments and suggestions, as well as to participants in the seminar at Concordia University, the Adam Smith asset pricing conference 2011, Financial Management Association conference 2011 at Denver, Mathematical Finance Days workshop in Montreal 2012, and AIDEA 2013 conference in Lecce, Italy. Appendix A In this appendix we analyze portfolio selection problems for investors and market makers. The value function of an agent depends on his wealth and the aggregate state of economy which in turn is described by average wealth of log-investors, X1, and average share holdings by market makers, N3. Let Z ¼ ðX 1 ; N 3 ÞT be a vector of macro-state variables. We solve the problem faced by investor 1 by using the Hamilton– Jacobi–Bellman (HJB) equation for his value function V i1 ðt; X i1 ; ZÞ:  2 1
1 max V i1t þ Ni1 σ S V i1X i X i þ ðN 1 σ S Þ2 V i1X 1 X 1 þ Ni1 σ 2S N1 V i1X i X þ N i1 μS V i1X i i 1 1 1 1 1 2 2 N1 A R o ðA:1Þ þu3 V iN3 þN 1 μS V i1X 1 ¼ 0; V i1 ðT; X i1 ; ZÞ ¼ ln X i1 ; where N1 is a number of shares hold by aggregate investor 1. It follows that Ni1 ¼ 

μS V i1X i þ σ 2S N 1 V i1X i X 1 1

1

σ 2S V i1X i X i 1

:

1

We conjecture that V i1 ¼ ln X i1 þ g 1 ðt; ZÞ, then Ni1 ¼ X i1

μS σ 2S

ðA:2Þ

and g 1 ðt; ZÞ solves the following PDE:   1 μS 2 1 g 1t þ þ ðN 1 σ S Þ2 g 1X 1 X 1 þ u3 g 1N3 þ N1 μS g 1X 1 ¼ 0; 2 σS 2

g 1 ðT; Z Þ ¼ 0:

In equilibrium N1 ¼ X 1 σμ2S . Therefore the last PDE can be rewritten as S  2   μS 1 1 2 þ X 1 g 1X 1 X 1 þX 1 g 1X 1 þ u3 g 1N3 ¼ 0; g 1 ðT; Z Þ ¼ 0: g 1t þ 2 2 σS The HJB equation for the value function of an investor 2, V i2 ðt; X i2 ; ZÞ, is  2 1
1 max V i2t þ Ni2 σ S V i2X i X i þ ðN 1 σ S Þ2 V i2X 1 X 1 þ Ni2 σ 2S N1 V i2X i X þ N i2 μS V i2X i i 2 2 2 1 2 2 2 N2 A R o
 i 1 þu3 V iN3 þN 1 μS V i2X 1 ¼ 0; V i2 T; X i1 ; Z ¼  e  γ 2 X 2 :

γ2

It follows that Ni2 ¼ 

μS V i2X i þ σ 2S N 1 V i2X i X 1 2

2

σ 2S V i2X i X i 2

2

:

ðA:3Þ

ðA:4Þ

ðA:5Þ

S. Isaenko / Journal of Economic Dynamics & Control 60 (2015) 73–94

We conjecture that V i2 ¼  γ1 e  γ 2 X 2 g 2 ðt; Z Þ, then 2   X 1 g 2X 1 μS N 1 g2X 1 μS i þ ¼ 1 þ N2 ¼ g2 γ 2 σ 2S γ 2 g2 γ 2 σ 2S

89

i

ðA:6Þ

and g 2 ðt; ZÞ solves the following PDE:  2 1 μS N 1 σ S g 2X 1 1 þ g 2 þ ðσ S N 1 Þ2 g 2X 1 X 1 þ u3 g 2N3 þ N1 μS g 2X 1 ; 0 ¼ g 2t  2 σS 2 g2 g 2 ðT; ZÞ ¼ 1: In equilibrium the last PDE can be rewritten as #  2 "   X 1 g 2X 1 2 μS 1 2 1 X 1 g 2X 1 þ X 1 g 2X 1 X 1  1 þ g 2 þ u3 g 2N3 : 0 ¼ g 2t þ 2 2 σS g2

ðA:7Þ

ðA:8Þ

m The indirect utility function of a market maker V m 3 ðt; X 3 ; X 1 ; N 3 Þ solves  1 1 m 2 m 2 m 2 m þ ðσ S N 1 Þ V þ u3 V m 0 ¼ max V m 3t þ ðσ S N 3 Þ V 3X m 3X 1 X 1 þ σ S N 3 N 1 V 3X m 3N 3 3 X3 3 X1 u3 A R 2 2     1 m þN 1 μS V m exp  γ 3 X m Vm þ ðN3 μS  αju3 j1 þ ε ÞV m 3X 1 ; 3 T; X 3 ; X 1 ; N 3 ¼  3 : 3X m 3

Vm 3



t; X m 3 ; X 1 ; N3



γ3

  ¼  γ exp  γ 3 X m 3 f ðt; X 1 ; N 3 Þ and find the equation for f: 1

We assume that 3  1 1 2 2 0 ¼ min f t þ ðγ 3 σ S N 3 Þ f þ ðσ S N1 Þ f X 1 X 1  γ 3 σ 2S N 1 N3 f X 1 þ u3 f N3 u3 A R 2 2 1þε Þf þ N1 μS f X 1 ; f ðT; X 1 ; N 3 Þ ¼ 1:  γ 3 ðN 3 μS  αju3 j The first order condition provides the optimal rate of trading:  1=ε   jf N3 j : u3 ¼  sgn f N3 αð1 þ εÞγ 3 f

Substitution of the last result into Eq. (A.9) gives the PDE for f ðt; X 1 ; N3 Þ:    1 1 f t þ ðγ 3 σ S N3 Þ2  γ 3 N 3 μS f þ ðσ S N1 Þ2 f X 1 X 1 þN 1 μS  γ 3 σ 2S N 3 f X 1 2 2 !1=ε

jf N3 j

f ¼ 0; f ðT; X 1 ; N 3 Þ ¼ 1: ε N3 αð1 þ εÞ1 þ ε γ 3 f The latter can be rewritten after replacing N1 with X 1 μS =σ 2S as the following: !   2  1 1 μS μS  X 1 f X 1 X 1 þX 1 μS  γ 3 σ 2S N3 f X 1 f t þ ðγ 3 σ S N3 Þ2  γ 3 N 3 μS f þ 2 2 σS σ 2S !1=ε

jf N3 j

f N ¼ 0; f ðT; X 1 ; N3 Þ ¼ 1: ε 3 1þε αð1 þ εÞ γ 3 f The last equation depends on

ðA:9Þ

ðA:10Þ

ðA:11Þ

ðA:12Þ

μS and σS which will be found endogenously as functions of t; N3 , and X1.

Appendix B In this appendix we prove Proposition 1. First, we find μS and σS. The stock market clearing condition reads      μS 1 X 1 g 2X 1 λ X þ λ þ þ 1  λ1  λ2 N 3 ¼ 1: σ 2S 1 1 2 γ 2 γ 2 g2 Therefore,

μS ¼ σ 2S Ω;

ðB:1Þ

where 1 ð1  λ1  λ2 ÞN 3   : Ω¼ 1 X 1 g 2X 1 λ1 X 1 þ λ2 þ γ 2 γ 2 g2 Moreover, because the set of state variables for the stock price in this economy includes D; X 1 and N3, we find from Itô's

90

S. Isaenko / Journal of Economic Dynamics & Control 60 (2015) 73–94

formula that σ S ¼ σ D SD þ σ S N 1 SX 1 , or

σS ¼

SD σ D ; 1  ΩSX 1 X 1

ðB:2Þ

where we took into account that N1 ¼ X 1 σμ2S ¼ X 1 Ω. S Let ξt be the pricing kernel in our economy with dynamics   μSt dW t ; ξ0 ¼ 1: dξt ¼  ξt

σ St

ðB:3Þ

  Since the process ξt St is a martingale, its drift is zero. Therefore, St ¼ ξ1 Et ξT DT and we find the PDE for S: t   μ 1 1 0 ¼ St þ μD  S σ D SD þu3 SN3 þ σ 2D SDD þ N21 σ 2S SX 1 X 1 þ σ D σ S N 1 SDX 1 ; 2 2 σS SðT; D; X 1 ; N3 Þ ¼ D: Let us conjecture that Sðt; D; X 1 ; N 3 Þ ¼ D þ hðt; X 1 ; N3 Þ then the last PDE results in the PDE for h given by Eq. (21) which has a solution by our assumption. Finally, we verify that u3 is independent from D. Since σS and μS are independent from D, we do not have to include D as a state variable in the HJB for V3. Therefore, u3 depends on only t; N 3 and X1.□ Appendix C In this appendix we describe a computational algorithm for finding equilibrium in the illiquid economy and derive the boundary conditions at X 1 ¼ 0. The equilibrium can be found by simultaneously solving the PDE's for g 2 ðt; X 1 ; N 3 Þ, f ðt; X 1 ; N3 Þ, and hðt; X 1 ; N3 Þ. We use a finite difference approach and solve the PDE's backwards. We apply Crank–Nicholson scheme for variables X1 and N3 in functions f and g2, and for variable N3 in function h. We approximate a first order derivative by a left-hand side Fðx  ΔxÞ  ΔxÞ  2FðxÞ difference FðxÞ  Δ while the second-order derivative in X1 is approximated by Fðx þ ΔxÞ þ Fðx . We use absorbing x Δx2 boundary conditions for variable X1 at the boundary X 1 ¼ X max and for variable N3 at the boundary N3 ¼ N3min . The ranges of the state variables are X 1 A ½0; 5; N3 A ½ 4; 8 while the number of grid points is 100 in X1-dimension, 100 in N3-dimensions, and 50,000 in t-dimension. We find that the convergence of numerical results deteriorates if α or λ1 þ λ2 decreases. Now let us find the boundary conditions at X 1 ¼ 0. In the limit of a very small X1, logarithmic investors do not hold stock shares and the equilibrium in the stock market is extrapolated to that without these investors. It is easy to see from μ Appendix A that stock holding by exponential investors becomes N 2 ¼ γ σS 2 so that the stock market clearing condition 2 S 2 implies that μS ¼ γ 2 N i σ S =λ2 . Moreover, Appendix B suggests that σ S ¼ σ D and h is a solution of the PDE: 0 ¼ ht þ μD þhN3 u3  γ 2 σ 2D Ni =λ2 ;

hðT; N3 Þ ¼ 0;

ðC:1Þ

where u3 is found from Eq. (A.10) and the solution of the PDE for function f: !1=ε
γ 

jf N3 j

f j; 0 ¼ f t þ γ 3 σ 2D 3 N 23  γ 2 N i N3 =λ2 f  ε N3 2 αγ 3 ð1 þ εÞ1 þ ε f f ðT; N 3 Þ ¼ 1:

ðC:2Þ

Similarly, in the absence of logarithmic investors g2 solves the PDE:   1 μS 2 0 ¼ g 2t  g 2 þ u3 g 2N3 ; g 2 ðT; N3 Þ ¼ 1: 2 σD

ðC:3Þ

The last equilibrium can be solved in the closed form in the special case of the calibration used in the paper. Lemma 3. Assume that ε ¼ 1 and X 1 ¼ 0, then f ðt; N 3 Þ ¼ expðAf ðtÞ þ Bf ðtÞN3 þ C f ðtÞN23 Þ;

ðC:4Þ

g 2 ðt; N 3 Þ ¼ expðAg ðtÞ þBg ðtÞN3 þ C g ðtÞN23 Þ;

ðC:5Þ

hðt; N3 Þ ¼ Ah ðtÞ þBh ðtÞN3 ;

ðC:6Þ

where functions Af ðtÞ; Bf ðtÞ; C f ðtÞ; Ag ðtÞ; Bg ðtÞ; C g ðtÞ; Ah ðtÞ, and Bh ðtÞ are given by formulas (C.15), (C.13), (C.11), (C.25), (C.23), (C.21), (C.31) and (C.29), respectively. Proof of Lemma 3. If ε ¼ 1 then Eq. (C.2) becomes 0 ¼ f t þ γ 3 σ 2D N3


γ

2  f N3 N3  γ 2 Ni =λ2 f  ; 2 4αγ 3 f 3

f ðT; N 3 Þ ¼ 1;

ðC:7Þ

S. Isaenko / Journal of Economic Dynamics & Control 60 (2015) 73–94

91

which, after replacing f with (C.4), is reduced to the following three equations: 0 ¼ C 0f þ γ 3 σ 2D


γ

3

2

 C2   f þ γ 2 1  λ1  λ2 =λ2  ;

αγ 3

0 ¼ B0f  γ 2 γ 3 σ 2D =λ2 

0 ¼ A0f 

B2f

4αγ 3

;

Bf C f

αγ 3

;

C f ðT Þ ¼ 0;

ðC:8Þ

Bf ðT Þ ¼ 0;

ðC:9Þ

Af ðT Þ ¼ 0:

ðC:10Þ

Eq. (C.8) is a Riccati equation which solution is given by C f ðtÞ ¼ αβγ 3 tanhðβðT  tÞÞ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i σ 2D

where β ¼

γ3 2

þ

γ 2 ð1  λ1  λ2 Þ λ2

α

0 ¼ B0f

ðC:11Þ

. It follows that Eq. (C.9) becomes

 γ 2 γ 3 σ 2D =λ2  βBf tanhðβðT  tÞÞ;

Bf ðTÞ ¼ 0:

ðC:12Þ

It is easy to verify that the solution of the last equation is given by Bf ðt Þ ¼ 

  γ 2 γ 3 σ 2D tanh βðT t Þ : λ2 β

ðC:13Þ

Hence, Eq. (C.10) becomes A0f ¼

 γ 22 γ 3 σ 4D 2 tanh βðT  t Þ ; 2 2 4αλ2 β

Af ðT Þ ¼ 0;

ðC:14Þ

which solution is given by  γ 2 γ σ 4 tanhðβðT tÞÞ þt T : Af ðt Þ ¼ 2 32 D2 β 4αλ2 β

ðC:15Þ

It follows that the trading rate is given by u3 ðt; N3 Þ ¼ 

f N3

2αγ 3 f

¼

    γ 2 σ 2D Bf ðtÞ þ2C f ðtÞN 3 ¼ tanh βðT  t Þ  βN3 : 2αγ 3 2αλ2 β

ðC:16Þ

The last formula implies that market makers switch the direction of trading when N3 ¼

γ 2 σ 2D γ2 : ¼ 2 ½γ 3 λ2 þ 2γ 2 ð1  λ1  λ2 Þ 2αλ2 β

Next, we find g 2 ðt; N3 Þ by solving (C.3). The latter can be written as  2      γ 2 σ 2D 1 γ 2 ½1  ð1  λ1  λ2 ÞN 3 σ D 0 ¼ A0g þ B0g N 3 þC 0g N23  þ  βN 3 tanh βðT  t Þ Bg þ 2C g N3 ; 2 λ2 2αλ2 β which can be reduced to the following three equations:  2   1 γ 2 ð1  λ1  λ2 Þσ D  2β C g tanh βðT  t Þ ; 0 ¼ C 0g  2 λ2 

C g ðT Þ ¼ 0;

ðC:18Þ



0 ¼ B0g þ

  γ 22 ð1  λ1  λ2 Þσ 2D γ 2 σ 2D þ C g  βBg tanh β ðT t Þ ; 2 αλ β 2 λ2

0 ¼ A0g 

    1 γ 2 σ D 2 γ 2 σ 2D þ tanh βðT  t Þ Bg ; 2 λ2 2αλ2 β

Bg ðT Þ ¼ 0;

Ag ðT Þ ¼ 0:

The solution of Eq. (C.18) is     1   γ ð1  λ1  λ2 Þσ D 2 2 ðT t Þsech βðT t Þ þ tanh βðT  t Þ ; C g ðt Þ ¼  2 2λ2 β implying that equation for function Bg becomes 0 ¼ B0g þ

ðC:17Þ

  γ 22 ð1  λ1  λ2 Þσ 2D  β Bg tanh β ðT t Þ 2 λ2

ðC:19Þ

ðC:20Þ

ðC:21Þ

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S. Isaenko / Journal of Economic Dynamics & Control 60 (2015) 73–94

 

γ 2 ð1  λ1  λ2 Þσ 2D 2λ2

2





 1     γ2 2 ðT  t Þsech βðT  t Þ þ tanh β ðT  t Þ  tanh βðT  t Þ ; αλ2 β β

Bg ðT Þ ¼ 0:

ðC:22Þ

The solution of the last equation is given by " #  2   γ 2 ð1  λ1  λ2 Þσ 2D γ 2 1 γ 22 ð1  λ1  λ2 Þσ 2D Bg ðt Þ ¼  tanh β ðT  t Þ 2 β 2 βλ αλ 2 2 λ 2

 þ

γ 2 ð1  λ1  λ2 Þσ 2D 2βλ2

2

 γ2 2 ðT  t Þsech βðT  t Þ : αλ2

ðC:23Þ

Finding function g2 is finished after integration of Eq. (C.20) which now can be rewritten as " #    2   γ 32 σ 4D  1 γ2 σD 2 ð1  λ1  λ2 Þσ D γ2 2 0 tanh βðT  t Þ  1  λ1  λ2  Ag ¼ 3 2 2 λ2 2 β αλ 2 2αλ2 β 

   γ 42 σ 6D 2 ð1  λ1  λ2 Þ2 ðT  t Þtanh β ðT t Þ sech βðT  t Þ : 4 3 2 8α λ2 β

ðC:24Þ

The integration reveals " #    2 γ 32 σ 4D 1 γ2σD 2 ð1  λ1  λ2 Þσ D γ2 Ag ¼  ðT  t Þ  1  λ1  λ2  3 2 2 λ2 2β αλ2 2αλ2 β    γ 42 σ 6D 1  t T þ tanh βðT  t Þ  ð1  λ1  λ2 Þ2 4 4 β 16α2 λ2 β   1   2  ðT t Þsech β ðT t Þ  tanh βðT t Þ :

ðC:25Þ

β

Finally, we find hðt; N3 Þ by solving (C.1). The latter can be written as     γ 2 σ 2D      βN 3  γ 2 σ 2D 1  1  λ1  λ2 N3 =λ2 ; 0 ¼ A0h þ B0h N3 þ μD þ Bh tanh βðT t Þ 2αλ2 β

hðT; N3 Þ ¼ 0;

ðC:26Þ

which can be reduced to the following two equations: 0 ¼ B0h Bh β tanhðβðT  tÞÞ þ γ 2 σ 2D ð1  λ1  λ2 Þ=λ2 ;   γ σ2 0 ¼ A0h þ μD  γ 2 σ 2D =λ2 þ Bh tanh βðT t Þ 2 D ; 2αλ2 β

Bh ðTÞ ¼ 0; Ah ðT Þ ¼ 0:

ðC:27Þ ðC:28Þ

Eq. (C.27) is similar to Eq. (C.13) and has the solution given by Bh ðt Þ ¼

  γ 2 σ 2D ð1  λ1  λ2 Þ tanh βðT  t Þ : λ2 β

ðC:29Þ

It follows that Eq. (C.28) can be written as 0 ¼ A0h þ μD  γ 2 σ 2D =λ2 þ

 γ 22 σ 4D ð1  λ1  λ2 Þ 2 tanh βðT t Þ ; Ah ðT Þ ¼ 0: 2 2 2αλ2 β

ðC:30Þ

The solution of the last equation is given by    γ 2 σ 4 ð1  λ1  λ2 Þ tanhðβðT  tÞÞ þ t  T :□ Ah ðt Þ ¼ μD  γ 2 σ 2D =λ2 ðT  t Þ  2 D 2 2 β 2αλ2 β

ðC:31Þ

Appendix D In this appendix we will prove Proposition 2 and derive a few related results. Proof of Proposition 2. An equilibrium in the economy with no trading costs is found by using a social planner. The preferences of the social planner are given by U ðD T Þ ¼

sup

λ1 C 1T þ ð1  λ1 ÞC 2T ¼ DT

z0 lnðC 1T Þ  ð1 z0 Þ

e  γ 3 C 2T

γ3

;

ðD:1Þ

S. Isaenko / Journal of Economic Dynamics & Control 60 (2015) 73–94

93

where z0 is a constant depending on the initial endowments of the agents and proportion It follows that the terminal consumption of the aggregate investor, C 1T , solves z0

λ1.27

1 λ1  ðγ 3 =ð1  λ1 ÞÞðDT  λ1 C 1T Þ ¼ ð1  z0 Þ e : C 1T 1  λ1

ðD:2Þ

The last equation has a unique solution since its right side is monotonically decreasing with the range of values ð0; þ 1Þ, while its left side is monotonically increasing starting from the positive value ð1  z0 Þ1 λ1λ e  ðγ 3 =ð1  λ1 ÞÞDT . 1 We find the terminal consumption of market makers from the constraint λ1 C 1T þ ð1  λ1 ÞC 2T ¼ DT and then we obtain 0 0 UðDT Þ from (D.1). The pricing kernel is given by ξt ¼ Et ½ξT  ¼ Et ½U ðDT Þ=E0 ½U ðDT Þ. Because νt ¼ Et ½U 0 ðDT Þ is a martingale and D is the only state variable, νðt; DÞ solves

νt þ 12 σ 2D νDD þ μD νD ¼ 0;

νðT; DÞ ¼ U 0 ðDÞ:




After solving the last PDE we find ξt ¼ νt =ν0 . Because dξt ¼  ξt μσStSt

μS νD σ D ¼ : σS ν

ðD:3Þ dW t and dνt ¼ νD ðt; DÞσ D dW t , we derive ðD:4Þ

Since the process ξt St is a martingale, its drift is zero. The latter allows us to derive the PDE for stock price Sðt; DÞ which is given by Eq. (25). Then, the stock return volatility is found from σ S ¼ σ D SD . Since the process ξt X 1t is a martingale, it has zero drift allowing us to derive the PDE for the wealth of agent 1:   μ 1 ðD:5Þ 0 ¼ X 1t þ μD  S σ D X 1D þ σ 2D X 1DD ; X 1 ðT; DÞ ¼ C 1T : 2 σS The wealth of agent 2 is found from the bond market clearing condition X 2 ¼ S1λ1λX 1 . Finally, the allocations to the stock market of agent 1 and agent 2 are given by N 1 ¼ σ DσXS1D and N 2 ¼ σ DσXS2D , respectively.□

1

The equations above can be solved only numerically. Therefore, we first find C 1T numerically and obtain the terminal condition for process ν. Then we solve (D.3) by using an explicit finite difference approach. Then we solve PDE's (25) and (D.5) by using the same approach and the conditional Sharpe ratio found from (D.4). Finally, we consider an economy in the limit of a large negative D. This economy is dominated by market makers so that h i γ D the representative agent has utility function at equilibrium given by  γ1 exp  1 3 λ1 . It follows that the pricing kernel in this 3
 h  i economy is proportional to νðt; DÞ ¼ exp  1γ3 Dλ expf12 ðγ 3 σ D =ð1  λ1 ÞÞ2  γ 3 μD = 1  λ1 ðT t Þg and the Sharpe ratio is equal 1

to γ 3 σ D =ð1  λ1 Þ. Solving the PDE for the stock price results in Sðt; DÞ ¼ D  ðγ 3 σ 2D =ð1  λ1 Þ  μD ÞðT tÞ. Therefore, σ S ¼ σ D and

μS ¼ γ 3 σ 2D =ð1  λ1 Þ. Appendix E

In this appendix we prove Lemmas 1 and 2. Assuming that dμS ¼ μμS dt þ σ μS dW; dσ S ¼ μσ S dt þ σ σ S dW, one can write dSt þ dt ¼ μS ðt þ dtÞ dt þ σ S ðt þ dtÞ dW t þ dt ¼ μS ðtÞdt þ μμS ðtÞ dt þ σ μS ðtÞ dW t dt þ σ S ðtÞ dW t þ dt 2

þ μσ S ðtÞ dt dW t þ dt þ σ σ S ðtÞ dW t þ dt dW t :

Because dW t and dW t þ dt are independent, the conditional covariance between two sequential returns is equal to 2 Covt ðdSt ; dSt þ dt Þ ¼ σ S ðtÞσ μS ðtÞ dt , while the conditional variances are var t ðdSt Þ ¼ σ 2S ðtÞ dt; vart ðdSt þ dt Þ ¼ ½ðσ S ðtÞ þ μσ S ðtÞ dtÞ2 2 2 2 þ σ μS ðtÞ dt þ σ σ S ðtÞ dt dt. It follows that the autocorrelation coefficient of the stock returns can be written as

σ μS ðtÞΔt ffi; Corr ΔS ðt Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½σ S ðtÞ þ μσ S ðtÞΔt2 þ σ 2μS ðtÞΔt 2 þ σ 2σ S ðtÞΔt

ðE:1Þ

1 2 small. If the market where Δt is very   is liquid then from Itô's formula σ S ¼ SD σ D  σ D ; μS ¼ St þSD μD þ 2SDD σ D . Therefore, σ μS ¼ μσS ¼ σ D StD þSDD μD þ 12SDDD σ 2D : For the time increment to be used, σ S ⪢jμσS jΔt; σ S ⪢jσ μS jΔt and Lemma 1 follows. In the illiquid market, we apply Itô's formula to Sðt þ dt; D; X 1 ; N3 Þ ¼ D þ hðt þ dt; X 1 ; N 3 Þ and obtain μS ðt þ dt; X 1 ; N3 Þ; σ S ðt þdt; X 1 ; N3 Þ. Then we reapply Itô's formula to find σ μS and μσ S . In particular, we derive

27

σ σS ¼ X 1 Sp Υ ;

ðE:2Þ

σ μS ¼ X 1 Sp Ψ ;

ðE:3Þ

μσS ¼ X 1 Sp Φ;

ðE:4Þ

If

δ1 and δ2 are the Lagrange multipliers of agents 1 and 2, respectively, then one can show that z0 ¼ δ λ

2 1

δ2 λ 1

.

þ δ1 ð1  λ1 Þ

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S. Isaenko / Journal of Economic Dynamics & Control 60 (2015) 73–94

where

Υ ¼ hX 1 X 1 X 1 Sp ;

ðE:5Þ h

i

h

i

Ψ ¼ htX 1 þhN3 X 1 u3 þhN3 u1X 1 þhX 1 X 1 2X 1 S2p þ 12 X 21 ðS2p ÞX 1 þ hX 1 S2p þ X 1 ðS2p ÞX 1 þ 12 X 21 S2p hX 1 X 1 X 1 ; h

Φ ¼ htX 1 þ hN3 X 1 u3 þ hX 1 ðSp Þt =Sp þ hX 1 ðSp ÞN3 u1 =Sp þ hX 1 X 1 2X 1 S2p þ 12 X 21 ðS2p ÞX 1 h i þ hX 1 S2p þ X 1 ðS2p ÞX 1 þ 12 X 21 Sp ðSp ÞX 1 X 1 þ 12 X 21 S2p hX 1 X 1 X 1

ðE:6Þ

i ðE:7Þ

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