Equilibrium Interaction of Borrowing and Short-Sale Constraints and Asset Pricing

Transcription

1 Equilibrium Interaction of Borrowing and Short-Sale Constraints and Asset Pricing Nam Huong Dau Current Draft: October 26, 2014 I would like to especially thank Paul Ehling, Bernard Dumas, Michael Gallmeyer and Christian Heyerdahl-Larsen for extensive discussions and invaluable comments. I would also like to especially thank Rui Albuquerque and Benjamin Holcblat for invaluable comments and suggestions. I would like to thank Ilan Cooper, Salvatore Miglietta, Andreea Mitrache, Håkon Tretvoll, Stathis Tompaidis, Costas Xiouros and seminar participants at CAPR lunch seminars for helpful comments and suggestions. I thank Paul Armand, Ben (Chunyu) Yang and Eduard Dubin for their insightful discussions on technical issues. I thank the Leiv Eiriksson Mobility Program of the Research Council of Norway for funding. Parts of this paper were written during my stay as a Visiting Scholar at the University of Texas at Austin, whose hospitality I m gratefully acknowledging. BI Norwegian Business School, nam.h.dau@bi.no. 1

2 Abstract I study borrowing and short-sale constraints and their compound impacts on asset pricing quantities in a dynamic general equilibrium economy with heterogeneity in beliefs and risk aversions. The constraints virtually never bind simultaneously as they generate negative effects on the binding of each other. For intermediate values of consumption shares that balance the constraints relative strength, the constraints bind intermittently. In the case of intermittent binding, under plausible parameters, the stock price lies between the levels of the single-constraint economies, and the compound impact of the constraints on the return volatility is most profound and can push the volatility below any single-constraint economy s level. In other cases, the binding of one constraint implies that the other is inactive and manifests no effects on equilibrium properties. The model also produces a speculative premium, which is due to the presence of both constraints and is characterized by the equilibrium stock price exceeding all investors marginal valuations. Keywords: Short-sale Constraint; Borrowing Constraint; General Equilibrium; Heterogeneous Beliefs; Heterogeneous Risk Aversions; Asset Price; Return Volatility; Speculative Premium JEL Classification: G10; G11; G12 2

3 1 Introduction Borrowing and short-selling constraints are two among the most common trading restrictions faced by investors in financial markets. For instance, margin requirements, a form of borrowing constraints, have been permanently imposed in the US financial markets through the Securities Exchange Act of Short-sale constraints in the form of short-selling bans are often imposed when financial markets are in turmoil, such as the time after the outburst of the financial crisis Other forms of short-selling restrictions faced more frequently by market participants include lending fees and the New York Stock Exchange s (NYSE s) uptick rule, among others. Further, as another important fact, borrowing and short-sale constraints are frequently imposed concurrently. Despite that, these two constraints are often studied separately, leaving a gap in what we know about their mutual impacts on each other as well as their compound impact on financial markets. My paper investigates this aspect of the constraints effects in a dynamic general equilibrium setting and shows that they indeed have first order effects on one another in equilibrium. Modeling portfolio constraints to study their general equilibrium effects is challenging in general as their presence limits investors ability to smooth consumption and gives rise to market incompleteness. Specifically, given a consumption profile, upon the event of one binding constraint, an investor is unable to form a portfolio that replicates optimal consumption in every state of nature. 3 Previous papers in the literature [Detemple and Murthy (1997), Gallmeyer and Hollifield (2008), Chabakauri (2012, 2013), Rytchkov (2014), among others] primarily study constraints in continuous time settings and employ extensions of the duality method developed in Cvitanic and Karatzas (1992) to solve the models. Due to technical limitation, these papers often focus on cases in which either the constrained investors have logarithmic preferences or there is only one con- 1 The Securities Exchange Act of 1934 transferred the authority to set minimum margins from the New York Stock Exchange and other private-sector exchanges to the Federal Reserve. Between 1934 and 1974, regulators were active in adjusting the initial margin requirement. They changed the requirement 22 times. In 1974 they set it at 50% and started to keep it stable. 2 To exemplify, in September 2009, the US Securities and Exchange Commission (SEC) imposed an emergency ban on nearly 1,000 financial stocks. 3 Effectively, constraints distort optimal risk-sharing between investors. For this reason they are deemed to have strong implications on the behavior of equilibrium asset pricing moments. 1

4 straint in the economy at a time. A drawback of the logarithmic preference is that it suppresses the effects of the constraints on asset prices and volatility. In this paper, to study both borrowing and short-selling constraints in a unified framework and to capture effects of the constraints on broad asset pricing moments, I construct a model economy in discrete-time and solve the model with the method developed in Dumas and Lyasoff (2012). More precisely, the economy under consideration is a dynamic, discrete-time exchange economy in the spirit of Lucas (1978). The model economy is populated by two groups of investors represented by two agents with power preferences. The agents have heterogeneous risk aversions. To smooth consumption, the agents trade in financial markets with one bond and with one risky asset (stock). As a central feature of the model, the agents cannot be at full freedom to trade as they wish because they may face borrowing and short-sale constraints. 4 In equilibrium, the less risk-averse agent borrows from the more risk-averse one to invest in the stock market. Therefore, the borrowing constraint imposed on the less risk-averse can bind endogenously. 5 In addition to preference heterogeneity, the agents can also differ in their beliefs about the dividend growth rate of the risky asset. 6 The more pessimistic agent of the two may, at some points in time, have a demand for shorting the stock. In such cases, the short-sale constraint binds. 7 In the model economy, each of the constraints can bind endogenously. A novel contribution of this paper is to show how the constraints affect the binding of each other. Numerical results demonstrate that the constraints do not bind simultaneously most of the times. In each of different intervals of the state variable, the optimist s consumption share, over the life-time of the economy, either only one of the constraints binds or both constraints bind intermittently. More specifically, 4 Subject to the borrowing constraint, an agent can borrow only up to a limited fraction of his desirable amount of stock investment to finance that investment. The short-sale constraint requires the constrained agent to hold a non-negative position of the risky asset subject to the constraint. 5 As, effectively, in equilibrium the borrowing constraint affects only the less risk-averse agent, it suffices to impose a sole borrowing constraint on the less risk-averse agent. 6 Specifically, the case of our interest is that the less risk averse agent is more optimistic, while the agent with higher risk aversion is more pessimistic and faces a short-sale constraint. 7 Heterogeneous risk aversions are modeled as the key ingredient in papers such as Bhamra and Uppal (2009), Chabakauri (2012), Rytchkov (2014) to study equilibrium impacts of borrowing constraints. In important papers that study general equilibrium impacts of short-sale constraints such as Lintner (1969), Miller (1977), Jarrow (1980), Detemple and Murthy (1997), Gallmeyer and Hollifield (2008), belief heterogeneity is also the key model ingredient. 2

5 it is only the borrowing constraint that binds when the pessimist dominates the economy; it is only the short-sale constraint that binds when the optimist dominates the economy. Intermittent binding occurs when the consumption shares are in the range that balances the relative strength of the constraints. 8 Through out this paper, intermittent binding is referred to with two features. First, the constraints bind but not in all times and all states of nature. In addition, when one constraint binds, the other is most likely does not. 9 I stress that this novel finding is robust and holds regardless of the interplay between beliefs and risk aversion. The results on binding behavior of the constraints presented above are a direct corollary of the finding that a binding borrowing constraint reduces the likelihood of the short-sale constraint to bind, and vice versa. Mechanically, the borrowing constraint, when it binds, has a positive effect on perceived market prices of risk of all agents in the economy. This occurs as a channel to reequilibrise the stock market after a fall in the optimist s stock holding, which is because he is not allowed to borrow to fully finance his demanded stock investment. The pessimist s market price of risk is therefore less likely to be negative, effectively alleviating his incentive to sell the stock short. The pessimist s market price of risk increases with the frequency of binding borrowing constraints. The borrowing constraint binds more frequently when the pessimist dominates the economy as it is more likely feasible for the optimist, the borrowing-constrained agent, to lever up. To the contrary, a binding short-sale constraint causes a fall in the stock supply, which in turn reduces the optimist s stock holding. Consequently, the demand for borrowing is cut down, reducing the rate with which the borrowing constraint binds. The short-sale constraint is more active when the optimist dominates the economy, as the aggregate risk aversion in such a case is lower, which implies lower market prices of risk in general. The preceding paragraph also discusses many effects of the constraints on agents investment behavior. To summarize, in general, in a constrained economy, asset holdings of agents are moderated, relative to the unconstrained economy s. To illustrate, a binding borrowing constraint forces 8 As a general result, a tighter borrowing constraint is active in a wider range of the optimist s consumption share, and conversely. 9 In fact, the constraints can bind at the same time, but this occurs with a very small probability and for only a few values of the state variable as shown in the bottom plot of Figure 2. 3

6 the pessimist to cut back his bond holding and turns his stock holding from negative to positive. The intuition for the latter effect was given earlier. The former occurs as the demand for funds is reduced. Similarly, a binding short-sale constraint prevents the pessimist from shorting the stock, and thus lowers his bond holding. Next, I discuss insights into equilibrium behavior of asset pricing quantities in the presence of both borrowing and short-sale constraints. It is well known that with CRRA preferences, one can not quantitatively match empirical facts, however one can still replicate many important empirical patterns qualitatively. Below, my paramertrization reflects such an objective of matching empirical patterns including procyclical stock price and countercyclical market prices of risk and return volatility, which are documented in the literature (e.g. Ferson and Harvey (1991), Schwert (1989).) Specifically, under our parametrization, one investor is less risk averse and more optimistic than the other. Note that cases such as one in which one investor is less risk averse and more pessimistic than the other, and hence either constraint is not binding are not of our interest. First, we assess the compound effect of the constraints on the stock price. When the borrowing constraint is sufficiently strict, the borrowing constraint s effects dominate, and the stock price is increased in the constrained economy. In contrast, when the borrowing constraint is weak, the short-sale constraint s effects dominate. The compound effect of the constraints on stock price thus primarily exhibits the short-sale constraint s, which increases the stock price when the optimist dominates the economy, and decreases the stock price otherwise. When both constraints bind intermittently, the stock price lies between the levels of the single-constraint economies. To understand the above results, we first discuss the price effects of the single constraints. Effects of the constraints on the stock price are driven by the relative strength of the classical income and substitution effects, which are in turn determined by the effects of the constraints on investment opportunities of the investors (Gallmeyer and Hollifield (2008)). We start with the borrowing constraint. The constraint forces the optimist to reduce his stock holding. To clear the market, the stock price must adjust for the pessimist to adjust his stock holding. The investment opportunities of the pessimist worsen as the constraint lowers the interest rate and turns the market price of risk 4

7 perceived by the pessimist from negative to positive but with a lowered absolute value. The pessimist s intertemporal elasticity of substitution (IES) is lower than 1 as his risk aversion coefficient γ o > 1. Accordingly, the income effect dominates, leading to a higher wealth-consumption ratio. We can draw the conclusion on the effect of the constraint on the wealth-consumption ratio of the optimist in a similar way. Overall, the aggregate wealth-consumption ratio is higher with a borrowing constraint, with a bolder effect when the pessimist dominates the economy. As the aggregate consumption equals the aggregate dividend, which is fixed, the aggregate wealth is larger with a borrowing constraint. Finally, as the aggregate wealth equals the stock market value, the stock price must increase. By the same reasoning noticing the two-way effect of the short-sale constraint on the interest rate, we can obtain the intuition for the two-way price effect of the short-sale constraint. 10 Overall, when the borrowing constraint is relatively strict, its price effect suppresses and replaces the negative price effect of the single short-sale constraint. Therefore, over the entire range of the state variable, when at least one constraint is active, the stock price in the two-constraint economy is higher, compared to the price in the unconstrained economy. When the two constraints are alternately active, the effects of the borrowing constraint on the interest rate and the market prices of risk are attenuated. As a result, the price level is lower than in the borrowing-constrained economy. Similar argument applies to explain why the price level in the two-constraint economy is higher than in the short-sale constrained economy. The binding behaviors of the constraints and their price effects discussed so far have two important implications. First, imposing one constraint, one needs to take into account the effect of the incumbent on the imposition as present contemporaneously, one of the constraints is likely inactive. Second, the choice of the benchmark economy can modify theoretical predictions of single-constraint models. 11 To illustrate, let us take the unconstrained economy as benchmark, the only choice in single-constraint economies, the short-sale constraint manifests it effects most of the 10 See Gallmeyer and Hollifield (2008) for a thorough analysis of the effect of intertemporal elasticity of substitution (IES) on a short-sale constraint s impact on the stock price and volatility. 11 For instance, Miller (1977) predicts that the stock price is inflated in the presence of a short-sale constraint. 5

8 times. However, as presented above, taking instead the borrowing-constrained economy as benchmark, which is allowed in this multiple-constraint model, there are intervals of the state variable in which one cannot observe any price effect of the short-sale constraint. This prediction on the price effect of the short-sale constraint is consistent with the empirical finding in Beber and Pagano (2013) that the price effect of short-selling constraints is at most neutral, a conclusion drawn from a thorough empirical investigation of short-selling bans imposed in several countries during the financial crisis The second asset pricing quantity of interest is the return volatility. Overall, both constraints reduce the volatility, as predicted by the single-constraint models. The novel finding of the paper is that the joint impact of the constraints on volatility is most profound when both constraints bind intermittently. Put another way, their compound impact is more significant than the impacts from the single constraints. Intuitively, when both constrains are in place, the agents exposure to financial market risk is cut off both on supply and demand sides, while the cut-off is only on one of the sides in a single-constraint economy. The third asset pricing property I study is the speculative premium. As agents in my model do not change their identity over time, following Detemple and Murthy (1997), I define the premium for both agents: the speculative premium of the stock with respect to an agent is the gap between the spot price of that stock and the agent s marginal stock price valuation. This approach is contrasted with one in settings such as Harrison and Kreps (1978), Morris (1996), and Scheinkman and Xiong (2003), which specify speculative premium from the perspective of the agent with the highest valuation at a time. The model results demonstrate that, when the two constraints are alternately active, the speculative premium strictly exists as in Harrison and Kreps (1978) s sense. That is, the spot price of the stock exceeds stock marginal valuations of all agents in the economy. The speculative premium arises as an agent can hold the stock and resell in the future at a price higher than his long-term valuation when one of the constraints binds next. When only the borrowing constraint binds, the premium exists for the optimist, and when only the short-sale constraint binds, the premium exists 6

9 for the pessimist. In the former case, the highest valuation is the pessimist s, and it equals the spot price. In the other case, the optimist s valuation is higher and equals the spot price. The speculative behavior arises for the same reason as above. These findings imply that, even though the constraints can lower the stock market volatility as expected by regulators, they could also be a source of speculative bubbles. 2 Literature Review This paper is most closely related to the literature investigating general equilibrium impacts of financial constraints. Detemple and Murthy (1997) and Basak and Cuoco (1998) study general equilibrium properties of asset pricing quantities assuming that investors have logarithmic preferences. More specifically, Detemple and Murthy (1997) study impacts of short-sale and borrowing constraints when both constraints simultaneously appear in a continuous-time setting where agents are heterogeneous in beliefs. Basak and Cuoco (1998) examine restricted stock market participation, a special case of the limited stock market participation constraint, and find that this trading constraint has potential to explain asset pricing puzzles such as equity premium and excess volatility puzzles. In these papers, due to the logarithmic preferences, income and substitution effects perfectly offset each other, and hence portfolio constraints do not affect stock prices and stock return volatilities. Recent papers in the literature relax the assumption that agents preferences are logarithmic. Gallmeyer and Hollifield (2008) characterize equilibrium quantities in the presence of a shortsale constraint in a setting with two heterogeneous agents where the pessimist, who is short-sale constrained, is logarithmic, while the other agent, the optimist, has power preference. Bhamra and Uppal (2009) study a model in which two agents have power preferences and are borrowingconstrained. They find that borrowing constraints increase the stock market volatility. Recently, Chabakauri (2012, 2013) study the impacts of various trading constraints in more general settings where agents in the economy can adopt power preferences. Rytchkov (2014), on the other hand, focuses on time-varying borrowing constraints. Chabakauri (2013) and Rytchkov (2014) put em- 7

10 phasis on the economies with two Lucas-trees. Similar to Gallmeyer and Hollifield (2008) and Chabakauri (2012, 2013), while allowing for heterogeneous power preferences, my model also allows for belief heterogeneity, a model ingredient crucial for the binding of the short-sale constraint. However, while Chabakauri (2012, 2013) study borrowing and short-sale constraints separately, my model accommodates both constraints, faced by different agents, and examines the constraints interaction and their compound impacts on asset pricing quantities in equilibrium. At the international level, Pavlova and Rigobon (2008), Schornick (2009) and Schornick (2010) study models with constrained logarithmic investors with two Lucas trees. In Schornick (2009) s model of international asset markets, investors in different countries face constraints on portfolio choice. Namely, the home investor faces a leverage constraint, while the foreign investor faces a limited participation constraint. She finds that the lifting of constraints can increase or decrease the liberalized stock market s volatility, depending on how severely the constraint was binding before being removed. Schornick (2010) solves a model with foreign investors facing a leverage constraint and shows that restricting investors leverage can lead to rising interest rates. There is also an extensive strand of literature, of which examples are Basak and Croitoru (2000, 2006), Garleanu and Pedersen (2011), Hugonnier (2012), among others, that investigates the role of financial constraints on limited arbitrage. In this literature, it is shown that in the presence of constraints on portfolios, mispricings are required for equilibrium. Gromb and Vayanos (2010) provide a survey of this literature. This paper is also related to a large literature on incomplete markets and transaction costs. Heaton and Lucas (1996) endogenize asset prices in a setting where agents face idiosyncratic labour income risk and trade, partially due to borrowing constraints, short-sale constraints and transaction costs, in the financial market to buffer that risk. They find that the model can produce a sizable equity premium only if transaction costs are large or the assumed quantity of tradable assets is limited. 12 Vayanos (1998) and Vayanos and Vila (1999) also endogeneize asset prices in 12 In Heaton and Lucas (1996), short-sale constraints rarely bind. 8

11 the presence of proportional transaction costs, and obtain the prices in closed-form. Specifically, Vayanos (1998) finds that prices of assets carrying transaction costs increase in the costs, while Vayanos and Vila (1999) find that the prices of an illiquid asset may increase or decrease depending on its supply. Buss et al. (2011) study the equilibrium impact of proportional transaction costs in a setting in which agents have recursive preferences and find that transaction costs have small effects on stock prices and expected returns. Buss and Dumas (2012) study how transaction costs endogenously generate stochastic liquidity, which in turn, is shown to affect assets prices either through direct costs associated with trading assets or through the pricing kernel. 3 The Model I study a general equilibrium endowment economy in the spirit of (Lucas, 1978), on a finite, discrete time horizon [0, T] and a discrete state space, with a single perishable consumption good. 13 I consider the economy with one risky asset (stock). There are two price-taking groups of investors in the economy, who may differ in beliefs about the growth rates of the dividend process and in risk aversions. The two groups of investors are modeled as two representative agents. The agents may face borrowing and short-sale constraints when trading in financial markets. 3.1 Financial Markets Two financial assets, a bond and a stock, are traded. The stock is characterized by its dividends. The dividend dynamics is captured by the binomial process δ(t) = δ(0)u t k d k, where k, t [0, T], 0 k t. Note that s = u, d are, respectively, the step sizes to the "up" and "down" nodes, that follow a generic node ξ at time t. I follow Cox et al. (1979) in specifying the magnitudes of u and d. 14 In this model, the stock is in fixed supply of one share. The stock price of the stock, S (t), is induced backward in my model. I assume that, at the terminal date, the ex-dividend price is S (T) = 0. The bond pays a risk-free interest every period and its price, B t 13 Uncertainty in the economy is represented through a filtered probability space (Ω, F, {F t }, P) on which is defined a binomial process. 14 The binomial tree proposed in Cox et al. (1979) is a discrete-time approximation of a Geometric Brownian Motion. Note that the approximation is not unique. There are different ways to specify the magnitudes of u and d, and with each specification, there exists a corresponding probability, p, that the dividend will go up. 9

12 satisfies B t = r t (1) where r t is the one-period riskless interest rate in the economy. The bond is in zero net supply. 3.2 Agents Agents are heterogeneous in beliefs about the expected growth rates of the dividend process. Namely, the first agent is optimistic and overestimates the rates, while the second agent is pessimistic and underestimates the rates. The introduction of belief heterogeneity is crucial, as it is the channel for the short-sale constraint to bind endogenously in the current setup. Indeed, in an economy without belief heterogeneity, the agents perceived market prices of risk have the same sign, inducing all agents to either short or long the stock at the same time. To clear the market every agent must have a long position in the stock. Agents Beliefs In this model, agents derive all information from observations of the dividend process and agree on all states of nature. In the dividend binomial tree, the magnitudes u and d are common knowledge among agents. What make the agents heterogeneous are their perceptions of the probabilities of occurrence of the states. Agent i, i {1, 2}, assigns probability p i that the dividend will go up. The connection between the agent i s perceived mean of the dividend process, µ i, and the probability of the "up" state under his belief, p i, is characterized as follows: 15 p i = eµ i t d u d, (2) where t is the length of the time step in the tree. I use p 0 to denote the transition probability under the "true" probability measure. Agents Preferences 15 The connection between p i and µ i is established generally. It holds regardless of how u and d are specified. To obtain (2), we can let the expectations of the log-normally distributed dividends written in continuous-time (a Geometric Brownian Motion) and in the form of a binomial-tree identical. Some simple manipulation of the continuous-time with the use of Ito lemma is needed to get us the connection above. 10

13 The two agents have heterogeneous preferences, with local power utility functions. The accumulative utility of agent i as of time τ is T T U i,τ = E iτ β t u i,t (c i,t ) = E c 1 γ i iτ β t i,t 1 γ i, (3) t=τ where, u i,t is the local utility function of agent i; c i,t is the consumption allocated to agent i at time t; γ i is the risk aversion coefficient of agent i; and β is the time preference, which is identical among agents. The subscript i in the expectation operator is the agent index. The expectation of agent i is taken with respect to his information structure (Ω, F i, {F i τ}, P i ), and hence agent-specific. t=τ Portfolio Constraints At each time t, agent i chooses a combination (c i,t, ( α i,t, θ i,t )) of consumption and investments in the bond and the stock, respectively, to maximize his life-time utility. The agents may face borrowing and short-sale constraints when trading in financial markets. Notice that a borrowing constraint affects only the less risk-averse agent as he is the one who levers up. Similarly, a shortsale constraint affects only the more pessimistic agent. Accordingly, it suffices to consider a setup in which the first agent (the optimist) faces a borrowing constraint, and the second agent (the pessimist) faces a short-sale constraint. 16 The short-sale constraint requires the pessimist to hold a non-negative position in the stock, and is given by 17 θ2,t 0. (4) The borrowing constraint limits the amount of fund borrowed to finance stock investment, and is defined through α 1,t Bt mθ 1,t S t, (5) where m [0, 1] is the margin requirement. To be more specific, this constraint stipulates that 16 Technically, it is not more complicated to have all agents face both constraints in my model. However, because some of the constraints can be redundant for the reason given above, the current specification renders the clarity to the model setup as well as the subsequent equilibrium analysis with no loss of generality. 17 In Detemple and Murthy (1997), the short-sale constraint is more flexible. Specifically, agents in their model are allowed to short a limited amount of stock. The short-sale constraint is less strict in that case. 11

14 the first agent cannot borrow to finance more than the fraction m of the value of his investment in the stock. Hence, a lower m implies a tighter borrowing constraint. 18 It is worth emphasizing that, the borrowing constraint is specified endogenously and is related to the market condition as its determinants are asset prices, the endogenous quantities in the model. To understand the borrowing constraint formulation better, note that (5) can be rewritten as α 1,t B t mθ 1,t S t. Under the assumption that the first agent is optimistic about the economy, one would expect that his position for the stock is non-negative. Hence, the quantity on the right hand side of the inequality above is non-negative given that S t and m are non-negative. Therefore, the borrowing constraint can only bind when the first agent borrows ( α 1,t <= 0 ). When he lends ( α1,t >= 0 ), the constraint does not matter, since the inequality is obviously satisfied. 3.3 Investors Optimization Problems Agent i maximizes his life-time utility T ( ) max u i,0 (c i,0 ) + E i0 c i,t,α i,t,θ i,t β t u i,t ci,t, (6) subject to his budget and to trading constraints at each time t. As specified in the previous subsection, agent 1 is subject to is the borrowing constraint (5), while agent 2 is subject to the short-sale constraint (4). The budget constraint at a time t is t=1 c i,t + α i,t B t + θ i,t S t = α i,t 1 + θ i,t 1 (S t + δ t ), t = 0,...T. (7) In the budget constraint equation (7), on the left hand side is the sum of the consumption (c i,t ) and the financial investments (α i,t, θ i,t ) of agent i at time t, and on the right hand side is his wealth from investments in the financial assets in the previous period. Note that, in the current model, 18 In the literature, the widely adopted approach is that the investment portfolio of an agent is characterized by the portfolio weights, in terms of fractions of his wealth, of alternative financial assets. Thus, the borrowing capacity of an agent is related to his wealth [e.g., Detemple and Murthy (1997), Chabakauri (2012)]. In contrast, instead of wealth weights, I use asset holdings to characterize agents investment portfolios and relate an agent s borrowing directly to his stock investment. Consistent with this specification, wealth is not employed as a state variable in our model. 12

15 agents have no exogenous endowments other than the dividends. Following Dumas and Lyasoff (2012), to facilitate the numerical computation of equilibrium, I define agents exiting wealths and the entering wealths. 19 The exiting wealth of agent i at time t is his total financial investment at the time: X i,t = α i,t B t + θ i,t S t. (8) The entering wealth of agent i at time t is his wealth from financial investments at t 1: W i,t = α i,t 1 + θ i,t 1 (S t + δ t ). (9) Then, the budget constraint of agent i at time t can be rewritten as c i,t + X i,t = W i,t. (10) The value function of agent i is V i,t (W i,t ) sup { U i,t (c i,t ) }, (11) where, U i,t is the preferences over the consumption of agent i at time t, and is defined in (3). Specifically, the Lagrangian for agent 1 is L 1,t = u 1,t (c 1,t ) + βe t [ V1,t+1 (W 1,t+1 ) ] φ 1,t (c 1,t + θ 1,t S t + α 1,t B t W 1,t ) + κ t ( α1,t B t + mθ 1,t S t ), (12) and the Lagrangian for agent 2 is L 2,t = u 2,t (c 2,t ) + βe t [ V2,t+1 (W 2,t+1 ) ] φ 2,t (c 2,t + θ 2,t S t + α 2,t B t W 2,t ) + λ t θ 2,t, (13) where, φ i (t), κ t, and λ t are the Lagrange multipliers associated with the budget constraint of agent 19 Casting the financial investments at time t into the exiting wealth, X i,t, allows one to establish a one-to-one mapping between agent i s consumption c i,t at time t and his financial wealth, W i,t. That is, given a pair (c i,t, W i,t ), one can always find a "required exiting wealth" that satisfies the budget constraint at time t. 13

16 i, the borrowing constraint, and the short-sale constraint, respectively. One can then obtain the system of first order conditions that solves the individual optimization problems of the agents as follows. First, the system consists of equations related to the first derivatives of the Lagrangian with respect to the variables: L i,t c i,t = 0 :u i,t ( ci,t ) = φi,t, (14) L i,t = 0 :c i,t + θ i,t S t + α i,t B t = W i,t, (15) φ i,t L [ 1,t ( )] = 0 :βe 1t V 1,t+1 W1,t+1 = ( ) φ 1,t κ t Bt, (16) α 1,t L [ 1,t ( ) = 0 :βe 1t V 1,t+1 W1,t+1 (S t+1 + δ t+1 )] = ( ) φ 1,t mκ t S t, (17) θ 1,t L [ 2,t ( )] = 0 :βe 2t V 2,t+1 W2,t+1 = φ 2,t B t, (18) α 2,t L [ 2,t ( ) = 0 :βe 2t V 2,t+1 W2,t+1 (S t+1 + δ t+1 )] = φ 2,t S t λ t. (19) θ 2,t In addition, there are complementary slackness conditions, which arise due to the presence of the inequality constraints: λ t θ 2,t = 0, (20) κ t ( α1,t B t + mθ 1,t S t ) = 0. (21) Finally, the inequality constraints and the requirement that the Lagrange multipliers λ t and κ t be non-negative complete the characterization of first order conditions. Notice that from equations (16)-(19), we can obtain the expressions of the prices of the bond and the stock: B t = βe [ ] 1t φ1,t+1 = βe [ ] 2t φ2,t+1, (22) φ 1,t κ t φ 2,t S t = βe [ 1,t φ1,t+1 (S t+1 + δ t+1 ) ] = λ [ t + βe 2,t φ2,t+1 (S t+1 + δ t+1 ) ]. (23) φ 1,t mκ t φ 2,t 14

17 The equations imply that if the agents trade, they agree on the prices. As in Dumas and Lyasoff (2012), I name equations in the second group as "kernel conditions". 20 I cast the system of equations into four groups. The first group specifies the state price densities of the agents in the economy. The second group gathers the budget constraints faced by the agents. The third group characterizes the model feature that all agents in the economy observe the same price processes of the stock and the bond. The last group are complementary slackness conditions. Specifically, groups 1 consists of equations (14); group 2 consists of equations (15); group 3 consists of equations (22)-(23); and group 4 consists of equations (20)-(21). From the kernel conditions (22)-(23), one can establish preliminary insights into the trading constraints effects. 21 First, note that the Lagrange multipliers associated with inequalities, κ t and λ t, are positive by construction. Thus, when the borrowing constraint binds, it increases assets prices. Accordingly, it is expected that the interest rate decreases in the presence of the borrowing constraint. Further, the tighter the borrowing constraint, the larger the likelihood that the constraint binds. Hence, one would expect that increasing the tightness of the borrowing constraint reduces the interest rate. It is also expected that the tighter the borrowing constraint, the larger the increase in the stock price. Regarding the short-sale constraint, imposing the constraint introduces a gap between the stock price and the pessimist s marginal stock price valuation. In the case where only the short-sale constraint is present, the magnitude of the gap is exactly the shadow price of the short-sale constraint, hence when the constraint binds, the price of the stock exceeds the marginal valuation of the pessimist. When a borrowing constraint is also present, it affects the frequency with which the short-sale constraint binds, and effectively, it modifies the magnitude of the gap between the stock price and its valuation by the pessimist. In particular, due to the increase in the pessimist s perceived market price of risk, the demand for shorting stock by the pessimist decreases. Consequently, in the presence of an additional borrowing constraint, it is is less likely that 20 Mathematically, the conditions require the state prices, φ i,t+1, to lie in a linear subspace (Dumas and Lyasoff (2012)). 21 In a continuous-time setting with the duality approach of Cvitanic and Karatzas (1992), one can establish preliminary insights into the portfolio constraints impacts by judging the signs of the adjustment parameters, which are used to embed a constrained agent s optimization problem in an incomplete-market economy into an equivalent fictitious complete-market one. For instance, see Chabakauri (2012) on this issue. 15

18 the short-sale constraint binds, leading to a decrease in the gap. While the first order conditions provide certain insights into effects of the single constraints on the interest rate and the stock price, they keep silent on the constraints compound effects on those quantities and on the volatility. These effects are addressed in section Equilibrium A financial market equilibirum is defined as a set of securities prices, securities holdings and consumption allocations in the population such that the securities markets clear. Definition 1. An equilibrium is a price system (r(t), S (t)) and consumption-portfolio schedule (ci(t), (α i (t), θ i (t))) such that 1) given (r(t), S (t)) and beliefs, investors choose (ci(t), (α i (t), θ i (t))) that optimize expected utility at all times and in all states of the economy, 2) markets (consumption good market, stock markets and the bond market) clear, i.e., C(t) = δ(t), (24) θ 1 (t) + θ 2 (t) = 1, (25) α 1 (t) + α 2 (t) = 0. (26) Therefore, equilibrium is characterized by the system of equations (14)-(21), and the market clearing conditions (24)-(26). In the numerical implementation, I use equation (22)-(23) in replacement of equations (16)-(19). 22 Note that, in the system above, if we use the budget constraints at 22 In this paper, I characterize the equilibrium by gathering all first order equations from individual optimization problems. In closely related papers such as Gallmeyer and Hollifield (2008), Chabakauri (2012), a representative agent is formed, and his optimization problem is solved to characterize the equilibrium. The utility function of the constructed representative agent is given by: u(c; ψ) = max c 1 +c 2 =C u 1(c 1 ) + ψu 2 (c 2 ), (27) where ψ is the stochastic weight defined the the ratio of the two agents state price densities. Such an aggregation was introduced in Cuoco and He (1994), and has been extensively used in settings with incompleteness. 16

19 time t ci,t+1 + Xi,t+1 = αi,t + θi,t (S t+1 + δt+1), (28) in replacement of those at time t, and consistently shift the equations regarding state prices one period forwards at the same time, we obtain a new system of equations that characterizes the artificial fact that at time t, each agent optimally chooses his investment portfolio at that time and consumptions at time t + 1, instead of time t. That is, the consumption at time t is not a decision variable. Hence, as from Dumas and Lyasoff (2012), if we use the consumptions at time t as state variables, the new system of equations becomes purely backward. Reorganized this way, one can then solve the original system of equations backward all the way down to time 1, with a simple forward system of equations left at the initial date, t = 0. The appendix A elaborates this important point. The appendix B provides details of the numerical algorithm to compute equilibrium. 4 Equilibrium Analysis In this section, I perform an economic analysis of the equilibrium impacts of the borrowing and short-sale constraints at the aggregate level. I solve numerically the model with one Lucas-tree and study how the constraints interact and how such an interaction affects the asset pricing implications of the constraints. For completeness, I will also present insights into the effects of the single constraints. However, the ultimate goal of this section is to deliver the insights into the compound impact of the constraints in equilibrium. The baseline economy is one with both heterogeneities in beliefs and risk aversions. These two forms of heterogeneity effectively sustain the possibility of endogenous binding of the short-sale constraint and the borrowing constraint, respectively. It is well-known that with power preferences, it is not feasible to quantitatively match the empirical patterns regarding the equity premium and excess volatility. However, one can still qualitatively reproduce the empirical patterns with heterogeneous risk aversions. Our parametrization reflects this pattern-matching objective. 23 These conditions are named "marketability" in Dumas and Lyasoff (2012), as they impose that given a consumption and investment plan at time t + 1, there always exists an investment portfolio at time t that makes the plan feasible. 17

20 Specifically, in the baseline model, the optimist is less risk-averse and his belief about the dividend growth rate is correct, while the pessimist is more risk-averse and underestimates the growth rate of the economy. In that spirit, in the numerical exercises throughout this subsection, I set the model parameters as follows: The dividend growth rates under the true probability measure and the optimist s belief are µ δ = µ δ1 = 1.80%; the dividend growth rates under the pessimist s belief is µ δ2 = 1.20%; the dividend volatility is σ δ1 = σ δ2 = 3.20%; the risk aversion coefficients are γ 1 = 1, γ 2 = 3; and finally, the time preference is ρ = 1%. The parameter values set for dividend growth rates and dividend volatility are consistent with the numbers reported in Campbell (2003). Below, to understand the equilibrium effects of the constraints, I plot the quantities of interest against the single state variable of the model the optimist s consumption share, ω. A small ω means that the economy is dominated by the pessimist; and when ω increases from 0 to 1, the optimist gradually takes over the economy. 4.1 Properties of the Unconstrained Economy I first discuss, in the unconstrained economy context, the agents perceived market prices of risk, and their implications on the agents stock holdings. As the agents adopt different beliefs about the dividend growth rate, they perceive different market prices of risk for the stock. 24 The solid lines in plots (b) and (c) of Figure 1 respectively show that, in the unconstrained economy, the market price of risk perceived by the optimist is positive, while the pessimist s perceived market price of risk is negative, except for a narrow range of the optimist s consumption share, when it is small. Correspondingly, as plot (b) of Figure 2 indicates, when the pessimist perceives a negative price of risk he sells the stock short, and the optimist consistently holds a positive position for the stock to clear the market. The sign of an agent s perceived market price of risk, therefore, determines whether he goes long or short the stock. Bond holdings of the two agents in the unconstrained economy are also shown in Figure 2. The 24 Specifically, risk is transferred from the pessimist to the optimist, and the market price of risk of the optimist undergoes an upward deviation from the level of the economy with homogeneous beliefs, and is higher than the pessimist s. See Basak (2005) for details of a mathematical decomposition of an agent s perceived market price of risk into two components: (1) the market price of risk in homogeneous belief economy; (2) the difference-in-beliefs factor. 18

21 solid curve in plot (a) of the Figure shows that, in this economy, the pessimist always holds a positive amount of the bond. This makes him be a constant loan supplier. It must therefore be that the optimist is always a borrower, and hence constantly the supply side of the bond market. When the pessimist dominates the economy, the optimist holds a small fraction of the economy, and hence the borrowing demand is small. Accordingly, the bond supply is low, reflected through a modest bond holding of the pessimist. 25 When the optimist dominates the economy, the pessimist can only provide a limited supply of loans, hence his small bond holding. The bond holding of the pessimist peaks when the agents consumption shares are relatively balanced so that the pessimist s wealth is sufficiently large to fulfil, in full or in part, the loan demand. Finally, it is worth noting that the state variable, the optimist s consumption share, is procyclical. As discussed above, the optimist s exposure to the stock market is larger than the pessimist s. Therefore, a positive shock to the stock market shift the consumption to the optimist and conversely. Accordingly, in this economy the stock price and is procyclical, while the market prices of risk and the return volatilities are countercyclical. The patterns of those equilibrium quantities are consistent with the empirical facts Binding of the Constraints and Asset Holdings In a constrained economy, how the constraints bind is fundamental for further insights into properties of equilibrium quantities. Importantly, I find that a binding borrowing constraint reduces the likelihood of the short-sale constraint to bind, and vice versa. Hence, the constraints almost never bind at once. There exist different ranges of the optimist s consumption share, ω, in which either only one of them binds or they bind intermittently and alternately. Figure 3 depicts the binding of the constraints. Each plot in the figure presents the expected numbers of bindings of the constraints, Nb, for three different economies with different degrees of tightness of the borrowing constraint: m = (dashed line), m = (dash-dotted line), and 25 The bond is issued by the optimist, as he wants to borrow from the pessimist. As the net supply of the bond is zero, at a time, the bond holding of the pessimist amounts to the bond supply. 26 This is a general property of an economy with heterogeneous risk aversions. For instance, see Chan and Kogan (2002) and Chabakauri (2012). 19

22 m = 0.9 (dotted line), with a larger m for a less stringent borrowing constraint. Specifically, plots (a) and (b) respectively present the expected numbers of binding of the borrowing constraint and the short-sale constraint; plot (c) presents the expected number of times the two constraints simultaneously bind. Note that, for a constraint, Nb = 1 means that the constraint always binds; Nb = 0 means that the constraint never binds; and Nb (0, 1) means that the constraint binds intermittently and alternately. Plot (c) confirms the conclusion that the two constraints almost never bind at the same time in any of the economies considered. I consider an economy with a moderately tight borrowing constraint, m =, to elaborate the binding behaviour of the constraints. Plots (a) and (b) of Figure 3 together show that: in this economy, when the pessimist dominates the economy (ω < 0.4), only the borrowing constraint binds; when the optimist dominates the economy (ω > 5), only the short-sale constraints binds; and when the presences of two agents are relatively balanced (0.40 < ω < 5), the two constraints bind intermittently. Intuitively, the borrowing constraint is more effective when it is relatively easy for the optimist to lever up. This occurs when the economy is dominated by the pessimist, the loan supplier. In the same line of reasoning, when the consumption share of the optimist grows large, the borrowing constraint is less effective. The short-sale constraint s effect is therefore dominant for large values of the optimist s consumption share, while both constraints are active when the consumption share is around. I now further elaborate the mechanism through which a binding borrowing constraint dampens the binding of a short-sale constraint. The former constraint, once it binds, inflates perceived market prices of risk of all agents in the economy, as shown in plots (b) and (c) of Figure 1. This is a response to the decrease in the demand for the stock, which arises because the binding borrowing constraint dampens the optimist s ability to lever up. To clear the stock market, all the perceived market prices of risk are increased. Specifically, as the pessimist now perceives a positive market price of risk, he increases his stock holding to a positive amount. Accordingly, the short-selling constraint turns inactive. How a binding short-sale constraint reduces the likelihood of binding of the borrowing constraint can be understood in a similar way. When the short-sale constraint binds, 20

23 the optimist cannot hold more than one share of the stock. The demand for the bond is reduced, and the borrowing constraints is therefore less likely to be effective. The discussion above holds when we change the degree of tightness of the borrowing constraint, with the effective domains of the constraints shifting either to the left or to the right on the optimist s consumption share grid. For m = 0.9, the borrowing constraint is weak and almost always inactive, while the short-sale constraint is effective in the majority of the entire domain of the consumption share, excluding the part with very small values of the share. In contrast, for m =, the borrowing constraint is strictly tight and its effects are predominant for most values of the optimist s consumption share, while the short-sale constraint is dormant most of the times. 4.3 Key Equilibrium Asset Pricing Quantities Having knowledge of the constraints behavior in equilibrium, I now proceed to investigate the properties of the key asset pricing quantities: the interest rate, the stock price, the return volatility, and the speculative premium. I also further discuss the agents perceived market prices of risk. Figure 1 presents the interest rate and the market prices of risk. Figure 4 exhibits the other quantities. Each plot in the figures shows four economies: one with no constraint, and three others with two constraints. The constrained economies are differentiated by the degrees of tightness of the borrowing constraints: m =, m =, m = 0.9, as explained in subsection 4.2. As concluded previously, for m = 0.9, the borrowing constraint is almost inactive, the thick dotted line therefore mainly presents the quantities of the economy with a single short-sale constraint. Similarly, the dashed line for m = primarily shows the borrowing constraint s effects The Interest Rate and Market Prices of Risk Plot (a) of Figure 1 shows, relative to the unconstrained economy, a binding borrowing constraint always lowers the interest rate. In contrast, a binding short-sale constraint increases the interest rate when the pessimist dominates the economy and slightly reduces the interest rate otherwise. These results are consistent with the discussion of the constraints effects in subsection 3.3. The intuition is as follows. When the borrowing constraint binds, the demand for loans is reduced. Consistently, 21