General Equilibrium Under Convex Portfolio Constraints and Heterogeneous Risk Preferences

TL;DR

This paper develops a continuous-time equilibrium model with heterogeneous agents facing convex portfolio constraints, explaining observed leverage cycles and risk premiums in financial markets.

Contribution

It introduces a novel model incorporating heterogeneity and convex constraints, linking them to leverage dynamics and risk premiums, supported by empirical evidence.

Findings

Margin constraints increase market risk premiums

Heterogeneity and constraints produce pro- and counter-cyclical leverage cycles

Leverage cycles are both pro- and counter-cyclical, as predicted by the model

Abstract

This paper characterizes the equilibrium in a continuous time financial market populated by heterogeneous agents who differ in their rate of relative risk aversion and face convex portfolio constraints. The model is studied in an application to margin constraints and found to match real world observations about financial variables and leverage cycles. It is shown how margin constraints increase the market price of risk and decrease the interest rate by forcing more risk averse agents to hold more risky assets, producing a higher equity risk premium. In addition, heterogeneity and margin constraints are shown to produce both pro- and counter-cyclical leverage cycles. Beyond two types, it is shown how constraints can cascade and how leverage can exhibit highly non-linear dynamics. Finally, empirical results are given, documenting a novel stylized fact which is predicted by the model, namely that the leverage cycle is both pro- and counter-cyclical.

Tables

Table 1. Table 1: Examples of constraint sets, support functions, and effective domain of the adjustment ν i subscript 𝜈 𝑖 \nu_{i} .

	${\Pi }_i$	${\delta }_i\left(\nu \right)$	${𝒩}_i$
Unconstrained	$ℝ$	$0$	$0$
Margin Constraint	$\{\pi :\pi \le m_i;m_i\ge 0\}$	$-\nu m_i$	${ℝ}_{-}$
Short-Sale Constraint	$\{\pi :\pi \ge s_i;s_i\le 0\}$	$-\nu s_i$	${ℝ}_{+}$

Table 2. Table 2: Regression results for dependent variable Δ ​ l ​ n ​   ​ L ​ e ​ v Δ 𝑙 𝑛   𝐿 𝑒 𝑣 \Delta ln\text{ }Lev for different sectors of the economy. A positive and significant coefficient on Δ ​ l ​ n ​   ​ G ​ D ​ P Δ 𝑙 𝑛   𝐺 𝐷 𝑃 \Delta ln\text{ }GDP implies procyclicality, while a negative and significant coefficient on the interaction with S / D 𝑆 𝐷 S/D implies counter-cyclicality when the price dividend ratio is high. Note: Variables are normalized using z-score.

	Nonfinancial Corporations	Nonfinancial Private Business	HH’s and Nonprofits	All Sectors
	(1)	(2)	(3)	(4)
Intercept	$-0.0042$	$-0.0078$	$-0.0103$	$-0.0046$
	$\left(0.0618\right)$	$\left(0.0610\right)$	$\left(0.0599\right)$	$\left(0.0590\right)$
$\Delta ln$ $GDP$	$0.1034$	$0.2117\ast \ast$	$0.2827\ast \ast \ast$	$0.2397\ast \ast$
	$\left(0.0994\right)$	$\left(0.0980\right)$	$\left(0.0963\right)$	$\left(0.0949\right)$
$S/D$	$-0.0045$	$-0.0119$	$-0.0209$	$-0.2107\ast \ast$
	$\left(0.0874\right)$	$\left(0.0862\right)$	$\left(0.0846\right)$	$\left(0.0834\right)$
$\Delta ln$ $GDP\ast S/D$	$-0.1727$	$-0.2726\ast \ast$	$-0.3610\ast \ast \ast$	$-0.5157\ast \ast \ast$
	$\left(0.1070\right)$	$\left(0.1055\right)$	$\left(0.1036\right)$	$\left(0.1021\right)$

Table 3. Table 3: Regression results for dependent variable Δ ​ l ​ n ​   ​ L ​ e ​ v Δ 𝑙 𝑛   𝐿 𝑒 𝑣 \Delta ln\text{ }Lev for different sectors of the economy. A negative and significant coefficient on Δ ​ l ​ n ​   ​ G ​ D ​ P Δ 𝑙 𝑛   𝐺 𝐷 𝑃 \Delta ln\text{ }GDP implies counter-cyclicality, while a positive and significant coefficient on the interaction with r 𝑟 r implies pro-cyclicality when the interest rate is high. As opposed to Table   2 , r 𝑟 r is high when the risk averse agent dominates, exactly when the price-dividend ratio is low. Note: Variables are normalized using z-score.

	Nonfinancial Corporations	Nonfinancial Private Business	HH’s and Nonprofits	All Sectors
	(1)	(2)	(3)	(4)
Intercept	$0.0170$	$0.0838$	$0.0948$	$0.0227$
	$\left(0.0674\right)$	$\left(0.0638\right)$	$\left(0.0674\right)$	$\left(0.0672\right)$
$\Delta ln$ $GDP$	$-0.5144\ast \ast \ast$	$-0.4412\ast \ast$	$-0.5330\ast \ast \ast$	$-0.9689\ast \ast \ast$
	$\left(0.1907\right)$	$\left(0.1808\right)$	$\left(0.1908\right)$	$\left(0.1904\right)$
$r$	$0.0164$	$-0.0847$	$0.0018$	$0.0549$
	$\left(0.0789\right)$	$\left(0.0748\right)$	$\left(0.0790\right)$	$\left(0.0788\right)$
$\Delta ln$ $GDP\ast r$	$0.4121\ast \ast$	$0.4095\ast \ast$	$0.5131\ast \ast \ast$	$0.7603\ast \ast \ast$
	$\left(0.1700\right)$	$\left(0.1611\right)$	$\left(0.1701\right)$	$\left(0.1697\right)$

Equations

$$\frac{d{D}_t}{D_t}={\mu }_{D}dt+{\sigma }_{D}d{W}_t$$

$$\frac{d{S}_t}{S_t}$$

$$\frac{d{S}_t^0}{S_t^0}$$

$$u_i(c)=\frac{c^{1-{\gamma }_i}}{1-{\gamma }_i}$$

$$d{X}_{it}=\left[X_{it}\left(r_t+{\pi }_{it}\left({\mu }_t+\frac{D_t}{S_t}-r_t\right)\right)-c_{it}\right]dt+X_{it}{\pi }_{it}{\sigma }_{t}d{W}_t$$

$$\underset{\{c_{it},{\pi }_{it}{\}}_{t=0}^{\infty }}{\max }$$

$$d{X}_{it}=\left[X_{it}\left(r_t+{\pi }_{it}\left({\mu }_t+\frac{D_t}{S_t}-r_t\right)\right)-c_{it}\right]dt+X_{it}{\pi }_{it}{\sigma }_{t}d{W}_t$$

$${\pi }_{it}\in {\Pi }_i$$

$$\sum _i{c}_{it}=D_t\text{ , }\sum _i(1-{\pi }_{it})X_{it}=0\text{ , }\sum _i{\pi }_{it}{X}_{it}=S_t$$

$$\mathbf{\omega }=[{\omega }_{1t},\dots ,{\omega }_{(N-1)t}{]}^T=[c_{1t}/D_t,\dots ,c_{(N-1)t}/D_t{]}^T$$

$$\mathbf{\Omega }=[{\omega }_{1t},\dots ,{\omega }_{Nt}{]}^T=[c_{1t}/D_t,\dots ,c_{Nt}/D_t{]}^T$$

$$\frac{d{S}_{it}^0}{S_{it}^0}$$

$$\frac{d{S}_{it}}{S_{it}}$$

$${\delta }_i(\nu )=\underset{\pi \in {\Pi }_i}{\sup }(-\nu \pi )$$

$$\frac{d{H}_{it}}{H_{it}}=-(r_t+{\delta }_i({\nu }_{it}))dt-\left({\theta }_t+\frac{{\nu }_{it}}{{\sigma }_t}\right)d{W}_t$$

$$c_{it}={\left({\Lambda }_i{e}^{\rho t}{H}_{it}\right)}^{-\frac{1}{{\gamma }_i}}$$

$$\frac{c_{it}^{-{\gamma }_i}}{c_{jt}^{-{\gamma }_j}}=\frac{{\Lambda }_i{H}_{it}}{{\Lambda }_j{H}_{jt}}$$

$${\theta }_t$$

$$r_t$$

$$-\frac{1}{2}\sum _i\frac{1+{\gamma }_i}{{\gamma }_i^2}{\left({\theta }_t+\frac{{\nu }_{it}}{{\sigma }_t}\right)}^2{\omega }_{it})$$

$$r_t$$

$$\frac{d{\omega }_{it}}{{\omega }_{it}}$$

$${\mu }_{\omega it}$$

$$+{\sigma }_D^2-{\mu }_D$$

$${\sigma }_{\omega it}$$

$$d\mathbf{\omega }={\mathbf{\mu }}_{\mathbf{\omega }}dt+{\mathbf{\sigma }}_{\mathbf{\omega }}d{W}_t$$

$${\gamma }_i=\frac{1}{{\xi }_t}-\frac{{\Xi }_t}{{\xi }_t}\frac{1}{{\sigma }_D}+\frac{{\nu }_{it}}{{\sigma }_D{\sigma }_t}\text{where}{\xi }_t=\sum _i\frac{{\omega }_{it}}{{\gamma }_i}\text{ , }{\Xi }_t=\sum _i\frac{{\omega }_{it}{\nu }_{it}}{{\gamma }_i}$$

$$\frac{1}{{\gamma }_i}\left[(1-{\gamma }_i)(r(\mathbf{\omega })+{\delta }_i({\nu }_i(\mathbf{\omega })))-\rho +\frac{1-{\gamma }_i}{2{\gamma }_i}{\left(\theta (\mathbf{\omega })+\frac{{\nu }_i(\mathbf{\omega })}{\sigma (\mathbf{\omega })}\right)}^2\right]V_i(\mathbf{\omega })+$$

$$\left[\frac{1-{\gamma }_i}{{\gamma }_i}\left(\theta (\mathbf{\omega })+\frac{{\nu }_i(\mathbf{\omega })}{\sigma (\mathbf{\omega })}\right){{\mathbf{\sigma }}_{\mathbf{\omega }}}^T+{{\mathbf{\mu }}_{\mathbf{\omega }}}^T\right]\nabla V_i(\mathbf{\omega })+\frac{1}{2}{{\mathbf{\sigma }}_{\mathbf{\omega }}}^{T}H{V}_i(\mathbf{\omega }){\mathbf{\sigma }}_{\mathbf{\omega }}+1=0$$

$$\underset{{\omega }_i\to 1}{\lim }{V}_j(\mathbf{\omega })=\frac{{\gamma }_j}{\rho -(1-{\gamma }_j)\left(\frac{{\left(\theta (\mathbf{\omega })+{\nu }_i(\mathbf{\omega })/\sigma (\mathbf{\omega })\right)}^2}{2{\gamma }_j}+r(\mathbf{\omega })+{\delta }_j({\nu }_j(\mathbf{\omega }))\right)}$$

$${\pi }_i(\mathbf{\omega })$$

$${\nu }_i(\mathbf{\omega })=\min \left\{0;m{\gamma }_i\sigma (\mathbf{\omega }{)}^2\left(1-\frac{1}{m\sigma (\mathbf{\omega })}\left(\frac{\theta (\mathbf{\omega })}{{\gamma }_i}+\frac{{\mathbf{\sigma }}_{\mathbf{\omega }}(\mathbf{\omega }{)}^T\nabla \mathbf{V}(\mathbf{\omega })}{V_i(\mathbf{\omega })}\right)\right)\right\}$$

$${\pi }_{it}\le m_i\iff \frac{{\alpha }_{it}{S}_t}{{\alpha }_{it}{S}_t+b_{it}{S}_t^0}\le m_i\iff \frac{1}{1+\frac{b_{it}{S}_t^0}{{\alpha }_{it}{S}_t}}\le m_i\iff -\frac{b_{it}{S}_t^0}{{\alpha }_{it}{S}_t}\le \frac{m_i-1}{m_i}$$

$$ER{P}_t={\mu }_t+\frac{D_t}{S_t}-r_t={\theta }_t{\sigma }_t$$

$$Leverage=\frac{Liabilities}{NetWealth}=\frac{Liabilities}{Assets-Liabilities}$$

Full text

Keywords: Asset Pricing, Heterogeneous Agents, General Equilibrium, Financial Economics.

Introduction

Market incompleteness and individual heterogeneity are two important characteristics of financial markets. Many markets exhibit incompleteness, in the sense that one cannot freely choose their portfolio choices either because of constraints imposed by lenders, because of regulatory constraints, or simply because of a true incompleteness in the market. At the same time, in order to generate trade among individuals they must differ in some form. If all agents were identical then market prices would make them indifferent to their portfolio holdings and there would never be any trade. This paper seeks to combine these two facts about financial markets by combining portfolio constraints and preference heterogeneity, with a particular application to margin constraints.

Margin constraints increase the market price of risk and decrease the interest rate, contributing to a higher equity risk premium. The interest rate is low because the constraint limits the supply of risk free bonds to the market. This limit in supply pushes up the bond price and down the interest rate. On the other hand, the market price of risk is high because constrained agents are unable to leverage up to take advantage of high returns. On the opposite side of this constraint are risk averse agents who would like to sell their risky assets to reduce the volatility of their consumption. They are unable to do so, given the counter-party is the constrained agent. Thus margin constraints create an implicit liquidity constraint which allows the market price of risk to remain high in order to compensate risk averse agents for having a riskier consumption stream.

Asset prices are higher or lower than in an unconstrained equilibrium depending on whether the income effect or the substitution effect dominates. When agents are constrained, other agents are forced to hold more risky assets. These unconstrained agents hold both risky and risk-free assets, implying that an increase in the market price of risk and a decrease in the risk-free rate represent an ambiguous change in the investment opportunity set. However, the effect tends to increase the equity risk premium. This has the effect of increasing the discount rate and simultaneously makes agents wealthier today and makes consumption tomorrow less expensive. The first effect (an income effect) causes agents to desire to consume more today. The second effect (a substitution effect) causes agents to desire to consume less today and more tomorrow. Which effect dominates depends on the EIS of unconstrained agents. If EIS is less than one then the substitution effect dominates: individuals consume less today, pushing up their wealth and thus increasing asset prices. If EIS is greater than one then the income effect dominates: individuals consume more today, pushing down their wealth and thus reducing asset prices. In this way we can see either an increase or a decrease in asset prices when some portion of agents are constrained, depending on whether the EIS is greater or less than one.

Margin constraints and preference heterogeneity generate both pro- and counter-cyclical leverage cycles. Less risk averse agents dominate the economy and the price of risky assets is high when aggregate production is high. High asset prices increase individual wealth and reduce leverage. On the contrary, risk averse agents dominate when aggregate production is low, reducing asset prices. Low asset prices cause individual wealth to be low and individual leverage to be high. With the introduction of a margin constraint less risk averse agents eventually run into a borrowing limit. Not only is borrowing reduced, but, as discussed before, asset prices can be higher under constraint. In turn, total leverage falls. In this way heterogeneous preferences and margin constraints produce both pro- and counter-cyclical leverage cycles.

Financial leverage has become an important policy variable since the crisis of 2007-2008. In particular leverage allows investors to increase the volatility of balance sheet equity, producing the possibility of greater returns. At the same time leveraged investors are exposed to larger down-side risk. In the face of negative shocks, constrained investors must sell assets to reduce their leverage. This is known as the ”leverage cycle”. The associated credit contraction produces large volatility in asset prices and has been the target of regulation in the post-crisis era (e.g. the Basel III capital requirement rules). However, leverage cyclicality remains a topic of debate. Leverage cyclicality is both pro- and counter- cyclical in the model presented here, depending on the aggregate state of the economy and the marginal agent. In section 4 I document in the data that cycles are both pro- and counter-cyclical depending on the level of aggregate asset pricing variables which can be interpreted as proxies for marginal preferences. This fact could reconcile some of the empirical debates about the cyclicality of leverage and re-enforces the study of preference heterogeneity as a driver of financial trade.

Convex portfolio constraints arise quite naturally in finance. A convex constraint simply states that the portfolio weights must lie in a convex set containing zero (see Stiglitz and Weiss, (1981) for an example of micro-foundations to credit constraints). In macroeconomics there are countless examples of particular models with market incompleteness which can be described in this setting of convex constraints, such as Aiyagari, (1994); Kiyotaki and Moore, (1997); Krusell and Smith, (1998); Bernanke et al., (1999) and many others. A margin constraint essentially states that an agent cannot borrow infinitely against their equity. This type of constraint is seen in consumer finance when borrowing money to purchase a home: one must almost always put up a down payment. In financial markets margin constraints arise in repo markets and other lending vehicles (see Hardouvelis and Peristiani, (1992); Hardouvelis and Theodossiou, (2002); Adrian and Shin, 2010a for empirical studies of margins). In fact real world experience motivated the theoretical study of leverage cycles initiated by Geanakoplos, (1996). In addition limits to arbitrage and financial bubbles have been studied under margin constraints in the context of liquidity (see e.g. Brunnermeier and Pedersen, (2009)). Many of these phenomena arise in the model presented in this paper, but the predictions for leverage cycles are emphasized because of their novelty.

In theoretical models leverage cyclicality depends greatly on the underlying assumptions producing trade. In his foundational work on the topic, Geanakoplos, (1996) shows how the combination of belief heterogeneity and margin constraints produce a pro-cyclical leverage cycle. However, this finding is in opposition to the contemporary paper by Kiyotaki and Moore, (1997), where participation constraints force agents to invest through intermediaries, whose credit constraints produce a counter-cyclical leverage cycle. More recently He and Krishnamurthy, (2013) and Brunnermeier and Sannikov, (2014) also produce counter-cyclical leverage cycles by including intermediaries through whom constrained agents can profit from risky assets. In fact, He and Krishnamurthy, (2013) even points out the debate in the applied literature and the fact that, ”[Their] model does not capture the other aspects of this process, … that some parts of the financial sector reduce asset holdings and deleverage.” These models imply that the mechanism producing trade determines leverage’s cyclicality.

The empirical literature has noted this ambiguity over the cyclicality of leverage in different. Korajczyk and Levy, (2003) study the capital structure of firms and find that leverage is counter-cyclical for unconstrained firms and pro-cyclical for constrained firms. However, Halling et al., (2016) contradict this by showing that target leverage is counter-cyclical once you account for variation in explanatory variables, pointing out that the effect in Korajczyk and Levy, (2003) is only the ”direct effect”. In the cross section of the economy Adrian and Shin, 2010b find that leverage is counter-cyclical for households, ambiguous for non-financial firms, and pro-cyclical for broker dealers. However, the authors study the relationship between leverage and changes in balance sheet assets. This comparison produces a mechanical correlation which somewhat disappears when assets are replaced by GDP growth as a proxy for the business cycle (see section 4). Ang et al., (2011) point out that when accounting for prices broker dealer leverage is counter-cyclical, but that hedge fund leverage is pro-cyclical. These contrary studies can be reconciled when controlling for financial variables such as the price/dividend ratio or the interest rate. In fact, for several sectors studied (see section 4) the leverage cycle is both pro- and counter-cyclical. This ambiguity is predicted by the model of preference heterogeneity and margin constraints presented here.

Many authors have criticized the assumption of a representative, constant relative risk aversion agent since Mehra and Prescott, (1985) posited the equity risk premium puzzle. The definition of new utility functions was the first major response to this puzzle, in particular Epstein-Zin preferences (Epstein and Zin, (1989); Weil, (1989)) and habit formation (Campbell and Cochrane, (1999)) have been used to explain the equity premium puzzle. However, several papers have studied preferences across individuals and found them to be heterogeneous and constant in time (Brunnermeier and Nagel, (2008); Chiappori and Paiella, (2011); Chiappori et al., (2012)), contradicting both of these new branches of the theoretical literature. In addition, Epstein et al., (2014) pointed out that the assumptions necessary to match the risk premium using Epstein-Zin preferences produce unrealistic preference for early resolution of uncertainty. Beyond these criticisms, one needs heterogeneity in order to generate trade at all in any market model. In a representative agent setting one looks for the prices which make the agent indifferent to not trading. Risk preference heterogeneity has succeeded in partially responding to these issues.

Heterogeneity in risk preferences has been used to generate trade in financial models since the foundational paper of Dumas, (1989). Since then many authors have studied the problem from different angles, assuming different levels of market completeness, utility functions, participation constraints, information structures, etc., but almost always under the assumption of only two preference types (Basak and Cuoco, (1998); Coen-Pirani, (2004); Guvenen, (2006); Kogan et al., (2007); Guvenen, (2009); Cozzi, (2011); Garleanu and Pedersen, (2011); Hugonnier, (2012); Rytchkov, (2014); Longstaff and Wang, (2012); Prieto, (2010); Christensen et al., (2012); Bhamra and Uppal, (2014); Chabakauri, (2013, 2015); Gârleanu and Panageas, (2015); Santos and Veronesi, (2010)). Cvitanić et al., (2011) studies the problem of $N$ agents with several dimensions of heterogeneity and focuses on the dominant agents, characterizing portfolios via the Malliavan calculus. Abbot, (2017) studies a setting with $N$ heterogeneous CRRA agents in a complete financial market using a value function approach and shows how changes in the number of types can produce substantially different quantitative results and how the variance in preferences provides an additional degree of freedom for explaining the equity risk premium puzzle. However, that work produces large amounts of aggregate leverage and high individual margins. This observation points towards the need to introduce some degree of constraint or incompleteness to better match the real world. To that end, this paper studies the same type of economy with $N$ heterogeneous CRRA agents under convex portfolio constraints with an application to margin constraints.

A fundamental paper by Cvitanić and Karatzas, (1992) studied the general case of convex portfolio constraints in partial equilibrium. The authors developed an ingenious way to embed the agent in a series of fictitious economies, parameterized via a sort of Kuhn-Tucker condition, and then to select the appropriate market to make the agent just indifferent. However, their approach was to use convex duality to characterize the solution, which relies on a strict assumption that the relative risk aversion be bounded above by one. This limitation led others to look to solve the primal problem directly, such as He and Pages, (1993); Cuoco and He, (1994); Cuoco, (1997); Karatzas et al., (2003). These works use dense and complex mathematical techniques which may or may not provide tractable solutions for calculation. The present paper takes a more direct approach to solve the primal problem by noticing that homogeneous preferences are associated to a value function which factors into a function of wealth and a function of the aggregate state, under the appropriate ansatz. Using this ansatz, the Hamilton-Jacobi-Bellman equation becomes a PDE over consumption weights.

1 A Model of Preference Heterogeneity

1.1 Financial Markets

Consider a continuous time, infinite horizon Lucas, (1978) economy with one consumption good. This consumption good, denoted $D_{t}$, is produced by a tree whose dividend follows a geometric Brownian motion (GBM):

[TABLE]

where $W_{t}$ is a standard Brownian motion and $(\mu_{D},\sigma_{D})$ are constants. Agents can trade in a (locally) risk-free and a risky security, whose prices are denoted $S^{0}_{t}$ and $S_{t}$ respectively. These prices are assumed to follow an exponential and an Itô process, respectively:

$\displaystyle\frac{dS_{t}}{S_{t}}$ $\displaystyle=\mu_{t}dt+\sigma_{t}dW_{t}$

(1)

$\displaystyle\frac{dS^{0}_{t}}{S^{0}_{t}}$ $\displaystyle=r_{t}dt$

(2)

where $(\mu_{t},\sigma_{t},r_{t})$ are determined in equilibrium. Individuals are initially endowed with a share in the per-capita tree, $\alpha_{i0}$, and a position in the risk-free asset, $\beta_{i0}$.

1.2 Preferences and Wealth

The economy is populated by an arbitrary number $N$ of atomistic agents indexed by $i\in\{1,\dots,N\}$. Agents have constant relative risk aversion (CRRA) preferences and differ in their rate of relative risk aversion, $\gamma_{i}$, such that their instantaneous utility is given by

[TABLE]

Denote by $X_{it}$ an individual’s wealth at time $t$ and note that initial wealth is given by $X_{i0}=\alpha_{i0}S_{0}+\beta_{i0}S^{0}_{t}$. Denote by $\pi_{it}$ the share of an individual’s wealth invested in the risky stock, which implies $1-\pi_{it}$ is the share invested in the bond. Assuming that trading strategies are self financing, an individual’s wealth evolves as

[TABLE]

1.3 Portfolio Constraints and Individual Optimization

Individual investors solve a utility maximization problem subject to their self-financing budget constraint and a portfolio constraint:

[TABLE]

where $\Pi_{i}\subseteq\mathbb{R}$ represents a closed, convex region of the portfolio space which contains $\{0\}$. For example $\Pi_{i}=\mathbb{R}$ is the unconstrained case, $\Pi_{i}=\mathbb{R}^{+}$ is a short sale constraint, $\Pi_{i}=\{\pi:\pi\leq m_{i}\hskip 5.0pt|\hskip 5.0ptm_{i}\geq 0\}$ is a margin constraint. This set is allowed to differ across agents, as implied by the subscript. This paper focuses on an application to margin constraints, but the approach is applicable to any constraint which can be written as a function of the aggregate state.

1.4 Equilibrium

Investors are considered to be atomistic and thus I consider a Radner, (1972) type equilibrium.

Definition 1**.**

*An equilibrium in this economy is defined by a set of processes

$\{r_{t},S_{t},\{c_{it},X_{it},\pi_{it}\}_{i=1}^{N}\}\hskip 5.0pt\forall\hskip 5.0ptt$, given preferences and initial endowments, such that $\{c_{it},X_{it},\pi_{it}\}$ solve the agents’ individual optimization problems and the following set of market clearing conditions is satisfied:*

[TABLE]

I study Markovian equilibria such that equilibrium quantities can be written as functions of some state vector. That is for some equilibrium process $Y_{t}$, I look for functions $f(\cdot)$ such that $Y_{t}=f(z_{t})$ for some process $z_{t}$. I will look for a particular equilibrium in the vector of consumption weights defined by

[TABLE]

This is in the spirit of Chabakauri, (2013, 2015), where given two agents we can take the consumption weight of a single agent as the state variable. I’ve only included $N-1$ consumption weights because the last is determined by market clearing. However, for some equations the full vector of weights is useful, so I will define

[TABLE]

The following section will describe how to characterize equilibrium processes in terms of these quantities.

2 Equilibrium Characterization

To solve this problem I begin with the approach of Cvitanić and Karatzas, (1992). This method uses a fictitious, unconstrained economy and a shadow cost of constraint, or Lagrange multiplier, to find the correct pricing process. Unlike in their work I do not use a duality approach, but show how the primal problem admits a Markov representative. The reason this works is because when preferences are homothetic they can be represented by a utility function which is homogeneous of some degree. In this case the value function factors and the resulting ODE is no longer a function of individual wealth. This approach will likewise work for any homogeneous utility function, including Epstein-Zin111In particular, first order conditions from a dynamic program give consumption as $c=u^{\prime-1}\left(\partial_{X}J(X,Y)\right)$, where $Y$ is any arbitrary, aggregate state vector and $X$ an individual’s wealth. We would like to find $c=X/V(Y)$. Equate these and rearrange to find $\partial_{X}J(X,Y)=u^{\prime}(X/V(Y))$. When the utility function is homogeneous of degree $k+1$, $u^{\prime}(\cdot)$ is homogeneous of degree $k$. Thus $\partial_{X}J(X,Y)=u^{\prime}(1)V(Y)^{-k}X^{k}$. By integrating with respect to $X$ one finds a proposal for the value function such that consumption is a linear function of wealth.. This process will be described in the following subsections.

2.1 Optimality in Fictitious Unconstrained Economy

In order to find the constrained equilibrium, we define new processes for individual prices, which are ”adjusted” by a process $\nu_{it}$, considered the shadow cost of constraint:

[TABLE]

The function $\delta_{i}(\cdot)$ is the support function of $\Pi_{i}$, which is defined as

[TABLE]

In addition, this gives rise to the effective domain of $\nu_{it}$ defined by $\mathcal{N}_{i}=\{\nu\in\mathbb{R}:\delta_{i}(\nu)<\infty\}$ (for examples see Table 1). Finally, we have a complimentary slackness condition which states $\nu_{it}\pi_{it}+\delta_{i}(\nu_{it})=0$. Each agent solves their optimization problem in the face of their individual, fictitious financial market.

Define the stochastic discount factor (SDF) of an individual agent as an Itô process which evolves as a function of the individual’s adjustment:

[TABLE]

By a straight-forward application of the martingale approach (Karatzas et al., (1987)) in this fictitious economy one finds individual consumption as a function of individual SDF’s:

[TABLE]

for all $i$, where $\Lambda_{i}$ is the Lagrange multiplier associated to the static budget constraint. In the case where $\Pi_{i}=\mathbb{R}$ $\forall$ $i$, the SDF’s coincide and the ratios of marginal utilities are constant. However, when agents are constrained in their portfolio choice this is not the case and we have

[TABLE]

These ratios of SDF’s, which are proportional to ratios of marginal utilities, are very familiar in the theory of incomplete market equilibria. In Cuoco et al., (2001), a representative agent with state dependent preferences is studied, where the preferences are a weighted average of individual preferences. The stochastic weights are exactly equal to the ratio of marginal utilities. This is also seen in Basak and Cuoco, (1998) and Hugonnier, (2012).

2.2 General Equilibrium Characterization

Equilibrium is characterized by first assuming the existence of a Markovian equilibrium, deriving a system of ODE’s for wealth-consumptions ratios, then recovering the adjustments $\nu_{it}$ using the complimentary slackness conditions. Given this it is possible to prove optimality of the value functions222This remains a claim at this point. The proof is ongoing.. First, consider the interest rate and market price of risk:

Proposition 1**.**

The interest rate and market price of risk can be shown to be functions of weighted averages of individuals’ consumption weights, preference parameters, and adjustments such that

[TABLE]

The interest rate and market price of risk take a typical form, but are augmented by the adjustment to individuals’ marginal utilities. First notice that the market price of risk (Eq. 6) is determined by the fundamental volatility $\sigma_{D}$ divided by the weighted average of elasticity of intertemporal substitution (EIS), exactly as in complete markets (Abbot, (2017)). In addition the constraint will either increase or reduce the market price of risk, depending on the domain of $\nu_{it}$. In the case of margin constraints $\nu_{it}\leq 0$, so the market price of risk will be weakly higher under constraint. This is driven by an implicitt liquidity constraint. Constrained agents are unable to take advantage of high returns. In addition, the effect of volatility implies that in times when stock price volatility is low, greater constraint implies greater returns. This correlation is again driven by the fact that agents cannot borrow to take advantage of the returns, producing the same type of liquidity effect described in the limits-to-arbitrage literature (e.g. Brunnermeier and Pedersen, (2009) or Hugonnier, (2012)). Risk neutral agents would arbitrage away the high returns, but cannot because of their margin constraint.

The interest rate similarly exhibits a familiar shape. We see a rate of time preference term, an intertemporal smoothing term, and a prudence or risk preference term:

[TABLE]

Both the intertemporal smoothing and prudence terms are augmented by the constraint. Under a homogeneous margin constraint, $\delta_{i}(\nu_{it})=-m\nu_{it}$, but recall that $\nu_{it}\leq 0$, which together imply that the constraint reduces interest rates through the intertemporal smoothing term. Constrained agents are unable to supply bonds to the market in order to transfer consumption and wealth from the future to today. A lower supply of bonds pushes up the price and down the interest rate. At the same time constraint affects the interest rate through the prudence motive by changing the demand for precautionary savings. Individuals demand more precautionary savings when their SDF is more volatile (Kimball, (1990)). When agents are constrained, their SDF is less volatile as they are unable to increase their exposure to fundamental risk. Ceterus paribus, this reduces the demand for precautionary savings and increases the interest rate, counteracting the intertemporal motive. Together these forces produce an equity risk premium which depends on the shape of heterogeneity, the degree of constraint, and the state variable, all driven by the individual consumption weights which determine the marginal agents.

How consumption weights evolve over time is important not only from an economic perspective, but also in order to derive the solution of the model. We can study the dynamics of consumption weights by applying Itô’s lemma and matching coefficients to find their drift and diffusion:

Proposition 2**.**

Consumption weights follow an Itô process whose dynamics are given by:

[TABLE]

This implies that the state variable $\bm{\omega}$ follows an Itô process such that

[TABLE]

where $\bm{\mu_{\omega}}=[\mu_{\omega 1t}\omega_{1t},\dots,\mu_{\omega(N-1)t}]^{T}$ and $\bm{\sigma_{\omega}}=[\sigma_{\omega 1t}\omega_{1t},\dots,\sigma_{\omega(N-1)t}]^{T}$

These equations are very similar to those one finds in the complete markets case (Abbot, (2017)), but augmented by the constraint. In particular, consider the volatility of consumption weights given in Eq. 10. An agent’s consumption volatility is exactly zero when their preference parameter satisfies

[TABLE]

We can think of this as the marginal preference level in the market for consumption. However, it is possible that this preference level is not unique. Consider the case where some agents face a margin constraint, but others do not. Amongst the unconstrained agents, the marginal preference level corresponds to the first two terms, while among the constrained agents all of the terms matter. Given $\nu_{it}\leq 0$ under margin constraints, there could very well exist both a constrained and an unconstrained agent who have zero consumption volatility. This is driven by the constrained agents being unable to leverage up to gain more exposure to aggregate risk.

Since $\theta_{t}$ and $r_{t}$ are functions of $\{\omega_{it}\}_{i=1}^{N}$, it remains to show that $\{\nu_{it}\}_{i=1}^{N}$ are as well. First, one can derive a system of PDE’s for individual wealth/consumption ratios, from which one can determine the adjustments.

Proposition 3**.**

Given Propostions 1 and 2 and assuming adjustments and volatility are functions of $\bm{\omega}$ such that $\nu_{i}(\bm{\omega})=\nu_{it}$ and $\sigma(\bm{\omega})=\sigma_{t}$, it is possible to define the interest rate and market price of risk as functions of $\bm{\omega}$ such that $r_{t}=r(\bm{\omega})$ and $\theta_{t}=\theta(\bm{\omega})$. Assuming there exists a Markovian equilibrium in $\bm{\omega}$, the individuals’ wealth-consumption ratios, $V_{i}(\bm{\omega})=X_{it}/c_{it}$, satisfy PDE’s given for each $i$ by

[TABLE]

*where $\nabla$ and $H_{\bm{\omega}}$ represent the gradient and hessian operators, and where $\bm{\mu_{\omega}}$ and $\bm{\sigma_{\omega}}$ are given in Proposition 2.

Boundary conditions when a single agent dominates are given by the autarkical case, where

[TABLE]

Boundary conditions when an agent’s weight goes to zero are given by the solution to an $N-1$ agent problem.

The boundary conditions when a single agent dominates represent the vertices of the state space. On the other hand, when an agent’s weight goes to zero, the economy solution is equivalent to a two agent economy, with the zero agent’s wealth/consumption ratio still satisfying LABEL:eq:prop:pde, but their choices having no effect on aggregate variables. These partial differential equations represent the shape of individuals’ wealth/consumption ratios over the state space. Unlike in complete markets, however, the system is highly non-linear, since the coefficients depend in a complicated way on the solution itself.

Next, consider the portfolios of individuals, given in Proposition 4.

Proposition 4**.**

Assuming adjustments and volatility can be written as functions of $\bm{\omega}$ such that $\nu_{i}(\bm{\omega})=\nu_{it}$ and $\sigma(\bm{\omega})=\sigma_{t}$, it can be shown that portfolios are functions of $\bm{\omega}$ such that $\pi_{i}(\bm{\omega})=\pi_{it}$, where

[TABLE]

where $\bm{\sigma}_{\bm{\omega}}(\bm{\omega})=[\sigma_{\omega i}(\bm{\omega})\omega_{i}]_{i}^{T}$ is the vector of diffusions of $\bm{\omega}$.

One can see right away that portfolios take the typical ICAPM form (Merton, (1971)). There is first a myopic term, represented by the market price of risk scaled down by risk aversion and volatility, which gives the instantaneous portfolio demand of an individual given the market price of risk. Next is a hedging term, determined by the co-movement of an individual’s wealth with the aggregate state. Finally, there is a constraint term, which compensates the individual’s portfolio such that they are within the constraint set.

On an aggregate level, we can derive asset pricing variables from an application of Itô’s lemma and from market clearing for wealth.

Proposition 5**.**

Assuming adjustments can be written as functions of \bm{\omega}suchthat$ν_i(**ω**) = ν_it$,itcanbeshownthatvolatilityandthepricedividendratioarefunctionsof$**ω$suchthat$**σ(**ω**) = σ_t$and$*S*(**ω**) = S_t/D_t$,where\begin{equation}\begin{aligned} \sigma(\bm{\omega})=\sigma_{D}+\frac{\bm{\sigma}{\bm{\omega}}(\bm{\omega})^{T}\left(\nabla\bm{V}(\bm{\omega})+\bm{J{V}}(\bm{\omega})^{T}\bm{\Omega}\right)}{\mathcal{S}(\bm{\omega})}\end{aligned}\end{equation}where$**J_V**(**ω**)$representstheJacobianmatrixandwhere\begin{equation}\begin{aligned} \mathcal{S}(\bm{\omega})=\sum_{i}\omega_{i}V_{i}(\bm{\omega})\end{aligned}\end{equation}representsthepricedividendratio$S_t/D_t. Volatility in Eq. 15 is driven by the fundamental volatility, the shape of wealth consumption ratios, and the volatility of consumption weights. When agents have high volatility in consumption weights, the volatility of asset prices will be higher. At the same time, individuals’ wealth will be less volatile under constraint. This will produce a reduction in volatility. We will see these two forces in the numerical simulations in section 3.

We need to derive an expression for $\{\nu_{it}\}_{i=1}^{N}$ in order to close the model. The functional form depends on the type of constraint. To that end, I will focus from here only on margin constraints. The following proposition gives the functional form for the adjustments under homogeneous margin constraints when $\pi_{it}\leq m$ for all $i$, where $m\geq 0$, which implies an effective domain of $\mathcal{N}_{i}=\{\nu:\nu\leq 0\}$ and a support function of $\delta_{i}(\nu)=-m\nu$.

Proposition 6**.**

Under margin constraints, adjustments can be written as functions of $\bm{\omega}$ such that $\nu_{i}(\bm{\omega})=\nu_{it}$, where

[TABLE]

Finally, we need a verification argument for optimality of the value functions. In particular, we would like to be sure that solution to the PDE’s in Proposition 3 are indeed the wealth/consumption ratios associated to the individuals optimal choices. If we are willing to make the assumption that the wealth/consumption ratios are twice continuously differentiable, then we can easily show using Itô’s lemma that the value functions are indeed optimal (this proceeds as in Chabakauri, (2015)). However, it would be preferable to relax this assumption. To do so we can make use of a powerful new result in Confortola et al., (2017), namely that under certain conditions the value function implied by Proposition 3 is indeed optimal333This is left as a claim, as only an outline of a proof has been completed..

Claim 1**.**

Assuming that individual wealth/consumption ratios $V_{i}(\bm{\omega})$ are $C^{1}$ with bounded first derivative, then there exists a unique solution to LABEL:eq:prop:pde (in the viscosity sense) and this solution corresponds to the value functions in Eq. 22. Furthermore this represents a Markovian equilibrium satisfied by Propostions 1, 2, 4, 3, 6 and 5.

This claim relies only on a single degree of differentiability444I believe this condition can be relaxed to simply Lipschitz continuity..

3 Numerical Solution

This section presents numerical results for several assumptions about the distribution of preferences. First, the case of two types is evaluated and the cyclicality of the leverage cycle is emphasized. The leverage cycle is pro- or counter-cyclical depending on the marginal agent. Second, results are presented for three agents. Two key features which are not observed in the two agent case are the possibility of cascading constraints and a highly non-monotonic leverage. When one agent is constrained, other agents tend to hold more leverage. This pushes the intermediate agent closer to their own constraint. At the same time, this increase in individual leverage can partially or even fully offset the reduction in total leverage generated by the first agent’s constraint. Over all simulations I hold fixed $(\mu_{D},\sigma_{D},\rho)=(0.01,0.032,0.02)$, chosen to compare to Chabakauri, (2015)**.

3.1 Two Types and Leverage Cycles

Consider the case of two agents555The two agent model represents the boundary of the three agent problem, so is its solution is necessary to treat the three agent case. In addition, understanding the shape of functions in this simple case will help to fix ideas in the more complex case of arbitrary number of types.* with relative risk aversion $(\gamma_{1},\gamma_{2})=(1.1,5.0)$ who face a margin constraint such that the share, $\pi_{it}$, of their wealth invested in the risky asset is less than some constant $m_{i}$. This is equivalent to a leverage constraint:*

[TABLE]

In particular, take $m_{i}=m=1.2$.

There will exist a region of the state space over which this constraint binds for the less risk-averse agent. In this region, the more risk averse agent holds a larger share of their wealth in the risky asset. In order to achieve these portfolio weights, the constrained agent holds fewer risky shares and the unconstrained agent holds more risky shares. By reducing their risky shares, the less risk-averse agent’s constraint actually tightens, causing them to sell more risky-shares, making the effect more than proportional. This corresponds to a substantial decline in leverage and a tightening of credit demand, pushing down the interest rate. In addition, the market price of risk is high in order to compensate the risk averse investor for holding a larger share. Whether these two effects combine to make asset prices higher or lower depends on whether the income or wealth effect dominates.