Extension: Two Risky Assets

III. Extension: Two Risky Assets

So far, I assume that the economy contains only one risky asset. In this sec- tion, I extend the analysis to an economy with two Lucas trees and explore how the imposition of margin requirements changes the volatilities of the assets and

Asset Pricing with Dynamic Margin Constraints

425 the correlation between their returns. 13 I assume that the dividends of risky

assets i = 1, 2 follow geometric Brownian motions with identical parameters:

To avoid unnecessary complications, the innovations of the standard Brownian motions dB 1t and dB 2t are assumed to be uncorrelated. In matrix notation, they

form a vector dB t = [dB 1t dB 2t ] ′ . The excess returns on the risky assets evolve according to

(18) where the 2 × 1 matrix of risk premia μ Q and the 2 × 2 matrix of diffusions Q

dQ t =μ Q dt + Q dB t ,

are determined in the equilibrium. As before, one of the state variables is the consumption share of type B agents s t =C B t /(D 1t +D 2t ), which evolves according to

(19) where μ s and s are determined by equilibrium conditions. Following the lit-

ds t =μ s dt + s dB t ,

erature on exchange economies with multiple Lucas trees (e.g., Menzly, San- tos, and Veronesi ( 2004 ), Cochrane, Longstaff, and Santa-Clara ( 2008 ), Martin ( 2013 )), the other state variable is chosen to be the first asset’s share in the aggregate dividend: u t =D 1t /D t , where D t =D 1t +D 2t . Using u t , the dynamics of the aggregate dividend can be described as

where D (u) = [σ D uσ D (1 − u)]. The stochastic equation for u t immediately follows from equation (17) ,

(21) where μ u (u) = σ 2 D u(1 − u)(1 − 2u),

du t =μ u (u t ) dt + u (u t ) dB t ,

u (u) = [σ D u(1 − u) − σ D u(1 − u)]. (22) Note that the dynamics of u t are fully determined by model assumptions, so

only the evolution of s t is an equilibrium outcome, just as in the model with a single Lucas tree.

Again, I assume that there are two types of CRRA agents (type A and type

B) with different coefficients of risk aversion. For the sake of tractability, type

B agents are assumed to have logarithmic preferences. Although the latter

13 A model with multiple trees and margin constraints is also considered in Garleanu and Pedersen ( 2011 ). However, the authors focus on changes in expected returns and asset valuations

produced by constraints, whereas my objective is to study the volatilities and the correlations of returns.

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assumption eliminates the hedging demand of type B investors, it substantially simplifies the characterization of the equilibrium.

The key ingredient of the model is margin requirements imposed on agents. In the model with one risky asset, the requirements constrain only long- leveraged positions (effectively, short positions in the risk-free asset). Although constraints on short positions in the stock could also be imposed, they are ir- relevant in the equilibrium because no one wants to short a risky asset. The situation is drastically different when there are multiple stocks in the econ- omy. If margin requirements are imposed only on long positions, agents can circumvent them by shorting one of the stocks instead of the risk-free asset. Al- though in such a case the equilibrium properties would be different from those in the unconstrained economy, the lack of margin requirements on the short side is unrealistic. This point is emphasized in Brunnermeier and Pedersen ( 2009 ) and Garleanu and Pedersen ( 2011 ). These papers argue that it is more appropriate to assume that the constraints are imposed on both long and short positions and have the form

i , i = 1, 2, are portfolio weights of risky assets, and ω i = max(ω i , 0) and ω i = max(−ω i , 0) are long and short positions, respectively. In general, the pa- rameters of the constraints m + i and m − i can be state dependent and determined by various endogenous variables like the volatility of returns. Effectively, equa- tion (23) imposes four linear constraints on portfolio weights that can be written as

ω ′ m K ≤ 1, K = 1, . . . , 4,

1 m 2 ] ,m = [m 1 m 2 ] , and m = [m 1 m 2 ] . Motivated by this observation, I assume that the margin re-

where ω = [ω 1 ω 2 ] ′ and m 1 = [m + 1 m + 2 ] ′ ,m 2 = [m +

quirements have the form of equation (24) with

(25) where

mσ ¯ D 1 mσ ¯ D −1

mσ ¯ D 1 mσ ¯ D −1 (26)

and ¯ m is a constant that determines the average tightness of the constraints. The suggested specification of margin requirements deserves several com- ments. First, it represents a generalization of the one-asset constraint from equation (5) with the threshold from equation (16) and α = 1. As demonstrated

in Section I , the equation for σ s (s) simplifies when the tightness of constraints is inversely proportional to volatility, and the two differential equations that

427 describe the equilibrium dynamics can be solved separately. A very similar

Asset Pricing with Dynamic Margin Constraints

simplification occurs in the multiasset case, and, together with the assump- tion of logarithmic preferences, it results in a closed-form solution for s (see

Proposition 4 ). This makes the model more tractable. Second, margin requirements have the same functional form for both assets. Although it is easy to incorporate cross-sectional heterogeneity of margin re- quirements in the model, I deliberately avoid it to ensure that all cross-sectional effects are purely endogenous and not produced by the assumed differences in margins. Note that in the vast majority of states the tightness of margin con- straints is not the same for the two assets because it is determined by asset volatilities, which vary across stocks.

Third, the constraint in the form of equation (25) implies that the exposures of the portfolio to the individual shocks dB 1 and dB 2 are collateralized sepa- rately. When returns are uncorrelated, the matrix Q is diagonal and separate collateralization of shocks reduces to separate collateralization of individual assets.

The definition of the equilibrium is standard and mimics its analog from Section I : it states that (i) all investors maximize their utilities subject to the budget and margin constraints, taking investment opportunities as given, (ii) aggregate consumption is equal to aggregate dividend, and (iii) all asset mar- kets clear. Because type B investors are less risk averse, they are likely to use leverage in the equilibrium and may be constrained by margin requirements in some states of the economy. Moreover, it is conceivable that type A investors may also be bound by margin constraints when the equilibrium properties of the risky assets are so different that these investors want to short one asset

and invest more in the other one. 14 In this case, the market becomes incom- plete for all investors, which substantially complicates the characterization of the equilibrium.

To preserve tractability, I consider only those equilibria in which type A investors are unconstrained in all states of the economy. Although such an equilibrium may not exist for some extreme combinations of model parame- ters, this is unlikely to limit the generality of my analysis. The constraint can bind type A investors if one of their portfolio weights is highly negative. For this to happen, either a myopic component or a hedging component should be negative. However, the former possibility is ruled out because in this case type

B investors would also short the same asset (due to their logarithmic prefer- ences, their asset demand contains only a myopic component) and no one would hold that asset in the equilibrium. Hence, a negative portfolio weight of type

A investors can be produced only by the hedging component. Although it is hard to characterize the relative size of the hedging demand in general (it is determined by derivatives of marginal utility with respect to state variables), for realistic parameters it tends to be smaller than the myopic demand, so the total demand is likely to be positive. This makes the constraint irrelevant for type A investors. In the numerical example presented below, I find the

14 I am very grateful to an anonymous referee for pointing this out.

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equilibrium assuming that type A investors are unconstrained and then con- firm that their optimal portfolios do indeed satisfy margin requirements.

Proposition 4 formally describes the equilibrium in the model.

P ROPOSITION

4: If an equilibrium in which type A investors are unconstrained in all states of the economy exists, it is characterized by (i) the drift μ s (s, u) and diffusion s (s, u) of the endogenous state variable s, (ii) the risk-free rate r(s, u) and market price of risk η(s, u), and (iii) the price–dividend ratio functions

S 1 /D 1 =f 1 (s, u) and S 2 /D 2 =f 2 (s, u). The functions μ s (s, u), s (s, u), r(s, u), and η(s, u) are given by equations (A23) to (A27) in Appendix A . The functions f 1 (s, u) and f 2 (s, u) are solutions to the second-order partial differential equations (A27) . The expected excess returns μ Q (s, u), the diffusion of returns Q (s, u), and the optimal portfolio weights of type B investors ω B (s, u) are algebraically linked to the functions characterizing the equilibrium as stated in equations (A28) to (A30) . The optimal portfolio weights of type A investors ω A (s, u) are given by equation (A31) .

Proof. See Appendix A .

Although analytic formulas are available for many variables, the properties of the equilibrium are more evident when all variables are plotted as functions of u and s. To facilitate comparison of the results with those in a one-asset

economy, I keep the parameters the same as before (except γ B = 1 instead of γ B = 2). As in Section II , the economy without margin requirements serves as a bench- mark, and its equilibrium variables are presented in Figure 3 . To save space, I plot several characteristics (expected returns, volatility of returns, beta, port- folio weights) for the first asset only. Because the assets have identical funda- mentals and the state variable u measures the contribution of the first asset to the aggregate dividend, the plots for the second asset can be obtained from the plots for the first one as a mirror image with respect to the plane u = 1/2 (i.e., by substituting 1 − u for u in all formulas).

The relations between the majority of equilibrium characteristics and the state variable s resemble the patterns observed in the one-asset case (see Figure 1 ), and as u → 1, all variables converge to their counterparts in a one- asset economy, in which only the first risky asset exists. As before, investor heterogeneity makes returns more volatile than fundamentals (recall that the

volatility of dividends is σ D = 0.032) and gives the risk-free rate and market price of risk a downward-sloping shape. Figure 3 also demonstrates how the equilibrium statistics depend on the rel- ative size of the first asset u, and shows that many observed relations appear to

be similar to those identified by Cochrane, Longstaff, and Santa-Clara ( 2008 ) in an economy with two Lucas trees but homogeneous investors. In particu- lar, the expected excess returns, the volatility of returns, and the volatility of the state variable s have a U-shaped form, whereas the risk-free rate has a hump-shaped form for any fixed s. These patterns are explained by better di- versification opportunities available to investors when trees have comparable

Asset Pricing with Dynamic Margin Constraints

drift of state variable ( µ s ) risk−free rate (r) expected excess return ( µ Q1 ) 0.01

volatility of state variable ( σ(ds)) market price of risk ( η 1 )

volatililty of returns ( σ(dQ 1 ))

price−dividend ratio (S /D )

portfolio weight ( ω 1 A 1 1 )

portfolio weight ( ω B 1 ) 1 10

correlation of returns ( ρ(dQ 1 , dQ 2 ))

0 0.5 u 0.5 s u 0 s Figure 3. The equilibrium with two risky assets (no margin constraints). This figure

presents various variables in the unconstrained equilibrium in an economy with two risky assets and heterogeneous agents. All variables are functions of the consumption share s of type B investors

and the share u of the first dividend D 1 in the aggregate dividend D. μ s is the drift and σ (ds) is the conditional volatility of the state variable s, r is the risk-free rate, η 1 is the market price of risk of the shock dB 1 ,μ Q1 is the drift and σ (dQ 1 ) is the volatility of excess returns on the first risky asset, S 1 /D 1 is the price–dividend ratio of the first asset, ω A 1 and ω B 1 are the portfolio weights of the first risky asset held by type A and type B investors, β 1 is the market beta of the first asset, and ρ(dQ 1 , dQ 2 ) is the conditional correlation between stock returns. The model parameters are

as follows: μ D = 0.018, σ D = 0.032, β = 0.01, γ A = 10, and γ B = 1.

size, that is, when u is close to 1/2. As u approaches one, the relative size of the first asset as well as the risk associated with it increases, which leads to a

higher market price of risk η 1 . Figure 3 also shows that the weights of the first asset in the portfolios of type A and type B investors grow with the relative

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size of the asset. This is intuitive because investors have to put more wealth in the first asset in the absence of a sufficient amount of the second one as u → 1. Finally, note that the impact of agents’ heterogeneity has a stronger impact than the presence of two trees. In particular, diversification makes returns less volatile than fundamentals as in Cochrane, Longstaff, and Santa-Clara ( 2008 ), but in the majority of states this effect is offset by the increase in the volatility produced by heterogeneity in investor risk preferences.

Note that there are two new variables that do not appear in the one-asset model: the market beta of the first asset β 1 and the correlation between re- turns on the risky assets. 15 The bottom-right panel of Figure 3 shows that the correlation is positive in all states of the economy, even though the assets have independent fundamentals. This pattern obtains for two reasons. First, the correlation between returns is positive even in an economy with two trees but homogeneous investors (Cochrane, Longstaff, and Santa-Clara ( 2008 )). The intuition of this effect is relatively straightforward: a negative shock to divi- dends on the first asset lowers its price and increases the relative share of the second asset. Because investors want to maintain the optimal diversification of their portfolios, they start selling the second asset. As a result, its price also decreases and stock returns become positively correlated.

Second, heterogeneity in investor preferences contributes to the positive cor- relation of returns. A negative shock to the dividend on the first asset decreases the price of that asset as well as investors’ wealth and causes a reallocation of risky assets from type B to type A investors (this effect is also present in the one-asset case). Hence, the aggregate risk aversion of investors in the economy increases, and the prices of all risky assets, including the second asset, fall. As a result, returns on the assets become positively correlated. Together with the increase in the volatility of returns, this is another consequence of risk

sharing among heterogeneous investors and the wealth effect. 16 In the major- ity of states, the correlation between returns increases as the share of type B investors decreases, that is, it is a decreasing function of s.

The bottom-left panel of Figure 3 shows that the beta of the first asset β 1 is largely determined by two factors: the correlation of the assets and the relative size of the first asset. In particular, the beta is close to one when the assets are

highly correlated, which happens when s is relatively low. Also, β 1 is high when the first asset is large and constitutes a substantial share of market value (i.e., u is close to one). In the limit u = 1, there is only one asset in the economy, and its beta mechanically equals one.

To illustrate the impact of margin requirements, I consider the same econ- omy as before but with the constraints specified by equations (24) to (26) . For consistency with the analysis of the one-asset economy, I set ¯ m = 1.3. Figure 4

15 The market betas of stocks are computed as β i = Cov t (dQ it , dQ mt )/Var t (dQ mt ), where the market excess return can be represented as dQ mt =w 1 dQ 1t +w 2 dQ 2t , and w 1 = uf 1 /(uf 1 + (1 −

u) f 2 ) and w 2 = (1 − u) f 2 /(uf 1 + (1 − u) f 2 ).

16 Contagion caused by the wealth effect has also been studied by Kyle and Xiong ( 2001 ) and Ehling and Heyerdahl-Larsen ( 2012 ), among others.

Asset Pricing with Dynamic Margin Constraints

drift of state variable ( µ s ) risk−free rate (r) expected excess return ( µ Q1 )

volatility of state variable ( σ(ds)) market price of risk ( η 1 )

volatililty of returns ( σ(dQ 1 )) 0.01

price−dividend ratio (S 1 /D 1 )

portfolio weight ( ω A 1 )

portfolio weight ( ω B 1 )

correlation of returns ( ρ(dQ 1 , dQ 2 ))

0 0.5 u 0.5 s u 0 s Figure 4. The equilibrium with two risky assets and margin constraints. This figure

presents various variables in the economy with two risky assets, heterogeneous agents, and mar- gin constraints. All variables are functions of the consumption share s of type B investors and

the share u of the first dividend D 1 in the aggregate dividend D. μ s is the drift and σ (ds) is the volatility of the state variable s, r is the risk-free rate, η 1 is the market price of risk of the shock dB 1 ,μ Q1 is the drift and σ (dQ 1 ) is the conditional volatility of excess returns on the first risky asset, S 1 /D 1 is the price–dividend ratio of the first asset, ω A 1 and ω 1 B are the portfolio weights of the first risky asset held by type A and type B investors, β 1 is the market beta of the first asset, and ρ(dQ 1 , dQ 2 ) is the conditional correlation between stock returns. The model parameters are as follows: μ D = 0.018, σ D = 0.032, β = 0.01, γ A = 10, γ B = 1, and ¯ m = 1.3.

shows the same variables as Figure 3 , but in the economy with time-varying margin requirements. Comparison of Figures 3 and 4 yields several important observations. First, margin requirements decrease the volatility of the state variable s as well as

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the volatility of stock returns in all states of the economy where they bind (this also follows from equations (A23) and (A28) ). The intuition for this result is the same as for its one-asset analog: margin requirements force type B investors to hold less leveraged portfolios, so the difference in portfolios of type A and type

B investors becomes smaller, and the excessive volatility produced by investor heterogeneity decreases.

A second and more interesting observation is that the magnitude of the volatility reduction varies across states of the economy. When the trees have comparable size (these states have u close to 1/2 and belong to the region A 1 defined in equation (A23) ), the graph of σ (ds) has a saddle shape that resembles

the shape of the volatility graph in the unconstrained economy (see Figure 3 ), but with a lower average volatility. As follows from the proof of Proposition 4 , only the borrowing constraint binds type B investors in those states. However, two constraints bind in states of the economy with low values of s and relatively

low or high values of u (these states belong to the regions A 12 and A 13 defined in equation (A23) ). This exacerbates the decline in volatility, which appears to

be stronger for more extreme values of u, and explains why the “valley” graph for σ (ds) in Figure 4 has additional “slope cuts” compared to its analog in Figure 3 . The effect is qualitatively different for the volatility of returns. The middle- right panel of Figure 4 implies that the returns on a large asset (e.g., the first asset when u is close to one) are more volatile than fundamentals, whereas the volatility of returns on a small asset (e.g., the first asset when u is close to zero) can be even lower than the volatility of fundamentals. Given that at almost any moment one asset is large and the other asset is small, margin constraints produce strong heterogeneity in asset volatilities in the cross-section. Note that this pattern of volatilities is in stark contrast with what happens without con-

straints: the middle-right panel of Figure 3 shows that the graph for volatility is almost symmetric relative to the plane u = 1/2, implying that both stocks at each moment have very similar volatilities.

To understand the intuition behind the dispersion of volatilities, consider the graph for the market beta of the first asset β 1 presented in the bottom- left panel of Figure 4 . It shows that the large asset (the first asset when u is high) has a much higher beta than the small one (the first asset when u is low). This difference results from two effects. First, the asset with a higher dividend represents a larger share of the market, so its beta is closer to one almost mechanically. This is observed in both constrained and unconstrained

economies (see Figure 3 ). Second, margin constraints decrease the correla- tion of returns (as I discuss below), thereby further increasing the gap in betas.

As in the one-asset economy, a less risk-averse investor prefers to hold a leveraged portfolio. Without margin constraints, this portfolio is constructed by shorting a risk-free asset and investing the proceeds in a diversified portfolio of the risky assets. However, such a portfolio is unattainable in the presence of constraints. Instead, it becomes optimal to overweight the asset with the highest beta, sacrificing the benefits of diversification for the sake of having a

Asset Pricing with Dynamic Margin Constraints

433 portfolio with the highest possible leverage. 17 This is clearly illustrated by the

B 1 of type B investors (again, the weight of the second asset ω in the portfolio

panel of Figure 4 that presents the weight of the first asset ω B

2 can be obtained from the graph for ω B 1 by flipping it around the plane u = 1/2): in the majority

of states, type B investors build a leveraged position by using only the risky asset that has the largest dividend share. For example, when u is large (the first tree dominates the second one), the portfolio weight of the second asset is positive but close to zero, and almost all capital is invested in the first asset. Given that type B investors nearly ignore the smallest asset (e.g., the first asset when u is substantially below 1/2), investor heterogeneity plays little role in amplifying its volatility. As a result, the diversification effect dominates, and the returns on the smallest asset can be even less volatile than fundamentals. However, the increase in volatility produced by investor heterogeneity and risk sharing is unaffected for the large asset, and this explains why the volatilities of returns may be so different in the cross-section in the presence of margin constraints.

Figure 4 also presents the graph for the weight of the first asset ω A 1 in the portfolio of type A investors. In contrast to the weights of type B investors, type

A investors always maintain unleveraged long positions in both assets, which means that type A investors never hit margin constraints in the equilibrium even though these constraints have been imposed on them. This result justifies the construction of the equilibrium using the assumption that type A investors are unconstrained in all states of the economy.

Finally, margin requirements have a strong effect on the correlation between asset returns. Comparing Figures 3 and 4 , it is easy to see that the correlation in all states decreases relative to the unconstrained economy, and this diminishes the asset beta when the asset constitutes a relatively small fraction of the market. Moreover, the decline in the correlation is more pronounced when the trees have comparable size (u is around 1/2). This effect has an intuitive explanation. When there is a positive shock to the dividend on the first asset, its price increases. Moreover, the share of the first asset u increases, so, according

to Figure 4 , the optimal weight of this asset in the portfolio of type B investors also increases. Hence, there is selling pressure on the second asset, which drives its price down. Because the prices move in opposite directions, the correlation between assets declines.