Margin Requirements And Equilibrium Asset Prices
Margin Requirements and Equilibrium Asset PricesDaniele Coen-PiraniGraduate School of Industrial Administration, Carnegie Mellon University,Pittsburgh, PA 15213-3890, USAAbstractThis paper studies the e ect of margin requirements on asset prices and trading volume ina general equilibrium asset pricing model where Epstein-Zin investors di er in their degree ofrisk aversion. Under the assumptions of unit intertemporal elasticity of substitution and zeronet supply of riskless assets, I show analytically that binding margin requirements do not a ectstock prices. This result stands in contrast to previous partial equilibrium analysis where xedmargin requirements increase the volatility of stock prices. In this framework, binding marginrequirements induce a fall in the riskless rate, increase its volatility, and increase stock tradingvolume.Keywords: Margin Requirements, General Equilibrium, Asset Prices, Stock Trading Volume,VolatilityJEL Classi cation: G11; G12; G18Telephone: (412) 268-6143; Fax: (412) 268-7064; E-mail: coenp@andrew.cmu.edu. This paper builds on materialfrom my Ph.D. dissertation at the University of Rochester. I thank Rui Castro, Larry Epstein, Darrell Du e, myadvisor Per Krusell, Marla Ripoll, Alan Stockman, Tony Smith, Chris Telmer, Stan Zin, an anonymous referee,and seminar participants at Carlos III, Carnegie Mellon, Florida State at Tallahassee, ITAM, Montreal, the FederalReserve Bank of Richmond, Rochester, Rutgers, and the Stanford Institute for Theoretical Economics for helpfulcomments and suggestions. I thank the W. Allen Wallis Institute of Political Economy at the University of Rochesterfor nancial help. The usual disclaimer applies.1
1IntroductionA variety of recent papers has argued that leverage constraints such as margin requirements in thecash or futures market for stock might increase, rather than stabilize, the volatility of stock prices.1Chowdry and Nanda (1998) show how xed margin requirements might induce rational changes instock prices even when economic fundamentals do not change. Aiyagari and Gertler (1999) andMendoza and Smith (2000) argue that binding margin requirements might induce stock prices to“overreact” to changes in economic fundamentals.While these results have been obtained in the context of di erent models, the economic mechanism behind them is the same. A negative shock to stock prices reduces the value of the collateralpledged by levered investors. A su ciently large fall in stock prices forces levered investors toliquidate stock in order to meet margin calls. Therefore, the price of stock has to fall even more inorder to induce more risk-averse investors to absorb the excess supply of risky assets.In this paper I argue that this result relies crucially on these models’assumption that the risklessrate is exogenously xed.2 Since the only price that is allowed to adjust when margin requirementsare binding is the stock price, these models might fail to properly account for the e ects producedby these constraints. The purpose of this paper is not to claim that margin requirements do notplay an important role in exacerbating the e ect of exogenous shocks on stock prices. It claims,instead, that when the riskless rate is allowed to adjust in response to shocks, “overreaction” ofstock prices will occur only in some well de ned circumstances, which have not been emphasizedby the literature mentioned above: i) a su ciently large positive net supply of riskless assets; ii) arelatively high intertemporal elasticity of substitution in consumption.To illustrate these points, I introduce a general equilibrium model where two types of EpsteinZin investors, heterogeneous in their degree of risk aversion, trade in stock and a riskless asset1The empirical evidence does not support the view that in the U.S. o cial stock margin requirements havesigni cantly contributed to diminish the monthly volatility of stock returns. See Schwert (1989) and Hsieh and Miller(1990). A major limitation of empirical studies is the fact that since 1934 the Federal Reserve Board has modi edthem only 22 times.2Chowdry and Nanda (1998) assume the existence of a storage technology with given return. Aiyagari and Gertler(1999) impose a market clearing condition for the riskless asset, but, since they assume that the household does notface any costs of adjusting riskless debt in its portfolio, the determination of the riskless rate in their model is thesame as in the representative-agent Lucas (1978) model. Mendoza and Smith (2000) assume an exogenous risklessrate because they model a small open economy.2
in order to share aggregate dividend risk. In equilibrium, less risk-averse investors buy stock onmargin and a margin requirement sets a limit to their use of leverage. Under the assumptionsthat investors have the same unit elasticity of intertemporal substitution and that the net supplyof riskless assets is zero, I show analytically that only the price of the riskless asset adjusts whenmargin requirements are binding. Under these assumptions the price of stock is una ected bymargin requirements and, in contrast to the partial equilibrium models mentioned above, does not“overreact” nor displays multiple equilibria.The intuition for this result is as follows: the market-clearing conditions for the stock and theriskless asset jointly imply that changes in the stock price-dividend ratio can only be due to changesin the aggregate propensity to save or to changes in the value of the net supply of riskless assets.Since the latter is, by assumption, equal to zero, margin requirements can a ect the price of stockonly by inducing a change in the aggregate propensity to save. By restricting leverage, marginrequirements set an upper bound to the ratio between the value of the stock an investor buysand the value of his wealth. With homothetic preferences and only capital income, consumptionsavings decisions can be separated from portfolio decisions, and are therefore not directly a ectedby this portfolio constraint. A unit elasticity of substitution implies that an investor’s consumptiondecision is independent of the risk-corrected mean return on his portfolio. Therefore, when marginrequirements bind, the investors’ savings decisions do not change and the price of stock is nota ected.In this model, when constrained levered investors are forced to sell stock, the rate of returnon the riskless asset has to fall to induce more risk-averse investors to absorb its excess supply.Consequently, binding margin requirements increase the volatility of the return on the riskless assetas well as stock trading volume. Thus, also in this economy, even if stock prices are una ected,margin requirements contribute to the instability of asset prices.I also consider two extensions of the basic model to examine how binding margin requirementsa ect stock prices when the riskless asset is in positive net supply or the intertemporal elasticity ofsubstitution is di erent from one.First, I allow for a positive net supply of riskless assets by introducing government bonds in themodel. In this case, aggregate wealth in the economy is equal to the value of stock plus the valueof government bonds. With a unit intertemporal elasticity of substitution, the market clearing3
condition for consumption implies that aggregate wealth is constant over time. It follows that,when margin requirements bind and the value of government-supplied bonds increases, the price ofstock must fall proportionally more than dividends.Second, I argue that in this general equilibrium model, in the empirically plausible case wherethe elasticity of intertemporal substitution is less than one, binding margin requirements increase,rather than decrease, the price of stock. When less risk-averse investors become constrained, infact, the risk-adjusted mean return on their portfolio decreases. If the elasticity of intertemporalsubstitution is less than one, their propensity to save out of current wealth increases, inducing ahigher price of stock. Thus, even when the assumptions of zero net supply of riskless assets andunit intertemporal elasticity are relaxed, the general equilibrium analysis undertaken in this papersuggests a more complex relationship between asset prices and margin requirements than the oneimplied by a partial equilibrium approach.The mechanism that induces a fall in stock prices when government bonds are in positive netsupply is similar to the one emphasized by Kiyotaki and Moore (1997). They consider a productioneconomy with two types of agents that are heterogeneous in terms of the production technologiesthey operate. More productive agents borrow from less productive ones to buy the productioninput, land. The latter plays the double role of input in production and collateral for borrowing.They show how an unanticipated low productivity shock can cause a downward spiral in landprices, exacerbated by the binding collateral constraint. The key to their result is the fact thatthere is more than one asset in positive net supply in the economy: land and the technology of lessproductive agents. As explained in more detail in section 5.1, in their model the “overreaction” ofland prices to a low productivity shock is the counterpart of the “underreaction”of the value of thetechnology used by less productive agents to the same shock. Similarly in my model, stock pricesfall more than dividends when margin requirements are binding only if the value of governmentbonds increases.3Another related paper is Detemple and Murthy (1997) who study the e ect of portfolio constraints on the relationship between asset prices and investors’ intertemporal marginal rates of3An important di erence between this paper and Kiyotaki and Moore (1997) is that while they focus exclusivelyon one-time unanticipated shocks, the dividend shocks considered here do not occur with zero probability. Thus,investors are aware of the fact that dividend shocks can occur, and can plan their consumption and portfolio decisionsaccordingly.4
substitution. They consider an economy where logarithmic expected utility agents trade assets because of di erent beliefs about the aggregate dividend process. The intuition about the relationshipbetween stock prices and margin requirements provided at the beginning of this introduction alsoapplies to their setup where portfolio constraints do not a ect equity prices. An important di erence between their paper and this one is that they consider a nite-horizon economy. Consequently,they do not address the issue of whether portfolio constraints will ever be binding in a stationaryequilibrium.4 In the model of this paper, instead, the long-run distribution of wealth across agentsis non-degenerate and margin requirements are occasionally binding in a stationary equilibrium.A less directly related literature is the one on asset pricing with incomplete markets (seeHeaton and Lucas (1996) and their references). These papers typically analyze general equilibrium economies where nancial markets are incomplete and agents face uninsurable labor incomerisk. This paper di ers from the ones in this literature in two important dimensions. First, thereis no idiosyncratic risk in the model of this paper, because the literature cited at the beginning hasmainly emphasized the role of margin requirements in exacerbating the e ect of aggregate risk onasset prices. Second, the literature on incomplete markets and asset prices focuses on a di erentkind of borrowing constraint than the one considered here. In those models, investors face xedshort-selling constraints on both riskless and risky assets. In particular, the amount that can beshorted does not depend on the endogenous variables of the model, such as the price of risky assets. With a margin requirement, instead, the maximum amount that an investor can short-sell isproportional to his current wealth, and is therefore state-dependent.The rest of the paper is organized as follows. Section 2 describes the model. Section 3 considersan individual investor’s portfolio and savings problems. Section 4 states and proves the main resultand discusses the e ects of margin requirements on investors’ portfolios, the volatility of risklessreturns, and trading volume. Section 5 discusses the robustness of this result when the assumptionsof unit intertemporal elasticity and zero net supply of riskless assets are relaxed, and in the presenceof multiple stocks. Section 6 concludes. The appendix contains proofs of the propositions and adescription of the numerical algorithm used to solve the model.4This is the case in Aiyagari and Gertler (1999) where margin requirements bind only during the transition to astationary equilibrium and, over time, investors move towards non-levered positions.5
2The ModelIn this section I introduce a discrete time dynamic general equilibrium model where two types ofinvestors trade in stock and riskless assets in order to share aggregate endowment uncertainty. Thetwo types di er in their degree of risk aversion. A margin requirement limits the amount of leveragean investor can use when purchasing or short selling stock.PreferencesThe economy is populated by two types of investors denoted by i 1; 2: The measure of eachtype is normalized to one. An investor’s preferences are represented by an Epstein-Zin recursiveutility function (Epstein and Zin (1989)). In contrast to the standard expected utility preferences, this speci cation allows for the independent parameterization of attitudes toward risk andintertemporal substitutability. I denote by Uit investor i’s utility from time t onward. This isde ned recursively as5Uit h(1) Cit exp (1iEt Uit 1) log Cit iii1, if 0 6 ilog Et Uit 1 1; if(1) 0:(2)Current utility is obtained by aggregating current consumption, Cit , and the certainty equivalentof random future utility, computed using the information available at time t:Both types of investors are assumed to discount the future at the same rate2 (0; 1) andto have the same attitude toward intertemporal substitutability. The intertemporal elasticity ofsubstitution between current consumption and the certainty equivalent of future utility is (1The parameterii)1: 1 measures the degree of risk aversion of an investor of type i; with a higherdenoting lower risk aversion. I make the following assumption regarding investors’types:Assumption 1. Investors di er only according to their degree of risk aversion. By conventioninvestors of type 1 are characterized by the highest degree of risk aversion, i.e.,1 2:Aggregate Endowment5The expression for the utility function when 0 (unit intertemporal elasticity of substitution) is easily obtainedusing de L’Hospital’s rule to compute the limit of (Ut1) for6! 0: The casei 0 can be similarly handled.
The aggregate endowment, in each period, consists of a perishable dividend Dt that evolvesaccording to the processDt 1 The growth ratet 1t 1 Dt :(3)satis es the following assumption:Assumption 2. The growth rate of the dividend follows an i.i.d. two-state Markov processf l;hg ;withl hand P r (t 1 l) Pr (t 1 h)t 12 0:5:The assumption of statistical independence of dividend growth across periods allows me tohighlight the endogenous mechanisms of the model since it implies that all changes in the conditionalmoments of asset returns are due to the dynamics of the distribution of wealth among investors.Markets and AssetsAt each point in time there exists a spot market for the consumption good, which I take tobe the numeraire, a market for a riskless asset (“bond”), and a market for a risky asset (“stock”).A share of the stock pays the dividend Dt at time t, while the bond trades for one period, witheach unit paying one unit of the consumption good. I denote by Bit the quantity of bonds held byinvestor i at the beginning of time t and by qt their price in terms of the consumption good. Forconvenience, I also de ne bit 1 to be the ratio Bit 1 Dt .I denote by sit the share of the stock held by investor i at the beginning of time t and by Ptthe ex-dividend price of one share in terms of the consumption good. I also denote by pt Pt Dtthe stock price-dividend ratio. The total number of shares outstanding is normalized to one.6GovernmentThe main role of the government in this economy is to supply one-period riskless bonds. LetBtg denote the quantity of outstanding government bonds at the beginning of time t. De ning bgt 1gto be the ratio Bt 1 Dt , the government budget readsqt bgt 16 bgtt(4)tIt is simple to introduce futures contracts in the model, along the lines of Aiyagari and Gertler (1999) andChowdry and Nanda (1998). In this version of the model, if margin requirements in the futures market are lowerthan the ones in the cash market, investors would leverage only by trading in futures contracts. It is easy to showthat the model with futures contracts is formally analogous to the model of this section.7
wheretTt Dt denotes the government’s tax revenue standardized by the aggregate dividend.For convenience, I assume that the government follows the policy of keeping the outside supply ofriskless bonds proportional to the aggregate dividend:bgt bgfor t 0; 1; 2:::;for some bg 0.7 Substituting bg for bgt and bgt 1 in (4) it is easy to see that government revenuemust adjust in the following way:t bg1qt :(5)tFinally, I assume that the government raises revenue by means of a proportional tax on dividends, so thattcan be interpreted as the dividend tax rate at time t.8Investors’ Budget ConstraintsBoth types of investors maximize their time-zero utility Ui0 subject to the sequence of budgetconstraints(1t pt ) sit bit cit qt bit 1 pt sit 1for t 0; 1; 2:::(6)twhere cit denotes the ratio Cit Dt : Without loss of generality, I assume that at time zero si0 2 [0; 1]and bi0 0 for i 1; 2.Margin RequirementsEach investor is subject to the following margin requirement(1t pt ) sit bitcitpt jsit 1 jtforfor t 0; 1; 2:::2 [0; 1] : This equation states that an investor must nance a fraction(7)of his stock purchases(or short sales) out of his own savings. Using equations (6) and (7) it is easy to see that theborrowing limit for an investor that buys or sells stock is proportional to his savings. Formally, for7This assumption guarantees that the supply of government bonds does not become a state variable of the model,which would signi cantly complicate the analysis.8I ignore taxation of interest income because it can be shown that in this setting it does not a ect the price ofequity.8
sit 1 0; we have1qt bit 1(1t pt ) sit bitcit ;twhile for sit 1 0 the constraint implies that1pt sit 1(1t pt ) sit bitcit :tThe parameteris exogenously given. I assume, with Aiyagari and Gertler (1999), that it is setby a government agency, like the Federal Reserve in the U.S.9 Notice that the borrowing constraint(7) di ers from the one commonly used in the asset pricing literature with incomplete markets,according to which the amount an investor can borrow is constant over time (see, e.g., Heaton andLucas (1996)). A margin requirement, instead, implies that this amount depends positively on hissavings and therefore on his wealth. As an investor’s wealth changes endogenously over time, sodoes his borrowing constraint.Recursive EquilibriumIn this economy aggregate demand for assets and the consumption good depends, in general,on the way aggregate wealth is distributed among investors, and so do bonds and stock prices. Lettdenote aggregate wealth at a given point in time. This can be obtained by adding the investors’budget constraints (6), imposing the stock and bonds market-clearing conditions, and taking intoaccount the government’s budget constraint (5), to obtain:(1tt pt ) bg 1 pt q t bg :tDenote by2 [0; 1] the fraction of aggregate wealth held by investors of type 1. Since thereare only two types of investors,denote by0 H;0summarizes the distribut
advisor Per Krusell, Marla Ripoll, Alan Stockman, Tony Smith, Chris Telmer, Stan Zin, an anonymous referee, . nism behind them is the same. A negative shock to stock prices reduces the value of the collateral . an individual invest
the full-cost price emerges from an equilibrium condition. Income of the rm at the equilibrium price equals, by de nition, equilibrium income. Rearranging this identity, equilibrium price equals variable cost plus equilibrium contribution margin per unit (the latter term being the xed cost plus equilibrium income, all divided by quantity).
Initial Margin Calculation Guide Page 8 of 36 3. CALCULATION OF TOTAL MTM AND MARGIN REQUIREMENT 3.1 Required Inputs 3.1.1 Risk Parameters and Margin Adjustments To derive total MTM and margin requirement, the risk parameters (including market risk and other risk components) and margin adjustments below are required. The sources are
the same margin group are calculated based on data from the previous 30 and 250 business days, respec-tively. An offsetting effect between products of two margin classes within the same margin group is only granted if min(ρ30, ρ250) 0.5. The details of netting at margin class and margin group level for the forward-looking margin are described
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the asset pricing implications of margin-induced trading. In this paper, we provide novel analysis of investor margin trading behavior using detailed account-level data in China that tracks hundreds of thousands of margin investors' borrowing and trading activities. The Chinese economy and its nancial markets have experienced tremendous
