financial leverage and market volatility with diverse. wu(et2011).pdf · econ theory (2011)...

Econ Theory (2011) 47:337–364DOI 10.1007/s00199-010-0548-8

SYMPOSIUM

Financial leverage and market volatility with diversebeliefs

Wen-Chung Guo · Frank Yong Wang ·Ho-Mou Wu

Received: 15 November 2009 / Accepted: 25 May 2010 / Published online: 29 July 2010© Springer-Verlag 2010

Abstract We study a general equilibrium model of asset trading with financialleverage, where the investors can engage in speculative trading with diverse beliefsabout the asset’s fundamental value. We show that an increase in the leverage ratiocauses the stock price to rise in the current period through a “leverage effect”, andwill result in more borrowing and more stock purchase that pumps the stock pricehigher in the subsequent period, known as the “pyramiding effect”. There can alsobe a “depyramiding effect” when the price falls because lenders issue margin callsand force stock sales, contributing to further stock price plummeting. Price changesfrom depyramiding effect, however, may not take effect when margin calls are nottriggered. We demonstrate that, under certain conditions, decreasing leverage ratiosleads to lower stock price volatility, measured by the variation of prices caused by anexogenous shock, when the shock is unanticipated. The influences of dispersion ofbeliefs and available investment funds on the relation between financial leverage andmarket volatility are also examined. When the shock is anticipated, we demonstrate

We thank Mordecai Kurz, Kenneth Arrow, Peter Hammond, Maurizio Motolese, Carsten Nielsen, GeorgeEvans, Hiro Nakata and participants of Stanford Institute of Theoretical Economics (SITE) Conference inthe summer of 2009 for many helpful suggestions.

W.-C. GuoDepartment of Economics, National Taipei University, Taipei, Taiwan

F. Y. WangSchool of International Trade and Economics, University of International Business and Economics,Beijing, China

H.-M. Wu (B)China Center for Economic Research, National School of Development,Peking University, Beijing, Chinae-mail: [email protected]

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338 W.-C. Guo et al.

that reducing leverage ratios may not lower stock price volatility, which poses animportant challenge to future studies on this issue.

Keywords Financial leverage · Diverse beliefs · Stock price volatility · Leverageeffect · Pyramiding/depyramiding effect

JEL Classification D84 · G12 · G18

1 Introduction

The financial sector became highly leveraged before the 2008 global financial crisisoccurred. The margins (or haircuts) in the repurchase market of various securitieswere in the range of 1–15%, implying a leverage ratio of about 7–100. As for theinvestment banks involved in the crisis, their leverage ratios were also very high. Suchhigh leverage ratios were often mentioned as one of the main causes for price volatilityduring the recent episode of dramatic fluctuations (For related works on the dynam-ics of recent fluctuations, see Fostel and Geanakoplos (2008), Geanakoplos (2009),and Brunnermeier and Pedersen (2009)). However, the microeconomic mechanismbetween financial leverage and price volatility has not been thoroughly studied in ageneral equilibrium framework with heterogenous agents. In this paper, we will studyan economy with exogenously given margin requirements where speculative tradingbecomes possible with diverse beliefs as in Kurz (1994), Kurz and Wu (1996) and Wuand Guo (2003, 2004).1 Our aim is to develop a theoretical framework to examine theimpact of leverage ratios (or margin requirements) on the level and variation of stockprices.

The relationship between margin requirements and price volatility has been a majorconcern in many empirical studies. However, there has been almost no conclusion fromthe empirical literature on the effects of margin requirements.2 In addition, there arefew theoretical conclusions on the relation between stock price volatility and marginrequirements. In one of the few analytical works, Kupiec and Sharpe (1991) study anoverlapping generations model in which there are two types of investors with differentrisk tolerances. Adopting a two-state Markovian framework, they analyze the particu-lar situation in which margin requirement binds in only one of two states. An increasein margin requirements will affect the demand of that particular state and lower its

1 For the discussion about speculation, see Keynes (1936), Feiger (1976), Hirshleifer (1975), Kohn (1978),Leach (1991), etc. Other related works on speculation with heterogeneous expectations include Miller(1977), Harrison and Kreps (1978), Varian (1985, 1989), Harris and Raviv (1993), Hart and Kreps (1986),Morris (1996) and Scheinkman and Xiong (2003). In addition to theoretical development, recently empir-ical studies such as Anderson et al. (2003) and Kurz and Motolese (2010) also provide supports for theinfluence of heterogeneous expectations on asset prices.2 While Moore (1966), Hsieh and Miller (1990) find no significant influence of the margin requirementon price volatility from the US data, Hardouvelis (1990), Hardouvelis and Peristiani (1992) and Seguin(1990) discover some negative impacts from an increase in the margin requirement on price volatility.Further discussions about the policy debate can be found in Kupiec (1998) and Hardouvelis and Theodossiu(2002).

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Financial leverage and market volatility with diverse beliefs 339

state price. It is shown that there is no unique theoretical relationship between marginrequirements and stock price volatility.3

This paper is distinguished from the literature by its completely rigorous frame-work and its emphasis on the role played by speculative trading and diverse beliefs.In this paper, margin requirements are not restricted to bind only in one state as inKupiec and Sharpe (1991).4 Instead, we adopt a broader approach relying on theheterogeneity of beliefs as in Kurz (1994, 2009) and Wu and Guo (2003, 2004) (alsosee Miller 1977; Harrison and Kreps 1978), which serves as an interesting channel forinvestigating speculation and price volatility. In our model the heterogenous investorsare allowed to trade with an exogenously given margin requirement. We introduce anexogenous shock in the second period and consider the difference of prices in a goodstate and in a bad state as a measure of market price volatility. Our results demonstratethat decreasing the leverage ratio may not necessarily reduce market volatility, whichposes an important challenge to any simple-minded policy prescription for controllingmarket volatility.

In our three-period framework, margin trades are found to contribute to stockprice volatility through a dynamic channel. A “pyramiding effect” may occur whenstocks continue to appreciate, allowing more borrowing and more stock purchases,which pumps the stock prices higher in the subsequent period. There can also be a“depyramiding effect” when prices fall because lenders may issue margin calls andforce stock sales, contributing to further stock price plummeting (see, for example,Luckett 1982; Hardouvelis 1990).

More especially, we find that the total effect of a price change can be decomposedinto three components: leverage effect, pyramiding effect, and depyramiding effect.These changes can be grouped into two parts. The first part, the leverage effect, refersto the amount of trades when investors are not forced to change their margin account.The second part, the pyramiding or depyramiding effect, is used to characterize theinfluence when investors have to rebalance their margin accounts. For instance, a lossappears after a decrease in asset prices, and therefore, investors sell shares to rebalancethe account. Furthermore, we utilize this decomposition to analyze whether increasingmargin requirement can, under a certain set of conditions, stabilize the market.

In the next section we present the basic model of stock trading with diverse beliefs.We demonstrate the existence of a unique equilibrium with speculative trading usingmargin accounts. The equilibrium stock prices can be characterized by using a“marginally optimistic investor,” constructed systematically from the belief structureof investors. In Sect. 3 we present the decomposition into leverage and pyramid-ing/depyramiding effects and analyze the impact of margin requirements on thesecomponents when the shock is unanticipated. Our analytical results demonstrate thatreducing financial leverage may stabilize the stock market only under a set of restrictive

3 The result of Kupiec and Sharpe (1991) is, however, constrained by their restricted framework of onlytwo states and two types of investors. Our framework is different from Kupiec and Sharpe’s because inour framework there will always be some investors facing the binding constraints imposed by the marginrequirement.4 Coen–Pirani (2005) further points out a binding margin requirement may not affect stock price wheninvestors have an Epstein–Zin preference.

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conditions. In Sect. 4 we study the influences of belief structure and investment fundson the relation between margin requirement and price volatility. We show that a moreoptimistic belief structure is associated with higher stock prices. The heterogeneity ofbelief is also shown to be positively associated with stock price volatility. In Sect. 5we extend our model to study market volatility when the shock is anticipated by theinvestors. Section 6 concludes the paper.

2 The model with financial leverage

The model has three periods (t = 0, 1, 2) and a risky asset (stock) realizing its finalpayoff V in period 2. The final payoff V is either high (VH ) or low (VL), with VH >

VL = 0 for simplicity to avoid technical complexity. However, the investors in theeconomy form different beliefs about the likelihood of a high payoff. In periods t =0, 1, investors trade on the basis of diverse beliefs. The total supply of this risky assetis normalized to unity. To model a large economy with perfectly competitive market,let there be a continuum of risk-neutral investors indexed by their types x , whichis assumed to be uniformly distributed on the interval [0, 1], as in Aumann (1964),Aumann (1966).5

2.1 The structure of diverse beliefs

In our model each type of investors has a different belief about the likelihood of highdividends. The type x investors have a subjective probability B0(x) in period 0 aboutobtaining a high payoff VH and 0 ≤ B0(x) ≤ 1. Putting all individual beliefs together,we can identify a belief structure {B0(x)}x∈[0,1] for the whole economy. Without lossof generality, we can reshuffle the investor types so that a higher value for x alwayscorresponds to the more optimistic investor type with a higher B0(x). We assume thatthe belief structure has a continuous, differentiable, and monotonically non-decreasing(B ′

0(x) ≥ 0) function from [0, 1] to [0, 1]. Here we consider a linear belief structureB0(x) = a + b · x , as shown in Fig. 1, where a > 0 measures the degree of optimismof the whole economy, and b ≥ 0 is used to characterize the extent of differences ininvestor beliefs.6 For beliefs B0(x) to be probabilities, it is assumed that a + b ≤ 1.Each investor type has an equal number of investors, which is normalized to be equalto one.

To simplify our discussion, all investors are assumed to have a limited investmentfund W0 in period 0, which is sufficiently large such that default will not occur.

5 In our model the properties of an equilibrium depend on the risk-neutrality assumption, which is alsoadopted in previous studies such as Harrison and Kreps (1978), Harris and Raviv (1993) and Morris (1996).When investors are risk averse, margin requirements may affect the stock trading behavior of only thoseoptimistic investors who face a binding constraint.6 The beliefs of investors can be made to be compatible with observed date or to be “rational” as in Kurz(1994), Wu and Guo (2003), Wu and Guo (2004) and Kurz and Motolese (2010). In our model these beliefscan also be given exogenously as in Miller (1977), Harrison and Kreps (1978) or Geanakoplos (2009).

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Fig. 1 An example of belief structure {B0(x)}x∈[0,1] : B0(x) = 0.25 + 0.5x

2.2 The shocks and market volatility

In period t = 1, a random shock occurs to affect investor expectations. The shock isassumed to be unanticipated here, and in Sect. 5 we will consider the case of anticipatedshock. Investors become either optimistic or pessimistic in period 1. In particular, anup state (u) and a down state (d) are considered for simplicity. We assume that thesubjective probabilities of obtaining a higher final payoff VH for any investor x canbe described as a parallel shift in B0(x):

Bu(x) = Probu(VH |x) = a + b · x + u, (1)

Bd(x) = Probd(VH |x) = a + b · x − d, (2)

where u > 0 and d > 0 are the changes in subjective probability for each investor,with a + b + u ≤ 1 and a − d ≥ 0 in order to have well-defined probabilities.7

The prices associated with these two states will be denoted by pu and pd . Marketvolatility, measured by pu − pd , is determined endogenously in our framework, withexogenously given belief structure, investment fund, and margin requirement.

We assume that in period 0 the initial margin requirement is exogenous and denotedby m0. The margin buyer needs to pay only a proportion m0 of his total purchase. Wedo not consider other lending instruments except margin trades. The shares purchasedwith margin trading are retained by financial intermediaries as collateral for borrowing.

7 One may consider a rational learning model to analyze the changes in subjective probabilities, for exam-ple, with the setting similarly to Biais and Gollier (1997). Then Bayesian rule can be applied and Bu(x) andBd (x) can be calculated, but forming nonlinear equations. We adopt the assumption of parallel movementsfor simplicity.

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Fig. 2 Belief, information structure, and trading periods. Investors trade in margin in period 0 with differentpriors. In period 1, investors learn form the realized state variable as up (u) or down (d), and trade withrebalancing to satisfy the margin requirement. A final payoff is realized in period 2

It is assumed that there is no default risk, and the parameter m0 is bounded away fromzero, i.e., 0 < m0 ≤ 1.8 We will study the fluctuations in endogenous prices pu andpd with exogenously given margin requirement m0 and shocks u and d.

In period 1, optimistic investors enjoy capital gains in the up state and thereforeincrease their positions using margin trades. However, in the down state, those inves-tors who suffer a capital loss have to watch out at the maintenance margin mT , which issmaller than the initial margin requirement. Once the values of their positions decreaseto reach the maintenance margin, a margin call will be issued. Then the investors haveto rebalance their position and are forced to sell part of their shares to get their positionback to the initial margin level. If the value of their positions are still higher than mT ,optimistic investors will not rebalance their position. Financial institutions serve asthe intermediary for margin trading, and we assume that these institutions function ina competitive and frictionless environment. To focus on the essentials, we assume thatthe riskless rate of return is zero.

2.3 Asset market equilibrium with margin requirements

We analyze the equilibrium in periods 0 and 1 when the investors speculate by mar-gin trading. In this economy with short sale constraints optimistic investors take longposition on the stock with margin, and pessimistic ones take a zero position.

For each investor x , in period 0 his investment decision is to consider whether therisky asset is worthwhile related to his subjective valuation, which is B0(x) · VH +(1 − B0(x)) · VL = (a + bx)VH (Fig. 2).

The beliefs Bu(x) and Bd(x) in period 1 are described by Eqs. (1) and (2). Similarto the work of Miller (1977), the willingness to pay for the stock is non-decreasingin x (since B ′(x) = b ≥ 0). Since there is a continuum of traders in equilibrium, a“marginally optimistic investor” x∗

0 must exist such that the willingness of investor

8 A sufficient condition to prevent the default from occurring in period 1 is that the price at down state isgreater than (1 − m0) multiplying the price in period 0.

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type x∗0 to pay is equal to the equilibrium price in period 0 p∗

0:

p∗0 = B0

(x∗

0

)VH = (

a + bx∗0

)VH . (3)

For investors of type x ∈ [x∗0 , 1], their willingness to pay is higher than the equilibrium

price. These investors will purchase stock on margin. We assume that short sales ofstock are prohabited.9 The market clearing condition can be written as

E0 = (1 − x∗

0

) W0

m0 p∗0

− 1 = 0,

where E0 is denoted as the excess demand for this risky asset, because optimistic inves-tors will use all of their wealth to purchase stocks on margin, giving initial holdingshares as D0(x) = W0/(p∗

0 m0). The above equation could be rewritten as

p∗0 =

(1 − x∗

0

)W0

m0,

which is a strictly decreasing function of x∗0 . From Eq. (3) we can derive the unique

equilibrium in period 010:

p∗0 = W0(a + b)VH

W0 + bm0VH, (4)

x∗0 = W0 − am0VH

W0 + bm0VH. (5)

In period 1, either up state or down state is realized and all investors adjust their beliefsas in Eqs. (1) and (2). Let p∗

u and p∗d be the equilibrium prices for the up and down

state, respectively. When the up state occurs, investors become more optimistic whichincrease their willingness to pay for the stock. As a result, the stock price appreciates,i.e., p∗

u > p∗0 . Therefore, for those investors x > x∗

0 , they make capital gains fromtrading on margin and their wealth increases to

Wu(x) = W0

m0 p∗0

p∗u −

(1

m0− 1

)W0 > W0.

In the above equation the first item is the position value which is equal to D0 p∗u .

The second item is the borrowing on margin. Henceforth, each one of the optimistic

investors (x ≥ x∗u ) will hold Du(x) = Wu(x)

m0 p∗u

> D0(x) shares of stock, and by the

9 The short sales constraint assumption is also adopted in Harrison and Kreps (1978), Kupiec and Sharpe(1991), Harris and Raviv (1993) and Morris (1996). However, this assumption can be relaxed in our frame-work. If investors are allowed to sell short with margin requirement ms , then those pessimistic investorswho originally did not hold any stock will take a short position, resulting in additional supply. The market

clearing condition can be extended in this framework as E0 = x∗0

W0p∗

0

1ms

, where the right hand side indicates

the aggregated short position.10 W0 is large enough such that W0 − am0VH > 0.

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market clearing condition, the type of marginally optimistic investor x∗u at the up state

is bigger than x∗0 . And the equilibrium conditions become

p∗u = (

a + bx∗u + u

)VH , (6)

Eu = (1 − x∗

u

) Wu(x)

m0 p∗u

− 1 = 0 , (7)

where Eu is the excess demand at the up state. In equilibrium we have

x∗u =

−k2 +√

k22 − 4k1k3

2k1, (8)

which is a solution to the following quadratic function:

k1x2u + k2x2

u + k3 = 0 , (9)

where

k1 = 1

m20

W0

p∗0

bVH ,

k2 = bVH + VH1

m20

W0

p∗0

(a + u − b) +(

1

m20

− 1

m0

)

W0,

k3 = (a + u)VH − VH1

m20

W0

p∗0

(a + u) +(

1

m20

− 1

m0

)

W0.

Next we consider the case of stock price decline. If the price depreciates, margintraders are not required to add funds to their margin account unless the margin callsare triggered. When a down state is observed in period 1, all investors become morepessimistic and stock price decreases, i.e., p∗

d < p∗0 . For those investors x > x∗

0 , theirwealth decreases to

Wd(x) = W0

m0 p∗0

p∗d −

(1

m0− 1

)W0 < W0.

In the down state, there are two possible outcomes, depending on whether margin callsare triggered. The first is the case when a decrease in price results in a loss such thatthe margin is still greater or equal to the maintenance margin, that is,

m0 p∗0 − (

p∗0 − p∗

d

)

p∗d

≥ mT ,

and investors who suffer a capital loss are not required to rebalance their position, sothe stock price is

p∗d = Bd

(x∗

0

)VH = (

a + bx∗0 − d

)VH . (10)

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The first case can be characterized by the following inequality:

d ≤ d∗m = m0 − mT

1 − mT

W0(a + b)

W0 + bm0VH, (11)

which means that the downward shock d should be smaller than the level d∗m .

When d > d∗m , a margin call is triggered, and we have the second case. Investors are

forced to rebalance their position to the level of the initial margin m0. Those investorsx > x∗

0 who purchase equities in period 0, however, suffer short-term capital losses.It can also be shown later that the marginal investors x∗

d are less than x∗0 . Since all

investors become pessimistic and those investors x > x∗0 suffer a loss of wealth as the

price decreases, some of those investors who do not own stock in period 0 may usetheir available funds to purchase that stock. Their types are x ∈ [x∗

d , x∗0 ), so they have

x∗0 − x∗

d population. Therefore, the equilibrium conditions become

p∗d = (

a + bx∗d − d

)VH , (12)

Ed = (x∗

0 − x∗d

) W0

m0 p∗d

+ (1 − x∗

0

) Wd(x)

m0 p∗d

− 1 = 0, (13)

where Ed is the excess market demand. It is assumed that W0 is large enough to avoidthe degenerated case x∗

d ≤ 0. By further calculations we can solve for the equilibrium.

x∗d = x∗

0 − (1 − m0)dVH

W0 − (1 − m0)bVH. (14)

Hence we establish our first result:

Proposition 1 [Existence and Uniqueness of the Equilibrium] A unique equilibriumexists in the economy with margin requirements and diverse beliefs {p∗

0, x∗0 , p∗

u, x∗u ,

p∗d , x∗

d } such that p∗d < p∗

0 < p∗u, x∗

d ≤ x∗0 < x∗

u .

Proof The equilibrium conditions (4), (5), (6), (8), (10), (12) and (14) of p∗0, x∗

0 , p∗u,

x∗u , p∗

d , x∗d can be solved uniquely. The fact p∗

0 < p∗u is obtained simply because each

investor becomes more optimistic and their willingness to pay for the stock increase.Because Wu > W0 the demands of shares Du(x) increase for those investors x ≥ x∗

0who have capital gains from the increase in stock price. Using the market clearingcondition (1 − x∗

u )Du(x) = 1, we can prove that x∗u > x∗

0 . The property of x∗d and p∗

dcan be proved similarly. ��The above result demonstrates how a shock in the period 1 may affect stock prices. Inperiod 1, if an up shift in expectation (u) is observed, an increase in demand occurs,pushing the price p∗

u up with an endogenously chosen x∗u . Conversely, the price p∗

d andcorresponding x∗

d are determined endogenously when the down state (d) is observed.

3 Market volatility and the influences of margin requirements

In this section we provide a framework to analyze the impact of margin require-ments on market volatility. We decompose the total effect into “leverage effect” and

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pyramiding/depyramiding effects. A change in the margin requirement initially influ-ences the credit for purchasing stocks. A simple example may help illustrate these twoeffects. Consider that an investor with $1,000 may purchase the stock up to $2,000 orwith 50% margin. If the margin is reduced to 40%, the investor can purchase up to$2,500. That is, there is an additional $500 credit for investments, namely the leverageeffect. Suppose that in the next period the price either increases or decreases 10%.When the price goes up, the investor enjoys a capital gain of $250 and may purchaseadditional shares, keeping the margin requirement satisfied. An increase in additionalcash from capital gain may push the price higher, which is the pyramiding effect.Conversely, the investor suffers a loss when the price goes down, and is required tosell their shares to satisfy the margin requirement. The price may decrease further,which is the depyramiding effect.

The above example intuitively suggests that a decrease in margin requirementresults in a positive effect on the initial stock price. More precisely, when the assetprice moves from the original level, before the margin call occurs and the investorsrebalance their position, we call it a leverage effect in this paper. However, investorsmay change their positions due to a capital gain or loss in the subsequent period.If the stock price increases, because the investors have additional credit to purchasenew shares and maintain the margin requirement, the stock price may increase further.This phenomenon is called the pyramiding effect. Conversely, the depyramiding effectdescribes the case when investors suffer a loss from stock depreciation and then reducetheir position. Hence, the stock price decreases further. While the above argument isaccepted for the margin requirement discussion, the effect of stock prices in a gen-eral equilibrium framework is more complicated. For instance, a decrease in marginrequirement results in an increase in initial price, reduces further capital gain, andtherefore affects the magnitude of pyramiding/depyramiding effect described earlier.Our general equilibrium framework will provide a complete analysis of these effects.

3.1 Leverage effect and static results

We first focus on the current equilibrium stock price. The static result of a marginrequirement here is consistent with the previous single-period model of Luckett (1982).The following two propositions characterize how changes in several parameters mayaffect the stock price in period 0:

Proposition 2 [Leverage Effect on the Current Stock Price] The equilibrium price p∗0

is negatively related to the margin requirement m0.

Proof It is simply observed from

∂p∗0

∂m0= − W0(a + b)bV 2

H

(W0 + bm0VH )2 < 0.

��

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In the aforementioned result, an increase in the margin requirement restrains the lever-age that investors can use, and therefore, decreases the stock demand and equilibriumprice.

Proposition 3 [Comparative Statics on the Current Stock Price] The equilibrium pricep∗

0 is positively related to available investment funds W0, the market fundamentals VH ,investors’ optimism a, and belief heterogeneity b.

Proof It is proven using the following comparative statics:

∂p∗0

∂W0= (a + b)bm0V 2

H

(W0 + bm0VH )2 > 0,

∂p∗0

∂VH= W 2

0 (a + b)

(W0 + bm0VH )2 > 0,

∂p∗0

∂a= W0VH

W0 + bm0VH> 0,

∂p∗0

∂b= W0 (W0 − am0VH ) VH

(W0 + bm0VH )2 > 0.

��When the available fund is higher, optimistic investors have greater wealth to purchase,which boosts the equilibrium price. It suggests that with an expansionary monetarypolicy more investment funds (W0) are injected into a speculative market, the opti-mistic investors will put more cash into the market and therefore boost the priceshigher. The equilibrium price is also positively related to the final payoff because ofthe market fundamentals (VH ). Moreover, the equilibrium price increases as investorsbecome more optimistic (a) or more diverse (b).

3.2 Decomposition of the total effect of changes in margin requirements

Because the influence of a change in margin requirement on price volatility can goeither way, we propose a decomposition to help our understanding. The leverage effectof margin requirements on the subsequent stock prices is examined first. Denote Fs(x∗

0 )

as the “unbalanced stock price,” which is the willingness to pay at state s, s ∈ {u, d},when investors do not rebalance their margin account:

Fu(x∗

0

) = Bu(x∗

0

)VH = (

a + bx∗0 + u

)VH ,

Fd(x∗

0

) = Bd(x∗

0

)VH = (

a + bx∗0 − d

)VH .

It leads to

Fu(x∗

0

) − Fd(x∗

0

) = [Bu

(x∗

0

) − Bd(x∗

0

)] · (VH − VL) = (u + d)VH .

When the up state occurs, the stock price can be described as

p∗u = Fu

(x∗

0

) + (p∗

u − Fu(x∗

0

)). (15)

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Fig. 3 Marginally optimistic investors and stock prices. In this example the parameters are given that VH =20, W0 = 20, B0(x) = 0.25 + 0.5x, Bu(x) = 0.45 + 0.5x, Bd (x) = 0.1 + 0.5x, m0 = 0.5, and mT =0.4. The solutions are as follows: marginally optimistic investors as (x∗

0 , x∗u , x∗

d ) = (0.70, 0.77, 0.60);equilibrium stock prices as (p∗

0 , p∗u , p∗

d ) = (12, 16.66, 8); unbalanced prices as (Fu(x∗0 ), Fd (x∗

0 )) =(16, 9); pyramiding effect is equal to 0.66, and depyramiding effect is equal to 1

The first item in the right part of the above equation can be indicated as the leverageeffect which does not entail a further increase or decrease in credit. The second item,therefore, is the pyramiding effect. Similarly, we have the equation in the down state:

p∗d = Fd

(x∗

0

) − (Fd

(x∗

0

) − p∗d

). (16)

The leverage and depyramiding effects are, respectively, the first and second items inthe right part of the above equation.

From this setting we can analyze the pyramiding/depyramiding effects. The pyr-amiding effect (p∗

u − Fu(x∗0 )) represents the impacts when investors rebalance their

credit using capital gains to purchase more shares. Conversely, the depyramiding effect(Fd(x∗

0 ) − p∗d) is the impacts when investors reduce their position further because of

a capital loss. We represent pyramiding and depyramiding effects as follows:

p∗u − Fu

(x∗

0

) = Bu(x∗

u

)VH − Bu

(x∗

0

)VH = (

x∗u − x∗

0

)bVH , (17)

Fd(x∗

0

) − p∗d = Bd

(x∗

0

)VH − Bd

(x∗

d

)VH = (

x∗0 − x∗

d

)bVH . (18)

We now decompose the total effect of margin requirement on asset price volatility intothree components, as shown in Fig. 3:

p∗u − p∗

d = [Fu

(x∗

0

) − Fd(x∗

0

)] + (p∗

u − Fu(x∗

0

)) + (Fd

(x∗

0

) − p∗d

). (19)

With such decomposition of Eq. (19), we can obtain the following characterization:

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Proposition 4 [Pyramiding/depyramiding] Pyramiding always occurs in equilibriumwhen the up state occurs, that is, p∗

u is greater than the unbalanced price Fu(x∗0 ) =

Bu(x∗0 )VH . However, depyramiding occurs only when the down change is large enough

to result in margin calls, i.e., d > d∗m. In the depyramiding case, p∗

d is less thanFd(x∗

0 ) = Bd(x∗0 )VH .

Proof Pyramiding occurs because x∗u > x∗

0 , which results in p∗u = Bu(x∗

u )VH >

Bu(x∗0 )VH . The price in the down state is observed in Eqs. (10) and (12). Therefore,

depyramiding appears only when the down change in expectation is large enough totrigger margin calls. ��The asymmetrical impact from a price increase and a price decrease is also foundin empirical observations in Hardouvelis and Theodossiu (2002). They point outthe asymmetric property of pyramiding/depyramiding effects that increasing marginrequirement is more effective in a bull market and less effective in a bear market. Ourresults further suggest that the depyramiding effect may occur only when the decreasein price is sufficiently large, causing margin calls.

Proposition 5 [Influence of Margin Requirements on Unbalanced Stock Prices] Anincrease in margin requirement m0 results in a negative leverage effect on the stockprice (p∗

u and p∗d). With the assumption of parallel movement in the belief structure,

the unbalanced stock price volatility is not associated with a margin requirement. Thatis,

1. ∂ Fs(x∗0 )/∂m0 < 0, s ∈ {u, d},

2. ∂[Fu(x∗0 ) − Fd(x∗

0 )]/∂m0 = 0.

Moreover, Fs(x∗0 ), s ∈ {u, d}, are positively related to W0, VH , a, and b.

Proof It is proven obviously by Fu = (a + bx∗0 + u)VH and Fd = (a + bx∗

0 − d)VH

and Proposition 2. ��The above result suggests that for the leverage effect, an increase in margin reducesthe credit investors can use, and therefore, decreases the stock price. However, becauseof the parallel shift in expectation in our framework, a margin requirement does notinfluence volatility in “unbalanced stock price”. These results can be reconsidered ina more general framework with non-parallel shifts of belief structure.

Proposition 6 [Influence of Margin Requirements on Pyramid/DepyramidingEffects] An increase in margin requirement results in a negative (positive) effect onpyramiding or depyramiding if the stock ownership becomes more (less) concentrated.That is,

1. ∂[p∗u − Fu(x∗

0 )]/∂m0 < 0, if ∂(x∗u − x∗

0 )/∂m0 < 0;2. ∂[Fd(x∗

0 ) − p∗d ]/∂m0 < 0, if ∂(x∗

0 − x∗d )/∂m0 < 0.

Proof It is obvious from (17) and (18). ��The second statement of Proposition 6 will be analyzed further in Proposition 9.

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Proposition 7 [Influence of Margin Requirements on Stock Prices] The equilibrium

price p∗u is decreasing related to the margin requirement m0, i.e., ∂p∗

u∂m0

< 0, and p∗d is

decreasingly related to the margin requirement m0 if the change in margin requirement

is small and d is not large enough to trigger margin calls, that is,∂p∗

d∂m0

< 0.

Proof Note that ∂x∗u

∂m0≥ 0 if ∂p∗

u∂m0

≥ 0. From∂x∗

0∂m0

< 0 it leads to a contradiction.Because x∗

d = x∗0 when margin calls are not triggered, and using the fact that x∗

0 isnegatively related to m0, we can verify that x∗

d and p∗d are negatively associated with

m0. ��

This proposition suggests that an increase in the margin requirement m0 reduces theamount of speculation by optimistic investors; therefore, prices decline as favorableinformation is observed. Hardouvelis (1990) provided empirical evidence on the mar-gin requirements effect on stock market volatility to examine the pyramiding/depy-ramiding effect argument. This paper, however, demonstrates how the pyramidingeffect may arise in a general equilibrium model with speculative trading and marginrequirements.

The next result summarizes the impacts of margin requirement on price volatilityin the subsequent period.

Proposition 8 [Influence of Margin Requirements on Price Volatility] An increase inmargin requirement m0 results in a negative (positive) effect on price volatility if thestock ownership becomes more (less) concentrated. That is,

∂[

p∗u − p∗

d

]/∂m0 < 0, if ∂

(x∗

u − x∗d

)/∂m0 < 0. (20)

Proof This is proven using Propositions 5 and 6. ��

Our theoretical results demonstrate that increasing margin requirement may not reducestock price volatility. We provide the conditions under which increasing marginrequirements (reducing financial leverage) can stabilize the stock market. Our resultscall for more detailed study of whether these conditions can be satisfied so that reducingleverage ratios can lead to the desired results in the economy.

4 Comparative statics of market volatility with unanticipated shocks

In the framework studied so far, we assume that the investors in period 0 do not antic-ipate the possible changes of beliefs described by Eqs. (1) and (2). Therefore, theycompute the payoffs according to Eq. (3) and do not purchase stocks to benefit fromintermediate trading in period 1. This is called the case of unanticipated shocks. Thissection presents more comparative static results for unanticipated shocks. The case ofanticipated shocks will be considered in the next section.

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4.1 The effects of margin requirements

From Proposition 4 we know that depyramiding effect is present once margin callsare triggered, i.e., when d > d∗

m . And we can obtain an analytical solution for thedepyramiding effect.

Proposition 9 When d > d∗m, the depyramiding effect increases with shock d and

heterogeneity b, and decreases with available investment fund W0 and initial marginrequirement m0.

Proof When margin calls are triggered, the depyramiding effect can be explicitlyderived as in Eq. (14). Direct calculation gives us the proposition. ��However, for the pyramiding effect an explicit solution is not derived, and numericalanalysis is necessary.

Next we present an example for the impacts of margin requirement on the equi-librium. It shows that equilibrium prices are negatively associated with the marginrequirement m0, and so is price volatility p∗

u − p∗d , under the conditions as specified

above. This example provides partial support for the policy prescription of reducingleverage ratios to stabilize the market.

Example 1 We consider a set of parameters VH = 20, W0 = 20, belief structureB0(x) = 0.25 + 0.5x, u = 0.2, d = 0.15, and maintenance margin mT = 0.8m0. In

Table 1 The impact of margin requirement on stock prices with unanticipated shocks

m0(%) x∗0 p∗

0 x∗u p∗

u x∗d p∗

d Fu Fd p∗u − Fu Fd − p∗

d p∗u − p∗

d

35 0.78 12.77 0.85 17.51 0.63 8.32 16.77 9.77 0.75 1.44 9.19

40 0.75 12.50 0.82 17.23 0.62 8.21 16.50 9.50 0.73 1.29 9.01

45 0.72 12.24 0.79 16.94 0.61 8.11 16.24 9.24 0.70 1.14 8.84

50 0.70 12.00 0.77 16.66 0.60 8.00 16.00 9.00 0.66 1.00 8.66

55 0.68 11.76 0.74 16.37 0.59 7.89 15.76 8.76 0.61 0.87 8.48

60 0.65 11.54 0.71 16.09 0.58 7.79 15.54 8.54 0.55 0.75 8.30

65 0.63 11.32 0.68 15.81 0.63 8.32 15.32 8.32 0.49 0.00 7.49

70 0.61 11.11 0.65 15.53 0.61 8.11 15.11 8.11 0.42 0.00 7.42

75 0.59 10.91 0.63 15.26 0.59 7.91 14.91 7.91 0.36 0.00 7.36

80 0.57 10.71 0.60 15.00 0.57 7.71 14.71 7.71 0.29 0.00 7.29

85 0.55 10.53 0.57 14.74 0.55 7.53 14.53 7.53 0.21 0.00 7.21

90 0.53 10.34 0.55 14.49 0.53 7.34 14.34 7.34 0.14 0.00 7.14

95 0.52 10.17 0.52 14.24 0.52 7.17 14.17 7.17 0.07 0.00 7.07

100 0.50 10.00 0.50 14.00 0.50 7.00 14.00 7.00 0.00 0.00 7.00

This illustration considers the set of parameters as VH = 20, W0 = 20, belief structure B0(x) = 0.25 +0.5x, u = 0.2, d = 0.15, and maintenance margin mT = 0.8m0. When margin requirement is bigger than64%, margin calls are not triggered; and to exclude the case of default, margin requirement is assumed tobe bigger than 35%. The following table present the results for the effects of margin requirements on theequilibrium

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A

B

C

Fig. 4 Margin requirements on price volatility, pyramiding, and depyramiding effects. The following threefigures illustrate the impacts of margin requirements on price volatility (p∗

u − p∗d ), pyramiding (p∗

u −Fu(x∗0 ))

and depyramiding (Fd (x∗0 ) − p∗

d ), given the parameters VH = 20, W0 = 20, B0(x) = 0.25 + 0.5x , andmaintenance margin mT = 0.8m0. The extent of price volatility, pyramiding, and depyramiding effectsare negatively associated with margin requirements and become greater with larger shocks in investors’expectations (u = 0.2, d = 0.15). a Price volatility and margin requirements. b Pyramiding and marginrequirements. c Depyramiding and margin requirements

Proposition 7 we analyze the margin requirement impact on prices p∗u , and p∗

d . Theexample as shown by Table 1 illustrates negative associations between them. Pyram-iding (p∗

u − Fu) and depyramiding (Fd − p∗d ) are also shown in Table 1. In Fig. 4 we

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illustrate how price volatilities, pyramiding, and depyramiding are negatively asso-ciated with margin requirements. A smaller shock (u = 0.15, d = 0.1) providesless volatility and pyramiding/depyramiding effects. For this example, market pricevolatility decreases as m0 increases.

Example 1 demonstrates the functioning of speculation when margin requirementsincrease. This result provides a theoretical explanation of speculative trading on sev-eral empirical evidences observed in the US and other markets (see, for example,Hardouvelis 1990; Hardouvelis and Peristiani 1992). It should also shed some lighton the policy issue of changing leverage ratios. If the government sets out to coola overheated stock market, an increase in margin requirement may pull the pricesdown and reduce stock price volatility. In contrast, if the government wants to spur onan undervalued stock market, a decrease in margin requirement may push prices up.However, such an action can increase stock market volatility.

4.2 The effects of dispersions of beliefs

With a similar set of parameters including VH = 20, W0 = 20, B0(x) = 0.25 +b · x, u = 0.2, d = 0.15, and maintenance margin mT = 0.8m0, Fig. 5 shows howbelief dispersion may affect price volatility (p∗


u − Fu(x∗0 )),

and depyramiding (Fd(x∗0 ) − p∗

d ). Generally speaking, price volatility, pyramiding,and depyramiding effects are positively associated with belief dispersion, except inthe case when depyramiding moves from positive to zero.

Figure 5a and b suggest that both price volatility and pyramiding increase as beliefsbecome more dispersed among investors. Depyramiding, as shown in Fig. 5c, how-ever, may drop to zero when margin calls are not triggered. Besides confirming theresult of Miller (1977), we also obtain a complete characterization that an increasein the divergence of belief will increase market price. Here, the marginally optimis-tic investors play an important role in characterizing such an equilibrium. When theheterogeneity of belief structure increases, equilibrium stock price increases while themarginally optimistic investors become relative pessimistic in order to clear the mar-ket. Furthermore, such decrease in the optimism of the marginally optimistic investorsis negatively related to the degree of the optimism in the economy.11 In other words,the optimism of the marginally optimistic investors in the up state decreases with theheterogeneity of belief structure, but the extent is less than that of date 0; thus pyramid-ing effects are positively associated with belief dispersion. A similar argument appliesto the depyramiding effect when margin calls are triggered. However, the marginallyoptimistic investor is the same with that of date 0 when margin calls are not triggered,resulting a zero depyramiding effect.

11 We can derive this property from Eq. (4): ∂x∗0 /∂b < 0 and ∂2x∗

0 /∂b∂a > 0.

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B

C

A

Fig. 5 Dispersion of beliefs on price volatility, pyramiding, and depyramiding effects. The following threefigures illustrate the impacts of dispersion of belief on price volatility (p∗


u − Fu(x∗0 ))

and depyramiding (Fd (x∗0 ) − p∗

d ), given the parameters VH = 20, W0 = 20, B0(x) = 0.25 + bx,

u = 0.2, d = 0.15, and maintenance margin mT = 0.8m0. Generally speaking, price volatility, pyramid-ing, and depyramiding effects are positively associated with dispersion of beliefs, except the case when thedepyramiding move from positive to zero. a Price volatility and dispersion of opinion. b Pyramiding anddispersion of opinion. c Depyramiding and dispersion of opinion

4.3 The effects of investment funds

Figure 6 illustrates the impact of investment funds on price volatility (p∗u − p∗

d ),pyramiding (p∗

u − Fu(x∗0 )), and depyramiding (Fd(x∗

0 ) − p∗d ), given the parameters

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A

B

C

Fig. 6 Investment funds on price volatility, pyramiding, and depyramiding effects. The following threefigures illustrate the impacts of margin requirements on price volatility (p∗


u −Fu(x∗0 ))

and depyramiding (Fd (x∗0 )− p∗

d ), given the parameters VH = 20, B0(x) = 0.25+0.5x, u = 0.2, d = 0.15,and maintenance margin mT = 0.8m0. The extent of price volatility, pyramiding, and depyramiding effectsare in general negatively associated with investment funds. In c, depyramiding is 0 when m0 = 0.8. a Pricevolatility and investment funds. b Pyramiding and investment funds. c Depyramiding and investment funds

VH = 20, B0(x) = 0.25 + 0.5x, u = 0.2, d = 0.15, and maintenance margin mT =0.8m0. The price volatility, pyramiding, and depyramiding effects are in general nega-tively associated with investment funds. In Fig. 6c, depyramiding is 0 when m0 = 0.8.The amount of investment funds is positively associated with the stock prices, but

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negatively associated with price volatility. Our model predicts that an increase inmoney invested in the market may lead to higher market returns and lower volatility(Fig. 6).

5 Market volatility with anticipated shocks

In the previous sections the investors are assumed not to realize that there will a shockin the next period. In this section, investors are assumed to anticipate a shock whichlet them become either more optimistic or pessimistic. The changes in beliefs arecommon knowledge for all investors. Therefore, the investors who purchase the stockin period 0 may want to sell the stock later for profits, similar to speculative tradingin Harrison and Kreps (1978) and Wu and Guo (2003, 2004).

In this section, the model setup is the same as that of Sect. 2 except that all inves-tors anticipate there will be a shock affecting their expectations at t = 1. Thus for allinvestors, B0(x) = d

u+d Bu(x) + uu+d Bd(x). That is to say, at t = 0 all investors are

aware that an up state will be realized at t = 1 with the same probability, althoughthey have different beliefs once the up is realized. Let θ = d

u+d denote the probabilityof the up state.

5.1 Leveraged trading with speculations

Here, we use the upper case to denote stock prices P0 and P1 (including Pu and Pd )for the case with anticipated shocks. Investors in period 0 choose their demands D0(x)

and D1(x) (including Du(x) and Dd(x)) to maximize their expected wealth:

W (x) = W0 + D0(x)(

P1 − P0

)+ D1(x)

(V − P1

).

Because the shock is anticipated by all investors, backward induction is applied. Att = 1, the analysis is just the same as in Sect. 2 since there is no more trading opportu-nities. There exist critical values xu and xd ∈ (0, 1) (marginally optimistic investor)such that equilibrium prices are characterized as follows:

Pu = Bu(xu)VH = (a + bxu + u)VH , (21)

Pd = Bd(xd)VH = (a + bxd − d)VH . (22)

Investors who are more optimistic than the marginally optimistic investor buy the stockon margin, and those who are more pessimistic are inactive in the market. When theup state is realized and stock price appreciates, for investors who are more optimistic,i.e., x > xu , their wealth becomes Wu(x) = W0 + D0(x)(Pu − P0) ≥ W0, and themarket clearing condition is

∫ 1xu

Wu(x)m0 Pu

dx = 1.But things become more complicated when the down state occurs. Investors who

buy the stock in period 0 suffer a capital loss. For investors who are more optimisticthan xd and purchase stock in period 0, when m0 P0−(P0−Pd )

Pd< mT , margin calls are

triggered, and they have to rebalance their position to the initial margin level. Therefore

123


for those investors, Dd(x) = Wd (x)m0 Pd

; otherwise, they continue to hold D0(x). For inves-tors who are more optimistic than xd and do not purchase stock in period 0, there isno problem of margin call; thus Dd(x) = W0

m0 Pd.

Then we study investors’ demand in period 0. By backward induction, investorsknow that D1 are functions of D0. The optimal demand D0(x) is chosen to maximizeinvestors’ expected final wealth:

E[W (x)] = W0 + D0(x)[θ Pu + (1 − θ)Pd − P0]+{θ Du(x)(Bu(x)VH − Pu) + (1 − θ)Dd(x)(Bd(x)VH − Pd)}, (23)

where E is the expectation operator. In a competitive equilibrium with prices P0, Pu ,and Pd , E[W (x)] is a linear function of D0. Thus D0(x) = W0/m0 P0 if and only if∂E[W (x)]

∂ D0≥ 0, otherwise D0(x) = 0 due to short sale constraint.

For the most pessimistic investors x < min{xu, xd}:

E[W (x)] = W0 + D0(x)(E(P1) − P0). (24)

If P0 ≤ E(P1),∂E[W (x)]

∂ D0= E(P1) − P0 ≥ 0, implying that they will purchase the

stock on margin, even though they do not hold any stock at t = 1. According to Kaldor(1939), they are the “speculators” who purchase the stock in period 0 with the purposeto sell it in period 1. However, if Pd < E(P1) < P0 < Pu , for these investors who are

most pessimistic, ∂E[W (x)]∂ D0

< 0; thus they hold zero position at both t = 0 and t = 1.Next we look for equilibria with Pu > Pd , especially those equilibria with Pu >

P0 > Pd . It is obvious that, given Pu and Pd , P0 cannot be higher than Pu in equilibria.And to exclude the trivial case where P0 ≤ Pd in equilibrium, we assume that investorshave enough fund.12 In the next proposition, we show that Pd < P0 ≤ E(P1) < Pu inequilibrium; therefore, speculative trading by the most pessimistic investors is alwayspresent.

Proposition 10 [Properties of Equilibrium with Anticipated Shocks] An equilibriumexists in the economy with anticipated shocks such that Pd < P0 ≤ E(P1) < Pu.

Proof See Appendix A. ��

5.2 The properties of equilibrium prices

When Pd < P0 ≤ E(P1) < Pu , the most pessimistic investors x < min{xd , xu} pur-chase stocks for speculation. Then we look for a specified equilibrium where investorsx ∈ (0, x0)(x0 > max{xd , xu}) purchase stock at t = 0. Other kinds of equilibria arediscussed in Appendix B.

To simplify our discussion, suppose that margin calls are always triggered at thedown state, or maintenance margin mT is very close to the initial margin requirement

12 It can be shown that there exists a unique equilibrium with Pu > Pd > P0 when W0/m0 ≤ (a −d)VH .

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m0. With the help of the assumption, we get derive the first derivative of investors’expected wealth with respect to their demand at t = 0 when xu > xd ,:

∂E[W (x)]∂ D0

=

⎧⎪⎪⎨

⎪⎪⎩

E(P1)−P0+θ Pu−P0m0 Pu

(x − xu)bVH +(1 − θ)Pd−P0m0 Pd

(x − xd)bVH , if x ≥ xu,

E(P1) − P0 + (1 − θ)Pd−P0m0 Pd

(x − xd)bVH , if xd ≤ x < xu,

E(P1) − P0, if x < xd .

(25)

And when ∂E[W (x)]∂ D0

|x=xu > 0 and ∂E[W (x)]∂ D0

|x=1 < 0, there exists x0 > xu such that∂E[W (x)]

∂ D0|x=x0 = 0 (Fig. 7).

And if xu < xd , the first derivative is

∂E[W (x)]∂ D0

=

⎧⎪⎪⎨

⎪⎪⎩

E(P1) − P0+θ Pu−P0m0 Pu

(x − xu)bVH +(1 − θ)Pd−P0m0 Pd

(x − xd)bVH , if x ≥ xd ,

E(P1) − P0 + θ Pu−P0m0 Pu

(x − xu)bVH , if xu ≤ x < xd ,

E(P1) − P0, if x < xu .

(26)

Since Pu > E(P1) > P0,∂E[W (x)]

∂ D0|x=xd must be positive, and when ∂E[W (x)]

∂ D0|x=1 < 0,

there exists x0 > xd such that ∂E[W (x)]∂ D0

|x=x0 = 0 (Fig. 8).Due to linearity, investors purchase stock on margin if and only if the first deriva-

tive of expected wealth to demand is positive. The conditions for equilibria are statedbelow no matter whether xu ≥ xd or xu < xd :

1. x0W0

m0 P0= 1, (27)

2. (x0 − xd)

W0m0 P0

(Pd − P0)

m0 Pd+ (1 − xd)

W0

m0 Pd= 1, (28)

1

Fig. 7 The first derivative of expected wealth with respect to demand when xd < xu

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Fig. 8 The first derivative of expected wealth with respect to demand xu < xd

3. (x0 − xu)

W0m0 P0

(Pu − P0)

m0 Pu+ (1 − xu)

W0

m0 Pu= 1, (29)

4.∂E[W (x)]

∂ D0|x=x0 = E(P1) − P0 + θ

Pu − P0

m0 PubVH (x0 − xu)

+(1 − θ)Pd − P0

m0 PdbVH (x0 − xd) = 0, (30)

5. max{xu, xd} < x0 < 1, (31)

where Eq. (27) is the market clearing condition in period 0, Eqs. (28) and (29) arethe market clearing conditions of the down and up state in period 1, respectively. Inaddition, x0 is determined by Eq. (30), and Eq. (31) is feasible condition of x0.

Next, we present an example to show the impact of margin requirement on stockprice levels and volatility. It is shown that the price levels P0, Pu and Pd are all decreas-ing with margin requirement. However, price volatility, measured by Pu − Pd , doesnot vary monotonically with margin requirement, which is different with the case ofunanticipated shock.

Example 2 Suppose that a = 0.25, b = 0.5, u = 0.2, d = 0.15, VH = 20, W0 = 5,so θ = 3/7, and Pu = 9 + 10xu, Pd = 2 + 10xd .

We can check that when m0 = 40%,

xu = 0.39 < xd = 0.41 < x0 = 0.60,

and prices

Pd = 6.13, Pu = 12.94, P0 = 8.63,

constitute an equilibrium.And when m0 = 80%,

xd = 0.13 < xu = 0.16 < x0 = 0.85,

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Table 2 The impact of margin requirement on price levels and volatility with anticipated shocks

m0(%) x0 P0 xu Pu xd Pd Pu − Pd

35 0.60 8.63 0.39 12.94 0.41 6.13 6.81

40 0.63 7.87 0.37 12.75 0.35 5.47 7.28

45 0.66 7.30 0.35 12.54 0.30 4.98 7.56

50 0.69 6.86 0.33 12.32 0.26 4.60 7.72

55 0.71 6.49 0.31 12.08 0.23 4.29 7.79

60 0.74 6.19 0.28 11.81 0.20 4.03 7.78

65 0.77 5.92 0.25 11.53 0.18 3.82 7.72

70 0.80 5.70 0.22 11.24 0.16 3.64 7.60

75 0.82 5.49 0.19 10.93 0.15 3.48 7.45

80 0.85 5.31 0.16 10.61 0.13 3.34 7.27

85 0.87 5.14 0.13 10.29 0.12 3.23 7.06

90 0.90 4.98 0.10 9.96 0.11 3.12 6.84

95 0.92 4.84 0.06 9.63 0.10 3.03 6.60

100 0.94 4.70 0.03 9.30 0.09 2.94 6.35

The following table presents the effects of margin requirements on the equilibrium with anticipated shocksgiven parameter: VH = 20, W0 = 5, B0(x) = 0.25 + 0.5x, u = 0.2, d = 0.15

and prices

Pd = 3.34, Pu = 10.61, P0 = 5.31,

constitute an equilibrium (Table 2).The price level decreases with margin requirement, or increases with leverage ratio,

which is very intuitive: the higher the leverage ratio, the more leveraged fund the inves-tor can invest in stock, and more demand drives the stock price up. But the impact ofleverage ratio on up-state and down-state prices are asymmetric, and we can get theeconomic reasons from the changes of xu and xd . Both xu and xd decrease with marginrequirement, and there exists m∗

0 (= 55% in Example 2) such that when m < m∗0, xd

decrease with margin requirement more quickly than xu , and when m > m∗0, xu

decrease more quickly than xd . Since ∂(Pu−Pd )∂m0

depends on ∂(xu−xd )∂m0

, price volatilityincreases with margin requirement when margin level is relative low, and decreaseswith it when margin is relative high. Therefore, a reduction in leverage ratios may notlead to lower price volatility, which poses an important challenge to any simple-mindedpolicy prescription for reducing market volatility.

6 Concluding remarks

In this paper we have developed a theoretical framework to analyze the impacts ofadjusting margin requirements on market prices and volatilities in the presence ofdiverse belief and speculative trading on the margin. Margin trading is demonstratedto enhance speculation and stock price volatility in our model because investors can

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obtain additional funds to finance their strategies through a leverage effect. The decom-position of total effect into the leverage effect and pyramding/depyramiding effectsare illustrated. Our results provide partial support for the claim that reducing financialleverage can stabilize the market when the shock is unanticipated. Dispersion of beliefis shown to increase the price in a good state but decreases the price in a down state.The effects of investment funds and policy implications are also discussed. Further-more, we demonstrate that when the shock is anticipated the investors may adoptspeculative trading strategies and a reduction in leverage ratio may not reduce pricevolatility.

The pyramiding/depyramiding effects analyzed in this paper will become clearerin an extended framework with multiple periods. Consider the framework with Tdates of trading and final payoff realizing in period T + 1, in which identical andindependent signals are observed at each date between 2 and T − 1. We can thensolve the equilibrium using iterations similar to equations obtained in this paper.The pyramiding mechanism is still effective because when an up state is observed,investors purchase more stocks because they are profiting from margin trading andrebalancing their positions, and therefore the stock prices will increase. The depy-ramiding effect, however, may not be effective when the margin calls are not trig-gered.

Although the impacts of reducing leverage ratios are often believed to be stabilizingthe market, our theoretical results demonstrate that, in spite of some partial supportfrom our model, such a belief still lacks a solid foundation in theory. Our paper pro-vides a rigorous general equilibrium foundation to study this issue more carefully.Further studies with risk-averse investors, non-parallel movement of belief structures,or more sophisticated trading strategies are needed to clarify this issue.

Appendix A: Proof of Proposition 10

We will show that in equilibrium Pd < P0 ≤ E(P1) < Pu by contradictions.Suppose that Pd < E(P1) < P0 < Pu in equilibrium, then for the most pessimistic

investors x < min{xd , xu}, E[W (x)] = W0+D0(x)[E(P1)−P0], which is decreasingin D0. Thus they hold zero position in period 0 and the speculation identified in thetext disappears.

First, we show that it must be xu < xd with Pd < E(P1) < P0 < Pu . Otherwise,if xu ≥ xd , by Eq. (25), ∂E

[W (x)

]/∂ D0(x) are definitely negative for investors

x ∈ (0, xu). And when x > xu, ∂E[W (x)]/∂ D0(x) is decreasing with x , so for all

x > xu,∂E[W (x)]∂ D0(x)

<∂E[W (x)]∂ D0(x)

|x=xu < 0. Therefore, ∀x ∈ [0, 1], ∂E[W (x)]∂ D0(x)

≤ 0 ⇒D0(x) = 0, which is a contradiction with the market clearing condition.

When xu < xd , by Eq. (26), we know that ∂E[W (x)]/∂ D0 takes its maximum atx = xd , so ∂E[W (x)]/∂ D0

∣∣x=xd

must be strictly positive; otherwise all investors donot hold any stock at t = 0, contradicting the market clearing condition again.

Since ∂E[W (x)]/∂ D0 is continuous with respect to x , there exists a marginalinvestor x0 ∈ (xu, xd) such that ∂E[W (x)]/∂ D0

∣∣x=x0

= 0, and for investor x <

x0 : ∂E[W (x)]/∂ D0 < 0. Thus, in period 0, at most there are as many investors as

123


(x0, 1) ⊂ (xu, 1) purchasing stock on margin, which is impossible due to the pyr-amiding effect of margin trading. Intuitively, when the up state is realized in period1, investors who purchase stock on margin get additional capital gains from margintrading, which further increases their demand at the up state. By the market clearingcondition, there must be less investors holding stock at the bull market. The samereasoning applies when down state is realized in period 1.

By contradictions of two complete case xu ≥ xd and xu < xd , equilibrium pricesmust be Pd < P0 ≤ E(P1) < Pu .

Appendix B

In this appendix, we analyze other possible equilibria with Pd < P0 ≤ E(P1) < Pu .One possible equilibrium is with xd < x0 < xu as shown in Fig. B1. The left part isrelated with Pu Pd

(1−θ)Pu+θ Pd< P0 < E(P1), and the right one is to P0 <

Pu Pd(1−θ)Pu+θ Pd

≤E(P1). For both cases, there exist x0 ∈ (xd , xu) such that ∂E[W (x)]

∂ D0

∣∣∣x=x0

= 0. Condi-

tions for such equilibria are

1. x0W0

m0 P0= 1 (market clearing condition at t = 0);

2. (x0 − xd)W0 + W0

m0 P0(Pd − P0)

m0 Pd+ (1− x0)

W0

m0 Pd= 1 (market clearing condition

at down state);3. (1 − xu) W0

m0 Pu= 1 (market clearing condition at up state);

4. E(P1) − P0 + (1 − θ)Pd−P0m0 Pd

bVH (x0 − xd) = 0;

5. ∂E[W (x)]/∂ D0|x=1 < 0, and ∂E[W (x)]/∂ D0|x=xu < 0.

However, condition 2 and condition 3 are contradictory: (1 − x0)W0

m0 Pd> (1 −

xu)W0

m0 Pd> (1 − xu)

W0

m0 Pu= 1, where the first inequality comes from x0 < xu and

the second inequality comes from Pd < Pu . So equilibrium of Fig. B1 does not exist.Another kind of equilibrium is also possible: there exists x01 ∈ (xd , xu), and

x02 ∈ (xu, 1) such that ∂E[W (x)]∂ D0

|x=x01= ∂E[W (x)]

∂ D0|x=x02

= 0. In equilibrium, investorsx ∈ (0, x01)

⋃(x02, 1) buy stock on margin at t = 0.

Fig. B1 Possible equilibria when xd < x0 < xu

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Fig. B2 Possible equilibrium when xd < x01 < xu < x02

And conditions for such equilibria are

1. (1 − x02 + x01)W0

m0 P0= 1 (market clearing condition at t = 0);

2. [(x01 − xd)+ (1− x02)] W0+ W0m0 P0

(Pd−P0)

m0 Pd+ (x02 − x01)

W0m0 Pd

= 1 (market clearingcondition at down state);

3. (x02 − xu) W0m0 Pu

+ (1 − x02)W0+ W0

m0 P0(Pu−P0)

m0 Pu= 1 (market clearing condition at

up state);4. E(P1) − P0 + (1 − θ)

Pd−P0m0 Pd

bVH (x01 − xd) = 0;

5. E(P1) − P0 + θ Pu−P0m0 Pu

bVH (x02 − xu) + (1 − θ)Pd−P0m0 Pd

bVH (x02 − xd) = 0;

6. ∂E[W (x)]/∂ D0∣∣x=1 > 0, and ∂E[W (x)]/∂ D0

∣∣x=xu

< 0.

We do not find any contradiction for such an equilibrium as in Fig. B2.

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