Andre Levy - Thesis - Revised - Nov 2015.pdf

7/24/2019 Andre Levy - Thesis - Revised - Nov 2015.pdf

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The price is not always right:the effects of liquidity costs and constraints on prices and allocations

Andr Levy

A thesis in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Banking and Finance

UNSW Business School

November 2015


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Originality Statement

I hereby declare that this submission is my own work and to the best of my

knowledge it contains no materials previously published or written by another person,

or substantial proportions of material which have been accepted for the award of any

other degree or diploma at UNSW or any other educational institution, except where

due acknowledgement is made in the thesis. Any contribution made to the research by

others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in

the thesis. I also declare that the intellectual content of this thesis is the product of my

own work, except to the extent that assistance from others in the project's design and

conception or in style, presentation and linguistic expression is acknowledged.

Signed ..............

Date ..............


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Acknowledgements

I am thankful for the financial support of the Sasakawa Foundation though their

Young Leaders Fellow Program. Funding towards our research was also provided bythe Australian Research Council (ARC).

I would like to thank the Australian Prudential Regulation Authority for their

financial and moral support in my PhD candidacy.

I am also thankful for the many opportunities that the Australian School of Business

at the University of New South Wales has offered me in both participating in seminars

and presenting our research at conferences.

I thank Ashley Coull, Stephanie Osborne and particularly Shirley Webster for easing

my pain in getting through the paperwork, often fixing my mistakes, and always with a

soothing smile.

I am most thankful for Professor David Feldmans guidance and support, who was

always open and available to provide valuable feedback.

I am also most thankful to Professor Peter Pham for going far beyond the call of duty

in helping me to navigate and comply with the universitys rules and regulations.

Needless to say, I am deeply indebted for the unwavering guidance and support of

my thesis supervisor, Professor Peter Swan AM FASSA, with whom I have enjoyed

many discussions on the topics of my research. Truly, I would not have made it without

his wholehearted support.

Last, but definitely not least, I would like to acknowledge the infinite patience and

confidence of my wife Leila in this long endeavour.


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Abstract

The main argument of this thesis is that price and value are not always the same. I

investigate the effects of impediments to trade and funding on prices and allocations in

three different settings.

In the first setting, I examine a market with transaction costs and short selling

constraints. I find that, when transaction costs differ across assets, funding constraints

create a shadow price for liquidity. Thus, it is the combination of asset and liability

illiquidity that generate illiquidity premiums. My first main chapter sheds light on the

long standing equity premium puzzle.

In the second setting, I show that, in a market with a small number of informed,

uninformed and noise traders, the demand for financial assets can be nonlinear on

prices. The theoretical literature thus far generally assumes these to be linear, or makes

assumptions that lead to linearity. I take a departure from those assumptions,

reconciling the model with the empirical literature, which has identified nonlinearities

in a variety of settings.

Finally, I demonstrate that financial access instability produces cycles of financial

bubbles and bursts. In this setting, I examine an exchange economy with three classes of

participants: lenders, borrowers and workers. I show that, if workers access to credit is

unstable, their entrance and exit into the financial market will generate price

fluctuations even when there are no exogenous risks to the economy. Further, and most

importantly, I demonstrate how their entry and exit is endogenously generated by a


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coordination failure amongst borrowers, with whom the new entrants compete for credit

and investments.

My results in this thesis point to the understanding that when liquidity is a scarce

resource, much like any other scarce resource, it commands a price. Such price of

liquidity is embedded in traded prices in the markets in which transactions and funding

are not costless, creating deviations away from the intrinsic value of traded assets.

Illiquidity can then substantially impair the market price system from optimally

allocating the economys resources. The results in this thesis thus have substantial

implications in public policy as well as in management compensation incentives.


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Table of Contents

Originality Statement ................................................................................................................ ii

Acknowledgements .................................................................................................................. iii

Abstract .................................................................................................................................... iv

1 Introduction ............................................................................................................. 1

1.1 Linearity and Nonlinearity of Limit Orders in Thin Markets ...................................... 3

1.2 Perfectly Rational Financial Bubbles ........................................................................... 5

2 Wither Equity Premium Puzzle? .......................................................................... 9

2.1 Introduction ................................................................................................................ 10

2.2 The Benchmark Case: Perfect Competition ............................................................... 17

2.3

Quadratic Transaction Costs ...................................................................................... 19

2.4 Imperfect Competition (Oligopsony) ......................................................................... 20

2.5 Illiquid Security with a Liquid Substitute .................................................................. 21

2.6 Model Calibrations ..................................................................................................... 29

2.7 Conclusions ................................................................................................................ 40

Appendix ................................................................................................................................. 41

3 Linearity and Nonlinearity of Limit Orders in Thin Markets ......................... 45

3.1 Introduction ................................................................................................................ 47

3.2 Literature .................................................................................................................... 48

3.3 Model ......................................................................................................................... 52

3.4 Conclusions ................................................................................................................ 57

Appendix A ............................................................................................................................. 59

Appendix B ............................................................................................................................. 68

4

Perfectly Rational Financial Bubbles .................................................................. 75

4.1 Introduction ................................................................................................................ 76

4.2

Capturing a Bubble .................................................................................................... 78

4.3 Literature .................................................................................................................... 79

4.4 Model ......................................................................................................................... 82

4.5 Results ........................................................................................................................ 88

4.6 Interpretation .............................................................................................................. 90

4.7 Prudential Regulation ................................................................................................. 91

4.8 Conclusions ................................................................................................................ 91

References ...................................................................................................................... 93


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1

Introduction

The main argument of this thesis is that liquidity is a scarce resource and, as any

scarce resource, it commands a price. Thus, when markets are not perfectly liquid, when

there are costs or impediments to trade and finances, prices reflect not only the

dividends they produce, but also the liquidity they provide. In other words, the market

values assets not only as means of production but also as means of exchange.

This has significant impacts on the market price systems ability to optimally allocate

resources in the economy. When liquidity is heterogeneous across assets, investments

may not concentrate in the most productive assets, and flock towards the most liquid.

This is essentially the core engine of financial bubbles, which typically create incentives

for entrepreneurs to make malinvestments.

Skewed incentives may also be present even when a bubble is not present. Corporate

executives remunerated with stock options or holdings may a have an incentive to

concentrate efforts in raising stock performance via improved liquidity rather than via

company productivity.


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The finance literature has mixed results in explaining the impact of liquidity in asset

prices. Divided in three separate chapters, I offer three models in which illiquidity

impacts equilibrium prices.

Chapter 2. Wither Equity Premium Puzzle

Chapter 3. Linearity and Nonlinearity of Limit orders in Thin Markets

Chapter 4. Perfectly Rational Financial Bubbles

Chapter 2 attempts to explain the equity premium puzzle, a long standing conundrum

in the finance literature. While previous attempts in the literature have had limited

success, I show that a liquidity premium is attained by combining impediments to

funding liquidity as well as to market liquidity. Intuitively, if traders face transaction

costs, but can fund their losses limitlessly, they can simply wait indefinitely for optimal

trading conditions to arise. Thus, while a bid-ask spread will be present, the shift from

the equilibrium mid-price in a perfectly liquid market is negligible. This has been

shown, though not always argued in this way, by a number of authors, and most

prominently by Constantinides (1986).

On the other hand, if they face no constraints or costs in funding, or equivalently on

short selling, they will freely shift their holdings between securities without ever

coming to a funding bind. Thus, funding constraints become immaterial, and again

equilibrium prices again remain undisturbed. What I show in this chapter is that one

may generate a liquidity premium by combining funding and market illiquidity. Again,

this is intuitive. As long as traders are free to trade their way out through at least one


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side of their balance sheet, they can always make illiquidity practically immaterial, and

thus keeping equilibrium prices unchanged. However, liquidity becomes a scarce

commodity with a substantial shadow price if traders face impediments both through

their assets and liabilities. Thence, a liquidity premium emerges in this setting, and the

price of a security is no longer purely a reflection of its intrinsic value.

1.1

Linearity and Nonlinearity of Limit Orders in Thin Markets

The main result from Chapter 3 is a market microstructure model of securities

trading that fits the empirical literature, in which it has been observed that the price

impact of traded volume is nonlinear. Heretofore the theoretical literature, following

Kyle (1985) and Kyle (1989) typically assumed price impact to be linear, and thus

inconsistent with empirical evidence. I provide a model that is consistent.

1.

I first show that two of the assumptions in Kyle (1989) paper, which has

been the cornerstone for much of the market microstructure literature, are

logically equivalent and therefore redundant;

2. I then solve a problem Kyle left unresolved by demonstrating that the two

assumptions in (1) may be derived from the other assumptions in his model;

3. Finally by changing one of those other assumptions I obtain an alternate

result which fits much more closely with what has been observed in the

empirical literature.

Kyle (1989) model is that of a thin market, i.e.with a limited number of traders that

can wield market power. In it, he (and much of the market microstructure literature),


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assumes risks to be normally distributed and that traders aversion to risk is independent

of their wealth. Furthermore, he assumes that traders investment demands are inversely

proportional to security prices and that they are all equally price sensitive. I first show

that these last two assumptions are logically equivalent, and then can be derived from

the first two. This is true provided I exclude aberrations such as demand curves with

kinks, discontinuities or singularities. That is a problem Kyle (1989) left unresolved,

though he correctly intuits to be the case. I close this gap when all traders are equally

informed, and when some are more informed than others.

Then I tackle the fact that the assumption of linear proportionality is not supported

by the empirical literature. What the empirical literature has recorded, by various

means, is that demand curves, rather than being linear, are convex. I present a setting

that yields exactly this result, and can be used to fundament empirical estimation. This

result solves yet another problem. An upshot of Kyles model is that in a thin market, in

which liquidity is compromised by gaming market power, traders are price sensitive, i.e.

they do not take them as given. However, as their price sensitivity is constant across

prices, the equilibrium price ends up being the same as if they were in a perfectly

competitive market. Hence, illiquidity has no such impact in this setting. However,

when price sensitivity is not constant across prices (i.e. demand curves are nonlinear),

equilibrium prices vary from perfectly competitive ones. Again, as in the previous

chapter, traded prices will include a liquidity premium or discount in addition to their

intrinsic value.


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1.2 Perfectly Rational Financial Bubbles

A microeconomic foundation for credit cycles has long been an unresolved problem

in the finance and economics literature. I demonstrate that the boom-bust credit cycle is

an equilibrium in a market where credit access is unstable. I show that asset bubbles are

the consequence of credit creation and destruction, simply by the entry and exit of new

borrowers into the market. This has a close parallel to the 2008 subprime crisis in the

US that eventually spread internationally.

I believe that the results hereby obtained are both relevant to theory as well as to

public policy.

My intent in developing this line of research has always been to produce results that

are relevant not only to finance theory, but also to finance practice.

1.1.1

Prudential Regulation

One of the consequences that emerged from the 2008 global financial crisis is the

need for prudential regulation reform. Basel II had already been heavily criticised for

being procyclical well before the crisis. Basel III then introduced liquidity

considerations into capital provisioning so as to counterbalance the procyclicality of

Basel II. What I demonstrate in Chapter 4 indicates that further reform is needed to

account for financial access instability as a cause of the booms and busts of credit

cycles. What it further indicates is that banking, as a service to the public, ought to be

regulated in a similar manner to other utilities, in which service licenses are conditioned


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on providing access to people at the fringes of society, even if subsidised by others in

the interest of their own safety.

1.1.2 Monetary Policy

Debates about asset inflation have entered central banking circles ever since the

dotcom bubble. The debates peaked at the 2008 financial crisis. There was, and still is,

much controversy between proponents of quantitative easing and fiscal austerity. At the

core of that discussion is the question of whether asset bubbles indeed exist. While they

are apparent to a lot of people, and are talked about in mainstream media, many

recognised experts have claimed, and still do, that market fluctuations are a reflection of

inherent risks of the economy. The latter understanding is a reflection of the lack of

conceptual tools to interpret observed phenomena. The results obtained in my research

provide those very tools, first in Chapter 2 that the difference in economic and financial

risks may be explained via liquidity considerations and then in Chapter 4 that liquidity

may indeed generate asset bubbles and detrimental credit cycles. These tools will then

provide policymakers the conceptual tools with which to frame policy debates and

decisions.

1.1.3

Executive Incentives

Another hot topic during the 2008 financial meltdown was that of executive

incentives. Much has been debated on the issue of fairness. While my research makes

no contribution to this question, it does raise the question on whether market based

incentive schemes maximise economic value. Ever since the expansion of securitisation

in the 1980s, executives became increasingly more compensated with either direct share


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holdings or stock options for the performance of their companies stock prices. Since

the essential result of my research is that stock prices reflect more than fundamental

firm value, then it stands to reason that executive compensation schemes do not only

maximise their firm value, but also its stock liquidity. Much of that has been

anecdotally observed. Executives seemed to be making decisions that were more

aligned with improving their ability to flip their stock in the market than in creating

actual economic value. This is yet another field in which my research may provide a

framework through which empirical research may be interpreted and oriented.

1.1.4 Automated Market Making

The insights provided by the results of my research around the role of liquidity in

financial markets may be a valuable tool for income generation. In particular, Chapter 3

derived explicit optimal investment demand curves which can be translated into limit

order schedules. These may be used to identify trading opportunities when market

discrepancies imply large enough deviation from reasonable assumptions. In doing so,

one may transfer liquidity across assets, asset classes, time and circumstance. The

results of my research provide a practical framework for pricing liquidity and thus

enable market makers to buy liquidity where and when it is cheap and sell it where and

when it is expensive. While that has always been the role of market makers, my results

add science to their art by way of automated into trading algorithms.

1.1.5 Securities Regulation

As with market making, the framework offered by the results of this research may

improve the ability of securities regulators to identify anomalies in the market,


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discerning liquidity dynamics from inside information or fraud. Again, large enough

discrepancies from explicit theoretical results around the shape and dynamics of limit

order demand schedules, may indicate significant departures from model assumptions in

which criminal activity is absent.

I suspect that the research I have produced may be just the beginning of practical

applications and further research that it may enable. In a sense, it provides a paradigm

shift from mainstream views about how financial markets operate as characterised by

the current finance literature. Moreover, it does this with the same traditional tools that

have been developed and honed in the literature, e.g. not resorting to heterodox

arguments. I feel this is important because the tools already in place have provided

paramount understanding thus far. I have just pushed them a little further.


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2

Wither Equity Premium Puzzle?

Chapter Summary

The meaning of this chapters title is twofold: I propose a solution to the puzzle, while

also predicting its demise. The puzzle emerges from the attempt to explain the

differences in yield between equities and Treasury bills solely from their differences in

risk, requiring absurdly high levels of risk aversion (Mehra and Prescott (1985)). By

acknowledging that equities and Treasury bills differ not only by their levels of risk, but

also by their liquidity, I show that, when transaction costs differ across assets, funding

constraints create a shadow price for liquidity. Thus, it is the combination of asset andliability illiquidity that generate illiquidity premiums. Further, I argue that the returns in

equities markets observed in the last 25 years are a reflection of dramatic reductions in

their transaction costs, and are likely to taper off as soon as their liquidity ceases to

expand.


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2.1 Introduction

Between 1896 and 1994 the yearly simple geometric mean equity premium for the

New York Stock Exchanges (NYSE) value weighted stocks was six percent (Campbell,

Lo, and MacKinlay (1997)) and has been approximately eight percent for the last 50

years (Cochrane (2005)). In their cornerstone paper, Mehra and Prescott (1985) attempt

to account for this premium using simulations of an intertemporal equilibrium growth

model with a representative consumer/investor, abstracting from transaction costs,

security market microstructure, liquidity considerations, and other frictions. They are

able to account for only a negligible proportion of this premium with a maximum of

0.4% explained by risk aversion.1

Mehra and Prescott (1985) and the subsequent literature surveyed by Cochrane

(2005) and Campbell, Lo, and MacKinlay (1997) focus on representative agent

equilibriums with agents identical in all respects, including endowments. These are

models of perfectly competitive market equilibriums with no market frictions, such that

Pareto optimality is achieved, with relative prices determined strictly from aggregate

risk preferences and irrespective of trading activity amongst market participants. Once

market frictions are introduced, risk preferences need to be balanced with illiquidity

costs, and hence relative prices are impacted by relative liquidity across investment

assets. Clearly, this impact can only be observed when investors have an incentive to

trade. For that, investors must differ in at least one respect. In the case here they differ

1 For a review of the puzzle see Constantinides (2004), Heaton and Lucas (2005) and other papers

presented at the UC Santa Barbara Equity Premium Conference, October, 2005.


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in endowments. Essentially, I replace the representative investor with investors with

differing endowments to motivate trading while retaining the assumption of identical

preferences. I show that preserving all the standard assumptions of rationality, utility

maximization, and even abstracting from the effects of information and beliefs a simple

exchange model with quite modest barriers in trade explains the major stylized and

empirical facts about equity, bond market returns, and trading turnover over the last 100

years.

I establish in a variety of settings within an exchange economy that the midpoint

price of an asset is unaffected by impediments in to trade such as a small number of

participants (oligopsony) or quadratic transaction costs. We dont explicitly assume the

possibility of final liquidation costs, but that is without loss of generality, as investor

factor those in expectation, treating them as just another transaction cost, though

incurred in the future. It questions the standard assumption that even if only a single

asset is traded its midpoint must fall by the present value of transaction costs.

Superficially, this suggests that illiquidity and trade impediments can never impact asset

returns. Heumann (2005), in a trading model akin to Kyle (1989) with market impact

costs also finds that illiquidity due to oligopsony is not priced. The trading price is

independent of the number of traders, which is a result I also obtained. He attributes

what he calls a surprising result to the two-sided nature of trading: Buyers demand a

price discount and sellers demand a price premium, and these effects cancel each other

out Kyle (1989, p. 5). This result, however, hinges upon the assumption of free

leverage. I show that in the presence of transaction costs, borrowing constraints generate


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a shadow price for liquidity which is not only important but can also dominate the price

of risk.

This apparent cancelling out of what are mutual harms due to illiquidity is puzzling

as both trader welfare and liquidity are clearly improving in the number of participants,

despite the implication of the finding that the number of participants is irrelevant to the

outcome. In Kyle's (1989) model, a smaller number of participants make all parties

worse off since the optimal (first best) level of trading occurs when an infinite number

of participants is possible. This means there is no cancelling out of welfare losses. If

there is a choice between regimes, for the less liquid regime there must be a

compensating fall in the asset price relative to the liquid regime. Grleanu and Pedersen

(2004) obtain a similar puzzling cancelling out for informed and liquidity trades with an

intuitive explanation similar to Heumann (2005).

Indeed, this cancelling out and irrelevance of transaction costs for a variety of taxes,

trade barriers induced by market power, and to market clearing prices is something that

has been under investigation by economists for decades in a variety of guises.1 The

irrelevance of market imperfections for midpoint exchange prices, as well as the

irrelevance or who notionally pays the tax or bears the transactions cost, is a

consequence of the fact that barriers in transacting become a tax wedge. It does not

mean that if the illiquid and identical liquid varieties of the same asset trade in the same

market that the prices and returns will be the same. In fact, it is entirely to the contrary.

1For example, Samuelson (1964) showed that if an income tax is applied uniformly to cash earnings and

capital gains and losses then asset prices are unaffected by such a taxation regime. However, his result

does not imply that if one traded asset is taxed and an identical one is not, that asset prices are unaffected.


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One purpose of this chapter is to show that the price differential between the two

otherwise identical assets can explain the equity premium puzzle of Mehra and Prescott

(1985) as almost entirely due to illiquidity rather than risk. I then extend the model by

allowing any degree of correlation between the liquid and illiquid asset. For example,

the liquid asset could represent T-Bills, rather than equity identical to the costly to trade

asset. I also investigate the implications of introducing traders with different investment

horizons on portfolio choices and returns.

This chapter initially considers a market with perfectly substitutable assets (for the

sake of simplicity, and without loss of generality) with different liquidity. I prevent the

use of the liquid asset as a perfect hedge for the illiquid asset by imposing short sale

constraints. This is reasonable in these circumstances as there are no other players in the

market from whom investors could borrow stock to be sold short. One could argue that

it would be perfectly possible for a seller to, in addition to selling the asset to the buyer,

borrow it from him, and sell it some more, under a repurchase agreement, i.e. a repo, to

buy it back in period 1. In the presence of a liquidity differential, the demand for this

transaction would drive up the equilibrium repo rate to the point of making the

borrowing uneconomic. Alternatively, investors could trade a derivative contract (e.g.

swap, forward, or future) that replicated the payoff of the risky asset. This contract,

however, would carry a transaction cost in its own right, potentially higher than that of

the underlying asset, as in a no arbitrage equilibrium its price should reflect the costs of

replicating its payoff with the latter.


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As noted above and in previous work by other authors (e.g. Heumann (2005),

Constantinides (1986), and Vayanos (1998)), I find little to no impact on mid-point

prices as a result of transaction costs while acknowledging that transaction costs can

widen the spread. The key to the result in this chapter that of a significant liquidity

spread is the recognition that a liquidity spread is generated by differences in liquidity

across assets in the market and not simply by illiquidity alone, while investors

(particularly sellers) are unable to circumvent these differences by short selling the most

liquid asset.

Detemple and Murthy (1997) examine the effect of trading constraints on no

arbitrage pricing. More recently, Basak (2008) derives multiplicity of price equilibriums

under portfolio constraints, although that is only the case with more than one such

constraint. Detemple and Serrat (2003) go one step further in considering an

intertemporal economy, albeit with a narrower class of constraints, namely constraints

on liquidity.

These simple models describe a world in which the single asset for which there is a

motive for trade is subject to barriers in either transaction costs, or to illiquidity due to

oligopsony power, in a symmetric fashion. They do not address the question posed by

Constantinides (1986) in his seminal contribution. His concern lay with the rate of

return when expected utility comparisons are made across equilibriums.1His focus was

the impact on the equilibrium rate of return of a proportional transaction cost when

1We thank Kerry Back for making this point strongly.


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investors are guaranteed a minimum utility level fully incorporating all gains from

trade. This is provided by a comparison across equilibriums, or in two parallel

economies, with the ability to trade an otherwise identical completely liquid asset with

no transaction costs. Speculators cannot directly arbitrage across equilibriums except in

expected utility terms, preferences matter, and an asset with the same risk differing in

terms of a tradability factor. He defined the illiquidity (or liquidity) premium as the

increase in the assets mean return, or dividend, which combined with the introduction

of transaction costs leaves unchanged the investors expected utility across the two

equilibriums, one with and the other without transaction costs.

In contrast, here I derive an illiquidity premium by comparing rates of return in the

same economy rather than across iso-util economies. My model can explain the cross

sectional returns on the NYSE over the 30 years to 1992 as a function of stock turnover

(Datar, Naik, and Radcliffe (1998)), and the 20% return on letter stock (Silber

(1991)). It can also help to explain changes in the 600% premium on A relative to B

stock prices in China when the assets are otherwise identical, the very small returns on

A relative to B stock and the relative daily returns Chen and Swan (2008).

In a pioneering contribution Amihud and Mendelson's (1986) model expected

discounted cash flow maximization by risk neutral agents with different investment

horizons and hence different intrinsic portfolio turnover rates. There is an illiquidity

premium arising from the existence of multiple securities with differing spreads,

together with a chain of indifferent investors linking the returns on these securities.

Amihud (2002) provides further cross-sectional and time series evidence that the


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excess equity return at least partly represents an illiquidity premium. Brennan and

Subrahmayam (1996) find evidence of a significant effect, due to the variable cost of

trading after controlling for factors such as firm size and market to book ratios.

Extensions in the same vein are provided by Easley, Hvidkjaer, and OHara (2002),

Pastor and Stambaugh (2003), and Easley and OHara (2004).

Constantinides (1986) computes the illiquidity premium using numerical simulations

based on Merton's (1971) and Merton's (1973) intertemporal asset pricing model of a

single agent able to rebalance his portfolio of the risky and riskless asset at a specified

cost, with constant relative risk aversion (CRRA) preferences and an infinite horizon.

Davis and Norman (1990), Aiyagari and Gertler (1991), Bansal and Coleman (1996),

Heaton and Lucas (1992), Heaton and Lucas (1996), and Heaton and Lucas (2005) use

the same setting (the latter also models idiosyncratic labour income shocks). Vayanos

(1998) models turnover as endogenously generated by investors with CARA

preferences based on life cycle considerations.1 Huang (2003) also finds a relatively

small liquidity effect.

Lo, Mamaysky, and Wang (2004) consider fixed rather than proportional transaction

costs, unlike Constantinides (1986) and the others mentioned above. Their economy is

composed of agents maximizing exponential utility, while hedging a nontraded risky

endowment with a traded asset whose dividends it perfectly correlates. Liu (2004)

extends that to multiple assets with different transaction costs, in a continuous time, on

1See also Vayanos and Vila (1999).


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an infinite horizon CARA setting. Jang et al. (2007) finds that by introducing stochastic

regime switching into Constantinides' (1986) model, trade demand increases, and

transaction costs can have a first order impact.

I present the risky security exchange model in Section 2. Simulations of equity and

bond markets from 1896 to 1994 are in Section 3, while Section 4 concludes. All proofs

are provided in the Appendix.

2.2

The Benchmark Case: Perfect Competition

I introduce our notation via the benchmark case of perfect competition. In this

economy all investors are identical with the same risk preference and they differ only in

their initial endowments. Since efficiency requires equal risk-sharing with respect to the

risky asset, investors will trade to the point at which their terminal period holdings of

the risky asset are identical.

There is a single risky asset and no impediments to trading. A pure exchange

economy with two periods, 0 and 1, is composed of N constant absolute risk aversion

(CARA) investors indexed by i , with the same CARA positive coefficient !, each with

0

ix consumption units in a savings account remunerated at the positive riskless rate r ,

and the remaining initial wealth0

iw invested in

0

iy units of a risky security, with

aggregate positive supply of Yunits. This riskless rate need not be the same as the yield

on a riskless Treasury bond that might benefit from a liquidity premium. The risky


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security yields a normally distributed (terminating) dividend payoff of v consumption

units, with mean and variance 2! . Each investor i chooses the amounti

! of the

risky security to trade from his initial wealth0

iw , measured in consumption units, in

period 0 , in order to maximize expected utility over consumption in the terminal

period, period 1 . Consumption in the initial period 0 is irrelevant to the model with the

investor choosing his period 1 consumption equal to his terminal portfolio consisting of

a combination of the riskless and risky asset with all benefits and outlays measured in

consumption units.

Proposition 2.1: Under perfect competition with Nprice taking CARA investors in

this economy, the market clearing equilibrium price is given by the present value of the

terminating dividend payoff in the second period, period 1, , minus a risk discount

that depends on the mean number of investment units per investor,Y

N, their risk

preference CARA coefficient, !, and variance, 2! :

21

1

Yp

r N !"

# $= %& '

+

. (2.1)

Each investors post-trade holding of the security in period 1 is denoted by1

iy . Investors

share the total stock of the risky security equally amongst themselves in the next

(terminating) period so as to perfectly share the risk in a Pareto efficient manner. As

there are no barriers to trade, each one holds the same quantity,1

iy , in equliibrium:

1 ,

i Yy i

N= ! , (2.2)


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with a purely competitive trade size (denoted by subscript C), along with no

impediments to trade such that post-trade holdings are equalized:

0,

i i

C

Yy i

N! = " # , (2.3)

and consumption in period 1 is given by:

( )

( ) ( )( ) ( ) ( )

22

0 0 0

ln 1 11 ,

2 2

i i i i i i i

C C C C

rc r x p y y i

r

! ! "# !

"

$ += + + $ + + $ + %

+

. (2.4)

This expression (2.4) for the value of consumption of the ith investor in the second

(terminal) period, period 1, consists of the valuation of the ith investors two assets, his

liquid money holdings,0

i i

Cx p!" , and his risky asset holdings, 0i i

Cy !+ , after his trades

have occurred. These components make up his after-trade portfolio in the terminal

period. Its primary determinants are the terminating mean dividend (i.e., valuation in

consumption units) per unit of the risky asset and each investors identical terminal

period holdings of the risky asset,1 1 0

i i i

C

Yy y y

N!= = = + , as perfect risk-sharing is

accomplished in the absence of trade barriers. The final term in the expression,

( )2

2

0

1

2

i i

Cy!" #+ , reflects risk-bearing costs that have been minimized across this

economy of otherwise identical investors, differing only in initial endowments.

2.3

Quadratic Transaction Costs

Proposition 2.2: With quadratic costs denoted by Tran that are defined in numeraire

units of consumption, with


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20

( )21

2Tran k != , (2.4)

where!

denotes the trade size and the quadratic transaction cost coefficient0k>

, the

midpoint price is the same as under perfect competition (equation (2.1) above). Trade

size is given by:

( )

2

021

i i

Q

Yy

r k N

!"#

!"

$ %= &' (

+ + ) *, (2.5)

and thus is diminishing in the quadratic transaction costs parameter, k.

2.4

Imperfect Competition (Oligopsony)

In the previous Sections 2.1 to 2.3 models investors are nave in the sense that they

take prices as given and do not take into account the adverse market impact that their

own actions will have upon them. In order to incorporate strategic behaviour, which in

turn justifies our quadratic cost approach, we assume that investors conjecture each

others demand functions to be linearly decreasing in price:

,

h h hp h i! " #= $ % & . (2.6)

Proposition 2.3: The Nash equilibrium market clearing price, in a two-period

economy of N CARA investors under oligopsony, is the same as in the perfectly

competitive case (2.1), 21

1

Yp

r N !"

# $= %& '

+ ( ), but allocations depend on the purely

competitive Pareto efficient allocationY

N, on the initial endowments

0

iy , and on the

number of strategic investors:


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21

0

1

2

1 1

ii yN Y

yN N N

!

= +

! !

, (2.7)

with the optimal strategic trade size:

0

2

1

i i

IC

N Yy

N N!

" # $= "% &" ' (

, (2.8)

where iIC

! denotes the optimal trade size under Imperfect Competition (oligopsony).

This is so because where investors have market power due to a limited number of

market participants, market impact costs act as a barrier to trade precisely like a form of

(quadratic) transaction costs. Barriers to trade are a form of transaction cost (tax

wedge) that does not distort the midpoint price but buyer and seller prices differ by the

spread induced by the transaction cost wedge. In the Appendix for this chapter, we

establish that this strategic effect of market power is equivalent to the quadratic

specification for transaction costs given above in equation (2.4). Moreover, quadratic

transaction costs are the only form consistent with equilibrium when agents differ in

size or market power.

2.5

Illiquid Security with a Liquid Substitute

Consider a one-period market of two types of CARA investors, buyer and seller,

denoted by superscripts B and S respectively. Both buyer and seller hold cash on a

savings account remunerated by a riskless rate r, but with only the seller holding a risky

asset endowment. We assume this for the sake of simplicity of notation, and without

loss of generality, as the absence of wealth effects from CARA preferences leaves any


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22

equity held in common immaterial to trade. There are two types of risky assets, liquid

and illiquid, denoted respectively by subscripts L and I, traded at the expense of a

quadratic transaction cost, Tran , with coefficients Lk and Ik (with L Ik k< ),

respectively, where ( )2

! denotes the square of the trade size1:

( ) ( )2 21 1

2 2L L I I

Tran k k ! != + (2.9)

I provide motivation for quadratic transaction costs in the Appendix. It stems from

strategic, rather than price taking, nature of trading. Each asset has liquidation valuesI

v

andLv , respectively, that are normally distributed with respective means

I and

L ,

variances 2I

! and2

L! , and covariance

I L!" " .

Neither of the assets can be short sold. One could argue that it would be perfectly

possible for a seller to, in addition to selling the asset to the buyer, borrow it from him,

and sell it some more, under a repurchase agreement, i.e. a repo, to buy it back in period

1. In the presence of a liquidity differential, the demand for this transaction would drive

up the equilibrium repo rate to the point of making the borrowing uneconomic. Thus

assuming a hard constraint is without loss of generality. Moreover, there is no derivative

contract available to mimic the assets payoff at the lower cost. As we will see below,

sellers will use the liquid asset as a partial substitute to the illiquid one to reduce their

risk. Thus if the supply of the liquid asset is smaller than that of the illiquid one, sellers

1 As we specify the mean payoffs to the liquid and illiquid asset, as well as their marginal transaction

costs, our definition of the relative supply of the two securities is immune to simple stock-split

redefinitions.


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23

will not be able to completely reduce their risky holdings, and the short selling

constraint will be binding. Later, we look into varying the proportions of liquid and

illiquid assets in the market.

If liquid and illiquid assets are partial substitutes and the supply of the liquid asset

sufficiently smaller than that of the illiquid one in relation to how much risk investors

wish to share, then they will trade as much as they can of the liquid asset first, and then

trade the remainder risk through the illiquid one.

We denote the buyer by superscript B and the seller by superscript ,S and their

wealth, riskless and risky holdings by w, x and y, respectively, denoting liquid and

illiquid securities byLandI, respectively:

, and

.

S S S S

I I L L

B B B B

I I L L

w x y p y p

w x y p y p

= + +

= + +

(2.10)

We also denote buyer and sellers post-trade holdings of securities and cash,

respectively, by y and x :

, ,

,

, ,

S B

I I I I I

I LS B

L L L L L

y Y yY Y

y Y y

! !

! !

= " =

>

= " =

(2.11)

and

( ) ( )

( ) ( )

2 2

2 2

1 1, and

2 2

1 1

2 2

S S S S S S

I L I I L L

B B B B B B

I L I I L L

x x c p k k

x x c p k k

! ! !

! ! !

= " + " "

= " + " "

(2.11)


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24

Given the smaller aggregate supply of the liquid asset, its short sale constraint will be

binding (as argued above), i.e. . Hence, buyer and sellers final wealth are:

( )

( )

1 , and

1 .

S S S S

I I L L

B B B B

I I L L

w r x y v y v

w r x y v y v

= + + +

= + + +

(2.11)

The seller and the buyer then choose their respective consumption levels and trading

volume to maximize their expected exponential utility of terminal wealth:

c

S

,!!I

S

,!!L

S"#$

%&' = argmaxc,!

I,!L( )s.t. !I,!L( ) < YI,YL( ) (exp ()c( )( E exp ()w

S

( )*+ ,-{ }, and

cB ,!!I

B

,!!LB"

#$ %

&' = argmax

c,!I,!L( )s.t. !I,!L( ) < YI,YL( )

(exp ()c( )( E exp ()wB( )*+ ,-{ }, (2.12)

where [ ]E denotes the expectation and !denotes the constant absolute risk aversion

(CARA) coefficient common to both buyer and seller.

This optimization yields two first order conditions with respect to the final period

consumption of the seller and buyer, respectively:

0 =ln 1+ r( )

!"rcS+ 1+ r( ) x +pI#

!I

S

+pLYL" 1

2k

I #!I

S$%&

'()

2

" 12k

L #!L

S$%&

'()

2*

+,,

-

.//

+ I Y

I"#!I

S$%&

'()" 1

2!0

I

2 YI"#!I

S$%&

'()

2

, and

0 =ln 1+ r( )

!"rcB + 1+ r( ) x"pI#

!I

S

"pLYL" 1

2k

I #!I

B$%&

'()

2

" 12k

L #!L

B$%&

'()

2*+,,

-.//

+ I#!I

B

+ L#!L

B

" 12!0

I

2 #!IB$

%& '()

2

"!10I0

L#!I

B

YL" 1

2!0

L

2YL

2,

(2.13)

L LY! =


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25

with the remaining four of the six (in total) conditions specifying the optimal trade-size

conditions for the seller of the illiquid and liquid assets, respectively, and the buyer of

the illiquid and liquid assets, respectively:

0 = 1+ r( ) pI!kI"!I

S#$%

&'(!

I + )*

I

2 YI!"!I

S#$%

&'(,

0 < += 1+ r( ) pL!kLYL( )! L +),*I*L YI!"!I

S#$%

&'(,

0 =! 1+ r( ) pI + kI"!I

B#$%

&'(+

I! )*

I

2"!IB

! ),*I*

LYL,and

0 =! 1+ r( ) pL + kLYL( )+ L! ),*I*L"!I

B

! )*L2

YL ,

(2.13)

where ! is the shadow price of the sellers short selling constraint, and superscript !!

indicates that the condition is evaluated at the optimum trade size value. Market clearing

requires that purchases and sales net to zero. Hence:

!!I

S

" #!!I

B

" !!I =

$%I

$%I

2+ 1+ r( )kI

%IYI

# &%LYL

2 . (2.14)

This result is different from the frictionless result, in which both investors equally

share the aggregate supply of risky assets. In this case, we regard the stock of the liquid

assetL

Y transferred from seller to buyer as being outside of the risk sharing problem,

due to the lack of wealth effects generated by the investors CARA preferences.


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26

Figure 2.1 The seller transfers to the buyer his liquid stock (lightgrey), and then some illiquid stock (dark grey).

One can easily verify that the trade volume above converges to the frictionless

market case, when 02

I L

I I

Y Yk!

"= # , and that the market converges to autarky, i.e.

0 whenI I

k! = "# .

By substituting the market clearing trading volume into the first order conditions

(2.13) we can solve for the equilibrium prices:

( )2

2

1, and

1 2

1.

1 1 2

I I L LI I I

I I I L LL L I L L L L L

I I

Y Yp

r

Y Yp Y k Y

r r k

! "! #!

#! ! "! #"! ! #!

#!

+$ %= &' (

+ ) *$ %&

= & & &' (+ + +) *

(2.15)

Given the analysis above, it is no surprise that the transaction cost does not change

the price of the illiquid asset, while only commanding a liquidity premium over the

price of the liquid asset.

S B

Liquid stock

Iliquid stock


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27

Figure 2.2 Equilibrium prices for liquid and illiquid assets with varyingtransaction costs and volatility

( 2%; 10.0%; 2.0%; 0.15 0.0015; 0.0;5;I I L L LI kr p p k ! = == = = =)18.78%; 1.02%; 1.29%I I L Lp p! ! "= = =

Carrying out comparative static analysis of increases on the two transaction cost

regimes for the price of the relatively liquid asset yields:

0, and 0L LL

I L

p pY

k k

! !> =" "' () *

++ , (2.44)

where I is the indicator function and ( )0! gives the maximum number of securities

traders are willing to buy at a zero price. This is not unlike the CARA-Normal case, in

which the demand curve also intercepts the quantity axis at a finite number. The second

order condition of (2.41) implies that ( )p! is downward sloping; hence, ( )0! must be

positive.

Further, it establishes that most traders are willing to quote to one another, even as

the price approaches zero. This is counterintuitive at first, given that traders are risk

neutral. Once we consider that traders act strategically in this setting, and thus are

cognisant of the impact their quote has in the traded price, it becomes clear this is so.

To illustrate this effect I plot in Figure 3.1 both the risk-neutral and the CARA-

Normal cases with 4N = traders, 0r = , 1 = , 1! = and ( )0 0.63! = with price p on


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55

S-shape is very much in line with the hyperbolic tangent price impact function Huang

and Ting (2008) confirmed of the stocks traded in the NYSE. In effect the S-shape of

the demand schedule produces narrower spreads for small volumes than in the CARA

case of linearity, and wider spreads for large volumes. In the example above the

transition occurs when 0.5p = and 1.5p = . I reproduce Figure 3.1 in Figure 3.2 below,

adding the spreads in the three cases. CARA are marked spreads with a dashed line.

Notice that the dashed box is outside the solid one for small volumes and outside it for

larger spreads. At 0.5p = and 1.5p = the solid and dashed boxes overlap.

Figure 3.2 Spread differences between aggregate pricing schedules of CARA and risk

neutral traders, ( )4, 0 0.63, 0.25, 2, 0%.N r! " #= = = = =

Further, by taking the first derivative of the solution we observe the slope trends to

negative infinity, meaning that traders are price insensitive around the mid-point price

1 r

+

. This holds close resemblance to what we observe in practice, where quoted bid-

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

q(p)

p($)

N= 4

Bid-ask spread


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56

ask spreads only become wider for large volumes which contrasts to the linear price

impact suggested by CARA preferences.

This contrast is greater as the number of players increases. As N increases, the

demand schedule ( )p! approaches the step function equal to ( )0! for1

pr

+

+

, and !" in the interval1 1

,1 1

N Ne e

r r

! !" #! +$ %+ +

.

AsN increases, the interval narrows around1 r

+

, and obtain the step function. As the

market approaches a competitive one, traders become price takers and bid-ask spreads

collapse to zero.


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59

Appendix A

3.4.1

Kyles Conjecture

I now trace back to the introduction on the assumption of linearity, and how it

propagated especially throughout the literature. In his seminal 1985 paper, Kyle

demonstrates the existence of a linear equilibrium in a market, where a risk neutral

informed trader and uninformed market makers trade strategically amongst themselves

and with noise traders. Solutions are confined to be linear by assumption through the

informed traders demand curve. Kyle (1989) extends this framework by making

explicit the assumption about traders limit order schedules, assuming risk preferences

to be of the CARA type, and all exogenous risk factors, (the asset liquidation value, the

noise traders demand, and informed traders uncertainty) to be jointly normal. He thus

shows there to be a unique Nash equilibrium of investment demand curves amongst

linear demand schedules. Indeed, in Kyle (1989), he writes:

In the existence part of the result, the strategy functions are not merely best linear

strategies. Instead, the linearity of the strategies is a derived result in the sense that

for each speculator, the equilibrium linear strategy dominates all nonlinear

alternatives. In the uniqueness part of the result, linearity is a constraint. This paper

does not investigate whether equilibriums with nonlinear strategy functions exist.

In the uniqueness part of the theorem, symmetry is also a constraint. We conjecture

that the theorem could be generalized to remove this constraint by constructing a

proof which allows traders to conjecture linear strategy functions which differ from

trader to trader, then prove that they are all the same in equilibrium, but this is not

attempted here.

To restate, what he means by the equilibrium linear strategy dominates all nonlinear

alternatives while confirming that [the] paper does not investigate whether

equilibriums with nonlinear strategy functions exist is that a traders optimal trading

strategy is indeed linear, when faced with a linear residual supply curve. This follows


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from market clearing, rather than from optimality. It is also a coherent assumption. If all

traders best response strategies are linear, then the residual supply curves they face

must indeed be linear. Kyles characterization of what his 1989 model does not

investigate is equally true of his 1985 paper and the recent extension of information

acquisition by agents (Kyle, Ou-Yang, and Wei (2011)). In the latter, the authors make

explicit the assumption that traders conjecture exclusively linear strategies. Nonlinear

equilibriums are ruled out by assumption.

It remains for one to show that, in this setting, equilibriums with nonlinear strategy

functions do not exist, and that the resulting linear equilibrium is symmetrical by

consequence of the models assumptions. Kyles intuition for his conjecture is clear and

convincing. Given the assumption that traders have CARA preferences and that all

exogenous risk factors are jointly normal, the traders expected utility turns out to be

quadratic on their trade quantities. This yields linear marginal utilities, or equivalently

linear demand schedules. Given these assumptions, I show Kyle's (1989) conjecture to

be correct for all smooth functions.12

Much of both the empirical and theoretical literature that followed from Kyles

framework assumes investment demand functions to be linear in prices, or equivalently

price impact functions to be linear in volume. Typically, the argument goes along the

lines of first assuming that each trader conjectures the other traders to be holding linear

demand schedules, or equivalently that the residual demand schedule is linear. It then

goes to show that each traders optimal schedule, in reaction to the residual demand

schedule is also linear. The argument does not actually show that equilibrium demand

curves ought to be linear amongst a wider class of alternative conjectures. It rather

12We thank Alex Boulatov for pointing out a limitation of our original proof in an earlier version.


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63

Proposition 3.2: If all traders are equally price sensitive, then their investment

demand is linear.

This result shows the converse of the previous one. If one assumes symmetry,

linearity emerges. Both results together show the logical equivalence of symmetry and

linearity in this setting.

Proposition 3.3: If traders take account only of their own security holdings in setting

their optimal demand schedule, then they are equally price sensitive. Therefore,

equilibrium demand schedules are linear.13

Proposition 3.3 establishes that traders who take account only of private holdings,

and not of aggregate supply, are equally price sensitive. Their equilibrium demand

schedules are therefore linear, by Proposition 3.2.

Proposition 3.4: If demand schedules are smooth, then they are linear in equilibrium.

This is the main result of this section. It connects linearity and thus also symmetry to

the assumptions on CARA preferences and joint normality. In doing so, we identify

what assumptions need to be modified to better explain the nonlinearity observed

throughout the empirical literature.

3.4.3

CARA and Normality with Information Asymmetry

In this Section, I transfer the above results to the exact model setup and notation of

Kyle (1989). Consider a two-period pure exchange economy for a single risky security

with an uncertain value , traded by N informed traders indexed by

13This proposition and its proof are inspired by a suggestion made by Alex Boulatov, to whom we are

indebted.

( )1~ 0, vv N !"

!


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64

, M uninformed market makers indexed by , and noise

traders. Traders and market makers act as liquidity suppliers by submitting a schedule

of limit orders to the market, while noise traders act as liquidity demanders by

submitting random market orders that are independent of the assets

value. Both informed and uninformed traders are risk averse, and quote quantitiesxn(p)

and ym(p), respectively, for each given trade price p to maximize their exponential

utility over their trade profits, each with CARA coefficients and respectively:

(2.52)

Each informed trader n receives a private signal before they submit their

orders. All informed traders signal errors are normally distributed with mean

zero and variance , and independent amongst themselves and of and . Since

traders have CARA preferences, their initial endowments are irrelevant to pricing, and

thus normalize to zero.

At this point, I depart from Kyle (1989) by refraining to assume linearity of residual

supply curves and symmetry of limit order schedules. Instead, I introduce the following

notation to denote residual supply curves for both informed and uninformed traders:

(2.53)

{ }1, ,n N! ! { }1, ,m M! !

( )2~ 0,N !! "

I!

U!

( ) ( )( )

( ) ( )( )

arg max E exp , ,

arg max E exp .

n I nx

m Uy

x p v p x p i

y p v p y p

!

!

" #= $ $ $% &

" #= $ $ $% &

!! !

!

n ni v e= +! !

1, ,

Ne e! !"

1

e!

"

v z

( ) ( ) ( )

( ) ( ) ( )

,

.

n j m

j n m

m n k

n k m

x p x p y p

y p x p y p

!

!

"

#

"

#

= + +

= + +

$ $

$ $

! !

! !


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65

Let be the market clearing price, so that:

(2.54)

As I will show, equilibrium demand schedules are elastic at any price point, and

residual supply curves thus yield unique market clearing prices through the following

inversion:

(2.55)

Since both the true asset value and the error in each of the informed traders private

signal are normally and independently distributed, all traders utility maximization

reduces to a simple mean-variance quadratic optimization:

(2.56)

While market clearing only holds at the equilibrium price, both informed and

uninformed traders design their order schedules for each price point conditional on it

turning out to be the market clearing price. We can thus replacepin (2.55) above forp

in (2.56):

(2.57)

*p

( ) ( )( ) ( )

* *

* *

0, ,

0, .

n n

m m

x p x p n

y p y p m

!

!

+ = "

+ = "

! ! ! !

! ! !

( )( ) ( )( )* 1 * 1 * 0, , .n n m mp x x p y y p m n! !! != ! = ! = "! ! ! ! ! !

( ) ( )( ) ( )

212

212

arg max E , var , ,

arg max E var .

n n I nx

m Uy

x p v p i p x v p i x

y p v p p y v p y

!

!

" # " #= $ $% & % &

= " # $ $ " #% & % &

! !! ! !

! !

( ) ( )( )

( ) ( )( )

1 212

1 212

arg max E , var , ,

arg max E var .

n n n I nx

m m Uy

x p v p i x x x v p i x

y p v p y y y v p y

!

!

""

""

# $ # $= " " "% & % &

= # $ " " " # $% & % &

! !! ! ! !

! !


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66

Using the Inverse Function Theorem, and denoting the markets aggregate demand

curve by , the first order conditions for the above imply:

(2.58)

The next two propositions establish the logical equivalence between linearity and

symmetry of demand schedules, in spite of information asymmetry and heterogeneity.

Proposition 3.5:If all traders demand schedules are linear, then they are symmetric

amongst informed and uninformed traders.

This is a striking result, although the literature commonly assumes symmetry. It

means that all informed and uninformed traders are equally price sensitive amongst

themselves, regardless of their information. This, in part, follows from them having

CARA preferences. The other part is due to joint normality of exogenous risks. This

implies that their expectations of the liquidation value, conditional on the market

clearing price, are linear on that price, and that their conditional variances are constant.

By furthermore assuming their demand curves to be linear, we solve for the curves

slope coefficients. We find that the latter are independent of each traders information.

They are thus the same amongst each group. Therefore, Proposition 3.1 establishes that

in the presence of CARA preferences and joint normality, linearity of demand schedules

implies symmetry. Proposition 3.6 below establishes that the converse also holds.

Proposition 3.6: If all informed and uninformed traders demand schedules are

symmetric amongst each group, then their investment demand is linear.

( )z p

( ) ( ) ( )( )( ) ( )

( ) ( ) ( )( )( ) ( )

11

' '

11

' '

var , E , ,

var E .

n I n n n

m U m

x p v p i z p x p v p i p

y p v p z p y p v p p

!

!

""

""

# $ # $= " " "% & % &

= # $ " " # $ "% & % &

! !! ! ! !

! ! !


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67

When we do not assume symmetry explicitly, differently from what is done in Kyle

(1989), we derive it from linearity, as demonstrated by Proposition 3.5. Proposition 3.6

shows that one may as well assume symmetry and derive linearity from it. Together

with Proposition 3.5, Proposition 3.6 completes the equivalence between linearity and

symmetry. The argument here is that to make demand schedules symmetric, the slope of

the traders demand curves must be independent of their information, as this is what

distinguishes them from one another. We then substitute into the optimality condition a

demand schedule that is separable as the sum of a price function and an information

function. When we match coefficients, we find that the price functions must be linear.

Symmetry and linearity are therefore logically equivalent in this setting. This allows us

to interpret any contrary empirical findings, as possible counterevidence of Kyle

models base assumptions, namely on risk preferences and distributions. In fact, the

next result establishes that these assumptions are sufficient to yield linearity and thus

also symmetry.

Proposition 3.8:If investment demands are smooth around the market clearing price,

then they will be linear in equilibrium.

Proposition 3.8 further grounds the equivalence established by Propositions 3.5 and

3.6 into the models risk preferences and distributional assumptions. Essentially, it

demonstrates Kyles seminal intuition that quadratic preferences, resulting from CARA

and joint normality, would yield linear demand schedules. These are the formers

marginals implied from the first order condition for optimality.

Next, I demonstrate this to be also the case within the realm of all smooth functions.

These are certainly more general than the assumptions of the previous literature, namely

linearity. I demonstrate it by induction. I first take the quadratic case to show that


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70

for some constant &, and equilibrium demands are therefore symmetric and linear.

Proof of Proposition 3.4: Taken across all traders, equation (2.43) amounts to a

system of nonlinear ordinary differential equations. In order to solve it, I make use of

the assumption that investment demands are infinitely differentiable at every price level

to represent them by a Taylor series around an arbitrary price point p :

( ) ( ) ( )( )0

1 .

!

k k

i i

k

p p p pk

! !

"

=

= #$ (2.63)

To show that investment demand is linear is to show that all terms ai,kfor k "{0, 1}

are equal to zero. I prove this by induction on a series of Taylor polynomials. Take first

the quadratic case:

( ) ( ) ( ) ( ) ( ) ( )2' "1

2 .

i i i ip p p p p p p p! ! ! ! = + " + "

(2.64)

By substituting into and rearranging terms, we have

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( )( )

( ) ( ) ( )

2' "1

2 2

2' "1

2

2 ' "

1

1.

i i i i

i i i

i i

r pp p p p p p p y

p p p p p p pr

p p p p

! ! !

"#

! ! !

"# ! ! $ $

$ +

+ $ + $ + $

+ $ + $+=

+ $ (2.65)

By inspection, one can readily see in (2.65), that if coefficients on both sides match,

then both ( )" i p! and ( )"

i p!

"

must be zero. The induction step is noting that, if for any

M,{ ( )" i p! , , ( )( 1) Mi

p! " } are all identically equal to zero, then

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )( )( ) ( ) ( )( )

' ( )1! 2

' ( )1!

12 ' ( )11 !

1

1,

MM

i i i iM

MMi i iM

MM

i iM

r pp p p p p p p y

p p p p p p pr

p p p p

! ! !

"#

! ! !

"# ! ! $

$ $$

$ +

+ $ + $ + $

+ $

+ $

+=

+ $ (2.66)


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73

( ) ( ) ( )

( ) ( )( ) ( )( )

( ) ( ) ( ) ( )( ) ( )( )

11 11

1

1 11

1 ' 1 ,

1 ' 1 .

n n

I I I Ip Ii n

m U U U Up

x p f p g i

N f p p i

y p h p M h p p

! " # # #

! " # #

$

$ $$

$

$ $$

= +

= $ $ + $ +

= = $ $

+ $

!!

!

(2.75)

For both and , the only possible solution is given by constant ( )'f p

and ( )'h p respectively, i.e. ( )n

x p! and ( )m

y p are linear onp.!

Proof of Proposition 3.7:Taken across all traders, equations in (2.43) amount to a

system of nonlinear ordinary differential equations. In order to solve the system, I make

use of the assumption that investment demands are infinitely differentiable almost

everywhere in order to represent them by their Taylor series around an arbitrary price

point p :

( ) ( )( )

( ) ( )( )

( )

0

( )

0

1 ,

!

1 .

!

kk

n nk

kk

m m

k

x p x p p pk

y p y p p pk

!

=

!

=

= "

= "

#

#

! !

(2.76)

To show that investment demand is linear is to show that all terms an,kfor k "{0, 1}

are equal to zero. We prove this by induction on a sequence of polynomials. The first

step of the induction is the quadratic case:

( ) ( ) ( )( ) ( )( )

( ) ( ) ( )( ) ( )( )

( ) ( ) ( )( ) ( )( )

( ) ( ) ( )( ) ( )( )

2' "1

2

2' "1

2

2' "1

2

2' "1

2

,

,

,

.

n n n n

n n n n

m m m m

m m m m

x p x p x p p p x p p p

x p x p x p p p x p p p

y p y p y p p p y p p p

y p y p y p p p y p p p

! ! ! !

! ! ! !

= + ! + !

= + ! + !

= + ! + !

= + ! + !

! ! ! !

! ! ! !

(2.77)

( )nx p! ( )my p


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By substitution and rearranging terms, we have:

( ) ( )( ) ( )( )( )

( ) ( )( )( )( ) ( )( ) ( )( ) ( )( )( )

( ) ( )( )( )( ) ( )

2' "1

2

11 ' "

2' "1

2

11 ' "

1 ,

1 .

n n n

I I n n I Ip Ii n

m m m

U U m m U Up

x p x p p p x p p p

x p x p p p p i

y p y p p p y p p p

y p y p p p p

! " # # #

! " # #

$

$

$ $

$

$

$ $

+ $ + $ %

$ + $ = + $ +

+ $ + $ %

$ + $ = + $

! ! !

!! !

(2.78)

By inspection, one can readily see, from equations in (2.78) above, that if polynomial

coefficients forpon both sides are to match, then ( )"

n

x p! , ( )"

n

x p!! , ( )

"

my p , ( )

"

my p! must

be zero. The induction step is to note that, if for any degree L,

{ ( )" n

x p! ,,( ) ( )L

nx p! , ( )"

my p , ...,

( ) ( )L

my p } are all identically equal to zero, then:

( ) ( )( ) ( ) ( )( )( )

( ) ( )( )

( )( )( ) ( )( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( )( )( ) ( )

' 1

!

111 ' 1

1 !

' 1

!

111 ' 1

1 !

1 ,

1 ,

LL

n n nL

LL

I I n n I Ip Ii nL

LL

m m mL

LL

U U m m U UpL

x p x p p p x p p p

x p x p p p p i

y p y p p p y p p

y p y p p p p

! " # # #

! " # #

$$$

$ $$

$$$$ $$

+ $ + $ %

& '$

+

$ = +

$ +

( )* ++ $ + $ %

& '$ + $ = + $( )* +

! ! !

!! !

(2.79)

for which the same argument used in the quadratic case also applies for ( )( ) Lnx p! ,

( )( ) Ln

x p!

! , ( )( ) Lm

y p , ( )( ) Lm

y p!

.

Thus, all coefficients, other than intercept and slope, are zero, i.e. traders

equilibrium demands are linear:

( ) ( ) ( )( )

( ) ( ) ( )( )

'

'

,

.

n n n

m m m

x p x p x p p p

y p y p y p p p

= + !

= + !

! ! !

(2.80)


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4

Perfectly Rational Financial Bubbles

Chapter Summary

I show how credit cycles can emerge without any exogenous shocks, simply by the

effects of endogenous money creation. Cycles of bubbles and crashes generate when

part of the population has unstable financial access. As this population at the margin

enters the credit market, by funding their investments, new money is created spurring anasset bubble. Once in, their exit withdraws that additional liquidity, causing the market

to crash. A key contribution of this chapter is making entry and exit into the credit

market endogenous. As such, no exogenous risk is required in this model for the market

to remain in a perpetual cycle of booms and busts. The model provides key insights into

the asset and credit bubbles at the turn of the 21stcentury, and how to prevent them.


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4.1 Introduction

The Austrian Business Cycle Theory (ABCT) was first put forward by Mises (1953

[1912]). It was later fleshed out and presented by Hayek (1932). ABCT postulates that

credit cycles are generated by excessive credit creation, which artificially lowers interest

rates, thereby inflating asset prices. The artificially inflated asset prices and low cost of

capital stimulate investors in making malinvestments which inevitably do not produce

the income to sustain the bubble. A market crash ensues.

A main criticism of ABCT is that it seems to presume that investors are not fully

rational; for if they were, they would not commit the folly of joining the bubble. In

equilibrium, the expectation of credit creation would simply raise asset prices today,

and the effect would simply be a change of numeraire. In other words, if everyone acts

rationally, money is neutral, i.e.its quantity does not affect allocations in the economy.

What Mises and Hayek wished to demonstrate was that, in fact, money is not neutral,

and that monetary policy has a material effect in the economy.

Friedman and Schwartz (1963) later agreed with Mises on the impact of monetary

policy in the economy. In their view, what caused the Great Depression was a

substantial reduction of the money supply. They find that what prolonged the Great

Depression, which ensued after the 1929 Crash, was the lack of prompt and decisive

action from the US Federal Reserve in providing liquidity when it was lacking in the

market (Friedman and Schwartz (1963)). Their view was that while the feedback loop of

illiquidity is cut, a liquidity injection has no lasting effects. Prices eventually rebalance

themselves, albeit at different nominal levels.


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78

In this sense, our model is akin to Minsky's (2008 [1986]) Financial Instability

Hypothesis, in which, like in ABCT, credit cycles may emerge without exogenous

shocks from the production sector, simply as a coordination failure in the financial

markets.

This Section introduces the general context, background history, and motivation for

the chapter. Section 2 grapples with the problem of defining financial bubbles and

makes explicit the definition in this chapter. Section 3 reviews the literature. Section 4

presents the model. Section 5 illustrates it with a sample calibration. Section 6 interprets

the model. Section 7 provides policy recommendations, and Section 8 concludes.

4.2

Capturing a Bubble

The very notion of a financial bubble is not yet well defined in the literature.

Different authors seem to apply the term differently. This is because of the difficulty in

demonstrating that a financial bubble can exist when agents in the economy are

perfectly rational. To some authors, financial bubbles can only be identified in hindsight

after they crash, as pre-crash prices are then clearly seen as abnormally elevated against

post-crash prices. I call this the backward-looking definition of financial bubbles. One

subscriber to this view is former US Federal Reserve Chairman, Alan Greenspan, who

testified in 2005 at the House Budget Committee that:

As events evolved, we recognized that, despite our suspicions, it was very

difficult to definitively identify a bubble until after the fact that is, when

its bursting confirmed its existence. (March 2, 2005, to House Budget

Committee)

Another view is that financial bubbles are defined as assets being traded at negative

risk premiums in spite of full knowledge of the buyer. This is the forward-looking


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79

definition of financial bubbles, for it relates to assets future returns rather than past

history of prices. This is the definition put forward by DeMarzo, Kaniel, and Kremer

(2008), and the one I adopt here.

4.3 Literature

Financial bubbles are indeed difficult to explain in an environment where investors

behave rationally. It stands to reason that no sane investor would wilfully join a market

bubble, only to have her investment wiped out by collapsing asset prices. Thus, a first

class of financial bubble models resorts to the assumption that investor rationality is

bounded. Amongst these, most notable are Scheinkman and Xiong (2003) and Abreu

and Brunnermeier (2003). Scheinkman and Xiong (2003) divide the market into

optimists and pessimists. A short selling constraint prevents pessimists from betting

against an overpriced asset, whom then reasonably decide to ride the bubble in hope of

offloading the asset soon after to an optimist. As with the arrival of new information,

optimists become pessimists, and pessimists become optimists. The continual role

reversal allows the bubble to perpetuate and keep floating asset prices above consensual

values. In Abreu and Brunnermeier (2003), rational arbitrageurs take advantage of

mispricing caused by the optimists, but their failure to synchronize their actions

prevents arbitrageurs from pulling their collective weight against an asset bubble. The

bubble thus grows until it bursts when a critical mass of arbitrageurs sells out.

An alternative explanation focuses on the word wilfully used above. A market

which investors can only access through financial intermediaries acting as investment

managers introduces agency problems. This is the cornerstone of a second class of

financial bubble models. Notably, Scharfstein and Stein (1990) demonstrate that

investment managers concerned with their reputations might choose to mimic the


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80

behaviour of other managers and ignore their own information. Allen and Gorton (1993)

show how fund managers can churn bubbles, knowingly investing their less informed

clients money in overvalued assets, for as long as their skill remains unknown to them.

Not only will investment managers knowingly herd into a bubble due to the information

asymmetry between their clients and themselves, but Shleifer and Vishny (1997) also

demonstrate that their deflection from the herd, in attempting to benefit their clients

from long-run arbitrage opportunities, is severely limited, since funds will flow out of

their hands, if the bubble continues to grow against their bets.

In both of these cases, however, the presence of client investors in the market is

taken as a given, presumably on the assumption that they are oblivious of a bubble, or

that they have no other choice. The first of these assumptions is indeed reasonable:

retail investors are typically less informed about investment assets than professional

money managers. However, even sophisticated investors get burned by agency

misrepresentation, if not outright fraud, as evidenced in 2008 by Madoffs $50 billion

swindle. Some instances, as Swan, Lu, and Westerholm (2015) show to be the case in

the Finnish market, are simply due to the misalignment of institutional in

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