The Dynamic Properties of Financial‐Market Equilibrium with Trading Fees

• Author: ADRIAN BUSS,BERNARD DUMAS
• Published date: 01 April 2019
• Date: 01 April 2019
• DOI: http://doi.org/10.1111/jofi.12744

THE JOURNAL OF FINANCE •VOL. LXXIV, NO. 2 •APRIL 2019

The Dynamic Properties of Financial-Market

Equilibrium with Trading Fees

ADRIAN BUSS and BERNARD DUMAS∗

ABSTRACT

We incorporate trading fees into a dynamic, multiagent general-equilibrium model in

which traders optimally decide when to trade. For that purpose, we propose an inno-

vative algorithm that synchronizes the traders. Securities prices are not so much af-

fected by the payment of the fees itself, but rather by the trade-off that the traders face

between smoothing consumption and smoothing holdings. In calibrated examples, the

interest rate and welfare decline with trading fees, while risk premia and volatilities

increase. Liquidity risk and expected liquidity are priced, leading to deviations from

the consumption-CAPM. With trading fees, capital is slow-moving, generating slow

price reversal.

∗Adrian Buss is with INSEAD and CEPR. Bernard Dumas is with INSEAD, University of

Torino, NBER, and CEPR. Previous versions of this article were circulated and presented under

the titles “The Equilibrium Dynamics of Liquidity and Illiquid Asset Prices” and “Financial-Market

Equilibrium with Friction.” Work on this topic was initiated while Dumas was at the University

of Lausanne and Buss was at Goethe University Frankfurt. Dumas’s research has been supported

by the Swiss National Center for Competence in Research “FinRisk,” by the INSEAD research

fund, and by the AXA chair of the University of Torino. He is thankful to Collegio Carlo Alberto

and to BI Norwegian Business School for their hospitality. The authors thank the Editor (Bruno

Biais), an Associate Editor, and three referees for helpful comments. They are also grateful, for

useful discussions and comments, to Beth Allen; Yakov Amihud; Andrew Ang; Laurent Barras;

S´

ebastien B´

etermier; Bruno Biais; John Campbell; Georgy Chabakauri; Massimiliano Croce; Mag-

nus Dahlquist; Sanjiv Das; Xi Dong; Itamar Drechsler; Philip Dybvig; Thierry Foucault; Kenneth

French; Xavier Gabaix; Nicolae Gˆ

arleanu; Stefano Giglio; Francisco Gomes; Amit Goyal; Harald

Hau; John Heaton; TerrenceHendershott; Julien Hugonnier; Ravi Jagannathan; Ely `

es Jouini; An-

drew Karolyi; Felix Kubler; David Lando; John Leahy; Francis Longstaff; Abraham Lioui; Edith

Liu; Hong Liu; Pascal Maenhout; Fr´

ed´

eric Malherbe; Ian Martin; Alexander Michaelides; Ste-

fan Nagel; Stavros Panageas; Paolo Pasquariello; ˇ

Luboˇ

sP

´

astor; Patrice Poncet; Tarun Ramado-

rai; Scott Richard; Barbara Rindi; Jean-Charles Rochet; Leonid Ros¸u; Olivier Scaillet; Norman

Sch¨

urhoff; Chester Spatt; Roberto Steri; Raman Uppal; Dimitri Vayanos; Pietro Veronesi; Grigory

Vilkov; VishViswanathan; Ingrid Werner; Jeffrey Wurgler; Fernando Zapatero; and participants at

workshops given at the Amsterdam Duisenberg School of Finance, INSEAD, the CEPR’s European

Summer Symposium in Financial Markets at Gerzensee, the University of Cyprus, the University

of Lausanne, Bocconi University, the European Finance Association, the Duke-UNC Asset Pric-

ing Workshop, the National Bank of Switzerland, the Adam Smith Asset Pricing workshop, the

YaleUniversity General Equilibrium workshop, the Center for Asset Pricing Research/Norwegian

Finance Initiative Workshop at BI, the Indian School of Business Summer Camp, Boston Univer-

sity, Washington University in St. Louis, ESSEC Business School, HEC Business School, McGill

University,the NBER Asset Pricing Summer Institute,the Universityof Zurich, the Universityof

Nantes, the Western Finance Association, and the Frankfurt School of Finance and Management.

The authors do not have any potential conflicts of interest to disclose as identified in the Journal

of Finance’s disclosure policy.

DOI: 10.1111/jofi.12744

795

796 The Journal of Finance R

THE FINANCIAL SECTOR OF THE ECONOMY issues and trades securities. But, more

importantly, it provides a service to clients, namely access to the financial

market and trading. This service is provided for a fee. Our objective in this

paper is to shed light on the impact of these trading fees on trading strategies,

asset prices, return moments, and liquidity premia—in a general-equilibrium

economy.

In particular, we first seek to identify the economic mechanisms through

which endogenous (il)liquidity, arising from trading fees, influences securities

prices, their returns, and their volatilities. We next seek to understand how

liquidity and liquidity risk are priced in equilibrium and how these premia

fluctuate over time. Finally, we wish to determine whether the slow movement

of investment capital that arises naturally in the presence of trading fees can

explain a slow reversal of the corresponding securities prices.

To address the above questions, we incorporate trading fees into a dynamic

general-equilibrium economy in which traders optimally and endogenously de-

cide when and how much to trade. Traders are endowed with an every-period

motive for trading, over and above the long-term need to trade for lifetime plan-

ning. That is, they receive endowments that are not fully hedgeable because,

even without trading fees, the financial market is incomplete. Hence, whenever

a trader’s realized endowment differs from the amount he has hedged, he has

a desire to adjust his portfolio positions. However, transactions are subject to

trading fees.

As a way of constructing a simple model, we bypass intermediaries and their

pricing policy, and instead let traders serve as dealers for, and pay trading fees

to, each other.1We take the fee function as given, but choose the functional form

in such a way that it reflects one special and important feature of the cost of

financial services: financial-market traders sometimes buy and at other times

sell the very same security, paying a positive fee irrespective of the direction of

the trade. Specifically, we assume that, to an approximation, the trading fee is

proportional to the value of the shares traded, and thus introduces a kink at

zero in the trading-fee function.

As a result, liquidity in our model is endogenous. Within a so-called “no-

trade region,” it is optimal for a trader not to trade, thereby preventing other

traders from trading with him. A trader’s reluctance to trade creates a negative

(pecuniary) externality for other traders, and, as a result, investment capital

in the economy is slow-moving. This is an additional endogenous, stochastic,

and quantitatively important consequence of the trading fee. Illiquidity begets

illiquidity. Hence, when purchasing a security, a trader must anticipate his,

and other people’s, desire and ability to resell the security in the marketplace

at a later time.

We first demonstrate the impact of this endogenous process of (il)liquidity on

equilibrium prices. We show analytically that differences between equilibrium

1In practice, traders trade through brokers or intermediaries. However, since the end users are

the traders, access to a financial market is ultimately a service that traders make available to each

other.

Financial-Market Equilibrium with Trading Fees 797

securities prices in the presence of trading fees and those in the absence of

trading fees result from changes in the state prices and the consumption of

individual agents. In other words, the differences do not result so much from

the payment of the trading fees itself as they do from a trade-off that the traders

face between smoothing consumption and smoothing holdings (to reduce the

cost of trading).

Second, we provide quantitative results for an illustrative setting with two

traded securities: (i) a fully liquid bond and (ii) a stock that is subject to

trading fees. We document that the resulting higher consumption growth

volatility of the individual traders pushes both securities’ prices up—due to

a precautionary-savings effect. As a result, even the stock price is increasing

in the level of its fees. Moreover, the higher consumption growth volatility has

a considerably negative impact on traders’ welfare. The same economic mech-

anism affects rates of return. While the rate of interest is reduced due to the

precautionary-savings effect, the equity premium, the stock’s return volatility,

and its Sharpe ratio increase.

Next, we show how, in the presence of trading fees and endogenous trad-

ing decisions, various liquidity (risk) premia arise, relative to the classic

consumption-CAPM (CCAPM). In particular, in our economy, not only liquid-

ity risk but also expected future liquidity generate a premium over and above

consumption risk. We next document the variation of each term over time.

While the unconditional average value of the CCAPM deviation is due mostly

to the liquidity risk premium, its unconditional variance is driven mostly by

fluctuations of the conditionally expected liquidity term.

In an extension of our basic model to the case of three traders, we show that

deviations from the frictionless equilibrium are not reduced when more traders

are present in the market. For example, the welfare loss and the waiting time

between trades increase relative to the two-trader economy. Intuitively, while

more traders in a market implies more people potentially ready to trade, it also

implies more people who need to trade (for partially idiosyncratic reasons). In-

terestingly,a sizeable fraction of trades in the stock market take place between

two traders only (bilateral trade).

Finally, we document that the slow movement of investment capital that

arises naturally in our model because of traders’ reluctance to trade can explain

slow price reversal. In the model with trading fees, when a shock occurs, traders

(partially) adjust immediately to that impulse, but they also react later on; that

is due to hysteresis. Indeed, the impulse moves a fee-paying trader closer to a

trade boundary, so that when subsequent shocks arrive in the same direction,

he will act to a further extent than he would have acted in the absence of

the earlier impulse. As a result, when a shock occurs, the price of a security

(over)reacts, while quantities do not adjust completely at first. When quantities

do adjust later, the price movement is reversed, leading to a pattern that is

known as “slow price reversal.”

The paper also makes a significant methodological contribution to the solu-

tion of equilibrium models with trading fees. We define a form of Walrasian

equilibrium that is the limit of a sequence of equilibria in which each trader’s