Anshumana And Kalay-Can Splits Create Market Liquidity - Theory And Evidence.pdf - Stock books 002 - Ryjufka

Journal of Financial Markets 5 (2002) 83–125

Can splits create market liquidity?

Theory and evidence $

V. Ravi Anshuman a, *, Avner Kalay b,c

FinanceandControl,IndianInstituteofManagement,BannerghattaRoad,Bangalore560076,India

The Leon Recanati Graduate School of Business Administration, Tel Aviv University,

P.O.B. 39010, Ramat Aviv, Tel Aviv 69978, Israel

c Department of Finance, David Eccles School of Business, University of Utah, Salt Lake City,

UT 84112, USA

Abstract

We present a market microstructure model of stock splits in the presence of minimum

tick size rules. The key feature of the model is that discretionary trading is endogenously

determined. There exists a tradeoﬀ between adverse selection costs on the one hand and

discreteness related costs and opportunity costs of monitoring the market on the other

hand. Under certain parameter values, there exists an optimal price. We document an

inverse relation between the coe3cient of variation of intraday trading volume and the

stock price level. This empirical evidence and other existing evidence are consistent with

JEL classiﬁcation: G12; G18; G32

Keywords: Stock splits; Liquidity; Tick size; Discreteness; Trading range; Optimal price

$ This paper draws on the Ph.D. dissertation of V. Ravi Anshuman and an earlier joint working

paper. We have received helpful comments from Larry Glosten, Ishwar Murty, Avanidhar

Subrahmanyam (the editor) and anonymous referees. We would also like to thank J. Coles, T.

Callahan, S. Ethier, R. Lease, U. Loewenstein, S. Manaster, J. Suay, E. Tashjian, S. Titman, Z.

Zhang, and seminar participants at Ben Gurion University, Boston College, Carnegie Mellon

University, Cornell University, Hebrew University, Hong Kong University of Science and

Technology, Rutgers University, Tel Aviv University, University of Utah and the European

Finance Association meetings for their helpful comments. The ﬁrst author acknowledges support

from the Global Business Program, University of Utah and the Recanati Graduate School of

Business, Tel Aviv University, Hong Kong University of Science and Technology, and the

University of Texas at Austin. We take responsibility for any remaining errors.

*Corresponding author. Tel.: +91-80-699-3104; fax: +91-80-658-4050.

E-mail address: anshuman@iimb.ernet.in (V.R. Anshuman).

PII: S 1 3 8 6 - 4181(01)00020-9

V.R. Anshuman, A. Kalay / Journal of Financial Markets 5 (2002) 83–125

1. Introduction

U.S. ﬁrms split their stocks quite frequently. In spite of inﬂation, positive

real interest rates, and signiﬁcant risk premiums, the average nominal stock

price in the U.S. during the past 50 years has been almost constant. Why would

ﬁrms keep on splitting their stocks to maintain low prices?This behavior is

puzzling since, by doing so, ﬁrms actively increase their eﬀective tick size (i.e.,

tick size/price), potentially exposing their stockholders to larger transaction

costs.

This paper presents a value maximizing market microstructure model of

stock splits. Our model joins practitioners in predicting that ﬁrms split their

stocks to move the stock price into an optimal trading range in order to

improve liquidity. 1,2 The driving force of the model stems from the fact that

prices on U.S. exchanges are restricted to multiples of 1/8th of a dollar. 3 This

restriction on prices creates a wedge between the ‘‘true’’ equilibrium price and

the observed price. 4 Thus a portion of the transaction costs incurred by traders

is purely an artifact of discreteness.

Anshuman and Kalay (1998) show that discreteness related commissions

depend on the location of the ‘‘true’’ equilibrium price on the real line. In other

words, whether the discrete pricing restriction is binding or not depends on the

location of the ‘‘true’’ equilibrium price relative to a legitimate price (tick) in a

discrete price economy. It may so happen that the ‘‘true’’ equilibrium price

(plus any transaction cost) is close to a tick. Discreteness related commissions

would be low in such a period. As information arrives in the market, the

location of the ‘‘true’’ equilibrium price changes, and discreteness related

commissions would, therefore, vary over time. They could be as low as 0 or as

high as the tick size.

Interestingly, liquidity traders can take advantage of the variation in

discreteness related commissions by timing their trades. Of course, such

1 Academicians have mostly relied on signaling models to explain stock splits (Grinblatt et al.,

1984). More recently, Muscarella and Vetsuypens (1996) provide evidence consistent with the

liquidity motive of stock splits. Practitioners, however, have all along held the belief that stock

splits help restore an optimal trading range that maximizes the liquidity of the stock (see Baker and

Powell, 1992; Bacon and Shin, 1993).

2 Independent of our work, Angel (1997) has also presented a model of optimal price level that

explains stock splits. In his model, the optimal price provides a tradeoﬀ between ﬁrm visibility and

transaction costs. In contrast, our model examines the behavior of liquidity traders in the presence

of discrete pricing restrictions.

3 There are exceptions to this restriction and more recently the NYSE has initiated a move

toward decimal trading.

4 The ‘‘true’’ equilibrium price is the market value of the asset conditional on all publicly

available information in an otherwise identical continuous-price economy without any frictions

(transaction costs).

V.R. Anshuman, A. Kalay / Journal of Financial Markets 5 (2002) 83–125 85

strategic behavior is not costless. It involves close monitoring of the market

to take advantage of periods with low discreteness related commissions.

In general, liquidity traders diﬀer in terms of their opportunity costs of

monitoring the market. Some liquidity traders may prefer not to time the

market because the beneﬁts from timing trades do not oﬀset their opportunity

costs of monitoring. In contrast, other liquidity traders who are endowed with

low opportunity costs of monitoring may ﬁnd it beneﬁcial to time their trades.

Such discretionary traders would trade together in a period of low discreteness

related commissions. The presence of additional liquidity traders in this period

(a period of concentrated trading) forces the competitive market maker to

charge a lower adverse selection commission than otherwise. Thus, discre-

tionary liquidity traders save on execution costs – adverse selection as well as

discreteness related commissions.

Because the tick size is ﬁxed in nominal terms (at 1/8th of a dollar), the

economic signiﬁcance of the savings in discreteness related commissions

depends on the stock price level. At low stock price levels, the savings in

execution costs due to timing of trades may be signiﬁcant enough to oﬀset the

opportunity costs of monitoring of most liquidity traders. There would be

highly concentrated trading at low price levels as most liquidity traders would

exercise the ﬂexibility of timing trades. Conversely, at high stock price levels,

few liquidity traders would time trades because the potential savings in

execution costs are economically insigniﬁcant.

The key implication of the model is that the stock price level aﬀects the

distribution of liquidity trades across time, and consequently, the transaction

costs incurred by them. In particular, we show that there exists an optimal

stock price level that induces an optimal amount of discretionary trading. This

optimal price results in the lowest (total) expected transaction costs incurred by

all liquidity traders.

Because investors desire liquidity (Amihud and Mendelson, 1986; Brennan

and Subrahmanyam, 1995), a value-maximizing ﬁrm should choose a stock

price level that maximizes liquidity (minimizes the total transaction costs

incurred by all liquidity traders). By splitting (or reverse splitting) its stock, a

ﬁrm can always reset its stock price to the optimal price level.

We present numerical solutions of the model to show that, under certain

parameter values, an optimal price exists. The numerical solutions show that

the optimal price is increasing in the volatility of the underlying asset and

decreasing in the fraction of liquidity traders. We also show that the

optimal price is (linearly) increasing in the tick size. Finally, using intraday

transaction data, we document a cross-sectional inverse relation between the

coe 3 cient of variation of time-aggregated trading volume (a measure of the

degree of concentrated trading in a stock) and the stock price level.

This empirical evidence and other existing evidence are consistent with the

model.

V.R. Anshuman, A. Kalay / Journal of Financial Markets 5 (2002) 83–125

The paper is organized as follows. Section 2 discusses a numerical example

that illustrates the key features of the model. The model is developed in

Section 3. Section 4 presents numerical solutions of the model. Section 5

discusses empirical evidence relevant to the model, and Section 6 concludes the

paper.

2. A numerical example

Consider the following example that illustrates the central theme of the

model – endogenization of discretionary trading. We make the following

simplifying assumptions in the numerical example. (i) There are two trading

opportunities (Periods 1 and 2). (ii) Discreteness related commissions in each

period are either $0.02 or $0.10 with equal probability. 5 (iii) Firms are

restricted to choose between two base prices ($50 or $100) – the base price

could be thought of as the oﬀer price in an initial public oﬀering. (iv) Liquidity

traders are of two types: 80 liquidity traders face very low opportunity costs of

monitoring ($0.01 per dollar of trade) and 40 liquidity traders face extremely

high opportunity costs of monitoring. (v) In each period, there are a ﬁxed

number of informed traders who speculate on information that is revealed at

the end of the period.

Before the market opens, liquidity traders face a strategic choice. They know

that monitoring the market can help them time their trades into the period with

low discreteness related commissions ($0.02). Not only would they be saving on

discreteness related commissions but also on adverse selection commissions

because of the concentration of liquidity trades in a single period.

However, monitoring the market is not costless. Among the liquidity traders,

those with extremely high monitoring costs would not ﬁnd timing trades

worthwhile. Such liquidity traders (40) behave like nondiscretionary traders.

Assuming that there are negligible waiting costs, these traders would be

indiﬀerent between trading in Period 1 or trading in Period 2. Let equal

number of nondiscretionary traders (40/2=20) arrive in the market in each

period.

The interesting question is with regard to the 80 liquidity traders with low

monitoring costs. Should they incur monitoring costs and time their trades or

join the bandwagon of nondiscretionary traders?If they choose not to monitor

(and, therefore, act as nondiscretionary traders), then each trading period

would consist of (80+40)/2=60 liquidity traders, assuming that the arrival

rate of nondiscretionary traders is constant (equal) in both periods. On the

other hand, if these liquidity traders choose to monitor, one of the trading

5 This assumption is purely for illustration purposes. In reality, there exists a probability

distribution of discreteness related commissions over the interval (0, tick size).

V.R. Anshuman, A. Kalay / Journal of Financial Markets 5 (2002) 83–125 87

periods would have 100 (80 discretionary and 20 nondiscretionary) liquidity

traders, and the other period would have only 20 nondiscretionary liquidity

traders. Hence the distribution of liquidity traders across the two periods

would be one of the following: (60, 60) if they choose not to monitor the market

and either (20, 100) or (100, 20) if they monitor the market.

Liquidity traders with low monitoring costs would think as follows. Their

choice to monitor or not depends on the total (per dollar) transaction costs

they face under each scenario. Total transaction costs are composed of adverse

selection commissions, discreteness related commissions, and monitoring costs.

Table 1 presents these costs at the two base prices in this economy.

Consider Panel A of Table 1 for the case when the base price is $50. Suppose

liquidity traders with low monitoring costs choose to monitor the market. Then,

in the period they trade, the adverse selection commissions would be low

because of the presence of 100 liquidity traders. In contrast, when they choose

not to monitor the market, the adverse selection commissions are going to be

higher because there would be only 60 liquidity traders. Assume that the adverse

selection commissions are $0.046 when there are 100 liquidity traders and

$0.535 when there are 60 liquidity traders (in the model, we derive the adverse

selection commissions endogenously). Monitoring the market and concentrat-

ing trades in a single period results in savings of ($0.535 $0.046)=$0.489 in

adverse selection commissions, or 0.978% of the base price of $50.

Panel B of Table 1 shows the adverse selection commissions when the base

price is $100. These numbers are scaled up versions of the adverse selection

commissions when the base price is $50. However, as shown in the (%) adverse

selection commission column, the adverse selection commissions (given a ﬁxed

number of liquidity trades) are identical at both base prices in percentage

terms. Therefore, the beneﬁt of concentrated trading (in terms of savings in

adverse selection commissions) is 0.978%, which is invariant to the base price.

Now consider discreteness related commissions when the base price is $50

(Panel A). If liquidity traders with low monitoring costs choose to monitor,

they would incur lower discreteness related commissions because they can time

their trades in the period with low discreteness related commissions ($0.02).

Note that they would incur expected discreteness related commissions of $0.04

(this is higher than $0.02 because it is always possible that both trading periods

have a realized discreteness related commission of $0.10). 6 In contrast, when

such liquidity traders choose not to monitor, they incur a higher expected

discreteness related commission of $0.06 (an average of $0.02 and $0.10). These

commissions ($ values) stay the same at the higher base price of $100 (Panel B).

6 The probability of both trading periods having high discreteness related commissions ($0.10) is

0.5 0.5=0.25. The probability of at least one period having low discreteness related commissions

($0.02) is 1 0.25=0.75. Therefore, the expected discreteness related commissions is 0.25

$0.10+0.75 $0.02=$0.04.

Anshumana And Kalay-Can Splits Create Market Liquidity - Theory And Evidence.pdf

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: