The Cross Section of Expected Returns and Amortized Spreads

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Review of Pacific Basin Financial Markets and PoliciesVol. 9, No. 4 (2006) 597638c World Scientific Publishing Co.

and Center for Pacific Basin Business, Economics and Finance Research

The Cross Section of Expected Returns andAmortized Spreads

Zhongzhi (Lawrence) He

Faculty of Business, Brock University500 Glenridge Ave., St. Catharines, ON, Canada, L2S 3A1

[email protected]

Lawrence KryzanowskiJohn Molson School of Business, Concordia University

1455 De Maisonneuve Blvd. West, Montreal, QC, Canada, H3G 1M8

[email protected]

The cross-sectional relationship between expected returns and amortized spreadsis studied in an overlapping-generations economy with an average investor. Thecommonality in liquidity is directly incorporated into the asset-pricing relation. Ina static equilibrium, the amortized spread of an asset is related to its expectedreturn through four channels; namely: the equilibrium zero-beta rate, the marketrisk premium, a level effect, and an incremental sensitivity effect. Although bothare present over the entire period, their relative importance shifts from a significantlevel to a significant sensitivity effect from the earlier to most recent sub-period inthe Canadian stock market.

Keywords: Amortized spread; asset pricing; liquidity commonality; clientele effect;share turnover.

JEL Classification: G11, G12.

1. Introduction

The seminal work of Amihud and Mendelson (1986) establishes the relation

between the required return on an asset and trading costs as measured by

the proportional bid-ask spread. The most prominent result of their work isthe clientele effect of liquidity; namely, given the same level of risk, assets

with higher (lower) proportional spreads are allocated to portfolios with

Corresponding author.

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598 Zhongzhi (Lawrence) He & Lawrence Kryzanowski

longer (shorter) expected holding periods. Amihud and Mendelson (1986)

test and report support for their main hypothesis that the market-observed

expected return is an increasing and concave piecewise-linear function of theproportional spread. If the clientele effect holds in the real economy, then a

description of the spread-return relation is readily obtained for each clientele

of investors. Specifically, after first observing an assets proportional spread,

this spread can be mapped with its required return using the estimated

piecewise-linear function.

The clientele effect of liquidity is generally supported empirically. For

example, Atkins and Dyl (1997) directly support the clientele effect by show-

ing that the length of the holding periods of investors is positively relatedto the bid-ask spreads for NYSE stocks. Datar, Naik and Radcliffe (1998)

test and support an implication of the clientele effect; namely, that asset

returns are negatively related to the turnover rates of assets. More recently,

Kryzanowski and Rubalcava (2005) report empirical findings that support a

generalized version of the investor clientele hypothesis, which they call the

international trade-venue clientele effect hypothesis.

In summary, the clientele effect of liquidity is an important and widely

supported result that prescribes the return-spread relation for different clien-teles of investors. However, the clientele effect does not tell us how asset

returns are related to spreads on average (i.e., across investor clienteles).

The average return-spread relation is an important issue in its own right

because it provides an analysis of the incremental required rate of return

for a unit increase in the level and/or the risk of trading costs, not for any

particular investor type, but for the market as a whole.

By examining how asset returns are related on average to transaction

costs, we complement the findings on the clientele effect in the return-spread literature. We start with the static economy set up used by Amihud

and Mendelson (1986). However, instead of theorizing a positive association

between the holding periods of different investor types and proportional

spreads, we aggregate across all the investor types to form a market-average

expected holding period that reflects the view of a representative investor.

For a given asset, the proportional spread divided by its average hold-

ing period gives rise to the notion of the amortized spread (Chalmers

and Kadlec, 1998), which is a measure of transaction costs for an averageinvestor. Thus, this paper aims to investigate the cross-sectional relationship

between expected returns and amortized spreads.

As in Amihud and Mendelson (1986), we take the bid-ask spread of each

asset as given. Nevertheless, instead of assuming a deterministic spread,

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A Cross Section Of Expected Returns And Amortized Spreads 599

the spread herein is treated as a random process driven by some underly-

ing liquidity factors. A recent body of literature that documents system-

atic liquidity motivates this treatment. Using a large transaction databaseconsisting of NYSE stocks, Chordia, Roll and Subrahmanyam (2000) find

highly significant spread correlations among individual securities, and even

stronger spread correlations for portfolios. Huberman and Halka (2001) doc-

ument the existence of commonality in the bid-ask spreads for 240 size-sorted

NYSE stocks. Using different statistical techniques, Hasbrouck and Seppi

(2001) find that the common factors in liquidity are relatively small. Given

the new evidence of systematic liquidity, Chordia, Roll and Subrahmanyam

(2000, p. 6) make the following statement regarding the impact of tradingcosts on asset pricing:

Hence, there are potentially two different channels by which trading

costs influence asset pricing, one static and one dynamic: a static

channel influencing average trading costs and a dynamic channel

influencing risk. In future work, it would be of interest to deter-

mine whether the second channel is material and, if so, its relative

importance.

Thus, the main contribution of this paper is to derive an asset pricing

relation that formally incorporates the commonalities in liquidity identi-

fied in the literature and to formally demonstrate and empirically test the

existence and relative importance of the two channels in asset pricing over

time.

The theoretical part of this paper develops the cross-sectional

relationship between expected returns and amortized spreads within an

overlapping-generations economy. In equilibrium, our parsimonious cross-sectional equation explains expected returns by three components: the zero-

beta rate that includes both the risk-free rate and the market-average

expected amortized spread; the market risk premium that incorporates com-

pensations for both fundamental and spread risks; and the asset-specific

term that captures the difference between the expected amortized spread

of a given asset and that of the market portfolio. To reflect the evidence of

systematic liquidity directly, we impose a liquidity factor structure on the

expected amortized spread. Our cross-sectional equation with the embed-ded liquidity factor explicitly indicates the co-existence of two effects of

the amortized spread: the level effect (static channel) that views the amor-

tized spread as an asset characteristic; and an incremental sensitivity effect

(dynamic channel) that views the amortized spread as a risk factor.

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The primary focus of our empirical examination is on cross-sectional tests

of the level and sensitivity effects of the amortized spread. We form 25 size-

spread portfolios following the approach of Fama and French (1992) usingmonthly data for a large cross section of Canadian stocks from April 1977 to

December 2002. To perform these cross-sectional tests, we use pooled cross-

section and time series GLS regressions, and we test the level and sensitivity

effects with a variety of control variables. We find evidence that the level

and sensitivity effects of the amortized spread are present in our whole test

period. This suggests that investors not only command higher compensation

for stocks with higher levels of trading costs, but that they also demand an

incremental risk premium for stocks that are more sensitive to broad liquid-ity shocks. We also examine the two effects over two sub-periods and find

an intertemporal shift in the relative importance of the level and sensitivity

effects. The level effect dominates the sensitivity effect during the first sub-

period, while the sensitivity effect is more prominent and significant during

the second sub-period. This implies that when facing both a higher level

and higher volatility of transaction costs that are induced by more active

trading, investors seem to care more about an assets sensitivity to market-

wide liquidity factors than the level of transaction costs. This finding isconsistent with the information-trading hypothesis of Wilcox (1993), which

relates trading strategies with the value and rate of decay of information

held by the representative investor. Our results are robust across different

sets of control variables and are not the artifact of the well-documented

seasonal behavior of liquidity. With regard to the control variables, we find

that conditional betas (i.e., conditioning on up- or down-markets as in Pet-

tengill, Sundaram and Mathur, 1995) can explain the negative beta-return

relation that is observed when the unconditional beta is used. Furthermore,counter to the literature (e.g., Scruggs and Glabadanidis, 2003; Chalmers

and Kadlec, 1998), we find a significantly positive return-volatility relation,

which subsumes the beta effect during up-markets and much of the size

effect.

We also empirically examine the time-series behaviors of the market-

average amortized spreads. Preliminary evidence shows that the market-

average amortized spread becomes increasingly important over time, both

in determining the level and volatility of average trading costs, and in influ-encing the equilibrium zero-beta rate. Furthermore, we find that the increase

in the level and volatility of the amortized spread during the late 1990s is

mainly due to increased trading activities rather than to a widening of the

proportional effective spread.

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Our paper extends the study of Chalmers and Kadlec (1998) who conduct

the first empirical examination of amortized spreads. Chalmers and Kadlec

(1998) test the level effect aloneof the amortized spread for US stocks dur-ing 1983 to 1992 and find weak evidence for a level effect. In comparison,

we jointly test both the level and sensitivity effects for a longer period (i.e.,

19822002) and find that the two effects are present, and that the relative

importance of the two effects differs for different time periods. Pastor and

Stambaugh (2003) examine the sensitivity effect of asset returns to liquid-

ity risk for the US stock market over the period between 1965 and 1998.

Using volume-related return reversals to construct an aggregate illiquidity

measure, Pastor and Stambaugh (2003) find that the sensitivities of returnsto fluctuations in aggregate liquidity command a significant risk premium.

However, Pastor and Stambaugh (2003) do not control for the level effect

of illiquidity, whereas we examine the joint effect and find that both effects

are significant.

Our paper is most closely related to the study of Acharya and Pedersen

(2005) who develop a liquidity-adjusted CAPM and examine three sources of

liquidity risk on the monthly returns of US stocks between 1964 and 1999. It

is important to highlight different model construction, illiquidity measures,and therefore empirical results between our study and theirs. Also in an over-

lapping generations economy, Acharya and Pedersen (2005) attribute liquid-

ity risk to three sources of liquidity betas: cov(Lj , Lm), which is the covaria-

tion between an assets illiquidity and the markets illiquidity; cov(Rj , Lm),

which is the covariation between an assets return and the markets illiquid-

ity; and cov(Lj , Rm), which is the covariation between an assets illiquidity

and the market return. Using a calibrated measure of illiquidity proposed by

Amihud (2002), Acharya and Pedersen (2005) find that the commonality ofliquidity, as captured by cov(Lj , Lm), is not important in asset pricing. How-

ever, this finding is limited by their model restriction that the three sources

of liquidity risk have the same risk premium in order to side step the severe

co-linearity problem that occurs when the three liquidity betas are jointly

estimated. The finding may not hold if cov(Lj , Lm) bears a higher premium.

In contrast, we develop a cross-sectional model under a mild and widely

adopted assumption (i.e., Amihud, 2002) that the firms cash flow process is

uncorrelated with liquidity shocks. Thus, our model focuses on the relativeimportance between the level and commonality of liquidity without the co-

linearity problem induced by cov(Rj , Lm) and cov(Lj , Rm). In terms of the

illiquidity measure, unlike Acharya and Pedersen (2005) who use an empiri-

cally calibrated illiquidity measure, the amortized spread used in our study

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is a direct illiquidity measure that is theoretically grounded (Amihud and

Mendelson, 1986) and empirically supported (Chalmers and Kadlec, 1998).

Furthermore, unlike the findings of Acharya and Pedersen (2005), our empir-ical results suggest that cov(Lj , Lm) plays an important asset-pricing role

beyond that contained in the level of illiquidity. Furthermore, our subpe-

riod analysis finds a predominant asset-pricing role for cov(Lj , Lm) after the

early 1990s. The different results obtained by Acharya and Pedersen may

be attributed to differences in model assumptions, illiquidity measures, and

the studied financial markets.

The remainder of this paper is organized as follows. Section 2 sets up

the economy and defines the amortized spread. Section 3 formally derivesthe cross-sectional model in equilibrium. In Section 4, we impose a factor

structure on the amortized spread and discuss the various asset-pricing chan-

nels of the amortized spread. Section 5 describes the data and measures for

empirical examination. Section 6 explains the estimation and testing proce-

dures, and presents and discusses the test results. In Section 7, we provide

concluding remarks.

2. The Amortized Spread

The initial setup of our economy is based on the static economy of Amihud

and Mendelson (1986). Consider I investor types (i = 1, . . . , I ) who invest

and trade J risky assets (j = 1, . . . , J ) and one risk-free asset. Each risky

assetj , with a market capitalization denoted by Vj , pays out a random cash

flow dj per unit of time. The round-trip trading cost for risky asset j is

measured by a random proportional spread sj per unit of time. The trading

cost for the risk-free asset is zero. Each investor-type iholds his assets for arandomly distributed time with meanE[Ti] = 1/i. In other words,i is the

expected turnover rate for investor-type i. Under such an economic setup,

the return-spread relation is developed (Amihud and Mendelson, 1986) as

the following:

rij = dj/Vj isj = Rj isj, (1)

where rij is the spread-adjusted (i.e., net) return of asset j to investor-type

iper unit of time, Rj

dj/Vj is the fundamental (gross) return on asset j,andisj is the spread-adjustment, or liquidation cost of asset j to investor-

type i.

The above equation is the same as Eq. (4) of Amihud and Mendel-

son (1986, p. 227), except that the proportional spread is modeled as a

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random process instead of a deterministic term. The random spread sjreflects the recently documented evidence of systematic liquidity, and it

intends to capture the sensitivity effect of illiquidity in a cross section.The clientele effect of liquidity hypothesizes that assets with higher

expected proportional spreadsE[sj ] are held by investors who plan a longer

holding period 1/i (a lower turnover i). Thus, the expected return of an

asset can be related to its proportional spread without observing different

investor types, i.e., the return-spread relation is an increasing and concave

piecewise-linear function. However, because the return-spread relation in

Eq. (1) is prescribed for a given investor clientele i, it does not tell us how

asset returns are on average related to spreads, i.e., when averaged acrossthe different holding periods of investors. Since this issue remains largely

unresolved, we hereby take a different route to examine the cross-sectional

relation of returns and spreads. Instead of prescribing the clientele effect or

observing different investor types, we aggregate across all investor types to

form a market-average expected turnover that reflects the view of a repre-

sentative investor. Formally, let wij denote the proportional wealth invested

in firm j by investor type i. For each risky asset j, we aggregate across

investor-type i from 1 to Iand obtain:

Rj i

wij rij =i

wijRji

wijisj = Rj j sj, j = 1, . . . J (2)1

In the above equation, Rj

iwij rij is a weighted average net return of

assetj , with the weight given by investor is proportional wealth invested in

asset j. Rj =

iwij Rj stems from the aggregation condition

iwij = 1.

2

j i

wiji is the expected turnover rate of asset j for an average or

representative investor. In Eq. (2), j sj is the expected turnover multiplied

by the proportional spread of assetj . Chalmers and Kadlec (1998) interpret

this product as the amortized spread of asset j, and they argue that the

amortized spread is more relevant than the proportional spread in asset

pricing.

1Equation (2) follows the intuition of Atkins and Dyl (1997). The authors argue that

the hypothetical average investor is an appropriate construct to investigate the relationbetween transaction costs and the time horizon of investors, which is the reciprocal of theturnover rate.2Given the definition of wij , if we aggregate across all the investor types (over i) for theproportional wealth invested in firm j, the total market proportional wealth on firmjmustsum to unity, i.e.,

Piwij = 1. This is referred to as the aggregation condition.

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To proceed, we use the definitions of expected turnover (j) and propor-

tional spread (sj) to obtain:

j sj = vjnj

sdj

pmj=

SjVj

Saj , (3)

where vj is the expected share volume of asset j for the next unit period,

nj is total number of shares outstanding of firm j, sdj is the random dollar

spread per share for the next unit period,pmj is the bid-ask midpoint, which

is used as the observed share price of asset j , Sj vj sdj is the total spread

of trading asset j, and Vj nj pm

j is the observed market capitalizationof firmj.

From Eq. (3), the amortized spread of the jth asset Saj is equivalently

defined as the assets total spread (as a proxy for the total trading costs

of the asset per unit of time) divided by the assets market capitalization.3

Substituting Saj into Eq. (2) yields:

Rj =Rj S

aj . (4)

Thus, the net return of asset j is the fundamental (gross) return net of its

amortized spread.

3. A Cross-Sectional Model of Expected Returns

and Amortized Spreads

In this section, we develop a capital market equilibrium within a static

economy as illustrated in the previous section. The trading motive of our

model can be illustrated by an overlapping-generations economy in which a

representative agent lives for one period from t to t+ 1.4 The agent has an

initial endowment W, with which he acquires J risky assets (j = 1, . . . , J )

and one risk-free asset at time t. At the end of the period (i.e., t+ 1), the

agent sells all his assets to the representative agent of his next generation

and derives utility from consumption.

3

Following Chalmers and Kadlec (1998), this definition of the amortized spread is imple-mented in our empirical examination.4The overlapping-generations economy has been used to motivate trades in the economyof Acharya and Pederson (2005). According to the authors, the overlapping-generationseconomy is a valid assumption to capture the large turnover observed in markets such asthe Canadian market studied herein.

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The return from holding the risk-free asset is the market observed risk-

free rate denoted byRf. The amortized spread for the risk-free asset is zero.5

The fundamental (gross) return from holding the jth asset is given by:

Rj =E[Rj ] + j, (5)

where E[Rj ] is the assets expected gross return to be determined in equi-

librium and j N(0, 2j ) is the random shock to the fundamental return

process that follows a normal distribution.

In addition, the dollar spread process (Sj sdj vj) for the jth asset is

given by:

Sj =Sj(E[Lj ] + j), (6)

where

Sj sdj vj is the currently observed total dollar spread for asset j,

Lj Sj/Sj is asset js spread ratio that measures the proportional

increase or decrease in dollar spread for the next unit period,

E[Lj ] is the assets expected spread ratio to be specified by a liquidity

factor model, and

j N(0, 2j ) is the random shock to the spread ratio process that

follows a normal distribution.

For any given asset j, we assume that the fundamental return process is

uncorrelated with the spread ratio process, such that cov(Rj ,Lj) = 0. This

assumption is adopted in the literature under a variety of economic settings.

In Amihud and Mendelson (1986), investors trade for liquidity rather than

informational reasons. Another source of systematic liquidity may be due to

a revolutionary market-making technology that systematically facilitates thetrading process. For example, Barber and Odean (2002) report that, other

things being equal, the on-line market-making technology induces more trad-

ing activities. A market-wide shock induced by liquidity and technological

reasons is likely not to be correlated with the return-generating process that

drives corporate cash flows. Furthermore, Amihud (2002) explicitly makes

this assumption in his investigation of the return-illiquidity relation.6

5

This assumption reflects the common observation that the treasury market is much moreliquid than the stock market (Amihud, 2002).6The assumption that the return process and the spread process are uncorrelated is neces-sary to derive the separate roles of the level and sensitivity effects of liquidity in our model.However, other studies indicate that the two processes may be correlated. For example,Baker and Stein (2004) find that liquidity can be used as a sentiment indicator given

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Given the above specifications, the portfolio problem of the representa-

tive agent is to consider both the fundamental value and trading costs of

firms. At each unit of trading interval, the agent optimally allocates assets bytaking both dimensions into his portfolio decision. If the amortized spread is

an important consideration for the agent to choose his optimal portfolio, he

will rationally require higher compensations for holding stocks with higher

levels of amortized spread (the level effect), as well as for stocks with higher

sensitivity to market-wide liquidity shocks (the sensitivity effect). These two

components of compensation are reflected in the equilibrium gross return of

a given asset.

Let W be the representative agents random wealth at the end of theunit period. W is determined by the dollar return on the risk-free asset,

plus the netdollar return on the risky portfolio. Specifically,

W =

W

j

Wj

Rf+

j

Wj Rj, (7)

where

W is the total wealth (dollar) of the representative agent at the begin-

ning of the period,

Wj is the dollar amount invested in the jth asset, and

Rj =Rj S

aj is given by Eq. (4).

The representative agents objective is to maximize the expected utility

derived from his end-of-period wealth, i.e.,

Max{Wj} E[U(W)]. (8)

The solution of the agents portfolio problem within the static market equi-

librium framework is provided in the appendix. The derivations of the solu-

tion are in the spirit of Fama (1976, Ch. 8) and Merton (1987). Our main

results are presented as follows:

E[Rj ] =Rz+wj (E[

Rm] Rz) + (E[Saj] E[S

am]) , (9)

the existence of a class of irrational investors that boost liquidity and can dominate themarket. To the extent that the assumption does not hold, the empirical results presentedin this paper on the separate roles of the two liquidity effects should be interpreted withcaution.

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where

Rz Rf+ E[Sam] is interpreted as the market equilibrium zero-beta

rate; (9.1)

wj vvj +

ssj is the weighted average of the fundamental and

spread betas; (9.2)

vj cov[Rj , Rm]/var[Rm] is the fundamental beta; (9.3)

sj cov[Saj ,

Sam]/var[Sam] is the spread beta; (9.4)

v var[Rm]/(var[Rm] + var[Sam]) is the proportional weight of the

fundamental beta; (9.5)s var[Sam]/(var[Rm] + var[S

am]) is the proportional weight of the

spread beta; (9.6)

E[Saj ] and E[Sam] are the expected amortized spreads of asset j and the

market portfolio, respectively; and

Rf is the market observed risk-free rate.

Given the above definitions, the following proposition explains an assets

expected (gross) return.

Proposition 1. An assets equilibrium (gross) return is composed of the

following three components:

(i) A zero-beta rate that captures both the risk-free rate and the market-

average expected amortized spread;

(ii) A market risk premium that incorporates compensations for both funda-

mental and spread risks, with the sensitivity given by the fundamentaland spread betas, and the proportion of the premiums given by the rel-

ative variance between these two types of risks; and

(iii) An asset-specific term that is measured by the deviation of an assets

expected amortized spread from the market-average expected amortized

spread.

Since Rz collects all the market-related terms, it is termed the equi-

librium zero-beta rate. The first component of Rz is the market-observedrisk-free rate Rf. The second component captures the expectation of the

market-average amortized spread. Hence, this endogeneous component is a

zero-beta premium that compensates for the expected adverse investment

environment that reduces the overall wealth due to higher trading costs.

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When the representative agent expects an overall deteriorating liquidity con-

dition as measured by a widening amortized spread, this investor responds

by increasing the zero-beta rate such that all the assets in the economy earnhigher illiquidity premia.

The second part of Proposition 1 shows that the cross-sectional model

preserves the risk-return relation as prescribed by the zero-beta version of

the CAPM (Black, 1972). One distinctive feature of our model is that the

beta is a weighted average of the fundamental beta and the spread beta

(as defined in Eqs. 9.3 and 9.4, respectively), with the weights given by

the proportional variance of the fundamental return (as defined in Eq. 9.5)

and the proportional variance of the spread ratio (as defined in Eq. 9.6)for the market portfolio. Whether the fundamental beta or spread beta has

more weight in the determination of the weighted average beta is an issue

to be examined empirically. If we combine the weights with the market risk

premium, the combined termsv(E[Rm]Rz) ands(E[Rm]Rz) then rep-

resent the compensation for fundamental risk and spread risk, respectively.

Hence, we can empirically assess the relative importance of their respec-

tive premiums by examining their respective proportional variances (i.e., v

versus s

).With regard to the third part of the proposition, Eq. (9) indicates that

the asset-specific term is determined cross-sectionally by the deviation of

the assets expected amortized spread from that of the market portfolio.

A firm earns a higher (lower) spread premium if its amortized spread is

expected to be higher (lower) than the market-average amortized spread. A

firm with the market-average expected amortized spread requires no addi-

tional spread premium. Thus, the asset-specific term of Eq. (9) formalizes the

intuition of Sharpe (1984) for an extended CAPM where assets with higher(lower) liquidity earn lower (higher) expected returns. The implications of

the expected amortized spread are formally examined in the following sec-

tion of the paper.

4. A Factor Model of the Amortized Spread

To reflect the evidence of systematic liquidity reported in the literature (dis-

cussed earlier), we formally impose a factor structure on the expected amor-tized spread term E[Saj ]. With no loss of generality, consider the following

one-factor model on the spread ratio process given by Eq. (6):

Lj Sj/Sj =E[Lj ] +lj

Y +j, (10)

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where

Y denotes a market-wide liquidity factor that follows a normal distri-

bution Y N(0, 2),

j is the idiosyncratic risk of the spread ratio with E[j |Y ] = 0 and

cov(j ,j) = 0 when j =j, and

lj cov(Lj ,Y)/var(Y ) is interpreted as the liquidity beta that mea-

sures the sensitivity of assetj s spread ratio Lj to the common liquidity

factor Y.

To establish the expected spread ratio E[Lj ], consider forming a zero-

cost portfolio of assets with the same level of market risk but with differentspread ratios Lj (j = 1, . . . , J ). If the zero-cost portfolio has zero sensitivity

to the common liquidity factor Y and is diversified of idiosyncratic risk

j, then the expected spread ratio for that portfolio must be zero, which

implies:

E[Lj ] =0+lj1, (11)

where

0 is the zero-beta illiquidity premium and1 is the risk premium for the common liquidity factor.

Then, from Eqs. (10) and (11), we have

E[Sj ] =Sj(0+lj1) (12)

E[Sm] =Sm(0+lm1), (13)

where lm is the sensitivity of the spread ratio of the market portfolio

to the common liquidity factor. In the special case where the commonliquidity factor happens to be the spread ratio of the market portfolio,

lm = 1.

Plugging Eqs. (12) and (13) into Eq. (9) and using the substitutions

Saj Sj/Vj and Sam Sm/Vm yields the following result:

E[Rj ] =Rz+wj (E[Rm] Rz) + (S

aj S

am)0+ (S

aj

lj S

am

lm)1, (14)

where

Rz Rf+ (0+lm1)Sam. (15)

Unlike the concave relation between expected returns and proportional

spreads as developed in Amihud and Mendelson (1986), Eq. (14) suggests

that the expected return of an asset is linearly related to its amortized

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spread. The last two terms of Eq. (14) explicitly indicate the co-existence of

two effects of the amortized spread on expected returns. Specifically, the 0

term measures the level effect that views the observed amortized spread Sajas an asset characteristic. The reason is that the level effect is free of both

fundamental risk and liquidity risk by the construction of0in Eq. (11), and

the level effect measures the zero-beta illiquidity premium. The loadings on

0 are determined by the deviation of an assets observed amortized spread

from the market-average amortized spread. Assets with above- (below-) aver-

age amortized spreads require higher (lower) equilibrium returns. Further-

more, the amortized spread can be viewed as containing anincrementalrisk

component as captured by the1 term in Eq. (14). Observe that the loadingon1 is jointly determined by the amortized spread level S

aj and the spread

beta lj . At a given level of amortized spread (i.e., fixing Saj ), the illiquid-

ity risk premium 1 is solely attributable to the cross-sectional variation in

spread betas. Thus, assets with higher sensitivity to market-wide liquidity

shocks require risk premia that are beyond the premia for the given level of

illiquidity.

To summarize, a re-examination of Eqs. (14) and (15) reveals that the

amortized spread may affect an assets equilibrium return through four chan-nels as given by the following proposition:

Proposition 2.

(i) Both the level(0)and sensitivity(1)effects of the overall market influ-

ence the equilibrium zero-beta rate, which, in turn, affects the required

returns of all the assets in the economy

(ii) The spread beta(s

j ) of a given asset exerts its pricing role through anembedded weighted component of the total beta (wj ), where its weight

determines the relative importance of the spread premium in the market

risk premium

(iii) The level effect affects the assets expected return through the zero-beta

illiquidity premium(0)

(iv) At a given level of the amortized spread(i.e., at a fixedSaj ), the sensi-

tivity effect plays an incremental pricing role through the illiquidity risk

premium(1).

In the next section of this paper, we empirically examine the impor-

tance of the above four channels using a large cross section of Canadian

stock market data. For Channel 1, we report the market-average amortized

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spread across our entire sample period (26 years) and two equal-length sub-

periods. We present evidence that the amortized spread plays an increasingly

important pricing role over time, both in determining the level and volatil-ity of market-average trading costs and as a component of the equilibrium

zero-beta rate. For Channel 2, we examine the proportional variance ofv

relative tos in order to test whether the fundamental betavj or the spread

beta sj is the dominant component in the weighted average beta wj . Since

we find that Channel 2 is not very important, our primary focus is on cross-

sectional tests of Channels 3 and 4. If both the level and sensitivity effects

are important, we expect their respective premia 0 and 1 to be signifi-

cantly positive. Furthermore, we examine the estimates and significance of0 and 1 in two sub-samples and test the relative importance of0 and 1across the two equal-length sub-periods.

5. Data Description

In the empirical part of our research, we define the unit of trading interval as

one month. Therefore, we examine monthly data on common stocks listed in

the CFMRC database from April 1977 to December 2002, for a total of 309time series observations.7 Compiled by the University of Western Ontario

and the TSX, the CFMRC for Canadian stocks is analogous to the CRSP

for US stocks. The primary advantage of using the CFMRC is that the

dataset is, relatively speaking, free of the well-known data-snooping biases

(Lo and MacKinlay, 1990). While the spread-return relation in the US stock

markets has been under investigation for at least the past two decades,

relatively little testing of such an important relation has been conducted

for the well-developed Canadian stock market. Therefore, findings from thisrelatively fresh dataset regarding the spread-return relation are less likely

to be spurious.

The monthly database of the CFMRC records monthly prices, returns,

betas, shares outstanding, share volume, and dividends for stocks listed on

the Toronto Stock Exchange (TSX) since January 1950. The daily database

contains the transaction prices, closing bid and closing ask quotes of listed

stocks for each trading day starting from March 1, 1977. From the daily

database, we extract the month-end closing bid and ask quotes for eachstock that passes through a screening process that controls for data errors,

7Chalmers and Kadlec (1998) examine the amortized spread for US stocks for a shortertime period, i.e., 19831992.

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adjusts for thin trading, and removes stale and invalid quotes. Furthermore,

firms with market values below CAD $5 million or firms with a market

price per share less than CAD $2 are deleted from further analysis. We thenmatch the month-end bid and ask quotes from the daily database with the

monthly shares outstanding and share volume for each selected stock. Our

data collection process results in a total of 309 cross sections from April

1977 to December 2002, with the monthly number of records ranging from

331 to 769 for each cross-section.

For each stockj at montht, its fundamental return is computed from the

closing bid-ask midpoints of consecutive months, i.e., Rjt = (pmjt p

mjt1+

djt)/pmjt1, wherepmjt is the bid-ask midpoint anddjt is the cash or cash equiv-alent dividend paid during month t for stockj. Due to the well-documented

serial correlation in the time series of transaction prices (e.g., Roll, 1984),

we do not take the database-provided returns that are transaction price

based. For the purpose of this study, it is important to isolate the effect

of fundamental returns from that of amortized spreads. Since transaction

prices may be contaminated by the spurious correlation induced by bid-

ask bounce, these two effects are intermingled when traded price returns

are used. Because the midpoints of bid-ask spreads are free from bid-askbounce, our calculated fundamental returns based on bid-ask midpoints are

free from the spurious correlation caused by bid-ask bounce.

For each stock j at month t, we calculate the effective spread as sdjt

2 |pjt pmjt |, where pjt and p

mjt are respectively the transaction price and

bid-ask midpoint.8 The total spread of trading stock j at month t is then

calculated as the product of the effective spread (sdjt) and the number of

shares traded (vjt), i.e., Sjt sdjt vjt .

9 Intuitively, Sjt roughly measures

the total amount of transaction fees paid to making the market for firm jduring the montht. Accordingly, the proportional change in the spread cost

is computed as Ljt (Sjt Sjt1)/Sjt1, which measures the proportional

change of total trading costs from month to month. Given theLjt series, we

estimate the liquidity beta of asset j or lj by regressing Ljt on Lmt, where

Lmt is the proportional change in spread costs of the market portfolio.

8

This is consistent with Chalmers and Kadlec (1998). They use the effective spread insteadof quoted bid-ask spread to calculate the amortized spread. The superscript ofsdjt indicatesthat the effective spread is in dollar terms.9As a robustness check, we also calculate the effective spread on a daily basis then sumup the values over the month to obtain the monthly effective spread. Doing such does nothave a material impact on our empirical results.

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For each stock j at month t, the market value is computed as Vjt

pmjt njt, wherepmjt is the bid-ask midpoint and njt is the number of shares

outstanding. Having defined Sjt and Vjt, the amortized spread is the ratioof the two, i.e., Sajt Sjt/Vjt. Thus, in our empirical implementation of

Eq. (14), we useSaj to proxy the level of the amortized spread, and Saj

lj to

proxy the sensitivity of the amortized spread.

At each month t, the above measures for the market portfolio are taken

as the equally weighted cross-sectional means, following the approach of

Amihud (2002), Chalmers and Kadlec (1998), and Chordia, Roll and Sub-

rahmanyam (2000). Similarly, the above measures for any constructed port-

folios are computed by taking the cross sectional means at each month t.For the risk-free rate, we convert the 90-day Canadian Treasury bill series

into a 30-day series for the same period.

6. Empirical Procedures and Test Results

6.1. The amortized spread of the market portfolio

Table 1 reports the time series statistics for the market portfolio that is con-structed by taking the cross-sectional averages of the investigated variables.

Our primary focus is on the amortized spread Sam and its relations with

the other variables. Consistent with the findings of Chalmers and Kadlec

(1998), the magnitude of the market-average amortized spread is small.

From Table 1, the monthly average amortized spread for our whole sample

is 0.0651%, annualized to 0.78%, of equity value. This percentage is higher

than but comparable to the annualized amortized spread of 0.5% for the US

stock market for the time period (i.e., 19831992) examined by Chalmersand Kadlec (1998).

We also divide our sample time period into two equal halves (1977:04

1989:12 and 1990:012002:12), and compute the same time series statistics

for each half. The market amortized spread exhibits in its own scale a signif-

icantly higher level and volatility in the second sub-period. From Table 1, we

observe that the average annualized amortized spread increases from 0.62%

(i.e., 0.0516% 12) in the first sub-period to 0.94% (i.e., 0.0782% 12) in

the second sub-period, and that its standard deviation increases by morethan 50% (i.e., from 2.19% to 3.36%). The increase in the level and volatil-

ity of the amortized spread is intuitively more evident from an examination

of Fig. 1, where the time series of the cross-sectionally averaged amortized

spread is plotted.

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Table 1. Time series statistics for the market portfolio.

This table reports the time series statistics (mean, median, standard deviation, 5% per-

centile, 95% percentile, proportional variance) for the investigated variables of the marketportfolio. The market portfolio is constructed by taking the equal-weighted average of allthe eligible stocks for each month. Panels AC report the statistics for the whole sam-ple period (1977:042002:12), first sub-period (1977:041989:12), and second sub-period(1990:012002:12), respectively. Sam is the amortized spread of the market portfolio, con-structed by taking the cross-sectional mean of the amortized spreads of the individualstocks. A stocks amortized spread is estimated as the product of the effective spread andshare volume divided by the market capitalization. Rm is the fundamental (raw) returnof the market portfolio, constructed by taking the cross-sectional mean of the raw returnsof the individual stocks. Rf is the risk-free rate, converted from 90-day Canadian T-billrates. ProSpd is the proportional effective spread of the market portfolio, constructed

by taking the cross-sectional mean of the proportional effective spreads of the individualstocks. Turnover is the turnover rate of the market portfolio, constructed by taking thecross-sectional mean of the turnovers of the individual stocks. The proportional variancefor Rm is given by

v var[Rm]/(var[Rm] + var[Sam]); and the proportional variance for

Sam is given by s var[Sam]/(var[Rm] + var[S

am]).

Variable Mean Median Std. Dev. 5% Pct. 95% Pct. Prop. Var.

Panel A: Whole time period: 1977:042002:12Sam (100) 6.51% 5.74% 3.14% 3.57% 11.92% 99.9%

Rf 0.69% 0.69% 0.32% 0.21% 1.23%ProSpd 2.38% 2.28% 0.46% 1.86% 3.33%Turnover 3.53% 3.07% 1.76% 1.63% 6.62%

Panel B: First sub-period: 1977:041989:12Sam (100) 5.16% 4.72% 2.19% 3.19% 8.82% 99.9%Rf 0.91% 0.86% 0.25% 0.56% 1.41%ProSpd 2.43% 2.27% 0.56% 1.86% 3.55%Turnover 2.52% 2.43% 0.83% 1.40% 3.87%

Panel C: Second sub-period: 1990:012002:12Sam (100) 7.82% 7.20% 3.36% 3.92% 13.92% 99.9%Rf 0.48% 0.41% 0.22% 0.16% 1.04%ProSpd 2.32% 2.29% 0.34% 1.90% 2.89%Turnover 4.51% 4.33% 1.86% 2.08% 7.32%

Since the amortized spread is jointly determined by the proportional

effective spread (as a measure for unit transaction costs) and turnover rate(as a proxy for trading activities), we investigate whether the increase in the

amortized spread over time is due to a higher turnover rate or to a widening

of the proportional spread. The result is plotted in Fig. 2, where the pro-

portional effective spread (the dashed line) remains fairly steady across the

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0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

7704

7807

7910

8101

8204

8309

8412

8603

8706

8809

8912

9103

9206

9309

9412

9603

9706

9809

9912

0103

0206

This figure plots the amortized spread of the market portfolio at a monthly frequency overthe period from 1977:04 to 2002:12. The amortized spread of the market portfolio (Samt)is computed by taking the cross-sectional mean of the amortized spreads for all eligiblestocks (Sait) for each month. A stocks amortized spread is estimated as the product of theeffective spread and share volume divided by its market capitalization.

Fig. 1. Monthly amortized spread of the market portfolio (1977:042002:12).

whole period, whereas the turnover rate (the solid line) experiences a dra-

matic increase in both its level and variability during the second sub-period.

Quantitatively, we find from Table 1 that the monthly proportional spread

drops, on average, by 11 basis point (from 2.43% to 2.32%) and the stan-

dard deviation decreases by 22 basis points (from 0.56% to 0.34%) during

the second sub-period. These figures indicate a significant decrease in boththe level and volatility of the unit transaction costs during the second sub-

period. On the other hand, the average turnover rate increases to 4.51% in

the second sub-period from 2.52% in the first sub-period, and the standard

deviation more than doubles (from 0.83% to 1.86%). Our findings suggest

that it is the increase in trading activities rather than in unit transaction

costs that have caused the intertemporal increase in the market-average

amortized spread.

In addition, our model (i.e., Eq. 9) predicts that the equilibrium zero-beta rate Rz is composed of the risk-free rate Rf and the market expected

amortized spread E[Sam]. We now investigate the relative importance of the

realized amortized spread of the market portfolio compared to the observed

risk-free rate. We calculate the ratio of these two components over time

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0%

2%

4%

6%

8%

10%

12%

7704

7805

7906

8007

8108

8209

8312

8501

8602

8703

8804

8905

9006

9107

9208

9309

9410

9511

9612

9801

9902

0003

0104

0205

This figure plots the proportional effective spread and turnover rate of the market portfolioat a monthly frequency over the period from 1977:04 to 2002:12. The solid line plots theturnover rate of the market portfolio, which is computed by taking the cross-sectionalmean of the turnover rates for all eligible stocks for each month. The dashed line plotsthe proportional effective spread of the market portfolio, which is computed by taking thecross-sectional mean of the proportional effective spreads for all eligible stocks for eachmonth.

Fig. 2. Monthly effective spread and turnover rate of the market portfolio (1977:042002:12).

(i.e., Samt/Rft). To do so, we convert the 90-day T-bill rate to proxy for

Rft and obtain Samt by taking the cross-sectional mean of the amortized

spread for all eligible stocks at month t. The time series behavior of the

ratio is plotted in Fig. 3. We observe a generally increasing trend for the

ratio over time, which suggests that the market-average amortized spreadplays an increasingly important role in the equilibrium zero-beta rate during

the later period of our sample.

The preliminary evidence of our time series analysis suggests that the

market-average amortized spread seems to exert an increasingly important

role in expected returns, both in determining the level and volatility of

market-average trading costs and in influencing the equilibrium zero-beta

rate. Furthermore, we find that it is the increase in trading activities rather

than a widening in the proportional spread that causes the increase in theamortized spread over time.

Table 1 also reports a test of the asset-pricing role of the spread beta.

According to the second channel as described in Proposition 2, we test

the relative significance between the fundamental and the spread betas by

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0%

10%

20%

30%

40%

50%

60%

70%

7704

7807

7910

8101

8204

8309

8412

8603

8706

8809

8912

9103

9206

9309

9412

9603

9706

9809

9912

0103

0206

This figure plots the ratio (in percentage) of the market-average amortized spread to themarket observed risk-free rate at a monthly frequency over the period from 1977:04 to2002:12. The market-average amortized spread (Samt) is computed by taking the cross-sectional mean of the amortized spreads (Sait) of all the eligible stocks at a monthlyfrequency. A stocks amortized spread is estimated as the product of the effective spreadand share volume divided by its market capitalization. The monthly risk-free rate (Rft)is converted from the Canadian 90-day T-bill rate.

Fig. 3. Monthly ratio of the market average amortized spread to the market observedrisk-free rate.

examining their proportional weights (v ands). The proportional variance

column in Table 1 indicates that the variance of the fundamental returns

(whose weight is more than 99.9%) completely dominates the variance of

the market-average amortized spread (whose weight is less than 0.1%). This

evidence infers that the fundamental beta vj is the predominant component

of the weighted average beta wj , and that the impact of the spread beta

sj is negligible. Because the time series of amortized spreads is of a much

smaller scale (as measured by its proportional weight) relative to the fun-damental return time series, the systematic component of the amortized

spread is unlikely to have a significant role in determining the market risk

premium.

6.2. Cross-sectional tests of the amortized spread

The relation between stock returns, fundamental betas, and amortized

spreads is tested over the period 1977:04 to 2002:12. The main testablehypothesis is that a level effect and sensitivity effect exists in the amor-

tized spread. Following the two-step portfolio approach developed by Fama

and MacBeth (1973), we divide our data into portfolio formation period,

portfolio estimation period, and cross-sectional test period.

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Fama and French (1992) find that the beta effect is highly correlated

with the size effect. Elfakhani, Lockwood and Zaher (1998) find that beta

does not explain returns for Canadian stocks after controlling for firm size.Thus, in the portfolio formation stage, we first sort stocks into five size

portfolios in December of each year starting from 1974. Within each of the

five size portfolios, we further classify firms into one of five sub-portfolios

on the basis of the amortized spreads of the firms.10 These 5 5 size-spread

portfolios are constructed at the end of each year T, where T = 1977 to

2001. For each of the twelve months in year T+ 1, we assign each firm

to one of the 25 portfolios formed at the end of the previous year. We then

compute equal-weighted fundamental returnsRit, market capitalizationsVit,amortized spreadsSait, and proportional changes in spreadsLitfor portfolioi

at montht. Finally, we subtract the proxy of the zero-beta rate from portfolio

is raw return to obtain the excess return of the portfolio. Specifically, Reit

RitRzt, whereRzt Rft+ Samt,Rft is obtained from the Canadian T-bill

series, and Samt is the cross-sectional mean of the amortized spread for all

the eligible stocks at month t.

In the portfolio estimation stage, the fundamental betavi (defined in Eq.

9.3) and the liquidity betali (defined in Eq. 9.4) are estimated for portfolioi. We useRmtto proxy for the market portfolio fundamental return, and Lmtto proxy for the realization of the common liquidity factor. In doing so, we

assume that the spread ratio of the market portfolio is the single underlying

liquidity factor. Following the approach of Fama and French (1992), we esti-

mate each portfolios fundamental beta and liquidity beta using 60-month

time series regressions. Specifically, in December of each year T, T= 1981

to 2001, we run the following time series regression for each portfolio i:

Reit=+viRemt+it, t= 1, . . . , 60 (16.1)

Lit=+liLmt+it, t= 1, . . . , 60, (16.2)

where

Reit RitRzt andRemt RmtRzt are the excess return for portfolio

iand the market portfolio, respectively, and

Lit and Lmt are the cross-sectional mean of Ljt (Sjt Sjt1)/Sjt1for portfolio i and the market portfolio, respectively.

10The double-sort procedure is due to Amihud and Mendelson (1986) in their test ofthe spread-return relation. Since firm size and amortized spread tend to be negativelycorrelated, controlling for firm size ensures that our findings on the amortized spread arenot due to the well-documented size effect.

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The estimates vi andli are used for portfolio i in year T + 1 up to

year 2001. At the end of the estimation stage, we have a test dataset of 25

cross-sections (i= 1 to 25) by 252 time series (t= 1982:01 to 2002:12).In the test stage, the spread-return relation is examined using pooled

cross-section time series GLS regressions, a methodology adopted by Ami-

hud and Mendelson (1986) and Brennan and Subrahmanyam (1996). The

explanatory variables of our analysis include the estimated fundamental

betas vit1, the level (Sait1 S

amt1), and the sensitivity (S

ait1

lit1

Samt1lmt1) of amortized spreads. The aforementioned regressors are pre-

scribed by Eq. (14). Two additional regressors are added as control variables

based on the extant literature. Fama and French (1992) argue that firmsize might proxy for loadings on other fundamental risk factors. Elfakhani,

Lockwood and Zaher (1998) find a significant size effect for Canadian

stocks. Therefore, the natural logarithm of portfolio is market capitaliza-

tion ln(Vit1) is included in our tests. Since Chalmers and Kadlec (1998)

find a strong contemporaneous association between the amortized spread

and return volatility, the volatility of excess returns is included in our

cross-sectional equation. Similar to the estimation procedure of fundamen-

tal betas, portfolio is return volatility denoted by it1 is estimated fromthe portfolios excess return time series of the past 60 months. The pur-

pose of including the size and volatility factors is to gauge whether these

well-documented effects subsume any of the level or sensitivity effects of the

amortized spread.11

In Table 2, we report the time-series means of the five independent

variables that include beta (vi), size (ln(Vi)), volatility (i), level (Sai)

and sensitivity (Sai li) of amortized spreads across the 5 5 size-spread

sorted portfolios. The results indicate that our sorting procedure pro-duces the desired portfolio characteristics, i.e., sizes (amortized spreads)

are monotonically increasing across each size (amortized spread) quintile.

More importantly, the results demonstrate that there exists a large spread

for both the level and sensitivity of the amortized spreads across the 25

11We do not include the book-to-market ratio as a control variable for the following

reasons: (1) Chalmers and Kadlec (1998) find that the book-to-market ratio is highlyinsignificant in their tests of the amortized spread; (2) Elfakhani, Lockwood and Zaher(1998) do not find a highly significant book-to-market effect for Canadian stocks in theirwhole sample; (3) Kothari, Shanken, and Sloan (1995) document survivorship biases inthe book-to-market ratio; and (4) Matching book-to-market ratios from COMPUSTAT toCFMRC for Canadian stocks drastically reduces our sample size.

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Table 2. Time series means of the independent variables across the 25 portfolios.

This table reports the time-series means of beta (vi), size (ln(Vi)), volatility (i), level

(Sai) and sensitivity (S

ai

li) of amortized spreads across the 5 5 size-spread sorted port-

folios. A stocks amortized spread is estimated as the product of the effective spread andshare volume divided by market capitalization. The 25 portfolios are constructed followingthe approach of Fama and French (1992). At December of each year starting from 1977,eligible stocks are sorted first into five size portfolios. Then within each size portfolio,stocks are sorted into five sub-portfolios based on the rank of the amortized spreads ofstocks. For each of the twelve months in the following year, each stock is assigned to oneof the 25 portfolios and market capitalizations Vit, amortized spreads S

ait, and propor-

tional change in spreadLit for portfolio i are computed for each montht. Each portfoliosfundamental beta vit1, liquidity beta

lit1 and standard deviation of excess returnsit1 are then estimated using 60-month time series regressions. The beta, size, returnvolatility, and the level and sensitivity of amortized spreads are the time-series averagesfor each portfolio for the sample period from 1982:01 to 2002:12.

Amortized Spread

Size Low 2 3 4 High

Market BetaSmall 0.75 0.95 1.01 0.99 1.16

2 0.86 0.90 1.19 1.30 1.163 0.76 1.06 1.12 1.10 1.28

4 0.81 0.99 1.03 1.16 1.33Big 0.77 0.95 0.92 0.95 1.16

Size (logarithm of market capitalization)Small 17.31 17.30 17.36 17.34 17.31

2 18.09 18.06 18.07 18.09 18.093 18.77 18.84 18.77 18.80 18.804 19.75 19.71 19.75 19.66 19.68

Big 21.58 21.74 21.54 21.34 21.03

Volatility (standard deviation of returns)Small 0.068 0.065 0.068 0.072 0.069

2 0.058 0.059 0.070 0.076 0.0733 0.052 0.063 0.065 0.065 0.0664 0.051 0.058 0.056 0.066 0.069

Big 0.049 0.056 0.054 0.058 0.055

Level (%)Small 0.048 0.057 0.104 0.170 0.254

2 0.043 0.063 0.080 0.095 0.1083 0.038 0.052 0.067 0.074 0.0824 0.029 0.041 0.050 0.062 0.073

Big 0.011 0.021 0.029 0.037 0.044

Sensitivity (%)Small 0.079 0.025 0.038 0.066 0.082

2 0.081 0.035 0.041 0.089 0.0193 0.036 0.040 0.089 0.158 0.2024 0.050 0.021 0.030 0.026 0.036

Big 0.003 0.010 0.012 0.018 0.013

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portfolios (i.e., the level ranges from 0.011 to 0.254, and the sensitiv-

ity ranges from 0.003 to 0.202). This is important, as the cross-sectional

dispersions of the level and sensitivity are necessary to provide explana-tory power of expected returns beyond that explained by the other inde-

pendent variables. The cross-sectional dispersion of the amortized spreads

implies that the liquidity clientele hypothesis may not hold in our sam-

ple of Canadian stocks. This is consistent with other studies (Brennan

and Subrahmanyam, 1996; Chalmers and Kadlec, 1998) that do not find

the clientele effect for US stocks.12 Furthermore, the correlation coefficient

between the level and sensitivity effects is moderate ( = 0.32). This sug-

gests that the two effects may play different roles in explaining expectedreturns.

Prior to testing the cross-sectional relation between expected returns and

amortized spreads, we run pooled GLS regressions without including the

amortized spread as explanatory variables. Doing such allows us to examine

the well-documented relation between returns, betas, sizes, and volatilities

in the Canadian stock markets. Specifically, we run the following regression

and report the results in Table 3.

Reit =0+1vit1+2ln(Vit1) +3it1+it. (17)

In Table 3, panels AC report results for the whole test period (1982:01

to 2002:12), first half-period (1982:01 to 1992:06), and second half-period

(1992:07 to 2002:12), respectively. With regard to the size effect, we confirm

the finding of Elfakhani, Lockwood and Zaher (1998) that firm size is neg-

atively related to stock returns in the Canadian market. The size effect is

significant and robust for both the whole period and the two sub-periods.

We also find a positive relation between volatilities and returns. This associ-ation is significant for the first sub-period (3 = 0.347 and t-value = 2.39),

but becomes insignificant for the second sub-period (3 = 0.127 and

t-value = 0.87). As for the risk-return relation, we find that betas are nega-

tively but insignificantly related to returns during the first sub-period. This

finding is consistent with that of Elfakhani, Lockwood and Zaher (1998) for

12If the liquidity clientele hypothesis holds, then the cross-sectional dispersion of amortized

spreads should be small since high (low) spread portfolios tend to have low (high) turnoverrates. Brennan and Subrahmanyam (1996) do not find a concave relation between thefixed costs of transacting and investment horizons for NYSE stocks. Chalmers and Kadlec(1998) find a positive association between spreads and turnovers, which is in contrast tothe prediction of the clientele effect. These findings further justify our examination of thereturn-spread relations without presuming the clientele hypothesis of liquidity.

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Table 3. The cross-sectional relation between returns and betas, sizes, and volatilities.

This table reports the test results for the cross-sectional relations between portfolio returns

and portfolio betas, sizes, and volatilities. Panels AC are the results for the wholetest period (1982:012002:12), first sub-period (1982:011992:06), and second sub-period(1992:072002:12), respectively. The test portfolios are constructed following the approachof Fama and French (1992). At December of each year starting from 1977, eligible stocksare sorted first into five size portfolios. Then within each size portfolio, stocks are sortedinto five sub-portfolios based on the rank of the amortized spreads of stocks. For eachof the 12 months in the following year, each stock is assigned to one of the 25 portfoliosand equal-weighted excess returns Reit (in excess of the proxy of the zero-beta rate Rzt)and market capitalizations Vit for portfolio i are then computed for each month t. Wethen estimate each portfolios fundamental beta vit1 and standard deviation of excessreturns it1 using 60-month time series regressions. Asset pricing tests are conducted

using the pooled cross-sectional and time series regressions for the whole test period andtwo sub-periods. Specifically, the coefficients (0, 1, 2, 3) are estimated from the fol-lowing model:

Reit =0+ 1vit1+ 2ln(Vit1) + 3it1+ it (i= 1 25; t= 1982:012002:12).

T-statistics are reported in parentheses. , and indicates significance based on atwo-tailed test at the 1%, 5% and 10% levels, respectively.

Run Intercept (0) Beta (1) Log of Size (2x100) Volatility (3)

Panel A: Whole test period (t= 1982:012002:12)

(1) 0.013 0.009(2.15) (2.56)

(2) 0.020 0.011 0.199(2.73) (3.42) (4.11)

(3) 0.018 0.019 0.156 0.191(2.56) (3.69) (3.04) (1.82)

Panel B: First sub-period (t= 1982:011992:06)(1) 0.002 0.002

(0.23) (0.45)(2) 0.015 0.006 0.211

(1.49) (1.39) (3.86)

(2) 0.016 0.009 0.173 0.347(1.72) (1.70) (2.80) (2.39)

Panel C: Second sub-period (t= 1992:072002:12)(1) 0.015 0.015

(2.36) (3.44)

(2) 0.023 0.015 0.196(3.18) (3.96) (3.16)

(3) 0.021 0.020 0.169 0.127(3.10) (2.93) (2.48) (0.87)

a somewhat similar time period (i.e., their data is from 1975:07 to 1992:12).

However, we find that the negative beta-return relation becomes highly sig-

nificant for the second sub-period and the whole period, even after size and

volatility are controlled for. The negative beta effect is counterintuitive, but

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is neither a new finding nor unique to our study. Elfakhani, Lockwood and

Zaher (1998) document negative risk premia for beta (although insignifi-

cant) in all test samples of Canadian stocks. Chalmers and Kadlec (1998),Datar, Naik and Radcliffe (1998), and Eleswarapu and Reinganum (1993)

report a significantly negative beta effect for the US stock market for tests

of the return-liquidity relation.

To further investigate the possible causes of this negative beta effect,

we employ the conditional beta approach proposed by Pettengill, Sundaram

and Mathur (1995). The authors argue that when using realizedreturns to

test the CAPM, beta should be positively related to returns when the excess

market returns are positive, but a negative relation should be expected whenthe excess market returns are negative. Therefore, we use conditional betas

in our subsequent analysis. The most comprehensive specification of our

model is given below:

Reit=0++1

+

it1+1(1 )

it1+2ln(Vit1)

+ 3it1+0(Sait1)

+1(Sait1

lit1)

+it

(i= 1 to 25 and t= 1982:01 to 2002:12). (18)

where

is a dummy variable:= 1 ifRmtRzt 0 and= 0 ifRmtRzt < 0,

(Sait 1) Sait 1S

amt 1 is the cross-sectional mean-adjusted loadings

on the level effect (0), and

(Sait 1lit 1)

Sait 1lit 1S

aim 1

lmt 1 is the cross-sectional mean-

adjusted loadings on the sensitivity effect (1).

The primary focus of the test is to examine the magnitude and signifi-

cance of0 and 1 after controlling for the conditional betas (+1 and

1),

size (2), and volatility (3) effects. If investors do command a higher pre-

mium to hold stocks with higher levelsof the amortized spread, a positive

and significant 0 is expected. Furthermore, if investors also require incre-

mental compensations for stocks with higher sensitivities to market-wide

liquidity shocks, then 1 should be positive and significant.

Pooled cross-section time series GLS regressions are used to estimate

Eq. (18). Results for the whole test period are reported in Table 4. The

estimates and significance of the level effect (0) and the sensitivity effect

(1) are reported in the last two columns of Table 4.

Results from panels A and B of Table 4 provide preliminary evidence

on the level and sensitivity effects of the amortized spread, when the two

effects are tested separately. Observe from Panel A that when tested alone,

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Table4.Thecross-section

alrelationbetweenreturnsandamortizedspreadsforth

ewholetestperiod.

Thistablerepo

rtsthetestresultsforthelevelandsensitivityeffectsfortheamortizedspreadforthe

testperiodfrom1982:01to2

002:12.

PanelsAandBreporttheresultsforthelevelandsensitivityeffectsalone,respectively.

PanelCreportstheresultsforthejointtestof

thelevelandsensitivityeffects.

Astocksamortizedspreadisestimated

astheproductoftheeffectivespreadandsharevolume

divided

bymarketcapitalization.

Thetestportfoliosareconstructedfollowing

theapproachofFamaandF

rench(1992).

AtDecember

ofeach

yearstartingfrom1977,eligiblestocksaresortedfirstinto5sizeportfolios.

Thenwithineachsizeportfolio,stocksaresortedinto5sub-

portfoliosbasedontherankoftheamortizedspreadsofstocks.Foreach

ofthe12monthsinthefollowingyear,eachstockisassignedto

oneofthe25portfoliosandequal-weightedexcessreturnsReit(inexcessoftheproxyofthezero-betarateRzt),marketcapitalizationsVit,

amortizedspreadsSait,andproportionalchangeinspreadLitforportfolioiarecomputedforeachmontht.Eachportfoliosfundamental

betavit

1,

liquiditybetal it

1

andstandarddeviationofexcessreturns

it1

arethenestimatedusing60-monthtimeseriesregr

essions.

Assetpricingt

estsareconductedusingthe

pooledcross-sectionalandtimeseriesregressionsforthewholetestperiod.

Specifically,the

coefficients(0,

+1,1,2,3,0,1)areestimatedfromthefollowin

gmodel:

Reit=0+

+1+it

1+1(1

)it1+2ln(Vit1)+

3it1+0(Sait

1)+1(Sait

1

l it

1)+it

(i=125;t=1982:012002:1

2),

whereisadummyvariable:=1ifRm

t

Rzt

0and=0ifR

mt

Rzt

The Cross Section of Expected Returns and Amortized Spreads

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