Research Article Frank Torchio and Sunita Surana* Effect ...€¦ · Research Article Frank Torchio and Sunita Surana* Effect of Liquidity on Size Premium and its Implications for

Research Article

Frank Torchio and Sunita Surana*

Effect of Liquidity on Size Premium and itsImplications for Financial Valuations

Abstract: Courts are often required to determine a stock’s “fair value,” which bydefinition excludes any reduction to value because of a lack of liquidity. Themethod of computing fair value most frequently used by practitioners is thediscounted cash flow analysis, which requires calculating the cost of equity.Over the last decade, many practitioners have included a size premium in thecomputation of the cost of equity based on the finding that historic returns forfirms with lower market capitalization are greater than the returns implied bythe standard capital asset pricing model. Our findings show that a substantialfraction of the measurement of size premiums reflects a lack of liquidity, whichdisproportionately affects smaller sized companies. Because a reduction to valuefrom illiquidity should not be reflected in the measurement of fair value, thisfinding has implications for assessments of fair value that employ the commonlyused size premiums. Specifically, our findings suggest that valuations of smallcapitalization stocks that reflect these size premiums will cause the fair value tobe underestimated because of the effect of lower liquidity. The smaller the size,the greater is the underestimation in value.

Keywords: fair value, business valuation, liquidity, size premium

DOI 10.1515/jbvela-2013-0022

1 Size premiums and fair value

Discounted cash flow (DCF) analysis is one of the key valuation methods taughtby academics and used by practitioners. A critical parameter of a DCF analysis is

*Corresponding author: Sunita Surana, Forensic Economics Inc., Rochester, NY, USA, E-mail:[email protected] Torchio, Forensic Economics Inc., Rochester, NY, USA; Simon Business School, Rochester, NY,USA, E-mail: [email protected]

J. Bus. Val. Econ. Loss Anal. 2014; 9(1): 55–85

Authenticated | [email protected] author's copyDownload Date | 8/7/14 11:29 AM

the weighted average cost of capital (used to discount expected cash flows)which comprises in part the cost of equity. The cost of equity is generallycomputed using a version of the capital asset pricing model (CAPM), for whichthe equation is:

Cost of Equity ¼ Risk-Free Rateþ Beta� Equity Risk Premiumð Þ ½1�Many valuation practitioners generally consider it appropriate to include in

the calculation of the cost of equity a premium based on the market capitaliza-tion of equity or size of the firm being valued. Empirical studies, most notablypublished in the Ibbotson SBBI Valuation Yearbooks (Ibbotson SBBI orIbbotson), have shown that the CAPM alone does not fully account for thehigher historical returns earned by smaller companies.1 These studies showthat historical returns for small firms are systematically greater than the returnsimplied by their betas (beta-adjusted returns) from the standard CAPM in eq. [1].

That is, the greater risk of smaller sized firms is not fully accounted for inthe standard beta calculations for these firms. To account for this size-relatedeffect, one of the variations of the CAPM equation includes a size premium,defined as:

Cost of Equity ¼ Risk-Free Rateþ Beta� Equity Risk Premiumð Þ þ Size Premium

½2�The higher the size premium, the higher is the cost of equity, and conse-

quently the lower is the DCF value, all else the same. Ibbotson SBBI hasmeasured historic size premiums by constructing portfolios of traded stocks bysize. The size premiums are computed as the average returns for each sizeportfolio less the average of the returns predicted by CAPM in eq. [1] for thestocks in each portfolio. Ibbotson has constructed both size-quartile portfoliosand size-decile portfolios. In 2001, Ibbotson refined its size analysis by dividingdecile 10, the smallest stock decile, into 10a and 10b (Ibbotson SBBI ValuationYearbook 2001, 122–3). In 2010, Ibbotson further divided the tenth decile intofour size categories: 10w, 10x, 10y, and 10z (Ibbotson SBBI Valuation Yearbook2010, 91). As can be seen from Table 1, the size premiums increase as thecompany size decreases.

Figure 1 plots the size premiums for the ten deciles (diamonds shapes) andalso shows the size premiums for categories 10w, 10x, 10y, and 10z (circleshapes) contained in the Ibbotson SBBI Valuation Yearbook (2011). As can beseen in Figure 1, the increase in the size premium is approximately linear for

1 See Ibbotson SBBI Valuation Yearbook (2011, 87–90). Banz (1981) first presented evidence thatsmaller firms earned higher risk-adjusted returns.

56 F. Torchio and S. Surana


decile 1 (–0.38%) through decile 9 (2.94%). But for the smallest size decile(decile 10), the premium increases substantially to 6.36%, which is above thelinear trend line based on the size premiums for deciles 1 through 9. Withindecile 10, the increase in size premiums is even more dramatic and ranges from3.99% for size category 10w to 12.06% for the smallest size category, 10z.2

The substantial size premiums measured for very small companies has beencriticized and its appropriateness continues to be debated among practitionersand researchers.3

Table 1: Size premiums for size-quartile and size-decile portfolios

Quartile groups Sizepremium

Decilegroups

Market capitalization of largestcompany in decile ($ millions)

Size premium

Large cap (1 and 2) n/a 1 314,623 −0.38%2 15,080 0.81%

Mid cap (3–5) 1.20% 3 6,794 1.01%4 3,711 1.20%5 2,509 1.81%

Low cap (6–8) 1.98% 6 1,776 1.82%7 1,212 1.88%8 772 2.65%

Micro-cap (9–10) 4.07% 9 478 2.94%

10w 236 3.99%10x 179 4.96%10y 143 9.15%10z 86 12.06%

Notes: Source of data is Ibbotson SBBI Valuation Yearbook (2011). Since the data in ouranalysis cover the time period 1926–2010, for comparability purposes, throughout this paperwe report the statistics from the 2011 Yearbook that uses data also from 1926 to 2010. Marketcapitalization in each decile is as of September 30, 2010.

2 Using selective time periods, some studies have documented higher abnormal returns forlarge firms than for small firms, i.e. effectively a large-firm effect (see, e.g. Al-Rjoub, Varela, andHassan (2005)). However, demonstrating the cyclicality of size premiums and the varyingrelative magnitudes of the size premiums for firms of different sizes over rolling 5-year periods,Ibbotson has cautioned against the use of selective periods for the computation of sizepremiums. The argument for using a longer time series of data is that it results in more stableestimates of the size premiums as unique events are not weighted heavily. See Ibbotson SBBIValuation Yearbook (2011, 95–96).3 For example, some argue that portfolios of smallest size companies contain a disproportion-ate number of financially distressed firms or contain the so-called “fallen angels.” See IbbotsonSBBI Valuation Yearbook (2011, 94–102) for a comprehensive review of the criticisms andIbbotson’s responses.

Effect of Liquidity on Size Premium 57


One argument raised by critics concerns the effects from the relative lack ofliquidity that disproportionately affects stocks of smaller sized companies. Theargument is that the lack of liquidity for many small-sized firms causes transac-tions costs of trading a share of stock to be greater, which in turn results ingreater observed historic returns to properly compensate investors for holdingthese stocks relative to more liquid stocks.

Ibbotson does not disagree that lower liquidity will contribute to the mag-nitude of the calculated size premiums. Ibbotson correctly responds that it isirrelevant whether or not the computed size premium also reflects the lowerliquidity in smaller sized companies when computing a stock’s fair marketvalue; the return to equity used to compute fair market value should includethe additional return required to compensate investors for holding less liquidstocks.4 This is because these transactions costs are real, can be substantial, and

–2%

0%

2%

4%

6%

8%

10%

12%

14%

0 1 2 3 4 5 6 7 8 9 10

Size

pre

miu

m

Decile

10w10x

10y

10z

Figure 1: Size premiums for size-decile portfolios and sub-groups of the tenth decileSource: Ibbotson SBBI Valuation Yearbook (2011).

4 According to valuation literature, “fair market value” is generally defined as: “…the amountat which property would change hands between a willing seller and a willing buyer whenneither is acting under compulsion and when both have reasonable knowledge of the relevantfacts.” See Pratt (2008, 41–42).



will affect the prices paid for a share of stock.5 For example, if an investor werecontemplating a purchase of 10 shares of a privately held company, the investorwould certainly take into account in the purchase price she pays the transactioncosts she would have to incur in order to sell those shares at a later time. So, tothe extent that the computed size premium includes the effects of less liquidstocks is largely irrelevant because it is generally appropriate that a fair marketvaluation reflect any illiquidity effect on the expected return.

In certain circumstances, however, the purpose of valuation is not to assessthe fair market value, but rather to analyze and compute “fair value.” Forexample, the appropriate measure of value in an appraisal proceeding for amerger is not fair market value but rather “fair value.” The key differencesbetween fair market value and fair value are that fair value requires that therebe: (1) no discount – either direct or implied – for the lack of liquidity; and (2) nodiscount for minority interest.6

Therefore, there is general agreement that the fair value of even a comple-tely illiquid, privately held stock in an appraisal proceeding should not reflectany discount to the valuation that would obtain for the same stock that traded ina completely liquid market. But, if a key valuation parameter – the cost ofequity – includes a premium for low liquidity, then the valuation obtained willnecessarily reflect a discount for illiquidity.

Is the value obtained from such an analysis the “fair value” of the stock ifthat value reflects an implicit discount for illiquidity? Because fair value is alegal concept, the answer to this question is left to the courts and legal scholars.This paper, however, provides an economic context to assist in answering thisquestion by quantifying the effect of liquidity reflected in the size-decile pre-miums that are currently used by many practitioners. Because the Ibbotson SBBIYearbooks are by far the most commonly cited and used source for size pre-miums, we use the methods suggested by Ibbotson for computing size premiumsand for measuring liquidity.

Our findings show that a substantial fraction of the measurement of sizepremiums reflects a lack of liquidity, which disproportionately affects smallersized companies. This finding has implications for computing fair value.Specifically, our findings suggest that valuations of small capitalization stocksthat reflect the commonly used size premiums will cause the fair value to be

5 Under the standard of fair market value, the focus must be on the specific property (owner-ship interest) being valued, basically “as is,” including control and marketability characteris-tics. Therefore, minority interests in closely held corporations or partnerships are valued toreflect lack of control and lack of marketability characteristics. See Pratt (2009, 10).6 See Laro and Pratt (2011, 12–13).



underestimated because of the effect of lower liquidity. The smaller the size, thegreater is the underestimation in value.

The rest of the paper is organized as follows. The next section discusses therelationship between liquidity and asset pricing. The third section describes thedata. Liquidity premiums, without accounting for size, are calculated in thefourth section. The fifth section discusses the replication of size premiumscontained in the Ibbotson SBBI Yearbook (2011). The sixth section shows theamount of liquidity premium subsumed in the calculation of size premiums forsize-decile portfolios and for the sub-groups of the tenth decile. Finally, the lastsection provides concluding remarks.

2 Liquidity and asset pricing

The marketability or liquidity of an asset refers to the degree to which it can beconverted to cash quickly without incurring large transaction costs or priceconcessions. Financial theory reasons that liquidity affects asset prices becauseinvestors price securities according to their returns net of trading costs, such astransactions costs and expected price concessions, and consequently investorsrequire a greater return for higher expected costs of achieving liquidity, all elsethe same. Thus, given two assets with the same expected cash flows but withdifferent liquidity, investors will pay less (demand a higher return) to hold themore illiquid asset.

Over the last 15 years, liquidity has been the subject of considerable researchin the financial literature. Many of these studies are discussed by Amihud,Mendelson, and Pedersen (2005) who review the literature concerning the effectsof liquidity on asset prices.

The consensus in the liquidity literature is that theory and empirical evi-dence strongly support three findings. First, investors require returns that com-pensate for the level of illiquidity of an investment.7 Thus, financial economistsexpect that asset and security prices will differ systematically depending on themarketability characteristics of the securities, all else equal. For example,restricted stock should be priced at a discount from the unrestricted stock’straded “market” price on a liquid exchange. Indeed, many empirical studies ofrestricted stock, of the relationship between stock returns and bid-ask spreads,of a company’s block transactions, and of private sales of a company’s stock

7 See, for example, Amihud and Mendelson (1986), Datar, Naik, and Radcliffe (1998), andAmihud, Mendelson, and Pedersen (2005).



prior to the company’s initial public offering have confirmed that high transac-tion costs and other restrictions generally cause securities to be priced atsignificant discounts from the market prices of comparable (often otherwiseidentical) liquid securities.

Table 2 summarizes the illiquidity discounts measured in several restrictedstock studies. These studies report median illiquidity discounts for restrictedstock of 9–45% and means of 13–42% based on hundreds of transactions.Because the restricted stock studies provide a direct measure of the costs ofilliquidity, many experts and courts have chosen to rely on these studies as anempirical guide in selecting illiquidity discounts to apply to the computed valuefrom a DCF analysis to arrive at a fair market value measure.

Second, stocks in publicly traded equity markets can have substantiallydifferent degrees of liquidity; liquidity is not a binary variable. That is, onecannot simply divide stocks into two categories of liquidity based solely onwhether or not the stock is publicly traded. While stocks that are not publiclytraded are generally characterized as illiquid, even among publicly tradedstocks, there can be important and substantial differences in the degrees ofliquidity.8

Third, illiquidity is correlated with size. That is, across all publicly tradedstocks, more small stocks tend to have lower liquidity than do large stocks.Practitioners and academics are in general agreement that the negative relation-ship between returns and liquidity is stronger for smaller stocks.9 In a recentstudy, Ibbotson et al. (2013) empirically studied the effect on returns fromdiffering levels of liquidity across all size-quartile portfolios of publicly tradedstocks between 1972 and 2011. Ibbotson et al. find that within each size-quartileportfolio, low liquidity portfolios generally earned higher returns than the highliquidity portfolios and that this effect is strongest among micro-cap stocks anddeclines from micro- to small- to mid- to large-cap stocks.

Thus, a key finding implied by Ibbotson et al. (2013) is that for smaller sizedcompanies the historic returns are substantially different between low liquidityand high liquidity stocks. If the CAPM returns are not similarly different acrossthe various liquidity groups, then it implies that a substantial portion of themeasure of what is generally referred to as size premium subsumes the premiumthat compensates investors for holding low liquidity stocks.

In the following sections, we investigate whether the size premiums purelyreflect the effect of size or if differences in liquidity can explain some of theobserved premiums. If the size premiums subsume a substantial component that

8 See, for example, Amihud and Mendelson (1986).9 See, for example, Amihud (2002) and Ibbotson et al. (2013).



Table2:

Studies

ofrestricted

stockmarke

tabilitydiscou

nts

Study

Timepe

riod

Sam

ple

size

Med

ian

Average

Price

discou

ntGroup

averag

ePrice

discou

ntGroup

averag

e

Pre-1990

stud

ies

SEC

overallaverag

e(197

1)1

Jan1966–Jun

e1969

398

n/a

30.9%

25.8%

31.6%

Gelman

(197

2)1968–1

970

89

33.0%

33.0%

Trou

t(197

7)2

1968–1

972

60

n/a

33.5%

Moron

ey(197

3)3

n/a

146

33.0%

35.6%

Mah

er(197

6)4

1969–

1973

n/a

n/a

35.4%

Pittockan

dStryker

(198

3)5

Oct

1978

–Jun

e1982

2845

.0%

n/a

Willam

ette

Man

agem

ent

Associates6

Jan1981–

May

1984

3331.2%

n/a

Wruck

(198

9)7

July

1979

–Dec

1985

3712.2%

13.5%

Hertzel

andSmith(199

3)8

Jan1980–M

ay1987

18n/a

42.0%

Silb

er(199

1)1981–

1988

69

n/a

33.8%

Long

-horizon

stud

ies

Hallan

dPo

lacek(199

4)1979

–Apr

1992

100þ

n/a

22.1%

23.0%

24.1%

Oliver

andMeyers(2000

)9Jan1980–D

ec1996

5325

.0%

27.1%

Roba

kan

dHall(2001)

1980–A

pr1997

230

20.1%

22.3%

Hall(2003

)10

1980–1

997

238

21.3%

n/a

1990

–199

7stud

ies

Hallan

dPo

lacek(199

4)May

1991–A

pr1992

17n/a

18.7%

21.0%

22.1%

Bajaj

etal.(2001)11

Jan1990

–Dec

1995

5126

.5%

28.1%

John

son(199

9)1991–1

995

72n/a

20.2%

Finn

erty

(2003)

Jan1991–F

eb1997

101

15.5%

20.1%

Aschw

ald(2000)

Jan1996

–Apr

1997

2314.0%

21.0%



1997

–2007Studies

(1-yea

rho

ldingpe

riod

)

Aschw

ald(2000)

Jan1997

–Dec

1998

159.0%

14.6%

13.0%

16.7%

Colum

biaFina

ncialAdv

isors,

Inc.12

June

1997

–Jun

e20

00

3211.7%

13.5%

Colum

biaFina

ncialAdv

isors,

Inc.12

Jan1999

–Jun

e20

00

2411.7%

13.7%

Hall(2003

)1997

–2000

182

25.9%

n/a

FMVOpinion

s13

1997

–2003

187

n/a

22.5%

FMVOpinion

s14

1997

–2007

311

n/a

20.6%

Post-2007stud

ies

(6-m

onth

holding

period

)

FMVOpinion

s15

2008

43n/a

n/a

12.6%

12.6%

Notes:

1 Discoun

tsInvolved

inPu

rcha

sesof

Com

mon

Stock

(196

6–1

969),Institutiona

lInvestor

Study

Repo

rtof

the

Securities

and

Exch

ange

Com

mission

,H.R.Doc.No.

64,

Part

5,92

ndCon

gress,

1stSession

(197

1,24

44–5

6),

citedin

Pratt,

Reilly,

andSch

weihs

(2000),

pp.39

6–39

8,

404;

2Cited

inPratt,Re

illy,

andSch

weihs

(2000

,39

8–9

9,40

4);3Cited

inPratt,Re

illy,

andSch

weihs

(2000

,39

9,40

4);4Cited

inPratt,Re

illy,

and

Sch

weihs

(2000,40

0,40

4);5 Cited

inPratt,Re

illy,

andSch

weihs

(2000,40

0,40

4);6Unp

ublis

hedstud

y,citedin

Pratt,Re

illy,

andSch

weihs

(2000,

400,

404);7Discoun

tforprivateplacem

entof

unregistered

shares.W

ruck

also

repo

rtsan

averag

eprem

ium

of4.1%

andmed

iandiscou

ntof

1.8%

for

36privateplacem

ents

ofregistered

shares;8Discoun

tforprivateplacem

entof

restricted

shares.Hertzel

andSmithalso

repo

rtan

averag

ediscou

ntof

15.6%

for88privateplacem

ents

ofno

n-restricted

shares;9Cited

inPratt,Re

illy,

andSch

weihs

(2000

,40

1–04).Med

ianis

approx

imate;

10Th

e1980–1

997timepe

riod

isassu

med

tomatch

theprevious

stud

ywithad

dition

altran

sactions

upda

ted;

11Unreg

isteredSha

res.

Bajaj

etal.also

repo

rtdiscou

ntsfor37

privateplacem

ents

ofregistered

shares

of14%

(average

)an

d10%

(med

ian);12Study

obtained

from

Colum

biaFina

ncialAdv

isors,

Inc;

13Unp

ublis

hedstud

y,citedin

Laro

andPratt(2005,

289);14Unp

ublis

hedstud

y,citedin

Laro

andPratt(2011,28

7);15Unp

ublis

hedstud

y,citedin

Laro

andPratt(2011,28

7).



compensates investors for holding less liquid stocks, then inclusion of such asize premium in the cost of equity will necessarily result in an illiquiditydiscount to the stock’s fair value. There is no economic distinction between aDCF in which the cost of capital is increased by a liquidity premium and a DCFthat uses a cost of capital with no liquidity premium, but to which an illiquiditydiscount is then applied.10

3 Data description

We use monthly common stock data from 1926 through 2010 compiled by theCenter for Research in Security Prices (CRSP) at the University of Chicago BoothSchool of Business. All common stocks traded on the New York Stock Exchange(NYSE), American Stock Exchange (AMEX),11 and NASDAQ stock markets areused. From 1926 through 2010, over 3 million monthly-level observations areavailable.

Monthly returns on the Standard & Poor’s (S&P) 500 index and 30-day UStreasury bill total return from 1926 through 2010 are also obtained from CRSP.Finally, long-term mean income return component of 20-year government bondsand long-term equity risk premiums are obtained from Ibbotson SBBI ValuationYearbook (2011).

For the analysis of adjustment to NASDAQ volume to account for thepotential over counting of traded volume, we use daily common stock datafrom 1990 through 2012 from CRSP. For this time period, over 30 million daily-level observations are available.

3.1 Adjustment to NASDAQ volume

Volumes in quote-driven dealer markets like NASDAQ were historically higherthan order-driven markets like NYSE, because public buyers and sellers tradedthough the intermediation of dealers on NASDAQ, leading to an over count oftrades among public traders. On NYSE, public buyers and sellers mostly traded

10 See Ibbotson SBBI Valuation Yearbook (2011, 101) for an explanation and a simple examplethat demonstrates the equivalence of using a liquidity premium in the cost of capital versusapplying an illiquidity discount to the value obtained from a DCF analysis.11 In October 2008, AMEX was acquired by NYSE Euronext.



among themselves. The differing market structures caused higher volumesfor NASDAQ stocks, all else equal. However, regulatory interventions by theSEC (e.g. the 1997 SEC mandated order handling rules at NASDAQ12) as well asthe rapid growth of electronic trading mechanisms have caused the tradingpatterns on different platforms to converge.

Ibbotson et al. (2013) divide volumes of NASDAQ stocks by an adjustmentfactor based on the analysis in Anderson and Dyl (2005). Based on stocksswitching from NASDAQ to NYSE from 1997 through 2002, Anderson and Dylfind that the mean daily volume declined an average of 24.7% and the mediandecrease was 37.9%. Further, based on the finding in Anderson and Dyl (2007)that the relative over-reporting of NASDAQ stocks has not lessened during 2003–2005 relative to the 1990–1996 time period, Ibbotson et al. (2013) apply theadjustment factor throughout their study period of 1972–2011. While Andersonand Dyl (2007) found no evidence that the over-reporting has lessened forNASDAQ stocks, using data from 1993 to 2010, Harris (2011) reports that overtime volumes between NYSE and NASDAQ stocks have become more and moresimilar leading to the homogenization of US equity markets.

In light of the differences in the findings in Harris (2011) and Anderson andDyl (2007) and the vast changes in the stock trading landscape, we analyzed thevolume of common stocks switching from NASDAQ to NYSE starting in1990 (since the regulatory changes likely to impact the over counting of tradedvolume and the rapid growth of electronic trading mechanisms occurred after1990). We studied volume on 60 trading days before and after the switch(with day 1 being the day of the switch). For each company switching fromNASDAQ to NYSE, average volume during 60 trading days before the switch isdivided by average volume during 60 trading days after the switch. The ratioof the two volumes is shown in Figure 2. The horizontal lines represent themedian ratios for consecutive 5-year periods beginning in 1990. As is evidentfrom the figure and consistent with Harris (2011), the ratio has been decreasingover time.

We use the median ratios over 5-year intervals to adjust NASDAQ volume.Specifically, the ratio of 2.07 is used to divide NASDAQ volume prior to 1994;1.75 is used to divide NASDAQ volume between 1995 and 1999; 1.38 is used todivide NASDAQ volume between 2000 and 2004; 1.32 is used to divide NASDAQvolume between 2005 and 2009; and 1.21 is used to divide NASDAQ volume after2009.

12 See, for example, McInish, Van Ness, and Van Ness (1998).



4 Liquidity premium

We measure liquidity using average monthly share turnover in each quarter.13

Share turnover is calculated as the volume of shares traded each month dividedby the number of shares outstanding at each month-end. This measure ofliquidity is similar to the one used in Ibbotson et al. (2013), except that whereasIbbotson et al. (2013) create annual portfolios of stocks and hence measureliquidity annually, we create portfolios of stocks that are rebalanced quarterlyin keeping with the size premium methodology in the Ibbotson SBBI Yearbooks,

0

1

2

3

4

1985 1990 1995 2000 2005 2010 2015

2.07

1.75

1.38 1.321.21

Median

Figure 2: Average volume on NASDAQ before the switch divided by average volume on NYSEafter the switchNote: In order to focus on presenting the medians, the figure shows ratios between 0 and 4.A few outliers (greater than 4) are not shown in the figure but are used in the computation ofthe median ratios.

13 Turnover rates and bid-ask spreads are typically used as measures of liquidity. See Amihud,Mendelson, and Pedersen (2005). Another measure of liquidity is the price impact such as theratio of absolute stock return to its dollar volume, averaged over some period (Amihud 2002).



and hence, we update the liquidity measure each quarter. The steps we take tocreate the liquidity-based portfolios of stocks are described below.

We first rank companies with primary listings on the NYSE based on theirliquidity at the end of each quarter. As mentioned above, liquidity at the end ofeach quarter is measured as average monthly share turnover in that quarter.

Second, based on these rankings, the companies are divided into twoequally populated groups based on the median liquidity at the end of eachquarter. Group “H” contains companies with liquidity greater than or equal tothe median liquidity, or the high liquidity companies. Group “L” containscompanies with liquidity lower than the median liquidity, or the low liquiditycompanies. The ranking of the stocks thus yields a liquidity cutoff demarcatingthe “H” and “L” groups at the end of each quarter. These liquidity cutoffsobtained from NYSE stocks are then used to assign common stocks listed onAMEX and NASDAQ to one of the two liquidity groups based on the end ofquarter liquidity measure for the AMEX and NASDAQ stocks.

Third, the liquidity groupings are rebalanced quarterly. Each month, everycompany is assigned a liquidity group based on its liquidity categorization in theprevious quarter. For example, the liquidity category that a company falls underas of the quarter ending in March is used to assign its liquidity group for April,May, and June of that year. Thus, each stock remains in the same liquidity groupfor each of the 3 months that follow its assignment to a liquidity group using theliquidity measure from the end of the previous quarter. This methodology issimilar to the methodology used by CRSP and the Ibbotson SBBI Yearbooks forthe creation of the size-decile portfolios (discussed further below).

Fourth, we compute monthly portfolio returns as the average returns of thestocks in each liquidity group from 1926 to 2010. Annual portfolio returns arecomputed by compounding the monthly returns.14

Fifth, tocompute the risk-adjustedportfolioreturns,wecompute thebeta foreachliquidityportfoliousingthesingle-factorCAPMmodel.FollowingtheSBBIYearbooks,we use the following single-factor regression equation to estimate each portfolio’ssystematic risk (commonly referred to in the literature as the portfolio’s beta).

ðrl � rf Þ ¼ αl þ βlðrm � rf Þ þ "l ½3�where,rl represents monthly return on portfolio l;rf represents 30-day US treasury bill total return; andrm represents monthly return on the market, measured by the S&P 500 index.

14 Annual return is computed when monthly return data are available for each of the 12months.



The slope of the regression, βl, in eq. [3] measures the portfolio’s sensitivityto variations in the market return, or its exposure to systematic risk.

Using the beta for each liquidity portfolio, βl, a CAPM portfolio return (inexcess of the riskless rate) is computed as the product of the estimated beta forthat portfolio and the equity risk premium (the difference between the meantotal return of the S&P 500 index and the mean income return component of 20-year government bonds for the time period 1926–2010).

Finally, for each liquidity portfolio of stocks, we calculate the differencebetween the portfolio’s actual average annual return in excess of the risk-freerate (measured by the mean income return component of 20-year governmentbonds for the time period 1926–2010) and the CAPM return also in excess of therisk-free rate. We refer to this difference as the liquidity premium. Table 3presents the liquidity premiums for the “H” and “L” liquidity groups. We findthat while the high liquidity stocks have a liquidity premium of less than 1%, theliquidity premium for the low liquidity stocks is over 7%.

Using the same methodology described above, we also categorize the datainto liquidity quartiles. The results are presented in Table 4. Again, we find thatliquidity premium increases as the stocks get less liquid. These findings suggestthat the CAPM underestimation of returns is a function of stock liquidity.

The analysis so far does not distinguish liquidity premiums for stocksstratified by their size. We next study how liquidity impacts the premiums foreach of the size-decile portfolios and for the sub-groups of the tenth size decile.

Table 3: Long-term returns in excess of estimated CAPM returns for high and low liquiditycategories

Liquiditygroup

Beta Actualarithmetic

mean return

Actual return inexcess of

riskless rate

CAPM return inexcess of riskless

rate

Liquidity premium(return in excess of

CAPM return)

[1] [2] [3] [4] ¼ [3] – 5.17% [5] ¼ β*(11.88% – 5.17%)

[6] ¼ [4]–[5]

H 1.36 14.95% 9.78% 9.09% 0.69%L 1.05 19.50% 14.33% 7.02% 7.31%

Notes: Historical riskless rate is measured by the 85-year (1926 through 2010) arithmetic meanincome return component of the 20-year government bond (5.17%). Long-horizon equity riskpremium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus thearithmetic mean income return component of the 20-year government bond (5.17%) from 1926to 2010.



But before we assess the impact of liquidity for these stock portfolios, we createsize-decile portfolios in a manner similar to CRSP.

5 Replication of Ibbotson size premiums

Using the monthly stock data, we next create size-based portfolios of stocks in amanner similar to CRSP and used in the Ibbotson SBBI Yearbooks. This involvesthe following steps (that are similar to the steps discussed above to create theliquidity-based portfolios).

Using monthly data from 1926 to 2010, we first rank the companies withprimary listings on the NYSE based on their market capitalizations at the end ofeach quarter. Market capitalization is calculated as the product of the closingprice on the last trading date of the quarter and the shares outstanding.15

Second, based on these rankings, the companies are divided into equally popu-lated deciles (decile 1 contains the largest companies, and decile 10 the smallest).Thus, therankingoftheNYSEstocksyieldsizecutoffs foreachdecile,wherethecutoffsare thehighest and lowestmarket capitalizationswithin each sizedecile. Thesedecile

Table 4: Long-term returns in excess of estimated CAPM returns for high, mid-high, mid-low,and low liquidity categories

Liquiditygroup

Beta Actualarithmetic

mean return

Actual returnin excess ofriskless rate

CAPM returnin excess ofriskless rate

Liquidity premium(return in excess of

CAPM return)

[1] [2] [3] [4]¼ [3] – 5.17% [5]¼β*(11.88% – 5.17%)

[6]¼ [4]–[5]

H 1.40 13.03% 7.86% 9.42% −1.56%MH 1.30 17.20% 12.03% 8.73% 3.31%ML 1.17 18.90% 13.73% 7.87% 5.86%L 0.95 19.61% 14.44% 6.35% 8.10%

Notes: Historical riskless rate is measured by the arithmetic mean income return component ofthe 20-year government bond (5.17%). Long-horizon equity risk premium is estimated by thearithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income returncomponent of the 20-year government bond (5.17%). MH denotes mid-high, and ML denotesmid-low.

15 In the calculation of the liquidity portfolios discussed above we calculated the averagemonthly share turnover over a quarter instead of just using the last month in the quarter inorder to preserve observations in the analysis that otherwise would be lost due to missingvolume data in the last months of the quarters.



cutoffs obtained from NYSE stocks are then used to assign common stocks listed onAMEX and NASDAQ to one of the size deciles based on the end of quarter marketcapitalization for the AMEX and NASDAQ stocks.

Third, size-decileportfolios are constructed for the stocksbasedon the size-decilerankings. Each month, every company is assigned a portfolio based on its decileranking in the previous quarter. For example, the decile ranking of a stock based onits market capitalization as of the quarter ending in March is used to determine thestock’s size decile for themonths of April,May, and June. Thus, each stock remains inthe same size decile for each of 3 months that follow its assignment to a size decileusing themarket capitalization from the end of the previous quarter.

Fourth, monthly portfolio returns are computed as the weighted average returnsof the stocks in each size decile, using market capitalizations based on the sharesoutstandingandclosingprice for thelast tradingdayof thepreviousmonthasweights.Annual portfolio returns are computed by compounding themonthly returns.

Fifth, to compute the risk-adjusted portfolio returns, we compute the betafor each size-decile portfolio using the single-factor CAPM model. Again, follow-ing the SBBI Yearbooks, we use the following single-factor regression equationto estimate each portfolio’s systematic risk.

ðrs � rf Þ ¼ αs þ βsðrm � rf Þ þ "s ½4�where,rs represents monthly return on portfolio s;rf represents 30-day US treasury bill total return; andrm represents monthly return on the market, measured by the S&P 500 index.

The slope of the regression, βs, in eq. [4] measures the portfolio’s sensitivityto variations in the market return, or its exposure to systematic risk.

Using the beta for each size-decile portfolio, βs, a CAPM portfolio return (inexcess of the riskless rate) is computed as the product of the estimated beta forthat portfolio and the equity risk premium.

Finally, for each size-decile portfolio of stocks, we calculate the differencebetween the portfolio’s actual average annual return in excess of the risk-freerate (measured by the mean income return component of 20-year governmentbonds for the time period 1926–2010) and the CAPM return also in excess of therisk-free rate. This difference is what is referred to as the size premium in theIbbotson SBBI publications.

Table 5 presents the size premiums by size decile. As the table shows, ourcomputed size premiums are very similar to the size premiums reported in theIbbotson SBBI 2011 Yearbook.



Table5:

Com

pariso

nof

Ibbo

tson

SBBIan

dTo

rchio–

Suran

along

-term

returnsin

excess

ofestimated

CAP

Mreturnsforsize-decile

portfolio

s

Decile

Beta

Actua

larithm

etic

mea

nreturn

Actua

lreturn

inex

cess

ofrisk

less

rate

CAPM

return

inex

cess

ofrisk

less

rate

Sizeprem

ium

(returnin

excess

ofCA

PMreturn)

[1]

[2]

[3]

[4]¼

[2]–[3]

[5]

[6]

[7]¼

[5]–[6]

[8]¼

[5]–5.17%

[9]¼

[2]*

(11.88%–5

.17%

)[10]

¼[8]–[9]

[11]

[12]

¼[10]–[11]

TSSBBI

TS–SBBI

TSSBBI

TS–SBBI

TSTS

TSSBBI

TS–SBBI

10.92

0.91

0.01

10.87%

10.92%

−0.05%

5.70

%6.15%

−0.44%

−0.38%

−0.06%

21.03

1.03

0.00

13.02%

12.92%

0.10%

7.85%

6.89%

0.96%

0.81%

0.15%

31.10

1.10

0.00

13.44%

13.56%

−0.12%

8.27%

7.40

%0.88%

1.01%

−0.13%

41.13

1.12

0.01

14.04%

13.91%

0.13%

8.87%

7.57%

1.30

%1.20

%0.10%

51.17

1.16

0.01

14.66%

14.75%

−0.09%

9.49

%7.82%

1.67%

1.81%

−0.14%

61.19

1.19

0.00

14.93%

14.95%

−0.02%

9.76

%7.97

%1.79

%1.82%

−0.03%

71.23

1.24

−0.01

15.24%

15.38%

−0.14%

10.07%

8.25%

1.82%

1.88%

−0.06%

81.30

1.30

0.00

16.20%

16.54%

−0.34%

11.03%

8.70%

2.33

%2.65%

−0.32%

91.34

1.35

−0.01

17.01%

17.16%

−0.15%

11.84%

8.99%

2.85%

2.94

%−0.09%

101.40

1.41

−0.01

21.49%

20.97%

0.52%

16.32%

9.40

%6.93%

6.36%

0.57%

Notes:Historicalrisklessrate

ismea

suredby

thearithm

etic

mea

nincomereturn

compo

nent

ofthe20

-yea

rgo

vernmen

tbo

nd(5.17%

).Long

-horizon

equity

risk

prem

ium

isestimated

bythearithm

etic

mea

ntotalreturn

oftheS&P50

0(11.88%)minus

thearithm

etic

mea

nincomereturn

compo

nent

ofthe20

-yea

rgo

vernmen

tbo

nd(5.17%

).TS

deno

tesTo

rchio–

Suran

aresu

lts.



As discussed above, Ibbotson divides the smallest size decile, decile 10, intofour size sub-groups (10w, 10x, 10y, and 10z) using the same methodology thatis used to construct the 10 size-decile portfolios. In Table 6, we replicate thatanalysis and show the computed size premiums for sub-size groups w, x, y, andz for decile 10 are quite close to the size premiums reported in the Ibbotson SBBIYearbook (2011). One reason for the differences between our estimates and thosereported in the Ibbotson SBBI Yearbook is that Ibbotson SBBI uses its internaldatabase of companies in addition to the companies reported by CRSP for thesub-groups of the tenth decile (based on email communication withMorningstar). Notwithstanding the differences, the results are consistent withthat from Ibbotson.

5.1 Missing volume data

Because the monthly stock data contain missing volume data for some months,the number of observations used to compute the liquidity premiums is less thanthe number of observations used to compute the size premiums. Table 7 com-pares the number of observations used to compute the liquidity premiums andthe number of observations used to compute the size premiums. The table alsocompares the mean annual returns and the standard deviation of annual returnsfor the two data samples.

Using the sub-set of monthly return data used to compute liquidity pre-miums, we re-compute size premiums for each size decile and for the sub-groupsof the tenth decile and compare the resulting premiums to the premiums pre-viously computed. As can be seen in Tables 8 and 9, the loss of data has deminimis effects on the results of the computed size premiums.

6 Liquidity versus size effect

As mentioned, several studies have shown that less liquid stocks earn higherreturns than more liquid stocks. Since liquidity risk is priced in stock returns, weexamine whether and to what extent stock liquidity accounts for the commonlyused size premiums. We use the methodology described above to separatelyestimate premiums (the difference between the actual returns in excess of theriskless rate and the estimated returns based on CAPM also in excess of theriskless rate) for low and high liquidity portfolios for each of the ten size deciles



Table6:Com

pariso

nof

Ibbo

tson

SBBIan

dTo

rchio–

Suran

along

-term

returnsin

excess

ofestimated

CAPM

returnsforsu

b-grou

psof

thetenthsize

decile

Size

grou

pBeta

Actual

arithm

etic

mea

nreturn

Actual

return

inex

cess

ofrisk

less

rate

CAPM

return

inex

cess

ofrisk

less

rate

Sizeprem

ium

(returnin

excess

ofCA

PMreturn)

[1]

[2]

[3]

[4]¼

[2]–[3]

[5]

[6]

[7]¼

[5]–[6]

[8]¼

[5]–5.17%

[9]¼

[2]*

(11.88%

–5.17%)

[10]

¼[8]–[9]

[11]

[12]

¼[10]–[11]

TSSBBI

TS–SBBI

TSSBBI

TS–SBBI

TSTS

TSSBBI

TS–SBBI

10w

1.40

1.39

0.01

18.67%

18.52%

0.15%

13.50%

9.36

%4.14%

3.99

%0.15%

10x

1.44

1.45

−0.01

21.42%

19.88%

1.54

%16.25%

9.69%

6.55%

4.96

%1.59

%10y

1.41

1.40

0.01

23.61%

23.72%

−0.11%

18.44%

9.43

%9.01%

9.15%

−0.14%

10z

1.34

1.34

0.00

28.37%

26.25%

2.12%

23.20%

9.00%

14.20%

12.06%

2.14%

Notes:Historicalriskless

rate

ismea

suredby

thearithm

etic

mea

nincomereturn

compo

nent

ofthe20

-yea

rgo

vernmen

tbo

nd(5.17%

).Long

-horizon

equity

risk

prem

ium

isestimated

bythearithm

eticmea

ntotalreturnof

theS&P50

0(11.88%)m

inus

thearithm

eticmea

nincomereturn

compo

nent

ofthe20

-yea

rgo

vernmen

tbo

nd(5.17%

).TS

deno

tesTo

rchio–

Suran

aresu

lts.



and also for the sub-groups of the tenth size decile. Results for the size-decileportfolios are presented in Table 10.

Clearly, higher liquidity stocks have significantly smaller sizepremiums than their less liquid counterparts within each size decile. Infact, there is virtually no size premium for the higher liquidity groups ofthe first eight size deciles. For the two smallest size deciles, the ninth and thetenth deciles, the premiums for the high liquidity group are only 2% and2.5%, respectively.

Table 7: Number of observations, mean annual returns, and standard deviation of annualreturns

Decile Total number ofobservations

Arithmetic mean of annualreturns

Standard deviation of annualreturns

Size-based Partial Size-based Partial Size-based Partial

1 125,566 125,027 10.87% 10.88% 19.47% 19.46%2 129,233 128,360 13.02% 13.06% 22.31% 22.33%3 137,118 134,914 13.44% 13.59% 23.72% 23.80%4 144,318 140,964 14.04% 14.14% 26.13% 26.17%5 156,495 150,852 14.66% 14.97% 26.83% 26.98%6 180,032 170,476 14.93% 15.20% 27.45% 27.64%7 208,066 194,967 15.24% 15.48% 29.84% 30.13%8 253,749 236,337 16.20% 16.45% 34.14% 34.37%9 372,783 344,372 17.01% 17.31% 36.48% 36.60%10 1,360,963 1,168,794 21.49% 21.67% 46.18% 46.27%

10w 131,871 121,312 18.67% 18.88% 42.67% 42.84%10x 171,610 157,854 21.42% 21.71% 46.36% 46.48%10y 263,696 240,817 23.61% 23.83% 54.63% 54.71%10z 793,786 648,811 28.37% 28.54% 53.63% 53.72%

10w H 43,843 15.41% 46.65%10w L 77,469 22.05% 50.05%10x H 50,427 20.57% 54.13%10x L 107,427 24.63% 46.46%10y H 65,780 20.34% 55.68%10y L 175,037 26.79% 59.96%10z H 131,697 21.07% 53.45%10z L 517,114 30.96% 59.49%

Notes: Liquidity categories are defined independently of the size groups. Since several obser-vations have stock price return data but do not have volume data, the liquidity analysis usesfewer observations than the purely size-based analysis. “Partial” denotes that some observa-tions are lost during the liquidity categorization.



Table8:Im

pact

ofloss

ofda

taon

size-decile

portfolio

s

Decile

Beta

Actual

arithm

etic

mea

nreturn

Actual

return

inex

cess

ofrisk

less

rate

CAPM

return

inex

cess

ofrisk

less

rate

Sizeprem

ium

(returnin

excess

ofCA

PMreturn)

[1]

[2]

[3]

[4]¼

[2]–[3]

[5]

[6]

[7]¼

[5]–[6]

[8]¼

[5]–5.17%

[9]¼

[2]*

(11.88%

–5.17%)

[10]

¼[8]–[9]

[11]

[12]

¼[10]–[11]

Partial

Size-

based

Partial–

size-bas

edPa

rtial

Size-

based

Partial–

size-bas

edPa

rtial

Partial

Partial

Size-

based

Partial–

size-bas

ed

10.92

0.92

0.00

10.88%

10.87%

0.00%

5.71%

6.14%

–0.44%

–0.44%

0.00%

21.03

1.03

0.00

13.06%

13.02%

0.04%

7.89%

6.89%

1.00%

0.96%

0.04%

31.10

1.10

0.00

13.59%

13.44%

0.15%

8.42%

7.39

%1.03%

0.88%

0.15%

41.13

1.13

0.00

14.14%

14.04%

0.10%

8.97%

7.58

%1.39

%1.30

%0.09%

51.17

1.17

0.00

14.97%

14.66%

0.31%

9.80%

7.83%

1.97

%1.67%

0.30%

61.19

1.19

0.00

15.20%

14.93%

0.27%

10.03%

7.99

%2.04%

1.79

%0.25%

71.24

1.23

0.01

15.48%

15.24%

0.24%

10.31%

8.30%

2.02%

1.82%

0.19%

81.30

1.30

0.01

16.45%

16.20%

0.25%

11.28%

8.74%

2.55%

2.33

%0.22%

91.35

1.34

0.01

17.31%

17.01%

0.30%

12.14%

9.04%

3.10%

2.85%

0.25%

101.41

1.40

0.01

21.67%

21.49%

0.18%

16.50%

9.45

%7.05%

6.93%

0.12%

Notes:See

thesu

mmarystatistics

fortheda

taus

edin

theliq

uidity

analysis

andthat

used

inthepu

rely

size-based

analysis.“P

artial”de

notesthat

someob

servations

arelost

during

theliq

uidity

catego

rization

.Historicalrisklessrate

ismea

suredby

thearithm

etic

mea

nincomereturn

compo

nent

ofthe20

-yea

rgo

vernmen

tbo

nd(5.17%

).Long

-horizon

equity

risk

prem

ium

isestimated

bythearithm

eticmea

ntotalreturnof

theS&P50

0(11.88%)

minus

thearithm

etic

mea

nincomereturn

compo

nent

ofthe20

-yea

rgo

vernmen

tbo

nd(5.17%

).



Table9:

Impa

ctof

loss

ofda

taon

sub-grou

psof

thetenthde

cile

Size

grou

pBeta

Actua

larithm

etic

mea

nreturn

Actua

lreturn

inex

cess

ofrisk

less

rate

CAPM

return

inex

cess

ofrisk

less

rate

Sizeprem

ium

(returnin

excess

ofCA

PM)

[1]

[2]

[3]

[4]¼

[2]–[3]

[5]

[6]

[7]¼

[5]–[6]

[8]¼

[5]–5.17%

[9]¼

[2]*

(11.88%–5

.17%

)[10]

¼[8]–[9]

[11]

[12]

¼[10]–[11]

Partial

Size-

based

Partial–

size-bas

edPa

rtial

Size-

based

Partial–

size-bas

edPa

rtial

Partial

Partial

Size-

based

Partial–size-

based

10w

1.40

1.40

0.01

18.88%

18.67%

0.21%

13.71%

9.41%

4.29

%4.14%

0.16%

10x

1.45

1.44

0.01

21.71%

21.42%

0.30%

16.54%

9.74

%6.81%

6.55%

0.25%

10y

1.41

1.41

0.01

23.83%

23.61%

0.21%

18.66%

9.47

%9.18

%9.01%

0.17%

10z

1.35

1.34

0.01

28.54%

28.37%

0.16%

23.37%

9.07%

14.30%

14.20%

0.10%

Notes:See

thesu

mmarystatistics

fortheda

taus

edin

theliq

uidity

analysis

andthat

used

inthepu

rely

size-based

analysis.“P

artial”de

notesthat

someob

servations

arelost

during

theliq

uidity

catego

rization

.Historicalrisklessrate

ismea

suredby

thearithm

etic

mea

nincomereturn

compo

nent

ofthe20

-yea

rgo

vernmen

tbo

nd(5.17%

).Long

-horizon

equity

risk

prem

ium

isestimated

bythearithm

etic

mea

ntotalreturnof

theS&P50

0(11.88%)

minus

thearithm

etic

mea

nincomereturn

compo

nent

ofthe20

-yea

rgo

vernmen

tbo

nd(5.17%

).



Results for each of the sub-groups of the tenth decile (groups 10w, 10x, 10y, and10z) are shown in Table 11. As before, for each of the sub-groups of the tenthdecile, premiums for the portfolios of high liquidity stocks are considerablysmaller than the premiums for their lower liquidity counterparts. Indeed, thepremium is not greater than 5% in any of the sub-groups’ high liquidity portfo-lios of stocks.

One observation from Tables 10 and 11 is that within each size group theestimated beta is greater for the high liquidity stocks than for the low liquidity

Table 10: Long-term returns in excess of estimated CAPM returns for size-decile portfolios splitinto high and low liquidity categories

Group Beta Actualarithmetic

mean return


CAPM return in excessof riskless rate

Premium (returnin excess of

CAPM return)

[1] [2] [3] [4]¼ [3] – 5.17% [5]¼β* (11.88%–5.17%) [6]¼ [4]–[5]

1 H 1.12 11.32% 6.15% 7.50% –1.35%L 0.81 10.71% 5.54% 5.41% 0.13%

2 H 1.18 12.92% 7.75% 7.90% –0.16%L 0.87 13.28% 8.11% 5.87% 2.25%

3 H 1.25 13.49% 8.32% 8.37% –0.05%L 0.93 14.28% 9.11% 6.22% 2.88%

4 H 1.31 14.04% 8.87% 8.80% 0.07%L 0.90 14.49% 9.32% 6.07% 3.25%

5 H 1.33 14.66% 9.49% 8.92% 0.57%L 0.97 15.68% 10.51% 6.50% 4.01%

6 H 1.37 14.04% 8.87% 9.21% –0.33%L 0.98 16.62% 11.45% 6.55% 4.90%

7 H 1.41 14.68% 9.51% 9.45% 0.06%L 1.06 16.61% 11.44% 7.10% 4.34%

8 H 1.46 15.19% 10.02% 9.83% 0.19%L 1.13 18.18% 13.01% 7.61% 5.40%

9 H 1.51 17.27% 12.10% 10.10% 1.99%L 1.20 18.49% 13.32% 8.08% 5.25%

10 H 1.62 18.52% 13.35% 10.90% 2.46%L 1.31 25.11% 19.94% 8.76% 11.18%

Notes: Liquidity categories are defined independently of the size groups. Historical riskless rateis measured by the arithmetic mean income return component of the 20-year government bond(5.17%). Long-horizon equity risk premium is estimated by the arithmetic mean total return ofthe S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-yeargovernment bond (5.17%).



stocks. Since higher liquidity stocks by definition trade more frequently than lessliquid stocks, their prices more quickly reflect the movements of the broadermarket. Thus, the high liquidity stocks are less prone to the underestimation ofbeta that low liquidity stocks suffer because of a lagged response to the marketmovements. Hence, the betas and therefore the estimated CAPM returns arehigher for more liquid stocks than less liquid stocks within the same sizeportfolio of stocks. Beta estimates for low liquidity stocks can be improved byaccounting for their lagged response to market movements. We discuss one suchmethodology in the Appendix.

The second observation from Tables 10 and 11 is that within each sizegroup, the average actual returns for high liquidity stocks are lower than thatfor low liquidity stocks. This is consistent with the findings in the generalliquidity literature that investors require compensation for holding low liquid-ity stocks, even within the same size group. The higher actual returns com-bined with the lower CAPM returns explain the significantly higher premiumsfor the less liquid stocks relative to their more liquid counterparts within thesame size group.

Table 11: Long-term returns in excess of estimated CAPM returns for sub-groups of the tenthsize decile split into high and low liquidity categories

Group Beta Actualarithmetic

mean return


CAPM return in excessof riskless rate


CAPM return)

[1] [2] [3] [4]¼ [3]–5.17% [5]¼β* (11.88%–5.17%) [6]¼ [4]–[5]

10w H 1.58 15.41% 10.24% 10.61% –0.37%L 1.31 22.05% 16.88% 8.80% 8.08%

10x H 1.61 20.57% 15.40% 10.84% 4.57%L 1.35 24.63% 19.46% 9.06% 10.40%

10y H 1.76 20.34% 15.17% 11.83% 3.34%L 1.31 26.79% 21.62% 8.77% 12.85%

10z H 1.84 21.07% 15.90% 12.33% 3.57%L 1.23 30.96% 25.79% 8.24% 17.55%

Notes: Liquidity categories are defined independently of the size groups. Historical riskless rateis measured by the arithmetic mean income return component of the 20-year government bond(5.17%). Long-horizon equity risk premium is estimated by the arithmetic mean total return ofthe S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-yeargovernment bond (5.17%).



6.1 Comparison of size premiums of higher liquidity stocks toIbbotson SBBI size premiums

We next compare the size premiums of higher liquidity stocks to the frequentlyused size premiums published in Ibbotson SBBI. Table 12 shows that for all sizeportfolios, the size premiums for higher liquidity stocks are substantially smallerthan the size premiums published in Ibbotson SBBI.

In summary, our research shows that, within each size portfolio, highliquidity stocks have substantially smaller size premiums than the size pre-miums computed for all stocks within the size portfolio. This finding holdsacross all size portfolios. It is a well-documented finding that liquidity is posi-tively correlated with size and that the lowest size portfolios contain a dispro-portionately greater number of low liquidity stocks. Thus, the large-sizepremiums computed by Ibbotson and others for low size portfolios are theconsequence of the disproportionately greater number of low liquidity stockspopulating the small-size portfolios.

The main inference from our findings is that the commonly used sizepremiums from Ibbotson SBBI, which are based on all (high and low liquidity)

Table 12: Comparison of size premiums of higher liquidity stocks toIbbotson SBBI size premiums

Decile Size premium

Ibbotson SBBI High liquidity

1 –0.38% –1.35%2 0.81% –0.16%3 1.01% –0.05%4 1.20% 0.07%5 1.81% 0.57%6 1.82% –0.33%7 1.88% 0.06%8 2.65% 0.19%9 2.94% 1.99%10 6.36% 2.46%10w 3.99% –0.37%10x 4.96% 4.57%10y 9.15% 3.34%10z 12.06% 3.57%

Notes: Ibbotson SBBI size premiums are from Ibbotson SBBI ValuationYearbook (2011). See tables 10 and 11 for the high liquidity sizepremiums.



stocks within the size category, overestimate size premiums for the high liquiditystocks. This finding has significant implications when assessing fair value,because fair value requires no reduction to value for lack of liquidity. Theimplicit illiquidity discount from using the commercial size premiums can besubstantial. For example, for sub-group 10z the difference between IbbotsonSBBI size premium of 12.06% and the 3.57% size premium for higher liquiditystocks in sub-group 10z is 8.5%. This 8.5% difference results in an implicitilliquidity discount of one-third for the average stock in sub-decile 10z (assum-ing a discount rate of 17.5%). Hence, the effect is non-trivial.

7 Conclusions

Courts are often required to determine a stock’s fair value, which by definitioneliminates any reduction to value because of a lack of marketability or liquidity.The method of computing fair value most frequently used by practitioners is theDCF analysis. A critical parameter of a DCF analysis is the computation of thecost of equity. Over the last decade, many practitioners have included in thecomputation of the cost of equity, a premium based on the finding that historicreturns for firms with lower market capitalizations are greater than the returnsimplied by the standard CAPM. This difference between the observed returns forsmall-sized firms and the computed CAPM returns is called a size premium.Ibbotson SBBI is the most common source of size premiums used bypractitioners.

In theory, the size premium compensates investors for the systematic risk ofholding small capitalization companies. Researchers have recognized that themeasurement of size premiums can also include the effects from the lack ofliquidity that disproportionately affects stocks of smaller sized companies. Thelack of liquidity for small-sized firms causes transactions costs of trading a shareof stock to be greater, which in turn results in a premium to properly compen-sate investors for holding these stocks relative to more liquid stocks.

Our research builds on the general research on liquidity premiums. Westratify the stocks used by Ibbotson SBBI to compute size premiums by ameasure of liquidity used by Ibbotson et al. (2013). For each size groupingused by Ibbotson SBBI, we divide the size group into high and low liquiditygroups. Our findings show that the frequently used measure of size premiumsincludes a substantial fraction that is explained by illiquid trading. For example,the size premium in the smallest size grouping by Ibbotson SBBI (sub-group 10z)



is 12.06%. When we stratify the stocks by liquidity, the size premium for the highliquidity stocks in sub-group 10z is reduced to 3.57%. Thus, the majority of themeasure of commonly used size premiums is attributable to the lack of liquidity.

This finding has implications for computing fair value which is meant toabstract from reductions due to illiquidity. Specifically, valuations of smallcapitalization stocks that reflect the Ibbotson SBBI size premium will cause thefair value to be underestimated because of the effect of illiquidity. The smallerthe size, the greater is the underestimation due to illiquidity from using theIbbotson SBBI size premiums.

Is the value obtained from such a DCF analysis that uses the standard sizepremium really the “fair value” of the stock if that value reflects an implicitdiscount for illiquidity? Because fair value is a legal concept, the answer to thisquestion is left to the courts and legal scholars. This paper, however, providesan economic context to assist in answering this question by quantifying theeffect of liquidity reflected in the size-decile premiums that are currently used bymany practitioners.

Acknowledgments: We would like to thank James Hoffman (the editor), threeanonymous reviewers, Kevin Brady, and Donald Puglisi for helpful commentsand suggestions on an earlier draft. The views expressed are those of the authorsand do not necessarily reflect those of Forensic Economics, Inc.

Appendix

Effect of the sum beta methodology

One method suggested to provide a better estimate of beta, one that reduces theunderestimation problem in less liquid stocks, is by accounting for the laggedresponse of small-sized companies to market movements by including in theregression a lagged market return in addition to the current market return. Weuse the method suggested by Ibbotson, Kaplan, and Peterson (1997) and calcu-late a current and a lagged beta coefficient and then sum the two coefficients toarrive at the beta estimate (called the sum beta). Tables 13 and 14 present thepremiums for the high and low liquidity groups for the ten size-decile portfoliosand for the sub-groups of the tenth decile, respectively, using the sum betamethodology. As expected, by better capturing the response to market move-ments, the sum beta methodology lowers the estimated premiums.



The results show that using sum betas, the beta estimates increase ascompared to the beta estimates from the single beta models used in Tables 10and 11. Interestingly, the higher betas that obtain from using the sum betaapproach almost eliminate or significantly reduce the size premiums for sizedeciles 1–10 for the higher liquidity stocks. Within decile 10, the results aresomewhat mixed with the size premium for higher liquidity stocks virtually zerofor sub-groups 10w and 10z, but positive 2.61% and 2.13% for sub-groups 10xand 10y, respectively.

Table 13: Long-term returns in excess of estimated returns using the sum beta methodology forsize-decile portfolios split into high and low liquidity categories

Group Sumbeta

Actualarithmetic

mean return


Estimated return inexcess of riskless rate


estimated return)

[1] [2] [3] [4]¼ [3]–5.17% [5]¼β* (11.88%–5.17%) [6]¼ [4]–[5]

1 H 1.12 11.32% 6.15% 7.54% –1.39%L 0.78 10.71% 5.54% 5.25% 0.29%

2 H 1.19 12.92% 7.75% 8.00% –0.25%L 0.91 13.28% 8.11% 6.08% 2.03%

3 H 1.26 13.49% 8.32% 8.44% –0.11%L 0.99 14.28% 9.11% 6.64% 2.46%

4 H 1.35 14.04% 8.87% 9.06% –0.20%L 1.01 14.49% 9.32% 6.75% 2.56%

5 H 1.37 14.66% 9.49% 9.17% 0.32%L 1.09 15.68% 10.51% 7.33% 3.18%

6 H 1.45 14.04% 8.87% 9.73% –0.85%L 1.13 16.62% 11.45% 7.60% 3.85%

7 H 1.48 14.68% 9.51% 9.96% –0.45%L 1.25 16.61% 11.44% 8.39% 3.05%

8 H 1.63 15.19% 10.02% 10.94% –0.92%L 1.39 18.18% 13.01% 9.33% 3.68%

9 H 1.67 17.27% 12.10% 11.22% 0.88%L 1.46 18.49% 13.32% 9.82% 3.50%

10 H 1.82 18.52% 13.35% 12.20% 1.15%L 1.67 25.11% 19.94% 11.23% 8.71%

Notes: Sum betas are estimated based on the methodology proposed by Ibbotson, Kaplan, andPeterson (1997). Liquidity categories are defined independently of the size groups. Historicalriskless rate is measured by the arithmetic mean income return component of the 20-yeargovernment bond (5.17%). Long-horizon equity risk premium is estimated by the arithmeticmean total return of the S&P 500 (11.88%) minus the arithmetic mean income return componentof the 20-year government bond (5.17%).



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