.lournal of Financial lkooomics 34 (1993) 373~3%.

asonal behavior m in asset prici

t R. Eleswarapu huhan Institute of Management. Bangaore. India

Marc R. Reinganum University of Iowa, ima City, IA 52242, USA

Received October 1992, fina! version recz.ved February 1993

This paper empirically investigates the seasonal behavior of the liquidity premium in asset pricing The evidence suggests a strong seasonal component. In the 1961-1990 period, the liquidity premium is reliably positive only during the month of January. For the non-January months, c le cannot detect a positive liquidity premium. The impact of the relative bid-ask spreads on asset pricing in non- January months cannot be reliably distinguished from zero. In contrast to An&d and Mendelson (1986), howclver, our evidence sugges,s that the size effect is signi&ant, even after controlling for spreads.

Key words: Bid-ask spreads; Asset pncing tiasonality; Liquidity premium

1 1) htmduetion

An imp Jrtant link between asset pricing and market microstructure is de- veloped by Amihud and Mendelson (1986). In their model, rational investors price securities in such a Yriay that the expected return is higher (firm value is lower) for stocks with larger bid-ask qreads.’ Amihud and Mendelson

Grrespondence fo: Marc R. Reinganum, Qepartment of Finance/CB& University of Iowa, 565 Philhps Hall, Iowa City, IA, 52242, USA.

*We wish to thank Paul Fellows, Narayana Kocherlakota, Paul Schultz, Richa.i-d Stevenson, Susan Watts, Bill Schwert (the editor), and Eugene F. Fanma (the referee) for their hsipful comments and suggestions. We would like to thank Hans Stoll ancl Robert Whalev for providing us with their spread data. We gratefully acknowledge the secretarial support of __._,a Herring and finan&_: support from the Financial Markets Research Institutt: at the University of lowa. Any remaining errors are ours.

‘Numerous r esearchers have adclressecl thti issue of ‘how the market maker sets the spread and what determines the level of spreads. For example, see Demsetz (i 968% Ragehot (1971), Tinic ( f 9721, Tinic and West (1972), Benston and Hagerman (1974), Garman (1976), Stoll(1978), Amihud and Mendelson (1980), Ho and Stoll(198 I)* Copeland and Galai (1983), Glosten and Milgrom (i985), O’Hara and Oldfield (19 t Er&y and G’Hara (1987j,, and Glosten and Harris (1988).

0304-405X/93,306. sevier Science Publishers B.V. All rights deserved

374 V I,?‘. Eleswarapu cAnd M. R. Reinganwz. Seasona&* of liquidit v pwmiumz

Mj provide empirrcal sup c nmdei in tests using portfolios of New York Stock Exch;r*?ge (NYSE) stocks during the ?961-80 period. They find a positive association between annual portfolio returns and bid-ask spreads.2 The design of their empirical tests, however, does not per& the explortition of potential monthly seasonality in the relation between ex- pected returns and bid-ask spreads.

The purpose of this paper is twofold: 1, to investigate the relation bs-:.- tween average returns and bid-ask spreads in January and in non-January months, and 2) to determine if A&M’s empirical resufts are sensitive to ,heir restrictive portfolio selection criteria. Previous research documsnts January effects [e.g., Rozeff and Kinney (1976), Meim (1983), Reinganum ( 1983 j, and Roll (198311. Tinic and West (1984, 1986) find that beta risk in the capital asset pricing model is priced only in the month of January. Chang and Pinegar (1988) report that the market return is reliably greater than the return on one-month Treasury bills only during January. This paper explores whether such seasonality also characterizes the behavior of liquidity premiums. In addition, this paper explores whether the restrictive portfolio selection tech- nique of Amihud and Mendelson could lead to spurious empirical conclusions.

The paper is organized as follows. Section 2 describes the data (which are updated through 1990) and t?ile portfolio formation method. Section 3 presents the empirical results, which suggest a strong January component in liquidity premiums. However, unlike Amihud and IVIendelson (1986), the results also suggest a significant size effect, even after accounting for the liquidity premium. The final section offers some conclusions.

2. Data and iuitial portfolio formation technique

The cross-sectional relation between monthly returns, betas, and the relative bid-ask spread is tested over the 1961-98 period using NYSE firms. Monthly NYSE stock returns are obtained from tapes provided by the Center for Research in Security Prices. The relative spread of a stock is the callar t&-ask spread divided by the average of the bid and ask grices. As in A&G/J, the average spread for stock i in year n, Sin, is the average of the br;ginning and end-of-year relative spreads in the preceding year n - 1. For 1960-79, the relative spread data are those used in Stoll and Whaley (1983);3 ,Fbr ;;lle 1980-89 period, the year-end spread data are obtained from Fitch Investors Service, Inc. h

2There is some disagreement about the existence of this liquidity premium. ConstivnGnides (1986), in an inte:temporal portColio selection model, show tlzat transactions costs should have only very small effects on the expected returns. Empiricaliy, Reinganum (1990) concluder that t%e NASDAQ provides better liquidity than the NYSE for small stocks using the A&M model, although Loughran (1992) , 91 ec.!ons the magnitude of this effect. J . t’

3Stoll and Whaley kindly provided us with these data.

V.R. Elewarapu and M R. Reinganm, Seasonality of liquidity premium 375

The return-spread relation is initially tested using 49 equally-weighted port- folios formed using the criteria of A&M. Stocks are olaced inta portfolios based L on their spread and estimated beta. The exact fcmaticc r;ro&ure requires eleven years of complete return data for a stock. In an iniGa1 6ve-year period, betas of individual stocks are estimated using market model regressions base on monthly rebams:

Rjt = ai + Mnt + &it 9 t=1,...,60,

where Ri, and R,,,? are the month t excess returns (over the correspondhg one-month Treasury bill return) on stock i and on the rng z=k zt index, respectively [following Amihud and Mendelson (1986) and Fama and Macbeth (1973), the market index is the equally-weighted portfolio of all NYSE stocks]. Stocks are ranked and divided into seven equal groups based on their average spread, &, in the last year of the second five-year period (see paragraph below). Each of these seven groups is divided further into seven equal subgroups by ranking the stocks according to their estimated beta ‘coefficients. There are thus 49 test portfolios with approximately equal numbers of stocks. [Sorting by size and beta, instead of the spread and beta, yields similar results (available on request). The potential impact of portfolio formation techniques is nicely discussed by Jegadeesh (1992).]

In the second five-year period, ihe betas of the 45 portfolios are estimated using the market model regression ;snd monthly portfolio returns. The average spread &,,) of a portfolio is computed by averaging the spreads of all the stocks in portfolio p in the last year of this five-year period. For example, for the test-period year 1961, the initial estimation period is 1951-55 and portfolio betas are estimated in the 1956-60 period. The spreads used in the test period would be from 1960. Thus, for each portfolio p (p = 1, . . . ,49) and each year n (n = 1, . . . ,30), there is an estimated beta, &,,, and an average spread, SPn.

These estimated betas and average spreads are compared against excess portfolio returns during the following year. For a stock to be included in the analysis, it needs eleven years of return data and a relative spread in the last year of the second five-year period. This estimation and ranking procedure is re- peated each year, resulting in 30 test-year periods (1961-90). The number of firms included in each test period ranges from 654 to 929.

Table 1 presents descriptive statistics for the 49 portfolios formed on the basis of spreads and estimated betas using the A&M criteria. Each cell of the table contains the value of the variable averaged over the 30 periods. Average portfolio relative spreads range from 0.454% to 3.530%. The average portfolio betas range between 0.517 and 1.470. The average market value of equity in each of these portfolios is also presented. Average equity values, calculated each year by averaging the equity valuec across all the stocks in that portfolio at the end of

376 V.R. Eleswarapu and MR. Rebaganum, Seasonalrty of liquidity premiww

Table 1

Averqe relative bid-ask spread (in percent), betas, and rzarket va!ne clfequity (in millions of dollars) for the d9 portfolios of NYSE firms based , _ 0~ Aruihud and Mendelson (1986) sample selection

criteria, requiring return data for ten years preceding each test year, 1961-90.

The portfolios are formed each year preceding the test year by ranking the stocks into seven equal groups, based on their spread, and then dividing them into seven equal subgroups according to their estimated beta coefficients. The number of firms included in each test year ranges from 654 to 929. Each cell contains three entries. The top number is the relative bid-ask spread of the portfolio in percent. The purtfolio spread is Pe average spread of the stocks in the year preceding the test year. The second number is the portfolio txtsr, estimated over the five years preceding the test-period year. The bottom number is the market value oithe equity in millions of dollars, where the equity value of the firm is computed at the end of the year prt+ing the test year. Each entry is averaged over the 30

years of this study, 1961-90.

Spread, [Beta), (Equity value)

Beta group Spread group Lowest 2 3 4 5 6 Highest Mean

Lowest 0.4541 [O.S57] (5 167)

2 0.7077 [0.570] (1492)

3 0.9030 co.5393 (1120)

4 1.1@45 [o.son]

(607)

5 1.3564 [0.564-J

(392)

6 1.7407 CO.5691

(215) f%i& SF 2.9874

L3.739-j (214)

Mean 1.3220 CO.5861 (1315)

0.4536 [0.701] (7328)

0.7089 [0.?17] (1816)

0.9006 CO.6773 (1044)

1.1036 CO.6631

(771)

1.3470 [0.704-J

(456)

1.7319 CO.7281

(228)

3.1041 LO.9353

(91)

1.3360 CO.7321 (1676)

0.4572 iO.760) (5730)

0.7127 CO.8 191 (1713)

0.9029 co.7991

(984)

1.1044 CO.8451

(579)

1.3562 CO.85 13

(374)

1.7568 CO.8881

(204)

3.2513 [ 1.0353

(81)

1.3630 [0.857] (1381)

0.4555 [0.806-J (453)

0.7079 CO.8861 (1304)

0.9003 [0.908] (1352)

1.1040 co.9341

(526)

1.3517 CO.9691

(305)

1.7554 co.9733

(204)

3.2458 Cl.1351

(81)

1.3600 co.9451 (1118)

0.4638 CO.8721 (2879)

0.7126 CO.9361 (1402)

0.9004 CO.9761

(757)

1.1019 CO.9961

(435)

1.1512 [Cl433

(317)

1.7541 Cl.1023

(162)

3.4209 [ 1.232)

(60)

1.3864 [ 1.0223

(859)

0.4665 LO.9951 (2811)

0.7110 [ 1.0303 (1108)

0.8994 [1.043]

(622)

1.1027 [ 1.088]

(448)

fi -3537 [1X3]

(266)

1.7593 Cl.1543

(133)

3.4191 F1.3313

(71)

1.3874 [l.lOS]

(780)

0.468 1 [1.104] (1470)

0.7090 [1,241]

(662)

0.9002 [ 1.2291

(531)

1.1010 Cl.2531

(322)

1.3580 Cl.3211

(207)

1.7331 [ 1.325)

(125)

3.5304 [ 1.4703

(55)

1.4057 [ 1.2771

(482)

0.4600 CO.8221 (4205)

0.7100 CO.8861 (1357)

0.9010 CO.8821

(916)

1.1032 [0.9&]

(527)

1.3535 [0.941]

(331)

1.7530 CO.9631

(181)

3.2799 [1.125]

(93)

1.3660 CO.9321 (1087)

-- -

the last year of the second five-year period, range between $55 million and $7.328 billion. Spreads, betas, and market value of equity are not independent. Low-spread stocks tend to be low-beta stocks, and vice versa. Spreads and market value of equity tend to be inversely related - the smaller the spread, the larger the firm value.

V.R. Eleswarapu and M.R. Reinganm, Seamnalir)? of liquidity premiums 377

3. Empirlcsl results

3.1. A&M portfolio formation technique

The test-period data comes from 1961-90, extending the original A&M sample by ten years. Data from the 1980s can be viewed as a validation period for the seasonJ effects investigated in this paper, since it roughly postdates the Keim (1983) evide ce on the January effect. What is the liquidity premium in January and i non-January months during the entire 1961-90 period? As a starting point, fig. 1 depicts the average January and non-January returns of the 49 NYSE portfolios classified by spread and beta. The graphs are dramatic. In January, as one moves from low-spread to high-spread securities, the average returns increase, which is consistent with a positive liquidity premium. On the other hand, the average non-January returns do not exhibit any such relation. The relation between average returns and spread appears virtually flat in the non-January months.

The formal empirical tests bv Amihud and Mendelson (1986) are based on the pooled cross-section and time:series methodology. Chen and Kan (1989) raise concerns with this methodology; they argue that since it constrains the market

Fig. la. Average January returns.

Average portfolio returns for the 49 portfolios of NYSE firms based on the Anrihud and Mendelson portfolio technique, requiring return data for ten years preceding each test ysar, 1961-90. The portfolios are formed each year preceding the test year by ranking the stocks into seven equal groups, based on their spread, and then dividing them into seven equal subgroups according to their estimated beta coeficients. The number of firms included in each test year ranges from 654 ice 929. The portfolio return is the arithmetic average of the monthly returns on the stocks in the portfolio in

excess of that month’s Treasury bill return.

378 V. R. Eleswarapu and M. R. Rhganum, Seasngvmlity oJr iiquidity premiums

Fig. lb. Average non-January returns.

Average portfolio returns for the -%9 portfo!ios of ‘NYSE firms based on the Amihud and Mendelson portfolio technique, requn A..0 .+c return data for ten years preceding each test yea;. 1961-90. The portfolios are formed each year preceding the test year by ranking the stocks into, seven equal groups, based on their spread, and then dividing them into seven equal subgroups acccrding to their estimated beta coefficients. The number of firms included in each test year ranges from 654 to 929. The portfolio return is the arithmetic average of the monthly returns on the stocks in the portfolio in

excess of that month’s Treasurv bill return.

risk premium to be constant over the SO-year period, it may induce a spurious spread effect. Chen and Kan recommend the use of cross-section regressions as in Fama and MacBeth (1973). Therefore, in this paper, only results from these monthly cross-section regressions are reported.4

The liquidity premiums estimated in a Fama--Macbeth fra. rework are pre- sented in table 2. The returns of the 49 portfolios are regre: sed each month against beta, spread, and size. The time-series averages of the m nthiy regression coefficients are reported in table 2. When all months from the 1961-90 period are lumped together (panel A of table 2), the liquidity premium coefficient of the spread variable) is positive but less than two standard errs rs from zero in regressions that include beta. When only the January months are considered, however, the liquidity premium and beta-risk premium are &imated io ‘cw’: positive and reliably different from zero. Thus, in January both reta and spread seem to be priced. IIowever, size does not anpear to be priced ir, January in the 1 . presence of spread and beta,

By contrast, in the non-January months, the point estimates of the liquidity premium are negative, although within two standard errors of zero. In fact, none

‘For the sake of completeness, we also use the A&M pooled cross-section and t%e-series regression methodology (available on request). We look at their 1961-80 time period, the overall test period of 1961-90, and the subperiod 1981-90. We investigate possible seasonality in the liquidity premium using January dummies. We find that the liquidity premium is reliably positive only in the month of January.

Table 2

Estimates of coefficients for Fama-Macbeth type regressions of excess returns for the portfolios of NYSE firms based on Amihud and Mendelson (1986) sample selection criterk, requiring reun data

for ten years preceding each test year. Standard errors in parentheses.

(A): R, = a0 + ad& + ep

0% R, = bo + b& + eFt

(C): R, = co + c&g -I- c& + @pc

(W: &f = do + dl iog(Si3) + e,

(E): R, = tfo + eJ.&, + e-S, + ejLog(Size) + e,

There is one obse+v t’ _ .s :m for each month of a test-period year for WC% e.3’ the 49 portfolios. As in Amihud and Mcndelson (1986), a stock is required to have eleven years of complete return data for inclusion in a portfolio. Excess return (R,,,) is the average excess return (over the corresponding orre-month Treasury bill return) for portfolio p in month t. The cross-sectional regression is fit in each month t of the test-period years. The coefficients are the time-series means with corresponding

standard errors.

Regression

Variable (A) (B) (C) (D) (E)

Beta

Spread

Log (Size)

N

Beta

Spread

Log (Size)

N

Beta

Spread

Log (Size)

N

0.0063 (0.0032)

360

0.0643 (0.0143)

30

0.0010 (0.0032)

330

Panel A: Test period 1961-90

All months

0.0053 (0.003 1)

0.1424 0.0659 (0.0741) (0.0647)

360 360

January

0.0374 (O.iH33j

2.306 1.8816 (0.3540) (0.3460)

30 30

Non-January

0.0024 (0.03 1)

- 0.0542 - 0.099 (0.0642) (0,055)

33Q 330

- 0.0011 (0.0005)

360

- 0.0026 (0.002 1)

30

0.0048 (0.0028)

0.0458 (0.0665)

- 0.0003 (0.0005)

360

0.0333 (0.0139)

1.617 (0.2400)

- 0.0026 (0.002!)

30

.._ -----

380 V. R. Eleswatapu and M. R. Reiriganum, Seasonality of liquidity premi?zFm

Table 2 (mntinued) -- -- _ - -

Regression -

Variable (A) (B) (C) (W (El

Beta

Spread

Log (Size)

N

Beta

Spread

Log(Size)

N

Beta

Spread

Log (Size)

N

‘- 0.0045 (O.OOS3)

Panel B: Test period 1981-W

All months *

- 0.0050 (0.005 1)

- 0.0366 0.0017 (O.lW73 (O.1063)

120

0.0202 (im66)

120 120

January

0.002 1 (O.!ml)

f.4233 1.4250 (0.4696) (O.S! 72)

10 10 10

Non-Jahuary

- 0.0067 - 0.0057 (0.0055) (0.0054)

- 0.1693 - 0.1277 (0.1049) (0.0906]

110 110 110

0.0005 (O,ooo8,

1211)

- 0.0077 (0.0033)

10

0.0012 (0.0008)

110

- c.oo60 (0.0052)

- 0.0524 (0.1157)

- - 0.0003 (0.0007)

120

0.0020 (0.0194)

1.5486 (0.3783)

0.0014 (0.0035)

10

- 0.0067 (0.0054)

- 0.1980 (“3.1121)

- omO5 (OMIOS)

110

of the prem_ium estimates in ihe Ann-January n~~&s FR reliably diflerent from zero. Thus, using the A&M sample selection criteria, the liquidity premium appears to be reiiably positive only in the month of January during 1961-90. [Of course, the magnitude of the liquidity preinium in January months may be overstated; Keim (1989) documents that the observed January returns are biased upwards for stocks with high spreads due to the trading patterns at the end of the year.]

Panel B of table 2 reports the results for the subperiod 1981-90, which post-da@ the original A&M sample period. In this subperiod, the liquidity premium is still positive’and more than two standard errors from zero in January. In regressions with beta, spread, and size, only spread appears to be statistically significant in January. The liquidity premium in non-January months is negative.

V.R. Eleswarapu and MR. Reingantm, Seasona/ity of liquidity pemiwns 381

Unlike the overall 1961-90 period, however, the point estimate of the liquidity premium in this subperiod is negative for all months combined.

3.2. Mod@ed portfolio formation technique ‘

The stringent data requirements of the A&&I selection criteria raise the possibility that the results are an art&&a& ;rnnt of' a limited sample rather than the manifestation of a true effect. For example, in the &WI framazork, the size variable does not appear to matter, even in January: after coatrollrng for spread. Yet the requirement that there be eleven years of return data may systematically exclude smaller filrms from the sample and: hence, bias the resu!ts against finding a size effect.

The restrictive A&M criteria are not necessary to test the cross-set: .nal relation between returns and spreads, In this section, portfolios are formed using just three years of return data. (Similar results are obtained with portfolios requiring five years of prefomation data instead of three.) Assignment afa stock to a particular beta/spread portfolio in a given test year depends on two criteria* 1) the stock’s spread in the previous year and 2) the stock’s ordinary least- squares beta estimated with 36 montils of preceding returns. Thus, only three years of preformation returns are needed for inclusion in the tests. Also, in contrast to the previous section, firms can be delisted in the middle of a test year, avoiding a potential survivorship bias.

For the test period, stocks are again ranked into seven equal groups based on their spread in the previous year, as in A&M. They are further divided into seven equal subgroups by ranking the stocks according to their estimated beta coefficients, computed with the preceding ‘4 monthly returns. The average portfolio excess returns in the test year a r~ obtained by averaging the excess returns of the stocks in each of the 49 portfolios. This procedure is repeated in each of the 30 test-year periods, 1961-90.

The portfolio formation technique dramatically increases the number of stocks included in the analysis. For example, the average number of firms in a year increases by 45 %, from 795 to b,I 53. The average market capitalization of a firm in the enlarged sample is 16% smaller than that of the restricted sample. In this analysis, the number of firms included in each test period ranges from 895 to 1,346 (versus a range of 654 to 929 for the restricted sample).

Another change from the A&M framework is that unconditional g~rrfolio Betsy are used in the cross-sectional regressions. The unconditional portfolio betas are computed using the monthly portfolio returns during the test-period years. In other words, portfolio betab are computed with 360 months of portfolio return data (1961-90). Table 3 presents the descriptive statistics for the 49 pdolios form using the modified portfolio technique. In general, they have the same character- istics as those formed using the A&M portfolio formation techniqtte. However, the portfolios now have smaller-size Grms with larger average bid-ask spreads.

382 VA Eleswarapu and MR. Reinganum, Seasonality of liquidity premiums

Table 3

Average relative bid-ask spread (in percent), betas, and market value of equity (in millions of dollars) for the 49 portfolios of NYSE firms based on modified tfolio formation technique, requiring

return data for three years preceding e test year, 1961-90.

The portfolios are formed each year preceding t king the stocks into seven equal groups, based on their spread, and then dividing ual subgroups according to their estimated beta coefficients. The mu year ranges from 895 to 1,346. Each cell contains three entries. Th asP .read of the portfolio in percent. The portfolio spread is the a plea, Treceding the test year. The second number is the estimated wrth c 41 months of portfolio return data (1961-90). The bottom number i eqway in millions of dollars, where the equity value of the firm is compute ece~o~,~ the test year. Portfolio spread

and market value of equity are averaged over the 30 yetrs ct this study, 1961-90.

Spread, [Beta], (Equity value)

Beta group Spread group Lowest 2 3 4 5 6 Highest Mean

Lowest 0.48 14 IO.5353 (5785)

2 0.762 1 [0.561] (1062)

3 0.9733 co.5773

(673)

4 1.2015 CO.565 ]

(459)

5 1.4880 co.5951

(267)

6 1.9184 co.5941

(130)

Highest 3.5117 [0.818-J

1(3W

Mean 1.4770 [0.607] (1240)

0.4700 CO.6831 (7081)

0.7605 co.7131 (2345)

0.9710 CO.69 l]

(869)

1.1994 co.75 13

(411)

1.4767 co.7411

(351)

1.92OC [0.833-j

(179)

3.3452 Cl.0321

(99)

1.4490 CO.7781 (1619)

0.478 1 co.7311 (4739)

0.7617 [0.790-J (1189)

0.9738 CO.8473

(788)

1.1986 [0.853]

(465)

1.4832 CO.8981

(283)

1.9298 [0.954]

(172)

3.5429 Cl.1831

(74)

1.4812 CO.8941 (1101)

0.4930 CO.8063 (2737)

0.7602 CO.85 13 (1169)

0.97 14 CQ.9491

(612)

1.1962 CO.9381

(394)

1.4783 P.~l

(233)

1.9296 rt.o60] - (146)

3.6285 [1.204-J

(64)

1.4939 co.9743

(765)

0.4973 CO.8451 (2291)

0.765 1 CO.9581

(946)

8.976C [0.979]

(574)

1.2035 [l.OlOl

(36

1.483; [ 1.0871

(226)

1.9288 [1.188]

(119)

3.5985 [ 1.3671

(54)

1.4932 [ 1.0621

(657)

0.5066 CO.98 l] (1958)

0.7654 Cl.0761 (9301)

6.9749 [1.058]

(547)

1.1983 [l.llO]

$39)

1.4832 Cl.1433

(197)

1.9401 [ 1.2391

(133)

3.7800 [ 1.437)

(54)

1.5212 Cl.1493

(594)

0.5064 [1.141] (1347)

0.7663 [ 1.2481

(674)

0 9z.W [ 1:234]

(387)

1.2017 Cl.3103

(262)

1.4906 Cl.3811

(161)

1.9627 [1.407]

(118)

3.9154 [r,ssl]

(53)

1.5462 Cl.3241

(429)

0.4904 CO.8171 (3705)

0.763 1 [0.885] (1188)

0.9?44 [0.905]

$36)

1 1999 [C r.9341

(388)

1.4834 co.9793

(245)

1.9328 Cl*0403

(142) 3.6175

[ 1.227) (100)

1.4950 [0.970]

(915)

The cross-sectional regression is fit in each aaronth of the test-period years using three independent variables: 1) unconditional portfolio betas, 2) average spread, and 3) size. The estimates of these ( -Macbeth) coefficients and standard errors are reported in table 4. The toe ts are the time-series means with corresponding standard errors.

W?. Eleswarapu and M. R Reinganum, Seasonality of liquidity premiums 383

Table 4

Estimates of Fama-Macbeth coefficients obtained with unconditional betas and modified portfolio formation technique, requiring return data for three years preceding each test year. Standard errors

in parentheses.

(A): R, = a0 + a~& + ept

(B): R, = b0 + b,S, + ept

(C): R, = co + Cl&M + c2 S +ept p?

0%: R, = do + &Log(Size) + ep

(E): R, = e. + e,S,, + e2Spr + e3 Log (Size) -+ epl

Assignment of a stock to a particular beta/spread portfolio in a given test year depends on two criteria: (1) the spread in the previous year and (2) a stock’s OLS beta estimated with 36 months of preceding returns. In the cross-sectional regression, the portfoiio beta is the unconditional beta; that is, the beta is computed using the monthly portfolio returns from all of the test-period years. The cross-sectional regression is fit in each month t of the test-peeod years. The coefficients are the

time-series means with corresponding standard errors. -

Regression

Variable (A) (B) (C) (D) (E)

Beta

Spread

Log (Size)

0.0024 (0.0035)

Panel A: Test period i961-90

All months

- 0.0001 (0.0035)

0.1339 0.1477 (0.0689) (0.0636)

N

Beta

Spread

Log (Size)

N

Beta

Spread

Log (Size)

N _.-_.- -

360 360 360

0.0642 (0.0161)

January

0.0319 (0.0149)

2.0964 i .7023 (0.3415) (0.3477)

30

- 0.0032 (0.0034)

30 30

Non-January

- 0.0029 (0.0035)

- 0.0445 0.0063 (0.0596) (0.0557)

330 330 330

- 0.0011 - 0.0008 (O-000@ (O.Ooos)

360 360

- 0.0158 (0.0032)

30

o.WO3 (0.0005)

330

- 0.0012 (0.0036)

0.0875 (0.0626)

0.0247 (0.0151)

1.3072 (0.28 14)

- 0.050 (0.0020)

30

- 0.0036 (0.0036)

- 0.0234 (0.0599)

- 0.0004 (0.W)

330

384 V. R. Eleswarapu and M. R. Reinganum, Seasonality of L#iity premiums

Table 4 (continued)

Regression

Vqri?51e (A) (B) C) D) (E) --. _ -- - --

Spread

Log (Size)

lv

B&a

Spread

Log, Size)

N

Beta

Spread

Log (Si2 2)

IJ

- 0.0075 (0.005 51

Panel B: Test period 1981-90

AfI months

-- 9.CO68 (CLASS)

- 0.0914 ti.0375 (0.097 1) (0.0918)

120

0.0125 (0.02 14)

120 120

January

0.0019 (0.023 1)

1.1940 1.1903 (0.4165) (0.4654)

10 10 10

Non-January

- 0.0094 - 0.0075 (0.0057) (0.0056)

- 0.2085 - 0.1491 (0.092 1) (0.082 3)

110 110 110

0.0 08 (OSM 07)

121

- 0.0071 (0.0030)

10

0.0016 (0.0007)

110

- O.Q’I78 (O.GOSS:,

- 0.0759 (0.0916)

- 0.0003 (O.rn)

t20

- wOO3 (e.0238)

1.0584 (0.42 18)

- 0.0014 (0.0027)

10

- 0.0085 (@.Om)

- 9.1790 (0.@865)

- 0.0002

(O.OOOw

110

Pane1 A of table 4 shows the results for the 1961-90 period. Overall, the basic result that the liquidity premium is present only in January still appears to hold. However, the size eflrect is now very much present. In fact, over all the months taken together, size is the only variable which appears to be significant in the presence of the other two variables. This conclusion differs from the inferences drawn based on the restricted A&M sample.

With the enlarged sample, the evidence for a positive beta-risk premium is diminished and perhaps nonexistent. The b eta-risk premium is not significant in the presence of spread and size, ever in January. (Using Dimson’s aggregated- coefficiems betas with one monthly 1td.d and one monthly lag yields the same result.) Th,2se results on he pricing of beta risk seem consistent with Fama and French (1992). Apparently, requiring only three years of data mstead of eleven

CI R. Eleswarapu and M. R. Reinganm. Seasma/@ qf liquidiry premiums 385

years makes a difference for inferences about the pricing ti! beta risk as well. The pricing of beta nsk may also be &ected by the relaxing t>f th? requirement that firms survive for the whole test-period year following the portfolio formation.

Panel B of table 4 shows the results for the subperiod l%I-90. The point estimates of the liquidity premium are ncgctive for all t& months combined. In non-January months, the liquidity pne.mium is negative and about two standard errors below zero In January, ti=o@~, the liquidity premium is still reliably positive. Thus, the predictions of t& e .A&M model are confirmed for January but not for the non-January months. Of course, in the non-January m<.n*hs for the overall period 1961-90, none of the variables seem to be reliably 2; iceal.

4. Summary and conclusions

This paper empirically investigates the season21 b&~&r of the liquidity premium as modeled by Amihud and Mendelson (1986). The evidence suggests a strong seasonal component. In the 1961-90 period, the liquidity premium is reliably positive only during the month of January. For the non-January months, one cannot detect a positive liquidity premium. That is, the impact of the relative bid-ask spread on asset priciug in non-January months cannot be reliably distinguished from zero.

Unlike the original A&M study, the evidence in this paper suggests a signifi- cant size effect even after controlling for spreads and beta. The restrictive sampie selection criteria of A&M tend to systematically exclude smaller fims and hence bias the results against finding a size effect. By modifying the portfolio formation technique, the number of firms included in the analysis increases by 45%.

The evidence reveals a positive relation between bid-ask spreads and average returns, but only during the month of January. The lack of such a positive relation between spreads and average returns outside of January may well be part of a broader puzzle. Tinic and West (1984) report that beta risk is reliably priced only during January. The broader issue may well be why the empirical representations of asset pricing models work well only in January* Apart from January, the empirical support for asset pricing mo&ls seems tenuous at best, as documented by the evidence in this paper for the model of A&M (1986). While liquidity may very well be a determinant of asset pricing, the reason that its effect on asset pricing seems limited to January is not clear.

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