liquidity changes following stock splits

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Liquidity Changes Following Stock SplitsAuthor(s): Thomas E. CopelandSource: The Journal of Finance, Vol. 34, No. 1, (Mar., 1979), pp. 115-141Published by: Blackwell Publishing for the American Finance AssociationStable URL: http://www.jstor.org/stable/2327148Accessed: 26/07/2008 23:10

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THE JOURNAL OF FINANCE * VOL. XXXIV, NO. 1 * MARCH 1979

Liquidity Changes Following Stock Splits

THOMAS E. COPELAND*

I. Introduction

THERE HAS BEEN CONSIDERABLE empirical research on the return behavior of common stocks in calendar intervals surrounding stock splits. A partial list includes work by Barker [2]; Johnson [25]; Hausman, West and Largay [21]; Fama, Fisher, Jensen, and Roll [16]; and Bar-Yosef and Brown [1]. However, little or no evidence has been collected about stockholder trading behavior in split-up securities. This is surprising because it is often alleged that stocks split because they provide "better" markets for trading. This study presents evidence about the liquidity effects of stock splits.

There are numerous rationales for stock splits, and many are related to the liquidity of trading. For example, one often hears on Wall Street that there is an "optimal" price range for securities. Stocks which trade in this range are presumed to have lower brokerage fees as a percent of value traded and therefore appear to be more liquid. This "optimal" range is considered to be a compromise between the desires of wealthy investors and institutions who will minimize brokerage costs if securities are high-priced, and the desires of small investors who will minimize odd-lot brokerage costs if securities are low-priced. Implicitly, there is a trade-off between diversification benefits and the lower transactions costs of round-lot trading.' One difficulty with the argument is that small investors can economize on odd-lots by forming investment clubs or by buying no-load mutual funds. A second difficulty is that odd-lot trades are simply not a significant fraction of trading activity. Finally, if the price of a security becomes "too high," the market obliges by making ten shares a round lot.

Another explanation for splits, also heard on Wall Street, is that they create "wider" markets. Following a split, the number of shareholders may increase simply because an individual, who holds one round lot and who is likely to sell it to one buyer before a two-for-one split, may sell two round lots to two people after the split.2 If the number of shareholders increases after the split, then trading volume increases. Demsetz [11] shows that higher volume results in lower

* Assistant Professor of Financial Economics, The University of California at Los Angeles. I would like to express my gratitude to Harry DeAngelo, David Mayers, Vinay Marathe, Ron Masulis, Keitlv Smith, and Richard Roll for helpful comments on earlier drafts of this paper. Financial assistance was generously provided by the Cantor Fitzgerald Fund for Financial Research. Also I would like to thank Josephine Cheng and Tsutomu Nakamura for computer programming. All errors in the paper are, of course, solely my responsibility.

' For a good exposition of this point see Goldsmith [19]. 2 Barker [2] compared a sample of 90 companies whose stocks had split with an equal sample of

nonsplit companies in the same industry groups. He found that between December 31, 1950 and December 31, 1953, the average gain in the number of shareholders for the split-up stocks was 30% as compared with 6% for the companies whose stocks had not split.

115

116 The Journal of Finance

bid-ask spreads. This would imply greater liquidity. However, there are param- eters other than simply the number of shareholders which affect volume. Models developed by Copeland [8, 9] and Epps [14] show that volume is determined by the rate of message arrival per unit time (also called information flow), the total number of shareholders, the percent of traders who view new information opti- mistically, the number of shares outstanding, and transactions costs. Any of these might change, and if the net effect is to reduce trading activity, then bid-ask spreads might increase rather than decrease.

Because there are good counterarguments for every argument in favor of higher liquidity in the post-split period, liquidity changes following stock splits is an empirical question. For a definition of liquidity, we shall adopt Hicks [22, pp. 163-170]. Liquidity is the imperfect "moneyness" of very short-term bills. The price of liquidity is the discount caused by the "trouble" of investing short-term bills which are not generally acceptable. A transactions cost must first be incurred in order to convert the bill into money for investment or consumption. If the bill were generally acceptable as a medium of exchange or if transactions costs were zero, then the bill and money would be perfect substitutes.

Two measures of liquidity are used in this p'aper: 1) changes in the proportional share volume of trading, and 2) changes in transactions costs as a percent of value traded. We use these two measures rather than looking at return residuals because factors other than liquidity effects may coincide with a stock split announcement. For example, Fama, Fisher, Jensen, and Roll [16] suggest that splits may be interpreted by investors as a favorable signal that the firm is able to maintain a new, higher level of earnings. Therefore, it is possible to observe positive return residuals upon the announcement of a split even though investors anticipate lower post-split liquidity. This is because stock returns reflect only the net effect of splits. If the benefit of the split message exceeds anticipated liquidity costs, return residuals will be useless for testing liquidity hypotheses. However, it will be argued that permanent changes in 1) the volume of trading and 2) the transactions costs associated with trading can be unambiguously interpreted as liquidity effects.

Part II of the paper develops an empirical model of trading in the jth security which is well-specified. As a side issue, the empirical results provide an estimate of the maximum length of time it takes new information to be completely incorporated into trading activity. Part III examines the stationarity of trading activity in the interval surrounding a split. Part IV estimates the effect on brokerage revenues and bid-ask spreads as a percent of value traded in the postsplit market. Part V summarizes and concludes.

H. An Empirical Model of Volume

If one wishes to test for changes in trading activity around an economic event, it is desirable to construct a time-series model which is based on reasonable assumptions about the underlying causal factors and which has desirable distributional properties of residuals. The earliest model of volume developed by Beaver [5] was ad hoc and, as we shall show, misspecified. More recently, Schwert

myy

下划线

Stock Splits 117

[34] has used a fourth-order moving-average model with the rate of change in share volume as the dependent variable. While this model has distributional properties superior to the one developed below, it concerns the rate of change in volume which is hard to interpret, and no inferences, other than those revealed by patterns of residuals, can be drawn from it. The finite time-series model developed here is based on assumptions about the effect of information arrival on trading volume and allows an estimate of the maximum time required for the full impact of a new message to be realized.

First, assume that market volume in the tth time period, Vmt, is a function of current information arrival, It, and previously generated news, It-,, It_-2, .

Vmt = f (It, It-1, * * * , It_n ) . 1 A reasonable question, which will be determined by the data, asks what weights should be applied to messages which arrived in past time intervals.

It is necessary to establish a model which allows estimation of the impact of information arrival on volume even though it is not possible to provide a metric for unobservable information, It, It-, ... . Assume that It represents the total impact of a message on aggregate market volume and is measured in millions of shares. Further, assume that the full impact of a message on market volume is felt in a finite number of calendar intervals ini a set of linearly declining weights, at >- 0, constrained to add to one (E at = 1).

at=a-bt; t=O,1, ,N. (2)

The weights are constructed to add to one because it is reasonable to assume that only one hundred percent of the information's impact should be counted, i.e., (E at) It = It.3 The linear constraint is written as

N N

> at=1=(N+1)a-bE t. (3) t=O t=O

Also, in order to constrain the weights to be positive, the weight in the last time period (and all following time periods) is set equal to zero.

aN= a-bN= O. (4)

The value of the information impact in the first time period, "a," winl be estimated and used to calculate the number of time periods, N (a real number), for the message to have full impact.

Solving the system of equations (2), (3), and (4) for the time interval, N, and the slope parameter, b, both written in terms of "a," we have

a 2 2- a b =2_ and N= . (5)

3By contrast, geometrically declining weights whose sum can be written as

1 Lim(a+a2'+ +a')= 1 >1 for O<a<1 n- +oo1- a

have a sum greater than 100%.


Also, market volume can be written as a function of "a":

vwt = aIt + a(2 - 2a) It_, + . + a(2 -

na)LnI (6) Vmtalt+~ 2-a 2 -a

In order to look at trading in individual securities, assume that the adjustment coefficient for an individual security is a constant fraction, aj, of the market adjustment coefficient, "a."4

Volume in the jth security can be written as:5

aj(2 - 2a1) aj(2 - naj) Vjt = a.It + 2 - aj It_i + *- + 2-aj

Market volume is the sum of individual volumes

Vmt Vjt- (8)

By disaggregating the jth security, we obtain

_____2a 2a - 3a? Vjt = vmt-it , ai-It- , I t2 (9)

ioi i+, 2-ai is 2-a

In order to simplify, let

E aj= a i#j

2ai - 2a K i,2 -ai

2ai - 3a, fi ~= K2

i,j 2-ai

and rewrite Vjt as

V= Vmt - aIt - KlI -K2t2- - K. (10)

Beaver [5] uses a regression equation (with a constant term) which is analogous to the market model used for security returns. We will refer to this as the volume market model (VMM) and write it as:

Vjt = -Cao + /1 Vmt + Ejt. (11)

By comparing (10) with (11), one can argue that the omitted variables, It, It-, * are likely to be highly inversely correlated (because of their negative sign) with the independent variable, Vmt, thereby causing a misspecification problem which will bias ,81 downward.

4A more complex model would assume that volume in the j-th security has three components. One fraction would relate the general market information, another to industry news independent of market news, and a third component would be specific to the firm.

'This is a time-series model which assumes that other factors which affect volume are constant. Factors other than market news might include the number of shareholders or changes in the schedule of transactions costs. These are assumed to be constant over time.

Stock Splits 119

The model can be improved by multiplying Vjt-1 by the adjustment coefficient,

"a," and subtracting the result from (10). The result is an equation which allows estimation of the first-period adjustment coefficient and which has less specification bias.

vjt= Vmt - aVmt-i + aVjt-i - a(It - aIt-i) - K1(It- - aIt-2) -K2(It-2 - aIt3)-* . (12)

A regression equation used to estimate (12) is a finite-adjustment time-series model (FTSM):

Vt= - ao + 81Vmt - 82Vmt-1 + /83Vjt-1 + Ejt (13)

If the first-period adjustment coefficient, "a," were equal to one, there would be no misspecification bias at all because AIt = It - It-, is uncorrelated with the independent variables in equation (12). For example,

COV(Vmt, It -It-,) = E[(Vmt - Vmt)(It - It-i - (I It- ]

and if a = 1, then Vmt = It, and we have

COV(Vmt, It - It-,) = E[(It - it)2 (L- )(It-It_1)

- VAR(It) - COV(It, It-1) = 0.

However, if "a" is less than one, then

COV(Vmt, It - It-) = COV(Vmt, It) - COV(Vmt, It-) > ?, (14)

and there would be misspecification bias in FTSM. The omitted variables in FTSM are of the form a(It - aIti1). Consequently, the magnitude of the misspecification error will be

aCOV(Vmt, It- aIt-i) = a[COV(Vmt, It) - aCOV(Vmt, It0)]. (15)

Equation (15) is less than (14) if 0 < a < 1, and all covariances are positive. Therefore, the transformation used to arrive at FTSM considerably reduces the

misspecification bias caused by the fact that It is not observable. In fact, the bias

approaches zero as the first-period adjustment coefficient, "a," approaches one. To the extent that bias does exist, the coefficients of the dependent variables should be biased downward because the unobservable variables are subtracted in

the FTSM and have positive covariance with Vmt, Vmt-1, and Vjt-1. There are several testable implications of the FTSM which predict that: (a)

the coefficient of Vmt should be equal to one (31 = 1); (b) the coefficients of Vmt-, and Vjt-1 should be equal and opposite in sign (-/32 = ,33), and less than or equal to one ( 1,82 = 18313

- C 1); (C) the constant term should be negative since it is a

proxy for the expected value of the unobservable variables (ao < 0); and (d) there

should be negative serial correlation in the residuals. In order to compare the volume market model (VMM) with the finite time-

series model (FTSM), a random sample of 25 companies was taken from a

complete list of all New York Stock Exchange companies which split between

January 1, 1963 and January 1, 1974. Companies were excluded for the following reasons: if they split more than once during the period, if they were financial


corporations, if the split was less than 1.25 for 1, or if the split occurred either before January 1, 1964 or after January 1, 1972. A list of the company names, split factors, announcement dates, and split dates is given in Appendix Table A- 1. For each sample company, weekly volume and the number of trading days per week were collected from Standard and Poor's ISL manuals. Also aggregate market volume on the NYSE was compiled.

Table 1 shows the mean, variance, skewness, and kurtosis for weekly volume divided by the split factor and the number of trading days per week. Because of the obvious nonnormality, the natural logarithm of the data was used for the FTSM. However, even in logarithmic form there is significant positive skewness in 16 out of 25 cases.6

Table 2 compares the volume market model (VMM), and the finite time-series model (FTSM), using all 574 observations. A quick look at the average coefficients presented at the bottom of Table 2 shows the predicted downward bias in ,1 in the VMM. By inspection, the FTSM model provides a better fit,7 but how well does it stand up to the predictions about the sign and magnitudes of the coefficients and the negative serial correlation of the residuals?

A two-tailed t-test with the null hypothesis that the coefficient of market volume, f13, is not significantly different from one was run using the coefficient and standard error estimates provided by the FTSM. At the 98% confidence level (t - 2.34), the null hypothesis could not be rejected in 17 out of 25 cases.

A second prediction of the FTSM is that -,82 = 33. This proposition was tested by using an F-test.8 Due to potential nonstationarities (which are demonstrated to exist later in the paper), the F-test was run separately on data up to 72 weeks before the split and on data from 72 weeks after the split to the end of the data. In only three of the 25 cases before the split and five of 25 after the split was there a significant difference between -,/2 and /3. Again, the empirical FTSM model is reasonably close to its theoretical specification.

Finally, the theoretical specification predicts that the intercept term, a, will be less than zero. Table 2 shows that it is less than zero in 14 out of 25 cases and that the average is less than zero. Also, the theory predicts negative serial correlation

6 Pearson [32] has developed confidence limits to test for nonnormality based on skewness. For 574 observations, the 98% confidence limit is .247.

7An attempt was made to use volume turnover instead of volume, but it provided worse fits because the shares outstanding for individual firms were available only quarterly and because shares outstanding for the entire NYSE were not available at all.

'The F-test is computed as follows: first, calculate a "constrained" sum of squared residuals from

ln V1t = a + ,811n Vmt + f82 (ln Vmttl -ln Vt-1) + EJt.

Next, using the above "constrained" sum of squared residuals and the unconstrained sum of squared residuals from the FTSM (using the natural logarithm of each variable), calculate an F-ratio

F (constrained SSR - unconstrained SSR)/n (unconstrained SSR)/(T - k)

where:

n = the number of constraints = 1 k = the number of independent variables plus the constant term = 4 T = the number of observations

Stock Splits 121

Table 1

Mean,

Variance,

Skewness,

and

Kurtosis

for

Average

Daily

Volume

(First

Row)

and

Log

Volume

(Second

Row)

Co.

Co.

No.

Mean

Variance

Skewness

Kurtosis

No.

Mean

Variance

Skewness

Kurtosis

1

66.92

21,663.15

3.46

18.98

13

103.15

18,542.21

11.31

177.81

1.82

4.58

1.00

2.42

4.40

.34

1.24

6.39

2

14.29

149.19

4.64

37.40

14

105.27

7,047.26

3.33

20.29

2.46

.35

.59

3.97

4.45

.36

.64

3.23

3

27.24

401.09

3.18

19.15

15

48.52

1,116.18

1.80

7.26

3.12

.34

.38

3.31

3.67

.43

.04

2.67

4

19.58

702.10

4.33

27.32

16

300.99

60,704.73

2.23

9.45

2.50

.83

.43

3.20

5.46

.46

.44

2.64

5

29.41

758.87

7.03

79.76

17

14.32

173.72

3.50

21.29

3.18

.35

.61

4.22

2.38

.53

.07

3.50

6

133.76

13,679.80

2.33

10.03

18

179.41

17,096.34

1.80

6.75

4.60

.58

.11

2.79

4.98

.40

.41

2.42

7

22.73

501.99

3.97

26.69

19

65.96

1,701.61

3.66

25.63

2.82

.58

.08

3.45

4.06

.22

.69

4.01

8

279.05

38,384.46

1.74

7.63

20

177.17

16,479.90

4.27

39.76

5.40

.49

-.23

2.84

5.01

.31

.49

3.35

9

49.62

1,121.32

4.81

45.20

21

64.64

1,575.05

1.49

5.67

3.77

.24

.53

4.26

4.00

.35

-.06

2.69

10

15.54

167.79

7.77

85.26

22

62.97

3,941.96

2.51

11.41

2.61

.23

.93

6.38

3.74

.83

-.07

2.64

11

13.26

128.56

3.24

17.58

23

195.98

15,886.78

2.41

12.90

2.36

.41

.49

3.60

5.12

.29

.42

2.90

12

288.95

187,710.76

6.22

54.20

24

39.54

741.53

1.84

7.67

5.25

.67

.74

3.83

3.48

.40

.08

2.70

25

278.53

20,956.99

3.47

34.49

5.52

.21

.06

3.02

Table 2

Comparison of Two-Volume Modelsa Co. No. /I t(/31) /3 t(33) /2 t(a2) a t(a) F r2/DW

1 -3.25 -25.47 31.65 26.99 648.0 .53/0.41 .61 3.21 .87 43.50 -1.06 -5.57 4.36 5.19 1717.9 .90/2.65

2 .14 2.72 1.17 2.48 7.4 .01/1.08 .69 4.60 .47 11.75 -.62 -4.13 .67 1.56 59.1 .24/2.16

3 .72 17.71 -3.52 -9.37 313.5 .35/1.27 .40 3.33 .37 9.25 .07 .58 -2.35 -6.18 152.8 .45/2.10

4 1.00 14.90 -6.71 -10.83 222.1 .28/1.17 .83 4.15 .41 10.25 -.23 -.87 -3.99 -6.33 127.9 .40/2.15

5 .53 11.31 -1.64 -3.84 128.0 .18/1.35 .71 5.07 .33 8.25 -.36 -2.57 -1.07 -2.61 70.2 .27/2.11

6 -.46 -7.13 8.78 14.97 50.9 .08/0.62 1.28 9.85 .75 25.00 -1.43 -11.00 2.50 5.56 329.1 .63/2.34

7 .79 13.64 -4.39 -8.29 186.0 .25/0.99 .81 5.06 .50 12.50 -.43 -2.69 -2.11 -4.31 147.7 .44/2.22

8 .51 8.88 .76 1.45 78.8 .12/0.69 1.32 10.15 .67 22.33 -1.18 -8.43 .46 1.18 214.1 .53/2.38

9 .21 5.06 1.81 4.68 25.6 .04/0.99 .98 8.91 .54 13.50 -.90 -7.50 .99 3.00 98.6 .34/2.22

10 .39 10.25 -.97 -2.79 105.1 .16/1.06 .45 4.09 .47 11.75 -.23 -2.09 -.59 -1.84 97.8 .34/2.24

11 .22 4.05 .32 .63 16.4 .03/1.20 .62 3.88 .41 10.25 -.49 -3.06 .25 .53 45.4 .19/2.19

12 -.47 -6.84 9.54 15.17 46.7 .08/0.44 .75 5.77 .81 40.50 -.84 -6.46 1.81 4.02 418.1 .69/2.28

13 .50 10.99 -.22 -.53 120.9 .17/0.96 .94 7.23 .53 13.25 -.70 -5.38 -.14 -.39 130.4 .41/2.24

14 .75 18.07 -2.45 -6.40 326.4 .36/0.83 .85 7.73 .59 19.67 -.53 -4.82 -1.09 -3.30 261.7 .58/2.32

15 .33 6.00 .63 1.23 36.0 .06/1.29 1.03 5.06 .38 9.50 -.84 -4.94 .54 1.13 48.3 .20/2.09

16 .97 22.27 -3.40 -8.53 495.8 .46/0.60 .82 8.20 .70 23.33 -.53 -5.30 -1.06 -3.42 501.0 .73/2.39

17 -.15 -2.41 3.79 6.50 5.8 .01/1.15 .64 3.56 .44 11.00 -.73 -4.05 2.17 3.95 50.9 .21/2.20

18 -.04 -.65 5.31 10.44 .4 .00/0.56 .85 7.08 .74 24.67 -.88 -7.33 1.60 4.32 245.2 .56/2.35

19 .09 2.13 3.26 8.68 4.6 .01/0.68 .66 7.33 .68 22.67 -.64 -6.60 1.05 3.50 171.3 .47/2.33

20 .83 24.72 -2.60 -8.43 611.0 .52/1.37 .77 7.70 .31 7.75 -.19 -1.72 -1.81 -5.84 244.9 .56/2.08

21 .79 19.64 -3.20 -8.72 385.8 .40/0.99 .62 5.64 .51 12.75 -.23 -2.09 -1.61 -4.74 238.6 .56/2.22

22 1.29 22.19 -8.09 -15.15 492.2 .46/1.20 .97 5.71 .40 10.00 -.18 -1.00 -4.97 -8.84 230.0 .55/2.13

23 .66 17.42 -.95 -2.73 303.3 .35/0.74 .76 7.92 .63 21.00 -.52 -5.25 -.33 -1.17 239.9 .60/2.35

24 .53 10.58 -1.41 -3.05 112.0 ,16/1.27 .68 4.53 .37 9.25 -.34 -2.27 -.85 -1.93 72.5 .28/2.11

25 .70 25.65 -.94 -3.72 658.1 .54/0.99 .31 10.13 .51 12.75 -.46 -5.75 -.45 -2.04 360.7 .66/2.21

Ave .303 1.08 215.2 .25/0.96 .794 .533 -.664 -.24 265.4 .50/2.24

The average difference between /82 and -,83 in the FTSM equals -.0488. a VMM: In [Vjt/(f1t)(dt)] = a + /31 In (Vmt/dt).

FTSM: In [Vjtl(fjt)(dt)]= a + ,8 In (Vmt/dt) + ,82 In (Vmti/dti1) + /83 In [Vjt_,1(fjt_j)(dt_j)]. where: fit = split factor for the jth security on the tth week.

dt = the number of trading days in the tth week.

Stock Splits 123

which occurs when the Durbin-Watson statistic is greater than two. This is true in all 25 cases.

The above tests lead to the conclusion that the empirical FTSM model fits its theoretical specification surprisingly well. There is, however, evidence of a downward bias caused by the omitted variable problem. Although only eight out of 25 of the concurrent market volume coefficients were significantly different from their predicted value of one, all but three were less than one. Other possible difficulties with the FTSM model are multicollinearity and heteroscedasticity. Multicollinearity was rejected because of 1) the large sample size, 2) the simple r2 between the collinear variables, ln Vmt, and ln Vmtl was only .9, not particularly high for time series data, 3) when ln Vmt-l was omitted, the adjusted r2 declined 18 out of 25 cases and did not change in the remaining 7, 4) even though multicollinearity biases the t-statistic downward, both 82 and 8 were significant in 21 out of 25 cases and finally, 5) the average partial correlation coefficient was .93. Heteroscedasticity was tested by dividing the time-series data for each company into halves. The ratio of the sums of squared residuals from each half is an F-test, which if shown to be significant, indicates heteroscedasticity. No evidence of heteroscedasticity was found.

Having compared the VMM and the FTSM, it is clear that the latter provides a much better fit of the data and considerably reduces the bias caused by the omitted variable problem. Furthermore, the coefficients of the lagged terms provide estimates of the first-period adjustment coefficient, "a." The estimates average .533 and .664 and range between .31 and .87 for 83 and -.07 and 1.43 for -f82. The larger standard errors for 82 account for its greater range. Using equation (5), an average adjustment coefficient of .533 implies that the full impact of information is felt within 2.75 weeks, while .664 implies 2.01 weeks. However, because of the downward bias caused by the omitted variable problem, we should consider the estimates of "a" to be lower bounds. That is, two to three weeks may be interpreted as the maximum time required for the full impact of a new message to be felt. A second possible interpretation is that two to three weeks measures the average length of message activity. Merton [29] has argued that new information arrival may be "lumpy" in the sense that there are short periods of rapid activity in a stock preceding a significant economic event followed by longer periods of relative quiescehee.

In the next section, the FTSM is used to perform residual analysis on volume before and after stock splits.

III. The Effect of Stock Splits on Trading Volume

There is reason to expect that there are nonstationarities in trading behavior such that the FTSM model must be estimated separately for the pre- and post- split data. Causal factors might include differences in transactions costs as a percentage of the value of assets traded or an increase in the number of shareholders. If nonstationarities exist and data before the split is pooled with data afterward, then residual analysis could be biased. Table 3 shows the result of using the FTSM on data up to 48 weeks before the split data and from 48 weeks


Table 3

Separate

Regressions A)

up to 48

Weeks

before

the

Split

and B)

from 48

Weeks

after

the

Split to

the

end of

the

Data In

Vft = a + /81 In

Vmt + /32

In

Vm,t-1

+

/83 In

Vjt-1 + Ejt

No.

Co.

No.

fI

t(fi1)

02

t(fl2)

3

t(3)

a

t(a)

Obs

lEjt2

1

-48

-.40

-.73

-.09

-.16

.44

4.40

7.21

1.72

80

27.16

+48

1.01

5.32

-.63

-3.32

.37

7.40

-3.17

-3.37

396

119.91

2

-48

.27

.84

.34

1.09

.58

7.25

-4.11

-1.79

125

18.12

+48

1.08

5.40

-.41

-1.95

.35

7.00

-4.76

-3.78

351

109.52

3

-48

.09

.29

-.20

-.65

.33

3.67

2.74

1.21

130

21.29

+48

.48

2.82

-.00

-.00

.34

6.80

-2.29

-2.16

346

77.16

4

-48

.54

1.20

.88

1.91

.31

3.44

-10.56

-2.93

131

50.87

+48

1.42

5.26

-.15

-.54

.34

6.80

-10.22

-5.62

345

187.55

5

-48

.92

3.83

-.61

-2.44

.50

6.25

-1.15

-.63

137

16.20

+48

.91

4.55

-.17

-.81

.21

4.20

-4.40

-3.44

339

103.09

6

-48

1.34

5.36

-.90

-3.46

.63

10.50

-1.85

-1.52

181

37.71

+48

1.62

9.53

-.92

-5.11

.51

10.20

-4.66

-3.79

295

53.81

7

-48

1.17

3.66

-.19

-.59

.45

6.43

-7.08

-4.21

182

58.61

+48

1.01

4.39

-.02

-.09

.35

5.83

-7.85

-4.64

294

98.57

8

-48

1.60

6.15

-.88

-3.14

.66

11.00

-4.36

-3.01

187

42.33

+48

1.58

9.29

-.55

-2.89

.37

7.40

-6.37

-5.13

289

54.75

9

-48

.81

4.76

-.41

-2.41

.33

4.71

-.87

-1.10

207

21.52

+48

1.31

7.71

-.36

-1.89

.35

5.83

-6.68

-5.39

269

47.08

10

-48

.03

.17

.12

.67

.16

2.29

.74

.99

227

30.96

+48

.50

3.13

-.44

-2.75

.59

11.80

.66

.65

249

35.41

11

-48

.99

3.54

-.35

-1.21

.27

3.85

-3.88

-3.18

228

77.01

+48

.64

2.91

-.22

-.96

.25

4.17

-2.29

-1.60

248

70.15

12

-48

.74

3.36

-.59

-.95

.83

20.75

-.28

-.31

235

51.38

+48

1.03

6.06

-.48

-2.66

.53

10.60

-2.99

-2.58

241

41.51

13

-48

.69

3.83

-.48

-2.66

.44

7.33

.56

.80

245

35.32

+48

1.13

5.95

-.52

-2.60

.27

4.50

-2.51

-1.99

231

46.35

14

-48

.31

2.07

-.25

-1.67

.34

5.67

2.22

3.47

246

25.82

+48

1.05

5.83

-.70

-3.68

.61

12.20

-1.47

-1.20

230

42.97

15

-48

.86

3.44

-.58

-2.32

.53

10.60

-.68

-.72

253

72.43

+48

1.51

5.81

-.74

-2.74

.24

3.43

-4.63

-2.60

223

79.18

Stock Splits 125

16

-48

.33

2.06

-.26

-1.63

.55

11.00

1.62

2.61

256

29.84

+48

1.05

7.50

-.46

-3.07

.56

9.33

-2.93

-2.84

220

22.41

17

-48

.97

4.04

-.77

-3.08

.51

10.20

-.46

-.53

275

85.54

+48

.90

3.21

-.25

-.86

.30

4.29

-4.64

-2.33

201

79.46

18

-48

1.12

5.89

-.92

-4.84

.72

18.00

-.32

-.49

282

53.13

+48

.66

3.67

-.79

-4.39

.65

13.00

2.95

2.15

194

29.93

19

-48

.60

4.62

-.54

-4.15

.60

12.00

1.11

2.36

283

26.54

+48

.89

5.24

-.19

-1.05

.49

8.17

-4.62

-3.55

193

24.65

20

-48

.53

3.79

-.15

-1.07

.38

6.33

-.44

-.94

283

29.32

+48

1.18

6.94

-.13

-.68

.18

2.57

-5.63

-4.20

193

27.33

21

-48

.63

4.20

-.12

-.75

.56

11.20

-2.85

-4.75

284

35.72

+48

.64

3.76

-.14

-.78

.42

6.00

-2.38

-1.87

192

26.48

22

-48

.56

2.24

.11

.42

.41

8.20

-4.05

-3.19

287

97.73

+48

1.24

4.28

-.72

-2.40

.39

5.57

-2.40

-1.12

189

72.61

23

-48

.45

3.46

-.32

-2.46

.52

10.40

1.19

2.64

289

27.28

+48

.88

5.50

-.61

-3.59

.65

10.83

-.64

-.54

187

22.25

24

-48

.57

2.85

-.15

-.75

.38

7.60

-1.63

-2.67

347

88.90

+48

1.13

3.32

-.47

-1.38

.28

3.11

-3.73

-1.29

129

38.91

25

-48

.75

7.58

-.33

-3.00

.49

12.25

-.94

-2.94

392

29.66

+48

.83

4.15

-.59

-2.81

.56

6.22

.22

.14

84

4.94

Ave

-48

.72

-.31

.48

-1.12

+48

1.03

-.43

.43

-3.50


after the split to the end of the sample.9 For 21 of the 25 companies, the intercept term decreased after the split, for 19 the coefficient of market volume increased, and for 17 the coefficient of lagged volume for the jth company decreased. This suggests that nonstationarity in the volume relationship follows the split. Note, however, that the average -,82 is approximately equal to the average f3, and that the average Ih is close to one. Also a > 0 and 82 < 0. These are al conditions predicted by the theoretical specification of the FTSM.

The Chow test can be used to test the null hypothesis that the before-split coefficients are not different from the after-split coefficients.'0 It was used on time-series data for each company excluding +48 weeks around the split. F-tests were obtained for each of the following four null hypotheses:

H(a): the intercept terms do not change;

H(f8i): 8i does not change (i = 1, 2, 3).

The results are summarized below:

Summary of Chow F-Test Number of Significant

Changes a /I /2 /3

Positive 1 7 0 2 Negative 10 0 2 5

Total 11 7 2 7

'The choice of 48 weeks was not completely arbitrary. However, because of the slight negative serial correlation, a runs test could not be used to let the data decide how large the interval before the split should be. Instead, we looked at patterns of cumulative average residuals for various intervals. These are shown in Figure 1. Deviations from a random walk do not appear until around 35 weeks before the split. Therefore, 48 weeks was selected as a "safe" interval which would exclude abnormal behavior caused by the split itself.

"' A brief description follows; however, see Chow [7] or Rao and Miller [33, pp. 148-152] for a detailed description. Let Q2 be the sum of squared residuals from the FTSM (using the natural logarithm of the variables) regressed on data up to n weeks (where n = 48) before the split and let Q3 be the sum of squared residuals from n weeks after the split to the end of the data. Next, pool the data from the two separate regressions and run the following equation.

ln Vt = Di, + D2, + ,B,ln Vmt + /l2D,,,_, ln Vmt-i + (/2D2, ,1 ln Vmt-i

+ /33Di, 1 ln V,t-I + 3,,-1 ln IVt-i + Ejt

where

Di,= for i= 1, ***,s- n = {Ofori=s+n, ...,T

where

D {= fori= 1, .., s-n

D2 f1 for i= s +n, *. * , T

n = the number of weeks before or after the split s = split week T = end of data

Stock Splits 127

At the 95% confidence level [for (476, 4) degrees of freedom], the intercept term decreased in ten out of the eleven cases where there was a significant change, and all seven significant changes in Ih (the coefficient of market volume) were increases. Evidence for nonstationarity in the relationship with lagged market volume was nonexistent, and while the evidence suggests that the coefficient of lagged individual security volume decreases, it is inconclusive.

Although it is not possible to draw strong conclusions from the results of equations run on weekly data for 25 companies, the evidence does suggest that stock splits result in nonstationarities in trading behavior. Therefore, the residual analysis was conducted by regressing the FTSM model on data up to -n weeks before the split date. Evidence of nonstationarity precluded use of pooling data before the split with data afterward which was the technique used in FFJR [16].11

Figure 1 shows the plots of cumulative average residuals against time for the FTSM model fit on data up to -24, -48, and -72 weeks before the split. The residual, Ejt, for the jth company in the tth week is the difference between the actual volume and the predicted volume which are both adjusted by the number of trading days per week and the stock split factor.

Ejt= ln Vjt-ln Vjt. (16)

The average and cumulative average residuals are defined respectively as12

1 25

ARt - E Ejt. (17) 25 j=1

T

CARt= E ARt; t= 1, *., n. (18) t=1

Since the volume data have been adjusted by the split factor and because the residual analysis is centered around the split data, a random pattern in the cumulative average residuals would imply that splits have no effect on trading behavior.

The evidence in Figure 1 is quite clear. Post-split volume is proportionately

The procedure is equivalent to constraining ,B, to apply to the entire period set of data. The sum of squared residuals from the above equation is Ql. The F-test is

F= QlQ-(Q2+Q3)/1 (Q2 + Q3)/(m+ n-2k)

where: m + n = the total number of observations = 476 and k = 4. " For the sake of curiosity, the residual analysis was run using the FFJR technique of pooling data

before and after the event and using the estimated coefficients to predict volume ?72 weeks around the split. The result for normalized cumulative average residuals is shown in Appendix Figure A-1. Had nonstationarities not been accounted for, the conclusion would have been that volume rises before the split and falls afterward. This, of course, is incorrect.

12 The same procedure was duplicated using normalized cumulative average residuals instead of cumulative average residuals. The effect was that the scale of the changes became exaggerated, but the pattern of residuals was unaffected. Normalized residuals are obtained by dividing the residual by the standard error of estimate from the FTSM equation used to predict the dependent variable.


'O -'6 .40 -40 -to to 20 50 42 L *7Q2l

?S3 2 1 WEE KI

-'.0 . r \ \ *~72 WUEEEKS

IL.o

A. - 4M)Ovs \

Figure 1. Cumulative average residuals +24, ?48, and ?72 weeks around the split date (FTSM Model)

lower after a split as indicated by continuing negative residuals. There is also evidence that volume begins to decrease before the split. However, it is more difficult to interpret pre-split evidence for two reasons. First, as shown in appendix Table A-1, the announcement date precedes the split date by an average of 12.65 weeks with the earliest announcement date falling 25.5 weeks before its split. Second, we cannot determine at what point the pattern of residuals significantly departs from its pre-split distribution. Apparently, trading activity decreases somewhat in anticipation of the split. However, it is prudent to refrain from drawing any strong conclusions from the pattern of residuals immediately prior to the split date.

Stock Splits 129

IV. The Effect of Stock Splits on Brokerage Revenues and Bid-Ask Spreads

As noted in the introduction, proportionately lower volume following a stock split is consistent with decreased liquidity but is not conclusive because other param- eters may affect volume (e.g., the number of shareholders, the rate of message arrival, and transactions costs). This section of the paper focuses directly on transactions costs. If brokerage revenues and bid-ask spreads increase as a percentage of value traded, we will have further evidence of liquidity decreases in the post-split period.

By using historic price and volume data in conjunction with published brokerage rates, one can simulate the probable effect of stock splits on transactions costs paid by individuals to brokerage houses. During the sample time period, brokerage schedules were fixed for small investors. Both buyer and seller paid brokerage commissions while, in addition, the seller paid New York state transfer taxes, an SEC fee, and federal taxes. Appendix Table A-2 summarizes the transactions costs.13

Weekly brokerage fees and taxes paid in a round-trip transaction were simulated from weekly volume data, average weekly price levels and knowledge of the transactions cost table. The simulation was necessary because the only way to reproduce actual transactions costs is to have continuous transactions data for each stock and to have complete knowledge of the various mechanisms used to avoid scheduled transactions costs. Examples of such mechanisms were give-up

13 Between 1963 and 1975, there were actually three different commission eras. During each, the commission is calculated by means of a simple linear formula:

Commission = B + A (Price).

From the beginning of 1963 until April 6, 1970, the coefficients in Table A-2 labeled Al and Bi were applicable. From April 6, 1970 until March 24, 1972, there was a surcharge added to the commission from the rate schedule. The amount was $15 or 50% of the commission, whichever was less. Finally, from March 24, 1972 until May 1, 1975, when fixed commission rates were abandoned, the coefficients labeled A2 and B2 were used for single round lots. The computation for multiple round lots was somewhat different and can be found at the bottom of Table A-2.

Of the three transactions costs eras, only the last is sufficiently complex to require explanation. Suppose we are dealing with a $36.00 stock which has undergone a two-for-one split. We assume that each trade involves a 200-share block, therefore the amount of money involved in the order is $7,200.00. If volume is 20,900 shares, then the one-way brokerage commission is

$22 + .009($7,200) = $86.80

on the multiple lot order plus $6.00 per round lot. Total round-trip brokerage commissions on 104 two hundred share lots are $20,550.40. The remaining 100 shares pays a round-trip commission of

[$22 = .009(3,600)] = $108.80.

Therefore, total brokerage commissions for a 20,900 share-day are $20,659.20. In addition to the brokerage costs, there are New York state taxes of $1,045.00 and SEC fees of $15.05. This brings the total transactions costs to $21,719.25.

In addition to brokerage commissions, the seller pays New York state taxes, an SEC fee, and federal taxes. The New York state tax had two eras: 1963 to July 1, 1966 and July 1, 1966 to May 1, 1975. The SEC fee is a trivial percentage of the transactions cost. The federal tax was levied from 1963 to January 1, 1966. Table A-2 contains the appropriate rates for all taxes.

130 The Journal of Finance arrangements, "regular-way reciprocity," "institutional membership," and "four- way tickets." Apparently between 40 and 80 percent of the supposedly fixed commissions were remitted to insitutional traders in the late 1960's.14 Therefore, it is possible only to simulate the approximate pattern of brokerage commissions and taxes paid by individual investors who presumably would not escape paying scheduled transactions costs.

For each trading week, transactions costs and taxes were simulated by using the average weekly price and volume. Before splits, all trades were assumed to be 100-share round lots, and after the split the assumed number of round lots per trade were:

No. of Before Split Factor After Companies

100 Shares 2 for 1 200 Shares 19 100 Shares 3 for 1 300 Shares 2 100 Shares 2-1/2 for 1 300 Shares 2 10 Shares 10 for 1 100 Shares 1

100 Shares 1-? /2for 1 200 Shares 1

A transactions cost index for the New York Stock Exchange was computed by assuming that the entire market volume was traded in 100-share lots at the average price of the Dow Jones Industrials. There is no doubt that the index overestimates the level of actual transactions costs. However, for our purposes, it is only necessary to argue that the changes in the level from week to week are highly correlated with the "true" index of transactions costs.

A residual analysis similar to that which was used to investigate the effect of splits on trading volume is applied to simulated transactions costs, Tjt. The regression equation is

ln Tjt = a + 81 ln Tnt + 82 ln Tm t_ + 83 In T, t_1 + Ejt. (19) The above model is used only for the purpose of obtaining the pattern of transactions costs residuals, Ejt. Therefore, although it has no theoretical specification per se, none is needed so long as the usual OLS assumptions are reasonably well approximated. Table 4 shows the results of regressions using data up to 48 weeks before the split and using data from 48 weeks after the split to the end. Note that 521 weekly observations of transactions costs were available for each security. Figure 2 shows cumulative average residuals from values predicted by equations using only data up to n weeks before the split.15

Figure 2 can be compared to Figure 1. Both show that on average the reaction to the anticipated split begins at approximately the 35th week before the split. However, while trading volume increases less than proportionately (a result which has been inferred from the continuing negative residuals in Figure 1) transactions costs increase more than proportionately because residuals are positive following the split.

14 Alfred E. Kahn, The Economics of Regulation: Institutional Issues (New York: Wiley, 1971), p. 196.

1 The pattern of the residual analysis remained unchanged when the sample included only the 19 companies which split two-for-one.

Stock Splits 131

In order to estimate the magnitude of the residual transactions costs following a split, residuals were calculated using parameter estimates based on regressions which used data up to 72 weeks before the split.

E]t = Tjt- & - 1 Tmt - I2 Tmt-_ - i3 Tjt-i (20)

In order to obtain dollar residuals, the equation uses transactions data rather than its logarithmic transformation. Figure 3 plots the cumulative average residuals. From the split date until 72 weeks afterward, the increase in cumulative average residuals is $1,478,000. This is $20,528 per week per stock.

For the split week and each of the following 72 weeks, the ratio of the residual transactions costs to the predicted level of transactions was computed.

Percent residual = E jj 1, 25. Pit

Then the mean and median percent residuals were calculated across all 25 sample companies in a given time period. Across all time periods, the average increase in transactions costs was 27.51%. The range was from -2.99% (43rd week following the split) to 94.47% (49th week). However, because of skewness in the residuals, there is reason to believe that the median may be a better measure of location. The average of the medians across all time periods was 7.92%.16 The range was from -20.58% (51st week) to 56.30% (7th week). This evidence, along with Figures 2 and 3 leads to the result that post-split brokerage revenues paid by individual traders increased in the post-split period. This is true in spite of the earlier results which showed that proportional volume decreased."7

The second major transactions cost is the bid-ask spread."8 Unfortunately, the bid-ask spreads for the sample of NYSE stocks used in the study were never recorded. Papers by Demsetz [11], West and Tinic [38], Tinic [36], and Benston and Hagerman [6] show that the bid-ask spread as a percentage of value is inversely correlated with volume and the number of competing dealers, and positively correlated with price variance. If the latter two variables are unaffected by splits, the fact that volume increases less than proportionately would imply higher bid-ask spreads in the post-split era. In order to test this proposition, a

16 The SEC fee as a percentage of transactions costs is trivial and can be ignored. Figure 4 shows transactions costs as a percent of the value of a round block, and the New York State Transfer Tax as a percentage of transactions costs for a round-trip transaction. New York state taxes are never more than 6.25% of the total cost. Consequently, at least 90 percent of the observed 7.92% increase in transactions costs is attributable to increased brokerage revenues. In other words, brokerage revenues increased by at least 7.1%.

17 If accurate cost data for clearing and depository expenses of brokerage houses were available, it might be possible to estimate the effect of stock splits on brokerage industry profitability at the margin. Then one might speculate on the motivation of the brokerage industry with regard to stock splits. The only statement we could find was that "Clearing charges and related depository charges ... represent about 5 percent of operating costs" (Weinberg et al. [37]). If this is true, then it appears that brokerage industry revenue changes exceed cost changes following stock splits.

18 Demsetz [11] provides a clear definition of transactions costs on the NYSE. "On the NYSE two elements comprise almost all of transaction cost-brokerage fees and bid-ask spreads. Transfer taxes could [also] be included...." The bid-ask spread is the transactions cost which is paid for predictable immediacy of exchange in organized markets. Although Garbade and Silber [19] show that the expected liquidity cost may not equal the bid-ask spread in markets where there are competing specialists, we assume that the bid-ask spread is monotonically related to the liquidity cost.


Table 4

Separate

Regressions A)

up to 48

Weeks

before

the

Split

and B)

from 48

Weeks

after

the

Split to

the

end of

the

Data In

Tjt = a +

P3i In

Tit + /?2

In

Tit-i + /83

In

Th,t-i + Ejt

Co.

No.

',

t(/31)

/2

t(/2)

/3

t(/3)

a

t(a)

r2/DW

obs

1

+48

.06

.13

-.35

-.84

.47

4.74

8.41

2.68

.23/2.14

81

-48

1.09

7.32

-.67

-4.32

.62

14.31

.01

.02

.57/2.40

344

2

+48

.97

4.17

-.14

-.57

.54

7.03

-2.09

-1.42

.52/2.09

126

-48

.85

5.85

-.35

-2.27

.37

6.90

1.45

1.99

.33/2.03

299

3

+48

.58

2.54

-.67

-2.98

.34

3.97

6.49

3.64

.15/2.16

131

-48

.65

4.87

.03

.22

.35

6.37

.81

1.22

.44/2.11

294

4

+48

1.34

4.03

.31

.87

.26

3.08

-6.30

-2.71

.39/2.00

132

-48

1.08

5.39

-.19

-.93

.26

4.48

-.67

-.67

.32/2.08

293

5

+48

.72

3.73

-.41

-2.08

.48

6.27

2.44

1.79

.29/1.92

138

-48

.61

3.84

.01

.07

.23

3.96

2.69

3.30

.25/2.07

287

6

+48

1.38

7.06

-.87

-4.14

.71

13.60

-.74

-.80

.73/2.10

182

-48

1.29

9.07

-.92

-6.04

.64

12.93

.76

1.04

.61/2.28

243

7

+48

1.27

5.51

-.31

-1.24

.43

6.49

-2.39

-2.20

.50/2.15

183

-48

.88

5.03

-.12

-.67

.26

4.12

.48

.54

.35/2.10

242

8

+48

1.14

5.41

-.43

-1.92

.70

13.01

-1.95

-1.87

.75/2.30

188

-48

1.09

7.81

-.35

-2.28

.44

7.52

.27

.71

.59/2.29

237

9

+48

.76

6.08

-.45

-3.44

.35

5.38

4.05

5.71

.30/2.07

208

-48

1.17

8.87

-.34

-2.28

.30

4.64

.05

.08

.56/2.00

217

10

+48

.14

1.04

.04

.29

.16

2.35

5.59

7.41

.05/1.99

228

-48

.64

5.25

-.53

-4.29

.59

10.03

2.89

3.77

.39/2.30

197

11

+48

1.15

5.54

-.46

-2.12

.28

4.43

.57

.63

.30/2.13

229

-48

.64

3.72

-.21

-1.18

.21

2.93

3.32

3.38

.25/2.11

179

12

+48

1.27

7.75

-1.08

-6.26

.86

26.14

.10

.14

.81/2.20

236

-48

.72

4.99

-.50

-3.34

.64

11.36

2.23

2.59

.50/2.24

189

13

+48

.70

5.35

-.51

-3.78

.42

7.24

4.48

6.25

.26/2.17

246

-48

.97

5.24

-.50

-2.55

.27

3.75

3.91

3.88

.25/2.11

179

14

+48

.64

5.38

-.56

-4.66

.35

5.64

6.07

7.95

.16/2.09

247

-48

.96

5.76

-.54

-3.06

.49

7.38

2.50

2.85

.45/2.26

178

15

+48

1.01

5.64

-.68

-3.67

.53

9.78

1.83

2.38

.38/2.13

254

-48

1.31

5.39

-.72

-2.83

.21

2.81

3.24

2.54

.23/1.96

171

Stock Splits 133

16

+48

.65

5.88

-.56

-4.78

.54

9.92

4.44

6.25

.32/2.33

257

-48

1.07

8.55

-.51

-3.56

.44

6.31

2.21

3.15

.61/2.06

168

17

+48

.92

5.22

-.65

-3.53

.53

10.31

1.89

2.59

.38/2.20

276

-48

.67

2.64

-.26

-1.02

.28

3.48

2.81

1.82

.14/2.09

149

18

+48

.86

6.32

-.55

-3.87

.74

18.23

.52

.98

.71/2.24

283

-48

.74

4.21

-.83

-4.86

.61

8.98

5.60

3.78

.41/2.28

142

19

+48

.68

7.26

-.54

-5.58

.60

12.55

2.93

5.78

.45/2.14

284

-48

.89

5.17

-.11

-.55

.44

5.76

-1.02

-.86

.49/2.22

141

20

+48

.72

7.45

-.32

-3.06

.33

5.90

4.03

8.03

.43/2.04

284

-48

1.16

6.71

-.19

-.99

.13

1.56

2.02

1.68

.39/2.04

141

21

+48

.80

7.51

-.32

-2.73

.54

10.87

.73

1.75

.65/2.19

285

-48

.93

5.27

-.30

-1.53

.39

4.96

1.03

.83

.38/2.20

140

22

+48

.78

4.48

-.12

-.68

.39

7.17

.62

.91

.42/2.10

288

-48

.90

3.29

-.31

-1.09

.33

4.01

2.13

1.10

.22/2.16

137

23

+48

.61

6.30

-.48

-4.85

.54

10.30

4.03

6.62

.33/2.22

290

-48

1.02

6.46

-.84

-5.08

.70

11.60

1.98

1.65

.57/2.54

135

24

+48

.62

4.75

-.24

-1.76

.36

7.24

3.03

5.34

.29/2.06

348

-48

.84

2.19

-.49

-1.25

.19

1.63

5.69

1.71

.09/2.06

77

25

+48

.83

12.67

-.46

6.15

.48

10.70

3.09

8.39

.63/2.17

393

-48

1.10

4.52

-.64

-2.19

.52

3.25

2.07

1.03

.59/1.71

32

Ave

+48

.82

-.43

.48

2.08

-48

.93

-.41

.40

1.94


8.0

7.0

K 72 LJEEKS

510

L1.0

_i.b~OO~O-o - io -50 '0 /t 50 607q S 0

Figure 2. Cumulative average residuals of brokerage revenues and taxes ?24, +48, and +72 weeks around the split date

random sample of 162 stock splits of OTC stocks between 1968 and 1976 was collected. The sample was stratified to select five splits from each quarter, and if there were fewer than five splits in a given quarter, the sample in that quarter was exhaustive. In each of three comparisons, the bid-ask spread as a percentage of the bid price on -n trading days before the split was compared with the same statistic +n days after the split (where n = 1, 20, 40). Table 5 summarizes the results. The bid-ask spread increased as a percentage of the bid price in 89.5%, 79.0%, and 74.34% of the paired observations for one, twenty, and forty days respectively. Also the average bid-ask spread increased from 4.85% of the bid price one day before the split to 7.03% one day afterward, from 4.73% twenty trading days before to 6.54% twenty days afterward, and from 4.95% to 6.79% in the forty trading-day interval.

A small-sample one-tailed t-test for the difference in means iS also presentea in Table 5.19 Although significant positive skewness in the observations biases the

'9 The t-test used was the small sample t-test where

t = Al - A2-

/2 '22

Vni n2

Stock Splits 135

Millions of $

1.5

4611 41

40 ..5o .40-t 40 .tO -1 e 1 s0 sio s0 60 T0

Figure 3. Cumulative average (dollar) brokerage revenues and taxes ?72 weeks around the split date

t-test downward for observations within a given year, the pooled data (across all years) have less bias and show significant differences at the 5% confidence level for each of the three comparisons. A nonparametric signs test, which is not biased by the skewness in the observed bid-ask spreads, shows that in 17 out of the 23 within-years' comparisons, the likelihood of observing the results is less than 5%.

a = sample variance

n = number of observations

A = sample mean

The adjusted degrees of freedom (see Hoel [23], pp. 275-279) is computed as follows:

( 2 A 2 2

cni n2

df= - 2 (2)2 ( 2)2

ni+1 n2+ 1


Table 5

Bid

Ask

Spreads as a

Percentage of

the

Bid

Price

Number of

One

Tail

Binomial

Avg.

Bid-Ask +

Bid

Decreased

Spreads

Probabilityb

Before,

After

t-test/Adjusted

Df.

Year

?1

+20

?40

?1

+20

?40

?1

?20

?40

?1

?20

?40

1968

3/20

-

-

.002

-

-

3.62,

4.69

-

-

1.43/40

-

-

1969

3/20

7/20

6/17

.002

.132

.174

4.64,

5.89

4.94,

6.01

4.75,

5.68

1.79/40

1.22/38

.98/32

1970

2/13

4/13

2/12

.011

.133

.019

5.55,

7.58

4.66,

5.50

4.80,

6.96

1.18/22

.51/27

1.24/20

1971

2/18

7/18

8/17

.001

.240

.500

2.34,

3.36

2.46,

3.30

4.95,

3.24

1.75/31

1.37/33

.19/35

1972

2/20

2/18

3/18

.000

.001

.004

2.64,

3.91

2.67,

3.56

2.50,

3.80

2.19/36a

1.55/38

1.61/28

1973

2/20

2/20

3/20

.000

.000

.002

3.81,

5.59

3.44,

5.85

3.46,

6.34

1.78/37

2.08/32a

2.59/32a

1974

0/11

0/11

0/11

.001

.000

.000

5.27,

10.04

5.34,

8.95

5.14,

9.00

2.54/16a

2.25/22a

2.08/21a

1975

2/20

3/18

7/18

.000

.004

.252

9.52,

14.13

9.94,

13.38

10.98,

13.19

1.17/35

.74/35

.45/34

1976

1/20

-

-

.000

-

-

6.45,

9.26

-

-

1.49/35

-

-

17/162

25/118

29/113

4.85,

7.03

4.73,

6.54

3.11,

6.79

Pooled

data: Normal

Approximation to

Binomial

Probability

T-test/DF

?1

10.06a

3.24/161a

?20

6.33a

2.15/119a

?40

5.17a

1.97/113a

a

Significant at

5%

confidence

level.

b

For m

decreases

out of N

observations,

this is

N

q

pxqN-xwhere p = q = .5

under

the

null

hypothesis.

x=o

\

/

Stock Splits 137

% of 100 Share Block Value

6~~~I A .<~ '~~~~

'V - -- 2

'3 t 3

2 4

Price Per

to to 30 10 SO Share

Figure 4. Transactions costs on a round-trip transaction

1. NYS Tax as of % of transactions cost 1/1/66 2. NYS Tax as of % of transactions cost 1/1/73 3. Transactions cost 1/1/73 4. Transactions cost 1/1/66

A normal approximation to the binomial for the pooled data shows that the null hypothesis, that the pre-split and post-split distributions are identical, can be rejected in every case. Although the evidence is limited by the lack of extensive time-series data, cross-section comparisons at levels +1, 20 and 40 trading days from the split date suggest that there are statistically significant increases in the bid-ask spread as a percent of the bid price and that the increase persists (up to at least two months following the split). This is consistent with proportionately lower volume and higher brokerage revenues.

V. Conclusion

A finite time-series model (FTSM) of trading volume for individual securities was developed from the assumption that trading during the current time period depends on messages arriving during the current and recent calendar intervals. When tested on approximately 11 years of weekly volume data for a random sample7 of 25 NYSE companies whose stocks had split, the FTSM model proved to be superior to the volume market model (VMM). After the FTSM model was


shown to satisfy the properties expected from its theoretical specification, it was used to estimate the speed of adjustment to new information. One hundred percent of the response of volume to new information took place in less than two to three weeks.

The FTSM model was then used to show that (a) nonstationarities in trading behavior follow stock splits, and (b) volume increases less than proportionately after stock splits. Next, brokerage -commissions and taxes paid by small investors were simulated for actualtrading behavior. Results show that brokerage revenues increased by at least 7.1% following splits. Finally, a sample of 162 OCT bid-ask spreads were collected for 1, 20 and 40 days before a two-for-one stock split and compared with the post-split spreads. The results showed that post-split bid-ask spreads increase significantly as a percentage of the value of the stock.

Taken together, these results lead to the conclusion that there is a permanent decrease in relative liquidity following the split. Evidence for a permanent rather than a temporary effect consists of three results: (1) Chow tests which show significant differences between pre- and post-split relationships, (2) patterns of volume residuals which do not change as the pre-split interval is increased from 24 to 72 weeks, and (3) patterns of brokerage cost residuals which are similar to volume in the pre-split period, but move in the opposite direction during the post- split period. These results would not obtain if abnormal liquidity were a temporary phenomenon confined to a short interval around the split date.

That brokerage fees and bid-ask spreads are higher after splits is consistent with abnormally favorable pre-split liquidity followed by normal post-split liquidity or the opposite, namely normal pre-split liquidity followed by abnormally low post-split liquidity. However, either way shareholders may interpret a split as a message of relatively lower liquidity following the split event. This directly contradicts the hypothesis that splits are motivated by a desire for "wider" or more liquid markets.

If liquidity is relatively lower in the post-split era, why do shareholders agree to splits? Surely, they will not inflict net losses upon themselves. However, it is not possible to conclude from the results of this study that they do. One can argue that benefits may exist from splitting which equal or exceed the liquidity costs documented here. For example, it is somewhat cheaper to diversify after a split because smaller values can be traded at round-lot transactions costs. Also, splits may have a value as messages which forecast anticipated dividend increases.

Finally, it is interesting (but not necessarily germane) to speculate why liquidity is relatively lower following a split. One possibility is that the rate of information arrival is higher before the split because companies which decide to split were doing comparatively well relative to the market. Lower post-split rates of information arrival might imply lower volume. Another possibility is that the portion of volume motivated by portfolio rebalancing may decrease after the split because individuals may buy and sell proportionately fewer shares after the split in order to achieve desired portfolio weights. However, these remain open questions in the interesting problem of why stocks split.

Stock Splits 139

-10 -60-50 -4O -3O .20 -0o t IO 0o )30 at So 6 70

Figure A-1. Cumulative average residuals ?72 weeks around the split date using the FTSM model and the FFJR technique which pools data before and after the split

Appendix

Table A-1

25 Sample Companies Date of:

Split Announce- Name Factor ment Split A Days A Weeks

1. Superior Oil 10 for 1 4/5/65 6/16/65 72 10.29 2. Koppers Co. 2 for 1 2/1/66 4/27/66 85 12.14 3. Standard Brands 2 for 1 1/28/66 6/1/66 124 17.71 4. Hammermill Paper 2 for 1 2/18/66 6/9/66 111 15.85 5. Allied Stores 5 for 2 4/20/66 7/18/66 89 12.71 6. Continental Airlines 3 for 1 1/31/67 5/22/67 111 15.85 7. Revere Copper & Brass 2 for 1 3/1/67 6/1/67 92 13.14 8. Eastern Airlines 2 for 1 5/24/67 7/3/67 40 5.71 9. Borg Warner 2 for 1 9/18/67 11/24/67 67 9.57

10. Adams Express 2 for 1 2/28/68 4/8/68 40 5.71 11. Eagle Picher 2 for 1 2/27/68 4/15/68 18 2.57 12. Texasgulf 3 for 1 12/8/67 6/4/68 178 25.43 13. Goodrich 3 for 2 6/20/68 8/12/68 53 7.57 14. Intl. Nickel 5 for 2 5/7/68 8/19/68 104 14.86 15. Hilton Hotels 2 for 1 8/5/68 10/8/68 64 9.14 16. Gulf Oil 2 for 1 7/24/68 10/28/68 96 13.71 17. Neptune Meter 2 for 1 1/23/69 3/11/69 47 6.71 18. Burroughs 2 for 1 1/16/69 4/29/69 103 14.71 19. Ntl. Distillers 2 for 1 2/24/69 5/6/69 71 10.14 20. Goodyear 2 for 1 2/12/69 5/7/69 84 12.00 21. Ntl. Gypsum 2 for 1 NA 5/16/69 NA NA 22. Mead Corp. 2 for 1 4/25/69 6/3/69 98 14.00 23. Phillips Petroleum 2 for 1 2/11/69 6/16/69 125 17.86 24. Owens Corning 2 for 1 5/5/70 7/28/70 84 12.00 25. General Electric 2 for 1 12/21/70 6/8/71 169 24.14


Table A-2

Taxes and Commission Rates on a Round Lot (100 Shares) Commission = A (Price) + B

New York State Tax 1/1/63-4/6/70 3/24/72-5/1/75

Price/Share 63-66 66-73 Al Bi B2 B2

3.99 $1.00 $1.25 2% $ 3.00 2% $ 6.40 4.00-4.99 1.00 1.25 1% 7.00 2% 6.40 5.00-7.99 2.00 2.50 1% 7.00 2% 6.40 8.00-9.99 2.00 2.50 1% 7.00 1.3% 12.00

10.00-19.99 3.00 3.75 1% 7.00 1.3% 12.00 20.00-23.99 4.00 5.00 1% 7.00 1.3% 12.00 24.00-24.99 4.00 .5.00 5% 19.00 1.3% 12.00 25.00-49.99 4.00 .5.00 5% 19.00 9% 22.00

50.00 4.00 5.00 .1% 39.00 .9% 22.00

Notes: 1. New York State tax changed on July 1, 1966. 2. Between April 6, 1970 and March 24, 1972, a commission surcharge (for 100 shares or less) is

added to the schedule applicable to the period January 1, 1963 to April 6, 1970. The surcharge is 50% of the commission or $15, whichever is less.

3. Between March 24, 1972 and May 1, 1975, the multiple round-lot computation is as follows:

Money Involved in Order Minimum Commission

$ 100- 2,500 1.3% plus $12.00 2,500- 20,000 .9% plus $22.00

20,000- 30,000 .6% plus $82.00 30,000-500,000 .4% plus $142.00

Plus (for each round lot)

1st to 10th round lot $6 per round lot 11th round lot and above $4 per round lot

4. The SEC fee paid by the seller (1963 to 1973) is one cent for each $500 or fraction thereof of the money involved.

5. Federal tax paid by the seller (1963 to January 1, 1966) is four cents on each $100 of the money involved. The minimum tax is 4,, and the maximum tax is 8, per share on stocks worth more than $200/share.

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Stock Splits 141

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liquidity changes following stock splits

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