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Title: Market Trading Volume
Description: One of the most fundamental notions of economics is the determination of prices through the interaction of supply and demand. The remarkable amount of information contained in equilibrium prices has been the subject of countless studies, both theoretical and empirical, and with respect to financial securities, several distinct literatures devoted solely to prices have developed.1 Indeed, one of the most well-developed and most highly cited strands of modern economics is the asset-pricing literature. However, the intersection of supply and demand
Description: One of the most fundamental notions of economics is the determination of prices through the interaction of supply and demand. The remarkable amount of information contained in equilibrium prices has been the subject of countless studies, both theoretical and empirical, and with respect to financial securities, several distinct literatures devoted solely to prices have developed.1 Indeed, one of the most well-developed and most highly cited strands of modern economics is the asset-pricing literature. However, the intersection of supply and demand
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Stock Market Trading Volume
Andrew W
...
However, while many economic models of financial markets have been developed
to explain the behavior of prices—predictability, variability, and information content—far
less attention has been devoted to explaining the behavior of trading volume
...
Our theoretical contributions include: (1) an economic
definition of volume that is most consistent with theoretical models of trading activity; (2)
the derivation of volume implications of basic portfolio theory; and (3) the development of an
intertemporal equilibrium model of asset market in which the trading process is determined
endogenously by liquidity needs and risk-sharing motives
...
∗
MIT Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142–1347, and NBER
...
SBR–9709976) is gratefully acknowledged
...
1 Notation
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2 Motivation
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3 Defining Individual and Portfolio Turnover
2
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3
5
5
8
9
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3 The Data
10
4 Time-Series Properties
4
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4
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11
14
19
5 Cross-Sectional Properties
5
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5
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5
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27
33
37
42
6 Volume Implications of Portfolio Theory
6
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6
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6
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46
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51
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7 Volume Implications of Intertemporal Asset-Pricing Models
7
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7
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7
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7
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7
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59
63
68
74
85
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The remarkable amount of information contained in
equilibrium prices has been the subject of countless studies, both theoretical and empirical,
and with respect to financial securities, several distinct literatures devoted solely to prices
have developed
...
However, the intersection of supply and demand determines not only equilibrium prices
but also equilibrium quantities, yet quantities have received far less attention, especially in
the asset-pricing literature (is there a parallel asset-quantities literature?)
...
Through theoretical and empirical analysis, we
seek to understand the motives for trade, the process by which trades are consummated,
the interaction between prices and volume, and the roles that risk preferences and market
frictions play in determining trading activity as well as price dynamics
...
g
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We argue that turnover—shares traded
divided by shares outstanding—is a natural measure of trading activity when viewed in the
context of standard portfolio theory and equilibrium asset-pricing models
...
Using weekly turnover data for individual
securities on the New York and American Stock Exchanges from 1962 to 1996—recently made
available by the Center for Research in Securities Prices—we document in Sections 4 and 5
the time-series and cross-sectional properties of turnover indexes, individual turnover, and
portfolio turnover
...
5% in 1962, reaching a high of 4% in October 1987, and dropping to just over
1% at the end of our sample in 1996
...
1
fairly concentrated in the early 1960’s, much wider in the late 1960’s, narrow again in the mid
1970’s, and wide again after that
...
To investigate
the cross-sectional variation of turnover in more detail, we perform cross-sectional regressions
of average turnover on several regressors related to expected return, market capitalization,
and trading costs
...
6% to 44
...
This suggests the possibility of a parsimonious linear-factor representation of
the turnover cross-section
...
To
investigate these implications empirically, we perform a principal-components decomposition
of the covariance matrix of the turnover of ten portfolios, where the portfolios are constructed
by sorting on turnover betas
...
Finally, to investigate the dynamics of trading volume, in Section 7 we propose an intertemporal equilibrium asset-pricing model and derive its implications for the joint behavior
of volume and asset returns
...
2 As a result, investors wish to hold two
distinct portfolios of risky assets: the market portfolio and a hedging portfolio
...
In equilibrium, investors trade
in only these two portfolios, and expected asset returns are determined by their exposure to
these two risks, i
...
, a two-factor linear pricing model holds, where the two factors are the
returns on the market portfolio and the hedging portfolio, respectively
...
From the trading volume of individual stocks, we
2
One example of changes in market conditions is changes in the investment opportunity set considered
by Merton (1973)
...
We find that the hedging-portfolio returns
consistently outperforms other factors in predicting future returns to the market portfolio,
an implication of the intertemporal equilibrium model
...
We conclude with suggestions for future research in Section 8
...
The literature on trading activity in financial markets is extensive and a number
of measures of volume have been proposed and studied
...
Other studies use aggregate turnover—the total number of shares traded divided by the total number of shares outstanding—as a measure of volume (see Campbell, Grossman, Wang
(1993), LeBaron (1992), Smidt (1990), and the 1996 NYSE Fact Book)
...
Studies
focusing on the impact of information events on trading activity use individual turnover as a
measure of volume (see Bamber (1986, 1987), Lakonishok and Smidt (1986), Morse (1980),
Richardson, Sefcik, Thompson (1986), Stickel and Verrecchia (1994))
...
And even the total number of trades (Conrad, Hameed, and Niden (1994)) and the number
of trading days per year (James and Edmister (1983)) have been used as measures of trading
activity
...
These differences suggest that different applications call for
different volume measures
...
After developing some basic notation in Section 2
...
2 and provide some economic motivation for turnover as a canonical measure of
3
See Karpoff (1987) for an excellent introduction to and survey of this burgeoning literature
...
4
trading activity
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3–2
...
Theoretical justifications
for turnover as a volume measure are provided in Sections 6 and 7
...
1
Notation
Our analysis begins with I investors indexed by i = 1,
...
, J
...
For each stock j, let N jt
be its total number of shares outstanding, Djt its dividend, and Pjt its ex-dividend price at
date t
...
e
...
, J
...
Let
Pt ≡ [ P1t · · · PJt ] and St ≡ [ S1t · · · SJt ] denote the vector of stock prices and shares
held in a given portfolio, where A denotes the transpose of a vector or matrix A
...
Finally, denote by Vjt the total
number of shares of security j traded at time t, i
...
, share volume, hence
Vjt =
where the coefficient
1
2
1
2
I
i
i
|Sjt − Sjt−1 |
(1)
i=1
corrects for the double counting when summing the shares traded
over all investors
...
2
Motivation
To motivate the definition of volume used in this paper, we begin with a simple numerical
example drawn from portfolio theory (a formal discussion is given in Section 6)
...
For concreteness, assume that
security A has 10 shares outstanding and is priced at $100 per share, yielding a market value
of $1000, and security B has 30 shares outstanding and is priced at $50 per share, yielding
a market value of $1500, hence Nat = 10, Nbt = 30, Pat = 100, Pbt = 50
...
Specifically, let investor 1 hold 1 share of A and 3
5
shares of B, and let investor 2 hold 9 shares of A and 27 shares of B
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Now suppose that investor 2 liquidates $750 of his portfolio—3 shares of A and 9 shares
of B—and assume that investor 1 is willing to purchase exactly this amount from investor 2
at the prevailing market prices
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What kind
of trading activity does this transaction imply?
For individual stocks, we can construct the following measures of trading activity:
•
•
•
•
•
Number of trades per period
Share volume, Vjt
Dollar volume, Pjt Vjt
Relative dollar volume, Pjt Vjt /
Share turnover,
j
Pjt Vjt
τjt ≡
Vjt
Njt
• Dollar turnover,
νjt ≡
Pjt Vjt
= τjt
Pjt Njt
where j = a, b
...
If investor
1 is unwilling to purchase these shares at prevailing prices, prices will adjust so that both parties are willing to
consummate the transaction, leaving two-fund separation intact
...
5
Although the definition of dollar turnover may seem redundant since it is equivalent to share turnover,
it will become more relevant in the portfolio case below (see Section 2
...
6
Volume Measure
A
B
Aggregate
Number of Trades
1
1
2
Shares Traded
3
9
12
Dollars Traded
$300
$450
$750
Share Turnover
0
...
3
0
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3
0
...
3
Share-Weighted Turnover
—
—
0
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3
Value-Weighted Turnover
—
—
0
...
• Value-weighted turnover,
τtV W ≡
Vat
Vbt
Pbt Nb
Pat Na
+
Pat Na + Pbt Nb Na Pat Na + Pbt Nb Nb
= ωat τat + ωbt τbt
...
Though these values vary considerably—
2 trades, 12 shares traded, $750 traded—one regularity does emerge: the turnover measures
are all identical
...
If all
investors hold the same relative proportions of risky assets at all times, then it can be shown
that trading activity—as measured by turnover—must be identical across all risky securities
(see Section 6)
...
For this reason, we will use turnover as the
measure of volume throughout this paper
...
7
2
...
Although we define the turnover ratio using the total number of shares traded, it is obvious
that using the total dollar volume normalized by the total market value gives the same result
...
But even after settling
on turnover as the preferred measure of an individual stock’s trading activity, there is still
some ambiguity in extending this definition to the portfolio case
...
6 For the specific purpose of investigating the
implications of portfolio theory and ICAPM for trading activity (see Section 6 and 7), we
propose the following definition:
p
p
Definition 2 For any portfolio p defined by the vector of shares held S tp = [ S1t · · · SJt ] with
p
non-negative holdings in all stocks, i
...
, Sjt ≥ 0 for all j, and strictly positive market value,
p
p
i
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, Stp Pt > 0, let ωjt ≡ Sjt Pjt /(Stp Pt ) be the fraction invested in stock j, j = 1,
...
Then its turnover is defined to be
τtp
J
≡
p
ωjt τjt
...
e
...
Such diversity in the trading of portfolios makes it difficult to
define single measure of portfolio turnover
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, J
...
While τtV W and τtEW are relevant to the theoretical implications
derived in Section 6 and 7, they should be viewed only as particular weighted averages of
individual turnover, not necessarily as the turnover of any specific trading strategy
...
Suppose, for example, shortsales
are allowed so that some portfolio weights can be negative
...
We
can modify (3) to account for short positions by using the absolute values of the portfolio
weights
τtp ≡
p
|ωjt |
p τjt
k |ωkt |
J
j=1
(5)
but this can yield some anomalous results as well
...
If the turnover of both stocks are identical and equal
to τ , the portfolio turnover according to (5) is also τ , yet there is clearly a great deal more
trading activity than this implies
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Neither (3) or (5) are completely satisfactory measures of trading activities of a portfolio
in general
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2
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Although there are several other alternatives, e
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, summing share
9
volume and then dividing by average shares outstanding, summing turnover offers several
advantages
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7 Therefore, we shall adopt this measure of
time aggregation in our empirical analysis below
...
3
(6)
The Data
Having defined our measure of trading activity as turnover, we use the University of Chicago’s
Center for Research in Securities Prices (CRSP) Daily Master File to construct weekly
turnover series for individual NYSE and AMEX securities from July 1962 to December 1996
(1,800 weeks) using the time-aggregation method discussed in Section 2
...
8 We choose a weekly horizon as the best compromise
between maximizing sample size while minimizing the day-to-day volume and return fluctuations that have less direct economic relevance
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9 We also omit NASDAQ stocks altogether since the differences between NASDAQ and the NYSE/AMEX (market structure, market capitalization, etc
...
However, stock splits can have non-neutral effects on trading activity
such as enhancing liquidity (this is often one of the motivations for splits), and in such cases turnover will
be affected (as it should be)
...
We have also prepared a set of access routines to read our extracted datasets via either
sequential and random access methods on almost any hardware platform, as well as a user’s guide to MiniCRSP (see Lim et al
...
More detailed information about MiniCRSP can be found at the website
http://lfe
...
edu/volume/
...
For example, on January 2, 1980, the entire
10
implications for the measurement and behavior of volume (see, for example, Atkins and Dyl
(1997)), and this should be investigated separately
...
Finally, in addition to the exchange and sharecode selection criteria imposed, we also
discard 37 securities from our sample because of a particular type of data error in the CRSP
volume entries
...
11 These
characteristics are presented in Figure 1 and in Tables 3 and 4
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20% to over 1% per week
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However, equal-weighted turnover behaves somewhat differently: Figure 1b shows that it reaches a peak of nearly 2% in 1968, then declines
until the 1980’s when it returns to a similar level (and goes well beyond it during October 1987)
...
NYSE/AMEX universe contained 2,307 securities with sharecode 10, 30 securities with sharecode 11, and
55 securities with sharecodes other than 10 and 11
...
10
Briefly, the NYSE and AMEX typically report volume in round lots of 100 shares—“45” represents 4500
shares—but on occasion volume is reported in shares and this is indicated by a “Z” flag attached to the
particular observation
...
In some instances, we have discovered daily share volume increasing by a
factor of 100, only to decrease by a factor of 100 at a later date
...
See Lim et al
...
11
These indexes are constructed from weekly individual security turnover, where the value-weighted index
is re-weighted each week
...
Note that these return indexes do not correspond exactly to the time-aggregated CRSP valueweighted and equal-weighted return indexes because we have restricted our universe of securities to ordinary
common shares
...
11
Since turnover is, by definition, an asymmetric measure of trading activity—it cannot
be negative—its empirical distribution is naturally skewed
...
Although a trend is still present, there is more evidence for cyclical behavior in both indexes
...
Over the entire sample
the average weekly turnover for the value-weighted and equal-weighted indexes is 0
...
91%, respectively
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48% and 0
...
62 for the valueweighted turnover index and 0
...
In contrast, the
coefficients of variation for the value-weighted and equal-weighted returns indexes are 8
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91, respectively
...
12
Value-Weighted Turnover Index
Equal-Weighted Turnover Index
4
4
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1965
1970
1975
1980
1985
1990
1995
1965
1970
1975
Year
1980
1985
1990
1995
Year
(a)
(b)
13
Log(Value-Weighted Turnover Index)
Log(Equal-Weighted Turnover Index)
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-2
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1995
Table 4 illustrates the nature of the secular trend in turnover through the five-year
subperiod statistics
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25%
and 0
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25% and 1
...
At the beginning of the sample, equalweighted turnover is three to four times more volatile than value-weighted turnover (0
...
07% in 1962–1966, 0
...
08% in 1967–1971), but by the end of the sample
their volatilities are comparable (0
...
23% in 1992–1996)
...
These anomalies are consistent with the extreme outliers associated with
the 1987 crash (see Figure 1)
...
1
Seasonalities
In Tables 5–7b, we check for seasonalities in daily and weekly turnover, e
...
, day-of-theweek, quarter-of-the-year, turn-of-the-quarter, and turn-of-the-year effects
...
The dependent variable for each regression is either turnover
or returns and the independent variables are indicators of the particular seasonality effect
...
Table 5 shows that, in contrast to returns which exhibit a strong day-of-the-week effect,
daily turnover is relatively stable over the week
...
023% for value-weighted
turnover and 0
...
Table 5 also shows that turnover is relatively stable over quarters—the third quarter has
the lowest average turnover, but it differs from the other quarters by less than 0
...
Turnover tends to be lower at the beginning-of-quarters, beginning-of-years,
and end-of-years, but only the end-of-year effect for value-weighted turnover (−0
...
074) are statistically
significant at the 5% level
...
Dev
...
78
0
...
66
0
...
91
0
...
38
−0
...
23
1
...
41
3
...
32
2
...
46
6
...
13
0
...
26
0
...
64
1
...
44
1
...
06
0
...
37
0
...
59
0
...
20
1
...
55
3
...
64
−3
...
14
−0
...
33
1
...
37
3
...
81
−18
...
44
−2
...
80
0
...
53
2
...
42
13
...
25
88
...
62
87
...
03
86
...
22
86
...
92
84
...
73
81
...
30
78
...
47
74
...
16
72
...
06
68
...
39
−0
...
27
−2
...
18
1
...
13
−7
...
22
−2
...
63
10
...
34
4
...
11
4
...
69
−5
...
54
−2
...
0
(0
...
0
(0
...
0
(0
...
1
(0
...
Turnover and returns are measured in
percent per week and p-values for Box-Pierce statistics are reported in parentheses
...
15
Statistic
τ VW
τ EW
RVW
REW
1962 to 1966 (234 weeks)
Mean
Std
...
Skewness
Kurtosis
0
...
07
1
...
80
0
...
21
1
...
04
0
...
29
−0
...
02
0
...
54
−0
...
50
1967 to 1971 (261 weeks)
Mean
Std
...
Skewness
Kurtosis
0
...
08
0
...
42
0
...
32
0
...
26
0
...
89
0
...
52
0
...
62
0
...
19
1972 to 1976 (261 weeks)
Mean
Std
...
Skewness
Kurtosis
0
...
10
0
...
57
0
...
20
1
...
59
0
...
39
−0
...
35
0
...
78
0
...
12
τ VW
τ EW
RVW
REW
1982 to 1986 (261 weeks)
1
...
30
0
...
14
1
...
29
0
...
28
0
...
01
0
...
33
0
...
93
0
...
19
1987 to 1991 (261 weeks)
1
...
35
2
...
88
1
...
27
2
...
81
0
...
43
−1
...
85
0
...
62
−2
...
44
1992 to 1996 (261 weeks)
1
...
23
−0
...
21
1
...
22
−0
...
24
0
...
37
−0
...
00
0
...
41
−0
...
30
1977 to 1981 (261 weeks)
Mean
Std
...
Skewness
Kurtosis
0
...
18
0
...
58
0
...
22
0
...
05
0
...
97
−0
...
31
0
...
08
−1
...
72
Summary statistics for weekly value-weighted and equal-weighted turnover and return indexes of NYSE and AMEX ordinary common shares (CRSP share codes
10 and 11, excluding 37 stocks containing Z-errors in reported volume) for July
1962 to December 1996 (1,800 weeks) and subperiods
...
Table 4: Summary Statistics for Weekly Turnover and Return Indexes (Subperiods)
...
147
(0
...
178
(0
...
061
(0
...
095
(0
...
164
(0
...
192
(0
...
044
(0
...
009
(0
...
170
(0
...
196
(0
...
112
(0
...
141
(0
...
167
(0
...
196
(0
...
050
(0
...
118
(0
...
161
(0
...
188
(0
...
091
(0
...
207
(0
...
842
(0
...
997
(0
...
369
(0
...
706
(0
...
791
(0
...
741
0
...
018)
0
...
232
(0
...
201
0
...
107)
0
...
023)
0
...
024)
(0
...
928
(0
...
095)
0
...
099)
(0
...
019
(0
...
062
(0
...
074
(0
...
153
(0
...
070
(0
...
008
(0
...
010
(0
...
243
(0
...
373
(0
...
109
(0
...
053
(0
...
179
(0
...
962
(0
...
189
(0
...
085
(0
...
755
(0
...
337
(0
...
Q1–Q4 are quarterly indicators, BOQ and
EOQ are beginning-of-quarter and end-of-quarter indicators, and BOY and EOY
are beginning-of-year and end-of-year indicators
...
17
the patterns in Table 6 are robust across subperiods: turnover is slightly lower on Mondays
and Fridays
...
12 In particular, in the 1992–1996 subperiod average Monday-returns for the
value-weighted index is positive, statistically significant, and the highest of all the five days’
average returns
...
050
(0
...
116
(0
...
092
(0
...
073
(0
...
224
(0
...
212
(0
...
030
(0
...
107
(0
...
053
(0
...
119
(0
...
046
(0
...
012
(0
...
251
(0
...
231
(0
...
070
(0
...
040
(0
...
054
(0
...
122
(0
...
124
(0
...
142
(0
...
262
(0
...
239
(0
...
093
(0
...
117
(0
...
054
(0
...
121
(0
...
032
(0
...
092
(0
...
258
(0
...
236
(0
...
111
(0
...
150
(0
...
051
(0
...
117
(0
...
121
(0
...
191
(0
...
245
(0
...
226
(0
...
122
(0
...
226
(0
...
080
(0
...
192
(0
...
157
(0
...
135
(0
...
246
(0
...
221
(0
...
040
(0
...
132
(0
...
086
(0
...
200
(0
...
021
(0
...
001
(0
...
269
(0
...
241
(0
...
119
(0
...
028
(0
...
087
(0
...
197
(0
...
156
(0
...
204
(0
...
276
(0
...
246
(0
...
150
(0
...
193
(0
...
090
(0
...
205
(0
...
039
(0
...
072
(0
...
273
(0
...
246
(0
...
015
(0
...
108
(0
...
084
(0
...
198
(0
...
127
(0
...
221
(0
...
273
(0
...
237
(0
...
050
(0
...
156
(0
...
069
(0
...
102
(0
...
123
(0
...
122
(0
...
232
(0
...
249
(0
...
117
(0
...
033
(0
...
080
(0
...
110
(0
...
010
(0
...
031
(0
...
261
(0
...
276
(0
...
009
(0
...
003
(0
...
081
(0
...
111
(0
...
066
(0
...
063
(0
...
272
(0
...
283
(0
...
080
(0
...
105
(0
...
081
(0
...
111
(0
...
087
(0
...
122
(0
...
266
(0
...
281
(0
...
050
(0
...
138
(0
...
076
(0
...
106
(0
...
056
(0
...
215
(0
...
259
(0
...
264
(0
...
026
(0
...
164
(0
...
118
(0
...
153
(0
...
104
(0
...
127
(0
...
131
(0
...
160
(0
...
029
(0
...
007
(0
...
135
(0
...
166
(0
...
116
(0
...
166
(0
...
134
(0
...
164
(0
...
018
(0
...
143
(0
...
126
(0
...
158
(0
...
136
(0
...
277
(0
...
Table 6: Seasonality (II) in Daily and Weekly Turnover and Return Indexes
...
See, for instance, Cross (1973), French (1980),
Gibbons (1981), Harris (1986a), Jaffe (1985), Keim (1984), and Lakonishok (1982, 1988)
...
Regressor
τ VW
τ EW
RVW
REW
τ VW
1962 to 1966 (234 weeks)
τ EW
RVW
REW
1972 to 1976 (261 weeks)
Q1
0
...
011)
0
...
030)
0
...
192)
0
...
224)
0
...
012)
0
...
025)
0
...
325)
1
...
355)
Q2
0
...
010)
0
...
029)
0
...
184)
0
...
215)
0
...
012)
0
...
024)
0
...
308)
−0
...
337)
Q3
0
...
009)
0
...
026)
0
...
165)
0
...
193)
0
...
012)
0
...
023)
−0
...
306)
−0
...
335)
Q4
0
...
010)
0
...
027)
0
...
173)
0
...
202)
0
...
012)
0
...
024)
0
...
319)
−0
...
349)
BOQ
−0
...
017)
−0
...
049)
0
...
310)
0
...
364)
−0
...
021)
−0
...
042)
−0
...
543)
−0
...
593)
EOQ
0
...
017)
0
...
048)
−0
...
304)
−0
...
357)
0
...
021)
−0
...
042)
0
...
554)
0
...
606)
BOY
−0
...
037)
−0
...
107)
0
...
674)
2
...
790)
−0
...
042)
−0
...
084)
1
...
098)
4
...
200)
EOY
−0
...
030)
−0
...
087)
0
...
548)
0
...
642)
−0
...
040)
−0
...
081)
0
...
055)
1
...
153)
1967 to 1971 (261 weeks)
1977 to 1981 (261 weeks)
Q1
0
...
010)
0
...
042)
0
...
258)
0
...
355)
0
...
024)
0
...
030)
−0
...
269)
0
...
280)
Q2
0
...
010)
1
...
041)
−0
...
247)
−0
...
341)
0
...
023)
0
...
029)
0
...
255)
0
...
266)
Q3
0
...
010)
0
...
040)
0
...
245)
0
...
338)
0
...
023)
0
...
029)
0
...
253)
0
...
264)
Q4
0
...
010)
0
...
042)
0
...
255)
0
...
352)
0
...
024)
0
...
030)
0
...
265)
−0
...
276)
BOQ
−0
...
017)
−0
...
070)
0
...
425)
0
...
586)
−0
...
042)
−0
...
052)
−0
...
458)
−0
...
478)
EOQ
−0
...
018)
−0
...
073)
0
...
442)
0
...
610)
−0
...
041)
−0
...
051)
−0
...
449)
−0
...
469)
BOY
−0
...
037)
0
...
151)
−0
...
919)
0
...
269)
−0
...
083)
0
...
103)
0
...
912)
1
...
952)
EOY
−0
...
033)
0
...
133)
0
...
811)
1
...
119)
−0
...
079)
−0
...
098)
1
...
868)
1
...
906)
Seasonality regressions (III) for weekly value-weighted and equal-weighted turnover and return indexes of NYSE or
AMEX ordinary common shares (CRSP share codes 10 and 11, excluding 37 stocks containing Z-errors in reported
volume) for subperiods of the sample period from July 1962 to December 1991
...
Table 7a: Seasonality (IIIa) in Weekly Turnover and Return Indexes
...
2
Secular Trends and Detrending
It is well known that turnover is highly persistent
...
Unlike
returns, turnover is strongly autocorrelated, with autocorrelations that start at 91
...
73% for the value-weighted and equal-weighted turnover indexes, respectively, decaying
very slowly to 84
...
59%, respectively, at lag 10
...
258
(0
...
177
(0
...
389
(0
...
524
(0
...
362
(0
...
432
(0
...
388
(0
...
687
(0
...
173
(0
...
115
(0
...
313
(0
...
356
(0
...
253
(0
...
302
(0
...
328
(0
...
292
(0
...
188
(0
...
058
(0
...
268
(0
...
164
(0
...
170
(0
...
223
(0
...
521
(0
...
570
(0
...
320
(0
...
190
(0
...
625
(0
...
526
(0
...
298
(0
...
353
(0
...
322
(0
...
219
(0
...
123
(0
...
132
(0
...
329
(0
...
336
(0
...
058
(0
...
078
(0
...
890
(0
...
705
(0
...
042
(0
...
052
(0
...
222
(0
...
158
(0
...
036
(0
...
006
(0
...
567
(0
...
840
(0
...
202
(0
...
114
(0
...
395
(0
...
033
(0
...
149
(0
...
102
(0
...
012
(0
...
857
(0
...
280
(0
...
158
(0
...
477
(0
...
160
(0
...
348
(0
...
220
(0
...
204
(0
...
753
(0
...
416
(0
...
254
(0
...
823
(0
...
202
(0
...
317
(0
...
159
(0
...
424
(0
...
305
(0
...
252
(0
...
105
(0
...
099
(0
...
081
(0
...
317
(0
...
160
(0
...
228
(0
...
787
(0
...
108
(0
...
060
(0
...
117
(0
...
316
(0
...
003
(0
...
013
(0
...
548
(0
...
655
(0
...
293
(0
...
207
(0
...
118
(1
...
379
(1
...
326
(0
...
104
(0
...
259
(1
...
037
(1
...
Q1–Q4 are quarterly indicators, BOQ and EOQ are beginning-of-quarter and end-of-quarter indicators, and BOY and EOY are
beginning-of-year and end-of-year indicators
...
20
kind of nonstationarity in turnover—perhaps a stochastic trend or unit root (see Hamilton
(1994), for example)—and this is confirmed at the usual significance levels by applying the
Kwiatkowski et al
...
13
For these reasons, many empirical studies of volume use some form of detrending to induce
stationarity
...
To gauge the impact of various methods of detrending on
the time-series properties of turnover, we report summary statistics of detrended turnover
in Table 8 where we detrend according to the following six methods:
d
τ1t = τt −
ˆ
α 1 + β1 t
ˆ
d
τ2t = log τt −
(7a)
ˆ
α 2 + β2 t
ˆ
(7b)
d
τ3t = τt − τt−1
d
τ4t =
(7c)
τt
(τt−1 + τt−2 + τt−3 + τt−4 )/4
d
τ5t = τt −
(7d)
ˆ
ˆ
α4 + β3,1 t + β3,2 t2 +
ˆ
ˆ
ˆ
ˆ
ˆ
β3,3 DEC1t + β3,4 DEC2t + β3,5 DEC3t + β3,6 DEC4t +
ˆ
ˆ
ˆ
ˆ
β3,7 JAN1t + β3,8 JAN2t + β3,9 JAN3t + β3,10 JAN4t +
ˆ
ˆ
ˆ
β3,11 MARt + β3,12 APRt + · · · + β3,19 NOVt
d
ˆ
τ6t = τt − K(τt )
(7e)
(7f)
where (7) denotes linear detrending, (7) denotes log-linear detrending, (7) denotes firstdifferencing, (7) denotes a four-lag moving-average normalization, (7) denotes linear-quadratic
detrending and deseasonalization (in the spirit of Gallant, Rossi, and Tauchen (1994)),14 and
13
In particular, two LM tests were applied: a test of the level-stationary null, and a test of the trendstationary null, both against the alternative of difference-stationarity
...
41 (level)
and 1
...
88 (level) and 1
...
The 1% critical values for these two tests are 0
...
216, respectively
...
(1992) for further details concerning unit root tests, and Andersen (1996) and Gallant,
Rossi, and Tauchen (1992) for highly structured (but semiparametric) procedures for detrending individual
and aggregate daily volume
...
, DEC4t and JAN1t ,
...
, NOVt denote monthly indicator variables for the months of March through November (we have omitted February to avoid perfect
21
(7) denotes nonparametric detrending via kernel regression (where the bandwidth is chosen
optimally via cross validation)
...
For example, the skewness of
detrended value-weighted turnover varies from 0
...
77 (kernel), and the kurtosis varies from −0
...
38 (kernel)
...
However, first-differenced value-weighted turnover has an autocorrelation coefficient of −34
...
In contrast, kernel-detrended value-weighted turnover has an
autocorrelation of 23
...
Similar disparities are also observed for the various detrended equal-weighted
turnover series
...
This does not correspond exactly to the Gallant, Rossi, and Tauchen (1994) procedure—they
detrend and deseasonalize the volatility of volume as well
...
MA(4)
Ratio
GRT
Kernel
Raw
Linear
Value-Weighted Turnover Index
Log
Linear
First
Diff
...
8
78
...
6
78
...
6
72
...
6
—
Mean
Std
...
Skewness
Kurtosis
0
...
48
0
...
21
0
...
26
1
...
84
0
...
31
0
...
20
0
...
20
0
...
75
1
...
20
0
...
02
0
...
25
1
...
38
0
...
16
1
...
38
0
...
37
0
...
09
0
...
30
0
...
80
0
...
35
0
...
44
0
...
19
0
...
21
1
...
20
0
...
51
0
...
28
1
...
32
0
...
17
0
...
67
0
...
22
0
...
37
0
...
19
1
...
57
4
...
69
−0
...
29
−0
...
01
0
...
30
0
...
95
−0
...
51
−0
...
21
−0
...
23
0
...
50
1
...
55
−0
...
19
−0
...
00
0
...
20
0
...
45
0
...
69
0
...
89
1
...
12
1
...
35
2
...
61
−0
...
28
−0
...
02
0
...
29
0
...
91
−0
...
26
−0
...
06
0
...
06
0
...
23
2
...
24
0
...
44
0
...
91
1
...
41
1
...
16
−0
...
44
−0
...
19
−0
...
16
0
...
55
2
...
09
−0
...
43
−0
...
00
0
...
46
0
...
11
−0
...
32
−0
...
09
−0
...
09
0
...
32
1
...
44
0
...
76
0
...
01
1
...
25
1
...
44
−0
...
38
−0
...
20
−0
...
16
0
...
54
2
...
59
−0
...
20
−0
...
01
0
...
21
0
...
73
91
...
59
87
...
44
87
...
17
87
...
57
85
...
63
70
...
21
58
...
10
56
...
25
58
...
30
54
...
45
74
...
17
63
...
86
62
...
37
60
...
83
57
...
57
−34
...
70
−4
...
35
2
...
96
9
...
10
3
...
95
22
...
48
−19
...
41
−6
...
35
4
...
78
−2
...
46
70
...
70
60
...
96
60
...
78
61
...
39
59
...
85
23
...
54
−6
...
78
−7
...
93
−1
...
29
−7
...
86
86
...
89
79
...
07
76
...
14
74
...
95
71
...
59
79
...
46
67
...
84
63
...
95
60
...
45
55
...
93
83
...
27
74
...
60
70
...
29
66
...
76
62
...
81
−31
...
69
−5
...
45
2
...
79
4
...
52
2
...
05
29
...
54
−13
...
97
−4
...
23
0
...
37
−2
...
48
77
...
60
66
...
14
62
...
03
59
...
62
56
...
06
39
...
95
8
...
80
−0
...
54
−3
...
71
−10
...
59
81
...
9
37
...
6
71
...
Six detrending methods are used: linear, log-linear, first differencing,
normalization by the trailing four-week moving average, linear-quadratic and seasonal detrending proposed by Gallant, Rossi,
and Tauchen (1992) (GRT), and kernel regression
...
6% to 88
...
15 To visualize the impact that
various detrending methods can have on turnover, compare the various plots of detrended
value-weighted turnover in Figure 2, and detrended equal-weighted turnover in Figure 3
...
The moving-average series looks like white noise, the log-linear series seems to
possess a periodic component, and the remaining series seem heteroskedastic
...
To address the
problem of the apparent time trend and other nonstationarities in raw turnover, the empirical
analysis in the rest of the paper is conducted within five-year subperiods only (the exploratory
data analysis of this section contains entire-sample results primarily for completeness)
...
From a purely statistical point of view, a nonstationary time series is nonstationary over
any finite interval—shortening the sample period cannot induce stationarity
...
However, from an empirical point of view, confining our attention to five-year subperiods
is perhaps the best compromise between letting the data “speak for themselves” and imposing
sufficient structure to perform meaningful statistical inference
...
(τt − τ )2
t
d
t (τjt
Note that the R2 ’s for the detrended equal-weighted turnover series are comparable to those of the valueweighted series except for linear, log-linear, and GRT detrending—evidently, the high turnover of small stocks
in the earlier years creates a “cycle” that is not as readily explained by linear, log-linear, and quadratic trends
(see Figure 1)
...
17
However, we acknowledge the importance of stationarity in conducting formal statistical inferences—it
is difficult to interpret a t-statistic in the presence of a strong trend
...
24
3
2
1
Turnover (%/week)
4
...
+
Raw VWT
Loglinear DT
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
0
3
2
1
Raw VWT
Linear DT
-1
0
Turnover (%/week)
4
...
+
0
500
1000
1500
0
500
Observation number
...
2
3
+
Raw VWT
Moving Avg DT
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
Raw VWT
Gallant et al
...
4
4
...
1000
Observation number
0
500
1000
Observation number
1500
0
500
1000
1500
Observation number
Figure 2: Raw and Detrended Weekly Value-Weighted Turnover Indexes, 1962 to 1996
...
+
Raw EWT
Loglinear DT
...
...
...
...
...
...
...
...
...
...
...
...
-1
-1
0
500
1000
1500
0
500
4
+
3
2
1
...
Raw EWT
First Diff DT
...
...
...
...
...
...
...
...
...
...
Raw EWT
Moving Avg DT
...
...
...
...
...
...
...
...
...
...
...
Raw EWT
Kernel DT
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
2
Raw EWT
Gallant et al
...
2
1500
0
1
2
3
+
0
Turnover (%/week)
...
0
...
...
...
...
...
...
...
...
...
...
...
...
1
...
+
0
500
1000
Observation number
1500
0
500
1000
1500
Observation number
Figure 3: Raw and Detrended Weekly Equal-Weighted Turnover Indexes, 1962 to 1996
...
Rather than filter our data
through a specific trend process that others might not find as convincing, we choose instead
to analyze the data with methods that require minimal structure, yielding results that may
be of broader interest than those of a more structured analysis
...
For example, we must assume that the mechanisms governing turnover is relatively stable
over five-year subperiods, otherwise even the subperiod inferences may be misleading
...
5
Cross-Sectional Properties
To develop a sense for cross-sectional differences in turnover over the sample period, we
turn our attention from turnover indexes to the turnover of individual securities
...
Figures 4a–b show that while the median turnover (the horizontal bars with vertical
sides in Figure 4b) is relatively stable over time—fluctuating between 0
...
The range of turnover is relatively narrow in the early 1960’s, with
90% of the values falling between 0% and 1
...
The cross-sectional variation of
turnover declines sharply in the mid-1970’s and then begins a steady increase until a peak
in 1987, followed by a decline and then a gradual increase until 1996
...
Andersen (1996) uses two
methods: nonparametric kernel regression and an equally weighted moving average
...
27
of log-turnover is similar over time up to a location parameter
...
To explore the dynamics of the cross section of turnover, we ask the following question:
if a stock has high turnover this week, how likely will it continue to be a high-turnover stock
next week? Is turnover persistent or are there reversals from one week to the next?
28
Deciles of Weekly Turnover
Deciles of Weekly Turnover (Averaged Annually)
6
6
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
0
4
3
0
1
2
Weekly Turnover [%]
5
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
-4
2
1
0
-1
-2
1995
Deciles of Log(Weekly Turnover (Averaged Annually))
-4
Log (Weekly Turnover [%])
1990
(b)
Deciles of Log(Weekly Turnover)
-3
1985
Year
1965
1970
1975
1980
Year
(c)
1985
1990
1995
1965
1970
1975
1980
1985
1990
Year
(d)
Figure 4: Deciles of weekly turnover and the natural loarithm of weekly turnover, 1962 to 1996
...
For example, the first entry of the first row—54
...
74% of the stocks that have turnover in the first decile this week will, on
average, still be in the first turnover-decile next week
...
51—implies that
21
...
Turnover
Transition
Next Week Decile
10–20
20–30
30–40
40–50
50–60
60–70
70–80
80–90
90–100
0–10
54
...
12)
21
...
06)
9
...
05)
5
...
04)
3
...
03)
2
...
03)
1
...
02)
0
...
02)
0
...
01)
0
...
01)
10–20
22
...
06)
28
...
10)
19
...
06)
11
...
05)
6
...
05)
4
...
04)
2
...
03)
1
...
03)
1
...
02)
0
...
02)
20–30
10
...
05)
20
...
07)
22
...
09)
17
...
06)
11
...
05)
7
...
05)
4
...
04)
3
...
03)
2
...
03)
1
...
02)
30–40
5
...
04)
11
...
05)
17
...
07)
19
...
08)
16
...
06)
11
...
05)
7
...
05)
4
...
04)
3
...
03)
1
...
02)
40–50
3
...
04)
7
...
05)
12
...
05)
16
...
06)
18
...
08)
15
...
06)
11
...
05)
7
...
05)
4
...
04)
2
...
03)
50–60
1
...
03)
4
...
04)
7
...
05)
12
...
05)
16
...
06)
18
...
08)
16
...
06)
11
...
05)
7
...
04)
3
...
03)
60–70
1
...
02)
2
...
03)
4
...
04)
8
...
05)
12
...
05)
16
...
07)
19
...
07)
16
...
06)
11
...
05)
5
...
04)
70–80
0
...
02)
1
...
03)
3
...
03)
5
...
04)
8
...
05)
12
...
05)
18
...
07)
21
...
08)
18
...
07)
10
...
05)
80–90
0
...
01)
1
...
02)
1
...
03)
2
...
03)
4
...
04)
7
...
05)
13
...
05)
20
...
07)
27
...
09)
20
...
06)
90–100
This
Week
0–10
0
...
01)
0
...
01)
0
...
02)
1
...
02)
1
...
03)
2
...
03)
5
...
04)
10
...
05)
23
...
07)
53
...
12)
Transition probabilities for weekly turnover deciles (in percents), estimated with weekly turnover of NYSE or AMEX
ordinary common shares (CRSP share codes 10 and 11, excluding 37 stocks containing Z-errors in reported volume)
from July 1962 to December 1996 (1,800 weeks)
...
, 10, and then normalized by the number of
consecutive pairs of weeks
...
Standard errors, computed under the assumption of independently and identically distributed transitions,
are given in parentheses
...
For example, there is
only a 18
...
18% and 11
...
For purposes of comparison, Tables 9b and 9c report similar transition probabilities esti30
mates for market capitalization deciles and return deciles, respectively
...
However, returns are considerably less persistent—indeed, Table 9c provides strong evidence
of reversals
...
50% probability of being in the tenth return-decile next week; stocks in the tenth return-decile this
week have a 20
...
These weekly
transition probabilities are consistent with the longer-horizon return reversals documented
by Chopra (1992), DeBondt (1985), and Lehmann (1990)
...
75
(0
...
18
(0
...
01
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
31
(0
...
61
(0
...
01
(0
...
01
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
09
(0
...
61
(0
...
23
(0
...
01
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
01
(0
...
26
(0
...
36
(0
...
31
(0
...
01
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
01
(0
...
29
(0
...
80
(0
...
85
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
01
(0
...
77
(0
...
77
(0
...
39
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
01
(0
...
31
(0
...
76
(0
...
86
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
78
(0
...
01
(0
...
14
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
08
(0
...
38
(0
...
48
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
00
(0
...
45
(0
...
49
(0
...
Each week all securities with nonmissing returns are sorted into market-capitalization deciles and the frequencies of transitions from decile i in one
week to decile j in the next week are tabulated for each consecutive pair of weeks and for all (i, j) combinations,
i, j = 1,
...
The number of securities with nonmissing returns in any given week varies between 1,700 and 2,200
...
Table 9b: Transition Probabilities of Weekly Market Capitalization Deciles
...
31
Return
Transition
Next Week Decile
10–20
20–30
30–40
40–50
50–60
60–70
70–80
80–90
90–100
0–10
12
...
09)
8
...
06)
7
...
06)
7
...
07)
7
...
07)
7
...
07)
8
...
07)
9
...
06)
12
...
07)
19
...
11)
10–20
9
...
06)
9
...
06)
9
...
05)
9
...
05)
9
...
05)
9
...
05)
9
...
05)
10
...
06)
11
...
06)
10
...
06)
20–30
8
...
06)
9
...
05)
10
...
05)
10
...
06)
10
...
06)
10
...
06)
10
...
06)
10
...
06)
10
...
06)
8
...
06)
30–40
7
...
06)
9
...
05)
10
...
06)
10
...
07)
11
...
07)
11
...
07)
11
...
07)
10
...
06)
9
...
05)
7
...
06)
40–50
7
...
07)
9
...
05)
10
...
06)
10
...
07)
11
...
08)
11
...
07)
11
...
07)
10
...
06)
9
...
05)
7
...
06)
50–60
7
...
07)
9
...
05)
10
...
06)
11
...
07)
11
...
07)
11
...
07)
11
...
07)
10
...
06)
9
...
06)
7
...
06)
60–70
7
...
06)
9
...
05)
10
...
06)
11
...
07)
11
...
07)
11
...
07)
11
...
07)
10
...
06)
9
...
06)
7
...
05)
70–80
8
...
06)
10
...
06)
10
...
06)
10
...
07)
10
...
06)
10
...
06)
10
...
06)
10
...
06)
9
...
06)
8
...
05)
80–90
10
...
07)
11
...
06)
10
...
06)
9
...
06)
9
...
06)
9
...
06)
9
...
06)
9
...
06)
9
...
06)
9
...
06)
90–100
This
Week
0–10
20
...
11)
12
...
06)
9
...
06)
8
...
06)
7
...
05)
6
...
05)
6
...
05)
7
...
05)
8
...
05)
12
...
08)
Transition probabilities for weekly return deciles (in percents), estimated with weekly returns of NYSE or AMEX
ordinary common shares (CRSP share codes 10 and 11, excluding 37 stocks containing Z-errors in reported volume)
from July 1962 to December 1996 (1,800 weeks)
...
, 10, and then normalized by the number of
consecutive pairs of weeks
...
Standard errors, computed under the assumption of independently and identically distributed transitions,
are given in parentheses
...
32
5
...
In Section
6 and 7, we propose formal models for the cross-section of volume
...
In particular, we wish to examine the explanatory power of several economically
motivated variables such as expected return, volatility, and trading costs in explaining the
cross section of turnover
...
Also, within each five-year period we exclude all stocks that are missing turnover
data for more than two-thirds of the subsample
...
ˆ
βr,j :
Slope coefficient from the time-series regression of stock j’s
return on the value-weighted market return
...
vj :
Average of natural logarithm of stock j’s market capitalization
...
dj :
Average of dividend yield of stock j, where dividend yield in
week t is defined by
djt = max
0 , log (1 + Rjt )Vjt−1 /Vjt
and Vjt is j’s market capitalization in week t
...
γr,j (1)
ˆ
First-order autocovariance of returns
...
The motivation for the first three regressors comes partly from linear asset-pricing models
such as the CAPM and APT; they capture excess expected return (αr,j ), systematic risk
ˆ
ˆ
(βr,j ), and residual risk (ˆ ,r,j ), respectively
...
Although a higher premium from lower liquidity should be inversely related to turnover, a
higher premium from heterogeneous information can lead to either higher or lower turnover,
ˆ
depending on the nature of information heterogeneity
...
Given that realized returns often generate portfolio-rebalancing
needs, the volatility of returns should be positively related to turnover
...
On the
theoretical side, the role of market capitalization in explaining volume is related to Merton’s
(1987) model of capital market equilibrium in which investors hold only the assets they are
familiar with
...
The motivation for log-price is related to
trading costs
...
This suggests that volume should be positively related to prices
...
g
...
If size and price are genuine factors driving
expected returns, they should drive turnover as well (see Lo and Wang (1998) for a more
formal derivation and empirical analysis of this intuition)
...
20 Often induced by differential taxation of dividends versus
capital gains, dividend-capture trading has been linked to short-term increases in trading
activity, e
...
, Karpoff and Walking (1988, 1990), Lakonishok and Smidt (1986), Lakonishok
and Vermaelen (1986), Lynch-Koski (1996), Michaely (1991), Michaely and Murgia (1995),
Michaely and Vila (1995, 1996), and Stickel (1991)
...
The effects of membership in the S&P 500 have been documented in many studies, e
...
,
Dhillon and Johnson (1991), Goetzmann and Garry (1986), Harris and Gurel (1986), Jacques
(1988), Jain (1987), Lamoureux and Wansley (1987), Pruitt and Wei (1989), Shleifer (1986),
Tkac (1996), and Woolridge and Ghosh (1986)
...
The purpose of the non-negativity restriction is to
ensure that inflows, e
...
, new equity issues, are not treated as negative dividends
...
The obvious motivation
for this variable is the growth of indexation by institutional investors, and by the related
practice of index arbitrage, in which disparities between the index futures price and the
spot prices of the component securities are exploited by taking the appropriate positions
in the futures and spot markets
...
Indexation began its rise in popularity with the advent
of the mutual-fund industry in the early 1980’s, and index arbitrage first became feasible in
1982 with the introduction of the Chicago Mercantile Exchange’s S&P 500 futures contracts
...
Another motivation for S&P 500 membership is its effect on the publicity
of member companies, which leads to more diverse ownership and more trading activity in
the context of Merton (1987)
...
In that model,
Roll shows that in the absence of information-based trades, prices bouncing between bid and
ask prices implies the following approximate relation between the spread and the first-order
return autocovariance:
s2
r,j
≈ − Cov[Rjt , Rjt−1 ] ≡ − γr,j (1)
4
(8)
where sr,j ≡ sj / Paj Pbj is the percentage effective bid/ask spread of stock j as a percentage
of the geometric average of the bid and ask prices Pbj and Paj , respectively, and sj is the
dollar bid/ask spread
...
Of course, using γr,j (1) does not eliminate this problem,
ˆ
which is a symptom of a specification error, but rather is a convenient heuristic that allows
us to estimate the regression equation (complex observations for even one regressor can yield
complex parameter estimates for all the other regressors as well!)
...
21
21
In a parenthetical statement in footnote a of Table I, Roll (1984) writes “The sign of the covariance was
36
Under the trading-cost interpretation for γr,j (1), we should expect a positive coefficient
ˆ
in our cross-sectional turnover regression—a large negative value for γ r,j (1) implies a large
ˆ
bid/ask spread, which should be associated with lower turnover
...
These eight regressors yield the following regression equation to be estimated:
ˆ
τj = γ0 + γ1 αr,j + γ2 βr,j + γ3 σ ,r,j + γ4 vj + γ5 pj + γ6 dj +
˜
ˆ
ˆ
γ7 SP500j + γ8 γr,j (1) + j
...
2
(9)
Summary Statistics For Regressors
Table 10 reports summary statistics for these regressors, as well as for three other variables
relevant to Sections 6 and 7:
ατ,j :
ˆ
Intercept coefficient from the time-series regression of stock j’s
turnover on the value-weighted market turnover
...
σ ,τ,j :
ˆ
Residual standard deviation of the time-series regression of
stock j’s turnover on the value-weighted market turnover
...
e
...
Table 10 contains means, medians, and standard deviations for these variables over each
of the seven subperiods
...
0 on average, with a cross-sectional standard deviation of about 0
...
Observe that return betas
have approximately the same mean and median in all subperiods, indicating an absence of
dramatic skewness and outliers in their empirical distributions
...
2 in the first
subperiod (1962–1966) to an all-time high of 3
...
7 (1987–1991) and 0
...
Also, the means and
preserved after taking the square root”
...
249
1
...
965
0
...
058
0
...
175
0
...
380
−2
...
851
8
...
442
1
...
685
0
...
042
0
...
178
0
...
382
−1
...
623
4
...
823
0
...
890
0
...
063
0
...
162
0
...
369
−3
...
007
8
...
074
1
...
805
0
...
086
0
...
176
0
...
381
−1
...
622
5
...
143
1
...
873
0
...
063
0
...
181
0
...
385
−1
...
573
8
...
908
1
...
097
0
...
062
0
...
191
0
...
393
−5
...
386
44
...
081
1
...
032
0
...
042
0
...
182
0
...
386
−3
...
136
21
...
576
0
...
641
0
...
272
0
...
009
0
...
065
2
...
725
5
...
646
0
...
889
0
...
064
0
...
900
0
...
827
0
...
361
0
...
128
0
...
954
3
...
948
3
...
910
0
...
940
µ
m
s
0
...
420
0
...
359 −0
...
291 0
...
292 0
...
472
1
...
595
0
...
403
0
...
780
0
...
561
0
...
449
0
...
043
0
...
638
1
...
818
1
...
749
0
...
643
µ
m
s
1
...
998
0
...
833
0
...
605
0
...
031
0
...
957
0
...
018
1
...
902
0
...
255
0
...
039
0
...
708
0
...
333
0
...
393
0
...
505
1
...
256 −0
...
899 0
...
272 0
...
419
1
...
208
1
...
834
0
...
379
0
...
637
0
...
511
1
...
378
0
...
480
1
...
002
0
...
562 17
...
893 17
...
406 1
...
086
0
...
383
1
...
225
0
...
367 17
...
104 17
...
991 1
...
085
0
...
319
0
...
955
0
...
252 17
...
825 17
...
619 1
...
254
0
...
356
0
...
936
0
...
081 18
...
737 18
...
097 1
...
113
0
...
455
0
...
863
0
...
419 18
...
813 18
...
581 1
...
977
0
...
414
6
...
847
5
...
778
5
...
013
1992 to 1996 (261 weeks)
0
...
113
0
...
851
0
...
520
5
...
407
4
...
450
3
...
007
Summary statistics of variables for cross-sectional analysis of weekly turnover of NYSE or AMEX ordinary common
shares (CRSP share codes 10 and 11, excluding 37 stocks containing Z-errors in reported volume) for subperiods of
the sample period from July 1962 to December 1996
...
The statistics are: µ (mean); m (median); and s (standard deviation)
...
g
...
2
mean versus 0
...
1 mean versus 1
...
Turnover betas are also
more variable than return betas, with cross-sectional standard deviations that range from
twice to ten times those of return betas
...
The means and medians vary from subperiod to subperiod in a manner also
consistent with the trading-cost interpretation—the higher the median of median turnover
τj , the closer to 0 is the median autocovariance
...
851 to −0
...
272 to 0
...
Between
the second and third subperiods, median autocovariance increases (in absolute value) from
−0
...
007 while median turnover decreases from 0
...
291, presumably due
to the Oil Shock of 1973–1974 and the subsequent recession
...
449 while median autocovariance decreases (in absolute value) to
−0
...
During the 1982–1986 subperiod when S&P 500 index futures begin trading, median autocovariance declines (in absolute value) to −0
...
704
...
708 versus 0
...
573 in the previous subperiod to −0
...
627 in the previous subperiod to −5
...
We have also estimated correlations among the variables in Table 10, which are reported
in Table 11a and 11b
...
We shall
address this issue more formally in Section 6
...
39
τj
τj
˜
ατ,j
ˆ
ˆ
βτ,j
σ
ˆ
,τ,j
αr,j
ˆ
ˆ
βr,j
vj
pj
dj
SP500
j
−62
...
1
−27
...
2
−63
...
7
13
...
1
31
...
7
32
...
7
4
...
9
10
...
6
−68
...
1
−30
...
7
77
...
0
47
...
7
28
...
2
43
...
3
18
...
3
−65
...
1
−41
...
2
−57
...
7
19
...
5
32
...
0
37
...
6
2
...
8
11
...
4
−70
...
3
−28
...
6
80
...
9
51
...
5
13
...
1
52
...
8
14
...
5
σ
ˆ
,r,j
1962 to 1966 (2,073 stocks)
τj
˜
ατ,j
ˆ
ˆ
βτ,j
σ ,τ,j
ˆ
αr,j
ˆ
ˆ
βr,j
σ ,r,j
ˆ
vj
pj
dj
SP500
j
γr,j (1)
ˆ
93
...
6
56
...
8
14
...
3
36
...
2
−7
...
4
−5
...
6
τj
˜
ατ,j
ˆ
ˆ
βτ,j
σ ,τ,j
ˆ
αr,j
ˆ
ˆ
βr,j
σ ,r,j
ˆ
vj
pj
dj
SP500
j
γr,j (1)
ˆ
96
...
9
77
...
2
10
...
2
56
...
5
−19
...
2
−14
...
7
τj
˜
ατ,j
ˆ
ˆ
βτ,j
σ ,τ,j
ˆ
αr,j
ˆ
ˆ
βr,j
σ ,r,j
ˆ
vj
pj
dj
SP500
j
γr,j (1)
ˆ
96
...
5
67
...
9
8
...
3
22
...
6
8
...
9
1
...
0
τj
˜
ατ,j
ˆ
ˆ
βτ,j
σ ,τ,j
ˆ
αr,j
ˆ
ˆ
βr,j
σ ,r,j
ˆ
vj
pj
dj
SP500
j
γr,j (1)
ˆ
96
...
7
61
...
0
10
...
8
28
...
3
8
...
4
2
...
2
1
...
9
70
...
7
59
...
4
−11
...
7
−9
...
6
3
...
9
−11
...
0
−15
...
5
9
...
6
9
...
8
5
...
1
16
...
8
34
...
5
−16
...
2
−6
...
1
14
...
2
45
...
9
−20
...
2
−10
...
6
1
...
3
−3
...
6
0
...
6
−14
...
2
1
...
2
−17
...
4
1
...
0
70
...
7
8
...
4
49
...
3
−11
...
2
−10
...
8
−83
...
2
4
...
2
−36
...
7
35
...
8
11
...
7
77
...
9
55
...
0
−40
...
3
−35
...
1
−12
...
5
50
...
7
−41
...
1
−35
...
2
−11
...
6
−1
...
1
16
...
0
2
...
8
61
...
7
−22
...
9
−11
...
9
1972 to 1976 (2,084 stocks)
8
...
2
69
...
2
54
...
7
12
...
4
−18
...
6
3
...
0
−5
...
7
−16
...
9
3
...
8
7
...
5
6
...
6
11
...
4
17
...
7
−1
...
8
−0
...
2
7
...
7
35
...
7
−11
...
9
−13
...
6
−14
...
3
5
...
6
9
...
2
5
...
9
12
...
8
−34
...
1
−8
...
0
55
...
4
2
...
8
18
...
7
17
...
2
8
...
0
−72
...
5
−8
...
0
−8
...
7
11
...
8
−0
...
8
54
...
9
47
...
6
−2
...
6
−15
...
5
−1
...
7
35
...
8
−16
...
8
−14
...
9
−5
...
2
30
...
8
−9
...
4
−19
...
6
24
...
4
12
...
9
8
...
3
Table 11a: Correlation Matrix for Weekly Turnover Regressors
40
τj
τj
˜
ατ,j
ˆ
ˆ
βτ,j
σ
ˆ
,τ,j
αr,j
ˆ
ˆ
βr,j
vj
pj
dj
SP500
j
−55
...
1
−21
...
7
−39
...
3
16
...
3
15
...
5
37
...
6
8
...
1
5
...
1
−62
...
9
−20
...
1
80
...
9
58
...
4
15
...
1
23
...
9
2
...
4
−61
...
8
−14
...
2
−23
...
5
13
...
7
19
...
4
37
...
3
11
...
0
4
...
2
−12
...
3
80
...
4
46
...
4
19
...
0
−6
...
5
5
...
1
17
...
8
79
...
1
45
...
1
20
...
3
−1
...
1
4
...
8
6
...
8
79
...
8
46
...
6
10
...
8
−9
...
6
2
...
6
64
...
8
−10
...
6
7
...
7
16
...
6
22
...
6
−77
...
8
−14
...
6
12
...
3
−12
...
1
−2
...
9
64
...
2
38
...
7
18
...
3
−0
...
1
9
...
4
24
...
2
−5
...
9
−2
...
6
4
...
5
−17
...
1
22
...
5
−3
...
9
15
...
6
10
...
6
18
...
4
1987 to 1991 (2,471 stocks)
25
...
2
56
...
1
49
...
6
31
...
0
−1
...
4
5
...
0
−1
...
8
5
...
7
3
...
4
−1
...
4
2
...
0
−9
...
5
2
...
4
2
...
8
11
...
5
9
...
3
12
...
0
−5
...
6
−3
...
4
−15
...
4
5
...
8
2
...
4
−39
...
6
22
...
2
−4
...
1
11
...
8
49
...
6
−6
...
1
6
...
8
17
...
3
15
...
9
−78
...
0
−13
...
0
5
...
1
−3
...
5
−8
...
3
43
...
6
28
...
4
12
...
1
−4
...
5
3
...
8
27
...
3
−18
...
4
−9
...
3
−3
...
4
24
...
7
−8
...
4
−9
...
2
4
...
8
16
...
4
17
...
1
Correlation matrix of variables for cross-sectional analysis of weekly turnover of NYSE or AMEX ordinary common
shares (CRSP share codes 10 and 11, excluding 37 stocks containing Z-errors in reported volume) for subperiods of
the sample period from July 1962 to December 1996
...
Table 11b: Correlation Matrix for Weekly Turnover Regressors (continued)
41
Median turnover is not particularly highly correlated with S&P 500 membership during the first four subperiods, with correlations ranging from −10
...
6%
(1972–1976)
...
7% in 1982–1986, 25
...
9% in 1992–1996
...
5% (1987–1991) to 55
...
Not surprisingly, log-price p j is highly
positively correlated with log-market-capitalization vj , with correlations exceeding 75% in
every subperiod
...
This may seem counterintuitive at first but recall that these are cross-sectional correlations, not time-series correlations,
and the level of dividends per share varies cross-sectionally as well as average log-price
...
3
Regression Results
Tables 12a and 12b contain the estimates of the cross-sectional regression model (9)
...
Since the log price and log market-capitalization regressors are so highly correlated (see Lim
et al
...
The exclusion of either variable does not affect the qualitative
features of the regression—no significant coefficients changed sign other than the constant
term—though the quantitative features were affected to a small degree
...
064) and pj has a positive coefficient (0
...
When vj is omitted the coefficient of pj is still positive
but smaller (0
...
028), and in both these cases the coefficients retain their
significance
...
This can be seen heuristically in the time-series plots of
Figure 1—compare the value-weighted and equal-weighted turnover indexes during the first
two or three subperiods
...
This begins to change in the 1977–1981 subperiod: the size coefficient is negative but
42
not significant, and when price is excluded, the size coefficient changes sign and becomes
significant
...
One
explanation of this change is the growth of the mutual fund industry and other large institutional investors in the early 1980’s
...
Therefore, the natural economies of scale in investment management coupled with the increasing concentration of investment capital make small stocks
less actively traded than large stocks
...
The first-order return autocovariance has a positive coefficient in all subperiods except the
second regression of the last subperiod (in which the coefficient is negative but insignificant),
and these coefficients are significant at the 5% level in all subperiods except 1972–1976 and
1992–1996
...
Membership in the S&P 500 also has a positive impact on turnover in all subperiods
as expected, and the magnitude of the coefficient increases dramatically in the 1982–1986
subperiod—from 0
...
091—also as expected given the growing
importance of indexation and index arbitrage during this period, and the introduction of
S&P 500 futures contracts in April 1982
...
029, perhaps because of the interactions between this indicator
variable and size and price (all three variables are highly positively correlated with each
other; see Lim et al
...
When size is omitted, S&P 500 membership
becomes more important, yet when price is omitted, size becomes more important and S&P
500 membership becomes irrelevant
...
23
ˆ
Both systematic and idiosyncratic risk—βr,j and σ ,r,j —have positive and significant imˆ
pact on turnover in all subperiods
...
Of course, her analysis is not directly comparable to ours
because she uses a different dependent variable (monthly relative dollar-volume divided by relative marketcapitalization) in her cross-sectional regressions, and considers only a small sample of the very largest
NYSE/AMEX stocks (809) over the four year period 1988–1991
...
742
(0
...
306
(0
...
378
(0
...
059
(0
...
068
(0
...
111
(0
...
354
(0
...
344
(0
...
401
(0
...
289
(0
...
797
(0
...
172
(0
...
134
(0
...
152
(0
...
209
(0
...
448
(0
...
434
(0
...
507
(0
...
437
(0
...
249
(0
...
188
(0
...
102
(0
...
111
(0
...
141
(0
...
345
(0
...
320
(0
...
367
(0
...
315
(0
...
344
(0
...
810
(0
...
059
(0
...
058
(0
...
008
(0
...
508
(0
...
508
(0
...
534
(0
...
048
(0
...
006
(0
...
048
(0
...
004
(0
...
006
(0
...
005
(0
...
8
0
...
025)
−0
...
024)
0
...
026)
0
...
002)
0
...
002)
0
...
002)
44
...
031
(0
...
007
(0
...
020
(0
...
001
(0
...
002
(0
...
003
(0
...
0
0
...
019)
0
...
018)
−0
...
020)
0
...
002)
0
...
002)
0
...
002)
44
...
043
(0
...
053
(0
...
013
(0
...
064
(0
...
028
(0
...
150
(0
...
070
(0
...
071
(0
...
130
(0
...
119
(0
...
8
38
...
095
(0
...
112
(0
...
057
(0
...
062
(0
...
009
(0
...
249
(0
...
173
(0
...
027
(0
...
117
(0
...
108
(0
...
7
41
...
027
(0
...
032
(0
...
008
(0
...
041
(0
...
008
(0
...
171
(0
...
114
(0
...
031
(0
...
058
(0
...
072
(0
...
5
32
...
057
(0
...
057
(0
...
040
(0
...
001
(0
...
037
(0
...
139
(0
...
137
(0
...
015
(0
...
015
(0
...
001
(0
...
2
42
...
The explanatory variables are:
ˆ
αr,j , βr,j , and σ ,r,j (the intercept, slope, and residual, respectively, from the time-series regression of an
ˆ
ˆ
individual security’s return on the market return); vj (natural logarithm of market capitalization), pj (natural
logarithm of price); dj (dividend yield); SP500j (S&P 500 indicator variable); and γr,j (1) (first-order return
ˆ
autocovariance)
...
385
(0
...
193
(0
...
602
(0
...
051
(0
...
018
(0
...
080
(0
...
543
(0
...
583
(0
...
562
(0
...
662
(0
...
313
(0
...
968
(0
...
155
(0
...
153
(0
...
171
(0
...
791
(0
...
831
(0
...
795
(0
...
004
(0
...
310
(0
...
025
(0
...
087
(0
...
095
(0
...
025
(0
...
689
(0
...
708
(0
...
711
(0
...
091
(0
...
187
(0
...
085
(0
...
006
(0
...
005
(0
...
006
(0
...
6
0
...
041)
0
...
036)
0
...
041)
0
...
001)
0
...
001)
0
...
001)
31
...
029
(0
...
087
(0
...
005
(0
...
000
(0
...
001
(0
...
000
(0
...
6
1982 to 1986 (261 weeks, 2,644 stocks)
0
...
007)
0
...
007)
0
...
005)
0
...
010)
—
0
...
009)
0
...
023)
0
...
020)
—
−0
...
081)
−0
...
081)
−0
...
081)
30
...
3
1987 to 1991 (261 weeks, 2,471 stocks)
0
...
005)
0
...
005)
0
...
005)
0
...
013)
—
0
...
010)
0
...
024)
0
...
019)
—
−0
...
097)
−0
...
098)
−0
...
097)
30
...
7
1992 to 1996 (261 weeks, 2,520 stocks)
0
...
007)
0
...
007)
0
...
006)
0
...
016)
—
0
...
012)
0
...
033)
0
...
026)
—
−0
...
164)
−0
...
164)
−0
...
166)
29
...
8
Table 12b: Cross-sectional regressions of median weekly turnover of NYSE and AMEX ordinary common
shares (CRSP share codes 10 and 11, excluding 37 stocks containing Z-errors in reported volume) for fiveyear subperiods of the sample period from July 1982 to December 1996
...
45
positive and significant in the others
...
In these two subperiods, the coefficient is negative, which contradicts the notion
that dividend-capture trading affects turnover
...
The explanatory power of these cross-sectional regressions—as measured by R 2 —range from
29
...
7% (1967–1971), rivaling the R 2 ’s of typical cross-sectional return
regressions
...
In order to
further analyze the cross-section of turnover, additional economic structure is needed
...
6
Volume Implications of Portfolio Theory
The diversity in the portfolio holdings of individuals and institutions and in their motives
for trade suggests that the time-series and cross-sectional patterns of trading activity can
be quite complex
...
e
...
In this case, all investors trade only in these separating funds and simpler
cross-sectional patterns in trading activity emerge, and in this section we derive such crosssectional implications
...
g
...
In particular, in this section we focus primarily on the cross-sectional properties of volume,
we assume nothing about the behavior of asset prices, e
...
, a factor structure for asset returns
may or may not exist
...
1 and 6
...
However, in Section 7, we provide a specific
intertemporal capital asset pricing model, in which mutual-fund separation holds and the
separating funds are linked with the underlying risk structure of the stocks
...
For example, mutual-fund separation is often derived in static
settings in which the motives for trade are not explicitly modeled
...
Furthermore, these models tend to
focus on a rather narrow set of trading motives—changes in portfolio holdings due to changes
in return distributions or preferences—ignoring other factors that may motivate individuals
and institutions to adjust their portfolios, e
...
, asymmetric information, idiosyncratic risk,
transactions costs, taxes and other market imperfections
...
A detailed discussion of these concerns is beyond the scope of this paper
...
Nevertheless, before deriving these implications, it is important to consider how some of the
limitations of mutual-fund separation may affect the interpretation of our analysis
...
For example, extending mutual-fund separation results to dynamic settings is
possible
...
However, in
a continuous-time setting—which has its own set of restrictive assumptions—Merton (1973)
shows that mutual-fund separation holds for quite general preferences and return processes
...
The CAPM is a well-known example of
mutual-fund separation in a static equilibrium setting
...
24
24
Tkac (1996) also attempts to develop a dynamic equilibrium model—a multi-asset extension of Dumas
(1990)—in which two-fund separation holds
...
Moreover,
if it is in the spirit of Dumas (1990) in which risky assets take the form of investments in linear production
technologies (as in Cox, Ingersoll and Ross (1985)), the model has no volume implications for the risky assets
since changes in investors’ asset holdings involve changes in their own investment in production technologies,
not in the trading of risky assets
...
g
...
Therefore, it is difficult to argue that current levels
of trading activity are too high to be justified by rational portfolio rebalancing
...
Moreover, from an empirical standpoint, little effort has
been devoted to calibrating the level of trading volume within the context of a realistic
asset-market model (see Lo, Mamaysky and Wang (2001) for more discussions)
...
One compelling reason is the fact that mutual-fund separation has
become the workhorse of modern investment management
...
Thus, it seems natural to begin with such models
in an investigation of trading activity in asset markets
...
If it does not, then this suggests the possibility of important weaknesses
in the theory, weaknesses that may have implications that extend beyond trading activity,
e
...
, preference restrictions, risk-sharing characteristics, asymmetric information, and liquidity
...
Another reason for focusing on mutual-fund separation is that it can be an important
benchmark in developing a more complete model of trading volume
...
Using mutual-fund separation as a benchmark allows us to gauge how important
other trading motives may be in understanding the different aspects of trading volume
...
Factors such as asymmetric information, idiosyncratic risks, transaction costs, and other
48
forms of market imperfections are also likely to be relevant for determining the level and
variability of trading activity
...
To examine their importance in explaining volume,
we need a more general and unified framework that can capture these factors
...
For all these reasons, we examine the implications of mutual-fund separation for trading
activity in this section
...
We
view this as the first step in developing a more complete understanding of trading and pricing
in asset markets and we hope to explore these other issues in future research (see also Section
7)
...
1 we consider the case of two-fund separation in which one fund is the riskless
asset and the second fund is a portfolio of risky assets
...
2 we investigate the
general case of (K +1)-fund separation, one riskless fund and K risky funds
...
We assume the existence of a riskless asset mainly to simplify
the exposition, but for our purposes this assumption entails no loss of generality
...
6
...
e
...
, J, and we begin by assuming two-fund
separation, i
...
, all investors invest in the same two mutual funds: the riskless asset and a
stock fund
...
Given
our normalization, the market portfolio S M —measured in shares outstanding—is simply a
vector of one’s: S M = [ 1 · · · 1 ]
...
= ht
...
1
i = 1,
...
2 for K = 2 (since
two of the three funds are assumed to contain risky assets)
...
His
t
i
holding in stock j is then Sjt = hi , j = 1,
...
Over time, investor i may wish to adjust
t
his portfolio
...
His trading in stock j, normalized by shares outstanding,
i
i
i
i
i
i
is: Sjt − Sjt−1 = hi − hi , i = 1,
...
But this, in turn, implies Sjt − Sjt−1 = Sj t − Sj t−1 ,
t
t−1
j, j = 1,
...
Thus, if two-fund separation holds, investor i’s trading activity in each
stock, normalized by shares outstanding, is identical across all stocks
...
, J
(11)
i=1
which is given by the following proposition:
Proposition 1 When two-fund separation holds, the turnover of all individual stocks are
identical
...
From the
definition of Section 2
...
, J
...
This
is not surprising given that individual stocks have identical values for turnover
...
For reasons that
becomes apparent in Section 6
...
, J
(12)
˜
where Ft = τtV W and bj = 1
...
Another implication is that each security’s relative
50
dollar-volume is identical to its relative market-capitalization for all t: Pjt Vjt /
Pjt Nj /
j
j
Pjt Vjt =
Pjt Nj
...
Tkac (1996) derives this result
in the context of a continuous-time dynamic equilibrium model with a special form of heterogeneity in preferences, but it holds more generally for any model that implies two-fund
separation
...
2
(K +1)-Fund Separation
We now consider the more general case where (K + 1)-fund separation holds
...
, SJt ) , k = 1,
...
The stock
holdings of any investor i are given by
i
S1t
K
...
=
hi Stk ,
kt
...
, I
...
Therefore, the turnover of stock j
kt
at time t is
τjt =
1
2
I
i
i
|Sjt − Sjt−1 | =
i=1
1 I
2 i=1
K
k
k
hi Sjt − hi Sjt ,
kt
kt−1
j = 1,
...
(14)
k=1
We now impose the following assumption on the separating stock funds:
Assumption 1 The separating stock funds, Stk , k = 1,
...
Given that, in equilibrium,
I
i=1
Si,t = SM for all t, we have
K
I
hi S k = S M
...
Following Merton (1973), we
call the remaining stock funds hedging portfolios
...
27
In addition, we can assume that all the separating stock funds are mutually orthogonal, i
...
, S k S k = 0,
51
To simplify notation, we define ∆hi ≡ hi −hi
kt
kt
kt−1 as the change in investor i’s holding
of fund k from t−1 to t
...
, K, and i = 1,
...
, hi have a continuous joint probability
˜
˜
where |hkt
1 and h1t 2t
Jt
density
...
, J
(15)
i=1
and the n-th absolute moment of the approximation error is o(λn )
...
, K
...
, J
...
Then the turnover of each stock has an approximate K-factor structure
...
k
k = 1,
...
, K, k = k
...
, K, hence the total
j=1
number of shares in each of the hedging portfolios sum to zero under our normalization
...
Moreover, i=1 hi = 1 and i=1 hi = 0, k = 2,
...
1t
kt
52
6
...
If turnover is driven by a linear K-factor model, the first
K principal components should explain most of the time-series variation in turnover
...
...
0
0
θ2
···
(18)
···
...
0
0
...
...
Since Σ is a covariance matrix, it is positive semidefinite hence all the eigenvalues are nonnegative
...
If (17) holds, it can be shown that as the size N of the cross section increases
without bound, exactly K normalized eigenvalues of Σ approach positive finite limits, and
the remaining N − K eigenvalues approach 0 (see, for example, Chamberlain (1983) and
Chamberlain and Rothschild (1983))
...
The only obstacle is the fact that the covariance matrix Σ must be estimated, hence we
encounter the well-known problem that the standard estimator
Σ≡
1
T
T
(τt − τ )(τt − τ )
t=1
53
is singular if the number of securities J in the cross section is larger than the number of time
series observations T
...
We do this by following the common practice of forming a
small number of portfolios (see Campbell, Lo, and MacKinlay (1997, Chapter 5)), sorted
by turnover beta to maximize the dispersion of turnover beta among the portfolios
...
For purposes of comparison and interpretation, we perform a parallel analysis for returns,
using ten return-beta-sorted portfolios
...
28
Singularity by itself does not pose any problems for the computation of eigenvalues—this follows from
the singular-value decomposition theorem—but it does have implications for the statistical properties of
estimated eigenvalues
...
We thank Bob Korajczyk
and Bruce Lehmann for bringing some of these issues to our attention and plan to investigate them more
thoroughly in ongoing research
...
This motivation should not be taken literally in our context
because the theoretical implications of Sections 6
...
However, given
the factor structure implied by (K +1)-fund separation (see Section 6
...
54
θ9
θ10
Period
θ1
θ2
θ3
0
...
0)
0
...
0)
0
...
0)
1967 to 1971
85
...
5)
5
...
5)
2
...
2)
1
...
1)
1
...
1)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
1972 to 1976
90
...
9)
3
...
3)
1
...
2)
1
...
1)
0
...
1)
0
...
1)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
1977 to 1981
85
...
5)
4
...
4)
4
...
4)
1
...
1)
1
...
1)
2
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
1982 to 1986
86
...
6)
6
...
5)
2
...
2)
1
...
1)
2
...
2)
1
...
1)
1
...
1)
1
...
1)
1
...
1)
0
...
1)
0
...
0)
1987 to 1991
91
...
0)
2
...
3)
1
...
1)
2
...
3)
1
...
2)
1
...
1)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
1992 to 1996
72
...
3)
11
...
0)
4
...
4)
0
...
0)
0
...
0)
0
...
0)
1967 to 1971
87
...
7)
4
...
4)
2
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
1972 to 1976
91
...
0)
4
...
4)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
1977 to 1981
91
...
0)
3
...
3)
1
...
1)
0
...
1)
0
...
1)
2
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
1)
0
...
0)
1982 to 1986
88
...
8)
4
...
4)
2
...
2)
1
...
1)
2
...
2)
2
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
1)
0
...
1)
1987 to 1991
92
...
1)
3
...
3)
1
...
1)
2
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
1992 to 1996
76
...
7)
10
...
9)
3
...
3)
θ1
θ2
θ3
θ4
θ5
θ6
θ7
85
...
5)
8
...
7)
3
...
3)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
82
...
3)
7
...
6)
4
...
4)
2
...
2)
1
...
1)
0
...
1)
83
...
3)
8
...
8)
2
...
2)
2
...
2)
1
...
1)
78
...
9)
7
...
7)
3
...
3)
2
...
3)
80
...
0)
6
...
5)
5
...
5)
71
...
3)
15
...
4)
4
...
4)
86
...
6)
7
...
7)
3
...
3)
1
...
1)
0
...
0)
0
...
0)
0
...
0)
82
...
3)
6
...
5)
5
...
5)
2
...
3)
1
...
1)
1
...
1)
79
...
9)
8
...
7)
5
...
5)
2
...
2)
1
...
1)
78
...
8)
10
...
9)
3
...
3)
2
...
2)
82
...
2)
4
...
4)
3
...
3)
79
...
9)
8
...
7)
4
...
4)
θ8
Turnover-Beta-Sorted Turnover Portfolios (τ VW )
θ4
θ5
θ6
θ7
θ8
θ10
0
...
1)
0
...
0)
0
...
0)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
1
...
1)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
3
...
3)
2
...
2)
1
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
2
...
2)
1
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
1)
0
...
1)
Return-Beta-Sorted Return Portfolios (RVW )
Turnover-Beta-Sorted Turnover Portfolios (τ EW )
55
θ9
Return-Beta-Sorted Return Portfolios (REW )
ˆ
Table 13: Eigenvalues θi , i = 1,
...
Turnover portfolios
are sorted by out-of-sample turnover betas and return portfolios are sorted by out-of-sample return betas, where the symbols “τ VW ” and “RVW ”
indicate that the betas are computed relative to value-weighted indexes, and “τ EW ” and “REW ” indicate that they are are computed relative to
equal-weighted indexes
...
Table 13 contains the principal components decomposition for portfolios sorted on outof-sample betas, where the betas are estimated in two ways: relative to value-weighted
indexes (τ V W and RV W ) and equal-weighted indexes (τ EW and REW )
...
For example, the upper-left
subpanel of Table 13 shows that in the second five-year subperiod (1967–1971), 85
...
6% is
captured by the first two principal components
...
The importance of the second principal component grows steadily through time for the
value-weighted case, reaching a peak of 15
...
3% of the variation in turnover in the last subperiod
...
However, the lower left
subpanel of Table 13 shows that for turnover portfolios sorted by betas computed against
equal-weighted indexes, the second principal component explains approximately the same
variation in turnover, varying between 6
...
4% across the six subperiods
...
Despite the fact that we have limited our analysis
to five-year subperiods, within each subperiod there is a certain drift in turnover; might
this account for the first principal component? To investigate this conjecture, we perform
eigenvalue decompositions for the covariance matrices of the first differences of turnover for
the 10 turnover portfolios
...
The second principal component is typically
30
In particular, the portfolios in a given period are formed by ranking on betas estimated in the immediately
preceding subperiod, e
...
, the 1992–1996 portfolios were created by sorting on betas estimated in the 1987–
1991 subperiod, hence the first subperiod in Table 13 begins in 1967, not 1962
...
And in one case—in-sample sorting on betas relative to
the equal-weighted index during 1987–1991—the third principal component accounts for an
additional 10%
...
In summary, the results of Tables 13 and 14 indicate that a one-factor model for turnover
is a reasonable approximation, at least in the case of turnover-beta-sorted portfolios, and
that a two-factor model captures well over 90% of the time-series variation in turnover
...
g
...
As compelling as these empirical results are, several qualifications should be kept in
mind
...
In particular, the asymptotic standard errors reported in Tables 13 and 14
were computed under the assumption of IID Gaussian data, hardly appropriate for weekly
US stock returns and even less convincing for turnover (see Muirhead (1982, Chapter 9)
for further details)
...
Monte Carlo simulations should also be conducted to check the finite-sample properties of
our estimators
...
More structure must be imposed on the data—in particular, an intertemporal
model of trading—to obtain a better understanding for the sources of turnover variation,
and we present such structure in the next section
...
We first develop an intertemporal equilibrium model
of stock trading and pricing with multiple assets and heterogeneous investors
...
We show that both volume and returns are driven by the
57
Period
ˆ
θ1
ˆ
θ2
ˆ
θ3
ˆ
θ4
ˆ
θ5
ˆ
θ6
ˆ
θ7
ˆ
θ8
ˆ
θ9
ˆ
θ10
Out-of-Sample Turnover-Beta-Sorted Turnover-Differences Portfolios (τ VW )
1967 to 1971
82
...
2)
7
...
6)
5
...
5)
2
...
2)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
1972 to 1976
81
...
1)
6
...
6)
4
...
4)
2
...
2)
2
...
2)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
1977 to 1981
85
...
5)
4
...
4)
2
...
3)
2
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
1982 to 1986
81
...
1)
5
...
4)
3
...
3)
2
...
2)
2
...
2)
1
...
2)
1
...
1)
0
...
1)
0
...
1)
0
...
1)
1987 to 1991
73
...
4)
10
...
0)
4
...
4)
3
...
3)
2
...
2)
1
...
2)
1
...
1)
1
...
1)
1
...
1)
0
...
1)
1992 to 1996
78
...
9)
8
...
8)
4
...
4)
2
...
2)
2
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
Out-of-Sample Turnover-Beta-Sorted Turnover-Differences Portfolios (τ EW )
1967 to 1971
82
...
2)
8
...
7)
4
...
4)
2
...
2)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
0
...
0)
1972 to 1976
79
...
0)
7
...
7)
4
...
4)
4
...
4)
1
...
2)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
0
...
0)
1977 to 1981
80
...
0)
5
...
5)
4
...
4)
3
...
3)
2
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
1982 to 1986
82
...
3)
5
...
4)
3
...
3)
2
...
2)
2
...
2)
1
...
1)
1
...
1)
0
...
1)
0
...
1)
0
...
0)
1987 to 1991
77
...
8)
5
...
5)
4
...
4)
2
...
2)
2
...
2)
2
...
2)
1
...
2)
1
...
1)
1
...
1)
1
...
1)
1992 to 1996
80
...
1)
6
...
6)
4
...
4)
2
...
2)
1
...
1)
1
...
1)
1
...
1)
0
...
1)
0
...
0)
0
...
0)
ˆ
Table 14: Eigenvalues θi , i = 1,
...
Turnover betas are calculated in two ways: with respect to a value-weighted turnover index
(τ VW ) and an equal-weighted turnover index (τ EW )
...
58
underlying risks of the economy
...
7
...
Several generalizations of the model
are discussed in Lo and Wang (2001b)
...
There are J
risky stocks, each pays a stream of dividends over time
...
Without loss of generality, in this section we assume that the total number of
shares outstanding is one for each stock
...
SJ ], where Sj is the number of stock j shares in the portfolio (j = 1,
...
A
portfolio of particular importance is the market portfolio, denoted by S M , which is given by
SM = ι
(20)
where ι is a vector of 1’s with rank J
...
In addition to the stocks, there is also a risk-free bond that yields a constant, positive
interest r per time period
...
Each investor is endowed with equal shares of the
stocks and no bond
...
, I, maximizes his expected utility of
the following form:
i
i
i
Et −e−Wt+1 −(λX Xt +λY Yt )DM t+1 −λZ (1+Zt )Xt+1
(21)
i
where Wt+1 is investor i’s wealth next period, Xt , Yti , Zti are three one-dimensional state
variables, and λX , λY , λZ are non-negative constants
...
We further assume
I
I
Yti =
i=1
Zti = 0
(22)
i=1
where t = 0, 1,
...
, I},
are IID over time with zero means
...
(23)
Without loss of generality, σDD is assumed to be positive definite
...
One feature of the model is that
investors are assumed to have a myopic, but state-dependent utility function in (21)
...
The state dependence
of the utility function is assumed to have the following properties
...
When the aggregate dividend
goes up, the marginal utility of wealth goes down
...
This utility function can be interpreted as the equivalent
of a value function from an appropriately specified dynamic optimization problem (see, for
example, Wang (1994) and Lo and Wang (2001a))
...
Another feature of the model is the IID assumption for the state variables
...
This impression, however, is false
since the state-dependence of investors’ utility function introduces important dynamics over
time
...
The particular form of the utility function and the normality of distribution for the state
variables are assumed for tractability
...
But we hope with
60
some confidence that the qualitative predictions of the model that we explore in this paper
are not sensitive to these assumptions
...
This is a modelling choice we have made in order to simplify our
analysis and to focus on the stock market
...
Equilibrium
i
i
Let Pt ≡ [P1t
...
; SJt ] be the (column) vectors of (ex-dividend) stock
prices and investor i’s stock holdings respectively
...
Definition 4 An equilibrium is given by a price process {Pt : t = 0, 1,
...
, I; t = 0, 1,
...
Sti solves investor i’s optimization problem:
Sti = arg max
s
...
i
i
i
E −e−Wt+1 −(λX Xt +λY Yt )DM t+1 −λZ (1+Zt )Xt+1
(24)
i
Wt+1 = Wti + Sti [Dt+1 + Pt+1 − (1+r)Pt ]
2
...
(25)
i=1
The above definition of equilibrium is standard, except that the bond market does not clear
here
...
For t = 0, 1,
...
(26)
Thus, Qjt+1 = Djt+1 + Pjt+1 − (1 + r)Pjt gives the dollar return on one share of stock j
in excess of its financing cost for period t + 1
...
Dollar return
Qjt+1 differs from the conventional (excess) return measure Rjt+1 which is the dollar return
normalized by the share price: Rjt+1 ≡ Qjt+1 /Pjt
...
We can now state the solution to the equilibrium in the following theorem:
Theorem 1 The economy defined above has a unique linear equilibrium in which
Pt = −a − bXt
(27)
Sti = I −1 −λY Yti ι − λZ Zti + λY (b ι)Yti (σQQ )−1 σQX
(28)
and
where
2
σQQ = σDD − b σXD − σDX b + σX b b
2
σQX = σDX − σX b
a =
1
(ασQQ ι + λZ σQX
¯
r
b = λX [(1+r)+λZ σXD ι)]−1 σDD ι
and a = 1/I
...
In our model, an investor’s utility function
depends not only on his wealth, but also on the stock payoffs directly
...
Such a “market spirit”
affects his demand for the stocks, in addition to the usual factors such as the stocks’ expected
returns
...
When (λX Xt +λZ Yti )
is positive, investor i extracts positive utility when the aggregate stock payoff is high
...
When
(λX Xt +λY Yti ) is negative, he has a negative attachment to the market, which makes holding
stocks more attractive
...
Given
62
the particular form of the utility function, Xt affects the equilibrium stock prices linearly
...
However, they do affect individual investors’ stock holdings
...
Since the aggregate utility variable Xt is driving the stock prices, it is also driving the
stock returns
...
The form of the utility function further states that the
investors utility function directly depends on Xt , which fully characterizes the investment
opportunities investors face
...
In our setting, however, we assume that investors optimize myopically but
insert such a dependence directly into the utility function
...
In particular,
they prefer those portfolios whose returns can help them to smooth fluctuations in their
utility due to changes in investment opportunities
...
7
...
For the stocks, their dollar return vector can be re-expressed
as follows:
˜
Qt+1 = ra + (1+r)bXt + Qt+1
(29)
˜
where Qt+1 ≡ Dt+1 − bZt+1 denotes the vector of unexpected dollar returns on the stocks,
which are IID over time with zero mean
...
In particular, they are driven by a single state variable X t
...
, I
(30)
where hi t ≡ I −1 − λY Yti , hi ≡ λZ (b ι)Yti − λY Zti , and
M
Ht
S H ≡ (σQQ )−1 σQX
...
That is, all investors’ portfolios can be viewed as investments in three common funds:
the risk-free asset and two stock funds
...
Moreover, in our current model, these two portfolios, expressed in
terms of stock shares, are constant over time
...
We present these predictions
through a set of propositions
...
The turnover of stock j is given by
τjt ≡
1 I
2 i=1
H
hi t − hi t−1 + hi − hi
Ht−1 Sj
Ht
M
M
∀ j = 1,
...
(32)
Let τt denote the vector of turnover for all stocks
...
In the special case when two-fund separation holds (when Xt = 0 ∀ t), turnover would have
an exact one-factor structure, τt = ιFM t
...
It is important to note that the loading of stock j’s
turnover on the second factor is proportional to its share weight in the hedging portfolio
...
In our empirical
analysis, we explore this information that the cross-section of volume conveys
...
Merton (1971) has discussed the properties of hedging portfolios in a
continuous-time framework as a characterization of equilibrium
...
Time Series Implications for the Hedging Portfolio
By the definition of the hedging portfolio in (31), it is easy to show that its current return
gives the best forecast of future market return
...
Then,
QM t+1 = ι Qt+1
and QHt+1 = S H Qt+1
...
The predictive power of S is measured by the R2 of the above regression
...
The solution, up to a scaling constant, is the hedging
portfolio
...
In other words, if we regress the market dollar return on the lagged dollar return of any
portfolios, the hedging portfolio gives the highest R2
...
For
expositional simplicity, we introduce some additional notation
...
Qpt+1 ≡ Qpt+1 − Et [Qpt+1 ] then denotes its unexpected
¯
˜
˜
dollar return and Qp its unconditional mean
...
It is easy to show that
2
σM = ι σQQ ι,
2
−1
σH = σXQ σQQ σQX ,
σM H = ι σQX
where σQQ and σQZ are given in Theorem 1
...
¯
(35c)
Equation (35) characterizes the cross-sectional variation in the stocks’ expected dollar returns
...
In this case, returns are IID over time
...
We have the following result:
Proposition 5 When Xt = 0 ∀ t, we have
˜
˜
˜
E Qt+1 |QMt+1 = βM QM t+1
where
˜
˜
˜
βM ≡ Cov[Qt+1 , QM t+1 ]/Var[QM t+1 ] = σDD ι/(ι σDD ι)
66
(36)
is the vector of the stocks’ market betas
...
Obviously in this case, the CAPM holds for the dollar returns
...
In the general case when Xt changes over time, there is an additional risk due to changing
market conditions (dynamic risk)
...
In this case, the risk of a stock is measured by its risk with
respect to the market portfolio and its risk with respect to the hedging portfolio
...
The expected returns of the stocks are then determined by
their exposures to these two risks and the associated risk premia
...
Moreover, The stocks’ expected
dollar returns satisfy
¯
¯
¯
Q = β M QM + βH QH
(39)
2
¯
¯
where QM = ασM + λY σM H and QH = ασM H + λY σH
...
The expected dollar return on the market portfolio gives the premium of the
market risk and the expected dollar return on the hedging portfolio gives the premium of
the dynamic risk
...
The pricing relation we obtain in Proposition 6 is in the spirit of Merton’s Intertemporal
CAPM in a continuous-time framework (Merton, 1971)
...
In contrast,
our pricing relation is derived from a dynamic equilibrium model
...
If we can identify the hedging portfolio empirically, its return provides the second risk
factor
...
7
...
Our first step is to empirically identify the hedging portfolio using the turnover data
...
In principle, this specifies the hedging portfolio
...
First, the exact two-factor specification
(33) is, at best, an approximation for the true data-generating process of turnover
...
We address both of these problems in
turn
...
, J
(40)
where FM t and FHt are the two factors that generate trading in the market portfolio and
the hedging portfolio, respectively, θHj is the percentage of shares of stock j in the hedging
portfolio (as a percentage of its total number of shares outstanding), and εjt is the error
68
term, which is assumed to be independent across stocks
...
Specifically
...
The two turnover indexes are
τtEW ≡
1
J
J
τjt = FM t + nEW FHt + εEW
t
(41a)
Nj
τjt = FM t + nSW FHt + εSW
t
N
j=1
(41b)
j=1
J
τtSW ≡
where
n
EW
1
=
J
J
J
θHj and n
SW
j=1
=
Nj
θHj
j=1 N
are the average percentage of shares of each stock in the hedging portfolio and the percentage
of all shares (of all stocks) in the hedging portfolio, respectively, and εEW and εSW are the
t
t
error terms for the two indexes
...
For the
remainder of our analysis, we shall ignore them
...
t
...
j = 1,
...
If, for example, there are other
common factors in addition to FM t and FHt , then εjt is likely to be correlated across stocks
...
32
To avoid degeneracy, we need Nj = Nk for some j = k, which is surely valid empirically
...
nEW − nSW
n
− nSW
EW
SW
Using the MiniCRSP volume database, we can empirically estimate {βτ j } and {βτ j }
EW
by estimating (42)
...
(43)
However, there are two remaining parameters, nEW and nSW , that need to be estimated
...
When the two common factors are not observed, the parameters {θH j} are only
identified up to a scaling constant and a rotation
...
In addition, when the two factors
are replaced by their linear combinations, (40) remains formally the same as long as {θ Hj } is
also adjusted with an additive constant
...
4)
...
(45)
The normalization nSW = 1 sets the total number of shares in the portfolio to a positive
value
...
Nonzero values of φ represent deviations from the market
33
For example, for any a, we have ∀ j:
˜
˜
τjt = FM t + θHj FH t + εjt = (FM t + aFHt ) + (θHj − a)FH t + εjt = FM t + θHj FH t + εjt
˜
˜
where FM t = FM t + aFHt and θHj = θHj − a
...
EW
SW
To estimate {βτ j } and {βτ j }, we first construct the two turnover indexes
...
34
Therefore, we estimate constrained linear regressions of the weekly turnover for each stock
on equal- and share-weighted turnover indexes in each of the seven five-year subperiods of
our sample
...
To
provide a clearer sense of the dispersion of these regressions, we first sort them into deciles
EW
based on {βτ j }, and then compute the means and standard deviations of the estimated
¯
coefficients {β EW } and {β SW }, their t-statistics, and the R2 s within each decile
...
585 and 6
...
The R2 s also look impressive, however, they must be
interpreted with some caution because of the imposition of the constraint (42), which can
¯
yield R2 greater than unity and less than zero
...
For comparison, we estimate the unconstrained version of (42) and compute the same
summary statistics, reported in Tables 16a and 16b, along with the mean and standard
deviation within each decile of p-values corresponding to the statistic that (42) holds
...
For
example, in the first subperiod, the average p-values range from a minimum of 4
...
4% in decile 6, and with a value of 19
...
However, in
the last subperiod, the average p-value is less than 5% deciles 2–6, and close to significance
¯
for most of the other deciles, which explains the negative R2 s in Tables 15a and 15b
...
First, given the large number of stocks in our sample,
imposing a global constraint like (42) requires a large amount of computer memory, which was unavailable
to us
...
35
EW
SW
¯
For example, a negative R2 arises when the variance of βτ j τtEW + βτ j τtSW exceeds the variance of
the dependent variable τjt , which can happen when the constraint (42) is imposed
...
D
...
D
...
D
...
D
...
D
...
906
−0
...
432
−0
...
107
0
...
927
1
...
568
6
...
119
0
...
064
0
...
097
0
...
145
0
...
434
4
...
783
−0
...
315
−0
...
264
0
...
110
1
...
661
5
...
134
0
...
068
0
...
087
0
...
154
0
...
330
2
...
013
−1
...
697
−0
...
015
0
...
084
1
...
161
7
...
845
0
...
096
0
...
114
0
...
187
0
...
501
4
...
096
−2
...
654
−1
...
394
0
...
330
2
...
913
10
...
347
0
...
208
0
...
189
0
...
308
0
...
809
4
...
394
−26
...
917
−3
...
273
4
...
749
8
...
410
11
...
023
12
...
956
2
...
243
1
...
258
2
...
491
3
...
906
1
...
432
1
...
893
0
...
073
−0
...
568
−5
...
119
0
...
064
0
...
097
0
...
145
0
...
434
4
...
944
65
...
879
22
...
365
4
...
639
−2
...
292
−9
...
755−2520
...
4
30
...
5
19
...
907
55
...
4
10
...
1
17
...
570
51
...
0
2
...
6
16
...
190
50
...
5
1
...
2
15
...
401
49
...
2
3
...
1
15
...
725
−18
...
905
−1
...
269
6
...
367
10
...
249
13
...
343
8
...
099
1
...
482
2
...
719
3
...
120
4
...
783
1
...
315
1
...
736
0
...
110
−0
...
661
−4
...
134
0
...
068
0
...
087
0
...
154
0
...
330
2
...
302
53
...
431
18
...
228
3
...
735
−4
...
609
−10
...
946 −175
...
871
58
...
771
56
...
855
55
...
260
54
...
871
53
...
178
54
...
630
53
...
149
54
...
383
52
...
2
16
...
3
14
...
2
13
...
0
13
...
0
14
...
755
21
...
600
10
...
620
2
...
322
−3
...
060
−9
...
319−1147
...
045
25
...
619
44
...
044
50
...
466
53
...
354
52
...
870
51
...
421
52
...
431
52
...
965
52
...
9
44
...
1
22
...
2
15
...
5
14
...
0
13
...
341
22
...
861
10
...
387
2
...
894
−3
...
038
−9
...
815 −872
...
846
32
...
167
48
...
304
54
...
655
55
...
438
55
...
971
53
...
560
54
...
487
55
...
311
55
...
8
23
...
8
18
...
1
16
...
7
15
...
5
13
...
276
−10
...
014
−2
...
062
2
...
684
6
...
894
11
...
901
3
...
466
1
...
765
1
...
801
2
...
311
4
...
013
2
...
697
1
...
985
0
...
084
−0
...
161
−6
...
845
0
...
096
0
...
114
0
...
187
0
...
501
4
...
164
−15
...
524
−5
...
833
1
...
864
6
...
860
11
...
591
4
...
628
2
...
180
1
...
519
2
...
983
3
...
096
3
...
654
2
...
394
0
...
330
−1
...
913
−9
...
347
0
...
208
0
...
189
0
...
308
0
...
809
4
...
Turnover over individual stocks is
regressed on the equally-weighted and share- weighted turnover indices, subject to the restriction that the
EW
SW
two regression coefficients, βτ
and βτ , must add up to one
...
The summary statistics are then reported for each decile
...
D
...
D
...
D
...
D
...
D
...
968
−2
...
640
0
...
357
2
...
754
3
...
168
5
...
038
0
...
380
0
...
231
0
...
196
0
...
237
1
...
487
−2
...
843
0
...
502
2
...
389
4
...
836
5
...
040
0
...
494
0
...
317
0
...
234
0
...
212
0
...
275
−1
...
245
0
...
520
0
...
303
2
...
271
8
...
858
0
...
155
0
...
098
0
...
159
0
...
498
9
...
636
−3
...
223
1
...
540
5
...
402
9
...
354
14
...
328
1
...
967
0
...
655
2
...
342
2
...
905
5
...
968
3
...
640
0
...
357
−1
...
754
−2
...
168
−4
...
038
0
...
380
0
...
231
0
...
196
0
...
237
1
...
525
4
...
180
1
...
954
−2
...
710
−6
...
630
−11
...
577
46
...
199
52
...
667
45
...
903
55
...
786
41
...
216 −19
...
531
28
...
922
3
...
248 −163
...
405 −348
...
9
20
...
9
22
...
7
686
...
8
101
...
6
1027
...
082
6
...
097
1
...
693
−5
...
567
−11
...
572
−17
...
137
50
...
224
54
...
537
56
...
268
57
...
583
57
...
582
51
...
325
42
...
513
23
...
673 −27
...
229 −921
...
8
18
...
0
19
...
8
18
...
6
19
...
1
4682
...
097
2
...
944
1
...
729
0
...
400
−1
...
117
−3
...
342 −423
...
369 −147
...
899 −14
...
534 −135
...
330−1353
...
177 −197
...
260 −130
...
480 −58
...
769 −24
...
190 −219
...
7
2631
...
2
899
...
2
669
...
7
684
...
8
1145
...
093
−4
...
832
1
...
887
8
...
139
15
...
370
21
...
763
2
...
512
1
...
062
4
...
615
4
...
580
5
...
487
3
...
843
0
...
502
−1
...
389
−3
...
836
−4
...
040
0
...
494
0
...
317
0
...
234
0
...
212
0
...
409
−1
...
371
0
...
779
1
...
725
2
...
061
3
...
092
0
...
301
0
...
313
0
...
641
0
...
027
1
...
275
2
...
245
0
...
480
0
...
303
−1
...
271
−7
...
858
0
...
155
0
...
098
0
...
159
0
...
498
9
...
Turnover over individual stocks is
regressed on the equally-weighted and share- weighted turnover indices, subject to the restriction that the
EW
SW
two regression coefficients, βτ
and βτ , must add up to one
...
The summary statistics are then reported for each decile
...
Clearly the two-factor model of turnover accounts for a
significant amount of variation in the weekly turnover of individual stocks
...
D
...
D
...
D
...
D
...
D
...
D
...
556
2
...
054
2
...
103
1
...
111
1
...
670
2
...
4
57
...
5
61
...
0
54
...
0
51
...
7
46
...
7
16
...
5
17
...
2
19
...
4
14
...
2
14
...
7
2
...
7
23
...
2
19
...
0
26
...
0
9
...
0
8
...
8
29
...
8
28
...
4
31
...
1
19
...
303
3
...
170
3
...
537
1
...
617
1
...
760
2
...
1
59
...
3
60
...
5
56
...
0
56
...
5
55
...
3
14
...
8
15
...
6
13
...
0
12
...
3
11
...
0
10
...
8
16
...
5
20
...
6
18
...
8
10
...
3
20
...
7
27
...
2
29
...
1
28
...
9
21
...
930
4
...
129
1
...
942
2
...
475
2
...
194
3
...
0
66
...
7
58
...
1
54
...
5
56
...
0
59
...
7
17
...
6
17
...
5
14
...
3
13
...
8
11
...
2
11
...
4
5
...
1
7
...
7
9
...
6
13
...
7
24
...
6
16
...
7
18
...
9
23
...
2
25
...
749
−1
...
730
−0
...
195
0
...
508
1
...
400
11
...
337
0
...
110
0
...
055
0
...
206
0
...
875
8
...
121
−3
...
500
−2
...
603
−0
...
554
2
...
685
5
...
109
−0
...
409
−0
...
086
0
...
096
1
...
275
7
...
101
0
...
078
0
...
087
0
...
201
0
...
556
3
...
306
−4
...
600
−2
...
628
2
...
379
4
...
533
6
...
908
−0
...
237
0
...
308
0
...
107
1
...
846
6
...
364
0
...
094
0
...
091
0
...
141
0
...
497
3
...
116
−3
...
949
0
...
249
3
...
929
6
...
823
10
...
174
2
...
466
1
...
262
0
...
080
1
...
817
2
...
761
3
...
398
1
...
967
0
...
337
−0
...
639
−15
...
451
1
...
786
0
...
644
0
...
058
1
...
059
12
...
608
5
...
394
4
...
810
1
...
015
−1
...
392
−4
...
689
3
...
229
2
...
123
1
...
886
2
...
246
2
...
966
2
...
534
1
...
851
0
...
383
−1
...
417
−9
...
908
1
...
634
0
...
968
0
...
931
1
...
223
5
...
280
7
...
725
5
...
003
0
...
096
−2
...
202
−4
...
584
3
...
632
0
...
733
2
...
814
3
...
678
4
...
371
1
...
120
0
...
480
0
...
361
−0
...
054
−5
...
731
0
...
481
0
...
659
0
...
611
0
...
894
3
...
313
8
...
085
3
...
076
−0
...
996
−3
...
950
−7
...
Turnover over individual stocks are
regressed on the equally-weighted and share- weighted turnover indices, giving two regression coefficients,
EW
SW
EW
βτ
and βτ
...
The summary
statistics are reported for each decile
...
7
...
In particular, in this section
we focus on the degree to which the the hedging portfolio can predict future stock returns,
especially the return on the market portfolio
...
D
...
D
...
D
...
D
...
D
...
D
...
251
2
...
876
0
...
823
1
...
161
1
...
482
1
...
0
61
...
0
56
...
5
56
...
5
54
...
6
54
...
0
17
...
7
19
...
9
14
...
9
13
...
8
13
...
4
13
...
7
8
...
0
11
...
1
13
...
9
20
...
0
23
...
3
21
...
3
21
...
8
24
...
8
29
...
497
2
...
427
0
...
118
1
...
842
1
...
197
2
...
6
57
...
7
57
...
9
57
...
3
54
...
8
54
...
2
19
...
5
16
...
0
16
...
8
14
...
7
13
...
6
7
...
9
5
...
0
4
...
7
11
...
0
4
...
1
19
...
0
15
...
4
16
...
7
24
...
4
15
...
656
1
...
156
1
...
000
1
...
105
1
...
231
1
...
5
57
...
2
55
...
5
56
...
9
53
...
3
47
...
3
21
...
5
20
...
8
17
...
8
16
...
3
15
...
7
10
...
0
4
...
1
6
...
8
14
...
5
4
...
0
22
...
5
16
...
0
19
...
8
26
...
3
12
...
595
1
...
396
0
...
951
0
...
176
1
...
088
1
...
4
61
...
8
56
...
6
55
...
3
51
...
4
45
...
6
21
...
6
20
...
0
18
...
0
17
...
1
14
...
5
4
...
3
2
...
5
2
...
4
6
...
7
6
...
3
16
...
7
13
...
8
10
...
8
18
...
7
17
...
622
−0
...
364
−0
...
218
0
...
081
1
...
993
7
...
777
0
...
101
0
...
093
0
...
184
0
...
398
4
...
975
−2
...
559
−0
...
780
1
...
241
3
...
819
4
...
038
−0
...
342
0
...
349
0
...
204
1
...
020
6
...
819
0
...
123
0
...
101
0
...
178
0
...
459
3
...
588
−1
...
021
0
...
247
2
...
278
3
...
012
6
...
153
−0
...
098
0
...
479
0
...
177
1
...
972
6
...
353
0
...
093
0
...
086
0
...
146
0
...
452
3
...
997
−1
...
307
0
...
443
2
...
344
2
...
104
3
...
894
−0
...
197
0
...
344
0
...
018
1
...
720
7
...
074
0
...
093
0
...
085
0
...
130
0
...
454
9
...
174
−1
...
623
0
...
064
1
...
028
2
...
624
3
...
647
1
...
160
0
...
585
0
...
194
1
...
593
1
...
749
1
...
186
0
...
504
0
...
209
−0
...
995
−6
...
675
0
...
539
0
...
673
0
...
641
0
...
784
5
...
516
4
...
553
2
...
781
0
...
727
−1
...
326
−3
...
102
1
...
686
0
...
917
1
...
885
1
...
350
2
...
377
1
...
045
0
...
340
0
...
396
−0
...
754
−4
...
097
0
...
477
0
...
465
0
...
486
0
...
663
2
...
014
3
...
945
1
...
548
−0
...
647
−2
...
369
−4
...
036
0
...
417
0
...
873
1
...
068
1
...
371
1
...
278
1
...
673
0
...
150
−0
...
338
−0
...
531
−4
...
325
0
...
440
0
...
438
0
...
474
0
...
610
2
...
224
2
...
899
0
...
013
−0
...
053
−1
...
921
−2
...
107
0
...
612
0
...
731
0
...
151
1
...
170
1
...
563
1
...
924
0
...
261
0
...
224
−0
...
477
−6
...
659
0
...
534
0
...
441
0
...
578
0
...
830
9
...
498
2
...
192
1
...
281
−0
...
836
−1
...
616
−2
...
Turnover over individual stocks are
regressed on the equally-weighted and share- weighted turnover indices, giving two regression coefficients,
EW
SW
EW
βτ
and βτ
...
The summary
statistics are reported for each decile
...
75
portfolio in Section 7
...
4 and 7
...
Hedging-Portfolio Returns
To construct the return on the hedging portfolio, we begin by calculating its dollar value
and dollar returns
...
, 7, Vjt (k) denote the total market
capitalization of stock j at time period t (the end of week t) in subperiod k, Qjt (k) denote its dividend-adjusted excess dollar return for the same period, and Rjt (k) denote the
dividend-adjusted excess return, and θj (k) the estimated share (as fraction of its total shares
outstanding) in the hedging portfolio in subperiod k
...
Among the stocks
satisfying this criteria, we eliminate those ranked in the top and bottom 0
...
36 We let Jt (k) denote the set of stocks that survive these two filters and
that have price and return data for week t of subperiod k
...
QHt (k) ≡
(47)
j
and the (rate of) return of the hedging portfolio is given by
RHt (k) ≡
36
QHjt (k)
VHt (k)
See Lo and Wang (2000a) for the importance of outliers in volume data
...
The procedure outlined above yields the return and the dollar return of the hedging
portfolio up to the parameter φ, which must be calibrated
...
2)
...
37 In all cases, there is a unique
global maximum, from which we obtain φ
...
Therefore, we eliminate these values from consideration, and for all
subperiods except subperiod 4 and 7 (i
...
, subperiods 2, 3, 5, 6), the omitted values of φ do
not seem to affect the choice of φ for the maximum R2 (see Lo and Wang (2001) for more
discussions on the choice of φ)
...
25, 4
...
75,
47, 38, and 0
...
Using QHt , the values of φ are
1
...
25, 2, 20, 27, and 0
...
With these values of φ in hand, we have fully
specified the hedging portfolio, its return and dollar return
...
Optimal Forecasting Portfolios (OFPs)
Having constructed the return of the hedging portfolio in Section 7
...
According to Proposition 4, the returns of the
hedging portfolio should outperform the returns of any other portfolios in predicting future
market returns
...
77
Sample Period
Statistic
Entire
67–71
72–76
77–81
82–86
87–91
92–96
0
...
477
10
...
476
0
...
070
0
...
182
−0
...
072
0
...
111
0
...
055
0
...
013
−0
...
945
−0
...
028
−0
...
010
−0
...
020
0
...
130
0
...
035
3
...
906
2
...
286
0
...
015
−0
...
066
−0
...
057
0
...
016
0
...
082
0
...
845
0
...
048
−0
...
095
0
...
014
−0
...
014
0
...
075
−0
...
031
Hedging Portfolio Return RHt
Mean
S
...
Skewness
Kurtosis
ρ1
ρ2
ρ3
ρ4
ρ5
ρ6
ρ7
ρ8
ρ9
ρ10
0
...
198
24
...
809
0
...
058
0
...
184
−0
...
079
0
...
098
0
...
044
0
...
029
0
...
479
0
...
018
−0
...
070
0
...
003
0
...
002
−0
...
017
0
...
039
0
...
597
0
...
006
−0
...
043
0
...
258
0
...
124
−0
...
174
0
...
045
−0
...
727
0
...
071
−0
...
045
−0
...
089
−0
...
008
−0
...
037
0
...
046
0
...
347
0
...
036
0
...
113
−0
...
093
−0
...
006
0
...
117
Hedging Portfolio Dollar Return QHt
Mean
S
...
Skewness
Kurtosis
ρ1
ρ2
ρ3
ρ4
ρ5
ρ6
ρ7
ρ8
ρ9
ρ10
2
...
836
0
...
082
0
...
082
0
...
021
0
...
010
−0
...
046
0
...
001
0
...
639
0
...
085
0
...
014
0
...
061
0
...
044
0
...
005
−0
...
030
1
...
059
−0
...
500
0
...
148
0
...
084
0
...
127
0
...
055
0
...
042
2
...
495
−0
...
286
0
...
052
0
...
127
−0
...
094
−0
...
028
−0
...
026
5
...
423
−0
...
537
0
...
125
0
...
037
0
...
053
−0
...
127
0
...
014
Table 17: Summary statistics for the returns and dollar returns of the hedging portfolio constructed from
individual stocks’ volume data using weekly returns and volume data for NYSE and AMEX stocks from
1962 to 1996
...
It is impractical to compare (50) to all possible portfolios, and uninformative to compare it to random portfolios
...
The following proposition shows
how to construct optimal forecasting portfolios (OFPs) (see Lo and Wang, 2001 for details):
Proposition 7 Let Γ0 and Γ1 denote the contemporaneous and first-order autocovariance
matrix of the vector of all returns
...
; wqN ), define A ≡ Γ0 −1 Γ1 wq wq Γ1
...
Since Γ0 and Γ1 are unobservable, they must be estimated using historical data
...
However, it is possible to construct OFPs from a much smaller
number of “basis portfolios”, and then compare the predictive power of these OFPs to the
¯
hedging portfolio
...
We form several sets of basis portfolios by sorting all the J stocks into K groups of
equal numbers (K ≤ J) according to: market capitalization, market beta, and SIC codes,
and then construct value-weighted portfolios within each group
...
Based on the estimated autocovariance matrices, the
OFP can be computed easily according to Proposition 7
...
As a compromise, for the OFPs based market capitalization and market betas,
we choose K to be 10, 15, 20, and 25
...
38
It is important that we use value-weighted portfolios here so that the market portfolio, whose return
we wish to predict, is a portfolio of these basic portfolios (recall that the target portfolio ω q that we wish
to forecast is a linear combination of the vector of returns for which Γk is the k-th order autocovariance
matrix)
...
For the OFP based on 10 market-capitalizationsorted portfolios, which we call “CAP10”, we construct 10 value-weighted portfolios each
week, one for each market-capitalization decile
...
To compute the OFP, we also require the weights
ωq of the target portfolio, in this case the market portfolio
...
The weights of the OFP for the basis portfolios CAP10 follow
immediately from Proposition 7
...
The OFPs of market-beta-sorted basis portfolios are constructed in a similar manner
...
We consider 10, 15, 20 and 25 groups, denoted by
“Beta10”, “Beta15”, and so on
...
Finally, the industry portfolios are based on SIC-code groupings
...
On the other, the first digit yields
only eight broad industry categories
...
80
#
1
2
3
4
5
6
7
8
9
10
11
12
13
SIC Codes
1–14
15–19, 30, 32–34
20–21
22, 23, 25, 31, 39
24, 26–27
28
29
35–36, 38
37, 40–47
48–49
50–59
60–69
70–89, 98–99
Description
Agriculture, forest, fishing, mining
Construction, basic materials (steel, glass, concrete, etc
...
Each week, stocks are sorted according to their SIC codes into the 13 categories defined above,
and value-weighted returns are computed for each group, yielding the 13 basis portfolios
which we denote by “SIC13”
...
Hedging Portfolio Return as A Predictor of Market Returns
Tables 18a and 18b reports the results of the regressions of RM t on various lagged OFP
returns and on the hedging portfolios RHt and QHt
...
40 Tables 18a and 18b show that the
hedging portfolios outperforms all of the other competing portfolios in forecasting future
market returns in three of the six subperiods (subperiods 2, 4, and 6)
...
And in subperiod 7,
the equal-weighted CRSP index return outperforms the hedging portfolio
...
First, in
40
We also considered nine other interest-rate predictors (six-month and one-year Treasury bill rates,
three-month, six-month, and one-year off-the-run Treasury bill rates, one-month and three-month CD
and Eurodollar rates, and the Fed Funds rate (all obtained from the Federal Reserve Bank of St
...
stls
...
org/fred/data/wkly
...
Each of these variables produced results similar to those for
the three-month constant-maturity Treasury bill return, hence we omit those regressions from Tables 18a
and 18b
...
Second, there is no consistent winner in these subperiods
...
Moreover, the best
performers in these subperiods performed poorly in the other subperiods, raising the possibility that their performance might be due to sampling variation
...
Indeed, among all of the regressors, the hedging portfolios were the most consistent across all six subperiods, a remarkable
confirmation of the properties of the model of Sections 7
...
2
...
Increasing the number of basis
portfolios should, in principle, increase the predictive power of the OFP
...
82
Parameter
Beta10
Beta15
Beta20
Beta25
Cap10
Cap15
Cap20
Cap25
SIC13
RH
QH
log(Cap−1 )
VW
EW
TBill
January 1967 to December 1971 (261 Weeks)
Intercept
t-Stat
Slope
t-Stat
R
2
0
...
330
0
...
810
0
...
360
−0
...
550
0
...
150
−0
...
890
0
...
430
0
...
780
0
...
240
−0
...
900
0
...
520
0
...
079
0
...
400
−0
...
240
0
...
380
−0
...
070
0
...
920
−0
...
860
0
...
270
0
...
460
0
...
200
0
...
900
0
...
330
0
...
330
0
...
240
0
...
130
0
...
250
0
...
080
—
—
—
—
0
...
001
0
...
012
0
...
005
0
...
005
0
...
045
0
...
021
0
...
016
—
January 1972 to December 1976 (261 Weeks)
Intercept
t-Stat
Slope
t-Stat
R
2
0
...
650
0
...
120
0
...
640
0
...
150
0
...
560
−0
...
630
0
...
670
0
...
580
0
...
830
0
...
660
0
...
640
0
...
660
0
...
730
−0
...
180
0
...
630
0
...
430
0
...
001
0
...
630
0
...
760
0
...
054 −0
...
430 −1
...
900
0
...
410
0
...
410
0
...
700
−0
...
060
0
...
640
0
...
910
—
—
—
—
0
...
005
0
...
001
0
...
002
0
...
001
0
...
008
0
...
008
0
...
003
—
January 1977 to December 1981 (261 Weeks)
83
Intercept
t-Stat
Slope
t-Stat
R
2
0
...
750
0
...
040
0
...
600
0
...
870
0
...
800
0
...
460
0
...
640
0
...
510
0
...
770
0
...
230
0
...
760
0
...
010
0
...
800
−0
...
850
0
...
530
−0
...
990
0
...
749
0
...
130
0
...
500
0
...
810
0
...
370
0
...
760
0
...
720
0
...
710
0
...
570
0
...
110
0
...
380
0
...
370
—
—
—
—
0
...
003
0
...
001
0
...
000
0
...
004
0
...
013
0
...
002
0
...
007
—
Table 18a: Forecast of weekly market-portfolio returns by lagged weekkly returns of the beta-sorted optimal forecast portfolios (OFPs), the
market-capitalization-sorted OFP’s, the SIC-sorted OFP, the return and dollar return on the hedging portfolio, minus log-market-capitalization,
the lagged returns on the CRSP value- and equal-weighted portfolios, and lagged constant-maturity (three-month) Treasury bill rates from 1962 to
1981 in five-year subperiods
...
25 for the return RH and 1
...
Parameter
Beta10
Beta15
Beta20
Beta25
Cap10
Cap15
Cap20
Cap25
SIC13
-Cap
VW
0
...
130
0
...
110
0
...
690
0
...
100
0
...
010
2
...
860
0
...
053
0
...
212
0
...
005
0
...
003
0
...
003
0
...
800
2
...
050
0
...
014 −0
...
320 −4
...
490
0
...
460
0
...
460
0
...
820
0
...
930
0
...
10
1
...
098
0
...
598
0
...
810
0
...
008
0
...
001
RH
QH
EW
TBill
January 1982 to December 1986 (261 Weeks)
Intercept
t-Stat
Slope
t-Stat
R
2
0
...
150
−0
...
030
0
...
150
0
...
910
0
...
130
−0
...
990
0
...
180
0
...
180
0
...
150
−0
...
470
0
...
160
−0
...
220
0
...
110
0
...
740
0
...
150
0
...
530
0
...
003
0
...
005
0
...
000
0
...
001
0
...
004
0
...
640
3
...
190
−0
...
047 −0
...
890 −1
...
490
0
...
012
0
...
003
1
...
294
1
...
002
1
...
353
−2
...
003
1
...
120
0
...
003
1
...
130
0
...
003
1
...
540
−2
...
003
1
...
062
−0
...
003
1
...
072
0
...
003
1
...
033
−0
...
010
0
...
002
0
...
021
0
...
000
0
...
073
0
...
003
January 1992 to December 1996 (261 Weeks)
84
Intercept
t-Stat
Slope
t-Stat
R
2
0
...
170
0
...
060
0
...
120
−0
...
080
0
...
110
0
...
930
0
...
060
−0
...
850
0
...
060
−0
...
090
0
...
130
−0
...
270
0
...
120
0
...
240
0
...
170
−0
...
550
0
...
000
0
...
003
0
...
000
0
...
001
0
...
003
0
...
130
3
...
510
−0
...
194 −0
...
700 −2
...
410
0
...
032
0
...
107
0
...
003 −0
...
780
3
...
000 −0
...
004 −0
...
192
7
...
800 −2
...
320
1
...
003
0
...
041
0
...
The value of φ is 1
...
5 for the dollar return QH on the hedging portfolio, respectively
...
5
The Hedging-Portfolio Return as a Risk Factor
To evaluate the success of the hedging-portfolio return as a risk factor in the cross section
of expected returns, we implement a slightly modified version of the well-known regression
tests outlined in Fama and MacBeth (1973)
...
However, in contrast to Fama and MacBeth (1973), we
use weekly instead of monthly returns, and our portfolio-formation, estimation, and testing
periods are five years each
...
Using the estimated coefficients {βiM } and {βiH }, we perform a double sort
among the individual securities in the estimation period, creating 100 portfolios corresponding to the deciles of the estimated market and hedging-portfolio betas
...
, 100,
M
constructed from the double-sorted rankings of the portfolio-estimation period, and βpt and
42
Our first portfolio-formation period, from 1962 to 1966, is only four and a half years because the CRSP
Daily Master file begins in July 1962
...
43
This induces a certain degree of survivorship bias, but the effects may not be as severe given that we
apply the selection criterion three periods at a time
...
85
H
βpt are the market and hedging-portfolio returns, respectively, of portfolio p obtained from
the estimation period
...
Summary statistics for these coefficients and their diagnostics are then reported, and this
entire procedure is repeated by incrementing the portfolio-formation, estimation, and testing
periods by five years
...
Because we use weekly instead of monthly data, it may be difficult to compare our
results to other cross-sectional tests in the extant literature, e
...
, Fama and French (1992)
...
4
...
Tables 20a–20c summarizes the results of all of these cross-sectional regression tests for
each of the five testing periods from 1972 to 1996
...
45 For example, the first three rows show that
the time-series average of the market-beta coefficients, {γ1t }, is 0
...
348 and an average R2 of 10
...
46 When the hedging-portfolio beta β H is added to the
t
¯
regression, the R2 does increase to 14
...
002
γ
with a t-statistic of −0
...
The average market-beta coefficient is still insignificant, but
44
Specifically, the SMB portfolio return is constructed by taking the difference of the value-weighted returns
of securities with market capitalization below and above the median market capitalization at the start of
the five-year subperiod
...
Therefore, the OFP returns are not available in the first portfolio-formation
period
...
However, since this has become the standard method for reporting
the results of these cross-sectional regression tests, we follow this convention to make our results comparable
to those in the literature
...
0
88
...
2
15
...
0
−26
...
6
8
...
7
100
...
8
4
...
5
−25
...
6
5
...
2
−15
...
0
40
...
7
−11
...
9
16
...
6
4
...
3
100
...
3
−6
...
5
9
...
0
53
...
7 −13
...
0
−4
...
6
−5
...
9
−25
...
0
−6
...
8
100
...
9
−2
...
6
12
...
9
7
...
6
−4
...
0
86
...
1
5
...
8
9
...
8
−2
...
2
100
...
0
92
...
6
91
...
7
−76
...
1
26
...
6
100
...
3
88
...
5
−71
...
8
36
...
6
92
...
0
97
...
7
−65
...
0
29
...
5
88
...
4
100
...
8
−60
...
9
29
...
7
69
...
0
−46
...
7
38
...
0 −60
...
6
100
...
5
−10
...
7
84
...
2
−71
...
1
32
...
0
22
...
7
−7
...
0
93
...
3
36
...
6
29
...
2
−10
...
1
100
...
0
27
...
0
84
...
5
100
...
3
−11
...
3
14
...
9
−59
...
1
−45
...
3
35
...
0
86
...
2
−8
...
8
−1
...
6
−4
...
5 −18
...
2
−18
...
0
−70
...
6
−4
...
9
44
...
2 −70
...
0
15
...
3
16
...
5
−45
...
3 −11
...
0
100
...
7
−12
...
0
24
...
8
−4
...
3
−6
...
0
87
...
6
35
...
9
−4
...
3
−12
...
3
100
...
0
90
...
4
82
...
8
22
...
6
15
...
2
100
...
5
82
...
3
12
...
6
8
...
4
88
...
0
87
...
2
9
...
6
8
...
3
82
...
1
100
...
0
10
...
0
12
...
8
59
...
2
49
...
0
−16
...
3
−12
...
6
12
...
3
10
...
7
100
...
7
10
...
6
7
...
6
11
...
3
10
...
0
94
...
7
8
...
6
12
...
7
10
...
9
100
...
87
RV W t
REW t
RHt
QHt
RSM Bt
ROF P t
EW
τt
SW
τt
January 1982 to December 1986 (261 Weeks)
RV W t
100
...
1
REW t
92
...
0
−17
...
1
RHt
6
...
2
RSM Bt
−2
...
6
ROF P t
QHt
6
...
8
−23
...
1
28
...
1 −10
...
6
−30
...
0
31
...
5
−12
...
8
−17
...
0
73
...
5
73
...
0
−41
...
0
1
...
2
−54
...
1
100
...
9
19
...
5
−23
...
6
13
...
0
−15
...
0
−20
...
9
EW
τt
27
...
0
−12
...
3
19
...
7
100
...
2
SW
τt
28
...
6
−7
...
2
6
...
9
93
...
0
January 1987 to December 1991 (261 Weeks)
RV W t
100
...
2
−40
...
0
8
...
9
−15
...
0
REW t
91
...
0
−44
...
5
44
...
3
−16
...
9
RHt
−40
...
3
100
...
1
−23
...
2
43
...
7
QHt
−36
...
5
58
...
0
−37
...
8
25
...
0
RSM Bt
8
...
6
−23
...
1
100
...
1
−11
...
9
ROF P t
18
...
3
−26
...
8
45
...
0
−18
...
7
EW
τt
−15
...
7
43
...
3
−11
...
5
100
...
7
SW
τt
−17
...
9
43
...
0
−16
...
7
94
...
0
15
...
4
January 1992 to December 1996 (261 Weeks)
100
...
3
REW t
84
...
0
73
...
5
46
...
2
18
...
4
RHt
95
...
2
100
...
8
−19
...
7
15
...
2
84
...
5
40
...
2
46
...
5
66
...
2
−13
...
0
−41
...
2
12
...
2
−19
...
6
100
...
3
3
...
1
−3
...
1
−5
...
7
0
...
3
100
...
0
EW
τt
15
...
2
15
...
0
3
...
0
100
...
7
SW
τt
10
...
4
11
...
2
−10
...
3
92
...
0
Table 19b: Correlation matrix for weekly returns on the CRSP value-weighted index (RV W t ), the CRSP
equal-weighted index (REW t ), the hedging-portfolio return (RHt ), the hedging-portfolio dollar-return (QHt ),
the return of the small-minus-big capitalization stocks portfolio (RSM Bt ), the return return ROF P t of the
optimal-forecast portfolio (OFP) for the set of 25 market-beta-sorted basis portfolios, and the equal-weighted
and share-weighted turnover indices (τtEW and τtSW ), using CRSP weekly returns and volume data for NYSE
and AMEX stocks from 1962 to 1996 and in five-year subperiods
...
The results for the two-factor model with the hedging-portfolio
dollar-return factor and the two-factor model with the SMB factor are similar
...
012 and −3
...
564 and
−4
...
In contrast, the marketbeta coefficients are not significant in either of these specifications, and are also of the wrong
sign
...
299 for the SMB factor and
a t-statistic of 4
...
For the three remaining test periods, the only specifications with any statistically significant factors are the SMB and MPP two-factor models in the 1992–1996 testing period
...
Overall, the results do not provide overwhelming support for any factor in explaining the
cross-sectional variation of expected returns
...
47 However, the point estimates of the cross-sectional regressions
show that the hedging-portfolio factor is comparable in magnitude and in performance to
other commonly proposed factors
...
Both volume and prices are driven by underlying economic forces,
and thus convey important information about the workings of the market
...
In this article, we hope to have made a contribution towards this goal
...
89
Model
Statistic
γ0t
ˆ
γ1t
ˆ
γ2t
ˆ
2
R (%)
January 1972 to December 1976 (261 Weeks)
M
Rpt = γ0t + γ1t βp +
Mean:
S
...
:
t-Stat:
pt
M
HR
Rpt = γ0t + γ1t βp + γ2t βp +
(φ = 1
...
50)
pt
pt
M
SM
Rpt = γ0t + γ1t βp + γ2t βp B +
pt
0
...
015
1
...
000
0
...
348
10
...
9
Mean:
S
...
:
t-Stat:
0
...
035
2
...
002
0
...
047
−0
...
037
−0
...
3
10
...
D
...
004
0
...
162
−0
...
034
−1
...
104
3
...
442
15
...
9
Mean:
S
...
:
t-Stat:
0
...
014
1
...
000
0
...
217
0
...
142
0
...
1
10
...
D
...
75)
M
HQ
Rpt = γ0t + γ1t βp + γ2t βp +
(φ = 4
...
001
0
...
166
0
...
022
2
...
7
12
...
D
...
003
0
...
748
−0
...
020
−0
...
012
0
...
712
13
...
4
Mean:
S
...
:
t-Stat:
0
...
013
3
...
001
0
...
754
−1
...
104
−4
...
5
12
...
D
...
001
0
...
251
0
...
017
−0
...
299
1
...
433
14
...
4
Mean:
S
...
:
t-Stat:
0
...
018
2
...
001
0
...
843
0
...
036
0
...
1
11
...
The five linear-factor models are: the standard CAPM (βp ), and four two-factor models in
which the first factor is the market beta and the second factors are, respectively, the hedging portfolio return
HR
HQ
beta (βp ), the hedging portfolio dollar-return beta (βp ), the beta of a small-minus-big cap portfolio
SM B
return (βp
), and the beta of the optimal forecast portfolio based on a set of 25 market-beta-sorted basis
OF P
portfolios (βp
)
...
D
...
75)
M
HQ
Rpt = γ0t + γ1t βp + γ2t βp +
(φ = 2
...
006
0
...
169
−0
...
019
−1
...
4
11
...
D
...
006
0
...
390
−0
...
020
−0
...
006
0
...
732
9
...
4
Mean:
S
...
:
t-Stat:
0
...
011
8
...
002
0
...
297
−0
...
874
−0
...
4
9
...
D
...
005
0
...
451
−0
...
019
−1
...
038
1
...
531
10
...
4
Mean:
S
...
:
t-Stat:
0
...
011
7
...
001
0
...
818
0
...
021
0
...
7
10
...
D
...
002
0
...
649
0
...
023
0
...
9
8
...
D
...
002
0
...
254
0
...
019
0
...
000
0
...
132
5
...
1
Mean:
S
...
:
t-Stat:
0
...
016
2
...
000
0
...
147
0
...
194
0
...
0
6
...
D
...
003
0
...
101
0
...
020
0
...
075
1
...
979
7
...
2
Mean:
S
...
:
t-Stat:
0
...
015
2
...
001
0
...
385
0
...
021
−0
...
4
7
...
The five linear-factor models are: the standard CAPM (βp ), and four two-factor models in
which the first factor is the market beta and the second factors are, respectively, the hedging portfolio return
HR
HQ
beta (βp ), the hedging portfolio dollar-return beta (βp ), the beta of a small-minus-big cap portfolio
SM B
return (βp
), and the beta of the optimal forecast portfolio based on a set of 25 market-beta-sorted basis
OF P
portfolios (βp
)
...
D
...
002
0
...
679
0
...
020
1
...
7
7
...
D
...
002
0
...
785
0
...
020
1
...
004
0
...
650
6
...
8
Mean:
S
...
:
t-Stat:
0
...
015
3
...
000
0
...
178
−1
...
992
−1
...
2
6
...
D
...
002
0
...
653
0
...
019
0
...
154
1
...
147
6
...
0
Mean:
S
...
:
t-Stat:
0
...
016
0
...
002
0
...
236
0
...
015
2
...
9
7
...
The five linear-factor models are: the standard CAPM (βp ), and four two-factor models in
which the first factor is the market beta and the second factors are, respectively, the hedging portfolio return
HR
HQ
beta (βp ), the hedging portfolio dollar-return beta (βp ), the beta of a small-minus-big cap portfolio
SM B
return (βp
), and the beta of the optimal forecast portfolio based on a set of 25 market-beta-sorted basis
OF P
portfolios (βp
)
...
Indeed, by exploiting the relation
between prices and volume in our dynamic equilibrium model, we are able to identify and
construct the hedging portfolio that all investors use to hedge against changes in market conditions
...
Although our model is purposefully parsimonious so as to focus attention on the essential
features of risk-sharing and trading activity, it underscores the general point that quantities,
together with prices, should be an integral part of any analysis of asset markets, both theoretically and empirically
...
Although this
is an old theme that has its origins in Black (1972), Mayers (1973), and Merton (1973),
it has become less fashionable as competing approaches such as the statistical approach of
Roll and Ross (1980) and Chamberlain and Rothschild (1983) and the empirical approach
of Fama and French (1992) have become more popular
...
An important direction for future research is to incorporate more specific aspects of the
market microstructure in the analysis of trading volume
...
For example, for most
individual investors, financial markets have traditionally been considered close to perfectly
competitive, so that the size of a typical investment has little impact on prices
...
But as institutional investors have grown in size and
sophistication over the past several decades, and as frictions in the trading process have
become more important because of the sheer volume of trade, it has become clear that
securities markets are not perfectly competitive, at least not in the short run
...
93
For example, if a large pension fund were to liquidate a substantial position in one security,
that security’s price would drop precipitously if the liquidation were attempted through a
single sell-order, yielding a significant loss in the value of the security to be sold
...
48 This suggests that there is information to
be garnered from quantities as well as prices; a 50,000-share trade has different implications
than a 5,000-share trade, and the sequence of trading volume contains information as well
...
Finally, the presence of market frictions such as transactions costs can influence the
level of trading volume, and serves as a bridge between the market microstructure literature
and the broader equilibrium asset-pricing literature
...
Some have even argued that additional trading frictions or “sand in the gears” ought to be
introduced in the form of a transactions tax to discourage high-frequency trading
...
e
...
An equilibrium model with fixed transactions costs,
e
...
, Lo, Mamaysky, and Wang (2001), may reconcile these two disparate views of trading
volume
...
See, for example, Admati and Pfleiderer (1988), Bagehot (1971), Easley and O’Hara (1987), Foster and
Viswanathan (1990), Kyle (1985), and Wang (1994)
...
49
94
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Title: Market Trading Volume
Description: One of the most fundamental notions of economics is the determination of prices through the interaction of supply and demand. The remarkable amount of information contained in equilibrium prices has been the subject of countless studies, both theoretical and empirical, and with respect to financial securities, several distinct literatures devoted solely to prices have developed.1 Indeed, one of the most well-developed and most highly cited strands of modern economics is the asset-pricing literature. However, the intersection of supply and demand
Description: One of the most fundamental notions of economics is the determination of prices through the interaction of supply and demand. The remarkable amount of information contained in equilibrium prices has been the subject of countless studies, both theoretical and empirical, and with respect to financial securities, several distinct literatures devoted solely to prices have developed.1 Indeed, one of the most well-developed and most highly cited strands of modern economics is the asset-pricing literature. However, the intersection of supply and demand