Relationship between Trading Volume and Security Prices and Returns

Relationship between Trading Volume and Security Prices and Returns
Walter Sun
Area Exam Report
MIT Laboratory for Information and Decision Systems
Technical Report P-2638
Exam Date:
Thursday, February 6th, 2003
Exam Time: 4:15PM - 6:15PM
Exam Room: 35-338 (Osborne Room)
Abstract
The relationship between trading volume and securities prices is a complex one which, when
understood properly, can lead to many insights in portfolio theory. Over the past forty years,
much work has been done trying to understand this relationship. In this document, we will
attempt to introduce and discuss some of these papers. First, we introduce basic topics of
finance theory, such as the Capital Asset Pricing Model and two-fund separation. With this
knowledge, we proceed to discuss how volume and price move together, how unusual volume
can be a predictive measure of future price changes, and also how volume can allow us to infer a
hedging portfolio. In each case, we present theoretical models which support empirical results.
Finally, we analyze some sample price and volume data around the most recent quarter of
earnings announcements.

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CONTENTS
i
Contents
List of Tables
iii
1
Introduction and Motivation
1
1.1
Gazing into the Crystal Ball - Predicting Price Movements . . . . . . . . . . . . . .
1
1.1.1
Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2
Trading Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.3
Interpretation of Information . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Initial Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
Volume’s Effect on Variability of Returns . . . . . . . . . . . . . . . . . . . .
3
1.2.2
Volume’s Predictive Nature for Price Changes . . . . . . . . . . . . . . . . . .
4
1.3
Document Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2
Overview of Portfolio Theory
6
2.1
Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.1
Why Everyone Holds the Market Portfolio . . . . . . . . . . . . . . . . . . . .
6
2.1.2
Beta of a Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1.3
Two-fund separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2.1
Using APT to Justify Diversification . . . . . . . . . . . . . . . . . . . . . . .
8
2.2.2
Multi-factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3
Efficient Frontier and Markowitz’s Portfolio Selection Model . . . . . . . . . . . . . .
8
2.4
Short Selling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.5
Lemons Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3
The Volume-Price Relationship
11
3.1
Volume is Positively Correlated with Absolute Price Changes . . . . . . . . . . . . . 11
3.2
Probabilistic Model for Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1
Consensus on Interpretation of Information . . . . . . . . . . . . . . . . . . . 11
3.2.2
General Case of Information Interpretation . . . . . . . . . . . . . . . . . . . 14
3.3
Volume is Heavy in Bull Markets, Light in Bear Markets . . . . . . . . . . . . . . . . 16
4
Serial Correlation of Returns with Abnormal Volume
18
4.1
Price Movements on Private Information . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2
Mean Reversion from Non-Informational Trading . . . . . . . . . . . . . . . . . . . . 19
4.2.1
Reasons for Non-Informational Trading . . . . . . . . . . . . . . . . . . . . . 19
4.2.2
Risk-Averse Investors as Market Makers . . . . . . . . . . . . . . . . . . . . . 19
4.2.3
Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2.4
Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2.5
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5
Inferring the Hedging Portfolio from Prices and Volume
24
5.1
Definitions and the Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2
Two-factor Turnover Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3
Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3.1
Estimating the Hedging Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3.2
Forecasting Market Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.3.3
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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ii
CONTENTS
6
An Analysis of Current Data
30
6.1
Testing the Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.2
Dataset Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.3
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7
Conclusions
34
A Statistics Review and Overview
35
A.1 Transforms of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A.1.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A.1.3 Moment-Generating Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A.2 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
A.2.1 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
A.2.2 p-Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.2.3 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.3 t-Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.4 Chi-square Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A.5 F-tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A.6 Linear Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A.6.1 First-Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A.6.2 R2 - The Coefficient of Determination . . . . . . . . . . . . . . . . . . . . . . 39
A.6.3 Higher-Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A.6.4 F-Test to Determine Significance in Regressions . . . . . . . . . . . . . . . . . 40
B Miscellaneous Details
41
B.1 Hedging Portfolio Forecasts Market Returns . . . . . . . . . . . . . . . . . . . . . . . 41
B.2 Details of Ying’s Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
B.3 Source Code - MixedTraders.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Bibliography
43

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LIST OF TABLES
iii
List of Tables
1
Expected number of trades, standard deviation, and coefficient of variation (CV(T))
for different number of investors/traders. . . . . . . . . . . . . . . . . . . . . . . . . . 13
2
Papers which test positive correlation between price change and volume (Karpoff [22]). 17
3
Tests for serial correlation of returns (Morse [33]). . . . . . . . . . . . . . . . . . . . 19
4
Cross-sectional regression tests of market and hedging portfolio β’s for five-year
subperiods. ¯
R2 is average R2 over the 100 portfolios. . . . . . . . . . . . . . . . . . . 28
5
Components of the Dow Jones Industrial Average (Co = Company, Corp = Corpo-
ration, Inc = Incorporated). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6
Average daily volume, 5-day pre-earnings average volume, percentage increase of
volume near earnings, and adjusted returns over five trading days, before and after
the earnings announcement for DJIA stocks. . . . . . . . . . . . . . . . . . . . . . . . 33

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1
1 Introduction and Motivation
When investors today read the business section of the paper, or obtain online quotations of their
favorite stocks, one of the statistics which usually goes unnoticed is volume data. After seeing the
price of a security, which is usually of primary interest, an investor may look next at data such as
yield, price-to-earnings ratio, market capitalization, or ex-dividend date, before even considering the
volume statistic. Despite being ignored by many investors, trading volume does have a relationship
to price data, returns, and other aspects of portfolio theory.
The following report provides an overview of research in the area of trading volume within the
wider discipline of stochastic finance. The analysis of trading volume and its relationship with
security prices and changes in price is a topic that has been considered for over 40 years. Its roots
are generally credited to the work of Osborne [36]. In his seminal work, he modeled price changes
according to a diffusion process that had a variance dependent on the quantity of transactions
on that particular issue. With this, he began a long line of work that considered the possible
relationship between returns and the volume of trading. Before these works are discussed and
analyzed, we shall try to motivate research in this area and hopefully answer the question ”Why is
the relationship between price and volume of any interest?”
1.1 Gazing into the Crystal Ball - Predicting Price Movements
One of the most sought after results of financial economics research is the predictability of asset
prices. As the reader might expect, one aspect of financial economics involves modeling future
prices of equities, bonds, or other derivative securities. Early research focused on attemping to
predict future prices based on historical prices alone.
1.1.1 Martingales
In 1565, Girolamo Cardano wrote in his text Liber de Ludo Aleae (The Book of Games of Chance)
that in a ”fair game” the total winnings, represented by Pt, is a stochastic process satisfying the
following condition:
E(Pt+1|Pt,Pt−1,...,P0) = Pt
meaning that the best estimate of what you will have tomorrow is what you have today. This process
is often called a martingale [5]. Exactly four hundred years later, Samuelson [40] showed that
efficient markets exhibit this behavior and described this condition as weak-form market efficiency.
The Efficient Market Hypothesis, introduced by Fama [12], states that in an efficient market there
are a large number of ”rational profit-maximizers” that actively compete, each trying to predict
future market values. The interaction of these participants causes the current price to fully reflect
the expectation of the future price of the security. In essence, the more efficient a market, the more
unpredictable future pricing will be, with the expected value of future prices equal to the current
price, as indicated by the equation above. A reader at this point may argue that historical equity
returns in the domestic market have been positive over the long run, thus making this hypothesis
questionable. In fact, it was shown in the 70’s [20, 24] that the efficient market hypothesis does
not hold for asset prices, but with the proper adjustment for risk and the prevailing risk-free
rate [8, 16, 20], Cardano’s equation does hold and weak-form market efficiency exists. From seeing
that price alone was insufficient to predict future prices, researchers sought other factors to aid in
their analysis1.
1It should be noted that recent research has shown that prices do not follow a purely random walk [26], but the
point of this discussion is to motivate why researchers have considered alternative factors, such as trading volume, to
predict future price trends.

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2
1
INTRODUCTION AND MOTIVATION
1.1.2 Trading Volume
One factor many have considered in the prediction of prices is trading volume. Volume is a measure
of the quantity of shares that change owners for a given security. For instance, on the New York
Stock Exchange (NYSE), the average daily volume for 2002 was 1.441 billion shares, contributing
to 40.9 billion dollars of securities traded each day among the roughly 2800 companies listed on the
NYSE [18]. The amount of daily volume on a security can fluctuate on any given day depending
on the amount of new information available about the company, whether options contracts are set
to expire soon2, whether the trading day is a full or half day, and many other possible factors.
Of the many different elements affecting trading volume, the one which correlates the most to the
fundamental valuation of the security is the new information provided. This information can be
a press release or a regular earnings announcement provided by the company, or it can be a third
party communication, such as a court ruling or a release by a regulatory agency pertaining to the
company. For example, McDonald’s Corporation (NYSE:MCD) has an average trading volume of
7.58 million shares per day. On December 17, 2002, they announced a warning and reduction of
expected earnings. The news led to trading of 35.17 million shares that day, about five times the
average, and a drop in price of 8%. The abnormally large volume was due to differences in the
investor’s view of the valuation after incorporating the new information. Because of what can be
inferred from abnormal trading volume, the analysis of trading volume and associated price changes
corresponding to informational releases has been of much interest to researchers.
Returning to the question of, why consider trading volume and its relationship to prices, Kar-
poff [22] suggests the following four possible reasons. First, it adds insight to the structure of
financial markets. The correlations which are found can provide information regarding rate of in-
formation flow in the marketplace, the extent that prices reflect public information, the market
size, and the existence of short sales and other market constraints. Second, studies that use a
combination of price and volume data to draw inferences need to properly understand this rela-
tionship. For example, trading volume is often used to determine whether or not a price change
was due to any informational content, and also whether investor interpretations of information are
consistent or differing. Some researchers [38] have used volume and price changes to determine
that shareholders hold securities primarily because of dividend yields. Beaver [2] asserts that the
volume corresponding to a price change due to new information indicates how much investors differ
in the interpretation of the new data. As one can imagine, the validity of many of these inferences
rely on the relationships between price and volume.
Third, understanding the price-volume relationship in futures and other speculative markets is
vital for one to determine why the distribution of rates of return appear kurtotic3. One theory is
that rates of return are characterized by a class of distributions with infinite variance, known as the
stable Paretian hypothesis. Another theory is that the data comes from a mixture of distributions
which each have different conditional variances, known as the mixture distribution hypothesis.
Research has shown that price data is generated by a stochastic process with changing variances
which can be predicted or estimated by volume data. These price/volume analyses support the
mixture of distribution hypothesis in the following way. If we measure changes in volume data as a
proxy for variances at different events, we observe that the distribution of returns exhibit different
conditional variances. As a result, our return data follows a mixture of distributions.
Fourth, price variability affects trading volume in futures contracts. This interaction determines
whether speculation is a stabilizing or destabilizing factor on futures prices. The time to delivery
2Options contracts expire on the third Friday of each month, unless that date is a holiday.
3The degree of peakedness of a distribution. It is a normalized form of the fourth central moment of a distribution
[14].

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1.2
Initial Work
3
of a futures contract affects the volume of trading, and possibly also the price.
1.1.3 Interpretation of Information
As Beaver noted, volume is a useful tool in determining how much disagreement exists with the
arrival of new information. Anything that causes investors to act can be described as information,
whether or not it truly has any fundamental impact on the underlying valuation of the company. For
example, a University of Michigan study found that, ”in the absence of clear financial information,
investor decisions are swayed by the aesthetics of financial reports” [42]. It is doubtful that there
exists any significant correlation between the aesthetics of a company report and its future earnings,
yet the study shows that some individuals attribute value to an organization which produces an
aesthetically pleasing report.
Sometimes, information on a company can impact the volume and price of another unrelated
company due to the sheer similarity of the ticker symbol. In particular, Rashes [37] discussed an
example where information releases on MCI Communications (Nasdaq:MCIC4) led to increased
volume on Massmutual Corporate Investors (NYSE:MCI), a case of co-movement due to ticker
confusion. MCI Communications was a large telecommunications firm that was acquired for more
than 20 billion dollars, while Massmutual Corporate Investors is a closed-end fund which trades
with net assets of roughly $200 million. Rashes found that Massmutual’s top volume days between
11/1/1996 and 11/13/1997 all occurred on days when there was merger news on MCI Communica-
tions, showing that Massmutual’s volume was correlated with MCI Communications’ trading vol-
ume, but not those of other telecommunications companies. The latter is certainly to be expected
because Massmutual, during that period, did not hold any major telecommunications company
stock, while the former can only be attributed to investor confusion.
1.2 Initial Work
Following is a summary of some of the initial work in this area of discipline. First, we analyze
Osborne’s seminal work, and then briefly discuss a work by Ying.
1.2.1 Volume’s Effect on Variability of Returns
In 1959, Osborne [36] hypothesized that securities prices could be modeled as a lognormal distribu-
tion with the variance term dependent on the trading volume. In particular, if y(τ ) = ln( P (t+τ) ),
P (t)
where P (t + τ ) and P (t) are the price of some issue at times t + τ and t, respectively, then the
steady state distribution of y may be expressed as
1
φ(y) = √
e −y2
2σ2τ ,
2πσ2τ
where σ is the dispersion that is positively correlated to the amount of trading volume. By verifying
that this model held empirically, Osborne concluded that the log return process was a Brownian
motion process. Furthermore, if we examine the probability distribution of price itself, we see that
P
dY
1
ln( P )2 1
f
P0
P (P ) = φ(y = ln(
))
=
exp(
)
.
P
√
−
0
dP
2πσ2τ
2σ2τ
P
When we compute the expectation, we obtain
∞
σ2τ
E(P ) =
P fP (P )dP = P0eσ2τ/2 ≈ P0(1 +
)
p=0
2
4MCI merged with WorldCom, Inc.; so, this ticker is no longer active.

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4
1
INTRODUCTION AND MOTIVATION
through a change of variables, completion of square, and Taylor series approximation. From the
empirically calculated values of σ, Osborne was able to predict a 3 to 5% annual return from this
model.
The fact that greater activity on a security will produce more variance in the price may seem
reasonable, even intuitive. However, what was remarkable was the fact that the annual drift
predicted by this Brownian motion process could partially explain the annual returns that the
market actually bore.
1.2.2 Volume’s Predictive Nature for Price Changes
Seven years later, in 1966, Ying produced a paper [43] which applied a series of statistical tests to a
six-year daily series of price and volume. As is the case for most of the other analyses discussed in
this document, Ying normalized the trading volume by the number of shares outstanding to avoid
any biases from issues with larger number of outstanding shares5. Similarly, prices were adjusted
to reflect quarterly dividends.
To apply the test of his hypotheses, Ying used as volume data the New York Stock Exchange
(NYSE)6 daily percentage volume (also called turnover), and Standard and Poor’s 500 index returns
from January 1957 to December 1962 for price data. Critics to his research have argued that the
underlying issues for volume and price data were not exactly the same, as well as the fact that
his adjustments for price were over quarterly dividend data, while the daily volume data was
adjusted by monthly total share data. Furthermore, it was found that some of these conclusions
were inconsistent with weak form of market efficiency7, although this fact alone might somewhat
be expected, as any relationship that can be found between volume and future prices will reject
the weak form of market efficiency.
Details of his analysis may be found in Appendix B.2, but his main conclusions were
• A small volume is usually accompanied by a fall in price
• A large volume is usually accompanied by a rise in price
• A large increase in volume is usually accompanied by a large price change.
• A large volume is usually followed by a rise in price
• If the volume has decreased (increased) five straight trading days, the price will tend to fall
(rise) over the next four trading days.
1.3 Document Outline
The previous two examples illustrate the origins of research in trading volume. In this report, we
will summarize and analyze three more recent papers in this field [4,22,28], and the author will then
provide some new analysis on the serial relationship between volume and price changes around 4th
quarter 2002 earnings announcements on the thirty stocks that comprise the Dow Jones Industrial
Average. This paper will be outlined as follows:
Basic Finance Background – Section 2 will provide a quick summary of some of the main
concepts of basic portfolio theory.
5Volume divided by Shares Outstanding is called turnover, and is often a preferred indicator for analysis.
6Most of the volume in domestic trading then occurred on the NYSE; the NASDAQ did not exist until 1971.
7The weak form of market efficiency states that stock prices reflect all information that can be found from historical
prices, trading volume, and short interest. Mathematically, one would say that the price satisfies a Markov process.

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1.3
Document Outline
5
Volume-Price Relationship – Section 3 will look at how volume and price move together and
will present a probabilistic model for trading.
Serial Correlation of Returns – Section 4 will consider how abnormal trading volume affects
the subsequent returns on the market.
Inferring the Hedging Portfolio – Section 5 will examine how we can use volume data to infer
the hedging portfolio.
Analysis of Current Market Data – Section 6 will discuss what kind of relationship exists
between price and volume in our current market.
Conclusion – Section 7 will conclude with a summary of this document. The appendix has
background on the mathematics and statistics required for this paper.

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6
2
OVERVIEW OF PORTFOLIO THEORY
2 Overview of Portfolio Theory
During the analysis of current finance papers, some basic concepts will be referenced. This sec-
tion provides a quick overview of portfolio theory and some financial terms of interest. A reader
knowledgeable on the topic of finance may skip this section without any loss of continuity.
2.1 Capital Asset Pricing Model
The Capital Asset Pricing Model (CAPM) was developed by Sharpe, Lintner, and Mossin [25, 34,
41]. Despite its restrictive constraints and assumptions, the model provides a general idea of the
relationship between the expected return of an asset and the riskiness of it. Details of CAPM
follow.
For CAPM to hold, the following assumptions which basically generalize investors as a homo-
geneous group of individuals must be true [3]. Namely,
• no single investor has a significant portion of the total wealth in the world, thus making all
investors price-takers who do not affect the price of securities through their own actions.
• we have a single-period horizon
• investments are limited to publicly traded financial assets and risk-free borrowing or lending.
This eliminates the investments of human capital, private companies, and government-funded
projects.
• there are no taxes or transaction costs
• all investors are rational, investing according to Markowitz [30]
• all investors have the same economic expectations of the world and its future financial devel-
opment
If these six conditions hold, the following can be concluded:
• every person will hold a market-weighted portion of each asset in their portfolio
• the market portfolio will be on the efficient frontier
• the risk premium on the market portfolio will be proportional to its risk and the amount of
risk aversion of a representative investor
• the risk premium on each individual asset will be proportional to the risk premium on the
market portfolio and the beta of that security.
2.1.1 Why Everyone Holds the Market Portfolio
At first glance, it might not be clear why everyone chooses to hold the market portfolio. If everyone
is rational, has the same investment options and time horizon, and possesses the same future
economic expectations, then each person will have a portfolio which is a linear combination of the
risk-free bond and the same market portfolio (depending on the amount of risk each individual
wants to bear). Now that we know that all investors will hold the same market portfolio, what
guarantees us that every stock will be in the portfolio. Suppose a certain stock does not belong in
the portfolio. That is, no one wants to buy the security. By the principle of supply and demand,
the price of this stock will drop until it becomes an attractive investment. Once it reaches this
point, it will become included in the market portfolio.

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