(Thesis) 2011 - The Technical Analysis Method of Moving Average Trading. Rules That Reduce The Number of Losing Trades - Marcus C. Toms
(Thesis) 2011 - The Technical Analysis Method of Moving Average Trading. Rules That Reduce The Number of Losing Trades - Marcus C. Toms
(Thesis) 2011 - The Technical Analysis Method of Moving Average Trading. Rules That Reduce The Number of Losing Trades - Marcus C. Toms
Marcus C. Toms
Newcastle University
August 2011
Abstract
A general issue with moving average trading is the assumption that all buy/sell signals result
in a trading action. The argument that such trading rules are representative of trading practice
is highly questionable. This thesis proposes two new moving average trading rules designed
to capture trading practice. The first trading rule is the trade reduction rule and is based on the
idea of allowing a trade to run. The second trading rule is the positive autocorrelation rule and
is based on the idea of only trading if it is believed to be profitable to do so. The trading rules
are tied to moving average trading via the buy/sell signal generating mechanism and alter the
way the price crossover rule responds to the buy/sell signals. Simulations of portfolios of UK
equities find that the trading rules uncover information that is missed by the price crossover
rule and there is evidence that this information is financially exploitable. This motivates the
argument that the information needed for trading to be economically viable is observable in
the price. The trading rules also establish a link with the market microstructure literature. The
trading rules uncover issues of informed trading (asymmetric information), liquidity, adverse
selection and price impact. The strongest interpretation that can be applied to the trading rules
in this context is that they are examples of informed trading. Compared to the price crossover
rule, the trading rules are better able to extract meaning from or are better able to understand
I would like to thank my supervisor Dr Graeme Chester for his continuing support. I would
also like to thank my advisor Professor Robert Hudson for offering words of encouragement
at the right time. Similar thanks go to my family. I would particularly like to thank my father
whose help in getting through the final year was invaluable. Special thanks are for Yu-Chun
List of Figures iv
1 Introduction 1
1.1 Background 1
1.5 Motivation 12
1.8 Scope 18
1.9 Contributions 18
2 Literature Review 20
2.3 Conclusion 27
3.1 Data 29
i
3.1.4 Reconstructing the spread 33
3.4 Summary 49
4.2.1 Long 69
4.2.2 Short 79
4.3 Conclusion 89
ii
6 Conclusion and Further Work 127
Appendix 137
(
A.11 VP(25,1) = VP(25) RW e1 ) 157
References 158
iii
List of Figures
4.1 Example of the moving average and its mapping to the bounded moving
average 54
4.3 Example trades for the long only trade reduction rule in terms of the path of the
4.4a Number of winning and losing trades for the long only price crossover and
trade reduction rules for the FTSE 100 portfolio for the period 01-January-
1965 to 30-June-2009 65
4.4b Number of winning and losing trades for the short only price crossover and
trade reduction rules for the FTSE 100 portfolio for the period 01-January-
1965 to 30-June-2009 66
4.5 Sum of the winning and losing trades for the long only and short only price
crossover and trade reduction rules for FTSE 100 portfolio for the period 01-
January-1965 to 30-June-2009 67
4.6a Mean return per trade for the long only price crossover and trade reduction
4.6b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the long only price crossover and trade reduction rules for the
4.7a Mean return per trade for the long only price crossover and trade reduction
iv
rules for the FTSE 250 portfolio 73
4.7b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the long only price crossover and trade reduction rules for the
4.8a Mean return per trade for the long only price crossover and trade reduction
4.8b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the long only price crossover and trade reduction rules for the
4.9a Mean return per trade for the long only price crossover and trade reduction
4.9b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the long only price crossover and trade reduction rules for the
4.10a Mean return per trade for the short only price crossover and trade reduction
4.10b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the short only price crossover and trade reduction rules for the
4.11a Mean return per trade for the short only price crossover and trade reduction
4.11b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the short only price crossover and trade reduction rules for the
4.12a Mean return per trade for the short only price crossover and trade reduction
v
rules for the FTSE Small Cap portfolio 85
4.12b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the short only price crossover and trade reduction rules for the
4.13a Mean return per trade for the short only price crossover and trade reduction
4.13b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the short only price crossover and trade reduction rules for the
autocorrelation and random walk dynamics and the return to trading the price
autocorrelation and random walk dynamics and their variance ratios 101
5.3 The first three eigenvectors for VP(100) estimated using 1000 random walk
5.4 The first three eigenvectors for VP(25) estimated using 1000 random walk
5.5 Variance explained by the first 10 principal components for VP(25) 108
5.6 Cumulative frequency distribution for VP(25,1) estimated using another 1000
5.7 Equity curves for the 10-day moving average price crossover and positive
5.8 Return for the price crossover and positive autocorrelation rules for Yule Catto 118
5.9 Return for the price crossover and positive autocorrelation rules for the FTSE
vi
100, FTSE 250, FTSE Small Cap and FTSE Fledgling portfolios 120
5.10 Ratio of the capital needed to fund trading for the price crossover and positive
6.1 Trading and the traders perception of the relationship in information flow 129
vii
List of Tables
viii
Chapter 1
Introduction
A trading system is a systematic method for buying and selling financial instruments with a
view to consistently making money. As such, not only do trading systems require that prices
are predictable but also that the predictable component is financially exploitable. This thesis
considers the technical analysis method of moving average trading as the basis for a simple
stock trading system. The question asked is whether remodelling the trading rules to reduce
the number of losing trades increases the mean return per trade to the extent that the trading
1.1 Background
patterns in prices, volume and other market statistics. Technical analysis usually proceeds by
recording market activity in graphical form and then deducing the probable future trend from
the pictured history. The premise is that prices exhibit various geometric regularities, which,
once identified, inform the trader what is likely to happen next. This in turn allows the trader
to run a profitable trading strategy. Technical analysis is prevalent in financial markets and is
readily accessible in practitioner texts such as Pring (2002), in the form of tools provided by
1
Of the technical analysis methods studied in the literature, it is moving average trading that is
perhaps the most compelling. Moving average trading has been shown to uncover predictable
behaviour in the first and second moments of the returns distribution and this result has since
been replicated for many different markets and asset classes. However, there is little evidence
that this implies a market beating trading strategy. Taylor (2005) discusses moving average
trading in detail. The conclusion is that while there is plenty of evidence that moving average
trading has been able to uncover predictable behaviour in the returns distribution and where
this has sometimes been sufficiently precise to allow risk adjusted profits of several per cent
per annum, the successful application of moving average trading is, in general, restricted to
An issue that arises in response to this is that the moving average trading rules discussed in
Taylor (2005) have not changed since their introduction in Brock et al. (1992). The trading
rules suffer from various problems, the most significant of which is that they are either not
profitable or borderline profitable. By profitable, it is meant that the mean return per trade is
greater than the expenditure on costs such that, on average, each trade earns the mean return
per trade minus costs. The background to this thesis is to ask whether remodelling the trading
rules to reduce the number of losing trades increases the mean return per trade to the extent
that the trading rules are profitable and, if so, whether this is economically significant.
Moving average trading refers to the practice of systematically buying and selling whenever
the price crosses its average. The idea is that prices move in trends such that at each point in
2
rising prices and a downtrend is defined as a period of falling prices. When the price cuts up
through its average from below, because recent prices are higher than older prices, the price is
said to be in an uptrend and a buy signal occurs. Similarly, when the price cuts down through
its average from above, because recent prices are lower than older prices, the price is said to
be in a downtrend and a sell signal occurs. The response following a buy signal is to buy and
the response following a sell signal is to sell. If the change in the price level in between buy
and sell signals is sufficient to cover costs, moving average trading is profitable. Conversely,
if the change in the price level in between buy and sell signals is not sufficient to cover costs,
500
price
moving average Sell
400 Sell
Sell Sell
Sell
Price
300
Buy
Buy Buy
Buy Buy
200
Jan-94 Jul-94 Jan-95 Jul-95 Jan-96 Jul-96 Jan-97
Time
Moving average trading is usually defined in the form of a trading rule. A trading rule is a
numerical method that maps the price onto investment decisions. A typical decision variable
at time t is the quantity of an asset qt +1 that is held from the time of the price observation at
3
time t until the time of the next price observation at time t + 1 . Let matn ( pt ) denote the n-day
1 n 1
matn ( pt ) = pt i
n i=0
(1.1)
The simplest moving average trading rule is the price crossover rule:
q1 = 0
///// RULE
qt +1 = qt
The rule starts with a zero investment position. The moving average is calculated at each time
step and the quantity qt +1 is set to 1 when the price is above its average, 1 when the price is
below its average and is unchanged when the price equals its average. The quantity qt +1 is the
investment position at time t + 1 and acts as a multiplier whereby the rule earns qt +1 units of
the time t to time t + 1 return where the return rt is the log return or log price first difference
rt = ln( pt ) ln( pt 1 ) .1 The investment position is said to be long when qt +1 is positive, short
when qt +1 is negative and neutral when qt +1 is zero. Thus, the rule defines a one-step-ahead
predictive classification scheme that divides the price into long, short and neutral investment
1
See Chapter 1 of Campbell et al. (1997) for a discussion of returns.
4
positions that earn positive, negative or zero multiples of the succeeding price change.
A more general version of the price crossover rule is the moving average crossover rule. The
moving average crossover rule is meant to reduce sensitivity to noise by first smoothing the
price. The rule uses two moving averages, a shorter-term average mats ( pt ) of length s and a
longer-term average matl ( pt ) of length l where s < l. The difference to before is that qt +1 now
q1 = 0
///// RULE
qt +1 = qt
A more general version still is the moving average crossover with percentage band rule. The
moving average crossover with percentage band rule is meant to compensate for non-trending
dynamics by introducing a percentage band b 0 offset around the longer-term average such
that the shorter-term average now has to cross the offset before trading takes place. The idea
is that in the absence of a trend the shorter-term average tends to wander around repeatedly
criss-crossing the longer-term average but where the cumulative price changes between each
the presence of a trend however, the shorter-term average is expected to cross the longer-term
5
q1 = 0
///// RULE
qt +1 = qt
The moving average crossover with percentage band rule is often used to represent a family
of trading rules parameterised as (s, l, b). The parameters are chosen according to the type of
trading rule. For example, (1, 50, 0) defines the 50-day price crossover rule. Note also that the
trading rules are always in the market. They do not take neutral positions. A neutral position
is said to occur when the price is too close to the moving average to form a view about the
trend. To allow for neutral positions, the moving average crossover with percentage band rule
q1 = 0
///// RULE
qt +1 = qt
IF (mats ( pt ) < matl ( pt )(1 + b) AND mats ( pt ) > matl ( pt )(1 b)) THEN qt +1 = 0 (1.5)
A further refinement is to specify that once a trade is opened, it is held for a fixed period and
then closed. Trading rules of this type are defined as having a fixed length and are denoted by
FMA(s, l, b) . For example, if the holding period is fixed at 10 days, once a trade is opened it
6
is held for 10 days and then closed. Any buy/sell signals that occur after the trade is opened
and before it is closed are ignored. If instead the holding period varies according to the time
separating the buy/sell signals, the trading rules are defined as having a variable length and
are denoted by VMA(s, l, b) . For example, the trading rules defined by (1.2) to (1.5) are all
variable length. The moving average trading rules just described are the most common in the
literature. Other moving average trading rules studied in the literature and that are rooted in
technical analysis but which receive much less attention are the triple moving average (Wong
et al. (2003)) and the adaptive moving average (Ellis and Parbery (2005)). Kaufmann (2005)
discusses these and other moving average trading rules from a technical analysis perspective.
past price information. For this reason, studies of technical analysis usually appear as tests of
the weak form efficient market hypothesis (EMH). The EMH states that a market is efficient
with respect to the information set t if it is not possible to use the information contained in
t to formulate buy/sell decisions that earn a higher return than is normal for the same level
of risk. A key implication of the EMH is that it is not possible for moving average trading to
successfully exploit past prices. This is because markets are efficient. An efficient market is
defined as a market where the price fully reflects all available information. The definition of
an efficient market as a market where the price fully reflects all available information is due
to Fama (1970) who defines three levels of efficiency, namely, weak form, semi-strong form
and strong form, each of which varies according to the information contained in t :
7
In a weak form efficient market, t comprises all past price information. In a
weak form efficient market, it is not possible to predict future prices using past
price information.
not possible to predict future prices using any kind of information whatsoever.
Each form of efficiency is progressively less restrictive and includes the information set of its
predecessor(s). Thus, if a market is strong form efficient, it is also semi-strong form efficient
and weak form efficient. Similarly, if a market is weak form inefficient, it is also semi-strong
2
Price information is information relating to the price. Examples of price information are the
price itself, whether it is rising or falling and how far it has risen or fallen. Public information
is information in the public domain. Examples of public information are news, published end
of year company accounts and analysts earnings forecasts. Private information is information
that is known but not by the market. Examples of private information are knowledge of fraud
before it is discovered, knowledge of a new order win before it is announced and knowledge
acquired through analysis and research. Note that private information can also be interpreted
as inside information. Trading inside information is illegal. Inside information is distinct from
private information acquired through analysis and research. In market microstructure, private
information is used to denote informed trading (see, for example, De Jong and Rindi (2009)).
Fama (1991) revises the classification scheme. Tests for weak form efficiency are referred to
as tests for return predictability. Tests for semi-strong form efficiency are referred to as event
studies. Tests for strong form efficiency are referred to as tests for private information.
8
form inefficient and strong form inefficient.
A problem with the definition of Fama (1970) is that to test if prices fully reflect all available
information it is necessary to test prices against a model that defines precisely what to expect
when prices do fully reflect all available information. This is known as the joint hypothesis
problem. Tests of whether prices fully reflect all available information are joint tests of the
hypotheses that (1) the market is efficient and (2) the model against which market efficiency
is judged is correct. Tests can fail because one of the two hypotheses is false or because both
Jensen (1978) avoids the joint hypothesis problem and stresses the importance of profitability
in testing for market efficiency. If it is not possible for a trader to profit financially, evidence
of market inefficiency is economically insignificant. Jensen (1978, p. 96) defines the EMH as:
Economic profits are defined as risk adjusted returns net of costs. This is a far more practical
definition and provides guidance on method. To test for market efficiency, it is sufficient to
consider the net risk-return profile of trading rules that trade information set t . This is made
No trading rule has an expected, risk adjusted, net return greater than that provided
9
The difference between the definition of Fama (1970) and those of Jensen (1978) and Taylor
(2005) is that the former does not allow prices to convey predictable information whereas the
latter do. While this might seem to be counter intuitive, predictability and profitability are not
the same thing. It is possible for prices to be predictable but where the predictable component
is not financially exploitable. All definitions are the same in this respect and express the idea
that in an efficient market, it should not be possible to systematically outperform the market
The benchmark reference for moving average trading is Brock et al. (1992) who test various
moving average trading rules applied to the Dow Jones Industrial Average (DJIA) from 1897
to 1986. Ten parameter combinations are evaluated for both fixed and variable length trading
rules with 1 s 5 , 50 l 200 and b = 0% or 1%. Days when qt +1 = 1 are classified as buy
days and days when qt +1 = 1 are classified as sell days. The following results are reported:
Returns on buy days are consistently higher than returns on sell days. The mean
return on buy and sell days across all parameter combinations is 12% and 7%
per annum respectively. Tests of the difference in means for buy and sell days
are statistically significant for each parameter combination. Results are robust
3
See Lim and Brooks (2011) for a systematic review of the empirical literature on the EMH.
10
Returns on sell days are negative. The mean return on sell days is negative for
all four sub-periods. Returns are not explained by a positive risk premium.
Returns on buy days are less volatile than returns on sell days. The standard
deviation of returns for sell days is less than the standard deviation of returns
for buy days for all four sub-periods. If volatility measures risk, the difference
in the level of risk does not explain the higher mean return for buy days than
GARCH pricing models cannot explain the results. The information uncovered
by the trading rules is not explained by linear or time varying volatility models.
Brock et al. (1992) view the results as economically significant. The difference in the mean
buy sell spread is 19% per annum compared to a buy and hold return of 5%. They do not
allow for costs however. Bessembinder and Chan (1998) test the same trading rules as Brock
et al. (1992) applied to dividend adjusted DJIA prices from 1926 to 1991. For the full sample
period, the mean buy sell spread across all trading rules is 4.4% per annum, giving one-way
breakeven costs of 0.39% per trade. Although significant compared to trading costs estimated
at 0.24% 0.26% per trade, breakeven costs decline over time. For the most recent sub-period
of 1976 1991, one-way breakeven costs are 0.22% per trade. It is unlikely that traders would
be able to use the trading rules to generate profits after costs. Similarly, Hudson et al. (1996)
test the same trading rules as Brock et al. (1992) applied to the FT30 from 1935 to 1994. For
the full sample period, the mean round-trip breakeven costs across all trading rules are 0.8%
per trade. This is compared to trading costs estimated at 1% upwards. As before, it is unlikely
that traders would be able to use the trading rules to generate profits after costs.
11
1.5 Motivation
The motivation for this thesis is as follows. It is clear from the results of Brock et al. (1992)
that moving average trading picks up information of some kind. Further, this information is
profitable albeit economically insignificant after costs. However, the trading rules are poorly
defined. If the buy/sell signals are thought of as defining the space of trading opportunities,
the trading rules fail to capture how a trader might react to a trading opportunity in practice.
Rather, they simply buy and sell every buy/sell signal regardless of whether this is the right
thing to do. The motivation is to address this issue by remodelling the trading rules to better
reflect what it is for a trading opportunity to be financially exploitable assuming the resulting
trading rules will have more power as a test of the weak form EMH. For example, doubling
the breakeven costs of the papers discussed in the previous section would cast doubt on the
EMH. The approach is innovative and is to remodel the trading rules to include a description
of trading practice. This is fitted to the moving average such that the information that would
normally be input to the trading decision is substituted with the information contained in the
moving average buy/sell signals. The next section describes the approach to remodelling the
Let buySignalt denote a buy signal and let sellSignalt denote a sell signal. A general model
of the trading rules of Section 1.2 but which does not take neutral positions is:
12
q1 = 0
///// RULE
qt +1 = qt
IF (buySignalt ) THEN qt +1 = 1
The question is whether (1.6) can be made to be profitable. By profitable, it is meant that the
mean return per trade is greater than the expenditure on costs. Let R denote the mean return
per trade and let c denote costs. At the end of trading, on average, each trade earns:
Rc (1.7)
Rc>0 (1.8)
The problem with trying to make (1.6) profitable is that to do so with any degree of certainty
requires an accurate model of future returns. This is a far from trivial task and is beyond the
scope of this thesis for that reason. The approach instead is as follows. As a trend following
strategy, a property of moving average trading is that it tends to generate a small number of
large winning trades offset against a large number of small losing trades. A winning trade is
defined as one where the return is positive. All other trades are defined as losing trades. In
general, a property of the winning trades is that the mean return per winning trade is more
than high enough to cover costs. However, there are many more losing trades than winning
13
trades. The losing trades act to reduce the mean return per winning trade, which reduces the
combined mean per trade to the point where it is either not profitable or borderline profitable.
To see this, Table 1.1 lists the mean return per trade for the winning trades, the losing trades
and the combined winning and losing trades for various instances of the price crossover rule
applied to the daily closing price of Yule Catto (a FTSE Small Cap chemicals company) for
the period 03-January-1972 to 30-June-2009. To allow for serial dependence in the order of
the trades, the mean return per trade is calculated as the continuously compounded mean. A
trade is defined as a sequence of 1s or 1s. Prices include the spread but do not include the
dividend. Costs are not included. From this, the mean return per winning trade is more than
high enough to cover costs. However, when the winning trades are combined with the losing
trades, the combined mean return per trade is loss making. Clearly, the ideal scenario is one
where there are winning trades only and no losing trades at all.
Table 1.1
Mean return per trade for Yule Catto
MA Winning Trades Winning Mean Losing Trades Losing Mean Combined Mean
MA is the moving average length n. Winning Trades and Losing Trades are the number of winning trades and
losing trades. Winning Mean, Losing Mean and Combined Mean are the mean return per trade for the winning
trades, losing trades and combined winning and losing trades.
14
Let T denote the set of trade that results from applying (1.6) and let this comprise the set of
winning trades W and the set of losing trades L such that W L = and T = W L . The
method underpinning this thesis is to use (1.6) to generate T whereby W and L can be said to
exist and to then assume that the nature of W and L is consistent with Table 1.1 such that in
the limit as the number of losing trades tends to zero, (1.6) is almost surely profitable. The
approach is to then remodel (1.6) to keep as many winning trades as possible at the same time
as reducing the number of losing trades. This transforms T into T * = f (T ) where f (T ) acts
on the mean return per trade of T in an attempt to satisfy (1.8). Define W and L as the mean
return per winning trade and the mean return per losing trade:
W = { x T x > 0} (1.9)
L = { x T x 0} (1.10)
1
W=
nw
x, nw = # {W } (1.11)
xW
1
L=
nl
x, nl = # { L} (1.12)
xL
R=
(n w ) (
W + nl L ) (1.13)
nw + nl
This thesis proposes two new trading rules designed in response to (1.13). The first trading
rule is the trade reduction rule. The trade reduction rule is motivated by the observation that
15
when a trade is closed, it is known if it is a winning trade or a losing trade. This information
can be used to explicitly transform (1.13) by responding differently to the losing trades. The
trade reduction rule follows from the idea of allowing a trade to run. Logically, in overview,
END IF (1.14)
The second trading rule is the positive autocorrelation rule. The positive autocorrelation rule
is motivated by the observation that moving average trading profits from persistence in sign
or, equivalently, from persistence in direction. A model that exhibits persistence in sign is to
assume returns are positively autocorrelated. By testing for positive autocorrelation, it is less
likely that a trade will be a losing trade since it is simultaneously more likely to be a winning
trade. This information can be used to implicitly transform (1.13) by responding differently to
the buy/sell signals. The positive autocorrelation rule follows from the idea of only trading if
it is believed to be profitable to do so. Logically, in overview, the trading rule can be written
as:
16
IF (buy / sell signal) THEN
END IF (1.15)
Both trading rules extend the price crossover rule to include a do nothing response. Due to its
design, the trade reduction rule is limited to remodelling the price crossover rule. It does not
generalise to allow for the other types of trading rule defined in Section 1.2. This is because
the buy/sell signals generated by the price crossover rule are ordered as minima and maxima
and this property is exploited by the trade reduction rule. There is no equivalent for the other
types of trading rule. The positive autocorrelation rule does generalise although it is difficult
to imagine what value there is in this. Consequently, the positive autocorrelation rule is also
limited to remodelling the price crossover rule. Both trading rules are exposed to exactly the
same buy/sell signals as the price crossover rule. The difference between the trade reduction
and positive autocorrelation rules is that the trade reduction rule applies after a trade is open
whereas the positive autocorrelation rule applies before a trade is open. The trading rules can
The research objectives are threefold. First, they are to build the proposed trading rules. The
trading rules are straightforward, easy to replicate and are tied to the price crossover rule via
the buy/sell signal generating mechanism. They do not deviate from moving average trading
to the point where it is no longer recognisable as such. Second, they are to test if the trading
17
rules are profitable and if they improve the price crossover rule. Third, they are to determine
1.8 Scope
Inevitably with work on trading systems, there is the question of method. It must be stressed
that the thesis is limited to the technical analysis method of moving average trading only and
that methods/strategies/technologies outside this are not considered. There is also no attempt
to build a working system. Related problems such as how to choose which stocks to trade and
how to choose between the trading rules are not addressed empirically. The scope is limited to
testing the trading rules as defined by the research objectives only. Testing is on a trading rule
1.9 Contributions
There are three contributions. First, there are the trading rules. Both trading rules haves been
designed to capture trading practice and include elements of decision-making as found in the
real world. Second, there is evidence that the trading rules uncover information that is missed
by the price crossover rule and that this information is financially exploitable. This motivates
the argument that the information needed for trading to be economically viable is observable
in the price. This is a challenge to market efficiency as defined by Fama (1970). Third, there
is an empirical link with market microstructure. The trading rules uncover issues of informed
trading (asymmetric information), liquidity, adverse selection and price impact. The strongest
interpretation that can be applied to the trading rules in this context is that they are examples
of informed trading. It is not known of any work that explicitly links moving average trading
18
1.10 Thesis outline
The remainder of the thesis is organised as follows. Chapter 2 reviews the literature. Chapter
3 describes the data and test method used to test the trade reduction rule. Chapter 4 presents
the trade reduction rule. Chapter 5 presents the positive autocorrelation rule. Chapter 6 offers
19
Chapter 2
Literature Review
This chapter reviews the literature on technical analysis. Particular attention is paid to moving
average trading. The chapter starts by defining technical analysis and provides an overview of
the main techniques. These are charts, trading rules and cycle analysis. This is followed by a
review of the survey literature on the use of technical analysis by professional traders in the
foreign exchange and equity markets and which is used to motivate the point that the trading
rules studied in the literature bear little resemblance to trading practice. The remainder of the
chapter is organised as follows. Section 2.1 introduces technical analysis and summarises the
main techniques. Section 2.2 reviews the survey literature. Section 2.3 offers conclusions.
patterns in prices, volume and other market statistics. The philosophy underpinning technical
analysis is that future prices are predictable from past prices as long as prices reflect changes
in supply and demand. The approach is to detect trends as soon as possible and to trade in the
20
towards a variety of economic, monetary, political and psychological forces. The
early stage and ride on that trend until the weight of the evidence shows or proves
Technical analysis differs from fundamental analysis. Fundamental analysis uses economic
variables such as interest rates, valuation ratios and industry trends to predict future returns
based on an economic model. Technical analysis uses past prices and other measures of the
price to predict future returns based on extrapolating the price into the future. Fundamental
analysis is usually concerned with predicting the long term and guides investment decisions.
In contrast, technical analysis is usually concerned with predicting the short term and guides
trading decisions. Fundamental analysis and technical analysis have little in common in this
respect. See Welch and Goyal (2008) for a survey of return predictability based on economic
variables.
To this end, technical analysis employs a number of techniques, the most common of which
are charts, trading rules and cycle analysis. Charting relies on detecting graphical patterns in
the price. Patterns are usually defined as reversal and continuation patterns. Reversal patterns
include the head and shoulders, double tops/bottoms and rounded tops/bottoms. Continuation
patterns include flags, pennants, wedges and rectangles. Studies of charting are often limited
by the need to design a pattern recognition algorithm to extract the patterns although studies
of charting are becoming increasingly common. See, for example, Lo et al. (2000), Dempster
and Jones (2002), Dawson and Steeley (2003), Wang and Chan (2007, 2009) and Leigh et al.
(2008). The general result is that there is evidence of predictive ability. It is not clear to what
21
Because they are mathematically tractable, the majority of studies of technical analysis are
presented in the form of trading rules. As defined in Chapter 1, a trading rule is a numerical
method that maps the price onto investment decisions. Trading rules are based on technical
indicators where a technical indicator is a quantitative function of the price state. Technical
indicators include moving averages, momentum oscillators and volume indicators. Example
momentum oscillators are the %K stochastic and the relative strength index (RSI). Example
volume indicators are the on balance volume and the money flow index. See Achelis (2001)
for an overview of the more popular technical indicators. Trading rule studies include Mills
(1997), Sullivan et al. (1999), Day and Wang (2002), Kwon and Kish (2002), Olson (2004),
Marshall and Cahan (2005) and Marshall et al. (2009). The general result is that while there
is evidence of predictive ability, the trading rules are rarely profitable after costs.
Cycle analysis decomposes the price into cycles or trends with different frequencies. Cycle
analysis includes Dow Theory, Elliot Wave Theory and Kitchin Waves. It also worth noting
that cycle analysis includes seasonalitys such as the January effect (Atanasova and Hudson
(2010)). Cycle analysis is the least represented in the literature due to its esoteric nature. For
example, Dow Theory decomposes the price into a primary trend, secondary trend and minor
trend. Primary trends are further decomposed into phases (Achelis (2001)). An exception is
Brown et al. (1998) who find that Dow Theory as defined by the recommendations made by
William Peter Hamilton during his tenure as editor of the Wall Street Journal for the period
1902 1929 results in positive risk adjusted returns. This is before costs however.
Park and Irwin (2007) review the profitability of technical analysis and discuss charting and
trading rule approaches as well as approaches based on genetic algorithms, neural networks
22
and time series models. The conclusion is that despite evidence of predictability, improved
profits after adjusting for risk. This is attributed to a lack of understanding of how technical
Historically, technical analysis has been treated with scepticism in the academic literature.
This can be linked to (1) influential and widely cited early studies of technical analysis such
as Alexander (1961, 1964), Fama and Blume (1966), Van Horne and Parker (1967, 1968) and
Jensen and Benington (1970) all of whose findings are negative and which do not support
technical analysis as having predictive value, (2) the dominance of the EMH as the prevailing
theoretical paradigm resulting in proponents of the EMH such as Malkiel (1996) dismissing
technical analysis as worthless, (3) the fact that much of technical analysis lacks a strictly
logical explanation and (4) the lack of evidence from practitioners of technical analysis to
Nevertheless, technical analysis remains widely used. Survey studies find strong evidence of
the continued use of technical analysis in practice. Taylor and Allen (1992) survey the use of
technical analysis amongst chief foreign exchange dealers in London in 1988. They find that
at least 90% of respondents place some weight on technical analysis in their decisions. The
weight given to technical analysis is highest for short forecast horizons of intra-day to three
months and declines for longer forecast horizons of six moths to a year, where greater weight
is given to fundamental analysis. The results indicate that 64% of respondents use moving
averages and/or trend following systems and that 40% use rate of change indicators and/or
23
oscillators. Technical analysis and fundamental analysis are seen as complementary with 60%
Anecdotal evidence implies that technical analysis is used to confirm fundamental analysis
rather than to contradict it. This suggests that technical analysis is used mainly in a decision
support capacity and not as a trading vehicle in its own right. However, 2% of respondents
rely exclusively on technical analysis and do not appear to use fundamental analysis at all.
Menkhoff (1997) surveys foreign exchange dealers and fund managers in Germany in 1992.
Technical analysis is used extensively with 87% of respondents giving at least a 10% weight
to technical analysis in their decisions. As in Taylor and Allen (1992), technical analysis and
fundamental analysis are seen as complementary. The weight given to technical analysis does
not decline so rapidly as the forecast horizon increases however and is relatively constant for
all forecast horizons. Menkhoff (1997) also surveys the use of flow analysis (the information
content in the order flow).1 The weight given to flow analysis is highest for forecast horizons
of intra-day and declines as the forecast horizon increases. The mean weights attached to the
importance of flow analysis, technical analysis and fundamental analysis are 18%, 37% and
45% respectively. Users of technical analysis are not differentiated by seniority or education.
There is a slight preference for technical analysis among younger respondents but this is not
statistically significant.
Lui and Mole (1998) survey foreign exchange dealers in Hong Kong in 1995. Questions on
how technical analysis is used are very specific. Technical analysis and fundamental analysis
1
Flow analysis is the analysis of information contained in the order flow where order flow is
signed transaction volume indicating purchases and sales. See Gehrig and Menkhoff (2004).
24
are seen as complementary with 85% of respondents confirming their use. The weight given
to technical analysis is highest for short forecast horizons and declines as the forecast horizon
seen as more useful in predicting trends whereas technical analysis is seen as more useful in
predicting turning points. The most popular technical analysis methods are moving averages
and/or trend following systems. The typical length of historic data is 12 months. Daily data is
Oberlechner (2001) surveys foreign exchange traders and financial journalists in Frankfurt,
London, Vienna and Zurich in 1996. Although difficult to interpret, for both groups, greater
weight is given to technical analysis for short forecast horizons and decreases as the forecast
horizon increases, where greater weight is given to fundamental analysis. Traders attach the
highest weight to technical analysis for forecast horizons of up to three months. Journalists
attach the highest weight to technical analysis for forecast horizons of up to one month. The
weight given to fundamental analysis by journalists is always higher than that of traders. This
suggests that those inside the market see it differently from those outside the market although
this can also be explained by a journalistic need to rationalise events. The majority of traders
use both technical analysis and fundamental analysis with 3% relying exclusively on one of
the two methods. When the results for traders in London are compared with Taylor and Allen
(1992), the importance attached to technical analysis increases across all forecast horizons.
Users of technical analysis are not differentiated by age, gender, market type or seniority.
However, traders with a less than US$50 million limit demonstrate a statistically significant
preference for technical analysis compared to traders with a greater than US$50 million limit.
25
Cheung and Chinn (2001) survey foreign exchange traders in the United States between 1996
and 1997. Respondents were asked to best describe their trading practice. Technical trading
best describes 30% of trading practice. Fundamental analysis best describes 25% of trading
practice. The remainder is characterised as customer order driven (22%) and jobbing (23%).
The percentage of respondents who best describe their trading practice as technical trading
also appears to have increased. When asked to best describe their trading practice five years
ago, 19% of respondents describe themselves as technical traders. The increase in technical
trading is at the expense of jobbing. The percentage of respondents who best describe their
Similarly, Cheung et al. (2004) survey foreign exchange traders in the United Kingdom in
1998. As in Cheung and Chinn (2001), respondents were asked to best describe their trading
practice. Technical trading best describes 33% of trading practice. Fundamental analysis best
describes 34% of trading practice. The remainder is characterised as customer order driven
(37%) and jobbing (36%). When asked to best describe their trading practice five years ago,
14% of respondents describe themselves as technical traders. The increase in technical trading
is at the expense of jobbing. The percentage of respondents who best describe their trading
Gehrig and Menkhoff (2006) survey foreign exchange dealers and fund managers in Austria
and Germany in 2001. The authors compare the results to Menkhoff (1997). Overall, the use
of technical analysis has gained ground. Technical analysis is used extensively with 97% of
respondents giving at least a 10% weight to technical analysis compared to 87% before. The
mean weights given to the importance of flow analysis, technical analysis and fundamental
analysis by dealers are 26%, 42% and 32%. This is compared to 21%, 37% and 42% before.
26
For fund managers, the mean weights are 17%, 37% and 46%. This is compared to 9%, 37%
and 54% before. The importance of flow analysis and technical analysis has increased while
the importance of fundamental analysis has declined. Users of technical analysis are also not
The survey studies discussed so far concentrate on the foreign exchange. The only known
survey study of the equity markets is Menkhoff (2010) who surveys mutual, pension, bond
and equity fund managers in the United States, Germany, Switzerland, Italy and Thailand in
2003/2004. Technical analysis is used extensively with 55% 87% of respondents giving at
least a 10% weight to technical analysis in their decisions. The mean weights given to flow
analysis, technical analysis and fundamental analysis are 10%, 23% and 67%. The greatest
weight is given to flow analysis for forecast horizons of intra-day to days, technical analysis
for forecast horizons of weeks and fundamental analysis for forecast horizons of months to
years. An average of 20% of respondents prefer technical analysis relative to other forms of
analysis. This is attributed to the use strategies that rely to a large degree on technical input
such as momentum.
2.3 Conclusion
The main point to take from the literature review is the simplest one the trading process is
information rich and highly complex. The different weights given to flow analysis, technical
analysis and fundamental analysis suggest different classes of information, all of which act
as input to the trading decision. Further, the variation in forecast horizon implies that traders
are likely to fit this information to a number of different models that are not only specific to
the type of trade but also to the perceived implications of the information content. Given the
survey literature, to dismiss technical analysis on the back of the argument that a trading rule
27
such as the moving average trading rules defined in Section 1.2 is sufficiently representative
To develop this point, the trading rules proposed in this thesis are designed to capture trading
practice. Specifically, they are designed to capture the trading process as driven by direction.
That is, the idea of prices going up and down and trading as an attempt to profit from this. To
achieve this, it is necessary to model the price as having direction. This is modelled using the
moving average. The trading rules extend moving average trading to include a description of
trading practice and are motivated by the desire to explore the nature of trading as typified by
the survey papers just discussed. The remainder of the thesis expands on the trading rules in
more detail.
28
Chapter 3
This chapter describes the data and test method used to analyse the trade reduction rule in the
next chapter. The data comprises 45 years of daily close and bid-ask prices for stocks listed
on the London Stock Exchange from 01-January-1965 to 30-June-2009. The stocks are drawn
from the FTSE 100, FTSE 250, FTSE Small Cap and FTSE Fledgling indices. An issue with
the data is that the bid-ask prices are often inconsistent with the close price in that they do not
always conform to the ordering bidt closet askt . This means it is necessary to recreate the
spread despite the availability of bid-ask prices and the majority of this chapter is directed at
addressing this issue. The test method takes the form of hypothesis tests. All hypothesis tests
are bootstrap tests. The remainder of the chapter is organised as follows. Section 3.1 describes
the data and the approach to recreating the spread. Section 3.2 provides an outline of relevant
simulation issues. Section 3.3 defines the hypothesis tests. Section 3.4 concludes with a short
summary.
3.1 Data
This section describes the data. Section 3.1.1 discusses the requirements of the data. Section
3.1.2 describes the data content and format. Section 3.1.3 describes the pre-processing applied
to the close price. After the close price has been pre-processed, the data is considered useable.
However, a general issue with the data is that the bid-ask prices are often inconsistent with the
29
close price. Section 3.1.4 states that the reason for this is the sampling of bid-ask prices after
the mandatory quote period and describes the approach to recreating the spread when dealing
with inconsistent or missing bid-ask prices. Section 3.1.5 describes the breakdown of the data
The requirements of the data are twofold. Firstly, it is to provide a data set by which to test
the trading rules and, secondly, it is to provide a measure of cost. With reference to the latter
requirement, an issue with moving average trading is that in the absence of costs the trading
rules overstate the return. This is because the trading rules assume trending behaviour in the
underlying price dynamics but when the price does not trend the trading rules are sensitive to
noise. This can sometimes generate small positive returns of, say, 1% but which are negative
after costs. If costs are not allowed for, some trades are misclassified as winning trades when
they are in fact losing trades. This overstates the return and can lead to erroneous conclusions
Costs in this case equate to the spread.1 Note however that the spread is subject to estimation
error. There are two reasons for this. Firstly, if available, the spread is not guaranteed to be
error free and, secondly, there are a large number of prices where the spread is not available.
Both cases require the spread to be estimated resulting in estimation error. Underlying this is
the question of experiment design. Given that prices are available from 1965 to 2009, which
is a large sample, it appears sensible to test all of them. However, the spread is not available
until 1986 at the earliest. If prices before 1986 are not tested on the basis that the spread is not
1
The spread is an execution cost and is the only cost considered. See Section 3.1.4.
30
available before this, half the data is abandoned. If prices before 1986 are tested on the basis
that it is possible to estimate the spread before this, half the data is subject to estimation error.
This has to be balanced against questions of do the trading rules work, are the results robust in
the presence of costs and are the results robust over time.
Because the concern is to test if the trading rules work, which raises the issue of whether the
trading rules work for all prices for all time, the approach is to test all of the data with prices
before 1986 seen as a robustness check. Consequently, the data is split into three sub-periods
of fifteen years each with 1986 as the middle of the middle sub-period. This gives three data
sets where the change in the availability of the spread is phased as opposed to stepped. Note
however that it is still necessary to estimate the spread due to it not being error free. It is of
course possible to avoid this by using different data. Given that it is now possible to obtain
high frequency time-stamped records of daily order flow and quote revisions, this data would
eliminate a lot of error. However, the data sets are large and require significant filtering (see,
for example, Dacorogna et al. (2001)). This is a considerable undertaking and is unnecessary
at this stage. The requirement is not to analyse the spread in itself, but is instead to derive a
reasonable estimate of costs. The test data is sufficient for this purpose.
The data consists of 45 years of daily close and bid-ask prices for stocks listed on the London
and comes as a sequence of date-stamped records of the form (datet , bidt , closet , askt ) where
datet is the date for which prices are available, bidt is the end of day bid price, closet is the
end of day close price and askt is the end of day ask price. Dividends are not included. Prices
31
are at daily intervals and exclude weekends but include public holidays. The exchange closes
on public holidays and prices for public holidays are duplicates of the previous days prices.
Prices for public holidays are not treated differently and are retained along with other prices.
The close price is always available and starts from 01-January-1965. The bid-ask prices are
available from 27-October-1986 onwards following the so-called Big Bang. The bid-ask
prices do not always start from this date though, 27-October-1986 is the earliest date from
which they are available. It is common to find the close price starting before this date and for
the bid-ask prices to start after this date. There is one price file per stock and there are 567
The data is available in two different formats. The master price in both formats is the close
price. The first format is unadjusted and is the official close price quoted on the exchange.
The second format is adjusted for capital events such as stock splits and rights issues. A
historical adjustment factor that accumulates capital events in reverse chronological order
back to the base date is also available. In general, it is easier to work with adjusted prices
since there is no need to manage changes in the price level attributable to changes in capital
structure. Prices are adjusted prices for this reason. Returns are not affected but it does mean
The first step in pre-processing is to check that there are no missing close prices. All trading
rules use the close price to generate buy/sell signals and this can cause problems if the close
price is missing. The first check is to delete all price files where the close price is missing. In
general, DataStream is a respected data vendor and it is unusual to find missing close prices.
The main reason for missing close prices is when a stock is suspended. The next check is to
32
test whether the adjusted (bidt , closet , askt ) prices equal the unadjusted (bidt , closet , askt )
prices multiplied by the adjustment factor. Testing is to zero decimal places. If the adjusted
and unadjusted prices match, the price file is accepted. If they do not match, the price file is
inspected manually. If it is clear why the test fails, the error is fixed and the test is repeated.
Examples of why a test fails are rounding errors and a delay in the update of the adjustment
factor. If it is not clear why a test fails or the error persists in some way, the price file is
deleted.
The next step in pre-processing is to check whether the (bidt , closet , askt ) prices conform to
the ordering bidt closet askt . When trading takes place it takes place at one of two prices,
namely, the bid price or the ask price. The bid price is the price received when selling and the
ask price is the price paid when buying. The difference is the bid-ask spread. In percent, the
The spread defines the costs incurred for each round-trip transaction and is the cost paid by
the trading rules for being able to trade immediately at the prevailing price. Equation (3.1) is
also known as the quoted spread. It should be mentioned that there is a substantial literature
dedicated to analysing the spread wherein the spread arises to compensate market makers or
costs. Bessembinder and Venkataraman (2010) discuss this literature in more detail. For the
purposes of the thesis however, the spread is a cost levied against the trading rules assuming
33
all trades are executed as market orders. The trading rules demand liquidity and the spread is
There are two problems with the spread. The first problem is that the bid-ask prices are only
available from 27-October-1986 onwards. This is not overly critical but it does mean that it is
necessary to reconstruct the spread before this. The second problem is that the bid-ask prices
are often inconsistent with the close price. The close price is always the official close price
and is always the master price. The problem is that the bid-ask prices appear to be sampled
after the mandatory quote period ends at 16:30 GMT. The mandatory quote period runs from
08:00 to 16:30 GMT and market makers are obliged to offer firm two-way prices during this
time. Outside the mandatory quote period, market makers are not obliged to offer prices if
they do not wish to do so. This leads to greater price uncertainty and the bid-ask prices can
move away from the close price because of this. A straightforward test of the consistency of
the bid-ask prices is to check if they conform to the ordering bidt closet askt . All price
files fail this test at least once. This means that it is necessary to recreate the spread despite
the availability of the bid-ask prices for a particular date. However, it can be assumed that
errors in the bid-ask prices are due to sampling after the mandatory quote period. A more
general consequence of sampling the bid-ask prices after the mandatory quote period is that
the spread is likely to be overestimated. In a study of intra-day data for example, Abhyankar
et al. (1997) find that the spread widens up to 30% after the mandatory quote period ends at
16.30 GMT. In a similar study, Cai et al. (2004) exclude data outside the mandatory quote
The first step in reconstructing the spread is to zero each bid price greater than the close price
and each ask price less than the close price. Zeroing is on an individual basis. If the bid price
34
is greater than the close price and the ask price is not less than the close price for example, the
bid price is set to zero and the ask price is left unchanged. The next step is to zero all bid-ask
prices that equal one another. Each price entry then conforms to either bidt closet < askt or
bidt < closet askt but not both. Because the bid-ask prices are sampled after the mandatory
quote period, the next step is to identify bid-ask prices that have moved so far away from the
close price that they appear as outliers. The problem is that it is difficult to define precisely
what an outlier is. For example, it is perfectly reasonable for a stock to trade with a spread of
1%, for the business to fail in some way, for the share price to crash and for the stock to end
up trading as a penny share with a spread of 100%. A spread of 100% is then unusual at the
start of the price file but not at the end. This suggests a method based on moving windows.
To expedite this process, the method is to adopt a youll know it when you see it approach
and to manually inspect the bid-ask prices of each price file for deviation from their nearest
neighbours whenever the spread is greater than five times the mean. The choice of five times
the mean is firstly to ensure that bid-ask prices less than this are not filtered and, secondly, to
identify bid-ask prices that are reasonably distant from the mean. The mean is preferred over
the median since it is less robust to outliers and is less likely to result in bid-ask prices being
removed.
Because either the bid price, the ask price or both the bid and ask price can be in error, the
method is to compute separate bid-ask spreads for the non-missing bid-ask prices. This is
always relative to the close price. The close price is then used as the reference price with
which to reconstruct the spread. The bid spread s(b)t relative to the close price at time t is:
closet bidt
s(b)t = (3.2)
closet
35
The ask spread s(a)t relative to the close price at time t is:
askt closet
s(a)t = (3.3)
closet
Bid-ask prices greater than five times the mean are inspected manually. If the bid-ask price is
inconsistent with its nearest neighbours and can be attributed to sampling after the mandatory
quote period, it is set to zero, the mean is recalculated and the process is repeated. Zeroing is
on an individual basis. The process repeats until either there are no bid-ask prices greater than
five times the mean or, if there are, where the bid-ask prices are either consistent with their
nearest neighbours or cannot be attributed to sampling after the mandatory quote period.
The next step is to go through each price file and to set each zeroed bid-ask price to have the
same spread as its immediate predecessor. Bid-ask prices are reset on an individual basis. The
objective is to preserve local asymmetry in the spread. Zeroed bid prices are estimated by:
closet 1 bidt 1
bidt = closet 1 (3.4)
closet 1
askt 1 closet 1
askt = closet 1 + (3.5)
closet 1
36
Note that as stated in Section 3.1.1, the requirement is to derive a reasonable estimate of the
costs incurred by the trading rules given that they will otherwise overstate the return. It is not
to study the spread in itself. The straightforward nature of (3.4) and (3.5) reflect this. Methods
such as, say, differential equations or Rolls (1984) estimator introduce a level of complexity
that is difficult to justify assuming that if the results are significant in any way, the next step
would be to turn to the high frequency data at which point the trading rules can be simulated
The final step is to estimate missing bid-ask prices back to the base date. Whatever method is
used will introduce a bias since the spread is unknown before this. The method is to calculate
the mean of (3.2) and (3.3) using the first available 260 bid-ask prices where a bid-ask price is
only included in the calculation if its spread is different to its immediate predecessor. This is
to guard against reusing previously missing bid-ask prices as well the situation where prices
do not change for long periods, as is sometimes the case with smaller stocks. If there is not
enough data to calculate the mean using 260 entries, all the bid-ask prices are used provided
their spread is different to their immediate predecessors. Let s(b) denote the resulting mean
bid spread. Missing bid prices back to the base date are estimated by:
(
bidt = closet 1 s(b) ) (3.6)
Let s(a) denote the resulting mean ask spread. Missing ask prices back to the base date are
estimated by:
(
askt = closet 1 + s(a) ) (3.7)
37
Table 3.1 lists the mean bid-ask spread defined in (3.1). The bid-ask spread is calculated after
the spread has been reconstructed and where the price files have been grouped into portfolios
and sub-periods. The grouping of the price files into portfolios and sub-periods is described in
the next section. The mean bid-ask spread is the mean of all the price files for each portfolio
for each sub-period. The spreads for FTSE AIM 100 and FTSE AIM All Share portfolios are
also shown for comparison. The AIM portfolios are not part of the test data however.
Table 3.1
Mean bid-ask spreads
Period 1 refers to the test period 01-Jan-1965 to 31-Dec-1979. Period 2 refers to the test period 01-Jan-1980 to
31-Dec-1994. Period 3 refers to the test period 01-Jan-1995 to 30-Jun-2009. Period All refers to the test period
01-Jan-1965 to 30-Jun-2009. Variances are not calculated since the spread is bounded below by zero and hence
the distributions are likely to be right skewed implying negative spreads.
The spread increases as company size decreases and is lowest for the FTSE 100 and highest
for the FTSE AIM All Share. The FTSE AIM 100 also trades on a tighter spread than might
be expected. Figure 3.1 plots the mean annual bid-ask spread. The spread before 1990 is less
variable than the spread after 1990 although, in general, it does not appear to be significantly
distorted. However, it does not capture the overall variability of the spread and it is unlikely
that the bid-ask prices would have been achievable in practice. It is also worth noting that in
October 1997 the London Stock Exchange introduced a limit order book for the FTSE 100
38
and more liquid members of the FTSE 250 via the Stock Exchange Trading System (SETS).
The limit order book allows public traders (non-market makers) to post prices at which they
are prepared to trade and hence to act as counterparties in the supply and demand of liquidity.
A limit order book was subsequently introduced for non-SETS members of the FTSE 250 in
November 2003 via SETSmm (SETS with market makers), which was extended to cover the
FTSE Small Cap and FTSE AIM 50 indices in July 2005 and December 2005 respectively. It
is noticeable from Table 3.1 that this appears to have resulted in a decrease in the spread from
test period 2 to test period 3 and is indicative of a more cost efficient trading mechanism. A
20%
ftse 100 ftse 250
ftse small cap ftse fledgling
15% ftse aim 100 ftse aim all share
Bid-Ask Spread
10%
5%
0%
1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005 2009
Year
In general, the decrease in the spread would also tend to imply an increase in liquidity. It is
common in the literature to use the spread as a proxy for liquidity where liquidity is defined
as the ability to trade quickly at low cost and with minimal price impact and where liquidity
decreases as the spread increases (alternatively, illiquidity increases as the spread increases).
39
There are problems with the spread as a proxy for liquidity however. First, the spread has to
be available. If not, it is necessary to rely on other measures. Lesmond (2005) discusses this
issue for emerging markets. Second, for daily spreads, the spread is likely to refer to the last
trade of the day only, which may or may not be representative of liquidity during the course
of the day. Spreads obtained at daily frequency are noisy in this respect. Third, the spread is
likely to reflect the detail of the underlying market structure. Cai et al. (2008) find that the
spread decreases for SETS traded stocks following its introduction in October 1997, which
suggests that the spread is likely to be wider for quote driven markets than for order driven
markets. Fourth, the spread is open to manipulation by market makers, a famous example of
which is the NASDAQ controversy (Christie and Schultz (1994), Christie et al. (1994)).
Fifth, the spread acts an invitation to trade. It does not convey information about whether a
trade will take place inside or outside the spread. The spread conceals the depth of liquidity.
Last, while the spread is at least indicative of liquidity in that it conveys information on the
cost of liquidity, it does not convey information on the difficulty of executing a trade or the
magnitude of its price impact. This is typical of the view of liquidity as a multi-dimensional
variable that incorporates speed (time taken to execute a trade), tightness (low trading costs),
depth (the volume possible without affecting the price), breadth (the number of participants
actively engaged in the market) and resilience (the speed at which price fluctuations due to
the trading process die out) (Hasbrouck (2007)). Goyenko et al. (2009) discuss liquidity and
Given the one-dimensional nature of the spread, there are also issues regarding its ability to
predict liquidity in the long term. First, the spread is asset specific. If liquidity is driven by
factors exogenous to the spread, whether the spread is sufficient to characterise liquidity is
debatable and there is now a growing literature that studies the determinants of liquidity by
40
testing for co-movement or commonality across alternative measures of liquidity (see, for
example, Chordia et al. (2000), Korajczyk and Sadka (2008) and Brockman et al. (2009)).
Second, liquidity is dynamic. For example, suppose there is a perceived increase in risk and
that this causes liquidity to flow from illiquid stocks to liquid stocks, which in turn causes the
spread for the illiquid stocks to increase and the spread for the liquid stocks to decrease. As a
minimum, this suggests that the information content in the spreads of assets of the same type
Last, markets are linked. If liquidity flows out of one market or asset class, it is reasonable to
expect it to flow into another. In other words, the relationship between measures of liquidity
across different types of market and asset class might be a better predictor of future liquidity
compared to measures of the market or asset class on their own. Chordia et al. (2005) and
Goyenko and Ukhov (2009) find that the liquidity of US bond and stock markets are linked
with a change in the liquidity of one market affecting liquidity in the other. The authors also
find evidence that the bond market acts as a channel for the transmission of monetary policy
variables via changes in liquidity. Overall, the preceding discussion implies that predicting
liquidity is non-trivial and is likely to involve economic and microstructure variables, either
directly or indirectly, as well as some function of their variation caused by changes in capital
movements.
Once the data is pre-processed, it is divided into portfolios. Portfolios are differentiated by
company size. The portfolios comprise stocks drawn from the FTSE 100, FTSE 250, FTSE
Small Cap and FTSE Fledgling indices. The FTSE 100 index comprises the 100 most highly
capitalised companies listed on the main market and represents approximately 80% of UK
market capitalisation. The FTSE 250 index comprises the next 250 most highly capitalised
41
companies and represents approximately 15% of UK market capitalisation. The FTSE Small
Cap index comprises the remaining companies listed as members of the FTSE All Share and
represents approximately 2% of UK market capitalisation. The FTSE All Share index is the
aggregation of the FTSE 100, FTSE 250 and FTSE Small Cap indices. The FTSE Fledgling
index comprises companies listed on the main market but which are too small to be included
The classification scheme is that if a stock is a member of the FTSE 100 index, it is allocated
to the FTSE 100 portfolio. If it is a member of the FTSE 250 index, it is allocated to the FTSE
250 portfolio and so on. The indices are supplied by the FTSE Group (www.ftse.com) and are
correct as of August 2009. It should be noted that the FTSE indices are revised on a quarterly
basis and are updated to reflect changes in market capitalisation. The portfolios do not reflect
these changes. Instead, they reflect the indices as of August 2009 and do not vary with time.
The reason for this is practical. An issue with moving average trading is that it suffers from a
stock selection problem. If the price dynamics exploited by moving average trading are not in
the price, the trading rules will not find them. Therefore, for the trading rules to have practical
value, it is necessary to solve the stock selection problem. The stock selection problem in this
case is seen as a search problem. Given an investment universe, the stock selection problem is
to search the investment universe and to select those stocks most suited to the trading rules. A
potential solution to this problem is proposed in Chapter 5. The classification scheme mirrors
this perspective in that it splits the investment universe into pseudo arbitrary search spaces at
the same time as providing guidance on where to search. While it can be argued that a ranking
scheme that explicitly controls for size also addresses this issue and simplifies the analysis in
general, the decision not to use a ranking scheme is attributable to the proposed solution to the
stock selection problem. That said however, an investigation of the stock selection problem is
42
further work and there is no reason why a ranking scheme or cross-sectional study cannot be
The portfolios, then, are limited to the main market and comprise 99 stocks for the FTSE 100
portfolio, 197 stocks for the FTSE 250 portfolio, 159 stocks for the FTSE Small Cap portfolio
and 112 stocks for the FTSE Fledgling portfolio. A full listing of each portfolio including start
and end dates is given in the appendix. AIM stocks are not included on the basis that the AIM
constitutes a different market although this is not to say that the trading rules cannot be tested
The data is also divided into sub-periods of 15 years each. This is to allow for robustness and
to test whether the returns to the trading rules vary with time. The first sub-period is referred
to as test period 1 and is from 01-January 1965 to 31-December-1979. The second sub-period
is referred to as test period 2 and is from 01-January 1980 to 31-December-1994. The third
sub-period is referred to as test period 3 and is from 01-January 1995 to 30-June-2009. The
last period is referred to as test period All and covers all of the data from 01-January 1965 to
30-June-2009. Table 3.2 summarises the breakdown into portfolios and test periods in terms
of the number of stocks in each portfolio for each test period. All test periods are reasonably
well populated. Understandably, the number of stocks in each test period increases with time.
43
Table 3.2
Number of stocks in each portfolio and sub-period
FTSE 100 40 70 99 99
Period 1 refers to the test period 01-Jan-1965 to 31-Dec-1979. Period 2 refers to the test period 01-Jan-1980 to
31-Dec-1994. Period 3 refers to the test period 01-Jan-1995 to 30-Jun-2009. Period All refers to the test period
01-Jan-1965 to 30-Jun-2009.
Smaller companies are of specific interest. The literature on moving average trading applied
to UK smaller companies is tiny. Belaire-Franch and Opong (2005) apply the variance ratio
test of Lo and MacKinlay (1988, 1989) and the non-parametric variance ratio test of Wright
(2000) to the value weighted FTSE 100, FTSE 250, FTSE 350 and FTSE All Share indices.
The null hypothesis of a random walk is rejected for the FTSE 250 index from January-1986
to September-1997 and for the FTSE All Share index from January-1978 to September-1997.
The null hypothesis is not rejected for the FTSE 100 index. Evidence against the FTSE 350
index is mixed. Rejections are due to positive serial dependence. More recently, Hung et al.
(2009) apply the same tests along with the multiple variance ratio test of Chow and Denning
(1993) to the value weighted FTSE 250 and FTSE Small Cap indices from January-1986 to
October-2005. The null hypothesis of a random walk is not rejected for the FTSE 250 index
but is rejected for the FTSE Small Cap index. As before, rejections are due to positive serial
dependence. Failure to reject the null hypothesis for the FTSE 250 index is hard to interpret.
An intuitive explanation is that the additional data is so negatively serially dependent that it
cancels out the positive serial dependence of the previous data. Why this does not also apply
to the FTSE Small Cap index is not clear however. Nevertheless, albeit extremely thin, there
44
is at least some evidence that the price dynamics of smaller companies are different to those
of larger companies and that these dynamics might be suited to moving average trading.
The only study known to apply moving average trading to UK smaller companies is Bokhari
et al. (2005). The authors test various trading rules using a random sample of stocks drawn
from the FTSE 100, FTSE 250 and FTSE Small Cap indices for the period Januray-1987 to
July-2002 and find that predictive ability increases as company size decreases. There is also
evidence that the predictive ability of the FTSE Small Cap companies has not declined over
time. There is no evidence of profitability however. Simulations in the presence of costs are
not profitable. The authors also note that for the FTSE Small Cap companies, the dominant
factor stopping the trading rules from being profitable is the size of the bid-ask spread. The
trading rules are likely to be profitable if the predictability exhibited by the FTSE Small Cap
companies was combined with the bid-ask spread for larger companies. A typical spread for
the FTSE Small Cap is quoted as 10.7%. It is not clear if this is representative of the spread
used to simulate the trading rules. If it is, it is 3 times larger than in Table 3.1. However, the
authors qualify this by noting firstly that the spread is stock dependent and, secondly, that it
The moving average used to simulate the trading rules is the exponentially weighted moving
ma0n ( p0 ) = p0
45
All simulations include the spread. All buys occur at the ask price and all sells occur at the
bid price. Buys and sells take place on the same day as the buy/sell signals. A problem with
moving average trading is that it is prone to drift if the price does not change for prolonged
periods, which sometimes results in false buy/sell signals. To counteract this, all simulations
strip out duplicate prices before simulating the trading rules. The buy/sell signals are always
refitted to the original price series however. The moving average is always calculated using
the close price. Hence, all buy/sell signals are in response to changes in the close price. It is
also assumed that there is sufficient liquidity such that each trade executes immediately, pays
A general problem with the trading rules is how to test for statistical significance given their
trade distributions. The trade distribution was defined in Chapter 1 as the set of trades T that
result from applying a trading rule. The problem is that the theoretical distribution of the test
statistics underlying the trade distributions is unknown. This also holds for measures derived
as functions of the trade distributions. A popular approach in this situation is to bootstrap the
test statistics. The bootstrap is due to Efron (1979) and is a computationally intensive method
that estimates the distributional properties underlying a sample by re-sampling the empirical
For hypothesis tests, the advantage of the bootstrap is that provided it is possible to specify
the distribution under the null hypothesis, statistical inference is straightforward in that the
significance of the test statistics can be estimated from the data without having to know the
underlying data generating process. All tests for statistical significance are hypothesis tests
46
and all hypothesis tests are bootstrap tests. The tests are described in detail below. For each
test, the number of bootstrap replications B is 500 and the random number generator is ran2
This is the test used for the win rate in Section 4.1.
Method
The method is to simulate each stock using the price crossover and trade reduction rules and
moving averages in the range n = 2, 3, K, 250 . The trades for each stock for each n are then
pooled to give the trade distribution for that n. The trade distribution is the set of trades T that
Null hypothesis
Let F and G denote the trade distributions for the price crossover and trade reduction rules.
The null hypothesis is that the trade distributions for the price crossover and trade reduction
Test statistic
The test statistic is the difference in the win rates. Under the null hypothesis of no difference,
the win rate is the same for the price crossover and trade reduction rules. The win rate is:
nw nl
= 1 (3.9)
nw + nl nw + nl
Let p and q denote the win rates for the price crossover and trade reduction rules. The test
statistic is t(x) = q p .
47
Computation of the bootstrap test statistic
Denote the pooled sample as x = F + G and let f and g denote the number of trades in F and
1. Draw B samples of size f + g with replacement from x. Calculate the win rate for the
first f samples and for the remaining g samples. Denote the win rate for each sample
as p and q * .
{
3. Calculate the bootstrap p-value as # t(xb ) t obs } B where t obs = t(x) , the observed
This is the test used for the mean return per trade in Section 4.2.
Method
The method is to simulate each stock using the price crossover and trade reduction rules and
moving averages in the range n = 2, 3, K, 250 . The trades for each stock for each n are then
pooled to give the trade distribution for that n. The trade distribution is the set of trades T that
Null hypothesis
Let F and G denote the trade distributions for the price crossover and trade reduction rules.
The null hypothesis is that the trade distributions for the price crossover and trade reduction
48
Test statistic
The test statistic is the difference in means. Under the null hypothesis of no difference, the
mean return per trade is the same for the price crossover and trade reduction rules. The mean
return per trade is defined in Chapter 1. Let p and q denote the mean return per trade for the
price crossover and trade reduction rules. The test statistic is t(x) = q p .
Efron and Tibshirani (1998) stress the importance of sampling under the null hypothesis. To
test the null hypothesis it is necessary for F and G to have the same mean. Let f and g denote
the number of trades in F and G. The method to compute the bootstrap test statistic is:
and g are the means of F and G and z is the mean of the combined sample.
2. Draw B samples of size f with replacement from f%1 , f%2 , K, f%f and B samples of size g
with replacement from g%1 , g% 2 , K, g% g . Denote the mean of each sample as p and
q* .
{
4. Calculate the bootstrap p-value as # t(xb ) t obs } B where t obs = t(x) , the observed
3.4 Summary
This chapter has introduced the data and described the bootstrap hypothesis tests used to test
the trade reduction rule in the next chapter. The main concern has been to derive an estimate
of the spread as a measure of cost. The spread is important when simulating the trading rules
49
since failure to account for costs overstates the return and can lead to erroneous conclusions
regarding profitability. More specifically, failure to account for costs overstates the win rate
and the mean return per trade, both of which are tested next.
50
Chapter 4
This chapter introduces the trade reduction rule. The trade reduction rule is based on the idea
of allowing a trade to run and extends the price crossover rule to increase the mean return per
trade by keeping as many winning trades as possible at the same time as reducing the number
of losing trades. Results for buying and selling are different. The mean return per trade for the
long only trade reduction rule is consistently higher than for the price crossover rule. This is
true for all portfolios for all test periods. The average increase in the mean return per trade is
84%. The mean return per trade for the short only trade reduction rule is consistently higher
than for the price crossover rule for the FTSE Small Cap and FTSE Fledgling portfolios only.
Evidence against the null hypothesis of no difference in the means of the trades generated by
the trade reduction and price crossover rules is mixed. It is not possible to conclusively reject
the null hypothesis for all portfolios for all test periods. However, there are large numbers of
trading rules that reject the null hypothesis and where failure to reject the null hypothesis is
otherwise marginal. Overall, there is sufficient reason to conclude that the trade reduction rule
behaves as expected even if it does not follow that the difference in the mean return per trade
Section 4.1 derives the trade reduction rule. Section 4.2 presents the test results for the null
hypothesis of no difference in the means of the trades generated by the trade reduction and
51
4.1 Trade reduction rule
This section defines the trade reduction rule. The trade reduction rule has two parts. The first
part is the bounded moving average. The bounded moving average maps the moving average
onto the range [1, 1] . This has a number of applications including testing the distribution of
buy/sell signals for deviation from a null model, testing the position of the buy/sell signals for
differences in profitability as well as allowing for trading rules that respond to patterns in the
path of the moving average. This latter application underpins the trade reduction rule and the
second part describes how knowing the order and position of minima and maxima can be used
to generalise the price crossover rule in such a way as to transform the mean return per trade.
Section 4.1.1 defines the bounded moving average. Section 4.1.2 describes the trade reduction
algorithm.
The intuition behind bounding the moving average is that the more the price falls, the more
the moving average falls and the more the price and the moving average are low relative to
before. Similarly, the more the price rises, the more the moving average rises and the more
the price and the moving average are high relative to before. Mapping the moving average
onto [1, 1] where 1 corresponds to low and 1 corresponds to high constrains the moving
average to move within a fixed range. This makes it possible to measure the position of the
moving average as defined by its location within [1, 1] . The bounding algorithm is:
52
1. Compute the n-day moving average of the price matn ( pt )
3. Set the maximum of the price and matn ( pt ) to bt+ = max( pt , matn ( pt ))
Since bt pt bt+ for all t, it holds that matn (bt ) matn ( pt ) matn (bt+ ) :
4. Compute the n-day moving average of bt and set this to the lower bound matn (bt )
5. Compute the n-day moving average of bt+ and set this to the upper bound matn (bt+ )
Figure 4.1 plots an example. The minima and maxima of Figure 4.1 are especially important.
Mathematically, when the price crosses its average, the moving average changes direction. If
the price cuts up through its average from below, the moving average changes direction from
falling to rising. Similarly, if the price cuts down through its average from above, the moving
average changes direction from rising to falling. This means that the minima and maxima in
the moving average of Figure 4.1 are identical to the buy/sell signals generated by the price
crossover rule defined in Chapter 1. Further, the minima and maxima in the moving average
are identical to the minima and maxima in the bounded moving average. Hence, the position
of the buy/sell signals is known. This information can be used to remodel the price crossover
rule by changing the way it responds to the position of the buy/sell signals and is the approach
53
underlying the trade reduction rule. Given that their buy/sell signals are not generated by the
price crossing its average, there is no equivalent for the other types of trading rule defined in
Chapter 1. For the other types of trading rule, the bounded moving average could also follow
a path similar to Figure 4.1 as the result of a single trade. This means that it is hard to identify
patterns that can be defined as general case. However, it is possible to capture something of
the nature of the other types of trading rule provided not all minima and maxima result in a
buy/sell signal. A feature of the trade reduction rule is that it achieves this without a second
moving average.
300 0.0
-0.5
200
-1.0
100 -1.5
Jan-94 Jul-94 Jan-95 Jul-95 Jan-96 Jul-96 Jan-97 Jan-94 Jul-94 Jan-95 Jul-95 Jan-96 Jul-96 Jan-97
Time Time
Figure 4.1 Example of the moving average and its mapping to the bounded moving average.
Figure 4.2 plots an example frequency distribution of the position of the minima and maxima
within [1, 1] . The distribution is non-normal and heavily weighted in the left and right tails.
Minima tend to occur close to 1 and maxima tend to occur close to 1. The reason for this is
that the bounded moving average is defined self-referentially. The bounded moving average
measures the extent to which the price goes up and down in terms of the similarity between
54
the moving average and itself. As a result, the clustering in the tails is due to persistence in
direction. The longer the price continues in the same direction, the more likely it is for the
moving average to resemble itself. The more likely it is for the moving average to resemble
itself, the more likely it is for the bounded moving average to approach and settle on 1. The
U-shape is due to the bounding algorithm. The same U-shape is found in a random walk with
the difference in dynamics reflected in differences in frequency. Random walks are found to
be less dense in the left and right tails and more heavily weighted through the middle.
Minima Maxima
35 35
30 30
25 25
% Frequency
% Frequency
20 20
15 15
10 10
5 5
0 0
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
Figure 4.2 Example frequency distribution of the position of minima and maxima.
To explain the trade reduction rule, it helps to first define the price crossover rule. As stated,
the minima and maxima in the moving average are identical to the buy/sell signals generated
by the price crossover rule. Define a local minimum lmint and a local maximum lmaxt as:
55
lmint = matn 2 ( pt ) matn1 ( pt ) < matn ( pt ) (4.1)
The long only price crossover rule can then be written as:
q1 = 0
///// RULE
qt +1 = qt
IF (lmint ) THEN qt +1 = 1
Similarly, the short only price crossover rule can be written as:
q1 = 0
///// RULE
qt +1 = qt
IF (lmint ) THEN qt +1 = 0
The nature of the price crossover rule is that a trade always comprises consecutive minima
and maxima. The trade reduction rule generalises the price crossover rule to allow a trade to
comprise multiple minima and maxima. This is achieved by exploiting the ordering and the
position of the minima and maxima. The objective is to transform the mean return per trade
by keeping as many winning trades as possible at the same time as reducing the number of
56
losing trades. The mean return per trade that results from applying the price crossover rule
R=
(n w ) (
W + nl L ) (4.5)
nw + nl
More generally, denote the mean return per trade by k / n . The approach is to transform the
k k
> (4.6)
m n
For positive k, (4.6) follows whenever m < n. Let Rt denote the return for the current trade.
q1 = 0
previousLMin0 = 0
///// RULE
qt +1 = qt
previousLMint = previousLMint 1
57
In the event of a local minimum, the trade reduction rule opens a new long trade if it has not
already done so and saves the position of the local minimum for when it is needed later. The
next event is a local maximum. When a local maximum occurs, the trade is either profitable
or it is not. If it is profitable, it is closed. This keeps as many winning trades as possible. The
trading process then restarts from the next local minimum. The sequence of events and their
If the trade is not profitable, rather than take a loss, the trade reduction rule waits for the next
buy/sell signal event. The reason this is said to be a buy/sell signal event is that normally the
next buy/sell signal would be a local minimum. The trade reduction rule also generates sell
signals whenever the bounded moving average cuts down through the position of the previous
local minimum. Hence, the source of the next buy/sell signal is not known until it occurs. If
the bounded moving average cuts down through the position of the previous local minimum
before the next local minimum occurs, the trade is closed and is expected to result in a loss. If
the trade was not profitable at the price of the previous local maximum, it is unlikely to be
profitable after the moving average has fallen from this. The trading process then restarts
from the next local minimum. The sequence of events is as shown below where a struck out
58
The reasons for closing the trade when the bounded moving average cuts down through the
position of the previous local minimum are twofold. Firstly, because the buy/sell signals of
the price crossover rule are ordered as minima and maxima, if a local minimum does not
occur before the bounded moving average cuts down through the position of the previous
local minimum, it is known that the next local minimum will occur at a position less than this.
Since the objective is to keep as many winning trades as possible, closing the trade at this
point ensures that the trading process restarts from the next local minimum. This means that
not too many buy signals are ignored and that the next trade will be kept should it turn out to
be a winning trade. Secondly, there is an open long trade when the moving average is falling.
In theory, the moving average could fall forever resulting in bankruptcy. This is undesirable.
More generally, if the moving average continues falling, this could result in significant losses.
Closing the trade when the bounded moving average cuts down through the position of the
previous local minimum is a simple method by which to control the magnitude of the losses.
However, the best that can be achieved is that the loss does not get any bigger than it already
is.
The remaining situation is where a local minimum occurs before the bounded moving average
cuts down through the position of the previous local minimum. The trade reduction rule does
nothing in this case other than to update the position of the previous local minimum to that of
the current local minimum. The trading process then continues from the next local maximum.
Because there has been a local minimum followed by a local maximum followed by a local
minimum but where the latter two signals have not been acted on, the effect is to reduce the
number of losing trades relative to the price crossover rule by one. This is also reflected in the
sum of the losing trades. The sequence of events and their relationship with the mean return
59
lmint j , lmaxt i , lmint nl = nl 1, x = x R t (4.10)
xL xL
There are several elements at work here. Firstly, updating the position of the previous local
minimum to the current local minimum ensures that all trades are closed relative to the most
recent local minimum. This has some similarities to a trailing stop loss where the stop loss is
adjusted to be the most recent local minimum. If the trade ends up as a losing trade, updating
the position of the local minimum ensures that the trade will be exited earlier, which helps to
control the magnitude of the losses. Secondly, the trade reduction rule is defined recursively.
Each time the pattern defined by (4.10) occurs, it is succeeded by either (4.8), (4.9) or (4.10).
If succeeding events are unbiased in that (4.8) sometimes occurs, some trades will end up as
winning trades. Not only does the trade reduction rule reduce the number of losing trades, it
also increases the number of winning trades. It is not possible to create winning trades from
nothing however. Consequently, the increase in the number of winning trades is likely to be
small. Lastly, the position of the previous local minimum is always less than the position of
the current local minimum. Because both minimums are associated with the prices at which
they occur, there is also the possibility that the price level of the previous local minimum is
less than the price level of the current local minimum. If so, there is the possibility that the
difference in price level will bias the outcome in favour of a winning trade. Figure 4.3 plots
some example trades (with annotation) for the long only trade reduction rule in terms of the
60
Winning Trade 1 Winning Trade 2
1.5 1.5
Bounded Moving Average Value
0.5 0.5
-0.5 -0.5
-1.5 -1.5
Figure 4.3 Example trades for the long only trade reduction rule in terms of the path of the
61
The short only trade reduction rule is the converse:
q1 = 0
previousLMax0 = 0
///// RULE
qt +1 = qt
previousLMaxt = previousLMaxt 1
The question then is what to expect from the trade reduction rule. Firstly, the trade reduction
rule should result in fewer trades than the price crossover rule. Secondly, there should also be
more winning trades and less losing trades. The trade reduction rule should have a higher win
rate where the win rate is the ratio of winning trades to the total number of trades. Tests find
that this is in fact the case. On average, the trade reduction rule generates 7% more winning
trades and 46% fewer losing trades than the price crossover rule. Bootstrap tests of the null
hypothesis of no difference in the win rate are soundly rejected. This is true for all portfolios
for all test periods long and short. Figure 4.4 plots a typical example of the test results for the
number of winning and losing trades using the FTSE 100 portfolio for the period 01-January-
1965 to 30-June-2009. Similar results are found for all portfolios for all test periods. The trade
reduction rule significantly transforms the trade distribution of the price crossover rule due,
primarily, to the decrease in the number of losing trades. While the increase in the number of
62
winning trades is also a contributory factor, the decrease in the number of losing trades is the
dominant element.
This is an encouraging result and suggests that the trade reduction rule should have an effect
on the mean return per trade. However, it can be seen from Figure 4.4 that the trade reduction
rule has little impact on the distribution of winning trades. This is to be expected. Given that
the winning trades are due to trending behaviour in the underlying price dynamics, the trade
reduction rule cannot be expected to uncover new trends not already uncovered by the price
crossover rule. On the other hand, the trade reduction rule does have a significant impact on
the distribution of losing trades. The problem is that to facilitate this, the losses incurred by
the trade reduction rule are different to the losses incurred by the price crossover rule. For the
price crossover rule, losses are always incurred at the most recent local maximum and their
magnitude is always determined by the difference in the price level relative to the previous
local minimum. For the trade reduction rule, losses are allowed to include multiple minima
and maxima and are incurred after the moving average has fallen from the most recent local
maximum. It is not clear if this difference is significant. Analysis finds that the difference is
not significant. For the long only trade reduction rule, the sum of the losing trades is always
less than for the price crossover rule. This is true for all portfolios for all test periods. For the
short only trade reduction rule, the sum of the losing trades is sometimes greater than for the
price crossover rule although this is true only for the FTSE 100 and FTSE 250 portfolios for
test period 2. The difference is also negligible. Figure 4.5 plots a typical example of the test
results for the sum of the winning and losing trades using the FTSE 100 portfolio for the
period 01-January-1965 to 30-June-2009. Similar results are found for all portfolios for all
test periods. The trade reduction rule does not generate significantly higher losses than the
price crossover rule and nor does it generate significantly lower profits. The trade reduction
63
rule can therefore be expected to have a similar return to the price crossover rule. Assuming
the return is positive, given the overall decrease in the number of trades, the trade reduction
rule can also be expected to have a higher mean return per trade. A higher mean return per
trade also implies higher breakeven costs, which may or may not be significant in terms of
64
FTSE 100 - Long, Test Period All FTSE 100 - Long, Test Period All
36 80
price crossover rule price crossover rule
trade reduction rule trade reduction rule
Winning Trades (1000's)
18 40
9 20
0 0
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 100 - Long, Test Period All FTSE 100 - Long, Test Period All
100% 1.25
price crossover rule
trade reduction rule 1.00
75%
0.75
Win Rate
P-Value
50% 0.50
0.25
25%
0.00
0% -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.4a Number of winning and losing trades for the long only price crossover and trade
reduction rules for the FTSE 100 portfolio for the period 01-January-1965 to 30-June-2009.
The p-values are the bootstrap results for the null hypothesis of no difference in the win rate.
65
FTSE 100 - Short, Test Period All FTSE 100 - Short, Test Period All
36 80
price crossover rule price crossover rule
trade reduction rule trade reduction rule
Winning Trades (1000's)
18 40
9 20
0 0
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 100 - Short, Test Period All FTSE 100 - Short, Test Period All
100% 1.25
price crossover rule
trade reduction rule 1.00
75%
0.75
Win Rate
P-Value
50% 0.50
0.25
25%
0.00
0% -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.4b Number of winning and losing trades for the short only price crossover and trade
reduction rules for the FTSE 100 portfolio for the period 01-January-1965 to 30-June-2009.
The p-values are the bootstrap results for the null hypothesis of no difference in the win rate.
66
FTSE 100 - Long, Test Period All FTSE 100 - Long, Test Period All
1400 600
Cumulative Sum Winning Trades
700 -600
350 -1200
0 -1800
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 100 - Short, Test Period All FTSE 100 - Short, Test Period All
1400 600
Cumulative Sum Winning Trades
700 -600
350 -1200
0 -1800
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.5 Sum of the winning and losing trades for the long only and short only price
crossover and trade reduction rules for FTSE 100 portfolio for the period 01-January-1965 to
30-June-2009.
67
4.2 Results for the mean return per trade
This section presents the first set of simulation results for the trade reduction rule. The aim is
to test the null hypothesis of no difference in the means of the trades generated by the price
crossover and trade reduction rules. The method is to simulate each stock individually using
moving averages in the range n = 2, 3, K, 250 and to combine the trades for each stock for
each n to obtain the set of trades for that n. The trade distributions are then bootstrapped to
test the null hypothesis. The test is described in Chapter 3.1 Note that the test is indicative of
whether the trade distributions have the same mean only. It does not test the actual return to
the trading rules. Results for the return are presented later. Section 4.2.1 presents the results
for the long only price crossover and trade reduction rules. The mean return per trade for the
trade reduction rule is consistently higher than for the price crossover rule. This is true for all
portfolios for all test periods. There is also evidence that the mean return per trade is lowest
post-1990 and that the mean return per trade increases as company size decreases. However,
evidence against the null hypothesis is mixed. In general, there are large numbers of trading
rules that reject the null hypothesis and where failure to reject the null hypothesis is otherwise
marginal. Overall, there is sufficient reason to conclude that the trade reduction rule behaves
as expected even though it does not follow that the difference in the mean return per trade is
consistently statistically significant. Section 4.2.2 presents the results for the short only price
crossover and trade reduction rules. Results for the short only trading rules are more variable
and there is evidence that the returns to selling are different. The mean return per trade for the
trade reduction rule is consistently higher than for the price crossover rule for the FTSE Small
Cap and FTSE Fledgling portfolios only. The mean return per trade for the FTSE Small Cap
and FTSE Fledgling portfolios is also unusually high. This may be indicative of survivorship
1
See Section 3.3 (page 46).
68
bias. As before, evidence against the null hypothesis is mixed. It is also weaker. Nevertheless,
in general, the trade reduction rule works much the same for selling as it does for buying with
the differences due to the difference in the behaviour of the underlying price dynamics.
4.2.1 Long
Figures 4.6 to 4.9 plot the mean return per trade for the long only price crossover and trade
reduction rules.2 Bootstrap p-values for the null hypothesis of no difference in the means are
also shown. The mean return per trade for the trade reduction rule is consistently higher than
for the price crossover rule. This is true for all portfolios for all test periods. The mean return
per trade is also lowest in test period 3. This is consistent with the literature where the returns
to moving average trading have declined post-1990. The mean return per trade also increases
as company size decreases and suggests that prices are more likely to trend as company size
decreases.
With reference to the p-values, evidence against the null hypothesis of no difference in the
means is mixed. In general, it is not possible to conclusively reject the null hypothesis for all
portfolios for all test periods. However, there are large numbers of trading rules that reject the
null hypothesis and where failure to reject the null hypothesis is otherwise marginal. This is
particularly evident when testing all of the data. The p-values in this case are strong evidence
against the null hypothesis although their variability across sub-periods would suggest that a
note of caution is perhaps more appropriate. Even so, as a whole, there is sufficient reason to
2
A long trade refers to buying in the expectation that prices will rise. The stock is then sold at
a higher price than the price for which it was bought. The profit is the sell price minus the buy
price. See Section 1.2 (page 4).
69
conclude that the trade reduction rule behaves as expected even if it does not follow that the
difference in the mean return per trade is consistently statistically significant. This suggests
that to consistently reject the null hypothesis, the trade reduction rule needs to transform the
distributions of both the winning and losing trades together and not just the losing trades on
their own.
70
FTSE 100 - Long, Test Period 1 FTSE 100 - Long, Test Period 2
10% 10%
price crossover rule price crossover rule
8% trade reduction rule 8% trade reduction rule
Mean Return Per Trade
4% 4%
2% 2%
0% 0%
-2% -2%
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 100 - Long, Test Period 3 FTSE 100 - Long, Test Period All
10% 10%
price crossover rule price crossover rule
8% trade reduction rule 8% trade reduction rule
Mean Return Per Trade
6% 6%
4% 4%
2% 2%
0% 0%
-2% -2%
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.6a Mean return per trade for the long only price crossover and trade reduction rules
71
FTSE 100 - Long, Test Period 1 FTSE 100 - Long, Test Period 2
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 100 - Long, Test Period 3 FTSE 100 - Long, Test Period All
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.6b Bootstrap p-values for the null hypothesis of no difference in the mean return per
trade for the long only price crossover and trade reduction rules for the FTSE 100 portfolio.
72
FTSE 250 - Long, Test Period 1 FTSE 250 - Long, Test Period 2
20% 20%
price crossover rule price crossover rule
16% trade reduction rule 16% trade reduction rule
Mean Return Per Trade
8% 8%
4% 4%
0% 0%
-4% -4%
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 250 - Long, Test Period 3 FTSE 250 - Long, Test Period All
20% 20%
price crossover rule price crossover rule
16% trade reduction rule 16% trade reduction rule
Mean Return Per Trade
12% 12%
8% 8%
4% 4%
0% 0%
-4% -4%
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.7a Mean return per trade for the long only price crossover and trade reduction rules
73
FTSE 250 - Long, Test Period 1 FTSE 250 - Long, Test Period 2
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 250 - Long, Test Period 3 FTSE 250 - Long, Test Period All
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.7b Bootstrap p-values for the null hypothesis of no difference in the mean return per
trade for the long only price crossover and trade reduction rules for the FTSE 250 portfolio.
74
FTSE Small Cap - Long, Test Period 1 FTSE Small Cap - Long, Test Period 2
25% 25%
price crossover rule price crossover rule
20% trade reduction rule 20% trade reduction rule
Mean Return Per Trade
10% 10%
5% 5%
0% 0%
-5% -5%
0 50 100 150 200 250 0 50 100 150 200 250
FTSE Small Cap - Long, Test Period 3 FTSE Small Cap - Long, Test Period All
25% 25%
price crossover rule price crossover rule
20% trade reduction rule 20% trade reduction rule
Mean Return Per Trade
15% 15%
10% 10%
5% 5%
0% 0%
-5% -5%
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.8a Mean return per trade for the long only price crossover and trade reduction rules
75
FTSE Small Cap - Long, Test Period 1 FTSE Small Cap - Long, Test Period 2
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
FTSE Small Cap - Long, Test Period 3 FTSE Small Cap - Long, Test Period All
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.8b Bootstrap p-values for the null hypothesis of no difference in the mean return per
trade for the long only price crossover and trade reduction rules for the FTSE Small Cap
portfolio.
76
FTSE Fledgling - Long, Test Period 1 FTSE Fledgling - Long, Test Period 2
20% 20%
price crossover rule price crossover rule
16% trade reduction rule 16% trade reduction rule
Mean Return Per Trade
8% 8%
4% 4%
0% 0%
-4% -4%
0 50 100 150 200 250 0 50 100 150 200 250
FTSE Fledgling - Long, Test Period 3 FTSE Fledgling - Long, Test Period All
20% 20%
price crossover rule price crossover rule
16% trade reduction rule 16% trade reduction rule
Mean Return Per Trade
12% 12%
8% 8%
4% 4%
0% 0%
-4% -4%
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.9a Mean return per trade for the long only price crossover and trade reduction rules
77
FTSE Fledgling - Long, Test Period 1 FTSE Fledgling - Long, Test Period 2
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
FTSE Fledgling - Long, Test Period 3 FTSE Fledgling - Long, Test Period All
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.9b Bootstrap p-values for the null hypothesis of no difference in the mean return per
trade for the long only price crossover and trade reduction rules for the FTSE Fledgling
portfolio.
78
4.2.2 Short
Figures 4.10 to 4.13 plot the mean return per trade for the short only price crossover and trade
reduction rules.3 Bootstrap p-values for the null hypothesis of no difference in the means are
also shown. The returns to selling are clearly different from the returns to buying. The mean
return per trade for the trade reduction rule is consistently higher than for the price crossover
rule for the FTSE Small Cap and FTSE Fledgling portfolios only. With the exception of test
period 1, there is little evidence that shorting the FTSE 100 portfolio is likely to be profitable.
The same observation applies to the FTSE 250 portfolio with the exception of test periods 1
and 3. On the other hand, the opposite is likely to be true for the FTSE Small Cap and FTSE
Fledgling portfolios, most notably for test periods 2 and 3. The mean return per trade for the
FTSE Small Cap and FTSE Fledgling portfolios during these periods is unusually high. This
is a difficult result to interpret. Because the mean return per trade increases as company size
decreases, as with buying, it might be that prices for smaller companies are more likely to
trend. If so, this appears to be true for prices on the way down as well as on the way up. The
FTSE index history of the portfolio constituents is not known though. Since the FTSE Small
Cap and FTSE Fledgling portfolios comprise the smallest 2% 3% of companies that make
up the main market but where the portfolios were not constructed until August 2009, looking
back in time, the portfolios might be biased by companies whose businesses are in decline but
which continue to maintain a listing and are still trading. In other words, the FTSE Small Cap
and FTSE Fledgling portfolios are likely to include a number of stocks that are present in the
portfolios for the reason that their value has fallen to the point where there is nowhere else for
3
A short trade refers to selling in the expectation that prices will fall. The stock is then bought
back at a lower price than the price for which it was sold. The profit is the sell price minus the
buy price. See Section 1.2 (page 4).
79
them to go. If so, the high mean return per trade is at least partly explained by survivorship
bias. This tends to be supported by the results for test period 1 where there are fewer stocks in
the portfolios and where the general shape of the mean return per trade is consistent with that
of the FTSE 100 and FTSE 250 portfolios. A further complication is that some stocks trade as
penny shares whereby a small change in the price can generate significant returns. Results for
shorting the FTSE Small Cap and FTSE Fledgling portfolios need to be treated with caution
particularly with respect to conclusions based on company size alone and to the existence of a
negative risk premium. Then again, if it were possible to acquire or at least approximate the
FTSE Small Cap or FTSE Fledgling portfolios in real-time, assuming similar results, such a
As with buying, evidence against the null hypothesis of no difference in the means is mixed.
In general, it is not possible to conclusively reject the null hypothesis for all portfolios for all
test periods. Evidence against the null hypothesis is also weaker than for buying. However,
there are large numbers of trading rules that reject null hypothesis and where failure to reject
the null hypothesis is otherwise marginal. This is most apparent for test period 3. This is an
acceptable if not decisive result and it is reasonable to conclude that the trade reduction rule
works much the same for selling as it does for buying with the differences attributable to the
80
FTSE 100 - Short, Test Period 1 FTSE 100 - Short, Test Period 2
8% 8%
price crossover rule price crossover rule
6% trade reduction rule 6% trade reduction rule
Mean Return Per Trade
2% 2%
0% 0%
-2% -2%
-4% -4%
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 100 - Short, Test Period 3 FTSE 100 - Short, Test Period All
8% 8%
price crossover rule price crossover rule
6% trade reduction rule 6% trade reduction rule
Mean Return Per Trade
4% 4%
2% 2%
0% 0%
-2% -2%
-4% -4%
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.10a Mean return per trade for the short only price crossover and trade reduction rules
81
FTSE 100 - Short, Test Period 1 FTSE 100 - Short, Test Period 2
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 100 - Short, Test Period 3 FTSE 100 - Short, Test Period All
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.10b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the short only price crossover and trade reduction rules for the FTSE 100
portfolio.
82
FTSE 250 - Short, Test Period 1 FTSE 250 - Short, Test Period 2
8% 8%
price crossover rule price crossover rule
6% trade reduction rule 6% trade reduction rule
Mean Return Per Trade
2% 2%
0% 0%
-2% -2%
-4% -4%
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 250 - Short, Test Period 3 FTSE 250 - Short, Test Period All
8% 8%
price crossover rule price crossover rule
6% trade reduction rule 6% trade reduction rule
Mean Return Per Trade
4% 4%
2% 2%
0% 0%
-2% -2%
-4% -4%
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.11a Mean return per trade for the short only price crossover and trade reduction rules
83
FTSE 250 - Short, Test Period 1 FTSE 250 - Short, Test Period 2
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
FTSE 250 - Short, Test Period 3 FTSE 250 - Short, Test Period All
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.11b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the long short price crossover and trade reduction rules for the FTSE 250
portfolio.
84
FTSE Small Cap - Short, Test Period 1 FTSE Small Cap - Short, Test Period 2
35% 35%
price crossover rule price crossover rule
28% trade reduction rule 28% trade reduction rule
Mean Return Per Trade
14% 14%
7% 7%
0% 0%
-7% -7%
0 50 100 150 200 250 0 50 100 150 200 250
FTSE Small Cap - Short, Test Period 3 FTSE Small Cap - Short, Test Period All
35% 35%
price crossover rule price crossover rule
28% trade reduction rule 28% trade reduction rule
Mean Return Per Trade
21% 21%
14% 14%
7% 7%
0% 0%
-7% -7%
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.12a Mean return per trade for the short only price crossover and trade reduction rules
85
FTSE Small Cap - Short, Test Period 1 FTSE Small Cap - Short, Test Period 2
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
FTSE Small Cap - Short, Test Period 3 FTSE Small Cap - Short, Test Period All
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.12b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the short only price crossover and trade reduction rules for the FTSE Small Cap
portfolio.
86
FTSE Fledgling - Short, Test Period 1 FTSE Fledgling - Short, Test Period 2
8% 50%
price crossover rule price crossover rule
6% trade reduction rule 40% trade reduction rule
Mean Return Per Trade
2% 20%
0% 10%
-2% 0%
-4% -10%
0 50 100 150 200 250 0 50 100 150 200 250
FTSE Fledgling - Short, Test Period 3 FTSE Fledgling - Short, Test Period All
250% 250%
price crossover rule price crossover rule
200% trade reduction rule 200% trade reduction rule
Mean Return Per Trade
150% 150%
100% 100%
50% 50%
0% 0%
-50% -50%
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.13a Mean return per trade for the short only price crossover and trade reduction rules
87
FTSE Fledgling - Short, Test Period 1 FTSE Fledgling - Short, Test Period 2
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
FTSE Fledgling - Short, Test Period 3 FTSE Fledgling - Short, Test Period All
1.25 1.25
1.00 1.00
0.75 0.75
P-Value
P-Value
0.50 0.50
0.25 0.25
0.00 0.00
-0.25 -0.25
0 50 100 150 200 250 0 50 100 150 200 250
Figure 4.13b Bootstrap p-values for the null hypothesis of no difference in the mean return
per trade for the short only price crossover and trade reduction rules for the FTSE Fledgling
portfolio.
88
4.3 Conclusion
A problem with moving average trading, and this applies to technical analysis in general, is
whether the buy/sell signals can be said to constitute true information events. At first glance,
the answer would appear to be no. Given that the trading rules extract information from past
prices and assuming the market is efficient in response to changes in information, the trading
rules can only reveal what has happened and not what will happen. True information events
precede the buy/sell signals and the trading rules do no more than respond to price changes
This is clearly not the case. If the buy/sell signals were not true information events then (1)
the return to trading the buy/sell signals should be random in which case the mean return per
trade should be zero or negative after costs and (2) the return to buying and selling should be
the same. Further, the trade reduction rule shows that not only do the buy/sell signals contain
information but also that the sequence in which they occur contains information. Hence, the
evolution of the price path contains information. Furthermore, this information is financially
exploitable. This is an extremely challenging result. Since the moving average is a smoothed
version of the price, the implication is that if this information is visible in the moving average,
potentially at least, it is visible for all to see. Assuming trading is the fundamental mechanism
by which information is imparted into the price, for trading to be economically viable, it has
to be profitable. For trading to be profitable, the information in the price has to be financially
exploitable. For the information in the price to be financially exploitable, it has to map to the
perception of a trading opportunity where for a trading opportunity to exist, there has to be a
feedback mechanism by which the decision to trade can be judged as correct. This feedback
mechanism is embedded in the future evolution of the price path. Trading cannot take place
without it for the reason that it is otherwise impossible to learn (acquire private information)
89
about the expected benefits of acting on the information in the price. The trade reduction rule
shows that this mechanism exists and is observable in the price path. Once a trade is opened,
information about profitability is fed back via the state of the price path and this information
is exploited to manipulate the mean return per trade. Chapter 6 expands on this result in more
detail.
This has implications for moving average trading as a test of the weak form EMH. An issue
with moving average trading is that the trading rules are liable to over trade, which reduces
the breakeven costs. The trade reduction rule addresses this issue by reducing the number of
losing trades. With reference to Figures 4.6 to 4.9 for example, the mean return per trade for
the long only trade reduction rule averaged across all moving averages for all portfolios for
the full test period is 84% higher than for the price crossover rule. Assuming this is reflected
in a similar increase in the breakeven costs, the trade reduction rule should have more power
as a test of the weak form EMH. However, the trade reduction rule is in the market for longer
and so takes on more risk. This makes it hard to judge the extent to which the increase in the
mean return per trade will be reflected in the breakeven costs after adjusting for risk. A more
This leaves the question of what information the trading rules are picking up. With reference
to Table 3.1 on page 38, excluding the AIM portfolios, the spread increases as company size
decreases. The theoretical explanation for this is that market makers are exposed to adverse
selection costs. From the market makers perspective, adverse selection arises when market
makers trade with informed traders who have private information about the true value of the
stock. Informed traders buy when the price is below its true value and sell when the price is
above its true value. Informed trading imparts information into the price whereby informed
90
buying causes the price to rise and informed selling causes the price to fall. Market makers
lose to informed traders since they trade at the wrong price and suffer losses when the price
moves against them. The spread allows market makers to recover these losses when trading
with uninformed traders. Uninformed traders do not have private information about the true
value of the stock and instead trade for reasons other than information. Uninformed trading
does not impart information into the price and so uninformed traders pay the spread, which
goes to the market makers as profit (Kyle (1985), Glosten and Milgrom (1985), Easley and
OHara (1987)). Market makers increase the spread to protect themselves against informed
trading and thus the spread increases as the probability of informed trading increases.
With reference to Figures 4.6 to 4.13, the mean return per trade also increases as the spread
increases. Because informed trading moves the price by virtue of imparting information into
the price, this suggests that the trading rules are picking up informed trading. The difference
between buying and selling also supports this. One explanation for this difference is that the
buy side trades on information whereas the sell side need not. Buy trades indicate an interest
in a stock and so are likely to convey firm specific information. Sell trades can be motivated
by the need for liquidity, as when selling to raise capital for example, and so need not convey
information. Sell trades are also likely to be limited to the range of stocks held in a portfolio
and are unlikely to reflect the range of choice available when buying. Buying and selling are
asymmetric with the buy side conveying more information than the sell side.
While this argument holds for the FTSE 100 and FTSE 250 portfolios, with the exception of
test period 1, it does not hold for the FTSE Small Cap and FTSE Fledgling portfolios where
the mean return per trade when selling is greater than the mean return per trade when buying.
One explanation is that this is a data artefact attributable to the bias induced by the portfolio
91
classification scheme and which may or may not disappear when strictly controlling for size.
Another explanation is that the information environment is wanting in some way and where
this attracts informed traders due to the possibility of an increase in the number of profitable
trading opportunities (Grossman and Stiglitz (1980)). If the stocks also trade on low volume,
it is likely that informed trading is highly visible either in terms of trade size or according to
some other measure of order flow. One reason the mean return per trade is so high is that the
portfolios are subject to significant idiosyncratic risk as well as to significant systematic risk
in the form of market wide sell-outs due to events such as the technology crash and the sub-
prime debt crisis. If informed trading is highly visible during these times and the price falls,
there is an incentive to piggyback these trades assuming other traders will see the price fall,
note the order flow and also sell. This exposes market makers to unwanted inventory, which
reduces prices, which causes more selling, which reduces prices, which causes more selling
and so on. The high mean return per trade might be explained by a momentum effect where
buying has little impact and selling dominates because everyone is selling. If so, this should
be evident in the intra-day order flow data where there should be a lack of buy side interest.
92
Chapter 5
This chapter presents the positive autocorrelation rule. Before proceeding, it should be noted
that historically, the approach was to first derive the positive autocorrelation rule and to then
derive the trade reduction rule at the same time as formalising the test method and analytical
framework needed to address the tradability issues raised by the positive autocorrelation rule.
The intention being that the positive autocorrelation rule could then be plugged in and tested
accordingly. However, the work did not get this far and so it remains in its raw experimental
form. First, the data is different. The data is from a different vendor, the test period is shorter
and the spread is not available. Second, the test method is different. While the test method of
the previous chapter was to fix the capital value of each trade and to test the mean return per
trade, the test method in this chapter fixes the trade size and tests the return. The simulation
method is also different and assumes a margin account. Questions of statistical significance
are not addressed. Last, the positive autocorrelation rule is parametric. Without a method to
choose between them, the number of possible trading rules expands considerably. Only one
parameter set is tested. The parameters do not follow from a particular method. That said, the
positive autocorrelation rule is intended to complement the trade reduction rule and is based
equivalent to testing for positive autocorrelation. The explanation for positive autocorrelation
in the market microstructure literature is asymmetric information. Informed traders move the
93
price in the direction of their trades and thereby induce serial correlation. The results in this
chapter support this interpretation. For stocks outside the FTSE 100, there is evidence of an
information spillover from time t to time t + 1 and which appears across a cluster of trading
rules, all of which are short term and roughly adjacent in timing. This indicates that positive
autocorrelation is prevalent and persistent at this time scale. The positive autocorrelation rule
has also been designed to exploit a specific pattern expressed in terms of serial correlation in
order flow that occurs in response to evidence of price impact and which persists for several
days. However, whether this pattern is solely responsible for inducing the correlation in price
changes is not clear. The positive autocorrelation rule can be used as a method to investigate
this and other issues at the microstructure level. The remainder of the chapter is organised as
follows. Section 5.1 defines the positive autocorrelation rule. Section 5.2 presents the results.
This section defines the positive autocorrelation rule. As stated, the positive autocorrelation
rule is based on the idea of only trading if it is believed to be profitable to do so. One way to
model this is to think of the trading act as conditional on the presence of a profitable trading
opportunity. The intuition behind the positive autocorrelation rule is that the trading act can
positive autocorrelation. Section 5.1.1 motivates the need to test for positive autocorrelation.
Section 5.1.2 discusses the variance ratio test and highlights some of its weaknesses. Section
5.1.3 introduces the method of Burgess (2000) as a technique for measuring the deviation of
the joint distribution of the variance ratio from the joint distribution for a random walk. The
magnitude of the deviation from a random walk is indicative of the extent to which positive
autocorrelation is observable in the price. Section 5.1.4 uses this as the basis for defining the
94
positive autocorrelation rule.
A time series is a sequence of data points or observations arranged in time order. A common
time series in finance is the daily close price { pt }Tt =1 = { p1, p2 , p3 , , pt } where t denotes
the time index at which a price is recorded. Define the return rt as the log return or log price
1 T
rt = rt r , r = rt (5.2)
T t =1
Perhaps the simplest time series model of the price is to assume returns {rt }t =1 follow a first
T
rt = rt 1 + et (5.3)
Where 1 < < 1 and et is the innovation or error term with mean and variance (0, e2 ) and
which is uncorrelated at all leads and lags. Equation (5.3) states that rt is a linear function of
95
1. For 0 < < 1 , rt exhibits persistence in sign. Positive returns tend to be followed by
positive returns and negative returns tend to be followed by negative returns. Returns
exhibit positive autocorrelation in that rt and rt 1 tend to have the same sign and so
tend to move in the same direction. This causes the price to trend.
2. For 1 < < 0 , rt exhibits reversion in sign. Positive returns tend to be followed by
negative returns and negative returns tend to be followed by positive returns. Returns
exhibit negative autocorrelation in that rt and rt 1 tend to have the opposite sign and
so tend to move in the opposite direction. This causes the price to mean revert and
3. For = 0 , rt equals et and returns are uncorrelated. This causes the price to follow
a random walk.
With respect to profitability, moving average trading requires that price changes follow one
another in sign. For example, because buying requires the moving average to rise, which in
turn requires the price to rise, and because buying is profitable if and only if the sell price is
higher than the buy price, which again requires the price to rise, buying requires that rising
prices follow rising prices. Similarly, selling requires that falling prices follow falling prices.
One way to capture these dynamics is to test for positive autocorrelation. To see this, Figure
5.1 plots three artificial price series simulated according to (5.3) and which exhibit positive
autocorrelation, negative autocorrelation and random walk dynamics respectively. The return
to trading the price series using a 10-day price crossover rule is also shown. The spread is not
simulated and the trading rule is allowed to go both long and short. The return to trading the
positive autocorrelation price series is consistently positive, the return to trading the negative
autocorrelation price series is consistently negative and the return to trading the random walk
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price series is consistently close to zero.1 In general, assuming the absence of drift, running a
moving average trading strategy without testing for positive autocorrelation is likely to result
Return
Price
100 1
0 0
0 500 1000 1500 2000 0 500 1000 1500 2000
Figure 5.1 Artificial price series that exhibit positive autocorrelation, negative autocorrelation
and random walk dynamics and the return to trading the price series using a 10-day price
crossover rule.
1
The return is the final value of each 1s worth of starting capital. A positive return greater
than one is profitable and a negative return less than one is loss making.
97
5.1.2 Variance ratio tests
A common test for serial dependence in time series data is to test whether the autocorrelation
coefficients are significantly different from zero. If the coefficients are significantly different
from zero, future values of the series are related to previous values and hence the presence of
lag k is:
(r r )(r
t tk r)
1 T
(k) = t = k +1
T , r = rt (5.4)
T t =1
(r r )
t
2
t =1
If (k) is significantly greater than zero, there is evidence of positive autocorrelation at lag k.
If (k) is significantly less than zero, there is evidence of negative autocorrelation at lag k. If
(k) is not significantly different from zero, there is no evidence of autocorrelation. Popular
tests designed to detect departures from zero autocorrelation in either direction at all lags are
the Box-Pierce test (Box and Pierce (1970)) and the Ljung-Box test (Ljung and Box (1978)).
A more recent test that is particularly powerful against positive and negative autocorrelation
as alternatives to a random walk is the variance ratio test. The idea behind the variance ratio
test is that the variance of a random walk is linear in the sampling interval. If the variance of
the one-period return is 2 then (1) the variance of the k-period return should be k 2 and (2)
the ratio of the variance of the k-period return to k times the variance of the one-period return
98
rt (k) = ln( pt ) ln( pt k ) (5.5a)
= rt + rt 1 + L + rt k +1 (5.5c)
The k-period variance ratio VR(k) is the k-period variance divided by k times the one-period
variance:
Lo and MacKinlay (1988) derive asymptotically standard normal test statistics. The variance
1 T
Tk t = k
( rt + rt 1 + L + rt k +1 kr )
2
1 T
VR(k) =
1 T
, r = rt (5.7)
( rt r )
2 T t =1
T t =1
To test for a random walk, Lo and MacKinlay (1988) show that the statistics Z1 (k) and Z 2 (k)
are distributed as N(0,1) under the null hypotheses of a homoskedastic and a heteroskedastic
2
For a homoskedastic random walk, the variance of the error term is constant whereas for a
heteroskedastic random walk, the variance of the error term changes with time. Estimates of
standard errors that assume constant variance are biased by heteroskedasticity and statistical
inference is invalid. See, for example, Wright (1980). The Z 2 (k) statistic corrects for this.
99
VR(k) 1 2(2k 1)(k 1)
Z1 (k) = , (k) = (5.8)
(k) 3kT
VR(k) 1 k 1
2(k j)
2 (r r ) (r
t
2
t j r )2
, * (k) =
t = j +1
Z 2 (k) = ( j), ( j) = (5.9)
* (k) j =1 k T 2
2
(rt r )
t =1
Lo and MacKinlay (1988) also show that the variance ratio is equivalent to a linear function
2 k 1
VR(k) = 1 + (k q)(q)
k q =1
(5.10a)
2
= 1+
k
[(k 1)(1) + (k 2)(2) + L + (k 1)] (5.10b)
Hence, variance ratio tests test the null hypothesis H 0 : (q) = 0 q = 1, 2, , k 1 against
the alternative H 1 : (q) 0 for some q. If the variance ratio is significantly greater than one,
there is evidence of positive autocorrelation. If the variance ratio is significantly less than one,
there is evidence of negative autocorrelation. If the variance ratio is not significantly different
from one, there is no evidence of autocorrelation. Figure 5.2 plots the artificial price series of
Figure 5.1 along with their variance ratios as defined in (5.7). For the positive autocorrelation
price series, the variance increases at a faster rate than the sampling interval and the variance
ratio rises above one. For the negative autocorrelation price series, the variance increases at a
slower rate than the sampling interval and the variance ratio falls below one. For the random
100
walk price series, the variance increases linearly with the sampling interval and the variance
Variance Ratio
200 2
Price
100 1
0 0
0 500 1000 1500 2000 0 10 20 30 40
Time Step k
Figure 5.2 Artificial price series that exhibit positive autocorrelation, negative autocorrelation
There are three problems with variance ratio tests. First, the Lo and MacKinlay (1988) tests
are asymptotic tests in that for a finite sample the sampling distributions of the test statistics
are approximated by their limiting distributions. Lo and MacKinlay (1989) analyse the finite
sample performance of the variance ratio and find that for small samples the null distribution
is right skewed and under rejects in the left tail. This can result in misleading inference if the
sample size is too small to justify the asymptotic approximations (Cecchetti and Lam (1994)).
Wright (2000) extends the Lo and MacKinlay (1988) tests to a non-parametric setting using
ranks and signs. This allows the exact simulation of the null distribution for any sample size.
Wright (2000) evaluates the rank and sign tests using the same time series benchmarks as Lo
and MacKinlay (1989) and concludes firstly that the rank and sign tests are better at detecting
101
departures from a random walk and, secondly, that they are especially powerful if the returns
Second, it is customary to test several different k values and to reject the null hypothesis if it
is rejected for some k. The problem is that the variance ratio test is an individual test whereas
testing several different k values is a joint test. Chow and Denning (1993) and Wright (2000)
stress that treating individual variance ratio tests as a joint test results in an oversized testing
strategy and can lead to over rejection of the null hypothesis. Joint tests that address this test
for some ki . Joint tests include the maximum modulus test of Chow and Denning (1993), the
subsampling test of Whang and Kim (2003), the Wald-type test of Chen and Deo (2006) and
Last, given the proliferation of different variance ratio tests, there is little consensus as to the
choice of test. It is common in this situation to apply a range of tests with the usual approach
being the Lo and MacKinlay (1988) and Wright (2000) tests and one or more joint tests. See,
for example, Chang et al. (2004), Hoque et al. (2007) and Charles and Darne (2009).
102
5.1.3 Variance ratio profiles
Burgess (2000) treats the joint variance ratio VP(k) = [VR(1), VR(2), , VR(k)] as having a
shape or profile as in the variance ratios of Figure 5.2 and projects this onto the eigenvectors
that result from principal component analysis of the covariance matrix of the corresponding
(
VP(k,i) = VP(k) RW ei ) (5.11)
Where RW is the mean profile for a random walk and ei are the eigenvectors or principal
components. The magnitudes of the projections onto the eigenvectors are then a measure of
the extent to which the profile as a whole deviates from (or is similar to) the archetype for a
random walk. This has several advantages. First, (5.11) is a general case test and addresses
the problems of the previous section. Second, testing for positive autocorrelation reduces to
testing for deviation from the archetype for a random walk. To see this, Figure 5.3 plots the
first three eigenvectors for VP(100) estimated using 1000 random walk time series and the
variance ratio defined in (5.7). The magnitude of the projection onto the first eigenvector e1
3
Before it is possible to calculate VP(k,i) , it is necessary to define RW and ei . Both terms
are effectively constant. To define them, it is necessary to simulate a number of random walk
time series, to calculate VP(k) for each time series and to save the VP(k) in the matrix RW.
The mean of RW is calculated as RW . This is subtracted from RW and principal component
analysis is applied to the covariance matrix of RW to obtain ei . The covariance matrix in this
case is calculated by the PCA tool. See Jolliffe (2002) for a rigorous introduction to principal
component analysis. Note that Burgess (2000) developed the variance ratio profile to test for
negative autocorrelation within a statistical arbitrage setting. It is used here to test for positive
autocorrelation within a moving average trading setting.
103
measures the extent to which the profile exhibits positive or negative autocorrelation. For a
random walk where each VP(k) equals one, assuming the mean profile for a random walk
RW also equals one, VP(k,1) maps to zero. For positive autocorrelation where each VP(k)
is greater than one, VP(k,1) maps to a number greater than zero. For negative autocorrelation
where each VP(k) is less than one, VP(k,1) maps to a number less than zero. The remaining
eigenvectors do not allow this interpretation due to their mixed positive and negative nature.
for VP(k,1) using a random walk and to use this to define a threshold above which deviation
from a random walk is deemed significant and thus above which trading can take place. This
agrees with Burgess (2000) who performs a Monte Carlo simulation and concludes that as a
test for predictability, the first principal component is the best test overall with the remaining
tests including autocorrelation, unit root and variance ratio tests. Third, it is not necessary to
limit the archetype to a random walk. For example, it is possible to define (5.11) in terms of
previously profitable trades and so (5.11) can be used to test for the similarity to a profitable
archetype. This may or may not provide insight into the stock selection problem discussed in
Chapter 3. Last, trading rules based on (5.11) are parametric in the variance ratio, the length
of data used to calculate the variance ratio, the profile length k, the eigenvector weights, the
mean of the random walk archetype and the threshold above which trading takes place. It is
possible to imagine these as variables within a genetic algorithm (Goldberg (1989), Mitchell
(1998)) and which optimises, say, the risk adjusted mean return per trade measure suggested
by Masters (1998). The parametric nature of the trading rules is also a disadvantage however
104
0.3
0.2
0.1
Coefficient
0.0
-0.1
e(1)
-0.2 e(2)
e(3)
-0.3
0 10 20 30 40 50 60 70 80 90 100
Figure 5.3 The first three eigenvectors for VP(100) estimated using 1000 random walk time
series.
Given the discussion of the previous section, for the first principal component, the long only
q1 = 0
///// RULE
qt +1 = qt
Where VP(k,1) is the projection of the variance ratio profile onto the archetype for a random
walk and x is the threshold above which trading takes place. Similarly, the short only positive
105
autocorrelation rule is:
q1 = 0
///// RULE
qt +1 = qt
IF (lmint ) THEN qt +1 = 0
The positive autocorrelation rule introduces two additional problems. Namely, how best to
define VP(k,1) and what is a suitable value for x. As discussed previously, because VP(k,1)
is highly parametric and because there is no optimisation, the problem of how best to define
VP(k,1) is not addressed. Rather, the choice is to use the R1 (k) test of Wright (2000) with a
profile length of k = 25 calculated over a 130-day moving window. The choice of R1 (k) test
is motivated by Wright (2000) who finds that it dominates all other tests in terms of power
and does not suffer size serious distortions in the presence of conditional heteroskedasticity.
1 T T +1
Tk t = k
( r1t + r1t 1 + L + r1t k +1 )
2
f (rt )
VR(k) = , r1t = 2 (5.14b)
1 T 2 (T 1)(T + 1)
r
T t =1 1t 12
106
Where f (rt ) is the rank of rt amongst the rt s. For k = 25, VP(k,1) is then:
( )
VP(25,1) = VP(25) RW e1 , VP(25) = [ R1 (1), R1 (2), , R1 (25)] (5.15)
Figure 5.4 plots the first three eigenvectors for VP(25) estimated using 1000 random walk
time series. The main difference between this and Figure 5.3 is that given a shorter profile
length, the weight given to each k is higher. The fact that both sets of eigenvectors exhibit
similar shapes indicates that the Burgess (2000) method is robust to the choice of measure.
Figure 5.5 plots the variance explained together with the cumulative variance for the first ten
principal components. The first principal component explains 85% of the total variance and
indicates that not too much information is lost despite limiting the positive autocorrelation
0.75
0.50
Coefficient
0.25
0.00
e(1)
-0.25
e(2)
e(3)
-0.50
0 5 10 15 20 25
Figure 5.4 The first three eigenvectors for VP(25) estimated using 1000 random walk time
series.
107
100%
variance
80% cumulative
Variance Explained
variance
60%
40%
20%
0%
1 2 3 4 5 6 7 8 9 10
Principal Component
Figure 5.5 Variance explained by the first 10 principal components for VP(25).
Figure 5.6 plots the cumulative frequency distribution for VP(25,1) estimated using another
1000 random walk time series. The 99th percentile occurs at x = 10. For x = 10 there is a 1%
chance of a random walk and for x > 10 there is evidence of positive autocorrelation. This is
the value used for the threshold x. It should be noted that for x > 10 it does not automatically
follow that the price is driven by positive autocorrelation alone. It is possible for the price to
dynamics but where the positive autocorrelation term is sufficiently dominant for VP(25,1) to
be greater than 10. All that can be said in this situation is that while a positive autocorrelation
term is likely to exist and so is likely to have an impact, the exact form of the price dynamics
remains unknown.
108
1.0
0.8
Cumulative Frequency
0.6
0.4
0.2
0.0
-12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5
VP(25,1)
Figure 5.6 Cumulative frequency distribution for VP(25,1) estimated using another 1000
Despite the intuition of combining moving average trading with a variance ratio test, it is not
known of any work in this area. The only work known to apply the method of Burgess (2000)
is Lindemann et al. (2005) who leave out the subtraction of the mean in the variance ratio of
(5.7) in order to allow for a random walk with drift. The first two principal components are
then plotted in (x, y) space to give a 2-dimensional structure visualisation tool. Trading rules
are not discussed. The only work known to explicitly combine moving average trading with
an autocorrelation model in a way similar to the positive autocorrelation rule is Fang and Xu
(2003) who estimate a rolling AR(1) model for the Dow Jones Industrial, Transportation and
Utilities indices. The model forecasts are combined with moving average trading rules such
that trading only takes place when both the AR(1) model and the trading rule emit a buy/sell
signal. This doubles the breakeven costs relative to the moving average trading rules on their
own. However, time variation in the model coefficient is not discussed and so the exact form
of the AR(1) models is not known. Nevertheless, this is an interesting result since Brock et al.
109
(1992) find that the returns to moving average trading are not explained by an AR(1) model
The positive autocorrelation rule can also be thought of as a nonlinear model that switches
trading on/off whenever the price enters/exits a positive autocorrelation regime. Brock et al.
(1992) suggest that the difference in the returns uncovered by moving average trading might
be explained by nonlinear models with asymmetric dynamics. Gencay (1998) finds evidence
of nonlinear predictability when the buy/sell signals from moving average trading rules are
used as inputs to a single-layer feedforward neural network. Gencay and Stengos (1998) find
that predictability increases significantly when volume is used an additional input. Self and
Mathur (2006) estimate a momentum threshold autoregressive (MTAR) model and use this
together with the asymmetric stationarity test of Enders and Granger (1998) to identify non-
stationary, symmetric stationary and asymmetric stationary regimes in the G7 national stock
indices. There is evidence that the return to moving average trading varies with each regime,
trading strategy. Choe et al. (2011) estimate nonlinear autoregressive (NAR) models for the
G7 national stock indices. Following Nam et al. (2005), trading rules that exploit patterns in
previous returns are found to be significant but only when nonlinearity results in a consistent
asymmetric pattern in the model coefficients. Chong and Lam (2010) estimate rolling AR(1)
and self-exciting threshold autoregressive (SETAR) models for the DJIA, NASDAQ, NYSE
and S&P 500 indices. The model forecasts generate higher returns than comparable moving
average trading rules in all cases. However, this is before costs. It is not clear if the models
110
It should also be noted that there is a literature that tests the weak form EMH based on tests
applied within rolling windows. Examples include rolling variance ratio tests (Tabak (2003),
Kim and Shamsuddin (2008)), rolling unit root tests (Phengpis (2006)), rolling bicorrelation
tests (Lim (2007)), rolling Hurst exponents (Cajueiro and Tabak (2006)) and rolling AR(1)
models (Lo (2004), Ito and Sugiyama (2009)). The general result is that there is evidence of
time varying deviation from a random walk. This suggests that market efficiency is not an all
or nothing condition and that there are times when there is evidence of inefficiency.
5.2 Results
This section presents the results for the positive autocorrelation rule. Section 5.2.1 describes
the data. Section 5.2.2 discusses the simulation method. The simulation method assumes an
execution only margin account and fixes the trade size. Section 5.2.3 presents an illustrative
example where it is shown that the positive autocorrelation rule is profitable and that profits
appears across a cluster of trading rules, all of which are short term and roughly adjacent in
timing. This captures the intuition of positive autocorrelation as a short term dynamic and is
evidence that positive autocorrelation is prevalent and persistent at this time scale. Section
5.2.4 presents the results for the FTSE 100, FTSE 250, FTSE Small Cap and FTSE Fledgling
portfolios. The results are the same. This prompts a discussion of the strategy underlying the
positive autocorrelation rule and it is suggested that this should be observable in the adverse
selection component of spread decomposition models. The positive autocorrelation rule can
be used to investigate this at the microstructure level. There is also evidence of price impact.
111
5.2.1 Data
As stated in the introduction, due to data limitations, the test data is different to that used for
the trade reduction rule. The data comprises 13 years of daily close prices for stocks listed on
the London Stock Exchange from 06-January-1994 to 06-January-2007. The data is from the
Sharescope and Updata Technical Analyst software packages, both of which are supplied with
historical price databases. The data comes as a sequence of date-stamped records of the form
(datet , closet ) where datet is the date for which the price is available and closet is the end of
day close price. Prices are adjusted for stock splits and other capital events but do not include
the dividend. Prices exclude weekends but include public holidays. Prices for public holidays
are duplicates of the previous days prices. Prices for public holidays are not treated differently
The first step in pre-processing is to check that the price is available for the whole of the test
period. If not, the price file is deleted. The next step is to check that the prices from the two
data sources equal one another. If not, the price file is deleted. This gives 253 stocks in total.
Investment trusts are not included. Once the data is pre-processed, the stocks are sorted into
portfolios. There is one portfolio for each of the FTSE 100, FTSE 250, FTSE Small Cap and
FTSE Fledgling indices. If a stock is a member of the FTSE 100 index, it is allocated to the
FTSE 100 portfolio. If it is a member of the FTSE 250 index, it is allocated to the FTSE 250
portfolio and so on. The indices are supplied by the FTSE Group (www.ftse.com). The date
at which the indices are correct is not available. A complete listing of each portfolio is given
in the appendix. There are 43 stocks in the FTSE 100 portfolio, 77 in the FTSE 250 portfolio,
74 in the FTSE Small Cap portfolio and 59 in the FTSE Fledgling portfolio.
112
5.2.2 Simulation method
Simulation assumes a retail client, execution only margin account. Such accounts are readily
available to the public and where the underlying financial instrument is either a contract for
difference or a spread bet. Margin allows for taking a position without needing the capital to
fund the full value of the position. For example, going long or short 10,000 worth of shares
with margin of 10% requires initial capital of 10,000 10% = 1,000. The effect is to gear
the return by reducing the capital commitment. Margin needs to be financed however. Long
positions pay interest on the overnight value of the position whereas short positions receive
interest on the overnight value of the position. The interest rates used to simulate the trading
rules are 7.5% for long positions and 2.5% for short positions. For example, the interest paid
on a long position with an overnight value of 10,000 is 10,000 (7.5% / 365.25) = 2.05
and which is debited from the account at the start of the next day. The interest received on a
short position with an overnight value of 10,000 is 10,000 (2.5% / 365.25) = 0.68 and
For simulation purposes, the margin requirement is 100%. There has to be sufficient capital
available to fund the full value of each trade. The reason for this is to determine whether the
positive autocorrelation rule has value as a real world decision support tool assuming trading
is leveraged and is in units of one normal market size. To simulate the costs associated with
leveraging the trading rules, they are subject to financing. It is then possible to determine if
the trading rules are stable in the presence of financing independently of gearing the return.
To simulate trading in units of one normal market size, the trade size is fixed at 1000 shares.
The problem with this is that because the trading rules are not fully invested, it is not known
how much capital is needed to fund trading until the end of the simulation. Similarly, fixing
113
the trade size introduces an element of self-financing in that the profits from previous trades
can be used to pay for future trades. The approach in this case is to first simulate the trading
rules and to calculate the cumulative cash profit. Once the cumulative cash profit is known,
the minimum capital needed to guarantee that each trade is executed is added in. This means
that the return is calculated ex-post and is the maximum return achievable for the minimum
capital needed to fund trading to the end of the simulation. This is calculated on a trading rule
by trading rule basis and is different for each trading rule for each moving average. While this
might appear to bias the return, this is not the intention. Rather, it is to highlight evidence of
profitability. For example, previewing the results, compared to the price crossover rule, the
positive autocorrelation rule needs less capital to fund trading. Given that uninvested capital
does not earn interest, equalising the capital across the trading rules suppresses the return to
the positive autocorrelation rule due to the high proportion of uninvested capital. Calculating
the return as the maximum achievable for the minimum capital needed to fund trading does
not suppress the return and instead highlights evidence of profitability. This is true for both
Because the spread is not available, round trip transaction costs are levied at 2.5% per trade.
For portfolios, the minimum capital needed to fund trading is calculated for the whole of the
portfolio. Stocks within the portfolio are traded individually and the minimum capital needed
to fund trading is the minimum capital needed to trade all of the stocks at the same time. This
includes margin calls. The return to a trading rule is then a cash equity curve that starts from
the minimum capital. Uninvested capital does not earn interest. Dividends are not simulated.
The moving average is as described in Section 3.2. Table 5.1 lists the parameters used for the
simulations.
114
Table 5.1
Simulation parameters
Margin Long Interest Short Interest Trade Size Round Trip Costs
Margin of 100% means that the return is subject to financing but is not geared.
Figure 5.7 plots the equity curves using a 10-day moving average for the price crossover and
positive autocorrelation rules applied to Yule Catto (a FTSE Small Cap chemicals company)
for the period 06-January-1994 to 06-January-2007. The simulation parameters are listed in
Table 5.1. Note that long and short trades are not treated separately. The trading rules can go
q1 = 0
///// RULE
qt +1 = qt
IF (lmint ) THEN qt +1 = 1
115
The positive autocorrelation rule is:
q1 = 0
///// RULE
qt +1 = qt
IF (lmint ) THEN qt +1 = 0
IF (lmaxt ) THEN qt +1 = 0
For the price crossover rule, the equity curve rises from January-1994 until September-2002
indicating that the price crossover rule is profitable and that the price is trending. The equity
curve then falls after September-2002 indicating that the price crossover rule is loss making
and that the price is no longer trending. The change in the price dynamics causes all previous
gains to disappear resulting in a net loss. For the positive autocorrelation rule, trading simply
switches off after September-2002 resulting in a net profit. In terms of performance, the price
crossover rule generates 298 trades and has a win rate of 24% with a mean return per trade of
0.4%. The positive autocorrelation rule generates 77 trades and has a win rate of 43% with a
mean return per trade of 2.6%. The price crossover rule also needs 3 times more capital than
116
Yule Catto Equity Curve
600 3.0
price crossover rule
500 2.5 positive autocorrelation rule
400 2.0
Return
Price
300 1.5
200 1.0
100 0.5
0 0.0
Jan-94 Mar-96 May-98 Jul-00 Sep-02 Nov-04 Jan-07 Jan-94 Mar-96 May-98 Jul-00 Sep-02 Nov-04 Jan-07
Time Time
Figure 5.7 Equity curves for the 10-day moving average price crossover and positive
Figure 5.8 plots the return for moving averages in the range n = 2, 3, K, 100 . The return to
the positive autocorrelation rule is consistently higher than for the price crossover rule. The
positive autocorrelation rule is also profitable for a range of moving averages,4 all of which
cluster together and capture the intuition of positive autocorrelation as a short term dynamic.
Although an isolated example, this is a fundamental result. As the moving average increases,
so the lag of the trading rules increases. As the lag of the trading rules increases, so the time
delay in the buy/sell signals increases. Consequently, there is little to separate neighbouring
trading rules other than the delay in the timing of the buy/sell signals. Because profits appear
4
From the data in Chapter 3, the mean spread for Yule Catto for the period 06-January-1994
to 06-January-2007 is 2.1%. Round trip costs transaction costs of 2.5% are therefore likely to
overestimate the spread. The positive autocorrelation rule is also subject to financing. This is
an additional cost. Conclusions regarding profitability are robust in this respect.
117
across a cluster of trading rules all of which are roughly adjacent in timing, there is evidence
that positive autocorrelation is both prevalent and persistent at this time scale. This takes the
form of an information spillover from time t to time t + 1 and which is sufficiently prevalent
for the positive autocorrelation rule to profit from it and which is sufficiently persistent for it
3.0
price crossover rule
2.5 positive autocorrelation rule
2.0
Return
1.5
1.0
0.5
0.0
0 10 20 30 40 50 60 70 80 90 100
Moving Average
Figure 5.8 Return for the price crossover and positive autocorrelation rules for Yule Catto.
118
5.2.4 Portfolio simulations
Figure 5.9 plots the results for the price crossover and positive autocorrelation rules for the
FTSE 100, FTSE 250, FTSE Small Cap and FTSE Fledgling portfolios. The trading rules are
the same as in the previous section and are defined by (5.16) and (5.17). Moving averages are
in the range n = 2, 3, K, 100 . The return for the positive autocorrelation rule is consistently
higher than for the price crossover rule.5 However, with regard to profitability, the results are
difficult to interpret. This is because it is not clear to what extent the results are robust to the
spread. For example, with reference to Table 3.1 on page 38, round trip transaction costs of
2.5% are likely to overestimate the spread for the FTSE 100 and FTSE 250 portfolios and to
underestimate the spread for the FTSE Small Cap and FTSE Fledgling portfolios. This limits
discussion of profitability to the FTSE 100 and FTSE 250 portfolios. Nevertheless, in general,
the positive autocorrelation rule is able to identify trades that are more profitable than for the
price crossover rule. For the FTSE 250 portfolio, these trades are also financially exploitable.
Although limited, this result implies that profitable trading opportunities are observable in the
price.
5
The apparent discrepancy in the results for the price crossover rule for this chapter and the
previous chapter is explained by the difference in strategy. The strategy for this chapter is to
fix the trade size. This does not break up serial dependence in the order of the trades and so
losses are compounded. Trades are also subject to financing which further reduces the return.
The strategy in the previous chapter was to fix the capital value of each trade. This breaks up
serial dependence in the order of the trades so that each trade earns the mean return per trade
by definition. Trades are also not financed and so the return is not reduced. Because it breaks
up serial dependence in the order of the trades, the better strategy is to fix the capital value of
each trade.
119
FTSE 100 FTSE 250
4 4
price crossover rule price crossover rule
positive autocorrelation rule positive autocorrelation rule
3 3
Return
Return
2 2
1 1
0 0
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
Return
2 2
1 1
0 0
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
Figure 5.9 Return for the price crossover and positive autocorrelation rules for the FTSE 100,
120
It is also noticeable that the return increases as company size decreases. This suggests that as
with the trade reduction rule, the positive autocorrelation rule is picking up informed trading.
This is a significant result. The positive autocorrelation rule is designed to exploit a specific
pattern. For example, suppose the price is pt and suppose the trader believes the correct price
The optimal strategy is to trade pt and for the price to immediately change to pt whereupon
a profit is realised. However, for the trader, it is unrealistic to expect the price to immediately
change to pt in response to a single trade. Suppose then that the trader knows other traders
will also recognise pt as a profitable trading opportunity and so instead of trading, waits for
these other traders to appear in the order flow. Once these traders start to appear in the order
flow, if there is evidence that pt is sensitive to their trades in that pt starts to move toward
pt , the trader also trades. Suppose there are many traders following this strategy. They also
respond to the sensitivity of the price to the order flow and trade as well. This induces serial
correlation in the order flow. Trades tend to be in the same direction and the price changes
accordingly. If this persists for several days, there is an information spillover from time t to
time t + 1 and the positive autocorrelation rule is profitable. The positive autocorrelation rule
is specifically designed to exploit this pattern by searching out times when it is likely to hold.
With reference to Figure 5.9, not only is there evidence to suggest that it exists but also that it
The significance of this is that the trading patterns and/or information dynamics underlying
the positive autocorrelation rule should be observable in the intra-day order flow data. More
specifically, assuming the positive autocorrelation rule is profitable, it should identify times
when there is variation in adverse selection costs. Popular spread decomposition models that
121
measure the adverse selection component of the spread include Glosten and Harris (1988),
George et al. (1991), Lin et al. (1995), Huang and Stoll (1997) and Madhaven et al. (1997).
However, a general issue with these models is that they often result in different estimates of
the adverse selection component and which are sometimes implausible (Neal and Wheatley
(1998), Van Ness et al. (2001), Chelley-Steeley and Park (2008)). A potential application of
the positive autocorrelation rule is to use it to investigate the adverse selection component of
the spread by splitting the price into times when the positive autocorrelation rule is profitable,
not profitable and not trading. Estimates of adverse selection should vary accordingly. If not,
either the adverse selection component of the models is misspecified or there is an unknown
source of asymmetric information. It is not known of any work that explicitly links moving
Similarly, apart from the FTSE 100 portfolio, the return for the positive autocorrelation rule
exhibits a distinct hump shape. Returns are maximised in the short term and decrease as the
length of the moving average increases. This implies that the information uncovered by the
positive autocorrelation rule is short lived. Empirical studies such as Hasbrouck (1991) and
Easley et al. (1997) suggest that informed traders with short lived information will prefer to
trade large trade sizes. If so, variation in trade size should also be observable in the intra-day
order flow data and the positive autocorrelation rule can be used to investigate this. It is also
worth noting that in London, market makers are only obliged to quote prices good for orders
up to one normal market size. If informed traders do prefer to trade large sizes, a measure of
informed trading might be the price impact of trades normalised by the normal market size.6
6
Price impact refers to the correlation between a trade and the subsequent price change. See
Hasbrouck (2007).
122
This measures the sensitivity of quote revisions to trades in units of normal market size. The
sensitivity of quote revisions to trades in units of normal market size is thought to be a trigger
There is also evidence of price impact. The fact that the price crossover rule is profitable for
the FTSE Fledgling portfolio indicates that prices exhibit a degree of persistence. Additional
evidence can be found by looking at the ratio of the capital needed to fund the trading rules.
For moving average n, let capital( pc)n denote the capital needed to fund the price crossover
rule and let capital( pa)n denote the capital needed to fund the positive autocorrelation rule.
Figure 5.10 plots the ratio of the capital needed to fund the price crossover rule relative to the
capital( pc)i
ratio = i=2
n (5.18)
capital( pa) i
i=2
As company size decreases, the capital needed to fund the price crossover rule tends to that
needed to fund the positive autocorrelation rule. This cannot be explained by volatility since
volatility increases as company size decreases. This implies that as company size decreases,
prices exhibit greater persistence. Trades are more likely to have a permanent impact, which
is likely to be a contributory factor in the return for the positive autocorrelation rule.
123
80
70
60
50
Ratio
40
30
20
10
0
FTSE 100 FTSE 250 FTSE Small Cap FTSE Fledgling
Portfolio
Figure 5.10 Ratio of the capital needed to fund trading for the price crossover and positive
autocorrelation rules.
5.3 Conclusion
The main result to come from this chapter is that in general, the positive autocorrelation rule
shows that profitable trading opportunities are observable in the price. For stocks outside the
FTSE 100, there is evidence of an information spillover from time t to time t + 1 . The hump
shape seen in the return indicates that this information is short lived. It also indicates that this
information is sufficiently prevalent for the positive autocorrelation rule to profit from it and
sufficiently persistent for it to appear across a cluster of trading rules, all of which are roughly
adjacent in timing. However, evidence of profitability is limited to the FTSE 250 portfolio. It
is not clear to what extent the results for the FTSE Small Cap and FTSE Fledgling portfolios
are robust to the spread. It is also not clear whether simulating long and short trades together
adds to or takes away from the return. Similarly, with reference to Figure 5.8, it is not clear if
trading is continuous for the whole of the test period. To draw stronger conclusions regarding
124
profitability it is necessary to simulate the positive autocorrelation rule using the data and test
method of the trade reduction rule as well as to investigate whether trading is continuous.
It is also noticeable that the return increases as company size decreases. This suggests that as
with the trade reduction rule, the positive autocorrelation rule is picking up informed trading.
However, underlying informed trading is the argument that it enhances market efficiency by
making prices more informative. This is in conflict with the evidence that the price dynamics
uncovered by the positive autocorrelation rule are profitable. The trading patterns underlying
the positive autocorrelation rule are therefore of great interest. A contribution of the positive
autocorrelation rule is to provide a method by which to peel back the price and to investigate
these issues at the microstructure level. Of particular note is that the positive autocorrelation
rule has been designed to exploit a specific pattern. This is expressed as serial correlation in
order flow that occurs in response to evidence of price impact and which persists for several
days. However, whether this pattern is solely responsible for inducing the correlation in price
changes is not known. Biais et al. (1995) suggest that serial correlation in order flow can be
explained by (1) the splitting of large orders, (2) piggybacking whereby traders follow what
other traders are doing and (3) similar reactions to the same events. Which of these is able to
explain the correlation in price changes should be observable in the intra-day order flow data.
The same argument holds for spread decomposition models. The positive autocorrelation rule
splits the price into times when it is profitable, not profitable and not trading. Estimates of the
model parameters should vary accordingly. To summarise, the positive autocorrelation rule is
clearly able to identify times when the price behaves in a way that is different from the norm.
The reasons for this difference should be observable in the intra-day order flow data. A more
detailed analysis of the positive autocorrelation rule at this level is further work.
125
As discussed in Chapter 3, moving average trading suffers from a stock selection problem. If
the price dynamics exploited by moving average trading are not in the price, the trading rules
will not find them. A potential solution to this problem is to record VP(k,i) at each time step
whereby each stock has a VP(k,i) signature. Ranking each stock at each time step according
to some function of its signature is a method by which to solve the stock selection problem.
Given their ranking, it is then possible to conduct a market wide search for those stocks most
suited to moving average trading. The maximally ranked stocks might be those with maximal
evidence of positive autocorrelation for example. There are two points. First, if the stocks are
ranked and sorted into portfolios, if the results for the positive autocorrelation rule vary with
the portfolios, there is the basis for a deeper investigation into the reasons for this. Second, it
is important to be aware that the positive autocorrelation rule is a functional decision support
tool. Ranking each stock at each time step allows the causes for their ranking to be assessed.
126
Chapter 6
The question asked in this thesis is whether remodelling moving average trading to reduce the
number of losing trades increases the mean return per trade to the extent that the trading rules
are profitable and, if so, whether this is economically significant. This is motivated by market
efficiency. A general issue with market efficiency is that it is extremely difficult to rationalise
trading if markets are efficient as defined by Fama (1970). If the price impounds all available
information, other than in response to new information, why is it that traders trade? There are
four possibilities. First, markets are efficient in the sense of Jensen (1978) and Taylor (2005).
Profitable trading opportunities exist but where it is not possible for traders to outperform the
market on a risk adjusted basis. However, traders might leverage their trades and outperform
the market if measured by the return on capital employed. Second, markets are inefficient in
the sense of Grossman and Stiglitz (1980). It is possible for traders to outperform the market
but where outperformance is limited by the cost of acquiring information. Third, markets are
all of the above hold in some way. Market efficiency as defined is not rich enough a concept
To examine these issues further, the approach is to adopt the perspective of the trader. Two
trading rules are proposed, both of which are designed to capture trading practice. The trade
127
reduction rule is based on the idea of allowing a trade to run and the positive autocorrelation
rule is based on the idea of only trading if it is believed to be profitable to do so. The trading
rules follow from an understanding of what it is to trade, are independent of the data and are
supported mathematically. The problem with modelling trading practice is that it implies the
need to model the trading decision. To capture this, the method is to define trading as driven
by price direction and to replace the information that would normally be input to the trading
decision with the information contained in the moving average buy/sell signals. This allows
the trading decision to be modelled in terms of buy/sell actions that transform the underlying
trade distribution. The advantage is that the modelling problem reduces to describing how a
trader might exploit price direction and where the moving average renders this testable. The
disadvantage is that the moving average is a crude tool. However, this crudity is intentional.
What is being modelled is not the price but the traders response to the price. If it is possible
for the trader to exploit the information in the moving average, not only is there evidence of
market inefficiency but given that the moving average is little more than a smoothed version
of the price, there is also evidence that this is observable in the price. The implication being
that if market inefficiency is observable in the price, potentially at least, it is visible for all to
see.
The easiest way to put this in context is to consider the following scenario. Suppose a trader
opens a trade. Suppose also that this is solely for the purposes of making a profit. For this to
occur, the price needs to convey two pieces of information. First, it needs to convey that the
price represents a trading opportunity. Second, it needs to convey that in the event of acting
on a trading opportunity, all else being equal, a profit will be realised. Figure 6.1 illustrates
128
Decision To Trade (Buy/Sell Actions)
Trading Opportunities
Figure 6.1 Trading and the traders perception of the relationship in information flow.
Note that Figure 6.1 is independent of the actual trading strategy. It is sufficiently general to
capture all trading strategies irrespective of their origin, complexity or implementation. The
first contribution of this thesis is that the trading rules recover the model of Figure 6.1. The
trade reduction rule relates to the feedback on the decision to trade. After a trade is opened,
information on profitability is fed back via the state of the price path and this information is
used to manipulate the mean return per trade. The positive autocorrelation rule relates to the
this information is used to manipulate the return. Both trading rules uncover information that
is missed by the price crossover rule. The information uncovered by the trade reduction rule is
long lived and the information uncovered by the positive autocorrelation rule is short lived.
The second contribution of this thesis is to show that the information uncovered by the trade
reduction and positive autocorrelation rules is financially exploitable. At its most basic level,
this means that the information needed for trading to be economically viable is observable in
129
the price. A general issue with trading is whether it is elitist. Can the average trader expect to
profit or is profitability the preserve of the select few? The choice of moving average trading
addresses this issue and there is evidence that the information underlying Figure 6.1 is visible
for all to see. This implies that first, for the test data at least, the market cannot be efficient in
the sense of Fama (1970). The results do not support this perspective. Rather, second, there is
support for learning. At the microstructure level, this implies an explanation for uninformed
traders. A problem with uninformed traders is why they continue to trade if they continue to
lose (OHara (2003)). Uninformed traders can be defined as attempting to learn the model of
Figure 6.1. If successful, they become informed traders. If unsuccessful, they can be expected
to stop trading and to leave the market. This supports the adaptive market hypothesis (AMH)
Third, given the support for learning, it should be possible to acquire expertise. This implies
that there should exist traders who consistently generate profits. There is some evidence that
such traders exist. Puckett and Yan (2011) find that the trades of institutional fund managers
generate an average excess of 20 to 26 basis points per annum after costs. While this appears
to be a small number, for average net assets of US$22 billion, this is a large number in cash
terms. When ranked by previous trading performance, for the top quintile, there is evidence
that performance persists from quarter to quarter. Barber et al. (2011) find that the trades of
the top ranked day traders in Taiwan generate an average excess of 28.1 basis points per day
after costs. At the portfolio level, this reduces to 2 basis points per day after costs. As before,
previous trading performance is the best predictor of future performance. Neither study finds
evidence to support the hypotheses that returns are explained by liquidity provision or inside
information. Results such as these are important and indicate that to fully understand market
efficiency it is necessary to study the individual trading records of skilled traders. This thesis
is a step in that direction. Fourth, there is support for technical analysis. Technical analysis is
130
used to visualise information. What is required of the trader is to decide if this information is
meaningful. Last, there is support for trading system design. The trade reduction and positive
autocorrelation rules are functional trading tools. However, the theme to the results is that by
modelling elements of decision making as found in the real world it is possible to show that
architectural issue. The majority of trading systems automate the trading process and design
the trader out. A different approach and which allows greater variation in decision-making is
The third contribution of this thesis is the relationship with market microstructure. A slightly
different way to think of Figure 6.1 is that market microstructure looks out of the market and
in the direction of the trader. The approach in this thesis is to look out of the trader and in the
direction of the market. In doing so, the trading rules are the glue that binds the two together.
The strongest interpretation that can be applied to the trading rules in this context is that they
are in fact examples of informed trading. Compared to the price crossover rule, they are more
able to extract meaning from (or more able to understand) the same price information. Given
this interpretation, it is perhaps not surprising to find that the trading rules uncover issues of
informed trading, liquidity, adverse selection and price impact. Intuitively, this is correct. Of
particular note is that the positive autocorrelation rule is a method by which to peel back the
price and to investigate these issues using high frequency data. The positive autocorrelation
rule has the potential to be a fruitful test bed in this respect. It is not known of any work that
explicitly links moving average trading and market microstructure in this way.
This leaves the question of economic significance. This is a difficult question and one that it
is not possible to address without further testing. However, there is evidence that the market
131
information environment is financially exploitable. It is reasonable to assume that traders are
aware of this and are able to profit from it. A testable prediction of this thesis is that the high
frequency data should contain evidence of traders who consistently generate profits. Whether
this also implies economic significance is not clear however. The issue with data of this type
is that each identifier needs to relate to a single trader and where the data needs to include all
of the traders activity. Clearly, there will always be some doubt about this. It is also the case
that the traders capital is an unobservable variable. This is perhaps the bigger problem since
the traders capital is the variable of interest. The advantage of trading rules is that economic
they generally fail to capture the broader complexity of the trading process. The work in this
thesis is intended to address this and the results suggest that economic significance is not by
of financial markets and is one that is necessary for them to exist given that it is the promise
of economic significance, or at least the perception of this promise that attracts traders.
It is thought that the market efficiency test most likely to generalise in the sense of also being
suitable for large cap stocks is to extend the trade reduction rule to include an autocorrelation
test but where trading switches off in the presence of negative autocorrelation. This needs to
be investigated. The trading rules can also be extended to include the location of the buy/sell
signals within [1, 1] . For example, for 1 y, z 1 , this gives trading rules of the type:
132
It is extremely difficult to develop intuition for these trading rules. By definition, there is an
element of path dependence. However, for the price crossover rule, the resulting trading rule
does not transform the trade distribution in any way. Rather, it samples it differently. For the
trade reduction rule, it is feasible that the information uncovered by the resulting trading rule
might vary according to the location of the buy/sell signals. This is because the location of the
buy/sell signals captures something of the extent to which the price has become increasingly
cheap or expensive relative to before. Since the trade reduction rule allows a trade to stay in
the market for longer, this allows this information to persist for longer and so there is time to
have an effect on the price. With reference to Figure 4.2 on page 55 for example, this means
that there might be significant predictive information in the extremes of the trade distribution.
More generally, the probability of ending up in the extremes of the trade distribution is lower
the closer the trade entry point is to 1 . Thus, the probability of a trade being exited due to a
loss is also lower. Because of this, the results for the trade reduction rule might be explained
by a subset of trades that dominate the rule. While unlikely it cannot be discounted. This can
be investigated by incrementally removing trades where the threshold for removing trades is
stepped in, say, 0.1 increments. It is also desirable to have a sense of how the trade reduction
rule sequences the buy/sell signals. One approach is to construct a decision tree and to relate
the sequencing of the buy/sell signals to the probability of an event occurring. This makes it
possible to define trading rules that, say, trade if and only if there is a 60% chance of a win.
As stated, moving average trading suffers from a stock selection problem. This is a difficult
problem and one that limits the viability of moving average trading as a trading system. The
proposed solution is to record VP(k,i) for each stock at each time step such that each stock
has a VP(k,i) signature. Ranking each stock at each time step according to some function of
133
its signature is a potential method by which to solve the stock selection problem. A possible
function is to weight the most recent VP(k,i) values by a Gaussian or Epanechnikov kernel.
However, implicit in this is the assumption that the maximally ranked stocks are those most
suited to moving average trading. This needs to be investigated. It is also implicit that if the
maximally ranked stocks are those most suited to moving average trading, they are the most
likely to exhibit persistence in price direction. Consequently, the ranking scheme might be a
proxy for price impact. This can be tested by ranking each stock based on a measure of price
impact such as the Amihud Ratio (Amihud (2002)). Both ranking schemes need to be tested
using the trade reduction and positive autocorrelation rules. This should provide insight into
It is also necessary to investigate the stock selection problem along with a trading strategy. It
was concluded during the course of the thesis that the preferred strategy is to hold the capital
value of each trade constant so that each trade earns the mean return per trade by definition.
This breaks up serial dependence in the order of the trades and so losses are not compounded.
Assuming the trading rules are profitable, the strategy is one where in the limit as the number
of trades tends to infinity, E[Rt ] > 0 . The cash profit is then knR where k is the capital value
of each trade, n is the number of trades and R is the mean return per trade. The advantage is
that the strategy scales up and is massively parallel. Given unlimited capital, it is possible to
trade the whole of the investment universe using multiple instances of the same trading rule
parameterised in the moving average length. However, the problem is how to grow the scale
of the strategy given limited capital. This is a non-trivial problem and needs to be addressed.
134
As a test for market efficiency, the variable of interest is the breakeven costs. It is necessary
to develop a test to test for the difference in the breakeven costs between trading rules. For a
bootstrap test, this is equivalent to specifying the distribution under the null hypothesis of no
difference. The breakeven cost is also the preferred way to test the statistical significance of
the trading rules relative to the price crossover rule. It is also necessary to test the breakeven
costs after adjusting for risk. Risk adjustment needs to be integrated into the breakeven costs
test procedure.
A general issue that arises in response to the positive autocorrelation rule is whether positive
autocorrelation dominates the price. One way to test this is to extend the trade reduction rule
to include a test for positive autocorrelation and to test the difference between the resulting
trading rule and the positive autocorrelation rule. This is equivalent to testing the difference
between the price crossover and trade reduction rules where both trading rules include a test
for positive autocorrelation. If positive autocorrelation dominates the price in that it induces
consistently winning trades, the relationship between the price crossover and trade reduction
rules of Chapter 4 might break down. If so, potentially at least, there is evidence that positive
autocorrelation dominates the price. However, it is not clear what to expect in the event of a
losing trade. If the price continues to trend, since the price crossover rule exits trades earlier
than the trade reduction rule, the price crossover rule might perform better. The relationship
of Chapter 4 might reverse. This should provide insight into the preferred choice of trading
rule. There is also some similarity between this and testing for price impact.
A major contribution of this thesis is to establish a link between moving average trading and
market microstructure. There is evidence that the positive autocorrelation rule is picking up
asymmetric information. The positive autocorrelation rule is a method by which to peel back
135
the price and to identify events where there should be some change in the trading patterns in
the high frequency data. It is thought that there should be evidence of adverse selection and
price impact. This needs to be investigated. It might even be possible to recover the anatomy
of a profitable trading opportunity at the microstructure level. There is also evidence that the
price crossover and trade reduction rules are picking up changes in liquidity on the sell side
when shorting the FTSE Small Cap and FTSE Fledgling portfolios. In general, naked shorts
are the most difficult trades to execute consistently and successfully. Whether liquidity is a
significant factor in determining the success of a short trade needs to be investigated. If it is,
On a more speculative note, the structure of the positive autocorrelation rule is ideally suited
to optimisation using genetic algorithms (Goldberg (1989), Mitchell (1998)). This can also be
extended to include the trade reduction rule in the sense that the choice of trading rule is also
a parameter. In the best case, the trading rule and stock selection problem can be treated as a
joint optimisation problem that optimises the trading rule together with the optimal portfolio.
Alternatively, optimising the trading rules on their own should provide some insight into the
optimal parameters and hence into the type of trading rule most suited to testing for market
efficiency. There may also be some value in combining the positive autocorrelation rule with
time series forecasts (see, for example, Tsay (2010)). It is also worth noting that the bounded
moving average reduces to a fuzzy Takagi-Sugeno model (Takagi and Sugeno (1985)). This
may or may not provide a route in fuzzy modelling and associated technologies such as fuzzy
neural networks.
136
Appendix
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
3I Group 29-Mar-94 30-Jun-09
Admiral Group 22-Sep-04 30-Jun-09
Amec 22-Dec-82 30-Jun-09
Anglo American 31-Dec-90 30-Jun-09
Antofagasta 05-Jul-82 30-Jun-09
Associated British Foods 01-Jan-65 30-Jun-09
Astrazeneca 28-May-93 30-Jun-09
Autonomy Corporation 30-Oct-00 30-Jun-09
Aviva 01-Jan-65 30-Jun-09
BAE Systems 19-Feb-81 30-Jun-09
Balfour Beatty 01-Jan-65 30-Jun-09
Barclays 01-Jan-65 30-Jun-09
BG Group 05-Dec-86 30-Jun-09
BHP Billiton 25-Jul-97 30-Jun-09
BP 01-Jan-65 30-Jun-09
British Airways 10-Feb-87 30-Jun-09
British American Tobacco 01-Jan-65 30-Jun-09
British Land 01-Jan-65 30-Jun-09
British Sky Broadcasting 07-Dec-94 30-Jun-09
BT Group 30-Nov-84 30-Jun-09
Bunzl 01-Jan-65 30-Jun-09
Cable & Wireless 04-Nov-81 30-Jun-09
Cadbury 01-Jan-65 30-Jun-09
Cairn Energy 21-Dec-88 30-Jun-09
Capita Group 24-Apr-89 30-Jun-09
Carnival 20-Oct-00 30-Jun-09
Centrica 14-Feb-97 30-Jun-09
Cobham 01-Jan-65 30-Jun-09
Compass Group 01-Feb-01 30-Jun-09
Diageo 01-Jan-65 30-Jun-09
137
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Eurasian Natural Resources 06-Dec-07 30-Jun-09
Experian 06-Oct-06 30-Jun-09
Fresnillo 08-May-08 30-Jun-09
Friends Provident Group 06-Jul-01 30-Jun-09
G4S 10-Jun-96 30-Jun-09
Glaxosmithkline 01-Jan-65 30-Jun-09
Hammerson 01-Jan-65 30-Jun-09
Home Retail Group 01-Jan-65 30-Jun-09
HSBC Holdings 09-Jul-92 30-Jun-09
ICAP 16-Nov-98 30-Jun-09
Imperial Tobacco Group 30-Sep-96 30-Jun-09
Inmarsat 16-Jun-05 30-Jun-09
Intercontinental Hotels Group 28-Mar-03 30-Jun-09
International Power 11-Mar-91 30-Jun-09
Intertek Group 23-May-02 30-Jun-09
Invensys 29-Mar-72 30-Jun-09
Johnson Matthey 01-Jan-65 30-Jun-09
Kazakhmys 06-Oct-05 30-Jun-09
Kingfisher 24-Nov-82 30-Jun-09
Land Securities Group 01-Jan-65 30-Jun-09
Legal & General Group 01-Jan-65 30-Jun-09
Liberty International 29-Jul-92 30-Jun-09
Lloyds Banking Group 28-Dec-95 30-Jun-09
London Stock Exchange Group 21-Jul-00 30-Jun-09
Lonmin 01-Jan-65 30-Jun-09
Man Group 06-Oct-94 30-Jun-09
Marks & Spencer Group 01-Jan-65 30-Jun-09
Morrison (Wm) Supermarkets 01-Jan-69 30-Jun-09
National Grid 08-Dec-95 30-Jun-09
Next 01-Jan-65 30-Jun-09
Old Mutual 09-Jul-99 30-Jun-09
Pearson 20-Aug-69 30-Jun-09
Pennon Group 11-Dec-89 30-Jun-09
Petrofac 03-Oct-05 30-Jun-09
Prudential 01-Jan-65 30-Jun-09
Randgold Resources 30-Jun-97 30-Jun-09
Reckitt Benckiser Group 01-Jan-65 30-Jun-09
Reed Elsevier 01-Jan-65 30-Jun-09
Rexam 01-Jan-65 30-Jun-09
138
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Rio Tinto 01-Jan-65 30-Jun-09
Rolls-Royce Group 19-May-87 30-Jun-09
Royal Bank Of Scotland Group 01-Jan-65 30-Jun-09
Royal Dutch Shell B 01-Jan-65 30-Jun-09
Rsa Insurance Group 01-Jan-65 30-Jun-09
Sabmiller 26-Feb-99 30-Jun-09
Sage Group 13-Dec-89 30-Jun-09
Sainsbury (J) 18-Jul-73 30-Jun-09
Schroders 01-Jan-65 30-Jun-09
Schroders N-V 08-May-86 30-Jun-09
Scottish & Southern Energy 17-Jun-91 30-Jun-09
Serco Group 11-May-88 30-Jun-09
Severn Trent 11-Dec-89 30-Jun-09
Shire 14-Feb-96 30-Jun-09
Smith & Nephew 01-Jan-65 30-Jun-09
Smiths Group 01-Jan-65 30-Jun-09
Standard Chartered 01-Jan-65 30-Jun-09
Standard Life 07-Jul-06 30-Jun-09
Tesco 01-Jan-65 30-Jun-09
Thomas Cook Group 30-Dec-04 30-Jun-09
Thomson Reuters 01-Jun-84 30-Jun-09
Tui Travel 06-Jan-82 30-Jun-09
Tullow Oil 04-Oct-89 30-Jun-09
Unilever 01-Jan-65 30-Jun-09
United Utilities Group 11-Dec-89 30-Jun-09
Vedanta Resources 04-Dec-03 30-Jun-09
Vodafone Group 25-Oct-88 30-Jun-09
Wolseley 01-Jan-65 30-Jun-09
Wpp 14-Apr-71 30-Jun-09
Xstrata 19-Mar-02 30-Jun-09
Period 1 refers to the test period 01-Jan-1965 to 31-Dec-1979. Period 2 refers to the test period 01-Jan-1980 to
31-Dec-1994. Period 3 refers to the test period 01-Jan-1995 to 30-Jun-2009. Period All refers to the test period
01-Jan-1965 to 30-Jun-2009. A symbol indicates inclusion in the test period.
139
A.2 FTSE 250 Portfolio (Chapters 3 and 4)
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
3I Infrastructure 26-Feb-07 30-Jun-09
888 Holdings 28-Sep-05 30-Jun-09
Aberdeen Asset Management 27-Mar-91 30-Jun-09
Aegis Group 21-Jan-83 30-Jun-09
Aggreko 26-Sep-97 30-Jun-09
Amlin 25-Nov-93 30-Jun-09
Aquarius Platinum 04-Oct-99 30-Jun-09
Arm Holdings 23-Apr-98 30-Jun-09
Arriva 01-Jan-69 30-Jun-09
Ashmore Group 11-Oct-06 30-Jun-09
Ashtead Group 04-Dec-86 30-Jun-09
Atkins (WS) 24-Jul-96 30-Jun-09
Aveva Group 04-Dec-96 30-Jun-09
Babcock International Group 11-Aug-89 30-Jun-09
Barr (AG) 27-Aug-69 30-Jun-09
Barratt Developments 01-Jan-69 30-Jun-09
BBA Aviation 01-Jan-65 30-Jun-09
Beazley 11-Nov-02 30-Jun-09
Bellway 16-May-79 30-Jun-09
Berkeley Group Holdings 18-Jul-84 30-Jun-09
Big Yellow Group 05-May-00 30-Jun-09
Bluebay Asset Management 16-Nov-06 30-Jun-09
Bodycote 12-Jan-72 30-Jun-09
Bovis Homes Group 08-Dec-97 30-Jun-09
Brewin Dolphin Holdings 08-Jun-94 30-Jun-09
Brit Insurance Holdings 27-Oct-95 30-Jun-09
Britvic 08-Dec-05 30-Jun-09
Brown (N) Group 01-Apr-70 30-Jun-09
BSS Group 01-Jan-65 30-Jun-09
BTG 05-Jul-95 30-Jun-09
Burberry Group 11-Jul-02 30-Jun-09
Carillion 29-Jul-99 30-Jun-09
Carpetright 22-Jun-93 30-Jun-09
Carphone Warehouse Group 13-Jul-00 30-Jun-09
Catlin Group 31-Mar-04 30-Jun-09
Chaucer Holdings 23-Nov-93 30-Jun-09
Chemring Group 01-Jan-69 30-Jun-09
140
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Chloride Group 01-Jan-65 30-Jun-09
Close Brothers Group 01-Jan-65 30-Jun-09
Computacenter 20-May-98 30-Jun-09
Connaught 27-Nov-98 30-Jun-09
Cookson Group 01-Jan-65 30-Jun-09
Cranswick 04-Dec-85 30-Jun-09
Croda International 01-Jan-65 30-Jun-09
CSR 25-Feb-04 30-Jun-09
Daejan Holdings 01-Jan-65 30-Jun-09
Daily Mail & General Trust 01-Jan-65 30-Jun-09
Dairy Crest Group 27-Aug-96 30-Jun-09
Dana Petroleum 05-Jun-95 30-Jun-09
Davis Service Group 01-Jan-65 30-Jun-09
De La Rue 01-Jan-65 30-Jun-09
Debenhams 03-May-06 30-Jun-09
Dechra Pharmaceuticals 20-Sep-00 30-Jun-09
Derwent London 10-Aug-84 30-Jun-09
Dignity 01-Apr-04 30-Jun-09
Dimension Data Holdings 18-Jul-00 30-Jun-09
Domino Printing Sciences 01-May-85 30-Jun-09
Dominos Pizza 23-Nov-99 30-Jun-09
Drax Group 14-Dec-05 30-Jun-09
DSG International 01-Jan-65 30-Jun-09
Dunelm Group 18-Oct-06 30-Jun-09
Eaga 06-Jun-07 30-Jun-09
Easyjet 14-Nov-00 30-Jun-09
Electrocomponents 05-Jul-67 30-Jun-09
Emerald Energy 15-Nov-93 30-Jun-09
Enterprise Inns 03-Nov-95 30-Jun-09
Euromoney Institutional Investors 23-Jun-86 30-Jun-09
Evolution Group 25-Jun-97 30-Jun-09
F&C Asset Management 02-Sep-83 30-Jun-09
Ferrexpo 14-Jun-07 30-Jun-09
Fidessa Group 06-Jun-97 30-Jun-09
Filtrona 03-Jun-05 30-Jun-09
First Group 15-Jun-95 30-Jun-09
Fisher (James) & Sons 01-Jan-65 30-Jun-09
Forth Ports 20-Mar-92 30-Jun-09
Galiform 16-Jul-92 30-Jun-09
141
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Game Group 05-Jul-85 30-Jun-09
Genus 05-Jul-00 30-Jun-09
GKN 01-Jan-65 30-Jun-09
Go-Ahead Group 06-May-94 30-Jun-09
Great Portland Estates 01-Jan-65 30-Jun-09
Greene King 01-Jan-65 30-Jun-09
Greggs 03-May-84 30-Jun-09
Halfords Group 02-Jun-04 30-Jun-09
Halma 19-Jan-72 30-Jun-09
Hargreaves Lansdown 14-May-07 30-Jun-09
Hays 25-Oct-89 30-Jun-09
Helical Bar 01-Jan-69 30-Jun-09
Henderson Group 22-Dec-03 30-Jun-09
Hikma Pharmaceuticals 31-Oct-05 30-Jun-09
Hiscox 30-Jun-95 30-Jun-09
HMV Group 08-May-02 30-Jun-09
Hochschild Mining 02-Nov-06 30-Jun-09
Homeserve 26-Nov-91 30-Jun-09
Hunting 29-Jul-70 30-Jun-09
IG Group Holdings 27-Apr-05 30-Jun-09
IMI 06-Apr-66 30-Jun-09
Inchcape 01-Jan-65 30-Jun-09
Informa 16-Apr-98 30-Jun-09
Intermediate Capital Group 31-May-94 30-Jun-09
Interserve 01-Jan-65 30-Jun-09
Investec 19-Jul-02 30-Jun-09
ITV 01-Jan-65 30-Jun-09
Jardine Lloyd Thompson Group 16-Oct-87 30-Jun-09
JKX Oil & Gas 11-Jul-95 30-Jun-09
Keller 04-May-94 30-Jun-09
Kesa Electricals 04-Jul-03 30-Jun-09
Kier Group 11-Dec-96 30-Jun-09
Ladbrokes 04-Oct-67 30-Jun-09
Lancashire Holdings 12-Dec-05 30-Jun-09
Logica 02-Nov-83 30-Jun-09
Marstons 01-Jan-65 30-Jun-09
McBride 06-Jul-95 30-Jun-09
Meggitt 01-Jan-69 30-Jun-09
Melrose 27-Oct-03 30-Jun-09
142
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Melrose Resources 17-Dec-99 30-Jun-09
Michael Page International 27-Mar-01 30-Jun-09
Micro Focus International 11-May-05 30-Jun-09
Millennium & Copthorne Hotels 24-Apr-96 30-Jun-09
Misys 11-Mar-87 30-Jun-09
Mitchells & Butlers 28-Mar-03 30-Jun-09
Mitie Group 01-Jan-69 30-Jun-09
Mondi 29-Jun-07 30-Jun-09
Moneysupermarket Dot Com 25-Jul-07 30-Jun-09
Morgan Crucible 01-Jan-65 30-Jun-09
Morgan Sindall 01-Jan-69 30-Jun-09
Mothercare 07-Jan-86 30-Jun-09
Mouchel Group 27-Jun-02 30-Jun-09
National Express Group 09-Dec-92 30-Jun-09
Northern Foods 01-Jan-65 30-Jun-09
Northumbrian Water Group 22-May-03 30-Jun-09
Novae Group 24-Nov-93 30-Jun-09
Pace 26-Jun-96 30-Jun-09
Partygaming 24-Jun-05 30-Jun-09
Paypoint 20-Sep-04 30-Jun-09
Persimmon 26-Apr-85 30-Jun-09
Peter Hambro Mining 26-Apr-02 30-Jun-09
Premier Farnell 01-Jun-66 30-Jun-09
Premier Foods 19-Jul-04 30-Jun-09
Premier Oil 21-Feb-73 30-Jun-09
Provident Financial 01-Jan-65 30-Jun-09
Punch Taverns 21-May-02 30-Jun-09
PV Crystalox Solar 05-Jun-07 30-Jun-09
PZ Cussons 01-Jan-65 30-Jun-09
Qinetiq Group 09-Feb-06 30-Jun-09
Rank Group 01-Jan-65 30-Jun-09
Rathbone Brothers 24-Sep-84 30-Jun-09
Redrow 16-May-94 30-Jun-09
Regus 16-Oct-00 30-Jun-09
Renishaw 02-Jun-83 30-Jun-09
Rentokil Initial 19-Mar-69 30-Jun-09
Restaurant Group 01-Jan-65 30-Jun-09
Rightmove 09-Mar-06 30-Jun-09
Robert Wiseman Dairies 25-Mar-94 30-Jun-09
143
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Rotork 24-Jul-68 30-Jun-09
RPS Group 28-Jul-87 30-Jun-09
Salamander Energy 29-Nov-06 30-Jun-09
Savills 20-Jul-88 30-Jun-09
SDL 06-Dec-99 30-Jun-09
Segro 01-Jan-65 30-Jun-09
Shaftesbury 19-Oct-87 30-Jun-09
Shanks Group 26-Feb-88 30-Jun-09
SIG 17-May-89 30-Jun-09
Smith (DS) 01-Jan-69 30-Jun-09
Soco International 28-May-97 30-Jun-09
Spectris 28-Nov-88 30-Jun-09
Spirax-Sarco Engineering 01-Jan-65 30-Jun-09
Spirent Communications 01-Jan-65 30-Jun-09
Sports Direct International 26-Feb-07 30-Jun-09
SSL International 13-Jul-90 30-Jun-09
St James Place 23-Aug-96 30-Jun-09
Stagecoach Group 26-Apr-93 30-Jun-09
Sthree 10-Nov-05 30-Jun-09
Stobart Group 26-Feb-02 30-Jun-09
Synergy Health 17-Aug-01 30-Jun-09
Tate & Lyle 01-Jan-65 30-Jun-09
Taylor Wimpey 01-Jan-65 30-Jun-09
Telecity Group 23-Oct-07 30-Jun-09
Tomkins 01-Jan-69 30-Jun-09
Travis Perkins 18-Sep-86 30-Jun-09
Tullett Prebon 13-Dec-06 30-Jun-09
UK Commercial Property Trust 29-Aug-06 30-Jun-09
Ultra Electronics Holdings 02-Oct-96 30-Jun-09
United Business Media 01-Jan-65 30-Jun-09
Vectura Group 01-Jul-04 30-Jun-09
Venture Production 18-Mar-02 30-Jun-09
Victrex 20-Dec-95 30-Jun-09
VT Group 16-Mar-88 30-Jun-09
Weir Group 01-Jan-65 30-Jun-09
Wellstream Holdings 25-Apr-07 30-Jun-09
Wetherspoon (JD) 29-Oct-92 30-Jun-09
WH Smith 01-Jan-65 30-Jun-09
Whitbread 01-Jan-65 30-Jun-09
144
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
William Hill 14-Jun-02 30-Jun-09
Wood Group (John) 28-May-02 30-Jun-09
Xchanging 24-Apr-07 30-Jun-09
Yell Group 09-Jul-03 30-Jun-09
Period 1 refers to the test period 01-Jan-1965 to 31-Dec-1979. Period 2 refers to the test period 01-Jan-1980 to
31-Dec-1994. Period 3 refers to the test period 01-Jan-1995 to 30-Jun-2009. Period All refers to the test period
01-Jan-1965 to 30-Jun-2009. A symbol indicates inclusion in the test period.
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
AGA Rangemaster Group 01-Jan-65 30-Jun-09
Air Partner 03-Nov-89 30-Jun-09
Alpha Pyrenees 28-Nov-05 30-Jun-09
Alphameric 03-Aug-84 30-Jun-09
Alterian 19-Jul-00 30-Jun-09
Anglo Eastern Plantations 17-May-85 30-Jun-09
Anglo Pacific Group 29-Mar-84 30-Jun-09
Anite 27-Jun-73 30-Jun-09
Antisoma 15-Dec-99 30-Jun-09
Arena Leisure 15-Nov-72 30-Jun-09
Ark Therapeutics Group 02-Mar-04 30-Jun-09
Ashley (Laura) Holdings 04-Dec-85 30-Jun-09
Assura Group 29-Oct-03 30-Jun-09
Avis Europe 03-Apr-97 30-Jun-09
Axis-Shield 22-Sep-93 30-Jun-09
Bloomsbury Publishing 22-Jun-94 30-Jun-09
Braemar Shipping Services 26-Nov-97 30-Jun-09
Brammer (H) 01-Jan-65 30-Jun-09
British Polythene Industries 07-Apr-65 30-Jun-09
Brixton 01-Jan-65 30-Jun-09
Business Post Group 02-Jul-93 30-Jun-09
Camellia 01-Jan-69 30-Jun-09
Capital & Regional 12-Dec-86 30-Jun-09
Care UK 16-Jul-86 30-Jun-09
Castings 01-Jan-69 30-Jun-09
Centaur Media 09-Mar-04 30-Jun-09
145
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Charles Stanley Group 01-Jan-69 30-Jun-09
Charles Taylor Consulting 09-Oct-96 30-Jun-09
Chesnara 19-May-04 30-Jun-09
Chime Communications 05-Jan-90 30-Jun-09
Chrysalis 05-Mar-69 30-Jun-09
Cineworld Group 26-Apr-07 30-Jun-09
Clarke (T) 01-Jan-69 30-Jun-09
Clarkson 27-Jun-86 30-Jun-09
CLS Holdings 26-May-94 30-Jun-09
Collins Stewart 23-Oct-00 30-Jun-09
Communisis 24-Jun-94 30-Jun-09
Consort Medical 24-Nov-82 30-Jun-09
Costain Group 01-Jan-65 30-Jun-09
Delta 01-Jan-65 30-Jun-09
Development Securities 07-May-85 30-Jun-09
Devro 29-Jun-93 30-Jun-09
Diploma 01-Jan-69 30-Jun-09
DTZ Holdings 29-Jul-87 30-Jun-09
E2V Technologies 19-Jul-04 30-Jun-09
Elementis 01-Jan-65 30-Jun-09
Emblaze 17-Oct-96 30-Jun-09
Fenner 01-Jan-65 30-Jun-09
Fiberweb 16-Nov-06 30-Jun-09
Findel 01-Jan-65 30-Jun-09
Fortune Oil 27-Sep-89 30-Jun-09
French Connection Group 07-Nov-83 30-Jun-09
Fuller Smith & Turner 05-Nov-80 30-Jun-09
Future 17-Jun-99 30-Jun-09
Galliford Try 01-Jan-69 30-Jun-09
Gem Diamonds 13-Feb-07 30-Jun-09
Gleeson (MJ) Group 01-Jan-65 30-Jun-09
Goldenport Holdings 31-Mar-06 30-Jun-09
Goldshield Group 11-Jun-98 30-Jun-09
Goodwin 01-Jan-69 30-Jun-09
Grainger 02-Mar-83 30-Jun-09
Hampson Industries 01-Jan-69 30-Jun-09
Hardy Oil & Gas 06-Jun-05 30-Jun-09
Hardy Underwriting Bermuda 27-Dec-96 30-Jun-09
Headlam Group 01-Jan-69 30-Jun-09
146
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Helphire Group 21-Mar-97 30-Jun-09
Hill & Smith Holdings 02-Apr-69 30-Jun-09
Hilton Food Group 16-May-07 30-Jun-09
Hogg Robinson Group 06-Oct-06 30-Jun-09
Holidaybreak 09-Jul-91 30-Jun-09
Hornby 17-Dec-86 30-Jun-09
Hyder Consulting 09-Aug-72 30-Jun-09
Imagination Technologies Group 05-Jul-94 30-Jun-09
Innovation Group 01-Jun-00 30-Jun-09
Intec Telecom Systems 09-Jun-00 30-Jun-09
International Ferro Metals 29-Sep-05 30-Jun-09
International Personal Finance 13-Jul-07 30-Jun-09
IP Group 14-Oct-03 30-Jun-09
IRP Property Investments 28-Apr-04 30-Jun-09
Isis Property Trust 26-Sep-03 30-Jun-09
ITE Group 10-Jan-94 30-Jun-09
JD Sports Fashion 21-Oct-96 30-Jun-09
JJB Sports 17-Nov-94 30-Jun-09
Johnston Press 28-Apr-88 30-Jun-09
Kcom Group 09-Jul-99 30-Jun-09
Kewill 16-Sep-85 30-Jun-09
Kofax 02-Apr-96 30-Jun-09
Laird 01-Jan-65 30-Jun-09
Lamprell 10-Oct-06 30-Jun-09
Lavendon Group 09-Oct-96 30-Jun-09
Lookers 27-Jun-73 30-Jun-09
Low & Bonar 01-Jan-65 30-Jun-09
LSL Property Services 15-Nov-06 30-Jun-09
Luminar Group Holdings 17-May-96 30-Jun-09
Management Consulting Group 16-Feb-87 30-Jun-09
Marshalls 01-Jan-69 30-Jun-09
McKay Securities 01-Jan-69 30-Jun-09
Mears Group 03-Oct-96 30-Jun-09
Mecom Group 22-Mar-05 30-Jun-09
Medicx Fund 27-Oct-06 30-Jun-09
Menzies (John) 01-Jan-65 30-Jun-09
Mucklow (A & J) Group 01-Jan-65 30-Jun-09
MWB Group Holdings 02-Apr-08 30-Jun-09
NCC Group 08-Jul-04 30-Jun-09
147
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Northgate 01-Jan-69 30-Jun-09
Optos 09-Feb-06 30-Jun-09
Oxford Biomedica 12-Dec-96 30-Jun-09
Oxford Instruments 18-Oct-83 30-Jun-09
Paragon Group Of Companies 25-Sep-85 30-Jun-09
Pendragon 10-Nov-89 30-Jun-09
Phoenix IT Group 10-Nov-04 30-Jun-09
Photo-Me International 01-Jan-65 30-Jun-09
Primary Health Properties 18-Mar-96 30-Jun-09
Prostrakan Group 13-Jun-05 30-Jun-09
Psion 11-Mar-88 30-Jun-09
Quintain Estates & Development 22-Jul-96 30-Jun-09
REA Holdings 01-Jan-69 30-Jun-09
Real Estate Opportunities 29-May-01 30-Jun-09
Renovo Group 06-Apr-06 30-Jun-09
Rensburg Sheppards 06-Apr-88 30-Jun-09
Ricardo 01-Jan-69 30-Jun-09
RM 13-Dec-94 30-Jun-09
Robert Walters 05-Jul-00 30-Jun-09
Rok 19-Aug-81 30-Jun-09
RPC Group 27-May-93 30-Jun-09
Safestore Holdings 08-Mar-07 30-Jun-09
Scott Wilson Group 14-Mar-06 30-Jun-09
Senior 01-Jan-65 30-Jun-09
Sepura 30-Jul-07 30-Jun-09
Severfield-Rowen 01-Jul-88 30-Jun-09
Smiths News 29-Aug-06 30-Jun-09
Southern Cross Healthcare 06-Jul-06 30-Jun-09
Speedy Hire 21-Jun-89 30-Jun-09
Spice 25-Aug-04 30-Jun-09
Sportech 28-Feb-86 30-Jun-09
Spring Group 01-Jan-69 30-Jun-09
St Ives 02-Oct-85 30-Jun-09
St Modwen Properties 25-Apr-86 30-Jun-09
STV Group 01-Jan-69 30-Jun-09
Ted Baker 23-Jul-97 30-Jun-09
Telecom Plus 16-Oct-98 30-Jun-09
Thorntons 23-May-88 30-Jun-09
Topps Tiles 30-May-97 30-Jun-09
148
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Town Centre Securities 01-Jan-69 30-Jun-09
Tribal Group 22-Feb-01 30-Jun-09
Trinity Mirror 01-Jan-65 30-Jun-09
TT Electronics 01-Jan-69 30-Jun-09
UK Coal 04-Jun-93 30-Jun-09
Umeco 07-Jul-89 30-Jun-09
Unite Group 31-May-99 30-Jun-09
UTV Media 19-Dec-86 30-Jun-09
Vitec Group 13-Dec-72 30-Jun-09
VP 11-Apr-73 30-Jun-09
Wilmington Group 05-Dec-95 30-Jun-09
Wincanton 17-May-01 30-Jun-09
Wolfson Microelectronics 15-Oct-03 30-Jun-09
Workspace Group 14-Dec-93 30-Jun-09
WSP Group 28-Sep-87 30-Jun-09
Yule Catto & Co 20-Oct-71 30-Jun-09
Period 1 refers to the test period 01-Jan-1965 to 31-Dec-1979. Period 2 refers to the test period 01-Jan-1980 to
31-Dec-1994. Period 3 refers to the test period 01-Jan-1995 to 30-Jun-2009. Period All refers to the test period
01-Jan-1965 to 30-Jun-2009. A symbol indicates inclusion in the test period.
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
4Imprint Group 01-Jan-65 30-Jun-09
600 Group 01-Jan-65 30-Jun-09
Abbeycrest 21-May-85 30-Jun-09
Acal 15-Jun-88 30-Jun-09
Accident Exchange Group 01-May-02 30-Jun-09
Advantage Property Income Trust 07-Feb-05 30-Jun-09
AEA Technology 25-Sep-96 30-Jun-09
Alexandra 30-Jan-85 30-Jun-09
Alexon Group 01-Jan-65 30-Jun-09
Alumasc Group 29-May-86 30-Jun-09
Anglesey Mining 03-Aug-88 30-Jun-09
Arc International 20-Sep-00 30-Jun-09
Associated British Engineering 01-Jan-65 30-Jun-09
Asterand 28-Jul-00 30-Jun-09
149
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Avon Rubber 01-Jan-65 30-Jun-09
Axa Property Trust 29-Mar-05 30-Jun-09
Beale 10-Mar-95 30-Jun-09
Berkeley Technology 08-Jan-85 30-Jun-09
Bisichi Mining 01-Jan-69 30-Jun-09
Blacks Leisure Group 28-Oct-70 30-Jun-09
Caffyns 01-Jan-65 30-Jun-09
Carclo 01-Jan-69 30-Jun-09
Cardiff Property 01-Jan-69 30-Jun-09
Carrs Milling Industries 24-May-72 30-Jun-09
Celsis International 05-Jul-93 30-Jun-09
City Of London Group 01-Jul-88 30-Jun-09
Clinton Cards 04-May-88 30-Jun-09
CML Microsystems 07-Feb-84 30-Jun-09
Coral Products 12-Apr-95 30-Jun-09
Corin Group 08-May-02 30-Jun-09
Cosalt 23-Jun-71 30-Jun-09
Creightons 05-Sep-86 30-Jun-09
Creston 01-Jan-69 30-Jun-09
Dawson Holdings 16-Jun-95 30-Jun-09
Dee Valley Group 16-Dec-94 30-Jun-09
Dialight 08-Nov-93 30-Jun-09
DRS Data & Research Services 04-May-94 30-Jun-09
Dyson Group 01-Jan-65 30-Jun-09
Electronic Data Processing 27-Sep-85 30-Jun-09
Filtronic 21-Oct-94 30-Jun-09
GB Group 28-May-93 30-Jun-09
Gresham Computing 28-Jun-84 30-Jun-09
Harvard International 30-Sep-87 30-Jun-09
Harvey Nash Group 02-Apr-97 30-Jun-09
Havelock Europa 30-Mar-84 30-Jun-09
Haynes Publishing Group 05-Dec-79 30-Jun-09
Heywood Williams Group 01-Jan-69 30-Jun-09
Highcroft Investments 17-Jun-70 30-Jun-09
Highway Capital 17-Mar-95 30-Jun-09
HR Owen 06-Feb-89 30-Jun-09
Jarvis 01-Jan-69 30-Jun-09
Jessops 28-Oct-04 30-Jun-09
Litho Supplies 19-Nov-93 30-Jun-09
150
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Local Shopping REIT (The) 26-Apr-07 30-Jun-09
London & Associated Properties 01-Jan-69 30-Jun-09
London Finance & Investment 01-Jan-69 30-Jun-09
Macfarlane Group 20-Jun-73 30-Jun-09
Mallet 19-Mar-87 30-Jun-09
Manganese Bronze Holdings 01-Jan-65 30-Jun-09
Microgen 14-Jan-83 30-Jun-09
Minerva 27-Nov-96 30-Jun-09
Molins 28-Jul-76 30-Jun-09
Morse 19-Mar-99 30-Jun-09
Moss Bros Group 01-Jan-65 30-Jun-09
MS International 01-Jan-69 30-Jun-09
Narborough Plantations 01-Jan-69 30-Jun-09
Nestor Healthcare Group 02-Dec-87 30-Jun-09
Network Technology 29-Jul-96 30-Jun-09
Norcros 13-Jul-07 30-Jun-09
North Midland Construction 01-Jan-69 30-Jun-09
Northamber 08-Jun-84 30-Jun-09
NXT 01-Jan-69 30-Jun-09
Office2Office 28-Jun-04 30-Jun-09
Parity Group 29-Jun-87 30-Jun-09
Parkwood Holdings 09-Dec-96 30-Jun-09
Phytopharm 24-Apr-96 30-Jun-09
Pochins 01-Jan-69 30-Jun-09
Porvair 04-May-88 30-Jun-09
Puricore 29-Jun-06 30-Jun-09
Queens Walk Investment 07-Dec-05 30-Jun-09
Raymarine 03-Dec-04 30-Jun-09
Renold 01-Jan-65 30-Jun-09
Ross Group 01-Jan-69 30-Jun-09
Rugby Estates Investment Trust 14-May-07 30-Jun-09
S&U 01-Jan-65 30-Jun-09
Sinclair Pharma 10-Dec-03 30-Jun-09
Skyepharma 23-Oct-87 30-Jun-09
Smart (J) & Co 01-Jan-69 30-Jun-09
Source Bioscience 01-Jan-69 30-Jun-09
Stanelco 14-Apr-88 30-Jun-09
Styles & Wood Group 06-Nov-06 30-Jun-09
Superglass Holdings 11-Jul-07 30-Jun-09
151
Stock Start Date End Date Period 1 Period 2 Period 3 Period All
Tex Holdings 01-Dec-71 30-Jun-09
Titon Holdings 01-Feb-88 30-Jun-09
Torotrak 24-Jul-98 30-Jun-09
Total Systems 30-Mar-88 30-Jun-09
Trafficmaster 30-Mar-94 30-Jun-09
Treatt 21-Jun-89 30-Jun-09
Triad Group 20-Mar-96 30-Jun-09
Trifast 15-Feb-94 30-Jun-09
Uniq 01-Jan-65 30-Jun-09
Vernalis 30-Jun-92 30-Jun-09
Victoria 02-Oct-68 30-Jun-09
Vislink 18-Apr-79 30-Jun-09
Volex Group 01-Jan-65 30-Jun-09
Walker Crips Group 21-Aug-96 30-Jun-09
Warner Estate Holdings 01-Jan-69 30-Jun-09
Waterman Group 23-May-88 30-Jun-09
White Young Green 28-Apr-86 30-Jun-09
Worthington Group 01-Jan-69 30-Jun-09
Xaar 16-Oct-97 30-Jun-09
Zotefoams 27-Feb-95 30-Jun-09
Period 1 refers to the test period 01-Jan-1965 to 31-Dec-1979. Period 2 refers to the test period 01-Jan-1980 to
31-Dec-1994. Period 3 refers to the test period 01-Jan-1995 to 30-Jun-2009. Period All refers to the test period
01-Jan-1965 to 30-Jun-2009. A symbol indicates inclusion in the test period.
A.5 Sharescope
Ionic House
http://www.sharescope.co.uk
152
A.6 Updata Technical Analyst
Updata
Hutchison House
5 Hester Road
http://www.updata.co.uk
153
A.8 FTSE 250 Portfolio (Chapter 5)
154
A.9 FTSE Small Cap Portfolio (Chapter 5)
155
A.10 FTSE Fledgling Portfolio (Chapter 5)
156
(
A.11 VP(25,1) = VP(25) RW e1 )
VP(25) is the variance ratio profile of the R1 (k) statistic VP(25) = [ R1 (1), R1 (2), , R1 (25)].
The first principal component (eigenvector) e1 and the mean profile RW are:
157
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Achelis, S.B. (2001) Technical analysis from A to Z, McGraw-Hill, New York, 2nd edition.
Alexander, S.S. (1961) Price movements in speculative markets: trends or random walks,
Alexander, S.S. (1964) Price movements in speculative markets: trends or random walks,
Amihud, Y. (2002) Illiquidity and stock returns: cross-section and time-series effects, Journal
Atanasova, C.V. and Hudson, R.S. (2010) Technical trading rules and calendar anomalies
Barber, B.M., Lee, Y-T., Liu, Y-J. and Odean, T. (2011) The cross-section of speculator skill:
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Belaire-Franch, J. and Opong, K. (2005) A variance ratio test of the behaviour of some FTSE
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Bessembinder, H. and Chan, K. (1998) Market efficiency and the returns to technical analysis,
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