-week high is a useful predictor of fu-Introduction 9
ture relative performance.” The indicator is an even more potent pre-dictor for Australian stocks.”°
¢ The head-and-shoulders chart pattern has limited forecasting powerwhen tested in an objective fashion in currencies. Better results canbe had with simple filter rules. The head-and-shoulders pattern, whentested objectively on stocks, does not provide useful information.””Traders who act on such signals would be equally served by followinga random signal.
¢ Trading volume statistics for stocks contain useful predictive infor-mation”® and improve the profitability of signals based on large pricechanges following a public announcement.”
¢ Computer-intensive data-modeling neural networks, genetic algo-rithms, and other statistical learning and artificial-intelligence meth-ods have found profitable patterns in technical indicators.”
WHO AM I TO CRITICIZE TA?
 
My interest in TA began in 1960 at the age of 15. During my high-schooland college years I followed a large stable of stocks using the Chartcraftpoint and figure method. I have used TA professionally since 1973, firstas a stock broker, then as managing partner of a small software com-pany, Raden Research Group Inc.—an early adopter of machine learningand data mining in financial market applications—and finally as a propri-etary equities trader for Spear, Leeds & Kellogg.*! In 1988, I earned theChartered Market Technician designation from the Market TechniciansAssociation. My personal TA library has over 300 books. I have pub-lished approximately a dozen articles and have spoken numerous timeson the subject. Currently I teach a graduate-level course in TA at theZicklin School of Business, Baruch College, City University of New York.I freely admit my previous writings and research do not meet EBTA stan-dards, in particular with regard to statistical significance and the data-mining bias.
My long-standing faith in TA began to erode in response to a verymediocre performance over a five-year period trading capital for Spear,Leeds and Kellogg. How could what I believed in so fervently not work?Was it me or something to do with TA in general? My academic training inphilosophy provided fertile grounds for my growing doubts. My concernscrystallized into full fledged skepticism as a result of reading two books:How We Know What Isn’t So by Thomas Gilovich and Why People BelieveWeird Things, by Michael Shermer. My conclusion: Technical analysts,10 INTRODUCTION
including myself, know a lot of stuff that isn’t so, and believe a lot ofweird things.
TECHNICAL ANALYSIS: ART, SCIENCE,OR SUPERSTITION?
 
There is a debate in the TA community: Is it an art or a science? The ques-tion has been framed incorrectly. It is more properly stated as: Should TA bebased on superstition or science? Framed this way the debate evaporates.
Some will say TA involves too much nuance and interpretation to ren-der its knowledge in the form of scientifically testable claims. To this I re-tort: TA that is not testable may sound like knowledge, but it is not. It issuperstition that belongs in the realm of astrology, numerology, and othernonscientific practices.
Creativity and inspiration play a crucial role in science. They will beimportant in EBTA as well. All scientific inquiries start with a hypothesis,a new idea or a new insight inspired by a mysterious mixture of priorknowledge, experience and a leap of intuition. Yet, good science balancescreativity with analytical rigor. The freedom to propose new ideas mustbe married to an unyielding discipline that eliminates ideas that proveworthless in the crucible of objective testing. Without this anchor to real-ity, people fall in love with their ideas, and magical thinking replaces crit-ical thought.
It is unlikely that TA will ever discover rules that predict with the pre-cision of the laws of physics. The inherent complexity and randomness offinancial markets and the impossibility of controlled experimentationpreclude such findings. However, predictive accuracy is not the definingrequirement of science. Rather, it is defined by an uncompromising open-ness to recognizing and eliminating wrong ideas.
I have four hopes for this book: First, that it will stimulate a dialogueamongst technical analysts that will ultimately put our field on a firmer in-tellectual foundation; second, that it will encourage further research alongthe lines advocated herein; third, that it will encourage consumers of TAto demand more “beef” from those who sell products and services basedupon TA; and fourth, that it will encourage TA practitioners, professionaland otherwise, to understand their crucial role in a human-machine part-nership that has the potential to accelerate the growth of legitimate TAknowledge.
No doubt some fellow practitioners of TA will be irritated by theseideas. This can be a good thing. An oyster irritated by a grain of sandsometimes yields a pearl. I invite my colleagues to expend theirIntroduction fi
energies adding to legitimate knowledge rather than defending theindefensible.
This book is organized in two sections. Part One establishes themethodological, philosophical, psychological, and statistical foundationsof EBTA. Part Two demonstrates one approach to EBTA: testing of 6,402binary buy/sell rules on the S&P 500 on 25 years of historical data. Therules are evaluated for statistical significance using tests designed to copewith the problem of data-mining bias. 
 
Methodological,Psychological,Philosophical,and Statistical
Foundations
 
  
Objective Rulesand TheirEvaluation
 
and a methodology for their rigorous evaluation. It defines an evalua-tion benchmark based on the profitability of a noninformative signal.It also establishes the need to detrend market data so that the perfor-mances of rules with different long/short position biases can be compared.
T his chapter introduces the notion of objective binary signaling rules
THE GREAT DIVIDE: OBJECTIVE VERSUS SUBJECTIVETECHNICAL ANALYSIS
 
Technical analysis (TA) divides into two broad categories: objective andsubjective. Subjective TA is comprised of analysis methods and patternsthat are not precisely defined. As a consequence, a conclusion derivedfrom a subjective method reflects the private interpretations of the analystapplying the method. This creates the possibility that two analysts apply-ing the same method to the same set of market data may arrive at entirelydifferent conclusions. Therefore, subjective methods are untestable, andclaims that they are effective are exempt from empirical challenge. This isfertile ground for myths to flourish.
In contrast, objective methods are clearly defined. When an objectiveanalysis method is applied to market data, its signals or predictions areunambiguous. This makes it possible to simulate the method on historicaldata and determine its precise level of performance. This is called backtesting. The back testing of an objective method is, therefore, a repeatable
516 METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONS
experiment which allows claims of profitability to be tested and possiblyrefuted with statistical evidence. This makes it possible to find out whichobjective methods are effective and which are not.
The acid test for distinguishing an objective from a subjective methodis the programmability criterion: A method is objective if and only if itcan be implemented as a computer program that produces unambigu-ous market positions (long,' short,” or neutral*). All methods that cannotbe reduced to such a program are, by default, subjective.
TA RULES
 
Objective TA methods are also referred to as mechanical trading rules ortrading systems. In this book, all objective TA methods are referred tosimply as rules.
A rule is a function that transforms one or more items of information,referred to as the rule’s input, into the rule’s output, which is a recom-mended market position (e.g., long, short, neutral). Input(s) consists ofone or more financial market time series. The rule is defined by one ormore mathematical and logical operators that convert the input time se-ries into a new time series that consists of the sequence of recommendedmarket position (long, short, out-of-the-market). The output is typicallyrepresented by a signed number (e.g., +1 or —1). This book adopts the con-vention of assigning positive values to indicate long positions and nega-tive values to indicate shorts position. The process by which a ruletransforms one or more input series into an output series is illustrated inFigure 1.1.
A rule is said to generate a signal when the value of the output serieschanges. A signal calls for a change in a previously recommended marketposition. For example a change in output from +1 to -1 would call forclosing a previously held long position and the initiation of a new shortposition. Output values need not be confined to {+1, —1}. A complex rule,whose output spans the range {+10, —-10}, is able to recommend positionsthat vary in size. For example, an output of +10 might indicate that 10 longpositions are warranted, such as long 10 contracts of copper. A change inthe output from +10 to +5 would call for a reduction in the long positionfrom 10 contracts to 5 (i.e., sell 5).
Binary Rules and Thresholds
The simplest rule is one that has a binary output. In other words, its out-put can assume only two values, for example +1 and -1. A binary ruleObjective Rules and Their Evaluation 17
INPUT OUTPUTMarket Time SeriesTime ofSeries Market PositionL________»Time
Mathematical wees pees
 
 
 
&Logical. +\—_____» : ! | TimeTime Operators heeeb OkeeeEtc.
FIGURE 1.1 TA rule transforms input time series into a time series of mar-ket position.
could also be designed to recommend long/neutral positions or short/neu-tral positions. All the rules considered in this book are binary long/short{+1,-l}.
An investment strategy based on a binary long/short rule is always ineither a long or short position in the market being traded. Rules of thistype are referred to as reversal rules because signals call for a reversalfrom long to short or short to long. Over time a reversal rule produces atime series of +1’s and -1’s that represent an alternating sequence of longand short positions.
The specific mathematical and logical operators that are used to de-fine rules can vary considerably. However, there are some commonthemes. One theme is the notion of a threshold, a critical level that distin-guishes the informative changes in the input time series from its irrelevantfluctuations. The premise is that the input time series is a mixture of infor-mation and noise. Thus the threshold acts as a filter.
Rules that employ thresholds generate signals when the time seriescrosses the threshold, either by the rising above it or falling beneath it.These critical events can be detected with logical operators called in-equalities such as greater-than (>) and less-than (<). For example, if the18 METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONS
time series is greater than the threshold, then rule output = +1, otherwiserule output = -1.
A threshold may be set at a fixed value or its value may vary over timeas a result of changes in the time series that is being analyzed. Variablethresholds are appropriate for time series that display trends, which arelarge long-lasting changes in the level of the series. Trends, which makefixed threshold rules impractical, are commonly seen in asset prices (e.g.,S&P 500 Index) and asset yields (AAA bond yield). The moving averageand the Alexander reversal filter, also known as the zigzag filter, are exam-ples of time series operators that are commonly used to define variablethresholds. The operators used in the rules discussed in this book are de-tailed in Chapter 8.
The moving-average-cross rule is an example of how a variablethreshold is used to generate signals on a time series that displays trends.This type of rule produces a signal when the time series crosses from oneside of its moving average to the other. For example;
If the time series is above its moving average, then the rule outputvalue = +1, otherwise the rule output value = -1.
This is illustrated in Figure 1.2.
Because it employs a single threshold, the signals generated by themoving-average-cross rule are, by definition, mutually exclusive. Given asingle threshold, there are only two possible conditions—the times seriesis either above or below! the threshold. The conditions are also exhaus-tive (no other possibilities).> Thus, it is impossible for the rule’s signals tobe in conflict.
PriceLevel Moving Time Series
 
Rule TimeOutput
+1 t:
|
FIGURE 1.2 Moving-average-cross rule.
    
 Objective Rules and Their Evaluation 19
Rules with fixed value thresholds are appropriate for market time se-ries that do not display trends. Such time series are said to be stationary.There is a strict mathematical definition of a stationary time series, buthere I am using the term in a looser sense to mean that a series has a rela-tively stable average value over time and has fluctuations that are con-fined to a roughly horizontal range. Technical analysis practitioners oftenrefer to these series as oscillators.
Time series that display trends can be detrended. In other words, theycan be transformed into a stationary series. Detrending, which is de-scribed in greater detail in Chapter 8, frequently involves taking differ-ences or ratios. For example the ratio of a time series to its movingaverage will produce a stationary version of the original time series. Oncedetrended, the series will be seen to fluctuate within a relatively well-defined horizontal range around a relatively stable mean value. Once thetime series has been made stationary, fixed threshold rules can be em-ployed. An example of a fixed threshold rule using a threshold of value of75 is illustrated in Figure 1.3. The rule has an output a value of +1 whenthe series is greater than the threshold and a value of —1 at other times.
Binary Rules from Multiple Thresholds
As pointed out earlier, binary rules are derived, quite naturally, from a sin-gle threshold because the threshold defines two mutually exclusive andexhaustive conditions: the time series is either above or below threshold.However, binary rules can also be derived using multiple thresholds, butemploying more than one threshold creates the possibility that the input
100755025
 
Rule + IOutput | a
 
  
FIGURE 1.3 Rule with a single fixed threshold.20 METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONS
time series can assume more than two conditions. Consequently, multiplethreshold rules require a more sophisticated logical operator than the sim-ple inequality operator (greater-than or less-than), which suffices for sin-gle threshold rules.
When there are two or more thresholds, there are more than two pos-sible conditions. For example, with two thresholds, an upper and lower,there are three possible conditions for the input time series. It can beabove the upper, below the lower, or between the two thresholds. To cre-ate a binary rule in this situation, the rule is defined in terms of two mutu-ally exclusive events. An event is defined by the time series crossing aparticular threshold in a particular direction. Thus, one event triggers oneof the rule’s output values, which is maintained until a second event,which is mutually exclusive of the first, triggers the other output value.For example, an upward crossing of the upper threshold triggers a +1, anda downward crossing of the lower threshold triggers a -1.
A logical operator that implements this type of rule is referred to as aSlip-flop. The name stems from the fact that the rule’s output value flipsone way, upon the occurrence of one event, and then flops the other way,upon the occurrence of the second event. Flip-flop logic can be used witheither variable or fixed threshold rules. An example of a rule based on twovariable thresholds is the moving average band rule. See Figure 1.4. Here,the moving average is surrounded by an upper and lower band. The bandsmay be a fixed percentage above and below the moving average, or the de-viation of the bands may vary based on of the recent volatility of thetimes, as is the case with the Bollinger Band.° An output value of +1 is trig-gered by an upward piercing of the upper threshold. This value is retained
PricePrice
Upper Band
Moving Average
 
 
FIGURE 1.4 Moving average bands rule.Objective Rules and Their Evaluation 21
until the lower threshold is penetrated in the downward direction, causingthe output value to change to -1.
Obviously, there are many other possibilities. The intent here hasbeen to illustrate some of the ways that input time series can be trans-formed into a time series of recommended market positions.
Hayes‘ adds another dimension to threshold rules with directionalmodes. He applies multiple thresholds to a stationary time series such as adiffusion® indicator. At a given point in time, the indicator’s mode is de-fined by the zone it occupies and its recent direction of change (e.g., up ordown over the most recent five weeks). Each zone is defined by an upperand lower threshold (e.g., 40 and 60). Hayes applies this to a proprietarydiffusion indicator called Big Mo. With two thresholds and two possibledirectional modes (up/down), six mutually exclusive conditions are de-fined. A binary rule could be derived from such an analysis by assigningone output value (e.g., +1) to one of the six conditions, and then assigningthe other output value (i.e., -1) to the other five possibilities. Hayes as-serts that one of the modes, when the diffusion indicator is above 60 andits direction is upward, is associated with stock market returns (ValueLine Composite Index) of 50 percent per annum. This condition has oc-curred about 20 percent of the time between 1966 and 2000. However,when the diffusion indicator is > 60, and its recent change is negative, themarket's annualized return is zero. This condition has occurred about 16percent of the time.”
TRADITIONAL RULES AND INVERSE RULES
 
Part Two of this book is a case study that evaluates the profitability of ap-proximately 6,400 binary long/short rules applied to the S&P 500 Index.Many of the rules generate market positions that are consistent with tradi-tional principles of technical analysis. For example, under traditional TAprinciples, a moving-average-cross rule is interpreted to be bullish (outputvalue +1) when the analyzed time series is above its moving average, andbearish (output value of —-1) when it is below the moving average. I referto these as traditional TA rules.
Given that the veracity of traditional TA maybe questionable, it is de-sirable to test rules that are contrary to the traditional interpretation. Inother words, it is entirely possible that patterns that are traditionally as-sumed to predict rising prices may actually be predictive of fallingprices. Alternatively, it is possible that neither configuration has any pre-dictive value.
This can be accomplished by creating an additional set of rules whose22 METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONSPrice”
MovingAverage
 
 
Traditional Ru
 
  
   
 
 
 
 
 
 
 
Inverse Rule
FIGURE 1.5 Traditional rules and inverse rules.
output is simply the opposite of a traditional TA rule. I refer to these asinverse rules. This is illustrated in Figure 1.5. The inverse of the moving-average-cross rule would output a value of —1 when the input time seriesis above its moving average, and +1 when the series is below its movingaverage.
There is yet another reason to consider inverse rules. Many of therules tested in Part Two utilize input series other than the S&P 500, for ex-ample the yield differential between BAA and AAA corporate bonds. It isnot obvious how this series should be interpreted to generate signals.Therefore, both up trends and down trends in the yield differential wereconsidered as possible buy signals. The details of these rules are taken upin Chapter 8.
THE USE OF BENCHMARKS IN RULE EVALUATION
 
In many fields, performance is a relative matter. That is to say, it is perfor-mance relative to a benchmark that is informative rather than an absolutelevel of performance. In track and field, competitors in the shot-put arecompared to a benchmark defined as best distance of that day or the bestever recorded in the state or world. To say that someone put the shot 43feet does not reveal the quality of performance, however if the best prioreffort had been 23 feet, 43 feet is a significant accomplishment!
This pertains to rule evaluation. Performance figures are only infor-mative when they are compared to a relevant benchmark. The isolatedObjective Rules and Their Evaluation 23
fact that a rule earned a 10 percent rate of return in a back test is mean-ingless. If many other rules earned over 30 percent on the same data, 10percent would indicate inferiority, whereas if all other rules were barelyprofitable, 10 percent might indicate superiority.
What then is an appropriate benchmark for TA rule performance?What standard must a rule beat to be considered good? There are a num-ber of reasonable standards. This book defines that standard as the per-formance of a rule with no predictive power (i.e., a randomly generatedsignal). This is consistent with scientific practice in other fields. In medi-cine, a new drug must convincingly outperform a placebo (sugar pill) tobe considered useful. Of course, rational investors might reasonablychoose a higher standard of performance but not a lesser one. Some otherbenchmarks that could make sense would be the riskless rate of return,the return of a buy-and-hold strategy, or the rate of return of the rule cur-rently being used.
In fact, to be considered good, it is not sufficient for a rule to simplybeat the benchmark. It must beat it by a wide enough margin to excludethe possibility that its victory was merely due to chance (good luck). It isentirely possible for a rule with no predictive power to beat its benchmarkin a given sample of data by sheer luck. The margin of victory that is suffi-cient to exclude luck as a likely explanation relates to the matter of statis-tical significance. This is taken up in Chapters 4, 5, and 6.
Having now established that the benchmark that we will use is the re-turn that could be earned by a rule with no predictive power, we now faceanother question: How much might a rule with no predictive power earn?At first blush, it might seem that a return of zero is a reasonable expecta-tion. However, this is only true under a specific and rather limited set ofconditions.
In fact, the expected return of a rule with no predictive power can bedramatically different than zero. This is so because the performance of arule can be profoundly affected by factors that have nothing to do with itspredictive power.
The Conjoint Effect of Position Bias andMarket Trend on Back-Test Performance
In reality, a rule’s back-tested performance is comprised of two indepen-dent components. One component is attributable to the rule’s predictivepower, if it has any. This is the component of interest. The second, and un-wanted, component of performance is the result of two factors that havenothing to do with the rule’s predictive power: (1) the rule’s long/short po-sition bias, and (2) the market’s net trend during the back-test period.This undesirable component of performance can dramatically influence24 METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONS
back-test results and make rule evaluation difficult. It can cause a rulewith no predictive power to generate a positive average return or it cancause a rule with genuine predictive power to produce a negative averagereturn. Unless this component of performance is removed, accurate ruleevaluation is impossible. Let’s consider the two factors that drive thiscomponent.
The first factor is a rule’s long/short position bias. This refers to theamount of time the rule spent in a +1 output state relative to the amountof time spent in a -1 output state during the back test. If either outputstate dominated during the back test, the rule is said to have a positionbias. For example, if more time was spent in long positions, the rule has along position bias.
The second factor is the market’s net trend or the average daily pricechange of the market during the period of the back test. If the market’s nettrend is other than zero, and the rule has a long or short position bias, therule’s performance will be impacted. In other words, the undesirable com-ponent of performance will distort back-test results either by adding to orsubtracting from the component of performance that is due to the rule’sactual predictive power. If, however, the market's net trend is zero or ifthe rule has no position bias, then the rule’s past profitability will bestrictly due to the rule’s predictive power (plus or minus random varia-tion). This is demonstrated mathematically later.
To clarify, imagine a TA rule that has a long position bias but that weknow has no predictive power. The signals of such a rule could be simu-lated by a roulette wheel. To create the long position bias, a majority ofthe wheel’s slots would be allocated to long positions (+1). Suppose thatone hundred slots are allocated as follows; 75 are +1 and 25 are -1. Eachday, over a period of historical data, the wheel is spun to determine if along or short position is to be held for that day. If the market’s averagedaily change during this period were greater than zero (i.e., net trend up-ward), the rule would have a positive expected rate of return even thoughthe signals contain no predictive information. The rule’s expected rate ofreturn can be computed using the formula used to calculate the expectedvalue of a random variable (discussed later).
Just as it is possible for a rule with no predictive power to produce apositive rate of return, it is just as possible for a rule with predictivepower to produce a negative rate of return. This can occur if a rule has aposition bias that is contrary to the market’s trend. The combined effect ofthe market's trend and the rule’s position bias may be sufficient to offsetany positive return attributable to the rule’s predictive power. From thepreceding discussion it should be clear that the component of perfor-mance due to the interaction of position bias with market trend must beeliminated if one is to develop a valid performance benchmark.Objective Rules and Their Evaluation 25
At first blush, it might seem as if a rule that has a long position biasduring a rising market trend is evidence of the rule’s predictive power.However, this is not necessarily so. The rule’s bullish bias could simply bedue to the way its long and short conditions are defined. If the rule’s longcondition is more easily satisfied than its short condition, all other thingsbeing equal, the rule will tend to hold long positions a greater proportionof the time than short positions. Such a rule would receive a performanceboost when back tested over historical data with a rising market trend.Conversely, a rule whose short condition is more easily satisfied than itslong condition would be biased toward short positions and it would get aperformance boost if simulated during a downward trending market.
The reader may be wondering how the definition of a rule can inducea bias toward either long or short positions. This warrants some explana-tion. Recall that binary reversal rules, the type tested in this book, are al-ways in either a long or short position. Given this, if a rule’s long (+1)condition is relatively easy to satisfy, then it follows that its short condi-tion (-1) must be relatively difficult to satisfy. In other words, the condi-tion required for the -1 output state is more restrictive, making it likelythat, over time, the rule will spend more time long than short. It is just aspossible to formulate rules where the long condition is more restrictivethan the short condition. All other things being equal, such a rule wouldrecommend short positions more frequently than long. It would be con-trary to our purpose to allow the assessment of a rule’s predictive powerto be impacted by the relative strictness or laxity of the way in which itslong and short conditions are defined.
To illustrate, consider the following rule, which has a highly restric-tive short condition and, therefore, a relatively lax long condition. Therule, which generates positions in the S&P 500 index, is based on the DowJones Transportation Average.!° Assume that a moving average withbands set at +3 percent and -3 percent is applied to the DJTA. The rule isto be short the S&P 500 while the DJTA is below the lower band, by defin-ition a relatively rare condition, and long at all other times. See Figure 1.6.Clearly, such a rule would benefit if the S&P were in an uptrend over theback-test period.
Now let’s consider the back test of two binary reversal rules whichare referred to as rule 1 and rule 2. They are tested on S&P 500 data overthe period January 1, 1976 through December 2004. During this period ofapproximately 7,000 days, the S&P 500 had an average daily return of+0.035 percent per day compounded, or +9.21897 percent annualized. As-sume that rule 1 was in a long state 90 percent of the time and rule 2 wasin a long state 60 percent of the time. Also, suppose that neither rule haspredictive power—as if their output values were determined by a roulettewheel with 100 slots. The output for rule 1 is based on a roulette wheel26 METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONS
Price
Moving Average
 
 
 
FIGURE 1.6 Rule with restrictive short condition and long position bias.
with 90 slots assigned a value of +1 and the remaining 10 assigned a valueof -1. The output for rule 2 is based on a wheel with 60 slots assigned a +1value and 40 a value of -1. By the Law of Large Numbers,!' it is reasonableto expect that over the 7,000 days, rule 1 will be long very close to 90 per-cent of the time and rule 2 will be long approximately 60 percent of thetime. Although the rules have different long/short biases, they have equalpredictive power—none. However, their expected rates of return will bequite different over this segment of market history.
The expected return of a rule depends upon three quantities; (1) theproportion of time the rule spent in long positions, (2) the proportion oftime spent in short positions (1 minus the proportion of time long) and (3)the market’s average daily price change during the historical test period.The expected return (ER) is given by the following equation.
Expected ReturnER = [p(L) x ADC] — [p(S) » ADC]
Wherep(L) - probability of long position (proportion long)p(S) - probability of short position (proportion short)ADC: average daily change in market traded
Based on this calculation, the expected return for rule 1 is .028 per-cent per day or 7.31 percent annualized.!* The expected return for rule 2 isObjective Rules and Their Evaluation 27
0.007 percent per day or 1.78 percent annualized.'® This demonstrates thatthe rules’ historical performance misleads us in two ways. First, bothrules generate positive returns, yet we know that neither has any predic-tive power. Second, rule 1 appears to be superior to rule 2 even though weknow they have equal predictive power—none.
When testing actual trading rules, one way to remove the deceptive ef-fect due to the interaction of position bias and market trend would be todo the following: Subtract the expected return of a nonpredictive rulewith the same position bias as the rule tested from the observed return ofthe tested rule. For example, assume that we did not know rules 1 and 2had no predictive power. Simply by knowing their historical position bias,90 percent long for rule 1 and 60 percent for rule 2, and knowing the mar-ket’s average daily return over the back-test period, we would be able tocompute the expected returns for rules with no predictive power havingthese position biases using the equation for the expected return alreadyshown. The expected returns for each rule and would then be subtractedfrom