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. c04 JWBK136-Harris March 20, 2008 10:54 Char Count=PA RT I IProfitabilityand Riskystematic trading methodologies offer great potential for consistentand robust performance if all the parameters that affect profitabil-Sity are well understood and rigorously quantified.An equation thatinvolves some of these parameters, the profitability rule, is discussed inChapter 4.This equation provides, among other things, valuable insight intosome of the constraints imposed on profitability by the realities of marketsand trading system operation.Risk and money management must be an integral part of every suc-cessful trading methodology.In Chapter 5, the fundamental law of riskand money management is presented.This is a simple rule every tradershould understand and apply constantly; however, in reality many pay lit-tle attention to its straightforward results.Any advanced risk and moneymanagement techniques used in trading system development can be suc-cessful provided that the basic requirements set by this simple law are metand it is effectively used to calculate risk and position size.An understanding of the meaning and implications of the profitabilityrule and the fundamental law of risk and money management is essentialfor the success of systematic trading methodologies.43 c04 JWBK136-Harris March 20, 2008 10:54 Char Count= c04 JWBK136-Harris March 20, 2008 10:54 Char Count=CHA PT E R 4The ProfitabilityRulehe profitability rule is a simple yet extremely important equation thatrelates the profitability of a trading system also referred to as theTsuccess rate to its profit factor and average winning to average los-ing trade.In this chapter, I present a derivation of the profitability rule andthen discuss some of the tradeoffs and limitations it imposes on trading sys-tem design and performance.I also discuss the impact of the profitabilityrule on the various trading time frames and derive a more general equationthat takes into account commissions and slippage.QUALITATIVE VERSUS QUANTITATIVEThose who decide to adopt a systematic approach to trading end up allocat-ing significant time and resources toward the development of mechanicalsystems.When developing such systems, one can easily get lost in a mazeof technical analysis indicators that are known to perform poorly, not onlybased on historical testing but also in actual trading.The poor performanceof most technical trading systems, often developed using platforms avail-able to retail traders, is mainly due to the limited ability of most techni-cal analysis methods to time price direction in an accurate and consistentmanner.Another factor contributing to poor performance is the lack of anunderstanding of the fundamental tradeoffs imposed on trading systemdesign by the relation of profitability, profit factor, and ratio of average45 c04 JWBK136-Harris March 20, 2008 10:54 Char Count=46 PROFITABILITY AND SYSTEMATIC TRADINGwinning to average losing trade.The result of this lack of understanding ofthe basic relationships that govern the behavior of trading systems is theimposition of unnecessary constraints during the development process.There are several references in the trading literature to the fact thatthe ratio of average winning to average losing trade, the profit factor,and the profitability of a trading system are related quantitatively.Onecan find such references in articles and books dealing mainly with posi-tion sizing, because some of the formulas presented there involve someof the abovementioned parameters.An example is the well-known Kellyformula.At the same time, one might have difficulty finding an explanation asto why these parameters are related and how to derive an equation thatcan be used by trading system developers.Even worse, qualitative refer-ences to such a mathematical relationship lacking any quantitative contentseem more to generate confusion than to illuminate the problems inher-ent in trading system development.Statements like It s good to have anaverage win/loss ratio of 2:1 or greater, or Let your profits run and cutyour losses short, provide no explanation about the underlying deductionprocess used to arrive at them.Furthermore, vague guidelines of this sort may impose unnecessarylimitations on the development of trading methodologies and often turnout to be false or bad advice.It is known that the literature in the trad-ing system development field, other than a few exceptions, is plagued withvague and unjustifiable statements that are not backed by any mathemati-cal derivation.However, readers should know that any proposed rule thatis not backed by a quantitative derivation is potentially false and advicebased on it is ultimately damaging or, at best, unnecessarily limiting.Thiswill become evident from the discussion on the derivation of the profitabil-ity rule.DERIVATION OF THE PROFITABILITY RULEA trading system is profitable over a period of time T, if the amount ofwinning trades is greater than the amount of losing trades over that period.If we denote the amount of winning trades by the sum of winning tradesand the amount of losing trades by the sum of losing trades, the followingmust hold for all profitable systems:W - L > 0 (4.1)T T c04 JWBK136-Harris March 20, 2008 10:54 Char Count=The Profitability Rule 47The average winning trade is defined as the sum of winning trades di-vided by their number:WTW = (4.2)NWwhere NW is the number of winning trades.Similarly, the average losingtrade is defined as:LTL = (4.3)NLwhere NL is the number of losing trades.By combining equations 4.1, 4.2,and 4.3, we obtainWNW - LNL > 0 (4.4)The number of winning plus the number of losing trades equals thetotal number of trades N, by definition.Therefore:NL = N - NW (4.5)By combining equations 4.4 and 4.5 and after dividing through by N > 0weobtain:NW N - NWW - L > 0 (4.6)N NNext, we define the profitability P, also referred to as the success rate,as the ratio of the number of wining trades NW to the total number of tradesN.As a result, P is a fraction that ranges from 0 to 1 (or from 0 to 100% whenexpressed as a percentage):NWP = (4.7)NAfter introducing equation 4.7 into equation 4.6, we obtain:WP - L(1 - P) > 0 (4.8)If the profitability P is assumed to be equal to the probability of win,then equation 4.8 tells us that the expected gain of a profitable system whena signal is generated is always greater than zero.Specifically, the probabil-ity of win times the average win minus the probability of loss times the c04 JWBK136-Harris March 20, 2008 10:54 Char Count=48 PROFITABILITY AND SYSTEMATIC TRADINGaverage loss is the expected gain E(g) of a trading system:E(g) = WP - L(1 - P) (4.9)The expected gain E(g) of a profitable trading system is always greaterthan zero [ Pobierz całość w formacie PDF ]
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. c04 JWBK136-Harris March 20, 2008 10:54 Char Count=PA RT I IProfitabilityand Riskystematic trading methodologies offer great potential for consistentand robust performance if all the parameters that affect profitabil-Sity are well understood and rigorously quantified.An equation thatinvolves some of these parameters, the profitability rule, is discussed inChapter 4.This equation provides, among other things, valuable insight intosome of the constraints imposed on profitability by the realities of marketsand trading system operation.Risk and money management must be an integral part of every suc-cessful trading methodology.In Chapter 5, the fundamental law of riskand money management is presented.This is a simple rule every tradershould understand and apply constantly; however, in reality many pay lit-tle attention to its straightforward results.Any advanced risk and moneymanagement techniques used in trading system development can be suc-cessful provided that the basic requirements set by this simple law are metand it is effectively used to calculate risk and position size.An understanding of the meaning and implications of the profitabilityrule and the fundamental law of risk and money management is essentialfor the success of systematic trading methodologies.43 c04 JWBK136-Harris March 20, 2008 10:54 Char Count= c04 JWBK136-Harris March 20, 2008 10:54 Char Count=CHA PT E R 4The ProfitabilityRulehe profitability rule is a simple yet extremely important equation thatrelates the profitability of a trading system also referred to as theTsuccess rate to its profit factor and average winning to average los-ing trade.In this chapter, I present a derivation of the profitability rule andthen discuss some of the tradeoffs and limitations it imposes on trading sys-tem design and performance.I also discuss the impact of the profitabilityrule on the various trading time frames and derive a more general equationthat takes into account commissions and slippage.QUALITATIVE VERSUS QUANTITATIVEThose who decide to adopt a systematic approach to trading end up allocat-ing significant time and resources toward the development of mechanicalsystems.When developing such systems, one can easily get lost in a mazeof technical analysis indicators that are known to perform poorly, not onlybased on historical testing but also in actual trading.The poor performanceof most technical trading systems, often developed using platforms avail-able to retail traders, is mainly due to the limited ability of most techni-cal analysis methods to time price direction in an accurate and consistentmanner.Another factor contributing to poor performance is the lack of anunderstanding of the fundamental tradeoffs imposed on trading systemdesign by the relation of profitability, profit factor, and ratio of average45 c04 JWBK136-Harris March 20, 2008 10:54 Char Count=46 PROFITABILITY AND SYSTEMATIC TRADINGwinning to average losing trade.The result of this lack of understanding ofthe basic relationships that govern the behavior of trading systems is theimposition of unnecessary constraints during the development process.There are several references in the trading literature to the fact thatthe ratio of average winning to average losing trade, the profit factor,and the profitability of a trading system are related quantitatively.Onecan find such references in articles and books dealing mainly with posi-tion sizing, because some of the formulas presented there involve someof the abovementioned parameters.An example is the well-known Kellyformula.At the same time, one might have difficulty finding an explanation asto why these parameters are related and how to derive an equation thatcan be used by trading system developers.Even worse, qualitative refer-ences to such a mathematical relationship lacking any quantitative contentseem more to generate confusion than to illuminate the problems inher-ent in trading system development.Statements like It s good to have anaverage win/loss ratio of 2:1 or greater, or Let your profits run and cutyour losses short, provide no explanation about the underlying deductionprocess used to arrive at them.Furthermore, vague guidelines of this sort may impose unnecessarylimitations on the development of trading methodologies and often turnout to be false or bad advice.It is known that the literature in the trad-ing system development field, other than a few exceptions, is plagued withvague and unjustifiable statements that are not backed by any mathemati-cal derivation.However, readers should know that any proposed rule thatis not backed by a quantitative derivation is potentially false and advicebased on it is ultimately damaging or, at best, unnecessarily limiting.Thiswill become evident from the discussion on the derivation of the profitabil-ity rule.DERIVATION OF THE PROFITABILITY RULEA trading system is profitable over a period of time T, if the amount ofwinning trades is greater than the amount of losing trades over that period.If we denote the amount of winning trades by the sum of winning tradesand the amount of losing trades by the sum of losing trades, the followingmust hold for all profitable systems:W - L > 0 (4.1)T T c04 JWBK136-Harris March 20, 2008 10:54 Char Count=The Profitability Rule 47The average winning trade is defined as the sum of winning trades di-vided by their number:WTW = (4.2)NWwhere NW is the number of winning trades.Similarly, the average losingtrade is defined as:LTL = (4.3)NLwhere NL is the number of losing trades.By combining equations 4.1, 4.2,and 4.3, we obtainWNW - LNL > 0 (4.4)The number of winning plus the number of losing trades equals thetotal number of trades N, by definition.Therefore:NL = N - NW (4.5)By combining equations 4.4 and 4.5 and after dividing through by N > 0weobtain:NW N - NWW - L > 0 (4.6)N NNext, we define the profitability P, also referred to as the success rate,as the ratio of the number of wining trades NW to the total number of tradesN.As a result, P is a fraction that ranges from 0 to 1 (or from 0 to 100% whenexpressed as a percentage):NWP = (4.7)NAfter introducing equation 4.7 into equation 4.6, we obtain:WP - L(1 - P) > 0 (4.8)If the profitability P is assumed to be equal to the probability of win,then equation 4.8 tells us that the expected gain of a profitable system whena signal is generated is always greater than zero.Specifically, the probabil-ity of win times the average win minus the probability of loss times the c04 JWBK136-Harris March 20, 2008 10:54 Char Count=48 PROFITABILITY AND SYSTEMATIC TRADINGaverage loss is the expected gain E(g) of a trading system:E(g) = WP - L(1 - P) (4.9)The expected gain E(g) of a profitable trading system is always greaterthan zero [ Pobierz całość w formacie PDF ]