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.9 that when tossing a fair coin,in which case P = 0.5, with an average win equal to the average loss, theexpected gain E(g) is zero.For a positive expected gain, it is the relationof profitability to the ratio of average win to average loss that matters, nottheir specific values.As a matter of fact, high profitability does not sufficeto have a winning trading system, as is evident from equation 4.9.Thus, wecan proceed to derive a more general expression that reveals the uniquerelationship of the parameters involved.The average losing trade, as given by equation 4.3, cannot be zero un-less the trading strategy is 100 percent profitable.Since in practice we arealways dealing with trading systems with P 0 (4.10)LNext, we define the ratio of average winning to average losing trade,RWL, as follows:WRWL = (4.11)LBy combining equations 4.10 and 4.11 and after solving for P we obtain theresult:1P > (4.12)1 + RWLI call equation 4.12 the profitability rule.Equation 4.12 relates the minimum profitability required to generate anet profit, over a period of time T, to the ratio of average winning to averagelosing trade calculated over the same period.If, instead of the greater-thansign in 4.12, the equality sign is used, that denotes the profitability, or suc-cess rate, of break-even performance, assuming commissions and otherfees are included in the calculation of the average win and average loss.Thus, the minimum profitability of a trading system so that it breaks evenover a period of time T is given by1P = (4.13)1 + RWL c04 JWBK136-Harris March 20, 2008 10:54 Char Count=The Profitability Rule 49Minimum Profitabilityas a Function of theTABLE 4.1Ratio of Average Winto Average LossRWL Minimum P (×100)10 9.09%5 16.67%2 33.33%1 50.00%0.5 66.67%0.25 80.00%0.125 88.88%Table 4.1 shows the minimum profitability for various values of RWLcomputed using equation 4.13.It is clear that there is an inverse relationbetween the profitability P and the ratio RWL.As RWL increases, the mini-mum profitability of a trading system required for break-even performancedecreases.Thus, in regard to the value of the RWL parameter, the followingtwo observations can be made:1.Trading systems with low RWL values must have high profitability,which implies a much larger number of winning trades than losingtrades.2.Trading systems with high RWL values can sustain lower profitability;thus they can generate fewer winning trades than losing trades and stillbe profitable.Equation 4.13 can be solved for the minimum RWL value required inorder for a trading system to maintain the minimum profitability P:1 - PRWL = (4.14)PEquation 4.14 can be used to estimate the minim risk/reward ratio re-quired given the minimum (break-even) profitability P.Table 4.2 shows some examples of the minimum required value of theparameter RWL for various values of profitability P, computed using equa-tion 4.14.It is clear from the table that the profitability increases as thevalue of the parameter RWL decreases. c04 JWBK136-Harris March 20, 2008 10:54 Char Count=50 PROFITABILITY AND SYSTEMATIC TRADINGMinimum Required Ratio ofTABLE 4.2 Average Win to Average Lossas a Function of ProfitabilityMinimum P (×1100) Required RWL20 430 2.3350 167 0.4970 0.4275 0.3390 0.11THE FUNDAMENTAL LAW OFTRADING STRATEGIESEquation 4.12, the profitability rule, is an expression of the minimum prof-itability required in order for a trading system s equity performance tobreak even over a time period T, as a function of the ratio of average win-ning to average losing trade RWL calculated over the same period.Thisequation is useful in terms of obtaining a lower bound on profitabilityP, but it does not tell us anything about obtaining a desired profit fac-tor Pf.The profit factor is defined as the ratio of the amount of winningtrades divided by the amount of losing trades.This parameter is probablythe most important to consider when designing and evaluating a tradingsystem.Experienced trading system designers know that it is the profit factorthat ultimately tells us whether a trading system performs, regardless ofthe values of profitability and average win to average loss ratio.In otherwords, the profit factor is a measure of the potential of a trading system togenerate profit besides just being profitable.Traders prefer systems with aprofit factor at least equal to 2, meaning that the amount of winners is twiceas large as the amount of losers over a period of time.Intraday trading sys-tems tend to have lower profit factors and trend-following systems usuallyhave much higher values.Short-term, position, and swing trading systemsfall somewhere in between regarding the values of the profit factor that canbe achieved in that time frame.In general, the higher the profit factor, thebetter the system performance.Traders often get the impression that the profit factor is an ad-hoc mea-sure, again by virtue of the vague and often confusing way these topics havebeen covered in the trading literature by and large.On the contrary, thisimportant parameter is hidden in equation 4.1 and it is revealed when we c04 JWBK136-Harris March 20, 2008 10:54 Char Count=The Profitability Rule 51divide through the equation by the second term, the sum of losing trades(always a positive quantity in reality), and then rearrange to yield:WTPf = > 1 (4.15)LTEquation 4.15 is the definition of the profit factor Pf, and this parametermust be greater than one in order for a trading system to be profitable.Byusing equations 4.2, 4.3, 4.5, and 4.11, equation 4.15 becomesNWPf = RWL (4.16)N - NWNow, we can divide both the numerator and denominator of equation4.16 by N > 0 and use the definition of profitability P of equation 4.7 to get:PPf = RWL (4.17)1 - PEquation 4.17 is an expression for the profit factor as a function of theprofitability and ratio of average winning to average losing trade.This equa-tion tells us that the three parameters involved are connected by a uniquefunctional relationship.This implies that given any two of the parameters,the third is uniquely determined.Solving equation 4.17 for profitability Pyields the final result:PfP = (4.18)Pf + RWLEquation 4.18 is more general than equation 4.12, because it is an ex-pression of the profitability P as a function of the profit factor Pf and ratioof average winning to average losing trade RWL.I call equation 4.18 the fun-damental law of trading strategies.Given the expected profit factor andthe expected ratio of average winning to average losing trade, this funda-mental law determines the profitability of the system.Alternatively, for adesired profit factor and an expected ratio of average winning to averagelosing trade, the minimum profitability P of a trading system must satisfythe following:PfP e" (4.19)Pf + RWLTable 4
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