Improve Your System With The Profitability Rule

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by Michael Harris

What are the capabilities and limitations of your trading system or investment strategy? You can ensure your system or strategy has a high probability of meeting your profit objectives by using this simple model as a measure of profitability.

understand the fundamental tradeoffs present in trading system design, or even worse, under-estimate the influence of those tradeoffs.

The profitability rule I present in this article sheds some light on the capabilities and limitations of trading systems, especially when commissions are factored in. It provides insight into systems operating in different time frames such as intraday or short-term, as well as trend-following systems. It would be worthwhile for every trader to understand how this simple rule can be used to realistically analyze the profitability of a system.

T

HEPROFITABILITYRULE

A trading system is profitable over a period of

time if the sum of all winning trades, ∑W, is greater than the sum of all losing trades, ∑L:

∑W – ∑L > 0 (Equation 1)

The average winning trade, AvgW, is defined as: AvgW = ∑W/NW or ∑W = AvgW * NW (2) where NW is the number of winning trades. Similarly, the average losing trade, AvgL, is defined as:

AvgL = ∑L/NL or ∑L =AvgL * NL (3) where NL is the number of losing trades. Combining equations 1, 2 , and 3 yields:

ost traders spend time trying to develop mechanical trading systems and often end up lost in a maze of indicators and techniques that produce dismal results. They may not fully

Improve Your System With

The Profitability Rule

DOUG TILLER

AvgW * NW – AvgL * NL > 0 (4) It is also true that:

NW = N – NL (5)

where N is the total number of trades. Combining equations 4 and 5 and dividing them by N yields:

AvgW NW

N – AvgL

N – NW

N > 0 (6)

The profitability (P) of a trading system, or investment strategy in general, is defined here as the ratio of the number of winning trades or investments to the total number of trades or investments made, a number ranging from zero to 1 (zero to 100%, in percentage terms):

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P =NW

N (7)

Combining equations 6 and 7 yields: AvgW * P – AvgL(1 – P) > 0 (8) Dividing equation 8 by AvgL (AvgL is not zero unless a “Holy Grail” technique is involved, in which case the issue of profitability is moot because P = 100%) gives:

AvgW

AvgL P – 1 – P > 0 (9)

RWL> 1 – P

P

This equation may be used to determine the level of risk/reward parameters required, as expressed by the RWL parameter, given the expected profitability P of a trading system or investment strategy.

P

ROFITABILITY

ANDTRADINGTIMEFRAMES

Short-term or intraday trading systems usually exhibit lower values for parameter RWL, as compared to trend-following systems. In practice, typical values for short-term and intraday trading systems for the RWL parameters lie within the range

The profitability rule shows you the

minimum profitability that is required to

generate a net profit in a trading system

or investment strategy.

In trend-following systems, the objective is to minimize losing trades during sideways markets and maximize winning trades during price trends. However, it is uncertain how future RWL values will compare with historical ones, since the future average winning trade size is unknown. This is because the size of future price trends is unpredictable and, at the same time, exit technique performance depends highly on volatility (such as trailing stops).

If you assume that more or less the same magnitude of trends and the same or lower associated trend volatility will exist in the future, the profitability of a trend-following system based on historical testing will guarantee future profits. This assumption exemplifies the stochastic nature of the profitability rule in the case of trend-following systems, where the parameter RWL is a random variable.

A practical implication of the nature of the profitability rule is that trend-following systems must have a greater historical percent profitability than the minimum required based on historical RWL values, because future RWL values may be drastically reduced by smaller trend sizes and higher RWL Minimum required profitability P (x100) 10 9.09% 5 16.67% 2 33.33% 1 50.00% 0.5 66.67% 0.25 80.00% 0.125 88.88%

The ratio of average winning to average losing trade, RWL, is defined as follows:

RWL= AvgW

AvgL (10)

Combining equations 9 and 10 and solving for P yields:

P > 1

RWL+ 1 (11)

Equation 11 is the profitability rule. It relates the minimum profitability of a trading system or investment strategy that is required to generate a net profit to the ratio of average winning to average losing trades or investments. The rule provides a simple model for profitability as a function of a single parameter. It is of extreme importance to the trading system or investment strategy development process and has several implications, some already known to trading system developers from an empirical standpoint.

Figure 1 shows the required minimum profitability of a system for various values of the RWL parameter, computed using equation 11. There is an inverse relationship between profitability P and the ratio RWL. As the RWL parameter increases, the minimum profitability of a trading system required in generating a net profit decreases. With respect to the value of the RWL parameter, you can draw the following conclusions:

• Trading systems with low RWL values must have a high profitability, which implies a much higher number of winning trades than losing trades.

• Trading systems with high RWL values may have a low profitability — that is, they can generate lower numbers of winning trades than losing trades and still be profitable. Alternatively, you may solve equation 11 for the minimum value of the RWL parameter that is required in order for a trading system or investment strategy to produce a certain level of profitability P. Solving equation 11 for RWL yields:

of 0.25 to 2, and true trend-following system values are far above 3. This is because in short-term and intraday trading systems the trading activity is based on price volatility. Specifically, as it relates to the profitability rule, it is likely that future RWL values willclosely match values determined during historical testing. This is true because in shorter-term time frames, trade exit techniques result in comparable ratios of profits to losses over time (especially when using targets and stops based on fixed percentages or increments of the entry price). Therefore, if the profitability of an investment strategy is maintained above the minimum required by the profitability rule, the system will always generate a profit.

FIGURE 1: MINIMUM REQUIRED PROFITABILITY.Minimum required prof-itability is expressed as a function of the ratio of average winning to average los-ing trades.

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market volatility. This imposes additional constraints in the design of trend-following systems, which must be robust enough to account for future market conditions resulting in much lower RWL values, and in comparison a demand for investment strategies exhibiting higher than expected profitability.

Thus, trend-following strategies will remain profitable for extended periods if they can maintain the minimum required profitability as determined by the minimum expected values of the RWL parameter. So does that mean trend-following systems can have lower profitability levels than short-term or intraday systems and still be profitable? This also gives a clue about the difficulties involved in designing high-profitability trend-following systems.

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HECOMMISSIONEFFECT

The profitability rule presented in equation 11 considers trading commissions implicitly by subtracting them from both the winning and losing trades. The effect of trading commissions can be considered explicitly by defining C as the round-turn commission charged per trade. Then, equation 1 is modified to include commissions as follows:

∑W – ∑L – N * C > 0 (12)

The commission paid for N trades is subtracted from the sum of winners, in addition to subtracting the sum of losers. Combining equations 2, 3, and 12 yields:

AvgW * NW – AvgL * NL – N * C > 0 (13) Combining equations 5 and 13 and dividing by N yields:

AvgW NW

N – AvgL

N – NW

N – C > 0 (14)

Combining equations 7 and 14 gives:

AvgW * P – AvgL * (1 – P) – C > 0 (15) Rearranging the terms of equation 15 yields:

AvgW * P + AvgL * P > C + AvgL or

P > C + AvgL

AvgW + AvgL (16)

Dividing both the numerator and denominator of equation 16 by AvgL (again, AvgL is not zero unless a Holy Grail is involved, in which case the issue of profitability is moot, because P = 100%), and using equation 10, results in:

PC>

1 + C AvgL 1 + RWL

(17) Equation 17 is the modified profitability rule that incorporates flat round-turn commissions. Note that equation 17 is identical to equation 11 with the exception of the term C/AvgL, which is added to the numerator. To distinguish this profitability from the one derived from equation 11, I denote it as PC.

The ratio C/AvgL is the commission paid per trade divided by the average losing trade, a factor denoted here as Cf. Then equation 17 may be written as:

PC> 1 + Cf

1 + RWL (18)

with: Cf=_AvgLC

Alternatively, equation 18 can be written as follows:

PC>_{1 + R}1 WL + Cf 1 + RWL or PC> P + Cf 1 + RWL = P +∆PC

The minimum required profitability including commissions (PC) is equal to that in equation 11, which does not account for commissions, plus a term equal to ∆PC,the value of which depends on the factor Cf and RWL.

As long as the factor Cf is kept low and the parameter RWL attains a high value, the value of the term ∆PC is kept low. Low values for the factor Cf are possible as long as the commission paid per trade is low compared to the average losing trade. This may hardly affect trading strategies with high average losers, such as trend-following systems, but it may adversely affect the required minimum profitability of intraday or short-term trading systems, depending on the value of the parameter RWL.

Often, traders take intraday or short-term positions based on price volatility design strategies that have low RWL values and require a high profitability. The low RWL values are the result of higher average losing trades compared with the average winning trades. This is because stop-losses are set higher than profit targets to avoid having the stops “run” by the market. Consider the following cases:

Case 1: RWL = 2, commission per trade = $30 (round turn) The minimum required profitability without considering commissions is determined by equation 11:

P > 1

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By getting a measure of the minimum

required profitability, you have an idea of

the minimum success rate that must be

maintained in order to be profitable.

If the average losing trade is $1,000, then from equation 17 you get:

PC>

1 + 30 1000

1 + 2 = 1.033 = 0.3433 or 34.33%

If the average losing trade is $200, you get:

PC> 1 + 30

200

1 + 2 = 1.153 = 0.3833 or 38.33%

In this example, the effect of commissions on the required minimum profitability does not affect its range significantly. It can still be considered a low-profitability requirement. But in case 2:

Case 2: RWL = 0.5, commission per trade = $30 (round turn) The minimum required profitability without considering commissions (equation 11) results in:

P > 1

0.5 + 1= 0.6666 or 66.66%

PC>

1 + 30 1000

1 + 0.5 = 1.031.5 = 0.6866 or 68.66%

When the average losing trade is lowered to $200, then from equation 17 you get:

PC> 1 + 30

200

1 + 0.5 = 1.151.5 = 0.7666 or 76.66%

In this case also — when the average losing trade is $200 — the required profitability increases from 66.66% to 76.66%. Although the increase, percentage-wise, is the same as in case 1, keep in mind that any gain in profitability at this higher range is much more difficult to obtain than in low ranges. Therefore, many would argue that graduating from 66.66% to 76.66% is more difficult than from 33.33% to 38.33%. As a final example, in case 3:

Case 3: RWL = 0.25, commission per trade = $30 (round turn)

Using the same series of equations, the minimum required profitability without considering commissions is:

P > 1

0.25 + 1 = 0.80 or 80%

PC>

1 + 30 1000

1 + 0.5 = 1.031.25= 0.8240 or 82.40%

When the average losing trade is lowered to $200, then from equation 17 you get:

PC> 1 + 30

200

1 + 0.5 = 1.151.25= 0.92 or 92%

In this last case — when the average losing trade is $200 — the required profitability increases from 80% to 92%. Although the percentage increase is the same as in the other two cases, no one would deny that 92% profitability is more difficult to achieve than 80% profitability.

You can determine from equation 17 that in order to minimize the commission effect on profitability you must minimize the factor Cf, which for fixed commissions means maximizing the average losing trade. You need a high-average winning trade from your system, and this is where both the difficulty and tradeoff lie. As you attempt to increase the average winners, more trades fail to meet the profit objectives and turn into losers, decreasing system profitability.

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HESCALPINGGAME

Scalpers — intraday traders trying to profit from small price moves — have a profit objective of covering commission costs and making a small profit. They also place very tight stops. It is a highly intensive trading process and requires special skills as well as an in-depth understanding of market operation. Obviously, such trading is not suitable to everyone. I will try to analyze the effect of commissions in scalping by assuming that the average losing trade and average winning trade are multiples of the commission rate C, as follows:

AvgW = m * C (19)

AvgL = k * C

where k and m are integer numbers. Combining equations 17 and 19 gives: PC> 1 + C k C 1 +m C k C

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or PC> 1 + 1 k 1 + m k

which further transforms to:

PC> 1 + k_{m + k} (20)

The required minimum profitability as a function of the integers m and k (multiples, in terms of the commission rate, of the average winning and average losing trades, respectively) is presented in equation 20. In this form, neither the commission size nor the magnitude of the average losing trades is required to determine the minimum required profitability.

Case 1: m = 2, k = 4 (the average losing trade is four times the commission and the average winning trade twice that). Applying equation 20 results in:

PC> 1 + 4_{2 + 4} = 0.8333 or 83.33%

From equation 11 you get: P > 1

1 +2 C 4 C

= 1

1.5= 0.6666 or 66.66%

The minimum required profitability with commissions was increased from 66.66% to 83.33%.

Case 2: m = 10, k = 10 (the average losing trade and average winning trade are both 10 times the commission rate). PC> 1 + 10_{10 + 10}= 0.55 = 55%

From equation 11 you also get:

P > 1 1 +10 C

10 C = 1

2= 0.50 or 50%

In this case the minimum required profitability, in the presence of commissions, was increased from 50% to 55%. This is a more realistic situation than in case 1, and offers a higher probability of devising a technique that will meet profitability objectives.

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SINGTHE

PROFITABILITYRULE

The profitability rule can be useful in understanding the limitations of various trading systems and in designing strategies with more possibility of trading success. Intraday and short-term traders can use the rule to estimate a priori the required minimum profitability based on their profit/loss objectives and commission structure.

The profitability rule is useful when a trading technique cannot be put in a form suitable for backtesting, or there is no historical data on which to test it. By getting a measure of the minimum required profitability, you have an idea of the minimum success rate that must be maintained in order to be profitable.

If the RWL parameter cannot be estimated in advance, as is true in the case of trend-following systems, you could use historical data analysis to determine the worst possible value for this parameter and then calculate the required minimum profitability. Understanding the fundamental tradeoffs imposed by the profitability rule may result in a more realistic approach to the system development process and expectation for trading performance.

Michael Harris is president of TradingPatterns.com, which specializes in the development and sale of mechanical trading systems for futures and equities, such as Automatic Pattern Search. He is also managing director of Harrison Investments (www. harrisontrading.com), an offshore money management and hedge fund consulting firm. He has been a trader and system developer for 15 years.

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UGGESTEDREADING

Harris, Michael [2001]. Stock Trading Techniques Based On Price Patterns, Traders Press.