The Trading System Inversion Paradox

This paradox arises because being a consistent loser in the markets is as hard as being a consistent winner. The resolution of this paradox is easy and requires high school level math. This paradox has some important ramifications about trading. As far as I know, this is the first time this paradox is explicitly formulated.

The mathematical expectation of a trading system, or trading method in general, is also known as the expected gain, expectancy or expected/average profit per trade. It is defined mathematically as the expected value of the random variable T, the P/L of trades, as follows:

E[T] = w × avgW – (1-w) × avgL                   (1)

where w is the win rate, avgW is the average winning trade and avgL is the average losing trade (in absolute value). Note that technically this is not the expectation but the average trade and this is explained in my book “Fooled By Technical Analysis.”

Let us assume that we have a consistently losing system such that over a sufficiently long period of time, the expectation is negative. From (1) we get:

w × avgW – (1-w) × avgL  < 0                       (2)

Next let us consider the following transformation T:

T:  {z = (1-w),  avgW’ = AvgL,  avgL’ = avgW}      (3)

In T above we define a loss rate that is equal to 1-w and corresponding average win and average loss quantities:

After applying the transformations (3) on (2) we get:

(1-z) × AvgL’ – z × AvgW’ < 0

or, after rearranging:

z × AvgW’ – (1-z) × AvgL’ > 0      (4)

Equation (4) gives the expectancy of a new system, where the win rate of the losing system w in (1) is now the loss rate (1-z) of the new system and the average winner and average loser become the new average loser and new average winner, respectively. This is an inverted system with positive expectation necessarily. Thus, if the original system in (1) was consistently losing over a sufficiently long period of time, the new system described by (4) is a consistent winner over the same period.

Can we then find consistent losers and fade them to make money? The answer is that finding a consistent loser is as hard as finding a consistent winner because of the tranformation defined in (3). Thus, finding a consistently winning system as in (4) by any method you like, technicals or fundamentals or both combined, is equivalent to finding a consistently losing system as in (1), because they are related by a simple transformation. This result does not take into account trading friction because one can always find a consistent loser due to high commission and slippage.

This result shows that becoming a consistent winner is as hard as becoming a consistent loser, even intentionally (without achieving that through high trading friction). The most important ramification is that fading a loser will work only if he is a consistent loser. But you never know that. He may be a temporary loser and his method may reverse sign soon. The other important ramification is that most traders that got ruined were probably not consistent losers but just unfortunate victims of bad risk and money management or were not able to overcome commission cost and slippage.

It should be noted that under special conditions a winning strategy may result from the interaction of two losing strategies, as briefly described in this post.  However, that is a totally different approach where the systems are not inverted but let to interact in such a way so that the small losses of one cancel the small winners of the other and we are left with the big winners of one or the other or both systems.


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