In my last post I made a reference to a popular definition of a trading edge that can be found in the trading literature and dicussed in various forums. This definition of a trading edge corresponds to what is usually referred to as expected gain, expectancy or expected profit by various authors. The trading edge, E, is thus 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 > 0 (1)
where w is the win rate, avgW is the average winning trade and avgL is the average losing trade (absolute value).
I also demonstrated in my last post that this definition of the trading edge is mathematically equivalent to the trivial claim that the profit factor is greater than 1. The profit factor is the ratio of the sum of winning trades to the sum of losing trades. The equivalence was demonstrated using equation (1) as follows:
E[T] > 0 => w × avgW – (1-w) × avgL > o => w × ∑W/Nw – (1-w) × ∑ L/NL > 0 => (Nw/N) × ∑W/Nw – (NL/N) × ∑ L/NL > 0 or
∑W/∑L > 1 (2)
where Nw and NL are the number of winning and losing trades, respectively, and ∑W and ∑L the sum of winning and losing trades respectively. Recall that w = Nw/N and 1-w = NL/N.
Next, I will show that the definition of a trading edge given by equation (1) is also equivalent to the trivial claim that the net profit is greater than zero. Equation (2) can be re-written as follows:
∑W – ∑L > 0 (3)
Equation (3) simply states that the net profit is greater than zero. Thus, Equation (1) is mathematically equivalent to equation (3). You do not believe me? All it takes is high school algebra to prove it. Start with equation (3):
∑W – ∑L > 0 (3) => avgW × Nw – avgL × NL > 0 (4)
simply because avgW = ∑W/Nw and avgL = ∑L/NL.
Now, recall that Nw + NL = N, or that winners and losers sum to the total trades. Then equation (4) becomes:
avgW × Nw – avgL × (N – Nw) > 0 (5)
Now, assuming that N > 0, divide equation (5) through by N:
avgW × Nw/N – avgL × (N – Nw)/N > 0 =>
=> avgW × Nw/N – avgL × (1 – Nw/N) > 0 (7)
Recall that w = Nw/N, i.e. the win rate, is equal to the number of winning trades divided by the number of total trades. Then equation (7) can be written as follows:
avgW × w – avgL × (1 – w) > 0 (8)
which is identical to equation (1). This means that, mathematically, equations (1) and (3) mean the same thing.
Next, it is fairly easy to show that the same initial equation (1) is equivalent to the statement that the average trade is greater that zero. The average trade is given by:
AvgT = (∑W – ∑L)/ N > 0 => ∑W/N – ∑L/N > 0 (9)
Now, multiply through by N >0 and we get equation (3)
∑W – ∑L > 0 (10)
We can go about proving the equivalence as before or we can try a different way. After multiplying the first term of equation (10) by (Nw/Nw) and the second by (NL/NL) and re-arranging, we obtain
(Nw/N) × ∑W/Nw + (NL/N) × ∑L/NL > 0 (11)
We now make use of the fact that Nw + NL = N => NL = N – Nw and (11) becomes
(Nw/N) × ∑W/Nw + ((N – Nw)/N) × ∑L/NL > 0 => (Nw/N) × ∑W/Nw + (1 – Nw/N) × ∑L/NL > 0 (12)
Recall that avgW = ∑W/Nw, avgL = ∑L/NL and w = Nw/N, then equation (12) becomes identical to equation (8)
avgW × w – avgL × (1 – w) > 0 (13)
which is identical to equation (1). This means that mathematically, equations (1) and (10) mean the same thing.
Thus, I have proved using high school math skills that the definition of a trading edge given by
w × avgW -(1-w) × avgL > 0
is equivalent to the trivial statements that
(A) The Profit Factor is greater that 1
(B) The Net Profit is greater than zero
(C) The Average Trade is greater than zero
There is no advantage in using any of the definitions in defining a trading edge. The only advantage may arise from specific uses, like for example, equation (1) is useful in determining the optimal bet size for geometric equity growth. The optimal bet size has been shown to be equal to the expected gain divided by the average winner. This is the well-know Kelly ratio discussed here in more detail.
To conclude, I really do no see any advantage in defining the trading edge either as the mathematical expectation of the trade P/L, as in (1), or by any other mathematically equivalent way as given by (A), (B) or (C). They all mean the same thing, i.e. that a trading method is making money. Equation (1) is fancier but it is mathematically equivalent to the other simpler definitions. Let us keep it simple then: a trading edge is what allows one to make money for an extended period of time. The longer the time, the stronger and the more reliable the edge is.