There are many references in trading literature to expectation as the “trading edge”. However, there is no sufficient sample to determine the expectation and what they refer to as an edge is the historical average trade.
The trading edge, E, is often defined 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.
Some authors in the trading system literature have claimed that there is a trading edge if E[T] > 0. However, this is mathematically equivalent to the trivial, or even naive, claim that the average trade is greater than 1.
The first problem is that the expectation E[T] in equation (1) is approximately equal to the average trade AvgT only for sufficiently large samples (Papoulis, p. 138). But what is a sufficiently large sample? That depends on many factors.
The second even more important problem is that E[T] is not stationary in general but can vary as a function of trade count N. If E[T] varies, we want the value to stay positive throughout the course of trading activity but also remain large enough so that it results in a satisfactory return above what we get from “risk-free” interest rate. We also want the variance to be small so that the resulting equity curve is as smooth as possible and without large swings.
It is easy to show that equation (1) is equivalent to the statement that the average realized trade is greater that zero. The average trade is given by:
AvgT = (∑W – ∑L)/ N > 0 => ∑W/N – ∑L/N > 0 (2)
Now, multiply through by N >0 and we get the following:
∑W – ∑L > 0 (3)
After multiplying the first term of equation (3) by (Nw/Nw) and the second by (NL/NL) and re-arranging, we obtain
(Nw/N) × ∑W/Nw + (NL/N) × ∑L/NL > 0 (4)
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. Reminder that w = Nw/N and 1-w = NL/N.
We next make use of the fact that Nw + NL = N => NL = N – Nw and (4) becomes
(Nw/N) × ∑W/Nw + ((N – Nw)/N) × ∑L/NL > 0 => (Nw/N) × ∑W/Nw + (1 – Nw/N) × ∑L/NL > 0 (5)
Recall that avgW = ∑W/Nw, avgL = ∑L/NL and w = Nw/N, then equation (5) becomes identical to equation (1)
E[T]=avgW × w – avgL × (1 – w) > 0 (6)
In essence then E[T] is the average realized trade and it is equal to the expectation (mean of the distribution) only in the limit of sufficient samples.
Therefore, what some authors present as a fancy formula for the trading edge, equation (1), reduces to the trivial and intuitive statement that having an edge is equivalent to having positive average trade. However, that regards only the past. Whether the edge will remain positive is not guaranteed by past data and in non-stationary markets E[T] will be a random variable.
What is a sufficient sample? It depends on the strategy and timeframe. Usually thousands of trades in daily timeframe are required for sufficient sample but E[T] may still become negative if there is a market regime change.
More information can be found in Chapter 2 of the book, Fooled By Technical Analysis.
Ref Papoulis, A., Probability, Random Variables and Stochastic Processes, 1965, McGraw-Hill Inc.