Expectancy & Expectancy Uncertainty 
August 9th, 2011, 05:02 PM  #1 (permalink) 
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Expectancy & Expectancy Uncertainty
As I develop and refine several discretionary and automated strategies, I have encountered the nagging question of "How do I know I have enough data to support going live with this method?" I am sure most everyone here at Big Mike's would say to look at the expectancy. Recent conversations with RM99 made me think more.
Expectancy is a good metric. But, like all statistical measures, expectancy is easy to misuse. A profitable system must necessarily have a positive expectancy. But, how do you know your expectancy is positive? There is uncertainty in all the parameters you use to compute expectancy, and thus, there is necessarily uncertainty in the expectancy metric itself. Below, I am going to derive a simple measure of expectancy uncertainty. I am sure others could do better, as statistics is not my strong point. This is the takehome idea: If your expectancy uncertainty is greater than your expectancy, you cannot say for certain that your expectancy is not negative. The expectancy, E, of a set of trades with a gain/loss set of T = {x1, x2, x3, ..., xN}, is: E = Pwin * Xwin  Plose * Xlose, where P is the probability of a winning trade (number of positive T entries divided by N, the length of T), Xwin is the average win gain (average of all positve entries in T), Plose is the probability of losing (Plose = 1 Pwin), and Xlose is the average loss (average of all negative entries in T). Note that in this notation, Xlose is a positive number because I have used a negative number before Xlose in the equation above. Now, Pwin, Xwin, and Xlose are actually estimates. We do not know their "true" values because we have a finite trade history. The less data we have (the shorter data set T), the more uncertainty we have of these estimates. The total uncertainty of expectancy is determined via the propagation of uncertainty of these point estimates. We'll call the uncertainty of expectancy UE. You calculate UE using calculus (the sum of the absolute values of the product of point estimate untertainties and the derivative of E with respect to each variable) that I won't show here unless somebody really wants to see it (if you actually care then you can probably do it yourself). When I do the math, I get: UE = (Xwin + Xlose + Pwin(Swin  Slose) + Slose)/ Sqrt(N) where Swin and Slose are the standard deviations of the profit/losses of the winning and losing trades, respectively. I am using the uncertainty of Pwin to be 1/sqrt(N), and the uncertainty of the means to be the sample standard deviations over sqrt(N). The idea is that your expectancy must be greater than the expectancy uncertainty. We can equate a breakeven point by assuming that the uncertainty always goes against you: E == UE From this equation, we can solve for N, the minimum number of trades to prove positive expectancy: N = (Xwin + Xlose + Pwin*Swin + Plose*Slose) / (Pwin*Xwin + Plose*Xlose) Note that the signs before Plose*Slose and Plose*Xlose are both positive (this means that the denominator isn't quite expectancy itself). Let's say you have a swing system with the following performance: Pwin = 75% profit factor = 0.5 (Xwin/Xlose), and for simplicity, let us say Swin = Slose = 0 (we're lazy). This profit factor and Pwin yields an expectancy of 12.5%. But, to prove that this estimate of expectancy is actually greater than zero, you need (using the formula above) N = 144 trades. If Swin and Slose are nonzero (and we know they are), your N balloons into a larger number. To say it a different way, For N < 144, UE > 12.5%, for N >=144, UE <= 12.5% Letting Swin = 2 and Slose = 1, we get N = 676 (it pays to keep track of your sigmas!). So, to show that you have positive expectancy, you a) need to show positive expectancy, and b) show that you have a sufficient number of trades. Hopefully this helped somebody? 