Mathematics

By now you must have see the projection curves coming off my EA during the live feed, but what does it mean?  How am I using that to enter trades?  That’s what this section is all about.

Consider the following:

Where:

x = Projection mean of the sample population for the current cycle

y = One standard deviation below the mean for the current cycle

We’ll further define PN as the ratio of the Projection mean to the sum of the mean and the Negative standard deviation such that:

PN = x/(x+y)

Thus, by defining the current cycle’s result as a ratio of the sample mean and standard deviation, we can now simplify our calculations based on a normal distribution with mean 0 and variance of 1.  This of course, assumes that the sample population is normally distributed.  Therefore, the probability of a buy entry resulting in a win as defined by the exit price being greater than the entry price, is:

Furthermore, let us determine the average amount won in a win and lost in a loss.  If we set the TP at x, then it follows that our maximum win is x, but the position could also be closed by a change of direction, or after the forecast time has expired.  In this case, the win could be somewhere in the range (0,x).  We average these two unitless quantities as follows:

And similarly for the averge lost amount:

And thus, the ratio of our average winning amount to average losing amount becomes:

It should be noted, however, that if the maximum positive sample standard deviation is used as the TP, this ratio becomes:

At this point you might be saying “why do I care”? — and rightfully so.  Where I’m going with this is the fact that I’m using the Kelly Criterion as a way of determining the theoretical profitability of given entry points (PN values).  Kelly’s percentage  is defined by Winning Probability + [(Winning Probability-1) / Win-to-Loss Ratio], hence the need to calculate the aforementioned values.

What Kelly represents is the optimum betting amount as a percent of our total pot.  In other words, if we were to bet any higher, we’re guaranteed to lose in the long term.  Another important note is that if the Kelly % is negative, that means a losing proposition no matter what.

With this in mind, let’s take a look at what the Kelly values are for different PN values in our EA:

Kelly Values for Mean TP

PN P(W) W/L Kelly
0.2 0.57926 0.350705 -0.62044
0.25 0.598706 0.453612 -0.28596
0.3 0.617911 0.56568 -0.05754
0.35 0.636831 0.68921 0.109895
0.4 0.655422 0.827237 0.238881
0.45 0.673645 0.98387 0.341939
0.5 0.691462 1.164837 0.426586
0.55 0.70884 1.378384 0.497608
0.6 0.725747 1.636859 0.558198
0.65 0.742154 1.959642 0.610576
0.7 0.758036 2.379021 0.656329
0.75 0.773373 2.953098 0.69663

Kelly Values for St. Dev TP

PN P(W) W/L Kelly
0.2 0.57926 1.376081 0.273507
0.25 0.598706 1.495602 0.330391
0.3 0.617911 1.629588 0.383442
0.35 0.636831 1.781306 0.432953
0.4 0.655422 1.955107 0.479177
0.45 0.673645 2.156921 0.522339
0.5 0.691462 2.395048 0.562639
0.55 0.70884 2.681475 0.600258
0.6 0.725747 3.034191 0.635359
0.65 0.742154 3.481468 0.668091
0.7 0.758036 4.070439 0.698592
0.75 0.773373 4.885966 0.726989

So we observe that if we chose the mean as the TP target, we’ll need to use a PN ratio of at least 0.35 to have a theoretically profitable system.  On the other hand, if we chose the upper standard deviation as the TP target, our system is theoretically profitable for a PN as low as 0.2 or even lower.

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