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Trading Systems And Positive Expectancy

Are you using a trading system? Well, actually you do, even if you are not aware of it. It doesn't matter if it's a complex set of rules or just a coin toss procedure. You are using some criteria in order to initiate and exit a trade. So, the proper question should be, do you know basic elements of your trading system such as its expectancy, the variability of returns, if it's good for reinvestment of returns, its risk of ruin, etc.?

With this series of posts we will examine how to develop and implement a winning trading system.

A cardinal rule is that, the trading system must have a positive expectation. The concept of mathematical expectation is necessary to be understood:

where: Pi = Probability of winning or losing

Y = Amount won or lost

N = Number of outcomes

What the above equation says is that, multiply each possible gain or loss by the probability of that gain or loss and sum those products together. So, if we are going to flip a coin and we will win €3 if it comes up tails but we will lose €1 if it comes up head, our mathematical expectation is:

Mathematical Expectation = (0.5*3) + (0.5*(-1)) = 1.5-0.5=1

In other words we are expecting to make €1 on average each flip.

In the above example, instead of winning €3 if it comes tails assume that we win €0.5 but lose €1 if it comes head. Our mathematical expectation now is:

Mathematical Expectation = (0.5*0.5) + (0.5*(-1)) = 0.25-0.5=-0.25

which means, that for every €1 we bet we would expect to lose on average 25 cents. If we bet €5 our expectation is:

ME = (0.5*0.5)*5 + (0.5*(-1))*5 = 1.25 - 2.5 = 1.25

(losing €1.25 on average per toll).

What we notice is that different amount of bets have different mathematical expectations in terms of € amounts, but the expectation as a percent of the amount bet is always the same.

Now let's bet €1 at the first toss, then €10 and then €20. Our total expectation is:

ME = (-0.25*1) + (-0.25*10) + (-0.25*20) = -0.25 - 2.5 - 5 = -7.75

We would therefore expect to lose on average €7.75. The above example illustrates why systems that try to change the size of their bets while having a negative expectation are doomed to fail. Money management or risk control cannot overcome its inherent limitations.

When trading, the expectation refers to the € profit of the average trade including all available winning and losing trades. The following formulations are identical:

Expectation(€) = NetProfit(€)/TotalNumberOfTrades

Expectation(€) = (%Win)*AverageWin - %Loss*AverageLoss


NetProfit = gross profit - gross loss (over the entire period)

%Win = probability of winning

%Loss = probability of losing

AverageWin = average € profit of all winning trades

AverageLoss = average € loss of all our losing trades

We should keep in mind that our estimate of the expectation is limited by the available data. Different data sets, will give different estimates of the average trade. Thus the expectation of a trading system is not a constant, but changes over time, markets and data sets. Hence, we should use as long a time period as possible to calculate our expectation.

Consider the following sequence of trades:

116.83, 90.08, 292.64, 104.94, 321.35, -196.72, -356.72, 38.73, -110.66, -130.48, 45.78

(they are derived from actual trading, concerning a couple of months back in 2010)

The mathematical expectation is 19.61, meaning that on average and per trade we will win €19.61 (commissions and slippage included). The above result creates a false confidence that we have a winning system and sound money management will exploit its potential to a maximum degree. But the data are too few and they cover just a couple of months of market action (biased due to a bullish, bearish or neutral environment). Subsequent trades proved that it was a rosy picture since the above system has a negative expectation of -72.86.

Expectation does not provide any measure of the variability of returns. So, risk is not fully quantified and other measures must be used, such as standard deviation.

In summary, our system must have a positive expectation, the value of which is not fixed but changes over time. For its calculation we must use as long a time period as possible.

In our next post, we will prove why mathematical expectation below zero leads to a loss of the entire stake (no matter how big it is) and i will share some thoughts about the calculation of expectancy.