Z-Score Optimization in Forex Trading (2024)

This guide sheds light on avery important but often overlooked concept inEA optimization— Z-score. But before wedescribe the actual use case of Z-score optimization, it would be wise toexplain abit more about this interesting trading system performance indicator.

What is Z-score?

A Z-score (orstandard score) measures thedistance between themean ofsome set ofthestatistical results andthegiven observation.

InForex systems, traders are interested in Z-score not of a trade's return (profit/loss size) but rather in a Z-score of the outcome— was it a profitable one or a losing one? A Z-score, calculated using winning and losing streaks, measures thedependence between theoutcome of the previous position andtheoutcome of the next position.

If we consider profitable positions aspositive results andlosing positions asnegative statistical results, counting thetotal number ofall wins, losses, overall trades, aswell asthenumber ofwin andloss streaks, we can calculate a Z-score foragiven trading strategy orexpert advisor.

Usually, a Z-score fluctuates between -3 to+3, but sometimes, can go above andbelow these "limits". A Z-score value of0 means that we are dealing with completely random results.

Each Z-score value has also aprobability ofdependence (P) associated with it, which informs us ofhow probable dependence between thetrades is. P values below -2 andabove +2 have high (>95%) probability ofdependence between trades:

• A positive Z-score means that profitable position is likely tobe followed byalosing one, while alosing one should probably be followed byawinning one. Basically, it means that the probability of long winning and losing streaks is low.
• A negative Z-score means that profitable positions are likely tobe followed bymore profitable positions, andlosing positions are tobe followed bymore losing positions. Basically, it means that winning or losing streaks are probable.

The concept of Z-score is illustrated in this image of two samples. The first sample is taken from a completely random trading system (50% chance of winning and 50% chance of losing), and the second sample is taken from a trading system with significant edge and dependence on previous trades:

As you can see, their Z-scores are totally different, with a large positive Z-score of the second system signaling high likelihood of switching between profitable and losing trades.

How to calculate Z-score of Forex system

To calculate a Z-score of your system, you need a rather large sample of trade outcomes recorded in chronological order. Using these outcomes, you can calculate Z-score with this formula:

where:

• Z is the Z-score you are calculating;
• N is the total number of trades in the sample;
• R is the total number of losing and winning streaks;
• P equals 2 × W × L;
• W is the total number of winning trades in the sample;
• L is the total number of losing trades in the sample.

Example of Z-score calculation

Let's look at the following example calculation process of a system's Z-score.

Let's assume a sample of 100 trading outcomes with 40 streaks (winning and losing), 50 winning trades, and 50 losing trades; then:

• N = 100;
• R = 40;
• P = 2 × 50 × 50 = 5000.

The negative Z-score value of -2.11 means that the system's trading outcomes are likely non-random and that losing positions will likely be followed by more losing positions while profitable trades will likely be followed by more profitable trades (trades are likely to come in streaks).

Z-score probability

Each Z-score value has a certain probability associated with it. This probability tells us how likely is it that the system that we analyze produces outcomes that are dependent on the previous outcome.

You can look at this table to quickly find the probability of such a connection between your trading results if you already know your Z-score:

If we consider the Z-score value of -2.11 from the example above, we can see that the probability of positive dependence between the system's outcomes is between 95.45% and 97.22%, which is quite significant.

How touse Z-score inForex?

If you know the Z-score ofyour expert advisor ortrading system andits value is above +2 orbelow -2, you can "skip" atrade when alosing position is expected. Nevertheless, theprofit onthis "skipped" position should be tracked (virtually), toknow when tostart trading again.

It should also be noted that Z-score calculation makes sense only for sufficiently large samples. Math literature suggests a sample size of no less than 51 to get a reliable Z-score value.

Z-score optimization

For example, you backtested your expert advisor and found out that after more than a hundred trades, the resulting Z-score is +3.02, which means that profitable andlosing positions are likely toappear inanalternating order (long streaks of losses or wins are not probable). The probability for Z-score value of +3.02 is greater than 99.73%.

It would make sense to modify thecode ofsuch anexpert advisor tostop sending live orders when aprofitable position is closed (the next one is very likely to be a loss). TheEA would then enter asort of a virtual trading mode, where position is opened andtracked only virtually (using MQL4 variables). When such position is closed (virtually of course), its profit/loss is considered— if it is aloss, live trading becomes enabled once again; if it is a win, operation invirtual mode should be continued. This is how it should be done forpositive Z-scores.

Ofcourse, it can be easily modified towork with negative Z-score.

You can download an exampleMT5 EA with positive Z-score optimization implemented:

• ATC Z-score Optimization Test Expert Advisor

Further improvement

This scheme ofZ-score optimization implies that you measure your Z-score onabacktest andthen consider that it won't change much during thelive run infuture. An alternative plan would be toimplement aconstant measurement ofZ-score insideyour EA's code andto adjust theoptimization on-the-fly. Theproblem is that it is quite difficult toimplement this ina live expert advisor.

When the shares ofthewinning andlosing positions are very unequal, Z-score optimization may become abit tricky. In this case, it is probably better tomake it asymmetrical (favoring less rare outcomes) orto skip it completely, even if Z-score is above/below +2/-2.

One notable case where Z-score optimization would be very inefficient and even ruinous is when your expert advisor opens more than one trade at once and these trades have nearly identical chance of ending up in loss or in profit. If your EA trades like that, you need to count such a group of nearly simultaneous trades as one outcome when computing a Z-score.

If you have any questions regarding theuse ofa Z-score metric inexpert advisor optimization orif you want toshare some idea forusing Z-score values in Forex trading, you can start a discussion on our forum.