In a previous article called “System Performance and Confidence Interval“, I showed how a statistical method could be used to analyze historical trading results to give us an idea if the system would likely fail in the future. In this article I would like to introduce a mathematical formula which can be applied to any trading system and used as an objective score to compare and rank different trading systems.

When it comes to trading systems, no two are alike. There is a vast number of different trading styles that cover the range from simple tick scalping to multi year investment models. Of course, there is a huge number of different instruments and markets to trade. How would one determine if two different trading systems that trade different markets with different trading styles make an informed choice on which system was more profitable? How do you compare two trading systems’ performances? If two trading systems are both profitable and both seem like good systems, is there a single metric that can be used to compare the profitability between each?

Expectancy is a concept that was described in Van Tharp’s book *Trade Your Way To Financial Freedom*. Expectancy tells you on average how much you expect to make per dollar at risk. For example, if you have a trading system that has a .50 Expectancy that means for every dollar you risk the trading system returns $.50.

There are two ways to compute Expectancy. Both methods are simple but one requires a little more explanation but does give a more conservative answer. The second method is a bit more straightforward to explain but only give an approximation of Expectancy. However, this approximation is good enough for what we are attempting to accomplish here.

Often you will find expectancy is computed with the following formula:

```
Expectancy = (AW * PW + AL * PL ) / | AL |
Where:
AW = Average winning trade in dollars
PW = Probability of winning trades
| AL | = Absolute value of the average losing trade in dollars
PL= Probability of losing trades
```

But if you look carefully at the calculation within the parenthesis you will see the value is nothing more than the trading system’s average net profit per trade. Substituting this value gives us the simplified formula of:

Expectancy = Average Net Profit Per Trade / | AL |

The two values you plug into the Expectancy formula can be found on most (if not all) strategy performance reports generated by backtesting. This is certainly true for TradeStation‘s strategy reports.

Now that we have our trading system’s Expectancy value are we ready to use this value to compare to other trading systems? Not yet. There is another step we must first take. Our Expectancy value simply tells us our historical profit per dollar risked for each trade. But we are missing something. Let’s imagine we have two trading systems that have two different Expectancy values:

**Trading System #1 has an Expectancy of .25**

** Trading System #2 has an Expectancy of .50**

Based on what we know it appears Trading System #2 produces more profit per dollar risked on each trade. In fact, it produces twice as much profit per dollar risked. Thus, if we risked $500 on each trade, Trading System 1 would generate $125 dollars while Trading System #2 would generate $250. But this not the complete picture. We are missing the frequency at which each trading system operates. For example, maybe Trading System #1 trades once per day while Trading System #2 trades once per week. We need to take into account the number of times the trading system trades over the number of days the system was tested. In Van Tharp’s book he described that as Expectancy multiplied by opportunity. Opportunity is nothing more than how often does a given trading system trade. Opportunity times Expectancy leads us to our final calculation for Expectancy Score.

This value is an annualized Expectancy value which produces an objective number that can be used in comparing various trading systems. In essence the Expectancy Score factors in a trading system’s trade frequency. The higher the Expectancy Score the more profitable the system. This final score allows you to compare very different trading systems.

Expectancy Score = Expectancy * Number of Trades * 365 / Days in historical test

The Number of strategy trading days is nothing more than the number of days your backtesting was performed.

With the Expectancy Score in hand we have a metric to aid us in comparing different trading systems. Other uses for Expectancy Score might also include using this value as a target for optimization. Often optimization is performed on net profit, profit factor, Sharpe ratio or other metrics. Using Expectancy may also be something worth pursuing. But how would you do that using TradeStation? Unfortunately, there is no easy way to do it with TradeStation. However, I’m working on some EasyLanguage code that will help with a manual process. More on that in a future issue.

Jeff is the founder of System Trader Success - a website and mission to empowering the retail trader with the proper knowledge and tools to become a profitable trader the world of quantitative/automated trading.