Reward vs. risk? You can throw it out on the window. Winning percentage? You can toss that one away, too.
OK. Maybe we’re overstating that a bit. However, it is fair to say that what probably are the two most popular trading statistics basically are useless — by themselves. Unfortunately, that exactly is how many people evaluate them. Many traders will jump into a trade with a reward-to-risk ratio of three-to-one, just because they presume this means they will win $3 for every $1 they lose.
Likewise, other traders are drawn to 90% winning percentage strategies like moths to a flame. They must enjoy the frequent sound of the cash register, ringing after each win. No doubt, making money on nine out of every 10 trades does sound pretty good.
In both cases, however, the traders following these approaches are missing a critical piece of the puzzle, and it’s a piece that can make a major difference in their bottom lines. “Questionable curves” (below) shows two hypothetical equity curves. The blue curve represents a three-to-one reward-to-risk ratio system. The green curve represents a 90% winning percentage system. Both systems are appealing according to their selected standalone statistics, yet both systems are losers. Why?
Simply put, those trading metrics by themselves are insufficient. Alone, they tell you only part of the story, and you easily can miss important facts about your trading system. All is not lost, though, as the key is combining these figures.
One plus one are three
When the reward-to-risk ratio and winning percentage are evaluated together, it becomes easy to determine if a trading strategy has promise — or if it’s doomed to financial failure. The equation for combining these numbers is:
(Win % / 100) * (Reward:Risk) - [(100 – Win%) / 100]
As long as this equation is greater than zero, the strategy will be profitable. Additionally, the higher the resulting number, the better the system. Many good, tradable systems can be found with values between 0.10 and 0.50.
The number is commonly referred to as “expectancy,” and it is an excellent statistic. In plain English, it tells you the amount of money you would win per dollar risked. For example, a value of 0.20 indicates that in the long run you will make 20¢ for every dollar you risk trading this system. In other words, it is a really good system.
One word of warning, however. Another incorrect formula for expectancy, at least in the context we’re interested in, is prevalent, particularly if you simply search this topic on the internet. The following is the incorrect formula: (probability of win * average win) – (probability of loss * average loss). Often, this will be described as expectancy; it is not. This is the average trade value. In our correct formula, we divide the average trade value by the negative of the average loss. This is the extra step that gives us the amount of money we would win per dollar risked.
Let’s evaluate our initial simple example using the correct formula for expectancy. In “Questionable curves,” the blue curve shows a system that has a reward-to-risk ratio of three-to-one, but the winning percentage (not initially disclosed) was only 20%. Plugging these numbers in, we get:
(20 / 100) * (3) - [(100 – 20) / 100] = -0.2
Because this is less than zero, the strategy is a losing one, which is obvious from the chart. Any trader using such a strategy eventually will, and inevitably, go broke trading it.
Also in “Questionable curves,” the green curve tracked the equity of a system that won nine out of every 10 trades. However, its reward-to-risk ratio was only 0.091. Plugging those numbers in, we get another losing strategy:
(90 / 100) * (0.091) - [(100 – 90) / 100] = -0.018
The lesson here is that you can’t evaluate a trading system with just the reward-to-risk ratio or just the winning percentage. You need both figures to determine the viability of your trading system. “Pick your system” (below) shows the expectancy of various reward-to-risk ratios with respect to their winning percentage. This chart provides a quick comparison of a number of values for each statistic. For example, a system with a reward-to-risk ratio of three-to-one and a winning percentage of 60% would be superior to a system with a reward-to-risk ratio of 1.5-to-one but a winning percentage of 90%. (And both of these would be exceptional systems, by the way.)
As we can see, many different combination sets of reward-to-risk and winning percentage produce winning strategies. You can make money with a low reward-to-risk, and you also can be profitable with a small winning percentage.
While by themselves these trading statistics don’t provide a complete picture of a trading system’s value, traders can’t ignore the psychological benefit they may offer. We are, after all, conditioned since grade school to evaluate ourselves in terms of winning percentage. Everyone wanted to score a 95% and bring home an “A,” after all. And who didn’t want to hit a game-winning home run in playground baseball?
If you are one of these traders, then having a low winning percentage might be impossible to stomach, and never banking a big win might be unfulfilling. The risk here is an unfulfilled trader might become an undisciplined trader. Although you shouldn’t let emotions affect your trading, not recognizing your personal tolerance for financial risk would ignore reality.
To that end, if you have an idea of your desired winning percentage, the equation presented earlier can be arranged to tell you the minimum reward-to-risk needed.
For positive expectancy, R:R > [(100 – win%) / 100] / (win % / 100)
In the same manner, the equation can be rearranged to return the necessary winning percentage for a given reward-to-risk ratio:
For positive expectancy, win % > 100 / (R:R +1)
Using these two modified equations, it is easy to determine the minimum requirements for profitability. This is shown in “Rewarding relationships” (below). Any strategies above the line will be profitable, while those below the line will be losers and should be avoided.
Using the concepts presented thus far, it is easy to evaluate real-world trading strategies. The table below presents four futures trading strategies. All have some appealing aspects, whether it is high winning percentage or a high reward-to-risk ratio:
System Win % R:R
1 25.3% 3.8
2 43.7% 2.9
3 44.7% 1.5
4 54.6% 0.5
But of these four systems, which are profitable? Which one is the “best?” Plotting the systems on the reward-to-risk vs. winning percentage curve gives the answers. This is shown in “Comparing systems” (below). With the four trading strategies plotted, it is easy to eliminate system No. 4 because it has a negative expectancy. This is a good example of a high winning percentage system not necessarily being a profitable system.
The other three strategies all have positive expectancy, but is one better than the others? Some traders will favor system No. 1 because it has the highest reward-to-risk ratio. The thrill of the occasional jackpot trade entices some traders, even if the winning percentage is low.
Other people will like system No. 3 the most because it has the highest winning percentage of the three profitable systems. These traders like to hear the cash register ring often from winning trades, even if those wins are relatively small. These are the traders who usually seem to win, and may even attract followers because of their seemingly prescient predictions.
Psychological (or marketing) reasons aside, the best system to trade is system No. 2. As shown on the plot, it is the furthest away from the breakeven line. Therefore, the expectancy of system No. 2 is the highest. It does not have the best reward-to-risk ratio, and it does not have the best winning percentage, but it does have the best combination of the two.
By themselves, reward-to-risk and winning percentage have little value. The same can be said about other trading statistics, such as net profit (which needs drawdown to quantify risk), consecutive winners/losers (which needs overall winning percentage for perspective) and average trade profit (which needs number of trades to gauge overall profitability). In each of these cases, the single statistic does not tell you enough about the strategy.
Even expectancy, by itself, does not tell the whole story. A lower expectancy system that trades frequently may indeed be preferable to a higher expectancy, but infrequently traded strategy. Again, focusing on just one number may lead to incorrect conclusions. The key is to evaluate all the pertinent statistics of a strategy at the same time. That way, some bad statistics can be overcome with the help of complementary statistics. In the end analysis, a thorough evaluation of all trading statistics, taken together, is the best approach for a good decision.
Kevin J. Davey has been trading for more than 20 years. Prior to trading full time starting in 2008, Kevin was a quality assurance and engineering executive for an aerospace company. You can reach him via his website www.kjtradingsystems.com.