Finding your Real Win Rate with Binomials and Exa Score
This article will provide a brief and simple introduction to using Binomial Proportion Confidence Intervals in your trading system so that you don’t fool yourself with false positive results.
STATISTICS, EXPLAINED BY PLAYING CARDS
Imagine I present you with a pack of playing cards, and I shuffle them up. The game is to have you draw a card, and if it’s black, you win $1, if it’s red, you lose $1.50.
You draw 5 cards off the top of the shuffled deck. Four came out black, one came out red. You celebrate with your friends about your 80% WIN RATE!!!
You did the math, and know that with a 1:1.5 Reward to Risk with an 80% win rate is fantastic. A money printer! With these statistics, across 100 games (or trades) you’ll have an average total gain of $80, and an average total loss of $30. This is great news, a winning system!!!
But, just hold on a minute here. That 80% win rate is only the results you got after 5 trades (or card draws). How reliable is that data, really?
As you’ve probably already figured out, with 26 red cards and 26 black cards, the odds of this game are 50/50. Your true win rate is 50% over the long run. 80% is only your observed win rate. Some quick mental math tells you that if we run through the whole deck, I’ll walk away with more money than you. This is a losing game for you, and you shouldn’t play. Even if the Reward : Risk was 1:1, at best this game is a waste of your time, and more likely, you’ll get eaten by spread and fees. (Remember, 1:1 at 80% win rate would still be amazing, but you know that’s not the case here)
- Observed Win Rate: The percentage that you’ve won over the last X number of trades. This number changes, and is known at all times.
- True Win Rate: The actual percentage of the time you can win with your current strategy if you don’t change anything. This number never changes, and can never be fully known. Binomials get you close.
Now, since I can’t convince you to play my game when you know that the odds are 50/50, let’s mix it up a bit.
The Card Game With Different Odds
Let’s start fresh. I open 9 decks of cards, throw all the cards in a pile and shuffle them thoroughly. Then I pick cards at random and assemble a 52 card deck without looking at them.
The Reward : Risk is still the same as the last game. But now, what’s the win rate? At this point, it’s anyone’s guess. Maybe all 52 are black, and you’ll win every trade. That’s highly unlikely, but now, until you’ve gone through every card, the true win rate will be unknown.
So, maybe you draw the first 5 cards, and you observe an 80% win rate. You know that only 5 cards into the game, you can’t trust that data. But, when can you start trusting it? That is the purpose of this article.
Game Theory Limits vs Trading
With the deck of 52 cards, there’s a limit to the number of cards you can actually draw: 52. But with trading, there’s no limit to the number of trades you could potentially make.
So with the cards, you might find that after all 52 cards are revealed, the deck that was comprised of 9 other decks has a 68% win rate. At 1:1.5 or 1:1, the odds are in your favor. That’s a game you’ll profit at if you just keep playing. But you only found out the true win rate after there were no more cards to draw.
Key Point: Your Observed Win Rate grows closer to your True Win Rate with every new data point. (Each card draw, or each trade)
But remember, with trading- there’s always another trade.
The Problem
Where do you draw the line? It’s obvious by now that 5 trades doesn’t tell you enough about your system to know if you’re going to be a winner over the long haul. Most people probably knew that before this page even finished loading.
Some people suggest 100 trades, some suggest 3 months, some 6 months. Those are all arbitrary!
3-6 months may be much longer than needed if you’re a scalping day trader, making 25 trades a day. And that’s way too short if you’re swing trading and make 2 trades a month.
At what point do you say, "That’s enough trades with this system to believe in it with enough conviction that I want to risk real money on it…(is it now? how about now?)"
The Solution: Binomial Proportion Confidence Intervals
Using some moderately fancy math, you can look back at your previous data (trade results: win/lose) and start to narrow in on your true win rate. Remember, you’ll never know 100% what your true win rate is until you’re done trading. But your observed win rate will grow closer and closer with every new data point. (Convergence)
What you can know, is how confident you should be that your observed win rate is close to your true win rate.
I’m going to first introduce the chart, then explain it. Have a look:
Total number of trades here is irrelevant, what’s important is how tight the red and blue bands are.
This chart is the same as the one above, but it is a snapshot of only the 7 most recent trades, zoomed in and scaled logarithmically, for readability.
The black line is the observed win rate, and there’s a Seven 9s (99.9999999%) guarantee that the (unknowable) True Win Rate falls somewhere between 56% and 62%, as of the most recent trade on the far right. The Current Observed Win Rate is 59%. This particular graph is also taking into account the Reward to Risk Ratio of the past results, and plotting that as the green line.
With the current R:R, a 53% win rate would be break-even. The current observed win rate is 59% (Profitable!) but there’s still room for error. We don’t have enough data yet. With more trades, the chart will continue to plot and the red and blue lines will continually tighten inward toward the black line. Hopefully when they’re within just a few percent of each other, they’re still well above that green line!
When you first start tracking your trades with this system, the red and blue will be at 0% and 100% for a while. If we checked it after only 5 trades (4 winners 1 loser like before), your black line will be at 80%, but the red and blue will still be at 0 and 100. There’s a 99.9999999% guarantee that your win rate is between 0% and 100%!
And now that you’re using this tool, you’ll know not to take that 80% win rate seriously. You know to wait until those bands tighten up and you’re profitable within a very narrow range of possible error if nothing changes in your system.
Exa Score
You shouldn’t trade your system live, risking real capital, until you have confidence in your system. A Binomial Proportion Confidence Interval Tool will provide you with a non-arbitrary means of mathematically gaining the confidence you need to trust that your system works and is worth risking real money on.
I use a final tool, which is derived from the Binomial Proportion Confidence Interval- called Exa Score. This essentially crunches all the numbers into a single metric, if you will.
The Exa Score is a combination of your Confidence Level, which is defined by subtracting the red line from the blue line, on the graph. Subtract that value from 100. That number is added to your Edge, which is defined by subtracting the green line from the black line. This value is multiplied by the Confidence Level once again. This is your Exa Score. I've laid it out as a math formula below for more context.
Confidence = 100 - (Blue - Red)
Edge = Black - Green
Exa = (Confidence + Edge) * Confidence
If your Exa Score is below 100, you usually don't have enough trade data to determine whether or not your system is profitable. If your confidence is above 99% and your Exa is still below 100, you have an unprofitable system. If your Exa Score is above 100, you have a profitable system!
Here’s a Link to a free spreadsheet that you can use to manually enter your trade P&L. I recommend inputting dollars gained/lost on each individual trade.
Good luck out there!
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