# The Math of Winning

## What Should Your Win Rate Be?

Most forex traders (and traders in general) think that their win percentage is the most important statistic of their trading. The relieving truth is that this is not the case.

Trading is one of the most counter-intuitive skills you will ever learn. Often what is right seems wrong. The idea that you can lose more trades than you win and still make money *feels *wrong.

Human beings are used to absolutes. We are not good at dealing with probabilities and vagueness. We like black and white scenarios. We like to think that winning equals good, and losing equals bad.

In trading this couldn’t be further from the truth and is in fact a dangerous attitude to have.

It Doesn’t Really Matter

You should not build your trading plan around your win percentage. Instead, you need to build it around your average risk to reward.

Using a simple mathematical formula you can easily calculate exactly what your win percentage *needs *to be in order to break-even. This is the only win-rate statistic that truly matters.

Stay above this hit rate and you will be making money.

How much money depends entirely on how good your strategy is for the current market conditions, but whatever you’re making, you can sleep knowing that you are on the right side of the fence.

The formula is this:

Minimum Win % = Risk ÷ (Risk + Reward) x 100

So, for example, if you have a 2:1 average reward-to-risk ratio then you only need to be winning 33% of your trades over the long-term in order to break-even.

1 ÷ (1 + 2) x 100 = 33%

Exceed that win rate and you will be making money.

For a 3:1 average reward-to-risk ratio you only need to win 25% of the time to *not* lose money (excluding commissions of course).

1 ÷ (1 + 3) x 100 = 25%

This means the higher your average reward, the lower your win percentage needs to be in order to make money.

Sounds good right?

It is. But the catch is that the more reward you shoot for, the less likely you are to get it.

So it’s not as simple as just shooting for higher reward.

When developing strategies you must balance your risk:reward profile with your win percentage until you find a combination that you feel comfortable with.

The Psychology of Losing

Your win percentage is almost meaningless on its own in terms of making you money. Almost.

This theoretically means that if you find a strategy with a huge reward:risk ratio that only wins 10% of the time but still makes you money, then technically that is a feasible way to trade.

But even though you would make money, the chances of you having the discipline to stick to such a strategy is slim.

The problem is our ancient psychological structures inherent in our brains.

A computer algorithm could trade a strategy like that no problem. It can take 9 losses in a row and not feel any trepidation whatsoever about taking the 10th trade.

Human beings on the other hand are an entirely different story.

*What Do Rats and Traders Have In Common?*

During a lecture I watched recently by Canadian psychologist Dr. Jordan Peterson he mentioned a very interesting fact about game ethics in rats.

It turns out that rats – which share 97% of genetic and primitive behavioral characteristics with humans – have an inbuilt maximum loss tolerance.

Rats wrestle, just like human beings. And they even pin each other, and they love that.

However, if you repeatedly pair them, and the big rat doesn’t let the little rat win at least 30% of the time, then the little rat won’t invite him to play anymore.

Dr. Jordan B. Petersonon Dr. Jaak Panksepp’s Research

Jordan goes on to summarize what this basically means.

First of all, everything in life is a game. Some games are more serious than others, but they are games nonetheless.

Trading is a game. It’s a high-stakes strategic game that requires a particular kind of skill and dedication to be successful at it.

But just like rats, human beings have a maximum pain tolerance for losing when it comes to playing games. If we engage in a competitive activity with someone or something that has superior talent to us (ie. the markets) and we don’t win at least *some *of the time, we tend to give up.

Human beings enjoy a challenge. In fact we thrive on challenge. Challenge imbues our lives with meaning. Without anything to challenge us to improve and get better, we stagnate. We atrophy. We lose interest.

However, if the challenge is literally impossible and no matter how hard we try we keep experiencing loss then that is arguably worse than no challenge at all.

It demoralizes us. It discourages us from committing to the game. It makes it difficult for us to develop confidence in our ability to perform.

Of course there are exceptions to this. For everyone, their personal loss tolerance will be different. There *are *traders out there with abysmal win rates that still kill it in the markets.

But be honest with yourself. There’s a big difference between backtesting and trading live. A strategy with a low win rate that makes lots of money might look fantastic on paper.

But if you are losing more often than you can psychologically endure, then you are going to make mistakes. You are going to lack conviction in your trades, and you are going to sabotage yourself.

Winning Is Subjective

Over time you will discover what your personal loss tolerance is. And then once you have that number you will know which strategies are more likely to work for you and which aren’t.

With this information you can craft your perfect trading plan.

I was very attracted to trend-following when I first began trading. It didn’t bother me to know that big winners were rare and that I would be taking a lot of losses. I always thought that I could handle losing lots of trades if it meant I still made money.

And to a large degree this is true. I have no ego when it comes to trading. I have nothing to prove to anyone. I *am *just trying to make money from the markets in any way possible.

But I still feel much more comfortable if I am winning at least 45% of the time. Whenever I get below that ratio, doubts begin to creep in.

I begin to wonder how long the losing streak will go on for and if I ought to be doing anything different to prevent it from getting worse.

I suspect that I am not the only one who feels this way during rough times.

Having a low win rate can work. But it opens you up to a host of potential psychological obstacles and traps. So keep that in mind when you are designing your strategies.

Forex trading coach Akil Stokes discussed this topic in depth in his 28th podcast episode. If you are interested in learning more about this subject then I highly recommend giving it a listen.

*The Trading Coach Podcast – Why I Prefer NOT To Win Big*

“Simple Math”

In the process of researching reference material I stumbled across this interesting post by Singaporean trader Rayner.

Rayner is a well-known trend-follower and trading educator with a great reputation among the trading community. I have watched hundreds of his YouTube videos and I can personally vouch for his authenticity and credibility.

In this post he breaks down the basics of risk:reward. He also lays out the algebraic formula for calculating your expected average return based on your past trades and your win percentage.

I know, I know. Math sucks.

Or maybe it doesn’t – if you’re someone who is great at math and enjoys doing it, then good for you! You are definitely going to excel as a trader so long as you can keep your psychology in check.

Personally I am not great at math, and it is a handicap. I appreciate and respect math, but I suck at it. Luckily for traders like me, this math is very simple. If I can do this math, anyone can do this math.

The formula is this:

Expectancy = ((1 + (Average_Win ÷ Average_Loss)) x Win Rate) - 1

For those traders reading this who are more like myself and need a little bit more help to understand this concept, I’ll lay out an example of how to use this math to calculate my own personal minimum win percentage and expectancy.

I am going to convert Rayner’s calculations from $ amounts into % amounts as I believe that is a better way to analyze your trading results.

Calculating Your Expectancy & Minimum Win %

At the time of writing this, the year hasn’t gone very well for me.

Last year was fantastic. After three years of trading actively I finally broke through my period of consistent losing and had four profitable months in a row for the first time ever.

It was exhilarating, and it was what inspired me to create this website and blog. I knew that I was very close to attaining consistent profitability and I wanted to share that journey with the trading community.

Fast-forward to April 2019: I am at break-even for the year and in a 6% drawdown after 3 months of trading and struggling to get to new equity highs.

It has been rough, but I have learned a lot about my strategy and myself. (Edit 20th June, 2019: I have since recovered from this drawdown).

Along the way I’ve been collecting as much statistical data about my trading as I can manage so that I can identify areas in which I need to improve.

Using this information I can plug my numbers into these mathematical formulas to calculate my personal “danger zone” and my theoretical expectancy.

If my strategy’s win rate falls below this danger zone then I am going to lose money. If it stays above it, I will make money. And the expectancy formula will tell me how much I can expect to make (or lose) on average over the long term.

*How to Calculate Your Win Percentage*

I have taken 58 trades this year (since February, when I began my **public trading journal**).

22 won, 31 lost, 5 broke even. That means 9.43% of my trades were break-even trades, but we’ll ignore those for the purposes of this example.

This gives me a total of 53 trades that either won or lost and a win rate on those trades of 41.51%.

Win_Rate = (Winners ÷ Total) x 100 Win_Rate = (22 ÷ 53) x 100 = 41.51%

*How to Calculate Your Average Gain/Loss*

My average winner has been 1.45%. My average loser has been 1.03%.

I calculated my average winner by adding up all of my winners (as a percentage of my total capital) and dividing the result by the total quantity of winning trades. I did the same for losing trades.

As my actual numbers are far too long to display like this, here is a theoretical example:

Average_Win = Winning_Total ÷ Total_Winning_Trades Average_Win = (0.50 + 1.99 + 2.08 + 0.50 + 1.01) ÷ 5 Average_Win = 1.27% Average_Loss = Losing_Total ÷ Total_Losing_Trades Average_Loss = (1.01 + 1.01 + 1.00 + 0.99) ÷ 4 Average_Loss = 1.03%

*How to Calculate Your Expectancy*

Now that I know my win rate, my average gain and my average loss, I can calculate my average long-term expectancy.

This is how much I can expect to make *on average *per winning trade over the long-term. Obviously some trades will be larger, some will be smaller, but the average gives you an idea of what is the “mean”.

We’ve all heard of “mean reversion” in markets, and our trades are no different. Calculating our mean (aka. average) win and loss will give us an idea of when we are under-performing or over-performing our strategy.

The formula for calculating my expectancy based on my numbers is this:

Expectancy = ((1 + (Average_Win ÷ Average_Loss)) x Win_Rate) - 1 Expectancy = ((1 + (1.45 ÷ 1.03)) x 0.4151) - 1 Expectancy = 0.99946 - 1 Expectancy = -0.00054%

What does this mean? It means I’m toying with my danger zone. My win rate is too low for my average win size.

For every 1% that I risk, I am losing -0.00054%.

Which means I am basically breaking even at this rate, which is what my results show.

*How to Calculate Your Danger Zone*

Now that I know I am trading within my “danger zone”, how do I work out exactly what win % that danger zone is?

Using the formula that I mentioned at the beginning of this post, I can plug in my numbers and find out.

Minimum_Win_Rate = Risk ÷ (Risk + Reward) x 100 Minimum_Win_Rate = 1.03 ÷ (1.03 + 1.45) x 100 Minimum_Win_Rate = 41.53%

This means that based on my average risk of 1.03% and reward of 1.45%, I need to win 41.53% of the time in order to *not lose money.*

My current win rate is 41.51%. Which means I have a slightly negative expectancy, which means I need to lift my game. I am not expressing my edge effectively.

I should have a win rate much closer to 50% according to my backtesting results. But I am a developing trader with a lot still left to learn so it’s not particularly surprising that I am under-performing right now.

I have also only taken 58 trades in this sample size. I really need to have taken upwards of 100 before I get a definitive reading on my statistics.

During backtesting my strategy endured drawdowns that lasted up to three months. But over hundreds of trades it had a strong positive expectancy. So as long as I do my best to stick to my plan then I have no reason to panic yet.

I just have to keep on keeping on and putting in the work – then reassess when I hit 100 trades.

Theoretical Examples

The previous expectancy example involves real numbers from my personal trading which makes the math convoluted.

Let’s do some quick examples with theoretical clean numbers for easier reference.

*Break-Even Expectancy*

**Average Win:** 3%**Average Loss:** 1%**Win Rate:** 25%

Minimum_Win_Rate = Risk ÷ (Risk + Reward) x 100 Minimum_Win_Rate = 1 ÷ (1 + 3) x 100 Minimum_Win_Rate = 25% Expectancy = ((1 + (Average_Win ÷ Average_Loss)) x Win_Rate) - 1 Expectancy = ((1 + (3 ÷ 1)) x 0.25) - 1 Expectancy = 1 - 1 Expectancy = 0%

If you win 3% and lose 1% on your average trade and you have a win rate of 25%, then for every 1% you risk over the long-term you can expect to make 0%. Which means you break even.

*Negative Expectancy*

**Average Win:** 2%**Average Loss:** 1%**Win Rate:** 30%

Minimum_Win_Rate = Risk ÷ (Risk + Reward) x 100 Minimum_Win_Rate = 1 ÷ (1 + 2) x 100 Minimum_Win_Rate = 33% Expectancy = ((1 + (Average_Win ÷ Average_Loss)) x Win_Rate) - 1 Expectancy = ((1 + (2 ÷ 1)) x 0.30) - 1 Expectancy = 0.9 - 1 Expectancy = -0.1%

If you win 2% and lose 1% on your average trade and you have a win rate of 30%, then for every 1% you risk over the long-term you can expect to lose -0.1% (on average).

*Positive Expectancy*

**Average Win:** 1.5%**Average Loss:** 1%**Win Rate:** 50%

Minimum_Win_Rate = Risk ÷ (Risk + Reward) x 100 Minimum_Win_Rate = 1 ÷ (1 + 1.5) x 100 Minimum_Win_Rate = 40% Expectancy = ((1 + (Average_Win ÷ Average_Loss)) x Win_Rate) - 1 Expectancy = ((1 + (1.5 ÷ 1)) x 0.5) - 1 Expectancy = 1.25 - 1 Expectancy = 0.25%

If you win 1.5% and lose 1% on your average trade and you have a win rate of 50%, then for every 1% you risk over the long-term you can expect to make +0.25% (on average).

*Increased Risk Expectancy*

**Average Win:** 5%**Average Loss:** 2%**Win Rate:** 45%

Minimum_Win_Rate = Risk ÷ (Risk + Reward) x 100 Minimum_Win_Rate = 2 ÷ (2 + 5) x 100 Minimum_Win_Rate = 29% Expectancy = ((1 + (Average_Win ÷ Average_Loss)) x Win_Rate) - 1 Expectancy = ((1 + (5 ÷ 2)) x 0.45) - 1 Expectancy = 1.58 - 1 Expectancy = 0.58%

If you win 5% and lose 2% on your average trade and you have a win rate of 45%, then for every 1% you risk you can expect to make +0.58% (on average).

*Reduced Risk Expectancy*

**Average Win:** 2%**Average Loss:** 0.5%**Win Rate:** 50%

Minimum_Win_Rate = Risk ÷ (Risk + Reward) x 100 Minimum_Win_Rate = 0.5 ÷ (0.5 + 2) x 100 Minimum_Win_Rate = 20% Expectancy = ((1 + (Average_Win ÷ Average_Loss)) x Win_Rate) - 1 Expectancy = ((1 + (2 ÷ 0.5)) x 0.5) - 1 Expectancy = 2.5 - 1 Expectancy = 1.5%

If you win 2% and lose 0.5% on your average trade and you have a win rate of 50%, then for every 1% you risk over the long-term you can expect to make +1.5% (on average).

Conclusion

The point of this post is to demonstrate that focusing on your win percentage alone is pointless.

Unless you are balancing your average gain and loss *against *your win rate then you are focusing on the wrong metric. Increasing your win rate won’t help if it’s not the problem.

Determining what *is *the problem is much easier if you know your danger zone and your average expectancy.

If your backtesting results show a positive expectancy but your live trading shows a negative expectancy, then you know that there’s a high chance that you are not executing your plan properly and consistently, or that your edge may be waning.

For example – if you are winning a lot but not making any money, then that means you are not allowing your winners to run and you are suffocating your profitability by cutting trades too early.

Trying to win even *more *won’t help that type of problem. In fact I can guarantee it will make things worse.

Likewise, if your live trading results are going through a drawdown, but your overall expectancy is positive, and you are striking above your “danger zone”, then you have nothing to worry about.

Keep executing as normal. Don’t change* anything.*

This is our job as traders. We devise a plan, we execute the plan, we analyze our results, and then we improve the plan – but only when necessary.

Do that long enough and it’s simply a matter of time before you start making consistent returns as a trader.