Math. Can it make us better traders?

Here is the part most traders never internalize: you can be wrong more than half the time and still make money. The variable that decides whether you are profitable is not your win rate in isolation — it is the relationship between your win rate and your reward-to-risk factor. Once you see the math, the next problem becomes obvious: your own brain is wired to sabotage it.

When an apple stem breaks from the tree branch, the apple falls to the ground (Newton’s law of universal gravitation). This happens every single time without deviation unless an unbalanced external force (Newton’s second law of motion), for example a hand plucking the apple and holding it, acts upon the apple. These are universally accepted certainties. The apple falls without thought, without planning but with very good reason…mathematics.

Mathematics is the study of relationships, properties, measurements, interactions and quantifications of numbers and objects. It governs nearly every aspect of life, occurs with certainty, and (possibly most importantly for our context) is devoid of emotional influence. It is this last piece — emotional influence — that I want to dive into further today.

What is the traders’ paradigm? Most traders fail because they book small wins and take large losses. Why do traders do this if it is self-sabotaging? Great question. I would venture to guess this phenomenon is best explained by the unconscious bias toward loss aversion. A trader enters a position, and they want to be right. Being right means they made money, and in that moment, making money feels good regardless of how much money it is relative to their overall financial position. Being right is a little rush of dopamine that tells the brain — yep, that’s right, I did that, I am a badass taking money from the big boys. Inversely, being wrong sucks. The money lost is likely immaterial; however, it dings the confidence.

Being wrong can make you question your setup, your ability, and your execution. If we know traders like being right and hate being wrong, how can we use this to our advantage by stopping it from being a detriment? As sentient beings, we are acutely aware of our environments, and our brain will act both consciously and unconsciously to protect us from perceived threats. When a trader is in profit on a position, he or she has a strong urge to close that position, lock in the profit, and declare victory. Inversely, when a trader is in the red, he or she is tempted to give the position “room to work,” hoping the position comes back and ends up in the green. This is our brain’s natural inhibition to protect ourselves. By understanding the mathematics behind this defensive mechanism, we can consciously work to override it and follow a more calculated, methodical approach to risk management. For a deeper look at the psychology side of this, see Minding Your Ps & Qs.

Now that we understand the challenges, let’s dive into the data. In Exhibit 1 we show the simulated results of 100 trades in S&P500 E-Minis (ES) using a 2-contract positioning with a 4-point fixed stop loss. We have simulated these results against various win rates and reward-to-risk (R:R) factors. For those not familiar, a win rate is simply the probability or percent you are correct — for example, a 40% win rate implies 40 winning trades and 60 losing trades in a sample of 100. The reward-to-risk factor is the multiple of the fixed stop loss at which you’re exiting your position. For a 2:1, you are exiting at 8 points when risking 4 points.

When reviewing the simulated results, you can clearly see the inverse correlation between win rate and R:R factor. As the R:R increases — meaning more reward for each unit of risk — the win rate required to break even decreases, as highlighted by the green bars.

Exhibit 1: Simulated Results of 100 trades at various R:R and win-rates

Lesson 1: Contrary to popular belief on Twitter, you do not need to be right every time to make money. In fact, it is very possible to make money being wrong MORE THAN you are right, if your risk-management is in line.

This is the same logic that drives the case against chasing extended moves and revenge-buying selloffs — see Knife Catching Usually Ends With Bloody Hands for the same R:R argument applied at the trade-entry level.

Now that we understand how win rate and reward-to-risk factor are inversely correlated, let’s look at what this can do for our equity curve. If we take a trader who is right 40% of the time and captures on average a 2:1 R:R (2x reward with 1x risk), we would expect to see a profit of 160 points, or $8,000, over 100 trades. Using a weighted-probability random-number generator, we simulated 100 trades, which resulted in 42 wins and 58 losses. The simulated results were a profit of 208 points, or $10,400.

Exhibit 2: Equity curve 42% win-rate 2:1 R:R in ES Points

As you can see, each incremental win results in a +24-point expectancy. This is important to note because your win rate will certainly fluctuate with market conditions. More importantly, we can use this understanding to avoid a common pitfall. When talking with traders, I often hear the phrase “trading is like taking two steps forward and five steps back.” Why is that the case? If we think back to the opening, where we talked about the subconscious mind being risk-averse, we know the mind does not like to book losers. Undisciplined traders have a detrimental habit of booking winners too quickly and allowing losers to run. If we take the same equity curve for a 42% win rate and we change just 2% of the trades from regular stop-outs (-4 points per lot) to “large losers” of -20 points per lot — and then we assume the trader makes another fatal mistake of averaging down with an add for a total loss of -80 points on the trade — what happens? As you can see in Exhibit 2B, if the trader allows just 2% of their trades to become large drawdowns, the equity curve goes from +208 to +48. If the trader allows a third large drawdown, the trader has gone from highly profitable to a loser in just three trades.

Exhibit 2B: Equity curve 42% win-rate 2:1 R:R in ES Points but taking 2 large (80 point) drawdowns

Lesson 2: Avoiding the booking of losses due to loss-aversion is counterproductive. A large loss destroys your equity curve. You need to increase win rate by 8% to counterattack the negativity of two large losses.

Now that we understand the math, how can we put this into practice? First off, let’s be clear: risk management is highly personal. What works for me may not work for Leo, and what works for Leo may not work for Job. With that said, there are some best practices that can be adapted to almost any style:

1. Aim for a 2:1 reward-to-risk factor. As traders, we need to be compensated for our risk. By aiming for 2:1, you allow yourself enough room to weather some bad days here and there while still being comfortably profitable.
2. Avoid large losses both in single trades and in days. No, losing does not feel good, but it is always better to stop out and reposition than to dig out of a large hole by averaging down. Just look at the math.
3. Protect yourself — everyone has bad days. Set daily loss limits or have shutdown rules. I have adopted Job’s three-strike rule: if I take three losing trades in a row, I take at least a 30-minute break away from the screens. I also use a daily loss rule that is the equivalent of about six losing trades without a winner. If I have three losers, take a break, and come back with three more losers, then I am just out of touch with the market on that day. No sense in continuing to give away capital. Shut it down, return the next day.
4. Don’t rely on discipline when you can have hard stops. I worked with my broker (EdgeClear) to incorporate my loss rules into my account settings on the server side. This takes away the emotion, as it is fixed by the broker’s risk-management department. In the days of $400 ES mini margins, you can be trading 20+ lots on a $10K account. Just because a broker will allow you to do it doesn’t mean it is a good idea.

Sizing is the other half of the equation. The math above assumes a fixed 2-contract position, but in practice the right size depends on what the market is offering you on a given day. For one approach to scaling size with conditions rather than emotion, see Using Relative Volume Effectively.

If you do these four steps, you’ll help mitigate human emotions out of your trading and rely on the factual data points shown above. This will help you stop sabotaging yourself and avoid the sensation of two steps forward and five steps back. Once the rules are in place, the next step is measuring whether you’re actually following them — Jared Tendler’s Dialing In on Trading Performance is the framework I lean on for that.

Remember, trading is hard. Emotions are harder. Follow these simple steps to give yourself the highest probability of success. Rome wasn’t built in a day, and your account won’t be either. If you teach yourself to be appropriately compensated for risk and to avoid large losers, you’ll build wealth that can change lives for generations to come.

Happy trading,

Cap

@capturetheta135