Risk-Reward — The Mathematics of Long-Term Profitability

You have two coin flips available. The first pays $200 if heads and costs you $100 if tails. The second pays $100 if heads and costs you $200 if tails. The probability of heads or tails is exactly equal on both, 50 to 50.

Which do you choose?

Obviously the first. Over ten flips you expect five wins and five losses. On the first coin you make $1,000 and lose $500, net $500 profit. On the second you make $500 and lose $1,000, net $500 loss.

The probability was identical on both. The outcome over time was completely different because the ratio between what you win when you are right and what you lose when you are wrong was different.

This is the risk-reward ratio. And it is the mathematical engine that determines whether any trading approach is profitable over time regardless of win rate. Arjun from Chapter 1 had a 60% win rate. He was choosing the second coin every trade without realising it.

A risk-reward ratio of 1:2 means you risk $1 to make $2. If your stop is 30 pips away and your target is 60 pips away, you are risking 30 to make 60.

A risk-reward ratio of 1:3 means you risk $1 to make $3. If your stop is 20 pips away and your target is 60 pips away, you are risking 20 to make 60.

The minimum risk-reward ratio worth trading is generally considered to be 1:2. At 1:2 you only need to be right on 34% of your trades to break even over time. At 1:3 you only need to be right on 25% of trades to break even. Even with a win rate well below 50%, a consistent 1:2 or better risk-reward ratio produces a profitable approach over time.

This is the single most important mathematical insight in all of trading. You do not need to be right most of the time to make money. You need your wins to be bigger than your losses.

You are asking the market: where is the next place that sellers are likely to appear and cap this move? That is where your target goes. Not at a round number. Not at a point that produces a nice-looking ratio on paper. At the level where the chart tells you the move is likely to stall or reverse.

Once you have identified the logical target, you compare it to your stop distance to calculate the natural risk-reward of the trade. If the natural risk-reward is at least 1:2, the trade has acceptable expectation. If the natural risk-reward is 1:1 or worse, the trade does not have positive expectation regardless of how good the technical setup looks and you should not take it.

This filter alone, rejecting trades where the natural risk-reward is below 1:2, eliminates a significant proportion of losing trades from most retail traders'' records without requiring any additional analytical skill.

Expectancy equals win rate multiplied by average win, minus loss rate multiplied by average loss.

For a trader with a 45% win rate, an average win of $200, and an average loss of $100: expectancy equals (0.45 x $200) minus (0.55 x $100) equals $90 minus $55 equals $35 per trade. This trader makes an average of $35 per trade. Over 100 trades that is $3,500 positive expectancy from a 45% win rate.

For a trader with a 65% win rate, an average win of $100, and an average loss of $200: expectancy equals (0.65 x $100) minus (0.35 x $200) equals $65 minus $70 equals minus $5 per trade. This trader loses an average of $5 per trade despite winning 65% of the time.

Calculating your expectancy from your trading journal gives you a completely objective measure of whether your approach has an edge. It removes all emotional interpretation and tells you in precise mathematical terms what the numbers actually say.

Closing a winning trade early because of nervousness before the target is reached converts a planned 1:3 trade into a de facto 1:1 trade. The analysis was right. The target was correct. But the emotional discomfort of watching the trade fluctuate caused an early exit that reduced the expected gain. Over many trades this habit systematically destroys positive expectancy.

Moving a stop to breakeven after the trade has moved a certain distance in your favour is a legitimate technique. It removes the risk of a winning trade turning into a loss without prematurely closing a good position. This is not the same as moving a stop against yourself. This is moving a stop in your favour.

The principle is this. Let winners develop toward their target. Cut losers at the stop. The ratio between your wins and losses over time is the primary determinant of whether your approach makes money. Any behaviour that systematically reduces your wins or increases your losses works against your long-term profitability even if it feels comfortable in the moment.