The math behind trading
Day 09 of 26 · 10 min read · By DT Terminal
Whether a set of trades makes or loses money comes down to a few numbers: how often you win, how big the wins are against the losses, and what each trade costs. This lesson covers risk to reward, the breakeven win rate, expectancy in R, and why a run of losses is normal even for a sound approach.
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Worked examples, the quiz and the interactive tools are in the app. Free account.
One trade tells you almost nothing
Every trade ends at the stop or the target. Price is moved by things you cannot see, so one trade is close to a coin flip, even when the coin is slightly weighted your way.
The weighting only shows across many trades: the law of large numbers. Flip a fair coin 10 times and 7 heads is ordinary. Flip it 1,000 times and 700 heads would be extraordinary.
All three lines are the same strategy, with a true win rate of 40%. After 10 trades they ranged from 10% to 70%. By 200 trades, only from 39.0% to 41.5%.
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Price is moved by thousands of other traders, news and order flow you cannot see. No amount of preparation tells you which way the next trade will end.
The law of large numbers says the more often you repeat something with a fixed chance, the closer the results come to that chance. A trader looking at the cold run in the figure would quit a strategy that works, and a trader looking at the hot one would think they had found something far better than they had.
One loss says nothing about whether an approach works, and neither does one win. Many traders wait for a sample of at least 50 to 100 trades taken the same way before drawing conclusions, and even then they treat the results as estimates.
Win rate and reward to risk
Win rate is the share of trades that won: 4 winners out of 10 is a 40% win rate.
Reward to risk (RR) is the average win compared with the average loss, both in R. As How to choose your risk explains, 1R is what you lose when the stop is hit. A trade that risks 20 pips to make 40 is 1:2, and a win pays +2R.
Neither number means anything on its own. The account only cares about the two together, and the number that combines them is the expectancy. Here are two traders, each judged by the wrong one:
| Win rate | Reward to risk | Result per trade | |
|---|---|---|---|
| Trader A | 70% | 1:0.4 | −0.02R |
| Trader B | 35% | 1:2.4 | +0.19R |
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Trades closed at breakeven are often left out of the win rate, so they neither help nor hurt it. A high win rate feels good and a big RR looks impressive, but neither one alone tells you whether the account grows.
Expectancy in R
Expectancy is what a strategy makes or loses per trade on average, over many trades. In R it works on any account size:
expectancy = win rate x avg win (R) minus loss rate x avg loss (R)
Positive means the strategy has an edge, before costs. Negative means it loses over time, however it feels trade to trade. The table assumes every loss is 1R:
| Win rate | 1:0.5 | 1:1 | 1:1.5 | 1:2 | 1:3 |
|---|---|---|---|---|---|
| 30% | −0.55R | −0.40R | −0.25R | −0.10R | +0.20R |
| 40% | −0.40R | −0.20R | 0.00R | +0.20R | +0.60R |
| 50% | −0.25R | 0.00R | +0.25R | +0.50R | +1.00R |
| 60% | −0.10R | +0.20R | +0.50R | +0.80R | +1.40R |
| 70% | +0.05R | +0.40R | +0.75R | +1.10R | +1.80R |
Worked example
Example 01: A 40% win rate at 1:2
The worked example, with every number walked through, is in the app.
Example 1 in dollars
Say each trade is a long on EURUSD with the entry at 1.08500, the stop at 1.08300 (20 pips) and the target at 1.08900 (40 pips). On a $10,000 account risking 1%, 1R is $100, so the size is 0.50 lots ($5 a pip). A win pays 40 x $5 = $200, which is +2R.
Over 10 trades with 4 winners: 4 x $200 = $800 won, 6 x $100 = $600 lost, net +$200, or +2R across 10 trades. The formula gives the same answer per trade.
You do not have to work these out by hand once you have trades to work from: a journal can read the win rate, average R, profit factor and expectancy off your trades. R needs a stop or a 1R value on each trade.
Costs come off every trade
Spread, commission and slippage come off every trade, winners and losers alike, so they come straight off the expectancy. A thin edge can disappear.
Worked example
Example 02: A 60% win rate at 1:0.8
The worked example, with every number walked through, is in the app.
Example 2 in pips and dollars
A short on GBPUSD with the entry at 1.27000, the stop at 1.27250 (25 pips) and the target at 1.26800 (20 pips). Risking $100, the size is 0.40 lots ($4 a pip), and a win pays 20 x $4 = $80, which is +0.8R.
Over 10 trades with 6 winners: 6 x $80 = $480 won, 4 x $100 = $400 lost, net +$80. If spread, commission and slippage come to 2.5 pips a trade, that is $10, or 0.1R.
The breakeven win rate
Every reward to risk has a win rate that breaks even. Set the expectancy to zero:breakeven win rate = 1 / (1 + RR)
At 1:1 you need to win half your trades; at 1:2, one in three; at 1:3, one in four. Bigger targets are also reached less often.
Costs move the line up. With a cost of c in R a trade, breakeven becomes (1 + c) / (1 + RR). The table uses 0.1R:
| Reward to risk | Breakeven win rate | With 0.1R costs |
|---|---|---|
| 1:0.5 | 66.7% | 73.3% |
| 1:0.8 | 55.6% | 61.1% |
| 1:1 | 50.0% | 55.0% |
| 1:1.5 | 40.0% | 44.0% |
| 1:2 | 33.3% | 36.7% |
| 1:3 | 25.0% | 27.5% |
Find your breakeven on practice trades
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With a cost of c in R, a win nets RR minus c and a loss costs 1 + c, which is where the formula comes from. A 45% win rate is comfortably profitable at 1:2 and a losing record at 1:1. Compare your win rate with the breakeven for the reward to risk you actually achieve, after costs.
Replay a few months of one instrument in the backtester and take every setup your plan allows, with the spread and commission set in the session's Customize step so costs are counted. Your real reward to risk is your average win divided by your average loss; set 1 / (1 + RR) next to the win rate in the Backtester's Analytics tab.
Losing streaks are normal
If a strategy loses a share q of its trades, the chance the next N all lose is q to the power N. At a 50% win rate, five losses next is about 3.1%.
But a run can start on any of the hundreds of trades you take, so one turning up somewhere is likely:
| Win rate | Losing run | Next trades all lose | Somewhere in 100 trades |
|---|---|---|---|
| 60% | 5 in a row | 1.0% | 45.9% |
| 50% | 6 in a row | 1.6% | 54.6% |
| 40% | 8 in a row | 1.7% | 49.0% |
| 40% | 10 in a row | 0.6% | 20.5% |
| 30% | 10 in a row | 2.8% | 58.0% |
Worked example
Example 03: Six losses in a row at a 50% win rate
The worked example, with every number walked through, is in the app.
Write your streak rule down
What the run costs, and why it is normal
At 50%, the chance the next 5 trades all lose is 0.5 x 0.5 x 0.5 x 0.5 x 0.5. Risking 1% of the current balance each time, six losses take a $10,000 account through 10,000.00, 9,900.00, 9,801.00, 9,702.99, 9,605.96, 9,509.90, 9,414.80. That is −6R, or −5.85% of the account. Painful, and fully recoverable. The same run at 5% a trade would cost more than a quarter of it.
The table in How to choose your risk shows what longer runs do at different risk levels. The point here is the other side of it: a run of losses is part of trading a strategy with an edge, so it is not, by itself, a sign the edge has gone. Work out the losing run your win rate makes likely over the next 100 trades and size your risk so you can sit through it.
A rule in Trading Rules can be flagged as broken on any trade, so whether you kept it shows up in your analytics instead of in your memory.
Run 100 trades yourself
Pick a win rate and a reward to risk to play out 100 trades. The solid line is the account in R; the gold dashes are what the expectancy predicts. New sample replays it with different luck.
Below is sample 1 as an account. It won 43 of 100 and finished at +29R, above the +20R expected, after losing 7 trades in a row.
DT Terminal ChartsSample account · 1 candle = 1 tradeIllustrationInteractive tool
Expectancy simulator
This interactive tool runs in the app, with your own numbers.
Things to try:
- At 40% and 1:2, the expectancy is +0.20R a trade, yet the first 10 samples finish anywhere from +5R to +41R.
- Set 60% and 1:0.5. A high win rate, and every one of the first 10 samples finishes below zero.
The chart in detail, and more to try
The chart is an illustration: each candle is one trade on a $10,000 account risking $100 (1R) a trade, green for a +$200 win and red for a $100 loss. Its deepest drawdown, −7R, ran from trade 20 to trade 36, wins included.
- At 40% and 1:2, one of the first 10 samples drops to −8R before it recovers, and every one of them has a losing run of at least 5.
- Set 50% and 1:1. The expectancy is zero, but samples still finish anywhere from −6R to +12R. That is how a trader with no edge can feel they have one.
- Set 30% and 1:3. The same +0.20R expectancy as 40% at 1:2, with longer losing runs: up to 13 in a row in the first 10 samples.
Check yourself
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