Risk/Reward Ratio: The Math Behind Every Trade

Computing R:R for a Single Trade

R:R = (Target − Entry) ÷ (Entry − Stop)

A planned long: entry $25.00, stop $23.50, target $29.50. Risk per share is $25.00 − $23.50 = $1.50. Reward per share is $29.50 − $25.00 = $4.50. The ratio is $4.50 / $1.50 = 3:1, which is the same statement as: the trade is denominated in units of R = $1.50, with a target of +3R and a stop of −1R. On 1,000 shares, −1R is −$1,500 and +3R is +$4,500.

The R-unit framing is what generalises. Once a trade is denominated in R, position size, account equity, and the underlying price level all factor out, and what remains is a sequence of multiples: −1R, −1R, +3R, −1R, +2.7R, and so on. That sequence is the object you actually evaluate. Reasoning in dollars per share confuses scale with edge.

Why Win Rate Alone Is Not a Statistic

The recurring beginner error is treating p as a proxy for profitability. It is not. Two traders, 100 trades each:

Trader A wins more often and loses money. Trader B wins less often and earns $8,000. Neither column — p alone, or average win alone — is sufficient to recover the sign of P&L. You need both, multiplied together. This is one reason Paul Tudor Jones organises his risk discipline around the magnitude of losses rather than the count of winners: bounding the size of the down moves dominates raising the count of the up ones, in the regime most discretionary traders operate in.

The asymmetry is arithmetic. A single oversized loss has the same effect on the running total as several normal-size wins. Cap the loss and you collapse the variance contribution from the bad tail. Hit rate is what the ego wants to optimise. Loss size is what the equation rewards. I will admit the seduction of a high p is real — the account curve looks smoother for a while, and that smoothness is mistaken for edge until the first real loss arrives and undoes a quarter of the recent run.

Expectancy as the Edge Function

Combine win rate and R:R into a single per-trade number:

E = p · W − (1 − p) · L

If E > 0, the strategy is profitable in the long run, in expectation. If E < 0, it is not. The magnitude is what you earn, on average, per trade taken.

Trader B's expectancy: (0.40 × $500) − (0.60 × $200) = $200 − $120 = +$80 per trade.

Over 100 trades, +$8,000. Over 1,000 trades, +$80,000. Assuming stationarity of p, W, and L — meaning the underlying generating distribution does not shift — expectancy scales linearly with sample size. The realised P&L for any finite sample is a draw from a distribution centred on N · E with standard deviation that grows like √N, so even a positive-E system goes through long underwater stretches that say nothing about whether the edge has decayed. Position sizing exists in part to keep you solvent through those stretches.

Expectancy in Units of R

A cleaner per-trade form, dimensionless in dollars:

E_R = p · (R:R) − (1 − p)

For Trader B: (0.40 × 2.5) − 0.60 = 1.0 − 0.60 = +0.40R per trade.

If R = $200 per trade, E = $80. If R = $500, E = $200. The dollar expectancy scales linearly with R, which is why position sizing and edge are orthogonal questions. Edge sets the multiplier; sizing sets the unit. Confuse them and you either undersize a real edge or oversize a fragile one. Both end badly. The second ends faster.

Break-even and Profit-target Thresholds

Setting E = 0 and solving for R:R as a function of p gives the minimum R:R consistent with break-even: R:R_min = (1 − p) / p. Numerically:

Two operational notes on this table. The break-even R:R column is tight to the math but loose in practice: it ignores commissions, slippage, and the fact that p is itself estimated with error. Modulo a sample-size discount on p and realised fills a few percent worse than the mid, the working threshold is the right column, not the middle one. The standard rule for discretionary equity strategies — do not take a trade with R:R below 2:1 — is calibrated so a 40% hit rate, which is a realistic ceiling for most competent operators, still clears expectancy with a margin. Druckenmiller and Jones target 5:1 on conviction trades. At 5:1 the break-even hit rate drops to 16.7%, which is what makes a low-frequency, high-conviction process workable as a business model.

Constructing a Trade With This Math

Before entry, three numbers are required, in this order. Entry price — where the position opens. Stop price — where the position closes if the thesis is wrong. The stop must reference a technical level: below structural support for longs, above resistance for shorts. An arbitrary percentage stop, decoupled from chart structure, is a parameter chosen to feel comfortable rather than to invalidate the trade. Those are different objects. Target price — where the position closes if right, also at a technical level: the next overhead resistance for longs, the next structural support for shorts.

Compute R:R. If it is below 2:1, do not enter. The arithmetic does not care that you spent forty minutes building the thesis. Skip it; the next trade is along.

An analogy from photography helps me, at least. A trade plan is a metered exposure: aperture, shutter speed, and ISO are three parameters fixed before the shutter opens, and once you press the release the frame is what it is. Entry, stop, and target are the analogous fixed parameters. R:R is the meter reading that tells you whether the exposure is in range. If the meter reads two stops under, you do not press the shutter and hope; you wait for better light, or you walk to a different scene. The instinct to take a marginal R:R trade because the chart is “set up” is the same instinct that makes a beginner photographer release the shutter on a backlit subject metered for the sky — the result is the underexposed frame the meter already warned about.

The Implementation Gap

Knowing the math is the easy part. Executing it under live P&L pressure is where the strategy actually gets tested. Two recurring failure modes, both of which corrupt realised expectancy.

Cutting winners early. Price reaches $27 on the $25 / $23.50 / $29.50 plan, and the position is closed because the unrealised gain feels at risk. The realised R-multiple is +1.33R rather than the +3R the plan was sized for. Repeat this across a sample and the empirical R:R collapses from 3:1 to roughly 1.3:1, which moves the strategy across the break-even line. Premature profit-taking is the most common single corruption of a positive-expectancy system. I have done it. I ran a mean-reversion book in 2022 with a planned +2.5R target and shipped most exits between +0.8R and +1.4R because the unrealised gains felt unstable on the screen. The backtest was +0.35R per trade. The realised was +0.05R. The system was fine; the operator was the leak.

Letting losers run past the stop. Price hits $23.50 and the stop is moved to $22, then $20.50, on the rationalisation that the move is overdone. The planned −1R loss becomes a realised −3.3R loss, and one instance offsets several winning trades at the original 3R target. The stop is the empirical realisation of the model assumption that the thesis is wrong; moving it edits the assumption after the fact, which means the backtest you are running in your head is no longer the system you originally signed off on.

Both errors share a structure: an unplanned modification, mid-trade, of a parameter the plan had already committed to. The fix is mechanical rather than psychological — bracket orders or stop-and-target tickets entered at trade open, so the exit rules execute without requiring a discretionary decision while the price is moving.

R-multiples in a Real Track Record

Look at any real book over a long enough horizon and the R-multiple distribution dominates the analysis. Best trades typically produce multiples in the +10R to +20R range or higher, depending on how tightly the initial stop was placed. Worst trades produce negative multiples in the −3R to −7R range — larger in magnitude than the planned −1R, because gaps and overnight moves regularly bypass stops, particularly in illiquid names where the next bid below the stop level is several ticks away. Net positive P&L comes from the right tail dominating the left tail in magnitude, despite the left tail occurring more often. That is the structural property a positive-expectancy book needs to have, and it is observable directly from the R-multiple histogram — not from the win rate, which can flatter or mislead almost arbitrarily.

Caveats on the Stationarity Assumption

The expectancy formula assumes p, W, and L are stable across the sample. They are not, in general. Win rates drift with regime; average wins compress when realised volatility falls; average losses widen during gap events that bypass stops. The 2:1 minimum R:R rule is robust to small drifts but not to regime change. A strategy that produced E = +0.4R in one volatility regime can produce E = −0.1R in the next, with no mistake on the operator's part — or rather, the mistake was upstream, in assuming the sample that calibrated the system was representative of what came next. Tracking realised R-multiples on a rolling basis — not a static backtest — is how you detect the drift before it shows up as a drawdown that consumes the account.

Assumptions, stated for this article: stops fill at the planned price (no gap-through), p and average R-multiples are estimated from a sample large enough to be informative (typically 30+ trades for a coarse estimate, 100+ for a usable one), and transaction costs are small relative to the R-unit. When any of those breaks — stops gapping in low-liquidity names, sample size below 30, R-units small enough that commissions are a meaningful fraction — the model's output is degraded in ways the formula does not warn you about. Size for that.

See also: Position Sizing and Risk Management · Trading Psychology: Fear and Greed · Paul Tudor Jones · Support and Resistance · How to Build a Watchlist