Expectancy: win rate, payoff and why they trade off

The first question most people ask about a trading method is "how often is it right?" It is close to the least informative question available. A method that is right 90% of the time can lose money relentlessly, and a method that is right 30% of the time can compound for decades. What separates them is not accuracy but the relationship between how often you are right and how much you collect when you are.

That relationship has a name — expectancy — and it is not a slogan. It is an arithmetic identity that constrains every trading system that has ever existed. Once you can write it down, a large amount of folklore about "high-probability setups" collapses on contact.

The identity

Express every outcome as a multiple of the amount you put at risk. If you buy a share at CHF 84.00 with a stop at CHF 78.00, your risk per share is CHF 6.00. That CHF 6.00 is one R — one unit of risk. A trade that exits at CHF 96.00 gained CHF 12.00 per share, or +2R. A trade stopped out gained −1R. Normalising to R lets you compare a small position in a volatile instrument with a large position in a quiet one, and it lets you add trades across different account sizes.

Expectancy is the mean R-multiple per trade:

E = (p × b) − (1 − p) × 1

where p is the fraction of trades that win and b is the average size of a winner in R. The loss term is 1R by construction, because R is defined as the loss. If you win 40% of the time and your winners average 2.5R:

E = (0.40 × 2.5) − (0.60 × 1.0) = 1.00 − 0.60 = +0.40R per trade

Two hundred trades a year at 0.40R is 80R of gross expected profit. If each R is 0.5% of equity, that is 40% of equity gross, before costs and before the compounding effects covered in the position sizing lesson.

The trade-off, made explicit

Set E = 0 and solve for the payoff you need at a given win rate. The breakeven payoff is:

b = (1 − p) / p

That single expression is the whole trade-off. Some values worth memorising:

• Win 30% of the time, you need winners averaging 2.33R just to break even.
• Win 40%, you need 1.50R.
• Win 50%, you need 1.00R.
• Win 60%, you need 0.67R.
• Win 70%, you need 0.43R.

Nothing in that table is a preference or a style. It is division. And it explains why the phrase "high win rate strategy" carries no information on its own: a 70% win rate with an average winner of 0.35R is a losing system (E = 0.70 × 0.35 − 0.30 = 0.245 − 0.30 = −0.055R), while a 30% win rate with 3R winners makes money comfortably (E = 0.90 − 0.70 = +0.20R).

Why the two cannot be raised together

The reason win rate and payoff trade off is mechanical, not psychological. Both are functions of the same two decisions: where you place the stop and where you take profit.

Move the profit target closer and the probability of reaching it before the stop rises — you have shortened the distance the price must travel in your favour while leaving the distance against you unchanged. Win rate goes up, payoff goes down. Widen the stop and the same thing happens from the other side: the price now has more room to wander against you without ejecting you, so fewer trades are stopped out, but each R is larger, so the same CHF profit is a smaller R-multiple.

In a diffusive price process, these effects are close to symmetric. For a driftless random walk, the probability of touching a target at distance T before a stop at distance S is approximately S / (S + T). If the target is at 1R (T = S), that is 50%. If the target is at 3R, it is 1/4 = 25%, and the expectancy is 0.25 × 3 − 0.75 = 0. If the target is at 0.5R, the probability is 2/3, and expectancy is 0.667 × 0.5 − 0.333 = 0. Every combination gives zero. That is the point: in the absence of an edge, moving the stop and the target around cannot manufacture expectancy — it only redistributes it between the two terms. Any positive expectancy has to come from somewhere else, from a genuine departure of the price process from the driftless case.

This is the most useful thing expectancy tells you. When you tune a system and the win rate improves, check the payoff. If it fell by the amount the arithmetic predicts, you have discovered nothing; you have only chosen a different point on the same indifference curve. If the product genuinely improved, you have found an edge, and it is worth understanding why before trusting it.

Same expectancy, very different lives

Consider two systems with essentially the same expectancy:

• System A: wins 65% of the time, average winner 0.80R. E = 0.65 × 0.80 − 0.35 = 0.52 − 0.35 = +0.17R
• System B: wins 30% of the time, average winner 2.90R. E = 0.30 × 2.90 − 0.70 = 0.87 − 0.70 = +0.17R

Identical expected value. Now the dispersion. The standard deviation of the per-trade R-multiple is about 0.86R for System A and about 1.79R for System B — more than double. That difference propagates into everything: drawdown depth, the length of losing runs, and how long you must trade before you can tell the system apart from noise.

Losing runs first. In System A the chance of six consecutive losses is 0.35⁶ ≈ 0.18% — you may never see one. In System B the chance of ten consecutive losses is 0.70¹⁰ ≈ 2.8%, which over a few hundred trades means you should expect to live through one or two such runs. A trader who has not done this arithmetic in advance will conclude, correctly by their own lights but wrongly in fact, that the system has stopped working.

Sample size next. The standard error of your estimated expectancy after n trades is σ/√n. To get a standard error equal to half your edge — a weak standard, roughly a two-sigma result — you need n ≈ (2σ/E)². For System A that is (2 × 0.86 / 0.17)² ≈ 102 trades. For System B it is (2 × 1.79 / 0.17)² ≈ 442 trades. Same edge; four times as much evidence required to see it. The drawdown and risk-of-ruin lessons take this further, but the seed is here: expectancy alone never describes a system. Expectancy and dispersion together do.

Where expectancy leaks

Everything above is gross. Costs come out of every trade, winner and loser alike, and they are charged on notional value rather than on risk — which means the tighter your stop, the more your costs matter as a fraction of R.

For a Swiss-resident investor, the cost stack on a share transaction includes brokerage commission, the bid-ask spread, slippage against your intended price, and Swiss federal stamp duty (Umsatzabgabe) where a Swiss securities dealer is a party to the trade: broadly 0.075% of turnover on Swiss securities and 0.15% on foreign securities, charged on the purchase and again on the sale. Derivatives, futures and foreign-exchange transactions are generally outside its scope, which is one reason the cost profile of a futures-based approach differs materially from a cash-equity one.

Work an example. Equity of CHF 200,000, risking 0.75% (CHF 1,500) per trade, entry at CHF 84.00, stop at CHF 78.00, so R = CHF 6.00 per share and the position is 250 shares — CHF 21,000 of notional. If round-trip friction totals 0.30% of notional, that is CHF 63, or 4.2% of your R. A system with a gross expectancy of +0.17R nets +0.128R. You have just given away a quarter of your edge to friction, and you would never have noticed it by looking at win rate.

Two consequences follow. Costs scale with notional, not with risk, so halving your stop distance while holding CHF risk constant doubles your position and doubles your cost in R terms. And trade frequency multiplies the leak: a system that trades 500 times a year at 4% of R in costs surrenders 20R annually before it does anything else.

Expectancy per unit of opportunity

One last refinement. Expectancy per trade is the wrong denominator for comparing systems that trade at different rates or tie up capital for different lengths of time. A system earning 0.5R per trade over 20 trades a year produces 10R. A system earning 0.12R per trade over 300 trades produces 36R. The second is a better business despite looking worse on every per-trade statistic — provided its costs and its capacity allow it.

There is a second denominator problem, and for a Swiss household it is the one that usually decides the question. Expectancy has to be compared against what the same capital would have earned somewhere else, and the most reliable competing use of a franc in Switzerland is a voluntary buy-in into the occupational pension fund. Within the gap between your accumulated pension capital and the maximum your plan allows, a buy-in is deductible from taxable income in the year it is paid. Whatever your marginal rate is — a function of your income, your canton and your commune — the buy-in returns that fraction immediately and with certainty. The money is then locked away until retirement, and there are conditions attached: benefits arising from a buy-in may not be drawn as a lump sum within three years of paying it in, and the lump sum itself is taxed separately when it is eventually withdrawn, at a reduced rate that again depends on where you live. But the comparison is unforgiving. A certain, immediate, one-off tax saving is a very high bar for an uncertain edge measured in fractions of R, and any honest expectancy calculation for a modest trading strategy has to acknowledge that the first few thousand francs of surplus capital each year have a better-paid job elsewhere.

What to take away

• Expectancy is (win rate × average winner in R) minus (loss rate × 1R); the breakeven payoff at any win rate p is exactly (1 − p)/p, which is why a high win rate on its own tells you nothing about whether a method makes money.
• Win rate and payoff are two ends of the same lever — moving the stop or the target trades one for the other, and in a driftless process every combination gives zero expectancy, so a real edge must come from the price process itself, not from where you put your orders.
• Two systems with identical expectancy can have wildly different dispersion; the lower win rate system will produce longer losing runs and needs several times more trades before its edge is statistically visible.
• Costs are charged on notional while your edge is measured in R, so tightening stops raises the fraction of your edge consumed by spreads, commission and Swiss stamp duty even when the expectancy in R looks better.
• Compare systems on expectancy multiplied by opportunity — R per year, not R per trade — and benchmark the result against the certain alternative uses of the same francs, such as a deductible pension-fund buy-in, before concluding that the strategy is worth running at all.

Sources

• Van K. Tharp, Trade Your Way to Financial Freedom, McGraw-Hill — the standard reference for R-multiples and expectancy as a system-evaluation framework.
• Ralph Vince, The Mathematics of Money Management: Risk Analysis Techniques for Traders, Wiley, 1992 — expectancy, holding-period returns and the geometry of repeated bets.
• Edward O. Thorp, A Man for All Markets, Random House, 2017 — on the difference between an edge and the appearance of one.
• Daniel Kahneman and Amos Tversky, "Prospect Theory: An Analysis of Decision under Risk", Econometrica, 1979 — why traders systematically prefer high win rates to high expectancy.
• Federal Act on Occupational Retirement, Survivors' and Disability Pension Plans (BVG/LPP), the provisions on voluntary purchase of benefits and the restriction on lump-sum withdrawal in the three years following a buy-in.
• Swiss Federal Act on Stamp Duties (Bundesgesetz über die Stempelabgaben) — the statutory basis for the Umsatzabgabe on securities transfers.
• SIX Swiss Exchange, trading and fee documentation — for the transaction-cost components applicable to Swiss-listed instruments.