Risk/Reward & Expectancy

The question that decides everything

Before you trade a system, the question is not "does it work on this chart?" It's "does it have positive expectancy?" Every other question — entries, exits, sizing, risk management — assumes the answer to this one is yes.

Expectancy compresses three variables into one number:

E = P_{win} \times \overline{Win} - P_{loss} \times \overline{Loss}

It's the average profit (or loss) per trade if you repeated the system infinitely. Positive = the system makes money on average. Negative = it loses. Zero = break-even. There is no other question to ask first.

Play with it

Expectancy & Kelly calculator

Win rate: 50%Avg winner ($)Avg loser ($)Trades simulated: 100

Expectancy / trade$50.00

Win/loss ratio2.00

Total P&L (100 trades)$5000

Kelly %25.0%

Expectancy = (win_rate × avg_win) − (loss_rate × avg_loss). Positive = the system makes money on average. A 30% win rate with $300 winners vs $100 losers has positive expectancy; a 70% win rate with $50 winners vs $200 losers has negative expectancy. Win rate alone tells you nothing. Kelly % is the theoretically-optimal fraction of capital to risk per trade — most practitioners use half Kelly (12.5%) for smoother drawdowns.

Drag the sliders and watch expectancy flip signs. Observe the three classic scenarios:

50% win rate, $200 winners, $100 losers → +$50/trade. Profitable.
30% win rate, $300 winners, $100 losers → +$20/trade. Profitable, despite losing more often than winning.
70% win rate, $50 winners, $200 losers → −$25/trade. Losing money despite winning 70% of the time.

The third is the scenario that ruins retail traders. It feels good — you're winning 7 of 10 — but the 3 losers are so much larger than the 7 winners that the math bleeds out anyway.

See it on a chart

Win rate ≠ profitability

Win rate70 %

Payoff ratio0.25×

Expectancy−0.125R / trade

P&L (10 trades)$-500

Toggle between the two scenarios. In the first, you win 7 of 10 trades — all those green arrows — but the equity curve bleeds downward because the 3 red exits are each 2R losses dwarfing the 0.5R wins. In the second, you lose 6 of 10 — mostly red exits — but the 4 winners at 3R each pull the equity curve upward. Watch the equity pane, not the win count.

R-multiples — the universal language

Serious traders don't talk in dollars. They talk in R — multiples of the fixed risk per trade.

1R = your per-trade risk (e.g., $500 on a $50k account at 1%)
A trade that hits its stop = −1R
A trade that hits a 2× reward target = +2R
A runner that triples = +3R

Expectancy in R:

E_R = P_{win} \times \overline{R_{win}} - P_{loss} \times \overline{R_{loss}}

Since losers typically = −1R (you stopped out at your risk amount) and winners vary, this simplifies in most real systems to:

E_R \approx P_{win} \times \overline{R_{win}} - P_{loss}

Example: 40% win rate, average winner 2.5R, losers 1R.

E_R = 0.4 \times 2.5 - 0.6 \times 1 = 1.0 - 0.6 = 0.4R

You earn 0.4R per trade on average. If 1R = $500, you earn $200/trade on average. Over 100 trades, that's $20,000 — regardless of how the individual trades are distributed.

The R framework is powerful because it normalizes across instruments. A 2R winner on Apple is the same success as a 2R winner on a futures contract — both moved your account by 2× your per-trade risk. You can compare systems that trade totally different markets in the same units.

Win rate vs payoff — the frontier

For every win rate, there's a minimum payoff (win/loss ratio) required for profitability:

Win rate	Required payoff to break even
20%	4.0×
30%	2.33×
40%	1.5×
50%	1.0×
60%	0.67×
70%	0.43×
80%	0.25×

Read this as: "if my win rate is X%, my average winner must be at least Y× my average loser just to break even." To be profitable, you need to clear that threshold.

Two canonical system shapes:

Trend-following lives in the 30–40% win rate / 2–3× payoff region. Few wins, big wins. Trades of months-to-years. The Turtles, CTAs, momentum strategies.
Mean reversion lives in the 65–80% win rate / 0.5–0.8× payoff region. Many small wins, occasional large losers. Pairs trading, fade-the-gap, VWAP reversion.

Both can be profitable. Both require discipline. The mistake is expecting a 75% win rate with 2× payoff — it exists, but not sustainably. The market doesn't give you both.

The Kelly connection

Once you have expectancy, Kelly tells you how much to bet. Revisited from the Position Sizing lesson:

f^* = \frac{bp - q}{b}

The calculator above shows Kelly % for the inputs you've chosen. Observe:

Negative expectancy → Kelly returns a negative number (don't trade this system)
Break-even system → Kelly returns zero (no bet size produces growth)
High-win-rate system with 1× payoff → Kelly is small even when win rate is 70%
Low-win-rate system with high payoff → Kelly can be aggressive despite most trades losing

Practical rule: if expectancy is positive but Kelly is < 1%, the edge is too thin to trade profitably after slippage, commission, and taxes.

The full expectancy profile

Expectancy alone is a mean. You also want the distribution around that mean. Kaufman's performance profile for a system should include:

E (expectancy) — the mean per trade
Standard deviation of trades — noise around the mean
Sharpe — mean / stdev, annualized
Max adverse excursion distribution — how bad does a typical losing trade get before stopping?
Profit factor — gross winners / gross losers. A profit factor of 1.5 means $1.50 of wins for every $1 of losses
% profitable in bull regimes vs bear regimes — does the expectancy survive different market environments?

A system with expectancy $50/trade and profit factor 1.2 is weaker than one with expectancy $30/trade and profit factor 1.8, because the second is less exposed to bad streaks.

Streaks and ruin

With 60% win rate, the probability of 10 consecutive losses is $(0.4)^{10} = 0.01\%$ — which sounds rare until you realize in 10,000 trades you'll see it happen. With 40% win rate, the probability of 10 consecutive losses is $(0.6)^{10} = 0.6\%$ — which means in 1000 trades you'll see six such streaks.

Expectancy is a long-run average. Any individual subsequence can diverge arbitrarily from it. Your position sizing must survive the worst streak that's likely in your sample.

Kaufman's practical estimate: for a 40%-win system, expect to see a 15-consecutive-loss streak sometime in your first 1000 trades. For a 30%-win system, 25 losses in a row. Size accordingly.

Half-Kelly in practice

Given a measured expectancy, the actionable sizing rule most retail traders use:

Compute full Kelly from your verified stats
Cap at 25% (no one should use 25%+ raw Kelly regardless of math)
Halve or quarter it for estimation-error insurance
That's your position-size fraction

Example: 50% win, $300 winners, $100 losers.

b = 3, \quad p = 0.5, \quad f^* = \frac{3 \times 0.5 - 0.5}{3} = \frac{1}{3} \approx 33\%

Capped at 25% → 25% full Kelly → 12.5% half-Kelly → 6.25% quarter-Kelly.

A 6.25% bet on a $50k account is $3,125 at risk per trade. That's already aggressive by retail standards but defensible from the math. Most retail traders would still scale down from there.

Quick check

Question 1 / 20 correct

System: 50% wins at 3R, 50% losses at 1R. What's expectancy in R?

What you now know

Expectancy = P(win) × avg_win − P(loss) × avg_loss — the single most important number about any system
R-multiples normalize across instruments and systems — everything is counted in units of per-trade risk
Win rate without payoff is meaningless — 80% win rate with 0.2× payoff loses money
Trend systems live at ~35% win, 2.5× payoff; mean-reversion at ~75% win, 0.5× payoff — both can work
Kelly converts a positive-expectancy system into a sizing fraction; use quarter-Kelly for estimation insurance
Always budget for the worst losing streak the statistics make likely, not the mean

Next: Diversification — why one good strategy isn't enough, and why correlation determines portfolio risk far more than the number of positions.