Retail traders are obsessed with Win-Rate. Social media gurus market systems claiming "90% Win-Rate!" to attract beginners.
However, in professional quantitative finance, win-rate in isolation is a meaningless statistic.
A trader with a 90% win-rate who risks £500 to make £50 will lose their entire account the moment a string of 2 normal losses occurs. Conversely, a systematic trader with a 40% win-rate who risks £100 to make £250 will generate compounding wealth over time.
In this lesson, we break down the Mathematics of Account Expectancy and Risk of Ruin.
1. The Mathematical Expectancy Formula
Statistical expectancy determines your long-term edge per unit of risk:
E = (W × R) - (L × 1)
Where:
- W = Win Rate (e.g., 0.55 for 55%)
- R = Risk-to-Reward Ratio (Average Win ÷ Average Loss)
- L = Loss Rate ($1 - W$, e.g., 0.45 for 45%)
3. High Win Rate vs. High Reward: The Math
Let's compare two traders:
Trader A (High Win-Rate Scalper)
- Win Rate ($W$): 80% (0.80)
- Average Win: £35
- Average Loss: £100
- Risk-to-Reward ($R$): 0.35:1
Expectancy = (0.80 × £35) - (0.20 × £100) = £28 - £20 = +£8.00 per trade
Trader B (Trend Follower)
- Win Rate ($W$): 40% (0.40)
- Average Win: £250
- Average Loss: £100
- Risk-to-Reward ($R$): 2.5:1
Expectancy = (0.40 × £250) - (0.60 × £100) = £100 - £60 = +£40.00 per trade
Result over 100 Trades:
- System A Total Net Profit: £800.00
- System B Total Net Profit: £4,000.00 (5x Higher Profit despite 40% Win-Rate!)
3. The Risk of Ruin Table
Risk of Ruin measures the probability that your account balance will reach zero (or hit a non-recoverable threshold) based on your risk per trade and win-rate:
┌───────────┬─────────────┬──────────────┬──────────────┐
│ Win Rate │ R:R Ratio │ 2.0% Risk │ 1.0% Risk │
│ │ │ Ruin Prob. │ Ruin Prob. │
├───────────┼─────────────┼──────────────┼──────────────┤
│ 40% │ 1 : 1.5 │ 14.3% │ 1.2% │
│ 40% │ 1 : 2.0 │ 3.1% │ 0.1% │
│ 50% │ 1 : 1.5 │ 0.8% │ 0.0% │
│ 50% │ 1 : 2.0 │ 0.0% │ 0.0% │
└───────────┴─────────────┴──────────────┴──────────────┘
Notice that simply reducing your risk per trade from 2.0% to 1.0% reduces your Risk of Ruin from 14.3% to 1.2% on a 40% win-rate strategy!
Core Rules for Positive Expectancy
- Never take trades with a Risk-to-Reward ratio lower than 1:1.5.
- Focus on maximizing Average Win size rather than fighting for high win-rates.
- Cap risk at 1.0% per trade to guarantee near-zero mathematical Risk of Ruin.