Uncertainty and variance: why good decisions can still lose

What you will learn

• Why a rule set with a positive expectancy can still lose money over weeks or months.
• How to calculate the spread (standard deviation) of results in R for a series of trades.
• How many trades you need before your measured results say something reliable.
• How to set realistic expectations so that normal variance does not break your discipline.

Explanation

In the previous lesson you learned to judge decisions by process, not by outcome. This lesson explains the mathematics behind that idea. Uncertainty means you cannot know the result of the next trade. Variance is a measure of how widely results spread around their average. A strategy with a positive average can still produce long periods of losses simply because results are spread widely.

You already know R-multiples and expectancy from the risk management course (1R = the amount you planned to risk; expectancy = average result per trade in R). Expectancy tells you the centre. It does not tell you how far a real series of trades can drift away from that centre. For that you need the standard deviation (SD), the typical distance of a single result from the average.

Three formulas you can use on paper

• Variance per trade = (average of the squared results) − (expectancy)². SD = square root of the variance.
• For a series of n independent trades: expected total = n × expectancy; SD of the total = SD per trade × √n.
• Standard error (SE) of your measured expectancy = SD per trade ÷ √n. A rough 95% range for the true expectancy is the measured value ± 2 × SE.

The key insight: the expected total grows with n, but the spread grows only with √n. In the short run, the spread dominates. In the long run, the edge (if real) slowly becomes visible. This is why a losing month is not proof that a strategy is broken, and a winning month is not proof that it works.

Losing-streak and recovery mathematics were covered in the risk management course (lesson on drawdowns). Here the focus is on what variance does to your judgement: people tend to see patterns in random sequences and change their rules after a few results. Knowing the expected spread in advance protects you from that.

Worked example

Hypothetical example: a rule set wins 40% of trades at +2R and loses 60% at −1R. Assume every trade is independent and results are exact (no partial exits). Fees, spreads, slippage and taxes are excluded.

1. Expectancy = 0.40 × 2 + 0.60 × (−1) = 0.80 − 0.60 = +0.20R per trade.
2. Average of squared results = 0.40 × 2² + 0.60 × (−1)² = 1.60 + 0.60 = 2.20.
3. Variance = 2.20 − 0.20² = 2.20 − 0.04 = 2.16. SD per trade = √2.16 ≈ 1.47R.
4. After 20 trades: expected total = 20 × 0.20 = +4R. SD of total = 1.47 × √20 ≈ 6.6R. With binomial probabilities, the total is negative whenever 6 or fewer of the 20 trades win: probability ≈ 25%.
5. After 100 trades: expected total = +20R. SD ≈ 1.47 × 10 = 14.7R. The total is negative with 33 or fewer wins: probability ≈ 9%.

With 1R = €50, the expected result after 20 trades is +€200, but one in four traders following exactly these rules would still be down. That is variance, not a mistake.

How many trades until the edge is statistically visible? You need 2 × SE < expectancy, so 2 × 1.47 ÷ √n < 0.20. That gives √n > 14.7, so n > about 216 trades.

Common mistake

Practice activity

On paper: a hypothetical rule set wins 50% of trades at +1.5R and loses 50% at −1R. (a) Calculate the expectancy. (b) Calculate the SD per trade. (c) After 50 trades, what is the expected total and the SD of the total? (d) Your journal shows a measured expectancy of +0.25R after 50 trades. Give a rough 95% range for the true expectancy.

Key takeaways

• Expectancy describes the centre; standard deviation describes how far real results can drift from it.
• The expected total grows with n, the spread only with √n, so short samples are dominated by luck.
• A positive rule set can easily show a losing month; judge it only after a pre-defined number of trades.
• Historical results are an observation, not a promise; the true edge may be smaller or may disappear.

Next step

In the next lesson you will build a daily routine for preparation, focus and an end-of-day review that keeps your process stable while variance does its work.