The statistics that matter: R, expectancy, standard error, drawdown and Monte Carlo

You have a sample of trades: now what? Statistics serves to answer three questions: does the system earn on average? (expectancy), how sure are we of this estimate? (uncertainty) and how much can I suffer along the way? (drawdown). Many traders look only at the win rate, the least informative measure. Here we look at the right statistics, with formulas and numbers.

Measuring in R

R is the initial risk of a trade. If you risk 50, a 100 win is +2R and a 50 loss is −1R. Recording everything in R makes trades of different size and pairs comparable and separates the quality of the system from the size of the position. Everything else is calculated from the list of results in R.

Expectancy

Expectancy is the average gain per trade, in R:

E = (W × R_win) − (L × R_loss)

where W = win percentage, L = loss percentage, R_win = average R of wins, R_loss = average R (in absolute value) of losses. It equals simply the arithmetic mean of the results in R.

Profit factor and other measures

• Profit factor (PF): sum of gains ÷ sum of losses (in absolute value). Above 1 the system earns. Values of 1.3-2 are good; very high values on little data are suspicious. In the example: gains = 42 × 2.1 = 88.2R; losses = 58R; PF = 1.52.
• Payoff ratio: average R of wins ÷ average R of losses (here 2.1).
• Win rate: useful but not on its own.
• Average R per trade = expectancy.
• Sharpe and Sortino: ratios between return and volatility of results. Useful for comparing systems, but need care (unreliable on small samples).

Uncertainty: the standard error

The sample gives you an estimate of expectancy, not its true value. The statistical error depends on the dispersion of results and the number of trades:

Standard error (SE) = σ ÷ √n

where σ is the standard deviation of results in R and n the number of trades. The 95% confidence interval is approximately E ± 1.96 × SE.

For an observed win percentage p, the standard error is √(p(1−p) ÷ n): with p = 0.50 and n = 100, SE = 5 points; the 95% interval is 40%-60%. A 60% win rate on 30 trades has SE ≈ 9 points: it could very well be a real 45%.

Beware: multiple comparisons

If you try 20 variants of your system and keep the best, the "best" will almost certainly have an expectancy inflated by pure luck (the data snooping problem). Remedies: declare beforehand what you'll test, adjust down the results of the "chosen" variants and always verify on new data (out-of-sample).

Drawdown and losing streaks

• Maximum drawdown (MDD): the maximum loss from the capital peak (in R or %). It measures the pain you'll have to endure.
• Drawdown duration: how long (or how many trades) to get back to the high.
• Maximum losing streak: the longest consecutive losses.

With a loss percentage q, the probability of k consecutive losses is about q^k per "attempt". With q = 0.58: 5 in a row = 6.6%, 8 in a row = 1.3%, 10 in a row = 0.4%. Over 300 trades, a streak of 8-10 losses is far from rare.

Practical rule: choose risk per trade so that the expected drawdown (not the best, the typical) stays within what you're willing to bear — both in money and emotionally.

Monte Carlo simulation: your history is one of many possible

The order of your trades is random: with the same results (the same list of R) the order might have been different, with very different drawdowns. The Monte Carlo (or bootstrap) simulation works like this:

1. Take the list of your n results in R.
2. Shuffle the order (or draw n values with replacement) and rebuild the equity curve.
3. Repeat 1,000-10,000 times.
4. Look at the distribution of results: maximum drawdown, final capital, losing streaks.

It tells you the realistic range of what you might experience: for example "the maximum drawdown is below 12% in 90% of simulations and above 20% in 5%". Your historical sequence is just one case.

In module 5 you'll see the Python code to do it. With a spreadsheet it can be done with the random function and a table of 1,000 columns, but Python is more convenient.

Risk of ruin

The risk of ruin is the probability of losing an unacceptable share of capital. It depends on expectancy, dispersion and risk per trade. For fixed-risk systems, one qualitative rule always holds: the risk of ruin grows explosively with risk per trade. Halving risk drastically reduces the probability of nasty surprises, at the cost of slower growth.

Equity curves: what to look at

Plot the equity chart (cumulative R) and observe:

• Constant slope or in jumps? A system that earns everything in a few "lucky" trades is fragile.
• Recovery from drawdowns: quick or very slow?
• Trend over time: improving, worsening or stable? A decline in the last period may signal that the regime has changed.
• Concentration of gains: what happens when you remove the 3 best trades? If expectancy becomes negative, the system depends on a few events.

Typical mistakes

• Looking only at the win rate.
• Ignoring the standard error: thinking +0.3R on 40 trades is "proven".
• Choosing the best variant among many without verification.
• Underestimating losing streaks.
• Not simulating risk with Monte Carlo before raising size.

In short

• Measure everything in R; expectancy is the mean of results.
• The standard error (σ ÷ √n) tells you how much to trust the estimate: with little data the interval can include zero.
• Profit factor, payoff, maximum drawdown and losing streak complete the picture.
• Monte Carlo shows the realistic range of drawdown and results.
• Risk per trade decides drawdown and probability of ruin.

Practical exercise

1. Calculate on 100 trades: win rate, R of wins/losses, expectancy, PF, maximum drawdown and losing streak.
2. Calculate the standard error and the 95% confidence interval of expectancy. Does it include zero?
3. Shuffle the order of your R 20 times on a sheet and note the maximum drawdown of each sequence.