Modeling Trading Expectancy with a Simple Binomial FrameworkOverview
Many traders focus only on win rate and average reward/risk, but real expectancy also depends on costs, position sizing, and the natural monthly variability of outcomes.
Let’s explore this with a transparent example: a strategy that averages 6 independent trades per month, with a 55% win rate.
We can model the possible monthly results using the binomial distribution — the same math behind coin flips, but applied to trading wins and losses.
Step 1: Understanding the Binomial Model
Imagine a biased coin where Heads = Win (55% probability), Tails = Loss (45%).
The probability of exactly k wins in 6 trades is:
P(k)=(6¦k)×(0.55)^k×(0.45)^(6-k)
P(k) = binom{6}{k} * (0.55)^k * (0.45)^{6-k}
The binomial coefficient
(6¦k) : binom{6}{k}
("6 choose k") counts how many ways you can get exactly k wins: (6¦k)=6!/(k!(6-k)!)
Examples:
k = 0 wins (probability ≈ 0.83%)
(6¦0)=1 {6!} / {0!(6-0)!}
k = 1 win (probability ≈ 6.09%)
(6¦1)=6 {6!} / {1!(6-1)!}
(The peak is at 3 wins with 30.3% probability.)
Step 2: Realistic Trade Parameters
Boundary conditions:
- Position size: 40% of capital per trade
- Win: Take Profit +4.5%
- Loss: Stop Loss -2%
- Costs (both wins and losses): Commissions (0.38% round-trip) + Tobin tax (0.2% at entry)
Net per trade (on the positioned capital):
• Win: +3.92% → +1.568% impact on total capital
• Loss: -2.58% → -1.032% impact on total capital
Step 3: Monthly Outcomes
Here are all possible results (without any stopping rule):
- 0 wins : probability 0.83% (return -6.19%)
- 1 win : probability 6.09% (return -3.59%)
- 2 wins : probability 18.61% (return -0.99%)
- 3 wins : probability 30.32% (return +1.61%)
- 4 wins : probability 27.80% (return +4.21%)
- 5 wins : probability 13.59% (return +6.81%)
- 6 wins : probability 2.77% (return +9.41%)
Weighting each outcome by its probability yields an average monthly return of +2.39%.
Adding a Drawdown Protection Rule: Stop Trading After 4 Losses
To reduce downside risk, add this rule: stop taking new trades for the month once 4 losses are reached (trades are sequential). This caps maximum losses at 4 per month.
We now need to recalculate probabilities and returns, because in bad sequences we stop early and avoid the 5th/6th loss.
How to calculate the new probabilities and expected return:
There are two mutually exclusive scenarios:
Never reach 4 losses → complete all 6 trades
This happens if losses ≤3 (i.e., wins ≥3).
Probability: ~74.47% (sum of original binomial probabilities for k=3 to 6 wins).
Returns remain the same as in the original table for these cases.
Reach exactly the 4th loss on trade t (t=4,5, or 6) → stop immediately after trade t
These use the negative binomial distribution: probability that the 4th loss occurs exactly on trial t.
Formula for each t:
P("stop at " t)=((t-1)¦3)×(0.45)^4×(0.55)^(t-4)
P({stop at } t) = binom{t-1}{3} * (0.45)^4 * (0.55)^{t-4}
(Choose 3 positions for the previous losses in the first t-1 trades, then loss on t.)
Detailed breakdown:
Stop after 4 trades (0 wins, 4 losses): Probability 4.10%, Return -4.13%
Stop after 5 trades (1 win, 4 losses): Probability 9.02%, Return -2.56%
Stop after 6 trades (2 wins, 4 losses; 4th loss exactly on the 6th): Probability 12.40%, Return -0.99%
Total probability of stopping: ~25.53% (equals the original cases with ≥4 losses).
To get the new expected return:
Weight the returns by these adjusted probabilities (full-6-trade cases + stopping cases).
Result:
expected monthly return ≈ +2.32% (slightly lower than the original +2.39%, because we sometimes miss potential recovering wins in bad months).
Key benefits:
- Maximum monthly loss capped at -4.13% (vs. -6.19% originally)
- Downside range significantly narrowed
- Small trade-off in average return for much better capital protection
Key Insight
Small, consistent edges can compound powerfully over time, but only if we:
- Account properly for all costs
- Size positions conservatively
- Prepare mentally for normal monthly swings
The binomial view helps us see both the average and the variability — essential for long-term survival and growth.
Adding simple rules like loss limits can further enhance the risk profile, often with minimal impact on expectancy.
What do you think — does your strategy show similar positive expectancy when modeled this way?
