Overview
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?
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?
Penafian
Maklumat dan penerbitan adalah tidak bertujuan, dan tidak membentuk, nasihat atau cadangan kewangan, pelaburan, dagangan atau jenis lain yang diberikan atau disahkan oleh TradingView. Baca lebih dalam Terma Penggunaan.
Penafian
Maklumat dan penerbitan adalah tidak bertujuan, dan tidak membentuk, nasihat atau cadangan kewangan, pelaburan, dagangan atau jenis lain yang diberikan atau disahkan oleh TradingView. Baca lebih dalam Terma Penggunaan.