A 45% win rate at 1.5R still loses 1 month in 6: binomial odds for 80 trades a month

[Fairman]

A 45% win rate at 1.5R still loses 1 month in 6: binomial odds for 80 trades a month

Say your setup wins 45% of the time and pays 1.5R on winners, losing 1R on the rest. Expectancy is 0.45 x 1.5 minus 0.55 = 0.125R per trade. Over 80 trades a month that is 10R, so $1,000 on a $10,000 account risking 1% a trade.

Net result for a month is 1.5 x wins minus the number of losses, which with W wins out of 80 is 1.5W minus (80 minus W), or 2.5W minus 80. That is negative when wins are 31 or fewer. The expected number of wins is 36, and the standard deviation of the win count is the square root of 80 x 0.45 x 0.55, about 4.4. So 31 wins is only one standard deviation under the average.

The exact binomial probability of 31 wins or fewer is 15.6%. That is a losing month about one time in six and a half, with nothing wrong with the system. Thirty wins gives minus 5R, 28 gives minus 10R, and 25 gives minus 17.5R.

The tail fills in with fewer trades. At 20 trades a month the same edge loses 25.2% of the time, at 40 trades 21.4%, at 160 trades 8.8%. More trades make the month closer to the average, but cost per trade has to stay small for that to help.

My rule of thumb is to decide before the month what a normal bad result looks like for my numbers, and to compare the actual month with that.

[Fairman]

Here is how deep a bad month can go with the same 80 trades. A month with 28 wins or fewer ends at minus 10R or worse, and the exact chance of that is 4.5%. At 1% risk that's a 10% drop in one month, from nothing but sampling. Twenty-five wins or fewer, the minus 17.5R case, happens 0.8% of the time. So if the plan is built on 1% risk, a 10% monthly loss is not proof the edge is gone. It's a result that shows up about once every 22 months. I'd use those percentages to set the review threshold ahead of time, before the pain starts to change the decision.

[Fairman]

Back to back losing months are the thing that tests discipline. If one month has a 15.6% chance of finishing negative and months are independent, two in a row is 0.156 x 0.156 = 2.4%. Over a six month stretch, the chance of at least one losing month is 1 minus 0.844 to the power 6, or 64%. Over twelve months it's 87%. So a trader with a real 0.125R edge should expect more than one down month in a typical year, probably several. Plan the cash and the head for that, and don't treat the first red month as a warning sign by itself.

[Fairman]

The calculation assumes the 45% and the 1.5R are true values. They're estimates from a sample, and a sample of 100 trades gives a win rate with an error of about 5 points either way, so the real figures could be a lot worse than you think. That makes the table above optimistic, if anything. A fair test is to rerun the numbers at 42% and at 40%: at 40% with 1.5R the expectancy is 0.4 x 1.5 minus 0.6 = 0, so the edge is gone. The gap between 45% and 40% is about one standard error, which is why an apparent edge needs a big sample.

[Fairman]

A practical way to use this is a monthly checklist with three lines. First, the number of trades. Second, the wins needed to finish at breakeven, which is trades divided by 2.5 for this payoff: 80 trades needs 32 wins. Third, the actual wins. If the month came in under that mark, check whether the trades followed the rules and whether costs matched the plan. If both are fine, a loss like that is within the range for the system, and a drawdown that size should not change position size. That check takes five minutes at month end and replaces a lot of guessing.