Trade-count horizon for equity-curve survival

The sample, not the calendar

Expected value is the average account change implied by win fraction and win-loss ratio after a planned number of trades. It is used to judge whether a loss path is still compatible with a positive edge.

Win fraction, denoted p, is the share of trades counted as winners. Win-loss ratio, denoted alpha, is average winner size divided by average loser size, scaled so one loss equals one unit.

Normalized to a one-unit loss, expected result after N trades equals N times (alpha times p minus 1 plus p). A non-positive value means this two-parameter model does not imply a long-run gain. With p equal to 0.60 and alpha equal to 1.2, that identity equals 9.60 units after 30 trades and implies 18 expected winners and 12 expected losers.

The evaluation horizon is the number of trades, not calendar duration.

Winner count and the unit-profit tail

If consecutive trades are independent with a constant win chance, winner count is binomial. A binomial probability model is a discrete law for how many winners occur in a fixed sequence of independent trades that share one constant win chance.

For 30 trades and p of 0.60 the mean is 18, the illustrated mass still spans about 13 to 23 winners, and the all-win and all-loss probabilities are about 0.00000022 and 0.0000000000012. When N is about 30 or larger, a normal curve that shares the binomial mean and standard deviation is treated as a close stand-in for winner-count probabilities.

Mapping winner count into unit profit produces another normal-shaped density centered on expected profit and bounded between minus N and alpha times N. In the 30-trade example the center is 9.60 units and a zero or small loss remains visible in the left tail.

Minimum trade count

Setting expected profit minus two scaled standard deviations to zero yields a minimum trade count T equal to 4p(1-p)(1+alpha) squared over (alpha p minus 1 plus p) squared. That T equals 45 when p is 0.60 and alpha is 1.2, and the identity is defined only when per-trade expectation is positive.

Minimum trade count is the smallest trade sample at which the modeled lower two-sigma profit bound is non-negative. Replacing p times (1-p) by its maximum of one-fourth tightens the same bound to T equal to one over (p minus 1 divided by (1+alpha)) squared.

Monte Carlo paths inside a sigma envelope

Monte Carlo simulation draws repeated random equity paths from a fixed win chance and payoff ratio so path dispersion can be compared with a theoretical band. A sigma envelope is a theoretically drawn band around expected equity formed by adding and subtracting two standard deviations.

Ten simulated 200-trade equity paths with p of 0.60 and alpha of 1.2, started at 100 units, rise on average while remaining mostly inside theoretically drawn mean-plus-or-minus-two-sigma envelopes about 95 percent of the time.

The path model also assumes fixed win and loss sizes, which holds cleanly only when exits are capped by a fixed target and stop.