The Number That Says Scale

The cost, in the same units as the edge

Module 13 charged 0.1230 a share for a round trip and module 14 has been counting in R, so the two have to be reconciled before anything on this page means anything. Lesson 63 measured one bar’s standard deviation at 1.5443 a share, and lesson 67 defined one R as one standard deviation of a trade. Take those as the same quantity, which is the assumption this page rests on and the one to attack first: the cost of a round trip is 0.1230 divided by 1.5443, which is 0.0796 of an R.

That is 79.65 per cent of a tenth-of-an-R edge. It is not a rounding. A trader with a genuine, real, measured edge of a tenth of an R a trade keeps 0.0204 of it, and hands the rest to the spread, the slippage and the commission that lesson 11 itemised.

Now put both numbers through the same machine. Lesson 19 turns an edge into the number of observations a decision needs, and the number is 7.85 divided by the square of the edge-to-noise ratio. The edge is in R and the noise is one R, so the ratio is the edge itself.

Squaring is what does it. Cutting the edge to a fifth multiplies the required sample by twenty-five, and the pace does not move, so the decision moves from a working life to a geological one. Lesson 67’s own figure, 589 trades from a sequential test rather than a fixed sample, is smaller and lands in the same place: 10.07 years at this pace, against fourteen and three-quarter months at the forty a month lesson 65 assumed.

Almost nobody you trade against has computed this, because the two halves live on different pages. The sample size is a statistics question and the round trip is a broker question, and the only place they meet is a division that takes ten seconds and nobody performs.

The way out, and why it closes

The escape is obvious and everybody reaches for it: take more trades. The losing-month rate falls with the square root of the pace, so a business that trades enough becomes legible. Lesson 96’s table said so.

Price it at both edges. Nine winning months in ten means a 10 per cent losing-month rate, which needs the edge multiplied by the square root of the monthly trade count to reach 1.2816 standard deviations.

The middle column is a business. Forty-five trades a month is two a day, it is roughly what lesson 65 assumed, and it buys three winning months in four. The right-hand column is not a business, it is a market-making desk: 1,098 trades a month is fifty-two a day, and 3,965 is one hundred and eighty-nine.

And the trap closes here. The right-hand column assumes the cost stays at 0.0796 of an R as the pace rises, and it does not. Lesson 93 measured what happens across the grid: the fastest family of rules trades eight and a third times in twenty-eight moves and loses more than a quarter of its gross return doing it, while everything from a fifteen-bar slow average outward trades once and loses three per cent. A faster rule holds smaller moves, so its R shrinks while the 0.1230 does not, and the cost in R goes up. The right-hand column is therefore the optimistic version of a column that is already impossible.

Scale is not a way around cost. It is a way of buying more of it. Every trade added to make the record legible is another 0.0796 of an R subtracted from the thing being measured, which is why the required sample and the achieved pace move in the same direction rather than towards each other.

What is left is the honest set of exits, and there are only three. Trade an instrument where the round trip is a smaller fraction of a trade’s R, which is lesson 11’s point that the same round trip runs from a fraction of a basis point to hundreds. Hold longer, so that R grows while the 0.1230 stays fixed. Or accept that the business is not measurable on a human timescale and stop describing it as one, which is the option lesson 100 puts on the card.

So divide your own round-trip cost by your own R, subtract it from your edge, and square the ratio before you tell anybody how long your record needs to be.

What this does not settle

That one R is one bar’s standard deviation. This page needs the two to be the same to convert 0.1230 into 0.0796, and they are not the same thing. Lesson 63’s 1.5443 is the spread of one bar’s move; lesson 67’s R is the spread of one trade’s outcome, and a trade held for several bars has a wider one. Holding longer therefore shrinks the cost in R faster than this page shows, which is the third exit above and the only one this course can recommend without qualification.

That 7.85 is the right constant. It is lesson 19’s, and it belongs to a specific test at specific error rates. Lesson 67’s sequential test gets to a verdict in 589 rather than 785 on the same edge, so the constant is not a law of nature; it is the price of one particular pair of mistakes. What survives any choice of test is the squaring, and the squaring is the whole finding.

That the edge is stable while you wait. Lesson 68 measured what happens when it is not: an edge that fades linearly to nothing over exactly the 589 trades the test needs is declared dead 65.46 per cent of the time and alive 34.53, on a median of 627 trades. A decision that takes 324 years is not a slow decision, it is no decision, because nothing survives unchanged for that long.

That trading more is the only way to more trades. It is not: trading the same rule on more instruments multiplies the count without shortening the holding period, which is what module 9 was about. Lesson 71 measured what that buys on rules from one family, 0.9472 of correlation, and lesson 91 found the whole 253-rule grid earns 4.16 against a single rule’s 4.10. Breadth is the answer that works arithmetically and it needs genuinely different objects, which this course has not got data for.

And the concession that costs most: this page has said the record cannot settle the question and has not said what to do in the meantime. A trader who cannot prove the edge in 324 years still has to decide, every month, whether to keep going, and that decision is made against an account balance rather than a test statistic. Lesson 99 runs the balance and finds that at this net edge the median ten-year path ends at 46,526 with 58.67 per cent of paths halving on the way.