What the Whole Grid Earns

The half of diversification nobody measures

The case for running several rules is made in one direction and checked in the other. It is made on risk: several records wobble less together than any one of them does alone, so you learn sooner and you sleep better. Lesson 71 measured exactly that and found the wobble barely moved, because the rules were 0.9472 alike. What is almost never checked is the other side of the same ledger, which is what the book makes. Risk is the price of a portfolio. Return is what you bought with it, and a page that prices something without saying what it bought has done half a job.

So here is the whole job on lesson 63’s grid, with nothing changed. Fast averages from 2 to 12, slow from 3 to 30, slow always longer, which is 253 rules. The signal is read at the close of bar i and the position is held over the move that starts at bar i plus one. Strict inequality, ties carry. Every rule is defined on the moves from bar 31 to bar 59, which is the twenty-eight lesson 71 correlates on, and every figure below is on those twenty-eight and gross of the turnover each rule generates, which lesson 93 prices separately.

The first thing the grid says is not about portfolios at all. Those 253 rules produce 28 distinct totals between them. Not 253 opinions about the market: 28, several of which are held by forty rules at once. The commonest total on the grid, 3.10 a share, is what 43 different configurations arrive at, and a trader who ran all 43 would have run one rule 43 times.

The book at every size

Take a book of N rules drawn from the grid at random, weight them equally, and record what it makes and how much it moves. Do that a thousand times for each N from seed 20260905, and report the median, the fifth and the ninety-fifth of the thousand. At N of 253 there is only one book and it is drawn once. The last column is how much smaller the book’s standard deviation is than the average of the standard deviations of the rules inside it, which is the number the case for diversifying is made in.

Read the median column down. It starts at 4.10 and ends at 4.16, and the six pennies between them are the difference between the median rule and the mean rule rather than anything a portfolio did. Read the outer columns instead and the table is doing something violent: the ninety-fifth falls from 6.20 to 4.16 while the fifth rises from 3.10 to 4.16. Every rule you add takes width off both ends at once, and it takes more off the top than off the bottom, because the top had further to fall.

The last column is the entire case for doing it, and at 253 rules it is 4.21 per cent. That is the ceiling. There is no book of this family, at any size, that wobbles more than four and a quarter per cent less than the rules inside it, because lesson 71’s 0.9472 is a floor under how alike they can be. One divided by that correlation is 1.06, and a book of 1.06 bets is a book of one.

Almost nobody you trade against has run this table on their own rules, and it costs an evening, because it needs each rule’s daily return series rather than each rule’s summary line. The summary line is what a platform gives you and it is exactly the thing that cannot answer the question.

Four rules, chosen not to look alike

Random draws are the fair test of the family, but nobody picks at random. Lesson 71 picked four rules the way a careful person would, spreading the speeds so no two look alike: a 2-bar average against a 5, a 3 against a 10, a 5 against a 20, and an 8 against a 30. That care is worth something and it is worth measuring rather than assuming.

Choosing them to be unalike did work. The cut in the wobble is 8.10 per cent against the 2.94 a random four gets, which is nearly three times the benefit for no extra rules. That is the strongest result on this page in favour of doing this at all, and it is worth stating before the arithmetic that follows takes it back.

What the care cost

The four-rule book made 4.10 a share over the twenty-eight moves. The best of the four, the 2-bar against the 5, made 4.90 on its own. So the book kept 83.7 per cent of the best rule’s return, and in exchange for the 16.3 per cent it gave up it received 8.10 per cent off its standard deviation.

Put those two numbers side by side and the trade is bad in the direction nobody checks. You paid 16.3 per cent of the return for 8.10 per cent of the wobble, which is two units of return for one unit of calm. A portfolio is supposed to do the opposite. It is supposed to buy more calm than it costs in return, and on a family this correlated it cannot, because the return of the book is the average of the rules and the wobble of the book is very nearly the average too.

The comparison that matters more is the one this course fixed in lesson 66. Over the same twenty-eight moves, buying at the first close and selling at the last makes 6.60 a share. The four-rule book makes 4.10. The best single rule makes 4.90. The best rule in the entire grid of 253, a 9-bar average against a 10-bar one, makes 7.00, and it is the only rule of the 253 that beats holding at all. Every book on this page, at every size, sits below the benchmark by more than a third.

A book of one family is one bet, priced as though it were four.

It is worth being exact about what that does and does not say, because the sentence is easy to over-read. It does not say diversification is a mistake. It says that diversifying inside one family buys almost nothing, and this course has only ever measured one family: moving-average crossovers on one instrument over one window. Two rules that are genuinely different objects would correlate less and the table would look different. What the table settles is that spreading the dial settings of one idea is not that, however many settings you spread it over.

So before adding a rule to your book, compute what the book of the ones you already have made, and put it beside your best single rule.

What this does not settle

That twenty-eight moves are enough to say any of this. They are not, and the course has been explicit about it since lesson 19. Twenty-eight moves is the window on which all 253 rules exist at once, which is the only reason it is the window, and lesson 74 already showed that the correlation this whole module spends is a property of the fortnight it is measured in: in the first fourteen moves 9.50 per cent of the pairs are correlated at exactly one, and in the last fourteen 72.23 per cent are. Every figure on this page inherits that fragility.

That equal weighting is the right way to hold a book. It is the simplest way, and lesson 73 showed that the standard machinery does better on the same data, cutting the standard deviation of the day by 10.65 per cent where equal weights cut it by 8.10. Lesson 74 is then the reason that improvement is not a promise. The equal-weight book is used here because it is the one a reader can reproduce without inverting anything.

That holding is a strategy. It is not, it is a benchmark, and it is the benchmark lesson 66 fixed before any of this was measured. Its 6.60 over these twenty-eight moves is one number from one window on one instrument, and quoting it as though it were an alternative would be exactly the error lesson 63 exists to prevent.

That the returns on this page are what a book would have earned. They are gross. Every position change costs 0.1230 a share and the four rules make thirteen of them over the twenty-eight moves, which nobody has subtracted yet.

And the concession that costs most: the whole page is one family. Moving-average crossovers on one instrument are as alike as trading rules get, and a book built out of genuinely different objects would not be pinned at 0.9472. This course cannot show you that, because it has measured one family, and a page that concluded “diversification does not work” from one family would be doing the thing lesson 63 was written to stop. What it can show is where the result actually came from, and that is worse than the average suggests: lesson 92 finds that three of the four-rule book’s twenty-eight moves are 80.5 per cent of everything it made.