The Window Decides

Why a long-or-flat rule cannot disagree for long

Start with the mechanism, because it is not subtle and it explains everything that follows. A rule in this family can be long or it can be flat. Two such rules can disagree about exactly one thing: when to be out. While both are in, they are not two rules at all, they are the same instrument twice, and the instrument is one thing.

So count how often they are all in. Over the first fifteen moves, an average of 71.9 per cent of the 253 rules are holding a position on a given day, and on the quietest of those days it is 17.0 per cent. Over the last fourteen it is 97.3 per cent on the average day and never below 91.7 per cent. That is the whole story: in the second fortnight the grid is not a set of rules with different opinions, it is 253 copies of a long position with a few of them occasionally stepping out.

Which is what the correlations say when you measure them separately.

The median pair in the last fourteen moves is correlated at one. Not close to one, at one: more than half of the 31,878 pairs in the grid produce the same fourteen numbers as each other. The middle row’s 8.93 per cent is lower than the first row’s 9.37 for a dull reason worth stating: agreeing on twenty-nine numbers is harder than agreeing on fifteen, so the longer window finds fewer exact ties even though it contains the fortnight in which almost everything tied. The full-window figure of 0.9464 that lesson 71 measured, and that lessons 72 and 73 spent, is an average of a fortnight in which a tenth of the grid was identical and a fortnight in which nearly three quarters of it was.

Before treating that as a finding, rule out the obvious objection: fifteen observations and fourteen observations measure a correlation badly, so perhaps the two halves differ because any two halves would. That is testable. Take each pair’s twenty-nine-move correlation, hold it fixed as the truth, and simulate what two halves of fifteen and fourteen draws would show. Under that null the median pair’s correlation moves by 0.0219 between halves and moves by more than 0.2 in 6.06 per cent of cases. What actually happens is that it moves by 0.0972, four and a half times as much, and moves by more than 0.2 in 21.4 per cent of pairs and by more than 0.4 in 9.4 per cent.

And the direction settles it. Sampling error is symmetric: half the pairs should rise and half should fall. In the measurement, 87.5 per cent of the 31,878 pairs rose. That is not noise, it is one thing happening to the whole grid at once.

Almost nobody has measured their own correlation twice. It is the same two columns of your own record you used in lesson 71, split down the middle and correlated twice instead of once, and the difference between the two answers is the number this page is about.

Module 9’s card gains a fourth column, and it is the highest correlation the rule has shown against the rest on any window you have measured, rather than the correlation it shows across all of them. Lesson 75 is where the whole card gets spent at once.

The same four rules, priced three ways

Take lesson 71’s four rules and lesson 72’s book, four positions at two per cent each, so the heat is eight per cent in every row of what follows. Nothing about the book changes. The only thing that changes is which fortnight the correlation was measured in.

Read the last column. The same four positions, sized identically, hand you a day on which everything goes against you 1.54 times more often when the correlation is measured on the second fortnight than on the first. Lesson 72’s finding was that heat is a maximum and correlation is the schedule; this table says the schedule is itself a function of when you looked.

The second column is worse than it looks. In the first fortnight the four rules are 1.43 independent bets, which is thin and is at least more than one. In the second they are 1.0989, which rounds to one bet in any language. You did not stop diversifying. The market stopped letting you.

And the third row is where lesson 73 stops working. Three of the six pairs are correlated at exactly one over those fourteen moves, so the covariance matrix has a determinant of zero, no inverse, and no minimum-variance weights of any kind. The optimiser does not return a bad allocation for that fortnight, it returns nothing at all. On the first fifteen it returns 0.448, 0.273, minus 0.498 and 0.777; on the full twenty-nine it returns 0.746, 0.078, minus 0.672 and 0.848; on the last fourteen the question has no answer.

So the practical rule is not about weights, it is about size. Price the book on the worst window you have measured rather than on the average of your windows, which on these four rules means planning for 1.10 bets and one bad day in 2.8 rather than 1.43 and one in 4.3.

So split your own record in half tonight, correlate your rules twice, and size on the worse of the two answers.

What this does not settle

That the comparison between the two windows is clean. It is not, and there is a known bias pointing the same way as the finding. The second fortnight is more volatile than the first, with the instrument’s own standard deviation 1.240 times higher, and measuring a correlation inside a more volatile stretch inflates it mechanically even when the underlying relationship has not moved at all. Apply the standard correction for that and the 0.8800 becomes 0.8310. Which still sits far above the first fortnight’s 0.5966, so the bias explains part of the move and not most of it, but a page that quoted 0.8800 without saying this would be overstating its own case.

That fifteen observations and fourteen observations measure anything. On their own they measure very little, and a single pair split that way is worth nothing. What makes the movement credible is that the same split was applied to 31,878 pairs and compared against a simulation of what chance alone would do, which is a test the individual pair cannot pass and the grid can.

That this is a fact about markets. It is a fact about long-or-flat rules on one instrument, which is the situation this course has been building since lesson 63. A rule that can be short has a second way to disagree; two instruments have a third. What generalises is the shape of the problem rather than the size of the number, and the shape is that the correlation you measured is conditional on a market condition you did not record.

That the worst window is the worst there is. It is the worst of two windows inside sixty closes, which is a very small sample of market conditions and almost certainly not the worst one that exists. A stretch in which every rule is long is not the frightening case; the frightening case is a stretch in which every rule is long into a reversal, which these sixty closes do not contain and which the sizing rule above is a poor substitute for having seen.

That this changes your weights. It does not, and lesson 73’s answer survives intact: the constrained optimiser is still the one to run, because a constraint is exactly what you want when the input is unstable. What changes is the input you feed it and the size you carry afterwards. A covariance measured on the worst window you have is a better argument than a covariance measured on all of it.

And the concession that costs most: this module has now produced four columns of a card and not one instruction about what to do when a number on it goes wrong. It says how many bets you carry, how often they all lose together, how much goes in each and how badly that answer moves. It has not said what happens at nine in the morning when the answer to one of those has changed, who is allowed to override it, or what gets shut down first. Lesson 75 spends the whole card, and it is about the part that is not arithmetic.