The Second Way to Disagree

What a second instrument buys

Lesson 71’s divisor takes a count of positions and an average correlation between them and returns the number of independent bets they amount to: the count divided by one plus the count less one times the average. Two positions at 0.88 give 1.0638, which is the 1.06 lesson 75 licensed and lesson 81 graded. The formula does not care what the positions are. It cares only how alike they move.

So put the same two rules on a second instrument and run it again. Four positions, six pairs. Two of those pairs are within an instrument and carry lesson 74’s 0.88. The other four cross between instruments and carry whatever the two instruments correlate at, which is a number this page will not choose for you.

The top row is the best case and it is not four. Doubling the positions at a cross-correlation of zero doubles the bets, and doubling is the ceiling because the within-instrument 0.88 is still there underneath, unchanged, on two of the six pairs. Then read down, and find the row your own two series put you in, because that is the only one that applies to you. At 0.6 the four positions carry 1.30 bets. You bought two more positions, two more spreads and two more lots of impact, and you bought a quarter of a bet.

The last row is the one to keep. At a cross-correlation equal to the within-instrument one, four positions on two instruments carry 1.10 bets, which is exactly the number lesson 74 measured for four rules on one instrument. The divisor cannot tell a second instrument from a fourth rule. Only the correlation goes in.

Almost nobody has run the divisor before adding an instrument rather than after. It is one line of arithmetic on a correlation you can measure in ten minutes, and it prices the whole of what the addition is worth before you pay for it.

Module 11 keeps a list of the degrees of freedom this course did not have, and this page opens it with one: a rule that may be short.

There is a second thing you can do with two instruments, and lesson 74 named it in passing without pricing it. Every rule in this course is long or flat, so two rules can disagree about exactly one thing, which is when to be out. A rule that may be short can disagree about direction, and the arithmetic of that is not the divisor, it is the variance of a difference. Two equal positions each with standard deviation one: hold both long and the pair’s standard deviation is the square root of one plus the correlation over two. Hold one short and the plus becomes a minus.

The two middle columns move in opposite directions down the same column of correlations, and that is the whole of the page. Every argument module 9 made was an argument for a smaller correlation, because a long-only book is worse the more alike its positions are. A spread is better the more alike they are, and at the 0.88 lesson 74 measured on this course’s own rules the spread’s standard deviation is a quarter of the long-only pair’s.

The same two positions, priced twice

Take the book lesson 81 graded and give it the second instrument it was missing, at the same 0.88 the rules already carry, because that is the number this course has actually measured rather than assumed.

Four positions, all long: 1.10 independent bets, against 1.06 for two positions on one instrument.

The second instrument therefore bought 0.04 of a bet, for two more positions.

Two positions, one of them short: a standard deviation of 0.2449 against 0.9695, which is 3.96 times smaller.

The short leg therefore bought more, on half as many positions, out of the same correlation.

Read those four lines against each other and the ranking is not close. Adding an instrument and keeping every leg long is the expensive repair and it buys four hundredths of a bet. Turning one leg over is the cheap one and it divides the day by four. Module 9 spent five lessons trying to find rules that were less alike, and the arithmetic says the profitable use of rules that are alike was available the whole time and needed one permission the course never granted.

Correlation is what a long-only book loses to and what a spread is made of.

Now price the permission, because it is not free and this course has already measured most of the bill. Two legs is two of lesson 12’s spreads on the way in and two on the way out, and two of lesson 59’s impact rather than one. The gross position is twice the net, so lesson 73’s warning applies exactly: every cost that scales with what you trade doubles while the standard deviation falls by 3.96. And the short leg carries a bill this course has never priced at all, which is what it costs to borrow the thing you are selling.

So measure the correlation between two instruments you already follow, put it into both tables, and compare what the second instrument buys against what turning one leg over buys.

What this does not settle

That the cross-correlation is yours to choose. Every row is printed because the number is a measurement you have to make on your own two series, and you have to make it on both halves of them, which is lesson 74’s finding standing behind this one. A page that picked one number here would be doing the thing lesson 71 was written to stop.

That a smaller standard deviation is a smaller risk. It is a smaller standard deviation on twice the gross position. Lesson 73 showed that the gross figure is where the costs live, and the second table changes the numerator of that ratio without touching the denominator. A spread at 0.88 with a quarter of the daily movement and twice the bill is a better trade only if the edge survives the doubling, and nothing on this page establishes that it does.

That a correlation of 0.88 stays 0.88. Lesson 74 measured the median pair of this course’s own rules moving 0.0972 between two halves of one record, against 0.0219 under a null that says nothing changed, which is 4.44 times as much movement as chance produces. Price a spread at 0.88 and live in a stretch where it is 0.5966, and its standard deviation rises from 0.2449 to 0.4491, which is 1.83 times. The trade that looked four times quieter is then twice as loud as you sized it for, and it happens without any rule changing.

That this course measured a pair. It did not. There are sixty closes of one instrument in this course, and the cross-correlation is a parameter on every row of both tables. What is measured here is the divisor and the variance of a difference, which are arithmetic and hold for any two series, and what is borrowed is lesson 74’s 0.88, which is a within-instrument number standing in for a between-instrument one. A reader who wants the finding on their own pair has to supply the pair.

And the concession that costs most: the short leg is a cost this course never priced. Lesson 12 priced a spread, lesson 59 priced impact and lesson 21 sized a stop, and none of them contains what it costs to borrow a security, what happens when the lender recalls it, or the fact that a short position’s loss has no ceiling to size against. The entire gain in the second table is bought with a leg this course cannot bill you for, which is the most honest thing this page can say about an elective that begins by granting a permission. Lesson 83 keeps the list going with the next missing degree of freedom, which is a second place to trade the same instrument, and asks what the 0.1230 a share this course has charged for every round trip since lesson 63 becomes when the two places disagree.