How Many Bets You Are Carrying

What the second rule actually buys

State the case for running several rules the way its believers state it, because it is arithmetic rather than folklore and most of it is right. Each rule’s record wobbles. Average several records and the wobbles partly cancel, so the average settles down faster than any one of them, and you learn whether you have an edge sooner. That is true, it is the whole reason a desk runs more than one thing, and the size of the effect is one line: the variance of the average of N records, each with the same variance and an average pairwise correlation of r, is the single-record variance multiplied by one plus N minus one times r, all over N.

Turn that line over and it is lesson 45’s divisor. The reciprocal, N over one plus N minus one times r, is the number of records you are effectively carrying, and lesson 45 already spent it on risk: twenty positions at 0.3 behave like three, at 0.8 like 1.2. The same quantity, read the other way up, is a multiplier on how long everything takes. Two faces, one number, and the whole of this page is about which number goes into it.

Lesson 45 handed you 0.3 and 0.8 as illustrations. Nobody trades an illustration, so measure it. The convention is lesson 63’s, unchanged: the signal is read at the close of bar i and the position is held over the move from bar i plus one, so a rule turns into 59 numbers, one per bar-to-bar move, each the position it held multiplied by what the price did. One thing has to be settled before any two of those can be compared. A rule whose slow average is thirty closes long cannot hold a position until bar thirty, and the numbers before that are not returns of zero, they are a rule that does not exist yet. Leave them in and one series sits pinned at zero while the other moves, which drags the measured correlation toward zero for every pair containing a slow rule. So drop the first thirty moves. What is left is twenty-nine on which all 253 configurations are defined, every pair is measured on the same window, and the 31,878 pairs are comparable with each other.

The mean is 0.9092, 99.40 per cent of the pairs are above six tenths, and the count of negative pairs is zero. Two figures in that table are worth stopping on. The last row is not a rounding: 2,847 of the pairs, 8.93 per cent of them, are the same series to the last decimal, two parameter settings that produced identical positions on every one of the twenty-nine moves. And the least alike pair in the whole set, out of a family wide enough to contain a two-bar average against a three-bar one and a twelve-bar average against a thirty-bar one, is 0.4611. There is no pair anywhere in the grid that offsets another even slightly. They are all long the same instrument some of the time, and being long the same instrument is most of what any of them do.

Now put the measured number through the divisor and read what it costs. Lesson 67 needed 589 trades to settle whether a tenth of an R was real, which at the forty trades a month lesson 65 fixed is 14.73 months of waiting. Running the same rule family in parallel does not divide that by the number of rules. It barely touches it.

The last row is the finding and it is not a rate of diminishing returns, it is a wall. As the number of rules grows without limit the divisor goes to one over the correlation, which at 0.9464 is 1.06, and the wait goes to the correlation multiplied by the original: 0.9464 of 14.73 months is 13.94. There is no number of moving-average rules on this instrument that gets you below fourteen months, and there never will be, because the limit does not contain N at all. Twenty rules take 589 trades each down to 559 and the wait from 14.73 months to 13.98, and the twentieth of them buys two ten-thousandths of a bet.

Read the fourth column against the fifth to see what is usually being sold. Eight rules ought to be 1.84 months and they are 14.03. Twenty ought to be three weeks and they are more than thirteen months. The gap between those two columns is the entire distance between diversification as it is described and diversification as it arrives, and it is set by one number that almost nobody measures on their own rules.

Almost nobody running several rules has correlated their return series against each other. It is two columns of your own daily record and one division, the same division lesson 45 spent an entire page on, and it decides whether the second rule was worth writing.

Module 9 starts a card here, and it has one row per rule you run. The first column is the highest correlation that rule has against anything already on the card — the highest and not the average, because the highest is the one you can act on. Lesson 75 is where the whole card gets spent at once.

Four rules, and the one that takes a bet away

Pick four rules the way a careful person would, spreading the speeds so that 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. On the face of it that is a fast rule, a medium one, a slow one and a very slow one. Here are the six correlations between them, on the same sixty closes.

2-and-5 against 3-and-10: 0.7326. Against 5-and-20: 0.6783. Against 8-and-30: 0.6254.

3-and-10 against 5-and-20: 0.8287. Against 8-and-30: 0.7870.

5-and-20 against 8-and-30: 0.9648.

Five of the six are between 0.62 and 0.83, which is a genuinely mixed set by the standards of the table above. The sixth is 0.9648. The slow rule and the very slow rule are the same trade wearing two parameter pairs, and nothing about the numbers 5, 20, 8 and 30 says so in advance. The average of the six is 0.7695, and four positions at that correlation are 1.2090 bets.

Now do the thing that feels like giving something up. Drop 5-and-20 and keep the other three. The three surviving correlations are 0.7326, 0.6254 and 0.7870, whose average is 0.7150, and three positions at that correlation are 1.2346 bets.

Three positions carry more independent bets than four positions did. Not the same, more. Closing a position added 0.0256 of a bet, about a fortieth, and it did so because the pair it broke up was contributing almost nothing and dragging the average correlation up for everything else. Which of the twins to delete is decided by the rest of the book rather than by the pair, because the pair is tied at 0.9648 with itself: 5-and-20 averages 0.7535 against the two rules that are not its twin and 8-and-30 averages 0.7062, so 5-and-20 is the one carrying less that the others do not already have. Delete 8-and-30 instead and you get 1.2033, below the four you started with. Delete the fast rule, the one that actually differs from the rest, and you fall to 1.1028.

However many you add, you end up holding one divided by the correlation.

Which reframes what to do about a crowded book. The reflex is to add: another instrument, another timeframe, another parameter set, on the theory that more names is more diversification. The arithmetic says the divisor does not care how many names you have, only how alike they are, and that a name whose highest correlation is 0.9648 is worse than nothing because it raises the average for the whole book. The move that raises the divisor is replacement, not addition, and the second move that raises it is subtraction.

So compute the six correlations of your own four rules tonight, and drop the one whose highest correlation is the highest, and if two rules share that highest with each other, the one that is more like everything else.

What this does not settle

That one average correlation describes a book. It does not, and the divisor is a summary rather than a model. A book with four genuinely independent rules and one redundant pair can have the same average correlation as a book in which everything is uniformly middling, and the two behave nothing alike on a bad day. The honest object is the whole matrix and what its eigenvalues do; this page uses the average because it is the version a reader can compute in a spreadsheet, and because the worked example shows that the average is enough to find the pair worth deleting.

That 0.9464 is a constant, or even that it is well measured. Dropping the warm-up is the right correction and it is expensive: it leaves twenty-nine moves, half the data, and those twenty-nine are one stretch of one instrument rather than a sample of market conditions. Rules look most alike in a stretch that trends, because a trend is when every one of them is simply long, and a reader is entitled to suspect that is part of what the number is measuring. Two checks are worth having. Correlate each pair instead on its own defined window, so a fast pair keeps fifty-four moves rather than twenty-nine, and the median comes back 0.9353. Leave the warm-up in, as a first pass at this did, and the median falls to 0.6142, which is the size of the error the correction removes. A single pair over twenty-nine observations at this level carries a standard error near 0.02, so an individual pair is worth about plus or minus 0.04; the median of 31,878 is far steadier than any pair in it, which is the only reason this page quotes four figures at all.

That the correlation you measure is the one you will get. Longin and Solnik found that equity market correlations rise in the tail rather than staying put, so the number that matters is the one on the day everything moves together, and it is higher than the calm-market number this page measured. Every figure in the second table is therefore optimistic in the direction that costs you, and the wall at 1.63 bets is a ceiling on a good day rather than a promise on a bad one.

That this is a fact about correlation rather than about one instrument. The 253 rules are all long or flat the same series, so they cannot be uncorrelated: the only thing they can disagree about is when to be in. Run the same family across four unrelated instruments and the average would fall, which is the actual argument for trading more than one thing and is lesson 39’s. What this page measures is the situation most readers are in, which is several rules on the instrument they know.

That more bets is the objective. It is not. The objective is expectancy, and a second rule with no edge lowers yours however uncorrelated it is, which is why lesson 70’s threshold and lesson 64’s bar both apply to each rule before this page’s divisor applies to the set. A book of twenty worthless rules with an average correlation of zero carries twenty independent bets and no reason to place any of them.

And the concession that costs most: this page counted bets and said nothing whatever about what a bad day costs, which is the question a reader actually arrived with. Four positions risking two per cent each are described everywhere as eight per cent of exposure, and that figure is right only if the four move as one. Lesson 72 puts the measured correlation into the other face of the same divisor and finds that the four are 7.84 per cent in the sense that matters, which is 98 per cent of the eight everybody quotes and nothing at all like the four the independent story would give you.