Correlation
Two series whose whole-run correlation is 0.45 read anywhere from -0.41 to 0.94 across ten-bar windows of this module’s own sixty bars, and from -0.03 to 0.89 across twenty-bar ones, which is most of the range the measure can take. Correlation is one division: the covariance of two return series over the product of their standard deviations, and what comes back is not a property of the pair. Three exact results survive that noise, and they are why the division is worth doing at all. Correlation and sensitivity are the same fact seen twice, because a regression slope is the correlation multiplied by the ratio of the two standard deviations — so a hedge can carry the right sign and the wrong size. A hedge removes the square of the correlation from the variance rather than the correlation itself, so 0.7 leaves 71 per cent of the original standard deviation standing. And a book of twenty positions with an average pairwise correlation of 0.3 carries the risk of three independent ones; at 0.8 it carries the risk of 1.2. Not one position size changed.
Prerequisites: Lesson 44, which computed the quantity correlation is a ratio of, lesson 38, whose window problem turns up here for the third time, and lesson 20, whose arithmetic sizes one position at a time and needs this lesson the moment there are two.
One division, and what it divides
Correlation takes lesson 44’s first two steps and adds one. Turn both price series into returns. Take the covariance, which is the average product of the two series’ deviations from their own means. Divide by the product of the two standard deviations. That last division is the whole point: covariance is measured in return-squared and cannot be compared across pairs, while the ratio is dimensionless and confined between minus one and one, so a number computed on two currencies can sit beside one computed on two shares.
Everything below runs on two series, and the second needs disclosing before it is used. The first is the sixty closes this module has worked with since lesson 38. The second is built from them by a published rule, so every figure on this page can be reproduced: each of its returns is half the first series’ return for that bar, plus 0.866 of the first series’ return 29 bars later, wrapping round at the end of the run. Those weights are chosen to give the second series roughly the volatility of the first and a correlation with it of one half. What actually comes out is 0.45, because the rotated series is not quite uncorrelated with the original — it reads -0.07 — and that miss is the first thing worth noticing. A pair built to hit a target on sixty bars misses it by five points.
Correlation and sensitivity are the same fact
Ask how far the second instrument moves when the first moves one per cent and you are asking for a regression slope, not a correlation. The two are not alternatives. The slope is the correlation multiplied by the ratio of the standard deviations, the second over the first, and that identity is exact rather than approximate. On these two series the covariance divided by the first series’ variance gives 0.4374, and the correlation of 0.4518 multiplied by the volatility ratio of 0.9682 gives 0.4374 as well. Same number, two routes.
The identity is worth having because the two halves fail independently. A correlation near one tells you two things move together; it tells you nothing about how far. If the second instrument is a third as volatile as the first, a correlation of 0.95 still means one unit of it offsets only a third of a unit of the first, and holding them one for one leaves two thirds of the exposure uncovered. That is how a hedge ends up with the right sign and the wrong size, and it is why the sensible instruction is to compute both numbers and multiply, rather than to read a correlation and feel reassured.
The second half of the arithmetic is less comfortable. Hedge one instrument with another in the best possible proportion and the variance that survives is one minus the square of the correlation. Squaring is brutal at the levels people actually work with. A correlation of 0.3 removes 9 per cent of the variance and leaves 95 per cent of the standard deviation. At 0.5 it removes 25 per cent and leaves 87. At 0.7, which almost anyone would call a strong relationship, it removes 49 per cent and still leaves 71 per cent of the original width. Only at 0.9 does the residual fall below half, and only at 0.95 below a third. On the module’s two series, whose correlation is 0.4518, the best hedge available removes 20 per cent of the variance. Everything else in that position is still there.
The window, for the third time
Lesson 38 established that a measurement belongs to its window, and lesson 44 found the effect twice over in volatility. Correlation is worse, because it is a ratio of three estimates rather than one, and the errors do not cancel. Below, the correlation between the module’s two series computed over every window of each length the run allows.
| Bars in the window | Windows available | Lowest | Median | Highest | Highest less lowest |
|---|---|---|---|---|---|
| 10 | 50 | -0.41 | 0.41 | 0.94 | 1.35 |
| 20 | 40 | -0.03 | 0.38 | 0.89 | 0.92 |
| 30 | 30 | 0.05 | 0.52 | 0.83 | 0.78 |
| 59 | 1 | 0.45 | 0.45 | 0.45 | 0.00 |
Read the last column first. The correlation coefficient can only take values two units apart, from minus one to one, and a ten-bar estimate of this pair covers 1.35 of those two units. It is not that the estimate is imprecise; it is that a ten-bar window can report a strong negative relationship and a strong positive one about the same pair, in the same run, a few bars apart. Nothing in the construction changed. The same rule generated every return in the series.
The sampling arithmetic says exactly how much of that is noise. Around an observed correlation of 0.50, the 95 per cent interval from 20 observations runs from 0.07 to 0.77. From 60 observations it runs from 0.28 to 0.67. From a full year of 252 it still runs from 0.40 to 0.59, an interval a fifth of a unit wide after a year of daily data. High correlations are measured more precisely — an observed 0.90 on 60 observations sits inside 0.84 to 0.94 — which is a small mercy, because the high readings are the ones that matter for a book.
The sign is not a property of the pair
The commonest thing said about two instruments is that they are inversely correlated, as though the sign were a fact about them. It is not. Take the most quoted pair of all, shares and government bonds, and give the market two independent things to be frightened of: growth and interest rates. Good growth news lifts shares and hurts bonds, because it raises the rates expected later. A rise in rates hurts both, because it discounts every future cash flow more heavily. Write shares as growth minus rates and bonds as minus growth minus rates, and the covariance between them is the variance of the rates shock minus the variance of the growth shock, while each has a variance equal to their sum.
The correlation is therefore the difference of the two variances over their sum, and nothing else. Give the growth shock three times the variance of the rates shock and it is -0.50; give the rates shock three times the variance of the growth shock and it is 0.50. Nine times in either direction gives -0.80 and 0.80. When they are equally important it is exactly zero. Nobody’s behaviour changed across that range and no relationship was broken; the mix of what the market was worried about changed, and the correlation is a readout of that mix.
Two things follow that are easy to get wrong. First, a correlation of zero between shares and bonds does not mean they are unrelated: on this account it means two strong relationships of opposite sign are in balance, which is a completely different situation from two instruments that genuinely have nothing to do with each other, and it will not stay at zero. Second, a sign flip is not evidence that something broke. It is evidence that the dominant shock changed, which is a thing markets do routinely and which you can often see coming from what is on the calendar.
Why it rises in a crisis, and how much of that rise is real
The observation that correlations go to one when everything is falling is true enough to be worth taking seriously and is usually explained by fear, which explains nothing. There are three mechanisms and none of them is fear.
The first is pure arithmetic. Suppose every instrument’s return is a common factor plus something of its own. The correlation between any two of them is the variance of the common factor divided by the sum of that and the idiosyncratic variance. Now let the common shock get bigger without anything else changing: no loading changes, no participant behaves differently, nothing about any instrument is altered. If the common shock’s variance was 1 against an idiosyncratic 3, the correlation was 0.25. Triple the common shock’s standard deviation, which takes its variance to 9, and the correlation is 0.75. That is the entire move, and it is a consequence of correlation being a ratio whose numerator a crisis inflates.
The second mechanism is real and is not about the factor structure at all. A leveraged holder meeting a margin call sells what they own rather than what is falling, so instruments that have nothing in common start moving together because one balance sheet contains all of them. That is a new common factor, created by the liquidation and lasting exactly as long as it does. It is the reason a crisis correlation can exceed anything the ordinary factor structure would produce, and the reason it can vanish within days.
The third is that the measurement is biased, and this one is worth arithmetic because it is large. A correlation computed only over the volatile stretch comes out higher than the same pair’s correlation measured over everything, even when nothing whatever has changed, because conditioning on big moves in one series selects the observations where the shared component dominated. The correction divides the observed figure by the square root of one plus the proportional rise in variance times one minus the observed correlation squared. Take a stretch whose variance quadrupled, which is volatility doubling, and an observed correlation of 0.80 corresponds to an unconditional 0.55. An observed 0.60 corresponds to 0.35. An observed 0.90 corresponds to 0.72.
That correction is contested, and the disagreement is the honest state of the question rather than a footnote. Adjusting for volatility removes much of the apparent rise, but work on the extremes of the distribution finds that correlation still rises in falling markets by more than volatility alone accounts for, and does not rise the same way in rallies. So some of the increase is an artefact of measuring during a violent period and some of it is a real property of falling markets. What the arithmetic settles is that the artefact is big enough to matter: if you quote a crisis correlation without saying what the variance did, you are quoting a number that is inflated by an unknown and substantial amount.
What it does to a book
Everything above is preparation for one calculation, which is the reason correlation is on the syllabus. Hold two positions of the same size in instruments of the same volatility, and the pair’s standard deviation is the square root of twice one plus the correlation, in units of one position. At a correlation of zero that is 1.41, not 2: two independent bets of the same size carry less than double the risk, and the difference is the whole of what diversification is. At 0.3 it is 1.61. At 0.8 it is 1.90. At 1.0 it is exactly 2, which is one position with two tickers on it.
Extend that to an equal-weight book of any size and the arithmetic stays exact. A book of instruments with the same volatility and an average pairwise correlation carries the risk of a smaller number of independent positions, and that number is the count divided by one plus the count less one times the average correlation. It depends on the average pairwise correlation and on nothing else about the structure, which is worth knowing, because it means you cannot fix a correlated book by rearranging which names it holds.
| Average pairwise correlation | 2 names | 5 names | 20 names | 50 names |
|---|---|---|---|---|
| 0.0 | 2.00 | 5.00 | 20.00 | 50.00 |
| 0.1 | 1.82 | 3.57 | 6.90 | 8.47 |
| 0.3 | 1.54 | 2.27 | 2.99 | 3.18 |
| 0.5 | 1.33 | 1.67 | 1.90 | 1.96 |
| 0.8 | 1.11 | 1.19 | 1.23 | 1.24 |
Every figure in the table is the number of independent positions the book is actually carrying. Read across the 0.3 row: going from five names to fifty, a tenfold increase in the work of running the thing, takes the count from 2.27 independent bets to 3.18. Read down the 20-name column and the collapse is faster still, from twenty at zero correlation to 6.90 at 0.1 and 2.99 at 0.3. The first tenth of correlation costs more than half the diversification in the book. That is not a warning about crises; 0.1 is a quiet market.
The floor is worth naming because it is exact. As the number of names grows without limit, the book’s standard deviation approaches the square root of the average correlation times a single name’s. At 0.3 that floor is 0.55, so no number of positions will take a book with that average pairwise correlation below 55 per cent of the volatility of one of them. At 0.8 the floor is 0.89. Adding names past the point where the count stops rising is work that buys nothing.
Now put the crisis into it. A book of twenty names at an average correlation of 0.3 is carrying 2.99 independent bets and a standard deviation of 0.58 of one name. Move the correlation to 0.75, which is what the arithmetic above says a tripled common shock does, and the same book carries 1.31 independent bets and a standard deviation of 0.87. Its risk rose by 51 per cent and nobody traded. To get back to the risk you had before, every position has to come down by 34 per cent — and that cut is the answer to the question lesson 44 left open, because it is a cut that volatility sizing alone will not make for you. Lesson 44’s arithmetic sizes each position against its own volatility, one at a time, and is blind to the fact that they have started being the same position.
What this does not settle
That the correlation you measured is the correlation. It is an estimate, and a noisy one. Around an observed 0.50, the 95 per cent interval from 20 observations runs 0.07 to 0.77, from 60 observations 0.28 to 0.67, and from a full year of 252 observations 0.40 to 0.59. Every rolling correlation on every platform is a number of this kind. Before treating a change in one as a change in the market, check whether the two readings’ intervals overlap, because at the window lengths people use they almost always do.
That the average pairwise correlation is all you need to know about a book. It is all you need for the variance, exactly, which is why the count above depends on nothing else. It is not all you need for anything else. Take the same twenty names and the same average of 0.3, but arrange them as two blocks of ten correlated 0.75 inside each block and -0.105 across, which is a legitimate correlation matrix and averages to 0.3 to the digit. The book still shows 2.99 independent bets and a standard deviation of 0.58, because those depend on the average alone. But ten of its twenty names are a block carrying 1.29 independent bets between them, so half the book is one position, and no figure in the table above can see that. The sentence above about not being able to fix a correlated book by rearranging its names is true of the variance and of nothing wider. Run the count on your blocks as well as on the whole.
That correlation measures how related two things are. It measures how well a straight line fits, and nothing else. Two series can be perfectly determined by one another and read zero, which happens whenever the relationship is symmetric about a turning point — a position that loses on a big move in either direction has exactly that shape. Everything on this page applies to instruments whose relationship is roughly linear over the range you hold them, and options are not among them.
That these two series are anybody’s. The second one was constructed from the first by a rule printed above, precisely so the arithmetic could be checked, and the module’s sixty bars were built to be legible on a small chart in the first place. The window table’s shape transfers — short windows really do give wild correlations on real data, and for the same reason. The specific figures do not, and neither does the 0.45.
That the book arithmetic covers a real book. The effective count assumes every position is the same size and every instrument the same volatility. A book with one position three times the size of the others is closer to being that one position than any count of tickers suggests, and the fix is to run the same calculation on the positions weighted by what they actually risk. The average-correlation result stays exact under equal weights and stops being exact the moment they are not.
That a correlation is stable enough to hedge on. The hedge ratio is a correlation multiplied by a ratio of two volatilities: three estimates, each carrying its own error, combined into one number that then decides a size. And the quantity moves most in exactly the conditions where the hedge is being asked to work. A hedge sized on a calm-market correlation is sized for the market that is not happening when you need it.
Problems
- Count the independent bets in your own book. Take the positions you hold, compute the correlation of each pair over the last 60 bars, average those correlations, and put the average into the count divided by one plus the count less one times the average. Write the answer next to the number of tickers on your screen. If you hold fifteen things and the arithmetic says two and a half, you now know what your diversification is worth, and the number was not available any other way.
- Check every hedge by its two numbers rather than by its story. For anything you hold because it is supposed to offset something else, compute the correlation between the two, compute the ratio of their standard deviations, and multiply. That product is the size the hedge should be, per unit of the thing hedged. Compare it with the size you actually hold. Then square the correlation and read off how much of the variance the hedge removes even when it is sized perfectly. At 0.7 the answer is under half.
- Split your own history and apply the adjustment. Take a pair you care about, split its history into the calmer half and the more volatile half by the volatility of the first instrument, and compute the correlation separately in each. Then take the ratio of the two variances and apply the correction above to the volatile-half figure. Whatever difference survives that adjustment is the part that is not an artefact of measuring during a violent stretch, and it is usually a good deal smaller than the raw comparison suggested.
Sources. Francis Galton, “Co-relations and their Measurement, chiefly from Anthropometric Data” (Proceedings of the Royal Society of London, 1888), for the coefficient itself, introduced to compare limb lengths rather than markets and defined then exactly as it is computed now. Ronald A. Fisher, “Frequency Distribution of the Values of the Correlation Coefficient in Samples from an Indefinitely Large Population” (Biometrika, 1915), for the sampling distribution and the transformation that produces the intervals quoted above. Harry Markowitz, “Portfolio Selection” (The Journal of Finance, 1952), for the portfolio variance identity that turns a pairwise correlation into a statement about a book, which is the only reason any of this is on a trading syllabus. Kristin J. Forbes and Roberto Rigobon, “No Contagion, Only Interdependence: Measuring Stock Market Comovements” (The Journal of Finance, 2002), for the conditioning bias and the adjustment applied above. François Longin and Bruno Solnik, “Extreme Correlation of International Equity Markets” (The Journal of Finance, 2001), for the other side of that argument: correlation rises in falling markets by more than the volatility adjustment explains, and does not rise the same way in rising ones.
Correlation rose in the third mechanism above because one balance sheet held everything, which points at a quantity this module has measured nothing about. Every number in lessons 38 to 45 was computed from prices. None of them says who is holding the position on the other side of yours, how much of it there is, or how much of it was borrowed. Lesson 46 takes up the positioning data that is actually published: what the reports contain, the lag between the date they describe and the date they appear, why a large position is not the same thing as a crowded one, and what can and cannot be concluded from knowing that somebody else is long.
Volatility as a Quantity
The quantity correlation is a ratio of, and the sizing this lesson corrects.
Read Lesson →What a Timeframe Is
The window problem in its general form, before correlation made it worse.
Read Lesson →Position Sizing
The arithmetic that sizes one position at a time and cannot see the second.
Read Lesson →Positioning Data
Who is holding the other side, and how much of it was borrowed.
Read Lesson →Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.
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