Guide · Risk measurement · Updated 2026-09-19

A correlation is an estimate over a window you chose, and it moves most in the conditions you bought it for

What the number actually is, before anything else

Write the calculation out and the problem announces itself. A Pearson correlation is a sum of paired deviations divided by the product of two standard deviations, computed over a set of observations, and that set is not given by the market. The quantity has three inputs, not two, so "these two sectors have a correlation of 0.4" is missing the third. That is why two competent people measuring the same markets on the same day produce different numbers and are both right. A price series has a history rather than a fixed distribution, which is the subject of stationarity in a price series.

The usual defence is that a long enough window settles it. It does not. It replaces a noisy answer about now with a stable answer about an average of conditions, most of which are not the ones the number is about to be used in. Length buys precision about the wrong quantity.

Measured, not asserted: eleven years of Indian sector indices

The data is the daily index close file the exchange publishes for every trading day, carrying every published index at once. Eight sector indices were used: information technology, banks, fast moving consumer goods, pharmaceuticals, metals, automobiles, energy and realty. Daily logarithmic changes in the closing value give 2,686 observations from 2015-11-10 to 2026-09-18, once the two changes that span a session missing from the archive are dropped, and eight indices give 28 pairs.

Taken over the whole period as a single figure, the 28 pairwise correlations average +0.494, from +0.341 for IT against Realty to +0.690 for Bank against Auto. Those are the numbers a risk model is usually handed. Now roll the same calculation through a 60 day window and see what they were summarising.

Read the last column first. The narrowest range of any pair is 0.672 and the widest 1.289, mean 0.904. In 17 of the 28 pairs the rolling figure went negative at some point, meaning the two sectors spent at least one quarter moving in opposite directions while their whole period figure sat comfortably positive.

The pair used for the rest of this page is the information technology index against the bank index, the one a portfolio builder is most likely to lean on: one sector earns largely abroad, the other is domestic and rate sensitive. Its whole period figure is +0.369, its 60 day windows ran from -0.231 to +0.819, and only 29 per cent of them sat within a tenth of that figure. The single number is accurate and it is not descriptive.

Half the movement is the instrument, and the rest is not

This is where most writing on the subject stops, and where it should not. A rolling estimate wobbles even when the quantity beneath it is fixed, because each window is a small sample, so showing that a rolling correlation moves proves nothing on its own. The claim has content only when set against what a constant correlation would produce through the same instrument.

So that comparison was run. Eight synthetic series were generated with a correlation fixed at 0.441, the observed average, over the same 2,686 observations, and the same 60 day calculation applied, 150 times with a fixed seed.

The control does real work. A single pair's rolling estimate has a standard deviation of 0.120 under a constant correlation, so a good part of any single pair's apparent instability is the instrument. What settles it is that the observed figure of 0.197 was reached by 0 of the 150 control runs, and the observed range of 1.050 by 0.

The average across 28 pairs is the cleaner test, because averaging cancels most of the independent estimation error. Its control variability collapses to 0.056 while the observed figure stays at 0.116, leaving genuine movement of about 0.102, roughly 77 per cent of the observed variance. Non-overlapping blocks give the same verdict. That is the honest version of the claim: correlation between Indian sectors genuinely moves, by several times the noise in measuring it, and anyone repeating the claim without this control has not established it.

One end date, six defensible windows, six answers

The point becomes concrete when the period is held fixed and only the look back varies. Every window below ends on the same final trading day. Nothing in the market differs between them.

The one month answer is -0.170 and the whole sample answer is +0.369. They do not share a sign. A risk report quoting either would be telling the truth, and a reader given only the number could not tell which truth had been chosen. Across the 28 pair average the disagreement is still 0.264 wide.

Window length also changes the apparent stability, in the direction that flatters whoever picks the long one.

A one year window makes the same data look far more settled. It has not removed the movement, it has averaged over it, which is only useful if the decision also averages over a year. The period is a choice in the same way: cut the sample into calendar years and the whole period figure dissolves.

For the two sector indices the yearly figures run from +0.046 to +0.604, a span of 0.558; for the 28 pair average, from +0.300 to +0.655. A page written in any one of those years, quoting that year's figure as the relationship between two Indian sectors, would have been accurate and would now be wrong.

The asymmetry: the number moves most in the conditions you bought it for

Instability alone would be manageable, because a quantity wandering at random is as likely to help as to hurt. The problem is that it does not wander at random.

To test it, every day was ranked by the realised standard deviation of the broad index over the twenty one trading days before it, so the conditioning variable carries no information from the day being classified. That detail is not fussiness, because the obvious design is wrong.

The first version of this test ranked days by a volatility window that included the day itself. Run against synthetic data whose true correlation never changes, that design produced an apparent stress lift averaging +0.126: roughly half of what it would have reported as a finding was an artefact of selecting days on a quantity containing the day's own move. Rebuilt to use strictly prior days, the same control reads -0.002, highest across 150 runs +0.140. Only the second design can support a claim.

The prior volatility measure ranges from 4.9 per cent at its calmest to 87.3 per cent at its most extreme, median 11.7 per cent, and the 267 most volatile days sit above 20.6 per cent.

On the most volatile tenth of days the average pairwise correlation is 0.643, against 0.448 on the other nine tenths. For the two sector indices the lift is larger, from +0.275 to +0.592. Every one of the 28 pairs moves the same way, with lifts from +0.069 for Pharma against Realty to +0.409 for IT against Energy, median +0.186. Under the control, 0 of 150 runs reached the observed average lift and 0 the pair lift.

Two qualifications. The progression across the ten bands rises overall but reverses between adjacent bands in places, with the bottom five averaging 0.378 against 0.527 for the top five. And the lift is a conditional average, not a rule about any particular day: the measurement supports the direction and rough size, not a prediction.

The direction is what matters. A diversification assumption is not used uniformly across conditions; it is used hardest when conditions are worst, because that is when the second position is supposed to be doing its work, and the measurement says that is when it does least of it.

What a point estimate does to a risk number

None of this matters until the number enters a calculation. Take two ordinary ones. The first is portfolio variance, where correlation enters through a cross term, so the effect on a two position book is real but damped. Using the measured annualised standard deviations, a half and half book prices out as follows across the measured spread.

The book figure moves from 13.3 per cent to 20.4 per cent across the measured spread, against 17.7 per cent at the point estimate: a swing of about 40 per cent of the quoted risk on identical holdings, from nothing but the choice of correlation input. Across the eight sector book the same substitution takes the standard deviation from 11.8 per cent to 20.7 per cent.

The second calculation is worse. A hedge ratio is proportional to the correlation rather than to the square root of a sum, so it inherits the movement one for one. The final column above runs from -0.230 to +0.815: not two versions of one hedge with a tolerance around them, but one short the second instrument and one long it. A hedge sized on the whole period figure and applied in a quarter resembling the calm end of the spread is not slightly wrong, it is the wrong trade.

This is the mechanism behind a familiar and unwelcome pattern. A book is sized on a correlation measured in calm conditions. Conditions deteriorate, the correlation rises, the book's true variance rises with it, and the position is now larger in risk terms than it was ever authorised to be, at the moment there is least appetite to carry it. Nothing was mismanaged: a point estimate of a moving quantity was fed into a calculation that assumed it was fixed. The same reasoning runs through position sizing from first principles.

What a sector index is in India, which is not what it sounds like

A structural point specific to this market changes how these numbers should be read: a published sectoral index here is not a broad basket. Under the sectoral methodology, constituents are weighted by free float market capitalisation subject to caps allowing one constituent up to 33 per cent of the index and the top three up to 62 per cent between them at rebalancing, with the series reconstituted twice a year on data ending January and July.

Two consequences follow. A correlation between two sector indices is substantially a correlation between a handful of large constituents, so it inherits their idiosyncratic behaviour rather than averaging it away. And composition changes on a schedule: a constituent crossing a cap at a rebalance changes what the index measures, and the correlation series registers that as a change in relationship when nothing in the economy has moved. The same concentration runs through the broad index, where financial services has recently been the largest sector weight by some distance, in the region of a third, so a book holding the broad index and adding a bank position for balance is more concentrated than its two line items suggest.

The regulator now measures the same thing at the holdings level

This is no longer only a portfolio construction concern. On 26 February 2026 the market regulator issued a circular on the categorisation and rationalisation of mutual fund schemes, superseding the corresponding clause of the earlier master circular, and put a number on overlap. Sectoral and thematic schemes must keep portfolio overlap with other schemes in that category, and with other equity categories apart from large cap, to no more than 50 per cent, and value and contra schemes from one fund house face the same limit against each other.

The construction of that limit is the part worth reading twice. Overlap is computed from daily portfolio values, averaged across a quarter, with monthly disclosure and a phased realignment period for existing schemes. Rather than measure once at a point in time, the regulator specified a frequency, an averaging period and a reporting cadence, because a co-movement statistic measured on one day is a statement about that day.

Overlap is the holdings level cousin of correlation: the same question about whether two exposures are actually distinct, asked of the portfolio rather than the price series, and a better instrument than inferring it from returns. The mechanics are in mutual fund portfolio overlap.

Correlation sees straight lines and nothing else

One limit sits underneath everything above and is not about instability at all. A correlation measures the strength of a linear association, so two series can be tightly bound together and still return a correlation near zero. The broad index demonstrates it on its own data.

The correlation between the index's change on one day and its change on the next, across 2,686 observations, is -0.035. On that evidence one day tells you nothing about the next. Now measure the same two days by the size of the move rather than its direction. The correlation between the magnitudes is +0.280, and between the squared changes +0.199.

Days in the largest fifth are followed by days averaging 1.6 times the size of those following the smallest. The two days are plainly dependent, and a correlation of -0.035 sees none of it, because the dependence lives in the magnitude rather than the direction.

So a low correlation is not evidence of independence. Two positions can show a correlation near zero in ordinary conditions and fail together, because what links them is a shared sensitivity to the size of a common shock rather than its direction. That is the most expensive thing a correlation can hide, and it is structurally incapable of showing it.

What to do instead

Four changes, each cheap, none requiring a more sophisticated model.

Record a range, never a point. The output should be the lowest window, the middle half and the highest, on a stated length. It costs the same computation, and any conversation about a risk number improves the moment the spread sits beside it.

Condition on state. Two figures, one for calm conditions and one for stressed, fall out of the measurement above anyway. Rank days by prior volatility, split, and use the stressed figure when sizing anything whose purpose is to survive stress. That a pooled figure is an average over states is the subject of regime conditional performance.

Stress the assumption rather than the position. The usual stress test moves prices and holds the correlation matrix fixed, which tests the wrong input. Re-run the book at the high end of the measured spread with prices unchanged. If it only works at the average correlation it does not work, because the average is not what is delivered in the conditions that make the test matter.

State the window and the period wherever the number appears. A correlation quoted without them is not a measurement, it is an opinion with a decimal point. Two figures of text beside every correlation would prevent most of the misuse described here and cost nothing.

None of this makes the quantity stable. It cannot be, because it was never a constant. What these changes do is make the calculation honest about what it carries, so that when the number moves, which the data above says it will and says when, the sizing was already built for it.