Guide · Risk and position sizing · Updated 2026-09-19

Two correlated positions are not two positions, and nine Indian sector positions measured 1.80 independent bets

Every number on this page was computed from the exchange's own session files at build time, with the method stated so the work can be repeated. The order matters: the mechanism first, then the measurement that gives it a size, then the adjustment, then the reason the adjustment is least dependable in the conditions that make it matter.

Risk is a quadratic form, which is why correlation is not a refinement

A book's variance is the sum, over every ordered pair of positions, of the first weight times the first standard deviation times the second weight times the second standard deviation times the correlation between them. The terms where a position is paired with itself are the standalone variances. Every other term carries a correlation. With nine positions there are nine of the first kind and 72 of the second, so most of the arithmetic that sets a book's risk is co-movement rather than individual volatility. Add a tenth position and you add one standalone term and eighteen cross terms.

The consequence has a direction. Hold two positions the same way and a positive correlation makes the pair riskier than the independence arithmetic says. Hold them opposite ways and the same positive correlation makes the pair less risky. Same two positions, same sizes, same period, and the risk differs by whichever direction you are holding them in. Take the most correlated pair in the measured set, Auto and Bank, with annualised standard deviations of 21.84 and 21.49 per cent and a measured correlation of 0.690. Held half and half the same way, the pair measured 19.92 per cent a year. Held half and half opposite ways, 8.52 per cent. That is a factor of 2.34 between two books containing the identical two positions at identical sizes. Independence would have priced both at 15.32 per cent and been wrong about each in opposite directions.

Two positions, and how many bets they actually are

The count that follows from this is the effective number of independent bets. Take the sum of the positions' standalone risks, square it, and divide by the book variance. For two equally sized positions of equal volatility it reduces to a single expression: two divided by one plus the correlation. At a correlation of zero it returns two. At a correlation of one it returns one, which is right, because the two positions are then one position held twice.

The steepness is the part worth internalising. Going from a correlation of zero to 0.49, which is the average of the 36 measured sector pairs, costs 0.66 of the two bets. Going from 0.49 to 0.69, the highest measured pair, costs only a further 0.16. Most of the diversification a pair can lose is lost at correlations people describe as moderate. The measured extremes of the set are IT against Media at 0.319, which is 1.50 bets, and Auto against Bank at 0.690, which is 1.18 bets. The most diversifying pair available among nine Indian sectors was still barely more than one and a half positions.

Nine Indian sector positions measured 1.80 bets

The same arithmetic runs on any number of positions. The universe here is the nine published NSE sector indices with a complete daily history under the current index names, so no series is spliced across the November 2015 renaming of the index family. The sample is 2,686 daily changes over 10.9 years.

Held in equal size, those nine positions measured 1.80 effective bets. The closed form for an equicorrelated book at the measured average correlation of 0.489 gives 1.83, which is the arithmetic check the build runs before writing anything. So a book an allocator would describe as spread across nine sectors of the Indian market carried slightly less independent content than two uncorrelated positions would have.

Two things in that table are worth more than the headline. The first is that the count stalls. It passes 1.68 at five positions and then moves by less than a fifth of a bet across the next four. The second is that it does not only stall, it reverses: adding Metal as the seventh position and Realty as the ninth each lowered the measured count, because each arrived more correlated with what was already held than the book's own average. A position can be added in good faith, carry its own distinct label, and remove diversification.

The hindsight column makes the ceiling explicit. The best combination available at any size peaked at 1.91 bets, with five positions, and that combination was picked after seeing the whole period. Nobody could have chosen it in advance, and it still did not reach two independent bets from nine available Indian sectors.

The obvious response, hold more positions, was tested directly. Adding the two other published financial sector indices to the nine takes the book to 11 positions. Its measured effective bets figure is 1.78, against 1.80 for the nine, and its volatility is 17.58 per cent against 16.98. Two extra positions, two extra sets of costs, two extra things to monitor, and slightly less independent content than before. The same arithmetic at the holdings level is why two funds with different names are often one fund.

Sizing by a position's own volatility answers a different question

The standard first improvement on equal rupee sizing is to size each position inversely to its own volatility, so that a position twice as volatile is held half as large. It is a real improvement and it is not a correlation adjustment. It is the correlation adjustment for one specific assumption.

Set every off-diagonal correlation to one and solve for weights that give each position the same contribution to book risk. The solution is inverse volatility, exactly. The build verifies this rather than asserting it: solving the equal risk problem on a matrix of all ones reproduces the inverse volatility weights to better than a millionth of a percentage point. So the naive rule is the correlation-adjusted rule under the assumption that everything moves together perfectly. That assumption is conservative about the book's total risk and wrong about its distribution, and the second error is the one that bites.

The direction of the error is not a matter of opinion here, it is measured. Rank the nine sectors by their average correlation with the other eight, rank them by how much the correlation adjustment changes their weight, and the two rankings line up at -0.998. Monotone, across every position. The sector with the lowest average correlation was raised by 20.5 per cent of its own weight, and the one with the highest was cut by 12.0 per cent. Sizing on own volatility alone systematically over-sizes the positions that are most redundant and under-sizes the ones actually doing the diversifying, because own volatility contains no information about redundancy. Volatility and average correlation are only loosely related in this set, correlating at 0.38, so one cannot stand in for the other.

The adjustment, computed

The adjustment is to size so that every position contributes the same amount to the book's risk. That is well defined: a position's risk contribution is its weight times the derivative of book risk with respect to that weight, and those contributions sum exactly to the book's standard deviation, not approximately. On this book they sum to 1.006427 against a book standard deviation of 1.006427 in daily per cent, which the build asserts before anything is written. The weights are solved by coordinate descent, converging here in 19 passes, and the solution is checked by confirming that the largest and smallest contributions agree to within a billionth.

Read the volatility column honestly. The correlation adjustment took the book from 16.98 per cent a year to 15.98, a reduction of 1.00 of a percentage point or 5.9 per cent. Inverse volatility on its own already captured 4.3 per cent of that, so the correlation part contributed 1.6 per cent beyond it. Expressed the other way, at a common volatility target the equally sized book has to be carried at 94.1 per cent of the correlation-adjusted book's gross exposure to run the same risk. These are real but modest numbers, and a page claiming the adjustment transforms a book's volatility would be overselling what the data shows.

What the adjustment does change decisively is the third and fourth columns. Equal rupee sizing put 15.5 per cent of the book's risk in one position and 6.9 per cent in another, a ratio of 2.25 to one, without anybody choosing that. Inverse volatility narrowed it to 1.45 to one. The correlation-adjusted book is flat by construction. A book whose risk splits 2.3 to one is making a concentrated bet it never wrote down, which is the part a sizing rule exists to prevent.

Out of sample, with the estimate made before the period it sizes

Everything above is in sample: the correlation matrix and the weights were computed from the same 2,686 sessions the result is reported on. That is the right way to show a mechanism and the wrong way to claim a benefit. So the same rules were run causally. Estimate the covariance from the trailing 250 sessions, set weights, hold them for the next 21 sessions, re-estimate. Every input to a holding period comes from sessions strictly before it.

Three readings. The adjustment survived the move out of sample: equal risk contribution on the trailing matrix measured 15.78 per cent against 16.97 for equal sizing, a reduction of 7.0 per cent, close to the in sample figure. Inverse volatility delivered 5.2 per cent on its own, so again most of the benefit was volatility scaling rather than correlation. And the worst single session barely moved: -11.94 per cent for the equally sized book against -11.66 per cent for the adjusted one, a difference of 0.28 of a percentage point on a day that cost more than eleven. The adjustment manages the ordinary distribution of outcomes. It does not manage the tail, and the tail is where books end.

The number you size on moves most when it matters most

The adjustment above rests on a correlation matrix, and that matrix is a point estimate of something that moves. The measurement of how much it moves is a separate piece of work on the same series: every sector pair travelled through a range of at least two thirds of the available scale on a rolling quarterly window. The consequence for sizing is direct. A sizing rule built on a point estimate of a moving quantity inherits every bit of that movement, and inherits it silently, because the weights look just as precise either way.

The rolling count makes the inheritance visible. Recomputed on a 250 session window, the same nine equally sized positions held between 1.38 and 2.50 effective bets across this record, and sat below two bets in 60 per cent of the windows. The low reads 1.38 at 2020-04-08. Nothing in the book changed on that date. The diversification did.

Effective bets fell from 2.23 in the calmer half of conditions to 1.46 in the most volatile tenth, a loss of 35 per cent of the book's independent content, while its volatility rose from 12.6 per cent to 28.7. Both movements run the wrong way at the same time. The book grows larger in risk terms exactly as it becomes less diversified, and neither shows up in a position report.

The failure this produces is not that the total risk estimate breaks. It is that the risk split breaks. Fit the equal risk weights on the calmer half of sessions and carry that book into the more volatile half, and its volatility comes out at 19.34 per cent against 19.38 for weights fitted on that state, which is almost no difference. But its largest risk share comes out at 15.55 per cent instead of the 11.11 per cent it was built for. The book that was carefully equalised is no longer equalised, and the position carrying 1.4 times its intended share of risk is not flagged anywhere.

Size on a correlation you chose, not the one you measured

The practical response is to stop feeding the measured matrix in unmodified. Raise every measured correlation a fixed fraction of the way toward one, then solve. The result is a valid correlation matrix for any fraction between zero and one, because it is a blend of two valid ones, and the two ends of the range are the two rules already on this page: at zero it is the measured matrix, at one it is inverse volatility.

The cost is small and the bargain is favourable. Raising every correlation 30 per cent of the way to one moved the book's measured volatility by 0.107 of a percentage point and widened the largest risk share from 11.11 to 11.71 per cent. Out of sample it cost 0.13 of a percentage point of realised volatility and bought a fall in average weight variation from 1.63 to 1.46 percentage points and in turnover from 2.64 to 2.30 percentage points a rebalance. The conservative book is closer to the one you would have built in stressed conditions, changes hands less, and gives up a fraction of a point of the thing being optimised.

State it plainly. A sizing rule should lean on the correlation matrix only as hard as the matrix can bear, and the measurement of how much it moves is the evidence on how hard that is. The most aggressive version of the adjustment, the one that squeezes the last basis point out of a measured matrix, is the version that most depends on the measured matrix being right about the next quarter.

Correlation-diversified is not exposure-diversified

One limit sits underneath all of this and no amount of adjustment reaches it. A correlation measures the strength of a straight line relationship in ordinary conditions. A book can be sized so that every correlation has been accounted for, every risk contribution equalised, and still sit almost entirely on one exposure.

Measure it. Decompose the measured covariance into its principal directions. The largest single direction accounts for 55 per cent of the correlation matrix on its own, and the effective dimension of the matrix, measured by the participation ratio of its eigenvalues, is 2.99 out of nine. The equally sized book placed 99.0 per cent of its variance in that one direction. The correlation adjustment moved it to 96.4 per cent. Even the hindsight-optimal five sector combination still sat at 84 per cent. The equally sized nine sector book correlates 0.923 with the broad index and the adjusted one 0.932. Nine sector positions, carefully weighted, are a slightly expensive way of holding the market.

All nine sectors closed lower on the same session 205 times in 2,686, which is 7.6 per cent of sessions. Multiply out the nine individual frequencies of a lower close and independence predicts about 3 such sessions, a factor of roughly 64. On the ten worst sessions for the equally sized book, an average of 8.8 of the nine sectors closed lower. There is no sizing rule that turns that into diversification, because the exposure being expressed on those days is not nine sector views, it is one view about Indian equity risk.

Two further gaps sit outside what a correlation can see. It is a linear measure, so two positions can show a low correlation in ordinary conditions and still fail together, when what links them is a shared sensitivity to the size of a shock rather than its direction. And it is a statistical measure computed on price history, so it knows nothing about a shared settlement mechanism, a shared source of funding, a shared regulatory exposure or a shared counterparty. A book can be beautifully correlation-diversified across nine sectors and be one policy change, one margin rule or one funding event away from behaving as a single position. The matrix will not tell you. Only reading what the positions actually are will.

What the number is for

Not for optimisation. The measured gains from optimising against the correlation matrix are modest, they shrink further once costs and turnover are counted, and the part of the matrix carrying them is the least stable part. Treating the effective bets figure as an objective to maximise would be fitting a number to the sample it was measured on, and the hindsight column earlier shows how flattering that exercise looks.

It is for calibration. A book described as nine positions, measured at 1.80 bets, is carrying a concentration its position count actively conceals, and every risk figure built on the position count is wrong by a knowable factor. Knowing that number changes what a position limit means, what a drawdown of a given size should be read as, and whether adding a tenth position is diversification or just another way of expressing the same view. That is a judgement question, which is why it sits alongside how sector exposures actually behave in this market rather than inside a formula.

And it is for scepticism about the word itself. A book is not diversified because it holds many names. It is diversified to the extent that its positions carry independent information, and that extent is measurable, small, and smallest exactly when it is being counted on. Size for the number you measured, check it against the number you would have measured in the worst quarter of the record, and size on the more conservative of the two.