Guide · Research method · Updated 2026-09-19

A pooled result is an average over market states, weighted by how often each one happened to occur

Every figure here was computed from the exchange's own session files rather than quoted, with the definitions stated so the work can be redone. That matters more than usual, because a conditional study is the easiest analysis to fake by accident, and the fake version is the more persuasive one.

The pooled figure is an identity, and the identity names its own weakness

Take any performance statistic that is an average over sessions and split the sessions into states, whatever the states are. The overall average is then the sum, across states, of each state's average multiplied by the fraction of sessions that fell into it. Not approximately. Exactly. On the record used here the two sides agree to the last digit the arithmetic carries, and the build asserts it before the page is written.

So a pooled number carries two things at once. One is conditional behaviour, a claim about the system. The other is a set of frequencies, a claim about the past decade of market conditions and nothing else. A single quoted figure is a product of the two, presented as if it were only the first.

Two consequences follow. A system whose conditional behaviour is stable can show a pooled figure that moves a great deal, because the mix of conditions moved. A system whose conditional behaviour is wildly unstable can show a pooled figure that looks steady, because the instability averaged out at the frequencies that happened to occur. Neither is visible from the pooled number.

Defining the regime without looking forward

Everything below rests on one definition, stated before any result. For each session, take the standard deviation of the previous 21 daily log returns of the broad index and annualise it. Those 21 returns all come from sessions strictly before the one being labelled. Sort the session into a third: calm, middle or stressed. The thresholds separating the thirds are the one third and two thirds points of every earlier volatility reading, re-estimated each session, after a burn in of 750 readings. That burn in is why labelling starts at 2016-03-01 although the data starts at 2013-01-01: a threshold is assigned only once a person could actually have held it.

Two things in that picture do real work. The first is clustering. The volatility reading does not wander randomly; it sits low for long stretches and then jumps. Measured on this record, a session carries the same regime label as the session before it 92.3 per cent of the time, and the 2,613 labelled sessions contain only 201 distinct regime episodes, with a median episode of 5 sessions and one episode running 271 sessions. That persistence is what makes a regime label usable at all. It is also why the record holds far less information than its session count suggests.

The second is that the thresholds drift. They begin at 12.75 and 15.60 and end at 10.47 and 14.40, because volatility in this market fell across the period and the expanding estimate followed it down. Had the thresholds been fixed at the full period values of 9.91 and 13.81, the labels would differ on 467 of the 2,613 sessions, which is 17.9 per cent of them. A fixed threshold looks more rigorous and is less honest: those values were unknowable when the early labels had to be assigned.

A causal definition also stops producing thirds. Out of sample the calm bucket took 45.6 per cent of sessions, the middle 26.6 per cent and the stressed 27.8 per cent. The imbalance is a finding rather than a defect: the market spent much of the period below what its own earlier history called quiet.

The label you could not have had at the time

This is the failure generic treatments leave out, and it is worth measuring rather than describing. Keep every other element identical and change one thing: let the volatility window start at the session being labelled and run forward instead of backward. That is what an eye does when it reads a finished chart and marks a stretch as volatile. It is also unavailable in advance.

The honest labelling separates the trend rule's conditional means by 7.53 basis points from best to worst. The peeked labelling separates them by 16.74, a factor of 2.2. For the index itself the spread goes from 10.20 to 17.77. No session's result changed. Only the label moved, and it had been allowed to see the answer.

The circularity is the point. If a label is fitted so that the conditional table looks clean, the table reports the fit rather than the market, and nothing inside the study can separate the two. The discipline is unglamorous: state the definition first, on grounds defensible before any conditional result was seen, then leave it alone when a small adjustment would sharpen the picture. The same applies to anyone eyeballing a chart and marking which stretches were trending, a judgement always made with the whole chart visible.

The conditional table for one rule

The rule is deliberately plain, so nothing in the result is an artefact of cleverness. Be present in the broad index for the next session when the previous close sat above the mean of the previous 200 closes, and be absent otherwise. Every input is a prior close. Results are the index's own log return in basis points per session, gross of every cost, tax and execution effect: a measurement of an index and a rule, not of a tradable outcome.

Read the last column first. The calm regime produced 87.9 per cent of everything this rule did across the decade, on 45.6 per cent of the sessions, and the middle regime took 31.7 per cent back off it. A reader shown only the pooled +2.41 cannot tell that this is a quiet market rule, paying its way in low volatility and giving some of it back in the middle band.

The presence column is where most conditional tables mislead. The rule stood in the market on 93.7 per cent of calm sessions and only 55.4 per cent of stressed ones, because the stressed regime is where the index sits below its own long mean. The stressed row is therefore not mainly a statement about how the rule behaves under stress. It is a statement about a rule that is mostly absent during it.

Two rules, paid in different states

Both rules show a positive pooled figure, and those figures say almost nothing about how differently the two were produced. The reversion rule, present only after a run of lower sessions, drew 93.9 per cent of its total from the calm regime and handed 56.3 per cent of it back in the middle one. The index, always present, drew 53.7 per cent of its total from the stressed third on 27.8 per cent of sessions.

That last figure unsettles people, so be precise about it. The stressed regime is not where the index did badly in this record. It is where the index did most of its violent work in both directions, and the net of that work was positive over this particular decade. A rule that sits out the stressed regime is not avoiding a loss zone. It is trading a smoother record for absence from the sessions that moved most.

Where the record was actually made

The largest 10 sessions account for 41.2 per cent of the index's entire cumulative record over the labelled period, and every one of them fell inside the stressed third. So did every one of the ten largest falls. The familiar advice about missing the best days is usually imported from another market and quoted with no regime attached. Attach one and it stops being a slogan: the best sessions and the worst sessions are the same regime, so a rule that cuts exposure to one cuts exposure to both. It is also why the stressed row of any conditional table carries the widest interval on this page. It averages over the bucket holding the fattest tail in the record.

The weights are history, not law

Now hold the conditional behaviour fixed. Take the three measured means exactly as they are and change only the weights, using the regime mix each calendar year actually delivered.

The reweighted figure for the trend rule runs from +1.61 under the 2024 mix to +4.63 under the 2017 mix, a year in which all but one labelled session fell in the calm third, against an actual pooled +2.41. That is a factor of about 2.9 between the ends, produced by the composition of the decade and not at all by the rule. Anybody reporting a single figure is forecasting that the next period's mix will resemble the sample's, and almost nobody says so.

A calendar regime that a rule change redefined

Market conditions are not the only thing that sets regime frequencies. On 26 May 2025 the market regulator issued a circular on the final settlement day for equity derivatives contracts, requiring each exchange to run expiries on either Tuesday or Thursday, to notify its choice and transition plan by 15 June 2025, and to keep one weekly benchmark index options contract on its chosen day, with other contracts moving to at least a one month tenor expiring in the last week of the month. The larger exchange moved contracts expiring on or after 1 September 2025 from Thursday to Tuesday, and the other took Thursday. Every page written before that change which calls Thursday the expiry day, and every conditional study treating a day of week effect as a fixed feature, describes a calendar that no longer exists.

Read that table carefully rather than triumphantly. Thursday sat at 0.97 of the period's own average absolute move before the change and 0.80 since, while Tuesday is unchanged at 0.92. The direction is what the rule change would suggest. It is also not the largest change in the table, which belongs to Wednesday and which no rule change predicts, and the sample is 50 sessions per weekday with 95 per cent half widths wide enough to swallow every one of these movements. The honest reading: the calendar regime was redefined by rule, and there is not yet enough data on the new one to measure it. That is exactly the situation in which confident conditional claims get published.

Splitting the sample is not free

Dividing a record into regimes buys specificity and pays in precision, at a rate fixed by arithmetic. The standard error of a mean falls with the square root of the number of observations, so a bucket holding a fraction of the sessions carries a standard error inflated by the square root of the reciprocal of that fraction. An exact third costs a factor of 1.73.

The largest conditional gap for the trend rule is 7.53 basis points between the calm and middle regimes, carrying a test statistic of 2.34. That clears the conventional threshold and should still not be treated as settled. A test with a four in five chance of detecting a gap of that size needs about 1,226 sessions in each regime, which at the rarer regime's frequency is roughly 19 years of sessions against the 10.6 years the record holds. When an underpowered test does cross the line, the estimate that crossed it is on average larger than the thing it estimates, because only the larger estimates get across. For the index the same calculation asks for 2,279 sessions per regime, about 35 years.

The harder version is the one the persistence figure set up earlier. That arithmetic treats sessions as independent observations, and they are not. With a label repeating 92.3 per cent of the time, the 2,613 labelled sessions hold 201 distinct episodes, and the independent content of a conditional estimate sits far closer to the episode count. Split three ways, that is a few dozen episodes each. A conditional claim has to survive that number, not the comfortable one.

Sectors do not share one regime sensitivity

The same labels, computed on the broad index, applied to the published sector indices answer a different question: how much does each sector's own daily movement change between calm and stressed?

The banking index expands by a factor of 2.13, the widest in the set, while the realty index expands by 1.34. The realty figure is not a sign of calm: that index moves 1.181 per cent on an average calm session against the banking index's 0.631 per cent, so it is already loud when the market is quiet and has less room to get louder. A ratio measures sensitivity, not level, and reporting only one of the two invites the wrong reading.

The practical content is that a position sized on pooled volatility is sized for a blend of states. When the regime turns, the sectors do not expand by the same amount, so the risk profile of a mixed book shifts without a single position being touched.

What the conditional view is for

Nothing above establishes that the conditional differences are real. This record cannot settle them, and a page claiming otherwise from ten years of sessions would commit the error it warns about. What the conditional view changes is what a performance figure means to the person holding it.

It names the state you are being paid in. A rule that drew 88 per cent of its historical total from quiet conditions has a concentrated and identifiable exposure, and the thing to monitor is the frequency of quiet stretches rather than the headline figure. A pooled number cannot deliver that, because the frequency it depends on has been folded into it and cannot be recovered.

It also tells you how to read a bad run. If the middle regime is where a rule historically gives some back, a poor stretch there is ordinary and a poor stretch inside the calm regime is the one deserving attention. On an equity curve the two look identical.

And it tells you what your sample actually is. A few thousand sessions holding a couple of hundred regime episodes, split three ways, is a small study, and knowing that sets the weight every conditional number can bear. The arithmetic of how much data would settle the question and the mechanics of classifying a market state are the two neighbouring pieces, and the standard backtesting errors are where a fitted regime label usually enters a study without being noticed.