Guide · Risk method · Updated 2026-09-19

A volatility number is an average over conditions; a scenario is a question that has an answer

Everything below was computed from the exchange's own session files rather than recalled, and no episode is given a cause. A price file cannot establish why a market fell, so each window here is described by its dates and its measured magnitude. That is a restriction and it is also an improvement: a scenario named after an event invites the reader to decide whether that event will recur, which is the wrong question. A scenario named by its shape invites the reader to ask whether a move of that shape would be survivable, which is the right one.

What a scenario asks that a statistic cannot

A volatility number is a single moment of a distribution estimated over a window. It is the average size of a session's move across whatever conditions that window contained, and its usefulness depends entirely on the conditions repeating in the same proportions. Measured on the illustrative book defined below, annualised volatility across the whole 3,385 session record is 16.1 per cent. Inside the ten windows this page selects it is 36.3 per cent and outside them 14.6 per cent. The headline figure is a weighted blend of a market that behaves one way for about nineteen sessions in twenty and another way for the twentieth.

The concentration is sharper than the volatility figures alone suggest. The 124 sessions inside those ten windows are 3.7 per cent of the record and carry 20.1 per cent of the book's total squared daily variation across thirteen years. A statistic that averages over that is not lying. It is answering the question of what a typical session looks like, and nobody who is worried about a portfolio is asking about a typical session.

A scenario asks something narrower and therefore answerable. Given this specific joint move across these specific indices, what is this specific book worth at the end of it? There is no estimation in that question once the move and the book are stated. The arithmetic is exact. The judgement moves entirely to the choice of move, which is where judgement belongs and where it can be examined.

Choosing the windows by rule, not by recall

Scenario selection is the step that decides everything downstream, and it is usually done from memory. Memory is a poor screen. It selects for how much an episode was discussed, how recently it happened and how neatly it fits a story, and none of those quantities is the amount of money it would have cost a book. The correction is to state a screen before looking, run it over the record and accept what it returns.

Two screens run here over the broad index, both stated in full before any result was computed.

The depth screen ranks every 20 session window in the record by the index's move from the close before it to the close at its end, takes the worst, then takes the next worst that does not overlap it, and continues until five windows are held. The volatility screen computes, for each session, the standard deviation of the last 10 daily log returns divided by the standard deviation of the 60 returns before those, ranks sessions by that ratio, and takes the five highest that sit at least 60 sessions apart. One screen looks for the largest fall. The other looks for the sharpest change in how far the index moves each session, which is a different quantity and is not required to be a fall at all.

One eligibility rule applies to both: a window is admitted only if the record holds 250 sessions before it, because the repricing below uses those sessions to build the estimate the scenario is compared against. That rule removes one 20 session decline of 12.09 per cent ending 2013-08-21, which would otherwise have ranked third on depth. It is stated here rather than quietly dropped, because an eligibility rule that removes the inconvenient cases is the oldest way to make a stress test look kind.

The two screens agree on three of five windows and disagree on the rest. Two appear only on depth. Two appear only on volatility, and both of those are windows over which the broad index rose, by +3.94 and +4.15 per cent. No human scenario library contains them. They are not remembered as stress because the index ended higher, and one of them turns out to hold the largest hedged loss in the entire set, which the section on offsets returns to.

That is the mechanical case against selecting from memory, and it is not a matter of taste. A screen that looks only for falls is structurally blind to an episode in which everything moves violently and the index nets out flat. A book is not held in the index. It is held in its own positions, and those can be destroyed by a move that leaves the benchmark unchanged.

The book, stated so the repricing can be checked

Every stress test needs a book, and an honest one states it completely. The book below is illustrative. It holds 100 units of capital across the eight published sector indices, with the weights chosen to look like an ordinary diversified Indian equity allocation rather than to produce any particular result. It is held unchanged through each window, with no trading, no rebalancing and no costs of any kind, which makes the repricing a clean measurement of the market move rather than a claim about what anybody could have achieved.

Two sleeves are named. The defensive sleeve is the pharmaceuticals and consumer staples indices, 20 units in total, and that designation is not a story: those two carry the two lowest measured betas in the set, 0.61 and 0.67, and the build refuses to run if that stops being true of the data. The main sleeve is the other six. On top of the book sits one more position, an index hedge: a short of the broad index sized at the book's beta measured over the 250 sessions before each window, held at that fixed notional through the window. That is the hedge a risk report would actually have specified on the morning the window opened.

The repricing method is three lines, and a reader with the same files can redo it. First, for each index, compute the simple return from the close before the window to the close at its end. Second, multiply each by the units held in it and sum, which gives the book's value change, because a buy and hold book's value is linear in its weights. Third, add the hedge, which contributes the negative of the pre window beta multiplied by the broad index's move. Nothing in that sequence is estimated. Every input is a published close.

The first result is the one that answers the question in the title. In all five depth windows the measured loss on the book exceeded the ninety nine per cent estimate carried into it, by between 1.01 and 3.92 times. That estimate is the standard construction: the standard deviation of the book's daily moves over the prior 250 sessions, multiplied by 2.3263 for the ninety nine per cent point of a normal distribution, then scaled to the window by the square root of its length. It is not a straw man. It is what a great many risk reports carry, and it was beaten every time.

What the sectors did together

A diversified book is a bet on a correlation. Not on a number in a report, but on the proposition that the things it holds will not all move the same way at the same time. That proposition is testable, and a window of stress is exactly where to test it.

All eight sector indices fell in four of the five depth windows, and seven of the eight in the fifth, opening 2015-08-05, where only pharmaceuticals rose, by 2.1 per cent. In 1 of the 3 declining windows the volatility screen returned, all eight fell as well. The spread between best and worst sector inside the deepest window was still wide, from -45.3 to -23.1 per cent, so the sectors did not become identical. They simply all pointed the same way, which is the only part of the diversification claim that matters when a book is being marked.

The correlation column shows the mechanism. Mean pairwise correlation across the eight sector indices rose from 0.44 in the 250 sessions before the deepest window to 0.88 inside it. There is an honest exception in the table and it is worth naming: in the window opening 2018-09-06, correlation did not rise at all. A page that claimed correlation always rises in stress would be overstating what this record supports, and the exception is why the measurement is run per window rather than assumed. The general instability of these estimates is the subject of the separate piece on correlation instability, and a stress test is the applied form of that argument.

The positions that were supposed to offset

Every book contains something held because it is expected to do well, or at least less badly, when the rest does badly. Those positions are the reason a book is considered safe, so they are the first thing a scenario should interrogate, at the size they were actually held.

The defensive sleeve cushioned and never offset. In all 8 declining windows it lost money, without exception. In the deepest it fell 23.7 per cent while the main sleeve fell 39.6 per cent. A sleeve that loses 23.7 per cent is not a hedge, it is a smaller loss, and a plan that counted it as the source of liquidity for a drawdown had already mis-stated its own position before the window opened.

The diversification ratio makes the same point as a single number. It is the weighted average volatility of the eight sector indices divided by the volatility of the book that holds them, so it measures how much the mixing actually removed. Before the deepest window it stood at 1.37, meaning the mixture removed about 27 per cent of the weighted average. Inside the window it fell to 1.06, or about 6 per cent. The diversification did not disappear, and most of it did.

The index hedge is the honest surprise in the table, and it goes the other way. Sized at the pre window beta it worked in the falling windows, leaving residuals of a per cent or so either side, because this book is broad enough that its beta is close to one and reasonably stable. That is a measured null result and it stays in. The interesting failure is elsewhere. In the window from 2019-09-13 to 2019-09-27, the broad index rose +3.94 per cent, the book rose +2.13 per cent, and the hedge therefore lost. The residual was -1.65 per cent, the worst of all ten windows, and only 3 of the eight sector indices fell. That is the window no memory based library would ever contain, and it is the one in which the offsets cost the most.

The general lesson is not that hedges fail. It is that a hedge is a position with its own behaviour, and a scenario is the only instrument that reveals what that behaviour is in the states that matter. A volatility number for the hedged book reports an average over states in which the hedge helped and states in which it did not, which is the same defect one level up.

Where the loss arrived, and how fast

The second thing a scenario shows and a statistic cannot is the shape of the loss in time. A twenty session decline of eleven per cent can be eleven sessions of one per cent, which almost any plan survives, or it can be three sessions and then nothing, which most plans do not.

Three sessions out of twenty carried between 49.5 and 109.7 per cent of each window's entire move. Where the share exceeds one hundred per cent, the other seventeen sessions were net positive and the whole of the damage sat in three of them. A monthly risk review sees none of this. It sees a month.

The breach column is where the one session statistic fails most plainly. Across the whole record, single session falls crossed a rolling ninety nine per cent threshold on 57 of 3,125 sessions, a rate of 1.82 per cent against the one per cent the construction implies, and 46 per cent of those crossings came within five sessions of another crossing. Independence is the assumption that fails, and it fails in the direction that hurts. 19 of the 57 crossings fell inside the ten windows, which cover 3.7 per cent of the record.

One window in the table has zero breaches and still cost the book 11.44 per cent. It opened 2022-04-11 and closed 2022-05-12, and not one of its sessions was extreme enough to trouble a daily threshold. That is the case a daily risk number is structurally unable to see: a book can be taken apart by twenty ordinary sessions pointing the same way, and every one of them will pass the test.

The exit is priced worst when it is wanted

The third thing the exercise reveals is that the assumption sitting underneath every plan to reduce exposure, that a position can be sold at something close to the price on the screen, is weakest exactly when the plan is invoked.

This data contains no bid, no offer and no depth, so nothing here measures a spread or market impact, and any claim that it does would be false. What it does contain is every session's open, high, low and close, and those support two honest observations. The first is the range multiple: inside these windows the broad index's average daily high to low range ran between 1.22 and 4.03 times its own average over the prior 250 sessions. The width of the band within which an exit price would have fallen expanded by that much, which means the penalty for choosing the wrong moment inside the session grew by the same factor.

The second is where the close sat within that band. Across ordinary sessions the close averaged 0.52 of the way from the session's low to its high, which is close to the middle. Inside the depth windows it averaged between 0.38 and 0.46. A close nearer the low means that, on average, a seller who waited was worse off than one who did not, and that the intraday path was against anyone reducing exposure during the session.

The third observation concerns the part of the move no execution can touch. Decomposing each window into the part that arrived between the previous close and the open, and the part that arrived during the session, the overnight component carried 71 per cent and 67 per cent of the whole decline in two of the five depth windows. In two others it carried none of it and was positive. That variation is itself the finding: gap risk is not a constant property of stress, it is a property of a particular episode, and a book has to be tested against both kinds rather than against an average of them. The liquidity dimension of the same problem, in the form of a fund facing redemptions it cannot meet at quoted prices, is worked through in the piece on liquidity stress testing a fund.

What the exercise cannot tell you

Three limits, each of which invalidates a common use of this work.

The next event is not in the sample. These ten windows are drawn from 14 years of one market. The deepest of them took the book down 36.4 per cent, and nothing in the record establishes that as a bound. A scenario library is a set of shapes that have occurred, which is a different object from a set of shapes that can occur, and a book sized to survive the worst thing in a thirteen year file has been sized against an arbitrary cut of history. The worst historical drawdown is itself a sample statistic, and it grows with the length of the sample for reasons that have nothing to do with risk changing.

The selection rule is a choice, and it can flatter. Two screens here produced ten windows and disagreed about half of them. A third screen would produce a third answer. Any of them can be adjusted, consciously or otherwise, until the book looks acceptable, and nothing inside the exercise detects that. The defence is to fix the screens before looking at any repricing, state them in full as they are stated above, and publish the windows a rule excluded along with the ones it kept.

A book that survives everything has been fitted to history. This is provable here rather than merely warned about, because the repricing is linear in the weights. Consider the weighting that maximises the worst outcome across the five depth windows. For any weighting, the worst of the five is at most that weighting's result in the deepest window, which is at most the best single sector return inside it. A book holding one hundred per cent of that single sector attains the bound, so no mixture can beat it. The optimiser's answer is therefore not a portfolio at all: it is one hundred per cent of pharmaceuticals.

The fitted book improves the worst case from 36.44 to 23.10 per cent, which is the entire point of fitting and is also the entire problem with it. Turn to the two windows it never saw, both from the volatility screen. In the one opening 2019-09-13, the fitted book lost 4.20 per cent where the stated book gained +2.13 per cent. Optimising against a scenario set converts that set into a target, and the resulting book is a statement about 3,385 sessions of the past rather than a position anyone would hold. The same failure in its more familiar form, where a strategy rather than a book is fitted, sits in the piece on conditional results.

What the exercise is actually for

Not for producing a number to put in a report. The number it produces is precise and conditional on a scenario nobody promised, and treating it as a loss forecast repeats the error the whole exercise exists to correct.

It is for finding which assumptions in a book are load bearing. The repricing above did not mainly reveal that the book can lose 36.4 per cent. It revealed that the defensive sleeve was a cushion rather than an offset, that eight sector holdings behaved like fewer than eight when it mattered, that the diversification ratio fell from 1.37 to 1.06, that a beta hedge sized on ordinary conditions held up in falls and cost the most in a window where the index rose, that in one window three sessions carried the entire loss, and that in another no single session was extreme enough to register on a daily measure. Each of those is a specific, checkable statement about a specific book, and none of them is recoverable from a volatility figure.

It is also for setting the size of a position honestly. A holding sized so that a one session move at the ninety ninth percentile is tolerable has been sized against a threshold this record crossed at 1.82 per cent rather than one, in clusters. A holding sized so that the worst measured window is tolerable has at least been sized against something that happened. Neither is a guarantee, and the second is a better question.

The work that makes this useful is unglamorous and it is mostly bookkeeping: state the book completely, fix the screens before looking, reprice without adjusting anything, and write down the windows the rules excluded. That discipline, rather than any single number the exercise produces, is the part worth learning, and it is the part generic treatments leave out. Confidence intervals on the same kind of measurement are handled in the piece on resampling an equity curve.