Guide · Research method · Updated 2026-09-19

An equity curve is one draw from a family of curves, and the width of that family is the number nobody reports

Every figure below was computed by the script that built this page, from real index closes held locally, and from resampling runs that execute each time it is rebuilt. The scheme, the replication count and the random seed are stated so the numbers can be reproduced rather than taken on trust.

One path is a sample of size one

An equity curve looks like a measurement and behaves like an anecdote. What it records is the cumulative product of a particular sequence of daily results, in the particular order in which they arrived. Change nothing about the strategy, the costs or the market, and simply run history again, and a different sequence would have produced a visibly different curve.

That is not a philosophical point. It is arithmetic about compounding. The terminal value of a path depends only on the multiset of returns, but everything else people read off a curve depends on the order: the depth of the worst drawdown, how long it lasted, how smooth the ascent looked, whether the bad stretch arrived early enough to make a person stop. Those are the properties that decide whether a record is acted on, and they are the properties a single ordering cannot pin down.

The bootstrap is the machinery for putting a number on that. It makes one substitution: where the true distribution of daily results is unknown, use the observed sample as if it were that distribution, and generate as many fresh histories from it as you like. Everything the method can and cannot do follows from that substitution, including the failures set out later on this page.

The resample is the whole method, and it is four steps

Take the 2,588 daily results the rule produced. Draw 2,588 of them with replacement, so some days appear twice and others not at all. Compound the draw into a path and record whatever statistics matter. Repeat 5,000 times, sort each statistic, and read the percentiles. That is the independent bootstrap in full, and the whole apparatus fits in four lines of code.

The demonstration series here is a long or flat rule on a sector index: hold when the 20 day average of the closing level sits above the 100 day average, hold nothing otherwise. The signal is read on the close of day t and applied to the return of day t plus one, so no value in the series uses a price it could not have seen. The rule switched 31 times across the record and was in the market 71.4 per cent of the days. It is a demonstration object chosen for being simple enough to state in a sentence. It is not a recommendation, and results are before costs, which the switch count lets a reader apply for themselves.

Three readings from that table are worth separating. The first is the terminal spread: the same distribution of daily results, reordered, could have ended anywhere between 0.90 and 5.85 times capital. Anyone presenting the single realised figure as the property of the rule is presenting one draw.

The second is the ratio. The observed annualised mean to standard deviation ratio is 0.611, which on a record of this length reads as a solid result. Its interval runs from -0.002 to 1.232. The lower bound is fractionally below zero, so after more than ten years of daily data the record does not separate from no edge at the conventional level. This is the ordinary situation rather than a special one, and it is the same arithmetic that governs how many trades are needed to tell an edge from luck.

The third is the statistic that fails usefully. The interval on the longest stretch below a prior peak runs from 287 to 2,120 days against a sample of 2,588 days. That interval covers most of the range the quantity could occupy, which is the bootstrap telling you plainly that the sample supports no claim about it. A point estimate for that statistic should not be quoted at all, and only running the interval reveals that.

What the independent resample destroys

Drawing days independently assumes each day is a fresh draw unrelated to the day before it. For the direction of daily index moves that assumption is close to true: the correlation between consecutive daily returns of this index is 0.019, which is negligible. For the size of those moves it is badly false.

The correlation between the size of one day's move and the size of the move ten days later is still 0.23. It barely decays across the whole range, which is the standard signature of volatility clustering: quiet periods and violent periods each persist. Measured another way, this series contains 37 days beyond three standard deviations, and the median gap between consecutive ones is 3.5 trading days. Extreme days do not arrive spread evenly across a decade. They arrive next to each other.

An independent resample scatters them. The consequence is not academic. The worst 21 day stretch this index actually delivered inside the sample is -45.1 per cent. Ask the independent bootstrap how bad a 21 day stretch could get inside a one year path, and its 2.5 per cent point is -22.5 per cent. The method assigns almost no probability to something that happened in the very data it was built from, and it does so because the only way to reach a fall of that size is to put the bad days next to one another, which is exactly the structure the resample removed.

The block bootstrap, and why the block length is a judgement

The repair is to resample runs rather than days. A moving block bootstrap picks a block length L, forms every block of L consecutive observations in the record, draws blocks with replacement, and joins them end to end until the path is the required length. Dependence survives intact inside each block and is broken only at the joins, so a path of n days built from blocks of L has roughly n divided by L places where the structure is artificially cut.

A variant known as the stationary bootstrap draws block lengths from a geometric distribution with a chosen mean instead of fixing them, which removes the dependence of the result on where the blocks happen to start and makes the resampled series stationary. It trades one judgement for another: the mean block length still has to be chosen.

The median hardly moves across the whole sweep, because the middle of the distribution is governed by the marginal distribution of returns, which every scheme preserves. The tail moves from -22.5 per cent to -45.1 per cent, a factor of 2.0. The block length is therefore close to irrelevant for a central estimate and decisive for a tail estimate, which matters because the tail is what risk work is about.

There is no length that is correct by derivation. A widely used starting point scales the block with the cube root of the sample size, which for a series of this length lands in the low teens, and that result was derived for estimating a variance rather than for a path statistic like a drawdown. The honest procedure is the sweep above: run a range, publish the range, and state which length was used for the headline figure and why. A length long enough to reproduce the worst real stretch is also long enough that the blocks are no longer independent of each other, and at the extreme the resample stops being a resample and simply reproduces the original ordering.

Whether the scheme matters is a property of your series, not a rule you can look up

The usual advice stops at the previous section, and it is incomplete in a way that is easy to check. On the illustrative rule used here, the choice of scheme barely moves anything at all.

On the raw sector index the drawdown interval widens sharply when blocks are used. On the rule built from that same index it hardly changes. The reason is visible in the tails. The index's worst single day is -16.73 per cent; the rule's worst day is -7.95 per cent. Across the 45 trading days from 20 February 2020 to 30 April 2020, the rule was in the market 2.2 per cent of the time, having gone flat on 2020-02-24, and it did not lose money on a single day inside that window, against the index's worst day of -16.73 per cent. The clustered extremes that make block structure matter were largely not in the rule's return series to begin with.

The general statement is the useful one. Whether dependence changes your answer depends on how much of the dependence survives into the series you are actually resampling, and a rule that spends time out of the market has already removed some of it. You cannot know which case you are in without running both and comparing. When the two agree, say so, because that is evidence. When they disagree, the dependence is doing the work, and the independent figure is the wrong one to publish.

What widens the interval and what narrows it

Four things set the width. The number of observations narrows it, and it does so with the square root, so quadrupling the sample halves the interval. Per observation variability widens it directly. Tail weight widens it, because a sample containing rare large moves produces replicates that sometimes contain several of them. Dependence widens it for any statistic that depends on ordering, which is the mechanism of the previous two sections.

The square root relation is visible rather than assumed. The one year width is 3.784 and the full sample width is 1.233, a factor of 3.07, against the 3.20 that the square root of the length ratio predicts. The observed ratio also wanders from -0.727 to 0.611 across those windows, which is a reminder that the point estimate on a short window is not a stable thing that the interval is merely decorating.

Now the part that gets stated backwards everywhere. A narrow interval on a short sample is not reassurance, because the width is computed from the variability observed inside that sample. A quiet year produces low observed variability, and low observed variability produces a narrow interval, regardless of how representative the year was.

Annualised variability across these adjacent years runs from 11.7 per cent to 25.3 per cent, a factor of 2.17, and the correlation between a window's own variability and the variability of the year that followed it is 0.03 across these 9 windows. The width a window produces therefore says almost nothing about the conditions the next one will bring. The window ending 2020-05 produced the second narrowest interval of the 9, at 3.322, and was followed by a year 2.05 times as variable as itself.

Treating a narrow interval as confirmation therefore inverts the meaning of the number. What it says is that this particular sample was calm, and calm samples are precisely the ones whose estimate of variability is most likely to be too low.

A bootstrap cannot see the overfitting that produced the returns

This is the failure mode generic explanations omit, and it is not a subtlety. The bootstrap resamples the returns a strategy produced. If the strategy was selected because it produced good returns on this data, the selection is baked into those returns, and every replicate inherits it faithfully. The method is being asked whether the winner's record could have arisen by chance from the winner's own returns, when the question that matters is whether a search of that size produces a winner like this when there is nothing to find.

Run it on the same index. Take every pair of a fast and a slow moving average across a grid, giving 986 candidate rules on an aligned sample of 2,400 days, and keep the best by the same ratio used throughout.

The correct null is built by applying the same block of dates to every candidate at once, so the correlations between candidates survive, after each candidate has had its own observed mean removed, so that nothing in the resampled world has an edge by construction. The statistic recorded from each replicate is the best ratio across all 986 candidates. That distribution has a median of 0.319 and a 95th percentile of 0.845. For a single rule fixed in advance the corresponding bar is 0.505. Searching raised the bar by 67 per cent, and nothing about the data changed.

The selected rule scored 0.901. Its own bootstrap interval is 0.300 to 1.477, with only 0.18 per cent of replicates at or below zero, which any reader would take as decisive. Against the search corrected distribution it clears the bar with a p value of 0.038. The same data, the same resampling machinery, and a difference of about 21 times in the apparent strength of the evidence, purely from counting what was tried.

One honest caveat on that experiment: all 986 candidates had a positive ratio, because the sector index rose across the period and every long or flat rule inherits some of that drift. The demonstration is about the machinery of inference, not about the merits of moving average rules, and it would run the same way on any candidate set. The practical consequence is simple. The number of variants tried is part of the evidence, it is never recoverable afterwards, and a record of what was discarded is worth more than a record of what survived.

India now verifies past risk and return, and the framework cannot verify a backtest

On 29 April 2026 the market regulator issued a circular recognising the first Past Risk and Return Verification Agency, a credit rating agency, and designating the National Stock Exchange as the verification data centre. The framework became operational on 4 May 2026 after a pilot that began on 8 December 2025. It covers investment advisers, research analysts and algorithmic trading providers, and it verifies risk metrics alongside return rather than return alone.

The detail that matters for this topic is the one most commentary passes over. Verification runs on a prospective basis only, from the effective date on which a person opts into the service. Performance generated before that date is outside it. As an interim arrangement, certified performance from the period before the framework may be communicated to clients only until 3 May 2028. The enrolment deadline for advisers and analysts, originally 3 August 2026, was extended to 3 September 2026 by a circular issued on the original date.

A backtested curve therefore cannot be verified under this regime at all, and neither can a live record that predates enrolment. That is worth reading as a statement about evidence rather than as paperwork. A single historical path, however long, is one realisation whose width was never reported, produced by a process whose search history nobody kept. Everything on this page is an explanation of why a regime built to certify performance would decline to certify it, and why forward metrics collected under a standard procedure are a different kind of object from a curve drawn after the fact.

What the width is actually for

The bootstrap does not certify a strategy and no arrangement of it ever will. What it does is convert a single curve into a statement about path variation, and there are four places where that statement changes a decision.

It sizes the drawdown to prepare for. The observed maximum drawdown of this rule is 23.2 per cent, which sits at the 26th percentile of its own resampled family against a median of 27.6 per cent, and the interval runs to 49.2 per cent. Planning around the observed figure plans around an ordering that was gentler than most of what this distribution produces, and position size is the decision that interval should feed.

It decides whether a gap between two curves is worth acting on. Two variants whose intervals overlap almost entirely have not been separated by the data, and choosing between them on the point estimates is choosing on noise.

It refuses claims. When the interval on the headline statistic contains the null value, as it does here after more than a decade, the correct report is that the sample does not establish the result, whatever the curve looks like.

It sets sample policy before the data is seen. Fixing the sample length, the resampling scheme, the block length and the replication count in advance removes the degrees of freedom that let a result be tuned after the fact, which is the same discipline as recording the number of variants tried.

What remains outside its reach is everything about whether the mechanism should have worked: why the effect exists, whether the conditions that produced it still hold, what would have to change for it to stop. That judgement is not a weaker substitute for the statistics. Given how wide these intervals are on the data lengths anyone actually has, it is the part of the work that carries the most weight, and it is the part that has to be taught rather than computed.