Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s.

On 31 July 2026 the KOSPI rose 17.91% in a single session, the largest one-day gain in its history, days after a circuit breaker halted trading and inside the second-worst month the index has ever had. Under a Gaussian at 1.5% daily volatility that is an 11.9-sigma move with a return period of 10^30 years. Fat tails fix that in one line. What they do not fix is that the best day landed next to the worst — which under any i.i.d. model has a probability of exactly 2/n, or 1.3% for this year.

A day that no risk model contains

On 31 July 2026 the KOSPI closed up 17.91% — 1,001.89 points, to 6,595.45. It is the largest one-day gain in the index’s history, ahead of the 11.95% it managed on 30 October 2008. It happened days after trading was halted by a circuit breaker for the eighth time this year, and inside a month that fell 22.4% — the second-worst month the index has ever had, behind October 1997.

Take that 17.91% and ask the question a risk system asks. If daily returns were Gaussian with a calm-regime standard deviation of 1.5% — a typical figure for a large equity index outside a crisis — the move is 11.9 standard deviations. The Gaussian probability of that is 10^-32, which works out to a return period of 10^30 years. The universe is about 10^10 years old.

So the model is wrong, and everybody already knows the first answer: fat tails. Replace the normal with a Student-t and the same move stops being absurd. At four degrees of freedom, matched to the same 1.5% volatility, the return period falls from 10^30 years to about 110 years — from unimaginable to something a long-lived institution has already lived through.

That fix is real and it is one line of code. It is also the smaller half of the problem, and this post is about the larger half.

Four falling lines against assumed daily volatility. The Gaussian line runs from 10 to the 69 years down to 10 to the 16, and the three Student-t lines sit tens of orders of magnitude below it, near or under the dashed reference lines for the age of the universe and the length of the index’s own history.

Fig 1. The vertical axis is a power of ten, so the gap between the Gaussian line and the Student-t lines is tens of orders of magnitude. This is the easy half of the problem: choosing a fat-tailed marginal moves a day like this from impossible to merely rare, and costs nothing but a parameter.

Fat tails are the easy half

The table below is the whole fat-tail argument, and I want it out of the way early because it is not in dispute. Each row assumes a different calm-regime daily volatility and reports the return period of a +17.91% day under three distributions.

assumed daily volthe move, in σGaussianStudent-t, df=4Student-t, df=3
1.00%17.9σ10^69550108
1.25%14.3σ10^4422755
1.50%11.9σ10^3011032
2.00%9.0σ10^163514

Two things to notice. The Gaussian column is absurd everywhere in the range, so no amount of arguing about the right volatility rescues it — the failure is the shape, not the parameter. And — this one surprised me — the monthly figure is not extreme at all. Scale 1.5% daily by the square root of 21 sessions and 22.4% in a month is 3.3 standard deviations, a Gaussian return period of about 7 years. Nothing to report.

Which is its own small lesson about reporting frequency. Aggregate to months and the central limit theorem quietly does its work: the sum of twenty-one wild days looks almost well-behaved, and a monthly risk report would have shown a bad-but-ordinary month where the daily data was screaming. If your tail statistics are computed on monthly data, they are not tail statistics.

Fine. Use a Student-t, or an extreme-value tail, or a jump. All of them are improvements, all of them are standard, and every risk textbook written since 1963 says so. Now here is what none of them do.

The best day of the century, one session after the worst

The KOSPI’s record gain did not arrive on a quiet Tuesday. It arrived in the middle of the crash, in the same week as a trading halt, immediately after the index had fallen roughly 41% from its June peak. The previous record — that 11.95% in October 2008 — also arrived in the middle of a crash. Both of the two largest one-day gains in this index’s history happened while it was falling apart.

That pattern has a probability attached, and it is worth computing, because the computation does not depend on any distribution at all.

Suppose daily returns are i.i.d. — drawn independently from the same distribution, whatever it is. Then every ordering of this year’s returns is equally likely, so the position of the best day and the position of the worst day are just a random ordered pair of distinct days. Counting the pairs that are neighbours gives a memorably simple answer: 2/n, where n is the number of sessions. For 2026 up to the end of July, 151 business days, that is 1.3%. Widen it to within three sessions and it is 3.9%. (I checked the formula against 200,000 simulated years: 1.38% against the exact 1.32%.)

Notice what is not in that expression. No volatility. No degrees of freedom. No tail index. Switching from a normal to a Student-t changes how large the moves are and changes nothing whatsoever about when they arrive. You can fit the fattest tail in the literature and your model will still put the best day of the decade on a random calendar date, uncorrelated with the worst.

Real markets do not work that way, and the mechanism is not mysterious: volatility clusters. A shock raises tomorrow’s expected volatility, which raises the chance of a large move tomorrow, in either direction. Fit that and the pattern appears. In a simulated series with clustering and nothing else — a plain GARCH(1,1) with Student-t shocks — the chance that today is a top-1% up day, given that yesterday was a bottom-1% down day, is 13.3%, against an unconditional 1.0%. That is 13 times the base rate, with no skew, no asymmetry and no narrative attached — just persistence in the variance. Group by yesterday’s decile instead and the shape is a U, not a slope: the bottom decile gives 3.3% and the top decile 3.7%, while the six middle deciles sit at or below the unconditional rate. What predicts a big move is another big move, in either direction.

A U-shaped line: after a bottom-decile day the chance of a top-1% day today is 3.3 percent against the unconditional 1 percent dashed line, it falls below the line for the middle deciles, and rises to 3.7 percent for the top decile.

Fig 2. After a bottom-decile day the chance of a top-1% day is 3.3 times its unconditional rate; condition on a bottom-1% day instead and it is 13 times. The curve is U-shaped, not sloped: what predicts a big move is another big move, in either direction. No i.i.d. model, however fat its tails, can produce this shape.

Which quietly demolishes an argument you have heard

“Stay invested. Miss the ten best days of the last twenty years and you lose most of your return.” That chart is in every fund brochure, and it is arithmetically true. It is also half of a sentence.

On my simulated 24 years: removing the ten best days costs 59% of final wealth. Removing the ten worst days adds 113%. In compounding terms those are 0.90 and 0.76 in logs — the same order of magnitude, the mirror argument if anything the larger one, and only one of the two lines has ever been drawn for a retail investor.

The clustering result tells you why you cannot have one without the other. 90% of those ten best days occur while the index is more than 10% below its previous peak, with an average drawdown of 32% at the moment they happen. The best days are not scattered through the good times. They are inside the crashes, because that is where the volatility is. 31 July 2026 is the cleanest possible illustration: to have captured the largest single-day gain in the index’s history, you had to be holding through a week that included two consecutive circuit breakers.

So “stay invested to capture the best days” is not a claim about upside. It is a claim about being able to tolerate the downside, and it should be argued on those terms.

Two lines from zero: removing the best days drops final wealth steeply into large negative values, removing the worst days raises it by a comparable amount. The two are near mirror images.

Fig 3. The left-hand argument is always shown and the right-hand one almost never is, and they are comparable in size — the one nobody draws being the larger. The honest reading is not “stay invested”: it is that 90% of those best days happen while the index is more than 10% below its peak, so capturing them and sitting through the crash are one decision, not two.

How a risk model fails the test it passes

This is the part that matters if you own a model rather than a portfolio, and it is why I care more about clustering than about tails.

On the same simulated series I built two static value-at-risk models at 1%, both estimated on the first half and both judged on the second, and ran the two standard backtests on each.

The first is the naive one: a Gaussian VaR from the in-sample standard deviation. It fails Kupiec’s proportion-of-failures test, which asks only whether the number of breaches is right — 1.90% observed against 1% expected, 57 breaches where 30 were expected, p = 1.1e-05. No surprise: thin tail, too many exceptions. This is the failure the fat-tail literature exists to fix.

So I fixed it, in the cheapest possible way: take the empirical 1st percentile of the in-sample returns instead of a Gaussian quantile. No distributional assumption at all, the fat tail handled non-parametrically. And it works — on the count:

  • breaches 1.07% against 1% expected, Kupiec p = 0.72. Passes comfortably.
  • Christoffersen’s independence test, which asks whether the breaches arrive spread out or together: p = 1.3e-05. Fails outright.
  • The probability of a breach tomorrow given a breach today is 16%, against the 1% the model implies. The longest run of consecutive breaches is 3.

Same data, same model, two tests, opposite verdicts — and the fat-tail fix moved the first verdict without touching the second. A capital buffer sized for 30 breaches spread across 12 years is not sized for 3 of them in 3 consecutive days. The count was never the risk. The arrival pattern was.

The remedy is not exotic — let the volatility move (GARCH, EWMA, a realised estimator, anything conditional) and report expected shortfall next to the quantile so the size of a breach counts and not just the fact of it. But the diagnostic matters more than the model: run the independence test, and put its p-value beside the coverage number. A dashboard reporting “VaR exceptions: 32, expected 30” is reporting the test the model passes.

Where this argument is thinner than it looks

Four things I am not claiming.

I did not estimate the volatility. The repo holds no KOSPI price series — the figures here are quoted from published reports, so the calm-regime volatility is an assumption I swept across a range rather than a number I measured. That is why Table 1 has four rows instead of one. The conclusion is stable across the range, but “σ was really 3% by late July” is a legitimate objection, and the honest answer is that a conditional model would have known that, which is the post’s point rather than a hole in it.

One index, one episode. Two record gains inside two crashes is a pattern with two observations in it. The clustering evidence in Figs 2 and 3 is simulated, not sampled from the KOSPI: it shows that a standard clustering model reproduces the pattern, not that the KOSPI’s parameters are the ones I chose.

GARCH is not the truth either. It reproduces volatility clustering and fat tails, and it still assumes a fixed structure that no market has agreed to. It has no jumps, no regime changes, and a symmetric response to good and bad news that equity indices are known to violate.

2/n assumes continuous i.i.d. returns. With ties, or with a deterministic calendar effect, the counting changes slightly. It does not change by anything that matters at n ≈ 151: at 141 sessions the figure is 1.42% rather than 1.32%.

What survives all four is narrow and, I think, useful: the arithmetic that makes a +17.91% day impossible is about the shape of the distribution, and the arithmetic that makes it land next to the crash is about dependence. The first is what everyone reaches for. The second is where the money is.

Next in this series, back to dynamics: what a neural network is doing when it looks like it has learned physics, and how to tell that apart from having memorised a trajectory.

Data

Reproducibility

  • seed: 20260804
  • environment: standarderror=0.1.0, python=3.11.15, numpy=2.4.4, scipy=1.17.1
  • closed forms: Gaussian and rescaled Student-t survival functions evaluated in logs (the probabilities underflow float64 by tens of orders of magnitude); adjacency probability (2kn - k(k+1)) / (n(n-1))
  • adjacency check: exact 0.01325 against 200,000 Monte Carlo draws 0.01377
  • sessions: 151 business days from 2026-01-02 to 2026-07-31 (ignoring Korean market holidays; at 141 sessions the adjacency probability is 1.42% instead of 1.32%)
  • simulation: GARCH(1,1), omega=0.02, arch=0.1, garch=0.88, t(5) shocks, 6,000 sessions (24 years), unconditional daily sd 1.19%
  • VaR backtest: two static VaR models at 1% — a Gaussian quantile and the empirical 1st percentile — both estimated on the first 3,000 sessions and tested on the next 3,000

Code: https://github.com/jongha-jeon-dev/standarderror