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What 517 Series Tell You: Fat Tails Are Universal, and So Are Data Artifacts

The first piece in this series looked at one series, the S&P 500, and found daily returns far from normally distributed. The natural next question: is that a quirk of the index, or does it hold everywhere? Running the same battery of tests (returns, skewness, kurtosis, normality, stationarity) across 517 series, 502 individual stocks, 5 major indices, and 10 Treasury/credit series, answers that question, and turns up a second finding nobody was looking for.

Fat tails: essentially universal

516 of 517 series (99.8%) show excess kurtosis above 1.0, meaning fatter tails than a normal distribution predicts. This isn't an S&P 500 property. It's close to a market property. Whatever generates large price moves more often than a bell curve expects, it operates almost everywhere in this universe: individual stocks, broad indices, and interest rate series alike.

The finding nobody was looking for

Rank the 502 individual stocks by excess kurtosis and the top of the list looks almost unbelievable: values in the hundreds and, in one case, over 6,000. That's not "fat tails." That's a red flag.

Checking the single worst trading day for each of the top names confirms it. These aren't crashes:

| Ticker | Date | One-day move | What actually happened | |---|---|---|---| | NVR | 1993-10-01 | +2,633% | Corporate restructuring | | HUBB | 1994-10-31 | +886% | Data discontinuity, not a recorded split | | KDP | 2018-07-10 | -82% | Keurig / Dr Pepper merger listing | | HST | 1993-10-12 | -81% | REIT conversion | | MO | 2008-03-31 | -70% | Philip Morris International spinoff |

Every one of these is a real corporate event that changed the share count, share structure, or the company itself overnight, not a day where the market violently repriced risk. The underlying data uses raw (non-dividend/split-adjusted) closing prices, and the standard data field that flags stock splits doesn't capture mergers, spinoffs, or restructurings. A statistical test has no way to distinguish "the stock became worth 27x more overnight" from "the stock was replaced by something else overnight." Both look identical to a kurtosis calculation.

Why this matters more than it might seem

It would have been easy to report "NVR, HUBB, and KDP are the most volatile stocks in the S&P 500" and move on. That claim would be confidently stated, quantitatively precise, and wrong. The lesson generalizes: any ranking built on raw statistical moments, across any dataset with corporate actions in its history, needs a screen for exactly this kind of artifact before the ranking means what it appears to mean. That screen doesn't exist yet in this pipeline. It's now a known, documented gap rather than a silent one, which is the more useful state to be in.

The quieter, textbook-confirming half

Set the outliers aside and the rest of the universe behaves exactly as classical time-series theory predicts:

• 496 of 517 series (96%) have stationary returns. Day-to-day changes

don't systematically drift.

• 503 of 517 series (97%) have non-stationary price or yield levels.

The levels themselves wander, the classic random-walk signature.

• One clean, meaningful exception: the VIX's own level is stationary,

unlike every other index. That's consistent with how the VIX is constructed, a bounded, mean-reverting volatility measure, not a price that can drift arbitrarily far from where it started. A statistical test independently rediscovering a known structural property of the index is a good sign the underlying methodology is sound.

• The 3-month Treasury yield is the one genuinely ambiguous case.

Standard stationarity tests disagree on it, most likely because it spent years pinned near zero during two different policy periods (2009 to 2015, and 2020 to 2022), which confuses tests built for continuously varying series.

The takeaway

Two findings, same dataset. Fat tails are close to a universal property of this market, not an S&P-500-specific one. And the loudest signal in any raw ranking is often the data, not the market. Both are useful to know before drawing conclusions from either.

*This is a Phase 1 finding from an ongoing statistical research project covering stocks, bonds, and indices as far back as available data allows. Data: Yahoo Finance (502 stocks, 5 indices) and FRED (10 Treasury/credit series). Full methodology and reproducibility details available on request.*