Market Reality vs. Gaussian Illusion: The Truth Behind Distributions

It’s no secret that market returns don’t follow a normal distribution. In fact, they are especially leptokurtic, with the famous “fat tails” characteristic of extreme events. A lesser-known fact is that many market returns also exhibit “thick centers,” with an unusually large number of small movements. That’s why the phrase “picking up pennies in front of a steamroller” is so popular when describing returns from risky assets — you usually make a little bit every day, and then lose a lot all at once. Of course, not every market behaves the same, and the behavior of a given asset can change significantly over time. If you’ve ever wondered which major stock index has generated the most extreme tails — wonder no more, though the answer might surprise you.

One of the first things any finance novice learns is that market returns have unusual properties. Although their distribution resembles a Gaussian bell curve, the resemblance is far from exact. Most famously, markets tend to show many more extreme events than a normal distribution would predict, even if the absolute frequency remains low. While most people intuitively understand that returns from things like credit or short volatility strategies have a large number of small outcomes, the same is true for equities. The chart below shows the normalized return distribution of the S&P 500 since 1928, using full sample volatility and overlaid with a Gaussian curve.

Image (S&P 500 Return Distribution Since 1928):

Of course, a given market can go through different volatility regimes — just compare the SPX during Donald Trump’s first year in office to that in 2025. In many ways, it’s more interesting to compare how market return distributions change across different volatility regimes. For example, a 3% move carried a very different “weight” in April compared to the end of 2023, when the VIX was under 13… even if the financial impact on portfolios was the same.

For most financial assets, an active trading day accounts for just under 0.4% of all trading days in a year. This percentage corresponds to the expected frequency of a ±2.9 standard deviation move in a normal distribution. In other words, under Gaussian assumptions, you would expect one daily return of 3 standard deviations (rounded from 2.9) per year. This provides a useful benchmark to assess how “extreme” an asset’s returns really are.

I went back to 1900 and calculated the frequency of 3σ moves in the U.S. stock market (using the Dow from 1900–1928, and SPX thereafter), comparing each day’s return to the annualized standard deviation calculated from the prior day. I used the absolute value of the daily return, since it doesn’t matter for this analysis whether the move is positive or negative. The chart below shows a rolling one-year count of 3σ days across the whole period — and as you can see, they’re often far more than just one!

Image (Rolling Count of 3σ Days, 1900–2025):

Unsurprisingly, the current count is relatively high, though lower than during the financial crisis and COVID. Interestingly, the SPX hadn’t seen fewer than 10 3σ days in a rolling year from the early 1970s until 2007; after that, the “tails” became significantly more extreme. Does this reflect a build-up of leverage in the financial system or the influence of volatility-selling strategies that increase the volatility of volatility itself? A thought-provoking question.

Another point of curiosity: how does the frequency of 3σ days in the Dow/SPX compare with other U.S. and global indices? So I ran the same analysis for 22 major global markets to determine:
a) which have the highest all-time frequency of 3σ days, and
b) which had the most such days over the past year.

Methodological note: Since holidays vary by region, I used a 5-day “holiday-free” calendar that yields approximately 263 observations per year. Counting holidays slightly lowers measured volatility, as there are about 11 zero-return days per year. The effect is minimal, but you may get slightly different results if you repeat the analysis with an exact holiday calendar for each market. The table below shows the results:

You might be surprised to learn that the NDX has a lower frequency of extreme tail events than the SPX — both historically and over the past year. In absolute terms, it’s more volatile… but within that, its extreme movements are actually more limited relative to its own sample behavior. Even more unexpectedly, the Canadian market turns out to have the highest frequency of 3σ days; ironically, its frequency is nearly 3 standard deviations above the average across all indices in the table.

Recently, Taiwan has shown the highest ratio of large moves over the past year — which makes sense, given its concentrated market and its central role in a heated long-term investment narrative. The fact that the Taiwan dollar itself experienced several massive normalized moves last month doesn’t hurt either. That said, I’m not entirely sure how much informational value we get from the distribution of 3σ days over just the last year — what does it really tell us that the SMI had 10 and Chinese equities had only five?

In any case, I never would have guessed that Canadian stocks had the most extreme tails — at least not at first glance. I plan to expand this analysis in the coming days and weeks — it will be interesting to see how the return profiles of currencies, bonds, and commodities compare to each other and to the stock market.