A normal curve says a day 4 standard deviations from the mean should arrive once in 15,787 trading days. From 29 January 1993 to 26 October 2026, SPY (the S&P 500 ETF, a fund that tracks the index) had 54 such days, with dividends counted. That is out of 8,492 daily returns. The normal curve expected 0.54.
Those days are the fat tails of the return distribution.
What are fat tails in investing? Fat tails means that extreme returns arrive more often than a normal distribution, the bell curve, allows. SPY had 54 days at least 4 standard deviations from its mean since 1993, against 0.54 expected.
From 2 January 2014 to 26 October 2026, all 14 assets we tested had far more days 4 standard deviations (4 sigma) from the mean than the normal curve expects. SPY had 22 and Bitcoin 32. Ten stocks together had 193, against 1.9 expected.
SPY's 4-sigma days cluster. 30 of its 54 fell in 2008 and 2020.
From 2 January 2014 to 26 October 2026, SPY's daily loss reached the normal curve's 1-in-1,000 point on 8.4 days in 1,000.
What are fat tails? What is a sigma move?
A return is the percentage change in a price, here over one day. The standard deviation measures how far returns typically sit from their mean. Our explainer on standard deviation works it by hand.
Statisticians write the standard deviation as sigma. A sigma move is a return measured in sigmas. For SPY from 2014, one sigma is 1.077% in a day, so a 4-sigma move sits about 4.3 percentage points from the mean.
A normal distribution is the bell curve. About 68.27% of days land within 1 sigma of the mean, 95.45% within 2 sigma and 99.73% within 3 sigma. Beyond 4 sigma the curve leaves 0.0063% of days, or 1 in 15,787.
Fat tails means the real tails hold more days than that. Excess kurtosis scores the tail weight. A higher score means more days far from the mean. A normal curve scores 0. SPY's daily returns from 2014 score 14.42.
How did we count 4-sigma days?
We used SPY, GLD (the gold ETF), TLT (the long-term US Treasury bond ETF) and Bitcoin. We added ten single stocks: Advanced Micro Devices, Amazon, Alphabet, Intel, Meta, Microsoft, Netflix, Nvidia, Palantir and TSMC (Taiwan Semiconductor Manufacturing Company). Prices count dividends, except Bitcoin, which pays none.
For each asset we took the daily returns and found the mean and the standard deviation (sigma) over the window. A 4-sigma day is a day whose return sits at least 4 sigma from the mean, up or down. A normal curve expects the number of daily returns times 0.0063%.
The main window runs from 2 January 2014 to 26 October 2026. SPY has 3,224 closes from 31 December 2013 and 3,223 daily returns in it. Palantir joins in October 2020, so it has 1,524 returns.
Where we use SPY's full history, we say so. That history has 8,493 closes and 8,492 daily returns. We use SPY's closing market price, not the fund's net asset value.
How often did SPY have 4-sigma days?
From 29 January 1993 to 26 October 2026, SPY had 8,492 daily returns and a sigma of 1.166%. It had 54 days at 4 sigma or more, 28 down and 26 up. A normal curve expects 0.54, so the count is about 100 times higher.
At 3 sigma the count was 125 against 22.9 expected. Chart 1 plots those 125 days in sigmas.
Source: YX Insights price data
The extra days arrive in bunches. SPY had 4-sigma days in 12 of the 34 calendar years in the file, in 23 different months. Of the 54, 18 fell in 2008 and 12 in 2020, so 30 fell in those two years.
The worst day was 16 March 2020, when SPY lost 10.94%, or 9.42 sigma below the mean. A normal curve gives a day that bad odds of less than 1 in a trillion trading days. This is SPY's closing price, not the index.
Do gold, Treasuries, Bitcoin and stocks have fat tails too?
Yes. All 14 assets had more 4-sigma days than the normal curve expects. The table gives the count for each.
4-Sigma Days By Asset, 2 January 2014 To 26 October 2026
Asset | Daily returns | Sigma, a day | 4-sigma days | Normal curve expects | Times more |
|---|---|---|---|---|---|
SPY | 3,223 | 1.08% | 22 | 0.20 | 108 |
GLD | 3,223 | 1.02% | 13 | 0.20 | 64 |
TLT | 3,223 | 0.91% | 10 | 0.20 | 49 |
Bitcoin | 4,677 | 3.52% | 32 | 0.30 | 108 |
AMD | 3,223 | 3.68% | 16 | 0.20 | 78 |
Amazon | 3,223 | 2.08% | 22 | 0.20 | 108 |
Alphabet | 3,223 | 1.80% | 25 | 0.20 | 122 |
Intel | 3,223 | 2.66% | 28 | 0.20 | 137 |
Meta | 3,223 | 2.38% | 19 | 0.20 | 93 |
Microsoft | 3,223 | 1.70% | 21 | 0.20 | 103 |
Netflix | 3,223 | 2.67% | 19 | 0.20 | 93 |
Nvidia | 3,223 | 2.94% | 19 | 0.20 | 93 |
Palantir | 1,524 | 4.39% | 12 | 0.10 | 124 |
TSMC | 3,223 | 2.07% | 12 | 0.20 | 59 |
Days at least 4 sigma from each asset's mean, up or down, 2 January 2014 to 26 October 2026, with dividends counted (Bitcoin pays none). Palantir starts on 1 October 2020. Source: YX Insights price data
Chart 2 puts the four funds and assets and the ten stocks together on one scale, 4-sigma days per 10,000 days. A normal curve gives 0.63.
Source: YX Insights price data
Gold had 13, with 8 down and 5 up. Its worst day was 30 January 2026, down 10.27% or 10.12 sigma. TLT had 10. Of those, 8 fell in March 2020.
The ten stocks had 193 together, 128 up and 65 down. The count runs from 12 for Palantir and TSMC to 28 for Intel.
How do we compare Bitcoin with stocks?
Bitcoin trades every day of the year, while stocks trade about 252 days a year. We did not compare annualised volatility. Each sigma comes from the asset's own returns, so each count is set against its own number of days.
Bitcoin had 32 4-sigma days in 4,677 daily returns, or 68.4 per 10,000 days. SPY had 68.3 per 10,000 of its own trading days, so the two rates are almost equal. Our guide to Bitcoin volatility compares it with SPY on weekly returns.
Does the result depend on how we measure sigma?
We ran three other versions with the same final code and window.
First, we measured each day against the sigma of the previous 252 daily returns (365 for Bitcoin), which a trader would have had before that day. This version measures each return from zero, not from the mean.
All 14 assets still had more 4-sigma days than expected. SPY had 22. The ten stocks had 212 against 1.9.
Second, we used weekly returns. Each of the 14 assets had at least one 4-sigma week, against 0.02 to 0.04 expected per asset. SPY had 5 in 623 weeks, GLD 3, TLT 1 and Bitcoin 2 in 666. The ten stocks had 28 in 5,900 weeks, against 0.4 expected.
Third, we removed the 4-sigma days from the sigma estimate and recounted. SPY had 31, GLD 18, TLT 10 and Bitcoin 39. All 14 stayed above expected.
What do fat tails do to a risk measure?
Value at Risk, or VaR, is a loss level that a stated share of days should breach. We built a normal-curve version at 99.9%. It is the mean minus 3.09 sigma, the loss a normal curve breaches on 1 day in 1,000.
For SPY from 2014 that loss is 3.27%. SPY returned that or worse on 27 of 3,223 days, or 8.4 per 1,000. Chart 3 shows the rate for each asset.
Source: YX Insights price data
GLD had 7.1 days per 1,000, Bitcoin 7.5, the ten stocks 5.1 and TLT 2.8. In SPY's full history the rate was 7.2 per 1,000, on 61 days.
The normal-curve limit was breached more than 1 day in 1,000 for each of the 14 assets.
What does this test assume?
The sample is 14 assets over 12.8 years, with SPY also shown over 33.7 years.
Sigma comes from the same window as the count. The trailing check removes that overlap.
The ten stocks are not a random sample. The numbers describe the past and make no forecast.
Our guide to volatility covers the measure behind them.
Across 14 assets and four ways of counting, 4-sigma days arrived far more often than a normal curve allows. SPY's 4-sigma days arrived in clusters.
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Common questions about fat tails
What does fat tails mean in finance?
Fat tails means extreme returns happen more often than a normal distribution predicts. In SPY from 1993 to 2026, 54 days sat at least 4 standard deviations from the mean, against 0.54 expected. The count is about 100 times higher, so a normal curve understates how often the largest moves arrive.
What does 4 sigma mean in investing?
A 4-sigma move is a return 4 standard deviations from the mean, with sigma standing for the standard deviation. For SPY from 2014, one sigma is 1.077% in a day. A 4-sigma move therefore sits about 4.3 percentage points from the mean. A normal curve puts 0.0063% of days beyond 4 sigma.
Are stock returns normally distributed?
No. Daily returns for all 14 assets we tested had more 4-sigma days than a normal curve allows from 2014 to 2026. Ten stocks together had 193 against 1.9 expected. SPY's excess kurtosis, which is 0 for a normal curve, was 14.42 over the same window.
How often should a 4-sigma day happen?
A normal curve expects a 4-sigma day, up or down, once in 15,787 days. That is about once in 63 years of 252 trading days. SPY had one 4-sigma day every 157 days from 1993 to 2026, with 54 in 8,492 returns.
Why do fat tails matter for risk?
Fat tails mean large losses arrive more often than a normal-curve risk number says. A normal curve puts a 3.27% SPY loss at 1 day in 1,000. SPY returned that or worse on 8.4 days per 1,000 from 2014 to 2026. Our guide to volatility covers the basics.
DISCLAIMER: This article is intended for educational purposes only. Any information or analysis in this note does not constitute an offer to sell or a solicitation of an offer to buy any securities. Nothing in this note is intended to be investment advice, nor should it be relied upon to make investment decisions. Any opinions, analyses, or probabilities expressed in this note are those of the author as of the note's date of publication and are subject to change without notice.
