Lesson 3 of 20 · 7 min
Why returns are not actually normal
Real markets produce far more extreme days than a bell curve allows, and the excess is not small.
The normal distribution fails on market returns in three documented, specific ways. Each one has a name, and each one matters in a different situation.
The three failures:
- Fat tails. Extreme moves occur far more often than the bell curve permits. This is the failure that matters for leverage.
- Skew. Declines tend to be sharper and faster than advances, so the distribution is not symmetric.
- Volatility clustering. Turbulent days follow turbulent days, so the width of the distribution is itself changing rather than fixed.
Fat tails can be quantified, and the scale of the discrepancy is what makes the point. At 25 percent annual volatility, a daily standard deviation is about 1.6 percent. On that basis, a seven percent daily decline is a 4.6 standard deviation event, which a normal distribution places at roughly once in five hundred years. Declines of that size have occurred several times in the past century.
Push further and the model stops being merely wrong. A twenty percent single-day decline is about fourteen standard deviations, which a normal distribution assigns a probability on the order of one in ten to the forty-fifth power. That is not a small number; it is a number with no physical interpretation — far less likely than picking a specific atom out of the observable universe. It happened in October 1987.
Volatility clustering explains part of the mechanism. Because turbulence arrives in runs, a period of calm has a genuinely narrow distribution and a period of stress has a genuinely wide one. Fitting one fixed width to both produces something that underestimates the stressed regime badly, which is exactly the regime in which a maintenance threshold gets tested.
Key terms
- Fat tails (excess kurtosis)
- The property that extreme outcomes are more common than a normal distribution predicts. Measured as kurtosis above the normal value of three.
- Skew
- Asymmetry in the distribution. Equity returns are typically negatively skewed: the large moves are more often downward.
- Regime
- A persistent state of market behaviour, such as calm or turbulent. Volatility clustering means regimes exist and one width does not fit both.
Worked example
The size of the discrepancy
Assume 25% annual volatility, so a daily standard deviation is 1.6% (25% ÷ √252).
- −3% day
- 1.9 sigma
- −7% day
- 4.6 sigma
- −20% day
- 14 sigma
- Observed −20% days in US equities
- At least one
- Error at the centre of the distribution
- Modest
- Error in the tail
- Dozens of orders of magnitude
Normal model: a few times a year. Roughly matches reality.
Normal model: about once in 500 years.
Normal model: probability near 1 in 10⁴⁵.
19 October 1987.
The bell curve is roughly right about ordinary days and catastrophically wrong about the days that trigger forced sales.
Check yourself
The normal distribution assigns a −20% single-day move a probability near one in 10⁴⁵, and such a day has occurred. What is the right conclusion?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.