Most statistics courses assume financial returns follow a normal distribution — the classic bell curve. In practice, this assumption is dangerously wrong.

The normal distribution predicts: - A 3-sigma daily move: once every ~3 years - A 5-sigma move: once every ~14,000 years - A 10-sigma move: essentially impossible

Reality: Black Monday 1987 (-22% in one day) was a ~25-sigma event under the normal model. The S&P 500 has experienced multiple 5-sigma moves in every decade.

Financial returns have excess kurtosis — heavier tails than the normal distribution predicts. This means extreme events ('black swans') are far more frequent than normal-distribution models suggest.

Excess kurtosis = kurtosis 3. Normal distribution has excess kurtosis = 0. Daily equity returns typically show excess kurtosis of 5–10.

Negative skew: Most equity strategies also exhibit negative skew — frequent small gains but occasional catastrophic losses. Think of it as 'picking up pennies in front of a steamroller.'

Risk models built on normality will systematically underestimate tail risk. Always size positions as if the worst historical loss can be exceeded.