Normal distribution in finance

Almost every quant model starts by assuming returns follow a normal distribution. What the bell curve is trying to tell us.

Play with it

Bell curves and fat tails. Compare the normal distribution with fat-tailed and skewed returns, and count the impossible days.

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Mean daily return0.05%

The center of the curve.

Standard deviation (volatility)1.00%

How spread out the days are.

Within 1σ (normal says 68%)68.6%

Within 2σ (normal says 95%)95.5%

Within 3σ (normal says 99.7%)99.8%

Two numbers describe this whole curve: a mean of 0.05% and a standard deviation of 1.00%. Of 5,000 simulated days, 68.6% land within one standard deviation, 95.5% within two and 99.8% within three. Under this model a move four or five standard deviations out is almost unbelievable.

Try this

Check the 68–95–99.7 rule: do the three measured shares match what the normal distribution says?
Raise the standard deviation. The center stays put, but the days swing around much more. That's what higher volatility means.
Draw new days a few times. How often do you see anything in the coral bars beyond three standard deviations?

The math

f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}}

$\mu$: the mean, the center of the curve
$\sigma$: the standard deviation, or volatility
$f(x)$: how likely a return near x is

Quick check

Two questions to see if it clicked. Answers explain themselves.

Under a normal distribution, roughly what share of observations fall within two standard deviations?

In finance, the standard deviation of returns is usually called…