Lesson 2 of 20 · 6 min

The normal distribution, and why finance uses it

The bell curve is convenient, mathematically tractable, and the default for reasons that are mostly not about accuracy.

The normal distribution is the bell curve: symmetric, single-peaked, and fully described by two numbers — its centre and its width. Finance uses it almost everywhere, and it is worth being clear-eyed about why.

The honest reasons:

Two parameters describe it completely, so a whole distribution can be summarised by an average and a standard deviation.
Sums of normal variables are normal, which makes portfolio mathematics tractable rather than intractable.
The central limit theorem says that averages of many independent effects tend toward normality, which suggests it as a default.
It produces closed-form answers. Black-Scholes exists in a formula rather than only as a simulation because of it.

The properties worth memorising are the coverage figures. About 68 percent of outcomes fall within one standard deviation of the centre, about 95 percent within two, and about 99.7 percent within three. Those three numbers let you convert any normal distribution into a rough answer without a calculator.

Finance usually applies normality to logarithmic returns rather than to prices, which is a meaningful refinement. It prevents prices from going negative and makes a 50 percent gain and a 33 percent loss symmetric, which they are in proportional terms. That adjustment fixes a real problem and does not fix the tail problem, which is the next lesson.

The central limit theorem deserves one caution, since it is the most-cited justification. It applies to averages of independent effects. Market returns during a crisis are neither independent of one another nor averages of many unrelated things — they are dominated by one shared cause. Precisely when the assumption matters most, its justification is weakest.

Key terms

Normal distribution: The symmetric bell curve, described entirely by a mean and a standard deviation. About 68% of outcomes lie within one standard deviation.
Log returns: Returns measured as the logarithm of the price ratio. Applying normality to these keeps prices positive and makes gains and losses symmetric.
Central limit theorem: The result that averages of many independent effects tend toward a normal distribution. It requires independence, which crises remove.

Worked example

The three coverage numbers, applied

A portfolio with an expected annual return of 7% and a standard deviation of 20%, assumed normal.

One standard deviation band: −13% to +27%
Two standard deviations: −33% to +47%
Three standard deviations: −53% to +67%
Chance of a year worse than −33%: About 2.3%
Implied frequency: About once in 44 years

Two numbers produced an entire set of answers. That convenience is the real reason the assumption persists.

Check yourself

A portfolio is assumed normal with a 7% expected return and 20% standard deviation. Roughly how often should a year worse than −33% occur, and why treat that answer with suspicion?

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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.