The normal distribution, and where it lies

The normal distribution is the most useful wrong assumption in finance. It needs to be understood in both directions: why it is used, and precisely how it fails.

The shape

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

Two parameters fully describe it: the mean $\mu$ and the standard deviation $\sigma$. Nothing else. That is the appeal — summarise a mess of data in two numbers and every question has a closed-form answer.

It is symmetric about the mean, so it assigns identical probability to a 5% gain and a 5% loss.

The rule worth memorising

Within how many standard deviations of the mean:

Range	Probability
$\mu \pm 1\sigma$	68.3%
$\mu \pm 2\sigma$	95.4%
$\mu \pm 3\sigma$	99.7%

A z-score expresses any observation in these units:

$z = \frac{x - \mu}{\sigma}$

So a return 2.5 standard deviations below the mean has $z = -2.5$, and the normal distribution says about 0.6% of observations fall below that.

Why finance reaches for it anyway

Three honest reasons, none of them "returns are normal":

1. The central limit theorem (chapter 8) says sums of many independent effects tend to normality. A day's return is the aggregate of many trades, so the normal is a plausible first guess.
2. It is closed under addition. Add two normals and you get a normal, which means portfolio maths stays tractable. Almost no other distribution is this well behaved.
3. Everything downstream assumes it. Black-Scholes, value-at-risk, the efficient frontier, most risk systems. Changing the assumption means rebuilding the tools.

How badly it fails

Market returns have fat tails: extreme moves occur far more often than a normal distribution permits, and the discrepancy is not a modest correction.

Work the problem above. With $\sigma = 1.1\%$ daily, a 6% fall is $z = -5.45$. The normal probability of that or worse is roughly $2.5 \times 10^{-8}$ — about one day in 40 million, or once every 160,000 years at 252 trading days a year.

Indian markets have had several such days in living memory. March 2020 alone produced more than one.

The model is not slightly off. It is wrong by a factor of thousands, in the direction that matters.

The error is systematically one-sided: the normal understates the probability of large losses, so every risk number built on it is an underestimate of exactly the event it exists to warn about.

The regulator's answer is not statistical

SEBI's market-wide circuit breakers halt trading at index movements of 10%, 15% and 20% in either direction. Those thresholds are not derived from a distribution — they are administrative limits set because the distribution cannot be trusted at the extreme. A system that genuinely believed in normality would need no such switch.

Why 'six sigma' is a warning, not a boast

When a fund or a risk desk describes a loss as a "six sigma event", there are two possible readings:

• An event of probability $10^{-9}$ occurred, which would be remarkable; or
• $\sigma$ was the wrong measure and the distribution was never normal.

The second explanation is almost always correct. Repeated extreme events are evidence against the model, not evidence of extraordinary bad luck. The honest response to a six-sigma day is to stop using the model that called it that.

What to do with the distribution instead

It is still the right first approximation, used with its limits stated:

• Treat it as a description of the middle, not the tails. Within one or two $\sigma$ it is reasonable.
• Never size a position from a normal tail probability. Use a scenario — "what if this falls 40%" — rather than a probability.
• Prefer historical and stress testing for extremes. What actually happened beats what the formula permits.
• Remember the asymmetry. Negative skew (chapter 3) means the left tail is both fatter and longer.

Chapter 7 of the Risk subject's advice to think in drawdowns rather than volatility is this chapter's practical conclusion.

The point

The normal distribution describes a dataset with two numbers and gives closed-form answers to everything, which is why finance is built on it. It also assigns a 6% single-day fall a probability of once in 160,000 years when such days occur within most investors' memory. Use it for the middle of the distribution, never for sizing against the tail, and treat a "six sigma event" as evidence the model was wrong rather than the luck extraordinary.