Stylised facts: what financial returns actually look like

A handful of properties hold across almost every asset, market and decade. They are worth knowing precisely, because they are the constraints any model has to satisfy — and because an interviewer will check whether you know which of them can be traded and which cannot.

Fact	Statement	Exploitable?
Fat tails	Kurtosis far above 3; six-sigma days are not rare	Yes — options, tail hedging
No return autocorrelation	Near zero at daily frequency	No, by construction
Volatility clustering	Squared returns strongly autocorrelated	Yes — volatility forecasting
Leverage effect	Falls raise volatility more than rises	Yes — skew
Aggregational normality	Monthly returns are closer to normal than daily	Indirectly
Volume–volatility correlation	They move together	Yes — as a feature
Intraday seasonality	U-shaped volume and volatility through the day	Yes — execution

Table 6.1 · The stylised facts. The second row is the only "no", and it is not an accident: any exploitable autocorrelation in returns is traded away, while the others survive because exploiting them requires taking risk.

Figure 6.2 · What the normal says about a three-sigma day. The two tails hold

0.27\%

between them — about one day in

370

, or roughly

0.7

days a year. Equity indices deliver several, which is the entire content of "returns have fat tails": not that the shape is wrong in the middle, but that the part you are paid to survive is many times thicker than this.

Proposition 6.3

How fat the tails are

Daily equity index returns have kurtosis typically between 5 and 20 against the normal’s 3, and the tails decay as a power law with an exponent around 3 to 5 rather than exponentially. The practical consequence is that a normal model understates the frequency of large moves by orders of magnitude — a six-sigma daily move should happen once every few million years and happens every few years.

Holds when

A tail exponent near 3 means the fourth moment may not exist, so sample kurtosis is unstable and grows with the sample.
Tails thin out as you aggregate: monthly returns are much closer to normal than daily ones.
Value-at-risk computed from a normal assumption is systematically too small in the tail.

Equation 6.4

The variance ratio test

Under a random walk, variance grows linearly with the horizon, so the ratio is one. Above one means trending; below means mean reversion.

VR(q) = \frac{\mathrm{Var}(r_t^{(q)})/q}{\mathrm{Var}(r_t)}

$r_t^{(q)}$: The $q$ -period return.
$VR > 1$: Positive autocorrelation — momentum.

Stylised facts: what financial returns actually look like

After this lesson you should be able to

How fat the tails are

The variance ratio test

The rest of this lesson is in Premium