Stylized Facts of Asset Returns

Suppose someone asks how much your portfolio could lose next month. To answer, you need a model of how returns behave: how big they get, how often the big ones arrive, and whether today's move says anything about tomorrow's. The natural first move is to reach for the bell curve. Markets do not cooperate. Look at a century of daily prices from stocks, bonds, currencies and commodities across four continents and the same handful of departures from the textbook picture appears every time. Those recurring departures are the stylized facts of asset returns. They are not a theory anyone invented — they are the checklist a model has to pass before it is describing a market that exists.

An analogy: markets are weather, not dice

Roll a die all day and each roll is a clean draw from the same jar: yesterday's six tells you nothing, and a roll of forty is not rare but impossible. Weather is different. Storms come in runs — a wild week is usually followed by another wild week, a still week by another still one. You cannot forecast which direction tomorrow's wind will blow, but you can forecast quite well how rough it will be. And once a lifetime something arrives that dwarfs everything else in the record.

Markets behave like weather. Every stylized fact below is a version of that sentence.

Fact 1: the tails are far too fat

Extreme days happen far more often than a bell curve permits. The standard measure is kurtosis, an average of fourth powers:

$$\text{kurtosis} = \frac{\frac{1}{n}\sum_{i=1}^{n} (r_i - \bar{r})^4}{s^4} .$$

In words: take each day's return $r_i$, measure how far it sat from the average return $\bar{r}$, raise that distance to the fourth power, average over all $n$ days, then divide by the Standard Deviation $s$ to the fourth so the units cancel. The fourth power is the trick — it lets far-out days dominate the total. A five-sigma day contributes $5^4 = 625$ times as much as a one-sigma day.

A normal distribution has kurtosis exactly 3. Daily equity returns typically land between 5 and 30 — the difference between "this crash was impossible" and "this crash was overdue."

Fact 2: volatility clusters

Big moves are followed by big moves of either sign; quiet days are followed by quiet days. Markets switch between calm and violent months in runs, not at random. To detect it, correlate the size of today's move with the size of a move $k$ days ago:

$$\rho_k = \text{corr}\big(|r_t|,\ |r_{t-k}|\big) .$$

In words: strip the signs off, so a $-3\%$ day and a $+3\%$ day count the same, then ask how strongly today's size tracks the size from $k$ days back. For daily equities this runs 0.2 to 0.3 at one day's lag and stays positive for months. Volatility is one of the most forecastable quantities in finance.

Fact 3: direction is close to unpredictable

Run the same correlation on returns with their signs and it collapses to roughly zero beyond a few minutes' lag. This is the one stylized fact that matches the textbook: if today's direction predicted tomorrow's, the trade would be obvious and someone would already have removed it. Size survives arbitrage because trading it requires taking real risk; direction does not.

Fact 4: down is not the mirror image of up

Return distributions lean left. Large falls are bigger and faster than large rises, and volatility jumps after a drop far more than after an equal-sized rally — the leverage effect. A symmetric model therefore understates the loss side of every risk number you compute.

Fact 5: zoom out and it starts to look normal

Daily returns are wildly non-normal, monthly returns much less so, annual returns close to a bell curve. Adding many draws together washes out the individual extremes — the The Central Limit Theorem at work. Pick a skewed parent in the explorer below and drag the sample size: the raw shape is lopsided and spiky, the averages of it are not.

Worked example 1: how often should a five-sigma day happen?

Take the bell curve at its word. The chance of landing more than five standard deviations from the mean, either side, is

$$P(|Z| > 5) \approx 0.000000573 .$$

Turn that into a waiting time. One over that probability is

$$\frac{1}{0.000000573} \approx 1{,}745{,}000 \text{ days} .$$

Markets trade about 252 days a year, so

$$\frac{1{,}745{,}000}{252} \approx 6{,}900 \text{ years} .$$

Under the normal model, one five-sigma day should arrive roughly every seven thousand years. The S&P 500 has delivered several in a single lifetime — October 1987 alone was past twenty sigma on the prevailing volatility estimate. The model is not slightly off; it is off by orders of magnitude exactly where the money is.

Worked example 2: predictable size, unpredictable sign

Here are ten consecutive daily returns, in percent, taken as a toy series:

$$-2.6,\ -1.9,\ +2.2,\ +1.5,\ -1.1,\ +0.2,\ +0.3,\ -0.1,\ -0.2,\ +0.1$$

Step 1 — split the days by size. Call a day "big" if its size is at least 1.0: days 1 to 5. Days 6 to 9 are "small" (day 10 has no follower).

Step 2 — average the size of the next day in each group. After a big day the followers are 1.9, 2.2, 1.5, 1.1 and 0.2:

$$\frac{1.9 + 2.2 + 1.5 + 1.1 + 0.2}{5} = \frac{6.9}{5} = 1.38 .$$

After a small day the followers are 0.3, 0.1, 0.2 and 0.1:

$$\frac{0.3 + 0.1 + 0.2 + 0.1}{4} = \frac{0.7}{4} = 0.175 .$$

A day following turbulence is about eight times larger than one following calm. That is Fact 2 in two lines of arithmetic.

Step 3 — now do the same for direction. Of the days that followed an up day, two were up and two down. Of those following a down day, three were up and two down. Roughly a coin flip either way — Fact 3, from the very same data that made size look so predictable.

What this means in practice

• Normal-based risk models are optimistic by design. A 99% Value at Risk read off a bell curve is breached far more often than one day in a hundred — hence stress tests on top of VaR.
• Forecast variance, not returns. Size clusters and sign does not, so the profitable modelling target is volatility. That is why GARCH exists and why vol-targeted sizing works.
• Square-root-of-time scaling is a shortcut, not a law. Daily volatility times $\sqrt{252}$ assumes independent, identically behaved days; clustering breaks the independence and fat tails break the rest.
• Backtests are shorter than they look. With turbulence bunched into episodes, ten years of daily data may hold only three or four independent stress events.

Nothing here says why markets behave this way; the facts are older and firmer than any explanation of them. Treat them as an acceptance test. Simulate from your model, then check three numbers: kurtosis, the autocorrelation of absolute returns, and the skew. If they do not look like the market you are modelling, the model has not earned the right to price your risk.