The empirical distribution of daily returns

Building intuition: what do daily returns actually look like?

For SPY since inception (1993):

• Mean: ~0.04% per day (~10% annualized).
• Standard deviation: ~1.2% per day (~19% annualized).
• Skewness — a number measuring asymmetry; negative means the loss tail reaches further than the gain tail: ~ −0.2 (slightly negative).
• Kurtosis — a number measuring tail heaviness, i.e. how common extreme days are; the normal bell curve scores exactly 3: ~10–15 depending on the sample window — several multiples of normal.
• Worst day in the SPY sample: October 15, 2008, about −9.8% (the S&P 500 index itself had a −20% day on October 19, 1987 — before SPY existed).
• Best days in the SPY sample: October 13, 2008 (+11.6%) and October 28, 2008 (+10.8%) — both during the financial crisis, not during rallies. Extreme up-days cluster inside crises too.

Now compare that to a normal distribution with the same mean and standard deviation. It would predict that a single-day move beyond ±4σ — more than four standard deviations from the mean, in either direction — should happen once every 16,000 days (about once every 60 years). In reality, SPY has seen many such days in just 30 years.

Why histograms can fool you

A histogram of daily returns looks bell-shaped at a glance. The center mass fits the normal nicely. The deception happens in the tails. There are so few observations beyond 3σ (three standard deviations from the mean) that those histogram bins are almost empty either way. The eye sees "matches normal." In reality, under a true normal those bins should be many orders of magnitude emptier — factors of 10 stacked on top of each other. The eye can't tell "rare" from "insanely rare."

The right tool to see this is a Q-Q plot (quantile-quantile). A quantile is a cut point in the data: the 5% quantile of returns is the return that only 5% of days fall below. On a Q-Q plot, you compare the empirical quantiles — the ones computed from your actual data — against the theoretical quantiles a normal distribution would produce. If returns were normal, the two sets would agree, and the plot would be a straight diagonal line.

For real stock returns, the line is straight in the middle and bends sharply away from the diagonal at the extremes — both tails curve outward. This bending is the visual signature of fat tails: the data's extreme values sit further out than the normal's extreme values.

Stylized facts (the "always true" patterns)

Rama Cont (2001) catalogued the stylized facts of asset returns — properties that show up across nearly all liquid markets:

1. Absence of autocorrelation in returns. Autocorrelation — the correlation of a series with its own past values — is near zero for returns: you can't predict tomorrow's return from today's at short horizons in efficient markets.
2. Strong autocorrelation in squared/absolute returns (volatility clustering — covered in A2-02).
3. Heavy tails (always).
4. Aggregate Gaussianity — Gaussian is another name for the normal distribution. Return distributions become more normal as you look at longer aggregation periods (monthly, yearly). Daily is the most non-normal regime.
5. Asymmetry between up and down moves (negative skew).
6. Leverage effect — vol rises after price falls (downside moves trigger higher vol than upside moves of the same size).
7. Volume/volatility correlation — high volume days are usually high volatility days.

These are not surprises in any equity market we'll consider. If your model assumes returns are normal and iid — independent and identically distributed, meaning every day is a fresh draw from the same unchanging bell curve — you are missing every single one of these.

Why this matters for the curriculum

Almost every topic from this point forward connects back to these properties:

• Backtest methodology (D4) — fat tails mean a small number of extreme outcomes dominate strategy returns. Subsampling and resampling — re-testing on many different slices of the data — fix this; ignoring it doesn't.
• Risk management (D6) — Value-at-Risk (a loss threshold like "the worst loss on 95% of days") computed under normality systematically underestimates extreme losses. Expected shortfall (ES) — the average loss on the bad days beyond that threshold — and stress testing fix this.
• Portfolio theory (D3) — mean-variance optimization implicitly assumes elliptical (normal-ish) returns. Real portfolios need correction.
• Options (B4 + D8) — the implied volatility skew (options that protect against crashes trading at higher implied volatility than others) exists precisely because the market knows returns aren't normal.

Code snippet (for laptop later)

import yfinance as yf
import numpy as np
import scipy.stats as stats

spy = yf.download("SPY", start="1993-01-29")["Adj Close"]
returns = np.log(spy / spy.shift(1)).dropna()

print(f"Mean: {returns.mean():.5f}  Std: {returns.std():.5f}")
print(f"Skew: {returns.skew():.3f}  Kurt (excess): {returns.kurt():.3f}")
print(f"Worst: {returns.min():.3f}  Best: {returns.max():.3f}")

# Q-Q plot
import matplotlib.pyplot as plt
stats.probplot(returns, dist="norm", plot=plt)
plt.show()

Draw it (your generative activity)

Today's generative activity is draw: sketch (in your head or on paper) what a Q-Q plot looks like for normal-distributed data, versus for fat-tailed data. The shape of the deviation from the diagonal is informative — get this picture lodged in your memory and you'll never confuse the two again.

Try it

Measure the shape of a distribution yourself — skewness (asymmetry) and excess kurtosis (tail weight):