Stylized facts of asset returns

Interactive exploration of the empirical regularities observed in financial asset returns

Financial risk models are built on stylized facts — empirical regularities observed across virtually all financial asset returns (see Christoffersen 2012, chap. 1). This page provides interactive illustrations of the most important stylized facts using daily Tesla stock returns. Understanding these facts is essential for choosing appropriate risk models: any model that ignores them will systematically misestimate risk.

Loading the data

The dataset contains daily log returns for Tesla stock. We load it and compute basic quantities that will be used throughout the illustrations.

1. Simple vs log return comparison

Risk models use log returns rather than simple (arithmetic) returns. The two are related by \(R = \ln(1 + r)\), where \(R\) is the log return and \(r\) is the simple return. For small returns, \(\ln(1+r) \approx r\), so they are nearly identical. But as returns grow larger, the approximation breaks down.

Tip

How to experiment

Move the slider to see how the gap between simple and log returns widens. At \(\pm 1\%\) the difference is negligible; at \(-30\%\) or \(+50\%\) it becomes substantial. Notice the asymmetry: the gap is larger for negative returns of the same absolute magnitude.

Key properties

Property Simple return \(r\) Log return \(R\)
Portfolio aggregation \(r_{PF} = \sum w_i r_i\) (additive across assets) Not additive across assets
Time aggregation Product of \((1+r)\) terms \(R_{t+1:t+K} = \sum_{k=1}^{K} R_{t+k}\) (additive across time)
Price non-negativity May imply negative prices if \(r < -1\) Always positive: \(S_{t+1} = e^R \cdot S_t > 0\)
Note

Convention in risk management

Log returns are preferred because: (1) multi-period returns are simple sums, making time aggregation straightforward; and (2) they automatically guarantee positive prices regardless of return magnitude.

2. Fat tails explorer

The unconditional distribution of daily returns has fat tails — extreme observations occur far more frequently than the normal distribution predicts. This is one of the most consequential stylized facts for risk management, because it means the normal distribution systematically underestimates the probability of large losses.

Tip

How to experiment

Increase the sigma threshold to focus on more extreme events. Compare the number of events the normal distribution predicts with what actually occurred. At 4–6 sigma, the discrepancy is dramatic.

3. Volatility clustering visualizer

Daily returns are nearly unpredictable (near-zero autocorrelation), but squared returns show strong positive autocorrelation. This means volatility is persistent: large moves tend to be followed by large moves, and calm periods tend to persist.

Tip

How to experiment

Adjust the maximum lag to see how far the persistence extends. Compare the ACF of returns (essentially noise) with the ACF of squared returns (strongly significant). This is the empirical foundation for models like RiskMetrics and GARCH.

4. RiskMetrics volatility model

The RiskMetrics model (JP Morgan, 1994) captures volatility clustering using a simple exponential smoothing formula:

\[ \sigma_{t+1}^2 = \lambda \, \sigma_t^2 + (1 - \lambda) \, R_t^2 \]

where \(\lambda = 0.94\) is the standard decay factor. Higher \(\lambda\) means the model relies more on past variance (smoother); lower \(\lambda\) means it reacts more strongly to new returns.

Tip

How to experiment

  1. Start with \(\lambda = 0.94\) (the RiskMetrics default) and observe how the volatility bands widen after large moves.
  2. Try \(\lambda = 0.80\) to see a much more reactive model — the bands “breathe” faster.
  3. Try \(\lambda = 0.99\) to see an almost flat, unresponsive estimate.
  4. Set \(\lambda = 1.00\) to get a constant volatility equal to the initial sample variance — this is equivalent to assuming volatility never changes, the assumption made by the normal distribution. Compare with lower values to see why time-varying volatility matters.

5. The sigma event calculator

Under the normal distribution, extreme events are astronomically rare. A “6-sigma” event should occur once every 4 million years. Yet financial markets experience such moves multiple times per decade. This calculator shows the dramatic gap between theory and reality.

Tip

How to experiment

Slide the threshold from 1σ to 10σ. Watch how the “expected waiting time” under the normal distribution grows from days to billions of years — while actual events keep occurring. The timeline below shows exactly when these extreme events happened.

6. Leverage effect

The leverage effect refers to the negative correlation between returns and subsequent volatility changes: price drops tend to increase volatility more than equally large price increases. This asymmetry is especially important for equity risk management — risk increases precisely when portfolios are losing value.

Tip

How to experiment

Adjust the decay factor λ to control how quickly the RiskMetrics volatility reacts to new returns. With the default λ = 0.94, observe how the next-day volatility forecast \(\sigma_{t+1}\) responds asymmetrically to negative vs positive returns of the same magnitude.

7. Horizon effect and the CLT

As the return horizon increases, the distribution of returns becomes closer to normal. This is a consequence of the Central Limit Theorem: multi-period returns are sums of daily returns, and sums of many random variables tend toward normality regardless of the underlying distribution.

Tip

How to experiment

Increase the horizon from 1 day to 60 or 120 days. Watch the QQ-plot straighten out and the kurtosis decline toward 3 (the normal value). At daily frequency, the tails are very fat; at quarterly frequency, the distribution is much closer to normal.

References

Christoffersen, Peter F. 2012. Elements of Financial Risk Management. 2nd ed. Academic Press.