Stylized Facts of Financial Returns

What Are Stylized Facts?

The term "stylized fact" refers to an empirical finding that is so consistent and widespread it's accepted as a general truth, even if it's a qualitative rather than a precise quantitative statement. These facts act as constraints; any realistic model of financial returns must be able to replicate them.

Let's begin with a short video that introduces this concept.

Now, let's explore the three most critical stylized facts one by one.

Fact 1: The Distribution of Returns is Not Normal

In our last lesson, the histogram of log returns may have looked bell-shaped at first glance, but it deviates from a true normal (Gaussian) distribution in crucial ways.

Financial returns exhibit two key properties that defy normality:

• Excess Kurtosis (Fat Tails): The distribution of returns is leptokurtic, meaning it's more peaked at the center and has "fatter" tails than a normal distribution. In practical terms, this means extreme events—both large gains and large losses (crashes)—occur far more frequently than a normal distribution would predict. A model based on the normal distribution would severely underestimate the risk of a market crash.
• Negative Skewness: Often, the left tail is slightly longer and fatter than the right tail. This reflects the reality that large negative returns (crashes) tend to be more frequent and severe than large positive returns (rallies).

For a deeper look at these properties, the following article provides clear visual and statistical explanations.

To see how you would perform this analysis yourself in Python, let's turn to a practical demonstration. The following video shows how to use histograms and Quantile-Quantile (Q-Q) plots to visually identify these fat tails. We'll cover Q-Q plots in more detail in the next lesson, but for now, the key is to see how the data points (blue dots) deviate from the theoretical line (red line) at the extremes, which is the signature of fat tails.

Fact 2: Low Autocorrelation in Raw Returns

Autocorrelation is the correlation of a time series with a lagged version of itself. A high positive autocorrelation at lag 1 would mean that a positive return today makes a positive return tomorrow more likely.

However, for raw asset returns, this is not the case. The autocorrelation is typically negligible for almost all time lags. This aligns with the Efficient Market Hypothesis (EMH), which posits that all available information is already reflected in prices. If returns were significantly autocorrelated, one could easily create a profitable strategy by just looking at past returns. Such an opportunity would be quickly arbitraged away.

The statistical evidence for this is seen in an Autocorrelation Function (ACF) plot.

Fact 3: Volatility Clustering

This is perhaps the most important stylized fact for quantitative trading. While returns themselves are unpredictable, their volatility (the magnitude of returns) is not.

Volatility clustering refers to the observation that "volatility is sticky": periods of large price moves (high volatility) are followed by more large price moves, and periods of small price moves (low volatility) are followed by more small price moves.

You can see this phenomenon clearly in market data. The plot below shows the price of the SPY ETF in the top panel and its measured volatility in the bottom panel. Notice the distinct clusters of high volatility, especially the massive spike in early 2020 corresponding to the COVID-19 market crash.

Since volatility isn't directly observable, we use proxies to measure it statistically. The most common proxies are squared returns ($r_t^2$) and absolute returns ($|r_t|$). By taking the square or absolute value, we focus purely on the magnitude of the return, not its direction.

If we plot the ACF of these squared or absolute returns, we see a starkly different picture than for raw returns. The autocorrelation is positive, significant, and decays slowly over many lags. This is the statistical signature of volatility clustering.

This slow decay of autocorrelation in absolute/squared returns is a sign of long-range dependence in volatility, which we can exploit to build predictive models.

Conclusion

In this lesson, you've learned about the three foundational stylized facts of financial returns. These are not just trivia; they are the guiding principles for any serious quantitative modeler.

Here are the key takeaways:

1. Non-Normality: Financial returns have "fat tails" (high kurtosis), meaning extreme events are more common than a normal distribution suggests. Models must account for this heightened risk.
2. Low Autocorrelation: Raw returns are largely uncorrelated with their past values, consistent with the Efficient Market Hypothesis. Simple models based on past returns are unlikely to be profitable.
3. Volatility Clustering: The magnitude of returns is autocorrelated. High-volatility events cluster together, as do low-volatility events. This predictability in volatility is a primary source of "alpha" (predictive edge) that quantitative strategies aim to capture.

These facts tell us that while predicting the direction of the market is incredibly difficult, predicting its level of risk or volatility is much more feasible.

In our next lesson, we will put this knowledge into practice. Building on the concept of volatility clustering, you will implement a reusable function to compute rolling volatility. This will be the first entry in your practical toolkit and our first step toward engineering features that can be used in a predictive model.