Empirical Properties of Financial Returns

Financial returns exhibit several stylized facts, or empirical properties, that distinguish them from data generated by simple statistical models. Understanding these properties is crucial for selecting appropriate econometric models. These facts relate to the distribution of returns, their time dependency, and their relationships across different assets.

1. Distribution of Asset Returns

Non-Normality: A key characteristic of financial returns is their departure from the normal distribution. While a normal distribution is often assumed for simplicity in many statistical models, real-world financial returns tend to have:
- Fat Tails (Excess Kurtosis): This means that extreme returns (both positive and negative) occur more frequently than predicted by a normal distribution. The tails of the distribution are "fatter" than those of a normal distribution. This implies a higher probability of large losses or gains.
- Skewness: Returns can exhibit skewness, meaning the distribution is not symmetric. Negative skewness (a longer left tail) is often observed, indicating a higher probability of large negative returns compared to large positive returns.
- Leptokurtosis: The distribution is more peaked than a normal distribution.

Implications of Non-Normality: The non-normality of returns has significant implications for risk management and portfolio optimization. Models that assume normality may underestimate the probability of extreme events and lead to inadequate risk assessments.

2. Time Dependency

Volatility Clustering: One of the most prominent features of financial returns is volatility clustering. This refers to the tendency for large returns (either positive or negative) to be followed by other large returns, and for small returns to be followed by other small returns. In other words, volatility tends to be persistent. This means periods of high volatility are clustered together, as are periods of low volatility.

Autocorrelation in Squared or Absolute Returns: While returns themselves often exhibit little or no autocorrelation, squared returns (or absolute returns) often show significant positive autocorrelation. This is another manifestation of volatility clustering. The autocorrelation in squared returns suggests that the magnitude of returns is predictable, even if the direction is not.

Little or No Autocorrelation in Returns: Linear autocorrelation in returns is often weak or non-existent, especially for short horizons. This suggests that it's difficult to predict future returns based on past returns alone, using simple linear models. However, this does not mean that returns are completely unpredictable, only that linear models may not capture the predictability.

3. Linear Dependency across Asset Returns

Correlation: Asset returns often exhibit correlation with each other. This means that the returns of different assets tend to move together to some extent. Correlation is a key input in portfolio diversification, as combining assets with low or negative correlations can reduce overall portfolio risk.

Beta: Beta measures the systematic risk of an asset, or its sensitivity to the overall market. It represents the expected change in an asset's return for a given change in the market return.

Factor Models: More generally, asset returns may be related through common factors, such as macroeconomic variables or industry-specific factors. Factor models, such as the Capital Asset Pricing Model (CAPM) and multifactor models, attempt to explain asset returns based on these underlying factors.

Summary

The stylized facts of financial returns – non-normality, time dependency (volatility clustering), and linear dependency across assets – are important considerations in financial econometrics. These properties motivate the use of more sophisticated models that can capture these features, such as GARCH models for volatility clustering and models based on non-normal distributions. Ignoring these properties can lead to inaccurate risk assessments, poor portfolio decisions, and unreliable model predictions.