Exploring Returns: Beyond Normal with Generalized Error Distribution

Recently, I engaged with an intriguing article by Graham Giller, which critically examines a prevalent assumption in financial analysis: the 'null hypothesis' that daily returns of major market indices, such as S&P500, conform to a normal distribution. This exploration led me to reconsider and explore various fundamental concepts in financial statistics, particularly the potential of alternative distributions like the generalized error distribution in describing market price changes

In both theory and practice, the rate of return is a pivotal concept in finance, playing a crucial role in investment decision-making. Typically, investors' decisions to buy or sell securities are based on the probability of achieving a certain profit level or incurring a loss. In practice, it's common to assume that rates of return follow a normal distribution. This assumption is not merely theoretical; it's embedded in various financial models and decision-support tools, such as the Black-Scholes stock option valuation method, the Capital Asset Pricing Model (CAPM), and certain Value at Risk (VaR) methodologies.

What's fascinating is that moving away from this assumption puts into question the reliance on two sigma as a significant deviation from the mean, as well as the rationality of the commonly used 95% confidence level criteria in financial analysis.

My Analysis Using the Generalized Error Distribution (GED):

In my analysis, following Graham Giller's article, I decided to fit the index and stocks data to the Generalized Error Distribution (GED), which is a statistical distribution that generalizes the normal (Gaussian) distribution.

The GED offers a more flexible approach for modeling financial returns, particularly for data that exhibit 'fat tails' - a common characteristic in financial markets.

Why use GED for Financial Returns?

1. Fat Tails: Financial market returns often exhibit fat tails, meaning extreme values are more common than what would be predicted by a normal distribution. The GED can capture this characteristic more effectively,
2. Flexibility: The additional parameter ν provides flexibility in modeling the distribution of returns. It allows the GED to adapt to different levels of kurtosis (peakiness or flatness) observed in real-world data,
3. Risk Management: For risk management purposes, accurately modeling the tails of the distribution is crucial. The GED's ability to capture tail risk better than the normal distribution makes it valuable for assessing financial risk,
4. Empirical Fit: Empirical studies often find that the GED provides a better fit for financial return data compared to the normal distribution, especially during periods of market stress or volatility.

Here, the parameters s, ν, and λ are defined as follows:

• s: This parameter is related to the scale of the distribution. It affects the spread of the distribution. A higher value of s leads to a wider distribution, while a lower value of s results in a narrower distribution.

• ν (nu): Also known as the shape parameter, ν allows the distribution to adjust its tail thickness. A higher value of ν results in heavier tails, allowing the distribution to capture fat tails more effectively. When ν = 2, the GED reduces to a normal distribution, and when ν = 1, it reduces to a Laplace distribution.

• λ (lambda): This parameter is related to the rate of decay of the tails of the distribution. A higher value of λ leads to faster decay of the tails, while a lower value of λ results in slower decay, allowing the distribution to capture extreme values more effectively.

This flexibility is crucial for capturing the fat tails and varying levels of kurtosis observed in real-world financial data. For risk management, accurately modeling the tails of the distribution is vital, and the GED's ability to capture tail risk more effectively than the normal distribution is particularly valuable.

The GED application:

The easiest way to utilize the GED is through the scipy.stats package and its gennorm function, which includes just one more parameter than the Normal distribution (be careful with a slightly different naming).

Testing for Normality:

To investigate the normality of rates of return, I set up two hypotheses: the null hypothesis (conformity with a normal distribution) and the alternative hypothesis (non-conformity). To perform parametric tests for normality on stock data, you can use either the Wald test or other tests such as Shapiro-Wilk, Anderson-Darling, or Kolmogorov-Smirnov. These tests have different assumptions and methodologies. The K-S and A-D tests are used to test the goodness of fit of a distribution (e.g., testing for normality), while the Wald test is used for hypothesis testing in statistical models.

The Wald Test:

The Wald test relies on the asymptotic properties of maximum likelihood estimators. It tests whether a parameter of the population distribution is equal to a specified value. The Wald Test Statistic is a measure of how much the estimated parameter (like the mean) of your data deviates from a hypothesized value under the null hypothesis (typically the population parameter).?

A low Wald test P-Value typically indicates strong evidence against the null hypothesis. In the context of normality testing, it suggests that the data does not follow a normal distribution. A p-value close to 0 (usually below a threshold like 0.05) suggests that the observed data significantly deviate from the hypothesized distribution—in this case, a normal distribution.

The Kolmogorov-Smirnov (K-S) test:

The Kolmogorov-Smirnov (K-S) test is a non-parametric test used to determine if a sample comes from a specific distribution. In the context of testing for normality in financial returns, the K-S test can be applied to assess whether the observed data deviates significantly from a normal distribution. The test statistics measures the maximum difference between the empirical distribution function of the data and the cumulative distribution function of a normal distribution. A higher value of the test statistic indicates a greater deviation from normality. Similarly, a low p-value resulting from the K-S test indicates a rejection of the null hypothesis of normality, providing evidence that the observed data significantly deviates from a normal distribution.

The results from the analysis:

Here are the plots of S&P500 (^GSPC) and some growth stocks:

From the plots, one can see that the results seem quite convincing. Rejecting the null hypothesis is not enough; as usual, we need a suitable alternative. The GED, while not perfect, provided a much better fit for describing the data, especially during periods of market stress or volatility.

It's important to remember that it's not enough to reject a Null Hypothesis; we also have to have something to replace it with. The GED, while not perfect, clearly does a much better job of describing the data.

Therefore, it's crucial to consider significant changes in the values of the parameters describing the GED, depending on the period under analysis and its length. The analysis of changes in developing parameters of GED may be used as an element of technical analysis, within the scope of predicting changes in stock market cycles.

Observations in Market Trends:

I have read other scientific papers and observed the GED applicability for different market regimes, noting discrepancies. During bear market periods, the level of the p-value was significantly higher than in bull market periods. This observation could be crucial for future market predictions and risk assessments.

The learning and future use:

• If using GED, significant changes in values of s and λ parameters should be considered, depending on the period under analysis and its length.

• Analysis of changes in developing s and λ parameters of GED may be used as an element of technical analysis, within the scope of predicting changes in stock market cycles.

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