Assuming Normality in Stock and Hedge Fund Returns: Real World Dilemmas

Assuming a normal distribution for stock or hedge fund returns is a common practice in finance, but is it realistic? This article explores the advantages and limitations of this assumption, along with alternative approaches. Understanding these nuances is crucial for accurate risk assessment and informed investment decisions.

Advantages of Normal Distribution Assumption

Simplicity: The normal distribution is mathematically tractable, making it easier to apply various statistical methods and models, such as the Black-Scholes model for options pricing. Central Limit Theorem: For large samples, the Central Limit Theorem suggests that the distribution of sample means will tend to be normal, supporting its use in certain contexts. Risk Measurement: Many risk management techniques, such as Value at Risk (VaR), often assume normality to estimate potential losses.

Limitations of Normal Distribution in Finance

The normal distribution, while convenient, has several limitations when applied to financial returns:

1. Fat Tails

Financial returns often exhibit 'fat tails', meaning there is a higher frequency of extreme positive and negative returns than would be expected under a normal distribution. This is particularly evident during periods of high market volatility or financial crises.

2. Skewness

Financial return distributions can be skewed, meaning they do not have the same probability of extreme positive and negative returns. For example, stock returns may be negatively skewed, reflecting the potential for significant losses.

3. Market Behavior

Market conditions can lead to non-normal behavior, such as the increase in correlations between assets during financial crises. These joint extreme movements are not captured by a normal distribution, leading to potential underestimation of risk.

Empirical Evidence

Numerous studies have demonstrated that stock returns often deviate significantly from normality, especially during periods of high volatility or market stress. For instance, during the 2008 financial crisis, the assumption of normal returns was severely challenged as market conditions became highly non-normal.

Alternative Approaches

Given the limitations of the normal distribution, many practitioners opt for alternative distributions or models that can better capture the characteristics of financial returns:

Log-normal Distribution: While not perfect, the log-normal distribution is often used for modeling stock prices due to its non-negativity and the logarithmic returns assumption. Student's t-Distribution: This distribution accounts for heavier tails, allowing for a better modeling of extreme events compared to the normal distribution. Generalized Pareto Distribution: This distribution is useful for modeling the tails of distributions, particularly when dealing with extreme values in financial returns.

Conclusion

While the normal distribution can be a useful approximation in some cases, it is essential to be aware of its limitations. Practitioners should consider alternative distributions or models that better capture the characteristics of financial returns, especially when making critical investment decisions or risk assessments.