The Distribution Of Financial Returns Made Simple

The distribution of financial returns is a fundamental concept in finance that has puzzled investors for decades. Despite its importance, the distribution of financial returns remains misunderstood by many investors. In this article, we will delve into the world of statistical distributions to explore why returns are often normally distributed and how this can impact investment decisions.

The Normal Distribution

Returns on investments have a stable distribution known as a normal distribution, also referred to as the Gaussian distribution or bell curve. This distribution is characterized by two key parameters: mean (μ) and standard deviation (σ). The mean represents the average return of an investment, while the standard deviation measures the volatility or dispersion of returns.

That said, it's essential to understand that log returns have a normal distribution. To see why, consider the following example:

Assume we have a stock with a daily return of 2%. If we take the natural logarithm (ln) of this return, we get an approximately 0.23% annualized return. This is because the natural logarithm function is monotonic increasing, meaning that as the input increases, the output also increases.

To illustrate how this works, consider a simple example:

Let's say we have a portfolio with three assets: C, BAC, and MS. The returns for each asset are as follows:

C: 2%, 3%, and 4% per year BAC: 1%, 1%, and 0% MS: 5%, 6%, and 7%

If we take the log of these returns, we get:

C: ln(0.02) ≈ -2.48%, ln(0.03) ≈ -3.04%, and ln(0.04) ≈ -4.17% BAC: ln(0.001) = -6.63% and ln(0.002) = -8.47% MS: ln(0.05) = -2.45% and ln(0.06) = -3.13%

As you can see, the log returns are approximately normally distributed. This is because we are applying a monotonic increasing function to each return.

The Normal Distribution in Action

The normal distribution has numerous implications for investment decisions. For instance:

Risk Management: By understanding that returns have a stable distribution, investors can better manage risk. A normal distribution allows for the calculation of confidence intervals and the identification of outliers. Investment Strategies: Investors can use statistical models to identify patterns in return distributions. This enables the development of more effective investment strategies, such as portfolio rebalancing and market timing. * Asset Allocation: By understanding the underlying distribution of returns, investors can create more informed asset allocation decisions. For example, a long-term investor with a normal distribution may allocate their portfolio accordingly.

Volatility Clustering

While the normal distribution is stable over time, there is one aspect that can affect its behavior: volatility clustering. This phenomenon refers to the tendency for periods of high volatility to coincide with extreme returns. To illustrate this concept:

Imagine a stock experiencing a 20% increase in value during an unusually volatile period. Without volatility clustering, we might expect the stock's return to follow a normal distribution over time.

However, if there is significant volatility clustering involved, we may see a pattern of extreme returns followed by periods of high volatility. This can lead to a skewed distribution, where the majority of data points are concentrated around the peak return.

The Jarque-Bera Test

To verify that our investment strategy follows a normal distribution, we can use the Jarque-Bera test. This statistical test is sensitive to outliers and can detect deviations from normality. If the p-value for this test is greater than 10^-8, we can conclude that our data follow a normal distribution.

For example:

Let's say we run the Jarque-Bera test on our portfolio returns. The results show that the p-value is approximately 2.5 × 10^-8, indicating that our data follows a normal distribution.

In conclusion, the distribution of financial returns has been shown to be normally distributed due to the properties of logarithmic transformations. This understanding can have significant implications for investment decisions and risk management strategies. By recognizing the patterns in return distributions, investors can make more informed decisions and potentially improve their investment performance.