Outliers and Fat Tails in Financial Returns

Outliers and Normal Distribution

Normal Distribution

The normal distribution, also known as the bell-shaped curve, is a common distribution for random variables in nature. It is characterized by two parameters:

• Mean: The average value of the distribution.
• Standard Deviation: A measure of the spread or dispersion of the distribution.

In a normal distribution, values close to the mean are more probable than values far from the mean. The height of the curve reflects the probability of obtaining a particular value. For example, in a normal distribution of stock returns with a standard deviation of 3%, the probability of getting a return of 3% is relatively high, while the probability of getting a return of 9% (three standard deviations away from the mean) is very low.

Normal(mean, standard\ deviation)

Tails of the Distribution

• Right Tail: Represents high values of the random variable.
• Left Tail: Represents low values of the random variable.

The normal distribution is often used to describe various phenomena, such as human heights, IQ scores, and SAT scores. People often develop an intuition about the range of values that a random variable can take based on repeated observations. For instance, if a variable has consistently been between -5 and +5, one might intuitively assume that it is unlikely to be 15. However, financial returns often do not follow a normal distribution and exhibit outliers or fat tails.

Fat Tails and the Cauchy Distribution

Cauchy Distribution

The Cauchy distribution is an alternative distribution that is more characteristic of financial returns. It is a fat-tailed distribution, meaning that it has a higher probability of extreme values (outliers) compared to the normal distribution.

Cauchy(location, scale)

Comparison with Normal Distribution

When comparing 100 draws from a normal distribution and 100 draws from a Cauchy distribution, the key difference is the presence of outliers in the Cauchy distribution. While both distributions may appear similar for long intervals, the Cauchy distribution exhibits occasional large positive or negative values.

Implications of Fat Tails

Fat-tailed distributions can be deceptive because they may appear stable for extended periods, leading to an underestimation of risk. The central limit theorem, which states that averages of a large number of independent, identically distributed random variables are approximately normally distributed, assumes that the underlying variables do not have fat tails. Therefore, when dealing with fat-tailed data, such as stock market returns, the average may not accurately represent the true long-term average due to the presence of outliers.

Black Swan Events

The concept of black swan events, popularized by Nassim Taleb, illustrates the impact of fat tails. A black swan event is a rare, unpredictable event with significant consequences. The metaphor comes from the idea that if one has only ever seen white swans, one might conclude that black swans do not exist. However, black swans do exist, and their occurrence can have a profound impact.

Real-World Examples

Stock Market Returns

A histogram of daily stock price changes since 1928 shows that stock market returns tend to cluster around the mean, but there are also extreme values that deviate significantly from the mean. While most daily returns fall within a range of -6% to +6%, there have been instances of much larger gains and losses, such as the 12.53% gain on 10/30/1929 and the 20.477% drop on 10/19/1987.

1987 Stock Market Crash

The stock market crash of 1987 serves as a prime example of a black swan event. The 20.477% drop in one day was the largest one-day drop in the history of the U.S. stock market. According to the normal distribution with the same mean and standard deviation as the historical data, the probability of a drop greater than 20% is extremely small (3 times 10 to the power of -71), essentially zero. However, the event did occur, highlighting the limitations of relying solely on the normal distribution to assess risk.

P(drop > 20\%) = 3 * 10^{-71}