Normal distribution

What Is Normal Distribution?

Normal distribution, also known as the Gaussian distribution, is a fundamental probability distribution in statistics and quantitative finance. It describes a continuous random variable whose values tend to cluster around a central point, with observations tapering off symmetrically as they move further away from that center. When plotted, the normal distribution forms a symmetrical, bell-shaped graph, commonly referred to as a "bell curve". This distribution is characterized by its mean (average) and standard deviation, which together define its location and spread. In the context of financial analysis, the normal distribution is often assumed for various financial phenomena due to its mathematical tractability and the insights provided by the Central Limit Theorem.

History and Origin

The concept of the normal distribution has a rich history, evolving from the work of several prominent mathematicians. It was first described by French mathematician Abraham de Moivre in 1733 as an approximation to the binomial distribution for a large number of trials, particularly in the context of games of chance.²⁵,²⁴,²³,²² Later, in the early 19th century, Pierre-Simon Laplace and Carl Friedrich Gauss independently developed the concept further in their work on the theory of errors and the method of least squares, primarily for analyzing astronomical data and measurement errors.²¹,²⁰,¹⁹ Gauss's significant contributions led to the distribution often being called the "Gaussian distribution."¹⁸,¹⁷ The term "normal" itself was later popularized by British polymath Francis Galton and mathematician Karl Pearson in the late 19th century, implying a standard or typical distribution.¹⁶

Key Takeaways

• The normal distribution is a symmetrical, bell-shaped probability distribution characterized by its mean and standard deviation.
• In a perfect normal distribution, the mean, median, and mode are all equal, located at the center of the curve.
• The empirical rule (68-95-99.7 rule) states that approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
• The Central Limit Theorem states that the distribution of sample means will approximate a normal distribution, regardless of the population's original distribution, given a sufficiently large sample size.
• Despite its widespread use, particularly in financial models, the normal distribution has limitations in capturing extreme events (fat tails) often observed in financial markets.

Formula and Calculation

The probability density function (PDF) of a normal distribution is given by the formula:

$f(x | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Where:

• ( f(x) ) represents the probability density at a given value ( x ).
• ( \mu ) (mu) is the mean of the distribution, which determines the location of the peak.
• ( \sigma ) (sigma) is the standard deviation, which measures the spread or dispersion of the data. ( \sigma^2 ) is the variance.
• ( \pi ) (pi) is a mathematical constant approximately equal to 3.14159.
• ( e ) is Euler's number, a mathematical constant approximately equal to 2.71828.

This formula allows for the calculation of the relative likelihood of a given value ( x ) occurring within a normal distribution defined by specific mean and standard deviation parameters.

Interpreting the Normal Distribution

Interpreting the normal distribution involves understanding its shape and the implications of its parameters. The peak of the bell curve represents the mean, median, and mode, indicating the most probable outcome. As one moves away from the mean, the probability of observing a value decreases, with the tails of the distribution representing less likely, more extreme outcomes. The standard deviation dictates the width of the curve; a smaller standard deviation means data points are tightly clustered around the mean, while a larger standard deviation indicates a wider spread.

For example, in statistical inference, understanding how many standard deviations an observation is from the mean helps in assessing its rarity or significance. The empirical rule, also known as the 68-95-99.7 rule, provides a quick way to understand data distribution:

• Approximately 68% of data falls within one standard deviation of the mean.
• Approximately 95% of data falls within two standard deviations of the mean.
• Approximately 99.7% of data falls within three standard deviations of the mean.¹⁵,¹⁴

This rule is crucial for setting confidence intervals and performing various forms of statistical inference.

Hypothetical Example

Consider a hypothetical investment fund whose annual asset returns are assumed to be normally distributed with a mean of 8% and a standard deviation of 12%.

1. Mean Return: The fund's average annual return is 8%. This is the most likely single return.
2. One Standard Deviation: According to the empirical rule, about 68% of the time, the fund's annual return will fall between ( 8% - 12% = -4% ) and ( 8% + 12% = 20% ). This means there's a 68% chance the return will be between -4% and 20%.
3. Two Standard Deviations: Approximately 95% of the time, the annual return will be between ( 8% - (2 \times 12%) = -16% ) and ( 8% + (2 \times 12%) = 32% ). So, there's a 95% chance the return will be between -16% and 32%.
4. Three Standard Deviations: About 99.7% of the time, the annual return will fall between ( 8% - (3 \times 12%) = -28% ) and ( 8% + (3 \times 12%) = 44% ). This implies a very high probability that the return will be within this wider range.

This example illustrates how the normal distribution can be used to understand the probable range of investment outcomes based on historical data.

Practical Applications

The normal distribution finds extensive practical applications across various areas of quantitative finance, investing, and risk management.

• Option Pricing: One of the most prominent uses is in the Black-Scholes model for option pricing. This widely used model assumes that the returns of the underlying asset follow a log-normal distribution, which means the natural logarithm of the asset prices are normally distributed.,
• Portfolio Theory: In modern portfolio theory, the normal distribution is often assumed for asset returns when calculating portfolio volatility and risk-adjusted returns. This simplifies the mathematical treatment of portfolios and allows for analytical solutions in many optimization problems.
• Risk Management: Financial institutions frequently use the normal distribution to estimate Value at Risk (VaR), a measure of potential financial loss. While its limitations in capturing extreme events are acknowledged, it forms a foundational component of many financial models used for risk assessment.
• Statistical Analysis of Market Data: Beyond specific models, the normal distribution is a key tool for general statistical analysis of market data, such as analyzing stock price movements, commodity prices, or interest rate fluctuations. The Central Limit Theorem is particularly relevant here, suggesting that even if individual data points aren't normally distributed, the average of large samples will tend towards a normal distribution.¹³,¹²,¹¹ This characteristic allows for the application of normal distribution-based statistical tests and analyses to a wide array of financial data.

Limitations and Criticisms

While the normal distribution is a powerful and widely used tool, it has significant limitations, particularly in finance, leading to considerable criticism. The primary critique revolves around its inability to accurately model the "fat tails" observed in real-world financial data.

• Fat Tails: Financial asset returns often exhibit "fat tails," meaning extreme positive or negative events occur more frequently than the normal distribution would predict.¹⁰,⁹,⁸ The normal distribution assigns a very low probability to events that fall many standard deviations away from the mean, understating the likelihood of market crashes or exceptionally large gains. For example, a decline of more than three standard deviations in monthly S&P 500 returns, which should be very rare under a normal distribution, has occurred much more often historically.⁷
• Leptokurtosis: This phenomenon is captured by a statistical measure called kurtosis. Distributions with higher kurtosis than the normal distribution are described as leptokurtic and possess fatter tails.⁶ The normal distribution assumes a kurtosis of 3 (excess kurtosis of 0), while financial returns often show significantly higher kurtosis.
• Impact on Risk Models: The assumption of normality can lead to an underestimation of risk in financial models, such as Value at Risk (VaR) calculations. If extreme events are more probable than the model assumes, then actual losses could significantly exceed predicted losses during periods of market stress. The 2008 global financial crisis is often cited as an example where models assuming normal distributions failed to adequately capture the systemic risks present in the market.⁵,⁴
• Constant Volatility: Models like Black-Scholes, which rely on the normal distribution, assume constant volatility, an assumption often violated in real markets, where volatility tends to fluctuate.³,²

These criticisms highlight the need for financial professionals to consider alternative distributions and advanced risk management techniques that better capture the nuances of real-world financial markets.

Normal Distribution vs. Fat-Tailed Distribution

The normal distribution and fat-tailed distribution represent two fundamentally different ways of describing the likelihood of outcomes, particularly important in finance.

The key difference lies in the likelihood of extreme outcomes. While the normal distribution suggests that events far from the mean are exceedingly rare, a fat-tailed distribution acknowledges that these "tail events" occur with greater frequency in many real-world phenomena, including financial markets. This distinction has profound implications for how risk is assessed and managed in investing.

FAQs

What are the key characteristics of a normal distribution?

The normal distribution is defined by two parameters: its mean (( \mu )) and its standard deviation (( \sigma )). It is perfectly symmetrical around its mean, with the mean, median, and mode all coinciding at the center. The curve never touches the x-axis, theoretically extending to infinity in both directions, though the probability quickly approaches zero further from the mean.¹

Why is the normal distribution so important in finance, given its limitations?

Despite its limitations, the normal distribution remains important in finance primarily due to the Central Limit Theorem and its mathematical simplicity. The Central Limit Theorem suggests that the sum or average of a large number of independent random variables will tend towards a normal distribution, regardless of the individual distributions. This property makes it a useful approximation for various financial aggregates and allows for the development of tractable financial models, even if they are imperfect.

Does the normal distribution predict stock market crashes?

No, the normal distribution does not accurately predict stock market crashes. Its symmetrical nature and thin tails mean it assigns extremely low probabilities to the very large, sudden movements (both up and down) that characterize market crashes. These events are better described by fat-tailed distributions, which account for a higher likelihood of extreme outcomes.