Theoretical distribution

Theoretical distribution is a fundamental concept in the quantitative finance category, providing a mathematical framework to understand the likelihood of various outcomes in a random process. It describes how values of a random variable are expected to be distributed across a range of possibilities, often represented by a probability function or a curve. Unlike an empirical distribution, which is derived from observed data, a theoretical distribution is derived from mathematical assumptions and principles. This statistical tool is crucial for financial modeling, risk management, and making informed decisions under uncertainty.

History and Origin

The concept of theoretical distributions has roots in the 17th and 18th centuries with the development of probability theory. Early pioneers like Abraham de Moivre, Pierre-Simon Laplace, and Carl Friedrich Gauss made significant contributions. De Moivre first derived the mathematical formula for what is now known as the normal distribution in the 18th century, initially as an approximation for the binomial distribution when the number of trials is large.¹⁵,¹⁴,¹³,¹² Laplace later expanded on this work, proving the Central Limit Theorem, which demonstrates that the sum or average of a large number of independent random variables will tend to be normally distributed, regardless of the original distribution of the variables.¹¹ Gauss further popularized the normal distribution through his work on measurement errors in astronomy, leading to its alternative name, the Gaussian distribution.¹⁰,⁹ The University of York provides further details on the historical evolution of this foundational concept.⁸

Key Takeaways

A theoretical distribution is a mathematical model describing the probability of different outcomes for a random variable.
It is distinct from an empirical distribution, which is based on observed data.
Theoretical distributions, such as the normal distribution, are foundational to quantitative finance, risk management, and statistical inference.
Key parameters like mean and standard deviation define the shape and characteristics of these distributions.
While powerful, theoretical distributions have limitations, especially when applied to complex financial phenomena that exhibit "fat tails" or skewness.

Formula and Calculation

A theoretical distribution is defined by a probability function, which assigns a probability to each possible outcome (for discrete distributions) or a probability density to a range of outcomes (for continuous distributions).

For instance, the Probability Density Function (PDF) of a normal distribution is given by:

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

Where:

(x) is the value of the random variable.
(\mu) (mu) is the mean of the distribution, representing its central tendency.
(\sigma^2) (sigma squared) is the variance, which measures the spread of the data.
(\sigma) (sigma) is the standard deviation, the square root of the variance.
(e) is Euler's number (approximately 2.71828).
(\pi) is pi (approximately 3.14159).

This formula allows for the calculation of the probability density at any given point (x) for a normally distributed variable, defined solely by its mean and variance.

Interpreting the Theoretical distribution

Interpreting a theoretical distribution involves understanding its shape and parameters to draw conclusions about the underlying random process. For example, a narrow normal distribution (small standard deviation) suggests that outcomes are tightly clustered around the mean, implying lower risk or variability. A wider distribution indicates greater dispersion and higher variability.

Analysts assess the skewness and kurtosis of a theoretical distribution to understand its symmetry and the likelihood of extreme events (tails). A perfectly symmetrical distribution, like the normal distribution, has zero skewness. Positive skewness indicates a longer tail on the right, suggesting a higher probability of large positive outcomes, while negative skewness indicates a longer tail on the left, suggesting a higher probability of large negative outcomes. High kurtosis, or "fat tails," implies that extreme events are more likely than what a normal distribution would predict.

Hypothetical Example

Consider an investment firm attempting to model the annual returns of a hypothetical diversified portfolio. Based on historical data and market expectations, the firm assumes the portfolio's annual returns follow a normal distribution with a mean ((\mu)) of 8% and a standard deviation ((\sigma)) of 15%.

Using this theoretical distribution, the firm can make probabilistic statements:

There's a 50% chance the portfolio return will be greater than 8% (the mean).
Approximately 68% of the time, the annual return will fall within one standard deviation of the mean, i.e., between -7% ((8% - 15%)) and 23% ((8% + 15%)).
Approximately 95% of the time, the annual return will fall within two standard deviations of the mean, i.e., between -22% ((8% - 2 \times 15%)) and 38% ((8% + 2 \times 15%)).

This theoretical framework helps in estimating potential gains or losses and forms the basis for various financial modeling and risk assessment techniques.

Practical Applications

Theoretical distributions are indispensable in various areas of finance and investing:

Risk Management: They are central to calculating metrics like Value at Risk (VaR) and Conditional Value at Risk (CVaR), which quantify potential losses over a specific period and confidence level. Regulators also leverage statistical models and theoretical distributions for supervisory stress tests. For instance, the Federal Reserve utilizes quantitative models for its Comprehensive Capital Analysis and Review (CCAR) stress tests to assess the resilience of large banks to adverse economic conditions.⁷
Portfolio Management: Modern Portfolio Theory (MPT) heavily relies on the assumption of normally distributed asset returns to optimize portfolio allocation, aiming to achieve the highest expected return for a given level of risk or the lowest risk for a given expected return.
Derivative Pricing: Models like the Black-Scholes option pricing model assume that the returns of the underlying asset follow a log-normal distribution (which implies the log-returns are normally distributed), allowing for analytical solutions to option prices.
Quantitative Analysis: They underpin Monte Carlo simulation, a widely used technique for modeling complex systems, forecasting financial variables, and valuing derivatives by simulating thousands or millions of possible scenarios.
Hypothesis testing: In data analysis, theoretical distributions provide the null hypothesis against which observed data is compared to determine if deviations are statistically significant.

Limitations and Criticisms

Despite their widespread use, theoretical distributions, particularly the normal distribution, face significant limitations in finance:

"Fat Tails" and Extreme Events: Financial market returns often exhibit "fat tails" (leptokurtosis) and skewness, meaning extreme positive or negative events occur more frequently than predicted by a normal distribution.⁶,⁵,⁴ Nassim Nicholas Taleb, author of The Black Swan, heavily criticizes the over-reliance on the normal distribution in financial modeling, arguing it creates a false sense of security and underestimates the probability of rare, high-impact events.,³ Research Affiliates has also discussed whether the normal distribution adequately describes returns.
Dynamic Nature of Markets: Financial asset returns are not always static and can exhibit changing volatility and correlations over time, which fixed theoretical distributions may not capture.²
Bounded Outcomes: While the normal distribution assumes an infinite range of possible values, many financial variables (like asset prices) are bounded by zero, making a normal distribution an imperfect fit. Log-normal distributions are often used for prices to address this.
Model Risk: Over-reliance on any single theoretical distribution can lead to "model monoculture," where all firms use similar models, potentially missing idiosyncratic risks and increasing systemic vulnerability, as highlighted by discussions around Federal Reserve stress tests.¹
Simplification of Reality: Theoretical distributions are simplifications. Real-world financial data are complex, influenced by human behavior, unforeseen events, and non-linear relationships that may not conform to predetermined mathematical forms.

Theoretical distribution vs. Empirical distribution

The distinction between a theoretical distribution and an empirical distribution is crucial in statistics and finance. A theoretical distribution is a probability distribution derived from a mathematical formula or a set of assumptions about a random phenomenon. It represents how data should behave under ideal or assumed conditions. Examples include the normal distribution, binomial distribution, or Poisson distribution, each defined by specific parameters and properties. These distributions provide a generalized model for a population.

Conversely, an empirical distribution (also known as a frequency distribution) is constructed directly from a set of observed data. It represents the actual frequencies or probabilities of values occurring in a specific dataset. For instance, plotting the historical daily returns of a stock would yield an empirical distribution. While an empirical distribution reflects past observations, it may not perfectly conform to any known theoretical distribution due to randomness, measurement errors, or the unique characteristics of the observed data. Analysts often compare an empirical distribution to a theoretical one (e.g., a normal distribution) to determine if the observed data broadly align with theoretical expectations or if there are significant deviations, such as "fat tails" or skewness. This comparison helps in validating models and understanding the inherent properties of financial data.

FAQs

Q: What is the primary purpose of using a theoretical distribution in finance?
A: The primary purpose is to model and understand the underlying random processes that drive financial phenomena, such as asset price movements or investment returns. This allows for statistical inference, risk assessment, and the development of predictive models.

Q: Can real-world financial data perfectly match a theoretical distribution?
A: Rarely. While theoretical distributions provide useful approximations and frameworks, real-world financial data often exhibit characteristics like "fat tails," skewness, and non-stationarity that deviate from the assumptions of many common theoretical distributions, such as the normal distribution.

Q: Why is the normal distribution so frequently used in finance despite its known limitations?
A: The normal distribution is widely used due to its mathematical tractability and the Central Limit Theorem, which suggests that sums or averages of many random variables tend towards normality. This simplifies calculations in complex financial models. However, practitioners are increasingly aware of its limitations and often employ more sophisticated models or adjusted distributions to account for real-world complexities.

Q: How do analysts choose which theoretical distribution to use?
A: The choice depends on the nature of the data, the financial problem being addressed, and the assumptions that can reasonably be made. Analysts examine historical data analysis to understand its characteristics (e.g., symmetry, tail behavior) and select a theoretical distribution that best fits these observed properties and the context of the problem. Sometimes, multiple distributions might be considered, or hybrid models are developed.