Glossary

13.4 Statistical Distributions

3 min read•june 18, 2024

and probability distributions are crucial tools in finance for analyzing data and making informed decisions. They help organize large datasets, identify patterns, and calculate probabilities of various financial outcomes.

Normal, exponential, and other distributions play key roles in modeling financial variables like stock returns and asset prices. Understanding these distributions allows finance professionals to estimate risks, calculate probabilities, and make predictions about market behavior.

Frequency Distributions and Probability Distributions in Finance

Frequency distributions for financial data

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  • Frequency distributions organize and summarize large financial datasets by dividing the data into classes or intervals (stock prices, investment returns) and counting the frequency of observations in each class
  • Constructing frequency distributions involves determining the number of classes, calculating the class width, setting the class boundaries (0-10%, 10-20%, 20-30%), and counting the frequency of observations in each class
  • Analyzing frequency distributions helps identify the shape of the distribution (symmetric, , or ), determine the central tendency (mean, median, and mode), and calculate the dispersion (range, variance, and )
    • measures the "tailedness" of the distribution, indicating the presence of extreme values
  • Applications in finance include analyzing stock price movements (Apple, Amazon), examining the distribution of returns on investments (mutual funds, ETFs), and assessing the risk and volatility of financial assets (bonds, derivatives)

Normal distributions in finance probabilities

  • is a symmetric, bell-shaped curve characterized by its mean (μ\mu) and (σ\sigma) used to model many financial variables (stock returns, portfolio returns)
  • 68-95-99.7 rule states that 68% of data falls within 1σ\sigma, 95% within 2σ\sigma, and 99.7% within 3σ\sigma of the mean in a
  • Standard normal distribution is a normal distribution with μ=0\mu=0 and σ=1\sigma=1
    • measures the number of standard deviations an observation is from the mean calculated as Z=X−μσZ = \frac{X - \mu}{\sigma}
  • Calculating probabilities using normal distributions involves converting data to z-scores and using standard normal distribution tables or software to find probabilities (Excel, R)
  • Applications in finance include estimating the probability of stock price movements (Microsoft, Google), calculating for investment portfolios (hedge funds, pension funds), and determining the likelihood of exceeding or falling below certain return thresholds (beating the market, underperforming benchmarks)
  • The states that the sampling distribution of the mean approaches a normal distribution as the sample size increases, regardless of the underlying population distribution

Exponential distributions for financial modeling

  • models the time between events in a characterized by a single parameter, the rate (λ\lambda), and has the where the probability of an event occurring is independent of the time since the last event
  • (PDF) of the is f(x)=λe−λxf(x) = \lambda e^{-\lambda x} for x≥0x \geq 0
  • (CDF) of the exponential distribution is F(x)=1−e−λxF(x) = 1 - e^{-\lambda x} for x≥0x \geq 0
  • Calculating probabilities using exponential distributions involves using the PDF or CDF to find the probability of an event occurring within a specific time interval (next trade within 5 minutes, default within 1 year)
  • Applications in finance include modeling the time between stock trades (high-frequency trading), estimating the probability of default for credit risk management (corporate bonds, loans), and analyzing the inter-arrival times of financial transactions or events (customer deposits, insurance claims)

Additional Distributions in Finance

  • is used to model asset prices and returns, as it assumes that logarithmic returns are normally distributed
  • is applied in hypothesis testing and constructing confidence intervals for variance estimates in financial analysis
  • is employed when working with small sample sizes or when the population standard deviation is unknown, often used in analyzing stock returns and other financial data

Key Terms to Review (22)

Bimodal: Bimodal refers to a statistical distribution or data set that has two distinct peaks or modes, indicating the presence of two separate subgroups or populations within the overall distribution. This characteristic is often observed in various fields, including finance, biology, and social sciences.
Central Limit Theorem: The central limit theorem is a fundamental concept in probability and statistics that states that as the sample size increases, the sampling distribution of the sample mean will approach a normal distribution, regardless of the underlying distribution of the population. This theorem is crucial in understanding the behavior of sample statistics and making inferences about population parameters.
Chi-Square Distribution: The chi-square distribution is a probability distribution used in statistical hypothesis testing to determine the likelihood of observing a particular set of data given a specific null hypothesis. It is a continuous probability distribution that is derived from the sum of the squares of independent standard normal random variables.
Cumulative Distribution Function: The cumulative distribution function (CDF) is a statistical function that describes the probability that a random variable takes a value less than or equal to a given value. It is a fundamental concept in probability theory and is closely related to the concepts of statistical distributions and probability distributions.
Exponential distribution: Exponential distribution is a continuous probability distribution used to model the time between events in a Poisson process. It is characterized by its constant hazard rate, meaning the event rate is consistent over time.
Exponential Distribution: The exponential distribution is a continuous probability distribution that models the time between independent events occurring at a constant average rate. It is commonly used to describe the waiting time between Poisson distributed events.
Frequency distribution: A frequency distribution is a tabular or graphical representation of data that shows the number of observations within specified intervals. It helps in understanding the distribution and patterns in a dataset.
Frequency Distributions: Frequency distributions are a statistical tool used to organize and summarize data by grouping observations into meaningful categories or bins based on their frequency or count. This technique allows for a clear visualization and understanding of the underlying patterns and characteristics of a dataset.
Kurtosis: Kurtosis is a statistical measure that describes the shape of a probability distribution. It quantifies the peakedness or flatness of a distribution relative to a normal distribution. Kurtosis provides information about the tails of a distribution, indicating whether the tails contain more or less data than expected for a normal distribution.
Lognormal Distribution: The lognormal distribution is a continuous probability distribution where the logarithm of the random variable follows a normal distribution. This means that the random variable itself is not normally distributed, but its logarithm is. The lognormal distribution is commonly used to model variables that are the product of many independent random variables, such as the size of particles, the concentration of pollutants, or the incomes of individuals.
Memoryless Property: The memoryless property, also known as the Markov property, is a characteristic of certain probability distributions and stochastic processes. It states that the future state of a system depends only on its current state and not on its past states or history.
Normal distribution: A normal distribution is a bell-shaped curve where most of the data points cluster around the mean, and probabilities for values taper off symmetrically towards both extremes. It is characterized by its mean and standard deviation.
Normal Distribution: The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetrical and bell-shaped. It is one of the most important and widely used probability distributions in statistics, with applications across various fields.
Poisson Process: A Poisson process is a statistical model that describes the occurrence of independent events over time or space. It is commonly used to analyze and predict the frequency of rare or random events, such as the number of customers arriving at a store or the number of radioactive particles emitted from a source.
Probability Density Function: The probability density function (PDF) is a mathematical function that describes the relative likelihood of a continuous random variable taking on a particular value. It provides a complete description of the probability distribution of a continuous random variable.
Skewed: Skewness is a measure of the asymmetry or lack of symmetry in the distribution of a dataset. A skewed distribution indicates that the data is not evenly distributed around the central tendency, with the bulk of the values concentrated on one side of the mean or median.
Standard deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. It is used to assess the risk and volatility of an investment's returns in finance.
Standard Deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of data values around the mean or average. It provides a way to understand how spread out a group of numbers is from the central tendency.
T-distribution: The t-distribution, also known as the Student's t-distribution, is a probability distribution used in statistical inference when the sample size is small, and the population standard deviation is unknown. It is a symmetric, bell-shaped curve that is similar to the normal distribution but has heavier tails, allowing for greater variability in the data.
Value at Risk (VaR): Value at Risk (VaR) is a statistical measure that quantifies the level of financial risk within a firm or investment portfolio over a specific time horizon. It estimates the maximum potential loss that could be incurred on an investment, given a certain probability, under normal market conditions. VaR is a key concept in the context of statistical distributions, probability distributions, and commodity price risk, as it provides a standardized way to measure and manage these types of risks.
Z-score: A z-score measures how many standard deviations a data point is from the mean. It helps determine the position of a value within a distribution.
Z-Score: A z-score is a standardized measure that expresses a data point's relationship to the mean of a dataset in terms of standard deviations. It is a fundamental concept in statistics that provides insight into the position and relative standing of a value within a distribution.
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