FIN7028: Advanced Financial Data Analytics

Statistics and Probability Primer

Barry Quinn CStat

2024-01-25

Introduction to Statistics in Financial Analytics

  • Statistics: Essential in decision-making and scientific inquiry.
  • Key role in financial analytics: Modeling market behaviors, risk assessment, and trend forecasting.
  • Integration with machine learning and big data in modern finance.

Statistical Modeling as an Iterative Process

  • Inspired by George Box: Statistical models are continually refined and improved.
  • Emphasizes the iterative nature of statistics in financial modeling.
  • Models are approximations of reality: Need for humility and vigilance.

The Iterative Process in Financial Analysis

  • Data Collection and Inductive Reasoning: Identifying patterns in financial data.
  • Deductive Reasoning and Model Testing: Assumptions and empirical testing.
  • Example: Understanding market trends through iterative modeling.

Model Refinement and Iteration

  • Flexibility and Parsimony: Adapting models to new information.
  • Continuous improvement based on empirical evidence.

Insights from Academic Sources

Bayesian Visualization and Workflow

  • Importance of visualization in Bayesian data analysis.
  • Role of visualization throughout the statistical workflow.

Andrew Gelman’s Perspective on Iterative Modeling

  • Continuous model refinement based on empirical evidence.
  • Embracing imperfection in models: A key to effective financial analysis.

Implications for Financial Modeling and Decision-Making

  • Financial markets’ complexity requires iterative approaches.
  • Benefits of iterative modeling: Improved responsiveness and effective communication.

Strategies for Implementing Iterative Approaches

  • Embedding iterative thinking: Cross-functional collaboration and open feedback.
  • Periodic audits and version control for continuous improvement.

George Box’s Vision in Financial Decision-Making

  • Iterative pursuit of excellence in financial modeling.
  • Emphasis on self-evaluation, reflection, and revision for sustained success.

Scalar Quantities in Finance

  • Scalars: Numerical values without direction, e.g., returns, exchange rates.
  • Real-world application: Computing annualized return of a stock.
  • R Example: Annualized Return Computation.

R Code: Annualized Return Computation

current_price <- 100
initial_price <- 80
holding_period <- 180 # Days
annualized_return <- (current_price / initial_price)^(365 / holding_period) - 1
print(annualized_return)
[1] 0.5722151

Vectors and Matrix Algebra Basics

  • Vectors: Arrays of numbers; matrices: Rectangular arrays.
  • Example: Comparing monthly returns across different assets.
  • R Example: Monthly Returns Comparison.

R Code: Monthly Returns Comparison

monthly_returns <- c(0.02, -0.01, 0.03)
asset_names <- c("Asset A", "Asset B", "Asset C")
returns_dataframe <- data.frame(Asset = asset_names, Return = monthly_returns)
print(returns_dataframe)
    Asset Return
1 Asset A   0.02
2 Asset B  -0.01
3 Asset C   0.03

Functions in Finance

  • Functions: Essential tools mapping inputs to outputs.
  • Application: Calculating compound interest.
  • R Example: Compound Interest Function.

R Code: Compound Interest Function

compound_interest <- function(principal, rate, periods) {
  return_amount <- principal * (1 + rate)^periods
  return(return_amount)
}
# Example usage
print(compound_interest(1000, 0.05, 5))
[1] 1276.282

Introduction to Probability Theory

  • Probability theory: Quantifying uncertainty and randomness.
  • Essential for hypothesis testing, parameter estimation, and prediction.

Basic Principles and Tools of Probability Theory

  • Introduction to key concepts: Sample Space, Events, Conditional Probability.
  • Foundational elements for more advanced statistical analysis.

Multiplicative Property and Chain Rule for Conditional Probability

  • Understanding joint probabilities and conditional dependencies.
  • Multiplicative Property: P(A ∩ B) = P(A) × P(B | A).
  • Chain Rule: Breaking down complex probabilities into simpler components.

Bayes’ Formula and Independence of Events

  • Bayes’ theorem: A powerful tool for updating probabilities.
  • Independence: When the occurrence of one event does not affect another.

Total Probability Theorem and Bayesian Extensions

  • Total Probability Theorem: Summing probabilities over a partition.
  • Extended Bayes’ Theorem: A cornerstone of Bayesian inference.

Probability Schools of Thought

  • Overview of Classical, Frequentist, and Bayesian methods.
  • Each school’s impact on financial analytics and decision-making.

Frequentism in Financial Time Series Econometrics

  • Emphasis on long-run frequencies and parameter estimation.
  • Frequentist methods in academic research and their applications.

Bayesian Methods in Market Analysis

  • Bayesian inference: Incorporating prior knowledge and new evidence.
  • Flexibility of Bayesian methods in adapting to new market information.

Classical Probability in Financial Analytics

  • Basics of classical probability theory.
  • Applications in risk assessment and modeling market events.

Connections Between Probability Paradigms

  • How Classical, Frequentist, and Bayesian methods interrelate.
  • Integration and overlaps in financial modeling approaches.

Impact in Financial Analytics

  • Holistic problem-solving through diverse probabilistic approaches.
  • Innovation and flexibility in financial analysis and modeling.

Conclusion

  • Recap of the significance of statistics and probability in financial analytics.
  • The importance of a multi-faceted approach to understanding financial data.
  • For more details and some sample exercises on this material see https://q-rap.connect.qub.ac.uk/AFDA/primer.html