Lecture Notes on Probability in Finance

Introduction to Probability

• Importance of probability theory in finance.
• Financial crisis context: 2007-present, compared to the Great Depression.
• Different perspectives on financial crises: Narratives vs. Probability models.

Overview of the Financial Crisis

• Historical Narrative:
  • Bubbles in stock, housing, and commodities markets.
  • Stock market collapse in 2000, recovery, then collapse again.
  • Institutional collapses:
    • Failures of companies investing in home mortgages (2007).
    • Bank run in the UK (Northern Rock).
    • US bank failures and international cooperation to prevent disaster.
    • Government bailouts.
• Probability Perspective:
  • Crisis as an accumulation of many small events rather than a few big events.
  • Importance of understanding underlying probabilities.

Key Concepts in Probability

• Core Concepts:
  • Probability, variance, covariance, regression, idiosyncratic risk, systematic risk.
• Breakdowns in Financial Theory:
  • Failure of independence.
  • Fat-tailed distributions and outliers.
• Concept of Probability:
  • Developed in the 1600s.
  • Complexity of the world and incremental shocks.

Financial Models and Statistical Concepts

• Return on Investment:
  • Defined as the capital gain + dividends.
  • Gross Return: Always positive (between 0 and infinity).
• Measures of Central Tendency:
  • Expected Value: Weighted sum of possible values of a random variable.
  • Average/Mean: Sum of x observations divided by n.
  • Geometric Mean: Used for estimating average return of investments.

Variance and Risk

• Variance: Measure of variability in returns.
  • Calculated as the average of squared deviations from the mean.
• Covariance: Measure of how two random variables move together.
  • Positive covariance indicates both move in the same direction, negative indicates opposite.
• Correlation: Scaled covariance, ranges from -1 to +1.

Independence and Risk Management

• Core Concept of Independence:
  • Independence is crucial for risk assessments and financial predictions.
• Value at Risk (VaR):
  • Measures the risk of loss in value of an asset or portfolio.
• Law of Large Numbers:
  • As the number of observations increases, the average becomes more stable and predictable.

CoVar and Breakdowns in Independence

• CoVar:
  • New concept that recognizes portfolios can co-vary more than previously thought during crises.
• Historical Examples:
  • October 30, 1929: Market rebounds after a crash.
  • October 19, 1987: Dramatic stock market drop which challenges the norm.

Outliers and Distribution Assumptions

• Normal Distribution vs. Fat-Tailed Distribution:
  • Normal distribution is often assumed in finance but doesn't capture extreme events well.
  • Fat-tailed distributions indicate a higher probability of extreme outlier events.
• Historical Data Analysis:
  • Example of stock price movements over time and the implications for financial predictions.

Conclusion

• Importance of understanding probability and its concepts in analyzing financial crises.
• Recognizing the limitations of traditional financial theories due to the breakdown of independence and the prevalence of fat-tailed distributions.