Lecture on Probability Trees and Conditional Expectations

Introduction

• Instructor: Jim
• Course Level: Level 1 CFA Program
• Topic: Quantitative Methods
• Learning Module: Probability Trees and Conditional Expectations

Key Concepts Discussed Previously

• Standard Deviation, Skewness, and Kurtosis: Measures of dispersion and risk.

Main Topics

Investment Decisions

• Uncertainty: Investment decisions are made under uncertainty.
• Understanding dispersion with the addition of future probability estimates.

Expected Value, Variance, and Standard Deviation with Probabilities

1. Expected Value (Mean):

   • Definition: The probability-weighted average of possible outcomes of a random variable (e.g., stock returns, dividend yields).
   • Calculation: Sum of each possible outcome weighted by its probability.

2. Variance and Standard Deviation:

   • Variance: Measures dispersion by averaging squared differences from the mean, weighted by probabilities.
   • Standard Deviation: Square root of variance; measures risk. Higher values signify greater risk.

Probability Trees

1. Decision Tree Fundamentals:

   • Start at time period zero and move through different branches with assigned probabilities.
   • Used for valuing options and bonds with embedded options.
   • Example: Tossing a coin with heads or tails showcases probability calculation.

2. Applications:

   • Stock price predictions (e.g., Proctor & Gamble): Multiple scenarios associated with changing probabilities.
   • Real-world events influencing probabilities (e.g., new product lines, economic changes).

3. Complex Examples:

   • Understanding risk through various economic scenarios affecting bond recovery rates.
   • Detailed calculation using the decision tree for defaulted bond issues.

Conditional Expectations

1. Definition and Importance:

   • Expected value of investment given specific real-world events.
   • Example: Recovering principal for defaulted bonds in different scenarios with assigned probabilities.

2. Application of Bayes' Formula:

   • Updating probabilities with new information to refine expected values.
   • Real-world example: Estimating probabilities of an actor being cast as James Bond considering new events.
   • Probability calculations using prior and posterior probabilities with Bayes' formula.

Example Problems

• Examples in the Learning Module: Step-by-step calculations for expected values and variances under different scenarios.
• Exam Tips: Understanding of probability trees, conditional expectations, and their applications in various financial contexts.
• Practice Problems: End-of-module problems focused on mean and standard deviation calculations.

Conclusion

• Wrap-Up: Embrace and understand quantitative methods as essential tools in finance.
• Next Steps: Work through the learning module examples and practice problems.
• Final Encouragement: Stay engaged and keep practicing to apply quantitative methods effectively in future CFA levels.

Thank you for watching and good luck with your studies!