FM 5031/2 - Financial Mathematics Practitioner Sequence

Module - Risk & Asset Allocation

Instructor - John Dodson

Fall 1 - Random Variables

  • Introduction
    • Summary
  • Random Variables
  • Characterizations
  • Transformations
  • Expectation
    • Law of Large Numbers
    • Monte Carlo
  • Summary Statistics
    • Central Moments
  • Normal Distribution
    • Central Limit Theorem
  • Entropy
  • Mixtures

  • exercise

Fall 2 - Dependence

  • Independence
  • Conditioning & Margining
  • Bivariate Normal
  • Dependence
    • Copulae
    • Concordance
  • Bayes' Rule
  • Covariance
    • Cholesky Decomposition
    • Spectral Decomposition
    • Mahalanobis Distance

  • case: regression as conditional expectation
  • exercise

Fall 3 - Common Characterizations

  • Topological Taxonomy
    • Finite Support
    • Countable Support
    • Interval Support
    • Half-line Support
    • Unbounded Support
  • Common Transforms
  • Common Mixtures
  • Non-Parametric
  • Multivariate Distributions

  • exercise

Fall 4 - Estimators

  • Motivation
  • Sample
    • Sufficient Statistic
  • Estimator
    • Loss
  • Maximum Likelihood Estimator
    • Standard Error
  • Admissibility

  • exercise

Fall 5 - Heteroskedasticity

  • GARCH Specification
  • Parameter Estimation
    • Variance Targeting
    • Forecasting

  • exercise

Fall 6 - Real-World Data

  • Robustness
  • M-Estimators
  • Missing Data
  • Overlapping Data
  • Other Techniques
  • Implied Characterization

  • case: option-implied standard deviation
  • assignment

Spring 1 - Prices as Random Variables

  • Modeling the Market
    • Securities Markets
    • Market Conventions
    • Investment Horizon
  • Quest for Invariance
    • Identifying Invariants
    • Projecting Invariants
    • Mapping Invariants

  • case: scaling between the sampling period and the investment period
  • exercise

    • Spring 2 - Bayesian Estimation

    • Bayes' Rule
    • Bayesian Estimation
      • Prior
      • Pseudo-Data
    • Bayesian Estimator
    • Determining the Prior

    • case: equilibrium allocation under normal markets and exponential utility
    • exercise

    Spring 3 - Investor Objective & Satisfaction

    • Review
    • Motivation
    • Objectives
      • Delta-Gamma
    • Satisfaction
      • Properties
      • Value-at-Risk
      • Expected Shortfall
      • Expected Utility

    • case: expected shortfall example
    • exercise

    Spring 4 - Mean-Variance Optimization

    • Constraints
    • Optimization
      • Dimension Reduction
      • Analytic Solution
      • Log-Normal Model
      • Investment Horizon
      • Benchmarks

    • case: optimal portfolio example
    • exercise

    Spring 5 - Bayesian Optimization

    • Estimation Risk
    • Robust Decisions
    • Decision Strategies
      • Prior Strategy
      • Sample-based Strategy
    • Motivation
    • Von Neumann-Morenstern
    • Robust Optimization
    • Michaud Resampling
    • Black-Litterman

    • case: quadratic programming
    • exercise

    Spring 6 - Practical Topics

    • Rebalancing & Normality
    • Allocation implied prior & CAPM
    • Measuring Skewness
    • Black-Litterman & the NIW Prior
    • Michaud & the NIW
    • SeDuMi & Practical Robust Optimization

    • project