FM 5031/2 - Financial Mathematics Practitioner Sequence

Module - Risk & Asset Allocation

Instructor - John Dodson

Fall 1 - Introduction & Random Variables

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

  • exercise: Jensen's inequality

Fall 2 - Common Characterizations

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

  • exercise: simulating and working with samples of random variates

Fall 3 - Dependence

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

  • case: regression as conditional expectation
  • exercise: non-linear dependence from a non-normal copula

Fall 4 - Market 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: graphical tests for invariants

Fall 5 - Estimators

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

  • exercise: estimating drift and volatility

Fall 6 - Heteroskedasticity

  • Specification
  • Estimation
    • Variance Targeting
    • Forecasting

  • case: comparing MLE's
  • exercise: estimating ARCH parameters

Fall 7 - Real-World Data

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

  • case: option-implied standard deviation
  • assignment: estimating a copula from de-volatized residuals

Spring 1 - Investor Objective & Satisfaction

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

  • case: expected shortfall example
  • exercise: VaR and ES for a generalized Pareto

Spring 2 - Mean-Variance Optimization

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

  • case: optimal portfolio example
  • exercise: prospect Theory optimal portfolio

Spring 3 - Bayesian Estimation

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

  • exercise: pseudo-data

Spring 4 - Allocations as Decisions

  • Estimation Risk
  • Robust Decisions
  • Decision Strategies
    • Prior Strategy
    • Sample-based Strategy

  • case: quadratic programming
  • exercise: opportunity cost

Spring 5 - Bayesian Optimization

  • Motivation
  • Von Neumann-Morenstern
  • Robust Optimization
  • Michaud Resampling
  • Black-Litterman

  • exercise: adjusting for a view

Spring 6 - Practical Topics in Optimization

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