FM 5031/2 - Financial Mathematics Practioner Sequence

Module - Risk & Asset Allocation

Instructor - John Dodson

Fall 0 - Preview: Classical Statistics

  • Introduction
  • Estimating an Unknown Quantity
    • with A Single Dependent Variable
    • with Several Dependent Variables

  • case: OLS

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

  • case: evaluating MLE's
  • exercise: estimating drift and volatility

Fall 6 - Heteroskedasticity

  • Specification
  • Estimation
    • Variance Targeting

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

Fall 7 - Real-World Data

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

  • exercise: estimating a copula from de-volatized residuals

Spring 1 - Investor Objective & Satisfaction

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

  • case: analytic expected shortfall
  • exercise: EVT expected shortfall

Spring 2 - Mean-Variance Optimization

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

  • case: example with Cornish-Fisher expected shortfall
  • exercise: example with Arrow-Pratt prospect utility

Spring 3 - Bayesian Estimation

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

  • case: coded example for analytic mean-variance optimization
  • exercise: pseudo-data sample size

Spring 4 - Allocations as Decisions

  • Estimation Risk
  • Bias & Efficiency
    • prior-based allocation strategy
    • sample-based allocation strategy

  • case: coded example of NIW prior
  • exercise: opportunity cost relative to a market-implied allocation

Spring 5 - Bayesian Optimization

  • Motivation
  • Von Neumann-Morenstern
  • Robust Optimization
  • Michaud Re-sampling
  • Black-Litterman

  • exercise: Black-Litterman in a broad investment universe

Spring 6 - Practical Topics in Optimization

  • Cash
  • Rebalancing & normality
  • Forecasting GARCH