FM 5031/2 - Financial Mathematics Practioner Sequence

Module - Data Analysis, Simulation, and Portfolio Optimization

Instructor - John Dodson

Week 14

• Summary of Meucci's Program
• objective: develop a practical scheme to identify an optimal asset allocation
• some key assumptions
• fixed universe of assets
• fixed investment horizon; no re-balancing considerations
• well-defined investor objective, constraints, and index of satisfaction
• quantifiable manager expertise
• Robust Extensions
• minimax decision formulation for aversion of estimation risk
• explicit market vector parameter uncertainty set
• allocate to minimize the maximum of the opportunity cost
• self-adjusting calibration mechanism
• uncertainty set defined in terms of the bayesian location-dispersion ellipsoid
• practicable implementation

Week 13

• Allocation Models
• Michaud (re-sampled)
• Von Neumann-Morgenstern (subjective expected utility)
• predictive distribution of the market
• classical-equivalent bayesian allocation
• Black-Litterman
1. determine parameters for the distribution of the market (assumed to be normal)
2. define investor's expertise (assumed to be linear)
3. specify confidence in investor's expertise, hence conditional distribution of view given outcome (assumed to be normal)
4. obtain investor's view and apply Bayes' Theorem to determine parameters of subjective distribution of the market (normal)
• consistency and marginal consistency of an investor's view
• Lab Exercise
• Application of Michaud re-sampling to Russell sector allocations
• Assignment

Week 12

• Opportunity Cost
• portfolio allocations as decisions
• the cost of sub-optimality is a loss functional for an estimator
• evaluating allocations
• classical optimization leverages estimation error
• it is not practical to extend satisfaction theory to include risk-aversion to estimation error
• illustrative examples
• prior allocation; e.g. equal-weighting
• small sample limit
• highly efficient but biased
• sample-based allocation
• unbiased but inefficient
• Lab Exercise
• 2nd step of optimization introduced in previous week
• prospect theory certainty-equivalent satisfaction
• Arrow-Pratt approximation

Week 11

• Bayesian Estimation
• parameters as random variables
• estimates as distributions
• prior and posterior density
• updating the prior using Bayes' Theorem
• classical-equivalent estimators
• posterior mean or mode
• uncertainty ellipsoid
• conjugacy
• interpretation of the prior in terms of a random pseudo-subsample
• normal mixture
• conjugate prior for a multinormal sample with unknown mean and covariance is Normal-Inverse Wishart
• explicit formulae for parmameter updates
• Lab Exercise
• mean-variance optimization with a benchmark-relative objective
• allocating a long-only equity portfolio amongst industrial sectors
• expressing a prior with a pseudo-subsample

Week 10

• Two-step Portfolio Optimization
• dimension reduction
• maximize expected value for fixed objective variance
• dual problem
• minimize objective variance for fixed expected value
• maximize satifaction along capital market line
• Analytic Solution to Mean-Variance Optimization
• affine constraint
• two-fund separation theorem
• Pitfalls to Mean-Variance Result
• only valid locally and for short horizons
• objective is not a valid index of satisfaction
• Lab Exercise
• optimal re-balancing of a portfolio
• Assignment

Week 9

• Investor Objective
• Stochastic Dominance
• Index of Satisfaction
• properties
• money-equivalent
• estimable
• risk averse
• homogeneous
• Euler decomposition
• coherent
• super-additive
• co-monotonic
• examples
• expected value
• certainty equivalent (utility)
• quantile (value at risk)
• spectral (expected shortfall)
• Lab Exercise
• introduction to MATLAB Optimization Toolkit

Week 8

• Introduction to Asset Allocation
• overview of financial securities and indexes
• problems with the Capital Asset Pricing Model
• homogenous and static objectives of the representative agent
• single period; no interperiod risk
• ambiguity about the Market Portfolio; i.e. Fama-French
• Brief Introduction to Linear Programming
• Concepts in Classical (non-Bayesian) Estimation
• information set: r.v.'s or i.i.d. observations of an invariant
• "unknown truths" about an invariant
• unknown population density
• unknown statistic value
• estimator as a function of information
• goodness
• loss
• error
• bias (accuracy)
• inefficiency (precision)
• Sample Size Regimes
• medium sample: maximum likelihood estimation
• large sample: non-parametric estimation
• small sample: shrinkage estimation
• Maximum Likelihood Estimation
• Fisher information
• Cramér-Rao lower bound
• Non-Parametric Estimation
• empirical density function
Glivenko-Cantelli: convergence of the empirical density
• Robustness
• noisy data
• jackknife
• trimming
• maximum likelihood: introduction to M-estimation
• non-parametric: introduction to kernel regularization

Week 7

• Background on Timeseries Analysis
• Inferring Independent Increments
• AR and MA models
• not applicable for univariate asset returns
• ...but useful for timeseries with seasonal patterns
• ...or non-simultaneous multivariate observations
• Inferring I.I.D. Increments (Invariants)
• Zakoian's TGARCH
• empirical basis
• asymmetrically variable conditional volatility
• ...models spikes, persistence, and crashes
• Estimating Timeseries Model Parameters
• maximum likelihood estimator
• Lab Exercise
• fitting TGARCH to S&P 500 index returns
• fitting GED to invariants
• simulating paths
• Assignment
• selection of shorter problems

Week 6

• Monte Carlo
• integration as expectation
• probability as expectation
• sample mean as an estimator
• scaling properites of the standard error
• Variance Reduction
• motivation
• classical techniques
• anithetic sample
• stratified sample
• control variate
• importance sampling
• Brownian Bridge
• application to treatment of missing data
• path stratification
• Likelihood Ratio for Brownian Motion Paths
• application to importance sampling
• introduction to change of measure
• Lab Exercise
• demonstration of variance reduction techniques for path-dependent problems
  in the solution to the assignment

Week 5

• Modeling The Market
1. identifying i.i.d. invariants
2. fitting a distribution
3. projecting to the horizon(s)
4. mapping back into asset values
• Statistical Considerations
• serial correlation
• non-simultaneity
• time inhomogeneity
• conditional heteroskedasticity
• Projections
• time-scaling with the characteristic function
• Dimension Reduction
• latent factors
• principal component analysis
• explicit factors
• systematic / idiosyncratic factors
• Lab Exercise
• working with equity stock timeseries
• interpreting Eigen-portfolios
• large-scale normal-copula simulation
• failure of the Cholesky method
• historical simulation method

Week 4

• Multi-variate Location and Scale
• location-dispersion ellipsoid
• Chebyshev inequality
• Conditional Probability
• Bayes' rule
• iterated expectation
• normal example: ordinary least-squares regression as conditional expectation
• Dependence
• Venn diagrams
• linear correlation
• Spearman's rho
• copulas
• marginal-copula factorization
• Kendall's tau
• Lab Exercise
• demonstration and discussion of solution to assignment
• matrix math in MATLAB
• generating correlated random variates
• Cholesky decomposition

Week 3

• Maximum Likelihood Estimator
• density of a sample as a function of unknown parameters
• mode as a point estimator
• Brownian Motion
• limit of a random walk
• role of the Central Limit Theorem
• characterization
• initial value
• continuity
• density of increments
• simulating paths
• functional form of geometric brownian motion
• Random Variates
• building blocks: standard uniforms and normals
• quantile method
• rejection method
• Modern High-Level Programming
• compiled vs. interpreted
• imperative vs. functional
• sequential vs. concurrent
• scalar vs. array
• Lab Exercise
• fitting a fat-tailed distribution to equity index returns
• simulating paths
• estimating statistical properties of portfolio values
• Assignment
• optimizing a portfolio with a maximum drawdown constraint

Week 2

• Distribution of the Parameter from the previous Lab Exercise
• probability associated with winning interval estimate
• risk / reward discussion
• efficiency
• Applications of Simulation
• experimental modeling
• integration (Monte Carlo)
• Taxonomy of Classical Univariate Distributions
• diagram from Casella & Berger
• Summary Statistics (Classical and Robust)
• location
• dispersion / scale
• standardization
• Chebyshev inequality
• Transformations of Random Variables
• general increasing function
• affine transformation
• sum of i.i.d. draws
• Lab Exercise
• sample statistics of the random walk
• terminal value, excursion, range
• review of anonymous functions in MATLAB

Week 1

• Introduction
• review of Sequence topics and objectives for this module
• syllabus
• technology
• Probability Basics
• probability triple
• random variables
• Characterizations of Distributions
• density (PDF)
• distribution (CDF)
• quantile function
• characteristic function
• Statistics Basics
• samples
• sufficient statistics
• Estimation Basics
• estimators as functions of random samples
• likelihood function example
• Lab Exercise
• guess an interval estimate for the unknown parameter of a uniform random variable given a sample
• loading data into MATLAB and working with arrays
• Practice Quiz