Multivariate Statistics

• Representations of distributions
  - analytical (pdf, cdf, quantile, cf)
  - Monte Carlo simulations
• Copula-marginal factorization
  - marginals/grades
  - pdf, cdf, simulations of copulas
  - special copulas
• Dependence/concordance statistics
  - Schweizer-Wolffmeasure
  - Kendall tau
  - Spearman rho
• Summary statistics
  - location-dispersion ellipsoid
  - principal component factorization
  - higher order statistics
• Correlation: theory, practice and pitfalls
• Multivariate distributions for the markets
  - (matrix-variate) normal
  - Student t and elliptical
  - Log-distributions
  - Wishart distribution
  - order statistics
  - mixture distributions

Multivariate Estimaton

• Estimators: definition and evaluation
  - loss, bias, inefficiency, error
  - generalized hypothesis testing
• Non-parametric estimators
  - order statistics and VaR estimator
  - sample mean/covariance: best-fitting ellipsoid
  - sample factor loadings: ordinary least squares
• Multivariate MLE: location, scatter, loadings
  - normal hypothesis: sample estimators
  - non-normal hypothesis: outlier rejection
• Multivariate shrinkage: location, scatter, loadings
  - Stein mean
  - Ledoit-Wolf covariance
• Multivariate robust: locations, scatter, loadings
  - assessing robustness: the influence function
  - M-robust estimators
  - outlier detection and high-breakdown ellipsoid
• Multivariate Bayesian: locations, scatter, loadings
  - analytically tractable examples
  - numerical techniques
• Missing observations and unbalanced panels
  - EM algorithm
  - ML marginalization

Market Modeling

• The quest for invariance
  - equities: log-returns
  - fixed-income: changes in yield to maturity
  - derivatives: changes in ATM implied volatility
• Advanced dynamics
  - Levy processes
  - ARMA, long-memory processes
  - GARCH, stochastic volatility, subordination
  - multivariate dynamics
• Projection to horizon: the FFT technique
• Pricing
  - analytical
  - second-order(gamma/convexity)
  - full Monte Carlo
• Dimension reduction
  - pincipal component analysis
  - explicit factors

Risk Management

• Dimension reduction, notable examples
  - Capital Asset Pricing Model
  - Multi-factors models
  - PCA of the swap market
• Investor's objectives
  - total return
  - benchmark allocation
  - net profits
• Global portfolio evaluation: stochastic dominance
• Summary portfolio evaluation: satisfaction
  - non-dimensional indices (Sharpe, info ratio)
  - expected utility and certainity-equivalent
  - quantiles and value at risk (VaR)
  - coherent measures and exp. shortfall (CVaR)
  - spectral measures of performance
• Volatility/VaR/CVaR/Risk decomposition
  - elliptical markets: semi-analytical
  - generic markets: Monte Carlo panel smoothing

13:45-14:00 "Meet the Quant": A conversation with Sebastián Ceria, CEO and Founder, Axioma

Portfolio Management I

• Constrained optimization: tractable problems
  - linear and quadratic programming
  - second order and semi-definite cone programming
• Mean-variance optimization
  - analytical: two-fund theorem
  - numerical: quadratic programing
  - pitfalls of the mean-variance approach
• Total return vs. benchmark allocation
• Market asymmetries: mean-CVaR optimization
• Estimation risk: allocations as decisions
  - oppurtunity cost
  - allocation decisions evaluated as a estimators
• Simple allocation techniques
  - general equilibrium/benchmark implied allocation
  - prior allocation
  - sample-based allocation

Portfolio Management II

• Bayesian allocation
  - predictive return
  - classical-equivalent
• Black-Litterman allocation
  - views on parameters
  - views on market
• Beyond Black-Litterman: non-normal markets
• Robust allocation (SOCP)
  - elliptical uncertainity sets
  - box uncertainity sets
• Robust/Bayesian allocation
• Liquidity
  - trading costs (fixed, execution, opportunity)
  - implimentation shortfall: temporary vs permanent impact
  - optimal execution of one-security trades
  - trading portfolios: the multivariate case