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0 . Executive summary
Executive summary
1 . Risk drivers identification
Risk drivers identification
View in Lab:
Risk drivers identification
Equities
View in Lab:
Equities
Currencies
View in Lab:
Currencies
Exchange rates
Contracts
Fixed-income
View in Lab:
Fixed-income
Zero-coupon bond
Rolling value
Yield to maturity
Alternative representations
Parsimonious representations
Spreads
Derivatives
View in Lab:
Derivatives
Call option
Rolling value
Implied volatility
Alternative representations
Parsimonious representations
Pure volatility products
Credit
View in Lab:
Credit
Obligor-level risk drivers
Aggregate risk drivers
Structural models
Merton’s structural credit model
Insurance
View in Lab:
Insurance
High frequency
View in Lab:
High frequency
Market microstructure
Activity time
Time-changed variables
Strategies
View in Lab:
Strategies
2 . Quest for invariance
Quest for invariance
View in Lab:
Quest for invariance
Efficiency: random walk
View in Lab:
Efficiency: random walk
Continuous-state random walk
Discrete-state random walk
Flexible combinations
Mean-reversion (continuous state): ARMA
View in Lab:
Mean-reversion (continuous state): ARMA
AR(1) process
AR(p) process
MA(q) process
ARMA(p,q) process
ARIMA(p,d,q) process
Mean-reversion (discrete state): Markov chains
View in Lab:
Mean-reversion (discrete state): Markov chains
Time-homogeneous Markov chains
Time-inhomogeneous Markov chains
Invariant and next-step function
Applications to credit risk
Connection with structural model
Long memory: fractional integration
View in Lab:
Long memory: fractional integration
Volatility clustering
View in Lab:
Volatility clustering
GARCH
Extensions of GARCH
Stochastic volatility
Multivariate quest
View in Lab:
Multivariate quest
Vector autoregression
Alternative models
Cointegration
View in Lab:
Cointegration
Modeling
Detection
Fit
Relationships among processes
View in Lab:
Relationships among processes
Inversion of first-degree lag polynomial
ARMA(p,q) processes as products of lag polynomials
Relationships between ARMA, MA and AR
AR(p) as VAR(1)
Univariate processes as VAR(1)
Relationships between VARMA, VMA and VAR
VAR(p) as VAR(1)
Multivariate processes as VAR(1)
Stationary as MA(∞): Wold representation
Toward machine learning
View in Lab:
Toward machine learning
Exogenous quest
Hidden quest
3 . Estimation
Estimation
View in Lab:
Estimation
Setting the flexible probabilities
View in Lab:
Setting the flexible probabilities
Exponential decay and time conditioning
Kernels and state conditioning
Joint state and time conditioning
Statistical power of flexible probabilities
Historical
View in Lab:
Historical
From historical distribution to flexible probabilities
Location-dispersion: HFP ellipsoid
Kernel estimation with flexible probabilities
Maximum likelihood
View in Lab:
Maximum likelihood
From maximum likelihood to flexible probabilities
Exponential family invariants
Location-dispersion: normal MLFP ellipsoid
Location-dispersion: t MLFP ellipsoid
Bayesian
View in Lab:
Bayesian
Exponential family invariants
Normal invariants
Shrinkage
View in Lab:
Shrinkage
Mean shrinkage: James-Stein
Covariance shrinkage: Ledoit-Wolf
Correlation shrinkage: random matrix theory
Covariance shrinkage: sparse eigenvector rotations
Covariance shrinkage: glasso
Generalized method of moments
View in Lab:
Generalized method of moments
Method of moments
Generalized method of moments - exact specification
Generalized method of moments - over specification
Robustness
View in Lab:
Robustness
Local robustness
Global robustness
Missing data
View in Lab:
Missing data
Randomly missing data
Times series of different length
Missing series
(Dynamic) copula-marginal
View in Lab:
(Dynamic) copula-marginal
Static copula
Dynamic copula
Points of interest, pitfalls, practical tips
View in Lab:
Points of interest, pitfalls, practical tips
Unconditional estimation
Standardization
Non-synchronous data
High-frequency volatility/correlation
Outlier detection
Exponential moving moments and statistics
Backward/forward exponential decay
Combining estimation techniques
4 . Projection
Projection
View in Lab:
Projection
One-step historical projection
View in Lab:
One-step historical projection
Analytical projection
View in Lab:
Analytical
Univariate analytical projection
Multivariate analytical projection
Monte Carlo projection
View in Lab:
Monte Carlo
Direct Monte Carlo
Copula-marginal
General-step Monte Carlo
Historical projection
View in Lab:
Historical
Historical bootstrapping
Hybrid Monte Carlo-historical
Application: multivariate Markov chains
View in Lab:
Application: multivariate Markov chains
Univariate Markov chain
Connection with structural credit model
Multivariate Markov chains
Connection with structural credit model
Square-root rule and generalizations
View in Lab:
Square-root rule and generalizations
Thin-tailed random walk
Thick-tailed random walk
Multivariate random walk
General processes
5 . Pricing at the horizon
Pricing at the horizon
View in Lab:
Pricing at the horizon
Exact repricing
View in Lab:
Exact repricing
Equities
Currencies
Fixed-income
Derivatives
Credit
High frequency
Strategies
Carry
View in Lab:
Carry
Equities
Currencies
Fixed-income
Derivatives
Other asset classes
Taylor approximations
View in Lab:
Taylor approximations
Equities
Fixed-income
Derivatives
Other asset classes
6 . Aggregation
Aggregation
View in Lab:
Aggregation
Stock variables
View in Lab:
Stock variables
Portfolio
Value
Exposure
Leverage
Static market/credit risk
View in Lab:
Static market/credit risk
Standardized holdings and weights
Scenario-probability distribution
Elliptical distribution
Quadratic-normal distribution
Enterprise risk management
View in Lab:
Enterprise risk management
Portfolio: balance sheet
Performance: income statement
Operational risk
Banking simplified regulatory framework
Insurance simplified framework
Points of interest and pitfalls
View in Lab:
Points of interest and pitfalls
Solvency and collateral
Credit value adjustment
CreditRisk+ approximation
7 . Ex-ante evaluation
Ex-ante evaluation
View in Lab:
Ex-ante evaluation
Stochastic dominance
View in Lab:
Stochastic dominance
Satisfaction/risk measures
View in Lab:
Satisfaction/risk measures
Mean-variance trade-off
View in Lab:
Mean-variance trade-off
Mean
Variance
Standard deviation
Mean-variance trade-off
A strange success story
Expected utility and certainty-equivalent
View in Lab:
Expected utility and certainty-equivalent
Common utility functions
Computation
Quantile (value at risk)
View in Lab:
Quantile (value at risk)
Definition
Computation
Spectral satisfaction measures/Distortion expectations
View in Lab:
Spectral satisfaction measures/Distortion expectations
Definition
Common spectral/distortion measures
Computation
Coherent satisfaction measures
View in Lab:
Coherent satisfaction measures
Definition
Common coherent measures
Computation
Non-dimensional ratios
View in Lab:
Non-dimensional ratios
Information ratio
Downside ratios
Additional measures
View in Lab:
Additional measures
Beta
Correlation
Buhlmann expectation
Esscher expectation
Pitfalls, points of interest and practical tips
View in Lab:
Pitfalls, points of interest and practical tips
The Arrow-Pratt approximation of the certainty-equivalent
Utility versus quantile
Utility versus spectrum functions
Spectral/distortion versus coherent measures
The Buhlmann and Esscher expectations are not distortion expectations
Satisfaction measures under normality
8a . Ex-ante attribution: performance
Ex-ante attribution: performance
View in Lab:
Ex-ante attribution: performance
Bottom-up exposures
View in Lab:
Bottom-up exposures
Pricing factors
Style factors/smart beta
Top-down exposures: factors on demand
View in Lab:
Top-down exposures: factors on demand
Analytical computation
Cardinality constraints
Joint distribution
View in Lab:
Joint distribution
Elliptical distribution
Scenario-probability distribution
Applications
View in Lab:
Application: hedging
8b . Ex-ante attribution: risk
Ex-ante attribution: risk
View in Lab:
Ex-ante attribution: risk
Risk attribution/risk budgeting: general criteria
View in Lab:
General criteria
Isolated/“first in”proportional attribution
“Last in”proportional attribution
Sequential attribution
Shapley attribution
Homogenous measures and Euler decomposition
View in Lab:
Homogenous measures and Euler decomposition
Standard deviation
Variance
Certainty-equivalent
Distortion/spectral measures
Economic capital
A compromise: two-step mean-variance
View in Lab:
A compromise: two-step mean-variance
Analytical solutions of the mean-variance problem
View in Lab:
Analytical solutions of the mean-variance problem
9b . Construction: cross-sectional strategies
Construction: cross-sectional strategies
View in Lab:
Construction: cross-sectional strategies
Simplistic portfolio construction
View in Lab:
Simplistic portfolio construction
Backtesting
Advanced portfolio construction
View in Lab:
Advanced portfolio construction
Signal-induced factor
Flexible factor
Backtesting
Relationship with FLAM and APT
View in Lab:
Relationship with FLAM and APT
Signal-induced moments of the P&L
Fundamental law of active management
APT assumption
Multiple portfolios
View in Lab:
Multiple portfolios
Characteristic portfolios
Signal-induced factors
Flexible factors
Relationship with FLAM and APT
Points of interest, pitfalls, practical tips
View in Lab:
Points of interest, pitfalls, practical tips
Machine learning
Generalized fundamental law of active management
9c . Construction: time series strategies
Construction: time series strategies
View in Lab:
Construction: time series strategies
The market
View in Lab:
The market
Risky investment
Low-risk investment
Strategies
Expected utility maximization
View in Lab:
Expected utility maximization
The objective
Optimization
Option-based portfolio insurance
View in Lab:
Option based portfolio insurance
Payoff design
Partial differential equation
Budget
Policy
A unified approach
Rolling horizon heuristics
View in Lab:
Rolling horizon heuristics
Constant proportion portfolio insurance
Drawdown control
Signal induced strategy
View in Lab:
Signal induced strategy
10 . Execution
Execution
View in Lab:
Execution
Market impact modeling
View in Lab:
Market impact modeling
Liquidity curve
Exogenous impact
Endogenous impact
Order scheduling
View in Lab:
Order scheduling
Trading P&L decomposition
Model P&L
Moments of model P&L
Model P&L optimization
Quasi-optimal P&L distribution
Order placement
View in Lab:
Order placement
Step 1: order scheduling
Step 2: order placement
11 . Executive summary
Executive summary
Overview
View in Lab:
Overview
Dominant-residual models
Systematic-idiosyncratic models
Regression LFM's
View in Lab:
Regression LFM’s
Definition
Solution: factor loadings
Recovery and fit
Residuals features
Natural scatter specification
Equivalent linear formulation for regression
Regularized regression
Principal component LFM's
View in Lab:
Principal component LFM’s
Definition
Principal component analysis
Solution: factor loadings and factor-construction matrix
Recovery and fit
Residuals features
Natural scatter specification
Identification issues
Regularized principal component LFM’s
Pitfall: principal factors are not principal components
Systematic-idiosyncratic LFM's
View in Lab:
Systematic-idiosyncratic LFM’s
Definition
Implications on parameters
Implications on covariance
Principal axis factorization
Loadings unearthing and factor approximation
Recovery and fit
Identification issues
Cross-sectional LFM's
View in Lab:
Cross-sectional LFM’s
Definition
Solution: factor-construction matrix
Recovery and fit
Residuals features
Natural scatter specification
Regularized cross-sectional models
Systematic-idiosycratic assumption
Overview
View in Lab:
Overview
Time series models
Data fit
Regression LFM's
View in Lab:
Regression LFM’s
Principal component LFM's
View in Lab:
Principal component LFM’s
Systematic-idiosyncratic LFM's
View in Lab:
Systematic-idiosyncratic LFM’s
Cross-sectional LFM's
View in Lab:
Cross-sectional LFM’s
Factor construction
Problems with pure cross-sectional estimation
Hybrid cross-sectional estimation
Truncation
View in Lab:
Truncation
Dynamic regression
View in Lab:
Wiener-Kolmogorov filtering
The solution in population
The solution in sample
Dynamic principal component
View in Lab:
Dynamic principal component
The solution in population
Computational issue
The solution in sample
Linear state space model
View in Lab:
Linear state space models
Model
Static case
Dynamic case
Application: principal component analysis of the swap market
View in Lab:
Finite set of times to maturity
Maximum likelihood
View in Lab:
Maximum likelihood
The model
Normal assumption
Student t assumption
Bayesian
View in Lab:
Bayesian
Normal conditional likelihood
Normal-inverse-Wishart prior distribution
Normal-inverse-Wishart posterior distribution
Student t predictive distribution
Classical equivalent
Uncertainty
Regularization
View in Lab:
Regularization
Step-wise regression selection
Lasso regression
Ridge regression
Foundations
View in Lab:
Valuation foundations
Instruments
Value
Cash-flows
Re-invested cash-flows
Cash-flow adjusted value
Profit-and-loss (P&L)
Payoff
Points of interest and pitfalls
View in Lab:
Points of interest and pitfalls
Value versus price
Multi-currency conversions
Actual versus simple P&L
Fundamental axioms
View in Lab:
Fundamental axioms
Law of one price
Linearity
No arbitrage
Stochastic discount factor
View in Lab:
Stochastic discount factor
Identification
Misidentification
Fundamental theorem of asset pricing
View in Lab:
Fundamental theorem of asset pricing
Risk-neutral pricing
View in Lab:
Risk-neutral pricing
General case
No rebalancing limit: forward measure
Continuous rebalancing limit
Capital asset pricing model framework
View in Lab:
Capital asset pricing model framework
Maximum Sharpe ratio portfolio
Security market line
Alternative derivation: linear factor model for stochastic discount factor
Covariance principle
View in Lab:
Covariance principle
Completeness
View in Lab:
Completeness
Definition
Pricing
Arrow-Debreu securities
Stochastic discount factor
Equilibrium: pure capital asset pricing model
View in Lab:
Equilibrium: pure capital asset pricing model
Arbitrage pricing theory
View in Lab:
Arbitrage pricing theory
Standard derivation: linear factor model for instruments
Alternative derivation: linear factor model for stochastic discount factor
Intertemporal consistency
View in Lab:
Intertemporal consistency
Continuous time variables
Martingales
Heuristic for stochastic discount factor time consistency
Heuristic for numeraire martingale
Fundamental axioms
View in Lab:
Fundamental axioms
Law of one price
Non-linearity
Arbitrage
Valuation as evaluation
View in Lab:
Valuation as evaluation
Variance and other shift principles
Certainty-equivalent principle
Distortion principles
Esscher principle
Fixed income
View in Lab:
Fixed-income
Vasicek
Other models
Valuation recipe
Holding P&L of a portfolio
View in Lab:
Holding P&L of a portfolio
Returns
View in Lab:
Returns
Basic definitions
Standard linear returns and weights
Generalized linear returns
Generalized weights and aggregation
Investments with capital injection
Log-returns
Excess performance
View in Lab:
Excess performance
Benchmark
Excess return
Representations of a distribution
View in Lab:
Representations of a distribution
Univariate distributions
Multivariate distributions
Marginalization
View in Lab:
Marginalization
Elliptical distributions
View in Lab:
Elliptical distributions
Fundamental concepts
Stochastic representations
Moments and dependence
Affine equivariance
Notable elliptical distributions
Generation of elliptical scenarios
Scenario generation with dimension reduction
Scenario-probability distributions
View in Lab:
Scenario-probability distributions
Types of scenario-probability distributions
Probability density function
Transformations and generalized expectations
Cumulative distribution function
Quantile
Smooth quantile
Moments and other statistical features
Categorical variables
Exponential family distributions
View in Lab:
Exponential family distributions
Normal distribution
Scenario-probability distribution
Other special classes of distributions
View in Lab:
Other special classes of distributions
Stable distributions
Infinitely divisible distributions
Notable specific distributions
View in Lab:
Notable specific distributions
Uniform distribution
Quadratic-normal distribution
Wishart distribution
Maximum likelihood
View in Lab:
Maximum likelihood
General theory
Hidden variables
Relevant cases
Numerical methods
Bayesian statistics
View in Lab:
Bayesian statistics
General theory
Bayesian estimation
Bayesian prediction
Analytical results
Numerical methods
Probabilistic prediction assessment for invariants
View in Lab:
Probabilistic prediction assessment for invariants
Estimators as decisions
Frequentist approach
Bayesian approach
Analytical results
Monte Carlo simulations
Cross-validation
Minimum relative entropy
View in Lab:
Minimum relative entropy
Base distribution and view variables
Point views
Distributional views
Partial views
Partial views on generalized expectations
Sanity check
Confidence
Relationship with Bayesian updating
Analytical implementation
View in Lab:
Analytical implementation
Base distribution
Views
Updated distribution
Confidence
Relevant special cases
Prior predictive distribution
View in Lab:
Equilibrium prior distribution
Prior distribution of expected returns
Prior predictive distribution
Active views
View in Lab:
Active views
Setting the parameters
Sanity check
Posterior predictive distribution
View in Lab:
Posterior distribution
Posterior distribution of expected returns
Posterior predictive distribution
Limit cases
View in Lab:
Limit cases
High confidence in prior
Low confidence in views
Full confidence in views
Generalizations
View in Lab:
Generalizations
From linear returns to risk drivers
From stock-like to generic asset classes
From normal to non-normal markets
From linear equality views to partial flexible views
The r-squared
View in Lab:
The r-squared
Definition
Relation to distance
Distributions geometry
View in Lab:
Distributions geometry
Fisher metric: length and volume
Flatness and geodesics
Duality: potentials and Legendre transformations
Distance and divergence
40 . Decision theory with model uncertainty
Decision theory with model uncertainty
View in Lab:
Decision theory with model uncertainty
Foundations of decision theory
View in Lab:
Foundations of decision theory
Fundamental concepts
Frequentist approach
Bayesian approach
Model risk, estimation risk
Univariate results
View in Lab:
Univariate results
Definition and properties of copulas
View in Lab:
Definition and properties of copulas
Grades
Copula
Sklar’s theorem
Copula invariance
Special classes of copulas
View in Lab:
Special classes of copulas
Elliptical copulas
Archimedean copulas
Copula-marginal separation
View in Lab:
Copula-marginal separation
Copula-marginal combination
View in Lab:
Copula-marginal combination
Univariate location-dispersion
View in Lab:
Univariate location-dispersion
Z-score and affine equivariance
Taxonomy of location-dispersion features
Location and dispersion as variational problems
Multivariate location-dispersion
View in Lab:
Multivariate location-dispersion
Tentative visualizations in low dimension
Location-dispersion ellipsoid
Affine equivariance
Taxonomy of multivariate location-dispersion features
Expectation and covariance
View in Lab:
Expectation and covariance
Definitions
Generalized affine equivariance
Connections with calculus
Connections with probability
44 . Invariance tests
Invariance tests
View in Lab:
Invariance tests
Simple tests
View in Lab:
Simple tests
Efficiency: Lévy processes
View in Lab:
Efficiency: Lévy processes
Infinite divisibility
Continuous state: Brownian diffusion
Discrete state: Poisson jumps
Lévy-Khintchine representation
Subordination
Mean-reversion (continuous state)
View in Lab:
Mean-reversion (continuous state)
Ornstein-Uhlenbeck process
Square-root process and other generalizations
Mean-reversion (discrete state)
View in Lab:
Mean-reversion (discrete state)
Time-homogeneous generator
Time-inhomogeneous generator
Projection of a Markov chain
Long memory: fractional Brownian motion
View in Lab:
Long memory: fractional Brownian motion
Fractional Brownian motion
Volatility clustering
View in Lab:
Volatility clustering
Stochastic volatility
Time change
Connection between time-changed Brownian motion and stochastic volatility
AR(1)
View in Lab:
AR(1)
Conditional distribution of AR(1)
Stationarity and unconditional distribution of AR(1)
VAR(1)
View in Lab:
VAR(1)
Conditional distribution of VAR(1)
Stationarity and unconditional distribution of VAR(1)
Cointegrated VAR(1)
Ornstein-Uhlenbeck
View in Lab:
Ornstein-Uhlenbeck
Conditional distribution of OU
Stationarity and unconditional distribution of OU
Multivariate Ornstein-Uhlenbeck
View in Lab:
Multivariate Ornstein-Uhlenbeck
Conditional distribution of MVOU
Stationarity and unconditional distribution of MVOU
Geometrical interpretation∗
Cointegrated Ornstein-Uhlenbeck
Rel. between (V)AR and (MV)OU
View in Lab:
Relationship between (V)AR and (MV)OU
MVOU is VAR(1)
VAR(1) is MVOU
Carry signals
View in Lab:
Carry signals
Fixed-income
Foreign exchange
Value signals
View in Lab:
Value signals
Book
Pricing
Technical signals
View in Lab:
Technical signals
Momentum
Filters
Cointegration
Microstructure signals
View in Lab:
Microstructure signals
Trade autocorrelation
Order imbalance
Price prediction
Volume clustering
Fundamental and other signals
View in Lab:
Fundamental and other signals
Signal processing
View in Lab:
Signal processing
Smoothing
Scoring
Ranking
Strong dominance (order zero dominance)
View in Lab:
Strong dominance (order zero dominance)
Weak dominance (first order stochastic dominance)
View in Lab:
Weak dominance (first order stochastic dominance)
Second order stochastic dominance
View in Lab:
Second order stochastic dominance
Order q stochastic dominance
View in Lab:
Order q stochastic dominance
Convex programming
View in Lab:
Convex programming
Conic programming
Second-order cone programming
Semidefinite programming
Quadratic programming
Linear programming
Integer n-choose-k selection
View in Lab:
Integer ¯n-choose-k selection
Naive selection
Forward step-wise selection
Backward step-wise selection
Lasso
Factor analysis algorithms
View in Lab:
Factor analysis algorithms
Pay attention, please:
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