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Open ToC

Overview

Theory

Introduction

About the ARPM Lab

Organization of the ARPM Lab

Learning the ARPM Lab by topic

Learning the ARPM Lab by channel

Audience and prerequisites

About quantitative finance: P and Q

Differences between P and Q

Commonalities between P and Q

Key notation tenets

Operators and special functions

Probability and general distribution theory

Summary statistical features

Stochastic processes

Time conventions and counting indexes

Factor models and learning

Views processing

Investor preferences/profile

I. The “Checklist”

Executive summary

[ 1 ] Risk drivers identification

[ 1.1 ] Equities

[ 1.2 ] Currencies

[ 1.2.1 ] Exchange rates

[ 1.2.2 ] Contracts

[ 1.3 ] Fixed-income

[ 1.3.1 ] Zero-coupon bond

[ 1.3.2 ] Rolling value

[ 1.3.3 ] Yield to maturity

[ 1.3.4 ] Alternative representations

[ 1.3.5 ] Parsimonious representations

[ 1.3.6 ] Spreads

[ 1.4 ] Derivatives

[ 1.4.1 ] Call option

[ 1.4.2 ] Rolling value

[ 1.4.3 ] Implied volatility

[ 1.4.4 ] Alternative representations

[ 1.4.5 ] Parsimonious representations

[ 1.4.6 ] Pure volatility products

[ 1.5.1 ] Obligor-level risk drivers

[ 1.5.2 ] Aggregate risk drivers

[ 1.5.3 ] Structural models

[ 1.5.4 ] Merton’s structural credit model

[ 1.6 ] Insurance

[ 1.7 ] Operations

[ 1.8 ] High frequency

[ 1.8.1 ] Market microstructure

[ 1.8.2 ] Activity time

[ 1.8.3 ] Time-changed variables

[ 1.9 ] Strategies

[ 1.10 ] The final output

[ 1.11 ] Points of interest, pitfalls, practical tips

[ 1.11.1 ] Spurious heteroscedasticity

[ 1.11.2 ] Spot, par, yield, forward, zero, risk-free,...which rate and curve?

[ 2 ] Quest for invariance

[ 2.1 ] Efficiency: random walk

[ 2.1.1 ] Continuous-state random walk

[ 2.1.2 ] Discrete-state random walk

[ 2.1.3 ] Flexible combinations

[ 2.2 ] Mean-reversion (continuous state): ARMA

[ 2.2.1 ] AR(1) process

[ 2.2.2 ] AR(p) process

[ 2.2.3 ] MA(q) process

[ 2.2.4 ] ARMA(p,q) process

[ 2.2.5 ] ARIMA(p,d,q) process

[ 2.3 ] Mean-reversion (discrete state): Markov chains

[ 2.3.1 ] Time-homogeneous Markov chains

[ 2.3.2 ] Time-inhomogeneous Markov chains

[ 2.3.3 ] Invariant and next-step function

[ 2.3.4 ] Applications to credit risk

[ 2.3.5 ] Connection with structural model

[ 2.4 ] Long memory: fractional integration

[ 2.5 ] Seasonality

[ 2.6 ] Volatility clustering

[ 2.6.1 ] GARCH

[ 2.6.2 ] Extensions of GARCH

[ 2.6.3 ] Stochastic volatility

[ 2.7 ] Multivariate quest

[ 2.7.1 ] Copula-marginal models

[ 2.7.2 ] Efficiency: multivariate random walk

[ 2.7.3 ] Mean reversion (continuous state): VAR(I)MA

[ 2.7.4 ] Vector autoregression of order 1: VAR(1)

[ 2.7.5 ] Mean reversion (discrete state): multivariate Markov chains

[ 2.7.6 ] Volatility clustering

[ 2.7.7 ] Toward machine learning

[ 2.8 ] Cointegration

[ 2.8.1 ] Modeling

[ 2.8.2 ] Detection

[ 2.9 ] Relationships among processes

[ 2.9.1 ] Inversion of first-degree lag polynomial

[ 2.9.2 ] ARMA(p,q) processes as products of lag polynomials

[ 2.9.3 ] Relationships between ARMA, MA and AR

[ 2.9.4 ] AR(p) as VAR(1)

[ 2.9.5 ] Univariate processes as VAR(1)

[ 2.9.6 ] Relationships between VARMA, VMA and VAR

[ 2.9.7 ] VAR(p) as VAR(1)

[ 2.9.8 ] Multivariate processes as VAR(1)

[ 2.9.9 ] Stationary as MA(∞): Wold representation

[ 2.10 ] The final output

[ 2.10.1 ] Model for groups

[ 2.10.2 ] Fit and invariants extraction for groups

[ 2.10.3 ] Marginal test for invariance

[ 2.10.4 ] Green light toward joint estimation

[ 2.11 ] Points of interest, pitfalls, practical tips

[ 2.11.1 ] Model-free invariance extraction

[ 2.11.2 ] Generalized Wold’s theorem

[ 2.11.3 ] Information set

[ 2.11.4 ] Returns are not invariants

[ 2.11.5 ] Sampling step size

[ 3 ] Estimation

[ 3.1 ] Setting the flexible probabilities

[ 3.1.1 ] Exponential decay and time conditioning

[ 3.1.2 ] Kernels and state conditioning

[ 3.1.3 ] Joint state and time conditioning

[ 3.1.4 ] Statistical power of flexible probabilities

[ 3.2 ] Historical

[ 3.2.1 ] From historical distribution to flexible probabilities

[ 3.2.2 ] Location-dispersion: HFP ellipsoid

[ 3.2.3 ] Kernel estimation with flexible probabilities

[ 3.3 ] Maximum likelihood

[ 3.3.1 ] From maximum likelihood to flexible probabilities

[ 3.3.2 ] Exponential family invariants

[ 3.3.3 ] Location-dispersion: normal MLFP ellipsoid

[ 3.3.4 ] Location-dispersion: t MLFP ellipsoid

[ 3.4 ] Bayesian

[ 3.4.1 ] Exponential family invariants

[ 3.4.2 ] Normal invariants

[ 3.5 ] Shrinkage

[ 3.5.1 ] Mean shrinkage: James-Stein

[ 3.5.2 ] Covariance shrinkage: Ledoit-Wolf

[ 3.5.3 ] Correlation shrinkage: random matrix theory

[ 3.5.4 ] Covariance shrinkage: sparse eigenvector rotations

[ 3.5.5 ] Covariance shrinkage: glasso

[ 3.6 ] Generalized method of moments

[ 3.6.1 ] Method of moments

[ 3.6.2 ] Generalized method of moments - exact specification

[ 3.6.3 ] Generalized method of moments - over specification

[ 3.7 ] Robustness

[ 3.7.1 ] Local robustness

[ 3.7.2 ] Global robustness

[ 3.8 ] Missing data

[ 3.8.1 ] Randomly missing data

[ 3.8.2 ] Times series of different length

[ 3.8.3 ] Missing series

[ 3.9 ] (Dynamic) copula-marginal

[ 3.9.1 ] Static copula

[ 3.9.2 ] Dynamic copula

[ 3.10 ] The final output

[ 3.10.1 ] Historical

[ 3.10.2 ] Analytical

[ 3.10.3 ] Copula-marginal

[ 3.11 ] Points of interest, pitfalls, practical tips

[ 3.11.1 ] Unconditional estimation

[ 3.11.2 ] Standardization

[ 3.11.3 ] Non-synchronous data

[ 3.11.4 ] High-frequency volatility/correlation

[ 3.11.5 ] Outlier detection

[ 3.11.6 ] Exponential moving moments and statistics

[ 3.11.7 ] Backward/forward exponential decay

[ 3.11.8 ] Combining estimation techniques

[ 4 ] Projection

[ 4.1 ] One-step historical projection

[ 4.2 ] Univariate analytical projection

[ 4.3 ] Multivariate analytical projection

[ 4.4 ] Monte Carlo

[ 4.4.1 ] Markov models

[ 4.4.2 ] State-space models

[ 4.4.3 ] Copula-marginal models

[ 4.4.4 ] Generalizations

[ 4.5 ] Historical

[ 4.5.1 ] Historical bootstrapping

[ 4.5.2 ] Hybrid Monte Carlo-historical

[ 4.6 ] Application: multivariate Markov chains

[ 4.6.1 ] Multivariate Markov chains

[ 4.7 ] Square-root rule and generalizations

[ 4.7.1 ] Thin-tailed random walk

[ 4.7.2 ] Thick-tailed random walk

[ 4.7.3 ] Multivariate random walk

[ 4.7.4 ] General processes

[ 4.8 ] The final output

[ 4.8.1 ] Analytical

[ 4.8.2 ] Scenarios

[ 4.9 ] Points of interest, pitfalls, practical tips

[ 4.9.1 ] Consecutive (non-)overlapping sequences

[ 4.9.2 ] Semi-analytical random walk projection via fast Fourier transform

[ 4.9.3 ] GARCH generalized next-step

[ 4.9.4 ] Projection step at a later stage of the Checklist

[ 4.9.5 ] Pitfalls for square-root rule

[ 4.9.6 ] Thick tails from thin tails

[ 4.9.7 ] Martingales

[ 4.9.8 ] Linear versus compounded returns

[ 4.9.9 ] FFT for moving averages

[ 4.9.10 ] Scenario projection enhancements by probability twisting

[ 4.9.11 ] Jumps versus diffusion

[ 5 ] Pricing at the horizon

[ 5.1 ] Exact repricing

[ 5.1.1 ] Equities

[ 5.1.2 ] Currencies

[ 5.1.3 ] Fixed-income

[ 5.1.4 ] Derivatives

[ 5.1.5 ] Credit

[ 5.1.6 ] High frequency

[ 5.1.7 ] Strategies

[ 5.2.1 ] Equities

[ 5.2.2 ] Currencies

[ 5.2.3 ] Fixed-income

[ 5.2.4 ] Derivatives

[ 5.2.5 ] Other asset classes

[ 5.3 ] Taylor approximations

[ 5.3.1 ] Equities

[ 5.3.2 ] Fixed-income

[ 5.3.3 ] Derivatives

[ 5.3.4 ] Other asset classes

[ 5.4 ] Hybrid Taylor/repricing approximation

[ 5.4.1 ] Global quadratic approximation

[ 5.4.2 ] Interpolated residual

[ 5.5 ] The final output

[ 5.5.1 ] Analytical approach

[ 5.5.2 ] Scenario-based approach

[ 5.6 ] Testing the pricing function

[ 5.7 ] Pitfalls and practical tips

[ 5.7.1 ] Pricing at the horizon and arbitrage

[ 5.7.2 ] Path dependence

[ 5.7.3 ] “Pricing at the horizon”versus “asset pricing/valuation theory”

[ 5.7.4 ] Black-Merton-Scholes is exactly correct!

[ 5.7.5 ] Greeks for intra-day updates

[ 5.7.6 ] Greeks at the horizon

[ 5.7.7 ] Bond carry versus accrued interest

[ 5.7.8 ] Option carry versus theta

[ 6 ] Aggregation

[ 6.1 ] Stock variables

[ 6.1.1 ] Portfolio

[ 6.1.2 ] Value

[ 6.1.3 ] Exposure

[ 6.1.4 ] Leverage

[ 6.2 ] Liquidity P&L/risk

[ 6.3 ] Static market/credit risk

[ 6.3.1 ] Standardized holdings and weights

[ 6.3.2 ] Scenario-probability distribution

[ 6.3.3 ] Elliptical distribution

[ 6.3.4 ] Quadratic-normal distribution

[ 6.4 ] Dynamic market/credit risk

[ 6.5 ] Stress-testing

[ 6.5.1 ] Theory

[ 6.5.2 ] Why have stress-tests

[ 6.5.3 ] Panic copula

[ 6.5.4 ] Extreme copula

[ 6.6 ] Enterprise risk management

[ 6.6.1 ] Portfolio: balance sheet

[ 6.6.2 ] Performance: income statement

[ 6.6.3 ] Operational risk

[ 6.6.4 ] Banking simplified regulatory framework

[ 6.6.5 ] Insurance simplified framework

[ 6.7 ] The final output

[ 6.7.1 ] Analytical approach

[ 6.7.2 ] Scenario-based approach

[ 6.8 ] Points of interest and pitfalls

[ 6.8.1 ] Solvency and collateral

[ 6.8.2 ] Credit value adjustment

[ 6.8.3 ] CreditRisk+ approximation

[ 7 ] Ex-ante evaluation

[ 7.1 ] Stochastic dominance

[ 7.2 ] Satisfaction/risk measures

[ 7.3 ] Mean-variance trade-off

[ 7.3.2 ] Variance

[ 7.3.3 ] Standard deviation

[ 7.3.4 ] Mean-variance trade-off

[ 7.3.5 ] A strange success story

[ 7.4 ] Expected utility and certainty-equivalent

[ 7.4.1 ] Common utility functions

[ 7.4.2 ] Computation

[ 7.5 ] Quantile (value at risk)

[ 7.5.1 ] Definition

[ 7.5.2 ] Computation

[ 7.6 ] Spectral satisfaction measures/Distortion expectations

[ 7.6.1 ] Definition

[ 7.6.2 ] Common spectral/distortion measures

[ 7.6.3 ] Computation

[ 7.7 ] Coherent satisfaction measures

[ 7.7.1 ] Definition

[ 7.7.2 ] Common coherent measures

[ 7.7.3 ] Computation

[ 7.8 ] Non-dimensional ratios

[ 7.8.1 ] Information ratio

[ 7.8.2 ] Downside ratios

[ 7.9 ] Additional measures

[ 7.9.2 ] Correlation

[ 7.9.3 ] Buhlmann expectation

[ 7.9.4 ] Esscher expectation

[ 7.10 ] Enterprise risk management

[ 7.11 ] The final output

[ 7.12 ] Pitfalls, points of interest and practical tips

[ 7.12.1 ] The Arrow-Pratt approximation of the certainty-equivalent

[ 7.12.2 ] Utility versus quantile

[ 7.12.3 ] Utility versus spectrum functions

[ 7.12.4 ] Spectral/distortion versus coherent measures

[ 7.12.5 ] The Buhlmann and Esscher expectations are not distortion expectations

[ 7.12.6 ] Satisfaction measures under normality

[ 8 ] Ex-ante attribution

[ 8a ] Ex-ante attribution: performance

[ 8a.1 ] Bottom-up exposures

[ 8a.1.1 ] Pricing factors

[ 8a.1.2 ] Style factors/smart beta

[ 8a.2 ] Top-down exposures: factors on demand

[ 8a.2.1 ] Analytical computation

[ 8a.2.2 ] Cardinality constraints

[ 8a.3 ] Relationship between bottom-up and top-down exposures

[ 8a.3.1 ] Subportfolios

[ 8a.3.2 ] Style factors/smart beta

[ 8a.4 ] Joint distribution

[ 8a.4.1 ] Elliptical distribution

[ 8a.4.2 ] Scenario-probability distribution

[ 8a.5 ] Application: hedging

[ 8a.6 ] The final output

[ 8a.7 ] Pitfalls and practical tips

[ 8a.7.1 ] Estimation versus attribution

[ 8a.7.2 ] The ex-ante attribution is not a regression on past data

[ 8b ] Ex-ante attribution: risk

[ 8b.1 ] General criteria

[ 8b.1.1 ] Isolated/“first in”proportional attribution

[ 8b.1.2 ] “Last in”proportional attribution

[ 8b.1.3 ] Sequential attribution

[ 8b.1.4 ] Shapley attribution

[ 8b.2 ] Homogenous measures and Euler decomposition

[ 8b.2.1 ] Standard deviation

[ 8b.2.2 ] Variance

[ 8b.2.3 ] Certainty-equivalent

[ 8b.2.4 ] Distortion/spectral measures

[ 8b.2.5 ] Economic capital

[ 8b.3 ] Esscher expectation

[ 8b.4 ] Buhlmann expectation

[ 8b.5 ] Minimum-torsion bets attribution of variance

[ 8b.5.1 ] Effective number of bets

[ 8b.5.2 ] Minimum-torsion bets

[ 8b.5.3 ] Traditional marginal contributions versus minimum-torsion contributions

[ 8b.6 ] The final output

[ 9 ] Construction

[ 9a ] Construction: portfolio optimization

[ 9a.1 ] A compromise: two-step mean-variance

[ 9a.2 ] Analytical solutions of the mean-variance problem

[ 9a.3 ] The final output

[ 9b ] Construction: cross-sectional strategies

[ 9b.1 ] Simplistic portfolio construction

[ 9b.1.1 ] Backtesting

[ 9b.2 ] Advanced portfolio construction

[ 9b.2.1 ] Signal-induced factor

[ 9b.2.2 ] Flexible factor

[ 9b.2.3 ] Backtesting

[ 9b.3 ] Relationship with FLAM and APT

[ 9b.3.1 ] Signal-induced moments of the P&L

[ 9b.3.2 ] Fundamental law of active management

[ 9b.3.3 ] APT assumption

[ 9b.4 ] Multiple portfolios

[ 9b.4.1 ] Characteristic portfolios

[ 9b.4.2 ] Signal-induced factors

[ 9b.4.3 ] Flexible factors

[ 9b.4.4 ] Relationship with FLAM and APT

[ 9b.5 ] Points of interest, pitfalls, practical tips

[ 9b.5.1 ] Machine learning

[ 9b.5.2 ] Generalized fundamental law of active management

[ 9c ] Construction: time series strategies

[ 9c.1 ] The market

[ 9c.1.1 ] Risky investment

[ 9c.1.2 ] Low-risk investment

[ 9c.1.3 ] Strategies

[ 9c.2 ] Expected utility maximization

[ 9c.2.1 ] The objective

[ 9c.2.2 ] Optimization

[ 9c.3 ] Option based portfolio insurance

[ 9c.3.1 ] Payoff design

[ 9c.3.2 ] Partial differential equation

[ 9c.3.3 ] Budget

[ 9c.3.4 ] Policy

[ 9c.3.5 ] A unified approach

[ 9c.4 ] Rolling horizon heuristics

[ 9c.4.1 ] Constant proportion portfolio insurance

[ 9c.4.2 ] Drawdown control

[ 9c.5 ] Signal induced strategy

[ 9c.6 ] Convexity analysis

[ 9d ] Construction: estimation and model risk

[ 9d.1 ] Allocation (=output) uncertainty: Robust allocation

[ 10 ] Execution

[ 10.1 ] Market impact modeling

[ 10.1.1 ] Liquidity curve

[ 10.1.2 ] Exogenous impact

[ 10.1.3 ] Endogenous impact

[ 10.2 ] Order scheduling

[ 10.2.1 ] Trading P&L decomposition

[ 10.2.2 ] Model P&L

[ 10.2.3 ] Moments of model P&L

[ 10.2.4 ] Model P&L optimization

[ 10.2.5 ] Quasi-optimal P&L distribution

[ 10.3 ] Order placement

[ 10.3.1 ] Step 1: order scheduling

[ 10.3.2 ] Step 2: order placement

[ 10.4 ] The final output

[ 10.5 ] Points of interest, pitfalls, practical tips

[ 10.5.1 ] Mean-variance optimization in complex models

[ 10.5.2 ] Price manipulation

[ 10.5.3 ] Testing

II. Factor models and learning

[ 11 ] Executive summary

[ 12 ] Linear factor models: foundations

[ 12.1 ] Overview

[ 12.1.1 ] Dominant-residual models

[ 12.1.2 ] Systematic-idiosyncratic models

[ 12.2 ] Regression LFM’s

[ 12.2.1 ] Definition

[ 12.2.2 ] Solution: factor loadings

[ 12.2.3 ] Prediction and fit

[ 12.2.4 ] Residuals features

[ 12.2.5 ] Natural scatter specification

[ 12.2.6 ] Symmetric regression

[ 12.2.7 ] Equivalent linear formulation for regression

[ 12.3 ] Principal component LFM’s

[ 12.3.1 ] Definition

[ 12.3.2 ] Principal component analysis

[ 12.3.3 ] Solution: factor loadings and factor-construction matrix

[ 12.3.4 ] Prediction and fit

[ 12.3.5 ] Residuals features

[ 12.3.6 ] Natural scatter specification

[ 12.3.7 ] Identification issues

[ 12.3.8 ] Pitfall: principal factors are not principal components

[ 12.4 ] Systematic-idiosyncratic LFM’s

[ 12.4.1 ] Definition

[ 12.4.2 ] Implications on parameters

[ 12.4.3 ] Implications on covariance

[ 12.4.4 ] Factor analysis: principal axis factorization

[ 12.4.5 ] Loadings unearthing and factor approximation

[ 12.4.6 ] Prediction and fit

[ 12.4.7 ] Identification issues

[ 12.5 ] Cross-sectional LFM’s

[ 12.5.1 ] Definition

[ 12.5.2 ] Solution: factor-construction matrix

[ 12.5.3 ] Prediction and fit

[ 12.5.4 ] Residuals features

[ 12.5.5 ] Natural scatter specification

[ 12.5.6 ] Systematic-idiosycratic assumption

[ 13 ] Linear factor models: estimation

[ 13.1 ] Overview

[ 13.1.1 ] Time series models

[ 13.1.2 ] Data fit

[ 13.2 ] Static regression LFM’s

[ 13.3 ] Static principal component LFM’s

[ 13.4 ] Static systematic-idiosyncratic LFM’s

[ 13.5 ] Static cross-sectional LFM’s

[ 13.5.1 ] Factor construction

[ 13.5.2 ] Problems with pure cross-sectional estimation

[ 13.5.3 ] Hybrid cross-sectional estimation

[ 13.6 ] Truncation

[ 13.7 ] Wiener-Kolmogorov filtering

[ 13.7.1 ] Solution

[ 13.7.2 ] Estimation

[ 13.8 ] Dynamic principal component

[ 13.8.1 ] Solution

[ 13.8.2 ] Computational issue

[ 13.8.3 ] Estimation

[ 13.9 ] Linear state space model

[ 14 ] Linear factor models: enhancements

[ 14.1 ] Performance of regression versus principal component

[ 14.1.1 ] Regression revisited

[ 14.1.2 ] Principal component revisited

[ 14.1.3 ] The comparison

[ 14.2 ] Factor selection for regression

[ 14.2.1 ] Step-wise regression selection

[ 14.2.2 ] Lasso regression

[ 14.2.3 ] Ridge regression

[ 14.3 ] Sparse principal component

[ 14.4 ] Conditional principal component

[ 14.5 ] More general constraints

[ 15 ] Linear factor models: pitfalls

[ 15.1 ] LFM’s are not a regression on past data(1) [⋆⋆]

[ 15.2 ] LFM’s are not about returns(2-3) [⋆⋆]

[ 15.3 ] LFM’s are not about stocks(4) [⋆⋆]

[ 15.4 ] LFM’s “factors”are not “factors returns”(5) [⋆]

[ 15.5 ] LFM’s are not systematic-idiosyncratic(6-7) [⋆⋆⋆]

[ 15.6 ] LFM’s are not horizon-independent(8) [⋆⋆]

[ 15.7 ] LFM’s are not a dimension reduction technique(9) [⋆]

[ 15.8 ] LFM’s are not APT and CAPM(10-11-12) [⋆⋆⋆]

[ 15.9 ] LFM’s do not include factor analysis(14) [⋆⋆]

[ 15.10 ] LFM’s do not extract premia-generating factors(16)[⋆⋆]

[ 15.11 ] LFM’s are not always necessary(13-15-17-18) [⋆⋆⋆]

[ 16 ] Machine learning: foundations

[ 16.1 ] Key ideas from linear factor models

[ 16.1.1 ] Supervised linear predictors

[ 16.1.2 ] Unsupervised linear autoencoders

[ 16.1.3 ] Conditionally uncorrelated linear models

[ 16.2 ] Key concepts for machine learning

[ 16.2.1 ] Supervised, unsupervised, reinforcement learning

[ 16.2.2 ] Regression, classification, clustering, structures

[ 16.2.3 ] Point and probabilistic prediction

[ 16.2.4 ] Discriminant and generative probabilistic models

[ 16.3 ] Supervised point prediction: regression

[ 16.3.1 ] Mean regression

[ 16.3.2 ] Linear mean regression

[ 16.3.3 ] Analysis of variance

[ 16.3.4 ] Quantile regression

[ 16.3.5 ] Linear quantile regression

[ 16.4 ] Supervised point prediction: classification

[ 16.4.1 ] Mode classification

[ 16.4.2 ] Linear classification

[ 16.4.3 ] Receiver operating characteristic (ROC)

[ 16.5 ] Supervised probabilistic prediction

[ 16.5.1 ] Regression

[ 16.5.2 ] Binary classification

[ 16.5.3 ] Multiple classification

[ 16.5.4 ] Generalized linear models

[ 16.6 ] Unsupervised learning: autoencoders

[ 16.6.1 ] Principal component linear factor models

[ 16.6.2 ] Cross-sectional linear factor models

[ 16.6.3 ] k-means clustering

[ 16.6.4 ] Independent component analysis

[ 16.7 ] Probabilistic graphical models

[ 16.7.1 ] Probabilistic factor analysis

[ 16.7.2 ] Mixture models

[ 16.7.3 ] Naive Bayes models

[ 16.7.4 ] Graphs

[ 16.7.5 ] Markov random fields

[ 16.7.6 ] Bayes networks

[ 17 ] Machine learning: estimation

[ 17.1 ] Probabilistic linear state-space models

[ 17.1.1 ] Model

[ 17.1.2 ] Estimation

[ 17.2 ] Probabilistic state space models

[ 17.2.1 ] Hidden Markov models

[ 18 ] Machine learning: enhancements

[ 18.1 ] Feature engineering

[ 18.1.1 ] Interaction

[ 18.1.2 ] Encoding

[ 18.1.3 ] Trees

[ 18.1.4 ] Segmentation

[ 18.1.5 ] Feature basis

[ 18.1.6 ] Exponential family

[ 18.2 ] Deep learning

[ 18.2.1 ] Point prediction

[ 18.2.2 ] Probabilistic prediction

[ 18.3 ] Gradient boosting

[ 18.4 ] Regularization

[ 18.4.1 ] Step-wise features selection

[ 18.4.2 ] Ridge, lasso, elastic nets

[ 18.4.3 ] Glasso

[ 18.4.4 ] Bayesian prior

[ 18.5 ] Ensemble learning

[ 18.5.1 ] Bagging

[ 18.5.2 ] Flexible probabilities as random-variables

[ 18.5.3 ] Flexible probabilities through conditioning

[ 18.5.4 ] Ensemble weighting

[ 19 ] From probabilistic regression to Gaussian processes

[ 19.1 ] Time series models

[ 19.2 ] Maximum likelihood

[ 19.2.1 ] The model

[ 19.2.2 ] Normal assumption

[ 19.2.3 ] Student t assumption

[ 19.3 ] Bayesian

[ 19.3.1 ] Normal conditional likelihood

[ 19.3.2 ] Normal-inverse-Wishart prior distribution

[ 19.3.3 ] Normal-inverse-Wishart posterior distribution

[ 19.3.4 ] Student t predictive distribution

[ 19.3.5 ] Classical equivalent

[ 19.3.6 ] Uncertainty

[ 19.4 ] Mixed approach

[ 20 ] Application: principal component analysis of the yield curve

[ 20.1 ] Cross-sectional structure of the yield curve covariance

[ 20.2 ] Finite set of times to maturity

[ 20.3 ] The continuum limit

[ 21 ] Application: credit default classification

[ 21.1 ] Background

[ 21.2 ] Fit and assessment

[ 21.3 ] Logistic regression

[ 21.4 ] Interactions

[ 21.5 ] Encoding

[ 21.6 ] Regularization

[ 21.8 ] Gradient boosting

[ 21.9 ] Cross-validation

[ 22 ] Application: clustering

[ 22.1 ] k-means clustering

[ 22.2 ] Shrinkage

III. Valuation

[ 23 ] Executive summary

[ 24 ] Background definitions

[ 24.1 ] Valuation foundations

[ 24.1.1 ] Instruments

[ 24.1.2 ] Value

[ 24.1.3 ] Cash-flows

[ 24.1.4 ] Re-invested cash-flows

[ 24.1.5 ] Cash-flow adjusted value

[ 24.1.6 ] Profit-and-loss (P&L)

[ 24.1.7 ] Payoff

[ 24.2 ] Points of interest and pitfalls

[ 24.2.1 ] Value versus price

[ 24.2.2 ] Multi-currency conversions

[ 24.2.3 ] Actual versus simple P&L

[ 25 ] Linear pricing theory

[ 25a ] Linear pricing theory: core

[ 25a.1 ] Fundamental axioms

[ 25a.1.1 ] Law of one price

[ 25a.1.2 ] Linearity

[ 25a.1.3 ] No arbitrage

[ 25a.2 ] Stochastic discount factor

[ 25a.2.1 ] Identification

[ 25a.2.2 ] Misidentification

[ 25a.3 ] Fundamental theorem of asset pricing

[ 25a.4 ] Risk-neutral pricing

[ 25a.4.1 ] General case

[ 25a.4.2 ] No rebalancing limit: forward measure

[ 25a.4.3 ] Continuous rebalancing limit

[ 25a.5 ] Capital asset pricing model framework

[ 25a.5.1 ] Maximum Sharpe ratio portfolio

[ 25a.5.2 ] Security market line

[ 25a.5.3 ] Alternative derivation: linear factor model for stochastic discount factor

[ 25a.6 ] Covariance principle

[ 25a.7 ] Point of interest and pitfalls

[ 25a.7.1 ] Relationships among fundamental axioms

[ 25b ] Linear pricing theory: further assumptions

[ 25b.1 ] Completeness

[ 25b.1.1 ] Definition

[ 25b.1.2 ] Pricing

[ 25b.1.3 ] Arrow-Debreu securities

[ 25b.1.4 ] Stochastic discount factor

[ 25b.2 ] Equilibrium: pure capital asset pricing model

[ 25b.3 ] Equilibrium: Buhlmann pricing equation

[ 25b.4 ] Arbitrage pricing theory

[ 25b.4.1 ] Standard derivation: linear factor model for instruments

[ 25b.4.2 ] Alternative derivation: linear factor model for stochastic discount factor

[ 25b.5 ] Intertemporal consistency

[ 25b.5.1 ] Continuous time variables

[ 25b.5.2 ] Martingales

[ 25b.5.3 ] Heuristic for stochastic discount factor time consistency

[ 25b.5.4 ] Heuristic for numeraire martingale

[ 26 ] Non-linear pricing theory

[ 26.1 ] Fundamental axioms

[ 26.1.1 ] Law of one price

[ 26.1.2 ] Non-linearity

[ 26.1.3 ] Arbitrage

[ 26.2 ] Valuation as evaluation

[ 26.2.1 ] Variance and other shift principles

[ 26.2.2 ] Certainty-equivalent principle

[ 26.2.3 ] Distortion principles

[ 26.2.4 ] Esscher principle

[ 26.3 ] Intertemporal consistency

[ 26.3.1 ] Continuous time variables

[ 26.3.2 ] Non-linear "martingales"?

[ 26.4 ] Point of interest and pitfalls

[ 26.4.1 ] Linear (mis)uses of non-linear pricing

[ 27 ] Valuation implementation

[ 27.1 ] Portfolio value

[ 27.1.1 ] Long positions

[ 27.1.2 ] Short positions

[ 27.1.3 ] Generic positions

[ 27.1.4 ] Funds

[ 27.1.5 ] Sum-of-parts

[ 27.1.6 ] Valuation recipe

[ 27.2 ] Equities

[ 27.2.1 ] Discounted cash-flows

[ 27.2.2 ] Multiples

[ 27.3 ] Options

[ 27.3.1 ] Bachelier

[ 27.3.2 ] Black-Scholes

[ 27.3.3 ] Heston

[ 27.3.4 ] Valuation recipe

[ 27.4 ] Fixed-income

[ 27.4.1 ] Vasicek

[ 27.4.2 ] Other models

[ 27.4.3 ] Valuation recipe

[ 27.5 ] Insurance

[ 27.5.1 ] Life insurance

[ 27.5.2 ] Non-life insurance

[ 27.6 ] Real assets

IV. Performance analysis

[ 28 ] Executive summary

[ 29 ] Performance definitions

[ 29.1 ] Holding P&L of a portfolio

[ 29.2 ] Trading P&L

[ 29.2.1 ] Single transaction

[ 29.2.2 ] Multiple transactions in one position

[ 29.2.3 ] Portfolio rebalancing

[ 29.3 ] Implementation shortfall

[ 29.4 ] Returns

[ 29.4.1 ] Basic definitions

[ 29.4.2 ] Standard linear returns and weights

[ 29.4.3 ] Generalized linear returns

[ 29.4.4 ] Generalized weights and aggregation

[ 29.4.5 ] Investments with capital injection

[ 29.4.6 ] Log-returns

[ 29.5 ] Excess performance

[ 29.5.1 ] Benchmark

[ 29.5.2 ] Excess return

[ 29.6 ] Path analysis

[ 29.7 ] Pitfalls and practical tips

[ 29.7.1 ] Linear versus compounded returns

[ 30 ] Performance attribution

V. Quant toolbox

[ 32 ] Distributions

[ 32.1 ] Representations of a distribution

[ 32.1.1 ] Univariate distributions

[ 32.1.2 ] Multivariate distributions

[ 32.2 ] Marginalization

[ 32.3 ] Conditioning

[ 32.3.1 ] Conditional variables

[ 32.3.2 ] Conditional features

[ 32.3.3 ] Deterministic versus stochastic conditioning

[ 32.4 ] Elliptical distributions

[ 32.4.1 ] Fundamental concepts

[ 32.4.2 ] Stochastic representations

[ 32.4.3 ] Moments and dependence

[ 32.4.4 ] Affine equivariance

[ 32.4.5 ] Notable elliptical distributions

[ 32.4.6 ] Generation of elliptical scenarios

[ 32.4.7 ] Scenario generation with dimension reduction

[ 32.5 ] Scenario-probability distributions

[ 32.5.1 ] Types of scenario-probability distributions

[ 32.5.2 ] Probability density function

[ 32.5.3 ] Transformations and generalized expectations

[ 32.5.4 ] Cumulative distribution function

[ 32.5.5 ] Continuous cumulative distribution function

[ 32.5.6 ] Quantile

[ 32.5.7 ] Continuous quantile

[ 32.5.8 ] Moments and other statistical features

[ 32.5.9 ] Probabilities parametrization

[ 32.6 ] Mixture distributions

[ 32.7 ] Exponential family distributions

[ 32.7.1 ] Normal distribution

[ 32.7.2 ] Scenario-probability distribution

[ 32.8 ] Other special classes of distributions

[ 32.8.1 ] Stable distributions

[ 32.8.2 ] Infinitely divisible distributions

[ 32.9 ] Notable specific distributions

[ 32.9.1 ] Uniform distribution

[ 32.9.2 ] Quadratic-normal distribution

[ 32.9.3 ] Wishart distribution

[ 33 ] Geometry of distributions

[ 33.1 ] Distributions geometry

[ 33.1.1 ] Fisher metric: length and volume

[ 33.1.2 ] Flatness and geodesics

[ 33.1.3 ] Duality: potentials and Legendre transformations

[ 33.1.4 ] Distance and divergence

[ 33.2 ] Exponential distributions geometry

[ 33.3 ] Scenario-probability distribution geometry

[ 34.1 ] Univariate results

[ 34.2 ] Definition and properties

[ 34.2.1 ] Grades

[ 34.2.2 ] Copula

[ 34.2.3 ] Sklar’s theorem

[ 34.2.4 ] Copula invariance

[ 34.3 ] Special classes of copulas

[ 34.3.1 ] Elliptical copulas

[ 34.3.2 ] Archimedean copulas

[ 34.4 ] Implementation

[ 34.4.1 ] Copula-marginal separation

[ 34.4.2 ] Copula-marginal combination

[ 35 ] Correlation and generalizations

[ 35.1 ] Measures of dependence

[ 35.1.1 ] Schweizer-Wolff measure

[ 35.1.2 ] Mutual information

[ 35.2 ] Measures of concordance

[ 35.2.1 ] Kendall’s tau

[ 35.2.2 ] Spearman’s rho

[ 35.3 ] Correlation

[ 35.4 ] Points of interest, pitfalls, practical tips

[ 35.4.1 ] Schweizer and Wolff measure via simulations

[ 36 ] Stochastic dominance

[ 36.1 ] Strong dominance (order zero dominance)

[ 36.2 ] Weak dominance (first order stochastic dominance)

[ 36.3 ] Second order stochastic dominance

[ 36.4 ] Order q stochastic dominance

[ 36.5 ] Points of interest, pitfalls, practical tips

[ 37 ] Location and dispersion

[ 37.1 ] Univariate location-dispersion

[ 37.1.1 ] Z-score and affine equivariance

[ 37.1.2 ] Taxonomy of location-dispersion features

[ 37.1.3 ] Location and dispersion as variational problems

[ 37.2 ] Multivariate location-dispersion

[ 37.2.1 ] Tentative visualizations in low dimension

[ 37.2.2 ] Location-dispersion ellipsoid

[ 37.2.3 ] Affine equivariance

[ 37.2.4 ] Taxonomy of multivariate location-dispersion features

[ 37.3 ] Expectation and covariance

[ 37.3.1 ] Definitions

[ 37.3.2 ] Generalized affine equivariance

[ 37.3.3 ] Connections with calculus

[ 37.3.4 ] Connections with probability

[ 38 ] Geometry of random variables

[ 38.1 ] Length, distance and angle

[ 38.2 ] Multivariate case: covariance product

[ 38.3 ] An alternative: expectation product

[ 38.4 ] Generalization of the non-central length: p-norms

[ 38.5 ] Orthogonal projection

[ 38.6 ] The r-squared

[ 38.6.1 ] Definition

[ 38.6.2 ] Relation to distance

[ 38.7 ] Visualization

[ 38.8 ] Geometry of portfolios

[ 38.8.1 ] Length, angle and distance

[ 38.8.2 ] Visualization

[ 38.8.3 ] Orthogonality and collinearity

[ 39 ] Spectral analysis

[ 39.1 ] The Fourier transform

[ 39.1.1 ] Linear algebra perspective

[ 39.1.2 ] Functional analysis perspective

[ 39.2 ] Spectrum

[ 39.3 ] Spectral representation

[ 39.3.1 ] Spectral representation restated

[ 39.3.2 ] Finite sample principal component analysis

[ 39.3.3 ] Infinite Toeplitz matrices

[ 39.3.4 ] Infinite sample principal component analysis

[ 39.4 ] Filtering

[ 39.5 ] Estimation

[ 40 ] Decision theory with model uncertainty

[ 40.1 ] Foundations of decision theory

[ 40.1.1 ] Fundamental concepts

[ 40.1.2 ] Frequentist approach

[ 40.1.3 ] Bayesian approach

[ 40.1.4 ] Model risk, estimation risk

[ 41 ] Estimation techniques

[ 41.1 ] Maximum likelihood

[ 41.1.1 ] General theory

[ 41.1.2 ] Hidden variables

[ 41.1.3 ] Relevant cases

[ 41.1.4 ] Numerical methods

[ 41.2 ] Bayesian statistics

[ 41.2.1 ] General theory

[ 41.2.2 ] Bayesian estimation

[ 41.2.3 ] Bayesian prediction

[ 41.2.4 ] Analytical results

[ 41.2.5 ] Numerical methods

[ 42 ] Views processing

[ 42.1 ] Minimum relative entropy

[ 42.1.1 ] Base distribution and view variables

[ 42.1.2 ] Point views

[ 42.1.3 ] Distributional views

[ 42.1.4 ] Partial views

[ 42.1.5 ] Partial views on generalized expectations

[ 42.1.6 ] Sanity check

[ 42.1.7 ] Confidence

[ 42.1.8 ] Relationship with Bayesian updating

[ 42.2 ] Analytical implementation

[ 42.2.1 ] Base distribution

[ 42.2.2 ] Views

[ 42.2.3 ] Updated distribution

[ 42.2.4 ] Confidence

[ 42.2.5 ] Relevant special cases

[ 42.3 ] Flexible probabilities implementation

[ 42.3.1 ] Base distribution

[ 42.3.2 ] Views

[ 42.3.3 ] Updated distribution

[ 42.3.4 ] Confidence

[ 42.4 ] Factor-based implementations

[ 42.5 ] Copula opinion pooling

[ 42.5.1 ] Base distribution

[ 42.5.2 ] Views

[ 42.5.3 ] Updated distribution

[ 42.5.4 ] Confidence

[ 42.5.5 ] The algorithm

[ 42.6 ] Generalized shrinkage

[ 42.6.1 ] Intuition

[ 42.6.2 ] Classical shrinkage

[ 42.6.3 ] Bayesian updating

[ 42.6.4 ] Minimum relative entropy

[ 42.6.5 ] Shrinkage

[ 42.6.6 ] Regularization

[ 43 ] Black-Litterman

[ 43.1 ] Equilibrium prior distribution

[ 43.1.1 ] Prior distribution of expected returns

[ 43.1.2 ] Prior predictive distribution

[ 43.2 ] Active views

[ 43.2.1 ] Setting the parameters

[ 43.2.2 ] Sanity check

[ 43.3 ] Posterior distribution

[ 43.3.1 ] Posterior distribution of expected returns

[ 43.3.2 ] Posterior predictive distribution

[ 43.4 ] Limit cases

[ 43.4.1 ] High confidence in prior

[ 43.4.2 ] Low confidence in views

[ 43.4.3 ] Full confidence in views

[ 43.5 ] Generalizations

[ 43.5.1 ] From linear returns to risk drivers

[ 43.5.2 ] From stock-like to generic asset classes

[ 43.5.3 ] From normal to non-normal markets

[ 43.5.4 ] From linear equality views to partial flexible views

[ 44 ] Estimation and assessment

[ 44.1 ] Background

[ 44.1.1 ] Panels

[ 44.1.2 ] Assumptions

[ 44.1.3 ] Estimation

[ 44.1.4 ] Testing

[ 44.2 ] Probabilistic prediction assessment for invariants

[ 44.2.1 ] Estimators as decisions

[ 44.2.2 ] Frequentist approach

[ 44.2.3 ] Bayesian approach

[ 44.2.4 ] Analytical results

[ 44.2.5 ] Monte Carlo simulations

[ 44.2.6 ] Cross-validation

[ 44.3 ] Bias versus variance

[ 44.4 ] Point prediction assessment

[ 44.4.1 ] Estimators as decisions

[ 44.4.2 ] Frequentist approach

[ 44.4.3 ] Bayesian approach

[ 44.4.4 ] Historical with flexible probabilities estimators

[ 44.4.5 ] Cross-validation

[ 44.5 ] Probabilistic prediction assessment

[ 44.5.1 ] Estimators as decisions

[ 44.5.2 ] Frequentist approach

[ 44.5.3 ] Bayesian approach

[ 44.5.4 ] Maximum likelihood with flexible probabilities

[ 44.5.5 ] Cross-validation

[ 45 ] Hypothesis testing

[ 45.1 ] Hypothesis testing for invariants

[ 45.1.1 ] Statistics

[ 45.1.2 ] P-value

[ 45.1.3 ] Univariate testing: the z-statistic

[ 45.1.4 ] Multivariate testing: the Hotelling statistic

[ 46 ] Stochastic processes cheat sheet

[ 46.1 ] Main definitions

[ 46.1.1 ] Weak white noise

[ 46.1.2 ] White noise

[ 46.1.3 ] Stationary process

[ 46.1.4 ] Integrated processes

[ 46.1.5 ] Ergodic process

[ 46.1.6 ] Cointegrated process

[ 46.1.7 ] Martingale process

[ 46.1.8 ] State process and Markov process

[ 46.1.9 ] Random fields

[ 46.1.10 ] Orthogonal increments process

[ 46.2 ] Relationships among processes

[ 46.2.1 ] White noise (with finite variance) ⇒ weak white noise

[ 46.2.2 ] White noise ⇒ stationary

[ 46.2.3 ] White noise ⇒ ergodic

[ 46.2.4 ] Ergodic ⇒ stationary

[ 46.2.5 ] Stationary ⇒ covariance-stationary

[ 46.2.6 ] Weak white noise ⇒ integrated

[ 46.2.7 ] Weak white noise ⇒ covariance-stationary

[ 46.2.8 ] Covariance-stationary ⇒ integrated

[ 46.3 ] Pitfalls

[ 47 ] Invariance tests

[ 47.1 ] Simple tests

[ 47.2 ] Refinements and pitfalls

[ 47.2.1 ] Circle-like covariance (not data)

[ 47.2.2 ] Stronger tests based on copulas

[ 48 ] Continuous time processes

[ 48.1 ] Efficiency: Lévy processes

[ 48.1.1 ] Infinite divisibility

[ 48.1.2 ] Continuous state: Brownian diffusion

[ 48.1.3 ] Discrete state: Poisson jumps

[ 48.1.4 ] Lévy-Khintchine representation

[ 48.1.5 ] Subordination

[ 48.2 ] Mean-reversion (continuous state)

[ 48.2.1 ] Ornstein-Uhlenbeck process

[ 48.2.2 ] Square-root process and other generalizations

[ 48.3 ] Mean-reversion (discrete state)

[ 48.3.1 ] Time-homogeneous generator

[ 48.3.2 ] Time-inhomogeneous generator

[ 48.3.3 ] Projection of a Markov chain

[ 48.4 ] Long memory: fractional Brownian motion

[ 48.4.1 ] Fractional Brownian motion

[ 48.5 ] Volatility clustering

[ 48.5.1 ] Stochastic volatility

[ 48.5.2 ] Time change

[ 48.5.3 ] Connection between time-changed Brownian motion and stochastic volatility

[ 49 ] First order autoregression

[ 49.1.1 ] Conditional distribution of AR(1)

[ 49.1.2 ] Stationarity and unconditional distribution of AR(1)

[ 49.2 ] VAR(1)

[ 49.2.1 ] Conditional distribution of VAR(1)

[ 49.2.2 ] Stationarity and unconditional distribution of VAR(1)

[ 49.2.3 ] Cointegrated VAR(1)

[ 49.3 ] Ornstein-Uhlenbeck

[ 49.3.1 ] Conditional distribution of OU

[ 49.3.2 ] Stationarity and unconditional distribution of OU

[ 49.4 ] Multivariate Ornstein-Uhlenbeck

[ 49.4.1 ] Conditional distribution of MVOU

[ 49.4.2 ] Stationarity and unconditional distribution of MVOU

[ 49.4.3 ] Geometrical interpretation∗

[ 49.4.4 ] Cointegrated Ornstein-Uhlenbeck

[ 49.5 ] Relationship between (V)AR and (MV)OU

[ 49.5.1 ] MVOU is VAR(1)

[ 49.5.2 ] VAR(1) is MVOU

[ 49.6 ] VAR(1)/MVOU fit

[ 50.1 ] Carry signals

[ 50.1.1 ] Fixed-income

[ 50.1.2 ] Foreign exchange

[ 50.2 ] Value signals

[ 50.2.1 ] Book

[ 50.2.2 ] Pricing

[ 50.3 ] Technical signals

[ 50.3.1 ] Momentum

[ 50.3.2 ] Filters

[ 50.3.3 ] Cointegration

[ 50.4 ] Microstructure signals

[ 50.4.1 ] Trade autocorrelation

[ 50.4.2 ] Order imbalance

[ 50.4.3 ] Price prediction

[ 50.4.4 ] Volume clustering

[ 50.5 ] Fundamental and other signals

[ 50.6 ] Signal processing

[ 50.6.1 ] Smoothing

[ 50.6.2 ] Scoring

[ 50.6.3 ] Ranking

[ 51 ] Optimization primer

[ 51.1 ] Convex programming

[ 51.1.1 ] Conic programming

[ 51.1.2 ] Semidefinite programming

[ 51.1.3 ] Second-order cone programming

[ 51.1.4 ] Quadratic programming

[ 51.1.5 ] Linear programming

[ 51.2 ] Integer ¯n-choose-k selection

[ 51.2.1 ] Naive selection

[ 51.2.2 ] Forward step-wise selection

[ 51.2.3 ] Backward step-wise selection

[ 51.2.4 ] Lasso

[ 52 ] Useful algorithms

[ 52.1 ] Factor analysis algorithms

[ 52.2 ] Matrix transpose-square-root

[ 52.2.1 ] Spectrum

[ 52.2.2 ] Riccati

[ 52.2.3 ] LDL-Cholesky

[ 52.2.4 ] Gram-Schmidt

[ 52.3 ] Markov chain Monte Carlo sampling

[ 52.3.1 ] Metropolis-Hastings

[ 52.4 ] Moment-matching scenarios

[ 52.4.1 ] Twisting scenarios

[ 52.4.2 ] Twisting probabilities

[ 52.5 ] Fast Fourier transform for projection

[ 52.5.1 ] Normalized empirical histogram

[ 52.5.2 ] Approximating the pdf and the characteristic function

[ 52.5.3 ] Using the discrete Fourier transform

[ 52.6 ] Minimum-torsion optimization algorithm

[ 53 ] Matrix manipulations

[ 53.1 ] Spectral theorem

[ 53.1.1 ] The eigenvalue problem

[ 53.1.2 ] Recursive solution

[ 53.1.3 ] Matrix notation

[ 53.1.4 ] The continuum limit

[ 53.2 ] Matrix algebra

[ 53.2.1 ] Key operators

[ 53.2.2 ] Useful identities

[ 53.3 ] Matrix calculus

[ 53.3.1 ] First order derivatives

[ 53.3.2 ] Second order derivatives

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