Contents

Contents

PART  I – OVERVIEW

Chapter 0 – Preface

0.1  What We’re About

0.2  Voldemort and the Second Edition

0.3  How to Read This Book

0.4  Notation

Chapter 1 – Value-at-Risk

1.1  Measures

Exercises

1.2  Risk Measures

Exercises

1.3  Market Risk

Exercises

1.4  Value-at-Risk

Probabilistic Metrics of Market Risk (PMMRs)

Value-at-Risk as a PMMR

Exercises

1.5  Risk Limits

Market Risk Limits

Stop-Loss Limits

Exposure Limits

VaR Limits

Summary Comparison

1.6  Other Applications of Value-at-Risk

Risk Reporting and Oversight

Bank Regulatory Capital Requirements

Economic Capital Calculations

Corporate Disclosures

Risk Budgeting

Other Applications

1.7  Examples

Example: The Leavens PMMR

Exercises

Example: Industrial Metals

Exercises

Example: Australian Equities

Example: Australian Equities (Monte Carlo Transformation)

Example: Australian Equities (Linear Remapping)

Example: Australian Equities (Quadratic Transformation)

1.8  Value-ar-Risk Measures

Risk Factors

Portfolio Holdings

Portfolio Mappings

Inference Procedures

Transformation Procedures

Exercises

Portfolio Remappings

Exercises

Summary

1.9  History

Regulatory Value-at-Risk Measures

Proprietary Value-at-Risk Measures

Portfolio Theory

Emergence of Risk Management

RiskMetrics

Publicized Losses

1.10  Further Reading

PART  II – ESSENTIAL MATHEMATICS

Chapter 2 – Mathematical Preliminaries

2.1  Motivation

2.2  Mathematical Notation

Cartesian Products

Vectors

Matrices

Functions

Polynomials

2.3  Gradient and Gradient-Hessian Approximations

Univariate Approximations

Multivariate Approximations

Taylor Polynomials

Exercises

2.4  Ordinary Interpolation

Example: Linear Interpolation

Example: Quadratic Interpolation

Ordinary Interpolation Methodology

Exercises

2.5  Complex Numbers

The number i

Complex Operations

Complex Functions

Exercises

2.6  Eigenvalues and Eigenvectors

Theory

Intuitive Example

Exercises

2.7  Cholesky Factorization

Positive Definite Matrices
Matrix “Square Roots”
Cholesky Factorization
Computational Issues
Exercises

2.8  Minimizing a Quadratic Polynomial

Exercises

2.9  Ordinary Least Squares

Example

Ordinary Least Squares Methodology

Example (Continued)

Exercises

2.10  Cubic Spline Interpolation

Exercises

2.11  Finite Difference Approximations of Derivatives

Exercises

2.12  Newton’s Method

Univariate Newton’s Method

Example: Univariate Newton’s Method

Caveats

Secant Method

Multivariate Newton’s Method

Line Searches

Exercises

2.13  Change of Variables Formula

Exercises

2.14  Numerical Integration: One Dimension

Riemann Sums

Example: Riemann Sums

Trapezoidal Rule

Example: Trapezoidal Rule

Simpson’s Rule

Example: Simpson’s Rule

Exercises

2.15  Numerical Integration: Multiple Dimensions

Quadrature

The Product Rule

Example: Product Rule

Curse of Dimensionality

Exercises

2.16  Further Reading

Chapter 3 – Probability

3.1  Motivation

3.2  Prerequisites

3.3  Parameters

Expectation

Expectation of a Function of a Random Variable

Variance and Standard Deviation

Skewness

Kurtosis

Quantiles

Moments

Exercises

3.4  Parameters of Random Vectors

Expectation of a Function of a Random Vector

Joint Moments

Covariance

Correlation

Exercises

3.5  Linear Polynomials of Random Vectors

Exercises

3.6  Properties of Covariance Matrices

Singular Random Vectors

Exercises

Multicollinear Random Vectors

Exercises

3.7  Principal Component Analysis

Example: European Currencies

Definition of Principal Components

Exercises

Choice of Weights With Principal Components

3.8  Bernoulli and Binomial Distributions

Bernoulli Distribution

Binomial Distribution

Exercises

3.9  Uniform and Related Distributions

Uniform Distribution

Multivariate Uniform Distribution

Exercises

3.10  Normal and Related Distributions

Normal Distributions

Lognormal Distributions

Chi-Squared Distributions

Joint-Normal Distributions

Exercises

3.11  Mixtures of Distributions

Parameters of Mixed Distributions

Mixed-Normal Distributions

Mixed Joint-Normal Distributions

Exercises

3.12  Moment-Generating Functions

Exercises

3.13  Quadratic Polynomials of Joint-Normal Random Vectors

Simplified Representation

Example

Moments

Example (Continued)

Other Parameters

Exercises

3.14  The Cornish-Fisher Expansion

Cumulants

Cornish-Fisher Expansion With Five Cumulants

Example

Exercises

3.15  Central Limit Theorem

Exercises

3.16  The Inversion Theorem

Complex Random Variables

Characteristic Functions

Inversion Theorem

Exercises

3.17  Quantiles of Quadratic Polynomials of Joint-Normal Random Vectors

The CDF of a Quadratic Polynomial of a Joint-Normal Random Vector

Quantiles of a Quadratic Polynomial of a Joint-Normal Random Vector

Example

Exercises

3.18  Further Reading

Chapter 4 – Statistics and Time Series Analysis

4.1  Motivation

4.2  From Probability to Statistics

4.3  Estimation

Samples

Estimators

Sample Estimators

Bias

Standard Error

Mean Squared Error

Estimators for Random Vectors

Exercises

4.4  Maximum Likelihood Estimators

ML Estimators

ML Estimates of Scalar Parameters

ML Estimates of Non-Scalar Parameters

Example: Mixed Normal Distribution

Exercises

4.5  Hypothesis Testing

Test Statistics

Example: Coin Tossing

Power and Significance Level

Exercises

Likelihood Ratio Tests

4.6  Stochastic Processes

Exercises

Conditional vs. Unconditional Distributions

Correlations, Autocorrelations and Cross Correlations

Stationarity Stochastic Processes

Differencing

Returns

Heteroskedasticity

Exercises

4.7 Testing for Autocorrelations

Ljung and Box Test

Exercises

4.8  White Noise, Moving-Average and Autoregressive Processes

White Noise Processes

Moving-Average Processes

Autoregressive Processes

Autoregressive Moving-Average Processes

Exercises

Properties

Estimation

4.9  Garch Processes

CCC-Garch

Orthogonal Garch

Properties of CCC-GARCH and Orthogonal GARCH

4.10  Regime-Switching Processes

Properties

4.11  Further Reading

Chapter 5 – Monte Carlo Method

5.1  Motivation

5.2  The Monte Carlo Method

Stanislaw Ulam

Example: Approximating

Example: Approximating a Standard Deviation

5.3  Realizations of Samples

Exercises

5.4  Pseudorandom Numbers

Linear Congruential Generators

Multiple-Recursive Generators

Inversive Generators

5.5  Testing Pseudorandom Number Generators

The Spectral Test

Discrepancy

Period

5.6  Implementing Pseudorandom Number Generators

The Number 0

Integer Calculations

Implementation Verification

Choosing a Generator

5.7  Breaking the Curse of Dimensionality

Crude Monte Carlo Estimator

Implications of Standard Error

Exercises

5.8  Pseudorandom Variates

Inverse Transform Method

Joint-Normal Pseudorandom Vectors

Exercises

Monte Carlo Simulation – Directly Modeling Relevant Random Vectors

5.9  Variance Reduction

Control Variates

Example: Control Variates

Control Variates for Monte Carlo Estimators of Quantiles

Stratified Sampling

5.10  Further Reading

PART  III – VALUE-AT-RISK

Chapter 6 – Historical Market Data

6.1  Motivation

6.2  Forms of Historical Market Data

Types of Prices

Collecting Data

Data Sources

6.3  Nonsynchronous Data

Causes of Nonsynchronous Data

Impact of Nonsynchronous Data

6.4  Data Errors

Errors

Data Filtering

Data Cleaning

6.5  Data Biases

6.6  Futures Prices

Nearbys

Nearbys and Distortions

Constant-Maturity Futures Prices

Exercises

6.7  Implied Volatilities

Expirations

Strikes

6.8  Further Reading

Chapter 7 – Covariance Matrix Construction

7.1  Motivation

7.2  Selecting Key Factors

Data Availability

Stationarity and Homoskedasticity

Structural Relationships

Consistency Over Time

7.3  Current Practice

Choice of Distribution

Conditional Mean Vectors

White Noise

Uniformly Weighted Moving Average Estimates

Example: Aluminum Prices

Exercises

Implicit Assumptions

Exponentially Weighted Moving Average Estimates

Exercises

Covariance Matrices That are Not Positive Definite

Exercises

Roll-Off Effect

7.4  Unconditional Leptokurtosis and Conditional Heteroskedasticity

An Experiment With Conditional Heteroskedasticity

Modeling Unconditional Leptokurtosis

Implications for Value-at-Risk Measures

Exercises

7.5  Further Reading

Chapter 8 – Primary Portfolio Mappings

8.1  Motivation

8.2  Day Counts

Actual Days, Basis Days, and Trading Days

Time Versus Maturity

Exercises

Value Dates

Example: Cash Valuation Discount Curve

Example: 2nd-Day Valuation Discount Curve

Example: Random Discount Curve

Exercises

8.3  Primary Mappings

Specifying 1S and ω

Specifying 1R and θ

Example

Exercises

Practical Examples

8.4  Example: Equities

Exercises

8.5  Example: Forwards

Procedure

Specific Portfolio

8.6  Example: Options

8.7  Example: Physical Commodities

8.8  Further Reading

Chapter 9 – Portfolio Remappings

9.1  Motivation

Facilitating Transformations

Facilitating Inference

Facilitating Another Remapping

Forms of Remappings

9.2  Holdings Remappings

Example: Holdings Remappings of Fixed Cash Flows

Example: Holdings Remapping of Interest Rate Caps

Designing Holdings Remappings

Exercises

9.3  Function Remappings

Selecting a Polynomial Form

Linear Remappings

Linear Remappings with Gradient Approximations

Example: Gradient Approximations

Quadratic Remappings

Quadratic Remappings with Gradient-Hessian Approximations

Interpolation and The Method or Least Squares

Selecting Realizations for Interpolation of Least Squares

Example: Cocoa options

Exercises

9.4  Variables Remappings

Example: Coffee Spreads

Modeling Curves

Exercises

Example: Implied Volatilities

Principal-Component Remappings

Example: US Treasury Securities

Exercises

9.5  Further Reading

Chapter 10 – Transformation Procedures

10.1  Motivation

10.2  Linear Transformation Procedures

Exercises

10.3  Quadratic Transformation Procedures

Example: Platinum Derivatives

Example Continued: Linear Polynomial Representation

Example Continued: Standard Deviation of Portfolio Value

Example Continued: Cornish-Fisher Expansion

Example Continued: Inverting the Characteristic Function

Exercises

10.4 Monte Carlo Transformation Procedures

Monte Carlo Standard Error

Empirical Analysis of Standard Error for Value-at-Risk

Exercises

10.5  Variance Reduction

Control Variates

Example: Control Variates

Stratified Sampling to Calculate Standard Deviation of Loss

Stratified Sampling to Calculate Value-at-Risk

Selective Valuation of Realizations

Exercises

10.6  Further Reading

Chapter 11 – Historical Simulation

11.1  Motivation

11.2  Generating Realizations Directly From Historical Market Data

Example

Mirror Values

11.3  Calculating Value-at-Risk With Historical Simulation

11.4  Origins of Historical Simulation

11.5  Flawed Arguments for Historical Simulation

Exercises

11.6  Shortcomings of Historical Simulation

Large Standard Errors

Stale Historical Data

11.7  Further Reading

PART  IV – IMPLEMENTATION AND VALIDATION

Chapter 12 – Implementing VaR

12.1  Motivation

12.2  Preliminaries

Go Slowly

Commitment

Implementation Team

12.3  Purpose

12.4  Functional Requirements

Requirements Format

Prototypes

Instrument Coverage

Frequency of Calculation

Inputs

Outputs

Interfaces

Exercises

12.5  Build vs. Buy

Vendor Software

Choosing Vendor Software

12.6  Implementation

Internally Developed Software

Agile Software Development Methods

Vendor Software

12.7  Further Reading

Chapter 13 – Model Risk, Testing and Validation

13.1  Motivation

13.2  Model Risk

Type A: Model Specification Risk

Type B: Model Implementation Risk

Type C: Model Application Risk

Exercises

13.3  Managing Model Risk

Personnel

Standard Assumptions and Modeling Procedures

Design Review

Testing

Parallel Testing

Backtesting

Ongoing Validation

Model Inventory

Vendor Software

Communication and Training

13.4  Further Reading

Chapter 14 – Backtesting

14.1  Motivation

14.2  Backtesting

14.3  Backtesting with Coverage Tests

A Recommended Standard Coverage Test

Kupiec’s PF Coverage Test

The Basel Committee’s Traffic Light Coverage Test

Exercises

14.4  Backtesting with Distribution Tests

Framework for Distribution Tests

A Graphical Distribution Test

A Recommended Standard Distribution Test

Exercises

14.5  Backtesting with Independence Tests

Christoffersen’s (1998) Exceedence Independence Test

A Recommended Standard Loss-Quantile Independence Test

Exercises

14.6  Example: Backtesting a One-Day 95% EUR VaR Measure

Example: Applying Coverage Tests

Example: Applying Distribution Tests

Example: Applying Independence Tests

Exercises

14.7  Backtesting Strategy

Backtesting as Hypothesis Testing

Alternatives

Joint Tests

Designing a Backtesting Program

Failing a Backtest

Backtesting Other PMMRs

14.8  Further Reading

Back Matter

Endnotes

References

Appendix – Normal Table