# Contents

## PART  I – OVERVIEW

### Chapter 1 – Value-at-Risk

Exercises

Exercises

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)

#### 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

## PART  II – ESSENTIAL MATHEMATICS

### Chapter 2 – Mathematical Preliminaries

#### 2.2  Mathematical Notation

Cartesian Products

Vectors

Matrices

Functions

Polynomials

Univariate Approximations

Multivariate Approximations

Taylor Polynomials

Exercises

#### 2.4  Ordinary Interpolation

Example: Linear 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

Exercises

#### 2.9  Ordinary Least Squares

Example

Ordinary Least Squares Methodology

Example (Continued)

Exercises

Exercises

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

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

The Product Rule

Example: Product Rule

Curse of Dimensionality

Exercises

### Chapter 3 – Probability

#### 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

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

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

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

### Chapter 4 – Statistics and Time Series Analysis

#### 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

Properties

### Chapter 5 – Monte Carlo Method

#### 5.2  The Monte Carlo Method

Stanislaw Ulam

Example: Approximating

Example: Approximating a Standard Deviation

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

## PART  III – VALUE-AT-RISK

### Chapter 6 – Historical Market Data

Types of Prices

Collecting Data

Data Sources

#### 6.3  Nonsynchronous Data

Causes of Nonsynchronous Data

Impact of Nonsynchronous Data

Errors

Data Filtering

Data Cleaning

#### 6.6  Futures Prices

Nearbys

Nearbys and Distortions

Constant-Maturity Futures Prices

Exercises

Expirations

Strikes

### Chapter 7 – Covariance Matrix Construction

#### 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

### Chapter 8 – Primary Portfolio Mappings

#### 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

Exercises

#### 8.5  Example: Forwards

Procedure

Specific Portfolio

### 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

Interpolation and The Method or Least Squares

Selecting Realizations for Interpolation of Least Squares

Example: Cocoa options

Exercises

#### 9.4  Variables Remappings

Modeling Curves

Exercises

Example: Implied Volatilities

Principal-Component Remappings

Example: US Treasury Securities

Exercises

### Chapter 10 – Transformation Procedures

#### 10.2  Linear Transformation Procedures

Exercises

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

### Chapter 11 – Historical Simulation

Example

Mirror Values

Exercises

#### 11.6  Shortcomings of Historical Simulation

Large Standard Errors

Stale Historical Data

## PART  IV – IMPLEMENTATION AND VALIDATION

### Chapter 12 – Implementing VaR

#### 12.2  Preliminaries

Go Slowly

Commitment

Implementation Team

#### 12.4  Functional Requirements

Requirements Format

Prototypes

Instrument Coverage

Frequency of Calculation

Inputs

Outputs

Interfaces

Exercises

Vendor Software

Choosing Vendor Software

#### 12.6  Implementation

Internally Developed Software

Agile Software Development Methods

Vendor Software

### Chapter 13 – Model Risk, Testing and Validation

#### 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

### Chapter 14 – 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