Extreme Value Theory
Duration: 2 days
 Introduction to EVT
 Block Maxima Models
 PeaksoverThreshold Models
 Applying EVT to Financial Data
 Estimation of VaR and Conditional VaR
 Stress Testing Using EVT
 Risk Management Using EVT
Extreme Value Theory is a branch of statistics dealing with the extreme deviations from the mean of
probability distributions. Extreme Value Theory has been widely used for assessing risk for highly
unusual events, such as 100year floods. The objective of this seminar is to give the participants
a good understanding of how Extreme Value Theory can be used as a practical tool in sophisticated
financial risk management. We will start with a general introduction to Extreme Value Theory,
explaining how apparently unexpected phenomena are actually happening according to well defined
rules. We will also discuss the areas where the theory can be applied, including the forecasting of
extreme weather, insurance events and the estimation of tail risks in different financial markets.
We will then present the two main approaches to estimating tail distributions: the “Block Maxima”
and the “Peaks over Threshold” groups of models. However, the emphasis will be on the practical
daytoday applications of these models, rather than on their theoretical mathematical properties.
We will demonstrate how a “Generalized Pareto Distribution” can be fitted to reallife financial
data (stock prices, etc.), and we will use graphics to display the results. We will then turn to
look at how EVT can be used in financial risk management. We will discuss the opportunities and
pitfalls of using EVT. The extreme value theory will be used to calculate conditional and
nonconditional VaR, and these measures will be compared to the VaR measures obtained using normal
distribution assumptions. Finally, we will discuss the use of EVT in “Stress Testing” and in
quantifying different operational risks.
Day One
09.00  09.15 Welcome and Introduction
09.15  12.00 Introduction to "Extreme Value Theory"
 What is "Extreme Value Theory"?
 Explaining Rare and Unexpected Events Using EVT

Examples of Catastrophic Losses
 Barings, Orange County, Daiwa, …
 Overview of Uses of EVT in Finance
 Limitations and Strengths of EVT in Risk Management
Basic EVT Tools
 A Brief Review of Probability Theory
 Statistical Analysis of Historical Data
 Quantiles vs. Tail Distributions
 Modelling and Measuring Extreme Values

Mathematical Foundation of EVT
 Extreme value limit laws (Fisher and Tippet, Gnedenko)
 Three fundamental types of extreme limit laws
 Generalized extreme value distribution
 Small Exercises
12.00  13.00 Lunch
13.00  16.30 Models for Extreme Values
 General Theory and Overview of Models
 Block Maxima Models

PeakoverThreshold Models
 Semiparametric models (Hill estimator)
 Parametric models (Generalized Pareto)

The Generalized Pareto Distribution
 Making efficient use of limited data
 Estimating excess distributions
 Estimating tails of distributions
 Using maximum likelihood inference to obtain parametric formula
 Optimal choice of cutoff point
 Time aggregation
 Fitting the GDP to typical financial data
 Modelling Predictive Distributions Using Baysian Methods
 Modelling Multivariate Extremes
 Exercises
Day Two
09.00  09.15 Brief recap
09.15  12.00 Measuring Risk Using EVT
 Overview of Risk Measures and their Strengths and Limitations
 Estimating and Interpreting "ValueatRisk"
 Estimating Expected Shortfall
 Extreme Market Risk

Estimating VaR Using EVT
 VaR for fully aggregated position
 VaR for position decomposed on risk factors
 VaR for positions with derivatives

Stress Testing Using EVT
 Analysis of stress losses with block maxima models

EVT and Stochastic Volatility Models
 Fitting a GARCH model to the historical data using a (pseudo) maximum likelihood
method
 Fitting the EVT distribution to the scaled residuals
 Verification by backtesting
 Examples, Simulations and Exercises
12.00  13.00 Lunch
13.00  16.00 Using EVT in Risk Management and Asset Management
 Calculating Regulatory Capital Using EVT

Modelling and Measuring Operational Risk
 Estimating the loss distribution using EVT
 Economic capital for operational risk
 Pricing operational risk
 Developing Scenarios for Future Extreme Losses

Asset Allocation Using EVT
 Asset allocation using different measures of risk
 Asset allocation based upon Extreme VaR
 Example: Two assets
 Using an approximation procedure for more assets

Applications of EVT to Insurance
 Overview of applications in insurance
 Case study: Extreme value statistics and Wind Storm Losses
Summary, Evaluation and Termination of the Seminar
