# Extreme Value Theory

Duration: 2 days
• Introduction to EVT
• Block Maxima Models
• Peaks-over-Threshold 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 100-year 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 day-to-day applications of these models, rather than on their theoretical mathematical properties. We will demonstrate how a “Generalized Pareto Distribution” can be fitted to real-life 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 non-conditional 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

### 09.15 - 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

### 13.00 - 16.30  Models for Extreme Values

• General Theory and Overview of Models
• Block Maxima Models
• Peak-over-Threshold Models
• Semi-parametric 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 cut-off point
• Time aggregation
• Fitting the GDP to typical financial data
• Modelling Predictive Distributions Using Baysian Methods
• Modelling Multivariate Extremes
• Exercises

## Day Two

### 09.15 - 12.00  Measuring Risk Using EVT

• Overview of Risk Measures and their Strengths and Limitations
• Estimating and Interpreting "Value-at-Risk"
• 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 back-testing
• Examples, Simulations and Exercises

### 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