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Quantitative Risk Measurement 2: Multivariate Statistics and Extreme Value Modelling

Duration: 2 days
  • Basics of Multivariate Modelling
  • Correlation Analysis, Regression Analysis and Discriminant Analysis
  • Estimating VaR from Multivariate Normal Distributions
  • Estimating Non-Normal Multivariate Distributions Using GARCH Modelling
  • Measuring VaR Using Principal Components Analysis
  • Measuring Risks Using Extreme Value Theory
  • Using EVT for Stress Testing and Economic Capital Planning
The objective of this seminar is to give you a good understanding of the use of multivariate statistics and Extreme Value modelling in quantifying and managing risk.

We start with a general introduction to multivariate statistics and analysis. We give an overview of the applications of multivariate modelling in finance, and we explain the basics of correlation and correlation analysis.

We then explain and demonstrate how you can use multiple regression analysis to determine relationships between economic and financial variables, and we explain the use “discriminant analysis” to compute linear predictors from sets of normally distributed data to allow for classification of new observations.

Further, we explain and show how sampling from multivariate return distributions can be performed and how “Value-at-Risk” can be derived from a total portfolio loss distribution that is generated using simulation techniques. We also explain how you can overcome the assumptions about normally distributed returns by using GARCH techniques to project volatilities from historical data.

We also explain and demonstrate how principal components analysis can be used to determine a smaller set of “synthetic” variables that could explain the original set (for example variations in the yield curve).

We then introduce Extreme Value Theory and explain and demonstrate its applications in finance.

We present the two main approaches to estimating tail distributions: the “Block Maxima” and the “Peaks over Threshold” groups of models. Emphasis will be on the practical day-to-day applications of these models, rather than on their theoretical mathematical properties. We demonstrate how a “Generalized Pareto Distribution” can be fitted to real-life financial data (stock prices etc.), and we visualize results using graphical tools.

We then turn to look at how EVT can be used in financial risk management. We discuss the opportunities and pitfalls of using EVT. We use extreme value theory to calculate conditional and non-conditional VaR, and we compare these measures with the VaR measures obtained using e.g. normal distribution assumptions. Finally, we discuss the use of EVT in “Stress Testing”, in quantifying operational risks, and in asset allocation.

Day One

09.00 - 09.15 Welcome and Introduction

09.15 - 12.00 Measuring Risk Using Multivariate Statistical Analysis

  • Basics of Multivariate Modelling
    • The use of multivariate modelling in finance
    • Correlation analysis
    • Multivariate correlation analysis
    • Partial, serial and canonical correlation
  • Regression Analysis
    • The regression line and the regression model
    • Multiple regression
    • Applications of multiple regression in finance
    • Collinearity and other problems
    • Examples of the use of regression analysis in finance
  • Discriminant Analysis
    • The discriminate function
    • Discriminant vs. regression analysis
    • Examples of the use of discriminant analysis in finance
  • Examples, Simulations and Exercises

12.00 - 13.00 Lunch

13.00 - 16.30 Measuring Risk Using Multivariate Statistical Analysis (Continued)

  • The Multivariate Normal Distribution
    • Sampling from multivariate normal distribution
    • Estimating VaR from multivariate normal distribution
    • Testing normality and multivariate normality
  • Estimating VaR from Non-Normal Multivariate Distributions
    • GARCH modelling and forecasting of volatility and correlation
  • Principal Components Analysis
    • Overview of multi-factor interest rate risk models
    • Eigenvalues, eigenvectors and the yield curve
    • Calculating and interpreting factor loadings
    • Using the factor model to calculate VaR
    • Factor immunization for hedging yield curve fluctuations
    • Monte Carlo simulation using PCA
  • Examples, Simulations and Exercises

Day Two

09.00 - 09.15 Brief recap

09.15 - 12.00 Measuring and Managing Risk Using Extreme Value Theory

  • General Introduction to Extreme Value Analysis
    • Explaining rare and unexpected events using EVT
    • Examples of catastrophic losses
  • Basic EVT Tools
    • Statistical analysis of historical data
    • Quantiles vs. tail distributions
    • Mathematical foundation of EVT
  • Models for Extreme Values
    • General theory and overview of models
    • Block Maxima models
    • Peak-over-Threshold models
    • The Generalized Pareto Distribution
    • Modelling predictive distributions using Bayesian methods
    • Modelling multivariate extremes
    • Multivariate extreme value copulas
  • Exercises

12.00 - 13.00 Lunch

13.00 - 16.30 Measuring and Managing Risk Using Extreme Value Theory (continued)

  • Measuring Risk Using EVT
    • Estimating and interpreting “Value-at-Risk” using EVT
    • Estimating expected shortfall
    • Extreme market risk
    • Stress testing using EVT
    • EVT and stochastic volatility models (GARCH)
    • Examples, simulations and exercises
  • Using EVT in Risk Management and Asset Management
    • Calculating regulatory capital using EVT
    • Modelling and measuring operational risk
    • Developing scenarios for future extreme losses
    • Asset allocation using EVT
    • Examples, simulations and exercises

Evaluation and Termination of the Seminar