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Actual standardized stress testing scenarios for interest rates are based on historical realizations of interest rate changes and measure the impact of these shocks from the past on the actual portfolio value. We suggest in this paper a parameterized stress testing procedure for the term structure of interest rates. After identifying the factors driving the term structure we select the tail of the empirical distribution of each risk factor using state-of the-art approaches and model the probability distribution of these observations using Extreme-Value- Theory(EVT). Finally, we simulate extreme events in the risk factor and record the influence on the term structure. Our model can simulate various shapes of the term structure and we can approximate the impact of the stress event on portfolio values without performing expensive simulation procedures. Our approach is quite general. It can be performed for different types of term structures like the zero curve, the yield-to-maturity curve, the par curve, or the swap curve.

Banking supervisory authorities require stress testing as a key component of a bank’s assessment of its capital position. Stress Testing is quite a broad concept from a regulatory point of view. The Basel Committee on Banking Supervision (1997) requires that “bank’s stress scenarios need to cover a range of factors that can create extraordinary losses or gains in trading portfolios, or make control of risk in those portfolios very difficult”. Blaschke et al. (2001) consider stress testing as “a range of techniques used to asses the vulnerability of a portfolio to major changes in the macroeconomic environment or to exceptional but plausible events”. There is no strict guideline on how to perform stress testing. Therefore, every bank can have its own approach.

Actual standardized stress testing scenarios for interest rates are based on historical realizations of interest rate changes and measure the impact of these shocks from the past on the actual portfolio value. This approach has the advantage that the defined shocks have been observed and have a positive probability of occurring a second time. One shortcoming is precisely this historical constraint. In this approach, we are not able to define the magnitude and level of probability for a future stress scenario, which has not been observed.

This paper deals with another approach. We suggest a parameterized stress testing procedure for the term structure of interest rates, based on a multifactor model of returns and results from the Extreme-Value-Theory (EVT) to model the tail behavior of the risk factors. It allows us to model changes in the yield curve and to simulate in a consistent way possible, but unobserved, extreme movements in the yield curve. We employ a multifactor model similar to those in Litterman and Scheinkman (1988), Knez et al. (1994) or Bliss (1997) to identify the factors driving the daily changes of the swap curve. After identifying the factors driving the term structure we select the tail of the empirical distribution of each risk factor using state-of the-art approaches and justified only by the statistical properties of the tail observations. The factors are extracted from daily observations of the swap curve using factor analysis. The observed factor realisations are uncorrelated and, in the best case, standard normal distributed, and thus independent. We can model the tail of the factor distributions and simulate extreme independent events based on the tail distributions.

Contents
List of Tables
List of Figures
Abbreviations
1 Introduction

1.1 Motivation of the Paper
1.2 Importance of Stress Tests in Risk Management
1.3 Stress Tests for Interest Rate Risk
1.4 Outline of the Thesis

2 Theory and Methods

2.1 Stress Testing Approaches
2.1.1 Formal Definition of a Stress Test
2.1.2 Unified Framework
2.1.3 Identifying Extreme Events

2.2 Term Structure Models
2.3 Modelling Extreme Events in the Pricing Factors

2.3.1 Parametric Framework of the Extreme Value Theory
2.3.2 Tail Estimation

3 Empirical Analysis

3.1 A Multifactor Model of Money Market and Bond Returns
3.1.1 Data Description
3.1.2 Results of Factor Analysis

3.2 Identifying Extreme Events

3.2.1 Imposing a Threshold
3.2.2 Mean Excess Return
3.2.3 Testing for a Truncated Normal Distribution
3.2.4 Overview of Extreme Events in the Risk Factors
3.3 Fitting the Generalized Pareto Distribution (GPD)

4 Simulation Results of Stress Scenarios

4.1 Ex-post Simulation Results
4.2 Extreme Movements of the Yield Curve

5 Conclusions
Bibliography

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