Concentrated Advanced Course on
Statistical Methods for Financial Risk Management
Main Lectures by
Alexander McNeil (ETH Zurich and RiskLab Zurich)
Further Lectures by:
Henrik Hult (Stockholm), Thomas Mikosch (Copenhagen), Catalin Starica
(Chalmers
University Gothenburg), Murad Taqqu (Boston)
Monday, May 26, 2003 - Friday, May 30, 2003
University of Copenhagen,
The Concentrated Advanced Course will be given at the Institute
of Mathematical Sciences, University of Copenhagen,
HC Ørsted Institute, Auditorium 4. See the Infomation below
how to get to the Institute.
There will be 6 hours of lectures per day, 4 given by Alexander
McNeil. The course is
organized by Søren Asmussen (University of Aarhus),
Jeff Collamore (University of Copenhagen),
Martin Jacobsen (University of Copenhagen),
Thomas Mikosch (University of Copenhagen) and
Michael Sørensen (University of Copenhagen)
Abstract
Quantitative methodology is an increasingly important component of
risk management in financial institutions. Financial risk management
presents an extremely interesting area of application for statistics
with many new challenges. Whereas much of traditional statistics
concerns the average, the normal and the expected, risk management has
more to do with the extreme, the abnormal and the unexpected. Central
technical issues will be modelling the volatility of financial return
time series, modelling extreme values and modelling dependent risks.
We will examine methods relevant for both market and credit risk
management.
Contents
- The Basics of Quantitative Risk Management
- financial risks and losses, risk measures, VaR, expected
shortfall or conditional VaR, risk factors and mappings
- Standard Statistical Methods
- variance-covariance, historical simulation, Monte Carlo, limits
of standard methods
- Fundamentals of Modelling Dependent Risks
- basic multivariate statistics, multivariate normal
distribution, multivariate normal mixture models, elliptical
distributions, hyperbolic distributions
- Modelling Financial Time Series
- basic time series concepts, empirical properties (stylized
facts) of financial time series, arguments for stochastic volatility,
ARCH and GARCH models
- Basic Topics in Extreme Value Theory
- maxima and worst case losses, extreme value distributions,
generalised Pareto distribution (GPD), peaks-over -thresholds (POT)
method, modelling excess losses and heavy tails, estimation of
quantiles (VaR) and expected shortfall
- Advanced Topics in EVT and Time Series
- Outperforming historical simulation with EVT, EVT for dependent
time series, EVT in a stochastic volatility framework
- Copulas, Correlation and Dependent Extreme Values
- introduction to copulas, useful copula families, drawbacks and
fallacies of ordinary correlation, rank correlation, tail dependence
- Multivariate Models: Calibration and Simulation
- efficient correlation estimation, tests for multivariate
normality and ellipticity, fitting copulas to data, Monte Carlo
simulation of dependent risk factors
- Portfolio Credit Risk: Models
- models for dependent defaults (latent variable models and
mixture models), industry examples(KMV/Moodys, CreditMetrics,
CreditRisk+), mapping between models
- Portfolio Credit Risk: Calibration and Model Risk
- understanding sources of model risk, the role of copulas in
standard models, statistical issues in default modelling
- Advanced Multivariate Market Risk Models
- multivariate risk factor properties, multivariate time series
models, multivariate GARCH models
The following distinguished researchers have agreed to give supplementary
lectures on topics related to finance, risk, insurance mathematics,
extremes: Henrik Hult (Stockholm), Thomas Mikosch
(University of Copenhagen), Catalin Starica (Chalmers University
Gothenburg), Murad Taqqu (Boston)
Lectures by Catalin Starica
-
We argue that the classical theory of statistical curve estimation offers
the right setup for consistent, non-parametric inference of time-changing
expected return, volatility and covariance in the analysis of financial
returns.
-
1. Risk-return dynamics in stock indexes.
An unconditional, non-parametric approach to simultaneous estimation of
volatility and expected return is discussed. We show by means of a
detailed analysis of the returns of the Standard \& Poors 500 composite
stock index over the last fifty years, how theoretical results and
methodological recommandations from the statistical theory of
non-parametric curve inference allow to consistently estimate both
expected return and volatility. Unlike the existing literature, we need
not postulate an {\it a priori} relationship risk-return nor specify the
evolution of first two moments through covariates. Our analysis gives
statistical evidence that the expected excess return (return minus the
risk free interest rate) of the Standard \& Poors 500 composite stock
index as well as the market price of risk (the ratio excess return over
volatility) vary through time both in size and sign. In particular, the
periods of negative (positive) excess return and market price of risk
seem to coincide with the bear (bull) markets of the index. A complex
relationship between risk and expected return emerges, far from the common
assumption of a positive, time invariant, linear relation.
-
2. Multivariate dynamics of stock returns.
A simple non-stationary model for multivariate returns is proposed. Unlike
most of the multivariate econometric models for financial returns, this
model supposes the volatility to be exogenous. The vector of returns are
independent and have a slowly changing unconditional covariance structure.
The methodological frame is that of non-parametric regression with
non-random equidistant design points, where the regression function is the
evolving unconditional covariance. Special attention is payed to the
accurate description of the tails of the innovations. As an application
the model is fit to a multivariate data set of returns on three different
financial instruments: a foreign exchange rate, an index and an interest
rate. The 1-day ahead multivariate distributional forecast performance is
evaluated.
Lectures by Thomas Mikosch: Multivariate regular variation
and the GARCH model
- Multivariate regular variation is a theoretical concept which
extends the notion of power law tail of a one-dimensional distribution
to higher dimensions. It is a useful concept for describing the
dependence of multivariate extremal events. In this case, it is
not useful
to use linear correlations or related dependence
measures. Indeed, the correlation of two random variables
measures the degree to which they
are linearly dependent; it is not
suitable for describing the dependence of the components of
a vector which assumes values far away from the origin.
-
It turns out that various financial time series models such as the
popular GARCH model exhibit regular variation in their
finite-dimensional
distributions. We discuss the consequences for the extremes of
such time series and for the extrapolation of the distribution of the
random vectors into regions where only a few or no observations
are available.
Lecture by Henrik Hult: Multivariate regular variation
for processes with independent increments
-
A natural class of multivariate dynamic stochastic models useful in
finance and insurance is the class of continuous-time
stochastic processes with independent
increments which, at each point in time, satisfies a multivariate regular
variation condition. The class includes certain Levy jump processes
which are considered as more natural models in finance and insurance
than Brownian motion.
-
Using the simple structure of these processes one
can derive the tail behavior of interesting vectors of functionals of
the process. Those include the
largest jump of the process in a given time interval and the supremum of the
process in this interval.
Lectures by Murad Taqqu:
-
1. An introduction to stable processes.
The familiar Gaussian models are named after the German mathematician
Carl Friedrich Gauss. Based on the usual bell-shaped probability
curve, they do not allow for large deviations and are thus often
inadequate for modelling heavy probability tails. In the last decades
however, data with heavy "probability tails" have been collected in
fields as diverse as economics and telecommunications suggesting the
use of non-Gaussian stable processes as possible models. These models
always have infinite variance and, in some cases, infinite mean as
well. We will provide an introduction to stable processes with
particular emphasis on integral representations. These representations
provide a physical way for understanding the structure of the stable
processes. A number of examples will be presented.
-
2. Local contagion in financial markets.
One commonly believes that international markets are more dependent
during a crisis than they are are in more tranquil times. This extra
dependence is often referred to as contagion between markets. We
present a definition of contagion between financial markets based on
local correlation and propose a test to detect the presence of
contagion. The test does not require the specification of crisis and
normal periods. As such, it avoids difficulties associated with
testing for correlation breakdown, such as hand picking subsets of
the data, and it provides a better understanding of the degree of
dependence between financial markets. Using this test, we find
evidence of contagion between developed and U.S. equity markets and
evidence of flight to quality from the U.S. equity market to the
U.S. government bond market.
The Concentrated Advanced Course aims at the graduate student
in probability theory, statistics, finance, economics, insurance mathematics
and the researcher who wants to get an overview of methods
and techniques on modeling, as well as on
practitioners from the insurance industry, banks and regulatory
authorities.
The course will be accessible
for Masters students with a background in statistics and extreme
value theory. The interested Masters student can receive 2.5 ECTS for
participation in the course and work on a 2-3 day project
which consists of a practical piece of data analysis with S-Plus
or R, such
as fitting a bunch of GARCH models to financial data, or doing some
practical extreme value theory or copula fitting.
In order to do this,
the participants should have access to S-Plus and the
S+FinMetrics module. This can be
arranged for the period of the course.
Programme
Adept Scientific and Alexander McNeil will give a presentation of their
software products (S+FinMetrics,...).
It is preferable if all participants
will bring a laptop.
Monday, May 26: Introduction to Risk Management and Financial Time
Series
09.30-10.15 Registration and Coffee
10.15-11.00 McNeil: Basics of Quantitative Risk Management
11.15-12.00 McNeil: Standard Methods for Risk Management
12.00-14.00 Lunch Break
14.00-14.45 Henrik Hult: Multivariate regular variation
for processes with independent increments
14.45-15.15 Coffee
15.15-16.00 McNeil: Financial Time Series
16.15-17.00 Knudsen + McNeil: Short Introduction to
S-Plus/Insightful products + first hands-on experience with S-Plus and
S+Finmetrics
Tuesday, May 27: Extreme Values in Financial Time Series
09.15-10.00 McNeil: Basics of EVT
10.15-11.00 McNeil: The POT Method and Tails of Loss distributions
11.00-11.30 Coffee
11.30-12.15 Starica: Risk-return dynamics in stock indexes
12.15-14.00 Lunch Break
14.00-14.45 Taqqu: An introduction to stable processes
14.45-15.15 Coffee
15.15-16.00 McNeil: EVT and Financial Applications
16.15-17.00 McNeil: Practical Session: - Financial Time Series + EVT
Wednesday, May 28: Multivariate Models and Copulas
09.15-10.00 McNeil: Basic Multivariate Models
10.15-11.00 McNeil: Copulas and Extremal Dependence
11.00-11.30 Coffee
11.30-12.15 Taqqu: Local contagion in financial markets
12.15-14.00 Lunch Break
14.00-14.45 Starica: Multivariate dynamics of stock returns
14.45-15.15 Coffee
15.15-16.00 McNeil: Fitting Copulas to Data
16.15-17.00 McNeil: Practical Session: Copulas in S+Finmetrics
Thursday, May 29: Credit Risk
09.15-10.00 McNeil: Introduction to Portfolio Credit Risk Models
10.15-11.00 McNeil: Modelling Dependent Defaults Credit Risk Models
11.00-11.30 Coffee
11.30-12.15 Mikosch: Multivariate extremes and multivariate regular
variation
12.15-14.00 Lunch Break
14.00-14.45 Mikosch: Extremes in financial time series
14.45-15.15 Coffee
15.15-16.00 McNeil: Practical Issues in Credit Risk
16.15-17.00 McNeil: Practical Session - Modelling Default Data
Friday, May 30: Multivariate Dynamic Models for Market Risk
09.15-10.00 McNeil: Multivariate Financial Time Series I
10.15-11.00 McNeil: Multivariate Financial Time Series II
11.00-11.30 Coffee
11.30-12.15 McNeil: Practical Session - Multivariate Time Series
Models in S-Plus
12.15 Closing and Lunch Break
Registration
There will be a regular
registration fee of 500 DKK for all participants, except Masters
students, and the
participants are expected to have their expenses covered by their
home institutions or from other sources.
The course will also be open to participants
from the financial,
insurance and other industries at an additional fee. For those participants,
the regular fee of 500 DKK and the additonal fee
will be charged by Adept Scientific.
Please register via
Adept Scientific
if you work in industry and are interested in attending this course.
Masters students pay a fee of 100 DKK.
This fee does not include lunches.
Please register via the registration form
at your earliest convenience before May 20, 2003.
The programme of the Course will be on the web after May 20.
The Course starts on Monday, 26 May, 10 a.m. and finishes on
Friday, 30 May, 12 a.m.
More Information
We have a page with information
on how to get to the HC Ørsteds Institute, where the Course
will be given.
Do not hesitate to contact the MaPhySto secretariat
(maphysto@maphysto.dk)
, the secretaries
of the Laboratory of Actuarial Mathematics (Actuarial@act.ku.dk)
or the local organizers
Thomas Mikosch (mikosch@math.ku.dk)
and
Jeff Collamore
(collamore@math.ku.dk)
for more information.
This document was last modified April 29,
2003. Questions or comments to the contents of this document should
be directed to
Actuarial@act.ku.dk.