# NMAK13005U Introduction to Extreme Value Theory (IntroExtremValue)(AAM)

MSc Programme in Actuarial Mathematics

In this course the student will learn about the basics of modern
extreme value theory.

These include the classical asymptotic theory about the weak limits
of standardized maxima and order statistics (Generalized Extreme
Value Distribution) and of the excesses above high thresholds
(Generalized Pareto Distribution) for sequences of iid random
variables. An important part occupies the classification of
distributions in different Maximum Domains of Attraction of the
limiting extreme value distributions. Based on this theory,
statistical tools and methods for detecting extremes and estimating
their distributions are considered. These include estimators

of the tail index of a Pareto-like distribution, the extreme
value index of a distribution, the parameters of an extreme value
distribution and the estimation of high/low quantiles
of a distribution and tail probabilities, possibly outside
the range of the data. We discuss notions such as Value-at-Risk and
Expected Shortfall which ate relevant for Quantitative Risk
Management and their relation with extreme value theory. In the end
of course, we discuss how the classical theory for independent
variables can be extended to dependent observations. Such
observations typically contain clusters of extreme values. We
will learn about the extremal index which measures the size of a
cluster and about the extremogram which measures lag-wise extremal
dependence in a time series. The theory will be illustrated by
various data sets from finance, insurance and
telecommunications.

In this course, the student will learn about the basics of
modern extreme value theory.
**Knowledge:**

In particular, he/she will know about the following topics:

Classical limit theory for sequences of iid observations and their
excesses above high thresholds.

Exploratory statistical tools for detecting and classifying
extremes.

Standard statistical methods and techniques for handling extreme
values, including estimation for extreme value distributions and in
their domains of attraction, the Peaks over Threshold (POT)
method for excesses above high thresholds.

Standard notions from Quantitative Risk Management such as
Value-at-Risk, Expected Shortfall, return period, t-year event, and
their relation with extreme value theory.

The notion of cluster of extremes for dependent data and how to
measure the size of clusters.
**Skills:**

At the end of the course, the student will be able to read
books, articles and journals which are devoted to topics of modern
extreme value theory and extreme value statistics.
**Competences:**

The student will be competent in modeling extremes of independent
and weakly dependent observations and be able to apply software
packages specialized for analyzing extreme values.

Example of course litterature:

C. Klueppelberg, P. Embrechts, T. Mikosch: Modelling Extremal Events for Insurance and Finance.

"Statistik 2 (Stat2)" or a similar course is recommended for the necessary knowledge of statistics.

In addition, two take home written assignments (mid term and final term tests) in which the student will solve some theoretical problems and get estimation experience with simulated and real-life financial and insurance data.

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- Credit
- 7,5 ECTS
- Type of assessment
- Oral examination, 30 minutes without preparation timeContinuous assessment, Two written assignmentsTwo written assigments during the course: Mid Term and Final Term.

Oral exam (30 minutes) without preparation time.

The total grade will be determined by the overall results for both the assignments and the oral presentation. All parts of the exam must be passed in order to pass the course. - Aid
- Only certain aids allowed
The oral final exam is without aids.

All aids are allowed for the two written assignments. - Marking scale
- 7-point grading scale
- Censorship form
- No external censorship
Several internal examiners for oral exam. One internal examiner for the written assignments.
- Re-exam
Oral examination (30 minutes) with internal censors without preparation time and without aids.

If the two assignments were not pass in the ordinary examination, the student must re-submit the two written assignments no later than two weeks before the beginning of the re-examination week.

##### Criteria for exam assesment

In order to obtain the grade 12 the student should convincingly and accurately demonstrate the knowledge, skills and competences described under Learning Outcome.

- Category
- Hours
- Lectures
- 35
- Course Preparation
- 56
- Exam Preparation
- 55
- Project work
- 60
- Total
- 206

### Course information

- Language
- English
- Course code
- NMAK13005U
- Credit
- 7,5 ECTS
- Level
- Full Degree Master
- Duration
- 1 block
- Placement
- Block 2
- Schedule
- A
- Course capacity
- No limit
- Continuing and further education
- Study board
- Study Board of Mathematics and Computer Science

##### Contracting department

- Department of Mathematical Sciences

##### Course responsibles

- Thomas Valentin Mikosch (7-726e707478686d457266796d33707a336970)

Office. 04.3.10