# High Risk Scenarios and Extremes: A Geometric Approach

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*Guus Balkema; Paul Embrechts*

A publication of the European Mathematical Society

Quantitative Risk Management (QRM) has become a field of research of
considerable importance to numerous areas of application, including
insurance, banking, energy, medicine, and reliability. Mainly motivated by
examples from insurance and finance, the authors develop a theory for
handling multivariate extremes. The approach borrows ideas from
portfolio theory and aims at an intuitive approach in the spirit of the
Peaks over Thresholds method. The point of view is geometric. It leads
to a probabilistic description of what in QRM language may be referred
to as a high risk scenario: the conditional behaviour of risk factors
given that a large move on a linear combination (portfolio, say) has
been observed. The theoretical models which describe such conditional
extremal behaviour are characterized and their relation to the limit
theory for coordinatewise maxima is explained.

The first part is an elegant exposition of coordinatewise extreme value
theory; the second half develops the more basic geometric theory.
Besides a precise mathematical deduction of the main results, the text
yields numerous discussions of a more applied nature. A twenty page
preview introduces the key concepts; the extensive introduction provides
links to financial mathematics and insurance theory.

The book is based on a graduate course on point processes and extremes.
It could form the basis for an advanced course on multivariate extreme
value theory or a course on mathematical issues underlying risk.
Students in statistics and finance with a mathematical, quantitative
background are the prime audience. Actuaries and risk managers involved
in data based risk analysis will find the models discussed in the book
stimulating. The text contains many indications for further research.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students interested in applications.