the modeling of rare events: f rom methodology to practice a nd back
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The Modeling of Rare Events: f rom Methodology to Practice a nd Back. Paul Embrechts Department of Mathematics Director of RiskLab , ETH Zurich Senior SFI Chair www.math.ethz.ch/~embrechts. Summary:. A bit of history A bit of theory An application Further work. - PowerPoint PPT PresentationTRANSCRIPT
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The Modeling of Rare Events:from Methodology to Practice
and Back
Paul EmbrechtsDepartment of Mathematics
Director of RiskLab, ETH Zurich Senior SFI Chair
www.math.ethz.ch/~embrechts
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Summary:
• A bit of history• A bit of theory• An application• Further work
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Perhaps the first:
Nicolaus Bernoulli (1687 – 1759)who, in 1709, considered the actuarial problem of calculating the mean duration of life of the lastsurvivor among n men of equal age who all die within t years. He re-duced this question to the following: n points lie at random on a straight line of length t,calculate the mean largest distance from the origin.
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Often quoted as the start:
Simon Denis Poisson (1781 – 1840)(however Cotes (1714), de Moivre (1718), ... )
1837 (p 206)
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Ladislaus J. von Bortkiewicz(1868 – 1931)
However, the real start with relevance to EVTwas given by:
The Law of Small Numbers
1898
(Prussian army horse-kick data)
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Developments in the early to mid 20th century:
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R.A. Fisher L.H.C. Tippett
M.R. Fréchet E.H.W. Weibull E.J. Gumbel
B.V. Gnedenko . . . R. von Mises
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With an early textbook summary:
(1958)
Emil Julius Gumbel(1891 – 1966)
Statistical Theory of Extreme Values and SomePractical Applications.National Bureau of Standards, 1954
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Then the later-20th Century explosion:
Laurens de HaanSidney I. Resnick
Richard L. Smith M. Ross Leadbetter
and so many more ...
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Reflected in:
o o o
and Black Swanary
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Recall the Central Limit Theorem:
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The basic 1-d EVT set-up,the «Extreme Value Theorem»:
EVT = Extreme Value Theory1-d = one dimensional
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Power lawBeware!
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An interludium on Regular Variation, more in particular, on the Slowly Varying L in Gnedenko’s Theorem:
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One further name and a book:
Jovan Karamata (1902 -1967)
by N.H. Bingham, C.M. Goldie and J.L. Teugels(1987): contains ALL about L-functions!
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EVT and the POT method
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Some isues:Practice is too often frequency oriented ... - every so often (rare event) - return period, 1 in x-year event - Value-at-Risk (VaR) in financial RM... rather than more relevant severity orientation - what if - loss size given the occurence of a rare event - Expected Shortfall E[X I X > VaR]This is not just about theory but a RM attitude!
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The Peaks Over Threshold (POT) Method
Crucial point!
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POT u
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Start an EVT-POT analysis:• First diagnostic checking• Statistical techniques• Graphical techniques• Standard software in all relevant hard- and
software environments: R, S-Plus, ...• In our example, McNeil’s QRM-LIB from: A.J. McNeil, R.Frey and P. Embrechts (2005)
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99%-quantile99%-conditional excess
99%-quantile with 95% aCI (Profile Likelihood): 27.3 (23.3, 33.1) 99% Conditional Excess: E( X I X > 27.3) with aCI
27.3u= (!)
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A warning on slow convergence!
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L matters!
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Some issues:• Extremes for discrete data: special theory• No unique/canonical theory for multivariate extremes
because of lack of standard ordering, hence theory becomes context dependent
• Interesting links with rare event simulation, large deviations and importance sampling
• High dimensionality, d > 3 or 4 (sic)• Time dependence (processes), non-stationarity• Extremal dependence ( financial crisis)• And finally ... APPLICATIONS ... COMMUNICATION !!
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Thank you!