a short introduction to extreme value theory · a short introduction to extreme value theory paddy...

A Short Introduction toExtreme Value Theory

Paddy Paddam

GIRO/CAS Convention 2001

Email: [email protected] Tel: +44 (0)20 7804 0830

Contents

• Introduction and Context

• Theory

• Short example and application issues

• Comments and discussion

Intro Context EVT Example Discussion

Loss severity

• The distribution of the size of a loss

• Common problem:

– Pricing

– Reinsurance pricing and modelling

– Catastrophe models

– Securitisation

– Capital Management/DFA

– Operational Risk Management

Intro Context EVT Example Discuss

Loss severity - Central Limit Theorem

• Use the normal distribution for modelling sample means

• Common problem:

– Pricing - Property damage claims on motor book

– Reinsurance pricing and modelling - low layers

– Securitisation - whole account portfolios

– Capital Management - attritional losses


Context - The problem


0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

1 10 100 1000 10000 100000 1000000 10000000

Empirical DF

Loss Severity (Log Scale)

Cum

ulative probability

The problem - The Tail


Loss Severity

Cum

ulative probability

97.50%

98.00%

98.50%

99.00%

99.50%

100.00%

0 500000 1000000 1500000 2000000 2500000

Which distribution would you use tomodel the extreme losses and why?Which distribution would you use tomodel the extreme losses and why?

Lognormal?

Pareto?

Gamma?

Weibull?

Lognormal?

Pareto?

Gamma?

Weibull?

7 PricewaterhouseCoopers

What is Extreme Value Theory?

• Statistical Theory of Extreme Events

• Fisher-Tippet Theorem

– For many loss distributions, the distribution of themaximum value of a sample is a generalised extreme valuedistribution.

• Generalised extreme value distributions are

– Heavy tailed => Frechet

– Medium tailed => Gumbel

– Short tailed => Weibull


Useful result in HydrologyUseful result in Hydrologyand and ClimatologyClimatology

Not so useful inNot so useful inInsurance?Insurance?

PBH Theorem

• Pickands-Balkema-de Haan theorem:

– For many loss distributions, the distribution of losses abovea high threshold is a Generalised Pareto Distribution.


0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

1 10 100 1000 10000 100000 1000000 10000000

Empirical DF

GPD

Even moreGPD

Normal

Peaks Over Threshold Method

For many loss distributions, the distribution of losses above a high

threshold is a Generalised Pareto Distribution

Pickands-Balkema-de Haan theorem

Peaks Over Threshold Method

For many loss distributions, the distribution of losses above a high

threshold is a Generalised Pareto Distribution

Pickands-Balkema-de Haan theorem

11

Generalised Pareto Distribution

• P(X<x|X>u) = 1 - [1+g(x-m)/s]^(-1/g) for g <> 01 - exp[-(x-m)/s] for g = 0

• Parameters:

– m = location

– s = spread

– g = shape

– u = threshold


0%

20%

40%

60%

80%

100%

0.1 1 10 100

Cum

ulat

ive

Dist

ribut

ion

Shape = 0 Shape = 0.5 Shape = 1

Fitting EVT distributions

• Maximum likelihood methods

• Probability weighted moments

• Variety of methods and software


Applying EVT - ExampleReinsurance modelling

• Considered a portfolio of 11 classes including Liability andProperty accounts

• Performed a standard stochastic claims frequency and severityanalysis

• In addition attempted to fit a GPD to the claims severity

• In our exercise, for 9 out of the 11 classes, the GPD was aboutas good or better than a standard loss distribution in modellingthe extreme tail values of the loss severity distributions.


90%

92%

94%

96%

98%

100%

0 100000 200000 300000 400000

Claim Size

Cum

ulat

ive P

roba

bilit

y

GammaGPDData

Fit to Claims Severity of a Non-MarineLiability Class


Choosing the ThresholdA Balancing Act


Lower ThresholdLarger Model error

More DataSmaller Parameter error

Higher ThresholdSmaller Model error

Less DataLarger Parameter error

One approach- Focus on shape parameter

• Shape parameter is the most important

• Higher shape parameter => Thicker tailed => Morelosses

• Compare the fitted shape parameter with the changein threshold (or equivalently the number ofexceedances).

• Identify a stable plateaux

• Threshold < Excess attachment point


Fitted GPD Shape Parameter Comparedto chosen threshold

00.20.40.60.8

11.21.41.6

0 2000 4000 6000 8000 10000 12000 14000Exceedences

Shap

e Par

amete

r


Decreasing ThresholdIncreasing Model Error

Decreasing Parameter Error

Simple Conclusion

• Fit a GPD to the tails of claims severity distributionsbecause it reduces Model error

• However Parameter error & Random fluctuations risksremain


Discussion

• Applications?

• Concerns?

• Questions?


References and Further Reading

• McNeil AJ Estimating the Tails of Loss Severity Distributions using Extreme Value Theory ASTIN Bulletin,Vol. 27 no 1, pp 117-137. http://www.casact.org/library/astin/vol27no1/117.pdf.

• J. Beirlant, G Matthys, G Dierckx, Heavy Tailed Distributions and Rating,ASTIN Bulletin, Vol 31, No 1, pp 37-58, May 2001 (Comprehensive list of references)

• Patrik P and Guiahi F, An Extrememly Important Application of Extreme Value Theory to Reinsurance Pricing,1998 CAS Spring Meeting Florida (A presentation of the analysis of ISO claims severity)

• McNeil AJ and Saladin T, The Peaks over Thresholds Method for Estimating High Quantiles of LossDistributions, 28 ASTIN Colloquium 1997 (POT method used in pricing reinsurance)

• Embrechts P, Extremes and Insurance, 28 ASTIN Colloquium 1997 (More examples)

• Embrechts P, Kluppelberg C, and Mikosh T, Modelling Extremal Events, Springer Verlag, Berlin 1997 (Themajor text book on the subject)

• Reiss R, and Thomas M, Statistical Analysis of Extreme Values, Birkhauser, Basel, Boston, 1997

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