Extreme Value Statistics (applied to Actuarial and Financial data)

72 Pages·2013·1.6 MB·English

One-dimensionalExtremeValueStatistics
TailDependence
MarginalExpectedShortfall
ExtremeRiskRegions
Extreme Value Statistics
(applied to Actuarial and Financial data)
John H.J. Einmahl
Tilburg University
ASTIN Colloquium
The Hague, May 22, 2013
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One-dimensionalExtremeValueStatistics
TailDependence
MarginalExpectedShortfall
ExtremeRiskRegions
Outline
One-dimensional Extreme Value Statistics
Tail Dependence
Marginal Expected Shortfall
Extreme Risk Regions
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One-dimensionalExtremeValueStatistics
TailDependence
MarginalExpectedShortfall
ExtremeRiskRegions
Introduction: How high, how long, how fast?
(cid:73) We often encounter questions about extreme events.
(cid:73) How high should the dikes in the Netherlands be in order
to protect us against extreme water levels?
(Disregarding “once in ten thousand years” storms.)
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One-dimensionalExtremeValueStatistics
TailDependence
MarginalExpectedShortfall
ExtremeRiskRegions
(cid:73) How much can we lose on one day on a certain portfolio
of shares?
(Disregarding “once in thousand” losses.)
(cid:73) How long can humans live?
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One-dimensionalExtremeValueStatistics
TailDependence
MarginalExpectedShortfall
ExtremeRiskRegions
(cid:73) How strong can an induced earthquake in the province of
Groningen be?
(cid:73) What can we say about extreme precipitation levels,
extreme wind speeds?
(cid:73) How fast can we, humans, run (the 100 meters)?
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TailDependence
MarginalExpectedShortfall
ExtremeRiskRegions
(cid:73) What is the maximal possible production of a firm, given
certain inputs?
(Estimate the efficient production frontier.)
(cid:73) How large should be the capital buffer of a financial
institution?
(cid:73) What will be the total (insured) loss of the first hurricane
topping Katrina?
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One-dimensionalExtremeValueStatistics
TailDependence
MarginalExpectedShortfall
ExtremeRiskRegions
Top 40 Insurance Losses
Tables showing the major losses 1970–2012
Table 9
The 40 most costly insurance losses (1970–2012)
Insured loss27
(in USD m, Date
indexed to 2012) Victims28 (start) Event Country
76 25429 1 836 25.08.2005 Hurricane Katrina; floods, dams burst, US, Gulf of Mexico, Bahamas,
damage to oil rigs North Atlantic
3355 70305030 19 213375 1214..0130..22001112 EHaurrtrhicqaunaek eS a(MndWy 9; f.l0o)o tdrisggers tsunami; aftershocks JUaSp eatn al
26 180 43 23.08.1992 Hurricane Andrew; floods US, Bahamas
24 349 2 982 11.09.2001 Terror attack on WTC, Pentagon and other buildings US
21 685 61 17.01.1994 Northridge earthquake (M 6.6) US
21 585 136 06.09.2008 Hurricane Ike; floods, offshore damage US, Caribbean: Gulf of Mexico et al
15 672 124 02.09.2004 Hurricane Ivan; damage to oil rigs US, Caribbean; Barbados et al
15 315 815 27.07.2011 Floods caused by heavy monsoon rains Thailand
1154 371752 13851 2129..0120..22001015 EHaurrtrhicqaunaek eW (MilmWa 6; f.3lo)o, dafstershocks NUSew, M Zeexaiclaon, dJamaica, Haiti et al
11 869 34 20.09.2005 Hurricane Rita; floods, damage to oil rigs US, Gulf of Mexico, Cuba
11 00031 123 15.07.2012 Drought in the Corn Belt US
9 784 24 11.08.2004 Hurricane Charley; floods US, Cuba, Jamaica et al
9 517 51 27.09.1991 Typhoon Mireille/No 19 Japan
8 467 71 15.09.1989 Hurricane Hugo US, Puerto Rico et al
88 422015 59652 2275..0021..21091900 EWairnthteqru satokerm (M DWa r8ia.8) triggers tsunami CFrhainlece, UK, Belgium, Netherlands et al
7 994 110 25.12.1999 Winter storm Lothar Switzerland, UK, France et al
7 453 354 22.04.2011 Major storm with wind up to 340km/h, United States (Alabama et al)
over 355 tornadoes
7 198 155 20.05.2011 Major tornado outbreak, United States (Missouri et al)
storms with winds up to 405km/h
6 748 54 18.01.2007 Winter storm Kyrill; floods Germany, UK, Netherlands, Belgium et al
6 264 22 15.10.1987 Storm and floods in Europe France, UK, Netherlands et al
6 255 38 26.08.2004 Hurricane Frances US, Bahamas
5 952 55 22.08.2011 Hurricane Irene, extensive flooding United States et al
5 607 64 25.02.1990 Winter storm Vivian Europe
5 568 26 22.09.1999 Typhoon Bart/No 18 Japan
54 296732 600- 0240..0099..21091908 EHaurrtrhicqaunaek eG (eMorWg e7s.0; )f,l ooovdesr 300 aftershocks NUSew, C Zaeriablbaenadn
4 673 41 05.06.2001 Tropical storm Allison; floods US
4 622 3 034 13.09.2004 Hurricane Jeanne; floods, landslides US, Caribbean: Haiti et al
4 357 45 06.09.2004 Typhoon Songda/No 18 Japan, South Korea
4 000 45 02.05.2003 Thunderstorms, tornadoes, hail US
3 890 70 10.09.1999 Hurricane Floyd; floods US, Bahamas, Columbia
3 775 59 01.10.1995 Hurricane Opal; floods US, Mexico, Gulf of Mexico
3 724 6 425 17.01.1995 Great Hanshin earthquake (M 7.2) in Kobe Japan
3 489 25 24.01.2009 Winter storm Klaus, wind up to 170km/h France, Spain
3 308 45 27.12.1999 Winter storm Martin Spain, France, Switzerland
3 119 246 10.03.1993 Blizzard, tornadoes, floods US, Canada, Mexico, Cuba
2 947 38 06.08.2002 Severe floods UK, Spain, Germany, Austria et al

2728293031
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One-dimensionalExtremeValueStatistics
TailDependence
MarginalExpectedShortfall
ExtremeRiskRegions
Rank Insured loss Event
1 76254 Katrina
2 35735 Japan
3 35000 Sandy
4 26180 Andrew
5 24349 9/11
··· ···
10 15315
··· ···
20 7453
··· ···
30 4673
··· ···
40 2947
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Cauchy data −− Hill estimator
8
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8 Hill
0. true
0 200 400 600 800
k
One-dimensionalExtremeValueStatistics
DTaailnDiespehnd efnircee data −− Hill estimator
MarginalExpectedShortfall
ExtremeRiskRegions
0
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-p.23/37
Large Danish fire insurance losses; threshold of 1 million DK.
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TailDependence
MarginalExpectedShortfall
ExtremeRiskRegions
(cid:73) For these data we can, as usual, compute the sample
mean and the sample variance.
(cid:73) However, for these loss data γˆ ≈ 0.7. This γˆ estimates
the important parameter γ, the extreme value index. The
extreme value index describes the tail heaviness of the
claims distribution. Again, these losses are from a
heavy-tailed distribution.
(cid:73) If X denotes a loss, then EXr = ∞ for r > 1/γ.
(cid:73) The estimate 0.7 strongly indicates that VarX might not
exist. Such a portfolio constitutes a real danger for the
insurer.
(cid:73) Because of this alarming message based on rather usual
insurance data, we need to understand the parameter γ
and its estimator γˆ.
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