Post on 16-Dec-2015
“Advanced Topics in Finance and Engineering: Extreme Value Theory (EVT), Risk Management,
and Applications” Econ. & Mat. Enrique Navarrete
Palisade Risk ConferenceRio de Janeiro 2009
CONFERENCE
Extreme Value Theory
TOPICS:
• Introduction and motivation;• Use of the Gumbel distribution (Extreme Value
Distribution);• Use of the Generalized Extreme Value Distribution (GEV);
– Parameter estimation by Maximum Likelihood (MLE);– Identification of the tail parmeter (Hill’s method);– Estimation of extreme loss percentiles;
Examples
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Extreme Value Theory
Motivation:• Maximum insurance claims (monthly maxima, N = 90
months)
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Frauds (x)1 $256.9132 $150.0193 $151.5634 $154.1555 $156.4776 $158.5537 $161.5148 $162.8659 $166.021
10 $169.75311 $170.93012 $173.78613 $176.82814 $178.99315 $182.07316 $184.28817 $186.02418 $187.93719 $192.36920 …
What is the “maximimun” claims level we can expect ?
By simulation methods, could we expect to get a number larger than
the historical maximum?
70 …71 $407.37172 $419.05373 $421.36874 $430.99475 $444.76476 $446.24077 $455.95378 $463.73079 $474.91580 $487.68781 $494.44782 $507.04083 $518.97384 $533.72385 $550.38486 $557.65087 $577.51288 $585.97489 $606.91590 $633.334
Fraudes sobre $ 150,000
$0
$50.000
$100.000
$150.000
$200.000
$250.000
$300.000
$350.000
$400.000
$450.000
$500.000
1 8 15 22 29 36 43 50 57 64 71
Pérdidas
Val
or
de
la P
érd
ida
Extreme Value Theory
Motivation:• = RiskWeibull(1,2171;172469;RiskShift(144825))
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99,0% 1,0%
−∞ 0,749
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
Values in Millions ($)
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
Val
ue
s x
10
^-6
Weibull
Weibull
Minimum$144844,0368Maximum$1366916,6887Mean $306486,7506Std Dev $133527,4049Values 10000
Weibull
p x0,99 749.000
0,995 823.0000,999 987.000
Extreme Value Theory
Motivation:• = RiskWeibull(1,2171;172469;RiskShift(144825))
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Weibull
p x0,99 749.000
0,995 823.0000,999 987.000
For monthly data, how often should we expect to see the values at the 99,5 % and 99,9 % levels ?
Extreme Value Theory
Related Question:
• If the chance of volcanic eruption today is 0,006 %, how do we interpret this small probability ?
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Extreme Value Theory
Related Question:
• If
* N
then:
N = 1/ (0,006 %) = 16,666 days
= 45,6 years®Scalar Consulting, 2009
N (time window to see an event) = 1 / Probability
N (number of days) * Daily probability = 1 event
Extreme Value Theory
Back to Problem:
The percentiles we have calculated indicate possible claim values that can actually occur, therefore these are the minimum monthly reserves to be held to cover possible claims at these confidence levels VAR
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ConfidenceLevel: VAR Provision Capital
99% $ 749.000 $306.470 $442.53199,5% $ 823.000 $306.470 $516.53199,9% $ 987.000 $306.470 $680.531
Extreme Value Theory
Back to Problem:
Now these confidence levels have failure rates:
Example: By setting up a monthly reserve of $ 823,000 (VAR 99,5 %), we would expect to cover all claims approximately 199/200 months (= 99,5 %) and will not be able to cover claims approx. 1 every 200 months
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How many How manyCovers Fails months years
VAR 99 % 99% 1% 100 8,3VAR 99,5 % 99,5% 0,5% 200 16,7VAR 99,9 % 99,9% 0,1% 1000 83,3
Extreme Value Theory
Application:
How do we set an appropriate level of monthly reserves that fails (falls short of claims) approximately once every 2 years ?
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Extreme Value Theory
Application:
How do we set a appropriate level of monthly reserves that fail (fall short of claims) approximately once every 2 years ?
Failure rate = (1/24 ) months = 4,2 %
Confidence level = (1 - 1/24) = 95,8%
VAR 95,8% = $ 590,000.
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Extreme Value Theory
More Applications:
• How high should we build a dam that fails (allows flooding) once every 40 years ?
• How strong to build homes to support hurricanes and collapse every 80 years ?
• How resistant to build antennae in presence of very strong winds ?
• How strong to build materials in general?
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Extreme Value Theory
Generalized Extreme Value Distribution (GEV):
• Under certain conditions, the GEV distribution is the limit distribution of sequences of independent and identically distributed random variables.
= location parameter; = scale parameter = shape (tail) parameter
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Extreme Value Theory
Fisher-Tippett-Gnedenko Theorem:
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Loss Distribution
Pro
bab
ilit
y
$
Only 3 possible families of distributions for the maximumm depending on the parameter !
> 0 (Fréchet)
= 0 (Gumbel)
< 0 (Reversed Weibull)
Extreme Value Theory
Generalized Extreme Value Distribution (GEV):
• For modeling maxima, the case < 0 is not interesting (“thin tails”);
• For the case (Gumbel), we can take shortcuts and avoid estimating the tail parameter;
use Gumbel (Extreme Value Distribution);
• For the case > 0 (“fat tails”), we have to use the GEV Distribution and estimate the tail parameter (Hill’s Plot).
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Extreme Value Theory
Location and Scale parameters (MOM):
1) Obtain sample mean ( ) and sample standard deviation (s) from the series of maxima;
2) We are assuming initially that the distribution is Gumbel ( = 0);
3) Estimate location ( ) and scale parameters ( ) using formulas from Method of Moments (MOM);
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x
x
s 6
Formulas apply to Gumbel distribution
Extreme Value Theory
Location and Scale parameters (MOM):
where = Euler´s Constant :
.
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577216,0
Limiting difference between the harmonic series and the natural
logarithm
Extreme Value Theory
Example 1: (MOM)• Maximum losses (monthly, N = 60)
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Sample
sample mean 69.117sample std dev 51.935
Gumbel MOM Percentiles:p x
0,99 232.0200,9917 239.6000,995 260.1900,999 325.444
Loss Plot Position 11 225.500 99,17%2 200.000 97,50%3 190.000 95,83%4 185.000 94,17%5 150.000 92,50%6 140.000 90,83%7 135.000 89,17%8 130.000 87,50%9 120.000 85,83%
10 118.000 84,17%11 113.000 82,50%12 110.000 80,83%13 … 79,17%
MOM (Gumbel)
location 45.743scale 40.494
Extreme Value Theory
Location and Scale parameters (MLE):
• As an alternative to MOM, we can calculate the location
and scale parameters by Maximum Likelihod Estimation
ie. and that maximize the function:
N
i
iN
i
i xxN
11
expln
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Extreme Value Theory
Example 1: (MLE)• Maximum losses (monthly, N = 60)
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Loss Plot Position 11 225.500 99,17%2 200.000 97,50%3 190.000 95,83%4 185.000 94,17%5 150.000 92,50%6 140.000 90,83%7 135.000 89,17%8 130.000 87,50%9 120.000 85,83%
10 118.000 84,17%11 113.000 82,50%12 110.000 80,83%13 … 79,17%
MLE MOM (Gumbel) 46.170 location 45.743 37.286 scale 40.494
Gumbel MLE Percentiles: Gumbel MOM Percentiles:p x p x0,99 217.690 0,99 232.020
0,9917 224.669 0,9917 239.6000,995 243.628 0,995 260.1900,999 303.712 0,999 325.444
Extreme Value Theory
Example 1:• @RISK: =RiskExtvalue(46170;37285)
p x0,995 244.0000,999 302.000
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Extreme Value Theory
Example 1:• When distribution is Gumbel ( = 0), we can use the @RISK
Extreme Value distribution:
@RISK: =RiskExtvalue(46170;37285)
p x0,995 244.0000,999 302.000
MLE MOM (Gumbel) 46.170 location 45.743 37.286 scale 40.494
Gumbel MLE Percentiles: Gumbel MOM Percentiles:p x p x0,99 217.690 0,99 232.020
0,9917 224.669 0,9917 239.6000,995 243.628 0,995 260.1900,999 303.712 0,999 325.444
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Extreme Value Theory
Generalized Extreme Value Distribution (GEV):
• Since in general , we need to estimate this parameter by Hill’s Method.
• Graph of:
0
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Extreme Value Theory
Example 2: (MLE)• Maximum losses (monthly, N = 60)
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MLE MOM (Gumbel) 49.265 location 31.400 46.396 scale 87.560
To get the loss percentiles we need to estimate the shape parameter
Loss Plot Position 11 795.000 99,17%2 400.000 97,50%3 190.000 95,83%4 185.000 94,17%5 150.000 92,50%6 140.000 90,83%7 135.000 89,17%8 130.000 87,50%9 120.000 85,83%
10 118.000 84,17%11 113.000 82,50%12 110.000 80,83%13 … 79,17%
Location and scale parameters are very different, suggesting distribution
is not Gumbel
Extreme Value Theory
Example 2:• Hill’s Diagram
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Hill Plots
0,000
0,500
1,000
1,500
2,000
2,500
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57
Hill Plot 1
Hill Plot 2
= 0,4
Extreme Value Theory
Example 2: (MLE)• Maximum losses (monthly, N = 60)
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Loss Plot Position 11 795.000 99,17%2 400.000 97,50%3 190.000 95,83%4 185.000 94,17%5 150.000 92,50%6 140.000 90,83%7 135.000 89,17%8 130.000 87,50%9 120.000 85,83%
10 118.000 84,17%11 113.000 82,50%12 110.000 80,83%13 … 79,17%
p x p x0,99 262.692 0,99 732.769
0,9917 271.377 0,9917 799.8840,995 294.968 0,995 840.5930,999 369.732 0,999 1.612.214
MLEGumbel EV Percentiles: GEV Percentiles:
MLE
We obtain very different GEV percentiles since the distribution is
not Gumbel ( ).0
Extreme Value Theory
Example 2:
• Since , we cannot use the Gumbel distribution; either estimate and use EVT or use a @RISK distribution, (not the Extreme value Distribution ie.Gumbel) since it will stay short.
0
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Extreme Value Theory
Example 2:• @RISK: =RiskPearson5(2,2926;124899;RiskShift(-12413))
p x0,995 767.0000,999 1.593.000
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Extreme Value Theory
Example 1 (Gumbel):• Hill’s Plot
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= 0.06 (ie. for all practical purposes the
distribution is Gumbel)
Hill Plots
0,000
0,500
1,000
1,500
2,000
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57
Hill Plot 1
Hill Plot 2
Extreme Value Theory
Example 3:• Hill’s Plot
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Hill Plots
0.000
0.200
0.400
0.600
0.800
1.000
1 3 5 7 9 11 13 15 17 19 21
Hill Plot 1
Hill Plot 2
= 0.01 (Gumbel)
Extreme Value Theory
Example 4:• Hill’s Plot
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= 0.38 (not Gumbel,
use GEV)
Hill Plots
0,000
0,500
1,000
1,500
2,000
2,500
1 2 3 4 5 6 7 8 9 10 11 12 13
Hill Plot 1
Hill Plot 2