Transcript Modelos Avanzados - Palisade Corporation: Maker of Risk

CONFERENCE
“Advanced Topics in Finance
and Engineering: Extreme Value
Theory (EVT), Risk Management,
and Applications”
Econ. & Mat. Enrique Navarrete
Palisade Risk Conference
Rio de Janeiro 2009
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
®Scalar Consulting, 2009
INTRODUCTION TO
EXTREME VALUE
THEORY (EVT)
$500.000
Extreme Value Theory
$450.000
$400.000
$350.000
Motivation:
•
$250.000
Maximum insurance claims (monthly maxima, N = 90
$200.000
$150.000
months)
Frauds (x)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$300.000
$256.913
$150.019
$151.563
$154.155
$156.477
$158.553
$161.514
$162.865
$166.021
$169.753
$170.930
$173.786
$176.828
$178.993
$182.073
$184.288
$186.024
$187.937
$192.369
…
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
$100.000
…
$407.371
$419.053
$421.368
$430.994
$444.764
$446.240
$455.953
$463.730
$474.915
$487.687
$494.447
$507.040
$518.973
$533.723
$550.384
$557.650
$577.512
$585.974
$606.915
$633.334
®Scalar Consulting, 2009
$50.000
$0
What is the “maximimun” claims
level we can expect ?
By simulation methods, could we
expect to get a number larger than
the historical maximum?
Extreme Value Theory
Motivation:
•
= RiskWeibull(1,2171;172469;RiskShift(144825))
Weibull
p
0,99
0,995
0,999
®Scalar Consulting, 2009
x
749.000
823.000
987.000
Extreme Value Theory
Motivation:
•
= RiskWeibull(1,2171;172469;RiskShift(144825))
Weibull
p
0,99
0,995
0,999
x
749.000
823.000
987.000
For monthly data, how often should we expect to
see the values at the 99,5 % and 99,9 % levels ?
®Scalar Consulting, 2009
Extreme Value Theory
Related Question:
•
If the chance of volcanic eruption today is
0,006 %, how do we interpret this small
probability ?
®Scalar Consulting, 2009
Extreme Value Theory
Related Question:
•
If
N (number of days) * Daily probability = 1 event
*
N
then:
N (time window to see an event) = 1 / Probability
N = 1/ (0,006 %) = 16,666 days
= 45,6 years
®Scalar Consulting, 2009
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
Confidence
Level:
99%
99,5%
99,9%
VAR
$ 749.000
$ 823.000
$ 987.000
®Scalar Consulting, 2009
Provision
$306.470
$306.470
$306.470
Capital
$442.531
$516.531
$680.531
Extreme Value Theory
Back to Problem:
Now these confidence levels have failure rates:
VAR 99 %
VAR 99,5 %
VAR 99,9 %
Covers
99%
99,5%
99,9%
Fails
1%
0,5%
0,1%
How many
months
100
200
1000
How many
years
8,3
16,7
83,3
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
®Scalar Consulting, 2009
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 ?
®Scalar Consulting, 2009
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.
®Scalar Consulting, 2009
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?
®Scalar Consulting, 2009
EXTREME VALUE THEORY
AND APPLICATIONS
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
®Scalar Consulting, 2009
Extreme Value Theory
Fisher-Tippett-Gnedenko Theorem:
Probability
Only 3 possible families
of distributions for the
maximumm depending
on the parameter  !
Loss Distribution
®Scalar Consulting, 2009
$

> 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).
®Scalar Consulting, 2009
Gumbel Distribution
(Extreme Value Distribution)

=0
Extreme Value Theory
Location and Scale parameters (MOM):
1) Obtain sample mean ( x ) 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);
  x   
®Scalar Consulting, 2009

6 s

Formulas apply to
Gumbel distribution
Extreme Value Theory
Location and Scale parameters (MOM):
where
 = Euler´s Constant :
Limiting difference between the
harmonic series and the natural
logarithm
.
  0,577216
®Scalar Consulting, 2009
Extreme Value Theory
Example 1: (MOM)
•
Maximum losses (monthly, N = 60)
1
2
3
4
5
6
7
8
9
10
11
12
13
Loss
225.500
200.000
190.000
185.000
150.000
140.000
135.000
130.000
120.000
118.000
113.000
110.000
…
Plot Position 1
99,17%
97,50%
95,83%
94,17%
92,50%
90,83%
89,17%
87,50%
85,83%
84,17%
82,50%
80,83%
79,17%
®Scalar Consulting, 2009
sample mean
sample std dev
Sample
69.117
51.935
MOM (Gumbel)
location
45.743
scale
40.494
Gumbel MOM Percentiles:
p
x
0,99
232.020
0,9917
239.600
0,995
260.190
0,999
325.444
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:
  xi   
 xi    N
  N  ln    

exp

 


  i 1
i 1 
   
N
®Scalar Consulting, 2009
Extreme Value Theory
Example 1: (MLE)
•
Maximum losses (monthly, N = 60)
1
2
3
4
5
6
7
8
9
10
11
12
13
Loss
225.500
200.000
190.000
185.000
150.000
140.000
135.000
130.000
120.000
118.000
113.000
110.000
…
Plot Position 1
99,17%
97,50%
95,83%
94,17%
92,50%
90,83%
89,17%
87,50%
85,83%
84,17%
82,50%
80,83%
79,17%
®Scalar Consulting, 2009


MLE
46.170
37.286
Gumbel MLE Percentiles:
p
x
0,99
217.690
0,9917
224.669
0,995
243.628
0,999
303.712
MOM (Gumbel)
location
45.743
scale
40.494
Gumbel MOM Percentiles:
p
x
0,99
232.020
0,9917
239.600
0,995
260.190
0,999
325.444
Extreme Value Theory
Example 1:
•
@RISK: =RiskExtvalue(46170;37285)
p
0,995
0,999
®Scalar Consulting, 2009
x
244.000
302.000
Extreme Value Theory
Example 1:
•
When distribution is Gumbel (  = 0), we can use the @RISK
Extreme Value distribution:


MLE
46.170
37.286
MOM (Gumbel)
location
45.743
scale
40.494
Gumbel MLE Percentiles:
p
x
0,99
217.690
0,9917
224.669
0,995
243.628
0,999
303.712
®Scalar Consulting, 2009
@RISK: =RiskExtvalue(46170;37285)
Gumbel MOM Percentiles:
p
x
0,99
232.020
p
0,9917
239.600
0,995
0,995
260.190
0,999
0,999
325.444
x
244.000
302.000
Generalized Extreme
Value Distribution (GEV)

>0
Extreme Value Theory
Generalized Extreme Value Distribution (GEV):
•
Since in general   0 , we need to estimate this
parameter by Hill’s Method.
•
Graph of:
®Scalar Consulting, 2009
Extreme Value Theory
Example 2: (MLE)
•
Maximum losses (monthly, N = 60)
1
2
3
4
5
6
7
8
9
10
11
12
13
Loss
795.000
400.000
190.000
185.000
150.000
140.000
135.000
130.000
120.000
118.000
113.000
110.000
…
Plot Position 1
99,17%
97,50%
95,83%
94,17%
92,50%
90,83%
89,17%
87,50%
85,83%
84,17%
82,50%
80,83%
79,17%
®Scalar Consulting, 2009


MLE
49.265
46.396
MOM (Gumbel)
location
31.400
scale
87.560
Location and scale parameters are
very different, suggesting distribution
is not Gumbel
To get the loss percentiles we need to
estimate the shape parameter 
Extreme Value Theory
Example 2:
•
Hill’s Diagram
Hill Plots
2,500
2,000
1,500
Hill Plot 1
1,000
Hill Plot 2
0,500
0,000
1
5
9 13 17 21 25 29 33 37 41 45 49 53 57
®Scalar Consulting, 2009
 = 0,4
Extreme Value Theory
Example 2: (MLE)
•
Maximum losses (monthly, N = 60)
1
2
3
4
5
6
7
8
9
10
11
12
13
Loss
795.000
400.000
190.000
185.000
150.000
140.000
135.000
130.000
120.000
118.000
113.000
110.000
…
Plot Position 1
Gumbel EV Percentiles:
99,17%
97,50%
MLE
95,83%
p
x
94,17%
0,99
262.692
92,50%
0,9917
271.377
90,83%
0,995
294.968
89,17%
0,999
369.732
87,50%
85,83%
84,17%
82,50%
80,83%
79,17%
®Scalar Consulting, 2009

GEV Percentiles:
MLE
p
x
0,99
732.769
0,9917
799.884
0,995
840.593
0,999
1.612.214
We obtain very different GEV
percentiles since the distribution is
not Gumbel (   0 ).
Extreme Value Theory
Example 2:
•
Since   0 , 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.
®Scalar Consulting, 2009
Extreme Value Theory
Example 2:
•
@RISK: =RiskPearson5(2,2926;124899;RiskShift(-12413))
p
0,995
0,999
®Scalar Consulting, 2009
x
767.000
1.593.000
Extreme Value Theory
Example 1 (Gumbel):
•
Hill’s Plot
Hill Plots
2,000
1,500
Hill Plot 1
1,000
Hill Plot 2
0,500
0,000
1 5
9 13 17 21 25 29 33 37 41 45 49 53 57
®Scalar Consulting, 2009

= 0.06 (ie. for
all practical
purposes the
distribution is
Gumbel)
Extreme Value Theory
Example 3:
•
Hill’s Plot
Hill Plots
1.000
0.800
0.600
Hill Plot 1
0.400
Hill Plot 2

0.200
0.000
1
3
5
7
9
11
13
®Scalar Consulting, 2009
15
17
19
21
= 0.01
(Gumbel)
Extreme Value Theory
Example 4:
•
Hill’s Plot
Hill Plots
2,500
2,000
1,500
Hill Plot 1
1,000
Hill Plot 2

0,500
0,000
1
2
3
4
5
6
7
8
®Scalar Consulting, 2009
9 10 11 12 13
= 0.38 (not
Gumbel,
use GEV)
Enrique Navarrete , Scalar Consulting
•
•
•
•
•
•
www.grupoescalar.com
[email protected]
MSc. University of Chicago
BS. Economics, BS. Mathematics, MIT
Risk Software, Consulting and Auditing
Risk courses offered jointly with Universidad
Iberoamericana, several countries