Transcript Extreme Value Theory: A useful framework for modeling extreme OR events

Extreme Value Theory:
A useful framework for
modeling extreme OR
events
Dr. Marcelo Cruz
Risk Methodology Development
and Quantitative Analysis
abcd
Operational Risk Measurement
Agenda





Database Modeling
Measuring OR: Severity, Frequency
Using Extreme Value Theory
Causal Modeling: Using Multifactor Modeling
Plans for OR Mitigation
Operational Risk Database Modelling
Doubtful Legislation
ABSTRACT
PROBLEMS
Systems Problems
Human Errors
Poor Controls
PROCES
S
OBJECTIVE
PROBLEMS
Process Failures
Legal
suits
Failures in
the process
Interest expenses Booking errors
Consequence = -$$$!
(P&L Adjustments)
Data Model
CEF’s
Volumes
Sensitivity
Data Quality
Control Gaps
Organization
Automation
Levels
Business
Continuity
IT Environment
Process &
Systems Flux
Control
Measure
KCI’s
Nostro Breaks
Depot Breaks
Intersystembreaks
Intercompany breaks
Interdesk breaks
Control Account
breaks
Unmatched confirmations
Fails
Operations loss data
Market Risk adjustments
 Error financing costs
 Write offs
 Execution Errors

Risk Optimization
Operational Risk
Market Risk
Earnings
Credit Risk
Volatility
P&L
(Revenue)
Operational Risk
(Costs)
For the first time banks are considering impacts on the
P&L from the cost side!
Measuring Operational Risk
Building the Operational VaR
1) Estimating Severity
2) Estimating Frequency
Choosing the distribution
Estimating Parameters
Testing the Parameters
PDFsandCDFs
Quantiles
3) Aggregating Severity and Frequency
Monte Carlo Simulation
Validation and Backtesting
Measuring Operational Risk
Losses
sizes
(in $)
120
80
52
36
25
24
Location = Average = 34.6
Scale = St Deviation= 32.2
2 22 20
18
1 15
10
7
Time
f (x) =
1
2ps
e
(x-m)2
2s
f(x) = 1.08% (PDF - probability dist function)
= 30.3% (CDF - cumulative dist function)
Measuring Operational Risk
What number will correspond to 95% of the CDF?
(How do I protect myself 95% of the time?)
Quantile Function = (CDF)-1--> the inverse of the
CDF (Solves the CDF for x)
In Excel, Normal Quantile function = NORMINV function
Lognormal Quantile function = LOGINV function
In our example:
=NORMINV(95%,34.6,32.2) = 87.6
Heavier tail !
=LOGINV(95%,3.2,.78) = 92.7
(Not heavy enough as our “VaR” would have 1 violation!)
Measuring Operational Risk
EXTREME VALUE THEORY
Losses
sizes
(in $)
80
36
120
52
threshold
25
24
2 22 20
18
101 15
7
Time
A model chosen for its overall fit to all database may not provide
a particular good fit to the large losses. We need to fit a distribution
specifically for the extremes.
Measuring Operational Risk
Broadly two ‘types’ of Extremes:
Losses
sizes
(in $)
Losses
sizes
(in $)
120
80
36 52
25
24
2 22 20
18
10 15
7
Threshold
Time
Peaks over Threshold (P.O.T.)
Fits Generalised Pareto
Distribution (G.P.D.)
80
36
24
120
52
25
2 22 20
18
101 15
7
Time
Distribution of Maxima over
a certain period - Fits the
Generalised Extreme Dist (GEV)
Measuring Operational Risk
Extreme Value Theory
Losses
sizes
(in $)
1 k
ˆ
 =  ln x - ln k
k k =1
120
80
36 52
25
24
2 22 20
18
10 15
7
Hill Shape
Threshold
Graphical Tests
Time
QQ and ME-Plots
Choose distribution
Measuring Operational Risk
Back to the example, comparing the results:
=NORMINV(95%,34.6,32.2) = 87.6
=LOGINV(95%,3.2,.78)
= 92.7
1 violation
(largest event
= 120)
Using GEV (95%,3-parameter) =143.5
No violations !
Extreme Value Theory
Example: Frauds in a British Retail Bank
1
2
3
4
5
6
7
8
9
10
11
12
1992
1993
1994
1995
1996
907,077
845,000
734,900
550,000
406,001
360,000
360,000
350,000
220,357
182,435
68,000
50,000
1,100,000
650,000
556,000
214,635
200,000
160,000
157,083
120,000
78,375
52,049
51,908
47,500
6,600,000
3,950,000
1,300,000
410,061
350,000
200,000
176,000
129,754
109,543
107,031
107,000
64,600
600,000
394,672
260,000
248,342
239,103
165,000
120,000
116,000
86,878
83,614
75,177
52,700
1,820,000
750,000
426,000
423,320
332,000
294,835
230,000
229,369
210,537
128,412
122,650
89,540
Extreme Value Theory
Hill method for the estimation of the shape parameter:
gˆk , n
(H )
1
2
3
4
5
6
7
8
9
10
11
12
=( k-1
1
-1
ln Xj , n ln Xk , n)

i =1
1995
600,000.34
394,672.11
260,000.00
248,341.96
239,102.93
165,000.00
120,000.00
116,000.00
86,878.46
83,613.70
75,177.00
52,700.00
LogLosses
13.3046855
12.8858106
12.46843691
12.42256195
12.38464941
12.01370075
11.69524702
11.66134547
11.37226541
11.33396266
11.22760061
10.87237073
Hill Plot
Hill
0.418875
0.626811
0.463749
0.385724
0.679528
0.884727
0.792239
0.982289
0.911449
0.926666
1.197653
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
Extreme Value Theory
QQ-Plot 1995
QQ-Plots:
1
0.9
0.8
0.7
0.6
Plotting:
0.5
0.4
{Xk , n, F - ( pk , n) : k = 1,..., n}
0.3
0.2
0.1
0
n - k  0.5
pk , n =
n
0
0.2
0.4
0.6
0.8
1
1.2
where
Approximate linearity suggests
good fit
Uses:
1) Compare distributions
2) Identify outliers
3) Aid in finding estimates for the parameters
Extreme Value Theory
Parameter Estimation
Methods :
1) Maximum Likelihood (ML)
2) Probability Weighted Moments (PWM)
3) Moments
PWM works very well for small samples (OR case!) and it is simpler.
ML sometimes do not converge and the bias is larger.
Extreme Value Theory
PWM Method:
(Based on order statistics)
Plotting Position
pn , k =
n - j  0.5
k
GEV
1
ˆ r ( ) =
w
n
n
X
Ur
j, n
j, n
, r = 0,1,2
j =1
 = 7.8590c  2.9554c 2
Auxiliaries
c=
2 w2 - w1
log 2
3w3 - w1
log 3

e
o
Location = w1 Scale =
scale{1 -  (1   )}
w2
(1 - 2 ) (1   )
-

-u
u t -1du, t  0
Extreme Value Theory
1994 Plot Position w1
PP^2
w2
1 6,600,000.00 0.958333333
6325000 0.918403 6061458.333
2 3,950,000.00
0.875
3456250 0.765625 3024218.75
3 1,300,000.00 0.791666667 1029166.667 0.626736 814756.9444
4 410,060.72 0.708333333 290459.6767 0.501736 205742.271
5 350,000.00
0.625
218750 0.390625
136718.75
6 200,000.00 0.541666667 108333.3333 0.293403 58680.55556
7 176,000.00 0.458333333 80666.66667 0.210069 36972.22222
8 129,754.00
0.375
48657.75 0.140625 18246.65625
9 109,543.00 0.291666667 31950.04167 0.085069 9318.762153
10 107,031.20 0.208333333 22298.16667 0.043403 4645.451389
11 107,000.00
0.125
13375 0.015625
1671.875
12
64,600.00 0.041666667 2691.666667 0.001736 112.1527778
w0
w1
w2
1,125,332.41
968,966.58
864,378.56
c
Shape
Scale
Location
-0.07731282 Hill
-0.5899362 Gamma
612,300.60
1,101,869.17
1.56577
1.06
Extreme Value Theory
Parameter Estimation (PWM and Hill)
Parameter
1992
1993
1994
1995
1996
 Shape Parameter
0.959265
0.994119
1.56577
0.679518
1.07057
m Location Parameter
410,279.77
432,211.40
1,101,869.17
215,551.84
445,660.38
 Scale Parameter
147,105.40
298,067.91
612,300.60
25,379.83
361,651.03
The shape parameter was estimated by the Hill method and the scale and location by the PWM.
Testing the Model - Checking the Parameters
Based on simulation, techniques like Bootstrapping and Jackknife helps find confidence intervals and bias in the parameters
Jacknife Test for Model GEV
Shape Std Err = 0.4208, Scale Std Err = 116,122.0647,
Location Std Err = 126,997.6469
Shape
Scale
Location
1.2
350000
300000
Jackknife =>
Parameter Value
1
250000
0.8
200000
0.6
150000
0.4
100000
0.2
50000
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Loss Number Removed(Descending)
Let  be the estimate of a parameter vector  based on a
sample of operational loss events x = (x1 , …,xn). An
approximation to the statistical properties can be
obtained by studying a sample of B bootstrap estimators
 m(b) (b = 1,…,B), each obtained from a sample of m
observations, sampling with replacement from the
observed sample x. The bootstrap sample size, m, may be
larger or smaller than n. The desired sampling
characteristic is obtained from properties of the sample {
m(1),…,  m(b)}.
<= Bootstrapping
Frequency Distributions
Poisson Distribution:
= 102
Number of Frauds
January
February
March
April
May
June
July
95
82
114
74
79
160
110
x
August
115
118
126
91%
95%
99%
f ( x) = 
k =0
e- k
k!
Poisson
Poisson CDF
Poisson PDF
4.50%
100.00%
4.00%
90.00%
3.50%
80.00%
3.00%
70.00%
Other popular
distributions to
estimate frequency
are the geometric,
negative binomial,
binomial, weibull, etc
60.00%
2.50%
50.00%
2.00%
40.00%
1.50%
30.00%
1.00%
20.00%
0.50%
10.00%
0.00%
0
50
100
150
200
0.00%
0
20
40
60
80
100
120
140
160
Measuring Operational Risk
Severity
Frequency
Prob
Prob
Number of Losses
Losses sizes
Prob
Aggregated Loss Distribution
Need to be solved
by simulation

Aggregated losses
Alternatives:
1) Fast Fourier Transform
2) Panjer Algorithm
3) Recursion
pF
n
n =0
*n
X
( x)
No analytical
solution!
Model Backtesting and Validation
Currently for Market / Credit Risks
1 59
MRC mt  1 = max[ VaRmt (10,1); Smt x  VaRmt - i (10,1)]  CreditCh arg e
60 i =1
Multiplier based on Backtests (Between 3 and 4)
Model Backtesting and Validation
 n x
Pr( x) =   p * (1 - p) n - x
 x
LR = 2[ln(  x (1 -  ) n - x ) - ln( p x * (1 - p) n - x )]
Kupiec Test
Exceptions can be modelled as independent draws from a binomial distribution
Interval Forecast
Method
1 if t  1  VaR mt
Im t  1 = 
0 if t  1  VaR mt
Series must exhibit the property of correct conditional coverage (unconditional)
and serial independence
Regulatory Loss
Functions
Cmt
1
Cm =
 f (t
=
g (t
, VaRmt ) if t  1  VaR mt
 1, VaR mt ) if t  1  VaR mt
n
C
i =1
mt  i
1
Define benchmarks
(some subjectivity)
Under very general conditions, accurate VaR estimates will generate the
lowest possible numerical score
Understanding the Causes - Multifactor
Modeling
Try to link causes to loss
events
For Example:
January
February
March
April
May
June
July
We are trying to explain the frequency and severity of
frauds by using 3 different factors.
Number of Op Errors Losses ($$)
System Downtime N. of Employees No. of Transactions
95
1,200,000
20
16
1,003
82
920,000
17
16
910
114
1,770,987
30
14
1,123
74
652,000
15
17
903
79
710,345
16
17
910
160
2,100,478
41
13
1,250
110
1,650,000
33
14
1,196
N. of Op Errors = 88.88 + 6.92 System Downtime + 5.32 Employees - 0.22 N. of transactions
R2 = 95%, F-test = 20.69, p-value = (0.01)
Losses = 4,597,086.21 - 7,300.01 System Downtime - 286,228 .59 Employees + 1,193 N.of Tr.
R2 = 97%, F-test = 42.57, p-value = (0.00)
Understanding the Causes - Multifactor
Modeling
Benefits of the Model
1) Scenario Analysis / Stress Tests
Ex: Using confidence intervals (95%) of the parameters to estimate
the number of frauds and the losses ($$) for the next month.
2) Cost / Benefit Analysis
Ex: If we hire 1 employee costing 100,000/year the reduction in
losses is estimated to be 286,228.
Developing an OR Hedging Program
OPERATIONAL RISK
(MEASURED)
Internal
Capital Allocation
MITIGATION
(Non financial)
Risk Transfer
Insurance
Securitization
• Specific coverage
• General coverage rather
• Immediate protection
than specific risks
• It would not pay immediately against catastrophes
after catastrophe (although
some new products claim to do
so)
Developing an OR Hedging Program
AGENT
INSTRUMENT
FINANCIAL
INSTITUTION
RISK TRANSFER
COMPANYor SPV
Insurance policy
offered by RTC
Takes the Risk and issues
Bonds linked to
operational event at the
financialinstitution
Buy the bond
FINANCIAL RESULTS Paid a premium
RISKS
None up to the limit
insured
CAPITAL MARKET
Receives a commission
Recieves high yield
None
If the operational event
described in the bond
happens in the financial
institution, loss of some
or all the principal or
interest
Developing an OR Hedging Program
Retain
Insurance
ORL Bond
(OR insurance)
CDF
Optimal point
OpVar
Conclusion
• It is possible to use robust methods to measure OR
•
OR-related events does not follow Gaussian patterns
•
More than just finding an Operational VaR, it is necessary to relate the losses to
some tangible factors making OR management feasible
•
Detailed measurement means that product pricing may incorporate OR
•
Data collection is very important anyway!
My e-mail is
[email protected]