extreme value theory: a useful framework for modeling extreme or events dr. marcelo cruz risk...

Extreme Value Theory:

A useful framework for modeling

extreme OR events

Dr. Marcelo CruzRisk Methodology Development

and Quantitative Analysisabcd

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

PROCESS

Legalsuits

Interest expensesBooking errors (P&L Adjustments)

Failures inthe process

Consequence = -$$$!

Human Errors

Systems Problems Poor Controls

Process FailuresABSTRACTPROBLEMS

OBJECTIVEPROBLEMS

Doubtful Legislation

Data Model

Market Risk adjustments Error financing costs Write offs Execution Errors

Market Risk adjustments Error financing costs Write offs Execution Errors

Operations loss dataOperations loss data

MeasureControl

Risk OptimizationRisk Optimization

Data Quality

Control Gaps

Organization

Volumes Sensitivity

Automation Levels

Business Continuity

CEF’sCEF’s

IT Environment

Process & Systems Flux

KCI’s

Nostro Breaks Depot Breaks Intersystem-

breaksIntercompany -

breaksInterdesk breaksControl Account

breaksUnmatched -

confirmationsFails

Operational Risk

P&L

Earnings

Volatility

Market Risk

Credit Risk

Operational Risk

(Revenue)

(Costs)

For the first time banks are considering impacts on theP&L from the cost side!

Measuring Operational Risk

Building the Operational VaR

1) Estimating Severity

2) Estimating Frequency

3) Aggregating Severity and FrequencyMonte Carlo SimulationValidation and Backtesting

Choosing the distributionEstimating ParametersTesting the Parameters

PDFs and CDFsQuantiles


10

2436

22

7

120

15

Lossessizes(in $)

Time

52

2021 18

80

25

Location = Average = 34.6Scale = St Deviation= 32.2

2)( 2x

2

1)( exf

f(x) = 1.08% (PDF - probability dist function) = 30.3% (CDF - cumulative dist function)


What number will correspond to 95% of the CDF?(How do I protect myself 95% of the time?)

In Excel, Normal Quantile function = NORMINV functionLognormal Quantile function = LOGINV function

Quantile Function = (CDF)-1--> the inverse of the CDF (Solves the CDF for x)

=NORMINV(95%,34.6,32.2) = 87.6=LOGINV(95%,3.2,.78) = 92.7

(Not heavy enough as our “VaR” would have 1 violation!)

Heavier tail !

In our example:


EXTREME VALUE THEORY

10

24

36

22

7

120

15

Lossessizes(in $)

Time

52

2021 18

80

25threshold

A model chosen for its overall fit to all database may not providea particular good fit to the large losses. We need to fit a distributionspecifically for the extremes.


Broadly two ‘types’ of Extremes:

Peaks over Threshold (P.O.T.)

Fits Generalised Pareto Distribution (G.P.D.)

Distribution of Maxima over a certain period - Fits the

Generalised Extreme Dist (GEV)

10

2436

22

7

120

15

52

2021 18

80

25

10

24

36

22

7

120

15

Time

52

20218

80

25

Threshold

Lossessizes(in $)

Time

Lossessizes(in $)


Extreme Value Theory

10

24

36

22

7

120

15

Time

52

20218

80

25

Threshold

Lossessizes(in $)

Hill Shape

Graphical TestsQQ and ME-Plots

Choose distribution

k

k

kxk 1

lnln1̂


=NORMINV(95%,34.6,32.2) = 87.6=LOGINV(95%,3.2,.78) = 92.7

1 violation(largest event = 120)

No violations !

Back to the example, comparing the results:

Using GEV (95%,3-parameter) =143.5


1992 1993 1994 1995 19961 907,077 1,100,000 6,600,000 600,000 1,820,000

2 845,000 650,000 3,950,000 394,672 750,000

3 734,900 556,000 1,300,000 260,000 426,000

4 550,000 214,635 410,061 248,342 423,320

5 406,001 200,000 350,000 239,103 332,000

6 360,000 160,000 200,000 165,000 294,835

7 360,000 157,083 176,000 120,000 230,000

8 350,000 120,000 129,754 116,000 229,369

9 220,357 78,375 109,543 86,878 210,537

10 182,435 52,049 107,031 83,614 128,412

11 68,000 51,908 107,000 75,177 122,650

12 50,000 47,500 64,600 52,700 89,540

Example: Frauds in a British Retail Bank


1

1

,,, )lnln(ˆ)(

i

nknjnk XXk-1

H

Hill method for the estimation of the shape parameter:

1995 LogLosses Hill1 600,000.34 13.30468552 394,672.11 12.8858106 0.4188753 260,000.00 12.46843691 0.6268114 248,341.96 12.42256195 0.4637495 239,102.93 12.38464941 0.3857246 165,000.00 12.01370075 0.6795287 120,000.00 11.69524702 0.8847278 116,000.00 11.66134547 0.7922399 86,878.46 11.37226541 0.982289

10 83,613.70 11.33396266 0.91144911 75,177.00 11.22760061 0.92666612 52,700.00 10.87237073 1.197653

Hill Plot

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 2 3 4 5 6 7 8 9 10 11 12

1


QQ-Plot 1995

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2

QQ-Plots:

Plotting:

},...,1:)(,{ ,, nkpFX nknk

Approximate linearity suggestsgood fit

n

knp nk

5.0,

where

1) Compare distributions2) Identify outliers3) Aid in finding estimates for the parameters

Uses:


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.

Parameter Estimation


PWM Method:

(Based on order statistics)

)1()21(

)}1(1{

3log

2log

3

2

2.9554c7.8590c

0,1,2r , 1

)(ˆ

2

1

13

12

2

1

,,

wScale

scalewLocation

ww

wwc

UXn

wn

j

njr

njr

Plotting Position

Auxiliaries

e u du tu t

o

1 0,

k

jnp kn

5.0,

GEV


1994 Plot Position w1 PP^2 w21 6,600,000.00 0.958333333 6325000 0.918403 6061458.3332 3,950,000.00 0.875 3456250 0.765625 3024218.753 1,300,000.00 0.791666667 1029166.667 0.626736 814756.94444 410,060.72 0.708333333 290459.6767 0.501736 205742.2715 350,000.00 0.625 218750 0.390625 136718.756 200,000.00 0.541666667 108333.3333 0.293403 58680.555567 176,000.00 0.458333333 80666.66667 0.210069 36972.222228 129,754.00 0.375 48657.75 0.140625 18246.656259 109,543.00 0.291666667 31950.04167 0.085069 9318.762153

10 107,031.20 0.208333333 22298.16667 0.043403 4645.45138911 107,000.00 0.125 13375 0.015625 1671.87512 64,600.00 0.041666667 2691.666667 0.001736 112.1527778

c -0.07731282 Hill 1.56577w0 1,125,332.41 Shape -0.5899362 Gamma 1.06w1 968,966.58 Scale 612,300.60w2 864,378.56 Location 1,101,869.17


Parameter Estimation (PWM and Hill)

Parameter 1992 1993 1994 1995 1996

Shape Parameter 0.959265 0.994119 1.56577 0.679518 1.07057

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 ParametersBased on simulation, techniques like Bootstrapping and Jack-knife helps find confidence intervals and bias in the parameters

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)}.

Jackknife =>

<= Bootstrapping

Jacknife Test for Model GEV

Shape Std Err = 0.4208, Scale Std Err = 116,122.0647,Location Std Err = 126,997.6469

0

0.2

0.4

0.6

0.8

1

1.2

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)

Par

amet

er V

alu

e

0

50000

100000

150000

200000

250000

300000

350000

Shape Scale Location

Frequency Distributions

Number of Frauds = 102

January February March April May June July August

95 82 114 74 79 160 110 115 91%118 95%126 99%

Poisson

Poisson PDF

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

0 50 100 150 200

Poisson CDF

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0 20 40 60 80 100 120 140 160

!)(

0 k

exf

kx

k

Poisson Distribution:

Other popular distributions toestimate frequencyare the geometric,negative binomial,binomial, weibull, etc


No analyticalsolution!

Need to be solvedby simulation

Prob

Number of Losses

Prob

Frequency

Losses sizes

Prob

Aggregated losses

Aggregated Loss Distribution

)(0

* xFpn

nXn

Alternatives:1) Fast Fourier Transform2) Panjer Algorithm3) Recursion

Severity

Model Backtesting and Validation

59

1

1 arg)]1,10(60

1 x );1,10(max[

i

imtmtmtmt eCreditChVaRSVaRMRC

Multiplier based on Backtests (Between 3 and 4)

Currently for Market / Credit Risks

Model Backtesting and Validation

)])1(*ln())1([ln(2

)1(*)Pr(

xnxxnx

xnx

ppLR

ppx

nx

Kupiec Test

Exceptions can be modelled as independent draws from a binomial distribution

Interval Forecast Method

Regulatory Loss Functions

mt1t

mt1t1m

VaR if 0

VaR if 1I

t

Series must exhibit the property of correct conditional coverage (unconditional)and serial independence

n

1i

imtm

mt1tmt1t

mt1t11

CC

VaR if )VaR,( g

VaR if ),(

mtt

mtVaRf

C

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: 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 TransactionsJanuary 95 1,200,000 20 16 1,003 February 82 920,000 17 16 910

March 114 1,770,987 30 14 1,123 April 74 652,000 15 17 903 May 79 710,345 16 17 910 June 160 2,100,478 41 13 1,250 July 110 1,650,000 33 14 1,196

Losses = 4,597,086.21 - 7,300.01 System Downtime - 286,228 .59 Employees + 1,193 N.of Tr.

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)

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

• Specific coverage• Immediate protection against catastrophes

OPERATIONAL RISK(MEASURED)

Capital Allocation

Internal Risk Transfer

Insurance Securitization

• General coverage rather than specific risks• It would not pay immediately after catastrophe (although some new products claim to do so)

MITIGATION(Non financial)


AGENTFINANCIAL INSTITUTION

RISK TRANSFER COMPANY or SPV CAPITAL MARKET

INSTRUMENT 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 Receives a commission Recieves high yield

RISKSNone up to the limit insured None

If the operational event described in the bond happens in the financial institution, loss of some or all the principal or interest

OpVar

CDF

Insurance ORL Bond(OR insurance)

Retain

Optimal point


• 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!

Conclusion

My e-mail is [email protected]

extreme value theory: a useful framework for modeling extreme or events dr. marcelo cruz risk...

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