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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

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

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  1. Extreme Value Theory: A useful framework for modeling extreme OR events Dr. Marcelo Cruz Risk Methodology Development and Quantitative Analysis abcd

  2. Operational Risk Measurement • Agenda • Database Modeling • Measuring OR: Severity, Frequency • Using Extreme Value Theory • Causal Modeling: Using Multifactor Modeling • Plans for OR Mitigation

  3. Failures in the process Operational Risk Database Modelling Doubtful Legislation Process Failures ABSTRACT Systems Problems PROBLEMS Human Errors Poor Controls PROCESS OBJECTIVE Legal PROBLEMS Interest expenses Booking errors suits (P&L Adjustments) Consequence = -$$$!

  4. KCI’s Nostro Breaks Depot Breaks Intersystem- breaks Intercompany - breaks Interdesk breaks Control Account breaks Unmatched -confirmations Fails Data Model Control Measure CEF’s Volumes Sensitivity Data Quality Operations loss data Control Gaps Organization • Market Risk adjustments • Error financing costs • Write offs • Execution Errors Automation Levels Business Continuity IT Environment Risk Optimization Process & Systems Flux

  5. Operational Risk Market Risk Earnings Volatility Credit Risk (Revenue) P&L Operational Risk (Costs) For the first time banks are considering impacts on the P&L from the cost side!

  6. Choosing the distribution Estimating Parameters Testing the Parameters PDFs and CDFs Quantiles 3) Aggregating Severity and Frequency Monte Carlo Simulation Validation and Backtesting Measuring Operational Risk Building the Operational VaR 1) Estimating Severity 2) Estimating Frequency

  7. 2 - m ( ) x 1 = s f ( x ) e 2 ps 2 Measuring Operational Risk Losses sizes (in $) Location = Average = 34.6 Scale = St Deviation= 32.2 120 80 52 36 25 24 22 21 20 18 15 10 7 Time f(x) = 1.08% (PDF - probability dist function) = 30.3% (CDF - cumulative dist function)

  8. 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 =NORMINV(95%,34.6,32.2) = 87.6 =LOGINV(95%,3.2,.78) = 92.7 Heavier tail ! (Not heavy enough as our “VaR” would have 1 violation!) Measuring Operational Risk What number will correspond to 95% of the CDF? (How do I protect myself 95% of the time?) In our example:

  9. Measuring Operational Risk EXTREME VALUE THEORY Losses sizes (in $) 120 80 52 36 threshold 25 24 22 21 20 15 18 10 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.

  10. Losses sizes (in $) 120 80 52 36 25 24 22 Threshold 2 20 15 18 10 7 Time Measuring Operational Risk Broadly two ‘types’ of Extremes: Losses sizes (in $) 120 80 52 36 25 24 22 20 21 15 18 10 7 Time 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)

  11. Losses sizes (in $) 120 80 52 36 25 24 22 Threshold 2 20 15 18 10 7 Time Measuring Operational Risk Hill Shape Extreme Value Theory Graphical Tests QQ and ME-Plots Choose distribution

  12. Measuring Operational Risk Back to the example, comparing the results: 1 violation (largest event = 120) =NORMINV(95%,34.6,32.2) = 87.6 =LOGINV(95%,3.2,.78) = 92.7 Using GEV (95%,3-parameter) =143.5 No violations !

  13. Extreme Value Theory Example: Frauds in a British Retail Bank

  14. Extreme Value Theory Hill method for the estimation of the shape parameter: 1 = å - - 1 ln X ln X ) ( H ) g ˆ ( j , n k , n k , n k-1 = i 1

  15. Extreme Value Theory QQ-Plots: Plotting: where Uses: 1) Compare distributions 2) Identify outliers 3) Aid in finding estimates for the parameters Approximate linearity suggests good fit

  16. 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.

  17. Extreme Value Theory PWM Method: (Based on order statistics) Plotting Position GEV Auxiliaries

  18. Extreme Value Theory

  19. Extreme Value Theory

  20. 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 1 250000 0.8 Parameter Value 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) Testing the Model - Checking the Parameters Based on simulation, techniques like Bootstrapping and Jack-knife helps find confidence intervals and bias in the parameters Jackknife => 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

  21. Frequency Distributions = 102 Number of Frauds January February March April May June July August 95 82 114 74 79 160 110 115 91% 118 95% 126 99% Poisson Poisson PDF 4.50% 4.00% 3.50% 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 0 50 100 150 200 Poisson Distribution: Poisson CDF Other popular distributions to estimate frequency are the geometric, negative binomial, binomial, weibull, etc 100.00% 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 0 20 40 60 80 100 120 140 160

  22. Need to be solved by simulation Alternatives : No analytical solution! 1) Fast Fourier Transform 2) Panjer Algorithm 3) Recursion Measuring Operational Risk Severity Frequency Prob Prob Number of Losses Losses sizes Aggregated Loss Distribution Prob Aggregated losses

  23. Model Backtesting and Validation Currently for Market / Credit Risks Multiplier based on Backtests (Between 3 and 4)

  24. Series must exhibit the property of correct conditional coverage (unconditional) and serial independence Define benchmarks (some subjectivity) Model Backtesting and Validation Kupiec Test Exceptions can be modelled as independent draws from a binomial distribution Interval Forecast Method Regulatory Loss Functions Under very general conditions, accurate VaR estimates will generate the lowest possible numerical score

  25. 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) 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. 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)

  26. 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.

  27. Developing an OR Hedging Program OPERATIONAL RISK (MEASURED) MITIGATION (Non financial) Internal Risk Transfer Insurance Capital Allocation Securitization • Specific coverage • Immediate protection • against catastrophes • General coverage rather • than specific risks • It would not pay immediately • after catastrophe (although • some new products claim to do • so)

  28. Developing an OR Hedging Program

  29. Developing an OR Hedging Program ORL Bond (OR insurance) Retain Insurance CDF Optimal point OpVar

  30. 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 marcelo.cruz@ubsw.com

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