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

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

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

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

7. Measuring Operational Risk 10 24 36 22 7 120 15 Losses sizes (in $) Time 52 20 2121 18 80 25 Location = Average = 34.6 Scale = St Deviation= 32.2  2  )( 2  x  2 1 )(  exf f(x) = 1.08% (PDF - probability dist function) = 30.3% (CDF - cumulative dist function)

8. Measuring Operational Risk What number will correspond to 95% of the CDF? (How do I protect myself 95% of the time?) In Excel, Normal Quantile function = NORMINV function Lognormal 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:

9. Measuring Operational Risk EXTREME VALUE THEORY 10 24 36 22 7 120 15 Losses sizes (in $) Time 52 20 2121 18 80 25 threshold 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. Measuring Operational Risk 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 24 36 22 7 120 15 52 20 2121 18 80 25 10 24 36 22 7 120 15 Time 52 20 2 18 80 25 Threshold Losses sizes (in $) Time Losses sizes (in $)

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

12. Measuring Operational Risk =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

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

14. Extreme Value Theory 1 1,,, )ln ( ˆ )(      i nknj nk XX k-1 H  Hill method for the estimation of the shape parameter: 1

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

16. Extreme Value Theory 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

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

18. Extreme Value Theory

20. Testing the Model - Checking the Parameters Based 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 = (x 1, …,x n ). 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

21. Frequency Distributions Poisson CDF 0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% 80.00% 90.00% 100.00% 020406080100120140160 Poisson Distribution: Other popular distributions to estimate frequency are the geometric, negative binomial, binomial, weibull, etc

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

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

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

25. 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. N. of Op Errors = 88.88 + 6.92 System Downtime + 5.32 Employees - 0.22 N. of transactions R 2 = 95%, F-test = 20.69, p-value = (0.01) R 2 = 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 Specific coverage Immediate protection against catastrophes OPERATIONAL RISK (MEASURED) Capital Allocation InternalRisk 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)

28. Developing an OR Hedging Program

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

30. 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 marcelo.cruz@ubsw.com