Modelling Operational Risk Using Bayesian Inference

Transcription

1 Pavel V. Shevchenko Modelling Operational Risk Using Bayesian Inference 4y Springer

2 1 Operational Risk and Basel II Introduction to Operational Risk Defining Operational Risk Basel II Approaches to Quantify Operational Risk Loss Data Collections LDCE LDCE LDCE LDCE General Remarks Operational Risk Models 17 2 Loss Distribution Approach Loss Distribution Model Operational Risk Data A Note on Data Sufficiency Insurance Basic Statistical Concepts Random Variables and Distribution Functions Quantiles and Moments Risk Measures Capital Allocation Euler Allocation Allocation by Marginal Contributions Model Fitting: Frequentist Approach Maximum Likelihood Method Bootstrap Bayesian Inference Approach Conjugate Prior Distributions Gaussian Approximation for Posterior Posterior Point Estimators Restricted Parameters 47

3 2.9.5 Noninformative Prior Mean Square Error of Prediction Markov Chain Monte Carlo Methods Metropolis-Hastings Algorithm Gibbs Sampler Random Walk Metropolis-Hastings Within Gibbs ABC Methods Slice Sampling MCMC Implementation Issues Tuning, Burn-in and Sampling Stages Numerical Error MCMC Extensions ' Bayesian Model Selection Reciprocal Importance Sampling Estimator Deviance Information Criterion 68 Problems 69 Calculation of Compound Distribution Introduction Analytic Solution via Convolutions Analytic Solution via Characteristic Functions Compound Distribution Moments Value-at-Risk and Expected Shortfall Monte Carlo Method Quantile Estimate Expected Shortfall Estimate Panjer Recursion Discretisation Computational Issues Panjer Extensions Panjer Recursion for Continuous Severity Fast Fourier Transform Compound Distribution via FFT Aliasing Error and Tilting Direct Numerical Integration Forward and Inverse Integrations Gaussian Quadrature for Subdivisions Tail Integration Comparison of Numerical Methods Closed-Form Approximation Normal and Translated Gamma Approximations VaR Closed-Form Approximation 106 Problems 108

4 Bayesian Approach for LDA Ill 4.1 Introduction Ill 4.2 Combining Different Data Sources Ad-hoc Combining Example of Scenario Analysis Bayesian Method to Combine Two Data Sources Estimating Prior: Pure Bayesian Approach Estimating Prior: Empirical Bayesian Approach Poisson Frequency The Lognormal CJ\f(ix, a) Severity with Unknown \x The Lognormal CJ\f(ix, a) Severity with Unknown pu and a Pareto Severity Estimation of the Prior Using Data The Maximum Likelihood Estimator Poisson Frequencies Combining Expert Opinions with External and Internal Data Conjugate Prior Extension Modelling Frequency: Poisson Model Modelling Frequency: Poisson with Stochastic Intensity Lognormal Model for Severities Pareto Model Combining Data Sources Using Credibility Theory Buhlmann-Straub Model Modelling Frequency Modelling Severity Numerical Example Remarks and Interpretation Capital Charge Under Parameter Uncertainty Predictive Distributions Calculation of Predictive Distributions General Remarks 175 Problems 177 Addressing the Data Truncation Problem Introduction Constant Threshold - Poisson Process Maximum Likelihood Estimation Bayesian Estimation Extension to Negative Binomial and Binomial Frequencies Ignoring Data Truncation Threshold Varying in Time 196 Problems 200 V

5 Modelling Large Losses Introduction EVT - Block Maxima EVT - Threshold Exceedances A Note on GPD Maximum Likelihood Estimation EVT - Random Number of Losses EVT - Bayesian Approach Subexponential Severity Flexible Severity Distributions g-and-h Distribution GB2 Distribution Lognormal-Gamma Distribution Generalised Champernowne Distribution a-stable Distribution 230 Problems 232 Modelling Dependence Introduction Dominance of the Heaviest Tail Risks A Note on Negative Diversification Copula Models Gaussian Copula Archimedean Copulas f-copula Dependence Measures Linear Correlation Spearman's Rank Correlation Kendall's tau Rank Correlation Tail Dependence Dependence Between Frequencies via Copula Common Shock Processes Dependence Between Aggregated Losses via Copula Dependence Between the k-th Event Times/Losses Modelling Dependence via Levy Copulas Structural Model with Common Factors Stochastic and Dependent Risk Profiles Dependence and Combining Different Data Sources Bayesian Inference Using MCMC Numerical Example ' Predictive Distribution 266 Problems 269

6 xv A List of Distributions 273 A.I Discrete Distributions 273 A. 1.1 Poisson Distribution, Poissonik) 273 A.1.2 Binomial Distribution, Bin(n, p) 274 A. 1.3 Negative Binomial Distribution, NegBin(r, p) 274 A.2 Continuous Distributions 275 A.2.1 Uniform Distribution, U(a, b) 275 A.2.2 Normal (Gaussian) Distribution, Af(/x, a) 275 A.2.3 Lognormal Distribution, CN(fx., a) 275 A.2.4 t Distribution, T(v, iu,,a 2 ) 276 A.2.5 Gamma Distribution, Gamma(a, /3) 276 A.2.6 Weibull Distribution, Weibull(a, 0).." 276 A.2.7 Pareto Distribution (One-Parameter), Pareto{%, XQ) 277 A.2.8 Pareto Distribution (Two-Parameter), Paretoi(a, P) 277 A.2.9 Generalised Pareto Distribution, GPD($, 0) 278 A.2.10 Beta Distribution, Beta(a, 0) 278 A.2.11 Generalised Inverse Gaussian Distribution, GIG(co, <p,v) A.2.12 ^-variate Normal Distribution, Mdifi, S) 280 A.2.13 ^-variate r-distribution, T d (v, /i,x) 280 B Selected Simulation Algorithms 281 B.I Simulation from GIG Distribution 281 B.2 Simulation from a-stable Distribution 282 Solutions for Selected Problems 283 References 289 Index 299