Stochastic Modeling. Theory and Reality from an Actuarial Perspective

Transcription

1 Stochastic Modeling Theory and Reality from an Actuarial Perspective

2 Table of Contents Page Introduction (*v) I. General Methodology I.A Stochastic Models vs. Non-stochastic Models 1-1 When should stochastic models be used? i-1 When should use of stochastic models be questioned? 1-3 Alternatives to stochastic models 1-4 Disadvantages of stochastic models '-5 Guidance on stochastic model implementation 1-5 I.B Risk-neutral vs. Real-world 1-6 I.B.1 Risk-neutral Scenarios 1-6 Background 1-6 Uses '-7 Calibration and parameterization '-8 Other considerations 1-8 I.B.2 Real-world Scenarios 1-9 Background Uses I_9 '-9 I.B.3 Techniques I-10 I.B.3.a Monte Carlo Simulation 1-10 Variance reduction 1-12 Antithetic-variable technique Control variate technique '-13 '-13 Stratified sampling 1-13 Importance sampling 1-14 I.B.3.b Lattices 1-15 Binomial lattice simulations 1-15 One-step binomial tree 1-15 Multi-step binomial trees 1-16 Trinomial lattice simulation I_18 I.B.3.C Regime-switching Models '-19 I.B.4 Nested Stochastic Projections 1-24 Nested stochastic solutions to practical applications 1-25 Nested stochastic modeling and the principles-based approach 1-26 Nested stochastic modeling and other international accounting standards 1-27 Managing the nested stochastic modeling process 1-27 Reducing the number of model points 1-28 Reducing the number of outer scenarios 1-28

3 Matching (ii) Reducing the number of inner paths 1-28 Reducing the number of nodes '~29 I.B.5 Deflators 1-29 Introduction 1-29 Mathematical definition 1-29 Properties 1-30 Applications '~3u Practical considerations 1-31 Illustrative example '"32 I.B.5.a Copulas 1-34 Fitting copulas 1-36 Simulating from copulas 1-38 References for Section I.B '~39 I.C Distributions and Fitting 1-40 I.C.1 Stochastic Models M1 I.C.2 Empirical vs. Model Distributions 1-42 I.C.3 A Simple Approach - of Moments 1-43 I.C.4 A Richer - Approach Maximum Likelihood 1-47 References for Section I.C '-48 I.D Random Number Generation 1_48 I.D.1 True and Pseudo Random Number Generators 1-49 I.D.2 Linear Congruential Generators D.3 Non-linear PRNGs 1-53 I.D.3.a Inversive Congruential Generators 1-53 I.D.3.b Binary Shift Register Generators 1-54 I.D.4 Empirical Tests for Random Numbers 1-54 I.D.4.a Kolmogorov-Smirnov Test 1-54 I.D.4.b Poker Test (partition test) 1-55 I.D.4.C Permutation Test 1-56 I.D.5 Methods of Sampling Non-uniform Distributions 1-57 I.D.5.a Inversion Method 1-57 I.D.5.b Acceptance/rejection Method 1-59 I.D.5.C Composition Method 1-60 I.D.5.d Switching Method 1-61 I.D.5.e Ratio of Uniforms Method 1-61 I.D.5.f Tabular Method 1-63 I.D.5.g Sampling Without Replacement 1-63 I.D.5.h Other Techniques and Special Cases 1-64 Gamma distribution Stable distributions Substitution method

4 (iii) I.D.6 Summary 1-67 References for Section I.D 1-67 I.E. Risk Measures 1-69 I.E.1 VaR 1-70 I.E.1.a Variance-covariance Method 1-70 I.E.I.b Monte Carlo Simulation 1-71 I.E.1.c Historical Simulation 1-71 I.E.2 Conditional Tail Expectation (CTE) 1-72 I.E.3 Note on the Confidence Level and Time Horizon 1-73 I.E.4 Multi-period Risk Measure 1-74 Time consistency 1-75 I.E.5 Note on the Aggregation of Risk 1-76 I. E.6 Other Risk Measures 1-76 References for Section I.E '-77 II. General Applications 11-1 II.A Economic Scenario Generators 11-1 II.A.1 Interest Rates 11-1 Realistic yield curve dynamics 11-1 HJM/BGM framework for generating arbitrage-free interest rate scenarios II-4 Realistic scenarios over longer time scales 11-5 Calibration of the interest rate generator ll-5 Key rate analysis of yield curve changes and associated calibration 11-7 Combination of interest rate scenarios with other risk factors 11-8 Lognormally vs. normally distributed interest rate scenarios, revisited 11-9 II.A.2 Exchange Rates H-9 FX models with deterministic interest rates H-10 FX models with stochastic Interest rates FX model with deterministic interest rates vs. FX model with stochastic interest rates Validating FX models II.A.3 Equity Returns "-12 An overview of equity scenario generation Arbitrage-free equity scenario generation Stylized facts of equity index returns Extensions of the Black-Scholes framework Realistic equity scenarios Risk-neutral equity model calibration H-19 Calibration function H~19 Optimization "~19 Data n-20

5 (iv) II.A.4 Credit Risks Modeling default risks Structural models Reduced form models 1,-22 Conclusions II.A.5 Inflation Models based on past inflation Models based on Phillips curve References for Section II.A It.B Life and Health Models II.B.1 Catastrophic Mortality Modeling Overview of the model Non-modeled items II.B.I.a Baseline Model Modeling process Model results II.B.I.b Disease Model Overview of recent pandemics Influenza AIDS SARS Other diseases General modeling approach Modeling the frequency of disease pandemics 1-37 Potential of repeating a pandemic I'-37 Data points Modeling the severity of disease pandemics l'-40 Severity curve: Fitting the main component Severity curve: Fitting the extreme component H-42 Other supporting assumptions H-43 Model results II.B.1.c Terrorism Model 1-44 Model design Data Modeling the frequency of terrorist events H-45 Defining levels H-45 Defining probabilities H-47 Other assumptions U-47 Model results "-47 II.B.1.d Combined Model Results H-48 II.B.2 Dynamic Policyholder Behaviour H-48 II.B.2.a Traditional Non-par Life Products II.B.2.b Traditional Par Life Products H-49 II.B.2.C Universal Life and Fixed Annuity 11-49

6 (v) II.B.2.d Variable Annuity Dynamic lapse in VA l\-5q Summary U-50 II.B.3 Morbidity and Claims Experience 1,-51 Incidence rates Severity of claim Inflation Utilization Voluntary terminations \l-s3 Claim continuance References for II.B II.C Non-life Claim Models II.C.1 Aggregate Triangle-based Models Stochastic loss development model Hoerl curve U-5B Mack's distribution-free model Bootstrap model 1I-59 Schnieper 1,-60 Generalized linear modeling framework II.C.2 Individual Claims Frequency/Severity Models Collective risk model Collective risk model by layer Transition matrix Generalized linear modeling applied to unpaid claim estimation Wright's model II.C.3 Catastrophe Modeling References for II.C II.D Non-life Financial Models II.D.1 Types of Models 11-7 II.D.I.a The Evolution of Models II.D.1.b Uses of Dynamic Risk Models II.D.2 Description of a Non-life Dynamic Risk Model II.D.2.a General Model Description II.D.2.b Economic Scenarios II.D.2.C Asset Scenarios II.D.2.d Underwriting Operations Premium Loss payments and liabilities Expense payments and liabilities Reinsurance Investment operations Accounting and taxation I'-77

7 (vi) Management response II.D.3 Parameterization and Correlation References for II.D II.E Country-and Region-specific Issues II.E.1 Regulatory Reporting II.E.2 Liability Valuations H.E.3 Financial Reporting and Embedded Values H-85 H.E.4 Product Design and Pricing II. E.5 Economic Capital Management References for II.E III. Evaluating and Discussing Results hm References for III lll.a Calibrating the Model Two approaches to model calibration Calibration to historical experience Calibration to current market conditions III.3 Validating the Model 111"5 III.C Conducting a Peer Review 'I1-11 III.D Communicating the Results III. E Auditing the Process IV. Case Studies iv-1 IV.A Development and Management of a Variable Annuity Product IV-1 IV.A.1 Introduction ]V-1 IV.A.I.a Variable Annuity IV-1 IV.A.1.b Embedded Guarantees 'V-2 IV.A.1.c Revenues and Expenses IV-2 IV.A.1.d Risks IV-3 IV.A.2 Product Specifications and Pricing Assumptions IV-3 IV.A.3 Economic Scenarios 'V-6 IV.A.3.a Deterministic or Stochastic IV-6 IV.A.3.b Risk-neutral vs. Real-world IV-6 Risk-free world Real-world IV-6 IV-7 IV.A.4 Developing Mortality and Expense Fee IV-7

8 (vii) IV.A.5 Developing GLWB Charge IV"7 IV.A.5.a Cost of GLWB IV-7 IV.A.5.b Charge of GLWB IV-6 IV.A.5.C Adequacy of Charge IV_8 Number of scenarios Lapse sensitivity Mortality sensitivity Fund allocation sensitivity Scenario sensitivity Decision of charge level Multiple stochastic variables IV-15 IV"8 IV-9 IV-12 IV-12 IV-14 IV-15 IV.A.6 Assessing Profitability of the Entire Contract IV-16 IV.A.6.a Profitability of Simple Requirements IV-17 IV.A.6.b Profitability of U.S. Statutory Requirements IV-18 IV.A.6.C Hedging Economic Liability IV-20 Hedge modeling Hedge results on year-by-year basis Hedge results on ROA basis IV-22 IV-21 IV-22 IV.A.7 Financial Reporting of Variable Annuities in the United States IV-23 IV.AJ.a U.S. Statutory IV-23 IV.A.7.b U.S. GAAP JV-24 IV.A.8 Risk Management of Variable Annuity IV-25 IV.A.8.a Product Design Risk IV-25 IV.A.8.b Market Risk IV-25 IV.A.8.c Risk with Reinsurance IV-25 IV.A.8.d Risk with Dynamic Hedge IV-26 IV.A.S.e Policyholder Behaviour Risk IV-26 IV.A.9 Development on an International Platform IV-26 IV.A.9.a Market Needs and Product Design IV-26 lv.a.9.b Economic Model and Data 1V-27 IV.A.9.C Liability and Capital Requirements IV-27 IV.A.9.d Financial Reporting IV-27 IV.B Economic Capital for a Multi-line Life Insurance Company IV-28 IV.B.1 The Case Study Company: Background on XYZ Life Insurance Company IV-28 IV.B.2 Fundamental Concepts of an Economic Capital Framework IV-28 IV.B.3 Modeling Risks IV-29 IV.B.4 General Methodology IV-30 IV.B.4.3 Risk Metrics IV-30 IV.BAb Confidence Level IV-30

9 (viii) IV.B.4.C Time Horizon iv_3 IV.B.4.d Projection Techniques,v-30 IV.B.5 Scenario Generation 1V-31 IV.B.5.a Economic Scenario Generator IV_32 Equity model Interest rate model Spot exchange rate model IV-32 IV-33 IV-33 Model parameterization IV-33 Starting interest rates (foreign and domestic) IV-33 Duration parameters (for bond fund calculations) IV-35 Equity returns IV-35 Currency returns IV-36 Money market IV-36 Domestic bond IV-36 Foreign bond IV-37 IV.B.5.b Credit Risk Model lv~37 Description of the model IV-37 Potential simplifications to the model IV-38 Calculating cost of a credit event 1V-39 Results IV-40 IV.B.5.C Mortality 'V-41 IV.B.5.d Morbidity 'V-41 Probability distributions for new claim costs IV-41 Probability distribution of claim runoff IV-43 Pricing risk 'V-43 IV.B.5.e Lapses IV"44 Operational and strategic risks IV-45 IV.B.6 Presentation of Results IV-46 PVFP risk metric GPVLrisk metric IV-46 IV-47 IV.B.6.a Calibration, Validation, and Review IV-48 Calibration Validation Peer review and checking IV-48 IV-48 IV-48 IV.C Embedded Value for a Multi-national Multi-line Life Insurance Company 1V-51 IV.C.1 Introduction IV-51 IV.C.I.a Brief History of Embedded Value Analysis IV-51 IV.d.b Time Value of Options and Guarantees IV-52 IV.C.1.c Balance Sheet Approach IV-53 IV.C.2 Current Embedded Value Analysis IV-54 IV.C.2.a Stochastic Models per Company IV-55 IV.C.2.b Economic Assumptions per Company IV-55

10 (ix) IV.C.2.C Results per Company IV-56 IV.C.3 Sample Embedded Value Analysis for a Single Product of a Life Insurance Company IV-57 IV.C.3.a Introduction IV-57 IV.C.3.b Economic Scenario Generator and Assumptions IV-58 IV.C.3.C Certainty Equivalent Present Value of Future Profits IV-58 IV.C.3.d Time Value of Options and Guarantees IV-59 IV.C.3.6 Sensitivities 'V-59 IV.C.3/ Review IV-61 IV.C.4 Future Embedded Value Analyses: Non-financial Stochastic Calculations? 1V-62 IV.D Unpaid Claim Variability for a Multi-line Non-life Insurance Company 1V-63 IV.D.1 Introduction 1V-63 IV.D.I.a Model Selection IV-63 IV.D.1.b Bootstrap Modeling 'V-65 IV.D.2 Building a Model IV-65 IV.D.2.a Diagnostic Testing IV-66 Residual graphs IV-66 Normality test 'V-70 Outliers IV-71 IV.D.2.b Model Results >V-72 Estimated unpaid results Estimated cash flow results Estimated ultimate loss ratio results IV-73 IV-74 IV-75 Estimated incremental results IV-76 Distribution graph IV-77 IV.D.2.c Combining Model Results 1V-78 IV.D.3 Aggregation of Results IV-81 IV.D.3.a Calculating Correlation IV-83 IV.D.3.b Correlation Process 1V-83 IV.D.4 Communication IV-85 IV.D.5 Components of the Capital Model IV-85 IV,D.5.a Required Capital 'V-86 References for IV.D IV-89 IV.E Stochastic Liability and Capital Calculations IV-103 IV.E.1 Background IV-103 Methods for calculating stochastic liabilities IV-103 IV.E.2 Detailed Example: Illustration of Stochastic Liabilities IV-105 Policy characteristics and baseline assumptions IV-105 Stochastic assumptions IV-106

11 (x) Lapse rates Mortality Pre-tax net investment earned rates Overview of calculating stochastic liabilities Development of aggregate scenarios Results for stochastic liabilities Results: Stochastic risk-based capital IV-106 IV'-106 IV-106 IV-106 IV-107 IV-108 IV-110 IV.E.3 Assumption Setting: Use of Margins IV-110 IV.E.4 Peer Review and Audit of Results IV-110 IV.F Economic Capital for a Multi-line Non-life Insurance Company IV-111 IV.F.1 Risk Metric and Time Horizon IV-111 IV.F.2 Description of the Model IV-111 Economic scenario module Underwriting module Asset module Accounting and taxation module IV-111 IV-112 IV-112 IV-113 IV.F.3 Inputs and Parameterization IV-113 Premium Expenses Losses Runoffof starting liability Future claims Individual large losses Aggregate small losses Catastrophe losses Investment model IV-113 IV-114 IV-114 IV-114 IV 115 IV-115 IV-116 IV-116 IV-117 IV.F.4 Correlation IV-117 IV.F.5 Validation and Peer Review IV-117 IV.F.6 Communication of Results IV-118 References for IV.F IV-120 IV.G Combining Economic Capital Results for Life and Non-life Companies IV-121 IV.G.1 Background IV-121 IV.G.2 Considerations for Companies with Both Life and Non-life Business IV-121 Covariance effects Cross-line subsidization Consistency of scenarios IV-122 Logistical considerations IV-121 IV-122 IV-122 IV.G.3 Combining the EC Models IV-122 IV.G.4 Presentation of Results IV-124 IV.G.5 Peer Review of Results IV-125

12 CFO Country (xi) V. References, Abbreviations and Author Biographies v-1 References v_1 Abbreviations v~5 Author Biographies v_11 Appendices - Appendix A Appendix B - Forum A-1 Practice A-3 B.1 Europe A-3 Solvency II A-3 What are the key risk modules of the standard SCR formula? A-4 How are risk modules aggregated in the standard formula? A-4 What are the criteria for an undertaking to use an internal model? A-5 Non-life technical provisions A-5 Equalization reserves A-6 B.2 Germany A-6 Non-life regulatory reporting A-6 Case loss and loss adjustment expense reserves A-6 IBNR A-6 Claim handling costs A-7 B.3 France A-7 Non-life regulatory reporting A-7 B.4 Switzerland A-7 Capital adequacy A-7 Reserves A-8 B.5 United Kingdom A-8 Individual Capital Assessment System A-8 Individual Capital Assessments A-9 Professional guidance A-9 B.6 Italy A-10 Non-life regulatory reporting A-10 B.7 The Netherlands A-10 Non-life regulatory reporting A-10 B.8 Romania A-11 Non-life regulatory reporting A-11 B.9 Australia A-11 Non-life regulatory reporting: Liability A-11 Non-life regulatory reporting: Capital adequacy A-12

13 Illustrative Bootstrap (xii) B.10 New Zealand A"12 B.11 Asia A~13 Singapore A"13 Malaysia A~13 Hong Kong A~13 China A"13 Chinese Taipei A-14 Japan A-14 Thailand A-14 South Korea A-15 India A""15 B.12 The Americas A"15 Canada A-15 Reserves A-16 Dynamic capital adequacy testing A-16 United States of America A"16 Non-life regulatory reporting A-16 Reserves A-17 Latin America A"1? Non-life regulatory reporting A-17 Unearned premium reserves A-18 IBNR A-18 Judicial claims A-18 Catastrophe reserves A-18 JLAE A-19 Data considerations A-19 Currency risk A-19 Asset/liability matching A-19 Inflation A-19 Reinsurance A-19 B.13 South Africa A-20 Appendix C - Appendix D - Capital adequacy A-20 References for Appendix B A-20 Additional Resources A-25 Market-Consistent Assumptions for Section III A-27 Model A-29 D.1,a A Simple Paid Loss Chain Ladder Simulation A-29 D. 1.b A Walk-through of the Basic Calculation, Based on Paid Loss Data A-30 References for Appendix D A-34 Appendix E- Correlation A-35 E. 1.a The Correlation Matrix A-35 E.1.b Measuring Correlation A-36

14 normal grouped loss-forecast censored (xiii) E. 1.C Modeling Correlation A-39 Appendix F-Maximum Likelihood Estimation A-43 F. 1 - MLE example model A-44 F.2 - MLE example lognormal model A-49 F.3 MLE estimates - F.4 MLE example - F.5 MLE example - and truncated data A-51 data A-54 models A-57 F.6 Evaluation of models A-61 F.7 MLE, "exponential fit," and generalized linear models A-65 F.8 Some simple stochastic models plus their mixture A-66 Poisson Model A-67 Negative Binomial Model A-67 Gamma Model A-68 Lognormal Model A-68 Pareto Model A-69 A Mixture of Models A-69 F.9 Bayesian methods A-70