Stochastic Modeling
Theory and Reality from an Actuarial Perspective
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 I_9
Uses '-9
I.B.3 Techniques I-10
I.B.3.a Monte Carlo Simulation 1-10
Variance reduction 1-12
Antithetic-variable technique '-13
Control variate technique '-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 ofouter scenarios 1-28
(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 - Matching 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 1-50
1.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 1-64
Stable distributions 1-65
Substitution method 1-66
(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 11-11
FX model with deterministic interest rates vs. FX model with stochastic interest rates 11-11
Validating FX models 11-12
II.A.3 Equity Returns "-12
An overview of equity scenario generation 11-12
Arbitrage-free equity scenario generation 11-13
Stylized facts of equity index returns 11-14
Extensions of the Black-Scholes framework 11-17
Realistic equity scenarios 11-18
Risk-neutral equity model calibration H-19
Calibration function H~19
Optimization "~19
Data n-20
(iv)
II.A.4 Credit Risks
Modeling default risks 11-21
Structural models 11-21
Reduced form models 1,-22
Conclusions 11-24
II.A.5 Inflation 11-24
Models based on past inflation 11-24
Models based on Phillips curve 11-26
References for Section II.A 11-27
It.B Life and Health Models 11-29
II.B.1 Catastrophic Mortality Modeling 11-29
Overview of the model 11-29
Non-modeled items 11-30
II.B.I.a Baseline Model 11-32
Modeling process 11-33
Model results 11-35
II.B.I.b Disease Model 11-35
Overview of recent pandemics 11-35
Influenza 11-36
AIDS 11-36
SARS 11-36
Other diseases 11-37
General modeling approach 11-37
Modeling the frequency of disease pandemics 1|-37
Potential of repeating a 1918-1920 pandemic I'-37
Data points 11-38
Modeling the severity of disease pandemics l'-40
Severity curve: Fitting the main component 11-40
Severity curve: Fitting the extreme component H-42
Other supporting assumptions H-43
Model results 11-43
II.B.1.c Terrorism Model |1-44
Model design 11-44
Data 11-45
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 11-49
II.B.2.b Traditional Par Life Products H-49
II.B.2.C Universal Life and Fixed Annuity 11-49
(v)
II.B.2.d Variable Annuity 11-49
Dynamic lapse in VA l\-5Q
Summary U-50
II.B.3 Morbidity and Claims Experience 1,-51
Incidence rates
Severity of claim 11-52
Inflation 11-52
Utilization 11-53
Voluntary terminations \l-S3
Claim continuance 11-54
References for II.B 11-54
II.C Non-life Claim Models 11-55
II.C.1 Aggregate Triangle-based Models 11-57
Stochastic loss development model 11-57
Hoerl curve U-5B
Mack's distribution-free model 11-58
Bootstrap model 1I-59
Schnieper 1,-60
Generalized linear modeling framework 11-61
II.C.2 Individual Claims Frequency/Severity Models 11-62
Collective risk model 11-63
Collective risk model by layer 11-64
Transition matrix 11-65
Generalized linear modeling applied to unpaid claim estimation 11-66
Wright's model 11-66
II.C.3 Catastrophe Modeling 11-67
References for II.C 11-68
II.D Non-life Financial Models 11-70
II.D.1 Types of Models 11-7°
II.D.I.a The Evolution of Models 11-71
II.D.1.b Uses of Dynamic Risk Models 11-71
II.D.2 Description of a Non-life Dynamic Risk Model 11-72
II.D.2.a General Model Description 11-73
II.D.2.b Economic Scenarios 11-73
II.D.2.C Asset Scenarios 11-74
II.D.2.d Underwriting Operations 11-74
Premium 11-74
Loss payments and liabilities 11-76
Expense payments and liabilities I'-77
Reinsurance 11-78
Investment operations 11-79
Accounting and taxation 11-79
(vi)
Management response 11-79
II.D.3 Parameterization and Correlation 11-80
References for II.D 11-80
II.E Country-and Region-specific Issues 11-81
II.E.1 Regulatory Reporting 11-81
II.E.2 Liability Valuations 11-83
H.E.3 Financial Reporting and Embedded Values H-85
H.E.4 Product Design and Pricing 11-87
II. E.5 Economic Capital Management 11-88
References for II.E 11-91
III. Evaluating and Discussing Results hm
References for III 111-2
lll.A Calibrating the Model 111-2
Two approaches to model calibration 111-3
Calibration to historical experience 111-3
Calibration to current market conditions 111-5
III.3 Validating the Model 111"5
III.C Conducting a Peer Review 'I1-11
III.D Communicating the Results 111-12
III. E Auditing the Process 111-14
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 IV-6
Real-world IV-7
IV.A.4 Developing Mortality and Expense Fee IV-7
(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 IV"8
Lapse sensitivity IV-9
Mortality sensitivity IV-12
Fund allocation sensitivity IV-12
Scenario sensitivity IV-14
Decision of charge level IV-15
Multiple stochastic variables 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 IV-21
Hedge results on year-by-year basis IV-22
Hedge results on ROA basis 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
(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 IV-32
Interest rate model IV-33
Spot exchange rate model 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 IV-46
GPVLrisk metric IV-47
IV.B.6.a Calibration, Validation, and Review IV-48
Calibration IV-48
Validation IV-48
Peer review and checking 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
(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 IV-73
Estimated cash flow results IV-74
Estimated ultimate loss ratio results 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
(x)
Lapse rates IV-106
Mortality IV'-106
Pre-tax net investment earned rates IV-106
Overview of calculating stochastic liabilities IV-106
Development of aggregate scenarios IV-107
Results for stochastic liabilities IV-108
Results: Stochastic risk-based capital 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 IV-111
Underwriting module IV-112
Asset module IV-112
Accounting and taxation module IV-113
IV.F.3 Inputs and Parameterization IV-113
Premium IV-113
Expenses IV-114
Losses IV-114
Runoffof starting liability IV-114
Future claims IV—115
Individual large losses IV-115
Aggregate small losses IV-116
Catastrophe losses IV-116
Investment model 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 IV-121
Cross-line subsidization IV-122
Consistency of scenarios IV-122
Logistical considerations 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
(xi)
V. References, Abbreviations and Author Biographies v-1
References v_1
Abbreviations v~5
Author Biographies v_11
Appendices
Appendix A - CFO Forum A-1
Appendix B - Country 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
(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
Capital adequacy A-20
References for Appendix B A-20
Additional Resources A-25
Appendix C - Illustrative Market-Consistent Assumptions for Section III A-27
Appendix D - Bootstrap 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
(xiii)
E. 1.C Modeling Correlation A-39
Appendix F-Maximum Likelihood Estimation A-43
F. 1 MLE example - normal model A-44
F.2 MLE example - lognormal model A-49
F.3 MLE estimates - censored and truncated data A-51
F.4 MLE example - grouped data A-54
F.5 MLE example - loss-forecast 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