Stochastic modeling : theory and reality from an … · Control variate technique '-13 Stratified...

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


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


B.7 The Netherlands A-10


B.8 Romania 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


Reserves A-17

Latin America A"1?


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

Stochastic modeling : theory and reality from an … · Control variate technique '-13 Stratified...

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