Table Of Content

Stochastic Modeling Theory and Reality from an Actuarial Perspective ISBN 978-0-9813968-1-1 2nd printing Association Actuarielle Internationale International Actuarial Association 150 Metcalfe Street, Suite 601 Ottawa, Ontario Canada K2P 1P1 www.actuaries.org Tel: 1-613-236-0886 Fax: 1-613-236-1386 Email: [email protected] © 2010 International Actuarial Association / Association Actuarielle Internationale Table of Contents Page Introduction .......................................................................................................................................................... (xv) I. General Methodology ................................................................................................................................. I–1 I.A Stochastic Models vs. Non-stochastic Models ..................................................................................................... 1 When should stochastic models be used? ............................................................................................ I–1 When should use of stochastic models be questioned? ....................................................................... 3 I– Alternatives to stochastic models .......................................................................................................... 4 I– Disadvantages of stochastic models ..................................................................................................... 5 I– Guidance on stochastic model implementation ..................................................................................... 5 I– I.B Risk-neutral vs. Real-world .................................................................................................................................. I–6 I.B.1 Risk-neutral Scenarios ....................................................................................................................................... I–6 Background ........................................................................................................................................... I–6 Uses...................................................................................................................................................... 7 I– Calibration and parameterization .......................................................................................................... 8 I– Other considerations ............................................................................................................................. 8 I– I.B.2 Real-world Scenarios ......................................................................................................................................... I–9 Background ........................................................................................................................................... I–9 Uses...................................................................................................................................................... 9 I– I.B.3 Techniques ...................................................................................................................................................... I1–0 I.B.3.a Monte Carlo Simulation ......................................................................................................................... I–10 Variance reduction .............................................................................................................................. I–12 Antithetic-variable technique ............................................................................................................... 13 I– Control variate technique .................................................................................................................... 13 I– Stratified sampling .............................................................................................................................. 13 I– Importance sampling........................................................................................................................... 14 I– I.B.3.b Lattices .................................................................................................................................................. 15 I– Binomial lattice simulations ................................................................................................................. I–15 One-step binomial tree........................................................................................................................ 15 I– Multi-step binomial trees ..................................................................................................................... 16 I– Trinomial lattice simulation .................................................................................................................. 18 I– I.B.3.c Regime-switching Models ...................................................................................................................... 19 I– I.B.4 Nested Stochastic Projections ......................................................................................................................... I–24 Nested stochastic solutions to practical applications .......................................................................... I–25 Nested stochastic modeling and the principles-based approach ........................................................ 26 I– Nested stochastic modeling and other international accounting standards ........................................... 27 I– Managing the nested stochastic modeling process ............................................................................ 27 I– Reducing the number of model points ................................................................................................ 28 I– Reducing the number of outer scenarios ............................................................................................ 28 I– I– (ii) Reducing the number of inner paths ................................................................................................... 28 Reducing the number of nodes ........................................................................................................... 29 I– I.B.5 Deflators .......................................................................................................................................................... I–29 Introduction ......................................................................................................................................... I–29 Mathematical definition ....................................................................................................................... 29 I– Properties ........................................................................................................................................... 30 I– Applications ........................................................................................................................................ 30 I– Practical considerations ...................................................................................................................... 31 I– Illustrative example ............................................................................................................................. 32 I– I.B.5.a Copulas ................................................................................................................................................. 34 I– Fitting copulas ..................................................................................................................................... I–36 Simulating from copulas ...................................................................................................................... 38 I– References for Section I.B .................................................................................................................. 39 I– I.C Distributions and Fitting ..................................................................................................................................... I–40 I.C.1 Stochastic Models ........................................................................................................................................... I–41 I.C.2 Empirical vs. Model Distributions ..................................................................................................................... I–42 I.C.3 A Simple Approach – Matching of Moments .................................................................................................... I–43 I.C.4 A Richer Approach – Maximum Likelihood ...................................................................................................... I–47 References for Section I.C ................................................................................................................... I–48 I.D Random Number Generation ............................................................................................................................. I–48 I.D.1 True and Pseudo Random Number Generators ............................................................................................... I–49 I.D.2 Linear Congruential Generators ...................................................................................................................... I–50 I.D.3 Non-linear PRNGs ........................................................................................................................................... I–53 I.D.3.a Inversive Congruential Generators ........................................................................................................ I–53 I.D.3.b Binary Shift Register Generators ........................................................................................................... I–54 I.D.4 Empirical Tests for Random Numbers ............................................................................................................. I–54 I.D.4.a Kolmogorov-Smirnov Test ..................................................................................................................... I–54 I.D.4.b Poker Test (partition test) ...................................................................................................................... I–55 I.D.4.c Permutation Test ................................................................................................................................... I–56 I.D.5 Methods of Sampling Non-uniform Distributions .............................................................................................. I–57 I.D.5.a Inversion Method ................................................................................................................................... I–57 I.D.5.b Acceptance/rejection Method ................................................................................................................ I–59 I.D.5.c Composition Method .............................................................................................................................. I–60 I.D.5.d Switching Method .................................................................................................................................. I–61 I.D.5.e Ratio of Uniforms Method ...................................................................................................................... I–61 I.D.5.f Tabular Method ...................................................................................................................................... I–63 I.D.5.g Sampling Without Replacement ............................................................................................................ I–63 I.D.5.h Other Techniques and Special Cases ................................................................................................... I–64 Gamma distribution ............................................................................................................................. I–64 Stable distributions ............................................................................................................................. 65 I– Substitution method ............................................................................................................................ 66 I– I– (iii) I.D.6 Summary ......................................................................................................................................................... 67 References for Section I.D ................................................................................................................. I–67 I.E. Risk Measures ................................................................................................................................................... I–69 I.E.1 VaR ........................................................................................................................................................... I–70 I.E.1.a Variance-covariance Method ................................................................................................................. I–70 I.E.1.b Monte Carlo Simulation ......................................................................................................................... I–71 I.E.1.c Historical Simulation .............................................................................................................................. I–71 I.E.2 Conditional Tail Expectation (CTE) .................................................................................................................. I–72 I.E.3 Note on the Confidence Level and Time Horizon ............................................................................................ I–73 I.E.4 Multi-period Risk Measure ............................................................................................................................... I–74 Time consistency ............................................................................................................................... I–75 I.E.5 Note on the Aggregation of Risk ...................................................................................................................... I–76 I.E.6 Other Risk Measures ....................................................................................................................................... I–76 References for Section I.E ................................................................................................................. I–77 I– II. General Applications ................................................................................................................................II–1 II.A Economic Scenario Generators .......................................................................................................................... 1 II.A.1 Interest Rates .................................................................................................................................................. II–1 Realistic yield curve dynamics ............................................................................................................. II–1 HJM/BGM framework for generating arbitrage-free interest rate scenarios ................................................. 4 II– Realistic scenarios over longer time scales ......................................................................................... 5 II– Calibration of the interest rate generator.............................................................................................. 5 II– Key rate analysis of yield curve changes and associated calibration ................................................... 7 II– Combination of interest rate scenarios with other risk factors .............................................................. 8 II– Lognormally vs. normally distributed interest rate scenarios, revisited ................................................ 9 II– II.A.2 Exchange Rates .............................................................................................................................................. II–9 FX models with deterministic interest rates ........................................................................................ II1–0 FX models with stochastic interest rates ............................................................................................ 11 II– FX model with deterministic interest rates vs. FX model with stochastic interest rates ...................... 11 II– Validating FX models ......................................................................................................................... 12 II– II.A.3 Equity Returns ............................................................................................................................................... II–12 An overview of equity scenario generation ........................................................................................ II–12 Arbitrage-free equity scenario generation .......................................................................................... 13 II– Stylized facts of equity index returns.................................................................................................. 14 II– Extensions of the Black-Scholes framework ...................................................................................... 17 II– Realistic equity scenarios .................................................................................................................. 18 II– Risk-neutral equity model calibration ................................................................................................. 19 II– Calibration function ............................................................................................................................ 19 II– Optimization ....................................................................................................................................... 19 II– Data ................................................................................................................................................... 20 II– II– (iv) II.A.4 Credit Risks ................................................................................................................................................... 21 Modeling default risks ........................................................................................................................ II–21 Structural models ............................................................................................................................... 21 II– Reduced form models ........................................................................................................................ 22 II– Conclusions ....................................................................................................................................... 24 II– II.A.5 Inflation .......................................................................................................................................................... II–24 Models based on past inflation .......................................................................................................... II–24 Models based on Phillips curve ......................................................................................................... 26 II– References for Section II.A ................................................................................................................ 27 II– II.B Life and Health Models ..................................................................................................................................... II–29 II.B.1 Catastrophic Mortality Modeling ..................................................................................................................... II–29 Overview of the model ....................................................................................................................... II–29 Non-modeled items ............................................................................................................................ 30 II– II.B.1.a Baseline Model .................................................................................................................................... 32 II– Modeling process ............................................................................................................................... II–33 Model results ..................................................................................................................................... 35 II– II.B.1.b Disease Model ..................................................................................................................................... 35 II– Overview of recent pandemics ........................................................................................................... II–35 Influenza ............................................................................................................................................ 36 II– AIDS .................................................................................................................................................. 36 II– SARS ................................................................................................................................................. 36 II– Other diseases ................................................................................................................................... 37 II– General modeling approach ............................................................................................................... 37 II– Modeling the frequency of disease pandemics .................................................................................. 37 II– Potential of repeating a 1918-1920 pandemic ................................................................................... 37 II– Data points ......................................................................................................................................... 38 II– Modeling the severity of disease pandemics ...................................................................................... II–40 Severity curve: Fitting the main component ...................................................................................... II–40 Severity curve: Fitting the extreme component .................................................................................. 42 II– Other supporting assumptions ........................................................................................................... 43 II– Model results ..................................................................................................................................... 43 II– II.B.1.c Terrorism Model ................................................................................................................................... 44 II– Model design ..................................................................................................................................... II–44 Data ................................................................................................................................................... 45 II– Modeling the frequency of terrorist events ......................................................................................... 45 II– Defining levels ................................................................................................................................... 45 II– Defining probabilities.......................................................................................................................... 47 II– Other assumptions ............................................................................................................................. 47 II– Model results ..................................................................................................................................... 47 II– II.B.1.d Combined Model Results ..................................................................................................................... 48 II– II.B.2 Dynamic Policyholder Behaviour ................................................................................................................... II–48 II.B.2.a Traditional Non-par Life Products ........................................................................................................ II–49 II.B.2.b Traditional Par Life Products ............................................................................................................... II–49 II.B.2.c Universal Life and Fixed Annuity.......................................................................................................... II–49 II– (v) II.B.2.d Variable Annuity .................................................................................................................................. 49 Dynamic lapse in VA .......................................................................................................................... II–50 Summary ........................................................................................................................................... 50 II– II.B.3 Morbidity and Claims Experience .................................................................................................................. II–51 Incidence rates .................................................................................................................................. II–51 Severity of claim ................................................................................................................................ 52 II– Inflation .............................................................................................................................................. 52 II– Utilization ........................................................................................................................................... 53 II– Voluntary terminations ....................................................................................................................... 53 II– Claim continuance ............................................................................................................................. 54 II– References for II.B ............................................................................................................................. 54 II– II.C Non-life Claim Models ....................................................................................................................................... II–55 II.C.1 Aggregate Triangle-based Models ................................................................................................................. II–57 Stochastic loss development model ................................................................................................... II–57 Hoerl curve ........................................................................................................................................ 58 II– Mack’s distribution-free model ........................................................................................................... 58 II– Bootstrap model ................................................................................................................................. 59 II– Schnieper ........................................................................................................................................... 60 II– Generalized linear modeling framework............................................................................................. 61 II– II.C.2 Individual Claims Frequency/Severity Models ............................................................................................... II–62 Collective risk model .......................................................................................................................... II–63 Collective risk model by layer ............................................................................................................ 64 II– Transition matrix ................................................................................................................................ 65 II– Generalized linear modeling applied to unpaid claim estimation ....................................................... 66 II– Wright's model ................................................................................................................................... 66 II– II.C.3 Catastrophe Modeling ................................................................................................................................... II–67 References for II.C ............................................................................................................................ II–68 II.D Non-life Financial Models ................................................................................................................................. II–70 II.D.1 Types of Models ............................................................................................................................................ II–70 II.D.1.a The Evolution of Models ...................................................................................................................... II–71 II.D.1.b Uses of Dynamic Risk Models ............................................................................................................. II–71 II.D.2 Description of a Non-life Dynamic Risk Model ............................................................................................... II–72 II.D.2.a General Model Description .................................................................................................................. II–73 II.D.2.b Economic Scenarios ............................................................................................................................ II–73 II.D.2.c Asset Scenarios ................................................................................................................................... II–74 II.D.2.d Underwriting Operations ...................................................................................................................... II–74 Premium ............................................................................................................................................ II–74 Loss payments and liabilities ............................................................................................................. 76 II– Expense payments and liabilities ....................................................................................................... 77 II– Reinsurance ....................................................................................................................................... 78 II– Investment operations........................................................................................................................ 79 II– Accounting and taxation..................................................................................................................... 79 II– II– (vi) Management response ...................................................................................................................... 79 II.D.3 Parameterization and Correlation .................................................................................................................. II–80 References for II.D ............................................................................................................................ II–80 II.E Country- and Region-specific Issues ................................................................................................................ II–81 II.E.1 Regulatory Reporting ..................................................................................................................................... II–81 II.E.2 Liability Valuations ......................................................................................................................................... II–83 II.E.3 Financial Reporting and Embedded Values ................................................................................................... II–85 II.E.4 Product Design and Pricing ........................................................................................................................... II–87 II.E.5 Economic Capital Management ..................................................................................................................... II–88 References for II.E ............................................................................................................................ II–91 II– III. Evaluating and Discussing Results ................................................................................................. III–1 References for III ............................................................................................................................... 2 III.A Calibrating the Model ........................................................................................................................................ III–2 Two approaches to model calibration ................................................................................................. III–3 Calibration to historical experience ..................................................................................................... 3 III– Calibration to current market conditions.............................................................................................. 5 III– III.B Validating the Model ......................................................................................................................................... III–5 III.C Conducting a Peer Review ............................................................................................................................. III1–1 III.D Communicating the Results ............................................................................................................................ III–12 III.E Auditing the Process ....................................................................................................................................... III–14 III– IV. Case Studies .............................................................................................................................................. 1 IV.A Development and Management of a Variable Annuity Product .................................................................IIVV––1 IV.A.1 Introduction .................................................................................................................................................... 1 IV.A.1.a Variable Annuity .................................................................................................................................. IV–1 IV.A.1.b Embedded Guarantees ....................................................................................................................... IV–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 ...................................................................................................................................... IV–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 ........................................................................................................................................... 7 IV– IV.A.4 Developing Mortality and Expense Fee ......................................................................................................... IV–7 IV– (vii) IV.A.5 Developing GLWB Charge ............................................................................................................................ 7 IV.A.5.a Cost of GLWB ..................................................................................................................................... IV–7 IV.A.5.b Charge of GLWB ................................................................................................................................. IV–8 IV.A.5.c Adequacy of Charge ............................................................................................................................ IV–8 Number of scenarios ........................................................................................................................... IV–8 Lapse sensitivity ................................................................................................................................. 9 IV– Mortality sensitivity ............................................................................................................................ 12 IV– Fund allocation sensitivity ................................................................................................................. 12 IV– Scenario sensitivity ........................................................................................................................... 14 IV– Decision of charge level .................................................................................................................... 15 IV– Multiple stochastic variables ............................................................................................................. 15 IV– 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 ................................................................................................ 22 IV– Hedge results on ROA basis ............................................................................................................. 22 IV– IV.A.7 Financial Reporting of Variable Annuities in the United States......................................................................... IV–23 IV.A.7.a U.S. Statutory .................................................................................................................................... IV–23 IV.A.7.b U.S. GAAP ........................................................................................................................................ IV–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.8.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 IV.A.9.b Economic Model and Data ................................................................................................................ IV–27 IV.A.9.c Liability and Capital Requirements .................................................................................................... IV–27 IV.A.9.d Financial Reporting ........................................................................................................................... IV–27 IV– 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 ................................................. 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.a Risk Metrics ....................................................................................................................................... IV–30 IV.B.4.b Confidence Level............................................................................................................................... IV–30 IV– (viii) IV.B.4.c Time Horizon ..................................................................................................................................... 30 IV.B.4.d Projection Techniques ....................................................................................................................... IV–30 IV.B.5 Scenario Generation ................................................................................................................................... IV–31 IV.B.5.a Economic Scenario Generator .......................................................................................................... IV–32 Equity model ..................................................................................................................................... IV–32 Interest rate model ............................................................................................................................ 33 IV– Spot exchange rate model ................................................................................................................ 33 IV– Model parameterization .................................................................................................................... 33 IV– Starting interest rates (foreign and domestic) ................................................................................... 33 IV– Duration parameters (for bond fund calculations) ............................................................................. 35 IV– Equity returns .................................................................................................................................... 35 IV– Currency returns ............................................................................................................................... 36 IV– Money market ................................................................................................................................... 36 IV– Domestic bond .................................................................................................................................. 36 IV– Foreign bond ..................................................................................................................................... 37 IV– IV.B.5.b Credit Risk Model .............................................................................................................................. 37 IV– Description of the model ................................................................................................................... IV–37 Potential simplifications to the model ................................................................................................ 38 IV– Calculating cost of a credit event ...................................................................................................... 39 IV– Results .............................................................................................................................................. 40 IV– IV.B.5.c Mortality ............................................................................................................................................. 41 IV– IV.B.5.d Morbidity ............................................................................................................................................ IV–41 Probability distributions for new claim costs ...................................................................................... IV–41 Probability distribution of claim runoff ............................................................................................... 43 IV– Pricing risk ........................................................................................................................................ 43 IV– IV.B.5.e Lapses ............................................................................................................................................... 44 IV– Operational and strategic risks ........................................................................................................ IV–45 IV.B.6 Presentation of Results ............................................................................................................................... IV–46 PVFP risk metric ............................................................................................................................... IV–46 GPVL risk metric ............................................................................................................................... 47 IV– IV.B.6.a Calibration, Validation, and Review ................................................................................................... 48 IV– Calibration ......................................................................................................................................... IV–48 Validation .......................................................................................................................................... 48 IV– Peer review and checking ................................................................................................................. 48 IV– IV– IV.C Embedded Value for a Multi-national Multi-line Life Insurance Company .............................................IV–51 IV.C.1 Introduction ................................................................................................................................................. 51 IV.C.1.a Brief History of Embedded Value Analysis ........................................................................................ IV–51 IV.C.1.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 IV–

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Stochastic Modeling Theory and Reality from an Actuarial Perspective ISBN 978-0-9813968-1-1 2nd printing Association Actuarielle Internationale International Actuarial Association 150 Metcalfe Street, Suite 601 Ottawa, Ontario Canada K2P 1P1 www.actuaries.org Tel: 1-613-236-0886 Fax: 1-613-236-1386 Email: [email protected] © 2010 International Actuarial Association / Association Actuarielle Internationale Table of Contents Page Introduction .......................................................................................................................................................... (xv) I. General Methodology ................................................................................................................................. I–1 I.A Stochastic Models vs. Non-stochastic Models ..................................................................................................... 1 When should stochastic models be used? ............................................................................................ I–1 When should use of stochastic models be questioned? ....................................................................... 3 I– Alternatives to stochastic models .......................................................................................................... 4 I– Disadvantages of stochastic models ..................................................................................................... 5 I– Guidance on stochastic model implementation ..................................................................................... 5 I– I.B Risk-neutral vs. Real-world .................................................................................................................................. I–6 I.B.1 Risk-neutral Scenarios ....................................................................................................................................... I–6 Background ........................................................................................................................................... I–6 Uses...................................................................................................................................................... 7 I– Calibration and parameterization .......................................................................................................... 8 I– Other considerations ............................................................................................................................. 8 I– I.B.2 Real-world Scenarios ......................................................................................................................................... I–9 Background ........................................................................................................................................... I–9 Uses...................................................................................................................................................... 9 I– I.B.3 Techniques ...................................................................................................................................................... I1–0 I.B.3.a Monte Carlo Simulation ......................................................................................................................... I–10 Variance reduction .............................................................................................................................. I–12 Antithetic-variable technique ............................................................................................................... 13 I– Control variate technique .................................................................................................................... 13 I– Stratified sampling .............................................................................................................................. 13 I– Importance sampling........................................................................................................................... 14 I– I.B.3.b Lattices .................................................................................................................................................. 15 I– Binomial lattice simulations ................................................................................................................. I–15 One-step binomial tree........................................................................................................................ 15 I– Multi-step binomial trees ..................................................................................................................... 16 I– Trinomial lattice simulation .................................................................................................................. 18 I– I.B.3.c Regime-switching Models ...................................................................................................................... 19 I– I.B.4 Nested Stochastic Projections ......................................................................................................................... I–24 Nested stochastic solutions to practical applications .......................................................................... I–25 Nested stochastic modeling and the principles-based approach ........................................................ 26 I– Nested stochastic modeling and other international accounting standards ........................................... 27 I– Managing the nested stochastic modeling process ............................................................................ 27 I– Reducing the number of model points ................................................................................................ 28 I– Reducing the number of outer scenarios ............................................................................................ 28 I– I– (ii) Reducing the number of inner paths ................................................................................................... 28 Reducing the number of nodes ........................................................................................................... 29 I– I.B.5 Deflators .......................................................................................................................................................... I–29 Introduction ......................................................................................................................................... I–29 Mathematical definition ....................................................................................................................... 29 I– Properties ........................................................................................................................................... 30 I– Applications ........................................................................................................................................ 30 I– Practical considerations ...................................................................................................................... 31 I– Illustrative example ............................................................................................................................. 32 I– I.B.5.a Copulas ................................................................................................................................................. 34 I– Fitting copulas ..................................................................................................................................... I–36 Simulating from copulas ...................................................................................................................... 38 I– References for Section I.B .................................................................................................................. 39 I– I.C Distributions and Fitting ..................................................................................................................................... I–40 I.C.1 Stochastic Models ........................................................................................................................................... I–41 I.C.2 Empirical vs. Model Distributions ..................................................................................................................... I–42 I.C.3 A Simple Approach – Matching of Moments .................................................................................................... I–43 I.C.4 A Richer Approach – Maximum Likelihood ...................................................................................................... I–47 References for Section I.C ................................................................................................................... I–48 I.D Random Number Generation ............................................................................................................................. I–48 I.D.1 True and Pseudo Random Number Generators ............................................................................................... I–49 I.D.2 Linear Congruential Generators ...................................................................................................................... I–50 I.D.3 Non-linear PRNGs ........................................................................................................................................... I–53 I.D.3.a Inversive Congruential Generators ........................................................................................................ I–53 I.D.3.b Binary Shift Register Generators ........................................................................................................... I–54 I.D.4 Empirical Tests for Random Numbers ............................................................................................................. I–54 I.D.4.a Kolmogorov-Smirnov Test ..................................................................................................................... I–54 I.D.4.b Poker Test (partition test) ...................................................................................................................... I–55 I.D.4.c Permutation Test ................................................................................................................................... I–56 I.D.5 Methods of Sampling Non-uniform Distributions .............................................................................................. I–57 I.D.5.a Inversion Method ................................................................................................................................... I–57 I.D.5.b Acceptance/rejection Method ................................................................................................................ I–59 I.D.5.c Composition Method .............................................................................................................................. I–60 I.D.5.d Switching Method .................................................................................................................................. I–61 I.D.5.e Ratio of Uniforms Method ...................................................................................................................... I–61 I.D.5.f Tabular Method ...................................................................................................................................... I–63 I.D.5.g Sampling Without Replacement ............................................................................................................ I–63 I.D.5.h Other Techniques and Special Cases ................................................................................................... I–64 Gamma distribution ............................................................................................................................. I–64 Stable distributions ............................................................................................................................. 65 I– Substitution method ............................................................................................................................ 66 I– I– (iii) I.D.6 Summary ......................................................................................................................................................... 67 References for Section I.D ................................................................................................................. I–67 I.E. Risk Measures ................................................................................................................................................... I–69 I.E.1 VaR ........................................................................................................................................................... I–70 I.E.1.a Variance-covariance Method ................................................................................................................. I–70 I.E.1.b Monte Carlo Simulation ......................................................................................................................... I–71 I.E.1.c Historical Simulation .............................................................................................................................. I–71 I.E.2 Conditional Tail Expectation (CTE) .................................................................................................................. I–72 I.E.3 Note on the Confidence Level and Time Horizon ............................................................................................ I–73 I.E.4 Multi-period Risk Measure ............................................................................................................................... I–74 Time consistency ............................................................................................................................... I–75 I.E.5 Note on the Aggregation of Risk ...................................................................................................................... I–76 I.E.6 Other Risk Measures ....................................................................................................................................... I–76 References for Section I.E ................................................................................................................. I–77 I– II. General Applications ................................................................................................................................II–1 II.A Economic Scenario Generators .......................................................................................................................... 1 II.A.1 Interest Rates .................................................................................................................................................. II–1 Realistic yield curve dynamics ............................................................................................................. II–1 HJM/BGM framework for generating arbitrage-free interest rate scenarios ................................................. 4 II– Realistic scenarios over longer time scales ......................................................................................... 5 II– Calibration of the interest rate generator.............................................................................................. 5 II– Key rate analysis of yield curve changes and associated calibration ................................................... 7 II– Combination of interest rate scenarios with other risk factors .............................................................. 8 II– Lognormally vs. normally distributed interest rate scenarios, revisited ................................................ 9 II– II.A.2 Exchange Rates .............................................................................................................................................. II–9 FX models with deterministic interest rates ........................................................................................ II1–0 FX models with stochastic interest rates ............................................................................................ 11 II– FX model with deterministic interest rates vs. FX model with stochastic interest rates ...................... 11 II– Validating FX models ......................................................................................................................... 12 II– II.A.3 Equity Returns ............................................................................................................................................... II–12 An overview of equity scenario generation ........................................................................................ II–12 Arbitrage-free equity scenario generation .......................................................................................... 13 II– Stylized facts of equity index returns.................................................................................................. 14 II– Extensions of the Black-Scholes framework ...................................................................................... 17 II– Realistic equity scenarios .................................................................................................................. 18 II– Risk-neutral equity model calibration ................................................................................................. 19 II– Calibration function ............................................................................................................................ 19 II– Optimization ....................................................................................................................................... 19 II– Data ................................................................................................................................................... 20 II– II– (iv) II.A.4 Credit Risks ................................................................................................................................................... 21 Modeling default risks ........................................................................................................................ II–21 Structural models ............................................................................................................................... 21 II– Reduced form models ........................................................................................................................ 22 II– Conclusions ....................................................................................................................................... 24 II– II.A.5 Inflation .......................................................................................................................................................... II–24 Models based on past inflation .......................................................................................................... II–24 Models based on Phillips curve ......................................................................................................... 26 II– References for Section II.A ................................................................................................................ 27 II– II.B Life and Health Models ..................................................................................................................................... II–29 II.B.1 Catastrophic Mortality Modeling ..................................................................................................................... II–29 Overview of the model ....................................................................................................................... II–29 Non-modeled items ............................................................................................................................ 30 II– II.B.1.a Baseline Model .................................................................................................................................... 32 II– Modeling process ............................................................................................................................... II–33 Model results ..................................................................................................................................... 35 II– II.B.1.b Disease Model ..................................................................................................................................... 35 II– Overview of recent pandemics ........................................................................................................... II–35 Influenza ............................................................................................................................................ 36 II– AIDS .................................................................................................................................................. 36 II– SARS ................................................................................................................................................. 36 II– Other diseases ................................................................................................................................... 37 II– General modeling approach ............................................................................................................... 37 II– Modeling the frequency of disease pandemics .................................................................................. 37 II– Potential of repeating a 1918-1920 pandemic ................................................................................... 37 II– Data points ......................................................................................................................................... 38 II– Modeling the severity of disease pandemics ...................................................................................... II–40 Severity curve: Fitting the main component ...................................................................................... II–40 Severity curve: Fitting the extreme component .................................................................................. 42 II– Other supporting assumptions ........................................................................................................... 43 II– Model results ..................................................................................................................................... 43 II– II.B.1.c Terrorism Model ................................................................................................................................... 44 II– Model design ..................................................................................................................................... II–44 Data ................................................................................................................................................... 45 II– Modeling the frequency of terrorist events ......................................................................................... 45 II– Defining levels ................................................................................................................................... 45 II– Defining probabilities.......................................................................................................................... 47 II– Other assumptions ............................................................................................................................. 47 II– Model results ..................................................................................................................................... 47 II– II.B.1.d Combined Model Results ..................................................................................................................... 48 II– II.B.2 Dynamic Policyholder Behaviour ................................................................................................................... II–48 II.B.2.a Traditional Non-par Life Products ........................................................................................................ II–49 II.B.2.b Traditional Par Life Products ............................................................................................................... II–49 II.B.2.c Universal Life and Fixed Annuity.......................................................................................................... II–49 II– (v) II.B.2.d Variable Annuity .................................................................................................................................. 49 Dynamic lapse in VA .......................................................................................................................... II–50 Summary ........................................................................................................................................... 50 II– II.B.3 Morbidity and Claims Experience .................................................................................................................. II–51 Incidence rates .................................................................................................................................. II–51 Severity of claim ................................................................................................................................ 52 II– Inflation .............................................................................................................................................. 52 II– Utilization ........................................................................................................................................... 53 II– Voluntary terminations ....................................................................................................................... 53 II– Claim continuance ............................................................................................................................. 54 II– References for II.B ............................................................................................................................. 54 II– II.C Non-life Claim Models ....................................................................................................................................... II–55 II.C.1 Aggregate Triangle-based Models ................................................................................................................. II–57 Stochastic loss development model ................................................................................................... II–57 Hoerl curve ........................................................................................................................................ 58 II– Mack’s distribution-free model ........................................................................................................... 58 II– Bootstrap model ................................................................................................................................. 59 II– Schnieper ........................................................................................................................................... 60 II– Generalized linear modeling framework............................................................................................. 61 II– II.C.2 Individual Claims Frequency/Severity Models ............................................................................................... II–62 Collective risk model .......................................................................................................................... II–63 Collective risk model by layer ............................................................................................................ 64 II– Transition matrix ................................................................................................................................ 65 II– Generalized linear modeling applied to unpaid claim estimation ....................................................... 66 II– Wright's model ................................................................................................................................... 66 II– II.C.3 Catastrophe Modeling ................................................................................................................................... II–67 References for II.C ............................................................................................................................ II–68 II.D Non-life Financial Models ................................................................................................................................. II–70 II.D.1 Types of Models ............................................................................................................................................ II–70 II.D.1.a The Evolution of Models ...................................................................................................................... II–71 II.D.1.b Uses of Dynamic Risk Models ............................................................................................................. II–71 II.D.2 Description of a Non-life Dynamic Risk Model ............................................................................................... II–72 II.D.2.a General Model Description .................................................................................................................. II–73 II.D.2.b Economic Scenarios ............................................................................................................................ II–73 II.D.2.c Asset Scenarios ................................................................................................................................... II–74 II.D.2.d Underwriting Operations ...................................................................................................................... II–74 Premium ............................................................................................................................................ II–74 Loss payments and liabilities ............................................................................................................. 76 II– Expense payments and liabilities ....................................................................................................... 77 II– Reinsurance ....................................................................................................................................... 78 II– Investment operations........................................................................................................................ 79 II– Accounting and taxation..................................................................................................................... 79 II– II– (vi) Management response ...................................................................................................................... 79 II.D.3 Parameterization and Correlation .................................................................................................................. II–80 References for II.D ............................................................................................................................ II–80 II.E Country- and Region-specific Issues ................................................................................................................ II–81 II.E.1 Regulatory Reporting ..................................................................................................................................... II–81 II.E.2 Liability Valuations ......................................................................................................................................... II–83 II.E.3 Financial Reporting and Embedded Values ................................................................................................... II–85 II.E.4 Product Design and Pricing ........................................................................................................................... II–87 II.E.5 Economic Capital Management ..................................................................................................................... II–88 References for II.E ............................................................................................................................ II–91 II– III. Evaluating and Discussing Results ................................................................................................. III–1 References for III ............................................................................................................................... 2 III.A Calibrating the Model ........................................................................................................................................ III–2 Two approaches to model calibration ................................................................................................. III–3 Calibration to historical experience ..................................................................................................... 3 III– Calibration to current market conditions.............................................................................................. 5 III– III.B Validating the Model ......................................................................................................................................... III–5 III.C Conducting a Peer Review ............................................................................................................................. III1–1 III.D Communicating the Results ............................................................................................................................ III–12 III.E Auditing the Process ....................................................................................................................................... III–14 III– IV. Case Studies .............................................................................................................................................. 1 IV.A Development and Management of a Variable Annuity Product .................................................................IIVV––1 IV.A.1 Introduction .................................................................................................................................................... 1 IV.A.1.a Variable Annuity .................................................................................................................................. IV–1 IV.A.1.b Embedded Guarantees ....................................................................................................................... IV–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 ...................................................................................................................................... IV–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 ........................................................................................................................................... 7 IV– IV.A.4 Developing Mortality and Expense Fee ......................................................................................................... IV–7 IV– (vii) IV.A.5 Developing GLWB Charge ............................................................................................................................ 7 IV.A.5.a Cost of GLWB ..................................................................................................................................... IV–7 IV.A.5.b Charge of GLWB ................................................................................................................................. IV–8 IV.A.5.c Adequacy of Charge ............................................................................................................................ IV–8 Number of scenarios ........................................................................................................................... IV–8 Lapse sensitivity ................................................................................................................................. 9 IV– Mortality sensitivity ............................................................................................................................ 12 IV– Fund allocation sensitivity ................................................................................................................. 12 IV– Scenario sensitivity ........................................................................................................................... 14 IV– Decision of charge level .................................................................................................................... 15 IV– Multiple stochastic variables ............................................................................................................. 15 IV– 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 ................................................................................................ 22 IV– Hedge results on ROA basis ............................................................................................................. 22 IV– IV.A.7 Financial Reporting of Variable Annuities in the United States......................................................................... IV–23 IV.A.7.a U.S. Statutory .................................................................................................................................... IV–23 IV.A.7.b U.S. GAAP ........................................................................................................................................ IV–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.8.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 IV.A.9.b Economic Model and Data ................................................................................................................ IV–27 IV.A.9.c Liability and Capital Requirements .................................................................................................... IV–27 IV.A.9.d Financial Reporting ........................................................................................................................... IV–27 IV– 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 ................................................. 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.a Risk Metrics ....................................................................................................................................... IV–30 IV.B.4.b Confidence Level............................................................................................................................... IV–30 IV– (viii) IV.B.4.c Time Horizon ..................................................................................................................................... 30 IV.B.4.d Projection Techniques ....................................................................................................................... IV–30 IV.B.5 Scenario Generation ................................................................................................................................... IV–31 IV.B.5.a Economic Scenario Generator .......................................................................................................... IV–32 Equity model ..................................................................................................................................... IV–32 Interest rate model ............................................................................................................................ 33 IV– Spot exchange rate model ................................................................................................................ 33 IV– Model parameterization .................................................................................................................... 33 IV– Starting interest rates (foreign and domestic) ................................................................................... 33 IV– Duration parameters (for bond fund calculations) ............................................................................. 35 IV– Equity returns .................................................................................................................................... 35 IV– Currency returns ............................................................................................................................... 36 IV– Money market ................................................................................................................................... 36 IV– Domestic bond .................................................................................................................................. 36 IV– Foreign bond ..................................................................................................................................... 37 IV– IV.B.5.b Credit Risk Model .............................................................................................................................. 37 IV– Description of the model ................................................................................................................... IV–37 Potential simplifications to the model ................................................................................................ 38 IV– Calculating cost of a credit event ...................................................................................................... 39 IV– Results .............................................................................................................................................. 40 IV– IV.B.5.c Mortality ............................................................................................................................................. 41 IV– IV.B.5.d Morbidity ............................................................................................................................................ IV–41 Probability distributions for new claim costs ...................................................................................... IV–41 Probability distribution of claim runoff ............................................................................................... 43 IV– Pricing risk ........................................................................................................................................ 43 IV– IV.B.5.e Lapses ............................................................................................................................................... 44 IV– Operational and strategic risks ........................................................................................................ IV–45 IV.B.6 Presentation of Results ............................................................................................................................... IV–46 PVFP risk metric ............................................................................................................................... IV–46 GPVL risk metric ............................................................................................................................... 47 IV– IV.B.6.a Calibration, Validation, and Review ................................................................................................... 48 IV– Calibration ......................................................................................................................................... IV–48 Validation .......................................................................................................................................... 48 IV– Peer review and checking ................................................................................................................. 48 IV– IV– IV.C Embedded Value for a Multi-national Multi-line Life Insurance Company .............................................IV–51 IV.C.1 Introduction ................................................................................................................................................. 51 IV.C.1.a Brief History of Embedded Value Analysis ........................................................................................ IV–51 IV.C.1.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 IV–

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