Stochastic Modeling Theory and Reality from an Actuarial

418 Pages·2012·5.76 MB·English

Stochastic Modeling
Theory and Reality from an Actuarial Perspective
ISBN 978-0-9813968-1-1

2nd printing

Association Actuarielle Internationale
International Actuarial Association
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Ottawa, Ontario
Canada K2P 1P1
www.actuaries.org
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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–