Actuarial Science - Theory and Methodology

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收藏 2008-10-16

Actuarial Science - Theory and MethodologyHanji Shang复旦大学英文版Chapter 1 Risk Models and Ruin Theory 1 1.1 On the Distribution of Surplus Immediately after Ruin under Interest Force 2 1.1.1 The Risk Model 2 1.1.2 Equations for Gs (u, y) 3 1.1.2.1 Integral Equations for Gs (u, y), Gs (u, y) and Gs{u,y) 3 1.1.2.2 The Case 6 = 0 6 1.1.3 Upper and Lower Bounds for Gs(0,y) 8 1.2 On the Distribution of Surplus Immediately before Ruin under Interest Force 11 1.2.1 Equations for Bs(u,y) 12 1.2.1.1 Integral Equations for B$(u,y) 12 1.2.1.2 The Case 6 = 0 14 1.2.1.3 Solution of the Integral Equation 15 1.2.2 Bg(u,y) with Zero Initial Reserve 17 1.2.3 Exponential Claim Size 19 1.2.4 Lundberg Bound 20 1.3 Asymptotic Estimates of the Low and Upper Bounds for the Distribution of the Surplus Immediately after Ruin under Subexponential Claims 21 1.3.1 Preliminaries and Auxiliary Relations 22 1.3.2 Asymptotic Estimates of the Low and Upper Bounds 27 1.4 On the Ruin Probability under a Class of Risk Processes . . 341.4.1 The Risk Model 34 1.4.2 The Laplace Transform of the Ruin Probability with Finite Time 34 1.4.3 Two Corollaries 40 Chapter 2 Compound Risk Models and Copula Decomposition 47 2.1 Introduction 47 2.2 Individual Risk Model and Compound Risk Model 48 2.2.1 The Link between the Compound Risk Model and the Individual Risk Model 49 2.2.2 One Theorem on Excess-of-loss Reinsurance 50 2.3 Recursive Calculation of Compound Distributions 52 2.3.1 One-dimensional Recursive Equations 53 2.3.2 Proofs of Theorems 2.2-2.3 58 2.3.3 Bivariate Recursive Equations 63 2.4 The Compound Poisson Random Variable's Approximation to the Individual Risk Model 65 2.4.1 The Existence of the Optimal Poisson r.v 66 2.4.2 The Joint Distribution of (N°(6),Nn) 69 2.4.3 Evaluating the Approximation Error 70 2.4.4 The Approximation to Functions of the Total Loss . 73 2.4.5 The Uniqueness of the Poisson Parameter to Minimizing Hn{6) 74 2.4.6 Proofs 75 2.5 Bivariate Copula Decomposition 82 2.5.1 Copula Decomposition 83 2.5.2 Application of the Copula Decomposition 88 Chapter 3 Comonotonically Additive Premium Principles and Some Related Topics 93 3.1 Introduction . . 93 3.2 Characterization of Distortion Premium Principles 94 3.2.1 Preliminaries 95 3.2.2 Greco Theorem 98 3.2.3 Characterization of Distortion Premium Principles . 101 3.2.4 Further Remarks on Additivity of Premium Principles 1073.2.4.1 Representation of Strictly Additive Premium Principles 107 3.2.4.2 Relationship among Additivities 109 3.3 Natural Sets of Distortion Premium Principles Ill 3.4 Ordering Risks by Distortion Premiums 119 3.4.1 n-ordered Orders of Real-valued Random Variables. . 121 3.4.2 n-ordered Dual Orders of Real-valued Random Variables 124 3.5 Final Remarks 129 Chapter 4 Fuzzy Comprehensive Evaluation and Fuzzy Information Processing for Risks 133 4.1 Introduction 133 4.2 Fuzzy Comprehensive Evaluation for Risks 134 4.2.1 Basic Concepts and Process 134 4.2.1.1 Construct Factor Set 134 4.2.1.2 Construct Weight Set 134 4.2.1.3 Construct Evaluation Set 134 4.2.1.4 Single Factor Fuzzy Evaluation 135 4.2.1.5 Fuzzy Comprehensive Evaluation 135 4.2.2 An Example of Risk Evaluation 136 4.2.2.1 Determination of Main Risk Factors 136 4.2.2.2 Evaluation of the Risk 137 4.2.2.3 Applications 139 4.3 Fuzzy Information Distribution in Risk Evaluation and Analysis 140 4.3.1 Concept of Fuzzy Information Distribution 140 4.3.2 Information Distribution Method 141 4.3.3 Improving IDM 144 4.3.4 Applications 145 4.4 Information Diffusion and Its Application to Risk Analysis . 146 4.4.1 Mechanism of Information Diffusion 146 4.4.2 An Example of Application — ID Problem 148 4.4.2.1 Large Sample 149 4.4.2.2 Small Sample — Statistical Approach . . . . 149 4.4.2.3 Small Sample — UIDM 150 4.4.3 An Example of Application — 2D Problem 151 4.4.3.1 Large Sample 151 4.4.3.2 Small Sample — Statistical Approach . . . . 1514.4.3.3 Small Sample — UIDM 152 4.4.4 Optimized Information Diffusion Method (OIDM) . . 153 4.4.4.1 OIDM in ID Case 154 4.4.4.2 OIDM in 2D Case 156 4.5 Conclusion 157 Chapter 5 Application of Fuzzy Mathematics to Actuarial Science 159 5.1 Introduction 159 5.2 Some Basic Notions of Fuzzy Set Theory 160 5.3 Application of FST in Life Insurance Game 162 5.3.1 Background 162 5.3.2 Some Relative Concepts and Theorems 162 5.3.3 Model of Game 166 5.3.4 The Example of Application 167 5.3.4.1 The Example 167 5.3.4.2 Conclusion 171 5.4 Decision-Making Method Applied in Life Insurance Companiesl71 5.4.1 Background 171 5.4.2 The Passive Decision—Two-stage Fuzzy Comprehensive Valuation 172 5.4.3 The Initiative Decision—Multi-object Fuzzy Group Decision 174 5.4.4 Synthetic Decision 178 5.5 The Risk Analysis of Complications for Some Diseases . . . 179 5.5.1 Background 179 5.5.2 The Risk of Complications 180 5.5.2.1 Determining the Variable 180 5.5.2.2 Define the Similar Matrix R 180 5.5.2.3 The Transitive Closure t(R) 182 5.5.2.4 Optimum Fuzzy Equivalent Matrix i?min • • 183 5.5.2.5 Some Results 183 5.5.3 The Illness Degree of Diseases 184 5.5.3.1 Basic Concept and Method 184 5.5.3.2 £h: Illness Degree of Hypertension(IDOH) . 185 5.5.3.3 £c: Illness Degree of Coronary Heart Disease (IDOC) 186 5.5.3.4 The Relationship between Hypertension and Coronary Heart Disease 1865.5.3.5 The Application to Insurance 188 5.6 Regression Forecasting Model with Fuzzy Factors 190 5.6.1 Background 190 5.6.2 Regression Forecasting Model with Crisp Factors . . 190 5.6.3 Regression Forecasting Model with Crisp Factors and Fuzzy Factors 191 5.6.3.1 Some Concepts, Methods and an Application Example 192 5.6.3.2 Regression Forecasting Model with Crisp Factors and Fuzzy Factors 194 5.6.4 Example and Comparison of Two Kinds of Regression Model 195 5.6.5 Conclusion 198 Chapter 6 Some Applications of Financial Economics to Insurance 201 6.1 Introduction 201 6.2 General Framework of the Valuation of Unit-linked Insurance Policy 203 6.2.1 Differential Equation Models for the Valuation of Policies without Surrender Option 204 6.2.2 P.D.E. Model for the Valuation of Policies with Surrender Option 206 6.2.3 Generalized Expected Discounted Value Approach . 208 6.3 Fair Valuation of First Kind of Unit-linked Policy 210 6.3.1 The Case a(t, A) = 0 210 6.3.2 The Case a(t, A) ^ 0 212 6.4 Fair Valuation of Second Kind of Unit-linked Policy without Surrender Option 214 6.4.1 P.D.E. Approach 215 6.4.2 G.E.D.V. Approach 219 6.5 Fair Valuation of Second Kind of Unit-linked Policy with Surrender Option 221 6.5.1 Analysis of Parameters 223 6.5.2 Local Analysis of Free Boundary near the Expiry Date 226 6.5.3 Integral Equation on v(t, A) 229 6.5.4 Numerical Results 232 6.5.4.1 Linear Complementary Problem and Projected SOR Method 2326.5.4.2 Solving Integral Equation (6.131) 240 Chapter 7 Exploring on the Risk Profile of China Insurance for Setting Appropriate Solvency Capital Requirement 245 7.1 Introduction 245 7.2 Toward a Risk-oriented Approach of Solvency Supervision System for China Insurers 247 7.2.1 Internal Control 248 7.2.2 Solvency Capital Requirement 248 7.2.3 On Site Inspection 250 7.2.4 Investment Control 250 7.2.5 Guarantee Fund 251 7.3 Risk Construction of Chinese Insurers 251 7.3.1 Risk Concepts 251 7.3.2 Identification of Methodologies 252 7.3.2.1 Normative Studies: International Comparisons and Case Analysis 253 7.3.2.2 Statistical Analysis 254 7.3.2.3 Field Study and Cases Analysis on China Insurers 255 7.3.2.4 Combined Approach 256 7.3.3 Keeping Up an Overall and Historical View on the Evolution of Risk Profile of China Insurance 256 7.3.3.1 Period 1: 1980 — 1995 257 7.3.3.2 Period 2: 1995 — End of 2003 258 7.3.3.3 Period 3: 2004 — Near Future 259 7.3.4 Risk Characteristics and Proposed Principles for Solvency Capital Requirement 261 7.3.4.1 Main Characteristics of Risk Profile . . . . 261 7.3.4.2 Guiding Principles for Setting Capital Requirement 262 Index 265