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Actuarial Science

Theory And Methodology

By: Hanji Shang (Editor)

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Hardcover | 4 July 2006

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282 Pages

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Since actuarial education was introduced into China in the 1980s, Chinese scholars have paid greater attention to the theoretical research of actuarial science. Professors and industry experts from well-known universities in China recently worked together on the project "Insurance Information Processing and Actuarial Mathematics Theory and Methodology", which was supported by the Chinese government. Summarizing what they achieved, this volume provides a study of some basic problems of actuarial science, including risk models, risk evaluation and analysis, and premium principles. The contributions cover some new applications of probability and statistics, fuzzy mathematics and financial economics to the field of actuarial practices. Discussions on the new insurance market in China are also presented.

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Non-Fiction Accounting & Finance Finance Insurance & Actuarial Studies Business & Management Business Mathematics & Systems Mathematics Probability & Statistics Economics Labour Economics Booktopia Publisher Services World Scientific Publishing

Econometrics

Preface
Risk Models and Ruin Theory	p. 1
On the Distribution of Surplus Immediately after Ruin under Interest Force	p. 2
The Risk Model	p. 2
Equations for G[subscript delta] (u, y)	p. 3
Integral Equations for G[subscript delta] (u, y), G[subscript delta](u, y) and G[subscript delta](u, y)	p. 3
The Case [delta] = 0	p. 6
Upper and Lower Bounds for G[subscript delta](0, y)	p. 8
On the Distribution of Surplus Immediately before Ruin under Interest Force	p. 11
Equations for B[subscript delta](u, y)	p. 12
Integral Equations for B[subscript delta](u, y)	p. 12
The Case [delta] = 0	p. 14
Solution of the Integral Equation	p. 15
B[subscript delta](u, y) with Zero Initial Reserve	p. 17
Exponential Claim Size	p. 19
Lundberg Bound	p. 20
Asymptotic Estimates of the Low and Upper Bounds for the Distribution of the Surplus Immediately after Ruin under Subexponential Claims	p. 21
Preliminaries and Auxiliary Relations	p. 22
Asymptotic Estimates of the Low and Upper Bounds	p. 27
On the Ruin Probability under a Class of Risk Processes	p. 34
The Risk Model	p. 34
The Laplace Transform of the Ruin Probability with Finite Time	p. 34
Two Corollaries	p. 40
Compound Risk Models and Copula Decomposition	p. 47
Introduction	p. 47
Individual Risk Model and Compound Risk Model	p. 48
The Link between the Compound Risk Model and the Individual Risk Model	p. 49
One Theorem on Excess-of-loss Reinsurance	p. 50
Recursive Calculation of Compound Distributions	p. 52
One-dimensional Recursive Equations	p. 53
Proofs of Theorems 2.2-2.3	p. 58
Bivariate Recursive Equations	p. 63
The Compound Poisson Random Variable's Approximation to the Individual Risk Model	p. 65
The Existence of the Optimal Poisson r.v.	p. 66
The Joint Distribution of ([Characters not reproducible]([Theta]), N[subscript n])	p. 69
Evaluating the Approximation Error	p. 70
The Approximation to Functions of the Total Loss	p. 73
The Uniqueness of the Poisson Parameter to Minimizing H[subscript n]([Theta])	p. 74
Proofs	p. 75
Bivariate Copula Decomposition	p. 82
Copula Decomposition	p. 83
Application of the Copula Decomposition	p. 88
Comonotonically Additive Premium Principles and Some Related Topics	p. 93
Introduction	p. 93
Characterization of Distortion Premium Principles	p. 94
Preliminaries	p. 95
Greco Theorem	p. 98
Characterization of Distortion Premium Principles	p. 101
Further Remarks on Additivity of Premium Principles	p. 107
Representation of Strictly Additive Premium Principles	p. 107
Relationship among Additivities	p. 109
Natural Sets of Distortion Premium Principles	p. 111
Ordering Risks by Distortion Premiums	p. 119
n-ordered Orders of Real-valued Random Variables	p. 121
n-ordered Dual Orders of Real-valued Random Variables	p. 124
Final Remarks	p. 129
Fuzzy Comprehensive Evaluation and Fuzzy Information Processing for Risks	p. 133
Introduction	p. 133
Fuzzy Comprehensive Evaluation for Risks	p. 134
Basic Concepts and Process	p. 134
Construct Factor Set	p. 134
Construct Weight Set	p. 134
Construct Evaluation Set	p. 134
Single Factor Fuzzy Evaluation	p. 135
Fuzzy Comprehensive Evaluation	p. 135
An Example of Risk Evaluation	p. 136
Determination of Main Risk Factors	p. 136
Evaluation of the Risk	p. 137
Applications	p. 139
Fuzzy Information Distribution in Risk Evaluation and Analysis	p. 140
Concept of Fuzzy Information Distribution	p. 140
Information Distribution Method	p. 141
Improving IDM	p. 144
Applications	p. 145
Information Diffusion and Its Application to Risk Analysis	p. 146
Mechanism of Information Diffusion	p. 146
An Example of Application - 1D Problem	p. 148
Large Sample	p. 149
Small Sample - Statistical Approach	p. 149
Small Sample - UIDM	p. 150
An Example of Application - 2D Problem	p. 151
Large Sample	p. 151
Small Sample - Statistical Approach	p. 151
Small Sample - UIDM	p. 152
Optimized Information Diffusion Method (OIDM)	p. 153
OIDM in 1D Case	p. 154
OIDM in 2D Case	p. 156
Conclusion	p. 157
Application of Fuzzy Mathematics to Actuarial Science	p. 159
Introduction	p. 159
Some Basic Notions of Fuzzy Set Theory	p. 160
Application of FST in Life Insurance Game	p. 162
Background	p. 162
Some Relative Concepts and Theorems	p. 162
Model of Game	p. 166
The Example of Application	p. 167
The Example	p. 167
Conclusion	p. 171
Decision-Making Method Applied in Life Insurance Companies	p. 171
Background	p. 171
The Passive Decision-Two-stage Fuzzy Comprehensive Valuation	p. 172
The Initiative Decision-Multi-object Fuzzy Group Decision	p. 174
Synthetic Decision	p. 178
The Risk Analysis of Complications for Some Diseases	p. 179
Background	p. 179
The Risk of Complications	p. 180
Determining the Variable	p. 180
Define the Similar Matrix R	p. 180
The Transitive Closure t(R)	p. 182
Optimum Fuzzy Equivalent Matrix R[subscript min]	p. 183
Some Results	p. 183
The Illness Degree of Diseases	p. 184
Basic Concept and Method	p. 184
[Zeta][subscript h]: Illness Degree of Hypertension(IDOH)	p. 185
[Zeta][subscript c]: Illness Degree of Coronary Heart Disease (IDOC)	p. 186
The Relationship between Hypertension and Coronary Heart Disease	p. 186
The Application to Insurance	p. 188
Regression Forecasting Model with Fuzzy Factors	p. 190
Background	p. 190
Regression Forecasting Model with Crisp Factors	p. 190
Regression Forecasting Model with Crisp Factors and Fuzzy Factors	p. 191
Some Concepts, Methods and an Application Example	p. 192
Regression Forecasting Model with Crisp Factors and Fuzzy Factors	p. 194
Example and Comparison of Two Kinds of Regression Model	p. 195
Conclusion	p. 198
Some Applications of Financial Economics to Insurance	p. 201
Introduction	p. 201
General Framework of the Valuation of Unit-linked Insurance Policy	p. 203
Differential Equation Models for the Valuation of Policies without Surrender Option	p. 204
P.D.E. Model for the Valuation of Policies with Surrender Option	p. 206
Generalized Expected Discounted Value Approach	p. 208
Fair Valuation of First Kind of Unit-linked Policy	p. 210
The Case [sigma](t, A) = 0	p. 210
The Case [sigma](t, A) [Characters not reproducible] 0	p. 212
Fair Valuation of Second Kind of Unit-linked Policy without Surrender Option	p. 214
P.D.E. Approach	p. 215
G.E.D.V. Approach	p. 219
Fair Valuation of Second Kind of Unit-linked Policy with Surrender Option	p. 221
Analysis of Parameters	p. 223
Local Analysis of Free Boundary near the Expiry Date	p. 226
Integral Equation on v(t, A)	p. 229
Numerical Results	p. 232
Linear Complementary Problem and Projected SOR Method	p. 232
Solving Integral Equation (6.131)	p. 240
Exploring on the Risk Profile of China Insurance for Setting Appropriate Solvency Capital Requirement	p. 245
Introduction	p. 245
Toward a Risk-oriented Approach of Solvency Supervision System for China Insurers	p. 247
Internal Control	p. 248
Solvency Capital Requirement	p. 248
On Site Inspection	p. 250
Investment Control	p. 250
Guarantee Fund	p. 251
Risk Construction of Chinese Insurers	p. 251
Risk Concepts	p. 251
Identification of Methodologies	p. 252
Normative Studies: International Comparisons and Case Analysis	p. 253
Statistical Analysis	p. 254
Field Study and Cases Analysis on China Insurers	p. 255
Combined Approach	p. 256
Keeping Up an Overall and Historical View on the Evolution of Risk Profile of China Insurance	p. 256
Period 1: 1980 - 1995	p. 257
Period 2: 1995 - End of 2003	p. 258
Period 3: 2004 - Near Future	p. 259
Risk Characteristics and Proposed Principles for Solvency Capital Requirement	p. 261
Main Characteristics of Risk Profile	p. 261
Guiding Principles for Setting Capital Requirement	p. 262
Index	p. 265
Table of Contents provided by Ingram. All Rights Reserved.

Actuarial Science

Theory And Methodology

At a Glance

Hardcover

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