Statistical and probabilistic methods in actuarial science:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2007
|
| Series: | Chapman & Hall/CRC interdisciplinary statistics series
|
| Subjects: | |
| Online Access: | Table of contents only Publisher description Klappentext Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 304-307) and index |
| Physical Description: | XVI, 351 S. graph. Darst. 25 cm |
| ISBN: | 1584886951 9781584886952 |
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MARC
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| 490 | 0 | |a Chapman & Hall/CRC interdisciplinary statistics series | |
| 500 | |a Includes bibliographical references (p. 304-307) and index | ||
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| adam_text | Covering many of the diverse methods in applied probability and statistics, Statistical
and Probabilistic Methods in Actuarial Science builds on readers existing knowledge
of probability and statistics by establishing a solid and thorough understanding of these
methods. It also emphasizes the wide variety of practical situations in insurance and
actuarial science where these techniques may be used.
Although some chapters are linked, several can be studied independently from the
others. The first chapter introduces claims reserving via the deterministic chain ladder
technique. The next few chapters survey loss distributions, risk models in a fixed period
of time, and surplus processes, followed by an examination of credibility theory in
which collateral and sample information are brought together to provide reasonable
methods of estimation. In the subsequent chapter, experience rating via no claim
discount schemes for motor insurance provides an interesting application of Markov
chain methods. The final chapters discuss generalized linear models and decision and
game theory.
Statistical and Probabilistic Methods in Actuarial Science is the ideal text for
students aspiring to careers in insurance, actuarial science, and finance. In addition,
the book serves as a valuable reference for insurance analysts who commonly use
probabilistic and statistical techniques in practice.
Features
•
Features practical applications to general insurance, including loss distributions
and collective risk models, reserving and experience rating, credibility estimation,
and security measures of risk
•
Presents useful introductions to generalized linear models, credibility theory, and
game theory
•
Includes material on computing, such as simulation techniques
•
Contains numerous worked numerical examples and problems with selected
answers in an appendix
•
Provides appendices on Bayesian statistics and basic tools in probability and
statistics
Contents
Dedication v
Preface vii
Introduction ix
1 Claims Reserving and Pricing with Run-Off Triangles 1
1.1 The evolving nature of claims and reserves 1
1.2 Chain ladder methods 4
1.2.1 Basic chain ladder method 5
1.2.2 Inflation-adjusted chain ladder method 8
1.3 The average cost per claim method 11
1.4 The Bornhuetter-Ferguson or loss ratio method 14
1.5 An example in pricing products 19
1.6 Statistical modeling and the separation technique 26
1.7 Problems 27
2 Loss Distributions 35
2.1 Introduction to loss distributions 35
2.2 Classical loss distributions 36
2.2.1 Exponential distribution 36
2.2.2 Pareto distribution 39
2.2.3 Gamma distribution 43
2.2.4 Weibull distribution 45
2.2.5 Lognormal distribution 47
2.3 Fitting loss distributions 51
2.3.1 Kolmogorov-Smirnoff test 52
2.3.2 Chi-square goodness-of-fit tests 54
2.3.3 Akaike information criteria 58
2.4 Mixture distributions 58
2.5 Loss distributions and reinsurance 61
2.5.1 Proportional reinsurance 62
2.5.2 Excess of loss reinsurance 62
2.6 Problems 68
xiii
xiv CONTENTS
3 Risk Theory 77
3.1 Risk models for aggregate claims 77
3.2 Collective risk models 78
3.2.1 Basic properties of compound distributions 79
3.2.2 Compound Poisson, binomial and negative binomial
distributions 79
3.2.3 Sums of compound Poisson distributions 85
3.2.4 Exact expressions for the distribution of S 87
3.2.5 Approximations for the distribution of S 92
3.3 Individual risk models for S 94
3.3.1 Basic properties of the individual risk model 95
3.3.2 Compound binomial distributions and individual risk
models 97
3.3.3 Compound Poisson approximations for individual risk
models 98
3.4 Premiums and reserves for aggregate claims 99
3.4.1 Determining premiums for aggregate claims 99
3.4.2 Setting aside reserves for aggregate claims 103
3.5 Reinsurance for aggregate claims 107
3.5.1 Proportional reinsurance 109
3.5.2 Excess of loss reinsurance Ill
3.5.3 Stop-loss reinsurance 116
3.6 Problems 120
4 Ruin Theory 129
4.1 The probability of ruin in a surplus process 129
4.2 Surplus and aggregate claims processes 129
4.2.1 Probability of ruin in discrete time 132
4.2.2 Poisson surplus processes 132
4.3 Probability of ruin and the adjustment coefficient 134
4.3.1 The adjustment equation 135
4.3.2 Lundberg s bound on the probability of ruin ip(U) . . 138
4.3.3 The probability of ruin when claims are exponentially
distributed 140
4.4 Reinsurance and the probability of ruin 146
4.4.1 Adjustment coefficients and proportional reinsurance . 147
4.4.2 Adjustment coefficients and excess of loss reinsurance 149
4.5 Problems 152
5 Credibility Theory 159
5.1 Introduction to credibility estimates 159
5.2 Classical credibility theory 161
5.2.1 Full credibility 161
5.2.2 Partial credibility 163
5.3 The Bayesian approach to credibility theory 164
CONTENTS xv
5.3.1 Bayesian credibility 164
5.4 Greatest accuracy credibility theory 170
5.4.1 Bayes and linear estimates of the posterior mean . . . 172
5.4.2 Predictive distribution for Xn+ 175
5.5 Empirical Bayes approach to credibility theory 176
5.5.1 Empirical Bayes credibility - Model 1 177
5.5.2 Empirical Bayes credibility - Model 2 180
5.6 Problems 183
6 No Claim Discounting in Motor Insurance 191
6.1 Introduction to No Claim Discount schemes 191
6.2 Transition in a No Claim Discount system 193
6.2.1 Discount classes and movement in NCD schemes . . . 193
6.2.2 One-step transition probabilities in NCD schemes . . . 195
6.2.3 Limiting distributions and stability in NCD models . . 198
6.3 Propensity to make a claim in NCD schemes 204
6.3.1 Thresholds for claims when an accident occurs .... 205
6.3.2 The claims rate process in an NCD system 208
6.4 Reducing heterogeneity with NCD schemes 212
6.5 Problems 214
7 Generalized Linear Models 221
7.1 Introduction to linear and generalized linear models 221
7.2 Multiple linear regression and the normal model 225
7.3 The structure of generalized linear models 230
7.3.1 Exponential families 232
7.3.2 Link functions and linear predictors 236
7.3.3 Factors and eovariates 238
7.3.4 Interactions 238
7.3.5 Minimally sufficient statistics 244
7.4 Model selection and deviance 245
7.4.1 Deviance and the saturated model 245
7.4.2 Comparing models with deviance 248
7.4.3 Residual analysis for generalized linear models .... 252
7.5 Problems 258
8 Decision and Game Theory 265
8.1 Introduction 265
8.2 Game theory 267
8.2.1 Zero-sum two-person games 268
8.2.2 Minimax and saddle point strategies 270
8.2.3 Randomized strategies 273
8.2.4 The Prisoner s Dilemma and Nash equilibrium in variable-
sum games 278
8.3 Decision making and risk 280
xvi CONTENTS
8.3.1 The minimax criterion 283
8.3.2 The Bayes criterion 283
8.4 Utility and expected monetary gain 288
8.4.1 Rewards, prospects and utility 290
8.4.2 Utility and insurance 292
8.5 Problems 295
References 304
Appendix A Basic Probability Distributions 309
Appendix B Some Basic Tools in Probability and Statistics 313
B.I Moment generating functions 313
B.2 Convolutions of random variables 316
B.3 Conditional probability and distributions 317
B.3.1 The double expectation theorem and E(X) 319
B.3.2 The random variable V(X Y) 322
B.4 Maximum likelihood estimation 324
Appendix C An Introduction to Bayesian Statistics 327
C.I Bayesian statistics 327
C.I.I Conjugate families 328
C.I.2 Loss functions and Bayesian inference 329
Appendix D Answers to Selected Problems 335
D.I Claims reserving and pricing with run-off triangles 335
D.2 Loss distributions 335
D.3 Risk theory 337
D.4 Ruin theory 338
D.5 Credibility theory 338
D.6 No claim discounting in motor insurance 340
D.7 Generalized linear models 340
D.8 Decision and game theory 341
Index 345
|
| adam_txt |
Covering many of the diverse methods in applied probability and statistics, Statistical
and Probabilistic Methods in Actuarial Science builds on readers' existing knowledge
of probability and statistics by establishing a solid and thorough understanding of these
methods. It also emphasizes the wide variety of practical situations in insurance and
actuarial science where these techniques may be used.
Although some chapters are linked, several can be studied independently from the
others. The first chapter introduces claims reserving via the deterministic chain ladder
technique. The next few chapters survey loss distributions, risk models in a fixed period
of time, and surplus processes, followed by an examination of credibility theory in
which collateral and sample information are brought together to provide reasonable
methods of estimation. In the subsequent chapter, experience rating via no claim
discount schemes for motor insurance provides an interesting application of Markov
chain methods. The final chapters discuss generalized linear models and decision and
game theory.
Statistical and Probabilistic Methods in Actuarial Science is the ideal text for
students aspiring to careers in insurance, actuarial science, and finance. In addition,
the book serves as a valuable reference for insurance analysts who commonly use
probabilistic and statistical techniques in practice.
Features
•
Features practical applications to general insurance, including loss distributions
and collective risk models, reserving and experience rating, credibility estimation,
and security measures of risk
•
Presents useful introductions to generalized linear models, credibility theory, and
game theory
•
Includes material on computing, such as simulation techniques
•
Contains numerous worked numerical examples and problems with selected
answers in an appendix
•
Provides appendices on Bayesian statistics and basic tools in probability and
statistics
Contents
Dedication v
Preface vii
Introduction ix
1 Claims Reserving and Pricing with Run-Off Triangles 1
1.1 The evolving nature of claims and reserves 1
1.2 Chain ladder methods 4
1.2.1 Basic chain ladder method 5
1.2.2 Inflation-adjusted chain ladder method 8
1.3 The average cost per claim method 11
1.4 The Bornhuetter-Ferguson or loss ratio method 14
1.5 An example in pricing products 19
1.6 Statistical modeling and the separation technique 26
1.7 Problems 27
2 Loss Distributions 35
2.1 Introduction to loss distributions 35
2.2 Classical loss distributions 36
2.2.1 Exponential distribution 36
2.2.2 Pareto distribution 39
2.2.3 Gamma distribution 43
2.2.4 Weibull distribution 45
2.2.5 Lognormal distribution 47
2.3 Fitting loss distributions 51
2.3.1 Kolmogorov-Smirnoff test 52
2.3.2 Chi-square goodness-of-fit tests 54
2.3.3 Akaike information criteria 58
2.4 Mixture distributions 58
2.5 Loss distributions and reinsurance 61
2.5.1 Proportional reinsurance 62
2.5.2 Excess of loss reinsurance 62
2.6 Problems 68
xiii
xiv CONTENTS
3 Risk Theory 77
3.1 Risk models for aggregate claims 77
3.2 Collective risk models 78
3.2.1 Basic properties of compound distributions 79
3.2.2 Compound Poisson, binomial and negative binomial
distributions 79
3.2.3 Sums of compound Poisson distributions 85
3.2.4 Exact expressions for the distribution of S 87
3.2.5 Approximations for the distribution of S 92
3.3 Individual risk models for S 94
3.3.1 Basic properties of the individual risk model 95
3.3.2 Compound binomial distributions and individual risk
models 97
3.3.3 Compound Poisson approximations for individual risk
models 98
3.4 Premiums and reserves for aggregate claims 99
3.4.1 Determining premiums for aggregate claims 99
3.4.2 Setting aside reserves for aggregate claims 103
3.5 Reinsurance for aggregate claims 107
3.5.1 Proportional reinsurance 109
3.5.2 Excess of loss reinsurance Ill
3.5.3 Stop-loss reinsurance 116
3.6 Problems 120
4 Ruin Theory 129
4.1 The probability of ruin in a surplus process 129
4.2 Surplus and aggregate claims processes 129
4.2.1 Probability of ruin in discrete time 132
4.2.2 Poisson surplus processes 132
4.3 Probability of ruin and the adjustment coefficient 134
4.3.1 The adjustment equation 135
4.3.2 Lundberg's bound on the probability of ruin ip(U) . . 138
4.3.3 The probability of ruin when claims are exponentially
distributed 140
4.4 Reinsurance and the probability of ruin 146
4.4.1 Adjustment coefficients and proportional reinsurance . 147
4.4.2 Adjustment coefficients and excess of loss reinsurance 149
4.5 Problems 152
5 Credibility Theory 159
5.1 Introduction to credibility estimates 159
5.2 Classical credibility theory 161
5.2.1 Full credibility 161
5.2.2 Partial credibility 163
5.3 The Bayesian approach to credibility theory 164
CONTENTS xv
5.3.1 Bayesian credibility 164
5.4 Greatest accuracy credibility theory 170
5.4.1 Bayes and linear estimates of the posterior mean . . . 172
5.4.2 Predictive distribution for Xn+\ 175
5.5 Empirical Bayes approach to credibility theory 176
5.5.1 Empirical Bayes credibility - Model 1 177
5.5.2 Empirical Bayes credibility - Model 2 180
5.6 Problems 183
6 No Claim Discounting in Motor Insurance 191
6.1 Introduction to No Claim Discount schemes 191
6.2 Transition in a No Claim Discount system 193
6.2.1 Discount classes and movement in NCD schemes . . . 193
6.2.2 One-step transition probabilities in NCD schemes . . . 195
6.2.3 Limiting distributions and stability in NCD models . . 198
6.3 Propensity to make a claim in NCD schemes 204
6.3.1 Thresholds for claims when an accident occurs . 205
6.3.2 The claims rate process in an NCD system 208
6.4 Reducing heterogeneity with NCD schemes 212
6.5 Problems 214
7 Generalized Linear Models 221
7.1 Introduction to linear and generalized linear models 221
7.2 Multiple linear regression and the normal model 225
7.3 The structure of generalized linear models 230
7.3.1 Exponential families 232
7.3.2 Link functions and linear predictors 236
7.3.3 Factors and eovariates 238
7.3.4 Interactions 238
7.3.5 Minimally sufficient statistics 244
7.4 Model selection and deviance 245
7.4.1 Deviance and the saturated model 245
7.4.2 Comparing models with deviance 248
7.4.3 Residual analysis for generalized linear models . 252
7.5 Problems 258
8 Decision and Game Theory 265
8.1 Introduction 265
8.2 Game theory 267
8.2.1 Zero-sum two-person games 268
8.2.2 Minimax and saddle point strategies 270
8.2.3 Randomized strategies 273
8.2.4 The Prisoner's Dilemma and Nash equilibrium in variable-
sum games 278
8.3 Decision making and risk 280
xvi CONTENTS
8.3.1 The minimax criterion 283
8.3.2 The Bayes criterion 283
8.4 Utility and expected monetary gain 288
8.4.1 Rewards, prospects and utility 290
8.4.2 Utility and insurance 292
8.5 Problems 295
References 304
Appendix A Basic Probability Distributions 309
Appendix B Some Basic Tools in Probability and Statistics 313
B.I Moment generating functions 313
B.2 Convolutions of random variables 316
B.3 Conditional probability and distributions 317
B.3.1 The double expectation theorem and E(X) 319
B.3.2 The random variable V(X \Y) 322
B.4 Maximum likelihood estimation 324
Appendix C An Introduction to Bayesian Statistics 327
C.I Bayesian statistics 327
C.I.I Conjugate families 328
C.I.2 Loss functions and Bayesian inference 329
Appendix D Answers to Selected Problems 335
D.I Claims reserving and pricing with run-off triangles 335
D.2 Loss distributions 335
D.3 Risk theory 337
D.4 Ruin theory 338
D.5 Credibility theory 338
D.6 No claim discounting in motor insurance 340
D.7 Generalized linear models 340
D.8 Decision and game theory 341
Index 345 |
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| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
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| id | DE-604.BV022886792 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:52:10Z |
| indexdate | 2024-07-09T21:07:46Z |
| institution | BVB |
| isbn | 1584886951 9781584886952 |
| language | English |
| lccn | 2007060501 |
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| oclc_num | 78893769 |
| open_access_boolean | |
| owner | DE-N2 DE-739 DE-91G DE-BY-TUM DE-384 DE-824 DE-11 |
| owner_facet | DE-N2 DE-739 DE-91G DE-BY-TUM DE-384 DE-824 DE-11 |
| physical | XVI, 351 S. graph. Darst. 25 cm |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Chapman & Hall/CRC |
| record_format | marc |
| series2 | Chapman & Hall/CRC interdisciplinary statistics series |
| spelling | Boland, Philip J. Verfasser aut Statistical and probabilistic methods in actuarial science Philip J. Boland Boca Raton [u.a.] Chapman & Hall/CRC 2007 XVI, 351 S. graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC interdisciplinary statistics series Includes bibliographical references (p. 304-307) and index Assurance - Mathématiques Mathematik Insurance Mathematics Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 s DE-604 http://www.loc.gov/catdir/toc/fy0709/2007060501.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0707/2007060501-d.html Publisher description Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091692&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091692&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Boland, Philip J. Statistical and probabilistic methods in actuarial science Assurance - Mathématiques Mathematik Insurance Mathematics Versicherungsmathematik (DE-588)4063194-1 gnd |
| subject_GND | (DE-588)4063194-1 |
| title | Statistical and probabilistic methods in actuarial science |
| title_auth | Statistical and probabilistic methods in actuarial science |
| title_exact_search | Statistical and probabilistic methods in actuarial science |
| title_exact_search_txtP | Statistical and probabilistic methods in actuarial science |
| title_full | Statistical and probabilistic methods in actuarial science Philip J. Boland |
| title_fullStr | Statistical and probabilistic methods in actuarial science Philip J. Boland |
| title_full_unstemmed | Statistical and probabilistic methods in actuarial science Philip J. Boland |
| title_short | Statistical and probabilistic methods in actuarial science |
| title_sort | statistical and probabilistic methods in actuarial science |
| topic | Assurance - Mathématiques Mathematik Insurance Mathematics Versicherungsmathematik (DE-588)4063194-1 gnd |
| topic_facet | Assurance - Mathématiques Mathematik Insurance Mathematics Versicherungsmathematik |
| url | http://www.loc.gov/catdir/toc/fy0709/2007060501.html http://www.loc.gov/catdir/enhancements/fy0707/2007060501-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091692&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091692&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bolandphilipj statisticalandprobabilisticmethodsinactuarialscience |