Risk modelling in general insurance: from principles to practice
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2012
|
| Edition: | 1. publ. |
| Series: | International series on actuarial science
|
| Subjects: | |
| Online Access: | Cover image Inhaltsverzeichnis |
| Physical Description: | XIV, 393 S. graph. Darst. |
| ISBN: | 9780521863940 |
Staff View
MARC
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| 245 | 1 | 0 | |a Risk modelling in general insurance |b from principles to practice |c Roger J. Gray ; Susan M. Pitts |
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Record in the Search Index
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| adam_text | IMAGE 1
CONTENTS
PREFACE INTRODUCTION 1.1 1.2
THE AIM OF THIS BOOK NOTATION AND PREREQUISITES 1.2.1 1.2.2
1.2.3 1.2.4
PROBABILITY STATISTICS SIMULATION THE STATISTICAL SOFTWARE PACKAGE R
MODELS FOR CLAIM NUMBERS AND CLAIM SIZES 2.1
2.2
2.3
DISTRIBUTIONS FOR CLAIM NUMBERS 2.1.1 2.1.2 2.1.3
2.1.4 2.1.5
POISSON DISTRIBUTION NEGATIVE BINOMIAL DISTRIBUTION GEOMETRIC
DISTRIBUTION
BINOMIAL DISTRIBUTION A SUMMARY NOTE ON R DISTRIBUTIONS FOR CLAIM SIZES
2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.2.9 2.2.10
A FURTHER SUMMARY NOTE ON R NORMAL (GAUSSIAN) DISTRIBUTION EXPONENTIAL
DISTRIBUTION GAMMA DISTRIBUTION FAT-TAILED DISTRIBUTIONS LOGNORMAL
DISTRIBUTION PARETO DISTRIBUTION WEIBULL DISTRIBUTION BURR DISTRIBUTION
LOGGAMMA DISTRIBUTION MIXTURE DISTRIBUTIONS
PAGE XIII
1 1 2 2 9 9 9
11 12 13
16 18 20 22 23 24 24 25 28 31 35 40 45 48 51 54
VLL
IMAGE 2
VIII CONTENTS
2.4 FITTING MODELS TO CLAIM-NUMBER AND CLAIM-SIZE DATA 58 2.4.1 FITTING
MODELS TO CLAIM NUMBERS 60
2.4.2 FITTING MODELS TO CLAIM SIZES 65
EXERCISES 83
3 SHORT TERM RISK MODELS 90
3.1 THE MEAN AND VARIANCE OF A COMPOUND DISTRIBUTION 91 3.2 THE
DISTRIBUTION OF A RANDOM SUM 93
3.2.1 CONVOLUTION SERIES FORMULA FOR A COMPOUND DISTRIBUTION 95
3.2.2 MOMENT GENERATING FUNCTION OF A COMPOUND DISTRIBUTION 98
3.3 FINITE MIXTURE DISTRIBUTIONS 100
3.4 SPECIAL COMPOUND DISTRIBUTIONS 103
3.4.1 COMPOUND POISSON DISTRIBUTIONS 103
3.4.2 COMPOUND MIXED POISSON DISTRIBUTIONS 108 3.4.3 COMPOUND NEGATIVE
BINOMIAL DISTRIBUTIONS 110 3.4.4 COMPOUND BINOMIAL DISTRIBUTIONS 114
3.5 NUMERICAL METHODS FOR COMPOUND DISTRIBUTIONS 115 3.5.1 PANJER
RECURSION ALGORITHM 116
3.5.2 THE FAST FOURIER TRANSFORM ALGORITHM 119
3.6 APPROXIMATIONS FOR COMPOUND DISTRIBUTIONS 124
3.6.1 APPROXIMATIONS BASED ON A FEW MOMENTS 125 3.6.2 ASYMPTOTIC
APPROXIMATIONS 126
3.7 STATISTICS FOR COMPOUND DISTRIBUTIONS 128
3.8 THE INDIVIDUAL RISK MODEL 134
3.8.1 THE MEAN AND VARIANCE FOR THE INDIVIDUAL RISK MODEL 136
3.8.2 THE DISTRIBUTION FUNCTION AND MOMENT GEN- ERATING FUNCTION FOR THE
INDIVIDUAL RISK MODEL 137
3.8.3 APPROXIMATIONS FOR THE INDIVIDUAL RISK MODEL 139 EXERCISES 140
4 MODEL BASED PRICING - SETTING PREMIUMS 147
4.1 PREMIUM CALCULATION PRINCIPLES 148
4.1.1 THE EXPECTED VALUE PRINCIPLE (EVP) 148
4.1.2 THE STANDARD DEVIATION PRINCIPLE (SDP) 149 4.1.3 THE VARIANCE
PRINCIPLE (VP) 149
4.1.4 THE QUANTILE PRINCIPLE (QP) 149
4.1.5 THE ZERO UTILITY PRINCIPLE (ZUP) 150
IMAGE 3
CONTENTS IX
4.1.6 THE EXPONENTIAL PREMIUM PRINCIPLE (EPP) 150 4.1.7 SOME DESIRABLE
PROPERTIES OF PREMIUM CALCULATION PRINCIPLES 152
4.1.8 OTHER PREMIUM CALCULATION PRINCIPLES 154
4.2 MAXIMUM AND MINIMUM PREMIUMS 155
4.3 INTRODUCTION TO CREDIBILITY THEORY 156
4.4 BAYESIAN ESTIMATION 157
4.4.1 THE POSTERIOR DISTRIBUTION 158
4.4.2 THE WIDER CONTEXT OF DECISION THEORY 159
4.4.3 THE BINOMIAL/BETA MODEL 161
4.4.4 THE POISSON/GAMMA MODEL 163
4.4.5 THE NORMAL/NORMAL MODEL 165
4.5 BAYESIAN CREDIBILITY THEORY 169
4.5.1 BAYESIAN CREDIBILITY ESTIMATES UNDER THE POISSON/GAMMA MODEL 170
4.5.2 BAYESIAN CREDIBILITY PREMIUMS UNDER THE NORMAL/NORMAL MODEL 172
4.6 EMPIRICAL BAYESIAN CREDIBILITY THEORY: MODEL 1 - THE BIIHLMANN MODEL
176
4.7 EMPIRICAL BAYESIAN CREDIBILITY THEORY: MODEL 2 - THE
BIIHLMANN-STRAUB MODEL 185
EXERCISES 196
RISK SHARING - REINSURANCE AND DEDUCTIBLES 205
5.1 EXCESS OF LOSS REINSURANCE 206
5.1.1 REINSURANCE CLAIMS 210
5.1.2 SIMULATION RESULTS 212
5.1.3 AGGREGATE CLAIMS MODEL WITH EXCESS OF LOSS REINSURANCE 213
5.2 PROPORTIONAL REINSURANCE 221
5.3 DEDUCTIBLES (POLICY EXCESSES) 223
5.4 RETENTION LEVELS AND REINSURANCE COSTS 226
5.5 OPTIMISING THE REINSURANCE CONTRACT 228
5.6 OPTIMISING REINSURANCE CONTRACTS BASED ON MAXIMISING EXPECTED
UTILITY 228
5.6.1 EXCESS OF LOSS REINSURANCE 229
5.6.2 PROPORTIONAL REINSURANCE 231
5.7 OPTIMISING REINSURANCE CONTRACTS BASED ON MINIMISING THE VARIANCE OF
AGGREGATE CLAIMS 234
5.7.1 MINIMISING VAR[5/] SUBJECT TO FIXED E[5 ; ] 235
IMAGE 4
X CONTENTS
5.7.2 MINIMISING VAR[S FL ] SUBJECT TO FIXED VARFS/] 236 5.7.3 COMPARING
STOP LOSS AND EQUIVALENT PROPORTIONAL REINSURANCE ARRANGEMENTS 237 5.7.4
MINIMISING VARFS 1 /] + VAR[S R ] 238
5.7.5 MINIMISING THE SUM OF VARIANCES WHEN TWO INDEPENDENT RISKS ARE
SHARED BETWEEN TWO INSURERS 239
5.8 OPTIMISING REINSURANCE CONTRACTS FOR A GROUP OF INDE- PENDENT RISKS
BASED ON MINIMISING THE VARIANCE OF THE DIRECT INSURER S NET PROFIT -
FINDING THE OPTIMAL RELATIVE RETENTIONS 247
5.8.1 OPTIMAL RELATIVE RETENTIONS IN THE CASE OF EXCESS OF LOSS
REINSURANCE 247
5.8.2 OPTIMAL RELATIVE RETENTIONS IN THE CASE OF PROPORTIONAL
REINSURANCE 251
EXERCISES 253
6 RUIN THEORY FOR THE CLASSICAL RISK MODEL 267
6.1 THE CLASSICAL RISK MODEL 267
6.1.1 THE RELATIVE SAFETY LOADING 269
6.1.2 RUIN PROBABILITIES 270
6.2 LUNDBERG S INEQUALITY AND THE ADJUSTMENT COEFFICIENT 272 6.2.1
PROPERTIES OF THE ADJUSTMENT COEFFICIENT 272 6.2.2 PROOF OF LUNDBERG S
INEQUALITY 276
6.2.3 WHEN DOES THE ADJUSTMENT COEFFICIENT EXIST? 279 6.3 EQUATIONS FOR
IJ/{U) AND P(U): THE RUIN PROBABILITY AND THE SURVIVAL PROBABILITY 282
6.4 COMPOUND GEOMETRIC REPRESENTATIONS FOR FT(U) AND P(U): THE RUIN
PROBABILITY AND THE SURVIVAL PROBABILITY 291 6.5 ASYMPTOTICS FOR THE
PROBABILITY OF RUIN 296
6.6 NUMERICAL METHODS FOR RUIN QUANTITIES 303
6.6.1 NUMERICAL CALCULATION OF THE ADJUSTMENT COEFFICIENT 303
6.6.2 NUMERICAL CALCULATION OF THE PROBABILITY OF RUIN 305 6.7
STATISTICS FOR RUIN QUANTITIES 308
EXERCISES 310
7 CASE STUDIES 316
7.1 CASE STUDY 1: COMPARING PREMIUM SETTING PRINCIPLES 316 7.1.1 CASE 1
- IN THE PRESENCE OF AN ASSUMED MODEL 316
IMAGE 5
CONTENTS
XI
7.1.2 CASE 2 - WITHOUT MODEL ASSUMPTIONS, USING
BOOTSTRAP RESAMPLING 322
7.2 CASE STUDY 2: SHARED LIABILITIES - WHO PAYS WHAT? 332 7.2.1 CASE 1 -
EXPONENTIAL LOSSES 333
7.2.2 CASE 2 - PARETO LOSSES 338
7.2.3 CASE 3 - LOGNORMAL LOSSES 344
7.3 CASE STUDY 3: REINSURANCE AND RUIN 348
7.3.1 INTRODUCTION 348
7.3.2 PROPORTIONAL REINSURANCE 351
7.3.3 PROPORTIONAL REINSURANCE WITH EXPONENTIAL CLAIM SIZES 353
7.3.4 EXCESS OF LOSS REINSURANCE IN A LAYER 356
7.3.5 EXCESS OF LOSS REINSURANCE IN A LAYER WITH EXPONENTIAL CLAIM SIZES
360
APPENDIX A UTILITY THEORY 368
APPENDIX B ANSWERS TO EXERCISES 380
REFERENCES 386
INDEX 389
|
| any_adam_object | 1 |
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| building | Verbundindex |
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| callnumber-first | H - Social Science |
| callnumber-label | HG8054 |
| callnumber-raw | HG8054.5 |
| callnumber-search | HG8054.5 |
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| callnumber-subject | HG - Finance |
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| ctrlnum | (OCoLC)801775291 (DE-599)BVBBV040296576 |
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| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 368 - Insurance |
| dewey-raw | 368/.01 |
| dewey-search | 368/.01 |
| dewey-sort | 3368 11 |
| dewey-tens | 360 - Social problems and services; associations |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV040296576 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:21:02Z |
| institution | BVB |
| isbn | 9780521863940 |
| language | English |
| lccn | 2012010344 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025151661 |
| oclc_num | 801775291 |
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| owner_facet | DE-384 DE-11 DE-20 DE-19 DE-BY-UBM |
| physical | XIV, 393 S. graph. Darst. |
| publishDate | 2012 |
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| spelling | Gray, Roger J. 1946-2011 Verfasser (DE-588)1024282740 aut Risk modelling in general insurance from principles to practice Roger J. Gray ; Susan M. Pitts 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2012 XIV, 393 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier International series on actuarial science Mathematisches Modell Risk (Insurance) Mathematical models MATHEMATICS / Applied bisacsh Risikotheorie (DE-588)4135592-1 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Mathematische Modellierung (DE-588)7651795-0 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 s Risikotheorie (DE-588)4135592-1 s Stochastisches Modell (DE-588)4057633-4 s Mathematische Modellierung (DE-588)7651795-0 s DE-604 Pitts, Susan M. Sonstige oth http://assets.cambridge.org/97805218/63940/cover/9780521863940.jpg Cover image SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025151661&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gray, Roger J. 1946-2011 Risk modelling in general insurance from principles to practice Mathematisches Modell Risk (Insurance) Mathematical models MATHEMATICS / Applied bisacsh Risikotheorie (DE-588)4135592-1 gnd Stochastisches Modell (DE-588)4057633-4 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Mathematische Modellierung (DE-588)7651795-0 gnd |
| subject_GND | (DE-588)4135592-1 (DE-588)4057633-4 (DE-588)4063194-1 (DE-588)7651795-0 |
| title | Risk modelling in general insurance from principles to practice |
| title_auth | Risk modelling in general insurance from principles to practice |
| title_exact_search | Risk modelling in general insurance from principles to practice |
| title_full | Risk modelling in general insurance from principles to practice Roger J. Gray ; Susan M. Pitts |
| title_fullStr | Risk modelling in general insurance from principles to practice Roger J. Gray ; Susan M. Pitts |
| title_full_unstemmed | Risk modelling in general insurance from principles to practice Roger J. Gray ; Susan M. Pitts |
| title_short | Risk modelling in general insurance |
| title_sort | risk modelling in general insurance from principles to practice |
| title_sub | from principles to practice |
| topic | Mathematisches Modell Risk (Insurance) Mathematical models MATHEMATICS / Applied bisacsh Risikotheorie (DE-588)4135592-1 gnd Stochastisches Modell (DE-588)4057633-4 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Mathematische Modellierung (DE-588)7651795-0 gnd |
| topic_facet | Mathematisches Modell Risk (Insurance) Mathematical models MATHEMATICS / Applied Risikotheorie Stochastisches Modell Versicherungsmathematik Mathematische Modellierung |
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