Risk theory: the stochastic basis of insurance
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London [u.a.]
Chapman and Hall
1984
|
| Edition: | 3. ed. |
| Series: | Monographs on statistics and applied probability
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 408 S. |
| ISBN: | 0412242605 041225980X |
Staff View
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| 245 | 1 | 0 | |a Risk theory |b the stochastic basis of insurance |c R. E. Beard ; T. Pentikäinen ; E. Pesonen |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a London [u.a.] |b Chapman and Hall |c 1984 | |
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| adam_text | Contents
Preface ix
Nomenclature xiv
1 Definitions and notation 1
1.1 The purpose of the theory of risk 1
1.2 Stochastic processes in general 5
1.3 Positive and negative risk sums 6
1.4 Main problems 7
1.5 On the notation 11
1.6 The moment generating function, the character¬
istic function, and the Laplace transform 15
2 Claim number process 18
2.1 Introduction 18
2.2 The Poisson process 19
2.3 Discussion of conditions 20
2.4 Some basic formulae 22
2.5 Numerical values of Poisson probabilities 23
2.6 The additivity of Poisson variables 27
2.7 Time dependent variation of risk exposure 29
2.8 Formulae concerning the mixed Poisson
distribution 35
2.9 The Polya process 38
2.10 Risk exposure variation inside the portfolio 43
3 Compound Poisson process 47
3.1 The distribution of claim size 47
3.2 Compound distribution of the aggregate claim 50
3.3 Basic characteristics of F 52
vi CONTENTS
3.4 The moment generating function 59
3.5 Estimation of S 60
3.5.1 Individual method 60
3.5.2 Statistical method 62
3.5.3 Problems arising from large claims 65
3.5.4 Analytical methods 67
3.5.5 Exponential distribution 68
3.5.6 Gamma distribution 69
3.5.7 Logarithmic normal distribution 72
3.5.8 The Pareto distribution 74
3.5.9 The two parametric Pareto and the quasi
log normal distributions 76
3.5.10 The family of Benktander distributions 79
3.5.11 Other types of distribution 83
3.6 The dependence of the S function
on reinsurance 84
3.6.1 General aspects 84
3.6.2 Excess of loss reinsurance 85
3.6.3 Quota share reinsurance 85
3.6.4 Surplus reinsurance 88
3.6.5 Technique using the concept of degree of loss 90
3.7 Decomposition of the portfolio into sections 94
3.8 Recursion formula for F 100
3.9 The normal approximation 104
3.10 Edgeworth series 107
108
3.11 Normal power approximation
3.12 Gamma approximation
3.13 Approximations by means of functions belonging
to the Pearson family 123
3.14 Inversion of the characteristic function 124
3.15 Mixed methods 124
4 Applications related to one year time span 126
4.1 The basic equation 126
4.2 Evaluation of the fluctuation range of the annual
underwriting profits and losses 130
4.3 Some approximate formulae 138
4.4 Reserve funds 142
4.5 Rules for the greatest retention 145
CONTENTS vii
4.6 The case of several Ms 152
4.7 Excess of loss reinsurance premium 154
4.8 Application to stop loss reinsurance 156
4.9 An application to insurance statistics 159
4.10 Experience rating, credibility theory 162
5 Variance as a measure of stability 171
5.1 Optimum form of reinsurance 171
5.2 Reciprocity of two companies 175
5.3 Equitability of safety loadings: a link to theory of
multiplayer games 179
6 Risk processes with a time span of several years 183
6.1 Claims 183
6.2 Premium income P( 1, ?) 198
6.3 Yield of investments 205
6.4 Portfolio divided in sections 211
6.5 Trading result 214
6.6 Distribution of the solvency ratio u 220
6.7 Ruin probability ^(u), truncated convolution 227
6.8 Monte Carlo method 233
6.8.1 Random numbers 233
6.8.2 Direct simulation of the compound
Poisson function 239
6.8.3 A random number generator for the
cycling mixed compound Poisson variable
X(0 241
6.8.4 Simulation of the solvency ratio u(/) 245
6.9 Limits for the finite time ruin probability ^r 250
7 Applications related to finite time span T 258
7.1 General features of finite time risk processes 258
7.2 The size of the portfolio 263
7.3 Evaluation of net retention M 265
7.4 Effect of cycles 266
7.5 Effect of the time span T 266
7.6 Effect of inflation 267
7.7 Dynamic control rules 272
7.8 Solvency profile 278
viii CONTENTS
7.9 Evaluation of the variation range of u(t) 281
7.10 Safety loading 284
8 Risk theory analysis of life insurance 288
8.1 Cohort analysis 288
8.2 Link to classic individual risk theory 292
8.3 Extensions of the cohort approach 295
8.4 General system 300
9 Ruin probability during an infinite time period 308
9.1 Introduction 308
9.2 The infinite time ruin probability 311
9.3 Discussion of the different methods 315
10 Application of risk theory to business planning 319
10.1 General features of the models 319
10.2 An example of risk theory models 322
10.3 Stochastic dynamic programming 330
10.4 Business objectives 336
10.5 Competition models 345
Appendixes 349
A Derivation of the Poisson and mixed Poisson processes 349
B Edgeworth expansion 355
C Infinite time ruin probability 357
D Computation of the limits for the finite time ruin
probability according to method of Section 6.9 367
E Random numbers 370
F Solutions to the exercises 373
Bibliography 396
Author index 403
Subject index 405
|
| adam_txt |
Contents
Preface ix
Nomenclature xiv
1 Definitions and notation 1
1.1 The purpose of the theory of risk 1
1.2 Stochastic processes in general 5
1.3 Positive and negative risk sums 6
1.4 Main problems 7
1.5 On the notation 11
1.6 The moment generating function, the character¬
istic function, and the Laplace transform 15
2 Claim number process 18
2.1 Introduction 18
2.2 The Poisson process 19
2.3 Discussion of conditions 20
2.4 Some basic formulae 22
2.5 Numerical values of Poisson probabilities 23
2.6 The additivity of Poisson variables 27
2.7 Time dependent variation of risk exposure 29
2.8 Formulae concerning the mixed Poisson
distribution 35
2.9 The Polya process 38
2.10 Risk exposure variation inside the portfolio 43
3 Compound Poisson process 47
3.1 The distribution of claim size 47
3.2 Compound distribution of the aggregate claim 50
3.3 Basic characteristics of F 52
vi CONTENTS
3.4 The moment generating function 59
3.5 Estimation of S 60
3.5.1 Individual method 60
3.5.2 Statistical method 62
3.5.3 Problems arising from large claims 65
3.5.4 Analytical methods 67
3.5.5 Exponential distribution 68
3.5.6 Gamma distribution 69
3.5.7 Logarithmic normal distribution 72
3.5.8 The Pareto distribution 74
3.5.9 The two parametric Pareto and the quasi
log normal distributions 76
3.5.10 The family of Benktander distributions 79
3.5.11 Other types of distribution 83
3.6 The dependence of the S function
on reinsurance 84
3.6.1 General aspects 84
3.6.2 Excess of loss reinsurance 85
3.6.3 Quota share reinsurance 85
3.6.4 Surplus reinsurance 88
3.6.5 Technique using the concept of degree of loss 90
3.7 Decomposition of the portfolio into sections 94
3.8 Recursion formula for F 100
3.9 The normal approximation 104
3.10 Edgeworth series 107
108
3.11 Normal power approximation
3.12 Gamma approximation
3.13 Approximations by means of functions belonging
to the Pearson family 123
3.14 Inversion of the characteristic function 124
3.15 Mixed methods 124
4 Applications related to one year time span 126
4.1 The basic equation 126
4.2 Evaluation of the fluctuation range of the annual
underwriting profits and losses 130
4.3 Some approximate formulae 138
4.4 Reserve funds 142
4.5 Rules for the greatest retention 145
CONTENTS vii
4.6 The case of several Ms 152
4.7 Excess of loss reinsurance premium 154
4.8 Application to stop loss reinsurance 156
4.9 An application to insurance statistics 159
4.10 Experience rating, credibility theory 162
5 Variance as a measure of stability 171
5.1 Optimum form of reinsurance 171
5.2 Reciprocity of two companies 175
5.3 Equitability of safety loadings: a link to theory of
multiplayer games 179
6 Risk processes with a time span of several years 183
6.1 Claims 183
6.2 Premium income P( 1, ?) 198
6.3 Yield of investments 205
6.4 Portfolio divided in sections 211
6.5 Trading result 214
6.6 Distribution of the solvency ratio u 220
6.7 Ruin probability ^(u), truncated convolution 227
6.8 Monte Carlo method 233
6.8.1 Random numbers 233
6.8.2 Direct simulation of the compound
Poisson function 239
6.8.3 A random number generator for the
cycling mixed compound Poisson variable
X(0 241
6.8.4 Simulation of the solvency ratio u(/) 245
6.9 Limits for the finite time ruin probability ^r 250
7 Applications related to finite time span T 258
7.1 General features of finite time risk processes 258
7.2 The size of the portfolio 263
7.3 Evaluation of net retention M 265
7.4 Effect of cycles 266
7.5 Effect of the time span T 266
7.6 Effect of inflation 267
7.7 Dynamic control rules 272
7.8 Solvency profile 278
viii CONTENTS
7.9 Evaluation of the variation range of u(t) 281
7.10 Safety loading 284
8 Risk theory analysis of life insurance 288
8.1 Cohort analysis 288
8.2 Link to classic individual risk theory 292
8.3 Extensions of the cohort approach 295
8.4 General system 300
9 Ruin probability during an infinite time period 308
9.1 Introduction 308
9.2 The infinite time ruin probability 311
9.3 Discussion of the different methods 315
10 Application of risk theory to business planning 319
10.1 General features of the models 319
10.2 An example of risk theory models 322
10.3 Stochastic dynamic programming 330
10.4 Business objectives 336
10.5 Competition models 345
Appendixes 349
A Derivation of the Poisson and mixed Poisson processes 349
B Edgeworth expansion 355
C Infinite time ruin probability 357
D Computation of the limits for the finite time ruin
probability according to method of Section 6.9 367
E Random numbers 370
F Solutions to the exercises 373
Bibliography 396
Author index 403
Subject index 405 |
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| discipline | Mathematik Wirtschaftswissenschaften |
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| edition | 3. ed. |
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| id | DE-604.BV021868906 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:03:19Z |
| indexdate | 2024-07-09T20:46:22Z |
| institution | BVB |
| isbn | 0412242605 041225980X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015084999 |
| oclc_num | 10207821 |
| open_access_boolean | |
| owner | DE-706 DE-83 DE-11 DE-91G DE-BY-TUM DE-188 |
| owner_facet | DE-706 DE-83 DE-11 DE-91G DE-BY-TUM DE-188 |
| physical | 408 S. |
| psigel | TUB-www |
| publishDate | 1984 |
| publishDateSearch | 1984 |
| publishDateSort | 1984 |
| publisher | Chapman and Hall |
| record_format | marc |
| series2 | Monographs on statistics and applied probability |
| spelling | Beard, Robert E. Verfasser aut Risk theory the stochastic basis of insurance R. E. Beard ; T. Pentikäinen ; E. Pesonen 3. ed. London [u.a.] Chapman and Hall 1984 408 S. txt rdacontent n rdamedia nc rdacarrier Monographs on statistics and applied probability Actuariat Assurance - Mathématiques Assurance - Mathématiques ram Processus stochastiques Processus stochastiques ram Risicotheorie gtt Risque (Assurance) Risques (assurance) ram Verzekeringswiskunde gtt Mathematik Insurance Mathematics Risk (Insurance) Stochastic processes Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Versicherung (DE-588)4063173-4 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Risikotheorie (DE-588)4135592-1 gnd rswk-swf Risikotheorie (DE-588)4135592-1 s Versicherung (DE-588)4063173-4 s 1\p DE-604 Versicherungsmathematik (DE-588)4063194-1 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s 2\p DE-604 3\p DE-604 Pentikäinen, Teivo 1917-2006 Verfasser (DE-588)170324826 aut Pesonen, Erkki Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015084999&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Beard, Robert E. Pentikäinen, Teivo 1917-2006 Pesonen, Erkki Risk theory the stochastic basis of insurance Actuariat Assurance - Mathématiques Assurance - Mathématiques ram Processus stochastiques Processus stochastiques ram Risicotheorie gtt Risque (Assurance) Risques (assurance) ram Verzekeringswiskunde gtt Mathematik Insurance Mathematics Risk (Insurance) Stochastic processes Versicherungsmathematik (DE-588)4063194-1 gnd Versicherung (DE-588)4063173-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Risikotheorie (DE-588)4135592-1 gnd |
| subject_GND | (DE-588)4063194-1 (DE-588)4063173-4 (DE-588)4079013-7 (DE-588)4135592-1 |
| title | Risk theory the stochastic basis of insurance |
| title_auth | Risk theory the stochastic basis of insurance |
| title_exact_search | Risk theory the stochastic basis of insurance |
| title_exact_search_txtP | Risk theory the stochastic basis of insurance |
| title_full | Risk theory the stochastic basis of insurance R. E. Beard ; T. Pentikäinen ; E. Pesonen |
| title_fullStr | Risk theory the stochastic basis of insurance R. E. Beard ; T. Pentikäinen ; E. Pesonen |
| title_full_unstemmed | Risk theory the stochastic basis of insurance R. E. Beard ; T. Pentikäinen ; E. Pesonen |
| title_short | Risk theory |
| title_sort | risk theory the stochastic basis of insurance |
| title_sub | the stochastic basis of insurance |
| topic | Actuariat Assurance - Mathématiques Assurance - Mathématiques ram Processus stochastiques Processus stochastiques ram Risicotheorie gtt Risque (Assurance) Risques (assurance) ram Verzekeringswiskunde gtt Mathematik Insurance Mathematics Risk (Insurance) Stochastic processes Versicherungsmathematik (DE-588)4063194-1 gnd Versicherung (DE-588)4063173-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Risikotheorie (DE-588)4135592-1 gnd |
| topic_facet | Actuariat Assurance - Mathématiques Processus stochastiques Risicotheorie Risque (Assurance) Risques (assurance) Verzekeringswiskunde Mathematik Insurance Mathematics Risk (Insurance) Stochastic processes Versicherungsmathematik Versicherung Wahrscheinlichkeitstheorie Risikotheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015084999&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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