Loss models: from data to decisions
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken, NJ
Wiley Interscience
2004
|
| Edition: | 2. ed. |
| Series: | Wiley series in probability and statistics
|
| Subjects: | |
| Online Access: | Publisher description Table of contents Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references (p. 671-680) and index |
| Physical Description: | XXII, 688 S. Ill., graph. Darst. 24 cm |
| ISBN: | 0471215775 |
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| 100 | 1 | |a Klugman, Stuart A. |d 1949- |e Verfasser |0 (DE-588)1044735155 |4 aut | |
| 245 | 1 | 0 | |a Loss models |b from data to decisions |c Stuart A. Klugman ; Harry H. Panjer ; Gordon E. Willmot |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Hoboken, NJ |b Wiley Interscience |c 2004 | |
| 300 | |a XXII, 688 S. |b Ill., graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Wiley series in probability and statistics | |
| 500 | |a Includes bibliographical references (p. 671-680) and index | ||
| 505 | 8 | |a Erg. bildet: Klugman, Stuart A.: Solutions manual to accompany Loss models | |
| 650 | 4 | |a Assurance - Mathématiques | |
| 650 | 4 | |a Assurance - Modèles mathématiques | |
| 650 | 4 | |a Assurance - Méthodes statistiques | |
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Contents
Preface
xvii
Acknowledgments
xxi
Part I Introduction
1
Modeling
3
1.1
The model-based approach
3
1.1.1
The modeling process
3
1.1.2
The modeling advantage
5
1.2
Organization of this book
6
Part II Actuarial models
2
Random variables
Ц
2.1
Introduction
И
2.2
Key functions and four models
13
2.2.1
Exercises
23
vi
CONTENTS
3
Basic distributional quantities
25
3.1
Moments
25
3.1.1
Exercises
33
3.2
Percentiles
3Ą
3.2.1
Exercises
35
3.3
Generating functions and sums of random
variables
36
3.3.1
Exercises
37
4
Classifying and creating distributions
зд
Ą.1
Introduction
39
4-2
The role of parameters
40
4-2.1
Parametric and scale distributions 41
4-2.2
Parametric distribution families
42
4-2.3
Finite mixture distributions
43
4-2-4
D ata-
dependent distributions
45
4-2.5
Exercises 4Q
4-3
Tail weight
48
4-3.1
Existence of moments
48
4-3.2
Limiting ratios
50
4-3.3
Hazard rate and mean residual life
patterns
50
4-3.4
Exercises
5Ą
4-4
Creating new distributions
5ß
4-Ą-1
Introduction
ßß
4-4-2
Multiplication by a constant
5
ρ
4-4-3
Raising to a power
57
4-4-4
Exponentiation
58
4-4-5
Mixing
59
4-4-6
Frailty models
β
2
4-4-7
Splicing
ß4
4-4-8
Exercises Q5
4-5
Selected distributions and their relationships
69
4-5.1
Introduction
β
g
4-5.2
Two parametric families
ßg
4-5.3
Limiting distributions
9
Ό
4-5-4
Exercises 7g
4-6
Discrete distributions
72
4-6.1
Introduction
72
CONTENTS
VII
4-6.2
The Poisson distribution
73
4-6.3
The negative binomial distribution
76
4-6-4
The binomial distribution
79
4-6.5
The (a,b,O) class
81
4-6.6
Truncation and modification at zero
83
4-6.7
Compound frequency models
88
4-6.8
Further properties of the compound
Poisson
class
95
4-6.9
Mixed frequency models
101
4.6.IO
Poisson
mixtures
103
4-6.11
Effect of exposure on frequency
108
4-6.12
An inventory of discrete distributions
109
4-6.13
Exercises
109
Frequency and severity with coverage modifications
115
5.1
Introduction
115
5.2
Déductibles
115
5.2.1
Exercises
120
5.3
The loss elimination ratio and the effect of
inflation for ordinary
déductibles
121
5.3.1
Exercises
123
5Ą
Policy limits
I24
5.4-1
Exercises
126
5.5
Coinsurance,
déductibles,
and limits
126
5.5.1
Exercises
128
5.6
The impact of
déductibles on
claim frequency
129
5.6.1
Exercises
133
Aggregate loss models
135
6.1
Introduction
135
6.1.1
Exercise
138
6.2
Model choices
139
6.2.1
Exercises
I40
6.3
The compound model for aggregate claims
ЦП
6.3.1
Exercises
1Ą8
6.4
Analytic results
153
6-4-1
Exercises
156
6.5
Computing the aggregate claims distribution
159
viii
CONTENTS
6.6
The recursive method
161
6.6.1
Applications to compound frequency
models
162
6.6.2
Underflow/overflow problems
165
6.6.3
Numerical stability
166
6.6.4
Continuous severity
166
6.6.5
Constructing arithmetic distributions
161
6.6.6
Exercises
170
6.7
The impact of individual policy modifications on
aggregate payments
174
6.7.1
Exercises
177
6.8
Calculations with approximate distributions
178
6.8.1
Arithmetic distributions
178
6.8.2
Empirical distributions
181
6.8.3
Piecewise linear cdf
182
6.8.4
Exercises
184
6.9
Inversion methods
184
6.9.1
Fast Fourier transform
185
6.9.2
Direct numerical inversion
188
6.9.3
Exercise
190
6.10
Comparison of methods
190
6.11
The individual risk model
192
6.11.1
Parametric approximation
192
6.11.2
Exact calculation of the aggregate
distribution
195
6.11.3
Compound
Poisson
approximation
201
6.11.4 Exercises
205
7
Discrete-time ruin models
209
7.1
Introduction
209
7.2
Process models for insurance
210
7.2.1
Processes
210
7.2.2
An insurance model
212
7.2.3
Ruin
213
7.3
Discrete, finite-time ruin probabilities
215
7.3.1
The discrete-time process
215
7.3.2
Evaluating the probability of ruin
216
7.3.3
Exercises
222
CONTENTS ¡x
Continuous-time ruin models
223
8.1
Introduction
8.1.1
The Poiss on process
8.1.2
The continuous-time problem
225
8.2
The adjustment coefficient and
Lundberg'
s
inequality
225
8.2.1
The adjustment coefficient
226
8.2.2
Lundberg
's inequality
230
8.2.3
Exercises
232
8.3
An integrodifferential equation
234
8.3.1
Exercises
239
8.Ą
The maximum aggregate loss
239
8.4.I Exercises
242
8.5
Cramer's asymptotic ruin formula and Tijms'
approximation
244
8.5.1
Exercises
250
8.6
The Brownian motion risk process
252
8.7
Brownian motion and the probability of ruin
256
Part III Construction of empirical models
9
Review of mathematical statistics
265
9.1
Introduction
265
9.2
Point estimation
266
9.2.1
Introduction
266
9.2.2
Measures of quality
267
9.2.3
Exercises
273
9.3
Interval estimation
275
9.3.1
Exercise
277
9.4
Tests of hypotheses
277
9.4.I Exercise
281
10
Estimation for complete data
283
10.1
Introduction
283
10.2
The empirical distribution for complete,
individual data
288
10.2.1
Exercises
291
χ
CONTENTS
10.3
Empirical distributions for grouped data
292
10.3.1
Exercises
295
11
Estimation for modified data
297
11.1
Point estimation
297
11.1.1
Exercises
304
11.2
Means, variances, and interval estimation
305
11.2.1
Exercises
315
11.3
Kernel density models
316
11.3.1
Exercises
321
11.
Ą
Approximations for large data sets
322
11.4.1 Introduction
322
11.4.2 Kaplan-Meier type approximations
323
11.4-3
Multiple-decrement tables
324
II.4.4 Exercises
325
Part IV Parametric statistical methods
12
Parameter estimation
331
12.1
Method of moments and percentile matching
331
12.1.1
Exercises
335
12.2
Maximum likelihood estimation
337
12.2.1
Introduction
337
12.2.2
Complete, individual data
З4О
12.2.3
Complete, grouped data
341
12.2.4 Truncated or censored data
341
12.2.5
Exercises
345
12.3
Variance and interval estimation
351
12.3.1
Exercises
358
12.4 Bayesian estimation
360
12.4-1
Definitions and
Bayes'
theorem
360
12.4-2
Inference and prediction
364
12.4.3 Conjugate prior distributions and the
linear exponential family
371
12-4-4 Computational issues
374
I2.4.5 Exercises
377
12.5
Estimation for discrete distributions
383
12.5.1
Poisson
383
CONTENTS
Xl
12.5.2 Negative
binomial
386
12.5.3
Binomial
389
I2.5.4
The (a,b,ï)
class
392
12.5.5
Compound models
396
12.5.6
Effect of exposure on maximum likelihood
estimation
398
12.5.7
Exercises
399
12.6
Bivariate models
402
12.6.1
Introduction
402
12.6.2
Copulas
4ОЗ
12.6.3
Exercise
405
12.7
Models with covariates
405
12.7.1
Introduction
405
12.7.2
Proportional hazards models
407
12.7.3
The generalized linear and accelerated
failure time models
413
I2.7.4 Exercises
4I6
13
Model selection
419
13.1
Introduction
419
13.2
Representations of the data and model
420
13.3
Graphical comparison of the density and
distribution functions
421
13.3.1
Exercises
426
13.4
Hypothesis tests
427
13.4-1
Kolmogorov-Smirnov test
428
13.4 -2
Anderson-Darling test
430
13.4-3
Chi-square goodness-of-fit test
432
13-4-4 Likelihood ratio test
436
13.4-5
Exercises
438
13.5
Selecting a model
440
13.5.1
Introduction
440
13.5.2
Judgment-based approaches
441
13.5.3
Score-based approaches
442
13.5.4
Exercises
449
14
Five examples
455
I4.I Introduction
455
I4.2 Time to death
455
xii CONTENTS
Ц.2.1
The data
455
14.2.2 Some calculations
457
14.2.3
Exercise
459
14-3
Time from incidence to report
459
14-3.1
The problem and some data
460
14-3.2
Analysis
460
14.4 Payment amount
461
14.4.1 The data
462
14.4.2 The first model
463
14-4-3
The second model
465
14.5
An aggregate loss example
466
14.6 Another aggregate loss example
471
14-6.1
Distribution for a single policy
471
14-6.2
One hundred policies
—
excess of loss
471
14-6.3
One hundred policies
—
aggregate stop-loss
473
14-6-4
Numerical convolutions
474
14-Ί
Comprehensive exercises
476
Part V Adjusted estimates and simulation
15
Interpolation and smoothing
483
15.1
Introduction
483
15.2
Polynomial interpolation and smoothing
485
15.2.1
Exercises
488
15.3
Cubic spline interpolation
489
15.3.1
Construction of cubic splines
491
15.3.2
Exercises
499
I5.4 Approximating functions with splines
500
15-4-1
Exercise
504
15.5
Extrapolating with splines
504
15.5.1
Exercise
505
15.6
Smoothing splines
505
15.6.1
Exercise
514
16
Credibility
515
16.1
Introduction
515
16.2
Statistical concepts
517
CONTENTS xiii
16.2.1
Conditional distributions
518
16.2.2
Conditional expectation
520
16.2.3
Nonparametric unbiased estimators
523
16.24
Exercises
529
16.3
Limited fluctuation credibility theory
530
16.3.1
Full credibility
532
16.3.2
Partial credibility
535
16.3.3
Problems with the approach
539
16.
3.4
Notes and References
539
16.3.5
Exercises
539
16.4
Greatest accuracy credibility theory
542
16.4.1
Introduction
542
16.4-2
The Bayesian methodology
545
16.4-3
The credibility premium
553
16.4-4
The
Buhlmann
model
. 557
16.4- 5
The Buhlmann-Straub model
560
16.4.6
Exact credibility
566
16.4-7
Linear versus Bayesian versus no
credibility
569
16.4.8
Notes and References
577
16.4.9
Exercises
578
16.5
Empirical
Bayes
parameter estimation
589
16.5.1
Nonparametric estimation
592
16.5.2
Semiparametric estimation
600
16.5.3
Parametric estimation
602
16.5.4
Notes and References
607
16.5.5
Exercises
607
17
Simulation
611
17.1
Basics of simulation
611
17.1.1
The simulation approach
612
17.1.2
Exercises
617
17.2
Examples of simulation in actuarial modeling
618
17.2.1
Aggregate loss calculations
618
17.2.2
Examples of lack of independence or
identical distributions
619
17.2.3
Simulation analysis of the two examples
620
17.2.4 Statistical analyses
622
17.2.5
Exercises
625
x/V
CONTENTS
Appendix A An inventory of continuous distributions
627
A.I Introduction
627
A.
2
Transformed beta family
631
A.
2.1
Four-parameter distribution
631
A.2.2 Three-parameter distributions
631
A,
2.3
Two-parameter distributions
633
A.3 Transformed gamma family
635
A.
3.1
Three-parameter distributions
635
A.
3.2
Two-parameter distributions
636
A.
3.3
One-parameter distributions
638
A.
4
Other distributions
638
A.
5
Distributions with finite support
6Ą0
Appendix
В
An inventory of discrete distributions
6Ą3
B.I Introduction
6A3
B.2 The (a,
6,0)
class
6ĄĄ
B.3 The(a,b,l) class
6Ą5
B.3.1 The zero-truncated subclass
645
B.3.
2
The zero-modified subclass
647
BĄ
The compound class
648
B.4-1 Some compound distributions
648
B.5 A hierarchy of discrete distributions
650
Appendix
С
Frequency and severity relationships
651
Appendix
D
The recursive formula
653
Appendix
E
Discretization of the severity distribution
655
E.
1
The method of rounding
655
E.2 Mean preserving
656
E.3 Undiscretization of a discretized distribution
657
Appendix
F
Numerical optimization and solution of systems
of equations
659
F.I Maximization using Solver
660
F.2 The simplex method
664
F.3 Using
Excelß
to solve equations
665
CONTENTS xv
References
Index
The First Edition of Loss Models was deemed "worthy of classical status" by
the Journal of the International Statistical Institute, While retaining its prede¬
cessor's thorough treatment of the concepts and methods of analyzing contin¬
gent events, this powerful Second Edition is updated and expanded to offer
even more complete and flexible coverage of risk theory, loss distributions, and
survival models.
Beginning with a framework for model building and a description of frequency
and severity loss data typically available, it shows readers how to combine
frequency, severity and loss models to build aggregate loss models and
credibility-based pricing models, and how to analyze loss over multiple time
periods, important features of this new edition include:
•
Thorough preparation for relevant parts of preliminary examinations
of the Society of Actuaries
(SOA)
and Casualty Actuarial Society (CAS)
•
Exercises based on past
SOA
and CAS exams
•
Examples using actual insurance data
•
Practical treatment of modern credibility theory
•
Data fifes and more from an ftp site
Loss Models, Second Edition is an important resource, providing a compre¬
hensive, practically motivated toolkit and an excellent reference, for actuaries
preparing for
SOA
and CAS preliminary examinations, students in actuarial
science who need to understand loss and risk models, and practicing
professionals involved in loss modeling.
STUART A. KLUGMAN, PhD, R>A, is Principal Financial Group Professor of
Actuarial Science at Drake University,
Des Moines,
iowa. HARRY
H-
PANJiR,
PhD,
F8A«
FOIA, HqììFIA,
ís Professor in
the
Department
of Statistics and
Actuarial Science at the University of Waterloo, Ontario, Canada, GORDON
E,
УУІШИОТ, РіФ,
FSA,.
FOIA,
is Munich Re Professor in the Department
of Statistics and
Actuaria!
Science at the University of Waterloo. |
| any_adam_object | 1 |
| author | Klugman, Stuart A. 1949- Panjer, Harry H. 1946- Willmot, Gordon E. 1957- |
| author_GND | (DE-588)1044735155 (DE-588)1044735732 (DE-588)122585860 |
| author_facet | Klugman, Stuart A. 1949- Panjer, Harry H. 1946- Willmot, Gordon E. 1957- |
| author_role | aut aut aut |
| author_sort | Klugman, Stuart A. 1949- |
| author_variant | s a k sa sak h h p hh hhp g e w ge gew |
| building | Verbundindex |
| bvnumber | BV019698600 |
| callnumber-first | H - Social Science |
| callnumber-label | HG8781 |
| callnumber-raw | HG8781 HG8781.K583 2004 |
| callnumber-search | HG8781 HG8781.K583 2004 |
| callnumber-sort | HG 48781 |
| callnumber-subject | HG - Finance |
| classification_rvk | QQ 630 SK 980 |
| classification_tum | WIR 190f MAT 902f WIR 522f |
| contents | Erg. bildet: Klugman, Stuart A.: Solutions manual to accompany Loss models |
| ctrlnum | (OCoLC)54988791 (DE-599)BVBBV019698600 |
| dewey-full | 368/.0122 368/.01 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 368 - Insurance |
| dewey-raw | 368/.01 22 368/.01 |
| dewey-search | 368/.01 22 368/.01 |
| dewey-sort | 3368 11 222 |
| dewey-tens | 360 - Social problems and services; associations |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV019698600 |
| illustrated | Illustrated |
| indexdate | 2024-10-28T11:01:21Z |
| institution | BVB |
| isbn | 0471215775 |
| language | English |
| lccn | 2004048066 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013026218 |
| oclc_num | 54988791 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-898 DE-BY-UBR DE-706 DE-1049 DE-521 DE-188 DE-N2 DE-355 DE-BY-UBR |
| owner_facet | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-898 DE-BY-UBR DE-706 DE-1049 DE-521 DE-188 DE-N2 DE-355 DE-BY-UBR |
| physical | XXII, 688 S. Ill., graph. Darst. 24 cm |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Wiley Interscience |
| record_format | marc |
| series2 | Wiley series in probability and statistics |
| spelling | Klugman, Stuart A. 1949- Verfasser (DE-588)1044735155 aut Loss models from data to decisions Stuart A. Klugman ; Harry H. Panjer ; Gordon E. Willmot 2. ed. Hoboken, NJ Wiley Interscience 2004 XXII, 688 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Includes bibliographical references (p. 671-680) and index Erg. bildet: Klugman, Stuart A.: Solutions manual to accompany Loss models Assurance - Mathématiques Assurance - Modèles mathématiques Assurance - Méthodes statistiques Assurance rasuqam Modèle mathématique rasuqam Méthode statistique rasuqam Statistische methoden gtt Verzekeringswezen gtt Verzekeringswiskunde gtt Waarschijnlijkheidstheorie gtt aInsurance xStatistical methods aInsurance xMathematical models Risikotheorie (DE-588)4135592-1 gnd rswk-swf Verlustrechnung (DE-588)4187883-8 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Versicherungsstatistik (DE-588)4235273-3 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Risikotheorie (DE-588)4135592-1 s Versicherungsmathematik (DE-588)4063194-1 s DE-604 Verlustrechnung (DE-588)4187883-8 s Mathematisches Modell (DE-588)4114528-8 s 1\p DE-604 Versicherungsstatistik (DE-588)4235273-3 s 2\p DE-604 Panjer, Harry H. 1946- Verfasser (DE-588)1044735732 aut Willmot, Gordon E. 1957- Verfasser (DE-588)122585860 aut http://www.loc.gov/catdir/description/wiley042/2004048066.html Publisher description http://www.loc.gov/catdir/toc/wiley041/2004048066.html Table of contents Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013026218&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013026218&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 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 |
| spellingShingle | Klugman, Stuart A. 1949- Panjer, Harry H. 1946- Willmot, Gordon E. 1957- Loss models from data to decisions Erg. bildet: Klugman, Stuart A.: Solutions manual to accompany Loss models Assurance - Mathématiques Assurance - Modèles mathématiques Assurance - Méthodes statistiques Assurance rasuqam Modèle mathématique rasuqam Méthode statistique rasuqam Statistische methoden gtt Verzekeringswezen gtt Verzekeringswiskunde gtt Waarschijnlijkheidstheorie gtt aInsurance xStatistical methods aInsurance xMathematical models Risikotheorie (DE-588)4135592-1 gnd Verlustrechnung (DE-588)4187883-8 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Versicherungsstatistik (DE-588)4235273-3 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4135592-1 (DE-588)4187883-8 (DE-588)4063194-1 (DE-588)4235273-3 (DE-588)4114528-8 |
| title | Loss models from data to decisions |
| title_auth | Loss models from data to decisions |
| title_exact_search | Loss models from data to decisions |
| title_full | Loss models from data to decisions Stuart A. Klugman ; Harry H. Panjer ; Gordon E. Willmot |
| title_fullStr | Loss models from data to decisions Stuart A. Klugman ; Harry H. Panjer ; Gordon E. Willmot |
| title_full_unstemmed | Loss models from data to decisions Stuart A. Klugman ; Harry H. Panjer ; Gordon E. Willmot |
| title_short | Loss models |
| title_sort | loss models from data to decisions |
| title_sub | from data to decisions |
| topic | Assurance - Mathématiques Assurance - Modèles mathématiques Assurance - Méthodes statistiques Assurance rasuqam Modèle mathématique rasuqam Méthode statistique rasuqam Statistische methoden gtt Verzekeringswezen gtt Verzekeringswiskunde gtt Waarschijnlijkheidstheorie gtt aInsurance xStatistical methods aInsurance xMathematical models Risikotheorie (DE-588)4135592-1 gnd Verlustrechnung (DE-588)4187883-8 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Versicherungsstatistik (DE-588)4235273-3 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Assurance - Mathématiques Assurance - Modèles mathématiques Assurance - Méthodes statistiques Assurance Modèle mathématique Méthode statistique Statistische methoden Verzekeringswezen Verzekeringswiskunde Waarschijnlijkheidstheorie aInsurance xStatistical methods aInsurance xMathematical models Risikotheorie Verlustrechnung Versicherungsmathematik Versicherungsstatistik Mathematisches Modell |
| url | http://www.loc.gov/catdir/description/wiley042/2004048066.html http://www.loc.gov/catdir/toc/wiley041/2004048066.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013026218&sequence=000003&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=013026218&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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