Actuarial models: the mathematics of insurance
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton, FL
Chapman & Hall
2007
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| Online-Zugang: | Publisher description Inhaltsverzeichnis Klappentext |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XXII, 633 S. |
| ISBN: | 9781584885863 1584885866 |
Internformat
MARC
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| 100 | 1 | |a Rotar', Vladimir I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Actuarial models |b the mathematics of insurance |c Vladimir I. Rotar |
| 264 | 1 | |a Boca Raton, FL |b Chapman & Hall |c 2007 | |
| 300 | |a XXII, 633 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 7 | |a Assurance - Mathématiques |2 ram | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Insurance |x Mathematics | |
| 650 | 0 | 7 | |a Versicherungsmathematik |0 (DE-588)4063194-1 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Versicherungsmathematik |0 (DE-588)4063194-1 |D s |
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Datensatz im Suchindex
| _version_ | 1804136548275322880 |
|---|---|
| adam_text | Contents
Preface
xv
Acknowledgments
xxi
Introduction
1
CHAPTER
0.
Some Preliminary Notions and Facts from Probability Theory,
the Theory of Interest, and Calculus1
7
1
PROBABILITY AND RANDOM VARIABLES
............... 7
1.1
Sample space, events, probability measure
............... 7
1.2
Independence and conditional probabilities
............... 9
1.3
Random variables, random vectors, and their distributions
....... 10
1.3.1
Random variables
........................ 10
1.3.2
Random vectors
......................... 11
1.3.3
Cumulative distribution functions
................ 14
1.3.4
Quantiles
............................ 17
1.3.5
Mixtures of distributions
.................... 17
2
EXPECTATION
................................ 18
2.1
Definitions
................................ 18
2.2
Integration by parts and a formula for expectation
........... 21
2.3
A general definition of expectation
................... 22
2.4
Can we encounter an infinite expected value in models of real
phenomena?
............................... 23
2.5
Moments of r.v. s. Correlation
...................... 24
2.5.1
Variance and other moments
.................. 24
2.5.2
The Cauchy-Schwarz inequality
................ 25
2.5.3
Covariance and correlation
................... 25
2.6
Inequalities for deviations
........................ 27
2.7
Linear transformations of r.v. s. Normalization
............. 28
3
SOME BASIC DISTRIBUTIONS
...................... 29
3.1
Discrete distributions
.......................... 29
3.1.1
The binomial distribution
.................... 29
3.1.2
The multinomial distribution
.................. 30
3.1.3
The geometric distribution
................... 30
3.1.4
The negative binomial distribution
............... 31
3.1.5
The
Poisson
distribution
.................... 32
3.2
Continuous distributions
........................ 33
3.2.1
The uniform distribution and simulation of r.v. s
........ 33
3.2.2
The exponential distribution
.................. 35
3.2.3
The r(gamma)-distribution
................... 36
3.2.4
The normal distribution
..................... 37
4
MOMENT GENERATING FUNCTIONS
.................. 38
4.1
Laplace transform
............................ 38
4.2
An example when a m.g.f. does not exist
................ 41
4.3
The m.g.f. s of basic distributions
.................... 41
4.3.1
The binomial distribution
.................... 42
4.3.2
The geometric and negative binomial distributions
....... 43
4.3.3
The
Poisson
distribution
.................... 43
4.3.4
The uniform distribution
.................... 43
4.3.5
The exponential and gamma distributions
........... 44
4.3.6
The normal distribution
..................... 44
4.4
The moment generating function and moments
............ 44
4.5
Expansions for m.g.f. s
......................... 46
4.5.1
Taylor s expansions for m.g.f. s
................. 46
4.5.2
Cumulants
............................ 46
5
CONVERGENCE OF RANDOM VARIABLES AND DISTRIBUTIONS
. 47
6
SOME FACTS AND FORMULAS FROM THE THEORY OF INTEREST
50
6.1
Compound interest
........................... 50
6.2
Nominal rate
............................... 53
6.3
Discount and annuities
......................... 54
6.4
Accumulated value
........................... 56
6.5
Effective and nominal discount rates
.................. 56
7
APPENDIX. SOME NOTATIONS AND FACTS FROM CALCULUS
... 57
7.1
The small
о
and big O notation
.................... 57
7.1.1
Smallo
............................. 57
7.1.2
Big
О
.............................. 59
7.2
Taylor expansions
............................ 59
7.2.1
A general expansion
...................... 59
7.2.2
Some particular expansions
................... 60
7.3
Concavity
................................ 61
CHAPTER
1.
Comparison of Random Variables. Preferences of Individuals
63
1
COMPARISON OF RANDOM VARIABLES
.
SOME PARTICULAR CRITERIA
...................... 63
1.1
Preference order
............................. 63
1.2
Several simple criteria
.......................... 66
1.2.1
The mean-value criterion
.................... 66
1.2.2
Value-at-Risk (VaR)
....................... 66
1.2.3
An important remark: risk measures rather than criteria
.... 69
1.2.4
Tail conditional expectation (TCE) or Tail-Value-at-Risk
(TailVaR)
............................ 69
1.2.5
The mean-variance criterion
.................. 73
1.3
On coherent measures of risk
...................... 76
COMPARISON OF R.V. S AND LIMIT THEOREMS OF
PROBABILITY THEORY
.......................... 79
2.1
A diversion to Probability Theory: two limit theorems
......... 80
2.1.1
The Law of Large Numbers (LLN)
............... 80
2.1.2
The Central Limit Theorem (CUT)
. . . ,........... 80
2.2
A simple model of insurance with many clients
............ 81
2.3
St. Petersburg s paradox
......................... 83
EXPECTED UTILITY
............................ 84
3.1
Expected utility maximization (EUM)
................. 84
3.1.1
Utility function
......................... 84
3.1.2
Expected utility maximization (EUM) criterion
........ 85
3.1.3
Some classical examples of utility functions
......... 88
3.2
Utility and insurance
.......................... 91
3.3
How we may determine the utility function in particular cases
..... 93
3.4
Risk aversion
.............................. 94
3.4.1
A definition
........................... 94
3.4.2
Jensen s inequality
....................... 95
3.4.3
How to measure risk aversion in the EUM case
........ 96
3.4.4
Proofs
.............................. 98
3.5
A new perspective: EUM as a linear criterion
............. 100
3.5.1
Preferences on distributions
................... 100
3.5.2
The first stochastic dominance
................. 101
3.5.3
The second stochastic dominance
................ 103
3.5.4
The EUM criterion
....................... 104
3.5.5
Linearity of the utility functional
................ 105
3.5.6
An axiomatic approach
..................... 108
NON-LINEAR CRITERIA
..........................
Ill
4.1
Allais
paradox
.............................
Ill
4.2
Weighted utility
............................. 112
4.3
Implicit or comparative utility
..................... 114
4.3.1
Definitions and examples
.................... 114
4.3.2
In what sense the implicit utility criterion is linear
....... 117
4.4
Rank Dependent Expected Utility
................... 118
4.4.1
Definitions and examples
.................... 118
4.4.2
Application to insurance
.................... 121
4.4.3
Further discussion and the main axiom
............. 122
4.5
Remarks
................................. 124
OPTIMAL PAYMENT FROM THE STANDPOINT OF THE INSURED
. 125
5.1
Arrow s theorem
............................. 125
5.2
A generalization
............................ 128
EXERCISES
................................. 129
CHAPTER
2.
An individual Risk Model for a Short Period
137
1
THE DISTRIBUTION OF AN INDIVIDUAL PAYMENT
......... 137
Ш
1.1
The distribution of the loss given that it has occurred
......... 137
1.1.1
Definitions. Characterization of tails
.............. 137
1.1.2
Some particular light-tailed distributions
............ 141
1.1.3
Some particular heavy-tailed distributions
........... 142
1.1.4
The asymptotic behavior of tails and moments
........ 145
pá
1.2
The distribution of the loss X
...................... 146
via
1.3
The distribution of the payment and types of insurance
........ 147
2
THE AGGREGATE PAYMENT
....................... 155
2.1
Convolutions
.............................. 155
2.1.1
Definitions and examples
.................... 155
2.1.2
Some classical examples
.................... 158
2.1.3
The analogue of the binomial formula for convolutions
.... 162
2.2
Moment generating functions
...................... 163
3
NORMAL AND OTHER APPROXIMATIONS
............... 165
3.1
Normal approximation
......................... 165
3.1.1
A heuristic approach
...................... 165
3.1.2
An important remark: the standard deviation principle
..... 170
3.1.3
A rigorous estimation
..................... 171
vó
3.1.4
The number of contracts needed to maintain a given security
level
............................... 176
Ы
3.2
How to take into account the asymmetry of S. The
Г
-approximation
.
178
vS
3.3
Asymptotic expansions and Normal Power (NP) approximation
.... 180
4
EXERCISES
................................. 182
CHAPTER
3.
Conditional Expectations
191
Ш
1
HOW TO COMPUTE CONDITIONAL EXPECTATIONS
.
THE CONDITIONING PROCEDURE
.................... 191
1.1
Conditional expectation given a r.v
.................... 191
1.1.1
The discrete case
........................ 191
1.1.2
The case of continuous distributions
.............. 193
1.2
Properties of conditional expectations
................. 196
1.3
Conditioning and some useful formulas
................ 198
1.3.1
A formula for variance
..................... 198
1.3.2
More detailed representations of the formula for total expecta¬
tion
............................... 198
1.4
Conditional expectation given
a r. vec
.................. 200
1.4.1
General definitions
....................... 200
1.4.2
On the case of an infinite-dimensional X
............ 202
1.4.3
On conditioning in the multi-dimensional case
......... 202
2
FORMULA FOR TOTAL EXPECTATION AND
CONDITIONAL EXPECTATION GIVEN A PARTITION
......... 204
Ш
2.1
Conditional expectation given an event
................. 204
2.2
The formula for total expectation
.................... 207
2.3
Expectation given a partition
...................... 208
3
CONDITIONAL EXPECTATIONS GIVEN RANDOM VARIABLES
OR VECTORS
................................ 209
3.1
The discrete case
............................ 209
3.2
The general case
............................. 211
4
ONE MORE IMPORTANT PROPERTY OF CONDITIONAL
EXPECTATIONS
............................... 213
4.1
Conditioning on partitions
....................... 213
4.2
Conditioning on r.v. s or r.vec. s
..................... 214
5
A GENERAL APPROACH TO CONDITIONAL EXPECTATIONS
.... 215
5.1
Conditional expectation relative to a
σ
-algebra
.............
215
5.2
Conditional expectation given a r.v. or
a r.vec
.............. 218
5.3
Properties of conditional expectations
................. 219
6
SOME PROOFS
............................... 220
6.1
Proofs of the properties stated in Section
1.2.............. 220
6.2
Proof of Proposition
2.......................... 222
7
EXERCISES
................................. 222
CHAPTER
4.
A Collective Risk Model for a Short Period
225
1
THREE BASIC PROPOSITIONS
...................... 225
2
COUNTING OR FREQUENCY DISTRIBUTIONS
............. 227
2.1
The
Poisson
distribution and Poisson s theorem
............ 227
2.1.1
A heuristic approximation
................... 227
2.1.2
The accuracy of the
Poisson
approximation
.......... 231
2.2
Some other counting distributions
.................. 233
2.2.1
The mixed
Poisson
distribution
................. 233
2.2.2
Compound mixing
....................... 237
2.2.3
The
(аДО)
and (a,b, ) (or Katz-Panjer s) classes
...... 239
3
THE DISTRIBUTION OF THE AGGREGATE CLAIM
.......... 241
3.1
The case of a homogeneous group
................... 241
3.1.1
The method of convolutions
.................. 241
3.1.2
The case when
N
has
a Poisson
distribution
.......... 245
3.1.3
The m.g.f. s method
...................... 246
3.2
The case of several homogeneous groups
............... 248
3.2.1
A general scheme and reduction to one group
......... 248
3.2.2
The significance of the weights
w¡
............... 251
4
NORMAL APPROXIMATION OF THE DISTRIBUTION OF
THE AGGREGATE CLAIM
......................... 253
4.1
A limit theorem
............................. 253
4.2
Estimation of premiums
........................ 257
4.3
The accuracy of normal approximation
................ 259
4.4
Proof of Theorem
10.......................... 260
5
EXERCISES
................................. 263
CHAPTERS. Random Processes. I. Counting and Compound Processes.
Markov Chains. Modeling Claim and Cash Flows
269
1
A GENERAL FRAMEWORK AND TYPICAL SITUATIONS
....... 269
1.1
Preliminaries
.............................. 269
1.2
Processes with independent increments
................. 271
1.2.1
The simplest counting process
................. 271
1.2.2
Brownian motion
........................ 271
1.3
Markov processes
............................ 274
2
POISSON
AND OTHER COUNTING PROCESSES
............ 276
2.1
The homogeneous
Poisson
process
................... 276
2.2
The non-homogeneous
Poisson
process
................ 281
2.2.1
A model and examples
..................... 281
2.2.2
Proof of Proposition
1...................... 284
2.3
The Cox process
............................ 285
3
COMPOUND PROCESSES
......................... 287
4
MARKOV CHAINS. CASH FLOWS IN THE MARKOV ENVIRONMENT
289
4.1
Preliminaries
.............................. 289
4.2
Variables defined on a Markov chain. Cash flows
........... 295
4.2.1
Variables defined on states
................... 295
4.2.2
Mean discounted payments
................... 296
4.2.3
The case of absorbing states
.................. 298
4.2.4
Variables defined on transitions
................. 301
4.2.5
What to do if the chain is not homogeneous
.......... 302
4.3
The first step analysis. An infinite horizon
............... 302
4.3.1
Mean discounted payments in the case of infinite time horizon
303
4.3.2
The first step approach to random walk problems
....... 305
4.4
Limiting probabilities and stationary distributions
........... 310
4.5
The ergodicity property and classification of states
.......... 314
4.5.1
Classes of states
......................... 314
4.5.2
The recurrence property
..................... 315
4.5.3
Recurrence and travel times
................... 318
4.5.4
Recurrence and ergodicity
................... 320
5
EXERCISES
................................. 321
CHAPTER
6.
Random Processes. II. Brownian Motion and Martingales.
Hitting Times
329
1
BROWNIAN MOTION AND ITS GENERALIZATIONS
......... 329
1.1
Further properties of the standard Brownian motion
.......... 329
1.1.1
Non-differentiability of trajectories
............... 329
1.1.2
Brownian motion as an approximation. The
invariance
principle
330
1.1.3
The distribution
of.wř,
hitting times, and the maximum value of
Brownian motion
........................ 331
1.2
The Brownian motion with drift
.................... 333
1.2.1
Modeling of the surplus process. What a Brownian motion with
drift approximates in this case
................. 334
1.2.2
A reduction to the standard Brownian motion
......... 335
1.3
Geometric Brownian motion
...................... 337
2
MARTINGALES
............................... 338
2.1
General properties and examples
.................... 338
2.2
Martingale transform
.......................... 343
2.3
Optional stopping time and some applications
............. 344
2.3.1
Definitions and examples
.................... 344
2.3.2
Wald s identity
......................... 348
2.3.3
The ruin probability for the simple random walk
....... 349
2.3.4
The ruin probability for the Brownian motion with drift
.... 350
2.3.5
The distribution of the ruin time in the case of Brownian motion
352
1¿J
2.3.6
The hitting time for the Brownian motion with drift
...... 353
2.4
Generalizations
............................. 354
2.4.1
The martingale property in the case of random stopping time
. 354
2.4.2
A reduction to the standard Brownian motion in the case of
random time
........................... 355
2.4.3
The distribution of the ruin time in the case of Brownian motion:
another approach
........................ 356
2.4.4
Proof of Theorem
11 ...................... 357
2.4.5
Verification of Condition
3
of Theorem
5............ 358
3
EXERCISES
................................. 359
CHAPTER?. Global Characteristics of the Surplus Process. Ruin Models.
Models with Paying Dividends
363
Ш
1
INTRODUCTION
.............................. 363
2
RUIN MODELS
............................... 366
2.1
Adjustment coefficients and ruin probabilities
............. 367
2.1.1
A preliminary condition: for large time horizons the aggregate
claim should be large
...................... 367
2.1.2
The main theorem
........................ 368
2.1.3
Proof of Lundberg s inequality
................. 370
2.2
Computing adjustment coefficients
................... 370
2.2.1
A general proposition
...................... 370
2.2.2
The discrete time case
...................... 374
2.2.3
The case of a homogeneous compound
Poisson
process
.... 377
2.2.4
The discrete time case revisited
................. 380
2.2.5
The case of the non-homogeneous compound
Poisson
process
381
2.3
Trade-off between the premium and the initial surplus
........ 381
v$
2.4
Three cases when the ruin probability may be computed precisely
. . 385
vž3
2.4.1
The case when the size of a separate claim is exponentially
distributed
............................ 385
2.4.2
The case of the simple random walk
.............. 386
2.4.3
The case of Brownian motion
................. 387
Ш
2.5
The martingale approach. A generalization of Theorem
1....... 388
2.6
The renewal approach
.......................... 390
2.6.1
The first surplus below the initial level
............. 390
2.6.2
The renewal approximation
................... 391
2.6.3
The
Cramér-Lundberg
approximation
............. 395
ΙΣ)
2.6.4
Proof of Theorem
5
from Section
2.6.1............. 395
2.7
Some recurrent relations and computational aspects
.......... 399
vM
3
CRITERIA CONNECTED WITH PAYING DIVIDENDS
......... 402
Ш
3.1
A general model
............................. 403
3.2
The case of the simple random walk
.................. 405
3.3
Finding an optimal strategy
....................... 408
4
EXERCISES
................................. 409
CHAPTERS. Survival Distributions
413
W
1
THE DISTRIBUTION OF THE LIFETIME
................. 413
1.1
Survival functions and force of mortality
................ 413
1.2
The time-until-death for a person of a given age
............ 418
1.3
Curtate-future-lifetime
......................... 422
1.4
Survivorship groups
........................... 423
1.5
Life tables and interpolation
...................... 424
1.5.1
Life tables
............................ 424
1.5.2
Interpolation for fractional ages
................. 429
1.6
Some analytical laws of mortality
.................... 431
2
A MULTIPLE DECREMENT MODEL
................... 434
Ш
2.1
A single life
............................... 434
2.2
Another view: net probabilities of decrement
.............. 438
2.3
A survivorship group
.......................... 442
2.4
Proof of Proposition
1.......................... 443
3
MULTIPLE LIFE MODELS
......................... 444
3.1
The joint distribution
.......................... 445
3.2
The lifetime of statuses
......................... 447
3.3
A model of dependency: conditional independence
.......... 451
3.3.1
A definition and the first example
................ 452
3.3.2
The common shock model
................... 453
4
EXERCISES
................................. 455
CHAPTERS Life Insurance Models
461
Ш
1
A GENERAL MODEL
............................ 461
1.1
The present value of a future payment
................. 461
1.2
The present value of payments to many clients
............. 464
2
SOME PARTICULAR TYPES OF CONTRACTS
.............. 467
2.1
Whole life insurance
.......................... 467
2.1.1
The continuous time case (benefits payable at the moment of
death)
.............................. 467
2.1.2
The discrete time case (benefits payable at the end of the year
of death)
............................. 467
2.1.3
A relation between Ax and Ax
.................. 470
2.1.4
The case of benefits payable at the end of the m-thly period
. . 471
2.2
Deferred whole life insurance
...................... 473
2.2.1
The continuous time case
.................... 473
2.2.2
The discrete time case
...................... 474
2.3
Term insurance
............................. 474
2.3.1
Continuous time
......................... 474
2.3.2
Discrete time
.......................... 476
2.4
Endowments
............................... 478
2.4.1
Pure endowment
........................ 478
2.4.2
Endowment
........................... 478
3
VARYING BENEFITS
............................ 480
3.1
Certain payments
............................ 480
3.2
Random payments
............................ 484
4
MULTIPLE DECREMENT AND MULTIPLE LIFE MODELS
...... 485
4.1
Multiple decrements
........................... 485
4.2
Multiple life insurance
......................... 488
5
ON THE ACTUARIAL NOTATION
..................... 491
6
EXERCISES
................................. 492
CHAPTER
10.
Annuity Models
499
1
INTRODUCTION. TWO APPROACHES TO COMPUTING ANNUITIES
499
1.1
Continuous annuities
.......................... 499
1.2
Discrete annuities
............................ 501
2
LEVEL ANNUITIES. A CONNECTION WITH INSURANCE
...... 504
2.1
Certain annuities. Some notation
.................... 504
2.2
Random annuities
............................ 504
3
SOME PARTICULAR TYPES OF LEVEL ANNUITIES. EXAMPLES
. . 506
3.1
Whole life annuities
........................... 506
3.2
Temporary annuities
........................... 509
3.3
Deferred annuities
............................ 512
3.4
Certain and life annuity
......................... 514
4
MORE ON VARYING PAYMENTS
..................... 516
5
ANNUITIES WITH m-thly PAYMENTS
.................. 518
6
MULTIPLE DECREMENT AND MULTIPLE LIFE MODELS
...... 521
6.1
Multiple decrements
........................... 521
6.2
Multiple life annuities
.......................... 523
7
EXERCISES
................................. 525
CHAPTER
11.
Premiums and Reserves
531
1
SOME GENERAL PREMIUM PRINCIPLES
................ 531
2
PREMIUM ANNUITIES
........................... 536
2.1
Preliminaries. General principles
.................... 536
2.2
Benefit premiums. The case of a single risk
.............. 537
2.2.1
Netratě
............................. 537
2.2.2
The case when
Ύ
is consistent with Z
............ 541
2.2.3
Variances
............................ 542
2.2.4
Premiums paid
m
times a year
................. 544
2.2.5
Combinations of insurances
................... 545
2.3
Accumulated values
........................... 546
2.4
Percentile premium
........................... 547
2.4.1
The case of a single risk
..................... 547
2.4.2
The case of many risks. Normal approximation
........ 549
2.5
Exponential premiums
......................... 552
3
RESERVES
.................................. 553
3.1
Definitions and preliminary remarks
.................. 553
3.2
Examples of direct calculations
..................... 554
3.3
Formulas for some standard types of insurance
............. 556
3.4
Recursive relations
........................... 557
4
EXERCISES
................................. 560
CHAPTER
12.
Risk Exchange: Reinsurance and Coinsurance
565
1
REINSURANCE FROM THE STAND POINT OF
A CEDENT
...... 565
1.1
Some optimization considerations
................... 565
1.1.1
Expected utility maximization
................. 566
1.1.2
Variance as a measure of risk
.................. 568
1.2
Proportional reinsurance. Adding a new contract to an existing portfolio
570
1.2.1
The case of a fixed security loading coefficient
......... 570
1.2.2
The case of the standard deviation premium principle
..... 573
1.3
Long-term insurance. Ruin probability as a criterion
......... 575
1.3.1
An example with proportional reinsurance
........... 575
1.3.2
An example with excess-of-loss insurance
........... 577
2
RISK EXCHANGE AND RECIPROCITY OF COMPANIES
....... 578
2.1
A general framework and some examples
............... 578
2.2
Two more examples with expected utility maximization
........ 586
2.3
The case of the mean-variance criterion
................ 590
2.3.1
Minimization of variances
................... 590
2.3.2
The exchange of portfolios
................... 593
3
REINSURANCE MARKET
......................... 598
3.1
A model of the exchange market of random assets
........... 598
3.2
An example concerning reinsurance
.................. 601
4
EXERCISES
................................. 604
Tables
607
References
611
Answers to Exercises
619
Subject Index
627
Ideal for students preparing for level
300
actuarial exams in the US. Actuarial
Modeîs:
The Mathematics of insurance provides a comprehensive exposition of insurance processes
models and presents mathematical setups and methods used in Actuarial Modeling.
Divided into three self-contained and explicitly designated parts of different levels of difficulty,
this book examines standard as well as advanced topics such as modern utility theory, martingale
technique, models with payments of dividends, reinsurance models, and classification of
distributions. It provides practical skills in analysis of insurance processes. This text discusses
a number of topics not commonly found in existing Actuarial Mathematics textbooks, including
achievements of the modern Risk Evaluation theory, premium principles, accuracy of normal
and
Poisson
approximation, and a reinsurance market model.
The book includes numerous examples, practice problems, and exercises on numerical
calculations using Excel . It includes also preliminary examination material for the Society of
Actuaries and the Casualty Actuarial Society (CAS), providing, in particular, real problems from
past CAS exams.
Features
Provides mathematical models for both non-life and life insurance
Uses real problems from past CAS exams and prepares students for level
300
actuarial exams in the US
Features a systematic presentation from a mathematical point of view/
Presents supplementary material containing main facts from
Probability
Theory,
Stochastic Processes, the Theory of Interest, and Calculus
Contains numerous examples, which may be viewed as a solution guide for
corresponding exams
Discusses many analytical procedures and contains many computational example
for the most part, with use of Excel
|
| adam_txt |
Contents
Preface
xv
Acknowledgments
xxi
Introduction
1
CHAPTER
0.
Some Preliminary Notions and Facts from Probability Theory,
the Theory of Interest, and Calculus1
7
1
PROBABILITY AND RANDOM VARIABLES
. 7
1.1
Sample space, events, probability measure
. 7
1.2
Independence and conditional probabilities
. 9
1.3
Random variables, random vectors, and their distributions
. 10
1.3.1
Random variables
. 10
1.3.2
Random vectors
. 11
1.3.3
Cumulative distribution functions
. 14
1.3.4
Quantiles
. 17
1.3.5
Mixtures of distributions
. 17
2
EXPECTATION
. 18
2.1
Definitions
. 18
2.2
Integration by parts and a formula for expectation
. 21
2.3
A general definition of expectation
. 22
2.4
Can we encounter an infinite expected value in models of real
phenomena?
. 23
2.5
Moments of r.v.'s. Correlation
. 24
2.5.1
Variance and other moments
. 24
2.5.2
The Cauchy-Schwarz inequality
. 25
2.5.3
Covariance and correlation
. 25
2.6
Inequalities for deviations
. 27
2.7
Linear transformations of r.v.'s. Normalization
. 28
3
SOME BASIC DISTRIBUTIONS
. 29
3.1
Discrete distributions
. 29
3.1.1
The binomial distribution
. 29
3.1.2
The multinomial distribution
. 30
3.1.3
The geometric distribution
. 30
3.1.4
The negative binomial distribution
. 31
3.1.5
The
Poisson
distribution
. 32
3.2
Continuous distributions
. 33
3.2.1
The uniform distribution and simulation of r.v.'s
. 33
3.2.2
The exponential distribution
. 35
3.2.3
The r(gamma)-distribution
. 36
3.2.4
The normal distribution
. 37
4
MOMENT GENERATING FUNCTIONS
. 38
4.1
Laplace transform
. 38
4.2
An example when a m.g.f. does not exist
. 41
4.3
The m.g.f.'s of basic distributions
. 41
4.3.1
The binomial distribution
. 42
4.3.2
The geometric and negative binomial distributions
. 43
4.3.3
The
Poisson
distribution
. 43
4.3.4
The uniform distribution
. 43
4.3.5
The exponential and gamma distributions
. 44
4.3.6
The normal distribution
. 44
4.4
The moment generating function and moments
. 44
4.5
Expansions for m.g.f.'s
. 46
4.5.1
Taylor's expansions for m.g.f.'s
. 46
4.5.2
Cumulants
. 46
5
CONVERGENCE OF RANDOM VARIABLES AND DISTRIBUTIONS
. 47
6
SOME FACTS AND FORMULAS FROM THE THEORY OF INTEREST
50
6.1
Compound interest
. 50
6.2
Nominal rate
. 53
6.3
Discount and annuities
. 54
6.4
Accumulated value
. 56
6.5
Effective and nominal discount rates
. 56
7
APPENDIX. SOME NOTATIONS AND FACTS FROM CALCULUS
. 57
7.1
The "small
о
and big O" notation
. 57
7.1.1
Smallo
. 57
7.1.2
Big
О
. 59
7.2
Taylor expansions
. 59
7.2.1
A general expansion
. 59
7.2.2
Some particular expansions
. 60
7.3
Concavity
. 61
CHAPTER
1.
Comparison of Random Variables. Preferences of Individuals
63
1
COMPARISON OF RANDOM VARIABLES
.
SOME PARTICULAR CRITERIA
. 63
1.1
Preference order
. 63
1.2
Several simple criteria
. 66
1.2.1
The mean-value criterion
. 66
1.2.2
Value-at-Risk (VaR)
. 66
1.2.3
An important remark: risk measures rather than criteria
. 69
1.2.4
Tail conditional expectation (TCE) or Tail-Value-at-Risk
(TailVaR)
. 69
1.2.5
The mean-variance criterion
. 73
1.3
On coherent measures of risk
. 76
COMPARISON OF R.V.'S AND LIMIT THEOREMS OF
PROBABILITY THEORY
. 79
2.1
A diversion to Probability Theory: two limit theorems
. 80
2.1.1
The Law of Large Numbers (LLN)
. 80
2.1.2
The Central Limit Theorem (CUT)
. . . ,. 80
2.2
A simple model of insurance with many clients
. 81
2.3
St. Petersburg's paradox
. 83
EXPECTED UTILITY
. 84
3.1
Expected utility maximization (EUM)
. 84
3.1.1
Utility function
. 84
3.1.2
Expected utility maximization (EUM) criterion
. 85
3.1.3
Some "classical" examples of utility functions
. 88
3.2
Utility and insurance
. 91
3.3
How we may determine the utility function in particular cases
. 93
3.4
Risk aversion
. 94
3.4.1
A definition
. 94
3.4.2
Jensen's inequality
. 95
3.4.3
How to measure risk aversion in the EUM case
. 96
3.4.4
Proofs
. 98
3.5
A new perspective: EUM as a linear criterion
. 100
3.5.1
Preferences on distributions
. 100
3.5.2
The first stochastic dominance
. 101
3.5.3
The second stochastic dominance
. 103
3.5.4
The EUM criterion
. 104
3.5.5
Linearity of the utility functional
. 105
3.5.6
An axiomatic approach
. 108
NON-LINEAR CRITERIA
.
Ill
4.1
Allais'
paradox
.
Ill
4.2
Weighted utility
. 112
4.3
Implicit or comparative utility
. 114
4.3.1
Definitions and examples
. 114
4.3.2
In what sense the implicit utility criterion is linear
. 117
4.4
Rank Dependent Expected Utility
. 118
4.4.1
Definitions and examples
. 118
4.4.2
Application to insurance
. 121
4.4.3
Further discussion and the main axiom
. 122
4.5
Remarks
. 124
OPTIMAL PAYMENT FROM THE STANDPOINT OF THE INSURED
. 125
5.1
Arrow's theorem
. 125
5.2
A generalization
. 128
EXERCISES
. 129
CHAPTER
2.
An individual Risk Model for a Short Period
137
1
THE DISTRIBUTION OF AN INDIVIDUAL PAYMENT
. 137
Ш
1.1
The distribution of the loss given that it has occurred
. 137
1.1.1
Definitions. Characterization of tails
. 137
1.1.2
Some particular light-tailed distributions
. 141
1.1.3
Some particular heavy-tailed distributions
. 142
1.1.4
The asymptotic behavior of tails and moments
. 145
pá
1.2
The distribution of the loss X
. 146
via
1.3
The distribution of the payment and types of insurance
. 147
2
THE AGGREGATE PAYMENT
. 155
2.1
Convolutions
. 155
2.1.1
Definitions and examples
. 155
2.1.2
Some classical examples
. 158
2.1.3
The analogue of the binomial formula for convolutions
. 162
2.2
Moment generating functions
. 163
3
NORMAL AND OTHER APPROXIMATIONS
. 165
3.1
Normal approximation
. 165
3.1.1
A heuristic approach
. 165
3.1.2
An important remark: the standard deviation principle
. 170
3.1.3
A rigorous estimation
. 171
vó
3.1.4
The number of contracts needed to maintain a given security
level
. 176
Ы
3.2
How to take into account the asymmetry of S. The
Г
-approximation
.
178
vS
3.3
Asymptotic expansions and Normal Power (NP) approximation
. 180
4
EXERCISES
. 182
CHAPTER
3.
Conditional Expectations
191
Ш
1
HOW TO COMPUTE CONDITIONAL EXPECTATIONS
.
THE CONDITIONING PROCEDURE
. 191
1.1
Conditional expectation given a r.v
. 191
1.1.1
The discrete case
. 191
1.1.2
The case of continuous distributions
. 193
1.2
Properties of conditional expectations
. 196
1.3
Conditioning and some useful formulas
. 198
1.3.1
A formula for variance
. 198
1.3.2
More detailed representations of the formula for total expecta¬
tion
. 198
1.4
Conditional expectation given
a r. vec
. 200
1.4.1
General definitions
. 200
1.4.2
On the case of an infinite-dimensional X
. 202
1.4.3
On conditioning in the multi-dimensional case
. 202
2
FORMULA FOR TOTAL EXPECTATION AND
CONDITIONAL EXPECTATION GIVEN A PARTITION
. 204
Ш
2.1
Conditional expectation given an event
. 204
2.2
The formula for total expectation
. 207
2.3
Expectation given a partition
. 208
3
CONDITIONAL EXPECTATIONS GIVEN RANDOM VARIABLES
OR VECTORS
. 209
3.1
The discrete case
. 209
3.2
The general case
. 211
4
ONE MORE IMPORTANT PROPERTY OF CONDITIONAL
EXPECTATIONS
. 213
4.1
Conditioning on partitions
. 213
4.2
Conditioning on r.v.'s or r.vec.'s
. 214
5
A GENERAL APPROACH TO CONDITIONAL EXPECTATIONS
. 215
5.1
Conditional expectation relative to a
σ
-algebra
.
215
5.2
Conditional expectation given a r.v. or
a r.vec
. 218
5.3
Properties of conditional expectations
. 219
6
SOME PROOFS
. 220
6.1
Proofs of the properties stated in Section
1.2. 220
6.2
Proof of Proposition
2. 222
7
EXERCISES
. 222
CHAPTER
4.
A Collective Risk Model for a Short Period
225
1
THREE BASIC PROPOSITIONS
. 225
2
COUNTING OR FREQUENCY DISTRIBUTIONS
. 227
2.1
The
Poisson
distribution and Poisson's theorem
. 227
2.1.1
A heuristic approximation
. 227
2.1.2
The accuracy of the
Poisson
approximation
. 231
2.2
Some other "counting" distributions
. 233
2.2.1
The mixed
Poisson
distribution
. 233
2.2.2
Compound mixing
. 237
2.2.3
The
(аДО)
and (a,b,\) (or Katz-Panjer's) classes
. 239
3
THE DISTRIBUTION OF THE AGGREGATE CLAIM
. 241
3.1
The case of a homogeneous group
. 241
3.1.1
The method of convolutions
. 241
3.1.2
The case when
N
has
a Poisson
distribution
. 245
3.1.3
The m.g.f.'s method
. 246
3.2
The case of several homogeneous groups
. 248
3.2.1
A general scheme and reduction to one group
. 248
3.2.2
The significance of the weights
w¡
. 251
4
NORMAL APPROXIMATION OF THE DISTRIBUTION OF
THE AGGREGATE CLAIM
. 253
4.1
A limit theorem
. 253
4.2
Estimation of premiums
. 257
4.3
The accuracy of normal approximation
. 259
4.4
Proof of Theorem
10. 260
5
EXERCISES
. 263
CHAPTERS. Random Processes. I. Counting and Compound Processes.
Markov Chains. Modeling Claim and Cash Flows
269
1
A GENERAL FRAMEWORK AND TYPICAL SITUATIONS
. 269
1.1
Preliminaries
. 269
1.2
Processes with independent increments
. 271
1.2.1
The simplest counting process
. 271
1.2.2
Brownian motion
. 271
1.3
Markov processes
. 274
2
POISSON
AND OTHER COUNTING PROCESSES
. 276
2.1
The homogeneous
Poisson
process
. 276
2.2
The non-homogeneous
Poisson
process
. 281
2.2.1
A model and examples
. 281
2.2.2
Proof of Proposition
1. 284
2.3
The Cox process
. 285
3
COMPOUND PROCESSES
. 287
4
MARKOV CHAINS. CASH FLOWS IN THE MARKOV ENVIRONMENT
289
4.1
Preliminaries
. 289
4.2
Variables defined on a Markov chain. Cash flows
. 295
4.2.1
Variables defined on states
. 295
4.2.2
Mean discounted payments
. 296
4.2.3
The case of absorbing states
. 298
4.2.4
Variables defined on transitions
. 301
4.2.5
What to do if the chain is not homogeneous
. 302
4.3
The first step analysis. An infinite horizon
. 302
4.3.1
Mean discounted payments in the case of infinite time horizon
303
4.3.2
The first step approach to random walk problems
. 305
4.4
Limiting probabilities and stationary distributions
. 310
4.5
The ergodicity property and classification of states
. 314
4.5.1
Classes of states
. 314
4.5.2
The recurrence property
. 315
4.5.3
Recurrence and travel times
. 318
4.5.4
Recurrence and ergodicity
. 320
5
EXERCISES
. 321
CHAPTER
6.
Random Processes. II. Brownian Motion and Martingales.
Hitting Times
329
1
BROWNIAN MOTION AND ITS GENERALIZATIONS
. 329
1.1
Further properties of the standard Brownian motion
. 329
1.1.1
Non-differentiability of trajectories
. 329
1.1.2
Brownian motion as an approximation. The
invariance
principle
330
1.1.3
The distribution
of.wř,
hitting times, and the maximum value of
Brownian motion
. 331
1.2
The Brownian motion with drift
. 333
1.2.1
Modeling of the surplus process. What a Brownian motion with
drift approximates in this case
. 334
1.2.2
A reduction to the standard Brownian motion
. 335
1.3
Geometric Brownian motion
. 337
2
MARTINGALES
. 338
2.1
General properties and examples
. 338
2.2
Martingale transform
. 343
2.3
Optional stopping time and some applications
. 344
2.3.1
Definitions and examples
. 344
2.3.2
Wald's identity
. 348
2.3.3
The ruin probability for the simple random walk
. 349
2.3.4
The ruin probability for the Brownian motion with drift
. 350
2.3.5
The distribution of the ruin time in the case of Brownian motion
352
1¿J
2.3.6
The hitting time for the Brownian motion with drift
. 353
2.4
Generalizations
. 354
2.4.1
The martingale property in the case of random stopping time
. 354
2.4.2
A reduction to the standard Brownian motion in the case of
random time
. 355
2.4.3
The distribution of the ruin time in the case of Brownian motion:
another approach
. 356
2.4.4
Proof of Theorem
11 . 357
2.4.5
Verification of Condition
3
of Theorem
5. 358
3
EXERCISES
. 359
CHAPTER?. Global Characteristics of the Surplus Process. Ruin Models.
Models with Paying Dividends
363
Ш
1
INTRODUCTION
. 363
2
RUIN MODELS
. 366
2.1
Adjustment coefficients and ruin probabilities
. 367
2.1.1
A preliminary condition: for large time horizons the aggregate
claim should be large
. 367
2.1.2
The main theorem
. 368
2.1.3
Proof of Lundberg's inequality
. 370
2.2
Computing adjustment coefficients
. 370
2.2.1
A general proposition
. 370
2.2.2
The discrete time case
. 374
2.2.3
The case of a homogeneous compound
Poisson
process
. 377
2.2.4
The discrete time case revisited
. 380
2.2.5
The case of the non-homogeneous compound
Poisson
process
381
2.3
Trade-off between the premium and the initial surplus
. 381
v$
2.4
Three cases when the ruin probability may be computed precisely
. . 385
vž3
2.4.1
The case when the size of a separate claim is exponentially
distributed
. 385
2.4.2
The case of the simple random walk
. 386
2.4.3
The case of Brownian motion
. 387
Ш
2.5
The martingale approach. A generalization of Theorem
1. 388
2.6
The renewal approach
. 390
2.6.1
The first surplus below the initial level
. 390
2.6.2
The renewal approximation
. 391
2.6.3
The
Cramér-Lundberg
approximation
. 395
ΙΣ)
2.6.4
Proof of Theorem
5
from Section
2.6.1. 395
2.7
Some recurrent relations and computational aspects
. 399
vM
3
CRITERIA CONNECTED WITH PAYING DIVIDENDS
. 402
Ш
3.1
A general model
. 403
3.2
The case of the simple random walk
. 405
3.3
Finding an optimal strategy
. 408
4
EXERCISES
. 409
CHAPTERS. Survival Distributions
413
W
1
THE DISTRIBUTION OF THE LIFETIME
. 413
1.1
Survival functions and force of mortality
. 413
1.2
The time-until-death for a person of a given age
. 418
1.3
Curtate-future-lifetime
. 422
1.4
Survivorship groups
. 423
1.5
Life tables and interpolation
. 424
1.5.1
Life tables
. 424
1.5.2
Interpolation for fractional ages
. 429
1.6
Some analytical laws of mortality
. 431
2
A MULTIPLE DECREMENT MODEL
. 434
Ш
2.1
A single life
. 434
2.2
Another view: net probabilities of decrement
. 438
2.3
A survivorship group
. 442
2.4
Proof of Proposition
1. 443
3
MULTIPLE LIFE MODELS
. 444
3.1
The joint distribution
. 445
3.2
The lifetime of statuses
. 447
3.3
A model of dependency: conditional independence
. 451
3.3.1
A definition and the first example
. 452
3.3.2
The common shock model
. 453
4
EXERCISES
. 455
CHAPTERS Life Insurance Models
461
Ш
1
A GENERAL MODEL
. 461
1.1
The present value of a future payment
. 461
1.2
The present value of payments to many clients
. 464
2
SOME PARTICULAR TYPES OF CONTRACTS
. 467
2.1
Whole life insurance
. 467
2.1.1
The continuous time case (benefits payable at the moment of
death)
. 467
2.1.2
The discrete time case (benefits payable at the end of the year
of death)
. 467
2.1.3
A relation between Ax and Ax
. 470
2.1.4
The case of benefits payable at the end of the m-thly period
. . 471
2.2
Deferred whole life insurance
. 473
2.2.1
The continuous time case
. 473
2.2.2
The discrete time case
. 474
2.3
Term insurance
. 474
2.3.1
Continuous time
. 474
2.3.2
Discrete time
. 476
2.4
Endowments
. 478
2.4.1
Pure endowment
. 478
2.4.2
Endowment
. 478
3
VARYING BENEFITS
. 480
3.1
Certain payments
. 480
3.2
Random payments
. 484
4
MULTIPLE DECREMENT AND MULTIPLE LIFE MODELS
. 485
4.1
Multiple decrements
. 485
4.2
Multiple life insurance
. 488
5
ON THE ACTUARIAL NOTATION
. 491
6
EXERCISES
. 492
CHAPTER
10.
Annuity Models
499
1
INTRODUCTION. TWO APPROACHES TO COMPUTING ANNUITIES
499
1.1
Continuous annuities
. 499
1.2
Discrete annuities
. 501
2
LEVEL ANNUITIES. A CONNECTION WITH INSURANCE
. 504
2.1
Certain annuities. Some notation
. 504
2.2
Random annuities
. 504
3
SOME PARTICULAR TYPES OF LEVEL ANNUITIES. EXAMPLES
. . 506
3.1
Whole life annuities
. 506
3.2
Temporary annuities
. 509
3.3
Deferred annuities
. 512
3.4
Certain and life annuity
. 514
4
MORE ON VARYING PAYMENTS
. 516
5
ANNUITIES WITH m-thly PAYMENTS
. 518
6
MULTIPLE DECREMENT AND MULTIPLE LIFE MODELS
. 521
6.1
Multiple decrements
. 521
6.2
Multiple life annuities
. 523
7
EXERCISES
. 525
CHAPTER
11.
Premiums and Reserves
531
1
SOME GENERAL PREMIUM PRINCIPLES
. 531
2
PREMIUM ANNUITIES
. 536
2.1
Preliminaries. General principles
. 536
2.2
Benefit premiums. The case of a single risk
. 537
2.2.1
Netratě
. 537
2.2.2
The case when
Ύ
is consistent with Z"
. 541
2.2.3
Variances
. 542
2.2.4
Premiums paid
m
times a year
. 544
2.2.5
Combinations of insurances
. 545
2.3
Accumulated values
. 546
2.4
Percentile premium
. 547
2.4.1
The case of a single risk
. 547
2.4.2
The case of many risks. Normal approximation
. 549
2.5
Exponential premiums
. 552
3
RESERVES
. 553
3.1
Definitions and preliminary remarks
. 553
3.2
Examples of direct calculations
. 554
3.3
Formulas for some standard types of insurance
. 556
3.4
Recursive relations
. 557
4
EXERCISES
. 560
CHAPTER
12.
Risk Exchange: Reinsurance and Coinsurance
565
1
REINSURANCE FROM THE STAND POINT OF
A CEDENT
. 565
1.1
Some optimization considerations
. 565
1.1.1
Expected utility maximization
. 566
1.1.2
Variance as a measure of risk
. 568
1.2
Proportional reinsurance. Adding a new contract to an existing portfolio
570
1.2.1
The case of a fixed security loading coefficient
. 570
1.2.2
The case of the standard deviation premium principle
. 573
1.3
Long-term insurance. Ruin probability as a criterion
. 575
1.3.1
An example with proportional reinsurance
. 575
1.3.2
An example with excess-of-loss insurance
. 577
2
RISK EXCHANGE AND RECIPROCITY OF COMPANIES
. 578
2.1
A general framework and some examples
. 578
2.2
Two more examples with expected utility maximization
. 586
2.3
The case of the mean-variance criterion
. 590
2.3.1
Minimization of variances
. 590
2.3.2
The exchange of portfolios
. 593
3
REINSURANCE MARKET
. 598
3.1
A model of the exchange market of random assets
. 598
3.2
An example concerning reinsurance
. 601
4
EXERCISES
. 604
Tables
607
References
611
Answers to Exercises
619
Subject Index
627
Ideal for students preparing for level
300
actuarial exams in the US. Actuarial
Modeîs:
The Mathematics of insurance provides a comprehensive exposition of insurance processes
models and presents mathematical setups and methods used in Actuarial Modeling.
Divided into three self-contained and explicitly designated parts of different levels of difficulty,
this book examines standard as well as advanced topics such as modern utility theory, martingale
technique, models with payments of dividends, reinsurance models, and classification of
distributions. It provides practical skills in analysis of insurance processes. This text discusses
a number of topics not commonly found in existing Actuarial Mathematics textbooks, including
achievements of the modern Risk Evaluation theory, premium principles, accuracy of normal
and
Poisson
approximation, and a reinsurance market model.
The book includes numerous examples, practice problems, and exercises on numerical
calculations using Excel"". It includes also preliminary examination material for the Society of
Actuaries and the Casualty Actuarial Society (CAS), providing, in particular, real problems from
past CAS exams.
Features
Provides mathematical models for both non-life and life insurance
Uses real problems from past CAS exams and prepares students for level
300
actuarial exams in the US
Features a systematic presentation from a mathematical point of view/
Presents supplementary material containing main facts from
Probability
Theory,
Stochastic Processes, the Theory of Interest, and Calculus
Contains numerous examples, which may be viewed as a solution guide for
corresponding exams
Discusses many analytical procedures and contains many computational example
for the most part, with use of Excel |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Rotar', Vladimir I. |
| author_facet | Rotar', Vladimir I. |
| author_role | aut |
| author_sort | Rotar', Vladimir I. |
| author_variant | v i r vi vir |
| building | Verbundindex |
| bvnumber | BV022463927 |
| callnumber-first | H - Social Science |
| callnumber-label | HG8781 |
| callnumber-raw | HG8781 |
| callnumber-search | HG8781 |
| callnumber-sort | HG 48781 |
| callnumber-subject | HG - Finance |
| classification_rvk | QQ 600 QQ 630 |
| ctrlnum | (OCoLC)494346427 (DE-599)BVBBV022463927 |
| dewey-full | 368.01 368/.01 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 368 - Insurance |
| dewey-raw | 368.01 368/.01 |
| dewey-search | 368.01 368/.01 |
| dewey-sort | 3368.01 |
| dewey-tens | 360 - Social problems and services; associations |
| discipline | Wirtschaftswissenschaften |
| discipline_str_mv | Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV022463927 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T17:41:33Z |
| indexdate | 2024-07-09T20:58:09Z |
| institution | BVB |
| isbn | 9781584885863 1584885866 |
| language | English |
| lccn | 2006045558 |
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| publisher | Chapman & Hall |
| record_format | marc |
| spelling | Rotar', Vladimir I. Verfasser aut Actuarial models the mathematics of insurance Vladimir I. Rotar Boca Raton, FL Chapman & Hall 2007 XXII, 633 S. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Assurance - Mathématiques ram Mathematik Insurance Mathematics Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Versicherungsmathematik (DE-588)4063194-1 s 1\p DE-604 http://www.loc.gov/catdir/enhancements/fy0664/2006045558-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=015671544&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015671544&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 |
| spellingShingle | Rotar', Vladimir I. Actuarial models the mathematics of insurance Assurance - Mathématiques ram Mathematik Insurance Mathematics Versicherungsmathematik (DE-588)4063194-1 gnd |
| subject_GND | (DE-588)4063194-1 (DE-588)4123623-3 |
| title | Actuarial models the mathematics of insurance |
| title_auth | Actuarial models the mathematics of insurance |
| title_exact_search | Actuarial models the mathematics of insurance |
| title_exact_search_txtP | Actuarial models the mathematics of insurance |
| title_full | Actuarial models the mathematics of insurance Vladimir I. Rotar |
| title_fullStr | Actuarial models the mathematics of insurance Vladimir I. Rotar |
| title_full_unstemmed | Actuarial models the mathematics of insurance Vladimir I. Rotar |
| title_short | Actuarial models |
| title_sort | actuarial models the mathematics of insurance |
| title_sub | the mathematics of insurance |
| topic | Assurance - Mathématiques ram Mathematik Insurance Mathematics Versicherungsmathematik (DE-588)4063194-1 gnd |
| topic_facet | Assurance - Mathématiques Mathematik Insurance Mathematics Versicherungsmathematik Lehrbuch |
| url | http://www.loc.gov/catdir/enhancements/fy0664/2006045558-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015671544&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=015671544&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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