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Actuarial Models: The Mathematics of Insurance Book

Actuarial Models: The Mathematics of Insurance
Actuarial Models: The Mathematics of Insurance, Ideal for students preparing for level 300 actuarial exams in the US, Actuarial Models: The Mathematics of Insurance provides a comprehensive exposition of insurance process models and presents mathematical setups and methods used in Actuarial Modeling.
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Actuarial Models: The Mathematics of Insurance, Ideal for students preparing for level 300 actuarial exams in the US, Actuarial Models: The Mathematics of Insurance provides a comprehensive exposition of insurance process models and presents mathematical setups and methods used in Actuarial Modeling. <, Actuarial Models: The Mathematics of Insurance
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  • Actuarial Models: The Mathematics of Insurance
  • Written by author Vladimir I. Rotar
  • Published by Taylor & Francis, Inc., August 2006
  • Ideal for students preparing for level 300 actuarial exams in the US, Actuarial Models: The Mathematics of Insurance provides a comprehensive exposition of insurance process models and presents mathematical setups and methods used in Actuarial Modeling.
  • Ideal for students preparing for level 300 actuarial exams in the US, Actuarial Models: The Mathematics of Insurance provides a comprehensive exposition of insurance process models and presents mathematical setups and methods used in Actuarial Modelin
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Preface Acknowledgments Introduction SOME PRELIMINARY NOTATIONS AND FACTS FROM PROBABILITY THEORY, THE THEORY OF INTEREST, AND CALCULUS

Probability and Random Variables Sample space, events, probability measure Independence and conditional probabilities Random variables, random vectors, and their distributions

Expectation Definitions Integration by parts and a formula for expectation A general definition of expectation Can we encounter an infinite expected value in models of real phenomena?
Moments of r.v.'s. Correlation Inequalities for deviations Linear transformations of r.v.'s. Normalization

Some Basic Distributions Discrete distributions Continuous distributions

Moment Generating Functions Laplace transform An example when a m.g.f. does not exist The m.g.f.'s of basic distributions The moment generating function and moments Expansions for m.g.f.'s

Convergence of Random Variables and Distributions

Some Facts and Formulas from the Theory of Interest Compound interest Nominal rate Discount and annuities Accumulated value Effective and nominal discount rates

Appendix. Some Notations and Facts from Calculus The "small o and big O" notation Taylor expansions Concavity

COMPARISON OF RANDOM VARIABLES. PREFERENCES OF INDIVIDUALS

Comparison of Random Variables. Some Particular Criteria Preference order Several simple criteria On coherent measures of risk

Comparison of R.V.'S and Limit Theorems of Probability Theory A diversion to Probability Theory: two limit theorems A simple model of insurance with many clients St. Petersburg's paradox

Expected Utility Expected utility maximization (EUM)
Utility and insurance How we may determine the utility function in particular cases Risk aversion A new view: EUM as a linear criterion

Non-Linear Criteria Allais' paradox Weighted utility Implicit or comparative utility Rank Dependent Expected Utility Remarks

Optimal Payment from the Standpoint of the Insured Arrow's theorem A generalization

Exercises

AN INDIVIDUAL RISK MODEL FOR A SHORT PERIOD The Distribution of an Individual Payment The distribution of the loss given that it has occurred The distribution of the loss X The distribution of the payment and types of insurance

The Aggregate Payment Convolutions Moment generating functions

Normal and Other Approximations Normal approximation How to take into account the asymmetry of S. The G-approximation Asymptotic expansions and Normal Power (NP) approximation

Exercises

CONDITIONAL EXPECTATIONS

How to Compute Conditional Expectations. The Conditioning Procedure Conditional expectation given a r.v Properties of conditional expectations Conditioning and some useful formulas Conditional expectation given a r.vec.

Formula for Total Expectation and Conditional Expectation Given a Partition Conditional expectation given an event The formula for total expectation Expectation given a partition

Conditional Expectations Given Random Variables or Vectors The discrete case The general case

One More Important Property of Conditional Expectations Conditioning on partitions Conditioning on r.v.'s or r.vec.'s

A General Approach to Conditional Expectations Conditional expectation relative to a s-algebra Conditional expectation given a r.v. or a r.vec Properties of conditional expectations

Some Proofs

Exercises

A COLLECTIVE RISK MODEL FOR A SHORT PERIOD Three Basic Propositions

Counting or Frequency Distributions The Poisson distribution and Poisson's theorem Some other "counting" distributions

The Distribution of the Aggregate Claim The case of a homogeneous group The case of several homogeneous groups

Normal Approximation of the Distribution of the Aggregate Claim A limit theorem Estimation of premiums The accuracy of normal approximation Proof of Theorem10

Exercises

RANDOM PROCESSES. I. COUNTING AND COMPOUND PROCESSES. MARKOV CHAINS. MODELING CLAIM AND CASH FLOWS A General Framework and Typical Situations Preliminaries Processes with independent increments Markov processes

Poisson and Other Counting Processes The homogeneous Poisson process The non-homogeneous Poisson process The Cox process

Compound Processes

Markov Chains. Cash Flows in the Markov Environment Preliminaries Variables defined on a Markov chain. Cash flows The first step analysis. An infinite horizon Limiting probabilities and stationary distributions The ergodicity property and classification of states

Exercises

RANDOM PROCESSES. II. BROWNIAN MOTION AND MARTINGALES. HITTING TIMES Brownian Motions and its Generalization Further properties of the standard Brownian motion The Brownian motion with drift Geometric Brownian motion

Martingales General properties and examples Martingale transform Optional stopping time and some applications Generalizations

Exercises

GLOBAL CHARACTERISTICS OF THE SURPLUS PROCESS. RUIN MODELS. MODELS WITH PAYING DIVIDENDS.
Introduction

Ruin Models Adjustment coefficients and ruin probabilities Computing adjustment coefficients Trade-off between the premium and the initial surplus

Three cases when the ruin probability may be computed precisely The martingale approach.
The renewal approach Some recurrent relations and computational aspects

Criteria Connected with Paying Dividends

A general model The case of the simple random walk Finding an optimal strategy

Exercises

SURVIVAL DISTRIBUTIONS The Distribution of the Lifetime Survival functions and force of mortality The time-until-death for a person of a given age Curtate-future-lifetime Survivorship groups Life tables and interpolation Some analytical laws of mortality

A Multiple Decrement Model A single life Another view: net probabilities of decrement A survivorship group Proof of Proposition 1 .

Multiple Life Models The joint distribution The lifetime of statuses A model of dependency: conditional independence

Exercises

LIFE INSURANCE MODELS A General Model The present value of a future payment The present value of payments to many clients

Some Particular Types of Contracts Whole life insurance Deferred whole life insurance Term insurance Endowments

Varying Benefits Certain payments Random payments

Multiple Decrement and Multiple Life Models Multiple decrements Multiple life insurance

On the Actuarial Notation

Exercises

ANNUITY MODELS Introduction. Two Approaches to Computing Annuities Continuous annuities Discrete annuities

Level Annuities. A Connection with Insurance Certain annuities. Some notation Random annuities

Some Particular Types of Level Annuities. Examples Whole life annuities Temporary annuities Deferred annuities Certain and life annuity

More on Varying Payment

Annuities with m-thly Payments

Multiple Decrements and Multiple Life Models Multiple decrement Multiple life annuities

Exercises

PREMIUMS AND RESERVES Some General Premium Principles

Premium Annuities Preliminaries. General principles Benefit premiums. The case of a single risk Accumulated values Percentile premium Exponential premiums

Reserves Definitions and preliminary remarks Examples of direct calculations Formulas for some standard types of insurance Recursive relations

Exercises

RISK EXCHANGES: REINSURANCE AND COINSURANCE Reinsurance from the Standpoint of a Cedent Some optimization considerations Proportional reinsurance. Adding a new contract to an existing portfolio Long-term insurance. Ruin probability as a criterion

Risk Exchange and Reciprocity of Companies A general framework and some examples Two more examples with expected utility maximization The case of the mean-variance criterion

Reinsurance Market A model of the exchange market of random assets An example concerning reinsurance

Exercises Tables References Answers to Exercises Subject Index


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Actuarial Models: The Mathematics of Insurance, Ideal for students preparing for level 300 actuarial exams in the US, Actuarial Models: The Mathematics of Insurance provides a comprehensive exposition of insurance process models and presents mathematical setups and methods used in Actuarial Modeling.
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Actuarial Models: The Mathematics of Insurance, Ideal for students preparing for level 300 actuarial exams in the US, Actuarial Models: The Mathematics of Insurance provides a comprehensive exposition of insurance process models and presents mathematical setups and methods used in Actuarial Modeling.
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