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# Actuarial Models: The Mathematics of Insurance

We do not plan to review this book.

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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