bluuebunny/crossref_metadata_2025 huggingface dataset:
Frontier between Public and Private Insurance Schemes
Authors: Zweifel Peter
Published: September 2006

From an economic point of view, insurance coverage is very much like any other goods or service purchased by households and firms. One would therefore expect individually contracted insurance coverage provided by private firms to dominate. However, mandated coverage provided by public insurance is common for the provision for old age, disability, workers' compensation, health, long‐term care and unemployment. Contributions to public insurance amount to about 30% of the Gross Domestic Product (GDP) and are still rising in major industrialized countries whereas the premium revenue of personal lines of private insurance make up as little as 5% of GDP even in the United Kingdom, a stronghold of private insurance. This raises the question, what determines the frontier between public and private insurance? and, in a normative vein, would a shifting of this frontier be in the interest of the insured?
Fuzzy Set Theory
Authors: Derrig Richard A. , Ostaszewski Krzysztof M.
Published: September 2006

Traditional actuarial methodologies have been built upon probabilistic models. They are often driven by stringent regulation of the insurance business. Deregulation and global competition of the last two decades have opened the door for new methodologies, among them, being fuzzy methods. The defining property of fuzzy sets is that their elements possibly have partial membership (between 0 and 1), with degree of membership of zero indicating not belonging to the set considered, and degree of membership of one indicating certainty of membership.

This article provides a quick overview of developing fuzzy sets methodologies in actuarial science, and the basic mathematics of fuzzy sets.
Gamma Function
Authors: Drekic Steve
Published: September 2006

This article summarizes some of the key mathematical/computational properties associated with the gamma function and incomplete gamma function that find use in applications to probability theory and parametric claims modeling.
Gaussian Processes
Authors: Débicki Krzysztof
Published: September 2006

The class ofGaussian stochastic processesplays an important role in stochastic modeling and is widely used as a rich source of models in many areas of applied probability, including actuarial and financial mathematics.

There are many practical and theoretical reasons for using Gaussian processes to model actuarial problems. For instance, the family of Gaussian processes covers a large class of correlation structures and enables explicit analysis of models for which classical renewal‐type tools do not work.

On the other hand, theoretical results, mostly based on central limit theorem‐type argumentation, formally justify the use of Gaussian processes.
Graduation
Authors: Miller Robert B.
Published: September 2006

In actuarial science, graduation refers to the adjustment of a set of estimated quantities to make the adjusted quantities conform to a pattern suitable for insurance purposes. We distinguish between two broad categories of approaches to actuarial graduation: those that involve fitting curves defined by functions and those that involve other methods. Curve‐fitting approaches use popular functional forms such as Gompertz, Weibull, Makeham, and splines. Non–curve‐fitting approaches in this article are grouped under the headings Weighted Moving Average Methods, Kernel Methods, Whittaker Methods, and Bayesian Methods.
Graunt,John (1620–1674)
Authors: Teugels Jozef L.
Published: September 2006

John Graunt can be considered the first scholar who applied statistical thinking to demographic questions. After some information on his life, we deal with his major contribution and its influence on further developments in the field.
Heckman–Meyers Algorithm
Authors: Klugman Stuart A.
Published: September 2006

The collective risk model for aggregate losses states that total losses for a period comprise a random number of random payments where the payments have identical distributions and all variables are independent. Evaluating probabilities for this model can be difficult. The Heckman–Meyers Algorithm is one of several approaches to this problem. While computationally challenging, it is accurate and in many situations is extremely fast.
HiddenMarkov Models
Authors: Rydén Tobias
Published: September 2006

The outcomes of a stochastic process are often influenced by another, not fully observable process. For example, a financial time series depends on the hidden state of the market. In most cases, this hidden process is taken to be a Markov chain. We investigate the probabilistic modelling of such processes and the resulting statistical procedures.
History of Actuarial Science
Authors: Hickman James
Published: September 2006

Actuarial science is the constantly evolving set of ideas that is the intellectual foundation of actuarial practice. Actuarial practice, in turn, is devoted to designing and managing financial security systems. The ideas in this set have roots in many basic disciplines, including the mathematical sciences, economics, demography, and business management.
H
uygens,
C
hristiaan and
L
odewijck (1629–1695)
Authors: Teugels Jozef L.
Published: September 2006

Christiaan and Lodewijck Huygens played a role in the early development of probability theory and its application to insurance topics.
Individual Risk Model
Authors: Dhaene Jan , Vyncke David
Published: September 2006

In the individual risk model, the total claims amount on a portfolio of insurance contracts is the random variable of interest. The total claims is modeled as the sum of all claims on the individual policies, which are assumed independent. We present several techniques, such as convolution and recursions, to obtain results in this model.
Inflation Impact on Aggregate Claims
Authors: Garrido José , Léveillé Ghislain
Published: September 2006

Inflation can have a significant impact on insured losses. Insurance companies recognize this effect on their aggregate claims when setting premiums and reserves.

Interest earned by insurance companies on their reserves can, to some extent, offset the effect of inflation on claims. This article looks at the combined effect of these economical variables, inflation and interest, on the distribution of aggregate claims.
Information Criteria
Authors: Claeskens Gerda
Published: September 2006

The selection of an adequate model is central to any data analysis. Information criteria provide formalized ways of model selection.
Insurance Regulation and Supervision
Authors: Tess Daniel
Published: September 2006

Aside from normal regulatory activities, companies and other entities that sell or provide insurance products are regulated in two special ways. Solvency regulation concerns insurers' financial ability to pay future claims against policies sold, and price regulation concerns the level of premiums that are charged to policyholders. Actuaries generally play a key active role in both of these regulatory systems.
I
tô Calculus
Authors: Steele J. Michael
Published: September 2006

This entry for the
Encyclopedia of Actuarial Sciences
provides an introduction to the Itô calculus that emphasizes the definition of the Itô integral and the description of Itô's formula, the most widely used result in the Itô calculus.
Life Insurance
Authors: Macdonald Angus S.
Published: September 2006

This article describes the main features and types of life insurance, meaning insurance against the event of death. Life insurers also transact annuity business.
Life Insurance Mathematics
Authors: Norberg Ragnar
Published: September 2006

The subject matter and methodology of modern life insurance mathematics are surveyed. Standard insurance products with payments depending only on life history events are described and analyzed in the commonly used Markov chain model under the assumption of deterministic interest rates. The actuarial equivalence principle for determining premiums and reserves is motivated by risk diversification in large portfolios. Differential equations for reserves and other conditional expected values are obtained and proclaimed basic constructive and computational tools. The traditional method for managing nondiversifiable risk arising from changes in, for example, mortality rates and interest rates is outlined; premiums are calculated under prudent assumptions and systematic surpluses thus created are repaid as bonus. An alternative way of managing nondiversifiable risk is to link the contractual payments to the development of mortality, interest, and possibly other demographic or economic indexes. Examples of so‐called index‐linked products are given, and premiums and reserves are determined by combined use of classical actuarial principles and principles of pricing and hedging from financial mathematics. Transfer of nondiversifiable risk to the financial markets through creation of tradable insurance derivatives is outlined as an idea. Methods for estimation of mortality and other vital rates, formerly a major issue in life insurance mathematics, are fetched from modern statistical life history analysis and therefore only briefly described.
Long Range Dependence
Authors: Samorodnitsky Gennady
Published: September 2006

Long‐range dependence (synonymous with
long memory
) is a property of some stationary processes. Intuitively, the notion is clear: a stationary stochastic process that has long‐range dependence is a process whose memory lasts very long. However, the term ‘long‐range dependence’ means more than simply ‘longer memory’. It means memory so long that the behavior of a stochastic process changes dramatically, and so long that the usual statistical tools break down when applied to observations arising from a long‐range dependent process.
Long‐Term Care Insurance
Authors: Jones Bruce L.
Published: September 2006

It is well known that older adults become increasingly frail as they age. Even in the absence of a disease, one eventually loses the physical and/or mental ability to live independently. The kind of assistance required when this occurs is referred to as long‐term care (LTC). LTC is very different from the health care needed by individuals with injury or disease, and it requires a different set of services. Often, long‐term care services can be provided in a person's home, and this is the ideal situation. However, many cases require that the individual be institutionalized in an appropriate LTC facility.
Loss‐of‐Profits Insurance
Authors: Fry Martin
Published: September 2006

Loss‐of‐Profits insurance is intended to cover losses to a business following physical damage by fire or breakdown, until turnover has recovered to a similar level to that prior to the damage. It requires that a fire or machinery breakdown policy has been effected by the insured. Cover is for a maximum sum insured representing the insured's gross profit and the increased cost of working for the anticipated period that production is interrupted following the physical damage, and for a maximum indemnity period representing the anticipated period to restore the business to normal operations and profitability.
Lotteries
Authors: Pflug Georg
Published: September 2006

Lottery companies sell risk to customers. They are natural counterparts to insurance companies who buy risk from customers. Insurance contracts contain a positive risk premium, that is, the asked premium exceeds the expected losses to cover the administrative costs and to allow profit and a safety margin for the insurance company. Lottery contracts contain a negative risk premium, that is, the price of the lottery ticket exceeds the expected win, again to allow a profit for the lottery company or – in state‐owned lotteries – for the national finances.

It seems quite paradoxical that a person may hold some insurance contracts (thus seems to be risk‐averse) and at the same time buy lottery tickets (thus seem to be risk‐seeking). A preliminary explanation can be found in the nature of the two risks: While insurance protects against unwanted risks and risks that cannot be controlled (accident, fire, etc.), lotteries offer new, independent, maybe even thrilling risks.
Lundberg Approximations, Generalized
Authors: Lin X. Sheldon
Published: September 2006

In this article, we briefly discuss generalizations to compound distributions of the classical Cramér–Lundberg approximation and bounds for the ruin probability.
Marine Insurance
Authors: Seltmann Astrid
Published: September 2006

Shipping is a business with long‐standing traditions as well as great risks at stake. But although marine insurance is the oldest form of insurance, it seems historically to have been relatively little touched by actuaries. This article will give an overview of marine insurance and its characteristics, the markets, and actuarial aspects. It will also suggest some reasons for the historically low actuarial involvement as well as point out the increasingly interesting potential, not the least due to recent developments in shipping and the general trend towards more sophisticated technical risk analysis and loss prevention measures in the industry in general.
Market Models
Authors: Pelsser Antoon
Published: September 2006

Interest rate modelling is one of the crucial ingredients in the calculation of the market value of insurance products. The standard case is the LIBOR Market Model where all interest rates have log‐normal distributions. We deal with LIBOR rates and options on LIBOR rates (caplets and floorlets) first, then with swap rates and options on swap rates (swaptions).
Markov ChainMonteCarlo Methods
Authors: Roberts Gareth O. , Rosenthal Jeffrey S.
Published: September 2006

One of the simplest and most powerful practical uses of the ergodic theory of Markov chains is in Markov chain Monte Carlo (MCMC). Suppose we wish to simulate from a probability density π (which will be called thetargetdensity) but that direct simulation is either impossible or practically infeasible (possibly due to the high dimensionality of π). This generic problem occurs in diverse scientific applications, for instance, Statistics, Computer Science, and Statistical Physics.

Markov chain Monte Carlo offers an indirect solution based on the observation that it is much easier to construct an ergodic Markov chain with π as a stationary probability measure, than to simulate directly from π. This is because of the ingeniousMetropolis–Hastingsalgorithm, which takes an arbitrary Markov chain and adjusts it using a simple accept–reject mechanism to ensure the stationarity of π for the resulting process.
Martingales
Authors: Schmidli Hanspeter
Published: September 2006

An important concept in probability are martingales, stochastic processes that can be considered as fair games. Many results on stochastic processes can be proved via the optional stopping theorem and the martingale convergence theorem. We define in this article the martingale, state the important results, and illustrate applications of the theory by way of examples.
Maximum Likelihood
Authors: Duan Jin‐Chuan , Fulop Andras
Published: September 2006

We describe the maximum likelihood (ML) estimation and inference. Main theoretical results under the i.i.d. assumption are presented along with discussions. To demonstrate the ML inference, an example dealing with large medical claims and their determining characteristics is presented. We then turn to the ML estimation for stationary dynamic models (i.e. dependent data points) and introduce the quasi‐ML principle as a way of dealing with potential distributional misspecification. The US interest rate series is used to illustrate the quasi‐ML inference for dependent data.
Mixtures of Exponential Distributions
Authors: Cai Jun
Published: September 2006

This articles discusses the properties mixed exponential distributions and consider their examples and variations appearing often in insurance.
Moral Hazard
Authors: Sonnenholzner Michael , Wambach Achim
Published: September 2006

In an insurance context, moral hazard comprises the phenomenon that having insurance gives the insured an incentive to alter his behavior to the detriment of the insurer. This manifests itself by the insuree employing suboptimal precautionary effort to avoid the loss. We discuss the insurance‐specific moral hazard model and several extensions with regard to many effort levels, continuous loss distribution, multiperiod contracts, and renegotiation.
Mortality Laws
Authors: Forfar David O.
Published: September 2006

Since the time of De Moivre in 1725, various mathematical formulae have been suggested to represent the pattern of mortality rates with age. Some of the formulae have applied at adult ages, some at very young ages, and some over the whole span of life. Certain formulae have been used in the graduation of mortality data. The article gives a survey of the mathematical laws used and examines how mortality might look in the foreseeable future if it goes on improving.
Screening Methods
Authors: Malutov Michail , Wynn Henry P. , Zhigljavsky Anatoly
Published: September 2006

Screening is a term applied to the general approach of detecting active or significant factors, elements, units, and so on when there are a large number of inactive factors. The meaning of ‘active’ depends on the context: effective treatments, presence of a disease, fault, etc. The methods are of considerable importance in medical and industrial applications. The theoretical aspects, particularly finding optimal or efficient experimental designs are closely related to search methods and special combinatorial constructions.
Neural Networks
Authors: Kramer Bert
Published: September 2006

Neural networks, a machine learning technique, are especially good at pattern recognition tasks. They have been successfully applied to a large number of classification and prediction tasks, like insurer insolvency prediction.

Neural networks consist of a (large) number of relatively simple processing units, which are usually arranged in layers. The output of a neural network is determined by the strengths (the weights) of the connections between the units and by the activation function, which transforms the input of a specific unit into the output from this unit. Two types of networks can be distinguished: feedforward networks and recurrent networks. Feedforward neural networks are a subset of the class of nonlinear regression and discrimination models.

In order to obtain the desired characteristics, a neural network has to be trained. Training algorithms can be categorized as supervised or unsupervised learning. The most common supervised training method for feedforward neural networks is called
back‐propagation
. Unsupervised learning is most commonly used in the field of cluster analysis. A common unsupervised training method is Kohonen's self‐organizing map.
Numerical Algorithms
Authors: Van Horebeek Johan
Published: September 2006

We give an overview of several computer intensive numerical algorithms that have special relevance in applications with an intrinsic stochastic component.
Derivative Pricing, Numerical Methods
Authors: Vetzal K.R.
Published: September 2006

Numerical methods are needed for derivatives pricing in cases where analytic solutions are either unavailable or not easily computable.

The subject of numerical methods in the area of derivatives valuation and hedging is very broad. A wide range of different types of contracts are available, and in many cases there are several candidate models for the stochastic evolution of the underlying state variables. Many subtle numerical issues can arise in various contexts. A complete description of these would be very lengthy, so in this article we will only give a sample of the issues involved.

The article first presents a detailed description of the different methods available in the context of the Black – Scholes – Merton model for simple European and American‐style equity options. It then describes how the methods can be adapted to more general contexts, such as derivatives dependent on more than one underlying factor, path‐dependent derivatives, and derivatives with discontinuous payoff functions.
Disability Insurance, Numerical Methods
Authors: Pitacco Ermanno
Published: September 2006

This article describes calculation methods used in actuarial practice for pricing disability annuity products. The following categories are dealt with: inception‐annuity models, models based on the probability of being disabled, models based on the expected time spent in disability, implementations of the multistate approach.
Oligopoly in Insurance Markets
Authors: Sonnenholzner Michael , Wambach Achim
Published: September 2006

Although empirical evidence on oligopolistic behavior in some sectors of the insurance industry exists, relatively few theoretical models have been developed. We discuss the cause for this discrepancy, elaborate on the classical oligopoly models and their applicability to the insurance market, and introduce the few models that are specific to the insurance industry. The underlying mechanism in these models is that undercutting a rival makes the pool of risks unattractive and thus, oligopolistic market power can be sustained.
Operational Time
Authors: Albrecher Hansjörg
Published: September 2006

We give a brief discussion of the concept of operational time and some of its applications in actuarial science.
Options and Guarantees in Life Insurance
Authors: Hardy Mary R.
Published: September 2006

In the context of life insurance, an option is a guarantee embedded in an insurance contract. There are two major types – the financial option, which guarantees a minimum payment in a policy with a variable benefit amount, and the mortality option, which guarantees that the policyholder will be able to buy further insurance cover on specified terms.
Ordering of Risks
Authors: Kaas Rob
Published: September 2006

The theory of Ordering of Risks has many actuarial applications, and is in fact one of the most important items in the actuarial toolkit. The most important concept is
stop‐loss order
, which compares risks as to their variability.
Ornstein–Uhlenbeck Process
Authors: Sørensen Michael
Published: September 2006

The Ornstein–Uhlenbeck process solves a linear stochastic differential equation driven by a Brownian motion and is therefore analytically tractable. Transition densities, asymptotics, and likelihood inference can be treated explicitly. Also the more general Ornstein–Uhlenbeck type processes that solve linear stochastic differential equations driven by Lévy processes are presented.
Outlier Detection
Authors: Gather Ursula , Pawlitschko Jörg
Published: September 2006

The analysis of all kinds of statistical data usually includes the detection of outliers as a first exploratory step. The reason for this is twofold: on the one hand we want to screen for aberrant values firstly and on the other hand we are interested in identifying data points that are of special interest. This paper discusses methods of detecting outliers in statistical data for a couple of often‐arising data structures. It also aims at clarifying the notion of an outlier, which often is only vaguely described in the literature.
P&I Clubs
Authors: Fry Martin
Published: September 2006

P&I (short for Protection and Indemnity) clubs provide a mutual not‐for‐profit indemnity pool (
see
Pooling in Insurance, Pooling of Employee Benefits) for owners or operators of ships. P&I insurance primarily covers owners or operators of ships for liability to third parties, although the clubs have responded to members' needs and to legislative changes with a range of other policies. The clubs operate by ‘calls’ on members. These calls are an allocation of anticipated claims costs, with adjustments made either by further calls or by refunds.
P
areto Optimality
Authors: Dana Rose‐Anne
Published: September 2006

Any sequence of positive, independent and identically distributed random variables can be considered as a renewal process. Traditional examples are the interclaim times and the successive claims in a homogeneous portfolio. The article describes the most important quantities and results from renewal theory. Also defective, delayed and discrete‐type renewal processes are discussed and illustrated with examples from actuarial science.
P
areto Rating
Authors: Mata Ana J.
Published: September 2006

The Pareto family of distributions is widely used in insurance and reinsurance practice to model the individual claim cost in an insurance portfolio. One of the practical advantages of the Pareto‐type distributions is that it extrapolates better beyond the observed data, and therefore it is more appropriate when rating high excess layer.
Phase Method
Authors: Bladt Mogens
Published: September 2006

The method of stages (due to Erlang) refers to the assumption of – and techniques related to – a positive random variable
X
, which can be decomposed into (a possibly random number of) stages each of which having an exponentially distributed duration.
Point Processes
Authors: Rolski Tomasz
Published: September 2006

The theory of point processes originates from problems of statistical demography, insurance, and mortality tables, although the connection may seem remote at first sight. More recent are problems from risk theory, in which one encounters the problem of how to describe a stream of arriving claims in a mathematically formal way.

The simplest case of a point process is the Poisson process. In general, we understand a point process to be a method of randomly allocating points to intervals or rectangles, etc. Formalizing this into the language of mathematics, a (stochastic) point process is a random denumerable collection of points in a state space 𝔼 with some small condition not allowing clustering. It turns out that it is sometimes mathematically convenient to define point processes as random variables assuming values at a space of measures.
Pooling Equilibria
Authors: Boyer Marcel , Boyer M. Martin
Published: September 2006

We present in this paper, different contexts under which pooling equilibria emerge in insurance economics. A pooling equilibrium may emerge when it is not optimal (or not feasible or not allowed) to offer different contracts to different agents. As a result, agents with different risk types are ‘pooled’ together even though the low‐risk agents end up subsidizing the high‐risk agents. This subsidy, quite like an externality imposed by the high‐risk types on the low‐risk ones, may lead low‐risk insureds to try to identify themselves and insurers to search for different ways to bypass the pooling equilibrium by identifying and luring away good risk types, thereby destabilizing the pooling equilibrium.
Prediction
Authors: Schmidt Klaus D.
Published: September 2006

Rate making and loss reserving both require the prediction of future payments for claims. Whenever this is possible, the prediction of future payments is based on claims experience in the past and on assumptions describing the way in which future payments depend on those in the past.

Mathematically, it is convenient to describe all payments by random variables and to model dependencies between these random variables by an assumption on their joint probability distribution. The random variables describing the past are assumed to be observable and form the statistical basis for predicting the random variable describing the future.

The statistical basis can be used in various ways to construct predictors. Therefore, there is a need of criteria for the selection of the predictor to be used. The main criterion is the comparison of predictors by a suitable measure of quality; such a measure is not given by nature but has to be chosen by the actuary. In addition, a preselection can be made by the requirement of certain properties which the predictor to be selected should possess; one such property is unbiasedness which can be understood as a mathematical formulation of the actuarial principle of equivalence.
Present Values and Accumulations
Authors: Macdonald Angus S.
Published: September 2006

Money has a
time value
; if we invest $1 today, we expect to get back more than $1 at some future time as a reward for lending our money to someone else who will use it productively. Suppose that we invest $1, and a year later we get back $(1 +
i
). The amount invested is called the
principal
, and we say that
i
is the
effective rate of interest per year
. Evidently, this definition depends on the time unit we choose to use. In a riskless world, which may be well approximated by the market for good quality government bonds,
i
will be certain, but if the investment is risky,
i
is uncertain, and our expectation at the outset to receive $(1 +
i
) can only be in the probabilistic sense.

We can regard the accumulation of invested money in either a retrospective or prospective way. We may take a given amount, $
X
say, to be invested now and ask, as above, to what amount will it accumulate after
T
years? Or, we may take a given amount, $
Y
say, required in
T
years' time (to meet some liability perhaps) and ask, how much we should invest now, so that the accumulation in
T
years' time will equal $
Y
? The latter quantity is called the
present value
of $
Y
in
T
years' time. For example, if the effective annual rate of interest is
i
per year, then we need to invest $1/(1 +
i
) now, in order to receive $1 at the end of one year. In standard actuarial notation, $1/(1 +
i
) is denoted
v
, and is called the discount factor. It is immediately clear that in a deterministic setting, accumulating and taking present values are inverse operations.
P
rice,
R
ichard (1723–1791)
Authors: Teugels Jozef L.
Published: September 2006

Price was a great thinker from the second half of the eighteenth century. Apart from contributions to philosophy, religion, and politics, he produced a number of substantial publications on both theoretical and practical aspects of life insurance.
Probability Theory
Authors: Hoffman‐Jørgensen J.
Published: September 2006

This article collects the basic concepts needed in all articles with a probabilistic undertone. After giving some historical development of probability concepts in insurance, the axiomatic concepts from probability theory are treated succinctly. Having dealt with the most common types of convergence, the most fundamental limit theorems are formulated with an indication of their interrelationship.
Professionalism
Authors: Low C.W.F.
Published: September 2006

A brief history of actuarial professionalism in the United Kingdom, its scope and limitations, and the position in July 2002.
Profit Testing
Authors: Richards Stephen
Published: September 2006

Every business needs to know if its products are profitable. However, it is much harder to determine the profitability of long‐term life and pensions business than most other industries. Strictly speaking, one only knows if a tranche of business has been profitable when the last contract has gone off the books, which could take fifty years for certain kinds of contract. Clearly, some methodology is needed to assess likely profitability before writing the business in the first place. This article is devoted to the subject of profit testing, the process of assessing the profitability of an insurance contract in advance of it being written.
Property Insurance—Personal
Authors: Ward Kimberley A.
Published: September 2006

Personal Property insurance in the United States is written on a number of insurance forms. By far, the most common ones are home owners package forms and private automobile physical damage coverage parts. Other forms include farm owners packages, dwelling forms, personal inland marine, and watercraft/recreational vehicle coverages.
Queueing Theory
Authors: Sigman Karl
Published: September 2006

The mathematical theory of congestion is associated with delays while waiting in a line or
queue
for service in a
system
. Examples of such systems include banks, post offices, and supermarkets as well as telecommunication systems involving telephones, computer networks, the Internet/the World Wide Web, inventory, and manufacturing processes. The early development of the field was motivated by telephone systems and the ways in which they could be made to operate more efficiently.

Much of the theory relies heavily on the use of
probability theory
and
stochastic processes
, and as such is widely viewed as a subfield of stochastic modeling and applied probability. There is also significant interplay with other fields such as scheduling theory, inventory theory, and insurance risk theory; it is a central part of
operations research
.
Random Walk
Authors: Doney R.A.
Published: September 2006

Random walks are the simplest of all stochastic processes; they include amenable examples of both Markov processes and martingales, and indeed the genesis of these two important classes of processes can be found in their study. Their origins lie in the first attempts to model gambling situations, and they still have many useful modeling applications. In actuarial mathematics, random walks occur widely in ruin theory. In discrete time, the surplus process (claims minus premiums) may be modeled as a random walk. In continuous time, the surplus observed at claim times is a random walk, for example, for the Sparre Andersen model. In both cases, the ruin probability is
P(M > u)
, where
u
is the initial reserve and
M
denotes the all‐time maximum of the random walk. The study of
M
also occurs in queueing theory, since it has the same distribution as the waiting time in a
GI/G/1
queue.
Redington,FrankMitchell (1906–1984)
Authors: Teugels Jozef L.
Published: September 2006

Redington has been considered one of the most influential members of the actuarial community, especially for his theory of immunization.
Regenerative Processes
Authors: Wolff Ronald W.
Published: September 2006

A regenerative process is a stochastic process with the property that after some (usually) random time, it starts over in the sense that, from the random time on, the process is stochastically equivalent to what it was at the beginning.

In many applied fields such as telecommunications, finance, and actuarial science, important probability models turn out to be regenerative. Properties of regenerative processes and the mathematical methods used in their analysis are fundamental in the analysis and understanding of these models.

For
classical
regenerative processes, cycles and cycle lengths are i.i.d. (independent and identically distributed). This assumption has been weakened over time. Now, processes satisfying the first paragraph above are regarded as regenerative, without any independent assumptions.

This article introduces the classical case in the section ‘Classical Regenerative Processes’, presents their time‐average properties in the section ‘Time‐average Properties, Classical Case’, and briefly treats more advanced topics in the section ‘Existence of Limits; Combining Processes’.
Reliability Classifications
Authors: Cai Jun
Published: September 2006

In this article, we consider reliability classifications for life distributions and discuss closure properties of classes of life distributions under important risk models such as convolution, mixing, and compounding. Applications of the reliability classes in insurance are discussed.
Renewal Theory
Authors: Willmot Gordon E.
Published: September 2006

Any sequence of positive, independent and identically distributed random variables can be considered as a renewal process. Traditional examples are the interclaim times and the successive claims in a homogeneous portfolio. The article describes the most important quantities and results from renewal theory. Also defective, delayed and discrete‐type renewal processes are discussed and illustrated with examples from actuarial science.
Resampling
Authors: Verrall Richard
Published: September 2006

Resampling methods have become popular in statistics because of the advance in computer power. They can be used to simplify many of the commonly used statistical procedures. In actuarial science, they are particularly relevant to the increasing use of stochastic and Simulation of stochastic processes simulation‐based techniques and it is likely that their use will increase further. We give an introduction to resampling methods, in particular, bootstrapping, and provide some examples of their use in practice.
Reinsurance, Reserving
Authors: Radtke Michael
Published: September 2006

The article deals with actuarial reserving techniques used in reinsurance. The impact of reinsurance on the actuarial reserving approach is discussed with a focus on the actuarial methods. In reinsurance, very often, the standard methods, that is, Chain‐Ladder or Bornhuetter Ferguson have to be modified because of specific reinsurance requirements. We introduce some methodical adjustments and illustrate the practical circumstances under which they have been taken into account. The paper ends with some comments on practical applications of reserving methods in reinsurance.
RESTIN
Authors: Ferrara Giovanna
Published: September 2006

RESTIN is an informal association of actuaries aimed at exchanging views on the ways in which actuarial thinking could help in the conduct of reinsurance.
Retention and Reinsurance Programmes
Authors: Centeno Maria de Lourdes
Published: September 2006

In this article, we give an overview of the literature on the optimal forms of reinsurance as well as on results on how to fix retentions.
Retrospective Premium
Authors: Meyers Glenn
Published: September 2006

Insurance is about the transfer of risk. If an entity, either a person or a business, wants to transfer all of its risk of loss to an insurer and the insurer is able to quantify the risk that the entity wishes to transfer, the price of the risk transfer is readily established by market forces and a deal is struck.

In many cases, the insurer is not confident that it can quantify the risk that the entity wishes to transfer. An entity sensing difficulty in getting a full transfer of risk may be willing to settle for a partial transfer of risk. The insurer would be more comfortable with a partial risk transfer because the entity has an incentive to keep its loss at a minimum.

This article describes a particular kind of risk‐sharing agreement called a retrospective rating plan. In these plans, the premium for an insurance contract is a function of the actual loss paid under the contract. The final premium is not known until after the contract has expired. This explains the name retrospective rating.
Risk Measures
Authors: Albrecht Peter
Published: September 2006

The literature on risk is vast (the academic disciplines involved are, for instance, investment and finance, economics, operations research, management science, decision theory, and psychology). So is the literature on risk measures. This contribution concentrates on financial risks and on financial risk measures (this excludes the literature on perceived risk (the risk of lotteries perceived by subjects) the literature dealing with the psychology of risk judgments and also the literature on inequality measurement.
Risk Process
Authors: Dickson David C.M.
Published: September 2006

This article considers classical risk processes in both continuous and discrete time, and discusses how the basic assumptions underlying these processes may be varied.
Optimal Risk Sharing
Authors: Aase Knut K.
Published: September 2006

Optimal risk sharing is considered from the perspective of the risk sharing model introduced by Karl Borch in the late 50's. First we introduce, in a modern setting, the main concepts from this theory. These we apply on the risk sharing problem between an insurer and an insurance customer. We motivate the development through a simple example, illustrating some fine points of this theory. In order to explain deductibles, we introduce transaction costs that are paid out ex post. Moral hazard provides another example.
Risk‐Based Capital Requirements
Authors: Brender Allan
Published: September 2006

An insurer needs sufficient financial resources to survive an unfavorable experience, pay all policyholder claims and continue business. For a company, this capital might be different from that required by supervisors or regulators. We survey the major components needed in the construction of the capital requirements. An illustration from the NAIC‐requirements in the US is given.
Severity of Ruin
Authors: Tsai Cary Chi‐Liang
Published: September 2006

In this article, continuous‐time risk models are introduced; the time of ruin, and the surplus immediately before and at ruin caused by a claim are defined. Then the probability of ruin, the (discounted) moment of the severity of ruin caused by a claim, the joint moment of the severity of ruin and the time of ruin caused by a claim, and the (discounted) defective joint/marginal distribution and probability functions of the surplus immediately before and at ruin are studied.
Simulation of Risk Processes
Authors: Burnecki Krzysztof , Härdle Wolfgang , Weron Rafał
Published: September 2006

The simulation of risk processes is a standard procedure for insurance companies. The generation of aggregate claims is vital for the calculation of the amount of loss that may occur. Simulation of risk processes also appears naturally in rating triggered step‐up bonds, where the interest rate is bound to random changes of the companies' ratings.
Simulation of Stochastic Processes
Authors: Asmussen Søren
Published: September 2006

In order to gain insight into the role played by the stochastic ingredients in insurance and finance, one is often forced to rely on simulation as no explicit analytic results are available. There exist a few general methods, but the generation on computer for most models requires non‐trivial and complicated operations that heavily depend on the specific structure of the underlying stochastic elements.
Simulation Methods for Stochastic Differential Equations
Authors: Platen Eckhard
Published: September 2006

As we try to build more realistic models in risk management, the stochastic effects need to be carefully taken into account and quantified. This paper provides a very basic overview on simulation methods for stochastic differential equations (SDEs).
Stability
Authors: Rusaityte Deimante
Published: September 2006

We discuss the stability problem for stochastic models. The general stability concept was introduced by Zolotarev. Its basic statements with a view towards actuarial risk theory were reproduced by Kalashnikov. Particular attention is payed to the stability of the ruin probability. We provide two examples of stability bounds: for finite state Markov chains and the ruin probability in the Sparre Andersen risk model.
Stationary Processes
Authors: Lindgren Georg
Published: September 2006

Stationary processes form a prime example of sets of not necessarily independent random variables, usually in the form of a time series. We collect the most important ingredients, in particular the spectrum and the spectral representation of such processes.
Stop‐Loss Premium
Authors: Cai Jun
Published: September 2006

This article considers stop loss premiums and related topics appearing in insurance.
Subexponential Distributions
Authors: Klüppelberg Claudia
Published: September 2006

Subexponential distributions are a special class of heavy‐tailed distributions. The name arises from one of their properties that their tails decrease slower than any exponential tail (3). This implies that large values can occur in a sample with nonnegligible probability, which makes the subexponential distributions natural candidates for modeling situations in which some extremely large values occur in a sample compared with the mean size of the data. Such a pattern is often seen in insurance data, for instance, in fire, wind–storm, or flood insurance (collectively known as catastrophe insurance). Subexponential claims can account for large fluctuations in the surplus process of a company, increasing the risk involved in such portfolios.
Surplus in Life and Pension Insurance
Authors: O'Brien Christopher David
Published: September 2006

Surplus in life insurance is a key issue for participating business: actuarial control over the emergence and distribution of surplus is a major subject in the management of life insurance companies. Although we here concentrate on surplus in the context of participating life business, surplus is also relevant in any kind of fund in which long‐term liabilities may be valued on the basis of assumptions about the future. For example, shareholders have needs and expectations of surplus in nonparticipating life business, with premium rates and payouts being determined with a view to rewarding shareholders, who have provided capital and run risks. In pensions business there are similar issues concerning the sources, size, analysis, and uses of surplus.
Surplus Process
Authors: Schmidli Hanspeter
Published: September 2006

In this article, we discuss the ideas behind modeling actuarial surplus processes in continuous time. Several models that have been treated successfully in the literature are reviewed.
Surplus Treaty
Authors: Mata Ana J.
Published: September 2006

Surplus treaty is a type of proportional reinsurance in which the percentage ceded to the reinsurer varies by original policy limit in order to keep the maximum loss amount retained by the ceding company fixed. The surplus treaty provides the ceding company greater capacity to write higher limits and to reduce fluctuations in its financial results.
Survival Analysis
Authors: Veraverbeke Noël
Published: September 2006

Survival analysis deals with the statistical modeling and analysis of durations of time from some origin to some ‘failure’. Such lifetimes are usually subject to censoring and/or truncation. We survey the most important developments around the two main landmarks: the product‐limit estimator of Kaplan and Meier (1958) 58 and the proportional hazards regression model of Cox (1972) 23.
Reinsurance — Terms, Conditions, and Methods of Placing
Authors: Czapiewski Colin , Gates Paul
Published: September 2006

The placing of reinsurance business is changing, with the role of intermediaries and methods of placement being constantly assessed in light of the costs. Smaller lines with many reinsurers are being replaced by much larger lines with few reinsurers. New forms of reinsurance are being introduced to reflect the requirements of the reinsured more accurately. In view of the more legal examination of contracts, the accuracy of the terms, conditions, exclusions, and other aspects of policy wording are now more important than ever before.
Time of Ruin
Authors: Tsai Cary Chi‐Liang
Published: September 2006

In this article, both classical discrete‐ and continuous‐time risk models are introduced; then the time of ruin and the severity of ruin are defined. The probability of ruin and some quantities concerning the time of ruin, such as the expectation of the present value of the time of ruin, the (discounted) moment of the severity of ruin caused by a claim, the joint moment of the severity of ruin, and the time of ruin caused by a claim, and the moments of the time of ruin due to oscillation and claim, respectively, are studied.