A Functional Approach to Approximations for the Individual Risk Model

A functional approach is taken for the total claim amount distribution for the individual risk model. Various commonly used approximations for this distribution are considered, including the compound Poisson approximation, the compound binomial approximation, the compound negative binomial approximation and the normal approximation. These are shown to arise as zeroth order approximations in the functional set-up. By taking the derivative of the functional that maps the individual claim distributions onto the total claim amount distribution, new first order approximation formulae are obtained as refinements to the existing approximations. For particular choices of input, these new approximations are simple to calculate. Numerical examples, including the well-known Gerber portfolio, are considered. Corresponding approximations for stop-loss premiums are given.

Download Full-text

EXPANSION IN BELL POLYNOMIALS OF THE DISTRIBUTION OF THE TOTAL CLAIM AMOUNT WITH WEIBULL-DISTRIBUTED CLAIM SIZES

The ANZIAM Journal ◽

10.1017/s1446181108000187 ◽

2008 ◽

Vol 49 (4) ◽

pp. 495-501

Author(s):

RAMON LACAYO

Keyword(s):

Random Number ◽

Negative Binomial ◽

Claim Amount ◽

Bell Polynomials ◽

Total Claim Amount ◽

Binomial Distributions ◽

Probabilistic Distribution ◽

Satisfactory Model ◽

Total Claim ◽

Fixed Period

AbstractThe total claim amount for a fixed period of time is, by definition, a sum of a random number of claims of random size. In this paper we explore the probabilistic distribution of the total claim amount for claims that follow a Weibull distribution, which can serve as a satisfactory model for both small and large claims. As models for the number of claims we use the geometric, Poisson, logarithmic and negative binomial distributions. In all these cases, the densities of the total claim amount are obtained via Laplace transform of a density function, an expansion in Bell polynomials of a convolution and a subsequent Laplace inversion.

Download Full-text

Prediction of Outstanding Liabilities in Non-Life Insurance

Astin Bulletin ◽

10.2143/ast.23.1.2005103 ◽

1993 ◽

Vol 23 (1) ◽

pp. 95-115 ◽

Cited By ~ 83

Author(s):

Ragnar Norberg

Keyword(s):

Life Insurance ◽

Linear Prediction ◽

Basic Result ◽

Claim Amount ◽

Insurance Company ◽

Life Insurance Company ◽

Total Claim Amount ◽

Generalized Poisson ◽

Total Claim ◽

The Individual

AbstractA fully time-continuous approach is taken to the problem of predicting the total liability of a non-life insurance company. Claims are assumed to be generated by a non-homogeneous marked Poisson process, the marks representing the developments of the individual claims. A first basic result is that the total claim amount follows a generalized Poisson distribution. Fixing the time of consideration, the claims are categorized into settled, reported but not settled, incurred but not reported, and covered but not incurred. It is proved that these four categories of claims can be viewed as arising from independent marked Poisson processes. By use of this decomposition result predictors are constructed for all categories of outstanding claims. The claims process may depend on observable as well as unobservable risk characteristics, which may change in the course of time, possibly in a random manner. Special attention is given to the case where the claim intensity per risk unit is a stationary stochastic process. A theory of continuous linear prediction is instrumental.

Download Full-text

Conditional mean risk sharing in the individual model with graphical dependencies

Annals of Actuarial Science ◽

10.1017/s1748499521000166 ◽

2021 ◽

pp. 1-27

Author(s):

Michel Denuit ◽

Christian Y. Robert

Keyword(s):

Graphical Models ◽

Risk Sharing ◽

Risk Model ◽

Allocation Rule ◽

Individual Risk ◽

Claim Amount ◽

Conditional Mean ◽

Operational Risks ◽

The Individual ◽

Mean Risk

Abstract Conditional mean risk sharing appears to be effective to distribute total losses amongst participants within an insurance pool. This paper develops analytical results for this allocation rule in the individual risk model with dependence induced by the respective position within a graph. Precisely, losses are modelled by zero-augmented random variables whose joint occurrence distribution and individual claim amount distributions are based on network structures and can be characterised by graphical models. The Ising model is adopted for occurrences and loss amounts obey decomposable graphical models that are specific to each participant. Two graphical structures are thus used: the first one to describe the contagion amongst member units within the insurance pool and the second one to model the spread of losses inside each participating unit. The proposed individual risk model is typically useful for modelling operational risks, catastrophic risks or cybersecurity risks.

Download Full-text

Estimation of the probable maximum loss based on extreme value theory for stock returns

Equilibrium ◽

10.12775/equil.2009.005 ◽

2009 ◽

Vol 2 (1) ◽

pp. 51-59

Author(s):

Marcin Fałdziński

Keyword(s):

Stock Returns ◽

Extreme Value Theory ◽

Value Theory ◽

Claim Amount ◽

Total Claim Amount ◽

Maximum Loss ◽

Probable Maximum Loss ◽

Insurance Portfolio ◽

Total Claim ◽

The Individual

Probable maximum loss is a measure coming from the insurance market, where is applied to insurance portfolio analysis. This correspond to the 20-80 rule, which states that 20% of the individual claims are responsible for more than 80% of the total claim amount in a well defined portfolio. The main aim of the presented paper is estimation of the probable maximum loss for stock returns which are treated as portfolios of securities. It turns out that probable maximum loss is a useful tool for risk analysis or/and diagnostic purposes at capital markets, but we have to be aware of its drawbacks.

Download Full-text

On the total claim amount for marked Poisson cluster models

Advances in Applied Probability ◽

10.1017/apr.2019.15 ◽

2019 ◽

Vol 51 (2) ◽

pp. 541-569 ◽

Cited By ~ 1

Author(s):

Bojan Basrak ◽

Olivier Wintenberger ◽

Petra Žugec

Keyword(s):

Sufficient Conditions ◽

Arrival Rate ◽

Random Variable ◽

Cluster Models ◽

Infinite Variance ◽

Claim Amount ◽

Total Claim Amount ◽

Total Claim ◽

The Individual ◽

Poisson Cluster

AbstractWe study the asymptotic distribution of the total claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence the arrival rate of future claims. We find sufficient conditions under which the total claim amount satisfies the central limit theorem or, alternatively, tends in distribution to an infinite-variance stable random variable. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes process as our key example.

Download Full-text

Incorporating heterogeneity into the prediction of total claim amount

Hacettepe Journal of Mathematics and Statistics ◽

10.15672/hjms.2017.421 ◽

2017 ◽

Vol 47 (4) ◽

pp. 1-15

Author(s):

Aslihan Senturk Acar ◽

Ugur Karabey ◽

Dario Gregori

Keyword(s):

Claim Amount ◽

Total Claim Amount ◽

Total Claim

Download Full-text

The individual risk model

Modern Actuarial Risk Theory ◽

10.1007/978-3-540-70998-5_2 ◽

2008 ◽

pp. 17-40

Keyword(s):

Risk Model ◽

Individual Risk ◽

The Individual

Download Full-text

COMPARISON OF APPROXIMATIONS FOR COMPOUND POISSON PROCESSES

Astin Bulletin ◽

10.1017/asb.2015.7 ◽

2015 ◽

Vol 45 (3) ◽

pp. 601-637 ◽

Cited By ~ 7

Author(s):

Raffaello Seri ◽

Christine Choirat

Keyword(s):

Random Variable ◽

Cumulative Distribution ◽

Claim Amount ◽

Approximation Methods ◽

Total Claim Amount ◽

Aggregate Claim ◽

Time Period ◽

Total Claim ◽

The Difference ◽

Object Of Study

AbstractIn this paper, we compare the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this model, it is usually supposed that a portfolio is at risk for a time period of length t. The occurrences of the claims are governed by a Poisson process of intensity μ so that the number of claims in [0,t] is a Poisson random variable with parameter λ = μ t. Each single claim is an independent replication of the random variable X, representing the claim severity. The aggregate claim or total claim amount process in [0,t] is represented by the random sum of N independent replications of X, whose cumulative distribution function (cdf) is the object of study. Due to its computational complexity, several approximation methods for this cdf have been proposed. In this paper, we consider 15 approximations put forward in the literature that only use information on the lower order moments of the involved distributions. For each approximation, we consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure their accuracy as λ = μ t diverges to infinity. Using these expansions, several statements concerning the quality of approximations for the distribution of the aggregate claim process can find theoretical support. Other statements can be disproved on the same grounds. Finally, we investigate numerically the accuracy of the proposed formulas.

Download Full-text