scholarly journals An Appropriate Way to Switch from the Individual Risk Model to the Collective One

1993 ◽  
Vol 23 (1) ◽  
pp. 23-54 ◽  
Author(s):  
S. Kuon ◽  
M. Radtke ◽  
A. Reich
Keyword(s):  
Risk Model ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Individual Risk ◽  
Risk Models ◽  
Approximation Quality ◽  
Collective Risk Model ◽  
Collective Risk ◽  
The Individual ◽  
Necessary And Sufficient

AbstractFor some time now, the convenient and fast calculability of collective risk models using the Panjer-algorithm has been a well-known fact, and indeed practitioners almost always make use of collective risk models in their daily numerical computations. In doing so, a standard link has been preferred for relating such calculations to the underlying heterogeneous risk portfolio and to the approximation of the aggregate claims distribution function in the individual risk model. In this procedure until now, the approximation quality of the collective risk model upon which such calculations are based is unknown.It is proved that the approximation error which arises does not converge to zero if the risk portfolio in question continues to grow. Therefore, necessary and sufficient conditions are derived in order to obtain well-adjusted collective risk models which supply convergent approximations. Moreover, it is shown how in practical situations the previous natural link between the individual and the collective risk model can easily be modified to improve its calculation accuracy. A numerical example elucidates this.

scholarly journals Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

2007 ◽  
Vol 37 (1) ◽  
pp. 93-112 ◽  
Author(s):  
Jun Cai ◽  
Ken Seng Tan
Keyword(s):  
Value At Risk ◽  
Negative Binomial ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Individual Risk ◽  
Risk Models ◽  
Optimization Criteria ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Stop Loss

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

The individual risk model: a compound distribution

1989 ◽  
Vol 116 (1) ◽  
pp. 101-107 ◽  
Author(s):  
R. J. Verrall
Keyword(s):  
Risk Model ◽  
Individual Risk ◽  
Risk Models ◽  
Collective Models ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Compound Distribution ◽  
Compound Binomial ◽  
Mean And Variance ◽  
The Individual

The two approaches to modelling aggregate claims—the individual and the collective models—have been regarded as arising by considering a portfolio of policies in different ways. The individual risk model (IRM) is derived by considering the claims on individual policies and summing over all policies in the portfolio, while the collective risk model (CRM) is derived from the portfolio as a whole. This is sometimes held to be the main difference between the IRM and the CRM. In fact the IRM can be derived in exactly the same way as the CRM and can be regarded as a compound binomial distribution. This makes a unified treatment of risk models possible, simplifies the calculation of the mean and variance of the IRM, and facilitates the calculation of higher moments.

scholarly journals Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

2007 ◽  
Vol 37 (01) ◽  
pp. 93-112 ◽  
Author(s):  
Jun Cai ◽  
Ken Seng Tan
Keyword(s):  
Value At Risk ◽  
Negative Binomial ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Individual Risk ◽  
Risk Models ◽  
Optimization Criteria ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Stop Loss

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

Practical applications of risk theory. Seminar, 29–30 September 1992

1993 ◽  
Vol 120 (1) ◽  
pp. 211-214
Author(s):  
A. S. Macdonald
Keyword(s):  
Case Studies ◽  
Model Compound ◽  
Risk Model ◽  
Risk Theory ◽  
Risk Models ◽  
Compound Poisson ◽  
Practical Applications ◽  
Collective Risk Model ◽  
Actuarial Mathematics ◽  
Collective Risk

A seminar on ‘The Practical Applications of Risk Theory’ was held at Staple Inn on 29–30 September 1992, organised jointly by the Institute and the Department of Actuarial Mathematics and Statistics at Heriot-Watt University. The aim of the seminar was to combine introductory talks on several aspects of risk theory with detailed presentations of case studies by practitioners.The first three sessions dealt with risk models.Ms Mary Hardy gave a short survey of the classical collective risk model, compound Poisson distributions, and some simple approximations to such distributions.

scholarly journals Frequency and Severity Dependence in the Collective Risk Model: An Approach Based on Sarmanov Distribution

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1400 ◽  
Author(s):  
Catalina Bolancé ◽  
Raluca Vernic
Keyword(s):  
Random Number ◽  
Negative Binomial ◽  
Risk Model ◽  
Numerical Study ◽  
Random Variables ◽  
Data Set ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Insurance Portfolio ◽  
The Individual

In actuarial mathematics, the claims of an insurance portfolio are often modeled using the collective risk model, which consists of a random number of claims of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent cost per claim. To facilitate computations, there is a classical assumption of independence between the random number of such random variables (i.e., the claims frequency) and the random variables themselves (i.e., the claim severities). However, recent studies showed that, in practice, this assumption does not always hold, hence, introducing dependence in the collective model becomes a necessity. In this sense, one trend consists of assuming dependence between the number of claims and their average severity. Alternatively, we can consider heterogeneity between the individual cost of claims associated with a given number of claims. Using the Sarmanov distribution, in this paper we aim at introducing dependence between the number of claims and the individual claim severities. As marginal models, we use the Poisson and Negative Binomial (NB) distributions for the number of claims, and the Gamma and Lognormal distributions for the cost of claims. The maximum likelihood estimation of the proposed Sarmanov distribution is discussed. We present a numerical study using a real data set from a Spanish insurance portfolio.

scholarly journals A Multi-Year Microlevel Collective Risk Model

2021 ◽  
Author(s):  
Rosy Oh ◽  
Himchan Jeong ◽  
Jae Youn Ahn ◽  
Emiliano A. Valdez
Keyword(s):  
Risk Model ◽  
Collective Risk Model ◽  
Collective Risk

scholarly journals On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment

1984 ◽  
Vol 14 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Jean-Marie Reinhard
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Risk Models ◽  
Markovian Environment ◽  
Necessary And Sufficient ◽  
Natural Way ◽  
Quantities Of Interest

AbstractWe consider a risk model in which the claim inter-arrivals and amounts depend on a markovian environment process. Semi-Markov risk models are so introduced in a quite natural way. We derive some quantities of interest for the risk process and obtain a necessary and sufficient condition for the fairness of the risk (positive asymptotic non-ruin probabilities). These probabilities are explicitly calculated in a particular case (two possible states for the environment, exponential claim amounts distributions).

Reversibility of Markov chains with applications to storage models

1985 ◽  
Vol 22 (01) ◽  
pp. 123-137 ◽  
Author(s):  
Hideo Ōsawa
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Single Server ◽  
General State ◽  
Insurance Risk ◽  
General State Space ◽  
Necessary And Sufficient

This paper studies the reversibility conditions of stationary Markov chains (discrete-time Markov processes) with general state space. In particular, we investigate the Markov chains having atomic points in the state space. Such processes are often seen in storage models, for example waiting time in a queue, insurance risk reserve, dam content and so on. The necessary and sufficient conditions for reversibility of these processes are obtained. Further, we apply these conditions to some storage models and present some interesting results for single-server queues and a finite insurance risk model.

scholarly journals A Functional Approach to Approximations for the Individual Risk Model

2004 ◽  
Vol 34 (2) ◽  
pp. 379-397 ◽  
Author(s):  
Susan M. Pitts
Keyword(s):  
Negative Binomial ◽  
Risk Model ◽  
Order Approximation ◽  
Functional Approach ◽  
Individual Risk ◽  
Claim Amount ◽  
Total Claim Amount ◽  
Total Claim ◽  
Binomial Approximation ◽  
The Individual

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.

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