A Review of Modern Collective Risk Theory with Dividend Strategies

2008 ◽  
Author(s):  
Benjamin Avanzi
Keyword(s):  
Risk Theory ◽  
Collective Risk

scholarly journals Diffusion approximations in collective risk theory

1969 ◽  
Vol 6 (2) ◽  
pp. 285-292 ◽  
Author(s):  
L. Donald Iglehart
Keyword(s):  
Large Deviations ◽  
Risk Premium ◽  
Risk Theory ◽  
Insurance Company ◽  
Diffusion Approximations ◽  
Security Risk ◽  
Insurance Policies ◽  
Collective Risk ◽  
A Company ◽  
Random Times

Collective risk theory is concerned with the random fluctations of the total assets, the risk reserve, of an insurance company. Consider a company which only writes ordinary insurance policies such as accident, disability, fire, health, and whole life. The policyholders pay premiums regularly and at certain random times make claims to the company. A policyholder's premium, the gross risk premium, is a positive amount composed of two components. The net risk premium is the component calculated to cover the payments of claims on the average, while the security risk premium, or safety loading, is the component which protects the company from large deviations of claims from the average and also allows an accumulation of capital. When a claim occurs the company pays the policyholder a positive amount called the positive risk sum.

Probabilities of ruin when the safety loading tends to zero

2000 ◽  
Vol 32 (3) ◽  
pp. 885-923 ◽  
Author(s):  
Vsevolod K. Malinovskii
Keyword(s):  
Finite Time ◽  
Absolute Constant ◽  
Classical Result ◽  
Risk Theory ◽  
Premium Rate ◽  
Collective Risk ◽  
Time Period ◽  
Positive Absolute Constant ◽  
Uniform Approximations ◽  
Insurance Business

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ>0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero as u→∞.

scholarly journals Diffusion approximations in collective risk theory

1969 ◽  
Vol 6 (02) ◽  
pp. 285-292 ◽  
Author(s):  
L. Donald Iglehart
Keyword(s):  
Large Deviations ◽  
Risk Premium ◽  
Risk Theory ◽  
Insurance Company ◽  
Diffusion Approximations ◽  
Security Risk ◽  
Insurance Policies ◽  
Collective Risk ◽  
A Company ◽  
Random Times

Collective risk theory is concerned with the random fluctations of the total assets, the risk reserve, of an insurance company. Consider a company which only writes ordinary insurance policies such as accident, disability, fire, health, and whole life. The policyholders pay premiums regularly and at certain random times make claims to the company. A policyholder's premium, the gross risk premium, is a positive amount composed of two components. The net risk premium is the component calculated to cover the payments of claims on the average, while the security risk premium, or safety loading, is the component which protects the company from large deviations of claims from the average and also allows an accumulation of capital. When a claim occurs the company pays the policyholder a positive amount called the positive risk sum.

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.

Moderate- and large-deviation probabilities in actuarial risk theory

1989 ◽  
Vol 21 (04) ◽  
pp. 725-741 ◽  
Author(s):  
Eric Slud ◽  
Craig Hoesman
Keyword(s):  
Strong Approximation ◽  
Large Deviation ◽  
Age Distribution ◽  
Risk Theory ◽  
Renewal Processes ◽  
Collective Risk ◽  
Large Deviation Theorem ◽  
Large Deviation Probabilities ◽  
Actuarial Risk ◽  
Portfolio Size

A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio ‘age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csörg et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals.

Probabilities of ruin when the safety loading tends to zero

2000 ◽  
Vol 32 (03) ◽  
pp. 885-923 ◽  
Author(s):  
Vsevolod K. Malinovskii
Keyword(s):  
Finite Time ◽  
Absolute Constant ◽  
Classical Result ◽  
Risk Theory ◽  
Premium Rate ◽  
Collective Risk ◽  
Time Period ◽  
Positive Absolute Constant ◽  
Uniform Approximations ◽  
Insurance Business

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ>0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero as u→∞.

Research in collective risk theory

1955 ◽  
Vol 1955 (1-2) ◽  
pp. 53-100 ◽  
Author(s):  
G. Arfwedson
Keyword(s):  
Risk Theory ◽  
Collective Risk

Notes on collective risk theory

1957 ◽  
Vol 1957 (1-2) ◽  
pp. 46-59 ◽  
Author(s):  
G. Arfwedson
Keyword(s):  
Risk Theory ◽  
Collective Risk
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