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→∞.

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→∞.

scholarly journals Planning Problems Of A Gambling-House with Application to Insurance Business

1975 ◽  
Vol 8 (3) ◽  
pp. 279-283
Author(s):  
Harald Bohman
Keyword(s):  
Planning Horizon ◽  
Risk Theory ◽  
Probability Of Ruin ◽  
Long Run ◽  
Time Period ◽  
Short Period ◽  
Insurance Business ◽  
A Minor ◽  
Planning Problems ◽  
Infinite Planning Horizon

In the classical risk theory the interdependence between the security loading and the initial risk reserve is studied. It could be said that the purpose of these studies are to state how large the security loading must be in order to avoid ruin of the insurance business. It has often been said that the classical approach with an infinite planning horizon is unrealistic. The main reason for this attitude is that if the security loading is equal to zero then ruin is certain. Since in practice it is often difficult to estimate the true size of the security loading the whole problem of ruin or non-ruin seems to rest on a rather shaky foundation. This attitude to the problem is reflected in studies in risk theory performed in recent years. The infinite planning horizon is then often replaced by a finite time period. Since the probability of ruin during a short period of time depends to a minor extent on the size of the security loading these studies are concentrated mainly on the shape of the claim distribution, while the security loading is of minor interest.Let us think of a gambling-house, where coin-tossing is practised. Let us assume that the gambling-house for reason of fairness decides to pay two dollars to each winner who has staked one dollar. Probability theory tells us that however rich the gambling-house may be, it will be ruined in the long run. This simple example reminds us of the trivial fact that insurance business without a sufficient security loading in the premium is commercially impossible.

The Ruin Probability During a Finite Time Period

Risk Theory ◽  
1977 ◽  
pp. 132-136
Author(s):  
Robert Eric Beard ◽  
Teivo Pentikäinen ◽  
Erkki Pesonen
Keyword(s):  
Finite Time ◽  
Ruin Probability ◽  
Time Period

Small Solutions of the Congruence ax2 + by2 ≡ c(mod k)

1969 ◽  
Vol 12 (3) ◽  
pp. 311-320 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Asymptotic Formula ◽  
Absolute Constant ◽  
Log P ◽  
Positive Absolute Constant ◽  
Image Position

In 1957, Mordell [3] provedTheorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that(1.1)provided (abc, p) = 1.Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.

scholarly journals Approximations of the generalised Poisson function

1969 ◽  
Vol 5 (2) ◽  
pp. 213-226 ◽  
Author(s):  
Lauri Kauppi ◽  
Pertti Ojantakanen
Keyword(s):  
Distribution Function ◽  
Fixed Time ◽  
Risk Theory ◽  
Normal Distribution Function ◽  
Expected Number ◽  
Time Period ◽  
Poisson Function ◽  
Total Claim ◽  
Basic Functions ◽  
Image Position

One of the basic functions of risk theory is the so-called generalised Poisson function F(x), which gives the probability that the total amount of claims ξ does not exceed some given limit x during a year (or during some other fixed time period). For F(x) is obtained the well known expansion where n is the expected number of claims during this time period and Sk*(x) is the k:th convolution of the distribution function S(z) of the size of one claim. The formula (1) is, however, much too inconvenient for numerical computations and for most other applications. One of the main problems of risk theory, which is still partly open, is to find suitable methods to compute, or at least to approximate, the generalised Poisson function.A frequently used approximation is to replace F(x) by the normal distribution function having the same mean and standard deviation as F as follows: where α1 and α2 are the first zero-moments of S(z): SM(Z) is here again the distribution function of the size of one claim. To obtain more general results a reinsurance arrangement is assumed under which the maximum net retention is M. Hence the portfolio on the company's own retention is considered. If the reinsurance is of Excess of Loss type, then where S(z) is the distribution function of the size of one total claim.

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.

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