scholarly journals A Heuristic Review of some Ruin Theory Results

1985 ◽  
Vol 15 (2) ◽  
pp. 73-88 ◽  
Author(s):  
G. C. Taylor
Keyword(s):  
Size Distribution ◽  
Stochastic Dominance ◽  
Ruin Probability ◽  
Recursion Formula ◽  
Claim Size ◽  
Infinite Time ◽  
Ruin Theory ◽  
Interesting Connection ◽  
Claim Size Distribution ◽  
Aggregate Claims

AbstractThe paper deals with the renewal equation governing the infinite-time ruin probability. It is emphasized as intended to be no more than a pleasant ramble through a few scattered results. An interesting connection between ruin probability and a recursion formula for computation of the aggregate claims distribution is noted and discussed. The relation between danger of the claim size distribution and ruin probability is reexamined in the light of some recent results on stochastic dominance. Finally, suggestions are made as to the way in which the formula for ruin probability leads easily to conclusions about the effect on that probability of the long-tailedness of the claim size distribution. Stable distributions, in particular, are examined.

scholarly journals A class of risk processes with delayed claims: ruin probability estimates under heavy tail conditions

2006 ◽  
Vol 43 (4) ◽  
pp. 916-926 ◽  
Author(s):  
Ayalvadi Ganesh ◽  
Giovanni Luca Torrisi
Keyword(s):  
Size Distribution ◽  
Ruin Probability ◽  
Heavy Tail ◽  
Claim Size ◽  
Probability Estimates ◽  
Claim Size Distribution ◽  
Risk Processes

We consider a class of risk processes with delayed claims, and we provide ruin probability estimates under heavy tail conditions on the claim size distribution.

scholarly journals Excess of Loss Reinsurance with Reinstatements Revisited

2005 ◽  
Vol 35 (01) ◽  
pp. 211-238 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
Incomplete Information ◽  
Size Distribution ◽  
Claim Size ◽  
Inverse Gaussian ◽  
Insurance Risk ◽  
Distribution Free ◽  
Analytical Approximations ◽  
Claim Size Distribution ◽  
Aggregate Claims ◽  
Stop Loss

The classical evaluation of pure premiums for excess of loss reinsurance with reinstatements requires the knowldege of the claim size distribution of the insurance risk. In the situation of incomplete information, where only a few characteristics of the aggregate claims to an excess of loss layer can be estimated, the method of stop-loss ordered bounds yields a simple analytical distribution-free approximation to pure premiums of excess of loss reinsurance with reinstatements. It is shown that the obtained approximation is enough accurate for practical purposes and improves the analytical approximations obtained using either a gamma, translated gamma, translated inverse Gaussian or a mixture of the last two distributions.

scholarly journals An Extension of Panjer's Recursion

2002 ◽  
Vol 32 (2) ◽  
pp. 283-297 ◽  
Author(s):  
Klaus Th. Hess ◽  
Anett Liewald ◽  
Klaus D. Schmidt
Keyword(s):  
Size Distribution ◽  
Collective Model ◽  
Risk Theory ◽  
Claim Size ◽  
Similar Characterization ◽  
Claim Number ◽  
Claim Size Distribution ◽  
Aggregate Claims ◽  
Image Position

AbstractSundt and Jewell have shown that a nondegenerate claim number distribution Q = {qn}nϵN0 satisfies the recursionfor all n≥0 if and only if Q is a binomial, Poisson or negativebinomial distribution. This recursion is of interest since it yields a recursion for the aggregate claims distribution in the collective model of risk theory when the claim size distribution is integer-valued as well. A similar characterization of claim number distributions satisfying the above recursion for all n ≥ 1 has been obtained by Willmot. In the present paper we extend these results and the subsequent recursion for the aggregate claims distribution to the case where the recursion holds for all n ≥ k with arbitrary k. Our results are of interest in catastrophe excess-of-loss reinsurance.

scholarly journals Excess of Loss Reinsurance with Reinstatements Revisited

2005 ◽  
Vol 35 (1) ◽  
pp. 211-238 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
Incomplete Information ◽  
Size Distribution ◽  
Claim Size ◽  
Inverse Gaussian ◽  
Insurance Risk ◽  
Distribution Free ◽  
Analytical Approximations ◽  
Claim Size Distribution ◽  
Aggregate Claims ◽  
Stop Loss

The classical evaluation of pure premiums for excess of loss reinsurance with reinstatements requires the knowldege of the claim size distribution of the insurance risk. In the situation of incomplete information, where only a few characteristics of the aggregate claims to an excess of loss layer can be estimated, the method of stop-loss ordered bounds yields a simple analytical distribution-free approximation to pure premiums of excess of loss reinsurance with reinstatements. It is shown that the obtained approximation is enough accurate for practical purposes and improves the analytical approximations obtained using either a gamma, translated gamma, translated inverse Gaussian or a mixture of the last two distributions.

scholarly journals Ruin Probability Functions and Severity of Ruin as a Statistical Decision Problem

Risks ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 68 ◽  
Author(s):  
Emilio Gómez-Déniz ◽  
José María Sarabia ◽  
Enrique Calderín-Ojeda
Keyword(s):  
Size Distribution ◽  
Loss Function ◽  
Ruin Probability ◽  
Claim Size ◽  
Statistical Decision ◽  
Squared Error Loss Function ◽  
Probability Functions ◽  
Claim Size Distribution ◽  
Two Parameters ◽  
Severity Of Ruin

It is known that the classical ruin function under exponential claim-size distribution depends on two parameters, which are referred to as the mean claim size and the relative security loading. These parameters are assumed to be unknown and random, thus, a loss function that measures the loss sustained by a decision-maker who takes as valid a ruin function which is not correct can be considered. By using squared-error loss function and appropriate distribution function for these parameters, the issue of estimating the ruin function derives in a mixture procedure. Firstly, a bivariate distribution for mixing jointly the two parameters is considered, and second, different univariate distributions for mixing both parameters separately are examined. Consequently, a catalogue of ruin probability functions and severity of ruin, which are more flexible than the original one, are obtained. The methodology is also extended to the Pareto claim size distribution. Several numerical examples illustrate the performance of these functions.

scholarly journals A class of risk processes with delayed claims: ruin probability estimates under heavy tail conditions

2006 ◽  
Vol 43 (04) ◽  
pp. 916-926 ◽  
Author(s):  
Ayalvadi Ganesh ◽  
Giovanni Luca Torrisi
Keyword(s):  
Size Distribution ◽  
Ruin Probability ◽  
Heavy Tail ◽  
Claim Size ◽  
Probability Estimates ◽  
Claim Size Distribution ◽  
Risk Processes

We consider a class of risk processes with delayed claims, and we provide ruin probability estimates under heavy tail conditions on the claim size distribution.

Minimizing the ruin probability through capital injections

2011 ◽  
Vol 5 (2) ◽  
pp. 195-209 ◽  
Author(s):  
Ciyu Nie ◽  
David C. M. Dickson ◽  
Shuanming Li
Keyword(s):  
Size Distribution ◽  
Ruin Probability ◽  
Substantial Reduction ◽  
Claim Size ◽  
Fixed Amount ◽  
Claim Size Distribution ◽  
Premium Calculation ◽  
Capital Injections

AbstractWe consider an insurer who has a fixed amount of funds allocated as the initial surplus for a risk portfolio, so that the probability of ultimate ruin for this portfolio is at a known level. We consider the question of whether the insurer can reduce this ultimate ruin probability by allocating part of the initial funds to the purchase of a reinsurance contract. This reinsurance contract would restore the insurer's surplus to a positive level k every time the surplus fell between 0 and k. The insurer's objective is to choose the level k that minimizes the ultimate ruin probability. Using different examples of reinsurance premium calculation and claim size distribution we show that this objective can be achieved, often with a substantial reduction in the ultimate ruin probability from the situation when there is no reinsurance. We also show that by purchasing reinsurance the insurer can release funds for other purposes without altering its ultimate ruin probability.

Tail of compound distributions and excess time

1996 ◽  
Vol 33 (01) ◽  
pp. 184-195 ◽  
Author(s):  
Xiaodong Lin
Keyword(s):  
Size Distribution ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Claim Size ◽  
Compound Distributions ◽  
Fundamental Identity ◽  
Claim Size Distribution ◽  
Simple Bounds ◽  
New Better Than Used ◽  
Better Than

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

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