Numerical Evaluation of Continuous Time Ruin Probabilities for a Portfolio with Credibility Updated Premiums

2010 ◽  
Vol 40 (1) ◽  
pp. 399-414 ◽  
Author(s):  
Lourdes B. Afonso ◽  
Alfredo D. Egídio dos Reis ◽  
Howard R. Waters
Keyword(s):  
Continuous Time ◽  
Finite Time ◽  
Numerical Evaluation ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Probability Of Ruin ◽  
Experience Rating ◽  
Classical Risk Process ◽  
Major Consideration

AbstractThe probability of ruin in continuous and finite time is numerically evaluated in a classical risk process where the premium can be updated according to credibility models and therefore change from year to year. A major consideration in the development of this approach is that it should be easily applicable to large portfolios. Our method uses as a first tool the model developed by Afonso et al. (2009), which is quite flexible and allows premiums to change annually. We extend that model by introducing a credibility approach to experience rating.We consider a portfolio of risks which satisfy the assumptions of the Bühlmann (1967, 1969) or Bühlmann and Straub (1970) credibility models. We compute finite time ruin probabilities for different scenarios and compare with those when a fixed premium is considered.

Calculating Continuous Time Ruin Probabilities for a Large Portfolio with Varying Premiums

2009 ◽  
Vol 39 (1) ◽  
pp. 117-136 ◽  
Author(s):  
Lourdes B. Afonso ◽  
Alfredo D. Egídio dos Reis ◽  
Howard R. Waters
Keyword(s):  
Gamma Distribution ◽  
Continuous Time ◽  
Finite Time ◽  
Ruin Probability ◽  
Numerical Evaluation ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Aggregate Claims ◽  
Classical Risk Process ◽  
Major Consideration

AbstractIn this paper we present a method for the numerical evaluation of the ruin probability in continuous and finite time for a classical risk process where the premium can change from year to year. A major consideration in the development of this methodology is that it should be easily applicable to large portfolios. Our method is based on the simulation of the annual aggregate claims and then on the calculation of the ruin probability for a given surplus at the start and at the end of each year. We calculate the within-year ruin probability assuming a translated gamma distribution approximation for aggregate claim amounts.We illustrate our method by studying the case where the premium at the start of each year is a function of the surplus level at that time or at an earlier time.

MEASURING THE IMPACT OF A BONUS-MALUS SYSTEM IN FINITE AND CONTINUOUS TIME RUIN PROBABILITIES FOR LARGE PORTFOLIOS IN MOTOR INSURANCE

2017 ◽  
Vol 47 (2) ◽  
pp. 417-435 ◽  
Author(s):  
Lourdes B. Afonso ◽  
Rui M. R. Cardoso ◽  
Alfredo D. Egídio dos Reis ◽  
Gracinda Rita Guerreiro
Keyword(s):  
Finite Time ◽  
Classical Model ◽  
Real Data ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Probability Of Ruin ◽  
Insurance Portfolio ◽  
Motor Insurance ◽  
Large Motor ◽  
The Impact

AbstractMotor insurance is a very competitive business where insurers operate with quite large portfolios, often decisions must be taken under short horizons and therefore ruin probabilities should be calculated in finite time. The probability of ruin, in continuous and finite time, is numerically evaluated under the classical Cramér–Lundberg risk process framework for a large motor insurance portfolio, where we allow for a posteriori premium adjustments, according to the claim record of each individual policyholder. Focusing on the classical model for bonus-malus systems, we propose that the probability of ruin can be interpreted as a measure to decide between different bonus-malus scales or even between different bonus-malus rules. In our work, the required initial surplus can also be evaluated. We consider an application of a bonus-malus system for motor insurance to study the impact of experience rating in ruin probabilities. For that, we used a real commercial scale of an insurer operating in the Portuguese market, and we also work on various well-known optimal bonus-malus scales estimated with real data from that insurer. Results involving these scales are discussed.

On Numerical Evaluation of Finite Time Survival Probabilities

1999 ◽  
Vol 5 (3) ◽  
pp. 575-584 ◽  
Author(s):  
D.C.M. Dickson
Keyword(s):  
Finite Time ◽  
Numerical Evaluation ◽  
Risk Model ◽  
Classical Risk Model ◽  
Probability Of Ruin ◽  
Survival Probabilities ◽  
Computational Aspects ◽  
The Classical Risk Model

ABSTRACTIn this paper we review three algorithms to calculate the probability of ruin/survival in finite time for the classical risk model. We discuss the computational aspects of these algorithms and consider the question of which algorithm should be preferred.

scholarly journals Phase-type Approximations to Finite-time Ruin Probabilities in the Sparre-Andersen and Stationary Renewal Risk Models

2005 ◽  
Vol 35 (1) ◽  
pp. 131-144 ◽  
Author(s):  
D.A. Stanford ◽  
F. Avram ◽  
A.L. Badescu ◽  
L. Breuer ◽  
A. Da Silva Soares ◽  
...  
Keyword(s):  
Finite Time ◽  
Risk Process ◽  
Computational Effort ◽  
Ruin Probabilities ◽  
Risk Models ◽  
Deficit At Ruin ◽  
Random Horizon ◽  
Phase Type ◽  
The Deficit At Ruin ◽  
Renewal Risk

The present paper extends the “Erlangization” idea introduced by Asmussen, Avram, and Usabel (2002) to the Sparre-Andersen and stationary renewal risk models. Erlangization yields an asymptotically-exact method for calculating finite time ruin probabilities with phase-type claim amounts. The method is based on finding the probability of ruin prior to a phase-type random horizon, independent of the risk process. When the horizon follows an Erlang-l distribution, the method provides a sequence of approximations that converges to the true finite-time ruin probability as l increases. Furthermore, the random horizon is easier to work with, so that very accurate probabilities of ruin are obtained with comparatively little computational effort. An additional section determines the phase-type form of the deficit at ruin in both models. Our work exploits the relationship to fluid queues to provide effective computational algorithms for the determination of these quantities, as demonstrated by the numerical examples.

scholarly journals Finite Time Ruin Problems for Perturbed Experience Rating and Connection with Discounting Risk Models

1986 ◽  
Vol 16 (1) ◽  
pp. 33-43 ◽  
Author(s):  
F. Abikhalil
Keyword(s):  
Finite Time ◽  
Ruin Probability ◽  
Risk Model ◽  
First Passage Time ◽  
Compound Poisson Process ◽  
Passage Time ◽  
Risk Process ◽  
Risk Models ◽  
Parabolic Boundary ◽  
Experience Rating

AbstractWe consider a generalisation of a risk process under experience rating when the aggregation of claims up to time t is a Brownian motion (B.M.) with a drift. We prove that the distribution of ruin before time t is equivalent to the distribution of the first passage time of B.M. for parabolic boundary.Using Wald identity for continuous time we give an explicit formula for this distribution. A connection is made with discounting risk model when the income process is a diffusion.When the aggregation of claims is a mixture of B.M. and compound Poisson process, we give (using Gerber's result 1973) an upper bound for the distribution of finite time ruin probability.

Ruin probabilities for competing claim processes

2004 ◽  
Vol 41 (03) ◽  
pp. 679-690 ◽  
Author(s):  
Miljenko Huzak ◽  
Mihael Perman ◽  
Hrvoje Šikić ◽  
Zoran Vondraček
Keyword(s):  
Ruin Probability ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Ruin Probabilities ◽  
Classical Risk Process ◽  
Risk Processes

Let C 1, C 2,…,C m be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C i . Formulae for the probability that ruin is caused by C i are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

scholarly journals Phase-type Approximations to Finite-time Ruin Probabilities in the Sparre-Andersen and Stationary Renewal Risk Models

2005 ◽  
Vol 35 (01) ◽  
pp. 131-144 ◽  
Author(s):  
D.A. Stanford ◽  
F. Avram ◽  
A.L. Badescu ◽  
L. Breuer ◽  
A. Da Silva Soares ◽  
...  
Keyword(s):  
Finite Time ◽  
Risk Process ◽  
Computational Effort ◽  
Ruin Probabilities ◽  
Risk Models ◽  
Deficit At Ruin ◽  
Random Horizon ◽  
Phase Type ◽  
The Deficit At Ruin ◽  
Renewal Risk

The present paper extends the “Erlangization” idea introduced by Asmussen, Avram, and Usabel (2002) to the Sparre-Andersen and stationary renewal risk models. Erlangization yields an asymptotically-exact method for calculating finite time ruin probabilities with phase-type claim amounts. The method is based on finding the probability of ruin prior to a phase-type random horizon, independent of the risk process. When the horizon follows an Erlang-l distribution, the method provides a sequence of approximations that converges to the true finite-time ruin probability as l increases. Furthermore, the random horizon is easier to work with, so that very accurate probabilities of ruin are obtained with comparatively little computational effort. An additional section determines the phase-type form of the deficit at ruin in both models. Our work exploits the relationship to fluid queues to provide effective computational algorithms for the determination of these quantities, as demonstrated by the numerical examples.

RECURSIVE CALCULATION OF RUIN PROBABILITIES AT OR BEFORE CLAIM INSTANTS FOR NON-IDENTICALLY DISTRIBUTED CLAIMS

2014 ◽  
Vol 45 (2) ◽  
pp. 421-443 ◽  
Author(s):  
Anisoara Maria Raducan ◽  
Raluca Vernic ◽  
Gheorghita Zbaganu
Keyword(s):  
Continuous Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Recursive Algorithm ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Numerical Examples ◽  
Claim Number ◽  
Recursive Calculation ◽  
The Impact

AbstractIn this paper, we present recursive formulae for the ruin probability at or before a certain claim arrival instant for some particular continuous time risk model. The claim number process underlying this risk model is a renewal process with either Erlang or a mixture of exponentials inter-claim times (ICTs). The claim sizes (CSs) are independent and distributed in Erlang's family, i.e., they can have different parameters, which yields a non-homogeneous risk process. We present the corresponding recursive algorithm used to evaluate the above mentioned ruin probability and we illustrate it on several numerical examples in which we vary the model's parameters to assess the impact of the non-homogeneity on the resulting ruin probability.

scholarly journals Estimators and Bootstrap Confidence Intervals for Ruin Probabilities

1989 ◽  
Vol 19 (1) ◽  
pp. 57-70 ◽  
Author(s):  
Christian Hipp
Keyword(s):  
Confidence Intervals ◽  
Bootstrap Method ◽  
Monte Carlo Study ◽  
Risk Process ◽  
Claim Amount ◽  
Ruin Probabilities ◽  
Infinite Time ◽  
Bootstrap Confidence Intervals ◽  
Classical Risk Process ◽  
The Bootstrap Method

AbstractFor the infinite time ruin probability in the classical risk process, efficient estimators are proposed in cases in which the claim amount distribution is unknown. Confidence intervals are computed which are based on normal approximations or on the bootstrap method. The procedures are checked in a Monte-Carlo study.

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