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.

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.

scholarly journals Fourier/Laplace Transforms and Ruin Probabilities

2002 ◽  
Vol 32 (1) ◽  
pp. 91-105 ◽  
Author(s):  
Fátima D.P. Lima ◽  
Jorge M.A. Garcia ◽  
Alfredo D. Egídio Dos Reis
Keyword(s):  
Classical Model ◽  
Risk Process ◽  
Laplace Transforms ◽  
Ruin Probabilities ◽  
Ruin Theory ◽  
Easy Method ◽  
Arrival Times ◽  
The Real ◽  
Renewal Risk ◽  
Risk Processes

AbstractIn this paper we use Fourier/Laplace transforms to evaluate numerically relevant probabilities in ruin theory as an application to insurance. The transform of a function is split in two: the real and the imaginary parts. We use an inversion formula based on the real part only, to get the original function.By using an appropriate algorithm to compute integrals and making use of the properties of these transforms we are able to compute numerically important quantities either in classical or non-classical ruin theory. As far as the classical model is concerned the problems considered have been widely studied. In what concerns the non-classical model, in particular models based on more general renewal risk processes, there is still a long way to go. In either case the approach presented is an easy method giving good approximations for reasonable values of the initial surplus.To show this we compute numerically ruin probabilities in the classical model and in a renewal risk process in which claim inter-arrival times have an Erlang(2) distribution and compare to exact figures where available. We also consider the computation of the probability and severity of ruin in the classical model.

Ruin Problems: Simulation or Calculation?

1996 ◽  
Vol 2 (3) ◽  
pp. 727-740 ◽  
Author(s):  
D.C.M. Dickson ◽  
H.R. Waters
Keyword(s):  
Life Insurance ◽  
Ruin Probabilities ◽  
Recursive Procedure ◽  
Probability Of Ruin ◽  
Advantages And Disadvantages ◽  
Simulation Techniques ◽  
Time Period ◽  
Insurance Portfolio ◽  
Aggregate Claims

ABSTRACTIn this paper we use a case study of a non-life insurance portfolio to demonstrate how recent research in ruin theory can be applied to solvency problems. By approximating the aggregate claims distribution for the portfolio by a translated gamma distribution, we estimate ruin probabilities through a recursive procedure when the insurer earns investment income on its surplus. We also show the results of applying simulation techniques to this problem, and discuss some advantages and disadvantages of simulation as a means of assessing ruin probabilities. Finally we discuss the calculation of the probability of ruin at the end of a specified time period.

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 On a discrete-time risk model with claim correlated premiums

2015 ◽  
Vol 9 (2) ◽  
pp. 322-342 ◽  
Author(s):  
Xueyuan Wu ◽  
Mi Chen ◽  
Junyi Guo ◽  
Can Jin
Keyword(s):  
Discrete Time ◽  
Risk Model ◽  
Insurance Industry ◽  
Joint Probability ◽  
Ruin Probabilities ◽  
Deficit At Ruin ◽  
Probability Of Ruin ◽  
Numerical Examples ◽  
Car Insurance ◽  
The Impact

AbstractThis paper proposes a discrete-time risk model that has a certain type of correlation between premiums and claim amounts. It is motivated by the well-known bonus-malus system (also known as the no claims discount) in the car insurance industry. Such a system penalises policyholders at fault in accidents by surcharges, and rewards claim-free years by discounts. For simplicity, only up to three levels of premium are considered in this paper and recursive formulae are derived to calculate the ultimate ruin probabilities. Explicit expressions of ruin probabilities are obtained in a simplified case. The impact of the proposed correlation between premiums and claims on ruin probabilities is examined through numerical examples. In the end, the joint probability of ruin and deficit at ruin is also considered.

THE EFFICIENT COMPUTATION AND THE SENSITIVITY ANALYSIS OF FINITE-TIME RUIN PROBABILITIES AND THE ESTIMATION OF RISK-BASED REGULATORY CAPITAL

2016 ◽  
Vol 46 (2) ◽  
pp. 431-467
Author(s):  
Mark S. Joshi ◽  
Dan Zhu
Keyword(s):  
Finite Time ◽  
Risk Model ◽  
Structural Parameters ◽  
Classical Model ◽  
Management Strategies ◽  
Rare Event ◽  
Insurance Companies ◽  
Ruin Probabilities ◽  
Efficient Computation ◽  
Regulatory Capital

AbstractSolvency regulations require financial institutions to hold initial capital so that ruin is a rare event. An important practical problem is to estimate the regulatory capital so the ruin probability is at the regulatory level, typically with less than 0.1% over a finite-time horizon. Estimating probabilities of rare events is challenging, since naive estimations via direct simulations of the surplus process is not feasible. In this paper, we present a stratified sampling algorithm for estimating finite-time ruin probabilities. We further introduce a sequence of measure changes to remove the pathwise discontinuities of the estimator, and compute unbiased first and second-order derivative estimates of the finite-time ruin probabilities with respect to both distributional and structural parameters. We then estimate the regulatory capital and its sensitivities. These estimates provide information to insurance companies for meeting prudential regulations as well as designing risk management strategies. Numerical examples are presented for the classical model, the Sparre Andersen model with interest and the periodic risk model with interest to demonstrate the speed and efficacy of our methodology.

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 Exponential Behavior in the Presence of Dependence in Risk Theory

2006 ◽  
Vol 43 (1) ◽  
pp. 257-273 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Jef L. Teugels
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Finite Time ◽  
Risk Theory ◽  
Ruin Probabilities ◽  
Exponential Behavior ◽  
Exponential Estimates ◽  
Insurance Portfolio ◽  
Actual Size ◽  
Dependence Structures

We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.

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