Ruin probabilities for open portfolios with a Bonus Malus System.
23th International Congress on Insurance: Mathematics and Economics (IME) Munich 2019
Authors: Lourdes Afonso, Gracinda Guerreiro, Rui Cardoso, Alfredo Egídio dos Reis.
Abstract: We study the impact of experience rating in finite and continuous time ruin probabilities, considering an open portfolio formulation. We measure the impact on ruin probabilities of modeling Bonus-Malus Systems (BMS) for an automobile portfolio using the classical risk model on an open portfolio environment. An open environment scenario brings new challenges when compared to a classical BMS approach, and subsequently on ruin probabilities. The base model was taken from [2] and already summarized in [1]. As for the modeling of BMS in open portfolios we will follow the results on [4] and some developments obtained in [3]. We illustrate and discuss the impact of the proposed formulation on the initial surplus needed to target a given ruin probability. We show that there is a significant impact on capital requirements, by reducing the initial surplus needed to maintain the same level (a small figure) of the ruin probability.
[1] Afonso, L. B., Cardoso, R. C., Egídio dos Reis, A. D. and Guerreiro, G. R. (2017). Measuring the impact of a bonus-malus system in finite and continuous time ruin probabilities for large portfolios in motor insurance. [ ASTIN Bulletin, vol. 47(2), pp. 417-435.
[2] Afonso, L. B., Egídio dos Reis, A. D., and Waters, H. R. (2009). Calculating continuous time ruin probabilities for a large portfolio with varying premiums. ASTIN Bulletin, vol. 39(1), pp. 117-136.
[3] Esquível, M.L., Guerreiro, Fernandes, J.M., Guerreiro, G.R. (2014). On the Evolution and Asymptotic Analysis of Open Markov Populations: Application to Consumption Credit. Stochastic Models, vol. 30(3) pp. 365-389.
[4] Guerreiro, G.R., Mexia, J.T. and Miguens, M.F.(2014). Statistical approach for open Bonus Malus. ASTIN Bulletin, vol. 44(01), pp. 63-83.
Heuristic approach to evaluate the fire risk sub-module in Solvency II. 21th International Congress on Insurance: Mathematics and Economics (IME) Vienna 2017
Slides: IME2017
Authors: Afonso, L.B.; Fradinho, J, Chibeles Martins, N.; Gomes, I.
Abstract: The Commission Delegated Regulation (EU) 2015/35 of 10 October 2014 contain implementing rules for Solvency II. In order to apply the requirements set out in article 132 for Non-life catastrophe risk sub-module; Man-made catastrophe risk sub-module; Fire risk sub-module it will be necessary to evaluate the largest sum insured of all buildings that are partly or fully located within a radius of 200 meters.
Consequently, Operational Research techniques are getting increasingly important as a decision support tools for Actuarial Analysis. Consider the following problem: given a list of clients inside a territory it is necessary to nd the centre of the 200 meters radius circle that aggregates the largest sum insured considering all buildings located partly or fully inside that circle. That problem can be formulated as a Binary Linear Programming Problem which raises the usual computational complexity issues. In this work the authors propose a formulations for the mentioned problem and a heuristic approach for solving it.
Keywords: Solvency II; Fire risk sub-module; Binary Linear Programming; heuristics.
Bonus malus systems and finite and continuous time ruin probabilities in motor insurance , IME 2015, June 2015, Liverpool, England.
Slides: IME2015
Authors: Afonso, L.B., Cardoso, R., Egídio dos Reis, A., Guerreiro, G.R.
Abstract: In motor insurance, we consider usually two types of ratemaking, a priori and a posteriori. Here, premium calculation is based on past experience and volatility is higher when compared to classical procedures where premium are paid continuously at a constant rate. Typically, the ruin probabilities are computed according to the classical Cramér-Lundberg model. Afonso et al. (2009) consider a model applicable to large portfolios where a varying premium is used by means of a mix of calculation and simulation. That procedure differs from the usual literature and allow us to obtain fast and reliable results in a finite and continuous time horizon. Those ideas can be brought for application in motor insurance ratemaking (experience rating), for two main reasons: First because premium calculation is applied for large portfolios, common in motor insurance, second because premium calculation is based on the past claim record. However, the model needs to be changed to fit in the features common in motor insurance.
Common experience rating models produce variations in annual premiums as function of the past claim number record, and not as a function of the past aggregate claims. This is approached by a Markov chain procedure. Only the number of claims is essential to determine the next rating class and calculate the applicable premium. However, aggregate claims are necessary to compute ruin probabilities for the portfolio.
We will measure the impact of a bonus malus system (BMS) in the ruin probabilities, considering different known optimal scales (e.g. Norberg, Borgan et al., Gilde and Sundt and Andrade e Silva and Centeno), as well as real commercial scales.
In these scenarios we will use real data from automobile third-party liability portfolios of a Portuguese insurer.
The methodologies and tools to be used are Markov chains, compound Poisson process, translated gamma distributions approximations, optimal premium scales and simulation.Keywords: Ruin probabilities; bonus malus system; motor insurance.
Measuring the impact of a bonus malus system in finite and continuous time ruin probabilities, for large portfolios in motor insurance, 2nd European Actuarial Journal Conference, September 2014, Vienna, Austria.
Authors: Afonso, L.B., Cardoso, R., Egídio dos Reis, A., Guerreiro, G.R.,
Avaliação de Responsabilidades em Fundos de Pensões, Seminário Gestão do Risco na Banca & Seguros, Universidade da Beira, Moçambique, Julho de 2013. Presentation
Author: Afonso, L. B.
Using Weighted Distributions to Model Operational Risk Bruxells Feb 2012 AFMath Conf
Authors: Afonso, L.B.; Corte Real, P.
Abstract: The quantification of operational risk has, much more than other types of risk which banks and insurers are obliged to manage, to deal with various concerns regarding data. Several studies document some of those concerns. One of the main questions that worries both researchers and practitioners is the bias of data for the operational losses amounts recorded.
We support the assertions made by several authors and defend that the bias concern is a very serious problem when modeling operational losses data. The bias is presented in all the databases, not only in the commercial databases provided by various vendors, but also in databases where the data for operational losses is collected and compiled internally.
We show that it’s possible, based on mild assumptions on the internal procedures put in place to manage operational losses, to make parametric inference using loss data statistics. We estimate the parameters for the losses amounts, taking in consideration the bias that, not being considered, generates a twofold error in the estimators for the mean loss amount and the total loss amount, the former being overvalued and the last undervalued.
In this paper, we do not consider the existence of a threshold for which, all losses above, are reported and are available for analysis and estimation procedures. We follow a different approach to the parametric inference. We consider that the probability that a loss is reported and ends up recorded for analysis, increases with the size of the loss, what causes the bias in the database but, at the same time, we don’t consider that a threshold exists, above which, all losses are recorded and available for analysis, hence, no loss has probability one of being recorded, in what we defend is a realist framework. We deduce the general formulae, present some simulations for common theoretical distributions used to model (operational) losses amounts and estimate the impact for not considering the bias factor when estimating the value at risk.
Numerical evaluation of continuous time ruin probabilities for a risk process with credibility based premiums. Sesimbra Outubro 2009 – SPE2009
Abstract: pdf
Speaker: Afonso, L.B.
Credibilidade e Ruína. Lisboa Maio 2008 – 1º Congresso Ibérico de Actuários
Abstract: pdf
Speaker: Afonso, L.B. (in Portuguese)
Evaluation of ruin probabilities for surplus processes with credibility and surplus dependent premiums. Lisboa 11 Fevereiro 2008- ISEG-UTL
Abstract: PhD Viva.
Speaker: Afonso, L.B.
Credibilidade e Ruína. Lisboa 22 Novembro 2007 – CEMAPRE2008
Abstract: 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. This calculation can be approximated either through Brownian motion or a translated gamma by approximating the distribution of the aggregate claim amounts. We check the accuracy of our method by comparing the results applied to the classical risk process with those by Wikstad (1971) and Seal (1978) in finite and continuous time. We also check its accuracy in the case of exponential and mixed exponential claim amounts by choosing a very long time horizon and comparing results with exact results for infinite time ruin probability. We consider a model where the aggregate claims have a compound Poisson distribution with either a fixed or a variable Poisson parameter. Also, we consider a portfolio of risks which satisfy the assumptions of the Bühlmann credibility model, where the pure premium is updated each year in accordance with the past experience.
Speaker: Afonso, L.B. (in Portuguese)
Continuous time ruin probablilities for a portfolio with credibility-adjusted premiums.Piraeus 10-12 July 2007 – IME2007
Abstract:We present results for the ruin probabilities in continuous and finite time for a classical risk process where the annual premium are updated according to some standard credibility models. We calculate the ruin probability of the portfolio and for each one of the risks separately. The results are obtained numerically using simulated annual aggregate claims. The within-year ruin probability are calculated using a translated gamma distribution approximation for aggregate claim amounts.
Speaker: Afonso, L.B.
A model for numerical evaluation of continuous time ruin probabilities with a variable premium rate. Orlando 19-22 June 2006 – ASTIN 2007
Abstract: In this paper we present a method for the numerical evaluation of the ruin probability in continuous, finite or infinite time for a classical risk process where the premium can change from year to year. 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 first a Brownian motion approximation and, secondly, a translated gamma distribution approximation for the aggregate claim amount.
Speaker: Egídio dos Reis, A. D.
Numerical evaluation of continuous time ruin probabilities for a risk process with credibility based premiums Leuven 18-20 July 2006 – IME2006
Abstract: We present a method for the numerical evaluation of ruin probabilities in continuous and finite time for a classical risk process where the premium can change from year to year. 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 first a Brownian motion approximation and, secondly, a translated gamma distribution approximation for the aggregate claim amount.
We consider this approach in the case where the annual premium is updated according to one of the standard credibility models, such as the Buhlmann-Straub model. We also explore the case where the premium at the start of each year is a function of the surplus level at that time.
Speaker: Afonso, L.B.
