Anna Castañer

An Analysis of Black-Box Optimization Problems in Reinsurance: Evolutionary-Based Approaches

Black-box optimization problems (BBOP) are de ned as those optimization problems in which the objective function does not have an algebraic expression, but it is the output of a system (usually a computer program). This paper is focussed on BBOPs that arise in the eld of insurance, and more speci cally in reinsurance problems. In this area, the complexity of the models and assumptions considered to de ne the reinsurance rules and conditions produces hard black-box optimization problems, that must be solved in order to obtain the optimal output of the reinsurance. The application of traditional optimization approaches is not possible in BBOP, so new computational paradigms must be applied to solve these problems. In this paper we show the performance of two evolutionary-based techniques (Evolutionary Programming and Particle Swarm Optimization). We provide an analysis in three BBOP in reinsurance, where the evolutionary-based approaches exhibit an excellent behaviour, nding the optimal solution within a fraction of the computational cost used by inspection or enumeration methods.

Ruin problems for a discrete time risk model with non-homogeneous conditions

This paper is concerned with a non-homogeneous discrete time risk model where premiums are fixed but non-uniform, and claim amounts are independent but non-stationary. It allows one to account for the influence of inflation and interest and the effect of variability in the claims. Our main purpose is to develop an algorithm for calculating the finite time ruin probabilities and the associated ruin severity distributions. The ruin probabilities are shown to rely on an underlying algebraic structure of Appell type. That property makes the computational method proposed quite simple and efficient. Its application is illustrated through some numerical examples of ruin problems. The well known Lundberg bound for ultimate ruin probabilities is also reexamined within such a non-homogeneous framework.

Optimal Stop-Loss Reinsurance: A Dependence Analysis

The stop-loss reinsurance is one of the most important reinsurance contracts in the insurance market. From the insurer point of view, it presents an interesting property: it is optimal if the criterion of minimizing the variance of the cost of the insurer is used. The aim of the paper is to contribute to the analysis of the stop-loss contract in one period from the point of view of the insurer and the reinsurer. Firstly, the influence of the parameters of the reinsurance contract on the correlation coefficient between the cost of the insurer and the cost of the reinsurer is studied. Secondly, the optimal stop-loss contract is obtained if the criterion used is the maximization of the joint survival probability of the insurer and the reinsurer in one period.

Discrete Schur-constant models

This paper introduces a class of Schur-constant survival models, of dimension n, for arithmetic non-negative random variables. Such a model is defined through a univariate survival function that is shown to be n-monotone. Two general representations are obtained, by conditioning on the sum of the n variables or through a doubly mixed multinomial distribution. Several other properties including correlation measures are derived. Three processes in insurance theory are discussed for which the claim interarrival periods form a Schur-constant model.

Equilibrium distributions and discrete Schur-constant models

This paper introduces Schur-constant equilibrium distribution models of dimension n for arithmetic non-negative random variables. Such a model is defined through the (several orders) equilibrium distributions of a univariate survival function. First, the bivariate case is considered and analyzed in depth, stressing the main characteristics of the Poisson case. The analysis is then extended to the multivariate case. Several properties are derived, including the implicit correlation and the distribution of the sum.

Evaluación de las tarifas de las pensiones de accidentes de trabajo y enfermedades profesionales (2011-2015) (Are the Rates Levied to Cover the Cost of Pensions for Occupational Injury and Disease Actuarially Fair? An Analysis for the Period 2011-2015))

Spanish Abstract: Este trabajo analiza si las tarifas que se aplican para la determinacion del denominado “capital coste” de pensiones derivadas de los accidentes de trabajo (AT) y enfermedades profesionales (EP) son las adecuadas, y si la falta de actualizacion periodica ha producido distorsiones financieras entre los actores implicados. A partir de los datos de la Muestra Continua de Vidas Laborales (MCVL) y de la evolucion de la longevidad de la poblacion general en Espana, se estiman tablas de mortalidad para las contingencias de viudedad e invalidez, que combinadas con los parametros economicos y financieros apropiados, permiten construir las bases tecnicas actuarialmente justas para cada uno de los anos objeto de analisis. Los resultados obtenidos sugieren que las tarifas aplicadas en el periodo 2011-2014 han sido muy superiores a las que se deberian haber aplicado, lo que habria originado una transferencia encubierta de recursos de las Mutuas Colaboradoras de la Seguridad Social (MCSS) hacia la Tesoreria General de la Seguridad Social (TGSS). Se concluye que es urgente articular un procedimiento de revision periodica anual que adapte los elementos fundamentales de la tarifa a las cambiantes condiciones financieras, economicas y demograficas.English Abstract: This paper analyses whether the rates levied to cover the actuarial cost of pensions deriving from occupational injury and disease are the appropriate ones, or whether the absence of regular updating has given rise to financial imbalances between the parties involved. Using data from the Continuous Sample of Working Lives and the evolution of life expectancy for the general population in Spain, we estimate mortality tables for widowhood and invalidity pensions which, combined with the appropriate economic and financial parameters, will enable us to create an actuarially fair technical basis for each year to be analysed. The results obtained suggest that the rates levied during the period 2011-2014 were much higher than they should have been and resulted in a hidden transfer of resources from the Mutual Insurance Societies for Occupational Accidents and Diseases to the General Treasury of Social Security. We conclude that there is an urgent need for a periodic review procedure to ensure that the main elements of the rate established keep pace with changing financial, economic and demographic conditions.

Some optimization and decision problems in proportional reinsurance

Reinsurance is one of the tools that an insurer can use to mitigate the underwriting risk and then to control its solvency. In this paper, we focus on the proportional reinsurance arrangements and we examine several optimization and decision problems of the insurer with respect to the reinsurance strategy. To this end, we use as decision tools not only the probability of ruin but also the random variable deficit at ruin if ruin occurs. The discounted penalty function (Gerber & Shiu, 1998) is employed to calculate as particular cases the probability of ruin and the moments and the distribution function of the deficit at ruin if ruin occurs. We consider the classical risk theory model assuming a Poisson process and an individual claim amount phase-type distributed, modified with a proportional reinsurance with a retention level that is not constant and depends on the level of the surplus. Depending on whether the initial surplus is below or above a threshold level, the discounted penalty function behaves differently. General expressions for this discounted penalty function are obtained, as well as interesting theoretical results and explicit expressions for phase-type 2 distribution. These results are applied in numerical examples of decision problems based on the probability of ruin and on different risk measures of the deficit at ruin if ruin occurs (the expectation, the Value at Risk and the Tail Value at Risk).

Els grups GEI com un experiència docent orientada a estudiants repetidors: el cas de Matemàtiques I en Economia

En aquest treball estudiem les millores introduides en els Grups d’Estudi d’Intensificacio (grups GEI) durant el curs academic 2012-2013. La introduccio d’aquests grups ha estat una aposta de la Facultat d’Economia i Empresa de la Universitat de Barcelona amb l’objectiu de facilitar a l’estudiant el seguiment de les assignatures que repeteix, tenint en compte el perfil d’aquest alumnat i oferir-li una via atractiva que li faciliti el seguiment i superacio de la materia i la seva compaginacio amb les altres assignatures de primera matricula. Despres d’un primer any d’implantacio en el curs academic 2011-2012, es va redefinir el disseny d’aquests grups amb l’objectiu de corregir les ineficiencies observades. Estudiem aixi, per l’assignatura de Matematiques I del grau d’Economia, com aquests canvis han incidit en el seguiment i rendiment comparant els resultats amb els obtinguts durant el passat curs academic.