How many claims does it take to get ruined and recovered?

We consider in the classical surplus process the number of claims occurring up to ruin, by a different method presented by Stanford and Stroinski [Astin Bulletin 24 (2) (1994) 235]. We consider the computation of Laplace transforms (LTs) which ´ can allow the computation of the probability function. Formulae presented are general. The method uses the computation of the probability function of the number of claims during a negative excursion of the surplus process, in case it gets ruined. When initial surplus is zero this probability function allows us to completely define the recursion for the transform above. This uses the fact that in this particular case, conditional time to ruin has the same distribution as the time to recovery, given that ruin occurs. We consider also the computation of moments of the number of claims during recovery time, which with initial surplus zero allows us to compute the moments of the number of claims up to ruin. We end this work by giving some insight on the shapes of the two types of probability functions involved.info:eu-repo/semantics/publishedVersio

On the moments of ruin and recovery times

In this paper we consider the calculation of moments of the time to ruin and the duration of the first period of negative surplus.We present a recursive method by considering a discrete time compound Poisson process used by Dickson et al. [Astin Bull. 25 (2) (1995) 153]. With this method we will also be able to calculate approximations for the corresponding quantities in the classical model. Furthermore, for the classical compound Poisson model we consider some asymptotic formulae, as initial surplus tends to infinity, for the severity of ruin, which allow us to find explicit formulae for the moments of the time to recovery.info:eu-repo/semantics/publishedVersio

How long is the surplus below zero?

Assuming the classical compound Poisson continuous time surplus process, we consider the process as continuing if ruin occurs. Due to the assumptions presented, the surplus will go to infinity with probability one. If ruin occurs the process will temporarily stay below the zero level. The purpose of this paper is to find some features about how long the surplus will stay below zero. Using a martingale method we find the moment generating function of the duration of negative surplus, which can be multiple, as well as some moments. We also present the distribution of the number of negative surpluses. We further show that the distribution of duration time of a negative surplus is the same as the distribution of the time of ruin, given ruin occurs and initial surplus is zero. Finally, we present two examples, considering exponential and Gamma(2,β) individual claim amount distributions.info:eu-repo/semantics/publishedVersio

Cramér-Lundberg results for the infinite time ruin probability in the compound binomial model

The compound binomial model for the risk process was introduced by Gerber (1988) and is sometimes considered as a discrete time approximation to the classical compound Poisson model in continuous time; Dickson (1994) discusses this issue. After having introduced some notation in Section 2, we describe the model and set up some recursions for the in nite time ruin probability in Section 3. The core of the paper is Section 4. Here we present the Lundberg inequality and the Cramér-Lundberg approximation for the infinite time ruin probability in the compound binomial model and characterise the class of severity distributions for which the asymptotic expression is exact. Finally, in Section 5, we compare this characterisation with the analogous characterisation in the continuous time Poisson model. Although it is well known in the latter model, we give a deduction comparable with the one in Section 4.info:eu-repo/semantics/publishedVersio

Ciências actuariais : modelos para seguros

O Actuariado ou Ciências Actuariais compreende um conjunto de matérias com aplicação na actividade seguradora. Matérias como Matemática (cálculo de probabilidades, matemática financeira), Estatística, Economia, essencialmente. Por a Matemática ser uma matéria fundamental, o actuariado também é conhecido por Matemática Actuarial. A designação 'Ciências Actuariais’ parece uma designação mais apropriada por ser mais geral (pode no entanto gerar alguma controvérsia sobre o termo ‘ciência’…). Hoje em dia em dia o actuariado já não se confina à actividade seguradora, estendendo-se a áreas como as Finanças, particularmente operações de bolsa, falando-se em Actuariado Financeiro, passando pelos Planos de Pensões. Não é mais que a aplicação de métodos aplicados ao cálculo do risco na actividade seguradora a áreas como as Finanças: cálculo do risco nos mercados financeiros.info:eu-repo/semantics/publishedVersio

Ruin problems in the generalized Erlang (n) risk model

For actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (2005), by considering a general claim amount distributioninfo:eu-repo/semantics/publishedVersio

A public micro pension programme in Brazil : Heterogeneity among states and setting up of benefit age adjustment

Brazil is the 5th largest country in the world, despite of having a \High Human Development" it is the 9th most unequal country. The existing Brazilian micro pension programme is one of the safety nets for poor people. To become eligible for this benet, each person must have an income that is less than a quarter of the Brazilian minimum monthly wage and be either over 65 or considered disabled. That minimum income corresponds to approximately 2 dollars per day. This paper analyses quantitatively some aspects of this programme in the Public Pension System of Brazil. We look for the impact of some particular economic variables on the number of people receiving the benet, and seek if that impact signicantly diers among the 27 Brazilian Federal Units. We search for heterogeneity. We perform regression and spatial cluster analysis for detection of geographical grouping. We use a database that includes the entire population that receives the bene- t. Afterwards, we calculate the amount that the system spends with the beneciaries, estimate values per capita and the weight of each UF, searching for heterogeneity re ected on the amount spent per capita. In this latter calculation we use a more comprehensive database, by individual, that includes all people that started receiving a benet under the programme in the period from 2nd of January 2018 to 6th of April 2018. We compute the expected discounted benet and conrm a high heterogeneity among UF's as well as gender. We propose achieving a more equitable system by introducing `age adjusting factors' to change the benet age.info:eu-repo/semantics/publishedVersio

Text mining and ruin theory: a case study of research on risk models with dependence

This paper aims to analyze unstructured data using a text mining approach. The work was motivated in order to organize and structure research in Risk Theory. In our study, the subject to be analyzed is composed by 27 published papers of the risk and ruin theory topic, area of actuarial science. They were coded into 32 categories. For the purpose, all data was analyzed and figures were produced using the software NVivo 11 plus. Software NVivo is a specialized tool in analyzing unstructured data, although it is commonly used just for qualitative research. We used it for Quali-Quant analysisinfo:eu-repo/semantics/publishedVersio

Further developments in the Erlang(n) risk process

For actuarial aplications, we consider the Sparre–Andersen risk model when the interclaim times are Erlang(n) distributed. We first address the problem of solving an integro-differential equation that is satisfied by the survival probability and other probabilities, and show an alternative and improved method to solve such equations to that presented by Li (2008). This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and establishing a one–one relation between them and the solutions of the integro-differential equation mentioned before. Afterwards, we apply our findings above in the computation of the distribution of the maximum severity of ruin. This computation depends on the non-ruin probability and on the roots of the fundamental Lundberg’s equation. We illustrate and give explicit formulae for Erlang(3) interclaim arrivals with exponentially distributed single claim amounts and Erlang(2) interclaim times with Erlang(2) claim amounts. Finally, considering an interest force, we consider the problem of calculating the expected discounted dividends prior to ruin, finding an integro-differential equation that they satisfy and solving it. Numerical examples are also provided for illustrationinfo:eu-repo/semantics/publishedVersio

Ruin problems and dual events

Dickson (1992) uses dual events to explain results relating to the distribution of the surplus immediately prior to ruin in the classical surplus process. In this paper we show that dual events can be used to explain other results in ruin theory. In particular we prove and explain the relationship between the density of the surplus immediately prior to ruin, and the joint density of the surplus immediately prior to ruin and the severity of ruininfo:eu-repo/semantics/publishedVersio