Alternative estimation methods for survival models

In this thesis, we will address the problem of estimating the parameters of two types of three parameter survival models for the force of mortality, the Makeham model and the Weibull model. The existing methods for estimating these parameters by the method of least-squares include using a log-transformation on the force of mortality data. We will propose another method of estimating these parameters that uses linear regression, and finding the least-squares estimates of the parameters by using the golden section search procedure for computer minimization

A comparison of parametric models for mortality graduation. Application to mortality data of the Valencia Region (Spain)

The parametric graduation of mortality data has as its objective the satisfactory estimation of the death rates based on mortality data but using an age-dependent function whose parameters are adjusted from the crude rates obtainable directly from the data. This paper proposes a revision of the most commonly used parametric methods and compares the results obtained with each of them when they are applied to the mortality data for the Valencia Region. As a result of the comparison, we conclude that the Gompertz-Makeham functions estimated by means of generalized linear models lead to the best results. Our working method is of additional interest for being applicable to mortality data for a wide range of ages from any geographical conditions, allowing us to select the most appropriate life table for the case in hand.La gradualització paramètrica de dades de mortalitat té com a objectiu l'estimació correcta de taxes de mort a partir de les dades de mortalitat mitjançant una funció que depèn de l'edat, els paràmetres de la qual s'ajusten a partir de les taxes "brutes" de mortalitat obtingudes directament de les dades. Aquest article presenta una revisió dels models paramètrics més habituals i la seua aplicació a les dades de mortalitat del País Valencià. Com a conseqüència de la comparació dels resultats obtinguts amb els diferents models, es conclou que les funcions de Gompertz-Makeham estimades mitjançant models lineals generalitzats condueixen als millors resultats. El mètode de treball que es presenta té interès suplementari per ser aplicable a dades de mortalitat per a un ampli rang d'edats i per a qualsevol àmbit geogràfic, permetent en cada cas seleccionar la millor taula de vida

A comparison of parametric models for mortality graduation. Application to mortality data of the Valencia Region

[EN] The parametric graduation of mortality data has as its objective the satisfactory estimation of the death rates based on mortality data but using an age-dependent function whose parameters are adjusted from the crude rates obtainable directly from the data. This paper proposes a revision of the most commonly used parametric methods and compares the results obtained with each of them when they are applied to the mortality data for the Valencia Region. As a result of the comparison, we conclude that the Gompertz-Makeham functions estimated by means of generalized linear models lead to the best results. Our working method is of additional interest for being applicable to mortality data for a wide range of ages from any geographical conditions, allowing us to select the most appropriate life table for the case in hand.Debón Aucejo, AM.; Montes-Suay, F.; Sala-Garrido, R. (2005). A comparison of parametric models for mortality graduation. Application to mortality data of the Valencia Region. SORT. Statistics and Operations Research Transactions. 29(2):269-288. http://hdl.handle.net/10251/147780S26928829

Calculation of Tabarru Funds Using Makeham's Mortalita Law and Gompertz's Mortalita Law Using the Cost Of Insurance Method

Tabarru funds have the meaning of a collection of funds given by insurance participants as a virtue fund with sincere intentions for the purpose of helping one participant with another if one of them gets a disaster. The tabarru fund management mechanism in Indonesia uses two types of operational systems, namely the product saving system (savings) and the non-saving product system. Management with a saving product system uses a savings mechanism with 5% for management funds. Meanwhile, in the non-saving product system with a no-savings mechanism, the amount and its management are not yet known, which will cause confusion for the community in the calculation. The cost of insurance method is one method that can be used to calculate tabarru funds. This method calculates tabarru funds by multiplying the percentage of tabarru funds by the cost of coverage. The percentage of tabarru funds is searched through the mortality table, management fee, and investment level. From the management fee of 25% and the investment rate of 5%, the percentage of tabarru funds using Makeham's mortality law is  for men and for men. Meanwhile, Gompertz's law of mortality obtained  for men and for women . ©2020 JNSMR UIN Walisongo. All rights reserved

Calculation of Tabarru Funds Using Makeham's Mortalita Law and Gompertz's Mortalita Law Using the Cost Of Insurance Method

Tabarru funds have the meaning of a collection of funds given by insurance participants as a virtue fund with sincere intentions for the purpose of helping one participant with another if one of them gets a disaster. The tabarru fund management mechanism in Indonesia uses two types of operational systems, namely the product saving system (savings) and the non-saving product system. Management with a saving product system uses a savings mechanism with 5% for management funds. Meanwhile, in the non-saving product system with a no-savings mechanism, the amount and its management are not yet known, which will cause confusion for the community in the calculation. The cost of insurance method is one method that can be used to calculate tabarru funds. This method calculates tabarru funds by multiplying the percentage of tabarru funds by the cost of coverage. The percentage of tabarru funds is searched through the mortality table, management fee, and investment level. From the management fee of 25% and the investment rate of 5%, the percentage of tabarru funds using Makeham's mortality law is  for men and for men. Meanwhile, Gompertz's law of mortality obtained  for men and for women . ©2020 JNSMR UIN Walisongo. All rights reserved

Profitability study of the annuities of EY-Insurance

Mestrado em Ciências ActuariaisEste trabalho procura analisar a rentabilidade obtida com a venda de anuidades e produtos anuais renováveis, na seguradora Vida onde decorreu o estágio. As questões relacionadas com os desvios observados na mortalidade e a necessidade de encontrar um modelo de sobrevivência mais ajustado à experiência da companhia foram aspetos de crucial importância. Procurou assim encontrar-se bases técnicas mais adequadas para o cálculo de prémios e reservas, tanto para os produtos já em comercialização, como para novos produtos que venham a ser lançados, pois também a taxa de juro e as despesas foram afloradas, ainda que brevemente. Por motivos de confidencialidade de dados, procedeu-se a uma distorção dos valores reais. Isto não teve obviamente qualquer consequência do ponto de vista das metodologias e técnicas aplicadas no estudo. Estavam disponíveis dados para um período de quatro anos, na sua maioria relativos a rendas imediatas e rendas imediatas reversíveis. Com base nisso, foi possível detetar que a tábua de mortalidade mais adequada será 108.95% da GKF95, o que talvez permita eliminar a maior parte dos desvios. Em complemento, foi ainda feita uma análise de sensibilidade, com diferentes cenários, para se estudar o efeito sobre o nível das reservas das diferentes possibilidades consideradas. Um exercício final de profit testing revelou que as responsabilidades continuam insuficientemente cobertas, pelo que trabalho adicional é necessário para resolver o problema.This study aims to evaluate the profitability of the life annuities in the insurance company where the internship took place by concentrating on finding the best mortality table for the company portfolio to quote the price for the new annuity businesses and reserving for the ones already sold. The project is based on real data that was intentionally transformed for the purpose of this text because of confidentiality reasons. The distortion conceals reality in an appropriate manner and has obviously no effects on the methodologies applied. Data concerns immediate and immediate reversible life annuities for four years, since these products comprise the most significant part of the company population of policy holders. The best mortality table for this data is 108.95% of GKF95 table, by least square fitting. In order to forecast the future mortality, the Gompertz-Makeham mortality model was applied and there were no systematic evolution through time for the future mortality. A Sensitivity analysis was performed to show the effects of different scenarios on mathematical reserving. Finally, a profit testing revealed that the technical bases for the annuities are not enough to cover the liabilities. 108.95% of GKF 95 table can be assumed as the initial table and 104.29% of GKF 95 table can be assumed to hold extra reserve, in order to guarantee an adequate mathematical reserve