This article presents a dataset of anonymised motor insurance policies from a Spanish insurance company covering the period 2022–2024. The dataset contains 354,140 policy-year observations and >185,000 unique insured policies per year, with a total of 47 variables. The dataset includes detailed information on policy and insured characteristics, driver and vehicle characteristics, as well as financial information such as premiums, exposure and claims. A systematic data processing and validation procedure was applied to ensure internal consistency, including the removal of duplicate policy-year records, standardisation of categorical variables and validation of numerical ranges. A key feature of the dataset is the detailed disaggregation of premiums, exposures, claim counts and incurred amounts by insurance coverage, including third-party liability (material and bodily injury), property damage, theft, fire, glass, legal protection and occupants. This structure supports reuse in actuarial pricing, risk modelling and insurance research.

Since the Lee and Carter [57] model, many researchers have focused on forecasting age-specific mortality rates. Recent research highlights the use of mortality improvements to increase forecasting accuracy. We propose a mortality improvement model inspired by interest rate modeling, where the changes in the mortality curve depend linearly on a reduced number of age-specific or “key” age(s). These variables are directly observable and capture the general mortality trend. Previous versions used one key age, whereas we introduce a second factor to improve the explanatory and forecasting power. Across populations, the first key age is consistently around 85, while the second is near age 30. We compare the accuracy of the models with seven benchmark mortality models in six populations, including the pandemic period. The results show that including a second factor enhances performance only when it is correlated with a sufficiently wide range of age-specific rates.

Predicting and understanding longevity have always been a major concern in the fields of demography and actuarial science. Accurate mortality projections at the country level enable governments to address issues such as the structure of pension schemes, public healthcare policies, or even solve macroeconomics problems. Similarly, precise mortality forecast at the individual firm level are essential for insurance companies to effectively price their life insurance products, establish reserve funds, and calculate their Solvency Capital Requirement (SCR). However, understanding mortality at the company level involves challenges due to a lack of sufficient large datasets which makes impossible to fit the most popular mortality models in the actuarial literature. Consequently, insurance companies tend to employ simpler methods which often involves the underestimation (overestimation) of mortality (longevity) in saving products and hence, placing too high margins on their estimates, resulting in a higher cost of solvency capital (SCR) and lower profitability. The objective of this paper is to fit and forecast the mortality of the insured population using various procedures that take into account the differences between the insured and national populations. For this purpose, we will recover the piggy-back methodology, whilst introducing a novel alternative approach. The paper provide a numerical illustration of how these methodologies work using data from a Spanish life insurance company. Additionally, we assess the potential impact of applying these proposals on SCR for this particular portfolio.

Fitting and forecasting the age pattern of mortality has been of interest to researchers and practitioners for a century and a half. In fact, many researchers have focused on the decomposition of the life length lifespan into homogeneous age groups for which a similar pattern of mortality can be discerned, and then considered the secular dynamics of these age groups. In this paper, we propose to use a mortality improvement rate model where the changes in mortality rates are linearly related to changes in three target rates corresponding to three 'key ages'. These ages represent age groups where mortality evolves according to a specific pattern. Thus, we decide to fit and forecast mortality for the entire age range using these three key ages, chosen based on their demographic meaning. The key age model is fitted to seven populations and two genders, and the in-sample and out-of-sample outcomes are compared to two mortality models used as benchmark. Despite its small number of parameters, the model shows good in-sample and out-of-sample accuracy.

This article presents the CvmortalityMult R package, a novel tool designed for modelling, forecasting and evaluating mortality models for several populations. The package facilitates the fitting and forecasting of multipopulation mortality models, providing accurate projections in an increasingly interconnected world characterized by minimal or no borders between countries. By incorporating different cross-validation (CV) techniques, the package allows for the assessment of the forecasting accuracy of multipopulation mortality models for specific countries or regions within a country. Through an empirical application to Spanish regions, we demonstrate the efficacy and simplicity of the CvmortalityMult R package in selecting and evaluating multipopulation mortality models. By providing accessible tools for mortality modelling, forecasting and testing, this package stands out as a valuable resource for advancing the understanding and forecasting of mortality trends across diverse populations. Its contributions extend to enhancing decision-making in critical fields such as life insurance, public health, and pension plan sustainability.

An empirical question that has motivated demographers is whether there is convergence or divergence in mortality/longevity around the world. The epidemiological transition is the starting point for studying a global process of mortality convergence. This manuscript aims to provide an update on the concept of mortality convergence/divergence. We perform a comprehensive examination of nine different mortality indicators from a global perspective using clustering methods in the period 1990-2030. In addition, we include analyses of projections to provide insights into prospective trajectories of convergence clubs, a dimension unexplored in previous work. The results indicate that mortality convergence clubs of 194 countries by sex resemble the configuration of continents. These five clubs show a common steady upward trend in longevity indicators, accompanied by a progressive reduction in disparities between sexes and between groups of countries. Furthermore, this paper shows insights into the historical evolution of the convergence clubs in the period 1990-2020 and expands their scope to include projections of their expected future evolution in 2030.

The continuous growth in life expectancy, besides to the difficult economic and financial situation of the public pension system in Spain, makes reverse mortgages an attractive solution for providing additional income to retirees. However, despite being almost 20 years old, the Spanish market remains immature. Consequently, providers face significant risks, due to factors such as interest rates, housing prices, and longevity. Numerous tourists visit Spain, and many retire there, obtaining legal residence. Therefore, lenders could be interested in marketing reverse mortgages to foreign residents. Nevertheless, the longevity risk faced by these lenders may differ depending on the nationality of the borrower, and profits and losses could vary. Consequently, we propose a methodology for comparing the pricing of reverse mortgages in Spain by considering differences in longevity risk. Specifically, we calculate the amount offered by three types of reverse mortgages to customers of different nationalities, genders, and ages with contracts made in Spain. Our conclusions are pertinent to Spanish lenders since the results indicate that, in general, a Spanish lender would assume a slightly larger risk when lending reverse mortgages to borrowers of the selected nationalities, regardless of other considerations, such as legal issues, which are not addressed in this article.

In this paper, we propose a simple dynamic mortality model to fit and forecast mortality rates for measuring longevity and mortality risks. This proposal is based on a methodology for modelling interest rates, which assumes that changes in spot interest rates depend linearly on a small number of factors. These factors are identified as interest rates with a given maturity. Similarly, we assume that changes in mortality rates depend linearly on changes in a specific mortality rate, which we call the key mortality rate. One of the main advantages of this model is that it allows the development of an easy to implement methodology to measure longevity and mortality risks using simulation techniques. Particularly, we employ the model to calculate the Value-at-Risk and Conditional-Value-at-Risk of an insurance product testing the accuracy and robustness of our proposal using out-of-sample data from six different populations.

Several recent research papers have suggested using improvement mortality rates models instead of directly examining mortality rates to fit and forecast mortality. Modelling improvement mortality rates has been a common practice in actuarial companies, often used to construct new life tables or to assess longevity risk for insurers and pension schemes. Therefore, in this study, we align with this branch of literature by adapting the improvement mortality model proposed by Atance and Navarro (Financ Innov 10:61, 2024), where improvements in mortality rates are assumed to linearly depend on a small number of key age mortality rates. Specifically, we consider two alternative hypotheses for the number of deaths (Binomial and Poisson) to model mortality improvement rates. We employed the maximum likelihood criterion to estimate model parameters and identify the key mortality rate. Additionally, we propose using a Cox-Ingersoll-Ross (CIR) process to project the expected values of mortality rates. We present both in-sample and out-of-sample measures of fitting accuracy and forecasting ability for this model. We compare it with several alternative mortality models, using data from Spain and Italy for the age range 50–99 during the period 1975 to 2019.

We employ a GARCH-type model to jointly estimate returns, conditional variance and skewness and show that conditional skewness outperforms sample skewness and conditional and sample variance in predicting future Bitcoin returns. Interestingly, the results show that the relationship between conditional skewness and future Bitcoin returns is different depending on the sample period. In the first subsample (2018–2020), a period of relative calm in the Bitcoin market, the relationship is negative, which is in line with that found in the literature. However, in the second subsample (2021–2022), a period of major turmoil in the Bitcoin market, the relationship is positive, which is consistent with that found in previous papers on the relationship between conditional market skewness and future index returns during crisis periods. Based on these results, a dynamic buy and sell strategy of buying or selling Bitcoin based on the estimated conditional skewness is proposed. This dynamic strategy outperforms a static buy-and-hold strategy. The profitability of this strategy can be viewed as the reward that investors demand for bearing the risk associated with the changing conditions in the cryptocurrency market that generate time-varying expected returns.

A reversal in the trend seen since 2010 of a reduction in adult mortality holds significant economic and financial implications. Mortality and longevity models applied in 2010 projected central death rates that have not materialized, resulting in measurable economic effects. This study examines the economic consequences of two financial-actuarial products— a year-term life insurance and life annuities— in light of the worsening central death rates within the 55-70 age bracket. Conducted across twelve countries (Australia, Canada, Denmark, France, Greece, Germany, Italy, Japan, Spain, Sweden, the United Kingdom, and the United States), the research demonstrates substantial cost savings in annuity products and cost overruns in risk insurance products.

The demographic perspective in Spain highlights the need to incorporate new alternatives that allow the sustainability of the welfare state. Clearly, one of the main solutions will be the reverse mortgage, which allows the important real estate savings of the elderly to realese and to procure income complementary to public pensions. This article analyzes, from the point of view of longevity risk, the impact between the use of sex distinct mortality tables or unisex tables, showing he importance of global portfolio management by the bank.

La perspectiva demográfica en España pone de manifiesto la necesidad de incorporar nuevas alternativas que permitan la sostenibilidad del estado de bienestar. Claramente, una de las principales soluciones será la hipoteca inversa, que permite hacer líquido el importante ahorro inmobiliario de los mayores y, así, procurar ingresos complementarios a las pensiones públicas. El presente artículo analiza, desde el punto de vista del riesgo de longevidad, el impacto entre la utilización de tablas de mortalidad diferenciadas o tablas conjuntas, poniéndose de manifiesto la importancia de la gestión global de la cartera por parte de la entidad financiera.

We develop a model to construct dynamic life tables based on the idea that the behavior of whole life table can be explained by a reduced number of factors. These factors are identified with some mortality rates at specific ages. These key mortality rates and model parameters estimates are obtained by applying a maximum likelihood criteria under the hypothesis of a binomial distribution of the number of deaths. We develop the single factor version of the model, which is implemented to the male USA population. The model is compared with a set of alternative well-known life tables models. To test the forecasting ability of the model we apply a battery of tests using out of sample data. Despite its simplicity, the outcomes indicate that this model is not outperformed by other more complex mortality models. Other important advantage of this model is that it can be easily implemented to address some longevity risk linked problems in the context of Solvency II.

Given the number of mortality models that have been considered in the literature, it is difficult to choose one model to forecast the probabilities of deaths. In this paper, we use the resampling methods to meet the mortality model that has a better forecasting ability. These techniques are a statistical tool that allows assessing the predictive performance of different models and which have not been used to compare mortality models. We employ four resampling methods that test the forecasting ability of three variations of the original Lee-Carter model in several European countries. The aim of this paper to compare different mortality models in terms of forecasting ability in the population studied by applying the resampling methods.

The accuracy of the predictions of age-specific probabilities of death is an essential objective for the insurance industry since it dramatically affects the proper valuation of their products. Currently, it is crucial to be able to accurately calculate the age-specific probabilities of death over time since insurance companies’ profits and the social security of citizens depend on human survival; therefore, forecasting dynamic life tables could have significant economic and social implications. Quantitative tools such as resampling methods are required to assess the current and future states of mortality behavior. The insurance companies that manage these life tables are attempting to establish models for evaluating the risk of insurance products to develop a proactive approach instead of using traditional reactive schemes. The main objective of this paper is to compare three mortality models to predict dynamic life tables. By using the real data of European countries from the Human Mortality Database, this study has identified the best model in terms of the prediction ability for each sex and each European country. A comparison that uses cobweb graphs leads us to the conclusion that the best model is, in general, the Lee–Carter model. Additionally, we propose a procedure that can be applied to a life table database that allows us to choose the most appropriate model for any geographical area.

The paper deals with the mortality risk evolution and presents a one-factor model explaining the dynamics of all mortality rates. The selected factor will be the mortality rate at the key age, and an empirical study involving males and females in France and Spain reveals that the present approach is not outperformed by more complex factor models. The key age seems to reflect several advantages with respect to other factors available in the literature. Actually, it is totally observable, and the methodology may be easily extended so as to incorporate more factors (more key ages), a cohort effect, specific mortality causes or specific ages. Furthermore, the choice of a key age as an explanatory factor is inspired by former studies about the dynamics of interest rates which allows us to draw on the model in order to address some longevity risk-linked problems. Indeed, one only has to slightly modify some interest rate-linked methodologies. Illustrative examples will be given.