In the field of mortality, the Lee–Carter based approach can be considered the milestone to forecast mortality rates among stochastic models. We could define a “Lee–Carter model family” that embraces all developments of this model, including its first formulation (1992) that remains the benchmark for comparing the performance of future models. In the Lee–Carter model, the ${\kappa }_t$ parameter, describing the mortality trend over time, plays an important role about the future mortality behavior. The traditional ARIMA process usually used to model ${\kappa }_t$ shows evident limitations to describe the future mortality shape. Concerning forecasting phase, academics should approach a more plausible way in order to think a nonlinear shape of the projected mortality rates. Therefore, we propose an alternative approach the ARIMA processes based on a deep learning technique. More precisely, in order to catch the pattern of ${\kappa }_t$ series over time more accurately, we apply a Recurrent Neural Network with a Long Short-Term Memory architecture and integrate the Lee–Carter model to improve its predictive capacity. The proposed approach provides significant performance in terms of predictive accuracy and also allow for avoiding the time-chunks’ a priori selection. Indeed, it is a common practice among academics to delete the time in which the noise is overflowing or the data quality is insufficient. The strength of the Long Short-Term Memory network lies in its ability to treat this noise and adequately reproduce it into the forecasted trend, due to its own architecture enabling to take into account significant long-term patterns.

After the World War II, developed countries experienced a constant decline in mortality. As a result, life expectancy has never stopped increasing, despite an evident deceleration in developed countries, e.g. England, USA and Denmark. In this paper, we propose a new approach for forecasting life expectancy and lifespan disparity based on the recurrent neural networks with a long short-term memory. This type of neural network leads to predicting future values of longevity indexes while maintaining the significant influence of the past trend, but at the same time adequately reproducing the recent trend into forecasting. The model was applied to five countries for two fitting periods focusing on the forecasting life expectancy and lifespan disparity, both independently and simultaneously at birth and age 65. The results were compared to the projections obtained by four different models, namely, the Double Gap, ARIMA, CoDa and Lee-Carter in the independent case and the first-order Vector Autoregression model in the simultaneous case. Our predictions seem to be coherent with historical trends and biologically reasonable, providing a more accurate portrait of the future life expectancy and lifespan disparity.

Several countries worldwide are experiencing a continuous increase in life expectancy, extending the challenges of life actuaries and demographers in forecasting mortality. Although several stochastic mortality models have been proposed in the literature, mortality forecasting research remains a crucial task. Recently, various research works have encouraged the use of deep learning models to extrapolate suitable patterns within mortality data. Such learning models allow achieving accurate point predictions, though uncertainty measures are also necessary to support both model estimate reliability and risk evaluation. As a new advance in mortality forecasting, we formalize the deep neural network integration within the Lee-Carter framework, as a first bridge between the deep learning and the mortality density forecasts. We test our model proposal in a numerical application considering three representative countries worldwide and for both genders, scrutinizing two different fitting periods. Exploiting the meaning of both biological reasonableness and plausibility of forecasts, as well as performance metrics, our findings confirm the suitability of deep learning models to improve the predictive capacity of the Lee-Carter model, providing more reliable mortality boundaries in the long run.

In this paper, we come up with an original trading strategy on Bitcoins. The methodology we propose is profit-oriented, and it is based on buying or selling the so-called Contracts for Difference, so that the investor’s gain, assessed at a given future time t, is obtained as the difference between the predicted Bitcoin price and an apt threshold. Starting from some empirical findings, and passing through the specification of a suitable theoretical model for the Bitcoin price process, we are able to provide possible investment scenarios, thanks to the use of a Recurrent Neural Network with a Long Short-Term Memory for predicting purposes.

Human lifespan increments represent one of the main current risks for governments and pension and health benefits providers. Longevity societies imply financial sustainability challenges to guarantee adequate socioeconomic conditions for all individuals for a longer period. Consequently, modelling population dynamics and projecting future longevity scenarios are vital tasks for policymakers. As an answer, the demographic and the actuarial literature have been introduced and compared to several stochastic mortality models, although few studies have thoroughly tested the uncertainty concerning mortality projections. Forecasting mortality uncertainty levels have a central role since they reveal the potential, unexpected longevity rise and the related economic impact. Therefore, the present study poses a methodological framework to backtest uncertainty in mortality projections by exploiting uncertainty metrics not yet adopted in mortality literature. Using the data from the Human Mortality Database of the male and female populations of five countries, we present some numerical applications to illustrate how the proposed criterion works. The results show that there is no mortality model overperforming the others in all cases, and the best model choice depends on the data considered.

In this paper, we consider a multi-agent portfolio optimization model with life insurance for two players with random lifetime under a dynamic game approach. Each player is a price-taker and invests in the market to maximize her own utility for consumption and bequest. The market is complete and consists of n different assets, of which $$n-1$$ n - 1 are risky with prices driven by Geometric Brownian motion, while one is risk-free. We analyze both the non-cooperative and cooperative scenarios, and by considering the family of CRRA utility functions, we determine the closed-form expressions of the optimal consumption, investment, and life insurance for both players. A sensitivity analysis is provided both to illustrate the impact of the biometric and risk aversion parameters on the optimal controls and to compare the non-cooperative strategies with the cooperative ones. As a result, we suggest that cooperation favors the consumption optimality, while non-cooperation promotes the coverage of the risk of death.