A Fuzzy-Random Extension of the Lee-Carter Mortality Prediction Model

The Lee-Carter model is a useful dynamic stochastic model to represent the evolution of central mortality rates throughout time. This model only considers the uncertainty about the coefficient related to the mortality trend over time but not to the age-dependent coefficients. This paper proposes a fuzzy-random extension of the Lee-Carter model that allows quantifying the uncertainty of both kinds of parameters. As it is commonplace in actuarial literature, the variability of the time-dependent index is modeled as an ARIMA time series. Likewise, the uncertainty of the age-dependent coefficients is also quantified, but by using triangular fuzzy numbers. The consideration of this last hypothesis requires developing and solving a fuzzy regression model. Once the fuzzy-random extension has been introduced, it is also shown how to obtain some variables linked with central mortality rates such as death probabilities or life expectancies by using fuzzy numbers arithmetic. It is simultaneously shown the applicability of our developments with data of Spanish male population in the period 1970-2012. Finally, we make a comparative assessment of our method with alternative Lee-Carter model estimates on 16 Western Europe populations

A random forest algorithm to improve the Lee–Carter mortality forecasting: impact on q-forward

Increased life expectancy in developed countries has led researchers to pay more attention to mortality projection to anticipate changes in mortality rates. Following the scheme proposed in Deprez et al. (Eur Actuar J 7(2):337–352, 2017) and extended by Levantesi and Pizzorusso (Risks 7(1):26, 2019), we propose a novel approach based on the combination of random forest and two-dimensional P-spline, allowing for accurate mortality forecasting. This approach firstly provides a diagnosis of the limits of the Lee–Carter mortality model through the application of the random forest estimator to the ratio between the observed deaths and their estimated values given by a certain model, while the two-dimensional P-spline are used to smooth and project the random forest estimator in the forecasting phase. Further considerations are devoted to assessing the demographic consistency of the results. The model accuracy is evaluated by an out-of-sample test. Finally, we analyze the impact of our model on the pricing of q-forward contracts. All the analyses have been carried out on several countries by using data from the Human Mortality Database and considering the Lee–Carter model

Characterization of between-group inequality of longevity in European Union countries

Comparisons of differential survival by country are useful in many domains. In the area of public policy, they help policymakers and analysts assess how much various groups benefit from public programs, such as social security and health care. In financial markets and especially for actuaries, they are important for designing annuities and life insurance products. This paper presents a method for clustering information about differential mortality by country. The approach is then used to group mortality surfaces for European Union (EU) countries. The aim of this paper is to measure between-group inequality in mortality experience in EU countries through a range of mortality indicators. Additionally, the indicators permit the characterization of each group. It is important to take into account characteristics such as sex; therefore, this study differentiates between males and females in order to detect whether their patterns and characterizations are different. It is concluded that there are clear differences in mortality between the east and west of the EU that are more important than the traditional south-north division, with a significant disadvantage for Eastern Europe, and especially for males in Baltic countries. We find that the mortality indicators have evolved in all countries in such a way that the gap between groups has been maintained, both in terms of the differences in mortality levels and variability

Longevity Basis Risk A methodology for assessing basis risk

This technical report details the methodology developed on behalf of the LBRWG to assess longevity basis risk. A user-guide which provides a high level summary of this report has also been produced. Together these documents form the key outputs of the first phase of a longevity basis risk project commissioned and funded by the IFoA and the LLMA, and undertaken on our behalf by Cass Business School and Hymans Robertson LLP

Life settlement pricing with fuzzy parameters

Existing literature asserts that the growth of life settlement (LS) markets, where they exist, is hampered by limited policyholder participation and suggests that to foster this growth appropriate pricing of LS transactions is crucial. The pricing of LSs relies on quantifying two key variables: the insured's mortality multiplier and the internal rate of return (IRR). However, the available information on these parameters is often scarce and vague. To address this issue, this article proposes a novel framework that models these variables using triangular fuzzy numbers (TFNs). This modelling approach aligns with how mortality multiplier and IRR data are typically provided in insurance markets and has the advantage of offering a natural interpretation for practitioners. When both the mortality multiplier and the IRR are represented as TFNs, the resulting LS price becomes a FN that no longer retains the triangular shape. Therefore, the paper introduces three alternative triangular approximations to simplify computations and enhance interpretation of the price. Additionally, six criteria are proposed to evaluate the effectiveness of each approximation method. These criteria go beyond the typical approach of assessing the approximation quality to the FN itself. They also consider the usability and comprehensibility for financial analysts with no prior knowledge of FNs. In summary, the framework presented in this paper represents a significant advancement in LS pricing. By incorporating TFNs, offering several triangular approximations and proposing goodness criteria of them, it addresses the challenges posed by limited and vague data, while also considering the practical needs of industry practitioners

Hybrid Lee-Carter model with adaptive network of fuzzy inference system and wavelet functions

Mortality studies are essential in determining the health status and demographic composition of a population. The Lee–Carter (LC) modelling framework is extended to incorporate the macroeconomic variables that affect mortality, especially in forecasting. This paper makes several major contributions. First, a new model (LC-WT-ANFIS) employing the adaptive network-based fuzzy inference system (ANFIS) was proposed in conjunction with a nonlinear spectral model of maximum overlapping discrete wavelet transform (MODWT) that includes five mathematical functions, namely, Haar, Daubechies (d4), least square (la8), best localization (bl14), and Coiflet (c6) to enhance the forecasting accuracy of the LC model. Annual mortality data was collected from five countries (Australia, England, France, Japan, and the USA) from 1950 to 2016. Second, we selected gross domestic product (GDP), unemployment rate (UR), and inflation rate (IF) as input values according to correlation and multiple regressions. The input variables in this study were obtained from the World Bank and Datastream. The output variable was collected from the mortality rates in Human Mortality Database. Finally, the LC model’s projected log of death rates was compared with wavelet filters and the traditional LC model. The performance of the proposed model (LC-WT-ANFIS) was evaluated based on mean absolute percentage error (MAPE) and mean error (ME). Results showed that the LC-WT-ANFIS model performed better than the traditional model. Therefore, the proposed forecasting model is capable of projecting mortality rates

Fuzzy Markovian Bonus-Malus Systems in Non-Life Insurance

Markov chains (MCs) are widely used to model a great deal of financial and actuarial problems. Likewise, they are also used in many other fields ranging from economics, management, agricultural sciences, engineering or informatics to medicine. This paper focuses on the use of MCs for the design of non-life bonus-malus systems (BMSs). It proposes quantifying the uncertainty of transition probabilities in BMSs by using fuzzy numbers (FNs). To do so, Fuzzy MCs (FMCs) as defined by Buckley and Eslami in 2002 are used, thus giving rise to the concept of Fuzzy BMSs (FBMSs). More concretely, we describe in detail the common BMS where the number of claims follows a Poisson distribution under the hypothesis that its characteristic parameter is not a real but a triangular FN (TFN). Moreover, we reflect on how to fit that parameter by using several fuzzy data analysis tools and discuss the goodness of triangular approximates to fuzzy transition probabilities, the fuzzy stationary state, and the fuzzy mean asymptotic premium. The use of FMCs in a BMS allows obtaining not only point estimates of all these variables, but also a structured set of their possible values whose reliability is given by means of a possibility measure. Although our analysis is circumscribed to non-life insurance, all of its findings can easily be extended to any of the abovementioned fields with slight modifications.University of Barcelon