Actuarial Science Research Articles

A new asymmetric extended family: Properties and estimation methods with actuarial applications.

In the present work, a class of distributions, called new extended family of heavy-tailed distributions is introduced. The special sub-models of the introduced family provide unimodal, bimodal, symmetric, and asymmetric density shapes. A special sub-model of the new family, called the new extended heavy-tailed Weibull (NEHTW) distribution, is studied in more detail. The NEHTW parameters have been estimated via eight classical estimation procedures. The performance of these methods have been explored using detailed simulation results which have been ordered, using partial and overall ranks, to determine the best estimation method. Two important risk measures are derived for the NEHTW distribution. To prove the usefulness of the two actuarial measures in financial sciences, a simulation study is conducted. Finally, the flexibility and importance of the NEHTW model are illustrated empirically using two real-life insurance data sets. Based on our study, we observe that the NEHTW distribution may be a good candidate for modeling financial and actuarial sciences data.

Ratemaking territories and adverse selection for flood insurance

Flood insurance is key to increase resilience but remains challenging to price for actuaries. Despite recent advances in high-resolution flood risk modeling, there are serious obstacles to implementing perfect risk differentiation at the homeowner's level. Creating more homogeneous ratemaking territories or groupings is one obvious solution but they need to be designed with care to avoid introducing significant adverse selection. Adverse selection may occur in insurance whenever the pricing structure is such that it attracts too many homeowners with underpriced policies affecting in turn the viability of the product. In this paper we create ratemaking territories (contiguous or not) for flood insurance based upon clustering techniques (forcing contiguity or not) and compare their risk differentiation capabilities with postal/zip codes, geopolitical territories or floodplain boundaries. We investigate adverse selection by setting up markets made of rational homeowners seeking for flood insurance among two competing sellers using different ratemaking territories. Based upon public data from Calgary (Canada), we find that postal/zip codes and the annual flood probability do not sufficiently capture flood risk, leading to significant adverse selection. When an unsophisticated seller uses geopolitical boundaries, it attracts homeowners with expected losses 2-5 times larger than its competitors yielding a loss ratio that is twice as large. The paper thus highlights the significance of adverse selection when selling flood insurance, with implications in actuarial science, climate risk management and public policy.

On the Gompertz–Makeham law: A useful mortality model to deal with human mortality

The Gompertz–Makeham model was introduced as an extension of the Gompertz model in the second half of the 19th century by the British actuary William M. Makeham. Since then, this model has been successfully used in biology, actuarial science, and demography to describe mortality patterns in numerous species (including humans), determine policies in insurance, establish actuarial tables and growth models. In this paper, we derive some structural properties of the Gompertz–Makeham model in statistics, demography, and actuarial sciences, and present some other ones already introduced in the literature. All structural properties we provide are expressed in closed-form, which eliminates the need to evaluate them with numerical integration directly. In addition, we study the estimation of the Gompertz–Makeham model parameters through the discrete Poisson and Bell distributions. In particular, we verify that the recently introduced discrete Bell distribution can be an interesting alternative to the Poisson distribution, mainly because it is suitable to deal with over dispersion, unlike the Poisson distribution. On the basis of real mortality datasets, we compute the remaining life expectancy for several countries and verify that the Gompertz–Makeham model, especially under the Bell distribution, provides proper results to deal with human mortality in practice.

Structural models for fog computing based internet of things architectures with insurance and risk management applications

Cybersecurity risk modeling and pricing are becoming a spotlight in actuarial science and operational research. This paper pertains to the analysis of the cybersecurity risks involved in the fog computing technology which has been intensively deployed in assorted Internet of Things (IoT) applications. To this end, a class of structural models are established to study the inherent cyber risk propagation process. Under the smart home applications, we manage to compute the compromise probabilities of individual nodes explicitly. Applications of the proposed structural models in the context of cyber insurance pricing are thoroughly explored. Finally, we propose an interval method for estimating the compromise probabilities of fog network’s elements, which can be used to efficiently identify weak nodes for cybersecurity risk management.

“Numberless Little Risks”: ‘Tropical Exposure’ in Globalizing Actuarial Discourse, 1852–1947

I argue that life insurance and imperial meaning making are deeply implicated in each other. As life insurance expanded internationally in the nineteenth century, and as insurers became advocates of White settlement, they grappled with what actuarial science meant in the context of their orientalist conceptions of colonial populations. In particular, actuaries were concerned with tropical markets and the racialized/exceptionalized differences they perceived in those markets. To address these tensions, insurers attempted two strategies: (1) incorporating tropical rates as additional premiums designed to cover the “extra mortality” of tropical markets; and (2) advocating for social practices of “sanitary progress” related to public health and sanitation. These practices, framed in orientalist terms, were not adopted in any smooth manner but in fumbling and meandering ways as insurers tried to understand what kinds of lives tropical settlers might be and how those lives might be priced. They eventually liberalized life insurance rates for White settlers in tropical settings, but insurers then confronted questions on how newly socialized “native” lives might be rendered calculable. This story of tropical exposure in globalizing actuarial discourse reinforces the ways in which race and racialized/exceptionalized differences were at the core of life insurance and the calculative devices it assembled between 1852 and 1947.

Evaluating the Role of Maternal Income in Mediating the Effect of Mother's Education on under Five Child Mortality in Kenya

Aims/ Objectives: The study identified the role of maternal income in mediating the effect of a mother's education level on UFCM. Study Design: We sourced secondary data from a 2014 Kenya Health Demographic Health Survey (KHDS). We analyzed data to determine mortality among the under-five children. Place and Duration of Study: Department of mathematics and Actuarial science, the Catholic University of Eastern Africa, Nairobi, Kenya. Methodology: We then conducted regression with mediation to assess the effect of a mother's education on UFCM in the presence of a mediator, "mothers income." We also determined the direct, total and indirect effects of the independent variable in the presence of mediation. Results: The results indicate that maternal education is directly important on UFCM. Maternal education level also channeled its effects through maternal income. Change from no education level to primary education level would reduce the number of deaths by 1.9 deaths per 1000 children. Of this decrease, 0.5 fewer deaths were attributed to the maternal income pathway representing 26 percent of the total effect. Conclusion: If an intervention could improve, the maternal income of mothers at no education level to that of mothers at higher education level while keeping the other aspects of social deprivation constant we could eliminate 26% of the education level effect.

Bayesian inference for the parameters of mortality rate in the models of dependent lives with application in life insurance

In this paper, the Bayesian inference of the model of dependent lives is considered. We use the bivariate Gompertz (BGP) distribution. This new bivariate survival model and its extension have recently been proposed by Shoaee and Khorram. This model is more flexible and can be applied in the actuarial science of life insurance, and reliability theory in competing risks and shock models. As we know, the maximum likelihood estimates do not always exist and therefore cannot always be calculated. So, the Bayesian estimations are considered using the squared error loss function and a priori distributions that create a dependency between the hyper-parameters. But given the assumptions, one can see that explicit expressions cannot be obtained for Bayesian estimations. The importance sampling method is proposed to calculate the Bayes estimations and also to create the corresponding HPD credible intervals of the unknown parameters. The Monte Carlo simulation studies and analysis of a real data set are also performed to evaluate the proposed method for estimating parameters. Finally, a generalization of the dependent lives models is presented and the Bayesian estimate for this new bivariate family distribution is introduced and a comparison is made between the models of this new family.

Remote teaching in times of social distancing: a study in the accounting and actuarial sciences courses

Purpose: Considering the imposed social isolation measures to contain the spread of the Sars-Cov-2 virus, the cause of COVID-19 pandemic, the in-person classes started to be held remotely. Those measures had an impact on education, mainly because they had to be adopted quickly, with no time for preparation. In our current study, we investigate how remote teaching has been occurring in the undergraduate courses of Accounting and Actuarial Sciences at a Public Higher Education Institution (PHEI), in times of social distancing.
 Methodology: An e-survey was carried out with teachers and students, in order to obtain information on the subject. The collected data were processed using statistical techniques to generate the results.
 Results: It came across that the studied PHEI did not have resources such as a Flipped classroom, Digital Whiteboard or any other technological tools to support remote classes, since the remote teaching modality became a need during the COVID-19 pandemic. The PHEI launched, via public notices, programs for digital inclusion in order to acquire mobile data and equipment, such as computers/smartphones/tablets for students; however, only few of the students could have access to those resources.
 Contributions of the Study: Our research results contribute to the state-of-the-art regarding undergraduate courses, by investigating their evolution towards remote learning. Such contributions provide information and knowledge on this subject to the national scientific literature, and give support and guidance to other courses offered by public and/or private higher education institutions.

Quantum Mechanics Approach for Appropriately Chosen Hamiltonian

Risk theory has always played a significant role in mathematical finance and actuarial sciences. A novel approach to the risk theory of non-life insurance is quantum mechanics. To compute finite-time non-ruin probability, I introduce the quantum mechanics formalism in discrete space and continuous space with the appropriately chosen Hamiltonian. By using the quantum mechanics approach and the stochastic method, the non-ruin operator is defined, and tensor products of operator concepts are presented for several examples.
 In this paper, Dirac notations are operated to find the Hamiltonian matrix with the eigenvector basis for two and three-state cases, and its tensor product version with a change of basis.

Magnetohydrodynamic Jeffrey Nanofluid Flow Over a Vertical Sheet

Aim: This study focuses on the numerical analysis of Jeffrey nano-fluid bound with magnetic field in presence of convectively heated boundary.
 Study Design: Abstract, introduction, Equations formulation, numerical analysis and conclusion
 Place and Duration of Study: Department of Mathematics and Actuarial Science, Kenyatta University, between 2021-2022
 Methodology: This paper discusses the imposed magnetic field on Jeffrey fluid suspended with nanometre-sized particles moving over a vertical sheet with a convectively heated boundary. The partial differential equations are formulated by considering assumptions and the boundary conditions to describe the continuity, momentum, energy and concentration of the fluid. The similarity transformation technique was applied to convert the partial differential equations into first-order linear differential equations which were simulated in Matlab by invoking the Adam’s-Moulton predictor-corrector scheme in ode113.
 The graphs have been analysed with the effects of Deborah, Dufour-Lewis, Hartman, and Prandtl numbers respectively, solutal stratification, diffusion, thermophoresis, temperature Grashof, mass Grashof, relaxation-retardation parameters on the flow velocity, concentration, temperature, skin friction, heat and mass transfers looked into.
 Results: While Deborah number increased velocity, it reduced concentration, skin friction and thermal boundary layer at lower numbers hence improved mass and heat transfer. Solutal stratification, Retardation-relaxation parameter and diffusion raised temperature thus heat transfer.
 Conclusion: Deborah number, solutal stratification, retardation-relaxation parameter and diffusion improves heat and mass transfer.

Actuarial intelligence in auto insurance: Claim frequency modeling with driving behavior features and improved boosted trees

Usage-based insurance (UBI) is now a sought-after auto insurance product in the market. By using a wide range of telematics data, insurance companies can better understand the insured's driving behavior and capture the relationship between insurance loss and the relevant risk factors. This study examines the frequency of UBI claims and combines machine learning algorithms with classic actuarial distributions to establish the predictive model. More specifically, considering the large number of driving behavior features and their complex interactions, we replace generalized linear models with boosted trees, and synchronously update the estimation results of the zero-inflation probability and mean parameter under a zero-inflated Poisson or zero-inflated negative binomial assumption. We further discuss the role of regularization terms and “dropout” in dual-parameter boosted trees, and propose a general framework for insurance claim frequency modeling, which shows high prediction accuracy on both UBI and French motor third-party liability datasets, as well as the interpretability. The potential of extensive driving behavior features has been further verified on a Chinese insurance dataset, and the factors that have a significant impact on vehicle risk are identified and quantified on this basis. In addition, we discuss in detail the key points of applying boosted trees in actuarial science, which also promotes predictive insurance analytics.

Estimation of Life Insurance Asset Value From Household Survey

The omission of the asset value of life insurance policies in computing household assets would underestimate household wealth and therefore bias measures of wealth inequality. We propose a method to estimate the cash value of household life insurance assets by combining the Lee-Carter model, which forecasts mortality risk, and the concept of the net level premium reserve in actuarial science. This method is readily applicable using household survey data.

Achievement Goals Analysis in the Learning of Calculus Based on Fuzzy Number Conjoint Method

The conjoint method, which is based on fuzzy sets of numbers, is widely used to describe linguistic values for human preference in an uncertain environment. However, the fuzzy sets used to describe the membership function of linguistic value do not realistically represent the physical world, so the conjoint method can fill the gap and produce more meaningful results. The fuzzy numbers conjoint method is used in this paper to analyze the achievement goals of undergraduates in the learning of calculus. One hundred and seven selected Bachelor of Science (Hons) Mathematics and Bachelor of Science (Hons) Actuarial Science students from one public university in Klang Valley, Selangor, participated in this study. The data for this study, which was distributed via Google form, was based on a previous study's Achievement Goals Questionnaire. The fuzzy number conjoint method with similarity measure based on geometric distance, ambiguity, value, area, left and right height were used to calculate and analyze the data gathered from respondents' opinions of attributes for each linguistic value. The priority of the degree of agreement among undergraduates on the achievement goals in the learning of calculus is worrying as they may not learn all that they possibly could in this subject , getting better grades than most other students , followed by avoiding performing poorly compared to other students in this subject , and doing better than other students with an overall ranking as follows .. The findings of this study can be used to assist and guide academicians and mathematics educators in enhancing students' achievement goals for calculus learning.

Composite bias‐reduced Lp‐quantile‐based estimators of extreme quantiles and expectiles

AbstractQuantiles are a fundamental concept in extreme value theory. They can be obtained from a minimization framework using an asymmetric absolute error loss criterion. The companion notion of expectiles, based on asymmetric squared rather than asymmetric absolute error loss minimization, has received substantial attention from the fields of actuarial science, finance, and econometrics over the last decade. Quantiles and expectiles can be embedded in a common framework of ‐quantiles, whose extreme value properties have been explored very recently. Although this generalized notion of quantiles has shown potential for the estimation of extreme quantiles and expectiles, available estimators remain quite difficult to use: they suffer from substantial bias, and the question of the choice of the tuning parameter remains open. In this article, we work in a context of heavy tails and construct composite bias‐reduced estimators of extreme quantiles and expectiles based on ‐quantiles. We provide a discussion of the data‐driven choice of and of the anchor ‐quantile level in practice. The proposed methodology is compared with existing approaches on simulated data and real data.

New Exponential-Type Estimators of Finite Population Variance Using Auxiliary Information

The Sri Lankan Journal of Applied Statistics(SLJAS) is an open-access, international, double-blind peer-reviewed journal published by the Institute of Applied Statistics, Sri Lanka (IASSL). The main purpose of the journal is to publish the results of original work on applications of Statistics and on theoretical and methodical aspects of Statistics. The journal also welcomes critical reviews including conceptual discussions, opinions and book reviews. Applications of Statistics in the area of Agriculture & Forestry, Medical, Dental and Veterinary Sciences, Natural, Physical Sciences, Social Sciences, Economics and Actuarial Science fall within the scope of the journal. This journal does not charge any fee for article processing and submission.

What can we learn from telematics car driving data: A survey

We give a survey on the field of telematics car driving data research in actuarial science. We describe and discuss telematics car driving data, we illustrate the difficulties of telematics data cleaning, and we highlight the transparency issue of telematics car driving data resulting in associated privacy concerns. Transparency of telematics data is demonstrated by aiming at correctly allocating different car driving trips to the right drivers. This is achieved rather successfully by a convolutional neural network that manages to discriminate different car drivers by their driving styles. In a last step, we describe two approaches of using telematics data for improving claims frequency prediction, one is based on telematics heatmaps and the other one on time series of individual trips, respectively.

Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property

The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of models commonly used for survival data analysis. Other methods, however, usually do not provide a framework for recognizing or dealing with the issue and so the issue is often unintentionally ignored. The theoretical foundations of this work are set out, which presents new and somewhat surprising results for the first hitting time distributions of Lévy processes that have the Esscher property.

Multi-population mortality projection: The augmented common factor model with structural breaks

Multi-population mortality forecasting has become an increasingly important area in actuarial science and demography, as a means to avoid long-run divergence in mortality projections. This paper aims to establish a unified state-space Bayesian framework to model, estimate, and forecast mortality rates in a multi-population context. In this regard, we reformulate the augmented common factor model to account for structural/trend changes in the mortality indexes. We conduct a Bayesian analysis to make inferences and generate forecasts so that process and parameter uncertainties can be considered simultaneously and appropriately. We illustrate the efficiency of our methodology through two case studies. Both point and probabilistic forecast evaluations are considered in the empirical analysis. The derived results support the fact that the incorporation of stochastic drifts mitigates the impact of the structural changes in the time indexes on mortality projections.

Copula based Bayesian data analysis of loss reserving

Prediction of loss reserves corresponding to dependent lines of business is one of the most important problems in the actuarial sciences. In this paper, we propose a class of copula based multivariate distributions to model the losses with the heavy tailed distribution in the run-off triangles to predict unpaid losses. We set up ANOVA, ANCOVA, and state space models with four choices of copulas, Clayton, Frank, Gumbel, and Gaussian to provide a new procedure for analyzing run-off triangle tables. We use the Hamiltonian Monte Carlo sampler to perform a Bayesian analysis to estimate the parameters. We apply the proposed models to the data set consists of two lines of business of paid losses data from the Schedule P of the National Association of Insurance Commissioners (NAIC) database. Using some well known criteria, we compare the prediction accuracy of the mean models. As a result, the ANCOVA model with the Clayton copula dominates the other models.

How a probabilistic analogue of the mean value theorem yields stein-type covariance identities

AbstractA method for the construction of Stein-type covariance identities for a nonnegative continuous random variable is proposed, using a probabilistic analogue of the mean value theorem and weighted distributions. A generalized covariance identity is obtained, and applications focused on actuarial and financial science are provided. Some characterization results for gamma and Pareto distributions are also given. Identities for risk measures which have a covariance representation are obtained; these measures are connected with the Bonferroni, De Vergottini, Gini, and Wang indices. Moreover, under some assumptions, an identity for the variance of a function of a random variable is derived, and its performance is discussed with respect to well-known upper and lower bounds.