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21 pages, 846 KiB

Open AccessArticle

Long-Run Equilibrium in the Market of Mobile Services in the USA

by Anton Badev, Stanimir Kabaivanov, Petar Kopanov, Vasil Zhelinski and Boyan Zlatanov

Mathematics 2024, 12(5), 724; https://0-doi-org.brum.beds.ac.uk/10.3390/math12050724 - 29 Feb 2024

Viewed by 524

We develop an empirical model of the market for mobile services in the USA based on providers’ response functions. Guided by a duopoly model, we obtain our empirical response functions from an approximation of quarterly response data on smartphone subscriptions by sigmoid functions [...] Read more.

We develop an empirical model of the market for mobile services in the USA based on providers’ response functions. Guided by a duopoly model, we obtain our empirical response functions from an approximation of quarterly response data on smartphone subscriptions by sigmoid functions of time. The robustness analysis suggests that our model fits the data well and outperforms the regression model. Further, we demonstrate that our empirical response functions satisfy the conditions for semi-cyclic contractions which guarantee the existence, uniqueness and stability of long-run equilibrium. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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20 pages, 976 KiB

Open AccessArticle

A Comparison between Explainable Machine Learning Methods for Classification and Regression Problems in the Actuarial Context

by Catalina Lozano-Murcia, Francisco P. Romero, Jesus Serrano-Guerrero and Jose A. Olivas

Mathematics 2023, 11(14), 3088; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143088 - 13 Jul 2023

Cited by 4 | Viewed by 2077

Abstract

Machine learning, a subfield of artificial intelligence, emphasizes the creation of algorithms capable of learning from data and generating predictions. However, in actuarial science, the interpretability of these models often presents challenges, raising concerns about their accuracy and reliability. Explainable artificial intelligence (XAI) [...] Read more.

Machine learning, a subfield of artificial intelligence, emphasizes the creation of algorithms capable of learning from data and generating predictions. However, in actuarial science, the interpretability of these models often presents challenges, raising concerns about their accuracy and reliability. Explainable artificial intelligence (XAI) has emerged to address these issues by facilitating the development of accurate and comprehensible models. This paper conducts a comparative analysis of various XAI approaches for tackling distinct data-driven insurance problems. The machine learning methods are evaluated based on their accuracy, employing the mean absolute error for regression problems and the accuracy metric for classification problems. Moreover, the interpretability of these methods is assessed through quantitative and qualitative measures of the explanations offered by each explainability technique. The findings reveal that the performance of different XAI methods varies depending on the particular insurance problem at hand. Our research underscores the significance of considering accuracy and interpretability when selecting a machine-learning approach for resolving data-driven insurance challenges. By developing accurate and comprehensible models, we can enhance the transparency and trustworthiness of the predictions generated by these models. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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22 pages, 681 KiB

Open AccessArticle

Numerical Method for a Risk Model with Two-Sided Jumps and Proportional Investment

by Jiaen Xu, Chunwei Wang, Naidan Deng and Shujing Wang

Mathematics 2023, 11(7), 1584; https://0-doi-org.brum.beds.ac.uk/10.3390/math11071584 - 24 Mar 2023

Cited by 2 | Viewed by 893

Abstract

In this paper, we consider a risk model with two-sided jumps and proportional investment. The upward jumps and downward jumps represent gains and claims, respectively. Suppose the company invests all of its surplus in a certain proportion in two types of investments, one [...] Read more.

In this paper, we consider a risk model with two-sided jumps and proportional investment. The upward jumps and downward jumps represent gains and claims, respectively. Suppose the company invests all of its surplus in a certain proportion in two types of investments, one is risk-free (such as bank accounts) and the other is risky (such as stocks). Our aim is to find the optimal admissible strategy (including the optimal dividend rate and the optimal ratio of investment in risky assets), to maximize the dividend value function, and discuss the effects of a number of parameters on dividend payments. Firstly, the HJB equation of the dividend value function is obtained by the stochastic analysis theory and the dynamic programming method, and the optimal admissible strategy is obtained. Since the integro-differential equation satisfied by the dividend value function is difficult to solve, we turn to the sinc numerical method to approximate solve it. Then, the error between the exact solution (ES) and the sinc approximate solution (SA) is analyzed. Finally, the relative error of a special numerical solution and an ES is given, and some examples of sensitivity analysis are discussed. This study provides a theoretical basis for insurance companies to prevent risks better. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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13 pages, 1124 KiB

Open AccessArticle

Facing a Risk: To Insure or Not to Insure—An Analysis with the Constant Relative Risk Aversion Utility Function

by M. Mercè Claramunt, Maite Mármol and Xavier Varea

Mathematics 2023, 11(5), 1070; https://0-doi-org.brum.beds.ac.uk/10.3390/math11051070 - 21 Feb 2023

Viewed by 1146

Abstract

The decision to transfer or share an insurable risk is critical for the decision maker’s economy. This paper deals with this decision, starting with the definition of a function that represents the difference between the expected utility of insuring, with or without deductibles, [...] Read more.

The decision to transfer or share an insurable risk is critical for the decision maker’s economy. This paper deals with this decision, starting with the definition of a function that represents the difference between the expected utility of insuring, with or without deductibles, and the expected utility of not insuring. Considering a constant relative risk aversion (CRRA) utility function, we provide a decision pattern for the potential policyholders as a function of their wealth level. The obtained rule applies to any premium principle, any per-claim deductible and any risk distribution. Furthermore, numerical results are presented based on the mean principle, a per-claim absolute deductible and a Poisson-exponential model, and a sensitivity analysis regarding the deductible parameter and the insurer security loading was performed. One of the main conclusions of the paper is that the initial level of wealth is the main variable that determines the decision to insure or not to insure; thus, for high levels of wealth, the decision is always not to insure regardless of the risk aversion of the decision maker. Moreover, the parameters defining the deductible and the premium only have an influence at low levels of wealth. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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27 pages, 838 KiB

Open AccessArticle

Numerical Method for a Perturbed Risk Model with Proportional Investment

by Chunwei Wang, Naidan Deng and Silian Shen

Mathematics 2023, 11(1), 43; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010043 - 22 Dec 2022

Cited by 2 | Viewed by 1088

Abstract

In this paper, we study the perturbed risk model with a threshold dividend strategy and proportional investment. The insurance companies are allowed to invest their surplus in a financial market consisting of a risk-free asset and a risky asset in fixed proportions; the [...] Read more.

In this paper, we study the perturbed risk model with a threshold dividend strategy and proportional investment. The insurance companies are allowed to invest their surplus in a financial market consisting of a risk-free asset and a risky asset in fixed proportions; the risky assets are modeled by the jump-diffusion process. Firstly, using the theory of the stochastic process and stochastic analysis, we obtained the integro-differential equations satisfied by the expected discounted dividend payments and the discounted penalty function. Secondly, we obtained the numerical approximate solutions of the integro-differential equations through the sinc method, since the analytical solutions of them are not easy to obtain, and we found that the error is within a manageable range. Finally, we considered some numerical examples where the claim sizes follow an exponential distribution, a mixture of two exponential distributions or the lognormal distribution in detail, and explored how perturbations and proportional investment affect dividends and ruin probability. Moreover, sensitive analysis showed that the proportion of the risky investment, the diffusion coefficient, the distribution of the claims and the positive jump in the risky assets investment all have explicit impacts on dividends and ruin probability. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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21 pages, 4036 KiB

Open AccessArticle

A Deep Neural Network Approach to Solving for Seal’s Type Partial Integro-Differential Equation

by Bihao Su, Chenglong Xu and Jingchao Li

Mathematics 2022, 10(9), 1504; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091504 - 01 May 2022

Cited by 4 | Viewed by 1563

Abstract

In this paper, we study the problem of solving Seal’s type partial integro-differential equations (PIDEs) for the classical compound Poisson risk model. A data-driven deep neural network (DNN) method is proposed to calculate finite-time survival probability, and an alternative scheme is also investigated [...] Read more.

In this paper, we study the problem of solving Seal’s type partial integro-differential equations (PIDEs) for the classical compound Poisson risk model. A data-driven deep neural network (DNN) method is proposed to calculate finite-time survival probability, and an alternative scheme is also investigated when claim payments are exponentially distributed. The DNN method is then extended to the numerical solution of generalized PIDEs. Numerical approximation results under different claim distributions are given, which show that the proposed scheme can obtain accurate results under different claim distributions. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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12 pages, 495 KiB

Open AccessArticle

Optimal Claim Settlement Strategies under Constraint of Cap on Claim Loss

by Hong Mao and Krzysztof Ostaszewski

Mathematics 2021, 9(24), 3284; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243284 - 17 Dec 2021

Viewed by 1603

Abstract

In this paper, we examine the question of how to devise an optimal insurance claim settlement scheme under the constraint of a cap on the amount of the claim payment. We establish objective functions to maximize the net benefit due to exaggerated claims [...] Read more.

In this paper, we examine the question of how to devise an optimal insurance claim settlement scheme under the constraint of a cap on the amount of the claim payment. We establish objective functions to maximize the net benefit due to exaggerated claims while at the same time maximizing the total expected wealth of the insured. Then, we establish a dual objective function to minimize the total expected loss, including the perspective of the insurer. Finally, we illustrate applications of our work and provide numerical analysis of it along with an example. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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11 pages, 2433 KiB

Open AccessArticle

Hybrid of the Lee-Carter Model with Maximum Overlap Discrete Wavelet Transform Filters in Forecasting Mortality Rates

by Nurul Aityqah Yaacob, Jamil J. Jaber, Dharini Pathmanathan, Sadam Alwadi and Ibrahim Mohamed

Mathematics 2021, 9(18), 2295; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182295 - 17 Sep 2021

Cited by 3 | Viewed by 1913

Abstract

This study implements various, maximum overlap, discrete wavelet transform filters to model and forecast the time-dependent mortality index of the Lee-Carter model. The choice of appropriate wavelet filters is essential in effectively capturing the dynamics in a period. This cannot be accomplished by [...] Read more.

This study implements various, maximum overlap, discrete wavelet transform filters to model and forecast the time-dependent mortality index of the Lee-Carter model. The choice of appropriate wavelet filters is essential in effectively capturing the dynamics in a period. This cannot be accomplished by using the ARIMA model alone. In this paper, the ARIMA model is enhanced with the integration of various maximal overlap discrete wavelet transform filters such as the least asymmetric, best-localized, and Coiflet filters. These models are then applied to the mortality data of Australia, England, France, Japan, and USA. The accuracy of the projecting log of death rates of the MODWT-ARIMA model with the aforementioned wavelet filters are assessed using mean absolute error, mean absolute percentage error, and mean absolute scaled error. The MODWT-ARIMA (5,1,0) model with the BL14 filter gives the best fit to the log of death rates data for males, females, and total population, for all five countries studied. Implementing the MODWT leads towards improvement in the performance of the standard framework of the LC model in forecasting mortality rates. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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21 pages, 386 KiB

Open AccessArticle

Valuation of Cliquet-Style Guarantees with Death Benefits in Jump Diffusion Models

by Yaodi Yong and Hailiang Yang

Mathematics 2021, 9(16), 2011; https://0-doi-org.brum.beds.ac.uk/10.3390/math9162011 - 23 Aug 2021

Viewed by 1417

Abstract

This paper aims to value the cliquet-style equity-linked insurance product with death benefits. Whether the insured dies before the contract maturity or not, a benefit payment to the beneficiary is due. The premium is invested in a financial asset, whose dynamics are assumed [...] Read more.

This paper aims to value the cliquet-style equity-linked insurance product with death benefits. Whether the insured dies before the contract maturity or not, a benefit payment to the beneficiary is due. The premium is invested in a financial asset, whose dynamics are assumed to follow an exponential jump diffusion. In addition, the remaining lifetime of an insured is modelled by an independent random variable whose distribution can be approximated by a linear combination of exponential distributions. We found that the valuation problem reduced to calculating certain discounted expectations. The Laplace inverse transform and techniques from existing literature were implemented to obtain analytical valuation formulae. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

17 pages, 4283 KiB

Open AccessArticle

Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

by Yujuan Huang, Jing Li, Hengyu Liu and Wenguang Yu

Mathematics 2021, 9(9), 982; https://0-doi-org.brum.beds.ac.uk/10.3390/math9090982 - 27 Apr 2021

Cited by 4 | Viewed by 2460

Abstract

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the [...] Read more.

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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Review

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13 pages, 1556 KiB

Open AccessReview

Potential Applications of Explainable Artificial Intelligence to Actuarial Problems

by Catalina Lozano-Murcia, Francisco P. Romero, Jesus Serrano-Guerrero, Arturo Peralta and Jose A. Olivas

Mathematics 2024, 12(5), 635; https://0-doi-org.brum.beds.ac.uk/10.3390/math12050635 - 21 Feb 2024

Viewed by 730

Abstract

Explainable artificial intelligence (XAI) is a group of techniques and evaluations that allows users to understand artificial intelligence knowledge and increase the reliability of the results produced using artificial intelligence. XAI can assist actuaries in achieving better estimations and decisions. This study reviews [...] Read more.

Explainable artificial intelligence (XAI) is a group of techniques and evaluations that allows users to understand artificial intelligence knowledge and increase the reliability of the results produced using artificial intelligence. XAI can assist actuaries in achieving better estimations and decisions. This study reviews the current literature to summarize XAI in common actuarial problems. We proposed a research process based on understanding the type of AI used in actuarial practice in the financial industry and insurance pricing and then researched XAI implementation. This study systematically reviews the literature on the need for implementation options and the current use of explanatory artificial intelligence (XAI) techniques for actuarial problems. The study begins with a contextual introduction outlining the use of artificial intelligence techniques and their potential limitations, followed by the definition of the search equations used in the research process, the analysis of the results, and the identification of the main potential fields for exploitation in actuarial problems, as well as pointers for potential future work in this area. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

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