Actuarial Mathematics Research Articles

Special Issue “Interplay Between Financial and Actuarial Mathematics II”

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FEATURES OF USING VBA IN TEACHING ACTUARIAL MATHEMATICS

The article is devoted to the analysis of practice of application of specific computational techniques within MS Excel, including VBA, for teaching selected topics in actuarial mathematics. Formulation of the problem. With the evolution of financial and actuarial models towards complexity and reliance on machine learning and data science, there is a growing demand for modern mathematicians, particularly actuaries, to acquire new skills and knowledge. This demand is being driven by new fintech trends in banking and insurance, such as smart payments, real-time credit risk assessment, automated reputation management, customer analytics, fraud detection and cryptocurrency trading. As a result, the training of future actuaries must meet high standards to ensure they are equipped to handle the sophisticated mathematical calculations required by insurance companies and other financial institutions. Materials and methods. To yield results, both theoretical methods, which involve analyzing books and publications, in particular in professional journals in the domains of finance and actuarial mathematics, and empirical methods, entailing the observation of the educational process aimed at training prospective actuaries, are employed in this study. Results. A study aimed at tackling the challenges encountered in teaching actuarial mathematics topics, such as life expectancy and mortality tables, revealed that incorporating a balanced blend of theoretical concepts and hands-on applications, coupled with illustrating mathematical models through real-world examples and reinforcing diverse computational techniques, constitutes a cornerstone of successful training for aspiring actuaries. Leveraging a versatile and user-friendly tool like VBA (Visual Basic for Applications) integrated into MS Excel enables educators to customize examples and assignments to match the intricacy and emphasis of their syllabus, thereby offering students a more targeted learning journey. Conclusions. The teaching of mathematical disciplines, especially financial and actuarial mathematics, in academic programmes for future actuaries requires the adaptation of traditional methods and approaches. In particular, the integration of VBA is key to teaching specific actuarial mathematics concepts, which aims to equip students with the necessary practical competencies. VBA facilitates interactive learning by enabling students to manipulate variables and observe changes in calculations in real time, which improves their understanding of actuarial concepts. VBA also allows you to automate repetitive tasks involving complex mathematical calculations, which saves time and reduces errors. Therefore, its use is recommended when teaching topics involving tabular data, especially life expectancy.

Calculating premium principles from the mode of a unimodal weighted distribution

Abstract The theory of utility is a well-known method of constructing insurance premiums (see e.g., Newton et al. (1986) Actuarial Mathematics. Itasca, Illinois: The Society of Actuaries.). Furman and Zitikis ((2008) Insurance: Mathematics and Economics, 42, 459–465.) proposed an alternative method using the mean value of a weighted random variable. According to this approach, for various choices of weighting, popular premiums such as net premium, modified variance premium, Esscher premium, and Kamps premium are obtained. On the other hand, some premiums cannot be obtained with this method, such as the premium of the exponential principle. In this paper, we provide a complementary theory by introducing a family of unimodal weighted distributions for which the mode is a premium principle.

Developing “Guess-Huwhat?” Strategic Intervention Material to Engage Grade 12 STEM Students in Understanding the Sound Terminologies

Teaching of sound concepts must be improved and presented in new ways to lessen misconceptions. One of the efficient teaching methods for sciences is the use of strategic intervention material (SIM) because it assists students in enhancing their learning towards the least-learned subjects or competencies. This research aimed to develop a strategic intervention material, Guess HuWhat?, in teaching the concepts of sounds to STEM senior high school students. It also aimed to lessen the misconceptions about the concepts of sounds and improve the students’ conceptual understanding of sounds. The study adapted Branch’s ADDIE model, which stands for Assess, Design, Develop, Implement, and Evaluate, as the basis for developing the actuarial mathematics course materials. This developed strategic intervention material was evaluated by Science Content Experts and was further utilized by some of the Grade 12 STEM students at Muntinlupa National High School to find out its effectiveness as a learning supplementary material and based on the data gathered, the Guess HuWhat? Strategic Intervention Material improved grade 12 STEM students’ understanding of sound terminologies. By repetitive actions, the developed game-based SIM can be an effective intervention material to deepen students’ comprehension of sound concepts and strengthen camaraderie among learners. This study recommends Guess HuWhat? Game in grade 12 science classes for continuous enhancement of teachers’ craft as well as innovation of the game itself. Moreover, it is highly recommended that teachers design and develop other game-based activities containing concepts that they perceive students are struggling with promoting that learning is indeed fun.

Mathematical model for the analysis of environmental risk factors

The article studies the economic consequences of harmful effects on the environment. A mathematical model has been developed for the analysis of environmental risk factors. When constructing a model, the probability of an insured event depending on time is taken into account. It is shown that different probabilistic impact models can be used for different categories of risk factors. The development of a dynamic model is based on the methods of probability theory, actuarial mathematics and, accordingly, on the, numerical methods and simulation methods. Analytical expressions for the risk factor integral value of environmental management are obtained in the work. The mathematical calculation of the distribution function, or the survival function before the time period occurrence of an insured event in different cases is performed, namely: in the case of a risk factor with a constant intensity, in the case of gradually emerging risk factors with their constant intensities, in the case when the survival function is decreasing at a linear rate, as well as in the case when equipment wear and tear leads to an exponential increase in the risk intensity.

Special Issue “Actuarial Mathematics and Risk Management”

Special Issue “Actuarial Mathematics and Risk Management”

A Simple Estimation of Parameters for Discrete Distributions from the Schröter Family

One of the common challenges in actuarial mathematics is finding a model for the number of claims and claim severity. We focus on one of the suggested models, namely, on the Schröter family of discrete probability distributions. Furthermore, we introduce a simple and easy-to-compute parameter estimate for this family of distributions, which can be used especially as initial values in optimization algorithms that are needed to compute other estimates.

On the Devylder–Goovaerts Conjecture in Ruin Theory

The Devylder–Goovaerts conjecture is probably the oldest conjecture in actuarial mathematics and has received a lot of attention in recent years. It claims that ruin with equalized claim amounts is always less likely than in the classical model. Investigating the validity of this conjecture is important both from a theoretical aspect and a practical point of view, as it suggests that one always underestimates the risk of insolvency by replacing claim amounts with the average claim amount a posteriori. We first state a simplified version of the conjecture in the discrete-time risk model when one equalizes aggregate claim amounts and prove that it holds. We then use properties of the Pareto distribution in risk theory and other ideas to target candidate counterexamples and provide several counterexamples to the original Devylder–Goovaerts conjecture.

Designing an Insurance Pricing Model using a Mathematical Approach

Purpose: Every time drivers take to the road, and with each mile that they drive, exposes themselves and others to the risk of an accident. Insurance premiums are only weakly linked to mileage, however, and have lump-sum characteristics largely. The result is too much driving, and too many accidents. The purpose of carrying out this research was to determine a model for calculating the premiums for Pay-As-You-Drive in automobile insurances. Methodology: To price Pay-As-You-Drvie auto insurance, we define a discounted collective risk model while the total number of claim has non-homogeneous Poisson distribution. By applying non-homogeneous Poisson distribution we can enter the mileage to the discounted collective risk model to the premiums for Pay-As-You-Drive in Automobile insurances. We apply the double Double Stochastic Poisson Process for modeling the DCRM. The Double Stochastic Poisson Process provides flexibility by letting the intensity not only depend on time but also by allowing it to be a stochastic process. Findings: By applying the doubly stochastic Poisson process to review the driver’s mileage in the model, the study found the distribution of discounted collective risk model and present the expected value of total loss for calculating the premiums. Recommendation: The current auto insurance pricing systems are inequitable because low-mileage drivers subsidize insurance costs for high-mileage drivers, and low-income people drive fewer miles on average. The study recommends a more efficient pricing systems model to find a good model for calculating the fair auto insurance premium. Keywords: Cox process, martingales, aggregate risk models, PAYD, actuarial mathematics.

Artificial Intelligence Analysis of Outdoor Sports Risk Self-Assessment on Insurance Psychology.

The development potential of China's medical insurance market is huge, and the research on medical insurance demand has always been the focus of academic discussions. As a result, the discipline of behavioral economics is derived, which aims to explain the decision-making behavior of individual insurance consumption. Among them, the focus of this study was to investigate the influence of individual psychological characteristics and cognitive level on insurance behavior under the difference of reference points. This paper combined behavioral insurance, actuarial mathematics and the econometrics knowledge system, comprehensive theoretical analysis, and empirical tests and analyzed the impact mechanism of individual frame effect on medical insurance demand under different reference points at multiple levels. At the same time, based on the risk self-assessment of outdoor sports, the artificial intelligence of insurance psychology was analyzed. Based on the correlation vector machine algorithm and the theoretical basis combined with the dual perspective of insurance products, the expected utility model was established under the "guarantee framework", and the prospect theoretical model was established under the "profit and loss framework". The framing effect was used to measure the relative size of "guarantee utility" and "profit and loss utility", and a high-insurance-rate model and a low-insurance-rate model were established. The theoretical model analysis found that under the high insurance rate, because the "profit and loss utility" is positive, the size of the individual frame effect is positively correlated with the willingness to insure. Under the low insurance rate, because the "profit and loss utility" is negative, the size of the individual frame effect is negatively correlated with the willingness to insure. The research results of this paper show that insurance is an important beginning of insurance consumption behavior, which includes the complex mentality and emotion of consumers on insurance activities. The insurance demand of policyholders is formed by the joint action of external and internal incentives. Many factors such as income level and education level play an important role in insurance consumption decision making.

Effectiveness of the Classical Actuarial Mathematics: Establishment of the Valuation Practices in the 1770s for the Equitable Life in England

1770年代の英国エクイタブル生命において，プライスとモーガンによる世界最初の生命保険会社のバリュエーションが実施されるに至った。このバリュエーション実務の構築を通じて，責任準備金概念を使用した統一的な契約価値評価に基づく生命保険会社の全体の財政状況の検証とそれに基づく経営の枠組みが姿を現わしたことになるが，本稿では，それとともに，その枠組みを実行可能とするために必要な大量の計算について，当時の計算力でもタイムリーに算出可能とする計算の体系が考案されたことを示す。この計算の工夫において，純粋数学の活用が効率性につながり，体系的数学が有効に保険数理へ持ち込まれているため，ここで古典生保数理が完成したと言えると主張する。また，既存の研究によりながら，当時において，この古典生保数理の事例が組織運営における定量的評価に基づく管理やその管理における計算などの知的労働の効率化の効果を示す事例とみなされ，他分野へ影響したことも指摘する。

A risk model for Forest fires based on asymptotic results for multivariate collective models. Single models and structured families of models

In this paper we propose an new approach to multivariate collective models based on asymptotic distibutions, since the modeling problems posed have large samples. Collective risk models play an important part in Risk Theory and in Actuarial Mathematics. Inference based on these models is centered on claims totals. However, the new approach has in mind a different kind of risk, the risk of forest fires, where the variables of interest are the number fires and the total burnt areas. As a result, a special case bivariate risk model is derived with this intent. Besides single models, structured families whose models correspond to the treatments of a base linear model are considered. This leads to an ANOVA-like situation where we don’t have to estimate the error, which enable to test the influence of several factors on the multivariate collective models mean vectors. Instead of F tests, we use tests, availing us of asymptotic distributions that lightens the treatment. To illustrate the approach, an application to forest fires in Portugal with real data is presented.

Poisson processes directed by subordinators, stuttering poisson and pseudo-poisson processes, with applications to actuarial mathematics

Weconsider a PSI-process, that is a sequence of random variables (&), i = 0.1,…, which is subordinated by a continuous-time non-decreasing integer-valued process N(t): <K0 = ÇN(ty Our main example is when /V(t) itself is obtained as a subordination of the standard Poisson process 77(s) by a non-decreasing Lévy process S(t): N(t) = 77(S(t)).The values (&)one interprets as random claims, while their accumulated intensity S(t) is itself random. We show that in this case the process 7V(t) is a compound Poisson process of the stuttering type and its rate depends just on the value of theLaplace exponent of S(t) at 1. Under the assumption that the driven sequence (&) consists of i.i.d. random variables with finite variance we calculate a correlation function of the constructed PSI-process. Finally, we show that properly rescaled sums of such processes converge to the Ornstein-Uhlenbeck process in the Skorokhod space. We suppose that the results stated in the paper mightbe interesting for theorists and practitioners in insurance, in particular, for solution of reinsurance tasks.

Special Issue “Interplay between Financial and Actuarial Mathematics”

The Special Issue aims to highlight the interaction between actuarial and financial mathematics, which, due to the recent low interest rates and implications of COVID-19, requires an interlace between actuarial and financial methods, along with control theory, machine learning, mortality models, option pricing, hedging, unit-linked contracts and drawdown analysis, among others [...]

Current models in the field of agricultural crops insurance

Agricultural production is an important and at the same time the most risky type of economic activity. Its reproductive process is associated with natural and climatic, biological and financial factors, the action of which in many cases is difficult to forgive and control. Crop production is particularly affected by cumulative natural risks. One of the ways to minimize agricultural risks is to use crop insurance as an important means of ensuring the riskiness of agricultural production from probable natural and weather factors. The pricing mechanism for crop insurance services is substantiated, based on the improvement of the insurer’s tariff policy when conducting crop insurance, which is implemented through a system of actuarial calculations and taking into account the peculiarities of insurance in the process of crop production. The possibility for insurance companies to carry out high-quality selection of agricultural risks, stimulating agricultural producers to insure, is analyzed. Actuarial balance has been studied as a necessary condition for improving the tariff policy of crop insurance, which should be based on such principles as: equivalence of insurance relations; admissibility; stability of insurance rates over a long period of time and expansion of insurance liability. The calculation of insurance rates has been improved by taking into account the availability of prices for insurance services, depending on the number of possible risks transferred to insurance and the amount of financial resources accumulated by the insurer, which should be sufficient to reimburse crops under crop insurance contracts. Actuarial balance is identified as a necessary condition for improving the tariff policy of crop insurance. The principles of insurance tariff policy to improve the calculation of insurance rates and the application of actuarial models in the field of crop insurance. Key words: tariff policy, insurance tariff, actuary, actuarial mathematics, actuarial model of agricultural insurance.

Pengembangan Bahan Ajar Matematika Aktuaria untuk Meningkatkan Kemampuan Analisis Mahasiswa

This study aimed to develop teaching materials in actuarial courses at FMIPA State University of Medan to support the learning during the Covid-19 pandemic. This research used research and development method. The samples of this study were 32 students of FMIPA State University of Medan who took actuarial courses. The research instrument used in this study was pre and post test and observation sheets. The results showed that the t test was 20.64, in which the t count value was greater than the t table value (20.64> 1.462). Thus, learning using actuarial mathematics teaching materials has an effect on students' analytical skills. Hence, Ho was rejected and Ha was accepted, which means that there was significant influence of actuarial mathematics teaching materials on the students’ analytical skills at the State University of Medan.

Extinction Moment for a Branching Tree Evolution with Birth Rate and Death Rate Both Depending on Age

In this paper, a branching tree evolution is established, in which the birth rate and the death rate are both dependent on node’s age. The extinction probability and the t-pre-extinction (extinct before time t ) probability are studied, by which the distribution of the extinction moment can be given. The analytical formula and the approximation algorithm for the distribution of extinction moment are given; furthermore, the analytical formula and the approximation algorithm of extinction probability are given, and a necessary and sufficient condition of extinction with probability 1 is given. It is the first time to study the distribution of extinction time for the branching process with birth rate and the death rate both depending on node’s age, and the results will do great help in the theory of branching process. It is expected to be applied in the fields of biology, genetics, medicine, epidemiology, demography, nuclear physics, actuarial mathematics, algorithm, and data structures, etc.

Моделювання строго φ-субгауссового узагальненого дробового броунiвського руху

In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fractional Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = (|x|^p)/p, |x| ≥ 1, p &gt; 1. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment.

Про один з методiв побудови моделiстрогоφ-субгауссового узагальненогодробового броунiвського руху

In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fracti-onal Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = exp{|x|} − |x| − 1, x ∈ R. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment.

Напрямки наукових дослiджень Ю.В. Козаченка: статистичне моделювання

В роботi висвiтлюються науковi здобутки доктора фiзико-математичних наук професора Юрiя Васильовича Козаченка в галузi статистичного моделювання. Козаченко Ю.В. працював на кафедрi теорiї ймовiрностей, статистики та актуарної математики КНУ iменi Тараса Шевченка. Професор Козаченко Ю.В. стояв бiля витокiв статистичного моделювання в Київському унiверситетi. Козаченко Ю.В. та його учнями розробленi науковi основи теорiї моделювання гауссових та близьких до них випадкових процесiв i полiв в рiзних функцiональних просторах iз заданими точнiстю i надiйнiстю. При розробцi методiв статистичного моделювання значна увага придiлялась дослiдженню збiжностi статистичних моделей випадкових процесiв та полiв в рiзних функцiональних просторах. До результатiв наукової школи Козаченка Ю.В. належить i розробка теорiї функцiональних просторiв випадкових величин. Значне мiсце в цих дослiдженнях займають простори Орлiча.