Actuarial mathematics

The classical mortality models such as the Cairns-Blake-Dowd (CBD), Lee-Carter (LC), Linear Regression (LR) models are used to model Systematic Mortality Risk (SMR) for many developed countries populations for actuarial product valuations. This research study aims at incorporating the Bühlmann credibility approach (BCA) to improve the SMR models to fit sub-Saharan African populations like Kenya. Since the Kenyan population does not exhibit the Gaussian properties used in modeling the classical error terms, we proposed using Normal Inverse Gaussian distribution to model these error terms instead of a Gaussian distribution. We model the error terms of the classical models (LC, CBD, and LR) as a Normal Inverse Gaussian (NIG) distribution through the Bühlmann credibility approach. This novel approach demonstrates an improved precision of the predicted SMR as shown by the values of MAPE and RMSE measures compared to those under classical mortality risk models. Ultimately, we have done actuarial valuations of annuities and assurances using our determined SMR, thus concluding that this BCA approach improves the accuracy of actuarial products sold in the Kenyan market.

Estudio de sostenibilidad financiera y actuarial de la incorporación de las personas que realizan trabajo no remunerado del hogar a la seguridad social. La obra es fruto del trabajo de varios autores. Sus autores principales son Rosario Maldonado Cabrera y Fabián Vallejo. Colaboración de :  Andrea Toledo – Marco conceptual: análisis de hecho; y de Mónica Vinueza Flores – Marco conceptual: análisis de derecho.

This is a summary of the numbered formulas and end of chapter concepts for EXAM MLC - Models for Life Contingencies.

Details of Reference:
Book Title: Study Manual for SOA Exam MLC Life Contingencies
Edition: 14th
Author of Reference: Abraham Weishaus, PhD, FSA, CFA, MAAA

For erratum, please let me know.

Thanks!
Monica Revadulla

In Kenya, it has always been reported that many pupils who are enrolled in a given school at the same time do not eventually graduate together. Once a pupil is admitted in class one at the date of entry, he or she is expected to proceed with the studies until the end of the 8th year and eventually graduate within the required period. It is, therefore, important to monitor the pupil's general progress from the time of entry in class one to graduation after the expected eight years in Kenya's Primary school level of education. This study is proposed to be conducted in Kitui County, to demonstrate the use of Multiple Decrement Table Model in determining the life expectancy of a pupil enrolled in class one to, the probability that a pupil would proceed to the next class immediately after finishing the previous class and the adversely affected class. The target population for this study includes all the public primary schools in Kitui County, and the model will be generalized to all the public primary schools in the county. The sample for the study includes the County’s class one Public Primary school enrollment in the year 2008, which is purposively selected.

Setiap manusia hidup pasti memiliki rasa khawatir akan masa depannya,
dan setiap orang pasti memiliki resiko terburuk dalam hidupnya seperti
mendapat musibah kebakaran , kecelakaan , sampai kematian. Agar tidak
mengalami kerugian terlalu besar dan menanggulangi resiko yang ada ,
sebagian orang mengansuransikannya dengan berbagai kebutuhan.Untuk
menanggung semua resiko tersebut ada suatu lembaga yang bertugas atau
bertanggung jawab menangani  masalah tersebut dan pihak yang bertanggung
jawab tersebut adalah lembaga asuransi.

A typical problem faced by small business entrepreneurs is to estimate - with reasonable probability - if their business will succeed or fail. This computational model allows them to evaluate the risk of investing in a business project, using financial tools, statistical and simulation, to answer questions of interest to the future of a business. The model is designed to require some input data (INPUTS), key information to process the project feasibility analysis. Once entered, the model will perform a run process through computer tools simulating each variable and estimating the expected results of the business (OUTPUTS). Through dynamic simulation technique, the model will be able to calculate up to 10,000 scenarios and possible outcomes for the business, leading to a conclusion - with reasonable probability - if the project is financially viable or not. The model will provide the results of the scenarios at the OUTPUTS section. This will allow the entrepreneurs to appreciate more clearly the likelihood of success of their business and to evaluate – on a technical basis - whether or not starting a business or adjust their calculation assumptions, BEFORE DECIDING TO INVEST AND WITHOUT PUTTING THEIR MONEY AT STAKE.

The purpose of this study is to evaluate the impact of time series models in forecasting exchange rate. The study determined the time series model that is reliable for predicting the exchange rates of foreign currencies like Dollar, Great Britain Pound and Japanese Yen in Nigeria Naira by determining some salient features about the exchange rate data. The model performance indices were calculated and the graphical implications were also displayed. Together, this study has shown that the linear model and exponential smoothing model at any level of damping factor are not suitable for predicting the exchange rate of foreign currencies like Dollar Great Britain Pound and Japanese Yen in Nigeria Naira. While Naive Model and 5-PD Moving Average are the best for forecasting these but the choice of selection still depends on the model performance index put into use.

Tsao, Hung-ping (2020). Mathematics of Hung-ping Tsao. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors).  Volume 2, Number 11, November 2020; 336 pages. Lenox Institute Press, Newtonville, NY, 12128-0405, USA. No. STEAM-VOL2-NUM11-NOV2020; ISBN 978-0-9890870-3-2. http://dx.doi.org/10.17613/eb3v-gh05, ..........ABSTRACT:  I would like to share some of my ideas in Number Theory, Actuarial Mathematics,
Sudoku Solving and Optimization Teaching with college students and colleagues. ............KEYWORDS:  Polynomial expression, Natural sequence, Power-sum, Binomial coefficient, Product-sum, Stirling number,
Pascal triangle, Alhazen’s Problem, Bernoulli coefficient, Arithmetically progressive sequence, Linear operator, Commutative ring, Recursive formula, n-layer integration, Sorting, Permutation, Combination, Cycle, Subset, Eulerian number, Stirling polynomial, Worpitzky’s identity, q-Gaussian coefficient, Bell number, Ordered Bell Polynomial, Eulerian Bell polynomial,, Life insurance and annuity, Mortality rate, Survivorship function, Mortality, Interest rate, Premium, Reserve, Sudoku, Puzzle, Row, Column, Box.

This study aims to (i) Conduct studies and issue civil servants pension fund in
Indonesia. (ii) Getting a great value of pension benefits and the present value of
pension benefits, (iii) Obtain a plan termination liability and the value of actuarial
liabilities. (iv) Obtain normal tuition fees in the defined benefit pension plan and to
explain and illustrate the surcharge on funding defined benefit pension plan. Based on
this analysis it can be concluded that the calculation of pension funds based on the age
of employees is currently appointed as civil servants (y), Age of employees (x), limits
the retirement age of employees (r), full-time of employee (t), The remaining period of
the employment (rx), and a starting salary of employees can be using the concept
Accrued Benefit Cost. The government needs to do a review of employee pension
payment system should be given attention to the value of the interest rate, large
pension benefits, the value of pension benefits, the value of liabilities termination plan
and to consider interest rates that do not burden the state budget (APBN)

The use of batteries are widely used in electronic goods, so that consumers do not know how long the life span of a battery and the mass difference if the battery life of both types of the same manufacturer with different battery types. One objective of the analysis was to determine the survival probability of a component will operate well within a certain period of time that generates the data life time. Looking for life time data parameter values using the Weibull distribution parameters estimation of linear least squares method after the transformation of the logarithm. The parameter values will be used to search for survival functions , a function of hazard, mean and variance of the Weibull distribution. The data used is the battery life of the brand ABC type SuperPower and Dry-Cell each totaling 20 batteries, where the data is not censored. On the results and discussion for the time t = 0, 300, 500 and 700 minutes, the value of the parameter α = 362.4 and β = 11.25 type batteries SuperPower and α = 334.5 and β = 12.85 for Dry-Cell battery types. ABC brand battery types SuperPower has a value of S (t), h (t), and mean better than batteries Dry-Cell, while the variance of types of Dry-Cell is better for battery lifetimes are relatively similar.

Actuaries are professionals who apply mathematics to financial problems. They evaluate the financial implications of contingent events, in other words, events that are not certain to occur. They are often involved in managing the risks that can arise from undesirable contingent events. Actuaries evaluate the likelihood of future events. They also design ways to reduce the financial impact of undesirable events that do occur. To do their work, actuaries must have a high level of technical knowledge. For example, they need to understand the nature of insurance, the risks inherent in different types of assets, the ways in which statistical models can be used, and the legal and regulatory constraints that apply to the business. They must also have good business sense, problem solving skills, and the ability to communicate effectively with others. Their work often affects many stakeholders, so they must be able to balance different interests and observe high ethical standards in doing so...

España ha seguido la tendencia de otros países en cuanto a la cobertura de las personas dependientes, es decir, aquellas que necesitan ayuda para realizar las tareas básicas de la vida diaria, y lo ha hecho mediante la aprobación de la Ley de Promoción de la Autonomía Personal y Atención a las personas en situación de Dependencia, que se basa

This actuarial model deals with the “fair interest rate” issue, which is a weighted - balanced way for pricing money; where neither the lender nor the borrower gets more benefit than his counterparty. Thus, whatever the choice of time and down payment the client decides on, he will get the optimal price of the requested funding.

Equation Z = f (x,y) represents the variable “fair interest rate”, where both parties would have a "win – win relationship"; i.e., they mutually benefit from lending and none of them gets advantage to the detriment of its counterpart.

The fair price of a car loan is on the surface of a three-dimension graph whose points converge to the optimal vector (x,y,z), which determines the interest rate i, for a healthy credit.

Designing a computational model allow users to interact with variables of down payment and time to observe their influence in the debt; such as monthly payment, interest, taxes and insurance, offering a financial run to analyze. Thus, the actuarial model is fully dynamic and suitable to scenarios.

In conclusion, any point outside the optimal surface chart would imply an advantage / disadvantage to either party, so this model is appropriate to sensitivity analysis to make decisions BEFORE closing the operation, considering different scenarios for the variables (x,y), thus the model represents a PREDICTION tool and even a HEDGING STRATEGY for financial decision making process.

El objetivo de esta investigación es elaborar un análisis comparativo de los sistemas públicos de jubilación más grandes que existen actualmente en México y Argentina; tomando en cuenta que ambos países han emprendido una serie de reformas en materia de planes de retiro de los trabajadores, y cómo han impactado en sus finanzas públicas, en la solvencia de las instituciones de seguridad social, en el monto de las pensiones que recibirán los futuros pensionados, en la tasa de reemplazo del ingreso y su efecto en la calidad de vida de los jubilados, considerando el incremento en la esperanza de vida de las personas. La metodología que se sigue en dicho estudio comienza por comparar los esquemas de jubilación de ambos países, recabar estadísticas, efectuar cálculos actuariales para determinar los importes de las pensiones alcanzadas por un pensionado medio, determinar la tasa de reemplazo del ingreso de los jubilados en ambas naciones latinoamericanas, y evaluar el impacto en la calidad de vida de los futuros pensionados, para arribar a algunas conclusiones y recomendaciones.

The aim of this research is to develop a comparative analysis of the largest public retirement systems currently existing in Mexico and Argentina; considering that both nations have undertaken several reforms on retirement plans for workers, and how this has impacted on public finances, the solvency of social security institutions; as well as the amount of pensions received by future pensioners, the replacement rate of the income and its effect on the quality of life of senior citizens, considering the increase in life expectancy of people. The methodology used in this study begins by comparing retirement schemes in both countries, gathering statistics, performing actuarial calculations to determine the amounts of pensions reached by average pensioners, and to determine the replacement rate of income that retirees can get in both Latin-American nations, estimating the impact on the quality of life of future pensioners; so finally stating some conclusions and recommendations.

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

El presente modelo actuarial de tasa de interes justa, pondera de manera equilibrada el precio del dinero - tanto para un prestamista, como para el solicitante - en donde ninguna de las partes resulta mas beneficiada en perjuicio de la otra. De este modo, sea cual sea la eleccion del enganche y plazo que el cliente decida, siempre obtendra el precio optimo del financiamiento solicitado. La funcion Z = f(x,y) representa a la variable de tasa de interes justa, en donde ambas partes tendrian una relacion del tipo ganar - ganar; es decir, que ambas partes resultan mutuamente beneficiadas de la operacion crediticia y en la cual ninguna de ellas obtiene ventaja en perjuicio de su contraparte. El precio justo del credito automotriz se encuentra sobre la superficie de una grafica de tres dimensiones cuyos puntos optimos convergen en el vector (x,y,z); que determina la tasa de interes i, para un credito sano. El diseno de un modelo computacional permite que el usuario interactue con las vari...

This paper was published in the Year 1998 on how the customers can choose an insurance company when there will be competition in India after the opening of the insurance sector. The insurance sector in India opened in the year 2000, so this paper was written two years prior.
The method has been developed using different variables such as premium rates, lapse rates, customer satisfaction, bonus rates, etc. The method can work on any number of variables. 
It may be noted that during those days, the dot matrix printer was there, so there may be some issue of clarity.

This paper was presented at 8th Global Conference of Actuaries in Mumbai in the year 2006 received the second prize.
This paper gives the first cut rate of Indian incidence of critical illness of cardiovascular, Strokes and Cancer diseases. This paper also compares the critical illness rates currently used in India, CIBT 93 with the rates calculated in this paper. The paper concludes that CIBT 93 should be modified before using in Indian conditions. Summary 1. In India Coronary Heart Disease ("CHD") is more prevalent than cancer whereas in the UK the trend is reverse. 2. In the next ten years, CHD is expected to increase many folds in rural as well as in Urban India. The range may vary from 13% to 80% at different ages. A major increase is expected in the younger ages than older ages. 3. In India both males and females have higher incidence rates of CHD than corresponding rates in the UK. 4. In India females have higher heart attack rates than males, whereas, in the UK, the trend is opposite. 5. The incidence rates of strokes in India are very low as compared to UK. 6. Both in India and the UK, the incidence rates of cancer of females are heavier than males between the ages 20 to 60 years. 7. The cancer incidence rates in males and females in India are heavier up to age 60, after that, UK rates become heavier. 8. According to this paper, one of the options available with the profession is either to make some serious adjustments to CIBT 93 rates to make it useable in India or to construct Indian specific critical illness rates

Autonomic computing systems is a new paradigm and they are characterized with the self-management properties. Autonomic computing systems are emerging as a significant new strategic comprehensive type of information systems, aimed at the design of complex distributed computing systems. Their work was inspired by the functioning of the human nervous system and is intended for designing and building self-management systems. These systems are continually evaluated in terms of their optimization and automatically adjust to changing conditions. Meeting these challenges, autonomic computing systems requires scientific and technological achievements in various fields, also new architectures that will support the effective integration of technologies used in these systems.
For the realization of self-activity in the autonomic computing systems is required to provide appropriate systems, technologies and services. In recent decades is characteristic extended parallel and distributed computing that support network environment. The technologies of autonomic computing systems can integrate with existing programming systems, in order to realize self-management applications.
The aim of this paper will be to define systems, technologies and services that are necessary for realization of the basic concepts and functions in autonomic computing systems and applications. It also would include the implementation of these systems, in terms of techniques of autonomic computing system that can be used with dynamically allocated servers to provide maximum functional usefulness. Also, will be presented system with self-managed properties that will be capable for reallocation of servers in case when there are changes in overload or when an error occurs on the server. The end of the paper will be presented performance evaluation and metrics of the autonomic computing systems. Namely, it is necessary to determine the indicators through wich will evaluate the performance and effectiveness of autonomic computing system.

Regresi logistik multinomial merupakan regresi logistik yang digunakan saat variabel dependen mempunyai sifat polichotomous atau multinomial. Hasil analisis yang diperoleh menyatakan bahwa usia pemegang polis dan cara pembayaran premi secara bersama-sama berpengaruh terhadap status polis asuransi. Usia dikelompokan menjadi dua kelompok yaitu: usia dibawah 30 tahun dan usia antara 31-35 tahun, dengan 4 jenis cara pembayaran yaitu: sekaligus, tahunan, semesteran dan kuartalan. Diperoleh estimasi peluang terbesar pada kelompok usia 31-35 tahun dengan cara pembayaran sekaligus yaitu sebesar 0,630.

This study endeavours at measuring bank competition in Indian loan market using a new competitiveness index, the Augmented Relative Profit Difference (ARPD), which quantifies the impact of marginal costs on performance, measured in terms of market shares. This study yields two major insights. First, the theoretical foundation of the ARPD is very robust as compared to other conventional measures. Second, applying this unbiased competition indicator to Indian bank loan markets shows that financial reform indeed has contributed to significant improvements in competition. As regards, the competition among specific types of banks, we found public sector banks and private sector banks are comparatively more competitive than foreign banks. The empirical finding indicates that the Indian loan market is monopolistic in nature.

The actuarial profession in India was brought up by British rulers in pre-independent India to use actuarial knowledge and qualifications to run the private insurance Companies. Mr. L.S. Vaiydyanathan was first Indian qualified actuary from Institute of Actuaries UK. He was the first person from India to get published his paper “Mortality of Indian Assured Lives” in 1939. The foundation of Indian Actuarial Profession was laid down in 1944 with the formation of Actuarial Society of India (ASI). The profession further got admitted into the International Actuarial Association in 1979. The first initiative towards the development of examination through ASI started by conducting entrance examination for student membership of Institute of Actuaries, UK in 1975.

El principal objetivo del trabajo es la determinacion de las bases para obtener la rentabilidad financiero-fiscal, tanto teorica como aplicada para las tres modalidades de hipoteca inversa mas importantes. Para ello se considera la hipoteca inversa como parte de una operacion financiero-actuarial de ahorro-pension a largo plazo, mas amplia y compleja. La operacion comienza invirtiendo una unidad monetaria en la compra de una vivienda, para despues enlazarla con la hipoteca inversa. Esto obliga a utilizar una metodologia financiera, fiscal y actuarial, estudiando, ademas, tres modalidades de hipoteca inversa: temporal, vitalicia y asociada a un Plan de Prevision Asegurado, PPA. Tambien se comentan las diferencias en cuanto a rentabilidad financiero-fiscal con respecto a otras alternativas de ahorro-pension mas comunes en Espana. Se puede concluir que la utilizacion de la hipoteca inversa es altamente recomendable por la elevada rentabilidad financiero-fiscal obtenida. Tambien se ha p...

Life insurance and the banking sector are the two core sectors where customers keep money with the trust that their money is safe. Both the sectors protect the customer's money. This article looks at both the assets and liabilities of the banking and insurance sector to find out the similarities and differences. The article also looks at the risks of both sectors to find whether there are opportunities for cross-pollination of people working from both sides.
https://bfsi.economictimes.indiatimes.com/news/banking/how-can-risk-management-professionals-switch-between-banking-insurance/84111473

The classical mortality models such as the Cairns-Blake-Dowd (CBD), Lee-Carter (LC), Linear Regression (LR) models are used to model Systematic Mortality Risk (SMR) for many developed countries populations for actuarial product valuations. This research study aims at incorporating the Bühlmann credibility approach (BCA) to improve the SMR models to fit sub-Saharan African populations like Kenya. Since the Kenyan population does not exhibit the Gaussian properties used in modeling the classical error terms, we proposed using Normal Inverse Gaussian distribution to model these error terms instead of a Gaussian distribution. We model the error terms of the classical models (LC, CBD, and LR) as a Normal Inverse Gaussian (NIG) distribution through the Bühlmann credibility approach. This novel approach demonstrates an improved precision of the predicted SMR as shown by the values of MAPE and RMSE measures compared to those under classical mortality risk models. Ultimately, we have done actuarial valuations of annuities and assurances using our determined SMR, thus concluding that this BCA approach improves the accuracy of actuarial products sold in the Kenyan market.