Моделирање на актуарските елементи на животното осигурување со примена на матричната алгебра

Abstract

The subject of this doctoral dissertation research is modelling the actuarial elements of life insurance by implementation of the matrix algebra. This paper comprehensively analyzes the theoretical basis of the matrix algebra, which is largely applied in the advanced statistics, which will be taken as the basis for further elaboration on the subject. Here we will discuss the vector space. Then it will be given an appropriate definition of matrices, which is a rectangular pattern of numbers arranged in rows and columns, thus we will include various types of matrices. We will develop the operations with matrices, matrix polynomials, rank and transposition of matrices and characteristic numbers and vectors of matrices. After setting the stage on this topic we will focus on actuarial basis for the formation of life insurance tariff. We will work out different types of life rent as follows: anticipative and discursive constant life rent, variable periodic life rent, and under-annual and continuous life rent. We will also develop the combined models for insurance of capital. In terms of insurance of two individuals or mutual insurance of two individuals, we will include the likelihood of two individuals to be alive, commutative numbers, combined models for mutual life insurance rent and capital insurance. The insurer is legally obliged to set up a fund - mathematical reserve that is used to collect the funds from the paid premium of all insurers. The mathematical reserve is calculated on the basis of a predetermined percentage of premiums paid, using appropriate actuarial estimates. At the end we will elaborate the matrix defining the actuarial bases of life insurance, thus defining the matrix of the commutative functions, it will be calculated the one-off premium and annual probable present value of different types of life rent, annual and under-annual net and gross premium for insurance of capital and rental insurances and insurance of capital for two individuals. The aforementioned leads us to the conclusion that the life insurance is unique among the financial instruments. It is one, maybe the unique one among the financial instruments which is based on the care and love. While there may be no personal benefit from the life insurance, the real incentive is the love for those we care most about - to be certain that they are taken care of.