729 AMERICAN Journal of Public Diplomacy and International 
Studies 

www. grnjournal.us   

AMERICAN Journal of Public Diplomacy and 
International Studies 

Volume 01, Issue 10, 2023 ISSN (E): 2993-2157 
 
 
 

Basics of Life Insurance Development. Construction of Life 

Continuity Functions 

 
Khamroeva Dilsuz Yakhoevna 

PhD student of the Faculty of Economics of the National University of Uzbekistan, 

lecturer of Turon Zarmed University. 

Email: mohlaroyim261218@gmail.com 

 

Annotation: In this study, protection against unpredictable accidents that occur on the basis of 

insurance activity planning, reduction of losses from these events, the role of life insurance in the 

insurance market, table of life expectancy and the necessity of constructing life expectancy functions 

has been studied. 

Keywords: Life insurance, long term and short term life insurance, rent, GDP, GNP, actuarial 

mathematics, life expectancy curves. 

 

INTRODUCTION. 

Today, in our country, it is necessary to further develop the financial market, expand the scope 

of coverage of the population with quality financial services, support the activities of insurance 

organizations, create favorable conditions for protecting the rights and legal interests of consumers in 

this area, as well as , measures to expand insurance types and increase consumer confidence in 

insurance, thereby increasing insurance income, are being implemented consistently. 

As you know, insurance of investments, cash, property, life etc. is a common practice all over the 

world. The global cataclysms and crises of the last 10-15 years have shown how dangerous this business 

is and how important this sector is in the social sphere (social security, security, etc.). 

Actuarial science studies insurance and its mathematical and theoretical foundations are classified 

as actuarial mathematics. One of the main tasks of actuarial mathematics is to establish the optimal 

ratio between the insurance premium (the money the policyholder pays to the insurance company) and 

the payment (the money the company pays when a certain insured event occurs). Of particular interest 

is the part of insurance called "personal insurance". This type of insurance includes health and life 

insurance. 

In calculations related to life insurance, as well as in the formation and modeling of insurance 

and pension schemes, actuaries widely use tables called life tables (TLE-life expectancy tables). For 

example, using the table, it is possible to determine what reserve the insurance company should have 

when insuring a homogeneous group of people aged 25-55. An important feature of simple tables is 

that TLE is compiled as a range of whole values of human age. This, in turn, causes certain 

inconveniences in actuarial calculations. First, these tables have a large size with many parameters. 

Secondly, TLE does not have the possibility to be used directly in calculations related to non-integer 

values of a person's age. For this reason, it is important to construct lifetime functions. 

LITERATURE ANALYSIS AND METHODOLOGY. 

A number of conditions have been presented regarding the importance and necessity of finding a 

http://www/
mailto:mohlaroyim261218@gmail.com


730 AMERICAN Journal of Public Diplomacy and International 
Studies 

www. grnjournal.us   

simple analytical formula to describe population mortality [2]. The main points in this regard are as 

follows: 

1) Like many physical phenomena described by simple formulas, there are several biological 

arguments in favor of the existence of analytical laws for the process of population extinction;  

2) From a practical point of view, it is more convenient to work with a function with several 

parameters than to deal with TLEs that are large in size; 

3) It is often easier to reconstruct the Lifetime Function if there are assumptions about its 

parameters than to construct the TLE. 

Thus, it is natural to consider the problem of finding an adequate analytical law of mortality that 

fits the data observed in reality. However, when solving such problems, researchers usually follow the 

following general principles [4]: 

1) The principle of theoretical validity. Finding an equation that has theoretical foundations, i.e. 

connections arising from various theoretical concepts. 

2) The principle of universality. The desire to define a general function that applies to the widest 

range of natural phenomena. In accordance with this principle, the general laws of life span distribution, 

which apply to various organisms, including humans, are of particular importance. 

3) The principle of sufficient approximation with the least number of parameters. The formula 

that meets this principle gives the most compact view of the data, which allows to restore the 

distribution with a minimum number of observations. 

4) The principle of local description. If the proposed law of distribution of life expectancy is valid 

only for a limited age range, this is not yet a reason for a critical attitude towards it. The limited 

application of a law does not show that it is wrong, only that it is a special case of another, more general 

and as yet unknown law. 

Life expectancy curves 

The uncertainty or unpredictability of death, illness, or accident is not only a major risk factor but 

also a source of randomness in life insurance. It allows the use of random events, quantities, processes 

in mathematical analysis of various aspects of life, health, car insurance, etc. At the same time, the 

creation of an adequate theory of life insurance should begin with the development of a system of 

concepts and quantification, which allows making objective conclusions about life expectancy. The 

main conclusion here is as follows, it is difficult to say anything definite about the death of a person, 

as a rule. However, if we consider a sufficiently large homogeneous group of people without being 

interested in the fate of individual people in this group, we are within probability theory as the science 

of mass random events with the property of frequency stability (for example, normal or Poisson 

distribution laws approach etc.). Therefore, using the terminology of probability theory, the life 

expectancy X is entered as a random variable - a random variable. 

Let X- denote the life expectancy with P(X>0)=1. Distribution function for X tm describing the 

complete characteristic, nature of X tm 

F(x)=P(X≤x)       (1) 

let it be    

In actuarial mathematics, it is common to consider a function that complements (1) and is called 

the survival function: 

𝑆(𝑥)=P(X>x)=1-F(x). 

This is the probability that a person will live to (maturity) x years. 

The life expectancy function as an additional function to the distribution function of X has the 

following characteristic properties: 

1) 𝑆(𝑥) is not increasing and ;0 ≤ 𝑆(𝑥) ≤ 1 

http://www/


731 AMERICAN Journal of Public Diplomacy and International 
Studies 

www. grnjournal.us   

2) 𝑆(0)=1, 𝑆(+∞)=0; 

3) 𝑆(𝑥) continuous from the right. 

However, for the actual process of death, properties 1) and 3) are slightly modified. In fact, the 

function of life duration must strictly decrease, otherwise there will be a certain period in a person's 

life, for example, dx=xi-x , he will not die. Besides, 𝑆(𝑥) must be continuous, otherwise in human life 

it will be the instant of death x0 with non-zero probability: 

DR=S(x0-)-S(x ),  

𝑆(𝑥0−) = lim
𝑥→𝑥0−

𝑆(𝑥), 

𝑆(𝑥0) = lim
𝑥→𝑥0+

𝑆(𝑥). 

Also, since X has a limited real life span, it is usually assumed to be finite at age a (as a rule, 

a=100, 120 years), so  𝑆(𝑥) x>a for S(x)=0. 

The lifetime S(x) has a simple statistical meaning. It is equal to the average percentage of a certain 

constant group of newborns who survive to the age of x [6]. 

According to the general property of probability theory, according to the stochastic nature of a 

continuous random variable, its density is . 𝑓(𝑥) 

𝑓(𝑥) = 𝐹′(𝑥) = −𝑆′(𝑥) 

The intensity of death (life expectancy) is determined as follows: 

𝜇𝑥 =
𝑓(𝑥)

1−𝐹(𝑥)
=

𝑓(𝑥)

𝑆(𝑥)
= −𝑑ln(𝑆(𝑥)). 

Obviously, it has the following properties [7]: 

1)𝜇𝑥 ≥ 0 

2).∫ 𝜇𝑢du=+∞
∞

0
 

Together, these functions are commonly referred to as the mortality (life expectancy) curve.  

Since mortality curves provide a general description of the decline of a given group of observations in 

a population, great importance is attached to finding an adequate model [6]. These types of curves are 

hypothesized based on demographic hypotheses. 

Some analytical laws of death have been studied in several works, 

𝑆(𝑥) = 1 −
𝑥

𝛼
  De Moivre Model (1729) 

𝑆(𝑥) = exp[−𝐵(𝑒𝛼𝑥 − 1 ) 𝛼⁄ ]- Gomperts model (1825) 

𝑆(𝑥) = exp[−Ax − 𝐵(𝑒𝛼𝑥 − 1 ) 𝛼⁄ ] Makeham Model (1860) 

𝑆(𝑥) = exp[−Ax − Hx2 2⁄ − 𝐵(𝑒𝛼𝑥 − 1 ) 𝛼⁄ ]  Makeham model 2 

𝑆(𝑥) = exp[− kx𝛽 (⁄ 𝛽)]- Weibull model 

𝑆(𝑥) = exp[−(𝑥 𝛼⁄ )𝛽(𝑥) − Veon modeli (2003) 

Life expectancy functions have been proposed in various forms by Perks (Perks, 1932), Beard 

(Beard, 1959-71), Vaupel (Vaupel et al., 1979), Bras (Le Bras, 1976) and Cannisto (Cannisto, 1992). 

proposed models [1]. 

DISCUSSION AND RESULTS. 

We model the distribution of random variable life expectancy X on the basis of life expectancy 

tables compiled for the general population of Uzbekistan (men and women). In this case, it is taken as 

a=100. in cases where it is a constant and a function of time, the parameter b is considered separately. 

http://www/


732 AMERICAN Journal of Public Diplomacy and International 
Studies 

www. grnjournal.us   

In the first case, 𝛽≈0.445. If is a function of time, then the corresponding curve is:  𝛽 

𝛽𝑛(𝑥) =
-0.00000029𝑥5 + 0.0000674𝑥4-0.00516𝑥3 + 0.1674𝑥2-2.197𝑥 + 12.09

𝑥 + 7.45923
 

where are the corresponding coefficients of the rational polynomial 𝛽𝑛(𝑥) found by the method 

of least squares. A joint representation of the corresponding analytical model and the empirical 

distribution functions of the observations is as follows (Figure 1). 

 
Figure 1. Joint plot of the constructed model and empirical life expectancy distribution 

function. 

 
Figure 2. Combined observation histogram and model density plot. 

CONCLUSION. 

Actuarial-friendly life expectancy functions provide solutions to a wide range of problems in life 

insurance calculations. This is much more convenient than Life Duration Tables. TLE cannot be used 

http://www/


733 AMERICAN Journal of Public Diplomacy and International 
Studies 

www. grnjournal.us   

directly in calculations involving non-integer values of a person's age. For this reason, it is important 

to construct lifetime functions. 

Through the above results, it is possible to use this model not only in economics and mathematics, 

but also in demography and biology. Life expectancy curves are of great importance in finding an 

adequate model, as they provide a general description of the decline of a given set of population 

observations. 

LIST OF REFERENCES 

1. Bowers N., Gerber H., Hickman J., Jones D., Nesbit C. Actuarial Mathematics. -Itasca, IL: The 

Society of Actuaries, 1986, -624p 

2. Devroy L., Dorfi L. Non-parametric plotting. L1-podkhod. - M.: "Mir",1988., -408 str. 

3. Slud EV Actuarial Mathematics and Life-Table Statistics. - Internet eBook 

4. . Borovkov A.A. The theory is probable. 2-e izd., pererab. i dop. - M.: "Nauka".1986 g., -432 

str. 

5. Gavrilov L.A., Gavrilova N.S. Biology prodolzhitelnosti jizni. 2-e izd. - M.: "Nauka",1991., -

280 str. 

6. Gerber H. Mathematics strashovaniya jizni. - M.: "Mir",1995 g., -154 str. 

7. Kasimov Yu.F. Introduction to actuarial mathematics. Strakhovanie jizni i pensionnyx scheme. 

- M.: "ANKIL",2001., -172 str. 

8. Abdushukurov A., Hamidov R. On a new model of mortality. Proceed. of annual world conf. 

"Financial and Actuarial Mathematics". Krasnoyarsk: KSTEI, p.3, 2009. 

9. Kendall M. DJ, Stuart A. The theory is raspredeleny. - M.: "Nauka",1966 g., -587 str. 

 

 

 

 

 

http://www/