“Statistical model of risk assessment of insurance company’s functioning”

The article studies the basic problems of studying the risks of insurance companies’ functioning. It distinguishes the main types of risks of losses that occur during the functioning of insurers. Considering the fact that in practical activity of insurance companies not all types of risks are realized it is offered to use the algorithm of constructing a model for risk assessment of insurance companies’ operations. The model is based on the essential requirements and defining features of statistical models, including: risk assessment on the basis of probability theory.


Introduction
Problem statement. The period of active market transformations in the economies of different countries, the worsening of crisis phenomena in the international financial system had a significant impact on the activity of insurance companies marked by the need to address fundamentally new challenges related to the prevention, identification and elimination of a significant increase in the concentration of risks in the insurance business. In this regard, it is necessary to carry out a thorough research of the issues related to the risks of insurance companies' operations.

Analysis of the latest research and publications.
Theoretical and methodological aspects of assessing the risks of insurance companies are reflected in the numerous works of scholars, in particular: O. Zhabynets explores the problems of classification of risks of insurance companies under conditions of European integration [7]. O. Kozmenko, V. Royenko study the risks of insurance companies in the process of investment activities [4]. O. Kozmenko and O. Kuzmenko study the fundamental formalization of the term "risk" through reinsurance operations [5]. Researcher R. Sobol explores the risks of insurance companies in the context of impact of methods of risk management on the financial condition of insurers [6]. Investigation of management of insurance risks is presented in the works of P. Bondarenko [2]. A. Boyko, V. Royenko, O. Kozhukhovska consider the assessment of risks from the point of view of insurance companies' participation in scheme operations [1,3].
Earlier unsolved parts of the problem. The analysis of the scientific research shows the lack of a common approach to determining the risks of insurance companies. Thus, it is increasingly important to carry out the modeling of risks of Olga  insurance companies in Ukraine because of the lack of an appropriate, adequate and objective methodology for assessing such risks.
The goal of the article is to develop a model for assessing the risks of insurance companies' functioning.
Presentation of the main material. Today, the risks of insurance companies' functioning are considered through the prism of losses caused by inadequate and erroneous internal processes or certain uncontrollable external events. The risks of insurance companies are characterized by a certain probability of occurrence in different combinations. Figure 1 shows the necessity of building a statistical model of risk management of insurance companies' functioning by using the probability approach. Therefore, the model should take into account all the peculiarities of risks of insurance companies' functioning while meeting certain general requirements.
The key requirements to the building of statistical model of risk management for insurance company's functioning with the use of probability approach are: the model should include and describe the main risk management process characteristics for insurance company's functioning, avoiding significantly simplifying them; the obtained results should be similar to the actual results, for it cannot be very common to the real situation; the model is applicable in the making of managerial decisions; the model should include both quantitative and qualitative parameters of risk assessment; the model should be predictable and easily analyzed, letting to identify the most dangerous risk factors.
In addition to the outlined ones, the basic requirement to the model is its adequacy to the real process.  The goal of the survey is to receive answers to the questions regarding the presence of certain risks during the analyzed time period.
The relevance of this approach is caused by the fact that risk incidents have a qualitative nature. It is possible to quantitatively describe them only through singular and zero values.
Therefore, in order to receive the relevant information for the model of risk assessment of insurance companies' functioning it is necessary to survey respondents with an expert method. It is proposed to gather the opinions of all representatives of insurance companies in Ukraine both in the context of risk incidents and the quantitative description of possible losses.
Another important feature in the assessment of risks is the assessment of risks in the functioning of insurance companies in the context of classical approaches based on the theory of probability.
Traditionally, risk has a mathematically expressed probability of losses and can be calculated with a sufficiently high degree of reliability. Therefore, the probability is calculated on the basis of statistical data for a certain period of functioning of a particular insurance company for each risk factor.
Let us consider in detail the structure of the proposed statistical model for assessing the risks in the functioning of insurance companies.
To enter input data we will build two tables, the first of which contains -the absolute number of realizations of a certain risk factor for each insurance company, and the second contains Spossible losses of insurance companies in certain currencies for each risk factor in general.
The model for assessing risks in the functioning of insurance companies is shown in Figure 3.
, which also characterizes the dispersion of the random variable relative to mathematical expectation.
Output parameters of the model for assessing the risks of insurance companies' functioning include: relative potential losses of insurance company W i ; unit price of risk factor U i ; coefficient of variation.
A random event A i (i = n , 1 ) is considered the occurrence of factor for the risks of insurance company's functioning (n -a total number of risk factors in insurance company's functioning). The probability of random events A i (i = n , 1 ) is denoted as p i (i = n , 1 ). We assume that some risk factors are always realized. This means that the events A i (i = n form a complete group, that is n i i p 1 1 . The possible monetary losses of the insurance company from the occurrence of an event A i (i = n , 1 ) is denoted through x i (i = n , 1 ). Based on these data we can set up a series for the distribution of random variables x i (i = n , 1 ).
For a start, we assume that all risk factors in the functioning of insurance companies are equal. In the future, to take into account different importance of various factors it is possible to introduce the socalled weight coefficients.
It is possible to distinguish the following methods for assessing the risks of insurance company's functioning: 1. Probability p i (i = n , 1 ) of occurrence of random event A i (i = n , 1 ), that is the probability that one or another risk factor of the insurance company's functioning is realized.

Standard deviation
, which also characterizes the dispersion of the random variable relative to mathematical expectation. If the variance is measured in square units relative to the random variable x i , then the standard deviation is measured in the same units as x i .
Let us consider this model.
At the first stage it is necessary to enter the data for K i and S i .   (2) Thus, we need to determine the probability p s by means of statistical research.
There may be several ways: 1. Let us suppose that s is the profit of insurance company (or working capital) for a certain period, and V s is the real or potential loss of profits (or reduced working capital) during the same period. Then the statistical probability can be defined as Let us assume that A t is an event that is realized as risk factor i.
The possible losses from the occurrence of event B i will be x i , which in this case will have the following values: As in the classical probability theory the calculation of mathematical expectation makes it possible to assess, which losses from risks the insurance company will bear in relative units. In other words, these are the losses relative to the total amount of losses.
In addition, in making managerial decisions it is important not only to assess the risks quantitatively, but also to take into account the fluctuations of indicators, that is, to determine the extent of variability of the results obtained from the model's use.
Classically, the variance and standard deviation are used for this purpose.
In practice, in the calculation of the variance a simplified equation is used: The variance for each group of insurance companies is calculated with equation 6: Accordingly, standard deviation is calculated with equation 7: Variance and standard deviation of random variables demonstrate the variability of random values relative to mathematical expectations M 1 (X), M 2 (X), M 3 (X) and M 4 (X) for the groups of insurance companies under consideration.
To determine the level of risks in the functioning of insurance companies a variation coefficient is used (equation 8): Calculation of the variation coefficient makes it possible to analyze and approve the scale of risks in the functioning of insurance companies: from 0% to 10% minimal risk; from 10% to 25% low risk; from 25% to 50% acceptable risk; from 50% to 75% critical risk; from 75% to 100% catastrophic risk.
According to this well informed managerial decisions should be made relating to risk management of insurance companies' functioning.

Conclusions
We believe that this model meets all the stated requirements and organically describes risk assessment of insurance companies' functioning.
The whole model is based on the fact that insurance companies function in uncertain conditions. Before making managerial decisions the departments of risk management of insurance companies should conduct a preliminary analysis of different types of risks by using the probability approach. This approach makes it possible to identify the most significant risk factors and to take measures aimed at minimizing the occurrence of negative consequences for companies.
An additional advantage of this model is that the model takes into consideration the company's real activities where each risk factor may be unrealized or realized in different combinations with different levels of probability, which for one company might have negative consequences while giving advantages for another.