IMPROVEMENT OF METHODS OF THE PROPERTY VALUATION BASED ON MATHEMATICAL AND NUMERICAL MODELING

The conducted researches aimed to analyze modern models and methods of implementation of the comparative approach in property and real estate appraisal, to investigate and evaluate their effectiveness, to suggest ways of further improving the methods of property valuation. To achieve this aim, the authors carried out a series of computational experiments on a computer. The essence of these experiments was the comparative evaluation of the cost of test properties based on the following methods: expert evaluation, solving systems of linear algebraic equations, based on the normalized distances in the space of pricing factors and the correlation-regression method. An analysis was made of the influence of methods of digitizing the initial data (pricing factors) on the final cost of real estate. On the basis of the studies, prospective ways and directions for further improvement of the methods of property valuation are determined.


Introduction
In conditions of active reformation of property relations, the need for services for valuation of property and property rights is regularly growing. The problems of property valuation related to property alienation, property disputes, the problem of land reform, the formation of an alternative model of partnership between the public and private sectors, require objective decisions of the appraisal expertise [1−6]. Accordingly, there is a need to improve the toolkit of the procedure for conducting property valuation by applying modern mathematical models and methods to obtain an objective market value of property and to make effective management decisions.

Computer Sciences
To achieve this aim, the following tasks are solved in the work: − to conduct an analysis of methodological approaches to the valuation of property and real estate; − to analyze the methods of implementing a comparative approach in the valuation of property and real estate; − to conduct a comparative analysis of the calculation of the property value on the basis of expert valuation methods, solving the system of linear equations, the method of calculating corrective corrections based on the normalized distances in the space of pricing factors and on the basis of the correlation-regression method; − to carry out computational experiments to investigate the effect of the process of digitizing the original data on the final value of the real estate object when implementing the studied methods.

The essence and methods of research of modern methods of implementing a comparative approach in valuation of property and real estate 1. Object, subject and methods of research
The objects of this research are real estate objects. The subject of this research is methods for determination of the value of real estate objects in the framework of a comparative approach.
Research methods are mathematical and numerical modeling.
The research is carried out in two main directions: 1. Computational experiment on the study of existing methods for implementing a comparative approach in the valuation of property and real estate, the essence of which was the comparative valuation of test real estate objects obtained by different methods.
2. Computational experiment to study the sensitivity of the methods under consideration to implement a comparative approach in the valuation of property and real estate according to the procedure for digitizing the initial data (pricing factors).

2. Investigated methods for implementing a comparative approach in valuation of property and real estate
Expert method of determining corrective corrections The simplest and most popular method for implementing a comparative approach in real estate valuation is the expert method of determining corrective amendments. It is based on making corrections to the cost of analogical objects for certain pricing factors, by comparing them with the valuation object. Most often, amendments are introduced to: terms of sale, location, technical condition and others. At the same time, a certain step is chosen, based on the vision of a specific expert. Most often this step is 5 %. The final value of the valuation object is derived as the arithmetic mean, modal or median value [2][3][4]6].
It is clear that this method is very dependent on the qualification of the expert and the choice of the calculated step.

The method of calculating corrective corrections based on the solution of a system of linear algebraic equations (SLAE)
The method of calculating corrective corrections based on the solution of a system of linear algebraic equations (SLAE) is more substantiated and formalized [6][7][8]. According to this method, the value of the property is based on the formula: where V 0 -the value of the valuation object, P i -the price of the i-th counterpart; ∆P ij -adjustment of the price of the i-th analogue to the difference in the object of estimation by the j-th pricing factor. According to the model (1), it is necessary to compare the object in succession with each of the selected analogs. As a result, let's obtain a system of linear equations: ... V x P x P ... x P P .
It is more convenient to find the solution of this system by the matrix method. The main drawback of this method is that the number of analogs should exceed by 1 number of factors. This certainly causes certain difficulties in the implementation of this method, related to the need to search for additional analogues or a reduction in the number of pricing factors.
The method of calculating corrective corrections based on the normalized distances in the space of pricing factors Another formalized method has no limitations on the number of pricing factors and analogical objects is the method of calculating corrective corrections based on the normalized distances in the space of pricing factors [8]. According to it, the definition of the value of the property is carried out in 4 stages.
Preparatory stage. At this stage, the state matrix M is formed: where n is the number of analogical objects; 01 02 0m x , x , , x … -the object of valuation; n1 n 2 nm x , x , , x … -i-th analogue. The first stage is the formation of a matrix of congruences At this stage, let's form the matrix of comparisons ΔM by subtracting the value of the first row of the matrix from the corresponding elements of all subsequent rows: 11 12 1m n1 n 2 nm Thus, each element of the comparison matrix ij x ∆ characterizes the difference between the i-th analogue object and the object of estimation by the j-th factor. The second stage is the normalization of the matrix of congruences In order to move away from the different units of measurement of the columns of the matrix ΔM, each element of the matrix is normalized as follows: Thus, let's obtain a normalized comparison matrix +∆M.

The third stage is the determination of weight coefficients
On the basis of the matrix +∆M let's calculate the distance l i from the formula: In an n-dimensional Euclidean space, the object of estimation acts as the center of coordinates, and to each analogue object there corresponds a separate point, remote from the center of coordinates (the object of valuation) by the distance l i .

Computer Sciences
To each quantity l j . it is possible to compare the reciprocal quantity i i 1 P l = , which can be transformed into a weighting coefficient by the formula: The fourth stage is the determination of the value of the valuation object. Taking into account the found weight coefficients, the value of the valuation object is determined by the formula: where C i -the cost of the i-th analogue object. This method is guaranteed to give the result of the value of the valuation object, does not exceed the limits of the cost of used analogs.
The advantage of this method is that it takes into account the degree of proximity of the valuation object to the analogue objects.

Correlation-regression method for determining the value of real estate objects
If there is a significant amount of information on analogous objects, then it is reasonable to determine the market value of a real estate object within the framework of a multifactor regression analysis that allows one to assess the degree of influence on the investigated performance indicator of each of the factors introduced into the model with a fixed position at the middle level of other factors [9−18].
Mathematically, the task is reduced to finding an analytical expression, it would better reflect the connection of the factor characteristics with the resultant: The most difficult problem is choosing the form of communication. However, taking into account that any function of many variables can be reduced to a linear form by logarithmation or substitution of variables, the multiple regression equation can be expressed in a linear form: Y a a X a X ... a X , = + + + + where Y -the calculated values of the effective indicator; Х 1 , Х 2 , …, Х n -the factors; а 0 , а 1 , а 2 ,…а n -the equation parameters. The equation parameters are calculated by the least squares method. So, in the case of two-factor regression: (11) Each coefficient of the equation indicates the degree of influence of the relevant factor on the score for a fixed position of other factors.
In order to identify the relative strength of the influence of individual factors and their reserves, it is advisable to calculate the partial elasticity coefficients and beta coefficients by the formulas: -the standard deviation of the i-th factor; Ó σ Y -the standard deviation of the result.
Partial coefficients of elasticity show how many percent the result on the average will change with 1 % change of each factor for a fixed position of other factors.

Analysis of the results of a computational experiment on the methods of realizing the comparative approach in real estate valuation
In order to valuate and compare various methods of calculating the value of real estate in the comparative approach, a number of test cases are examined, a fragment of one of which is given below.
Let's suppose that it is necessary to evaluate the property (living quarters), which is characterized by the indicators shown in Table 2. For comparison, 5 analogical objects are selected, the characteristics of which are also given in Table 2. Let's assess the value of the object for such pricing factors: technical condition, location, availability of improvement, the presence of a parking slot. According to the expert method of calculating the cost of the valuated object, a standard step of 5 % was chosen. As a result of the calculations, the following results are obtained ( Table 3). Table 3 Result of calculation of cost of valuated object by a method of an expert estimation Further, the cost of the same property is calculated using the SLAU solution method. For this purpose, the initial data were digitized ( Table 4).
As a result of the implementation of this method, a solution is obtained in this form ( Table 5). Element V 0 -this is the market value of 1 square. m (for the estimated property -13362.5 UAH.).
Other elements of the obtained matrix V can be interpreted as follows: In order to reveal the dependence of the choice of the method of digitizing the initial data on the final cost of the valuation object, a computational experiment is carried out, a fragment of the results of which are given in Table 6. In the course of the experiment it turned out that the method of digitizing data in this method affects the final result of the cost. In this case, even corrections may occur with a negative sign, contrary to the physical meaning of the problem.   Further, the cost of this real estate object is calculated by determining the weight coefficients based on the normalized distances in the space of the pricing factors. For this purpose, data are digitized similar to the previous method and calculations are performed. The obtained results can be interpreted as the value of the object of valuation and the degree of remoteness of the object from the analogical objects (Table 7).
To determine the dependence of the choice of the method of digitizing data on the final cost of the valuation object, a computational experiment is conducted to study this method, a fragment of the results of which are given in Table 8. In the course of the experiment it turned out that the method of digitizing data in this method also affects the final result of the value of the valuation object ( Table 8).

Discussion of the research results on methods for implementing a comparative approach in real estate valuation
To date, the main methods that implement a comparative approach in the valuation of property and real estate is the method of expert evaluation, the method of calculating corrective corrections based on the solution of a system of linear equations, the method of calculating corrective corrections based on the normalized distances in the space of pricing factors, and the correlation-regression method.
Comparison of the results of calculations of the value of the object valuation by various methods has shown that the expert method, as the most subjective, gives the most overestimated value of real estate objects; the method of SLAE solution and the method of determining the weight coefficients on the basis of the normalized distances in the space of the pricing factors show a somewhat understated valuation (Fig. 1). In addition, the analysis of the results of a computational experiment on the influence of methods of digitizing the initial data on the final value of the real estate object when implementing various methods has shown that the method of calculating corrective corrections based on the solution of a system of linear equations and the method of calculating weight coefficients on the basis of the normalized distances in the space of pricing factors essentially depend on the method of digitizing the initial data, while the correlation-regression method does not exert addiction is without suspended its advantage.

Conclusions
Valuation of property and property rights is an important prerequisite for the functioning and development of civilized market relations. Therefore, the problems associated with the improvement of valuation and appraisal activity are always topical.
Modern practice of valuation of real estate has a significant methodological and calculating apparatus for making sound management decisions. However, the proposed economic and mathematical models in most are aimed at reflecting the general trends in the development of the real estate market, while the real estate market is characterized by variability and multifactoriness, which makes it necessary to take into account its stochastic nature in computational models.
On the basis of the research carried out in the work, it can be concluded that today the most expedient in valuation of real estate is the development and improvement of methods of correlation and regression analysis, it allows not only to determine the current market value of real estate objects, but also to establish investigative causality. Obligations between pricing factors , predicting qualitative and quantitative changes in the real estate market, is an integral part in making the most effective management solutions.
The methods and results of scientific research presented in this work are used in the evaluation of real residential and non-residential real estate LLC "Tasador" (Kyiv, Ukraine).