Factor analysis. Economic analysis. Anna sergeevna litvinyuk economic analysis Calculate the influence of factors generalizing indicator

All phenomena and processes of economic activity of enterprises are interconnected and interdependent. Some of them are directly related to each other, others are indirectly. Hence, an important methodological issue in economic analysis is the study and measurement of the influence of factors on the value of the studied economic indicators.

Under economic factor analysis a gradual transition from the initial factorial system to the final factorial system is understood, the disclosure of a full set of direct, quantitatively measurable factors that affect the change in the effective indicator.

By the nature of the relationship between the indicators, the methods of deterministic and stochastic factor analysis are distinguished.

Deterministic Factor Analysis is a methodology for studying the influence of factors, the relationship of which with the effective indicator is of a functional nature.

The main properties of the deterministic approach to analysis:
· Construction of a deterministic model by logical analysis;
· The presence of a complete (rigid) relationship between indicators;
· Impossibility of separating the results of the influence of simultaneously acting factors that cannot be combined in one model;
· Study of relationships in the short term.

There are four types of deterministic models:

Additive models represent an algebraic sum of indicators and have the form

Such models, for example, include cost indicators in relation to production cost elements and cost items; an indicator of the volume of production in its relationship with the volume of output of individual products or the volume of output in individual divisions.

Multiplicative models can be summarized by the formula

An example of a multiplicative model is the two-factor sales volume model

where H- the average number of employees;

CB- average output per employee.

Multiple models:

An example of a multiple model is the indicator of the turnover period of goods (in days). T OB.T:

where Z T- the average stock of goods; O P- one-day sales volume.

Mixed models are a combination of the above models and can be described using special expressions:

Examples of such models are cost indicators per ruble. commercial products, profitability indicators, etc.

To study the relationship between indicators and quantitatively measure many factors that influenced the effective indicator, we present general model transformation rules in order to include new factor indicators.

To detail the generalizing factor indicator into its components, which are of interest for analytical calculations, use the method of lengthening the factor system.

If the original factorial model is a, then the model will take the form .

To isolate a number of new factors and construct the factor indicators necessary for the calculation, the method of expanding factor models is used. In this case, the numerator and denominator are multiplied by the same number:

To construct new factor indicators, the method of reducing factor models is used. When using this technique, the numerator and denominator are divided by the same number.

The detail of factor analysis is largely determined by the number of factors, the influence of which can be quantified, therefore, multifactor multiplicative models are of great importance in the analysis. Their construction is based on the following principles:
· The place of each factor in the model should correspond to its role in the formation of an effective indicator;
· The model should be built from a two-factor full model by sequentially dividing factors, usually qualitative, into their components;
· When writing a formula for a multivariate model, the factors should be located from left to right in the order of their replacement.

Building a factorial model is the first stage of deterministic analysis. Next, a method for assessing the influence of factors is determined.

Chain substitution method consists in determining a number of intermediate values of the generalizing indicator by sequentially replacing the basic values of the factors with the reporting ones. This method is based on elimination. Eliminate- means to eliminate, exclude the influence of all factors on the value of the effective indicator, except for one. At the same time, proceeding from the fact that all factors change independently of each other, i.e. first, one factor changes, and all the others remain unchanged. then two change while the others remain unchanged, and so on.

In general, the application of the chain setting method can be described as follows:

where a 0, b 0, c 0 - the basic values of the factors influencing the generalizing indicator y;

a 1, b 1, c 1 - the actual values of the factors;

y a, y b, - intermediate changes in the resulting indicator associated with changes in factors a, b, respectively.

The total change D у = у 1 –у 0 consists of the sum of changes in the resulting indicator due to changes in each factor with fixed values of the remaining factors:

Let's consider an example:

table 2

Initial data for factor analysis

Indicators	Symbols	Basic values	The actual meaning	The change
Indicators	Symbols	Basic values	The actual meaning	Absolute (+, -)	Relative (%)
Volume of marketable products, thousand rubles
Number of employees, people
Production per worker,

The analysis of the effect of the number of workers and their output on the volume of marketable products will be carried out in the manner described above, based on the data in Table 2. The dependence of the volume of commercial products on these factors can be described using a multiplicative model:

Then the influence of changes in the number of employees on the generalizing indicator can be calculated using the formula:

Thus, the change in the volume of marketable products was positively influenced by a change in the number of employees by 5 people, which caused an increase in the volume of production by 730 thousand rubles. and a negative impact was exerted by a decrease in output by 10 thousand rubles, which caused a decrease in volume by 250 thousand rubles. The combined influence of two factors led to an increase in the volume of production by 480 thousand rubles.

The advantages of this method: versatility of use, simplicity of calculations.

The disadvantage of this method is that, depending on the chosen order of replacement of factors, the results of factorial decomposition have different meanings. This is due to the fact that as a result of the application of this method, a certain indecomposable residue is formed, which is added to the magnitude of the influence of the last factor. In practice, the accuracy of assessing factors is neglected, highlighting the relative importance of the influence of one factor or another. However, there are certain rules that govern the sequence of substitutions:
· In the presence of quantitative and qualitative indicators in the factor model, first of all, the change in quantitative factors is considered;
· If the model is represented by several quantitative and qualitative indicators, the sequence of substitution is determined by logical analysis.

Quantitative factors when analyzing, they understand those that express the quantitative certainty of phenomena and can be obtained by direct accounting (the number of workers, machines, raw materials, etc.).

Qualitative factors determine the internal qualities, signs and characteristics of the studied phenomena (labor productivity, product quality, average working day, etc.).

Method of absolute differences is a modification of the chain substitution method. The change in the effective indicator due to each factor by the method of differences is determined as the product of the deviation of the studied factor by the basic or reported value of another factor, depending on the chosen sequence of substitution:

Relative difference method is used to measure the influence of factors on the growth of the effective indicator in multiplicative and mixed models of the form y = (a - c) . with. It is used in cases where the source data contains previously determined relative deviations of factor indicators in percent.

For multiplicative models like y = a . v . c the analysis method is as follows:

Find the relative deviation of each factor indicator:

Determine the deviation of the effective indicator at due to each factor

Example. Using the data in the table. 2, we will analyze the method of relative differences. The relative deviations of the factors under consideration will be:

Let's calculate the impact on the volume of marketable products of each factor:

The calculation results are the same as when using the previous method.

Integral method allows you to avoid the disadvantages inherent in the method of chain substitution, and does not require the use of techniques for the distribution of the indecomposable remainder by factors, since it has a logarithmic law of redistribution of factor loadings. The integral method allows achieving a complete decomposition of the effective indicator by factors and is universal in nature, i.e. applicable to multiplicative, multiple and mixed models. The operation of calculating a definite integral is solved using a PC and is reduced to the construction of integrands that depend on the type of function or model of the factor system.
1. What management tasks are solved by means of economic analysis?
2. Describe the subject of economic analysis.
3. What are the distinctive features of the method of economic analysis?
4. What are the principles underlying the classification of techniques and methods of analysis?
5. What role does the comparison method play in economic analysis?
6. Explain how to build deterministic factor models.
7. Describe the algorithm for applying the most simple methods of deterministic factor analysis: the method of chain substitutions, the method of differences.
8. Describe the advantages and describe the algorithm for applying the integral method.
9. Give examples of problems and factor models to which each of the methods of deterministic factor analysis is applied.

This might be interesting (selected paragraphs):

In statistics, planning and analysis of economic activity, index models are the basis for a quantitative assessment of the role of individual factors in the dynamics of changes in generalizing indicators. The index method is one of the methods of elimination. It is based on relative indicators of dynamics, spatial comparisons, the implementation of the plan, expressing the ratio of the actual level of the analyzed indicator in the reporting period to its level in the base period (or to the planned, or for another object). Any index is calculated by comparing the measured (reported) value with the base one. Indices expressing the ratio of directly measurable quantities are called individual, and those characterizing the ratio of complex phenomena - group, or total.

Statistics operates with various forms of indices (aggregate, arithmetic, harmonic, etc.) used in analytical work.

An aggregate index is the basic form of any general index; it can be converted to both arithmetic mean and harmonic mean indices. Using aggregate indices, it is possible to identify the influence of various factors on the change in the level of effective indicators in multiplicative and multiple models.

The correctness of determining the size of each factor depends on:

1) the number of decimal places (at least four);

2) the number of factors themselves (the relationship is inversely proportional).

The principles of constructing indices: a change in one factor with the same value of all the others, while if the generalizing economic indicator is a product of quantitative (volumetric) and qualitative indicators-factors, then when determining the influence of a quantitative factor, the qualitative indicator is fixed at the basic level, and when determining the influence of a qualitative factor, the quantitative indicator is fixed at the level of the reporting period.

Let Y = a? B? C? D. Then:

Wherein: l Y = l a? l b? l c? l d.

The index method makes it possible to decompose not only relative, but also absolute deviations of the generalizing indicator by factors. In this case, the influence of individual factors is determined using the difference between the numerator and denominator of the corresponding indices, i.e., when calculating the influence of one factor, the influence of the other is eliminated:

Let Y = a? B, where a is a quantitative factor, ab is a qualitative one. Then:

a 1? b 0 -a 0? b 0 - absolute increase in the resulting indicator due to factor a;

a 1? b 1 -a 1? b 0 - absolute increase in the resulting indicator due to factor b;

a 1? b 1 -a 0? b 0 - an absolute increase in the resulting indicator due to the influence of all factors.

This principle of decomposition of the absolute increase (deviation) of the generalizing indicator by factors is suitable for the case when the number of factors is equal to two (one of them is quantitative, the other is qualitative), and the analyzed indicator is presented as their product.

The theory of indices does not provide a general method for decomposing the absolute deviations of the generalized indicator by factors when the number of factors is more than two. To solve this problem, the method of chain substitutions is used.

5.3. methods of quantitative analysis of the influence of factors on the change in the final indicator

In the analysis of economic activity, which is sometimes called accounting analysis, methods of deterministic modeling of factor systems prevail, which give an accurate (and not with some probability characteristic of stochastic modeling), balanced characterization of the influence of factors on the change in the result indicator. But this balance is achieved by different methods. Let's consider the main methods of deterministic factor analysis.

Differential calculus method. Differentiation is the theoretical basis for a quantitative assessment of the role of individual factors in the dynamics of the resultant generalizing indicator.

In the method of differential calculus, it is assumed that the total increment of a function (resulting indicator) is decomposed into terms, where the value of each of them is determined as the product of the corresponding partial derivative by the increment of the variable over which the given derivative is calculated. Let us consider the problem of finding the influence of factors on the change in the resulting indicator by the method of differential calculus using the example of a function of two variables.

Let the function z -fix, y) be given; then, if the function is differentiable, its increment can be expressed as

Dg = - Dx4 - Du + 0 (h / dx2 + D ;; 2), 5x 8y Y

where Az = (zi -Zo) change of function; Ax = (*! X0) change of the first factor; Ay = (yi -y0) change in the second factor;

0 (- / Dx + & y2) is an infinitely small quantity of a higher order than

This value is discarded in the calculations (it is often denoted r - epsilon).

The influence of the factor x and y on the change in z is determined in this case as

AZx = -Ax and AZv = -yAy "

and their sum is the main, linear with respect to the increment of the factor, part of the increment of the differentiable

functions. It should be noted that the parameter O (VA * 2 + Ay2) is small at

sufficiently small changes in factors and its values can differ significantly from zero with large changes in factors. Since this method gives an unambiguous decomposition of the influence of factors on the change in the resulting indicator, then this time

position can lead to significant errors in assessing the influence of factors, since it does not take into account the value of the

Consider the application of the method using a specific example

functions: Let the initial and final values

factors and the resulting indicator, then the influence of factors on the change in the resulting indicator is determined, respectively, by the formulas

It is easy to show that the remainder in the linear expansion of the function z xy is equal to AxAy. Indeed, the total change in the function was - and the difference between the total change (Azx + Azy) and Az is calculated by the formula

Δz Azx Azy = (xlyi XaYv) y0Ax x ^ Ay =

UM) - (* oYi - * oyo) = * i (Y. Yo) -ho (Yi ~ Yo) =

"(* Yi ~ JCqVo)" ki ~ xo) Ui (Yi "U = = (x #) y ^)) (x0yi Xou0) ~ ui ~ y0) x0 (yi Yo) ~~ = (Yi Y0) ^ xz) Ahay.

Thus, in the method of differential calculus, the so-called indecomposable remainder, which is interpreted as a logical error of the method of differentiation, is simply discarded. This is the "inconvenience" of differentiation for economic calculations, which, as a rule, require an exact balance of changes in the final indicator and the algebraic sum of the influence of all factors.

Index method for determining factors for a generalizing indicator. In statistics, planning and analysis of economic activity, index models are the basis for a quantitative assessment of the role of individual factors in the dynamics of changes in generalizing indicators.

So, studying the dependence of the volume of sales of products at the enterprise on changes in the number of employees and their productivity, you can use the following system

interrelated indices:

where ./* is the general index of the change in the volume of product sales;

Г - individual (factorial) index of changes in the number of employees;

1 ° factorial index of changes in labor productivity of workers;

D, Dy - average annual output per worker, respectively, in the baseline and reporting periods; RQ, RX average annual headcount, respectively, in baseline

and reporting periods.

The above formulas show that the overall relative change in the volume of production is formed as the product of relative changes in two factors: the number of workers and their labor productivity. The formulas reflect the practice of constructing factor indices adopted in statistics, the essence of which can be formulated as follows.

If the generalizing economic indicator is a product of quantitative (volumetric) and qualitative indicators-factors, then when determining the influence of a quantitative factor, the qualitative indicator is fixed at the basic level, and when determining the influence of a qualitative factor, the quantitative indicator is fixed at the level of the reporting period.

The index method makes it possible to decompose not only relative, but also absolute deviations of the generalizing indicator by factors.

In our example, the formula (1) allows you to calculate the value of the absolute deviation (increase) of the generalizing indicator - the volume of production of the enterprise:

dlg = id, * i-ІЗД).

where AN is the absolute increase in the volume of production in the analyzed period.

This deviation was formed under the influence of changes in the number of workers and their labor productivity. To determine what part of the total change in the volume of production dos

tied by changing each of the factors separately, it is necessary when calculating the influence of one of them to eliminate the influence of the other factor.

The increase in the volume of production due to changes in the productivity of workers is determined similarly by the second factor:

Formula (2) corresponds to this condition. In the first factor, the influence of labor productivity is eliminated, in the second - the number of workers, therefore, the increase in the volume of production due to a change in the number of workers is determined as the difference between the numerator and denominator of the first factor:

The stated principle of decomposition of the absolute increase (deviation) of the generalizing indicator by factors is suitable for the case when the number of factors is equal to two (one of them is quantitative, the other is qualitative), and the analyzed indicator is presented as their product.

The theory of indices does not provide a general method for decomposing the absolute deviations of the generalizing indicator by factors when the number of factors is more than two and if their relationship is not multiplicative.

Chain substitution method (difference method). This method consists in obtaining a number of intermediate values of the generalizing indicator by sequentially replacing the basic values of the factors with the actual ones. The difference between two intermediate values of the generalizing indicator in the substitution chain is equal to the change in the generalizing indicator caused by the change in the corresponding factor.

In general, we have the following system of calculations by the method of chain substitutions:

= / (af $ ya ...) - the basic value of the summarizing indicator; factors

Yo = / (in | A () C () D? D ...) - intermediate value; - intermediate value;

intermediate value;

actual value.

The general absolute deviation of the generalizing indicator is determined by the formula

The general deviation of the generalized indicator is decomposed into factors:

due to a change in factor a -

W ^ ya-yo - / (eoVo4> ->;

due to a change in factor b -

FiafactftQ ...)

The chain substitution method, like the index method, has drawbacks that you should be aware of when using it. First, the calculation results depend on the sequence of replacing factors; secondly, an active role in changing the generalizing indicator is unreasonably often attributed to the influence of a change in a qualitative factor.

For example, if the investigated indicator z has the form of a function, then its change over the period is expressed by the formula

where Az is the increment of the generalizing indicator; Ah, ay increment of factors; х№ у0 - basic values of factors;

the baseline and reporting periods, respectively.

Grouping the last term with one of the first in this formula, we obtain two different versions of chain substitutions. First option:

Second option:

Az = x ^ y + (y0 + Ay) Ax = XdAy + y) AX.

In practice, the first option is usually used, provided that x is a qualitative factor and y is a quantitative factor.

This formula reveals the influence of the qualitative factor on the change in the generalizing indicator, that is, the expression (y0 + Ay) Ax is more active, since its value is set by multiplying the increment of the qualitative factor by the reported value of the quantitative factor. Thus, the entire increase in the generalizing indicator due to the joint change in factors is attributed to the influence of only the qualitative factor.

Thus, the problem of accurately determining the role of each factor in changing the generalizing indicator is not solved by the usual method of chain substitutions.

In this regard, the search for ways to improve the precise unambiguous definition of the role of individual factors in the context of the introduction of complex economic and mathematical models of factor systems in economic analysis is of particular relevance.

The task is to find a rational computational procedure (method of factor analysis), which eliminates conventions and assumptions and achieves an unambiguous result of the magnitudes of the influence of factors.

Method of simple addition of indecomposable remainder. Not finding a sufficiently complete justification of what to do with the remainder, in the practice of economic analysis, they began to use the method of adding an irreducible remainder to a qualitative or quantitative (main or derived) factor, and also to divide this remainder among the factors equally. The last proposal is theoretically substantiated by S. M. Yugenburg 1104, p. 66 - 831.

Taking into account the above, you can get the following set of formulas.

First option

& ZX ^ & xy0 + AxAy + Yes "O" o + Ay) = Axy ^;

Wtppg> ™ ISYAPYANT

D? L = AxyQ; Azv = Auh $ + AxAy - Ay (xQ + Ax) = Auh ^.

The third option

There are other proposals that are rarely used in the practice of economic analysis. For example, refer AXAy to the second term with a coefficient equal to

Ahuo + Auchts

And add the remainder to the first

term. This technique was defended by V.E. Adamov. He believed that “despite all the objections, the only practically unacceptable, although based on certain agreements on the choice of index weights, would be the method of interrelated study of the influence of factors using the qualitative indicator of the weights of the reporting period in the index, and the basis weights in the volume index. period ".

The described method, although it removes the problem of "indecomposable residue", is associated with the condition of determining quantitative and qualitative factors, which complicates the task when using large factor systems. At the same time, the expansion of the total increase in the result indicator by the chain method depends on the sequence of substitution. In this regard, it is not possible to obtain an unambiguous quantitative value of individual factors without observing additional conditions.

Weighted finite difference method. This method consists in the fact that the magnitude of the influence of each factor is determined by both the first and the second order of substitution, then the result is summed up and the average value is taken from the resulting sum, which gives a single answer about the value of the influence of the factor. If more factors are involved in the calculation, then their values are calculated for all possible substitutions.

Let us describe this method mathematically using the notation used above.

As you can see, the method of weighted finite differences takes into account all variants of substitutions. At the same time, when averaging, it is impossible to obtain an unambiguous quantitative value of individual factors. This method is very laborious and, in comparison with the previous method, complicates the computational procedure, since it is necessary to sort out all possible variants of substitutions. Basically, the method of weighted finite differences is identical (only for the two-factor multiplicative model) to the method of simple addition of the indecomposable remainder when dividing this remainder between the factors equally. This is confirmed by the following transformation of the formula:

Likewise

It should be noted that with an increase in the number of factors, and hence the number of substitutions, the described identity of the methods is not confirmed.

Logarithmic method. This method, described by V. Fedorova and Yu. Egorov, consists in the fact that a logarithmically proportional distribution of the remainder is achieved by the two desired factors. In this case, no prioritization of the factors is required.

Mathematically, this method is described as follows.

The factorial system z - xy can be represented as Igz = lgx + lgy, then

where U = logx (+] g jv Igzo = IgXQ + 1

Expression (4) for Az is nothing more than its logarithmic proportional distribution over the two desired factors. That is why the authors of this approach called this method “the logarithmic method for decomposing the increment Az into factors”. The peculiarity of the logarithmic decomposition method is that it allows one to determine the non-residual influence of not only two, but also many isolated factors on the change in the result indicator, without requiring the establishment of a sequence of actions.

In a more general form, this method was described by A. Humal, who wrote: “Such a division of the increase in the work can be called normal. The name is justified by the fact that the resulting division rule remains in force for any number of factors, namely: the increase in the product is divided between variable factors in proportion to the log-

rhymes of their coefficients of change. " Indeed, in the case of the presence of a larger number of factors in the analyzed multiplicative model of the factor system (for example, z, the total increment of the effective indicator will be:

Decomposition of the increment into factors is achieved by introducing the coefficient k, which, in the case of equality to zero or mutual compensation of factors, does not allow using the specified method. Formula (4) for Лг can be written differently:

M = & + Mu = ■ Mkx + (5)

In this form, this formula (5) is currently used as a classical one, describing the logarithmic method of analysis. It follows from this formula that the total increment of the final indicator is distributed among the factors in proportion to the ratio of the logarithms of the factor indices to the logarithm of the final indicator. It does not matter which logarithm is used (natural mN or decimal IgN).

The main disadvantage of the logarithmic method of analysis is that it cannot be “universal”, it cannot be used in the analysis of any kind of models of factor systems. If, when analyzing multiplicative models of factor systems using the logarithmic method, obtaining the exact values of the influence of factors is achieved (in the case when Δr = 0), then with the same analysis of multiple models of factor systems, it is not possible to obtain exact values of the influence of factors.

So, if a short model of the factorial system is presented in the form

then a similar formula (5) can be applied to the analysis of multiple models of factorial systems, i.e.

Az = U + My + Aztx + Dg * yy

gae $ --k; th

This approach was used by D.I. Vainshenker and V.M. Ivanchenko when analyzing the implementation of the plan for profitability. When determining the magnitude of the increase in profitability due to the increase in profit, they used the coefficient k "x.

Having failed to obtain an exact result in the subsequent analysis, D.I. Vainschenker and V.M. Ivanchenko limited themselves to using the logarithmic method only at the first stage (when determining the factor Az "J. which is nothing more than the proportion of the increase in one of the factors in the total increase in the constituent factors.The mathematical content of the coefficient L is identical to the “method of equity participation” described below.

If in the short model of the factorial system Y

then, when analyzing this model, we get:

& Z = Z C = Azx + Azy = Azx + AZtAZql

Azx ~ Azkx = Az-Dyu = & z-Azxi

It should be noted that the subsequent division of the factor Az "y by the method of logarithm into the factors Az" c and Az "q cannot be carried out in practice, since the logarithmic method, in its essence, provides for obtaining logarithmic deviations, which for the separating factors will be approximately the same. and the disadvantage of the described method.Application of the "mixed" approach in the analysis of multiple models of factorial systems does not solve the problem of obtaining an isolated value from the entire set of factors influencing the change in the final indicator.The presence of approximate calculations of the values of factorial changes proves the imperfection of the logarithmic method of analysis.

Method of coefficients. This method, described by I. A. Belobzhetsky, is based on comparing the numerical value of the same basic economic indicators under different conditions.

IA Belobzhetskiy proposed to determine the magnitude of the influence of factors as follows;

The described method of coefficients captivates with its simplicity, but when substituting digital values into the formulas, the result of I. A. Belobzhetsky turned out to be correct only by chance. With the exact implementation of algebraic transformations, the result of the total influence of factors does not coincide with the magnitude of the change in the result indicator obtained by direct calculation.

The method of splitting the increments of factors. In the analysis of economic activity, the most common tasks are the tasks of direct deterministic factor analysis. From an economic point of view, such tasks include an analysis of the implementation of the plan or the dynamics of economic indicators, in which the quantitative value of the factors that influenced the change in the final indicator is calculated. From a mathematical point of view, the problems of direct deterministic factor analysis represent the study of a function of several variables.

A further development of the method of differential calculus was the method of splitting the increments of factor signs, in which one should split the increments of each of the variables into sufficiently small segments and recalculate the values of the partial derivatives for each (already sufficiently small) displacement in space. The degree of fragmentation is taken such that the total error does not affect the accuracy of economic calculations.

Hence, the increment of the function z -f (x, y) can be represented in general form as follows:

function change

due to the change in the factor x by the value Ax xx xih

due to the change in the factor y by the value of the Error e decreases with increasing n.

For example, when analyzing a multiple model of the factor system

of the form z = - by the method of splitting the increments of factorial recognition

kov we obtain the following formulas for calculating the quantitative values of the influence of factors on the resulting indicator:

e can be neglected if n is large enough. The method of splitting increments of factor signs has advantages over the method of chain substitutions. It allows you to determine unambiguously the magnitude of the influence of factors with a predetermined accuracy of calculations, is not associated with the sequence of substitutions and the choice of qualitative and quantitative indicators-factors. The splitting method requires compliance with the conditions for the differentiability of the function in the region under consideration.

An integral method for assessing factor influences. Further

logical development of the method of splitting increments of factor

features became an integral method of factor analysis. This

the method, like the previous one, was developed and justified by A.D. Sheremet and his students.It is based on summation

increments of a function defined as a partial derivative,

multiplied by the increment of the argument at infinitesimal intervals. In this case, the following conditions must be observed:

continuous differentiability of a function, where an economic indicator is used as an argument;

the function between the starting and ending points of the elementary period changes along the straight line Ge;

the constancy of the ratio of the rates of change of factors

In general form of the formula for calculating the quantitative values of the influence of factors on the change in the resulting indicator

where Ge is a straight-line oriented segment on the plane (x, y) connecting the point (xa, y) with the point (x1r y ().

In real economic processes, a change in factors in the field of definition of a function can occur not along a straight line segment Ge, but along a certain oriented curve G. But since the change in factors is considered for an elementary period (i.e., for a minimum period of time during which at least one of the factors will receive an increment), then the trajectory Г is determined in the only possible way - a straight-line oriented segment Ge, connecting the initial and final points of the elementary period.

Let us derive a formula for the general case.

The function of changing the resulting indicator from factors is set

y = f (xx, x2, ..., хт),

where Xj is the value of factors; j - 1, 2, ..., t;

y - the value of the resulting indicator.

The factors change in time, and the values of each factor at n points are known, that is, we will assume that n points are given in space:

Mx = (x, x, ..., X1m), M2 = * m)> Mi = (A> Ar- ^

where x | the value of the th indicator at the moment /.

Points Мх and М2 correspond to the values of the factors at the beginning and end of the analyzed period, respectively.

Suppose that the exponent y received an increment of Ay for

analyzed period; let the function y = f (xl, x2, ..., xm) be differentiable and y -fxj (xl xj is the partial derivative of

this function by argument xy.

Suppose L "is a segment of a straight line connecting two points M" and M * 1 (/ "= 1,2, n - D). Then the parametric equation of this straight line can be written in the form

Xj = x "j + Xі) f.j = 1, 2, m; 0< і < I.

Let us introduce the notation

AUi, = J / v (^ i ^ 2, ..., xm) (i> c (; Y = 1,2, ..., m.

Considering these two formulas, the integral over the segment i can be written as follows:

The element of this matrix characterizes the contribution of the th indicator to the change in the resulting indicator for the period

Summing up the values to the matrix tables, we get

the following line:

The value of any i-th element of this line characterizes the contribution of the y-th factor to the change in the resulting indicator Ay. The sum of all Ay, (/ = 1,2, ..., t) is the full increment of the resulting indicator.

There are two areas of practical use of the integral method in solving problems of factor analysis.

The first direction can be attributed to the tasks of factor analysis, when there is no data on changes in factors within the analyzed period, or it is possible to abstract from them, that is, there is a case when this period should be considered as elementary. In this case, the calculations should be carried out along the oriented straight line Ge. This type of factor analysis tasks can be conventionally called static, since the factors participating in the analysis are characterized by the invariability of position in relation to one factor, the constancy of the conditions for the analysis of the measured factors, regardless of their location in the model of the factor system. The increments of factors are measured in relation to one factor selected for this purpose.

The static types of tasks of the integral method of factor analysis should include calculations related to the analysis of the implementation of the plan or dynamics (if the comparison is made with the previous period) of indicators. In this case, there is no data on changes in factors within the analyzed period.

The second direction can be attributed to the tasks of factor analysis, when there is information about changes in factors within the analyzed period and it must be taken into account, that is, the case when this period, in accordance with the available data, is divided into a number of elementary ones. In this case, the calculations should be carried out along some oriented curve Г connecting the point (x0, y) and the point (x, y) for the two-factor model. The problem is how to determine the true form of the curve G, along which the movement of factors x y took place in time. the type of factor analysis problems can be conventionally called dynamic, since the factors involved in the analysis change in each period divided into sections.

The dynamic types of problems of the integral method of factor analysis should include calculations related to the analysis of time series of economic indicators. In this case, it is possible to choose, albeit approximately, an equation that describes the behavior of the analyzed factors in time for the entire period under consideration. In this case, in each elementary period being divided, an individual value, different from others, can be taken.

The integral method of factor analysis finds application in the practice of computer deterministic economic analysis.

The static type of problems of the integral method of factor analysis is the most developed and widespread type of problems in the deterministic economic analysis of the economic activity of controlled objects.

In comparison with other methods of a rational computational procedure, the integral method of factor analysis eliminated the ambiguity in assessing the influence of factors and made it possible to obtain the most accurate result. The results of calculations by the integral method differ significantly from those given by the method of chain substitutions or modifications of the latter. The greater the magnitude of changes in factors, the greater the difference.

The method of chain substitutions (its modifications), in its essence, takes into account the ratio of the values of the measured factors weaker. The greater the gap between the values of the increments of factors included in the model of the factor system, the more the integral method of factor analysis reacts to this.

In contrast to the chain method, the integral method has a logarithmic law of redistribution of factor loads, which indicates its great advantages. This method is objective as it excludes any suggestions about the role of factors prior to analysis. Unlike other methods of factor analysis, the integral method observes the provision on the independence of factors.

An important feature of the integral method of factor analysis is that it provides a general approach to solving problems of various types, regardless of the number of elements included in the model of the factor system and the form of connection between them. At the same time, in order to simplify the computational procedure for decomposing the increment of the resulting indicator into factors, one should adhere to two groups (types) of factor models: multiplicative and multiple. The computational procedure for integration is the same, and the resulting final formulas for calculating the factors are different.

Formation of working formulas of the integral method for multiplicative models. The use of the integral method of factor analysis in deterministic economic analysis most completely solves the problem of obtaining uniquely determined values of the influence of factors.

There is a need for formulas for calculating the influence of factors for many types of models of factor systems (functions).

It was established above that any model of a finite factorial system can be reduced to two types - multiplicative and multiple. This condition predetermines that the researcher deals with two main types of models of factor systems, since the rest of the models are their varieties.

The operation of calculating a definite integral for a given integrand and a given integration interval is performed according to a standard program stored in the memory of the machine. In this regard, the task is reduced only to the construction of integrands that depend on the type of function or model of the factor system.

To facilitate the solution of the problem of constructing integrands, depending on the type of model of the factor system (multiplicative or multiple), we propose matrices of initial values for constructing integrands of the elements of the structure of the factor system. The principle inherent in the matrices makes it possible to construct the integrands of the elements of the structure of the factorial system for any set of elements of the model of the finite factorial system. Basically, the construction of integrands for the elements of the structure of a factor system is an individual process, and in the case when the number of structural elements is measured in a large number, which is a rarity in economic practice, they proceed from specifically specified conditions.

When forming working formulas for calculating the influence of factors in the conditions of using a computer, the following rules are used that reflect the mechanics of working with matrices: the integrands of the elements of the structure of the factorial system for multiplicative models are constructed by the product of a complete set of elements of the values taken for each row of the matrix, referred to a certain element of the structure of the factorial system with the subsequent decoding of the values given to the right and at the bottom of the matrix of initial values (Table 5.2).

Let us give examples of constructing subintephalic expressions.

Example 1 (see table 5.2).

Type of models of the factor system / = xyzq (multiplicative model).

Factor system structure

Formation of working formulas of the integral method for multiple models. The integrand expression of the elements of the structure of the factor system for multiple models is constructed by entering under the integral sign the initial value obtained at the intersection of the lines, depending on the type of model and the elements of the structure of the factor system, with subsequent decoding of the values shown on the right and at the bottom of the matrix of initial values.

Example 2 (Table 5.3).

Du + Dg + d # +

■ A * + ^ + Az + ^ + Ap

4 o (y0 + zu +? O + kx) z

Lou + Az + Hell, & Az has Hell

- -; / = -; t = -; n = -H

Dx lx ah ah

The subsequent calculation of a definite integral over a given integrand and a given integration interval is performed using a computer according to a standard program in which Simpson's formula is used, or manually in accordance with the general rules of integration.

In the absence of universal computational tools, we will offer a set of formulas for calculating structure elements for multiplicative (Table 5.4) and multiple (Table 5.3) models of factor systems that are most often found in economic analysis, which were derived as a result of the integration process. Taking into account the need to simplify them as much as possible, a computational procedure was performed to compress the formulas obtained after calculating certain integrals (integration operations).

Let us give examples of constructing working formulas for calculating the elements of the structure of the factor system.

Example 1 (see table 5.4).

The type of the model of the factorial system f = xyzq (multiplicative model).

Factor system structure

a / = shtt shrt = A * + 4 + 4 + 4 Working formulas for calculating the elements of the structure of the factor system:

Factor system model view

Working formulas for calculating the elements of the structure of the factor system

The use of working formulas is significantly expanded in deterministic chain analysis, in which the identified factor can be stepwise decomposed into components, as it were, in another plane of analysis.

An example of a deterministic chain factor analysis can be an on-farm analysis of a production association, in which the role of each production unit in achieving the best result for the association as a whole is assessed.

The integral method gives accurate estimates of factor influences. The calculation results do not depend on the sequence of substitutions and the sequence of calculating factorial influences. The method is applicable for all types of continuously differentiable functions; it does not require prior knowledge of which factors are quantitative and which are qualitative.

The application of the integral method requires knowledge of the basics of differential calculus, integration techniques and the ability to find the derivatives of various functions. At the same time, in the theory of the analysis of economic activity for practical applications, the final working formulas of the integral method for the most common types of factor dependencies have been developed, which makes this method available to every analyst. Here are some of them.

1. Factor model of the type u = xy: Au = Aih + Aig

Ax-Ay, Aih = y0Ax + ---;

Auy = x0Ay + -; Au = Au + Aih.

2, Dm = Aih + Diu + Dmg;

Dm = l: 0 -ts -Ay + -l0 -Ay-Az + -Zq ■ Ax -Ay + -Ay ■ Az ■ Dx;

4. Factorial type model

The use of these models makes it possible to select factors, the purposeful change of which makes it possible to obtain the desired value of the result indicator.

1. The method of chain substitutions is used to calculate the influence of individual factors on the corresponding aggregate indicator. This method of analysis is used only when the relationship between the studied phenomena has a strictly functional nature, when it is presented in the form of direct or inverse proportional relationship. In these cases, the analyzed aggregate indicator as a function of several variables should be depicted as an algebraic sum, product or quotient from dividing some indicators by others.

When calculating, you must adhere to the following rules:

· First, the influence of quantitative and then qualitative factors is taken into account;

First of all, the factor of the first level changes, then the second, third, etc.

In general, we have the following system of calculations by the method of chain substitutions:

The basic value of the summary indicator;

y 0 = f (a 1 b 0 c 0 d 0 ...) is an intermediate value;

y 0 = f (a 1 b 1 c 0 d 0 ...) is an intermediate value;

y 0 = f (a 1 b l c] d 0 ...) is an intermediate value;

………………………………

y 0 = f (a l b] c l d l ...) is the actual value.

The general absolute deviation of the generalizing indicator is determined by the formula

The general deviation of the generalized indicator is decomposed into factors:

Due to a change in factor a

Due to changes in factor b

The chain substitution method has disadvantages that you should be aware of when using it. First, the calculation results depend on the sequence of replacing factors; secondly, an active role in changing the generalizing indicator is unreasonably often attributed to the influence of a change in a qualitative factor.

2. Index method is based on comparing the actual level of the object under study in the reporting period to its level in the base period. Planned values can be used instead of the value in the base period.

The index method is used to calculate the influence of factors in multiplicative and multiple models.

3. Method of absolute differences. It is used to calculate the influence of factors on the effective indicator in multiplicative models and combined models of the type:

In accordance with the method of absolute differences, it is necessary to calculate the absolute growth of each factor. Then the magnitude of the influence of one factor or another is determined by multiplying its growth by the planned value of the factors located in the model to the right of it, and by the actual value of the factors located to the left.

For example, the calculation algorithm for a multiplicative model of the type is:

4. Method of relative differences. Used in multiplicative and combined models. First, the relative gain of each factor should be calculated. Further, the magnitude of the influence of the factor on the effective indicator is determined by multiplying its relative growth by the planned value of the effective indicator.

So, for a multiplicative model of the type, the relative deviations of factor indicators are as follows:

The deviation of the effective indicator due to the influence of each factor is calculated by the formulas:

5. Method of differential calculus. Based on total differential formula. For a function of two variables, we have a complete increment of the function:

where are the factorial increments of the corresponding variables;

Partial derivatives;

An infinitely small quantity of a higher order than. This value is discarded in the calculations (it is often denoted as ε.

Thus, the influence of factor x on the generalizing indicator is determined by the formula:

The total increment of the resulting indicator is decomposed into terms, where the value of each of them is determined as the product of the corresponding derivative by the increment of the factor by which the given derivative is calculated.

6. Integral method of factor analysis.It is based on summing the increment of a function defined as a partial derivative multiplied by the increment of the argument over infinitesimal intervals.

In this case, certain conditions must be met:

· The integrand must be continuous and differentiable;

· The rate of change of factors should be constant, i.e. dx = const/

The integral method is based on the Euler-Lagrange integral, which establishes a relationship between the increment of a function and the increment of factor signs.

For the function, we have the following formulas for calculating factor influences:

Influence of factor x;

Influence of factor y.

Differential calculus method.

The theoretical basis for a quantitative assessment of the role of individual factors in the dynamics of the effective (generalizing) indicator is differentiation.

In the method of differential calculus, it is assumed that the total increment of functions (resulting indicator) differs into terms, where the value of each of them is determined as the product of the corresponding partial derivative by the increment of the variable over which this derivative is calculated. Let us consider the problem of finding the influence of factors on the change in the resulting indicator by the method of differential calculus using the example of a function of two variables. Let the function z = f (x, y) be given, then if the function is differentiable, its increment can be expressed as

where - change of functions;

Δx (x 1 - x o) - change in the first factor;

- change in the second factor;

Is an infinitely small quantity of a higher order than

The influence of factors x and y on the change in z is determined in this case as

and their sum is the main (linear with respect to the increment of factors) part of the increment of the differentiable function. It should be noted that the parameter is small for sufficiently small changes in factors and its values can differ significantly from zero for large changes in factors. Since this method gives an unambiguous decomposition of the influence of factors on the change in the resulting indicator, this decomposition can lead to significant errors in assessing the influence of factors, since it does not take into account the value of the remainder, i.e. .

Let's consider the application of the method using a specific function as an example: z = xy. Let the initial and final values of the factors and the resulting indicator (x 0, y 0, z 0, x 1, y 1, z 1) be known, then the influence of factors on the change in the resulting indicator is determined, respectively, by the formulas:

It is easy to show that the remainder in the linear expansion of the function z = xy is

Indeed, the total change in the function was, and the difference between the total change and is calculated by the formula

Thus, in the method of differential calculus, the so-called indecomposable remainder, which is interpreted as a logical error of the method of differentiation, is simply discarded. This is the "inconvenience" of differentiation for economic calculations, which, as a rule, require an exact balance of changes in the effective indicator and the algebraic sum of the influence of all factors.

Index method for determining the influence of factors on the generalizing indicator.

So, studying the dependence of the volume of output at an enterprise on changes in the number of employees and their labor productivity, one can use the following system of interrelated indices:

(5.2.1)

(5.2.2)

where I N is the general index of the change in the volume of production;

I R - individual (factor) index of changes in the number of employees;

I D - factorial index of changes in labor productivity of workers;

D 0, D 1 - the average annual output of marketable (gross) output per worker, respectively, in the baseline and reporting periods;

R 0, R 1 - the average annual number of industrial and production personnel, respectively, in the baseline and reporting periods.

The above formulas show that the overall relative change in the volume of output is formed as the product of relative changes in two factors: the number of workers and their labor productivity. The formulas reflect the practice of constructing factor indices adopted in statistics, the essence of which can be formulated as follows. If the generalizing economic indicator is a product of quantitative (volumetric) and qualitative indicators-factors, then when determining the influence of a quantitative factor, the qualitative indicator is fixed at the basic level, and when determining the influence of a qualitative factor, the quantitative indicator is fixed at the level of the reporting period.

The index method makes it possible to decompose not only relative, but also absolute deviations of the generalizing indicator by factors. In our example, the formula (5.2.1) allows you to calculate the value of the absolute deviation (increase) of the generalizing indicator - the volume of output of the commercial output of the enterprise:

where is the absolute increase in the volume of commercial output in the analyzed period.

This deviation was formed under the influence of changes in the number of workers and their labor productivity. To determine what part of the total change in the volume of output was achieved due to the change in each of the factors separately, it is necessary to eliminate the influence of the other factor when calculating the influence of one of them.

Formula (5.2.2) corresponds to this condition. In the first factor, the influence of labor productivity is eliminated, in the second - the number of employees, therefore, the increase in the volume of output due to a change in the number of employees is determined as the difference between the numerator and denominator of the first factor:

The increase in the volume of output due to changes in the productivity of workers is determined similarly by the second factor:

The theory of indices does not provide a general method for decomposing the absolute deviations of the generalized indicator by factors when the number of factors is more than two.

Chain substitution method.

This method consists, as already proved, in obtaining a number of intermediate values of the generalizing indicator by successive replacement of the basic values of the factors with the actual ones. The difference between two intermediate values of the generalizing indicator in the chain of substitutions is equal to the change in the generalizing indicator caused by a change in the corresponding factor.

In general, we have the following system of calculations by the method of chain substitutions:

- the basic value of the generalizing indicator;

- intermediate value;

………………………………………………..

…………………………………………………

Is the actual value.

The general absolute deviation of the generalizing indicator is determined by the formula

The general deviation of the generalized indicator is decomposed into factors:

due to a change in factor a

due to a change in factor b

The chain substitution method, like the index method, has drawbacks that you should be aware of when using it. First, the calculation results depend on the successive replacement of factors; secondly, an active role in changing the generalizing indicator is unreasonably often attributed to the influence of a change in a qualitative factor.

For example, if the investigated indicator z has the form of a function, then its change over the period is expressed by the formula

where Δz is the increment of the generalizing indicator;

Δx, Δy - increment of factors;

x 0 y 0 - basic values of factors;

t 0 t 1 - respectively the base and reporting periods of time.

Grouping the last term with one of the first in this formula, we obtain two different versions of chain substitutions.

First option:

Second option:

In practice, the first option is usually used (provided that x is a quantitative factor and y is a qualitative factor).

This formula reveals the influence of a qualitative factor on the change in the generalizing indicator, that is, the expression of a more active connection to obtain an unambiguous quantitative value of individual factors without observing additional conditions is not possible.

Weighted finite difference method.

This method consists in the fact that the magnitude of the influence of each factor is determined by both the first and the second order of substitution, then the result is summed up and the average value is taken from the resulting sum, which gives a single answer about the value of the influence of the factor. If more factors are involved in the calculation, then their values are calculated for all possible substitutions. Let us describe this method mathematically using the notation used above.

Likewise

It should be noted that with an increase in the number of factors, and hence the number of substitutions, the described identity of the methods is not confirmed.

Logarithmic method.

This method consists in the fact that a logarithmically proportional distribution of the remainder is achieved according to the two desired factors. In this case, no prioritization of the factors is required.

Mathematically, this method is described as follows.

The factorial system z = xy can be represented in the form lg z = lg x + lg y, then

Dividing both sides of the formula by and multiplying by Δz, we obtain

(*)

where

The expression (*) for Δz is nothing more than its logarithmic proportional distribution over the two desired factors. That is why the authors of this approach called this method “the logarithmic method for decomposing the increment Δz into factors”. The peculiarity of the logarithmic decomposition method is that it allows one to determine the non-residual influence of not only two, but also many isolated factors on the change in the effective indicator, without requiring the establishment of a sequence of actions.

In a more general form, this method was described by the mathematician A. Humal, who wrote: “Such a division of the increase in the product can be called normal. The name is justified by the fact that the resulting division rule remains valid for any number of factors, namely: the increment of the product is divided between variable factors in proportion to the logarithms of their coefficients of change. " Indeed, in the case of the presence of a larger number of factors in the analyzed multiplicative model of the factor system (for example, z = xypm), the total increment of the effective indicator Δz will be

where

In this form, this formula is currently used as a classical one, describing the logarithmic method of analysis. It follows from this formula that the total increment of the effective indicator is distributed among the factors in proportion to the ratio of the logarithms of the factor indices to the logarithm of the effective indicator. It does not matter which logarithm is used (natural ln N or decimal lg N).

The main disadvantage of the logarithmic method of analysis is that it cannot be “universal”, it cannot be used in the analysis of any kind of models of factor systems. If, when analyzing multiplicative models of factor systems using the logarithmic method, obtaining the exact values of the influence of factors (in the case when) is achieved, then with the same analysis of multiple models of factor systems, it is not possible to obtain exact values of the influence of factors.

So, if the multiple model of the factorial system is presented in the form

then ,

then a similar formula can be applied to the analysis of multiple models of factorial systems, i.e.

where

If in the multiple model of the factor system , then when analyzing this model we get:

It should be noted that the subsequent dismemberment of the factor Δz ’y by the method of logarithm into the factors ∆z’ c and ∆z ’q cannot be carried out in practice, since the logarithmic method in its essence provides for obtaining logarithmic ratios, which for the dismembering factors will be approximately the same. This is precisely the disadvantage of the described method. The use of a "mixed" approach in the analysis of multiple models of factor systems does not solve the problem of obtaining an isolated value from the entire set of factors that affect the change in the effective indicator. The presence of approximate calculations of the magnitudes of factorial changes proves the imperfection of the logarithmic method of analysis.

Method of coefficients. This method, described by the Russian mathematician I. A. Belobzhetsky, is based on comparing the numerical value of the same basic economic indicators under different conditions. A. Belobzhetsky proposed to determine the magnitude of the influence of factors as follows:

The method of splitting the increments of factors.

In the analysis of economic activity, the most common tasks are the tasks of direct deterministic factor analysis. From an economic point of view, such tasks include an analysis of the implementation of the plan or the dynamics of economic indicators, in which the quantitative value of the factors that influenced the change in the effective indicator is calculated. From a mathematical point of view, the problems of direct deterministic factor analysis represent the study of a function of several variables.

Hence, the increment of the function z = f (x, y) can be represented in general form as follows:

where n is the number of segments into which the increment of each factor is divided;

A x n = - a change in the function z = f (x, y) due to a change in the factor x by the value;

A y n = - a change in the function z = f (x, y) due to a change in the factor y by the value

The error ε decreases with increasing n.

For example, when analyzing a multiple model of a factorial system of the type by the method of splitting increments of factorial signs, we obtain the following formulas for calculating the quantitative values of the influence of factors on the resulting indicator:

ε can be neglected if n is large enough.

The method of splitting increments of factor signs has advantages over the method of chain substitutions. It allows you to determine unambiguously the magnitude of the influence of factors with a predetermined accuracy of calculations, is not associated with the sequence of substitutions and the choice of qualitative and quantitative indicators-factors. The splitting method requires compliance with the differentiability conditions of the function in the considered region.

An integral method for assessing factor influences.

The further logical development of the method of splitting the increments of factor signs was the integral method of factor analysis. This method is based on summing the increments of a function defined as a partial derivative multiplied by the increment of the argument over infinitesimal intervals. In this case, the following conditions must be observed:

continuous differentiability of a function, where an economic indicator is used as an argument;

the function between the start and end points of the elementary period changes in a straight line;

the constancy of the ratio of the rates of change of factors

In general form, formulas for calculating the quantitative values of the influence of factors on the change in the resulting indicator (for the function z = f (x, y) - of any kind) are derived as follows, which corresponds to the limiting case when:

where Гe is a straight-line oriented segment on the plane (x, y) connecting the point (x 0, y 0) with the point (x 1, y 1).

In real economic processes, the change in factors in the domain of the function definition can occur not along a straight line segment e, but along a certain oriented curve. But since the change in factors is considered for an elementary period (that is, for the minimum period of time during which at least one of the factors will gain an increment), then the trajectory of the curve is determined in the only possible way - a straight-line oriented segment of the curve connecting the initial and final points of the elementary period.

Let us derive a formula for the general case.

The function of changing the resulting indicator from factors is set

Y = f (x 1, x 2, ..., x t),

where x j is the value of factors; j = 1, 2, ..., m; y - the value of the resulting indicator.

The factors change in time, and the values of each factor at n points are known, i.e., we will assume that n points are given in an m-dimensional space:

where x ji is the value of the j-th indicator at the moment i.

Points M 1 and M p correspond to the values of the factors at the beginning and end of the analyzed period, respectively.

Suppose that the indicator y received an increment of Δy for the analyzed period; let the function y = f (x 1, x 2, ..., x m) be differentiable and f "xj (x 1, x 2, ..., x m) is the partial derivative of this function with respect to the argument x j.

Suppose Li is a line segment connecting two points M i and M i + 1 (i = 1, 2,…, n-1).

Then the parametric equation of this straight line can be written in the form

Let us introduce the notation

Taking into account these two formulas, the integral over the segment Li can be written as follows:

j = 1, 2, ..., m; I = 1,2, ..., n-1.

Calculating all the integrals, we obtain the matrix

The element of this matrix y ij characterizes the contribution of the j-th indicator to the change in the resulting indicator for the period i.

Summing up the values of Δy ij according to the matrix tables, we get the following row:

(Δy 1, Δy 2, ..., Δy j, ..., Δy m.);

The value of any j-th element of this line characterizes the contribution of the j-th factor to the change in the resulting indicator Δy. The sum of all Δy j (j = 1, 2, ..., m) is the full increment of the resulting indicator.

There are two areas of practical use of the integral method in solving problems of factor analysis. The first direction can be attributed to the tasks of factor analysis, when there is no data on changes in factors within the analyzed period, or it is possible to abstract from them, that is, there is a case when this period should be considered as elementary. In this case, calculations should be carried out along an oriented straight line. This type of factorial analysis problems can be conventionally called static, since the factors involved in the analysis are characterized by the invariability of position in relation to one factor, the constancy of the conditions for the analysis of the measured factors, regardless of their location in the model of the factor system. The increments of factors are measured in relation to one factor selected for this purpose.

The second direction can be attributed to the tasks of factor analysis, when there is information about changes in factors within the analyzed period and it must be taken into account, that is, the case when this period, in accordance with the available data, is divided into a number of elementary ones. In this case, the calculations should be carried out along some oriented curve connecting the point (x 0, y 0) and the point (x 1, y 1) for the two-factor model. The problem is how to determine the true form of the curve along which the movement of factors x and y took place in time. This type of factorial analysis problems can be conventionally called dynamic, since the factors involved in the analysis change in each period divided into sections.

The dynamic types of problems of the integral method of factor analysis should include calculations related to the analysis of time series of economic indicators. In this case, it is possible to select, albeit approximately, an equation that describes the behavior of the analyzed factors in time for the entire period under consideration. In this case, in each elementary period being divided, an individual value, different from others, can be taken. The integral method of factor analysis finds application in the practice of deterministic economic analysis.

In contrast to the chain method, the integral method has a logarithmic law of redistribution of factor loads, which indicates its great advantages. This method is objective, since it excludes any assumptions about the role of factors prior to analysis. Unlike other methods of factor analysis, the integral method observes the provision on the independence of factors.

An important feature of the integral method of factor analysis is that it provides a general approach to solving problems of various types, regardless of the number of elements included in the factor system model and the form of connection between them. At the same time, in order to simplify the computational procedure for decomposing the increment of the resulting indicator into factors, one should adhere to two groups (types of factor models: multiplicative and multiple.)

The computational procedure for integration is the same, and the resulting final formulas for calculating the factors are different. Formation of working formulas of the integral method for multi-plication models. The use of the integral method of factor analysis in deterministic economic analysis most completely solves the problem of obtaining uniquely determined values of the influence of factors.

There is a need for formulas for calculating the influence of factors for many types of models of factor systems (functions). It was established above that any model of a finite factorial system can be reduced to two types - multiplicative and multiple. This condition predetermines that the researcher deals with two main types of models of factor systems, since the rest of the models are their varieties.

To facilitate the solution of the problem of constructing integrands, depending on the type of model of the factor system (multiplicative or multiple), we propose matrices of initial values for - constructing integrands of the elements of the structure of the factor system. The principle inherent in the matrices makes it possible to construct the integrands of the elements of the structure of the factorial system for any set of elements of the model of the finite factorial system. Basically, the construction of integrands for the elements of the structure of a factor system is an individual process, and in the case when the number of structural elements is measured in a large number, which is a rarity in economic practice, they proceed from specifically specified conditions.

When forming working formulas for calculating the influence of factors in the conditions of using a computer, the following rules are used that reflect the mechanics of working with matrices: the integrands of the elements of the structure of the factorial system for multiplicative models are constructed by the product of a complete set of elements of the values taken for each row of the matrix, referred to a certain element of the structure of the factorial systems with the subsequent decoding of the values given to the right and at the bottom of the matrix of initial values (Table 5.1).

Table 5.1

The matrix of initial values for constructing the integrands of the structure elements of multiplicative models of factor systems

Elements of the structure of the factor system	Elements of the multiplicative model of the factor system							The integrand formula








The integrand formula		y / x = (y 0 + kx) dx	z / x = (z 0 + lx) dx	q / x = (q 0 + mx) dx	p / x = (p 0 + nx) dx	m / x = (m 0 + ox) dx	n / x = (n 0 + px) dx

Let us give an example of constructing integrands.

Example:

Type of models of the factorial system f = x y zq (multiplicative model).

Factor system structure

Construction of integrands

where

Formation of working formulas of the integral method for multiple models. The integrands of the elements of the structure of the factor system for multiple models are constructed by entering under the integral sign the initial value obtained at the intersection of the lines, depending on the type of the model and the elements of the structure of the factor system, with subsequent decoding of the values given to the right and down from the matrix of initial values.

The subsequent calculation of a definite integral over a given integrand and a given integration interval is performed using a computer according to a standard program in which the Simpson formula is used, or manually in accordance with the general rules of integration.

In the absence of universal computational tools, we propose a set of formulas for calculating structural elements for multiplicative and multiple models of factor systems, which were most often found in economic analysis, which were derived as a result of the integration process. Taking into account the need to simplify them as much as possible, a computational procedure was performed to compress the formulas obtained after calculating certain integrals (integration operations).

Let us give an example of constructing working formulas for calculating the elements of the structure of a factor system.

Example:

The type of the model of the factorial system f = xyzq (multiplicative model).

Factor system structure

Working formulas for calculating the elements of the structure of the factor system:

Rating analysis- one of the options for a comprehensive assessment of the financial condition of the enterprise. Rating analysis is a method of comparative assessment of the activities of several enterprises. The essence of the rating assessment is as follows: businesses line up(grouped) according to certain characteristics or criteria.

Signs or criteria reflect either individual aspects of the enterprise (profitability, solvency, etc.) or characterize the enterprise as a whole (sales volume, market volume, reliability).

When conducting rating analysis there are two main methods: expert and analytical. The expert method is based on the experience and qualifications of experts. Experts on the basis of available information, according to their methods, analyze the enterprise. The analysis takes into account both quantitative and qualitative characteristics of the enterprise.

Unlike the expert method, the analytical method is based only on quantitative indicators... The analysis is carried out according to formalized calculation methods. When applying the analytical method, three main stages can be distinguished:

primary "filtration" of enterprises. At this stage, enterprises are eliminated, about which, with a high degree of probability, we can say that their reporting causes great suspicion;

calculation of coefficients, according to a previously approved method;

There are several disadvantages that reduce the effectiveness of rating analysis when determining the financial condition of an enterprise:

The reliability of the information on which the rating is based. The rating analysis is carried out by independent agencies on the basis of public, official statements of the enterprise. The official reporting that enterprises publish in the media is the balance sheet. The imperfection of the Russian accounting system, gaps in Russian financial legislation, the large volume of the shadow economy - all this does not allow to fully trust the official reporting of enterprises. This problem can be partially solved by conducting an audit of the company's reporting.

Late rating analysis. As a rule, the rating is calculated based on the balance for the year. Annual balances are due until March 31 of the year following the reporting year. Then it takes some time to compile a rating. Thus, the rating appears on the basis of information that was relevant 3-4 months ago. During this time, the state of the enterprise could change significantly.

Subjectivity of expert opinion (with the expert method of rating analysis). It is difficult to formalize the opinions of experts, and the position of an enterprise in the rating largely depends on them.

The most complete and detailed study of the company's activities for assigning a rating assessment can be carried out by the company's employees. Since, in addition to official information, they can use inside information... However, enterprise employees may be subjective in assessing performance and are not always competent enough to conduct such an analysis.