Factor analysis of changes in the performance indicator. Factor analysis methods

In order to find out how profitable or unprofitable a company is, it is not enough to simply count money. To understand this for sure, and most importantly, to help increase profits, you need to regularly carry out the work of the enterprise as a whole. And for this you need to have some skills in the accounting field and certain information. It is worth considering that the company operated both during inflation and during the crisis. Prices changed constantly. Now do you understand why a banal counting of money does not make it possible to objectively assess the situation with profits or costs? After all, the price factor must also be taken into account.

So, many people find it difficult to do an example of our analysis, we hope it will help them do their own - by analogy, this type of diagnosis is compiled extremely quickly. It is presented in the form of a table. First, let's make the header of our factor analysis. Draw a table with 5 columns and 9 rows. Make the first column wider - it will contain the names of the enterprise's articles, not numbers. It will be called “Indicators”, which is what you should write in the first line of the column. In it, fill out all the lines according to the sample: 1 - name, 2 - put the number 1 - column numbering, in line 3 write - “Sales revenue”, 4 - “Cost”. In the fifth line of the first column, put the item “Business expenses”. In 6, write “Process management costs.” The seventh line is called - and 8 - “Index of price changes”, and the last, 9th line - “Sales at comparable prices”.

Next, we move on to the design of column 2: in line 1 we write - " Previous period, thousand rubles." (you can write other monetary units - euro, dollar, etc. - depends on what currency you will carry out the calculations in), and in the second line we write the number - 2. Go to the 3rd column - in it, 1 line has the name - "Reporting period", thousand rubles. And the second is filled in with the number 3. Next, we draw up our factor analysis of revenue and go to column 4. In the first line we enter - "Absolute change, thousand rubles." the second line contains a small formula: 4 = 3-2. This means that the indicators that you will write in the subsequent lines will be the result of subtracting the indicators of the second column from the indicators of the third. We proceed to the design of the last - 5 columns in it. you need to write: "Relative changes%", which means that in this column all data will be recorded in percentage. In the second line the formula is: 5=(4/2)*100%. That's it, we've designed the header, all that remains is to fill out each item in the table with the appropriate data. We carry out factor analysis, an example of which we give you. First of all, we calculate the price change index - this is perhaps the most important figure in our calculations. We write down the numbers of different periods in the corresponding columns. In columns 4 and 5 we carry out the necessary calculations. Factor analysis, an example of which you can view, assumes accuracy in numbers. Therefore, you only need to write reliable information in 3 lines of each column. In 4 and 5 we again carry out calculations. As you understand, factorization is mainly carried out in lines 5 and 6: try to add as real, not underestimated, numbers there as possible. In the 4th and 5th columns of these lines, again carry out calculations using the formulas. Next, we carry out a factor analysis of revenue in column 7 - profit. We write down reliable numbers in columns 2 and 3, and in columns 4 and 5 we again calculate everything using the formulas. And the last column remains: we write the data, we calculate. Bottom line: factor analysis, an example of which we give you, shows what the impact of each of the factors described in the articles is on profit or production costs. Now you see the weak points and can correct the situation in order to get as much profit as possible.

You have done all the calculations to perform factor analysis, but they will not help you at all unless you analyze the data thoroughly.

Introduction to Factor Analysis

During recent years factor analysis has found its use among a wide range of researchers mainly due to the development of high-speed computers and statistical software packages (for example, DATATEXT, BMD, OSIRIS, SAS and SPSS). This also affected a large group of users who did not have relevant mathematical training but were nevertheless interested in using the potential of factor analysis in their research (Harman, 1976; Horst, 1965; Lawley and Maxswel, 1971; Mulaik, 1972).

Factor analysis assumes that the variables under study are a linear combination of some hidden (latent) unobservable factors. In other words, there is a system of factors and a system of studied variables. A certain dependence between these two systems allows, through factor analysis, taking into account the existing dependence, to draw conclusions on the variables (factors) being studied. The logical essence of this dependence is that the causal system of factors (the system of independent and dependent variables) always has a unique correlation system of the studied variables, and not vice versa. Only under strictly limited conditions imposed on factor analysis is it possible to unambiguously interpret causal structures across factors for the presence of correlations between the variables under study. In addition, there are problems of a different nature. For example, when collecting empirical data, various kinds of errors and inaccuracies may be made, which in turn complicates the work of identifying hidden unobservable parameters and their further research.

What is factor analysis? Factor analysis refers to a variety of statistical techniques, the main task of which is to represent a set of studied characteristics in the form of a reduced system of hypothetical variables. Factor analysis is a research empirical method that primarily finds its application in social and psychological disciplines.

As an example of the use of factor analysis, we can consider the study of personality traits using psychological tests. Personality properties cannot be directly measured; they can only be judged on the basis of a person’s behavior, answers to certain questions, etc. To explain the collected empirical data, their results are subjected to factor analysis, which allows us to identify those personal properties that influenced the behavior of the subjects in the experiments.

The first stage of factor analysis, as a rule, is the selection of new features, which are linear combinations of the previous ones and “absorb” most overall variability in the observed data, and therefore convey much of the information contained in the original observations. This is usually done using principal component method, although other techniques are sometimes used (for example, the principal factors method, the maximum likelihood method).

The principal component method is a statistical technique that allows you to transform the original variables into their linear combination (GeorgH.Dunteman).

The purpose of the method is to obtain a reduced system of source data, which is much easier for understanding and further statistical processing. This approach was proposed by Pearson (1901) and, independently of him, was further developed by Hotelling (1933). The author tried to minimize the use of matrix algebra when working with this method.

The main goal of the principal component method is to isolate primary factors and determine the minimum number of common factors that satisfactorily reproduce the correlations between the studied variables. The result of this step is a matrix of factor loading coefficients, which in the orthogonal case are correlation coefficients between variables and factors. When determining the number of factors to be selected, the following criterion is used: only factors with eigenvalues ​​greater than the specified constant (usually unity) are selected. However, usually the factors obtained by the principal component method cannot be interpreted clearly enough. Therefore, the next step in factor analysis is to transform (rotate) the factors in such a way as to facilitate their interpretation. Rotation

factors consists in finding the simplest factor structure, that is, such a variant of assessing factor loadings and residual variances, which makes it possible to meaningfully interpret common factors and loadings.

The most commonly used rotation method by researchers is the varimax method.

This is a method that allows, on the one hand, by minimizing the spread of squared loadings for each factor, to obtain a simplified factor structure by increasing large and decreasing small factor loadings, on the other hand. So, the main goals of factor analysis are:

reduction number of variables (data reduction); structure definition.

Therefore, factor analysis is used either as a data reduction method or as a classification method.

Practical examples and advice on the use of factor analysis can be found in the book by Stevens (1986); a more detailed description is given by Cooley and Lohnes (1971); Harman (1976); Kim and Mueller (1978a, 1978b); Lawley and Maxwell (1971); Lindeman, Merenda and Gold (1980); Morrison (1967) and Mulaik (1972). An interpretation of secondary factors in hierarchical factor analysis as an alternative to traditional factor rotation is given by Wherry (1984).

Issues in preparing data for use

factor analysis

Let's look at a series of questions and short answers using factor analysis.

What level of measurement does factor analysis require or, in other words, in what measurement scales should data be presented for factor analysis?

Factor analysis requires that variables be presented on an interval scale (Stevens, 1946) and follow a normal distribution. This requirement also assumes that covariance or correlation matrices are used as input data.

Should a researcher avoid using factor analysis when the metric basis of the variables is not precisely defined, i.e. Is the data presented on an ordinal scale?

Not necessary. Many variables representing, for example, measures of subjects' opinions on a large number tests do not have a precisely established metric base. However, in general, it is assumed that many “ordinal variables” can contain numerical values ​​that do not distort and even preserve the basic properties of the characteristic being studied. The researcher’s tasks: a) correctly determine the number of reflexively identified orders (levels); b) take into account that the sum of the admitted distortions will be included in the correlation matrix, which is the basis for the input data of the factor analysis; c) correlation coefficients are fixed as “ordinal” distortions in measurements (Labovitz, 1967, 1970; Kim, 1975).

For a long time it was believed that distortions are assigned to the numerical values ​​of ordinal categories. However, this is unfounded, since for metric quantities distortions, even minimal ones, are possible during the experiment. In factor analysis, the results depend on the possibility of errors obtained during the measurement process, and not on their origin and correlation to data of a certain type of scale.

Can factor analysis be used for nominal (dichotomous) variables?

Many researchers argue that using factor analysis for nominal variables is very convenient. First, dichotomous values ​​(values ​​equal to “0” and “1”) exclude the choice of anything other than them. Secondly, as a result, the correlation coefficient is the equivalent of the Pearson correlation coefficient, which acts as the numerical value of the variable for factor analysis.

However, there is no clear positive answer to this question. Dichotomous variables are difficult to express within the framework of an analytical factor model: each variable has a weighting value of at least two main factors - general and specific (Kim, Muller). Even if these factors have two values ​​(which is quite rare in real factor models), then the final results in the observed variables must contain at least four different values, which, in turn, justify the inconsistency of using nominal variables. Therefore, factor analysis for such variables is used to obtain a number of heuristic criteria.

How many variables should there be for each hypothetically constructed factor?

It is assumed that for each factor there should be at least three variables. But this requirement is omitted if factor analysis is used to confirm a hypothesis. In general, researchers agree that it is necessary to have at least twice as many variables as factors.

One more point regarding this issue. The larger the sample size, the more reliable the criterion value CI-square. Results are considered statistically significant if the sample contains at least 51 observations. Thus:

N-n-150,(3.33)

where N is the sample size (number of measurements),

n – number of variables (Lawley, Maxwell, 1971).

This is, of course, only a general rule.

What is the meaning of the sign of the factor loading?

The sign itself is not significant and there is no way to assess the significance of the relationship between a variable and a factor. However, the signs of the variables included in the factor have a specific meaning relative to the signs of other variables. Different signs simply mean that the variables are related to the factor in opposite directions.

For example, according to the results of factor analysis, it was found that for a pair of qualities open-closed(multifactorial Catell questionnaire) there are positive and negative weight loads, respectively. Then they say that the share of quality open, in the selected factor there is more than the share of quality closed.

Principal components and factor analysis

Factor analysis as a data reduction method

Suppose a (somewhat "dumb") study is conducted in which the height of one hundred people is measured in meters and centimeters. So there are two variables. If we further investigate, for example, the effect of various nutritional supplements on growth, would it be advisable to use both variables? Probably not, because... Height is one characteristic of a person, regardless of the units in which it is measured.

Let's assume that people's satisfaction with life is being measured using a questionnaire containing various items. For example, questions are asked: are people satisfied with their hobby (item 1) and how intensively do they engage in it (item 2). The results are transformed so that average responses (for example, for satisfaction) correspond to a value of 100, while below and above the average responses are smaller and large values, respectively. Two variables (responses to two different items) are correlated. From the high correlation of these two variables, we can conclude that the two questionnaire items are redundant. This, in turn, allows the two variables to be combined into one factor.

The new variable (factor) will include the most significant features of both variables. So, in fact, the original number of variables has been reduced and two variables have been replaced by one. Note that the new factor (variable) is actually a linear combination of the two original variables.

An example in which two correlated variables are combined into a single factor shows the main idea of ​​factor analysis or, more precisely, principal components analysis. If the example with two variables is extended to a larger number of variables, the calculations become more complex, but the basic principle of representing two or more dependent variables as one factor remains valid.

Principal component method

Principal component analysis is a method of data reduction or reduction, i.e. by reducing the number of variables. A natural question arises: how many factors should be identified? Note that in the process of sequential selection of factors, they include less and less variability. The decision about when to stop the factor selection procedure depends largely on one's view of what constitutes small "random" variability. This decision is quite arbitrary, but there are some recommendations that allow you to rationally choose the number of factors (see section Eigenvalues ​​and number of allocated factors).

In the case where there are more than two variables, they can be considered to define a three-dimensional "space" in the same way that two variables define a plane. If there are three variables, then a three-dimensional scatterplot can be constructed (see Figure 3.10).

Rice. 3.10. 3D trait scatterplot

For the case of more than three variables, it becomes impossible to represent points on a scatterplot, but the logic of rotating the axes to maximize the variance of the new factor remains the same.

After the line for which the dispersion is maximum is found, some scatter of data remains around it and it is natural to repeat the procedure. In principal component analysis, this is exactly what is done: after the first factor highlighted, that is, after the first line is drawn, the next line is determined that maximizes the residual variation (the spread of data around the first line), etc. Thus, the factors are sequentially identified one after another. Since each subsequent factor is determined in such a way as to maximize the variability remaining from the previous ones, the factors turn out to be independent of each other (uncorrelated or orthogonal).

Eigenvalues ​​and number of allocated factors

Let's look at some standard results from principal component analysis. With repeated calculations, factors with less and less variance are identified. For simplicity of presentation, it is believed that work usually begins with a matrix in which the variances of all variables are equal to 1.0. Therefore, the total variance is equal to the number of variables. For example, if there are 10 variables and the variance of each is 1, then the largest variance that can potentially be extracted is 10 times 1.

Suppose that a study of life satisfaction included 10 items to measure different aspects of satisfaction with home life and work. The variance explained by the sequential factors is presented in Table 3.14:

Table 3. 14

Eigenvalue table

Cumulate. % In the second column of table 3. 14. (eigenvalues)

the variance of the new, just identified factor is presented. The third column for each factor gives the percentage of the total variance (in this example it is 10) for each factor. As you can see, the first factor (value 1) explains 61 percent of the total variance, factor 2 (value 2) explains 18 percent, etc. The fourth column contains the accumulated (cumulative) variance. So, the variances allocated by the factors are called eigenvalues

. This name comes from the calculation method used.

Once you know how much variance each factor contributed, you can return to the question of how many factors should be retained. As stated above, this decision is arbitrary in nature. However, there are some generally accepted recommendations, and in practice, following them gives the best results.

Criteria for selecting factors Kaiser criterion. First, only those factors are selected

eigenvalues of which there are more than 1. Essentially, this means that if a factor does not allocate variance equivalent to at least the variance of one variable, then it is omitted. This criterion was proposed by Kaiser (1960) and is the most widely used. In the above example (see Table 3.14), based on this criterion, only 2 factors (two main components) should be retained. Scree criterion

is

Both criteria have been studied in detail by Browne (1968), Cattell and Jaspers (1967), Hakstian, Rogers and Cattell (1982), Lynn (1968), Tucker, Koopman and Lynn (Tucker, Koopman, Linn, 1969). Cattel suggested finding a place on the graph where the decrease in eigenvalues ​​from left to right slows down as much as possible. It is assumed that to the right of this point there is only a “factorial scree” (“talus” is a geological term for rock fragments that accumulate at the bottom of a rocky slope). In accordance with this criterion, 2 or 3 factors can be left in the example considered.

Which criterion should still be preferred in practice? Theoretically, it is possible to calculate characteristics by generating random data for a specific number of factors. Then you can see whether the criterion used has detected a sufficiently accurate number of significant factors or not. Using this general method, the first criterion ( Kaiser criterion) sometimes retains too many factors, while the second criterion ( scree criterion) sometimes retains too few factors; however, both criteria are quite good under normal conditions, when there are a relatively small number of factors and many variables.

In practice, an important additional question arises, namely: when the resulting solution can be meaningfully interpreted. Therefore, several solutions with more or less factors are usually examined, and then the one that makes the most sense is selected. This issue will be further discussed within the framework of factor rotations.

Commonalities

In the language of factor analysis, the proportion of variance in a particular variable that belongs to common factors (and is shared with other variables) is called community. That's why extra work The challenge facing the researcher when applying this model is to estimate the commonalities for each variable, i.e. the proportion of variance that is common to all items. Then share of variance, for which each item is responsible, is equal to the total variance corresponding to all variables minus the communality (Harman and Jones, 1966).

Main factors and main components

Term factor analysis includes both principal component analysis and principal factor analysis. It is assumed that, in general, it is known how many factors should be identified. One can find out (1) the significance of the factors, (2) whether they can be interpreted in a reasonable way, and (3) how to do this. To illustrate how this can be done, we work backwards, that is, start with some meaningful structure and then see how that translates into results.

The main difference between the two factor analysis models is that in principal component analysis it is assumed that all variability of variables, whereas principal factor analysis uses only the variability of a variable that is common to other variables.

In most cases, these two methods lead to very similar results. However, principal component analysis is often preferred as a data reduction method, while principal factor analysis is better used to determine the structure of the data.

Factor analysis as a method of data classification

Correlation matrix

The first stage of factor analysis involves calculating the correlation matrix (in the case of normal sampling distribution). Let's return to the satisfaction example and look at the correlation matrix for the variables related to satisfaction at work and at home.

All processes occurring in business are interconnected. Both direct and indirect connections can be traced between them. Various economic parameters change under the influence various factors. Factor analysis (FA) allows you to identify these indicators, analyze them, and study the degree of influence.

The concept of factor analysis

Factor analysis is a multidimensional technique that allows you to study the relationships between the parameters of variables. In the process, the structure of covariance or correlation matrices is studied. Factor analysis is used in a variety of sciences: psychometrics, psychology, economics. The basics of this method were developed by psychologist F. Galton.

Objectives of the

To obtain reliable results, a person needs to compare indicators on several scales. In the process, the correlation of the obtained values, their similarities and differences is determined. Let's consider the basic tasks of factor analysis:

• Detection of existing values.
• Selection of parameters for a complete analysis of values.
• Classification of indicators for system work.
• Detection of relationships between resultant and factor values.
• Determining the degree of influence of each factor.
• Analysis of the role of each value.
• Application of the factor model.

Every parameter that affects the final value must be examined.

Factor analysis techniques

FA methods can be used both in combination and separately.

Deterministic Analysis

Deterministic analysis is used most often. This is due to the fact that it is quite simple. Allows you to identify the logic of the impact of the company’s main factors and analyze their impact in quantitative terms. As a result of the DA, you can understand what factors should be changed to improve the company's performance. Advantages of the method: versatility, ease of use.

Stochastic analysis

Stochastic analysis allows you to analyze existing indirect relationships. That is, there is a study of indirect factors. The method is used if it is impossible to find direct connections. Stochastic analysis is considered complementary. It is only used in certain cases.

What is meant by indirect connections? With a direct connection, when the argument changes, the value of the factor will also change. An indirect connection involves a change in the argument followed by a change in several indicators at once. The method is considered auxiliary. This is due to the fact that experts recommend studying direct connections first. They allow you to create a more objective picture.

Stages and features of factor analysis

Analysis for each factor gives objective results. However, it is used extremely rarely. This is due to the fact that complex calculations are performed in the process. To carry them out you will need special software.

Let's consider the stages of FA:

1. Establishing the purpose of the calculations.
2. Selection of values ​​that directly or indirectly affect the final result.
3. Classification of factors for complex research.
4. Detecting the relationship between the selected parameters and the final indicator.
5. Modeling of mutual relationships between the result and the factors influencing it.
6. Determining the degree of impact of the values ​​and assessing the role of each parameter.
7. Use of the generated factor table in the activities of the enterprise.

FOR YOUR INFORMATION! Factor analysis involves very complex calculations. Therefore, it is better to entrust it to a professional.

IMPORTANT! When carrying out calculations, it is extremely important to correctly select factors that influence the results of the enterprise. The selection of factors depends on the specific area.

Factor analysis of profitability

A profitability analysis is carried out to analyze the rationality of resource allocation. As a result, it is possible to determine which factors most influence the final result. As a result, we can retain only those factors that the best way affect efficiency. Based on the data obtained, you can change the company's pricing policy. The following factors may influence the cost of production:

• fixed costs;
• variable costs;
• profit.

Reducing costs provokes an increase in profits. In this case, the cost does not change. We can conclude that profitability is affected by existing costs, as well as the volume of products sold. Factor analysis allows us to determine the degree of influence of these parameters. When does it make sense to do it? The main reason for this is to reduce or increase profitability.

Factor analysis is carried out using the following formula:

Rв= ((W-SB -KRB-URB)/W) - (WB-SB-KRB-URB)/WB, Where:

VT – revenue for the current period;

SB – cost price for the current period;

KRB – commercial expenses for the current period;

URB – management expenses for the previous period;

VB – revenue for the previous period;

KRB – commercial expenses for the previous period.

Other formulas

Let's consider the formula for calculating the degree of impact of cost on profitability:

Rс= ((W-SBot -KRB-URB)/W) - (W-SB-KRB-URB)/W,

CBO is the cost of production for the current period.

Formula for calculating the impact of management expenses:

Rur= ((W-SB -KRB-URot)/W) - (W-SB-KRB-URB)/W,

URot is management expenses.

The formula for calculating the impact of business costs is:

Rк= ((W-SB -KRo-URB)/W) - (W-SB-KRB-URB)/W,

CR is commercial expenses for the previous time.

The total impact of all factors is calculated using the following formula:

Rob=Rv+Rс+Rur+Rk.

IMPORTANT! When making calculations, it makes sense to calculate the influence of each factor separately. Overall PA results are of little value.

Example

Let's consider the organization's indicators for two months (for two periods, in rubles). In July, the organization's income amounted to 10 thousand, production costs - 5 thousand, administrative expenses - 2 thousand, commercial expenses - 1 thousand. In August, the company's income amounted to 12 thousand, production costs - 5.5 thousand, administrative expenses - 1.5 thousand, commercial expenses - 1 thousand. The following calculations are carried out:

R=((12 thousand-5.5 thousand-1 thousand-2 thousand)/12 thousand)-((10 thousand-5.5 thousand-1 thousand-2 thousand)/10 thousand)=0.29-0, 15=0.14

From these calculations we can conclude that the organization’s profit increased by 14%.

Factor analysis of profit

P = RR + RF + RVN, where:

P – profit or loss;

РР – profit from sales;

RF – results of financial activities;

RVN is the balance of income and expenses from non-operating activities.

Then you need to determine the result from the sale of goods:

PP = N – S1 – S2, where:

N – revenue from the sale of goods at selling prices;

S1 – cost of products sold;

S2 – commercial and administrative expenses.

The key factor in calculating profit is the sales turnover of the company.

FOR YOUR INFORMATION! Factor analysis is extremely difficult to perform manually. You can use special programs for it. The simplest program for calculations and automatic analysis - Microsoft Excel. It has tools for analysis.

Factor analysis is understood as a method of complex and systematic study and measurement of factors for the value of effective indicators.

Distinguish following types factor analysis: deterministic (functional)

stochastic (probabilistic)

Deterministic factor analysis – this is a technique for assessing the influence of factors whose connection with the performance indicator is functional in nature, i.e. the effective indicator can be presented as a product, quotient or algebraic sum of factors.

Methods of deterministic factor analysis:

chain substitution method

index

integral

absolute differences

relative differences, etc.

Stochastic analysis – a methodology for studying factors whose connection with an effective indicator, unlike a functional one, is incomplete, probabilistic.

Methods of stochastic factor analysis:

correlation analysis

regression analysis

dispersive

component

modern multivariate factor analysis

discriminant

Basic methods of deterministic factor analysis

THE CHAIN ​​SUBSTITUTION METHOD is the most universal; it is used to calculate the influence of factors in all types of factor models: addition, multiplication, division and mixed.

This method allows us to determine the influence individual factors to change the value of the effective indicator by replacing the base value of each factor indicator with the actual value in the reporting period. For this purpose, a number of conditional values ​​of the performance indicator are determined, which take into account the change in one, then two, three, etc. factors, assuming that the rest do not change.

Comparing the value of an effective indicator before and after changing the level of one or another factor allows us to exclude the influence of all factors except one and determine its impact on the increase in the effective indicator.

The algebraic sum of the influence of factors must necessarily be equal to the total increase in the effective indicator. The absence of such equality indicates mistakes have been made.

INDEX METHOD is based on relative indicators of dynamics, spatial comparisons, plan implementation (indices), which are defined as the ratio of the level of the analyzed indicator in the reporting period to its level in the base period (or to the planned or other object).

Using indices, you can identify the influence of various factors on changes in performance indicators in multiplication and division models.

The INTEGRAL METHOD is a further logical development of the considered methods, which have a significant drawback: when using them, they assume that the factors change independently of each other. In fact, they change together, are interconnected, and from this interaction an additional increase in the effective indicator is obtained, which is added to one of the factors, usually the last one. In this regard, the magnitude of the influence of factors on the change in the performance indicator changes depending on the place in which one or another factor is placed in the model under study.

When using the INTEGRAL method, the error in calculating the influence of factors is distributed equally between them, and the order of substitution does not matter. The error distribution is carried out using special models.

Types of finite factor systems, the most frequently encountered in the analysis of economic activity:

additive models

multiplicative models

;

multiple models

;
;
;,

Where y– effective indicator (initial factor system);

x i– factors (factor indicators).

In relation to the class of deterministic factor systems, the following are distinguished: basic modeling techniques.

,

those. multiplicative model of the form
.

3. Factor system reduction method. Initial factor system
. If we divide both the numerator and denominator of the fraction by the same number, we get a new factor system (in this case, of course, the rules for selecting factors must be followed):

.

In this case we have a finite factor system of the form
.

Thus, the complex process of forming the level of the studied indicator of economic activity can be decomposed using various techniques into its components (factors) and presented in the form of a model of a deterministic factor system.

Modeling the return on capital indicator of an enterprise ensures the creation of a five-factor profitability model, which includes all indicators of intensification of the use of production resources.

We will conduct a profitability analysis using the data in the table.

CALCULATION OF KEY INDICATORS FOR THE ENTERPRISE FOR TWO YEARS

Based on the basic indicators, we will calculate the indicators of intensification of production resources (rub.)

Five-factor model of return on assets (advanced capital)

.

We will illustrate the methodology for analyzing the five-factor model of return on assets using the method of chain substitutions.

First, let's find the profitability value for the base and reporting years.

For base year:

For the reporting year:

The difference in the profitability ratios of the reporting and base years was 0.005821, and as a percentage - 0.58%.

Let's look at how the five factors mentioned above contributed to this increase in profitability.

In conclusion, we will compile a summary of the influence of factors on the deviation of profitability of the 2nd year compared to the 1st (base) year.

Total deviation, % 0.58

Including due to the influence of:

labor intensity +0.31

material consumption +0.28

depreciation capacity 0

Total cost: +0.59

capital intensity −0.07

working capital turnover +0.06

Total advance payment −0.01

The main types of models used in financial analysis and forecasting.

Before we start talking about one of the types financial analysis– factor analysis, let us recall what financial analysis is and what its goals are.

The financial analysis is an assessment method financial condition and performance efficiency of an economic entity based on studying the dependence and dynamics of indicators financial statements.

Financial analysis has several purposes:

• assessment of financial situation;
• identifying changes in financial condition in space and time;
• identification of the main factors that caused changes in financial condition;
• forecast of main trends in financial condition.

As you know, there are the following main types of financial analysis:

• horizontal analysis;
• vertical analysis;
• trend analysis;
• method of financial ratios;
• comparative analysis;
• factor analysis.

Each type of financial analysis is based on the use of a model that makes it possible to evaluate and analyze the dynamics of the main indicators of the enterprise. There are three main types of models: descriptive, predicative and normative.

Descriptive models also known as descriptive models. They are fundamental for assessing the financial condition of an enterprise. These include: construction of a system of reporting balance sheets, presentation of financial statements in various analytical sections, vertical and horizontal analysis of reporting, a system of analytical coefficients, analytical notes for reporting. All these models are based on the use of information financial statements.

At the core vertical analysis there is a different presentation of financial statements - in the form relative values, characterizing the structure of generalizing final indicators. An obligatory element of the analysis is the dynamic series of these quantities, which makes it possible to track and predict structural changes in the composition of economic assets and the sources of their coverage.

Horizontal analysis allows you to identify trends in changes in individual items or their groups included in the financial statements. This analysis is based on the calculation of the basic growth rates of balance sheet and income statement items.

System of analytical coefficients– the main element of financial analysis, used by various groups of users: managers, analysts, shareholders, investors, creditors, etc. There are dozens of such indicators, divided into several groups according to the main areas of financial analysis:

• liquidity indicators;
• financial stability indicators;
• business activity indicators;
• profitability indicators.

Predicative models These are predictive models. They are used to forecast a company's income and its future financial condition. The most common of them are: calculating the point of critical sales volume, constructing forecast financial reports, dynamic analysis models (strictly determined factor models and regression models), situation analysis models.

Normative models. Models of this type allow you to compare the actual results of enterprises with the expected ones calculated according to the budget. These models are used primarily in internal financial analysis. Their essence boils down to the establishment of standards for each item of expenditure for technological processes, types of products, responsibility centers, etc. and to the analysis of deviations of actual data from these standards. The analysis is largely based on the use of strictly deterministic factor models.

As we see, modeling and analysis of factor models occupy an important place in the methodology of financial analysis. Let's consider this aspect in more detail.

Basics of modeling.

The functioning of any socio-economic system (which includes an operating enterprise) occurs in conditions of complex interaction of a complex of internal and external factors. Factor- This is the reason, driving force any process or phenomenon that determines its character or one of its main features.

Classification and systematization of factors in the analysis of economic activity.

The classification of factors is their distribution into groups depending on common features. It allows you to gain a deeper understanding of the reasons for changes in the phenomena under study, and more accurately assess the place and role of each factor in the formation of the value of effective indicators.

The factors studied in the analysis can be classified according to different criteria.

By their nature, factors are divided into natural, socio-economic and production-economic.

Natural factors have a great influence on the performance results in agriculture, in forestry and other industries. Taking into account their influence makes it possible to more accurately assess the results of the work of business entities.

Socio-economic factors include living conditions workers, organization of health-improving work at enterprises with hazardous production, general level personnel training, etc. They contribute to a more complete use of the enterprise’s production resources and increase the efficiency of its work.

Production and economic factors determine the completeness and efficiency of use of the enterprise's production resources and the final results of its activities.

By degree of impact on results economic activity factors are divided into primary and secondary. The main ones include factors that have a decisive impact on the performance indicator. Those that do not have a decisive impact on the results of economic activity in the current conditions are considered secondary. It should be noted that, depending on the circumstances, the same factor can be both primary and secondary. The ability to identify the main ones from the entire set of factors ensures the correctness of the conclusions based on the results of the analysis.

Factors are divided into internal And external, depending on whether the activities of a given enterprise affect them or not.

The analysis focuses on internal factors that the enterprise can influence. Factors are divided into objective , independent of the will and desires of people, and subjective

subject to the influence of the activities of legal entities and individuals.

According to the degree of prevalence, factors are divided into general and specific. Common factors operate in all sectors of the economy. Specific factors operate within a particular industry or a specific enterprise. In the process of an organization's work, some factors influence the indicator under study continuously throughout the entire time. Such factors are called permanent. Factors whose influence appears periodically are called

variables (this is, for example, the introduction of new technology, new types of products). And Of great importance for assessing the activities of enterprises is the division of factors according to the nature of their action into intensive extensive, and not the qualitative characteristics of the enterprise’s functioning. An example is an increase in the volume of production due to an increase in the number of workers. Intensive factors characterize the qualitative side of the production process. An example would be an increase in production volume by increasing the level of labor productivity.

Most of the factors studied are complex in composition and consist of several elements. However, there are also those that cannot be broken down into their component parts. In this regard, factors are divided into complex (complex) And simple (elemental). An example of a complex factor is labor productivity, and a simple one is the number of working days in reporting period.

Based on the level of subordination (hierarchy), factors of the first, second, third and subsequent levels of subordination are distinguished. TO first level factors These include those that directly affect the performance indicator. Factors that influence the performance indicator indirectly, with the help of first-level factors, are called second level factors

etc. It is clear that when studying the influence of any group of factors on the operation of an enterprise, it is necessary to organize them, that is, carry out an analysis taking into account their internal and external relations

, interaction and subordination. This is achieved through systematization. Systematization is the placement of the phenomena or objects being studied in a certain order, identifying their relationship and subordination. Creation factor systems

is one of the ways of such systematization of factors. Let's consider the concept of a factor system.

Factor systems All phenomena and processes of economic activity of enterprises are interdependent. Relationship between economic phenomena

is a joint change in two or more phenomena. Among the many forms of regular relationships, an important role is played by cause-and-effect (deterministic), in which one phenomenon gives rise to another.

In addition, each phenomenon can be considered as a cause and as a consequence. For example, labor productivity can be considered, on the one hand, as the reason for changes in production volume and the level of its cost, and on the other hand, as a result of changes in the degree of mechanization and automation of production, improvement in labor organization, etc.

Quantitative characterization of interrelated phenomena is carried out using indicators. Indicators characterizing the cause are called factorial (independent); indicators characterizing the consequence are called effective (dependent). The set of factor and resultant characteristics related by cause and effect is called factor system.

Modeling any phenomenon is the construction of a mathematical expression of an existing relationship. Modeling is one of the most important methods scientific knowledge. There are two types of dependencies studied in the process of factor analysis: functional and stochastic.

A relationship is called functional, or strictly deterministic, if each value of a factor characteristic corresponds to a well-defined non-random value of the resultant characteristic.

A relationship is called stochastic (probabilistic) if each value of a factor characteristic corresponds to a set of values ​​of the resulting characteristic, i.e., a certain statistical distribution.

Model factor system is a mathematical formula that expresses real connections between the analyzed phenomena. IN general view it can be represented like this:

where is the resultant sign;

Factor signs.

Thus, each performance indicator depends on numerous and varied factors. At the core economic analysis and its section - factor analysis- identify, evaluate and predict the influence of factors on changes in the performance indicator. The more detailed the dependence of the performance indicator on certain factors is studied, the more accurate the results of the analysis and assessment of the quality of the enterprises’ work. Without a deep and comprehensive study of factors, it is impossible to draw informed conclusions about the results of operations, identify production reserves, and justify plans and management decisions.

Factor analysis, its types and tasks.

Under factor analysis understands the methodology for a comprehensive and systematic study and measurement of the impact of factors on the value of performance indicators.

IN general case the following can be distinguished main stages of factor analysis:

1. Setting the purpose of the analysis.
2. Selection of factors that determine the performance indicators under study.
3. Classification and systematization of factors in order to ensure comprehensive and systematic approach to the study of their influence on the results of economic activity.
4. Determination of the form of dependence between factors and the performance indicator.
5. Modeling the relationships between performance and factor indicators.
6. Calculation of the influence of factors and assessment of the role of each of them in changing the value of the performance indicator.
7. Working with the factor model (its practical use for managing economic processes).

Selection of factors for analysis of a particular indicator is carried out on the basis of theoretical and practical knowledge in a particular industry. In this case, they usually proceed from the principle: the larger the complex of factors studied, the more accurate the results of the analysis will be. At the same time, it is necessary to keep in mind that if this complex of factors is considered as a mechanical sum, without taking into account their interaction, without identifying the main, determining ones, then the conclusions may be erroneous. In business activity analysis (ABA), an interconnected study of the influence of factors on the value of performance indicators is achieved through their systematization, which is one of the main methodological issues of this science.

An important methodological issue in factor analysis is determining the form of dependence between factors and performance indicators: functional or stochastic, direct or inverse, linear or curvilinear. Here we use theoretical and practical experience, as well as methods for comparing parallel and dynamic series, analytical groupings of source information, graphical, etc.

Modeling economic indicators also represents a complex problem in factor analysis, the solution of which requires special knowledge and skills.

Calculation of the influence of factors- the main methodological aspect in ACD. To determine the influence of factors on the final indicators, many methods are used, which will be discussed in more detail below.

The last stage of factor analysis is practical use factor model to calculate reserves for the growth of the effective indicator, to plan and predict its value when the situation changes.

Depending on the type of factor model, there are two main types of factor analysis - deterministic and stochastic.

is a technique for studying the influence of factors whose connection with the effective indicator is functional in nature, that is, when the effective indicator of the factor model is presented in the form of a product, quotient or algebraic sum of factors.

This type of factor analysis is the most common, since, being quite simple to use (compared to stochastic analysis), it allows you to understand the logic of the action of the main factors of enterprise development, quantify their influence, understand which factors and in what proportion it is possible and advisable to change to increase production efficiency. We will consider deterministic factor analysis in detail in a separate chapter.

Stochastic analysis is a methodology for studying factors whose connection with a performance indicator, unlike a functional one, is incomplete and probabilistic (correlation). If with a functional (complete) dependence with a change in the argument there is always a corresponding change in the function, then with a correlation connection a change in the argument can give several values ​​of the increase in the function depending on the combination of other factors that determine this indicator. For example, labor productivity at the same level of capital-labor ratio may be different at different enterprises. This depends on the optimal combination of other factors affecting this indicator.

Stochastic modeling is, to a certain extent, a complement and deepening of deterministic factor analysis. In factor analysis, these models are used for three main reasons:

• it is necessary to study the influence of factors for which it is impossible to build a strictly determined factor model (for example, the level of financial leverage);

• influence needs to be studied complex factors, which cannot be combined into the same strictly deterministic model;
• it is necessary to study the influence of complex factors that cannot be expressed by one quantitative indicator (for example, the level of scientific and technological progress).

In contrast to the strictly deterministic approach, the stochastic approach requires a number of prerequisites for implementation:

1. the presence of a population;
2. sufficient volume of observations;
3. randomness and independence of observations;
4. uniformity;
5. the presence of a distribution of characteristics close to normal;
6. the presence of a special mathematical apparatus.

The construction of a stochastic model is carried out in several stages:

• qualitative analysis (setting the purpose of the analysis, defining the population, determining the effective and factor characteristics, choosing the period for which the analysis is carried out, choosing the analysis method);
• preliminary analysis of the simulated population (checking the homogeneity of the population, excluding anomalous observations, clarifying the required sample size, establishing distribution laws for the indicators being studied);
• construction of a stochastic (regression) model (clarification of the list of factors, calculation of estimates of the parameters of the regression equation, enumeration of competing model options);
• assessment of the adequacy of the model (checking the statistical significance of the equation as a whole and its individual parameters, checking the compliance of the formal properties of the estimates with the objectives of the study);
• economic interpretation and practical use of the model (determining the spatio-temporal stability of the constructed relationship, assessing the practical properties of the model).

In addition to dividing into deterministic and stochastic, the following types of factor analysis are distinguished:

• direct and reverse;
• single-stage and multi-stage;
• static and dynamic;
• retrospective and prospective (forecast).

At direct factor analysis The research is conducted in a deductive manner - from the general to the specific. Reverse factor analysis carries out the study of cause-and-effect relationships using the method of logical induction - from particular, individual factors to general ones.

Factor analysis can be single stage And multi-stage. The first type is used to study factors of only one level (one level) of subordination without detailing them into their component parts. For example, . In multi-stage factor analysis, factors are detailed a And b on constituent elements in order to study their behavior. The detailing of factors can be continued further. In this case, the influence of factors is studied different levels

subordination. It is also necessary to distinguish static And dynamic

factor analysis. The first type is used when studying the influence of factors on performance indicators on the corresponding date. Another type is a technique for studying cause-and-effect relationships in dynamics. Finally, factor analysis can be retrospective, which studies the reasons for the increase in performance indicators over past periods, and promising,

which examines the behavior of factors and performance indicators in perspective.

Deterministic factor analysis. has a fairly strict sequence of procedures:

• construction of an economically sound deterministic factor model;
• choosing a factor analysis technique and preparing conditions for its implementation;
• implementation of counting procedures for model analysis;
• formulating conclusions and recommendations based on the results of the analysis.

The first stage is especially important, since an incorrectly constructed model can lead to logically unjustified results. The meaning of this stage is as follows: any expansion of a strictly determined factor model should not contradict the logic of the “cause-effect” relationship. As an example, consider a model linking sales volume (P), headcount (H) and labor productivity (LP). Theoretically, three models can be explored:

All three formulas are correct from the point of view of arithmetic, however, from the point of view of factor analysis, only the first one makes sense, since in it the indicators on the right side of the formula are factors, i.e. the cause that generates and determines the value of the indicator on the left side (consequence ).

At the second stage, one of the methods of factor analysis is selected: integral, chain substitutions, logarithmic, etc. Each of these methods has its own advantages and disadvantages. Brief comparative characteristics We will look at these methods below.

Types of deterministic factor models.

The following deterministic analysis models exist:

additive model, i.e., a model in which factors are included in the form of an algebraic sum; an example is the commodity balance model:

Where R- implementation;

Inventory at the beginning of the period;

P- receipt of goods;

Ending inventory;

IN- other disposal of goods;

multiplicative model, i.e., a model in which factors are included in the form of a product; An example is the simplest two-factor model:

Where R- implementation;

H- number;

PT- labor productivity;

multiple model, i.e., a model that represents ratio of factors, For example:

where is the capital-labor ratio;

OS

H- number;

mixed model, i.e. a model in which factors are included in various combinations, for example:

,

Where R- implementation;

Profitability;

OS- cost of fixed assets;
About- cost of working capital.

A strictly deterministic model that has more than two factors is called multifactorial.

Typical problems of deterministic factor analysis.

In deterministic factor analysis, four typical problems can be distinguished:

1. Assessing the influence of relative changes in factors on the relative changes in the performance indicator.
2. Assessing the impact of an absolute change in the i-th factor on the absolute change in a performance indicator.
3. Determining the ratio of the change in the effective indicator caused by a change in the i-th factor to the base value of the effective indicator.
4. Determination of the share of the absolute change in the performance indicator caused by the change in the i-th factor in the total change in the performance indicator.

Let us characterize these problems and consider the solution to each of them using a specific simple example.

Example.

The volume of gross output (GP) depends on two main factors of the first level: the number of employees (NH) and average annual output (AG).

We have a two-factor multiplicative model: .

Let's consider a situation where both production and the number of workers in the reporting period deviated from the planned values.

Data for calculations are given in Table 1.

Table 1. Data for factor analysis of gross output volume.

Task 1.

The problem makes sense for multiplicative and multiple models.

;

.

Let's consider the simplest two-factor model.

Obviously, when analyzing the dynamics of these indicators, the following relationship between the indices will be fulfilled:

.

where the index value is the ratio of the indicator value in the reporting period to the base one.

Let's calculate the indices of gross output, number of employees and average annual output for our example: According to the above rule, the gross output index is equal to the product of the indices of the number of workers and average annual output, i.e. Obviously, if we calculate the gross output index directly, we will get the same value: We can conclude: as a result of an increase in the number of employees by 1.2 times and an increase in average annual output by 1.25 times, the volume of gross output increased by 1.5 times. Thus, relative changes in factor and performance indicators are related by the same relationship as the indicators in the original model. This problem is solved by answering questions like: “What will happen if

i-th indicator

will change by n%, and j-th indicator will change by k%?".

Task 2. Is y main task deterministic factor analysis; its general formulation has the form: Let

- a strictly determined model that characterizes the change in the performance indicator y is obliged to increase the i-th factor, i.e. write the following dependence:

where is the general change in the performance indicator, which develops under the simultaneous influence of all factor characteristics;

The change in the performance indicator is influenced only by the factor.

Depending on which method of model analysis is chosen, factor decompositions may differ. Therefore, in the context of this task, let us consider the main methods of analyzing factor models.

Basic methods of deterministic factor analysis.

One of the most important methodological factors in ACD is determining the magnitude of the influence of individual factors on the increase in performance indicators. In deterministic factor analysis (DFA), the following methods are used for this: identifying the isolated influence of factors, chain substitution, absolute differences, relative differences, proportional division, integral, logarithm, etc.

The first three methods are based on the elimination method.

Eliminate means to eliminate, reject, exclude the influence of all factors on the value of the effective indicator, except one. This method is based on the fact that all factors change independently of each other: first one changes, and all others remain unchanged, then two change, then three, etc., while the rest remain unchanged. This allows us to determine the influence of each factor on the value of the indicator under study separately. Let's give brief description

the most common methods. The chain substitution method is a very simple and visual method, the most universal of all. It is used to calculate the influence of factors in all types of deterministic factor models: additive, multiplicative, multiple and mixed. This method allows you to determine the influence of individual factors on changes in the value of the effective indicator by gradually replacing the basic value of each factor indicator

Let us recall that when using this method, the order in which the values ​​of the factors change is of great importance, since the quantitative assessment of the influence of each factor depends on this.

First of all, it should be noted that there is not and cannot exist a single method for determining this order - there are models in which it can be determined arbitrarily. Only for a small number of models can formalized approaches be used. In practice this problem does not have of great importance, since in retrospective analysis, trends and the relative importance of one or another factor are important, and not precise estimates of their influence.

Nevertheless, to maintain a more or less uniform approach to determining the order of replacement of factors in the model, general principles can be formulated. Let us introduce some definitions.

A sign that is directly related to the phenomenon under study and characterizes its quantitative aspect is called primary or quantitative.

These signs are: a) absolute (volumetric); b) they can be summed up in space and time. Examples include sales volume, headcount, cost of working capital, etc. Features that relate to the phenomenon under study not directly, but through one or more other features and characterize the qualitative side of the phenomenon being studied are called or secondary high quality

.

These signs are: a) relative; b) they cannot be summed up in space and time. Examples include capital-labor ratio, profitability, etc. The analysis identifies secondary factors of the 1st, 2nd, etc. orders, obtained by sequential detailing.

A strictly determined factor model is called complete if the effective indicator is quantitative, and incomplete if the effective indicator is qualitative. In a complete two-factor model, one factor is always quantitative, the second is qualitative. In this case, it is recommended to start replacing factors with a quantitative indicator. If there are several quantitative and several qualitative indicators, then you should first change the value of the factors of the first level of subordination, and then the lower one.

As you can see, the second indicator of gross output differs from the first in that when calculating it, the actual number of workers was taken instead of the planned one. The average annual output per worker in both cases is planned. This means that due to the increase in the number of workers, production output increased by 32,000 million rubles. (192,000 - 160,000).

The third indicator differs from the second in that when calculating its value, the output of workers is taken at the actual level instead of the planned one.

The number of employees in both cases is actual. Hence, due to increased labor productivity, the volume of gross output increased by 48,000 million rubles.

(240,000 - 192,000). Thus, exceeding the plan for gross output was the result of the influence of the following factors:

Algebraic sum

factors when using this method must be equal to the overall increase in the effective indicator:

The absence of such equality indicates errors in the calculations.

Other methods of analysis, such as integral and logarithmic, can achieve higher accuracy of calculations, but these methods have a more limited scope and require a large amount of calculations, which is inconvenient for conducting operational analysis. Task 3.:

.

In a certain sense, it is a consequence of the second standard problem, since it is based on the resulting factor decomposition. The need to solve this problem is due to the fact that the elements of factor decomposition are absolute values ​​that are difficult to use for spatio-temporal comparisons. When solving a problem, the 3 factor decomposition is supplemented

relative indicators α Economic interpretation: the coefficient shows by what percentage compared to the base level the performance indicator has changed under the influence of the i-th factor.

;

Let's calculate the coefficients

for our example, using the factor decomposition obtained earlier by the method of chain substitutions:

Thus, the volume of gross output increased by 20% due to an increase in the number of workers and by 30% due to an increase in output. The total increase in gross output was 50%.

.

Economic interpretation: the coefficient shows the share of the increase in the performance indicator due to the change in the i-th factor. There is no question here if all factor characteristics change unidirectionally (either increase or decrease). If this condition is not met, solving the problem may be complicated. In particular, in the simplest two-factor model, in such a case, the calculation according to the given formula is not performed and it is considered that 100% of the increase in the effective indicator is due to a change in the dominant factor characteristic, i.e., a characteristic that changes in the same direction as the effective indicator.

relative indicators γ for our example, using the factor decomposition obtained by the chain substitution method:

Thus, the increase in the number of workers accounted for 40% of the total increase in gross output, and the increase in output - 60%.