Scientific journal
European Journal of Natural History
ISSN 2073-4972


Bamadio B 1 Semenchin E.A. 1
1 University State of Kuban
Proposed technique of αi rate estimation of Beaver’s coefficient ki in the portfolio, formed by these coefficients, allows us to minimize the average squared error estimation of portfolio efficiency. This technique allows an expert to get additional information about credit solvency of the experimental company.

One of the indicators, characterizing the financial condition of the company, is its solvency, i.e. the ability to repay liabilities in time.

The necessity of implementation of borrower liability analysis is dictated by the lending policy and concerns of bank. The bank needs to be informed: will the borrower be able to pay back cash resources with a glance at interests fixed by the bank, does the bank have prospects for future development, and how high is the risk of not recouping the fixed sums?

The basic attention by borrower’s solvency estimation is concentrated on the activity, characterizing his ability to provide quickly his quick credit and interests’ repayment.

For bankruptcy forecast of a company financial analyst Williams Beaver offered a system of indicators, letting estimate the financial condition of a company to diagnose bankruptcy [1].

Target setting

The quantity of the existing bankruptcy threat of the company we can estimate (in the rough) by pentad model of W. Beaver based on calculation of company activities: a1 – net profit, a2 – depreciation of production assets, a3 – loan capital, a4 – working assets, a5 – current liabilities to juridical entities and individuals, a6 – internal circulating funds, a7 – non-circulating assets.

The rates ai, i = 1, ..., 7, let calculate the value of the coefficients:

Eqn1.wmf Eqn2.wmf Eqn3.wmf

Eqn4.wmf Eqn5.wmf (1)

On the basis of the values ki, i = 1, ..., 5, valued by W. Beaver for three types of companies: successful, bankrupt during a year and bankrupt during five years, we can make a conclusion about the bankruptcy risk of the experimental company. System of values and regulatory indicators for these three types of companies is presented in the Table 1 [2].


Table 1

Metrics of W. Beaver

Coefficient ki

Coefficient value ki

Standard values of calculated coefficients and quantities

Group 1, successful companies

Group2, 5 years before bankruptcy

Group 3, 1 year before bankruptcy

Coefficient of beaver, k1


k1 > 0,4

k1 ≈ 0,2

k1 < –0,15

Coefficient of current liquidity, k2


k2 > 2

1 ≤ k2 ≤ 2

k2 < 1

Return of assets, k3


k3 ≥ 0,06

0,01 ≤ k2 ≤ 0,06

–0,22 ≤ k2 ≤ 0,01

Coefficient of financial dependence, k4


k4 < 0,35

0,35 ≤ k4 ≤ 0,80

k4 ≥ 0,80

Share of own circulating funds in the assets, k5


k5 ≥ 0,4

0,1 ≤ k5 ≤ 0,4

k5 < 0,1


We will form a ‘portfolio’ from coefficients ki, i.e. we will form sum total (k1, ..., k5) from Beaver’s rates. Let αi – be share of coefficient ki in total (k1, ..., k5) (i.е. αi – weight or coefficient of value ki), αi ≥ 0, i = 1, ..., 5, α1 + ... + α5 = 1. We’ll suppose that k1, ..., k5 are random quantities. Let σi – be the root-mean-square deviation ki, R = α1k1 + ... + α5k5 –efficiency of sum total (k1, ..., k5) (R – sum points of all the rates of this totality).

The aim of this project – to work out the technique of shares estimation coefficients ki, i = 1, ..., 5,, in the portfolio, when the risk lets make mean-square error in the portfolio efficiency estimation, i.e. when by estimation R is minimum.

This technique allows an expert receive to additional information about credit solvency of the company.

The results of investigation offered in this project are logical continuation of investigations, the results of which are stated in works [3, 4].

The technique of portfolio optimization from Beaver’s values

According to suppositions from point 1 risk level to make mean-squared error evaluating credit solvency of the company is [5, 6]:

Eqn6.wmf (2)

where vij – covariance between ki, kj, i.e. vij = cov(ki, kj), i, j = 1, ..., 5.

The task of share defining αi, i = 1, ..., 5, of different Beaver’s values is brought to task solution of portfolio optimization:

Eqn7.wmf (3)

This task represents the task of minimizing of quadratic form from n variables α1, ..., αn, meeting the conditions Eqn8.wmf, αi ≥ 0, i = 1, ..., 5, i.e. the task of quadratic programming.

The solution of this task can be built with the usage of different instruments, for example, using software environment Excel.

Solving the equation (3), we’ll get different values of Eqn9.wmf, i = 1, ..., 5. The more value Eqn9.wmf, the more influence has the indicator i of ki on the risk level, i.e. lets allow mean-squared error evaluating the efficiency of portfolio totality from Beaver’s rates.


Experimental data of Beaver’s rates (see Table 1), calculated on the basis of balance sheet of the company, public corporation ‘Lenmoloko’ [7], presented in the Table 2.


Тable 2

Beaver’s values of public corporation «Lenmoloko»


Values numbers

On 12/31 2011

On 12/ 31 2010

On 12/ 31 2009

On 12/31 2008

On 12/31 / 2007

Beaver’s coefficient, k1






Coefficient of current liquidity, k2






Efficiency of assets, k3






coefficient of financial dependence, k4






Share of own circulating funds in the assets, k5







We’ll calculate arithmetic mean of Beaver’s i indicator using the formula:

Eqn10.wmf (4)

Using the data of Table 2 and formula (4) we’ll find that Eqn11.wmf, Eqn12.wmf, Eqn13.wmf, Eqn14.wmf and Eqn15.wmf.

The elements Vij of covariance matrix V of indicators ki, we’ll calculate by the formula:

Eqn16.wmf (5)

We’ll have:

Eqn17.wmf (6)

Solving the task (3) with the usage of the program environment Microsoft Excel, we’ll find Eqn9.wmf, i = 1, ..., 5:

Eqn18.wmf (7)

Minimum variance (minimum rate of error risk) is equal:


From these calculations we can make a conclusion that by evaluation of mean-squared error of R, k4 is more significant in comparison with k1, k2, k3 and k5, Eqn20.wmf, Eqn21.wmf, Eqn22.wmf and Eqn23.wmf is much less than Eqn24.wmf, that is k4 has more influence on risk level.


1. Beaver W.H. Financial Rations and Predictions of Failure // Empirical Research in Accounting Selected Studies, Supplement to Journal of Accounting Research, 1996.

2. Dontsova L.V., Nikiforova N.A. Analysis of financial statement: Textbook. – 3-rd edition., revised and supplemented edition. – Moscow: Publishing house «Delo I Servis», 2005. – 368 p.

3. Bamadio B. Credit solvency evaluation of companies-borrowers of Russia and Mali // Izvestiya of Kuban State University // Natural sciences. Edition. – 2013. – № 1 (2). – P. 57–61.

4. Bamadio B. Basic aspects of credit solvency evaluation of companies-borrowers of Russia and Mali // International journal of applied and fundamental researches. – 2013. – № 1. – P. 13.

5. Berezhnaya E.V., V.I.Berezhnoi. Mathematical methods of economical systems modelling. – М.: Finances and Statistics, 2003. – 368 p.

6. Mathematical methods and models of operation investigations / edited by V.A. Kolemayeva. – М.: Yunity-Dana, 2008. – 592 p.

7. Corporation public ‘Lenmoloko’ Document /