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 – noncirculating assets.
The rates ai, i = 1, ..., 7, let calculate the value of the coefficients:
(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 rootmeansquare 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 meansquare 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 meansquared error evaluating credit solvency of the company is [5, 6]:
(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:
(3)
This task represents the task of minimizing of quadratic form from n variables α1, ..., αn, meeting the conditions , 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 , i = 1, ..., 5. The more value , the more influence has the indicator i of ki on the risk level, i.e. lets allow meansquared error evaluating the efficiency of portfolio totality from Beaver’s rates.
Example
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»
Value 
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 
2,186 
1,271 
0,432 
0,315 
0,653 
Coefficient of current liquidity, k2 
0,238 
0,551 
2,967 
2,486 
2,054 
Efficiency of assets, k3 
0,902 
0,697 
0,127 
0,112 
0,241 
coefficient of financial dependence, k4 
0,413 
0,549 
0,294 
0,357 
0,369 
Share of own circulating funds in the assets, k5 
–0,314 
–0,246 
0,579 
0,531 
0,389 
We’ll calculate arithmetic mean of Beaver’s i indicator using the formula:
(4)
Using the data of Table 2 and formula (4) we’ll find that , , , and .
The elements Vij of covariance matrix V of indicators ki, we’ll calculate by the formula:
(5)
We’ll have:
(6)
Solving the task (3) with the usage of the program environment Microsoft Excel, we’ll find , i = 1, ..., 5:
(7)
Minimum variance (minimum rate of error risk) is equal:
From these calculations we can make a conclusion that by evaluation of meansquared error of R, k4 is more significant in comparison with k1, k2, k3 and k5, , , and is much less than , that is k4 has more influence on risk level.
References
1. Beaver W.H. Financial Rations and Predictions of Failure // Empirical Research in Accounting Selected Studies, Supplement to Journal of Accounting Research, 1996. http://www.defaultrisk.com.
2. Dontsova L.V., Nikiforova N.A. Analysis of financial statement: Textbook. – 3rd edition., revised and supplemented edition. – Moscow: Publishing house «Delo I Servis», 2005. – 368 p.
3. Bamadio B. Credit solvency evaluation of companiesborrowers 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 companiesborrowers 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.: YunityDana, 2008. – 592 p.
7. Corporation public ‘Lenmoloko’ Document / http://www.lenmoloko.spb.ru/documents/balance_2011_4.xls.