**MAIN DIRECTIONS OF ECONOMIC DEVELOPMENT IN THE 21 ^{ST} CENTURY**

**Monitoring and logical and probabilistic management of crediting process**

Allowance of credits to natural persons is essential part of the banking. All banks are individual because they serve various segments of population in different cities and regions, enterprises of different branches and sizes with various forms of property. Therefore, every bank has to apply proper credit risk model, monitoring and crediting process control system.

Advantages of the LP credit risk model are high accuracy, robustness and transparency (Solozhentsev (2012), (2013), Solozhentsev, Stepanova and Karasev (2005)). LP credit risk model has high accuracy in recognition of the "good" and "bad" credits and seven times more robustness in classification of the credits in comparison with other known models. Transparency of the LP credit risk model is provided by possibility to perform credit risk analysis, calculation of contributions of the parameters and their grades in credit risk of the bank and accuracy of credit's classification, optimization of the number of parameters and their grades, transparency of the risk scenario and criterion function.

The gist of the technology:

- determination of the minimal volume for learning sample;
- elimination of the part of incorrectly recognized "good", "bad" and outdated credits;
- crediting process monitoring;
- forming of the signal sets from finalized credits;
- replacement of LP risk models as forming of signal sets of credits;
- forming of learning sample for identification of the new LP risk model;
- analysis and bank's crediting process control.

Credits of the natural persons are described up to 20 parameters (table 1), every parameter has from 2 to 11 grades (Seitz and Stickel (1996)). Parameters of the credit and their grades are casual events-parameters and events-grades. Events-grades of every parameter form a group of incompatible events. Events lead to credit default with certain probability. Risk scenario of credit default is stated so: default occurs, if any one, or any two, ..., or all events-parameters occur.

Logical risk model of credit's default:

Y = Z_{1} ∨ Z_{2} ∨ ... ∨ Z_{n}. |
(1) |

Logical risk model of credit's default in equivalent orthogonal form:

Y = Z_{1} ∨ Z_{2}Ż_{1} ∨ Z_{3}Ż_{1}Ż_{2} ∨ ... ; |
(2) |

Probabilistic risk model of credit's default:

P(Y) = P_{1} + P_{2}(1 - P_{1}) + P_{3}(1 - P_{1})(1 - P_{2}) + ... . |
(3) |

where P_{1}, P_{2}, ... - probabilities of credit's default as a result of occurrences of events-parameters; Q_{1} = 1 - P_{1}, Q_{2} = 1 - P_{2}, ... In formula (3) values of probabilities for events-grades are placed. Risk is within {0,1} under any values of probabilities of events-grades.

The identification (learning) of the LP credit risk model is performed on the statistical data (Solojentsev and Karasev, (2002)) and the goal of this procedure is to calculate probabilities of the events-grades P_{jr}, r = 1, ..., N_{j}, j = 1, ..., n, the admitted credit risk P_{ad} and risks P_{i}, i = 1, 2, ..., N of the credits (fig. 1). The condition P_{i} > P_{ad} let us distinguish the following types of the credits: N_{gg} - are "good" both the LP-model and statistics; N_{gb} - are "good" by the LP-model but "bad" by statistics; N_{bg} - are "bad" by the LP-model but "good" by statistics; N_{bb} - are "bad" both the LP-model and statistics.

Criterion function is the maximal number correctly classified credits with correct classification:

F = N_{bb} + N_{gg} → max. |
(4) |

From (4) it follows the accuracy of the LP risk model in the classification of the "good" E_{g} and "bad" E_{b} credits and in classification of the whole set E_{m} is equal to:

E_{g} = N_{gb} / N_{g}; E_{b} = N_{bg} / N_{b}; E_{m} = (N - F) / N. |
(5) |

Restriction:

- probabilities Pjr have to satisfy the condition: 0 < P
_{jr}< 1, j = 1, ..., n; r = 1, ... , N_{j}; - average risks of credits on probabilistic model and on the statistics have to be equal in order to keep mathematical sense of probabilities;
- admitted risk Pad is determined so that mistakes of recognition of "good" and "bad' credits are equal (recognition asymmetry principle).

The probabilistic risk model identification has the following features:

- criterion function depends on the large number of real positive arguments Pjr (there are 94 parameters in credit risk of natural persons);
- criterion function is the number of correctly recognized "bad" and "good" credits (it is integer value and it is stepped);
- criterion function has local extreme ("plate");
- derivatives of the criterion function F with respect to parameters P
_{jr}cannot be computed analytically; - in the search for the optimum F
_{max}it is impossible to increment parameters P_{jr}by arbitrary positive or negative values because it would change the average risk.

The offered algorithmic iterative method of the LP risk model identification permits to optimize the LP risk model with any complexity and any number of the credits, parameters and grades (Solozhentsev (2012), (2013)).

Research results were obtained from statistical data about 1000 credits, where 700 are "good" and 300 are "bad" (Solozhentsev, Stepanova and Karasev, (2005)). Every credit has n = 20 parameters with the total number of the grades - 94. After identification function value F_{max} = 822 was obtained. LP credit risk model has essentially less mistakes in classification of the credits E_{m} = 0,155, E_{g} = 0,174, E_{b} = 0,162, in comparison with known techniques which have F_{max} = 750 - 720; E_{m} = 0,25 - 0,28.

We offer technology of the monitoring, learning of LP-model and crediting process control. Technology contains the following decisions: use statistical data about finalized credits of the bank as the learning sample, restriction of the learning sample volume, exception of the part incorrectly recognized "good", "bad" and outdated credits, forming of the signal part from finalized credits, forming of the learning sample for construction of the new LP risk model, replacement of the old LP-model with new LP-model of the credit risk, estimation of the crediting process quality on several indicators (criteria).

*Learning sample volume.* We have performed calculation research, how the learning sample volume influences on mistakes of the credit recognition. Recognition mistakes increase asymptotically with increasing of the sample volume. The probabilistic model has the limited number of probabilities for events-grades and LP-model recognizes credits better if the number of the credits in statistical data is not large. For LP-model learning when the number of the events-parameters is 20 and the number of the events-grades is 94 we can accept the minimal number of the credits in the learning sample is N_{min} = 1000 - 1200: if we will increase the number of credits in statistical data, the mistakes of the recognition of "good" and "bad" credits will not change practically, they will be constant. Suppose, it is important advantage of the LP credit risk model.

*Exception of the part of incorrectly recognized and outdated credits.* Credit is described by 20 parameters. LP-model can't provide absolute accuracy because it does not take into account other factors by reason of juridical and legal prohibitions. Borrower can keep money not in bank but home and does not inform bank about. Force-majeure events can occur. Borrower could not inform about his(her) relations with wife(husband), mother-in-law, children, about his(her) health and health of relatives. This latent information causes to the appearance of incorrectly recognized credits. Incorrectly recognized credits should be partially excepted from the database and process of the re-learning of LP credit risk model.

Economics is developed and changed permanently and information about outdated credits should be piecemeal excepted from the LP risk model re-learning process.

Let research increasing of the accuracy of LP-model in recognition of "good" and "bad" credits by the exception of the part of incorrect recognized credits. Incorrectly recognized "good" credits Ngb are within interval of the risks {P_{ad}, G} and incorrectly recognized "bad" credits N_{bg} are within interval of the risks {B, P_{ad}}.

The number of the credits in the learning sample after the exception of the part of outdated and incorrectly recognized credits are:

N = N - a_{1} N_{gb} - a_{2} N_{bg} - a_{3} N_{old} = N - N_{gb}* - N_{bg}* - N_{old}* , |
(6) |

where a_{1}, a_{2}, a_{3} - exception coefficients with values within interval [0, 1]: N_{gb}*, N_{bg}*, N_{old}* - credits that were excepted from statistical data.

From N_{gb} we are excepting the credits (figure 2) with most risk (that are in the "tail" of the distribution of "good" credits). From N_{bg} we are excepting credits with least risk (that are in the "tail" of the distribution of "bad" credits). Coefficients a_{1}, a_{2}, a_{3} depend on the number of credits, which were finalized during a year. It should accept a_{1}, a_{2}, a_{3} within interval {0.1 - 0,2}, i. e. move to optimal accuracy of the recognition of "good" and "bad" credits bit by bit. Using results of the analysis of learning indicators the coefficients a_{1}, a_{2}, a_{3} can be corrected.

*Signal part from finalized credits.* Let form signal part from the last finalized credits with volume N_{sign}. In signal part there are N_{g sign} "good" and Nbsign "bad" credits. The average risk of the credits in the signal part is equal to P_{m sign} = N_{b sign} / N_{sign}. and it is one of the most general estimations of crediting process quality. The number of credits in signal part N_{sign} = 100 - 200 is enough for the crediting process estimation - accuracy of the LP credit risk model and optimality of the exception of incorrect and outdated credits in statistical data.

*Periodicity of the re-learning of LP credit risk model.* The number of finalized credits N_{sign} = 100 - 200 in signal part we will accept as periodicity of the re-learning of LP credit risk model. The LP credit risk model is re-learned after accumulation of every following signal part of the finalized credits.

*Elaboration of the learning sample volume.* Taking into account the number of credits in signal part N_{sign}, the number of credits in learning sample after exception of part of outdated and incorrectly recognized credits is equal to:

N = N + N_{sign} - a_{1} N_{gb} - a_{2} N_{bg} - a_{3} N_{old} , |
(7) |

Taking into account the restrictions on the learning sample volume let write the condition for choice and correction of the exception coefficients a_{1}, a_{2}, a_{3} of outdated and incorrect credits from the learning sample:

N_{sign} = a_{1} N_{gb} + a_{2} N_{bg} + a_{3} N_{old} . |
(8) |

The goal of crediting process control is increasing the accuracy of the recognition of "bad' and "good" credits and, thereby, reducing of bank's losses. For control purposes, we calculate indicators (criteria) of all crediting process in a bank:

1) Coefficient of satisfied clients:

K_{s}^{t} = N^{t} / M^{t} |
(9) |

where: K^{t} - the number of credit applications, including K_{g}^{t} - the number of "good" borrowers, who has obtained credit, K_{b}^{t} - the number of "bad" borrowers, those applications were declined.

2) Average credit risk of a bank by the finalized credits:

P_{av}^{t} = Y_{b}^{t} / Y^{t}, |
(10) |

where: Y^{t} - the number of finalized credits, including Y_{g}^{t} - the number of "good" credits; Y_{b}^{t} - the number of "bad" credits (having problems).

3) Average credit risk of the signal set,

P_{av sign} = N_{b}^{t} _{sign} / N^{t} _{sign}. |
(11) |

For crediting process control, the learning sample is formed for the construction of the new LP risk model, further the new LP credit risk model is constructed, mistakes of recognition of "good" and 'bad" credits, average and admitted risks are calculated, old LP risk model is replaced by the new LP risk model.

*LP credit risk model control.* Parameters of the LP risk model control after learning and analysis of contributions of events-grades are following:

**credit risk of the borrower**is compared with the admitted risk and decision about credit issue is made;**coefficient of asymmetry**of the recognition of "good" and "bad" credits;**mistakes of the recognition**of "good", "bad" credits and in average;**price for the credit**depends on credit risk and difference between credit risk value and admitted risk value;**the number of parameters**describing credit application;**the number of grades**for every parameter.

Crediting process control. Parameters of crediting process control in bank by monitoring results are following:

- volume of the signal set of the credits and periodicity of learning and replacement of the LP-model;
- mistakes of recognition of "good", "bad" credits and in average;
- average credit risk in the signal set of the credits;
- average credit risk for all statistical data;
- coefficient of satisfied clients;
- exception coefficients a1, a2, a3 for the incorrectly recognized "good" and "bad" credits and outdated credits.

**References **

- Solozhentsev E.D., 2012. Risk management technologies (with logic and probabilistic models). Springer, 328 p.
- Solozhentsev E.D., 2013. Risk Management Technologies in Structural Complex Systems. Tutorial - SPb.: GUAP, 435 p. (In Russian)
- Solozhentsev E. D., Stepanova N. V., Karasev V.V., 2005. Transparency of Methods for Assessment of Credit Risks and Ratings. Saint Petersburg.: St.Petersburg University Press, 197 p., (In Russian)
- Seitz J., Stickel E., 1996. Consumer Loan Analysis Using Neural Network / In Proc. of the Bankai Workshop: Adaptive Intelligent Systems. Brussels, 14 - 19 Oct.
- Solojentsev E.D., Karasev V.V., 2002. Identification of Logic and Probabilistic Models of Risk in Structurally Complex Systems with Groups of Incompatible Events. - Automation and Remote Control, March 2002; Vol. 63, Issue 3, pp. 433-448.

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Intelligent Integrated Systems of Automated Designing Laboratory", IPME RAS

E-mail: esokar@gmail.com