Abstract: The object of study in the article is the credit, however it does not concern its economic essence regarding both the creditor, and the borrower. The key aspects of studу concern the mathematical foundations for modeling the  phenomenon of related borrowers. The author stars her research by analyzing economic maintenance of a phenomenon of being related, while directly giving strictly mathematical interpretation to the economic terms, which she employs. Further the author explicitly considers functional and stochastic aspects of coherence of borrowers, showing continuous connection among these aspects. The methodological basis of a research is formed with the set-theoretic and probability-theoretic approaches. Application-oriented aspects of the loan portfolio modeling are explained in terms of the algebraic theory of risk. The graphic illustration of the main results of operation is based on classical submissions of the Cartesian product of finite sets and on traditional (quasi) trees of classification. The key result of this  study is revealing of the complete mathematical list of possible options for the coherence of borrowers in the functional and probability-theoretic contexts. At the same time the article contains detailed descriptions of all possible relation options within the object field of study. This a  novel relation classification. Similar studies are not found among either Russian, or foreign studies so far. Another result of this study is a practical one, since some of the received ratios can be used in order to form efficient algorithms for the direct analysis of risk processes within the complicated systems.

Abstract: Evaluation of risks of system development is an urgent task for a for a variety of disciplines such as economics and sociology, technology and ecology, the system studied at the intersection of different disciplines. Often the parameters of such systems are discrete, set of possible states is bounded. The application package “MultiMIR” was designed to evaluate risks of development in such systems. An important difference of “MultiMIR” from other application is in achievement of polynomial computational complexity for some classes of systems, while most analogues offer only exponential complexity. The article describes: purpose of the application, main ideas used as a basis for algorithms, application architecture. The author gives an overview of ways of using the application. The conceptual basis of the theory used in the development of algorithms implemented in “MultiMIR” is in theoretical probabilistic approach. As a specific mathematical apparatus the author has chosen formalism of theory of multisets, which, in author’s opinion, has the richest expressive possibilities for the study in the described the subject area. As a programming system used in the development of the first version of the application the author used VBA-subsystem office with Microsoft Office. The selection of the programming system is dictated by the features and preferences of the primary target of the package: banking and financial analysts. Using “MultiMIR” allowed for the first time to provide accurate calculation of such non-linear measures of risk as expected utility, distorted probability measure, "Value at Risk", and so on for medium and large homogeneous portfolios term financial instruments without involving time-consuming analytical methods. Unlike traditionally used for this purpose Monte Carlo method, approached based on the described above application allows obtaining an exact solution using a comparable amount of CPU resource. “MultiMIR” application can also be used for verification of reliability of the results obtained using Monte Carlo methods considered classical in the financial risk management.