L.E. Mistrov, D.E. Milushev

Nowadays conditions in scientific and applied aspects the problem of maintenance of financial stability of functioning of various level financial systems in the market of competitive bank services is actual. Carrying out by a bank of various sort of financial operations is connected with decision-making process and depends upon influencing integrated indicators of efficiency of its work gross the income, net profit or profitability) values of volume of involved/placed resources, the sizes of interest rates and duration of terms of attraction of money resources. For substantiation of financial stability of bank as the basic property of its functioning with the set efficiency, it is represented on set system signs in the form of active organizational system (OS) characterized by its communications on management, information supply, execution and interaction. It allows to characterize financial stability of OS by its ability to come back in a balance condition in a small vicinity of change of indicators of efficiency. It can be responsed by a point or set of points in phase space of OS conditions to which all trajectories of its development from some area of possible decisions approach. Their occurrence is caused by dependence of integrated indicators of OS efficiency on possible values of resources parametres varied in range. As these indicators can accept critical (threshold) values, so at transition through which the area, of possible OS decisions undergoes qualitative reorganisation and, accordingly, dynamics of its trajectory changes sharply and its stability decreases. Proceeding from this, trajectories of OS development can aspire to mobile steady points (balance conditions), mobile steady limiting cycles (some curve) and unstable (critical) points in which the system can pass to the condition of balance or choose the way of formation of new system orderliness. As in OS the basic operating and executive element is a person its functioning is carried out in the modes far from the equilibrium. It is caused by the fact that OS application represents difficult interaction of set of elements, each of which can work not only on certain algorithms, but can also form the rules of functioning proceeding from values of external parametres indignations as well as intersystem changes on the basis of optimisation of proper responses, maneuver by actions strategy, structure, resources and technologies. Presence in OS of prominent features: a) of the existence conflict between its elements wanted for their functioning with the set efficiency of a resource, b) a competition or a conflict of "rivalry" type demanding at a substantiation ways of its application also presence of a resource, cause the decision of a problem on the basis of dynamic distribution of volume placed / involved resources and rendered services in financial operation for maintenance of transfer of bank to a mobile cycle of stable equilibrium on set of strategy of competitors counteraction. It has allowed to present OS functioning as consecutive optimum distribution of resources between stages of financial operation on the basis of application of the theory multistage bimatrix games on a survival with the nonzero sum. The search of criterial functions - maximinial value of probability of OS performance task in view in the financial operation non-less set, was carried out by solving the functional equation of its functioning on the basis of the method consecutive proximities for the search inside or on border of area of possible decisions by means of derivatives in a direction of extreme points or maximin values of criteria functions. Necessity and sufficiency of conditions of the decision search in the form of extremum points follows from absence of the general directions of cones of possible directions of resources distribution strategy competing OS on set of possible decisions, that is in a point of a maximum (minimum) their crossing is empty. Thus the search of an extremum of multistage game is defined by the vector of values of varied parametres of competing actions strategy the OS characterised by the interest rate of crediting (α), the interest rate under deposits (β) and the interest rate under specifications of profitability of other bank services (γ). As result of solving of an optimising problem at the initial stage of the multistage financial operation, values of parametres of resources distribution strategy are defined: α*=25,1%, β*=2,2% and γ*=10%, providing profitability of activity of OS at level of 1,7%. On the subsequent steps of operation possible strategy of OS are defined from an achievement condition maximinial values of criterial functions by corresponding updating of α, β and γ parametres values to a lesser or bigger degree, or preservation of operating strategy. The developed mathematical apparatus and algorithm of a consecutive substantiation of OS actions strategy has provided a finding of a critical (threshold) point of financial operation or the point of its break-even development, and to prove a mobile limiting cycle of its stable equilibrium in the market of the competitive bank services, providing maximinial value of criterial functions for reception of the set profit in financial operation on the basis of distribution of active and passive resources.