Economic and Financial Transactions Govern Business Cycles

Problem/Relevance - This paper presents new description of the business cycles that for decades remain as relevant and important economic problem.<br> <br>Research Objective/Questions - We propose that econometrics can provide sufficient data for assessments of risk ratings for almost all economic agents. We use risk ratings as coordinates of agents and show that the business cycles are consequences of collective change of risk coordinates of agents and their financial variables.<br><br>Methodology - We aggregate similar financial variables of agents and define macro variables as functions on economic space. Economic and financial transactions between agents are the only tools that change their extensive variables. We aggregate similar transactions between agents with risk coordinates x and y and define macro transactions as functions of x and y. We derive economic equations that describe evolution of macro transactions and hence describe evolution of macro variables.<br><br>Major Findings - As example we study simple model that describes interactions between Credits transactions from Creditors at x to Borrowers at y and Loan-Repayment transactions that describe refunds from Borrowers at y to Creditors at x. We show that collective motions of Creditors and Borrowers from safer to risky area and back on economic space induce frequencies of macroeconomic Credit cycles.<br><br>Implications - Our model can improve forecasting of the business cycles and help increase economic sustainability and financial policy-making. That requires development of risk ratings methodologies and corporate accounting procedures that should correspond each other to enable risk assessments of economic agents.


Introduction
Financial accounting and reporting are important tools for corporate management and for macroeconomic modelling. In this paper we show that risk ratings assessments mostly based on financial accounting and reporting of corporations, banks, small firms and companies establish ground for macroeconomic modelling of the business cycles. Assessments of risk ratings of economic agents utilize data delivered by accounting and reporting of numerous economic agents in different industries of the entire economics. Change of risk ratings of agents induce change of The rest of the paper is organized as follows. In Section 2 we present model setup and main definitions (Olkhov, 2017b. In Section 3 we introduce economic equations on macrotransactions and discuss their economic meaning. In Section 4 we argue economic assumptions that allow describe the business cycles. As example we study a model interactions between Credits transactions CL (t,x,y) from Creditors at with risk ratings x to Borrowers with risk ratings y and Loans-Repayments transactions LR(t,x,y) of refunds from Borrowers at y to Creditors at x. We model these transactions by a system of economic equations and describe their evolution in a selfconsistent manner. Starting with these equations we derive the system of ordinary differential equations (ODE) and derive simple solutions that describe the business cycles around the growth trend of Credits C(t) in the entire economics. Conclusions are in Section 5.

Model Setup
In this Section we introduce main definitions of our approach (Olkhov, 2016a-b;2017a-c). Let's regard any participants of economic and financial relations like banks, companies, households and etc., as economic agents. Agents have a lot of economic and financial variables like Assets, Credits, Consumption, Debts and Investment and etc. Aggregations of agents variables define macroeconomic variables. For example aggregation of agents Investment equals macro Investment, aggregation of agents Consumption defines macro Consumption and etc. Thus description of agents variables model evolution of macro variables like Investment, Credits, Working Hours and etc., and different properties of business cycles. Economic and financial variables of agents are changed due to corresponding transactions between agents. For example Banks provide Credits to Borrowers and such transactions change amount of Credits provided by Banks and amount of Loans received by Borrowers. Hence description of transactions between agents models evolution of agents variables. Thus modelling transactions helps model the business cycles.
It is obvious that any transactions between agents are performed under definite Expectations. Since Muth (1961) and Lucas (1972) importance and impact of Expectations on economic and financial evolution were studied in numerous papers and we refer (Kydland & Prescott, 1980;Brunnermeier & Parker, 2005;Greenwood & Shleifer, 2014;Manski, 2017) as only small part of these research. Macroeconomic evolution is very complex and any model of macroeconomics describes certain approximation only. In our paper we simplify description of the business cycles and neglect impact of Expectations on transactions between agents. Let's propose that transactions between agents depend on other transactions only. Such approximation allows describe transactions-driven business cycle model. We shall model impact of Expectations on transactions and the business cycles in forthcoming publications.
Let's regard macroeconomics as ensemble of numerous economic agents that are under action of different risks. There are many economic and financial risks that impact agents variables and their transactions. Impacts of different risks on agents economic evolution and risk assessment are studied by numerous of papers and we refer only few (Gupton, et al, 1997;Alvarez & Jermann, 1999;Diebold, 2012;Christiano et al, 2013;BIS, 2014;Skoglund & Chen, 2015;Engle, 2017). We don't argue here problems of risk assessments but show that risk assessment methodologies can become a ground for macroeconomic modelling. Let's outline that for decades international rating companies as Moody's, Fitch, S&P (Metz & Cantor, 2007;Chane-Kon, et.al, 2010;Kraemer & Vazza, 2012) provide risk assessment and attribute risk ratings like AAA, A, BB, C and etc. for global banks and international corporations. Let's propose that it is possible to assess risk ratings for all agents of entire economics -for global banks, corporations and for small companies and even households. That requires a lot of additional econometric and statistical data. We hope that quality, accuracy and granularity of current U.S. National Income and Product Accounts system (Fox, et al., 2014) give us confidence that all econometric problems can be solved. Let's propose that our assumptions are fulfilled and it is possible evaluate risk assessments for all agents of entire economics. Risk ratings take values of risk grades like AAA, A, BB, C and we regard these grades as points x1,…xm of discrete space. Usage of risks ratings allows distribute economic agents over points x1,…xm on discrete space. Let's further call the space that maps agents by their risk ratings as economic space. Ratings of single risk distribute agents over points of onedimensional discrete space. Assessments of two or three risks distribute agents on economic space with dimension two or three. It is obvious that number of risk grades, number of points AAA, A, BB, C… is determined by methodology of risk assessment. Let's assume that assessment methodology can be generalized to make risk grades continuous so, they fill certain interval (0,X) on space R. Let's take point 0 as most secure and point X as most risky grades. Value of most risky grade X always can be set as X=1 but we use X notation for convenience. Let's assume that risk assessments of n risks define coordinates of agents on space R n . Economic agents of economics under n risks fill economic domain (1.1): Below we study economic and financial transactions and develop the business cycle model for economics that is under the action of n risks on economic space R n . Transactions between agents change their economic and financial variables. For example agent A can provide Credits to agent B. This transaction will change Credits provided by agent A and Loans received by agent B. Each transaction takes certain time dt and we consider transactions as rate or speed of change of corresponding variables. For example Credits transactions from agent A at moment t define rate of change of total Credits provided by agent A at moment t. Let's call extensive economic or financial variables of two agents as mutual if output of one becomes an input of the other. For example, Credits as output of Creditors are mutual to Loans as input of Borrowers. Any exchange between agents by mutual variables is carried out by corresponding transactions. Any agent at point x may carry out transactions with agent at any point y on economic space. Different transactions define evolution of different couples of mutual variables. We regard agents as simple units of macroeconomics and treat agents alike to "economic particles" and economic or financial transactions between agents as "economic interactions" between "economic particles". For brevity let's further call economic agents as e-particles and economic space as e-space. Now let's present above considerations in a more formal manner.
As example let's study Credits transactions that provide Loans from Creditors to Borrowers and follow Olkhov (2017b-c). Let's take that Credits transactions cl1,2(t,x,y) describe Credits provided by from e-particle 1 as Creditor at point x to Borrower at e-particle 2 as at point y at moment t. Let's call Credits and Loans as mutual variables. Let's state that all extensive economic or financial variables can be allocated as pairs of mutual variables or can be describes by mutual variables. Thus transactions describe dynamics of all extensive economic and financial variables of e-particles and hence determine macroeconomic evolution and the business cycles.

Macro transactions between points on e-space
Let's assume that transactions between e-particles at x and y describe exchange of mutual variables like Credits and Loans. Description of transactions between separate e-particles is very complex problem and we replace it by rougher model. To do that let's define economic and financial transactions between points of e-space. Main idea: let's replace precise description of transactions between separate e-particles by rougher description of transactions associated with points of espace that don't distinguish separate e-particles. Such a roughening is already used in economics.
For example aggregation of all Credits between agents of entire economics define macro Credit C(t) (see 3.2) provided in macroeconomics at moment t and equal macro Loans L(t) received in macroeconomics at moment t. Modelling transactions between all separate agents at points x and y on e-space establish too detailed picture. On the other hand description of variables like macro Credits C(t) as aggregates all transactions between all agents of entire economics gives too simplified economic model. We develop intermediate description of economy that aggregate transactions between agents that belong to domains near points x and y on risk e-space. Such approximation neglect granularity of separate e-particles but allows take into account distribution of transactions on e-space. Such approach is similar to transition from kinetic description of multiparticles system to hydrodynamic approximation in physics (Landau & Lifshitz, 1981;1987;Resibois & De Leener, 1977).
Let's assume that e-particles on e-space R n at moment t have coordinates x=(x1,…xn) and risk velocities υ=(υ1,…υn). Risk velocities describe change of risk coordinates of e-particles. Let's rougher description of Credit transactions between e-particles by small unit volume dV(z) and replace it by transactions between points of e-space. Let's assume that Let's assume that a unit volume dV(x) at x and dV(y) at y contains many e-particles (agents) but scales dVi of a unit volume dV(x) are small to compare with scales Xi of economic domain (1.1):

Let's define Credit transaction CL(t,z) at point z=(x,y) as sum of all Credit transactions cl1,2(t,x,y)
between all e-particles i=1,..N(x) in unit volume dV(x) at x and j=1,..N(y) in unit volume dV(y) at y and average this sum during time term Δ as follows: We use ∈ ( ) to denote that coordinates x of e-particle i belong to unit volume dV(x). Let's underline that value o f Credit transaction CL(t,z) at z=(x,y) can change in time and due to motion of e-particles at x and y. Motion of e-particles at x and y induce motion of Credit transaction CL(t,z) alike to motion of continuous media and we outline parallels between Credit transaction CL(t,z) and fluids. To define motion or velocity υ(t,x,y) of Credit transaction let's introduce impulse pij of Credit transaction cl1,2(t,x,y) for couple of e-particles i and j at x and y that are involved into on 2n-dimensional e-space z=(x,y) respectively as Relations (1.4; 2.1-2.7) define Credit transactions CL(t,z) and impulse P(t,z) on 2n-dimensional espace z=(x,y). Integral of Credits transactions CL(t,x,y) by variable y over e-space R n defines rate of change all of Credits C(t,x) from point x at moment t.
Integral (3.1) also defines rate of change of all Loans L(t,y) received at point y. Integral of CL(t,x,y) by variables x and y on e-space describes rate of change of total Credits C(t) provided in economy and total Loans L(t) received in economy at time t: Relations (3.1; 3.2) show that Credits transactions CL(t,x,y) define evolution of Credits C(t,x) provided from point x and total Credits C(t) provided in economy at moment t and their mutual variables -Loans L(t,y) received at point y and total Loans L(t) received in macroeconomics at moment t.
As usual risk ratings are related with economic agents or their securities. Now let's introduce notion of mean risk for macroeconomic or financial variable. As example let's take macro Credits and Loans. Let's assume that e-particle 1 (Bank 1) with risk coordinate x at moment t issues Credits C1(t,x) and e-particle 2 (Bank 2) with risk coordinate y at moment t issues Credits C2(t,y). Coordinate x and y define risk ratings of Bank1 (e-particle1) and Bank 2 (e-particle 2). Let's state a question: What is the risk rating for group of two Banks? Group of two Banks issue Credits C1(t,x)+ C2(t,y). Let's define mean Credits risk XC1,2(t) for two Banks as: 1,2 ( ) = 1 ( , )+ 2 ( , ) 1 ( , )+ 2 ( , ) 1,2 ( )( 1 ( , ) + 2 ( , )) = 1 ( , ) + 2 ( , ) (3.3) Relations (3.3) define mean risk of Credits as average of risk coordinates of agents weighted by value of Credits they issue at time t. Similar relations for Loans L1(t,x) and L2(t,y) received by eparticles 1 and 2 at points x and y define Loans mean risk XL1,2(t) as: and mean Loan risk coordinates XL(t) as Mean Credits risk XC(t) equals mean risks of total Credits C(t) issued in economy. It is alike to coordinates XC(t) of center of total "mass" of Credits C(t) in economy with Credits mass density C(t,x). Let's remind that C(t,x) -amount of Credits provided from all agents at point x. Mean Loans risk XL(t) defines mean risk coordinates of total Loans L(t) received in economy.
Nevertheless that due to (3.2) total Credits C(t) equal total Loans L(t) mean Credits risk XC(t) is not equal to mean Loans risk XL(t). Different economic variables -Investment I(t,x), Assets A(t,x) and etc. define different values of their mean risks. Let's remind that all variables are determined by corresponding economic transactions due to relations (3.1). Credits transactions mean risk of CL(t,z=(x,y)) define mean risk of mutual variables for z=(x,y) as: Relations (3.5) show that macro transactions like Credits transactions CL(t,x,y) determine evolution of Credits mean risks XC(t) and Loans mean risks XL(t). The same statement is correct for mean risks determined by other macro transactions.
Why we attract attention to definition of mean risks of macro variables? We propose that evolutions of mean risks for different macro variables impact the business cycles of these variables. Let's take Credits C(t) as example. Mean Credits risk XC(t) is not a constant. It changes due to change of coordinates x and amount of Credits provided by e-particles. Growth of risks of eparticles can increase and decline of risks can reduce mean Credits risk XC(t). E-particles fill economic domain (1.1). Risk ratings of e-particles on economic domain (1.1) are bounded by minimum or most secure and maximum or most risky grades. Thus mean Credits risk XC(t) as well as mean risks of any other macro variable can't grow up or diminish steadily along each risk axes as their values are bounded on economic domain (1.1). Hence value of mean Credits risk XC(t) should oscillate along risk axes and these fluctuations of mean risk XC(t) can be very complex.
We propose that business cycles and fluctuations of mean risks of macro variables are highly associated. Growth of mean Credits risk XC(t) correspond with growth of total Credits C(t) provided in economy and decline of Credits mean risk correspond with total Credits contraction. Reasons for mean risk change can be exogenous or endogenous. Mean risk change can be induced by technology shocks, political or regulatory decisions and etc. Reasons can be different but outcome should be the same -business cycles are governed by change of mean risks. Relations between mean Credits risk XC(t) and value of total Credits C(t) are much more complex but we repeat main statement: business cycles and fluctuations of mean risks are linked very tightly. To avoid excess complexity we don't derive equations on mean risks here, but refer to (Olkhov, 2017d).
As we show in (3.5) Credits transaction CL(t,x,y) determine mean Credits XC(t) and Loans XL(t) risks. Below in Sec. 3, Sec.4 and in Appendix we introduce economic equations that describe model dynamics of Credits transaction CL(t,x,y) on e-space (5.1-5.5). Starting with these equations we derive the system of ODE (A.4; A.8.4-7; A.10.1-10.2) that describe the business cycles of macro Credits C(t) provided in economy and macro Loans L(t) received in economy. Due to (3.1) total value of Credits MC(t,x) provided from point x up to moment t equal:

Economic equations on macro transactions
Credit transactions between points x and y on e-space determine evolution of macro variables (3.1 -3.11) (Olkhov, 2017b;2017c). Electronic copy available at: https://ssrn.com/abstract=3417717 Thus left side of (4. P(t,z) in the left side and action of other factors that can induce these changes in the right side.

2) describes change of transaction impulses P(t,z)=(Px(t,z), Py(t,z)) (2.3-2.7) due to change in time ∂P/∂t and due to flux through surface of unit volume that equals divergence ∇ • ( ). Right hand side Q2 describes action of other factors on evolution of transaction impulses P(t,z). Economic equations (4.1; 4.2) present a balance relations between changes of transactions CL(t,z) and their impulses
To describe a particular economic model via equations (4.1; 4.2) let's determine direct form of right hand side Q1 and Q2. Macro transactions CL(t,z) and their impulses P(t,z) can depend on other transactions and on other economic factors like expectations, for example. In this paper we present the business cycle model in the approximation that takes into account interactions between different transactions only and neglects impact macroeconomic variables or expectations and other economic factors. We shall describe impact of expectations in forthcoming publications.
Here we propose that all extensive macro variables are determined by macro transactions or depend on variables that are described by macro transactions.
Equations (4.1; 4.2) allow describe evolution of transactions under action of Q1 and Q2 for two economic approximations. First approximation describes transactions and their mutual extensive variables under given exogenous impact determined by Q1 and Q2. In other words one studies evolution of transactions under given action of known exogenous factors Q1 and Q2. The second approximation permits describe self-consistent evolution of transactions under their mutual interaction due to equations (4.1; 4.2). Real economic and financial transactions depend on numerous factors and that makes description extremely complex. We propose to start with the simplest case that models mutual interactions between two transactions. For this case left side of (4.1; 4.2) describe transaction 1 and factors Q1 and Q2 are determined by transaction 2 and vice versa. Such approximation gives simple self-consistent model of mutual evolution of two interacting transactions and allows describe the business cycle model related to fluctuations of macro variables determined by these transactions. Below we study self-consistent model that describe mutual interaction between Credits CL(t,z) and Loan-Repayment LR(t,z) transactions. As consequences we describe the business cycle time fluctuations of macro Credits C(t) and macro Loans L(t).
Let's study simplest case and assume that Credits transactions CL (t,z) in the left side of (4.1;4.2) depend on Q1 and Q2 that determined by Loan-Repayment LR(t,z) transactions. Loan-Repayment LR(t,z) transactions describe payout on Credits by Borrowers from point y to Creditors at point x. Let's describe evolution of Loan-Repayment LR(t,z) transactions by left side of equations similar to (4.1;4.2) with Q1 and Q2 determined by Credits transactions CL(t,z). We propose that Credits from x to y and Loan-Repayments from y to x are made at same time t and vice versa. Such assumptions simplify mutual dependence between Credits transactions CL(t,z) and Loan-Repayment LR(t,z) and allow describe the business cycle fluctuations of macro Credits C(t) issued at time t.

How macro transactions describe the business cycles
In (Olkhov, 2017d-e) we proposed that agents perform only local transactions with agents at same point x. Such simplifications describe interactions between macro variables at point x by local operators. In this paper we model transactions that can occur between agents at arbitrary points x and y. Such transactions describe non-local economic and financial "action-at-a-distance" between e-particles (agents) at points x and y on e-space R n . Below we describe the business cycles determined by non-local Credit CL(t,z) and Loan-Repayment LR(t,z) transactions. Let's assume that CL(t,z) at point z=(x,y) on e-space R 2n depend on Loan-Repayment LR(t,z) transactions and

= • ( , ) = ( • ( , ) + • ( , ))
Loan-Repayment impulse D(t,z) and velocity u(t,z) are determined similar to (2.1-2.7). Let's assume that same relations define factor Q12 for equation ( in risky direction z and that can induce growth of Credits CL(t,z) to risky points. As well negative value of • ( , ) models Loan-Repayment flows from risky to secure domain and that can decrease Credits CL(t,z) as Creditors can prefer more secure Borrowers. This model simplifies Credit modelling as it neglect time gaps between providing Credits from x to y and Loan-Repayment received from Borrowers at y to Creditors at x and neglect other factors that can impact on Credits allocation. To determine Q21 factor for (4.2) on Credit impulses P(t,z) let's assume that Q21 is a linear operator and in a matrix form takes form: Let's assume that Q22 factor that define equations (4.2) on Loan-Repayment impulses L(t,z) is similar linear operator: ) and equations (4.2) for impulses P(t,z) and L(t,z) take form: Equations (5.4-5.5) describe simple linear mutual dependence between transaction impulses P(t,z) and D(t,z). Economic meaning of equations (5.4; 5.5) can be explained as follows. Let's mention that integral of each component of impulses P(t,z) or its components Pxi(t,z) and Pyi(t,z) along axes xi or yi over dz define total macro impulses P(t) and its components Pxi(t) or Pyi(t) along risk axis xi or yi and due to (2.7; A.6.3.1; A.6.3.2): . Business cycle fluctuations (6.1; 6.2) may happen about exponential growth trend exp(γt) (A.10.1-2) and we take coefficient γ =max(γx, γy). Thus γ describes maximum growth trend induced by (A.8.6-7; A.9.1-2; A.10.1-2). Factors (A.8.8-9) are proportional to product of total Credits C(t) and transactions velocity squared υ 2 (t) and we call them as Credits "energy" because they looks like kinetic energy of a body with mass C(t) and velocity squared υ 2 (t). However meaning of Credits "energy" have nothing common with energy in physics. Macro Credits MC(t) during time term [0,t] are described by (6.2). If the initial value C(0) is non zero then macro Credits MC(t) has linear and exponential growth trend and oscillations with same frequencies ω and ν about these trends. Solutions (6.1) for Credits transactions C(t) and for Loan-Repayment transactions R(t) present simplest form of Credit cycles under single risk and simple interactions between two macro transactions (Appendix). Action of several risks makes the Credit cycles more complex (A.11). If one neglect growth trend then Credit cycles C(t) under action of n risks can take form (A.11): Electronic copy available at: https://ssrn.com/abstract=3417717 Relations (6.3) with frequencies ωi reflect oscillations of Credit impulses P(t) along axes xi, and frequencies νi along axes yi, i=1,..n on 2n dimensional e-space (x,y) (Appendix)

Conclusions
Current business cycle models (Kiyotaki, 2011) are based on general equilibrium theory. "The economy is in general equilibrium when prices have fully adjusted so that supply equals demand in all markets." (Starr, 2011). We assume that economic processes are too diverse, complex and changeable to be described only by general equilibrium theory. Occam's razor (Baker, 2007) principle states that the less initial assumptions are made in the model -the better. Thus it is reasonable develop economic and business cycle theory on base of econometric data only and without ad hoc assumptions of general equilibrium. It is obvious that any economy is an open system and for sure any economic model should depend on numerous exogenous phenomena and factors. Meanwhile it is important to understand and describe internal, endogenous economic properties and relations that govern macroeconomic evolution and development. In this paper we study and model endogenous economic processes that induce and manage macroeconomic business cycles. We propose that econometrics provides sufficient data for risk assessments of all agents of entire economics and suggest use agent's ratings x as their coordinates. All extensive economic or financial variables are defined as sum of corresponding variables of agents near point x. Economic and financial transactions between agents are the only tools for change of agents variables. We aggregate similar transactions between agents at x and y and describe evolution of macro transactions by economic equations (4.1-4.2). Motion of transactions can be treated alike to motion of fluids and is determined by average collective velocity of agents. For example motion of Credit transactions is determined by collective risk velocity of Creditors at x and Borrowers at y (2.6; 2.7). Macro impulses and velocities (5.6-5.7) define motion of Creditors (3.2; 3.11) along risk axis x and Borrowers along y. Collective motions of Creditors and Borrowers occur on economic domain (1.1) that is bounded by minimum and maximum risk grades. Hence macro motion (5.6-5.7) of Creditors and Borrowers from safer to risky direction should change by opposite motion from risky to safer area. We show that oscillations of Creditors and Borrowers motion on economic domain from safer to risky direction and back induce macroeconomic Credit cycles. The same relations govern the business cycles of Investment and Consumption, Demand and Supply and etc. Motions of the same economic agents generate the business cycles of different macroeconomic variables and that explain coherence and interactions between cycles of different macroeconomic variables. Economic evolution under action of several risks and interactions between numerous economic and financial transactions makes description of the business cycles on multi-dimensional economic space rather complex problem. This paper describes the business cycles in the approximation that takes into account interactions between different transactions only and neglects action of expectations. Even such simplification uncovers rich and complex relations between transactions that govern the business cycles. We'll describe impact of expectations on the business cycles in forthcoming paper.
Econometric assessments of risk ratings of economic agents use corporate financial accounting and reporting. Thus unification of accounting methodologies becomes important as for macroeconomic forecasting as for corporate reporting itself. Unified corporate reporting establishes ground for correct risk assessments and macroeconomic forecasting on economic space. It helps define corporate risk ratings and risk motion and is important for corporate management and shareholders as tool for assessment of corporate risk trajectory. Assessments of Electronic copy available at: https://ssrn.com/abstract=3417717 To define equations on Pzxi(t), Pzyi(t), Dzxi(t), Dzyi(t) let's use equations (A.6.1 ; A.6.2). Let's multiply equations (A.6.1) by xi and take integral by dxdy Second integral equals zero due to same reasons as (A.2.1). Let's take first integral by parts: First integral in the right side equals zero and we obtain: Let's denote as Electronic copy available at: https://ssrn.com/abstract=3417717 Factors ECxi(t) and ECyi(t) (A.8.2-8.3) are components of EC(t) along each axes xi and yi. Factors ( ) = ( ) 2 ( ) (A.8.8 -8.9) are alike to kinetic "energy" of particle with mass C(t) and velocity squared υ 2 (t) but these similarities have no further analogies. Equations on ECxi(t,z) and ECyi(t,z) take form similar to (4.1): Economic meaning of (A.9.1-A.9.7) is as follows: "energies" ECxi(t), ECyi(t), ERxi(t), ERyi(t) grow up or decay in time by exponent exp(γxi t) and exp(γyi t) that can be different for each risk axis i=1,..n. Here γxi define exponential growth or decay in time of ECxi(t) induced by motion of Creditors along axes xi and γyi and same time describe exponential growth or decrease in time of ECyi(t) induced by motion of Borrowers along axes yi. The same valid for ERxi(t), ERyi(t) respectively. Let's underline that due to (A.8.8) velocity squared υ 2 (t) is not equals to square of velocity υ(t)=(υx(t), υy(t)) determined by ( Electronic copy available at: https://ssrn.com/abstract=3417717