Key rate and loan market in Russia

The paper proposes a mechanism for the impact of changes in the key rate on the volume of newly issued loans. The volume depends on the price (interest rates on loans), and the price depends on the key rate and the actual consumption of loans in the previous period (generalized cobweb cycle). The model was estimated by a Kalman filter, adequacy was confirmed by simulation. It is possible to forecast the average rate on loans for a month in advance according to the information published by the Central Bank of the Russian Federation (CB). By playing various scenarios for changing the key rate, it was found that in quiet periods of economic development, the usual laws of supply and demand operate in the loan market and by raising the key rate, you can reduce inflation. In the turbulent (overheated) state of the economy, an increase in the key rate can, on the contrary, provoke an increase in the issuance of loans and unconventional manipulations with the key rate are required.


INTRODUCTION
The picture of the loan market in Russia. As follows from the review of credit institutions in [1], the liabilities of banks on ruble deposits to non-financial organizations and the households (mainly arising from the issuance of loans and borrowings) amount to ~ 46 trillion rubles on 01.01.21. At the same time, new loans for ~ 4 trillion ₽ are issued monthly (the old ones are being repaid in the meantime). The increase in the issuance of new loans according to various estimates (see [2], [3]) is from 160 to 300 billion ₽ on average per month. This growth in the money supply is causing inflation, which is containing mainly by regulating the key rate at which banks borrow from the CB.
The objective of this work: modeling the mechanism of the impact of changes in the key rate on the volume of loans issued, playing out possible scenarios. Basic assumptions: consumption of loans depends on their prices, loans are supplied to the market at a price determined, mainly, by the key rate of the CB and real demand. Let us introduce the values: t p -the true weighted average interest rate on loans, t l -the true volume of loans, t pk -the key rate of the CB [5]. The values t pf , t p , t pk will be measured in units 1 Pk =7.75% -the key rate at the base moment t = 1 (02/01/2019), t lf and t l in units 1 Lf = 3.327 trillion ₽ -the volume at the same base moment.

MODEL AND DATA
We will assume that the dynamics of the loan market is determined by the dependence of the demanded volume (consumption) of loans on the interest rate and the price (interest rate) of the offer of loans for the next month depends on the real volume of loans consumption in the current month This leads to the well-known cobweb "pork supply" cycle, see Substitute (1) into (2), introduce the state vector of the system Then the model of the system in the state space has a form where the operator of evolution , and observation

KALMAN FILTER AND INTERPRETATION OF CALCULATION RESULTS
We choose the estimate of the initial state at time Tl-1 1| 1Tl Tl , and the covariance matrix of this estimate as 2 0 0 0 . We take the covariance matrix of the observation error vector t v equal to was carried out by minimizing the mean square weighted sum of errors in predicting rate and volume.
Here  With these parameters, demand and supply lines (between which a cobweb cycle develops) depicted in Fig. 2 for the moment t = 22 (01.11.2020). 2. From Tl = 22 (11.2020) to Tu = 31 (08.2021). During this period (unfavorable, turbulent), the volume of loans issued grew, the key rate was raised, loan rates increased. Large volumes of loans issued in July and August 2021 can be explained by the rush expectations of an increase in rates in the near future after an increase in the key rate (see Fig. 3).
The minimum RMSE is 0.235 (RMSERW = 0.452). to +1.37%. In other words, with an increase in interest rates, the volume of new loans will increase! 2  = 0.374. that is, the potential rate on loans at very small total volume tends to the key rate + 2.8%.
3  = -0.084 gives the average elasticity coefficient of the new interest rate on the volume of loans issued (relative) equal to -0.19%. This means weak (inverse!) dependence of the supply price on demand and its rigid binding to the key rate of the CB (with the average actual volume of loans, the offer price is equal to the key rate + 1. 66%).
With these parameters, demand and supply lines practically swap places in Fig. 2, but the convergence of the cobweb cycle (now counterclockwise) is preserved.

SIMULATION AND SCENARIOS
To verify the adequacy of the system model, it is necessary to compare its functioning with the estimated parameters (simulation) with the behavior of the real system. To do this, you can use the equations of dynamics (5) and (6) without noise elements [3]. Or, equivalently, use these parameters to build the transfer matrix of the system and, after the reverse z-transformation, get the output of the system [6]. We start with the same initial vector of state, controls -real (4) . In the same way, it is possible, by setting different types of control outside the sample, to play the system development scenarios. We will do this separately for the above-mentioned time periods.
For a quiet period, see Fig. 4. Here, a scenario with a 0.5% decrease in the key rate per month leads to an expected decrease in rates and an increase in loans.
For a turbulent period, see Fig. 5. Here there is an abnormal increase in the volume of new loans with an increase in rates, so a careless increase in the key rate can provoke a sharp increase in the money supply (at least in the short term) and a decrease in the key rate, on the contrary, would calm the market and reduce the volumes! Interestingly, if periodically raise the key rate by 1%, and, in a month, lower by 1%, then the interest rates on loans will fluctuate, but the volumes will be almost constant (both in calm and turbulent periods).
In general, a good coincidence of solid and dotted curves (excluding random peaks and dips) in Fig. 4, 5 allows us to conclude about the adequacy of the proposed model to the real system.