A simple model of interbank trading with tiered remuneration
11 A simple model of interbank trading with tiered remuneration
Toshifumi Nakamura a Bank of Japan b Abstract:
A negative interest rate policy is often accompanied by tiered remuneration, which allows for exemption from negative rates. This study proposes a basic model of interest rates formed in the interbank market with a tiering system. The results predicted by the model largely mirror actual market developments in late 2019, when the European Central Bank introduced, and the Switzerland National Bank modified, the tiering system.
JEL codes:
E43, E52, E58
Keywords: negative interest rate, interbank market, tiered remuneration a The views expressed in this paper are those of the author and should not be interpreted as those of the Bank of Japan or other institutions the author has worked for. b International Department; email: [email protected] . The author worked for ECB, DG-Market operation MML, when he wrote the first draft of this paper.
1. Introduction
Central banks that have imposed negative interest rates have recently either introduced a tiering system or modified the existing one, such that it exempts the burden of negative interest rate up to a predefined limit. Danmarks Nationalbank, the central bank of Denmark, has been using negative interest rates with a tiering system for more than five years and reduced the policy rate by 10 bps to β0.75% in September 2019. The European Central Bank (ECB) introduced a two-tier system from October 2019, following a 10-bps rate cut to β0.50% a month before. In November 2019, the Swiss National Bank modified its tiering system by increasing the exemption threshold for banks. This tiering system may introduce a few knock-on effects in the market. First, the system mitigates the cost burden of a negative interest rate for financial institutions. Second, it helps increase the interbank market trading activity, which is a result of the arbitrage opportunities arising from the differences in remuneration rates. For example, Bank A, which has exceeded the allocated exemption amount, may want to lend cash in the market if the market rate is higher than the negative rates charged by the central bank; at the same time, bank B, which has not reached the upper limit of the exemption amount will borrow cash from the market provided that the market rate is below the zero rate. This arbitrage activity may, however, lead to a negative effect by raising the market rate to a level higher than in a market without a tiering system. In other words, the tiering system weakens the transmission of the negative remuneration rate to the interbank market. This study formulates a basic model of an interbank market with tiering remuneration and aims to describe these three effects, that is, burden mitigation, trading volume, and rate formation. The simplicity of the model allows it to be tested with actual market data, but this simplicity also means that the model assumes high liquidity across all banks and negligible payment shocks. If the model were to include shocks, such as central bank lending mentioned in Pooleβs (1968) framework, the market rate would become the risk-weighted average of the central bankβs lending and deposit facility rate (Bindseil, 2004). The market rate in the model discussed in this paper will, however, be linked to the exemption size.
Other authors have also conducted research on this subject in recent years. Boutros and Witmer (2017) introduced a theoretical model with a tiering remuneration system but without testing actual market data. A year later, Afonso et al. (2018) successfully tested a model using market data, comparing their model of the United States Federal Funds market with individual bank-level data.
2. Model and Equilibrium
Setup in the morning
Bank π has a profile of excess liquidity π₯ π and cost regarding trade transactions (except for the trading rate) π π % . We assume uniform and mutually independent distributions of π₯ π ~(0,1) and π π ~(0,1) , for simplicity. Tiering remuneration in the afternoon
Bank π follows the following remuneration schedule: is applied to the reserve position in the afternoon π₯ aπ up to the individual limit of exemption π’ π and β1% is applied to the exceeding amount. We assume π’ π is the same for all institutions, i.e. π’ π = π’ . The share of the aggregated amount of the upper limit of exemption relative to the aggregated excess liquidity, Exemption Share (E), is calculated as: πΈ = β« π’ ππ dπ β« π₯ ππ dπβ = 2π’ (1) Trading decision during the day
Bank π decides to either lend π π , borrow π π , or do nothing. The decision is made by comparing the market rate βπ% , associated cost π π % , remuneration rate for the exemption amount, and β1% for the exceeding amount. π₯ aπ = { π₯ π β π π , π₯ π > π’ π and β π β π π β (β1) > 0 π₯ π + π π , π’ π > π₯ π and 0 β (βπ) β π π > 0π₯ π , otherwise (2) The concept modeled in (2) is that bank π ( π₯ π > π’ π ) will lend money if the income from lending βπ minus cost π π minus the opportunity cost to receive remuneration β1 is positive. Similarly, bank π ( π’ π > π₯ π ) will borrow money if the income from borrowing π minus cost π π is positive. Equilibrium market rate
Market rate βπ β should be decided such that it balances the aggregated lending and borrowing needs. This amount is known as the trading volume π . π = β« π ππ dπ = β« π ππ dπ (3) π = β« max(π₯ π β π’ π , 0) β π {βr β βc i +1>0}i di = β« max(π’ π β π₯ π , 0) β π {r β βc i >0}i di (4) π = 12 (1 β π’) (1 β π β ) = 12 π’ π β (5) Using equation (5) , we have βπ β = β(1 β π’) π’ + (1 β π’) (6) π = 12 π’ (1 β π’) π’ + (1 β π’) (7) Further, there are some banks that have reserves exceeding π’ π but avoid lending because of associated costs. Then the share of the aggregated amounts to which the negative rate is to be applied, relative to the aggregated excess liquidity, Negative Remuneration Share ( π ) is calculated as: π = β« max(π₯ π β π’, 0) β π {βπ β βπ π +1β€0}π dπ β« π₯ ππ dπβ = (1 β π’) π’ + (1 β π’) (8)
3. Testing with actual data
Figure 1 shows a plot of the average data for the 3 rd and 4 th quarters of 2019 of πΈ plotted against the observed market rate for four currencies with tiering systems. The observed rate is the overnight or next day rate standardized by dividing it by the degree of negative remuneration. The bold line shows the theoretical βπ β using (6) . Key changes in the negative interest rate policy and tiering, as mentioned in Section 1, were made between the 3 rd and 4 th quarters of 2019. Consequently, the relative position of the two points for each currency represent the impact of the changes in the market rate. The effect is pronounced for the Euro (EUR) and Swiss franc (CHF), as we exclude the term before these changes in the calculation of the average for the 4 th quarter. Sources: NASDAQ, Six Swiss Exchange, Danmarks Nationalbank, BoJ, SNB Figure 1: Relationship between πΈ and observed market rate Despite the oversimplification, the actual data do not differ greatly from the results of the model. The slope of CHF is steeper than the EUR and this is precisely the modelβs implication: as πΈ approaches 1.0, the sensitivity of the market rate to πΈ increases. Some deviation from the theoretical rate can be explained by the characteristics of the selected short-term rates. For example, a collateralized rate SARON (Swiss Average Rate Overnight) is used -1.20 -1.00 -0.80 -0.60 -0.40 -0.20 βπ β π’ πΈ JPY
DKKEUR CHF for CHF while an unsecured rate Denmark T/N (Tomorrow/Next) is used for the Danish krone (DKK). Therefore, the differences in the model can be interpreted as risk premium and collateral demand. In the case of EUR, the overnight rate selected here is the benchmark rate β¬STR. This is an unsecured rate but is composed of the deposit rate offered by the major banks. Hence, it tends to be lower than the negative remuneration rate set by the central bank. If we can use the EONIA, the lending rate offered by the major banks, the position of the points are above the model. However, the calculation method of the EONIA changed in October 2019 and only reflects the subsequent development of β¬STR. Another interesting observation is that of a relatively vertical line for the Japanese yen (JPY). The rate used here is the unsecured rate TONAR, and the observed rate is volatile around β0.5 . One of the reasons for this may be that πΈ is as high as 0.99; this situation may have risen in order to preserve the market function. Equation (7) explains how the market trading activity π is maximized at around πΈ =1.0 ; this is further proven by the directional changes in 2019 for both EUR and CHF currencies as they highlight high sensitivity of the market rate to E. Equation (8) predicts that the negative remuneration share for JPY is approximately ; in reality, it is close to 5%. This may be linked to the reserve neutral exemption scheme of Japan. The exemption limit in Japan is based on the actual reserve size in 2015 and the amount of the funding from the central bank, unlike the cases of the Euro and CHF (in the eurozone and Switzerland respectively), where the exemption amount is defined by the multiple of the reserve requirements. Therefore, the reserve position relative to the exemption is distributed more evenly such as π₯ π ~(0.3,0.7) compared to the default setup of the model π₯ π ~(0,1) . As a result, the slope of the curve modeled in (7) becomes roughly 2.5 times the original sensitivity.
4. Conclusions
This study proposes a simplistic model operating in an interbank market with tiering remuneration. One of the limitations of the model, rather an effect of the oversimplification, is that the model does not consider the actual demand of cash in the interbank market. The original purpose of the interbank trade is to adjust the reserve position by considering payment shocks, fulfilling reserve requirements, and real funding needs for investing activity. In the case of Denmark, where the ratio of the central bankβs balance sheet over GDP is relatively small compared to other jurisdictions, the actual requirement of the funding is not negligible in the formation of the market rate. In the case of Switzerland, where the negative remuneration is calculated daily, the payment shock of each day plays an important role. This study has given priority to simplicity to help in understanding the mechanism of the tiering system and its impact in the interbank market. The prediction of actual market data may not be perfect but it is acceptable in some cases. The author believes that the model proposed in this study offers a good balance of simplicity and efficiency within a practical approach.
References
Afonso, G., Armenter, R., Lester, B., 2019. A model of the federal funds market: yesterday, today, and tomorrow.
Review of Economic Dynamics , 33, 177-204. Bindseil, U., 2004. The operational target of monetary policy and the rise and fall of reserve position doctrine . European Central Bank Working Paper, No. 372 . Boutros, M., Witmer, J., 2017. Monetary policy implementation in a negative rate environment.
Bank of Canada Staff Working Paper, No. 25 . Poole, W., 1968. Commercial bank reserve management in a stochastic model: Implications for monetary policy.
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