In recent years, central banks have tested the limits of lowering monetary policy rates to expand economic activity. The ‘zero lower bound’ was challenged as many central banks went below zero, and the question arose of whether there was actually an ‘effective lower bound’. The important contribution of Brunnermeier and Koby (2018), henceforth ‘BK’, extends this debate and explores theoretically the possible existence of a ’reversal interest rate’ below which further reduction in policy rates are in fact counterproductive. The term ‘reversal rate’ has now become part of the jargon used by both central bankers and analysts to discuss the monetary policy stance.

In BK a reversal rate exists as lower rates have a negative effect on bank profitability which erodes the banks’ equity capital, and in the presence of a capital constraint leads to lower lending. In a recent paper (Repullo 2020), I present a critical review of this mechanism.

It should be noted that in BK’s model, a reduction in the policy rate has two opposite effects on bank capital. On the one hand, there is a positive effect due to the increase in the value of long-term mark-to-market assets. On the other hand, there is a negative effect on profitability. The negative profitability effect is key for the existence of a reversal rate, while the positive revaluation effect works in the opposite direction. Since the revaluation effect weakens BK's result, and in their model banks do not have such assets in their balance sheet, I focus my review on the profitability channel.

## Brunnermeier and Koby’s model of the reversal rate

BK’s model features a local monopoly bank which is the single provider of loans and deposits in a local market, but is a perfect competitor in an economy-wide securities markets. The bank has a given amount of equity capital and faces an upward sloping supply of deposits and a downward sloping demand for loans. The bank can also invest in debt securities whose interest rate is taken to be the monetary policy rate set by the central bank. BK assume that the bank's maximisation problem is subject to two financial frictions: a capital constraint and a liquidity constraint.

The liquidity constraint requires the bank to invest a fraction of their deposits in (liquid) debt securities. This constraint plays a key role for their results. In particular, a binding liquidity constraint makes lending equal to the sum of a proportion of its deposits (those not invested in debt securities) and the exogenous capital. This means that what happens to lending following a reduction in the policy rate is driven by what happens to deposit taking. I show that if the liquidity constraint is binding, lower policy rates lead to lower deposits and, hence, lower lending. However, this is not the narrative of the reversal rate in BK, which is linked to the effect of lower rates on bank profitability. For this reason, I focus my review on the effect of the policy rate on lending in the presence of only a capital constraint.

In B, the capital constraint requires the bank to back a fraction of its lending with equity capital. They argue that this constraint captures “economic and regulatory factors”. However, their constraint features the future value of the bank's capital, not the current value as it should be if it were to capture a regulatory capital requirement. Moreover, their constraint is not implied by a standard forward-looking collateral constraint.

At any rate, I review the possible existence of a reversal rate in the presence of BK's capital constraint, showing the following results. First, if the capital constraint is not binding, lower policy rates always lead to higher lending. Second, if the capital constraint is binding, a reversal rate exists if and only if the bank is a net investor in debt securities, a condition that is typically satisfied by high deposit banks (those with more deposits than loans). Third, there is no single reversal rate, since whenever it exists it depends on bank-specific characteristics, in particular their relative advantage in raising deposits versus granting loans.

These results are very much in line with the empirical results in Heider et al. (2019), who state that “[t]he introduction of negative policy rates by the European Central Bank in mid-2014 leads to (...) less lending by euro-area banks with a greater reliance on deposit funding.” They are also consistent with the results in Eggertsson et al. (2019) showing that Swedish banks that rely more heavily on deposit financing experienced lower credit growth after the policy rate became negative.

It should be noted that BK's specification of the capital constraint in terms of the future value of the bank's capital has a simple justification: no reversal would exist if lending is constrained by the current (exogenous) value of the bank's capital, since the constraint would just set an upper bound on lending. However, this misses their intuition that bank profitability should matter for lending, because if shareholders do not get an adequate return from their investment they might not want to contribute their capital to the bank or shift it to other uses.

## An alternative model

To capture this intuition, I then consider an alternative model in which the bank's equity capital is not exogenous, but it is endogenously provided by shareholders that demand a return on their investment that incorporates a positive excess cost of capital. In this model, there is a profitability constraint that requires the future value of the bank's capital to be greater than or equal to the opportunity cost of the funds provided by shareholders. The future value of the bank's capital is driven by two components: profits from lending, equal to the spread between the loan rate and the policy rate multiplied by the total amount of loans, and profits from deposit taking, equal to the spread between the policy rate and the deposit rate multiplied by the total amount of deposits. While the former are always positive, the latter can be negative when (i) the policy rate becomes negative, and (ii) there is a zero lower bound on deposit rates.

The question is: Is it possible to get a reversal rate when the losses from deposit taking become very large? Unfortunately, the answer is no. In this setup, the profitability constraint does not bring about a reversal rate, except in the extreme form of banks closing down – or stopping to take deposits, as in Ulate-Campos (2019) – when shareholders do not get the required return from their investment. The intuition for this result is straightforward. Lower policy rates always increase the bank's profits from lending, since they reduce the weighted average cost of deposits and capital. For the same reason, with a downward sloping demand for loans, they increase bank lending. At some point, the losses from deposit taking may exceed the profits from lending, but until we get to that point the bank will continue to expand its lending, as this maximizes its profits.

## Conclusion

The conclusion that follows from this analysis is that policy makers should not assume that lower negative rates will, at some point, be contractionary for lending. Negative rates may bring about some problems, in the banking system and the broader economy, but the expectation should be that they will increase lending.

I would like to end with a few remarks. First, I have used a setup in which the bank is a monopolist in lending and deposit taking in a local market but has access to a competitive debt market. The local monopoly assumption simplifies the analysis, but all the results can be obtained in a setup in which several banks compete (for example à la Cournot) in a local market and have access to a competitive debt market.

Second, in addressing the effects of low profitability on bank lending it is natural to think about a process that takes time, with shareholders gradually reducing their capital until the capital constraint becomes binding. The alternative model tries to capture this process in a reduced form manner by assuming that the initial bank capital is endogenously provided by bank shareholders. Still, dynamic models of banking that address these issues would be most welcome.

Finally, it is important to stress that partial equilibrium models like the ones in my paper have obvious limitations in capturing general equilibrium effects of monetary policy actions. However, they can be useful as building blocks for macroeconomic models with a solid microfoundation of the banking system.

## References

Brunnermeier, M and Y Koby (2018), “The Reversal Interest Rate,” NBER Working Paper No. 25406.

Eggertsson, G, R Juelsrud, L Summers and E Getz Wold (2019), “Negative Nominal Interest Rates and the Bank Lending Channel,” NBER Working Paper No. 25416.

Heider, F, F Saidi, and G Schepens (2019), “Life below Zero: Bank Lending under Negative Policy Rates,” *Review of Financial Studies* 32: 3728-3761.

Repullo, R (2020), “The Reversal Interest Rate: A Critical Review,” CEPR Discussion Paper No. 15367.

Ulate-Campos, M (2019), “Going Negative at the Zero Lower Bound: The Effects of Negative Nominal Interest Rates,” Federal Reserve Bank of San Francisco Working Paper 2019-21.