The Reversal Interest Rate: Abadi, Brunnermeier & Koby (2023)

What this is. The paper’s core results, the model it builds on (an infinite-horizon New Keynesian economy with imperfectly competitive banks and net-worth-constrained lending), and the key equations behind the reversal interest rate mechanism: enough to know what it found and how, without reading all 37 pages. To replicate or extend it, read the full source at the original.

TL;DR

The paper introduces and characterizes the reversal interest rate: the rate below which further monetary easing becomes contractionary for bank lending. The mechanism runs through two competing channels. An interest rate cut initially raises banks’ capital gains on long-term bond holdings (the capital gains channel), boosting net worth and expanding lending. Simultaneously, lower rates compress banks’ net interest income (the NII channel), reducing net worth over time. When banks face a binding capital constraint on lending, the NII effect eventually dominates, causing further cuts to shrink, rather than expand, credit supply. In a New Keynesian model calibrated to the euro area, the reversal rate is approximately -0.9 percent for aggregate investment and -1.4 percent for bank lending on impact. Extending rate cuts for prolonged periods (“low-for-long” policies) makes the NII losses cumulate, so policies that initially stimulate lending can eventually turn contractionary. The model also dampens the forward guidance puzzle: in the calibrated economy, an 8-quarter promise to hold rates at -1 percent generates investment and output peaks roughly half as large as in a standard New Keynesian model without bank capital constraints, because agents anticipate the long-run decline in lending. Contemporaneous models by Eggertsson et al. (2019) and Ulate (2021) study related rate-reversal mechanisms through deposit rate floors and bank profitability, but differ in channel and quantitative implications. The role of intermediary net worth in credit supply builds on Brunnermeier and Sannikov (2014); the bank optimization block follows the monopolistic banking firm of Klein (1971).

Core results

Magnitudes are as reported; \* is not applicable (no hypothesis tests in this calibration paper). Locators point to the source PDF.

#	Result	Locator	Magnitude
R1	Bank lending reversal rate on impact is approximately -1.4 percent in the euro area calibration	Figure 2 left, p. 2107; §III.C, p. 2106	Below -1.4%, a further 10bp Taylor rule cut reduces bank lending at t=0; above -1.4%, the same cut expands lending
R2	Aggregate investment reversal rate on impact is approximately -0.9 percent; higher than for bank lending because investment is forward-looking	Figure 2 right, p. 2107; §III.C, pp. 2106-2107	Below -0.9%, a further 10bp cut reduces aggregate investment on impact; the investment reversal rate lies above the bank-lending reversal rate
R3	A 10bp cut near the reversal rate reduces NII by approximately 6bp (as a fraction of assets); near the steady state the same cut reduces NII by only about 1bp	Figure 7 left, p. 2112; §IV.A, p. 2112	NII effect is approximately 6x stronger at the reversal rate than near the steady state; consistent with Borio, Gambacorta, and Hoffmann (2017)
R4	More deposit-dependent banks’ total loans grow roughly 2pp less than less deposit-dependent banks after a 30bp ECB rate cut in negative territory	Figure 7 right, p. 2112; §IV.A, pp. 2112-2113	Model prediction lies within 1 SD of Heider, Saidi, and Schepens (2019) DiD estimate of -2pp for banks with 15pp higher deposit-to-asset ratio
R5	Forward guidance (8 quarters at -1%) generates investment and output peaks roughly half as large as in a frictionless model without bank net-worth constraints	Figure 6, p. 2111; §III.E, pp. 2110-2111	Frictionless economy reaches investment peak of about 0.87 log points; benchmark model reaches about 0.5 log points; investment reverses by the 8th quarter in the benchmark
R6	The reversal rate for aggregate investment ranges from approximately -1.8 to -0.5 percent across the parameter sensitivity analysis, always exists within the calibrated range	Figure 8, p. 2115; §IV.B, pp. 2114-2116	-1.8% when capital constraints are tight (3.5bp loan rate sensitivity to 25bp capitalization rise); -0.5% when constraints are loose (10.5bp sensitivity); benchmark -0.9% is near center

Overall (paper’s conclusion). The reversal interest rate exists under empirically grounded conditions on bank loan and deposit demand: net interest income falls when rates are sufficiently low, and bank lending is constrained by net worth. In the calibrated New Keynesian model for the euro area, the reversal rate lies close to -1 percent for aggregate investment. Its level depends primarily on the tightness of banks’ capital constraints and the economy’s reliance on bank credit. “Low-for-long” rate environments can initially appear stimulative while eventually becoming contractionary, and this dynamic dampens the power of forward guidance relative to standard New Keynesian models.

Theory / model

The model is an infinite-horizon New Keynesian economy in discrete time, $t \in \{0,1,2,\ldots\}$. The main agents are: households; bank-dependent intermediate goods firms (fraction $\xi = 0.998$); non-bank-dependent intermediate goods firms (fraction $1-\xi$); a continuum of banks $j \in [0,1]$; capital goods producers; monopolistic retailers; and a central bank (Section I, pp. 2087-2092). Two key frictions distinguish the model from standard New Keynesian frameworks: banks have market power in setting deposit and loan rates, and bank lending is constrained by net worth.

Households. A household’s lifetime utility is (equation 1, p. 2088):

$$\sum_{t=0}^{\infty} \beta^t \left[ u(C_t, C_{t-1}, H_t) + \zeta \Phi(\mathcal{L}_t) \right] \tag{1}$$

where $C_t$ is consumption, $H_t$ hours worked, $\beta \in (0,1)$ the discount factor, $\mathcal{L}_t = \mathcal{L}(D_t, M_t)$ aggregates real deposits $D_t$ and cash $M_t$, and $\Phi$ is utility from liquid savings with a satiation point $\mathcal{L}^*$. The parameter $\zeta > 0$ scales liquid-asset demand. The flow utility uses habit formation:

$$u(C_t, C_{t-1}, H_t) = \frac{(C_t - hC_{t-1})^{1-\sigma}}{1-\sigma} - \chi \frac{H_t^{1+\varphi}}{1+\varphi}$$

with $h = 0.62$, $\sigma = 1$ (IES), and $\varphi = 2$ (inverse Frisch elasticity; Table 1, p. 2102). Cash earns zero net return, so the presence of cash as a substitute for deposits implies that deposit demand becomes unresponsive to the policy rate when rates are sufficiently low (Lemma 1, p. 2096). This floors the bank’s ability to pass rate cuts through to depositors, compressing margins.

Bank balance sheet and net worth. Banks extend loans $L_t$ and hold long-term government bonds $B_t^L$ at price $Q_t^B$, financed by deposits $D_t$ and net worth $N_t$. The balance sheet is (equation 4, p. 2089):

$$L_t + Q_t^B B_t^L = D_t + N_t \tag{4}$$

Long-term bonds mature with probability $1/\tau$ each period (expected maturity $\tau$; calibrated to 13.6 quarters to match average euro area bank bond holdings, p. 2104). The real bond price satisfies (equation 11, p. 2092):

$$Q_t^B = \frac{1}{\tau} \sum_{s=0}^{\infty} \left(\prod_{r=0}^{s} \frac{1}{1+i_{t+r}}\right) \left(1 - \frac{1}{\tau}\right)^s, \qquad Q^{B*} = \frac{1}{1 + \tau i^*} \tag{11}$$

Banks pay a fixed fraction $\gamma$ of net worth as dividends each period, so net worth accumulates according to (equation 5, p. 2090):

$$N_{t+1} = (1-\gamma)\left[\frac{1+i_t}{1+\pi_{t+1}}Q_t^B B_t^L + \frac{1+i_t^L}{1+\pi_{t+1}}L_t(i_t^L) - \frac{1+i_t^D}{1+\pi_{t+1}}D_t(i_t^D, i_t) - \Psi^L(N_t, L_t) - \Psi^D(Q_t^B B_t^L, D_t)\right] \tag{5}$$

where $\Psi^L(N_t, L_t)$ is the lending cost (homogeneous of degree one, decreasing in $N_t$, increasing in $L_t$) and $\Psi^D$ is the liquidity cost of deposit issuance. Lending is subject to a capital constraint $L_t \leq \psi^L N_t$: when this binds, credit supply is fully determined by net worth.

Capital gains and NII channels. An interest rate cut at $t=0$ affects bank net worth through two channels. Define $N_t(N_0, i)$ as net worth at time $t$ when the initial net worth is $N_0$ and the policy rate is cut to $i$. The total effect decomposes as (equation 17, p. 2097):

$$\frac{dN_t}{di} = \underbrace{\frac{\partial N_t}{\partial N_0}\frac{dN_0}{di}}_{\text{Capital gains channel}} + \underbrace{\frac{\partial N_t}{\partial i}}_{\text{NII channel}} \tag{17}$$

The capital gains channel captures the revaluation of existing bond holdings ($dN_0/di < 0$: a cut raises $Q_0^B$, boosting initial net worth). Lemma 3 (p. 2097) establishes that this channel weakens as the horizon $t \to \infty$ or as initial bond holdings $B^{L*} \to 0$: once long-term bonds mature, the initial capital gain no longer propagates. The NII channel captures the persistent compression of profit margins. Lemma 2 (p. 2097) shows that for $i_t \leq \underline{i}$, a further cut unambiguously reduces NII since deposit demand is unresponsive to the policy rate (cash dominates bonds as outside option). Lemma 4 (p. 2097) shows the NII channel depresses net worth at all future dates when rates are sufficiently low.

Characterization of the reversal rate. The reversal interest rate is defined formally in Definition 1 (p. 2093): the time-t reversal rate $i_t^{RR}$ is the highest interest rate such that bank lending $L_t(i)$ is increasing in $i$ for all $i < i_t^{RR}$. Proposition 1 (p. 2095) characterizes it: $i$ is $i_t^{RR}$ if (i) the capital constraint binds at all $i' \leq i$ and (ii) bank net worth is increasing in the interest rate, $dN_t(i)/di > 0$, for all $i' < i$. Proposition 2 (p. 2098) guarantees existence when initial bond holdings are small enough; Proposition 3 (p. 2098) guarantees existence when the horizon $t$ is long enough. Proposition 4 (p. 2098) establishes that $i_{t+1}^{RR} \geq i_t^{RR}$: the reversal rate is weakly increasing in the horizon, so an initially stimulative cut can become contractionary at longer horizons. Proposition 5 (p. 2099) formalizes the low-for-long result: for rates held below $i^*$ for long enough ($T > \bar{T}$), bank lending eventually falls below its steady-state level $L^*$.

Method

Solution method. The quantitative model is solved using Dynare’s full nonlinear algorithm rather than perturbation methods, since the model’s binding capital constraints create large nonlinearities and non-monotonicities near the reversal rate (§III.A, p. 2101). The economy begins at its steady state; an unanticipated monetary policy shock is announced at $t=0$, and the economy evolves deterministically thereafter under perfect foresight.

Bank’s problem. Banks maximize discounted dividends (equation 6, p. 2090):

$$\max_{B_t^L,\, i_t^L,\, i_t^D} \sum_{t=0}^{\infty} \beta^t \Lambda_t \gamma N_t \quad \text{subject to (4) and (5)} \tag{6}$$

where $\Lambda_t$ is the household’s marginal utility of consumption. In the analytical section (Section II), holding loan and deposit demand curves at their steady-state values, the bank’s problem reduces to maximizing net interest income period by period (equation 13, p. 2094):

$$\text{NII}(N_t, i_t) = \max_{B_t^L,\, i_t^L,\, i_t^D}\; i_t Q_t^B B_t^L + i_t^L L^*(i_t^L) - i_t^D D^*(i_t^D, i_t) \tag{13}$$

subject to the capital constraint $L_t \leq \psi^L N_t$ and liquidity constraint $Q_t^B B_t^L \geq \psi^D D^*(i_t^D, i_t)$.

Optimal loan and deposit rates. First-order conditions give (equations 14-15, p. 2094):

$$i_t^L = i_t + \frac{1}{\varepsilon_t^L} + \lambda_t + \mu_t \tag{14}$$ $$i_t^D = i_t - \frac{1}{\varepsilon_t^D} + (1-\psi^D)\mu_t \tag{15}$$

where $\varepsilon_t^L > 0$ and $\varepsilon_t^D < 0$ are the semielasticities of loan and deposit demand, $\lambda_t \geq 0$ is the Lagrange multiplier on the capital constraint, and $\mu_t \geq 0$ is the multiplier on the liquidity constraint. When constraints are slack, rates are set at the standard monopolistic markup/markdown over the policy rate. When the capital constraint binds, the loan spread widens, raising borrowing costs for bank-dependent firms and reducing their capital demand.

Calibration. The model is calibrated to the euro area; one period equals one quarter. Conventional DSGE parameters (Table 1, p. 2102) are set to standard values: Rotemberg price adjustment cost $\theta = 70.7$, capital share $\alpha = 0.36$, Taylor rule persistence $\rho^{mp} = 0.93$, and inflation coefficient $\phi^\pi = 2.74$ (from Coenen et al. 2019). Banking-sector parameters (Table 2, p. 2114) are calibrated to match euro area bank balance sheet moments: fraction of bank-dependent firms $\xi = 0.998$ (SME fraction in Eurostat); relative productivity $A^b/A^{nb} = 0.43$ (matching a 55.8% SME output share); loan elasticity $\varepsilon^L = 200$ (targeting a 2% loan spread from Freriks and Kakes 2021); deposit elasticity $\varepsilon^D = -275$ (targeting a 1% deposit spread from ECB MIR data); leverage cost $\kappa^L = 0.017$ (targeting 7bp loan rate increase per 25bp capitalization rise, from Macroeconomic Assessment Group 2010); bond maturity $\tau = 13.6$ quarters (matching average bank bond holding maturity from Hoffmann et al. 2019); and steady-state net worth-to-loan ratio $N^*/L^* = 0.155$ (from Altavilla, Boucinha, and Peydro 2018).

The monetary policy rule in the quantitative model is a Taylor rule with inertia (equation 19, p. 2105):

$$\frac{1+i_t}{1+i^*} = \left(\frac{1+i_{t-1}}{1+i^*}\right)^{\rho^{mp}} \left(\frac{1+\pi_t}{1+\pi^*}\right)^{\phi^\pi(1-\rho^{mp})} \exp(\epsilon_t^{mp}) \tag{19}$$

The benchmark results consider a single monetary shock $\epsilon_0^{mp}$ at $t=0$.

Capital goods producers. Capital goods producers solve (equation 8, p. 2091):

$$\max_{I_{t+1}^z} \sum_{t=0}^{\infty} \beta^t \Lambda_t \left\{ Q_t^{K,z} I_t^z \left[1 - \Xi\!\left(\frac{I_{t+1}^z}{I_t^z}\right)\right] - I_{t+1}^z \right\} \tag{8}$$

where $\Xi(\cdot)$ is a convex adjustment cost function (quadratic, $\kappa^I = 5$, implying an investment elasticity of 0.2 with respect to $Q_t^{K,z}$). This forward-looking structure explains why the reversal rate for aggregate investment (-0.9%) lies above the reversal rate for bank lending (-1.4%): investment internalizes the future deterioration of bank profits and declines before lending actually reverses.

Empirical specifications

The quantitative analysis does not use panel regressions. Instead the paper compares full nonlinear impulse response functions (IRFs) across initial interest rate levels, using the Dynare solution (§III.A, p. 2101).

Main results construction (R1, R2). For each initial policy rate $i \in \{-1.5\%, -1\%, 0\%, 1\%, 2\%\}$, the model first computes the economy’s response to a Taylor rule shock that reduces the time-0 rate to $i$, then adds a marginal 10bp Taylor rule innovation and computes a second IRF. The marginal IRF plotted in Figure 2 (p. 2107) is the difference between these two responses. The reversal rate is where this difference changes sign: at approximately $i = -1.4\%$ for bank lending and $i = -0.9\%$ for aggregate investment.

NII validation (R3). The left panel of Figure 7 (p. 2112) plots the model’s marginal NII response (as a fraction of steady-state assets) to a 10bp cut as a function of the initial policy rate, overlaid on the empirical estimate from Borio, Gambacorta, and Hoffmann (2017). The model predicts approximately -1bp near the steady state and approximately -6bp near the reversal rate, consistent in sign and order of magnitude with that study.

DiD comparison (R4). To compare model predictions with micro evidence, the paper introduces hypothetical banks calibrated to a deposit-to-asset ratio 15 percentage points below the benchmark. These banks coexist in equilibrium but do not affect aggregate dynamics. Their marginal loan rates and lending quantities are computed for a -30bp Taylor rule shock. The right panel of Figure 7 (p. 2112) plots the log difference in lending at impact as a function of the initial policy rate, comparing the model with the DiD estimate from Heider, Saidi, and Schepens (2019), who use euro area syndicated loan data over June 2014 to December 2015. The model prediction of approximately -2pp is within one standard deviation of the Heider, Saidi, and Schepens (2019) estimate.

Forward guidance (R5). The central bank commits to holding the policy rate at -1 percent for $T = 8$ quarters, then reverts to the Taylor rule. The impulse responses are compared across the benchmark model and a frictionless version in which banks face no leverage costs ($\kappa^L = 0$). The frictionless economy mimics the dynamics of standard New Keynesian models: investment and output peaks are approximately twice those in the benchmark. In the benchmark, the anticipation of future bank-profit deterioration dampens the initial stimulus, and investment reverses by quarter 8 (Figure 6, p. 2111). Eggertsson and Woodford (2003) derive the standard forward guidance channel; the reversal rate mechanism here attenuates it.

Sensitivity analysis (R6). Figure 8 (p. 2115) varies $\kappa^L$ (the leverage cost parameter, reported as the response of loan rates to a 25bp capitalization target increase) between 3.5bp and 10.5bp and the bank-dependent output share $A^b/A^{nb}$ between 0.45 and 0.65. Across this range, the on-impact reversal rate for aggregate investment lies between -1.8 and -0.5 percent, remaining within a narrow band around the benchmark estimate of -0.9 percent.

Datasets used

This is a theoretical and calibration paper. The quantitative results come from a Dynare solution of the calibrated structural model rather than direct econometric estimation. Calibration targets are drawn from published statistics and the literature:

Source	Role in paper	Wiki page
ECB MFI statistical data	Deposit-to-GDP ratio target for $\mathcal{L}^*$ (satiation point); equity issuance-to-asset ratio target for $\hat{N}$	No page yet
Eurostat (Structural Business Statistics)	Fraction of SMEs in the firm universe ($\xi = 0.998$) and SME share of output (55.8%)	No page yet
Altavilla, Boucinha, and Peydro (2018)	Tier-1 capitalization ratio target: 15.5% (steady-state $N^*/L^*$)	No page yet
Hoffmann et al. (2019)	Average bank bond holding maturity: 3.4 years ($\tau$) and loan-to-bond ratio ($L^*/B^{L*}$)	No page yet
ECB MIR database (Freriks and Kakes 2021)	Loan spread of 2% and deposit spread of 1% in steady state	No page yet

Sample: euro area, calibrated to the period surrounding the ECB’s introduction of negative interest rates in 2014. Model period: one quarter.

When to read the full paper

Use the original if you are: characterizing the effective lower bound on monetary policy when banking frictions are present; building or extending New Keynesian DSGE models with imperfectly competitive banks and balance-sheet constraints; studying the transmission of negative interest rate policy to bank lending and firm investment; evaluating the “low-for-long” forward guidance puzzle with financial intermediaries; or running the Dynare replication (code at https://doi.org/10.3886/E188611V1). The key tables and figures are: Figure 2 (reversal rates by initial rate level), Figure 6 (forward guidance dampening), Figure 7 (NII and DiD validation), and Figure 8 (sensitivity analysis).

Attribution and rights

Source: peer-reviewed, American Economic Review 113(8), 2023. Paywalled; no CC license. This distillation was extracted by an LLM on 2026-06-24 and is not human-verified or independently reproduced.

Abadi, Joseph, Markus Brunnermeier, and Yann Koby. “The Reversal Interest Rate.” American Economic Review 113, no. 8 (August 2023): 2084-2120. DOI: 10.1257/aer.20190150. Extract-only; all rights reserved by the American Economic Association.