IMES DISCUSSION PAPER SERIESThe Reversal Interest Rate

Markus K. Brunnermeier and Yann Koby
Discussion Paper No. 2019-E-6

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
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IMES Discussion Paper Series 2019-E-6
June 2019

The Reversal Interest Rate

Markus K. Brunnermeier* and Yann Koby**

Abstract
The reversal interest rate is the rate at which accommodative monetary policy
reverses and becomes contractionary for lending. Its determinants are 1) banks’
ﬁxed-income holdings, 2) the strictness of capital constraints, 3) the degree of passthrough to deposit rates, and 4) the initial capitalization of banks. Quantitative easing
increases the reversal interest rate and should only be employed after interest ratecuts are exhausted. Over time the reversal interest rate creeps up since asset
revaluation fades out as ﬁxed-income holdings mature while net interest income
stays low. We calibrate a New Keynesian model that embeds our banking frictions.

Keywords: Monetary Policy; Lower Bound; Negative Rates; Banking
JEL classification: E43, E44, E52, G21*Department of Economics, Princeton University (E-mail: markus@princeton.edu)**Department of Economics, Princeton University (E-mail: ykoby@princeton.edu)
We are grateful for comments by Joseph Abadi, Julien Bengui, Emmanuel Farhi, Lunyang Huang,
Florian Heider, Sam Langﬁeld, Cesaire Meh, Helene Rey, Philipp Schnabl, Frank Smets, Adi
Sundaram, Skander Van den Heuvel, Christian Wolf, and seminar participants at the European
Central Bank, the Bank for International Settlements, Princeton University, the Federal Reserve
Board, Banque de France, the Danish Central Bank, the Finnish Central Bank, the London
School of Economics, ETH Zurich, the Swiss Economists Abroad meetings, the Swiss National
Bank, the 2018 AEA meetings, Humboldt University, the Philadelphia Macro Workshop, theInternational Research Forum on Monetary Policy, the University of Chicago, and the Bank of
Canada. We acknowledge support from the Macro Financial Modelling Project. The views
expressed in this paper are those of the authors and do not necessarily reflect the official viewsof the Bank of Japan.

1Introduction

In most New Keynesian models, the economy enters a liquidity trap because of an exogenously assumed zero lower bound. This assumption has been called into question since a
growing number of central banks – the Swedish Riksbank, Danmarks Nationalbank, theSwiss National Bank, the European Central Bank, and the Bank of Japan – have led money
market rates into negative territory as a response to the Great Recession. In addition togoing negative, these rates have been kept low for a long period. Although unusual by historical standards, this economic environment is likely to prevail amid the persistent decline
in real and nominal interest rates over the last two decades.
This motivates the question: what is the effective lower bound on monetary policy? Wesuggest in this paper that it is given by the reversal interest rate, the rate at which accommodative monetary policy reverses its effect and becomes contractionary for lending. A
monetary policy rate decrease below the reversal interest rate depresses rather than stimulates the economy.
Importantly, the reversal interest rate is not necessarily zero, as commonly assumed. In
our model, when the reversal interest rate is positive, say 1%, a policy rate cut from 1% to0.9% is already contractionary. On the other hand, if the reversal interest rate is -1%, policy
rate cuts stay expansionary up to that point, even if their effectiveness might be impaired.
The reversal interest rate is endogenous, and its existence is guaranteed when banks’
gains from maturity mismatch are insufficiently large. We formally demonstrate this resultin a stylized model of monetary policy transmission through banks. Following an interest
rate cut, two opposing forces affect banks’ net worth. On the one hand, banks make capitalgains on assets with long-term fixed-rate coupon payments (e.g., bonds). On the other hand,
the rate cut shrinks banks’ net interest income going forward. The yield they obtain per unitof liability employed decreases. Note that our result does not require the elasticity of deposit
demand faced by banks to vary with the level of interest rates, even though properties ofthis elasticity matter in our quantitative application.
Our comparative statics reveal four key determinants of the reversal interest rate: 1)banks’ holdings of long-term fixed-income assets, 2) banks’ equity capitalization, 3) thetightness of capital constraints, and 4) the deposit supply elasticity faced by banks. Higher
initial holdings of long-term fixed-income assets imply a larger maturity mismatch, resulting
in a larger asset revaluation that decreases the reversal interest rate. Low initial bank capitalization and restrictive capital constraints both imply that the capital constraint tightens

1sooner following the drop in profitability caused by rate cuts, ceteris paribus. Finally, when
the deposit supply elasticity increases as rates fall – due to consumer awareness of spreads
or cash competition as rates approach zero – banks’ profits shrink faster and the reversal
interest rate is higher.
Quantitative easing (QE) increases the reversal interest rate, as it takes long-term fixedincome holdings out of bank balance sheets. Consequently, QE should only be employed
after interest rate cuts are exhausted.
Our multiple-period extension shows that the effectiveness of a given path of policy rates
depends on the duration of banks’ initial fixed-income holdings. The negative effects on
net interest income we described cumulate every period, while asset revaluation fades out asbank assets mature. In other words, a rate cut far in the future decreases banks’ profitability
but without revaluation gains if all initial fixed-income holdings have matured by then. In
other words, the reversal interest rate “creeps up”: said differently, the most stimulatory
path of policy rates is increasing over time. “Low for long” rate environments can depresslending relative to policies that generate a relatively larger decrease in the slope of the yield
curve while keeping the long end of that curve high.
The economics behind our results carry through in general equilibrium. After embedding
our banking model in a New Keynesian macro model, we calibrate it in order to study
quantitatively the importance of general equilibrium feedbacks. A new force emerges due
to sticky prices. A policy rate cut generates a demand for credit that increases banks’intermediation and hence profits. This force, in loose terms, decreases the reversal interest
rate. We find in our calibration that the monetary authority’s ability to stimulate lending
rates on impact declines with the size of the monetary shock and reverses at around -1% for
the euro area. Given the persistence of our monetary shock, the negative effects are evenmore pronounced on lending rates one or two years ahead. This is due to our earlier “creepingup” effect: banks are shielded from rate cuts on impact, but not later. Once the reversal inbank lending is crossed, the economy’s reliance on bank credit – the share of firms that arebank-dependent and the extent to which they are – dictates the aggregate implications forinvestment, output, and consumption. The reversal interest rate for these aggregate variables
is lower, as other channels through which monetary policy operates – nonbank-dependent
firms’ funding costs and the inter-temporal substitution channel – remain active. Finally,
we also show that a permanently lower steady-state real natural rate r∗ leaves less leeway
for monetary policy to be effective should the inflation target π ∗ stay unchanged.
The rest of this paper is organized as follows. First, we present a partial-equilibrium,2

two-period model where we provide explicit conditions for the existence of the reversal interest rate. In Section 2, we provide comparative statics and implications of our model for
quantitative easing. In Section 3, we extend the model to multiple periods while keeping
the analysis in partial equilibrium, and analyze how paths of policy rates transmit throughbanks. Finally, in Section 4 we introduce our simple banking model inside a New Keynesian model. We calibrate the model and illustrate its implications for impulse responses ofaggregate variables to a monetary shock. The last section concludes.
Literature Review. A long-standing literature developed the concepts of the “balance
sheet” and “bank lending” channels of monetary policy, emphasizing the importance of thebalance sheet structure and the net worth of intermediaries for the transmission of monetary
policy (Bernanke and Blinder, 1988; Bernanke and Gertler, 1995; Van den Heuvel, 2007). In
our model, these objects are key determinants of the transmission of monetary policy.
From a theoretical standpoint, our microeconomic modeling of banks stands on the shoulders of a literature formally started by Klein (1971) and Monti (1972).1 This literature emphasized the importance of market power when modeling banks.2 Sharpe (1997) provides
evidence of switching costs for depositors, and Kim et al. (2003) and Chodorow-Reich (2014)
demonstrate the existence of relationship costs. Hainz et al. (2017) offer suggestive evidence
that depositors’ switching costs might change in low-rate environments, which we exploit inour quantitative exercise.
Banks’ market power on their funding sources materializes in the impaired transmissionof money market rates to bank deposit rates, affecting banks’ interest rate risk exposure.
Saunders and Schumacher (2000) and Maudos and Fernandez de Guevara (2004), among
others, showed this fact empirically. Eggertsson et al. (2017) document a collapse of thepass-through of the monetary policy rate when the latter hits negative territory, owing tozero-interest-bearing cash becoming relatively more competitive. Work by Rognlie (2016)
also suggests that the elasticity of demand changes at zero and sub-zero rates, affecting thepass-through. Drechsler et al. (2017) showed that quantities also respond to the spread thatbanks’ charge. Importantly, this impaired transmission suggests a wedge between contractualand effective maturity of deposits, a fact long recognized by regulators when assessing banks’interest rate risk (Hoffmann et al., 2018).
1

Santomero (1984) provides a good survey of this early theoretical literature.
Petersen and Rajan (1995) in fact argue that banks need some monopoly power to sustain their businesses.
2

3

The interest rate exposure on banks’ liabilities due to market power is balanced by banks’
long-term fixed-income assets. Begenau et al. (2015) and Gomez et al. (2017) document thatbanks’ assets are exposed to interest rate risk, which banks do not fully hedge in derivative
markets (Abad et al., 2016; Hoffmann et al., 2018). Hoffmann et al. (2018) and Drechsler
et al. (2018) show that this asset exposure is rationalized by banks’ market power on deposits
as it provides a natural hedge to the resulting interest rate risk. Drechsler et al. (2018) show
effectively that due to this hedge, realized net interest income varies little with the level ofpolicy rates: banks optimally choose a maturity mismatch in order to hedge the interest
rate risk created by their market power on the liability side of their balance sheets. Weview our study as complementary: we argue that the long-lasting low/negative interest rate
period was largely unexpected and hence not hedged, and we study the implications of an
unexpected shock. Di Tella and Kurlat (2017) offer an alternative rationalization with asimilar result. In Brunnermeier and Sannikov (2016), banks also hold interest rate risk sinceappropriate monetary policy provides a “stealth recapitalization” in downturns. Importantly,
these hedging strategies work in expectation. Hence, upon unusual interest rate realizations,the valuations of banks can fluctuate significantly, as in our model.
A recent empirical literature has shown real effects of low/negative rate environments
on banks’ profitability. English et al. (2012), Ampudia and Heuvel (2018), Claessens et al.
(2017), Eisenschmidt and Smets (2018), and Wang et al. (2018) provide evidence that banks’net interest income and equity valuations vary with the level of interest rates, possibly in anonlinear way. In particular, Claessens et al. (2017) find that a 1% policy rate drop implies,
on average, a net interest margin 8 basis points, but that this magnitudes grows as rates
move lower. This effect carries through bank profitability. Moreover, for each additional
year of low rates, margins and profitability fall further. Altavilla et al. (2017) document that
the ECB’s introduction of negative interest rates was significantly detrimental to banks’net interest income, although increased intermediation activity as well as an improvementin the risk profile of banks’ assets helped sustain returns on assets. Evidence provided by
Ampudia and Heuvel (2018) suggest that banks’ profitability response to interest rate cuts is
non-monotonic: in normal times, interest rate cuts increase banks’ valuations, although thisdoes not hold in low-rate environments. Our DSGE model replicates this non-monotonicity.
Finally, we rely on a literature showing that the profitability of banks impacts theirlending activities and hence the level of intermediation in the economy. Brunnermeier andSannikov (2014) provide a theoretical foundation where intermediaries’ profitability is key
for the economy to function properly. Cavallino and Sandri (2018) obtain contractionary4monetary easing in their theoretical model and explore the implications in an open economy
context. Empirically, Chodorow-Reich (2014) shows real effects of bank lending frictions onfirm employment. Heider et al. (2017) employ a difference-in-difference analysis using syndicated loans in the euro area to document that banks with a high deposit base decreased theirlending relative to low-deposit, wholesale-funded banks following the ECB’s decision to implement negative interest rates.3 Importantly, Gropp et al. (2018) show that banks exposed
to higher capital requirements decrease their risk-weighted assets instead of recapitalizing,as in our model.

2

Two-Period Partial Equilibrium Setup

In this section, we analyze a two-period banking model in partial equilibrium. That is, wehold fixed any aggregate prices and quantities that are not directly determined by banks’decisions.
Each of a continuum of (ex ante) identical banks is initially endowed with equity fundingof E0 . In addition, banks can raise liabilities D in the form of bank deposits. On the asset
side of the balance sheet, banks have two investment opportunities: loans L to firms andfixed-income assets S. Banks compete for loans to firms, while they take the return on fixedincome assets as well as their initial equity as given. Figure 1 displays a stylized balance
sheet of a bank.

2.1

Timing of events

There are two periods, 0 and 1. We let i0 denote the interest rate between time 0 and 1 thatwas expected before the beginning of period 0. At the beginning of period 0, the central
bank sets the policy rate to i, which may differ from i0 . Observing i, banks then set newinterest rates to maximize their period 1 net worth.

2.2

Bank assets

Bank loans. Each bank grants loans to a unit measure of its customer firms. We assume
for simplicity that informational friction make firms locked in a relationship with their house3
Gomez et al. (2017) offer similar evidence, by studying two groups differentially exposed to interest raterisk. The group whose profitability is affected negatively (in relative terms) by a change in aggregate interest
rates decreases its lending.

5

Figure 1: Bank’s balance sheet: two-period model.

bank. The loan demand that bank j faces is denoted by L(iLj ), where iLj is the nominal rateon bank loans that bank j offers.4
Fixed-Income Assets. Each bank can also invest in fixed-income assets S. These assets
are available in perfectly elastic supply, with a yield i that banks take as given.5

2.3

Bank Liabilities

Deposits. As with loans, each bank is naturally associated with depositors over which
they have market power. We assume the demand schedule is given by D(iD ), where D(·)
is increasing. In the comparative statics section – as well as in our quantitative analysis –we’ll consider ways in which D(·) might also depend on the current level of rates: we might
expect, for example, competition for deposits to be tougher in low-rate environments.
Equity. Let E0 (i0 ) be the banks’ book equity before the surprise policy rate change. We
assume that banks’ book equity after the surprise change, E0 (i), is a function of the nominal
policy interest rate i. This captures the fact that the value of banks’ past assets and liabilities
might change after monetary policy changes its stance. This revaluation can take the form ofcapital gains on mark-to-market assets, but include in spirit any asset revaluation, including,
4

In our general equilibrium section, we micro-found the loan demand of firms and make them dependent
on aggregate conditions beyond iLj.
5
A broader interpretation of these assets would include equities, as banks take their returns essentiallyas given, and the risk-free component of these returns tracks the policy rate i.

6for example, changes in loan-losses provisions. We decompose the equity after the monetary
policy shock into E0 (i) = ē0 + e0 (i), where ē0 is the interest-insensitive component of initial
equity and e0 (i) the interest-sensitive part. We assume that ∂e0 (i)/∂i < 0, reflecting amaturity mismatch.6

2.4

Financial frictions

Banks face two forms of financial frictions. First, banks are subject to a capital constraintof the form
ψ L L + ψ S S ≤ N1 ,where ψ L , ψS ≥ 0 are risk weights and N1 denotes the bank’s (nominal) net worth (defined
below). That is, a weighted average of book assets must be covered by the value of the bank.
We set ψ S = 0 for concreteness.7 The capital constraint captures economic and regulatory
factors. Note that E0 does enter the constraint indirectly through N1 : ceteris paribus, alarger E0 leads to higher net worth.8
Second, banks face a liquidity constraint of the formS ≥ ψ D D,with ψ D > 0. That is, each bank’s fixed-income holdings must cover a certain fraction ofdeposits. Such a constraint captures the fact that banks need sufficient and easily accessible
funds to avoid run risk.

2.5

Banks’ problem

Finally, let L + S = D + E0 (i) be the balance sheet identity of the bank. Then, we can write
its problem asmax

iL ,iD ,L,D,S,N1

N1 = (1 + iL )L + (1 + i)S − (1 + iD )D

L + S = D + E0 (i)6For example, assume that banks enter the period with some equity ē0 and an interest rate swap with
0−inotional value A0 . Setting e0 (i) = i1+i
A0 is isomorphic to including this derivative in the analysis.
7

The analysis generalizes to the case ψS < ψL .
Regulation often has an object closer to E0 in the analysis. Such a constraint would not tighten in our
two-period model, but would if time between these two periods is split into more sub-periods.
87ψD D ≤ S
ψ L L ≤ N1
L = Lj (iLj )D
D = Dj (iD
j ; i−j , i).

This problem offers no particular mathematical difficulties, and hence we omit conditions forexistence and uniqueness of a symmetric equilibrium. In our micro-foundations of D(·), L(·)
in later sections, existence and uniqueness are straightforward to show.

3

Partial Equilibrium Reversal Interest Rate

3.1

Definition of the reversal interest rate

We now explicitly define the “reversal interest rate” as the rate at which a decrease in thenominal policy rate, i, stimulates lending if and only if the current level of the interest rate,i, is above the reversal interest rate iRR .
Definition 1 (reversal interest rate). Let iRR define the reversal interest rate such that∗< 0;
1. i > iRR implies dL
di∗

2. i = iRR implies dL
= 0;
di
∗

3. i < iRR implies dL> 0.
di
In what follows, we first derive the bank’s optimal setting rules. We then spell out
sufficient conditions under which a reversal interest rate obtains.

3.2

Banks’ rate-setting rules

Monetary policy affects the marginal investment opportunity of banks. Given that a bankcan earn a return of i from holding a fixed-income asset, i encodes the opportunity cost of
granting loans, and banks charge a mark up above it. Similarly, for deposits banks apply amark down on the marginal investment yield i.
The constraints limit banks’ portfolio choices. In particular, when the capital constraint
binds, banks charge higher-than-desired lending rates in order to decrease their leverage.

8

Similarly, when the liquidity constraint binds, banks offer higher-than-desired lending anddeposit rates in order to bring the liquidity ratio up.
Let εf ∗ denote the semi-elasticity of the function f with respect to the relevant rate,evaluated at the optimal pricing rules.9 The next lemma formally encodes these results.
Lemma 1 (rate-setting rules). The optimal rate on loans is given by
iL∗ =

i
|{z}

+

Marginal
opportunity cost

1
ψL
+
λL∗ .
εL∗
1 + ψL|{z}| {z }Mark upCapital constraint

1

ψD
λD∗
1 + ψL| {z }(1)

The optimal rate on deposits is given by
iD∗ =

i
−
|{z}

Marginal
benefitεD∗|{z}Mark down+(2)

Liquidity constraint

When constraints are slack, the Lagrangian multipliers are simply zero; when they dobind, the Lagrange multipliers are defined by the FOCs, and actual rates are given by theconstraints themselves.10

3.3

Existence of iRR

We now show how the constraints lead to a reversal of the bank lending channel. Rememberthat the capital constraint depends on how profitable the bank is. The next lemma showsthat profits of banks have two components: net interest income (NII) and capital gains (CG).
NII is defined as11
NII = i|L∗ L∗{z+ i S}∗ −Interest income

9

L)
That is, εL∗ = ∂ log∂iL(iL

i|D∗{zD}∗.

Interest expenses. Although mathematically these are semi-elasticities, economically theyiL =iL∗L D

are elasticities since the units of i , i are percentage points.
10
Smooth costs would enter in a similar way, increasing both lending and deposit rates.
11
Note that our definition differs from measurements of realized net interest income as given, for example
that in Drechsler et al. (2017). One can view our capital gains as the component of realized net interest
income which is “shielded” from interest rate fluctuations.

9Capital gains, on the other hand, are simply the change in initial equity (retained earnings)
created by the surprise change in interest rates:
CG = E0 (i) − E0 (i0 ).
The following lemma also shows that when there are no capital gains, that is, CG = 0, thenthe change in profits following an interest rate cut is strictly negative. We first state theresult when the liquidity constraint binds and then show that it only works to increase thederivative.
Lemma 2 (profit response). Assume that ψ D = 0. The change in profits following a change
in i is then given by
 dNII
dE0 (i) dN1∗= (1 + λL∗ )+ (1 + i).
| {z } | {zdidi } |{z di }
Amplification

NII>0

(3)

CG<0

dN ∗

Moreover, if dEdi0 (i) = 0, then di1 > 0.
dN ∗Also, if ψ D > 0, then the derivative is larger than di1 given in (3) above.
This result is intuitive. An interest rate cut depresses the return on new investments
in fixed-income assets. Because fixed-income assets are always held by banks (whether theliquidity constraint binds or not), net interest income, and hence profits, decreases. That
this is sufficient is a consequence of the envelope theorem – the fact that the first-orderconditions described above apply. However, an interest rate cut also leads to an increase in
E0 (i), on the basis of the maturity mismatch. Without such gains, profits unambiguously
decrease following an interest rate cut.
Moreover, an amplification occurs when the capital constraint binds (implying that λL∗ >0). From that point onward, banks are forced to divert loan investment into fixed-income
asset investment. Ceteris paribus, that makes further cuts more harmful to banks’ profits.
Returning to our main result, when the capital gains are sufficiently small – that is, thechange in E0 (i) is small enough – then the NII channel dominates. Hence, profits decrease
with a decline in i. Moreover, as long as the capital constraint does not bind, diL∗ /di > 0,so that an interest rate cut lowers the loan interest rate, leading to more loans. Both forcestighten the constraint. Eventually, the constraint inevitably binds: at that point the policy
rate hits the reversal interest rate, because any further decrease in i will decrease profits,
which through the constraint must decrease L∗ , so that dL∗ /di flips sign.
10

Proposition 1. (existence of iRR ) When capital gains E0 (i) are sufficiently low (uniformly
bounded by some number), there exists a finite reversal interest rate iRR .
Furthermore, nothing guarantees the reversal interest rate to be zero or any particularnumber. Instead, it has an endogenous number that depends on the state of the economy
and in particular that of the banking sector, as highlighted by our comparative statics.
Corollary 1. Generically, iRR 6= 0.
A numerical example. Figure 2 displays a numerical example of a reversal interest rate
triggered by a binding capital constraint for a baseline interest rate of i0 = 1.5%. An
interest rate cut lowers banks’ net worth (lower-left panel), as the decline in net interest
income exceeds the increase in capital gains (lower-right panel). The capital constraint
tightens until it inevitably binds. At this stage, i falls below the reversal interest rate anda further decrease in the policy rate lowers loan volume (top-left panel). Interestingly, theloan interest rate then rises below that interest rate (top-right panel).

3.4

Comparative statics results

We now derive the comparative statics results. Unsurprisingly, the reversal interest rate islower when capital constraints are looser or initial equity is low. This is consistent withCorbae and D’Erasmo (2014), who find in a structurally estimated banking model that anincrease in capital requirements leads to a decline in aggregate loan supply and an increasein the loan interest rate.12,13 Our third result in the next proposition states that, everythingelse equal, higher capital constraints on leverage make subsequent interest rate cuts below
the reversal interest rate even more harmful for lending.
Proposition 2 (capital constraint and equity). The reversal interest rate iRR has the following properties:
1. The reversal interest rate iRR increases in the risk weight of the capital constraint ψ L .
12

It is important to note that we neglect the potential risk-taking effects of decreasing interest rates,
which might be the basis for a constraint – see, for example, Di Tella (2013) or Klimenko et al. (2015). In
a theory encompassing both channels, a trade-off would emerge between the two; we are only modeling one
side of a trade-off, and hence our results are unambiguous here.
13
Note that for Propositions 2 and 3 below, the statements hold only weakly; however, for some reasonable
constellations of parameters they hold strictly.

11

Figure 2: A numerical example of a reversal interest rate (dashed vertical line) due to a binding
capital constraint. The constraint binds since the losses on net interest income (NII) are notsufficiently compensated by capital gains (CG). ∆CG and ∆N II refer to changes relative to theirrespective value at a baseline rate of i0 = 1.5%.

2. The reversal interest rate iRR decreases in the interest rate sensitivity of the initial
equity ∂e0 (i)/∂i (keeping E0 (i0 ) constant).
3. An interest rate cut below the reversal rate is more detrimental for lending in an economy with a tighter capital constraint. Specifically, consider two economies, A and B,
RR
that have the same reversal interest rate iRR
A = iB and that are identical in all respects
RR∗∗
except that ē0,A < ē0,B and ψAL < ψBL . Then, for any i < iRR
A = iB , LA (i) > LB (i).
Arguably, one of the most striking features of our reversal result is that it does not relyon stickiness of the deposit rate. We highlight that, more generally, what matters is the factthat banks have more market power on the liability side of their balance sheet.
Hence, it follows that decreases in their market power coming from either the extensive
or intensive margin of their supply of liabilities rise the reversal interest rate. Specifically,
the next proposition shows the higher intensive margin deposit supply, coming, for example,
from competition with other nominal saving vehicles such as cash also increases the reversal
12interest rate. We also show that if banks face a higher elasticity as rates fall due to heightenedbank competition, the reversal interest rate increases as well. Hainz et al. (2017) document
this phenomenon for Germany, where customers became more enclined to switch banks as
the policy rate declined. In contrast, if banks are funded to a large extent with wholesalefunding and the funding rate drops almost one-for-one with the policy rate, then the reversal
interest rate is lower. Madaschi and Nuevo (2017) and Erikson and Vestin (2019) show that
this was the case in Sweden. Finally, we note that larger liquidity constraints also work
toward making the reversal interest rate higher.
Proposition 3 (liquidity constraint and deposit rate pass-through). The reversal interest
rate iRR has the following properties:
1. For iD , such that d(iD ) = 0 for all iD ≤ iD , where d(·) is the intensive margin of thedeposit supply from depositors, iRR is increasing in iD . In other words, the reversal
interest rate is increasing with the lower bound on deposits.
2. Similarly, assume that the perceived elasticity ε∗D is decreasing in i, leaving D(·) unchanged – e.g., individual banks face a higher perceived elasticity for extensive margin
reasons. Then, the reversal interest rate is larger than with a constant ε∗D .
3. The reversal interest rate iRR increases with the tightness of the liquidity constraintψD .
We finally add an obvious but important result. Suppose that capital gains can be paidout as dividends: that is, equity is given by E0 (i) = ē0 + (1 − υ0 )e0 (i), υ0 is the dividendrate, and e0 (i) represents capital gains (asset revaluation). Certainly, the larger υ0 is, theless effective interest rate cuts are at recapitalizing gains for the purpose of making newloans, thus increasing the reversal interest rate. Our framework therefore rationalizes someof the recent policies implemented to restrict the dividend payments of banks.
Proposition 4 (dividends and the reversal interest rate). Suppose that equity is given by
E0 (i) = ē0 + (1 − υ0 )e0 (i), with υ0 the dividend rate. Then, iRR is increasing in υ0 .

3.5

Optimal sequencing of QE

Our model also implies an optimal sequencing of interest rate policy and other monetary
operations such as quantitative easing (QE). QE changes the bond holdings of the banking
sector and hence its interest rate risk exposure. QE reduces the banks’ holdings of long-term
13bonds; hence, after QE, the interest rate sensitivity of bank equity ∂e(i)/∂i is reduced, which
increases the reversal interest rate.
Proposition 5 (QE and capital gains). Quantitative easing, which lowers the interest ratesensitivity of banks’ initial equity ∂e(i)/∂i while leaving the overall level of E0 (i0 ) unchanged,
lowers potential capital gains from a subsequent interest rate cut and hence increases thereversal interest rate iRR .
The optimal sequence of stimulating monetary policy is to cut the interest rate all theway toward the reversal interest rate before conducting QE measures.
We emphasize that this is a partial equilibrium result. In general equilibrium, otherforces might pull toward an alternative sequencing.14

4The “Creeping up” Effect

In this section, we extend the model to a three-period setting. This allows us to study how
announcements about a path of policy rates impact the business of the bank, in particular
net interest income in the future and the feedback on lending today. Our main result isthat the optimal length of interest rate cuts should be related to the maturity of the banks’
existing assets. The reason is as follows. As in the two-period model, a cut in an interest
rate in the future has two effects: 1) fixed-income assets experience capital gains, while 2)
net interest income will be depressed. Since the fixed-income assets mature over time, thefirst force slowly fades out whereas the loss in margins on future business does not. Hence,
the interest rates that maximize lending “creep up” over time.
To make that intuition concrete, we consider banks in a setting similar to our two-period
model, except that banks enter the period with two assets on their books: a one-periodbond and a two-period bond. Moreover, their equity in the second-period is endogeneousand depends on profits that banks make in the first period. We then ask: what is the pathof interest rates that maximizes banks’ loan supply? Since fixed-income holdings of the first
period are larger than those of the second period, the case for cutting the interest rate isstronger in the first period, where the capital gains are higher, while the effect on net interest
income is similar across both periods. In that sense, the optimal path of (reversal) interest
14

In particular, the sequencing described above stealthily recapitalizes banks, meaning that losses othersectors – the government, households, foreigners, or others – could distort other economic decisions.

14

rates is increasing, and an exceedingly long-lasting environment of low interest rates might
hurt lending.

4.1

Three-period model extension

In our three-period model, the monetary authority controls a path of one-period interest
rates {i0,1 , i1,2 }: the rate between the first two periods, i0,1 , and the rate between the lasttwo periods, i1,2 . Banks enter the period with their past book, consisting of equity ē0 , oneperiod bonds B0,1 , and two-period zero-coupon bonds B0,2 .15 Bonds are priced competitively
1
1
B
at pB
0,1 = 1+i0,1 and p0,2 = (1+i0,1 )(1+i1,2 ) . Hence the bank’s equity entering the period is
B
E0 (i0,1 , i1,2 ) = ē0 + pB
0,1 B0,1 + p0,2 B0,2 : when rates and therefore prices change, so too does
equity entering the new period.
Each period, the bank is able to grant loans and deposits and to invest in fixed-income
assets. The demand for loans L(·) and the supply of deposits D(·) is the same in both
periods. We also assume that, before the policy experiment, the interest rates are equal inboth periods, denoted by i∗ . Hence, the two periods are identical in every aspect to repeating
our two-period model twice, except that we have one- and two-period bonds and the equity
level in the second period is now endogenously specified. As before, let N1 be the net worthof the bank after optimization in the first period. We assume that part of the earnings isretained. Specifically, we assume a dividend (payout) rate υ ∈ (0, 1) of the net worth, sothat E1 = (1 − υ)N1 . We further assume that υ is such that E0 (i∗ , i∗ ) = E1 . In sum, theenvironment is totally stationary when there are no policy changes, and the bank makes
similar decisions in both periods should rates stay at i∗ . Finally, we keep the analysis inpartial equilibrium, so there are no feedbacks from the policy changes and banks’ endogenous
responses.

4.2

Loan-maximizing policies

We define the loan-maximizing policies iP0,1 , iP1,2 as those that maximize the discounted sumof loans:(iP0,1 , iP1,2 ) = argmax L∗0,1 (i0,1 , i1,2 ) + β P L∗1,2 (i0,1 , i1,2 ),i0,1 ,i1,2

15More generally, these bonds should represent the duration structure of banks’ balance sheets as they
enter the period, in the spirit of Begenau et al. (2015).

15

where L∗0,1 , L∗1,2 are the optimal choices of banks’ loan supply given the interest rates, andβ P is a “policy specific” discount factor (of the social planner). We assume that β P ≤ 1,that is, the policy cares more about present loans than future loans.
Our goal in this section is to characterize the choices iP0,1 , iP1,2 .

4.3“Creeping up” result

Our main result is that, under mild conditions, iP0,1 < iP1,2 – that is, the interest rate path“creeps up.” The key reason for this result is that, although the loss in net interest income
following an interest rate cut is similar in both periods, the capital gains from cutting theshort-term rate are larger than from cutting the long-term rates since assets mature. In
other words, a long-lasting low-interest-rate environment is going to hurt banks’ flow profits
in every period, while generating low capital gains in the later periods. As a consequence, it
is optimal to cut the short-term rate more deeply than the long-term one.
One condition we need when capital gains on long-term assets are present, however, isthat the policy makers must care about loans in the second period. To see this, supposethat β P = 0, that is, the policy maker is myopic and cares only about current loan volume.
Suppose, moreover, that B0,2 > 0, that is, there are capital gains to be made on long-term
assets. The policy response will naturally be to decrease the long rate as much as possible,
so as to maximally boost capital gains on long-term assets, which will drive down long-termloans to very low levels. Consequently, bank net worth would tank and, with it, loan volume
in the second period. Hence, to avoid these myopic cases, we need β P to be sufficiently close
to one whenever B0,2 > 0.
Proposition 6. Assume that B0,1 and B0,2 are small enough such that the loan-maximizing
rates iP0,1 , iP1,2 are well defined. Then iP0,1 < iP1,2 .
Note that we have not assumed that a low net worth in the long term feeds back on
the banks’ ability to lend in the first period.16 This would make the case for cutting thelong rate even weaker, as there would be an additional motive to raise the long-term ratefurther in order to avoid the drop in long-term net worth that would feed back on the bank’s
risk-taking ability.
16
That is, we could have assumed that ψ L L0,1 ≤ N1 + βN2 , where β is some discount factor relevant toevaluating how much long-term net worth impacts a bank’s ability to take on risk.

16

5

Reversal in a New Keynesian DSGE Model

We now ask whether a reversal interest rate still exists in a quantitatively realistic generalequilibrium setting in which changes in the policy rate can stimulate aggregate demand due
to nominal price rigidities. As in our partial equilibrium model, the capital gains are notenough to offset the fall in the net interest income of banks, decreasing their profitability
and hence threatening their ability to provide productive lending to the economy. Moreover,the decreased pass-through to deposit rates weighs further on bank profitability. However,standard New Keynesian forces operating through inter-temporal substitution and pricerigidity generate an increase in loan demand and hence lift bank profitability. If such lift isstrong enough, it may overturn our partial equilibrium results.17 The lack of deposit ratepass-through mitigates this channel, very much like an economy entering a liquidity trap.
Our calibration quantitatively pins down these forces. We find that a general equilibrium
reversal interest rate still obtains. Moreover, even before the reversal interest rate is reached,
the effectiveness of monetary policy decreases as it approaches the reversal interest rate: inthat sense, our frictions smoothly affect monetary policy’s effectiveness.

5.1

Environment

Time is discrete and the horizon is infinite. Households choose consumption, savings, andlabor supply to maximize their lifetime utility over consumption and leisure. They own
banks and all three types of firms in the economy: intermediate goods producers, retailers,and final goods producers. Some intermediate goods producers require bank loans to sustain
their investment activities. Banks obtain deposits from households and invest these savingsin government fixed-income assets and loans to the bank-dependent intermediate goods producers. Intermediate goods producers competitively sell goods to retailers. These retailers
differentiate these goods at no cost and sell them to competitive final goods producers. Retailers are subject to price frictions, in a New Keynesian fashion. Final goods producers thenbundle retail goods into final goods usable for consumption and capital. Finally, a monetary authority (government) supplies nominal fixed-income assets to banks elastically at aparticular interest rate, taxing (or redistributing gains to) households lump-sum to financesuch assets.
17

With flexible prices, general equilibrium forces will work to alleviate the negative consequences of areversal interest rate, but they cannot undermine its existence.

17

Households. A unit continuum of identical households with separable preferences overconsumption and labor choose consumption Ct , labor supply lt , and deposits Dt in order tomaximize their lifetime utility:max{Ct ,lt ,Dt+1 }

E0

∞
X

β t U (Ct , Ct−1 , lt ),t=0where the per-period utility is given by(Ct − hCt−1 )1−γ
l1+ϕ
U (Ct , Ct−1 , lt ) =
−χ t
1−γ
1+ϕ
subject to their budget constraint
pt Ct + Dt+1 = pt wt lt + (1 + iD
t−1,t )Dt + pt Πt − pt Tt ,where Πt denotes profits coming from retailers, intermediate goods producers, and banks,
and Tt are government lump-sum transfers.
Final goods producers. Final goods producers purchase retail goods j ∈ [0, 1] at pricept (j) and aggregates them into the final good, with production function
Z 1
Yt (j)

Yt =

−1



 −1

dj

0

They then sell these final goods on competitive markets to households for consumption andto intermediate firms for capital investment.
Capital goods producers. Capital goods producers costlessly differentiate final output
goods in an investment good that can be used by intermediate firms as capital. The market
for such goods features perfect competition, with Qt denoting the price of investment. Capital goods producers face adjustment costs on the rate of change of investment. They use
the stochastic discount factor of households to discount profits. The capital accumulation
equation reads:
Kt+1 = (1 − δ)Kt + It (1 − Ξ(It+1 /It ))18

Where Ξ(It+1 /It ) is an adjustment cost function. Given this setting we can write the problemof capital goods producers as:
max E0∞X

β t Λt [Qt It (1 − Ξ(It+1 /It )) − It ]t=0

We choose the following quadratic specification for the adjustment cost function:
κACΞ(It+1 /It ) =
2



2
It+1
−1
It

Retailers. A unit continuum of retailers is indexed by j ∈ [0, 1]. Each produces its own
retail good variety j by costlessly transforming intermediate goods. They face the demand
function for their retail variety derived from the problem of the final goods producers. Retailers are subject to Rotemberg price adjustment costs. The problem of a single retailer j
is then summarized by
max Et{pt+s (j)}

"∞
X

h
− 
β Λt+s pt+s (j)1− p−1
t+s Yt+s − mct+s pt+s (j) pt+s Yt+s
s

s=0

θ
−
2



#
2
i
pt+s (j)1− −1− 1 pt+s (j) pt+s Yt+s ,pt+s−1 (j)where real marginal costs mct are equal to pIt , the price of intermediate goods, and Λt+s =UC (Ct+s ), where Ct+s is the consumption of the representative household.
Intermediate firms. Labor and capital are combined to produce intermediate goods.
These goods are then sold competitively to retailers. Labor is hired competitively at a wage
rate w. Capital is purchased a period in advance from final goods producers, and depreciates
slowly. There are two types of firms in the economy, which differ in two aspects: their accessto financial markets and their productivity.
Intermediate firms are of two types. A share 1 − ξ of firms are neoclassical and obtaincapital directly from households. Specifically, the problem of neoclassical firms is to maximize
their profits, given that they hire labor at the real wage wt and acquire capital a period in
advance at price Qt . In so doing, they value the opportunity cost of capital acquisition at19

the effective rental rate rt−1,t defined by
rt−1,t = QtΛt−1
− Qt+1 (1 − δ)
βΛt

(4)

where δ is the depreciation rate of capital and Qt+1 its resale price. They then produce
output using a Cobb-Douglas production function with TFP Ā and sell it at the given pricepIt . Hence their problem reads as
max pIt Ā ktα `t1−αν`t ,kt− wt `t − rt−1,t kt .

The second type of intermediate firms, the remaining share ξ, are bank dependent. They
produce by combining labor and capital in a Cobb-Douglas production function with productivity A. These firms are born without financial resources, so their entire capital purchasemust be financed with a bank loan; in subsequent periods, they can use retained earnings as asource of financing, which diminishes their need for bank loans. We introduce heterogeneous
duration of firms’ projects in order to obtain a maturity structure for bank loans. Specifically, when a bank-dependent firm is founded, it learns a specific duration τ ∈ {1, ..., T } ofits project, drawn from a distribution Γτ . In their founding period, these firms can choose toenter long-term bank contracts that fix their loan and rate schedules; changing the schedule,
once settled, requires paying a fixed cost Fb .18 Specifically, firms can commit to an interest∗
rate schedule iL∗0,...,τ and associated loan schedule L0,...,τ . Similarly, we assume that these firms
plan their entire investment path given prices as they draw a project. Deviating from thiscapital path in subsequent periods costs a fixed cost Fk .19
Finally, we assume that firms discount using the SDF of households, although frictions
prevent them from being able to draw funds from them.20,21
Given this setting, a firm that drew a project of duration τ at time 0 and commits to a18

This helps us micro-found capital gains from loans on banks’ balance sheets. Capital gains will alsocome from banks’ holding of bonds.
19
In a perfect foresight equilibrium, this is inconsequential. After an unexpected monetary shock, it acts
as an adjustment cost, by allowing only firms with new projects (as well as non-bank-dependent ones) toadjust their capital stock in response to price changes. On top of generating hump-shaped responses ininvestment and loans, this assumption also greatly simplifies solving the banks’ problem.
20
Note that banks are special in our model because they have the ability to grant loans to these firms.
21− 1 in equilibrium, the choice of the households’ SDF is inconsequential.
Since rtL > ΛΛt+1
t20

plan solves
max E0τX

β t Λ t xt

t=0∗
xt + Qt kt − Lt = pIt yt + Qt (1 − δ)kt−1 − wt `t − (1 + rtL )Lt−1 − Fb 1Lt 6=L∗t,0 − Fk 1kt 6=kt,0αyt = Akt−1`1−αt

1 + rtL =1 + iL∗iLt−1 − iL∗
t−1,0
t−1,0
+ 1Lt 6=L∗t,0
1 + πt1 + πtLt ≥ 0, xt ≥ 0,with k−1 = L−1 = 0. In our calibration, it will be the case that the optimal payout rate is
xt = 0 for all t < τ – that is, firms will direct all retained earnings toward investment andminimize the reliance on bank loans. Also, note that in a perfect-foresight equilibrium, firms
will be indifferent as to whether to commit to a plan or not.
Banks. A unit continuum of identical banks exists. We assume that banks pay a fixed dividend rate to their owners.22 Solving the problem of each bank is hence identical to solving
our two-period model repeatidely, using the dividend assumption to obtain the transition
rule for equity. To keep the banks’ problems differentiable, we replace our capital inequality
constraint with smooth leverage costs. We also assume that there is perfect competition forloans, so that loans are priced at marginal costs that include costs from leverage. Specifically, marginal costs have three components: 1) the return on fixed-income assets it , which
represent the marginal opportunity cost of lending; 2) a per-loan unit leverage cost %(γtL ),Ltwhere γtL is the aggregate leverage of the banking sector and γtL = N1,t+1
and Lt , N1,t+1 are,respectively, the aggregate loans and next-period aggregate net worth of the banking system;and 3) a time-invariant per-loan unit cost cL . The loan rate offered on new loans is thengiven byiLt = it + %(γtL ) + cL .
22

The fixed-dividend assumption makes certain that banks do not drive leverage costs %(·) to zero byborrowing from households. This is consistent with the empirical evidence in Gropp et al. (2018). Whenundercapitalized, banks do not recapitalize but instead deleverage by decreasing the size of their balance
sheets or rebalance their assets toward less risky ones. Debt overhang is one underpinning for this behavior(Admati et al., 2017).

21

However, as described in the previous sections, banks also offer firms long term loans withlocked-in rates and quantities. They price these according to marginal costs.23
Next, we describe the capital accumulation process of banks between periods. We assume
that banks accumulate real equity according to the following formula:
Et+1 = (1 − υ)1
Nt+1 ,
1 + πt+1

where υ is a fixed dividend rate, and Nt+1 is the nominal (in period t terms) net worth ofbanks. Dividends are rebated directly to households.
Next, we need to parametrize deposit demand in the model and, in particular, how themargins on deposit vary with the level of interest rates. The data suggests that deposits
rates are quite sticky, with a declining pass-through as money market rates decrease. We
micro-found one explanation for this phenomenon at the extensive margin in Appendix B.24
Simply put, we posit that households become more aware of spreads as rates approach zero or
negative territory. This makes the extensive margin dominate bank mark downs on depositDD
rates so that we can simply set iDt = it − η (it ) and parametrize η (·) to fit the observed
pass-through in the data. Importantly, once they post a rate, banks in our setting are willingto absorb as many deposits as are demanded by households.25
Finally, given the level of equity Et , the loan demand given prices Lt (iLt ), and deposit
demand Dt (iDt ), banks’ balance sheet equation means that the total level of fixed-income
assets invested by banks must be St = Et + Dt − Lt . We furthermore allow banks to invest inP
fixed-income assets of different maturities τ . Let total holdings be St = τ St,τ . These assets
are elastically supplied by the government at fair prices. In the perfect-foresight equilibriumthat we study, banks are indifferent between maturity choices. We pick the maturity choicesof banks to match their empirical maturity structure of assets.
Hence, the representative bank’s net worth isNt+1 = (1 + iLt )Lt (iLt ) +X

pt+1,τ St,τ − (1 + iDt )Dt ,

τ

where pt+1,τ is the price of the fixed-income asset with remaining maturity τ at time t + 1.
23

After a surprise shock, though, banks still pay the leverage costs coming from these fixed-interest loans.
The explanation at the intensive margin would be the presence of cash.
25
This is important, as it can be that iD
t < it . In our two-period model, the liquidity constraint played
the role of preventing banks from shedding their fixed-income assets. Here, banks hold on to their deposits.
24

22

Government and Monetary Authority. The government taxes (or transfers) a lumpsum amount Tt to households, which allows it to pay the nominal interest rates it on fixedincome assets. Ricardian equivalence holds in our economy, making the timing of taxes
irrelevant. The monetary authority follows a Taylor rule, which is assumed to take thefollowing common form:
1 + it=1 + i∗

1 + it−1
1 + i∗

ρi 

1 + πt1 + π∗

(1−ρi )φπ

m

eεt ,

where i∗ is the steady-state policy rate, π ∗ is the steady-state inflation rate, and εmt is amonetary policy shock.

5.2

Solution concept

We calibrate our model to the euro area, where negative rates have been implemented since2014.26 We assume that every agent in the economy has perfect foresight over the future andsolve the deterministic equilibrium after a one-time unexpected monetary policy shock εmt .
Our computational algorithm solves for the full nonlinear system of equations, and hence
does not rely on perturbation techniques. This is important, since our economy inherently
features large nonlinearities.

5.3

Calibration

We set the length of a period to one year. The calibration of the New Keynesian block of
the model is standard, and the parameters are summarized in Table 1.
We calibrate the remaining parameters to match key moments about banks and the
production sector’s dependence on bank lending. Table 2 displays the resulting parameters. We choose the nominal fixed-income asset rate and households’ deposit rates to be,respectively, 2.00% and 0.65%, corresponding to values prior to the low/negative rate environment. We use the EONIA for the fixed-income asset rate, and the deposit rate for
euro-area households in the ECB MIR data. Next, we need to match the pass-throughof fixed-income rates into deposit rates. We choose to do so using the flexible functional
form η D (i) = i − η1 − η2 exp(η3 i), which allows for a decaying pass-through. Concretely, we
fit the parameters η1 , η2 , and η3 such that 1) the steady-state values satisfy the equation
26

The euro area also has a higher share of small and medium-sized firms, and its firms are more bankdependent than firms in the U.S.

23

Table 1: Conventional DSGE Parameters.

Parameter

Description

Value

γ

IES parameter2hHabit formation

0.6

φ

Disutility of labor1δCapital depreciation

0.1

αCapital share

0.33

ν

Scale parameter

0.81



Retail price elasticity

6

θ

Rotemberg cost

60

π

φ

Taylor rule coefficient

1.5

ρi

Taylor rule persistence

0.8

D
iD
SS = iSS − η (iSS ); 2) the average deposit rate during the negative rate environment, which
was iD
neg = 0.1%, matches the average negative rate on fixed-income assets of ineg =-0.3%,
Dthat is iD
neg = ineg − η (ineg ); and 3) the pass-through at steady-state value is perfect, that is,∂iD= 1 in the steady state.27 Figure 8 in the Appendix depicts the resulting pass-through.
∂i
Next, we match banks’ profits to assets to equal the equivalent ratio in the data, whichis 1.80%.28 Given steady-state lending costs, cL adjusts to match this ratio. For leverageκLwith parameter κL and γ̄: it has an asymptote
costs, we choose the specification %(γ) = γ−γ̄as γ → γ̄, mimicking our hard constraint in the two-period model. We pick a maximum
leverage γ̄ of 10%, in accordance with the guideline provided by the ECB for the minimum
leverage ratio. Steady-state leverage costs depend on the capital ratio of banks in the steadystate: we pin it down to 15.5%, the average value reported by Altavilla et al. (2017). As
in Alpanda et al. (2014), we set the capital leverage cost parameter κL so that we matchthe elasticity of rates to changes in capital leverage in the MAG report of the Financial
Stability Board (Macroeconomic Assessment Group, 2010). This report estimates that a 1%
increase in capital leverage results in a 0.28% increase in capital funding costs. Next, we

27This is desirable, as IRFs in the steady state’s vicinity are not affected by the lack of deposit passthrough, which facilitates the comparison with the vanilla New Keynesian model.
28
We sum net interest income and net fee and commission income for euro-area domestic banks and dividethat by the outstanding amounts of assets for the years 2007-2013, e.g., prior to the introduction of negative
rates. The data is from the ECB’s statistics on consolidated banking data.

24

obtain information on the maturity structure of banks from Hoffmann et al. (2018). Within
fixed-income assets, we target a duration of 3.4 years, which we match using a geometricallydecaying structure of banks’ fixed-income asset purchases such that the share of fixed-income
assets expiring in every period is 1 − τS . Within loans, we set the firm project duration (Γτ )such that a firm can draw either a short-term (one year) or long-term (three year) projectupon its foundation, and set the fraction p for the former to 75% so that the average banks’loan maturity is 1.9 years. We set the fixed cost of renegotiating loans Fb and Fk large enoughthat no bank-dependent firms renegotiate their loans or investment plans in equilibrium.29
We set loans to represent 60% of banks’ assets, as in Hoffmann et al. (2018).30 Loan demandthen pins down the banks’ balance sheet size. Next, we pick the investment adjustment cost
parameter κAC to imply an elasticity of investment to a shock to the contemporaneous priceof capital of about 3, generating a value close to Smets and Wouters (2003). Finally, weneed to specify the relative characteristics of two intermediate producer types. We relate ourbank-dependent firms to small and medium enterprises, which represent more than 99% of
all enterprises in the euro area. We calibrate ξ to the value reported by Eurostat’s statistics
on SMEs. However, we adjust the relative productivity Ā/A so that bank-dependent firms
only end up producing 55.8% of output in steady state, consistent with the values reported
in Eurostat. Our neoclassical firms end up being about three times more productive; moreover,by enjoying cheaper borrowing rates, they end up producing significantly more output
than the average bank-dependent firm, which allows us to match the data.

5.4

Results

To study the impact of each marginal innovation on aggregate variables, we generate innovations εm
0 to the Taylor rule of increasing magnitudes. In an economy log-linearized aroundits steady state, the resulting impulse responses scale up proportionality. Our economy, incontrast, features a a nonlinear and non-monotonic response to each marginal innovations.31
Specifically, we first study the effects of a marginal – 10 basis points – negative Taylor
rule innovation in the vicinity of the steady state and report the resulting impulse response29

In the perfect-foresight equilibrium, firms are indifferent between locking in their loan interest rate andcapital plans or not.
30
A degree of freedom exists because total debt, and hence total assets, is not pinned down in equilibrium.
31
Our economy naturally has strong sign dependence: monetary expansions are less effective than contractions, as our frictions (bank leverage costs, deposit rate pass-through) are one-sided frictions. We focus
on expansions.

25

Table 2: Calibrated Parameters.

Parameter

Description

Value

iSS

Steady-state policy rate

2.00%

cL

Loan cost

0.73%

η1

Deposit pass-through parameter

8.2e-4

η2

Deposit pass-through parameter2.6e-4

η3

Deposit pass-through parameter

154

κAC

Investment AC parameter

0.15

κL

Leverage costs parameter

4.8e-3γ̄

Maximal equity-to-capital ratio

10%

Steady-state loans-to-safe-assets ratio

3/2υBank dividends

11.4%

τS

Fixed-income maturity parameter

0.7p

Probability of three-year project draw

75%ξ

Share of bank-dependent firms

99.8%

Relative firm productivity

2.83

LS S/SS S

Ā/Afunction. Then, we generate innovations of larger sizes and study the effect of the last 10
basis point innovation. In other words, we compute three impulse responses: that of a small
shock in the vicinity of the steady state, that of a large shock, and finally a small shockin addition to the large one; we then compare the first IRFs with the difference between
the last two IRFs. We use a Newton algorithm with automatic differentiation as a solution
procedure, iterating on 10 basis point innovations.
Before describing our results, we point out two subtleties of our analysis. First, a reversal
within our experiment necessarily applies to a particular variable at a particular horizon. In
our two-period model, we focused on the lending rate on impact. Due to the creeping up
effect, the reversal rate for the lending rate one or two periods ahead may differ. Similarly,
if one focuses on GDP, the reversal rate will also be different. Second, given that our
economy’s constraints are smooth – in contrast to the sharp constraints of our previous
sections’ stylized models – the economic mechanisms we highlight have consequences before
aggregate variables display a full reversal, and work toward dampening the effectiveness ofrate cuts before reversing it.

26

Figure 3: Impulse response of loan rates to Taylor rule innovations of increasing size, in deviations
from the steady state. The legend contains the size of the respective innovations to the Taylor rule.
The dashed line reflects a reversal of the response, including on impact. For the largest shock, thepolicy rate reaches about -1% on impact. The dotted line shows that a reversal occurs even forsmaller innovations but at a later horizon, in line with our “creeping up” result.

Figure 3 depicts the response of bank lending rates to negative Taylor rule innovations ofincreasing size, and Figure 4 displays the response of physical capital from bank-dependent
firms to the same innovations. As negative monetary innovations become larger, their ability
to stimulate lending rates diminishes, eventually reverting (dashed line) when the level ofnominal rates on impact reaches about -1%.32 Hence, in this particular sense, we estimate
the reversal interest rate to be about -1% in our calibration.
Figure 5 (left panel) reflects the same result, but instead computes the marginal responseof the lending rates to a 0.1% shock after the steady state and after an innovation large
enough to trigger a “reversal” of the impact response of lending rates.33 Figure 5 also contains
the response of loans, and Figures 6 and 7 depict the responses of additional variables.
Some variables experience reversals even at higher rates: in particular, the response ofbank net worth to a marginal shock reverses when policy rates arrive in the vicinity of 0%;loans rates and loan quantities two periods ahead also reverse before -1% is reached. In
contrast, some variables do not experience a reversal at -1%: output on impact, for example,
32

Due to the inflation response in the Taylor rule, a one percent negative innovation to the Taylor ruledecreases the actual nominal rate by less than one percent.
33
Alternatively, taking the first curve, as well as the two differences in the last two curves of Figure 3,
more or less replicates Figure 5 – except that the innovations have a size of 50 basis points.

27

Figure 4: Impulse response of log capital of new bank-dependent firms to Taylor rule innovationsof increasing size, in deviations from the steady state. The legend contains the size of the respective
innovations to the Taylor rule. The dashed line reflects a reversal of the response, including on
impact. For the largest shock, the policy rate reaches about -1% on impact. The dotted line showsthat a reversal occurs even for smaller innovations but at a later horizon, in line with our “creepingup” result.

still rises following a marginal shock, although it does decline two periods ahead. The rightpanel of Figure 5 shows that the response of loans closely follows that of the loan rate,
modulo changes in other prices.
Figure 6 confirms that the reversal in loan rates and loans has aggregate consequences for
output and investment. On impact, the increase in aggregate demand generates an increasein output, as loans do not play a direct role in the production of period 0 output. Investment,
moreover, still increases due to the response of neoclassical firms in the economy, which do
not rely on loans.34 However, in subsequent periods, the rise in loan rates results in depressed
investment in output, which persists for one to four years. These results reinforce the ideathat the contractionary aggregate effects of low or negative interest rate environments may
take some time to materialize.
Finally, Figure 7 shows that the reversal is due to poor bank profitability and risingleverage costs. Note that bank net worth actually increases following a shock around thesteady state, due to the intermediation boom that lifts banks’ net interest income. In
contrast, a cut into negative territory deeply depresses the net worth of banks. This is34

Costs of changing investments, as present in Christiano et al. (2005), for example, would likely weakenthat mechanism.

28

Figure 5: Marginal impulse responses of loan rates and loans to a 10 basis points innovation to the
Taylor Rule in two economies. In the baseline economy (plain line), the shock occurs in the vicinityof the steady state. In the second economy, the marginal shock occurs on top of an innovation
to the Taylor rule that, on its own, would depress the policy rate to about -1% on impact. The
reversal in loan rates has been crossed at this stage.

Figure 6: Marginal impulse responses of output and investment to a 10 basis points innovation
to the Taylor rule in two economies. In the baseline economy (plain line), the shock occurs in
the vicinity of the steady state. In the second economy, the marginal shock occurs on top of an
innovation to the Taylor rule that, on its own, would depress the policy rate to about -1% on
impact. The reversal in loan rates has been crossed at this stage.

29

Figure 7: Marginal impulse responses of bank net worth and leverage costs to a 10 basis points
innovation to the Taylor Rule in two economies. In the baseline economy (plain line), the shock
occurs in the vicinity of the steady state. In the second economy, the marginal shock occurs on topof an innovation to the Taylor rule that, on its own, would depress the policy rate to about -1% on
impact. The reversal in loan rates has been crossed at this stage.

consistent with the evidence documented in Ampudia and Heuvel (2018), who documentthat the response of banks’ stock valuations to monetary policy shocks changes sign as thelevel of interest rates decreases. Following their decrease in net worth, banks are forced
to increase their loan rates in order to lower their leverage costs. Quantitatively, three
key forces shape the response of banks’ net worth. First, banks’ net worth on impact iswell hedged from interest rate risk as they hold long-term assets. Without this hedge, thereversal rate would be substantially larger. However, these assets mature, making net worth
more sensitive in subsequent periods. Second, the impaired deposit rate pass-through aspolicy rates decrease substantially lowers bank profitability, especially as rates enter negative
territory. Finally, this impaired pass-through also weakens the inter-temporal substitution
channel. This decreases the intermediation gains that generated the positive net worth
response close to the steady state.

5.5

The effects of permanently lower rates

We now consider a permanent decrease in the natural rate r∗ and suppose that the steadystate inflation target π SS stays unchanged. This implies that the steady-state nominal

30

rate iSS decreases.35 This permanent shift in the level of steady-state rates reduces theeffectiveness of monetary policy for shocks of comparable magnitude. Concretely, for a shockof a similar magnitude to force our baseline economy into its reversal rate, the economy witha lower iSS has a lower response of aggregates to that shock. This indicates that the reversal
interest rate has not decreased one-for-one with the steady-state nominal rate change, leaving
less leeway for monetary policy.36

6Conclusion

We have shown the conditions for the existence of a reversal interest rate, the rate at whichmonetary policy stimulus reverses its intended effect and becomes contractionary. Its existence relies on the net interest income of banks decreasing faster than recapitalization gains
from banks’ initial holdings of fixed-income assets. We showed that its level depends on themagnitudes of these capital gains, the overall capitalization of banks, the strength of theleverage constraints faced by banks, and the deposit supply elasticity. The reversal interest
rate creeps up over time, making steep but short rate cuts preferable to “low for long” interest rate environments. Finally, we included our banking frictions in a New Keynesian modeland showed that the economics we described have meaningful effects for the transmission ofmonetary policy in general equilibrium.
For the sake of tractability, we have omitted other channels through which monetary
policy can affect banks as well as the real economy. In particular, policies such as ECB’slong term refinancing operations could have alleviated some of the low rates’ effect on bankmargins. Moreover, we have omitted the explicit modeling of risk; hence, we have remained
agnostic on how low rates change nonperforming loans and the associated responses in provisions. We see these as important quantitative refinements for future research. Finally, weview our results as driven by unusual surprise movements in interest rates: low-for-long andnegative rates were largely unforeseen events. It remains a question whether banks can andwill adjust to a permanently lower interest rates environment – for example, by increasing
their maturity mismatch. The competitive landscape faced by banks could also change, withdepositors growing accustomed to the possibility of negative interest rates, hence supporting
35

Effectively, β permanently shifts upward.
The maintained assumptions here are that 1) banks do not change their maturity structure in responseto that change, and 2) the pass-through of deposit rates stays as described by the data. Our analysis
here is purely positive. We leave it to future research to determine whether such responses have happened
empirically.
36

31

banks’ profitability in negative-rate environments.

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A

ProofsLemma 1
Proof. The Lagrangian of this problem is
L =(1 + iL )L + (1 + i)S − (1 + iD )D − µ(L + S − D − E0 (i)) + λD (ψ D D − S)− λL (ψ L L − (1 + iL )L − (1 + i)S + (1 + iD )D).
The first-order conditions with respect to S, iL and iD areµ = (1 + λL )(1 + i)


1
L L
L L
L
µ − (1 + λ ) 0 + λ ψ
1+i =1 + λL
L


1
L D
D D
D
µ − (1 + λ ) 0 + λ ψ .
1+i =1 + λL
D
0

0

D L
L
Define D
D =  , L = − . Rearrangement of these first-order conditions then yields the equations
in Lemma 1.

Lemma 2
Proof. We can write a bank’s problem as
N (i) = max (1 + iL )L + (1 + i)S − (1 + iD )D
iL ,iD ,Ss.t. L + S = D + E0 (i), ψ L L ≤ (1 + iL )L + (1 + i)S − (1 + iD )D, ψ D D ≤ S.
The envelope condition of this problem impliesdN= µE00 (i) + (1 + λL )S,diwhere the Lagrange multipliers µ and λL are as defined in Lemma 1. This yieldsdN= (1 + λL )(S + (1 + i)E00 (i)).
diWrite
CG = E00 (i), N II = iL L + iS − iD D.

35

Combining these definitions with the expression for dN
di , we obtain


dN
dN II
= (1 + λL )+ (1 + i)CGdidi
as desired.
Finally, note that when ψ D > 0, banks’ holding of fixed-income assets (weakly) increases,
making the derivative higher.

Proposition 1
Proof. Define N (i) as in the proof of Lemma 2. First, we show that N (i) is an increasing function.
0
L
From the previous lemma above, we know that dNdi ≥ µE0 (i) + (1 + λ )S. Given that we can boundthe first term by assumption, it suffices to show that S > 0. But that is immediate given ourassumption of a liquidity constraint.
Thus the derivative of N (i) is bounded away from 0 for i < i0 . That is, there exists some G(i0 )such that:
N (i) ≤ N (i0 ) − G(i0 )(i0 − i).
When the capital constraint does not bind, the quantity of loans made by the bank is given by
L(iL∗ ), where iL∗ satisfies the equationiL∗ = i +1εL (iL∗ ).

Note that L(iL∗ ) is decreasing in i. For sufficiently low i, then,ψ L L(iL∗ ) ≥ N (i0 ) − G(i0 )(i0 − i) ≥ N (i),meaning there exists a largest interest rate î such that the capital constraint binds for all i < î. In
1 dN
RR = î, since dL < 0 for all i < î andthis region, L(iL ) = ψ1L N (i), so dL
di = ψ L di < 0. Therefore ididLdi > 0 for i > î.

Main Lemma
Suppose i0 > iRR when the parameters of the bank’s problem are θ = (ψ L , ψ D , e0 , e0 (i)). Underan alternative set of parameters θ̂ = (ψ̂ L , ψ̂ D , ê0 , ê0 (i)) such that N (i, θ) > N (i, θ̂) for i ≤ i0 , thereversal interest rate is lower under parameters θ than under θ0 (so long as it is unique under bothsets of parameters).
Proof. Define iL∗ (i) implicitly as the solution to the equationiL −L(iL )= i.
L0 (iL )Note that iL∗ (i) is increasing in i, so L(iL∗ (i)) is decreasing in i. Furthermore, iL∗ does not depend36

on parameters. The reversal interest rate iRR is the solution to the equation
ψ L L(iL∗ (i)) = N (i, θ).
Let iRR (θ) be the reversal interest rate under parameters θ. With parameters θ0 , for any i ≤ iRR (θ)
we haveψ L L(iL∗ (i)) ≥ N (i, θ) > N (i, θ0 ).
This is because by the definition of the reversal interest rate, the function N (i, θ) must be increasing
in i in the region i ≤ iRR (θ). Thus it cannot be that iRR (θ0 ) ≤ iRR (θ).

Proposition 2
Proof.
1. Clearly, an increase in the capital constraint ψ L weakly decreases N (i) for all i. Then, by
the main lemma, the reversal interest rate must increase.
2. Consider a shift in the interest rate sensitivity of equity such that E0 (i0 ) remains constant0
but dedi is uniformly increased for all i. Then N (i) is uniformly increased for all i ≤ i0 ,since capital gains following an interest rate cut are larger. By the main lemma, the reversal
interest rate must decrease.
3. Note that the first-order condition for iD implies that iD is the same in both economies fora given level of i < iRR , so DA (i) = DB (i). Then the equation
Lj (i) + Sj (i) = Dj (i) + E0j (i)for j ∈ {A, B} implies LA (i) + SA (i) < LB (i) + SB (i), since E0A (i) < E0B (i) for all i.
Furthermore, note that when i = iRR , iL is the same in both economies, so it must be thatSA (i) < SB (i). ThusdN
= (1 + λL )(S(i) + (1 + i)E00 (i))
dimust be larger in economy B when evaluated at iRR because S(iRR ) is larger. Thus
dLA (i)dLB (i)
L dNA (i)L dNB (i)
= ψA
< ψB
=
di
di
di
di
A (i)
B (i)
at iRR , so the balance sheet constraints Lj (i) + Sj (i) = D(i) + E0j (i) yield dSdi
> dSdiat
RR
i = i . But clearly, then, we can keep iterating this argument to obtain SA (i) < SB (i) for
all i < iRR , which then implies LA (i) > LB (i) for all i < iRR through the equation for dN
diabove (using the fact that the constraint is tighter in economy B).

Proposition 3
Proof.

37

1. A higher lower bound on the deposit rate lowers N (i), so the main lemma implies that iRR
increases.
2. Lower markdowns as rates decrease must lower N (i) uniformly, so again the main lemma
applies.
3. An increase in the liquidity coefficient ψD can only lower N (i). Therefore, iRR must increaseby the main lemma.

Proposition 4
Proof. A larger υ0 must make N (i) (weakly) increase, so iRR must weakly decrease by the main
lemma.

Proposition 5
Proof. Here we consider a perturbation E0 (i) → Ẽ0 (i) such that Ẽ0 (i) ≥ E0 (i) if and only if i ≥ i0 .
In particular, for i < i0 , Ẽ0 (i) < E0 (i), so N (i) is shifted uniformly downward to below i0 . Hence,by the main lemma, iRR must increase.

Proposition 6
Proof. There are four possible cases: 1) the capital constraint does not bind in either period at an
optimum, 2) the capital constraint binds in both periods, 3) the capital constraint binds only inthe first period, and 4) the capital constraint binds only in the second period. We consider thesecases in turn.
1
1. If the bank is unconstrained in both periods, iL
t = it−1,t + εL for t = 1, 2. Thus it is possibletto increase the quantity of loans made in both periods by decreasing both i0,1 and i1,2 by asmall constant  > 0. Therefore, case 1) is never optimal for the central bank.

2. If the capital constraint binds in both periods, it is possible to increase the bank’s net
worth in both periods by increasing i0,1 as long as B0,1 and B0,2 are sufficiently low that
B0,1
B0,2
dN II1ddi0,1 > di0,1 ( 1+i0,1 + (1+i0,1 )(1+i1,2 ) ). This increase in interest rates at t = 1 then increases
the quantity of loans made at t = 1 and t = 2, so it is never optimal for the constraint tobind in both periods.
3. When the capital constraint binds only in the first period, the central bank can increase
the net worth of the bank (and thus the quantity of loans made) by cutting i1,2 , since thisB
B0,2
increases the value of equity E0 (i0,1 , i1,2 ) = 1+i0,1+ (1+i0,1 )(1+iand N1 is increasing in
0,1
1,2 )
equity. This cut in i1,2 does not change the quantity of loans made at t = 2 because thebank’s choice of iL
2 is unconstrained by assumption. Hence it cannot be that the capitalconstraint binds only in the first period.

38

4. Given the analysis of the three cases above, it must be that the capital constraint binds
only in the second period. We now argue that it must be exactly binding (in the sense thatL
λL
2 = 0) when B0,2 is sufficiently low. Suppose λ2 > 0. Then the quantity of loans made in
the second period is increasing in the bank’s net worth N2 in period 2, as ψ L L2 = N2 . Achange in the interest rate at t = 2 has two effects: it changes the value of bank equity at
t = 1, which feeds into t = 2 net worth, and it directly impacts net interest income at t = 2.
Formally,
N2 (i0,1 , i1,2 ) = max (1 + iL )L + (1 + i1,2 )S − (1 + iD )D
iL ,iD ,Ss.t. L + S = D + E1 (i0,1 , i1,2 ), ψ L L ≤ N2 , ψ D D ≤ S.
Then the envelope theorem implies


dN2
dE1
L
∗
= (1 + λ ) S + (1 + i1,2 )
.
di1,2
di1,2
Note that using the formula E1 = (1 − ν)N1 , we can write
dE1
dN1 dE0= (1 − ν)di1,2
dE0 di1,2
B0,2dN1= −(1 − ν).
dE0 (1 + i0,1 )(1 + i1,2 )2
1
Recall from the proof of Lemma 1 that dN
dE0 = 1 + i0,1 when the capital constraint does notbind. Thus


B0,2
dN2
L
∗
= (1 + λ ) S − (1 − ν),
di1,2
1 + i1,2

so net worth is increasing in i1,2 when B0,2 is sufficiently small. Therefore, when B0,2 is small
and the constraint in the second period binds, loans are increasing in i1,2 . Loans in the first
period do not depend on i1,2 because the capital constraint is slack at t = 1. The capitalconstraint must then bind exactly at t = 2.
The proof in part 1) to raise interest rates in the first period as long as the constraint in thefirst period is slack as well. Consider setting i0,1 = i0,2 = î such that î is the highest interest
rate for which the constraint binds in the second period. We now show that under theseinterest rates, the constraint in the first period will be slack. By assumption, B0,1 and B0,2
are small enough that N1 (î, î) < N1 (i∗ , i∗ ). Then
E1 (î, î) = (1 − ν)N1 (î, î) < (1 − ν)N1 (i∗ , i∗ ) = E1 (i∗ , i∗ ) = E0 (i∗ , i∗ ),so equity in the second period is lower than in the first. By the main lemma, the reversal
interest rate must be higher in the second period because net worth is increasing in equity.
Hence at î, the capital constraint in the first period must be slack, so by the argument above
it is never optimal to increase i0,1 from î, meaning that at an optimum i0,1 ≤ i1,2 .

39

B

A Micro-Foundation for Deposit StickinessWe describe in this section a micro-foundation for the lack of pass-through on deposits. Each bankis naturally associated with a continuum of depositor households. Their deposit supply is sticky inthe sense that the depositors shop around for better deposit rates if the spread between i and thedeposit rate iD
j from their associated bank j is larger than some threshold, the “activation level,”Dη (i). Hence, banks compete on prices, but only if the spread they charge relative to some baseline
rate i is large enough.37
Importantly, we assume that the search “activation level” η D (i) is decreasing in the interest
rate i. In other words, depositors become more sensitive to spreads when the policy rate is low.
This generates pressure on the extensive margin of banks’ deposit margins as rates decrease. For
example, depositors are more prone to switch banks if the interest rate is negative, as empirically
documented in Hainz et al. (2017). In addition, bank deposit rate choices are also driven by intensive
margin considerations. That is, conditional on keeping a customer, the bank might decide to offer
an attractive interest rate to ensure that the customer supplies a sufficient amount of deposits
instead of simply consuming his income or substituting to alternative savings vehicles like cash.
Concretely, each depositor household h ∈ [0, 1] in the continuum associated with bank j has
an activation level η D (i). He only considers looking at the rates offered by competing bank j 0 ifDD
the rate offered by his bank, iD
j , is below i − η (i). We assume that ∂η (i)/∂i ≥ 0, that is, theactivation level is increasing with the interest rate level, so that spreads are less tolerated at low
levels of the policy rate.
D
Let us denote iD−j ≡ {ij 0 }j 0 6=j as the vector of competitors’ deposit rates. Hence, the share of
costumers ϑD
j that actually stay with bank j is
D D
ϑD
j (ij ; i−j , i) ≡ 1{i−iD ≤η D (i) ∨ iD > maxj 0 6=j iD0 } .
jjj

We can then decompose the residual deposit supply faced by bank j as consisting of an extensive
and an intensive margin:D
D D D
Dj (iDj ; i−j , i) = ϑj (ij ; i−j , i)|{z}Extensive margin

×

d(iD
j )
| {z }

Intensive margin

Provided the intensive demand for deposits is sufficiently inelastic – which is true in our calibration– the extensive margin dominates and the rates are given byDiD
t = it − η (it )D
Figure 8 depicts the pass-through of it in iDt for our calibration of η (it ).

37

Varying markdowns at the extensive margin can be modeled in numerous ways. A large literature
focuses on switching costs (Klemperer, 1995), which is sometimes applied to banking (as in Sharpe (1997)).
Our goal here is to have a realistic yet parsimoniously parametrized model that easily fits in a New Keynesianmodel.

40

Figure 8: Pass-through of nominal rates into deposit rates given the calibrated values of η1 , η2 , η3 .
The pass-through is assumed to be one-for-one for policy rates above the steady state (i > iSS ).

41