Financial Cycles with Heterogeneous Intermediaries
Nuno Coimbra

Hélène Rey

Paris School of Economics

London Business School, CEPR and NBER

June 6, 2016
Abstract
This paper develops a dynamic macroeconomic model with heterogeneous financial intermediaries and endogenous entry. It features time-varying endogenous
macroeconomic risk that arises from the risk-shifting behaviour of financial intermediaries combined with entry and exit. We show that when interest rates are
high, a decrease in interest rates can stimulate investment and increase financial
stability. In contrast, when interest rates are low, further stimulus can increasesystemic risk and induce a fall in the risk premium through increased risk-shifting.
In this case, the monetary authority faces a trade-off between monetary stimulus
and financial stability.
Keywords: Banking, Macroeconomics, Monetary Policy, Risk-shifting, Leverage, Financial cycle.

1

Introduction

The recent crisis has called into question our modeling of the macro-economy andof the role of financial intermediaries. It has become more obvious that the financial
sector, far from being a veil, plays a key role in the transmission of shocks and in
driving fluctuations in aggregate risk. Macroeconomic models have long recognized theimportance of capital market frictions for the transmission and the amplification ofshocks.
In the literature featuring a collateral constraint (see e.g. Bernanke and Gertler(1989), Kiyotaki and Moore (1997)), agency costs between borrowers and lenders introduce a wedge between the opportunity cost of internal finance and the cost of externalfinance: the external finance premium. Any shock lowering the net worth of firms,
households or banks can cause adverse selection and moral hazard problems to worsen,
as the borrowers stake in the investment project varies, increasing the size of the externalfinance premium. As a result this leads to a decrease in lending and a fall in aggregate
demand. Other recent models where a financial market friction plays a key amplifying
role are Mendoza (2010), Mendoza and Smith (2014), Gertler and Kiyotaki (2015),
Gertler and Karadi (2011) who use a collateral constraint in quantifiable frameworks1 ;Brunnermeier and Sannikov (2014) and He and Krishnamurthy (2013) where intermediaries cannot raise more than a fixed amount of equity; and Adrian and Shin (2010),
Coimbra (2015) and Adrian and Boyarchenko (2015) where intermediaries face value atrisk constraints. Financial intermediation not only plays a role in transmitting shocks
but also in generating endogeneous increases in macroeconomic risk. In Brunnermeierand Sannikov (2014), for example, the economy may spend time in low asset price andlow investment states. As a consequence of the existence of such suboptimal paths,
macroeconomic risk may increase and will do so in periods where asset prices tend to
be depressed and financial intermediaries underinvest. He and Krishnamurthy (2014)
develop a model to quantify systemic risk, defined as the risk that financial constraints
bind in the future.
This paper develops a simple general equilibrium model of monetary policy transmission with a risk-taking channel, in which the risk of intermediary default increases inperiods of low volatility, low interest rate and compressed spreads, as observed during thepre-crisis period between 2003 and 2007. It provides a very precise definition of systemicrisk as a state that would trigger generalized solvency issues in the financial sector that
require government intervention. This is achieved by building a novel framework with a
1

See also Gertler et al. (2012), Curdia and Woodford (2010), Farhi and Werning (2016), Aoki et al.
(2016)

2

continuum of financial intermediaries heterogeneous in their value-at-risk constraints
and a moral hazard friction due to limited liability that leads to risk-shifting. Value-at
risk constraints are realistic features of the regulatory environment; they are embeddedin Basel II and Basel III. They also reflect the practice of internal risk management infinancial intermediaries, whether as a whole or for specific business lines within banks.
Their heterogeneity may reflect heterogenous risk attitudes on the demand side ordifferent implementations of regulatory constraints across institutions. We also assume
deposit guarantees which are a widespread institutional feature.2
Value-at-risk constraints to model financial intermediaries have been used in a
number of papers (see for example Danielsson, Shin and Zigrand (2010) or Adrianand Shin (2010, 2014)). Under a slightly different formulation of the value-at-risk
constraint, Adrian and Boyarchenko (2015) describe an intermediary leverage cycle
using a representative intermediary. In their model, leverage is highly procyclical andthere is a trade-off between capital requirements, which reduce the risk of intermediary
distress (defined as the constraint being binding), and the pricing of risk. Allen andGale (2000) have shown that current and future credit expansion can increase risk
shifting and create bubbles in asset markets, while Nuño and Thomas (2013) showthat the presence of risk-shifting creates a link between asset prices and bank leverage,
generating banking cycles. In Mendoza and Smith (2014) bankers determine their
exposure to systemic shocks by trading off the risk-shifting gains due to limited liabilityof banks with the value of preserving their capital after a systemic shock. Like us, Fostel
and Geanakoplos (2012) emphasize financial frictions and heterogeneity in investors
to generate fluctuations in asset prices. One of the contributions of our paper is to
study the interplay between risk-shifting and a heterogenous pool of intermediaries,
thereby generating macroeconomic risk fluctuations and movements in the risk premium.
Another contribution is to provide an intuitive and clearly defined measure of systemicrisk. Finally, our paper is closely related to the growing literature on the risk taking
channel of monetary policy (Borio and Zhu (2012), Bruno and Shin (2015)).
In the model, financial intermediaries collect deposits from households. Deposits
are guaranteed by the government. They also invest and hold shares in the aggregate
capital stock, which provide a risky return. Realistically, financial intermediaries have
limited liability, which introduces a risk-shifting motive for investment and mispricingof risk. Those with a looser value-at-risk constraint will have a higher option value of
default, which will generate pricing effects of entry and exit in risky financial markets.
Both the aggregate capital stock and the risk premium of the economy are determined2

We are therefore abstracting from the important literature on bank runs (see e.g. Diamond andDybvig (1983), Diamond and Kashyap (2016), Gertler and Kiyotaki (2015)).

3

by an extensive margin (which financial intermediaries lever) and an intensive margin(how much does each lever). This is the key novel feature of the model. Having variationin the intensive and the extensive margins generates both movements in aggregate
leverage and asset pricing implications which are unusual in our models but seem tobear some resemblance with reality. While output and consumption seem to varymonotonically with the interest rate in short horizons, the underlying financial structure(and systemic risk) is non-monotonic. We explain here the basic economic intuition
behind the workings of the model.
Our model features an endogenous non-linearity in the trade-off between monetary
policy and financial stability. When the level of interest rates is high, a fall in interest
rates leads to entry of less risk-taking intermediaries into the market for risky projects.
The average intermediary is then less risky, so a fall in interest rates (i.e. a monetary
expansion) has the effect of reducing systemic risk and expanding the capital stock.
There is no trade-off in this case between stimulating the economy and financial stability.
However, when interest rates are very low, a monetary expansion leads to the exit of
the least risk-taking active intermediaries, which are priced out of the market by alarge increase in leverage of the more risk-taking ones. This increases systemic risk inthe economy despite positive effects on the aggregate capital stock, which is always
increasing with a fall in interest rates. For this region, the intensive margin growthin leverage dominates the extensive margin fall as interest rates are reduced. In otherwords, the most risk-taking intermediaries increase their leverage so much that they
more than compensate for the exit of the least risk-taking ones. There seems to bea clear trade-off between stimulating the economy and financial stability. Because oflimited liability there will be over-investment, which will be even larger when the sector
is dominated by more risk-taking intermediaries. Of course, the level of the interest rate
is itself an outcome of the general equilibrium model and therefore a fixed point problem
has to be solved. This non-linearity constitutes a substantial difference from the existing
literature and is a robust mechanism coming from the interplay of the two margins. It
provides a novel way to model the risk-taking channel of monetary policy analysed in
Borio and Zhu (2012) and Bruno and Shin (2015). Recent empirical evidence on therisk-taking channel of monetary policy for loan books has been provided by Dell’Ariccia
et al. (2013) on US data, Jimenez et al. (2014) and Morais et al. (2015), exploiting
registry data on millions of loans of the Spanish and Mexican Central Banks respectively.
There are several important advantages of this novel set up to model financial
intermediation. First, it takes seriously the risk-taking channel in general equilibrium
and therefore allows the joint study of the usual expansionary effect of monetary policy-via a boost in investment - and of the macroeconomic financial stability risk, which isendogenous. Monetary policy is modeled as a reduction in the funding costs of financial
4intermediaries. An extension of the model featuring nominal variables is left for future
work. Second, it is able to generate periods of low risk premium which coincide withperiods of high endogenous macroeconomic risk. This happens when the market isdominated by more risk-taking intermediaries which also feature high levels of leverage.
These periods also correspond to high levels of over-investment and inflated asset prices
due to stronger risk-shifting motives. Thirdly, the model is highly tractable as it iscrafted in a way such that the financial intermediation building block, although rich,can be inserted in a general equilibrium macroeconomic model. Fourthly, because themodel introduces a simple way to model financial intermediary heterogeneity, it opens
the door to a vast array of empirical tests of the monetary transmission mechanism
based on microeconomic data on banks, shadow banks, asset managers, and so on.
Indeed the heterogeneity can be in principle matched in the data with actual companies
or business lines within companies and with their leverage behaviours.
Section 2 of the paper describes the model. Section 3 describes the main results inpartial equilibrium, thereby building intuition. Section 4 shows the general equilibrium
results and the response to monetary policy shocks. Section 5 looks at some empiricalevidence for the cross-sectional implications of the model. The case of financial crises
with costly intermediary default is analyzed in section 6 and section 7 concludes.

2

The Model

The general equilibrium model is composed of a representative household, a continuumof financial intermediaries, and a very simplified Central Bank and government. Thereis only aggregate risk, in the form of productivity shocks and monetary policy shocks.

2.1

Households and the production sector

The representative household has an infinite horizon and allocates her budget in order
to consume a final good CtH . She finances her purchases using labour income Wt andreturns from a savings portfolio. We assume that the household has a fixed labour
supply and does not invest directly in the capital stock Kt .3 It can either save using a
one-to-one storage technology StH and/or purchase deposits DtH from financial intermediaries at price qt . Intermediaries use deposits, along with inside equity ωt , to invest
3

Given households are risk-averse and intermediaries are not (and also engage in risk-shifting),
relaxing the assumption households cannot invest directly would make no difference in equilibrium
unless all intermediaries are active and constrained. There are also little hedging properties in theasset, since the correlation of the shock to returns with wage income is positive. In the numerical
exercises, it is never the case that all intermediaries are active and constrained, so to simplify notationand clarify the household problem we exclude this possibility by assumption.

5

in capital and storage. In Section 4 we will introduce monetary policy as a source ofwholesale funding. Monetary policy will therefore affect the weighted average cost offunds for intermediaries.
The production function combines labour and capital in a typical Cobb-Douglas
function. Since labour supply is fixed, we normalize it to 1. Output Yt is then produced
according to the following technology:θ
Yt = Zt Kt−1
log Zt = ρz log Zt−1 + εzt
εzt ∼ N (0, σz )

(1)
(2)
(3)

where Zt represents total factor productivity. θ is the capital share, while εzt is theshock to the log of exogenous productivity with persistence ρz and standard deviation
θ−1
σz . Firm maximization then implies that wages Wt = (1 − θ)Zt Kt−1
and returns on aθ−1Kunit of capital Rt = θZt Kt−1 + (1 − δ).
The household program can then be written as follows:max∞{Ct ,StH ,DtH }

E0

∞
X

β t u(CtH )

s.t.

t=0
H
H
H
H
H
+ St−1+ Wt − Tt
Ct + qt Dt + St = Dt−1

(4)

t=0

∀t

(5)

where β is the subjective discount factor and u(.) the period utility function. Tt arelump sum taxes and StH are savings invested in the one-to-one storage technology. Notethat the return on deposits is risk-free despite the possibility of intermediary default.
The reason is that deposits are guaranteed by the government, which may need toraise taxes Tt in the event intermediaries cannot cover their liabilities. Households
understand that the higher the leverage of intermediaries, the more likely it is forthem to be taxed in the future. However, they do not internalize this in their individual portfolio decisions since each household cannot by itself change aggregate deposits.
The return on storage is also risk-free, which implies that households will be
indifferent between deposits and storage if and only if qt = 1. Therefore, they will
not save in the form of deposits if qt > 1 and will not invest in storage if qt < 1.
In equilibrium, the price of deposits will be bounded by the unity return on storage,
implying that qt ≤ 1. In the case qt = 1, the deposit quantity will be given by financial
intermediary demand, with the remaining household savings being allocated to storage.

6

2.2

Financial intermediaries

The financial sector is composed of financial intermediaries which fund themselves
through inside equity and household deposits4 . They use these funds to invest in
the aggregate capital stock. They benefit from limited liability. Intermediaries arerisk neutral and maximize expected returns subject to a Value-at-Risk constraint. To
capture the diversity of risk attitudes among financial intermediaries, we assume thatthey are heterogeneous in the maximal probability of default αi , the key parameter inthe VaR constraint. The maximal probability of not being able to repay stakeholders(shareholders and depositors) varies across intermediaries and is distributed according
to measure G(αi ) with αi ∈ [α, α].
The balance sheet of intermediary i at the end of period t is as follows
Assets Liabilitieskitωti
qt ditwhere kit are the shares of the aggregate capital stock held by intermediary i; dit thedeposit amount which was contracted at price qt and ωti the inside equity.

2.2.1

Value-at-Risk constraint

Financial intermediaries are assumed to be constrained by a Value-at-Risk condition.
This condition imposes that intermediary i invests in such a way that the probability
its return on equity is negative must be smaller than an exogenous intermediary-specificparameter αi .5 The VaR constraint for intermediary i can then be written as:
K
Pr(Rt+1
kit − dit < ωti ) ≤ αi

(6)

The probability that net profit is smaller than starting equity must be less or equal
than αi . This constraint is not only in the spirit of the Basel Agreements (limiting
downside risk and preserving an equity cushion), but Value-at-Risk techniques arealso used by banks and other financial intermediaries to manage risk internally. It
also has the property of generating procyclical leverage, which can be observed in thedata as described in Geanakoplos (2011) and Adrian and Shin (2014) when equity is
4

We will extend the funding options to include wholesale funding in section 4.
Alternatively we could posit that the threshold is at a calibrated non-zero level of losses. There isa mapping between the distribution G(αi ) and such a level, so for any level we could find a G̃(αi ) thatwould make the two specifications equivalent given expected returns. We decide to use the current one
as it reduces the parameter space.
5

7

measured at book value. Using a panel of European and US commercial and investmentbanks Kalemli-Ozcan et al. (2012) also provide evidence of procyclical leverage and ofimportant cross-sectional variations across types of intermediaries. Heterogeneity in theparameter of the value-at-risk constraint can be rationalized in different ways. It could
be understood as reflecting differentiated preferences for risk-taking or differentiatedimplementation of regulatory requirements. For example, the Basel Committee undertook a review of the consistency of risk weights used when calculating how much capital
global banks put aside for precisely defined portfolio. When given a diversified test
portfolio the global banks surveyed produced a wide range of results in terms of modeled
value-at-risk and gave answers ranging from 13 million to 33 million euros in terms ofcapital requirement with a median of about 18 million (see Basel Committee on Banking
Supervision (2013) p.52). Some of the differences are due to different models used,some to different discretionary requirements by supervisors and some to different risk
appetites, as ”Basel standards deliberately allow banks and supervisors some flexibility
in measuring risks in order to accommodate for differences in risk appetite and local
practices” (p.7).

2.2.2

Intermediary investment problem

We assume that intermediaries live for two-periods, receiving an endowment of equity
ω in the first and consuming their net worth in the second, if it is positive. This is a
simplifying assumption, but in the data, book value equity is indeed very sticky. We
show in Figure (8) in the appendix the almost one-for-one correlation between size of
assets at book value and debt for a sample of banks, as well as the correlation withbook value equity. Balance sheet expansions and contractions tend to be done throughchanges in debt and not through movements in book value equity. Other papers in theliterature assume that a maximum amount of equity can be raised (Brunnermeier andSannikov (2014), He and Krishnamurthy (2014)) or that dividends pay outs are costly
as in Jermann and Quadrini (2012).
Net worth consumed by financial intermediaries is denoted by cit .6 When net profits
are negative, cit = 0 and the government repays depositors as it upholds deposit insurance. This is a pure transfer, funded by a lump sum tax on households. Hence, in ourmodel, households are forward-looking and do intertemporal optimization while most of
the action in the intermediation sector comes from heterogenous leverage and risk-taking
in the cross-section. This two-period modeling choice is made for simplicity and allows
us to highlight the role of different leverage responses across financial intermediaries.
When intermediary j is inactive, then cjt = ω as they consume the return of the storage technology
is one.
68Other papers in the literature have used related assumptions for example exogenous
death of intermediaries in Gertler and Kiyotaki (2015) or difference in impatienceparameters in Brunnermeier and Sannikov (2014).
Each intermediary will have to decide whether it participates or not in the market
for risky assets (participating intermediary versus non participating inetrmediary) and,conditionally on participating whether it uses deposits to lever up (active intermediary)or just invests its own equity (passive intermediary).
Each intermediary is assumed to be a risk-neutral price taker operating in a competitive environment. It maximizes net worth over the next period by picking kit , under
the VaR constraint, while taking deposit prices qt and asset return distributions RtK (ε)
as given. The program of a participating intermediary (i.e. an intermediary whichinvests in the risky capital stock) is given by: K
Vitp (Kt , Zt ) = max Eit Rt+1
kit − dit
(7)
K
s.t. Pr(Rt+1
kit − dit < ωti ) ≤ αi(8)where αi is the Value-at-Risk threshold, or the maximum probability of not being able
to repay stakeholders fully.
Intermediaries can also choose not to participate in risky financial markets. In thiscase, they simply invest all their equity in the storage technology and collect it at the
beginning of the next period. The value function of a non-participating intermediary issimply:Vitnp (Kt , Zt ) = ωti
2.2.3

(9)

Limited liability

The expectations operator Eit is indexed by i because of limited liability. The presenceof limited liability truncates the profit function at zero, generating an option value of
default that intermediaries can exploit. For a given expected value of returns, a higher
variance increases the option value of default as intermediaries benefit from the upside
but are insulated from the downside. This means that for a given kit and dit we have
that


 K

 K

K
Et max(0, Rt+1
kit − dit ) ≡ Eit Rt+1
kit − dit ≥ Et Rt+1
kit − dit(10)9

with the inequality being strict whenever the probability of default is strictly positive.
Financial intermediary consumption cit and deposit insurance transfers tit are given by:icit = 1πi ≥0 RtK kt−1
− di,t−1
(11)

i
K i
tt = 1πi <0 di,t−1 − Rt kt−1(12)iwhere πti ≡ RtK kt−1
− di,t−1 are total portfolio returns net of liabilities. The indicatorfunction selects the appropriate case depending on whether intermediary i can repay
its liabilities or not. Total intermediary consumption CtI and aggregate transfers/taxes
Tt are given by integrating over the mass of intermediaries:
Z
I
Ct = cit dG(αi )(13)Z
Tt = tit dG(αi )(14)For now we assume default is costless in the sense that there is no output cost when
the government is required to pay deposit insurance. In another section, we will drop
this assumption of costless default by having a more general setup that allows for alower return on assets held by distressed intermediaries.

2.3

Business model choice and financial market equilibrium

Financial intermediaries are price takers, therefore the decision of each one depends
only on the expected return on assets7 and the cost of liabilities. Since the mass of anintermediary is zero, balance sheet size does not affect returns on the aggregate capitalstock. Intermediary i will decide to participate in the market for risky assets wheneverVip ≥ V np . He will invest in storage otherwise and is then called a non-participatingintermediary. This condition determines entry and exit into the market for risky capitalendogenously.
There is however another important endogenous participation decision. Intermediaries which participate in the market for risky assets have to choose whether to leverup and, if they do, by how much. We will refer to the decision to lever up or not,i.e. to enter the market for deposits as the extensive margin. We will refer to the decision regarding how much to lever up conditional on levering up as the intensive margin.
Financial intermediaries which lever up are called active intermediaries. Active
intermediaries have a risky business model. Financial intermediaries which participate
7

Taking into account limited liability.

10

in the market for risky capital but do not lever up are called passive intermediaries.
They have a safe business model. If the intermediary decides to lever up, it will do soup to its value-at-risk constraint 8 . Hence all active intermediaries (with risky businessmodels) will be operating at their constraint.
Intensive margin (optimal investment for active intermediaries):z

Let Zte ≡ E(Zt+1 ) = Ztρ . For an active intermediary i deciding to lever up, the following
VaR condition will bind: iPr πt+1≤ ω = αi(15)After some straightforward algebra, we obtain the following:

 1−θ

Kt
Pr εt+1 ≤ log(1 − ω/kit (1 − qt ) − qt (1 − δ))= αiqt θZte(16)

The leverage of an active intermediary i (defined as assets over equity ) is given by:kit(1/qt − 1)=ω1/qt − (1 − δ) − θZte K θ−1 exp(F −1 (αi ))(17)Where F −1 (αi ) is the inverse cdf of the technology shock evaluated at probability αi .
Given the monotonicity of the cdf, leverage will be increasing in αi .
Extensive margin (endogenous participations):
An intermediary can decide to invest in risky capital markets or in the storage
technology. If they decide to not participate at all, then their value function is simply
ω given the unit returns on storage.
If an intermediary decides to participate in the market for risky projects it then has
to decide in favour of a risky business model (levered) or a safe business model (onlyinvest its equity).
Let V L denote the value function of active intermediaries (who decide to lever up)
and V N the one of passive ones.

8

K
VitL = Eit [Rt+1
kit − dit ](18)

K
VitN = Rt+1
kitN + ω − kitN

(19)

See Theorem 7.1 in Appendix B

11with kitN ∈ [0, ω]. Since there is no risk of defaulting on deposits if you have none,
then there is no option value of default for non-levered intermediaries. This N group
includes intermediaries who invest all their equity in capital markets (kitN = ω) andintermediaries who do so only partially9 . As shown in Theorem 6.1, all intermediaries
in the L group are levering up to their constraint.
We can then use the condition VitL = VitN to find the value αtj = αtL for whichintermediary j is indifferent between levering or not. Above αtL (looser value-at-risk
constraints), all intermediaries will be levered up. For any levered intermediary i, thefollowing condition holds:


 K 
K
Eit kit Rt+1
− dit ≥ ω Et Rt+1
(20)
where the left hand side is the expected payoff on the assets of intermediary i and theright hand side is the payoff when it invests only its equity ω in capital markets. Using
the balance sheet equation kit = qt dit + ω, we can substitute for deposits, which leads
to the following condition:



 K 
K
Eit kit Rt+1
− 1/qt + ω/qt ≥ ω Et Rt+1
(21)
For the marginal intermediary j, equation (21) holds with equality:



 K 
K
Ejt kjt Rt+1
− 1/qt + ω/qt = ω Et Rt+1

(22)

Since all active intermediaries will be at the constraint, we can combine equation
 K (22)with equation (17) evaluated at the marginal intermediary. Moreover, Et Rt+1 isea function of Zt+1
and Kt therefore this equation defines an implicit function of theethreshold VaR parameter αtL (= αj ) with variables (qt , Zt+1
, Kt ).
To close the financial market equilibrium, we need to use the market clearing
condition.
Z α
kit dG(αi )
(23)
Kt =
α

The integral has three main blocks corresponding to active levered intermediaries (above
αtL ), passive intermediaries who invest all their equity in the risky capital stock (between
αtN and αtL ) and passive intermediaries who invest only a fraction (possibly zero) of
their equity, the remainder being in storage (below αtN ).
9Which occurs only if the intermediary has a sufficiently tight VaR constraint12

We define αN ≡ FδθθZte Kt−1



as the marginal intermediary for whom the constraint

would bind exactly if he invests all of its equity in risky capital markets (kit = ω). AsK
long as E[Rt+1≥ 1], then ∀αi ∈ [αtN , αtL ], we have that kit = ω.
By plugging in the expressions for asset purchases and using the expression for αtN ,eeequation (23) defines an implicit function of (αtL , qt , Zt+1
, Kt ). Since Zt+1
is a functionof a state variable and intermediaries are price takers, these two implicit functions(market clearing (23) and extensive margin (22)) pin down the aggregate capital stock
Kt and the marginal levered intermediary αtL , for a given deposit price qt and expectedeproductivity Zt+1
. In general equilibrium, the deposit price qt will be determined inconjunction with the recursive household problem.
Systemic Risk
Very importantly, we can define in the model a precise and intuitive measure ofsystemic risk. A systemic crisis can be defined as a state of the world where all active
intermediaries fail to repay in full all of their stakeholders (deposits and equity). Since
all the risk in the model is aggregate, the probability of a systemic crisis is also theprobability that the least risk-taking levered intermediary is being distressed, which issimply given by αtL .10

3

Partial equilibrium results

We look at the distribution of intermediary assets conditional on the price of deposits
qt and on expected productivity Zte . In Figure (1), we show an example of the crosssectional distribution of assets for three different values of the deposit rate. The
calibration of the model is discussed in more detail in section 4.
cases, the area below each line11 is the aggregate capital stock Kt =R In the three
kit dG(αi ) and the vertical line showing a large drop in balance sheet sizes identifies
the marginal levered intermediary αtL . To the left of the cutoff αtL , intermediaries are
not levered, which corresponds to the more conservative VaR constraints. They have a”safe business model”. To the right of the cutoff, the balance sheet size kit increases
with αti . That is, the more risk-taking is the active intermediary, the larger will be
its balance sheet for a given qt and Zte . Those intermediaries have a ”risky business10

We could consider equally easily that there is a systemic crisis when a certain proportion of levered
financial intermediaries are unable to repay depositors or when a certain fraction of total assets is heldby distressed intermediaries.
11
Assuming a uniform distribution for G(αi ) as in the baseline calibration.

13

model”.

Figure 1: Cross-sectional distribution of assets

The graph also illustrates how the intensive and extensive margins affect the aggregate
capital stock as the deposit interest rate changes. For the three cases displayed, asdeposit rates fall (a rise in q), the intensive margin is always increasing. That is,for every intermediary that remains active, the balance sheet grows when the cost ofleverage falls. This is because a lower rate reduces the probability of default for a given
balance sheet size, as a lower rate reduces the cost of liabilities that needs repaying
next period. Intermediaries expand their balance sheet up to the new limit and active
intermediaries grow in size.
Perhaps less intuitively, the effect on the extensive margin is ambiguous. Dependingon the level of interest rates, a fall in interest rates (a rise in q) can lead to more or
fewer intermediaries choosing to lever. In Figure (1), for example, both the low and thehigh level of deposit rates have higher cutoffs than the medium level.
Non-linear trade-off between economic activity and financial stability.

14

Following a fall in interest rates, intermediaries expand their asset holdings raising
the aggregate capital stock (intensive margin effect). We have however very interesting
asymmetries depending on the level of the interest rate.
When the interest rate level is high, the lower cost of liabilities reduces the probability of default for a given balance sheet size. Hence all intermediaries with a riskybusiness model can lever more (intensive margin); there are also positive returns for themarginal intermediary. More intermediaries therefore can lever and enter the market
for deposits (extensive margin). In this case, the system becomes less risky since thoseintermediaries have a stricter value-at -risk constraint. There is therefore no trade-off
between lower interest rate and financial stability.
When the interest level is low, the intensive margin effect of a decrease in interest
rate is strong and the curvature of the production function leads to a decrease in
expected asset returns which is enough to price out the intermediaries at the cutoff.
The sign of the effect on αL depends on whether the fall in asset returns is stronger than
the fall in the cost of liabilities. In the case of initially low interest rates, a further fall(in those rates) leads to financial markets being characterized by fewer intermediaries
choosing to lever. Those intermediaries are also larger and more risk-taking on average.
There is a therefore a clear trade-off between a lower interest rate (which corresponds
in equilibrium to an expansionary monetary policy) and financial stability.
Hence, as shown in Figure (1), when interest rates rise from low to medium to high,balance sheets become less heterogeneous and the difference between the most levered
and the least levered intermediary falls. In contrast for low level of interest rates, we
see an increase concentration of risk and of the skewness of leverage in the cross-sectionof intermediaries.
In Figure (2), the left graph plots the cutoff αtL as a function of deposit prices
qt (moving to the right is equivalent t a decrease in interest rate) for three different
productivity levels, while the right graph does the same for the aggregate capital stock
Kt . As we can see, Kt is monotonically increasing with qt . As expected, the lower is theinterest rate, the higher will be aggregate investment. However, the change in financial
structure underlying the smooth increase in the capital stock is non-monotonic. As
we can see from the left graph, the cutoff αtL first decreases and then goes up. Wheninterest rates are high (qt is low) a fall in interest rates leads to entry by less risk-taking
intermediaries (a fall in the cutoff αtL ) into levered markets. On the other hand, wheninterest rates are low, a fall in interest rates leads to a rise in the cutoff αtL , whichmeans the least risk-taking intermediaries which were levered will now reduce theirbalance sheet size while more risk-taking intermediaries increase theirs.
15

Figure 2: Partial equilibrium in the financial sector

Hence, interestingly, and unlike in the earlier literature, there is a potential trade-off
between financial stability and monetary policy when interest rates are low, but notwhen they are high. During a monetary expansion, the cost of liabilities is reduced
and the partial equilibrium results described above follow. The fact that risk-taking
intermediaries are able to lever more can increase the capital stock and price out lessrisk-taking ones. This means that the financial sector becomes less stable, with risky
assets concentrated in very large, more risk-taking financial institutions. There is also
potentially large mispricing of risk12 , since these are the institutions that engage themost in risk-shifting. As a result, the effects of risk-shifting on over-investment describedby Malherbe (2015) are amplified through the change in the extensive margin.
We illustrate this point in our partial equilibrium setting by doing a 100 basis points
monetary expansion for different target rates. For this experiment, we assume a very
simple monetary policy rule:ν
Rt = Rt−1
R̄1−ν εRt(24)

where Rt = 1/qt is the deposit rate. εRt is a monetary policy shock, R̄ the long-run levelof interest rates and ν the persistence of the shock, calibrated13 to 0.24. For simplicity,
as we are not modeling the nominal side of the economy, we assume that the monetary
12

Defined here as the difference between the market price and the price investors would be willing to
pay in the absence of limited liability.
13
Annualized value as estimated by Curdia et al. (2015)16

Figure 3: Partial equilibrium IRF to a 100 basis points fall in deposit rates

authority can directly affect the deposit rate. We relax this assumption in section 4
and show how it can be mapped into this exercise.
Results can be seen in Figure (3), plotted as percentage changes from their respective
values at target rates R̄.14 The time period corresponds to one year and the state of
the economy when the shock hits is the one corresponding to the target rates.
In the left graph we see that the rise in output seems to be slightly larger whenrates are low, with the rise ranging from 3.4% to 4.5%. The monotonicity of Kt withrespect to qt ensures, as expected, that monetary policy stimulates investment and thecapital stock in all cases. The behaviour of the cutoff αtL is, however, very differentiated.
When the target rates are high, there is a small negative effect of a monetary expansion
on the cutoff. That means that less risk-taking intermediaries enter levered markets
and the average probability of intermediary default falls (although marginally so). In
this case, there is no trade-off between financial stability and monetary expansion.
This is definitely not the case when target interest rates are low. In that case, averageleverage increases massively by 25% and the cutoff also rises. The large increase inleverage by very risk-taking intermediaries then prices out the less risk-taking ones at
the margin, raising the average probability of default among levered intermediaries. Forintermediate levels, we see that this effect is muted, with leverage increasing slightly
more than in the first case and the effects on the average default probability being
positive but only marginally so.
Hence, in some cases, lowering interest rates may well stimulate the economy but
also contribute to an increase of systemic risk. In our model, this happens through a
14

Note that there is no truly dynamic aspect in the partial equilibrium model and it can be seen asa sequence of static problems. The general equilibrium model of section 4 will feature a fully dynamic
household problem which affects the banking problem via demand for deposits.

17

change in the composition of intermediaries. Less risk-taking intermediaries exit levered
markets and decrease their asset holdings as they are priced out by more risk-taking
institutions due to decreasing returns to capital. The latter use low interest rates toincrease their leverage significantly. Given that risk-shifting is larger in riskier intermediaries, this also generates more risk-taking on aggregate. But these effects happen
only for low levels of interest rates. At higher levels, there is no such trade-off between
monetary policy and financial stability. Our framework, appropriately enriched, should
ultimately help us quantify the importance of the risk-taking channel of monetary policy.
Even in the absence of monetary policy, these effects have implications for thecyclicality of leverage, systemic risk and aggregate risk-shifting. The cyclicality ofthe savings behaviour and its effect on equilibrium deposit prices will lead to cyclicalmovements in leverage and investment. In general equilibrium, we will look at how
shocks to productivity and intermediary default propagate in this model.

4

General Equilibrium

In this section, we solve the model in general equilibrium by joining the household andintermediary problems. The objective of this section is to show that the financial sector
equilibrium can be easily integrated in a standard general equilibrium framework, with
monetary policy and productivity shocks, as well as costly default.

4.1

Monetary policy as a change in funding cost

In this section we allow intermediaries to fund themselves through wholesale funding
lit . We assume that the monetary authority can control the price of wholesale funding
relative to deposits, by providing funds at a spread γt from deposits.15 . The price ofwholesale funding is qtL and we denote the price of deposits as qtD . We have that:qtL = qtD (1 + γt )(25)

Monetary policy is exogenous, akin to a funding subsidy γt which follows a simple
AR(1) process in logs.
log γt = (1 − ργ )µγ + ργ log γt−1 + εγtεγt ∼ N (0, σγ )
15

(26)
(27)

The monetary authority is assumed to be a deep-pocketed institution which can always fundwholesale funding. Like deposits, wholesale funds are always repayed (by bailout if necessary). To
avoid dealing with the monetary authority’s internal asset management, we assume the cost of fund isa deadweight loss (or gain).

18

where µγ is the central bank target subsidy, ργ the subsidy’s persistence and εγ aremonetary policy shocks with σγ standard deviation.
If the central bank were to provide unlimited funds to intermediaries at this rate,they would leverage using only wholesale funding. We assume that wholesale funding isgiven in a fixed proportion λ of other liabilites, which in this case are simply deposits.
Total wholesale funding for intermediary i is then:lit = λdit

(28)

The balance sheet of an intermediary i is then:Assets LiabilitieskitωD
qt ditqtL litGiven our assumptions, we can then define qtF as the total cost of a unit of funding andfit as total external funds of bank i.
qtF =1 + λ(1 + γt ) D
qt1+λ
fit = (1 + λ)dit

(29)
(30)

We can then write the balance sheet as:
Assets Liabilitieskitω
qtF fit
We obtain the same banking problem as before, replacing deposits by total funds
fit and the deposit price by the unit cost of funds qtF . Since γt is an exogenous variable,
we can map fit and qtF easily to deposits dit and their price qtD . By moving γt thecentral bank will be able to change qtF as long as changes in equilibrium qtD do not
offset perfectly the changes in the spread on the total cost of funding.

4.2

Solving the dynamic model

The financial sector equilibrium can be seen as a sequence of static problems given
funding costs qtF . We can then solve for the aggregate capital stock K and cutoff αtL as
a function of qtF and expected productivity Z e .
K = K ∗ (q F , Z e )

(31)

L

(32)

α =α

L,∗

F

e

(q , Z )19

By integrating balance sheet equations, we obtain an expression for total funds Ft anddeposit supply Dt :
Z α
F
(33)
q t Ft =kitL dG(αi ) − [1 − G(αtL )]ω
αL
t

Z α
Dt =
αL
t

dLit dG(αi ) =

Ft
1+λ

(34)

R
where Ft = fit dG(αi ) are total liabilities held by active intermediaries and Dt isaggregate deposit demand. Market clearing in the deposit market also requires supply
and demand to be equal.
DtH = Dt

(35)

The goods market clearing also requires that output is used in consumption of households
and intermediaries, investment and the accumulation of storage. The investment goodis the consumption good and there we allow for disinvestment.
H
I
St−1+ St−1+ Yt = CtH + CtI + StH + StI + It + Tt
(36)
R
where CtI = cit dG(αi ) and Tt = tit dG(αi ). Note that taxes here are equal to thecapital injections, which Rrequire real resources. StH are the holdings of storage heldby households and StI = sit dG(αi ) are aggregate storage holdings held by financial
intermediaries at t.

R

To find an equilibrium, we need to have the deposit price which, conditional on
exogenous variables and the financial sector equilibrium, is consistent with the householdproblem. We proceed by iterating on qtD , imposing the financial market equilibrium
results. For a given deposit price qtD , we can find the law of motion for household
wealth and consumption and use the Euler equation errors to update the deposit price.
A more detailed explanation of the algorithm used for our global solution method canbe seen in Appendix A.
H
To define formally the equilibrium, let S = {Dt−1 , St−1
, Kt−1 , Zt−1 , γt−1 , εzt , εγt }∞t=0
be the vector of state variables and shocks. Given a sequence of prices {qtD }∞t=0 andfinancial market rules K ∗ (S), αL,∗ (S), define the optimal decisions of the representative
household as C H (S), DH (S), S H (S). We can then define the equilibrium as follows.

Definition 1 An equilibrium is a sequence of prices {qt }∞t=0 , and policy rules
C (S), DH (S), S H (S), K ∗ (S), αL,∗ (S), such that:H• C(S), DH (S), S H (S), K ∗ (S), αtL,∗ (S) are optimal given {qt }∞t=0• Asset and consumption markets clear at every period t
20

4.3

Calibration

To solve the model numerically, we need to choose the period utility function, thedistribution of the Value-at-Risk probabilities of intermediaries and calibrate theremaining parameters. Given the interaction between extensive and intensive margineffects, the mass of intermediaries in a given section of the distribution could havean important role in determining which of the two effects dominates. To highlightthat the results described are not a consequence of this distribution, we assume thatG(αi ) is uniform between [0, α]. For the utility function, we assume a standard CRRA
representation.
u(C) =

C 1−ψ − 1
1−ψ

(37)

Table 1: Calibration of selected parameters

Parameter
ψ
β
ρz
σz
µγ
ργ
σγ
λ
1+λ

θ
δ
ω
α

Value
4
0.95
0.9
0.028
0.02
0.24
0.01
0.3
0.35
0.1
0.5
0.1

Description
Risk aversion parameter
Subjective discount factor
AR(1) parameter for TFPStandard deviation of TFP shock
Target spread over deposit rates
Spread persistenceStandard deviation of spread
Central Bank funding percentage
Capital share of output
Depreciation rate
Equity of intermediaries
Upper bound of distribution G(αi )The calibration can be seen in Table (1). For the utility function parameters, risk
aversion ψ and the subjective discount factor β, we chose standard values from theliterature. Similarly for θ, the capital share of output, and for δ the depreciation rateof the capital stock.
α is the probability of default of the riskiest intermediary in activity. Bali, Brown
and Caglayan (2014) report that the median lifespan of a hedge fund is slightly less than
5 years, which would imply a value of approximately 0.2. Given our specification of a21

uniform distribution and the fact that equity size is fixed, we use a more conservative(even if still very large) value of 0.1. This implies an average lifespan of 10 years for themost risky of financial intermediaries and 20 years for the median intermediary. ω waschosen so that the average leverage is close to 10. 16
Ideally, one would back out from the data the distribution of leverage and map it
into G(α). We could also infer a distribution for intermediate-specific ω from the size ofeach type of participant. We leave that for future work and focus for now on exploring
the effect of the change in the marginal intermediary, when intermediaries are identicalin all respect except their Value-at-Risk constraint. This helps to isolate the effects of
having different Value-at-Risk parameters, which is the core of the model.

4.4

Monetary policy shocks

Figure 4: Monetary policy shock of 100 basis points to γt

In Figure (4) we see the impact of a 100 basis points monetary policy shock in 3
different scenarios to illustrate the non-linear effects of monetary policy on systemicrisk. Scenario 1 features a low initial capital stock (corresponding to high equilibrium
levels of the interest rate), where there is no trade-off between monetary policy andfinancial stability and scenario 2 is for a larger capital stock (corresponding to a low
level of equilibrium interest rate) where we are in the trade-off zone. Scenario 3 is at therisky steady-state17 . As in Coeurdacier et al. (2011) we define the risky steady-state as
16

The value of α, the shape of the distribution and also ω will all contribute to determine the financial
sector reaction to changes in deposit rates. For that reason, we also conducted some comparative
statics on both α and ω to see how the model changes with those parameter calibrations. There is very
little effect on the first moments of real variables such as output and consumption, although there areimportant changes on equilibrium leverage as we change ω and on the risk premium as we change α.
17These three scenarios were chosen to illustrate the parallel with the partial equilibrium setting,since the solution of the model is such that there is, ceteris paribus, a negative correlation between the

22

the steady-state in which there are no shocks but economic agents take into account thefull stochastic structure of the model when they optimize (unlike in the deterministic
steady-state where they expect no shocks). Very interestingly, we can easily relate thegeneral equilibrium results to the partial equilibrium intuitions developed above. In thecase of a low initial capital stock (associated with a high equilibrium funding rate), a
positive monetary policy shock expands output, increases aggregate leverage (as thecapital stock goes up and equity is fixed) and at the same time it decreases systemic
risk, thanks to less risk-taking intermediaries levering more. We are in the ”no trade-offzone of monetary policy” where a decrease in the interest rate increases investment
and financial stability. In the case of a high initial capital stock (associated to a low
funding cost for intermediaries), an expansionary shock has a larger positive effect on
output and leverage but this time, risk averse intermediaries at the margin choose not
to lever, reducing their balance sheet size significantly. Less risk averse intermediaries
leverage a lot and financial stability is affected negatively. We are in the ”trade-off zoneof monetary policy” where there is a conflict between stimulus and financial stability.
This is a very different trade-off from the traditional Phillips curve which has been thebenchmark model driving monetary policy analysis for many years. Aggregate economic
variables behave smoothly but the underlying change in financial structure supportingthese macroeconomic outcomes can be dramatic depending on the level of the interest
rate.

5

Empirical evidence on the cross-section of intermediaries balance sheets.
Our model, unlike the previous literature which did not usually feature intermediaries heterogeneity, has interesting implications for the cross-section of intermediarybalance sheets, namely regarding leverage and how its distribution varies over the cycle.
First, aggregate leverage is not only monotonically decreasing with the interest rate,
its derivative is also monotonically increasing. So when interest rates are low, further
reductions lead to larger increases in leverage than when they are high. This is because
the sensitivity of leverage to the cost of funding is larger for intermediaries who are in
the upper range of the risk-taking distribution (conditional on remaining active). So
the more concentrated is capital in this upper range, the more sensitive will aggregate
leverage be to interest rate changes.
capital stock and the deposit rates as can be seen in figure (2)23We used banks balance sheet data to compute leverage at the bank level. Leverage isdefined as the ratio of assets over equity at book value. In Figure (10) we show the timeseries of leverage weighted by bank assets (a very similar picture can be obtained if one
uses equity weights). We separate the top of the distribution (75th percentile) from thebottom (25th percentile). Before 1984-85 there seems to be no noticeable correlation.
After that the correlation is striking, with a clear dichotomy between the two parts ofthe distribution. Ihe more leveraged banks increase their leverage sizably as the interest
rate goes down. In particular, there are two leverage spikes corresponding to very low
levels of the interest rate: during the pre-crisis period and after 2009. In contrast, after
an initial moderate increase in leverage (between 1984 and 1992, following a decline
in the Fed Funds rate), the less leveraged banks deleverage massively after 1992 as
the interest rate falls further down and keep a constant low leverage after that. We
interpret this fact as the less risk-taking intermediaries effectively exiting the risky
leveraged business model as they are priced out by the most risk-taking ones.
It would be interesting (but beyond the scope of this paper) to determine whether
the change in correlation between leverage and the Effective Fed Funds Rate is dueto the wave of the deregulation of the 1980s (in the US the Depository Institutions
Deregulation and Monetary Control Act of 1980 and the GarnSt. Germain Depository
Institutions Act of 1982 eroded the distinctions between banks and other financial
institutions).
Second, a very distinct implication of the model is that low levels of the interest
rate are associated with an increased skewness of leverage: risk gets concentrated inthe (endogenously) larger, more risk-taking players. As with aggregate leverage, whenrates are low, skewness is also more sensitive to interest rate movements. In Figure(5), we show the shape of cross-sectional skewness as a function of q for three different
levels of productivity. It is also apparent that the direct impact of productivity on
skewness, although positive, seems second-order relative to the impact of interest rates.18
We use banks balance sheet data to compute the time series of the skewness ofleverage. In Figure (11), we present three time series of skewness: unweigted, weighted
by equity and weighted by assets in parallel with the movements of the effective fed
funds rate. Again, there is no noticeable correlation before 1984-85 but after that thecorrelation is again remarkable. After 1984, lower rates are strongly correlated withincreased positive skewness. These results are striking and very encouraging for themechanism of our model. We are not aware of any paper studying the distribution and18

Although in general equilibrium productivity will also affect skewness indirectly via its impact on
deposit prices

24

Figure 5: Skewness as a function of deposit prices q

skewness of leverage and linking it to monetary policy.
The model also implies that in the cross-section (endogenously) larger more leveraged
intermediaries make higher profits in good states of the world but are more exposed toaggregate risk. Accordingly we analyse the returns of financial intermediaries in the
run up to the crisis and look at its correlation with leverage and with the exposure
to aggregate risk (measured by the world market beta). Figure (6) shows a positive
correlation between pre-crisis betas and returns. We also find a positive correlation
between returns and leverage, confirming the results of Miranda-Agrippino and Rey(2015).

6

Costly intermediary default

In this section we relax the assumption of costless intermediary default. As in theprevious section, levered intermediaries active in risky financial markets can potentiallydefault on depositors if the realisation of the productivity shock is low enough. Thisrequires intervention by the government to pay deposit insurance, which might be less25

Figure 6: Market beta and pre-crisis returns for Global Systemically Important Banks (G-SIBs)

benign than previously assumed.
We parameterize the cost of intermediary bailouts by assuming that the return on
capital held by defaulting intermediaries suffers a proportional loss ∆ relative to capital
held by non-defaulting intermediaries. This disruption can affect financial markets inthe following periods by creating an efficiency loss ∆t which is proportional to the massof capital held by defaulting banks µdt .19
So the loss of productivity is intermediary-specific during default (it affects only
the defauting intermediaries, not the others), but it can affect the whole sector moving
forward. We call this the crisis state. We model the persistence of the crisis statethrough a Poisson process, with a constant probability p of exiting the crisis at eachperiod. Depending on the process, variable ξt takes the value of one if the crisis carries
on to the next period or zero if it does not. Our specification nests both the case of
costless default (∆ = 0) and the case where there is no disruption of financial markets
in subsequent periods (p = 1). We then have:19

For example, if defaulting intermediares held 3% of total capital during default at t − 1, then if
the crisis persists ∆t = 0.03∆.

26

kit 1(πi <ω) dG(αi )Kt∆t = ξt−1 max(µdt−1 ∆, ∆t−1 )
R

µdt =

(38)
(39)
(40)

where the indicator function takes the value of 1 if intermediary i is in default or 0 if
not. If there are also defaults during a crisis state, then the max operator ensures thatthe largest penalty applies going forward. Productivity for all financial intermediaries
is scaled down (whenever the economy is in crisis) by a factor proportional to thepercentage of total capital held defaulting banks. ξt−1 is known to agents when they
make their investment decisions at period t − 1, so the uncertainty on the returns
on their capital investment is only on the realization of the exogenous productivity
process20 . This timing assumption allows us to keep tractability as the main differZeence in the financial sector block is that now Zt+1
= (1 − ∆t )Ztρ . Since both ∆t and
Zt are state variables, then we can still solve for the financial sector equilibrium as before.
This set up is tractable and allows us to parameterize crises of different severity
and length. Borio et al. (2016) present empirical evidence showing that there canbe substantial and long lasting productivity drops after financial crises. We calibrate
p = 0.5 which implies an average crisis length of 2 years and ∆ = 0.05 implying amaximal efficiency loss of 5% .

6.1

Productivity shocks and financial crises

In this section we study the impact of a financial crisis on the path of the economy,
following a large productivity shock. Figure (7) shows the impact of a large productivity
shock in 3 possible scenarios.
Scenario 1 is when the economy at the risky steady-state is hit at period t by thelargest possible shock that does not trigger defaults. Scenario 2 and 3 are when theeconomy is hit with the smallest shock such that all levered intermediaries default. The
difference between scenarios 2 and 3 is in the length of the crisis. Scenario 2 is the”lucky” scenario, where the crisis does not carry on to the next periods: ξt = 0. Scenario
3 is the ”unlucky” scenario, where the crisis carries on for an additional 4 periods:ξs = 1 for s = t to t + 3. Not surprisingly, when crisis hits there is a large decline in
20

There is still uncertainty on asset returns if the intermediary defaults but this penalty is notconsidered in the intermediary problem due to limited liability truncating the profit functions at zeroin those states.

27

Figure 7: Large shock to exogenous productivity

output. As productivity is low, only the most risk-taking intermediaries can operate.
The average leverage of active intermediaries first shoots up but then decreases and falls
below the pre-crisis state as more interimediaries find it worthwhile to enter the riskybusiness model again when productivity improves. Note that this initial rise is a purecomposition effect as total sector leverage falls21 . The increase in the average leverageof active banks is then purely due to the exit of the least levered active intermediaries from the risky business model leaving only the most risk-taking ones active initially.
The length of the crisis also has very interesting dynamic effects. Given thathouseholds expect to exit the crisis state with probability p, then when exit fails tomaterialize in Scenario 3 they are effectively running down their wealth. The deposit
rate grows as it becomes more costly for the household to save and fund bank leverage.
When eventually the economy exits the crisis state, household wealth is low and demand
for leverage rises, leading to a jump in deposit rates to again compensate households for
decreased consumption today. The economy then exits the crisis with a risk-premium
above the pre-crisis levels, due to high cost of funding. This effect is also present witha short crisis, but is particularly stark for the longer crisis.

7

Conclusion

This paper develops a novel framework for modeling a financial sector with heterogeneous financial intermediaries. The heterogeneity in the Value-at-Risk constraints
generates not only endogeneous entry and exit in risky capital markets, but also time
variation in leverage, risk-shifting and systemic risk.
21

See Figure 14

28

The interaction between the intensive and the extensive margins of investment
generates a rich set of non-linear dynamics where the level of interest rates plays a key
role. When interest rates are high, a monetary expansion (defined here as a decrease in
the cost of funding for intermediaries) increases both the intensive margin (the amountof leverage) and the extensive margin (which intermediaries leverage up). The intensive
margin grows because active intermediaries are able to lever more and the fall in thecost of funding leads to increased participation by less risk-taking institutions which
enter levered markets. The monetary authority is then able to stimulate the economy,
while at the same time decreasing systemic risk.
However, when interest rates are already initially low, a further reduction can leadto large increases in leverage by the most risk-taking institutions, pricing out previously
active intermediaries despite the fall in the cost of funding. As before, the intensive
margin grows but there is a fall in the extensive margin as intermediaries at the margin
exit levered markets. Importantly, the intermediaries who stop levering and decrease
their balance sheet size have lower probabilities of default than those that remain
levered, leading to an increase in systemic risk.
Limited liability plays a crucial role in these dynamics by creating an option value of
default. This induces risk-shifting by financial intermediaries and increases willingness
to pay for risky assets. Voluntary entry and exit will then happen at the lower range ofrisk-shifting from active intermediaries, generating a a potential increase in systemic risk.
In the model there is no idiosyncratic risk, so this effect on systemic risk is happening
because of the shift in business model of intermediaries and not because of a fall in thenumber of institutions in the market.
Because our framework has heterogeneity at its heart, it allows us to make us ofcross-sectional data on intermediary balance sheets. For example, we derive novel
implications linking the times series of the skewness of leverage and monetary policy.
These implications are borne out in the data, particularly since the mid-80s after which
there is a striking correlation.
Our financial block is easy to embed in a more standard general equilibrium framework, as the rich dynamics that arise from the composition of active financial intermediares can be described simply by the dynamics of the aggregate capital stock Kt andthe cutoff αtL . We plan to extend our model to environments with sticky prices and amore complex portfolio choice on the bank side. We also plan to apply it to explain the
dynamics of the real estate market, using detailed data. We will as well use it to study
boom and bust cycles in emerging markets as well as the endogenous dynamics of the
VIX.
29

The model can also be calibrated to fit a more realistic distribution of financial
intermediaries, as one could in practice back out the distribution of αi from leverage data
and map it to the ergodic distribution of leverage in the model. Given the numericalintegration approach, it is also possible to extend the model to have a distribution ofintermediary-specific equity ω i . That said, allowing for time variation in equity would
require the introduction of an additional state-variable in the financial sector problem
which would make the solution more computationally intensive.22 We leave these issues
for future research.

22
And having at the same time time-varying and intermediary-specific equity could require an
infinitely dimensional state-space without additional assumptions.

30

Figures

Figure 8: SIFIs Balance sheet changes in the crisis run up: total change in size versus change
in equity or debt.

31

Figure 9: Industry and SIFIs Balance sheet changes in the crisis run up: total change in size
versus change in equity or debt.

32

Figure 10: Time series of the leverage of banks and the Effective Fed Funds Rate. Leverage isweighted by assets. The 25 and 75 percentiles of banks are shown.

33

Figure 11: Time series of the skewness of leverage of banks and the Effective Fed Funds Rate.
Skewness is left unweighted or weighted by assets or by equity.

34

Figure 12: Monetary policy shock of 100 basis points to γt - Financial Variables: Deposit
rate, spread between the expected retruns to capital and the deposit rate, funding rate and total
leverage.

Figure 13: Monetary policy shock of 100 basis points to γt - Real variables: consumption,
wage income, capital stock.

35

Productivity shock

Figure 14: Large shock to exogenous productivity - Financial Variables: Deposit rate, spread
between the expected retruns to capital and the deposit rate, funding rate and total leverage.

Figure 15: Large shock to exogenous productivity - Real variables: consumption, wage income,capital stock.

36

Appendix A. Numerical solution method
The solution method is composed of two main blocks. The first block solves the partial
equilibrium problem for a grid of points for variables q f and Z e . We discretize thestate space using 100 nodes for Z e and 200 for q f . Given funding costs q f and expected
productivity Z e we can solve jointly for equations (21) and (23), plugging in equation(17) in the latter. This gives us policy functions K ∗ (q, Z e ) and αL,∗ (q f , Z e ).
The second block is the recursive one. First we define the household savings problem
as a function of disposable wealth Ωt , productivity Z̃t , efficiency adjustment ∆t andmonetary policy γt .
HHΩt = (1 − θ)Yt − Tt + Dt−1
+ St−1The procedure entails the following steps
1. Discretize the state space S for the variables (Ω, Z, ∆, γ). The process for Z andγ are approximated using a Tauchen and Hussey (1991) quadrature procedure
with 11 and 5 nodes respectively. The state space for the variable Ω is discretizedusing 500 nodes and we use 10 for ∆.
2. Iterate on prices q D and policy function C ∗ (S) starting with an initial guess q d (S)for deposit prices and C ∗ (S). For every point Sj ∈ S:(a) Using the state vector and qjd , calculate qjf and Zje .
(b) Solve for (Kj , αjL ) using K ∗ (qjf , Zje ) and αL,∗ (qjf , Zje ). Back out deposit supply
Dj from the balance sheet equations.
(c) Plug Dj in the budget constraint of the agent. Together with Cj = C ∗ (Sj )this pins down SjH .
(d) Calculate expectations of (S 0 |S) and update deposit prices and policy functions using the optimality conditions and numerical integration.
(e) Check for convergence. If ||(qj0 − qj )|| + ||(Cj∗ )0 − Cj∗ || is smaller than athreshold value stop. Else, go back to (a) and repeat.
To numerically integrate intermediary capital into aggregate capital, Gauss-Legendre
quadrature using 51 points is used. To calculate expectations of future net disposable
wealth, we also need to calculate taxes conditional on future shocks. For a given
productivity draw Z 0 |Zj we identify the threshold intermediary for which no bailout isneeded: (RK ki − di ) = ω. We can then calculate the amount Tt of taxes required bynumerical integration.
37

Appendix B. Proofs
K] > 1, intermediary i will either lever up to its Value-atTheorem 7.1 When E[Rt+1iRisk constraint: dit = dt , or not raise deposits at all : dit = 0.

The Value-at-Risk constraint bounds the maximum level of leverage of intermediary i,itherefore dit ∈ [0, dt ]. The profits of intermediary i as a function of deposits are:
Z ∞

 K
i
(41)
Rt+1 (ω + dit ) − RtD dit dF (ε)πt (dit ) =εit (dit )where εit is the max of 0 (the lower bound of the support for ε) and the shock for which
profits are zero)
RtD dit
−
1
+
δ
ω+dit

εit = max 0,(42)θZ ρZ Ktθ−1
Taking derivatives:
∂πti
=
∂dit

Z ∞


∂εi
K
Rt+1
(ε) − RtD dF (ε) − πti (εi ) i∂dt
εit (dit )

(43)

i

∂ε
Lemma 7.1 Given equations (41) and (42), then πti (εit ) ∂di = 0tω(1−δ)i i
i
This is easy to check. For any dit ≥ Rω(1−δ)
d −1+δ , then πt (εt ) = 0. For dt < Rd −1+δ , then
t

i

t

∂ε
ei = 0 and ∂di = 0 due to the max operator.
t

We then have as first and second derivative:
Z ∞

∂πti
K
=
Rt+1
(ε) − RtD dF (ε)
∂dit
εit
 i
 K
∂ 2 πti
D ∂εt
=
−
R
(ε(d
))
−
R
it
t+1
t
∂d2i,t
∂dit

(44)

∂π i
K
Given the monotonicity of Rt+1(ε), then ∀d˜ such that ∂ditt = 0, then it followsd˜
˜ − RD < 0 or all elements in the integral are non-negative and it cannot
that RK (ε(d))
t+1t˜ Then ∂ 2 π2ti
be zero for finite d.
∂d> 0 and if d˜ exists then it is a minimum. We then
i ihave that the maximum must be at the bounds: dit = arg max πt (0), πt (dt )
it

d˜

38

Appendix C: Data Description
Bank balance sheet data is from the Compustat Bank Fundamentals Quarterly database.
Banks returns data is from CRSP. MSCI World Index data is from Bloomberg. The
Effective Federal Funds Rate is from the Federal Reserve Economic Data. All data has
been accessed on April 11, 2016.
The leverage ratio is defined as the ratio of total assets to total equity. Total assets
and equity are the variables “atq” and “seqq” in Compustat, respectively, the latterbeing defined as described as Stockholders Equity Parent Quarterly. Alternatively one
can use “ceqq” for total equity, described as Common/Ordinary Equity. Results arevery similar. Total liabilities are defined as “lseq” minus “seqq” (total liabilities andequity minus Total equity). We drop negative equity and negative assets from thedataset.
For the leverage series, we compute both unweighted and weighted averages ofthe leverage ratio for each quarter. For the weighted average we use total assets or
total equity as weights. We also compute the 25th and the 75th percentiles of both
unweighted and weighted leverage, the former being just a special case of the latter
where the weighting vector is a vector of ones. We perform 4-period MA filtering on
the resulting series.
For the skewness of leverage, we compute the cross-sectional standard deviationand third moment of the leverage ratio for every period. And then compute the crosssectional sample skewness using a simple approach laid out below.
PN

3
i=1 (xit − x̄t )mt (3) =

N
s

st =
St =

PN

2
i=1 (xit − x̄t )N −1
mt (3)(st )3where xit is the leverage ratio of bank i in period t, x̄t is the period-specific crosssectional mean of leverage, St is the sample cross-sectional skewness in period t, st is
the period-specific sample cross-sectional variance and mt (3) the period-specific sample
third central moment of the cross-section. We use this approach to the unweighted serie
or weighted by either total assets or total equity.

39

Table 2: Descriptive statistics by year
Years
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015

Observations
Banks
Leverage
45
6
47
31
49
42
59
68
90
194
104
348
109
370
111
385
129
457
141
502
146
536
146
545
146
548
148
581
146
584
146
584
157
628
156
622
155
617
164
624
290
770
330
787
378
798
442
845
558
853
627
879
643
854
675
843
707
844
755
869
837
817
1015
1585
1037
3280
940
3237
885
3078
853
2976
901
2947
970
3160
931
3419
945
3380
919
3277
921
3302
899
3179
857
3128
822
3026
787
2908
756
2782
738
2662
735
2644
715
2587
742
2636
731
2590
707
2538
649
2270

Mean
2668.32509.32655.02681.52297.42300.12544.82819.02714.72947.73231.23938.54687.24642.74913.25549.85983.06906.6
7766.18695.88047.58547.69320.69862.210924.2
11644.713032.814558.9
15749.015728.7
16898.2
10215.2
5651.06379.87061.77952.310200.6
10872.2
11699.3
13527.1
16427.5
20190.922638.2
28285.7
33856.2
41236.447944.649063.6
50894.561020.2
64506.7
67815.1
70072.3
73028.8

Total Assets
MedianSt.Dev.
1313.72973.81268.42822.41359.83003.31386.73262.5989.3
3372.4
1014.23523.31124.83876.61197.04297.11107.44536.01279.24945.91479.15390.01742.06635.71844.88438.91846.68746.01964.19248.82204.810557.1
2340.811845.3
2623.813824.92929.915393.93303.6
17064.1
3144.716409.3
3453.116721.5
3782.1
17592.4
4041.4
18370.04715.5
19738.5
4922.221131.5
5099.725726.7
5243.430155.84967.633335.1
4676.834556.1
4740.637177.9
1002.831799.4
568.3
23574.0556.728817.7614.6
32239.1
681.2
36473.8677.049801.4
636.9
56150.4
607.0
61812.9637.471205.3
698.3
79420.4
778.2
97645.0
811.3113936.9
896.3
142986.0
991.3
172961.4
1046.2224693.9
1106.1268072.6
1195.5264906.0
1130.1267429.2
1117.8308083.9
1120.0315693.1
1198.5315063.8
1317.6317166.4
1592.0311377.5

40

Mean
550.8
296.4
376.4
295.7
155.3
154.3
157.9
170.3
160.5
166.4
174.4
190.4
207.5
226.2
245.4
268.7
283.9
318.3
356.5
401.6
386.0
437.8
478.2
524.0
607.0
617.3
702.0
789.7
866.6
915.2
1069.9
683.1
391.7
443.5
492.8
535.3
640.4
680.9
719.7
848.0
1026.71171.51365.11551.41835.42120.62326.42739.63028.53599.73896.84263.34619.05144.6

Total Equity
MedianSt.Dev.
466.7
263.3
303.9
253.3
313.4
324.6
87.6
330.7
58.5
238.7
70.9
210.2
74.2
214.5
80.3
222.3
78.1
218.6
82.4
225.1
89.4
234.2
106.1
253.5
113.6
279.6
120.2
316.6
129.0
352.4
138.8
394.2
144.4
426.7
158.5
479.7
178.4
538.4
195.9
609.8
190.4
640.0
207.3
731.9
223.6
783.6
247.3
853.6
291.4
979.2
301.3
950.9
308.5
1196.7297.71373.3302.2
1545.4280.3
1666.8324.2
1994.6
79.8
1812.751.4
1415.053.2
1720.958.7
1926.666.7
2084.768.0
2615.858.7
3008.653.8
3388.157.0
4190.862.7
4759.068.7
5401.472.2
6981.881.5
7602.691.1
9106.997.810491.0
99.111816.0
105.1
15110.0
102.2
15764.0
112.1
17769.1
115.4
18798.7
127.1
19779.0139.920997.3
178.2
22587.2

Mean
13.3
12.8
12.4
13.0
14.0
14.3
15.2
15.2
15.2
16.1
17.1
18.6
19.1
17.7
17.8
18.1
18.3
18.6
18.6
18.5
18.3
17.9
18.2
18.9
18.8
18.6
19.4
20.2
19.2
19.1
15.5
13.0
12.1
11.4
11.3
11.4
11.4
11.7
12.3
12.1
12.3
12.5
12.3
12.2
11.8
11.4
12.1
12.6
12.6
10.9
10.5
10.3
9.9
9.9

Leverage Ratio
MedianSt.Dev.
14.0
1.2
12.8
1.3
12.7
2.8
13.1
2.7
13.9
2.9
13.9
3.6
14.8
4.1
14.8
3.5
14.8
3.5
15.5
4.2
16.5
5.1
17.6
6.4
17.9
7.0
16.8
7.1
17.0
8.1
17.3
5.5
17.4
5.7
17.7
6.2
17.5
6.0
17.2
6.1
17.1
7.1
16.9
6.8
16.9
8.0
16.6
17.6
16.4
15.2
16.0
11.7
16.2
14.6
16.1
21.8
16.0
16.2
15.4
39.0
14.3
11.2
12.5
4.4
11.8
4.0
11.3
3.9
11.1
3.7
11.1
3.8
11.0
4.7
11.3
4.7
11.7
6.1
11.4
8.8
11.4
20.1
11.5
19.5
11.4
13.1
11.3
13.8
11.1
5.3
10.8
5.2
11.2
5.8
10.9
9.2
10.5
13.1
10.1
5.4
9.7
4.7
9.6
4.1
9.4
3.3
9.4
3.1

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44