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Finance and Economics Discussion SeriesDivisions of Research & Statistics and Monetary Affairs

Federal Reserve Board, Washington, D.C.

The Risk Channel of Monetary Policy

Oliver de Groot

2014-31

NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminarymaterials circulated to stimulate discussion and critical comment. The analysis and conclusions set forthare those of the authors and do not indicate concurrence by other members of the research staff or theBoard of Governors. References in publications to the Finance and Economics Discussion Series (other thanacknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

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The Risk Channel of Monetary Policy

Oliver de Groot y

April 9, 2014

Abstract

This paper examines how monetary policy aects the riskiness of the nancial sectorsaggregate balance sheet, a mechanism referred to as the risk channel of monetary policy . Istudy the risk channel in a DSGE model with nominal frictions and a banking sector that canissue both outside equity and debt, making banks exposure to risk an endogenous choice,and dependent on the (monetary) policy environment. Banks equilibrium portfolio choiceis determined by solving the model around a risk-adjusted steady state. I nd that banksreduce their reliance on debt nance and decrease leverage when monetary policy shocks areprevalent. A monetary policy reaction function that responds to movements in bank leverageor to movements in credit spreads can incentivize banks to increase their use of debt nanceand increase leverage, ceteris paribus, increasing the riskiness of the nancial sector for thereal economy.

Keywords: Financial intermediation, Porfolio choice, Debt and equity, Monetary policy,Risk-adjusted steady state

JEL classications: C63, E32, E44, E52, G11

I would like to thank P. Benigno (IJCB co-editor), J. Bianchi, G. Corsetti, E. de Groot, C. Giannitsarou, J.Matheron, M. Paustian, M. Ravn and J. Roberts for helpful comments and discussions as well as participantsat the Symposium of the Society of Nonlinear Dynamics and Econometrics, the Chicago Feds Annual Confer-ence on Bank Structures and Competition, the Banque de France - Deutsche Bundesbank Worshop on CurrentMacroeconomic Challenges, the Federal Reserve Boards Monetary Analysis Workshop, and the IJCB 2013 An-nual Conference. I am also grateful to A. Queralto for making his codes available. Disclaimer: The viewsexpressed in this paper are those of the author and do not necessarily reect those of the Federal Reserve Board.

y Federal Reserve Board, Washington D.C., Email: [email protected]

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1 Introduction

The recent nancial crisis highlighted the importance of nancial intermediaries balance sheets,demonstrating the extent to which nancial intermediaries leverage themselves and make useof debt nance, aects the probability of future nancial crises occurring and the amount of

damage a negative shock (either originating in the nancial sector or not) does to the economy.This paper assesses whether the monetary policy environment can meaningfully aect nancialintermediaries (privately) optimal mix of outside equity and debt nance, and as a consequencetheir balance sheets resilience to shocks.

As the literature on the balance sheet channel has made clear, the nancial accelerator isgreatest when borrowers leverage ratios and reliance on debt are high .1 ;2 Investment banksbalance sheets in the US in the run up to the nancial crisis displayed these key indicatorsof a powerful propagation channel, with historically high leverage ratios and heavy reliance onshort-term debt. Quantitative macroeconomic models have, however, largely remained silent onthe determination of the balance sheet of the nancial sector, often calibrating the steady stateof nancial friction models to match long-run averages of leverage and short-term debt ratios in

the data. In reality, a banks balance sheet composition is the product of an optimizing decisionby the banks owner(s) in which they face a trade-o between risk and return.In this paper, I explore a model in which banks face such an optimizing decision. In par-

ticular, this paper is concerned with the role that the design of monetary policy plays in thedeterminants of a banks balance sheet size and composition.

The design and implementation of monetary policy (as well as regulatory policy) in therun up to the crisis has received much criticism after the event. However, the link betweenmonetary policy and the likelihood or severity of a nancial crisis has been dicult to reconcilewithin standard macroeconomic models. This paper builds on the work of Gertler, Kiyotaki, andQueralto (2012) , which explicitly develops a real business cycle model in which banks balancesheet decisions are endogenously determined. The key innovation of this paper is to augment

their model with sticky prices to motivate standard monetary policy objectives. The questionis then to ask whether there exists a trade-o between the standard monetary-policy objectivesof a new-Keynesian model and the eect these objectives may generate on incentives for theendogenous structure of nancial institutions balance sheets.

To be precise, I present a quantitative new-Keynesian business cycle model in which banksintermediate funds between households and non-nancial rms. The banks hold a representativeasset, which arises from lending to fund physical capital purchases of the production sector.Importantly, these assets yield a risky return. The composition of the liability side of thebalance sheet is the main interest in this paper. I assume that banks have three sources of fundingavailable: inside equity (or internal net worth), which is the accumulation of retained earnings,external equity issuance (outside equity) and external debt nance (in this case, household

deposits).The Modigliani and Miller (1958) theorem tells us that in a frictionless market, the value of a rm is independent of its capital structure. To motivate a trade-o between outside equity anddebt nance, I introduce a simple agency problem that supposes bankers have an incentive toabscond with bank assets, so that at the margin, it is easier for a banker to expropriate funds if outside equity accounts for a larger share of her banks balance sheet. To prevent bankers fromabsconding with assets, households limit the ability of banks to leverage up their inside equity.

1 The nancial accelerator literature, which has emphasized the balance sheet channel can be traced back toBernanke and Gertler (1989) and Kiyotaki and Moore (1997) . The balance sheet channel for banks has beenstressed by Gertler and Karadi (2011) and Gertler and Kiyotaki (2010) .

2 Several dierent agency problems have been adopted in the macroeconomic literature to generate a balancesheet channel. These include costly state verication of borrowers (Townsend (1979) in Bernanke and Gertler(1989)), a hold up problem for lenders (Hart and Moore (1994) in Kiyotaki and Moore (1997) ) and coordinationfailure (Goldstein and Pauzner (2005) in de Groot (2012) ).

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However, there is also a benet to the bank of issuing outside equity. If a bank is heavily relianton debt, which is a non-state contingent claim on the bank, then any uctuations in the returnon assets will have to be absorbed by the banks net worth. Since the return on outside equityis state contingent and linked to the return on assets, it provides a valuable hedge for banksnet worth when uncertainty is high.

In this framework, the optimal balance sheet composition of the bank will depend on thestochastic nature of asset returns. And one of the determinants of the stochastic nature of theeconomy is the policy environment. 3 Banks would like to stabilize volatility in the shadowvalue of their net worth. If monetary policy acts to achieve this, banks have less incentive toresort to outside equity nance and will leverage up their balance sheets, thus partly osettingthe aims of the change in the monetary policy regime. It is this endogenous response of thebanking system to take on more risk when the asset return risk decreases that I refer to as therisk channel of monetary policy .

Investigating the endogenous portfolio structure of banks within a quantitative DSGE modelis, however, not without its technical challenges. The predominant use in the macroeconomicliterature of a rst-order approximation around the deterministic steady state is problematic for

what this paper wants to achieve. As is well known, altering the monetary policy rule doesnot alter the deterministic steady state of a DSGE model. Nor will it capture banks incentiveto alter their steady state balance sheet composition. To overcome this problem I solve themodel around a risk-adjusted steady state (in the spirit of Devereux and Sutherland (2011)and Coeurdacier, Rey, and Winant (2011) and developed in de Groot (2013) ), which explicitlyaccounts for uncertainty. In a prototypical real business cycle model, this amounts to capturingthe eect of household precautionary savings on steady state capital stocks. In the modelpresented in this paper, the risk-adjusted steady state also captures the eect of risk on bankssteady state balance sheet composition, which has important implications for the strength of thenancial accelerator mechanism. Computing the risk-adjusted steady state provides a challengeprecisely because it requires the steady state and the dynamics of the model around the steady

state to be determined jointly. It follows that because the design of monetary policy can alterthe risk-adjusted steady state (because of monetary policys eect on the second moments of variables in the model), and because the altered steady state itself aects the dynamic behaviorof the model around that steady state, we are able to capture the risk channel.

The paper makes two contributions to the literature. First, it shows that exogenous uncer-tainty (i.e., increases in the standard deviation of monetary and other shocks) can signicantlyalter banks (privately) optimal balance sheet composition. Increased uncertainty reduces theability of the banking sector to intermediate credit. Banks balance sheets are particularlysensitive to monetary and capital quality uncertainty because both these shocks have rst-ordereects on asset returns in the model. Second, the paper shows how the monetary policy regimecan also alter the determination of banks balance sheets. Within a restricted class of monetary-

policy reaction functions, I nd that altering the aggressiveness with which nominal interest ratesreact to ination and output deviations only weakly eects the composition of banks balancesheet. However, a reaction function that responds to deviations of banks leverage or creditspreads can generate signicant shifts in the composition of banks balance sheet and thereforechanges in the dynamic responses to shocks.

Many commentators have put forward the assertion that the conduct of monetary policy inthe late 1990s and early 2000s generated a low-risk environment that incentivized banks to takeon more risk, make greater use of short-term debt, and leverage up their balance sheets. Morerecently, several papers have provided theoretical models for such a risk channel. Diamondand Rajan (2009) , Farhi and Tirole (2012) and Chari and Kehoe (2009) among others focus

3 It is left to future research to incorporate into this framework issues regarding direct regulation of the nancialsector. Focus here is given to the indirect eect standard monetary policy has on asset returns and the balancesheet decisions of banks.

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on the moral hazard consequences of bailouts and credit market instruments. Farhi and Tirole(2012)s paper, for example, considers a three-period endowment economy with strategic com-plementarities between private leverage and monetary policy. When maturity transformation isprevalent, the central bank has little choice but to facilitate renancing. Equally, reducing pri-vate leverage lowers the return on equity. The key insight of this literature is that banks choose

to correlate their risk exposures, that optimal monetary policy can be time inconsistent, andthat macroprudential policy can therefore be welfare enhancing. Diamond and Rajan (2009) ,using a similar three-period endowment environment also show that monetary policy is timeinconsistent. Lowering interest rates when households demand funds prevents a damaging runon illiquid assets, but encourages banks to increase leverage and fund more illiquid projects.Optimal monetary policy under commitment in their environment involves raising interest rateswhen there is no liquidity crisis in order to punish banks that have chosen to be illiquid.

There is also a growing literature on macroprudential policy including, among others, Loren-zoni (2008) , Korinek (2011) , Stein (2012) , Bianchi (2011) and Nikolov (2010). Lorenzoni (2008) ,for example, is another three-period model with nancial frictions, via limited commitment innancial contracts, which results in excessive borrowing ex ante and excessive volatility ex post.

The friction generates a pecuniary externality that is not internalized in private contracts andprovides a framework to evaluate policies to prevent nancial crises. While providing importantinsights, most of these models are not rich enough to be provide a quantitative insights into theimportant trade-os policymakers may face.

The key extension of this paper, therefore, is that it studies the risk taking of banks within aquantitative macroeconomic model with nominal frictions, allowing for the joint examination of monetary policy design and banks balance sheet composition. The paper proceeds as follows:Section 2 presents the model. Section 3 sets out the parameterization and explains the solutiontechnique. Section 4 presents the results of the numerical experiments and Section 5 concludes.

2 Model

The baseline model is a DSGE model with investment costs, nominal rigidities, and nancialfrictions. There are ve types of agents: households, capital producers, manufacturers, retailersand bankers. The banking sector follows Gertler, Kiyotaki, and Queralto (2012) . In particu-lar, banks intermediate funds between households and manufacturers by raising both debt and(outside) equity. An agency problem between households and banks, however, limits how muchbanks are able to leverage their (inside) equity. The model is closed with a monetary policyreaction function. Of central interest to this paper is the interaction between the monetarypolicy environment and banks endogenous balance sheets composition.

2.1 Households

There is a unit measure of identical households. Each household consists of a fraction 1 f of bankers and f of workers. Workers supply labor to manufacturers and bring home wages to theirhousehold. Bankers manage banks and bring home any earnings. Within each family, there isconsumption insurance. Workers and bankers rotate over time, with a banker becoming a workerwith xed probability 1 . As (1 ) f bankers become workers, a proportion (1 ) f= (1 f )of workers become bankers, keeping the size of the two populations unchanged. The householdprovides its new bankers with a small start up fund.

Household preferences are given by

max E t

X1

i=0 i

1

1 C t + i hC t+ i 1

%

1 + #L1+ #t+ i

1

(1)

where E t (:) denotes the rational expectations operator, conditional on the time t information

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set, 2 (0; 1) is the subjective discount factor, C t is consumption, Lt is labour supply. Thefollowing parameter restrictions ensure well behaved preferences: h 2 [0; 1), %;# > 0.

Households have access to two nancial assets: bank debt (deposits), D t and bank (outside)equity, E t at relative price QE;t 1. Bank debt pays the non-state-contingent (risk-free) grossreal return R t from t 1 to t while bank equity pays a state-contingent gross real return, denoted

RE;t . Let W t be the real wage and t net payos to the household from ownership of nancialand non-nancial rms. The household budget constraint is given by

C t = W t L t + t + R t D t + QE;t 1RE;t E t D t+1 QE;t E t+1 . (2)

The households rst-order optimality conditions are given by

W t =U L;tU C;t

, E t ( t;t +1 ) R t+1 = 1 and E t ( t;t +1 RE;t +1 ) = 1 , (3)

wheret 1;t

U C;tU C;t 1

denotes the stochastic discount factor between periods t 1 and t, and U C;t and U L;t denote themarginal utility of consumption and the marginal (dis)utility of labour, respectively.

2.2 Manufacturers

A representative, perfectly competitive manufacturer produces intermediate goods that are soldto retailers. At the end of period t, the manufacturer purchases capital, K t+1 at price QK;t foruse in production in period t + 1 . The manufacturer purchases the capital using funds frombanks. By assumption, there is no friction in the process of obtaining funds from banks andthe manufacturer is therefore able to oer the bank a state-contingent security. In this regard,the banks are like private equity funds. Let "A;t and "K;t denote total factor productivity

and capital quality, respectively. At each time t, the manufacturer uses capital and labour toproduce output, Y t :

Y t = exp ( "A;t )(exp( "K;t ) K t ) L1t , (4)

where 2 (0; 1). "A;t and "K;t are exogenous stochastic processes of the form "s;t +1 = s "s;t +s s;t +1 for s = ( A; K ) and s;t +1 Niid (0; 1). Let X t =

P m;tP t be the ratio of the price

of intermediate good, P m;t to the aggregate price level, P t . The manufacturers rst-orderoptimality condition for labour demand is given by

W t = X t (1 ) Y tL t

. (5)

Since manufacturers are perfectly competitive, the gross real return on capital is:

RK;t = exp ( "K;t )X t Y texp (" K;t )K t + (1 ) QK;t

QK;t 1. (6)

2.3 Capital producers

At the end of period t, competitive capital producers buy the entire capital stock from manufac-turers, repair depreciated capital and build new capital. Production of capital involves convexadjustment costs. The capital producers then sell both the repaired and new capital back tomanufacturers. The objective of a capital producer is given by

max E t X1i=0 t;t + i QK;t + i I t+ i 1 + ' I 2 I t+ iI t+ i 1 12

!I t + i!, (7)5

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where I t is investment and ' I 0 scales the adjustment costs. The capital producers rst-orderoptimality condition determines the price of capital:

QK;t = 1 + ' I

2 I tI t 1

12

+ I tI t 1

' I I tI t 1

1 E t t;t +1I t+1I t

2' I

I t+1I t

1 . (8)

The aggregate capital stock in the economy evolves according to

K t+1 = (1 )exp( "K;t ) K t + I t . (9)

2.4 Retailers

Final output, Y t , is a CES aggregator of measure one of dierentiated retailers

Y t = Z 10 Y " 1"r;t dr "" 1

, (10)

where Y r;t is the output of retailer r and " > 1 denotes the intratemporal elasticity of substitutionacross dierent varieties of retail goods. From cost minimization of users of nal output,

Y r;t =P r;tP t

"

Y t and P t = Z 10 P 1 "r;t dr 1" 1

. (11)

Retailers costlessly brand intermediate output: One unit of intermediate output is used forone unit of retail output. Retailers enjoy monopolistic pricing power, but face a convex priceadjustment cost ( al a Rotemberg (1982) ), which generates nominal rigidities in the economy.The objective of retailers is given by

max E t X1

i=0 t;t + iP r;t + iP t+ i Y r;t + i X t+ i Y r;t + i

'2

P r;t + iP r;t + i 1 1

2

Y t+ i!, (12)where is the steady-state gross ination rate and ' 0 scales the adjustment costs. Notingthat the equilibrium will be symmetric (P r;t + i = P t+ i ) for all r and i, the retailers rst-orderoptimality condition is given by

' t 1 t = 1 " (1 X t ) + ' E t t;t +1t +1 1 t +1

Y t +1Y t

, (13)

where t is the gross ination rate from t 1 to t .Since price adjustment and investment adjustment costs are paid in real units, the economys

aggregate resource constraint is given by

1 '

2t 1

2!Y t = C t + 1 + ' I 2 I tI t 1 1 2!I t . (14)2.5 Banks

The model thus far is a conventional DSGE model. Frictionless nancial intermediation wouldensure that the following arbitrage condition should hold:

E t t;t +1 (RK;t +1 R t+1 ) = 0

Instead, this section develops a model of banking with agency problems, driving a wedge betweenE t t;t +1 RK;t +1 and E t t;t +1 R t+1 , and ensuring a non-trivial role for the composition of banksbalance sheets for economic outcomes.

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Banks lend funds, obtained from households, to manufacturers. Bank j s balance sheet is

QK;t K j;t +1 = N j;t + D j;t +1 + QE;t E j;t +1

where K j;t +1 is the quantity of nancial claims on manufacturers gross returns on capital. Sincethese claims are perfectly state contingent, it is possible to denominate one claim as one unitof capital, as I have done, implying that QK;t is also the relative price of each claim. N j;t isthe amount of net worth - or inside equity - that a bank has and D j;t +1 the deposits that thebank obtains from households. E j;t +1 is the quantity of outside equity that the bank issues tohouseholds and QE;t is the relative price of each claim. If one unit of outside equity is the claimon one unit of capital, then the gross real return on outside equity is given by

RE;t = exp ( "K;t )X t Y texp (" K;t )K t + (1 ) QE;t

QE;t 1.

The banks inside equity evolves as the dierence between earnings on assets and paymentson liabilities,

N j;t +1 = ( RK;t +1 RE;t +1 B j;t R t+1 (1 B j;t )) QK;t K j;t +1 + R t N j;t

where B j;t QE;t E j;t +1Q K;t K j;t +1 . Bank assets earn the state-contingent real gross return RK;t +1 . House-

hold deposits get paid the non-contingent real gross return Rt+1 and outside equity is paid thestate-contingent real gross return RE;t +1 . The banks objective is given by

V j;t = max E t P1i=0 (1 ) i t;t +1+ i N j;t +1+ i . (15)To motivate a limit on a banks ability to expand its balance sheet, I follow Gertler, Kiyotaki,and Queralto (2012) by introducing the a moral hazard problem: Bankers are able to abscondwith a fraction, of bank assets. This introduces an incentive compatibility constraint:

V j;t ( B j;t ) QK;t K j;t +1 (16)

Households will only provide funds up to the point at which bankers are still marginally bettero by not absconding with assets. To motivate a non-trivial choice for the composition of banksliabilities, I assume that the fraction of bank assets that bankers can abscond with is convex inthe share of assets funded by outside equity:

( B j;t ) = 0 1 + 1B j;t + 1

2 B2 j;t (17)

The rationale, that it is more dicult to abscond with assets funded by debt than by equity,comes from Calomiris and Kahn (1991) , and relies on the insight that debt requires the bank tomeet a non-contingent payment every period while dividend payments on equity are tied to theperformance of the banks assets and are therefore more dicult to monitor by outsiders .4

We can express V j;t as follows:

V j;t = K;t + B j;t E;t QK;t K j;t +1 + N;t N j;t (18)

with

K;t = E t ( t;t +1 t+1 (RK;t +1 R t+1 )) (19)

E;t = E t ( t;t +1 t+1 (R t+1 RE;t +1 )) (20)

N;t = E t ( t;t +1 t+1 ) R t+1 (21)4 If was independent of B , banks would strictly prefer to issue outside equity over debt. In this case, with

outside nancing coming from only equity, banks net worth would be completely shielded from movements inassets returns, thus rendering the nancial accelerator obsolete. There would however remain a credit spread insteady state that cannot be arbitraged away.

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where t = (1 )+ B t t . I assume that, in equilibrium, the incentive compatibility constraint,equation ( 16) binds .5 Rewriting equation ( 16) gives an expression for the inverse of the ratioof inside equity to total assets

j;t =N;t

( B j;t ) S;t + B j;t E;t

where j;t QK;t K j;t +1

N j;t . Combining the rst order conditions of the banks objection function,equation ( 15), subject to the incentive compatibility constraint, equation ( 16), for K j;t +1 andB j;t gives the following equilibrium condition:

E;t

K;t + B j;t E;t =

0(B j;t )( B j;t )

(22)

Symmetry of the equilibrium ensures that B j;t = Bt and j;t = t for all j . New banksreceive a start-up fund from households of !Q K;t K t . The evolution of aggregate net worth istherefore given by:

N t+1 = (RK;t RE;t B t 1 R t (1 B t 1)) t 1 + R t N t + !Q K;t K t

2.6 Monetary policy

Monetary policy is characterized by a simple reaction function

RN;tRN

= tX tX

Xt exp( S (S t S ))

1 R N (23)

RN;t 1RN

R N

exp( "M;t )

with the nominal interest rate, RN;t reacting only to deviations of observable variables fromtheir respective steady states (denoted by variables without time subscripts). In this set up,the central bank uses X t as an observable proxy of the output gap. The reaction function allowsfor the possibility that monetary policy reacts to two nancial indicators, bank leverage and thecredit spread, S t E t RK;t +1 R t+1 . Monetary policy shocks follow the exogenous stochasticprocess "M;t +1 = M "M;t + M M;t +1 with M;t Niid (0; 1). Finally, the link between nominaland real interest rates is given by the Fisher relation:

RN;t = R t+1 E t ( t+1 ) . (24)

2.7 Discussion of the model

The banking sector, and in particular banks balance sheet composition is of primary interest inthis paper; the rest of the model is relatively standard.

To understand the relationship between the monetary policy environment and the compo-sition of banks balance sheets, consider an expansionary monetary policy shock. Nominalrigidities in the economy means that a fall in the nominal risk-free rate generates a fall in thereal risk-free rate. In a model without nancial frictions, arbitrage ensures that the requiredexpected return on capital falls (to rst-order) one-for-one with a fall in the risk-free rate. Since

5 I choose parameter values such that, within the neighbourhood of the steady state, the incentive compatibilityconstraint does, in fact, bind.

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there are diminishing marginal returns to capital, a fall in the required expected return on cap-ital means that a larger set of investment projects have a positive net present value, generatinga boom in investment .6

The existence of an agency problem limits the amount of credit household are willing toextend as a function of banks net worth, as an over extension of credit could mean that bankers

have an incentive to forgo their accumulated retained earnings and abscond with a fraction of the banks assets instead. The limit on the creation of credit prevents arbitrage, thus driving awedge between the expected required return on capital and the risk-free rate.

When banks asset returns are below expectation, banks use their retained earnings to paytheir creditors. The fall in banks net worth therefore heightens the agency problem, causingthe wedge between the expected required return on capital and the real risk-free rate to movecountercyclically While in a frictionless nancial sector, the expected required return on capitaland the risk free rate move one-for-one, in the model with an agency problem, the expectedrequired return on capital moves by more than one-for-one. As a consequence, in response toan expansionary monetary policy shock, an even larger set of investment projects have a positivenet present value, generating an even greater boom in investment.

The extent to which banks are able to leverage themselves, and the extent to which banks networth is damaged by shocks, are crucial for the amplication and propagation of shocks throughthe nancial system. In particular, a bank heavily funded with non-contingent liabilities (debt)will experience high volatility in its net worth while for a bank which issues a lot of outside equity,a state-contingent claim on the bank, unexpected movements in asset returns are absorbed bythe concomitant movement in the return paid on outside equity, thus damping uctuations innet worth.

This begs the question, why dont banks issue only state-contingent claims? The insightfrom Calomiris and Kahn (1991) is that debt is a disciplining device for bankers. Banks,in choosing their (privately) optimal mix of short-term debt and outside equity nance, aretherefore an endogenous source of the amplication and propagation of shocks in the economy.

The banker maximizes the value of his bank, V subject to the incentive constraint binding.Given the net worth of the bank, the banker has two choice variables, the quantity of assets(capital) it invests in, QK;t K t+1 and the share of those assets funded by issuing outside equity,B t . The marginal benet of an additional unit of outside equity is given by E;t QK;t K t+1while the marginal benet of an additional asset is K;t + E;t B t . Thus, the marginal rate of substitution between outside equity and expanding the size of the balance sheet is E;t Q K;t K t +1

K;t + E;t B t.

The unconstrained optimum for the bank, all else equal, is to choose the highest feasible leverageand to raise external funds using only outside equity.

From the incentive compatibility constraint, the price of an additional unit of outside equityis d tdB t E;t QK;t K t+1 , while the price of an additional asset is t K;t + E;t B t . Both

these prices are assumed positive. The rst says that substituting debt with an additional unitof outside equity causes the ratio of assets that can be expropriated to rise more than the valueof the bank, causing the constraint to tighten. The second says something similar, that anadditional asset will raise the marginal quantity of assets that can be expropriated more thanit raises the value of the bank, again causing the constraint to tighten. The marginal rate of transformation between outside equity and an additional asset is therefore

ddB

= 0 E

( K + E B ) < 0, (25)

where 0 refers to the rst derivatives of equation ( 17) with respect to B. The marginal rateof transformation is strictly negative. In other words, the banker faces a trade-o since an

6 For simplicity in this discussion, I abstract from changes in other relative prices, like wages and the eect onthe labour market outcomes.

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increase in outside equity issuance must result in a lower capital-to-net-worth ratio, all elseequal. Equating the marginal rate of substitution and the marginal rate of transformationdelivers the equilibrium relation of equation ( 22).

In a risk-free environment, the benet of substituting outside equity for debt is zero ( E = 0 ),since the return on debt and outside equity is identical ( R = RE ). In addition, it follows that

0= 0 . We can show the following comparative statics in the neighborhood of the deterministicsteady state

dBd E

= B0

00( K + E B )dB

d EE =0

= 00K

> 0dB

d K=

0

E0 00( K + E B )

dBd K

E =0= 0

. (26)

The relations in the rst line hold because is convex by assumption. Thus, a marginalincrease in the value of outside equity leads to an increase in the share of outside equity issuance.The second line says that, at the margin, an increase in the excess value of assets over depositsdoes not generate any portfolio shifting.

d

d E = B

( K + E B ) 0 E

( K + E B ) dxd E

d

d E E =0 = B

( K + E B ) > 0d

d K= ( K + E B )

0 E( K + E B )

dxd K

dd K

E =0= ( K + E B ) > 0

dd N

= 1( K + E B ) 0 E

( K + E B ) dxd N

dd N

E =0= 1( K + E B ) > 0

(27)

The above comparative statics make use of the binding incentive compatibility constraint.The eect on leverage of a change in E , K , or N is the outcome of a direct and an indirecteect, as a result of a change in the liability mix of the bank. In the neighborhood of thedeterministic steady state, the indirect eect is zero.

To understand how risk aects this equilibrium relationship, we need to interpret E;t andK;t . E;t is the excess value of substituting outside equity for deposits while K;t is the excess

value of assets over deposits, and K;t + E;t B t is the excess value of assets over external nanceE;t E t ( t;t +1 t+1 (R t+1 RE;t +1 )) (28)

K;t + E;t B t E t ( t;t +1 t+1 (RK;t +1 R t+1 (1 B t ) RE;t +1 B t )) . (29)

Clearly, both E;t and E;t + E;t B t need to be non-negative for the bankers problem to be welldened. Notice also that both equations look like asset pricing equations, where the left-handside is the price, t;t +1 t+1 is the stochastic discount factor, and the remainder of the righthand side is the expected return. We can interpret et+1 t;t +1 t+1 as the bankers stochasticdiscount factor, which is dierent from the households stochastic discount factor, t;t +1 . t+1is the shadow marginal value of a unit of net worth. t+1 varies countercyclically becausethe incentive constraint becomes more binding in a downturn. Thus, the bankers stochastic

discount factor is countercyclical and more volatile than the households stochastic discountfactor.To be clear, I compare equation ( 28) to the rst order conditions of the households problem

(equations in ( 3)), which give 0 = E t ( t;t +1 (R t+1 RE;t +1 )) . At this stage, it is useful tointroduce the concept of the risk-adjusted steady state. As discussed earlier, at the deterministicsteady state, bankers and households are indierent between holding debt and outside equity.To ensure a determinate portfolio choice, we must consider a steady state that accounts forfuture uncertainty. The risk-adjusted steady state is dened as the portfolio that agents wouldhold if there was no risk today (the vector " t = 0), but they expected risk in the future( " t+ = Niid (0 ; I ) for all > 0). To compute the risk-adjusted steady state, it is possible totake a second-order approximation around the point " t +1 = 0 , which gives

0 = ( R RE ) cov t ( t;t +1 ; RE;t +1 ) (30)

E = e( R RE ) cov t et+1 ; RE;t +1 , (31)10

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where variables without time subscripts denote the variables at their risk-adjusted steady stateand cov t ( t;t +1 ; RE;t +1 ) is a covariance term conditional on time t information. Clearly, in-corporating expected risk at the risk-adjusted steady state allows monetary policy to aectsteady-state portfolio choices because the conduct of monetary policy can aect how variablescomove in the economy.

It follows that because cov t ( t;t +1 ; RE;t +1 ) and cov t et+1 ; RE;t +1 are both negative, but

with the second larger than the rst, and with > 0, the excess marginal value of substitutingoutside equity for debt for the banker is positive, E > 0. In other words, increased volatilityin the return on outside equity makes bankers more willing to issue outside equity, causingthe excess marginal value of substituting outside equity for debt to rise. Moreover, while thecredit spread may rise in a high uncertainty state, banks discounted value of that credit spread,

K;t , falls. This is because the realized spread between the return on capital and the costof borrowing is procyclical. When K;t + E;t B t falls, the value of the bank is lower as is themaximum leverage ratio that it can attain.

In Section 4 I validate this discussion by conducting numerical experiments to show howbanks balance sheet compositions are aected by the policy environment in which banks operate.

3 Parameterization and solution technique

3.1 Parameterization

Table 1 summarizes the structural parameter values used in the numerical experiments thatfollow. Many of the parameters are conventional in the literature, such as the income shareof capital, , the subjective discount factor, and the depreciation rate, , and all are setconsistent with time periods denoting quarters. Household preferences display habit formationand abstract from wealth eects on labour supply. The habit parameter, h = 0 :75 is settowards the upper end of the range of values seen in the literature. = 2 implies (holdinghours constant) an intertemporal elasticity of substitution of 0:5. # = 0 :33 implies a Frischelasticity of labour supply of 3. The weighting term, % = 0 :25 ensures that households allocate20% of their time to work (at the deterministic steady state). I follow Gertler and Karadi (2011)in choosing " = 4 :17 for the price elasticity of demand. Following Keen and Wang (2007) , Ichoose the Rotemberg price adjustment parameter, ' to match a Calvo (1983) -style modelin which rms face a 0:779 probability of keeping prices xed every quarter. The investmentadjustment parameter is set at 1.

The four parameters novel to the banking sectors agency problem are calibrated to match thesame steady-state moments as those in Gertler, Kiyotaki, and Queralto (2012) . In particular,they match a credit spread of 120 basis points, a ratio of total equity to assets, B + 1 = = 1=3,and a ratio of inside to outside equity of 2=3. Finally, they ensure that a rise in the standarddeviation of capital quality innovations from 0:69% to 2:07% reduces the inverse of the totalequity to asset ratio by 1=3. My method of solving for the risk-adjusted steady state diersfrom method preferred by Gertler, Kiyotaki, and Queralto (2012) , thus implying a need forthe parameters of the banking sector to be recalibrated from those in Gertler, Kiyotaki, andQueralto (2012) . In particular, the risk adjustment in my solution method is smaller for agiven standard deviation of the exogenous shock process. Thus, the key dierence is that my

2 = 8 :35 as opposed to 13:4.7 The smaller value of this parameter means that the incentive7

1 is negative in the calibration. This means that even in the deterministic steady state, banks issue anon-zero amount of outside equity. The economic rationale might be that there are some eciency gains inmonitoring the bank by having at least a small proportion of banks funding coming from outside equity. Moreimportantly though, the share of outside equity to assets responds to shocks. Therefore, the calibration of 1

and the standard deviation of shocks are chosen to ensure that the rst-order approximation of the model doesnot generate outside equity turning negative.

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Table 1: Parameterization

Description Value

Standard DSGE parameters Income share of capital 0:33

Quarterly subjective discount rate 0:99

h; ;%;# Preferences 11 C t hC t 1 %1+ # L

1+ #t

1 0:75; 2; 0:25; 0:33

" Price elasticity of demand 4:17 Quarterly depreciation rate 0:025' Adjustment cost ' 2 ( t = 1)

2 48:8' I Adjustment cost

' I 2 (I t =I t 1 1)

2 1

Banking sector Survival probability 0:9685

0; 1; 2 Agency cost: 0 1 + 1B + 22 B2 0:26; 0:75; 8:35

! Transfer to new bankers 0:0289

Exogenous shock processes

A ; 100 A Technology shock 0:95; 0:45K ; 100 K Capital quality shock 0; 1:5M ; 100 M Monetary policy shock 0:15; 0:24

Monetary policy

; X ; R N Standard monetary policy parameters 1:5; 0:5=4; 0; S Leverage, credit spread 0; 0

compatibility constraint tightens less in my model for a given change in the outside equity toasset ratio.

The model has three exogenous stochastic processes. The capital quality shock is calibratedin line with Gertler, Kiyotaki, and Queralto (2012) . In the baseline, I set K = 1 :5% and

K = 0 . The technology shock is considered to be highly persistent with the A = 0 :95. Themonetary policy persistence parameter is lower, M = 0 :15. In the baseline, the standard

deviation of the two shocks are A

= 0 :46% and M

= 0 :24%, respectively.The baseline reaction function is fairly conventional with the feedback coecient on inationand output set at 1:5 and 0:5=4 respectively.

3.2 Solution technique

The aim is to nd a rst-order approximation of the models dynamics in the neighborhoodof a risk-adjusted steady state. A formal statement of the solution technique is presented inAppendix A. Here I provide a description of the technique using a prototypical real businesscycle model. In so doing, I explain how the solution technique compares to others in theliterature.

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Consider the prototypical real business cycle model with equilibrium conditions 8

E t C t+1C t

exp( A A;t +1 ) K 1t = 1 (32)

exp( A A;t ) (K t ) K t+1 = C t , (33)

the exact solution of which is given by the decision rules, C t = g (K t ; A A;t ) and K t+1 =h (K t ; A A;t ). Since g and h do not, in general, have a closed-form representation, we wish tond a rst-order approximation of the functions g and h around the risk-adjusted steady state

C t C = gK (K; 0) (K t K ) + g (K; 0) A A;t (34)K t+1 K = hK (K; 0) (K t K ) + h (K; 0) A A;t , (35)

where C and K without time subscripts denote the risk-adjusted steady state values of consump-tion and capital, respectively, and gK ; g ; hK ; h are, as yet, undetermined coecients. Note,though, that the rst-order coecients are a function of the steady state, K .

Following Coeurdacier, Rey, and Winant (2011) , the risk-adjusted steady state is denedas the "point where agents choose to stay at a given date if they expect future risk and if the realization of shocks is 0 at this date." This implies setting C t = C , K t+1 = K t = K and

A;t = 0 in (32) and ( 33). The realization of C t+1 and A;t +1 are unknown at date t. Substitutingequation ( 34) into equation ( 32) leaves only the exogenous stochastic variable, A;t +1 :

1C

K 1E t f C + g (K; 0) A A;t +1 g exp( A A;t +1 ) = 1 .

We need to evaluate the object

E t f C + g (K; 0) A A;t +1 g exp( A A;t +1 ) ,

but because this object is unlikely to have a closed-form solution, we take a second-order ap-proximation around A;t +1 = 0 :

C 1 + (1 + ) C 2 (g (K; 0))2 2 C 1g (K; 0) + 12A2

.

Finally, the steady-state counterparts to equations ( 32) and ( 33) are as follows

K 10BB@

1 + (1 + ) C 2g2 2 C 1g + 12A2

| {z } risk-adjustment

1CCA

= 1

K K = C ,

where I have dropped the notation showing the explicit dependence of g on K . Notice thatwhen the risk-adjustment term is zero, the steady state coincides with the deterministic steadystate. The risk-adjustment term is a function of the conditional variance of consumption,g2 2A , the conditional covariance between consumption and the productivity shock, g 2A , andthe variance of the productivity shock, 2A . In a prototypical real business cycle model, thisrisk-adjustment raises K relative to the deterministic steady state due to precautionary savings.In the full model, in addition to taking account of precautionary saving, the risk-adjustment willalso alter banks portfolio choice, and in consequence the strength of the nancial accelerator.

8 This model is nested within the full model presented in Section 2. Financial and nominal frictions have beenremoved. Households are assumed to supply a xed quantity of labor, normalized to 1. In addition, h = 0 ,

= 1 , and A = K = 0 .

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Moreover, since changes in monetary policy alter the rst-order approximated coecients of thedecision rules, g (:), monetary policy is able to aect banks balance sheet composition in steadystate as well.

Since the steady state and the rst-order dynamics are jointly determined, the model issolved iteratively to nd a xed point. With an initial guess for the steady state, one can solve

for the rst-order dynamics using standard methods. The rst-order dynamics, however, areunlikely to be consistent with the initial steady state guess, prompting the steady state vectorto be updated. This procedure continues until convergence is achieved.

The rst-order solution of the model in the neighborhood of a risk-adjusted steady state is inthe spirit of Devereux and Sutherland (2011) and Coeurdacier, Rey, and Winant (2011) .9 Thereare, however, two important points of note. First, the risk-adjusted steady state as I use ithere is not exactly as Devereux and Sutherland (2011) or Coeurdacier, Rey, and Winant (2011)proposed it. The Devereux and Sutherland (2011) technique is to take an approximation aroundthe deterministic steady state. However, at the deterministic steady state, the internationalportfolio choice problem that concerns them is indeterminate. 10 They therefore apply a risk-adjustment to the equilibrium conditions pertaining to the portfolio choice problem only. I,

however, apply the risk-adjustment to all of the models forward looking equations. The mainquantitative eect of applying the risk-adjustment to the entire set equations is that I captureboth changes in bankers optimal portfolio but also changes in households precautionary savingsas a result of changes in exogenous uncertainty and changes in policy.

The Coeurdacier, Rey, and Winant (2011) method requires a second-order approximationof the steady state as well as evaluating the coecients of the rst-order approximation of theequilibrium conditions up to second-order. This second step is crucial for Coeurdacier, Rey,and Winant (2011) as it ensures that the net foreign asset position in their small open economymodel is no longer a unit root process. However, this additional step adds greatly to thecomputational burden yet alters the solution (in my case) only imperceptibly. I therefore solvethe model with the steady state up to second-order but evaluate the coecients of the rst-order

conditions only up to a zeroth-order.11

Finally, the Appendix details the calculation of welfare at the risk-adjusted steady state. Themeasure of welfare I present in the next section is conditional welfare, measured in consumptionequivalent units. In some cases, I present welfare relative to the baseline calibration, in othersimulations I present welfare relative to the deterministic steady state. A positive consumptionequivalent value indicates that household need to be compensated for moving from the baselinecalibration (or deterministic steady state) to the new environment.

4 Results

In this section, I present several numerical experiments that shed light on the interaction be-tween banks balance sheet composition and the monetary policy environment. The monetarypolicy environment in this model is characterized by the nominal interest rate instrument thatevolves as the product of an exogenous stochastic process and an endogenous reaction functionto observable economic conditions. I rst consider the eect on banks balance sheet composi-tion of changes in monetary and other exogenous uncertainty. I then consider the eect on bank

9 The DSGE literature accounting for risk and portfolio choices in solution methods are closely linked. Anon-exhaustive list of other important papers in the literature include Collard and Juillard (2001) , Evans andHnatkovska (2012) , Benigno, Benigno, and Nistic (2013) and Kliem and Meyer-Gohde (2013) . One dierencewith many of these papers is that they incorporate time-varying risk, which my solution method does not.

10 The banks portfolio choice problem in this model is uniquely pinned down at the deterministic steady state:B dss = 1 = 2 . This is because the optimality condition reduces to solving 0 (B ) = 0 as E;t = 0 . And that,in turn, is because households become indierent between holding a debt or a equity contract with the bank.

11 The risk-adjusted steady state I use also diers from the risk adjustment employed in Gertler, Kiyotaki, andQueralto (2012) .

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balance sheets of changes to the endogenous reaction function that the policy maker employs.

4.1 Eect of exogenous uncertainty on banks balance sheets

Figure 1 presents the behaviour of the economy in response to a change in monetary and otherexogenous uncertainty. The horizontal axis in each plot indicates the standard deviation of exogenous innovations (in percent). The red, green, and blue lines plot the eect of technology,capital quality, and monetary policy shocks, respectively. In each case, we alter the value of i ,holding j = 0 for j 6= i. The top four panels show the eect of uncertainty on the risk-adjustedsteady state of several variables. At the deterministic steady state, all these lines are at. Thebottom four panels show the eect of uncertainty on the standard deviation of a set of variableson interest.

The top left panel shows the risk-adjusted steady state for two balance sheet indicators, theoutside equity to asset ratio and total leverage respectively. In the face of increased uncertainty,banks issue more outside equity as a share of total assets (as well as in absolute terms).Thisis because there is greater value in hedging the asset return risk banks face relative to benetof higher leverage. Similarly, banks inside-equity-to-asset ratio also increases in the face of increased exogenous uncertainty. This is in part because of the increased use of outside equitywhich exacerbated the agency problem between banks and households, and limits the amountof funds households are willing to provide. But inside equity (in absolute terms) also rises.This is because the credit spread, E t (RK;t +1 R t +1 ) has risen (as seen in the top right panel),leading to a higher steady-state accumulation of retained earnings. However, the risk-adjustedvalue, E t t;t +1 t +1 (RK;t +1 R t +1 ) is lower for the bank in the more uncertain environment,which contributes to tightening the incentive compatibility constraint. As the standard devi-ation of capital quality shocks increases from zero to 3%, banks decrease total leverage fromapproximately 412 to 2, while the outside-equity-to-asset ratio rises from around 0:09 to 0:22.In parallel, credit spreads almost double, from 90 basis points to close to 180.

The second row of Figure 1 shows the eect on the steady-state capital stock and on householdwelfare. The capital stock plot is interesting as the eect of technology uncertainty is quitedierent from monetary and capital quality uncertainty. When technology uncertainty rises,the steady-state capital stock also rises. For the other two shocks, the capital stock falls.These dierential eects result from two separate mechanisms. Most of the discussion in thispaper has focussed on the eect of uncertainty on a banks balance sheet. As uncertainty rises,banks deleverage, resulting in a reduction of credit created for a given quantity of bank networth and therefore a contraction in the capital stock, which is 100% nanced by credit in themodel. However, uncertainty has an important second eect in the model in that it generatesprecautionary savings by households as in a standard business cycle model without nancialfrictions. Households wish to save more in an uncertain environment resulting in a build up

of savings in the form of capital. Thus, with constrained banks, uncertainty has two osettingeects on the capital stock in the economy. For technology shocks (at least at high levels of uncertainty), the precautionary saving motive dominates, while for monetary and capital qualityuncertainty, the eect on credit constraints dominates.

Despite technology shocks generating a rise in steady-state capital (and consumption), theright-hand panel shows that uncertainty is unambiguously bad for household welfare, mostlybecause households are risk averse. The measure of welfare shown in the plot is the consumptionequivalent stream required for the household to be indierent between the baseline level of uncertainty and the actual level of uncertainty as dened by the horizontal axis, conditionalon the economy being initially at the baseline risk-adjusted steady state. (For the case of thetechnology shock, this means that households, in the transition, initially reduce consumption in

order to build capital). Even without the transitional dynamics (i.e., using an unconditionalmeasure of welfare), higher uncertainty remains unambiguously bad for households. For ahousehold to be indierent between moving from the baseline calibration of the shocks to an

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environment in which A = 3% and K = M = 0 , the households stream of consumption in theinitial calibration would need to be 0:3% lower every period.

The change in the composition of banks balance sheets inuences the transmission of shocksthrough the nancial sector. Rows 3 and 4 plot the standard deviation of three key endogenousvariables - ination, output growth and credit spreads. In each case, the dotted line shows the

standard deviation, had the steady state been held unchanged. With a rst-order approximationof the model around an unchanged steady state, the standard deviation of endogenous variableswould increase linearly with the standard deviation of the exogenous shock process, as thedotted lines show. In all four plots, it is clear that the standard deviation of the endogenousvariables change with respect to the standard deviation of exogenous innovations at a slowerrate than implied a naive approximation of the model around an unaltered steady state (i.e.,the dotted lines). This should not be a surprise. In the face of greater uncertainty, banks haveshifted towards greater equity nance, severely dampening the nancial accelerator channel.The eect of changes in banks balance sheet on macroeconomic volatility can be quantitativelylarge. Suppose, for example, that the standard deviation of the capital quality shock innovationrose from 1.5% to 3%. If we believed that banks balance sheet determination was orthogonal

to this change in the environment, and total bank leverage remained at 312 , the model wouldexpect the standard deviation of output growth to increase from 34 % to 1

14 %. However, when

the model accounts for banks endogenous balance sheet adjustment, reducing total leverage to2 12 , the standard deviation of output growth rises from

12 % to only 0:6%. The eect of changes

in the stochastic environment through the balance sheet channel have the largest implicationsfor the volatility of credit spreads. The standard deviation of credit spreads does not increasemonotonically in the standard deviation of the exogenous innovations. Instead, the standarddeviation of credit spreads rises initially but then falls again and attens out at a standarddeviation of around 1 percent.

We can use these plots to draw implications about the relationship between banks balancesheets and exogenous shock processes in dierent periods of US history, like the pre-great moder-

ation, great moderation and post-great moderation periods. The diculty with calibrating themodel to dierent periods of US macroeconomic history using estimated shock processes frompapers like Smets and Wouters (2007) is that these estimated shock processes are naturally aproduct of a linear model and de-trended series. However, suppose we put these caveats asideand take the parameters from Smets and Wouters (2007) at face value Then we would nd littleevidence of a strong risk channel. Smets and Wouters (2007) nd estimates of A equal to 0:58and 0:35 for the periods 1966-75 and 1984-2004 respectively. 12 As can be seen from Figure 1,such a change induces a shift in total bank leverage of around one tenth, which is not enoughto account for the build up of leverage during the great moderation.

An alternative, back-of-the-envelope exercise, is to start with the (quarter-on-quarter) stan-dard deviation of output growth for the US before, during and after the great moderation.

These are 1.05% (1960-84), 0.49% (1985-2007), and 0.85% (2008-12), respectively, and takenfrom the US National Income and Product Accounts (NIPA tables). Next, assume that theonly structural change to have occurred in this time has been the standard deviation of technol-ogy innovations .13 From the third row, the model implies a standard deviation of technologyshock pre-great moderation of 1% and during the great moderation of 0.5%. In the absenceof the risk channel, the estimate for the pre-great moderation period would have been 0.9%.Thus, ignoring the risk channel and banks endogenous balance sheet choices can signicantlybias the estimates of the standard deviation of shocks.

12 Table 5., p606.13 We can repeat this exercise in the next subsection by backing out the balance sheet eect as a result of

monetary policy changes pre-Volcker and thereafter using estimated values of Taylor rule parameters from the

literature. The risk channel, however, in terms of historical changes in monetary policy ination activism is againquantitatively small.

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4.2 Eect of monetary reaction function on banks balance sheets

Figures 2-5 present the behaviour of the economy in response to changes in the monetary-policyreaction function. The horizontal axis in each plot indicates the reaction-function parameterthat is being adjusted. The remaining parameters are held at their respective baseline values.The information in the gures are similar to Figure 1. However, there are three noteworthydierences. First, for each policy experiment, I show three dierent exogenous stochastic states,a low, baseline, and high risk state corresponding to 100 K = 0 ; 1:5; and 2:5; respectively, withthe other shock processes held xed at the baseline calibration. Second, the consumption equiv-alent conditional welfare measure is relative to the deterministic steady state, an assumptionmade purely for clarity of the exposition that does not aect the welfare ranking of alternativepolicies. Third, the dotted lines in rows 3 and 4 have a dierent interpretation. In each case,the dotted lines show the standard deviation of the variables had the model been solved arounda risk-adjusted steady state that responds to changes in the exogenous stochastic environmentbut does not respond to changes in the policy environment (i.e., the changes in the policy para-meter values on the horizontal axis). The dotted lines can therefore be interpreted as the naivepolicymakers prediction about the eect of a change in his reaction function, unaware of therisk channel.

Figure 2 plots changes in the reaction functions responsiveness to deviation of ination, 2 [1:01; 2:5]. There are two unsurprising results. First, aggressively responding to deviations

of ination reduces the volatility of ination and second, aggressively responding to ination isstrictly welfare improving, in line with standard results in the literature. There are, however,three further eects of aggressive ination targeting that are of interest as they are the result of the risk channel of monetary policy. First, in a high risk environment, aggressively respondingto ination reduces volatility in nancial market credit spreads, while standard solution methodswould have predicted a rise in the volatility of credit spreads. Accompanied with this reductionin the volatility of credit spreads, the average credit spread also rises by about 5 basis pointsas the coecient increases from 1:5 to 2:5, a result of banks issuing modestly more outsideequity and reducing their leverage.

These eects are relatively small. The change in the risk environment from low to high riskhas a several orders of magnitude larger eect on banks balance sheet composition than doesincreasing the aggressiveness of ination activism from 1:01 to 2:5.

Figure 3 repeats the same experiment for the coecient X 2 [0; 0:25], the aggressivenesswith which the policymaker responds to deviations of its proxy for the output gap. The atnessof the curves in rows 1 and 2 and the closeness of the solid and dotted lines in row 3 indicatesthat the risk channel is relatively negligible along this dimension of policy. The only signicantdierences appear in the volatility of nancial sector credit spreads in the bottom panel of thegure.

Figures 2-3 suggest that variation in the conventional arguments of the reaction function havelittle eect on banks balance sheets. To the extent that the central bank has nancial stabilityconcerns, however, the eect can be much greater. And a discussion of whether central banksshould use their monetary policy tool, the nominal interest, for nancial stability objectives -leaning against assets bubbles, dampening credit cycles and preventing the build up of nancialstability - is at the centre of the current policy debate.

In this spirit, I consider the eect rst of the central bank responding with its nominalinterest rate to movements in bank leverage, shown in Figure 4.14 In Figure 5, I consider theresponse of the central bank to movements in credit spreads. Since both leverage and spreadsmove countercyclically in the model, the policymaker would respond countercyclically, loweringthe nominal interest rate when leverage rises and/or credit spreads rise. Thus and S areassumed to take negative values.

14 The central bank is assumed to respond to the ratio of assets to inside equity. The results are little aectedby changing the target variable to be total leverage.

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Figure 2: Eect of variation in monetary policy response to ination

0.1

0.2

O u t s i

d e e q . /

a s s e

t s ( l h s

, s q u a r e s

)

T o t a l

l e v e r a g e

( r h s

, l i n e )

Bank balance sheet

3

4

1

1.2

1.4

1.6

Credit spread (ann. %)

39

39.5

40

40.5

41

41.5

Capital stock

0.5

1

1.5

Conditional welfare(consumption compensation)

% r

e l a t

i v e

t o

d e t . s s

5

10

15

20

St. dev. inflation (%)

1.5 2 2.5

0.55

0.6

0.65

0.7

0.75

0.8

St. dev. output growth (%)

1.5 2 2.5

0.7

0.8

0.9

1

1.1

St. dev. credit spreads (ann. %)

Low risk stateBaselineHig h risk state

Note: The horizontal axis measures changes in a monetary policy parameter. In each case the monetary policyparameter is adjusted, holding the rest of the calibration unchanged. The top four panels plot risk-adjusted

steady state value of several key variables. The bottom three panels plot standard deviations of several keyvariables. The three dierent colors in each plot reect three dierent levels of exogenous uncertainty. Low,baseline and high refer to 100 K = 0 ; 1:5; 2:5, respectively. Dashed lines in the bottom three panels denote

the standard deviations with the steady state incorporating the state of exogenous uncertainty (low, baseline orhigh) but not incorporating changes in the monetary policy parameter.19

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Figure 3: Eect of variation in monetary policy response to output gap

0.1

0.2

O u t s i

d e e q . /

a s s e

t s ( l h s

, s q u a r e s

)

T o t a l

l e v e r a g e

( r h s

, l i n e )

Bank balance sheet

3

4

1

1.2

1.4


40

40.5

41

41.5

Capital stock

0.2

0.4

0.6

0.8

1

1.2


% r

e l a t

i v e

t o

d e t . s s

1

1.5

2


0 0.02 0.04 0.06 0.08

0.5

0.55

0.6

0.65

0.7

0.75


X

0 0.02 0.04 0.06 0.08

0.7

0.8

0.9

1


X





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Figure 4: Eect of variation in monetary policy response to leverage

0.1

0.2

O u t s i

d e e q . /

a s s e

t s ( l h s

, s q u a r e s

)

T o t a l

l e v e r a g e

( r h s

, l i n e )

Bank balance sheet

3

4

1

1.2

1.4


38

39

40

41

42Capital stock

0.2

0.4

0.6

0.8

11.2

1.4


% r

e l a t

i v e

t o

d e t . s s

5

10

15


-0.1 -0.05 0

0.4

0.5

0.6

0.7


-0.1 -0.05 0

0.6

0.8

1

1.2






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In Figure 4, 2 [ 0:1; 0]. Thus, at the extreme value of 0:1, a 10% rise in bank leverageis countered with a 1% reduction in the nominal interest rate. 15 Given the dominance of supplyshocks in the calibration, responding to leverage dampens the volatility of output growth atthe expense of greater ination volatility. Consider the baseline calibration (green line) and aswitch of policy from = 0 to = 0:1. Banks, perceiving this policy change to dampen the

volatility of their asset returns, decrease their share of funding from outside equity and expandtheir leverage. The outside-equity-to-asset ratio drops from 0.15 to 0.11 while total leveragerises from 3.1 to 3.9. In the absence of this balance sheet adjustment, the standard deviationof output would have halved, from 0.6% to 0.3%. However, due to banks willingness to takeon more balance sheet risk in response to the change in the policy environment, the standarddeviation drops only to 0.45%. In this example, therefore, the risk channel of monetary policyis quantitatively important. When we look at the welfare implications of this policy choice, themodel provides a mixed message. In benign or normal times, targeting leverage can be welfareenhancing. However, in a high risk environment, targeting leverage is bad for welfare.

Figure 5 experiments with S 2 [ 4; 0]. This range implies that, at its most extreme,S = 4, a 10 basis point rise in credit spreads results in a 40 basis point reduction in the

nominal interest rate. Again, the eects of the risk channel appear to be most powerful atthe baseline calibration, with less pronounced eects in high and low risk states. Targetingmovements in credit spreads, like targeting leverage, increases banks willingness to issue debtinstead of outside equity and leverage up. Credit spreads also fall. Perversely, the volatility of credit spreads rises. In the counterfactual experiment with no risk channel (the dotted line),a change in policy towards responding aggressively to movements in credit spreads would haveyielded a reduction in the standard deviation of the credit spread from 0.9% to below 0.8%.However, with the risk channel present, the increase in bank leverage results in the standarddeviation of credit spreads actually rising, to close to 1.1%. In contrast to targeting leverage,however, the risk channel amplies the standard deviation of ination while dampening thestandard deviation of output growth, relative to the model solution without the risk channel.

The results presented in this section suggest that the risk channel of monetary policy has,under certain policy prescriptions, meaningfully sized economic eects. Policymakers should beaware of this endogenous risk channel via endogenous changes in bank balance sheets, especiallyif they aim to redesign policy to target specic nancial sector indicators using the standardtools of monetary policy.

5 Conclusion

There is a popular view that the great moderation of the 1990s and early 2000s sowed the seed of the global nancial crisis in 2007. As macroeconomic outcomes became less uncertain, nancialintermediaries built up leverage and took on more risk. In turn, another literature has tried toexplain the causes of the great moderation, from which two main views have emerged. One isa good luck story, that the global economy simply enjoyed a period in which the shocks hittingthe economy were unusually modest. The other view is that central banks had a better designof monetary policy.

This paper explores the risk channel of monetary policy in a quantitative macroeconomicmodel by endogenizing the composition of banks funding. I nd that when central bankstarget nancial variables such as cyclical leverage or credit spreads, policy can alter banksbalance composition in a quantitatively meaningful way, and aect how shocks are ampliedand propagated through the nancial sector. The numerical experiments in this paper suggestthat central banks and nancial sector regulators should be vigilant of how periods of relativetranquillity (like the great moderation) can generate a potential build up of risks in the economy

15 Values of < 0:1 generate convergence problems for the solution algorithm.

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Figure 5: Eect of variation in monetary policy response to credit spread

0.1

0.2

O u t s i

d e e q . /

a s s e

t s ( l h s

, s q u a r e s

)

T o t a l

l e v e r a g e

( r h s

, l i n e )

Bank balance sheet

3

4

1

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1.4


39.5

40

40.5

41

41.5

Capital stock

0.2

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0.8

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1.2


% r

e l a t

i v e

t o

d e t . s s

1

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4

5


-4 -3 -2 -1 0

0.3

0.4

0.5

0.6

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S

-4 -3 -2 -1 0

0.6

0.8

1


S





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Goldstein, I. and A. Pauzner (2005). Demanddeposit contracts and the probability of bankruns. Journal of Finance 60 (3), 12931327.

Hart, O. and J. Moore (1994). A theory of debt based on the inalienability of human capital.Quarterly Journal of Economics 109 (4), 84179.

Keen, B. and Y. Wang (2007). What is a realistic value for price adjustment costs in new-Keynesian models? Applied Economics Letters 14 (11), 789793.

Kiyotaki, N. and J. Moore (1997). Credit cycles . Journal of Political Economy 105 (2), pp.211248.

Kliem, M. and A. Meyer-Gohde (2013). Monetary policy and the term structure of interestrates. Unpublished manuscript .

Korinek, A. (2011). Systemic risk-taking: Amplication eects, externalities, and regulatoryresponses. ECB Working Papers 1345 .

Lorenzoni, G. (2008). Inecient credit booms . The Review of Economic Studies 75 (3), pp.809833.

Modigliani, F. and M. Miller (1958). The cost of capital, corporation nance and the theoryof investment. The American Economic Review 48 (3), 261297.

Nikolov, K. (2010). Is private leverage excessive? Unpublished manuscript .

Rotemberg, J. (1982). Monopolistic price adjustment and aggregate output. The Review of Economic Studies 49 (4), 517531.

Schmitt-Groh, S. and M. Uribe (2004). Solving dynamic general equilibrium models usinga second-order approximation to the policy function. Journal of Economic Dynamics and Control 28 (4), 755775.

Smets, F. and R. Wouters (2007). Shocks and Frictions in US Business Cycles: A BayesianDSGE Approach. The American Economic Review 97 (3), 586606.

Stein, J. (2012). Monetary policy as nancial stability regulation. The Quarterly Journal of Economics 127 (1), 5795.

Townsend, R. (1979). Optimal contracts and competitive markets with costly state verica-tion. Journal of Economic Theory 21 (2), 26593.

A Risk-adjusted steady state and rst-order dynamics: Theory

This section explains how to solve a model as a rst-order approximation of the model arounda second-order approximation of the models risk-adjusted steady state.

Let the equilibrium conditions of the model be written as

E t [f (yt+1 ; yt ; x t+1 ; x t ; zt+1 ; zt )] = 0 (36)

zt+1 = z t + " t+1

where yt is an ny 1 vector of endogenous nonpredetermined variables, xt is an nx 1 vectorof endogenous predetermined variables, zt is an nz 1 vector of exogenous variables and "t isan nz 1 vector of exogenous i.i.d. innovations with mean zero and unit standard deviations.The matrices and are of order nz n z and is a scalar scaling the amount of uncertaintyin the economy.

Next, let the (unknown) decision rules that solve the system of equations in ( 36) be yt =

g (x t ; zt ) and x t+1 = h (x t ; zt ). The risk-adjusted steady state, xr

solvesxr = h (xr ; 0) with yr = g (xr ; 0) (37)

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Substituting the decision rules into ( 36) and evaluating at the (also as yet unknown) risk-adjustedsteady state gives

f (x r ; ) E t [f (g (xr ; " t+1 ) ; g (xr ; 0) ; xr ; xr ; " t+1 ; 0)] = 0

Note that " t+1 is not an argument but the variable of integration inside the expectations operator.Taking a second-order approximation of f around = 0 (but a rst-order approximation of g (:)and h (:)) gives

[f (x r ; )]i [f (x r ; 0)]i + 2

2 [f (xr ; 0)]i = 0 (38)

wheref (x r ; 0) = f (yr ; yr ; x r ; xr ; 0; 0)

and

[f (xr ; 0)]i = f y0y0i [gz ] [gz ]

[I ]

+ f y0z 0i

[gz ] [ ] [I ] + [ z 0z 0]i

[ ] [ ] [I ]i = 1 ;:::;n; ; = 1 ;:::;ny ; ; = 1 ;:::;n z ; ; = 1 ;:::;n"

The notation follows that in Schmitt-Groh and Uribe (2004) . The rst derivatives of thedecision rules, gx are solved using standard methods of rst-order approximation. The solutionis found by iterating between a set of steady state values (y; x) and a set of decision rulecoecients (gz ) until convergence is achieved .16

B Equilibrium conditions

The model presented in Section 2 has 23 endogenous variables, f C t ; I t ; K t ; N t ; t ; QK;t ; QE;t ; RN;t ;R t ; U C;t ; B t ; Y t ; t ; L t ; t 1;t ; X t ; S;t ; E;t ; N;t ; t ; RE;t ; RK;t ; S t g and 23 equilibrium equations:

Aggregate resource constraint:

1 '

2 ( t 1)2 Y t = C t + 1 +

' I 2

I tI t 1

12!I t

Capital accumulation:K t+1 = (1 )exp( "K;t ) K t + I t

Price of capital:

QK;t = 1 + ' I

2

I t

I t 11

2+

I t

I t 1' I

I t

I t 11 E t t;t +1

I t+1

I t

2' I

I t +1

I t1

Inverse of banking sectors inside equity to asset ratio:

t = QK;t K t+1

N t

Banking sectors inside equity accumulation:

N t = (RK;t RE;t B t 1 R t (1 B t 1)) t 1 + R t N t + !Q K;t K t

Household Euler equation and arbitrage condition:

Et t;t +1

Rt +1

= 1 and Et t;t +1

(RE;t +1

Rt +1

) = 116 Code to implement this solution algorithm and replicate the results in the paper is available from the author

on request.

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Household stochastic discount factor:

t 1;t = U C;tU C;t 1

Banking sectors binding incentive compatibility constraint:

t =N;t

0 1 + 1B t + 22 B2t S;t + B t E;t

Banking sectors optimal leverage-outside equity trade-o:

E;t

S;t + B t E;t=

1 + 2B t1 + 1B t + 22 B

2t

Return on capital and outside equity:

RK;t = exp ( "K;t )X t Y texp (" K;t )K t + (1 ) QK;t

QK;t 1

RE;t = exp ( "K;t )

X t Y texp (" K;t )K t + (1 ) QE;t

QE;t 1Labour market equilibrium:

X t (1 ) Y t U C;t = %L1+ #t C t hC t 1 %1 + #

L1+ #t

Marginal utility of consumption:

U C;t = C t hC t 1 %1 + #

L1+ #t

t h C t+1 hC t %1 + #

L1+ #t+1

Aggregate production function:

Y t = exp ( "A;t )(exp( "K;t ) K t ) L1tBanking sectors marginal value of an intermediating an additional unit of credit, of substitutingone unit of debt for one unit of outside equity and of an additional unit of net worth, respectively:

K;t = E t ( t;t +1 t+1 (RK;t +1 R t+1 ))

E;t = E t ( t;t +1 t+1 (R t+1 RE;t +1 ))

N;t = E t ( t;t +1 t+1 ) R t+1

Shadow value of net worth:t = (1 ) + B t t

Phillips curve:

' t 1 t = 1 " (1 X t ) + ' E t t;t +1t+1 1 t+1

Y t+1Y t

Monetary policy reaction function:

RN;tRN

= tX tX

Xt exp( S (S t S ))

RK;tRK

R K QK;tQK

Q K1 R N

RN;t 1RN

R N

exp( "M;t )

Credit spread:S t = E t RK;t +1 R t+1

Fisher relation:RN;t = R t+1 E t ( t+1 )

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M 5 12

RK Pn "i=1 gR Ki2 2

i

M 6 12

( RK R)Pn "i=1 gi 2 + 2 gi gi + gi 2 2iM 7 12 (RE R) gi

2 + 2 gi gi + gi 2 (RE R) + RE 2gi gR Ei + 2 gi gR Ei + gR Ei 2

M 8 12

RPn "i=1 gi 2 + 2 gi gi + gi 2 2iM 9

12

' Pn "i=1 3 gi 2 + 2 gi gi + 2 gi gY i 2iM 10

12

RPn "i=1 gi 2 2iwhere n " is the number of exogenous shocks and i = f A;K;M g indexes the shock. For example,g

C A is the rst-order response of consumption (on impact) of a shock to technology.

D Welfare

Welfare at time t is denoted

W t = E t X1i=0 i U (C t + i ; L t+ i )In recursive form, welfare can be written as:

W t = U (C t ; L t ) + E t W t +1

At the risk-adjusted steady state, this becomes

W = U (C; L) + M W

1

where the risk-adjustment takes the form:

M W W

2gWi

2 2i .

Given initial policy A and alternative policy B, the consumption equivalent compensation re-quired to be indierent about switching from A to B is u and is implicitly given by

W B = U (1 u ) C A ; LA + M AW

1 Given the rst-order approximation of the decision rule, W t = g (x t ) around the risk-adjustedsteady state with policy B

log W t log W B = gWx log x t log x

B

the welfare of policy B, conditional on initially being at the state vector xA (i.e., the risk-adjustedsteady state with policy A), denoted W B jA is

log W B jA log W B = gWx log xA log xB

To rewrite this level of welfare in terms of consumption equivalent units, c, I calculate

W B jA = U (1c) C A ; LA + M

AW

1 .

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