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Intermediary Leverage Cycles and Financial Stability Tobias Adrian and Nina Boyarchenko The views presented here are the authors’ and are not representative of the views of the Federal Reserve Bank of New York or of the Federal Reserve System

Introduction

Questions about Financial Stability Policy

Systemic distress of financial intermediaries raises questions aboutfinancial stability policies:

How does capital regulation affect the trade-off between the pricing ofcredit and the amount of systemic risk?

How does macroprudential policy function in equilibrium?

What are the welfare implications of capital regulation?

We develop a theoretical framework to address these questions

T. Adrian, N. Boyarchenko Intermediary Leverage Cycles May 8, 2013 2

Introduction

Our Approach

We use a standard macro model with a financial sector and add onekey assumption:

Funding constraints of financial intermediaries are risk based, sointermediaries have to hold more capital when the riskiness of theirassets increases

This assumption is empirically motivated from risk managementpractices and regulatory constraints

Equilibrium dynamics capture stylized facts:

Procyclical leverage of intermediary balance sheetsProcyclical share of intermediated creditRelationship between asset risk premia and intermediary leverage


The Model

Economy Structure

Producersrandom dividendstream, At, per unitof project financed bydirect borrowing fromintermediaries andhouseholds

Intermediariesfinanced by house-holds against capitalinvestments

Householdssolve portfolio choiceproblem betweenholding intermedi-ary debt, physicalcapital and risk-freeborrowing/lending

Atkht

it

Atkt

Cbtbht

1


The Model

Production

Aggregate amount of capital Kt evolves as

dKt = (It − λk)Ktdt

Total output evolves as

Yt = AtKt

Stochastic productivity of capital {At = eat}t≥0

dat = adt + σadZat

pktAt denotes the price of one unit of capital in terms of theconsumption good


The Model

Households

Household preferences are:

E[∫ +∞

0e−(ξt+ρht) log ctdt

]Liquidity preference shocks (as in Allen and Gale (1994) and Diamondand Dybvig (1983)) are exp (−ξt)

dξt = σξρξ,adZat + σξ

√1− ρ2

ξ,adZξt

Households do not have access to the investment technology

dkht = −λkkhtdt


The Model

Households’ Optimization

max{ct ,kht ,bht}

E[∫ +∞


]subject to

dwht = rftwhtdt + pktAtkht (dRkt − rftdt) + pbtAtbht (dRbt − rftdt)− ctdt

and no-shorting constraints

kht ≥ 0

bht ≥ 0


The Model

Intermediaries

Financial intermediaries create new capital

dkt = (Φ(it)− λk) ktdt

Investment carries quadratic adjustment costs (Brunnermeier andSannikov (2012))

Φ (it) = φ0

(√1 + φ1it − 1

)Intermediaries finance investment projects through inside equity andoutside risky debt giving the budget constraint

pktAtkt = pbtAtbt + wt


The Model

Intermediaries’ Risk Based Capital Constraint

Risk based capital constraint (Danielsson, Shin, and Zigrand (2011))

α

√1

dt〈ktd (pktAt)〉2 = wt

Implies a time-varying leverage constraint

θt =pktAtkt

wt=

1

α

√1dt

⟨d(pktAt)pktAt

⟩2

Note that the constraint is such that intermediary equity isproportional to the Value-at-Risk of total assets

This will imply that default probabilities vary over time

Microfoundation of the risk based capital constraint is provided inAdrian and Shin (2010)


The Model

Risk-based Capital Constraints

VaR is the potential loss in value of inventory positions due toadverse market movements over a defined time horizon with aspecified confidence level. We typically employ a one-day timehorizon with a 95% confidence level.

Source: Goldman Sachs 2011 Annual Report

More


The Model

Commercial Bank Tightening Standards

Q2−91 Q2−93 Q2−95 Q2−97 Q2−99 Q2−01 Q2−03 Q2−05 Q2−07 Q2−09 Q2−110

20

40

60V

IX

−50

0

50

100

Cre

dit T

ight

enin

g

ρ=0.68013


The Model

Systemic Distress

Solvency risk defined by

τD = inft≥0{wt ≤ ωpktAtKt}

Term structure of systemic distress

δt (T ) = P (τD ≤ T | (wt , θt))

In distress

Management changes

Intermediary leverage reduced to θ ≈ 1 by defaulting on debt

Intermediary instantaneously restarts with wealth

wτ+D

=θτDθ

wτD


The Model

Intermediaries’ Optimization

Intermediary maximizes equity holder value to solve

max{kt ,βt ,it}

E[∫ τD

0e−ρtwtdt

]subject to the dynamic intermediary budget constraint

dwt = ktpktAt (dRkt + (Φ (it)− it/pkt) dt)− btpbtAtdRbt

and the risk based capital constraint

α

√1

dt〈ktd (pktAt)〉2 = wt


The Model

EquilibriumAn equilibrium in this economy is a set of price processes {pkt , pbt ,Cbt}t≥0, aset of household decisions {kht , bht , ct}t≥0, and a set of intermediary decisions{kt , βt , it , θt}t≥0 such that:

1 Taking the price processes and the intermediary decisions as given, thehousehold’s choices solve the household’s optimization problem, subjectto the household budget constraint.

2 Taking the price processes and the household decisions as given, theintermediary’s choices solve the intermediary optimization problem,subject to the intermediary wealth evolution and the risk based capitalconstraint.

3 The capital market clears:

Kt = kt + kht .

4 The risky bond market clears:

bt = bht .

5 The risk-free debt market clears:

wht = pktAtkht + pbtAtbht .

6 The goods market clears:

ct = At (Kt − itkt) .


The Model

Related Literature

Leverage Cycles: Geanakoplos (2003), Fostel and Geanakoplos(2008), Brunnermeier and Pedersen (2009)

Amplification in Macroeconomy: Bernanke and Gertler (1989),Kiyotaki and Moore (1997)

Financial Intermediaries and the Macroeconomy: Gertler andKiyotaki (2012), Gertler, Kiyotaki, and Queralto (2011), He andKrishnamurthy (2012, 2013), Brunnermeier and Sannikov (2011,2012)


Solution

Solution Strategy

Equilibrium is characterized by two state variables, leverage θt andrelative intermediary net worth ωt

ωt =wt

wt + wht=

wt

pktAtKt

Represent state dynamics as

dωt

ωt= µωtdt + σωa,tdZat + σωξ,tdZξt

dθtθt

= µθtdt + σθa,tdZat + σθξ,tdZξt

Risk-based capital constraint implies

α−2θ−2t = σ2

ka,t + σ2kξ,t


Solution

Volatility Risk

−5 0 5−4

−2

0

2

4

Lagged Volatility Growth

Leve

rage

Gro

wth

−0.5 0 0.5 1−1

−0.5

0

0.5

1

Lagged VIX Growth

Leve

rage

Gro

wth

y = 0.00074 − 0.12xR2 = 0.013

y = 0.014 − 0.21xR2 = 0.053

Data Mean 5% Median 95%

β0 0.014 0.000 -0.003 0.000 0.003β1 -0.208 -0.105 -0.187 -0.104 -0.025R2 0.053 0.013 0.001 0.011 0.035

Simulation details


Solution

Intermediary Balance Sheets I

−4 −2 0 2 4−1

−0.5

0

0.5

Total Credit Growth

Inte

rmed

iate

d C

redi

t Gro

wth

−5 0 5

−5

0

5

Equity Growth

Leve

rage

Gro

wth

−1 −0.5 0 0.5 1−5

0

5

Debt Growth

Leve

rage

Gro

wth

−0.05 0 0.05 0.1 0.15 0.2−0.2

−0.1

0

0.1

0.2

Total Credit Growth

Inte

rmed

iate

d C

redi

t Gro

wth

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Equity GrowthLe

vera

ge G

row

th

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Debt Growth

Leve

rage

Gro

wth

y = 0.0086 + 0.56xR2 = 0.056

y = −0.071 + 0.76xR2 = 0.46


Solution

Intermediary Balance Sheets II

Table: Procyclicality of Intermediated Credit


β0 -0.071 -0.112 -0.203 -0.108 -0.040β1 0.756 0.434 0.190 0.433 0.680R2 0.460 0.048 0.009 0.045 0.101


Solution

Excess Returns

−5 0 5

−0.1

0

0.1

0.2

0.3

Leverage Growth

Equi

ty E

xces

s R

etur

n

−1 0 1 2−1

−0.5

0

0.5

1

Lagged Leverage GrowthFina

ncia

l Sec

tor E

quity

Ret

urn

y = 0.026 − 0.018xR2 = 0.052

y = 0.12 − 0.31xR2 = 0.17


β0 0.118 0.076 0.068 0.076 0.084β1 -0.310 -0.031 -0.038 -0.031 -0.024R2 0.167 0.100 0.064 0.100 0.143


Solution

Equilibrium Prices of Risk I

Shocks

dyt = σ−1a (d log Yt − Et [d log Yt ]) = dZat

d θt =(σ2θa,t + σ2

θξ,t

)− 12

(dθtθt− Et

[dθtθt

])=

σθa,t√σ2θa,t + σ2

θξ,t

dZat +σθξ,t√

σ2θa,t + σ2

θξ,t

dZξt .


Solution

Equilibrium Prices of Risk II

Price of risk of leverage

ηθt =

√√√√1 +(σka,t − σa)2

σ2kξ,t

(− 2θtωtpkt

β (1− ωt)σkξ,t + σξ

√1− ρ2

ξ,a

)

Price of risk of leverage is always positive (Adrian, Etula, and Muir(2013)), and depends on leverage growth in a nonmonotonic fashion(Adrian, Moench, and Shin (2010) find a negative relationship)


Solution

Equilibrium Prices of Risk III

−4 −2 0 2 4 6 8 10 12 14−4

−2

0

2

4

6

8

10

12

14

S1B1

S1B2 S1B3

S1B4

S1B5

S2B1

S2B2

S2B3 S2B4

S2B5

S3B1

S3B2 S3B3

S3B4

S3B5

S4B1 S4B2

S4B3

S4B4 S4B5

S5B1

S5B2 S5B3

S5B4

S5B5

Mom 1

Mom 2

Mom 3Mom 4Mom 5

Mom 6Mom 7

Mom 8Mom 9

Mom10

0−1yr

5−10y1−2yr

2−3yr3−4yr4−5yr

Predicted Expected Return

Rea

lized

Mea

n R

etur

n

Figure: Source: Adrian, Etula, and Muir (2013)


Solution

Equilibrium Prices of Risk IV

Price of risk of output

ηyt = σa + σξ

(ρξ,a −

σka,t − σaσkξ,t

√1− ρ2

ξ,a

)

Switches sign, consistent with insignificant estimates of price ofoutput risk

Usually becomes negative when exposure to liquidity shock is small


Distortions and Amplification

Term Structure of Systemic Risk

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Horizon

Dis

tres

s pr

obab

ility

α=2α=4



Volatility Paradox

0.05 0.1 0.15 0.2

0.2

0.3

0.4

0.5

Local volatility

Dis

tres

s pr

obab

ility

0.2 0.3 0.4

0.2

0.3

0.4

0.5

Price of leverage riskD

istr

ess

prob

abili

ty



Constant Leverage Benchmark

Constant expected output and consumption growth

But lower level of output and consumption growth

Constant investment and price of capital

Liquidity shocks have no impact on real activity



A Sample Path of the Economy

0 10 20 30 40 50 60 704

6

8

Year

Out

put

0 10 20 30 40 50 60 704

6

8

Cons

umpt

ion

0 10 20 30 40 50 60 70

0.20.40.60.8

Year

Wea

lth

0 10 20 30 40 50 60 70

5101520

Year

Leve

rage

0 10 20 30 40 50 60 70−0.2−0.1

00.1

Year

Equi

ty R

etur

n



Household Welfare

2 4 6 8 10α

Wel

fare

2 4 6 8 10

0.2

0.4

0.6

0.8

1

αD

istr

ess

prob

abili

ty

6 month1 year5 year



Conclusion

Dynamic, general equilibrium theory of financial intermediaries’leverage cycle with endogenous amplification and endogenoussystemic risk

Conceptual basis for policies towards financial stability

Systemic risk return trade-off: tighter intermediary capitalrequirements tend to shift the term structure of systemic downward,at the cost of increased prices of risk today

Model captures important stylized facts:1 Procyclical intermediary leverage2 Procyclicality of intermediated credit3 Financial sector equity return and intermediary leverage growth4 Exposure to intermediary leverage shocks as pricing factor



Tobias Adrian and Hyun Song Shin. Procyclical Leverage and Value-at-Risk.Federal Reserve Bank of New York Staff Reports No. 338, 2010.

Tobias Adrian, Emanuel Moench, and Hyun Song Shin. Financial Intermediation,Asset Prices, and Macroeconomic Dynamics. Federal Reserve Bank of NewYork Staff Reports No. 442, 2010.

Tobias Adrian, Erkko Etula, and Tyler Muir. Financial Intermediaries and theCross-Section of Asset Returns. Journal of Finance, 2013. Forthcoming.

Franklin Allen and Douglas Gale. Limited market participation and volatility ofasset prices. American Economic Review, 84:933–955, 1994.

Ben Bernanke and Mark Gertler. Agency Costs, Net Worth, and BusinessFluctuations. American Economic Review, 79(1):14–31, 1989.

Markus K. Brunnermeier and Lasse Heje Pedersen. Market Liquidity and FundingLiquidity. Review of Financial Studies, 22(6):2201–2238, 2009.

Markus K. Brunnermeier and Yuliy Sannikov. The I Theory of Money.Unpublished working paper, Princeton University, 2011.

Markus K. Brunnermeier and Yuliy Sannikov. A Macroeconomic Model with aFinancial Sector. Unpublished working paper, Princeton University, 2012.



Jon Danielsson, Hyun Song Shin, and Jean-Pierre Zigrand. Balance sheetcapacity and endogenous risk. Working Paper, 2011.

Douglas W. Diamond and Philip H. Dybvig. Bank runs, deposit insurance andliquidity. Journal of Political Economy, 93(1):401–419, 1983.

Ana Fostel and John Geanakoplos. Leverage Cycles and the Anxious Economy.American Economic Review, 98(4):1211–1244, 2008.

John Geanakoplos. Liquidity, Default, and Crashes: Endogenous Contracts inGeneral Equilibrium. In M. Dewatripont, L.P. Hansen, and S.J. Turnovsky,editors, Advances in Economics and Econometrics II, pages 107–205.Econometric Society, 2003.

Mark Gertler and Nobuhiro Kiyotaki. Banking, Liquidity, and Bank Runs in anInfinite Horizon Economy. Unpublished working papers, Princeton University,2012.

Mark Gertler, Nobuhiro Kiyotaki, and Albert Queralto. Financial Crises, BankRisk Exposure, and Government Financial Policy. Unpublished working papers,Princeton University, 2011.

Zhiguo He and Arvind Krishnamurthy. A Model of Capital and Crises. Review ofEconomic Studies, 79(2):735–777, 2012.



Zhiguo He and Arvind Krishnamurthy. Intermediary Asset Pricing. AmericanEconomic Review, 103(2):732–770, 2013.

Nobuhiro Kiyotaki and John Moore. Credit Cycles. Journal of Political Economy,105(2):211–248, 1997.


Empirical Evidence

Broker-Dealer Balance Sheets: Levels

5

7

9

11

13

15

17

19

21

23

25

0

0.5

1

1.5

2

2.5

3

3.5

2000 2002 2004 2006 2008 2010 2012

Dollars (trillions)

Source: Flow of Funds

Total Assets(left axis)

Total Liabilities(left axis)

Leverage(right axis)

Security Broker-Dealer: Assets, Liabilies, Equity, LeverageAssets to Equity

Back


Empirical Evidence

Broker-Dealer Balance Sheets: Annual Growth

-50

-40

-30

-20

-10

0

10

20

30

40

-50

-40

-30

-20

-10

0

10

20

30

40

2000 2002 2004 2006 2008 2010 2012

Percent

Source: Flow of Funds

Percent

Equity

Total Assets

Total Liabilities

Leverage

Security Broker-Dealer: Assets, Liabilies, Equity, Leverage

Back


Empirical Evidence

Broker-Dealer Balance Sheets: Adjustments

-1500

-1000

-500

0

500

1000

-1500 -1000 -500 0 500 1000

Security Broker Dealer Change of Assets as a Function of Change in Equity and in Liabilities (Annual)

Liabilities Changes

Equity Changes

Source: Federal Reserve Flow of Funds. Equity is at book value. -0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09

Equity Growth

Issuance Growth

Back


Empirical Evidence

Balance Sheet Adjustments

-400

-300

-200

-100

0

100

200

300

-400 -300 -200 -100 0 100 200 300

Cha

nge

in E

quity

& C

hang

es in

Deb

t (B

illio

ns)

Change in Assets (Billions)

Investment Banks

Equity

Debt

-600

-400

-200

0

200

400

600

800

-400 -200 0 200 400 600 800

Cha

nge

in E

quity

& C

hang

es in

Deb

t (B

illio

ns)

Change in Assets (Billions)

Commercial Banks (FDIC)

Equity

Debt

Back


Empirical Evidence

Broker-Dealers and Banks

Back


Empirical Evidence

Broker-Dealer VaR


Empirical Evidence

Broker-Dealer VaR

Back


Additional Results

Simulation Parameters

Parameter Value

a 0.0651σa 0.388ρ 0.06

ρh − σ2ξ/2 0.05

φ0 0.1φ1 20λk 0.03

ρξ,a 0σξ 0.0388α 2.5

Ref.: Brunnermeier and Sannikov (2012)

Monthly simulation frequency

10000 paths; 70 years Back


Additional Results

EquilibriumLemma 1

µRk,t = K0 (ωt , θt) +Ka (ωt , θt)σka,t + σξ

√1− ρ2

ξ,aσkξ,t

µRb,t = B0 (ωt , θt) + Ba (ωt , θt)σka,t + Bξ (ωt , θt)σkξ,t

µωt = O0 (ωt , θt) +Oa (ωt , θt)σka,t +Oξ (ωt , θt)σkξ,t

µθt = S0 (ωt , θt) + Sa (ωt , θt)σka,t −Oξ (ωt , θt)σkξ,t

rft = R0 (ωt , θt) +Ra (ωt , θt)σka,t

σba,t =2θtωtpkt + β (1− ωt)

βωt (θt − 1)σa −

2θtωtpkt + β (1− θtωt)

βωt (θt − 1)σka,t

σbξ,t = −2θtωtpkt + β (1− θtωt)

βωt (θt − 1)σkξ,t

σθa,t = −2θtωtpkt + β (1− ωt)

βωt(σka,t − σa)

σθξ,t = −2θtωtpkt + β (1− ωt)

βωtσkξ,t

σkξ,t = −

√θ−2t

α2− σ2

ka,t

σka,t =θ−2t

α2+ σ2

a

(1 +

1− ωt

ωt (2θtωtpkt + β (1− ωt))

).


Additional Results

Risk-free Rate

Household Euler equation

rft =

(ρh −

σ2ξ

2

)+

1

dtE[

dctct

]− 1

dtE

[〈dct〉2

c2t

+〈dct , dξt〉2

ct

]

Goods market clearing implies

dct = d (KtAt − itktAt)

= AtdKt + (Kt − itkt) dAt − Atktdit − At itdkt − kt 〈dit , dAt〉


Additional Results

Stress Tests

Inherent limitations to VaR include [. . . ] VaR does not estimatepotential losses over longer time horizons where moves may beextreme.


Additional Results

Stress TestsCould consider a forward-looking capital constraint

θ−1t ≥ ϑ

√Et

[∫ T

t

(σ2ka,s + σ2

kξ,s

)ds

].

Looks like a robust-control constraintRewrite intermediary optimization as

Vt (ϑ) = max{i ,β,k,αs}

Et

[∫ τD

te−ρ(s−t)wt (i , β, k) ds

]s.t.

θ−1s

αs≥√σ2ka,s + σ2

kξ,s

θ−1t = ϑ

√Et

[∫ T

t

θ−2s

α2s

ds

].

“Choose optimal capital plan”

Back


Extensions

Alternative Specification

Two financial sectors: banks and funds

Leveraged intermediaries have VaR constraint (as in the currentpaper) while funds have skin in the game constraint (as in He andKrishnamurthy (2012, 2013))

Bank managers, fund managers, and households have log utility

VaR constraint sometimes binds


Extensions

HouseholdsInvest in risk-free debt,non-bank financial sec-tor and bank financialsector

BanksCreate new capital; fi-nanced by debt issuanceto the households

FundsHold existing capi-tal; financed by profitsharing contracts withhouseholds

Producersproductivity At per unitof project; financed byfinancial sector

πbtwhtπftwht

CbtbhtπftwhtdR f

t

Atkt Atkht

Φ (it) kt

More


Extensions

Intermediaries

BanksCreate new capital;financed by debt is-suance to the house-holds

FundsHold existing capital;financed by profit shar-ing contracts withhouseholds

Two types of intermediaries: non-bank (“fund”) and bank

Unit mass of specialists manage funds; unit mass of bankers managebanks

Future work: interactions between different intermediary types


Extensions

Fund Sector

Modeled as in He and Krishnamurthy (2012, 2013)

Fund is formed each period t as a random match between a specialistand a household

Specialist contributes all of his wealth wft to the fund

Household contributes up to mwft to the fund

m: tightness of the specialists’ capital constraint

Specialists control the allocation of fund capital to holding capitalprojects and risk-free debt

Notice:

No new capital project creation

No risky debt

Back


Extensions

Specialists’ Optimization I

Specialists’ maximize expected consumption

max{cft ,θft}

E[∫ +∞

0e−ρt log cftdt

],

subject to the dynamic budget constraint

dwft

wft= θft (dRkt − rftdt) + rftdt − cft

wftdt


Extensions

Specialists’ Optimization II

Lemma 2

The specialists consume a constant fraction of their wealth

cft = ρwft ,

and allocate the fund’s capital as a mean-variance investor

θft =µRk,t − rftσ2ka,t + σ2

kξ,t

.

Back


Extensions

Banking Sector

Banks create new capital

dkt = (Φ(it)− λk) ktdt

Investment carries quadratic adjustment costs (Brunnermeier andSannikov (2012))

Φ (it) = φ0

(√1 + φ1it − 1

)Banks finance investment projects through inside equity and outsiderisky debt giving the budget constraint

pktAtkt = pbtAtbt + wt

Back


Extensions

Bankers’ Optimization I

The representative banker solves

maxθt ,it ,cbt

E[∫ +∞

0e−ρt log cbtdt

]subject to the dynamic budget constraint

dwt

wt= θt

(dRkt − rftdt +

(Φ (it)−

itpkt

)dt

)− (θ − 1) (dRbt − rftdt) + rftdt − cbt

wtdt,

and the risk-based capital constraint

θt ≤1

α√σ2ka,t + σ2

kξ,t

.


Extensions

Bankers’ Optimization IILemma 3

The representative banker optimally consumes at rate

cbt = ρwt

and invests in new projects at rate

it =1

φ1

(φ2

0φ21

4p2kt − 1

).

While the capital constraint in not binding, the banking system leverage is

θt =σ2ba,t − σka,tσba,t + σ2

bξ,t − σkξ,tσbξ,t − (µRb,t − rft)((σba,t − σka,t)2 + (σbξ,t − σkξ,t)2

)+

(µRk,t + Φ (it)− it

pkt− rft

)(

(σba,t − σka,t)2 + (σbξ,t − σkξ,t)2) .

Back


Extensions

Households

Household preferences are:

E[∫ +∞


]Liquidity preference shocks (as in Allen and Gale (1994) and Diamondand Dybvig (1983)) are exp (−ξt)

dξt = σξdZξt

Households allocate wealth between risky bank debt and contributionsto funds

Back


Extensions

Households’ Optimization IThe representative household solves

maxπkt ,πbt

ct

E[∫ +∞

0e−ξt−ρht log ctdt

],

subject to the dynamic budget constraint

dwht

wht= πktθft (dRkt − rftdt) + πbt (dRbt − rftdt) + rftdt − ct

whtdt,

the skin-in-the-game constraint

πktwht ≤ mwft ,

and no shorting constraints

πkt ≥ 0

bht ≥ 0.


Extensions

Households’ Optimization II

Lemma 4

The households’ optimal consumption choice satisfies

ct =

(ρh −

σ2ξ

2

)wht .

While the households are unconstrained in their wealth allocation, the households’optimal portfolio choice is given by[

πktπbt

]=

([θftσka,t θftσkξ,tσba,t σbξ,t

] [θftσka,t σba,tθftσkξ,t σbξ,t

])−1 [θft (µRk,t − rft)µRb,t − rft

]−[θftσka,t σba,tθftσkξ,t σbξ,t

]−1 [0σξ

].

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Extensions

EquilibriumAn equilibrium in the economy is a set of price processes {pkt , pbt , rft}t≥0, aset of household decisions {πkt , bht , ct}t≥0, a set of specialist decisions{kft , cft}t≥0, and a set of intermediary decisions {kt , it , bt , cbt}t≥0 such thatthe following apply:

1 Taking the price processes, the specialist decisions and the intermediarydecisions as given, the household’s choices solve the household’soptimization problem, subject to the household budget constraint, the noshorting constraints and the skin-in-the-game constraint for the funds.

2 Taking the price processes, the specialist decisions and the householddecisions as given, the intermediary’s choices solve the intermediary’soptimization problem, subject to the intermediary budget constraint, andthe regulatory constraint.

3 Taking the price processes, the household decisions and the intermediarydecisions as given, the specialist’s choices solve the specialist’soptimization problem, subject to the specialist budget constraint.

4 The capital market clears at all dates

kt + kft = Kt .

5 The risky bond market clears

bt = bht .

6 The risk-free debt market clears

wt + wft + wht = pktAtKt .

7 The goods market clears

ct + cbt + cft + Atkt it = KtAt .

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Extensions

Additional Research

Tradeoff between capital and liquidity regulation

Stress tests

Intermediation chains

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