Capital Requirements,
Risk Choice, and Liquidity Provision
in a Business Cycle Model
Juliane Begenau
Harvard Business School
July 11, 2015
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Motivation
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How to regulate banks?
Capital requirement: min equity/ risky assets
Literature: higher capital requirements
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limit banks' risk-taking
reduce lending and liquidity provision
This paper:
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develops general equilibrium model to quantify trade-os
derives optimal capital requirement
shows importance of eect of capital requirements on
endogenous prices
mechanism is based on household's demand for safe assets
Quantitative GE model
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Main features
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Quantied with data from the FDIC and NIPA
Consistent with business cycle facts
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banks are essential for a part of production
households' preference for safe & liquid bank debt
banks receive government subsidy
macroeconomic variables
banking sector aggregates
Findings
Optimal capital requirement
I 14% of risky assets
Increasing the capital requirement to 14%
I reduces the supply of liquid bank debt by 9%
I reduces volatility of income-risky-asset ratio by 6%
I increases loans by 2%
Main proposition:
Higher capital requirements leading to a reduction in the
supply of bank debt can in fact result in more lending
Core assumption:
Investors value safe and liquid assets in the form of bank debt
more the scarcer they are
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Corporate bond spread and bank debt
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1999
3.5
3
2005
2.5
2000
2006
Spread
2
2008
2004
2001
1.5
1
2007
0.5
2002
0
2003
2011 2013
2010
-0.5
-1
0.7
2009
0.75
0.8
0.85
0.9
2012
0.95
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Bank Debt-to-GDP
The gure plots the spread between the Aaa corporate bond rate and the implied
interest rate on bank debt against the bank debt-to-GDP ratio for 1999-2013.
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Related literature
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Banks' role for production
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Households' demand for liquid bank debt
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Kareken & Wallace 1978; Keeley & Furlong 1990; Gennotte & Pyle
1991; Schneider & Tornell 2004; Farhi & Tirole 2011; Admati et al 2013;
Allen, Carletti, Goldstein & Leonello 2014
Kelly, Lustig & Nieuwerburgh 2012; Gandhi & Lustig 2012; Marques,
Correa & Sapriza 2013; Duchin & Sosyura 2014
Quantication of capital requirements
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Diamond & Dybvig 1983; Gorton & Winton 1995; Diamond & Rajan
2000; Kashyap, Rajan & Stein 2002; Van Den Heuvel 2008; Williamson
2012; Dang, Gorton, Holmström & Ordonez 2013; DeAngelo & Stulz
2013; Allen, Carletti & Marquez 2014
Bansal & Coleman 1996; Krishnamurthy & Vissing-Jørgensen 2013;
Greenwood, Hanson & Stein 2014
Risk-shifting incentives
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Diamond 1984; Sharpe 1990; Boyd & Prescott 1986; Holmström &
Tirole 1997; Winton 2000; Van Den Heuvel 2002; Bolton & Freixas 2006
James 1978; Hoshi, Kashyap & Scharfstein 1991; Gertler & Gilchrist
1992; Dell'Ariccia, Detragiache & Rajan 2008; Iacoviello & Minetti 2008
Corbae & D'Erasmo 2011, 2012; Christiano & Ikeda 2013; Clerc et al
2014; Nguyen 2014; Martinez-Miera & Suarez 2014
Outline
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Mechanism
Model
Trade-o Description
Taking the model to the data
Welfare
Two-sector business cycle model
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Technology f : non-bank dependent
ytf
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=
Ztf
f
kt−1
α Ntf
Technology h : banking sector funded activities as part of
the overall output (collateralized lending)
h
yth = Zth kt−1
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1−α
Banks run h production directly
Choose investment and risk
v
Banks' risk-return trade-o
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Captures menu of investment choices v
h
Zth productivity level in yth = Zth kt−1
h
h
+ φ1 − φ2 σth σth +σth ht+1
σth = ρh log Zth σt−1
log Zt+1
Productivity maximizing σh
E[log(Zh)]
Risk choice of banks
σh
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σth determines mean and exposure to aggregate shock
Balance sheet and prots
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Balance sheet at the beginning of t
h
kt−1
+ bt−1 = et−1 + st−1
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Prots
πt =
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10
h
rt st−1
yth − δ h kt−1
+ rtB bt−1 −
| {z }
| {z }
|
{z
}
h
interest
exp.
interest
inc.
income from k
Capital requirement
et ≥ ξkth
Subsidy to banks
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Capture eects of guarantees on banks' liabilities
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Assume: government cannot commit to not bailout banks
Guarantees subsidize leverage and risk-taking
Unlimited liability and explicit subsidy

T Rt =





et−1 + πt
h


exp −ω1
+ ω2
σt

h
|{z}
|{z}
k
t−1


|
{z
}
risk-taking
size
h
ω3 kt−1
capitalization
ω1 , ω2 , ω3 > 0
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Implies complementarity between leverage and risk-taking
Banks' problem
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Maximize present value of dividend payout dt
subject to
dt = πt + T Rt − ∆et −
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2
κ
dt − d¯ − adjkh
2
Dividend smoothing motive captured with dividend
adjustment costs
Households
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Utility of households
U (ct , st ) = log ct + θ
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(st /ct ) 1−η
1−η
where η > 1.
Net worth of households
nt = (dt + pt ) Θt−1 + (1 + rt ) st−1
f
+ rtf + 1 − δ f kt−1
−
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Taxes
Maximize discounted expected utility subject to the
budget constraint
ct + st + ktf + pt Θt = nt + wtf Ntf
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Recursive competitive equilibrium
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State vars: capital stocks, productivity levels, households'
net worth, equity after prots
Exog. shocks to log Z h & log Z k
Given prices, households, rms, and banks optimize
Policies satisfy market clearing for
bonds, bank debt, capital stocks, labor, bank shares, and
consumption.
Gov budget constraint holds: T R + BrB = Taxes
Outline
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Mechanism
Model
Trade-o Description
Taking the model to the data
Welfare
Decisions relevant for trade-o
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Risk choice
Leverage choice
Lending choice
Risk choice
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Subsidy: source of excessive risk-taking
Productivity maximizing σh
E[log(Zh)]
Risk choice of banks
σh
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Higher ξ : reduce σ h and increase E log Z h
Leverage choice
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Households' FOC wrt liquid bank debt:
re − r
= Us (c, s) /Uc (c, s) > 0
1 + re
⇒ re =
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1
β
−1>r
Discount on bank debt → binding capital requirement:
e = ξk h
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What happens to r when s falls?
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debt is preferred
gov. subsidy adds to this
larger the scarcer s
reduction in s leads to reduction in r
re − r
Lending choice
yh
TR
1 + v h − δh + g
= ξ (1 + re ) + (1 − ξ) (1 + r)
|
{z
}
k
kh
|
{z
}
h
Funding
costs
of
k
Benet of kh
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Higher capital requirements
→ With rates xed: banks want to reduce scale
→ Reduction in debt reduces r through GE
→ Overall funding costs fall if ξ not too large
Fall in funding costs→ banks want to increase kh
Strength depends on key parameters: η and v
Welfare eects of capital requirements
Increase in the capital requirement
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reduces risk-taking
reduces liquidity through a reduction in bank debt
increases assets through lower funding costs
Optimal capital requirement
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trades-o reduction in liquidity against reduction in
risk-taking and an increase in consumption
Outline
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Mechanism
Model
Trade-o Description
Taking the model to the data
Welfare
Mapping from model to data
Model
y h : bank output
k h : bank capital
y f : rm output
k f : rm capital
c: consumption
s: bank debt
π : prots
r: rate on bank debt
σ h : risk choice
e: equity
NIPA and FDIC balance sheet & inc stat.
income− sec int. income
loans + trading assets
NIPA total GDP − bank output
NIPA K − kh
NIPA consumption
bank liabilities
net income + non interest expense
interest expenses / bank liabilities
std of BC component log (yh /kh )
tier 1 equity
Period: 1999q1:2013q4
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Quantication of Parameters
1. natural data counterpart
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e.g. persistences of productivity shocks
2. using steady state conditions of the model and targeting
moments one for one
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e.g. time preference rate of households
3. jointly
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e.g. parameters governing adjustment costs, liquidity
preference, and size of banking sector
Key parameters
Par. Governs
η
hh dislike for changes
in bank debt/ consum.
v
strength of ↓ r to ↑ ξ
conversion of risky
assets into bank output
strength of ↑ kh to ↑ ξ
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Target Moment
Value
3.15
vol(bank liab/cons)
0.30
inc/risky assets
Business cycle correlations
Investment
Consumption
Bank income
Bank risky assets
Bank liabilities
Interest rate on bank liabilities
Bank income-risky assets
Bank prot
Bank investment
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GDP
Data Model
0.97 0.98
0.94 0.87
0.66 0.64
0.37 0.32
0.31 0.32
0.68 0.64
0.61 0.62
0.34 0.64
0.46 0.55
Outline
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Mechanism
Model
Trade-o Description
Taking the model to the data
Welfare
Welfare
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Compute the optimal capital requirement
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use local approximation methods
simulate economy under benchmark tier-1-equity/risky
asset and under new requirements
compute value function of households
Welfare as a function of capital requirements
Optimal ξ = 14%
Welfare in % consumption equivalent units
0
−0.05
−0.1
−0.15
−0.2
−0.25
0.05
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0.1
0.15
ξ
0.2
0.25
Reduction in funding costs
TR
yh
yh
1 + v h − δ h + h 1 + ω1 (1 − v) h = ξ (1 + re ) + (1 − ξ) (1 + r)
{z
}
|
k
k
k
|
{z
}
h
Funding
costs
of
k
Benet of kh
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Benchmark: kh costs 2.57% with r = 1.5% and re = 10%
Increase requirement to 14%
→ With xed prices, cost would increase by 10%
→ Reduction in debt reduces r to 0.88%
→ Overall funding costs fall by 16% to 2.15%
Fall in funding costs → banks increase kh by 2%
Raising capital requirement during crisis or boom?
Welfare in % consumption equivalent units
0.1
0
−0.05
−0.1
−0.15
−0.2
−0.25
0
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Start in Recession
Start in Boom
0.05
0.05
0.1
0.15
ξ
0.2
0.25
0.3
0.35
Conclusion
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Mechanism:
Higher capital requirement can lead to more lending when
bank debt is valued for being safe and liquid
Optimal capital requirement about ξ = 14%
Caveat: potential for safe asset substitution from shadow
banks may mitigate channel
Thank you
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Subsidy to banks
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Capture eects of guarantees on banks' liabilities
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Assume: government cannot commit to not bailout banks
Guarantees subsidize leverage and risk-taking
Unlimited liability and explicit subsidy


T Rt =




et−1 + πt
h


exp −ω1
+
ω
σ
2
t

h
|{z}
|{z}
k
t−1


|
{z
}
risk-taking
size
h
ω3 kt−1
capitalization
ω1 , ω2 , ω3 > 0
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Back
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Implies complementarity between leverage and risk-taking
Key parameters (ctd)
Par. Governs
Target Moment
φ2
risk-return trade-o
strength of
Value
0.89
vol(inc/risky-assets)
↓ σ h & ↑ k h to ↑ ξ
ω2
risk-taking due to
subsidy
strength of
↓ σ h to ↑ ξ
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2.92
Var log yth /kt | log πt−1
h