A Model of Capital and Crises
Zhiguo He
Booth School of Business, University of Chicago
Arvind Krishnamurthy
Northwestern University and NBER
AFA, 2011
Introduction
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Intermediary capital can a¤ect asset prices.
We study the role of distressed intermediary capital in the crises of
market with complex asset classes (e.g. MBS).
Introduction
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Intermediary capital can a¤ect asset prices.
We study the role of distressed intermediary capital in the crises of
market with complex asset classes (e.g. MBS).
A General Equilibrium (GE) model where intermediaries, rather than
households, are marginal.
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Frictions are endogenously derived based on optimal contracting
considerations.
Mechanism: Intermediation capital a¤ects participation/risk-sharing.
In normal times households participate through intermediation;
When intermediaries su¤er losses,
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Distressed intermediary sector averse to hold risky positions, risk
premium goes up.
Households “‡y to quality,” drive down interest rate.
Model Structure (1)
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Unit supply of risky asset with dividend
riskless asset in zero-net supply.
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hR
∞
0
e
ρh t
i
ln cth dt .
Limited participation in risky asset market. They invest in
intermediaries.
Specialists E
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= gdt + σdZt , and
Risky asset price Pt and interest rate rt are determined in GE.
Households E
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dD t
Dt
R∞
0
e
ρt
ln ct dt , ρ < ρh . They run intermediaries.
Only intermediaries/specialists can invest in the risky asset. They are
marginal investors.
Derive Intermediation Constraint from moral hazard primitives.
Model Structure (2)
The economy.
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Intermediation: 1) Short-term contracting between agents; 2)
Equilibrium in competitive intermediation market;
Asset pricing: 3) Optimal consumption/portfolio decisions; 4) GE.
The Heart of the Model: (equity) Capital Constraint
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Say household with wealth Wth , and specialist with wealth Wt .
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Given specialist’s contribution Wt in the intermediary, household
contributes Tth as equity investment (for risk sharing).
Capital Constraint: Tth is capped at mWt so risk sharing is capped
at 1 : m.
Intermediation capacity mWt is increasing in the specialist’s
contribution Wt , as re‡ection of agency friction.
The Heart of the Model: (equity) Capital Constraint
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Say household with wealth Wth , and specialist with wealth Wt .
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Given specialist’s contribution Wt in the intermediary, household
contributes Tth as equity investment (for risk sharing).
Capital Constraint: Tth is capped at mWt so risk sharing is capped
at 1 : m.
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Intermediation capacity mWt is increasing in the specialist’s
contribution Wt , as re‡ection of agency friction.
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How to interpret m?
1. Intermediary capital requirement: outside/inside contribution ratio;
(Holmstrom-Tirole, QJE)
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O¢ cers/Directors inside holdings in …nancial industry around 18%.
2. Incentive contract— the performance share of hedge fund managers.
Think of “2 and 20.”
3. Mutual funds’‡ow-performance sensitivity. Specialist’s Wt tracks his
past gains and losses (Shleifer-Vishny, JF)
Intermediation Constraint: An Example (1)
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Say m = 1, Wth = 80. Comparing Wth to mWt .
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Unconstrained Region: Wt = 100. Then Tth = Wth = 80;
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Zero net debt. Risky asset price Pt = Wt + Wth = 180.
Fund’s total equity 180. Intermediary holds risky asset without
leverage, …rst-best risk sharing.
Intermediation Constraint: An Example (2)
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Constrained Region: Wt = 50. Then Tth = mWt = 50;
Intermediation Constraint: An Example (2)
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Constrained Region: Wt = 50. Then Tth = mWt = 50;
Intermediary’s total equity is 50 + 50 = 100. But Pt = 130.
In equilibrium, the intermediary borrows 30 from the debt market;
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It is supplied by households Wth
Specialist and household have equal shares in the intermediary;
Specialist’s leveraged position in risky asset:
α=
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Tth = 30.
130/2
specialist’s portion of asset
=
= 130%.
specialist’s equity
50
Risk premium has to adjust to make this high leverage optimal.
Risk Premium and Interest Rate
Interest Rate
Risk Premium
0.05
0.7
m=4
m=6
0.6
0
0.5
c
w (m=6)=9.07
0.4
c
w (m=6)=9.07
0.3
c
w (m=4)=13.02
-0.05
c
w (m=4)=13.02
m=4
m=6
0.2
-0.1
0.1
0
0
5
10
15
Scaled Specialist's Wealth w
20
25
0
5
10
15
Scaled Specialist's Wealth w
20
25
Intermediation Stage Game
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Short-term contracts only. At time t, contract from t to t + dt.
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Household with wealth Wth , and specialist with wealth Wt .
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Household contributes Tth , specialist Tt . TtI = Tth + Tt .
Intermediation Stage Game
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Short-term contracts only. At time t, contract from t to t + dt.
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Household with wealth Wth , and specialist with wealth Wt .
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Household contributes Tth , specialist Tt . TtI = Tth + Tt .
Specialist in charge of intermediary. Moral Hazard:
1. Unobserved due diligence action st = f0, 1g.
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Shirking (st = 1) reduce return by X t but brings private bene…t B t .
2. Unobserved portfolio choice EtI (dollar exposure to risky asset);
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Undoing activity. Not crucial.
Fund’s return EtI (dRt rt dt ) + TtI rt dt st Xt dt, private bene…t
st Bt dt. Focus on implementing working.
Intermediation Contract
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A¢ ne contracts for sharing returns.
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βt : specialist’s share; K̂t dt: transfer to specialist.
βt TtI
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Set Kt
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Dynamic budget constraint
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< dWt = rt Wt dt ct dt + βt EtI (dRt
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:
dWth
=
Tt rt + K̂t .
rt Wth dt
cth dt
+ (1
βt )EtI
rt dt ) + Kt dt,
(dRt
rt dt )
Kt dt.
Reduce contract to ( βt , Kt ). Sharing rule and fee.
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Specialist chooses Et = βt EtI . Household buys risk exposure
Eth = (1 βt )EtI from intermediary.
In competitive intermediation market, the fee will take some simple
linear form.
IC Constraint and Maximum Household’s Exposure
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IC constraint: specialist bears at least a certain fraction of risk.
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Incentive provision. Skin in the game.
Bt
No shirking: βt Xt Bt
0 ) βt
Xt
A lower bound on βt .
1
1 +m .
Specialist always chooses βt EtI = Et independent of βt .
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In the paper we show Et is independent of K .
IC Constraint and Maximum Household’s Exposure
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IC constraint: specialist bears at least a certain fraction of risk.
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Incentive provision. Skin in the game.
Bt
No shirking: βt Xt Bt
0 ) βt
Xt
A lower bound on βt .
Specialist always chooses βt EtI = Et independent of βt .
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t
In the paper we show Et is independent of K .
fund’s total risk exposure. S: Et = βt EtI , H: Eth = (1
I
βt ) EtI .
Household exposure from the contract, or exposure supply:
Eth = (1
I
1
1 +m .
As βt
1
1 +m ,
βt )EtI =
1
βt
βt
Et .
households maximum exposure Eth
m Et .
Key Intuition and Equity Implementation
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The households exposure is capped due to agency frictions
Eth mEt .
It caps a risk-sharing rule between households and specialists.
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Incentive provision implies that specialists have to bear su¢ cient risk.
In bad times this friction kicks in.
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Even if specialists wealth is low, they still have to bear
disproportionally large risk.
Key Intuition and Equity Implementation
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The households exposure is capped due to agency frictions
Eth mEt .
It caps a risk-sharing rule between households and specialists.
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Incentive provision implies that specialists have to bear su¢ cient risk.
In bad times this friction kicks in.
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Even if specialists wealth is low, they still have to bear
disproportionally large risk.
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Equity implementation: Households (outsiders) cannot hold more
m (equity) shares.
than 1 +
m
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Equity capital constraint: Given specialist’s equity Wt , households
can make at most mWt equity contributions.
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Recall contract ( βt , Kt ) . We have derived equilibrium βt . What
determines fee Kt ?
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Households pay competitive fees in the intermediation market.
Competitive Intermediation Market
At time t, specialists make o¤ers ( βt , Kt ) to households who can
accept/reject o¤ers. The intermediation market reaches equilibrium if: 1)
βt is incentive compatible; 2) no pro…table deviation coalitions.
Lemma: In equilibrium, households face a per-unit-exposure price of
kt
0: to purchase Eth , he has to pay Kt = kt Eth .
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Idea: equivalence between core and Walrasian equilibrium.
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Households and specialists form coalitions to chop o¤ the exposure
linearly.
Now we start studying agents’consumption/portfolio problems.
Households’ Consumption/Portfolio Rules
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Log investors. Simple consumption rule; myopic mean-variance
portfolio choice.
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Risky asset excess return dRt rt dt = π R ,t dt + σR ,t dZt .
hR
i
h
∞
Household max E 0 e ρ t ln cth dt subject to
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fc t ,Et g
dWth = Wth rt dt
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cth dt + Eth (dRt
rt dt )
kt Eth dt.
Standard problem; households achieve exposure Eth by paying
per-unit-cost of kt .
Optimal consumption cth = ρh Wth , optimal exposure
π
k
Eth = Rσ,t2 t Wth .
R ,t
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Optimal risk exposure is decreasing in exposure price kt .
Specialists’ Consumption/Portfolio Rules
1 βt
βt Et
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The specialist supplies an exposure
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kt , he gets intermediation fees Kt dt = kt
R∞
The specialist is solving: max E 0 e
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1 βt
βt Et
ρt
fc t ,Et , βt g
dWt = Et (dRt
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. Given exposure price
rt dt ) +
max
βt 2[ 1 +1m ,1 ]
1
βt
βt
dt.
ln ct dt subject to
kt Et dt + Wt rt dt
ct dt.
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h
βt = 1 +1m if kt > 0, otherwise βt 2 1 +1m , 1 if kt = 0. Exposure
supply schedule.
Et is the exposure expected by households, and must coincide with
the specialist’s actual optimal choice in REE.
Specialists’ Consumption/Portfolio Rules
1 βt
βt Et
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The specialist supplies an exposure
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kt , he gets intermediation fees Kt dt = kt
R∞
The specialist is solving: max E 0 e
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1 βt
βt Et
ρt
fc t ,Et , βt g
dWt = Et (dRt
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. Given exposure price
rt dt ) +
max
βt 2[ 1 +1m ,1 ]
1
βt
βt
dt.
ln ct dt subject to
kt Et dt + Wt rt dt
ct dt.
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h
βt = 1 +1m if kt > 0, otherwise βt 2 1 +1m , 1 if kt = 0. Exposure
supply schedule.
Et is the exposure expected by households, and must coincide with
the specialist’s actual optimal choice in REE.
π
Solution: ct = ρWt and Et = σ2R ,t Wt , and specialists receive fee
R ,t
of Kt = qt Wt where qt =
1 βt
βt
kt
π R ,t
.
σ2R ,t
Unconstrained vs. Constrained Regions (1)
Unconstrained vs. Constrained Regions (1)
Equilibrium Asset Prices: Solution
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We derive everything in closed form.
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State variables (Dt , Wt ). Scales with Dt .
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Uni-dimensional state variable wt
distribution.
Wt /Dt captures wealth
Equilibrium Asset Prices: Solution
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We derive everything in closed form.
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State variables (Dt , Wt ). Scales with Dt .
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Uni-dimensional state variable wt
distribution.
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Consumption rules ct = ρWth , cth = ρh Wth .
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Zero net debt Wt + Wth = Pt , goods clearing ct + cth = Dt . So
Pt
1
= h+ 1
Dt
ρ
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Wt /Dt captures wealth
ρ
ρh
Specialist’s risky (percentage) position αt =
constrained region.
wt .
Pt
(1 +m )W t
> 1 in
Asset Pricing (1)
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The economy is in constrained region whenever
wt = Wt /Dt < w c
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In unconstrained region, wt increases deterministically toward w c .
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1
.
mρh + ρ
Perfect risk sharing rule. Relative patience level ρ < ρh matters.
In constrained region, specialists take a higher leverage than
households. Therefore wt becomes stochastic and drops when
fundamental falls.
When (scaled) intermediary capital wt falls in constrained region,
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Risk premium rises;
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Interest rate falls;
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Volatility rises;
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Correlation endogenously rises.
Asset Pricing (2)
Uncon. Region
Con. Region
Et
Wt
1
1 +m Pt
αt
1
σR ,t
σ
π R ,t
σ2
kt
0
rt
1 +(ρh ρ)w t
(1 +m ) ρh w t
σ
1 +(ρh
σ2
w t (m ρh + ρ )
(1 +m ) ρh
m ρh + ρ
(1 +m ) ρh
m ρh + ρ
>σ
1
1 +(ρh ρ)w t
1 ( ρ +m ρh )w t
(1 +m ) ρh σ 2
(1 ρw t )(1 +(ρh ρ)w t ) w t (m ρh +ρ)2
ρh + g
+ρ ρ
ρ )w t
>1
σ2
ρh
wt
σ2
h
ρh + g + ρ ρ
ρ (1 +m )
1
wt
ρ
> σ2
>0
ρ h wt
m 2 ρh + (m ρh )
(1 ρw t ) ( ρ +m ρh )
2
2
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Risk Premium and Interest Rate
Interest Rate
Risk Premium
0.05
0.7
m=4
m=6
0.6
0
0.5
c
w (m=6)=9.07
0.4
c
w (m=6)=9.07
0.3
c
w (m=4)=13.02
-0.05
c
w (m=4)=13.02
m=4
m=6
0.2
-0.1
0.1
0
0
5
10
15
Scaled Specialist's Wealth w
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20
25
0
5
10
15
20
25
Scaled Specialist's Wealth w
Asymmetry. Crisis like.
When constraint binds wt < w c , specialist bears disproportionally
large risk, causing more volatile pricing kernel.
Flight to quality. 1) Specialists precautionary savings. 2) Household
‡y to debt market.
Comovement
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Consider an in…nitesimal asset with
dDt
d D̂t
=
+ σ̂d Ẑt .
Dt
D̂t
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ˆ t is:
The correlation between dRt and dR
ˆ t) = q
corr (dRt , dR
1
1 + (σ̂/σR ,t )2
.
Unconstrained region, since σR is constant, the correlation is
constant.
Constrained region, rising correlation.
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Market return volatility σR ,t rises, magnifying the common
component of returns.
Concluding Remarks (1)
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Canonical intermediation friction meets canonical GE asset pricing
models.
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Calibratable, easy to quantify e¤ects.
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We have another paper where specialists have general CRRA power
utility, with capital constraint as given.
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Calibrate the model to the MBS market;
Add in labor income, debt households (create leverage in
unconstrained region), and other necessary twists...
Study the crisis dynamics (especially recovery), government liquidity
injection policies, etc.
Concluding Remarks (2): Calibration result
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.12
25%
0.1
20%
0.08
15%
0.06
10%
0.04
5%
0.02
0
0
0.05
0.1
c
w /P=0.91
Transit from 12%
10%
7.5%
5%
4%
0.15
0.2
0.25
w/ P
0.3
0.35
0.4
0.45
0
0.5
Crisis Recovery
W/o Capital Infusion W. Capital Infusion (48bn)
0.17
0
0.66
0.31
2.72
2.20
5.88
5.06