Download A Macroeconomic Framework for Quantifying Systemic Risk Zhiguo He Arvind Krishnamurthy

A Macroeconomic Framework for
Quantifying Systemic Risk
Zhiguo He
University of Chicago & NBER
Arvind Krishnamurthy
Northwestern University & NBER
June 2012
Systemic Risk
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Systemic risk: risk (probability) of a state where
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…nancial intermediation is disrupted
small fundamental shocks to …nancial intermediaries can have
quantitatively large e¤ects on macro economy
Goal: Write down a non-linear macro model to assess systemic risk
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much of the time the link between …nancial intermediation and
macro economy is small
but in (crisis) states the e¤ects are greatly ampli…ed
Systemic Risk
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Systemic risk: risk (probability) of a state where
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…nancial intermediation is disrupted
small fundamental shocks to …nancial intermediaries can have
quantitatively large e¤ects on macro economy
Goal: Write down a non-linear macro model to assess systemic risk
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much of the time the link between …nancial intermediation and
macro economy is small
but in (crisis) states the e¤ects are greatly ampli…ed
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How well does the model match asymmetry (i.e. occasional e¤ects
of …nancial intermediation) in the data?
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How well can an intermediary shock channel explain patterns in
2007-2009?
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How likely is the economy, say unconditionally, to enter a systemic
risk episode?
Innovation Relative to Much of Literature
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We study a model with occasionally binding …nancial constraint
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Typical models (e.g., Kiyotaki-Moore (1997),...) linearize around
steady state where constraint binds.
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Cannot talk about 1) likelihood that intermediation is disrupted (its
always disrupted...) and 2) how severely it is disrupted
Our model solution has stochastic steady state, with fully solved
equilibrium prices and policies
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Main drawback: need to reduce state variables
Have to leave out some common DSGE elements
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Similar methodology to Mendoza (2010) and Brunnermeier-Sannikov
(2011)
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Model elements adopted from He-Krishnamurthy (2012), with real
investment and housing
Preview of model result
Sharpe ratio
interest rate
6
0.1
5
0.05
0
4
-0.05
3
-0.1
2
-0.15
1
0
0
-0.2
5
10
scaled intermediary reputation e
15
20
-0.25
0
5
investment I/K
10
scaled intermediary reputation e
15
20
steady state distribution
0.105
0.035
0.104
0.03
0.103
0.025
0.102
0.101
0.02
0.1
0.015
0.099
0.01
0.098
0.005
0.097
0.096
0
5
10
scaled intermediary reputation e
15
20
0
0
5
10
15
20
25
scaled intermediary reputation e
30
35
40
Preview of model result
Sharpe ratio
interest rate
6
0.1
5
0.05
0
4
-0.05
3
-0.1
2
-0.15
1
0
0
-0.2
5
10
scaled intermediary reputation e
15
20
-0.25
0
5
investment I/K
10
scaled intermediary reputation e
15
20
steady state distribution
0.105
0.035
0.104
0.03
0.103
0.025
0.102
0.101
0.02
0.1
0.015
0.099
0.01
0.098
0.005
0.097
0.096
0
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5
10
scaled intermediary reputation e
15
20
0
0
5
10
15
20
25
scaled intermediary reputation e
30
35
Crisis: ecrisis = 0.65, binding capital constraint
Distress: edistress = 4 so that Pr (e edistress ) = 33% as in data
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Strategy
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Crises are rare. How do we quantify model?
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Even if economy is currently not in a crisis state, the anticipation of
a crisis a¤ects decisions.
1. We match data on “distress" (33% of data) and “non-distress"
periods (67% of data).
2. We extrapolate to a crisis and ask how well the model can match
patterns from 2007-2009.
3. We compute conditional probabilities of triggering a crisis (measuring
“systemic risk" probabilities).
Evidence of Non-Linearity
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Excess bond premium (EBP): the risk premium part of credit
spread (removing default part), Gilchrist and Zakrajsek (2010).
Correlates with measures of intermediary health.
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Use EBP to classify distress periods (33%) and non-distress periods
(the rest)
Distress Periods
1973Q1 - 1975Q3
1982Q2 - 1982Q4
1985Q4 - 1987Q3
1988Q4 - 1990Q1
1992Q4 - 1993Q2
2001Q2 - 2003Q1
2007Q3 - 2009Q3
NBER Recessions
11/73 - 3/75
7/81 - 11/82
7/90 - 3/91
3/01 - 11/01
12/07 - 6/09
State-Dependent Covariances (1)
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Equity = Total market value of equity of …nance, insurance and real
estate sectors. (works as well if only include banks + broker/dealers)
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All variables are growth, except Sharpe ratio constructed from EBP
Distress
Non Distress
Cov
Corr
Cov
Corr
Equity , Investment
1.31%
51.48
0.07
5.79
Equity , Consumption
0.25%
45.85
0.03
14.74
Equity , Sharpe
-6.81%
-35.96
-0.14
-0.06
Equity , Landprice
4.06%
60.65
0.12
0.07
State-Dependent Covariances (2)
All variables are growth, except Sharpe ratio constructed from EBP
Distress
Non Distress
NBER+2
Excl-Crisis
NBER+2
Excl-Crisis
Equity , Investment
0.84%
0.37
-0.06
0.03
Equity , Consumption
0.13%
0.04
0.01
0.03
Equity , Sharpe
-7.57%
-2.12
-0.78
-0.19
Equity , Landprice
4.39%
-0.63
-0.31
-0.01
Note: Similar numbers if only use NBER dates, but distress sample is
only 20% of observations.
VAR Evidence of Non-Linearity (3)
VAR order: [intermediary equity, aggregate stock market, EBP,
investment]. Coe¢ cients depend on distress/non-distress state. Quarterly
growth rates.
PANEL A: DISTRESS PERIODS
Market to Equity
Equity to Equity
25
EB (credit risk premium) to Equity
0
20
Investment to Equity
5
-0.2
20
15
4
-0.4
15
3
-0.6
10
2
-0.8
10
5
5
1
-1
0
-1.2
0
0
2
4
6
8
0
0
2
4
6
8
-1.4
0
2
4
6
8
-1
0
2
4
6
8
PANEL B: NON DISTRESS PERIODS
Equity to Equity
Market to Equity
20
Investment to Equity
EB to Equity
12
0.2
2
10
0.1
1.5
15
0
8
1
-0.1
10
6
0.5
-0.2
4
2
0
0
2
4
6
8
0
0
-0.3
5
-0.5
-0.4
0
2
4
6
8
-0.5
0
2
4
6
8
-1
0
2
4
6
8
Road Map of the Rest of Talk
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Model, mechanism, and solution
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Calibration
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Baseline parameters
Prices and polices, comparative statics
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Matching data on distress and non-distress
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Systemic crisis
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Extrapolate to crisis state
Uncover fundamental shocks in the recent crisis
How likely are crises?
Agents and Technology
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Two classes of agents: households and bankers
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Households own the entire economy, but subject to frictions related
to bankers who control intermediaries (next slide)
Two types of capital: productive capital Kt and housing capital H.
Fixed supply of housing H 1
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Price of capital qt and price of housing Pt determined in equilibrium
Agents and Technology
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Two classes of agents: households and bankers
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Households own the entire economy, but subject to frictions related
to bankers who control intermediaries (next slide)
Two types of capital: productive capital Kt and housing capital H.
Fixed supply of housing H 1
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Price of capital qt and price of housing Pt determined in equilibrium
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Production Y = AKt , with A being constant
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Fundamental shocks: stochastic capital quality shock dZt
dKt
= it dt
Kt
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δdt + σdZt
Investment/Capital it , quadratic adjustment cost
Φ(it , Kt ) = it Kt +
κ
(it
2
δ)2 Kt
Aggregate Balance Sheet
'
Loans to Capital
Producers it
$
&
%
6
Intermediary Sector
Capital qt Kt
Housing Pt H
Household Sector
HH
Y
HH
Equity Et
Debt Wt
Et
Financial Wealth
HH
Wt = qt Kt + Pt H
Aggregate Balance Sheet
'
Loans to Capital
Producers it
$
Aggregate bank reputation Et
&
%
6
Intermediary Sector
Capital qt Kt
Equity Et
Household Sector
HH
Y
Constraint:HEH
t
Financial Wealth
Et
HH
Wt = qt Kt + Pt H
No constraint
Housing Pt H
Debt Wt
Et
Single Bank/Banker
Capital qt kt
Housing Pt ht
Equity et
Debt dt
Portfolio share in capital: αkt = qet kt t
Portfolio share in housing : αht = Pet tht
Borrowing (no constraint): dt = qt kt + Pt ht
et = (αkt + αht
Return on bank equity: d R̃t = αkt dRtk + αht dRth
(αkt + αht
Banker (log preference) solves: maxαk ,αh E [d R̃t
t t
rt dt ]
1)rt dt
m Var [d R̃ ]
t
t
2
1 ) et
Single Bank/Banker
Capital qt kt
Housing Pt ht
Equity et
Debt dt
Properties
(k, h) scales with e
(k, h) increasing in Et [dR r ]
(k, h) decreasing in Var [dR ]
Portfolio share in capital: αkt = qet kt t
Portfolio share in housing : αht = Pet tht
Borrowing (no constraint): dt = qt kt + Pt ht
et = (αkt + αht
Return on bank equity: d R̃t = αkt dRtk + αht dRth
(αkt + αht
Banker (log preference) solves: maxαk ,αh E [d R̃t
t t
rt dt ]
1)rt dt
m Var [d R̃ ]
t
t
2
1 ) et
General Equilibrium (1)
Intermediary Sector
Capital qt Kt
Equity Et
Housing pt H
Debt Wt
Household Sector
Financial Wealth
yXX
X
Constraint:
Et
XXXEt
XXX
Et
Wt = qt Kt + pt H
Portfolio share in capital: αkt = qEt Kt t
Portfolio share in housing: αht = PEt tH
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Given a particular state (Kt , Et ), the portfolio shares are pinned
down by GE
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Portfolio shares must also be optimally chosen by banks
max Et [d R̃ t
αkt ,αht
rt dt ]
m
Vart [d R̃ t ]
2
General Equilibrium (2)
Intermediary Sector
Capital qt Kt
Equity Et
Housing pt H
Debt Wt
Household Sector
Financial Wealth
yXX
X
Constraint:
E
XX Et
XXX t
X W = q K +p H
Et
t
t t
t
Portfolio share in capital: αkt = qEt Kt t
Portfolio share in housing: αht = PE tt
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Prices (returns) have to adjust for optimality:
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Et [dRth
rt dt ], Et [dRtk
rt dt ] ) equations for Et [dPt ], Et [dqt ]
Rewrite to get ODEs for P (K , E ) and q (K , E )
Scale invariance: De…ne e
E /K ; then P = Kp (e ) and q (e )
Capital Producers and Investment
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Capital goods producers (owned by households) undertake real
investment
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Producers must sell the capital stock to intermediaries at price qt
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Possible interpretations:
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Risk averse intermediaries bear aggregate fundamental shocks
Real investment is a¤ected by …nancial condition of intermediaries to
capture “credit crunch”
Entrepreneurs raise capital from VC/PE at the price of qt
Commercial banks makes collateralized loans
Investment decision
max qt it Kt
it
Φ(it , Kt ) ) it = δ +
qt
1
κ
Capital Constraint
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Single bank has reputation et linked to intermediary performance
(constant m)
d et
= mR̃t .
et
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Poor past returns reduce reputation
Households invest a maximum of et dollars of equity capital with
this banker
Capital Constraint
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Single bank has reputation et linked to intermediary performance
(constant m)
d et
= mR̃t .
et
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Poor past returns reduce reputation
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Households invest a maximum of et dollars of equity capital with
this banker
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Death rate η, and entry d ψt > 0 of new bankers in extreme states
(modeled later)
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Et : aggregate reputation. Identical banks, aggregate dynamics of Et
d Et
= md R̃t
Et
ηdt + d ψt
Capital Constraint
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Single bank has reputation et linked to intermediary performance
(constant m)
d et
= mR̃t .
et
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Poor past returns reduce reputation
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Households invest a maximum of et dollars of equity capital with
this banker
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Death rate η, and entry d ψt > 0 of new bankers in extreme states
(modeled later)
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Et : aggregate reputation. Identical banks, aggregate dynamics of Et
d Et
= md R̃t
Et
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ηdt + d ψt
Note: Et is like “net worth" in many other models.
Households’ Problem (1)
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Choose consumption cty and housing cth to maximize
E
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Z ∞
0
e
ρt
(1
φ) ln cty + φ ln cth dt
Equilibrium rental price Dt (housing asset dividend), FOC
cth D t
φ
=
cty
1 φ.
In equilibrium (Cth = H = 1)
Dt =
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φ
1
Cy
φ t
φ: expenditure share in housing, or the relative size of housing sector
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Households free to trade short-term debt.
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Interest rate rt = ρ + Et dCty /Cty
Vart dCty /Cty
Households’ Problem (2)
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Representative household enters time t with …nancial wealth Wt
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The household splits wealth: (1
λWt to “bond households”
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λ) Wt to “equity households,”
Equity households invest their portion of wealth as equity of
intermediaries, subject to capital frictions
Bond households invest in riskless bonds
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Once returns are realized, both members pool their wealth again (as
in Lucas 1990)
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The only role of bond households (i.e. parameter λ) is to introduce
intermediary’s leverage in normal time
Debt/Equity Ratio
'
Loans to Capital
Producers it
$
Aggregate bank reputation Et
&
%
6
Intermediary Sector
Capital qt Kt
Housing Pt H
Equity Et
Debt Wt
Household Sector
HH
Y
Constraint:HEH
t
Financial Wealth
Wt = qt Kt + pt H
Et
HH
( 1 λ ) Wt
No constraint
Et
λWt
Equity Capital Constraint
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Unconstrained capital structure: λWt of Debt, (1
Intermediary equity capital Et is given by
Et = min [Et , (1
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λ)Wt of Equity.
λ ) Wt ]
How can capital constraint come to bind, beginning in a state where
E t > ( 1 λ ) Wt ?
Suppose a 10% shock to real estate and price of capital, so that
Wt # 10% (Household wealth = aggregate wealth)
Reputation follows dEEt t = md R̃t + ... Two forces make Et # more
than 10%:
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Equity is levered claim on assets: Return on equity = d R̃t <
m > 1 in our calibration.
10%
Boundary Conditions
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When e = ∞, Et > (1
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We solve for p (∞), q (∞) analytically
As e ! 0, intermediaries’portfolio volatility, i.e. Sharpe ratio, rises
New bankers enter if e = e (Sharpe ratio hits γ, exogenous
constant)
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λ) Wt frictionless economy
Entry increases aggregate E but requires physical capital K at
conversion rate of β
e is a re‡ecting boundary
Boundary conditions at the entry point e
q 0 (e ) = 0, p 0 (e ) =
p (e ) β
, and Sharpe_Ratio (e ) = γ
1 + eβ
Calibration: Baseline Parameters
Parameter
Panel A: Intermediation
m Performance sensitivity
λ Debt ratio
η
Banker exit rate
γ Entry trigger
β
Entry cost
Panel B: Technology
σ
Capital quality shock
δ Depreciation rate
κ Adjustment cost
A Productivity
Panel C: Others
ρ Time discount rate
φ
Housing share
Choice
Target
2.5
0.5
13%
5.5
2.35
Average Sharpe ratio (38%)
Average intermediary leverage
Good model dynamics
Highest Sharpe ratio
Land price volatility
5%
10%
2
0.14
Investment and Consumption volatilities
Literature
Literature
Investment-to-capital ratio
2%
0.5
Literature
Housing-to-wealth ratio
Equilibrium Prices and Policies (1)
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ecrisis = 0.65: binding capital constraint
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edistress = 4 so that Pr (e
edistress ) = 33% as in data
p(e), s c aled hous ing pric e
q(e), c apital pric e
0.8
1.01
1.008
0.7
1.006
0.6
1.004
0.5
1.002
1
0.4
0.998
0.3
0.996
0.2
0.1
0
0.994
2
4
6
8
10
12
14
scaled intermediary reputation e
16
18
20
0.992
0
2
4
return v olatility of hous ing
6
8
10
12
14
scaled intermediary reputation e
16
18
20
16
18
20
return v olatility of c apital
1.5
0.0535
0.053
0.0525
1
0.052
0.0515
0.051
0.5
0.0505
0.05
0.0495
0
0
2
4
6
8
10
12
14
scaled intermediary reputation e
16
18
20
0.049
0
2
4
6
8
10
12
14
scaled intermediary reputation e
Equilibrium Prices and Policies (2)
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ecrisis = 0.65: binding capital constraint
edistress = 4 so that Pr (e edistress ) = 33% as in data
Sharpe ratio
interest rate
6
0.1
5
0.05
0
4
-0.05
3
-0.1
2
-0.15
1
0
0
-0.2
5
10
scaled intermediary reputation e
15
20
-0.25
0
5
investment I/K
10
15
scaled intermediary reputation e
20
steady state distribution
0.105
0.035
0.104
0.03
0.103
0.025
0.102
0.101
0.02
0.1
0.015
0.099
0.01
0.098
0.005
0.097
0.096
0
5
10
scaled intermediary reputation e
15
20
0
0
5
10
15
20
25
30
scaled intermediary reputation e
35
40
Matching State-Dependent Covariances: Baseline
vol (Eq )
vol (I )
vol (C )
vol (LP )
vol (EB )
cov (Eq, I )
cov (Eq, C )
cov (Eq, LP )
cov (Eq, EB )
Distress
Data
Baseline
31.48%
26.01
8.05%
5.73
1.71%
3.29
21.26%
22.87
60.14%
49.96
1.31%
0.80
0.25%
0.34
4.06%
4.56
-6.81%
-6.69
Non Distress
Data Baseline
17.54
6.77
6.61
5.39
1.28
3.94
9.79
9.38
12.72
6.32
0.07
0.36
0.03
0.26
0.12
0.63
-0.14
-0.09
Matching State-Dependent Covariances: lower σ
Distress
vol (Eq )
vol (I )
vol (C )
vol (LP )
vol (EB )
cov (Eq, I )
cov (Eq, C )
cov (Eq, LP )
cov (Eq, EB )
Non Distress
Baseline
σ = 4%
Data
Baseline
σ = 4%
31.48%
26.01
20.57
17.54
6.77
5.09
8.05%
5.73
4.40
6.61
5.39
4.18
Data
1.71%
3.29
2.67
1.28
3.94
3.36
21.24%
21.26
12.91
9.79
9.38
6.41
60.14%
48.96
36.52
12.72
6.32
4.33
1.31%
0.81
0.47
0.07
0.37
0.22
0.25%
0.35
0.22
0.03
0.26
0.17
4.06%
4.56
2.03
0.12
0.63
0.34
-6.81%
-6.69
-3.58
-0.14
-0.09
-0.04
Matching State-Dependent Covariances: No Housing
Distress
vol (Eq )
vol (I )
vol (C )
vol (LP )
vol (EB )
cov (Eq, I )
cov (Eq, C )
cov (Eq, LP )
cov (Eq, EB )
Non Distress
Baseline
φ=0
Data
31.48%
26.09
14.62
8.05%
5.73
5.14
1.71%
3.29
4.52
21.24%
21.26
60.14%
48.96
1.31%
0.25%
Data
Baseline
φ=0
17.54
6.77
5.00
6.61
5.40
5.01
1.28
3.92
4.94
9.79
9.38
9.42
12.72
6.32
0.03
0.81
0.53
0.07
0.37
0.25
0.35
0.44
0.03
0.26
0.25
0.12
0.63
-0.33
-0.14
-0.09
4.06%
4.56
-6.81%
-6.69
-0.00
Uncovering Shocks in the Recent Crisis
Data
Model
Uncovering Shocks in the Recent Crisis
Data
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Model
Based on realized equity return we uncover fundamental shocks to K
07QIII
07QIV
08QI
08QII
08QIII
08QIV
09QI
09QII
09QIII
09QIV
-3.77%
-7.24
-6.62
-2.85
-0.48
-3.10
-2.32
-1.15
-0.04
-0.77
Total -25%. Capital constraint binds after 08QII— systemic crisis
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In the model (data), land price fall by 71% (55%)
Probability of Crisis
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2007Q2, Prob(crisis occurs in the next 2 years)=0.09%, Prob(5
years) = 2.62%, Prob (10 years) = 10.05%
Probability of Crisis
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2007Q2, Prob(crisis occurs in the next 2 years)=0.09%, Prob(5
years) = 2.62%, Prob (10 years) = 10.05%
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Conditional probability of hitting crisis (left) or distress (right)
Probability of Entering D is tres s States
Probability of C apital C ons traint Binding
0.35
1.1
1
0.3
0.9
0.25
0.8
in nex t 5 y ears
in nex t 10 y ears
0.2
in nex t 5 y ears
in nex t 10 y ears
0.7
0.6
0.15
0.5
0.4
0.1
0.3
0.05
0.2
0
0
2
4
6
8
Initial C ondition e
init
10
12
0.1
0
2
4
6
8
Initial C ondition: e
10
init
Note: Probabilities are low, which suggests improved capital bu¤ers
would have limited e¤ects.
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VIX and Systemic Risk
e, s c aled i ntermediary reputation
20
15
10
5
0
0
50
100
150
200
250
300
tim e (quarters )
prob. of c apital c ons traint being c ons trained in nex t x y ears
350
400
450
1
nex t 1 y ear
nex t 5 y ears
nex t 10 y ears
0.5
0
0
50
100
150
50
100
150
200
tim e (quarters )
hous i ng pric e v olatility
250
300
350
400
250
300
350
400
0.4
0.3
0.2
0.1
0
0
200
tim e (quarters )
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Volatility in our model rises most sharply when the constraint binds
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Coincident indicator and no predictive content.
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What might work better? A VIX spread: Long-maturity VIX minus
short-maturity VIX
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Other indicators...
Conclusion
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We develop a fully stochastic model of systemic crisis, with two
major frictions:
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Equity capital constraint on intermediary sector
Intermediaries have substantial holdings in real assets (physical
capital or housing)
We …nd that the model
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not only qualitatively delivers the nonlinearity observed in the data
but also quantitatively matches the di¤erential comovements in
distress and non-distress periods
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Recent 07/08 crisis requires a cumulative negative shock around
-25%
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Things we are working on: more on model-based measure of
systemic risk