Debt Crises
For Whom the Bell Tolls
Harold Cole
UPenn
Daniel Neuhann
UPenn
Guillermo Ordoñez
UPenn
September 28, 2015
1 / 40
Debt Crises
I
Debt Crises = Spread jumps and Defaults
I
Regular occurrence, but come and go as surprises.
I
I
I
Fundamentals necessary but not sufficient.
Contagion = comovement of spreads common
I
No clear pattern of contagion related to fundamental linkages!
I
Maybe learning about a common factor. Herding.
I
Correlated structure of sunspots.
Consider new story: portfolio linkages + information acquisition.
2 / 40
EU Crisis
Source: Wright on Broner, Erce, Martin and Ventura (2014)
3 / 40
EU Crisis Regimes?
-5
0
residuals
5
10
15
Crisis: Double sensitivity to GDP growth
z }| {
2000q1
2005q1
2010q1
2015q1
tq
Residuals from
Debt
Y ieldsit = (β1 + β2 Ic )∆GDPit + (β3 + β4 Ic ) GDP
+ it
it
(with Country and Year FE)
4 / 40
This paper
I
Simple model of portfolio choice with information acquisition
(where investments are not fundamentally linked).
I
Capital flowing across economies generates contagion.
I
Information about economies generates multiple equilibria.
I
Informed: With participation of informed investors.
I
Uninformed: Without participation of informed investors.
5 / 40
This paper
I
Simple model of portfolio choice with information acquisition
(where investments are not fundamentally linked).
I
Capital flowing across economies generates contagion.
I
Information about economies generates multiple equilibria.
I
Informed: With participation of informed investors.
I
Uninformed: Without participation of informed investors.
I
No need for fundamental linkages for contagion
I
No clear pattern of contagion, as it sovereign spreads depend on
I
I
Own fundamentals and others’ fundamentals.
I
Own equilibrium and others’ equilibria.
Not only crises are contagious, also information regimes!
5 / 40
Model
I
Two periods.
I
Mass 1 of investors.
I
I
CRRA utility functions: u(c)
I
Endowment W in period 1.
Two countries i ∈ {1, 2}.
I
In Period 1, each country must repay debt Di (net of income).
I
I
Roll over at price pi , so new debt is bi ≡ Di /pi .
In Period 2, each country has income Yi and a cost of default θi
I
Repayment: Yi −
I
Repay if
Di
.
pi
Default: (1 − θi )Yi .
Yi > Ȳ (θi ) ≡
Di
pi θi
6 / 40
Default
f (Yi )
Default
&
Ȳ (θi ) ≡
Di
pi θi
Yi
7 / 40
Further Simplifying Assumptions
I
I
Two possible costs of default (realized in period 1):
I
θH w/prob. a.
I
θL w/prob. 1 − a.
I
Any investor can become informed about θ at utility cost K.
Three possible levels of income (realized in period 2):
I
YL w/prob. x.
I
YM w/prob. z.
I
YH w/prob. 1 − x − z.
8 / 40
Further Simplifying Assumptions
z
1−x−z
x
YL
Y (θH )
a
YM
Y (θL )
YH
1−a
9 / 40
Further Simplifying Assumptions
With probability a
the default probability is κH = x
z
↓
Good state
1−x−z
x
YL
Y (θH )
a
YM
Y (θL )
YH
1−a
9 / 40
Further Simplifying Assumptions
With probability 1 − a
the default probability is κL = x + z
z
Bad state
↓
1−x−z
x
YL
Y (θH )
a
YM
Y (θL )
YH
1−a
9 / 40
Auctions
I
In period 1 each country rolls over its debt using auctions
I
Investors submit pair-orders (p, b).
I
The country fills orders in decreasing order of p until it raises D.
I
If there is a marginal price p, no reason to bid any other p.
Uninformed do not learn from prices before bidding, but they know
which are the possible prices in equilibrium after the bidding!
10 / 40
A Single Country
11 / 40
Uninformed Equilibrium
I
Without knowing the state, κ
b = ax + (1 − a)(x + z)
I
A single price p, independent of the realized state.
I
Given the price p, investors bid b to maximize
max U = κ
bu(W − pb) + (1 − κ
b)u(W − pb + b)
b
FOC {b}
κ
bpu0 (W − pb) = (1 − κ
b)(1 − p)u0 (W − pb + b)
|
{z
} |
{z
}
M g. Costs
M g. Benef its
12 / 40
Uninformed Equilibrium
I
Without knowing the state, κ
b = ax + (1 − a)(x + z)
I
A single price p, independent of the realized state.
I
Given the price p, investors bid b to maximize
max U = κ
bu(W − pb) + (1 − κ
b)u(W − pb + b)
b
FOC {b}
u0 (W − pb + b)
pb
κ
=
u0 (W − pb)
(1 − p)(1 − κ
b)
I
=⇒
b( p , κ
b)
(−) (−)
If p = 1 − κ
b, then b = 0. Investors charge a risk premium!
12 / 40
Uninformed Equilibrium
I
Without knowing the state, κ
b = ax + (1 − a)(x + z)
I
A single price p, independent of the realized state.
I
Given the price p, investors buy b to maximize.
max U = κ
bu(W − D) + (1 − κ
b)u(W − D +
b
I
D
)
p
Imposing the resource constraint pb = D,
u0 W − D +
u0 (W
D
p
− D)
=
pb
κ
(1 − p)(1 − κ
b)
I
The utility of investors decreases with p.
I
FOC pins down p. Both sides increase in p and span [0, 1].
12 / 40
Price in Uninformed Equilibrium
1
High κ
b
0.9
Medium κ
b
Low κ
b
0.8
0.7
0.6
0.5
LHS
0.4
0.3
0.2
Debt Crisis Condition (when u(c) = log(c))
0.1
0
κ
b>1−
0
0.14
0.28
0.42
D
W
0.56
P
13 / 40
Price in Uninformed Equilibrium
RHS
Debt Crisis Condition (when u(c) = log(c))
1
D
W
0.9
>1−κ
b
0.8
Low
0.7
D
W
0.6
Medium
0.5
D
W
0.4
High
0.3
D
W
0.2
0.1
0
0
0.1
0.2
0.3
0.4
P
14 / 40
Incentives to Acquire Information
I
Information about θ can be acquired at utility cost K.
15 / 40
Incentives to Acquire Information
I
Information about θ can be acquired at utility cost K.
I
Benefits of deviating and acquiring information:
Buy more in the good state, since the price p is relatively low.
∆(UH ) ≡ U (b( κH , p)) − U (b(b
κ, p)) > 0
|{z}
x
Buy less in the bad state, since the price p is relatively high.
∆(UL ) ≡ U (b( κL , p)) − U (b(b
κ, p)) > 0
|{z}
x+z
I
There are no incentives to acquire information if
a∆(UH ) + (1 − a)∆(UL ) < K
|
{z
}
χU
15 / 40
Condition for Uninformed Equilibrium
2.2
·10−2
2
χU
1.8
1.6
1.4
1.2
1
0.8
0.6
0.105
0.123
0.141
0.159
0.177
z1
16 / 40
Condition for Uninformed Equilibrium
9.8
·10−3
χU
9.6
9.4
9.2
9
8.8
8.6
8.4
8.2
8
36.4
42.64
48.88
55.12
61.36
D1
17 / 40
Informed Investors
I
Informed investors know the state, κH = x and κL = x + z
I
Same maximization as before, but knowing the state s ∈ {L, H}.
Then
u0 (W − ps bIs + bIs )
ps κs
=
u0 (W − ps bIs )
(1 − ps )(1 − κs )
=⇒
bIs ( ps , κs )
(−) (−)
18 / 40
Informed Investors
I
Informed investors know the state, κH = x and κL = x + z
I
Assume all investors are informed.
Resource constraints are ps bs = D in both states. Hence
u0 W − D +
u0 (W
I
D
ps
− D)
=
ps κs
(1 − ps )(1 − κs )
Since κH < κL , then pH > pL .
18 / 40
Uninformed Investors
I
What if an investor decides to be uninformed (and save K)?
I
Uninformed investors do not know the state, but they know that
the country will always sell bonds to them at pH ≥ pL .
I
Maximization problem:
U
U
max U U = a κH u(W − pH bU
H ) + (1 − κH )u(W − pH bH + bH )
U
bU
L ,bH
U
U
U
U
U
+(1 − a) κL u(W − pH bU
H − pL bL ) + (1 − κL )u(W − pH bH − pL bL + bH + bL )
19 / 40
Uninformed Investors
I
FOC {bU
H } They always get to buy at pH .
a[pH κH u0 (W −pH bU
H )]
U
+(1−a)[pH κL u0 (W −pH bU
H −pL bL )]
= a[(1−pH )(1−κH )u0 (W −pH bUH +bUH )]+(1−a)[(1−pH )(1−κL )u0 (W −pH bUH −pL bUL +bUH +bUL )]
I
FOC {bU
L } When they get to buy at pL , they also buy part at pH .
U
pL κL u0 (W −pH bU
H −pL bL )
=
U
U
U
(1−pL )(1−κL )u0 (W −pH bU
H −pL bL +bH +bL )
19 / 40
Uninformed Investors
I
FOC {bU
H } They always get to buy at pH .
a[pH κH u0 (W −pH bU
H )]
U
+(1−a)[pH κL u0 (W −pH bU
H −pL bL )]
= a[(1−pH )(1−κH )u0 (W −pH bUH +bUH )]+(1−a)[(1−pH )(1−κL )u0 (W −pH bUH −pL bUL +bUH +bUL )]
I
p H bU
H < pH bH
Uninformed investors spend less in the good state!
In equilibrium bU
H ≥ 0! (otherwise info revelation!)
I
FOC {bU
L } When they get to buy at pL , they also buy part at pH .
U
pL κL u0 (W −pH bU
H −pL bL )
=
U
U
U
(1−pL )(1−κL )u0 (W −pH bU
H −pL bL +bH +bL )
U
I
p H bU
H + pL bL > pL bL
Uninformed investors spend more in the bad state!
19 / 40
Resource Constraints
Denote by n the fraction of informed investors.
npH bIH + (1 − n)pH bU
H
U
npL bIL + (1 − n)[pH bU
H + pL bL ]
= D
= D
20 / 40
Prices as a function of n
0.84
pH
0.82
0.8
0.78
0.76
pU
7−→
0.74
0.72
0.7
0.68
0.66
Uninformed only bid pL
0.64
7−→
0.62
0.6
pL
0
0.1
0.2
0.3
0.4
0.5
n
0.6
0.7
0.8
0.9
21 / 40
Utilities as a function of n
UI
U
(Uninformed Eq)
UU
0
0.1
0.2
0.3
0.4
0.5
n
0.6
0.7
0.8
0.9
22 / 40
Utilities as a function of n
Investors end up losing with information!
UI
U
(Uninformed Eq)
UU
0
0.1
0.2
0.3
0.4
0.5
n
0.6
0.7
0.8
0.9
22 / 40
Equilibrium Multiplicity
K
χU
χI ≡ U I − U U
0
0.1
0.2
0.3
0.4
0.5
n
0.6
0.7
0.8
0.9
23 / 40
Equilibrium Multiplicity
n∗ in Informed Equilibrium
. (where U I − K = U U )
K
χU
- Uninformed Equilibrium
χI ≡ U I − U U
0
0.1
0.2
0.3
0.4
0.5
n
0.6
0.7
0.8
0.9
23 / 40
How Information Depends on z
0.55
0.5
n∗ (z)
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
5 · 10−2
0.105
0.123
0.141
0.159
0.177
z1
24 / 40
How Information Depends on D
0.5
0.45
n∗ (D)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
5 · 10−2
0
36.4
42.64
48.88
55.12
61.36
D1
25 / 40
Importance of Information on Prices
Three Regions of Equilibria!
0.78
pH
0.76
pU
0.74
E(p)
0.72
0.7
0.68
0.66
0.64
0.62
pL
0.6
0.58
0.56
0.105
0.123
0.141
0.159
0.177
z1
Information induces price volatility and lower expected prices!
With a “conservative’ selection of equilibrium, past sins or virtues matter!26 / 40
Debt Burden Across Equilibria
112
E(b)
110
108
106
104
102
bU
100
98
0.105
0.123
0.141
0.159
0.177
z1
Small domestic shocks can create large changes in debt burden!
27 / 40
Different Components of κ
b
I
So far we have increased κ
b = x + (1 − a)z by increasing z.
I
What if we increase x?
I
There are less incentives to acquire information (reduction in the
relative difference between states).
I
What if we reduce a?
I
The incentives to acquire information are non-monotonic (at the
extremes there is no uncertainty about the state).
I
For information acquisition incentives, it matters how the expected
default probability increases!
28 / 40
Two Countries
29 / 40
Pure Contagion on Prices
I
Assume no investor knows the state in either country, κ
b1 and κ
b2 .
I
A single price in each country, p1 and p2 .
I
Given these prices, investors bid b1 and b2 to maximize.




max U = κ
b1 κ
b2 u(W − p1 b1 − p2 b2 ) +(1 − κ
b2 ) u(W − p1 b1 + (1 − p2 )b2 )
b1 ,b2
|
{z
}
|
{z
}
u(−−)
u(−+)




b2 ) u(W + (1 − p1 )b1 + (1 − p2 )b2 )
+(1 − κ
b1 ) κ
b2 u(W + (1 − p1 )b1 − p2 b2 ) +(1 − κ
{z
}
|
{z
}
|
u(+−)
u(++)
30 / 40
Pure Contagion on Prices
I
Bids (b1 and b2 ) and prices (p1 and p2 ) solve
I
FOC {b1 }
p1 κ
b1 [b
κ2 u0 (−−) + (1 − κ
b2 )u0 (−+)] = (1−p1 )(1−b
κ1 ) [b
κ2 u0 (+−) + (1 − κ
b2 )u0 (++)]
|
{z
}
|
{z
}
E(u0 (−•))
I
E(u0 (+•))
FOC {b2 }
p2 κ
b2 [b
κ1 u0 (−−) + (1 − κ
b1 )u0 (+−)] = (1−p2 )(1−b
κ2 ) [b
κ1 u0 (−+) + (1 − κ
b1 )u0 (++)]
|
{z
}
|
{z
}
E(u0 (•−))
I
E(u0 (•+))
Resource constraints: p1 b1 = D1 and p2 b2 = D2
31 / 40
Pure Contagion on Prices
I
Bids (b1 and b2 ) and prices (p1 and p2 ) solve
I
FOC {b1 }
I
I
FOC {b2 }
E(u0 (+•))
p1 κ
b1
=
E(u0 (−•))
(1 − p1 )(1 − κ
b1 )
E(u0 (•+))
p2 κ
b2
=
E(u0 (•−))
(1 − p2 )(1 − κ
b2 )
Resource constraints: p1 b1 = D1 and p2 b2 = D2
31 / 40
Pure Contagion on Prices
I
Assume an increase in κ
b1 .
I
We know
then
dp2
db
κ1
u0 (−+)−u0 (++)
1−b
κ1
|
dp1
db
κ1
< 0. Differentiating the FOC for country 2 bonds,
< 0 if
−
D1 ∂p1 00
κ1 u (++)
p21 ∂b
E(u0 (•+))
{z
Relative change in gains from b2
}
<
u0 (−−)−u0 (+−)
1−b
κ1
|
−
D1 ∂p1 00
κ1 u (+−)
p21 ∂b
E(u0 (•−))
{z
Relative change in losses from b2
}
This condition is satisfied under CRRA utility functions, as
they display prudence (that is, u000 (c) > 0).
Intuition
32 / 40
Pure Contagion on PricesRHS
p2 κ
b2
E(u0 (•+))
=
E(u0 (•−))
(1 − p2 )(1 − κ
b2 )
1
0.9
Coeff. RA = 0.5
Coeff. RA = 1
0.8
0.7
0.6
Coeff. RA = 1.5
0.5
0.4
Blue is low κ
b1
Red is high κ
b1
0.3
0.2
0.1
0
0
0.15
0.3
P1
0.45
33 / 40
Pure Contagion on PricesRHS
p2 κ
b2
E(u0 (•+))
=
E(u0 (•−))
(1 − p2 )(1 − κ
b2 )
1
0.9
Coeff. RA = 0.5
Coeff. RA = 1
0.8
0.7
0.6
Coeff. RA = 1.5
0.5
0.4
Blue is low κ
b1
Red is high κ
b1
0.3
0.2
Contagion is stronger with higher prudence!
0.1
0
Possible debt crisis even without domestic shock!
0
0.15
0.3
P1
0.45
33 / 40
Cross-Country Info Complementarities
6
·10−2
χI1 (allowing the other country to be symmetrically informed)
5
4
χI2
3
K
2
χU
1
0
χI (forcing the other country to be uninformed)
0
0.25
0.5
0.75
n
34 / 40
Cross-Country Info Complementarities
6
·10−2
If the other country is symmetrically informed
χI1
5
Uninformed and “very” informed equilibrium coexist!
4
χI2
3
K
2
χU
1
If the other country is uninformed
0
χI
Only uninformed equilibrium is feasible!
0
0.25
0.5
0.75
n
34 / 40
Contagion on Information Regime
0.8
pU
pH
0.75
0.7
E(p)
0.65
0.6
0.55
0.5
0.45
pL
0.1
0.13
0.16
0.19
0.22
z1
Absent information feedback from the other country, only U equilibrium!
35 / 40
Contagion on Information Regime
Remember those initial regressions?
0.8
0.75
pU
•
pH
•
•
•
•
•
Low sensitivity to fundamentals
•
Small errors when U equilibrium
0.7
E(p)
0.65
0.6
0.55
0.5
0.45
pL
0.1
0.13
0.16
0.19
0.22
z1
35 / 40
Contagion on Information Regime
Remember those initial regressions?
0.8
0.75
pU
•
pH
•
•
•
•
•
0.7
E(p)
0.65
0.6
. Greece?
0.55
•
0.5
0.45
pL
0.1
0.13
0.16
0.19
0.22
z1
35 / 40
Contagion on Information Regime
Remember those initial regressions?
0.8
0.75
•
•
•
pU
pH
•
0.7
E(p)
0.65
•
0.6
•
0.55
•
0.5
0.45
pL
0.1
0.13
0.16
0.19
0.22
z1
35 / 40
Contagion on Information Regime
0.8
0.75
Remember those initial regressions?

•
•

Larger positive errors

pU
•
•










0.7
Larger sensitivity of
↑
spreads to fundamentals
0.65
pH
E(p)
Larger negative errors
•
0.6
•






0.55
•
0.5
0.45
pL
0.1
0.13
0.16
0.19
0.22
z1
35 / 40
Magnification of Contagion
I
So far we have discussed contagion in its purest form, but there
are magnifying forces.
I
Endogenous probability of default (b
κi depends on pi ).
I
Fundamental linkages (b
κi depends on κ
b−i ).
I
Time-varying prudence (risk-aversion that changes with wealth).
I
Market segmentation concentrates contagion.
I
Structure of information costs across countries.
36 / 40
Suggestive Thoughts
I
How can Japan or the U.S. sustain very large debt/GDP ratios
with low and stable spreads?
Uninformed Equilibrium?
I
Why can many countries not raise their debt/GDP ratio without
triggering high volatility and increases in spreads?
Informed Equilibrium?
37 / 40
Extensions
I
Extension to K states and I countries.
I
I
Just add FOCs and resource constraints.
Extension to a continuous distribution of Y .
I
Thresholds Y (θk ) are endogenous and jointly determined with the
probability of default in each state.
38 / 40
Final Remarks
I
Simple model of portfolio choice with information acquisition.
I
For a single country
I
Information is more likely with high default probability and debt.
I
Information is “bad” both for investors (waste on information) and
countries (lower prices, more debt, and higher volatility).
I
Multiplicity implies that a small change in fundamentals can have
large effects on prices and debt.
I
With equilibrium hysteresis, prices in two countries with the same
parameters but different past can behave very differently.
39 / 40
Final Remarks
I
Simple model of portfolio choice with information acquisition.
I
For many countries
I
Given investor prudence, there is price and debt contagion.
I
Strong cross-country complementarities on the incentives to produce information.
I
Shocks in one country can cause changes of equilibrium in others.
I
Information regimes affect the strength of contagion.
I
Information regimes are also contagious.
39 / 40
The Effects of Prudence
u0 (c)
Assume NO prudence
0
E(u (•−))
(e.g. quadratic utility)
E(u0 (•+))
c(−−)
c(+−)
C
c(−+)
c(++)
39 / 40
The Effects of Prudence
u0 (c)
Prudence with CRRA
Back
E(u0 (•−))
E(u0 (•+))
c(−−)
c(+−)
C
c(−+)
c(++)
39 / 40