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Financial Networks and Contagion
Matthew Elliott
Benjamin Golub
Matthew O. Jackson
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Introduction
• Contagions make understanding network structure of financial interactions critical
• Need tools to help evaluate potential risk of contagion
• Need to understand effects of integration and diversification
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Our Contributions:
• Develop network measures related to cascades
• Illuminate and distinguish the effects of diversification and integration
• Examine policies to avoid initial failures and some associated moral hazard issues
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Outline
• Model
• Cascades: Diversification and Integration
• Endogenous Values and Moral Hazard
• An Illustration with European Debt Data
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Basics of the Model: • Organizations (firms, banks, countries, etc.) hold
• Assets
• Shares in each other
• Dropping below some level of value, an organization experiences a discontinuous drop:
• Cash flow problems disrupt production…
• Bankruptcy costs…
• Drop in value of one organization leads to drop in values of others they have financial arrangements with, cascades…
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Model
• {1, ..., n} Organizations (countries, firms, banks...)
• {1, ..., m} Assets (primitive investments)
• pk price of asset k
• Dik holdings of asset k by organization i
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Cross Holdings:
• Cij cross holdings: fraction of org j owned by org i
• Cii = 0 (don’t own yourself)
• Ĉii = 1 – Ʃj Cji fraction of org i privately held
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Value of an Organization
Vi = Ʃk Dik pk + Ʃk Cij Vj Book Value
Direct Asset Cross Holdings
Holdings
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Value of an Organization
Vi = Ʃk Dik pk + Ʃk Cij Vj
V = Dp + CV
V = (I – C)-1 Dp Book Value
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Value of an Organization
V = (I – C)-1 Dp Book Value
Market Value: value to final (private) investors
v = ĈV = Ĉ (I – C)-1 Dp Market Value
v = A Dp Market Value
(c.f. Brioschi et al 89, Fedenia et al 96)
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Example
• Two organizations n=2
• Each own half of each other
holdings by private investors
0 . 5
.5 0 C =
.5 0
0 .5 Ĉ =
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Example
• Two organizations n=2
• Each own half of each other
private investors’ claims on values
0 . 5
.5 0 C =
.5 0
0 .5 Ĉ = A = Ĉ (I-C)-1 =
2/3 1/3
1/3 2/3
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Example
2 1
Ĉ11= .5
C21=.5
C12=.5
Ĉ22= .5
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
1
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
.5
.5 1
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
.5
.5
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
.5
.5 .25
.25
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
.5 .25
.25
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
.5 .25
.25 .125
.125
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
.5 .25
.125
.125
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
.5 .25
.0625 .125
.125
.0625
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
.5 .25
.0625 .125 .0625
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
.5 .25
.0625 .125
.03125
.0625
.03125
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Example
2 1
.5
.5
.5
.5
What happens to 1$ of investment income to 1?
2/3 1/3
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Drops in Values
• If an organization’s value drops below some vi it incurs a cost bi
• Cash flow problems:
• E.g., Spanair unable to pay for fuel, forced to cancel flights
• Bankruptcy costs
• Legal costs, costs of reorganization
• Changes in production
• discontinuous changes in production decisions
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Bankruptcy/Liquidity Costs:
b(v) bankruptcy costs = bi if vi < vi
0 otherwise
v = A (Dp – b(v))
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Example
• Each organization starts with an asset worth pi
• Bankrupt if vi drops below 50, incurs cost of 50
v=A(Dp-b(v)) = 2/3 1/3
1/3 2/3
p1 - b1(v)
p2 - b2(v)
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Equilibria
• The equilibria form a complete lattice
• We focus on the unique ``best-case’’ equilibrium where the fewest organizations fail
• Easy Algorithm to find it:
• Identify organizations that fail even if no others do
• Identify those that fail due to the failures identified above
• Iterate...
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Outline
• Model
• Cascades: Diversification and Integration
• Endogenous Values and Moral Hazard
• An Illustration with European Debt Data
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Three Necessary Components of a Cascade:
• A first failure: some organization needs to fail
• Initial Contagion: some neighbors need to be sufficiently exposed to fail too
• Interconnection: to continue to cascade, the network must have sufficiently large components
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What Affects Cascades:
• Diversification: How many other organizations does a typical organization cross hold?
• Integration: How much of a typical organization is cross held?
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Diversification/Integration
• n=100 organizations
• Random network g with Pr( gij = 1 ) = d/(n-1)
• d = expected # other organizations that an organization cross holds (d = level of diversification)
• Fraction c of org cross-held (evenly split among those holding it), 1-c held privately (c = level of integration)
• So, Cij = cgij/dj Ĉii = 1-c
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Diversification/Integration
• One asset per organization (their investments), starts at value 1
• Pick one asset to devalue to 0
• Threshold is vi = θvi for all i
• Look at resulting cascade
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Diversification Preview: Dangerous Middle Levels
• Low diversification:
• fragmented network, no widespread contagion
• Medium diversification
• Connected network, contagion is possible
• Exposure to only a few others makes it easy to spread
• High diversification
• Little exposure to any single other organization
• Failures do not spread
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Diversification and Contagion: 93% threshold, c=.5
Percent of
Orgs
that Fail
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59
theta = .93
Degree: Expected # of cross-holdings
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0
20
40
60
80
100
120
0.3 1
1.7
2.3 3
3.7
4.3 5
5.7
6.3 7
7.7
8.3 9
9.7
10
.3 11
11
.7
12
.3 13
13
.7
14
.3 15
15
.7
16
.3 17
17
.7
18
.3 19
19
.7
Theta = .99
theta = .96
theta = .93
theta = .90
theta = .87
theta = .84
Percent of
Orgs
that Fail
Degree: Expected # of cross-holdings
Diversification and Contagion: various thresholds
.99
.96
.93
.90
.87
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Proposition 1: Diversification
Consider a regular random network where organizations have in and out degree (an integer adjacent to) d, and common threshold v and integration c, and asset values 1. Drop an asset to 0.
If c(1-c) < 1-v , then there is no contagion.
Otherwise,
If d < 1, then there is no limit contagion.
If 1 ≤ d < c(1-c)/(1-v), there is limit contagion.
If c/(1-v) < d, then there is no limit contagion.
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Diversification
• nonmonotonicity: middle ranges of connections maximize contagion
• Competing forces:
• Increased diversification increases component size
• Increased diversification decreases spread from one organization to neighbor
• Degree that maximizes contagion is increasing with threshold
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Integration
• Low integration: little exposure to others, failures don’t trigger others
• Middle integration: exposure to others substantial enough to trigger contagion
• High integration: difficult to get a first failure – failure of own assets does not trigger failure
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Integration: .93 threshold
0
20
40
60
80
100
120
0.3 1
1.7
2.3 3
3.7
4.3 5
5.7
6.3 7
7.7
8.3 9
9.7
10
.3 11
11
.7
12
.3 13
13
.7
14
.3 15
15
.7
16
.3 17
17
.7
18
.3 19
19
.7
c = .5
c = .4
c = .3
c = .2
c = .1
Percent of
Orgs
that Fail
Degree: Expected # of cross-holdings
.5
.4
. 3
.2
.1
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Integration: .93 threshold
0
10
20
30
40
50
60
70
80
0.3 1
1.7
2.3 3
3.7
4.3 5
5.7
6.3 7
7.7
8.3 9
9.7
10.3 11
11.7
12.3 13
13.7
14.3 15
15.7
16.3 17
17.7
18.3 19
19.7
c = .9
c = .7
c = .5
c = .3
c = .1
Percent of
Orgs
that Fail
Degree: Expected # of cross-holdings
.5 .7
.3
.9
.1
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High Integration - First Failures, threshold .8
0
20
40
60
80
100
120
0.3 1
1.7
2.3 3
3.7
4.3 5
5.7
6.3 7
7.7
8.3 9
9.7
10
.3 11
11
.7
12
.3 13
13
.7
14
.3 15
15
.7
16
.3 17
17
.7
18
.3 19
19
.7
c = .4
c = .6
c = .7
c = .8
c = .9
Frequency of first failures
Degree: Expected # of cross-holdings
.4 .6
. 7
.8
.9
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Integration
• Increases exposure to others
• Decreases exposure to own idiosyncratic risks
• Increases contagion, but can decrease first failures
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Proposition 3: Integration
Conditional upon a first failure, trades at `fair’ prices (determined by values relative to initial p’s) that weakly increase Aij for all i and j (i.e., integrations) weakly increase the number of organizations that fail in any cascade. Integration that decreases Aii decreases circumstances of own-asset induced first failures.
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Proposition 3: Integration
This is a general result - intuition: increasing integration increases the number of organizations exposed to bankruptcy costs of failing organizations trades at fair prices do not change any of their initial market values
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Summary on Cascades: Integration and Diversification:
• A first failure: some organization needs to fail
own risk with integration
• Initial Contagion: some neighbors need to be sufficiently exposed to fail too
exposure w integration exposure w diversification
• Interconnection: to continue to cascade widely, the network must have sufficiently large components
connectedness w diversification
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Outline
• Model
• Cascades: Diversification and Integration
• Endogenous Values and Moral Hazard (skip)
• An Illustration with European Debt Data
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Illustrating Application
• Consider 6 key countries in Europe that have substantial cross holdings of each other’s debt
• Treat them as an isolated system (illustrating exercise, not for policy...)
• See what happens if one of them fails
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Raw Cross Holdings of National Debt in Millions of $
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Derived Exposures:
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.18
.12
.13
.11
.11
.05
Greece
France Germany
Portugal
Italy Spain
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Cascades
• Set vi to be a fraction θ of 2008 GDP
• Look at 2011 GDP as initial pi
• Calculate vi
• Calculate cascades in best equilibrium
• Have bi = vi / 2 (could rescale everything to debt levels – here set to GDP levels)
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Normalized GDPs
2008 2011 Drop %
France 11.99 11.62 3
Germany 15.28 14.88 3
Greece 1.47 1.27 14
Italy 9.65 9.20 5
Portugal 1.06 1.00 6
Spain 6.70 6.25 7
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θ fraction .90 .93 .935 .94
First Failure
Greece Greece
Greece
Greece Portugal
Second Failure
Portugal Spain
Third Failure
Spain France
Fourth Failure
France Germany
Germany Italy
Fifth Failure
Italy
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European Debt
• Portugal fragile: little exposure, but close to threshold
• Portugal triggers Spain, triggers France, Germany
• Italy is last to cascade: held by others, but much less exposed to Spain than France, Germany (but exposed to Fr, G)
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Conclusions
• Values need to be derived from cross holdings carefully
• Diversification and Integration both face (different) competing effects, nonmonotonicities
• Model can serve as a foundation for studying bailouts and incentives...
• Can be taken to data...
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Summary on Cascades: Integration and Diversification:
• A first failure: some organization needs to fail
own risk with integration
• Initial Contagion: some neighbors need to be sufficiently exposed to fail too
exposure w integration exposure w diversification
• Interconnection: to continue to cascade widely, the network must have sufficiently large components
connectedness w diversification
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Extra Slides
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Outline
• Model
• Cascades: Diversification and Integration
• Endogenous Values and Moral Hazard
• An Illustration with European Debt Data
![Page 60: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/60.jpg)
Endogenous Values:
• Let p1=p2=10 (D=I) and so v1=v2=10
• What if v1=8 and v2=11 and b2 = 6 ?
• Without any intervention, 2 loses 6
0 . 5
.5 0 C = A = Ĉ (I-C)-1 =
2/3 1/3
1/3 2/3
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Bailout:
• Without any intervention, incur a cost 6 and so the values become: v1=8 and v2= 6
(2 bears 2/3 of the cost and 1 bears 1/3)
0 . 5
.5 0 C = A = Ĉ (I-C)-1 =
2/3 1/3
1/3 2/3
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Bailout:
• Without any intervention, incur a cost 6 and so the values become: v1=8 and v2= 6
(2 bears 2/3 of the cost and 1 bears 1/3)
• If instead 1 gave a $ to 2, then values are
v1=9 and v2= 11 !
0 . 5
.5 0 C = A = Ĉ (I-C)-1 =
2/3 1/3
1/3 2/3
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Moral Hazard
• Suppose that 2 can choose:
b2 either 2 or 6
v2 either 10 or 11
Only gets a transfer if b2 = 6 and v2 = 11
Best off by choosing highest bankruptcy cost and threshold
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Bailouts: No free lunch Proposition 4
Suppose at prices p, org i is closest to first failure and would just fail at prices λp.
For any fair trades of assets or cross holdings that change Ai , there exists some p’ in any neighborhood of λp such that i now fails at p’ but did not before.
So, avoiding i’s failure will require unfair trades if it covers any neighborhood of prices.
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.
• L
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Integration: .96 threshold
0
20
40
60
80
100
120
0.3 1
1.7
2.3 3
3.7
4.3 5
5.7
6.3 7
7.7
8.3 9
9.7
10
.3 11
11
.7
12
.3 13
13
.7
14
.3 15
15
.7
16
.3 17
17
.7
18
.3 19
19
.7
c = .5
c = .4
c = .3
c = .2
c = .1
Percent of
Orgs
that Fail
Degree: Expected # of cross-holdings
.5 .4 . 3
.2
.1
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Proposition 2: Integration
Suppose that 1-c of every organization is owned
privately for some c ≤ 1/2. Then:
Aii is decreasing in c, and
Aij is nondecreasing in c, and increasing if i
and j are path connected
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p2
p1 50
50
100 0
100 FF1
``First Failure
Frontier of
organization 1’’
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p2
p1 50
50
100 0
100 FF1
FF2
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p2
p1 50
50
100 0
100 FF1
FF2
First Failure
Frontier
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p2
p1 50
50
100 0
100 SF1
``Second Failure
Frontier’’:
If org 2 fails, 1
loses 50/3.
More fragile
FF1
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p2
p1 50
50
100 0
100
SF2
SF1 FF1
FF2
Prices for which
cascades occur
due to
organizations’
interdependencies
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p2
p1 50
50
100 0
100
SF2
SF1 FF1
FF2
Cascades in the
best case
equilibrium
1 and 2 fail
for sure
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p2
p1 50
50
100 0
100
SF2
SF1
Multiple Equilibria
FF1
FF2
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Example
• Two organizations n=2
• Each own half of each other
How much of j’s asset value
`belongs’ to investors of i
0 . 5
.5 0 C =
.5 0
0 .5 Ĉ = A = Ĉ (I-C)-1 =
2/3 1/3
1/3 2/3
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Multiple Equilibria
v = A (Dp – b(v))
Multiple solutions: multiple equilibria
![Page 77: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/77.jpg)
Outline
• Model
• Cascades: Diversification and Integration
• Endogenous Values and Moral Hazard
• An Illustration with European Debt Data
![Page 78: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/78.jpg)
Discontinuities:
• Costs need not be large:
• Small changes in one organization's value can trigger discontinuities in others’ values
• Cascades occur can occur with small changes in values, when margins are low
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Cascades
• Let us now examine the equilibria
• Examine when it is that failure of one organization leads to a cascade
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Example
.18 .13 .15
.77 .83 .66
.05 .04 .19
A = Ĉ (I-C)-1 =
0 .75 .75
.85 0 .10
.10 0 0
C =
.05 0 0
0 .25 0
0 0 .15
Ĉ =
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Example
• Two organizations n=2
• Each own half of each other 0 . 5
.5 0 C =
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Example
1
2 3
Holdings of 1
by 2: C21 = .85
Ĉ11 = .05 private investors
C31= .10
![Page 83: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/83.jpg)
Example
1
2 3
.75
.85
.75
.10
.10
.05
.25 .15
C + Ĉ
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Example
1
2 3
.75
.85
.75
.10
.10 .25 .15
1 dollar flows into 1 .05
.05
.10 .85
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Example
1
2 3
.75
.85
.75
.10
.10 .25 .15
Follow the .85 from 2 .05
.05
.10 .85
.6375
.2125
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Example
1
2 3
.77 .05
.66
.04
Iterating:
A=Ĉ (I-C)-1 A11= .18
A12= .13 A13= .15
.19 .83
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Example
1
2 3
.13
.77
.15
.05
.66
.46 = v1
2.26=v2
.28=v3
.04
v=Ĉ (I-C)-1 Dp
unit assets
![Page 88: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/88.jpg)
p1 > v1
1
survives
1 fails
p1 < v1
FF1
p2
p1 50
50
100 0
100
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p2 < v2
2 fails
p2 > v2
2
survives FF2
p2
p1 50
50
100 0
100
![Page 90: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/90.jpg)
FF1
p2
p1 50
50
100 0
100
FF2
First Failure
Frontier
![Page 91: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/91.jpg)
FF1‘=FF
1
p2
p1 50
50
100 0
100
FF2‘=
FF2 1 and 2 fail
for sure
No cascades and
no multiple
equilibria
![Page 92: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/92.jpg)
p2
p1 50
50
100 0
100
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p2
p1 50
50
100 0
100
![Page 94: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/94.jpg)
FF1
p2
p1 50
50
100 0
100
FF2
1 and 2 fail
for sure
![Page 95: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/95.jpg)
p2
p1
FF1
50
50
75 25
75
FF2
FF’2
FF’1
Cascades in the
best case
equilibrium
1 and 2 fail
for sure
![Page 96: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/96.jpg)
p2
p1
FF1
50
50
75 25
75
FF2
FF’2
FF’1
1 and 2 fail
for sure
Cascades in the
best case
equilibrium
![Page 97: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/97.jpg)
1 and 2 fail
for sure
Multiple
equilibria
p2
p1
After Trade
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p2
p1 0
FF1
FF2
Current
factor values
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p2
p1 0
FF1
FF2
Current
factor values
FF’1
FF’2
![Page 100: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/100.jpg)
p2
p1
A11 p1 + A12 p2 > v1
1
survives
1 fails
A11 p1 + A12 p2 < v1
FF1
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p2
p1
A21 p1 + A22 p2 < v2
2 fails
A21 p1 + A22 p2 > v2
2
survives
FF2
![Page 102: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/102.jpg)
p2
p1
First Failure
Frontier
FF2
FF1
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p2
p1
First Failure
Frontier
FF2
FF1
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p2
p1
First Failure
Frontier
FF3
FF1
FF2
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p2
p1
FF1
FF2
FF’1
FF’2
![Page 106: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/106.jpg)
p2
p1
FF3
FF’1
FF’’1
FF’3
![Page 107: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/107.jpg)
p2
p1
FF’3 FF’’3
FF’2
FF’’2
![Page 108: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/108.jpg)
p2
p1
FF’’2
FF’’3
FF’’1
New First
Failure Frontier
![Page 109: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/109.jpg)
p2
p1
FFFnew
FFFold
![Page 110: Financial Networks and Contagion · Our Contributions: • Develop network measures related to cascades • Illuminate and distinguish the effects of diversification and integration](https://reader030.vdocuments.us/reader030/viewer/2022040720/5e2bdfd59087857176140217/html5/thumbnails/110.jpg)
p2
p1
FF1’ FF1
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p2
p1
FF1’
FF2’
FF2
FF1
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p2
p1
FF1’
FF2’
FF2
FF1
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First Failure
Frontier
p2
p1
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New First Failure
Frontier
Old First Failure
Frontier
p2
p1
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p2
p1
1 and 2 fail
for sure
Multiple equilibria
Before Trade
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p2
p1
FF1’ FF1
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p2
p1
FF1’
FF2’
FF2
FF1
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p2
p1
FF1’
FF2’
FF1 FF1’|2 fails
FF1’|1 fails
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1 and 2 fail
for sure
Multiple
equilibria
p2
p1
After Trade
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1 and 2 fail
for sure
Multiple
equilibria
p2
p1
After Trade
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p2
p1
Old Multiple
Failure
Regions
New Multiple Failure Regions
Comparison
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A|B
B
A Price tomorrow with
probability 0.25
Price Today
Key p2
p1
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B
A
Trades at fair prices cannot save B
from bankruptcy with the bad price
realization.
p2
p1
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B
A’
A
B’
But A may be prepared to
engage
in trade favorable to B.
p2
p1
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A’
B’
A’|B
After this trade B looks fairly safe
again. p2
p1
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A’|B
A’
B’
However, if the bad price realization
occurs, we are back in the same situation
as before.
p2
p1
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A’ B’ B’’ A’’
Suppose A bails out B again. p2
p1
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A’’|B
B’’ A’’
B then looks fairly safe again. p2
p1
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A’’|B
B’’ A’’
But if the bad price realization
occurs, both organizations are now
at risk.
p2
p1
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A|B
B
A
In the counterfactual of no trade, B would
have gone bankrupt but A would be safe p2
p1
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p2
p1
First Failure
Frontier
FF3
FF1
FF2
Current
factor values
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p2
p1 FF2
FFFold
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p2
p1
FF1’
FF3
FF1
FF3’
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p2
p1
FFFold FF1’
FF3’ FF2
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A = Ĉ (I-C)-1 =
0 . 5
.5 0 C = .5 0
0 .5 Ĉ =
2/3 1/3
1/3 2/3
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.
• L
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.
• L
1/3 1/3 1/3
1/2 1/2 0
1/2 0 1/2
T =