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Weak & Strong Systemic Fragility
Casper G. de Vries
Erasmus University Rotterdam
& Tinbergen Institute
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Contents
• Current Credit Crisis• Fat Tails• Multivariate Fat Tails• Systemic Risk Measure• Bank Networks• Hedge Fund Application• Estimation• Conclusion
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Banks & Systemic Risk
A Reality Check
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Current Crisis Aspects
• History: Savings glut & Low interest produced Credit binge big time
• Basel I stimulates Conduits formation to shift mortgages off-balance & repackaging of risk
• Interest rate tightening triggers mortgage failures; Basel II 2008 start brings on-balance fears
• Asymmetric information about conduits, non-market over the counter instruments, turns money market into market for lemons
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Two Crucial Crisis Features
• I/ Asymmetric Information
• II/ Interconnectedness of Banks and other Vehicles
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I/ The Asymmetric Information Problem
• The Old Lady meets the Old Maid, or …
Payoff if win: W
Payoff if loose: -L
Willingness to play if not dealt the old maid if: 3/4W-1/4L>0
Probability to win: 3/4
Probability to loose: 1/4
Thus play if: W/L>1/3
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The Old Lady meets the Old Maid
Payoff if win: W
Payoff if loose: -L
Willingness to play if not dealt the old maid if: 3/4W-1/4L>0
Probability to win: 3/4
Probability to loose: 1/4
Subprime woes lead to risk reassessment such that L increased and: W/L<1/3
autarky
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II/ Bank Network System
Banks are highly interconnected:directly• Syndicated Loans• Conduits• Interbank Money Marketindirectly• Macro interest rate risk• Macro gdp risk
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daily returns 1991-2003
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
ING
AB
NA
MR
O
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simulated normal returns
-0.2
0
0.2
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
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Fat versus Normal:2 Features
• Univariate: More than normal outliers along the axes
• Multivariate: Extremes occur jointly along the diagonal, systemic risk is of higher order than if normal (market speak: market stress increases correlation)
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Basel Motivation
• Systemic Risk of banks is important due to the externality to the entire economy
• Motive for Basle Accords & why banks are stronger regulated than insurers (Solvency)
• Surprise is micro orientation of Basle II, rather than macro approach
• Improper information to supervisor; overregulation or too little?
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Fat Tails
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Normal versus Fat Tail
2/2
2
11}Pr{ ses
sY
ssX }Pr{
01
2/2
s
e s
Normal
Fat
Ratios
2/
1
2se
s
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Multivariate Fat Tails
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Normal
Normal
Sum / Fractal Nature (square root rule)
with n
Rate Declines with Sum
2/2
2
11}Pr{ se
ssY
nsn
ii e
s
nsY 2/
1
2
2
1}Pr{
n
s
ndn
ssnd
ndsYdn
ii 22
1
log
]2
2logloglog2
1[
log/}Pr{log2
2
1
4/21
2
2
12}2
Pr{}2Pr{}Pr{ ses
sYsYsYY
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Portfolio X+Y composed of indepedent normal returns X and YPortfolio failure probability is of lower order than marginal failure probabilities
X+Y
2s
2s
s
s)exp(
2
1
2
1}2Pr{ 2s
ssYX
portfolio failure probability
)2
1exp(
2
11}Pr{ 2ss
sX
marginal failure probability
Of Larger Order Than
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Feller Theorem
ssXP 1}{ ssYP 1}{
ssssssYsXP 2121)1)(1(},{ 2
Consider two independent Pareto distributed random variables X and Y
Their joint probability is
ssX }Pr{
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Fat Tail
Pareto
Sum / Fractal Nature
with n
Rate Independent of Summation
ssX }Pr{
ssXX 2}Pr{ 21
nssXn
ii }Pr{
1
1log
]log[loglog/}Pr{log
1
nd
sndndsXd
n
ii
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Portfolio X+Y composed of independent fat tailed returns X and YMarginal and portfolio failure probabilities are of the same order
X+Y
2s
2s
s
s
ssYX 12}2Pr{
Marginal failure probability
ssX }Pr{
Portfolio failure probability
Of Equal Order as
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Conclude
• With linear dependence, as between portfolios and balance sheets, the probability of a joint failure is:
• Of smaller order than the individual failure probabilities in case of the normal, hence systemic risk is unimportant
• Of the same order in case of fat tails, hence systemic risk is important
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Systemic Risk Measure
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Systemic Risk Measure
• Like marginal risk measure VaR
• Desire a scale for measuring the potential Systemic Risk
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1 + Pr{Min>s} / Pr{Max>s} = 1 +
Pr{Max>s}
Pr{Min>s}
Given that there is a bank failing, what is the probability the other bank fails as well?
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Answers Differ Radically
• Normal• Zero
• Fat Tails• Positive
Question: If bank exposures are linear in the risk factors, andbanks have some of these factors in common, then what is theexpected number of failures given that there is a failure?
Note: to see something under normality, we need a finer risk measure like the Trace of the covariance matrix / the Tawn measure
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x
y
-10 0 10 20
-15
-10
-50
51
0
Student-t (3 df.) independent asset returns
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• Note Plus Shape + is due to the Outliers
• Under normal, just get a Circular cloud
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• Consider Bank versus Market Neutral Hedge Fund
• Bank is Long in both X and Y
• Bank portfolio return X+Y
• Hedge fund is long in X, short in Y
• Hedge fund portfolio return X-Y
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x - y
y +
x
-4 -2 0 2 4
-4-2
02
4Normal independent portfolio returns
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x + y
x -
y
-10 0 10 20
-10
01
02
0Student-t (3 df.) non-correlated portfolio returns
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Systemic Risk Measure
• Do not know where systemic failure sets in• Take limits• Evaluate in limit and extrapolate back• Construct Multivariate VaR, in terms of
Failure Probability, rather than loss quantile
• Conditional Failure Measure:• Given that one bank fails, Probability the
other banks fail
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1 + Pr{Min>s} / Pr{Max>s} = 1 +
Pr{Max>s}
Pr{Min>s}
X
X
Y
Y
Systemic Risk Measure:
}),(Pr{
}),(Pr{1
},Pr{1
}Pr{}Pr{
sYXMax
sYXMin
sYsX
sYsX
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1 + Pr{Min>s} / Pr{Max>s} = 1 +
Pr{Max>s}
Pr{Min>s}
= 1+
Normal Case
= 1
R
Q
R
Q
aR+(1-a)Q
aR+(1-a)Q
(1-a)R+aQ
(1-a)R+aQ
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Ledford-Tawn measure
Need fine measure in case of normality since
Use instead
1}),(Pr{
}),(Pr{1
},Pr{1
}Pr{}Pr{
sYXMax
sYXMin
sYsX
sYsX
},Pr{log
}Pr{log}Pr{loglim2
1
sYsX
sYsXs
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Portfolios composed of indepedent returns R and Q
aR+(1-a)Q
(1-a)R+aQ
s/a
s/(1-a)
R
Q
(1-a)R+aQ=s
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1 + Pr{Min>s} / Pr{Max>s} = 1 +
Pr{Max>s}
Pr{Min>s}
= 1+
Fat Tail Case
> 1
Q
R
R
Q
aR+(1-a)Q
aR+(1-a)Q
(1-a)R+aQ
(1-a)R+aQ
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Systemic risk with fat tails
Since numerator and denominator are of the same order in case of fat tails, the measure
Is appropriate
}),(Pr{
}),(Pr{1
},Pr{1
}Pr{}Pr{
sYXMax
sYXMin
sYsX
sYsX
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Bank Network System
• Syndicated Loans
• Conduits
• Interbank Money Market
• 4 Banks with 4 Projects
• Each Project Divisible into 4 Parts
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Bank Networks
autarky Wheel 1 Wheel 2
Star Full DiversificationCycle
Banks are circles. Arrows indicate transfer of –part of- project, loan etc.
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Bank Networks
autarky Wheel 1 Wheel 2
Star Full DiversificationCycle
1/4
1/4
1/4
1/4
1/4
1/4
1/4 1/4
1/2
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Systemic Risk, normal
E=1, T=1/4 E=1, T=2/5 E=1, T=1/2
E=1, T=11/20 E=1, T=2/3 E=4, T=1
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Systemic Risk, fat tails, =3
E=1 E=1+1/27 E=1+1
E=1+3/41 E=1+1/4 E=1+3
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Conclude
• Pattern of Systemic Risk under fat tails differs from normal based covariance intuition
• Too much Diversification hurts Systemic Risk (slicing and dicing convexifies the exposures)
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Banks & Hedge Funds
application
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Cross plot Insurers versus Banks
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stock price in %
0
50
100
150
200
250
300
350
400
450
500n
ov-9
4
no
v-9
5
no
v-9
6
no
v-9
7
no
v-9
8
no
v-9
9
no
v-0
0
no
v-0
1
no
v-0
2
no
v-0
3
no
v-0
4
no
v-0
5
no
v-0
6
bank
tremont
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Monthly Returns 1994-2007
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
EU Banks Index
Trem
ont D
edic
ated
Sho
rt B
ias
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x + 2 * y
x -
2 *
y
-10 0 10 20
-10
01
0Student-t (3 df.) long versus dedicated short bias
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2 * y + x
x -
2 *
y
-6 -4 -2 0 2 4 6
-50
5
Normal long versus dedicated short bias
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Monthly Returns 1994-2007
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
EU Banks index
AB
NA
MR
O
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Monthly Returns 1994-2007
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
EU Banks Index
Trem
ont H
edge
Fun
d In
dex
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Cross plot Hedge Funds vs. Banks
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Cross plot Banks vs. HFR Equity Market Neutral
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Cross plot Banks vs. HFR Fixed Income High Yield Index
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CAPM Explanation
Hedge Funds
Banks11 bRB 22 bRB
2121 2 bRBB
2121 BB
Corr(B1+B2,B1-B2)=0
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Banks &Hedge Funds
• Banks are MORE Risky than Hedge Funds
• Little Systemic Risk effects of hedge funds for the Banking Sector
• If anything, hedge funds are grasshoppers wearing the bolder hat that provides the protection
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Multivariate Estimation
Count Measure
![Page 62: Weak & Strong Systemic Fragility Casper G. de Vries Erasmus University Rotterdam & Tinbergen Institute](https://reader035.vdocuments.net/reader035/viewer/2022062515/56649cf85503460f949c99d8/html5/thumbnails/62.jpg)
1 + Pr{Min>s} / Pr{Max>s} = 1 +
Pr{Max>s}
Pr{Min>s}
![Page 63: Weak & Strong Systemic Fragility Casper G. de Vries Erasmus University Rotterdam & Tinbergen Institute](https://reader035.vdocuments.net/reader035/viewer/2022062515/56649cf85503460f949c99d8/html5/thumbnails/63.jpg)
rr
0 100 200 300 400 500
0.0
0.2
0.4
0.6
Correlated Normal
![Page 64: Weak & Strong Systemic Fragility Casper G. de Vries Erasmus University Rotterdam & Tinbergen Institute](https://reader035.vdocuments.net/reader035/viewer/2022062515/56649cf85503460f949c99d8/html5/thumbnails/64.jpg)
Correlated Student-trr
0 100 200 300 400 500
0.0
0.2
0.4
0.6
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Bank Data: ABNAMRO & INGrr
0 100 200 300 400 500
0.0
0.1
0.2
0.3
0.4
0.5
0.6
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Interpretation
1 in 3 times one bank ‘fails’, the other bank fails as well
![Page 67: Weak & Strong Systemic Fragility Casper G. de Vries Erasmus University Rotterdam & Tinbergen Institute](https://reader035.vdocuments.net/reader035/viewer/2022062515/56649cf85503460f949c99d8/html5/thumbnails/67.jpg)
Estimated failure measureBanks and Insurers (bivariate normal)
0.01330.0063Insurer
0.00630.0082Bank
InsurerBank
Mean
0.01330.0063Insurer
0.00630.0082Bank
InsurerBank
Mean
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Estimated failure measureBanks and Insurers across EU
0.1070.0690.11700.0744Insurer
0.0690.0950.07440.1038Bank
InsurerBankInsurerBank
MedianMean
0.1070.0690.11700.0744Insurer
0.0690.0950.07440.1038Bank
InsurerBankInsurerBank
MedianMean
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Four Conclusions
• Asymmetric Information, market trade or OTC
• Linear dependence and normal risk cannot produce systemic risk
• Linear dependence and fat tails imply that systemic risk is always there
• Need for systemic risk scale like Richter scale, in order to impute correct capital requirements and signal potential stress
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Thank You, Until the next Crisis!
Check Eurointelligence, EMUMonitor
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Technical Appendix
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Fat Tail versus Normal
Some further results
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1 + Pr{Min>s} / Pr{Max>s} = 1 +
Pr{Max>s}
Pr{Min>s}
X
X
Y
Y
Systemic Risk Measure:
}),(Pr{
}),(Pr{1
},Pr{1
}Pr{}Pr{
sYXMax
sYXMin
sYsX
sYsX
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Portfolios composed of indepedent returns R and Q
aR+(1-a)Q
(1-a)R+aQ
s/a
s/(1-a)
R
Q
(1-a)R+aQ=s
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1 + Pr{Min>s} / Pr{Max>s} = 1 +
Pr{Max>s}
Pr{Min>s}
= 1+
Fat Tail Case
> 1
Q
R
R
Q
aR+(1-a)Q
aR+(1-a)Q
(1-a)R+aQ
(1-a)R+aQ
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1 + Pr{Min>s} / Pr{Max>s} = 1 +
Pr{Max>s}
Pr{Min>s}
= 1+
Normal Case
= 1
R
Q
R
Q
aR+(1-a)Q
aR+(1-a)Q
(1-a)R+aQ
(1-a)R+aQ
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Normal Details
))1(2
1exp(
2
1)1(})1({})1({
}{}],[{
)exp(2
12/1}2/1{}2{
}2{}],[{
22
22222
2
aa
s
s
aasRaaPsQaaRP
SXPsYXMaxP
ss
sRPsQRP
sYXPsYXMinP
01
1
2/1
1
2
1exp
2
1
}{
}],[{22
22
aa
aa
sXP
sYXMinP
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Fat Tail Details
a
ssYXMinP
12}],[{
a
ssYXMaxP 2}],[{
11
1}{
}{1
a
a
sMaxP
sMinP
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Conduit Runs
• Bank Return X; Market Risk M; Interest Rate Risk R; Idiosyncratic Risk E
• Bank 1:• Bank 2:• If: • Systemic Risk
(expected number of joint failures):
1111 ERMX
2222 ERMX ssEPsRPsMP }{}{}{
2
21}1|{
2121
2121
kkE