weak & strong systemic fragility

Weak & Strong Systemic Fragility

Casper G. de Vries

Erasmus University Rotterdam

& Tinbergen Institute

Contents

• Current Credit Crisis• Fat Tails• Multivariate Fat Tails• Systemic Risk Measure• Bank Networks• Hedge Fund Application• Estimation• Conclusion

Banks & Systemic Risk

A Reality Check

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

Two Crucial Crisis Features

• I/ Asymmetric Information

• II/ Interconnectedness of Banks and other Vehicles

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

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

II/ Bank Network System

Banks are highly interconnected:directly• Syndicated Loans• Conduits• Interbank Money Marketindirectly• Macro interest rate risk• Macro gdp risk

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

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

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)

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?

Fat Tails

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

Multivariate Fat Tails

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

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

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{

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

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

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

Systemic Risk Measure

Systemic Risk Measure

• Like marginal risk measure VaR

• Desire a scale for measuring the potential Systemic Risk

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?

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

x

y

-10 0 10 20

-15

-10

-50

51

0

Student-t (3 df.) independent asset returns

• Note Plus Shape + is due to the Outliers

• Under normal, just get a Circular cloud

• 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

x - y

y +

x

-4 -2 0 2 4

-4-2

02

4Normal independent portfolio returns

x + y

x -

y

-10 0 10 20

-10

01

02

0Student-t (3 df.) non-correlated portfolio returns

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

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

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

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

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

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

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

Bank Network System

• Syndicated Loans

• Conduits

• Interbank Money Market

• 4 Banks with 4 Projects

• Each Project Divisible into 4 Parts

Bank Networks

autarky Wheel 1 Wheel 2

Star Full DiversificationCycle

Banks are circles. Arrows indicate transfer of –part of- project, loan etc.

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

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

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

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)

Banks & Hedge Funds

application

Cross plot Insurers versus Banks

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

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

x + 2 * y

x -

2 *

y

-10 0 10 20

-10

01

0Student-t (3 df.) long versus dedicated short bias

2 * y + x

x -

2 *

y

-6 -4 -2 0 2 4 6

-50

5

Normal long versus dedicated short bias

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

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

Cross plot Hedge Funds vs. Banks

Cross plot Banks vs. HFR Equity Market Neutral

Cross plot Banks vs. HFR Fixed Income High Yield Index

CAPM Explanation

Hedge Funds

Banks11 bRB 22 bRB

2121 2 bRBB

2121 BB

Corr(B1+B2,B1-B2)=0

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

Multivariate Estimation

Count Measure

1 + Pr{Min>s} / Pr{Max>s} = 1 +

Pr{Max>s}

Pr{Min>s}

rr

0 100 200 300 400 500

0.0

0.2

0.4

0.6

Correlated Normal

Correlated Student-trr

0 100 200 300 400 500

0.0

0.2

0.4

0.6

Bank Data: ABNAMRO & INGrr

0 100 200 300 400 500

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Interpretation

1 in 3 times one bank ‘fails’, the other bank fails as well

Estimated failure measureBanks and Insurers (bivariate normal)

0.01330.0063Insurer

0.00630.0082Bank

InsurerBank

Mean

0.01330.0063Insurer

0.00630.0082Bank

InsurerBank

Mean

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

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

Thank You, Until the next Crisis!

Check Eurointelligence, EMUMonitor

Technical Appendix

Fat Tail versus Normal

Some further results

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

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

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

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

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

Fat Tail Details

a

ssYXMinP

12}],[{

a

ssYXMaxP 2}],[{

11

1}{

}{1

a

a

sMaxP

sMinP

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