Systemic risk with central clearing party

Cagin Ararat, Fatena El-Masri, Qi Feng, Xuwei Yang

MRC on Financial Mathematics

Snowbird, Utah

June 19, 2015

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Overview

1 Introduction

2 Framework

3 Computational Study

4 Future work

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Introduction

Central clearing

Over-the-counter derivatives market

2010 - Dodd-Frank Wall Street Reform Act:Regulate Credit Cards, Loans and Mortgages, Oversee Wall Street,Stop Banks from Gambling with Depositors’ Money, Regulate RiskyDerivatives, Bring Hedge Funds Trades into the Light, Oversee CreditRating Agencies, Increase Supervision of Insurance Companies, andReform the Federal Reserve

Intermediation by a central clearing party (CCP)

Prime responsibility of CCP is to provide e�ciency and stability to thefinancial markets that they operate in.

Two main processes that are carried out by CCPs: clearing andsettlement of market transactions.

Clearing relates to identifying the obligations of both parties on eitherside of a transaction.Settlement occurs when the final transfer of securities and funds occur.

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Introduction

Central clearing

“Without CCP”:

f

a

b

c

d

e

“With CCP”:

CCP

a

b

c

d

e

f

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Introduction

Central clearing

“Without CCP”:

f

a

b

c

d

e

“With CCP”:

CCP

a

b

c

d

e

f

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Introduction

CCP - systemic risk

How does CCP a↵ect systemic risk?

Amini, Filipovic, Minca ’14Aggregation mechanism insensitive to capital levelsScalar coherent risk measureSu�cient condition ensuring reduction in systemic risk

TodayAggregation mechanism sensitive to capital levelsSet-valued risk measure defined via the aggregation function and ascalar coherent risk measureVarying allocation levels for the CCPNumerical comparison of set-valued risks

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Framework

Network without the CCP

Variant of Eisenberg, Noe ’01

�i : the interbank assets holded by the bank i ,i = 1, · · · ,m.

Lij : nominal interbank liabilities. cash-amount bank i owes bank j .

Li =Pm

j=1

Lij : total nominal liabilities of bank i .

⇧ij =

(Lij/Li , ifLi > 0

0, otherwise.

Qi : foundamental value of external asset, Pi liquidation value of Qi

At time t = 1,

Assets: �i +Pm

j=1

Lji + Qi

Liabilities: Li

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Framework

Network without CCP

The clearing vector of payments L⇤ = (L⇤1

, · · · , L⇤m) is the fixed point of�(L⇤) = L

⇤, where

�(`)i = Li ^

0

@�i +mX

j=1

`j⇧ji + Pi

1

A , i = 1, · · · ,m.

The liquidation fraction of the external asset of bank i is:

Zi =

⇣�i +

Pmj=1

L

⇤j ⇧ji � Li

⌘�

Pi^ 1

The net worth of bank i at time t = 2

Ci = �i + Qi +mX

j=1

L

⇤j ⇧ji � Zi (Qi � Pi )� Li

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Framework

Systemic risk measure before CCP

Capital allocation: k 2 Rm

Bank i starts with �i + ki .

Net values after clearing: C k

Aggregation function A↵(x) = (1� ↵)Pm

i=1

x

+

i � ↵Pm

i=1

x

�i

Aggregate value A↵(C k)

Coherent risk measure: ⇢

Systemic risk measure: R↵(C k) = {k 2 Rm | ⇢(A↵(C k)) 0}

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Framework

E↵ect of central clearing

Same random liability matrix (Lij), external asset (Qi ,Pi )

Net exposure to bank i : ⇤i =Pm

j=1

Lji �Pm

j=1

Lij

All interbank liabilities goes through CCP now.

Capital of CCP: �0

Capital of bank i : �i

Up-front payment of bank i to CCP: gi �i

Proportional fee by CCP: f 2 [0, 1]

Nominal liabilities between CCP and bank i :

i CCP

Li0 = (⇤i + gi )�

L

0i = (1� fi )⇤+

i

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Framework

Systemic risk measure with CCP

Capital allocations: k0

for CCP, k 2 Rm for banks

Compute clearing vector L⇤.

Compute net worths C k . (C := C

0)

Challenge: C k is a nonlinear function of k .Sensitive to capital levels!

Aggregation function: A↵(x) = (1� ↵)Pm

i=0

x

+

i � ↵Pm

i=0

x

�i

Aggregate value: A↵(C k)

Systemic risk measure in m + 1 dimensions:R↵(C ) = {(k

0

, k) 2 Rm+1 | ⇢(A↵(C k)) 0}Systemic risk measure for fixed k

0

:R↵,k

0

(C ) = {k 2 Rm | ⇢(A↵(C k)) 0}

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Framework

Does CCP reduce systemic risk?

Without CCP: R↵(C ) ✓ Rm

With CCP, fixed k

0

: R↵,k0

(C ) ✓ Rm

Nice case: CCP reduces systemic risk in every direction, i.e.,R↵(C ) ✓ R↵,k

0

(C )

For large k

0

, this is the case.

Minimal cash requirement for CCP:k

0

= inf{k0

� 0 | R↵(C ) ✓ R↵,k0

(C )}What happens when k k

0

?

CCP reduces risk w.r.t. some directions w 2 Rm+

\ {0}:

infk2R↵,k

0

(

ˆC)

hk ,wi infk2R↵(C)

hk ,wi

Region for such w : yet to be studied.

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Computational Study

Numerical example (Amini, Filipovic, Minca ’14)

Credit default swaps (CDS) market

(Xij ,Xji ) bivariate Gaussian with correlation: Monte Carlo simulation

Liabilities Lij = (|Xij |� |Xji |)+, Lji = (|Xji |� |Xij |)+

m = 40 banksGrouping: 20 small banks, 20 large banksSame capital allocation within each group

Scalar risk measure: ⇢(Z ) = E[�Z ]

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Computational Study

Numerical example

Brown: High risk, Blue: Low risk

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Computational Study

Numerical example

CCP with k

0

= 0 CCP with k

0

= 10

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Future work

Su�cient condition for reduction in systemic risk

Fixed k

0

� 0.

When do we have R↵(C ) ✓ R↵,k0

(C )?

Su�cient: ⇢(A↵(C k)� A↵,k0

(C k)) 0 for every k 2 Rm.

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Future work

Su�cient condition for reduction in systemic risk

By the definition of the fixed point, we have the following

`i = Li ^ (�i + Pi +mX

i=1

`j⇡ji )

then

⇢(�`i ) ⇢(�Li ) ^ ⇢(��i � Pi �mX

i=1

`j⇡ji )

⇢(�`i ) ⇢(�Li ) ^

2

4�i⇢(�1) + ⇢(�Pi ) +mX

j=i

⇢(�`j)

3

5

now we have a system of inequalities, ⇢(`) ⇢(L) ^ Bm⇥m Here weassume ⇢ is coherent.

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Future work

Upper bound for the scalar risk measures

⇢(A↵(Ck)) ↵

mX

i=1

⇢(Li )+↵mX

i=1

⇢(�L

⇤i )+(1�↵)

mX

i=1

((�i+ki )⇢(�1)+⇢(�Qi ))

+(1� ↵)mX

i=1

⇢(ZiQi � ZiPi )��� without CCP

⇢(�A↵,k0

(C k)) ↵mX

i=0

⇢⇣�(Li0 � Pi � �i + gi )

+

⌘+(1�↵)

mX

i=0

((�i + ki )⇢(1))

+mX

i=0

⇢(Qi ) + (1� ↵)mX

i=0

⇢(Zi (Pi � Qi ))��� with CCP

Combining the above, we have

⇢(A↵(Ck)� A↵,k

0

(C k)) ⇢(A↵(Ck)) + ⇢(�A↵,k

0

(C k)) (1)

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Future work

Su�cient condition for reduction in systemic risk

Equation (1) implies that

⇢(A↵(Ck)� A↵,k

0

(C k)) h(k), (2)

where the h(k) depends on the k and the parameters. If h(k) 0 forevery k , then

R↵,k0

(C ) ◆ R↵(C ).

To do: Study h(k).Generalization: by considering a di↵erent aggregation function, and alsothe risk measure function ⇢ .

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Future work

Thank you.

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