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Systemic risk with central clearing party
Cagin Ararat, Fatena El-Masri, Qi Feng, Xuwei Yang
MRC on Financial Mathematics
Snowbird, Utah
June 19, 2015
Systemic risk with CCP MRC June 19, 2015 1 / 19
Overview
1 Introduction
2 Framework
3 Computational Study
4 Future work
Systemic risk with CCP MRC June 19, 2015 2 / 19
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.
Systemic risk with CCP MRC June 19, 2015 3 / 19
Introduction
Central clearing
“Without CCP”:
f
a
b
c
d
e
“With CCP”:
CCP
a
b
c
d
e
f
Systemic risk with CCP MRC June 19, 2015 4 / 19
Introduction
Central clearing
“Without CCP”:
f
a
b
c
d
e
“With CCP”:
CCP
a
b
c
d
e
f
Systemic risk with CCP MRC June 19, 2015 4 / 19
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
Systemic risk with CCP MRC June 19, 2015 5 / 19
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
Systemic risk with CCP MRC June 19, 2015 6 / 19
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
Systemic risk with CCP MRC June 19, 2015 7 / 19
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}
Systemic risk with CCP MRC June 19, 2015 8 / 19
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
Systemic risk with CCP MRC June 19, 2015 9 / 19
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}
Systemic risk with CCP MRC June 19, 2015 10 / 19
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.
Systemic risk with CCP MRC June 19, 2015 11 / 19
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 ]
Systemic risk with CCP MRC June 19, 2015 12 / 19
Computational Study
Numerical example
Brown: High risk, Blue: Low risk
Systemic risk with CCP MRC June 19, 2015 13 / 19
Computational Study
Numerical example
CCP with k
0
= 0 CCP with k
0
= 10
Systemic risk with CCP MRC June 19, 2015 14 / 19
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.
Systemic risk with CCP MRC June 19, 2015 15 / 19
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.
Systemic risk with CCP MRC June 19, 2015 16 / 19
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)
Systemic risk with CCP MRC June 19, 2015 17 / 19
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 ⇢ .
Systemic risk with CCP MRC June 19, 2015 18 / 19
Future work
Thank you.
Systemic risk with CCP MRC June 19, 2015 19 / 19