Systemic Risk and Central Counterparty Clearing
Damir Filipovi´c
(joint with Hamed Amini and Andreea Minca)
Swiss Finance Institute
Ecole Polytechnique F´
ed´
erale de Lausanne
Systemic Risk: Models and Mechanisms
Cambridge, 28 August 2014
Outline
Financial Network
Central Counterparty Clearing
CCP Impact
Does a CCP reduce systemic risk?
Pareto optimality analysis
2/37
Outline
Financial Network
Central Counterparty Clearing
CCP Impact
Does a CCP reduce systemic risk?
Pareto optimality analysis
Financial Network
3/37
Setup
I
Two periods t = 0, 1, 2
I
Values at t = 1, 2 are random variables on (Ω, F)
I
m interlinked banks [m] := {1, 2, . . . , m}
Financial Network
4/37
Instruments
I
I
Cash: zero return
External asset:
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I
I
Interbank liabilities:
I
I
I
fundamental value q at t = 0 and Q at t = 1, 2
liquidation value P at t = 1, 0 < P < Q
formation at t = 0
realization/expiration at t = 1
No external debt
Financial Network
5/37
Banks’ initial capital at t = 0
I
Bank i has initial capital
ci = γi + yi q > 0
I
γi ≥ 0 units of cash
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yi ≥ 0 units of external asset
Financial Network
6/37
Interbank liabilities realize at t = 1
I
I
I
Lij (ω) cash-amount bank i owes bank j
P
Li = j∈[m] Lij total nominal liabilities of bank i
P
j∈[m] Lji total nominal receivables from other banks
Financial Network
7/37
Bank i’s nominal balance sheet at t = 1
I
Assets
γi +
I
P
j∈[m] Lji
Liabilities
P
Li =
I
+ yi Q
j∈[m] Lij
Cash balance
γi +
Financial Network
P
j∈[m] Lji
− Li
8/37
Liquidation of external asset at t = 1
I
If bank i’s cash balance is negative,
P
γi + j∈[m] Lji < Li ,
it sells external assets at liquidation price P < Q
I
Bank i is bankrupt if
γi +
|
P
j∈[m] Lji
{z
+ yi P < Li ,
}
liquidation value of assets
and then bank j receives a proportion
Πij = Lij /Li
of liquidation value of bank i’s assets
→ interbank liabilities are of equal seniority
Financial Network
9/37
Interbank clearing equilibrium
Interbank clearing equilibrium is attained at the clearing total
liability vector L∗ = (L∗1 , . . . , L∗m ) determined as fixed point
Φ(L∗ ) = L∗
where Φ : [0, L] → [0, L] is given by
P
Φi (`) = Li ∧ γi + j∈[m] `j Πji + yi P , i ∈ [m]
Theorem 1.1 (Eisenberg and Noe (2001)).
There exists a unique interbank clearing equilibrium.
Financial Network
10/37
Bank i’s balance sheet at t = 2
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Units of liquidated external asset
+
P
Li − γi − j∈[m] L∗j Πji
∧ yi
Zi =
P
I
Assets
Ai = γi +
I
∗
j∈[m] Lj Πji
P
Liabilities
Li =
I
+ Zi P + (yi − Zi )Q
P
j∈[m] Lij
Capital
Ci = Ai − Li
Financial Network
11/37
Bankruptcy characterization
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Shortfall of bank i equals
Ci− = Li − L∗i
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Bank i is bankrupt if and only if
Ci < 0
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(or L∗i < Li )
If bank i is bankrupt then all its external assets are liquidated
Zi = yi
Financial Network
12/37
Outline
Financial Network
Central Counterparty Clearing
CCP Impact
Does a CCP reduce systemic risk?
Pareto optimality analysis
Central Counterparty Clearing
13/37
Central Clearing Counterparty (CCP)
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We label the CCP as i = 0
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All liabilities are cleared through the CCP
→ star shaped network
→ interbank clearing equilibrium is trivial (no fixed point
problem)
Central Counterparty Clearing
14/37
Capital structure of CCP
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The CCP is endowed with
I
I
external equity capital γ0
guarantee fund
Pm
i=1
gi
where gi ≤ γi is received from bank i at time t = 0
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Guarantee fund is junior to equity capital
I
Banks’ shares in the guarantee fund have equal seniority
Central Counterparty Clearing
15/37
Liabilities
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Bank i’s net exposure to CCP
Λi =
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Pm
j=1 Lji
−
Pm
Bank i’s nominal liability to the CCP
b
Li0 = Λ−
i − gi
I
j=1 Lij
+
CCP’s nominal liability to bank i
b
L0i = (1 − f )Λ+
i
→ CCP charges a volume based fee f on bank i’s receivables
f × Λ+
i
Central Counterparty Clearing
16/37
Guarantee fund
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Bank i’s nominal share in the guarantee fund:
Gi = (Λi + gi )+ − Λ+
i
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Linking facts:
Gi − b
Li0 = gi − Λ−
i ,
Gi × b
Li0 = 0
Li0 gi -‐gi Gi 0 Λi Figure: Gi and b
Li0 as functions of Λi
Central Counterparty Clearing
17/37
Clearing equilibrium
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bi
External assets liquidated Z
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bi
Terminal net worth of CCP and banks C
Central Counterparty Clearing
18/37
Sensitivity analysis
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bi does not depend on (f , g)
Number of liquidated assets Z
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Shortfall of bank i does not depend on (f , g)
b − = (Λi + yi P + γi )−
C
i
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For all i, j ∈ [m], we have
b+
bi
∂C
∂C
i
=
≤0
∂f
∂f (
b+ ≥ 0
bi
∂C
∂C
i
=
∂gj
∂gj ≤ 0
Central Counterparty Clearing
if i 6= j
if i = j
19/37
Aggregate surplus identity
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Convention: For comparison we set
C0 = γ0
Lemma 2.1.
CCP improves aggregate surplus by lower liquidation losses:
Pm b + Pm
Pm
+
b
i=0 Ci =
i=0 Ci + (Q − P)
i=1 (Zi − Zi ).
RHS does not depend on (f , g).
Central Counterparty Clearing
20/37
Aggregate sensitivity analysis
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Aggregate net worth of financial system is non-decreasing in g
∂
Pm
k=0 Ck
b
∂gi
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b−
∂C
0
≥ 0, for all i ∈ [m]
∂gi
Aggregate net worth of the banks is non-increasing in g
∂
Pm
k=1 Ck
∂gi
I
=−
b
=−
b+
∂C
0
≤ 0, for all i ∈ [m]
∂gi
Same for f
Central Counterparty Clearing
21/37
Outline
Financial Network
Central Counterparty Clearing
CCP Impact
Does a CCP reduce systemic risk?
Pareto optimality analysis
CCP Impact
22/37
Comparison study
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CCP Impact
Compare financial network with and without CCP
b
Write C = (C0 , . . . , Cm ), and similarly C
23/37
CCP state-wise impact
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CCP always reduces
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liquidation losses
bi ≤ Zi
Z
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bank shortfalls (bankruptcy cost)
b− ≤ C−
C
i
i
I
CCP always improves
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aggregate terminal bank net worth
Pm b
Pm
i=1 Ci ≥
i=1 Ci
I
aggregate surplus
Pm b + Pm
Pm
+
b
i=0 Ci =
i=0 Ci + (Q − P)
i=1 (Zi − Zi )
{z
}
|
≥0
I
CCP Impact
b− ≥ 0
CCP imposes shortfall risk C
0
24/37
CCP capital impact decomposition
Lemma 3.1.
Difference in capital of bank i ∈ [m] is given by
bi − Ci = T1 + T2 + T3
C
where . . .
CCP Impact
25/37
. . . difference in capital due to
I
counterparty default:
Λ+
T1 = − Pm i
b−
+ C0
i=1 Λi
I
+
Pm
j=1 (Lj
− L∗j )Πji
liquidation loss:
bi )(Q − P) ≥ 0
T2 = (Zi − Z
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fees and losses in guarantee fund:
T3 = −f Λ+
i −
CCP Impact
Gi
∗
(Gtot − Gtot
)≤0
Gtot
26/37
Difference in risk components with and without CCP
10
2
x 10
E[T1]
1.5
E[T2]
E[T3]
1
0.5
0
−0.5
−1
−1.5
−2
0
1
2
3
4
5
g
6
7
8
9
10
9
x 10
Figure: Expected capital difference components, for f = 0
CCP Impact
27/37
Outline
Financial Network
Central Counterparty Clearing
CCP Impact
Does a CCP reduce systemic risk?
Pareto optimality analysis
Does a CCP reduce systemic risk?
28/37
Systemic risk measure as in Chen et al. (2013)
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Generic coherent risk measure ρ(X )
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Aggregation function, α ∈ [0, 1],
Pm
P
−
C+
Aα (C) = α m
i=0 Ci − (1 − α)
{z i=0 i }
| {z } |
bankruptcy cost
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tax benefits
Systemic risk measure
ρα (C) = ρ (Aα (C))
Does a CCP reduce systemic risk?
29/37
Impact on aggregation function
Lemma 4.1.
b − Aα (C) = αC
b − − ∆α
Aα (C)
0
where
∆α = α
P
i∈[m]
b − + (1 − α)(Q − P) P
b
Ci− − C
Z
−
Z
i
i
i∈[m]
i
is nonnegative, ∆α ≥ 0, and does not depend on (f , g). Hence
b − ρα (C) = ρ Aα (C)
b − ρ (Aα (C)) ≤ ρ Aα (C)
b − Aα (C)
ρα (C)
b − + ρ(−∆α )
≤ αρ C
0
with equlity if ρ(X ) = E[X ]
Does a CCP reduce systemic risk?
30/37
Impact on systemic risk measure
Theorem 4.2.
The CCP reduces systemic risk if (and only if)
b − < −ρ (−∆α )
αρ C
0
(for ρ(X ) = E[X ]). The RHS does not depend on (f , g).
Does a CCP reduce systemic risk?
31/37
Outline
Financial Network
Central Counterparty Clearing
CCP Impact
Does a CCP reduce systemic risk?
Pareto optimality analysis
Pareto optimality analysis
32/37
CCP and banks’ utility function
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CCP and banks are risk neutral
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Utility function = expected surplus
h i
b+
ui (f , g) = E C
i
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Participation constraints: (f , g) is feasible if
u0 (f , g) ≥ γ0 competitive case
ui (f , g) ≥ E Ci+ , i ∈ [m], monopolistic case
Pareto optimality analysis
33/37
Symmetric case
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γi ≡ γ, yi ≡ y , gi ≡ g , and Lij are iid, Lij , P, Q independent
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Consequence:
u0 (f , g ) + mu1 (f , g ) = γ0 + E [≥ 0] ≡ constant
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Consequence: every feasible (f , g ) is Pareto optimal
Pareto optimality analysis
34/37
Numerical result: parameters
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Complete inter dealer network based on BIS 2010 data
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m = 14 banks
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γ0 = $5bn
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γ = $10bn
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total notional $16tn
Pareto optimality analysis
35/37
Numerical result: Pareto optimal policies
9
10
x 10
9
8
7
g
6
5
4
3
2
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
f
Figure: Feasible Pareto optimal policies, and systemic risk zero line
Pareto optimality analysis
36/37
Conclusion
Conclusion
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Simple general financial network setup with and without CCP
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CCP always improves aggregate surplus through lower asset
liquidation losses
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CCP always reduces banks’ bankruptcy cost
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CCP introduces tail risk, and may increase systemic risk
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Find sufficient (and necessary) condition for systemic risk
reduction
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Numerical example shows that CCP reduces systemic risk for
feasible fee and guarantee fund policies (open question: does
this hold in general?)
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