Portfolio Compressionin Financial Networks:

Incentives and Systemic Risk

Steffen Schuldenzucker𝟏,𝟐 and Sven Seuken1

Workshop on the Systemic Impact of Digitalization on FinanceUniversity of ZurichDecember 20, 2019

1Computation and Economics Research Group, University of Zurich

2Algorithms and Complexity Research Group, Goethe University Frankfurt

2

Financial interconnectedness gives rise to systemic risk via financial contagion

3

To reduce interconnectedness, eliminate cycles = Portfolio Compression

4

To reduce interconnectedness, eliminate cycles = Portfolio Compression

5

Research Questions:

1. When socially beneficial?

2. Incentives to do it?

Compression is mandatory today

6

— EMIR Regulations, 2017

When counterparties have more than 500 contracts outstanding with each other, [there is an] obligation to have procedures to analyse the possibility to conduct the exercise [of portfolio compression].

Portfolio Compression: Process

7

Financial Institution Financial Service Provider

Contracts

Find Cycles

Unwind Proposal

Accept / Reject

1

2

3

4 All involved banks need to accept for the compression to become effective!

Prior Work• Central Clearing and Systemic Risk: A lot

• Duffie and Zhu (2011), Duffie et al. (2015), Amini et al. (2015), …

• Portfolio Compression: Algorithms / Compressed Amount• O’Kane (2017), D’Errico and Roukny (2019)

• Portfolio Compression and Systemic Risk: Very little• Schuldenzucker, Seuken, Battiston (note, 2016): Can be socially detrimental• Veraart (WP, 2019): Can be detrimental; simple sufficient conditions

• Network Structure & Systemic Risk• Elliott ea. (2014), Acemoglu ea. (2015), Glasserman and Young (2015), …

• Change in Network Structure• Feinstein et al. (WP, 2017): Sensitivity to changes in network, keep absolute

liabilities the same (≠ compression)

8

Formal Model: Financial Network(Rogers and Veraart, 2013)

Financial System 𝑋 = (𝑁, 𝑒, 𝑙, α, 𝛽)

Set of banks 𝑁, single bank 𝑖 ∈ 𝑁

External Assets 𝑒 with 𝑒𝑖 ∈ ℝ+ (shocked)

Nominal Liabilities 𝑙 with 𝑙𝑖,𝑗 ∈ ℝ+= what i owes to j

Default Cost Parameters 𝛼, 𝛽 ∈ [0, 1]

9

Formal Model: Clearing Payments(Rogers and Veraart, 2013)Theorem (Rogers and Veraart, 2013): There is a unique point-wise maximal matrix 𝑝 of payments such that:

𝑝𝑖,𝑗 = ൞

𝑙𝑖𝑗 if 𝑎𝑖(𝑝) ≥ 𝑙𝑖 (No Default)

𝑙𝑖𝑗

𝑙𝑖⋅ 𝑎𝑖′(𝑝) if 𝑎𝑖 𝑝 < 𝑙𝑖 (Default)

10

Total Assets of i 𝑎𝑖 𝑝 = 𝑒𝑖 +𝑗𝑝𝑗𝑖

Total Assets after Default Costs 𝑎𝑖′ 𝑝 = 𝛼𝑒𝑖 + 𝛽

𝑗𝑝𝑗𝑖

Total Liabilities of i 𝑙𝑖 = 𝑙𝑖∗

where…

Formal Model: Portfolio Compression

A compression: 𝑐 = (𝑐𝑖𝑗) that is a circulation in 𝑙, i.e.:

1. 0 ≤ 𝑐𝑖𝑗 ≤ 𝑙𝑖𝑗 ∀𝑖, 𝑗

2. σ𝑗 𝑐𝑖𝑗 = σ𝑗 𝑐𝑗𝑖 ∀𝑖

Cf. D’Errico and Roukny (2019)

Compressed Financial System: 𝑋𝑐: = (𝑁, 𝑙 − 𝑐, 𝑒, 𝛼, 𝛽)

11

Research Questions, formal

Bank’s Utility := Equity = 𝐸𝑖: = max 0, 𝑎𝑖 𝑝 − 𝑙𝑖Social Welfare = Total Equity = 𝐸Σ ≔ σ𝑖 𝐸𝑖

Given 𝑋, 𝑐:

1. When is 𝑐 socially beneficial?i. Pareto improvement: 𝐸𝑖

𝑐 ≥ 𝐸𝑖 ∀𝑖ii. Welfare improvement: EΣ

𝑐 ≥ 𝐸Σ

2. When is 𝑐 incentivized for participating banks?𝐸𝑖𝑐 ≥ 𝐸𝑖 ∀𝑖 ∈ 𝑁(𝑐)

where N(c)= 𝑖 ∈ 𝑁 ∣ 𝑐𝑖 > 0 and 𝑐𝑖 ≔ σ𝑗 𝑐𝑖𝑗

12

Compression may be socially detrimental / not incentivized

13

Compression

𝛼 = 𝛽 = 0.5

2

240.625

0.625

𝐸Σ = 6.75

0.625

2

0 0.125 4

1.750.25

𝐸Σ = 5.75

2

2

0 0 1.75

3

3

1

1

The effect depends on the parameters in a complex and non-monotonic way

14

Relative Liabilities: 𝜋𝑖𝑗 ≔𝑙𝑖𝑗

𝑙𝑖

Difference in Relative Liabilities: Δπ𝑖𝑗 = π𝑖𝑗𝑐 − π𝑖𝑗

Sufficient conditions for Pareto Improvement: Relative Liability Change

15

Theorem: Assume for all i, j with Δ𝜋𝑖𝑗 > 0, we have at least one of:

Sufficient Conditions ⇒ Pareto Improvement

a) High Recovery: 𝑟𝑖 ≥ 𝛽

b) High capitalization wrt. compressed liabilities:

𝑒𝑖𝑙𝑖 − 𝑐𝑖

≥𝛽

𝛼

c) Full default costs on interbank liabilities: 𝛽 = 0

d) Sufficiently not leaving /uniform compression at (i, j):

𝜂𝑐𝑖𝑗

𝑙𝑖𝑗

𝜂𝑐𝑖𝑙𝑖

≥ 𝛽

16

𝜂 𝑥 =1

1𝑥− 1

Then c is a Pareto improvement.

Homogenous ⇒ Pareto Improvement(𝑋, 𝑐) homogeneous if all equal across 𝑖 ∈ 𝑁(𝑐):

Theorem: homogeneous ⇒ Pareto improvement

17

• 𝑒𝑖• σ𝑗:𝑐𝑗𝑖=0

𝑝𝑗𝑖

• σ𝑗:𝑐𝑗𝑖>0𝑙𝑗𝑖

• σ𝑗:𝑐𝑖𝑗=0𝑙𝑖𝑗

• σ𝑗:𝑐𝑖𝑗>0𝑙𝑖𝑗

• 𝑐𝑖

Conjecture: Degree of Homogeneity

18

Conclusion• Ex-post analysis of portfolio compression = cycle elimination

• Compression may be socially detrimental / not incentivized

• Feedback paths necessary for non-incentivized compression

• Sufficient Conditions for Pareto Improvement

• Homogeneity is good for compression

Future Work:

• Additional benefits 𝑏𝑖 ≥ 0 from compression ⇒ No qualitative change?

• Ex-ante view / distribution of shocks

19Thank You! Questions?