Interconnected Balance Sheets, Market Liquidity,and the Amplification Effects in a Financial

System

Nan Chen

(Joint Work with David Yao (Columbia) and Xin Liu (CUHK))

Department of Systems Engineering and Engineering ManagementThe Chinese University of Hong Kong

Bachelier Congress, Jun. 5, 2014

Systemic Risk

I Systemic risk: the risk of collapse of an entire financial systemdue to its ill structure.

I The risk starts from the failure of single entity or a cluster ofentities

I Interconnectedness among entities propagates suchidiosyncratic risk to the whole system

I The objective of our work attempts to understand theamplification effect of the financial system, especially thecontribution of two major factors to the systemic risk:

I interconnected balance sheets of the banking system (NetworkEffect)

I market liquidity (Liquidity Effect)

The Financial Network Becomes Complex: TransactionLevel

I Comparison between the traditional and modern bankingsystems

households Mortgage

bank households

mortgage deposits

Traditional Banking System

Modern Banking System

households

Money market

fund

Commercial

Bank Security firm CDO issuer

Mortgage pool

households

MMF shares

Commercial

papers

repo CDO

mortgage

Mortgage-backed

securities

The Financial Network Becomes Complex: InterbankMarket Level

I A diagram of the UK interbank exposure network (FinancialStability Report (2013))

Networks of large exposures between UK banks 2009:

The Financial Network Becomes Complex: Global Level

I The cross-border banking networks in 1980 and 2007 (Minoiuand Reyes (2011 IMF Report))

!

Figure : 1980 Figure : 2007

Market Liquidity

I Market liquidity constitutes another channel transmitting thesystemic risk.

I An unexpected loss causes problem in the liquidity of a firm. Itmay need to start asset sales to meet its obligation.

I However, other institutions may also be in stress at the sametime and rely on the market to liquidate their assets to gainaccess to additional cash. And the market has only limitedliquidity to absorb.

A Vicious Circle Amplified by the Market Liquidity

I The asset selling of many financial institutions creates aliquidity spiral (Brunnermeier (2009, JEP), Brunnermeier andPedersen (2009, RFS), Adrian and Shin (2010, JFI)):

Positions Liquidation

Price Decline Liquidity Problems Initial Losses

Loss on Existing Positions

Liquidity Situation of Bear Sterns before Its Near Failure

Contribution of the Current Paper

The contribution of our work are two-foldI Insights:

I Two multipliers, network multiplier and liquidity amplifier,are identified to characterize analytically the amplificationeffects.

I Liquidity plays a dominating role in triggering large scalecontagion under the current market environment.

I Capital as a metric of systemic resilience.I Policy assessment: direct asset purchase and capital injection

I Method:I Equilibrium-constrained optimization formulationI Partition algorithmI Sensitivity analysis

Literature Review on Financial Networks and Contagion

I Some early literature: Economic origin of financial contagionI Rochet and Tirole (1996), Kiyotaki and Moore (1997), Allen and Gale (2000), Freixas, Parigi, and

Rochet (2000), and Leitner (2005)

I Network effect:I Modelling: Eisenberg and Noe (2001), Gai and Kapadia (2010), Liu and Staum (2010),

Gai, Haldane, and Kapadia (2011), Haldane and May (2011), Acemoglu, Ozdaglar, and

Tahbaz-Salehi (2012), Amini, Cont, and Minca (2013), and Elliott, Golub, and Jackson (2014).I Empirical studies: Furfine (2003), Elsinger, Lehar, and Summer (2006), Degryse and

Nguyen (2007), and Glasserman and Young (2014).

I Liquidity and fire sale:I Shleifer and Vishny (1997), Gromb and Vayanos (2003), Allen and Gale (2004), Glaneanu and

Pedersen (2007), Brunnermeier and Pedersen (2009)

I Some related work:I Cifuentes, Ferrucci, and Shin (2005) and Nier et al. (2007): simulation

I Amini, Filipovic and Minca (2013): CCP

Review of Eisenberg and Noe (2001)

I Timeline:I Time 0: An interbank lending-and-borrowing system is

establishedI Time 1 (Armageddon, maybe?): All debts mature; the system

reaches a market-clear repayment equilibrium

Review of E-N: Bank’s Balance Sheets at Time 0

I There are N banks, indexed by 1, 2, · · · ,N, and anon-leveraged sector in the economy.

I For bank i , its balance sheet at time 0 is given by

The Balance Sheet of Bank i at Date 0

Assets Liabilities and EquityEnd-user Loan: yi Equity: ei

Interbank Loans: xi,k (some k 6= i) Interbank Claims: xj,i (some j 6= i)

External debt claim: xEi

I Let li indicate the nominal amount of total liabilities of bank i :

li =∑

j

xj ,i + xEi .

Let Π denote the relative liability matrix in which πji = xij/lj .

Review of E-N: Illustration

Bank 1

Bank 3

Bank 2

Banking system

Non-leveraged Sector

Date 0 Date 1

Bank 1

Bank 3 Bank 2

Banking system

Non-leveraged Sector

Review of E-N: Repayment Rules at Time 1

I Loan repayment at time 1 obeys the following rules:

A. Residual claimant: debt claims have priority than equity.B. Limited liabilities: each banks pays at most its available assets.C. In the default of a bank, all the interbank loans and the

external debt claims are with the same seniority.

I Given the loan repayments from the external borrowers arey = (y1, · · · , yN), yi ≤ yi , the interbank repayments inequilibrium for each bank is then determined by

pi = min{

li , yi︸︷︷︸Repayment fromnonleverage sector

+∑j 6=i

pjπji︸ ︷︷ ︸Repayment from

other banks

}.

Multiplicity of Market Equilibria and Fix-point FormulationI In a vectorial form, the equilibrium repayment p is the

solution to the following fixed-point problem:

p = min{l, y + pΠ}.I Equilibrium existence and multiplicity:

Bank 1

Bank 3

Bank 2

Banking system

Non-leveraged Sector

Date 0 Date 1

Bank 1

Bank 3 Bank 2 $1

$1

$1

Equilibrium I

$1

$1

$1

Bank 1

Bank 3 Bank 2

$C<1

$C $C

Equilibrium II

Our Proposal: Optimization with Equilibrium Constraints

I In light of this self-fulfilling complexity, we focus on thelargest equilibrium in this talk. Equivalently, we are interestedin the following optimization problem:

max∑

i

pi

s.t. p = min{l, y + pΠ}

I The above problem can be further reduced to a linearprogramming:

Primal : max p1 Dual : min yδ + lηs.t. p ≤ l (η) ⇔ s.t. (I − Π)δ + η ≥ 1

p ≤ y + pΠ (δ) δ ≥ 0, η ≥ 0

Our Proposal: Optimization with Equilibrium Constraints

I The optimal solution to the LP takes the following form:there exist a partition of the set of banks {1, 2, · · · , n} intotwo complementary subsets D and N , D = {i : pi < `i},N = {i : pi = `i}, such that

Primal sol : pN = lN ; pD = (yD + lNΠN ,D)(ID − ΠD)−1.

Dual sol : δD = (ID − ΠD)−11D, δN = 0N ;

ηD = 0D, ηN = 1N + ΠN ,D(ID − ΠD)−11D

I Network multiplier:

δD = (ID − ΠD)−11D = (ID + ΠD + Π2D + · · · )1D

Partition Algorithm

N

Round 1I Partition Candidate: N = Dc ,D = ∅

I Equilibrium Candidate:p = l

I Feasibility Checking:l ≤ y + lΠ?

(Equity := y + lΠ− l ≥ 0?)

I Yes: terminate

I No: Include the violators into D

Partition Algorithm

NI

Round 2I Partition Candidate: N = Dc ,D = I

I Equilibrium Candidate: pN = lN ,pD = yD + lNΠN ,D + pDΠD

I Feasibility Checking:lN ≤ yN + lNΠN + pDΠD,N ?

(Equity :=

yN + lNΠN + pDΠD,N − lN ≥ 0?)

I Yes: terminate

I No: Include the violators into D

Partition Algorithm

NI II

Round 3I Partition Candidate: N = Dc ,D = I + II

I Equilibrium Candidate: pN = lN ,pD = yD + lNΠN ,D + pDΠD

I Feasibility Checking:lN ≤ yN + lNΠN + pDΠD,N ?

(Equity :=

yN + lNΠN + pDΠD,N − lN ≥ 0?)

I Yes: terminate

I No: Include the violators into D

An Extended Model with Market LiquidityI Now we consider an extended model to incorporate the

influence of market liquidity. For bank i , its balance sheet attime 0 is given by

The Balance Sheet of Bank i at Date 0

Assets Liabilities and EquityEnd-user Loan: yi Equity: ei

Interbank Loans: xi,k (some k 6= i) Interbank Claims: xj,i (some j 6= i)

Liquid Assets: ai External debt claim: xEi

Illiquid Assets: si

I The selling price of illiquid assets is influenced by the marketdemand. Its inverse demand function is

q = Q(N∑

i=1

si ),

where q is the market price of one unit illiquid asset,∑N

i=1 si

is the aggregate supply, and Q(0) = 1, Q(·) monotonelydecreasing.

I Example: Q(y) = exp(−γy), γ: market depth

I Liquid asset can be sold at its face value.

Repayment Vector

I Given the realization of external repayment y = (y1, · · · , yN),the liquidation amounts of liquid and illiquid assetsa = (a1, · · · , aN) and s = (s1, · · · , sN), the income for eachbank is

yi︸︷︷︸Repayment fromnonleverage sector

+∑j 6=i

pjπji︸ ︷︷ ︸Repayment from

other banks

+ ai + si q︸ ︷︷ ︸Income from

asset liquidation

where q = Q(∑N

i=1 si ).

Market Equilibrium

DefinitionA quadruplet (p∗, a∗, s∗, q∗) is called a market equilibrium if for alli ,

1. (Limited liability) pi = min{li , yi +∑

j 6=i p∗j πji + a∗i + s∗i q∗};2. (No short sale of assets) a∗i ∈ [0, ai ] and s∗i ∈ [0, si ];

3. (Sale of liquid asset occurs first)

a∗i = min

max

p∗i −

yi +∑j 6=i

p∗j πji

, 0

, ai

and

s∗i = min

max{

p∗i −(

yi +∑

j 6=i p∗j πji

)− li , 0

}q∗

, si

.

4. (Market clearing) q∗ = Q(∑N

i=1 s∗i ).

Main Result I: Existence and Uniqueness of the Equilibria

I Multiple equilibria caused by limited liquidity

I We can formulate the following optimization problem withequilibrium constraints to solve for the largest equilibrium(p∗, a∗, s∗, q∗):

max q

s.t. (p, a, s, q) satisfy constraints 1-4.

I An extended partition algorithm is developed to solve theabove optimization problem. We have, p∗ ≥ p, a∗ ≤ a,s∗ ≤ s, and q∗ ≥ q, for any other equilibrium (p, a, s, q).

Revisiting Partition Algorithm in the Presence of Liquidity

N

Round 1I Partition Candidate: N = Dc ,D = ∅

I Equilibrium Candidate:p = l, q1 = 1

I Feasibility Checking:

Equity := y + a + s · q1 + lΠ− l ≥ 0?

I Yes: terminate

I No: Include the violators into D

Revisiting Partition Algorithm in the Presence of Liquidity

NInetwork (Π)

market (q)

Round 2I Partition Candidate: N = Dc ,D = I

I Equilibrium Candidate:pN , pD, q2

I Feasibility Checking:Equity :=

yN + aN + q2 sN + lNΠN +

pDΠD,N − lN ≥ 0?

I Yes: terminate

I No: Include the violators into D

Revisiting Partition Algorithm in the Presence of Liquidity

NI IInetwork (Π)

market(q)

Round 3I Partition Candidate: N = Dc ,D = I + II

I Equilibrium Candidate:pN , pD, q3

I Feasibility Checking:Equity :=

yN + aN + q3 sN + lNΠN +

pDΠD,N − lN ≥ 0?

I Yes: terminate

I No: Include the violators into D

Contagion Estimation and Capital

Define a shock hits bank 1, and its size is ∆y1 = y1 − y1.I Round 1:

P(Bank 1 Defaults) ≥ P(∆y1 ≥ e(1)1 )

where e(1)1 is the capital market value of bank 1 before this round:

e(1)1 := y1 + a1 + s1 +

n∑k=1

lkπk1 − l1.

I Round 2:

E(# of Defaults in this round|Bank 1 default) ≥∑j∼1

P(∆y1 ≥ e(2)1 +

e(2)j

π1j

),

where e(2)j is the capital market value of bank j before this round:

e(2)j := yj + aj + sj q2 +

n∑j=1

lkπk1 − l1.

Contagion Estimation and Capital

I A generic round:

E(# of Defaults in this round|All the banks in D default)

≥∑

j∈N(D)

P

(∆y1 >

eD(ID − PD)−1PD,j

e1(ID − PD)−1PD,j

+ej

e1(ID − PD)−1PD,j

).

Main Result II: Structure of the Largest Equilibrium

I The algorithm can help us classify the whole banking systemin equilibrium into four subsets:

I N = {i : pi = li , ai = 0, si = 0} (Banks which can repay theirliabilities without liquidating assets).

I M = {i : pi = li , 0 < ai < ai , si = 0} (Banks which have toliquidate part of their liquid asset to repay the liabilities).

I L = {i : pi = li , ai = ai , 0 < si < si} (Banks which have toliquidate part of their illiquid asset to repay the liabilities).

I D = {i : pi < li} (Banks in default)

I The repayment vectors of these three subsets of banks satisfythe following equations:

p∗D = [yD + aD + qsD + lDc ΠDc ,D][ID −ΠD]−1,

p∗L = yL + aL + qsL + pDΠD,L + lDc ΠDc ,L = lL,

p∗M = yL + aL + pDΠD,M + lDc ΠDc ,M = lM,

p∗N = lN .

Main Result III: Liquidity Amplifier

I For i ∈ L,

∂q∗

∂yi=

γ

1− γ(sL1 + sD[ID −ΠD]−1ΠD,L1)

I We refer to the above quantity as the liquidity amplifier. Tounderstand this, note

γ

1− γS= γ + γ(γS) + γ(γ(γS))S + · · · .

Main Result IV: Policy Assessment

Take two intervention policies as an example:

I Direct asset purchase:

I Capital injection

Assets   Liabili+es  

Debts  

Equity  

External  Investments  

Illiquid    Securi+es  

Liquid  Securi+es  

$Δ

Direct  Asset  Purchase  

Figure : Direct Asset Purchase

Assets   Liabili+es  

Debts  

Equity  

External  Investments  

Illiquid    Securi+es  

Liquid  Securi+es  

$Δ

Capital  Injec+on  

$Δ

Figure : Capital Injection

Main Result IV: Policy Assessment (Con’d)

I Asset Purchase at Market price

∂q∗

∂∆

∣∣∣Market

=γ

1− γ(sL1 + sD [ID − ΠD ]−1ΠD,L1)︸ ︷︷ ︸Liquidity Amplifier

,

∂p∗

∂∆

∣∣∣Market

= LA · sD∗ (ID∗ − PD∗ )−1.

I Capital Injection:

∂q∗

∂∆

∣∣∣Capital

= LA · ei (ID∗ − PD∗ )−1PD∗,L∗ 1,

∂p∗

∂∆

∣∣∣Market

= ei (ID∗ − PD∗ )−1 + LA · ei (ID∗ − PD∗ )−1PD∗,L∗ 1 · sD∗ (ID∗ − PD∗ )−1.

I Asset Purchase at Face Value

Main Result IV: Policy Assessment (Con’d)

I Policy implication:I Asset purchase focuses on liquidity improvement

I Total asset value

Total Asset Value (TAV) After Purchase

= TAV Before Purchase + ∆− q∗ ·∆

q∗

= TAV Before Purchase.

I Liquidity ratio:

LR After Purchase =Liquidity + ∆

TAV

I Capital injection eyes more on network effectI Total asset value increases by ∆I Liquidity ratio:

LR After Purchase =Liquidity + ∆

TAV + ∆

Numerical Example: Data

I We use the data set from the European Banking Authority’s2011 stress test.

Bank Number and Name Total AssetInterbank

Asset

Non-Interbank

AssetEquity Liability

DE017 DEUTSCHE BANK AG 1905630 194399 1711231 30361 1875269

DE018 COMMERZBANK AG 771201 138190 633011 26728 744473

DE019 LANDESBANK BADEN 374413 133906 240507 9838 364575

DE020 DZ BANK AG DT. ZEN 323578 135860 187718 7299 316279

DE021 BAYERISCHE LANDES 316354 97336 219018 11501 304853

DE022 NORDDEUTSCHE LAN 228586 91217 137369 3974 224612

DE023 HYPO REAL ESTATE H 328119 29084 299035 5539 322580

DE024 WESTLB AG, DüSSELD 191523 58128 133395 4218 187305

DE025 HSH NORDBANK AG, H 150930 9532 141398 4434 146496

DE027 LANDESBANK BERLIN 133861 49253 84608 5162 128699

DE028 DEKABANK DEUTSCH 130304 41255 89049 3359 126945

ℓ

 

 

Numerical Example: Three Network Configuration

I The detailed specification of network structures are notavailable in the raw data.

I We recover it using the relative-entropy based algorithm inUpper and Worms (2004, EER).

I To find the bilateral exposures between banks, let

L =

0 · · · l1j · · · l1N

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.li1 · · · 0 · · · liN...

.

.

.

.

.

.

.

.

.

.

.

.lN1 · · · lNj · · · 0

.

I We solve the following optimization problem:

min∑i,j

lij ln(lij/xij )

subject to∑

j lij = observed interbank asset and∑i lij = observed interbank liabilities

Numerical Example: Three Network Configurations(Con’d)

I Three kinds of network configurations for the core banks areexamined.

Figure : CompleteNetwork Figure : Ring-Like

Network

Figure :Core-Periphery

Numerical Example: Market Environment

I Q(s) = exp(−γs).I Baseline case:

I γ = 1× 10−7

I 30% of the total asset of each bank is illiquidUnder this choice, the market price of the illiquid asset will drop to $0.864 from its face value $1

when every bank in the system depletes its holdings.

Numerical Example: Liquidity ContagionWe let one of the banks receive shocks on its asset in the followingexperiment, i.e., y1 = (1− θ)y1, θ ∈ [0, 1].

Numerical Example: Liquidity Contagion (Con’d)

I No significant contagion in the absence of market liquidity.

I Introducing illiquid asset into the banking system amplifies thecontagion effect significantly.

I Network structure matters when contagion kicks off.

I Our results echo the concern of Andrea Enria, Chairperson ofthe European Banking Authority, in his opening statement forPublication of the 2011 EU-wide Stress Test Results:

Where a core tier one ratio falls above the benchmark but close to 5pc and a bank

has sizeable exposures to sovereigns under stress, the supervisor must oversee that it takes

specific steps to strengthen its capital position, including where necessary restrictions on

dividends, deleveraging, issuance of fresh capital or conversion of lower quality instruments

into core tier 1 capital.

Numerical Examples: Sensitivity of Market Depth γWe change the market-depth parameter γ in the system in thefollowing picture.

Numerical Examples: Sensitivity of Market Price relative tothe Shock Size

We consider the market-clearing price q and ∂q/∂θ

Concluding Remarks

I We present an equilibrium-constrained optimizationformulation to model the systemic risk.

I Our research identifies network and liquidity multipliersexplicitly and points out different roles played by these twofactors in systemic risk development.

Numerical Example: Three Network Configuration (Con’d)

I Liability matrices under different configurations:

DE017 DE018 DE019 DE020 DE021 DE022 DE023 DE024 DE025 DE027 DE028

DE017 0 23166 22360 22727 32673 30619 9763 19512 3200 16533 13848

DE018 23166 0 15020 15266 21948 20568 6558 13107 2149 11106 9302

DE019 22360 15020 0 14735 21184 19852 6330 12651 2075 10719 8979

DE020 22727 15266 14735 0 21532 20178 6434 12858 2109 10895 9126

DE021 32673 21948 21184 21532 0 0 0 0 0 0 0

DE022 30619 20568 19852 20178 0 0 0 0 0 0 0

DE023 9763 6558 6330 6434 0 0 0 0 0 0 0

DE024 19512 13107 12651 12858 0 0 0 0 0 0 0

DE025 3200 2149 2075 2109 0 0 0 0 0 0 0

DE027 16533 11106 10719 10895 0 0 0 0 0 0 0

DE028 13848 9302 8979 9126 0 0 0 0 0 0 0

DE017 DE018 DE019 DE020 DE021 DE022 DE023 DE024 DE025 DE027 DE028

DE017 0 38370 37181 37724 21012 19691 6278 12548 2058 10632 8906

DE018 38370 0 0 0 25854 24229 7725 15440 2532 13082 10958

DE019 37181 0 0 0 25052 23478 7486 14961 2453 12677 10618

DE020 37724 0 0 0 25418 23820 7595 15179 2489 12862 10773

DE021 21012 25854 25052 25418 0 0 0 0 0 0 0

DE022 19691 24229 23478 23820 0 0 0 0 0 0 0

DE023 6278 7725 7486 7595 0 0 0 0 0 0 0

DE024 12548 15440 14961 15179 0 0 0 0 0 0 0

DE025 2058 2532 2453 2489 0 0 0 0 0 0 0

DE027 10632 13082 12677 12862 0 0 0 0 0 0 0

DE028 8906 10958 10618 10773 0 0 0 0 0 0 0

DE017 DE018 DE019 DE020 DE021 DE022 DE023 DE024 DE025 DE027 DE028

DE017 0 64270 0 0 33704 31585 10071 20128 3301 17055 14285

DE018 0 0 49741 0 22909 21469 6845 13681 2243 11592 9710

DE019 0 0 0 49129 21958 20577 6561 13113 2150 11111 9307

DE020 63411 0 0 0 18765 17585 5607 11206 1838 9495 7953

DE021 33927 19146 21799 22464 0 0 0 0 0 0 0

DE022 31794 17942 20429 21052 0 0 0 0 0 0 0

DE023 10137 5721 6514 6712 0 0 0 0 0 0 0

DE024 20261 11434 13018 13415 0 0 0 0 0 0 0

DE025 3322 1875 2135 2200 0 0 0 0 0 0 0

DE027 17167 9688 11031 11367 0 0 0 0 0 0 0

DE028 14380 8115 9240 9521 0 0 0 0 0 0 0

liability matrix under complete network structure

liability matrix under star network structure

liability matrix under ring network structure

Multiple Equilibria under Limited Liquidity

I Two bank systemI Balance sheets at time 1

Bank 1External investments y1 = $0.1 External debt b1 = $1Iliquid Asset s1 = 1 Equity e1

Bank 2External investments y2 = $0.9 External debt b2 = $1Iliquid Asset s2 = 2 Equity: e2

I Q(s) = exp(−s)

I Equilibrium equations:I

s1 = 1 ∧ 0.9

q, s2 = 2 ∧ 0.1

q, and q = exp(−(s1 + s2))

I Two equilibria: (s1, s2) = (1, 0.4092) and (s1, s2) = (1, 2)