systemic stress-testing: fire sales, endogenous risk and ... ·...

Price-mediated contagion Fire sales Systemic stress test Indirect exposures Monitoring exposure to fire sales

Systemic stress-testing:fire sales, endogenous risk and

price-mediated contagion

Rama Cont and Eric Schaanning

Winter 2017 Meeting on Macro Financial Modeling, New York

Systemic Stress Testing and price-mediated contagion R Cont and E. Schaanning


Channels of loss amplification in the financial system

1 Counterparty Risk: balance sheet contagion through assetdevaluation

2 Funding channel: balance sheet contagion through withdrawalof funding (bank runs by depositors, institutional bank runs bylenders)

3 Feedback effects from fire sales: loss contagion throughmark-to-market losses in common asset holdings

Research on financial networks and their use in macroprudentialregulation has focused on direct contagion mechanisms (1+2).Regulatory measures have focused on 1 (large exposure limits,central clearing, CVA, ring-fencing) or 2 (LCR, NSFR).



Bank stress tests• Stress tests assume ’passive’ behavior by banks.BCBS 2015: “Stress tests conducted by bank supervisors stilllack a genuine macro-prudential component” .. “endogenousreactions to initial stress and feedback effects” are missing.

• How do financial institutions react when faced with stress?• Market stress can lead financial institutions to unwindpositions (constrained by capital, liquidity, leverage...):-large body of empirical evidence of deleveraging in stressscenarios (Shleifer 2010, Coval & Stafford 2007, Ellul et al2011, Cont & Wagalath 2016).-evidence from banks ‘living wills’: (Credit Suisse Report,2015): “If we are unable to raise needed funds in the capitalmarkets (...), we may need to liquidate unencumbered assetsto meet our liabilities [..] at depressed prices, which in eithercase could adversely affect our results of operations andfinancial condition."



A framework for systemic stress testing with endogenouseffects

We build on previous theoretical work on fire sales (Shleifer 2010,Coval & Stafford 2007, Ellul et al 2011, Kyle & Xiong 2005, Cont& Wagalath 2013,..) and recent empirical studies (Greenwood etal 2013, Eisenbach -Duarte 2014) to construct an operationalframework for quantifying fire sales spillovers and incorporating itin a system-wide stress test for financial institutions.

R Cont and Eric Schaanning (2016)Fire sales, indirect contagion and systemic stress-testing,http://ssrn.com/abstract=2541114.


http://ssrn.com/abstract=2541114


Systemic stress testing with endogenous effectsIngredients:

1 Portfolio holdings: holdings of financial institutions by assetclass: N institutions, K illiquid asset classes, M marketableasset classes → N × (M + K ) portfolio matrix (network)

2 Portfolio constraints: capital ratio, leverage ratio, liquidityratio,... → range of admissible portfolios ("safety zone").

3 Reaction function: reaction of a bank when its portfolio exitsthe admissible region (deleveraging/ rebalancing)

4 Market impact: market prices react to portfolio rebalancing5 Mark-to-market accounting: transmits market impact to all

institutions → may lead to feedback if market losses large

Initial shock

Deleveraging

Mark to market losses

Market impactto assets



Balance sheets: illiquid vs marketable assetsIlliquid assets

Residential mortgage exposuresCommercial real estate exposure

Retail exposures: Revolving credits, SME, OtherIndirect sovereign exposures in the trading book

Defaulted exposuresResidual exposuresMarketable assetsCorporate bondsSovereign debtDerivatives

Institutional client exposures: interbank, CCPs,...Table: Stylized representation of asset classes in bank balance sheets. (Data:European Banking Authority 2011)



• Illiquid holdings of institution i : Θi :=∑Kκ=1 Θiκ .

• Marketable Securities held by i : Πi :=∑Mµ=1 Πiµ .

• Equity (Tier 1 capital): C i

• Financial institutions are subject to various one-sidedportfolio constraints: leverage ratio, capital ratio, liquidityratio.

• Leverage ratio of i :

λi = Assets(i)C i = Θi + Πi

C i ≤ λmax

• Capital ratio of i :

λi = RWA(i)C i =

∑wκΘi ,κ +

∑µ Πi ,µwµ

C i ≤ Rmax

Basel III: λmax = 33, Rmax = 12.5 = 1/0.08• Banks maintain a capital/liquidity buffer (slightly) above theregulatory requirements → target leverage ratio λi

b < λmax,target capital ratio R i < Rmax.



Deleveraging• Observation: when portfolio constraints are breachedfollowing a loss in asset values, financial institutionsdeleverage their portfolio by selling some assets in order tocomply with the portfolio constraint.

Deleveraging assumption: if following a loss Li in asset values,leverage of bank i exceeds constraint,

λi = Θi + Πi − Li

C i − Li > λmax

bank deleverages by selling a proportion Γi ∈ [0, 1] of assets inorder to restore a leverage ratio λi

b ≤ λmax:

(1− Γi )Πi + Θi − Li

C i − Li = λib ≤ λmax ⇒ Γi = C i (λi − λi

b)Πi 1λi>λmax



Develeraging in response to a loss

Figure: Percentage of marketable asset deleveraged in response to a shock toassets (circles) for a leverage constraint of 20. Leverage targeting (dotted blue)would lead to a linear response.



Market impact and Feedback effectsTotal liquidation in asset µ at k-th round: qµ =

∑Nj=1 Πj,µ

k Γjk+1

Market impact : ∆SµSµ = −Ψµ(qµ),

Impact/ inverse demand function: Ψµ > 0,Ψ′µ > 0,Ψµ(0) = 0.

Price move at each iteration of fire sales:

Sµk+1 = Sµk

1−Ψµ

N∑j=1

Πj,µk Γj

k+1

,

Πi ,µk+1 =

(1− Γi

k+1

)︸︷︷︸

Non-liquidated assets

Previous value︷︸︸︷Πi ,µ

k

1−Ψµ

N∑j=1

Πj,µk Γj

k+1

︸︷︷︸Price impact on remaining holdings



CCapital

AssetsDeleveragingzone

Leve

rage

cons

traint

No deleveragingzone

max

b

leve

rage

targ

et

I()

+ I()

Illiquidity

Inso

lven

cy

Figure: Portfolio constraints define a set of admissible portfolios. A large lossmay take the portfolio outside this set, in which case banks deleverage in orderto revertto this set.



Portfolio overlaps as drivers of loss contagionWhen market impact is linear Ψµ(x) = x/Dµ (where Dµ = marketdepth) the mark-to-market loss of i resulting from fire sales is

Li ≈N∑

j=1

M∑µ=1

ΠiµΠjµ

Dµ︸︷︷︸Ωij

Γj =N∑

j=1ΩijΓj ,

where Ωij is the liquidity weighted overlap between portfolios iand j (Cont & Wagalath 2013):

Ωij =M∑µ=1

ΠiµΠjµ

DµDµ = market depth for asset µ

Ωij = exposure of marketable assets of i to 1% deleveraging by j .⇒ loss contagion = contagion process on network defined by [Ωij ]



The EU indirect contagion network



Portfolio overlaps across EU banks (EBA 2011)

Liquidity weighted overlap (EUR)

Per

cent

05

1015

20

0 10 102 103 104 105



2nd round overlaps across EU banks (EBA 2011)

Per

cent

05

1015

101 102 103 104 105 106 107 108 109 1010 1011

Distribition of elements of Ω2 representing 2nd round spillovereffects.



Market impact function



Market impact function and market depthThe impact of a total distressed liquidation volume q is modelledby a level-dependent market impact function

Ψµ(q,S) =(1− Bµ

S

)(1− exp

(− qδµ

)),

whereδµ = c ADVµ

σµ

√τ ,

• S ≥ Bµ is the price-floor• ADV : average daily volume, σµ: daily volatility• c ≈ 0.25, a coefficient to make Ψµ consistent with empiricalestimates of the linear impact model for small volumes q.

• τ is the liquidation horizon



Market impact function and market depthThe impact of a total distressed liquidation volume q is modelledby a level-dependent market impact function

Ψµ(q,S) =(1− Bµ

S

)(1− exp

(− qδµ

)),

whereδµ = c ADVµ

σµ

√τ ,

• S ≥ Bµ is the price-floor• ADV : average daily volume, σµ: daily volatility• c ≈ 0.25, a coefficient to make Ψµ consistent with empiricalestimates of the linear impact model for small volumes q.

• τ is the liquidation horizonSystemic Stress Testing and price-mediated contagion R Cont and E. Schaanning


Estimated market depth



Stress scenarios

• A stress scenario is defined by a vector ε ∈ [0, 1]K whosecomponents εκ are the percentage shocks to asset class κ.

• Initial/Direct loss of portfolio i : L0i (ε) = ε.Πi =∑κ Πiκεκ

• Gradual increase of the shock size εκ from 0% to 20%.• As an illustration we consdier the following stress scenarios:

1. Spanish residential and commercial real estate losses2. Northern Europe residential losses3. Southern Europe commercial real estate losses4. Eastern Europe commercial real estate losses



Fire sales losses



Fire sales losses and market depth



Endogenous losses modify stress test outcomes



Failures due to illiquidity and insolvency

Figure: The model allows to distinguish between failures due to insolvency (neg-ative equity - left) and failures due to illiquidity (zero liquid assets - right).



Indirect exposuresConsider two institutions (A) and (B).

• A and B hold a common financial asset (1). A holds anilliquid asset (2) that B does not hold. Notional exposure of Bto 2 is zero.

• However, in the event of a large shock to the value of theilliquid asset (2), A may be forced to sell some of its financialassets, pushing down its market price, resulting in a marketloss for the bank B.

• So: B experiences a loss following a large shock to the illiquidasset: B has an (indirect) exposure to an asset it does nothold!

• Magnitude of this indirect exposure is directly linked to theoverlap between B and institutions holding this asset.

• Large diversified institutions increase overlaps across systemand become nodes for price-mediated contagion.



Indirect exposures

The effective exposure of institution i to asset class κ is given by

E i ,κ(εκ) := Loss(i , εκ)εκ

= Θi ,κ︸︷︷︸Notional exposure

+ FLoss(i , εκ)εκ︸︷︷︸

Indirect exposure

,

where FLoss(i , εk) is the total fire sales loss that i suffers as aresult to the shock εκ to asset class κ.

The effective exposure isscenario dependent and accounts for losses that i would suffer in astress scenario.

→ it reflects the network dependent (and actual!) risk of i ’sportfolio.



Indirect exposures

The effective exposure of institution i to asset class κ is given by

E i ,κ(εκ) := Loss(i , εκ)εκ

= Θi ,κ︸︷︷︸Notional exposure

+ FLoss(i , εκ)εκ︸︷︷︸

Indirect exposure

,

where FLoss(i , εk) is the total fire sales loss that i suffers as aresult to the shock εκ to asset class κ. The effective exposure isscenario dependent and accounts for losses that i would suffer in astress scenario.

→ it reflects the network dependent (and actual!) risk of i ’sportfolio.



Indirect exposures: how large are they?

Figure: Losses of HSBC and Banco Santander as a function of depreciation ofSpanish housing sector.



Figure: Indirect exposures of UK banks to Spanish housing sector, 2011.



Monitoring



Portfolio overlaps as drivers of Indirect contagion

When market impact is linear, the mark-to-market loss of iresulting from fire sales is given by

Li ≈N∑

j=1

M∑µ=1

ΠiµΠjµ

Dµ︸︷︷︸Ωij

Γj =N∑

j=1ΩijΓj ,

where Ωij is the liquidity weighted overlap between portfolios i andj (Cont & Wagalath 2013).Thus: price mediated contagion can be modeled as a contagionprocess on a network whose nodes are financial institutions andwhose links are weighted with liquidity weighted overlaps.



European banking system: indirect contagion network



5 10 15 20

010

0000

2000

00

Rank

Eig

enva

lue

Figure: European banking system: Principal component analysis of liquidity-weighted overlaps. Source: EBA (public)



Liquidity weighted overlaps: 1st principal component

0 20 40 60 80

0.0

0.1

0.2

0.3

0.4

0.5

Component

Val

ue

Deutsche BankBNP

Soc.Gen.

HSBC

BarclaysLloyds

UniCredit

ING

DNB

Countries of EV plots

ESDEFRGB/IEITNO/SE/FI/DKPT/GRrest

Figure: European banking system: Liquidity weighted overlaps. Source: EBA(public)



Indirect Contagion Index

The principal eigenvector U = (Ui , i = 1...N) corresponding to thelargest eigenvalue of the matrix of liquidity-weighted overlapsprovides a measure of (eignevector) centrality of the node i in theindirect contagion network

Definition (Indirect Contagion Index (ICI))We define the Indirect Contagion Index (ICI) of a financialinstitution i as its component Ui in the (normalized) principaleigenvector of the matrix of liquidity weighted portfolio overlaps:

ICI(i) = Ui



Indirect Contagion Index as a measure of exposure to firesales loss

Figure: Regression of log(FLoss i ) on log(ICIi ) for a 13% shock at estimatedmarket depth. R2 = 0.89. Source: EBA (public) & authors calculations.



Comparison to “leverage targeting"models



Response functions

Figure: Leverage targeting response function (dashed) and two variants of thethreshold model (full and circles) response functions.



Distribution of fire sales losses

Figure: Fire sales loss for different scenarios and different model combinations.




fire_sales_loss_percentage_total_loss.png



Can the effect of price-mediated contagion be mimicked byscaling up macro shocks in stress tests?

No: scaling up the macro shocks can replicate the average bankloss but not the cross-sectional distribution of losses across banks.



SummaryQuantitative model for fire sales spillovers in a network ofinstitutions with common asset holdings subject to one-sidedportfolio constraints:

• Tipping point: Existence of critical macro shock level beyondwhich fire sales are triggered and significant contagion occurs.In EU banks: threshold large – but not extreme.

• Fire sales losses: Even with optimistic estimates of marketdepth, fire sales losses can amount to over 20% of systembank equity. This is significant enough to change the outcomeof stress tests.

• Heterogeneity of bank losses: The cross sectionaldistribution of losses due to fire sales cannot be replicated bysimply applying a larger initial macro-shock to all assets orbanks.



Summary• Illiquidity and insolvency: Our model allows to distinguishbetween failures due to insolvency and defaults due toilliquidity. Ignoring failures due to illiquidity may lead to asevere underestimation of the extent of contagion.

• Indirect exposures: Our model leads to a quantifiable notionof indirect exposure to an asset class. EU banks are shown tohave significant exposure to housing markets in otherEuropean countries.→ Calls for a re-thinking of macro-prudential regulation at thenational level.

• Indirect contagion index: Liquidity-weighted overlaps leadto a bank-level indicator that may be used for monitoring andfor quantifying the contribution (and vulnerability) of afinancial institution to price-mediated contagion;



Implications for macroprudential supervision and policy• Incorporating bank reactions greatly alters the outcome of thestress tests

• Capital adequacy should be examined in the light of systemicstress tests incorporating such endogenous effects andcontagion mechanisms

• Fire sales and the resulting price-mediated contagion leads tosignificant indirect exposures across sectors and countries.Systemic stress tests allow to evaluate these indirectexposures.

• Disseminating indirect exposures can help financialinstitutions manage and internalize this risk.

• Most failures occurs through illiquidity, not insolvency:suspension of mark-to-market accounting for illiquid assetsdoes not necessarily help this.



References

• R Cont and E Schaanning. Fire sales, indirect contagion andsystemic stress-testing,http://ssrn.com/abstract=2541114

• R Cont, L Wagalath (2013) Running for the Exit: DistressedSelling and Endogenous Correlation in Financial Markets,Mathematical Finance, Vol 23, Issue 4, p. 718-741, October2013.

• R Cont, L Wagalath: Fire sale forensics: measuringendogenous risk. Mathematical Finance, Volume 26, Issue 4(Oct. 2016) 835-866.


http://ssrn.com/abstract=2541114

systemic stress-testing: fire sales, endogenous risk and ... ·...

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