measuring credit risk in a large banking system: econometric modeling and empirics - andre lucas,...

Measuring credit risk in a large banking

system: econometric modeling and

empirics*

SYstemic Risk TOmography:

Signals, Measurements, Transmission

Channels, and Policy Interventions

F.E.B.S., 3rd annual conference

June 7, 2013

Andre Lucas, Bernd Schwaab, Xin ZhangVU University Amsterdam / ECB / Riksbank

*: Not necessarily the views of ECB or Sveriges Riksbank

Intro Model Empirics Conclusion

Outline

Intro

Model

Empirics

Conclusion

1 / 37


Motivation

Since 2007, financial stability surveillance and assessment have become

key priorities in central banks, in addition to monetary policy.

Prudential mandate entails a high-dimensional problem. For example,

FDIC oversees > 7000 U.S. banks. SSM ≈ 130 + 5800 European banks.

Objective: Develop framework to give model-based answers to what is

financial sector joint tail risk? Tail risk conditional on one default?

Useful for counterparty credit risk management, and assessing the impact

of monetary policy measures on euro area financial sector (tail) risk.

2 / 37


Contributions

We develop a novel non-Gaussian, high-dimensional framework to infer

conditional and joint measures of financial sector risk.

Derive a conditional LLN to compute risk measures without simulation.

Based on a multivariate skewed–t density, with tv volatilities and

dependence. Fits large cross-section due to a parsimonious factor

structure.

Model is sufficiently flexible for frequent re-calibration to market data.

Works well with unbalanced data/missing values.

Application to euro area financial firms from 1999M1 to 2013M3.

3 / 37


Two problems...and answers

P1: Financial sector comprises many firms. Joint risk assessment is a

high-dimensional & non-Gaussian problem.

A1: GHST handles non-Gaussian features and DECO the large cross

section. The cLLN facilitates computation of joint and conditional risk

measures.

P2: Stress dependence is time-varying and not directly observed. In bad

times, both uncertainty/volatility and dependence increase. Time varying

parameters required.

A2: Either a non-Gaussian state space model, using simulation methods,

or a observation driven/GAS model, using standard Maximum Likelihood.

Thus, high-dimensional non-normal time-varying parameter model, with

unobserved factors.

4 / 37


Literature

1. Portfolio credit risk and loss asymptotics: Vasicek (1977), Lucas,

Straetmans, Spreij, Klaasen (2001), Gordy (2003), Koopman, Lucas, Schwaab

(2011, 2012).

2. Market risk methods (volatility & NG dependence): Engle

(2002), Demarta and McNeil (2005), Creal, Koopman, and Lucas (2011),

Zhang, Creal, Koopman, Lucas (2011).

3. Observation-driven time-varying parameter models: Creal,

Koopman, and Lucas (2013), Creal, Schwaab, Koopman, Lucas (2013), Harvey

(2012), Patton and Oh (2013).

4. Financial sector risk assessment/systemic risk: Most related are

Hartmann, Straetman, de Vries (2005), Malz (2012), Suh (2012), and Black,

Correa, Huang, Zhou (2012).

5 / 37


Market risk - credit risk link

In a Merton (1974) model for i = 1, 2 firms,

dV i,t = Vi,t· (µidt+ σidWi,t) ,

yi,t = log (V i,t/V i,t−1) ∼ N(µi−σ2i /2, σ

2i ),

where Vi,t is the asset value firm i at time t, and dW1,tW2,t = ρdt.

In a Levy-driven model (Bibby and Sorensen (2001)),

dV i,t =1

2v(Vi,t) [log(f(Vi,t)v(Vi,t))]

′dt+

√v(Vi,t)dWi,t,

yi,t = log (V i,t/V i,t−1) ∼ GHST(σ2i , γi, υ),

where v(Vi,t) and f(Vi,t) are real-valued functions.

6 / 37


The GH (skewed t) copula model

Firm defaults iff its log asset value (yit) falls below a threshold (y∗it),where

yit = (ςt − µς)Litγ +√ςtLitεt, i = 1, ..., n,

εt ∼N(0, In) is a vector of risk factors,

Lit contains risk factor loadings, γ ∈ Rn determines skewness,

ςt ∼ IG(ν2 ,ν2 ) is an additional risk factor.

A default occurs with probability pit, where

pit = Pr[yit < y∗it] = Fit(y∗it) ⇔ y∗it = F−1it (pit),

where Fit is the GHST-CDF of yit.

Focus on conditional probabilities Pr[yit < y∗it|yjt < y∗jt], i 6= j, ...

7 / 37


A factor copula model

Consider a two-factor model with common factor κt ∼N(0, 1), common

tail risk factor ςt ∼ IG(ν2 ,ν2 ), and idiosyncratic εt ∼N(0, IN ),

yit = (ςt − µς)γit +√ςtzit, i = 1, ..., N.

zit = ηitκt + λitεit,

where γit = Litγ, E[ςt] = µς and Var[ςt] = σ2ς .

λit =√

1− ρ2it, and ηit=ρit.

Remark: vector ηt∈ RNx1and matrix Λt = diag(λit)∈ RNxN are

functions of ρt (to be estimated later).

8 / 37


The law of large numbers

LLN: In a large sample, empirical

averages are not far away from

their expected values.

9 / 37


The conditional Law of Large Numbers (1)

The portfolio default fraction at time t is

cN,t=1

N

N∑i=1

1{yi,t< y∗i,t}.

As 1{yi,t< y∗i,t|κt, ςt} are conditionally independent, as N →∞,

cN,t ≈1

N

N∑i=1

E[1{yi,t < y∗i,t|κt, ςt}

]=

1

N

N∑i=1

Pr(yi,t < y∗i,t|κt, ςt

):= CN,t.

10 / 37



Two remarks:

• CN,t= 1N

∑Ni=1 Pr

(yi,t < y∗i,t|κt, ςt

)is random because κt, ςt are

random, not because of εt or yi,t.

•

Pr(yi,t < y∗i,t|κt, ςt

)= Φ

((y∗i,t+µςγit−ςtγit)/

√ςt−ηi,tκt

λt

∣∣∣∣κt, ςt) ,where Φ(·) denotes the standard normal CDF.

Given this, a joint tail risk measure (TRMt) is

pt= Pr (CN,t(κt, ςt) > c),

i.e. the probability that the default rate in the portfolio exceeds a fixed

fraction c ∈ [0, 1].

11 / 37



CN,t(κt, ςt) is monotonically decreasing in κt for any fixed ςt.

We use this to efficiently compute threshold levels κ∗t,N (c, ς) for each

value of ς by solving CN,t(κ∗t,N (c, ς), ς) ≡ c.

As a result, we can compute the joint tail risk measure (TRMt) very

quickly based on 1-dimensional numerical integration

pt= Pr(CN,t > c) =

∫Pr (κt< κ∗t,N (c, ςt))p(ςt)dςt.

This is a cause for celebration: works within seconds!

12 / 37


The conditional Law of Large Numbers (4)A systemic influence measure (SIMi,t) is given by

Pr (C(−i)N−1,t> c(−i)|yi,t< y∗i,t)

= p−1it Pr(C(−i)N−1,t > c(−i), yi,t < y∗i,t)

= p−1it

∫Pr (κt< κ∗N−1,t(c

(−i), ςt), yi,t < y∗i,t|ςt)p(ςt)dςt

= p−1it

∫Φ2

(κ∗N−1,t(c

(−i), ςt), z∗i,t(·); ηi,t

)p(ςt)dςt,

where c(−i) is a fixed fraction in the portfolio abstracting from firm i,and z∗i,t(y

∗i,t, ςt) = (y∗i,t − (ςt − µς)γi,t)/

√ςt.

Remark: This is close to Hartmann, Straetman, de Vries (2005)’s Multivariate

Extreme Spillovers; but now time-varying at a high frequency.

13 / 37



Two final remarks:

1. “Connectedness” := 1N

∑Ni=1SIMi,t

2. SIMi,t without tail risk factor, only common factor exposure

= p−1it Pr(κt < κ∗t,N (c(−i), ςt), yi,t < y∗i,t|ςt)∣∣∣ςt≡1

The difference to full SIMi,t is dependence in excess of what is

implied by common factor exposure.

14 / 37


A flexible dynamic distribution

C:\RESEARCH\StabilityMeasure\TexAndOthers\Tex\Presentations\MySlides\graphics\DensGH.eps 12/10/13 17:07:58

Page: 1 of 1

Gaussian t GHST

-10 -8 -6 -4 -2 0 2 4 6 8 10

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Gaussian t GHST

GHST distribution: a

result of the factor

model...and fit the data.

Introduce the time

variation in parameters.

15 / 37


The dynamic GHST distribution

The GHST pdf nests symmetric-t and normal.

p(yt; ·) =υυ2 21−

υ+n2

Γ(υ2

)πn2

∣∣∣Σt

∣∣∣ 12 ·Kυ+n

2

(√d(yt) · (γ

′γ)

)eγ′L−1t (yt−µt)

(d(yt) · (γ

′γ))−υ+n

4 d(yt)υ+n2

,

where ...

d(yt) = υ + (yt−µt)′Σ−1t (yt−µt),

µt = −υ/(υ − 2) Ltγ,

Σt = LtL′t.

16 / 37


Time varying parameters

A score-driven model ...

Σt = DtRtDt

= Dt(f t)Rt(f t)Dt(f t)

ft+1 = ω+∑p−1

i=0Aist−i+

∑q−1

j=0Bjft−j ,

where st = St∇t is the scaled score

∇t = ∂ ln p(yt; Σ(f t), γ, υ)/∂f tSt = Et−1[∇t∇

′t|yt−1, yt−2, ...]

−1scaling matrix.

See CKL (JAE 2013) for overview and BKL (2012) for stationarity

discussion.

17 / 37


Impact curve: robust to outliers

Figure 1: News impact curve under the Student t score-based conditional variancemodel

−4 −2 0 2 4

05

1015

y

w[v

](y)

y2ν=∞ν=10ν=4

3.2 Model for the conditional copula density function

All assets considered here are major US deposit banks. For such a homogeneous

group of assets, it is reasonable to assume that the correlation parameter in the

copula density function is the same for all pairs of stocks. This so-called “dynamic

equicorrelation parameter” has been extensively studied by Engle and Kelly (2012),

under the assumption that there are no shifts in the unconditional equicorrelation.

In our application on financial institutions, this assumption is likely to be violated.

Because of the interconnectedness between financial institutions, episodes of high

systemic risk are characterized by dramatic changes in the volatility and correlations

of financial institutions. See e.g. the recent empirical evidence in Leonidas and

Italo de Paula (2011) and the review in Section 2. Ang and Chen (2002) find that

10

18 / 37


Volatility modeling

In the GARCH-GHST model, the steps st are:

st = St ·Ψ′tH ′tvec(

yty′t − Σt

).

In the score driven GHST model, the steps st are:

st = St ·Ψ′tH ′tvec(w1tyty

′t − Σt − w2tγy

′t

).

19 / 37


Volatility modeling: a comparison

C:\RESEARCH\First Paper\Mphil\2010GASGHST\MGASGHSTFinal\FourSeries\Output\ForSoFiE\VolCompare.gwg 06/13/11 18:09:43

Page: 1 of 1

2002 2003 2004 2005 2006

0.01

0.02

0.03

0.04

0.05

0.06

0.07DCC-GHST Volatility: 03/01/1989 to 01/01/2010

Coca-Cola Merck

IBM JP Morgan

2002 2003 2004 2005 2006

0.01

0.02

0.03

0.04

0.05

0.06

0.07DGH-GHST Volatility: 03/01/1989 to 01/01/2010

Coca-Cola Merck

IBM JP Morgan

20 / 37


A parsimonious correlation structure

If the dimension N is large, we assume N firms divided into mgroups. Group i contains ni firms with equicorrelation structure.

Rt =

(1− ρ21,t)In1 . . . . . . 0

0 (1− ρ22,t)In2 . . . 0

......

. . ....

0 0 . . . (1− ρ2m,t)Inm

+

ρ1,t`1ρ2,t`2

...ρm,t`m

· (ρ1,t`′1 ρ2,t`′2 . . . ρm,t`′m),

where `i ∈ Rni×1 is a column vector of ones and Ini an ni × niidentity matrix.

21 / 37


Speeding up the score computation

Even medium size dimensions are a severe problem for most multivariate

dependence models (think DCC and N = 30).

The matrix calculation in large dimension can be done analytically.

22 / 37


Proposition 1Let yt follow a GHST distribution p(yt; Σt, γ, υ) with zero mean, where

Σt(ft) is driven by the GAS transition equation. Then the dynamic score

is

ft+1 = ω +ASt∇t +Bft,

∇t = Ψ′tH′tvec

{wtyty

′t − Σt −

(1− υ

υ − 2wt

)Ltγy

′t

}Ht = a messy expression (in paper),

Ψt = ∂vech(Σt)′/∂f ′t ,

wt =υ + n

2 · d(yt)−k′v+n

2

(√d(yt) · (γ

′γ)

)√d(yt)/γ

′γ, k′a(b) =

∂ lnKa(b)

∂b.

23 / 37


Proposition 2If yt follows a GHST distribution and the time varying correlation matrix

has equicorrelation structure Rt = (1− ρt)I + ρtll′ and ρt = ft, then

the dynamic score is



(wt · yty′t − Σt −

(1− ν

ν − 2wt

)Ltγy

′t

),

Ht = (Σ−1t ⊗ Σ−1t )(Lt ⊗ Ik),

Ψt =exp(−ft)

(1 + exp(−ft))2(− ρ0,t√

1− ρ20,tvec(Ik)

+

ρ0,t

k√

1− ρ20,t+

(k − 1)ρ0,t

k√

1− ρ20,t + kρ20,t

`k2).

24 / 37


Proposition 3

If yt follows a GHST distribution where the covariance matrix Σt = R

contains m×m blocks with ρi = (1 + exp(−fi,t))−1 for i = 1, · · · ,m,

ft is a m× 1 vector driven by the dynamic score model



(wt · yty′t − Σt −

(1− ν

ν − 2wt

)Ltγy

′t

),

Ht = (Σ−1t ⊗ Σ−1t )(Lt ⊗ Ik),

Ψt =∂vec(Lt)

′

∂ft=∂vec(Lt)

′

∂ρt

dρ′tdft

,

where Ψt are certain block-structured matrix.

25 / 37


Proposition 3: continued

dρ′tdft

= diag

(exp(−f1,t)

(1 + exp(−f1,t))2, · · · ,

exp(−fm,t)(1 + exp(−fm,t))2

),

∂vec(Lt)′

∂ρt=

vec

I1 0 . . . 0

.

.

.

.

.

.. . .

.

.

.0 0 . . . 00 0 . . . 0

, · · · , vec0 0 . . . 0

.

.

.

.

.

.. . .

.

.

.0 0 . . . 00 0 . . . Im

· diag

−ρ1,t√1−ρ21,t

.

.

.−ρm,t√1−ρ2m,t

+

vec

J1,1 0 . . . 0

.

.

.

.

.

.. . .

.

.

.0 0 . . . 00 0 . . . 0

, · · · , vec0 0 . . . 0

.

.

.

.

.

.. . .

.

.

.0 0 . . . 00 0 . . . Jm,m

·

∂c′ii,t

∂ρt

+

vec

0 J1,2 . . . 0

J2,1 0 . . . 0

.

.

.

.

.

.. . .

.

.

.0 0 . . . 0

, · · · , vec0 0 . . . 0

.

.

.

.

.

.. . .

.

.

.0 0 . . . Jm−1,m0 . . . Jm,m−1 0

·

∂c′ij,t

∂ρt,

where cii,t, cij,t are certain scalars (in paper).

26 / 37


Illustration with a small dataset

10 banks in Euro Area:Bank of Ireland, Banco Comercial Portugues, Santander, UniCredito,National Bank of Greece,

BNP Paribas, Deutsche Bank, Dexia, Erste Group, ING.

Data: January 1999 - March 2013,

monthly equity returns and EDF observations.

Risk measures: 10,000,000 simulation based, and/or LLN approximations.

27 / 37


Joint tail risk

Pr(3 or more defaults from 10), DECO, simulated

1999 2001 2003 2005 2007 2009 2011 2013

0.05

0.10

0.15Pr(3 or more defaults from 10), DECO, simulated Pr(3 or more defaults from 10), Full Corr, simulated

1999 2001 2003 2005 2007 2009 2011 2013

0.05

0.10

0.15Pr(3 or more defaults from 10), Full Corr, simulated

Pr(3 or more defaults from 10), DECO, LLN approximation

1999 2001 2003 2005 2007 2009 2011 2013

0.05

0.10

0.15Pr(3 or more defaults from 10), DECO, LLN approximation Pr(3 or more defaults from 10), DECO, sim

Pr(3 or more defaults from 10), Full Corr, sim Pr(3 or more defaults from 10), DECO, LLN

1999 2001 2003 2005 2007 2009 2011 2013

0.05

0.10

0.15 Pr(3 or more defaults from 10), DECO, sim Pr(3 or more defaults from 10), Full Corr, sim Pr(3 or more defaults from 10), DECO, LLN

Equity and EDF measures for ten banks: Bank of Ireland, Banco Comercial Portugues, Santander, UniCredito,

National Bank of Greece, BNP Paribas, Deutsche Bank, Dexia, Erste Group, ING.

28 / 37


Conditional tail risk

Intro Financial framework LLN Estimation/DECO App1, N=10 App2, N=73 Conclusion

Systemic Risk Measuresin euro area financial sector, ten banks, 1999 onwards

SRM, Full Corr, SimSRM, DECO, SimSRM, DECO, LLN

2000 2005 2010

0.5

1.0 Bank of IrelandSRM, Full Corr, SimSRM, DECO, SimSRM, DECO, LLN

Banco Comr. PortuguesSIMFullBanco Comr. PortuguesSIMBanco Comr. PortuguesLLN

2000 2005 2010

0.5

1.0 Banco Com ercial PortuguesBanco Comr. PortuguesSIMFullBanco Comr. PortuguesSIMBanco Comr. PortuguesLLN

SantanderSIMFullSantanderSIMSantanderLLN

2000 2005 2010

0.5

1.0 Santander

SantanderSIMFullSantanderSIMSantanderLLN

UniCreditoSIMFullUniCreditoSIMUniCreditoLLN

2000 2005 2010

0.5

1.0 UniCreditoUniCreditoSIMFullUniCreditoSIMUniCreditoLLN

National Bank of GreeceSIMFullNational Bank of GreeceSIMNational Bank of GreeceLLN

2000 2005 2010

0.5

1.0 National Bank of GreeceNational Bank of GreeceSIMFullNational Bank of GreeceSIMNational Bank of GreeceLLN

BNP ParibasSIMFullBNP ParibasSIMBNP ParibasLLN

2000 2005 2010

0.5

1.0 BNP ParibasBNP ParibasSIMFullBNP ParibasSIMBNP ParibasLLN

DBSIMFullDBSIMDBLLN

2000 2005 2010

0.5

1.0 Deutsche BankDBSIMFullDBSIMDBLLN

DexiaSIMFullDexiaSIMDexiaLLN

2000 2005 2010

0.5

1.0 DexiaDexiaSIMFullDexiaSIMDexiaLLN

ERSTE GROUP BANKSIMFullERSTE GROUP BANKSIMERSTE GROUP BANKLLN

2000 2005 2010

0.5

1.0 Eerste Group BankERSTE GROUP BANKSIMFullERSTE GROUP BANKSIMERSTE GROUP BANKLLN

INGSIMFullINGSIMINGLLN

2000 2005 2010

0.5

1.0 INGINGSIMFullINGSIMINGLLN

Ten banks: Bank of Ireland, Banco Comercial Portugues, Santander, UniCredito, National Bank of Greece,

BNP Paribas, Deutsche Bank, Dexia, Erste Group, ING.

Equity and EDF measures for ten banks: Bank of Ireland, Banco Comercial Portugues, Santander, UniCredito,

National Bank of Greece, BNP Paribas, Deutsche Bank, Dexia, Erste Group, ING.

29 / 37


A study of 73 European financial firms

73 European large financial firms: European banks, insurance companies

and investment companies.

Data: January 1999 - March 2013,

monthly equity return and EDF.

Unbalanced Panel: longest time series contains 172 observations and the

shortest one has 10 observations.

Risk measures: Only LLN approximations.

30 / 37


Joint tail risk

5 or more defaults 7 or more defaults 10 or more defaults

1999 2001 2003 2005 2007 2009 2011 2013

0.1

0.2

0.35 or more defaults 7 or more defaults 10 or more defaults

31 / 37


Average SIM

Avg SIM: Pr[7 or more firms default | firm i defaults]

1999 2001 2003 2005 2007 2009 2011 2013

0.4

0.6

0.8Avg SIM: Pr[7 or more firms default | firm i defaults]

32 / 37


Observed and unobserved risk factors

Which common factors drive stock returns, idiosyncratic volatilities, as

well as stock return correlations, see for example Hou et al. (2011) and

Bekaert et al. (2010).

Observed factors:(i) Euribor-EONIA – measure of liquidity and credit risk,(ii) S&P index return – state of equity markets,

(iii) VSTOXX – indicator of market turbulence.

Equicorrelation handles large cross sections: ρt = ft + βXt.

33 / 37


Economic factors augmented score model

Euribor−EONIA

2000 2005 2010

0.0

0.5

1.0

1.5Euribor−EONIA S&P index

2000 2005 2010

−0.1

0.0

0.1S&P index

VSTOXX

2000 2005 2010

0.2

0.4

0.6 VSTOXX Equicorrelation

2000 2005 2010

0.25

0.50

0.75Equicorrelation

GAS factor GAS factor, with economic factors

2000 2005 2010

0.5

1.0GAS factor GAS factor, with economic factors

Equicorrelation Equicorrelation, with economic factors

2000 2005 2010

0.25

0.50

0.75Equicorrelation Equicorrelation, with economic factors

34 / 37


Estimation results

GAS-Eqcorrelation GAS-Factor(t-1) GAS-Factor(t)

A 0.406 0.517 0.451(0.103) (0.121) (0.115)

B 0.837 0.815 0.827(0.084) (0.083) (0.096)

ω 0.548 0.576 0.548(0.053) (0.099) (0.098)

ν 20.506 20.066 20.229(2.670) (2.654) (1.593)

γ -0.176 -0.181 -0.180(0.038) (0.040) (0.040)

Euribor-EONIA 0.340 0.042(0.129) (0.128)

S&P index -0.704 -0.323(0.344) (0.407)

VSTOXX -0.498 -0.040(0.328) (0.312)

Log-lik 2913.072 2922.797 2913.524

35 / 37


Conclusion

How to obtain estimates of financial sector joint tail risk, and tail risk

conditional on a default, if the cross section is very large?

A: GHST-GAS-DECO, a non-Gaussian high-dimensional framework.

Equicorrelation handles large cross sections; works with unbalanced data.

cLLN permits to compute risk measures quickly, without simulation.

Application to euro area financial firms from 1999M1 to 2013M3.

36 / 37

This project is funded by the European Union

under the 7th Framework Programme

(FP7-SSH/2007-2013) Grant Agreement n°320270

www.syrtoproject.eu

measuring credit risk in a large banking system: econometric modeling and empirics - andre lucas,...

Economy & Finance

conditional risk measures

tail risk conditional

joint risk assessment

portfolio credit risk

measuring credit risk

nancial sector joint

systemic risk tomography

modelbased answers