Measuring credit risk in a large banking system:econometric modeling and empirics ∗

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

Financial Stability Conference, Washington D.C.May 30-31, 2013

email: xin.zhang@riksbank.se

∗Not necessarily the views of ECB or Sveriges Riksbank.1 / 24

Motivation

Financial stability assessments are important: Dodd-Frank legislation in

U.S., EBA and EC/ECB stress tests in E.U..

Stress tests typically focus on a large cross section (say 15-100) of larger

banking groups.

Stress tests are expensive, thus infrequent, and subject to other issues.

Model-based risk/stability assessments are cheap and fast, can be done

weekly, though potentially less accurate.

2 / 24

Contributions

A novel non-Gaussian framework for financial stability assessment, based

on time-varying probabilities of simultaneous financial firm defaults.

Focus on extreme tail events.

Overcome substantial difficulties in the econometric modeling of

large-dimensional, unbalanced panels.

Small empirical study: which factors determine systemic correlation and

systemic risk?

3 / 24

Literature

1 Systemic risk: see e.g. Brownlees and Engle (2011),Acharya, Pedersen, Philippon, Richardson (2012), Billio,Getmansky, Lo, Pelizzon (2012),...

2 Observation-driven time-varying parameter models:Creal, Koopman, Lucas (2008), Harvey (2008), Lucas,Schwaab, Zhang (2011),...

3 Large dimensional covariance matrix models: Engle,Shephard, Sheppard (2008), Patton and Oh (2011, 2013),Engle and Kelly (2012),...

4 / 24

The firm asset value

Let yt = (y1,t, · · · , yN,t)′ be generated by a normal mean-variancemixture process

yt = (ςt − µς)Ltγ +√ςtLtεt, (1)

where1) εt ∼ N(0, IN ),2) ςt follows a InverseGamma(ν/2, ν/2) is an additional risk factor,e.g. for interconnectedness,3) Lt is related to the covariance Σt.

5 / 24

The Levy process driven Merton model

A simple Levy driven firm default model:

yt ∼ pghst(Σt, γ, ν), (2)

with Σt = LtL′t = DtRtDt.

The firm i and j’s joint probability of default pij,t:

pij,t = Fρij,t(y∗i,t, y

∗j,t), (3)

define ρij,t as one pairwise correlation element in Rt.

6 / 24

The GH (skewed t) distribution

pghst(yt; θt) =νν2 21−

ν+n2

Γ(ν2 )πn2 |Σt|

12

·K ν+n

2

(√d(yt) · (γ′γ)

)eγ′L−1t (yt−µt)

(d(yt) · (γ′γ))− ν+n4 d(yt)

ν+n2

,

θt = {µt, Σt, γ, ν},d(yt) = ν + (yt − µt)′Σ−1t (yt − µt),

µt = − ν

ν − 2Ltγ.

If γ = 0, the GH skewed t simplifies to a Student’s t density.

Time variation in Σt is driven by the Generalized AutoregressiveScore model.

7 / 24

Generalized Autoregressive Score modelRecall that yt ∼ pghst(Σt, γ, ν), now assume that Σt = G(ft) and thatthe time varying parameters follow

ft+1 = ω +Ast+Bf t,

where st = St∇t is the scaled score,

∇t = ∂ log pghst(yt|F t−1; f t, θt)/∂f t.

St = Et−1[∇t∇′t]−1.

If ft = σ2t , in a Normal distribution (GARCH):

ft+1 = ω +Ay2t+1 +Bft.

Consider the skewed t case:

st = St ·Ψ′tH ′tvec(wt · yty′t − Σt −

(1− ν

ν − 2wt

)Ltγy

′t

),

scaling matrix St is inverse Fisher information matrix; see Creal,

Koopman, Lucas (2013 JAE).8 / 24

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.

9 / 24

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.

A one equicorrelation correlation structure for the dynamic score model,benchmark and a special case:

Rt = (1− ρt)I + ρt``′.

10 / 24

The conditional Law of Large Numbers (1)

The mixture model (1) is a two-factor model with commonGaussian factor κt and a mixing factor ςt:

yt = (ςt − µς)γ +√ςtzt,

zt = ηtκt + Λtεt. (4)

The vector ηt ∈ RN×1 and the diagonal matrix Λt ∈ RN×N arefunctions of Rt, γ, ν.

11 / 24

The conditional Law of Large Numbers (2)

The percentage of defaults at time t

cN,t =1

N

N∑i=1

1{yi,t < y∗i,t|κt, ςt},

1{yi,t < y∗i,t|κt, ςt} are conditional independent, as N → +∞:

cN,t ≈1

N

N∑i=1

E(1{yi,t < y∗i,t|κt, ςt}) =1

N

N∑i=1

P(yi,t < y∗i,t|κt, ςt).

We can write the default common factor κt = κ∗t (cp,t, ςt), because

P(yi,t < y∗i,t|κt, ςt) = Φ

((y∗i,t + µςγi − ςtγi)/

√ςt − ηi,tκt

λi,t

∣∣∣∣κt, ςt) .

12 / 24

Two risk measures

The “Banking Stability Measure” (BSM):

pt = P(CN,t > cp,t) =

∫P(κt < κ∗t (cp,t, ςt))p(ςt)dςt.

The “Systemic Risk Measure” (SRM):

P(CN−1,t > c−ip,t|yi,t < y∗i,t) =P(CN−1,t > c−ip,t, yi,t < y∗i,t)

P(yi,t < y∗i,t)

=

∫Φ2(

z∗i,t√µς√

1−σ2ς γ′γ, κ∗t (c

i1,t, ςt), ηi,t)p(ςt)dςt∫

P(κt < κ∗t (ci1,t, ςt))p(ςt)dςt

,

where c−ip,t is the default proportion in the group excluding firm i. The

SRM is calculated as the average over N firms.

13 / 24

Illustration with a small dataset

10 banks in Euro Area:Bank of Ireland, BBVA, Santander, UniCredito, the National Bank ofGreece.

BNP Paribas, Commerzbank, Deutsche Bank, Societe Generale, ING.

Data: January 1994 - June 2010,

198 monthly equity returns and EDF observations.

Models:

1 Dynamic score model: two-step estimation, correlation targeting.

2 Dynamic equicorrelation model.

3 Dynamic two-block equicorrelation model.

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

14 / 24

Data descriptions

Std.Dev. Skewness Kurtosis Minimum MaximumBank of Ireland 1.309 -0.594 16.053 -113.917 106.153BBVA 0.710 -0.512 3.220 -38.894 37.003Santander 0.720 -0.725 3.758 -40.720 37.609BNP Paribas 0.675 -0.502 3.261 -34.001 32.959Commerzbank 0.940 -1.101 5.474 -67.779 45.536Deutsche Bank 0.760 -0.421 3.906 -46.588 45.444Societe Generale 0.777 -0.968 4.110 -53.679 29.201ING 0.896 -1.647 8.939 -73.367 45.187UniCredito 0.752 -0.048 3.282 -44.318 36.017National Bank of Greece 0.938 0.336 2.324 -48.178 53.652

15 / 24

Correlation estimationsC:\RESEARCH\StabilityMeasure\Program\PaperCode\Figure\CorrelationNewNew.png 09/10/12 14:17:13

Page: 1 of 1

2000 2010

0.60

0.65

0.70 Bank of Ireland-BBVA Correlation

2000 2010

0.45

0.50

0.55Bank of Ireland-Santander Correlation

2000 2010

0.5

0.6Bank of Ireland-BNP Paribas Correlation

2000 2010

0.4

0.5Bank of Ireland-Commerzbank Correlation

2000 2010

0.5

0.6

0.7Bank of Ireland-DB Correlation

2000 2010

0.45

0.55 Bank of Ireland-Societe Generale Correlation

2000 2010

0.3

0.4

0.5Bank of Ireland-ING Correlation

2000 2010

0.40

0.45

0.50

0.55Bank of Ireland-UniCredito Correlation

2000 2010

0.60

0.65

0.70 Bank of Ireland-National Bank of Greece Correlation

2000 2010

0.4

0.5

0.6

0.7Deco Correlation

2000 2010

0.25

0.50

0.75 Deco Correlation Rolling Window Ave Corr

2000 2010

0.4

0.6

0.8Deco 1 Deco3

Deco 2

16 / 24

The Banking Stability MeasureC:\RESEARCH\StabilityMeasure\Program\PaperCode\ShortSampleFinal\Fig\FinalBSM.eps 09/26/12 11:51:12

Page: 1 of 1

1995 2000 2005 2010

0.01

0.02

BSM-LLN

1995 2000 2005 2010

0.01

0.02

BSM-SIM

1995 2000 2005 2010

0.01

0.02

BSM-SIM(DECO)

1995 2000 2005 2010

0.01

0.02

BSM-LLN BSM-SIM(DECO)

BSM-SIM

17 / 24

The Systemic Risk MeasureC:\RESEARCH\StabilityMeasure\Program\PaperCode\ShortSampleFinal\Fig\FinalSRM.eps 09/26/12 14:42:40

Page: 1 of 1

2000 2010

0.25

0.50

0.75

1.00Bank of Ireland-LLN Bank of Ireland-SIM

2000 2010

0.50

0.75

1.00BBVA-LLN BBVA-SIM

2000 2010

0.50

0.75

1.00Santander-LLN Santander-SIM

2000 2010

0.50

0.75

1.00BNP Paribas-LLN BNP Paribas-SIM

2000 2010

0.25

0.50

0.75

1.00Commerzbank-LLN Commerzbank-SIM

2000 2010

0.50

0.75

1.00DB-LLN DB-SIM

2000 2010

0.50

0.75

1.00Societe Generale-LLN Societe Generale-SIM

2000 2010

0.6

0.8

1.0ING-LLN ING-SIM

2000 2010

0.50

0.75

1.00UniCredito-LLN UniCredito-SIM

2000 2010

0.25

0.50

0.75

1.00NatBank of Greece-LLN NatBank of Greece-SIM

2000 2010

0.60

0.65

0.70

0.75SRM-LLN

18 / 24

A study of 73 European financial firms

73 European large financial firms: European banks, insurance companies

and investment companies.

Data: January 1992 - June 2010,

monthly equity return and EDF.

Unbalanced Panel: longest time series contains 488 observations and the

shortest one has 10 observations.

Models:

1 Dynamic equicorrelation model.

2 Dynamic equicorrelation model, augmented with economic variables.

Risk measures: Only LLN approximations.

19 / 24

The risk MeasureC:\RESEARCH\StabilityMeasure\Program\PaperCode\LongSampleFinal\SARA\Fig\IndicatorCompareCheck.png 10/04/12 11:20:11

Page: 1 of 1

1995 2000 2005 2010

0.01

0.02

0.03

BSM indicator, GAS-Equicorrelation approx indicator

1995 2000 2005 2010

0.50

0.55

0.60SRM indicator, GAS-Equicorrelation approx indicator

20 / 24

Economic factorsC:\RESEARCH\StabilityMeasure\Program\PaperCode\Figure\ExpVarPlot.png 08/10/12 08:26:35

Page: 1 of 1

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0

1Eurobor-EONIA

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

-0.05

0.00

0.05EU Stock Index Return

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

0.2

0.4

0.6VSTOXX

21 / 24

Economic factors augmented score modelC:\RESEARCH\StabilityMeasure\Program\PaperCode\Figure\CorrelationDecoEcon.png 08/27/12 04:11:53

Page: 1 of 1

2000 2005 2010

0.2

0.3

0.4

0.5

0.6Deco Correlation

2000 2005 2010

0.2

0.3

0.4

0.5

0.6Deco+EconVar Correlation

2000 2005 2010

0.2

0.3

0.4

0.5

0.6Deco Correlation Deco+EconVar Correlation

2000 2005 2010

-1.4

-1.3

-1.2

-1.1

-1.0Deco Factor Deco+EconVar Factor

22 / 24

Conclusion and future research

An econometric framework for financial sector risk assessment.Dynamic model with parsimonious correlation structure.

Two risk measures are proposed for large dimensional financialdata.

Work in progress: systemic importance/risk contribution of eachbank, incorporating additional variables and block correlationstructure.

23 / 24

Thank you.

24 / 24