measuring credit risk in a large banking system: econometric modeling and empirics · 2020. 8....
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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: [email protected]
∗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.
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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?
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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),...
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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.
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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.
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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.
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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.
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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``′.
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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.
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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) .
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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.
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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.
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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
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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
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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
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The Systemic Risk MeasureC:\RESEARCH\StabilityMeasure\Program\PaperCode\ShortSampleFinal\Fig\FinalSRM.eps 09/26/12 14:42:40
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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
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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.
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The risk MeasureC:\RESEARCH\StabilityMeasure\Program\PaperCode\LongSampleFinal\SARA\Fig\IndicatorCompareCheck.png 10/04/12 11:20:11
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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
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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
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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
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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.
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Thank you.
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