This document presents a novel framework for modeling conditional and joint probabilities of sovereign default risk in the Euro area based on credit default swap data. The model uses a dynamic multivariate skewed-t distribution with time-varying volatility and correlations. The analysis finds that while large-scale asset purchase programs by central banks reduced joint default risks, they did not significantly impact perceived interconnectedness between countries.

1. Conditional Probabilities for Euro Area
Sovereign Default Risk*
SYstemic Risk TOmography:
Signals, Measurements, Transmission
Channels, and Policy Interventions
2nd Conference on Credit Analysis and Risk
Management
Basel, Sep 6, 2013
Andre Lucas, Bernd Schwaab, Xin Zhang
VU University Amsterdam / ECB / Riksbank
*: Not necessarily the views of ECB or Sveriges Riksbank

2. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Contributions
We propose a novel modeling framework to infer conditional and
joint probabilities for sovereign default risk from observed CDS.
Novel framework? Based on a dynamic GH skewed—t multivariate
density/copula with time-varying volatility and correlations.
Multivariate model is sufficiently flexible to be calibrated daily to credit
market expectations. Not an "official opinion".
Analysis is based on Euro area CDS data from 2008 to end-2012.
Event study: SMP/EFSF and OMT announcement & impact on risk.

3. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Literature
1. Sovereign credit risk: e.g. Pan and Singleton (2008), Longstaff,
Pan, Pedersen, and Singleton (2011), Ang and Longstaff (2011).
2. Risk contagion, see e.g. Forbes and Rigobon (2002), Caporin,
Pelizzon, Ravazzolo, Rigobon (2012).
3. Observation-driven time-varying parameter models, see Creal,
Koopman, and Lucas (2011, 2012), Zhang, Creal, Koopman, Lucas
(2011), Creal, Schwaab, Koopman, Lucas (2011), Harvey (2012).
4. Non-Gaussian dependence/copula/credit modeling, see e.g.
Demarta and McNeil (2005), Patton and Oh (2011).

4. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Empirical questions
(Q1) Financial stability information: Based on credit market
expectations, what is ...
Pr(two or more credit events in Euro area)?
Pr(i|j)-Pr(i), for any i,j?
Spillovers, e.g. Pr(PT|GR) - Pr(PT|not GR)?
Corrt (i,j) at time t?
(Q2) Model risk: For answering (a), how important are parametric
assumptions? Normal vs Student-t vs GH skewed-t.
(Q3) Event study: did the May 09, 2010 SMP/EFSF announcement
change risk dependence? How?
Did the Aug 02, 2012 OMT announcement change risk dependence?

5. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Copula framework
Sovereign defaults iff benefits (vit ) exceed a cost (cit ), where
vit = (ςt − µς)˜Lit γ +
√
ςt
˜Lit ǫt , i = 1, ..., n,
ǫt ∼N(0, In) is a vector of risk factors,
˜Lit contains risk factor loadings,
γ ∈ Rn determines skewness,
ςt ∼ IG is an additional scalar risk factor for, say, interconnectedness.
A default occurs with probability pit , where
pit = Pr[vit > cit ] = 1 − Fi (cit ) ⇔ cit = F−1
i (1 − pit ),
where Fi is the CDF of vit .
Focus on conditional probability Pr[vit > cit |vjt > cjt ], i 6= j.

6. Introduction Model Joint and cond risk SMP and EFSF Conclusion
GH skewed-t dependence
yt = µ + Lt et , t = 1, ..., T, et ∼ GHST, E[et e′
t ] = In,
p(yt ; ·) =
υ
υ
2 21− υ+n
2
Γ υ
2 π
n
2 ˜Σt
1
2
·
Kυ+n
2
p
d(yt ) · (γ′γ) eγ′ ˜L−1
t (yt −˜µt )
(d(yt ) · (γ′γ))− υ+n
4 d(yt )
υ+n
2
,
where
d(yt ) = υ + (yt − ˜µt )′ ˜Σ−1
t (yt − ˜µt ),
˜µt = −υ/(υ − 2) ˜Lt γ,
˜Σt = ˜Lt
˜L′
t is scale matrix
If γ = 0, then GH skewed-t simplifies to Student’s t density.
If in addition υ−1 → 0, then multivariate Gaussian density.
˜Σt (ft ) = ˜Lt (ft )˜Lt (ft )′ is driven by 1st and 2nd derivative of the pdf.

7. Introduction Model Joint and cond risk SMP and EFSF Conclusion
The model with time varying parameters
Assume that Σt = Dt Rt Dt = Lt (ft )Lt (ft )′ and that
ft+1 = ω+ ∑
p−1
i=0
Ai st−i + ∑
q−1
j=0
Bj ft−j ,
where st = St ∇t is the scaled score
∇t = ∂ ln p(yt ; ˜Σ(f t ), γ, υ)/∂f t
St = Et−1[∇t ∇′
t |yt−1, yt−2, ...]−1
,
Scaling matrix St is inverse conditional Fisher information matrix.

8. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Time varying parameters: score
Important: first two derivatives are available in closed form.
∇t = ∂ ln p(yt ; ˜Σ(f t ), γ, υ)/∂f t
=
∂vech(Σt )′
∂ft
∂vech(Lt )′
∂vech(Σt )
∂vec(˜Lt )′
∂vech(Lt )
∂ ln pGH (yt |ft )
∂vec(˜Lt )
= ...
= Ψ′
t H′
t vec wt yt y′
t − ˜Σt − 1 −
υ
υ − 2
wt
˜Lt γy′
t
where Ψt = ∂vech(Σt )/∂f ′
t
Ht = messy
wt =
υ + n
2 · d(yt )
−
k′
v +n
2
p
d(yt ) · (γ′γ)
p
d(yt )/γ′γ
; k′
a(b) =
∂ ln Ka(b)
∂b
.

9. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Extracting marginal pd’s from CDS
CDS fee equates a premium leg and a default leg given a default
intensity, see Duffie (1999), O’Kane and Turnbull (2003).
• We use a nonlinear solver to find the default intensity that matches
E[PV premium leg] = E[PV default leg].
• Intensity and annual pd are nonlinearly related.
• Use 25% recovery rate and interest rate flat at 1%, ignore
counterparty credit risk.
• Overall, not that difficult.

10. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Marginal pd’s from CDS
2008 2009 2010 2011 2012 2013
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
Austria
Belgium
Germany
Spain
France
Greece
Ireland
Italy
The Netherlands
Portugal

11. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Pr[ 2 or more credit events ]
Pr[2 or more credit events], one year horizon
2008 2009 2010 2011 2012 2013
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
For comparison: announcement of SMP and EFSF on 10 May 2010
26 July 2012
02 Aug 2012
06 Sep 2012
Pr[2 or more credit events], one year horizon

12. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Pr[ 2 or more credit events ]
Wessel, D: "ECB’s Tutorial in Successful Jawboning", WSJ, 09 Jan 2013.

13. Introduction Model Joint and cond risk SMP and EFSF Conclusion
The probability of k=0,1,2,... failures
0 default
1 default
2 defaults
3 defaults
4 defaults
5 defaults
6 defaults
7 defaults
8 defaults
9 defaults
10 defaults
2008 2009 2010 2011 2012 2013
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 default
1 default
2 defaults
3 defaults
4 defaults
5 defaults
6 defaults
7 defaults
8 defaults
9 defaults
10 defaults

14. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Conditional pds: Pr(country i|GR)
Austria
Belgium
Germany
Spain
France
Ireland
Italy
Netherlands
Portugal
2008 2009 2010 2011 2012 2013
0.1
0.2
0.3
0.4
0.5
Austria
Belgium
Germany
Spain
France
Ireland
Italy
Netherlands
Portugal

15. Introduction Model Joint and cond risk SMP and EFSF Conclusion
The SMP/EFSF package, 10 May 2010
Joint risk, Pr(i ∩ j)
Thu 06 May 2010 Tue 11 May 2010
PT GR ES PT GR ES
AT 1.1% 1.1% 1.0% 0.6% 0.7% 0.5%
BE 1.2% 1.4% 1.1% 0.9% 1.0% 0.8%
DE 1.0% 1.1% 0.9% 0.8% 0.8% 0.6%
ES 3.0% 3.3% 1.5% 1.6%
FR 1.0% 1.0% 0.8% 0.8% 0.9% 0.7%
GR 4.8% 3.3% 2.3% 1.6%
IR 2.6% 3.1% 2.2% 1.4% 1.8% 1.2%
IT 2.8% 2.9% 2.4% 1.4% 1.5% 1.3%
NL 0.9% 0.9% 0.7% 0.6% 0.7% 0.5%
PT 4.8% 3.0% 2.3% 1.5%
Avg 2.0% 2.2% 1.7% 1.1% 1.2% 0.9%

16. Introduction Model Joint and cond risk SMP and EFSF Conclusion
The SMP/EFSF package, 10 May 2010
Conditional risk, Pr(i | j)
Thu 06 May 2010 Tue 11 May 2010
PT GR ES PT GR ES
AT 17% 8% 26% 22% 10% 27%
BE 20% 10% 28% 32% 15% 40%
DE 16% 8% 23% 26% 12% 30%
ES 49% 25% 50% 23%
FR 16% 8% 22% 28% 12% 35%
GR 78% 87% 80% 81%
IR 43% 23% 57% 49% 26% 58%
IT 45% 22% 63% 49% 21% 65%
NL 14% 7% 19% 21% 10% 24%
PT 36% 79% 33% 74%
Avg 33% 16% 45% 40% 18% 48%
Bottom line: joint risks ↓↓, but dependence ↑ / → .

17. Introduction Model Joint and cond risk SMP and EFSF Conclusion
The OMT annoucement(s)
Joint risk, Pr(i ∩ j)
Thu 24 Jul 2012 Fri 07 Sep 2012
PT GR ES PT GR ES
AT 1.2% 1.4% 1.2% 0.8% 0.7% 0.7%
BE 1.8% 2.4% 2.0% 1.3% 1.5% 1.5%
DE 0.9% 1.1% 0.9% 0.8% 0.9% 0.9%
ES 3.8% 7.0% 2.5% 4.0%
FR 1.6% 2.1% 1.8% 1.1% 1.3% 1.2%
GR 9.5% 7.0% 5.6% 4.0%
IR 4.3% 6.4% 3.5% 2.7% 4.2% 2.6%
IT 3.5% 6.2% 5.0% 2.5% 3.5% 3.2%
NL 1.2% 1.4% 1.0% 1.0% 1.1% 1.1%
PT 9.5% 3.8% 5.6% 2.5%
Avg 3.1% 4.1% 2.9% 2.0% 2.5% 2.0%
Announcements on 26 Jul 12, 02 Aug 12, and 06 Sep 12.

18. Introduction Model Joint and cond risk SMP and EFSF Conclusion
The OMT annoucement(s)
Conditional risk, Pr(i | j)
Thu 24 Jul 2012 Fri 07 Sep 2012
PT GR ES PT GR ES
AT 11% 1% 16% 11% 1% 15%
BE 17% 3% 26% 20% 2% 31%
DE 8% 1% 11% 11% 1% 19%
ES 35% 8% 37% 5%
FR 15% 2% 23% 17% 2% 25%
GR 87% 90% 83% 83%
IR 40% 7% 45% 40% 5% 54%
IT 32% 7% 64% 37% 4% 66%
NL 11% 2% 13% 15% 1% 22%
PT 10% 48% 7% 52%
Avg 28% 5% 37% 30% 3% 41%
Bottom line: joint risks ↓↓, but dependence ↑ / → . "Firewall"-analogy?

19. Introduction Model Joint and cond risk SMP and EFSF Conclusion
Conclusion
We proposed a novel modeling framework to infer conditional and
joint probabilities for sovereign default risk from observed CDS.
Based on a dynamic skewed—t multivariate density with time-varying
volatility and correlations.
Central bank lage-scale asset purchase program announcements (such as
SMP, OMT) affected joint risk primarily through impact on marginal
risks, not perceived connectedness.

20. This project is funded by the European Union
under the 7th Framework Programme
(FP7-SSH/2007-2013) Grant Agreement n°320270
www.syrtoproject.eu