Introduction Model Joint and cond risk SMP and EFSF Conclusion
Conditional probabilities for Euro area sovereigndefault risk
Andre Lucas, Bernd Schwaab, Xin Zhang
Rare events conference, San Francisco, 28-29 Sep 2012email: [email protected]: www.berndschwaab.eu
Disclaimer: Not necessarily the views of ECB or ESCB.
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Contributions
We propose a novel modeling framework to infer conditional andjoint probabilities for sovereign default risk from observed CDS.
Novel framework? Based on a dynamic GH skewed—t multivariatedensity/copula with time-varying volatility and correlations.
Multivariate model is suffi ciently flexible to be calibrated daily to creditmarket expectations. Not an "offi cial opinion".
Analysis is based on Euro area CDS data from 2008M1 to 2011M6.Event study: SMP/EFSF announcement & initial impact on risk.
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. 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).
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Empirical questions
(Q1) Financial stability information: Based on credit marketexpectations, 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 parametricassumptions? Normal vs Student-t vs GH skewed-t.
(Q3) Event study: did the May 09, 2010 Euro area rescue packagechange risk dependence? How?
Introduction Model Joint and cond risk SMP and EFSF Conclusion
The GHST copula framework
Sovereign defaults iff benefits (vit ) exceed a cost (c it ), where
vit = (ςt − µς)L̃itγ+√
ςt L̃itεt , i = 1, ..., n,
εt ∼N(0, In) is a vector of risk factors,L̃it 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−1i (1− pit ),
where Fi is the CDF of vit .
Focus on conditional probability Pr[vit > cit |vjt > cjt ], i 6= j .
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Data: skewed, fat tailed, tv vol’s and correlation
ATBEDEESFRGRIEITNLPT
2008 2009 2010 2011
500
1500
2500 ATBEDEESFRGRIEITNLPT
Average of rolling window correlations
2008 2009 2010 2011
0.25
0.50
0.75
Average of rolling window correlations
squared differences, AT
2008 2009 2010 2011
2.55.07.5 squared differences, AT
squared differences, GR
2008 2009 2010 2011
51015 squared differences, GR
Austria
2008 2009 2010 20110.250.000.25
Austria
Greece
2008 2009 2010 2011
2.5
0.0
2.5 Greece
Introduction Model Joint and cond risk SMP and EFSF Conclusion
The GH skewed-t multivariate distribution
yt = µ+ Ltet , t = 1, ...,T , et ∼ GHST, E[ete ′t ] = In,
p(y t ; ·) =υ
υ2 21−
υ+n2
Γ(
υ2
)π
n2∣∣Σ̃t ∣∣ 12 ·
K υ+n2
(√d(yt ) · (γ′γ)
)eγ′L̃−1t (yt−µ̃t )
(d(yt ) · (γ′γ))−υ+n4 d(yt )
υ+n2
,
where
d(yt ) = υ+ (yt − µ̃t )′Σ̃−1t (yt − µ̃t ),
µ̃t = −υ/(υ− 2) L̃tγ,Σ̃t = L̃t L̃′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.Σ̃t (ft ) = L̃t (ft )L̃t (ft )′ is driven by 1st and 2nd derivative of the pdf.
Introduction Model Joint and cond risk SMP and EFSF Conclusion
The model with time varying parameters
Assume that Σt = DtRtDt = Lt (ft )Lt (ft )′ and that
ft+1 = ω+∑p−1i=0 Ai st−i+∑q−1
j=0 Bj ft−j ,
where st = St∇t is the scaled score∇t = ∂ ln p(y t ; Σ̃(f t ),γ, υ)/∂f tSt = Et−1[∇t∇′t |yt−1, yt−2, ...]−1,
Scaling matrix St is inverse conditional Fisher information matrix.
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(y t ; Σ̃(f t ),γ, υ)/∂f t
=∂vech(Σt )′
∂ft
∂vech(Lt )′
∂vech(Σt )∂vec(L̃t )′
∂vech(Lt )∂ ln pGH (yt |ft )
∂vec(L̃t )= ...
= Ψ′tH′tvec
{wtyty ′t − Σ̃t −
(1− υ
υ− 2wt)L̃tγy ′t
}where Ψt = ∂vech(Σt )/∂f ′t
Ht = messy
wt =υ+ n2 · d(yt )
−k ′v+n
2
(√d(yt ) · (γ′γ)
)√d(yt )/γ′γ
; k ′a(b) =∂ lnKa(b)
∂b.
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Extracting marginal pd’s from CDS
We equate the premium and default leg of CDS given a default intensity.
pit ≈sit (1+ rt )1− reci
, (∗)
where
sit = CDS annual fee, country i , time t
rt = LIBOR 1 year rate, flat
reci = 25% expected recovery, stressed.
Eqn (∗) is exact if the term structures for pd’s and interest rates are flat,sit is paid to the seller continuously, and there is no counterparty creditrisk, see Brigo and Mercurio (2007).
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Marginal pd’s from CDS
AustriaGermanyFranceIrelandNetherlands
Belg iumSpainGreeceItalyPortugal
2008 2009 2010 2011
0.05
0.10
0.15
0.20
0.25
0.30
0.35AustriaGermanyFranceIrelandNetherlands
Belg iumSpainGreeceItalyPortugal
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Volatility estimates
2008 2009 2010 2011
0.05
0.15 Austria CDS changes squaredAustria t Est. vol
Austria Gaussian Est. volAustria GHST Est. vol
2008 2009 2010 2011
0.025
0.075Belgium CDS changes squaredBelgium t Est. vol
Belgium Gaussian Est. volBelgium GHST Est. vol
2008 2009 2010 2011
0.005
0.010Germany CDS changes squaredGermany t Est. vol
Germany Gaussian Est. volGermany GHST Est. vol
2008 2009 2010 2011
1
2 Spain CDS changes squaredSpain t Est. vol
Spain Gaussian Est. volSpain GHST Est. vol
2008 2009 2010 2011
0.01
0.02 France CDS changes squaredFrance t Est. vol
France Gaussian Est. volFrance GHST Est. vol
2008 2009 2010 2011
5
15Greece CDS changes squaredGreece t Est. vol
Greece Gaussian Est. volGreece GHST Est. vol
2008 2009 2010 2011
0.25
0.50 Ireland CDS changes squaredIreland t Est. vol
Ireland Gaussian Est. volIreland GHST Est. vol
2008 2009 2010 2011
0.25
0.50 Italy CDS changes squaredItaly t Est. vol
Italy Gaussian Est. volItaly GHST Est. vol
2008 2009 2010 2011
0.025
0.050 Netherlands CDS changes squaredNetherlands t Est. vol
Netherlands Gaussian Est. volNetherlands GHST Est. vol
2008 2009 2010 2011
1
3 Portugal CDS changes squaredPortugal t Est. vol
Portugal Gaussian Est. volPortugal GHST Est. vol
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Dynamic correlations
2008 2009 2010 2011
0.25
0.50
0.75
Gaussian correlationRolling window correlationGaussian correlationRolling window correlation
2008 2009 2010 2011
0.25
0.50
0.75
t correlationRolling window correlationt correlationRolling window correlation
2008 2009 2010 2011
0.25
0.50
0.75
GHST correlationRolling window correlationGHST correlationRolling window correlation
2008 2009 2010 2011
0.25
0.50
0.75
Gaussian correlationt correlationGHST correlationrolling window correlation
Gaussian correlationt correlationGHST correlationrolling window correlation
Introduction Model Joint and cond risk SMP and EFSF Conclusion
The probability of two or more failures
Prob of 2 or more defaults, GaussianSymmetric tGHST
2008 2009 2010 2011
0.025
0.050
0.075
0.100
0.125
0.150 Prob of 2 or more defaults, GaussianSymmetric tGHST
Introduction Model Joint and cond risk SMP and EFSF Conclusion
The probability of k=0,1,2,... failures
0 default1 default2 defaults3 defaults4 defaults5 defaults6 defaults7 defaults8 defaults9 defaults10 defaults
2008 2009 2010 2011
0.1
0.2
0.3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
0 default1 default2 defaults3 defaults4 defaults5 defaults6 defaults7 defaults8 defaults9 defaults10 defaults
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Conditional pds: Pr(all i| all j)
2008 2009 2010 2011
0.5
1.0
2008 2009 2010 2011
0.5
1.0
2008 2009 2010 2011
0.5
1.0
2008 2009 2010 2011
0.5
1.0
2008 2009 2010 2011
0.250.500.751.00
2008 2009 2010 2011
0.5
1.0
2008 2009 2010 2011
0.5
1.0
2008 2009 2010 2011
0.5
1.0
2008 2009 2010 2011
0.5
1.0
AustriaGermanyFranceIrelandNetherlands
BelgiumSpainGreeceItalyPortugal
2008 2009 2010 2011
0.5
1.0AustriaGermanyFranceIrelandNetherlands
BelgiumSpainGreeceItalyPortugal
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Conditional pds: Pr(i|GR)
2008 2009 2010 2011
0.2
0.4
0.6Gaussian
2008 2009 2010 2011
0.2
0.4
0.6Symmetric t
2008 2009 2010 2011
0.2
0.4
0.6GH Skewed t Austria
BelgiumGermanySpainFranceIrelandItalyNetherlandsPortugal
2008 2009 2010 2011
0.2
0.4
0.6GH Skewed twith zero correlation
AustriaBelgiumGermanySpainFranceIrelandItalyNetherlandsPortugal
Introduction Model Joint and cond risk SMP and EFSF Conclusion
GHST spillovers: Pr(i|GR) - Pr(i|not GR)
2008 2009 2010 2011
0.25
0.50
0.75 Pr( i | GR)
AustriaGermanyFranceItalyPortugal
BelgiumSpainIrelandNetherlands
2008 2009 2010 2011
0.01
0.02
0.03
0.04 Pr( i | not GR)AustriaGermanyFranceItalyPortugal
BelgiumSpainIrelandNetherlands
2008 2009 2010 2011
0.25
0.50
0.75 Pr( i | GR) Pr( i | not GR)
Introduction Model Joint and cond risk SMP and EFSF Conclusion
The May 09, 2010 package
Joint risk, Pr(i ∩ j)
Thu 06 May 2010 Tue 11 May 2010
PT GR DE PT GR DE
AT 1.1% 1.1% 0.6% 0.6% 0.7% 0.4%
BE 1.2% 1.4% 0.7% 0.9% 1.0% 0.6%
DE 1.0% 1.1% 0.8% 0.8%
ES 3.0% 3.3% 0.9% 1.5% 1.6% 0.6%
FR 1.0% 1.0% 0.6% 0.8% 0.9% 0.6%
GR 4.8% 1.1% 2.3% 0.8%
IR 2.6% 3.1% 0.8% 1.4% 1.8% 0.6%
IT 2.8% 2.9% 0.9% 1.4% 1.5% 0.6%
NL 0.9% 0.9% 0.5% 0.6% 0.7% 0.5%
PT 4.8% 1.0% 2.3% 0.8%
Avg 2.0% 2.2% 0.8% 1.1% 1.2% 0.6%
Introduction Model Joint and cond risk SMP and EFSF Conclusion
The May 09, 2010 package
Conditional risk, Pr(i | j)Thu 06 May 2010 Tue 11 May 2010
PT GR DE PT GR DE
AT 17% 8% 53% 22% 10% 46%
BE 20% 10% 60% 32% 15% 61%
DE 16% 8% 26% 12%
ES 49% 25% 78% 50% 23% 63%
FR 16% 8% 58% 28% 12% 62%
GR 78% 99% 80% 86%
IR 43% 23% 75% 49% 26% 68%
IT 45% 22% 77% 49% 21% 64%
NL 14% 7% 49% 21% 10% 50%
PT 36% 91% 33% 81%
Avg 33% 16% 71% 40% 18% 64%
Bottom line: joint risks ↓↓, but dependence ↑. "Firewall"-analogy?
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Conclusion
We propose a novel modeling framework to infer conditional andjoint probabilities for sovereign default risk from observed CDS.
Novel framework? Based on a dynamic GH skewed—t multivariatedensity/copula with time-varying volatility and correlations.
Multivariate model is suffi ciently flexible to be calibrated daily to creditmarket expectations.
Analysis is based on Euro area CDS data from 2008M1 to 2011M6.Event study: SMP/EFSF announcement & initial impact on risk.
Introduction Model Joint and cond risk SMP and EFSF Conclusion
Thank you
