Modelling cross-border bank contagion using … contagion effects in banking with copulae Methodological proposal Empirical evidence Conclusions Modelling cross-border bank contagion

Estimating contagion effects in banking with copulaeMethodological proposal

Empirical evidenceConclusions

Modelling cross-border bank contagion usingMarshall-Olkin copula

Silvia Angela Osmetti(joint work with R. Calabrese)

[email protected] "Cattolica del Sacro Cuore" of Milan

Aug 28-30th, 2013Credit Scoring and Credit Control XIII Conference

University of EdinburghBusiness School

Silvia Angela Osmetti Modelling cross-border bank contagion



Outline

1 Estimating contagion effects in banking with copulaeCopulae and tail dependenceSome copula-based models for bank contagion

2 Methodological proposalThe Marshall-Olkin copulaEstimation procedureThe censored sampling

3 Empirical evidenceDatasetEstimation results

4 Conclusions




Outline




4 Conclusions




Outline




4 Conclusions




Outline




4 Conclusions




Copulae and tail dependenceSome copula-based models for bank contagion

Definition of Copula

Let FX (x) = P(X ≤ x) and FY (y) = P(Y ≤ y) be the PDs ofthe banks of two countries in a given time.Let H(x ,y) = P(X ≤ x ,Y ≤ y) the joint PDs.

The copula is the function C : [0,1]× [0,1]→ [0,1] such that

H(x ,y) = C(FX (x),FY (y))

The joint distribution is a function of two components:1] the marginal distributions of the banks PDs2] the dependence structure between the banks defaults of

the two countries





Definition of Copula

The copula is

C(u,v) = P(U ≤ u,V ≤ v) 0 < u < 1; 0 < v < 1

• The copula focus on the dependence between the PDs of thebanks of the two countries.• It is independent from parametric assumptions on themarginal models for bank default prediction.• It allow to identify different levels of dependence across thedistributions: i.e. the upper tail dependence

λu = limu→1−1

P[X > F−1X (u)|Y > F−1

Y (u))] = limu→1−1

P[Y > F−1Y (u)|X > F−1

X (u))]





Some drawbacks in copula-based models

Few authors applied copula-based models to analyse bankcontagion:

Weiss (2012) modelled the dependence of abnormal bankreturn by mixtures of Student’s t, Frank, Clayton, Gumbel;De Vries (2005) modelled the interbank deposit market bythe Farlie-Gumbel-Morgenstern copula.

SOME POSSIBLE DRAWBACKSmarket models cannot be applied to all bankscopulae without tail dependence are not suitableextreme value copulae are suitablethe impact of common shocks is underestimated




The Marshall-Olkin copulaEstimation procedureThe censored sampling

Our Proposal

In order to model cross-border bank contagion effects, wesuggest to apply the Marshall-Olkin (MO) copula to theempirical distribution functions of time to banks failure of twocountries. The failures are due to:

idiosyncratic characteristics (balance sheet data)common shocks (i.e. economic cycle)





Our Proposal

MAIN ADVANTAGES OF OUR PROPOSALSince marginal distributions are nonparametricallyestimated, the results do not depend on the models usedto estimate the probability of banks failure (PD).Since the MO copula is an extreme value copula, ourproposal is suitable to study association between extremevalues.MO copula shows an upper tail dependence: high valuesof PDs show strong dependence.MO copula shows a singularity, assigning non nullprobability to the event "in the two countries the banksdefault in the same period" due probably to the commonshocks.





Our Proposal






Our Proposal






The Marshall-Olkin (MO) copula

The MO copula comes from the non fatal-shock model ofMarshall and Olkin

F (X ,Y ) = exp(−(λ1 +λ3)x − (λ2 +λ3)y +λ3 min(x ,y))

Starting from MO model we obtain the MO copula

C(u,v) = uv min(u−θ ,v−θ ),

θ ∈ [0,1] is the tail dependence parameteru and v are the PDs of the banks of the two countries in agiven time.





The Marshall-Olkin (MO) copula

The MO copula has a singularity for u = v . Therefore:

C(u,v) =2−2θ

2−θCa(u,v)+

θ

2−θCs(u,v)

It shows the impact of θ on the model:� for θ = 0 there is only the absolutely continuous part� for θ = 1 there is only the singularity.

For high values of θ , common shock is the most importantcomponent of the dependence structure.





Observations from copula





Canonical Maximum likelihood (CML)

Step I the marginal PDs are estimated by the empiricalcumulative distribution function of the variables time to default

ui = FX (xi) and vi = FY (yi)

Step II the copula parameter is estimated by maximizing theconditional likelihood function

L(θ |u, v) =n

∏i=1

cθ (u, v)

with respect to θ ∈ (0,1) where c is the copula density function.





Canonical Maximum likelihood (CML)

We found the estimator of θ

θ = (1+exp(−ψ))−1

with

ψ =− ln

n−2n3−Smin +√

n2 +S2min−Smin(2n−4n3)

2n3

with n3 > 0 and Smin =

n∑

i=1min(− ln(ui),− ln(vi)).





I type censored sampling

(X ,Y ) are censored at T = (t∗, t∗)

All the information of non-defaulted banks can be used toestimate the parameters.





I type censored sampling

Step I the marginal PDs are estimated by the Kamplan-Maierestimator for censored data.

Step II the copula parameter is estimated by maximizing theconditional likelihood for censored sample

l(θ ,FX ,FY ) =n

∑i=1

{ln[c(FX (xi ),FY (yi ))]

∆Xi ∆Y

i + ln[C1(FX (xi ),FY (yi ))]∆

Xi ∆Y

i +

+ ln[C2(FX (xi ),FY (yi ))]∆

Yi ∆X

i + ln[C(FX (xi ),FY (yi ))]∆

Xi ∆

Yi

}




DatasetEstimation results

Dataset

The dataset comes from Bankscope and concerns data for264 Italian and UK banks for the period 1995-2012.We consider a bank in distress when it is dissolved, inliquidation or receivership.We estimate banks’ probabilities of failure by applying theBGEVA model (Calabrese, Marra and Osmetti, 2013) toI balance sheet data (idiosyncratic component): Liquid

assets/Tot Dep & Bor, Tier 1 Ratio, Total Capital Ratio,Equity/Liabilities, etc...;

I macroeconomic variables (systemic component): Growthrate GDP, Inflation rate, Unemployment rate, Interest rate,etc...





Marginal distributions in MO copula

1 Italian and UK banks are ordered on the basis of theirfailure probabilities.

2 The empirical cumulative distribution functions of the timesto bank failures for UK and Italy are the marginaldistributions in the MO copula (non-parametric approach).

3 We estimate the parameter θ of the MO copula for acomplete and a censored sampling.

4 We compare the Goodness of fit of the MO copula with theGaussian and the Gumbel copula.





Estimation result and comparison

Copula parameters estimates and goodness of fit test:

Copula parameter estimate p-valueGaussian ρ = 0.27 0.54Gumbel r = 1.001 0.73

MO θ = 0.45 0.91

sample parameter estimatecomplete sample θ = 0.45censoring sample θ = 0.76





MO copula estimate

Mo copula and contour lines estimate.




Conclusions

We have proposed to combine balance sheet models withcopula methodology for modeling cross-border bank contagion.

We have proposed to apply the MO copula to estimate thedependence between times to bank failures of twocountries. In this way defaults of banks are due toidiosyncratic (i.e. balance sheet information) andsystematic (i.e economic cycle) characteristics.We have proposed the maximum likelihood estimators ofthe copula parameter for complete and censored sample.The empirical evidence from UK and Italy has shown thatthe MO copula is the best-fit model for bank contagion.




Bibliography

Calabrese R., Osmetti S., Modelling small and medium enterprise loandefaults as rare events: the generalized extreme value regressionmodel, Journal of Applied Statistics. 40 (6), pp. 1172-1188 (2013).Crook J., Moreira F., Checking for asymmetric default dependence in acredit card portfolio: a copula approach, Journal of Empirical Finance11, pp. 728-742 (2011).Marshall A.W., Olkin I., A Multivariate Exponential Distribution, Journalof the American Statistical Association 62, pp. 30-40 (1967).Nelsen R.B., An Introduction to Copulas, Springer, New York (2006).Osmetti S.A. Maximum likelihood estimate of Marshall-Olkin copulaparameter:complete and censored sample, Italian Journal of AppliedStatistics 22(2), pp.211-240 (2012)

WeißG. N.F., Analysing contagion and bailout effects with copulae,Evidence from the subprime and Japanese banking crises, Journal ofEmpirical Finance 36, pp. 1-32 (2012).


Documents

Modelling cross-border bank contagion using … contagion effects in banking with copulae Methodological proposal Empirical evidence Conclusions Modelling cross-border bank contagion