analysis of copula functions and applications to credit risk management · 2005-09-26 · u2...

Analysis of Copula Functions and

Applications to Credit Risk

Management

Philipp KoziolMichael Kunisch

Financial Modelling Workshop Ulm, 2005

Philipp Koziol WHU - Otto Beisheim School of Management

Analysis of Copula Functions and Applications to Credit Risk Management

Motivation

Modelling dependent defaults is crucial in credit risk

Copula Functions are a very useful toolto model joint default distributions

But:

• Impacts of copulas on credit derivative prices?

• Which copula should be used?

Literature:

• Standard: Gauss Copula

• Li (2000) and Schonbucher/Schubert (2001):

=⇒ No Analysis of copulas

Goal: Influence of copulas on creditderivative prices

Philipp Koziol WHU - Otto Beisheim School of Management 1

Analysis of Copula Functions and Applications to Credit Risk Management

Idea of Copulas

joint distribution function:F (x1, . . . , xn)

↙ ↘

dependence betweenthe random variables:

C(u1, . . . , un)

MarginalDistributions:

F1(x1), ..., Fn(xn)

↘ ↙

F (x1, . . . , xn) = C(F1(x1), . . . , Fn(xn))

Philipp Koziol WHU - Otto Beisheim School of Management 2

Analysis of Copula Functions and Applications to Credit Risk Management

Basics in Copulas and Dependence (1)

• Global Dependence Measure:

◦ Rank Correlation: Kendalls tau

+ capture nonlinearities+ independent of marginal distr.+ only dependent of copula parameters

=⇒ sensitivities of τ

• Local Dependence Measure:

– Tail Dependence:

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

u1

u2

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Analysis of Copula Functions and Applications to Credit Risk Management

Basics in Copulas and Dependence (2)

Important Examples of Copulas

•Elliptical C.: Combination of elliptical distr.

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

u1

u2

(a) Gauss Copula

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

u1

u2

(b) t4-Copula

=⇒ Symmetrical Structure

•Archimedean C.: artificially generated by ϕ(t)

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

u1

u2

(c) Gumbel C.

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

u1

u2

(d) Clayton C.

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

u1

u2

(e) Frank C.

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

u1

u2

(f) Nelsen C.

=⇒ ”Non” Symmetrical Structure

•Farlie-Gumbel-Morgenstern C.: easy expression

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Analysis of Copula Functions and Applications to Credit Risk Management

Models

1. Li Copula Model:

• Marginal default distribution Fi(ti) = P(τi ≤ ti)=⇒ Poisson or Cox Processes

• Dependence Structure through a Copula Function:

P(τ1 ≤ t1, ..., τn ≤ tn) = CLi(F1(t1), ..., Fn(tn))

• Valuation: Monte-Carlo Simulation of (τ1, . . . , τn)

2. Schonbucher/Schubert Model:

• Extension of the Li Model:

=⇒ Dynamics of the default intensities

• Simulation of the default times is exactly the same

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Analysis of Copula Functions and Applications to Credit Risk Management

Data

Goal: Comparison of credit derivative prices=⇒ Influence of copulas

• Credit Portfolio: n = 2 identical credits

=⇒ 2 Turkey Zerobonds (B1 Rating), Maturity=8.6a

• Marginal distribution: Cox Process

dλi(t) = µi · λi(t) · dt+ σi · λi(t) · dWi(t), i = 1, 2,

• where µ1 = µ2 = 0,

• σ1 = σ2 ≈ 0.11,

• E(dW1(t) · dW2(t)) = 0.

• Default Dependency: Copula Functions

• Simulation of default times (τ1, τ2)

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Analysis of Copula Functions and Applications to Credit Risk Management

Applications in Credit Risk Manag. (1)

1. Valuation of nth-to-Default Swaps:

• First-to-Default Swap FtD(t,T):

FtD(t, T ) = E

(

e−∫ min(τ1,τ2)t rsds · 1{min(τ1,τ2)≤T}

∣

∣ Ht

)

=⇒ considered time horizon: T = 1year at t = 0

Distribution of τmin = min{τ1, τ2} crucial

Strong dependence in [0, T ] =⇒ Low FtD premia

⇓

Simulation Resultsin the Schonbucher/Schubert model

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Analysis of Copula Functions and Applications to Credit Risk Management

Archimedean Copulas

• Archimedean copulas have variable price behavior

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Analysis of Copula Functions and Applications to Credit Risk Management

Elliptical Copulas

• Elliptical copulas have equal price characteristics

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Analysis of Copula Functions and Applications to Credit Risk Management

Numbers

Fix Kendall’s tau

Kendall’s tau ∆FtD=max-min % of upper bound

0.2 0.0208 11.2%

0.5 0.034 20%

Fix FtD premium

Fix FtD premium τ -range ∆τ

0.18 [0.08; 0.4] 0.32

0.17 [0.13; 0.61] 0.48

⇓

Choice of the copula is important for FtD premia

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Analysis of Copula Functions and Applications to Credit Risk Management

Results for the FtD Swap

• Tail Dependence is crucial for FtD premia

• Archimedean copulas are more flexible

• t-copula differs strongly for low dependencies

⇓

Choice of the copula is important for FtD premia

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Analysis of Copula Functions and Applications to Credit Risk Management

Applications in Credit Risk Manag. (2)

2. Valuation of defaultable Zerobonds:

B1(t, T ) = E

(

e−∫ Tt rsds · 1{τ1>T}

∣

∣ Ht

)

• Defaultable Zerobond at t = 1• maturity T = 8.6 years• firm 2 has already defaulted

Distribution of τ1 important

⇓

Simulation Resultsin the Schonbucher/Schubert model

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Analysis of Copula Functions and Applications to Credit Risk Management

Archimedean Copulas

• Archimedean copulas have variable price behavior

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Analysis of Copula Functions and Applications to Credit Risk Management

Elliptical Copulas

• Elliptical copulas have equal price characteristics

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Analysis of Copula Functions and Applications to Credit Risk Management

Numbers

Fix Kendall’s tau

Kendall’s tau ∆CS=max-min % of upper bound

0.1 503bp 41.9%

0.4 1300bp 44.8%

Fix Credit Spread

Fix credit spread τ -range ∆τ

1250bp [0.075; 0.28] 0.205

1600bp [0.18; 0.4] 0.22

⇓

Choice of the copula is important for credit spreads

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Analysis of Copula Functions and Applications to Credit Risk Management

Results for defaultable Zerobonds

• probability mass in the center is crucial

=⇒ Tail Dependence is less important

• copulas without T. D. =⇒ highest spreads

• Archimedean copulas are more flexible

• t-copula differs strongly for low dependencies

Choice of the copula is important for credit spreads

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Analysis of Copula Functions and Applications to Credit Risk Management

Conclusion

• Choice of the copula has a high impact on prices

• Copula behavior varies for different creditderivatives

• Simultaneous Consideration of global and localDependencies is necessary

• Reduction of model risk by identification ofnecessary copula properties

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