copulas from infinitely divisible distributions-applications to credit value at risk

8/2/2019 Copulas From Infinitely Divisible Distributions-Applications to Credit Value at Risk

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Copulas from Infinitely Divisible Distributions:

Applications to Credit Value at Risk

Thomas Moosbrucker

Graduate School of Risk ManagementUniversity of Cologne

Albertus-Magnus-Platz50923 Koln, Germany

Phone: +49 (0)221 470 3967Fax: +49 (0)221 470 7700

[email protected]

First Draft: March, 2006This Draft: June, 2006

JEL classification: G13

Keywords: Credit Value at Risk, Credit Portfolio Models, LHP approximation,

Infinitely Divisible Distributions, Levy Processes


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Copulas from Infinitely Divisible Distributions:Applications to Credit Value at Risk

Abstract

This article proposes infinitely divisible distributions for credit portfolio modelling. First,

we show that these distributions are well suited to fit into a one factor copula approach. We

compare four different specifications (Normal Inverse Gaussian, Variance Gamma, Merton

Jump Diffusion and Kou Jump Diffusion), where the last two distributions are new to credit

portfolio modelling. The comparison in this article is done with respect to Value at Risk(VaR) measures. We find that the more extreme tail behaviour of the infinitely divisible

distributions leads to significantly different VaR measures. Therefore, we conclude that the

identification of the correct dependence structure is of major importance for the estimation

of VaR.

JEL classification: G13

Keywords: Credit Value at Risk, Credit Portfolio Models, LHP approximation,

Infinitely Divisible Distributions, Levy Processes


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1 Introduction

Besides single-name default probabilities, the dependence structure of defaults is of major

importance for credit portfolio risk management. The most important industry models

such as KMV and CreditMetrics assume that a Gaussian copula can describe these de-

pendencies. Furthermore, the Internal Ratings based approach of the latest Basel accord

relies on this assumption. The Gaussian model traces back to the work of Merton (1974),

who assumes that asset value processes are driven by Brownian motions.

One of the main advantages of the Gaussian assumption is that these models are

easy to implement and that they lead to fast computations. Vasicek (1987) derived an

analytical formula for the loss distribution of an infinitely large portfolio called the largehomogeneous portfolio (LHP) approximation. This allows practitioners to implement an

oversimplified but fast version before setting up a more sophisticated credit risk model.

Furthermore, a single parameter (correlation) determines the entire dependence structure

between two defaults. This number is easy to communicate.

In practice, there are several ways to estimate default correlations. One method is based

on historical default time series. However, more often than not, very little data is available.

Another method to estimate default correlations consists of using asset and equity returncorrelations as proxying variables. In this case, usually, more data is available. However,

log returns are not normally distributed. They are leptokurtic and often skewed. Thus,

the estimation of a Gaussian correlation parameter is inappropriate.

In the market of equity options, it is well-known that the assumption of normally

distributed log returns in the Black-Scholes model leads to a volatility smile. The implied

volatility of a low strike price option is higher than that of a high strike price option.

Since the work of Black and Scholes, researchers have developed several models to cover

the stylized facts of log returns for the valuation of options. In particular, models based

on Levy processes have proven to be successful. These models have a significantly better

fit to equity options than the approach via Brownian motions.

The volatility smile in equity options has its counterpart in credit derivatives. In liquid

CDS index tranches, there is a pronounced correlation smile when a Gaussian copula model

is assumed. This shows that market participants do not fully believe in the Gaussian copula

model. Instead, the risk-neutral distribution exhibits stronger extremal dependence than

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that captured by a Gaussian copula.

In this paper, we consider factor copulas resulting from infinitely divisible distributions

for credit risk management. These distributions arise from the before mentioned Levy

processes. Namely, we use Variance Gamma (VG) and Normal Inverse Gaussian (NIG)

distributions and distributions of the jump-diffusion models by Merton (1976) and Kou

(2002) (MJD and KJD). While NIG and VG factor copulas have been used for the valuation

of Collateralized Debt Obligations (CDOs) in a risk-neutral setting in Kalemanova et al.

(2005) and Moosbrucker (2006), the other distributions have not been used in a factor

copula approach for credit portfolios yet.

In our setting, infinitely divisible distributions have the following advantages: First,

their defining property of infinite divisibility makes them ideally suited for a factor ap-

proach. As a consequence, the resulting formulas for the distribution of the number of

defaults and the LHP approximation are simple. The original LHP approximation for the

Gaussian copula by Vasicek (1987) is a special case. Second, all copulas proposed in this

article contain the Gaussian copula as a special or limiting case. They may therefore be

seen as natural extensions. Third, these distributions are not limited to fitting to stylized

facts of log returns and lead to a flattening of the volatility smile in equity options. Fur-

thermore the VG and NIG copulas have proven to explain the correlation smile in CDS

index tranches (see Kalemanova et al. (2005) and Moosbrucker (2006)).

In this article, we investigate the consequences of the use of the alternative copulas for

Value at Risk (VaR) measures. We apply three different methods. First, we calculate VaR

measures for the different copulas using the same correlation parameters for the default

triggering variables. This method is similar to the one used by Frey and McNeil (2001)

in order to compare the Gaussian and the student t copula. Since identical correlations

of default triggering variables in different models result in different correlations of default

events, we also calibrate all copulas such that the default correlations are matched. This

second method leads to a more realistic comparison of resulting VaR measures. The

approach was proposed by Schonbucher (2002) and is also used by Schloegl and OKane

(2005). For a comparison between the Gaussian and the student t copula, Schloegl and

OKane (2005) found that there are only minor differences in VaR measures. Finally, we

apply the method used by Hamerle and Rosch (2005). In this case, we simulate default time

series based on one of the infinitely divisible copulas and estimate correlation parameters

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and default probabilities under the assumptions of the Gaussian copula model. Again,

there do not seem to be large differences in VaR measures between the Gaussian and the

student t copula. However, the results are different for the other one factor copulas.

The structure of this article is as follows. The next section presents the distributions

and the corresponding factor copulas. We derive the number of defaults distribution and

the LHP approximation. Section 3 gives the numerical results. Section 4 concludes the

article.

2 Infinitely divisible distributions and factor copulas

2.1 Infinitely divisible distributions

A probability distribution F is called infinitely divisible if for every n N there are n i.i.d.random variables such that their sum has distribution F.1 Infinitely divisible distributions

are in a one-one correspondence to Levy processes: If (Xt)t0 is a Levy process, then for

every t, Xt has an infinitely divisible distribution. Conversely, if F is an infinitely divisible

distribution then there is a Levy process (Xt)t0 such that the distribution of X1 is given

by F.An example of an infinitely divisible distribution is the normal distribution which

corresponds to Brownian motion. Since the distributions we use in this article are better

known through their process counterparts in the literature of option pricing, we introduce

the corresponding processes here. The distributional properties are given in the Appendix.

The first class of Levy processes we consider consists of time-changed Brownian mo-

tions. In this case, a Brownian motion (Wt)t0 is subordinated by an increasing Levy

process (Gt)t0, that isXt = t + Gt + WGt.

Throughout this article, we set = = 0 and we restrict the distributions to unit

variances.

If (Gt)t0 is an inverse Gaussian process, then (Xt)t0 is called a Normal Inverse Gaus-

sian process. This process was proposed by Barndorff-Nielssen (1997). Madan, Carr

and Chang (1998) model (Gt)t0 as a Gamma process. (Xt)t

0 is then called a Variance

1See for example Cont and Tankov (2003).

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Gamma process. We denote the variance of the subordinating process by . For 0,the limiting process is a Brownian motion.

The jump diffusion processes considered in the papers of Merton (1976) and Kou (2002)

can be written as

Xt = t + Wt +Nti=1

Yt.

In this equation, (Nt)t0 is a Poisson process with intensity and Yi is the distribution

of jumps. In the Merton (MJD) case, Yt is normally distributed, while Kou (KJD) uses

exponential distributions for the upward and downward jumps. In this article, we restrict

both the jump distributions Y1 and the distribution ofX1 to zero mean and unit variance.

We set = 0 as before. Thus, the jump intensity is the only additional parameter in

these models. For = 0, we see that the processes contain Brownian motion as a special

case.

2.2 Factor copulas

Let n N be the number of entities in a portfolio and for 0 < < 1 let X = (X1, . . . , X n)be a random vector where Xi =

M +

1 Zi for every i {1, . . . , n} and M and

all Zi are mutually independent random variables with zero mean and unit variance. We

then call the copula function of X a one factor copula.

The most important one factor copula is the Gaussian copula, where the factors M

and Zi (and thus Xi) are standard normally distributed. The Gaussian copula underlies

the approaches of KMV, CreditMetrics and the latest Basel accords. A good reference for

this model is Li (2000).

If M = M kX

and Zi = Zi kX

, where M and Zi = Zi are standard normallydistributed and X 2(k), then the resulting copula is a student t copula with k degreesof freedom.

In this article, we propose infinitely divisible distributions for factor copulas. The prop-

erty of infinite divisibility makes these distributions ideally suited for a factor approach:

if the parameters for the factors M and Zi are chosen suitably, then the distribution of

the default triggering variable Xi is of the same class. This property is shown in the

Appendix for VG, NIG, MJD and KJD distributions. Since the density functions of these

distributions are available (at least approximately) in closed form, this leads to simple and

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fast computations. Furthermore, the parameters have a clear economic interpretation.

2.3 Number of Defaults Distribution

As for the Gaussian copula, the number of defaults distribution is obtained by conditioning

on the common factor M. In the following, we assume that the portfolio is homogeneous

with respect to default probabilities. If the unconditional default probability of entity i is

given by p = P(Xi < C), then the default probability conditional on M = m is

p(m) = P(Xi < C|M = m) = P

Zi z) .

From this equation, the VaR of the loss distributions can be calculated in closed form.

It is given by

VaRz(L) = FZ

F1X .(p)

q1z(M)

1

,

where q1z(M) is the (1 z)-quantile of the systematic factor M.

3 Numerical Results

3.1 Identical correlation for default triggering variables

In this subsection, we compare the factor copulas with respect to VaR measures when the

marginal default probabilities and the correlations of the default triggering variables are

identical. In the Gaussian case, these inputs determine VaR measures. The other copulas

have additional parameters. We use the third and fourth moments (skewness and kurtosis)

to appoint these additional parameters.

All distributions in this article have zero skewness. For kurtosis, we consider two cases:

an excess kurtosis of 13

(T1, VG1, NIG1, MJD1, KJD1) and an excess kurtosis of 1 (T2,

VG2, NIG2, MJD2, KJD2). The higher excess kurtosis corresponds to a more extreme

scenario which differs more from the Gaussian model and exhibits larger tail dependence.

Table 1 summarizes the parameters used in this article.2Schloegl and OKane (2005) derive the cdf of this mixing variable.

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Insert Table 1 about here.

Throughout this study, we consider four different credit portofolios, two medium-risk

portfolios with marginal default probabilities of 1% and two high-risk portfolios with

marginal default probabilities of 7.5%. In both cases, we consider a case with low and a

case with high correlation (20% and 40%). We calculate all VaR measures by the LHP

formula.

Insert Table 2 and 3 and Figure 1 and 2 about here.

Table 2 and 3 and Figure 1 and 2 show the ratio of the VaR given by the alternative

copulas to that given by the Gaussian copula. Therefore, the specified model leads to a

higher VaR measure than the Gaussian model if and only if this ratio is larger than 1.

Typically, the student t copula results in higher VaR measures than the Gaussian copula

the only exceptions are low marginal distributions and VaR quantiles below 95%. The

other distributions show a different pattern: while VaR measures of the specified models

are lower for quantiles below 99%, the ratio to the Gaussian VaR increases dramatically

above 99%. In general, the use of copulas from infinitly divsible distributions therefore puts

more weight on extreme scenarios. Medium-size losses are less likely, whereas very large

losses are more likely to occur. This effect is larger for low marginal default probabilities

and low correlations. As Schloegl and OKane (2005) point out, the tail dependence

does manifest itself more in this case. Furthermore, this effect is more pronounced if the

distributions exhibit heavier tails. If we compare the different copulas, we find that the

NIG and VG copulas behave quite similarly. The difference to the Gaussian copula is

more pronounced for the MJD and especially the KJD copula.

3.2 Identical correlation of default events

Instead of using identical correlations for the default triggering variables, it may seem

more appropriate to compare the different models with identical default correlations. This

correlation is defined as the correlation of the Bernoulli distributed variables 1X1


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p12 = P(X1 < C X2 < C), the default correlation is

de =p12 p1p2

p1(1 p1)p2(1 p2)

.

We calculate default correlations resulting from the Gaussian copula approach and

determine the correlations of the default triggering variables for the alternative factor

copulas. Schloegl and OKane (2005) use this method for the student t copula. The

approach was originally proposed by Schonbucher (2002). The results of this calibration

are given in Table 4.


In almost all cases, the correlation of default triggering variables is smaller than in

the Gaussian model. The only exception is the MJD copula for lower correlations. Since

the VaR is increasing in correlation, the ratio of Gaussian and alternative VaR should

therefore decrease in comparison to section 3.1. We calculate the VaR ratios as in the

previous subsection, this time based on the calibration results of Table 4. The results are

given in Table 5 and 6 and Figure 3 and 4.


In line with Schloegl and OKane (2005), we find that the resulting VaR measures

are quite similar for the Gaussian and student t copulas. The ratio is between 0.982 and

1.100 in all cases. In contrast, the difference to the Gaussian copula is not wiped our forthe other copulas we have considered. Compared to the study in section 3.1, VaR ratios

decrease, but there are still significant differences between the Gaussian and the other

copulas. In particular, at the 99.9% level, all but two VaR ratios are larger than 1.

3.3 Model Risk in the dependence structure

In this subsection, we simulate a default time series based on the different copulas and es-

timate marginal default probabilities and correlation based on the incorrect assumption of

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a Gaussian copula. This is the method used in Hamerle and Rosch (2005) for a comparison

between the Gaussian and the student t copula.

We use the same sample portfolios and parameter sets as in the previous subsections.

As in subsection 3.1, we use correlations of default triggering variables of 20% and 40%.

We now assume that the portfolio consists of 1000 entities. For each portfolio, we simulate

1000 default time series for each copula. As in Hamerle and Rosch (2005), we set the time

series to length T = 10.

For estimation, we employ the maximum likelihood framework as in Hamerle and Rosch

(2005). In all cases except the student t copula, the log-likelihood function is

l(, ) =

T

t=1 ln

ndtp(m)dt(1 p(m))ndtdFM ,while for the student t copula it is

l(, ) =Tt=1

ln

0

n

dt

p(m)dt(1 p(m))ndtdd2(k)

.

For each model and each portfolio, we simulate a default time series and estimate the

parameters and under the (false) assumption of a Gaussian copula. Table 7 gives the

average parameters of this estimation. We find that default probabilities can be estimated

very well: results are between 0.990% and 1.142% for a true default probability of 1% and

between 7.404% and 7.631% for a true default probability of 7.5%. This is not a surprise

since the estimation of the default probability is based on the relative number of defaults.

This number does not depend on the correlation structure.

When analyzing correlation, the results are different. For the student t copula, the

Gaussian estimates are higher than the true correlation. The only exception is a default

probability of 7.5% and a correlation of 40%, where the real correlation and the Gaussianestimate are nearly identical. This finding is in line with both the result of Hamerle

and Rosch (2005) and the results of section 3.2. One might be tempted to conclude

that the estimation procedure in this subsection chooses parameters to match the default

correlation. However, when we compare the other models, we see that this is not true.

Instead of overestimating correlation (as one might expect form the results of section 3.2),

we find lower correlation under the misspecified Gaussian model. This behaviour may be

explained as follows: typically, the extreme tail of the distribution (quantiles above 99%)does not manifest itself in a simulated default time series of length T = 10. The Gaussian

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correlation estimation is therefore based on lower quantiles of the distribution. By the

results of the previous subsections, this Gaussian estimation is therefore lower than the

true parameter value.


As in the previous subsections, we calculate VaR ratios of the true and misspecified

model. The results are given in Table 8 and 9 and Figures 5 and 6. As in the previous

subsection, we see that the differences between the student t and the Gaussian copula are

wiped out. From this analysis, Hamerle and Rosch (2005) conclude that there is not muchmodel risk in the correlation structure. However, our findings for the alternative copulas

show that this is not true. The different copula assumptions lead to significantly different

VaR measures.


4 Conclusion

In this article, we have compared six different one factor copulas with respect to Value at

Risk. These copulas arise, if the normal, student t, and infinitely divisible distributions

are implemented into a one factor copula. From our analysis, we draw the following con-

clusions: First, there is substantial model risk in the dependence structure. While the

Gaussian and the student t copula lead to very similar VaR measures (if default corre-lations are identical), these figures are very different for the NIG, VG, MJD and KJD

copulas. Overall, these copulas result in higher risk measures for quantiles above 99%.

Second, when marginal default probabilities and correlation are estimated based on his-

torical default series, correlation is underestimated under the assumption of a Gaussian

copula. Therefore, the ratios of true and estimated VaR are even higher than their theo-

retical values. This behaviour is in sharp contrast to the student t copula, where the true

VaR and its Gaussian estimation are almost identical. Contrary to Hamerle and Rosch

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(2005), we therefore find that there is substantial model risk in the dependence structure

of a credit portfolio.

There is a variety of further research questions, both theoretically and empirically.

Obviously, one could try other infinitely divisible distributions in a one factor approach.

However, if one does not want to rely on Monte Carlo simulations, their density or distri-

bution functions should be known.

Empirically, it might be interesting to apply the copulas proposed in this article to

liquid CDS index tranches. While the NIG and VG distributions have shown to fit to the

correlation smile in liquid CDS index tranches, it might be interesting to check whether

the other distributions lead to a similar fit.

Appendix

This Appendix provides the properties of the infinitely divisible distributions used in this

article.

A.1 NIG Distributions

Definition 1. A random variable X is said to be Normal inverse Gaussian (NIG) dis-

tributed with parameters ,,, with 0, 2 2, > 0, R if its characteristicfunction is given by

X(z) = E[eizX ] = eiz+(0iz), z R,

where iz =

2 (+ iz)2. We write X N IG(,,,).

Proposition 1. LetX N IG(,,,). Then:

1. The density fX of X is

fX(x) =

e

K1(

2 + (x )2)2 + (x )2 e

(x).

K is the modified Bessel function of the third kind,

K

(x) =1

2

0

y1 exp12x(y + y1) dy.

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2. X is infinitely divisible. The corresponding Levy process is

Xt = t + Gt + WGt,

where = 2 , = and (Gt)t0 is an inverse Gaussian process with parameters

((0)1, 20). (Wt)t0 is a standard Brownian motion.

3. The first four centralized moments (mean, variance, skewness and kurtosis) are

E[X] = +

,

Var[X] =2

3,

S[X] = 32

5

,

K[X] = 3 + 32(2 + 42)7.

with = 0.

The transformation properties are given by the following proposition:

Proposition 2. 1. If X N IG(,,,) and c > 0, then

cX N IGc

, c

,c,c .2. If X1 N IG( , , 1, 1) and X2 N IG( , , 2, 2) then

X1 + X2 N IG ( , , 1 + 2, 1 + 2) .

Proof. This can be seen directly from the definition and the properties cX(z) = X(cz)

and X+Y(z) = X(z) Y(z) for c R and independent X and Y.

Corollary 1. If the factors distributions are

M N IG(, 0, , 0)

and

Zi N IG(

1 , 0,

1 , 0),

then the distribution of Xi =

M +

1 Zi is

Xi N IG(, 0, , 0).

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A.2 VG Distributions

Definition 2. A random variable X is said to be Variance Gamma (VG) distributed with

parameters ,

R and , > 0 if its characteristic function is given by

X(z) = E[eizX ] = eiz

1 iz + 1

22z2

1

, z R.

We write X V G(,,,).

Proposition 3. LetX V G(,,,). Then:

1. The density fX of X is given by

fX(x) =2exp

x21 2( 1)

(x )22

2

+ 21

2 14

K1

1

2 12(x )222

+ 2 .

K is the modified Bessel function of the third kind,

K(x) =1

2

0

y1 exp

12

x(y + y1)

dy.

2. X is infinitely divisible. The corresponding Levy process is given by

Xt = t + Gt + WGt,

where (Gt)t0 is a Gamma process with parameters (1

, ) and(Wt)t0 is a standardBrownian motion.

3. The first four centralized moments are

E[X] = + ,

Var[X] = 2 + 2,

S[X] = 32 + 22

(2 + 2)3/2,

K[X] = 3(1 + 2 4(2 + 2)2).

Proposition 4. 1. If X V G(,,,) and c > 0, then

cX V G(c,,c,c).

2. IfX1 V G(1, 1, 1, 1) and X2 V G(2, 2, 2, 2) are independent and such that211 =

222 and

121

= 222

then

X1 + X2 V G1 + 2, 121 + 2

,21 + 22, 1 + 2 .13


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M V G(0,

, 1, 0)

and

Zi V G(0, 1 , 1, 0),


M +

1 Zi is

Xi V G(0, , 1, 0).

A.3 MJD Distributions

Definition 3. We call a random variable X Merton-jump-diffusion (MJD) distributed

with parameters R, , , > 0 if its characteristic function is given by

X(z) = E[eizX ] = exp

2z2

2+

e

2z2/2+iz 1

.

We write X MJD(,,,).

Proposition 5. LetX MJD(,,,). Then:1. The density fX of X is

fX(x) = e

k=0

k exp (xk)2

2(2+k2)

k!

2(2 + k2).

2. X is infinitely divisible. The corresponding Levy process is given by

Xt = Wt +Nt

i=1Yi,

where Yi N(, 2) is normally distributed, (Nt)t0 is a Poisson process with inten-sity and (Wt)t0 is a standard Brownian motion.

3. For = 0, the first four centralized moments are

E[X] = 0,

Var[X] = 2 + 2,

S[X] = 0,

K[X] = 3(1 + 3).

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Proposition 6. 1. If X MJD(,,,) and c > 0 then

cX MJD(c,,c,c) .

2. If X1 MJD(1, 1, , ) and X2 MJD(2, 2, , ) are independent then

X1 + X2 MJD

21 + 22, 1 + 1, ,




M MJD1 , , 0, 1

and

Zi MJD

1 , (1 ), 0, 11

,


M +

1 Zi is

Xi M J D

1 ,, 0, 1

.

A.4 KJD Distributions

Definition 4. We call a random variable X Kou-jump-diffusion (KJD) distributed with

parameters (,,+, , p) if its characteristic function is given by

X(z) = E[eizX ] = exp

2z2

2+ iz

p

+ iz 1 p

+ iz

.

We write X KJD(,,+, , p).Proposition 7. LetX KJD(,,+, , p). Then:

1. The density fX of X is approximatively given by

fX(x) =1

x

+

p+e

(22+)/2e(x)+

x 2+

+ (1 p)e(22)/2e(x)x + 2+

.

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2. X is infinitely divisible. The corresponding Levy process is

Xt = Wt +Nt

i=1Yi,

where Yi N(, 2) is two-sided exponentially distributed with parameters + and, (Nt)t0 is a Poisson process with intensity and(Wt)t0 is a standard Brownian

motion.

3. The first four centralized moments are

E[X] =

p

+ 1 p

,

Var[X] = 2 + p2+

+ 1 p2

,S[X] =

p

3+ 1 p

3

,

K[X] =

p

4++

1 p4

.

Proposition 8. 1. If X KJD(,,+, , p) and 0 < c < 1, then

cX

KJDc, , +c , c , p2. If X1 KJD(1, 1, +, , p) and X2 KJD(2, 2, +, , p) are independent,

then

X1 + X2 KJD

21 + 22, 1 + 2, +, , p

.


M KJD1 2+

,, +, +, pand

Zi KJD

1 2+

, (1 ),

1 +,

1 +, p

,


M +

1 Zi is

Xi

KJD1

2+

, , +, +, p .

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Distribution mean variance skewness kurtosis jump copmponent

Gauss = 0 = 1

T1 = 0 = 1 df = 22

T2 = 0 = 1 df = 10

NIG1 = 0 = = 0 = 3

NIG2 = 0 = = 0 =

3

VG1 = 0 = 1 = 0 = 19

VG2 = 0 = 1 = 0 = 13

MJD1 = 0 =

1 2 = 19

= 1

MJD2 = 0 = 1 2 = 13 = 1KJD1 = 0 =

1

2+

p = 12

, + = = 112 + =12

KJD2 = 0 =

1 2+

p = 12

, + = = 14 + =12

Table 1: Parameters sets used in the numerical examples in section 3. Parameters of T1, NIG1, VG1,

MJD1 and KJD1 are chosen to lead to an excess kurtosis of 13

. T2, NIG2, VG2, MJD2 and KJD2 have

an excess kurtosis of 1. In MJD and KJD the additional parameters are chosen such that the distribution

of jumps has unit variance.

Correlation = 20% Correlation = 40%

Confidence Level Confidence Level

90% 95% 99% 99.5% 99.9% 90% 95% 99% 99.5% 99.9%

T1 1.052 1.168 1.348 1.268 1.468 0.933 1.032 1.165 1.193 1.217

NIG1 0.814 0.837 1.054 1.151 1.648 0.817 0.809 0.992 1.117 1.411

VG1 0.816 0.854 1.080 1.155 1.596 0.822 0.820 0.999 1.116 1.381

MJD1 0.854 0.836 0.889 1.023 1.876 0.823 0.807 0.927 1.048 1.568

KJD1 0.421 0.309 0.245 0.601 3.143 0.471 0.401 0.534 0.915 3.023

T2 1.045 1.304 1.720 1.563 1.970 0.829 1.042 1.345 1.408 1.447

NIG2 0.654 0.662 0.984 1.236 2.386 0.650 0.616 0.916 1.209 1.979

VG2 0.654 0.697 1.050 1.240 2.248 0.663 0.650 0.936 1.188 1.921

MJD2 0.655 0.591 0.773 1.336 2.870 0.614 0.527 0.867 1.335 2.119

KJD2 0.512 0.445 0.501 0.601 3.699 0.350 0.216 0.237 1.044 3.044

Table 2: VaR ratios of specified model over Gaussian model for identical correlations of default triggering

variables. Marginal default probabilities are 1% and VaR is calculated via LHP approximation.

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90% 95% 99% 99.5% 99.9% 90% 95% 99% 99.5% 99.9%

T1 1.054 1.077 1.104 1.108 1.111 1.024 1.040 1.051 1.050 1.042NIG1 0.916 0.965 1.152 1.240 1.405 0.920 0.977 1.123 1.161 1.176

VG1 0.913 0.975 1.169 1.251 1.391 0.914 0.979 1.133 1.167 1.172

MJD1 0.962 0.967 1.019 1.077 1.447 0.960 0.978 1.050 1.095 1.195

KJD1 0.919 0.924 0.989 1.080 1.955 0.910 0.940 1.092 1.262 1.332

T2 1.114 1.162 1.215 1.222 1.221 1.052 1.087 1.107 1.103 1.083

NIG2 0.815 0.904 1.319 1.519 1.762 0.812 0.937 1.297 1.352 1.281

VG2 0.795 0.915 1.378 1.576 1.756 0.787 0.932 1.344 1.372 1.275

MJD2 0.865 0.882 1.175 1.530 1.839 0.855 0.922 1.259 1.370 1.296

KJD2 0.691 0.691 1.291 2.225 2.062 0.631 0.728 1.781 1.638 1.331

Table 3: VaR ratios of specified model over Gaussian model for identical correlations of default triggering

variables. Marginal default probabilities are 7.5% and VaR is calculated via LHP approximation.

Marginal default probability

1% 7.5%

Gaussian 20% 40% 20% 40%

default de 2.413% 7.736% 7.024% 16.823%

T1 10.42% 32.89% 16.77% 37.75%

NIG1 14.60% 33.27% 18.12% 37.55%

VG1 15.07% 33.78% 18.01% 37.40%

MJD1 21.66% 36.35% 20.93% 38.77%

KJD1 12.54% 20.33% 17.03% 34.23%

T2 0.23% 23.34% 12.08% 34.18%

NIG2 9.62% 25.44% 16.02% 34.44%

VG2 8.98% 26.58% 15.23% 33.73%

MJD2 23.73% 35.05% 22.15% 36.33%

KJD2 17.77% 24.96% 13.71% 23.55%

Table 4: Correlations of default triggering variables of the specified models that lead to the the same

default correlations as the Gaussian copula.

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Gaussian Correlation = 20% Gaussian Correlation = 40%

Confidence Level Confidence Levelde 90% 95% 99% 99.5% 99.9% de 90% 95% 99% 99.5% 99.9%

T1 10.42% 1.017 1.020 1.016 1.012 1.001 32.89% 1.000 1.008 1.013 1.012 1.010

NIG1 14.60% 0.764 0.742 0.861 0.966 1.306 33.27% 0.841 0.775 0.853 0.938 1.177

VG1 15.07% 0.771 0.769 0.906 1.004 1.292 33.78% 0.844 0.790 0.873 0.953 1.164

MJD1 21.66% 0.860 0.860 0.947 1.058 2.005 36.35% 0.847 0.791 0.843 0.940 1.438

KJD1 12.54% 0.599 0.455 0.360 0.360 1.489 20.33% 0.559 0.387 0.275 0.297 1.715

T2 0.23% 1.078 1.100 1.081 1.058 0.994 23.34% 0.982 1.041 1.095 1.100 1.091

NIG2 9.62% 0.694 0.634 0.676 0.744 0.992 25.44% 0.694 0.634 0.676 0.744 0.992

VG2 8.98% 0.549 0.522 0.659 0.788 1.264 26.58% 0.673 0.593 0.704 0.838 1.312

MJD2 23.73% 0.659 0.618 0.916 1.535 3.215 35.05% 0.636 0.522 0.757 1.163 1.933KJD2 17.77% 0.426 0.310 0.225 0.340 2.358 24.96% 0.405 0.237 0.162 0.292 2.194

Table 5: VaR ratios of specified model over Gaussian model for identical default correlations. Marginal

default probabilities are 1% and VaR is calculated via LHP approximation.

Gaussian Correlation = 20% Gaussian Correlation = 40%


de 90% 95% 99% 99.5% 99.9% de 90% 95% 99% 99.5% 99.9%

T1 16.77% 1.010 1.012 1.014 1.014 1.013 37.75% 1.010 1.009 1.011 1.010 1.008

NIG1 18.12% 0.888 0.925 1.093 1.177 1.348 37.55% 0.888 0.941 1.074 1.117 1.150

VG1 18.01% 0.884 0.933 1.107 1.186 1.331 37.40% 0.894 0.942 1.081 1.120 1.144

MJD1 20.93% 0.974 0.986 1.049 1.110 1.054 38.77% 0.950 0.960 1.025 1.071 1.183

KJD1 17.03% 0.880 0.865 0.890 0.950 1.840 34.23% 0.880 0.857 0.951 1.100 1.328T2 12.08% 1.016 1.014 1.005 1.001 0.992 34.18% 1.009 1.010 1.008 1.007 1.003

NIG2 16.02% 0.856 0.879 1.026 1.106 1.281 34.44% 0.856 0.896 1.013 1.059 1.114

VG2 15.23% 0.732 0.813 1.186 1.376 1.636 33.73% 0.732 0.844 1.206 1.274 1.243

MJD2 22.15% 0.890 0.925 1.273 1.622 1.875 36.33% 0.828 0.866 1.174 1.311 1.283

KJD2 13.71% 0.651 0.607 0.712 1.519 2.017 23.55% 0.651 0.527 1.084 1.512 1.317

Table 6: VaR ratios of specified model over Gaussian model for identical default correlations. Marginal

default probabilities are 7.5% and VaR is calculated via LHP approximation.

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Marginal default probability

1% 7.5%

Correlation Correlation

20% 40% 20% 40% T1 1.100 26.930 1.142 41.654 7.497 21.280 7.565 39.081

NIG1 1.010 16.483 1.030 29.754 7.508 17.152 7.501 33.149

VG1 0.984 12.401 1.004 26.255 7.599 15.754 7.439 30.361

MJD1 0.966 12.058 0.927 22.722 7.476 16.262 7.557 31.741

KJD1 0.950 6.850 0.916 12.479 7.463 14.708 7.397 27.941

T2 1.076 35.945 1.089 46.627 7.404 25.197 7.631 41.991

NIG2 0.977 13.744 0.953 22.934 7.408 16.156 7.546 29.567

VG2 1.010 9.086 0.945 17.557 7.530 12.949 7.578 24.847

MJD2 0.927 7.309 0.909 15.016 7.454 13.386 7.430 25.840

KJD2 0.941 3.493 0.980 7.563 7.618 11.555 7.489 21.618

Table 7: Estimated parameters under the assumption of a Gaussian copula.



90% 95% 99% 99.5% 99.9% 90% 95% 99% 99.5% 99.9%

T1 0.918 0.948 0.988 0.999 1.013 0.812 0.894 1.012 1.042 1.079

NIG1 0.837 0.898 1.206 1.404 1.978 0.764 0.864 1.261 1.482 1.967

VG1 0.917 1.053 1.533 1.803 2.527 0.795 0.941 1.440 1.698 2.247

MJD1 0.984 1.060 1.305 1.501 3.086 0.876 1.057 1.597 1.930 3.132

KJD1 0.756 0.792 1.020 1.288 9.492 0.570 0.678 1.434 2.773 10.870

T2 0.935 0.972 1.001 1.001 0.993 0.801 0.926 1.086 1.117 1.140

NIG2 0.722 0.789 1.312 1.762 3.466 0.672 0.784 1.533 2.164 3.846

VG2 0.773 0.948 1.775 2.382 4.527 0.723 0.928 1.932 2.690 4.923

MJD2 0.885 0.941 1.593 2.889 7.295 0.718 0.832 2.067 3.534 6.478

KJD2 0.663 0.618 0.719 1.301 12.827 0.449 0.416 0.819 4.228 15.668

Table 8: VaR ratios of specified model over Gaussian model; Gaussian parameters estimated by maximum

likelihood. Marginal default probabilities are 1% and VaR is calculated via LHP approximation.

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90% 95% 99% 99.5% 99.9% 90% 95% 99% 99.5% 99.9%

T1 1.035 1.049 1.064 1.067 1.067 1.024 1.047 1.064 1.064 1.054

NIG1 0.957 1.028 1.255 1.358 1.547 0.981 1.083 1.284 1.329 1.329

VG1 0.967 1.063 1.324 1.429 1.605 1.011 1.143 1.383 1.427 1.408

MJD1 1.024 1.055 1.147 1.220 1.652 1.032 1.103 1.233 1.287 1.386

KJD1 1.009 1.050 1.175 1.297 2.375 1.040 1.148 1.413 1.637 1.695

T2 1.049 1.060 1.065 1.064 1.056 1.018 1.042 1.058 1.057 1.046

NIG2 0.877 0.997 1.499 1.737 2.029 0.895 1.096 1.598 1.670 1.557

VG2 0.896 1.081 1.737 2.017 2.291 0.914 1.180 1.847 1.898 1.735

MJD2 0.974 1.039 1.467 1.937 2.367 0.998 1.165 1.709 1.868 1.734

KJD2 0.793 0.842 1.707 3.001 2.854 0.774 0.991 2.683 2.493 1.994

Table 9: VaR ratios of specified model over Gaussian model; Gaussian parameters estimated by maximum

likelihood. Marginal default probabilities are 7.5% and VaR is calculated via LHP approximation.

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T1NIG1

VG1

MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T2NIG2

VG2

MJD2

KJD2

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T1

NIG1VG1

MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T2

NIG2VG2

MJD2

KJD2

Figure 1: VaR ratios for identical correlations of default triggering variables, PD= 1%.

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0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T1

NIG1

VG1

MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T2

NIG2

VG2

MJD2

KJD2

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T1

NIG1

VG1MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T2

NIG2

VG2MJD2

KJD2

Figure 2: VaR ratios for identical correlations of default triggering variables, PD= 7.5%.

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0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T1

NIG1

VG1

MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T2

NIG2

VG2

MJD2

KJD2

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T1

NIG1

VG1

MJD1KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T2

NIG2

VG2

MJD2KJD2

Figure 3: VaR ratios for identical default correlations, PD= 1%.

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T1

NIG1

VG1

MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T2

NIG2

VG2

MJD2

KJD2

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T1

NIG1VG1

MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T2

NIG2VG2

MJD2

KJD2

Figure 4: VaR ratios for identical default correlations, PD= 7.5%.

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0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T1

NIG1

VG1

MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T2

NIG2

VG2

MJD2

KJD2

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T1

NIG1

VG1

MJD1KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T2

NIG2

VG2

MJD2KJD2

Figure 5: VaR ratios between true and misspecified (Gaussian) copula, PD= 1%.

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T1

NIG1

VG1

MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=20%

T2

NIG2

VG2

MJD2

KJD2

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T1

NIG1VG1

MJD1

KJD1

0.9 0.92 0.94 0.96 0.98 10

1

2

3

=40%

T2

NIG2VG2

MJD2

KJD2

Figure 6: VaR ratios between true and misspecified (Gaussian) copula, PD= 7.5%.

copulas from infinitely divisible distributions-applications to credit value at risk

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