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SIMULATION OF CORRELATED DEFAULTS
USING COPULAS
Hradayesh Kumar and Sahil Barua
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Under the guidance of
Prof. Malay Bhattacharyya
Hradayesh Kumar and Sahil Barua are second year students at the Indian Institute of Management, Bangalore.
We are indebted to Prof. Malay Bhattacharyya, Quantitative Methods and Information Sciences Area, Indian
Institute of Management, Bangalore, for his invaluable guidance, insights and support throughout the duration
of this study. We are also grateful to Navneet Daga and Vivek Pabari for several illuminating discussions and
for their suggestions.
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Simulation of Correlated Defaults using Copulas
Abstract
The growth of the market for credit derivatives has been striking in terms of notional
principal and market value. In 2007, the total notional principal outstanding on credit
derivatives was $35.1 trillion with a gross market value of $948 billion (ISDA, 2007). The
modelling of default risk at the firm level is well known, with credit rating agencies and
banks applying several models for calculation of default probabilities. However, there has
been little or no work on the modelling of default risk at a portfolio level. This study
develops a methodology for the modelling of the joint distribution of default risk using
copulas. Due to bond market data being unavailable in India, the study makes use of equity
market data obtained from the Prowess and Business Beacon databases for the A-list
companies on the Bombay Stock Exchange to calculate default probabilities over the period
1991-2006. Estimation of the marginal distributions for default probabilities at the firm-level
and of the copulas, at the portfolio level, is then carried out. The choice of the best-fitting
multivariate distribution is based on the log-likelihood criterion. The model developed is
currently based on a bivariate distribution, for simplicity. It replicates the fat-tailed joint
distribution of defaults as is empirically observed. This is the first time that the calculation of
default probabilities using equity market data has been estimated in India, and we believe
that this is also the first paper to attempt the modelling of correlated defaults in India.
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TABLE OF CONTENTS
1. Introduction
2. Description of data and introduction to KMV-Merton model
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3. Introduction to Copulas
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4. Modelling of correlated defaults
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5. Simulation Results
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6. Scope for further study
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7. Conclusion
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8. References
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9. Appendix
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1. INTRODUCTION
Modelling of default risk for individual firms has been studied extensively using
models such as the KMV-Merton model. However, there have been relatively few
attempts to model default risk at the portfolio level. The dependence structure of
defaults by several firms is of particular interest to banks, credit rating institutions and
regulators of derivative markets. The effect of industry-wide factors, for example, plays
an important role in determining the probability of correlated default. The Basle
committee report has identified portfolio risk as one of the important risks to be
managed by banks and financial institutions. To banks lending to several firms within an
industry, the effect of a joint default event can be of unmanageable proportions. The
hedging of these credit risks therefore requires the development of models for joint
default and identification of the dependence structure of defaults. In derivative markets,
an entirely new class of products called credit derivatives has developed in recent years.
A number of credit derivatives such as collateralized debt obligations and credit default
swaps are structured to have payoffs depending on the occurrence of default events.
The accurate pricing of credit derivatives based on an underlying basket of securities
would also, therefore, require a thorough analysis of joint default and the dependence
structure of default. Traditional pricing methods have used linear correlations to model
joint default, but it is now easily proved that the use of linear correlations leads to gross
misspecification of the credit risk involved. This arises primarily due to the fact that
default probabilities are not normally distributed, and joint distributions of default
probabilities often indicate tail dependence significantly higher than that suggested by
the normal distribution. This requires a more accurate specification of the correlations
as well as the development of a model not based on the assumption that the underlying
distribution of default probability is necessarily normal. Similar studies have been
carried out for US data by Das and Geng [2003].
In this study, we use equity market data obtained from the Prowess and Business
Beacon databases for 205 A-list companies on the Bombay Stock Exchange to model
default probabilities for the individual firms. This is then used for simulation of the joint
default process. The objective of the paper is to obtain specific results for the level of
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default risk for individual firms as well as the tail-dependence of joint defaults as
observed empirically in the data.
Traditional credit rating and derivative pricing is based on the probability of
default of individual firms. There are several models used by banks, financial institutions
and credit rating agencies for the modelling of default probabilities. These include the
KMV-Merton model and the Moodys Risk Management System (MRMS). Default
probabilities are expressed in the form of hazard rates, denoted as i(t) for i = 1, 2.. N.
The use of equity data indicates that the hazard rates change over time, depending on
the values of debt and net-worth of the firm. The survival probability for a firm is given
by si(T) = E {exp[-T i(t)dt]}. The probability of default is therefore given by pi(t) = 1 –
si(t). This paper develops a model for the joint distribution for hazard rates i(t) for i = 1,
2.. N.
It is easily seen that the univariate distribution of default probability for an issuer
is not normal, and the first step in the study is therefore to obtain the best-fitted
marginal distribution for each issuer that captures the observed skewness and tails of
each individual default probability. Standard goodness of fit tests such as the
Kolmogorov, Anderson-Darling and log-likelihood tests are used to obtain the best fitting
marginal distributions. The joint distribution for two issuers is then developed using
copula functions.
We have allowed the marginal distributions to take on a number of forms with
different skewness and tail properties. We have considered seven different
distributional forms: a beta distribution, an exponential distribution, a generalized
extreme value distribution, a non-parametric distribution, a gamma distribution, an
extreme value distribution and a normal distribution. The best distribution for the
individual issuer is found using goodness of fit tests.
Copula functions have then been used to combine the marginal distributions of
the issuers into a joint distribution that captures the skewness and tail properties of the
distributions of the individual issuers. Six types of copulas are considered to model the
joint distribution of default and to replicate the observed joint occurrence of outliers.
These include the normal copula, the Clayton copula, the rotated Clayton copula, the
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Gumbel copula, the rotated Gumbel copula and the symmetrized Joe-Clayton copula. Of
these, only the normal copula does not incorporate any tail-dependence.
The fit of the copula functions to the observed joint distribution is measured
using the log-likelihood criterion. This requires the modelling of forty-nine different joint
distributions, and based on maximum log-likelihood, the best fitting joint distribution is
selected.
This paper focuses only on bivariate distributions for default risk to identify a
suitable joint distribution of the hazard rates of individual firms. This requires the
capturing of level of default risk for each individual issuer. More importantly, however, it
also involves capturing of the dependence structure of defaults, with particular
emphasis on tail-dependence, or the simultaneous occurrence of extreme events, which,
as mentioned earlier, would have catastrophic consequences for banks or for the pricing
of credit derivatives.
The increasing usage of credit derivatives for hedging purposes in Indian markets
as well as relaxed lending norms for banks and financial institutions make such a study
of correlated defaults particularly important. The joint distributions developed here may
be extended to multivariate cases easily and used for pricing of credit derivatives such as
collateralized default obligations (CDOs) and credit default swaps (CDSs). From a risk
management perspective, the joint distributions may be used to evaluate the relative
safety of different credit portfolios through the modelling of credit value at risk (CVaR).
Regulators may also use the joint distributions to assess intra-day trading limits for
credit derivatives.
This is the first study on modelling of correlated defaults attempted for Indian
markets. In addition, default probabilities are not available in the public domain in India
as they are in the US through the Moodys database and the lack of a well-developed
corporate bond market makes the calculation of default probabilities difficult. This is,
therefore, also the first time that equity market indicators are being used for calculation
of default probabilities. The growth of the credit derivative markets in the country also
necessitates the development of an accurate econometric model for pricing and hedging
strategies. Traditional pricing methods have relied on the assumptions of univariate
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normality and linear correlation measures that, we prove, do not hold good. In addition,
there is the issue of tail dependence. Tail-dependence refers to the simultaneous
occurrence of extreme events and has serious implications for portfolio management
and pricing. From a risk-management perspective, a joint distribution with taildependence is far riskier than one without. To model this tail-dependence, copula
functions have been used.
The choice of the copula function used depends on the marginal distributions of
default probability for individual issuers. We find that the generalized extreme value,
beta and exponential distributions fit the observed data well. The choice of copula is
however, dependent on the choice of marginal distributions as well. It is shown that the
best fitting copula function does not necessarily result from the best fitting marginal
distributions. This is proved by evaluating forty-nine joint distributions in terms of their
goodness of fit using the log-likelihood criterion.
The paper proceeds as follows. In section 2 we describe the data collected and
used for calculation of hazard rates and probabilities of default and introduce the
modelling procedure for hazard rates. Section 3 is a brief introduction to copulas and
section 4 contains the details of the study. Section 5 and section 6 contain the results
and the course of further study that would be required for a more accurate modelling of
correlated default. Section 7 concludes the paper.
2. DESCRIPTION OF DATA AND INTRODUCTION TO KMV-MERTON MODEL
The data-set used is obtained from the Prowess and Business Beacon databases and
comprises of equity market data obtained for 205 A-list companies on the Bombay Stock
Exchange for the period 1991-2006. For each issuer, calculation of the default
probability and hazard rate is based on the two-step KMV-Merton model. The first step
involves calculation of the firm’s ‘distance to default’, which is based on the difference
between the market value of the firm’s equity and debt and firm volatility. The second
step is based on the assumption that distance to default is standard normally distributed
and calculate the probability of default as the inverse standard normal of the distance to
default. The hazard rate is then calculated from the equation specified previously in this
paper. The default probabilities used for fitting of the marginal distributions and
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estimation of the joint distribution represent monthly data over a period of 13 years
from 1994 to 2006, leading to a total of 156 data points.
A reduced form of the KMV-Merton model as proposed by Bharath and Shumway
[2004] is used for calculation of distance to default and obtaining the default probability.
The expression is as below:
DD = {ln [(E+F)/F] + (r – 0.5)  σ2V)}/ (σVT)
A detailed discussion of the full and reduced form KMV-Merton models is carried out
in Section 4 of the paper.
The 205 different firms for which data has been obtained have been divided into a
number of industries since the objective of the paper is primarily to study default
correlations within industries. The methodology described in the following sections as
well as the results are presented for the petroleum industry – for the firms Chennai
Petroleum Corporation Ltd and IBP Ltd. Similar results have been obtained for the
manufacturing industry and the study may be extended easily to other industries as
well. Table 5 contains data on the mean and standard deviation of default probabilities
for various industries.
3. INTRODUCTION TO COPULAS
3.1 Definition A copula is a multivariate joint distribution defined on the n-dimensional
unit cube [0,1]n and the marginal distributions are uniform on the interval [0,1]. The
function C: [0,1]n  [0,1] is an n-dimensional copula if it satisfies the following
conditions:

C(u) = 0 when u  [0,1] has one component equal to 0

C(u) = ui when u  [0,1] has all components equal to 1 except the ith, which is ui

C(u) is n-increasing
The marginal distributions of the variates are specified by Fi(Xi) for i = 1, 2, .. ,N. The
joint distribution is given by the function F(X). The copula associated with F(X) can then
be defined as:
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C(u1, .., un) = F(F1-1(u1), .., Fn-1(un))
A more detailed description of copulas may be found in Nelson [1999].
Copula functions are particularly important since they allow the modelling of
marginal distributions separate from their dependence structure. The use of copulas in
credit risk analysis is important for two reasons, as studied by Das and Geng [2003],
[2001] and Koziol and Kunisch [2005]. Copulas provide for flexibility in the choice of
marginal distributions for individual issuers. It becomes possible, therefore, to vary
these marginal distributions and observe the effects on the joint default process for a
portfolio. Also, the use of copulas allows the modelling of tail dependence, which is
observed empirically, or the simultaneous occurrence of extreme events and the choice
of different copulas facilitates the study of different tail dependence structures and
copula parameters on portfolio risk.
3.2 Properties Let Ui = Fi(Xi) for all i = 1, .., n. Then U = (U1, .., Un) is a vector of random
variables with uniform marginal distributions. Copulas have the following two important
properties:

Every joint distribution may be written as a copula. This is Sklar’s Theorem, first
introduced in Sklar [1959] and [1973].

If all the marginal distributions, Fi(Xi) are continuous, then the associated copula is
unique.
There are other properties of copulas as well which are not mentioned in this paper. For
further discussion, the reader may refer to Nelson [1999] or any of the more detailed
studies on properties of copulas specified in the bibliography.
3.3 Examples In this paper we have used six types of copulas. The functional forms for
these copulas are discussed below.
Normal copula: The normal copula of the n-variate normal distribution with correlation
matrix  is defined as
C (u1,.., un) = n ( -1(u1), .., -1(un) )
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Gumbel copula: This was first introduced by Gumbel [1960] and can be expressed as
below
C(u1,.., un) = exp [ - ( ( - ln ui ) )1/ ]
 is a the parameter of the Gumbel copula that determines the tail of the distribution.
The rotated Gumbel copula can be obtained from the Gumbel copula as follows:
C’(u,v) = u + v – 1 + C(1-u,1-v;)
Clayton copula: This was first introduced by Clayton [1978] and can be expressed as
below
C(u1,.., un) = [ ( ui - - 1 ] 1/
 is a parameter that determines the tail-dependence structure of the distribution. The
rotated Clayton copula can be obtained from the Clayton copula as follows:
C’(u,v) = u + v – 1 + C(1-u,1-v;)
Symmetrized Joe-Clayton copula: The Joe-Clayton copula was first introduced by Joe
[1997] and can be expressed as below
C(u1,.., un) =1 – ( 1 – [  (1 – (1 - ui))- - 1 ] -1/ ) -1/
 = 1 / (log2(2 - u)) and  = -1/(log2L) where u and L are the upper and lower tail
dependence respectively.
3.4 Importance of Copulas Traditionally, Pearson’s correlation  has been used for
measuring the degree of association between random variables. However, this measure
has several deficiencies, as examined by Embrechts, et al [1999]. First of all, this is a
measure of only the degree of linear association between the random variables. Second,
a correlation of zero does not imply a lack of dependence. It merely implies a lack of
linear dependence. In such cases, a more robust measure of association must be used,
such as Kendall’s , which is a rank-correlation measure. In a bivariate case, Kendall’s 
is defined as below
 = P [(X2 – X1) (Y2 – Y1) > 0] - P [(X2 – X1) (Y2 – Y1) < 0]
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If (X2 – X1) (Y2 – Y1) > 0 the pair of observations is said to be concordant, otherwise it is
discordant. Kendall’s  is closely associated with copula parameters. For example, for a
normal copula,
[X,Y] = 2/ arcsin ((X,Y))
There are similar equations for different copulas. For example, for the Gumbel copula
we have  = 1 – 1/ and for a Clayton copula,  =  / ( + 2).
Copulas also permit the modelling of tail-dependence. Tail dependence refers to the
simultaneous occurrence of extreme events. There is evidence that tail-dependence
exists, for example, in US markets as proved by Das, Freed, Geng and Kapadia [2001].
Their study also proved that correlation levels are higher when default probabilities are
high and are lower when default probabilities are low. Of the copulas considered, only
the normal copula has zero tail dependence. The dependence structure in case of the
normal copula is therefore largely defined by the central observations. The modelling of
tail-dependence therefore requires the use of different copulas – such as the Gumbel
and Clayton family of copulas. Tail dependence may, in addition, be specified for both
the upper as well as the lower tails.
If (X1, X2) is a continuous random vector with marginal distributions F1 and F2, the
coefficient of upper tail dependence is
u =
P [ X2 > F2-1(z) | X1 > F1-1(z) ]
If u > 0 then upper tail-dependence is said to exist. Lower tail dependence is defined as
L =
P [ X2 < F2-1(z) | X1 < F1-1(z) ]
If L > 0 then lower tail-dependence is said to exist. Tail dependence can be expressed in
terms of the copula parameters as well. For example, the Gumbel copula has upper tail
dependence with u = 2 – 21/ and the Clayton copula has lower tail dependence given
by L = 21/.
Figure 3 illustrates the upper tail dependence obtained from the Gumbel copula.
This is done using 250 created data points with a parameter value of 1.5 for the copula
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to plot the probability distribution function for the Gumbel copula. Similarly, figure 4
illustrates the lower tail dependence obtained from the Clayton copula. This is also done
using 250 created data points with a parameter value of 0.4 for the copula to plot the
probability distribution function for the Clayton copula.
4. MODELLING OF CORRELATED DEFAULTS
The first step in the modelling of the joint distribution of default is the modelling of
individual probabilities of default. This is done based on the equity market data obtained
from the Prowess and Business Beacon databases for 205 companies on the A-list of the
Bombay Stock Exchange. Several models are available for the modelling of probability of
default, the most popular of which is the KMV-Merton model. Due to the lack of a
developed bond market in India, the use of equity market data necessitates the use of a
simplified version of the KMV Merton model. The model is described below.
4.1 KMV Merton Model The KMV-Merton model develops the default probability for an
individual issuer at any given point in time. This may be assessed using either bond market data
or equity market data, though bond market data is typically considered to be more accurate. As
per a framework suggested by Merton [1974], the equity of a firm is equivalent to a call option
on the underlying value of the firm with the strike price being the face value of the firm’s debt.
This model allows for the value of the firm and the firm volatility to be assessed from the equity
value, its volatility and other observed variables. Having obtained these values, the probability of
default is obtained as the normal cumulative density function of a z-score calculated using the
firm’s underlying value, the face value of firm’s debt and the firm’s volatility.
There are two principal assumptions in this model of default-probability. They are as follows:
1. The underlying value of a firm follows geometric Brownian motion.
2. Each firm issues just one zero coupon bond with maturity of T years.
The second assumption refers to the debt taken by the firm for a period of T years. In order to
obtain the default probability, a measure called the ‘distance to default’ is defined. The face
value of the firm’s debt is subtracted from the market value of the equity and the result is then
divided by the estimated volatility of the firm. This is referred to as the distance of default.
DD = {[Mkt. Value of Equity] – [Face Value of Debt]}/ (Estimated value of firm volatility)
The probability of default is then given by:
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PD = N(-DD)
where N(.) is the standard normal cumulative distributive function.
The figure shows the distribution of default probability and illustrates the concept of distance to
default:
The distance to default is nothing but the distance on the above plot between the expected
asset value and the default point. Here the asset value corresponds to the total value of the firm
and the default point refers to the face value of debt. EDF is the expected default frequency or
the probability of default. The distance to default narrows as the firm value declines and the
default occurs when it falls below the face value of the debt. It is important to note that the
default point also changes with time and this aspect is taken care of in the model.
The market value of the firm’s equity is estimated using the Black-Scholes formula as under:
E = V N(d1) – e-rT  F  N(d2)
(1)
where V is the value of the firm, F is the face value of the debt, r is the instantaneous risk-free
rate (since the Black-Scholes equation is derived under the assumption of risk-neutrality),
d1 = {ln (V/F) + (r + 0.5  σ2) T} / σT
(2)
d2 = d1 – σ T
(3)
σ is the firm volatility.
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Further, it can be shown through Ito’s lemma that
Equity Volatility = (Value of firm/Equity)  {(Equity)/(Value of firm)}  (Firm Volatility)
Equity Volatility = (Value of firm/Equity)  N(d1)  (Firm Volatility)
(4)
(1) and (4) must be solved to obtain the firm’s volatility. The final default probability is given by
PD = N [-{(ln(V/F) + (μ – 0.5* σ2 ))/(σ * sqrt T)} = N(-DD)
It is important to note that the most crucial inputs to the model are the market value of equity,
face value of debt and the equity volatility. The PD increases with a decline in the market value
of equity.
4.2 Reduced Model Bharath and Shumway [2004] describe a simpler version of the KMVMerton model that has significant predictive power. The advantage of this method is that it
avoids the task of solving the simultaneous equations (1) and (4), which is fairly challenging. The
modifications made are as follows:
1. The market value of the firm’s debt is taken as equal to its face value
2. Assuming the equity and the debt risks are strongly correlated, the debt volatility (σD ) is
estimated as (0.25  σE + 0.05), σE being the equity volatility
3. The volatility of the firm is then calculated as σV = E/(E+D)  σE + D/(E+D)  σD
4. The return on assets of the firm is estimated as the return on the firm’s stock over the
previous year
The distance to default is then obtained as
DD = {ln [(E+F)/F] + (r – 0.5)  σ2V)}/ (σVT)
Bharath and Shumway conclude that the probability calculated using this alternative model is
credible for the following reasons:

The structure of KMV Merton distance to default and expected default frequency is retained

The amount of information captured in the model is more or less the same
Computationally, this method is much easier to implement and this model lends itself easily
to the use of equity market data. This is the model adopted for modelling the default probability
in this paper.
4.3 Fitting of marginal distributions As described earlier, seven different distributions
are fitted to the raw data on default probability. For this case we consider two marginal
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distributions – for Chennai Petroleum Corporation Ltd. and IBP Ltd. The seven different
distributions fitted to the default probability data are compared based on log-likelihood
values and four different goodness of fit tests: the Kolmogorov, Anderson-Darling, L1 and
L2 distance tests as described in Berg and Bakken [2006].

The Kolmogorov distance is defined as the supremum over the absolute differences
between two cumulative density functions, Femp(x) and Fest(x)

The Anderson-Darling statistic is AD = max | Femp(x) - Fest(x) | /  Fest(x) [1 - Fest(x)]

The L1 distance is equal to the average of the absolute differences between the
empirical and statistical distributions

The L2 distance is equal to the root mean squared difference between the two
distributions
Table 1 in the appendix gives the best-fitting distribution according to the five goodnessof-fit criteria for the two marginal distributions. Figure 1 in the appendix is the marginal
distribution for IBP Ltd with the various fitted distributions superimposed and Figure 2 is
the marginal distribution for Chennai Petroleum Corporation Ltd with the various fitted
distributions superimposed.
4.4 Estimation and fitting of the appropriate copula function The estimated marginal
distributions are then combined using six different copulas and the best fitting copulamarginals combination is selected. It is important to note that the best fitted marginal
distributions do not necessarily produce the best fitting copula. It is necessary to choose
the best-fitting combination of copula and marginal distributions. In this case, forty-nine
different combinations of marginal distributions are considered.
The final step in the process is therefore the evaluation of the goodness of fit of the
joint distributions estimated above. The criterion used in this case is the log-likelihood of
each copula-marginals combination and it is found that the best combination is the
symmetrised Joe-Clayton copula with exponential marginal distributions.
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5. SIMULATION RESULTS
The results of the simulation process are discussed under two sections. The first
section discusses the fitting of the marginal distributions to the default probabilities of
individual issuers and the estimation of the best-fitting of the copula. The second section
discusses the implications of the best-fitted joint distribution on the dependence
structure and correlation of default probabilities.
Best-fitting marginal distributions and joint distribution Traditional approaches to pricing
of credit derivatives based on more than one underlying security use Pearson’s
correlation as a measure of association between the securities. We observe clearly from
the plot of the default probability data as well as from the results of the goodness-of-fit
tests that the marginal distributions of default probabilities are not normal. This can be
seen in Table 2 which contains the value of the Kolmogorov, Anderson-Darling, L2 and L1
statistics for each distribution for IBP Ltd. and Chennai Petroleum Corporation Ltd. As a
result, using Pearson’s correlation for pricing of these derivatives leads to incorrect
results. Table 3 contains the best fitting copula-marginals combinations based on the
log-likelihood criterion. It is seen that the four best fits are the Joe-Clayton, Reverse
Gumbel, Clayton and Gumbel copulas with exponential marginal distributions. Other
interesting results are obtained by assuming non-parametric marginal distributions,
yielding the normal copula as the best-fitting copula and by assuming GEV marginal
distributions yielding the reverse Clayton copula as the best fitting copula.
Dependence structure and correlation The degree of tail dependence depends on the
copula chosen to model the joint distribution. In this study we have used seven marginal
distributions and six copulas, leading to a total of 294 models, each having its own set of
parameters and dependence structure. The average log-likelihood values for each of the
copulas are given in Table 4. It is seen that the three copulas with upper tail
dependence, namely the Joe-Clayton, Gumbel and Reverse Clayton copulas have the
highest average log-likelihoods. As an illustration, we present the four best fitting
copula-marginals combinations in terms of log-likelihood in Table 3. The plots of the
cumulative distribution functions are shown for the Gumbel, Clayton and Normal
copulas in Table 3. The best-fitting combination is observed to be the symmetrized Joe-
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Clayton copula with exponential marginal distributions. The values of the parameters for
this copula are also equal to the upper and lower tail-dependence coefficients. For the
Joe-Clayton copula these values are given by 0.2742 and 0.1867, indicating both upper
as well as lower tail-dependence. Thus we see that the joint distribution of default
probabilities exhibits evidence of tail-dependence. This can also be verified from the
scatter plot for the default probabilities of the two firms as shown in figure 5, which
indicates clear tail dependence. As an example, we generate the cdf plots for the
Gumbel and Normal copulas with their parameters as estimated by the simulation. The
results are presented in figures 6 and 7.
The implications of these findings are interesting. First of all, the best-fitting JoeClayton copula indicates that default probabilities tend to cluster at both tails.
Therefore, correlated default and correlated survival are clear for these firms. This is
likely to be a consequence of industry-wide factors that affect the ability of both firms to
meet their debt obligations simultaneously, implying that default events are triggered by
common factors. This raises interesting questions for further studies on common factors
in correlated defaults. The best-fitting Gumbel copula has a parameter value of 1.6695
and once again, indicates significant upper tail-dependence as per the plot of its
cumulative distribution function. The cumulative distribution function for the best-fitting
normal copula with non-parametric marginals is plotted in figure seven. As pointed out
previously, the underlying marginal distributions for default probabilities are proved to
be non-normal and as a result, measures of association such as Pearson’s correlation
cannot be used when pricing credit derivatives or assessing credit risk for portfolio
management. The correct correlation parameter to be used would be the parameter
value obtained for the normal copula. The marginal distributions have a linear
correlation of 0.7582 whereas the parameter for the normal copula is 0.874, which is
much higher than the linear correlation. Thus, a hedging strategy based on the linear
correlation will be misleading and will lead to greater exposure to default risk than one
based on the findings from the normal copula.
6. SCOPE FOR FURTHER STUDY
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The lack of a developed bond market and credit rating data in the public domain in
India places severe constraints on the development of accurate hazard-rate models.
However, with the rapid development of the market for credit derivatives and the
evolution of the corporate debt market, such data is likely to be available in the future. It
would be essential to recalculate the default probabilities used in this study based on
credit rating or bond market yield curve data.
This paper discusses only a bivariate case – using marginal distributions for default
probabilities of only two issuers. However, for banks and financial institutions lending to
more than two firms within a sector or for credit derivatives such as collateralized
default obligations or credit default swaps based on more than two underlying
securities, it would be essential to develop a multivariate model for obtaining the joint
distribution of default probabilities. This paper, in addition, focuses on the joint default
distribution for default probabilities of issuers within the same industry. For banks with
lending mandates, it might be pertinent to develop models for correlated defaults across
industries.
It has been observed previously by Das, Geng, Freed and Kapadia [2001] that default
correlations are dynamic and exhibit asymmetry over time. There are periods of time
when defaults cluster while in other periods, this is absent. This would require a timeseries based approach for modelling of the joint distribution, which has not been
attempted in this paper.
7. CONCLUSION
This paper develops an approach for assessing credit default risk through modelling
of the joint distribution of default probabilities for individual issuers using copula
functions. This is of importance for portfolio risk management by banks and financial
institutions, pricing of credit derivatives based on a basket of securities and calculation
of VaR and regulation of the credit derivatives markets. We believe that this is the paper
to attempt to fit joint distributions to default probabilities using copulas in India. The
paper uses equity market data obtained from the Prowess and Business Beacon
databases for 205 A-list companies on the Bombay Stock Exchange over the period
1991-2006. This data is used to calculate the default probabilities for individual firms
19
which are divided into different industries and this leads to development of a bivariate
model for default probabilities using copula functions.
A total of forty-nine combinations are considered for specification of hazard
rates for the individual issuers and these are combined using six different copulas,
leading to a total of 294 models. Of these, the best-fitted marginal distributions are
identified using four different tests and the best-fitting four joint distribution models are
selected based on the log-likelihood criterion and their implications are discussed in
Section 5. Firstly it is seen that the marginal distributions are not normal, as assumed in
traditional pricing methods for credit derivatives. Therefore, traditional measures of
association such as the Pearson’s correlation cannot be used in pricing or credit risk
assessment applications. Secondly, the best-fitting copula-marginals combination is the
Joe-Clayton copula with exponential marginal distributions. This proves that default
probabilities for the issuers indicate significant upper and lower tail-dependence and has
important implications for pricing of credit derivatives and portfolio management.
Similarly, the best-fitting normal copula with non-parametric marginals has much higher
correlation than the linear correlation between the marginal distributions, which
reinforces the inappropriateness of the use of linear correlation as a measure of
association when modelling joint default. The results therefore illustrate the importance
of taking into account the dependence structure of the marginal distributions for default
probabilities through copula functions to correctly assess credit risk.
20
REFERENCES
[2003] Das, Sanjiv R., and Geng, Gary, (2003), “Simulating correlated default processes
using copulas: A criterion based approach,” working paper, Santa Clara University.
[2001] Das, S., G. Fong., and G. Geng, (2001), “Correlated Default Processes,” working
paper, Santa Clara University and Gifford Fong Associates.
[2001] Das, S., G. Fong., and G. Geng, (2001), “The impact of Correlated Default Risk on
Credit Portfolios,”Journal of Fixed Income, December, v11(3), 9-19.
[1999] Embrechts, Paul., Alexander McNeil, and Daniel Straumann (1999), “Correlation
and dependence in risk management: Properties and Pitfalls,” working paper, University
of Zurich.
[1999] Embrechts, Paul., Alexander McNeil, and Daniel Straumann (1999), “Correlation:
Pitfalls and Alternatives,” working paper, Department Mathematik, ETH Zentrum, Zurich.
[1999] Nelson, Roger B., (1999), “An introduction to Copulas,” Springer-Verlag, NewYork.
[2002] Patton, Andrew J., (2002), “Applications of Copula Theory in Financial
Econometrics,” dissertation, London School of Economics.
[2005] Koziol, Philipp, and Kunisch, Michael, (2005), “Analysis of Copula Functions and
applications to Credit Risk Management,” financial modelling workshop, Ulm.
[2002] Das, S., Freed, L., Geng, G., and Kapadia, N., (2002), “Correlated Default Risk,”
working paper, Santa Clara University.
[2006] Berg, Daniel, and Bakken, Henrik, “Copula Goodness-of-fit Tests: A comparative
Study,” Norwegian Research Council.
[2004] Bharath, Sreedhar T., and Shumway, Tyler, (2005), “Forecasting default with the
KMV-Merton model,” University of Michigan.
[2006] Kole, E., Koedijk, K. and Verbeek, M., (2006), “Selecting Copulas for Risk
Management,” working paper, Erasmus University.
21
22
APPENDIX
Table 1: Best-fit distribution according to various parameters
IBP Limited
Chennai Petroleum
Corporation Ltd
Log-likelihood
Beta
Beta
Kolmogorov
Gamma
Beta
Anderson-Darling
Gamma
EV
L1
Beta
Normal
L2
Gamma
Beta
Table 2: Value of test-statistics for goodness-of-fit test
IBP Ltd
CPC Ltd
K
AD
L
L2
K
AD
L
L2
Beta
0.171
0.39
0.053
0.071
0.149
0.688
0.068
0.079
Exponential
0.532
1.13
0.262
0.309
0.198
0.476
0.132
0.147
GEV
0.194
4.24
0.093
0.104
0.259
4.548
0.121
0.137
Gamma
0.168
0.38
0.054
0.070
0.150
0.815
0.072
0.083
Normal
0.389
0.91
0.139
0.170
0.200
0.402
0.065
0.089
EV
0.353
0.82
0.146
0.175
0.196
0.398
0.075
0.095
23
Table 3: Best-fitting copula-marginals combinations
Marginals
Best-fit copula
Log-likelihood
Exponential
Joe-Clayton
192.94
Exponential
Reverse Gumbel
187.61
Exponential
Clayton
174.8
Exponential
Gumbel
124.17
Non-parametric
Normal
112.68
GEV
Reverse Clayton
103.58
Table 4: Average log-likelihoods for different copulas
Copula
Average Log-likelihood
Joe-Clayton
56.450
Gumbel
49.288
Reverse Clayton
44.600
Reverse Gumbel
42.596
Clayton
39.452
Normal
12.136
24
Table 5: Average default probability by industry
Mean
Standard Deviation
Construction/Developers
0.2018
0.0797
Finance Corporations
0.0984
0.0926
Minerals
0.0663
0.0813
Hotels/Services
0.0074
0.0113
Textiles
0.0159
0.0233
Transportation
0.0247
0.0409
Consumer Industries
0.0150
0.0237
Entertainment
0.0134
0.0079
Foods
2.6  10-9
6.18  10-11
Chemical/Dyes/Paints
0.0224
0.0311
Technology
0.0251
0.0366
Manufacturing/Heavy Machinery
0.0817
0.0952
Telecom
0.2853
0.3198
Fertilizers/Chemicals
0.0711
0.0633
Cement
0.1134
0.2330
Power
0.0512
0.0605
Petroleum/Petrochemicals
0.1080
0.1407
Auto/Airlines
0.0473
0.1066
IT/Software
0.0894
0.1922
Banks
0.2177
0.1412
Pharmaceuticals/Healthcare
0.0146
0.0255
25
Figure 1: CDF for CPC Ltd
Figure 2: CDF for IBP Ltd
26
Figure 3: Gumbel copula PDF
Figure 4: Clayton copula PDF
27
Figure 5: Scatter plot for default probabilities
Figure 6: Best-fitting Gumbel copula plot
28
Figure 7: Best-fitting normal copula plot