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1 SIMULATION OF CORRELATED DEFAULTS USING COPULAS Hradayesh Kumar and Sahil Barua 1 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. 2 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. 3 TABLE OF CONTENTS 1. Introduction 2. Description of data and introduction to KMV-Merton model 4 8 3. Introduction to Copulas 9 4. Modelling of correlated defaults 13 5. Simulation Results 18 6. Scope for further study 20 7. Conclusion 21 8. References 23 9. Appendix 25 4 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 5 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 6 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 7 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 8 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)}/ (σVT) 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: 9 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) ) 10 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/(log2L) 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] 11 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 12 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: 13 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} / σT (2) d2 = d1 – σ T (3) σ is the firm volatility. 14 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)}/ (σVT) 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 15 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. 16 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- 17 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 18 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. 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[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