CREDIT RISK IN THE TRADITIONAL BANKING BOOK A VaR APPROACH UNDER

CREDIT RISK IN THE TRADITIONAL BANKING BOOK: A VaR APPROACH UNDER CORRELATED DEFAULT• by Cristiano Zazzara* JEL Classification: G11, G21, G28 1. Introduction Banks and financial institutions in Italy and many other countries are developing and enhancing methods to measure and manage the main risk inherent in their business operations: the credit risk of their loan portfolios. The specific direction that these efforts have taken is to draw on advances in financial engineering and statistics to create computer simulations and analytical methods. These techniques provide a more accurate measurement of risk, which can then be used in bank management (for example, to determine more accurately the pricing of financial instruments and effective credit limits, or even appropriate allocations of capital) and in the regulatory environment to determine capital charges. The measurement of the credit risk of loan portfolios usually entails the same basic procedure as the measurement of market risk, i.e. the VaR (Value at Risk) framework is used in a model that calculates the maximum potential loss or expected loss of the portfolio. However, there are several impediments to these measurements: a) Models are harder to handle than those for market risk. In other words, credit risk models deal with a default event for which one cannot assume simple (logarithmic) normality and b) Data are subject to many constraints that will reflect on many aspects of parameter estimation and setting, including default rate, recovery rate and default correlation. • Paper presented at the IX International Tor Vergata Conference on Banking and Finance (November 16, 2000). * Fondo Interbancario di Tutela dei Depositi (FITD) and University Luiss “Guido Carli” of Rome. The views expressed herein are the author’s and not necessarily those of the Fondo Interbancario di Tutela dei Depositi. Thanks to Mark Carey (Federal Reserve Board of Governors), Andrea Federico (Oliver, Wyman & Company), my colleagues Francesco Pistelli and Aurelio Maccario of the FITD for useful conversations and Marco Pellegrini for helpful research assistance. Correspondence should be addressed to the author at Fondo Interbancario di Tutela dei Depositi, Via del Plebiscito, 102, 00186 Rome. +39-06-699.86.402 (voice), +3906.67.98.916 (fax), czazzara@fitd.it. 331 In this paper we focus on the above problems and propose a method that roughly captures portfolio credit risks while minimizing the calculation loads. Finally, we consider the impact of this technique on credit risk management. The structure of this paper is as follows. In section 2, we review the related literature in this field. In sections 3 and 4, we outline the framework for simplified credit risk measurement. We proceed applying these techniques to a sample portfolio and demonstrate their applicability to the Italian case using the data on corporate default rates recently released by the Bank of Italy. Finally, we draw some conclusions about this method. 2. Related literature Considerable progress has been made over the past decade in the effort to quantify and manage the risk of loan portfolios1. While banks and financial institutions may have adopted different models for their own credit portfolios, three models, namely, CreditMetrics™2, Portfolio Manager™ 3 and CreditRisk+™4, have become publicly available and therefore have caught the most attention5. All three models focus on the same sets of important factors, namely default probabilities and default correlation. However, they differ significantly in how these factors are incorporated. Both CreditMetrics and Portfolio Manager assume that default probabilities are non-stochastic, while CreditRisk+ assumes that default probabilities are random variables themselves. Default correlation in both CreditMetrics and Portfolio Manager is derived from asset return correlation, but in CreditRisk+ it is introduced through the assets’ dependence on a common default probability. Both CreditMetrics and Portfolio Manager use Monte Carlo simulation to obtain loss statistics which can be time consuming, while CreditRisk+ uses a nonsimulation approach to calculate the portfolio loss distribution and is numerically rather efficient. We therefore see that the methodology adopted in CreditRisk+ seems rather different from those of CreditMetrics and Portfolio Manager, and several authors have tried to make comparison and reconciliation between CreditRisk+ and other models. Broeker and Rolfes (1998) showed how to incorporate the effect of rating migrations into the framework of CreditRisk+, while Koyluoglu and Hickman (1998) and Crouhy, Galai and Mark (2000) explored difference and common ground among various models. Gordy (2000) further examined in great details the 1 See Savona, Sironi (2000) for evidences on large Italian banks and Ong (1999) for a technical description of internal credit risk models components. 2 See Gupton, Finger, Bhatia (1997), downloadable from http://www.riskmetrics.com. 3 See KMV (undated). 4 See Credit Suisse Financial Products (1997), downloadable from http://www.csfp.csh.com/. 5 See Saunders (1999) for a comprehensive overview of current credit risk models. 332 structures and numerical results of CreditMetrics and CreditRisk+, and Finger (1998) showed in the one-factor case how to calibrate model parameters so that both models produce exactly the same mean and standard deviation of loss. Common to all these models is the fact that the models describe only a single period and therefore describe for a specific risk horizon whether each asset of interest defaults within the period (usually one year). Finger (2000) surveyed a number of extensions of the single horizon models that allow for a treatment of default timing over longer horizons. In this paper, we take a combined approach, including elements of several different models rather than trying to choose among them. Similar to CreditMetrics and KMV’s PortfolioManager, this paper employs a Monte Carlo aggregation strategy (although no individual bank would need to operate Monte Carlo programs to calculate the standard deviation of the loan portfolio in the presence of correlated defaults). Similar to CreditRisk+, assets are grouped according to certain characteristics, simplifying structure and computations. Differently from the above models, we directly take into account default correlations, estimating average default correlations (within and between industry sectors) from historical data through an actuarial-based approach6. Default correlations are significant when one considers the likelihood that multiple borrowers will fall into financial difficulties. The impact of macroeconomic fluctuations on credit losses is an essential consideration, but in this paper we assume such fluctuations are accounted for in estimating parameters. 3. Outline of the model In this section we present the characteristics of our model. 3.1. The loss approach: Default Mode (DM) versus Mark-To-Market (MTM) This paper’s framework embodies a Default Mode (DM) rather than a Mark-to-Market (MTM) philosophy. That is, only credit defaults are modeled; variations in the value of assets due to changes in credit quality short of default are assumed unimportant. Although a case can be made for MTM approaches, we chose a DM approach for its simplicity, feasibility, and because we believe it is more appropriate for traditional banking books (most credit risk borne by most banks still is in the traditional banking book, 6 As outlined in Federal Reserve Board of Governors (1998) and Banking Supervision (1999a). Basel Committee on 333 especially the loan portfolio7). A DM approach simplifies the framework because most fluctuations in value need not be modeled. Similarly, implementation is made more feasible by a DM approach because the modeling burden on banks is enormously reduced8. The vast majority of commercial banks’ assets rarely trade and the prices of any trades that do occur are usually unobservable. Thus, current “market” values of assets must be approximated. Given the lack of transaction price data, market value approximation algorithms are very difficult to develop and validate and would be a major source of operational error in MTM-based capital allocation procedures. 3.2. Inputs for credit risk measurement: a) exposure, recovery rate, probability of default and risk horizon In our simplified model, we use fixed values for the exposure amount and the recovery rate. In particular, we assume the exposure amount is the amount remaining when the amount recoverable from collateral and guarantees (conservatively estimated) is subtracted from the amount of the loan. The recovery rate for this remainder is assumed to be 0%. However, we recognize these assumptions are obviously unrealistic. For example, several studies9 find that recovery rates on defaults of subordinated debt are smaller than those for senior debt. Moreover, a bank’s exposure at the beginning of a period may differ substantially from exposure at the time of a default during the period. Amortization of principal may reduce exposure on a term loan, or a distressed borrower may make large drawdowns on its line of credit prior to default10. Similar to Creditmetrics, the default rate is assumed to be deterministic (i.e., it does not vary over time), remaining constant in the assumed one-year risk evaluation period (the choice of the one year horizon seems consistent As described in Savona, Sironi (2000), risk managers of large Italian banks confirm that credit risk accounts for approximately 70-80% of total risk deriving from banks’ business activities. 8 When a financial institution has tens or hundreds of thousands of credit exposures, simulations for complex models of credit risk management require enormous calculation loads. Even powerful computers may require a long calculation time before risk results are available. 9 See, for example, Carty, Lieberman (1996) and Altman, Kishore (1996) for evidences on the US bond market. Generale, Gobbi (1996) present the results of a survey on the practices of recovery in the Italian bank loan market. 10 Asanorw and Marker (1995) provide evidence on loan commitments for the Citibank and illuminate how such dynamics should be handled. Furthermore, Gupton et al. (1997) and Ong (1999) describe the concepts of adjusted exposure and usage given default for trading and banking book assets. 7 334 with banks’ accounting practices11). Moreover, we assume the bank is able to effectively discriminate its borrowers on the basis of their creditworthiness and can therefore assign a proper probability of default to them. Therefore, we will not focus our attention on the determination of the counterparties’ probability of default and in our model we will take this parameter as given12, even though we suggest default rates be estimated from historical default data, either using banks’ internal data or publicly available data. In this latter case, one can use for example rating data from Moody’s or S&P’s to determine average default rates and other relevant statistics for rated obligors in the portfolio. Unfortunately, most bank counterparties are private (even in the US) and this is especially true in Europe. For example, in the United Kingdom only 25 of the 100 companies listed in the London Stock Exchange and included in the FT-SE index receive a rating (and we are considering the second most developed financial market of the world, after the US); in Germany only 30 private companies are rated and in Italy 15 at the most. Thus, for European banks the Internal Risk Based (IRB) approach recently proposed by the Basel Committee on Banking Supervision (1999b) seems to be the only feasible solution13. b) default correlation Portfolio management of credit risk cannot be performed in isolation without understanding the full impact of correlation on the portfolio. As there is very strong evidence14 (at least in the US market) that movements in credit quality of different obligors are correlated, it is not prudent to set all correlations to zero, ignoring their implications for portfolio credit risk management. It must be kept in mind that the higher the degree of correlation, the greater is the volatility (i.e. unexpected loss) of a portfolio’s value attributable to credit risk. Since we are dealing with a DM approach, we are only interested in correlation of default occurrences. However, there is a main difficulty surrounding default correlation: the absence of direct empirical observation of “simultaneous defaulting events” from the market over a reasonable period of time. Faced with these quandaries – the scarcity of correlations data and the necessity of incorporating default correlation – 11 In this case this assumption implies the bank is able to replenish its capital level within a one year horizon. This issue is left as a further research topic. 12 See Carey (2000) for an empirical examination of the major methods currently used to estimate average default probabilities. 13 For a description and evidence on internal credit risk rating systems in the US bank market see Tracey, Carey (1998) and English, Nelson (1998). As to large Italian banks, De Laurentis (2000) provides the result of a survey (conducted by the Working Group on Credit Risk of the Fondo Interbancario di Tutela dei Depositi) on the features of internal credit risk rating systems for banking book assets. 14 See Moody’s Investor Services (1997). 335 we suggest, at the very least, that some fixed, constant level of default correlation be used in the internal credit risk model. Currently there exist three approaches to the estimation of default correlation: 1) The first approach uses the asset return correlation to quantify the default correlation, such as the Creditmetrics model (devoloped by Gupton et al. (1997)). In this latter case, using asset return correlation, we can obtain the default correlation of two discrete events over one year period based on the formula: ρ 1, 2 = p1, 2 − p1 p 2 p1 (1 − p1 ) p 2 (1 − p 2 ) (3.1) where p1 and p 2 are the probabilities of default for asset 1 and asset 2 respectively, while p1, 2 is the probability that both assets default. 2) The second approach utilizes historical loss datasets, either internally built by banks or publicly available, such as those available from rating agencies like Moody’s and Standard & Poor’s15 and from central banks and other consortiums16. 3) The last approach derives default correlation implicitly from the fact that common factors influence the variation of default rates for borrowers belonging to the same group or category. This is the approach used in the CreditRisk+ by Credit Suisse Financial Products (1997), which has been used in the actuarial science for insurance pricing. In this paper we take the second route outlined above. However, given the relative infrequency of default events there is in general no hope of achieving a complete estimation of all default correlation coefficients between individual loans in a portfolio from loss experience data. In fact, in a portfolio with N assets, the number of default correlations to be estimated should be (N2 - N)/2. Therefore, we resort to the solution of grouping obligors (this approach is similar in spirit to CreditRisk+) according to certain characteristics, simplifying structure and computations. Particularly, as an approximation we turn to indexing all the obligors in the portfolio by industry-specific information (industry sectors) and then calculating only the average default correlations within and between industry sectors. The advantage of our choice will be that of reducing the dimensionality of the 15 A common feature of such datasets is their focus on bond or commercial loan experiences for U.S. obligors. 16 Data availability varies in countries. It is our understanding that private firms, central banks, or other entities have built large databases for loan experiences in several countries (for example, Central Banks of Argentina, Italy and Japan have built similar datasets). 336 default correlation matrix (i.e., reducing the number of default correlations to be estimated). Consequently, this will result in faster calculations17. Even though this “top-down” approach to the estimation of default correlation captures only its average measure (and can lead to wide confidence level around these estimates), we believe it may be wiser to do this than to use zero default correlation. Let us now turn to a method that makes use of historical default data. As reported by Gupton, Finger, Bhatia (1997) and Basel Committee on Banking Supervision (1999a), average default correlation ( ρ ) for loans in the same group (in our case, industry sectors) is derived from the variance of the underlying default rates using the following formula: Nσ 2 −1 p (1 − p ) ρ= N −1 (3.2) When N (which is the number of observations in the time series) is large, Equation (2) can be approximated as: ρ≅ σ2 p (1 − p ) (3.3) Whereas, average default correlation between borrowers belonging to different groups (for example, ρ kl for groups k and l) is derived from the covariance between annual default rates for those groups, according to the following formula: ρ kl = 17 Covariance ( p k , p l ) p k (1 − p k ) p l (1 − p l ) (3.4) We remind you that in our model the probability of default for individual obligors is supposed to be known (in particular, we advocate the use of an internal risk rating system in order to assign to each obligor a reasonable estimate of default probability). Therefore, we will use the average default probability for obligors belonging to a specific industry sectors only to derive the relative average default correlation parameter. In fact, if we assigned to each obligor belonging to the same industry sector the same average default probability for that sector we would ignore their different creditworthiness. For example, two obligors classified AAA and C respectively have a different riskiness (and, therefore, a different probability of default) but -on average -- a similar default correlation profile. 337 With these formulae in hand, it is easy to derive the average level of default correlation within and between industry sectors for loans in the portfolio. These measures will prove useful later when calculating the degree of variability of the loan portfolio loss distribution. However, there are many caveats we must bear in mind before using this method with confidence. In fact, the main problem is the lack of long time series for corporate default rates supporting these estimates, that usually results in model parameters instability. In section 5, we will provide an implementation of this method to derive average levels of default correlation, indicating the width of the confidence levels around such estimates. 4. Approaches to credit risk modelling under the “Default-mode” philosophy: from the mean/standard deviation approach to the Monte Carlo simulation method In this section we first provide a mean/standard deviation approach to the estimate of the first two moments of the loan loss distribution under the assumption of correlated defaults. Then we show how to generate the full loss Probability Distribution Function (PDF) of the loan portfolio in a Monte Carlo setting. As pointed out in section 3.2, in the following discussion we assume the probability of default for each loan is given exogeneously to the model. 4.1. The mean/standard deviation approach under correlated defaults We consider a portfolio with N exposures. The default rate for exposure i -- up to some point in the future -- is pi ; the amount of the exposure is ei and the recovery rate at default is ri ( 0 ≤ ri ≤ 1 ) (all values are fixed)18. The portfolio loss L can be expressed using a random variable with either 1 or 0 as its value19: 1 Di =  0 ( Probability pi ) ( Probability 1 - p i ) 18 It is common to set up models so that these parameters are deterministic values, but ordinarily they will have some degree of uncertainty. 19 This is called a Bernoulli random variable. 338 Therefore, L = ∑ Di * ei * (1 − ri ) i =1 N (4.1) In Equation (5), the loss is a discrete value, but when N is sufficiently large and the interval between values is sufficiently small, it can be treated as continuously distributed. Assuming the recovery rate ri is equal to zero and ignoring correlation between default events, the expected value and the variance for L can be expressed respectively as: E[ L ] = ∑ p i * e i i =1 N (4.2) Var[ L] = ∑ p i * (1 − p i ) * ei i =1 N (4.3) Therefore, the standard deviation for L is equal to: St.dev[ L] = N ∑p i =1 i N i * (1 − p i ) * ei2 (4.4) = ∑ p (1 − p ) i =1 i N * ∑e i =1 N 2 i (4.5) From formula (8.1) we can derive the following equivalent expression: St.dev[L] = ∑ p i (1 − p i ) i =1 N N * ∑ ei * i =1 N ∑e ∑e i =1 i =1 N i N 2 i (4.6) The circled ratio above can be considered as an Index of Concentration (or Diversification) of the portfolio (IC), in the absence of correlation between 339 default events20. In this case, the IC ranges from 0 to 1 and depends on the amount and number of the exposures only. Therefore, this index will be lowest (diversified portfolio) when the exposures in the portfolio are of the same amount and their number is quite large. On the other hand, it will be highest (concentrated portfolio) when the exposure amounts are all different and there are few exposures in the portfolio. In the limit, when the portfolio is composed of only one exposure the IC is equal to 1. Since this index does not include any level of default correlation, the information it gives may be misleading. Thus, we can have a portfolio with a low IC but which is not diversified at all. We make clear this point with the following example: Suppose we have a portfolio of 10,000 loans of the same amount, and all these loans have been granted to borrowers belonging to the same industry-group (let’s say, for example, the “Transport” industry sector)21. In this case, the IC will be quite low even though, in practice, the loan portfolio is totally concentrated in only one industry sector. In this latter case, loans in the portfolio may be influenced by common background factors. Therefore, the IC seems to be of dubious meaning in terms of concentration risk. Hence, let us see what happens when including some level of default correlation between exposures. Before proceeding, we clarify that -- in accordance with the method previously described for estimating average default correlation within and between industry groups (see section 3.2) -- the correlation effect will be considered as the industry-sector level only. The extension to the correlated case is quite straightforward. In fact, the standard deviation of the loan loss random variable will be equal to: St.dev[ L] = ∑ p (1 − p ) ∗ e i i i 2 i + 2 ∗ ∑ ρ ij ∗St.dev[ Li ] ∗ St.dev[ L j ] i , j =1 i≠ j N (4.7) where St.dev[L i ] = e i * p i * (1 − p i ) 20 The IC is similar to the Herfindhal Index -- widely used to calculate concentration of market shares -- which is equal to the sum of the squared weights of the portfolio. Ford (1998) proposes a similar concentration ratio for managing the risk of a loan portfolio. 21 In addition to industry data, this case can be easily extended to consider geographic-specific information. 340 St.dev[L j ] = e j * p j * (1 − p j ) that, after a few passages, becomes: St.dev[L] = ∑ p (1 − p ) i =1 i i N N ∗ ∑ ei ∗ i =1 N ∑e ∑e i =1 i =1 N i N 2 i 2 * ∑ ρ ij ∗e i ∗ e j ∗ 1+ i , j =1 i≠ j N ∑e i =1 N 2 i (4.8) The circled ratio in formula (4.8) can be considered as an Index of Concentration in case of Correlated default (ICC). In fact, it is expressed as the product of the previous IC and a correlation factor (CF). Obviously, the CF collapses to 1 when ρ ij is equal to zero; in this latter case the IC and ICC will coincide. This extension is extremely important since the higher the default correlation between loans, the higher the degree of concentration in the portfolio, no matter the amount and the number of loans. In fact, when ρ ij is large, even though the number of loans in the portfolio is large, the ICC is high (in the limit, when ρ ij is equal to 1 also the ICC is equal to 1). In other words, the lower the default correlation between exposures, the more prevalent is the effect of diversification by number (and amount) of loans, as measured by the IC; conversely, the greater the default correlation, the less prevalent is the effect of diversification by number (and amount) of loans. In section 5, we will show these effects on a fictitious loan portfolio. 4.2. The estimation of the loan portfolio loss distribution as a whole through Monte Carlo Simulations Rather than assume the loss PDF of the loan portfolio conforms to some family of distributions that could be parameterized by the mean and standard deviation calculated above (in order to identify the multiple of the estimated standard deviation22), we resort to Monte Carlo simulation to generate the loss distribution, and then estimate the maximum loss at various percentile levels23. As reported in Basel (1999a), this is the approach used by some market practitioners. Ong (1999) provides a technical and tangible description of this method. 23 For example, this is the approach also taken by Gupton et al. (1997) in Creditmetrics. There is an analytical approach other than the simulation approach as well. This approach makes certain assumptions about the loss distribution for individual exposures, and then uses 22 341 It is easy to generate multivariate normal random numbers using the Cholesky decomposition of the variance/covariance matrix as long as the random variables exhibit a (logarithmic) normal distribution. However, the default mode approach used in this paper assumes a Bernoulli distribution of "default" and "non-default," so the process of generating multivariate Bernoulli random numbers that take account of the correlation is not a simple application of the Cholesky decomposition. However, the Cholesky decomposition can be used for normal distributions, so one method is to use the normal distribution as a medium for generating Bernoulli random numbers. Our approach, similar to that of Creditmetrics, uses the corporate asset value model of Merton (1974) that indicates default will occur if the value of a company's assets falls below a certain level. In other words, corporate asset value contains a threshold value that is the dividing line between default and non-default. We can therefore create a model that assumes that the rate of return on assets will have the standard normal distribution. In other words, if the (average) default rate of industry sector i is pi , then the threshold value for default/non-default is given by Φ −1 ( p i ) , where Φ −1 (⋅) is the inverse function of the cumulative density function of the standard normal distribution. We remind you that default rates for industry sectors will only be used for determining the average level of default correlation; therefore, each company will be assigned a specific probability of default by the bank. In Figure 1 we represent the asset value model. analytical techniques to obtain the loss distribution of the portfolio as a whole. CreditRisk+ is just an application of the analytical approach (see Credit Suisse Financial Products (1997)). 342 Figure 1. The Asset Value Model of Merton Following the above logic, we use the following method to generate multivariate Bernoulli random numbers, illustrating two cases: 1) when loans belong to the same industry sector and have therefore the same default correlation; 2) when loans belong to different industry sectors with a different default correlation. 1) We will consider a lending portfolio in which default correlation between individual exposures is constant. This happens when loans belong to the same industry sector. First we consider the random variable Di (i = 1,2, L , n) , which has a Bernoulli distribution. p) 1 ( Probability Di =  0 ( Probability 1 − p ) (4.9) 343 In other words, Di (i = 1,2, L , n) for exposure i in the portfolio (comprising n exposures) takes the value 1 (default) with probability p (average default probability for a particular sector) and 0 (non-default) with probability 1 − p . Also, the correlation coefficient of each Di is ρ and is constant within each industry sector (this is the average default correlation for a specific sector). As noted above, the process of generating multivariate Bernoulli random numbers that take account of the correlation is not a simple application of the Cholesky decomposition. However, the Cholesky decomposition can be used for normal distributions, so one method is to use the normal distribution as a medium for generating Bernoulli random numbers. Therefore, we consider a random variable X i (i = 1, 2, L , n) that follows the standard normal distribution with 0 for its mean and 1 for its variance. (However, individual variables are correlated rather than independent). At this time, Di is expressed as: 1 (−∞ < X i ≤ Φ −1 ( p )) Di =  −1 0 (Φ ( p ) < X i < ∞ ) (4.10) where Φ −1 (⋅) is the inverse function of the cumulative density function of the standard normal distribution. For the correlation coefficient of Di (i = 1,2, L , n) to be ρ (this is a default correlation coefficient), we need properly set a correlation coefficient ~ ρ (this is an asset return correlation coefficient) for X i (i = 1, 2, L , n) . The default correlation coefficient ρ can be expressed as: ρ= where E[ D i D j ] − p 2 p (1 − p) p (1 − p) (4.11) 1 (−∞ < X i ≤ Φ −1 ( p ), − ∞ < X j ≤ Φ −1 ( p)) Di D j =  (otherwise ) 0 ~ dimensional normal distribution with a correlation coefficient of ρ . 344 (4.12) Therefore, E[ Di D j ] is the cumulative density function of a two −1 Φ − 1 ( pi ) Φ ( p j ) E[ Di D j ] = −∞ ∫ −∞ ∫ 1 ~ 2π 1 − ρ 2   1 2 2 ~ exp − ~ 2 ) xi + x j − 2 ρxi x j  dxi dx j  2(1 − ρ  ( ) (4.13) This makes it possible to use simultaneously Equation (13) and Equation ~ (11) to obtain a ρ that will satisfy Equation (11). However, numerical calculations will be required to obtain the definite integral above. It is therefore possible to obtain multivariate Bernoulli random numbers Di by using Equation (10.1) after generating multivariate normal random numbers at the n -th dimension with mean of 0, variance of 1, and constant ~ correlation coefficient of ρ . 2) Also when loans belong to different industry sectors (with a different default correlation), we express default/non-default for exposure i within the portfolio using the Bernoulli random number Di , as was shown in Subsection 1. But, in this latter case, we cannot necessarily assume that the default rates p k , p l (for example for sectors k and l), will be equal, nor does it necessarily follow that the correlation coefficient between the default events of the exposures of different sectors ρ k ,l will be constant either. In this case, random numbers are generated as follows. Assuming loans in the portfolio are distributed between only two industry sectors, k and l, we then derive the average default correlation between these two sectors in the following way. First Di , k , D j , l ( i ≠ j and k ≠ l ) are expressed as shown below using random variables X i , k , X j ,l which follow the standard normal distribution. Thus, Di , k 1 (−∞ < X i , k ≤ Φ −1 ( p i , k )) = −1 0 (Φ ( p i , k ) < X i , k < ∞) (4.14) −1 1 (−∞ < X j ,l ≤ Φ ( p j ,l )) D j, l =  −1 0 (Φ ( p j , l ) < X j , l < ∞) 345 If the correlation coefficient of Di , k , D j ,l is ρ k ,l , and simplyfing notation omitting the subscripts i and j for the variables p , then the following relationship holds: ρ k ,l = E[ Di , k D j ,l ] − p k p l p k (1 − p k ) p l (1 − p l ) j ,l (4.15) Likewise, if the correlation coefficient of X i , k , X ~ is ρ k ,l , then through ~ E[ Di , k D j ,l ] , the relationship between ρ k ,l and ρ k ,l is: −1 Φ −1 ( p k ) Φ ( p l ) E[ Di ,k D j ,l ] = −∞ ∫ −∞ ∫ 1 ~2 2π 1 − ρ k ,l   1 2 2 ~ exp − ~ 2 ) xi ,k + x j ,l − 2 ρ k ,l xi , k x j ,l  dxi , k dx j ,l   2(1 − ρ k ,l   ( ) (4.16) ~ We can therefore find ρ k ,l for all i ≠ j to arrive at a correlation matrix of standard normal distribution variables X 1 , L , X n , such that we can get D1 , L , Dn . 5. Numerical Examples In this section we produce some numerical examples to illustrate how our method can be used to value a fictitious loan portfolio for the Italian case. We assume the portfolio is composed of 25 loans, distributed over 5 industry sectors, with average probabilities of default and default correlations estimated from the Bank of Italy’s loss experience datasets, recently released in the first quarter of 2000. We also assume that default probabilities are assigned to each borrower according to the following Table. Table 1. Rating Classes and Average Default Rates Rating Class AAA AA A BBB BB B CCC 346 Mean Default rate 0,10% 0,50% 1,00% 2,00% 5,00% 7,50% 11,00% In Table 2 below we give full details for the loan portfolio at issue, while in Table 3 we provide a description of Industry Codes reported in the last column of Table 2. Table 2. The Fictitious Loan Portfolio Loan 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Exposure (thousand $) 100 300 250 320 400 800 700 420 500 620 150 250 300 500 600 730 420 400 760 820 350 380 470 550 125 Rating Class AAA AA B CCC BBB BBB B AA AAA BBB BB CCC A B CCC BB AAA AA CCC A A BB B AA B Prob. Default 0,10% 0,50% 7,50% 11,00% 2,00% 2,00% 7,50% 0,50% 0,10% 2,00% 5,00% 11,00% 1,00% 7,50% 11,00% 5,00% 0,10% 0,50% 11,00% 1,00% 1,00% 5,00% 7,50% 0,50% 7,50% Industry Code 51 58 51 73 54 56 73 73 58 51 54 58 51 58 73 56 54 54 54 73 51 54 73 58 58 347 Table 3. Description of Industry Codes Industry Codes 65 73 63 66 57 58 59 60 53 54 61 55 51 52 64 56 62 71 68 69 70 67 72 Description Other industrial products Other sales-related services Paper, publishing, and printing products Construction Agricultural and industrial machinery Office, EDP Machinery and others Electric material Transport Iron and non iron material and ore Ores and products based on non-metallic minerals Food products, beverages and tobacco-based products Chemicals Agricultural, forestry, and fishing products Energy products Rubber and plastic products Metal products, apart from machinery and means of conveyance Textile, leather, shoes, and clothing products Transport related services Hotel and public firms products Internal transport services Sea and air transports Services trade and similar Communication services Source: Bank of Italy’s “Base Informativa Pubblica”, 2000-Q1. 5.1. Data description and estimation methodology The time series of annual default rates we employ come from the Bank of Italy’s “Base Informativa Pubblica”, a large and publicly available dataset that collect different kinds of data for Italian Banks and Financial Institutions as a whole (that is, data for individual Institutions are not available). In particular, we use the new time series of default data, recently released by the Bank of Italy in the first quarter of 2000, which contain yearly data (in terms of amount and number) of oustanding loans, current year adjusted doubtful 348 loans24 and default rates from 1985 to 1999. Default rates are calculated by dividing current year adjusted doubtful loans by outstanding loans of the previous year. These data are split up in many classification variables such as: industry, location of the borrower, amount and number of the loans, and others25. Table 4 provides descriptive statistics (including average default correlations as well) for the yearly default data, with reference to industry sectors (i.e., omitting other classification variables). Confidence levels for average default correlation within sectors are calculated at 95%. Table 4. Descriptive statistics of annual default rates for industry sectors (1985-1999) Statisti cs Industry Codes 65 73 63 66 57 58 59 60 53 54 61 55 51 52 64 56 62 71 68 69 70 67 72 Averag e 2,96% 3,52% 1,78% 5,12% 2,15% 1,86% 1,53% 1,67% 1,24% 2,61% 2,97% 1,05% 4,16% 0,35% 1,55% 2,76% 3,00% 1,36% 3,84% 0,91% 0,92% 3,00% 0,04% Standard Deviatio n 1,07% 1,14% 0,89% 1,77% 1,16% 1,78% 0,77% 1,44% 1,53% 0,79% 1,13% 0,57% 3,22% 0,66% 0,70% 1,92% 1,21% 0,99% 1,68% 0,34% 0,93% 0,86% 0,05% Varianc e 0,01% 0,01% 0,01% 0,03% 0,01% 0,03% 0,01% 0,02% 0,02% 0,01% 0,01% 0,00% 0,10% 0,00% 0,00% 0,04% 0,01% 0,01% 0,03% 0,00% 0,01% 0,01% 0,00% Average Default Correlation Lower level 0,21% 0,21% 0,24% 0,34% 0,34% 0,93% 0,21% 0,68% 1,03% 0,13% 0,24% 0,17% 1,39% 0,67% 0,17% 0,73% 0,27% 0,39% 0,41% 0,07% 0,51% 0,14% 0,03% (within sectors) - confidence 95% level Estimate Upper d 0,40% level 0,99% 0,38% 0,45% 0,64% 0,63% 1,73% 0,40% 1,27% 1,92% 0,24% 0,45% 0,32% 2,60% 1,25% 0,32% 1,37% 0,50% 0,73% 0,76% 0,13% 0,94% 0,26% 0,06% 0,96% 1,12% 1,60% 1,58% 4,31% 0,99% 3,15% 4,77% 0,60% 1,11% 0,79% 6,47% 3,11% 0,80% 3,41% 1,25% 1,81% 1,89% 0,33% 2,34% 0,64% 0,15% Source: Our elaborations on Bank of Italy’s “Base Informativa Pubblica”, 2000-Q1. Sectors 66 (Construction) and 51 (Agricultural, Forestry, and fishing products) showed the highest average default probability with 5.12% and 4.16%, respectively, during the period. Their riskiness is also confirmed by their levels of standard deviations (1.775% for the Construction sector and 24 Doubtful loans are adjusted to take account of possible difference of judgments (or possible omissions) on the part of banks. In this regard, see Ascenzo, Viviani (2000). 25 For a fuller description of this time series, see Ascenzo, Viviani (2000). 349 3.22% for the Agricultural et al. one). On the contrary, sectors 72 (Communication Services) and 52 (Energy products) proved to be almost riskless with average levels of default probabilities which are far below 1%. On the side of average default correlations, the latter -- for loans in the same industry sector -- have been estimated using formula (3), while 95% confidence levels have been calculated using the method described in Appendix 2. These figures confirm that average default correlation coefficients are all different from zero at 95% confidence level and corroborate our choice of considering some fixed level of default correlation. However, the range of variation of these levels seem quite large; therefore, this prevents us to arrive at firm conclusions about the magnitude and the impact of such default correlations on credit risk measures for our loan portfolio. For example, average default correlation for Sector 51 (Agricultural et al.) can vary from 1.39% to 6.47% with an average level of 2.60%. This should be kept in mind before putting too much confidence on these figures. Moreover, default rates fluctuate with business cycles and with time and thus their stability is another important research issue that should not be neglected for implementation purposes. 5.2. Applications and results As shown in Table 2, the 25 loans of our example portfolio are distributed over 5 sectors: 73, 58, 54, 51, 56. In Table 5 we summarize the loan portfolio composition. Table 5. Example Loan portfolio composition Sector 73 58 54 51 56 Total Exposure Absolute value 3.330 2.225 2.510 1.620 1.530 11.215 % 29,692% 19,840% 22,381% 14,445% 13,642% 100% First, we calculate the average default correlation matrix for the 5 sectors where the 25 loans of the fictitious portfolio are distributed. The default correlation coefficients between sectors are given by formula (4), and will be used later to derive the asset return correlation matrix. In Table 6 we summarize the average default rates and default correlation within the 5 sectors, while in Table 7 we report the average default correlation matrix. 350 Table 6. Average default rates and default correlation within sectors Table 7. Average Default Correlation Matrix Second, we determine the mean and the standard deviation of losses, and derive the Index of Concentration in the absence (IC) and under correlated defaults (ICC). Based on formulae (6) and (9.1), the mean and the standard deviation of portfolio losses are equal to $489,1450 and $519,3424, respectively. From formula (9.1.), we then isolate the ICC and split it up into two parts: the IC and the CF. The IC is equal to 0,2197 and the CF amounts to 1,0574. This latter components represents the adjustment to the IC for the presence of some level of default correlation. Consequently, the ICC (IC*CF) is equal to 0,2324. However, we point out the CF is not very high due to the low levels of default correlation, as appear in Table 7 above. Third, we generate the full loan loss PDF using the method outlined in section 4.2. Following the corporate asset value of Merton, we use the standard normal distribution as a medium to generate Bernoulli random numbers. This will enable us to apply the Cholesky decomposition to generate multivariate Bernoulli normal random numbers. In practice, this method infers the asset return correlations from the default correlations. In this regard, we distinguish two cases and analyze them separately: 1) when 351 loans belong to the same industry sector, and 2) when loans belong to different industry sectors. Let us consider the first one: Solving simultaneously Equation (13) and ~ Equation (11) we obtain the following ρ s that satisfy Equation (11). In the last row of Table 8 we also report the ratio of the asset return correlations to the default correlations. Table 8. Average Default Correlations and Asset Return Correlations within sectors As to the second case, solving simultaneously Equation (16) and Equation ~ (15) we can find the various ρ k ,l s between the 5 sectors, according to the default correlation coefficients previously calculated. In the last column of Table 9 we again report the ratio of the asset return correlations to the default correlations. Table 9. Average Default Correlations and Asset Return Correlations between sectors ∼ρk,l ρ56,73 ρ56,58 ρ56,54 ρ56,51 ρ51,73 ρ51,58 ρ51,54 ρ54,73 ρ54,58 ρ58,73 ∼ρk,l 3,00% 9,00% 5,00% 6,00% 2,00% 7,00% 1,00% 1,00% 2,00% 4,00% ρk,l 0,43% 1,43% 0,77% 1,12% 0,35% 1,17% 0,19% 0,21% 0,28% 0,52% Ratio 7,02 6,31 6,49 5,34 5,70 5,98 5,22 4,82 7,06 7,72 We then arrive at determining the average asset return correlation matrix (reported in Table 7), that through a Cholesky factorization can be used to simulate -- in a Monte Carlo setting -- the full loan loss PDF. In particular, 352 with this procedure we can obtain a set of correlated multivariate Bernoulli random numbers, D1 , L , D n , with value of 0 in case of non-default and value of 1 in case of default. Table 10. Average Asset Return Correlation Matrix Taking the values of the n (n=25) exposures (see Table 2), that is e1 , L , e n , the portfolio loss for the first trial D 1 is found as L1 = ∑ Di1 * e i . i =1 n For the second trial and beyond (our simulation consists of 20,000 runs26), we also derive L1 , L , Ln and finally draw a histogram for these values. We then assume this histogram represents the true loss distribution and calculate percentile levels for the maximum loss. The results of the simulation for the fictitious portfolio are reported in the following Table 8 and Figure 1. Table 11. Summary Results of the Example Portfolio Loss Distribution Total Exposures Expected Loss 11.215 489,15 Percentiles Max Loss 90,00% 1.233,65 95,00% 1.570,10 97,50% 1.794,40 99,00% 2.130,85 99,50% 2.355,15 99,90% 2.803,75 99,95% 3.028,05 99,97% 3.252,35 99,99% 3.588,80 Standard Deviation 519,34 Value at Risk 744,51 1.080,96 1.305,26 1.641,71 1.866,01 2.314,61 2.538,91 2.763,21 3.099,66 26 This is the same number of runs executed in the Creditmetrics Model. 353 Figure 1. – Loss Probability Density Function for the Example Portfolio 40,00% 35,00% 30,00% 25,00% Probability 20,00% The Expected Loss lies between these two values 15,00% 10,00% 5,00% 0,00% 0 11 , 0 0 2 22 ,15 4 33 ,30 6 44 ,45 8 56 ,60 0 67 ,75 2 78 ,90 5 89 ,05 10 7,2 0 0 11 9,3 2 5 12 1,5 3 0 13 3,6 4 5 14 5,8 5 0 15 7,9 7 5 16 0,1 8 0 17 2,2 9 5 19 4,4 0 0 20 6,5 1 5 21 8,7 3 0 22 0,8 4 5 23 3,0 5 0 24 5,1 6 5 25 7,3 7 0 26 9,4 9 5 28 1,6 0 0 29 3,7 1 5 30 5,9 2 0 31 8,0 4 5 32 0,2 5 0 33 2,3 6 5 34 4,5 7 0 35 6,6 8 5 37 8,8 00 0 ,9 5 Loss 6. Conclusions In this paper we propose a simplified method to evaluate credit risk in a bank’s loan portfolio that could be implemented using publicly available loss experience data (industry or rating-based datasets). The basic assumptions are rather restrictive since the exposures are considered fixed, the probabilities of default are deterministic and the recovery rate is set to zero (even though its inclusion in the framework is straightforward)27; on the other hand, correlations between default events are taken into account both within and between industry sector. Under the mean/standard deviation approach (see Basel, (1999a)), we analytically derive an index of concentration in case of correlated defaults 27 Even though our choice seems difficult to digest, we do not include variables whose data are not publicly available (this is just the case of the Italian market). In any case, since the resulting estimates are conservative, our assumption of zero-recovery rate fits with the scenario we are representing: losses under the occurrence of extreme events. 354 (ICC) (that just depends on the amount and number of the exposures and on default correlations) that can be easily calculated at the portofolio level. Rather than parameterize a specific loss distribution by the mean and standard deviation of the loan portfolio in order to estimate the portfolio maximum loss, we use Monte Carlo simulations to generate the full loan loss distribution (this is similar in spirit to Creditmetrics). In order to apply the Cholesky decomposition, we use the standard normal distribution to generate Bernoulli random numbers (our random variable has a Bernoulli distribution). This assumption makes possible to derive the asset return correlation matrix and consequently minimize the burdens of the simulation procedure. The use of Monte Carlo simulations is advocated since it makes the framework more flexible in view of future developments (that is, the inclusion of other assumptions in the model such as non-zero and variable recovery rates, correlation between probabilities of default and recovery rates, and other refinements). Even though this approach seems appealing there are several caveats that should be highlighted before putting too much stock on its results. Among them: • The parameters’ estimation is quite sensitive to the available data. In our case the relevant parameters are estimated with only 15 years of data that may not be representative of the future. This has been confirmed by the wide confidence level around estimates of the average default correlation within sectors. Moreover, default rates fluctuate with business cycles (bad/good years could be not reflected in the limited dataset) and with time. As a possible solution to this problem, we recommend that stress tests be performed on the data at issue in order to simulate and re-estimate the relevant model parameters. • More research is required as to the valuation of exposure. The implications of a wrong-way loan valuation may be considerable. • The choice of the time horizon is not trivial. Studies that illuminate this issue are advocated. Our simple and flexible framework for credit risk measurement aims to identify a level of economic capital that is neither an “ideal” to which banks should aspire, nor a standard for internal risk management. Our aim is only to provide a “raw” evaluation of the capital level that should buffer future losses, according to data availability and technical considerations. 355 REFERENCES ALTMAN, EDWARD I. AND VELLORE M. KISHORE (1996), Almost Everything You Wanted to Know about Recoveries on Defaulted Bonds, Financial Analysts Journal, November/December, 57-64. ASARNOW E., MARKER J. (1995), Historical Performance of the U.S. Corporate Loan Market: 1988-1993, Journal of Commercial Lending, pp. 1332, spring. ASCENZO M.P., VIVIANI U. (2000), Nuove basi informative realizzate dalla banca d’Italia, in “Modelli per la gestione del rischio di credito. I “ratings” interni”, Tematiche Istituzionali, Banca d’Italia, April. BASEL COMMITTEE ON BANKING SUPERVISION (1999a), Credit Risk Modelling: Current Practices and Applications, Document No. 49, April. BASEL COMMITTEE ON BANKING SUPERVISION (1999b), A New Capital Adequacy Framework, Document No. 50, June. BROEKER F., ROLFES B. (1998), Good Migration, Risk Magazine, November. CAREY M. (2000), Parameterizing Credit Risk Models With Rating Data, Finance and Economics Discussion Series, No. 47, Federal Reserve Board of Governors. CARTY L.V. AND D. LIEBERMAN (1996) Defaulted Bank Loan Recoveries, Moody’s Special Report, November. CREDIT SUISSE FINANCIAL PRODUCTS (1997), CreditRisk+: A credit risk management framework, Technical Document. CROUHY M., GALAI D., MARK R. (2000), “A comparative analysis of current credit risk models”, in Journal of Banking and Finance, vol. 24, n. 1/2. ENGLISH W.B., NELSON W.R. (1998), Bank Risk Rating of Business Loans, Finance and Economics Discussion Series, No. 51, Federal Reserve Board of Governors. FEDERAL RESERVE BOARD OF GOVERNORS (1998), Credit Risk Models at Major U.S. Banking Institutions: Current State of the Art and Implications for Assessments of Capital Adequacy, Federal Reserve System Task Force on Internal Credit Risk Models, May. FINGER C. C. (1998), Sticks and Stones, Working Paper Number 98-01, The Riskmetrics Group, New York (www.riskmetrics.com). FINGER C. C. (2000), A comparison of stochastic default rate models, Working Paper Number 00-02, The Riskmetrics Group, New York (www.riskmetrics.com). FORD J. K. (1998), Measuring Portfolio Diversification, The Journal of Lending and Credit Risk Management”, February. 356 GENERALE A. E G. GOBBI (1996) , Il recupero dei crediti: costi, tempi e comportamenti delle banche, Temi di discussione, Banca d’Italia, No. 151. GORDY M. (2000), A comparative anatomy of current credit risk models, in Journal of Banking and Finance, vol. 24, n. 1/2. GUPTON G. M., FINGER C. C., BHATIA M. (1997), CreditMetrics, technical document, J.P. Morgan & Co., New York (www.riskmetrics.com). KMV (undated), Portfolio Manager Model, San Francisco, KMV Corporation, manuscript. KOYLUOGLU, H. UGUR, AND ANDREW HICKMAN (1998), A generalized framework for credit risk portfolio models, Working Paper, Oliver, Wyman & Company, New York. MOODY’S INVESTOR SERVICES (1997), Rating Migration and Credit quality correlation, 1920-1996, Global Credit Research, July. ONG M. (1999), Internal Credit Risk Models. Capital Allocation and Performance Measurement, Risk Publications, London. SAUNDERS A. (1999), Credit Risk Measurement. New Approaches to Value at Risk and other Paradigms, Wiley Frontiers in Finance, New York. SAVONA P., SIRONI A (2000), La gestione del rischio di credito. Esperienze e modelli nelle grandi banche italiane, Edibank, Bancaria Editrice. TREACY W.E., CAREY M.S. (1998), Credit Risk Rating al Large U.S. Banks, Federal Reserve Bulletin, November. ZAZZARA C. (1999), Il ruolo del capitale nelle banche e la sua regolamentazione: dall’Accordo di Basilea del 1988 ad oggi, Rivista Minerva Bancaria, No. 5. 357 Appendix 1 – Derivation of the average default correlations A. Correlation within a group28 We first consider N companies with the same default rate (same rating). Di is a random variable with a value of 1 when Company i defaults and 0 when it does not. The average default rate is p , and the standard deviation of default is σ . This produces the following relationship: 1 (i : default ) Di =  0 (otherwise ) p= σ= 1 N (1) ∑D i =1 N i (2) (3) N p(1 − p) S is the total number of defaults so that S = ∑ Di . The variance of S i =1 is therefore: Var ( S ) = ∑∑ ρ ijσ 2 i =1 j =1 N N N N = ∑∑ ρ ij p (1 − p) i =1 j =1 N   = p (1 − p )  N + 2∑∑ ρ ij  i =1 j < i   (4) The default correlation between companies is ρ ij so that ρ ii = 1 and ρ ji = ρ ij . When one turns to the average default correlation ρ rather than the default correlation between companies ρ ij , one can define ρ as follows: ρ= 2∑∑ ρ ij i =1 j < i N N ( N − 1) (5) 28 We referred to Appendix F of Creditmetrics -- Gupton et al. (1997) -- for this section. 358 Using this to express the variance of S , we arrive at: Var ( S ) = p (1 − p )[N + N ( N − 1) ρ ] (6) The volatility of default σ is in the relationship σ 2 = Var ( S / N ) , so that  S  Var ( S ) σ 2 = Var   = N2 N 1 + ( N − 1) ρ = p(1 − p) N (7) This can be transformed to express the average default correlation ρ as: Nσ 2 −1 p (1 − p ) ρ= N −1 When N is large, Equation (16) can be approximated as: (8) ρ≅ σ2 p (1 − p ) (9) B. Correlation between different groups Similarly, for different groups k and l , random variables Dk ,i and Dl , j can be defined for Company i and Company j , so that their value is 1 in default and 0 otherwise. Assume that N companies are in the group k ; M companies are in the group l . The total numbers of defaults are defined as S k and S l , and the average default rates as p k and pl , as shown below. S k = ∑ Dk ,i , S l = ∑ Dl , j i =1 j =1 N M (10) 1 pk = N ∑D i =1 N k ,i , pl = 1 M ∑D j =1 M l, j (11) 359 The average default correlations ρ kl for different groups k and l are defined as: ρ kl = ∑∑ ρ i =1 j =1 N M ij NM (12) Like Equation (14), the covariance of S k and S l becomes as follows: Cov ( S k , S l ) = ∑∑ ρ ij i =1 j =1 N M pk (1 − p k ) pl (1 − pl ) (13) = p k (1 − p k ) pl (1 − pl ) ρ kl NM However, the following also holds true: Cov ( S k , S l ) S S  = Cov  k , l  = Cov ( p k , p l ) NM N M (14) Therefore, the average default correlation ρ kl between groups can be expressed as follows: ρ kl = Cov ( p k , p l ) p k (1 − p k ) p l (1 − p l ) (15) Appendix 2 – Calculation of the standard errors for variances and average default correlations within industry sector groups The sampling distribution of the sample variance is a form of gamma distribution known as the chi-square distribution, indicated as χ 2 . This distribution has a different shape for different degrees of freedom. It is necessary to standardize the sample variance in a manner analogous to the way in which the normal distribution is standardized. The standardization 2 takes the following form: χ n −1 = (n − 1) * s 2 , where the subscript “n-1” σ2 360 refers to the degrees of freedom, which, in the case of χ 2 are the number of observations minus 1. To develop confidence intervals for variances, we are not concerned about the sampling distribution of the variance itself, but of the sampling distribution of the standardized variable (n − 1) * s 2 2 ≈ χ n −1 . To produce a σ2 95% confidence interval for the point estimate of our variance, we need to determine the values of χ 2 that account for 2,5% in each tail of the distribution. Thus we need a value of χ 2 that assumes 97,5% of the values are to the right and another value which assumes 2,5% are to the right. If we refer to the degree of confidence as 1 − α , we need the values of χ 12−α / 2 e 2 χ α / 2 . If we are working to 95% confidence, the value of α will be 0,05 and 2 2 the relevant χ 2 parameters will be χ n −1,0 , 975 and χ n −1,0 , 025 . The confidence interval is given as (n − 1) * s 2 (n − 1) * s 2 ≤σ 2 ≤ 2 2 χ n −1,1−α / 2 χ n −1,α / 2 and the probability statement is  (n − 1) * s 2 (n − 1) * s 2  2 ≤σ ≤ p 2  =1− α 2 χ n −1,α / 2   χ n −1,1−α / 2   Now, we filter these lower and upper values of the variance through σ2 equation (3), ρ ≅ , to get 95% confidence levels for average p (1 − p ) default correlations within the same industry sectors, as reported in Table 4 (pag. 19). 361