Implementing the CyRCE model

Transcription

1 BANCO DE MEXICO Implementing the CyRCE model Structural simplifications and parameter estimation Fernando Ávila Embríz Javier Márquez Diez-Canedo Alberto Romero Aranda April 2002

2 Implementing the CyRCE model Structural simplifications and parameter estimation Fernando Avila Embríz Javier Márquez Diez-Canedo Alberto Romero Aranda I. Introduction Banco de México s research document describes a credit risk model that for diverse reasons is more appropriate for emerging markets than those designed in countries with developed markets. First, by assuming that loss distribution due to debtor default can be characterized by its mean and variance, closed form expressions for value at risk (VaR) are obtained without having to resort to very onerous numerical techniques in terms of computer resources and time. An interesting property of the model is that a measure of the concentration can be obtained, which enables its impact on portfolio credit risk to be evaluated and is associated with the individual limits loans should respect for either regulatory reasons or management decisions of each bank. Closed form expressions for risk measurement enable capital adequacy ratios to be obtained as well as individual loan limits where the relationship between these management parameters and credit risk is explicit. Another virtue is that the model allows for a totally arbitrary segmentation of the loan portfolio, which facilitates detecting riskier segments, determining individual limits on loans differentiated by segment, and the allocation of required capital so that each segment of the portfolio is adequately capitalized. Finally, as all of the elements which contribute to risk are parameterized and can therefore be determined exogenously, information weaknesses typical our markets can be overcome by making assumptions about their values. The efficiency of the model s calculation makes the sensitivity analysis or stress tests on uncertain parameters easy exercises to undertake and enriches the risk analysis, even in the absence of statistically valid estimates of the model s parameters. Despite all its virtues, the more general version of the original model presents a series of technical problems that leave a few loose ends and require a calculation effort that grows exponentially as the number of loans in the portfolio increases. In particular, the proposed change in the variable in order to manage the effect of co-variation between loan portfolio defaults results in the need to factorize the M variance-covariance matrix through a matrix S so that M = S T S, which implies solving several technically complex problems. First, since matrix S is not the only one, which do you choose? So far it has not been possible to categorically determine whether any representation of S is just as good or whether one of them is better than the rest. But also, owing to difficulties estimating the covariances, the first approximation of M often occurs in a matrix that is not positive definite, 1 See Márquez, April 2002.

3 without which it is impossible to undertake the required factorization. This implies that an additional numerical process called conditioning is needed to adjust co-variance estimates so that the matrix is positive definite, as a result of which many of the original estimates significantly change. Finally, as mentioned, even though everything is well conditioned and there is no theoretical impediment, obtaining matrix S requires a numerical procedure to be performed in which the number of operations geometrically increases with size N of matrix M. This paper reformulates the general original model in easier terms using the Rayleigh coefficient as the measure that summarizes the effect of variation-co-variation on credit risk by maintaining all of the model s desirable properties and making the factorization process of the co-variance matrix with all of its implied problems unnecessary. The new version of the model also leads to a new important theoretical result, as it enables a measure of risk concentration to be obtained, which indicates how the correlation between defaults impacts concentration in terms of number of loans and thus the loan portfolio s risk. The implementation of any credit risk model usually assumes that loans can be grouped so that all loans belonging to a group share the same characteristics and the parameters or elements that determine the potential losses the group s loans can cause are the same for all members. For example, in CreditMetrics TM loans are grouped by rating and so all loans with the same rating have the same probability of migrating to different quality (rating) states and the same probability of default. In CreditRisk +, all loans with the same rating and which likewise depend on the risk factors that determine them, have the same probability of default. Furthermore, in this latter paradigm, loans are grouped in buckets with the same number of loss units ; in other words: it is assumed that the loss that generates a borrower s default is the same for all borrowers in the same bucket. In CyRCE, the result of segmenting the portfolio is that it assumes that the likelihood of default is the same for all loans in a segment as well as the correlation between them. But also, the default correlation between loans in one segment and those in another is also the same. As this paper will show, this leads to a structure in which the matrix operations the model requires are done algebraically, obtaining expressions that result in considerable memory, programming and calculation savings. Finally, estimate techniques for main parameters are provided; namely: probabilities of default and correlations. Given that the available information presents serious deficiencies with respect to the rating of the loans by banks, the procedure used was designed to work directly with available data on borrower default within specific segments of a loan portfolio. As a result, the methodology adopted would in all events be more conceptually similar to the one that KMV uses. In our view this is even better than ratings, as in the final analysis all rating schemes claim to be indicative of default probabilities but in our environment and few exceptions they rarely achieve it. It is a fact that given its obvious inability to predict crisis and bankruptcies of important public companies, the rating agencies themselves which have been calculating migration and default probabilities associated with their respective rating systems for more than a decade, are aware of their limitations as vehicles for estimating default probabilities. More recently Moody s acquired KMV and a drastic change in methodology is anticipated in which its new rating system will be based on default probability estimates and not the other way around as has traditionally been the case. Likewise, Standard & Poors has just announced that it has hired a new team of experts in the subject which should also lead to a change in the way borrowers are rated.

4 Thus, the estimate techniques presented in this paper assume that the number of borrowers is known for each segment of the portfolio and that, although being up-to-date with their payments in the previous period, default in the second. The relationship between the number of borrowers who default versus the total number of borrowers that are up-to-date in the previous period is what is known as the default rate of debtors within the segments into which the loan portfolio is divided. Probabilities of default and correlations can be made based on this information. In this paper we present several estimate methods and discuss their drawbacks and virtues. The estimate methods presented specify requirements for estimating the necessary parameters for avoiding problems related to a poor conditioning of the covariance matrix. II. Rethinking the general model using the Rayleigh coefficient Assume that the loan portfolio loss distribution can be characterized by its mean and variance and that the default probabilities vector is π with a co-variances matrix between defaults M obtained exogenous to the model. On the basis of the original analysis, the inequality of VaR with respect to capital is: In the original model it was noted that as M is positively defined, there is a Q matrix such that where is the diagonal matrix of values characteristic of M, and Q is an orthogonal matrix of eigenvectors of M, with the property that =. 2 Let, where is the diagonal matrix with the square roots of the eigenvalues of M, such that. Making the change of variable we have. What is most important about this observation is that the change in the variable evidenced a resizing of the original loan vector through the square root of the covariance matrix S. This resizing means that loans with bigger default covariances with other loans in the portfolio grow compared to their original value for those with a lower covariance is the opposite. This in turn means that although a lot of credit in the hands of a few debtors can be risky, it is riskier to have a lot of risk concentrated among a certain group of loans regardless of whether they are many or few. This in turn means that at any given moment, a very diversified portfolio of small loans in which the individual loans carry high default probabilities and are very correlated may represent greater risk than a portfolio of a few large loans with low default probabilities and independence. This discussion is taken up and formalized in the next section. 2 Any intermediate text on linear algebra can be consulted; for example, Strang G. 1980, o Mirsky L

5 Unlike the original model which follows this line of reasoning and takes it as far as possible, another is followed here. Thus, multiplying and dividing by, and dividing by the following capital adequacy is obtained: where is a measure of the loss variance and represents the expected loss relative to the total value of the portfolio. Proceeding according to the original article, and applying premise 5.1, which states that implies the threshold for individual loan limits is obtained as follows: ( ) Ratios (2.3) and (2.6) basically have the same structure as those of the original model only now the loss variance is comprised of the concentration index and the Rayleigh ratio; in other words: Actually, results for value at risk and individual thresholds that this representation of the model produces are identical to those obtained from the original model, but as the change of variable is not involved, the calculation is considerably simpler. In the section below, this implication is researched obtaining of a risk concentration measure Risk concentration measure To evidence the way in which the correlation impacts concentration and increases risk, consider the case in which all loans have the same default probability and are identically correlated with peers through the correlation coefficient. The default covariance between any pair of loans is: ( ) This means that the covariance matrix has the following structure:

6 ( ) This is equivalent to the next matrix representation: { } This allows for calculating the portfolio loss variance using the following matrix calculation: { } As a result, the risk value expression is a follows: { } In this expression, the loss variance has two components. The first is the Bernoulli variance while the component that reflects the concentration effect is: If the correlation is positive, is a convex combination of the Herfindahl index of a fully concentrated portfolio and the portfolio. Evidently, increases with. Furthermore, while when we have if. In other words, if all loans in the portfolio are perfect and positively correlated in terms of risk they behave as if they were a single loan. Generally speaking, it can be said that the portfolio of correlated loans and concentrated in accordance with behaves exactly the same as one of independent loans but with a concentration index instead of. Thus, can be considered a correlation adjusted concentration index. Furthermore (2.11) is used to calculate this index. For this, of the expression (2.3) recall that the portfolio default variance is. Thus, if the portfolio variance is equal to the portfolio variance of the analyzed case, and must be found that satisfy: [ ]

7 Finally, if, solving for we obtain: ( ) ( ) [ ] [ ] This expression provides an equivalent correlation measure that summarizes the way in which all loans in the portfolio are correlated with peers. EXAMPLE 2.1 In order to appreciate the meaning of the previous results, below we include a small numerical example. Consider the following portfolio of 25 loans taken from the CreditRisk + manual: Table 2.1 N of loans RATING A B C D E F G 1 $4,728 $5,528 $3,138 $5,320 $1,800 $1,933 $ $7,728 $5,848 $3,204 $5,765 $5,042 $2,317 $1,090 $4,831 $20,239 $15,411 $2,411 $2,652 $4,912 $2,598 $4,929 $5,435 $6,467 $6,480 TOTAL $22,805 $30,994 $45,544 $12,439 $11,902 $6,480 TOTAL $12,456 $11,376 $21,520 $31,324 $22,253 $9,259 $21,976 $130,164 The default probabilities for the loans are taken from the following table: Table 2.2 Rating Default probability (%) A 1.65 B 3.00 C 5.00 D 7.50 E F G The covariances matrix is the same as that used in those examples and is shown in Appendix A. This matrix is classified in three groups as follows: [ ]

8 Assuming normality and a confidence level of 5%, the VaR portfolio is: From the examples given in the original document we know that, and after doing the corresponding calculations we obtain: The capital adequacy condition is: Assuming, such that. The ratio ( 2.6) provides the individual threshold for the loans ( ) ( ) In other words, Table 2.1 shows how there are only two loans that significantly surpass the threshold. We now examine the impact of the correlation on the concentration. From (2.13) we obtain the equivalent correlation of the portfolio: [ ] [ ] From (2.11), the correct concentration index by correlation is: Regardless of the fact that this portfolio is fairly poor, by adding a correlation of 22% to an average default probability for the 10.89% portfolio, we obtain a standard default deviation of, compared with if the loans were independent. Thus, the equivalent correlation of 22% duplicates the standard deviation of defaults with respect to the case of independent loans. It is interesting to compare the concentration index corrected by correlation, which is four times greater

9 than the concentration index. In terms of capital adequacy, the correlated portfolio requires, which is 59% more than the 27% required in the case of independent loans. 3. Handling different correlation sizes This section adapts the ratios of the original model using the Rayleigh coefficient. Following the original model, F is randomly partitioned into classes,, where is a vector containing loan balances belonging to the xxth segment. Next, the expected default probabilities vector and the variance-covariance matrix are partitioned as follows: a) ; Partition of the default probabilities vector where is the default probabilities vector of segment i; i = 1,2,3,...,h b) The variance-covariance matrix is partitioned as follows: [ ] Each sub-matrix corresponds to the idiosyncratic variance-covariance matrix of group i and has size ; where is the number of loans in the segment. All of these matrixes are positive, defined the same as and matrixes contain the covariances of default probabilities among the group of loans and. From here on, let the value of the portfolio associated with segment, and. Now, let, where is the percentage of capital assigned to segment i ; [ ]. For partition purposes a small change was made to the matrix which now has the following shape 3 : [ ] 3 This definition of the matrix differs from that proposed in the original piece of work and has the advantage described in the text.

10 Each matrix only takes into account correlations between defaults on loans in group with those in other groups but eliminates correlations between those in the other groups which have no direct impact on the segment being analyzed. Note how, as such, unlike the one defined in the original work, these matrixes have the property of: As in the original model, the constant that allows for individual VARs to be added to obtain the portfolio total is: Proceeding as usual, let us define Where y. It is easy to verify that. Now, dividing (3.3) by : { } Solving for we obtain ( ) { } where And for Theorem 5.2, ( ) { }

11 In the above, (3.4) establishes capital adequacy for each segment, (3.5) is the concentration threshold for the segment, and (3.8) is the expression for individual thresholds. Making ensures capital adequacy for the portfolio. EXAMPLE 3.1 To illustrate the results from the updated model, compared with the original article, the same portfolio as in example 7.2 will be used. Its segmentation is presented in Table 3.1. Table 3.1 Rating A1 C2 C4 D1 D3 F1 F4 G2 F1 $ 4,728 $ 3,204 $ 4,912 $ 5,320 $ 20,239 $ 1,933 $ 2,598 $ 1,090 Total= $ 44,024 Rating $ 5,528 $ 3,138 $ 4,831 $ 5,042 $ 15,411 $ 2,411 $ 358 $ 6,467 Total= $ 43,186 Rating $ 7,728 $ 5,848 $ 5,435 $ 5,765 $ 1,800 $ 2,317 $ 2,652 $ 4,929 $ 6,480 Total= $ 42,954 In terms of Appendix A, the matrixes S i for each segment are as follows: ( ) ( ) ( ) As for Table 3.2, the portfolio values, their concentration indexes, and the capital allocated to each segment are summarized. Table 3.2 Segment i 1 2 $ 44,024 $ 43, $ 20,293 $ 19,907 3 $ 42, $ 19,800

12 Of (3.2), the parameter which allows for the sum of adjusted VaRs is: Calculating with (3.3), using a confidence level of 5% and assuming normality, we obtain the following capital adequacy ratios: Note that the results differ from those obtained in example 7.2 of the original model, further evidencing how the third segment is the riskiest as the capital assigned to the last segment does not cover its risk. This is due to the new definition of matrix which yields a much higher value for the normality parameter. Note how, and therefore for all segments. The table below shows the expected default probabilities, the Rayleigh coefficient, and the correlation corrected Herfindahl-Hirschman index. Table 3.3 Segment i Correlation correction These values can be used to verify ratios (3.5) for all portfolio segments. As expected, the third group is the critical one:

13 ( ) { } Now, from (3.7) we obtain the same results from example 7.2 of the original model for individual thresholds that imply capital adequacy: Summing up, no loan in the first group exceeds its limit, while in the second group the loan for $15,411 exceeds its threshold. Finally, as expected, the third group is the most problematic as only the three smallest loans respect the threshold. It is interesting to analyze the effects of the correlation in this example as it establishes that the first group can contain loans of any size, but the riskiest segment must include only very small loans, which, although logical, is clearly not the case. As for capital adequacy, (3.8) is examined, and comparing, we obtain: with Note that the result is the same as in example , and that the model s new specification is congruent with the original, but simplifies the calculations considerably. However, there is an improvement due to the new definition of matrix,which further highlights the risk differences between segments. 4. Using the structure to simplify the calculation and application of the model with limited information In practice, any of the credit risk methodologies have either implicitly or explicitly portfolio segmentation criteria that assumes that each segment s loans share certain characteristics. For example, at CreditMetrics TM, the loan group depends on its rating and seniority, which implies that all loans in the segment have the same default probabilities and migration to a different quality status. It also implies that they have the same discount rate curves for markto-market effects and the same loss rates given default in the event of non-payment. Similarly, in CreditRisk + the grouping criteria is loss given default; in other words, it is assumed that for all loans in a bucket the loss is the same number of standard units in the event of default. It is also assumed that all loans with the same rating which respond in the same way to the risk factors that explain them, have the same default probability. This in turn carries the implicit assumption that groups of loans whose default correlation is the same with respect to loans of other groups can be identified.

14 4.1 Basic relations In CyRCE, although the model is totally general and there is no restriction on each loan having a different default probability and correlations between any pair of portfolio loans under analysis can be different, it is convenient to adopt some type of taxonomy otherwise the problem of estimating parameters becomes impossible in practice. In our view, as the grouping can be totally arbitrary, there is more flexibility for choosing a classification criteria either because it is known (or inferred) beforehand that the risk characteristics of each defined segment respond differently to the risk factors that determine them because the grouping allows for statistical reliability with respect to parameter estimates to be obtained, or a combination of both. Thus, for estimate effects it can be assumed that within each group default probabilities and correlation between any two loans within the group are the same. We can also assume that between groups, the default correlation between any pair of loans, where one belongs to group i" and the other group j, is always the same. Besides the meaning revealed in terms of detecting excessive risk concentrations already discussed in previous sections, this provides the model with a structure that has significant implications in terms of parameter estimates and computer resource savings. Thus, it is assumed that the default probabilities of the loans within each segment are identical and each pair of loans is similarly correlated such that: This in turn, of (2.9), means that the idiosyncratic covariance matrix of each segment has the following structure: { } Similarly, note that the sub-matrixes of matrix M have the following structure: ( ) ( ) As a result, the expression for the loss variance for individual segments is: { ( ) } ( )( ) }

15 Recalling that is obtained as follows: { } where is the concentration index adjusted for the correlation of the segment i. Note that in (4.4) all of the matrixes multiplications have been eliminated implying substantial savings in terms of both the memory required to store data and the calculations themselves. Recalling that: ( ) Once the loss variance calculation has been done for all segments, an analysis for each segment and the whole portfolio from expression (2.1) to (3.7) becomes very efficient. 4.2 Applying the model using limited information about the loan portfolio: obtaining the Herfindahl-Hirschman index Any risk model requires two types of information, namely, a description of the distribution (by size) of the loan portfolio, its default probabilities and default correlation between loans. The model developed presents several options for making calculations with limited information. From the paragraph above it should be clear that it is not strictly necessary to know the configuration of the loan portfolio in detail. Note that based on any segmentation of the portfolio, the only information required is: t he total value of the loans in each segment, some information about the distribution of the loans in the segment allowing the Herfindahl index to be estimated, default probability estimates, default correlations for loans in each segment, and default correlations between loans of different segments. The problem of estimating default probabilities and correlations is dealt with in the next section. In it we only discuss how estimates of the Herfindahl index can be obtained using some basic statistics. Thus, assume that the portfolio has segmented different groups into. First, if besides knowing the value of the loan portfolio of each segment,, the largest loan in each segment " " is known, then theorem 6.1 of the original article presented in appendix A states that:

16 Therefore, if, we obtain an approximation of the concentration index for each segment. The estimate is normally very conservative. If we obtain data on the number of loans per segment, the average loan and variance of the size of loans, we can obtain a precise index. To do this, first note that is the value of the loan portfolio of segment i, and that the value of the entire portfolio is ( ) ( ) [( ) ( ) ] ( ) { ( ) ( ) } ( ) ( ) { } { } ( ) Thus, the index can be solved: ( ) Now, having obtained the concentration index per segment, we can know that of the entire portfolio as follows: ( ) Summing up, expressions (4.7) and (4.8) are exact figures for the respective concentration indexes. They are obtained using fairly limited information and without needing to know the portfolio; in other words, only each segment needs to be known: the number of loans and the value of the portfolio or else the average value of the loans and the variance or standard deviation of the value of the loans. 5. Estimating default probabilities and idiosyncratic correlations Rating models are the most common vehicle for obtaining default probabilities. In fact, by design, accredited paradigms like CreditRisk + or CreditMetrics TM depend on the loan portfolio being rated. Obviously, if a bank has a good system for rating borrowers and groundwork has been done to obtain statistics on default rates associated with the rating, it is useful for obtaining default probabilities and their respective correlations. In the case of borrowers rated by one of the big rating agencies like Standard & Poors, Moodys or Fitch, information that they produce can be used. Although rating systems have an unquestionably practical value, it should be noted that using default probabilities associated with rating models presents some issues. First, it virtually forces the loan portfolio to be segmented according to ratings, which is not necessarily desirable for the purpose of identifying risky segments of the portfolio.

17 For example, borrowers who undertake activities in different sectors are frequently given the same rating and so their default probabilities correspond to different risk factors or respond differently to the same risk factors. For this reason, faced with the changing nature of risk factors, an erroneous appreciation of risk levels in different rating headings can occur. Thus, we should emphasize that a rating system is not a requisite for estimating default probabilities. In fact, the base of any estimate of default probabilities and correlations are borrowers default rates in different segments, independently of whether the segments are associated with ratings or not. KMV has clearly proven this, and currently a lot of research is taking place in this field which seeks to estimate default probabilities mainly based on the stochastic performance of risk factors that determine default probabilities. 4 Returning to the basic ratio of model (2.1), VaR and capital adequacy, the most important elements of the risk measurement are p, the default probabilities vector and M, the matrix of covariances between loan portfolio defaults. Throughout this paper we have assumed that the loan portfolio is segmented according to, where is a vector that contains loan balances belonging to the xxth segment. For the sake of consistency, it is also necessary to partition the default probabilities vector and the variance-covariance matrix as indicated in section 3; in other words: for each segment we have the vector of default probabilities of loans in segment ;. Furthermore an idiosyncratic covariance matrix corresponds to each segment of size ; where is the number of loans in the segment. In this section we deal with the issue of estimating these matrixes, assuming that within each segment the loans have the same default probabilities and are identically correlated to their peers. We should point out that all of these matrixes are positive and defined the same as. In particular, as will be seen in what follows, the problem of estimating correlation coefficients for loans inside each segment is technically complicated because it requires a functional way of distributing multi-variate and correlated binominal probability, and significantly differs from the way in which the correlation is usually considered, which is discussed in section 6. Various ways of estimating default probability and correlation coefficients for loans that comprise a portfolio segment are discussed below, assuming that data on borrower default rates for different segments of the portfolio is available. The relevant theoretical results on default probability and the correlation coefficient and the different proposed estimators are presented. The results obtained are shown at the end of each segment. Although the main interest in estimating the parameters is within the context of CyRCE, the authors believe that the same concepts are applicable to other paradigms. 5.1 Default probabilities As indicated in the previous introduction, the starting point for estimating loan default probabilities and correlations is the historical delinquency rate. 4 See Cossin and Pirotte, 2001.

18 It is currently accepted that default probabilities change over time according to the behavior of the risk factors that affect them; for example, interest rates, the GDPs of different economic sectors and the economy in general, the exchange rate, the employment level etc. To begin the discussion we will analyze a simple case in which it is assumed that the default probability of a group of loans is the same over a certain time horizon T divided into n periods. An estimate of such a probability, which is intuitively deemed reasonable, consists of taking a weighted moving average of delinquency rates observed in the n periods of the horizon under review. This implicitly leads to the assumption of independent historical default rates. A small additional complication is the changing nature of the loan portfolio itself; in other words, in practice different segments of bank loan portfolios change whatever segmentation criteria was used in terms of both the number of loans and their size. The changes can also be significant from one period to another A likely maximum estimator of the default probability under the assumption of independence All estimate methods assume that the following data is known: = Number of performing loans in the portfolio in the period t ; t = 0,1,2,...,n-1 = Number of performing loans in t-1 which default during the period t; t = 1,2,3,...,n By definition, the default rate for each period t; t = 1,2,3,...,n is: The, assuming that all loans in the portfolio have the same default probability p and that they are independent, the random variable denoting the number of defaults in each time period is distributed according to a binomial, that is: { } ( ) If defaults from one period to another are independent, the maximum likelihood estimator is obtained by maximizing the probability that succession,,... has been observed; in other words: { } ( )

19 Now, let ;. As ( ) are constant, the probability of default that maximizes (5.3) is the same as that maximizing expression. Taking the first derivative of this expression and taking it to zero we obtain the following expression: [ ] The previous expression is indeed zero both at and ; however, these are minimums and not maximums for (5.3). The solution of interest is therefore the one that satisfies, that is: In terms of default rates by applying (5.1) we obtain, The maximum likelihood estimator of default probability is therefore the weighted average of default rates where the weight in each period is the percentage representing the number of performing loans in each period as compared with the total sum of performing loans over the horizon considered Binomial distribution of the number of defaults when loans are correlated with peers In this section we propose a joint binomial probability associated with the number of defaults by relaxing the independence assumption. Actually what is being sought is an expression for a multi-variate binominal distribution in which the Bernoulli variables that comprise it are correlated. Developing some intuition for the problem, we first analyze the case in which there are only two Bernoulli variables with the same probability of occurrence that are correlated through the correlation coefficient ; in other words: where {

20 In this case we know that: We assume that [ ] exists, which by definition is: The problem resides in finding an expression for: { } { } By definition of covariance between random variables we know that: and substituting (5.7) and (5.8) we know that: Furthermore, Therefore { } { } Proceeding in the same way for the remaining cases, { } { }

21 From which we obtain: Expressions for the two other cases,, are obtained by developing and respectively. It is clear that and that: From this analysis we infer that the general expression for the bi-varied Bernoulli distribution is: { { }} We only need to verify if the above is a probability distribution. Based on that, it is easy to check that the probabilities total one. However, it is important to analyze if the correlation coefficient might be subject to some additional restriction in order to ensure that all probabilities are within interval [0,1]; in other words: { { }} In order to guarantee the non-negativity of (5.10), (5.11) and (5.12), we need: } These restrictions are summarized in the following expression: { } To conclude note that: And that If p = 0.5 then

22 From these observations we infer that, { } [ ] And that therefore there is no additional restriction on, except that To conclude the bi-varied and correlated binomial can be viewed as the sum of two correlated Bernoulli variables; in other words: { } ( ) [ ] 5.3 A generalization of the correlated multi-variate binomial distribution and estimation of parameters by maximum likelihood Although the distribution of the sum of two derived Bernoulli variables in the previous section is very direct, the generalization to variables presents a series of technical problems fairly complicated to resolve. In particular, there is no single solution for obtaining a general formula. 5 The following is the solution that apparently has greater practical application. 6 For this let: N t = Number of performing loans in the portfolio in the period t ; t = 0,1,2,...,n-1 k t = Number of performing loans in t-1 that default in the period t; t = 1,2,3,...,n = Correlation coefficient between a pair of Bernoulli variables. p = Probability of any of the variables assuming a unitary value. So a probability function associated with the sum of N t -1 correlated Bernoulli random variables is given by the following expression: { } ( ) [ ] [ ] [ ] 5 For a broad discussion of this topic, see Farah, JOE (1997).

23 where subject to the following restrictions, [ ] [ ] These two relationships are summarized by: ( ) { } The number of loans is generally significant and the probability of default is small so only the following restriction can be used: But this is simply equivalent to:. Note that if, the case is reduced to that of independence. If we assume that defaults are independent from one period to another, the maximum likelihood estimator of and is obtained by maximizing the following expression: { } [ ] [ ] [ ] subject to The previous expression is non-linear and difficult to maximize; however, the solution is facilitated by some transformations. Firstly, note that the previous problem is equivalent to the following: Subject to { } [ ] [ ] [ ]

24 Finally, for optimization purposes, the problem can be simplified further. Given that the function to maximize is a non-negative function, logarithmic function objectives can be taken, finally arriving at the next problem of equivalent optimization: { { }} [ ( ) ] Subject to: So using real data on and, and an appropriate optimization algorithm we obtain *, *, which solve the previous problem of y * = */(1+ *). 5.4 Estimation using normal approximation One alternative worth exploring is the use of a binomial approximation through Rule 1 distribution. For this purpose, recall from (2.8) and (2.9) that the covariance matrix associated with each period and each segment takes the following form: [ ] Note how in each period the size of the sum vectors changes and the matrix identity of the previous expression changes because its size corresponds to the number of borrowers in accordance with the default period. If the random vector measuring the number of simultaneous defaults is denoted by each component is given by a random Bernoulli variable; in other words: where { The expected value of the total number of defaults is given as:

25 Likewise, the variance of the total number of defaults is given as: [ ] [ ] [ ] [ ] Given that the total number of defaults is represented by the sum of the random variables a normal approximation can be used for the total number of defaults. ( ) where: [ ] In order to guarantee the non-negativity of, we need to The number of loans is large in general and only the restriction can be handled: Thus maximum likelihood estimators for and are obtained by solving the following optimization problem: ( ) As in the case of the multi-variate binomial, the objective function can be simplified: ( ) ( )

26 Using logarithms and doing, we arrive at the next expression: [ ] [ [ ]] } 5.5. Estimation by method of moments A technique that is used in many cases is the method of moments in which sample moments of a distribution are matched with the same population moments in order to obtain an estimator of associated parameters. The first population moment,, is defined as the expected value of an ongoing function of a random variable x, in other words, [ ] The median is the most common type of movement, in which is simply the identity function. The method of moments traditionally considers powers of the variable K. The mean is also known as the first moment of the distribution while is the second un-centered moment of the distribution. Note that the two previous moments enable us to express the variance of the random K variance as: [ ] So far only moments that are considered characteristics of the population have been described. From this viewpoint, the previous definitions are not very useful. It is therefore necessary to define sample counterparts of previous moments such as sample moments for a random variable. These are defined as: Once the population moments and the sample moments of the random variable distribution have been defined, a population moment is defined (or a function of population moments) using the corresponding sample moments (or functions of sample moments). Regarding the

27 case in point, the default rate, is used as a random variable, in other words: As the total number of borrowers is known, the expected value of (5.32) is given by: ( )

28 So, assuming that all loans in the portfolio have the same probability of default and are independent, the estimator using the method of moments is given by: Now, from (5.28) it is known that [ ] from which we obtain ( ) [ ] Furthermore, we know that: [ ] By applying the method of moments to (5.35), (5.33) and (5.34) we obtain, [ ] where [ ] We suggest using instead of, in order to eliminate the effect of the last period observation. Then the estimator for is given by

29 Default probability [ ] 5.6 Implementing: comparing estimation methods In the section above we expounded the relative theory of default probabilities and correlation and estimators found for estimating them. In this section we present a comparison of the estimates obtained using the estimation methods mentioned Results Historical default rates were used to estimate default probabilities and correlation coefficients, as observed in the database of the Credit Information List that banks must deliver to Banco de México each month with a moving window of 12 months; in other words, the default probability is assumed to be the same for each 12-month period. 7 Graphs 5.1 and 5.2 show estimates of the default probability made using the different methods presented here. It can be seen that the probability estimated by the maximum likelihood method assuming independence (simple Binomial) underestimates in all cases, basically because in the absence of correlation, the default rate observed must be fully explained by the default probability. 4.0% 3.5% Estimates of default probability 3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% Correlated binomial Normal approximation Binomial moments Simple Graph 5.1 Comparison of probability of default estimates 7 Only multiple bank loans were used so no information related to financial leasing companies, factoring companies and development banks was used.

30 Correlation between defaults Graph 5.2 shows a comparison between the estimated correlation coefficients for the different methods mentioned. It can be seen that the correlation coefficients show the same trend for all estimates studied. 2.5% Correlation coefficient estimates between defaults 2.0% 1.5% 1.0% Correlated binomial Normal approximation Method of moments 0.5% 0.0% Graph 5.2. Comparison of correlations between defaults The following table represents the results for the default approximations obtained using the estimation methods mentioned. Time Simple approximation Correlated approximation Normal approximation Method of moments Table 5.1

31 Likewise estimates for the correlation coefficient between defaults is presented. Time Correlated binomial Normal approximation Method of moments Table 5.2 Note that any of the three methods produces very similar sized results. Evidently, the easiest is the method of moments as it does not require any optimization procedure. As for the other two methods, it is important to mention that estimating by normal approximation is not easy to obtain because the associated density function becomes uncertain on the parameter restrictions border. 6. Completing the covariance matrix In section 5 above, we addressed the problem of estimating default probabilities and the idiosyncratic correlations of each segment of the portfolio. To complete the picture, the correlation between loan defaults of different segments must be estimated. This enables the elements of sub-matrixes which contain covariances between the loan defaults of the group i and j, to be estimated. The basic assumption in this case is that the correlation coefficient is equal for any pair of loans of two different segments. From (4.2) in section four, recall that: ( ) ( ) where y represent dimension of sum vectors and respectively. For estimate purposes, given that the probabilities of default are estimated in terms of idiosyncrasies using one of the techniques described in the previous section, we only need to estimate the correlations between defaults of borrowers of different segments. This is convenient for other reasons. First, for numerical stability reasons as the correlations are restricted to interval [-1, 1], which allows for obtaining more precise numbers that are

32 generally more normal than the covariances. A second benefit is that when normalized, results are comparable. The last reason is that it reduces the number of parameters to estimate as well as the degrees of freedom for hypothesis test considerations as the diagonal elements are always 1. In reality, the correlation estimation process is more in line with what text books usually cover The correlation coefficient A standard measure of dependency between two variables, let us say X and Y, is moment, known as the covariance of X and Y, defined as: { } { } { } { } In plain terms, it can said that the covariance represents a weighted average of the conditional means measured on the unconditional mean. This can be expressed as { [ {{ } } { }]} Although the covariance is a natural form of measuring relations, it is convenient to have a measure that is invariant under changes in scale or position. The correlation coefficient is thus stated as: ( ) Geometrically, one can think of the correlation coefficient as a measure of rectangularity. For this consider a real space with the following internal product: { } Thus, if vectors X and Y are measured around their respective means, the covariance is represented by the internal product of these two vectors while the correlation between them is simply the inner product of the normalized vectors. This is formally expressed as: } In order to see that (6.4) and (6.5) have the same functional form, recall that a vector norm is the square root of the inner product of the vector with itself. Recalling that in Euclidean

33 space (6.5) represents the cosine of the angle between two vectors, we verify that the measure is restricted to interval [-1, 1] and also gives a geometric interpretation of the independence between two vectors. Here we should point out that even though two vectors are rectangular (independent), if and only if their inner product is zero, in the field of probability an inner product (covariance) of zero does not necessarily imply the rectangularity of the vectors while their rectangularity leads to an inner product equal to zero. Figure 1: Correlation geometry Two considerations are fundamental when studying the correlation coefficient (6.5): the first is that it is a linear independent coefficient. Generally speaking, the joint variation is too complex to be summarized by one number. The second consideration is that the use of P as an interdependent measure is convenient only in the case of normal or near normal variation. 6.2 Estimators of Once the correlation coefficient has been studied (6.5) the problem of how it can be estimated needs to be addressed. Below we discuss different estimating techniques based on a sample Product moment correlation A first approximation when estimating the expression (6.5) is to estimate each of the moments separately and subsequently infer the sample correlation from them. Thus we will have: ( ) }

34 Although a priori the estimate seems heuristic, it can be shown that in the case of a binormal sample, (6.6) it is the maximum likelihood estimator of. This enables the use of (6.6) as an estimator of the correlation in most practical cases, especially when the sample is large enough Daniels correlation The Daniels estimator is a generalization of formula (6.6). This estimator is used to test the independence between the series observed; it is expressed as follows: ( ) In this expression, denotes and depend on ( ) and ( ), respectively. As shown, (6.6) is a special case of (6.7) which has the advantage of ( ) not having to be restricted to a function of the values of the series but admits, for example, measures based on ranges. This gives it a lot of versatility as the definition of both and de means the dependence pattern wished to be detected can be controlled. 6.3 Matrix estimation So far we have theoretically discussed the correlation between two random variables. As we said at the beginning of the section, we are interested in obtaining the interaction of a series of random variables, in particular borrower defaults, through the correlations matrix. In the credit risk model developed, given its special structure, we have seen that the segmented covariance matrix has the shape shown in (3.0), and only sub-matrices with the shape expressed in (6.1) need to be estimated. As shown in (6.7), the problem of estimating the matrix is summarized as a problem of estimating

35 correlations to peers, as correlations associated with the intrinsic covariance and variance matrices are estimated according to the previous section. By taking the same sample across estimates we ensure that the resulting matrix is positively defined. The estimator of each sub-matrix of correlations is thus given by the following expression: ( ) where Estimation When implementing the model, the estimation of the correlation is done with a moving window of periods. The estimator (6.7) is therefore as follows: ( ) ( ) where Results The data used are historically observed default rates associated with the different segments the portfolio was divided into. The Credit Information List is used to obtain default probabilities and the idiosyncratic correlations of section 5. Currently, as a result of an institutional agreement, the main segmentation criterion is by bank and borrower economic activity. The loan portfolio of each bank is divided into 22 groups as shown in Table 61. The results were obtained by applying the expression (6.9) with a moving window of months.