مقایسه رگرسیون خطی و تجزیه و تحلیل بقا با استفاده از روش های توزیع تک و مخلوط در مدل سازی LGD

Estimating the recovery rate and recovery amount has become important in consumer credit due to the new Basel Accord regulation and the increase in the number of defaulters as a result of the recession. We compare linear regression and survival analysis models for modelling recovery rates and recovery amounts, in order to predict the loss given default (LGD) for unsecured consumer loans or credit cards. We also look at the advantages and disadvantages of using single and mixture distribution models for estimating these quantities.

The New Basel Accord allows banks to calculate their credit risk capital requirements according to one of two approaches. The first, namely the standardized approach, requires a percentage of the risk weighted assets to be set aside, where the percentage is given in the regulations. The second, known as the internal ratings based (IRB) approach, allows banks to use internal estimates of the components of credit risk to calculate their credit risk capital. Institutions using IRB need to develop methods of estimating the following components for each segment of their loan portfolio: – PD (probability of default in the next 12 months); – LGD (loss given default); and – EAD (expected exposure at default). Modelling the probability of default (PD) has been the objective of credit scoring systems for fifty years, but modelling LGD is not something that was really addressed in consumer credit prior to the advent of the Basel regulations. Modelling LGD appears to be more difficult than modelling PD, for two reasons. Firstly, much of the data may be censored (debts still being paid) because of the long time scale of recovery. Linear regression does not deal very well with censored data, and even the Buckley-James approach (Buckley & James, 1979) does not cope well with this form of censoring. Second, debtors have different reasons for defaulting, which lead to different repayment patterns. For example, some people do not want to repay, and some people cannot repay because of permanent changes in their situation; while for others the reason for non-repayment may be temporary. One distribution may find it hard to model the outcomes of these different reasons. However, survival analysis can handle censored data, and segmenting the whole default population is helpful in modelling LGD for defaulters with different reasons for defaulting. Most LGD modelling research has concentrated on corporate lending, where LGD (or its opposite, the recovery rate (RR), where View the MathML sourceRR=1−LGD) was needed as part of the bond pricing formulae. Even in this case, LGD was assumed until a decade ago to be a deterministic value obtained from a historical analysis of bond losses or from bank experience (Altman, Haldeman, & Narayanan, 1977). Only when it was recognised that LGD was part of the pricing formula and that one could use the price of non-defaulted risky bonds to estimate the market’s view of LGD were models of LGD developed. If defaults are rare in a particular bond class, then it is likely that the LGD obtained from the bond price is essentially a subjective judgment by the market. The market also trades defaulted bonds, and thus one can obtain the market values of defaulted bonds directly (Altman & Eberhart, 1994). These values of the LGD, whether obtained from defaulted bonds or implied in the price of non-defaulted bonds, were used to build regression models that related LGD to relevant factors such as the seniority of the debt, country of issue, size of issue, size of the firm and industrial sector of the firm, but most of all to the economic conditions which determined where the economy was in relation to the business cycle. The most widely used model is Moody’s KMV model, LossCalc (Gupton, 2005). It transforms the target variable into a normal distribution using a Beta transformation, regresses the transformed target variable on a few characteristics, and then transforms the predicted values back, to get the LGD prediction. Another popular model, Recovery Ratings, was created by Standard and Poor’s Ratings Services (Chew & Kerr, 2005); it divides the loans into 6 classes which cover different recovery ranges. Descriptions of the models are given in several books and reviews (Altman et al., 2005, De Servigny and Oliver, 2004, Engelmann and Rauhmeier, 2006 and Schuermann, 2005). Such modelling is not appropriate for consumer credit LGD models, since there is no continuous pricing of the debt as there is on the bond market. The Basel Accord (Basel Committee on Banking Supervision, 2004, paragraph 465) suggests using the implied historic LGD as one approach for determining the LGD for retail portfolios. This involves identifying the realised losses (RL) per unit amount loaned in a segment of the portfolio and estimating the default probability PD for that segment, from which one can calculate LGD, since RL=LGD.PD. One difficulty with this approach is that it is often accounting losses that are recorded rather than the actual economic losses. Also, since LGD must be estimated at the segment level of the portfolio, if not at the individual loan level, in some segments there is often insufficient data segments to obtain robust estimates. The alternative method suggested in the Basel Accord is to model the collection or work out process. Such data were used by Dermine and de Carvalho (2006) for bank loans to small and medium sized firms in Portugal. They used a regression approach, albeit a log–log form of the regression, to estimate LGD. The idea of using the collection process to model LGD for mortgages was suggested by Lucas (2006). The collection process was split into whether the property was repossessed or not, and the loss if there was repossession. Thus, a scorecard was designed to estimate the probability of repossession, where Loan to Value was key, and then a model was used to estimate the percentage of the estimated sale value of the house that is actually realised at sale time. For mortgage loans, a one-stage model was built by Qi and Yang (2009). They modelled LGD directly, and found that LTV (Loan to Value) was the key variable in the model; they achieved an adjusted R2R2 value of 0.610, but this dropped to 0.15 if LTV was excluded. For unsecured consumer credit, the only available approach is to model the collection process, but now there is no security to be repossessed. The difficulty in such modelling is that the loss given default, or the equivalent recovery rate, depends both on the ability and willingness of the borrower to repay, and on decisions made by the lender as to how vigorously they will pursue the debt. This is identified at a macro level by Matuszyk, Mues, and Thomas (2010), who use a decision tree to model whether the lender will collect in house, use an agent on a percentage commission, or sell off the debts, with different actions putting different limits on the possible LGD. Even if one concentrates on one mode of recovery only (for example, in house collection), it is still very difficult to get good estimates. Matuszyk et al. (2010) look at various versions of regression, while Bellotti and Crook (2009) add economic variables to the regression. Somers and Whittaker (2007) suggest using quantile regression, but the results in terms of R2R2 are poor in all cases—between 0.05 and 0.2. Querci (2005) investigated data from an Italian bank on geographic location, loan type, workout process length and borrower characteristics, but concluded that none of them was able to explain LGD, though borrower characteristics were the most effective. In this paper, we use linear regression and survival analysis models to build predictive models for the recovery rate, and hence LGD. Both single distribution and mixture distribution models are built, and we compare the two approaches. This analysis will give an indication of how important it is to use models—survival analysis based ones—which can cope with censored debts, and will also investigate whether mixed distribution models give better predictions than single distribution models. The comparison will be made based on a case study involving data from an in-house collection process for personal loans. This consisted of collection data on 27,000 personal loans over the period from 1989 to 2004. In Section 2 we briefly review the theory of linear regression and survival analysis models. In Section 3 we explain the idea of mixture distribution models as they are applied in this problem. In Section 4 we build and compare single distribution models using linear regression and survival analysis based models, while in Section 5 we create mixture distribution models, to enable us to compare them. In Section 6 we summarise the conclusions reached.

Estimating the recovery rate and recovery amount has recently become much more important, both because of the new Basel Accord regulation and because of the increase in the number of defaulters due to the recession. This paper compares single distribution and mixture distribution models of predicting the recovery rate for unsecured consumer loans. Linear regression and survival analysis are the two main techniques used in this research, where survival analysis can cope with censored data better than linear regression. For survival analysis models, we investigated the use of proportional hazard models and accelerated failure time models, although the latter have certain problems that need to be addressed: they do not allow zeros to exist in the target variable and the recovery rate cannot be bounded above. This can be overcome by not defining View the MathML sourceRR>1 to be censored at 1 and by first using a logistic regression model to classify which loans have zero and non-zero recovery rates. Cox’s proportional hazard regression models can deal with zeros in the target variable and with the requirement that View the MathML sourceRR≤1 for all loans, so that approach was tried, both with logistic regression used first to split off the zero recoveries and without using logistic regression first. In all cases, the approaches were used to model both the recovery rate and the recovery amount, and for all of the models it proved to be better to model the recovery rate and then use this estimate to calculate the recovery amount, rather than modelling the recovery amount directly. In our comparison of the single distribution models, it has been shown that linear regression is better than survival analysis models in most situations. For recovery rate modelling, linear regression achieves a higher R2R2 value and Spearman rank coefficient than the survival analysis models. The Cox model without the logistic regression first is the best of the survival analysis models. This is surprising, given the flexibility of distribution that the Cox approach allows. Of course, one would expect the minimum MSE to be obtained by the linear regression on the training sample, because that is what the linear regression tries to do. However, the superiority of the linear regression also holds for the other measures, on both the training and test sets. One reason for this may be the need to separate the zero recovery rate cases in the accelerated failure time approach. This is obviously difficult to do, and the errors from this first stage result in a poorer model at the second stage. This could also be the reason why the mixture models do not provide any real improvement. Finding suitable segments is difficult, and the resultant subgroups are not as homogeneous as one would wish. Another reason for the survival analysis approach not doing so well is that in performing these comparisons we used test sets where the recovery rate was known for all of the debtors. That is, they had all been either paid off or written off. Thus, there was no opportunity to test the model’s predictions on those who were still paying, which is of course the type of data that are used by the survival analysis models, though not the regression based models. Finally, in the survival analysis approach there is the question of whether loans with View the MathML sourceRR=1 are really censored or not. Assuming that they are not censored would lead to lower estimates of RR, which might be more appropriate for the conservative philosophy of the Basel Accord. These results are based on the case study data set, which, though quite large, is from only one UK lender. The results require further validation from either the use of other data sets or some theoretical underpinning for them to be considered valid for all types of unsecured consumer credit LGD modelling.