Raffaella Calabrese

Modelling small and medium enterprise loan defaults as rare events: the generalized extreme value regression model

A pivotal characteristic of credit defaults that is ignored by most credit scoring models is the rarity of the event. The most widely used model to estimate the probability of default is the logistic regression model. Since the dependent variable represents a rare event, the logistic regression model shows relevant drawbacks, for example, underestimation of the default probability, which could be very risky for banks. In order to overcome these drawbacks, we propose the generalized extreme value regression model. In particular, in a generalized linear model (GLM) with the binary-dependent variable we suggest the quantile function of the GEV distribution as link function, so our attention is focused on the tail of the response curve for values close to one. The estimation procedure used is the maximum-likelihood method. This model accommodates skewness and it presents a generalisation of GLMs with complementary log–log link function. We analyse its performance by simulation studies. Finally, we apply the proposed model to empirical data on Italian small and medium enterprises.

Bankruptcy prediction of small and medium enterprises using a flexible binary generalized extreme value model

We introduce a binary regression accounting-based model for bankruptcy prediction of small and medium enterprises (SMEs). The main advantage of the model lies in its predictive performance in identifying defaulted SMEs. Another advantage, which is especially relevant for banks, is that the relationship between the accounting characteristics of SMEs and response is not assumed a priori (eg, linear, quadratic or cubic) and can be determined from the data. The proposed approach uses the quantile function of the generalized extreme value distribution as link function as well as smooth functions of accounting characteristics to flexibly model covariate effects. Therefore, the usual assumptions in scoring models of symmetric link function and linear or pre-specified covariate-response relationships are relaxed. Out-of-sample and out-of-time validation on Italian data shows that our proposal outperforms the commonly used (logistic) scoring model for different default horizons.

Estimating bank default with generalised extreme value regression models

The paper proposes a novel model for the prediction of bank failures, on the basis of both macroeconomic and bank-specific microeconomic factors. As bank failures are rare, in the paper we apply a regression method for binary data based on extreme value theory, which turns out to be more effective than classical logistic regression models, as it better leverages the information in the tail of the default distribution. The application of this model to the occurrence of bank defaults in a highly bank dependent economy (Italy) shows that, while microeconomic factors as well as regulatory capital are significant to explain proper failures, macroeconomic factors are relevant only when failures are defined not only in terms of actual defaults but also in terms of mergers and acquisitions. In terms of predictive accuracy, the model based on extreme value theory outperforms classical logistic regression models.

Downturn Loss Given Default: Mixture distribution estimation

The internal estimates of Loss Given Default (LGD) must reflect economic downturn conditions, thus estimating the “downturn LGD”, as the new Basel Capital Accord Basel II establishes. We suggest a methodology to estimate the downturn LGD distribution to overcome the arbitrariness of the methods suggested by Basel II. We assume that LGD is a mixture of an expansion and recession distribution. In this work, we propose an accurate parametric model for LGD and we estimate its parameters by the EM algorithm. Finally, we apply the proposed model to empirical data on Italian bank loans.

A comparative analysis of the UK and Italian small businesses using Generalised Extreme Value models

This paper presents a cross-country comparison of significant predictors of small business failure between Italy and the UK. Financial measures of profitability, leverage, coverage, liquidity, scale and non-financial information are explored, some commonalities and differences are highlighted. Several models are considered, starting with the logistic regression which is a standard approach in credit risk modelling. Some important improvements are investigated. Generalised Extreme Value (GEV) regression is applied in contrast to the logistic regression in order to produce more conservative estimates of default probability. The assumption of non-linearity is relaxed through application of BGEVA, non-parametric additive model based on the GEV link function. Two methods of handling missing values are compared: multiple imputation and Weights of Evidence (WoE) transformation. The results suggest that the best predictive performance is obtained by BGEVA, thus implying the necessity of taking into account the low volume of defaults and non-linear patterns when modelling SME performance. WoE for the majority of models considered show better prediction as compared to multiple imputation, suggesting that missing values could be informative.

The effectiveness of TARP-CPP on the US banking industry: A new copula-based approach

Following the 2008 financial crisis, regulatory authorities and governments provided distressed banks with equity infusions in order to strengthen national banking systems. However, the effectiveness of these interventions for financial stability has not been extensively researched in the literature. In order to understand the effectiveness of these bailouts for the solvency of banks this paper proposes a new model: the Longitudinal Binary Generalised Extreme Value (LOBGEV) model. Differing from the existing models, the LOBGEV model allows us to analyse the temporal structure of the probability of failure for banks, for both those that received a bailout and for those that did not. In particular, it encompasses both the flexibility of the D-vine copula and the accuracy of the generalised extreme value model in estimating the probability of bank failure and of banks receiving approval for capital injection. We apply this new model to the US banking system from 2008 to 2013 in order to investigate how and to what extent the Troubled Asset Relief Program (TARP)-Capital Purchase Program (CPP) reduced the probability of the failure of commercial banks. We specifically identify a set of macroeconomic and bank-specific factors that affect the probability of bank failure for TARP-CCP recipients and for those that did not receive capital under TARP-CCP. Our results suggest that TARP-CPP provided only short-term relief for US commercial banks.

'Birds of a feather' fail together: Exploring the nature of dependency in SME defaults

This article studies the effects of incorporating the interdependence among London small business defaults into a risk analysis framework using the data just before the financial crisis. We propose an extension from standard scoring models to take into account the spatial dimensions and the demographic characteristics of small and medium-sized enterprises (SMEs), such as legal form, industry sector, and number of employees. We estimate spatial probit models using different distance matrices based only on the spatial location or on an interaction between spatial locations and demographic characteristics. We find that the interdependence or contagion component defined on spatial and demographic characteristics is significant and that it improves the ability to predict defaults of non-start-ups in London. Furthermore, including contagion effects among SMEs alters the parameter estimates of risk determinants. The approach can be extended to other risk analysis applications where spatial risk may incorporate correlation based on other aspects.

A Generalized Additive Model for Binary Rare Events Data: an Application to Credit Defaults.

We aim at proposing a new model for binary rare events, i.e. binary dependent variable with a very small number of ones. We extend the Generalized Extreme Value (GEV) regression model proposed by Calabrese and Osmetti (Journal of Applied Statistics 40(6):1172–1188, 2013) to a Generalized Additive Model (GAM). We suggest to consider the quantile function of the GEV distribution as a link function in a GAM, so we propose the Generalized Extreme Value Additive (GEVA) model. In order to estimate the GEVA model, a modified version of the local scoring algorithm of GAM is proposed. Finally, to model default probability, we apply our proposal to empirical data on Italian Small and Medium Enterprises (SMEs). The results show that the GEVA model has a higher predictive accuracy to identify the rare event than the logistic additive model.

Optimal cut-off for rare events and unbalanced misclassification costs

This paper develops a method for handling two-class classification problems with highly unbalanced class sizes and misclassification costs. When the class sizes are highly unbalanced and the minority class represents a rare event, conventional classification methods tend to strongly favour the majority class, resulting in very low detection of the minority class. A method is proposed to determine the optimal cut-off for asymmetric misclassification costs and for unbalanced class sizes. Monte Carlo simulations show that this proposal performs better than the method based on the notion of classification accuracy. Finally, the proposed method is applied to empirical data on Italian small and medium enterprises to classify them into default and non-default groups.

Single-name concentration risk in credit portfolios: a comparison of concentration indices

For assessing the effect of undiversified idiosyncratic risk, Basel II has established that banks should measure and control their credit concentration risk. Concentration risk in credit portfolios comes into being through an uneven distribution of bank loans to individual borrowers (single-name concentration) or through an unbalanced allocation of loans in productive sectors and geographical regions (sectoral concentration). To evaluate single-name concentration risk in the literature concentration indices proposed in welfare (Gini Index) and monopoly theory (Herfindahl- Hirschman index, Theil entropy index, Hannah-Kay index, Hall-Tidemann index) have been used. In this paper such concentration indices are compared by using as benchmark six properties that ensure a consistent measurement of single-name concentration. Finally, the indices are compared on some portfolios of loans.For assessing the effect of undiversified idiosyncratic risk, Basel II has established that banks should measure and control their credit concentration risk. Concentration risk in credit portfolios comes into being through an uneven distribution of bank loans to individual borrowers (single-name concentration) or through an unbalanced allocation of loans in productive sectors and geographical regions (sectoral concentration). In this paper six properties that ensure a coherent measure of single-name concentration are identified. To evaluate single-name concentration risk in the literature, Herfindahl-Hirschman index has been used. This index represents a particular case of Hannah-Kay index proposed in monopoly theory. In this work the proof that Hannah-Kay index satisfies all the six properties is given. Finally, the impact of the elasticity parameter in Hannah-Kay index on the single-name concentration measure is analysed by numerical applications.