Ultimate recovery mixtures
Introduction
The economic value of debt in the event of default is a key determinant of the default risk premium required by a lender and the regulatory capital charged to limit exposure to losses. The pricing of default risk insurance (CDS contracts) and the emergence of distressed debt as an investment class add further incentive to better understand the distribution of payoffs in the event of default.2 Adding to market-driven incentives, Basel II and III provide regulatory incentives to the development of recovery models in financial institutions adopting an advanced internal ratings based (IRB) approach to computing capital requirements.
Recognizing the importance of capturing the behavior of recoveries in the event of default to quantitative models of credit risk, recent years have seen a wave of research from academics and industry professionals seeking to document the key empirical features of observed recovery outcomes. While payoffs to debt holders in the event of default depend on the interplay of many factors, often idiosyncratic, notable empirical regularities from prior research are evident.
- 1.
Recovery distributions tend to be bimodal, with recoveries either very high or low, implying as Schuermann (2004)3 observes, that the concept of average recovery is potentially very misleading.
- 2.
Collateralization and degree of subordination are the key determinants of recovery on defaulted debt. The (proportional) value of claims subordinate to the debt at a given seniority, known as the Debt Cushion, also seems to matter. The analysis of Keisman and Van de Castle (1999) suggests that all else equal, the larger the Debt Cushion, the higher the expected recovery outcome.
- 3.
Recoveries tend to be lower in recessions and other periods when the rate of aggregate defaults is high. Altman et al. (2005) demonstrate an association between default rates and the mean rate of recovery whereby up to 63% of the variation in average annual recovery can be explained by the coincident annual default rate. Further, Frye (2000) shows that a 10% realized default rate results in a 25% reduction in recoveries relative to its normal year average.
- 4.
Industry matters. Acharya et al. (2007) suggest that macroeconomic conditions do not appear to be significant determinants of individual bond recoveries after accounting for industry effects. More recently, Jankowitsch et al. (2012) find that the type of default, seniority of the bond and industry are as important determinants of recovery as balance sheet ratios motivated by structural credit risk models, macrovariables and transaction cost variables.
- 5.
Variability of recoveries is high, even intra-creditor-class variability, after categorization into sub-groups. For example, Schuermann (2004) notes that senior secured bond investments have a flat distribution – indicating that recoveries are relatively evenly distributed from 30% to 80%.
Clearly, the empirical features of historical recoveries suggest the need for caution in applying popular (parametric) tools of inference – such as OLS regressions and calibrated Beta distributions. While OLS regression models provide simple, intuitive summaries of data relationships, they make strong assumptions about the conditional distribution of recovery outcomes and focus attention on variation in the mean. Alternatively, Beta distributions calibrated to historical data are used in many commercial models of portfolio risk to characterize the distribution of loss outcomes.4 While Beta distributions offer a simple, parsimonious way of capturing a very broad range of distributional shapes over the unit interval, Servigny and Renault (2004) observe that they cannot accommodate bi-modality, or probability masses near zero and unity – important features of empirical recovery distributions.
While stylized models and a growing body of empirical evidence reveal much about the important influences on debt recovery outcomes, they also serve to highlight the challenges inherent in building a quantitative model to account for: characteristics specific to the defaulted instrument, borrower characteristics, macroeconomic conditions at the time of default, and the idiosyncrasies of recovery distributions’ shape. Building on insights from empirical research and the findings of recent studies documenting the relative merits of non-parametric and regression based approaches, we present in this paper a novel approach to modeling recoveries on defaulted debt using mixtures of Gaussian distributions.
More specifically, our paper makes three contributions to the literature. First, we present an approach to modeling recovery distributions that retains the flexibility of non-parametric methods while providing transparency with respect to the economic sources of variation in recovery outcomes. Second, we estimate and evaluate the out-of-sample performance of our model using Moody’s Ultimate Recovery Database spanning a 25 year sample period ending in 2011. As noted by Bastos, 2010, Qi and Zhao, 2011, very few studies to date have evaluated the predictive performance of alternative modeling methodologies. While they present tests of non-parametric approaches relative to regression-based alternatives, neither of the studies consider semi-parametric models. Third, our model provides further clarity on the role and importance of economic influences on recovery outcomes.
The remainder of our study proceeds as follows. We provide in Section 2 an overview of recent approaches to recovery modeling and an overview of the approach proposed in this paper. In Section 3 we describe the ultimate recoveries data used in this study and we detail the econometric approach in Section 4. We report model estimates and comparative performance metrics in Section 5 and summarize our findings in Section 6.
Section snippets
Recovery modeling approaches
Recent studies have investigated the forecasting performance of non-parametric estimation approaches relative to a variety of parametric regression specifications. Using loss data on defaulted Portuguese bank loans, Bastos (2010) finds that non-parametric regression trees tend to outperform parametric regression-based forecasts over shorter (annual) horizons. Similarly, using a larger US sample of defaulted loans and bonds, Qi and Zhao (2011) find that forecasts based on regression trees and
Data description
We use discounted ultimate recoveries from Moody’s Ultimate Recovery Database provided by Moody’s of New York. Moody’s database provides several measures of the value received by creditors at the resolution of default – usually upon emergence from Chapter 11 proceedings. Moody’s estimate of the discounted value of ultimate recovery is our choice of the measure of the economic value accruing to a creditor at the time of default. Moody’s calculates discounted ultimate recoveries by discounting
Econometric framework: some elaboration
As noted, we commence by assuming a Gaussian form for the approximating densities in Eq. (2), modeling the data y using a probability weighted mixture of m Gaussian likelihoods:where is the mean of mixture component j and its standard deviation.9 The sample
Results
Given that a primary objective of our empirical analysis is to estimate and evaluate the predictive performance of our proposed conditional mixture model, we note from the outset our split of the overall sample into 2307 estimation observations comprising the sample up to and including 2001, and 2413 test observations covering the period beginning 2002 to the end of 2011. As such, our results are based on true out of sample data. Unless stated otherwise, our discussion of the model focuses on
Conclusion
We present in this paper a new approach to modeling the distribution of recoveries on defaulted loans and bonds using mixtures of distributions. We take a Bayesian perspective and model (transformed) ultimate recoveries using a mixture of Gaussian distributions wherein the mixing probabilities are explicitly conditioned on borrower characteristics, debt features and the economic conditions prevailing at the time of default.
Our empirical findings suggest that our formulation delivers predictive
Acknowledgements
We thank David Keisman (Moody’s Investor Services) for providing us with access to Moody’s Ultimate Recovery Database. We thank participants at Macquarie University’s 2012 Financial Risk Day, the 2012 Methods in International Finance Conference (Sydney), and the 2013 International Risk Management Conference (Copenhagen) for helpful comments and suggestions. We are especially grateful to Philip Gray, Chris Heaton, Xinxin Shang, Pedro Sottile, and an anonymous referee for many constructive
References (28)
- et al.
Does industry-wide distress affect defaulted firms? Evidence from creditor recoveries
Journal of Financial Economics
(2007) Forecasting bank loans loss given default
Journal of Banking and Finance
(2010)- et al.
Benchmarking regression algorithms for loss given default modeling
International Journal of Forecasting
(2012) - et al.
Comparison of modeling methods for loss given default
Journal of Banking and Finance
(2011) - et al.
Comparisons of linear regression and survival analysis using single and mixture distributions approaches in modelling LGD
International Journal of Forecasting
(2012) - et al.
Almost everything you wanted to know about recoveries on defaulted bonds
Financial Analysts Journal
(1996) - Altman, E., Kuehne, B., 2013. Defaults and Returns in the High-Yield Bond and Distressed Debt Market: The Year 2012 in...
- et al.
The link between default and recovery rates: theory, empirical evidence, and implications
The Journal of Business
(2005) - et al.
A simple empirical model of equity implied probabilities of default
Journal of Fixed Income
(2011) - Bastos, J.A., 2013. Ensemble predictions of recovery rates, Journal of Financial Services Research...
Forecasting default with the Merton distance to default model
Review of Financial Studies
Explaining the gibbs sampler
The American Statistician
Collateral damage
Risk
Cited by (58)
Intertemporal defaulted bond recoveries prediction via machine learning
2022, European Journal of Operational ResearchCitation Excerpt :In particular, they show using a dataset for the years 1982 to 1999 that defaulted corporate bonds in distressed industries exhibit 10% to 15% lower average recovery rates. Altman & Kalotay (2014) introduce a modeling approach based on mixtures of Gaussian distributions conditioned on borrower characteristics, instrument characteristics, and credit market conditions. They show that the forecasts generated by this method are more accurate than parametric regression-based forecasts during out-of-time estimation.
Opening the black box – Quantile neural networks for loss given default prediction
2022, Journal of Banking and FinanceDeep learning for modeling the collection rate for third-party buyers
2022, International Journal of ForecastingCitation Excerpt :Prior studies focused on estimating recovery rates and improving the prediction accuracy of estimation. All the studies prior to Hurlin et al. (2018) used common performance measures such as MSE, MAE, TIC, and R-squared for comparing different parametric and non-parametric recovery rate models (see, for example, Altman and Kalotay (2014), Hartmann-Wendels et al. (2014), Kalotay and Altman (2017), Kaposty et al. (2020), and Nazemi and Fabozzi (2018)). Hurlin et al. (2018) suggested a new method for comparing LGD models based on loss functions.
Predicting loss given default using post-default information
2021, Knowledge-Based SystemsLocal logit regression for loan recovery rate
2021, Journal of Banking and FinanceCitation Excerpt :In order to accommodate some of the aforementioned atypical properties of recoveries, numerous studies introduced parametric models with various distributional assumptions. They include mixture Gaussian model (Altman and Kalotay, 2014), hybrid finite mixture models (Hartmann-Wendels et al., 2014), and Zero-adjusted Gamma regression (Tong et al., 2013). As said in the introduction, the transformation and back transformation of recovery data introduce bias into the model parameter estimates and hence into the prediction of recoveries.
The determinants of bank loan recovery rates in good times and bad – New evidence
2020, Journal of Economic Behavior and OrganizationCitation Excerpt :Furthermore, the current literature on corporate debt recovery determinants does not focus specifically on bank loans. Altman and Kalotay (2014) combine bank loans with corporate bonds, whereas Mora (2015) investigates corporate bonds. A bank loan is fundamentally different to other securities, as they are typically senior to corporate bonds, making bank loan recoveries more likely due to a different repayment hierarchy.
- 1
Tel.: +1 (212) 998 0709; fax: +1 (212) 995 4220