{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T11:31:39Z","timestamp":1789471899521,"version":"build-2803163510"},"reference-count":38,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,7]],"date-time":"2023-02-07T00:00:00Z","timestamp":1675728000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The objective of the present paper is to propose a new method to measure the recovery performance of a portfolio of non-performing loans (NPLs) in terms of recovery rate and time to liquidate. The fundamental idea is to draw a curve representing the recovery rates over time, here assumed discretized, for example, in years. In this way, the user can get simultaneously information about recovery rate and time to liquidate of the portfolio. In particular, it is discussed how to estimate such a curve in the presence of right-censored data, e.g., when the NPLs composing the portfolio have been observed in different time periods, with a method based on an algorithm that is usually used in the construction of survival curves. The curves obtained are smoothed with nonparametric statistical learning techniques. The effectiveness of the proposal is shown by applying the method to simulated and real financial data. The latter are about some portfolios of Italian unsecured NPLs taken over by a specialized operator.<\/jats:p>","DOI":"10.3390\/computation11020029","type":"journal-article","created":{"date-parts":[[2023,2,8]],"date-time":"2023-02-08T02:29:08Z","timestamp":1675823348000},"page":"29","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Measuring the Recovery Performance of a Portfolio of NPLs"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7248-1603","authenticated-orcid":false,"given":"Alessandra","family":"Carleo","sequence":"first","affiliation":[{"name":"Department of Business Economics, Roma Tre University, 00145 Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roberto","family":"Rocci","sequence":"additional","affiliation":[{"name":"Department of Statistical Sciences, Sapienza University, 00185 Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maria Sole","family":"Staffa","sequence":"additional","affiliation":[{"name":"Human Sciences Department, European University of Rome, 00163 Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,7]]},"reference":[{"key":"ref_1","unstructured":"Resti, A., and Sironi, A. (2007). Risk Management and Shareholders\u2019 Value in Banking, John Wiley & Sons."},{"key":"ref_2","unstructured":"Basel Committee on Banking Supervision (2001). The Internal Ratings-Based Approach, Bank for International Settlements. Available online: https:\/\/www.bis.org\/publ\/bcbsca05.pdf."},{"key":"ref_3","unstructured":"Basel Committee on Banking Supervision (2008). Principles for Sound Liquidity Risk Management and Supervision, Bank for International Settlements. Available online: http:\/\/www.bis.org\/publ\/bcbs144.pdf."},{"key":"ref_4","unstructured":"Basel Committee on Banking Supervision (2017). Basel III: Finalising Post-Crisis Reforms, Bank for International Settlements. Available online: https:\/\/www.bis.org\/bcbs\/publ\/d524.pdf."},{"key":"ref_5","unstructured":"European Banking Authority (2023, January 30). Guidelines on PD Estimation, LGD Estimation and the Treatment of Defaulted Exposure. Available online: https:\/\/www.eba.europa.eu\/sites\/default\/documents\/files\/documents\/10180\/2033363\/6b062012-45d6-4655-af04-801d26493ed0\/Guidelines%20on%20PD%20and%20LGD%20estimation%20%28EBA-GL-2017-16%29.pdf?retry=1."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hastie, T., Tibshirani, R., and Friedman, J. (2009). The Elements of Statistical Learning, Springer. [2nd ed.].","DOI":"10.1007\/978-0-387-84858-7"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1016\/j.ejor.2014.06.043","article-title":"Support Vector Regression for Loss Given Default Modelling","volume":"240","author":"Yao","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ye, H., and Bellotti, A. (2019). Modelling Recovery Rates for Non-performing Loans. Risks, 7.","DOI":"10.20944\/preprints201902.0135.v1"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1016\/j.ijforecast.2020.06.009","article-title":"Forecasting recovery rates on non-performing loans with machine learning","volume":"37","author":"Bellotti","year":"2021","journal-title":"Int. J. Forecast."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jbankfin.2018.08.014","article-title":"A reinforced urn process modeling of recovery rates and recovery times","volume":"96","author":"Cheng","year":"2018","journal-title":"J. Bank. Financ."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gambetti, P., Roccazzella, F., and Vrins, F. (2022). Meta-Learning Approaches for Recovery Rate Prediction. Risks, 10.","DOI":"10.2139\/ssrn.4067066"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/j.ijforecast.2019.05.009","article-title":"Predicting loss given default in leasing: A closer look at models and variable selection","volume":"36","author":"Kaposty","year":"2020","journal-title":"Int. J. Forecast."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Min, A., Scherer, M., Schischke, A., and Zagst, R. (2020). Modeling Recovery Rates of Small and Medium-Sized Entities in the US. Mathematics, 8.","DOI":"10.3390\/math8111856"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.ijforecast.2011.01.006","article-title":"Benchmarking regression algorithms for loss given default modeling","volume":"28","author":"Loterman","year":"2012","journal-title":"Int. J. Forecast."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2842","DOI":"10.1016\/j.jbankfin.2011.03.011","article-title":"Comparison of modeling methods for loss given default","volume":"35","author":"Qi","year":"2011","journal-title":"J. Bank. Financ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.jbankfin.2017.03.001","article-title":"Downturn LGD Modeling using Quantile Regression","volume":"79","year":"2017","journal-title":"J. Bank. Financ."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Gostkowski, M., and Gajowniczek, K. (2020). Weighted Quantile Regression Forests for Bimodal Distribution Modeling: A Loss Given Default Case. Entropy, 22.","DOI":"10.3390\/e22050545"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.jbankfin.2013.11.021","article-title":"Ultimate Recovery Mixtures","volume":"40","author":"Altman","year":"2014","journal-title":"J. Bank. Financ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1016\/j.ejor.2018.05.059","article-title":"Systematic Effects among Loss Given Defaults and their Implications on Downturn Estimation","volume":"271","author":"Betz","year":"2018","journal-title":"Eur. J. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.ejor.2014.01.043","article-title":"Downturn loss given default: Mixture distribution estimation","volume":"237","author":"Calabrese","year":"2014","journal-title":"Eur. J. Oper. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1093\/rof\/rfw028","article-title":"Intertemporal Forecasts of Defaulted Bond Recoveries and Portfolio Losses","volume":"21","author":"Kalotay","year":"2017","journal-title":"Rev. Financ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1111\/rssa.12466","article-title":"Modelling the Loss Given Default Distribution via a Family of Zero-and-one Inflated Mixture Models","volume":"182","author":"Tomarchio","year":"2019","journal-title":"J. R. Stat. Soc. Ser. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.ijforecast.2010.08.005","article-title":"Loss given default models incorporating macroeconomic variables for credit cards","volume":"28","author":"Bellotti","year":"2012","journal-title":"Int. J. Forecast."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1057\/jors.2014.9","article-title":"Modelling LGD for unsecured retail loans using Bayesian methods","volume":"66","author":"Bijak","year":"2015","journal-title":"J. Oper. Res. Soc."},{"key":"ref_25","first-page":"43","article-title":"Estimating credit risk parameters using ensemble learning methods: An empirical study on loss given default","volume":"12","author":"Sun","year":"2016","journal-title":"J. Credit Risk"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1057\/jors.2013.158","article-title":"Forecasting Loss Given Default models: Impact of account characteristics and the macroeconomic state","volume":"65","author":"Tobback","year":"2014","journal-title":"J. Oper. Res. Soc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3","DOI":"10.2478\/ngoe-2018-0018","article-title":"Modeling of Cash Flows from Nonperforming Loans in a Commercial Bank","volume":"64","author":"Devjak","year":"2018","journal-title":"Na\u0161e Gospod. Our Econ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1111\/rssc.12474","article-title":"Time matters: How default resolution times impact final loss rates","volume":"70","author":"Betz","year":"2021","journal-title":"J. R. Stat. Soc. Ser. C"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.1016\/j.jbankfin.2005.05.005","article-title":"Bank loan losses-given-default: A case study","volume":"30","author":"Dermine","year":"2006","journal-title":"J. Bank. Financ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6","DOI":"10.18267\/j.efaj.12","article-title":"Survival Analysis in LGD Modeling","volume":"7","author":"Witzany","year":"2012","journal-title":"Eur. Financ. Account. J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.ijforecast.2010.06.002","article-title":"Comparisons of linear regression and survival analysis using single and mixture distributions approaches in modelling LGD","volume":"28","author":"Zhang","year":"2012","journal-title":"Int. J. Forecast."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Kalbfleisch, J.D., and Prentice, R.L. (2002). The Statistical Analysis Failure Time Data, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9781118032985"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1111\/j.1540-6261.1989.tb02630.x","article-title":"Measuring Corporate Bond Mortality and Performance","volume":"44","author":"Altman","year":"1989","journal-title":"J. Financ."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Corazza, M., Perna, C., Pizzi, C., and Sibillo, M. (2022). Mathematical and Statistical Methods for Actuarial Sciences and Finance. MAF 2022, Springer.","DOI":"10.1007\/978-3-030-99638-3"},{"key":"ref_35","unstructured":"Rocci, R., Carleo, A., and Staffa, M.S. (2021). Working Paper n. 16, Universit\u00e0 degli Studi Roma Tre, Collana del Dipartimento di Economia Aziendale, Universit\u00e0 degli Studi Roma Tre. Available online: https:\/\/economiaziendale.uniroma3.it\/wp-content\/uploads\/sites\/9\/file_locked\/2021\/12\/WP16-Carleo.pdf."},{"key":"ref_36","unstructured":"Basel Committee on Banking Supervision (2005). Guidance on Paragraph 468 of the Framework Document, Bank for International Settlements. Available online: http:\/\/www.bis.org\/publ\/bcbs115.pdf."},{"key":"ref_37","unstructured":"R Core Team (2022). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing. Available online: https:\/\/www.R-project.org\/."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Wood, S. (2017). Generalized Additive Models: An Introduction with R, Chapman and Hall\/CRC. 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