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Jul 18, 2020

Stochastics and Financial Mathematics

Master Thesis

Credit risk and survival analysis: Estimation of Conditional Cure Rate

Author: Supervisor: Just Bajželj dr. A.J. van Es

Examination date: Daily supervisor: August 30, 2018 R. Man MSc

Korteweg-de Vries Institute for Mathematics

Rabobank

Abstract

Rabobank currently uses a non-parametric estimator for the computation of Conditional Cure Rate (CCR) and this method has several shortcomings. The goal of this thesis is to find a better estimator than the currently used one This master thesis looks into three CCR estimators. The first one is the currently used method. We analyze its performance with the bootstrap and later develop a method, with better performance. Since the newly developed and currently used estimators are not theoretically correct with respect to the data, a third method is introduced. However, according to the bootstrap the latter method exhibits the worst performance. For the modeling and data analysis the programing language Python is used.

Title: Credit risk and survival analysis: Estimation of Conditional Cure Rate Author: Just Bajželj, [email protected], 11406690 Supervisor: dhr. dr. A.J. van Es Second Examiner: dhr. dr. A.V. den Boer Examination date: August 30, 2018

Korteweg-de Vries Institute for Mathematics University of Amsterdam Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

Rabobank Croeselaan 18, 3521 CB Utrecht

https://www.rabobank.nl

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http://kdvi.uva.nl https://www.rabobank.nl

Aknowledgments

I would like to thank my parents who made possible for me to finish the two years Masters in Stochastics and Financial Mathematics in Amsterdam, that helped me to become the person I am today. I would also like to thank people from Rabobank and the department of Risk Analytics, thanks to whom I have written this thesis and, during my six-month internship, and showed me that work can be more than enjoyable. In particular, I would like to acknowledge all my mentors, Bert van Es who always had time to answer all my questions, Viktor Tchistiakov who gave me challenging questions and ideas, which represent the core of this thesis and, Ramon Man, who always showed me support and cared that this thesis was done on schedule.

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Contents

Introduction 5 0.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.2 Research objective and approach . . . . . . . . . . . . . . . . . . . . . . . 7

1 Survival analysis 8 1.1 Censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Competing risk setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Current CCR Model 20 2.1 Model implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.1 Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Performance of the method . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 The bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Cox proportional hazards model 28 3.1 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Estimation of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.2 Baseline hazard estimation . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Model implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Performance of the method . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Proportional hazards model in an interval censored setting 45 4.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.1 Generalized Linear Models . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Model implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.1 Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Performance of the method . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Popular summary 60

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Introduction

Under Basel II banks are allowed to build their internal models for the estimation of risk parameters. This is known as the Internal Rating Based approach (IRB). Risk parame- ters are used by banks in order to calculate their own regulatory capital. In Rabobank, the Loss Given Default (LGD), Probability of Default (PD) and Exposure at Default (EAD) are calculated with IRB. Loss given default describes the loss of a bank in case that the client defaults. Default happens when the client is unable to pay monthly payments for the mortgage for some time or one of other default events, usually connected with the client’s financial diffi- culties, happen. Missed payment is also known as arrear. After the client defaults his portfolio is non-performing and two events can happen. The event when the clients portfolio returns to performing one is known as cure. Cure happens if the costumer pays his arrears and has no missed payments during a three-months period or he pays the arrears after the loan restructuring and has no arrears in a twelve-months period. The event in which the bank needs to sell the client’s security in order to cover the loss is called liquidation. Two approaches are available for LGD modeling. The non-structural approach consists of estimating the LGD by observing the historical loss and recovery data. Rabobank uses another approach, the so-called structural approach. While the bank deals with LGD in a structural way, different probabilities and outcomes of default are considered. The model is split in several components that are developed separately and later combined in order to produce the final LGD estimation, as can be seen from Figure 0.1. In order to calculate the loss given default we firstly need to calculate the probability of cure, the loss given liquidation, the loss given cure and indirect costs. Rabobank assumes that loss given cure equals zero, since cure is offered to the client only if there is a zero-loss to the bank and any indirect costs that are suffered during the cure process are taken into account in the indirect costs component. Therefore, in this thesis sometimes the term loss is used instead of the term liquidation. Indirect costs are usually costs that are made internally by the departments of the bank that are involved into processing of defaults, e.g., salaries paid to employees and admin- istrative costs. The equation used for LGD calculation, can be seen in Figure 0.1.

The parameter probability of cure (PCure) is the probability that the client will cure before his security is liquidated. All model components, including PCure depend on co- variates from each client. A big proportion of cases, which are used to estimate PCure, are unresolved cases. A client that defaulted and was not yet liquidated nor cured is called unresolved. The easiest way to imagine an unresolved state is when every client needs some time after default in order to cure or sell their property. The non-absorbing state before cure or

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Figure 0.1: LGD = PCureLGC + IC + (1− PCure)LGL.

liquidation is called unresolved state. Unresolved cases can be treated in two ways:

• Exclusion: the cases can be excluded from the PCure estimation

• Inclusion: with expected outcome: we can include unresolved cases into the PCure estimation by assigning an expected outcome or Conditional Cure Rate to them.

If unresolved cases are excluded from the PCure estimation, it can happen that the parameter estimator will be biased. In other words PCure will be estimated on a sample where clients are cured after a short time. Consequently, clients, who would need more time to be cured would get a smaller value for PCure than they would deserve. Since PCure tells us the probability that a client will be cured after default and such a probability is not time dependent, treatment of unresolved cases with exclusion would be wrong. One approach within Rabobank is to treat unresolved cases by assigning them a value called Conditional Cure Rate. Conditional Cure rate (CCR) can be estimated with a non-parametric technique. This thesis will be about developing a new model able to eliminate the existing shortcomings of the currently employed model.

0.1 Background

CCR tells us the probability that a client will cure after a certain time point conditioned on the event that client is still unresolved at that point. Rabobank’s current model estimates CCR with survival analysis, which is a branch of statistics that is specialized in the distribution of lifetimes. The lifetime is the time to the occurrence of an event. In our case the lifetime is the time between the default of the client and cure or liquidation. Current CCR is a combination of Kaplan Meier and Neslon Aalen estimators, which are two of the most recognized and simple survival distribution estimators. The current CCR model has some

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