# Modelling Recovery Rates for Non-Performing Loans

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## Abstract

**:**

## 1. Introduction

- Suppose individual i has already defaulted on a loan, let $EA{D}_{i}$ be the exposure at default for this individual i.
- Let ${A}_{i}$ be the administration costs (e.g., letters, phone calls, visits, lawyers and legal work) incurred for individual i.
- Let ${R}_{i}$ be the amount recovered for individual i.

- A beta mixture model is parameterised by mean and precision based on two sets of predictor variables on the interval of (0, 1) in order to model the two modes located at just after 0 and around 0.55.
- A logistic regression model is used for the mode at boundary value 1.

## 2. Data

**Dataset****1**- provides 48 predictor variables of personal information including socio-demographic variables, Credit Bureau Score and debt status for 120,699 individuals for loans originating between January 1998 and May 2014 from several different financial institutions. Overall, 97.5% of them have credit card debt and only 2.5% are refinanced credit cards (product = “R”). Partial information was extracted from a Bad Debt Bureau. Each record corresponds to a bad loan and has a unique key Loan.Ref.
**Dataset****2**- records all the recoveries made by the bank before the debt collection company purchased the debt portfolio. It contains 15 predictor variables about historical collection information, which includes number of calls, contacts and visits made by the bank to collect the debt. It also includes repayments in the format of monthly summary. In total, there are 42,832 individuals’ records in Dataset 2, among which only 34,807 individuals can be matched to Dataset 1 by Loan.Ref. Numbers of calls, contacts, visits, repayment and some other monthly activities are aggregated by summing for each loan identified by Loan.Ref.
**Dataset****3**- records all the recoveries made by the debt collection company after they purchased the debt portfolio from the bank. It includes 12 predictor variables about the ongoing collection information. There are 8281 individuals in total, among which only 8237 individuals are from Dataset 1. Since only positive repayments are recorded, all the recovery rates we calculated are strictly greater than 0. Therefore, in the modelling section, we only focus on the recovery modelling in the interval (0, 1], which is slightly different from the usual RR defined in [0, 1]. The debt collection period recorded in this dataset is from January 2015 to end of November 2016.

#### Recovery Rate Calculation

## 3. Modelling Methodology

#### 3.1. Linear Regression with Lasso

#### 3.2. Multivariate Beta Regression

#### 3.3. Inflated Beta Regression

#### 3.4. Beta Mixture Model combined with Logistic Regression

#### Predictions Using the Beta Mixture Model

- Assign the new observation to the cluster that achieves the highest log-likelihood. This is a hard clustering approach, which assigns the observation to exactly one cluster (Fraley and Raftery. 2002).
- Assign the new observation to each cluster j with probability $P\left({M}_{j}\right)$. This is a soft clustering approach, which assigns the observation to a percentage weighted cluster (Leisch 2004).

**Approach 1: Maximum log-likelihood.**

**Approach 2: Prior Probability.**

## 4. Results

#### Model Performance

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IRB | Internal ratings based |

RR | Recovery rate |

LGD | Loss given default |

PD | Probability of default |

EAD | Exposure at default |

EM | Expectation-Maximisation (algorithm) |

MSE | Mean square error |

MAE | Mean absolute error |

MAAE | Mean absolute aggregate error |

## Appendix A

**Table A1.**Descriptive statistics. $n=8237$. For numeric variables: min, mean (standard deviation), max. For factors, frequency (%age) for each level. All predictive data is collected prior to servicing.

Variable | Type | Description | Statistics |
---|---|---|---|

RR post | numeric | Recovery rate (outcome variable) | 0.000508, 0.280 (0.283), 1 |

Product | factor | Type of loan | C:7468 (90.7%), R:769 (9.3%) |

Principal | numeric | Original loan amount | 0, 3120 (2330), 15000 |

Interest | numeric | Interest payments | 0, 551 (439), 3380 |

Insurance | numeric | Insurance fees | 0, 42 (84.6), 953 |

Late charges | numeric | Late charge fees | 0, 269 (109), 1470 |

Overlimit fees | numeric | Over credit limit fees | 0, 13.3 (24.6), 315 |

Creditlimit | numeric | Credit limit | 0, 4560 (2660), 13800 |

Sex | factor | Sex | F:3196 (38.8%), M:5041 (61.2%) |

Married | factor | Marriage status | 0:1201 (14.6%), D:518 (6.3%), M:3929 (47.7%), O:217 (2.6%), S:2230 (27.1%), W:142 (1.7%) |

Age | numeric | Age | 1, 48.7 (11.1), 87 |

DelphiScore | integer | Credit bureau score | 0, 298 (138), 443 |

Bureau Sub 1 | factor | Loan is in the servicer’s bureau (1 = True) | 0: 1520 (18.5%), 1: 6717 (81.5%) |

CustPaymentFreq | integer | Customer repayment frequency | 1, 7.56 (5.59), 29 |

Post Balance | numeric | Exposure amount at start of servicing | 0, 3130 (2630), 15900 |

Total paid amount | numeric | Total net paid amount | −275, 1200 (1100), 11200 |

Total calls | numeric | Total number of calls | 0, 104 (106), 911 |

Total contacts | numeric | Total number of contacts (except calls) | 0, 28.5 (26.5), 196 |

Bankreport Freq | numeric | Bank reporting frequency | 0, 11.6 (7.92), 26 |

Pre recovery rate | numeric | Recovery rate | −0.130, 0.258 (0.217), 2.89 |

Employer | factor | Employer known | EmployerProvided:8053 (97.8%), NoInfo:184 (2.2%) |

Total number | integer | Total number of loan accounts | 0, 2.3 (2.43), 68 |

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**Figure 1.**Histogram of recovery rates for 8237 loans after pre-preprocessing described in Section 2. The stack of 1s shows frequency of $RR=1$, but the stack at 0 shows frequency for small $RR>0$.

**Figure 3.**Beta distribution. (

**a**) Beta Distribution with Fixed $\varphi $; (

**b**) Beta Distribution with Fixed $\mu $.

**Figure 6.**$E\left(y\right|{M}_{j},{X}_{new})$ based on the Test dataset, for the two clusters ($m=2$). (

**a**) $E\left(y\right|{M}_{1},{X}_{new})$; (

**b**) $E\left(y\right|{M}_{2},{X}_{new})$.

**Figure 7.**Predicted RR on test data ($n=2746$) using beta mixture with four different priors, combined with logistic regression.

**Figure 8.**Predicted RR against true RR on test data ($n=2746$) using beta mixture with the indifferent prior, combined with logistic regression.

Approach 2 | $\mathit{P}\left({\mathit{M}}_{1}\right)$ | $\mathit{P}\left({\mathit{M}}_{2}\right)$ |
---|---|---|

${\pi}_{j}$ prior | Extract ${\pi}_{j}$ from the EM algorithm ${\pi}_{j}\left[1\right]$ | Extract ${\pi}_{j}$ from the EM algorithm ${\pi}_{j}\left[2\right]$ |

Prior based on training set cluster size ratio | $\frac{\mathrm{Cluster}\phantom{\rule{4.pt}{0ex}}1\phantom{\rule{4.pt}{0ex}}\mathrm{size}\phantom{\rule{4.pt}{0ex}}}{\mathrm{Total}\phantom{\rule{4.pt}{0ex}}\mathrm{sample}\phantom{\rule{4.pt}{0ex}}\mathrm{size}}$ | $\frac{\mathrm{Cluster}\phantom{\rule{4.pt}{0ex}}2\phantom{\rule{4.pt}{0ex}}\mathrm{size}}{\mathrm{Total}\phantom{\rule{4.pt}{0ex}}\mathrm{sample}\phantom{\rule{4.pt}{0ex}}\mathrm{size}}$ |

Indifferent Prior | $\frac{1}{2}$ | $\frac{1}{2}$ |

Variables | Beta Mixture Model in (0, 1) | Beta Regression in (0, 1) | ||||
---|---|---|---|---|---|---|

M1 Estimate | Pr(>|z|) | M2 Estimate | Pr(>|z|) | Betareg Estimate | Pr(>|z|) | |

$\mathit{\eta}$ | ||||||

(Intercept) | −0.67015 | <0.0001 | −2.62862 | <0.0001 | −1.80064 | <0.0001 |

Product R | −0.03376 | 0.47711 | −0.00766 | 0.59733 | 0.02270 | 0.41766 |

Principal | 0.00056 | NA | 0.00114 | NA | 0.00081 | 0.00000 |

Interest | 0.00065 | <0.0001 | 0.00118 | NA | 0.00097 | 0.00000 |

Insurance | 0.00082 | <0.0001 | 0.00116 | <0.0001 | 0.00086 | <0.0001 |

Late Charges | 0.00042 | 0.00578 | 0.00115 | <0.0001 | 0.00072 | <0.0001 |

Overlimit Fees | −0.00105 | 0.07594 | 0.00145 | <0.0001 | 0.00018 | 0.52533 |

Credit limit | 0.00004 | NA | −0.00001 | NA | −0.00003 | <0.0001 |

Sex = Male | 0.03659 | 0.17453 | −0.01412 | 0.13364 | 0.00969 | 0.43796 |

Marital status = | ||||||

Divorced | −0.01175 | 0.85305 | −0.01427 | 0.47359 | −0.03144 | 0.25840 |

Married | −0.06356 | 0.10819 | −0.01476 | 0.16836 | −0.03850 | 0.01957 |

Single | 0.00982 | 0.83178 | 0.00695 | 0.63324 | 0.00332 | 0.86926 |

Widow | −0.14627 | 0.19497 | 0.02311 | 0.51404 | −0.03869 | 0.45314 |

Other | 0.12328 | 0.17954 | −0.03476 | 0.22384 | 0.04570 | 0.24125 |

Age | −0.00273 | 0.05378 | −0.00038 | 0.42389 | −0.00115 | 0.07159 |

Credit Bureau Score | 0.00059 | 0.10337 | 0.00007 | 0.07890 | 0.00038 | 0.00222 |

Bureau bad debt | −0.32990 | 0.01290 | -0.06936 | <0.0001 | −0.24123 | 0.00000 |

Cust Payment Freq | 0.06530 | <0.0001 | 0.03506 | <0.0001 | 0.05046 | <0.0001 |

Post Balance | −0.00106 | NA | −0.00127 | NA | −0.00103 | 0.00000 |

Total Paid Amount | 0.00004 | NA | −0.00038 | NA | −0.00014 | <0.0001 |

Total Calls | −0.00044 | 0.00515 | −0.00023 | 0.00275 | −0.00032 | <0.0001 |

Total Contacts | −0.00136 | 0.03257 | 0.00040 | 0.08116 | −0.00031 | 0.28402 |

Bank report Freq | −0.01719 | <0.0001 | −0.00407 | <0.0001 | −0.01117 | <0.0001 |

Pre recovery Rate | 0.56850 | <0.0001 | 3.63447 | <0.0001 | 2.26212 | <0.0001 |

EmployerNoInfo | −0.04457 | 0.63820 | −0.01277 | 0.65487 | 0.03439 | 0.38375 |

Total Number | −0.00949 | 0.16151 | −0.00169 | 0.42951 | −0.00776 | 0.00651 |

$\mathit{\gamma}$ | ||||||

(Intercept) | 1.60514 | <0.0001 | 2.64737 | <0.0001 | 1.45450 | 0.00000 |

Pre recovery Rate | 0.49096 | 0.00025 | −2.11510 | <0.0001 | −0.18488 | 0.01538 |

Post Balance | 0.00039 | <0.0001 | 0.00018 | NA | 0.00031 | 0.00000 |

Cust Payment Freq | 0.02949 | <0.0001 | 0.17612 | <0.0001 | 0.07759 | 0.00000 |

Credit Bureau Score | −0.00058 | 0.00458 | −0.00033 | 0.09534 | −0.00028 | 0.01388 |

Model | MSE | MAE | MAAE |
---|---|---|---|

Linear Regression | |||

Linear regression | 0.024984 | 0.114268 | 0.025894 |

Stepwise linear regression | 0.024752 | 0.113621 | 0.025700 |

Linear regression with Lasso | 0.025228 | 0.114847 | 0.023739 |

Linear regression, excluding Dataset 2 | 0.026822 | 0.121385 | 0.026303 |

Beta regression | |||

Standard beta regression | 0.085630 | 0.260459 | 0.161366 |

Inflated beta regression | 0.076650 | 0.216374 | 0.048466 |

Beta mixture model combined with logistic regression | |||

Max log-likelihood | 0.018750 | 0.095432 | 0.030629 |

Prior based on R Flexmix ${\pi}_{j}$ | 0.018460 | 0.091833 | 0.023991 |

Prior based on training set cluster size ratio | 0.019325 | 0.092225 | 0.022594 |

Indifferent Prior | 0.018030 | 0.092399 | 0.026298 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ye, H.; Bellotti, A.
Modelling Recovery Rates for Non-Performing Loans. *Risks* **2019**, *7*, 19.
https://doi.org/10.3390/risks7010019

**AMA Style**

Ye H, Bellotti A.
Modelling Recovery Rates for Non-Performing Loans. *Risks*. 2019; 7(1):19.
https://doi.org/10.3390/risks7010019

**Chicago/Turabian Style**

Ye, Hui, and Anthony Bellotti.
2019. "Modelling Recovery Rates for Non-Performing Loans" *Risks* 7, no. 1: 19.
https://doi.org/10.3390/risks7010019