# A Multi-State Approach to Modelling Intermediate Events and Multiple Mortgage Loan Outcomes

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Modelling Intermediate Events and Multiple Loan Outcomes

#### 3.1. Competing Risks

#### 3.2. Multi-State Models

## 4. Statistical Analysis and Results

#### 4.1. Data

#### 4.2. Conceptual Framework

#### 4.3. Results and Discussion

#### 4.3.1. Descriptive Statistics

#### 4.3.2. Transition Matrix

#### 4.3.3. Cumulative Incidence Functions

#### 4.3.4. Prognostic Factors for Event Specific Transitions

#### Current to Prepayment Transitions

#### Current to Delinquency and Default Transitions

#### Delinquent to Current Transition (Cure or Recovery)

#### Default to Current Transition (Cure or Recovery)

#### 4.4. Model Validation

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | The recently approved BCBS (Basel IV) reforms of the standardised (CR-SA) approach (by making it more granular and risk sensitive) and of the CR-IRB approach for the calculation of risk weighted assets for credit risk will limit the extent to which banks can reduce capital requirements through the use of internal models (e.g., by eliminating the option to use any IRB approach for equity and advanced CR-IRB for institutions and large corporations). |

2 | The early repayment of a mortgage significantly impacts both the bank’s profitability, via reduced interest margins, the loss of foregone interest payments (less the risk of a default on the outstanding debt) and the bank’s interest rate and liquidity positions. Prepayment creates a reinvestment risk problem for banks that must be addressed by appropriate Asset–Liability Management (e.g., immunization) techniques (see, e.g., Bravo and Silva 2006). |

**Figure 1.**A competing risks scenario with three causes of failure. Notes: This is just for illustration reasons as there are many other possible causes of failure or transition from normal performance (current).

**Figure 2.**A multi-state model framework for analysing mortgage loans data. Source: Author’s preparation. Notes: The numbers on the arrows represent the direction of the transition. For example, 12 represents the transition from State 1 (current) to State 2 (delinquency). Similarly, number 37 means transition from State 3 (default) to State 7 (foreclosure and Dil) and so on.

Acquisition Variables | Description/Values |
---|---|

Loan Identifier | Mortgage loan unique identifier. |

Origination Date | MM/YYYY |

Loan Purpose | An indicator that denotes if a mortgage loan in a pool is either a purchase money mortgage or refinance mortgage. |

Product Type | Fixed-rate mortgage or adjustable-rate mortgage. |

Property Type | Single-Family (SF), Condo (CO), Co-Op (CP), Manufactured Housing (MH), PUD (PU) |

Relocation Mortgage Indicator | Yes/No |

Channel/ Origination Type | Retail (R), broker (B), correspondent (C) |

Borrower Credit Score | 300–850 |

Co-Borrower Credit Score | 300–850 |

Debt-to-Income (DTI) Ratio | Calculated at origination and is obtained by dividing the borrower’s total monthly obligations by stable monthly income. |

First-Time Homebuyer Indicator | Denotes if a borrower is a first-time homebuyer. |

Number of Borrowers | Number of individuals who are obligated to repay the loan. |

Number of Units | The number of units the mortgaged property has. |

Occupancy Status | Indicates how the borrower used the mortgaged property at the time of origination. |

Original Interest Rate | Mortgage original interest rate |

Original Loan Term | The number of months for which a borrower’s payments are due |

Original Loan-to-Value (LTV) | The Original LTV reflects the loan-to-value ratio of the loan amount secured by a mortgaged property on the origination date of the underlying mortgage loan. |

Original Unpaid Principal Balance (UPB) | The original amount of the mortgage loan as indicated by the mortgage documents. |

**Source:**Fannie Mae.

Performance Variables | Description/Values |
---|---|

Current Loan/Delinquency Status | The number of days (months) a borrower is delinquent, i.e., 0 = Current, or < 30 days overdue; 1 = 30–59 days; 2 = 60–89; 3 = 90–119; and so on. |

Modification Flag | Y = Yes, N = No |

Monthly Reporting Period | MM/DD/YYYY |

Loan Age | Number of months after origination |

Zero balance code | Reason mortgage loan’s balance was reduced to zero, e.g., 01= prepaid; 03 = Short Sale; and so on. |

**Source:**Fannie Mae.

State | Description |
---|---|

Current/normal performance | This is when a borrower is up to date with payments or overdue by less than 30 days. |

Delinquency | When payments are 30–59 days overdue. |

Default | This is when a borrower missed payments for 60 days or more consecutively. |

Prepayment | Occurs when a loan is paid in a shorter period than agreed contractually. |

Mortgage foreclosure | Occurs when a borrower fails to pay in time or in full instalments and the lender repossesses the property. |

Deed-in-Lieu, REO Disposition | This is when the borrower seeks release from the mortgage contract by voluntarily transferring the title of the property to the lender. |

Short sale | This is when a homeowner sells a home for less than the balance remaining on a mortgage and pays off all (or a portion of) mortgage balance with the proceeds. |

Recovery or cure | This is when a borrower once in delinquency or default resumes making payments. |

Class of Variables | Variable | Category | Percent |
---|---|---|---|

Loan and borrower characteristics | Channel | Broker | 13.6 |

Correspondent | 31.8 | ||

Retail | 54.6 | ||

First Time Homebuyer Indicator | Yes | 5.3 | |

No | 94.7 | ||

Property characteristics and purpose | Loan Purpose | Purchase | 15.3 |

No Cash-out Refinance | 53.4 | ||

Cash-out Refinance | 31.3 | ||

Occupancy Status | Principal | 93.2 | |

Second | 4.0 | ||

Investor | 2.9 | ||

Property Type | Condo | 6.8 | |

Co-op | 0.4 | ||

Manufactured Housing | 0.2 | ||

Planned Urban Development | 19.5 | ||

Single Family | 73.1 | ||

Number of Units | 1 | 98.7 | |

2 | 1.1 | ||

3 | 0.1 | ||

4 | 0.1 | ||

Behavioural variables | Relocation | Yes | 0.2 |

No | 99.8 | ||

Modification Flag | Yes | 0.3 | |

No | 99.7 |

**Source:**Author’s preparation.

Variable | Min | 1st Quantile | Median | Mean | 3rd Quartile | Max | Std Dev |
---|---|---|---|---|---|---|---|

Borrower characteristics | |||||||

Borrowers’ Credit Score | 508.0 | 741.0 | 773.0 | 763.0 | 793.0 | 850.0 | 40.06 |

Co-borrower’s Credit Score | 505.0 | 751.0 | 779.0 | 769.3 | 797.0 | 850.0 | 36.99 |

Debt-to-Income Ratio | 1 | 24 | 32 | 33.08 | 42.0 | 64.0 | 11.81 |

Number of Borrowers | 1.0 | 1.0 | 2.0 | 1.61 | 2.0 | 7.0 | 0.498 |

Loan characteristics | |||||||

Loan-to-Value | 3.0 | 60.0 | 73.0 | 68.77 | 80.0 | 97.0 | 15.65 |

Original Loan Term (months) | 301 | 360 | 360 | 359.9 | 360 | 360 | 1.88 |

Unpaid Principal Balance | 10,000 | 147,000 | 215,000 | 235,223.13 | 308,000 | 950,000 | 113,370.64 |

Original Interest Rate | 1.88 | 4.75 | 4.875 | 4.98 | 5.125 | 8.625 | 0.361 |

Behavioural variables | |||||||

Loan Age at estimation (months) | 0 | 30 | 41.0 | 45.77 | 58.0 | 92.0 | 24.63 |

Number of transitions | 0 | 1.0 | 1.0 | 1.27 | 1 | 63.0 | 2.156 |

Sojourned time (months) | 1.0 | 26 | 40.0 | 43.61 | 54 | 92.0 | 25.96 |

**Source:**Author’s preparation.

To | ||||||||
---|---|---|---|---|---|---|---|---|

Current | Delinquency | Default | Repurchase | Prepayment | Foreclosure and Dil | Short Sale | ||

Current | 0.116 | 0.169 | 0.0 | 0.001 | 0.714 | 0.0 | 0.0 | |

From | Delinquency | 0.706 | 0.0 | 0.271 | 0.0 | 0.023 | 0.0 | 0.0 |

Default | 0.749 | 0.0 | 0.0 | 0.011 | 0.054 | 0.127 | 0.06 |

**Source:**Author’s estimation.

Current to Prepaid Transitions (1) | Current to Delinquent/Default (2) | Delinquent to Current (3) | Default to Current (4) | |
---|---|---|---|---|

Covariate | HR [Estimate] (SE) | HR [Estimate] (SE) | HR [Estimate] (SE) | HR [Estimate] (SE) |

Modification Y | 2.37 [0.863 ***] (0.0145) | 5.57 [1.72 ***] (0.0299) | Omitted | Omitted |

Purpose P | 0.979 [−0.0208 ***] 0.00778 | 0.819 [−0.199 ***] (0.0136) | 1.16 [0.152 ***] (0.0147) | 1.36 [0.305 ***] (0.0367) |

Purpose R | 1.08 [0.08 ***] (0.00435 | 0.852 [−0.16 ***] (0.0136) | 1.06 [0.0564 ***] (0.00983) | 1.03 [0.0281] (0.0245) |

Property Type CP | 0.78 [−0.249 ***] (0.0517) | 1.13 [0.123] (0.147) | 1.23 [0.207 ***] (0.0862) | 0.989 [−0.0112] (0.228) |

Property Type MH | 0.654 [−0.424 ***] (0.0408) | 0.938 [−0.0636] (0.110) | 1.04 [0.0346] (0.0861) | 1.02 [0.0199] (0.223) |

Property Type PU | 1.10 [0.0988 ***] (0.00111) | 0.993 [−0.00682] (0.0372) | 0.95 [0.0515 **] (0.0212) | 1.27 [0.236 ***] (0.0535) |

Property Type SF | 1.12 [0.112 ***] (0.0104 | 1.21 [0.190 ***] (0.0343) | 0.927 [−0.0762 ***] (0.0188) | 1.27 [0.242 ***] (0.0417) |

Relocation Y | 1.29 [0.253 ***] (0.0442) | 0.952 [−0.0495] (0.135) | 1.08 [0.0773] (0.106) | 1.83 [0.604 ***] (0.216) |

Occupancy P | 1.49 [0.399 ***] (0.0128) | 1.10 [0.094 ***] (0.0335) | 0.818 [−0.201] (0.0216) | 1.04 [0.0404] (0.0573) |

Occupancy S | 1.32 [0.274 ***] (0.0155) | 1.02 [0.0218] (0.0441) | 0.925 [−0.0783 **] (0.0491) | 1.01 [0.00872] (0.0832) |

First time homebuyer U | 0.575 [−0.553 *] (0.2.36) | 0.92 [−0.11] (0.0113) | Omitted | Omitted |

First time homebuyer Y | 0.901 [−0.104 ***] (0.0124) | 0.791 [−0.234 ***] (0.0355) | 0.985 [−0.0172] (0.0202) | 0.927 [−0.0753] (0.0475) |

Units | 0.753 [−0.283 ***] (0.0143) | 0.854 [−0.1.58 ***] (0.0361) | 0.988 [−0.0116] (0.0216) | 0.828 [−0.189 ***] (0.0542) |

Channel C | 1.03 [0.0267 ***] (0.00635) | 0.897 [−0.109 ***] (0.0192) | 0.978 [−0.0219 *] (0.0129) | 1.03 [0.0266] (0.0315) |

Channel R | 0.911 [−0.0933 ***] (0.00587) | 0.785 [−0.242 ***] (0.0179) | 0.941 [−0.0861 ***] (0.0123) | 0.972 [−0.0284] (0.0303) |

Co-borrower credit score | 0.998 [−0.00163 ***] (0.0000712) | 0.993 [−0.00714 ***] (0.000197) | Omitted | Omitted |

Borrower credit score | 0.998 [−0.00174 ***] (0.0000706) | 0.989 [−0.0112 ***] (0.000199) | 1.0 1 [0.00202 ***] (0.0000895) | 1.03 [0.00174 ***] (0.000219) |

DTI | 0.999 [−0.000439 *] (0.000175) | 1.02 [0.0214 ***] (0.000558) | 0.992 [−0.0081 ***] (0.00038) | 0.996 [−0.00415 ***] (0.000983) |

Number of borrowers | 0.905 [−0.0999 ***] (0.0181) | 0.982 [−0.0186] (0.0518) | 1.04 [0.0378 ***] (0.00865) | 1.09 [0.0846 ***] (0.0219) |

Loan-to-Value (LTV) | 0.996 [−0.00358 ***] (0.000128) | 1.01 [0.012 ***] (0.000462) | 0.994 (−0.0059 ***) (0.000317) | 0.991 [−0.0094 ***] (0.00147) |

Loan term | 0.998 [−0.00234 *] (0.000977) | 1.03 [0.0294 ***] (0.00525) | 1.0 [0.000631] (0.00407) | 0.992 [−0.00842] (0.0091) |

Original principal balance | 1.01 [0.0000019 ***] (0.0000000177) | 1.01 [6.62e−07 ***] (0.0000000589) | 1.0 [−0.000000001] (0.000000067) | 1.01 [−0.000001 ***] (0.000000018) |

Original interest rate | 1.90 [0.644 ***] (0.00551) | 1.48 [0.39 ***] (0.048) | 0.868 [−0.141 ***] (0.00998) | 0.804 [−0.294 ***] (0.0401) |

**Notes**: HR represent hazard ratios and SE stands for standard errors. A variable is omitted in the model when it does not carry enough data to run that model.

Model | AUROC Curve | Standard Error | 95% Confidence Interval |
---|---|---|---|

Current to Prepaid | 0.7830 | 0.0021 | 0.77876–0.78718 |

Current to Delinquent | 0.7968 | 0.0015 | 0.79380–0.79980 |

Delinquent to Current | 0.6913 | 0.0083 | 0.68359–0.71604 |

Default to Current | 0.7110 | 0.0024 | 0.70639–0.71569 |

© 2020 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**

Chamboko, R.; Bravo, J.M. A Multi-State Approach to Modelling Intermediate Events and Multiple Mortgage Loan Outcomes. *Risks* **2020**, *8*, 64.
https://doi.org/10.3390/risks8020064

**AMA Style**

Chamboko R, Bravo JM. A Multi-State Approach to Modelling Intermediate Events and Multiple Mortgage Loan Outcomes. *Risks*. 2020; 8(2):64.
https://doi.org/10.3390/risks8020064

**Chicago/Turabian Style**

Chamboko, Richard, and Jorge Miguel Bravo. 2020. "A Multi-State Approach to Modelling Intermediate Events and Multiple Mortgage Loan Outcomes" *Risks* 8, no. 2: 64.
https://doi.org/10.3390/risks8020064