Mortgage forbearance: impact on the probability of default Mees Kruijer

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default

Mees Kruijer

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Supervisor RUG: Prof. dr. R.H. (Ruud) Koning

Supervisor RiskQuest: E.M.N. (Evert) de Beleir

Second Assessor: L. (Lingwei) Kong

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default

Mees Kruijer

January 6, 2021

Abstract

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1 Introduction

The Coronavirus, which started in 2019 in the city of Wuhan and spread over the whole world within 6 months, hurt the public health tremendously in 2020. It is considered the most crucial health disaster in decades and the greatest worldwide challenge since the 2nd World War. Nations introduced measures like travel bans, lock-downs, and social-distancing to overcome the spread of the virus and enhance public health. But where these measures helped to stop the spread of the virus, they plunged the economy. Thousands of firms expe-rienced among others diminishing turnover, export bans, a declining number of customers, with a resulting 31.4% decrease (with respect to the previous quarter) of the GDP and an unemployment rate of 13.0% in the United States in the second quarter of 20201_{. To help}

the population and enterprises, the US-government launched the Coronavirus, Relief, and Economic Security (CARES) Act2. This Act incorporates assistance for American workers, families, and small business, helps to preserve jobs in industries affected by the spread of the virus, and it provides help for local governments.

Imagine a group of people of which a small proportion will contract the Coronavirus. Ev-eryone who has symptoms related to the Coronavirus is treated with painkillers for a certain period and by giving the painkillers, the symptoms disappear. The treatment group is not medically examined during the treatment period, so it is unsure which patients of the treat-ment group contracted the Coronavirus and how the ones with symptoms physically develop. Furthermore, due to the lack of information about symptoms and corona, the prediction of the proportion of patients in this group will be difficult in the future. This process can be compared to the situation on the credit market in 2020, caused by a feature of the CARES Act. This feature provides the possibility for borrowers with a government-backed mortgage to request for forbearance for up to 180 days, starting in April 2020. In popular language, this is called a payment holiday.

A mortgage is a financial contract between a borrower and a lender and is used by the borrower to buy a property (the collateral). The lender and the borrower make agreements about the term, the level of interest, and the repayment period. The US mortgage market is one of the largest mortgage markets in the world; the total outstanding debt of households at the end of 2019 was 16.01 trillion3_{. Mortgages are provided by mortgage services as banks}

and other financial institutions. After that, most of the mortgages are sold to Government-sponsored enterprises (GSE’s) like Fannie Mae and Freddie Mac. These enterprises securitize the mortgages into mortgage-backed securities (MBS) and sell them to investors, like pension funds and insurance firms.

The payment holiday implies that if a borrower is not capable of paying his mortgage

pay-1_{https://tradingeconomics.com/united-states/} 2

https://home.treasury.gov/policy-issues/cares

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ments to his mortgage provider, he can request to reduce or pause his monthly payments for up to 6 months. The qualification for this regulation is simple and straightforward; no proof is needed, one just simply applies by stating that your financial situation is affected by the Corona crisis. These deferred payments can be compensated by three options, depending on the mortgage provider: lump-sum payment at the end of the term, extended-term, or increased payments over time.

The opportunity to defer payments can be supportive for many mortgage holders; they overcome their financial distress and will be able to fulfill their monthly obligation in the future. For instance, if someone who was fired in April 2020 did find a job in August 2020, this forbearance can be of great aid. For the other part of the group, this opportunity will not only result in deferred payments, but also in deferred financial distress, or even deferred default. To state briefly, the payment holiday could help a lot of people, but it also creates a problem; the payment holiday acts like a cloud hiding important information about the financial situation of the borrower.

This thesis addresses the following research question: What effect does the introduction of mortgage forbearance have on the probability of default of a mortgage portfolio on a one-year horizon?

To provide an answer to the research question, we follow three steps. In the first step, a logistic probability of default model (PD model) will be fitted on the US mortgage data from 2000 till 2019. This model predicts the probability of default for the portfolio in De-cember of each year for the following year. Both time consistent variables and time-varying variables are incorporated as risk drivers. Besides this, the payment history is also included as a risk driver, to examine the consequences of partly lacking this information due to the payment holiday.

Because the data for 2020 is not available yet, the required variables for each mortgage that are used in the PD model will be simulated for 2020 during the second step, as the focus of this thesis is on active mortgages in 2020. For the time-varying variables such as the housing price index, assumptions of official institutions will be used. The information about the payment structure in 2020 will be simulated using a Markov Chain. The delinquency status of the mortgages of December 2019 will be used as a starting point to simulate mort-gage statuses for all the months of 2020. Two different transition matrices will be used for the Markov Chain. For the first three months, a transition matrix based on ‘boom’ times is used. For the remaining part of 2020, a transition matrix based on the Global Financial Crisis is used, a period that shows resemblance to the period that started in April 2020.

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simu-lated status without payment holiday, the status could improve due to the payment holiday. The payment holiday that is used for analysis is a simplified payment holiday; it gives oblig-ors the opportunity to defer their interest payments and repayments during the period March 2020 - September 2020. To compare the results between a payment holiday and no payment holiday, different scenarios will be simulated.

In the last step, the results will be derived for the different scenarios. Firstly, using the results of the simulations, the probability of default for 2021 is predicted using the estimated logistic PD model. Secondly, mortgage statuses for 2021 are simulated, to determine the share of obligors that received support in 2020 and eventually defaulted after the holiday or in 2021. Finally, because the results are based on simulations using multiple assumptions, a broad sensitivity analysis is performed to test how the results vary when different assump-tions are used.

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2 Data

In this thesis, Fannie Mae’s Single-Family Historical Loan Performance Data Set4 _{is used.}

Fannie Mae, or Federal National Mortgage Association, is a government-sponsored enter-prise. It publishes this data set to provide a better understanding of the performance of these mortgages. Fannie Mae does not directly provide mortgage loans, but it purchases mortgage loans from mortgage services from all over the US. Fannie Mae packages the mort-gages loans and sells them to institutional investors like pension funds and insurance firms as mortgage-backed securities (MBS). Due to this process, mortgage services have access to capital. This increased liquidity could help to stabilize the mortgage market and decrease the level of interest rates5_{. In figure 2.1, this securitization is visualized.}

Figure 2.1: From mortgage to MBS.

The data set is a subset of Fannie Mae’s 30-year and less, fully amortizing, full documen-tation, single-family, conventional fixed-rate mortgages6_{. The data contains information of}

more than 35 million mortgages acquired between 2000 and 2019. For process convenience, the data set is split into quarters. For each mortgage, there is a monthly observation with the performance information for the months the mortgage was active. This monthly observation contains 108 variables, consisting of time-varying and time constant variables.

4_{https://capmrkt.fanniemae.com/portal/funding-the-market/data/loan-performance-data.}

html

5_{https://www.fhfa.gov/SupervisionRegulation/FannieMaeandFreddieMac}

6_{It does not contain information about adjustable-rate mortgage loans, balloon mortgage loans,}

interest-only mortgage loans, mortgage loans with prepayment penalties, government-insured mortgage loans, Home Affordable Refinance Program (HARP) mortgage loans, Refi PlusTM_{mortgage loans, or non-standard}

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The time constant variables are all known at origination and contain among others, the credit score (FICO), total debt, collateral value, and property type. The time-varying vari-ables change over time and are dependent on the actions of the borrower. These contain among others, the unpaid principal balance (UPB), the status of the mortgage (current, delinquent, prepayment, default), and the foreclosure costs at default.

The subset used in this study is randomly drawn, with an equal number of mortgages origi-nated each year; this to construct a diversified and representative subset of mortgages. If the subset would be constructed using an equal proportion each year, the subset will primarily contain mortgages originated in years in which many mortgage are originated (2002-2006), while the subset is used to estimate a predictive model for prediction of active mortgages in 2020. The subset contains monthly mortgage performance information up to and including 2019. At the start of this study, no data was available about the mortgages’ performance in 2020. Mortgages without a credit score, Loan-To-Value, or Loan-To-Income are removed from the data set and only fixed rate mortgages with an initial duration of 30 years are included. 30-year mortgages have on average higher interest rates than 15-year mortgages, but lower monthly payments. Consequently, a borrower’s financial situation with a 15-year mortgage could behave different to that of the financial situation of a borrower with a 30-year mortgage when a crisis appears.

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Figure 2.2: Historical Default Rate Figure 2.3: Historical Prepayment Rate

Figure 2.4: Historical Average Delinquency

The FICO-score, Loan-to-Value (LTV), interest rate, and Loan-to-Income (LTI) will be used as risk drivers in the PD model explained in section 3. The FICO-score is the credit score of the obligor, provided by the Fair Isaac Corporation7. The LTV is the ratio between the mortgage and the value of the collateral, and the LTI is the ratio between the mortgage and income. The distributions of these four variables at origination are plotted in figure A.1 till A.4 in the appendix. To give a first impression of the impact of these variables on the proba-bility of default, the historical yearly default rate of equally divided subgroups is calculated. The subgroups are based on the value of the FICO-score, LTV, interest rate, or LTI. In

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Figure 2.5: Historical default rate of four subgroups, based on the FICO score at origination.

Figure 2.6: Historical default rate of four sub-groups, based on the Loan-To-Value at origi-nation.

Figure 2.7: Historical default rate of

four subgroups, based on the interest rate at origination.

Figure 2.8: Historical default rate of four subgroups, based on the LTI at origination.

figure 2.5, the different default rates for the four subgroups based on the FICO-score groups are plotted. This graph shows a clear difference between the subgroups; this implies that the likelihood of default is lower when an obligor has a high FICO-score. Also for the subgroups based on Loan-To-Value and interest rate, the curves show a clear difference, which is shown in figures 2.6 and 2.7, respectively. The difference between the curve for the subgroups based on Loan-To-Income is small, which could imply that the LTI will not be a strong risk driver.

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Figure 2.9: The GDP-growth, Housing Price Index, and the unemployment rate for 2000-2019

This information is obtained from the Federal Reserve Bank of St. Louis 8. The three fac-tors show negative developments during and after the Global Financial Crisis, similar to the historical default rate. Furthermore, figure 2.9 shows that in 2019 these factors returned to pre-crisis values and that the economy was in boom times, with record-high housing prices, a low unemployment rate and strong GDP growth numbers. In short, at the end of 2019, everything was fine.

3 The Probability of default model

In this section, the theoretical framework for the estimation of the probability of default model (PD model) is explained. This PD model is estimated at the end of each year, and estimates the likelihood of default for each mortgage for the following year. In section 3.1, the model and its estimation method are stated. To assess the model’s predictive power, the Gini-Index and the KS-statistic are calculated as performance measures. Those will be discussed in section 3.2. In section 3.3, the risk drivers are proposed and previous literature concerning these risk drivers is discussed. The final model is summarized in section 3.4. The results of this model are discussed in section 3.5.

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3.1 PD model

The data for the PD model consists of a binary dependent variable, equalling 1 if a mortgage will default in the next year and 0 otherwise. This variable is notated by yit, where i

is the obligor and t the year. The independent variables, containing characteristics of a mortgage and macroeconomic factors, are captured in 3-dimensional matrix X, with row xit

the independent variables for mortgage i in year t. The goal is to find a model that predicts the probability of default as good as possible, given the characteristics of the mortgage. In this thesis, a binary choice regression model is used. Rather than modelling the response variable directly, this model describes the probability that the binary response variable yit

is equal to 1; in our case, the probability that a mortgage defaults in the following year. Summarized, a binary choice model models for each mortgage in each year:

pit = P r(yit = 1|xit) = f (xit, β) (3.1)

where xit is a vector containing the characteristics of obligor i at year t, and β the vector

with coefficients.

The logistic regression model, proposed by Cox (1958), is defined by

f (xit, β) =

1

1 + exp(−(β0+ xitβ))

(3.2)

The coefficients β are estimated using the maximum likelihood estimation. The log-likelihood of the logistic regression is

l(β; y, x) = n X i=1 T X t=1 yitln f (xit, β) + (1 − yit) ln(1 − f (xit, β)) (3.3)

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3.2 Performance Measures

Since the prediction of the probability of default is of interest, the model’s predictive power is essential. To examine the predictive power, the model is fitted on a training set. Thereafter, the probability of default is predicted for the test set and performance measures are calcu-lated. To construct the training and test set, the data set is partitioned into four subsets, of which one is the training set and one the test set. The training set consists of mortgages active in 2000-2015 and 70 % of the obligors. The test set is the part of data that has no matching obligors and no matching years with the training set. This process is visualized in figure A.5 in the Appendix.

Two performance measures are used to test the predictive power of the model. The first one is the Gini Index (Gastwirth, 1972), the second one is the KS-statistic (Fasano and Franceschini, 1987). The assessment for both statistics is not directly based on the proba-bility of default, but on the log odds, also known as the credit score in modelling default. The credit score equals

score = ln p 1 − p (3.4)

where p is the probability of default. In the logistic regression model, the score equals

score = β0+ xitβ (3.5)

which also shows the linear relationship mentioned in section 3.

Gini-index The Gini-index is the most common test statistic to appraise credit models. It is based on the Cumulative Accuracy Profile (CAP) curve, also known as the Life curve, the Power curve or the Dubbed curve (Řezáč and Řezáč, 2011). The CAP-curve captures how the defaulted mortgages can be separated from the non-defaulted mortgages, based on credit score of the mortgages. It shows how the conditional cumulative distribution of the score given default, F (score|yit = 1) relates to the cumulative distribution function of the score

F (score) (Sobhart et al., 2000). An example of a CAP-curve is visualized in figure 3.1. The proportion of defaults is on the vertical axis, the proportion of the ordered mortgages (from highest to lowest credit score) on the horizontal axis. The perfect model is a model that perfectly separates the defaulted from the non-defaulted, such that 100% of the defaulted mortgages are in the Pr(y = 1) part of the mortgages. The diagonal line corresponds to the CAP-curve of a random model, a model that randomly assigns default to a mortgage. The larger the area between the CAP-curve of the random model and the CAP-curve of the estimated model, the higher the Gini-index, i.e., the closer the Gini-index is to 1, the better the model:

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Figure 3.1: Visualization of the Gini-index. The perfect model in this example has a default rate of 20%.

KS-statistic The KS-statistic compares the conditional cumulative distribution function (CDF’s) of the credit score of the non-defaults F (score|yit = 0) and the cumulative

distri-bution function of the defaults F (score|yit = 1), where yit is the dependent binary variable.

If the model predicts well, the two CDF’s follow a different curve (Fasano and Franceschini, 1987). The KS-statistic is the maximum difference between the CDF’s at the same score

KS-statistic = max

score|F (score|y = 1) − F (score|y = 0)| (3.6)

The KS-statistic is between 0 and 1. The closer to 1, the better model and its predictive power.

3.3 Theoretical Framework

One of the main causes of the Global Financial Crisis was the dysfunctionality of the mort-gage market, fueled by the availability of subprime mortmort-gages. Borrowers with low credit score were able to buy mortgages of which the monthly obligations were too high for their income. These borrowers were teased by low interest rates, in the form of adjustable-rate mortgages9_{. Also, the value of the mortgage could be higher than the value of the collateral,}

with the expectation that the price could not decrease. This, in combination with declining housing prices and value decreasing MBS’s, led to the start of the Global Financial Crisis (Campbell and Cocco, 2011). Since then, the interest in understanding the housing market and the drivers of default has grown rapidly, especially in crisis times. The Global Finan-cial Crisis is seen as a crisis caused by the banks, while the Corona crisis could lead to a banking crisis; bankrupt enterprises and higher unemployment rates may lead to an increase in defaults and losses for banks. A default on a mortgage can be the consequence of many

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factors; those factors are known as risk drivers. In this section, risk drivers of default of previous research are explored, to compose a combination of risk-drivers that can predict the probability of default as good as possible.

A risk driver can be a time constant variable or a time-varying variable. A model con-sisting only of variables that do not vary over time, is called a ‘through the cycle’ (TTC) model (Gavalas and Syriopoulos, 2014). It is solely based on constant variables, which are known at the origination of the mortgage, and it does not take into account risk drivers that change over time. A model that does take into account changing risk drivers like macroe-conomic variables is called a ‘point-in-time’ (PIT) model (Gavalas and Syriopoulos, 2014). It predicts a different probability of default for different points in time and closely tracks the state of the economy. The disadvantage is that with varying circumstances, it is hard to predict the probability of default without making any assumptions. In this thesis, a PIT model with several time constant variables is used.

Another factor that could be of importance is the relation between the monthly obligation and the borrower’s level of income. The monthly obligation is determined by the interest rate and the mortgage size. Campbell and Cocco (2011) suggest that the Loan-To-Income (LTI) indirectly affects the probability of default, but less obvious as the LTV and usually combined with a high LTV. Furthermore, the LTI is observed at origination. A change in income (decline or unemployment) cannot be observed or predicted; the variable is not time-varying. That is the same case in this thesis; only the LTI at origination is available. The interest rate is the other factor that determines the monthly obligation. Logically, the higher the interest rate, the higher the monthly obligation, the higher the probability of delinquency and default. The data set used in this thesis only contains information about mortgage loans with fixed interest rates and hence the monthly obligations are deterministic.

In the US, the FICO score is the industry standard for consumers credit risk 10 and it is based on the credit history (for instance, student or car loans) of an obligor. The FICO is of great importance for applying for a loan; it determines the amount one can borrow, how long to repay, and the interest rate of the loan. A borrower with a high FICO score can save a significant amount of money due to a lower interest rate and lower service fees. The FICO score will be incorporated as a risk driver, as it is an estimation for the level of risk of a borrower11. Sengupta and Bhardwaj (2015) studied the stability of the FICO score; they concluded that indeed a low FICO score was related to high risk, and a high FICO score was related to low risk.

The most important risk driver in this thesis is the information about the payment structure of the borrower. The monthly reporting period in the data set contains valuable information about the financial status of a borrower each month. It serves as a monthly check of the

10

The FICO score was introduced by Fair Isaac Corporation. (https://www.fico.com/)

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financial health of the borrower. Marouani (2014) studied the relationship between delin-quency status and default rate of Small and Medium Enterprises. She states that historical payment behaviour can predict a default quite well and it makes the model more dependent on the current situation of the borrower than using financial ratios only. For this study, the information of the payment structure is divided into two parts. The first part concerns the information of the delinquent status at the end of the year, the moment of estimating the probability of default. If a borrower is not delinquent at the end of the year, this variable equals 0. In case of a delinquent obligor, this variable equals the number of months he or she is delinquent. A delinquent status of 1 corresponds to delinquency of 30 to 60 days. This continues to a status of 11, which corresponds to delinquency of 330 to 360 days. The delinquencies more than 360 days are binned in status 12. This is because the occurrences of the dimension higher than 360 are sparse since, in general, the obligor defaults before reach-ing this level of delinquency. The second part contains the information about the maximum delinquent status of the last year. This information is translated into 12 dummy variables.

To incorporate macroeconomic effects, three factors are included as risk driver. The first one is the unemployment rate. Employment is closely related to income and thus the ability to pay the monthly obligations. Gerardi et al. (2015) found that individual unemployment is a strong predictor for default. This could imply that the unemployment level could be a good predictor for the probability default on portfolio basis. The second factor is the Housing Price Index. The collateral of the mortgage is directly related to this index. When prices have increased, a borrower (in financial distress) could easily sell the house, repay the mortgage and use the profit for further housing (rent/buy), independently of his current Loan-To-Value. The GDP-growth is included to incorporate a general benchmark of the condition of the economy. All these macroeconomic factors have had extraordinary appear-ances in 2020. The unemployment rate and GDP-growth saw their largest increase in 2020 Q2, succeeded by its largest decline in 2020 Q312_{. The housing price index acts like nothing}

has happened this year and has grown for approximately 5% with respect to the previous year (at the end of 2020 Q3). This is probably caused by the huge housing shortage; the supply of houses have not met the demand for over 10 years, with a deficit of 2.5 million houses nowadays (Freddie Mac, 2018).

3.4 The Model

The model estimates the probability of default for the next year in the month December. December is chosen because the last available data at the start of this thesis is data from December 2019, so the most recent data available is incorporated. The monthly reporting data elaborated in section 2 is used to construct yearly data. The monthly indicator for default is used to construct a yearly binary variable that indicates whether a mortgage defaults in the next year. Mortgages that are in default in December are removed. The observations in the data set range from December 2000 to December 2018, as 2019 is the

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last year the defaults are known. Consequently, the data set consists of 618,665 yearly observations. In table 3.1, the risk drivers are summarized and their minimum, average and maximum values are shown.

Table 3.1: Risk drivers of default

Mean Min Max Description

Current LTV [%] 69.40 36 138 Original LTV corrected for repayments

and housing price index

LTI [%] 35.43 1 64 Loan-To-Income at origin

FICO-score 728.1 387 842 Credit score of the obligor

at origin

Interest Rate [%] 5.739 2 11 Interest rate at origin

Loan Age 37.85 0 238 Loan age in months at year end

Delinquent status year end * 0 12 Current delinquent status at year end.

1 equals 30-60 days, 12 equals 360 or more.

Delinquent 30-60 days 0.048 0 1 _{delinquent status of past year is 30-60 days}Dummy variable: 1 if maximum

Delinquent 60-120 days 0.0156 0 1 Dummy variable: 1 if maximum

delinquent status of past year 60-120 days

Delinquent 360+ days 0.0063 0 1 _{delinquent status of past year 360 days or more}Dummy variable: 1 if maximum Unemployment Rate [%] 6.120 3.892 9.608 National unemployment rate past year

Housing Price Index 160.3 104.8 202.6 National housing price index past year

(Jan 2000 = 100)

GDP growth [%] 3.832 −1.794 6.738 National GDP-growth past year, YOY

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3.5 Estimation results

This section contains the results of the logistic regression model. The coefficients of the variables of the model are shown in table 3.2. To assess the predictive power of this esti-mated model, the model is fitted on a training data set, consisting of 335,693 observations (≈ 50%). Consequently, this model is tested on a test data set consisting of 82,099 observa-tions (≈ 12%). This resulted in a Gini-index 0.91 and the KS-statistic of 0.86. Those values imply a highly predictive model. The within-sample prediction is plotted in A.6, which can be found in the Appendix.

The coefficients for the current LTV and the interest rate are positive, which means that a higher LTV or interest increases the likelihood of default. The coefficients for the Loan-To-Income and the Loan age are not significant.

The FICO-score coefficient is positive, which implies that a higher FICO-score results in a higher probability of default. It could be that when borrowers can take more risk because of their high FICO-score, i.e., they can borrow much, they take too much risk. Recall figure 2.5, which plotted the historical default rates for different groups based on the FICO-score. Based on this, and the fact that the Global Financial Crisis was fueled by the defaults of subprime mortgage13_{, this result seems somewhat odd. By estimating the probability of}

default solely based on the FICO-score, the coefficient turns out to be negative, i.e., a lower FICO-score predicts a higher probability, more in line with the figure. So, the FICO-score coefficient flips sign by including the other predictors, which could be caused by the multi-collinearity between the FICO-score and the other risk drivers.

The macroeconomic factors’ coefficients are all significant and explainable; a positive co-efficients for the unemployment rate and a negative coco-efficients for the Housing Price Index and the GDP-growth. The coefficients for the variables concerning the payment history are all extremely significant, implicating the importance of these variables for estimating the probability of default. Missing parts of this information in 2020 will have quite some conse-quence for predicting the default, which will be discussed later.

Now that the probability of default model is estimated, the next step is to predict the probability of default for 2021 by incorporating the possibility to defer payments. Before this can be completed, the variables that serve as risk drivers in the PD model have to be constructed for the year 2020, as the data for 2020 is not available in the data set. For the variables concerning the mortgage’s delinquent statuses, a simulation based on a Markov Chain will be used. The derivation of the data for 2020 will be discussed in the next section.

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Table 3.2: Results of the logistic regression model.

Estimate Std. Error z-value Pr(> |z|)

(Intercept) −11.5881 0.4912 −23.59 0.0000 Current LTV 2.4069 0.1665 14.46 0.0000 LTI −0.0024 0.0018 −1.31 0.1917 FICO-score 0.0024 0.0004 6.43 0.0000 Interest Rate 0.1267 0.0187 6.78 0.0000 Loan Age 0.0004 0.0007 0.54 0.5888

Delinquent status year end 0.5253 0.0225 23.31 0.0000

Delinquent 30-60 days 2.3286 0.1028 22.65 0.0000 Delinquent 60-120 days 3.5834 0.0983 36.44 0.0000 Delinquent 120-180 days 3.7903 0.1282 29.57 0.0000 Delinquent 180-240 days 3.2236 0.1682 19.16 0.0000 Delinquent 240-300 days 2.3047 0.2133 10.81 0.0000 Delinquent 300-360 days 1.5944 0.2583 6.17 0.0000 Delinquent 360+ days 0.8374 0.2828 2.96 0.0031 Unemployment Rate 0.0515 0.0161 3.20 0.0014

Housing Price Index −0.0034 0.0015 −2.21 0.0271

GDP growth −0.0224 0.0110 −2.04 0.0415

4 Simulating data for 2020

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4.1 Portfolio 2020

The portfolio for 2020 consists of 50,154 mortgages. The value of the portfolio (outstanding balance) is $ 9,956,778,871. The average age of the mortgages in the portfolio is 52 months and the average interest rate is 4.64%. In figure 4.1 till 4.4, the distributions of the current Loan-To-Value, interest rate, loan age, and FICO score are plotted. As shown in figure 4.3, the interest rates are significantly lower than the interest rates of all the mortgages used for model estimation, caused by the downward trend of the yield curve in the past 20 years. In table 4.1, the distribution of the delinquency status is summarized, showing that only 2.39% is delinquent at the start of 2020.

Table 4.1: Distribution of delinquency statuses portfolio 2019

0 1 2 3 4 5 6 7 8 9 10 11 12

Amount 48957 681 199 56 64 45 20 16 13 14 14 12 63 Proportion [%] 97.61 1.38 0.40 0.11 0.12 0.09 .04 0.03 0.03 0.03 0.03 0.02 0.12

Figure 4.1: FICO-score distribution of the portfolio at the end of 2019

Figure 4.2: Loan To Value distribution portfolio at the end of 2019

4.2 Time-varying variables

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Figure 4.3: Interest Rate distribution portfolio at the end of 2019.

Figure 4.4: Loan Age distribution portfolio at the end of 2019.

which is estimated at approximately an increase of 5% with respect to December 201914. The value of the loan age is derived by adding 12 months to the loan age.

For the unemployment level and GDP-growth, the forecasts of the International Monetary Fund are used. They expect the unemployment rate in 2020 in the US will increase to 6.9%, and the GDP will decrease by 4.3%15. These are unprecedented values, and certainly in this combination. This implies that the PD model is not estimated on similar values of variables we are experiencing today. Therefore, the predictive power of the PD model could be violated. As these three macroeconomic variables are not the only risk drivers and not the most important, we assume that using the estimated PD model for predicting the PD in 2021 is justified.

4.3 Simulating the payment structure

In this section, the simulation of the payment structure in 2020 is explored. The theoretical background of the Markov Chain is explained in section 4.3.1. Section 4.3.2 outlines previous research about the Markov Chain in credit risk modelling. The method concerning the use of two transition matrices and its corresponding assumption is outlined in section 4.3.3.

4.3.1 Theoretical Model

To simulate the delinquency status of each month in 2020, a Discrete-Time Markov Chain (Markov, 1960) is used. The Markov Chain is characterized by the memoryless property, also known as the Markov property. Let {Xn} be a Markov Chain, with Xn be the state of

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https://tradingeconomics.com/united-states/

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a mortgage in month n. The Markov property states that the distribution of the next state Xn+1 only depends on the current state Xn, and is independent of earlier states. Let the

set of possible states be defined by S = (s0, s1, ..., sm). Each step, the chain jumps from one

state to the next state. The transition probability of moving from state si to sj is noted by

pij and is equal to

pij = Pr(Xn+1 = sj|Xn= si) (4.1)

The transition probabilities of moving from one state to the other in month n are assembled in transition matrix P (n, n + 1). Let x(n) be a vector with the unconditional probabilities for each state for a mortgage. If the status in month n is known, x(n) is equal to a vector with one entry equal to one, and all the other entries equal zero. A forecast can be made for the next month n + 1, or as many months required. The forecast for n + 1 equals

x(n + 1) = x(n)P (n, n + 1) (4.2)

4.3.2 Literature review

The use of the Markov Chain is not frequently used in assessing the status of mortgages, it is more common in other departments of risk management. Credit rating agencies like Moody’s and Fitch publish credit ratings for enterprises and governments. Based on these historical ratings, a rating migration matrix can be constructed. This matrix contains all the probabilities of moving from one rating to another rating in the next period. One ma-jor problem of these rating migration matrices is that they are not constant over time and depend on the business cycle (Helwege and Kleiman, 1997). Bangia et al. (2000) divided the economy into two states and found differences in the distribution of defaults in credit portfolios. Nickell et al. (2000) found that transition matrices vary with the economy.

The first who proposed this method for mortgages’ credit behaviour, were Cyert et al. (1962). The idea behind this is that the mortgage’s status does not jump from current to default all of a sudden. The starting status is equal to current status, and if the borrower meets his obligations properly, the status of the mortgage the next month is still current. If he does not pay his monthly obligation for one month, the status changes to 30 to 60 days delinquent. The status of the mortgage might change every month, dependent on the ac-tions of the owner of the the mortgage. Frydman et al. (1985) tested the approach of Cyert et al. (1962) empirically and found that the assumptions of stationary and homogeneity of the Markov Chain are not satisfied; the transition matrix is not constant over time. Be-tancourt (1999) conducted a similar study using data from Freddie Mac16 _{and found that}

the predictive power of a Markov Chain with a constant transition matrix method is poor. Grimshaw and Alexander (2011) proposed two modifications to the estimation method of default of mortgage loans. The first one is to separate loans into different groups based on origination characteristics. This resulted in different transition matrices for different groups, but constant over time within groups. The second proposal incorporates the change of

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acteristics over time, such as the age of the loan and repayment history. This resulted in a transition matrix that changes over time.

4.3.3 The Model

The previous studies on using the Markov Chain for credit behaviour used the Markov Chain model to estimate the probability of default. This thesis uses this method to simulate the status of the mortgages of the months in 2020 and uses this information as input for the logistic regression model to estimate the probability of default for 2021. After that, the Markov Chain is used to simulate the mortgage status for the 12 months of 2021 to exam-ine which supported obligors eventually defaults after the payment holiday in 2020 or in 2021.

How to determine which transition matrix to use for which period? The determination of the transition matrices is essential; it determines a majority of the risk drivers of the PD model and determines the share of obligors that will make use of the payment holiday. A good method to determine the different transition matrices for 2020 would be to param-eterize the transition matrices by constructing a model that determines the time-varying transition probabilities based on several factors and characteristics. One disadvantage of this method is that the circumstances of the Corona crisis are unprecedented; calibrating an adequate transition matrix could lead to inconsistent results. In this thesis, a simplified approach is used; this approach uses the essential factor for being delinquent, namely a de-cline in income17_{. For business owners this is caused by diminishing turnover or bankruptcy,}

but for most people this is a consequence of being unemployed. In figure 4.5, the historical unemployment rate is plotted next to the average delinquency status per year. The two lines follow approximately the same path; this is confirmed by a correlation of 0.69.18

For the first three months of 2020, when business was booming and the Coronaviris had not reached the US, a transition matrix is calculated based on the active loans in 2015-2019. The unemployment rate in the first quarter of 2020 (3.8%) resembles the unemployment rate of 2015-2019. This matrix is hereafter referred as the ‘boom’ transition matrix. For the transition matrix for the second period of 2020, April till December, a transition matrix based on mortgage statuses during the Global Financial Crisis (2009-2010-2011) is used. The unemployment rate since April 2020 shows similar values as the unemployment rate during the Global Financial Crisis. Furthermore, by using this matrix, the dynamics of the obligors’ financial situation during a crisis is incorporated in the simulation. For instance, if an obligor was fired during the crisis and consequently became delinquent, this will be visible in the transition matrix. But if this obligor manages to find a job again, this is incorporated in the matrix. Therefore, this ‘corona’ transition matrix captures approximately all the possible

17_{Since only Fixed Rate Mortgages are considered, the monthly obligations are deterministic and a}

fluc-tuating interest rate does not change this.

18_{The correlation between the other macroeconomic factors Housing Price Index en GDP growth are}

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Figure 4.5: Historical unemployment rate and the historical average delinquency status.

migrations of an obligor’s financial situation during a crisis.

Combining the theory and the assumptions, the Markov Chain in this thesis is as follows. The Chain starts with a starting value equal to the delinquent status of the mortgage of December 2019. This starting value is noted as x(0). Let x(1), x(2), x(3) be the values for January, February, and March, and x(4), ..., x(12) the values from April till December. The set of possible states equals S = (s0, s1, ..., s12, sD, sP), where s0 is the current state, s1 to s12

are the delinquent states 1,2,...,12, sD is the default state, and sP is the prepayment state.

Let P1 be the matrix for the first three months of 2020 and P2 for the last nine months of

2020. A simplified example of a transition matrix is

Current Delinquent Default Prepayment       Current 0.9 0.05 0.01 0.04 Delinquent 0.5 0.45 0.03 0.02 Default 0 0 1 0 Prepayment 0 0 0 1

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Table 4.2: Simulating mortgages statuses for each month in 2020.

Starting Value Jan Feb Mar Apr ... Dec Notation x(0) x(1) x(2) x(3) x(4) ... x(12) Simulation - x(0) × P1 x(1) × P1 x(2) × P1 x(3) × P2 ... x(11) × P2

An example of the simulation can be found in table 4.3

Table 4.3: Example of the result of the simulation process. Note that this is not a represen-tative sample. Obligor ID Dec 2019 Jan 2020 Feb 2020 Mar 2020 Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sept 2020 Oct 2020 Nov 2020 Dec 2020 1 0 0 P P P P P P P P P P P 2 12 12 12 12 12 12 12 12 12 D D D D 3 1 2 3 4 4 5 6 1 1 1 0 0 0 4 10 11 12 D D D D D D D D D D 5 1 1 1 1 1 2 2 2 1 1 1 0 0 6 12 12 12 12 12 12 12 12 12 12 12 12 12 7 1 1 1 1 1 1 2 1 0 0 0 0 0 8 4 5 6 7 8 9 10 11 12 12 12 12 12 9 3 4 5 6 7 8 9 10 11 12 D D D 10 1 2 2 2 3 4 5 5 D D D D D

4.4 Transition matrices 2020

The estimated transition matrix for the first part of 2020, P1, can be found in table 4.4.

The matrix for the second part of 2020, P2, is plotted in table 4.5. The first difference

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‘boom’ matrix than in the ‘corona’ matrix.

Table 4.4: P1: the ’boom’ transition matrix. The rows correspond to the status in period n

and the columns the status in period n + 1.

Current 1 2 3 4 5 6 7 8 9 10 11 12 D P Current 0.982 0.007 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.011 1 0.392 0.449 0.146 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.012 2 0.140 0.193 0.375 0.278 0.002 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.011 3 0.086 0.040 0.126 0.295 0.434 0.004 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.014 4 0.085 0.015 0.019 0.052 0.240 0.579 0.001 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.009 5 0.105 0.015 0.011 0.014 0.033 0.243 0.560 0.002 0.001 0.000 0.000 0.000 0.000 0.005 0.010 6 0.098 0.016 0.008 0.007 0.012 0.036 0.159 0.632 0.004 0.000 0.001 0.000 0.001 0.013 0.015 7 0.090 0.014 0.007 0.001 0.004 0.005 0.026 0.180 0.641 0.002 0.002 0.000 0.000 0.010 0.018 8 0.074 0.011 0.001 0.004 0.000 0.004 0.009 0.037 0.176 0.656 0.000 0.000 0.001 0.013 0.013 9 0.064 0.012 0.005 0.002 0.002 0.002 0.007 0.005 0.033 0.172 0.659 0.000 0.003 0.017 0.016 10 0.066 0.007 0.000 0.004 0.004 0.002 0.004 0.004 0.007 0.033 0.135 0.702 0.004 0.022 0.004 11 0.042 0.011 0.000 0.003 0.003 0.000 0.000 0.003 0.000 0.005 0.024 0.137 0.707 0.042 0.024 12 0.027 0.004 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.001 0.000 0.002 0.901 0.050 0.010 D 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 P 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000

Table 4.5: P2: the ’corona’ transition matrix. The rows correspond to the status in period

n and the columns the status in period n + 1.

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5 Payment holidays

The possibility to defer payments in 2020 is the largest nationwide mortgage support measure in the mortgage market history. This section explores the implementation of a payment holiday from March 2020 till September 2020. The first question that arises is which obligors opt for the payment holiday, which is discussed in section 5.1. The second question is how to determine the mortgage status after the payment holiday for an obligor who participated in the mortgage forbearance, to construct adequate risk drivers to predict a probability of default for 2021. A simple method is to make use of the latest information available, i.e., the mortgage status just before the start of the payment holiday. In section 5.2, this method is explained, and a more advanced and optimistic one is proposed.

5.1 Which obligors make use of the payment holiday?

The CARES Act made it possible for mortgage owners to participate in the forbearance for up to 180 days19_{. This possibility is offered to everybody in need and no extra application}

has to be fulfilled. But which obligors opt for the opportunity to defer, and for how long? This is not a random choice, it is exogenous. In the real situation, the information asym-metry between the borrower and the lender leads to sample selection. The borrower knows his own (future) financial situation, while the lender only receives this information through the monthly payments. Borrowers with a positive job prospect probably will not use the payment holiday, whereas borrowers with an insecure employment perspective will use the payment holiday.

The CARES Act states that if a borrower uses the possibility of forbearance, this will not influence the borrower’s credit score (FICO). The FICO-score is important, since it deter-mines the terms of loans in the future. For instance, it is beneficial to take a car loan instead of paying cash in order to build a decent payment history to upgrade the FICO score and obtain more favourable terms for your mortgage loan in the future. So while applying for a payment holiday does not impact your FICO-score, negative credit history (delinquency) does. The more months a borrower is delinquent, the worse the effect on his credit score. Therefore, it seems reasonable to assume that everyone who is experiencing financial diffi-culties will apply for the payment holiday.

The duration of the holiday will be different for each borrower; this depends on the res-urrection of the financial situation. While the actual payment holiday option started in April and was still live in October, a simplified version is applied in this thesis: a payment holiday with a fixed duration: 180 days starting in April 2020. This means that if a bor-rower experiences difficulties in June, the forbearance applies for June, July, August, and September.

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5.2 The mortgage status after the payment holiday

Once an obligor has applied for the payment holiday, the question is how his financial sit-uation will develop during the payment holiday. Will he recover and be able to fulfill his monthly obligations again after the payment holiday? Or will the payment holiday lead to a postponement of financial distress, delinquency and even default? This information will be used as a starting value to simulate the mortgage statuses for the period after the payment holiday. These statuses will be simulated again using the ’corona’ transition matrix (P2),

explained in section 4.3.3.

The first method to incorporate the payment holiday in the mortgage statuses is to use the last information available on the mortgage status and assume this status did not change during the payment holiday. This method will be referred to as method 1. For example, suppose a borrower was delinquent 70 days before the introduction of the payment holiday (status 2). In that case, status 2 will be used to simulate the status of the mortgage for the remaining months of 2020 after the payment holiday. Applying this process assumes that borrowers will not benefit from the payment holiday, only that the status will not ex-acerbate. But the whole idea behind payment holidays is that a borrower in need could overcome financial distress. Hence, this option could be reasonable but is a bit pessimistic. The application of option 1 on the example of simulations of table 4.3 is displayed in table 5.1.

Table 5.1: Example of of an simulation with option 1 for incorporating the payment holiday. Note that this is not a representative sample.

Obligor ID Dec 2019 Jan 2020 Feb 2020 Mar 2020 Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sept 2020 Oct 2020 Nov 2020 Dec 2020 1 0 0 P P P P P P P P P P P 2 12 12 12 12 12 12 12 12 12 12 12 12 12 3 1 2 3 4 4 4 4 4 4 4 0 0 0 4 10 11 12 D D D D D D D D D D 5 1 1 1 1 1 1 1 1 1 1 1 0 0 6 12 12 12 12 12 12 12 12 12 12 12 12 12 7 1 1 1 1 1 1 1 1 1 1 0 0 0 8 4 5 6 7 7 7 7 7 7 7 6 7 8 9 3 4 5 6 6 6 6 6 6 6 7 8 9 10 1 2 2 2 2 2 2 2 2 2 1 0 0

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More specifically, it is assumed that every month during the payment holiday, the simulated delinquency status will be downgraded by 1 (one month). So if the borrower’s delinquency status would increase without payment holiday, the ‘holiday delinquent status’ remains the same (+1 − 1 = 0). If the the borrower’s delinquency status remains equal without payment holiday, the ‘holiday delinquent status’ decreases with 1 (+0 − 1 = −1). If the borrower’s delinquency status enhances without the holiday, the ‘holiday delinquent status’ decreases with 2 (−1 − 1 = −2). Since the information about delinquency status higher than a year (status 12) is pooled into one status, we assume that if the simulated status jumps from status 12 to status 12 without the payment holiday, the ‘holiday delinquent status’ does not increase or decrease. If the borrower would have defaulted without the payment holiday, he will not default with the payment holiday. If this is the case, the ‘holiday delinquent status’ does not increase or decrease. All the possible migrations and its application with payment holiday are shown in table 5.2.

Table 5.2: Possible migrations and its modifications of the simulation for option 2.

Simulation without holiday Migration size Payment holiday correction Payment holiday delinquent status Period 0 1 2 1 2 1 2 1 2 Simulation 1 2 3 4 +1 +1 −1 −1 2 2 Simulation 2 2 3 3 +1 0 −1 −1 2 1 Simulation 3 2 3 2 +1 −1 −1 −1 2 0 Simulation 4 11 12 12 +1 0 −1 0 11 11 Simulation 5 12 12 12 0 0 0 0 12 12 Simulation 6 11 10 9 −1 −1 −1 −1 9 7

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Table 5.3: Example of of a simulation with option 2 for incorporating the payment holiday. Note that this is not a representative sample.

Obligor ID Dec 2019 Jan 2020 Feb 2020 Mar 2020 Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sept 2020 Oct 2020 Nov 2020 Dec 2020 1 0 0 P P P P P P P P P P P 2 12 12 12 12 12 12 12 12 12 12 12 12 D 3 1 2 3 4 3 3 3 0 0 0 0 0 0 4 10 11 12 D D D D D D D D D D 5 1 1 1 1 0 0 0 0 0 0 0 1 0 6 12 12 12 12 12 12 12 12 12 12 12 12 12 7 1 1 1 1 1 0 0 0 0 0 0 0 0 8 4 5 6 7 7 7 7 7 7 7 8 8 9 9 3 4 5 6 6 6 6 6 6 6 7 8 9 10 1 2 2 2 2 2 2 2 2 2 3 2 0

6 Results

In this section, the results are discussed. Firstly, the probability of default for 2021 for the different scenarios is estimated and reviewed in section 6.1. To examine the sensitivity of the different assumptions used in this thesis, an extensive sensitivity analysis is performed in section 6.2.

6.1 Probability of default 2021

The results of the different scenarios and methods are compared in this section. For each scenario, the simulation is performed on the portfolio consisting of 50,154 mortgages. The simulation is replicated several times and it turns out that those replications produce very similar results as the results presented in this section. Therefore, we assume that the results presented below are close to consistent results. For the three scenarios, the same macroe-conomic factors and transition matrices will be used. The Housing Price Index is expected to grow by 5%, the GDP is expected to decrease by 3% and the unemployment rate is ex-pected to decrease to 8.9%. Transition matrices P1 and P2 showed in section 4.4 are used to

simulate mortgage statuses for the first and second part of 2020, respectively. Furthermore, a full participation of the financially distressed obligors in the payment holiday is assumed.

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holiday or in 2021, the mortgage statuses of the remaining mortgages at the end of 2020 are simulated for the 12 months of 2021 using the ’corona’ transition matrix. It turns out that 9% of the obligors who were financially distressed during the payment holiday period (which could have been supported), eventually default in 2020 or in 2021.

Payment holiday - option 1 The second scenario is the scenario with a payment holiday and option 1. This option uses the last information available before the holiday to simulate mortgage statuses for the period after the payment holiday. The default rate based on the simulation for 2020 is equal to 0.1%. The prepayment rate equals 15%. The prepaid and defaulted mortgages are not used in the prediction for default for 2021. The distribution of the delinquency statuses of the portfolio at December 2020 and the maximum delinquent statuses of 2020 of the portfolio for for scenarios 1 and 2 are plotted in table 6.1. This table shows that both the number and size of the delinquent statuses are lower for the second scenario compared to the first scenario. For instance, 776 borrowers (≈ 1.5%) have a maximum delinquent status of 2 in 2020 without a payment holiday, while only 410 borrowers (≈ 0.8%) experience this level of delinquency in 2020 with a payment holiday.

Table 6.1: Distribution of delinquency information of active mortgages in Dec 2020 of sce-nario 1 and 2.

0 1 2 3 4 5 6 7 8 9 10 11 12

No PH Del. Dec 2020 40413 879 299 172 126 108 72 77 55 34 36 23 107 No PH max. Del. 2020 37193 3239 776 296 202 170 109 104 63 49 41 29 130 PH opt 1 Del. Dec 2020 40912 806 240 145 83 42 43 35 28 13 14 8 72 PH opt 1 max. Del. 2020 39569 1829 410 202 109 65 58 39 33 19 15 11 82

Using the information of table 6.1 for the variables concerning the delinquency statuses in the PD model, the predicted probability of default of 2021 is estimated to be 0.32%. This implicates that the PD with a payment holiday is lower than the PD in the scenario without payment holiday (0.51%). Hence, offering the payment holiday and assuming option 1 yields in a PD that is approximately 40% lower. The mortgage statuses of the remaining mortgages are again simulated for 2021, using the ‘corona’ transition matrix. Now, 4% of the financially distressed obligors defaults after the payment holiday or in 2021.

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Furthermore, compared to scenario 2, the higher delinquency statuses occur, on average, less frequent. For instance, 202 mortgages have status 4 in December in scenario 2, while only 140 have status 4 in scenario 3.

Table 6.2: Distribution of delinquency information of active mortgages in December 2020 of scenarios 1 and 3.

0 1 2 3 4 5 6 7 8 9 10 11 12

No Holiday Del. Dec 2020 40413 879 299 172 126 108 72 77 55 34 36 23 107 No Holiday max Del. 2020 37193 3239 776 296 202 170 109 104 63 49 41 29 130 Holiday Opt 2 Del. Dec 2020 41281 757 174 48 25 22 17 14 11 10 11 1 59 Holiday Opt 2 max Del.2020 39576 1978 415 140 62 68 26 29 20 19 13 6 78

Using the information about the maximum delinquency status and the delinquency status at December 2020, the PD model predicts a probability in 2021 of 0.22%. This is again lower than the probability of default in 2021 of scenario 1 (0.51%), a difference of approximately 60%. The probability of 0.22% is also lower than PD of scenario 2 (0.32%). The mortgage statuses of the remaining mortgages are again simulated for 2021, using the ‘corona’ matrix. Using the simulation for 2021, a proportion of 2% of the supported borrowers eventually defaults after the payment holiday or in 2021. In table 6.3, the results for the three different scenarios are summarized.

Table 6.3: The results of the three simulated scenarios.

Scenario Defaults 2020 Supported

obligors Predicted PD 2021 Proportion of the supported group that defaults after holiday or in 2021* No payment holiday 0.2% - 0.51% 9%

Payment holiday - Option 1 0.1% 6.5% 0.32% 4%

Payment holiday - Option 2 0.07% 6.5% 0.22% 2%

*For the scenario without payment holiday, this is the proportion of the group that could have been helped during the holiday is.

6.2 Sensitivity analysis

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the participation share of financially distressed who will make use of the payment holiday is examined.

6.2.1 Macroeconomic factors

To examine the effect of variations the macroeconomic factors on the probability of default in 2021, four other (two better, two worse) values are used to predict the PD. The predicted PD changes as expected in each scenario, but the PD decrease due to the payment holiday remains significant for the different values for the macroeconomic factors, implicating that a payment holiday remains beneficial for the number of defaults. The results are shown in tables 6.4, 6.5, and 6.6.

Table 6.4: The probability of default in 2021 for the three scenarios for different values of the Housing Price Index, in %.

Scenario 0% 2.5% 5.0% 7.5% 10%

No holiday 0.56 0.52 0.51 0.49 0.46

Payment holiday - Option 1 0.36 0.34 0.32 0.31 0.30 Payment holiday - Option 2 0.25 0.24 0.22 0.21 0.21

Table 6.5: The probability of default in 2021 for the three scenarios for different values of the Unemployment rate, in %.

Scenario 11.9% 10.4% 8.9% 7.4% 5.9%

No holiday 0.55 0.54 0.51 0.48 0.46

Table 6.6: The probability of default in 2021 for the three scenarios for different values of the GDP-growth, in %.

Scenario -9.0% -6.0% -3.0% 0.0% 3.0%

No holiday 0.56 0.54 0.51 0.49 0.46

6.2.2 The use of the ‘corona’ transition matrix

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Therefore, the assumption of the transition matrix can be considered as one of the important assumption in this thesis and should be tested adequately, especially for the last nine months of 2020. First, the mortgage statuses for 2020 are all simulated using the ‘boom’ matrix (P1).

Different weighted average transition matrices of the ‘boom’ and ‘corona’ matrix are then calculated and used to construct the mortgage statuses for the last nine months of 2020. Using these mortgage statuses, the probability of default for 2021 is calculated. The results can be found in table 6.7. This table shows that the choice of the transition is matrix is essential for predicting the PD. If the last nine months’ transition probabilities are not as bad as assumed, the results vary a lot. For instance, if the equal-weighted matrix 0.5P1+ 0.5P2 is

used for the last nine months, the probability of default improves with only 14% for scenario 2, and 40% for scenario 3.

Table 6.7: The probability of default in 2021 for the three scenarios for different transition matrices for April till December 2020, in %.

Scenario P1 0.25P1+0.75P2% 0.5P1+0.5P2 0.75P1+0.25P2 P2

No holiday 0.28 0.33 0.35 0.45 0.51

Payment holiday - Option 1 0.25 0.27 0.30 0.31 0.32

Payment holiday - Option 2 0.17 0.19 0.20 0.21 0.22

6.2.3 Information asymmetries

In section 5.1, it was assumed that every obligor who experiences financial distress during the payment holiday will utilize the opportunity because of the benefit for their FICO-score. But before applying for the forbearance, the obligor needs to know about it. A survey by Fannie Mae shows that approximately 20% of the obligors in financial distress do not know about the option to defer payments20. Therefore, the assumption of 100% participation of the financial distressed obligors is adapted, and the corresponding PD are calculated. The result can be found in table 6.8.

Table 6.8: The probability of default in 2021 for the three scenarios for different assumptions about the participation rate of the payment holiday, in %.

Scenario 60% 70% 80% 90% 100%

No holiday 0.51 0.51 0.51 0.51 0.51

This table shows the importance of clear communications of supporting measures to the ones who need support. Suppose only 60% of the financially distressed people are aware of the

20

https://www.fanniemae.com/research-and-insights/perspectives/

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7 Conclusion

This thesis focuses on the introduction of a mortgage forbearance and the consequences for the probability of default. It finds that a mortgage forbearance approximately halves the predicted probability of default of a mortgage portfolio in the year following the forbearance, but still a small proportion of supported obligors defaults on the short term. The CARES Act and several other financial supporting programs in other countries including mortgage forbearance are still active, implicating the relevance of this thesis. Financial institutions could use this thesis to implement the different proposed methods and adapt to their situa-tion, and examine whether an introduction of a payment holiday will be beneficial for their mortgage portfolio on the short term.

First, a logistic regression model is used to determine the risk drivers of default. Secondly, a Markov Chain method is used to simulate mortgage status for 2020. Using two different methods, the mortgage forbearance of 180 days is incorporated in the simulated mortgage statuses. The first method uses the last information available of the mortgage status (before the payment holiday). The second method uses the simulated mortgage statuses during the period of the payment holiday and accounts for the fact that obligors without monthly obligations enhance their delinquency status. The actual data has to be examined, in order to determine which method is the most appropriate. Finally, the probability of default for 2021 without a payment holiday in 2021 is compared to the PD for 2021 with a payment holiday in 2021.

The results show that most of the independent variables of the estimated PD model are significant, with the payment history as the most crucial risk driver. The two transition matrices used for simulation show clear differences between the probabilities of moving to a higher or lower delinquent state, which emphasizes the difference between boom and cri-sis times. With full participation of financial distressed obligors in the payment holiday, approximately 6.5% of the obligors are supported by the payment holiday. Following the first method, the difference between the PD for 2021 with a payment holiday and without a payment holiday is approximately 40%. A proportion of 4% of the supported obligors eventually defaults after the holiday or in 2021. Following the second method, the difference between the probability of default is almost 60%. Of the supported obligors, 2% eventually defaults after the holiday or in 2021.

We conclude that the introduction of the mortgage forbearance is beneficial for the number of defaults on the short term.

7.1 Discussion and further research

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two options we used for incorporating a payment holiday resembles how the mortgage status actually developed after the payment holiday. Furthermore, in 2021 could be examined if the default rate decreased for mortgage portfolios of which the mortgage provider provided a payment holiday (compared to mortgages portfolios without a forbearance), i.e., if our findings will be confirmed.

Another interesting topic for further research is the change of risk in the long term. As the deferred payments have to be repaid, one way or another, supported obligors might ex-perience future financial distress, but for now it is unsure whether this increase in risk in the long term will outweigh the decreased risk on the short term.

One of the drawbacks of this thesis is the use of the transition matrix of the Global Financial Crisis as transition matrix for corona period of 2020. An important difference between the periods is the Housing Price Index, which could imply a different probability of moving to the prepayment status. The Housing Price Index in 2020 is still growing, while this was diminishing during the Global Financial crises. This means that that people can make a profit on their mortgage by selling the house and prepay the mortgage, instead of default-ing. Secondly, the interest rate is much lower in 2020 than during the Global Financial Crisis.

In this thesis, we estimated the logistic regression model as PD model, and used this model to predict the PD for 2021. As some values of the variables of 2020 are extraordinary, cali-brating the model for 2021 could lead an inconsistent estimate. In this case, a neural network model could deliver more consistent predictions.

An improvement could be to increase the number of replications of the simulation, and derive a more consistent result. The perfect simulation method would be to use the Monte Carlo method with 10,000 simulations. Using that approach, it would be possible to con-struct strong confidence intervals.

To make the model more time-varying, the probability of default model could be estimated on for a shorter time frame, quarterly or even monthly. As we have seen this year, a lot can happen in a year. By using a shorter time frame, the model would be more up to date and adapted to the current situation.

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A

Appendix

A.1 Distribution at origination

Figure A.1: Distribution of the FICO-scores at origin.

Figure A.2: Distribution of the Loan-To-Value at origin.

Figure A.3: Distribution of the Interest Rate at origin.

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A.2 Construction of the training and test set

Figure A.5: The training data set and the test data set. The training set will approximately contain 50 % of the observations, and the test set 10 %.

A.3 Within-sample prediction of the estimated PD model

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A.4 Markov Chain Simulation

To simulate a vector with one state equal to one, the cumulative transition matrix is multi-plied by 10000. Then a random number is picked in the range of 1 till 10000, and the first entry of the row of the current state of the cumulative transition matrix multiplied by 10000 that exceeds this random drawn number, will be the state of the next period. The simplified example multiplied by 10000 is

Current Delinquent Default Prepayment           Current 9000 9500 9600 10000 Delinquent 5000 9500 9800 10000 Default 0 0 10000 10000 Prepayment 0 0 0 10000

Mortgage forbearance: impact on the probability of default Mees Kruijer

default

Mees Kruijer

Supervisor RUG: Prof. dr. R.H. (Ruud) Koning

Supervisor RiskQuest: E.M.N. (Evert) de Beleir

Second Assessor: L. (Lingwei) Kong

default

Mees Kruijer

Contents

1

Introduction

2

Data

3

The Probability of default model

3.1

PD model

3.2

Performance Measures

3.3

Theoretical Framework

3.4

The Model

3.5

Estimation results

4

Simulating data for 2020

4.1

Portfolio 2020

4.2

Time-varying variables

4.3

Simulating the payment structure

4.4

Transition matrices 2020

5

Payment holidays

5.1

Which obligors make use of the payment holiday?

5.2

The mortgage status after the payment holiday

6

Results

6.1

Probability of default 2021

6.2

Sensitivity analysis

7

Conclusion

7.1

Discussion and further research

Bibliography

A

Appendix

A.1

Distribution at origination

A.2

Construction of the training and test set

A.3

Within-sample prediction of the estimated PD model

A.4

Markov Chain Simulation