Credit risk assessment of mortgage portfolios: considering macroeconomic information as well

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Credit risk assessment of mortgage portfolios:

considering macroeconomic information as well

M.R. van den Broek

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies Specialization: Actuarial Studies

University of Groningen

Supervisors:

Prof. Dr. T.K. Dijkstra (RuG) Drs. R. Spreij (PwC)

Co-assessor:

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Credit risk assessment of mortgage portfolios:

considering macroeconomic information as well

Mark van den Broek

August 21, 2015

Abstract

When banks sell Residential Mortgage-Backed Securities (RMBS), es-sentially portfolios of mortgage loans, they transfer the credit risk of the mortgages to the buyers. A proper assessment of this risk is of great importance to both seller and buyer.

The estimates that credit rating agencies supplied turned out to be of poor quality in the build-up of the financial crisis of 2007-8. To help assess default rates Central Banks now oblige sellers of RMBS portfo-lios to provide loan level data. But generally credit risk assessments fail to take macroeconomic information into account. This paper remedies that: Default rates on Dutch RMBS portfolios are estimated using tra-ditional loan-to-value and debt-to-income data as well as macroeconomic variables as unemployment and divorce rates. With Cox proportional haz-ards model the latter are shown to have a relatively large impact on default rates. A proper credit risk assessment can and should take macroeconomic variables into account.

JEL Classification: C01; C14; C23; C24; G12; G21; G33

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1 Introduction

During the last decade the market for mortgage loans has experienced an in-crease in the level of regulations. Triggered by the collapse of the US real estate market in 2007, institutions try to reduce credit risk on mortgage loans. In general mortgage markets are substantial in size. In the Netherlands the outstanding balance of residential loans was 632 billion euros in 2013 (EMF,

2014), which is slightly more than that year’s gross domestic product (GDP) of 603 billion euros (CBS, 2015). It is widely believed that the global crisis was largely triggered by incorrect valuation of mortgage-backed securities (Manola and Uroˇsevi´c, 2010).

From the perspective of a bank mortgage loans are an asset. Either the borrower pays back the mortgage or the bank exercises its right to sell the col-lateral, which is the house in case of residential mortgages. Banks can securitize part of their assets and give out bonds or notes. The payments on these bonds or notes depend on the underlying assets. This type of bonds or notes are called Asset-Backed Securities (ABS). In principal any asset with cash flows can be securitized. In this paper we will focus on a subset of ABS, namely Residential Mortgage-Backed Securities (RMBS). Banks can design a portfolio filled with a certain combination of mortgage loans, which forms the underlying pool of assets for the RMBS notes.

By selling the RMBS notes banks bring future cash flows to the present. In exchange the rights on the borrower’s payments and on the collateral are transferred to the buyers of the notes. Essentially banks transfer the credit risk of the loans in the portfolio to the buyers. Typically banks sell RMBS portfolios to investors, for example pension funds or life insurance companies. RMBS investments are attractive to them due to the longer maturity cash flows to match their longer liability profile, furthermore they have an appetite for less liquid assets in order to take advantage of the additional yield (NAIC, 2011). For valuation purposes it is of great importance to both seller and buyer to quantify the credit risk of an RMBS portfolio.

RMBS notes are divided into different levels of seniority. Payments on the mortgage loans, both interest and principal payments, are allocated by means of a ‘waterfall’ (or sequential) structure. That is, the most senior notes get paid their share first, then the second most senior notes and so forth. In other words, losses are first absorbed by the most junior notes. Only if the losses are greater than the total balance of the most junior notes, the second most junior notes will be harmed by further losses. The level of seniority is sometimes denoted by A up to E, A being the most senior notes and E the most junior notes. These names should not be confused with the risk score assigned by credit rating agencies (e.g. AA+). Nevertheless it holds that the A notes are the least risky followed by the B notes up to the E notes, assuming the notes are backed by the same assets.

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newly created A notes are not comparable with the original A notes. These new A notes are in fact ‘E-A’ notes, that is, within the original E notes the A notes are first in line. For such a clear re-securitization you could find a name which reflects the actual position. However, one could make up very complex combinations of securitizations of different notes. In that case it is not possible to give them an appropriate name. In practice notes are labeled in a way such that the label only indicates the level of seniority on the underlying assets, not necessarily the original assets. After a couple of consecutive securitizations it is not clear anymore what the actual risk on the underlying assets is. The mortgage crisis in the US has proven this to be very harmful to economies of entire countries.

On the asset side it is crucial to have realistic insights about the credit risk on the portfolio of mortgage loans. The performance of the loans determines the returns on the notes. Traditionally, credit rating agencies have made esti-mations on this credit risk. However, during the crisis they turned out to be not that adequate. This gave rise to all types of questions. In response, among other things, the European Central Bank (ECB) introduced the obligation for sellers of RMBS portfolios to provide loan level data. Then for each loan in the portfolio, all loan specific variables are observable over time. Note that due to the set-up of regular mortgages, some variables (e.g. income) are only measured at origination. Other variables like the current outstanding balance, current in-terest rate and the account status are updated every month. The availability of loan level data can increase the transparency in actual risks. Since both de-faulted and performing loans can be observed, it is possible to identify to what extend certain variables affect default rates. In this paper we will use survival analysis to get insights in default rates on Dutch RMBS portfolios.

Current survival models that quantify credit risk of mortgage portfolios take loan specific variables into account. The effect of seperate variables on sur-vival/failure times is estimated, where failure is defined as defaulting on the loan in this context. Some key variables used are debt-to-income (DTI) and loan-to-value (LTV) ratios (Igan and Kang, 2011). Portfolios with an above-average share of loans with high DTI and/or LTV ratios are expected to have higher default rates, ceteris paribus. Traditional valuation models only take these micro level, loan specific variables into account. Moreover, some of the explanatory variables do not get updated over time, making these models rather static. Taking dynamic macroeconomic variables like unemployment into ac-count might improve the traditional, more static credit risk models. In this paper we want to find the answer the the following question:

• What is the impact of including macroeconomic variables next to loan specific variables in the assessment of credit risk of Dutch RMBS portfo-lios?

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time-varying variables in order to estimate credit risk in a more dynamic way. Insights in the drivers of credit risk enables us to make forecasts on future credit risk as well.

The remainder of this paper is structured as follows. In section 2 we will compare existing literature on credit risk in the context of mortgage portfolios.

section 3 describes the datasets used in this paper. section 4 elaborates on the methodology and the mathematical specifications of the Cox proportional hazards model. Insection 5the results are presented and interpreted,section 6

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2 Literature Review

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put option in the US. However, in most European countries this is not a realistic approach. The value of an option should always be non-negative, either 0 or the positive difference between the value of the underlying asset and the strike price, which is the outstanding balance in this case. In most European countries the condition of a non-negative value does not hold by definition, since borrowers are still responsible for the residual debt after default (if there is any).

So whether applying option-pricing theory is appropiate depends, among other things, on the settlement of the residual debt. However, quite some ar-ticles on credit risk on mortgages do not state that explicitly. This has caused some confusion on this topic. For example,Simon(2009) applies option-pricing theory to embedded options in French mortgage loans. His conclusion is that a significant share of the put options is completely mispriced, defined as more than 40% off. However, the assumptions underlying option-pricing theory are not met in general, since French borrowers are usually responsible for the resid-ual debt. By not taking into account this difference, interpreting the results becomes rather ambiguous.

Instead of applying option-pricing theory, survival models can be used in Europe to estimate the probability of default. The expected loss (EL) depends both on the probability of default (PD) and the loss given default (LGD), most often defined as a percentage of the exposure at default (EAD). Under the as-sumption of independence between probability of default and loss given default, the expected loss is given by (Bluhm et al.,2010)

EL = PD · EAD · LGD

For two independent random variables X and Y it holds that the product of their expectations is equal to the expectation of their product (Miller and Miller,

2004):

E[XY ] = E[X] · E[Y ]

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3 Data

In order to apply survival analysis both loan level data and historical data on macroeconomic variables are needed. In subsection 3.1 we will briefly discuss the macroeconomic variables which are taken into account in the analyses. In

subsection 3.2a general description of the loan level data will be given. A more elaborate inspection on the loan level data gives rise to some concerns, these are discussed insubsection 3.3.

3.1 Macroeconomic variables

Next to loan specific information we have collected data and/or forecasts on macroeconomic variables (CBS,2015). We have Dutch data on:

• monthly (seasonily adjusted) unemployment rates • quarterly gross domestic product (growth) • yearly divorce rates

• monthly interest rates (Euribor) • monthly house price index

In our final dataset, which will be used for our analyses, the values for the macroeconomic variables are matched to the corresponding period.

3.2 Loan level data

By analysing loan specific variables for both defaulted and performing loans, one can get insights in the loan specific drivers of credit risk. Though the ECB collects loan level data since 2012, they are not publicly available. However, some stakeholders like investors and academic researchers might get access the database, which is managed by the European DataWarehouse. Every seller of RMBS is obliged to fill in the RMBS template designed by the ECB on a periodic basis, most often monthly. This template consists of 157 loan specific variables, partly mandatory and partly optional to fill in. Among them are income, original and current balance, valuation of the property, current interest rate, duration and account status. In Appendix A a list of all variables can be found. Every borrower has a unique borrower identitfier, which makes it possible to track a borrower over time and transform the monthly data into panel data.

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Netherlands. In our dataset we have mortgage loans ranging from originated 6 months ago (origination is our time 0) up to 360 months old and everything in between. After securitization, no new borrowers are added to the portfolio, hence the most recent loans are of the time of securitization. In our dataset we have RMBS portfolios securitized between 2007 and 2014.

Some borrowers have multiple loan parts. This might be the case when a mortgage loan consists of both an interest-only part and an annuity part. Our survival analysis is performed on borrower level, hence multiple loan parts per borrower need to be merged first. Total balances should be added, for interest rates a weighted average is taken and for the origination date the first of different loan parts is taken. Most other variables are identical for different loan parts.

3.3 Data issues

Having a closer look to the loan level data, the data quality turns out to be rather poor. Depending on each portfolio approximately 20% to 40% of all mandatory entries in the template are missing or non-disclosed. Some of these missing variables are not that relevant for our analysis in the first place, e.g. servicer identifier, but in most portfolios at least some crucial variables are missing. To give an example, more than half of the Dutch portfolios does not keep track of the loss on sale. The most likely reason for this is the delay in the recoveries. At default the current outstanding balance will change to zero. From a fair value point of view this makes sense: you do not expect the borrower to repay their loan. For defaulted loans only borrower identifiers and the fact that they are defaulted are registered. All the other variables, like (original) debt-to-income and loan-to-value are blank/non-disclosed from that moment on. However, when the recoveries from selling the property are known, the total loss on sale is known as well. In order to investigate loss given default and/or total losses, this information is required. Unfortunately the majority of Dutch portfolios lacks this information. Due to this informational gap we need to restrict our empirical research to the probability of default. The probability of default and loss given default are partly influenced by the same variables, e.g. current loan-to-value. This dependence has an impact on total expected losses, to what extend is not clear though. Data on actual losses would be needed.

Furthermore, some portfolios remove defaulted loans from their dataset, which makes them not suitable for survival analysis anymore. Missing data on relevant variables like total income at origination occurs as well for some of our portfolios. As a consequence every portfolio needs to be checked indi-vidually on certain necessary (basic) requirements. Portfolios that do not meet necessary requirements, like keeping track of defaulted loans, are not considered in our final/total dataset.

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the original/current balance divided by total year income. The possibly troubled outliers might influence our analysis. For that reason we used both the raw DTI ratios as well as adjusted DTI ratios in our analysis. For the adjusted DTI ratios loans with doubtful incomes are not taken into account. After some analyses, we decided to design an adjusted DTI ratio that only takes into account loans with a total year income of at least 5,000 euros.

Another issue in the data collection process is the availability of up to date information for (almost) defaulted loans. For defaulted loans, as mentioned, most loan specific variables are not available anymore. This can be resolved by replacing these variables in the month of default by the values of the previous month. However, for some (almost) defaulted loans, it is necessary to use values of more than 1 month ago due to availability. This causes for example the current loan-to-value to be not always exactly the actual current loan-to-value at default, the expected errors are expected to be very small though.

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4 Methodology

In this paper we will apply survival analysis to loan level data. Banasik et al.

(1999) have compared survival analysis with more standard approaches like logistic regression in a credit scoring context. They concluded that survival analysis outperformed the other methods used. Narain(1992) had already ap-plied survival analysis to loan level data earlier. Though the predicted results fitted the actual data quite well, no comparison with other techniques was made at the time. One of the main advantages of survival models is the ability to incorporate time-varying variables and to deal with censored data. For censored data points it is only known that the failure time is larger (or smaller) than a certain time, but it is not known by how much (Cox,1972). More on censoring insubsection 4.3.

In this section we will firstly introduce survival analysis in general. In sub-section 4.2we will introduce the Cox proportional hazards model, a specific type of model within the class of survival models. First we will assume no censoring nor ties in the data, which we will relax insubsection 4.3andsubsection 4.4. A brief summary of the implementation of the model in the software will be given insubsection 4.5.

4.1 Survival analysis

The time to default can be seen as the failure time T . There are several equiv-alent ways to characterize the probability distribution of a survival random variable (Kalbfleisch and Prentice, 2002):

• density function f (t) • survivor function S(t) • hazard function λ(t)

• cumulative hazard function Λ(t)

We will briefly discuss the relationship between these characterizations. Even though they are equivalent in a way that if you know one, you can derive the others, they have different interpretations.

The density function of the continuous random failure time T is given by: f (t) = lim

∆t→0

P (t < T ≤ t + ∆t)

∆t (4.1)

For a discrete random failure time, the density function gives the probability of a failure time at time t, that is, f (t) = P (T = t).

The survivor function has the interpretation of the probability of a failure time larger than t, hence surviving at least up to time t:

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For continuous random variables this implies S(t) = 1 − Z t 0 f (u) du = Z ∞ t f (u) du (4.2)

For discrete random variables this implies S(t) = 1 −X

u≤t

f (u) =X

u>t

f (u) (4.3)

The hazard function has the interpretation of the failure rate at time t, conditional on survival up to time t. Note that the hazard function should not be interpreted as a probability or density function, but as a rate defined per time unit. Its definition is given by:

λ(t) = lim

∆t→0

P (t < T ≤ t + ∆t|T > t)

∆t (4.4)

In order to identify the relationship between f (t), S(t) and λ(t), we assume without loss of generality a continuous failure time distribution for the moment. Then we have: λ(t) = lim ∆t→0 P (t < T ≤ t + ∆t|T > t) ∆t = lim ∆t→0 P (t < T ≤ t + ∆t)/∆t P (T > t) = f (t) S(t) (4.5)

For discrete random variables, this last expression for the hazard rate can be applied as well, as long as S(t) > 0. For the discrete case it is relatively easy to see that the hazard function cannot be interpreted as a probability. If P (T = t) > P (T > t), we have that f (t) > S(t) which leads to a hazard func-tion larger than 1.

There is not a clear intuitive interpretation for the cumulative hazard func-tion, although it does hold that a higher cumulative hazard function indicates a higher risk of failure by time t. Besides, it has nice mathematical properties. For continuous hazard functions the cumulative hazard function is defined as:

Λ(t) = Z t

0

λ(u) du (4.6)

In Equation 4.5 we have seen that we can rewrite the hazard function to λ(t) =_S(t)f (t). Furthermore we have

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Now we can define −d dt[log S(t)] = − 1 S(t)· S 0_(t) = −−f (t) S(t) = f (t) S(t) = λ(t) (4.7)

If we combine this result with the definition of the cumulative hazard function we get Λ(t) = Z t 0 λ(u) du = Z t 0 − d du[log S(u)] du = − log S(t) + log S(0) = − log S(t)

Now we can rewrite the relationship between the survival function and the cumulative hazard function

log S(t) = −Λ(t)

S(t) = e−Λ(t) (4.8)

4.2 Cox proportional hazards model

Within the class of survival models David Cox developed his Cox proportional hazards model, also referred to as the Cox model, which was published in the Journal of the Royal Statistical Society in 1972. This model was an extension of the results of Kaplan and Meier (1958), they developed the nonparametric Kaplan-Meier estimator for the survival function at time t. Cox introduced the use of explanatory variables, or covariates, in order to identify different effects on the hazard rate. By applying the techniques described insubsection 4.1we can compare the hazard rates by Cox with the survival functions estimates by Kaplan and Meier. The main idea of the Cox model is that the covariates have a multiplication effect on the hazard rate (Andersen and Gill, 1982). As the name already suggests, this effect is proportional. That is, for equal increases in a certain covariate value (in absolute size), the hazard rates will be multiplied with the same factor. Hence an increase in the covariate value from 90 to 100, has the same relative effect as an increase from 100 to 110.

The model defines the hazard rate as follows:

λ(t; X) = λ0(t) exp(Xβ) = λ0(t) exp(β1X1+ . . . + βpXp) (4.9)

In this expression is λ0(t) the baseline hazard function, X = (X1, . . . , Xp) an

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p the number of covariates, β = (β1, . . . , βp) a p × 1 vector of coefficients for

the p covariates, and λ(t; X) the resulting n × 1 vector of hazard rates. If a certain covariate Xi does not have an effect on the hazard rate, the

co-efficient βi will be 0. The baseline hazard rate λ0(t) will be multiplied with 1

and consequently not be affected by this covariate. Typically these covariates are removed from the model. It might also occur that for an individual the (binary) covariate values are all equal to 0 (which can be thought of as an all 0 row in the matrix X). In that case the hazard rate is equal to the baseline hazard rate for that individual. The only requirement for the baseline hazard is that λ0(t) > 0. The baseline hazard is nonparametric, it does not have any

specific shape. The covariates part is parametric, making the Cox model a semiparametric model (Zhang,2005).

One of the main advantages of the Cox model is the ability to incorporate time-varying covariates. If we let denote X1all static or constant covariates and

X2(t) all time-varying or dynamic covariates,Equation 4.9can be rewritten to

λ(t; X(t)) = λ0(t) exp (X1β1+ X2(t)β2) (4.10)

For the moment we assume no censored observations nor ties in the failure times. In the following subsections we will relax these assumptions. In order to fit the Cox model, we first need to introduce the concept of risk sets. The risk set at time t, R(t), is defined as the set of indices of all objects who have not failed before time t. We can order all n objects in ascending order of failure time. We define t(i)to be the failure time of the i-th failed object. Under the assumption of

no ties and censored observations it holds that 0 < t(1)< t(2)< . . . < t(n)< ∞.

Similarly we define λ(i)(t) to be the hazard rate of the i-th failed object.

Since the baseline hazard λ0(t) is left unspecified, ordinary likelihood

meth-ods cannot be applied straight away to estimate β (Zhang, 2005). The Cox model is fitted in two stages, first the coefficients for the covariates (β1, . . . , βp)

are fitted by means of maximizing the partial likelihood, whereafter the base-line hazard λ0(t) will be fitted by maximizing the full likelihood using the fitted

values ˆβ (Cox,1975). Note that in Cox(1972) the term conditional likelihood was used, but after revision the name was changed to partial likelihood inCox

(1975). The partial likelihood for the i-th failed object is given by Li(β) =

λ(i)(t; X)

λ(i)(t; X) + λ(i+1)(t; X) + . . . + λ(n)(t; X)

In the numerator we have the hazard rate of the i-th failed object and in the denominator the sum of hazard rates of all objects still at risk. We can rewrite this expression. Let X(i) denote the row of X corresponding to the i-th failed

object. In other words, let X(i)denote the p covariate values for the i-th failed

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Applying the concept of risk sets gives us

= _Pexp(X(i)β)

j∈R(t(i))

exp(X(j)β)

(4.11)

Under the assumption of no ties nor censored observations we can multiply the partial likelihood for all n objects to obtain the sample partial likelihood

L(β) = n Y i=1     exp(X(i)β) P j∈R(t(i)) exp(X(j)β)     (4.12)

Taking the natural logarithm gives us the partial log likelihood

`(β) = n X i=1   X(i)β − log    X j∈R(t(i)) exp(X(j)β)       (4.13)

This partial log likelihood can be maximized over β in order to find the fitted values ˆβ. Therefore we take the partial derivative with respect to β, which is the partial score function U (β), and solve U (β) = 0. This partial score function is given by U (β) ≡ ∂`(β) ∂β = n X i=1 X(i)− " P j∈R(t(i)) X(j)exp(X(j)β) P j∈R(t(i)) exp(X(j)β) #! (4.14)

The following derivation shows that the last term inEquation 4.14is indeed the partial derivative of the last term inEquation 4.13with respect to β:

∂ ∂β " log k X i=1 exp(xiβ) !# = ∂ ∂β h

log (exp(x1β) + . . . + exp(xkβ))

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U (β) = 0 has a unique solution, or put differently, `(β) has a unique maxi-mum partial likelihood estimator (MPLE), if the Hessian matrix (second par-tial derivatives of `(β)) is negative definite (Simon and Blume, 1994). It is equivalent to state that the Fisher information I(β) should be positive definite

I(β) ≡ −E ∂ 2 ∂β2`(β) β > 0

The inverse of the Fisher information evaluated at the MPLE is an estimator for the asymptotic covariance matrix. This can be used to approximate standard errors for ˆβ (Hayashi,2000).

By maximizing the partial likelihood function we find the estimated values for ˆ

β. These estimates ˆβ are taken as given when we maximize the full likelihood to fit the baseline hazard λ0(t). The baseline hazard is a mapping from time t

to some non-negative real number R+. As our time-axis we take the age of

the loan. Typically, default rates are not the same during the lifetime stages of a loan, even if all other factors are. Loans just after origination and close to maturity are less likely to default than loans in between (FitchRatings,2014). The baseline hazard is a time-dependent function. It can be interpreted as the rate of default for borrowers for which all covariates have value 0. The base-line hazard takes the natural development of default rates over the lifetime of mortgage loans into account. The fitted baseline hazard is given by (Breslow,

1974) ˆ λ0(ti) = di P j∈R(ti) exp(X(j)β)ˆ (4.15)

where di is the number of failures at time ti. Under the assumption of no ties

in the data di has value 1 for ti= t(1), . . . , t(k)and 0 otherwise.

4.3 Censoring

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risk on mortgage portfolios. Left-truncation refers to the phenomenon we can only observe loans which have not defaulted yet at the start of the observation period. We have to take this into account in order to model credit risk correctly due to a survivor bias, that is, the most risky loans might have defaulted before the start of the observation period (Lamarca et al.,1998). Ignoring this survivor bias will lead to an underestimation of credit risk. The Cox model enables us to correct for the fact that only a subset of mortgage loan can be observed. Only survival within the observation period is taken into account. InFigure 4.1and

Figure 4.2a hypothetical example is given to clarify the concept of censoring in the context of mortgage loans.

Figure 4.1: Censored data with year as time-axis

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Figure 4.2: Censored data with corresponding age as time-axis

4.4 Ties

So far we have assumed no ties in the failure times. For mortgage loans this is not realistic though. Usually information is updated on a monthly basis, but definitely in discrete time. As a consequence, multiple loans default ‘at the same time’. Insubsection 4.2we have defined the risk sets for the partial likelihood in such a way that each event can be ordered in terms of failure times. We have to adjust our partial likelihood expression in order to deal with ties. Furthermore, some of the n observations might be right-censored. Let l denote the number of distinct failure times, where l ≤ n. There are 3 different methods to account for ties: the approximation byBreslow(1974), the approximation byEfron (1977) and the exact method byKalbfleisch and Prentice(1980).

Even for relatively small datasets the exact method is from a computational point of view too complex (Cleves et al.,2008). This method takes all possible orderings of tied failure times into account. All of these orderings are then weighted by the probability of occurance, where the risk set is adjusted for each ordering. For 10 tied failure times, this already gives 10! = 3, 628, 800 possible orderings. This becomes computationally infeasible very quickly.

The approximation by Breslow uses the discrete version of the continuous partial likelihood. Here risk sets are not adjusted for the subsequent failures, which makes it computationally easier. The partial likelihood byBreslow(1974) is given by L(β) = l Y i=1    X(i+)β h P j∈R(t(i)) exp(X(j)β) idi    (4.16)

where di denotes the number of ties at t(i) and X(i+) is defined as the sum of

the covariates with tied failure times, that is, X(i+)=P_j∈D₍_t

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last expression D t(i) denotes the set of individuals who failed at t(i).

The approximation by Efron orders the ties in the data as of their failures were not exactly at the same time. This ordering is randomly done with equal probability for each tied failure, adjusting the risk sets for subsequent failures (Efron,1977). The partial likelihood is then given by

L(β) = l Y i=1   exp(X(i+)β) Qdi k=1 h P j∈R(t(i)) exp(X(j)β) − k−1 di P j∈R(t(i)) exp(X(j)β) i   (4.17) where di again denotes the number of ties at t(i) and X(i+) the sum of the

covariates with tied failure times.

It is shown that the approximation by Efron is more accurate than the approximation by Breslow (Cleves et al., 2008). Even though Efron’s method takes a bit more steps, from a computational point of view both are of the same order of complexity. We will use the approximation by Efron to account for ties in the data.

4.5 Implementation

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5 Results

5.1 Model comparison

In this section we will present some relevant part of our analysis. Different models are tested on their significance. For comparison purposes we use the Information Criterion by Akaike, commonly abbreviated as AIC. Originally this acronym stands for An Information Criterion (Akaike, 1974), but is nowadays widely used as Akaike’s Information Criterion. This criterion is defined by

Akaike(1973):

AIC = −2 log(L) + 2k = −2` + 2k (5.1) where L and ` are respectively the likelihood and log-likelihood and k the di-mension of the parameter space.

A model with a higher log-likelihood and less fitted parameters has a better fit than a model (based on the same data) with a lower log-likelihood and more fitted parameters. The AIC takes both log-likelihood and the number of param-eters into account. The higher the log-likelihood, the lower the AIC. Adding more parameters might lead to a higher log-likelihood, but not necessarily to a better model. Based on the likelihood, model selection would be biased towards on overfitted model. The AIC tries to corrects for this bias by ‘punishing’ the use of more parameters. The more parameters in the model, the higher the AIC. When comparing models, the model with the lowest AIC is preferred, provided they are based on the same data obviously. The absolute value of the AIC does not give us much information, it is used for comparison purposes only. Unfor-tunately the AIC is still a biased criterion for model selection. The good news though is that the probability of overfitting decreases as n increases (McQuarrie and Tsai,1998).

The Bayesian Information Criterion (BIC), or the equivalent Schwarz Infor-mation Criterion (SIC), can also be used for model comparison. These criteria were developed from a Bayesian perspective. For the BIC adding an extra vari-able to the model is not ‘punished’ with a factor 2, but with a factor log(n), n denoting the number of observations (Akaike, 1978):

BIC = −2 log(L) + k · log(n) = −2` + k · log(n) (5.2) where L and ` are respectively the likelihood and log-likelihood, k the dimension of the parameter space and n the number of observations.

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5.2 Model selection

As mentioned in subsectionData issues, the raw DTI ratios might be unrealistic for some borrowers. In the light of some unlikely low incomes, an adjusted DTI is designed as well. About 0.2% of the raw DTI ratios are not taken into account in the adjusted DTI when we consider all portfolios. However, for some individual portfolios up to 2% of the raw DTI are discarded in the adjusted DTI. First we will look at the differences for the total dataset. From the loan level data DTI and LTV ratios are expected to impact default rates. At macro level, we include unemployment and divorce rates for now. This leads to the following results.

coef exp(coef) se(coef) z p

Original DTI adj 0.0973 1.10 0.04650 2.09 3.6e-02 Current LTV 0.0373 1.04 0.00422 8.84 0.0e+00 Unemployment 0.7233 2.06 0.17747 4.08 4.6e-05

Divorce 0.6233 1.87 0.21374 2.92 3.5e-03

Original DTI raw 0.0687 1.07 0.03924 1.75 8.0e-02 Current LTV 0.0376 1.04 0.00421 8.92 0.0e+00 Unemployment 0.7210 2.06 0.17746 4.06 4.6e-05

Divorce 0.6247 1.87 0.21367 2.92 3.5e-03

Table 5.1: Raw versus adjusted DTI ratios

The column exp(coef) indicates the multiplicative effect of that covariate on the default rate. Since the coefficients are positive here, the multiplicative effects are all larger than 1. Consequently, for higher covariate values the default rate is expected to increase. The last two columns indicate the statistical significance. Column z is the Wald statistic, defined as

z = ˆ β − β0 se( ˆβ) = ˆ β se( ˆβ) (5.3)

Under the null hypothesis, the Wald statistic follows the standard normal dis-tribution. The last column states the p-value of H0: β0= 0 versus H1: β 6= 0.

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a relatively high share of loans not taken into account in the adjusted DTI. For these portfolios the statistical significance is even worse for the raw DTI. If we for example would include current DTI and unemployment rates, the raw DTI is not significant at all while the adjusted DTI is.

Current DTI adj 0.0403 1.04 0.0173 2.33 2.0e-02 Unemployment 0.8612 2.37 0.1676 5.14 2.8e-07

Current DTI raw -0.000186 1.00 0.001 -0.185 8.5e-01 Unemployment 0.857960 2.36 0.168 5.119 3.1e-07

Table 5.2: Example insignificant raw DTI ratios

We also tested whether including other loan specific covariates in the model would have a significant effect on default rates and/or would lead to a better fit based on the AIC and BIC. For some of the covariates this has proven not to be the case. Among those are age, original and current balance, total income and original property value. They are not statistically significant, even at α = 0.10, nor did the AIC/BIC went down by including them. However, portfolios do not have data on all loan specific variables. Which variables are missing, changes per portfolio though.

To illustrate the last point: not all portfolios specify whether the total income is coming from 1 or more than 1 income and/or the amounts of the primary and secondary income seperately. For individual portfolios (between 25,000 and 75,000 loans approximately) that do keep track of the primary and the secondary income seperately, we can add the following variables:

• Share main income: the largest income as a percentage of total income (hence ranging from 50% to 100%)

• Multiple incomes: a binary variable indicating whether total income is coming from more than 1 income

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one of them. Although for individual portfolios these variables can be statis-tically significant, they are not for the total dataset. About only half of the portfolios specify the primary and secondary income seperately, which might explain why the variables share highest income and multiple incomes are not statistically significant when all portfolios are considered.

Kau and Keenan (1995) suggest that house prices influence default rates. Though their model is focused on the US, we still want to investigate how that translates to the Dutch RMBS market. First we will state our reference model for which we will interpret the results. Afterwards we will investigate the consequences of including Dutch house price index in the model.

Since the variables share highest income and multiple incomes are not signif-icant when all portfolios are considered, we do not include them in our model. Furthermore, we will use the adjusted current DTI instead of the raw DTI ratios. For the macroeconomic variables both lagged and current version are significant. However, we cannot include both in the same model since the vari-able and its lagged version are strongly correlated. This phenomenon is called multicollinearity (Hayashi,2000). Unemployment can be modelled by means of an ARMA-process (Floros, 2005). We will add only one lag of unemployment and divorce rates to the model. Although most recent lags are significant, it depends per portfolio which lag gives the best fit. We tested all lags from 0 up to 9 and 12. Lags higher than 6 gave a worse fit, but for the lags between 0 and 6, there was not one lag superior to the others. For our total dataset we use the values of 3 months ago, the choice of 3 is not indisputable though. Having said that, the following results are obtained.

Current DTI adj 0.1225 1.13 0.04153 2.95 3.2e-03 Current LTV 0.0365 1.04 0.00428 8.53 0.0e+00 Unemployment 0.7218 2.06 0.17748 4.07 4.8e-05

Divorce 0.6326 1.88 0.21400 2.96 3.1e-03

Table 5.3: Default model

We can interpret the results as follows. Both unemployment and divorce rates are measured as a percentage. For (lagged) unemployment it holds that an increase of 1 percentage point in the unemployment rate causes default rates to be multiplied by e0.7218 _{= 2.06. This is equivalent of saying that default}

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of our model we get e10·0.0365 = 1.44, or equivalently 1.0410 = 1.44. Hence, borrowers with a 10 percentage points higher current LTV default at a 44% higher rate, ceteris paribus. Note that we have used Dutch RMBS portfolios. The value of the coefficients on the covariates, indicating the magnitude of the effect on default rates, can vary for other countries due to differences in tax system and/or regulatory framework. Especially LTV ratios have limited impact in the Netherlands (NVB,2014). In the Netherlands original LTV ratios larger than 100% are more common than in comparable countries. The risk associated with higher LTV ratios is relatively small in the Netherlands. So far we have discussed the impact of LTV ratios on default rates. However, LTV ratios have an impact on loss given default as well. Due to this dual effect, LTV ratios have a stronger effect on total expected losses than on default rates only. DTI ratios affect default rates as well. Typically, for Dutch mortgages loans the original DTI rarios are between 2.5 and 6.5, based on the 10th and 90th percentile. The association for home owners in the Netherlands, Vereniging Eigen Huis, advises first-time homebuyers to borrow at most 4.5 times your yearly income. Borrowers with a DTI ratio of 5.5 default at a 13% higher rate compared to borrowers with a DTI ratio of 4.5. For all four discussed covariates, the coefficients are statistically significant at the 1% level.

Having said that, note that the macroeconomic variables have a very strong impact on default rates. Surprisingly, these macroeconomic variables are not taken into account in some basic models on credit risk valuation. A possible explanation might be that buyers of RMBS portfolios are not always allowed to take loan loss provisions based on macroeconomic variable due to (interna-tional) accounting rules. A portfolio with relatively high LTV and DTI ratios is expected to show relatively high default rates. Therefore, the buyer of an RMBS portfolio is allowed to take a provision since they are expected to face losses in the future. Such provisions are called loan loss provisions (LLP). For macroeconomic variables this is not always allowed, therefore making it rather complicated to implement. For internal use, buyers and sellers of RMBS port-folios could still take provisions. However, explaining the difference between internal and external numbers is sometimes experienced as (too) complicated. Nonetheless, both buyer and seller of RMBS portfolios should be aware of the impact of macroeconomic variables on credit risk of RMBS portfolios.

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coef exp(coef) se(coef) z p Current DTI adj 0.1214 1.129 0.04159 2.92 0.0035 Current LTV 0.0368 1.037 0.00429 8.58 0.0000 Unemployment 0.4147 1.514 0.25949 1.60 0.1100

Divorce 0.8494 2.338 0.26544 3.20 0.0014

Houses -0.1457 0.864 0.09167 -1.59 0.1100

Table 5.4: Default model extended with house price index

Although the Dutch house price index was individually significant, jointly with the other 4 variables it is not. Note that especially the coefficient on lagged unemployment has changed and even has become insignificant. An increase in the unemployment 3 months ago might influence current house prices. Higher unemployment rates can cause a decrease in the demand of houses, which causes house prices to go down. In that case lagged unemployment and house prices are correlated and should not be used within the same model to avoid multi-collinearity. Indeed, when we replace the variable unemployment by the variable house prices (instead of adding them both), all 4 coefficients are statistically sig-nificant. However, the AIC and BIC indicate a better fit for the model including unemployment.

The same argument holds for the variable gross domestic product (GDP). GDP (growth) and unemployment rates are negatively correlated, the higher the level of GDP (growth), the lower unemployment rate. Hence including both leads to multicollinearity. Including unemployment rate leads to a better fit.

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coef exp(coef) se(coef) z p Current DTI adj 0.1119 1.118 0.04450 2.51 1.2e-02 Current LTV 0.0360 1.037 0.00449 8.02 1.0e-15 Unemployment 0.9178 2.504 0.18675 4.91 8.9e-07

Divorce 0.6260 1.870 0.22577 2.77 5.6e-03

Redemption -0.0926 0.912 0.20585 -0.45 6.5e-01 Table 5.5: Default model extended with redemption indicator

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6 Discussion

Even though loan level data are available, they are far from complete. We cannot take certain variables into account in our model due to the absence of proper information. For most borrowers information on credit history is not available, although negative credit history is expected to have a positive relationship with default rates. Whether total income is coming from 1 or more than 1 income might be relevant as well in estimating and forecasting default rates. Our model to estimate and forecast default rates can be improved if those variables can be included.

Furthermore, since most portfolios do not measure actual incurred losses after default, we cannot measure the dependence between the probability of default and the loss given default. Even though it is usually assumed that the PD and LGD are independent, they are both affected by partly the same variables, e.g. LTV ratios. This dependence can increase total credit risk on portfolio level. However, the impact of this dependence cannot be quantified.

Supervisors of financial markets, the ECB and/or national central banks can and should improve the information availability on a loan level basis. Most likely the information is already somewhere on a more local level, but at least on a European level, some crucial parts are not available. Regulatory decisions or opinions based on incomplete information can easily be improved by having more accurate underlying data available.

In this paper we have investigated the Dutch RMBS market. However, it might be of interest to investigate other European countries as well and compare those outcomes with the outcomes for the Netherlands. Including data from multiple countries in one dataset can lead to comparing apples with oranges. To give an example, in the Netherlands it holds that interest on mortgage loans is tax deductable. Therefore DTI ratios tend to be higher without lenders being more at risk. In most other countries interest is not tax deductible. Therefore comparing Dutch DTI ratios with German DTI ratios is not fair.

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7 Conclusion

Including macroeconomic variables leads to better estimations for default rates on Dutch RMBS portfolios. In our model we use the loan level variables current DTI ratios (adjusted) and current LTV ratios. The macroeconomic variables un-employment and divorce rates have a significant impact on default rates, hence are included as well. The impact of the macroeconomic variables on default rates is relatively large. A 1 percentage point increase in unemployment and divorce rates cause an increase in default rates of 106% and 88% respectively. Borrowers with a 10 percentage points higher current LTV ratio default at a 44% higher rate. Current DTI ratio is defined as a factor: current balance divided by total yearly income. Borrowers with a 1 point higher DTI ratio, default at a 13% higher rate. Note that the effect on default rates is relative, not in per-centage points. Since default rates strongly depend on certain macroeconomic variables, we recommend buyers and sellers of RMBS portfolios to take these dynamic macroeconomic variables into account as well.

Although in the US house prices are one of the main explanatory variables in estimating default rates, it is not statistically significant in the Netherlands. However, US borrowers are not responsible for the residual debt whereas Dutch borrowers are. Note that for loss given default, Dutch house prices do have an impact. Despite that default rates are not significantly affected by house prices, expected losses are still influenced by house prices through LGD. Hence, expected losses are still negatively correlated with house prices.

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A

List of variables RMBS template

Sellers of European RMBS portfolios, need to supply monthly loan level data to the European DataWarehouse. The European DataWarehouse is responsible for the data gathering process. They are supervised by both the European Central Bank and National Central Banks. In cooperation with the central banks, the following template is designed.

type commitment code variable

Identifiers Mandatory AR1 Pool Cut-off Date Identifiers Mandatory AR2 Pool Identifier Identifiers Mandatory AR3 Loan Identifier Identifiers Optional AR4 Regulated Loan Identifiers Mandatory AR5 Originator Identifiers Mandatory AR6 Servicer Identifier Identifiers Mandatory AR7 Borrower Identifier Identifiers Mandatory AR8 Property Identifier Borrower Information Optional AR15 Borrower Type Borrower Information Optional AR16 Foreign National

Borrower Information Optional AR17 Borrower Credit Quality Borrower Information Optional AR18 Borrower Year of Birth Borrower Information Optional AR19 Number of Debtors Borrower Information Optional AR20 Second Applicant Year of

Birth

Borrower Information Mandatory AR21 Borrower’s Employment Status

Borrower Information Optional AR22 First-time Buyer Borrower Information Optional AR23 Right to Buy Borrower Information Optional AR24 Right to Buy Price Borrower Information Optional AR25 Class of Borrower Borrower Information Mandatory AR26 Primary Income

Borrower Information Mandatory AR27 Income Verification for Primary Income

Borrower Information Optional AR28 Secondary Income

Borrower Information Optional AR29 Income Verification for Secondary Income

Borrower Information Optional AR30 Resident

Borrower Information Optional AR31 Number of County Court Judgements or equivalent - Satisfied

Borrower Information Optional AR32 Value of County Court Judgements or equivalent - Satisfied

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Borrower Information Optional AR34 Value of County Court Judgements or equivalent - Unsatisfied

Borrower Information Optional AR35 Last County Court Judge-ments or equivalent Year Borrower Information Optional AR36 Bankruptcy or Individ-ual Voluntary Arrange-ment Flag

Borrower Information Optional AR37 Bureau Krediet Regis-tratie 1 to 10 - Credit Type

Borrower Information Optional AR38 Bureau Krediet Regis-tratie 1 to 10- Registration Date

Borrower Information Optional AR39 Bureau Krediet Regis-tratie 1 to 10 - Arrears Code

Borrower Information Optional AR40 Bureau Krediet Regis-tratie 1 to 10 - Credit Amount

Borrower Information Optional AR41 Bureau Krediet Regis-tratie 1 to 10 - Is Coding Cured?

Borrower Information Optional AR42 Bureau Krediet Regis-tratie 1 to 10 - Number of Months Since Cured Borrower Information Optional AR43 Bureau Score Provider Borrower Information Optional AR44 Bureau Score Type Borrower Information Optional AR45 Bureau Score Date Borrower Information Optional AR46 Bureau Score Value Borrower Information Optional AR47 Prior Repossessions Borrower Information Optional AR48 Previous Mortgage

Ar-rears 0-6 Months

Borrower Information Optional AR49 Previous Mortgage Ar-rears 6+ Months

Loan Characteristics Mandatory AR55 Loan Origination Date Loan Characteristics Mandatory AR56 Date of Loan Maturity Loan Characteristics Optional AR57 Account Status Date Loan Characteristics Optional AR58 Origination Channel /

Ar-ranging Bank or Division Loan Characteristics Mandatory AR59 Purpose

Loan Characteristics Optional AR60 Shared Ownership Loan Characteristics Mandatory AR61 Loan Term

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Loan Characteristics Mandatory AR65 Loan Currency Denomi-nation

Loan Characteristics Mandatory AR66 Original Balance Loan Characteristics Mandatory AR67 Current Balance

Loan Characteristics Optional AR68 Fractioned / Subrogated Loans

Loan Characteristics Mandatory AR69 Repayment Method Loan Characteristics Mandatory AR70 Payment Frequency Loan Characteristics Mandatory AR71 Payment Due Loan Characteristics Mandatory AR72 Payment Type Loan Characteristics Optional AR73 Debt to Income

Loan Characteristics Optional AR74 Type of Guarantee Provider

Loan Characteristics Optional AR75 Guarantee Provider Loan Characteristics Optional AR76 Income Guarantor Loan Characteristics Optional AR77 Subsidy Received

Loan Characteristics Optional AR78 Mortgage Indemnity Guarantee Provider Loan Characteristics Optional AR79 Mortgage Indemnity

Guarantee Attachment Point

Loan Characteristics Optional AR80 Prior Balances Loan Characteristics Optional AR81 Other Prior Balances Loan Characteristics Optional AR82 Pari Passu Loans Loan Characteristics Optional AR83 Subordinated Claims Loan Characteristics Optional AR84 Lien

Loan Characteristics Optional AR85 Retained Amount Loan Characteristics Optional AR86 Retained Amount Date Loan Characteristics Optional AR87 Maximum Balance Loan Characteristics Optional AR88 Further Loan Advance Loan Characteristics Optional AR89 Further Loan Advance

Date

Loan Characteristics Optional AR90 Flexible Loan Amount Loan Characteristics Optional AR91 Further Advances

Loan Characteristics Optional AR92 Length of Payment Holi-day

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Loan Characteristics Optional AR101 Percentage of pre-payments allowed per year

Interest Rate Mandatory AR107 Interest Rate Type Interest Rate Mandatory AR108 Current Interest Rate

In-dex

Interest Rate Mandatory AR109 Current Interest Rate Interest Rate Mandatory AR110 Current Interest Rate

Margin

Interest Rate Mandatory AR111 Interest Rate Reset Inter-val

Interest Rate Optional AR112 Interest Cap Rate Interest Rate Mandatory AR113 Revision Margin 1 Interest Rate Mandatory AR114 Interest Revision Date 1 Interest Rate Mandatory AR115 Revision Margin 2 Interest Rate Mandatory AR116 Interest Revision Date 2 Interest Rate Mandatory AR117 Revision Margin 3 Interest Rate Mandatory AR118 Interest Revision Date 3 Interest Rate Mandatory AR119 Revised Interest Rate

In-dex

Interest Rate Optional AR120 Final Margin Interest Rate Optional AR121 Final Step Date

Interest Rate Optional AR122 Restructuring Arrange-ment

Property and Addi-tional Collateral

Optional AR128 Geographic Region List Property and

Addi-tional Collateral

Mandatory AR129 Property Postcode Property and

Optional AR130 Occupancy Type Property and

Mandatory AR131 Property Type Property and

Optional AR132 New Property Property and

Optional AR133 Construction Year Property and

Optional AR134 Property Rating Property and

Mandatory AR135 Original Loan to Value Property and

Mandatory AR136 Valuation Amount Property and

Mandatory AR137 Original Valuation Type Property and

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Optional AR139 Confidence Interval for Original Automated Valu-ation Model ValuValu-ation Property and

Optional AR140 Provider of Original Au-tomated Valuation Model Valuation

Mandatory AR141 Current Loan to Value Property and

Optional AR142 Purchase Price Lower Limit

Mandatory AR143 Current Valuation Amount

Mandatory AR144 Current Valuation Type Property and

Mandatory AR145 Current Valuation Date Property and

Optional AR146 Confidence Interval for Current Automated Valu-ation Model ValuValu-ation Property and

Optional AR147 Provider of Current Au-tomated Valuation Model Valuation

Optional AR148 Property Value at Time of Latest Loan Advance Property and

Optional AR149 Indexed Foreclosure Value Property and

Optional AR150 Ipoteca Property and

Optional AR151 Date of Sale Property and

Optional AR152 Additional Collateral Property and

Optional AR153 Additional Collateral Provider

Optional AR154 Gross Annual Rental In-come

Optional AR155 Number of Buy to Let Properties

Optional AR156 Debt Service Coverage Ratio

Optional AR157 Additional Collateral Value

Optional AR158 Real Estate Owned Property and

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Optional AR160 Time Until Declassifica-tion

Performance Informa-tion

Mandatory AR166 Account Status Performance

Informa-tion

Optional AR167 Date Last Current Performance

Informa-tion

Optional AR168 Date Last in Arrears Performance

Informa-tion

Mandatory AR169 Arrears Balance Performance

Informa-tion

Mandatory AR170 Number Months in Ar-rears

Mandatory AR171 Arrears 1 Month Ago Performance

Informa-tion

Mandatory AR172 Arrears 2 Months Ago Performance

Informa-tion

Optional AR173 Performance Arrange-ment

Mandatory AR174 Litigation Performance

Informa-tion

Mandatory AR175 Redemption Date Performance

Informa-tion

Optional AR176 Months in Arrears Prior Performance

Informa-tion

Mandatory AR177 Default or Foreclosure Performance

Informa-tion

Mandatory AR178 Date of Default or Foreclo-sure

Mandatory AR179 Sale Price lower limit Performance

Informa-tion

Mandatory AR180 Loss on Sale Performance

Informa-tion

Mandatory AR181 Cumulative Recoveries Performance

Informa-tion

Optional AR182 Professional Negligence Recoveries

Optional AR183 Loan flagged as Contencioso

Credit risk assessment of mortgage portfolios: considering macroeconomic information as well

Credit risk assessment of mortgage portfolios:

considering macroeconomic information as well

M.R. van den Broek

Credit risk assessment of mortgage portfolios:

considering macroeconomic information as well

Mark van den Broek

August 21, 2015

Contents

1

Introduction

2

Literature Review

3

Data

3.1

Macroeconomic variables

3.2

Loan level data

3.3

Data issues

4

Methodology

4.1

Survival analysis

4.2

Cox proportional hazards model

4.3

Censoring

4.4

Ties

4.5

Implementation

5

Results

5.1

Model comparison

5.2

Model selection

6

Discussion

7

Conclusion

References

A

List of variables RMBS template