CONDITIONAL DEFAULT RISK IN HOUSING ARMS: A
BIVARIATE PROBIT APPROACH
Alan Hwee Loon TEO
Email: teoalanhl@yahoo.com.sg
Seow Eng ONG
Email: seong@nus.edu.sg
April 30, 2005
Please direct comments to Alan Hwee Loon TEO at Department of Real Estate, National
University of Singapore, 4 Architecture Drive, Singapore 117566, or email:
teoalanhl@yahoo.com.sg
1
CONDITIONAL DEFAULT RISK IN HOUSING ARMS: A BIVARIATE
PROBIT APPROACH
Abstract
Mortgage risk studies largely ignore the sequential nature of delinquency as a necessary
decision preceding default. The strength of such a relationship, if any, is likely to render
existing default models as inefficient and misspecified. This paper provides the first rigorous
analysis of the delinquency and conditional default risks of residential adjustable mortgages in
a bivariate probit model. Utilizing 633 mortgage observations from 1980 to 1999, we found
that the disturbance correlation of delinquency and default is highly significant. This supports
the use of our model. We also found several variables to have signs different between the
bivariate probit model and the corresponding independent probit models. The implication is
that if we rely solely on the independent models, our predictions could be flawed. These
warrant the inclusion of the preceding delinquency decision in default risk models.
(136 words)
Key words
Default risk, Delinquency risk, Bivariate Probit
2
CONDITIONAL DEFAULT RISK IN HOUSING ARMS: A BIVARIATE
PROBIT APPROACH
1.0 Introduction
Mortgage Lending is an imperative component of the businesses of financial institutions
worldwide. This is accelerated by the growth of the private residential property markets in the
respective countries. In the US, almost half of the mortgages are securitized as Mortgagebacked Securities (MBS) (Deng, Quigley and van Order, 2000). Similar trends can be found
in Singapore. With Singapore’s de facto central bank, Monetary Authority of Singapore
recommending the securitization of real estate (Sing and Ong, 2004) and the provision of
favorable tax treatment for such issues (Ong, Ooi and Sing, 2000), there is enormous potential
for the secondary mortgage market to take off in Singapore.
Understanding the various types of mortgage risk is thus essential, for both practitioners and
academics. The credit risk management functions of financial institutions are essentially
geared towards assessing the credit or default risk of customers i.e. the possibility of a
borrower not being able to service a loan. To mitigate prepayment risk and to share interest
rate risk with borrowers, lenders are increasing moving towards adjustable-rate mortgages
(ARMs). In academia, default risk (e.g. Quigley and Van Order, 1990), prepayment risks (e.g.
Ong, 2000) and the competing relationship between them (e.g. Deng, et al., 2000; and Clapp,
Deng and An, 2004) are areas of intense study.
However, studies on mortgage delinquency and the incorporation of delinquency in overall
mortgage risk models are far and few. This can be attributed to a lack of suitable available
data (Von Furstenberg and Green, 1974), difficulties in modeling delinquency within the
prevailing option-based approach (Quercia and Stegman, 1992), and perception among
practitioners that the financial consequences of delinquency is less severe than default
(Quercia, et al., 1992).
In this paper, we postulate mortgage delinquency as a significant and necessary decision that
is taken by the borrower before the default decision. We further propose that the motivations
3
for delinquency exert a significant influence on the subsequent motivations for default such
that default risk models should incorporate the risk of delinquency.
It is crucial to define and differentiate delinquency and default at this early stage. There is a
lack of consensus on this aspect. Earlier scholarly works (von Furstenberg and Green, 1974;
Campbell and Dietrich, 1983; and Zorn and Lea, 1989) differentiate them according to the
eventual outcome. Loans with missed payments that are eventually repaid are defined as
delinquent while loans that are eventually foreclosed are defined as default. These papers
focus on studying the latter and assume that default is synonymous with foreclosure. This
differentiation is only possible with hindsight and is not useful for lenders and MBS
underwriters, as they cannot differentiate between the two at the time when missing the first
mortgage and thus undertake suitable strategies. These studies also do not differentiate
between different motivations of defaulters – those who miss payments with the intention of
giving up their properties and do eventually foreclose, and those who miss payments with the
intention of reinstating their mortgages but were eventually unable to prevent foreclosure.
The second group of studies, developed by Vandell (1993) and later in the works of Ambrose,
Buttimer and Capone (1997), Ambrose and Capone (1996, 1998 and 2000), Ambrose and
Buttimer (2000), Deng (1997) Deng, et al. (2000) and Phillips and VanderHoff (2004)
included the distinction between default and foreclosure in their models. Foreclosure is
regarded as one of the possible paths a defaulted loan can take. However, this group of papers
classifies loans as default once a scheduled payment is missed. Therefore, no distinction is
made between delinquency and defaults.
On the other hand, practitioners in the US and Singapore 1 differentiate delinquency and
default by the number of days of missed installments. Delinquency is defined as the
nonpayment of a mortgage payment due (e.g. Ambrose and Buttimer, 2000; and Holmes,
2003). Default occurs when a borrower has missed 90 days’ installment and the fourth
payment is due (Ambrose and Capone, 2000). Therefore, delinquency is a necessary precursor
1
This definition is supported by Section 25 of the Conveyancing and Law of Property Act in Singapore, which
states that "A mortgagee shall not exercise the power of sale conferred by this Act unless - notice requiring
payment of the mortgage money has been served on the mortgagor or one of several mortgagors, and default has
been made in payment of the mortgage money or part thereof for 3 months after the service…”
4
to default. During delinquency, the lender usually sends reminders to the borrower to make up
the missed payments. Although the lender has the right to foreclose the property as missing an
installment is tantamount to a breach of contract, he would usually refrain until default. Thus,
the borrower has the option to repay the missed installments and reinstate the mortgage. Once
the loan transited to default, the lender will issue a formal legal letter to the borrower
indicating the lender’s right to proceed with foreclosure proceedings any time from then on.
The commencement of foreclosure proceedings is significant because borrowers are generally
not able to reinstate their delinquent loans once the foreclosure sale occurs (except for some
states). Thus, the borrower faces real danger of losing his home with the transition to default.
Using this terminology, there are two unambiguous decision points i.e. 1) whether to
delinquent, and 2) once in delinquency, whether to default. We adopt this set of definitions in
our paper.
Besides the significance of the crossover from delinquency to default, this definition is also
useful in differentiating optimal and trigger-event defaulters via different initial motivations
of delinquency and default. Optimal defaulters refer to borrowers who delinquent to maximize
their wealth when their mortgages are in negative equity positions. They will transit to default
unless there are favorable changes in their equity positions2. Trigger-event defaulters refer to
borrowers who delinquent due to some exogenous events and have every intention of
reinstating their mortgages. They will avoid default unless the impacts of the trigger-events
have seriously affected their financial ability to reinstate. The differing motivations of the two
categories of defaulters are elucidated by the practitioners’ definition as shown in Exhibit 1.
Therefore, just as foreclosure was a two-step process whereby the borrower must default
before foreclosure can occur, we propose default to be modeled as a two-step process.
Based on the definitions adopted in this paper, there is a seemingly obvious relationship
between delinquency and default. Firstly, the essential preceding step of a defaulting
mortgage is delinquency. Secondly, as iterated in Exhibit 1, the motivation for default is
essentially originated from the motivation for delinquency. For instance, Waller (1989) found
2
This follows from Ambrose and Capone (1998) which offers similar arguments for the transition from default
to foreclosure.
5
that lengthy delinquency period might cause borrowers to accumulate so much back payments
that default becomes unavoidable. Thus, the relevant covariates for delinquency and default
are likely to be related. As a result, Quercia, et al. (1992) advocated the study of default
decision within a framework that incorporates the delinquency decision. However, no studies
have tested or taken into account the potentially influential relationship between default and
delinquency. Thus, the first research question of our paper contends with the significance of
such a relationship and whether the sequential nature of delinquency and default does produce
a significant impact on default decision modeling.
The significance of the research question is derived from the absence of the delinquency
decision in existing default models. If a significant relationship can be proven to exist
between the two decisions, such default models would be mis-specified and inefficient.
Incorporation of the delinquency option in future mortgage-pricing models would then be
essential. More practically, lenders often ignore delinquency and focus their attention on
controlling default risks. As delinquency is the essential preceding step of default and the
effects of their determinants may be different, it may be more efficient to engage different risk
mitigation tactics depending on whether they are in delinquency or have transited to default.
Thus, the first and second contributions of this paper are to verify the presence of a significant
relationship between mortgage delinquency and default, and to identify the influential factors
of both delinquency and conditional default in a bivariate probit model.
As mentioned, there is generally a lack of studies on mortgage delinquency. This represents a
missing piece of the puzzle of the nature of mortgage risks. Without a comprehension of the
motivation of each possible borrower action during the lifespan of a mortgage, we would not
be able to attain a complete picture of mortgage risk. Our third research question is thus
represented in this gap in existing knowledge on mortgage risk. In responding to the research
motivation, the third contribution of this paper is to attempt to fill the knowledge gap by
examining the nature of delinquency risk in a joint delinquency-default risk model.
6
In responding to the research questions, we examine the postulated sequential relationship
between delinquency and default, and model default decisions as a 2-stage process using the
bivariate probit methodology. We test three aspects of delinquency-default behavior:
1. the presence of a significant relationship between delinquency incidence and
subsequent default decisions;
2. the influential determinants of delinquency and of default in a conditional probability
model and the disparity with the corresponding independent probability models;
3. the potential reversal in signs of variables that may influence delinquency and
conditional default in opposite ways.
We found that a significant relationship exists between delinquency and the subsequent
default decisions, and that the signs of several variables in the bivariate probit model are
reversed when compared with the independent probit model. We also found evidence of
information asymmetry, and the significance of the temporal effects on default decisions.
The next section provided a review of related literature, followed by a description of the
research methodology for the intended analyses. The data and its descriptive statistics are then
explained before the findings and implications of the results are presented.
2.0 Literature Review
It is necessary to note that most literature on mortgage risks was originated from the US,
where Fixed Rate Mortgages (FRMs) are prevalent (Ong, 2000). Conversely, all mortgages
originated in Singapore are ARMs (Khor and Ong, 1998). The exogeneous and endogeneous
factors affecting both forms of mortgages may thus diverge. For instance, the prepayment risk
for Singapore mortgages is very low (Ong, Maxam and Thang, 2002) while the prepayment
risk for ARMs in US may be higher resulting from potential switches to FRMs to take
advantage of interest rate movements (Ambrose and LaCour-Little, 2001). However,
Campbell, et al. (1983) found that most determinants that affect default decisions influence
delinquency in the same way. Therefore, the methods and factors used in the literature to
rationalize mortgage risks in FRMs serve as a platform for our analysis.
2.1 Methodologies of Mortgage Risk Studies
7
Mortgage risk studies essentially started in the early 1960s and the main methodologies used
included regression (von Furstenberg and Green, 1974; Morton, 1975; and Campbell, et al.,
1983), logit (Vandell and Thibodeau, 1985), and multinomial logit (Zorn, et al., 1989; and
Cunningham and Capone, 1990). Such models often suffer from a lack of theoretical basis for
the borrower behaviors. This led to the development of the Borrower Payment Model and the
Option-based Model.
Mortgage risk studies that utilize the options-based theories to explain default and prepayment
behaviors focus on the net equity position i.e. the house price movement and the term
structure. These two factors are postulated to be the main determinants of such behaviors
(Kau, Keenan, Muller and Epperson, 1992; Kau, Muller and Epperson, 1993; Kau, Keenan
and Kim, 1993 & 1994; Ambrose, Buttimer and Capone, 1997; Ambrose and Buttimer, 2000).
However, earlier options-based studies failed to take into account the competing nature of
foreclosure and prepayment.
Thus, more recent studies utilize the competing risk model employed by Deng, et al. (2000) to
incorporate a wider range of default factors like borrower characteristics (Ambrose and
Capone, 2000; Ong, et al., 2002; Lambrecht, et al., 2003), macroeconomic characteristics
(Ambrose and Capone, 2000; Ambrose and LaCour-Little, 2001; Ong, et al., 2002) and loan
factors (Ambrose and LaCour-Little, 2001; Ong, et al., 2002).
In addition, Ambrose and Capone (1998) and Phillips, et al. (2004) use a multinomial logit
model to test the influence of borrower, mortgage and macroeconomic variables on the
conditional probability of foreclosure. The former relied upon the options-based approach to
identify the independent variables while the latter included the variables that they believe to
be significant.
2.2 Variables of Delinquency and Default Decisions
2.2.1 Determinants of Mortgage Delinquency
Ambrose and Capone (1996, 1998) and Waller (1988) described the aim of delinquency is
either to put the funds, originally intended to pay the installments, to other uses due to
financial difficulties, or to exercise the implicit put option to abandon the property. A third
8
cause of delinquency noted by Waller (1988) is the economic incentive borrowers can gain
from living in the house rent-free before foreclosure takes place.
Von Furstenberg, et al. (1974) found that the equity-value ratio possesses a significant
negative relationship with delinquency while the age of mortgages has a positive relationship.
In addition, mortgages of existing houses are more prone to delinquency than those taken on
new houses. Herzog and Earley (1970) and Morton (1975) also found income, occupation and
the number of children to be influential determinants.
Zorn, et al. (1989) argued that delinquency can be regarded as a form of borrowing from the
lender at the mortgage contract rate. Therefore, when interest rate increases, delinquency rate
will correspondingly rise as people “borrow” at the relatively cheaper source of fund to
finance other uses. Canner et al. (1991) found that the receipt of government assistance,
headed by a minority, and martial status have positive influences.
On a more somber note, Canner et al. (1991) pointed out that delinquency prediction consists
of a large unexplained random component as credit problems can arise from events that are
difficult to foresee. Thus, the use of ex-ante data has the ability to capture components that
systematically affect delinquency and are observable to the lender at loan origination but
ignores the more unpredictable ex-post components.
2.2.2 Determinants of Mortgage Default
Literature on mortgage loan specific characteristics traditionally focuses on the equity
position of the borrower. Several proxies are used including the loan-to-value ratio at
origination (Campbell, et al., 1983), current loan-to-value ratio (Campbell, et al., 1983;
Cunningham and Capone, 1990), value-to-total debt ratio (Waller, 1988; Zorn, et al., 1989)
and book value (Giliberto and Houston, 1989; Hendershott and Schultz, 1993). Other
mortgage loan specific factors used include the age of the mortgage (Waller, 1989 and
Schwartz and Torous, 1993), mortgage term (Bervokec et al., 1994) and mortgage rate (Zorn,
et al., 1989 and Ambrose and Capone, 1996 and 2000).
9
Property related factors examined include the price volatility of the property (Schwartz and
Torous, 1993; Capozza et al., 1998; Ambrose and Capone, 2000), age (Canner et al., 1991)
and neighborhood quality (Vandell and Thibodeau, 1985). Other significant factors consist of
the returns from property capital appreciation (Schwartz and Torous, 1993; Kau et al., 1994)
and the returns from rental yield (Capozza et al., 1997 and 1998).
With regards to borrower related characteristics, the payment-to-income ratio is a popular
ability-to-pay measure but yields inconsistent results. Vandell (1978) and Campbell and
Dietrich (1983) found a positive relationship while other studies found a negative relationship
(Springer and Waller, 1993; and Cunningham and Capone, 1990). Other studies focus on the
wealth of the individuals and household income (Canner et al., 1991; and Bervokec et al.,
1994), age (Capozza et al., 1997), and the number of years of job tenure (Cunningham and
Capone, 1990; Hakim and Haddad, 1999).
Exogenous factors include demographic or macroeconomic factors. Unemployment is the
more popular factor used by a number of studies that include Campbell and Dietrich (1983),
Lea and Zorn (1986), and Capozza et al. (1997).
2.3 Studies Incorporating the Delinquency-Default Relationship
To the best of our knowledge, this paper is the first to investigate the relationship between
delinquency and default but several past studies do deal with post-delinquency outcomes
without reference to the subsequent default decision.
Herzog and Earley (1970) examined the factors affecting the conditional risk of foreclosure
given delinquency and also the unconditional risk of foreclosure. The results show that after
taking into account the effect of delinquency, although the signs of the variables remain the
same, factors like the loan-to-value ratio, number of dependents, borrower age, and
occupation became insignificant. This constitutes preliminary evidence of the importance of
taking into account the conditional effect of delinquency.
Ambrose and Buttimer (2000) postulated to study lenders’ ability to influence borrower
behavior in post-delinquency situations through the use of deficiency judgments. It defined
10
“delinquency” as having missed one mortgage installment and studied the probability of
foreclosure, reinstatement and prepayment, conditional on delinquency. Ambrose and
Buttimer (1998) examined the conditional probability of foreclosure given default.
Holmes (2003) examined commercial mortgage borrower actions after delinquency occurs
using a game-theoretic model. It postulates that post-delinquency action is either
reinstatement or foreclosure. The paper thus investigates the probability of foreclosure and the
probability of reinstatement given delinquency. The paper also utilized a multinomial logit
specification in two sets of tests.
These papers do not consider the transition from delinquency to default, which is the main
focus of our study. In addition, our paper utilizes the bivariate probit model instead of the
multinomial logit model as we postulate a sequential decision-making framework.
3.0 Theory and Methodology
We motivate our model with the conventional options-pricing theory for pricing mortgages
and the subsequent competing risk methodology. In line with the objectives of the paper, we
develop and argue for a bivariate probit model of delinquency and of default conditional on
delinquency, allowing correlated disturbances.
3.1 Research Methodology and Theory
Competing Risk Methodology
Initially, studies using option models focus on modeling either default (Foster and Van Order,
1984; Quigley and Van Order, 1995) or prepayment (Schwartz, et al., 1989; Quigley and Van
Order, 1990) independent of the other. These studies focus on the role of stochastic processes
of house price and interest rates on borrower decisions. However, the importance of the joint
relationship between default and prepayment is increasingly underscored by Kau and Keenan
(1996), Kau, et al. (1992, 1995), and Schwartz, et al. (1993).
According to Deng, et al. (2000), the proportional hazard model originated by Cox and Oakes
(1984) provides a convenient framework for analyzing the exercise of options. In addition,
Han and Hausman (1990), Sueyoshi (1992) and McCall (1996) provided a maximum
11
likelihood estimation approach to model competing risks simultaneously. Thus, the competing
risks of the prepayment and default options, the joint survivor function conditional on η p ,
η d , r, H, Y, and X can be expressed in the following form:
(
S t p , t d r , H , Y , X ,η p ,η d
)
tp
⎡
'
−
η
⎢ p ∑ exp γ pk + g pk (r , H , Y ) + β p X
k =1
= exp ⎢
td
⎢
⎢− η d ∑ exp γ dk + g dk (r , H , Y ) + β d' X
k =1
⎣
)⎤⎥
(
(
)
⎥
⎥
⎥
⎦
(1)
where Y is a vector of option-related variables other than r and H that will be used to estimate
the market values of the options empirically, X is a vector of other non-option-related
variables to account for other effects on the function. γ
jk
are parameters of the baseline
function while η p and η d are the unobserved heterogeneities associated with the hazard
functions for prepayment and default respectively.
Based on the joint survival function, the explanatory variables are expanded to include a
proxy for the put option to default, a proxy for the call option to prepay, a control for
information asymmetry, and proxies for transaction costs and trigger-events.
The call-option for each loan can be defined as
Call_ Optioni , k =
Vi , m − Vi*,r
Vi , m
where Vi*,r =
TM i − k i
Pi
∑ (1 + r )
S =1
i
S
and Vi , m =
TM i − k i
Pi
∑ (1 + m )
τ
S =1
(2),(3),(4)
S
i + ki
and ri is the mortgage note rate, TM i is the mortgage term, k i is the mortgage duration after
origination at time τ i , mτ i + k i is the market interest rate, and Pi is the monthly mortgage
payment. It shows the fraction of the difference between the present values of the mortgage
balance payment stream at the mortgage rate and the market interest rate. However, since we
are working on ARMs, we expect ri = mτ i + k i . The call-option value, and thus the risk of
prepayment is effectively zero.
The put-option for each loan or the probability of negative equity can be defined as
⎛ (log Vi , m − log M i , k ) ⎞
⎟
Put_ Optioni , k = Φ⎜
⎟
⎜
2
ω
⎠
⎝
(5)
12
where Φ (.) is cumulative standard normal distribution function, M i , k is the market value of
the property and
ω2 is the estimated variance.
Deng, et al. (2000) described the presence of asymmetric information where the borrowers
know more about their own house price volatility than the lenders. Thus, risky houses will be
financed with higher initial loan-to-value (LTV) ratios. The paper uses the initial LTV ratio as
a covariate in an attempt to control for asymmetric information. In addition, we propose the
number of borrowers and mortgage term as proxies for asymmetric information.
The contingent claims approach utilized in pricing of mortgages requires the assumption of
perfect capital markets and borrowers act in an impersonal way on maximizing their financial
positions (Kau, et al., 1993). Thus, pure options-theoretic default models postulate default to
occur once the value of the property falls below the value of the mortgage (Kau, et al., 1992,
1995). However, it is well established that borrowers do not default straight away when their
mortgages are in negative equity due to transaction costs like default penalty, deficiency
judgment, cost of moving, etc. As a result, the frequency of actual defaults is under-predicted
by pure option-theoretic default models (Foster et al., 1984, 1985; Cunningham, et al., 1984).
In addition, the extent of this transaction costs can be more logically treated as differing from
individual to individual. Given a portfolio of mortgages with largely similar characteristics,
some borrowers will default once the mortgages falls into negative equity (“pure” rational
default) while other borrowers will not default even when their mortgages are deep in
negative equity due to the presence of “personalized” transaction costs (Kau and Keenan,
1995). Therefore, Stanton (1995) and Green and LaCour-Little (1999) modeled transaction
costs as varying across borrowers.
We postulate to motivate the use of observable personalized factors like borrower, property
and mortgage characteristics to proxy transaction costs in modeling delinquency/ default
decisions. Deng and Gabriel (2002) and Clapp et al. (2004) included these variables without
directly motivating them but also found them to be influential in estimating mortgage
termination risks.
13
In addition, the presence of a “trigger-event” may be necessary to force the borrower to
default on the mortgage. Since trigger-events are exogenous, we should incorporate macro
level variables to proxy for them. As the responses of these events are differentiated among
individuals, this is additional motivation to include the personalized factors.
Despite of the inclusion of proxies for transaction cost and trigger-events, estimations may not
be perfect due to unobserved/ unmeasured heterogeneity among borrowers as described by the
unobserved error terms, η p and η d . These include borrower tastes or abilities, and propertyspecific factors such as unexpected depreciation or appreciation. Our paper focuses on the
correlation of these unobserved factors. A strong correlation between the unexplained
tendency to delinquent and the unexplained tendency to default would substantiate the
inclusion of delinquency decision into existing default models.
3.2 Empirical Model
The Sample Selection Problem
Past studies that utilize the proportional hazard model (e.g. Deng, et al., 1996 and 2000)
multinomial logit methodology (e.g. Vandell and Thibodeau, 1985; Zorn, et al., 1989; and
Ambrose and Capone, 1998) that assumes that the borrower decides on the course of action at
a single point in time. However, it is likely that the borrower makes his decisions on a
sequential basis. The sequential nature of the borrower decisions is illustrated in Exhibit 2.
As can be seen, the sequential nature of borrower decisions presents us with the sample
selection or incidental truncation issue. The loans that are in default must have gone through
delinquency as a preceding step, i.e. the dependent variable of the default equation yields a
value only if the loan is delinquent. If we do not incorporate the influential variables of the
delinquency decision into the subsequent default decision, the latter will produce biased result
(unless ρ = 0). Inconsistency in the parameters estimated will make any inferences dubious.
As mentioned, when ρ is not equal to zero, the default equation and delinquency equation may
contain some common unexplained or omitted variables. In our research question, we expect
the variables that increase the probability of delinquency to increase the risk of default. We
14
also expect the unobserved factors of the two decisions to be positively correlated. The
presence of such unobserved similarities between the two decisions will make a one-level
decision-making framework like multinomial logit or conditional logit inappropriate. Ditto for
independent probit or logit models.
The delinquency decision and the subsequent default decision process constitute a bivariate
discrete dependent variable model that exhibits the selection bias. The multivariate probit
model (MVP) allows for simultaneous estimation of multiple equations to account for
different motivations for each equation and it allows interdependence between the multipleequations through the covariance matrix. We can correct for the selection bias by applying a
concept initially developed by Heckman (1979) for continuous variable. Wynand and Bernard
(1981) further the concept by applying it to discrete dependent variable analysis. Utilizing a
joint approach, the bivariate probit model (BVP) with selection accounts for the potential
correlation between the two decisions, and corrects for selection bias.
In addition, another advantage of utilizing the BVP over single-level models is that
independence from irrelevant alternatives (IIA) need not be assumed. IIA, which follows from
the assumption of independence of random errors, implies that the odds ratio does not depend
on other choices. However, studies on consumer behavior require the relaxation of the IIA
assumption (Greene, 2003) due to the presence of close substitutes for the choices.
The nested logit model (NL) is often presented as an alternative to the BVP for relaxing the
IIA assumption across clusters of choices. However, choices within a cluster have to maintain
the IIA, i.e. the individual unobserved disturbances at any given level and across levels of the
decision framework within a cluster are to be independently distributed (Hunt, 2000).
Consequently, as delinquency and default decisions belong to the same cluster (see Exhibit 2),
the NL does not allow for disturbance correlation between them3. Since we are interested in
the significance of including delinquency in existing default models, we should adopt a model
that allows correlation rather than assume zero correlation between their disturbances. In
addition, although the decision structure in Exhibit 2 looks like a nesting structure, it does not
3
It does, however, allow the composite disturbances that share an upper-level to be positively correlated [i.e.
Cov (ε1, ε2‫׀‬ε1) > 0].
15
assume a simultaneous decision-making process as is required by NL (Knapp, White and
Clark, 2001). The structure is intended to represent a sequential decision-making process.
The Bivariate Probit Model (BVP)
To examine the relationship between the probability of delinquency and the conditional
probability of default given delinquency, our empirical model requires an assumption of a
delinquency probability function and a default probability function where the ith borrower
maximizes a linear indirect utility function V ij* over j outcomes
Vij* = α j + β j X i + δ j W i + φ j Z i + ε ij , i = 1, ….., N, j = 1, ….., J
(6)
where Xi is a vector of option-related characteristics, Wi is a vector of individual borrower,
property and mortgage characteristics and Zi is a vector of macroeconomic variables other
than house price and interest rate.
Utilizing the utility theory or rational choice perspective on behavior as developed by
McFadden (1973), and using subscript 1 and 2 to represent steps 1 and 2 of the default
decision-making framework, the BVP specification for (6) would be
V i1*
=
x'i1 βi1 + ε1 ,
yi1 = 1 if V i1* > 0, 0 otherwise,
(7)
Vi *2
=
x'i2 βi2 + ε2 ,
yi2 = 1 if V i1* > 0, 0 otherwise,
(8)
E [ε1 | x1, x2] = E [ε2 | x1, x2] = 0
(9)
Var [ε1 | x1, x2] = Var [ε2 | x1, x2] = 1
(10)
Cov [ε1, ε2| x1, x2] = ρ.
(11)
where Vij represents the unobservable indirect utility developed above, y1 is the observable
actual decision for default incidence in the second step decision, y2 is the observable actual
decision for delinquency incidence in the first step decision, and x1, x2 represents the vector of
independent variables affecting y1 and y2 respectively. The relationship between delinquency
and default decisions is examined by testing the covariance of their disturbances, ρ.
The bivariate normal CDF is
P ( X 1 p x1 , X 2 p x 2 ) = ∫
x2
−∞
∫ φ (z , z
x1
−∞
1
2
, ρ )dz 1 dz 2
(12)
16
which we denote as Φ 2 (x1 , x 2 , ρ ) where φ (.) is a notation for standard normal distribution
and the subscript 2 indicates its bivariate nature. The density is
φ 2 ( x1 , x 2 , ρ ) =
e
( 2 )(x
− 1
2
2
1 + x 2 − 2 ρx1 x 2
(
2π 1 − ρ
2
)
) / (1− ρ )
1
2
(13)
2
Due to the selection bias, there are thus three types of observations in our sample:
a) Continue Payment (CP) (y2 = 0)
:
P( y 2 = 0 ) = 1 − Φ( x 2 β 2 )
(14)
b) Delinquent but Reinstated (DR) (y1 = 0, y2 = 1) :
P( y 2 = 1, y1 = 0) = Φ( x 2 β 2 ) − Φ 2 ( x 2 β 2 , x1 β 1 , ρ ) ,
(15)
c) Delinquent and Default (DD) (y1 = 1, y2 = 1)
:
P( y 2 = 1, y1 = 1) = Φ 2 ( x 2 β 2 , x1 β 1 , ρ ) .
(16)
where φ (.) is a notation for standard normal distribution and the subscript 2 indicates its
bivariate nature.
The likelihood function is
L = ∏ P (CP )× ∏ P ( DR ) × ∏ P ( DD)
CP
n
DR
(
=∏PV <0
*
2i
)
(1− y 2 i )
i =1
DD
(
n
× ∏ P V ≥ 0, V ≤ 0
*
2i
*
1i
)
( y 2 i × y1i )
i =1
n
(
× ∏ P V ≥ 0, V ≥ 0
*
2i
*
1i
)
( y 2 i × y1i )
.
(17)
i =1
Substitute (7) and (8) into (17)
n
n
i =1
i =1
log L = ∑ (1 − y 2i ) ln[P(ε 2i < − x 2i β 2 )] + ∑ y 2i (1 − y1i ) ln[P(ε 2i ≥ − x 2i β 2 ∩ ε 1i ≤ − x1i β 1 )] +
∑
n
i =1
y 2i y1i ln[P(ε 2i ≥ − x 2i β 2 ∩ ε 1i ≥ − x1i β 1 )]
(18)
With symmetry of the bivariate normal distribution,
P(ε 2i ≥ − x 2i β 2 ∩ ε 1i ≥ − x1i β 1 ) ⇔ Φ 2 ( x 2i β 2 , x1i β 2 , ρ )
(19)
In addition,
(
)
(
) (
P y 2*i ≥ 0, y1*i ≤ 0 = 1 − P y 2*i < 0 − P y 2*i ≥ 0, y1*i ≥ 0
)
(20)
17
Thus, the log-likelihood function is
n
n
i =1
i =1
log L = ∑ (1 − y 2i ) ln[1 − Φ( x 2i β 2 )] + ∑ y 2i (1 − y1i ) ln[Φ( x 2i β 2 ) − Φ 2 ( x 2i β 2 , x1i β 1 , ρ )] +
∑
n
i =1
y 2i y1i ln[Φ 2 (x 2i β 2 , x1i β 1 , ρ )]
(21)
The subsequent estimations are undertaken via the maximum likelihood estimation.
The relationship between delinquency and default decisions is examined by testing the null
hypothesis that the covariance of their disturbances, ρ, is equal to zero. The likelihood ratio
test, Wald test and the Lagrange multiplier statistic are utilized. We run (7) and (8) separately
via the independent probit models (where ρ = 0) to serve as a base from which to compare our
results of the bivariate probit model.
3.4 Covariates of Model
3.4.1 Option-related Variable
Proxies put option indicates the probability of negative equity and thus allowing foreclosure.
It is expected that higher the probability of negative equity would induce delinquency.
3.4.2 Mortgage Loan Specific Characteristics
There are two aspects to Price Premium, which is calculated as the ratio of the difference of
purchase price and valuation, over valuation. Firstly, the fact that borrower is willing pay an
amount excessive to the fair value implies high preference to the property. As such, his
transaction cost of foreclosure will be higher and he will try to keep the mortgage current, i.e.
the risk of delinquency will be lower. Secondly, paying a premium may cause the borrower’s
finances to suffer as more savings are spent on the property. This will increase the risk of
delinquency when trigger-event occurs.
It is highly probable that Central Provident Fund (CPF) funds will first be utilized to pay for
the property before borrowing the rest of the purchase price, subject to certain stipulated
limits. CPF is the mandatory savings scheme in Singapore where both the employer and
employee contributes to the fund. The use of CPF funds reduces the loan quantum and thus
18
the monthly mortgage payments. This increases the affordability of the installments and
contributes to the accumulation of financial resources, thus enhancing borrower’s ability to
meet any unexpected financial commitments.
It can be argued that borrowers’ decision on the mortgage term is a confluence of borrowers’
assessment of their financial abilities and financial commitments. Thus, Mortgage Term can
serve as a useful proxy for information asymmetry for borrowers’ financial abilities, which
may not be entirely identified by lenders.
The premium of mortgage rates over the prime rate is a measure of lenders’ perceived
riskiness of a borrower. The premium is to compensate the lenders for approving loans that
are perceived to be of higher risk. Thus, higher premium is reflective of the higher risk of the
borrower. We utilize the mortgage rates for each loan and the prime lending rate as at August
2002 to calculate the risk premium.
An interesting feature of the ARMs in Singapore is that most loans consist of a preferential
fixed rate for the first two to three years (Ong, 2002). The below-market rates enticed new
borrowers. On one hand, the preferential fixed rate improves the affordability of borrowers
and enables borrowers to accumulate more non-housing wealth for the initial period. On the
other hand, such schemes are anticipated to attract higher risk borrowers whom just qualify
for the ‘teaser rates’. Dummy variables are used to differentiate the loans where loans with
initial preferential rates allocated the value of 1, otherwise are allocated 0.
3.4.3 Property Specific Characteristics
The tenure of residential properties in Singapore is essentially categorized into either 99-year
leasehold or freehold properties. Holding affordability constant, it is documented that people
would prefer freehold properties for continuity. Higher preference generally implies higher
transaction costs and leads to lower risk of delinquency. Dummy variables are used to
differentiate the effect of the type of lease (99-leasehold properties are allocated the value of
1; and otherwise are allocated 0). The properties can also be classified as either low-rise or
high-rise, where people preferring the former. Dummy variables are again used where the
latter is allocated the value of 1; and otherwise is allocated 0.
19
An independent variable that is used for low-rise properties is the Land Area. It is expected
that with other things held constant, people would prefer larger land area. A determinant for
high-rise properties is the floor level the property is located (Ong and Koh, 2000). People
generally prefer to live on higher floors. Delinquency risks are anticipated to be lower for
properties that the purchasers fancy. Another variable used is the built-up area of the property.
3.4.4 Borrower Specific Characteristics
The Payment-to-Income ratio directly indicates the borrowers’ abilities to pay the mortgage
installments. The ratio at the time of origination is one of the criterions used by lenders to
assess the credit worthiness of potential borrowers. A higher ratio reflects lower affordability
and greater difficulties of keeping the mortgage current in the face of trigger financial events.
The initial mortgage payment and total household income at the time of origination is used.
Some studies suggest a higher number of co-borrowers would lead to lower mortgage risks.
This can be motivated as higher total household income (Bervokec, et al., 1994; and
Lambrecht, et al., 1997). Neo and Ong (2004) suggest risk sharing as a reason for the negative
relationship with foreclosure risks. However, we postulate that the number of co-borrowers
may be an indication of information asymmetry. Borrowers are aware of their wealth and
financial commitments. They would include more co-borrowers if their financial
circumstances are less favorable, i.e. information asymmetry. From the lenders’ point of view,
they would require more borrowers if the borrowers are perceived as risky. Thus, a higher
number of borrowers essentially imply higher risks.
We assume that older borrowers would accumulate financial resources that form a foundation
against any trigger financial commitments, thereby reducing the risk of delinquency.
Purpose of purchasing the property can be categorized into either for owner-occupation or for
investment. Owner-occupiers tend to have emotional attachment to the property as their
homes and hence higher transaction cost. They are thus more motivated to continue to pay the
mortgage when financial difficulties strike and require a more serious negative equity to
default. On the other hand, investors are motivated by the profit motive. They may take more
20
risks leading to an inability to pay the mortgages when financial difficulties occur. They may
also be dependent on the rents received to pay the mortgage. Dummy variables are utilized to
categorize the purchasers where investors are assigned the value of 1, and 0 if otherwise.
The certainty of future flows of income is proxied by the number of years the borrower with
the highest income has been in his current employment (Vandell, et al., 1985; and
Cunningham, et al., 1990). With the payment-to-income ratio to indicate affordability at
origination, certainty of future flows of income proxy the probability of continued ability to
upkeep the mortgage in the future.
A potential significant factor is occupation of the youngest borrower. Occupations as
professionals, executives and managers earn stable income while self-employed persons and
sales persons earn unstable income. The stability of future income streams proxy the
probability of continued financial ability. A dummy variable of 1 is allocated to unstableincome occupations, otherwise is allocated 0.
3.4.5 Environmental Characteristics
Market sentiments can proxy the returns on other investments (Zorn, et al., 1989). When
market sentiments are good, funds will be directed away from mortgage payment to other
more attractive investments. Conversely, poor sentiments imply a lack of good investments
that borrowers can park their money in. Accordingly, funds will be better used in repaying the
mortgages to prevent incurring late payment penalties. This is similar to the argument put
forth by Ong (2000) and Ong et al. (2002) although the research was on prepayment risk. It is
well noted that changes in property prices do reflect changes in fundamentals and sentiments
(Ong, 2000 and Ong et al., 2002). However, it is liable to lag the current market sentiments.
On the other hand, better market sentiments represent higher returns from the borrowers’
other investments. This will increase their financial wealth, which improves their ability to
withstand financial shocks. Market sentiments are proxied by change in the Straits Times
Index (STI) that is a price-weighted index consisting of 30 major stocks in Singapore.
Retrenchments will affect borrowers’ abilities to continue with the mortgage payments. The
threat of retrenchments and uncertainty of future income affects borrowers can be measured
21
by the change of unemployment rate from the origination date to the date of delinquency or if
there is no delinquency, the date of censor is used. On the other hand, changes in Gross
Domestic Product (GDP) can proxy the change in income of the borrowers since we do not
have the individual income progression of the individual borrowers.
3.4.6 Model Controls
Information Asymmetry
The first model control is to account for the presence of information asymmetry whereby
borrowers are more aware of their house price volatility. Deng, et al. (2000) argued for this
and utilized the initial Loan-to-Value ratio (LVR) at the time of origination as a proxy. In
addition, we also use the mortgage term and the numbers of borrowers of a mortgage can also
proxy information asymmetry.
Guaranteed Sum and Time to Wealth Realization
The presence of “guaranteed” equity is a unique feature in Singapore and should be
appropriately controlled in our model.
The equity component of a mortgage usually would be derived from the home purchaser’s
cash savings. Thus, for a loan-to-equity ratio of, say 0.8, the potential homeowner needs to
cough up cash and pay upfront the 20 per cent house value or the purchase price. This equity
component is usually one of the main loan criteria for the lender to control the default risk of
the mortgage. The equity component is used as a buffer against immediate default when the
value of the property falls. It is only when property value falls beyond the stipulated margin
of, say 20 per cent, which lenders will suffer a loss and face potential defaults.
In Singapore, to encourage homeownership, potential homeowners can use a combination of
cash and CPF savings (a compulsory retirement savings scheme which both employer and
employee have to contribute a stipulated percentage of the salary to) to pay for the equity
component of the purchase price upfront so as to reduce the loan quantum. The CPF can also
be utilized to pay the monthly installments of the mortgage.
22
In order to protect the retirement savings of Singaporeans, rulings stipulate that if the property
is sold by the lender under foreclosure, proceeds from the sale must first be used to repay the
entire amount of CPF funds utilized4. Therefore, the equity component that is derived from
CPF funds is “guaranteed” equity. We shall term the other component “cash equity”.
As the property value declines, the borrower will suffer loss in his cash equity component.
When the entire cash equity component is wiped out, the borrower does not suffer any more
losses as the CPF component of his equity is guaranteed. Any further losses are transferred to
the lender. This reduces the potential loss of borrower’s equity and makes it easier to default.
As a result, the distribution of the equity between cash and guaranteed equity will have a
differential impact on default decisions and loss aversion. Thus, we include a control variable
for the amount of guaranteed equity for each observation.
In addition, we expect a temporal effect on borrower behavior due to the use of CPF savings,
which can only be withdrawn at retirement age. As property value continue to decline after
when the cash equity component has been wiped out, the optimal borrower stands to gain by
default if he exchanges the lower valued property for the higher valued mortgage contract.
However, another implication of the use of CPF funds for mortgage payments is that such a
financially optimal gain cannot be realized immediately. The gain is deposited back to the
borrower’s CPF account upon foreclosure sale and can only be withdrawn upon reaching
retirement age. This gain is thus “deferred”. The period before this gain can be realized
depends on the length of time before retirement.
We postulate the time to retirement to be positively correlated to the risk of default. When the
time to retirement or the time to wealth realization is long, the borrower may be reluctant to
default although he may gain, as such a gain can only be realized in the future. An alternative
explanation is that the gain is insufficient for the borrower to default after discounting for the
temporal effect; the gain may no longer be attractive. The “deferred” gain behaves
increasingly like “current” gain as the time to wealth realization becomes shorter.
4
Since October 2002, this ruling has been amended to allow lenders first lien to the foreclosure proceeds. This
implies that the CPF component is no longer guaranteed. As before, the cash equity component is similarly
reduced when property falls. For the CPF equity component, the borrower will suffer a loss as the property value
continues to fall.
23
Thus, we expect a positive (negative) relationship between age of a borrower (time to
retirement) and the risk of default after controlling the potential amount of gain through the
guaranteed equity. It is important to note that AGE has been included in the model, and the
postulated effect is negative. Therefore, the effect of the temporal effect has to be very
significant before it can overcome the inherent negative effect to exhibit a positive sign.
CPF Contributions and its Role in Averting Delinquencies
As mentioned, the CPF can also be used to pay the monthly installments for the mortgage.
The monthly contribution to the borrower’s CPF account becomes a stabilizing source of fund
to upkeep the mortgage. When trigger-events strike, the borrower’s budget may be adversely
affected that he is unable to use cash to pay for the installments. He can then increase the
amount of CPF savings that are used. Thus, the monthly CPF contributions become a source
of funds from which the borrower can rely on in times of crisis.
We calculate the Mortgage Installment-to-CPF Contribution ratio, which is the ratio of the
monthly mortgage installments to the monthly contributions to the borrower’s CPF Ordinary
account as a control variable and it is expected that the excess capacity of the monthly CPF
contributions (lower installment-to-contribution ratio) has a negative (positive) relationship
with delinquency/ default risk. However, we will include an interactive variable of the ratio
multiplied by a dummy variable where value of 1 represents borrowers who are unable to pay
their entire mortgage installment using CPF savings. We exclude borrowers who are using
their CPF to pay for the entire monthly installments as trigger events that affect their cash
flows no longer affect their ability to upkeep the mortgage.
4.0 Data and Descriptive Statistics of Variables
The data used in the empirical estimation is based upon a major insurer in Singapore whose
business portfolio includes the issuances of residential mortgages. The database consists of
633 random samples of individual housing mortgages and the observations of delinquency
and default are taken monthly, from January 1999 to August 2002. A total of 133 cases have
become delinquent at certain times within the period of analysis and a total of 55 observations
have been in default. The dataset provides a rich variety of micro-level borrower, loan and
24
property characteristics for our model estimations. Exhibit 3 presents the determinants, codes
used and the expected signs of influence.
The summary descriptive statistics is shown in Exhibit 4. The origination dates of the sample
range from March 1980 to December 1999. Since only 14 cases originated before 1991, a
better measure of central tendency would be the median at 1998. The average loan amount is
$363,697 with standard deviation of $161,537.
The averages of PUTOP1 and PUTOP2 are 0.1260 and 0.1255 and their standard deviations
are 0.1165 and 0.1161 respectively. The average valuation is $670,357 with a higher range of
$447,000 to $3,400,000. PREMIUM is shown to range from –50% to 52.27% while the
average value is close to zero. The amount of CPF lump sum used by the borrowers range
from zero (not used) to $631,000. The resultant CPFPRICE ranges from zero to 92.21%. The
average CPFPRICE is 17.41%. The average mortgage term (MT) is 24.08 years, which range
from 3 to 33 years. The breach of the stipulated maximum loan term of 30 years and the odd
number of years is due to negotiations between the delinquent borrower and the lender after
loan origination to extend the period over which the loan shall be paid. As for RISKPRE, the
average value is –1.4365 with a standard deviation of 0.4712. 66.82% of the sample has initial
preferential rates (PRM).
73.52% of the sample cases are leasehold properties and the remainders are of freehold
tenure. Property type (TYPE) is dominated by high-rise properties. 546 (86.26%) of the
mortgages were backed by either condominium housing or apartments. Terraces, semidetached housing or detached housing, backed the remaining mortgages. The average land
area (LAREA) of the low-rise properties is 2,436 sq ft and it ranges from 1,317 sq ft to 8,256
sq ft depending on whether they are terraces, semi-detached or detached housing, in ascending
order of the level of land area. The floor levels (FLOOR), which the high-rise properties are
located, range from 1st to 33rd storey with the average level of 6.8163. The mean built-up area
(BUAREA) is 1515.85 sq ft.
Monthly mortgage installments payable has an average of $2097.94. The corresponding
PINRATIO ranges from 0.0095 to 0.8412 with an average of 0.2654. BORROWER varies
25
from 1 to 5 with a mean of 2.0695. PURPOSE is dominated by that of owner-occupation at
95.58% or 605 cases while the reminder is purchased for investment. The average YRSEMP
is 9.2914 years with standard deviation of 7.6898.
In the periods under study, the change in unemployment rate (CUNEMP) has range from 0.3231 to 1.750 with an average of 0.5988. Generally, unemployment rates have been
increasing due to the economic crisis in the region. However, due to the relatively large
standard deviations, the median can be a more precise measure of central tendency.
The mean CUNEMP1 for the period is 0.5229 while that for CUNEMP2 is 0.5147. Their
standard deviations are similar at around 0.58. The average of CSTI1 and CSTI2 are relatively
low at 0.02606 and 0.03351 respectively. The relatively large standard deviations when
compared with the corresponding mean values suggests that there have been reasonably large
fluctuations in the STI over the period from the origination dates of the loans to the
delinquency and default dates or the censor dates. Similarly in the changes in Gross Domestic
Product (CGDP1 and CGDP2), the means are 0.1781 and 0.1801 while the standard
deviations are 0.2147 and 0.2148 respectively.
Finally, the mean value of LVR is 0.56662. GUARSUM has an average value of $199,167.74,
which ranges from zero (for borrowers not using CPF savings for mortgage payments) to
$1,131,160.16. The mean value of MORTCPF is 1.6772 with a standard deviation of 2.6996.
5.0 Empirical Analysis
The summary and full results of the bivariate probit (Model 1) and the independent probit
(Model 2) models for delinquency and default are presented in Exhibit 5 and 6 respectively.
5.1 Disturbance Correlation of Delinquency and Default
Tests are carried out to assess the correlation of the disturbances of delinquency and default.
The null hypothesis is that ρ equals zero, i.e. the model consists of independent probit
equations, which can be estimated separately. If so, the bivariate probit model will not be
superior to the independent probit models.
26
The model estimate for ρ (0.9945 with p-value of 0.0000) 5 provides strong preliminary
evidence of a close relationship between the unobserved components of delinquency and
default. For robustness, we examine other tests of disturbance correlation.
The Lagrange multiplier statistic is 497.9805. The critical chi-square value at 1 degree of
freedom is 3.84, which rejects the null hypothesis. The Wald test yields (0.9945/0.1831)2 =
29.5245, which leads to the same conclusion. The Likelihood Ratio statistic, based on log
likelihood of the unrestricted and restricted model (calculations are not included in the paper),
is 2[-245.6257 – (- 716.3548)] = 941.4582. The null hypothesis is similarly rejected.
The rejection of the null hypothesis shows that after accounting for the observable
determinants, the unexplained components of the two risks are still highly correlated. The
closeness of the relationship suggests the importance of building delinquency into default or
mortgage models to efficiently and accurately examine mortgage risks. The high correlation
increases the efficiency of utilizing the bivariate probit model, and supports our model choice.
5.2 Significant Covariates of the Bivariate Probit Model
Option-related Variable
The results reveal that a higher PUTOP encourages both delinquency and default. The results
provide support for the option theory, which shows that stochastic movements in the interest
rates and house prices are important in determining delinquency risk, and are in line with
theoretical predictions.
However, there is a difference between Model 1 and Model 2. After taking into account the
correlation between the disturbances of the two decisions, PUTOP for Model 1 is no longer
significant. If we had relied solely on the results of Model 2, we would risk placing too much
emphasis on the probability of negative equity as a determinant of delinquency and default.
Mortgage Specific Characteristics
5
Although the value of p is quite close to 1, there are no problems with convergence for the algorithm. The
correlation coefficient also seems to be relatively stable or robust, varying no lesser than 0.95 depending on the
number and type of variables we included in our model.
27
The results from CPFPRICE show the importance of building the effects of delinquency in a
conditional default model. The expected negative sign in CPFPRICE is only found in default
equation of Model 1. If we only examine the independent model, we would be tempted to
conclude that the use of CPF funds actually elevates mortgage risks with the positive
relationships with delinquency and default incidence. The bivariate probit model enables us to
recognize the although using CPF funds to pay for part of the property value increases
delinquency incidence, once the loan is in delinquency, the risk of default is reduced.
Model 1 reveals consistent positive relationships between MT and delinquency and default,
while the delinquency equation for Model 2 shows a negative relationship. The presence of
information asymmetry can better justified when we take into consideration the disturbance
correlation. A higher MT may be reflective of information asymmetry where the borrower has
greater awareness of his limited financial capability as compared with his commitments.
PREMIUM yields consistent negative signs in both our models, contrary to our expectations.
One possible explanation is that given the borrower is willing to pay an amount excessive to
the fair value of the property, he must have a high preference for that property. This would
inflate his transaction cost of foreclosure thus reducing the risk of delinquency and default.
This suggests that a higher price premium paid may be more reflective of the preference for
the property, rather than the lower household finances that result.
We found significant negative relationships between PRM and both the models. This reveals
that the use of the initial “teaser rates” is able to reduce mortgage risks. The lower initial
monthly payments supposedly reduce the financial stress of borrowers such that the borrower
is able to continue upkeep the mortgage even after the preferential period. This provides a
strong imperative for the lender to propagate mortgages of such nature. Similarly, our results
reveal RISKPRE to be positively correlated to delinquency and default in both models, in
accordance to our prior.
Property Specific Characteristics
Generally, property specific characteristics exhibit relative consistency among between the
bivariate probit model and the independent probit models. TYPE and FLOOR both yield signs
28
in accordance to theoretical predictions. However, LAREA exhibit signs that are contrary
with our expectations.
Results from TENURE and BUAREA provide us with further rationalization of the use of the
bivariate probit model. Model 2 shows that the signs of both variables are contrary to our
predictions. However, after including the effects of correlation between delinquency and
default, the signs of the default equation of Model 1 exhibit the correct directions. Triggerevents may initially cause delinquency even though the borrower may incur higher transaction
costs. The borrower may feel the threat of incurring the transaction costs more strongly after
initial delinquency when default seems imminent and work towards reinstatement.
Borrower Specific Characteristics
Majority of the borrower specific variables are consistent between the bivariate probit model
and the independent probit models. BORROWER, PURPOSE and OCCUP yield signs in
accordance to theoretical predictions. In particular, the consistently positive relationship
found in BORROWER shows the information asymmetry effect is dominant over possible
risk sharing among the co-borrowers.
Contrary to our initial postulation, AGE yields positive relationships with delinquency and
default in both models. As mentioned, the implication of the positive influence of AGE may
be the result of the temporal effect of wealth change realization on borrower behavior.
PINCRATIO and YRSEMP show signs opposite to our expectations although they are not
significant. The former suggests that financial institutions’ reliance on ability-to-pay measures
to assess credit risks may be flawed. It may be the case that people with higher ability-to-pay
happen to possess other characteristics that may be more susceptible to default. The latter
indicates that the length of time in current job may not be a good indicator of continued
income stability.
Environmental Characteristics
29
CUNEMP is significant although the sign opposite with that predicted. This finding is
interesting but not without precedent. Cunningham and Capone (1990) found such a
relationship between regional unemployment rate and prepayment and default rate.
CGDP possess signs in accordance to our expectations and is significant except for the default
equation of Model 1. It seems that the contribution of the bivariate probit model here is a
reminder that a growing economy does not necessarily mean that all is good and well for a
mortgage portfolio. Improving economic growth may proxy a rise income for borrowers,
which discourages delinquency. However, for loans that go into delinquency, the probability
of transiting to default is higher. This is foreseeable as borrowers who delinquent despite the
good economic results are those who have suffered a major trigger-event, which makes the
transition to default inevitable. Lenders have to thus invest more attention to loans that
delinquent in good economy.
The positive signs found for CSTI shows the dominance of the variable to proxy returns on
other investments (Zorn and Lea, 1989). This suggests that borrowers may capitalize on good
market sentiments to increase their financial resources as a foundation against delinquency
and default. Significance at the 1 percent level is present in all equations except in the default
equation of Model 1. This lack of significance suggests that although the borrower may
transfer financial resources to exploit attractive alternative investments, he would generally be
prudent enough to avoid default. This inconspicuous inference can be easily overlooked if we
only examine the independent probit models.
Model Controls
The initial LVR is found to be positively correlated to the delinquency risk and negatively
correlated with default risk. The lack of significance and mixed signs suggest that the
argument of information asymmetry may not present in our model. Borrowers do not seem to
be able to increase their LVR when their house price volatility is higher. The reasons may
include a good knowledge of the local property market by the lenders, and the legal restriction
that caps the original LVR at 0.8. MT and BORROWER, potential variables to proxy for
information asymmetry are found to possess the correct signs. These provide some evidence
for the presence of information asymmetry.
30
We expect GUARSUM to be an impetus for borrowers to go into delinquency and default.
The results show the contrary. This is interesting as Singapore practitioners usually deem
guaranteed nature of the use of CPF funds to encourage delinquency and default. If it can be
established otherwise, a major impediment against securitization of mortgages in Singapore
can be reduced. We have to tread with caution due to the lack of significance for the default
equation for Model 1.
DMORTCPF exhibits consistent signs for both models and is significant for the delinquency
equations. Excess capacity of the borrowers’ CPF funds to pay for the monthly installments is
shown to be useful in averting delinquency.
5.3 Implications of Results
Firstly, our tests found a significant correlation between the disturbances of delinquency and
that of default. This indicates that importance and efficiency gains of constructing the effects
of delinquency into existing mortgage default and prepayment risk models. Although this
conclusion may seem logical since delinquency is defined as a necessary precursor to default,
the relationship has been ignored in past studies. By taking into account the two decision
nodes, the efficiency and accuracy of mortgage decision models can be improved. Practically,
lenders should engage in negotiations with the borrowers once delinquency occurs as the
unexplained components of delinquency push up the default risks as well. This contrasts with
certain traditional lenders’ practice of contacting the borrowers only after default occurs.
Secondly, we also examined the independent variables of delinquency and of conditional
default given delinquency. Our emphasis is on the disparity of the directions of influences of
the variables affecting delinquency and default decisions between the bivariate probit and
independent probit models. We also examine the differences of the directions of influence of
variables affecting delinquency and conditional default of the bivariate probit model.
The results show several variables to have signs different between the bivariate probit model
and the corresponding independent probit models. These include CPFPRICE, MT, TENURE
and BUAREA for default models. The difference in signs for the conditional default model as
31
compared to the independent default model shows that delinquency decision exerts an
influential effect on default decision. We also found several instances where there is a lack of
significance for the conditional default equation while being significant for the independent
default equation. These include CGDP2, CSTI2 and GURASUM. Therefore, if we rely solely
on the independent models, our predictions could be erroneous.
In addition, we found several instances where the signs of influence are opposite between
delinquency incidence and conditional default incidence for the bivariate probit model. These
include CPFPRICE, TENURE, BUAREA, LVR, and DCPF. These indicate that a same set of
variables may affect delinquency risk and default risk in different ways. Besides providing
further verification of the importance of including delinquency in mortgage risk assessment, it
also has implication for the lenders. Simply relying on conventional preconceptions on the
directions of influences of certain variables may cause lenders to oversight on the true risks of
their mortgage portfolio. They also have to understand that certain variables may exert a
trade-off relationship with different types of mortgage risks. By controlling some loan criteria,
the lender may decrease certain risks, but may inadvertently increased other risks. Moreover,
the presence of information asymmetry reveals lenders’ inabilities to uncover borrowers who
are inherently more risky.
The main limitation of this study was the limited sample size and period of study. Further
analysis using a larger sample size and a longer study period may be required to further verify
the results obtained in this study.
6.0 Conclusion
This study has contributed to existing mortgage literature by verifying the significance of the
relationship between delinquency and default, identifying the influential risk factors in a
bivariate probit model, and comparing the results with the independent probit models.
More practically, it has also simultaneously revealed critical implications for lenders and
investors in the mortgage and MBS market respectively. The presence of the close
relationship between delinquency and default decisions suggests that to reduce losses
attributed to default or foreclosure, lenders should attempt to minimize the risk of
32
delinquency. However, lenders must be aware that controlling for one type of mortgage risk
e.g. delinquency risk may cause another type of mortgage risk e.g. default risk to increase.
Thus, lenders have to adopt different strategies at different stages of a mortgage’s lifecycle.
Finally, due to the presence of information asymmetry, lenders may have to rely on
alternative yardsticks to identify riskier borrowers rather than conventional criterions.
33
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36
Exhibits
Exhibit 1 Differing Motivations of Delinquency and Default
Optimal
Trigger-event
Delinquency
Time
Default
Financial difficulties
in paying installments
due to exogenous
events
Financial position not
improved sufficiently
to enable reinstatement
Wealth maximizing to
delinquent due to
negative equity
position
No favorable changes
in net equity position
to support
reinstatement
Exhibit 2 Borrowers’ Sequential Decision-Making Framework
Borrower
Decisions
Continue
Payment
1
Delinquency
Reinstatement
Reinstatement
2
Default
Termination
3
37
Exhibit 3 List of Determinants, Codes and Expected Signs of Influence
Variable
Code
Option-related Variables
Put Option Value as at Delinquent Date or Censor Date
PUTOP1
Put Option Value as at Default Date or Censor Date
PUTOP2
Mortgage Loan Specific Characteristics
Price Premium
PREMIUM
CPF-to-Price Ratio
CPFPRICE
Mortgage Term
MT
Premium of Mortgage Rate over Prime Rate
RISKPRE
Preferential Rate Characteristics
PRM
Property Specific Characteristics
Tenure where Freehold = 0
TENURE
Type of Property where Low-rise = 0
TYPE
Land Area
LAREA
Floor Level
FLOOR
Built-up Area
BUAREA
Borrower Specific Characteristics
Payment-to-Income ratio
PINCRATIO
Number of Borrowers
BORROWER
Age of Youngest Borrower
AGE
Purpose of Purchase where Owner-occupation = 0
PURPOSE
Number of years in current employment
YRSEMP
Occupation where stable income = 0
OCCUP
Environmental Characteristics
Change in GDP as at Delinquent Date or Censor Date
CGDP1
Change in GDP as at Default Date or Censor Date
CGDP2
Change in STI as at Delinquent Date or Censor Date
CSTI1
Change in STI as at Default Date or Censor Date
CSTI2
Change in Unemployment Rate as at Delinquent Date
CUNEMP1
or Censor Date
Change in Unemployment Rate as at Default Date or
CUNEMP2
Censor Date
Model Controls
Initial Loan-to-Value Ratio
LVR
Guaranteed Sum
GUARSUM
Monthly CPF Contributions able cover entire mortgage
DCPF
installment = 0
Mortgage Installment-to CPF Contribution x DCPF
DMORTCPF
Exhibit 4 Descriptive Statistics
All Loans*
Mean
Std.Dev.
LOANAMT
364682
161589
PUTOP
0.125974
0.116484
PUTOP2
0.125535
0.116056
PREMIUM
0.00133478
0.0529033
CPFPRICE
0.174087
0.148288
MT
24.0774
6.29587
RISKPRE
-1.43649
0.266325
FRM
0.668246
0.471215
TENURE
0.729858
0.444384
TYPE
0.862559
0.344584
LAREA
334.849
914.556
Loans with Positive Equity#
Mean
Std.Dev.
361637
162796
0.110265
0.0949849
0.10983
0.0944695
0.00113918
0.0537147
0.180076
0.149153
23.9183
6.35287
-1.435
0.268443
0.693333
0.461495
0.72
0.449374
0.855
0.352395
353.266
935.936
Expected Signs
+
+
+
+
+
+/+
+
+
+/+
+
+/+/+
+
+
+
+
+
Loans with Negative Equity@
Mean
Std.Dev.
420038
127521
0.410659
0.103049
0.410125
0.102675
0.00489122
0.0353986
0.0651947
0.0703916
26.9697
4.27555
-1.46364
0.226134
0.212121
0.415149
0.909091
0.291937
1
0
0
0
38
FLOOR
5.89021
5.61821
5.86417
5.71224
6.36364
3.5162
BUAREA
1515.85
577.372
1528.35
587.125
1288.64
273.894
PINCRATI
0.265437
0.11331
0.260666
0.111416
0.352173
0.11411
BORROWER
2.06951
0.543869
2.07
0.555831
2.06061
0.242306
AGE
36.3924
7.09307
36.5643
7.11967
33.2668
5.84633
PURPOSE
0.0442338
0.205777
0.0466667
0.2111
0
0
OCCUP
0.229068
0.420565
0.228333
0.420109
0.242424
0.435194
YRSEMP
9.12317
7.68984
9.23411
7.7146
7.10606
7.0298
CUNEMP1
0.522937
0.583481
0.491607
0.57405
1.09257
0.449903
CUNEMP2
0.514662
0.582245
0.484109
0.573364
1.07018
0.453836
CGDP1
0.178057
0.21474
0.177477
0.21986
0.188597
0.076049
CGDP2
0.180145
0.214834
0.179455
0.219989
0.192685
0.0739465
CSTI1
0.0260575
0.276199
0.0354261
0.27918
-0.144282
0.126909
CSTI2
0.0335119
0.277631
0.0431259
0.280655
-0.141288
0.121038
LVR
0.56662
0.169857
0.557116
0.168416
0.739408
0.080006
GUARSUM
199168
144334
202748
146672
134068
64323.2
MORTCPF
1.67717
2.69959
1.68467
2.76697
1.53571
0.581734
CLTV
0.540007
0.328806
0.509555
0.297043
1.09368
0.387396
CLTV2
0.539082
0.328395
0.508646
0.296535
1.09246
0.388259
*Total sample size is 633 observations, where 133 are delinquent loans and 55 are defaulted loans.
# Sample size of loans with positive equity is 600, where 123 are delinquent loans and 49 are defaulted loans.
@ Sample size of loans with negative equity is 33, where 10 are delinquent loans and 6 are defaulted loans.
Exhibit 5 Summary Results of Bivariate Probit Model and Independent Probit Models
Determinant
PUTOP1
PUTOP2
PREMIUM
CPFPRICE
MT
RISKPRE
PRM
TENURE
TYPE
LAREA
FLOOR
BUAREA
PINCRATI
BORROWER
AGE
PURPOSE
OCCUP
YRSEMP
CUNEMP1
CGDP1
CSTI1
CUNEMP2
CGDP2
CSTI2
LVR
GUARSUM
Expected
Sign
+
+
+
+
+
+/+
+
+
+/+
+
+
+/+
+/+
+
Bivariate Probit Model (Model 1)
Delinquency
+
+
+
+
+
+
+
+
+
+
+
+
+
**
***
*
*
**
Default
+
+
+
- ***
+
+
+
+
+
+
+ ***
+
***
***
***
+
- ***
- ***
+
-
Independent Probit Model (Model 2)
Delinquency
+ **
+
+
+
+
+
+
+
+
+
+
+
***
***
***
*
**
Default
+ **
+
+
+
- ***
+ **
+ *
- *
+
+
+
+
+ ***
+
***
***
***
+
- ***
+
-
***
*
***
**
39
DCPF
+
+
DMORTCPF
+ *
+
+
*** significant at 1%; ** significant at 5%; * significant at 10%
+ **
+
+
Exhibit 6 Full Results of Bivariate Probit Model and Independent Probit Models
Determinant
Bivariate Probit Model (Model 1)
Independent Probit Model (Model 2)
Delinquency
Default
Delinquency
Default
ONE
3.7612
-0.82525
3.8926
-1.8321
(1.9774)
(4.3944)
(1.4320)
(1.8453)
PUTOP1
0.0028251
2.0056 **
(1.1348)
(0.81983)
PUTOP2
2.5722
2.1008 **
(2.1604)
(0.88129)
PREMIUM
-1.6973
-0.27016
-1.6405
-0.020842
(3.0190)
(2.6256)
(1.7193)
(1.8854)
CPFPRICE
4.1006 **
-0.047505
3.7129 ***
1.3606
(1.6363)
(2.7896)
(1.2776)
(1.5898)
MT
0.0085215
0.027366
-0.0060388
0.0090728
(0.024280)
(0.037575)
(0.017168)
(0.020699)
RISKPRE
0.26283
0.19104
0.26807
0.18418
(0.43611)
(0.60685)
(0.29683)
(0.37027)
PRM
-4.0710 ***
-2.2983 ***
-3.8076 ***
-2.1421 ***
(0.38155)
(0.49482)
(0.37574)
(0.35230)
TENURE
-0.48874 *
0.27742
-0.50522 ***
-0.045584
(0.28816)
(0.40379)
(0.19261)
(0.24758)
TYPE
0.93155 *
1.5396
1.0594
*
2.1807 **
(0.57372)
(3.6902)
(0.56709)
(0.98486)
LAREA
0.67525E-04
0.21847E-03
0.15620E-03
0.42879E-03 *
(0.15414E-03)
(0.00201607)
(0.15854E-03)
(0.23679E-03)
FLOOR
-0.028050
-0.034945
-0.022539
-0.038964 *
(0.025965)
(0.0386027)
(0.016902)
(0.0221686)
0.46845E-03 **
-0.37076E-03
0.47137E-03 **
8.4937E-05
BUAREA
(0.20352E-03)
(0.75760E-03)
(0.20320E-03)
(0.29460E-03)
PINCRATI
-0.36330
-0.11873
-1.1572
-0.18819
(1.1017)
(1.4806)
(0.81935)
(0.90011)
BORROWER
0.051976
0.12477
0.043978
0.070302
(0.15345)
(0.30629)
(0.13815)
(0.17238)
AGE
0.29757E-03
0.0081097
0.0010432
0.013098
(0.016296)
(0.027205)
(0.013095)
(0.016263)
PURPOSE
0.37319
0.47200
0.074747
0.36673
(0.34905)
(0.73858)
(0.34444)
(0.41277)
OCCUP
0.25763
1.1005 ***
0.25832
0.60672 ***
(0.21345)
(0.39951)
(0.17784)
(0.20570)
YRSEMP
0.0080586
0.018503
0.0066521
0.011807
(0.014990)
(0.017059)
(0.010299)
(0.011973)
CUNEMP1
-1.8649 ***
-1.7778 ***
(0.27370)
(0.26798)
CGDP1
-6.8326 ***
-7.21349 ***
(0.88273)
(1.2230)
CSTI1
1.5363
***
1.54346 ***
(0.44565)
(0.30716)
CUNEMP2
-1.1305 ***
-0.96514 ***
(0.40444)
(0.27133)
CGDP2
-2.2801
-2.1562 *
(2.0730)
(1.3426)
40
CSTI2
LVR
0.98586
(0.75870)
-2.8149
(1.9192)
-2.3063E-06
(0.46352E-05)
1.1036
(0.71121)
5.6725E-05
(0.67491E-03)
0.61800
(0.97721)
GUARSUM
-0.10245E-05 ***
(0.18086E-05)
DCPF
-0.43619
(0.31968)
DMORTCPF
0.90717E-03 *
(0.51318E-03)
ρ
0.99449 ***
(0.18310)
ln L
-245.6257
-414.3648
ln L0
McFadden R2
0.5928
*** significant at 1%; ** significant at 5%; * significant at 10%
Note: the t ratios are presented in parentheses.
0.13126
(0.76153)
-9.4588E-06 ***
(0.16757E-05)
-0.32128
(0.22369)
0.97088E-03 **
(0.43337E-03)
0
1.2356 ***
(0.39088)
-0.96339
(0.95708)
-4.0556E-06 **
(0.19487E-05)
0.45932
(0.32041)
0.35786E-03
(0.46979E-03)
0
-186.6854
-325.4273
0.4356
-123.0343
-186.9108
0.3131
41