We demonstrate, with a transition matrix, how real estate prices as well as "trigger events" can affect the likelihood of homeowners re-defaulting in loan modification programs.

1. A Statistical/Mathematical Approach to
Enhanced Loan Modification Targeting
Rising home prices may play a larger role than factors such as job
loss in determining which consumers are less likely to re-default on
modified home loans.
Executive Summary
When the housing bubble burst in 2007, millions of
homeowners entered foreclosure and default. To
reduce the burden of such defaults on borrowers,
lenders and the economy, many banks introduced
loan modification programs, such as writing down
a portion of the loan, reducing interest rates or
extending payment terms.
However, within six months of such loan modifi-
cations, a significant percent of borrowers had
defaulted.1
This is bad for all concerned. This white
paper describes a simple and effective combina-
tion of statistical and mathematical techniques
that more accurately determine when and for
whom it is most effective to modify loans. These
techniques accurately separate the effects of
various factors in predicting borrower behavior.
They suggest that rising home prices should be
given more weight in choosing which borrowers
will successfully repay their loans when given
loan modifications.
Loan Modification: The Challenges
Numerous business and market challenges make
banks and other lenders reluctant to offer loan
modifications. The first is identifying the type of
modification (e.g., a lower interest rate, writing off
a portion of the principal, a longer payment term
or a combination of these) that is most beneficial
for the lender and the borrower — and will be most
effective in assuring the borrower continues to
make loan payments on time.
The second challenge is determining whether
there are greater financial benefits to the lender
in allowing a borrower to default or whether
the cost of a lower interest rate or principal, or
longer payment terms, is justified by the eventual
repayment of the loan. The third challenge is to
estimate the risk that any given borrower will
re-default, and to identify those who have both
the ability and willingness to pay. This white paper
addresses this third question of identifying which
borrowers are most likely to re-default.
Prior research clearly shows the link between
a property with negative equity (i.e., being
“underwater”) and its move into foreclosure.2,3,4
If a borrower has sufficient equity in his home,
he has the option of a second mortgage to stay
current with the terms of his loan in the event of a
job loss or medical emergency. While the strength
of this link varies from borrower to borrower, we
assume for the sake of this paper that the value
of the property is one of the prime determinants
of whether borrowers will default.
We also, however, assume that “trigger events”
including loss of income, unemployment and
events such as economic recession also influence
• Cognizant 20-20 Insights
cognizant 20-20 insights | may 2014

2. 2
the default decision. So do borrower attributes
such as the borrower’s credit score, age, education,
marital status and type of loan (e.g., the interest
rate, adjustable vs. fixed, etc.)
Our Analytical Approach
Performance data on the aftermath of loan modi-
fications is scarce. However, using data about the
customer’s behavior before his loan was modified
is technically equivalent to using post-loan-mod-
ification data to determine the probability he will
default.
We perform our analysis by estimating a second
order Markov chain/transition matrix. This differs
from traditional prediction techniques in that it
considers only the borrower’s most recent state
(i.e., his demonstrated willingness and ability to
pay) rather than his long-term history in order
to predict future behavior and thus what sort of
loan modification, if any, he should receive. Our
hypothesis, which has been demonstrated by
application of this technique, is that this analysis
produces more accurate results than traditional
methods such as credit scoring or long-term
analysis of payment behaviors.
Conditional probabilities can be estimated by
either multinomial logistic regression (which
assumes the same set of variables influence all
state equations) or binomial logistic regression
(and rescaling so that the range of probabilities
add up to 1 by applying the Begg-Gray method).5
Either of these techniques allows the effects
of loans that end early due to prepayment to
be separated from those that end early due to
default.
Let’s illustrate with a numerical example. Using
loan level data we build a multinomial logistic
model. The dependent variables are (Current /
60+ Delq), (30+ Delq / 60+ Delq), (60+ Delq / 60+
Delq), (90+ Delq / 60+ Delq) and (Default / 60+
Delq). Note that (Current / 60+ Delq) describes
a loan with 60+ delq which was more than two
months delinquent but is now back to a current
state.
Figure 1 is a transition matrix estimated from
the logistic model equations for a loan with the
ID of “X.” In the top left corner, 0.6 signifies that
this loan with the ID of “X” was more than three
months delinquent but is now current, with the
probability the borrower will remain current
estimated at 0.60.
This table also shows how the second order
Markov chain is helpful in predicting a borrower’s
re-default risk. In other words, which combina-
tion of factor values is most likely to put them
into either a delinquent or nondelinquent state.
For example, we know that a borrower’s debt-to-
income (DTI) ratio is one of the determinants in
a default decision and is linearly related to the
probability of default (the higher the DTI, the
higher the probability of default). Keeping all
other factors constant we can thus estimate the
highest level of DTI at which the borrower will
remain nondelinquent. Here we have chosen 60+
delq as that level, based on observed trends.
This matrix can be used to predict either long-term
or average behavior of a borrower using simple
linear algebra. Rearranging the matrix so rows and
columns of the absorbing state come first provides
a canonical form, dividing the transition matrix
into four sub-matrices (as shown in Figure 2).
Note that In is the square identity matrix if there
is more than one observing state. Otherwise,
the scalar and 0 is zero matrix. The fundamen-
tal matrix is defined as F = (I - Q)-1
, which gives
the average number of months that the borrower
stays on the books before the borrower defaults
and the lender writes off the loan.
cognizant 20-20 insights
Second Order Transition Matrix
Figure 1
State Current 30+ Delq 60+ Delq 90+ Delq Default
(Current / 60+ Delq) 0.6 0.3 0.1 0 0
(30+ Delq / 60+ Delq) 0.5 0.4 0.05 0.05 0
(60+ Delq / 60+ Delq) 0.2 0.3 0.4 0.1 0
(90+ Delq / 60+ Delq) 0.05 0.05 0.1 0.5 0.3
(Default / 60+ Delq) 0 0 0 0 1

3. 3cognizant 20-20 insights
We can now answer how long a borrower will stay
current on their loan if he started with a current
state which is the sum of the first row (i.e., 30.23
+ 18.98 + 7.17 + 3.33 = 59.7 months). Similarly,
if he starts with 30+ Delq, 60+ Delq and 90+
Delq, he stays current for 58, 54 and 24 months
respectively.
Conclusion
Lenders originate loans according to the risk
associated with each borrower — in other words,
pricing the risk. However unforeseen factors such
as job loss, a decline in home prices or recession
can force large numbers of borrowers into
defaults.
The success of loan modification programs
designed to ensure borrowers continue to make
payments on time is highly dependent on the
timing of the loan modification. Our research
shows that trigger events such as job loss are
secondary factors compared with the direction of
property prices. While this paper’s sole goal was
to identify the average time a borrower will avoid
default given the current economic situation, a
similar technique can be used to assess how mac-
roeconomic volatility affects borrowers’ abilities
to keep current with the terms of loan modifica-
tion programs.
Sub-Matrix Assumptions
Figure 2
Absorbing Non-absorbing
Absorbing states In O
Non-absorbing
states
R Q
Canonical Form of Transition Matrix
Fundamental Matrix
Figure 3
Figure 4
State Default Current 30+ Delq 60+ Delq 90+ Delq
(Default/60+ Delq) 1 0 0 0 0
(Current/60+ Delq) 0 0.6 0.3 0.1 0
(30+ Delq/60+ Delq) 0 0.5 0.4 0.05 0.05
(60+ Delq/60+ Delq) 0 0.2 0.3 0.4 0.1
(90+ Delq/60+ Delq) 0.3 0.05 0.05 0.1 0.5
State Current 30+ Delq 60+ Delq 90+ Delq
(Current/60+ Delq) 30.2315 18.98148 7.175926 3.333333
(30+ Delq/60+ Delq) 28.287 19.53704 6.898148 3.333333
(60+ Delq/60+ Delq) 26.0648 17.31481 8.009259 3.333333
(90+ Delq/60+ Delq) 11.0648 7.314815 3.009259 3.333333

4. About Cognizant
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About the Authors
Mahesh Hunasikatti is a Senior Associate with Cognizant Analytics. With wide experience applying
statistics business consulting and analytics in mortgage retail banking, financial and insurance
industries, Mahesh has delivered analytics services in the areas of mortgage product develop-
ment, product pricing, claims and fraud management in both the banking and insurance industries.
He holds an M.Sc. from the University of Agricultural Sciences, Bangalore. Mahesh can be reached at
Mahesh.Hunasikatti@cognizant.com.
Manivannan Karuppusamy is a Manager with Cognizant Analytics. He has 11 years of experience
in analytics and has worked extensively in the mortgage domain for the last few years. Manivannan
is responsible for modeling and development, as well as testing and research, for U.S. mortgage
products such as pre-payment risk, credit risk and collateral risk-fraud. He can be reached at
Manivannan.Karuppusamy2@cognizant.com.
Footnotes
1
OCC and OTS Mortgage Metrics Report: Disclosure of National Bank and Federal Thrift Mortgage, Office
of the Comptroller of the Currency and Office of Thrift Supervision (OCC and OTS), 2008.
2
Foster, Chester, and Robert Van Order, “An Option-Based Model of Mortgage Default,”
Housing Finance Review 3 (4): 351–72; 1984.
3
Kau, James B., Donald C. Keenan, and Taewon Kim, “Transaction Costs, Suboptimal Termination and
Default Probabilities,” Journal of the American Estate and Urban Economics Association 21 (3): 247–63;
1993.
4
Vandell, Kelly D., “How Ruthless Is Mortgage Default? A Review and Synthesis of the Evidence,”
Journal of Housing Research 6 (2): 245–62; 1995.
5
Begg, C.B. and R. Gray, “Calculation of Polychotomous Logistic Regression Parameters Using,
Individualized Regressions,” Biometrika, 71(1):11-18; 1984.
Acknowledgement
The authors would like to thank Rajendra Anjanappa for his invaluable contributions and support in the
development of this white paper.
References
• Loan Data, Third Quarter 2008, Office of Thrift Supervision, Department of the Treasury,
Washington, D.C.
• Pearson’s, An Addison-Wesley product, “MARKOV CHAINS,” 2003,
http://www.aw-bc.com/greenwell/markov.pdf.