**Section VI. Summary of Methodology**

**A. Specification of FHA Mortgage Termination Models**

This Review applies statistical techniques consistent with the
literature and applicable to the FHA experience. The purpose of the
analysis is to estimate, for FHA loans on the books as of the end of FY
2007, their future probabilities of default and prepayment, so as to
compute future outstanding balances, cash flows, and capital ratios.
Using loan-level data, ordinary regression analysis breaks down, because
the dependent variable indicating default or prepayment is not
continuous, but rather is discrete: it is a "1" if either a prepayment
or a default occurs in any given quarter or a "0" otherwise (i.e., it is
an active loan). One of the problems for ordinary regression analysis in
this situation is that the estimated probability of default is not
constrained to be between 0 and 1. Binomial logit analysis is one widely
used technique to deal with this issue, and has been applied here.

Further complicating the statistical analysis is the fact that mortgage
borrowers possess two mutually exclusive options, one to prepay the loan
and the other to default on it. From a lender's or insurer's point of
view, these are "competing risks" in the sense that they are mutually
exclusive, and realization of one risk precludes the other. Prepayment
means cessation of cash flow in the form of any mortgage insurance
premiums, but thereafter eliminates any chance of default. Conversely,
defaulting means default costs are incurred and uncertainty about the
possibility and timing of prepayment is elimintated. These competing
risks present unique challenges for statistical estimation.

Multinomial logit regression is a general approach to deal with these
competing risks, but it is computationally difficult, even for today's
high-powered computers. An equivalent technique, based on separate
binomial logit estimates for default and prepayment can be used when
appropriate adjustments are made for competing risks. The adjustments
for competing risks are of two types: (1) to the data used for
estimation, and (2) to the resulting estimated coefficients from the
separate binomial logit regressions.

The adjustments needed are to the data used to estimate the equations:
for the default equation, eliminating---or "censoring"---the loan's
observations in the quarter of the prepayment and subsequently; and for
the prepayment equation, eliminating/censoring the observations during
the quarter the delinquency starts that leads to a claim and
subsequently (during that delinquency/default period, the risk of
prepayment becomes zero; if the delinquency is not followed by a claim,
the loan remains in the prepayment estimation database).

Once the separate default and prepayment logit equations are estimated,
the appropriate multinomial logit probabilities of default and
prepayment are computed mathematically from the separate estimates.
Another motivation to estimate separate binomial logit models is that
FHA mortgage risk exposure is ultimately determined by the timing and
frequency of claim events, rather than the default events that precede
insurance claims on the MMI Fund. Thus, the default events (ultimately
leading to a claim termination) that result in censoring of prepayment
occur with different timing than the claim events. Separating the
estimation of logit models of prepayment and claim events simplifies
accounting for this timing difference.

Following is an overview of the statistical approach used in this
Review. Appendix A provides additional details of these steps, as well
as a description of the variables used to "explain" prepayment and claim
terminations.

The general approach used in this Review is similar to the multinomial
logit models reported by Calhoun and Deng (2002) that were originally
developed for application to OFHEO's risk-based capital adequacy test
for Fannie Mae and Freddie Mac. The multinomial model recognizes the
competing risks nature of prepayment and claim terminations, while the
use of quarterly data aligns closely with key economic predictors of
mortgage prepayment and claims such as changes in interest rates and
housing values.

The multinomial logit models have several benefits over a traditional
linear regression. First, it ensures the sum of the three probabilities
is equal to 100 percent. This means that at any point in time, a loan
can only experience one of the three possible outcomes: prepay, claim,
or survive. Second, the possible value of each probability is
constrained to be between zero and one under this approach. There is no
possibility of estimating a negative probability or a probability
exceeding 100 percent. Third, as the probability of one risk increases,
the probability of the other risk would automatically be reduced,
reflecting the competing-risk nature between prepayment and default.
Finally, it allows us to estimate the conditional termination rates
using loan-level data. With loan-level observations, the possible
outcomes at each point in time are either only 0, the event did not
happen, or 1, the event happened. Typical multiple regression models are
unsuitable for estimating such discrete dependent variables. Logit
regression is specifically designed to handle these types of
observations.

Following an approach suggested by Begg and Gray (1984), we estimated
separate binomial logit models for prepayment and claim terminations,
and then mathematically recombined the parameter estimates to compute
the corresponding multinomial logit probabilities. This approach allowed
us to account for differences between the timing of FHA claim
terminations and the appropriate censoring of potential prepayment
outcomes at the onset of default episodes that ultimately lead to
claims.

The loan performance analysis was undertaken at the loan level. Through
the use of categorical explanatory variables and discrete indexing of
mortgage age---in effect classifying loan data into "strata"\-- it was
possible to achieve considerable efficiency in data storage and
estimation. In effect, the data were transformed into synthetic loan
pools, but without loss of detail on individual loan characteristics
beyond that implied by the categorization of the explanatory variables.
Sampling weights were used to account for differences in the number of
identical loans in each loan strata.

Conditional claim and prepayment rates increase quickly during the first
two years following mortgage origination before peaking, and then
decline slowly over the remaining life of the loan. We applied a series
of piece-wise linear spline functions to model the impact of mortgage
age on conditional claim and prepayment probabilities. This approach is
sufficiently flexible to provide a close fit during the first three
years following mortgage origination, including the peak years of claim
or prepayment risk, while limiting the number of model parameters that
have to be estimated.

#### B. Differences in the Timing of Borrower Default Episodes and Claim Terminations

For consistency with the available data on loss rates, the incidence and
timing of mortgage default-related terminations is defined specifically
according to FHA claim incidences. The Begg-Gray method of estimating
separate binomial logit models is particularly advantageous in dealing
with this requirement. In recognition of the potential censoring of
prepayment prior to the actual claim termination date, we used
information on the timing of the initiation of default episodes leading
to claim terminations to create a prepayment-censoring indicator that
was applied when estimating the prepayment-rate model, in effect
removing that observation from the prepayment equation database when it
was clear from the nature of the delinquency/default/claim path that the
probability of prepayment was zero during that time.

Similarly, a separate binomial logit claim-rate model was estimated
accounting for censoring of potential claim terminations by observed
prepayments, and the two sets of parameter estimates were recombined
mathematically according to the above equations to produce the final
multinomial model for prepayment and claim probabilities. This approach
facilitated unbiased estimation of the prepayment function, which would
not be possible in a joint multinomial model of claim and prepayment
terminations, since one could not simultaneously censor loans at the
onset of default episodes and still retain the observations for
estimating subsequent claim termination rates.

To summarize, estimation of the multinomial logit model for prepayment
and claim terminations involved the following steps:

-   Data on the start of a default episode that ultimately leads to an
    FHA claim was used to define a default-censoring indicator for
    prepayment.

-   A binomial logit model for conditional prepayment probabilities was
    estimated using the default-censoring indicator to truncate
    individual loan event samples at the onset of any default episodes
    (and all subsequent quarters).

-   A binomial logit model for conditional claim probabilities was
    estimated using observed prepayments to truncate individual loan
    event samples during the quarter of the prepayment event (and all
    subsequent quarters).

-   The separate sets of binomial logit parameter estimates were
    recombined mathematically (according to the above equations) to
    derive the corresponding multinomial logit model for the joint
    probabilities of prepayment and claim terminations accounting for
    the competing risks.

**C. Loan Event Data**

We used loan-level data to reconstruct quarterly loan-event histories by
relating mortgage origination information to contemporaneous values of
time-dependent factors. In the process of creating quarterly event
histories, each loan contributed an additional observed "transition" for
every quarter from origination up to and including the period of
mortgage termination, or until the last time period of the historical
data sample (if the loan remained active). The term "transition" is used
here to refer to any period in which a loan remains active or in which
claim or prepayment terminations are observed.

The FHA single-family data warehouse records each loan for which
insurance was endorsed and includes additional data fields updating the
timing of changes in the status of the loan. A dynamic event history
sample was constructed from the database of loan originations by
creating additional observations for each quarter that the loan was
active from the beginning amortization date up to and including the
termination date for the loan, or the first quarter of FY 2007 if the
loan had not terminated prior to that date.

Additional "future" observations were created for projecting the future
performance of loans currently outstanding, and additional future
cohorts were created to enable simulation of the performance of future
books of business. These aspects of data creation and simulation of
future loan performance are discussed in greater detail in Appendix C.

**D. Statistical Sampling**

The whole population of loan-level data from the FHA single-family data
warehouse was extracted for the FY 2007 analysis. This produced a
starting sample of approximately 22 million single-family loans
originated between FY 1975 and the first quarter of FY 2007. These data
were used to generate loan-level event histories for up to 120 quarters
(30 years) of loan life per loan (or until the scheduled age of maturity
of the loan).

Estimation and forecasting were undertaken separately for each of the
following six FHA mortgage product types: (1) FRM30 -- fixed-rate
30-year home purchase mortgages; (2) FRM15 -- fixed-rate 15-year home
purchase mortgages; (3) ARM -- adjustable-rate home purchase mortgages;
(4) FRM30_SR -- fixed-rate 30-year streamlined refinance mortgages;
(5) FRM15_SR -- fixed-rate 15-year streamlined refinance mortgages; and
(6) ARM_SR -- adjustable-rate streamlined refinance mortgages.

#### We used a 20-percent random sample of FRM30 mortgages and 100-percent samples for all other product types. Loan-level information on borrower FICO scores was introduced for the FY 2007 Review and some historical information on borrower FICO scores was obtained through a choice-based sampling scheme that impacted the sampling of FRM30 loans. Further discussion is provided in the description of loan-level borrower credit score data in Appendix A.

**E. Borrower Credit Scores**

Borrower credit scores at the loan level are included in the models
estimated for the FY 2007 Review. FHA now has relatively complete data
on borrower FICO scores for loans originated since FY 2004. In addition,
FHA has retroactively obtained borrower credit history information for
selected samples of FHA loan applications submitted as far back as FY
1992 and continuing up to FY 2004. These data provide an additional
source of loan-level information on borrower FICO scores that were used
in estimation. The application of loan-level data on borrower FICO
scores is described in greater detail in Appendix A.

**F. Cash Flow Model**

After the future claim and prepayment rates were projected by the
econometric models, the corresponding cash flows were computed. The cash
flow computation model includes the calculation of: 1) upfront mortgage
insurance premiums, 2) annual mortgage insurance premiums, 3) claim
losses, and 4) premium refunds. Two other cash flows were modeled in
previous reviews but are not included in our analyses. The
administrative expense was discontinued according to Federal credit
reform requirements, and distributive shares were suspended in 1990.
There is no indication that either of these will be resumed in the
foreseeable future. The Federal credit subsidy present value conversion
factors published by the Office of Management and Budget are used in
discounting these future cash flows to determine their present value as
of the end of FY 2007.