Appendix A: Econometric Analysis of Mortgages

This appendix describes technical details of the econometric models used
to estimate the historical and future performance of FHA single-family
loans for the FY 2005 Review. We first summarize the model specification
and estimation issues arising from the analysis of FHA claim and
prepayment rates. Then we describe for the specific explanatory
variables used in the analysis. The model estimation output statistics
and graphical comparisons of the overall within-sample fit of the models
are provided thereafter.

### I. Model Specification and Estimation Issues

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

For the FY 2005 Review, the TAC Team has developed and estimated
competing risk models for mortgage prepayment and claim terminations.
Prepayment and claim rates estimates were based on a multinomial logit
model for quarterly conditional probabilities of prepayment and claim
terminations. The general approach is based on 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. The use of
quarterly data aligns more closely with key economic predictors of
mortgage prepayment and claims such as changes in interest rates and
housing values.

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

The present analysis differs from the Calhoun-Deng (2002) study in two
important ways. First, following the 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 claim terminations and the censoring of potential prepayment outcomes
at the onset of default episodes that ultimately lead to claims. This
issue is discussed in greater detail below.

A second difference from the Calhoun-Deng (2002) study was the treatment
of mortgage age in the models. The traditional models apply quadratic
age functions for both mortgage default and prepayment terminations.
While the quadratic age function fits reasonably well for estimating
conventional mortgage defaults rates, it worked less well for
prepayments, as it failed to capture the more rapid increase in
conditional prepayment rates early in the life of the loans. FHA
conditional claim and prepay rates also show a more rapid increase
during the early part of the loan life. We found a quadratic
specification to be insufficiently flexible to capture the age patterns
of conditional claim and prepayment observed in the FHA data. The
approach we adopted was a series of piece-wise linear spline functions.
This approach is sufficiently flexible to fit the relatively rapid
increase in conditional claim and prepayment rates observed during the
first two to three years following mortgage origination, while still
providing a good fit over the later ages and limiting the overall number
of model parameters that have to be estimated. At the end of this
Appendix we present graphical comparisons showing the goodness of fit by
age of our final model estimates.

As indicated, the starting point for specification of the loan
performance models was a multinomial logit model of quarterly
conditional probabilities of prepayment and claim terminations. The
corresponding mathematical expressions for the conditional probabilities
of claim$(\pi_{C}(t))$, prepayment$(\pi_{P}(t))$, or remaining active
$(\pi_{A}(t))$over the time interval from $t$ to $t + 1$ are given by:

$\pi_{C}(t) = \frac{e^{\alpha_{C} + X_{C}(t)\beta_{C}}}{1 + e^{\alpha_{C} + X_{C}(t)\beta_{C}} + e^{\alpha_{P} + X_{P}(t)\beta_{P}}}$
(1)

$\pi_{P}(t) = \frac{e^{\alpha_{P} + X_{P}(t)\beta_{P}}}{1 + e^{\alpha_{C} + X_{C}(t)\beta_{C}} + e^{\alpha_{P} + X_{P}(t)\beta_{P}}}$
(2)

$\pi_{A}(t) = \frac{1}{1 + e^{\alpha_{C} + X_{C}(t)\beta_{C}} + e^{\alpha_{P} + X_{P}(t)\beta_{P}}}$
$$ (3)

where the constant terms $\alpha_{C}$ and $\alpha_{P}$ and the
coefficient vectors $\beta_{C}$ and $\beta_{P}$ are the unknown
parameters to be estimated. $X_{C}(t)$ is the vector of explanatory
variables for the conditional probability of a claim termination,
and$X_{P}(t)$ is the vector of explanatory variables for the conditional
probability of prepayment. Some elements of $X_{C}(t)$ and $X_{P}(t)$
are constant over the life of the loan and are not functions of$t$.

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

#### As mentioned above, timing differences between borrower default episodes and actual FHA claims led us to apply the Begg-Gray method of estimating separate binomial logit models for FHA prepayment and claim terminations and then recombine the parameter estimates to derive the corresponding multinomial logit model. The issue in this case is the time lag between the time that a borrower decides to cease payment on a loan, i.e., default, and when FHA actually receives the claim from the servicer. Because prepayments are unlikely to occur for defaulting loans on their way to becoming claim terminations, censoring of prepayments actually occurs prior to the observed claim termination date. Failure to account for this particular form of censoring could result in biased estimates of the parameters of the prepayment model.

#### The claim-rate model is best viewed as a reduced-form of a more complicated model with two components: (1) an option-based model of borrower payment behavior that determines the incidence and timing of default events that ultimately lead to FHA claims; and (2) a model for differences in the waiting time from borrower default until the claim is submitted to FHA. The second component can be properly addressed in conjunction with estimates of loss severity (or loss-given-default), and can vary significantly with differences in state laws on mortgage foreclosure, differences in lender loss-mitigation policies, and with current economic conditions that affect the values and time-to-sale of collateral properties.

For the FY 2005 Review, we apply average loss severity rates observed
between the FY 2000 and FY 2004 stratified by six mortgage product
types. 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.

A separate claim-rate model was estimated that accounted for the
censoring of potential claim terminations by observed prepayments. The
two sets of parameter estimates were recombined mathematically 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 cannot
simultaneously censor loans at the onset of default episodes and retain
the same observations for estimating subsequent claim termination rates.

The Begg-Gray methodology produces parameter estimates that are
theoretically equivalent to those in the multinomial logit model. By
estimating the prepayment and claim rate models separately, we can
isolate the issues associated with the timing of claims from the
estimation of the parameters of the prepayment function. Failure to
exclude defaulting loans from the sample of loans assumed to be at risk
of prepayment would result in downward bias in the estimates of
conditional probabilities of prepayment because loans with zero chance
of prepayment would be included in the sample in estimating conditional
prepayment rates.

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

1.  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.

2.  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 the default episodes
    and all subsequent quarters.

3.  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.

4.  The separate sets of binomial logit parameter estimates were
    recombined mathematically to derive the corresponding multinomial
    logit model for the joint probabilities of prepayment and claim
    terminations.

#### B. Computation of Multinomial Logit Parameters from Binomial Logit Parameters

Once the separate binomial claim rate and prepayment rate models have
been estimated, the parameter estimates must be combined to compute the
multinomial probabilities. The theory underlying the Begg-Gray method is
that the values of parameters$\alpha_{C}$,$\beta_{C}$,$\alpha_{P}$, and
$\beta_{P}$ from separate binomial logit (BNL) models are identical to
those in the corresponding multinomial logit (MNL) model. Assume that
conditional probabilities for claim and prepay terminations for separate
BNL models are given, respectively, by:

$\pi_{\text{BNL}}^{C} = \frac{e^{\alpha_{C} + X_{C}\beta_{C}}}{1 + e^{\alpha_{C} + X_{C}\beta_{C}}},\ \pi_{\text{BNL}}^{P} = \frac{e^{\alpha_{P} + X_{P}\beta_{P}}}{1 + e^{\alpha_{P} + X_{P}\beta_{P}}}$.
(4)

We have suppressed the time index *t* to simplify the notation. We can
rearrange terms to solve for $e^{\alpha_{C} + X_{C}\beta_{C}}$ and
$e^{\alpha_{P} + X_{P}\beta_{P}}$ in terms of binomial probabilities
$\pi_{\text{BNL}}^{C}$ and$\pi_{\text{BNL}}^{P}$, respectively,

$e^{\alpha_{C} + X_{C}\beta_{C}} = \frac{\pi_{\text{BNL}}^{C}}{(1 - \pi_{\text{BNL}}^{C})},\ e^{\alpha_{P} + X_{P}\beta_{P}} = \frac{\pi_{\text{BNL}}^{P}}{(1 - \pi_{\text{BNL}}^{P})}$.
(5)

Then we can substitute directly into the MNL probabilities for
$e^{\alpha_{C} + X_{C}\beta_{C}}$ and$e^{\alpha_{P} + X_{P}\beta_{P}}$:

$\pi_{\text{MNL}}^{C} = \frac{\frac{\pi_{\text{BNL}}^{C}}{(1 - \pi_{\text{BNL}}^{C})}}{1 + \frac{\pi_{\text{BNL}}^{C}}{(1 - \pi_{\text{BNL}}^{C})} + \frac{\pi_{\text{BNL}}^{P}}{(1 - \pi_{\text{BNL}}^{P})}},\ \pi_{\text{MNL}}^{P} = \frac{\frac{\pi_{\text{BNL}}^{P}}{(1 - \pi_{\text{BNL}}^{P})}}{1 + \frac{\pi_{\text{BNL}}^{C}}{(1 - \pi_{\text{BNL}}^{C})} + \frac{\pi_{\text{BNL}}^{P}}{(1 - \pi_{\text{BNL}}^{P})}}$.
(6)

These expressions for the MNL probabilities can be simplified
algebraically to:

$\pi_{\text{MNL}}^{C} = \frac{\pi_{\text{BNL}}^{C} \cdot (1 - \pi_{\text{BNL}}^{P})}{(1 - \pi_{\text{BNL}}^{C} \cdot \pi_{\text{BNL}}^{P})},\ \pi_{\text{MNL}}^{P} = \frac{\pi_{\text{BNL}}^{P} \cdot (1 - \pi_{\text{BNL}}^{C})}{(1 - \pi_{\text{BNL}}^{C} \cdot \pi_{\text{BNL}}^{P})}$.
(7)

Equations (7) were used to derive the corresponding MNL probabilities
directly from separately estimated BNL probabilities.

**C. Loan Event Data**

We used loan-level data to reconstruct quarterly loan event histories by
combining mortgage origination information with 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. 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. The data set used in this
Actuarial Review is based on an extract from FHA's database as of March
31, 2005. The data set was first filtered for loans with missing or
abnormal values of key variables in our econometric model. In addition,
lender information was not used in our econometric model, loans with
missing lender/servicer information were also excluded from our
analysis. Most of those loans were believed to have already been prepaid
but the records were not yet updated. Since FY 2004, HUD has been
investigating and updating the performance records of these loans.

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 end of the first
quarter of FY 2005 if the loan has 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. Random Sampling**

A full 100-percent sample of loan level data from the FHA single-family
data warehouse was extracted for the FY 2005 analysis. This produced a
starting sample of approximately 20 million single-family loans
originated between FY 1975 and the first quarter of FY 2005. At the
estimation stage a 10-percent random sample of loans is used to generate
loan-level event histories for up to 120 quarters (30 years) of loan
life per loan, or until the scheduled maturity date of the loan.

#### II. Explanatory Variables

Three main categories of explanatory variables were developed:

1.  Fixed loan characteristics, such as mortgage product type,
    amortization term, origination year and quarter, original
    loan-to-value (LTV) ratio, original loan amount, original mortgage
    interest rate, and geographic location (MSA, state, Census
    division);

2.  Dynamic variables based entirely on the loan information, such as
    mortgage age, season of the year, and scheduled amortization of the
    loan balance; and

3.  Dynamic variables derived by combining loan information with
    external economic data, such as interest rates and house price
    indexes.

In some cases the two types of dynamic variables are combined, as in the
case of adjustable rate mortgage (ARM) loans where external data on
changes in Treasury rates are used to update the original coupon rates
and payment amounts on ARM loans in accordance with standard FHA loan
contract features. This in turn affects the amortization schedule of the
loan.

Exhibit A-1 summarizes the specific explanatory variables that are used
in the statistical modeling of loan performance. All of the variables
listed in Exhibit A-1 were entered as 0-1 dummy variables in the
statistical models, with the exception of the mortgage age variables,
which were entered directly. The specification of each variable is
described in more detail as below.

*Mortgage Product Types*

Separate statistical models were estimated for 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.

6.  ARM_SR Adjustable-rate streamlined refinance mortgages.

*Specification of Piece-Wise Linear Age Functions*

Exhibit A-1 lists the series of piece-wise linear age functions that
were used for the six different mortgage product types. For example, we
create a piece-wise linear age function for FRM15 loans with knots (the
k's) at 2, 4, 8, and 12 quarters by generating 5 new age variables
*age1*-*age5* defined as follows:

$\begin{array}{r}
\text{age1} = \begin{Bmatrix}
\text{AGE} & \ \text{if AGE } \leq \text{ k}_{1} \\
k_{1} & \ \text{if AGE } > \text{ k}_{1}
\end{Bmatrix} \\
 \\
\text{age2} = \begin{Bmatrix}
0 & \text{if AGE } \leq \text{ k}_{1} \\
\text{AGE-k}_{1} & \text{if k}_{1} < \text{AGE } \leq \text{ k}_{2} \\
\text{AGE-k}_{2} & \text{if AGE } > \text{ k}_{2}
\end{Bmatrix} \\
 \\
\text{age3} = \begin{Bmatrix}
0 & \text{if AGE } \leq \text{ k}_{2} \\
\text{AGE-k}_{2} & \text{if k}_{2} < \text{AGE } \leq \text{ k}_{3} \\
\text{AGE-k}_{3} & \text{if AGE } > \text{ k}_{3}
\end{Bmatrix} \\
 \\
\text{age4} = \begin{Bmatrix}
0 & \text{if AGE } \leq \text{ k}_{3} \\
\text{AGE-k}_{3} & \text{if k}_{3} < \text{AGE } \leq \text{ k}_{4} \\
\text{AGE-k}_{4} & \text{if AGE } > \text{ k}_{4}
\end{Bmatrix} \\
 \\
\text{age5} = \begin{Bmatrix}
0 & \text{if AGE } \leq \text{ k}_{4} \\
\text{AGE-k}_{4} & \text{if AGE } > \text{ k}_{4}
\end{Bmatrix}
\end{array}$$$ (8)

Coefficient estimates corresponding to the slopes of the line segments
between each knot point and for the last line segment are estimated and
reported in Exhibit A-2. The overall AGE function (for this 5-age
segment example) is given by:

#### $\text{Age Function} = \beta_{1} \cdot \text{age1} + \beta_{2} \cdot \text{age2} + \beta_{3} \cdot \text{age3} + \beta_{4} \cdot \text{age4} + \beta_{5} \cdot \text{age5}$ (9)

#### 

Age functions with greater or fewer numbers of segments are developed in
a similar manner. The number of segments is determined by
trial-and-error estimation and review of in sample fit to the observed
age profiles of conditional claim and prepayment rates.

#### Loan Size

Loan size is defined relative to the average sized FHA loan originated
in the same state during the same fiscal year. The resulting values were
stratified into 5 levels based on direct examination of the data, with
the middle category, *category 3*, corresponding to average-sized loans
plus or minus 10 percent, *i.e.*, 90 to 110 percent of the size of the
average sized loan.

#### Loan-to-Value Ratio {#loan-to-value-ratio .Style-Heading-4-+-Bold}

Loan to value is recorded in the FHA's data warehouse. The LTV ratio
variable may exceed 100 percent due to FHA's practice of allowing the
financing of some closing costs, so a categorical outcome is included
for this possibility. Based on discussions with FHA, any LTV values
recorded for streamline refinance products were considered unreliable
for use in the analysis. We imputed original LTV values for these loans
for the purpose of establishing the starting point for tracking the
evolution of the probability of negative equity (see description of this
variable below). The imputed values were based on the mean LTV values
for FRM30, FRM15, and ARM loans stratified by product, beginning
amortization year and quarter, and geographic location (state and
county).

#### Season {#season .Style-Heading-4-+-Bold}

The season of an event observation quarter is defined as the season of
the year corresponding to the calendar quarter, where 1=Winter (January,
February, March), 2=Spring (April, May, June), 3=Summer (July, August,
September), and 4=Fall (October, November, December).

#### Probability of Negative Equity {#probability-of-negative-equity .Style-Heading-4-+-Bold}

Following the approach applied by Deng, Quigley, and Van Order (2000),
Calhoun and Deng (2002), and others, we computed the equity positions of
individual borrowers using *ex ante* probabilities of negative equity.
The probability of negative equity is a function of the current loan
balance and the probability of individual house price outcomes that fall
below this value during the quarter of observation. The distributions of
individual housing values relative to the value at mortgage origination
were computed using estimates of house price drift and volatility based
on OFHEO House Price Indexes (HPIs) published in the first quarter of
2005.

The probability of negative equity is computed as follows:

$\ \text{PNEQ} = \Phi\left\{ \frac{\text{ln}\text{}(\text{UPB}(t)) - \text{ln}\text{}(P(0) \cdot \text{HPI}(t))}{\sigma(t)} \right\}$
(10)

where![](media/image1.wmf) is the standard normal cumulative
distribution function evaluated at *x*, *UPB(t)* is the current unpaid
mortgage balance based on scheduled amortization, *P(0)* is the value of
the borrower's property at mortgage origination, *HPI(t)* is an index
factor for the percentage change in housing prices in the local market
since origination of the loan, and $\sigma(t)$is a measure of the
diffusion volatility for individual house price appreciation rates over
the same period of time. The values of *HPI (t)* are computed directly
from the house price indexes published by OFHEO, while the diffusion
volatility is computed from the following equation:

$\sigma(t) = \sqrt{a \cdot t + b \cdot t^{2}}$. (11)

The parameters "*a*" and "*b*" in this expression are estimated by OFHEO
when applying the three-stage weighted-repeat-sales methodology advanced
by Case-Sheller (1987, 1989). Further details on the OFHEO HPI
methodology are given in Calhoun (1996).

The resulting values of PNEQ were stratified into seven levels ranging
from less than 5-percent to more than 30-percent probability of negative
equity as listed in Exhibit A-1.

#### Mortgage Premium (Spread)

The financial incentive of a borrower to refinance is measured using a
variable for the relative spread between the current mortgage contract
interest rate and the current market mortgage rate:

$\text{MP}(t) = \left\{ \frac{C(t) - R(t)}{C(t)} \right\}$. (12)

Where *C (t)* is the current note rate on the mortgage and *R (t)* is
the current market average fixed-rate mortgage rate. This variable is as
an approximation to the call option value of the mortgage given by the
difference between the present value of the "anticipated" future stream
of mortgage payments discounted at the current market rate of interest,
*R(t)*, and the present value of the mortgage evaluated at the current
note rate, *C(t)*. Additional details are given in Deng, Quigley, and
Van Order (2000) and Calhoun and Deng (2002).

The relative mortgage premium values for ARMs and FRMs are derived in
exactly the same manner, except that the current coupon is always equal
to the coupon at origination for FRMs. ARM coupon rates are updated over
the life of the mortgage as described below.

#### ARM Coupon Rate Dynamics

To estimate the current financial value of the prepayment option for ARM
loans, we required the path of the coupon rate over the active life of
individual ARM loans. The coupon rate resets periodically to a new level
that depends on the underlying index, plus a fixed margin, subject to
periodic and lifetime caps and floors that specify the maximum and
minimum amounts by which the coupon can change on any one adjustment and
over the life of the loan. Accordingly, the ARM coupon rate at time
*t*,![](media/image2.wmf), was computed as follows:

$\begin{array}{r}
\ C(t) = \text{max} \\

\end{array}$ (13)

where![](media/image3.wmf) is the underlying rate index value at time
*t*, *S* is the "lookback" period, and *Margin* is the amount added to
![](media/image4.wmf) to obtain the "fully-indexed" coupon rate. The
periodic adjustment caps are given by
![](media/image5.wmf)and![](media/image6.wmf), and are multiplied by
dummy variable ![](media/image7.wmf) which equals zero except during
scheduled adjustment periods. Maximum lifetime adjustments are
determined by ![](media/image8.wmf)and![](media/image9.wmf),
and$\text{Life}_{\text{Min}}$ is the overall minimum lifetime rate
level.

*Yield Curve Slope*

Expectations about future interest rates and differences in short-term
and long-term borrowing rates associated with the slope of the Treasury
yield curve influence the choice between ARM and FRM loans and the
timing of refinancing. We use the ratio of the ten-year Constant
Maturity Treasury yield to the one-year Constant Maturity Treasury yield
to measure the slope of the Treasury yield curve.

#### Burnout Factor

A burnout factor is included to identify borrowers who have foregone
recent opportunities to refinance. The burnout factor is included to
account for individual differences in propensity to prepay, often
characterized as unobserved heterogeneity. In addition, unmeasured
differences in borrower equity at the loan level may give rise to
unobserved heterogeneity that can impact both prepayment and claim
rates. Borrowers with negative equity are less likely to prepay due to
the difficulty of qualifying and more likely to exercise the default
option.

Changes were introduced to the burnout factor for the FY 2005 Review.
The previous burnout factor, which was identical to that used in the
OFHEO risk-based capital stress test model, took the value one if the
mortgage note rate exceeds the market mortgage rate by 200 basis points
or more in any two of the preceding eight quarters. Empirical evidence
now suggests that borrowers who refinance tend to do so at much lower
thresholds. The new burnout factor measures the average of the number of
basis points the borrower was in the money, for all quarters during
which the borrower was in the money, anytime during the preceding 8
quarters. The resulting measure was categorized into smaller 50 basis
point categories that provide a more refined measure of burnout.

*Pre-1986 Origination*

An indicator for loans originated prior to FY 1986 Q3 is included to
account for improvements in FHA underwriting requirements.

*Post-1995 Origination*

An indicator for loans originated after FY 1995 is included to account
for a loosening of FHA underwriting requirements.

*Exposure Year/Quarter FRM Rate*

A variable measuring the market average FRM mortgage rate is included to
distinguish high-rate and low-rate market environments.

*Changes in Metropolitan Area Unemployment Rates*

For the FY 2005 Review we undertook to develop a measure of changes in
metropolitan area unemployment rates. Data on metropolitan area
unemployment rates were obtained from the Bureau of Labor Statistics and
converted into times series from which we computed a dynamic measure for
the percentage change in the unemployment rate over the preceding year.

The unemployment rate variables did not perform well in any of the
preliminary models that were estimated, and are not included in the
final model specifications. No consistent pattern was observed between
mortgage claims and increases in local area unemployment rates, in
contrast to the strong relationship between loan performance and
borrower equity. This outcome is consistent with prior experience using
this variable in loan-level models in which borrower behavior is more
strongly linked to changes in the borrower's equity position or changes
in the value of the mortgage instrument due to changes in interest
rates. Changes in these variables have a direct impact on property and
mortgage values, whereas the local area unemployment measure has a much
weaker connection to individual borrowers.

*ARM Payment Burden*

Another new variable considered for the FY 2005 Review was the ARM
payment burden. This variable measured the percentage change in the
monthly payment since origination on ARM loans. The percentage change
was categorized into 5 levels ranging from no increase to more than
30-percent increase.

The ARM payment burden variables did not perform well in the preliminary
models that were estimated and were generally not statistically
significant. This variable is highly collinear with the mortgage premium
(spread) and burnout variables (for loans that do not prepay),
particularly over the early years before there is substantial
amortization of the loan balance. As a result, this variable contributes
little to the explanation of loan performance once the other variables
are included.

*Source of Down Payment Assistance*

The FHA single-family program has experienced a significant increase in
the use of down payment assistance from relatives, non-profit
organizations, and government programs in the past two years, and loans
to borrowers utilizing down payment assistance have been observed to
generate higher claim rates.

For the FY 2005 Review we included a series of indicators for the use of
different types of down payment assistance.

+-----+-----+-----+----+-------+--------+------+---+--------------------+
| **E |     |     |    |       |        |      |   |                    |
| xhi |     |     |    |       |        |      |   |                    |
| bit |     |     |    |       |        |      |   |                    |
| A-  |     |     |    |       |        |      |   |                    |
| 1** |     |     |    |       |        |      |   |                    |
+=====+=====+=====+====+=======+========+======+===+====================+
| *   |     |     |    |       |        |      |   |                    |
| *Lo |     |     |    |       |        |      |   |                    |
| git |     |     |    |       |        |      |   |                    |
| Mo  |     |     |    |       |        |      |   |                    |
| del |     |     |    |       |        |      |   |                    |
| Ex  |     |     |    |       |        |      |   |                    |
| pla |     |     |    |       |        |      |   |                    |
| nat |     |     |    |       |        |      |   |                    |
| ory |     |     |    |       |        |      |   |                    |
| Va  |     |     |    |       |        |      |   |                    |
| ria |     |     |    |       |        |      |   |                    |
| ble |     |     |    |       |        |      |   |                    |
| s** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| *   |     |     |    | **Val |        |      |   | **Description**    |
| *Va |     |     |    | ues** |        |      |   |                    |
| ria |     |     |    |       |        |      |   |                    |
| ble |     |     |    |       |        |      |   |                    |
| Nam |     |     |    |       |        |      |   |                    |
| e** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| *   |     |     |    |       |        |      |   |                    |
| *Mo |     |     |    |       |        |      |   |                    |
| rtg |     |     |    |       |        |      |   |                    |
| age |     |     |    |       |        |      |   |                    |
| Age |     |     |    |       |        |      |   |                    |
| F   |     |     |    |       |        |      |   |                    |
| unc |     |     |    |       |        |      |   |                    |
| tio |     |     |    |       |        |      |   |                    |
| n** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     | FR  | FR  | A  | FRM   | FR     | AR   |   | Piece-wise linear  |
|     | M30 | M15 | RM | 30_SR | M15_SR | M_SR |   | age functions for  |
|     |     |     |    |       |        |      |   | ages up to         |
|     |     |     |    |       |        |      |   | specified knot     |
|     |     |     |    |       |        |      |   | points.            |
|     |     |     |    |       |        |      |   |                    |
|     |     |     |    |       |        |      |   | Estimated          |
|     |     |     |    |       |        |      |   | parameters give    |
|     |     |     |    |       |        |      |   | the slope of the   |
|     |     |     |    |       |        |      |   | age function for   |
|     |     |     |    |       |        |      |   | each segment.      |
|     |     |     |    |       |        |      |   |                    |
|     |     |     |    |       |        |      |   | Functions differ   |
|     |     |     |    |       |        |      |   | by mortgage        |
|     |     |     |    |       |        |      |   | product type as    |
|     |     |     |    |       |        |      |   | indicated.         |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   | 2   | 2   | 2  | 2     | 2      | 2    |   |                    |
| ge1 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   | 4   | 4   | 4  | 4     | 4      | 4    |   |                    |
| ge2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   | 8   | 8   | 8  | 8     | 8      | 8    |   |                    |
| ge3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   | 12  | 12  | 12 | 12    | 12     | 12   |   |                    |
| ge4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   | 16  | 16  | 16 | \> 12 | 16     | 16   |   |                    |
| ge5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   | 20  | \>  | 20 |       | 20     | 20   |   |                    |
| ge6 |     | 16  |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   | 24  |     | 24 |       | 24     | 24   |   |                    |
| ge7 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   | 28  |     | 28 |       | \> 24  | \>   |   |                    |
| ge8 |     |     |    |       |        | 24   |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   | 32  |     | 32 |       |        |      |   |                    |
| ge9 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ag  | 40  |     | 40 |       |        |      |   |                    |
| e10 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ag  | 60  |     | \> |       |        |      |   |                    |
| e11 |     |     | 40 |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ag  | 80  |     |    |       |        |      |   |                    |
| e12 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ag  | \>  |     |    |       |        |      |   |                    |
| e13 | 80  |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| **L |     |     |    |       |        |      |   |                    |
| oan |     |     |    |       |        |      |   |                    |
| Siz |     |     |    |       |        |      |   |                    |
| e** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| l   |     |     |    | 0 \<  |        |      |   | Relative loan size |
| oan |     |     |    | X ≤   |        |      |   | measured as        |
| cat |     |     |    | 60    |        |      |   | percent difference |
| _ca |     |     |    |       |        |      |   | from average size  |
| t_1 |     |     |    |       |        |      |   | loan originated in |
|     |     |     |    |       |        |      |   | same state in the  |
|     |     |     |    |       |        |      |   | same year.         |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| l   |     |     |    | 60 \< |        |      |   |                    |
| oan |     |     |    | X ≤   |        |      |   |                    |
| cat |     |     |    | 90    |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| l   |     |     |    | 90 \< |        |      |   |                    |
| oan |     |     |    | X ≤   |        |      |   |                    |
| cat |     |     |    | 110   |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| l   |     |     |    | 110   |        |      |   |                    |
| oan |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤ 140 |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| l   |     |     |    | X \>  |        |      |   |                    |
| oan |     |     |    | 140   |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| **  |     |     |    |       |        |      |   |                    |
| Loa |     |     |    |       |        |      |   |                    |
| n-t |     |     |    |       |        |      |   |                    |
| o-V |     |     |    |       |        |      |   |                    |
| alu |     |     |    |       |        |      |   |                    |
| e** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ltv |     |     |    | 0 \<  |        |      |   | Loan-to-value at   |
| cat |     |     |    | X ≤   |        |      |   | origination.       |
| _ca |     |     |    | 80    |        |      |   | Missing or zero    |
| t_1 |     |     |    |       |        |      |   | values replaced    |
|     |     |     |    |       |        |      |   | with update file   |
|     |     |     |    |       |        |      |   | provided by FHA.   |
|     |     |     |    |       |        |      |   | Additional missing |
|     |     |     |    |       |        |      |   | values imputed as  |
|     |     |     |    |       |        |      |   | mean LTV by state, |
|     |     |     |    |       |        |      |   | origination FY,    |
|     |     |     |    |       |        |      |   | and product type.  |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ltv |     |     |    | 80 \< |        |      |   |                    |
| cat |     |     |    | X ≤   |        |      |   |                    |
| _ca |     |     |    | 90    |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ltv |     |     |    | 90 \< |        |      |   |                    |
| cat |     |     |    | X ≤   |        |      |   |                    |
| _ca |     |     |    | 95    |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ltv |     |     |    | 95 \< |        |      |   |                    |
| cat |     |     |    | X ≤   |        |      |   |                    |
| _ca |     |     |    | 97    |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ltv |     |     |    | 97 \< |        |      |   |                    |
| cat |     |     |    | X ≤   |        |      |   |                    |
| _ca |     |     |    | 98    |        |      |   |                    |
| t_5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ltv |     |     |    | 98 \< |        |      |   |                    |
| cat |     |     |    | X ≤   |        |      |   |                    |
| _ca |     |     |    | 100   |        |      |   |                    |
| t_6 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ltv |     |     |    | X \>  |        |      |   |                    |
| cat |     |     |    | 100   |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_7 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| (   |     |     |    |       |        |      |   |                    |
| con |     |     |    |       |        |      |   |                    |
| tin |     |     |    |       |        |      |   |                    |
| ued |     |     |    |       |        |      |   |                    |
| on  |     |     |    |       |        |      |   |                    |
| fol |     |     |    |       |        |      |   |                    |
| low |     |     |    |       |        |      |   |                    |
| ing |     |     |    |       |        |      |   |                    |
| pa  |     |     |    |       |        |      |   |                    |
| ge) |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| *   |     |     |    |       |        |      |   |                    |
| *Se |     |     |    |       |        |      |   |                    |
| aso |     |     |    |       |        |      |   |                    |
| n** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| sea |     |     |    | X = 1 |        |      |   | Calendar quarter   |
| son |     |     |    |       |        |      |   | of mortgage        |
| _ca |     |     |    |       |        |      |   | origination.       |
| t_1 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| sea |     |     |    | X = 2 |        |      |   |                    |
| son |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| sea |     |     |    | X = 3 |        |      |   |                    |
| son |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| sea |     |     |    | X = 4 |        |      |   |                    |
| son |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| *   |     |     |    |       |        |      |   |                    |
| *Pr |     |     |    |       |        |      |   |                    |
| oba |     |     |    |       |        |      |   |                    |
| bil |     |     |    |       |        |      |   |                    |
| ity |     |     |    |       |        |      |   |                    |
| of  |     |     |    |       |        |      |   |                    |
| Ne  |     |     |    |       |        |      |   |                    |
| gat |     |     |    |       |        |      |   |                    |
| ive |     |     |    |       |        |      |   |                    |
| Eq  |     |     |    |       |        |      |   |                    |
| uit |     |     |    |       |        |      |   |                    |
| y** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| p   |     |     |    | 0.00  |        |      |   | Probability of     |
| neq |     |     |    | ≤ X ≤ |        |      |   | negative equity.   |
| cat |     |     |    | 0.05  |        |      |   | Based on OFHEO     |
| _ca |     |     |    |       |        |      |   | house price drift  |
| t_1 |     |     |    |       |        |      |   | and volatility     |
|     |     |     |    |       |        |      |   | estimates.         |
|     |     |     |    |       |        |      |   | MSA-level          |
|     |     |     |    |       |        |      |   | estimates used for |
|     |     |     |    |       |        |      |   | selected MSAs,     |
|     |     |     |    |       |        |      |   | otherwise, Census  |
|     |     |     |    |       |        |      |   | Division level     |
|     |     |     |    |       |        |      |   | estimates are      |
|     |     |     |    |       |        |      |   | used.              |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| p   |     |     |    | 0.05  |        |      |   |                    |
| neq |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤     |        |      |   |                    |
| _ca |     |     |    | 0.10  |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| p   |     |     |    | 0.10  |        |      |   |                    |
| neq |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤     |        |      |   |                    |
| _ca |     |     |    | 0.15  |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| p   |     |     |    | 0.15  |        |      |   |                    |
| neq |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤     |        |      |   |                    |
| _ca |     |     |    | 0.20  |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| p   |     |     |    | 0.20  |        |      |   |                    |
| neq |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤     |        |      |   |                    |
| _ca |     |     |    | 0.25  |        |      |   |                    |
| t_5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| p   |     |     |    | 0.25  |        |      |   |                    |
| neq |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤     |        |      |   |                    |
| _ca |     |     |    | 0.30  |        |      |   |                    |
| t_6 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| p   |     |     |    | X \>  |        |      |   |                    |
| neq |     |     |    | 0.30  |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_7 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| *   |     |     |    |       |        |      |   |                    |
| *Mo |     |     |    |       |        |      |   |                    |
| rtg |     |     |    |       |        |      |   |                    |
| age |     |     |    |       |        |      |   |                    |
| P   |     |     |    |       |        |      |   |                    |
| rem |     |     |    |       |        |      |   |                    |
| ium |     |     |    |       |        |      |   |                    |
| (   |     |     |    |       |        |      |   |                    |
| Spr |     |     |    |       |        |      |   |                    |
| ead |     |     |    |       |        |      |   |                    |
| )** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| spr |     |     |    | X ≤   |        |      |   | Mortgage premium   |
| ead |     |     |    | -30   |        |      |   | value measured as  |
| cat |     |     |    |       |        |      |   | difference between |
| _ca |     |     |    |       |        |      |   | current coupon     |
| t_1 |     |     |    |       |        |      |   | rate and average   |
|     |     |     |    |       |        |      |   | FRM market rate    |
|     |     |     |    |       |        |      |   | relative to        |
|     |     |     |    |       |        |      |   | current coupon     |
|     |     |     |    |       |        |      |   | rate.              |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| spr |     |     |    | -30   |        |      |   |                    |
| ead |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤ -20 |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| spr |     |     |    | -20   |        |      |   |                    |
| ead |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤ -10 |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| spr |     |     |    | -10   |        |      |   |                    |
| ead |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤ 0   |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| spr |     |     |    | 0 \<  |        |      |   |                    |
| ead |     |     |    | X ≤   |        |      |   |                    |
| cat |     |     |    | 10    |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| spr |     |     |    | 10 \< |        |      |   |                    |
| ead |     |     |    | X ≤   |        |      |   |                    |
| cat |     |     |    | 20    |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_6 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| spr |     |     |    | 20 \< |        |      |   |                    |
| ead |     |     |    | X ≤   |        |      |   |                    |
| cat |     |     |    | 30    |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_7 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| spr |     |     |    | X \>  |        |      |   |                    |
| ead |     |     |    | 30    |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_8 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| *   |     |     |    |       |        |      |   |                    |
| *Yi |     |     |    |       |        |      |   |                    |
| eld |     |     |    |       |        |      |   |                    |
| Cu  |     |     |    |       |        |      |   |                    |
| rve |     |     |    |       |        |      |   |                    |
| S   |     |     |    |       |        |      |   |                    |
| lop |     |     |    |       |        |      |   |                    |
| e** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| y   |     |     |    | 0.0 ≤ |        |      |   | Yield curve slope  |
| csl |     |     |    | X ≤   |        |      |   | measured as ratio  |
| ope |     |     |    | 1.0   |        |      |   | of 10‑year CMT to  |
| cat |     |     |    |       |        |      |   | 1-year CMT.        |
| _ca |     |     |    |       |        |      |   |                    |
| t_1 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| y   |     |     |    | 1.0   |        |      |   |                    |
| csl |     |     |    | \< X  |        |      |   |                    |
| ope |     |     |    | ≤ 1.2 |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| y   |     |     |    | 1.2   |        |      |   |                    |
| csl |     |     |    | \< X  |        |      |   |                    |
| ope |     |     |    | ≤ 1.5 |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| y   |     |     |    | X \>  |        |      |   |                    |
| csl |     |     |    | 1.5   |        |      |   |                    |
| ope |     |     |    |       |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    | (cont |        |      |   |                    |
|     |     |     |    | inued |        |      |   |                    |
|     |     |     |    | on    |        |      |   |                    |
|     |     |     |    | foll  |        |      |   |                    |
|     |     |     |    | owing |        |      |   |                    |
|     |     |     |    | page) |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| **B |     |     |    |       |        |      |   |                    |
| urn |     |     |    |       |        |      |   |                    |
| out |     |     |    |       |        |      |   |                    |
| Fa  |     |     |    |       |        |      |   |                    |
| cto |     |     |    |       |        |      |   |                    |
| r** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   | Burnout factor     |
|     |     |     |    |       |        |      |   | equal to the       |
|     |     |     |    |       |        |      |   | average number of  |
|     |     |     |    |       |        |      |   | basis points the   |
|     |     |     |    |       |        |      |   | prepayment option  |
|     |     |     |    |       |        |      |   | was in the money   |
|     |     |     |    |       |        |      |   | during those       |
|     |     |     |    |       |        |      |   | quarters the       |
|     |     |     |    |       |        |      |   | option was in the  |
|     |     |     |    |       |        |      |   | money during the   |
|     |     |     |    |       |        |      |   | preceding 8        |
|     |     |     |    |       |        |      |   | quarters.          |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| in  |     |     |    | X ≤ 0 |        |      |   |                    |
| _mo |     |     |    |       |        |      |   |                    |
| ney |     |     |    |       |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_1 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| in  |     |     |    | 0 \<  |        |      |   |                    |
| _mo |     |     |    | X ≤   |        |      |   |                    |
| ney |     |     |    | 50    |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| in  |     |     |    | 50 \< |        |      |   |                    |
| _mo |     |     |    | X ≤   |        |      |   |                    |
| ney |     |     |    | 100   |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| in  |     |     |    | 100   |        |      |   |                    |
| _mo |     |     |    | \< X  |        |      |   |                    |
| ney |     |     |    | ≤ 150 |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| in  |     |     |    | 150   |        |      |   |                    |
| _mo |     |     |    | \< X  |        |      |   |                    |
| ney |     |     |    | ≤ 200 |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| in  |     |     |    | X \>  |        |      |   |                    |
| _mo |     |     |    | 200   |        |      |   |                    |
| ney |     |     |    |       |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_6 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| *   |     |     |    |       |        |      |   |                    |
| *Pr |     |     |    |       |        |      |   |                    |
| e-1 |     |     |    |       |        |      |   |                    |
| 986 |     |     |    |       |        |      |   |                    |
| O   |     |     |    |       |        |      |   |                    |
| rig |     |     |    |       |        |      |   |                    |
| ina |     |     |    |       |        |      |   |                    |
| tio |     |     |    |       |        |      |   |                    |
| n** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| pr  |     |     |    | X ≥   |        |      |   | Post- or           |
| e_f |     |     |    | 1986  |        |      |   | pre-FY1986 Q3      |
| y86 |     |     |    |       |        |      |   | origination.       |
| _ca |     |     |    |       |        |      |   | Included to        |
| t_1 |     |     |    |       |        |      |   | account for        |
|     |     |     |    |       |        |      |   | changes in FHA     |
|     |     |     |    |       |        |      |   | underwriting       |
|     |     |     |    |       |        |      |   | standards.         |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| pr  |     |     |    | X \<  |        |      |   |                    |
| e_f |     |     |    | 1986  |        |      |   |                    |
| y86 |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| **  |     |     |    |       |        |      |   |                    |
| Pos |     |     |    |       |        |      |   |                    |
| t-1 |     |     |    |       |        |      |   |                    |
| 995 |     |     |    |       |        |      |   |                    |
| O   |     |     |    |       |        |      |   |                    |
| rig |     |     |    |       |        |      |   |                    |
| ina |     |     |    |       |        |      |   |                    |
| tio |     |     |    |       |        |      |   |                    |
| n** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| pos |     |     |    | X ≤   |        |      |   | Pre-or post-FY1995 |
| t_f |     |     |    | 1995  |        |      |   | origination.       |
| y95 |     |     |    |       |        |      |   | Included to        |
| _ca |     |     |    |       |        |      |   | account for        |
| t_1 |     |     |    |       |        |      |   | changes in FHA     |
|     |     |     |    |       |        |      |   | underwriting       |
|     |     |     |    |       |        |      |   | standards.         |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| pos |     |     |    | X \>  |        |      |   |                    |
| t_f |     |     |    | 1995  |        |      |   |                    |
| y95 |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| *   |     |     |    |       |        |      |   |                    |
| *Ex |     |     |    |       |        |      |   |                    |
| pos |     |     |    |       |        |      |   |                    |
| ure |     |     |    |       |        |      |   |                    |
| Yea |     |     |    |       |        |      |   |                    |
| r/Q |     |     |    |       |        |      |   |                    |
| uar |     |     |    |       |        |      |   |                    |
| ter |     |     |    |       |        |      |   |                    |
| FRM |     |     |    |       |        |      |   |                    |
| Rat |     |     |    |       |        |      |   |                    |
| e** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| e   |     |     |    | X ≤ 6 |        |      |   | FRM average        |
| y_r |     |     |    |       |        |      |   | mortgage rate      |
| ate |     |     |    |       |        |      |   | during exposure    |
| cat |     |     |    |       |        |      |   | year and quarter.  |
| _ca |     |     |    |       |        |      |   | Included to        |
| t_1 |     |     |    |       |        |      |   | distinguish        |
|     |     |     |    |       |        |      |   | high-rate and      |
|     |     |     |    |       |        |      |   | low-rate           |
|     |     |     |    |       |        |      |   | environments.      |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| e   |     |     |    | 6 \<  |        |      |   |                    |
| y_r |     |     |    | X ≤ 7 |        |      |   |                    |
| ate |     |     |    |       |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| e   |     |     |    | 7 \<  |        |      |   |                    |
| y_r |     |     |    | X ≤ 8 |        |      |   |                    |
| ate |     |     |    |       |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| e   |     |     |    | 8 \<  |        |      |   |                    |
| y_r |     |     |    | X ≤ 9 |        |      |   |                    |
| ate |     |     |    |       |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| e   |     |     |    | 9 \<  |        |      |   |                    |
| y_r |     |     |    | X ≤   |        |      |   |                    |
| ate |     |     |    | 10    |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| e   |     |     |    | X \>  |        |      |   |                    |
| y_r |     |     |    | 10    |        |      |   |                    |
| ate |     |     |    |       |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_6 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| (   |     |     |    |       |        |      |   |                    |
| con |     |     |    |       |        |      |   |                    |
| tin |     |     |    |       |        |      |   |                    |
| ued |     |     |    |       |        |      |   |                    |
| on  |     |     |    |       |        |      |   |                    |
| fol |     |     |    |       |        |      |   |                    |
| low |     |     |    |       |        |      |   |                    |
| ing |     |     |    |       |        |      |   |                    |
| pa  |     |     |    |       |        |      |   |                    |
| ge) |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| **  |     |     |    |       |        |      |   |                    |
| Met |     |     |    |       |        |      |   |                    |
| rop |     |     |    |       |        |      |   |                    |
| oli |     |     |    |       |        |      |   |                    |
| tan |     |     |    |       |        |      |   |                    |
| Une |     |     |    |       |        |      |   |                    |
| mpl |     |     |    |       |        |      |   |                    |
| oym |     |     |    |       |        |      |   |                    |
| ent |     |     |    |       |        |      |   |                    |
| R   |     |     |    |       |        |      |   |                    |
| ate |     |     |    |       |        |      |   |                    |
| s** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ue  |     |     |    | X ≤   |        |      | P |                    |
| chn |     |     |    | -30   |        |      | e |                    |
| gca |     |     |    |       |        |      | r |                    |
| t_1 |     |     |    |       |        |      | c |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | c |                    |
|     |     |     |    |       |        |      | h |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | g |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | o |                    |
|     |     |     |    |       |        |      | v |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | r |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | h |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | p |                    |
|     |     |     |    |       |        |      | r |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | c |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | d |                    |
|     |     |     |    |       |        |      | i |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | g |                    |
|     |     |     |    |       |        |      | y |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | r |                    |
|     |     |     |    |       |        |      | i |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | h |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | m |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | r |                    |
|     |     |     |    |       |        |      | o |                    |
|     |     |     |    |       |        |      | - |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | r |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | u |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | m |                    |
|     |     |     |    |       |        |      | p |                    |
|     |     |     |    |       |        |      | l |                    |
|     |     |     |    |       |        |      | o |                    |
|     |     |     |    |       |        |      | y |                    |
|     |     |     |    |       |        |      | m |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | r |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | . |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ue  |     |     |    | -30   |        |      |   |                    |
| chn |     |     |    | \< X  |        |      |   |                    |
| gca |     |     |    | ≤ -20 |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ue  |     |     |    | -20   |        |      |   |                    |
| chn |     |     |    | \< X  |        |      |   |                    |
| gca |     |     |    | ≤ -10 |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ue  |     |     |    | -10   |        |      |   |                    |
| chn |     |     |    | \< X  |        |      |   |                    |
| gca |     |     |    | ≤ 0   |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ue  |     |     |    | 0 \<  |        |      |   |                    |
| chn |     |     |    | X ≤   |        |      |   |                    |
| gca |     |     |    | 10    |        |      |   |                    |
| t_5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ue  |     |     |    | 10 \< |        |      |   |                    |
| chn |     |     |    | X ≤   |        |      |   |                    |
| gca |     |     |    | 20    |        |      |   |                    |
| t_6 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ue  |     |     |    | 20 \< |        |      |   |                    |
| chn |     |     |    | X ≤   |        |      |   |                    |
| gca |     |     |    | 30    |        |      |   |                    |
| t_7 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ue  |     |     |    | 30 \< |        |      |   |                    |
| chn |     |     |    | X ≤   |        |      |   |                    |
| gca |     |     |    | 50    |        |      |   |                    |
| t_8 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| ue  |     |     |    | 50 \< |        |      |   |                    |
| chn |     |     |    | X ≤   |        |      |   |                    |
| gca |     |     |    | 100   |        |      |   |                    |
| t_9 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| uec |     |     |    | 100   |        |      |   |                    |
| hng |     |     |    | \< X  |        |      |   |                    |
| cat |     |     |    | ≤ 150 |        |      |   |                    |
| _10 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| uec |     |     |    | X \>  |        |      |   |                    |
| hng |     |     |    | 150   |        |      |   |                    |
| cat |     |     |    |       |        |      |   |                    |
| _11 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| **  |     |     |    |       |        |      |   |                    |
| ARM |     |     |    |       |        |      |   |                    |
| P   |     |     |    |       |        |      |   |                    |
| aym |     |     |    |       |        |      |   |                    |
| ent |     |     |    |       |        |      |   |                    |
| Bu  |     |     |    |       |        |      |   |                    |
| rde |     |     |    |       |        |      |   |                    |
| n** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   |     |     |    | X ≤ 0 |        |      | P |                    |
| rm_ |     |     |    |       |        |      | e |                    |
| pay |     |     |    |       |        |      | r |                    |
| men |     |     |    |       |        |      | c |                    |
| tca |     |     |    |       |        |      | e |                    |
| t_1 |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | i |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | c |                    |
|     |     |     |    |       |        |      | r |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | s |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | i |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | m |                    |
|     |     |     |    |       |        |      | o |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | h |                    |
|     |     |     |    |       |        |      | l |                    |
|     |     |     |    |       |        |      | y |                    |
|     |     |     |    |       |        |      | p |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | y |                    |
|     |     |     |    |       |        |      | m |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | s |                    |
|     |     |     |    |       |        |      | i |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | c |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | o |                    |
|     |     |     |    |       |        |      | r |                    |
|     |     |     |    |       |        |      | i |                    |
|     |     |     |    |       |        |      | g |                    |
|     |     |     |    |       |        |      | i |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | i |                    |
|     |     |     |    |       |        |      | o |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | . |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   |     |     |    | 0 \<  |        |      |   |                    |
| rm_ |     |     |    | X ≤   |        |      |   |                    |
| pay |     |     |    | 10    |        |      |   |                    |
| men |     |     |    |       |        |      |   |                    |
| tca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   |     |     |    | 10 \< |        |      |   |                    |
| rm_ |     |     |    | X ≤   |        |      |   |                    |
| pay |     |     |    | 20    |        |      |   |                    |
| men |     |     |    |       |        |      |   |                    |
| tca |     |     |    |       |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   |     |     |    | 0 \<  |        |      |   |                    |
| rm_ |     |     |    | X ≤   |        |      |   |                    |
| pay |     |     |    | 30    |        |      |   |                    |
| men |     |     |    |       |        |      |   |                    |
| tca |     |     |    |       |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| a   |     |     |    | X \>  |        |      |   |                    |
| rm_ |     |     |    | 30    |        |      |   |                    |
| pay |     |     |    |       |        |      |   |                    |
| men |     |     |    |       |        |      |   |                    |
| tca |     |     |    |       |        |      |   |                    |
| t_5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| **  |     |     |    |       |        |      |   |                    |
| Sou |     |     |    |       |        |      |   |                    |
| rce |     |     |    |       |        |      |   |                    |
| of  |     |     |    |       |        |      |   |                    |
| D   |     |     |    |       |        |      |   |                    |
| own |     |     |    |       |        |      |   |                    |
| P   |     |     |    |       |        |      |   |                    |
| aym |     |     |    |       |        |      |   |                    |
| ent |     |     |    |       |        |      |   |                    |
| Ass |     |     |    |       |        |      |   |                    |
| ist |     |     |    |       |        |      |   |                    |
| anc |     |     |    |       |        |      |   |                    |
| e** |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| gif |     |     |    | None  |        |      | S |                    |
| t_l |     |     |    | Rec   |        |      | o |                    |
| tr_ |     |     |    | orded |        |      | u |                    |
| src |     |     |    |       |        |      | r |                    |
| _ca |     |     |    |       |        |      | c |                    |
| t_1 |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | o |                    |
|     |     |     |    |       |        |      | f |                    |
|     |     |     |    |       |        |      | d |                    |
|     |     |     |    |       |        |      | o |                    |
|     |     |     |    |       |        |      | w |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | p |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | y |                    |
|     |     |     |    |       |        |      | m |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | s |                    |
|     |     |     |    |       |        |      | s |                    |
|     |     |     |    |       |        |      | i |                    |
|     |     |     |    |       |        |      | s |                    |
|     |     |     |    |       |        |      | t |                    |
|     |     |     |    |       |        |      | a |                    |
|     |     |     |    |       |        |      | n |                    |
|     |     |     |    |       |        |      | c |                    |
|     |     |     |    |       |        |      | e |                    |
|     |     |     |    |       |        |      | . |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| gif |     |     |    | Rela  |        |      |   |                    |
| t_l |     |     |    | tives |        |      |   |                    |
| tr_ |     |     |    |       |        |      |   |                    |
| src |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_2 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| gif |     |     |    | Non-P |        |      |   |                    |
| t_l |     |     |    | rofit |        |      |   |                    |
| tr_ |     |     |    |       |        |      |   |                    |
| src |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_3 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| gif |     |     |    | Gover |        |      |   |                    |
| t_l |     |     |    | nment |        |      |   |                    |
| tr_ |     |     |    |       |        |      |   |                    |
| src |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_4 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
| gif |     |     |    | Other |        |      |   |                    |
| t_l |     |     |    |       |        |      |   |                    |
| tr_ |     |     |    |       |        |      |   |                    |
| src |     |     |    |       |        |      |   |                    |
| _ca |     |     |    |       |        |      |   |                    |
| t_5 |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+
|     |     |     |    |       |        |      |   |                    |
+-----+-----+-----+----+-------+--------+------+---+--------------------+

**III. Model Estimation Results**

Exhibit A-2 and A-3 present the coefficient estimates for the binomial
logit models for conditional claim and prepayment probabilities.

  --------------------------------------------------------------------------------------------------------------------------------
  **Exhibit A-2**                                                                                                             
  -------------------- ------------ ---- ------------ ---- ------------ ---- ------------ ---- ------------ ---- ------------ ----
  **Results for                                                                                                               
  Conditional Claim                                                                                                           
  Rate Model                                                                                                                  
  Estimation**                                                                                                                

  Variable             FRM 30            FRM 15            ARM               SR FRM 30         SR FRM 15         SR ARM        

  loancat_cat_2        -0.016732         -0.160244         -0.072824         0.273969          -0.008372         0.084128     \*

  loancat_cat_3        -0.128082         -0.307136         -0.270335         0.441777          -0.320789         0.249221      

  loancat_cat_4        -0.197559         -0.752877         -0.340092         0.470568          -0.546842         0.260696      

  loancat_cat_5        -0.249766         -0.490569         -0.423801         0.365607          -0.781124         0.066382     \*

  ltvcat_cat_2         0.469594          0.747209          0.559814                                                            

  ltvcat_cat_3         0.334264          0.902079          0.591527                                                            

  ltvcat_cat_4         0.371866          1.188054          0.591202                                                            

  ltvcat_cat_5         0.244715          0.947665          0.453571                                                            

  ltvcat_cat_6         0.272767          0.700217          0.866739                                                            

  ltvcat_cat_7         0.382445          0.581725     \*   0.647210                                                            

  season_cat_2         -0.045690         -0.135457         -0.086338         -0.021745         0.201557          0.030289     \*

  season_cat_3         -0.018125         -0.187107         -0.053147         -0.012551    \*   0.218215          -0.207372     

  season_cat_4         0.000286     \*   -0.172602         -0.091357         -0.005937    \*   -0.251634         0.076085      

  pneqcat_cat_2        0.477840          0.639057          0.355824          0.641124          0.970934          0.558920      

  pneqcat_cat_3        0.590987          0.883998          0.443204          1.021999          0.928073          0.990068      

  pneqcat_cat_4        0.764703          1.076217          0.622707          1.248694          0.431279          1.488170      

  pneqcat_cat_5        0.893551          0.732395          0.826702          1.444939          1.376952          1.575700      

  pneqcat_cat_6        1.062103          1.552369          1.010598          1.586848          1.592882          2.044242      

  pneqcat_cat_7        1.400141          1.844753          1.609487          2.399955          2.191905          2.762291      

  ycslopecat_cat_2     -0.081311         -0.126868         -0.147397         -0.426936         0.018613     \*   -0.020076    \*

  ycslopecat_cat_3     -0.011497         -0.034818    \*   -0.189458         -0.273201         0.225656          0.075550     \*

  ycslopecat_cat_4     -0.066557         -0.142677         -0.187546         -0.148120         0.144324     \*   -0.013022    \*

  spreadcat_cat_2      0.479155          0.340711          0.197414          -0.399250                           0.097840     \*

  spreadcat_cat_3      0.657314          -0.062450    \*   0.320994          -0.255871                           0.169858      

  spreadcat_cat_4      0.887306          0.215390          0.216201          0.169508                            -0.059099    \*

  spreadcat_cat_5      0.967931          0.266151          0.228617          0.406086                            -0.079644    \*

  spreadcat_cat_6      1.088967          0.391725          0.317799          0.708497                            -0.079644    \*

  spreadcat_cat_7      1.280974          0.379263          0.317799          0.891223                            -0.079644    \*

  spreadcat_cat_8      1.442082          0.582245                            1.021441                                          

  inmoneycat_cat_2     -0.046575         -0.128041         0.495487          -0.213929         0.473322          0.544196      

  inmoneycat_cat_3     0.142659          -0.013059    \*   0.630699          0.019480     \*   0.659423          0.480549      

  inmoneycat_cat_4     0.393570          0.202241          1.188455          0.253725          1.422897          0.480549      

  inmoneycat_cat_5     0.618083          0.504391          1.630496          0.420967          1.340174          0.480549      

  inmoneycat_cat_6     0.837551          0.575533          1.630496          0.594618          1.855513          0.480549      

  (continued on                                                                                                               
  following page)                                                                                                             

  gift_ltr_src_cat_2   0.343228          0.400345          0.342958                                                            

  gift_ltr_src_cat_3   1.007345          1.366716          0.906482                                                            

  gift_ltr_src_cat_4   0.662357          2.460486          0.921824                                                            

  gift_ltr_src_cat_5   0.726536                                                                                                

  pre_fy86_cat_2       0.699994          0.861123                                                                              

  post_fy95_cat_2      0.467352          0.316874          0.679368          0.562291          -0.046727    \*   1.119083      

  ey_ratecat_cat_2                                         0.075879     \*                                       -0.134383     

  ey_ratecat_cat_3                                         -0.151703                                             -0.615481     

  ey_ratecat_cat_4                                         -0.365531                                             -0.777643     

  ey_ratecat_cat_5                                         -0.194312                                             -1.314773     

  ey_ratecat_cat_6                                         0.014893     \*                                       -0.675084     

  age1                 1.611829          0.620501          2.492865          1.182654          0.555414          1.762337      

  age2                 0.707664          1.016088          1.041914          0.838917          0.487529          0.854395      

  age3                 0.196007          0.217370          0.257009          0.159520          0.358573          0.257120      

  age4                 0.026338          0.048903          0.105508          0.070478          0.056284          0.104720      

  age5                 0.005000          0.035729          0.008198          -0.025621         0.079832          -0.017089     

  age6                 -0.014888         -0.055026         -0.029256                           -0.042060                       

  age7                 -0.043340                           0.024139                            -0.094686                       

  age8                 -0.031681                           -0.057743                           -0.105049                       

  age9                 -0.035792                           -0.037102                                                           

  age10                -0.013896                           -0.058138                                                           

  age11                -0.045270                           -0.025721                                                           

  age12                -0.063514                                                                                               

  age13                -0.057111                                                                                               

  \_cons               -12.781940        -11.622790        -14.716460        -12.000470        -11.846360        -13.101790    

  Statistics           FRM 30            FRM 15            ARM               SR FRM 30         SR FRM 15         SR ARM        

  Log likelihood       -6487906.6        -74540.766        -506651.19        -486983.98        -31438.213        -46818.861    

  Number of obs        297610690         8403630           19682180          34420250          9547900           2381300       

  LR $\chi^{2}$        1053509.12        12291.26          81949.02          78761.55          3909.78           10043.75      

                                                                                                                              

   Prob \> $\chi^{2}$  0.0000            0.0000            0.0000            0.0000            0.0000            0.0000        

                                                                                                                              
  --------------------------------------------------------------------------------------------------------------------------------

\* Not significant for 0.05-level asymptotic normal test

  -------------------------------------------------------------------------------------------------------------------------------
  **Exhibit A-3**                                                                                                            
  -------------------- ------------ ---- ------------ ---- ------------ ---- ------------ --- ------------ ---- ------------ ----
  **Results for                                                                                                              
  Conditional Prepay                                                                                                         
  Rate Model                                                                                                                 
  Estimation**                                                                                                               

  Variable             FRM 30            FRM 15            ARM               SR FRM 30        SR FRM 15         SR ARM        

  loancat_cat_2        0.370862          0.256887          0.277454          0.321970         0.132616          0.317468      

  loancat_cat_3        0.644319          0.399674          0.477953          0.543630         0.225987          0.544409      

  loancat_cat_4        0.823667          0.513876          0.624842          0.671676         0.261625          0.620688      

  loancat_cat_5        0.965692          0.599617          0.686423          0.774034         0.389178          0.816958      

  ltvcat_cat_2         -0.167545         -0.020742         -0.093570                                                          

  ltvcat_cat_3         -0.116537         -0.030026         -0.021562                                                          

  ltvcat_cat_4         -0.019245         0.034244          0.081163                                                           

  ltvcat_cat_5         0.044646          0.106757          0.082819                                                           

  ltvcat_cat_6         -0.130374         -0.020729    \*   -0.055104                                                          

  ltvcat_cat_7         0.060673          0.094637     \*   0.048348                                                           

  season_cat_2         -0.041567         -0.066184         0.011142          0.022039         0.022357          0.106471      

  season_cat_3         0.011018          -0.045468         0.056560          0.056933         -0.081255         -0.017400    

  season_cat_4         -0.149179         -0.182362         -0.085513         -0.138809        -0.218921         -0.092460     

  pneqcat_cat_2        -0.187764         -0.244514         -0.271937         -0.248319        -0.215832         -0.188917     

  pneqcat_cat_3        -0.237237         -0.445660         -0.409251         -0.332121        -0.391674         -0.352293     

  pneqcat_cat_4        -0.307093         -0.606166         -0.537343         -0.438045        -0.574717         -0.420086     

  pneqcat_cat_5        -0.459252         -0.524166         -0.631915         -0.650952        -0.408219         -0.795630     

  pneqcat_cat_6        -0.594571         -0.691834         -0.820448         -0.801052        -0.783135         -0.795086     

  pneqcat_cat_7        -0.638595         -0.834639         -1.081210         -1.107349        -0.789614         -1.302474     

  ycslopecat_cat_2     -0.039655         -0.060205         -0.478498         -0.087066        0.224313          -0.212156     

  ycslopecat_cat_3     -0.077359         -0.102052         -0.376750         -0.297373        0.022689     \*   -0.286571     

  ycslopecat_cat_4     0.491302          0.331956          -0.526625         0.543495         0.722667          -0.290775     

  spreadcat_cat_2      0.686341          0.057182     \*   0.293635          -0.657686                          0.315479      

  spreadcat_cat_3      0.641292          0.442555          0.518244          -0.662849                          0.529468      

  spreadcat_cat_4      0.777440          0.723854          0.786113          -0.386373                          0.770454      

  spreadcat_cat_5      1.260226          1.064783          1.115632          0.049462                           1.027561      

  spreadcat_cat_6      1.871985          1.393165          1.231015          0.539379                           1.243956      

  spreadcat_cat_7      2.098209          1.523365          1.231015          0.755196                           1.243956      

  spreadcat_cat_8      2.094521          1.514091                            0.821478                                         

  inmoneycat_cat_2     0.140066          0.116783          0.012252          0.326640         0.433841          -0.169788     

  inmoneycat_cat_3     0.233292          0.118487          -0.193520         0.380531         0.514983          -0.299316     

  inmoneycat_cat_4     0.287868          0.142322          -0.163184    \*   0.334196         0.600402          0.020028     \*

  inmoneycat_cat_5     0.176030          -0.007592    \*   0.456368          0.241120         0.543969          0.020028     \*

  inmoneycat_cat_6     -0.008856         -0.193809         0.456368          0.112217         0.466013          0.020028     \*

  gift_ltr_src_cat_2   0.026838          -0.027810    \*   -0.062109                                                          

  gift_ltr_src_cat_3   0.032066          0.698979          -0.045678                                                          

  gift_ltr_src_cat_4   -0.225389         -0.075738    \*   -0.059354                                                          

  gift_ltr_src_cat_5   0.094901     \*                     0.452919     \*                                                    

  pre_fy86_cat_2       0.145497          0.027091                                                                             

  (continued on                                                                                                              
  following page)                                                                                                            

  post_fy95_cat_2      0.357813          0.322949          0.340555          0.479892         0.271979          0.6122433     

  ey_ratecat_cat_2                                         -0.329457                                            -0.2454614    

  ey_ratecat_cat_3                                         -0.602031                                            -0.6117676    

  ey_ratecat_cat_4                                         -1.067716                                            -0.9149861    

  ey_ratecat_cat_5                                         -1.622924                                            -1.045389     

  ey_ratecat_cat_6                                         -2.452726                                            -3.024939     

  age1                 0.559038          0.523853          0.847535          0.249767         0.439417          0.5152128     

  age2                 0.233554          0.331733          0.285669          0.069065         0.135992          0.0963906     

  age3                 0.036981          0.060857          0.034597          -0.034810        0.040098          -0.0369361    

  age4                 0.019908          0.043802          -0.018869         0.010536         -0.011580         -0.0299088    

  age5                 -0.008118         -0.048640         -0.037177         -0.009297        0.063282          -0.0004549   \*

  age6                 -0.028365         0.005899          -0.036166                          0.034037                        

  age7                 0.003968                            0.000685     \*                    -0.057383                       

  age8                 -0.000668    \*                     0.007282                           0.010947                        

  age9                 -0.001789                           0.012696                                                           

  age10                -0.001749                           -0.001236    \*                                                    

  age11                -0.024703                           -0.026716                                                         

  age12                0.002702                                                                                               

  age13                0.000637                                                                                               

  \_cons               -7.235682         -6.812974         -5.259501         -4.657524        -5.676237         -4.277679     

  Statistics           FRM 30            FRM 15            ARM               SR FRM 30        SR FRM 15         SR ARM        

  Log likelihood       -38554779         -1105290.8        -3549272.2        -6323305.4       -1273136.1        -509327.44    

  Number of obs        301615260         8607600           20139030          35759420         9821870           2481800       

  LR $\chi^{2}$        7546626.41        120436.49         518527.82         1016237.49       95378.97          56856.57      

                                                                                                                             

   Prob \> $\chi^{2}$  0.0000            0.0000            0.0000            0.0000           0.0000            0.0000        

                                                                                                                             
  -------------------------------------------------------------------------------------------------------------------------------

\* Not significant for 0.05-level asymptotic normal test

**IV. Graphical Comparisons of Goodness-of-Fit by Age of Loan**

Exhibits A-4 to A-15 present within-sample comparisons of the overall
goodness-of-fit of the binomial logit models for claim and prepayment
probabilities. Separate comparisons are given for each of the six
mortgage product types. The graphs compare the average value by mortgage
age of the observed and predicted conditional claim and prepayment
probabilities. The large fluctuations in the observed probabilities at
higher values of mortgage age are the result of sampling variation due
to the small numbers of surviving loans.

> **Exhibit A-4**
>
> ![](media/image10.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-5**
>
> ![](media/image11.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-6**
>
> ![](media/image12.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-7**
>
> ![](media/image13.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-8**
>
> ![](media/image14.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-9**
>
> ![](media/image15.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-10**
>
> ![](media/image16.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-11**
>
> ![](media/image17.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-12**
>
> ![](media/image18.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-13**
>
> ![](media/image19.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-14**
>
> ![](media/image20.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}
>
> **Exhibit A-15**
>
> ![](media/image21.wmf){width="5.661111111111111in"
> height="4.161111111111111in"}

**Literature Cited**

Begg, C.B. and R. Gray, "Calculation of Polychotomous Logistic
Regression Parameters Using Individualized Regressions,"
[Biometrika]{.underline}, 71(1):11-18, 1984.

Calhoun, C.A. and Y. Deng, "A Dynamic Analysis of Fixed- and
Adjustable-Rate Mortgage Terminations," [Journal of Real Estate Finance
and Economics]{.underline}, 24(1/2):9-33, 2002.

Calhoun, C.A., "OFHEO House Price Indexes: Technical Description,"
Washington, D.C., Office of Federal Housing Enterprise Oversight, April
1996.

Case, K.E. and Shiller, R.J., \"Prices of Single Family Real Estate
Prices,\" [New England Economic Review]{.underline}, 45-56, 1987.

Case, K.E. and Shiller, R.J., \"The Efficiency of the Market for
Single-Family Homes,\" [The American Economic Review]{.underline},
79:125-137, 1989.

Deng, Y., J. M. Quigley and R. Van Order, "Mortgage Termination,
Heterogeneity, and the Exercise of Mortgage Options,"
[Econometrica]{.underline}, 68(2):275-307, 2000.