Appendix C: Data for Loan Performance Simulation

This appendix describes the technical details of the application of the
estimated econometric models to produce forecasts of future loan
performance. We begin by describing how loan event data for future time
periods were generated to project future loan performance and
mortgage-related cash flows. This required creating future event data
both for existing books of business and for future loan cohorts not yet
originated. Then, we summarize how the future economic forecasts were
applied. The forecasts of the economic factors are discussed in Appendix
D. The derivation and application of the dispersion volatility estimates
for the national average house price forecast is also explained in
detail.

**I. Future Loan Event Data**

The development of future loan event data was closely integrated with
the development of the data used in the statistical estimation of loan
performance described in Appendix A. As described in Appendix A, the
process of building the historical loan event data entailed expanding
FHA loan origination records into dynamic quarter-to-quarter event data
from loan origination up to and including the period of loan
termination. The loan event data were augmented with external economic
data (house price indexes and interest rates) to compute a number of
time-varying predictors of conditional prepayment and claim rates.

For loans that did not terminate within the historical sample period (FY
1975 Q1 to FY 2005 Q2), the process of building the period-by-period
event data was continued for the scheduled remaining term of the loan.
The key difference in this case is that forecasted values of the
external economic scenarios are required. Otherwise, the data creation
process followed the same steps.

In addition, we were required to forecast loan performance of future FHA
loan books originated through FY 2012. We were required to base our
actuarial projections for future loan books on estimates of total loan
volume of FY 2005 through FY 2012 books of business provided by FHA from
their internal demand model. To establish a baseline set of loan
characteristics for computing predicted prepayment and claim rates, we
utilized historical loan level data for the last four complete quarters
of the historical data sample (FY 2004 Q2-Q4 and FY 2005 Q1). In effect,
we duplicated the loan origination records for these four quarters and
assumed that the same loan composition will be originated during the
ensuing quarters of FY 2005 to FY 2012. We then updated the initial
values of economic variables for future loan originations based on the
corresponding economic forecasts to reflect conditions at the time of
origination. After we project the future mortgage cash flows for the
detailed loan stratifications, we scaled the results to conform to the
projections of total future loan volume from the FHA demand model.

As described in Appendix A, the data used for statistical estimation
comprise detailed loan stratifications grouped by age of loan, all
possible combinations of the categorical outcomes for the explanatory
variables, and additional categories such as mortgage product types. The
data for future cohorts are organized in the identical manner. Although
we have discussed the development of historical and future loan event
data as a separate processes, in practice we undertook an integrated
approach whereby both the historical and future loan event data were
generated at the same time and stored in the same database. This greatly
simplified the projection of future conditional claim and prepayment
probabilities by taking advantage of internal routines for prediction
from estimation. Once a logit model has been estimated, the software can
automatically generate predicted values, in this case predicted logit
probabilities, for all records in the data, regardless of whether they
were used in estimation. Additional fields are automatically added to
the dataset that contain the predicted logit probabilities for all
records with non-missing values of the explanatory variable fields used
in estimation. Automatic generation of predicted probabilities in this
manner greatly reduced the chances for human error in computing the
predicted probabilities.

**II. Future Economic Forecasts**

FHA provided quarterly forecast data from Global Insight, Inc. These
data served as the source of future interest rates and appreciation in
housing values in our analysis.

For the projection of future changes in housing values, we used the
Global Insight's forecast of national average sales price of existing
single-family homes. Because the national-level house price series is an
average of regional house price performance, it tends to smooth out
differences in house price trends in the individual underlying regional
indexes. This implies an additional layer of uncertainty with regard to
the dispersion of individual house price appreciation rates around the
market average, now represented by a national-level house price index
(HPI). When using the national-level house price forecast to compute the
probability of negative equity, it was important to take into account
both sources of uncertainty.

To address this issue, we developed a methodology to estimate the
historical dispersion of the OFHEO regional (Census division) and
metropolitan HPIs in relation to the national average house price
series. This analysis is described in greater detail in the next section
of this Appendix. To summarize, estimates of additional dispersion were
combined with those for the underlying regional or metropolitan index to
increase the overall estimated dispersion of the appreciation rate of
individual houses within a geographical location. The additional
volatility is added only from the start of the future forecast of loan
performance, that is, at the point that the computation of the
probability of negative equity switches from using MSA-level indexes to
using the forecasted national level house price index. The additional
dispersion increases with time in a similar manner to the dispersion of
one single house from the MSA-level index.

Recall that the source of house price appreciation rates for historical
loans was HPIs obtained from OFHEO. OFHEO reports 28 MSA-level indexes.
The rule for assigning metro indexes is to use the MSA index if one
exists for the loan's Federal Information Processing Standards (FIPS)
code, otherwise use the Core Based Statistical Area (CBSA) index if the
population in the CBSA exceeds 1 million. Based on this criterion, we
chose a subset of 96 representative MSAs out of a possible 379 reported
by OFHEO. For all other loans outside these MSAs, we applied one of the
nine OFHEO Census division HPIs depending on the location of the
mortgage.

As described in Appendix A, the indexes are used in conjunction with
estimates of house price diffusion parameters to compute probabilities
of negative equity at each loan age for individual borrowers. The
volatility estimate reflects the uncertainty about the dispersion of
individual house price appreciation rates around the national average
appreciation rate.

**III. Dispersion of Local House Price Indices**

As also described in Appendix A, the distributions of individual housing
values relative to the value at loan origination were computed directly
pursuant to OFHEO HPIs in terms of estimates of house price drift and
diffusion volatility that were computed in the first quarter of 2005.

To forecast the future changes in housing values, we utilized Global
Insight's national-level HPI forecast. However, the Global Insight's
national average house price data and forecast do not provide estimates
of the diffusion volatility between a single property and the national
index. Although OFHEO publishes a national-level HPI, it is based on a
weighted average of indexes for the nine Census divisions, and no
separate diffusion parameters are provided at the national level. Thus,
direct estimates of house price dispersion around a national index do
not exist. Since we used Census-division-level and metropolitan-level
indexes for estimating historical loan performance models, we elected to
adopt an approach that would build upon the OFHEO local house price
volatility estimates and modify it to be consistent with the forecasting
period when a national-level index is used.

We estimated the variance in the geometric growth rates of housing
values implied by the division indexes around the geometric growth rates
of the national-level index. The following discussion uses the case of
MSA indexes as an example, but the same approach is applied in the case
of the Census division indexes.

The growth rate for property *i* between time periods *t* and *s*
relative to its MSA index is given by:

$\text{ln}(G_{i,t,s}) = \text{ln}(H_{i,t}) - \text{ln}(H_{i,s}) = \text{ln}(H_{\text{MSA},t}) - \text{ln}(H_{\text{MSA},s}) + \varepsilon_{i,t,s}$
(1)

where *H~i,t~* is the value of house *i* at time *t*, *H~MSA,t~* is the
house price index of the surrounding MSA at time *t*, and *ε~i,t,s~* is
the deviation of the growth rates of the house *i* and its surrounding
MSA index from time *s* to *t*. Similarly, the growth rate implied by
the MSA index relative to the national average forecast can be
decomposed as follows:

$\text{ln}(G_{\text{MSA},t,s}) = \text{ln}(H_{\text{MSA},t}) - \text{ln}(H_{\text{MSA},s}) = \text{ln}(H_{N,t}) - \text{ln}(H_{N,s}) + \varepsilon_{\text{MSA},t,s}$
(2)

where *H~N,t~* is the national house price index at time t, and
*ε~MSA,t,s~* is the deviation of the growth rates of the MSA index and
the national index from time *s* to *t*. Intuitively, one can think that
the growth rate of a particular MSA as the national average growth rate
plus a deviation term (*ε~MSA,t,s~*). Similarly, the growth rate of a
particular house is equal to the MSA level average growth rate plus
another deviation term (*ε~i,t,s~*).

Plug equation (2) into equation (1), we find that the individual house
price growth rate equals the national average house price index growth
rate and the sum of the dispersions of individual property around MSA
and the specific MSA around national average growth rates:

$\text{ln}(G_{i,t,s}) = \text{ln}(H_{N,t}) - \text{ln}(H_{N,s}) + \varepsilon_{i,t,s} + \varepsilon_{\text{MSA},t,s}$
(3)

Notice that the variance of the first component of dispersion error
given by $\varepsilon_{i,t,s}$ can be computed directly from the "*a*"
and "*b*" parameters estimated by OFHEO using the three-stage
weighted-repeat-sales methodology:

$E\left( \sigma^{2}\left( \text{ln}G_{\text{i,t,s}} - \text{ln}G_{\text{MSA,t,s}} \right) \right) = E\left( \sigma^{2}\left( \varepsilon_{i,t,s} \right) \right) = a \cdot (t - s) + b \cdot (t - s)^{2}$
(4)

where E(•) is the expectation operator, (*t*-*s*) is the number of
quarters since the loan origination time *s*. We estimated the variance
of the second component error $\varepsilon_{\text{MSA},t,s}$for the
dispersion of the MSA index growth rate around the national index growth
rate forecast by a linear regression in the following form:

$\sigma^{2}\left( \text{ln}G_{\text{MSA,t,s}} - \text{ln}G_{\text{N,t,s}} \right) = \sigma^{2}\left( \varepsilon_{\text{MSA,t,s}} \right) = c \cdot (t - s) + e$
(5)

where e is the residual of the regression. Because equation (4) was
estimated by OFHEO as a residual term when estimating the MSA house
price index using all houses within that location, the individual
property diffusion term must be orthogonal to the diffusion term between
the MSA and the national house price indices. That is, the noise term
$\varepsilon_{i,t,s}$ is independent of the
$\varepsilon_{\text{MSA},t,s}$, or:

$\rho\left( \varepsilon_{\text{i,t,s}},\ \varepsilon_{\text{MSA,t,s}} \right) = 0$
(6)

This implies the following model for the variance of individual house
price cumulative appreciation rates around the national average
forecast:

$E\left( \sigma^{2}\left( \text{ln}G_{i,t,s} \right) \right) = a \cdot (t - s) + b \cdot (t - s)^{2} + c \cdot (t - s)$
(7)

The parameter "*c*" required for projecting the additional dispersion of
the MSA index around the national average forecast was estimated as
follows. For each quarter *t* we computed the cross-sectional (across
MSAs) dispersion variance (MSA versus national) for each possible value
of $(t - s) > 0$, which corresponds to time since loan origination,
*i.e.*, mortgage age. We then computed the average dispersion variance
according to each age of loan:

$\sigma^{2}({\overline{\varepsilon}}_{t - s}) = \frac{1}{N}\text{  }\sum_{}^{}\left\lbrack \text{ln}(\frac{G_{\text{MSA},t,s}}{G_{N,t,s}}) \right\rbrack^{2}\ t = \ s + 1,s + 2,\ \text{.}\text{.}\text{.},\ T$
(8)

where *N* is the number of MSAs used in the estimation sample, and *T*
is the most current quarter that house price indexes are available
(first quarter of 2005 in this Review). This gives us a
cross-section/time-series sample of average MSA index dispersion
variance around the national average forecast that we assume is a linear
function of (*t-s*):

$\sigma^{2}({\overline{\varepsilon}}_{t - s}) = c \cdot (t - s) + u_{t - s}\ t = \ s + 1,s + 2,\ \text{.}\text{.}\text{.},\ T;\ s = 0,1,\ \text{.}\text{.}\text{.};\text{  }s < t\ $
(9)

where *u~t-s~* is the regression residual. We estimated the unknown
parameter "*c*" using a weighted least square regression with the number
of average variance observations at each value of *t-s* as weights. The
estimated quarterly standard deviations ($\sqrt{c}$) at age one, i.e.,
(*t-s*) = 1, were 2.7682% for MSA indexes and 2.7192% for Census
division indexes.

One of the following two formulas was applied depending on whether the
time period was historical or future:

$\left\{ \begin{matrix}
E\left( \text{ln}G_{i,t,s} \right) = \text{ln}G_{\text{MSA},t,s} \\
\sigma^{2}\left( \text{ln}G_{\text{i,t,s}} \right) = a \cdot (t - s) + b \cdot (t - s)^{2}
\end{matrix} \right.\ \ \text{if }s < t \leq T$ (10a)

$\left\{ \begin{matrix}
E\left( \text{ln}G_{i,t,s} \right) = \text{ln}G_{\text{MSA},T,t} + \text{ln}G_{N,t,T} \\
\sigma^{2}\left( \text{ln}G_{\text{i,T,s}} \right) = a \cdot \left( \text{t-s} \right) + b \cdot \left( \text{t-s} \right)^{2} + c \cdot (t - T)
\end{matrix} \right.\ \ \text{if }s < T \leq t$ (10b)

Equations (10a) were applied to historical sample time periods when the
MSA index was used to update expected housing values, and equations
(10b) were applied during future time periods when the national average
forecast was used to update expected housing values.

For future loan originations only a single formula set is required:

$\left\{ \begin{matrix}
E\left( \text{ln}G_{i,t,s} \right) = \text{ln}G_{N,t,s} \\
\sigma^{2}\left( \text{ln}G_{\text{i,t,s}} \right) = a \cdot \left( \text{t-s} \right) + b \cdot \left( \text{t-s} \right)^{2} + c \cdot (t - s)
\end{matrix} \right.\ \ \text{if }T \leq s < t$ (11)

Equations (11) were applied to future loan originations and only the
national average forecast was used to update expected housing values.

The additional term associated with dispersion of the growth rate of an
MSA house price index around the growth rate of the national house price
index increases the overall dispersion volatility and results in higher
probabilities of negative equity. This is counterbalanced by reduced
relative frequency of low expected HPI values when using a national
average house price forecast instead of the more volatile MSA house
price indexes.

**IV. A Numerical Example**

Exhibit C-1 presents an example to depict the expected value and the
dispersion of the value of a house growth over time. Let's assume that
the sales price of a underlying house of a mortgage loan originated in
Binghamton, NY at the first quarter of FY 1990 (*s*) was \$100,000.
Further assume that the last historical house price indexes available
are as of the first quarter of FY 2005 (*T*).

**Exhibit C-1**

![](media/image1.wmf){width="6.49375in" height="3.7576388888888888in"}

The expected value and the dispersion standard deviation of this house
at each exposure year (*t*) can be computed following Equations (10b).
The center line in Exhibit C-1 shows the expected value of the house at
each exposure year. This expected value was updated by using the OFHEO
house price index for the Binghamton, NY MSA up to *T*. After *T*, since
a forecast of the HPI growth rate for the Binghamton MSA is not
available, the future expected value of the house is then updated
according to the national house price growth rate forecasted by Global
Insight.

The house price growth rate of Binghamton MSA could be higher or lower
than the national average for the next 15 years. The red shaded area
represents the possible future value of the house if the Binghamton MSA
growth rate is 1 standard deviation above or below the national average
growth rates. Note that, prior to T, because the expected house price
was updated by using the MSA specific index, such MSA-level dispersion
did not exist in all historical distributions.

The gold shaded area represents the additional dispersion of 1 standard
deviation due to the individual house level volatility around the MSA
average. The size of this source of dispersion starts immediately after
the house was sold in the first quarter of 1990 and continues to grow
over the age of the mortgage loan. The OFHEO dispersion volatility
parameters were used in computing this source of dispersion.

As a result, the outer boundary of the shaded area in Exhibit C-1
provides a visual sense of the dynamics of the house price distribution
over the life of the mortgage. The probability of negative equity
variable is directly computed by the probability that the house price
may fall below the UPB of the mortgage at any point in time.