Technical Paper Series

Congressional Budget Office

The FHA Budget Subsidy Simulation System: 

A Dynamic Simulation Model of Budget Outcomes for FHA

Single-Family Mortgage Insurance

Charles A. Capone, Jr.

Microeconomic and Financial Studies Division

July 2003

2003-7

Technical papers in this series are preliminary and are circulated to stimulate discussion and critical

comment. They are not subject to CBO’s formal review and editing processes. The analysis and

conclusions expressed in them are those of the authors and do not necessarily represent the position

of the Congressional Budget Office. References in publications should be cleared with the authors.

Papers in this series can be obtained at www.cbo.gov (select Publications and then Technical

Papers).

For additional information about this paper, contact Charles Capone at (202) 226-2649

or by email at ccapone@cbo.gov. 
ii

Abstract

This paper introduces a dynamic simulation model of value to the government from FHA singlefamily mortgage insurance. The model, known as the FHA Budget Subsidy Simulation System

(FHA-BSSS), was built at CBO as a tool for analyzing the sensitivity to economic fluctuations of

net receipts from or outlays to FHA, and how that sensitivity could be used to inform initial

budget estimates and ongoing re-estimates.

The FHA-BSSS is a software system that utilizes loan-record databases and econometric models

to forecast FHA cash flows and resulting budget subsidy rates in a stochastic environment.

Separate econometric models are used to create the economic forecasts, relate mortgage

termination rates to economic conditions, and to determine post-default outcomes and costs.

FHA cash flows are computed from predicted mortgage events, and then are converted into

budget subsidy rates and dollars. 

This technical paper both describes the research entailed in building the FHA-BSSS, and

provides implementation details to allow replication of the model. The paper ends with some

basic results from using the FHA-BSSS to make current estimates of subsidy rates on 1992-2007

budget-year cohorts of FHA insured loans. The model version used was current as of March

2003, economic data was current as of December 2002, and FHA loan information was current as

of September 30, 2002.
iii

CONTENTS

I. INTRODUCTION ........................................................... 1

FHA AND ITS BUDGET ACCOUNTING .................................... 1

USE OF SIMULATION MODELS IN MORTGAGE VALUATION ............... 3

THE FHA BUDGET SUBSIDY SIMULATION SYSTEM (FHA-BSSS) ............ 4

OUTLINE OF REMAINING SECTIONS ..................................... 6

II. ECONOMIC FORECASTING MODELS ................................ 7

VECTOR AUTOREGRESSIVE MODEL OF ECONOMIC CONDITIONS .......... 8

FORECASTING THE VAR VARIABLES .................................. 11

HOUSE–PRICE GROWTH-RATE REGRESSION MODEL .................... 12

FORECASTING HOUSE PRICE GROWTH ................................. 13

ACCOUNTING FOR THE POSSIBILITY OF DEPRESSIONS .................. 16

RANGE OF SIMULATED ECONOMIC VALUES ............................ 17

ADDING LOCAL HOUSING CYCLES ..................................... 18

THE TREASURY YIELD CURVE AND DISCOUNT FACTORS ................ 21

III. MORTGAGE DEFAULT AND PREPAYMENT REGRESSION MODELS

............................................................... 24

GROUPING LOAN RECORDS FOR USE IN REGRESSION ANALYSIS AND

FORECASTING ................................................. 27

DEFINING MORTGAGE TERMINATION EVENTS ......................... 29

THE LOGISTIC REGRESSION APPROACH TO ESTIMATING DEFAULT AND

PREPAYMENT PROBABILITY EQUATIONS ........................ 31

EXPLANATORY VARIABLES USED IN REGRESSION ANALYSIS AND

PROBABILITY FORECASTS ...................................... 35

RESULTS OF LOGISTIC REGRESSION ESTIMATIONS ..................... 38

Analysis-of-Variance Tests ................. ................. ................. . . 39

Discussion of Coefficient Estimates ................. ................. ............. 44

USE OF REGRESSION EQUATIONS IN FORECAST SIMULATIONS .......... 64

Loan Performance in First Three Quarters of Loan Life ................. ............. 65

Loan Performance After Year Twenty ................. ................. ........... 65

Performance Regression Coefficient Adjustments ................. ................. . . 66

IV. POST-DEFAULT OUTCOME REGRESSION MODELS ................ 70

LOGISTIC REGRESSION OF WORKOUT ASSISTANCE OFFERS ............. 71

Regression Model ................. ................. ................. ......... 72

Regression Results ................. ................. ................. ......... 73

Use of Model in Forecast Simulations ................. ................. ........... 75

TIME-TO-FORECLOSURE REGRESSION ................................. 75

Regression Model ................. ................. ................. ......... 76

Regression Results ................. ................. ................. ......... 77

Use of Model In Forecast Simulations ................. ................. .......... 80

TIME-TO-DISPOSITION REGRESSION ................................... 81
Regression Model ................. ................. ................. ......... 81

Regression Results ................. ................. ................. ......... 82

Use of Model in Forecast Simulations ................. ................. ........... 83

REO RECOVERY RATE REGRESSION ................................... 83

Regression Model ................. ................. ................. ......... 83

Regression Results ................. ................. ................. ......... 86

Use of Model in Forecast Simulations ................. ................. ........... 87

V. FINAL FORECAST ASSUMPTIONS AND SUBSIDY RATE CALCULATIONS 

 .............................................................. 88

PREMIUM INCOME ................................................... 90

OTHER PARAMETERS NEEDED TO CALCULATE NET DEFAULT COSTS .... 90

FUTURE COHORT CHARACTERISTICS AND VOLUMES ................... 91

SUBSIDY CALCULATIONS ............................................. 94

VI. SIMULATION RESULTS FOR THE 1992-2007 COHORTS ............. 95

SUBSIDY RATE CONFIDENCE BOUNDS ................................. 96

Subsidy Rate Estimates as a Basis for Policy Making ................. ................ 97

FREQUENCY DISTRIBUTIONS AND MEAN-VALUE ESTIMATES ............ 99

BIASES IN CURRENT FHA SUBSIDY RATE ESTIMATES .................. 104

FREQUENCY DISTRIBUTIONS OF SUBSIDY RATE COMPONENTS ......... 104

VALUE AT RISK ..................................................... 105

FIGURES, BOXES, AND TABLES

Figure 1. Structure of the FHA Budget Subsidy Simulation System ............. 108

Figure 2. Mortgage Interest Rates, Historical Series and Forecast Confidence 

Bounds ..................................................... 109

Figure 3. House Price Growth Rates, Historical and Forecast Confidence Bounds . . 109

Figure 4. Confidence Bounds for Subsidy Rate Estimates, by Cohort ............ 110

Figure 5. Frequency Distribution of Subsidy Rate Estimate for FY2003 Loan 

Guarantees .................................................. 111

Figure 6. Frequency Distribution of Premium Revenue Component of Subsidy Rate

Outcomes for FY2003 ......................................... 112

Figure 7. Frequency Distribution of the Default Cost Component of Subsidy Rate

Outcomes for FY2003 ......................................... 113

Box 1. Transformation of Inflation Variable for Use in the Vector Autoregression 

Model ...................................................... 114

Box 2. Deriving Discount Factors By Interpolating Forward Rates ............ 116

Table 1. Vector Autoregression Equation ................................. 118

Table 2. National House-Price-Growth-Rate Regression ..................... 119

Table 3. Parameter Values for Stochastic Cumulative, Regional House-Price-Path 

Deviations from Cumulative, National House-Price Growth ........... 120

Table 4. Treasury Rates Used in Credit Subsidy Calculations ................. 122

Table 5. Computing Major Points on the Constant Maturity Treasury Yield Curve

From the 3-month and 10-year Rates and Regression Equations ........ 123

Table 6. Explanatory Variables Used in Mortgage Performance Regressions ..... 124
Table 7. ANOVA Tests of the Statistical Significance and Relative Influence of

 Explanatory Variables in Default-Rate Regression Analysis ........... 127

Table 8. ANOVA Tests of the Statistical Significance and Relative Influence of

 Explanatory Variables in Prepayment-Rate Regression Analysis ....... 128

Table 9. Results of Default- Rate Logistic Regressions ....................... 129

Table 10. Results of Prepayment-Rate Logistic Regressions .................... 132

Table 11. Default Related Cash-Flow Timing and Regression Models ............ 135

Table 12. Logistic Regression of the Probability of Receiving Loan-Workout 

Offers ...................................................... 136

Table 13. Foreclosure-Time Regression: Accelerated Failure Time with Weibull 

Error Distribution ............................................ 137

Table 14. Foreclosed-Property Disposition (Sales) Time Regression: Accelerated 

Failure Time with Weibull Error Distribution ...................... 139

Table 15. Foreclosed Property Recovery Rate Regression ..................... 141

Table 16. Miscellaneous Parameters Used to Calculate Insurance-Claim Cost in

Simulations ................................................. 142

Table 17. Confidence Bounds for Subsidy Rate Simulation Results, by Cohort 

Year ....................................................... 144
FHA-BSSS, I. Introduction Page 1

1

Several individuals read this manuscript at various stages: Kim Cawley, Ufuk Demiroglu, Robert Dennis,

Peter Fontaine, Robert McClelland, John McMurray, Susanne Mehlman, Albert Metz, and Marvin Phaup. The

assistance of these persons is greatly appreciated, yet the author maintains responsibility for the paper’s final content.

2

This and all other general comments in this paper refer to single-family mortgage insurance provided in

programs that are part of the Mutual Mortgage Insurance Fund. That Fund is required to be actuarially sound. It

accounts for around 90 percent of all FHA single-family insurance, and over 80 percent of all insurance (dollar

volume) provided by FHA.

I. INTRODUCTION1 

FHA AND ITS BUDGET ACCOUNTING

After nearly 70 years of existence, the Federal Housing Administration (FHA) remains an

important federal credit program. It insures nearly 10 percent of all mortgage loans

originated in the U.S. each year, and it accounts for half of all federal credit

activity–direct loans and guarantees of all types. FHA has a statutory mandate for its

major single-family mortgage insurance programs to remain actuarially sound.2 That

mandate requires that net budgetary receipts for the government remain positive under a

wide range of economic conditions. 

This analysis concludes that the balance of underwriting standards and premium fees now

used by FHA do not ensure actuarial soundness into the future. FHA’s principal

insurance fund is, therefore, not as sound as indicated by annual independent actuarial

reviews and financial audit reports. Those reviews and reports include as assets the

expected (future) net income from outstanding insurance and, according to this study, that

value is overstated. 
FHA-BSSS, I. Introduction Page 2

3

This type of accounting has been in place for federal credit programs since 1992. For summaries of the

FHA re-estimate revisions see, Office of Management and Budget, FY2004 Federal Credit Supplement, Table 8.

Expected net income is overstated because it is based on estimates of insured-loan

defaults and prepayments under unrealistic economic conditions, where house prices and

interest rates are very stable. In contrast, it is well known that normal economic

fluctuations themselves increase loan terminations, which then lowers net receipts from

loan guarantees. In addition, FHA faces an asymmetric profit/loss function so that budget

receipts estimated using average economic conditions will be larger than the average of

receipt levels across all possible economic conditions. 

The existing bias in initial budget estimates is seen when one decomposes annual budget

revisions required by the Office of Management and Budget (OMB). Those revisions,

called re-estimates, are based on actual, to-date program performance and new forecasts

of future default and prepayment rates of the insured mortgages. Budget receipt reestimates based on loan performance have been overwhelmingly downward for all annual

budget-year cohorts with measurable experience.3 The re-estimate numbers published by

OMB, however, do not show this result because they also include offsetting amounts

from one-time program changes and asset sales. When these offsets are removed, there

emerges a clear pattern of over-predicting expected budget receipts from each year’s

book-of-business and by a large percentage.

To understand the extent of the bias, and to correct for it, requires a simulation model
FHA-BSSS, I. Introduction Page 3

4A recent issue of the Journal of Banking and Finance (vol. 24, 2000) has numerous articles that detail

models currently in use at major banks. See also, Kaskowicz, et al, Best Practices in Mortgage Default-Risk

Management and Economic Capital, unpublished manuscript, Loan Performance Inc. (February 2002), available at

http://www.loanperformance.com/library/articles/best_practices_2002.pdf. In April, 2003, Moody’s Investor

Services introduced a proprietary simulation model for measuring mortgage credit risk and pool-level capital

requirements which is similar in nature to the FHA-BSSS.

capable of valuing an insurance portfolio under a wide range of plausible economic

conditions. This technical paper describes such a model for FHA single-family mortgage

insurance programs under the umbrella of the Mutual Mortgage Insurance Fund.

USE OF SIMULATION MODELS IN MORTGAGE VALUATION

Simulation models are in regular use by private investors in mortgages and mortgagebacked securities to develop risk-adjusted valuations. The most prominent of these are

option-adjusted spread (OAS) models that attempt to make default-free but prepaymentprone mortgage yields comparable to (non-callable) bond yields. Simulated yieldadjustment factors account for pool-level prepayment potential and are subtracted from

mortgage coupon rates to arrive at bond-like yields. Models that analyze credit-risk yield

spreads are less common, principally because mortgage credit risk in the U.S. is highly

concentrated in a small number of government agencies, government sponsored

enterprises, and private mortgage insurers. However, large commercial banks and thrifts

are now looking at developing such models to potentially decrease capital requirements

on whole-loan investments under pending revisions to the Basle capital accords.4

Simulation models face two primary challenges: developing reliable behavioral equations
FHA-BSSS, I. Introduction Page 4

5

Research behind the FHA-BSSS was accomplished with the help of many individuals. In particular, the

author thanks: Robert Dennis, Robert Arnold, John Peterson, and Ufuk Demiroglu of the CBO Macroeconomic

Analysis Division for building the stochastic economic-environment models; Malgorzata Klosek, visiting scholar, for

the model of regional house-price cycles; Jenny Au, Sean Corcoran, Lori Ellebracht, Erin Hirsch, Joseph Nichols,

DaRon Ross, Errick Simmons, and Aurora Swanson for research assistance; Carol Frost for SAS consulting; Georgia

Brown, Eric Guille, Guanli Lu, and Rick Williams for IT support; Judy May, Dominique Stasulli, and Edward

Szymanoski of FHA for access to data; and Debbie Lucas for research oversight.

of how mortgage borrowers respond to economic events; and simulating a reasonable

range of future economic outcomes with appropriate correlations between economic

measures/variables. When these two elements are pieced together, they must produce

reasonable results across a wide spectrum of economic conditions so as to provide a

probability distribution of ultimate outcomes that can accurately guide investor decisions. 

This study of FHA single-family mortgage insurance addresses these challenges directly.

Economic conditions are modeled in a vector autoregressive system of equations, where

movements in any one variable have effects on movements of all other variables in the

system, in future periods. Behavioral models are built for borrower termination decisions,

and for the events and time sequences triggered by borrower default. Adding the latter

allows for a full development of the probability distribution of default-related claim costs. 

THE FHA BUDGET SUBSIDY SIMULATION SYSTEM (FHA-BSSS)5

Performance of government loan guarantee programs like FHA are measured by

guarantee-contract values, as defined by current budget accounting rules. The question of

interest is, for each dollar of loans insured by FHA, what is the expected net present value

to the federal government? This value is called a subsidy rate. The FHA Budget Subsidy
FHA-BSSS, I. Introduction Page 5

6

12 USC 1711(f).

Simulation System (FHA-BSSS) described in this study attempts to calculate an unbiased

estimate of the true, but unknown subsidy rate that will reveal itself as the future economy

unfolds and actual program outcomes are known. 

The FHA-BSSS also measures the probability that expected net receipts from FHA could

rather turn out to be (unexpected) net outlays for the federal budget. Analysis of such

outcomes can lead to measures of economic capital: the level of capital reserves that

would be required to assure solvency of the FHA program against a loss-event defined in

the tail of the outcome probability distribution. Not that a federal government program

can or needs to hold capital reserves, but it is a risk-metric familiar to private investors.

Since 1990, the Congress has indicated an interest in FHA producing net receipts for the

federal budget, as defined by statutory “capital” requirements.6

In process, the FHA-BSSS takes pools of FHA-insured loans and projects their

experience, and the resulting cash flows of FHA, through 1000 different future economic

paths. The use of 1000 simulation trajectories is arbitrary. It was chosen to balance the

need for generating a sufficient number of outlier events with the need to control total

computer processing time. 

Simulation results shown in this paper (section VI) are for entire budget-year cohorts of

loan guarantees–FHA endorsements by fiscal year book-of-business. The structure of the
FHA-BSSS, I. Introduction Page 6

FHA-BSSS allows for further examination of subsidy rate outcomes for sub-groups based

upon values of nine different categorical variables, including product type and loan-tovalue class (see section III). The simulation system also was built to address a wide range

of policy issues involving how changes in administrative policies and program usage

affect the Federal budget. These ancillary matters are, however, beyond the scope of this

paper and are not addressed here.

OUTLINE OF REMAINING SECTIONS

Figure 1 provides a schematic outline of how the various parts of the FHA-BSSS work

together to create subsidy rate forecasts. The (downward) flow of that illustration is

matched by the order of material presented in the following sections of this study.

Relevant parts of Figure 1 are repeated at the beginning of each section, as a guide to

readers. Each segment of Figure 1 is mapped to sections of the study as follows: 

Section of Figure 1 Part of the FHA-BSSS Sections of this Study

Top Forecast economic inputs for

mortgage performance models 

II. Economic Forecasting Models

Center Forecast mortgage default and

prepayment rates

III. Regression Models of Mortgage

Default and Prepayment

Center Forecast default resolution

types and timing and losses for

foreclosures

IV. Post-Default Outcome

Regression Models

Bottom Calculate cash-flow time series

and discount to net present

values

V. Final Forecast Assumptions and

Subsidy Rate Calculations

(not in figure) Discussion of simulation results VI. Simulation Results for the 1992-

2007 Cohorts
The FHA-BSSS, II. Economic Forecasting Models page 7

II. ECONOMIC FORECASTING MODELS 

 

The first thesis of simulation analysis is that normality of probability density functions

cannot always be assumed. When the outcome of interest is the product of numerous

stochastic processes, each with its own unique distributional form, then the distribution of

final outcomes can be best understood through repeated-experiment simulations. The

mean outcome across a large number of simulated economic environments can provide an

unbiased estimate of the actual, but unknown outcome of the process in question. In this

case the outcome is an (average) predicted subsidy rate, which is the net present value of

cash flows from loan guarantees, per dollar of guarantees, for each budget year cohort or

book-of-business. 

For mean simulation outcomes to be unbiased estimators, the simulations of economic

environments must be realistic. Such realism includes the size and distribution of
The FHA-BSSS, II. Economic Forecasting Models page 8

potential values of economic variables, the correlations between their values, and their

movements over time. To create realistic economic environments, the FHA-BSSS uses

both regression analysis and stochastic (random) shocks to the values produced using the

regression equations. The various models and steps involved are: 

• A vector autoregressive (VAR) model of macro economic changes;

• Regression of national house price movements on the economic variables of the

VAR model;

• Addition of random shocks or disturbances to values predicted with the regression

models, to create the full range of possible variable values; 

• Addition of the (random) possibility of Great-Depression-like events;

• Generation of random local house price paths and cycles around national price

movements; and

• Interpolation of points along the Treasury yield curve to create interest rates for

discounting cash flows to produce subsidy estimates.

VECTOR AUTOREGRESSIVE MODEL OF ECONOMIC CONDITIONS

The FHA-BSSS projects economic conditions using a five-equation Vector

Autoregression (VAR) model. VAR techniques are especially suited to time-series data

where there are strong correlations among variables but the direction of causation is

difficult to determine. VAR is also known as impulse-response modeling because when

one variable changes, its effects ripple over time through all other variables. The
The FHA-BSSS, II. Economic Forecasting Models page 9

7

Longer lag structures were tested but provided no net benefits.

8

Core inflation excludes food and energy price movements.

magnitudes of these responses are modeled in a system of equations where

contemporaneous values of each variable are regressed on lagged values of all variables

in the system. 

The VAR used to project quarterly economic conditions for the FHA-BSSS uses a 3-

quarter lag structure in equations for the following variables:7

• 30-year Mortgage Interest Rate, in log form;

• National Civilian Unemployment Rate, in log form;

• Inflation, measured as the annual rate of change in the core component of the

urban consumer CPI each quarter, transformed as described below;8

• Ratio of the 30-year mortgage rate to the 10-year constant maturity Treasury

rate, in log form; and

• Ratio of the 10-year Treasury yield to 3-month Treasury bill rate , in log form.

Logarithmic (log) transformations yield regression equations based on percentage

changes in variables rather than on levels. This change is desirable for stochastic

simulations of how economic conditions change over time, and is also valuable for

creating regression variables with constant variances. Using the untransformed variables

(actual interest rates, unemployment rates, and inflation rates) directly in the regression

would produce sub-optimal results because error variances would be proportional to the
The FHA-BSSS, II. Economic Forecasting Models page 10

9

The mortgage rate series is the conventional 30-year mortgage rate reported by Freddie Mac to the Federal

Reserve Board.

values of the underlying variables, creating a heteroskedasticity problem. Log

transformations generally alleviate this problem; however, log transformations are also

restricted to positive variable values. While this constraint is not a problem for estimating

a regression model using recent history, it then restricts the values that can be forecast for

the future.

The possibility of negative forecast values is desirable for one variable, inflation. To

allow for negative forecast values while also managing the inherent heteroskasticity

problem, a transformation other than the logarithmic is required. The transformation used

here relies upon the volatility of the inflation time series to structure an appropriate

function. Its form is derived and shown in Box 1.

The VAR is estimated using quarterly data from 1968 to 2001, and results are in Table 1.

All interest rates come from historical series published by the Federal Reserve Board,

while unemployment and core inflation rate data are from the Bureau of Labor Statistics

of the U.S. Department of Commerce.9 The unemployment rate is the adult, noninstitutionalized population unemployment rate.
The FHA-BSSS, II. Economic Forecasting Models page 11

10Readers wanting to replicate this forecasting system, who do not have the residuals from the VAR

estimation, can generate random shocks by assuming they are normally distributed random variables with zero means

and standard deviations equal to the standard errors of the regression equations reported in Tables 1 and 2.

FORECASTING THE VAR VARIABLES 

The VAR equations are the basis for forecast simulations but, by themselves, do not

produce the values necessary for predicting mortgage performance and subsidy rates. To

generate forecast predictions, the VAR equations (Table 1) are seeded from recent

historical experience. Doing so provides lagged values for computing predictions in the

first quarter of the forecast period. Random shocks (or deviations) are next added to those

first quarter fitted values using re-sampling techniques. These shocks are random draws

from the regression residuals for each VAR equation. They provide deviations of actual

historical data from regression equation predictions.10

Once first quarter values are determined for VAR variables, they become one-quarter

lagged values when forecasting the second quarter and the re-sampling procedure is again

used to generate random shocks. This two-step process continues for the entire 30-year

forecast period. Once finished, the VAR variables are un-transformed to get values of the

economic variables of interest. For interest rates and unemployment, the inverse

transformation is exponentiation. For inflation, the inverse transformation function is

described in Box 1.

Quarterly forecasts for each variable are normalized so that mean values across all
The FHA-BSSS, II. Economic Forecasting Models page 12

11Results shown in this technical paper use the December 2002 Budget and Economic Outlook.

12While there is evidence that house price growth affects consumer expenditures through the cashing out of

home equity via second mortgages and cash-out refinancing, CBO does not predict house prices as part of its larger

macroeconomic model. Thus, making house price growth a recursive function to the VAR provides needed

correlations of house prices with economic conditions without violating the premises of the larger macroeconomic

model. There is no consensus in the broader research community as to whether housing has measurable feedback

effects into the general economy or not.

simulations match CBO baseline economic forecasts, for each quarter. This baseline

forecast is reported by CBO in its most recent Budget and Economic Outlook.

11 The

normalization is the additive difference between the simulation mean in any given

quarter, and the CBO baseline value. The normalization constant for each variable, in

each quarter, is added to predicted values in each simulation run. 

HOUSE–PRICE GROWTH-RATE REGRESSION MODEL

A national house-price growth equation is estimated using functions of economic

variables found in the VAR system. This creates one-way causation where the macro

economy affects housing prices but house-price growth does not affect the macro

economy. In forecast simulations, this approach allows house prices to be affected by

shocks in the macro economy, without temporal shocks in the housing market having

feedback effects on the economy.12

The estimated house-price-growth (least squares) regression equation is reported in Table

2. It models annual-equivalent rates of house price growth each quarter as a function of:

• The change in expected inflation, from previous quarter;
The FHA-BSSS, II. Economic Forecasting Models page 13

13The effect of this tactic is minor compared to the size of random shocks that are added to the predicted

growth-rate series. Yet allowing the regression coefficients to be random variables when forecasting the future

results in a more continuous distribution of outcomes and more accurate confidence intervals, when compared to

historical variations in house price growth rates. 

• The change in real mortgage interest rates, from previous quarter;

• One quarter lag in the change in real mortgage rates; and

• The change in the unemployment rate, from four quarters earlier.

Expected inflation is a weighted average of inflation in the previous 12 quarters. The

weights are given with Table 2 and come from the Phillips Curve model used in CBO’s

macroeconomic forecasts.

FORECASTING HOUSE PRICE GROWTH

Forecast values of the VAR economic variables, with the random shocks, provide inputs

into the house price growth equation. To increase the variability of house-price growth

forecasts, regression coefficient values are assumed normal (Gaussian) random variables

with means equal to the regression estimates, and standard deviations from those same

estimates (see Table 2).13 Random shocks are added to house-price growth predictions

using the same re-sampling techniques employed for the VAR equations.

A supplement to the house-price growth equation is added to account for the possibility

of a cyclical peak in national house price growth at the start of the forecast period (2002,

fourth quarter). At that time, real national house price growth had been above average for
The FHA-BSSS, II. Economic Forecasting Models page 14

14The house price series is the national index produced by the Office of Federal Housing Enterprise

Oversight (OFHEO). It is a repeat-transactions price index based on loan purchase information provided to OFHEO

from Fannie Mae and Freddie Mac.

15Previous peaks occurred in 1979 and in 1990.

over three years and was just beginning to show signs of slowing down. To look for the

evidence of a cyclical peak, we examined the time trend of the ratio of a national house

price index (HPI) to the Bureau of Labor Statistics residential rent price index (rent of

primary residence series).14 This ratio shows cycles of roughly ten years duration, with

current market conditions at a peak with house prices 12.8 percent above what would be

supported by their average relationship to rental prices (over 1975-2002). This 12.8

percent represents cumulative growth of house prices in excess of what would maintain

long-run parity between the purchase and rental markets.15

This excess growth may represent a housing market disequilibrium, rather than a

permanent shift in the demand for owner-occupied housing. To deal with this uncertainty,

a probabilistic approach is employed for when and by how much to dissipate the 12.8

percent excess growth . The dissipation in any simulated economic scenario is based a

first-order autocorrelation coefficient for the HPI-to-rental-price ratio series. Using

regression analysis, the quarterly autocorrelation coefficient of the series is estimated as

0.994, with a standard deviation of 0.018. Values used for the autocorrelation coefficient

in the economic simulations are randomly drawn from a normal distribution with mean

0.994 and standard deviation 0.018, but truncated at 1.00. When the randomly selected

coefficient is 1.00 (37 percent of the time) there is no dissipation of the 12.8 percent
The FHA-BSSS, II. Economic Forecasting Models page 15

16The dissipation process is not a correction for any implied housing price “bubble.” Indeed, even with low

values of r, dissipation of the 12.8 percent “excess” growth occurs very slowly. With a ri value two standard

deviations below its mean, one-half of the excess growth is dissipated in four years. At one standard deviation below

the mean, it takes seven years to remove one half of the excess growth, and at the mean value itself it takes 29 years

to reach the 50 percent dissipation mark. While this correction does not address the question of housing price

bubbles, the potential for local price bubbles is addressed in the next section.

“excess growth,” so that such growth is assumed to be due to a permanent shift in

demand for owner-occupied housing. Thus, the adjustment factor imbeds a probabilistic

estimate of whether or not there truly has been abnormally high real house price growth

and a disequilibrium in housing markets.

If house price growth predicted by the regression equation is DHPI(t,i), for forecast

quarter t= { 1,..., 120} in simulation i = {1,...,1000}, the final, adjusted growth rate with

the excess-growth adjustment factor, , is:

where e(i,t) is the random shock adjustment, r(i) ~ N(.994, (.018)2) is the random

correlation coefficient for simulation i (censored at 1.0), 12.8 (percent) is the excess

growth to be removed over time, and multiplying the right-hand-side by 4 converts the

quarterly dissipation effect to an annual rate. For all values of ri < 1.0, 

declines with t. Smaller values of ri result in faster dissipation, while larger values result

in slower dissipation.16
The FHA-BSSS, II. Economic Forecasting Models page 16

Because house-price growth-rate predictions are annual rates, they must be divided by

four to obtain quarterly rates needed for mortgage performance forecasts. Dividing by

four is appropriate here because, growth rates are transformed into house price indices by

taking the exponential of cumulative growth rates.

ACCOUNTING FOR THE POSSIBILITY OF DEPRESSIONS

Events like the Great Depression do not appear in the data available for regression

analysis of the economic environment. To add risk of depression-like events, a stochastic

depression generator is added to the VAR and house-price-growth equations. The

probability of a depression-like event occurring in any given time series simulation is

arbitrarily set to 2.5 percent. That is, one-in-forty simulations are likely to experience a

depression event, meaning 25 events in 1000 simulations. When a depression is triggered,

changes in unemployment, inflation, and house price growth are patterned after the 1930-

1942 experience in the U.S. Unemployment rates increase by 20 percentage points over

VAR simulation values in the first four years, and then decline back to their VAR

simulation values over the ensuing 8 years. Inflation drops by 7.7 percentage points per

year below VAR simulated values for each of 3 years, and annual HPI growth rates drop

by 11.1 percentage points below simulated values for each of 3 years. After year three,

both inflation and HPI growth return to their simulated values.

Because depression events create deviations from projections of economic conditions,
The FHA-BSSS, II. Economic Forecasting Models page 17

they do not necessarily result in economic environments with the conditions of the Great

Depression of the 1930s. Also, their impact on forecasts of mortgage performance

outcomes and FHA cash flows depends on when these events actually occur during the

30-year forecast horizon, vis-a-vis mortgage origination. Depression events generated in

the last ten years will have little effect on predictions of mortgage performance and

subsidy rates.

The occurrence of depression-like conditions is not wholly dependent on these add-on

events. In the VAR-and-house-price-growth system of equations, it is possible for a

congruence of interest rates, unemployment, and inflation to create recessionary and

depressionary housing market conditions apart from this depression-event add-on.

However, given the range of economic conditions existing over the historical period

available for regression analysis, the rate of naturally occurring housing recessions and

depressions coming out of the simulations is too small, without the add-on event

generator.

RANGE OF SIMULATED ECONOMIC VALUES

The range of final output values from the VAR and house-price growth equations, for use

in predicting mortgage performance and subsidy rates, is illustrated in Figures 2 and 3.

These Figures show historical, to-date movements in mortgage interest rates (Figure 2)

and national house price growth rates (Figure 3), along with one and 99 percent
The FHA-BSSS, II. Economic Forecasting Models page 18

17Statement of Alan Greenspan, Chairman, Federal Reserve Board, before the Joint Economic Committee

of Congress, April 17, 2002. ( http://www.federalreserve.gov/boarddocs/testimony/2002/20020417/default.htm)

18These data are available at www.ofheo.gov. The data were studied by Malgorzata Klosek, while a visiting

scholar at CBO. She also developed the Ornstein-Uhlenbeck process usage described in this section.

19Local cyclical patterns around long-term national trends, was also found by Joseph Gyouko and Richard

Voith, “Local Market and National Components in House Price Appreciation,” Journal of Urban Economics vol. 32

no. 1 (July), 1992, pp. 52-69.

confidence bounds for the forecast simulation series for each variable.

ADDING LOCAL HOUSING CYCLES

As Federal Reserve Board Chairman, Alan Greenspan, recently noted, housing markets

are, by nature, local: “A home in Portland, Oregon is not a close substitute for a home in

Portland, Maine, and the ‘national’ housing market is better understood as a collection of

small, local housing markets.”17 Study of Census Division and Metropolitan level (MSA)

house price growth rate series published by the Office of Federal Housing Enterprise

Oversight (OFHEO) reveals clear cyclical patterns around national trends.18 The

amplitude of those cycles was larger in the 1980s, yet cyclical patterns continued through

the 1990s.19 

The statistical characteristics of these patterns are replicated in two parts. First is a

process that generates serially correlated (random) growth paths, and second is a cyclegenerating process. Serial correlations are created using the Ornstein-Uhlenbeck (OU)
The FHA-BSSS, II. Economic Forecasting Models page 19

20G.E. Ornstein and L.S. Uhlenbeck, “On the Theory of the Brownian Motion,” in Selected Papers on Noise

and Stochastic Processes, N. Wax ed. New York: Dover Press, 1954, pp. 93-112. Reprinted from Physical Review

vol. 36 (September 1, 1930), pp. 823-841.

process, which generates exponentially correlated growth rates.20 The OU process is the

continuous version of a distributed lag function, and is modeled here in its discrete form.

The discrete formulation mimics daily price movements, from which end-of-quarter

values are drawn. The OU process allows for the type of variation seen in the historical

data, namely, that some regions and localities may cycle near the national growth rate

over time, while others will diverge. Adding a cycle-generating process to this forces each

house price series to have more pronounced cyclical movements, creating a selfcorrecting process for housing booms and busts. The phase of each cycle at the start of

the forecasting period is determined by recent deviations in each price series’ movements

from national house price movements.

Deviations of cumulative local house-price growth from the national level are computed

using the zero-mean OU process, h(t), which satisfies the differential equation: 

. 

Here, " is a time-series correlation parameter (affects half-life of correlations), T’(t) is

standard Brownian motion, and t represents quarterly time realizations of the Brownianmotion process. A discrete form of the OU process is used to simulate the cumulation of

daily fluctuations in house prices:
The FHA-BSSS, II. Economic Forecasting Models page 20

where k represents each (daily) realization of the process, there are 92 days in each

quarter, N(0,1) is a standard normal random variate, and h(t,0) = h(t-1, 92). The OU

process is seeded by 0(0,0) ~N(0,1/"), which assures stationarity in the resulting h(t,k)

time series. Adding the volatility or amplitude parameter, e, leads to quarterly realized

values of cumulative house price deviations, g(t):

g(t ) = h(t, 92)*e 

The parameters for these processes (a and e) are chosen separately for Census Divisions

and for MSAs to account for higher volatility in the historical MSA house price series

(see Table 3). The OU process uses random number generation, so that each specific

locality–MSA and Census Division–will undergo the full range of possible deviations

from national house price growth, across the 1000 simulated economic environments.

The cyclical effect added to the OU price-growth factor each quarter is:

where s is the amplitude of the cycle, W determines length of cycle (in quarters, t), and r

= (0, 2p ) indicates the phase of the cycle at time zero. Cycle phase, r, is a uniform

random variable whose range is zero to 2p, the length of the cosine wave. The starting

phase for each region in the forecast period is set using a uniform random variable within

a segment of the full (0, 2p) interval. The particular segment is chosen according to the

relationship between regional and national house price growth over the most recent fiveyear period. The difference between average annual growth at the regional and national
The FHA-BSSS, II. Economic Forecasting Models page 21

21Lognormal price indices follow from normally distributed cumulative house price growth rates, within any

given geographic area, over any specific time period.

levels determines whether the individual region will begin the forecast period in an

expansion or contraction, and in what part of that cycle phase (See Table 3).

Final Census Division and MSA house price indices used to predict mortgage

performance are calculated as the exponential of the sum of national (cumulative) houseprice growth rates and local deviations. Use of the exponential function follows from

assuming that house prices in any given locality are lognormally distributed, at any given

point in time.21 This assumption underlies the OFHEO house price indices used here for

statistical analysis of house price growth rates.

THE TREASURY YIELD CURVE AND DISCOUNT FACTORS

Following budget preparation rules established by OMB, a “basket-of-zeros” approach is

used to discount FHA cash flows and calculate net present value subsidy rate estimates.

This approach uses implied spot interest rates across the full length of the constant

maturity Treasury yield curve. Cash flows in year one are discounted with the one-year

spot rate, those in year two with the two-year spot rate, and so on. To be consistent with

budget practice in the Executive Branch, the FHA-BSSS obtains discount rates for new

budget cohorts from the forecast Treasury yield curve in the first quarter of each calendar
The FHA-BSSS, II. Economic Forecasting Models page 22

22Often OMB picks a yield curve from a week in December, rather than January, to give the Executive

Branch agencies more time to prepare their final budget numbers. The FHA-BSSS, however, only forecasts data on a

quarterly basis and, therefore, cannot pick individual weeks. Instead, first-quarter (January-March) predictions are

used to match the timing of the President’s Budget. Using the previous fourth quarter (October-December) would

have little impact on simulation results.

23The relationship of the 30-year (constant maturity) Treasury yield to the 10-year yield is based on analysis

of 1985-2000 data.

year. This matches the January timeframe for budget preparation.22 For existing cohorts,

OMB determines the yield curves to be used by federal agencies for discounting cash

flows for credit programs. These yield curves are averages across each fiscal year. Yields

used for 1992-2002 are replicated in Table 4.

The VAR system of equations predicts only the 3-month and 10-year Treasury yields. All

other rates are computed using interpolation techniques. First, the basic shape of the yield

curve is determined through relationships of other major points to the 3-month and 10-

year yields. These relationships are average, proportional shape factors derived with

regression analysis on monthly data, 1980-2000.23 Equations used to compute them are

reported in Table 5.

Further interpolations required to complete the entire yield curve are made assuming

logarithmic shapes for movements of implied interest rates on sub-intervals of the yield

curve, as described in Box 2. These sub-interval rates are called forward rates, and they

are computed over successive 6-month intervals between the yield curve points calculated

from the formulas in Table 5. Forward rates are first converted into spot-rate equivalents,

which are term-equivalent yields that match a series of forward rates over the entire term
The FHA-BSSS, II. Economic Forecasting Models page 23

of interest. The spot rates are then used for discounting cash flows from each year,

starting with the year the loan guarantees are made. All cash flows are treated as if they

occurred at the midpoint of each fiscal year, so the required spot rates and discount

multipliers are for securities with maturities of : 6, 18, 30,...,372 months.
FHA-BSSS, III. Mortgage Default and Prepayment page 24

24Prices of these options are effectively imbedded in mortgage interest rates. Those rates are the sum of the

risk-free rate, a credit-risk (default) premium, and an early-termination premium. For a recent exposition of the

options approach to mortgage performance analysis see, Deng, Quigley, and Van Order, “Mortgage Terminations,

Heterogeneity, and the Exercise of Mortgage Options,” Econometrica, vol. 14, no. 3 (2000), pp. 275-307. 

25In options markets, the agreed upon purchase price is the strike price. The ability of holders to exercise

their options at any time, up to maturity, is a feature associated with what are generally called American options.

III. MORTGAGE DEFAULT AND PREPAYMENT REGRESSION

MODELS 

Simulated economic conditions are used to predict future performance of guaranteed

loans, throughout the 30-year life of each budget year cohort. The first outcomes to be

measured are mortgage defaults and prepayments each quarter. Defaults lead to another

sequence of events (discussed in Section IV) to determine net costs. Prepayments can

both curtail premium income and trigger premium rebates to insured borrowers. 

Established economic theory on mortgage terminations uses a framework of financial

options, whereby borrowers are assumed to hold two valuable options imbedded in their

home mortgages.24 The first is a call option, which permits borrowers to buy mortgages

from lenders for par, at any time, and without penalty.25 This option identifies the

prepayment feature of nearly all home mortgages issued in the United States. 
FHA-BSSS, III. Mortgage Default and Prepayment page 25

The call option is in-the-money whenever current mortgage rates are lower than the

coupon rate on the outstanding mortgage by an amount exceeding transaction costs of

obtaining new financing. Being in-the-money means the mortgage liability–the present

value of future payments due on the mortgage–is greater than the par value (outstanding

loan balance). When market interest rates fall below the mortgage coupon rate, the

present value of future payments on the existing mortgage increase to an amount greater

than the outstanding balance. When this situation occurs, it is in the borrower’s best

interest to obtain new financing at the lower interest rate, using those funds to purchase

the original mortgage back from the lender. Exercise of the call option in this fashion is a

rational response to changes in market prices, as the homeowner attempts to maximize

lifetime consumption flows from expected lifetime income. Optimal exercise timing is a

function of expectations of future movements in interest rates–whether they might drift

downward or upward–and how long the homeowner expects to stay in the property. There

is no limit to how often homeowners can refinance their properties with new mortgages,

but there are transaction costs that influence optimal call option exercise. 

The second option imbedded in home mortgages is a default option. In most cases the

mortgaged property is the sole collateral for the loan, thus it can be in the borrower’s best

financial interest to put, or sell, the mortgage back to the lender when the value of that

collateral is less than the value of the mortgage liability. This transaction effectively takes

place in property foreclosure, where the borrower exchanges the property for release from

the debt obligation. This is not a costless transaction for the borrower because of
FHA-BSSS, III. Mortgage Default and Prepayment page 26

26Many states permit lenders to pursue deficiency judgments against borrowers to recover their loss in

foreclosure. However, FHA has a policy of not pursuing this option except on repeated foreclosures.

27In the pure options theory of mortgage borrower behavior, having in-the-money options does not mean

immediate exercise. Wealth-maximizing individuals will wait until the expected value of an option is maximized in

the present. This means looking for cyclical lows in house prices and interest rates. Not that borrowers can predict

these things with great accuracy but, rather, one expects to see greater rates of option exercise when their values are

greater. Empirical analysis focuses on probabilities of events, and increasing values of option exercise increase the

probability that any given borrower will chose to an immediate exercise.

impairments to future credit and costs associated with moving the household. Yet when

the value of the option is large enough to outweigh these costs, homeowners maximize

their lifetime consumption flows by exercising this put option.26

The value of the put option increases when the call option is in the money. A valuable

call option means the effective cost of the mortgage liability is greater than the

outstanding loan balance, which only enhances the value of the put option. Thus, as the

value of the excess mortgage cost increases (market mortgage rates decline), both the call

and the put options become more valuable. Which one, if either, is exercised depends on

the difference between property value and outstanding loan balance, and on expectations

of future movements in interest rates and house values.

27

Predicting mortgage terminations is not as easy as just measuring interest rate and house

price movements. The actual cost of option exercise varies by loan, property, and

household characteristics. In addition, while market interest rates are readily available

current values of individual properties are not. Because of these additional considerations,

empirical studies of mortgage termination rates use regression techniques that measure
FHA-BSSS, III. Mortgage Default and Prepayment page 27

28The regression analysis detailed in this study follows most closely from the following two works:

Ambrose, Capone, and Deng, “Optimal Put Exercise: An Empirical Examination of Conditions for Mortgage

Foreclosure,” The Journal of Real Estate Finance and Economics, vol. 23, no. 2 (2001), pp. 213-234; and U.S.

Department of Housing and Urban Development, Office of Federal Housing Enterprise Oversight, ”Risk Based

Capital,” Federal Register, vol. 64, no. 70 (April 13, 1999), pp. 18172-18192.

29These data were provided by FHA from its Single Family Data Warehouse. The request was for loans

originating on or after January 1, 1975. The data were re-classified to create a new data warehouse with an

organization that facilitates the statistical analysis and forecast simulations described here.

30Prior to this aggregation, there are a large number of screening and cleaning routines employed to

eliminate bad data and to replace bad or missing data elements using other available information, when possible.

Some classifications use characteristics that show up through values of other data fields. For example, FHA

contractors traditionally have used groupings of non-standard LTV ratios to label both streamline refinance loans and

investor loans. Those same criteria are employed here. 

average response rates of borrowers to changes in option values. Researchers also add to

their analyses various loan, borrower, and property characteristics that help identify

factors making option values different across borrowers.28

GROUPING LOAN RECORDS FOR USE IN REGRESSION ANALYSIS AND

FORECASTING

Mortgage default and prepayment forecasting equations used in the FHA-BSSS were

estimated with records on over 16 million FHA loan guarantee originations,1975-1999.29

To expedite data processing and statistical estimation, loans were first aggregated into

groups or “cells,” based on common values of nine essential characteristics.30 These

characteristics, and the value ranges (classes) used to determine which loans are

aggregated together are:

• Mortgage product type (fixed-rate 30-year, fixed-rate 15 year, adjustable interest

rate, graduated payment, graduated equity)
FHA-BSSS, III. Mortgage Default and Prepayment page 28

31Starting with roughly 16 million loans, the aggregations yield less than 1 million loan groups. For the

forecast simulations, property age is dropped as a classification variable, and fiscal year of loan endorsement

replaces calendar quarter of origination, yielding just over 300,000 records. Because many loan groups have no loans

surviving at the start of the forecast period, only 102,000 loan groups on historical cohorts actually enter the forecast

simulations. Approximately 3,000 additional loan groups are added for each future cohort year added to the

simulations. Tables A.9-10 show that the number of observations in the regression analyses can still be in the

millions. This is because each loan group provides up to 80 quarterly observations to the regression analysis. 

• Mortgage purpose (purchase, refinance)

• Loan-to-Value ratio ( 80% or less, 81-90%, 91-95%, 96-100%, over 100%)

• Owner type (occupant, non-occupant (investor))

• Interest rate (under 4% to over 18%, grouped in 50 basis point increments)

• Property location (9 Census Divisions and the 25 metropolitan areas where FHA

does its greatest volume of business)

• Property price (up to 50% of area median price, 51-100%, over 100%)

• Property age (new, 1-15 years, 16-30 years, over 30 years)

• Loan origination quarter (starting with 1975Q1=1)

Grouping loans according to common values of these nine characteristics maintains

essential information for measuring differences in loan performance while, at the same

time, the number of records to be processed is greatly reduced.31 The characteristics are

measured as discrete classification variables, producing a finite number of possible

combinations of values. Each unique combination of values of the nine variables defines

a record. Each record includes the total number and dollar volume of loans matching a

given classification, and the numbers of defaults and prepayments occurring each quarter.

Average values of some variables, across all loans in the group, are also maintained in
FHA-BSSS, III. Mortgage Default and Prepayment page 29

32The MSAs (with their identification numbers) used for location grouping in this study are, in order from

largest to smallest FHA volume: Chicago (1600), Riverside, CA (6780), Los Angeles (4480), Washington, DC

(8840), Atlanta (520), Phoenix-Mesa (6200), Denver (2080), Minneapolis (5120), Baltimore (720), Dallas (1920),

Philadelphia (6160), Detroit (2160), Salt Lake City (7160), Sacramento (6920), Las Vegas (4120), St. Louis (7140),

Seattle-Bellvue (7600), New York (5600), Houston (3360), Fort Worth (2800), Indianapolis (3480), Nashville

(5360), Norfolk-New Port News (5720), Orlando (5960), and Kansas City (3760).

each record (e.g, loan-to-value ratio and mortgage coupon rate).

The property location variable identifies whether a loan is in one of 25 Metropolitan

Statistical Areas (MSA) or not.32 The 25 are chosen because they had the highest volumes

of FHA insurance business in the 1990s. Those not in one of these MSAs are assigned to

their respective Census Divisions. This geographic mapping is used for assigning house

price growth rates.

DEFINING MORTGAGE TERMINATION EVENTS

The particular events studied in this analysis are prepayment terminations and default

events. Borrower default is defined here as a non-cured 90-day delinquency rather than a

loan termination. Defining default as non-cured 90-day delinquencies permits a secondary

analysis of whether borrowers lose their homes in foreclosure or not. This second step is

important now that FHA has a fully functioning loss mitigation program designed to

create workout options. Such options enable many defaulted borrowers to avoid

foreclosure and keep their homes and mortgages. Regression modeling of rates of

workout offers versus property foreclosure is discussed in Section IV. 
FHA-BSSS, III. Mortgage Default and Prepayment page 30

33Refinancings are actually of two types, but data are not available to distinguish them. These types are the

pure interest-rate refinancings done to lower monthly payments, and cash-out refinancings, which are often used to

consolidate debt. Cash-out refinancings are somewhat insensitive to mortgage interest rate changes because they are

driven by needs for large amounts of cash, either to payoff higher-cost consumer debt, or else to pay for large nonhousing expenditures.

Borrowers that self-cure 90-day delinquencies by making back payments are treated as

having chosen to continue servicing the mortgage rather than defaulting. If they instead

sell the property to cure a default, their choice is tagged as a payoff. If neither of these

occurs, then the loan is tagged as a (non-cured) default for this analysis. 

Loan prepayments arise from both property sales and refinancing. While it would be

beneficial to treat these as distinct events/choices, the required data are only available

beginning in 1991. Starting the analysis in 1991 would miss the richness of the 1980s

economic experience, which included wide swings in interest rates and deep regional

housing recessions. Therefore, both types of loan payoffs are combined together and

statistical analysis of loan performance begins in 1975.33

Loan prepayments due to property sale do not represent call option exercise. They still,

however, represent utility-maximizing choices of households, given expected lifetime

income, the price of housing relative to other goods, and the relative cost of owning

versus renting a home. The incentives for such mobility-induced loan terminations–both

defaults and payoffs–are not as easily measured with empirical data, as are the incentives

for option exercise, because they rely more upon unknown individual borrower

circumstances. Still, it is important to develop regression and forecasting models that use
FHA-BSSS, III. Mortgage Default and Prepayment page 31

34In the simulations, all open 90-day delinquencies without a recorded loan workout agreement at the end of

FY2002 are considered non-cured defaults that will result in insurance claims during the first year of the forecast

period (FY2003).

available data to capture systematic patterns in homeowner mobility.

THE LOGISTIC REGRESSION APPROACH TO ESTIMATING DEFAULT AND

PREPAYMENT PROBABILITY EQUATIONS

Forecast equations for mortgage performance in the FHA-BSSS predict probabilities of

borrower choices each quarter, given economic conditions and loan characteristics. The

statistical analysis used to generate these equations was performed on FHA-insured loans

originating between 1975 and 1999, with termination activity followed (quarterly)

through 2000 for defaults and 2001 for prepayments. The difference in final observation

dates for defaults and prepayments is necessary to give time to determine which 90-day

delinquencies were not cured by borrowers.34 In addition, loan termination activity is only

analyzed through the twentieth year of each insurance cohort. After that, there is little

available data on loan performance, and such latter-year events will have little effect on

subsidy rate estimates. Likewise, very few loans terminate in the first two quarters of loan

life, and so these observations also are excluded from the regression analysis. Section V

describes how default and prepayment rates are determined in the forecast simulations for

very young and very old loans.

Mortgage performance between the third-quarter and twentieth-year of loan life is
FHA-BSSS, III. Mortgage Default and Prepayment page 32

35See C.B. Begg and R. Gray, “Calculation of Polychotomous Logistic Regression Parameters Using

Individualized Regressions,” Biometrika vol. 71 (1984), pp. 11-18.

modeled with pairs of binomial-choice equations. In each equation, the choice is either to

make monthly mortgage payments due that quarter, or else enter a termination path. The

termination paths are payoff (prepayment) or default. The choice to enter either of these

paths is modeled using logistic regressions. Logistic or, simply, logit models are

consistent with consumer utility/wealth maximization, can be used with longitudinal data

(time series on individual observations), easily handle variables whose values change

over time, and have a peculiar property making them especially suited for this analysis.

That property, independence of irrelevant alternatives, means default and prepayment

decisions can be modeled separately and then recombined to form an equation used for

calculating probabilities of default and prepayment in each period (calendar quarter).35

Separate default and prepayment estimations are useful for testing different variable

specifications for default and prepayment equations. When the probabilities estimated

from these equations are applied to grouped data, they generate forecasted rates (and

numbers) of terminations each quarter.

 

To measure major differences in borrower responses to economic environments, separate

pairs of regression equations are estimated for: purchase-money fixed-rate mortgages,

refinance fixed-rate mortgages, purchase-money adjustable-rate mortgages, and refinance
FHA-BSSS, III. Mortgage Default and Prepayment page 33

36The different risks and risk-preferences of borrowers who choose adjustable-rate mortgages has been

discussed and analyzed in: Jan Breukner and James Follain, “The Rise and Fall of the ARM: An Econometric

Analysis of Mortgage Choice,” Review of Economics and Statistics, vol. 70 (February, 1988), pp. 93-102; Charles

Capone and Donald Cunningham, “Estimating the Marginal Contribution of Adjustable-Rate Mortgage Selection to

Termination Probabilities in a Nested Model,” Journal of Real Estate Finance and Economics, vol. 5 (1992), pp.

333-357. 

37Mortgages with shorter-terms (15 years) and those on investment properties exhibit unique termination

patterns, but there are too few of them to warrant separate statistical analyses. Mortgage performance forecasts on

15-year mortgages use the estimated equations for 30-year mortgages. The 15-year loans will have smaller predicted

default rates and larger predicted prepayment rates than comparable 30-year loans because they accumulate equity

more quickly. Investor-property loans have higher default and prepayment rates, which are measured in regression

models through dummy-variable scaling factors. Thus, the investor loans are included in the regression estimations

along with non-investor loans. 

38Logit estimation is designed to model “conditional” rates of event choices. That means the probabilities in

question are in reference to the number of loans in a group that actually survive and are current at the start of each

observation quarter. So, Pd is computed as the number of loans defaulting in a given quarter, divided by the number

of loans active at the start of the quarter less the number of loans prepaying during the quarter. Prepayments are

removed because the logit model estimates probabilities of events (default or prepay) versus the non-event

(continuing mortgage).

adjustable-rate mortgages.36, 37

These equations resemble:

 for default, and 

 for prepayment.

Where Pd is the probability of default, Pp the probability of prepayment, and ed and ep are

i.i.d. error terms with logistic distributions. In the FHA regression analysis, the two

probabilities are computed as relative frequencies within each loan group, in each

quarter.38 The left-hand-side construction in each equation is the log of the odds ratio for
FHA-BSSS, III. Mortgage Default and Prepayment page 34

39Such techniques choose the values of that maximize the joint probability that the observed choices are

the “correct” choices, given the values of the explanatory variables, X.

40These probability formulas result from rearranging the exponentials of the logit equations shown above.

each termination choice, which is assumed a linear function of explanatory variables

denoted by the X vectors. The estimated coefficient vectors, , provide weights

associated with each variable’s impact on incentives to enter a termination path. These

weights are estimated on the historical data using maximum likelihood techniques.39

When estimating probabilities for simulation forecasting, represent the weighted

sum of default incentives, and the weighted sum of prepayment incentives. X

values for each loan group include both fixed loan and property characteristics, and timevarying loan and property characteristics that are functions of economic conditions in the

forecast period. The weighted sums of incentives to default and prepay are re-combined

to calculate the two probabilities of interest:

40

and .

Estimated probabilities for each loan group are multiplied by numbers of surviving loans

in each quarter to arrive at numbers of predicted defaults and prepayments. These
FHA-BSSS, III. Mortgage Default and Prepayment page 35

41In the forecast simulations, the number of predicted defaulted-loan workouts in each quarter is first netted

from the number of defaults so that the new number of surviving loans includes the worked-out defaults. Analysis of

workouts is covered in Section IV. In the historical data, loan workouts were not a measurable phenomenon until

1997. Before that time, loan default, as measured here, included property foreclosure and the repurchase of defaulted

loans by FHA. That repurchase program (Single Family Mortgage Assignment Program) accounted for up to 25

percent of non-cured defaults before it was eliminated by the Congress in favor of direct loan workout assistance.

Other minor sources of loan defaults in the historical data include pre-foreclosure sales of properties, where

borrowers receive financial assistance to sell their homes, and voluntary deed transfers. These are both minor

programs. In the forecast simulations, deed transfers are ignored and pre-foreclosure sales are treated by adjusting

the cost of paying insurance claims on defaults that go to foreclosure. The exact treatment is discussed in Section V. 

42Loan groups only enter the regression analysis if they start with at least 10 loans in the quarter of

mortgage origination. For fixed-rate mortgages, where the longitudinal databases are quite large, a one-in-three

sample of loan groups is used in the regression analyses.

terminations are then subtracted from beginning-of-quarter survivors to calculate the

number of surviving loans used to predict defaults and prepayments in the next quarter.41

The statistical regression estimation procedure weights each loan group observation

according to the number of surviving loans in the group, at the start of each quarter. Thus,

final coefficient estimates ( ) match what would be obtained by using individual

loan data.42

EXPLANATORY VARIABLES USED IN REGRESSION ANALYSIS AND

PROBABILITY FORECASTS

Explanatory variables used to estimate the logit regressions (Xd and Xp) and, subsequently,

to predict probabilities of events in the forecast simulations, include loan-group

characteristics, economic condition indicators, and more direct measures of the financial

incentives to exercise imbedded call and put options (Table 6). All variables were chosen
FHA-BSSS, III. Mortgage Default and Prepayment page 36

for their expect ability to measure differences in incentives to terminate mortgages.

Motivations to prepay (call) or default (put) are determined primarily by movements in

property values and interest rates, along with the more general economic conditions that

affect household mobility. These two choices are substitutes for one another; once one is

chosen the other option is no longer available. The result of substitutability and common

determining factors is that variables used in the regression equations are mostly the same.

There are slight differences in the variables used for default and prepayment, and two

additional variables are used only for adjustable rate mortgages. 

Non-financial variables, like property price (PRICE_CLASS) and property age

(AGE_CLASS) are important proxies for differences in financial incentives by classes of

mortgages when those differences cannot be measured directly. For example, borrowers

who purchase older homes are more likely to encounter significant maintenance and

repairs costs that could precipitate default (as put option), and those purchasing older

homes are also more likely to see these as starter homes, building equity that will permit

trading-up at a future date. Thus, property age, though not a direct financial variable

itself, measures general class-level incentives tied to unmeasurable financial details and

borrower preferences that affect default and prepayment rates.

Most of the variables used in this analysis are in categorical rather than continuous form

(see Table 6). This approach has numerous advantages. First, the regression model is less

restricted because no functional form is imposed on the explanatory variables. With
FHA-BSSS, III. Mortgage Default and Prepayment page 37

43This approach requires that the underlying continuous variables be placed into value-classes in a

systematic and meaningful fashion. In the extreme, categorizing variables leads to kernel estimation where the

category sizes are as small as possible so that a non-parametric regression results.

44For categorical variable coefficients to give reasonable results, one then must have a wide variability in X

values in the regression data. This is why the 1980s and 1990s experience are both valuable in estimating a

regression model that can be used with forecast simulations of a wide range of economic conditions. During the

1980s, interest rates reached historical peaks, and house prices experiences strong regional cycles. During the 1990s,

interest rates experienced large declines while regional house-price cycles continued to appear. While use of

categorical variables caps the influence of outlier values of the X variables, the economic simulations used in the

FHA-BSSS are designed to provide X-values in the highest and lowest value classes. Thus, we balance limits on the

response of borrowers to extreme economic events, we make sure that such events are adequately represented in the

forecast simulations.

continuous variables, one must assume something about whether the X’s enter in linear or

some other form. Without some prior expectation of what that form should be, one is left

assuming a linear form and thus assuming that the incentive-weight ‘s are constant

across all values of each X. Categorical variables, on the other hand, allow the data to

speak for themselves with regard to how the response function behaves as X values

change.43 Each category will have its own value of . 

A second advantage is that use of categorical variables eliminates the chance of predicted

probabilities reaching unreasonably large levels when X values take on extreme values.

When continuous variables enter the regression in linear form, the ‘s are most accurate

when X values are near their mean-values across the regression sample, and can produce

unrealistic results when X variables take on outlying values.44 With categorical variables,

however, the ‘s produce fixed response effects for X-values in each defined range. 
FHA-BSSS, III. Mortgage Default and Prepayment page 38

45This is also known as a full-rank or centered-effects parameterization. The alternative, and more common

parameterization is the reference form. In that specification, the net effect of the coefficients on the (omitted) highest

value classes of all categorical variables are jointly imbedded in the constant term of the regression.. The

parameterization type affects how the data are organized for the regression and the interpretation of coefficient

estimates, but not the actual predicted probabilities that come from the estimated equation.

A third advantage of categorical variables is that the relative magnitudes of termination

incentive effects ( ‘s ) can be more easily compared both within and across the X

variables. Because the scale of categorical variables is identical for all X variables, the

relative size of the ‘s can be directly interpreted as relative magnitude of effects, both

within and across the X variables.

RESULTS OF LOGISTIC REGRESSION ESTIMATIONS

Results of the mortgage performance regressions are reported in Tables 7-10. First are

Tables of analysis-of-variance tests that measure the relative influence of each

explanatory variable on the estimated probabilities in the regression sample. These are

followed by Tables of the regression estimates of the ‘s . 

The regression equations use an effects parameterization. With this approach, the

coefficients for classes within each categorical variable measure effects relative to the

average across all value classes. Thus, the coefficients across all classes of any one

variable sum to zero.45 The last value-class for each variable is then dropped from the
FHA-BSSS, III. Mortgage Default and Prepayment page 39

regression estimation because their coefficients are merely the negative sum of all other

(estimated) coefficients. They are, however, reintroduced into Tables 9-10 and are

highlighted in italics. 

Several coefficients could not be estimated, or not reasonably estimated with the available

data. These omissions mostly affect adjustable-rate refinance loan equations for which

data are only available starting in 1992. Reasonable values are imputed from estimated

coefficients within and across equations. Imputed coefficients are highlighted in the

Tables in boldface. The procedures used for imputation are described in Section V.

Analysis-of-Variance Tests

Analysis-of-variance (ANOVA) tests reported in Tables 7 and 8 measure the influence of

adding each variable to the regression equation. Test statistics highlight the relative

influence of each explanatory variable in measuring differences in actual historical

outcomes. Because these rankings can be influenced by which variables had the most

variability in the historical period, one cannot say that the same rank ordering of influence

will hold in a forecast environment. However, the historical period in this analysis did

have good variability in all variables, and simulations of economic environments used for

forecasting do replicate that variability. Thus, this analysis point to which factors should

be most influential in predicting default and prepayment rates in the forecast period.
FHA-BSSS, III. Mortgage Default and Prepayment page 40

46The adjustable-rate, refinance-loan default equation deviates from the others at this point. The call-option

variable is fourth most influential, behind both property age and property price. This is not too surprising, given that

borrowers who refinance into adjustable-rate products have shown a lack of interest in locking into fixed-rate

products when interest rates were relatively low. In addition, the call-option value on adjustable-rate mortgages,

when it is in-the-money, is less than it is for fixed-rate mortgages because of the interest-rate adjustment features of

these loans. Also, this equation only has data beginning with 1992 loan originations, because there was no

measurable FHA refinance activity with adjustable-rate mortgages until that year.

47Separating these components in empirical/statistical analysis is reasonable because we do not have direct

measures of either one. Indeed, direct estimates of the values of these components can only be made using repeated

simulations of future (random) paths of interest rates and house prices from positions at the time of observations, and

using backward-solving algorithms from the time of mortgage maturity. The variables used here are common

representations of option value used in empirical research of mortgage terminations.

Rankings of Variables by Influence in Default Regressions. ANOVA test results

emphasize the importance of option-related financial variables. In the default regressions,

the number one influence by far is the percentage of mortgages in a loan group likely to

have loan balances below their property values (NEQ_EQ_CLASS). This is the most

direct measure of put-option value, and it is followed in importance by the call-option

value variable (SPREAD_CLASS).46 These two variables complete the put option value,

which in a direct measure would be the sum of (negative) property equity and excess

mortgage value (present value of the future payment stream less the mortgage balance).47

From there, variable influence across equations starts to diverge, yet the variables which

stand out most are property age (AGE_CLASS), origination year (COHORT_YEAR), and

property price (PRICE_CLASS). 

As seen in Table 9, the property age variable picks-up higher default incentives with new 

homes, below-average incentives on homes aged 1-30 years, and average incentives on

older homes (over 30-years of age). With new homes, builders often capitalize many
FHA-BSSS, III. Mortgage Default and Prepayment page 41

48Mortgagee Letter 86-15, Change to Single Family Programs, August 8, 1986.

buyer transaction costs into the price of the home. This practice makes it easier for the

buyer to purchase new homes, but it also increases the mortgage balance. Because the

homeowner cannot recoup these costs upon resale, new home values exhibit lower rates

of appreciation than do previously existing homes, for a period of time. This then leads to

higher put option values on these mortgages. The ‘s also indicate higher put option

value on older homes versus younger (but not versus new) homes because of the higher

incidence of high-cost repairs and maintenance in the early years of mortgage life. These

costs, when they exist, both lower the effective value of the property and require cash

outlays that the homeowner may not be able to provide.

The loan origination year/cohort variable is a catch-all variable that measures differences

in the quality of loans as underwriting standards tighten and loosen over time. In some

years, these effects can be quite dramatic. In the early 1980s, when interest rates were

high, lenders initiated many creative financing arrangements to qualify borrowers for

mortgages. These all entailed hidden costs which effectively increased loan-to-value

ratios of mortgages to property, increasing put option values. The mortgage industry, led

by Fannie Mae, Freddie Mac, and FHA, eliminated many of these practices in 1985 and

1986. FHA, in particular, restricted the use of prepaid interest buydowns and tightened

standards for other loan quality characteristics.48 Then, in 1995, FHA took deliberate

steps to lower underwriting standards to attract more first-time and minority homebuyers.
FHA-BSSS, III. Mortgage Default and Prepayment page 42

49Mortgagee Letter 95-7, Single Family Loan Production - Revised Underwriting Guidelines and Other

Policy Issues, January 27, 1995.

These changes included broader acceptance of alternative sources of income and

downpayment, and the use of non-standard means of verifying borrower

creditworthiness.49 The result has been a steady increase in the underlying default

incentives of successive insurance cohorts, as shown in the cohort year coefficients in

Table 9.

Property-price class is another variable which may be an indicator of a number of factors

contributing to differences in default incentives. The estimated coefficients in Table 9

show an almost symmetrical relationship between default incentives on low-valued

properties (below 50 percent of area median price) and on high-valued properties (above

100 percent of area median price), with those at the low-end being above and those at the

high-end below average. The low-value class represents roughly the bottom 25 percent of

homes in a given market. Homeowners in this price range will have lower income levels,

have fewer financial resources to draw on in emergency situations, and may be more

prone to job loss during economic downturns. In addition, the lower-and upper-ends of

the housing distribution are sub-markets with less activity and thus less stable house

prices than the broad mid-section of the distribution. FHA does not insure homes at the

upper-end, as mortgage limits set by FHA are generally somewhere between 95 and 150
FHA-BSSS, III. Mortgage Default and Prepayment page 43

50FHA establishes county-by-county mortgage limits within statutory bounds. By law, FHA mortgage

amounts must be at least 48 percent of the mortgage limit for Fannie Mae and Freddie Mac (the so-called

conforming loan limit). They also can be no more than 87 percent of that same limit. For calendar year 2002 these

limits were $144,336 and $261,609. FHA receives petitions from localities to adjust their mortgage limits above the

statutory minimum, based on evidence that the area median price is above the lower statutory bound. FHA can adjust

county-level loan limits up to the minimum of 95 percent of area median house price or the upper statutory bound.

percent of area median price.

50 However, FHA does a significant amount of its business

in the low end of local housing markets.

Rankings of Variables by Influence in Prepayment Regressions. The loan

payoff/prepayment regressions (Tables 8 and 10) use the same explanatory variables as do

the default regressions, with just two exception: first, the probability of having at least 20

percent positive equity (POS_EQ_CLASS) replaces probability of negative equity

(NEG_EQ_CLASS); and, second, the slope of the Treasury yield curve (10-year to 1-year)

is added as a new determinant. The probability of positive equity variable is more

appropriate here because it measures the ability to pay selling costs and move, to invest in

a more expensive trade-up home, refinance into the conventional market, or do a cash-out

refinance. The yield curve slope indicates changing incentives to refinance between fixedand adjustable-rate mortgages as economic conditions change.

ANOVA results show that call-option-value (SPREAD_CLASS) is the dominant influence

on fixed-rate loan payoff rates. It is roughly equal in predictive power with property

equity (POS_EQ_CLASS) for adjustable-rate (home) purchase mortgages, and second to

origination-year effects (COHORT_YEAR) for adjustable-rate refinance loans. The lesser
FHA-BSSS, III. Mortgage Default and Prepayment page 44

influence of the call-option value in adjustable-rate mortgage equations results from the

smaller incentive these borrowers have to refinance because, in any given quarter, many

borrowers are eligible for the annual allowable and automatic interest-rate adjustment of

up to one percentage point.

Property equity (POS_EQ_CLASS) is an important determinant of prepayment rates in all

equations. It signals the abilities of borrowers to sell their homes for mobility or trade-up

reasons, and it also indicates potential for cash-out refinancing. Both adjustable-rate

mortgage prepayment equations are also strongly influenced by the current market rate for

fixed-rate mortgages (CUR_MKT_RATE). When this rate is low, borrowers with

adjustable-rate mortgages have incentives to eliminate the future risk associated with such

instruments and secure fixed payments.

A more detailed discussion of each variable and its estimated regression coefficients

follows in the next section.

Discussion of Coefficient Estimates

Nonlinear models, like the logistic, do not lend themselves to direct interpretation of how

explanatory variable values influence loan termination rates. As discussed earlier, the

estimated Beta coefficients are weighting factors that indicate the relative influence of

each explanatory variable on the log-odds of a particular termination choice. Therefore

they are discussed as default and prepayment incentive effects. Actual changes in
FHA-BSSS, III. Mortgage Default and Prepayment page 45

51The change in event probability associated with changes in explanatory variables are also dependent on

the starting values of those variables. Thus, one cannot say, for example, that the increased in the probability of

default for borrowers in LTV_CLASS=3 (91-94% ) over borrowers in LTV_CLASS= 2 (81-90%) is a constant value. It

depends on the values of all other variables for the borrower (or loan group) in question.

52Anthony Pennington-Cross and Joseph Nichols, “Credit History and the FHA-Conventional Choice,” Real

Estate Economics, vol. 28, no. 2, 2000 (summer), pp. 307-336.

53Though the distinction appears small, Pennington-Cross and Nichols, Ibid., show data that also indicates

FHA borrowers in the 91-95 percent LTV class having the highest overall credit quality of all FHA borrowers. Those

researchers mix all mortgage types together. 

probabilities resulting from changes in variable values require a complex manipulation of

the coefficient estimates and variable values.51

In this section, the estimated coefficients reported in Tables 9-10 are reviewed for

direction of effects (positive or negative), relative magnitudes of effects (for categorical

variables), and the reliability of the measured effects (levels of statistical significance). As

each variable is introduced, some explanation of why it is included in the regression

equations is also provided.

Default Regression Results: Loan Group Characteristics.

Loan-to-value ratio. Downpayments made at loan origination are an indicator of

household abilities to generate liquid assets and avoid default. Researchers have also

found that borrowers who make larger downpayments tend to have stronger credit

histories.

52 Fixed-rate loans show greatest credit quality in the 91-95 percent

LTV_CLASS.

53 The best performing adjustable-rate refinance loans are those in the 81-90

percent LTV_CLASS, and the best performing adjustable-rate purchase loans are in the
FHA-BSSS, III. Mortgage Default and Prepayment page 46

54This assertion holds in this context because the regression models control for locality specific house price

movements.

55This adverse selection problem, whereby FHA insures loans that perform worse than would be expected

by observable measures, is mostly an issue with fixed-rate loans. FHA insurance pricing is more attractive for

adjustable-rate mortgages, compared with private mortgage insurance, making FHA more competitive for higher

credit-quality borrowers who want adjustable-rate products. In addition, there is less-upside default risk from FHA

adjustable-rate mortgages because their maximum annual rate changes are only one percent, compared with a

standard two percent annual change “cap” on conventional adjustable-rate mortgages.

combined 81-95 percent LTV_CLASS(es). 

The result for fixed-rate loans stands out because it does not conform with options theory:

loans with larger downpayments (lower LTV_CLASS) should have lower incentives to

default, and will find the put option in-the-money less often than borrowers with smaller

downpayments.54 However, among FHA insured loans, borrowers with 10 percent and

larger downpayments have other financial weaknesses that prevent them from obtaining

less costly conventional financing.55

There are relatively few loans with LTV ratios of 80 percent or less, so they are not

included in the statistical analysis. In the simulation forecasts, these loans receive an

implied LTV_CLASS effect value of zero (no positive or negative effect). They have small

default rates because of their larger equity positions, both in the historical data and in the

forecast simulations.

Price class, Property Age Class, Owner type, and Mortgage Age Class. All of the default

regressions show peaks in the underlying time patterns (AGE_CLASS) of default rates in
FHA-BSSS, III. Mortgage Default and Prepayment page 47

either the 3-4 or 5-6 year time period, and strong declines after year 10. The peak periods

correspond with a confluence of rising rates of household mobility with insufficient time

for property appreciation to provide home equity for paying selling and moving costs. By

year ten, loan amortization has reached roughly 10 percent, so as long as property value

has not declined, there will be enough equity to pay the transaction costs of selling and

moving and default rates should drop dramatically.

Investors (INVESTOR=YES) have measurably higher default incentives than do owneroccupants, resulting from the risks they take with occupancy and renter-induced property

depreciation. Owners of lower-priced properties (PRICE_CLASS=LOW) have higher

incentives than do owners of higher-priced homes (PRICE_CLASS=HIGH); and

borrowers purchasing new homes or refinancing them in the first year

(PROP_AGE_CLASS=NEW) have higher default incentives than do other borrowers.

These last two effects were discussed above in the context of the ANOVA Tests.

Cohort-year effects. FHA underwriting policies have undergone many changes over time.

Two of the most important in the study period occurred in 1986 and in 1995. These were

discussed above, with the ANOVA test results. Compounding the effects of the 1995

changes was a reduction in minimum downpayment requirement, under the rubric of
FHA-BSSS, III. Mortgage Default and Prepayment page 48

56Congress made this change in the FY1999 budget for the Department of Housing and Urban Development

(HUD), and FHA implemented it in Mortgagee Letter 98-29, Single Family Loan Production - Mortgage

Calculation Simplification, October 22, 1998.

57General Accounting Office, Mortgage Financing Changes in the Performance of FHA-Insured Loans,

GAO-02-773 (July 10, 2002). 

downpayment simplification.

56 The principal effect of this “simplification” was to permit

a minimum three-percent downpayment for all FHA loans. Previously, there was a tiered

structure that required three percent on the first $25,000, five percent on the next

$100,000, and 10 percent on any amounts above $125,000.

The trend of declining quality in insurance cohorts since 1995 was recently confirmed in

a report by the U.S. General Accounting Office (GAO).57 That report cites both reduced

underwriting stringency and problems in California as reasons for increased default rates

on insurance cohorts written between 1995 and 1999. It is possible, however, that this

trend may have stopped in 2001, when FHA lowered its insurance premiums to levels that

are now highly competitive with private insurance for borrowers with 5-9 percent

downpayments, and much less expensive than private insurance for borrowers with 3-4

percent downpayments. FHA premiums are also much lower than private insurance for

adjustable-rate and graduated-payment mortgages, and for so-called A-minus borrowers

whose credit is just below prime, conventional quality. Evidence that the deterioration in

loan quality in the late 1990s may have stopped comes from a recent shift in the loan-tovalue distribution of FHA loans. In the first quarter of fiscal year 2002 (the last data used

in this analysis), there was a shift in the distribution of loans by loan-to-value class, with
FHA-BSSS, III. Mortgage Default and Prepayment page 49

58Mortgagee Letter 98-1, Single Family Loan Production - Underwriting Adjustable Rate Mortgages,

Interest Buydowns, Homeownership Counseling and Other Credit Policy Issues, January 2, 1998. 

59For example, in an environment where initial rates on ARMs are six percent, a borrower that would

qualify for a $125,000 mortgage would, with the same monthly payments, qualify for a $139,600 mortgage with the

teaser-discounted rate of five percent. If interest rates were stable and the mortgage rate adjusted back to six percent

at the end of year one, then the borrower’s monthly payment would increase by $86 dollars. While all borrowers who

take out ARMs are taking interest rate risk, the ARM borrower that qualifies because of the teaser is taking an added

risk that even in an environment of stable interest rates, they will experience increased payments in year two.

over five percent moving from the above-95-percent loan-to-value class to the 91-95

percent class.

One additional, important change in underwriting occurred for adjustable-rate mortgages

(ARMs) in 1998. At that time, FHA eliminated the use of “teasers” on ARMs.58 Teasers

are interest-rate discounts in year one that allow borrowers to qualify for loans with

monthly payments lower than they are likely to be in the future. Prior to 1998, average

teasers were one percentage point. That one percent could make a big difference in

qualifying borrowers for loans, but it also increased the risk of default when borrowers

could not make the (expected) higher payments in the future.59 Eliminating teasers led to

a dramatic drop-off in ARM originations in 1998. ARMs, as a percent of all FHA

originations, dropped from over 30 percent (of dollar volumes), to just five percent in

1998, and have stayed below that level since then.

Default Regression Results: Economic Conditions.

House-price-cycle stage. House-price-cycle stage is added to the regression equations for

two reasons. First, the options theory of default suggests that borrowers looking to
FHA-BSSS, III. Mortgage Default and Prepayment page 50

60The application of this theory to empirical investigations of mortgage performance across housing-price

cycles is discussed in Brent W. Ambrose, Charles A. Capone, Jr., and Yongheng Deng, “Optimal Put Exercise: An

Empirical Examination of Conditions for Mortgage Foreclosure,” Journal of Real Estate Finance and Economics

vol. 23 no. 2 (September, 2001), pp. 213-234.

61In pure, theoretical, mortgage-pricing models, optimal default occurs when the value of exercising the

option today is greater than the present value of option exercise across all possible future dates. In empirical analysis,

however, given that housing cycles average around ten years in length, and given the larger levels of loan

amortization that exist after the tenth year of mortgage life, when a housing downturn does exist, the current cycle is

the only one of interest for modeling put option exercise. Borrowers will not necessarily anticipate cyclical troughs

with great accuracy. Yet, to the extent there are borrowers who desire to exercise put options on their mortgages,

there should be greater default activity around the trough of the cycle, as economic indicators start to point to

potential near-term stabilization in the housing market and an up-coming turning point. This effect is independent of

the dollar size of individual property negative equity positions: all borrowers with in-the-money put options, who

desire to exercise those options, will benefit most by doing so near the trough of the housing cycle. Therefore, houseprice-cycle stage is an important indicator variable distinct from property equity (NEQ_EQ_CLASS).

62Stages 1 and 2 represent the first and second halves of house-price downturns, measured by the actual

price declines. Stage 3 is that period of time in which prices are rising but remain in the range previously defined as

Stage 2.

exercise put options on their mortgages will want to do so when a cycle is approaching a

trough.60 At that point, the value of the option is at its maximum.61 Regression results

indicate that incentives to default do increase during housing recessions

(CYCLE_STAGE= { 1,2}) and, for fixed-rate-refinance loans, continue into the early

stages of expansions (CYCLE_STAGE= 3).62

A second reason for including a cycle-stage variable is that, in addition to price declines,

housing recessions cause illiquidity. Housing recessions are caused by contractions in

area employment, which lead to significant reductions in home buying activity. This

decrease in market liquidity means that many homeowners cannot sell their mortgaged

properties in a reasonable time frame without experiencing a substantial financial loss.

Difficulty in selling properties increases incentives to default, especially for borrowers
FHA-BSSS, III. Mortgage Default and Prepayment page 51

63Donald Cunningham and Charles Capone, “The Relative Termination Experience of Adjustable to FixedRate Mortgages,” Journal of Finance vol. XLV, no. 5 (December, 1991), pp. 1687-1703. That study used regional

unemployment rates and local house price indices.

64This mobility constraint introduces the issue of what researchers call “sub-optimal” option exercise. If an

unemployed worker can only find work in another city, then he/she may exercise the put option even if there is

positive equity, because the net equity position in the property is negative after selling costs. If, instead, the

homeowner were to find work locally, then the put option would not be exercised either because it is not directly inthe-money (positive equity before property selling costs) or else the timing of option exercise is not yet optimal.

who may need to move to a new area to obtain work.

National unemployment rate. Coefficient signs for the national unemployment rate

variable (US_UN_RATE) are consistently negative across all four default equations. This

counter-intuitive result–that increases in the national unemployment rate decrease default

incentives–also has been found in other mortgage research.63 Note that this effect is in

addition to the local economic conditions captured in both the probability of negative

equity (NEG_EQ_CLASS) and house-price-cycle-stage (CYCLE_STAGE) effects. Thus,

the national unemployment rate effect indicates a lack of household mobility when the

national economy is in recession and, likewise, an increase in mobility when jobs across

the nation are plentiful. When borrowers cannot move to other areas to obtain new work,

they will more likely find ways to maintain their current homes and mortgages even if the

put option is in-the-money.64

Market shift. There were many major changes in the mortgage market during the 1990s.

Because it is not possible to account for each one, a shift variable is added for all

observations starting in 1993Q1 (MKT_SHIFT). This variable shows a measurable
FHA-BSSS, III. Mortgage Default and Prepayment page 52

65The opposite effect shown for ARMs implies nothing of importance because ARM volumes did not reach

measurable levels until 1992, while the shift variable is turned on in 1993.

66Changes in underwriting rules and costs in the conventional market have marked effects on FHA

prepayments. Borrowers most often use FHA for qualifying for their first home, and then refinance with FHA only in

the early years of their mortgage, while property equity is still relatively low. There are virtually no refinances of

FHA loans into new FHA loans after the tenth year of being in a home.

reduction in default rates for fixed-rate mortgages as refinancing became an easier option

in the 1990s (MKT_SHIFT=ON).65

One of the most significant changes for borrowers was the movement toward zero-cost

refinancing that occurred in the 1993-1994 refinance period. Along with this was a

change by the private mortgage insurance industry to monthly premium payments, and the

opening of the conventional market to accept refinance originations with less than a 10

percent downpayment.66 These and other factors ushered in an era of frequent, if not

regular, refinancing of homes. This shift in market operations had the effect of lowering

the role of call-option value in determining put-option value. 

To see this result, note that FHA allows refinancing of insured mortgages without

property appraisals, so that even borrowers with negative equity positions can reduce their

mortgage liability and thus lower put-option values. However, if there is no appraisal to

prove sufficient property value, borrowers must pay cash for closing costs on the new

loan, rather than financing them in the insured mortgage. Changes in the mortgage market

that greatly reduced closing costs and eliminated the need for cash at closing allowed

more FHA borrowers to exercise refinance call options rather than to see the value of
FHA-BSSS, III. Mortgage Default and Prepayment page 53

67The primary means for eliminating cash requirements on new loans, without adding the transaction costs

to the loan balance, is by adding a premium on the mortgage interest rate of 0.25 to 0.50 percent.

unattainable call options enhancing the value of put options.67

Market rates on fixed-rate mortgages: ARM effects. Fluctuations in the long-term

mortgage rate (CUR_MKT_RATE) reflect market expectations of future movements in

short-term rates. Thus, higher long-term rates signal to ARM borrowers that their

payments are likely to only continue to increase in the future. For both purchase and

refinance equations, upward movements in CUR_MKT_RATE create increasing

incentives to default as it increases the present value of the mortgage liability. The

measured effect in the regression equations is four times as large for purchase mortgages

as it is for refinance mortgages. This variable is the predominant ARM-specific effect in

the purchase equation, indicating larger anticipated rises in mortgage payments in the

future causes borrowers to look for less costly housing alternatives in the present.

Default Regression Results: Financial Incentives

Probability of Negative Equity. The probability of negative equity (NEG_EQ_CLASS)

variable is the most direct measure of financial incentive to default, and it has the largest

influence on default incentives (See Table 7). For loan groups used in this analysis, the

probability represents the expected percentage of loans in a group with in-the-money put

option values (absent the call option effect) at each point in time. It is calculated with

three pieces of information: a to-date price-index level, representing house-price
FHA-BSSS, III. Mortgage Default and Prepayment page 54

68The standardized normal variable defined in the text is: , where HPI is the house price

index, LTV the loan to value ratio, and s the square root of the variance of cumulative house price growth rates

around the mean index level. All values are as of the time (t) of observation. This statistic is used as the lower bound

on an integration of the standard normal density function to compute the probability that an individual property

underlying a given mortgage could have depreciated in value enough to eliminate all equity.

appreciation (or depreciation) since loan origination; amortized loan-to-value (LTV) ratio,

representing the outstanding loan balance as a percent of the original house price; and a

measure of the variance of individual property appreciation rates around the index level.

The three are combined to yield a standard-normal statistic that is used to measure the

probability that the value of any one property could have fallen enough, compared to the

local market, to eliminate all market-generated and owner-invested equity.68 This is the

probability that there could be to-date depreciation of value on the property in question,

and that it could be large enough to create a negative equity position for the home owner. 

The house-price series and associated volatility measures are produced by OFHEO on a

quarterly basis. While not based on FHA loans, they are based on Fannie Mae and

Freddie Mac loan purchases, and so represent the broad mid-section of all housing

markets. As discussed above, under Grouping Loan Records for Use in Regression

Analysis and Forecasting, separate house price indices produced by OFHEO are used to

calculate NEQ_EQ_CLASS for the 25 MSAs where FHA business volumes are largest,

and for the nine Census Divisions. Loans not in one of these MSAs are assigned price

indices from their respective Census Divisions. The Census Division indices produced by
FHA-BSSS, III. Mortgage Default and Prepayment page 55

69The adjusted Division price series are computed as:

, where HPId is the OFHEO house price index for Census

Division, d, pm is the ratio of MSA population to Division population, and HPIm is the MSA price index. The

summations are over all MSAs within a particular Division. Population weights are from the 1990 Census.

OFHEO are, however, first adjusted by population weights to extract out the effects of

price changes unique to MSAs within their boundaries.

69

Interest rate spreads. The interest rate spread (SPREAD_CLASS) effects are in line with

predictions of option-pricing theory. When spreads are large and negative (market rates

below the note rate), higher default incentives result; when they are large and positive,

smaller default incentives exist. 

When market interest rates are high relative to the existing mortgage contract rate,

borrowers are more willing to service their existing mortgages, knowing that their present

housing is cheap relative to market-rate alternatives. Also, because FHA loans are fully

assumable by new property owners, borrowers can essentially sell the value of a belowmarket interest rate and thus capture the value of the negative-call option in sales price. In

both cases, the value of the put option is reduced because the value of the mortgage

liability is less than the outstanding loan balance.

Burnout variables: number of refinance opportunities and new opportunity flag. For

many years, mortgage researchers have been concerned with a phenomenon called
FHA-BSSS, III. Mortgage Default and Prepayment page 56

70Gordon Crawford and Hsiu-Wen Wu. “On the Rationality of Non-Refinancing Behavior,” unpublished

manuscript, presented at the 1998 ASSA/AREUEA Annual Conference, Chicago, IL. Washington, DC: Fannie Mae,

December 1997.

71Stavros Peristiani, Paul Bennett, Gordon Monsen, Richard Peach, and Jonathan Raiff, “Credit, Equity,

and Mortgage Refinancings,” Federal Reserve Bank of New York Economic Policy Review vol. 3 no. 2 (July 1997),

pp. 83-99.

72A variable that counted the number of times the mortgage spread was 100 basis points or more was also

tested in the regressions, but it proved to be a less valuable predictor of default and prepayment incentives than the

deep-in-the-money 200 basis point variable.

burnout. This is where pools of mortgages get burned out by successive refinance waves,

so that remaining loans are less sensitive to future refinance options. Borrowers whose

loans remain in the pools after these periods may be expecting to move soon anyway,70

others may be overwhelmed by the mortgage qualification and settlement processes, and

still others may have impaired credit and not qualify for a new loan.71 This last category

of borrowers left in burned-out pools is of most concern to mortgage investors. Their

presence implies that one can expect higher default rates from survivors in burned-out

pools. Two variables are used to test for burnout effects: BURN_SUM and NEW_REFI. 

BURN_SUM is the more important of the two measures (see Table 7). It counts the

number of past quarters in which market interest rates were at least 200 basis points

below the contract rate on existing mortgages.72 These were times when refinancing was

clearly in borrowers’ best interests. The regression results show positive burnout effects

(increased default incentives) for adjustable rate mortgages, but negative effects for fixedrate mortgage borrowers. It has long been argued that borrowers who take out ARMs are

less risk-averse than those who use fixed-rate mortgages, meaning that they are more
FHA-BSSS, III. Mortgage Default and Prepayment page 57

73Breukner and Follain , op. cit.; Capone and Cunningham, op. cit.

willing to chance default and foreclosure.73 The result found here indicates an increased

potential for ARM borrowers to impair their credit in ways that make qualifying for a

mortgage refinance more difficult. The negative effects found in fixed-rate loan equations

suggest that these concerns are not important there. Something else is going on with these

borrowers, perhaps as suggested by the authors cited above.

The second burnout variable is NEW_REFI, which is a flag (dummy) variable turned on

when: 1) the current period interest rate spread is at least -1.50 percentage points, and 2)

the spread was smaller (market interest rates higher) than -1.50 percentage points in

lagged quarters three through seven. The intent is to capture whether borrowers are in a

new refinance period, with enough lapsed time from the previous period to repair any

damaged credit and reverse any “burnout” type problems of poor credit quality in a

surviving loan pool. Such a reversal would be indicated by negative coefficients for

NEW_REFI. The regression results confirm that there is some reversal effect for

purchase-money ARMs. 

The combined negative coefficients on BURN_SUM and NEW_REFI in the fixed-rate

purchase equation suggest that poor quality borrowers leave FHA mortgage pools

quickly, leaving higher quality borrowers in “burned-out” pools. The strong influence of

the call-option variable (SPREAD_CLASS) on default rates, combined with these effects,
FHA-BSSS, III. Mortgage Default and Prepayment page 58

74This payment adjustment is based only on the principal and interest components of the mortgage payment.

75Following the rank order of ANOVA (Wald) test statistics in Table 7, payment adjustment is the fifth

most significant variable influencing ARM refinance-loan default, but the ninth most important variable for

determining ARM purchase-loan default incentives.

suggests that to the extent interest rate effects enhance put option values, they do so

independent of borrower credit-quality concerns.

ARM payment adjustment. Because adjustable-rate mortgages have changing payment

amounts over time, they face additional sources of incentives to default, beyond those

faced by fixed-rate, fixed-payment mortgages. The primary variable used to capture

ARM-specific effects is the relative size of the payment adjustment in the current year

(PMT_ADJ_CLASS). It is the percentage increase or decrease in monthly payments in the

present year, as compared with the previous year.74 Larger positive adjustments should

increase default incentives as payment burdens increase, and negative adjustments should

decrease default incentives. 

Regression results indicate that while this variable does not represent a major influence

on ARM default incentives (Table 7), it is relatively more important for refinance than for

purchase mortgages.75 The coefficient signs and sizes for refinance loans are as expected,

except for the largest payment-reduction class. The positive default response to large

negative adjustments for all ARMs likely indicates that large declines in short-term

interest rates are monetary policy responses to recession-level increases in national

unemployment rates. This unemployment-enhancement effect overwhelms what
FHA-BSSS, III. Mortgage Default and Prepayment page 59

76See Bruekner and Follain, op. cit., and Capone and Cunningham, op. cit.

otherwise should be a decrease in default incentives strictly based on payment burdens.

This effect clearly dominates all payment-adjustment size effects for purchase loans,

though not for refinance loans. 

Prepayment Regression Results: Loan Group Characteristics.

Loan-to-value ratio. Differences in prepayment incentives by original LTV_CLASS are

limited and mixed. While this variable has some predictive power in both default and

prepayment equations, it is never among the top five influencing variables. As an

indication of the abilities of borrowers to raise cash to manage financial set-backs, this

variable is more important in the default equations. Here, however, it only provides some

indication of differences in borrower mobility rates because borrowers can refinance with

FHA regardless of their equity position.

Price class, Property Age Class, Owner type, and Mortgage Age Class. Peak age-related

payoff incentives (AGE_CLASS) are in the 3-4 year age bracket for borrowers with fixedrate mortgages, and in the 1-2 year age bracket for those with adjustable-rate mortgages.

AGE_CLASS effects stay larger longer for fixed-rate mortgages, which is consistent with

the hypothesis that more mobile borrowers choose ARMs.

76

Investors (INVESTOR=YES) should be less mobile than owner-occupants, with lower
FHA-BSSS, III. Mortgage Default and Prepayment page 60

payoff incentives. In the FHA data, there is little evidence either way. The investor effect

is small but positive in the purchase-mortgage equation, and smaller and negative in the

refinance equation. FHA policy is to not insure investor ARMs, so there are no effects to

measure there.

The largest property age effects are for refinance loans, where borrowers who are

refinancing new homes (PROP_AGE_CLASS=NEW)–refinance in the first year of

owning the home--continue to have higher motivations to prepay again in the future.

Cohort-year effects. While cohort-year effects (COHORT_YEAR) for loan default are

quite clearly associated with major changes in underwriting procedures, no such patterns

exist for prepayments. Capturing idiosyncratic elements associated with each cohort of

business increases the efficiency of estimates on more pertinent loan characteristics and

economic environment coefficients.

Prepayment Regression Results: Economic Conditions

House-price-cycle stage. House-price-cycle stage (CYCLE_STAGE) models local

economic cycles. As such, the regression results reveal increased incentives for household

mobility during the first half of local recessions (CYCLE_STAGE=1), as laid-off workers

find work elsewhere and sell their homes. Mobility is then depressed as the cycle

approaches and passes through the trough (CYCLE_STAGE={2,3}). During these stages,

declines in housing market activity make it more difficult for laid-off workers to sell,
FHA-BSSS, III. Mortgage Default and Prepayment page 61

even if they do take jobs in other areas. Thus, we see declining incentives to prepay that

mirror increased incentives to default. 

The largest effects are for ARM purchase mortgages, with much larger payoff incentives

when CYCLE_STAGE=1, and much lower incentives when CYCLE_STAGE=3 (initial

phase of expansion from trough). ARM refinance loans do not show any statistically valid

results here because the data sample for that regression does not have good data on

transitions of loans through house-price-cycle-stages.

Market rates on fixed-rate mortgages: ARM effects. Current market interest rates for

fixed-rate mortgages (CUR_MKT_RATE) are the third most influential variable in both

the ARM-purchase and ARM-refinance regressions (Table 7). Low rates incent ARM

borrowers to refinance into fixed-rate products, and higher rates increase their incentive

to stay with the ARM.

National unemployment rate. As discussed earlier for default regressions, increases in the

national civilian unemployment rate (US_UN_RATE) depress overall borrower mobility,

which then decreases incentives either to default or to prepay mortgages. The largest

prepayment effect is in the ARM purchase-mortgage equation, where the coefficient is

more than twice the size of those in other equations. Overall, as seen in the ANOVA test

results, national unemployment rate changes are a less important influence on prepayment

rates than they are on default rates.
FHA-BSSS, III. Mortgage Default and Prepayment page 62

77FHA loans with initial periods of fixed interest rates, followed by annual adjustments, are called hybrid

ARMs. Lenders are not yet expecting that FHA hybrid ARMs, under current law, will contribute much to FHA

origination volumes because the interest-rate adjustment caps at the expiration of the fixed-rate term are too

restrictive. The lending community is working on technical changes to the implementing legislation that will make

these loans more attractive to mortgage investors.

Yield curve slope. The slope of the Treasury yield curve (YLD_SLOPE_CAT) reveals

information about incentives to refinance into adjustable rate products, independent of the

value of refinancing into a (new) fixed-rate loan. Regression results show that nearly all

of the positive incentive to refinance is associated with steep yield curves, with slopes

over 200 basis points, and especially over 300 basis points (YLD_SLOPE_CAT=5) . In

such environments, new ARM loans can both reduce monthly payments for a number of

years, and have low lifetime interest rate caps. 

Not only are new FHA ARMs attractive steep yield-curve environments, but conventional

market ARMs with initial fixed-rate terms of 3, 5, or 7 years are also attractive to

homeowners. FHA recently received statutory authority to offer such intermediate-term

products. When they become available, those products should decrease loss of business to

the conventional market during steep yield-curve environments.77

Market shift. The MARKET_SHIFT variable was discussed earlier for the default

regressions. Its effects on prepayment incentives are all large and they rank higher in

overall influence (ANOVA Wald tests) than do the MARKET_SHIFT effects in the

default equations.
FHA-BSSS, III. Mortgage Default and Prepayment page 63

Prepayment Regression Results: Financial Incentives.

Probability of (20 percent or higher) Positive Equity. This variable (POS_EQ_CLASS) is

analogous in construction to the probability of negative equity variable

(NEQ_EQ_CLASS) used in the default regressions. Here, however, it is not a direct

option-value variable but rather an indication of sufficient home equity for property sale

or cash-out refinancing. It has an important influence on overall prepayment incentives.

Interest rate spreads. As discussed earlier, prepayment incentives are highly influenced

by the value of the call option (SPREAD_CLASS).

Burnout variables: number of refinance opportunities and new opportunity flag. Analysis

here follows the earlier discussion of burnout issues as they relate to the default

regressions. The negative coefficient on BURN_SUM in the prepayment equations likely

reflects the existence of a core group of homeowners who do not respond to call-option

incentives. Because FHA has a large presence in the lower end of housing markets, there

is a core of borrowers for whom the financial benefits of refinancing can be small or

nonexistent because of low loan balances, even when the call option appears to be deeply

in-the-money for mortgage borrowers in general. However, the positive coefficients on

NEW_REFI=ON show some of the same increase-in-credit-quality effect seen for this

variable in the default equations.
FHA-BSSS, III. Mortgage Default and Prepayment page 64

ARM payment adjustment. Annual payment adjustments (PMT_ADJ_CLASS) can

influence borrower decisions to prepay, as well as to default. For ARM purchase

mortgages, large payment adjustments–either negative or positive–are associated with

larger prepayment incentives. Annual interest rate changes on FHA ARMs are limited to

one percentage point, so large downward movements in market interest rates only yield a

corresponding decrease in mortgage payments if the borrower refinances. That appears to

be what is seen in the largest (positive) coefficient being on PMT_ADJ_CLASS=1, where

downward payment adjustments are larger than 5 percent. When rates drop precipitously,

borrowers can get an even larger reduction in payments by refinancing.

USE OF REGRESSION EQUATIONS IN FORECAST SIMULATIONS

In the forecast simulations, data input into the logistic probability equations for default

and prepayment are loan-group characteristics and forecasted economic conditions (see

Section II). Economic conditions change quarterly and have different trajectories in each

of the simulation paths. The probability equations are used to predict termination events

only between the fourth quarter of loan-life and the end of the twentieth year of each

cohort. Outside these bounds, quarterly default and prepayment rates are predicted as

here. 

The regressions described earlier in this section also have missing coefficients on some

categories of the economic factors, due to relatively short loan-data histories. Most of
FHA-BSSS, III. Mortgage Default and Prepayment page 65

these are for ARMs, where FHA experience starts in 1991. There is also one instance in

which an effect coefficient must be imputed for fixed-rate refinance loans. These special

cases are detailed below, by variable. 

Loan Performance in First Three Quarters of Loan Life

Defaults in the first year of loan life are rare. They are generally due to faulty

underwriting so the costs are often borne by loan originators, rather than by FHA.

Prepayment activity in this period is also negligible. In the simulation model, termination

rates in the first three quarters of loan life are fixed at a 30 basis point annual rate

(0.0030) for defaults and a 100 basis point (0.0100) annual rate for prepayments. These

rates reflect an average experience of recent cohorts.

Loan Performance After Year Twenty

Assumptions used for post-year-twenty experience have very little effect on lifetime loan

performance and estimated subsidy rates. The minimal effect is because most

terminations have occurred by the end of year 20, and because discounting cash flows to

produce subsidy rates diminishes the effect of any variations in post-year-twenty

performance. Pre-1982 insurance cohorts enter the forecasts with 20 or more years of loan

seasoning. They are assigned termination rates in line with forecasts made by the FHA
FHA-BSSS, III. Mortgage Default and Prepayment page 66

78Deloitte & Touche, U.S. Department of Housing and Urban Development Annual Actuarial Review of the

Federal Housing Administration’s Mutual Mortgage Insurance Fund As of Fiscal Year 2000. Philadelphia, PA:

Deloitte Touche Tohmatsu, December 13, 2000, Appendix I.

actuarial-review contractor.78 Contractor projections of default rates on these loans show

a fairly uniform random pattern within the range of 5 to 15 basis points (.0005 - .0015)

per year, so a value of 10 basis points is used in the FHA-BSSS. Prepayment rates

predicted by the FHA contractor also appear as fairly uniform random variables, with a

range of 5 to 17 percent per year, so a value of 11 percent is used in the FHA-BSSS.

Post-year-twenty loan performance for cohorts entering the forecast period with less than

twenty years seasoning (post 1982 cohorts), follows from the year-twenty forecasts made

in the FHA-BSSS. They are fixed at the larger of 5 percent per year or the projected rate

in the fourth quarter of year twenty. Default rates in the post-year-twenty period start with

the projected rate in the final quarter of year twenty and decline with a 10 percent (not

percentage points) annual rate of decay after that. 

Performance Regression Coefficient Adjustments

There is no one single method of imputing values for missing regression coefficients. In

the regressions described above, most missing values are for highest-value classes of

categorical variables. Therefore, the primary imputation method used in the FHA-BSSS is

to extrapolate beyond the estimated class-level effects using a ratio of class-level effects

found in another regression. For example, if variable X has a complete set of seven
FHA-BSSS, III. Mortgage Default and Prepayment page 67

coefficient estimates in regression 1, say b1(1) - b1(7), but is missing class 7 in

regression 2, b2(7), then .

In a few cases, this type of imputation is not done because the value of the highest-class

effect estimated in the regression analysis does not correspond with the rest of the series,

nor to what is seen in the regression coefficient series in other equations. These are cases

in which the final class appears to be an outlier class with too few observations for

reasonable estimation of the effect.

The following discussion of imputation for individual coefficient values refers to

regression results reported in Tables 9 and 10. The imputed values are identified in those

Tables by boldfaced type.

Mortgage Age Class. The refinance regression for ARM defaults only measured effects

for the first five age classes (years 1-10). Given the large (negative) size of the category

five effect, category six and seven effects are simply set equal to that value. For the

purchase regression for ARM defaults, the category seven age effect is set to that same

value (from the ARM refinance equation).

For ARM prepayment regressions, the category seven age effect in the purchase equation

is extrapolated using the class-seven-to-six ratio found in the fixed-rate purchase
FHA-BSSS, III. Mortgage Default and Prepayment page 68

equation. Category six and seven age effects in the refinance equation are set equal to the

effects computed for the ARM purchase equation. 

House Price Cycle Stage. ARM refinance loans in the observation period did not pass

through any complete house price cycles. As a result, all observations are in

CYCLE_STAGE=4, leaving even that effect subsumed in the constant term of the

regression. Thus, when imputing values, instead of the four stage coefficients summing to

zero, as would be the case if all stages entered the regression, the CYCLE_STAGE=4

coefficient value must be set to zero. 

For default regressions, effects in the fixed-rate refinance equation are nearly identical to

those in the fixed-rate purchase equation. It is then reasonable to use effects from the

ARM purchase default equation directly in the ARM refinance default equation, for cycle

stages one through three. Effects for the prepayment equation start with those from the

ARM purchase equation, which are scaled according to the relative size of effects in the

fixed-rate refinance prepayment equation compared with the fixed-rate purchase

prepayment equation. Again, the stage four effect is set to zero.

Interest Rate Spreads. Effects are imputed for the last two interest rate classes in the

ARM-purchase-default equation as well as for both the first two and last three classes in

the ARM-refinance-default equation. In each case, missing class effects are extrapolated

from the existing (intermediate class) series using relative cross-classes effects from the
FHA-BSSS, III. Mortgage Default and Prepayment page 69

fixed-rate-default equations, purchase and refinance, respectively. However, before the

calculations can be performed for the ARM-refinance-default coefficients, the final

spread-class effect for fixed-rate refinance default equation is extrapolated from the crossclass relationship (ratio of class-14-to-class-12 effects) found in the fixed-rate-purchasedefault equation.

For prepayment equations, effects for the last two spread classes in the fixed-raterefinance and ARM-purchase equations are set equal to the effects in the fixed-ratepurchase equation. For ARM-refinance loans, effects for the first and last three classes are

set equal to one-half the effects of the fixed-rate-refinance equation. This choice follows

the relationship seen for classes in which both of these equations have coefficient

estimates .

Lowest Loan-To-Value Class. In all performance regressions, the lowest loan-to-value

(LTV) class–loans with original LTVs below 80 percent–is excluded from the estimation

because of low volumes. They average between one and two percent of total FHA

insurance per year. To allow for their inclusion in the simulations, the effect for LTV

class 1 is imputed as follows:

a. Prepayment equations: zero (average) effect for the two purchase equations

and class 2 (81-90 LTV) effect for the two refinance equations.

b. Default equations: class 2 effect used for class 1in all equations. 
FHA-BSSS, IV. Post-Default Outcomes page 70

IV. POST-DEFAULT OUTCOME REGRESSION MODELS 

Default rate regression models described in section III provide only part of what is needed

to measure timing and dollar amounts of credit-cost cash flows for FHA loan guarantees.

The actual cost of paying insurance claims, and time lags in the process, depend on many

additional factors. These factors include mortgage terms, state foreclosure laws, economic

conditions, and FHA policies. It is important to estimate statistical models that relate cost

factors to economic conditions, and to measure systematic differences across mortgage

types and property locations. Four such regression models are presented in this section.

Cost factors insensitive to such influencing factors are treated as fixed amounts and

percentages, and are discussed in section V.
FHA-BSSS, IV. Post-Default Outcomes page 71

The four regression models described here are for:

• Probability (relative frequency) of defaulted loans receiving workouts;

• Time-to-foreclosure for loans not receiving workouts, and those that fail in

attempted workouts;

• Time-to-disposition (sale) for foreclosed properties; and

• Recovery rates (net sale proceeds) on foreclosed properties.

A timeline showing how these and other cost factors fit together to determine amount and

timing of credit-cost cash flows is provided in Table 11.

LOGISTIC REGRESSION OF WORKOUT ASSISTANCE OFFERS

Once delinquent borrowers miss three payments and a fourth is due (90-days

delinquency), FHA loan servicers are required to evaluate each case for use of tools to

avoid foreclosure. Loan servicers themselves are evaluated annually by FHA on how well

they do in reinstating defaulted borrowers and in minimizing the costs of resolving all

defaults. Those that score high are rewarded with higher foreclosure-cost reimbursements

(75 percent versus 67 percent) and are given more flexibility in the administrative rules

governing eligibility of borrowers for workout assistance. 

This workout program, called Loss Mitigation, is relatively new, having just begun full
FHA-BSSS, IV. Post-Default Outcomes page 72

79This program was approved by Congress and announced by FHA April, 1996. Interim regulations were

issued in July, 1996. However, given the time necessary for FHA and lenders to retool information systems and

retrain staff, there was no measurable activity until 1997. 

80The term “workout” is used here, rather than loss mitigation, because workouts also include forbearances

offered by loan servicers, outside the FHA loss mitigation tracking system. These forbearances accounted for as

much as three-quarters of all workouts during the late 1990s. However, FHA now directs the servicers to report these

as loss-mitigation forbearances, so the rate of outside forbearances has declined.

operations in 1997, but having grown steadily since then.79 Loss mitigation activities have

significant effects on default-related insurance-claims costs. Whereas a full foreclosure

can cost FHA, on net, 35 percent or more of the outstanding loan balance, successful loss

mitigation cases have an average cost of just 1.5 percent of outstanding loan balances.

Therefore, it is important to understand how the program is being used by loan servicers,

and how rates of use vary across borrowers, loan types, and economic conditions.

Regression Model

The probability of loan workout (versus foreclosure) model is based on a binary logistic

regression using records on over 400,000 non-cured 90-day defaults, from 1998Q1 to

2001Q3.80 The regression estimation is on individual loan activity. In simulation forecasts

the estimated probability equation is used to predict the percent of defaults that receive

workout offers, by loan group and quarter. The regression estimation uses an effects

parameterization, as do the loan termination regressions (see section III). Coefficient

estimates from the logistic regression are reported in Table 12. 

The time-frame of the regression analysis is also the phase-in period of the FHA loss-
FHA-BSSS, IV. Post-Default Outcomes page 73

mitigation program. Workout offers increased dramatically from 18 percent of all noncured defaults in early 1998 to over 75 percent by mid-year 2001. Numerous ways of

capturing changes in the baseline rate of workout offers during the phase-in period were

tested here. These tests involved using a time-varying constant term in the regression, or

else adding some function of elapsed time since 1997Q1. While regressions with these

effects increased the statistical fit of the estimated model, none provided satisfactory

results when applied in forecast simulations; the variability of simulated outcomes was

too small, as forecast results were dominated by the program phase-in effects. In contrast,

industry analysts who work with conventional-market, default-workout programs indicate

that a large variance of outcomes across economic conditions should be expected. 

Estimating the regression with economic variables and without any phase-in effects

produced a model with larger coefficients on the economic variables, allowing for a wider

range of outcomes in the simulations. The final regression model then excludes any

phase-in effects, yet has a constant term that is consistent with program activity in 2001.

Regression Results

ANOVA test statistics calculated for this regression indicate that HPI_CLASS has the

largest effect on workout utilization rates, with PRICE_CLASS second, and

HPI_GROWH_CLASS third. The effect coefficient for HPI_CLASS ranges from -2.20 for

price index values below 0.90 (more than 10 percent cumulative, to-date property value
FHA-BSSS, IV. Post-Default Outcomes page 74

81These changes in workout probability (p) are calculated using the relationship between logit coefficients

and the workout odds-ratio. The odds-ratio at the sample mean is p/(1- p) = 0.33/(1-.33) = 0.50. When the effect for

HPI_CLASS =1 or 6 is turned on, the odds-ratio will change proportionately, according to the value of each

HPI_CLASS coefficient (b): p1/(1-p1) = exp{b}*(.33/.67), where p1 is the new probability of workout. Inserting the

values for b and solving for p1 gives the new probabilities. While the data sample is centered around a workout

probability of p=.33, simulations using the regression equation are centered around p=.50.

82However, in October, 2002, FHA issued new directives to loan servicers that allow long-term forbearance

plans for borrowers unemployed but with good prospects of near-term employment.

depreciation) to +1.26 for price index values above 1.50 (more than 50 percent

cumulative, to-date property value appreciation). Given an average probability of workout

of near 33 percent across the historical sample, the range of workout rates resulting just

from differences in house price growth is from 5 to 64 percent.

81

When viewing workout rates by loan characteristics, one sees the greatest rate of loan

workout offers for high-priced homes (PRICE_CLASS = 3), loans with high initial

downpayments (LTV_CLASS = 1), and loans with 15-year amortization periods

(PROD_CLASS = 2). These outcomes all suggest that households in higher income

classes who, in general, will have more resources to draw upon to effect a workout, do

indeed have higher rates of use of workouts. In general, workouts can only be offered if

the borrower has regained income sufficient for maintaining the mortgage and property

within a short period of time.82
FHA-BSSS, IV. Post-Default Outcomes page 75

Use of Model in Forecast Simulations

The regression results reported in Table 12 are missing effects for CYCLE_STAGE =

{1,2}. This omission is because the short history of the FHA loss mitigation program

yields no observations in these stages in any Census Division or MSA. In forecast

simulations, the effect coefficient for stage 4 is used also for stage 1, and the effect for

stage 3 is used also for stage 2. This choice is based on an assumption that ease of finding

new work will be more similar between stages 2 and 3, representing the bottom of a

recessionary period, and also between stages 1 and 4, which represent early recession and

expansion. 

TIME-TO-FORECLOSURE REGRESSION

Foreclosure is the legal process of taking property title from homeowners to satisfy

unpaid mortgage debts. As this process falls into the general category of property rights

and law, foreclosure rules are specific to each state. However, there are two principal

methods in use: power-of-sale and judicial proceedings. Power-of-sale, also known as

trustee sale, allows a trustee appointed at the time of mortgage origination to sell the

property in an orderly auction, as soon as the lender has complete homeowner notification

requirements. These sales can be done quickly. The usual time frame for an uncontested

power-of-sale foreclosure is between six and 16 weeks.

Judicial foreclosure, on the other hand, requires court filings and proceedings that can last
FHA-BSSS, IV. Post-Default Outcomes page 76

for many months. States that require use of judicial proceedings often also have statutory

post-foreclosure redemption periods, to give homeowners even more time to find the

funds necessary to reinstate their mortgages and reclaim their homes. Once a judge

decides that the lender is within its rights to obtain judgment, the property is auctioned by

a court magistrate. These are often referred to as Sheriff’s Sales.

Foreclosure times also vary according to economic conditions and the number of homes

in foreclosure. As recessions deepen, and defaults escalate, the institutional time

necessary to process individual cases increases. Also, as recessions deepen, more

borrowers file for bankruptcy court protection. Such filings cause an automatic stay on all

debt collections, including pending foreclosures. Attorneys for the lenders must then file

with the courts for release from the stay in order to proceed with the foreclosure. All of

these events tend to extend average foreclosure times as economic conditions worsen.

Regression Model

Because foreclosure times are inherently institutional processes, they are best modeled

with waiting-time theory. In waiting-time models, the focus of empirical estimation is the

distribution of times from entry into a queue until service. Here, the foreclosure time

distribution is modeled using an accelerated failure time (AFT) regression of months
FHA-BSSS, IV. Post-Default Outcomes page 77

83After studying the data and estimation results, all available data was used, dating back to defaults in 1975.

The only restriction placed on the data was that loans assigned to FHA prior to foreclosure were removed from the

sample. These loans could have extended foreclosure timeframes because of forbearances provided in assignment.

The assignment program is no longer operating, so whatever information could be gained from studying that history

is no long valid for forecasting the future.

84The AFT is related to the proportional hazard model, where the event of interest is the rate of exit from

the sample each period, due to foreclosure. Proportional hazard models estimate conditional rates of exit each

period, with the values of explanatory variables and effect parameters scaling those rates up or down. Proportionality

in the AFT comes from the way that values of explanatory variables scale time-to-exit up or down. This scaling

indirectly changes the hazard (exit) rates.

from default to foreclosure, using individual loan data.83 AFT is a semi-log regression

where time-to-foreclosure, in log form, is regressed on explanatory variables that act as

proportional adjustments to a baseline, stochastic foreclosure time.84 That underlying,

baseline stochastic distribution is found in the error term of the model. The distributional

form used here is the Weibull, which allows for periods of increasing and then decreasing

rates of foreclosures, over time. This simply means the foreclosure/waiting time

probability distribution has a single-peak at some point in time greater than zero. The

estimation procedure chooses the best-fit Weibull shape for the data.

Regression Results

Full regression results are reported in Table 13, which includes regression coefficients,

the standard errors of those estimates, and a column of proportionality factors. These

factors show the effects of changes in the explanatory variables on foreclosure times. In

the AFT model, a one unit change in a continuous variable, or turning on a categorical

effect, multiplies the base foreclosure time by exp{b}, where b is the regression

coefficient. Negative coefficients will have exp{b} values less than one, and positive
FHA-BSSS, IV. Post-Default Outcomes page 78

85The extremely large proportionality factors on some DEFAULT_YR effects, where those effects lack

statistical significance, only highlight that there were few observations with these effects and that their foreclosure

times differed markedly. Removing these years from the analysis did not change the regression results. The

underlying coefficient effects for these years are not used in the forecast simulations.

86All regression analysis is performed using the SAS System statistical software. The AFT procedures in

SAS (PROC LIFEREG) do not allow for an effects parameterization. The last value category of each variable is

automatically removed from the data used in estimation.

coefficients will have exp{b} values greater than one.85

Unlike logit regressions used for mortgage performance and default workouts, categorical

variables in the AFT regression do not use an effects parameterization.86 The effects of

the largest value-class for all categorical variables are combined into the constant term of

the regression. The effects shown in Table 13 are then all relative to that of the combined

highest-value classes of all variables.

The regression explanatory variables capture differences in foreclosure times by

economic conditions and state property laws. Economic conditions enter the model

through house-price cycle stage (CYCLE_STAGE), local house price growth in the first

four quarters after default (HPI_GROWTH), and the market interest rate for long-term

fixed-rate mortgages at the time of default (CUR_MKT_RATE). Property-law effects

enter the regression through a Census Division indicator (CENSUS_DIV). This measures

the average effect of the various state laws in that Division, weighted by actual FHA

activity in each state. The regression results show moderate differences across Divisions.
FHA-BSSS, IV. Post-Default Outcomes page 79

The last explanatory variable in the time-to-foreclosure regression is default year

(DEFAULT_YR). This variable controls for institutional factors affecting foreclosures.

Prior to 1998, foreclosure time including processing of assignment requests by HUD field

offices. The purpose of (mortgage) assignment was for HUD to buy defaulted mortgages

and give borrowers additional time to work out their financial difficulties. However, a

large percentage of applications were denied and they appear here through extended

foreclosure times. The DEFAULT_YR variable captures assignment processing delays

and differences in HUD rules on the time lenders have to initiate foreclosure, over time.

To avoid problems associated with outlier foreclosure time values that may represent data

errors rather than true event times, time-to-foreclosure on individual cases is censored at

36 months. All values greater than 36 are treated as though they were observed as open

cases at 36 months, with no additional information available. The Weibull shape

parameter is estimated assuming that the shape of the waiting time (foreclosure time)

function as it approaches the 36-month mark continues after that point.

The regression results in Table 13 show significant variations across Census Divisions,

and that foreclosure times are extended by 27 percent (average effect of 3 months) when

local housing cycles are passing through a trough (CYCLE_STAGE=3). Default year

effects show a downward trend over time. The DEFAULT_YR effect used in the

simulation forecasts is 0.50, which corresponds with the effects of 1996 and 1999.
FHA-BSSS, IV. Post-Default Outcomes page 80

87For more information on the Weibull distribution, see Merran Evans, Nicholas Hastings, and Brian

Peacock, Statistical Distributions, 2

nd ed. New York: John Wiley & Sons, 1993, chapter 41.

Though the coefficient for 2000 is lower, the volatility of this parameter over the final

years of the historical period calls for caution when choosing a value to use as a baseline

level for forecast simulations.

Use of Model In Forecast Simulations

When predicting foreclosure times on grouped data in the forecast simulations, all

estimates begin with the mean-value Weibull baseline foreclosure time, using the

estimated shape parameter, rather than randomly assigning values from the Weibull

distribution. Given the shape parameter, c, the mean of the Weibull distribution is

, where G is the Gamma function. 

Starting with the mean value saves computation time without sacrificing any portfoliolevel variability. The mean baseline time, W-bar, is scaled for each forecast observation

according to values of the explanatory variables.87 Forecast foreclosure time is then

, where W-bar is the mean-value Weibull variable, X is the vector of

explanatory variables and b is the vector of estimated coefficients. The exp(Xb) term is

known as the scale parameter for the Weibull distribution. 
FHA-BSSS, IV. Post-Default Outcomes page 81

TIME-TO-DISPOSITION REGRESSION

A second time-frame of interest for credit-cost cash flows is the delay between property

foreclosure and property sale. The purpose of foreclosure sales is ostensibly to liquidate

properties and payoff the defaulted loans. However, sales to third parties at foreclosure

auctions are rare because the properties are generally deteriorated to where they need

repairs and significant maintenance before they can be sold at a price close to the

outstanding loan balance. Third-party sales are also inhibited by the foreclosure sale

process, one in which potential buyers cannot inspect the property beforehand and must

provide large deposits at the time of auction. Thus, FHA generally takes title to foreclosed

properties by instructing its loan servicers to bid in the auction, on its behalf. Servicers

then file claims for insurance reimbursement for the loan balance and foreclosure

expenses, and HUD takes the properties and works with contractors to sell them. The

properties are known as real estate owned or, simply, REO.

Regression Model

Like time-to-foreclosure, time-to-disposition is modeled from waiting-time theory, using

an AFT regression on months from foreclosure to disposition. Time-to-disposition is

primarily a function of economic conditions at the time of foreclosure, but it has

institutional influences as well. Explanatory variables used in this regression are the same

as were used for time-to-foreclosure, only here, the economic variables are observed at

time of foreclosure, rather than time of default, and house price growth is in categorical
FHA-BSSS, IV. Post-Default Outcomes page 82

form (HPI_GROWTH_CLASS). In addition, time-to-disposition is censored at 24 months,

rather than 36. To account for various problems FHA has had with property-management

contractors over the years, a series of foreclosure-year variables are added to the analysis

(FORECLOSE_YR).

Regression Results

Estimation results are found in Table 14. The proportionality factors reported there

indicate that REO disposition times are extended during housing cycle downturns, as

prices and sales volumes fall throughout the market. The housing-cycle proportionality

effect alone is around 10 percent (stages 1 and 2), while the added effect of negative

appreciation rates (HPI_GROWTH_CLASS = 1) is another 14 to 19 percent. Together,

these results indicate that periods of housing market declines extend property sales times

by 25 to 30 percent, meaning 1.5 to 2 months, or one quarter.

Interest rates prevailing after foreclosure (CUR_MKT_RATE) also affect disposition

times, but the effect is modest. Each one percentage point increase in mortgage rates leads

to a 3 percent increase in disposition time. With an average disposition time of 6 months,

that amounts to just under one week. So, it takes a five percentage point increase in

interest rates to extend disposition times by one month.
FHA-BSSS, IV. Post-Default Outcomes page 83

Use of Model in Forecast Simulations

The only free parameter to be chosen for forecast simulations is the FORECLOSE_YR

effect. As with calendar year effects in the foreclosure time regressions, there has been a

downward trend over time, with some volatility in recent years. The value chosen is 0.40,

which equals the average value of the 1997-2000 effects.

REO RECOVERY RATE REGRESSION

The final stage in the process of settling accounts on loan defaults is disposing of

foreclosed properties. Disposition produces receipts that help offset the cost of paying full

insurance claims at the time of foreclosure. Just how large the receipts are, relative to the

unpaid balance of the loan at default, is a function of housing market conditions after

foreclosure, property and owner characteristics, and regional differences. The ratio of net

sales receipts to the unpaid loan balance is the foreclosed-property recovery rate.

Regression Model

Recovery rates are modeled with ordinary least-squares regression. The dependent

variable in the regression is sales price less sales expense, divided by the outstanding loan

balance. A recovery ratio measure net of property and holding expenses was tested, but it

fit the data less well. Likewise for a recovery ratio net out repairs expenses. These

additional disposition related expenses are then added in the forecast simulations as fixed
FHA-BSSS, IV. Post-Default Outcomes page 84

88An additional issue arises because HUD has programs where it sells properties at below-market prices to

non-profit organizations, local governments, policemen, and school teachers. Eliminating observations with very-low

sales prices did not change the regression results materially. Therefore, since below-market sales are expected to

continue, we allowed these loans to remain in the regression sample set. 

89Department of Housing and Urban Development, Office of Federal Housing Enterprise Oversight, Risk

Based Capital, Federal Register, Vol. 64 no. 70, Tuesday, April 13, 1999, pp. 18188-18192, esp. p. 18190.

90PRICE_DROP_STAT is computed as, , where HPIt

 is the market house price index value

from loan origination to default, LTVt is the original loan-to-value ratio, amortized to time of default, t, and st is the

standard deviation of cumulative house price growth rates around log(HPIt). 

ratios of loan balances (See section V, Table 16).

88

 

Following work by OFHEO on conventional-market recovery rates, the best variable

found here for modeling economic volatility in recoveries on FHA loans is a statistical

measure of the price decline necessary to make the default put option in-the-money

(PRICE_DROP_STAT).89 This variable is the same standardized normal statistic used to

calculate the probability of negative equity (NEG_EQ_CLASS) variable used in the

logistic default regressions. Only here, the statistic is the additive inverse of that used to

compute NEQ_EQ_CLASS.

90 PRICE_DROP_STAT is the sum of market generated

property equity (at the average appreciation rate) and borrower generated equity via the

downpayment and regular payments toward loan principal. It is then a measure of how

much property-value depreciation must be on an individual property to wipe out market

and borrower generated equity, and thus to make foreclosure an attractive option to the

borrower/homeowner.
FHA-BSSS, IV. Post-Default Outcomes page 85

91The large differences between property-price classes are clearly seen in frequency distributions of

recovery rates, by price class. The distributions are normally distributed with similar variances, but with means of

0.67, 0.79, and 0.88 for classes 1, 2 and 3, respectively.

PRICE_DROP_STAT indicates how good or bad are housing market conditions at the

time of foreclosure. A high value of PRICE_DROP_STAT indicates that market

conditions were good so that there had to be a large amount of property depreciation to

make foreclosure in the best financial interest of the mortgage borrower. In such cases,

property depreciation on foreclosed properties is likely to be just barely enough to make

the foreclosure beneficial to the borrower/homeowner so that property recovery rates are

likely to be high. On the other hand, a low (or negative) value of PRICE_DROP_STAT

indicates that housing market conditions are fair-to-poor, so that it does not take much

property-specific depreciation to make foreclosure in-the-money to the

borrower/homeowner. In such cases it is likely that typical property depreciation levels

will cause actual recovery rates to be low.

Additional effects found to be important in the regression analysis are the property price

class (PRICE_CLASS), owner-type (INVESTOR), and Census Division (CENSUS_DIV).91

Over time, some Divisions consistently had recovery rates that were either higher or

lower than expectations based on economic and property characteristics. Mortgage rates

were excluded from this regression because their influence was too small to merit

including them in the simulation forecasts.
FHA-BSSS, IV. Post-Default Outcomes page 86

92Declines in PRICE_DROP_STAT can be thought of as indicating that exercised put options will be deeper

in the money, yielding lower recovery rates. 

A control for disposition year (DISP_YEAR) was added to the regression to account for

variations in HUD procedures and contracting methods over time.

Regression Results

Regression results are reported in Table 15. Additional analysis-of-variance tables (not

shown) indicate that PRICE_DROP_STAT is the largest influence on recovery rate

outcomes, with PRICE_CLASS second and CENSUS_DIV third. The positive coefficient

on PRICE_DROP_STAT suggests that, in healthy housing markets (PRICE_DROP_STAT

>> 0), the average recovery rate will be greater than the 79 percent indicated by the

constant term of the regression. In declining markets (PRICE_DROP_STAT < 0), the

recovery rate will be lower. The 79 percent average recovery on REO exists when average

depreciation in the market would make the value of the average property equal to its

outstanding loan balance. Defaulted properties normally have appreciation/depreciation

worse than the market average, leading to the additional (average) 21 percent loss of

value in these circumstances.92 

Loans in the highest property price class (above 100 percent of area median price) have

recovery rates 23 percentage points higher than those in the lowest class (under 50

percent of area median price). Investment properties have recovery rates 11 percent below

owner-occupied properties. By region of the country, the lowest recovery rates over time
FHA-BSSS, IV. Post-Default Outcomes page 87

have been consistently in the New England and MidAtlantic Census Divisions. 

Use of Model in Forecast Simulations

To forecast recovery rates in the subsidy-rate simulations, the FORECLOSE_YR effect

chosen is an average of the most recent five-year period (0.040). 
FHA-BSSS, V. Final Forecast Assumptions and Subsidy Rate Calculations page 88

V. FINAL FORECAST ASSUMPTIONS AND SUBSIDY RATE

CALCULATIONS 

Forecasts of economic conditions and mortgage performance provide the foundation for

setting up calculations of government cash flows from FHA loan guarantees. The two

primary cash-flow streams that define subsidy estimates on FHA guarantees are premium

income and credit expenses. These, in turn, are sums of individual calculations for:

• Premium income inflow on new mortgage originations (up-front premium);

• Premium income each quarter on outstanding loans, within contractual time

frames (ongoing premiums are assessed annually, paid monthly);

• Partial refunds of up-front premiums due to borrowers that payoff mortgages in

the early years (based on premium earning rates and schedules determined by

FHA);

• Claim payments to loan servicers for workout efforts;

• Claim payments to loan servicers for foreclosed properties; and

• Recoveries on foreclosed properties, net of all property and sales expenses.
FHA-BSSS, V. Final Forecast Assumptions and Subsidy Rate Calculations page 89

Lifetime cash-flow time series for each loan group are the grafting of historical cash

flows with future projections. They are created for each simulated economic path.

Regression models outlined in Sections II-IV provide all of the meaningful variation in

subsidy rates, yet there are several additional cost and revenue factors needed to complete

the cash flows. These are discussed here. A final issue addressed here is determination of

volumes and characteristics of future insurance cohorts. 

All historical data on cash flows is maintained in databases used as starting positions for

the simulation analysis. These records are aggregations, following the same scheme used

for mortgage performance regression analysis data bases, as outlined in Section III. The

one difference here is that historical data used as forecast starting positions are aggregated

by cohort (fiscal) year, rather than by loan origination quarter. To make quarterly

predictions of mortgage events, all loans are assumed to have been originated in the

second quarter of the calendar year of their cohort (third quarter of fiscal year). The main

reason for quarterly forecasts is to model the effects of intra-year movements in interest

rates. Because the loan aggregation scheme separates loans by original interest-rate class,

the assumption of a common origination quarter is not restrictive. After quarterly

forecasts are made, the cash flows are aggregated by fiscal year of experience before

subsidy rates are calculated.
FHA-BSSS, V. Final Forecast Assumptions and Subsidy Rate Calculations page 90

93Readers who desire the exact details are referred to Deloitte & Touche LLP, Annual Actuarial Review of

the Federal Housing Administration’s Mutual Mortgage Insurance Fund Fiscal Year 2001. Philadelphia: Deloitte &

Touche, December 2001, Tables D.2 and D.3.

PREMIUM INCOME

Premium income is the more straightforward of the two primary cash-flow series. FHAinsured borrowers pay an up-front premium, which is nearly always financed with the

mortgage, and pay ongoing, monthly fees (based on an annually assessed premium) for a

certain length of time. The up-front fee is refundable, in part, if the borrower pays off the

mortgage early. Both premium and refund rates have changed over time and so are

specific to each origination cohort. Current rates are used for all future cohorts.93

Because the mean/average prepayment in any quarter will occur at the end of month two,

premium income on loans paid off in a given quarter is assumed for two of the three 

months. For loans defaulting in each quarter, no premium income is recorded. Loans that

receive workout assistance return to paying premiums (in the simulations) in the quarter

after default.

OTHER PARAMETERS NEEDED TO CALCULATE NET DEFAULT COSTS

There are numerous instances in which default cost components are not particularly

sensitive to economic conditions or loan characteristics. Recent trends in FHA business

activity provide reasonable values for these cost parameters in forecast simulations. Each
FHA-BSSS, V. Final Forecast Assumptions and Subsidy Rate Calculations page 91

94U.S. Department of Housing and Urban Development, Office of Policy Development and Research,

Evaluation of the Federal Housing Administration Preforeclosure Sale Demonstration. Washington, DC: U.S.

Department of Housing and Urban Development, June 1994.

parameter is listed and described in Table 16.

FHA has an additional default workout option that has seen relatively little use: the

preforeclosure sale. It permits borrowers to sell their properties, with FHA paying for any

loss-on-sale. These sales generally result in measurable savings to FHA, over the net cost

of foreclosure and property disposition. However, use of this tool has been limited to

about five percent of the number of foreclosures each year. Because it has not been an

important tool for FHA, and its usage has been fairly consistent over time, the FHABSSS focuses on other aspects of default resolution and cost. In the forecast simulations,

five percent of foreclosures are treated as preforeclosure sales, and are given a cost saving

of 8 percentage points. This cost saving rate was the average found in HUD’s evaluation

of its preforeclosure sale pilot.94

FUTURE COHORT CHARACTERISTICS AND VOLUMES

Forecast simulations provide estimates of experience on future loan cohorts, as well as

projections of lifetime performance for existing cohorts. Volumes and characteristics of

future cohorts are generated in four parts: 
FHA-BSSS, V. Final Forecast Assumptions and Subsidy Rate Calculations page 92

1. All key variables other than the interest rate are taken from a chosen historical cohort;

2. Interest rates are assigned based upon values generated in the vector autoregression

(section II), by quarter of loan origination; 

3. Refinance dollar volumes are fixed in the first two years of the forecast period and

adjustable rate mortgage (ARM) volumes are fixed throughout the forecast period as a

percent of all originations; and

4. Dollar volumes of purchase and refinance loans grow over time with population and

inflation factors.

In the FHA-BSSS, these four sets of factors are specified as follows:

1. The most recent FHA originations available at the time of FHA-BSSS development

were for FY2002. Those loans are used as a basis for starting future originations.

First, the loans are aggregated using eight of the nine classification variables outlined

in section III (interest rates are excluded). Numbers and dollars of loans are recorded

for each aggregate loan group. All future years use the distribution of loan-group

characteristics (i.e., percentage of dollar volume found in each loan group) from the

FY2002 cohort of FHA insurance endorsements.

2. Within each quarter’s originations, interest rates are assigned by product type. Three

product types are used for coupon interest rate determination: fixed-rate 30-year,

fixed-rate 15-year, and adjustable rate mortgages. No graduated payment or graduated
FHA-BSSS, V. Final Forecast Assumptions and Subsidy Rate Calculations page 93

95ARM shares have been higher in years with significant refinance activity. Since the current version of the

FHA-BSSS does not predict refinance waves, the lower rate of ARM activity is assumed to exist in future years.

equity loans are assumed to originate in the forecast period. The VAR model outlined

in section II generates a fixed-rate 30-year mortgage rate directly. A 50 basis point

deduction is taken to arrive at a 15-year fixed-rate. ARM contract rates are the

prevailing one-year Treasury rate plus an index margin of 2.75 percent. This margin is

the most common used for FHA loans. Because teaser rate discounts have not been

permitted since 1998, the index-adjusted Treasury rate provides the best estimate of

rates on newly originated ARMs. 

3. The dollar volume of refinance loans is fixed at $50 billion in FY2003, and at $20

billion in FY2004. These correspond with FHA predictions mid-year 2002. All loangroup dollar volumes and numbers are adjusted to reflect differences between what

was calculated based on 2002 origination shares of refinance and purchase loans and

these new totals. The ARM share of total originations, both purchase and refinance, is

fixed at 4 percent, which is representative of ARM shares in recent years.95

4. Population-growth induced increases in mortgage volumes (and numbers) are set at 2

percent per year and the inflationary change in mortgage volumes is set to 4 percent.

The 4 percent inflation factor begins for purchase loans in FY2003 and for refinance

loans in FY2004.

The resulting origination volumes for future cohorts are not tied to either house price

appreciation or prepayment rates generated by the forecast simulations. They are, rather,
FHA-BSSS, V. Final Forecast Assumptions and Subsidy Rate Calculations page 94

illustrative of continuing current business into the future. 

SUBSIDY CALCULATIONS

Cash flows resulting from mortgage performance projections are grafted to historical cash

flows associated with each loan group to complete life-of-loan cash-flow series. Net

premium income is subtracted from net credit expense in each year, and the difference is

discounted to the year of loan origination (insurance endorsement) to compute subsidy

amounts. The final subsidy rate is the (present value) subsidy amount divided by dollar

amount of loans insured. The subsidy rate is then a function of the dollar mix of loan and

property types in each cohort, as well as economic conditions in the forecast simulations.

Discounting procedures were discussed in section II. Each historical origination year

cohort uses one set of interest rates and discount factors for all simulations, coming from

the year of loan origination. These rates are determined and maintained by OMB. The

underlying interest rates used to calculate discount factors for 1992-2003 are provided

here in Table 4. For all future cohorts, discount rates are specific to each simulated

economic scenario. Because the President’s budget is formulated in the first quarter of

each calendar year, the FHA-BSSS uses those interest rates to calculate discount rate

factors in each simulation. Thus, interest rates prevailing in 2003Q1 are used for the

FY2004 cohort discount factors, and 2004Q1 rates are used for the FY2005 cohort

discount factors.
FHA-BSSS, VI. Simulation Results page 95

VI. SIMULATION RESULTS FOR THE 1992-2007 COHORTS 

The FHA-BSSS model identified in Figure 1 and described in sections II-V generates a

large number of outputs. These outputs range from databases with cash-flow summaries

to tables of descriptive statistics for subsidy-rate and mortgage-termination-rate

outcomes, and to graphs of frequency distributions of subsidy rate outcomes and their

component parts. However, the ultimate objective of the FHA-BSSS is to improve budget

preparation, so the results described here focus on the distribution of subsidy rate

outcomes by budget-year cohort of loan guarantees. 

 The simulations merge historical data through FY 2002 with forecasts that start in FY

2003. All historical loan origination, termination, and cash-flow data are current through

2002Q3 (FY 2002). Economic data (house price growth, interest rates, inflation rates, and

unemployment rates) are current through 2002Q4. The simulations described here were

performed in March 2003.

Simulations at a given point in time provide budget re-estimates of outstanding cohorts,

initial estimates for new cohorts, and projections for future cohorts. All of these can be

summarized together in graphic and tabular form. Figure 4 maps subsidy rate means and

confidence bounds (1, 5, 95, and 99 percentiles) by cohort year. The exact values mapped

in Figure 4 are provided in Table 17. 
FHA-BSSS, VI. Simulation Results page 96

SUBSIDY RATE CONFIDENCE BOUNDS

Figure 4 is most helpful for highlighting the rate at which confidence bounds tighten with

the seasoning of individual cohorts. Cohorts from the early and mid-1990s have very tight

confidence bounds today, which indicates that any future changes in mortgage

termination patterns will have little effect on the measured value of these loans in the

federal budget. In contrast, as we move toward more recent and then to future cohorts,

projections of confidence bounds increase dramatically, indicating there still could be

sizable future revisions to current budget estimates. 

For recent cohorts, this increased variance of potential outcomes reflects uncertainty with

respect to economic conditions during the crucial early years of mortgage life. Default

and prepayment rates tend to peak in the three-to-seven-year time frame, and what those

peak rates will be depend on house price growth and interest rates during the formative

years of cohort existence. For future cohorts, the continued increase in confidence bounds

is in proportion to the increase in range of mortgage coupon rates and existing house price

conditions that could exist when those loans originate. 

Following the time-line of cohorts in Figure 4, one also sees that mean-value subsidy

rates approach zero in future years. Indeed the mean subsidy rate for FY2007 resembles

the low rates of the 1995-1997 cohorts. This movement toward zero reflects both the

increasing uncertainty in economic conditions and the movement of the means of the
FHA-BSSS, VI. Simulation Results page 97

distribution of each economic variable to CBO’s long-run average predictions, over a

five-year period. Today, interest rates are at historical lows and real (national) house price

growth is at an historically high level. Taken together, they portend relatively low

prepayments and low defaults for the 2002-2004 cohorts. As mean levels of the economic

variables in each quarter move toward their long-run averages (7.43 percent 30-year

mortgage rates and 1.2 percent annual, real, national house price growth), conditions will

be less favorable for new FHA guarantees. The implication of this movement is that

subsidy rate estimates for outstanding FHA single-family loan guarantees should not be

used as a basis for long-run policy regarding premiums and underwriting standards.

Subsidy Rate Estimates as a Basis for Policy Making

Economic conditions in 2003 are very favorable for new mortgage guarantees–interest

rates are low and house-price growth remains healthy throughout most of the nation. This

favorable climate means that simulation results for 2003 do not provide an appropriate

basis to set policy on insurance premiums and underwriting standards. Simulation results

for 2007 are more instructive for policy making because they embed a long-range view of

plausible initial economic conditions. The 1000 simulations run for 2007 allow for initial

house-price growth and interest rates to be high or low (See Figures 2 and 3) and to be

moving either up or down.

The low mean-value subsidy rate (-0.34 percent) and high probability of positive values
FHA-BSSS, VI. Simulation Results page 98

(30 percent) for 2007 suggest that the current premium structure is marginally sufficient

for long-run solvency of Mutual Mortgage Insurance Fund programs. Actuarial soundness

generally requires a larger margin of error, meaning a long-run mean subsidy rate farther

from zero and a correspondingly lower probability of losses. 

While there is no one measure of actuarial solvency, the term means that the underlying

(insurance) business enterprise has a limited probability of sustaining cumulative losses

on outstanding insurance contracts large enough to jeopardize its viability. It is generally

calculated without consideration of the value of new business in the future. For a private

firm, actuarial soundness means controlling loss exposure so that the probability of firm

insolvency and failure are kept within bounds acceptable to investors. 

FHA enjoys the full faith-and-credit of the U.S. government and so does not face

insolvency or failure, as do privately owned firms. For FHA the issue is then two-fold:

the frequency at which losses might be experienced, and the potential size of those losses.

Policy makers have time horizons of one, five, and ten years when formulating the federal

budget. Actuarial solvency for FHA then must be defined as net receipts outweighing net

outlays within those time horizons. 

The 30-percent probability of positive subsidy rates estimated for 2007 means that, going
FHA-BSSS, VI. Simulation Results page 99

96Or a six-year loss experience, as occurred in 1980-1985, happening every twenty years.

97FHA’s statutory capital cumulates over time so that the twenty-year capital ratio can be expected to be

around 1.5 percent. The actual capital ratio reported by FHA today is a summary measure of expected net receipts

and net outlays on all past and presently outstanding insurance policies; it does not begin with a zero balance at any

given point in time. Therefore, it is not a good measure of the solvency or actuarial soundness of FHA’s insurance

programs as they exist today. See USC 1711(f) for a description of how capital is measured for FHA. 

forward, FHA can expect to have net losses on three-out-of-ten years books-of-business.96

The -0.34 percent average expected long-term subsidy rate implies that, over a ten-year

period, net receipts will outweigh net outlays by an amount equal to about 3.4 percent of

an average year’s business volume. The size of FHA’s total outstanding portfolio

averages between four and five times the size of any one year’s business volume, so the

implied “capital” ratio achieved over a ten-year period, as defined in law for FHA, will be

somewhere around 0.75 percent.97 Yet, that result too is an average that has a frequency

distribution around it. It is based on having the average expected number of years with net

losses (outlays), and the size of those losses also being of average expected size.

FREQUENCY DISTRIBUTIONS AND MEAN-VALUE ESTIMATES

The mean-value, simulated, subsidy rate outcome for each cohort yields an unbiased

estimate of the true dollar subsidy amount to be revealed once all loans have terminated

and all program cash flows are known. Unbiasedness means the expected value of the

sum of all future revisions, given the information available at the time the initial

estimate/forecast is made, is zero. At the same time, the mean is subject to

misunderstanding because there is a better-than-even chance that revisions will be
FHA-BSSS, VI. Simulation Results page 100

downward rather than upward. 

To see this potential for misunderstanding, let us start with an illustration of the

distribution of outcomes for the FY2003 cohort of FHA guarantees, shown here in Figure

5. If one thinks of Figure 4 as the top-view of a probability mapping (like a contour map),

then Figure 5 is a side-view, cross-section, or cut-away of that mapping, taken at one

horizontal pointSthe year 2003. Figure 5 is a density plot, with relative frequencies of

events (subsidy rate outcomes) measured vertically. The vertical dashed line marks the

mean of the distribution (-1.85 percent).

The right skewness of the distribution implies that the mean is to the right of the median.

The median is the estimate for which there is equal probability that the actual outcome

will be either higher or lower: half of all potential outcomes are higher and half are lower.

If the mean is used as the initial budget estimate, there is a better than even chance of

future downward revisions, as actual performance becomes known. In the distribution for

FY2003, the mean subsidy rate is at the 63rd percentile of outcomes, meaning there is a

63 percent probability that revisions will be downward and just a 37 percent probability

that revisions will be upward. 

If policy makers see a preponderance of downward revisions, they could mistakenly

conclude that subsidy rate estimates used in budget preparation are too conservative, so
FHA-BSSS, VI. Simulation Results page 101

that there is more room for lowering premium charges or loosening underwriting

standards. The mean estimate may appear overly conservative in the short run (underpredicting actual budget receipts from FHA), but that is only because it makes allowance

for the chance of large losses in future years. Choosing the mean rather than the median

then provides something like a self-insurance premium against the potential for bad

outcomes. Unlike the median, the mean considers the actual size of those bad outcomes

in addition to their relative frequency. 

This additional factor in mean-value calculations becomes meaningful when annual reestimates are made. Re-estimates of subsidy rates are made for all outstanding cohorts of

guarantees, and the netting of resulting adjustments are either passed on budget as new

receipts (net downward dollar adjustment to subsidies) or require new budget outlays to

provide for presently unfunded, but now expected, claim liabilities (net upward dollar

adjustment to subsidies). The calculation of the dollar budget effect involves multiplying

changes to subsidy rate estimates by dollars of loan originations. Even if the number of

cohorts with downward revisions to subsidy rates is greater than the number of cohorts

with upward revisions, it will likely be that the upward revisions are of a larger size and

thus will balance the net budget effects of the re-estimates back toward zero. 

Such balancing may not always occur in one set of (annual) re-estimates, but will more

likely occur over time. That is because both good and bad economic surprises will affect
FHA-BSSS, VI. Simulation Results page 102

all outstanding cohorts. Good economic surprises could mean a net of downward subsidy

rate re-estimates for outstanding cohorts, while bad economic surprises could mean a net

of upward re-estimates. The point of using the mean-value estimate (rather than the

median) when making re-estimate adjustments is that when bad surprises happen, they

will tend to be of larger size than are the good surprises when they occur, and thus budget

effects of re-estimates will tend to balance out over time even though bad surprises

happen less often than good surprises. 

Even then, the mean value estimate may still appear to be overly conservative over a

period of several years. Because by using mean-value estimates there will be more good

surprises than bad surprises, upward subsidy rate re-estimates may not fully offset

downward subsidy rate re-estimates over a five or ten year period. If that turns out to be

the case, it is because the economic forecasts used in the FHA-BSSS to generate the

distribution of subsidy rate outcomes include extremely bad events, meaning national

recessions and depressions (see section II). Because they are small probability events, by

definition, such worst-case outcomes may not appear for ten, twenty, thirty, or more

years. In current subsidy rate forecasts, those bad events serve to help pull the mean away

from the median. If they do not occur for many years, then net annual re-estimates over

any ten year period could provide more new budget receipts than new budget outlays. 

While it is valuable to include worst-case events to understand the shape of the entire
FHA-BSSS, VI. Simulation Results page 103

98Mean outcomes are reported for all years, 1992-2007, in Table 17.

frequency distribution of outcomes, it would also be a reasonable policy position to

ignore outlier events when making decisions on program design. Such a position would

lead to using the median- rather than the mean-value estimate when deciding on such

policy options as premiums and underwriting standards. Choice among these options can

be greatly influenced by the measured budget impact of any changes. Given the twin but

competing goals for FHA's single family insurance programs of encouraging affordable

homeownership opportunities while maintaining actuarial soundness, policy makers may

want to focus on median in addition to mean outcomes. 

While the distinction between mean and median can be important when budgets are

formulated, it disappears fairly quickly as loan cohorts age. At present, mean and median

predictions are virtually identical for the 1992-2001 cohorts. The differences for 2002-

2007 are:98

Cohort Year Mean Subsidy

Rate Estimate

Median Subsidy

Rate Estimate

2002 -1.73 -1.83

2003 -1.85 -2.07

2004 -1.53 -1.81

2005 -1.05 -1.38

2006 -0.65 -1.18

2007 -0.34 -0.94
FHA-BSSS, VI. Simulation Results page 104

99Office of Management and Budget, FY2004 Federal Credit Supplement, Table 8.

BIASES IN CURRENT FHA SUBSIDY RATE ESTIMATES

While mean and median subsidy rate outcomes are identical for pre-2002 cohorts, that

does not mean that budget estimates now in use for those years are accurate. As

mentioned in section I, current subsidy rates reported by FHA and published by OMB are

not based on the type of analysis used in the FHA-BSSS.99 While FHA uses a similar

model of mortgage default and prepayment rates, it projects loan terminations and cash

flows over one, smooth economic path. The result is that FHA consistently predicts

subsidy rates far to the left even of median outcomes and thus consistently overstates

expected net budget receipts from its loan guarantees. The annual re-estimation process

only produces a slow convergence to mean/median expectations because it uses the same

type of economic forecasts as used in initial budget-year estimates.

Other reasons for a downward bias in subsidy rate estimates currently used in FHA

budget formulation include using fixed factors for rates of use of loss mitigation

(foreclosure avoidance tools) and foreclosure losses. In the FHA-BSSS, these important

elements of default costs are dynamic with respect to mortgage characteristics and

economic conditions (see section IV).

FREQUENCY DISTRIBUTIONS OF SUBSIDY RATE COMPONENTS
FHA-BSSS, VI. Simulation Results page 105

100Figure 7 does not show the last few default cost component observations, which extend to +10.26

percent.

101To avoid loss of detail, the far right extreme of the tail of the subsidy rate distribution is not shown in

Figure 5. In 3 of 1000 outcomes, the subsidy rate exceeds the 4 percent maximum rate shown in Figure 5.

Subsidy rates are simply the sum of prepayment revenue and default cost component

rates. Though these two components are interdependent, meaning that default and

prepayment terminations are substitute outcomes for borrowers, it can be instructive to

look at their separate frequency distributions. The right-skewness of the subsidy rate

distribution comes principally from skewness of the default cost component, which itself

reflects the magnitude of potential bad outcomes. (Each subsidy rate is just the sum of

premium revenue and default cost components.) In contrast, the premium-revenue

component is very symmetrical. Distributions of these components for FY2003 are shown

in Figures 6 (premium revenue) and 7 (default cost). Following budget accounting rules,

default costs are positive values while premium revenues are negative. 

Not only is the premium revenue distribution more symmetrical, but it has a smaller

variance as well. The distribution of premium outcomes falls within a 2.35 percentage

point range, while the range of default cost outcomes is over 10 percentage points.100

When combined, the entire range of net subsidy rate outcomes in Figure 5 is from -3.8 to

+7.6, or over 11 percentage points.101

VALUE AT RISK
FHA-BSSS, VI. Simulation Results page 106

Private investors use various metrics to define risk of loss on a given investment. One

common measure used for credit risk on lending positions is value-at-risk (VaR). It is a

measure of the dollars that could be lost were a given bad event to occur within a defined

period of time. That event is defined by a critical value along the frequency distribution of

outcomes, where that critical value identifies a risk-tolerance level. For example, given

the long-run nature of FHA loan guarantees, it would be reasonable to start with the

critical value for the 99th percentile event. That value is the subsidy rate for which only 1

percent of possible outcomes for one year’s book-of-business are worse. The dollar

amount of outlays implied by that subsidy rate would be the implied capital requirement

for that percentile-rated risk tolerance. Holding such capital reserves would assure the

solvency of the enterprise so long as events worse than that did not happen. Capitalizing

against all possible eventualities is not practical, nor is it generally profitable, and so VaR

critical values are always something less than 100%.. 

The 99th percentile subsidy rate for 2003 is 2.07 percent. If FHA could and did hold

reserves of 2.07 times its guarantee volume then taxpayers would be protected from

losses with a 99 percent confidence level. There would be only a 1 percent chance that the

outcome of these guarantees could be so bad as to require net budget outlays to enable

FHA to fulfill its obligations to its lender partners, given current contractual terms for

premium payments and refunds. However, 2003 is a very good year for mortgage

insurance originations. If we go from the 2003 to the 2007 cohort, and thus examine a
FHA-BSSS, VI. Simulation Results page 107

long-run expectation of FHA outcomes under a wide range of initial economic

conditions, the VaR at the 99th percentile is nearly 7 percent.
Figures, Boxes, and Tables page 108

Figure 1. Structure of the FHA Budget Subsidy Simulation System
Figures, Boxes, and Tables page 109

Figure 2. Mortgage Interest Rates, Historical Series and Forecast Confidence Bounds

(percent)

Figure 3. House Price Growth Rates, Historical and Forecast Confidence Bounds

 (percent, annual rate)
Figures, Boxes, and Tables page 110

Figure 4. Confidence Bounds for Subsidy Rate Estimates, by Cohort
Figures, Boxes, and Tables page 111

Figure 5. Frequency Distribution of Subsidy Rate Estimate for FY2003 Loan 

Guarantees (percentage)
Figures, Boxes, and Tables page 112

Figure 6. Frequency Distribution of Premium Revenue Component of 

Subsidy Rate Outcomes for FY2003 (percent)
Figures, Boxes, and Tables page 113

Figure 7. Frequency Distribution of the Default Cost Component of 

Subsidy Rate Outcomes for FY2003 (percent)
Figures, Boxes, and Tables page 114

Box 1. Transformation of Inflation Variable for Use in the Vector Autoregression Model

Economic variables used in the vector autoregression (VAR) model (see Table 1) have larger variances

at higher levels. To achieve uniform variances in the regression, four of the five variables are log

transformed. Using a log transformation also eliminates negative values from the forecasts, which is too

restrictive for inflation. Thus, a new transformation is derived here to provide a regression variable with

a constant variance and which allows for negative values of inflation forecasts. The relationship

between actual inflation rates and transformed values is shown in the following figure:

Derivation of the transformation function begins with a regression of the variance of the quarterly

inflation time-series. A second-order regression of squared deviations of actual from expected inflation

is performed on expected inflation: 

(p - p

e

)

2

 = a + bpe + g (pe)

2 + e ,

where p is actual inflation (annual rate), p

e

 is expected inflation, and e is the residual (error) term.

Expected inflation is a weighted moving average of past inflation rates (see description in Table A.2). 

Generalized least squares estimates of the three regression parameters are:

= 0.3355 = -0.2780 = 0.0726

If we define s 

2

(p|pe) = E[(p - p

e

)

2

] as the estimated variance of possible inflation rates in each period,

conditional on recent experience as measured in p

e

, the regression measures how that variance changes

with the underlying level of inflation, indicated by p

e

.

The estimated regression equation has a minimum value at percent, and the standard

deviation of inflation at that minimum value is, percent. The variance is

held constant for values of p < p0, rather than allowing it to rise. Thus, the general equation for the

variance of inflation is: 

(Continued)
Figures, Boxes, and Tables page 115

Box 1 (continued)

The goal of the transformation process is to create a new variable, B’ , with a uniform variance for use in

the VAR model. Resultingly, the transformation function, f(B) = p’, should exhibit the property that

, where C is an arbitrary constant. f(B) would then neutralize changes in the variance

of p’ otherwise caused by changes in the variance of the underlying level of p.

Rearranging terms yields , which implies, . A closedform solution for f(p) is: , 

where D is the constant of integration. Values of C and D are chosen so that f(B0) = B0 and f’(B0) = 1,

which assures that the implied, complete transformation function across all values of inflation will be

continuous at p = p0 = 1.9. Thus, C = s0 and D=p0.

The final transformation function used to create the inflation variable in the VAR model is then:

The inverse transformation function needed to obtain inflation values from predictions of the VAR

(transformed) inflation equation is:

While values of p < p0 do not appear in the regression sample, they may appear in the VAR predictions,

especially after stochastic shocks are added (that is, p’ < p0 = 1.9). Applying no inverse transformation

to predictions in this range creates the one-to-one correspondence between p’ and p for values at or

below 1.9. The lack of any inverse transformation on low and negative values of p’ also imposes a

constant standard deviation on inflation forecasts below 1.9 percent at s0 = 0.29 percent. The standard

deviation of inflation forecasts above 1.9 percent will rise with p

e

, as indicated by the regression

equation shown near the top of this Box. 
Figures, Boxes, and Tables page 116

Box 2. Deriving Discount Factors By Interpolating Forward Rates

Discount factors used to create present-values of cash flows are specified as 1/(1+r)

t

, where r

is an appropriate interest rate and t is the number of (semi-annual) time periods. FHA cash

flows are accumulated annually, and discounted to year of insurance endorsement (loan

origination) as if all cash flows occurred at mid-year. Thus, discount rates are needed for 6,

18, 30, 42 months, and so on. The VAR equations (Table 1) provide 3-month and 10-year

rates, and the historical slope relationships (Table 5) provide rates for 6 months, 1 year, 2

years, 5 years, and 30 years. The intermediate points are arrived at by first calculating

successive 6-month yields that complete each missing interval. These yields are called

forward rates. Refer to these as f

1

, f

2

, and so on, each representing forward rates over

successive six month intervals. We start with f

1

 = r6m, where r6m is derived in Table 5. The

resulting discount factor for interval one is: d1 = . Successive discount factors

are products of the individual forward-rate present-value factors. Because constant maturity

Treasury (CMT) rates give yields for securities sold at par, the one-year yield (r1) must be

equivalent to what earnings on two successive 6-month investments, and f2 can be solved for

in the following equation:

1 = d2 + (r1/2)(d1 + d2) , where 

This equation says the par yield (1=100%) must equal the present value of the principal

payment after one year ($1 discounted with d2), plus the sum of the discounted value of two

semi-annual interest payments at the one-year CMT yield, r1. With r1 and d1 now known, we

can solve for the second period discount factor, d2 as:

and the forward rate for the second 6-month time period is: 

Beyond the second interval, multiple forward rates must be solved for simultaneously, each to

complete the next interval along the yield curve. For the one-to-two-year interval, there are

two forward rates, f

3

 and f

4

. Assuming forward rates follow a logarithmic relationship, we

have: .

This equation has two unknowns, f

4

 and f

3

, and is solved together with the market yield

equivalence condition:

, where 

(continued)
Figures, Boxes, and Tables page 117

Box 2 (continued)

An interval-bisection grid search is used to solve for values of f

3

 and f

4

 that best fit these two

equations. The same procedure is used to compute forward rates between two-and-five years,

five-and-ten, and ten-and-thirty. The series of forward rates is then used to compute spot rates

that will be used for discounting cash flows. For example, the 18-month spot rate, , is

computed from the first three forward rates: . 

And the resulting discount factor for 18-month cash flows is: 
Figures, Boxes, and Tables page 118

Table 1. Vector Autoregression Equations 

Mortgage Rates Unemployment Inflation Yield Curve Slope

Spread of Mortgage to

Treasury Rate

Explanatory Variables

Coefficient

Estimate

Standard

Error

Coefficient

Estimate

Standard

Error

Coefficient

Estimate

Standard

Error

Coefficient

Estimate

Standard

Error

Coefficient

Estimate

Standard

Error

Constant 0.0720*0.0492 0.0329 0.0468 0.8062** 0.3956 -0.0349 0.0992 0.0679* 0.0465

MORTGAGE_RATE{1} 1.1341*** 0.0946 0.1076 0.0900 0.6388 0.7603 -0.1163 0.1906 0.0555 0.0894

MORTGAGE_RATE{2} -0.1864*0.1377 -0.0738 0.1310 0.0927 1.1066 0.0606 0.2774 -0.1457 0.1302

MORTGAGE_RATE{3} -0.0370 0.0840 -0.0042 0.0798 -0.6750 0.6747 -0.0430 0.1692 0.1278*0.0794

US_UNEMP_RATE{1} 0.0759 0.1103 1.3698*** 0.1049 -2.3991*** 0.8862 0.6353*** 0.2222 0.0965 0.1042

US_UNEMP_RATE{2} -0.2587*0.1780 -0.3629** 0.1693 3.7319*** 1.4305 -0.7494*** 0.3586 -0.0556 0.1682

US_UNEMP_RATE{3} 0.2287** 0.1056 -0.0815 0.1005 -1.2539*0.8488 0.2310 0.2128 -0.0631 0.0998

INFLATION_RATE{1} 0.0112 0.0111 0.0143*0.0106 0.3435*** 0.0892 -0.0344*0.0224 0.0089 0.0105

INFLATION_RATE{2} 0.0067 0.0088 0.0012 0.0084 0.6794*** 0.0706 0.0075 0.0177 -0.0023 0.0083

INFLATION_RATE{3} -0.0032 0.0112 -0.0018 0.0107 -0.2203*** 0.0903 0.0311* 0.0226 -0.0241** 0.0106

YLD_CURVE_SLOPE{1} -0.0502 0.0479 0.0082 0.0456 0.4151 0.3850 1.1383*** 0.0965 -0.0568 0.0453

YLD_CURVE_SLOPE{2} -0.0494 0.0737 0.0347 0.0700 -0.8023*0.5918 -0.2685** 0.1484 0.0088 0.0696

YLD_CURVE_SLOPE{3} 0.0256 0.0542 -0.0663* 0.0515 -0.1698 0.4351 -0.0389 0.1091 0.0511 0.0512

SPREAD_TO_TREAS{1} -0.6220*** 0.1010 0.0735 0.0960 -1.7646** 0.8114 0.5650*** 0.2034 0.6857*** 0.0954

SPREAD_TO_TREAS{2} 0.6474*** 0.1311 -0.1103 0.1246 0.5877 1.0531 -0.8242*** 0.2640 -0.0297 0.1239

SPREAD_TO_TREAS{3} 0.0522 0.1138 0.0066 0.1082 -0.3371 0.9144 0.6609*** 0.2292 0.0683 0.1075

SUMMARY STATISTICS

Time Period 1968:1-2001:4

Adjusted Coefficient of

Determination (R-bar2 )

97.17 97.64 80.78 86.69 57.38

Standard error of regressions 0.0382 0.0363 0.3066 0.0769 0.0361

LEGEND

MORTGAGE_RATE 30-year fixed-rate mortgage coupon rate, Freddie Mac series, percent, log form

US_UNEMP_RATE Unemployment rate, civilian, noninstitutionalized population, age 16 and older, percent, log form.

INFLATION_RATE Quarterly change in the consumer price index, urban consumer, percent, transformed variable (see Box 1).

YLD_CURVE_SLOPE Ratio of 10-year Treasury yield to 3 month Treasury bill rate, log form

SPREAD_TO_TREAS Ratio of 30-year mortgage rate to10-year constant maturity Treasury yield, log form

{ i} Lagged values, i = {1,2,3} quarters

Statistical Significance in one-sided tests *** .01 level, ** .05 level, * .10 level
Figures, Boxes, and Tables page 119

Table 2. National House-Price-Growth-Rate Regression 

Variable

Name

Coefficient

Estimate

Standard

Error Description

Constant 1.2405*** 0.3816 The model is estimated with seasonal effect dummies. The results are

averaged to produce the intercept effect shown here.

)_EXPECTED_INFLATION -0.6899*** 0.1914 Expected inflation, forecast by CBO, change from previous quarter,

multiplied by 4 to get annual rate

)_REAL_MORT_RATE -0.6694*** 0.0994 Real mortgage rate, Freddie Mac commitment rate, 30-year mortgages,

less expected inflation, change from previous quarter, multiplied by 4

to match house price-growth which appears on the left-hand-side of

the equation in an annualized rate.

)_REAL_MORT_RATE{1} -0.2879*** 0.0778 lagged difference in the real mortgage rate, multiplied by 4

)_US_UNEMP_RATE -1.4522*** 0.3712 US civilian, noninstitutionalized population, unemployment rate, age

16 and older, change from value four quarters previous

SUMMARY STATISTICS

Time Period 1975Q2 - 2001Q1

Standard error of the regression 3.3740

Statistical significance in one-sided

tests *** .01 level, ** .05 level, *

 .10 level

Notes: The house price growth rate is four times the quarterly growth rate found in the OFHEO U.S. HPI series. All variables in regression

are in percent form. See Table 1 for more details on variables used to create the difference-variables in this regression. Expected

inflation comes from the CBO Philips Curve equation. It is a weighted average of the previous 12 quarters inflation (change in CPI),

where the weights (starting with the most recent quarter) are: 0.4014, 0.2425, 0.1296, 0.0555, 0.0136, -0.0032, -0.0019 ,0.0108, 

0.0277, 0.0420, 0.0468, 0.0351. All inputs for forecast simulations are generated using the vector autoregression model found in Table

1, along with the inverse transformation of inflation rates described in Box 1.
Figures, Boxes, and Tables page 120

Table 3. Parameter Values for Stochastic Cumulative, Regional House-Price-Path 

Deviations from Cumulative, National House-Price Growth

Parameter Value Conditions

Ornstein-Uhlenbeck Process: h(t, k) = h(t, k-1)*(1- a/92) + N(0, 2/92)

 h(0,0) = N(0,1/a) and h(t,0) = h(t-1,92)

Realized values of growth factors: g(t) = h(t, 92)*e, 

Definitions: t = quarters, k= daily increments (k = {1,...,92})

N( ) = normal (Gaussian) random number

a for Divisions .08 The MSA value is the most common value for

describing 1980-2000 data movements. Division

value increases impact of past values, smoothing the

series.

a for MSAs .10

e for Divisions .001 The MSA value is the most common for describing

1980-2000 data movements. Division value is set

equal to MSA and differences in volatilities are a

function of s, below.

e for MSAs .001

Cycle Phase add-on: 

s for Divisions .05*(1+ N(0, .01)) Scale amplitude of cosine function, which cycles

between -1 and +1. Scale factors are drawn as

normal random variates with means of .05 or .09,

and standard errors of one-tenth of those amounts.

Actual realized values of s are unique for each

simulation iteration and locality.

s for MSAs .09*(1+ N(0, .01))

W for Divisions .15*(1+ N(0, .01)) Average length of cycle is 10 years, with random

deviations having standard error of 10 percent (1.5

W for MSAs .15*(1+ N(0, .01)) years)

r U(0,p/2) + p

Uniform random variate

over the (0,p/2) interval,

and shifted by p. The

phase parameter, r, is

randomly chosen at the

start of each simulation

(iteration), within the

segments defined here for

each region. 

when 

 is average annual house price growth in

region/locality r, over the most recent five year

period (1997Q2- 2002Q1). is the same for

the US.

U(0,p/2) + 0.50*p when 

U(0,p/2) + 0.75*p when 

(continued)
Figures, Boxes, and Tables page 121

Table 3. Parameter Values for Stochastic Cumulative, Regional House-Price-Path 

Deviations from Cumulative, National House-Price Growth

(continued)

Parameter Value Conditions

r U(0,p/2) + 1.25*p when 

U(0,p/2) + 1.50*p when 

U(0,p/2) + 1.75*p when 
Figures, Boxes, and Tables page 122

Table 4. Treasury Rates Used in Credit Subsidy Calculations

Fiscal Year 1 year or less 

More than 1

year and less

than 5 years

5 years or more,

and less than 10

years

10 years or more, and

less than 20 years 20 years or more

1992 4.06 5.52 6.93 7.59 7.59 

1993 3.25 4.58 5.90 6.65 6.65 

1994 4.06 5.36 6.21 6.77 6.77 

1995 5.99 6.78 7.11 7.26 7.39 

1996 5.45 5.96 6.33 6.64 6.77 

1997 5.50 6.19 6.51 6.77 6.89 

1998 5.34 5.57 5.68 5.85 5.98 

1999 4.76 5.11 5.36 5.67 5.81 

2000 6.04 6.38 6.39 6.40 6.36 

2001 4.52 4.76 5.22 5.58 5.75 

2002 1.94 3.37 4.63 5.32 5.62

Source: U.S. Executive Office of the President, Office of Management and Budget (OMB).

Notes:

1. All rates are stated as simple annual rates, which differ from bond equivalent rates

and bank discount rates. The 10-20 year and the 20+ year intervals were a single

interval in the 1992-1994 Budgets. The separate intervals were effective with the

1995 Budget.

2. These actual fiscal-year averages derived from the prevailing market yields during

the fiscal year, excluding the last five business days, on outstanding fixed-rate

Treasury securities, categorized on the basis of their remaining maturity. The rates

for 2002 are actual fiscal year averages derived from the prevailing market yields

during the fiscal year, excluding the last ten business days, on outstanding fixed-rate

Treasury securities, categorized on the basis of their remaining maturity.

3. The rates above are used to calculate credit subsidy cost, to reestimate credit subsidy

cost, and to calculate the interest income and expense of the financing accounts. They

can be used in the OMB credit subsidy calculator without modification.
Figures, Boxes, and Tables page 123

Table 5. Computing Major Points on the Constant Maturity Treasury Yield Curve From 

the 3-month and 10-year Rates and Regression Equations

Yield Curve Point/Rate Equation Notes

3-month, bond equivalent

yield

The 3-month yield must first be

converted to a bond-equivalent form

(365 day year, rather than 360 days)

to compute longer-term yields.

Bound on the 10-year

yield

r10 = Max(r10, 0.0001) Assures that any negative or near

zero interest rates calculated in the

VAR model (Table A.1) are

converted to positive numbers.

Bound on the 3-month

yield

Assures nonnegative results and that

yield curves close to zero remain

upward sloping.

6-month yield 1.036 is the estimated coefficient

found by regressing the 6-month

yield on the 3-month yield.

1-year yield 1.054 is the estimated coefficient

found by regressing the 1-year yield

on the 3-month yield.

2-year yield 1.091 is the estimated coefficient

found by regressing the 2-year yield

on the 3-month yield.

5-year yield 0.83 1 is the estimated coefficient

found by regressing the 5-year yield

on the 10-year yield.

30-year yield r30 = 1.029*r10 1.029 is the estimated coefficient

found by regressing the 30-year

yield on the 10-year yield.

Source: U.S. Congressional Budget Office
Figures, Boxes, and Tables page 124

Table 6. Explanatory Variables Used in Mortgage Performance Regressions 

Variable Names class

Category

Values Descriptions

Loan Group Characteristics

LTV_CLASS 1 up to 80% Original loan-to-value ratio (LTV), where value is the lesser

of purchase price or appraised value. Streamline refinance

loans rarely have appraisals, and so LTVs are computed as

either 97 (single-family) or 100 (condominiums) plus the upfront premium. In the few cases where value is missing for

purchase mortgages, the same values are used, plus the 100

LTV is assigned to new construction. 

2 81-90%

3 91-95%

4 96-100%

5 over 100%

PRICE_CLASS 1 under 50% Based on ratio of house price to area (MSA) median house

2 51-100% price.

3 over 100%

PROP_AGE_CLASS 1 under 1 year Age of house at time of loan origination.

2 1-15 years

3 16-30 years

4 over 30 years

COHORT_YEAR 1975-2000 Fiscal year of loan insurance endorsement.

INVESTOR NO/YES Investor properties are identified in FHA data bases since

1992. Before that time, investor loans are identified by LTVs

between 84.5 and 85.5 because investors were required to

have a minimum 15 percent downpayment. This criterion

was first used in the original (1989) actuarial study of the

FHA MMIF portfolio, performed by PriceWaterhouse (now

PriceWaterhouseCoopers). Beyond this, if a loan cannot be

identified as an owner-occupied property using the FHA data

fields, then it is assigned to the investor class if the LTV is

under 90 percent.

AGE_CLASS 1 1-2 years Age of mortgage, from quarter of origination.

2 3-4 years

3 5-6 years

4 7-8 years

5 9-10 years

6 11-15 years

7 16+ years

Economic Conditions

CYCLE_STAGE 1 initial recession House price cycles are defined as periods in which prices

decline at least 3 percent, using a five-quarter movingaverage of the OFHEO HPI. The moving average removes

the significant quarterly volatility in the HPI series. Stage 1

includes all observation quarters during the first half of a

price decline, Stage 2 covers the time period of the second

half of the price decline, and Stage 3 is the time during which

prices recover out of Stage 2. Stage transitions are based on

movements in the smoothed HPI, not time. Cyclical peaks

and troughs are identified by comparing quarterly (smoothed)

HPI values with five quarters of leads and lags. Cycles are

based on HPI paths from quarter of loan origination, by MSA

and Census Division. Observations not in one of these three

stages are labeled as stage 4 (out-of-cycle).

(continued)
Figures, Boxes, and Tables page 125

Table 6. Explanatory Variables Used in Mortgage Performance Regressions 

(continued)

Variable Names class

Category

Values Description

CYCLE_STAGE 2 latter recession

3 early expansion

4 out of cycle

US_UNEMP_RATE National unemployment rate, civilian, non-institutionalized

population, age 16 and over (in percent).

CUR_MKT_RATE Median contract rate on FHA originations in the current

quarter, decimal form. Used only in ARM regressions to

model incentives to prepay/refinance into fixed-rate products.

YLD_SLOPE_CAT 1 up to 0% Ten-year constant maturity Treasury (zero coupon bond)

2 0.1 - 1% yield, minus the one-year constant maturity Treasury yield.

3 1.1 - 2%

4 2.1 - 3.0%

5 over 3.0%

MKT_SHIFT OFF/ON Observations starting 1993Q1.

Financial Incentives

NEG_EQ_CLASS 1 over 50% Probability of negative equity for properties in a loan group,

each quarter. It is the probability that an individual property

could have value erosion large enough to offset both the

original downpayment and additional loan amortization, over

time. Values are computed as integrals of the standard

normal density function, up to the limit defined by: the log of

the ratio of current loan balance to original-price-times-HPI,

divided by the standard deviation of log(HPI). The log(HPI)

is cumulative house price growth implied by the HPI, and its

standard deviation is computed from so-called volatility

parameters published by OFHEO in its quarterly HPI Report.

The ratio described here is a standard-normal random variate,

which follows from an assumption that house price growth

rates following a random walk.

2 26-50%

3 11-25%

4 2-10%

5 up to 1%

POS_EQ_CLASS 1 over 75% Probability of having at least 20% equity in a property, each

quarter. It is the probability that an individual property could

have experienced enough price appreciation, given the initial

downpayment and on-going loan amortization, to reach a

20% equity threshold. It is analogous to NEG_EQ_CLASS,

only measured as the density mass in the upper tail of the

cumulative house-price-growth distribution, and where the

limit of integration is: log of, HPI divided by loan-balanceover-original-house-price plus 0.20, divided by the standard

deviation of log(HPI).

2 51 - 75%

3 26 - 50%

4 11 - 25%

5 0 - 10%

SPREAD_CLASS 1 under -.04 Current rate on mortgages minus the mortgage coupon rate,

in decimal. Current rates are median values from FHA loan

originations in each calendar quarter. Separate median values

are found for (30-year) fixed-rate and adjustable-rate

mortgages. An implied median value for 15-year loans is

computed as the 30-year rate less 0.0050 (one-half percent).

2 -.04 - -.03

3 -.029 - -.02

4 -.019 - -.015

5 -.014 - -.010

6 -.090 - -.005

7 -.005 - 0.00

8 .001 - .005

9 .005 - .010

10 .011 - .015

(continued)
Figures, Boxes, and Tables page 126

Table 6. Explanatory Variables Used in Mortgage Performance Regressions 

(continued)

Variable Names class

Category

Values Description

SPREAD_CLASS 11 .016 - .020

12 .021 - .030

13 .031 - .040

14 over .04

BURN_SUM Number of past quarters of loan life in which

SPREAD_CLASS = {1,2,3}.

NEW_REFI OFF/ON Turned on when SPREAD_CLASS={1,2,3,4} in current

quarter, but had higher values in lagged quarters 3-7.

PMT_ADJ_CLASS 1 under -5% Percent change in monthly payments of principal and interest

for adjustable-rate mortgages, from previous year. Once

PMT_ADJ_CLASS is established, that value remains for the

entire 4-quarter rate period. Adjustments are based on interest

rates prevailing in the quarter preceding the adjustment quarter

(the fully indexed market rate), and annual and lifetime

interest rate change caps on FHA ARMs (1% and 5%,

respectively). 

2 -4 - 0%

3 1 - 5%

4 6 - 10%

5 over 10%
Figures, Boxes, and Tables page 127

Table 7. ANOVA Tests of the Statistical Significance and Relative Influence of Explanatory 

Variables in Default-Rate Regression Analysis

Wald 

P2

 Statistic Values, by Regression Equation

Variable DoF

Fixed Rate,

Purchase

Adjustable Rate, 

Purchase

Fixed Rate, 

Refinance

Adjustable

Rate, Refinance

AGE_CLASS 6 6958 793 1017 180

COHORT_YEAR 25 16631 1105 970 108

INVESTOR 1 890 - 178 -

MKT_SHIFT 1 925 9 117 2

LTV_CLASS 3 535 117 373 1

PRICE_CLASS 2 4705 1660 52 121

PROP_AGE_CLASS 3 575 315 147 100

NEG_EQ_CLASS 4 29928 7598 7251 1412

CYCLE_STAGE 3 2745 81 146 -

US_UNEMP_RATE 1 3302 910 1333 83

SPREAD_CLASS 13 18794 2611 2390 120

PMT_ADJ_CLASS 4 - 279 - 103

CUR_MKT_RATE 1 - 185 - 1

BURN_SUM 1 864 70 14 21

NEW_REFI_OPP 1 346 50 15 1

Notes: These are ANOVA Type III, partial sums of squares tests. Bold faced values are NOT

statistically significant at the 0.01 level. DoF represents degrees-of-freedom in the test. It is

the number of free parameters that are estimated for each variable. For classification variables,

this is the number of classes/categories minus one. For single-class (on/off or dummy) and

continuous variables, DoF equals one. Ranking of Wald Statistic values within each regression

equation indicates ranking of importance of each variable in measuring differences in default

rate outcomes. See Table 6 for a description of variables. 
Figures, Boxes, and Tables page 128

Table 8. ANOVA Tests of the Statistical Significance and Relative Influence of Explanatory

Variables in Prepaym ent-Rate Regression Analysis

Wald 

P2

 -Squared Statistic Values, by regression equation

Variable DoF

Fixed Rate,

Purchase

Adjustable

Rate, 

Purchase

Fixed Rate, 

Refinance

Adjustable Rate,

Refinance

AGE_CLASS 6 53000 2677 7236 462

COHORT_YEAR 25 94455 5219 2799 1486

INVESTOR 1 170 - 8 -

MKT_SHIFT 1 31114 360 7186 0

LTV_CLASS 3 1906 734 1615 113

PRICE_CLASS 2 48982 3952 4345 531

PROP_AGE_CLASS 3 2104 247 169 44

POS_EQ_CLASS 4 55365 17793 14274 641

CYCLE_STAGE 3 3241 89 60

US_UNEMP_RATE 1 2890 2715 344 59

SPREAD_CLASS 13 353507 17996 42135 943

YLD_SLOPE_CLASS 4 27579 3800 4714 537

PMT_ADJ_CLASS 4 - 2030 - 171

CUR_MKT_RATE 1 - 9232 - 723

BURN_SUM 1 27002 2590 730 55

NEW_REFI_OPP 1 8817 214 119 3

Notes: These are ANOVA Type III, partial sums of squares tests. Bold faced values are NOT

statistically significant at the 0.01 level. DoF represents degrees-of-freedom in the test. It is

the number of free parameters that are estimated for each variable. For classification variables,

this is the number of classes/categories minus one. For single-class (on/off or dummy) and

continuous variables, DoF equals one. Ranking of Wald Statistic values within each regression

equation indicates ranking of importance of each variable in measuring differences in

prepayment rate outcomes. See Table 6 for a description of variables. 
Figures, Boxes, and Tables page 129

Table 9. Results of Default- Rate Logistic Regressions

Coefficient Estimates By Loan Type

Variable Names class

Category

Values

Fixed Rate,

Purchase

Adjustable

Rate, 

Purchase 

Fixed Rate,

Refinance

Adjustable

Rate,

Refinance

CONSTANT 1 -4.0295*** -5.0167*** -3.9235*** -3.7708***

Loan Group Characteristics

LTV_CLASS 1 up to 80% 0.0450 -0.2235 0.1821 -0.1689

2 81-90% 0.0450**

* -0.2235*0.1821*** -0.1689

3 91-95% -0.1601*** -0.2501*** -0.0255 0.1840

4 96-100% 0.0477*** 0.2273*** 0.0424*** 0.0249

5 over 100% 0.0674*** 0.2463*** -0.1990*** -0.0400

PRICE_CLASS 1 under 50% 0.2368*** 0.2779*** 0.0874*** 0.3660***

2 51-100% 0.0100*** 0.0485*** -0.0143*0.0322

3 over 100% -0.2468*** -0.3264*** -0.0731*** -0.3982***

PROP_AGE_CLAS

S

1 under 1 year 0.1138***

0.1443*** 0.2195*** 0.3868***

2 1-15 years -0.0388*** -0.0184** -0.0006 0.0750

3 16-30 years -0.0550*** -0.0968*** -0.1793*** -0.2694***

4 over 30 years -0.0200 -0.0291** -0.0396 -0.1924***

COHORT_YEAR 1975 -0.1718*** -2.7644

1976 -0.289*** -2.8447

1977 -0.3044*** 0.9507

1978 -0.1080*** 1.1248***

1979 -0.0785*** 0.9082

1980 0.1224*** 0.2626

1981 0.3761*** 0.5844

1982 0.5486*** -0.0156

1983 0.1125*** 0.3017

1984 0.3703*** 0.8382***

1985 0.2634*** 0.5301*

1986 -0.0691*** -0.2151*** 0.1641

1987 -0.2956*** -0.1629*** 0.1511

1988 -0.3268*** -0.4212*** 0.1765

1989 -0.3813*** -0.3892*** 0.1994

1990 -0.4769*** -0.5009*** 0.2233

1991 -0.4364*** -0.1627*** 0.1266 -0.5174

1992 -0.4578*** -0.1243*** -0.4104 -0.2607*

1993 -0.3955*** -0.2053*** -0.4553 -0.1271

1994 -0.1973*** -0.1530*** -0.4001 -0.0522

1995 0.0759*** 0.2098*** -0.0235 0.3585***

1996 0.1882*** 0.3560*** 0.0013 0.1976*

1997 0.2929*** 0.3984*** 0.1039 0.3068**

1998 0.3560*** 0.3505*** -0.0452 -0.2340*

1999 0.5672*** 0.5188*** 0.0335 -0.0432

2000 0.7149*** 0.5011*** 0.2788 0.3717***

(continued)
Figures, Boxes, and Tables page 130

Table 9. Results of Default- Rate Logistic Regressions (continued)

Coefficient Estimates By Loan Type

Variable Names class

Category

Values

Fixed Rate,

Purchase

Adjustable

Rate, 

Purchase 

Fixed Rate,

Refinance

Adjustable

Rate,

Refinance

INVESTOR NO -0.3350*** -0.1588***

YES 0.3350*** 0.1588***

AGE_CLASS 1 1-2 years -0.2725*** 0.8015 -0.0104 0.2409***

2 3-4 years 0.1445*** 0.9904 0.3789*** 0.6148***

3 5-6 years 0.2165*** 0.7253 0.3075*** 0.1967***

4 7-8 years 0.1216*** 0.4653 0.2375*** -0.3973***

5 9-10 years 0.0705*** 0.2563 -0.0369 -0.6551

6 11-15 years -0.0103 0.0451 -0.3586*** -0.6551

7 16+ years -0.2703*** -0.6551 -0.5180*** -0.6551

Economic Conditions

CYCLE_STAGE 1 initial recession 0.1999*** 0.3381*0.1878*** 0.3381

2 latter recession 0.2004*** 0.5985*** 0.2123*** 0.5985

3 early expansion -0.1314*** -0.2289 -0.1037 -0.2289

4 out of cycle -0.2689*** -0.7077*** -0.2964*** 0

US_UNEMP_RATE..................................... -0.1497*** -0.4444*** -0.3106*** -0.6430***

CUR_MKT_RATE....................................... 17.1784*** 4.3519

MKT_SHIFT OFF 0.1210*** -0.0747*** 0.1515*** -0.4279

 ON -0.1210*** 0.0747*** -0.1515*** 0.4279

Financial Incentives

NEG_EQ_CLASS 1 over 50% 0.9087*** 1.1734*** 1.1452*** 1.3355***

2 26-50% 0.3191*** 0.1693*** 0.4761*** 0.0731

3 11-25% 0.0112*** -0.2680*** 0.0157 -0.2901***

4 2-10% -0.3141*** -0.2893*** -0.4798*** -0.5007***

5 up to 1% -0.9249*** -0.7854*** -1.1572*** -0.6178***

SPREAD_CLASS 1 under -.04 0.9473*** 1.2732*** 1.7767 1.509

2 -.04 - -.03 0.8160*** 1.0779*** 1.3324 1.2775

3 -.029 - -.02 0.6505*** 0.7143*** 0.9943 0.8466***

4 -.019 - -.015 0.5663*** 0.5343*** 0.9342 0.7196***

5 -.014 - -.010 0.3691*** 0.3629*** 0.7632 0.6618***

6 -.090 - -.005 0.1963*** 0.1857** 0.4903 0.4296***

7 -.005 - 0.00 -0.0188*** -0.0881 0.3082 0.3489***

8 .001 - .005 -0.1281*** -0.4622*** 0.1207 -0.0574

9 .005 - .010 -0.1987*** -0.5035*** -0.0147 -0.1815*

10 .011 - .015 -0.3444*** -0.6199*** -0.2222 -0.0652

11 .016 - .020 -0.4290*** -0.9807*** -0.4633 -0.7254***

12 .021 - .030 -0.4393*** -1.1820*** -0.7316 -1.1455

13 .031 - .040 -0.7759*** -2.0877 -0.4946 -0.7744

14 over .04 -1.2113*** -3.2592 -2.0173 -3.1585

BURN_SUM -0.0214*** 0.0214*** -0.0168*** 0.0513***

NEW_REFI OFF 0.0969*** 0.0567*** 0.0699*** -0.0285

 ON -0.0969*** -0.0567*** -0.0699*** 0.0285

(continued)
Figures, Boxes, and Tables page 131

Table 9. Results of Default- Rate Logistic Regressions (continued)

Coefficient Estimates By Loan Type

Variable Names class

Category

Values

Fixed Rate,

Purchase

Adjustable

Rate, 

Purchase 

Fixed Rate,

Refinance

Adjustable

Rate,

Refinance

PMT_ADJ_CLASS 1 under -5% 0.2276*** 0.0810

2 -4 - 0% -0.0976*** -0.3042***

3 1 - 5% -0.0090 -0.0328

4 6 - 10% -0.0416*** 0.0986***

5 over 10% -0.0794*** 0.1574***

Summary Statistics

Loan-group observations 1,977,847 458,595 357,064 60,347

Likelihood ratio P2

Degrees of Freedom (DoF)

166,722

(DoF=64)

26,757

(DoF=56)

18,039

(DoF=64)

3250

(DoF=43)

Notes: Italics represent coefficient values for classes omitted from regressions, calculated as the

negative sum of estimated class effects; boldface represent imputed coefficient values, as

described in Section III. Standard indicators of statistical significance are marked as: *** for

0.01 level (one-sided test), ** for 0.05 level, and * for 0.10 level. These indicators, however,

are less meaningful for categorical variables. What matters is the significance of differences

between the coefficients, and not differences of the coefficients from zero. Missing cells

represent a lack of loan records with variable values in these categories. The fixed-ratepurchase equation was estimated on a one-in-three sample of loan groups, where each started

with at least 10 loans. Each sampled loan group provides quarterly time series of observations,

through 2000Q4, the end of year twenty of loan life, or until all loans terminate, whichever

comes first. The number of actual loan quarters represented by the loan-group observations

listed in the Summary Statistics is typically very large. For example, for the fixed-ratepurchase equation, the number of loan quarters in the regression estimation is over 65 million,

so that, on average, each loan-group provides 33 quarters of observations.
Figures, Boxes, and Tables page 132

Table 10. Results of Prepayment-Rate Logistic Regressions

Coefficient Estimates By Loan Type

Variable Names class

Category

Values

Fixed Rate,

Purchase

Adjustable

Rate, 

Purchase 

Fixed Rate,

Refinance

Adjustable

Rate,

Refinance

CONSTANT 1 -3.4581*** 1.6821*** -3.409*** -0.0143

Loan Group Characteristics

LTV_CLASS 1 up to 80% 0.0000 0.0000 -0.1615 -0.2936

2 81-90% -0.0621*** -0.0052 -0.1615*** -0.2936***

3 91-95% 0.0199*** 0.0337*** -0.0382*** 0.2006***

4 96-100% 0.0553*** 0.0451*** 0.0746*** -0.0426

5 over 100% -0.0131 -0.0736 0.1251*** 0.1356**

PRICE_CLASS 1 under 50% -0.3430*** -0.2428*** -0.2352*** -0.2749***

2 51-100% 0.0215*** 0.0489*** -0.00206 0.0118

3 over 100% 0.3561*** 0.3164*** 0.1101*** 0.1393***

PROP_AGE_

CLASS

1 under 1 year 0.0572*** -0.0025 0.1484*** 0.2972***

2 1-15 years 0.0262*** 0.0269*** -0.0249*** -0.1091***

3 16-30 years -0.0555*** 0.0055** -0.0566*** -0.1164***

4 over 30 years -0.0279*** -0.0299*** -0.0669 -0.0717

COHORT_YEAR 1975 0.6621*** 0.2594

1976 0.6276*** 0.6529***

1977 0.6306*** 0.7489***

1978 0.3238*** 0.7621***

1979 -0.1602*** -0.0112

1980 -0.3728*** -0.0471

1981 -0.3676*** 0.6156***

1982 0.0207** 0.1851

1983 -0.2774*** 0.0572

1984 -0.2826*** -0.2459***

1985 -0.1790*** -0.148***

1986 -0.4499*** -0.4321***

1987 -0.3914*** 0.1718*** -0.4508***

1988 -0.3807*** 0.2944*** -0.5589***

1989 -0.2674*** 0.1352*** -0.4388***

1990 -0.1615*** 0.0065 -0.3724***

1991 -0.0971*** -0.1827*** -0.2693*** -0.4734

1992 0.0281*** -0.2265*** -0.1405*** 0.0849

1993 0.0505*** -0.2526*** -0.2093*** 0.0453

1994 -0.0213*** -0.3190*** -0.2276*** -0.017

1995 -0.0842*** -0.0599*** -0.2681*** 0.3498

1996 -0.0811*** 0.0024 -0.123*** 0.7846

1997 -0.0235*** -0.0556*** -0.00654 0.8410*

1998 -0.0833*** -0.1624*** 0.0377 0.8014

1999 -0.1192*** -0.0228 -0.1761*** 0.6762

2000 0.6925*** 0.1216*** 0.1024*** 0.6159

2001 0.7643*** 0.5496*** 0.7043*** -3.7087

(continued)
Figures, Boxes, and Tables page 133

Table 10. Results of Prepayment-Rate Logistic Regressions (continued)

Coefficient Estimates By Loan Type

Variable Names class

Category

Values

Fixed Rate,

Purchase

Adjustable

Rate, 

Purchase 

Fixed Rate,

Refinance

Adjustable

Rate,

Refinance

INVESTOR NO 0.0807*** -0.0137***

YES -0.0807*** 0.0137***

AGE_CLASS 1 1-2 years 0.1717*** 0.3795*** 0.8303*** 0.3247***

2 3-4 years 0.5187*** 0.3199*** 0.5625*** -0.0709

3 5-6 years 0.3698*** 0.0306 0.3548*** -0.2664***

4 7-8 years 0.1566*** -0.0066 0.1766*** -0.1374

5 9-10 years -0.1138*** -0.0947*** -0.3868*** -0.1897**

6 11-15 years -0.3387*** -0.3473*** -0.4611*** -0.3473

7 16+ years -0.7643*** -0.7837 -1.0763 -0.7837

Economic Conditions

CYCLE_STAGE 1 initial

recession

0.2717*** 0.8743*** 0.1716*** 0.5522

2 latter recession -0.0633*** 0.0490 -0.0418 0.0324

3 early

expansion

-0.1927*** -1.0322*** -0.1623*** -0.8694

4 out of cycle -0.0157 0.1089 0.0325 0.0000

US_UNEMP_RATE....................................

........

-0.1005*** -0.3813*** -0.1314*** -0.1926***

CUR_MKT_RATE

.............................................

-49.3482*** -45.4862***

YLD_SLOPE_CAT 1 up to 0% -0.3950*** -0.0557*** -0.3650*** -0.095

2 0.1 - 1% -0.3110*** -0.1514*** -0.4058*** -0.2171

3 1.1 - 2% -0.0629*** 0.0652*** -0.2977*** 0.0483

4 2.1 - 3.0% 0.1979*** -0.3956*** 0.1900*** -0.612

5 over 3.0% 0.5710*** 0.5375*** 0.8785*** 0.8758***

MKT_SHIFT OFF -0.3175*** -0.2638*** -0.4861*** -1.0753

 ON 0.3175*** 0.2638*** 0.4861*** 1.0753***

Financial Incentives

POS_EQ_CLASS 1 over 75% 0.5080*** 0.5615*** 0.6144*** 0.3686***

2 51 - 75% 0.1950*** 0.2469*** 0.2520*** 0.1672***

3 26 - 50% 0.0006 0.0302*** -0.0830*** -0.00451

4 11 - 25% -0.1284*** -0.1934*** -0.2764*** -0.1766***

5 0 - 10% -0.5752*** -0.6452*** -0.5070*** -0.3547***

SPREAD_CLASS 1 under -.04 2.0141*** 1.8872*** 1.8986*** 0.9493

2 -.04 - -.03 1.6354*** 1.0198*** 1.6058*** 0.4999***

3 -.029 - -.02 1.2572*** 0.9507*** 1.3326*** 0.5701***

4 -.019 - -.015 0.8084*** 0.8422*** 1.0571*** 0.5284***

5 -.014 - -.010 0.6776*** 0.6909*** 0.8225*** 0.3803***

6 -.090 - -.005 0.2162*** 0.4579** 0.4581 0.2787***

7 -.005 - 0.00 -0.2366*** 0.0701 -0.00741 -0.0263

8 .001 - .005 -0.4589*** -0.4550*-0.2512 -0.3826***

9 .005 - .010 -0.5883*** -0.4746** -0.4684 -0.4496***

10 .011 - .015 -0.6213*** -0.5769** -0.6588*** -0.3438***

(continued)
Figures, Boxes, and Tables page 134

Table 10. Results of Prepayment-Rate Logistic Regressions (continued)

Coefficient Estimates By Loan Type

Variable Names class

Category

Values

Fixed Rate,

Purchase

Adjustable

Rate, 

Purchase 

Fixed Rate,

Refinance

Adjustable

Rate,

Refinance

SPREAD_CLASS 11 .016 - .020 -0.6473*** -0.9166*** -0.8613*** -0.4333***

12 .021 - .030 -0.8449*** -0.8028*** -1.1047*** -0.5524

13 .031 - .040 -1.3468*** -1.3468 -1.3468 -0.6734

14 over .04 -1.8648*** -1.8648 -1.8648 -0.9324

BURN_SUM -0.0548*** 0.0214*** -0.0444*** -0.0267***

NEW_REFI OFF -0.1651*** 0.0567*** -0.0574*** 0.0154*

ON 0.1651*** -0.0567*** 0.0574*** -0.0154*

PMT_ADJ_CLASS 1 under -5% 0.1461*** -0.0125

2 -4 - 0% -0.1055*** -0.0930***

3 1 - 5% -0.0839*** -0.0369***

4 6 - 10% -0.0207*** 0.0205*

5 over 10% 0.0640*** 0.1219***

Summary Statistics

Loan-group observations 2,135,432 545,235 400,417 72,041

Likelihood ratio P2

Degrees of Freedom

1,460,078

(DoF=69)

328,091 

(DoF=61)

178,878

(DoF=69)

27,720

(DoF=50)

Notes: Italics represent coefficient values for classes omitted from regressions, calculated as the

negative sum of estimated class effects; boldface represent imputed coefficient values, as

described in section III. Standard indicators of statistical significance (one-sided test) are

marked as: *** for 0.01 level, ** for 0.05 level, and * for 0.10 level. These indicators,

however, are less meaningful for categorical variables. What matters is the significance of

differences between the coefficients, and not differences of the coefficients from zero. Missing

cells represent a lack of loan records with variable values in these categories. The fixed-ratepurchase equation was estimated on a one-in-three sample of loan groups, where each started

with at least 10 loans. Each sampled loan group provides quarterly time series of observations,

through 2000Q4, the end of year twenty of loan life, or until all loans terminate, whichever

comes first. The number of actual loan quarters represented by the loan-group observations

listed in the Summary Statistics is typically very large. For example, for the fixed-ratepurchase equation, the number of loan quarters in the regression estimation is over 65 million,

so that, on average, each loan-group provides 33 quarters of observations.
Figures, Boxes, and Tables page 135

Table 11. Default Related Cash-Flow Timing and Regression Models

Significant Points in

Time Events 

Use of Regression

Models in

Simulations Cash Flow Items

Mortgage payment

due but unpaid

Starts delinquency

period.

90-day delinquency

(default)

Three missed payments

and a fourth due-andpayable defines default,

the point at which

foreclosure of property

rights becomes a viable

option.

Fourth month of

delinquency

Determination of

eligibility of borrower

for workout option.

Administrative costs

paid by FHA to loan

servicers for workout

agreements with

borrowers.

Foreclosure of

borrower property

rights and transfer of

property title to

HUD/FHA

Path for all defaulted

loans without workout

agreements in place by

sixth month.

Time from default to

foreclosure.

Claim expense paid by

FHA for loan balance

and foreclosure

expenses.

Property

Disposition/Sale

Sale of property Time-to-Disposition

and Recovery Rate

on Sale

Net proceeds on sale,

as percent of

defaulting loan

balance; property

holding expenses.
Figures, Boxes, and Tables page 136

Table 12. Logistic Regression of the Probability of Receiving Loan-W orkout Offers

Variable Names class

Category

 Values

Estimated

Coefficients

Standard

Errors Descriptions

Intercept -1.0854*** 0.0732 Regression constant term

LTV_CLASS 1 up to 80% 0.3885*** 0.0258 Origination LTV

2 81-90% 0.0169 0.0163

3 91-95% 0.0729*** 0.0163

4 96-100% -0.1708*** 0.0092

5 over 100% -0.3075***

PROD_CLASS 1 FRM30 0.2161*** 0.0269 Fixed-rate 30-year

2 FRM15 0.6909*** 0.0345 Fixed-rate 15-year

3 ARM -0.0722* 0.0276 Adjustable rate

4 GPM -0.7964*** 0.0615 Graduated payment

5 GEM -0.0384 Graduated equity

PRICE_CLASS 1 under 50% -0.3224*** 0.0064 Purchase price as a percent of area

2 51-100% 0.0074 0.0046 (MSA) median house price

3 over 100% 0.3150***

CYCLE_STAGE 3 early expansion -0.8387*** 0.0602 See Table A.5 for description

4 out of cycle 0.8387***

HPI_CLASS 1 under .90 -2.2037*** 0.1370 Value of HPI at time of default,

where value is 1.00 at loan

origination.

2 .90 - .99 -1.3202*** 0.0476

3 1.0 - 1.09 0.1619*** 0.0301

4 1.10 - 1.24 0.5768*** 0.0292

5 1.25 - 1.49 1.2583*** 0.0293

6 1.5 or over 1.5269***

HPI_GROWTH_

CLASS

1 under 0 0.0109 0.0097 4-quarter HPI growth leading, prior

to quarter of loan default.

2 0 - .04 -0.3358*** 0.0071

3 .05 - .09 0.0526*** 0.0056

4 .10 and over 0.2723***

MORT_AGE_Q -0.0178*** 0.0004 Mortgage age at default, in quarters

Summary Statistics

Observations 405,539 (207,262 are reported in the FHA loss mitigation data base and another 198,277 are

lender forbearances reported in the FHA default monitoring system)

Pseudo Rsquared .085

Likelihood ratio

Chi-squared 26,688 (Degrees of Freedom=20)

Notes: Coefficients without standard errors are imputed values from estimated class effects and appear in

italics. Only CYCLE_STAGE = {3,4}in the regression data sample. In the forecast simulations, the effect

for CYCLE_STAGE = 3 is also used for CYCLE_STAGE = 2, and the effect for CYCLE_STAGE = 4 is

also used for CYCLE_STAGE = 1. Standard indicators of statistical significance (one-sided test) are

marked as: *** for 0.01 level, ** for 0.05 level, and * for 0.10 level. These indicators, however, are less

meaningful for categorical variables. What matters is the significance of differences between the

coefficients, and not differences of the coefficients from zero. Probability estimates (P) are computed

from variable values (X) and estimated effect coefficients (b) as: .
Figures, Boxes, and Tables page 137

Table 13. Foreclosure-Time Regression: Accelerated Failure Time with Weibull Error

Distribution 

Variable Names

Category 

Values

Estimated

Coefficients

Standard 

Errors

Proportionality

Factors Descriptions

CONSTANT 1 3.7011 0.0109 Regression intercept term.

CYCLE_STAGE 1 -0.1182 0.0032 0.89 1

st half of price decline

2 0.1126 0.0024 1.12 2

nd half of price decline

3 0.2421 0.0030 1.27 Recovery from stage 2

HPI_GROWTH -4.9764 0.0113 0.01 House price growth in first 4

quarters of default (decimal)

CUR_MKT_RATE -0.1860 0.0008 0.83 Current mortgage rates (%)

CENSUS_DIV NEW 0.0926 0.0044 1.10 New England

MAT 0.2773 0.0024 1.32 Middle Atlantic

SAT -0.0058 0.0016 0.99 South Atlantic

ENC 0.0957 0.0020 1.10 East North Central

WNC -0.0380 0.0022 0.96 West North Central

ESC -0.1367 0.0024 0.87 East South Central

WSC -0.2081 0.0016 0.81 West South Central

MTN -0.1507 0.0018 0.86 Mountain

DEFAULT_YR 1975 4.8376 338.80 126.16 Year of default

1976 4.5314 338.80 92.89

1977 5.8693 338.80 354.00

1978 2.2857 0.4377 9.83

1979 6.9335 46.86 1026.06

1980 2.7816 0.2190 16.14

1981 3.0678 0.0726 21.49

1982 2.2192 0.0164 9.20

1983 1.6955 0.0111 5.45

1984 1.2623 0.0100 3.53

1985 0.9656 0.0095 2.63

1986 1.0310 0.0095 2.80

1987 0.9176 0.0095 2.50

1988 0.8383 0.0094 2.31

1989 0.7965 0.0094 2.22

1990 0.6979 0.0093 2.01

1991 0.5987 0.0092 1.82

1992 0.5891 0.0092 1.80

1993 0.6414 0.0092 1.90

1994 0.6136 0.0093 1.85

1995 0.6427 0.0092 1.90

1996 0.5054 0.0092 1.66

1997 0.4383 0.0092 1.55

1998 0.4721 0.0092 1.60

1999 0.4951 0.0092 1.64

2000 0.3910 0.0093 1.48

WEIBULL Shape parameter

0.4376 0.0004 Defines shape of error

distribution

Summary Statistics

Observations 874790

Right-censored

values 24395 (reported foreclosure times are over 36 months)

Mean time to foreclosure 12.76 months

(Continued)
Figures, Boxes, and Tables page 138

Table 13. Foreclosure-Time Regression: Accelerated Failure Time with W eibull

Error Distribution (continued)

Notes: Left-hand-side (dependent) variable is months between default and foreclosure completion.

Coefficient values for CYCLE_STAGE=4 (no cycle), CENSUS_DIV=PACIFIC, and

DEFAULT_YR=2001 are jointly included in the constant term of the regression.

Proportionality factors are exp{b}, where b is the regression effect coefficient. Predicted

values of foreclosure time (Tf) are computed as: Tf = (e

Xb)

(-ln(U))c, where X represents a

vector of zeros and ones that indicate which effect coefficients are on(1) or off (0) for a

given observation, and b is the vector of estimated coefficients, U is a uniform random

number, and c is the Weibull shape parameter. The forecast simulations in this report use

the mean of the Weibull distribution, rather than randomly generated values, so that:

, where G represents the Gamma function.
Figures, Boxes, and Tables page 139

Table 14. Foreclosed-Property Disposition (Sales) Time Regression: Accelerated

Failure Time with Weibull Error Distribution 

Variable Names

Category

Values

Coefficient

Estimates

Standard

Errors

Proportionality factors Descriptions

CONSTANT 1.0825 0.0241 Regression intercept term

CYCLE_STAGE 1 0.1104 0.0102 1.117

Observation in 1st half of HPI

decline 

2 0.0945 0.0073 1.099

Observation in 2nd half of HPI

decline

3 -0.0210 0.0062 0.979

Observation in recovery from stage

number 2

HPI_GROWTH_

CLASS

under 0 0.1764 0.0065 1.193 House price growth in first 4 qtrs of

0.0 - .04 0.1351 0.0062 1.145 default

.05 - .09 0.1269 0.0058 1.135

CUR_MKT_RATE 0.0301 0.0027 1.031 Current mortgage rates (fixed-rate,

30-year, %)

FORECLOSE_YR 1981 0.8949 0.7651 2.447 Year of property foreclosure.

1983 1.7158 0.2893 5.561 (No observations in 1982)

1984 2.8698 0.1535 17.633

1985 2.4747 0.0452 11.878

1986 2.4506 0.0331 11.595

1987 2.3847 0.0215 10.856

1988 2.0125 0.0136 7.482

1989 1.7313 0.0100 5.648

1990 1.0122 0.0076 2.752

1991 0.8884 0.0072 2.431

1992 0.8296 0.0064 2.292

1993 0.7641 0.0063 2.147

1994 0.5880 0.0064 1.800

1995 0.4817 0.0066 1.619

1996 0.3966 0.0066 1.487

1997 0.4048 0.0063 1.499

1998 0.5000 0.0065 1.649

1999 0.3758 0.0069 1.456

2000 0.3191 0.0061 1.376

CENSUS_DIV NEW 0.2117 0.0082 1.236 New England

MAT -0.0192 0.0048 0.981 Middle Atlantic

SAT -0.1256 0.0034 0.882 South Atlantic

ENC -0.0587 0.0044 0.943 East North Central

WNC -0.1968 0.0048 0.821 West North Central

ESC -0.2093 0.005 0.811 East South Central

WSC -0.1565 0.0036 0.855 West South Central

MTN -0.1850 0.0042 0.831 Mountain

Weibull scale

parameter 0.7650 0.0007

Defines shape of the error

distribution

 Summary Statistics

Observations 610,704

Right-censored

values 20,991 (reported disposition times are over 24 months)

Mean sale time 5.7 months

(Continued)
Figures, Boxes, and Tables page 140

Table 14. Foreclosed-Property Disposition (Sales) Time Regression: Accelerated

Failure Time with Weibull Error Distribution 

(continued)

Note: Left-hand-side (dependent) variable is months between foreclosure completion and property

sale. Combined effects for CYCLE_STAGE=4, HPI_GROWTH > 0.10,

FORECLOSURE_YR=2001, and CENSUS_DIV=Pacific are jointly included in the constant

term of the regression. Proportionality factors are exp{b}, where b is the regression effect

coefficient. Predicted values of foreclosure time (Tf) are computed as: Tf =

, where X represents a vector of zeros and ones that indicate which effect

coefficients are on(1) or off (0) for a given observation, and b is the vector of estimated

coefficients, U is a uniform random number, and c is the Weibull shape parameter. The

forecast simulations in this report use the mean of the Weibull distribution, rather than

randomly generated values, so that: , where G represents the

Gamma function.
Figures, Boxes, and Tables page 141

Table 15. Foreclosed Property Recovery Rate Regression

Variables

Category

Values

Coefficient

Estimates

Standard

 Errors Descriptions

CONSTANT 0.7922 0.0025 Regression intercept term

PRICE_DROP_SCORE 0.0713 0.0003 Standard normal statistic measuring

distance between price decline needed to

make default in-the-money and the

median market price growth found in the

HPI.

PRICE_CLASS < 50% -0.2296 0.0011 Based on ratio of house price to area

51-100% -0.1010 0.0008 (MSA) median house price.

CENSUS_DIV NEW -0.2097 0.0026 New England

MAT -0.1624 0.0014 Middle Atlantic

SAT 0.0019 0.0010 South Atlantic

ENC -0.0698 0.0013 East North Central

WNC -0.0778 0.0015 West North Central

ESC -0.0449 0.0015 East South Central

WSC -0.0367 0.0011 West South Central

MTN 0.0514 0.0013 Mountain

INVESTOR NO 0.1133 0.0020 Investor owned property flag

FORECLOSURE_YR 1985 0.0022 0.0119 Calendar year of foreclosure completion

1986 -0.0565 0.0129

1987 -0.0673 0.0095

1988 -0.0924 0.0049

1989 -0.0506 0.0033

1990 -0.1195 0.0018

1990 0.0022 0.0024

1991 -0.1321 0.0016

1992 -0.1173 0.0015

1993 -0.0845 0.0015

1994 -0.0629 0.0015

1995 -0.0455 0.0016

1996 -0.0414 0.0016

1997 -0.0500 0.0016

1998 -0.0560 0.0015

1999 -0.0398 0.0015

2000 -0.0205 0.0015

Summary Statistics

Observations 610,322

R-Squared .1878

Mean of recovery rate .7918

Note: Left-hand-side (dependent) variable is the ratio of the sales price less sales expenses to the

unpaid loan balance at time of default. Combined effects for PRICE_CLASS=3 (over

100%), CENSUS_DIV = Pacific Division, INVESTOR=YES, and

FORECLOSURE_YR=2001 are jointly included in the regression constant. This is an

ordinary least squares regression, where predicted recovery rates are a linear function of the

explanatory variables and estimated coefficients.
Figures, Boxes, and Tables page 142

Table 16. Miscellaneous Param eters Used to Calculate Insurance-Claim Cost in

Simulations

Parameters Values Used in/when Descriptions

Workout Failure Rate 0.20 Computing

percent of default

workouts.

Applied to

number of loans

receiving workout

assistance.

Available data are not yet sufficient for

econometric analysis of workout success/failure

rates. The rate used here is consistent with recent

FHA experience.

Foreclosure time upper bound

of 10 quarters

Timing of claim

expenses and

ultimate recovery

on REO sales

This limit is placed on results of the foreclosure

time regression, simply to assure that no out-ofbounds results occur.

Interest expense in

claim

75% of

current

mortgage

note rate,

from quarter

of last-paidinstallment to

foreclosure

quarter

Insurance claim

payment on

foreclosure

FHA reimburses loan servicers for lost interest

on defaulted mortgages, up to the date of

foreclosure. This reimbursement is at a note rate

equivalent to the average 30-year Treasury yield

in the year of loan origination. CBO

approximates the mortgage-to-Treasury spread

rather than maintaining a lookup table of

Treasury rates in the simulation model.

Foreclosure-related

expenses

7.7% of

unpaid

principal

balance

Insurance claim

payment on

foreclosure

This represents the cost to FHA of attorney fees,

property taxes due, and other costs of securing

title to the property, net of any escrow funds

available. FHA generally reimburses servicers

for two-thirds of this net expense. The 7.7% rate

used here is the average expense rate booked by

FHA, 1996-2001.

Preforeclosure Sale

Benefits

8% on 5% of

foreclosures

Insurance claim

payment on

foreclosure

FHA maintains a little-used program of assisting

certain defaulted borrowers to sell their

properties prior to foreclosure. With little data

on this program, CBO uses the cost savings

gleaned by HUD in its 1994 pilot study

evaluation (Charles A. Capone, Jr., Evaluation

of the Federal Housing Administration

Preforeclosure Sale Demonstration.

Washington, DC: U.S. Department of HUD,

Office of Policy Development and Research,

June 1994.) This cost saving (8% of the unpaid

principal balance) is applied to 5% of

foreclosures in the simulations.

(continued)
Figures, Boxes, and Tables page 143

Table 16. Miscellaneous Parameters Used to Calculate Insurance-Claim Cost in

Simulations

(Continued)

Parameters Values Used in/when Descriptions

Cost of loan workouts 1.5% of

outstanding

loan balance

Credit cost

expense booked

in default quarter

FHA supports loan servicer workout initiatives

with incentive payments. These are generally a

few hundred dollars, averaging just 0.5% of the

outstanding loan balance (as of January, 2002).

However, when FHA pays a partial claim to

make the borrower whole (and placing a lien

against the property), the cost has been, on

average, between 7 and 8% of the loan balance,

over time. 

By applying the incentive payment rate (0.5%)

to all workouts, and partial claim cost (8%) to 10

percent of the cases, one arrives at a net charge

for each loan workout of 1.3%. CBO rounds this

up to 1.5% for use in the simulations.

REO property sale

time

upper bound

of 8 quarters

time between

foreclosure and

property sale. 

Limitation placed on results of regression

equation to prevent the possibility of out-ofbound results.

REO sale recovery

rate

constrained to

the (.30,

1.30) interval

revenue inflow

from property

sales, net of sales

expenses

Regression results are bounded to remove outof-bound events. HUD sales of some homes for

$1 to nonprofits and local governments is

accounted for in the data used to estimate the

sale-time regression equation. Thus, the

regression equation captures average results for

loans with given characteristics.

Other property

management expenses

6.25% adjusts property

sale proceeds

downward

Expense ratio–against unpaid principal

balance–for property management, repairs, and

taxes during the property holding period. The

6.45% rate is the average for properties sold in

1999 and represents an average experience of

the most recent 5 year period.

Premium Refund

Schedule

rates vary by

loan

origination

date and age

of mortgage.

Rebates to

borrowers who

payoff mortgages

early.

The rebate schedule corresponds with FHA’s

determination of how quickly the up-front

premium is actually earned. For loans insured

prior to 1994, the premium is considered earned

over the life of the loan. For 30-year loans, the

rebate declines to 50 percent (of the up-front

premium) by the end of the fifth year of loan

life, and 10 percent by the end of the

seventeenth year. For loans insured from

January 1994 through December 2000,

premiums are earned over just seven years, with

reimbursement rate falling to 50 percent after 40

months. Loans insured since January 2001 have

a five-year earning schedule, with rebate

percentages in years 1-5 being: 85, 65, 45, 25,

and 10, respectively.
Figures, Boxes, and Tables page 144

Table 17. Confidence Bounds for Subsidy Rate Simulation Results, by Cohort Year

(percent)

Cohort Year

Percentiles

1% lower

bound

5% lower

bound

mean 5% upper

bound

1% upper

bound

1992 -1.15 -1.14 -1.12 -1.10 -1.09

1993 -2.18 -2.16 -2.10 -2.06 -2.04

1994 -2.12 -2.09 -2.03 -1.96 -1.94

1995 -0.25 -0.21 -0.14 -0.07 -0.04

1996 -0.74 -0.69 -0.57 -0.47 -0.43

1997 -0.70 -0.64 -0.51 -0.39 -0.33

1998 -1.80 -1.70 -1.44 -1.20 -1.10

1999 -2.57 -2.41 -1.97 -1.53 -1.32

2000 -1.50 -1.34 -1.02 -0.70 -0.52

2001 -2.04 -1.79 -1.19 -0.55 -0.14

2002 -3.17 -2.81 -1.73 -0.26 0.77

2003 -3.37 -2.97 -1.85 0.10 2.07

2004 -3.32 -2.88 -1.53 0.86 3.36

2005 -3.06 -2.66 -1.04 1.90 4.31

2006 -3.04 -2.53 -0.64 3.08 6.16

2007 -3.10 -2.57 -0.32 4.05 6.92