Credit Risk in Residential Mortgages:
Measuring the Collateral Risk with House Price Indices
Che-Chun Lin
National Tsing Hua University
101, Sec. 2, Kuang-Fu Road
Hsin-Chu 30013, Taiwan
chclin@mx.nthu.edu.tw
And
Tyler T. Yang*
Integrated Financial Engineering Inc.
51 Monroe Street, Plaza E-6
Rockville, MD 20850
tyler.yang@ifegroup.com
(301) 309-6560
(301) 309-6562 (fax)
March 2005
*
Corresponding author.
Credit Risk in Residential Mortgage:
Measuring the Collateral Risk with House Price Indices
Abstract
This paper develops a model to properly capture the house price risk at the individual
house level. By decomposing the total volatility into the national volatility, regional
dispersion, and house level dispersion, risk-based pricing of residential mortgage credit
risk at the regional level is shown to be economically justifiable.
Residential mortgages are usually considered much safer than other consumer loans due
to the additional collateral protection. Most recent mortgage credit risk studies apply the
option concept by treating default as a put option that allows the borrower to sell the
collateral house to the lender at the price of the unpaid principal balance of the loan.
Calhoun and Deng (2002) introduced a variable “probability of negative equity” (PNEQ)
to better reflect the nature of this put option. In this paper, we develop a methodology to
properly estimate PNEQ using the publicly available house price indices published by the
Office of Federal Housing Enterprise Oversight (OFHEO).
The methodology properly captures the distribution of a typical house within a local
economy in terms of mean and volatility. Extended from the model developed by Buist,
Megbolugbe, and Yang (1998), we are able to properly estimate the future PNEG for
mortgages.
Numerical examples are provided by using OFHEO house price indices at the state level
in estimating the PNEG and their impact on mortgage default rates. With the different
HPI growth rate volatilities, it is anticipated that the projected PNEG will vary among
different states, indicating state level risk-based pricing of mortgages is feasible and
justifiable.
Credit Risk in Residential Mortgage:
Measuring the Collateral Risk with House Price Indices
Model Description:
OFHEO publishes repeat sales based house price indices at the national-level and for nine
Census divisions. It also publishes the dispersion parameters of the growth rate of one
single house versus the average of the specific Census division the house is located.
However, because the national level house price index is imputed from the weighted
average of the nine Census regions, no separate dispersion parameters are published at
the national level. Thus, direct estimate of house price dispersion around a national index
does not exist. Most mortgage lending institutions used regional-level and state-level
indexes for estimating historical loan performance models. When applying the models in
forward projection, they have to depend on some HPI forecast into the future.
Forecasting HPI at divisional or state level could be difficult. But the forecast of national
house price growth rates are generally available as one of the key macro economic
indicators. The model developed here allows a lender to properly estimate the dispersion
of one individual house against the national average HPI growth rate. The conditional
distribution will also allow the computation of the probability of negative equity, which is
one of the most significant variables in estimating the conditional default rate of
mortgages.
For consistency with the OFHEO estimates of house price volatility, we required
estimates of the variance in the geometric growth rates of housing values implied by the
regional indexes around the geometric growth rates implied by the national-level index.
The following discussion uses the case of state indexes as an example, but the same
approach is applied in the case of the Census division indexes.
The growth rate for property i between time t and s relative to its state index is given by:
ln( Gi )  ln( H i ,t )  ln( H i ,s )  ln( H State,t )  ln( H State,s )   i ,t ,s
(1)
Similarly, the growth rate implied by the state index relative to the national average
forecast can be decomposed as follows:
ln( GState )  ln( H State,t )  ln( H State,s )  ln( H N ,t )  ln( H N ,s )   State,t ,s
(2)
Therefore, relative to the national average forecast, the individual house price growth rate
equals the growth rate implied by the national index and the sum of the dispersion of
individual around state growth rates and the specific state around national average growth
rates:
ln( Gi )  ln( H N ,t )  ln( H N ,s )   i ,t ,s   State,t ,s
(3)
Recall that the variance of the first component of dispersion error given by  i ,t ,s can be
computed directly from the “a” and “b” parameters published by OFHEO:
 2 ln Gi lnG State    2  i ,t ,s   a  t  s   b  t  s 2
(4)
where (t-s) is the number of quarters since loan origination. For consistency with the
OFHEO formulation we required an estimate of the variance in the second component
error  State,t ,s for the dispersion of the state indexes around the national average forecast
that would also be linear in time, as follows:
 2 ln GState / ln GN    2  State,t ,s   c  t  s 
(5)
Because equation (4) was estimated by OFHEO as residual term when estimate the state
house price index using all houses within that location, it must be orthogonal to the
volatility of the state level house price indexes. That is the noise term  i ,t ,s is
independent of the  State,t ,s , or:
  i,t,s   State,t,s   0
(6)
This implies the following model for the variance of individual house price appreciation
rates around the national average forecast:
 2 ln Gi lnG N   a  t  s   b  t  s 2  c  (t  s)
(7)
The parameter “c” required for projecting the additional dispersion of the state index
around the national average forecast was estimated as follows: For each quarter t we
computed the cross-sectional (across state) average dispersion variance (state versus
national) for each possible value of (t  s)  0 , which corresponds to time since loan
origination, i.e., mortgage age:
 2 ( t ,s ) 
1
 ln( G
nState State
/ G N ,t ,s )
2
State,t , s
t  2, 3, ..., T ; s  t
(8)
This gives us a cross-section/time-series sample of average state index dispersion
variance around the national average forecast which we assume is a linear function of t-s
:
 2 ( t ,s )  c  (t  s)  ut ,s
t  s, s  1, ..., T ; s  1, 2, ...,
(9)
We estimated the unknown parameter “c” using a weighted least square regression using
the number of average variance observations at each value of t-s as weights. The
estimated quarterly standard deviation ( c ) values were 2.8903% for state indexes and
2.7192% for Census division indexes.
One of the following two formulas was applied depending on whether the time period
was historical or future:
 2 ln Gi lnG State   a  t  s   b  t  s 2
 2 ln Gi lnG N   a  t-s  b  t-s2  c  (t  T )
if t  T
if t  T
(10a)
(10b)
where T is the last historical time period (FY 2004 Q1). Equation (10a) was applied to
historical sample time periods when the state index was used to update expected housing
values; and equation (10b) was applied during future time periods when the national
average forecast was used to update expected housing values.
For future loan originations only a single formula is required:
 2 ln Gi lnG N   a  t-s  b  t-s2  c  (t  s)
if s  T
(11)
Equation (11) was applied to future loan originations and only the national average
forecast was used to update expected housing values.
The additional term associated with dispersion of an state or Census division index
around the national average forecast increases the overall dispersion volatility and results
in higher probabilities of negative equity. This is counterbalanced by reduced relative
frequency of low expected HPI values when using a national average house price forecast
instead of the more volatile local or regional indexes.