Valuing Mortgage Insurance Contracts
with Counterparty Risk and Capital Forbearance
Chia-Chien Chang and Min-Teh Yu*
November 11, 2009
Abstract
This study sets up a contingent-claim framework incorporating the counterparty
default risk and capital forbearance to derive a closed-form solution of mortgage
insurance contracts. This study further investigates how critical policy parameters
such as capital requirements, prompt closure, and other policy-instrument parameters
relate to the value of mortgage insurance by evaluating the partial derivatives of the
closed-form expression. The numerical analysis indicates that default risk premium
can be substantial in the presence of catastrophic risk of the housing price.
Furthermore, forbearance threshold and time to delay effects could be more
significant as initial leverage ratio increases.
Key Words: Mortgage insurance, Counterparty default risk, Forbearance, Capital
standard.
*
Chang: Department of Finance, National Kaohsiung University of Applied Science, Kaohsiung, 807,
Taiwan. Email: u7501247@yahoo.com.tw. Yu: Department of Finance, Providence University,
Taichung 43301, Taiwan. Fax: 886-4-26311170, Email: mtyu@pu.edu.tw. Corresponding author: Yu.
1
1. Introduction
In view of the occurrence of subprime mortgage crisis in the United States,
Mortgage Insurance Companies Association (MICA) reports that large mortgage
insurers of its members have reported $2.6 billion in losses in 2008, sparking concerns
that rising foreclosure rates of the borrowers could force the mortgage industry into a
money crunch. For example, Shares of Radian Guaranty, Triad, and PMI Mortgage
Insurance Group have lost 90 percent of their share value in 2007; Triad Guaranty
Insurance Corporation fails to meet capital requirement1 in March 31, 2008 and even
is going out of business. As a consequence, the phenomenon that the defaults of the
borrower will spill over into the default probabilities of the mortgage insurer needs to
be considered, and it is termed ‘counterparty default risk’. However, when the
mortgage insurer fails to meet the risk-to-capital ratio and the government is forced to
allow continuing their operations in order to avoid the systematic economic crisis,
capital forbearance occurs.2 It is essential to incorporate the counterparty default risk
and capital forbearance, generally not considered by the previous studies, into the
pricing model of mortgage insurance－particularly in the case of a mortgage crisis.
Mortgage insurance plays an important role in the functioning of housing finance
markets since it transfers the risk exposures on borrowers’ defaults from lenders to
mortgage insurers and facilitates the creation of secondary mortgage markets. Private
mortgage insurance, known as mortgage guaranty insurance, guarantees that, in the
event of a default, a mortgage insurer will pay the mortgage lender for any loss
resulting from a property foreclosure up to the 20 to 30 percent of the claim amount
1
As of March 31, 2008, Mortgage Insurance Companies Association (MICA) reports that Triad's
risk-to-capital ratio, 27.7-to-1, exceeded the maximum (25-to-1) generally permitted by insurance
regulations and Illinois insurance law.
2
Capital forbearance has been recognized in the literature as a major determinant of deposit insurance
(e.g. Kane (1986) and Nagarajan and Sealey (1995))
2
(Canner and Passmore, 1994). Mortgage insurers operate under state insurance laws3
and most states regulate the industry in three special ways: (1) A contingency reserve
equal to one-half of all premiums received must be maintained for 10 years; (2) A 4%
capital ratio applies to risk in force; (3) A monoline requirement forces a firm to write
only mortgage insurance.
Due to the subprime mortgage crisis, the rising foreclosure rates of the borrowers
results in the capital scarcity of some mortgage insurers. To help private mortgage
insurers, some policies of capital injection are published. For instance, Freddie Mac
announced that private mortgage insurers do not need to meet the increasing capital
requirements when the credit ratings of mortgage insurers have been downgraded
below AA. Moreover, on March 19, 2008, the regulator of Fannie Mae and Freddie
Mae, the Office of Federal Housing Enterprise Oversight (OFHEO) agreed to reduce
the existing 30 percent OFHEO-directed capital requirement to 20 percent. The
OFHEO estimates that this reduction should provide up to $200 billion of immediate
liquidity to the mortgage-backed securities market. The Housing and Economic
Recovery Act enacted on July 30, 2008, is intended to ensure the safe and sound
operation of Fannie Mae and Freddie Mae by injecting capital into the two large U.S.
suppliers of mortgage funding. Furthermore, to solve the short of capital and ensure
the operation, the Federal Reserve was forced to bail out American Insurance Group
(AIG), the largest insurance company in the world, by providing an emergency loan
of $85 billion for AIG to sell off assets to repay losses on September 16, 2008.
Voluminous previous studies of the mortgage insurance contracts, such as
Schwartz and Torous (1992), Dennis et al. (1997), and Bardhan et al. (2006), model
3
The chartering of the first post-Depression mortgage insurer, the Mortgage Guaranty Insurance
Corporation is allowed when Wisconsin passed a mortgage insurance law in 1956. California followed
with a comprehensive mortgage insurance act in 1961, and the California statute became the standard
for the mortgage insurance laws that followed in other states (Rapkin, 1973).
3
the unconditional probability of default exogenously with a constant interest rate. On
the other hand, Kau et al. (1992, 1993, 1995), and Kau and Keenan (1995, 1996)
improve the literature by using a structural approach with two state variables, interest
rate and housing price, to be more realistic to the market environment and to model
defaults endogenously as a put option. This study extends Kau and Keenan (1995,
1996) and sets up an option-pricing framework to price mortgage insurance contracts
with the considerations of counterparty default risk and regulatory (capital)
forbearance. Our mortgage insurance valuation problem is further complicated by a
number of practical considerations. The liabilities facing these mortgage insurers are
mainly the premiums of mortgage insurance contracts that are interest rate and
housing price sensitive. This fact makes the considerations of interest rate and housing
price risks particularly important in modeling the insurer’s liability. This paper uses
the square-root process of Cox et al. (1985) to describe the stochastic process of the
interest rate and adopts a jump-diffusion model to describe housing price dynamics to
reflect the catastrophic events of natural and financial disasters (such as subprime
mortgage crisis).
Our model and numerical analyses show that the intensity and severity of
catastrophic events for the housing price have identified a significant impact on the
default risk premiums of the mortgage insurance contract. The higher the mortgage
insurer’s initial asset–liability structure is, the lower the default risk of the mortgage
insurer and the higher the value of the mortgage insurance will be. Not surprisingly,
interest rate risk management practice of a mortgage insurer will affect the mortgage
insurance. Finally, we also demonstrate that a lower forbearance threshold increases
the impact of delay on the mortgage insurance, and that a longer time of delay
amplifies the impact of forbearance threshold on mortgage insurance.
4
The next section presents the model of interest rate, housing price, liability and
asset dynamics of a mortgage insurer. Section 3 specifies the payoffs of mortgage
insurance for various scenarios and derives closed-form formulae for the mortgage
insurance contracts. Section 4 provides the numerical analyses and discussion of the
results. The last section draws conclusions about our findings and their implications.
2. A model for the mortgage insurer
This study adopts a structural approach to calculate the default probability of
mortgage insurers issuing mortgage insurance contracts. Structural models provide a
link between the credit quality of a firm and the firm’s economic and financial
conditions such as the firm’s assets and capital structure. Hence, defaults are
endogenously generated within the model. Specifically, our structural model follows
from Cummins (1988), Duan et al. (1995), and Duan and Yu (2005), Lee and Yu
(2002, 2007) and it allows for the dynamics of interest rates and liabilities. This
structural model allows a mortgage insurer suffering from the default risk of
borrowers to bear higher default risk of its own.
Since the insurer’s asset–liability structure, interest rate and housing price
specification are important factors in determining the values of mortgage insurance,
this section begins by specifying the interest rate process, the borrower’s housing
price process, and the insurer’s asset and liability dynamics. And then we show their
corresponding processes under the risk-neutralized pricing measure.
2.1 The mortgage insurer’s liability process
In the literature the liability process is typically modeled by a lognormal
diffusion process, such as Cummins (1988) and Fries, et al. (1997). This modeling
5
fails to explicitly take into account the impact of stochastic interest rates and housing
prices. This shortcoming is particularly important for modeling the liability value of
the mortgage insurer, because falling house prices and rising interest rates are the
precipitating factors for the catastrophic nature of mortgage insurance. Following
Duan et al. (1995) and Lee and Yu (2007), we model the mortgage insurer’s total
liability value as consisting of other two risk components –interest rate and house
price risks captured by the Wiener components of the interest rate and house price
processes. Hence, the liability value of the mortgage insurer is governed by the
following process:4
dL(t )
dH (t )
  L dt  L ,r dr (t )  L , H
  L dWL ,t
L(t )
H (t )
(1)
where  L is the instantaneous drift of the liability value at time t ; WL ,t is the
Wiener process that pertains to idiosyncratic shocks to the capital market;
L , r 
 *
 L*  Lr
denotes the instantaneous interest rate elasticity, and L , H  L LH
H
 r (t )
represents housing price elasticity of the mortgage insurer’s liability.  L   L* ,

1   Lr2   LH 2 , and  L* is the volatility of the small shocks. From the
Equation (1), the total liability risk can be expressed as
L2,r 2  L2, H  H2   2 .
That is, the total instantaneous variance of mortgage insurer liabilities can be
decomposed into three components: interest rate risk, housing price risk and credit
risk.
2.2 The instantaneous interest rate process
The instantaneous interest rate is assumed to follow the square-root process of
4
The similar detailed derivation may refer to Duan et al. (1995) and Lee and Yu (2007).
6
Cox et al. (1985). This setting avoids the negative interest rate that appears in
Vasicek’s model (Vasicek, 1977). Thus, the instantaneous interest rate process can be
written as follows:
d r( t)  (  r ( t ) )d t
t
v rr, ,td Z
(2)
where  denotes the mean-reverting force measurement;  presents the long-run
mean of the interest rate, and  is the volatility parameter for the interest rate.
It is standard to use the device of risk-neutral pricing technique to price a
derivative contract such as mortgage insurance contracts. According to the Girsanov’s
theorem,
*     r ,  * 
 r (t )

dt ,
, dZ rQ,t  dZ r ,t  r

 r
where term  r is the market price of interest rate risk and is a constant under the
assumption of Cox et al. (1985), and ZrQ,t is a Wiener process under Q , then the
dynamics of the interest rate process under the risk-neutralized pricing measure Q
can be described as
dr (t )   * ( *  r (t ))dt  v r (t ) dZ rQ,t .
(3)
2.3 The housing price process
We follow Kau and Keenan (1996) and describe the housing price dynamics for
the borrower at time t as the combination of a Wiener process and a compound
Poisson process in the following:
dH (t )
= mH dt + s H dWH ,t +
H (t )
é N (t )
ù
êd
ú,
(
Y
1)
l
k
dt
n
ê
ú
êë n= 0
ú
û
å
(4)
where H (t ) is the housing price at time t and mH is the expected growth rate of
7
housing price.  H is the constant volatility of the Wiener component of the housing
price process. Term WH (t ) : t  0 is a standard Wiener process. The role of WH (t )
is the unanticipated instantaneous change of housing price, which reflects normal
events, but it may not work so well for abnormal events (such as the catastrophic
events, the government policy, or the mortgage crisis). Here, Yn : n  1, 2,... are
independent for the sequence and identically distributed nonnegative random
variables representing the nth jump size. The jumps in housing prices correspond to
the arrival of abnormal information, and this paper assumes that jump size is
lognormally distributed with mean q and variance d 2 . The last term,   dt ,
1
compensates for the jump process, where   E (Y  1)  exp (   2 ) , i.e., the
2
expected percentage changes in housing price if an abnormal event occurs.
 N (t ) : t  0 represents the total number of jumps during a time interval of (0, t ]
and it is based on a Poisson process with a intensity parameter  . Furthermore, all
three sources of randomness, WH (t ) standard Wiener process, N (t ) Poisson
process, and Yn the jump size, are assumed to be independent.
By the Girsanov’s theorem, dWHQ,t  dWH ,t   H dt , where term  H is the
market price of housing price risk, and assuming that the jump risk is nonsystematic
and diversifiable (i.e. a risk premium to the jump risk is unnecessary), we show that
the dynamic process of housing price under the risk-neutral probability measure Q :
dH (t )
= r (t )dt + s H dWHQ,t +
H (t )
é N (t )
ù
êd
ú.
(
Y
1)
l
k
dt
n
ê
ú
ú
ëê n= 0
û
å
(5)
Based on the risk-neutralized results for the interest rate in the Equation (3) and
housing price in the Equation (5), the dynamics of a mortgage insurer’s liabilities
8
under the risk-neutralized measure can be arranged as follows:
dL(t )
 r (t )dt  L ,r r (t )dZ rQt,  L H, dWHQ t ,  L dWLQt , ,
L(t )
(6)
where WLQ,t is a Wiener process for the risk-neutral probability measure Q and is
independent of ZrQ,t and WHQ,t .
2.4 The mortgage insurer’s asset process
We assume that the asset dynamics of the mortgage insurer follows a lognormal
diffusion process and consider the impact of stochastic interest rates on the asset value
as follows:
dA(t )
  A dt   A,r dr (t )   A dWA,t ,
A(t )
where  A is the instantaneous drift of the asset value at time t ;
(7)
WA,t is the
Wiener process that denotes the credit risk on the assets of the mortgage insurer;
 A,r 
 *A  Ar
is the instantaneous interest rate elasticity of the asset of the mortgage
 r (t )
insurer;  A   *A ,  
1   Ar2 , and  *A is the volatility of the credit risk. In a
risk-neutral world, the asset of the mortgage insurer can be risk neutralized as follows:
dA(t )
 r (t )dt  A,r r (t )dZ rQt,   AdWAQt ,,
A(t )
(8)
where the term WAQ,t is the Wiener process under the risk-neutral measure and is
independent with ZrQ,t .
9
3. Valuation of Mortgage insurance
Once the risk-neutral processes of interest rate, housing price, liability, and asset
dynamics are known, one can value the mortgage insurance by discounting expected
payoffs in the risk-neutral world. This section specifies the payoffs of mortgage
insurance under alternative considerations. We first present the base case, in which
mortgage insurers will not default (i.e., no counterparty risk exists), and then look into
the mortgage insurance with counterparty default risk. Finally, we evaluate how
regulatory actions and capital forbearance affect the values of mortgage insurance.
3.1 Payoffs of mortgage insurance
At time t  0 , i.e., at origination, the lender issues a T  month mortgage,
secured by the housing, for the amount of B(0)  LV H (0) . Let LV be the initial
loan-to-value ratio and H (0) is the initial housing price. We assume that the
mortgage loan has a fixed interest rate c (risk-free interest rate plus a spread,
r (t )  spread ) and that installments y are paid monthly. Hence, with no prepayment
or default prior to time t , the loan balance B (t ) at time 0  t  T is given by the
following expression:
B(t ) 
y
1
1 
c  (1  c)T t

.

(9)
In addition, at time t  0 , the mortgage insurer writes a mortgage insurance that
promises to compensate the lender only when the borrower defaults. The following
section presents the payoffs of mortgage insurance under alternative considerations.
10
3.1.1 Without counterparty default risk
In the case where the mortgage insurer will not default, we follow Kau and
Keenan (1995, 1996) and model mortgage insurance as a portfolio of American put
option. Thus, the payoffs of mortgage insurance contact without counterparty risk at
time t , M ND (t ) , is given as follows:
 LR B(t )

M ND (t )  ( B(t )  H (t ))

0

if H (t )  (1  LR ) B(t )
if (1  LR ) B(t )  H (t )  B(t )
(10)
if H (t )  B(t )
where LR denotes the coverage ratio. Equation (10) implies that if the housing price
H (t ) exceeds the remaining loan balance B (t ) at time 0  t  T , it implies that the
borrower would not default at time 0  t  T and thus the loss of the mortgage
insurer
is
zero.
On
the
other
hand,
if
(1  LR ) B(t )  H (t )  B(t )
or
H (t )  (1  LR ) B(t ) at time 0  t  T , it shows that the housing price is not sufficient
for a full repayment of the loan balance, and the possibility that the borrower defaults
increases such that the mortgage insurer cover the remaining loan balance up to some
maximum coverage LR in return for receiving the defaulted housing H (t ) .
3.1.2 With counterparty default risk
Given the each time interval is a month, the payments of the borrower must be
made at each month and the borrower has the opportunity to default at those times.
When the mortgage insurer itself is default-risky, the payoffs of mortgage insurance at
time t , M D (t ), is given as follows:
11
 LR B(t )
 L B(t ) A(t )
 R
L(t )


M D (t )  ( B(t )  H (t ))
 ( B(t )  H (t )) A(t )

L(t )


0

if H (t )  (1  LR ) B(t ) and A(t )  L(t )
if H (t )  (1  LR ) B(t ) and A(t )  L(t )
if (1  LR ) B(t )  H (t )  B(t ) and A(t )  L(t )
(11)
if (1  LR ) B(t )  H (t )  B(t ) and A(t )  L(t )
otherwise
The first and third terms of Equation (11) represent the loss of the mortgage
insurer when the borrower default at time 0  t  T and the mortgage insurer does
not default during the remaining life of the mortgage insurance; namely, during the
remaining life of the mortgage insurance, the value of the mortgage insurer’s total
asset A(t ) is greater than the expected default trigger L (t ) , and thus the default of
the mortgage insurer does not occur. The second and fourth terms of this equation
presents recovery values of the mortgage insurance when the mortgage insurer
defaults at time 0  t  T , i.e., A(t ) is less than L (t ) , and the borrower also default
at time 0  t  T during the remaining life of the mortgage insurance. The recovery
value is, therefore, equal to the nominal claim of the mortgage insurance multiplied by
the recovery rate, A(t ) L(t ) . Therefore, the setup of Equation (11) indicates that our
mortgage insurance contract embeds a portfolio of vulnerable American puts that may
be exercised not only when the mortgage borrowers default but also when the contract
is forced to be terminated by the default of mortgage insurers.
3.1.3 With counterparty default risk and capital forbearance
In the previous section, we assume an insolvent mortgage insurer will liquidate
its assets to meet its obligations. The mortgage insurance contact will be terminated
and the insurer will not be in operation. In reality, however, undercapitalized mortgage
insurers are not closed immediately and insolvent insurers are still allowed to operate.
12
This phenomenon resembles the capital forbearance observed in the banking industry.
We model capital forbearance along the line of Ronn and Verma (1986) and
Duan and Yu (1994). At the time 0  t  T , a mortgage insurance institution cannot
be taken over unless its asset value falls below the forbearance threshold,  L(t ) ,
where  is the forbearance parameter and is taken to be less than or equal to one.
Even if its asset value of the mortgage insurer cannot meet the capital standard, as
long as it does not fall below the forbearance threshold, the mortgage insurer will not
be forced to face an immediate intervention and can indeed extend its operation until
t   . Although capital forbearance alters the conditions for triggering a resolution,
the resolution will, if it takes place, fully restore the asset value to the mortgage
insurer’s outstanding liabilities, L (t ) . Hence, the cash flow of the mortgage insurance
with the counterparty default risk and capital forbearance at time t , M C (t ) , can be
characterized as:
 LR B(t )
 F (t )

 LR B(t ) A(t )

L(t )


M C (t )  ( B(t )  H (t ))
G (t )

 ( B(t )  H (t )) A(t )

L(t )


0
if H (t )  (1  LR ) B(t ) and A(t )  qL(t )
if H (t )  (1  LR ) B(t ) and  L(t )  A(t )  qL(t )
if H (t )  (1  LR ) B(t ) and A(t )   L(t )
if (1  LR ) B(t )  H (t )  B(t ) and A(t )  qL(t )
(12)
if (1  LR ) B(t )  H (t )  B(t ) and  L(t )  A(t )  qL(t )
if (1  LR ) B(t )  H (t )  B(t ) and A(t )   L(t )
otherwise
where the parameter q reflects the capital standard set by the regulatory authority,
which is the lower bound of the asset value of the mortgage insurer. Generally,
mortgage insurer must operate within a 25-to-1 ratio of risk to capital, which means
they set aside $1 of capital for every $25 of risk they insure. Insured risk is defined as
the percentage of each loan covered by an insurance policy. This capital standard can
be translated into q  (25  1) / 25  1.04 in our model. Furthermore, F (t ) and
13
G (t ) are the value from the extended operation under forbearance, and their cash
flow at the time of (t   ) can be described as:
if A(t   )  L(t   ),
 LR B(t   )

F (t   )   LR B(t   ) A(t   )
otherwise.

L(t   )

(13)
if A(t   )  L(t   ),
( B(t   )  H (t   ))

G (t   )   ( B(t   )  H (t   )) A(t   )
otherwise.

L(t   )

3.2 Valuation of mortgage insurance contracts
To understand the effect of capital requirement and capital forbearance on the
mortgage insurance premiums, in this section, we consider a single-period case of the
mortgage insurance (single payment loans, i.e., t  T )
with the following
simplifications. The interest rate simplifies to be constant, and the housing price of the
borrower and the liability of the mortgage insurer are independent. Based on these
conditions, we intend to ascertain the exact relationship of the mortgage insurance
premium under capital standard and forbearance.
Based on Equation (10), using Merton’s (1976) pricing results for European put
options with constant interest rate, the closed-form formula of mortgage insurance can
be represented, DL1 , as follows:
¥
DL1 =
å
e x p- (l k (+
m= 0
m
T1 )(l k) + ( T ) 1 )
[P( K 1 )- P K( 2],)
m!
where P( Ki ) = Ki exp(- rm T ) N (- d2m ( Ki )) - H (0) N (- d1m ( Ki )), i = 1, 2,
ln( H (0) Ki ) + (rm ±
d1m,2 m ( Ki ) =
s m2 T
s m2
)T
2
, K1 = B(T ), K2 = (1- LR ) B(T ),
14
(14)
rm = r - l k +
m q + 0.5m d2
md 2
, s m2 = s H2 +
.
T
T
DL1 , can be duplicated by a long position in a European put option with a strike
price K1 and a short position in a European put option with a strike price K 2 , both
with the time to maturity equal to the time T . If l = 0, it means that no abnormal
shock event occurs, and so the volatility of housing price process only captures the
normal volatility.
Furthermore, to derive the closed-form formula of the mortgage insurance in
response to considering the counterparty default risk and capital forbearance, we
directly assume that the asset-debt ratio of the mortgage insurance company
A(t ) L(t ) follows a geometric Brownian motion, which replaces the setup of A(t )
and L (t ) in Equations (5) and (7). Thus the dynamics of the asset-debt ratio of the
mortgage insurance company under the risk-neutrality are given as:
dR(t )
 rt dt   R dWRQ,t ,
R(t )
where  R denotes the constant volatility of the instantaneous rate of return of R (t ) ,
which represents the uncertainty sources of both the mortgage insurer’s total asset
value and total liability value. Hence, when involving counterparty default risk and
capital forbearance, the pricing formula of the mortgage insurance, DL2 , can be
described as follows:
é
æ
ö
æ
ö
DL2 = DL1 êR(0) ççN (a2 ) - N (b 1 )÷
+ R(0) ççN (d1 ( K3 ), a2 , m) - N (d1 ( K3 ), b2 , m )÷
÷
÷
÷
÷
ê
èç
ø
èç
ø
ë
+ N ( d2 ( K3 ) , a1 , m
- )
where
15
ù
N (2d (3K )1,bú ,m(15)
)
ú
û
1 2
s R (T + t )
2
,m=
s R2 (T + t )
ln(1 R (0)) ±
d1,2 ( K 3 ) =
T
T+ t
1
1
ln( q R (0)) m s R2 T
ln(r R(0)) m s R2 T
2
2
a1,2 =
, b1,2 =
.
2
2
sRT
sRT
We observe that the values of mortgage insurance under forbearance DL2 are greater
than those of a portfolio of Merton's put DL1 and the excess premiums are the
forbearance costs. The above closed-form solution can in fact be considered a general
form of mortgage insurance pricing under capital forbearance and one with a number
of special cases. For example, it yields the case of Bardhan et al. (2006) when we set
R(0) = ¥ ,   0 ,   q  1 and l = 0 .
Capital forbearance
The immediate impact of granting capital forbearance is best demonstrated by
the following derivative property:
n  b1 
DL2 DLC DL
( R(0)  1)   N  dk 2   N  d k1   0, (16)


 DL2



  R2 T
where
n  b1 
DLC
 DL2
( R(0)  1)  0 or  0;

  R2 T
n  b1 
DL
 DL2
   N  dk 2   N  dk1   0;

  R2 T
d k1 
 ln  
 R2
 R2
2

, d k 2  d k1   R2 ;
and n denotes the probability density function of a standard normal variable. The
16
sign of
DL2

is negative indicating that a higher value of  makes less
undercapitalized insurers will be forborne and then this will reduce the cost of
mortgage insurance. We also note that the impact of the forbearance on the mortgage
 DLC 
insurance includes a positive (or negative) capital-component effect 
 and a




 DL
negative time-component effect 
 

 . In other words, when the asset value is

larger (or less) than the liability value of the mortgage insurer, the positive (or
negative) capital-component effect reflects that less (or more) insolvencies will be
resolved at T , whereas the negative time-component effect shows that more
insolvencies will be resolved at T   . The negative aggregate effect indicates that
the time component dominates the capital component. This shows that the forbearance
policy can save the loss of mortgage insurer temporarily by delaying closures, but the
future resolution cost dominates the loss savings and rises up the cost of mortgage
insurer.
Capital Requirements
The impact of adjusting capital requirements is demonstrated by the following
derivative property:
DL2
1
 DL2
 R( 0 n) a(2
q
q  R2 T
)(N
1 d k 3 (  qn
) )a 2 N (d k )4 (
)(17)0 ,
where
dk 3 
The positive sign of
 ln q 
 R2
 R2
2

, d k 4  d k 3   R2 .
DL2
indicates that the higher level of required capital
q
17
increases the cost of mortgage insurance. This property makes intuitive that higher
capital requirement may let more undercapitalized, but solvent, insurers to extend
their coverage to time T   . The extension of coverage will increases the value of
mortgage insurance.
The mortgage insurance value is expected to be positively related to time delay,
and can be shown by the following equation:
æ a - md ( K )
s R2
b - md1 ( K3 )
¶ DL2
1
3
=
R(0) n(d1 ( K3 )) çççN ( 2
)- N( 2
)
¶t
4 (T + t )
èç
1- m 2
1- m 2
+N(
a1 - md 2 ( K3 )
1- m 2
)- N(
b1 - md 2 ( K3 ) ö
÷
)÷
> 0.
÷
÷
2
ø
1- m
This result indicates that extending the time delay to resolve the undercapitalized
mortgage insurer raises the cost of mortgage insurer, and the longer the time delay, the
higher the cost of the mortgage insurer.
Cross Effect of Forbearance and Time of Delay
The two key policy parameters of capital forbearance are the forbearance
threshold parameter   and the time of delay parameter   . We find that the
forbearance parameter is negatively related to the delay parameter based on the
following partial differential:
n  b1  n  d k1 
DL22
  DL2
 0.

2  T
The negative relation interprets that a lower capital forbearance threshold allows more
undercapitalized mortgage insurer to extend their operations, and increases the
 DL2 
positive impact of delay on the mortgage insurer cost 
 . Meanwhile, it shows




that a longer time of delay permits the undercapitalized mortgage insurer to operate
18
over a longer period of time, and therefore lengthens the negative impact of the capital
 DL2 
forbearance threshold on the cost of mortgage insurer 
.
  
4. Numerical Analysis
This section estimates the values of mortgage insurance for alternative scenarios
using the least-squares approach provided by Longstaff and Schwartz (2001). This
method is simple, but powerful, for valuing American-style options. To consistent
with the fact that payments must be made each month in practice, the simulations are
run on a monthly basis with 50,000 paths and then the mortgage insurance premiums
are monthly premiums. A base set of parameters is established and summarized in
Table 1.
[Table 1 is here]
Deviations from the base values provide insights into how changes in the
characteristics of the asset–liability structure of mortgage insurer, the frequency and
severity of catastrophe events of housing price, interest rate process and capital
regulatory policy affect the mortgage insurance values.
We first consider the case where the mortgage insurer may default, i.e., with
counterparty default risk, and then report the default free and default risky mortgage
insurance premium rates for alternative sets of catastrophic intensity of housing price
and the mean and variances of jump size of housing price. The mortgage insurance
premium rates are calculated by prices of mortgage insurance premium as a
percentage of the initial loan value. The differences between the default free and
19
default risky mortgage insurance premium rates are the default risk premiums, which
are shown in Table 2. As expected, default free mortgage insurance premium rates are
more valuable to the mortgage insurer and have higher values than their
corresponding default risky mortgage insurance premium rates. The higher the
mortgage insurer’s initial asset–liability structure is, the lower the default risk and the
higher the mortgage insurance premium rates will be.
[Table 2 is here]
Furthermore, observe that the default risk premiums increases with the
catastrophic intensity and the mean and variance of jump size of housing price. For
instance, in the case where the mortgage insurer’s initial asset–liability structure
( A L) is 1.3, mean of jump size ( ) is -0.01 and variance of jump size ( 2 ) is 1,
the default risk premium increases from 0.088 basis points to 0.136 basis points when
the catastrophic intensity ( ) increases from 0.5 to 1, and rises to 0.360 basis points
when the catastrophic intensity increases to 2. The default risk premium increases
with catastrophic intensity and the mean of jump size and volatility of housing price,
and the premium can be substantial in the presence of jump risk of housing price and
should not be neglected in the valuation of mortgage insurance policy. For example,
in the case where ( , ,  2 )  (2,0.01,0.2) , the default risk premium goes from 0.405
basis points to 0.551 basis points when ( A L) decreases from 1.5 to 1.1, an
increment of 14.6 basis points. The increment of default risk premium drops down to
about 7 basis points for the case of ( , ,  2 )  (0.5,0.01,0.05) .
Table 3 reports the premium rate of mortgage insurance contract while fixing
20
( , ,  2 )  (1, 0.01,1). We set (L , H )  2, 1, 0 to measure different scenarios of
housing sensitivity of asset, and set the interest rate elasticity pair to
( A,r , L ,r )  (2, 1) , ( A, r , L , r )  (1 L / A, 1) and ( A,r , L ,r )  (0, 0) , respectively to
measure different scenarios of interest rate elasticity gaps. We find that the housing
sensitivity of asset is an increasing function of mortgage insurance premium rate.
Moreover, in the case of ( A,r , L ,r )  (0, 0) , both assets and liabilities are interest rate
insensitive, whereas the scenario of (A, r , L, r )  (1 L / A, 1) has a zero size-adjusted
interest rate elasticity gap, which can be regarded as an active interest rate risk
management practice that strives to eliminate the interest rate risk facing the mortgage
insurer by adjusting the asset–liability mix. The case of ( A, r , L , r )  (2, 1) presents
a positive duration gap on the asset–liability structure. The difference in the premium
rates for ( A, r , L , r )  (2, 1) and that for (A, r , L, r )  (1 L / A, 1) reflects the interest
rate risk. The interest rate risk effect is quite substantial and is more obvious for a
lower leverage mortgage insurer. It is also not surprising to see no effect caused by
the interest rate risk premium when both assets and liabilities are interest rate
insensitive, i.e., for the case of ( A,r , L ,r )  (0, 0) . Note that the mortgage insurance
premium rates for the zero-gap and insensitivity scenarios are very similar. This
indicates that active interest risk management can achieve the same effect as sticking
exclusively to interest rate insensitive assets and liabilities, regardless of the housing
price sensitivity of asset. This indicates that a mortgage insurer that employs interest
rate management can effectively have a much lower interest rate risk and can increase
the values of its mortgage insurance contracts.
[Table 3 is here]
21
The premium rate of mortgage insurance under alternative levels of the
forbearance threshold
  ,
capital requirement
q
and time to delay   are
reported in Table 4. As forbearance extends the insurance coverage to the
undercapitalized mortgage insurer, the premium rate of mortgage insurance increases
with a lower forbearance threshold. For example, under the situations of
( ,  2 , ) =(2,0.2,6) and A / L =1.3, the premium rate of mortgage insurance increases
from 323.7 basis points to 359.0 basis points when the forbearance threshold  
decreases from 100% to 95%, and rises to 39.73 basis points when the forbearance
threshold   decreases from decreases to 90%. In addition, the sensitivities of the
forbearance threshold effect increase with their initial levels of leverages. For instance,
the mortgage premium rate for a borrower with initial leverage ratio at 1.1 increases
by 52.6 basis points when the forbearance threshold reduces from 100% to 90%, and
the changes are more substantial (86.5 basis points) for mortgage insurers with a
leverage at 1.5. Meanwhile, it shows that the premium rate of mortgage insurance can
be saved by delaying resolution of undercapitalized mortgage insurers, which
ultimately raises the cost of mortgage insurance. And the time to delay effect could be
more substantial as initial leverage ratio increases. For instance, the mortgage
premium rate with initial leverage ratio at 1.1 increases by 85 basis points when the
time to delay rises from 4 to 6 months, whereas the changes goes up to 103.4 basis
points for mortgage insurers with a leverage at 1.5. Moreover, increased capital
requirement will result in increased mortgage insurance premium rates. In brief,
counterparty default effect, forbearance threshold effect, time to delay effect, and
capital requirement effect are critical factors in the premium of mortgage insurance.
[Table 4 is here]
22
5. Conclusions
We have developed a model for measuring the mortgage insurance premium rate
by taking into account the default risk of the mortgage insurer, interest rate risks,
catastrophic intensity and severity of the housing price, and capital regulatory policy.
We treat forbearance as an option to delay the resolution of undercapitalized mortgage
insurer in the current regulatory framework and then derive a closed-form solution
with some specific conditions, to examine and measure its impacts, and to discuss its
possible implications.
The results indicate that the default risk premium can be substantial in the
presence of catastrophic risk of the housing price and should not be neglected in the
valuation of the mortgage insurance policy. Our results have interesting implications
for forbearance policy. We show that the forbearance policy can save the loss of
mortgage insurer temporarily by delaying closures, but the future resolution cost
dominates the loss savings and rises up the cost of mortgage insurer. Furthermore,
time to delay effects is positive related by the value of mortgage insurance. And a
lower forbearance threshold increases the impact of delay on the mortgage insurance
value, and that a longer time of delay amplifies the impact of forbearance threshold on
mortgage insurance value. And forbearance threshold and time to delay effects could
be more significant as initial leverage ratio increases. This implies that it would raise
the loss of mortgage insurer ultimately, instead of saving the cost of mortgage insurer,
when the capital forbearance policy is considered. In sum, counterparty default effect,
forbearance threshold effect, time to delay effect and capital requirement effect are all
critical factors on the mortgage insurance valuation.
23
References
1.
Bardhan, A., R. Karapandža, and B. Urošević, 2006, Valuing Mortgage Insurance
Contracts in Emerging Market Economies, Journal of Real Estate Finance and
Economics, 32, 9–20.
2.
Canner, G. B., and W. Passmore, 1994, Private Mortgage Insurance, Federal
Reserve Bulletin, 9, 883-899.
3.
Cox, J., Ingersoll, J., and S., Ross, 1985, The Term Structure of Interest Rates,
Econometrica 53, 385–407.
4.
Cummins, J. D., 1988, Risk-Based Premiums for Insurance Guaranty Funds,
Journal of Finance 43, 593–607.
5.
Dennis, B., C., Kuo, and T., Yang, 1997, Rationales of Mortgage Insurance
Premium Structures, Journal of Real Estate Research, 14(3), 359-378.
6.
Duan, J. C., Moreau, A., and C.W., Sealey, 1995, Deposit Insurance and Bank
Interest Rate Risk: Pricing and Regulatory Implications, Journal of Banking and
Finance, 19, 1091–1108.
7.
Duan, J. C., and M. T., Yu, 1994, Forbearance and Pricing Deposit Insurance In a
Multiperiod Framework, Journal of Risk and Insurance, 61(4), 575-591.
8.
Duan, J. C., and M. T., Yu, 2005, Fair Insurance Guaranty Premia in the Presence
of Risk-Based Capital Regulations, Stochastic Interest Rate and Catastrophe Risk,
Journal of Banking and Finance, 29 (10), 2435–2454.
9.
Fries, S. M., P. Mella-Barral and W., R. M. Perraudin, 1997, Optimal Bank
Reorganization and the Fair Pricing of Deposit Guarantees, Journal of Banking
and Finance, 21, 441-468.
10. Hendershott, P., and R., Van Order, 1987, Pricing Mortgages: Interpretation of
the Models and Results, Journal of Financial Services Research, 1(1), 19-55.
11. Kane, E. J., 1986, Appearance and Reality in Deposit Insurance-The Case for
Reform, Journal of Banking and Finance, 10, 175 -188.
12. Kau, J. B., and D. C. Keenan, 1995, An Overview of the Option-Theoretic
24
Pricing of Mortgages, Journal of Housing Research, 6(2), 217-244.
13. Kau, J. B., and D. C. Keenan, 1996, An Option-Theoretic Model of Catastrophes
Applied to Mortgage Insurance, Journal of Risk and Insurance, 63(4) 639-656.
14. Kau, J., B., D. C. Keenan, and J. E. Epperson, 1992, A Generalized Valuation
Model for Fixed-Rate Residential Mortgages, Journal of Money, Credit and
Banking, 24, 280-299.
15. Kau, J. B., D. C. Keenan, and W. J. Muller, 1993, An Option-Based Pricing
Model of Private Mortgage Insurance, Journal of Risk and Insurance, 60(2),
288-299.
16. Kau, J. B., D. C. Keenan, W. J. Muller, and J. E. Epperson, 1995, The Valuation
at Origination of Fixed-Rate Mortgages with Default and Prepayment, Journal of
Real Estate Finance and Economics 11, 3-36.
17. Longstaff, F., and E., Schwartz, 2001, Valuing American Options by Simulation:
A Simple Least-Squares Approach,
Review of Financial Studies, 14, 113-147.
18. Lee, J. P., and M. T., Yu, 2007, Valuation of Catastrophe Reinsurance with
Catastrophe Bonds, Insurance Mathematics and Economics, 41(2), 264-278.
19. Merton, R. C., 1976, Option Pricing when Underlying Stock Returns are
Discontinuous, Journal of Financial Economics, 3, 125–144.
20. Merton, R. C., 1977. An analytic derivation of the cost of deposit insurance and
loan guarantee, Journal of Banking and Finance, 1, 3–11.
21. Nagarajan, S., and C., W. Sealey, 1995, Forbearance, Prompt Closure, and
Incentive Compatible Bank Regulation, Journal of Banking and Finance, 19,
1109-1130.
22. Rapkin, C., 1973, The Private Insurance of Home Mortgages. Revised Edition.
23. Ronn, E. I. and A. K. Verma, 1986, Pricing Risk-Adjusted Deposit Insurance: An
Option-Based Model, Journal of Finance, 41, 4, 871-895.
24. Schwartz, E., S. and W., N. Torous, 1992, Prepayment, Default, and the
Valuation of Mortgage Pass-Through Securities, Journal of Business,
65(2):221-239.
25
25. Vasicek, O.A., 1977, An Equilibrium Characterization of Term Structure, Journal
of Financial Economics, 5, 177–188.
26
Table 1
Parameters definitions and base values
Asset parameters
Values
A(0)
Mortgage insurer’s asset
A/L=1.1,1.3 and 1.5
 A, r
Interest rate elasticity of asset
-1, 0
A
Volatility of credit risk
5%
Liability parameters
L(0)
Mortgage insurer’s liability
100
L , r
Interest rate elasticity of liability
-1, 0
L , H
Housing price elasticity of liability
-1, 0
L
Volatility of credit risk
5%
Interest rate parameters
r (0)
Initial instantaneous interest rate
5%

Magnitude of mean-reverting force
0.2

Long-run mean of interest rate
5%

Volatility of interest rate
10%
r
Market price of interest rate risk
-0.01
Housing price parameters
H (0)
Borrower’s housing price
200,000
sH
Volatility of housing price risk
5%
f
Interest rate elasticity of housing price
-1, 0
Mean of jump size
-1% and 1%
Variance of jump size
5%, 10% and 20%
Jump intensity
0.5, 1 and 2
H ,r
q
d
2

Other parameters
LR
Insurance coverage level
25%
y
Installments monthly
500
c  r  spread
Contract rate
Spread=2%
T
Loan maturity
360 months

Forbearance parameter
100%, 95% and 90%

The time of delay
2, 4 and 6 months
q
Capital standard
1.04, 1.05 and 1.06
27
Table 2
The mortgage insurance premium rates with counterparty default risk but no capital
forbearance
( ,  ,  2 )
Default risky MI rate
Default risk premiums
A/ L
A/ L
Default free
MI rate
1.1
1.3
1.5
1.1
1.3
1.5
(0.5,-0.01, 0.05)
0.811%
0.686%
0.723%
0.756%
0.125%
0.088%
0.055%
(0.5, -0.01, 0.1)
0.864%
0.698%
0.755%
0.785%
0.186%
0.109%
0.079%
(0.5, -0.01, 0.2)
1.567%
1.197%
1.273%
1.359%
0.370%
0.294%
0.208%
(1, -0.01, 0.05)
0.915%
0.718%
0.779%
0.814%
0.197%
0.136%
0.101%
(1, -0.01, 0.1)
1.086%
0.688%
0.765%
0.833%
0.398%
0.321%
0.253%
(1, -0.01, 0.2)
1.912%
1.480%
1.587%
1.644%
0.432%
0.325%
0.268%
(2, -0.01, 0.05)
1.193%
0.752%
0.833%
0.953%
0.441%
0.360%
0.240%
(2, -0.01, 0.1)
1.647%
1.180%
1.251%
1.360%
0.467%
0.396%
0.287%
(2, -0.01, 0.2)
2.980%
2.519%
2.562%
2.593%
0.461%
0.418%
0.387%
0.871%
0.706%
0.777%
0.796%
0.165%
0.094%
0.075%
0.934%
0.738%
0.797%
0.845%
0.196%
0.137%
0.089%
1.637%
1.258%
1.293%
1.419%
0.379%
0.344%
0.218%
0.975%
0.758%
0.801%
0.845%
0.217%
0.174%
0.130%
1.146%
0.738%
0.785%
0.855%
0.408%
0.361%
0.291%
1.982%
1.540%
1.611%
1.705%
0.442%
0.371%
0.277%
1.273%
0.812%
0.893%
1.015%
0.461%
0.380%
0.258%
1.737%
1.240%
1.311%
1.427%
0.497%
0.426%
0.310%
3.030%
2.479%
2.582%
2.625%
0.551%
0.448%
0.405%
(0.5,0.01, 0.05)
(0.5, 0.01, 0.1)
(0.5, 0.01, 0.2)
(1, 0.01, 0.05)
(1, 0.01, 0.1)
(1, -0.01, 0.2)
(2, 0.01, 0.05)
(2, 0.01, 0.1)
(2, 0.01, 0.2)
This table presents mortgage insurance premium rates and default risk premiums with counterparty default risk
but no capital forbearance. This table also presents the default risk premium for alternative interest rate and
housing sensitivity structure. The mortgage insurance premium rates are calculated by prices of mortgage
insurance premium as a percentage of the initial loan value and reported for alternative sets of catastrophe
intensities ( ) and catastrophe loss mean and volatilities ( ,  2 ) . A / L represents the initial asset–liability
structure of the mortgage insurers. All estimates are computed using 50,000 sample paths. Other parameter
values are specified in Table 1.
28
Table 3
The mortgage insurance premium rates with counterparty default risk for alternative
interest rate and housing sensitivity structure
A/ L
1.1
L , H
1.3
1.5
( A, r , L , r )  (0, 0)
0
1.498%
1.564%
1.789%
-1
0.709%
1.061%
1.341%
-2
0.228%
0.616%
0.918%
L , H
( A, r , L , r )  (1 L / A, 1)
0
1.505%
1.514%
1.768%
-1
0.694%
1.011%
1.311%
-2
0.208%
0.566%
0.885%
L , H
( A, r , L , r )  (2, 1)
0
1.225%
1.304%
1.629%
-1
0.514%
0.891%
1.231%
-2
0.112%
0.486%
0.834%
This table presents the mortgage insurance premium rates with counterparty default risk for alternative
interest rate and housing sensitivity structure. The mortgage insurance premium rates are calculated by
prices of mortgage insurance premium as a percentage of the initial loan value and reported for
alternative sets of housing sensitivity ( L , H ) of the mortgage insurer’s liabilities, interest rate
sensitivities ( A, r , L , r ) of the mortgage insurer’s assets and liabilities. A / L represents the initial
asset–liability structure or capital position of the mortgage insurers. All estimates are computed using
50,000 sample paths. Other parameter values are specified in Table 1.
29
Table 4
The mortgage insurance premium rates with counterparty default risk and capital
forbearance
Capital
1.04
requirement
Capital
forbearance
( ,  2 ,  )
100%
1.05
1.06
95%
90%
100%
95%
90%
100%
95%
90%
A / L  1.1
(0.5, 0.05, 2)
0.185%
0.221%
0.299%
0.204%
0.261%
0.330%
0.231%
0.290%
0.369%
(0.5, 0.1, 4)
0.194%
0.236%
0.310%
0.217%
0.279%
0.349%
0.244%
0.308%
0.379%
(0.5, 0.2, 6)
0.202%
0.255%
0.324%
0.219%
0.290%
0.368%
0.250%
0.321%
0.399%
(1, 0.05, 2)
0.281%
0.329%
0.386%
0.300%
0.358%
0.427%
0.317%
0.377%
0.458%
(1, 0.1, 4)
0.594%
0.686%
0.801%
0.650%
0.715%
0.796%
0.667%
0.730%
0.812%
(1, 0.2, 6)
0.924%
1.026%
1.147%
0.994%
1.125%
1.267%
1.107%
1.337%
1.579%
(2, 0.05, 2)
1.491%
1.594%
1.747%
1.590%
1.733%
1.892%
1.685%
1.868%
2.090%
(2, 0.1, 4)
2.514%
2.693%
2.900%
3.174%
3.354%
3.590%
3.434%
3.654%
3.924%
(2, 0.2, 6)
3.224% 3.477%
A / L  1.3
3.750%
3.413%
3.680%
3.976%
3.630%
3.860%
4.170%
(0.5, 0.05, 2)
0.204%
0.258%
0.319%
0.231%
0.289%
0.361%
0.254%
0.317%
0.399%
(0.5, 0.1, 4)
0.207%
0.262%
0.327%
0.224%
0.292%
0.362%
0.257%
0.321%
0.398%
(0.5, 0.2, 6)
0.216%
0.273%
0.342%
0.230%
0.296%
0.379%
0.263%
0.345%
0.429%
(1, 0.05, 2)
0.304%
0.361%
0.424%
0.317%
0.378%
0.449%
0.340%
0.405%
0.487%
(1, 0.1, 4)
0.607%
0.699%
0.818%
0.667%
0.733%
0.820%
0.680%
0.810%
0.842%
(1, 0.2, 6)
0.997%
1.099%
1.240%
1.007%
1.147%
1.298%
1.121%
1.419%
1.749%
(2, 0.05, 2)
1.506%
1.647%
1.879%
1.603%
1.749%
1.911%
1.738%
1.919%
2.140%
(2, 0.1, 4)
2.527%
2.750%
2.977%
3.187%
3.402%
3.627%
3.854%
4.067%
4.292%
(2, 0.2, 6)
3.237% 3.590%
A / L  1.5
3.973%
3.429%
3.699%
4.007%
3.915%
4.112%
4.430%
(0.5, 0.05, 2)
0.219%
0.275%
0.345%
0.242%
0.307%
0.386%
0.269%
0.336%
0.425%
(0.5, 0.1, 4)
0.224%
0.285%
0.364%
0.236%
0.307%
0.379%
0.274%
0.340%
0.425%
(0.5, 0.2, 6)
0.233%
0.298%
0.372%
0.247%
0.333%
0.420%
0.271%
0.357%
0.449%
(1, 0.05, 2)
0.321%
0.389%
0.461%
0.334%
0.399%
0.473%
0.360%
0.432%
0.519%
(1, 0.1, 4)
0.624%
0.721%
0.845%
0.684%
0.775%
0.881%
0.700%
0.772%
0.869%
(1, 0.2, 6)
1.050%
1.161%
1.301%
1.034%
1.189%
1.370%
1.148%
1.518%
1.899%
(2, 0.05, 2)
1.523%
1.674%
1.996%
1.720%
1.883%
2.055%
1.869%
2.070%
2.335%
(2, 0.1, 4)
2.544%
2.788%
3.095%
3.204%
3.492%
3.786%
3.899%
4.111%
4.354%
(2, 0.2, 6)
3.264%
3.637%
4.129%
3.486%
3.838%
4.299%
4.149%
4.312%
4.899%
This table presents the mortgage insurance premium rates with counterparty default risk and capital forbearance
for alternative capital parameters. The mortgage insurance premium rates are calculated by prices of mortgage
insurance premium as a percentage of the initial loan value and reported for alternative sets of capital requirement
( q ), capital forbearance (  ) and time to delay (  ). A / L represents the initial asset–liability structure or
capital position of the mortgage insurers. All estimates are computed using 50,000 sample paths. Other parameter
values are specified in Table 1.
30