Fixed Rate Mortgages by fanzhongqing

VIEWS: 1 PAGES: 72

• pg 1
```									           Topic 3:
Fixed Rate Mortgage Mechanics
Fixed Rate Mortgage
Mechanics
• Recall that to the investor (lender), the fixed rate
mortgage is a type of annuity.
– The investor pays the borrower an up-front amount in
return for a promised stream of future cash flows.
– At time zero (i.e. origination) the present value of the
annuity must equal the cash the investor pays the
borrower, i.e.
• Casho = PV0(Future Cash Flows)
• If the cash were worth more than the PV of the future cash flows,
the lender would not be willing to make the loan, since they would
be paying more for the annuity than it was worth.
• If the cash were worth less than the PV of the future cash flows, the
borrower would not be willing to accept the loan because they would
be taking on a liability that is higher than the asset they would
receive (the cash), thus reducing their wealth.
2
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics
• Therefore, at time 0, the only way the two parties
will come to an agreement is if the exchange is
equal: the lender must give the investor an
amount in cash that is equal to the present value
of the remaining future cash flows.
• After time 0, of course, this relationship does not
hold.

3
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics
• The mortgage contract specifies how to
calculate the various cash flows associated with
the mortgage. This includes:
– The “Principal” amount of the loan determines the
monthly payments. This is normally set to the amount
of cash the investor gives the borrower at time 0.
(unless the loan includes points);
– C - the contract rate of the mortgage;
– n - the number of monthly payments; and
– Pmt – the monthly payment on the mortgage.

4
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Balance
• At time 0 we know that the value of the mortgage
is equal to the cash received. For now, we will
assume that the principal is set to that same
amount.
• Thus, the value of the mortgage must have the
following relationship:
                                   
        1                          
 1                              n 

Prin  Pmt * 
1 c
12                      

c
12

5
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Balance
• Thus, we know that if the contract rate were 8%,
with a 20 year (240 payments) term and monthly
payments of \$850, the principal amount must be
101,621.15
              
       1      
1

 1  .08

240 

12 
              
101,621.15 850 *
.08
12

– You can confirm this on your financial calculator by
entering the following: N=240, i=8/12, Pmt=-850.00,
and FV=0, and then solving for PV.

6
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Balance
• Note that this formula actually works for any point during
the life of the mortgage – that is, if you know the remaining
term, the contract rate, and the monthly payment, this
formula tells you the current outstanding principal balance.

            
       1    
1

 1 c

n 

12 
                 
Prin  Pmt *
c
12

7
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Payments
• While knowing how to determine the principal
amount is important, it is perhaps more
interesting (from a potential homeowner’s
standpoint) to know how to calculate the
payment that will be required given a known
balance or loan amount.
• This just requires simple algebraic manipulation
of the balance formula.

8
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics – Payments

Pmt  Prin *
c / 12
          
    1     
1

 1 c


n 

12 
         
– So, for a \$100,000 loan at 10% for 30 years, the
payment is \$877.57.
877.57  100,000*
.10 / 12
              
       1      
1



 1  .10
360 

12 
        
– Again, you can confirm this on your calculator by
setting:
• N=360, i=10/12, PV=100,000, and FV=0.
9
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgages
Mechanics - Payments
Pmt  Prin *
c / 12
          
    1     
1

 1 c

n 

12 
       
• This formula also works at any point in time. That
is, if you know the balance, remaining term, and
contract rate, you can plug those numbers into
the above formula and determine the monthly
payment.

10
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Amortization
• The mortgage contract terms determine the order in
which payments are attributed to the account. The usual
way this occurs is:
•   Overdue interest and penalties are paid first;
•   Current interest is paid second;
•   Overdue principal is paid third;
•   Current principal is paid fourth; and
•   Any remaining cash pre-pays principal are paid last.
• Thus, normally (i.e. when scheduled payments are made
on time), the investor (lender) takes the interest out of
the payment first, and then takes the principal.
• The interest amount is found by multiplying the balance
at the beginning of the month by the monthly interest
rate:
• Interest due = Beginning Balance * c/12.
11
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Amortization
• The principal due can then be found by
subtracting the interest due from the payment:
• Principal Due = Pmt – Interest Due

• From this information we can create an
amortization chart or schedule, which is a table
detailing the periodic payments of the loan.

12
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Amortization
• For a 30 year, 9% mortgage, original balance of
\$200,000, the amortization chart is as follows:
Principal          \$200,000.00 Payments                                    360 Contract rate             9.00%

Beginning             Interest                 Ending
Month           Balance      Payment Due         Principal Due Balance
1     \$200,000.00 1609.245 \$1,500.00       \$109.25 \$199,890.75
2     \$199,890.75 1609.245 \$1,499.18       \$110.06 \$199,780.69
3     \$199,780.69 1609.245 \$1,498.36       \$110.89 \$199,669.80
4     \$199,669.80 1609.245 \$1,497.52       \$111.72 \$199,558.08
5     \$199,558.08 1609.245 \$1,496.69       \$112.56 \$199,445.52
6     \$199,445.52 1609.245 \$1,495.84       \$113.40 \$199,332.11
7     \$199,332.11 1609.245 \$1,494.99       \$114.25 \$199,217.86
8     \$199,217.86 1609.245 \$1,494.13       \$115.11 \$199,102.75
9     \$199,102.75 1609.245 \$1,493.27       \$115.97 \$198,986.77

Note that the above is an Excel spreadsheet – you should be able to “click” on it and actually use it.

13
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Amortization

• Notice the relationships between principal
payment, interest payment and total payment.

1800
1600
1400
1200                                                           Payment
1000
Interest Due
800
600                                                           Principal Due
400
200
0
0   100            200                300         400                   14
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Price
• At origination, the contract rate of the mortgage
equals the market interest rate for the type of
loan and creditworthiness of the borrower.
– It is the equality of the market and contract rates
which forces the balance and value of the mortgage
to be the same at time 0.
• Over time, since the contract rate is fixed, the
contract and mortgage rates will diverge. Thus,
the value and balance of the mortgage will
diverge over time.
– This means we have to concern ourselves with
determining the value (price) of the mortgage at times
other than time t.                                   15
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Price
• To do this we simply take the present value of
the remaining payments using the current
market rate, which is virtually the same formula
that we used to determine the balance:
                                                                      
        1                                                        1    
1                                                          1

Value  Pmt *



1 r   
n 

12 



 1 c


n 

12 

Balance  Pmt *
r                                                          c
12                                                         12

• The difference is that we use the market rate,
r, instead of the contract rate, C.

16
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Example
• At this point it might be useful to look at an extended
example.
•   Consider a borrower who originally took out a \$200,000 loan for 30 years at
9%. Five years have passed and the market rate is now 7%.
– What is the monthly payment on the loan?
– What is the balance of the loan?
– What is the value of the loan?
– The monthly payment is \$1,609.25:
\$1,609.25 200,000*
.09 / 12
              
       1      
1 


 1  .09
360 

12 
   
– Confirm using calculator:
N=360, i=9/12, PV = 200,000, FV = 0, solve for Pmt.

17
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Example
• Example (continued)
– After 5 years the balance is \$191,760:
              
       1      
1

 1  .09

300 

12 
            
191,760.27 1609.25*
.09
12

– You can confirm with a calculator in two ways:
• N = 60, i=9/12, PV = 200,000, Pmt = -1609.25, solve for FV
– Or
• N = 300, i=9/12, Pmt = -1609.25, FV=0, solve for PV.

18
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Example
• Example (continued)
– Also note that you can determine the payment from
ONLY the current balance, contract rate, and
remaining term. You do not need to know the
original term:
\$1,609.25 191,760*
.09 / 12
                 
         1       
1




1  .09
12
300 



– On a calculator:
• N=300, i=9/12, PV = 191,760, FV=0, solve for PMT.

19
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Example
• Example (continued)
– The value of the mortgage, at the 7% contract rate is:
\$227,687.12,
              
       1      
1

 1

.07
300 

12 
               
227,687.12 1609.25*
.07
12

• Confirm with calculator: N=300, i=7/12, PMT = 1609.25, FV=0, solve for PV.

– Contrast this with the balance which is still:

                   
            1      
1


 1  .09

300 

12 

191,760.27 1609.25*
.09            20
Copyright 2009 - Dr. Richard Buttimer
12
Fixed Rate Mortgage
Mechanics - Effective Yield
• Frequently, we will know the price of a mortgage, and
its contractual details, but we will not know the market
discount rate.
– Fortunately, we can use the present value of an annuity formula
to solve for the discount rate.
– We simply have to solve for the effective yield (y) in the
equation below. This can be done through a search algorithm
or by use of a financial calculator.

              
              
1       1    
  y      n

 1  12  
       
Known Price  Pmt * 
y
12
21
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Effective Yield
• Continuing with our example, let us say that a bank
could purchase the mortgage for \$180,000. What
would be the effective yield if a bank purchased it at
that price?
– It would be 9.79%
                
        1       
1

 1  .0979

300 

12 
             
\$180,000 1609.25*
.0979
12

– Confirming this is simple:
N=300, PV = 180,000, PMT = -1609.25, FV=0, solve for I.
Might have to multiply the found i by 12.

22
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Effective Yield
• Note that in the absence of prepayment
penalties or points, the effective yield on a
mortgage (to the borrower) is always the
contract rate of the loan.

23
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Prepayment
• This extended example raises an interesting
point.
• The borrower is scheduled to make payments that are worth, at
the current market rate of 7%, \$227,687.12.
• The mortgage contract, however, grants them the right to pay off
the loan at any time by repaying the balance, which is the
\$191,760.27.
– Thus, by taking out a new loan for \$191,760.27, at
the current market rate (7%) and using the
proceeds to pay off the original loan, the borrower
would increase its wealth by \$35,926.85.
• In essence they would be replacing one liability of \$227,687.12
with a new liability of \$191,760.27, therefore, reducing their
overall liabilities by \$35,926.85.
24
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Prepayment
• Discounting the remaining payments at the
market rate and comparing that figure to the
balance allows us to quantify the benefits of
prepaying the loan.
– Frequently it is costly to refinance a loan. There are
often closing costs and other fees charged to
refinance. Optimally, a borrower will not refinance if
the gain from refinancing is less than the
refinancing costs, i.e. (Value – Balance> Cost of
Refinance).

25
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Prepayment
• When we talk about the “value” of this loan to
the lender, we have to realize that they factor
in the borrower’s right to “call” the loan.
– In the previous example the “value” of the loan is
not really \$227,687.12 because the lender knows
the borrower is going to prepay it. They realize the
value is probably no more than \$191,760.27.

26
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Prepayment
• If we denote the value of the promised
payments as “A”, and the value of the call
option as “C”, and any transaction costs of
refinancing as “T”, then the true value of the
mortgage will be:
• V = A – (C-T).
– Since in the previous example we had no
transaction costs, i.e. T=0, then
• \$191,760.27 = 227,687.12 – 35,926.85

27
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Prepayment

• It is useful to examine what happens to the
value of the mortgage if rates changed
instantaneously.
– To do this let’s use the same data from our previous
example but assume it will cost the borrower \$2500
to refinance.
– We assume the borrower will only prepay when it is
financially beneficial to do so, i.e. when:
– A – Balance – T > 0

28
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Prepayment
– Graphically, the value of A, i.e. the PV of the
remaining payments, (V if you ignore the value of
C), looks like this to the lender:
\$450,000.00

\$400,000.00

\$350,000.00

\$300,000.00

\$250,000.00

\$200,000.00

\$150,000.00

\$100,000.00

\$50,000.00

\$0.00
0   0.02   0.04       0.06          0.08           0.1    0.12   0.14   0.16   29
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Prepayment
– Graphically, the value of C, i.e. the value of the
borrower exercising their call option, is given by
(again, to the lender!):
\$50,000.00

\$0.00
0   0.02   0.04   0.06           0.08             0.1      0.12   0.14   0.16

(\$50,000.00)

(\$100,000.00)

(\$150,000.00)

(\$200,000.00)

(\$250,000.00)
30
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Prepayment
– Combining these two shows the value of the
mortgage to the bank (V). Note the spike in value
just below the contract rate.
\$300,000.00

\$250,000.00

\$200,000.00

\$150,000.00

\$100,000.00

\$50,000.00

\$0.00
0.06   0.07         0.08             0.09             0.1   0.11   0.12

(\$50,000.00)

(\$100,000.00)

31
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Prepayment
– It may be easier to see this by looking only at the
graph of V:
\$200,000.00

\$190,000.00

\$180,000.00

\$170,000.00

\$160,000.00

\$150,000.00

\$140,000.00

\$130,000.00

\$120,000.00

\$110,000.00

\$100,000.00
0   0.02   0.04         0.06          0.08          0.1   0.12   0.14   0.16

32
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics – Prepay Penalties
• One idea to remember is that banks
understand, and explicitly build into mortgage
rates, the risk of prepayments.
• Some borrowers, primarily commercial
borrowers, but, some residential borrowers as
well (notably subprime borrowers) are (or
were) willing to contractually agree not to
prepay in order to secure a lower contract rate.
By doing so, they helped to lock in the
“annuity” for the lender, at the initial rate based
on the fixed-rate mortgage.
33
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics – Prepay Penalties
• A common way for the borrower to signal to the lender
their willingness to forgo the prepayment option is by
accepting a prepayment penalty.
– A prepayment penalty is simply an additional fee that the
borrower agrees to pay, in addition to the outstanding balance,
should they prepay the loan.
– Frequently, these prepayment penalties end after some
specified period of time (5, 10 or 15 years for example).
– Some common prepayment penalties include
• A flat fee;
• A percentage of the outstanding balance; and
• The sum of the previous six months interest.

34
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics – Prepay Penalties
• The real effect of the prepayment penalty is to
raise the borrower’s effective interest rate should
they prepay.
• Consider the following example.
– A borrower takes out a loan with a contract rate of
10%, a term of 30 years, and an initial balance of
\$100,000. The loan provides that there is a
prepayment penalty of 2% of the outstanding balance
if they choose to prepay the loan.

35
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics – Prepay Penalties
• If the borrower prepays after 5 years, what is
the effective interest rate on the loan?
– To determine this we must first determine the cash
flows related to the loan.
– The original payment is simply \$877.57/month.

877.57  100,000*
.10 / 12
              
       1      
1

 1  .10

 360 

12 
       

36
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics – Prepay Penalties
• After 5 years the balance will be 96,574.14.

                   
           1       
1

 1  .10
            12 

300 
        
96,574.14  877.57 *
.10 / 12

• Thus, to pay off the loan the borrower will have to pay
a lump sum of \$98,505.63.
– 98,505.63 = 96,574.14 * 1.02

37
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics – Prepay Penalties
• Therefore, to the borrower, the cash flows are:
• Positive cash flow of \$100,000 at time 0
• 60 negative cash flows of \$877.57
• One final negative cash flow at month 60 of 98,505.63

• Their effective yield is the yield that makes this
equation true, which is 10.30%.
         1                         
 1                                
 1  y / 12
60
100,000  877.57 *                                       98,505.63
    y / 12                          (1  y / 12) 60
                                   
                                   

– Confirm: N=60, PV = 100,000, PMT=-877.57,
FV=-98,505.63, solve for i.
38
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics – Prepay Penalties
• One has to be careful in this analysis,
however.
– To the borrower making the decision to prepay at
are sunk costs and must be ignored.
– Where this enters the borrower’s decision making is
at origination.
• A borrower that expects to prepay at month 60, would opt not to
take out a mortgage with a prepayment penalty.

39
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics – Prepay Penalties
• Consider at origination if the borrower
suspected that they would likely prepay within
5 years.
– If they were offered two loans, one at a contract rate
of 10% and a 2% prepayment penalty or one at
10.25% and no prepayment penalty, then they
should select the 10.25% loan.
– The reason lenders charge prepayment penalties is
to induce borrowers to reveal their expectations

40
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
• One unusual feature of the mortgage market related to
prepayment penalties is the practice of charging
borrowers “points”.
– Technically a point is a fee that the borrower pays the bank at
origination. For each point charged, the borrower pays one
percent (1%) of the initial loan balance to the bank.
• The effect of this, of course, is to reduce the actual
– Thus if a borrower took out a \$100,000 loan, but was charged
2 points, they would receive \$100,000 from the bank and then
write a check to the bank for \$2000, making their net proceeds
\$98,000. Therefore, the lender is actually only paying out
\$98,000 on the \$100,000 loan.

41
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
• Of course since the borrower only received,
net, \$98,000, at origination, the value of the
mortgage at origination can only be \$98,000.
– The payments, however, will be based on the
nominal principal amount of \$100,000. The only
way for the PV of the future payments to be worth
\$98,000, therefore, is to reduce the contract rate.
• This is, of course, exactly what happens – the more points you
pay, the lower your contract rate.
• Note, however, that the effective interest rate is not constant, it is
a function of when the loan is paid off.

42
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
• To calculate the effective interest rate on a
mortgage with points you must go through
multiple steps:
– First, use the contract parameters (i.e. contract
principal, term, and contract rate) to determine the
cash flows. You will use the contract balance to do
this.
– Second, find the effective yield which equates the
present value of the future cash flows to the amount
of cash (net) received at the closing. You will use
the cash received at closing to do this.

43
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
• Again, this may be best illustrated with an example.
– A borrower takes out a loan with a 10% contract rate, a balance of
\$150,000, and 30 year term. The bank charges two points.
• If the borrower never prepays, what is the effective interest
rate on the loan?
– Step 1: Determine the actual cash flows
• The monthly payment is 1313.36:

\$1,316.36 150,000*
.10 / 12
              
       1      
1



 1  .10
360 

12 
      
– Step 2: Since the borrower never prepays, just solve for the
effective interest rate using the cash received at closing.
44
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
• Since the bank charges 2 points on a \$150,000 loan,
• 147,000 = 150,000 * (1-.02)
• The final step is to determine the yield which equates
the cash received at time 0 with the present value of the
monthly payments:
1
1                     360
1  y 

        12 

\$147,000  1,316.36*
 y / 12

• This is 10.24%, which you can confirm on your
calculator as:
N=360, PV=147,000, PMT=-1,316.36, FV=0, solve for i.
45
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
• What would be the yield if the borrower prepaid after
10 years?
• Obviously the time 0 cash and the monthly payments are the same. The
only additional item we need to know is the balance of the loan after 10
years, which is given by discounting the remaining payments at the
contract rate.

1
1
\$136,407.0 2  1,316.36 *

1  .10
12

240

.10 / 12 

– You can confirm this in two ways on your calculator:
N=120, I = 10/12, PV=150,000, PMT = -1,316.36, Solve for FV, or
N=240, I = 10/12, PMT = -1,316.36, FV=0, and Solve for PV
46
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
• Now, we again determine the yield which sets
present value of the future cash flows equal to
the cash received at time 0:
1
1                      120
1  y 

        12 
             136,407.02
\$147,000  1,316.36*                                        
 y / 12               (1  y/12)120

• The answer is y = 10.33159%, which you can
confirm as:
N=120, PV = 147,000, PMT = -1,316.36, FV = -136,407.02 and
then solving for i.

47
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
• The effect of the points on the effective cost of
financing changes with the amount of time the
borrower holds the loan.
– The chart below illustrates the effective yield given the date at
which the borrower prepays the loan
35
30
Effective Yield

25
20
15
10
5
0
0   60       120               180               240   300   360
Prepayment Month                                     48
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
• Finally, consider if at origination the borrower
had a choice between two loans.
– Loan A is the mortgage with points we just
examined.
– Loan B is for \$147,000, at 10.3% and no points.
• Which loan should the borrower take?
– The answer to that depends upon the borrowers
expectations regarding their tenure in the mortgage.

49
Copyright 2009 - Dr. Richard Buttimer
Fixed Rate Mortgage
Mechanics - Points
– Clearly if the borrower expects to never prepay the
mortgage, they should take loan A, because the
effective rate on the loan will be 10.24%, well below
the 10.30% of loan B.
– If, however, the borrower expects to prepay after 10
years (or before), they should take loan B, since
with a 10 year prepayment horizon loan A has an
effective interest rate of 10.33%.

50
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Incremental Cost
• The final issue we will examine is the
incremental cost of financing.
• Frequently, borrowers of equal
creditworthiness will observe different interest
rates for different sized loans. That is, an 80%
loan to value (LTV) mortgage will have a lower
contract rate than a 90% LTV loan.
• The question is, what is the effective interest
rate on that marginal increment of 10%?

51
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Incremental Cost
• An example may be useful:
– In February of 2000, 80% LTV 30 year mortgages
had a contract rate of approximately 8.25% while
95% LTV mortgages had a contract rate of
approximately 8.75%.
• If you were a borrower with a \$200,000 house you could borrow
\$160,000 at 8.25% or \$190,000 at 8.75%.
– So to borrow the incremental \$30,000 your overall
interest rate goes up by .5%.
• One way of looking at this is that you are borrowing the first
\$160,000 at 8.25%, and the remaining \$30,000 at some effective
rate. The question is, what is that effective rate?
– To solve this, let’s consider the cash flows.
52
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Incremental Cost
– The payments on the 80% and 95% loans are
(respectively):

Payment80% loan  160,000*
.0825 / 12         1,202.02
1
1

1  .0825
12
360


Payment95% loan  190,000*
.0875 / 12         1,494.73
1
1

1  .0875
12
360


– Thus, there is a \$292.70/month differential between
the two loans.
53
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Incremental Cost
• One way to view this is that you are paying
\$292.70/month to borrow \$30,000 for 30 years. This
implies the following statement must be true:
1
1                     360
1  y 

        12 

30,000  292.70 *
 y / 12

Solving for y we find that the incremental cost of
financing is: 11.308%.
• Again, you can confirm this with your calculator by:
– N=360, PV=30,000, PMT = -292.70, FV=0,and solving for i.

54
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Incremental Cost
• What this means is that if you can borrow
\$30,000 for less than 11.308%, you should
take the 80% LTV loan and then borrow the
remaining funds from that other source. If you
cannot borrow \$30,000 for less than 11.308%,
you should take the 95% loan.

• A closely related issue is the idea of a second
mortgage.

55
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
• It is not at all uncommon for a borrower to take
out two mortgages on the same property. The
first will typically be for 80% or 90% LTV, with
the second being for 20%, 10% or 5%.
– Occasionally, it will be the case that it is cheaper, in
terms of the effective cost of financing, to take out
an 80% LTV first loan, and then a 10% second
loan, than it would be to take out a single 90% LTV
loan.
• In the past, borrowers could do this to avoid having to have a
Private Mortgage Insurance contract, since lenders only looked at
the first mortgage balance. Now, they are increasingly looking at
the Cumulative LTV (CLTV) when deciding whether to require
PMI.
56
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
• Calculating the effective cost of financing when
there are two mortgages is not particularly
difficult.
– You first determine each mortgage’s monthly
payments individually.
– You then combine their initial balances and monthly
payments and solve for the effective interest rate.
– An example may make this easy to see.

57
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
– Example: Bob wishes to buy a
house for \$100,000. He will put
10% down, take out an 80%
LTV first loan at 5.5%, and a                   Pmt1  80,000 *
.055 / 12           454.23
10% second loan at 7%. Each                                                  1          
1
mortgage is for 30 years. What                                       1  .055 / 12360 

will be his total cost of financing
if he keeps each mortgage for
Pmt2  10,000 *
.07 / 12           66.53
the full 30 year term?
           1         
– First, let’s calculate each                                       1
mortgage payment:                                                    1  .07 / 12360 


58
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
– Now we can combine the two mortgages, and find the interest
rate that sets the initial balances equal to the present value of
the total monthly payments.
            1                     
 1                               
1  y /12 
360

90, 000  (520.76) *                                   
      ( y /12)                    
                                  
                                  
Solving for y, yields:
y  5.67%

– You can confirm this on your calculator by:
N=360, PV = 90,000, PMT = -520.76, FV=0, and solving for i.

59
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
• Notice that the effective interest rate is not simply
the average, or even the arithmetically weighted
average of the two contract rates.
– You must use the procedure on the previous two slides to
determine the effective interest rate. If you try to take an
arithmetic average of the two contract rates you will get the
– This is because the mortgage payment equations are non-
linear due to the exponents in the formulas.

60
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
• Now, what happens if the two mortgages are for
unequal terms?
– You simply have to deal with two sets of cash flows. This
means you will have to use your cash flow keys on your
once you become familiar with that procedure its not too
difficult.
the second mortgage was only for 10 years.

61
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
– Of course this does not
change the first payment,
but it does change the
Pmt1  80,000 *
.055 / 12           454.23
second payment:                                                         1          
1
   1  .055 / 12360 


Pmt2  10,000 *
.07 / 12           116.11
           1         
1
   1  .07 / 12120 


62
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
– Once again, we find the interest rate that sets the present value
of the payments equal to the combined balances. Notice that we
now have to deal with two streams of cash flows:

                     1      
           1           240   
  454.23 *  1  r / 12   
           1                        (r / 12)     
1  1  r / 12120   
           
                 

90,000  (570.34) *                                                        
                     
120
(r / 12)                         r                
                                1                   
                                  12 
                                  
                                  

                                  

Solving for r, yields :
r  5.57%

63
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
– Note that the second annuity (the \$454.23/month one) starts in
121 months, so using the PVA formula tells us its value at month
120, so we have to discount that value back to time 0.

                     1      
           1           240   
  454.23 *  1  r / 12   
          1                      (r / 12)     
1
 1  r / 12120                              
                            
90,000  (570.34) *                                                     
                  
120
(r / 12)                       r                
                             1                   
                               12 
                                  
                                  

                                  

Solving for r, yields :
r  5.57%

64
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
• Its actually pretty easy to do this on your calculator.
Simply use your cash flow keys:

CF0 = -90,000
CF1= 570.34
N1=120
CF2= 454.23
N2=240
– And the solve for IRR. Note that you IRR will be in monthly
terms, don’t forget to multiply by 12. On an exam, you will
lose points if you forget to multiply by 12!

65
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
– What if Bob prepaid both mortgages after 5 years? Let’s go back
to the assumption that Bob had a 30 year second mortgage.
Remember that our two mortgage payments, then, are:
Pmt1=454.23, Pmt2 = 66.53.
– We need the balance of each mortgage after 60 months:

         1          
1              300 

Balance  454.23 *  1  .055 / 12   73,968.48
1
.055 / 12

          1        
1             300 

Balance2  66.53 *  1  .07 / 12   9,413.16
.07 / 12

66
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
– So now we can combine all of the cash flows and determine the
effective interest rate:

           1        
1 
1  r / 12 60  83,381 .64
90 ,000  520 .76 *                     
r / 12           1  r / 12 60
Solving for r yields :
r  5.67%

– Notice that you can use your time value of money keys for this:
n=60; PV=90,000; PMT=-520.76; FV=-83,381.64
and solve for r.

67
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages

• Finally, what if Bob had only paid off the second
mortgage after 5 years, but had held the first mortgage
for the full 30 years?
– Again, all we really have to do is lay out the cash flows on a
month by month basis:
                     1       

         1  1  r / 12300  

454.23 *                     
           1                                           r / 12       
1  1  r / 1259  520.76  9413.16          
                    

90,000  520.76 *                                       
      r / 12           1  r / 1260
1  r / 12 60


                   

solving for r yields :
5.569%

68
Copyright 2009 - Dr. Richard Buttimer
Rate Mortgage
Mechanics – Second Mortgages
– Again, its easiest to do this using your calculator’s cash flow
keys.
– The only real trick is to realize that you get 59 payments of
520.76, then one payment of (520.76+9,413.16) in month 60
when the second mortgage is paid off, followed by 300 payments
of 454.23.To enter this do the following:
CF0=-90,000
CF1=520.76         N1=59
CF2=9,933.92       N2=1
CF3=454.23         N3=300

• Note that we call this the effective cost of financing,
even though we express it as a rate!                69
Copyright 2009 - Dr. Richard Buttimer
Second Mortgages

• It is important to note what is really going on with
a second mortgage.
• The lender of the first mortgage makes a loan that
is valued based on the LTV and creditworthiness
of the borrower.
• The second lender makes a second loan, that is
subordinate to the first, which normally has a
higher interest rate higher to compensate for the
fact that it is increasing the overall LTV and is
subordinate to the first mortgage.
70
Copyright 2009 - Dr. Richard Buttimer
Second Mortgages

• The problem often comes in the fact that the
borrower may not be able to afford both
mortgages.
• The lender of the first mortgage is in “first
position” with respect to collateral priority.
• The lender of the second mortgage is just rolling
the dice, since because they are in a second
position, they aren’t very well protected.

71
Copyright 2009 - Dr. Richard Buttimer
Second Mortgages

• However, with two different lenders, operating
with different criteria to loan to the borrower, there
often isn’t any single person making sure that the
loans (collectively) should be structured this way,