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					           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




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                         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.



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                   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.
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                       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.
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                                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.


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                                   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.


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                       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.


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                       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
     time 60, the previous 59 payments already made
     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
     about their future prepayment patterns.


                                                             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
   cash received by the borrower.
    – 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,
  the borrower receives $147,000.
      • 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
      wrong answer.
    – 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
      calculator instead of your Time Value of Money keys, but
      once you become familiar with that procedure its not too
      difficult.
    – Let’s return to our previous example, but now assume that
      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
      And then solve for IRR, and multiply your answer by 12.


 • 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,
  or even made at all.
• Thus, by seperating the lenders, and maximizing
  the LTV, borrowers are often able to borrow more
  than they should.
• This helped lead to the real estate crisis we are
  are currently experiencing.
                                                             72
                     Copyright 2009 - Dr. Richard Buttimer

				
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