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fixed rate mortgages

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									   Fixed Rate Mortgage
   Mechanics
 Recall that to the investor, 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.
   Fixed Rate Mortgage
   Mechanics
 Casho = PV0(Future Cash Flows)
 If the cash were worth more than the PV of
  the future cash flows, the bank would not be
  willing to make the loan they would be
  paying more for the annuity than it was
  worth.
   Fixed Rate Mortgage
   Mechanics
 Casho = PV0(Future Cash Flows)
 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 was worth more
  than the asset they would receive (the cash),
  reducing their wealth.
   Fixed Rate Mortgage
   Mechanics
 Casho = PV0(Future Cash Flows)
 Thus, 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.
   Fixed Rate Mortgage
   Mechanics
 The mortgage contract specifies how to
  calculate the various cash flows associated
  with the mortgage. This will include:
 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).
   Fixed Rate Mortgage
   Mechanics
 The other terms specified in the mortgage
  contract include:
     r - the contract rate of the mortgage,
     n - the number of monthly payments,
     Pmt – the monthly payment on the mortgage.
   Fixed Rate Mortgage
   Mechanics - Balance
 At time 0 we know that the value of the
  mortgage is equal to the cash received. For
  now, assume that the principal is set to that
  same amount.
 Thus, the value of the mortgage must have
  this relationship:            
                                     1           
                              1               n 
                             
                Prin  Pmt * 
                                 1 c
                                       12        
                                                  
                                   c
                                     12
   Fixed Rate Mortgage
   Mechanics - Balance
 Thus, we know if the contract rate were 8%,
  with 20 years (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
    Fixed Rate Mortgage
    Mechanics - Balance
 Note that this formula actually works for any point
  during the life of the mortgage – that is, if you tell
  me the remaining term, the contract rate, and the
  monthly payment, this formula tells you the
  currently outstanding principal.
                                             
                                        1    
                                   1
                                 
                                    
                                  1 c
                                 
                                            
                                            n 

                                          12 
                                              
                    Prin  Pmt *
                                      c
                                        12
   Fixed Rate Mortgage
   Mechanics - Payments
 While knowing how to determine the
  principal amount is important, it is perhaps
  more interesting (from a potential
  homeowners standpoint) to know how to
  calculate the payment that will be required
  given a known balance.
 This just requires simple algebraic
  manipulation of the balance formula.
   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 
                                                  
   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.
   Fixed Rate Mortgage
   Mechanics - Amortization
 The mortgage contract will state 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,

      Any remaining cash pre-pays principal.
   Fixed Rate Mortgage
   Mechanics - Amortization
 Thus, normally (i.e. when scheduled
  payments are made on time), the investor
  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.
   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.
    Fixed Rate Mortgage
    Mechanics - Amortization
 For a 30 year, 9% mortgage, original balance
  of $200,000.
    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.
   Fixed Rate Mortgage
   Mechanics - Amortization
 Notice the relationship 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
   Fixed Rate Mortgage
   Mechanics - Price
 At origination the contract rate of the
  mortgage will equal 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.
   Fixed Rate Mortgage
   Mechanics - Price
 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.
   Fixed Rate Mortgage
   Mechanics - Price
 To do this we simply take the present value
  of the remaining payments using the current
  market rate:
                                        
                                   1    
                             1
                           
                           
                           
                               1 r   
                                       n 

                                     12 
                                         
             Value  Pmt *
                                 r
                                   12
   Fixed Rate Mortgage
   Mechanics - Price
 Of course this is the same basic formula as
  the one we used to calculate the balance,
  with the difference that we use the market
  rate, r, instead of the contract rate, c.

                                                             
                        1                               1    
                  1                                 1

  Value  Pmt *
                
                
                
                     
                     1  r  n 

                           12 
                               
                                                   
                                                      
                                                    1 c
                                                   
                                                              
                                                              n 

                                                            12 
                                                                
                                   Balance  Pmt *
                      r                                 c
                        12                                12
   Fixed Rate Mortgage
   Mechanics - Example
 At this point it might be useful to look at an
  extended example.
        Consider that a borrower 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?
      Fixed Rate Mortgage
      Mechanics - Example
 Example (continued)
     The monthly payment is $1,609.25:

              $1,609.25 200,000*
                                         .09 / 12
                                                      
                                              1       
                                      1
                                    
                                    
                                    
                                        1  .09
                                                12
                                                      
                                                   360 
                                                       
                                                       

     After 5 years the balance is: $191,760:
                                                  
                                           1      
                                     1
                                    
                                    
                                            
                                     1  .09
                                               300 

                                             12 
                                                         
               191,760.27 1609.25*
                                        .09
                                            12
      Fixed Rate Mortgage
      Mechanics - Example
 Example (continued)
     Note that I can determine the payment from
      ONLY the current balance, contract rate, and
      remaining term:
              $1,609.25 191,760*
                                         .09 / 12
                                                     
                                             1       
                                     1
                                    
                                    
                                    
                                        
                                        1  .09
                                               12
                                                      
                                                  300 
                                                      
                                                      
      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

     Contrast this with the balance, which is still
                                                    
                                             1      
                                       1
                                      
                                            
                                       1  .09
                                      
                                                     
                                                 300 

                                               12 
                                                     
                 191,760.27 1609.25*
                                          .09
                                              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.
   Fixed Rate Mortgage
   Mechanics - Effective Yield
 We simply have to solve for 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
   Fixed Rate Mortgage
   Mechanics - Effective Yield
 In the previous example, let us say the 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
   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.
   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 that loan at any time by
  repaying the balance, which is the
  $191,760.27.
   Fixed Rate Mortgage
   Mechanics - Prepayment
 Thus, by taking out a new loan for
  $191,760.27, at the current market rate (7%)
  and used the proceeds to pay off the original
  loan, they would increase their wealth by
  $35,926.85.
 In essence they would be replacing one
  liability worth $227,687.12 with one worth
  $191,760.27.
   Fixed Rate Mortgage
   Mechanics - Prepayment
 Discounting the remaining payments at the
  market rate and comparing that to the
  balance allows us to quantify the benefits to
  prepaying the loan.
 Frequently it is costly to refinance a loan.
  Optimally, one will not refinance if the gain
  to refinancing is less than the refinancing
  costs, i.e. (Value – Balance> Cost of Refi).
      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.
      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
      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
    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 bank!)
       $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
    Fixed Rate Mortgage
    Mechanics - Prepayment
   Graphically, the value of C, i.e. value of the
    borrower exercising their call option, is given
    (again, to the bank!):
        $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)
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)
Fixed Rate Mortgage
Mechanics - Prepayment
   It may be easier to see this by looking only at 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
   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 increasingly residential
  borrowers, are willing to contractually agree
  not to prepay in order to secure a lower
  contract rate.
   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.
      Fixed Rate Mortgage
      Mechanics – Prepay Penalties
 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,
     The sum of the previous six months interest.
   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 for with a contract rate of
         10%, a term of 30 years, and an initial balance of
         $100,000. There is a prepayment penalty of 2% of the
         outstanding balance if they prepay the loan.
      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.
     The original payment is simply $877.57/month.

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

                                            12 
                                                   
   Fixed Rate Mortgage
   Mechanics – Prepay Penalties
 After 5 years the balance will be 96,574.14.
                                                     
                                             1       
                                    1
                                  
                                        
                                   1  .10
                                  
                                                  
                                                  300 

                                               12 
                                                      
             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
   Fixed Rate Mortgage
   Mechanics – Prepay Penalties
 Thus, 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             60 
             100,000  877.57 * 
                                    1  y / 12   98,505.63
                                      y / 12       (1  y / 12)60
                                                  
                                                  
   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, would opt
  not to take out a mortgage with a prepayment
  penalty.
    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 borrower charge prepayment
    penalties is to induce borrower to reveal their
    expectations about their future prepayment
    patterns.
   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 of
  the initial loan balance to the bank.
      Fixed Rate Mortgage
      Mechanics - Points
 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.
   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.
   Fixed Rate Mortgage
   Mechanics - Points
 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.
      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.
     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.
      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?
   Fixed Rate Mortgage
   Mechanics - Points
 First, determine the actual cash flows
        The monthly payment is

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


 Since the borrower does not prepay the loan,
  the next step is to determine the yield based
  on the cash actually received at the closing.
   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.
   Fixed Rate Mortgage
   Mechanics - Points
 That is, determine:
                                               1
                                   1                   360
                                        1  y 
                                        
                                                12 
                                                    
            $147,000  1,316.36*
                                           y / 12



 The answer is y = 10.24%
      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 
   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 %
    Fixed Rate Mortgage
    Mechanics - Points
 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
      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.
    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 10.3% 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%.
   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.
      Rate Mortgage
      Mechanics – Incremental Cost
 The question is, what is the effective interest
  rate on that differential.
     For example 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%.
    Rate Mortgage
    Mechanics – Incremental Cost
   So to borrow the incremental $30,000 your
    overall interest rate goes up by .5%.
   One way of looking at this is that you 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.
    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
    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%.
    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.
    Rate Mortgage
    Mechanics – Second Mortgages
   It is not at all uncommon for a borrower to take
    out two mortgages. The first will typically be for
    80 or 90% LTV, with the second mortgage 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.
    Rate Mortgage
    Mechanics – Second Mortgages
   To calculate the effective cost of financing when
    there are two mortgages is not particularly
    difficult. You first determine each mortgage’s
    monthly payments individually, then you
    combine their initial balances and monthly
    payments and solve for the effective interest rate.
   An example may make this easy to see.
    Rate Mortgage
    Mechanics – Second Mortgages
   Example: Bob wishes to buy
    a house for $100,000. He
    will put 10% down, take out      Pmt1  80,000 *
                                                            .055 / 12          454.23
                                                                1          
    an 80% LTV first loan at                           1              360 
                                                        1  .055 / 12 
    5.5%, and a 10% second
    loan at 7%. Each mortgage is                             .07 / 12
    for 30 years. What will be       Pmt2  10,000 *                            66.53
                                                                1         
                                                       1             360 
    his total cost of financing if                      1  .07 / 12 
    he keeps each mortgage for
    the full 30 year term?
   First, let’s calculate each
    mortgage payment:
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  r / 12360 
                 90,000  (520.76) *                   
                                          (r / 12)     
                                      
                                                       
                                                        
                 Solving for r, yields :
                 r  5.67%
Rate Mortgage
Mechanics – Second Mortgages
   Notice that the effective interest rate is not simply the
    average, or even 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.
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.
Rate Mortgage
Mechanics – Second Mortgages
   Of course this does not
    change the first                                 .055 / 12
                              Pmt1  80,000 *                             454.23
    payment, but it does                        
                                                 1
                                                          1          
                                                                360 
    change the second                            1  .055 / 12 

    payment:                                         .07 / 12
                              Pmt2  10,000 *                            116.11
                                                         1         
                                                1             120 
                                                 1  .07 / 12 
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%
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%
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. You will
    lose points if you forget to multiply by 12!
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
Rate Mortgage
Mechanics – Second Mortgages
   So now we can combine all of the cash flows and
    determine the effective interest rate:
                                      1        
                              1            60 

          90 ,000  520 .76 *  1  r / 12    83,381 .64
                                  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.
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 
    90,000  520.76 *                  
                                           520.76  9413.16  
                                                                         
                                                                                            
                                                                                             
                           r / 12           1  r / 1260
                                                                        1  r / 12 60

                       
                                       
                                        
    solving for r yields :
    5.569%
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.

								
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