Fixed Rate Mortgage Mechanics

1
Fixed Rate MortgageMechanics

• 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.

2
Fixed Rate MortgageMechanics

• 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.

3
Fixed Rate MortgageMechanics

• 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.

4
Fixed Rate MortgageMechanics

• 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.

5
Fixed Rate MortgageMechanics

• 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).

6
Fixed Rate MortgageMechanics

• 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.

7
Fixed Rate MortgageMechanics - 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

8
Fixed Rate MortgageMechanics - 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

9
Fixed Rate MortgageMechanics - 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.

10
Fixed Rate MortgageMechanics - 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.

11
Fixed Rate MortgageMechanics Payments

• So, for a 100,000 loan at 10 for 30 years, the
  payment is 877.57.

12
Fixed Rate MortgagesMechanics - Payments

• 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.

13
Fixed Rate MortgageMechanics - 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.

14
Fixed Rate MortgageMechanics - 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.

15
Fixed Rate MortgageMechanics - 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.

16
Fixed Rate MortgageMechanics - Amortization

• For a 30 year, 9 mortgage, original balance of
  200,000.

Note that the above is an Excel spreadsheet you
should be able to click on it and actually use
it.
17
Fixed Rate MortgageMechanics - Amortization

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

18
Fixed Rate MortgageMechanics - 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.

19
Fixed Rate MortgageMechanics - 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.

20
Fixed Rate MortgageMechanics - Price

• To do this we simply take the present value of
  the remaining payments using the current market
  rate

21
Fixed Rate MortgageMechanics - 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.

22
Fixed Rate MortgageMechanics - 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?

23
Fixed Rate MortgageMechanics - Example

• Example (continued)
• The monthly payment is 1,609.25

• After 5 years the balance is 191,760

24
Fixed Rate MortgageMechanics - Example

• Example (continued)
• Note that I can determine the payment from ONLY
  the current balance, contract rate, and remaining
  term

25
Fixed Rate MortgageMechanics - Example

• Example (continued)
• The value of the mortgage, at the 7 contract
  rate is 227,687.12,

• Contrast this with the balance, which is still

26
Fixed Rate MortgageMechanics - 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.

27
Fixed Rate MortgageMechanics - 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.

28
Fixed Rate MortgageMechanics - 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

29
Fixed Rate MortgageMechanics - 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.

30
Fixed Rate MortgageMechanics - 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.

31
Fixed Rate MortgageMechanics - 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.

32
Fixed Rate MortgageMechanics - 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).

33
Fixed Rate MortgageMechanics - Prepayment

• When we talk about the value of this loan to
  the lender, we have to realize that they factor
  in the borrowers 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.

34
Fixed Rate MortgageMechanics - 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. T0, then
• 191,760.27 227,687.12 35,926.85

35
Fixed Rate MortgageMechanics - Prepayment

• It is useful to examine what happens to the
  value of the mortgage if rates changed
  instantaneously.
• To do this lets 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

36
Fixed Rate MortgageMechanics - 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!)

37
Fixed Rate MortgageMechanics - Prepayment

• Graphically, the value of C, i.e. value of the
  borrower exercising their call option, is given
  (again, to the bank!)

38
Fixed Rate MortgageMechanics - Prepayment

• Combining these two shows the value of the
  mortgage to the bank (V). Note the spike in
  value just below the contract rate.

39
Fixed Rate MortgageMechanics - Prepayment

• It may be easier to see this by looking only at
  graph of V.

40
Fixed Rate MortgageMechanics 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.

41
Fixed Rate MortgageMechanics 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.

42
Fixed Rate MortgageMechanics 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.

43
Fixed Rate MortgageMechanics Prepay Penalties

• The real effect of the prepayment penalty is to
  raise the borrowers 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.

44
Fixed Rate MortgageMechanics 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.

45
Fixed Rate MortgageMechanics Prepay Penalties

• After 5 years the balance will be 96,574.14.

• 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

46
Fixed Rate MortgageMechanics 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.

47
Fixed Rate MortgageMechanics 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 borrowers decision making is at
  origination. A borrower that expects to prepay,
  would opt not to take out a mortgage with a
  prepayment penalty.

48
Fixed Rate MortgageMechanics 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.

49
Fixed Rate MortgageMechanics - 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.

50
Fixed Rate MortgageMechanics - 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.

51
Fixed Rate MortgageMechanics - 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.

52
Fixed Rate MortgageMechanics - 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.

53
Fixed Rate MortgageMechanics - 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.

54
Fixed Rate MortgageMechanics - 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?

55
Fixed Rate MortgageMechanics - Points

• First, determine the actual cash flows
• The monthly payment is

• 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.

56
Fixed Rate MortgageMechanics - 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.

57
Fixed Rate MortgageMechanics - Points

• That is, determine

• The answer is y 10.24

58
Fixed Rate MortgageMechanics - 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.

59
Fixed Rate MortgageMechanics - Points

• Now we again determine the yield which sets
  present value of the future cash flows equal to
  the cash received at time 0

• The answer is y10.33159

60
Fixed Rate MortgageMechanics - Points

• The chart below illustrates the effective yield
  given the date at which the borrower prepays the
  loan

61
Fixed Rate MortgageMechanics - 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.

62
Fixed Rate MortgageMechanics - 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.

63
Rate MortgageMechanics 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.

64
Rate MortgageMechanics 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.

65
Rate MortgageMechanics 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, lets consider the cash flows.

66
Rate MortgageMechanics Incremental Cost

• The payments on the 80 and 95 loans are
  (respectively)Thus there is a
  292.70/month differential between the two

67
Rate MortgageMechanics 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
  trueSolving for y we find that the
  incremental cost of financing is 11.308.

68
Rate MortgageMechanics 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.

69
Rate MortgageMechanics 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.

70
Rate MortgageMechanics Second Mortgages

• To calculate the effective cost of financing when
  there are two mortgages is not particularly
  difficult. You first determine each mortgages
  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.

71
Rate MortgageMechanics 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 10 second loan at 7. Each
  mortgage is for 30 years. What will be his total
  cost of financing if he keeps each mortgage for
  the full 30 year term?
• First, lets calculate each mortgage payment

72
Rate MortgageMechanics 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.

73
Rate MortgageMechanics 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.

74
Rate MortgageMechanics 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.
• Lets return to our previous example, but now
  assume that the second mortgage was only for 10
  years.

75
Rate MortgageMechanics Second Mortgages

• Of course this does not change the first payment,
  but it does change the second payment

76
Rate MortgageMechanics 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

77
Rate MortgageMechanics 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.

78
Rate MortgageMechanics Second Mortgages

• Its actually pretty easy to do this on your
  calculator. Simply use your cash flow keysCF0
  -90,000CF1 570.34N1120CF2 454.23N2240
• And the solve for IRR. Note that you IRR will be
  in monthly terms, dont forget to multiply by 12.
  You will lose points if you forget to multiply by
  12!

79
Rate MortgageMechanics Second Mortgages

• What if Bob prepaid both mortgages after 5 years?
  Lets go back to the assumption that Bob had a 30
  year second mortgage. Remember that our two
  mortgage payments, then, are Pmt1454.23, Pmt2
  66.53.
• We need the balance of each mortgage after 60
  months

80
Rate MortgageMechanics Second Mortgages

• So now we can combine all of the cash flows and
  determine the effective interest rate

• Notice that you can use your time value of money
  keys for this n60 PV90,000 PMT-520.76
  FV-83,381.64and solve for r.

81
Rate MortgageMechanics 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

82
Rate MortgageMechanics Second Mortgages

• Again, its easiest to do this using your
  calculators cash flow keys.
• The only real trick is to realize that you get 59
  payments of 520.76, then one payment of
  (520.769,413.16) in month 60 when the second
  mortgage is paid off, followed by 300 payments of
  454.23.To enter this do the followingCF0-90,000
  CF1520.76 N159CF29,933.92 N21CF3454.23
  N3300And then solve for IRR, and multiply your
  answer by 12.