Your Federal Quarterly Tax Payments are due April 15th

# FRM by linxiaoqin

VIEWS: 0 PAGES: 74

• pg 1
```									Fixed-rate Mortgages, Mortgage
Yield, and Refinancing
Decisions

1
Mortgage Contract Rate

Generalized Mortgage Contract Rate

Rj = R* + (1-a)D + a E(P)
where:
Rj = contract interest rate on mortgage of type j.
R* = real rate of return
a = risk sharing parameter
P = pure interest rate risk component
j = term of loan
2
Mortgage Contract Rate
 a=1
Rj = R* + E(P)             Uncapped ARM or free floating rate.
 0<a<1
Rj = R* + (1-a)D + aE(P)     Capped ARM
 a=0
Rj = R* + D                  FRM

Contact rate = risk free rate + liquidity + default + prepayment + inflation +
interest rate risk + origination and servicing cost

3
Mathematics of level-payment
mortgages
• Mortgage investors must be able to
calculate scheduled cash flows associated
with mortgages.
• Servicers of mortgages must be able to
calculate servicing fee
• We also need to know cash flow from
mortgage pools to price MBS

4
Monthly Mortgage Payment
• Mortgage payment requires the application of PVA

• PVA = A[1-(1+i)-n]/i
– where:
– A = amount of annuity
– n = number of periods
– PVA = present value of annuity
– i = periodic interest rate
• The term in the outer bracket is called the present value of   5
annuity factor (PVAF)
• Redefine terms for level pay mortgage

• MB0 = DS([1-(1+i)-n]/i)
– where:
– DS = monthly mortgage payment
– n = amortization period or term or mortgage
– MB0 = original mortgage amount
– i = simple monthly interest (annual/12)
• Solving for DS gives
• DS = MB0{[i(1+i)n]/[(1+i)n -1 )]}
• The term in outer bracket is called mortgage constant or
payment factor
• So what is a mortgage constant (MC)?
Illustration

• Original mortgage balance (MB0) = \$100,000, term/amortization
period (n) = 360 mons., interest rate (i) = 9.5 or .095/12 = .0079167
• DS = MB0{[i(1+i)n]/[(1+i)n -1 )]}
• DS = \$1,000,000{[.0079167(1.007967)360]/[(1.0079167)360 - 1]}
•     = \$100,000(.0084085) = \$840.85
• Illustration using calculator:
-\$100,000 = PV ; 9.5/12 = I; 30x12 = n; PMT = ?

7
Mortgage Balance
• Mortgage Balance each period is given by the ff. formula
• MBt = MB0{[(1+i)n - (1+i)t]/[(1+i)n - 1]},
• where MB0 = mortgage balance after t months

• Example: Mortgage balance in 210th month is

• t = 210; n = 360; MB0 = \$100,000; i = .095/12 = .0079167

• MB210 = 100,000{[(1.0079167)360 - (1.0079167)210]/[(1.0079167)360 - 1]} =
\$73,668
• Check (calculator): \$840.85 = PMT 9.5/12 = i ; 150 = n PV =? \$73,668
8
Scheduled Principal Payment
• Scheduled principal payment (Pt) is
• Pt = MB0{[i(1+i)t-1]/[(1+i)n - 1]

•   Example: Scheduled principal payment for 210th month is
•   P210 = {[.0079167(1.0079167)210 - 1]/[(1.0079167)360-1]}
•   = 100,000{.0079167(5.19696) = \$255.62
•   CHECK:
•   840.85 = PMT ; 9.5/12 = i ; 13x12 = n ; PV = \$75,171.72
•   Balance at end of month 210 = \$73,667.78
•   Scheduled principal paid = \$75,171.72 - \$73,667.78 = \$1503.94

9
Scheduled Interest
•   Scheduled interest is as follows:
•   It = MB0{i[(1+i)n - (1+i)t-1]/[(1+i)n - 1]}
•   where It = interest in month t
•   Example: scheduled interest in month t is
• I210 = 100,000{.0079167[(1.0079167)360 - (1.0079167)210 - 1]/
[(1.0079167)360 - 1]}
• = 100,000{.0079167[(17.095 - 5.19696)]/[17.095 - 1]} = \$585.23

• CHECK
• Debt Service = 255. 62(p) + 585.23 (i) = \$840.85          10
Monthly Mortgage Cash flow
• If the mortgage investor services the mortgage the investor’s cash
flow is principal, interest payment
• If the investor sells the right to service the mortgage the interest
income is net of servicing fee
• Servicing fee = [MBt(servicing fee rate)]/12
• Example: assume servicing fee rate is .5%, then servicing fee for
month 211 is = [(73,668)(.005)]/12 = 368.34/12 = \$30.70
• Note the balance at end of month 210 (\$73,668)is the beginning
balance for month 211
• Net interest payment for month 211 = \$583.21 - 30.70 = \$552.51

11
Mortgage Amortization Schedule

Loan Amount = \$100,000
Interest Rate = 10%
Term of Loan or amortization period = 30 yrs.
Mortgage Constant = .10608
Yearly payment
Debt Service = Loan Amount x Mortgage Constant
= 100,000 x .10608
Yearly Payment = \$10,608

12
Amortization Schedule
A.   INTEREST RATE METHOD
BOY1 principal balance          = \$100,000
EOY1 interest (100,000 x .1)    = \$10,000
EOY1 principal repaid           =    \$608
(10,608 - 10,000)
EOY1 balance (100,000 - 608)    = \$99,392

BOY2 principal balance          = \$99,392
EOY2 interest (99,392 x .1)     = \$9,939.2
EOY2 principal repaid           = \$668.2
(10,608 - 9,939.2)
EOY2 balance (99,302 - 668.8)   = \$98,723.2
Amortization Schedule
Amount
Year   Outstanding   Payment   Interest   Principal

0      \$100,000
1      99,392        \$10,608    \$10,000   \$608
2      98,723.2      10,608     9,939.2   668.2
3      97,987.52     10,008    9,872.32   735.68

14
Amortization Schedule
B. PRESENT VALUE METHOD
Loan Amount = \$100,000
Annual Interest Rate = 10%
Frequency of Payments = Monthly
Term of Loan = 30 yrs. (360 months)
Monthly Mortgage Constant = .00877572
Monthly Debt Service = 100,000 x .00877572 = \$877.57
Annual Payment = 100,000 x .00877572 x 12 = \$10,530.86

15
Amortization Schedule
BOY1 principal balance                 =       \$100,000
EOY1 balance = [PVAF 10/12, 348] x 877.57 = 113.3174 x 877.57 = \$99,443.95
EOY1 prin. repaid = 100,000 - 99,443.95 = \$556.05
EOY1 interest = 10,530.86 - 556.05 = \$9,974.81

BOY2 principal balance         = \$99,443.95
EOY2 balance = [PVAF 10/12, 336] x 877.57 = 112.6176 x 877.57 = \$98,829.83
EOY2 prin. repaid = 99,443.95 - 98,829.83 = \$614.12
EOY2 interest = 10,530.86 - 614.12 = \$9,916.74

16
Amortization Schedule
Amount
Year   Outstanding   Payment      Interest   Principal

0      \$100,000
1      99,443.95     \$10,530.86   \$9,974.81 \$556.05
2      98,829.83     10,530.86     9,916.74 614.12
3      98,151.47     10,530.86     9,852.50 678.36

17
Alternatives For Determining Mortgage
Balance
1. Present value of annuity factor (PVAF)

PVAF i%, n - t
Proportion Outstanding = ---------------------------
PVAF i%, n
where
n = the period over which the loan is amortized
t = period in which balance is desired
n - t = remaining life of the loan

18
Alternative method of determining
mortgage balance
2. Mortgage Constant (MC)

MC i%, n
Proportion Outstanding = ---------------------------
MC i%, n - t

19
Example
What is the proportion outstanding at the end of 10th
year for a loan which is fully amortizing, with a term of
30 years, interest rate of 10%, monthly payments. The
original loan amount is \$100,000

PVAF 10/12%, 240 mon.                103.624619
PO = ------------------------------- =   --------------- = .909380195
PVAF 10/12%, 360 mon                  113.950820

Therefore balance outstanding = (.909380195)(100,000) = \$90,938.02

20
Example
Mortgage Constant Approach

MC 10/12%, 360 mon               .008776
PO = -------------------------- = ------------- = .909430051
MC 10/12%, 240 mon               .009650

Proportion paid off = (1 - .909430051) = .0905699
Outstanding loan amount = 100,000x.909430051 = \$90,943.0051

21
Alternatives for Determining %of Loan
Outstanding
3. Future value of annuity factor (FVAF):
FVAF t , i
PO    =   1 - ------------------------
FVAFn, i

204.844979
=     1 - --------------------- = .909380194
2260.487925
where:
FVAFt = future value of annuity factor in period t
FVAFn = future value of annuity factor in period n
t = year in which balance is desired                 22
n = term or amortization period of loan
Mortgage Pricing and Yield Conventions
The price of an asset is equal to the present value of its expected
cash flow.
Steps:
• estimate the expected cash flow
• discount the cash flow at an appropriate interest rate (or
set of interest rates)
• The result is the present value of the asset
• Mortgages are financial assets and their prices are
determined in similar fashion
23
Yield to Prepayment Model
DS1      DS2               (DSn + Bn)
P = -------- + ------- + ,...., + -------------
(1 + r)1 (1 + r)2             (1 + r)n
where:
DSt = monthly debt service (monthly cash flow)
Bt = balance outstanding at time prepayment
r = monthly yield.
n = prepayment year
P = price of mortgage
24
Market Price
• The price of loan depends on the market interest rate and
effective yield
• A loan of \$100,000 with a coupon of 10% will sell at a discount
when market rate has risen to 11%
• The same loan will sell at a premium if rates have declined to
9% since origination
• To determine market we need the following:
• coupon
• term of loan
• market interest rate
• The price should be the present value of the expected cash flow at the 25
market price
Effects of changing interest rate
• Market price of mortgage = PV of debt
service (DS) at the appropriate discount rate
(y).
• If market interest rate have not changed
since loan origination the appropriate
discount rate is simply the contract rate or
coupon rate on the mortgage instrument.
26
Mortgage Pricing: An example
Suppose a mortgage loan in the face amount of
\$100,000 is made and is to be amortized over 30 years
(360 months) and has a term of 30 years. The contract
rate is 10%, with monthly payments
The cash flow form the mortgage is

debt service
= 100,000 x (MC 10/12,360))
= (100,000)(.008776) = \$877.60
27
Mortgage price: no change in interest
rate.
A. AT THE INSTANT THE LOAN IS MADE THE PRICE (P)
OF THE LOAN IS DETERMINED AS FOLLOWS:

P = (PVAF 10/12%, 360 mo)(877.60)
= (113.950820)(877.60)
= \$100,000

Note: We use the contract rate as the discount rate.

28
Mortgage price at year 10 (no change in
interest rates)
B. PRICE OF THE MORTGAGE AT VARIOUS TIMES
1. Price of the mortgage at the end of 10 years, assuming
no change in market interest rate since mortgage
origination.

= (PVAF 10/12%,240 mo)877.60)
= (103.624619)(877.60)
= \$90,940.96
Note: market price will equal book value
29
Price of mortgage at year 10: Market rate
increases
2. Price of the mortgage at the end of 10th year assuming
market interest rate have increased to 12% since
origination of mortgage.

PRICE = (PVAF 12/12%, 240 mo)(877.60)
= (88.017279)(877.6)
= \$77,243.96

Note: there is now a capital loss to the lender
30
Market price at end of year 10: Market
rate decrease
3. Price of mortgage at the end of 10th year assuming market
rates have declined to 8% since mortgage origination.

PRICE = (PVAF 8/12%, 240mos)(877.60)
= (119.554292)(877.60)
= \$104,920.8466
Note: there is now a capital gain to lender: i.e a premium mortgage.
A buyer should be willing to pay more than face value because the
mortgage promises a coupon of 10% compared to market rate of
8%
31
Pricing balloon mortgages.
What is a balloon mortgage?
Suppose the term of the mortgage is 25 years rather than
30 years. The price will determined as follows:
PV of debt service + PV of balance at end of year 25.
The balance at end of year 25 is calculated as follows:
End of year 25 balance = (PVAF 10/12,60))(877.60) =
(47.065369)(877.60) = \$41,304.56783

32
Mortgage yield analysis
Yield to Prepayment Model (YTP)
• The YTP is the internal rate of return (IRR) on an
individual mortgage that will equate the present value of
the cash flow to lender to the price of mortgage.

In computing the yield the convention is to assume that the
borrower will prepay the loan before maturity

This yield measure is for an individual loan
33
Yield calculation
Effective Cost of Borrowing
• Statutory Cost
• Third-Party Charges
• Finance Charges

34
What is a point ?
A Point is simply a percentage that
indicates the extent to which a loan
has been discounted.

35
Consequences of Points

• The actual loan amount received by the borrower
is less than the amount requested
• The debt service or mortgage payment is based
on the face value of the loan not the discounted
amount
• The actual yield to the lender or the cost to the
borrower will be higher than the contract interest
rate
36
An Example
• A borrower requested a loan of
\$40,000 from a lender. The lender
agreed to make loan with the following
features: interest rate of 10%, monthly
payments (or monthly compounding),
with 5 points. The loan will be
amortized over 30 years or 360
months. The term of the loan is also 30
years.                                37
An Example
CASE 1: LENDER DEDUCTS POINT UP-FRONT
Amount received by borrower = \$40,000 - (.05x40,000)
= \$38,0000
Face value of loan              = \$40,000.
Monthly cash flow to lender = Monthly debt service
= [MC 10/12, 360]x 40,000 = .008776 x 40,000
= \$351.04
The investment made by the lender is \$38,000, but has the
right to receive cash flow based on \$40,000.

38
Expected Yield to the Lender
INTERPOLATING (IRR)
PV @ 10.5% = 109.320766 X 351.04 = 38,375.96169
PV @ 11% = 105.006346 X 351.04 = 36,861.42770
Difference                   1514.53399
PV @ 10.5% =                         38,375.96169
Desired PV =                         38,000.00000
Difference                          375.96169
375.96169
Expected Yield = 10.5 + ------------- (11 - 10.5)
1514.53399
= 10.62% > 10%                     39
Effect of Prepayment
Suppose the same loan is taken out but the borrower
prepays or calls the loan at the end of 5th year with no
• Cash Flow Pattern
(1) Annuity of 351.04/month from year 1 to year 5.
(2) Lump sum payment at the end of year 5 = the loan
balance.
= [PVAF 10/12, 300] X 351.04 =
10.047230X351.04 = 38,630.97961
40
Effect of Prepayment
(3) The amount invested by lender is still \$38,000
PV = -\$38,000, PMT = 351.04, n = 60,
FV = 38,630.97961, i = ?

Monthly yield = .944486

Annual yield = .94486x12 = 11.34%

IMPACT OF HOLDING PERIOD : 11.34% > 10.62 > 10%

41
Impact of Prepayment Penalty
Suppose the lender charges 2% penalty when the
loan is prepaid, then the cash flow will be:
•   \$351.04/month from 1 to 5
•   \$38,630.9791 at the end of year 5
•   .02x38,630.9791 = \$772.62 at the end of year 5
•   Lumpsum = \$39,403.59 = 38,630.9791 + 772.62
•   See if you can calculate the yield
42
The Effects of Points: Another View
• CASE 2: NOW ASSUME THE LENDER FINANCES
POINTS
• FACE VALUE OF LOAN = 40,000 + .05x40,000 = \$42,000
• ACTUAL LOAN AMOUNT = \$40,000
• DEBT SERVICE = (MC 10/12, 360) x 42,000
•              = .008776x42,000 = \$368.592

• PV = - \$40,000, PMT = 368.592, N = 360, i = ?
• Monthly yield = .882505932
•   Annual yield = 10.59% compare with yield when lender deducts points up-front
43
opportunities
• The monthly cash flow from the mortgage can be
reinvested
• Mortgage yield should be more than that earned on
similar corporate bond with semiannual coupon
payments
• The effective yield on the mortgage must incorporate
the monthly compounding effect
• Finally, to make the yield on a mortgage comparable to
that of Treasury or corporate bond of similar maturity,
calculate the Bond Equivalent Yield.                 44
Effective Yield on a mortgage
EY = (1+y/12)12 - 1
where y is annual yield

Nominal yield y = 10.59% or monthly yield r = 10.59/12
= .8825 form previous example

EY= (1.008825)12 - 1
= 1.111194372 - 1
= 11.11%                                       45
Bond Equivalent Yield
To make the yield on a mortgage comparable to that of
Treasury or corporate bond of the similar maturity
calculate the bond equivalent yield (BEY)
BEY = 2[(1+y/12)6 - 1]

ILLUSTRATION:
y = 10.59 or r = .8825
BEY = 2[(1.008825)6 - 1] = 2[1.054132047-1]
= 10.86%                                    46
Mortgages
• If P = PAR the YTP is independent of
when prepayment occurs
• The YTP will also be equal to the
contract rate on the mortgage (see
Exhibit 4-8, page 109 of text book)
• What are the underlying assumptions here?

47
Mortgage Refinancing Decision
• Falling interest rates create opportunities for refinancing
because the option is now “in the money”.
• “Out of the money” mortgage may be refinanced for
other reasons
• Refinancing even in the case of falling rates may or may
not be beneficial
• To make a sound refinancing decision we must consider
the benefit as well as the cost of refinancing

48
Variables Influencing Refinancing
Decision
Cf.: Existing loan versus new loan
1.Mortgage type
2.Interest rate
3.Mortgage maturity
4.Prepayment penalty
5.Financing costs
6.Holding period
49
An illustration: The Original loan
Assumptions
Existing Mortgage
·Original loan amount   \$50,000
·Interest Rate          12 percent
·Term                   30 years
·Age of mortgage        5 years
·Prepayment Penalty            5 percent
. Monthly payment

50
Parameters of new loan
NEW MORTGAGE
·Loan Amount         =   Outstanding balance on old
loan
·Interest rate       =   10½ percent
·Term                =   25 years
·Refinancing costs   =   3%
·Holding period      =   Hold mortgage until
maturity
·Borrower's          =   13 percent
opportunity rate
51
Refinancing Decision
Determine Outstanding Balance on Old Loan
First determine monthly payment
\$50,000    =     PV
12/12      =     i
30 x 12    =     n
PMT        =     \$514.30

52
Refinancing Decision
Outstanding balance at the end of 5th year
\$514.30      =     PMT
12/12        =     i
25 x 12      =     n
PV           =     \$48,831

Therefore the outstanding balance on the old loan
= \$48,831
53
Refinancing Decision
Determine Monthly Payment on New Loan
Payment:

\$48,831    =     PV
10.5/12    =     i
25 x 12    =     n
PMT        =     \$461.05

54
Refinancing Decision
Determine Cost of Refinancing
Prepayment Penalty on existing loan
= .05 x \$48,831
=\$2,442
Plus financing cost on new loan
= .03 x \$48,831
= \$1,465
TOTAL : \$3,907
55
Refinancing Decision
Determine Benefit of Refinancing

Monthly payments on existing loan =   \$514.30
Less Monthly payments on new loan=     461.05
TOTAL                              \$53.25 25

56
Refinancing Decisions
Determine Present Value of Savings
Present value of savings
\$53.25       =    PMT
13 /12       =    i
25 x 12      =    n
PV           =    \$4,721
Net present value=\$4,721-3,907=\$814.
The refinancing is worth undertaking under our assumption.
What is the yield on this mortgage refinancing investment?
57
Question:Shorter holding period
• What happens if we assume the borrower’s
holding period is only 10 years?
– 1.The cost of refinancing, \$3,907, will be the
same.
– 2.The benefit of refinancing will change.

58
Determine Benefit of Refinancing
• 1. Monthly savings on mortgage payment
Monthly payments on existing loan =            \$514.30
Less Monthly payments on new loan=            461.05
\$53.25
This amount is equal to that under the first example, but we
will receive this saving for only 120 months and not 300
months as was the case in the first example.

59
2. Lump sum saving
• Determine the outstanding balance on old loan at the end
of 15 years.
\$514.30     =      PMT
12/12       =      i
15 x 12     =      n
PV          =      \$42,852
• Determine the outstanding balance on new loan at the end
of 10 years.
\$461.05     =      PMT
10.5/12     =      i
15 x 12     =      n
PV          =      \$41,709
Lump sum saving=42,852-\$41,709=\$1,143                      60
Determine Present Value of Savings
1. PV of monthly savings
\$53.25 =       PMT
13 /12 =       i
10 x 12 =      n
PV             =\$3,566
2. PV of the lump sum savings
-\$1,143 =      FV
13 /12 =       i
10 x 12 =      n
PV             =\$313.68
Net present value=\$3,566+313.68-3,907=-\$27.32
Therefore, we should not refinance if we plan to hold the
mortgage for only 10 years.
61
Question: Find the holding period that has NPV=0.
Refinancing
•   Summary:
•   Find present value of savings
•   Subtract cost from PV of savings
•   If positive NPV refinance
•   If negative NPV do not refinance

62
Wraparound Mortgage
• Junior mortgage that wraparounds or includes an
existing first mortgage
• Permits a second lender to extend additional debt
to the borrower by lending an amount above the
balance outstanding on the existing first mortgage
• The additional amount is extended without the
borrower paying off the balance outstanding on
the first mortgage.

63
Some Important Features
• The face value of the loan = balance on existing first
mortgage + amount actually extended by the wrap
lender
• The debt service is based on face value of the wrap
loan.
• The interest rate is typically below market but above
that of the existing first loan
• Borrower makes one payment to the wrap lender
• The wrap lender keeps his/her share of payment and
passes on the remaining portion to the first lender.

64
An Example
An \$800,000 mortgage has 25-year maturity and calls for
level monthly payments based on an annual interest rate of
7%. At the end of the 15th year with market interest rate
now at 9% the borrower requests an additional amount of
\$200,000, but the first lender is unwilling or unable to
wraparound the outstanding balance on the existing first
loan the \$200,000 requested by the borrower, at an annual
interest rate of 8% for 10 years, with monthly payments.
Problem: Determine the expected yield to the wraparound
lender.
65
Steps
A. Calculate the debt service on the original loan
monthly debt service = .007068 * 800,000
= \$5654.4

B. Find the balance outstanding on first mortgage
PV. of \$5654.40/month @ 7% for 10 years
= 86.126354*5654.4
= \$486,992.8561
66
Steps
C. Find the debt service on wraparound
loan
face value = 486,992.8561 + 200,000
= \$686,992.8561
monthly debt service =
686,992.8561*.012133 = \$8335.28

67
Steps
D. Find the net debt service to the wrap lender
\$8335.28 - \$5654.4 = \$2680.88, this is the actual cash
flow that goes to wrap lender

E. Find the IRR to the wrap lender
PV = -200,000            CF = \$2680.88
n = 10*12 = 120
Monthly IRR = .861871 Annual IRR = 10.34%
Why is the IRR > than the contract rate?              68
Below Market Financing
Below market financing can arise if
• the seller holds a mortgage on the property that is
assumable and current market rates are high
• if the buyer is able to assume the existing
mortgage
• the seller may provide purchase money mortgage
at below market interest rate

69
Below Market Financing
• In an efficient market the seller will increase the selling
price of the house to reflect the value of the below
market financing.
• Therefore in a competitive market the seller should
capture the benefit of the below market financing in the
form of higher selling price.
• The effective cost of borrowing to the buyer should be
such that the buyer is indifferent between assuming the
old loan at below market rate and getting a new loan
with the same face value at current market rate.

70
An Example
Market value of property = \$100,000
Assumable mortgage = \$70,000
remaining term           = 15 years
contract rate             = 9%
selling price with BMF = \$105,000
Current mortgage terms: 11%, 15 years, \$70,000.
Is the house fairly priced at \$105,000?
71
Example
Selling price = PV of the debt + PV of equity
Payment on old loan = 70,000x.010143 = \$710.01
New loan payment = 70,000x.011366 = \$795.62
PV of the difference at market rate
= (795.62-710.01)x PVAF ( 11/12, 180.)
= \$85.61 x 87.981937      = \$7,532.133

72
Example
If the market value of the property = \$100,000
Then the seller can ask as much as \$10,532 =
\$100,000+\$7532
The buyer should be indifferent between paying
\$107,532 and obtaining a new loan of \$70,000 at 11%,
and buying the house at its market value of \$100,000
Note: At any price below \$107,532 the buyer gets a bargain
and at any price above \$107,532 the buyer is worse off. See if
you can demonstrate this.

73
Yield Maintenance Agreements and Lock-out
provisions in commercial Mortgages
• Commercial mortgages often contain either or both of the
following important provisions
– lock-out provision
– yield maintenance agreement
• Lock-out provision prohibits prepayments prior to
maturity or some stated period before maturity.
• Yield maintenance agreement is designed to discourage
prepayment and to compensate the lender when
prepayments occur during periods of falling interest rates.
74

```
To top