1. 課程六:Mortgage Markets

2. Alternative mortgage contracts
Fixed Rate Mortgage (FRM)
Adjustable Rate Mortgage (ARM)
Graduated Payment Mortgage (GPM)
Shared Appreciation Mortgage (SAM)
Reverse Annuity Mortgage (RAM)
Growing Equity Mortgage (GEM)
Balloon Mortgage

3. Other Mortgages Junior Mortgage Purchase Money Mortgage Land Contract
Wraparound Mortgage

4. Types of mortgage amortization
Interest only mortgage (bullet loans)
Partially amortizing or balloon mortgage
Fully amortizing

5. Risks faced by mortgage finance intermediaries
Credit risk:
risk that money borrowed might not returned timely
Default risk:
risk that money lent might not be repaid
Cash flow risk:
risk that market conditions will alter scheduled cash flows
prepayment risk
inflation risk
exchange risk
interest rate risk
Liquidity risk:
risk that money will be needed before it is due

6. 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
D = risk loading
P = pure interest rate risk component
j = term of loan

7. Mortgage Contract Rate
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

8. 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

9. 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 annuity factor (PVAF)

10. 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)?

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

12. 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 =
MB210 = 100,000{[( )360 - ( )210]/[( ) ]} = $73,668
Check (calculator): $ = PMT 9.5/12 = i ; 150 = n PV =? $73,668

13. 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 = {[ ( ) ]/[( )360-1]}
= 100,000{ ( ) = $255.62
CHECK:
= PMT ; 9.5/12 = i ; 13x12 = n ; PV = $75,171.72
Balance at end of month 210 = $73,667.78
Scheduled principal paid = $75, $73, = $

14. 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{ [( )360 - ( ) ]/ [( ) ]}
= 100,000{ [( )]/[ ]} = $585.23
CHECK
Debt Service = (p) (i) = $840.85

15. 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 = /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 = $ = $552.51

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

17. Amortization Schedule
A. INTEREST RATE METHOD
BOY1 principal balance = $100,000
EOY1 interest (100,000 x .1) = $10,000
EOY1 principal repaid = $608
(10, ,000)
EOY1 balance (100, ) = $99,392
BOY2 principal balance = $99,392
EOY2 interest (99,392 x .1) = $9,939.2
EOY2 principal repaid = $668.2
(10, ,939.2)
EOY2 balance (99, ) = $98,723.2

18. Amortization Schedule
Amount
Year Outstanding Payment Interest Principal
0 $100,000
1 99,392 $10, $10,000 $608
2 98, , ,
3 97, , ,

19. 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 =
Monthly Debt Service = 100,000 x = $877.57
Annual Payment = 100,000 x x 12 = $10,530.86

20. Amortization Schedule
BOY1 principal balance = $100,000
EOY1 balance = [PVAF 10/12, 348] x = x = $99,443.95
EOY1 prin. repaid = 100, , = $556.05
EOY1 interest = 10, = $9,974.81
BOY2 principal balance = $99,443.95
EOY2 balance = [PVAF 10/12, 336] x = x = $98,829.83
EOY2 prin. repaid = 99, , = $614.12
EOY2 interest = 10, = $9,916.74

21. Amortization Schedule
Amount
Year Outstanding Payment Interest Principal
0 $100,000
1 99, $10, $9, $556.05
2 98, , ,
3 98, , ,

22. 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

23. Alternative method of determining mortgage balance
2. Mortgage Constant (MC)
MC i%, n
Proportion Outstanding =
MC i%, n - t

24. 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
PO = = =
PVAF 10/12%, 360 mon
Therefore balance outstanding = ( )(100,000) = $90,938.02

25. Example Mortgage Constant Approach MC 10/12%, 360 mon .008776
PO = = =
MC 10/12%, 240 mon
Proportion paid off = ( ) =
Outstanding loan amount = 100,000x = $90,

26. Alternatives for Determining %of Loan Outstanding
3. Future value of annuity factor (FVAF):
FVAF t , i
PO =
FVAFn, i
= =
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
n = term or amortization period of loan