課程 6:Mortgage Markets

Other Mortgages • Junior Mortgage • Purchase Money Mortgage • Land Contract • Wraparound Mortgage

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

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

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

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

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

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)

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 = ?

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

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

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

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

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

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

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

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

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

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

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

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

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

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 n = term or amortization period of loan

課程 6:Mortgage Markets