課程六:Mortgage Markets

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

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 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 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 = .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