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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 D = risk loading 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 prepayment penalty or call premium. • 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 Adjusting for reinvestment 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 Additional Note on Yield From 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 advance the additional funds. A new lender agrees 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