1. 課程 9: ARM Payment Plans

2. What is an Adjustable Rate Mortgages (ARM) In an ARM the debt service can increase or decrease depending on fluctuations in a designated index. This is in sharp contrast to Fixed Rate Mortgage (FRM) where the payment stays constant over the duration or term of the loan. In an ARM, unlike FRM, the borrower is responsible for most or all of the interest rate risk. Remember in the case of FRM the lender is responsible for any interest rate risk.

3. Why the call to ARMs? –Interest Rate Risk –Asset-liability mismatch –Removal of Regulation Q –Asymmetric Risk Bearing between lenders and borrowers falling rates rising rates

4. Factors Affecting ARM Pricing Contract Rate = Index + Margin Rate Reset Timing –frquency with which rate is reset Payment Reset Timing –frequency with which payment is reset Payment Caps Teaser Rates or Initial Discount Negative Amortization

5. ARM Pricing Factors (continued) Type of index: –Treasury Index (T), – Cost of Funds (COFI) Periodic interest rate caps –Floor caps –Ceiling caps Life of loan interest rate caps None ARM factors –Slope of yield curve –Volatility of interest rate –Credit risk –Interest rate risk

6. Basic Principles of Valuation of ARMs Current Coupon Treasury Yield Curve Treasury Interest Rate Scenarios Coupon Reset Along Scenarios Prepayment Model Projected Cash Flows Discount at Treasury Rate Pertinent to Each Cash Plus Spread Market Price Non Treasury Indices Generated from Treasury Scenario Contractual Terms of ARM

7. An Example of ARM UNCAPPED ARM: To illustrate the mechanics of uncapped ARM consider the following ARM. The face value of loan is $60,000. The value of the index in years 1, 2, 3, 4 and 5 are 6%, 10%, 13%, 15% and 10%, respectively. The adjustment interval is one-year and the loan is amortized over 360 months (30years). The margin is 2%. The ARM has neither periodic interest cap nor life-of-loan cap. This means that contract rate can increase or decrease freely. The ARM has no teaser.

8. Illustration : ARM Pricing CALCULATIONS: Year 1: CR 1 = 6 + 2 = 8%, DS 1 = (60,000)(MC 8/12, 360 ) = $440.28 OB 1 = (440.28)(PVAF 8/12, 348 ) = $59,502 Year 2: CR 2 = 10+2 =12% DS 2 = (59,502)(MC 12/12, 348 ) = $614.30 OB 2 = (614.30)(PVAF 12/12, 336 ) = $59260 DS = debt service; OB = outstanding balance; PVAF = present value of annuity factor.

9. Contd. Year 3 : CR 3 = 13+2 = 15% DS 3 = (59260)(MC 15/12, 336 ) = $752.27 OB 3 = (752.27)(PVAF 15/12, 324 ) = $59,106 Year 4: CR 4 = 15+2 = 17 DS 4 = (59,106)(MC 17/12, 324 ) = $846.22 OB 4 = (846.22)(PVAF 17/12, 312 ) = $58,991.69 Year 5: CR 5 = 10+2 = 12 DS 5 = (58,991.69)(MC 12/12, 312 ) = $617.64 OB 5 = (617.64)(PVAF 12/12, 300 ) = $58,643.07

10. Example of ARM with interest rate cap To illustrate the mechanics of ARM with interest rate caps consider the following. The ARM is in the amount of $60,000. The interest rate is adjusted annually if the value of the underlying index changes. The loan is amortized over 360 months. The margin is 2% and the periodic rate cap is 2%. The loan also has a life-of-loan cap of 5%. The value of the index are as follows: Yr1 = 9%, Yr2 = 10%, Yr3 = 13, Yr4 = 15%, Yr5 = 10%

11. ARM with interest rate cap Year 1: 1 R 1 = t I i + m = 9+2=11 DS 1 =(MC 11/12, 360 )(60,000) = $571.39 EOY 1 =(571.39)(PVAF 11/12, 348 ) = $59,730 Year 2: Is interest rate cap binding? t-1 I j + c = 9 + 2 = 11; 11 > 2 I j = 10, So CAP is not binding  2 R 1 = t I j + m = 10 + 2 = 12 DS 2 = (MC 12/12, 348 )(59,730) = $616.63 EOY 2 = (616.6)(PVAF 12/12, 336 ) = $59,485

12. Contd. Year 3: Is interest rate cap binding? t-1 I j + c = 10 + 2 = 12 < 3 I j = 13  cap is binding *Therefore the most we can add is 2%  3 R 1 = t-1 I j + m + c = 10 + 2 + 2 = 14% DS 3 = (MC 14/12, 336 )(59,485) = $708.37 EOY 3 balance = (708.37)(PVAF 14/12, 324 ) = $59,301 Year 4: Is interest rate cap binding? (boundary conditions) t-1 I* j + c = 12* + 2 = 14 < 4 I j = 15  Cap is binding Note the effective value of index in year three is 12 or (13-1)

13. Contd. DS 4 = (MC 16/12, 324 )(59,301) = $801.65* EOY 4 Balance = (801.65)(PVAF 16/12, 312 ) = $59,159 Year 5: Since there are no floor caps there is no limit on how low the contract rate can be. The value of the index in year five is 10%. 5 R = 5 I + m = 10 + 2 = 12 DS 5 = (MC 12/12, 312 )(59,159) = $619.37 EOY 5 balance = (619.37)(PVAF 12/12, 300 ) = $58,807

14. Effects of life of loan caps Assume there is life-of-loan cap of 5% and we are at the end of year 4 and also that year 5 index = 16% Analysis: –Life of loan cap is now binding (1+2+2 = 5) –Contract rate in year 5 will be same as contract rate in year 4 = 16% DS 5 = (MC 16/12, 312)(59,159) = $801.64 Thus with life of loan cap at 5% and an increase in the index in year 5, the payment would have been $801.64 and not $619.37