Adjustable-rate Mortgages

Risk Sharing Arrangement Lender bears all risk Shift all interest rate risk to borrower Shift part of interest rate to borrower

Factors Affecting ARM Pricing Contract Rate = Index + Margin Rate Reset Timing frequency with which rate is reset, monthly, annually, etc Payment Reset Timing frequency with payment is reset, usually annually Payment Caps Teaser Rates or Initial Discount Points Negative Amortization

ARM Pricing Factors (continued) Type of index: Treasury (T), Cost of Funds (COFI ) Interest rate Caps –Periodic interest rate caps Floor caps Ceiling caps –Life of loan interest rate caps Other non ARM factors Slope of yield curve Volatility of interest rate Interest rate risk, Credit risk

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

Illustration : ARM Pricing See page 149 of text book for assumptions UNCAPPED ARM or Unrestricted ARM Year 1: CR 1 = 8%, DS 1 = (60,000)(MC 8/12, 360 ) = $ 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 ) = $ OB 2 = (614.30)(PVAF 12/12, 336 ) = $59260

Uncapped ARM (contd.) Year 3 : CR 3 = 13+2 = 15% DS 3 = (59260)(MC 15/12, 336 ) = $ 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 ) = $ OB 4 = (846.22)(PVAF 17/12, 312 ) = $58,991.69

ARM with payment cap and negative amortization Amortization Payment cap is set at 7.5% and negative amortization Year 1: CR 1 = 7+2 = 9% DS 1 = (60,000)(MC 9/12, 360 ) = $ Yr1 interest = (.09/12)(60,000) = $450 $482.77(DS 1 ) > $450 (I 1 ), therefore amortization is positive OB 1 = (482.77)(PVAF 9/12, 348 ) = $59,590

ARM with payment cap (Contd) Year 2: CR 2 = 10+2 = 12% DS 2 (uncapped) = (59,590)(MC 12/12, 348 ) = Since $ > $ by more than 7.5%, the payment cap is binding DS 2 (Capped at 7.5%) = (482.77)(1.075) = $ Because payment cap is binding there will be negative amortization YR2 interest = (.12/12)(59,590) = $ $ (DS 2 ) < $595.90, therefore there is negative amortization

ARM with payment cap (Contd). This is equal to $ $ = -$76.92 Future value = (76.92)(FVAF 12/12, 12) = (76.92)( ) = $ Beginning of year 3 balance = $59,590 + $ = $60,566 Year 3: CR = 15 DS 3 (uncapped) = (60,566)(MC 15/12, 336) = $ Since $ > $ by 48.2% > 7.5% payment cap is binding DS 3 (capped at 7.5%) = (518.98)(1.075) = $ YR3 interest = (.15/12)(60,566) = $757.90

ARM with payment cap (Contd). Since $ (DS 3 ) < $ (I 3 ) negative amortization is present This is equal to $ $ = -$ Future value of negative amortization = (199.18)(FVAF 15/12, 12 ) = (199.18)( ) = $ Therefore, beginning of year 4 balance = $60,566 + $ = $63,128

ARM with periodic and life of loan interest rate caps The following notation will be used: t R j = the contract rate for period t and adjustment length j t I j = index value in period t and adjustment for period j m = margin C = the value of the periodic interest rate cap j = length of the adjustment period e.g.. 1/2, 1yr, 2yr, etc

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

ARMs with interest rate caps (Contd). Year 3: Is interest rate cap binding? t-1 I j + c = = 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 = = 14% DS 3 = (MC 14/12, 336 )(59,485) = $ 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)

ARMs with interest rate caps (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 = = 12 DS 5 = (MC 12/12, 312 )(59,159) = $ EOY 5 balance = (619.37)(PVAF 12/12, 300 ) = $58,807

ARMs with interest rate caps (Contd). Effect of floor interest rate cap Assume there is floor cap of 2%. This means that even if the index declines by more than 2% the maximum reduction in rate will be 2% Year 5: Is floor cap binding? t-1 I > t I = 14 > 10 by 4 > c or 2  floor cap is binding 5 R = t-1 I + m - c = = 14 DS 5 = (MC 14/12, 312)(59,159) = $ Therefore, had there been a floor cap of 2% the payment for year 5 will be $709.20, not $619.37

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) = $ Thus with life of loan cap of 5% and an increase in the index in year 5, the payment would have been $ and not $619.37