1. 課程七 : Adjustable-rate Mortgages

2. Deficiencies of traditional fixed rate mortgage With high and variable inflation two problems are created: tilt problem mismatch problem The tilt problem refers to tendency of real burden of mortgage payments to “tilt “to initial years The result is that inflation places homeownership out the reach of many households The second, mismatch, problem refers to the mismatch between the assets and liabilities of lenders –Lenders borrow short and lend long –The difference between the lending rate and the borrowing rate causes lenders to become technically insolvent –Liabilities are related to face value of mortgage assets, but market value will be below face value of assets

3. The nature of the tilt problem The following figure illustrates the nature of the distortion or tilt in real payments For 9% rate of inflation, the path starts at more than twice the no inflation level. Notice how the 9% inflation path terminates at well below half of the no inflation path The high initial payment caused by inflation has the effect of foreclosing homeownership from large segment of population

4. Nature of the Tilt Problem: Real Value of Monthly Payments No Inflation 4% Inflation 9% Inflation $ 50 $ 100 $ 150 $ 200 Years Elapsed 5101520253035

5. Illustration of the tilt problem and mortgage affordability Consider, a household with a monthly income of $500. The family allocates 31% of its income to mortgage payment for a 30-year fixed rate mortgage. Scenario 1: no inflation; real rate = 7% = contract rate Payment =.31x500 = $155 [PMT] interest =7/12 = 0.5833 [i] term = 30x12= 360 [n] Present value = $23,297.67 [PV ] –Thus the household can afford a mortgage of $23,297.67, a figure which is 3.9 its annual income. –If you have a calculator with present value and mortgage payment buttons, just key in the numbers followed by the appropriate buttons to get the amount of $23,297.67..

6. The tilt problem: effect of inflation Consider, a household with a monthly income of $500. The family allocates 31% of its income to mortgage payment for 30-year fixed rate mortgage. Scenario 1: inflation at 9%; contract rate = 16% Payment = $155 [PMT] interest 16/12 = 1.3333 [i] term = 30x12 = 360 [n] Present value = $11,526.24 [PV] –with the same amount of income and inflation at a modest 9% the, household can afford a mortgage of only $11,526.24. This is figure is only 49% of the mortgage amount the household can afford in a world of no inflation. Alternatively for the household to afford a mortgage of $23,298, from the previous example, it must be willing to allocate $313.30, or 63% of its income to mortgage payment, if inflation is at 9%.

7. How rising inflation makes housing unaffordable Interest ratePaymentInterestPTI Ratio 25%$416.94$4999.6383.38% 20%$334.20$3998,9966.84% 15%$252.89$2997.5150.58% 10%$175.51$1995.0035.10% This table illustrates the tendency of FRM to redistribute real mortgage payments toward early years of the loan. This tilting of payments causes an increasing mismatch between real payments and the income capacity of households over the loan life. The result is that housing becomes less affordable.

8. Adjustable Rate Mortgages: 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

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

10. 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

11. 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

12. 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

13. 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 ) = $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

14. Uncapped ARM (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

15. ARM with payment cap and negative amortization Amortization (See page 151 to 152 of text book) 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 ) = $482.77 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

16. ARM with payment cap (Contd) Year 2: CR 2 = 10+2 = 12% DS 2 (uncapped) = (59,590)(MC 12/12, 348 ) = 615.18 Since $615.18 > $482.77 by more than 7.5%, the payment cap is binding DS 2 (Capped at 7.5%) = (482.77)(1.075) = $518.98.Because payment cap is binding there will be negative amortization YR2 interest = (.12/12)(59,590) = $595.90 $518.98 (DS 2 ) < $595.90, therefore there is negative amortization

17. ARM with payment cap (Contd). This is equal to $518.98-$595.90 = -$76.92 Future value = (76.92)(FVAF 12/12, 12) = (76.92)(12.682503) = $975.54 Beginning of year 3 balance = $59,590 + $975.54 = $60,566 Year 3: CR 3 13 + 2 = 15 DS 3 (uncapped) = (60,566)(MC 15/12, 336) = $768.91 Since $768.91 > $518.98 by 48.2% > 7.5% payment cap is binding DS 3 (capped at 7.5%) = (518.98)(1.075) = $557.90 YR3 interest = (.15/12)(60,566) = $757.90

18. ARM with payment cap (Contd). Since $557.90 (DS 3 ) < $757.08 (I 3 ) negative amortization is present This is equal to $557.90 - $757.08 = -$199.18 Future value of negative amortization = (199.18)(FVAF 15/12, 12 ) = (199.18)(12.860361) = $2521.53 Therefore, beginning of year 4 balance = $60,566 + $2561.53 = $63,128

19. 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

20. ARM with interest rate caps Year 1: 1 R 1 =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> 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

21. ARMs with interest rate caps (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)

22. 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 = 10 + 2 = 12 DS 5 = (MC 12/12, 312 )(59,159) = $619.37 EOY 5 balance = (619.37)(PVAF 12/12, 300 ) = $58,807

23. 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 + 2 - 2 = 14 DS 5 = (MC 14/12, 312)(59,159) = $709.20 Therefore, had there been a floor cap of 2% the payment for year 5 will be $709.20, not $619.37

24. 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 of 5% and an increase in the index in year 5, the payment would have been $801.64 and not $619.37