課程七 : Adjustable-rate Mortgages

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

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

Nature of the Tilt Problem: Real Value of Monthly Payments

No Inflation

4% Inflation

9% Inflation$ 50

$ 100

$ 150

$ 200

Years Elapsed

5 10 15 20 25 30 35

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 . .

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%.

How rising inflation makes housing unaffordable

Interest rate Payment Interest PTI Ratio

25% $416.94 $4999.63 83.38%

20% $334.20 $3998,99 66.84%

15% $252.89 $2997.51 50.58%

10% $175.51 $1995.00 35.10%

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.

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

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 TreasuryYield Curve

Treasury Interest RateScenarios

Coupon Reset AlongScenarios

PrepaymentModel

ProjectedCash Flows

Discount atTreasury Rate

Pertinent to Each CashPlus Spread

Market Price

Non Treasury IndicesGenerated 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: CR1 = 8%,

DS1 = (60,000)(MC 8/12, 360) = $440.28

OB1 = (440.28)(PVAF 8/12, 348) = $59,502

Year 2: CR2 = 10+2 =12%

DS2 = (59,502)(MC 12/12, 348) = $614.30

OB2 = (614.30)(PVAF 12/12, 336 ) = $59260

Uncapped ARM (contd.)

Year 3 : CR3 = 13+2 = 15%

DS3 = (59260)(MC 15/12, 336) = $752.27

OB3 = (752.27)(PVAF 15/12, 324) = $59,106

Year 4: CR4 = 15+2 = 17

DS 4 = (59,106)(MC 17/12, 324) = $846.22

OB 4 = (846.22)(PVAF 17/12, 312) = $58,991.69

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: CR1 = 7+2 = 9%

DS1 = (60,000)(MC 9/12 ,360) = $482.77

Yr1 interest = (.09/12)(60,000) = $450

$482.77(DS1) > $450 (I1), therefore amortization is positive

OB1 = (482.77)(PVAF 9/12, 348) = $59,590

ARM with payment cap (Contd)Year 2: CR2 = 10+2 = 12%

DS2 (uncapped) = (59,590)(MC 12/12, 348) = 615.18

Since $615.18 > $482.77 by more than 7.5%, the payment cap is binding

DS2 (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 (DS2) < $595.90, therefore there is negative amortization

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: CR3 13 + 2 = 15

DS3 (uncapped) = (60,566)(MC 15/12, 336) = $768.91

Since $768.91 > $518.98 by 48.2% > 7.5%

payment cap is binding

DS3 (capped at 7.5%) = (518.98)(1.075) = $557.90

YR3 interest = (.15/12)(60,566) = $757.90

ARM with payment cap (Contd).Since $557.90 (DS3) < $757.08 (I3) 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

ARM with periodic and life of loan interest rate caps

The following notation will be used:

tRj = the contract rate for period t and adjustment length j

tIj = 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 capsYear 1: 1R1 = 9+2=11

DS1 = (MC 11/12, 360)(60,000) = $571.39

EOY1 = (571.39)(PVAF 11/12, 348) = $59,730

Year 2: Is interest rate cap binding?

t-1 Ij + c = 9 + 2 = 11>2Ij = 10, So CAP is not binding

2R1 = tIj + m = 10 + 2 = 12

DS2 = (MC 12/12, 348)(59,730) = $616.63

EOY2 = (616.6)(PVAF 12/12, 336) = $59,485

ARMs with interest rate caps (Contd).Year 3: Is interest rate cap binding?

t-1 Ij + c = 10 + 2 = 12 < 3Ij = 13 cap is binding

*Therefore the most we can add is 2%

3R1 = t-1 Ij + m + c = 10 + 2 + 2 = 14%

DS3 = (MC 14/12, 336)(59,485) = $708.37

EOY3 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 < 4Ij = 15 Cap is binding

Note: the effective value of index in year three is 12 or (13-1)

ARMs with interest rate caps (Contd).

DS4 = (MC 16/12, 324)(59,301) = $801.65*

EOY4 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%.

5R = 5I + m = 10 + 2 = 12

DS5 = (MC 12/12, 312)(59,159) = $619.37

EOY5 balance = (619.37)(PVAF 12/12, 300) = $58,807

ARMs with interest rate caps (Contd).

• Effect of floor interest rate capAssume 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 > tI = 14 > 10 by 4 > c or 2 floor cap is binding

5R = t-1 I + m - c = 14 + 2 - 2 = 14

DS5 = (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

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%

• DS5 = (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