Solutions to Questions – Chapter 5
Adjustable and Floating Rate Mortgage Loans
Question 5-1
In the previous chapter, significant problems regarding the ability of borrowers to meet mortgage payments and
the evolution of fixed interest rate mortgages with various payment patterns were discussed. Why didn’t this
evolution address problems faced by lenders? What have lenders done in recent years to overcome these
problems?
Question 5-2
How do inflationary expectations influence interest rates on mortgage loans?
Question 5-3
How does the price level adjusted mortgage (PLAM) address the problem of uncertainty in inflationary
expectations? What are some of the practical limitations in implementing a PLAM program?
Question 5-4
Why do adjustable rate mortgages (ARMs) seem to be a more suitable alternative for mortgage lending than
PLAMs?
Question 5-5
List each of the main terms likely to be negotiated in an ARM. What does pricing an ARM using these terms mean?
Question 5-6
What is the difference between interest rate risk and default risk? How do combinations of terms in ARMs affect
the allocation of risk between borrowers and lenders?
Question 5-7
Which of the following two ARMs is likely to be priced higher, that is, offered with a higher initial interest rate?
Question 5-8
What are forward rates of interest? How are they determined? What do they have to do with indexes used to
adjust ARM payments?
Question 5-9
Distinguish between the initial rate of interest and expected yield on an ARM. What is the general relationship
between the two? How do they generally reflect ARM terms?
Question 5-10
If an ARM is priced with an initial interest rate of 8 percent and a margin of 2 percent (when the ARM index is
also 8 percent at origination) and a fixed rate mortgage (FRM) with constant payment is available at 11 percent,
what does this imply about inflation and the forward rates in the yield curve at the time of origination? What is
implied if a FRM were available at 10 percent? 12 percent?
Solutions to Problems – Chapter 5
Adjustable Rate and Variable Payment Mortgages
Problem 5-1
(a) Compute the payments at the beginning of each year of the PLAM.
CFj nj
-$89,300
Problem 5-2
(2)
(3)
(4)
(5)
(6)
(7)
(8)
EOY
Balance
(1) – (7)
Annual
Interest
Rate
Monthly
Interest
Rate (2)/12
Payments
Monthly
Interest
(3) x (1)
Monthly
Amort
Annual
Amort.
Year
(4) –(5)
0
1
6.00%
0.50%
$1,199.10
$1,000.00
$199.10
$2,456.02
$197,544
2
7.00%
0.58%
$1,327.75
$1,152.34
$175.41
$2,173.82
$195,370
(a)
(b)
(c)
(d)
(e)
Problem 5-3
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
EOY
Balance
(1) – (7)
Annual
Interest
Rate
Monthly
Interest Rate
(2)/12
Monthly
Interest
(3) x (1)
Monthly
Amort
Annual
Amort.
BOY
Balance
(4) –(5)
Year
Payments
0
1
$150,000
7.00%
0.58%
$997.95
$875.00
$122.95
$1,523.71
$148,476
2
148,525
7.00%
0.58%
$997.95
$866.11
$131.84
$1,633.86
$146,842
3
146,942
7.00%
0.58%
$997.95
$856.58
$141.37
$1,751.98
$145,090
4
145,244
6.00%
0.50%
$905.34
$725.45
$179.89
$2,219.06
$142,871
(a)
Monthly Payment = $997.95
Loan Balance EOY 3 = $145,090
(b)
New Monthly Payment = $906.30
(c)
Interest only monthly payment = $875
Problem 5-4
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
EOY
Balance
(1) – (7)
Annual
Interest
Rate
Monthly
Interest
Rate
(2)/12
Monthly
Interest
Monthly
Amort
Annual
Amort.
BOY
Balance
(4) –(5)
Year
Payments
0
1
$100,000
2.00%
0.17%
$423.85
$166.67
$257.19
$3,114.70
$96,885
2
96,885
6.00%
0.50%
$635.55
$484.43
$151.12
$1,864.15
$95,021
(a)
Monthly payment during 1 year = $423.85
(b)
Monthly payment in 2 year = $635.55
(c)
Percentage increase in monthly payment = 50%
(d)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
EOY
Balance
(1) – (7)
Annual
Interest
Rate
Monthly
Interest
Rate
(2)/12
Monthly
Interest (3)
x (1)
Monthly
Amort
Annual
Amort.
BOY
Balance
(4) –(5)
Year
Payments
0
1
$100,000
2.00%
0.17%
$423.85
$166.67
$257.19
$3,114.70
$96,885
2
96,885
2.00%
0.17%
$423.85
$161.48
$262.38
$3,177.57
$93,708
3
93,708
2.00%
0.17%
$423.85
$156.18
$267.67
$3,241.71
$90,466
4
90,466
6.00%
0.50%
$617.95
$452.33
$165.62
$2,043.02
$88,423
Monthly payments at beginning of year 4 = $ 617.95
Problem 5-5
Problem 5-6
Compute the payments, loan balance, and yield for an unrestricted ARM
Problem 5-7
Compute the payments, loan balances, and yield for an ARM that has a maximum 5% annual payment cap and allows
negative amortization.
Principal = $150,000
Term = 30 years
Problem 5-8
Compute the payments, loan balances, and yield for an ARM that has a 1% annual and 3% lifetime interest rate cap and
does not accumulate negative amortization.
Principal = $150,000
Points = 2.00%
Term = 30 years
Problem 5-9
(a) Compute the payments, loan balances, and yield for a Stable Home Mortgage which is comprised of a fixed and
adjustable rate component.
Loan Amount = $95,000
Points = 2.00%
Fixed Rate Portion: 75.00% of the loan balance
10.50% annual interest rate
30 year term
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4
93,459.69
860.99
92,793.03
5
82,793.03
930.15
92,155.73
Calculator: Calculator: IRR(CF1, CF2, ….CFn)
CFj nj
-$93,100
842.85 n = 12
895.26 n = 12
913.24 n = 12
860.99 n = 12
930.15 n = 11
930.15 + 92,155.73 n = 1
Solve for the IRR:
= 0.94% x 12 = 11.26% (annual rate, compounded monthly)
(b) Adjustable rate portion now has an initial rate of 9.5% and an annual interest rate cap of 1%
Loan Amount = $95,000
Points = 2.00%
Fixed Rate Portion: 75.00% of the loan balance
10.50% annual interest rate
30 year term
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Year
BOY
Balance
Annual
Interest
Rate
Monthly
Interest
Rate (2)/12
Payments
Monthly
Interest
(3) x (1)
Monthly
Amort (4)
-(5)
Annual
Amort.
EOY
Balance
(1) – (7)
0
1
$71,250
10.50%
0.88%
$651.75
$623.44
$28.31
$356.61
$70.893
2
70,893
10.50%
0.88%
651.75
620.32
31.43
395.91
70.497
3
70,497
10.50%
0.88%
651.75
616.85
34.90
439.54
70,058
4
70,058
10.50%
0.88%
651.75
613.01
38.74
487.98
69,570
5
69,570
10.50%
0.88%
651.75
608.74
43.01
541.75
69,028
Adjustable Rate Portion: 25.00% of the loan balance
9.00% initial interest rate
2.00% margin
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Year
BOY
Balance
Capped
Interest
Rate
Monthly
Interest
Rate (2)/12
Payments
Monthly
Interest
(3) x (1)
Monthly
Amort (4)
-(5)
Annual
Amort
EOY
Balance
(1) – (7)
0
1
$23,750
9.50%
0.791%
$199.70
See note.
See note.
See note.
$23,604
2
23,604
10.50%
0.875%
217.00
23,472
3
23,472
11.50%
0.958%
234.45
23,351
4
23,351
10.00%
0.833%
208.78
23,173
5
23,173
11.00%
0.917%
225.50
23,008
Problem 5-10
(a) Loan Balance at the end of year five is $116,333.93
Solution:
n = 5×12 or 60
i = 12/12 or 1
PV = -$100,000
Problem 5-11
Problem 5-12
Problem 5-13