Solution to Questions – Chapter 20
The Secondary Mortgage Market: CMOs and Derivative Securities
Question 20-1
What is a mortgage pay-through bond (MPTB)? How does it resemble a mortgage-backed bond (MBB)? How
does it differ?
Question 20-2
Are the overcollateralization requirements the same for mortgage pay-through bonds as for the mortgage-backed
bonds?
Question 20-3
Question 20-4
What is a CMO? Explain why a CMO has been called as much of a marketing innovation as a financial
innovation.
Question 20-5
What is meant by a derivative investment?
Question 20-6
Name the four major classes of mortgage-related securities. As an issuer, explain the reasons for choosing one
type over another.
Question 20-7
What is the major difference between a CMO and the other types of mortgage-related securities?
Question 20-8
Why are CMOs overcollateralized?
Question 20-9
What is the purpose of the accrual tranche? Could a CMO exist without a Z class? What would be the difference
between the CMO with and without the accrual class?
Question 20-10
Which tranches in a CMO issue are least subject to price variances related to changes in market interest rates?
Why?
Question 20-11
What is the primary distinction between mortgage-related securities backed by residential mortgages and those
backed by commercial mortgages?
Question 20-12
Name the major types of credit enhancement used for commercial-backed mortgage securities.
Question 20-13
What is a “floater”/”inverse–floater” tranche in a CMO offering?
Question 20-14
What is the role of the “scaler” in structuring an (F) and (IF) structure?
Question 20-15
Why would anyone want to purchase an (F) or (IF) derivative type of investment?
Question 20-16
What are (IO) and (PO) strips? Which tends to be more volatile in price? Why?
Question 20-17
In what ways is a CMBS structure different from a CMO backed by residential mortgages? Why is default risk
in a CMBS offering given more attention?
Question 20-18
How do CDOs differ from CMBS?
Solution to Problems – Chapter 20
The Secondary Mortgage Market: CMOs and Derivative Securities
Problem 20-1
(a) The Initial WAC is simply the coupon rate of each tranche weighted by the initial tranche balance
Weighted
Tranche
Balance
Weighting
Coupon Rate
Avg Coupon
A
40,500,000
37.50%
8.25%
3.09%
B
22,500,000
20.83%
9.00%
1.87%
Z
45,000,000
41.67%
10.00%
4.17%
Total
$108,000,000
WAC = 9.13%
(b) To calculate the maturity of each tranche, the yearly interest and principal paid on each tranche must be calculated.
Mortgage Pool
10
0.00
0.00
0.00
0.00
Tranche B
Amount
$22,500
Rate
9.00%
Year
Beg. Bal
Interest
Principal
End Bal
1
$22,500.00
$2,025.00
$0.00
$22,500.00
2
22,500.00
2,025.00
0.00
22,500.00
3
22,500.00
2,025.00
0.00
22,500.00
4
22,500.00
2,025.00
13,144.65
9,355.35
5
9,355.35
841.98
9,355.35
0.00
6
0.00
0.00
0.00
0.00
7
0.00
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
9
0.00
0.00
0.00
0.00
10
0.00
0.00
0.00
0.00
(c) The weighted average coupon each year is found by weighting the coupon rate for each class by the outstanding balance
of that class.
A
B
Z
Total
WAC
Coupon
8.25%
9.00%
10.00%
End of Year
Balance
Balance
Balance
Balance
0
40500
22500
45000
108000
9.14%
1
28941
22500
49500
100941
9.28%
2
16226
22500
54450
93176
9.45%
3
2240
22500
59895
84635
9.69%
4
0
9355
65885
75240
9.88%
5
0
0
64905
64905
10.00%
6
0
0
53537
53537
10.00%
7
0
0
41031
41031
10.00%
8
0
0
27276
27276
10.00%
9
0
0
12144
12144
10.00%
(d)
Tranche a
Cash
Year
Beg. Bal
Interest
Principal
End Bal
Flow
1
$40,500.00
3,341.25
$11,558.86
$28,941.14
14,900.11
2
28,941.14
2,387.64
12,714.74
16,226.40
15,102.39
3
16,226.40
1,338.68
13,986.22
2,240.18
15,324.89
4
2,240.18
184.82
2,240.18
0.00
2,425.00
5
0.00
0.00
0.00
0.00
0.00
6
0.00
0.00
0.00
0.00
0.00
7
0.00
0.00
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
9
0.00
0.00
0.00
0.00
0.00
10
0.00
0.00
0.00
0.00
0.00
P V at
8.50%
$40,309
Tranche B
Cash
Year
Beg. Bal
Interest
Principal
End Bal
Flow
1
$22,500.00
$2,025.00
$0.00
$22,500.00
2,025.00
2
22,500.00
2,025.00
0.00
22,500.00
2,025.00
3
22,500.00
2,025.00
0.00
22,500.00
2,025.00
4
22,500.00
2,025.00
13,144.65
9,355.35
15,169.65
5
9,355.35
841.98
9,355.35
0.00
10,197.33
6
0.00
0.00
0.00
0.00
0.00
7
0.00
0.00
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
9
0.00
0.00
0.00
0.00
0.00
10
0.00
0.00
0.00
0.00
0.00
P V at
9.50%
$22,110
Tranche Z
Total
Cash
Year
Beg. Bal
Interest
Payment
End Bal
Flow
0
($45,000)
1
$45,000.00
4,500.00
0.00
$49,500.00
0.00
2
49,500.00
4,950.00
0.00
54,450.00
0.00
3
54,450.00
5,445.00
0.00
59,895.00
0.00
4
59,895.00
5,989.50
0.00
65,884.50
0.00
5
65,884.50
6,588.45
7,567.98
64,904.97
7,567.98
6
64,904.97
6,490.50
17,858.86
53,536.61
17,858.86
7
53,536.61
5,353.66
17,858.86
41,031.42
17,858.86
8
41,031.42
4,103.14
17,858.86
27,275.70
17,858.86
9
27,275.70
2,727.57
17,858.86
12,144.42
17,858.86
10
12,144.42
1,214.44
13,358.86
0.00
13,358.86
IRR
10.00%
P V at
9.75%
$45,768
(e)
Residual Class
Total
Other
Year
in pool
Classes
Residual
0
($4,500.00)
1
$18,308.86
$16,925.11
1,383.75
2
$18,308.86
17,127.39
1,181.47
3
$18,308.86
17,349.89
958.96
4
$18,308.86
17,594.65
714.20
5
$18,308.86
17,765.30
543.55
6
$18,308.86
17,858.86
450.00
7
$18,308.86
17,858.86
450.00
8
$18,308.86
17,858.86
450.00
9
$18,308.86
17,858.86
450.00
10
$18,308.86
13,358.86
4,950.00
IRR
19.10%
(f)
Assuming 10% prepayment
Mortgage Pool
Year
Beg. Bal
Payment
Interest
Principal
End Bal
Prepayment
1
$112,500.00
$18,308.86
11,250.00
$18,308.86
$94,191.14
11250.00
2
94,191.14
$16,355.40
9,419.11
$16,355.40
$77,835.74
9419.11
3
77,835.74
$14,589.84
7,783.57
$14,589.84
$63,245.90
7783.57
4
63,245.90
$12,991.06
6,324.59
$12,991.06
$50,254.84
6324.59
5
50,254.84
$11,538.88
5,025.48
$11,538.88
$38,715.96
5025.48
6
38,715.96
$10,213.17
3,871.60
$10,213.17
$28,502.79
3871.60
7
28,502.79
$8,991.80
2,850.28
$8,991.80
$19,510.99
2850.28
8
19,510.99
$7,845.66
1,951.10
$7,845.66
$11,665.33
1951.10
9
11,665.33
$6,721.45
1,166.53
$6,721.45
$4,943.88
1166.53
10
4,943.88
$5,438.27
494.39
$4,943.88
$0.00
494.39
Tranche A
Cash
Year
Beg. Bal
Interest
Principal
End Bal
Flow
1
$40,500.00
3,341.25
$22,808.86
$17,691.14
26,150.11
2
17,691.14
1,459.52
17,691.14
0.00
19,150.66
3
0.00
0.00
0.00
0.00
0.00
4
0.00
0.00
0.00
0.00
0.00
5
0.00
0.00
0.00
0.00
0.00
6
0.00
0.00
0.00
0.00
0.00
7
0.00
0.00
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
9
0.00
0.00
0.00
0.00
0.00
10
0.00
0.00
0.00
0.00
0.00
P V at
8.50%
$40,369
Tranche B
Cash
Year
Beg. Bal
Interest
Principal
End Bal
Flow
1
$22,500.00
$2,025.00
$0.00
$22,500.00
2,025.00
2
22,500.00
2,025.00
3,614.26
18,885.74
5,639.26
3
18,885.74
1,699.72
18,885.74
0.00
20,585.46
4
0.00
0.00
0.00
0.00
0.00
5
0.00
0.00
0.00
0.00
0.00
6
0.00
0.00
0.00
0.00
0.00
7
0.00
0.00
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
9
0.00
0.00
0.00
0.00
0.00
10
0.00
0.00
0.00
0.00
0.00
P V at
9.50%
$22,232
Tranche Z
Total
Cash
Year
Beg. Bal
Interest
Payment
End Bal
Flow
0
($45,000)
1
$45,000.00
4,500.00
0.00
$49,500.00
0.00
2
49,500.00
4,950.00
0.00
54,450.00
0.00
3
54,450.00
5,445.00
1,149.10
58,745.90
1,149.10
4
58,745.90
5,874.59
18,865.65
45,754.84
18,865.65
5
45,754.84
4,575.48
16,114.37
34,215.96
16,114.37
6
34,215.96
3,421.60
13,634.77
24,002.79
13,634.77
7
24,002.79
2,400.28
11,392.08
15,010.99
11,392.08
8
15,010.99
1,501.10
9,346.76
7,165.33
9,346.76
9
7,165.33
716.53
7,437.99
443.88
7,437.99
10
443.88
44.39
488.27
0.00
488.27
IRR
10.00%
P V at
9.75%
$45,588
Residual
Total
Other
Year
in pool
Classes
Residual
0
($4,500.00)
1
$29,558.86
$28,175.11
1,383.75
2
$25,774.52
24,789.92
984.60
3
$22,373.42
21,734.56
638.86
4
$19,315.65
18,865.65
450.00
5
$16,564.37
16,114.37
450.00
6
$14,084.77
13,634.77
450.00
7
$11,842.08
11,392.08
450.00
8
$9,796.76
9,346.76
450.00
9
$7,887.99
7,437.99
450.00
10
$5,438.27
488.27
4,950.00
IRR
16.10%
(g)
10 percent price increase after issue
Tranche A
Cash
Year
Beg. Bal
Interest
Principal
End Bal
Flow
10% Price Increase
0
($44,550)
1
$40,500.00
3,341.25
$22,808.86
$17,691.14
26,150.11
26,150.11
2
17,691.14
1,459.52
17,691.14
0.00
19,150.66
19,150.66
3
0.00
0.00
0.00
0.00
0.00
0.00
4
0.00
0.00
0.00
0.00
0.00
0.00
5
0.00
0.00
0.00
0.00
0.00
0.00
6
0.00
0.00
0.00
0.00
0.00
0.00
7
0.00
0.00
0.00
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
0.00
9
0.00
0.00
0.00
0.00
0.00
0.00
10
0.00
0.00
0.00
0.00
0.00
0.00
P V at
8.50%
$40,369
YTM
1.18%
Tranche B
Cash
Year
Beg. Bal
Interest
Principal
End Bal
Flow
10% Price Increase
0
($24,750)
1
$22,500.00
$2,025.00
$0.00
$22,500.00
2,025.00
2,025.00
2
22,500.00
2,025.00
3,614.26
18,885.74
5,639.26
5,639.26
3
18,885.74
1,699.72
18,885.74
0.00
20,585.46
20,585.46
4
0.00
0.00
0.00
0.00
0.00
0.00
5
0.00
0.00
0.00
0.00
0.00
0.00
6
0.00
0.00
0.00
0.00
0.00
0.00
7
0.00
0.00
0.00
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
0.00
9
0.00
0.00
0.00
0.00
0.00
0.00
10
0.00
0.00
0.00
0.00
0.00
0.00
P V at
9.50%
$22,232
YTM
5.12%
Class Z
Total
Cash
Year
Beg. Bal
Interest
Payment
End Bal
Flow
10% Price Increase
0
($45,000)
($49,500)
1
$45,000.00
4,500.00
0.00
$49,500.00
0.00
0.00
2
49,500.00
4,950.00
0.00
54,450.00
0.00
0.00
3
54,450.00
5,445.00
1,149.10
58,745.90
1,149.10
1,149.10
4
58,745.90
5,874.59
18,865.65
45,754.84
18,865.65
18,865.65
5
45,754.84
4,575.48
16,114.37
34,215.96
16,114.37
16,114.37
6
34,215.96
3,421.60
13,634.77
24,002.79
13,634.77
13,634.77
7
24,002.79
2,400.28
11,392.08
15,010.99
11,392.08
11,392.08
8
15,010.99
1,501.10
9,346.76
7,165.33
9,346.76
9,346.76
9
7,165.33
716.53
7,437.99
443.88
7,437.99
7,437.99
10
443.88
44.39
488.27
0.00
488.27
488.27
YTM
8.18%
PV at
9.75%
$45,588
Problem 20-2
(a) Beginning Balance = $1,000,000
Prepayment Rate = 0.00
PV/Price at 6%
$387,480
$709,390
IO/Strip
PO/Strip
$221,902
Scale
Interest Rate
Interest Payable
(F) Floater
0.50
0.08
1,000,000
0.50
0.08
$160,000
Scale
Interest Rate
Interest Payable
(F) Floater
0.60
0.08
800,000
0.40
0.08
$160,000
Beginning
Interest
Principal
PO
Ending
Period
Balance
IO/Strip
PO/Strip
Prepayment
Balance
1
$1,000,000
$80,000
$69,029
$200,000
$730,971
2
730,971
58,478
58,536
146,194
526,241
3
526,241
42,099
49,474
105,248
371,518
4
371,518
29,721
41,637
74,304
255,578
5
255,578
20,446
34,839
51,116
169,623
6
169,623
13,570
28,913
33,925
106,785
7
106,785
8,543
23,698
21,357
61,730
8
61,730
4,938
19,015
12,346
30,369
9
30,369
2,430
14,601
6,074
9,695
10
9,695
776
9,695
0
0
The price of the 20% prepayment IO and PO strips is the PV of the cash flows at 6%
(c) Impact of 2% increase in interest rate under 50% proportions
Scale
Interest Rate
Interest Payable
(F) Floater
$1,000,000
0.50
0.10
$100,000
(IF) Inverse Floater
1,000,000
0.50
0.06
60,000
$160,000
F receives $100,000 and IF receives $60,000
Impact of 2% increase in interest rate under 60% / 40% proportions
Scale
Interest Rate
Interest Payable
(F) Floater
$1,200,000
0.60
0.10
$120,000
(IF) Inverse Floater
800,000
0.40
0.05
40,000
$160,000
Impact of 2% decrease in interest rate under 50% proportions
Scale
Interest Rate
Interest Payable
(F) Floater
$1,000,000
0.50
0.06
$60,000
(IF) Inverse Floater
1,000,000
0.50
0.10
100,000
$160,000
Impact of 2% decrease in interest rate under 60% / 40% proportions
Scale
Interest Rate
Interest Payable
(F) Floater
$1,200,000
0.60
0.06
$72,000
(IF) Inverse Floater
800,000
0.40
0.11
88,000
$160,000
Summary
Yield
2% increase
2% decrease
Proportions
of Issue
in interest rate
in interest rate
Case (a)
(F)
50%
8%
10
6
(IF)
50%
8%
6
10
Case (b)
(F)
60%
8%
10
6
(IF)
40%
8%
5
11
In case (a) there is an equal impact of changing interest rates on the F and IF yields, that is, each either increases or decreases
by 2%. In case (b) however, IF investors will experience greater volatility in yield. This is because the proportion of each
class comprising the tranche is now 60 – 40. Therefore, for each 1% change in the underlying interest rate, IF investors will
realize a change in yield of 1.5%.
Problem 20-4
See table below:
Prepayment
Rate
IRR on
Residual
0.00%
19.10%
5.00%
17.33%
10.00%
16.10%
15.00%
15.53%
20.00%
14.99%
25.00%
14.74%
30.00%
14.74%
Problem 20-5
See below:
Libor
(F) Rate
(IF) Rate
F interest
IF Interest
Total interest
0%
0.00%
24.00%
$0
$1,200,000
$1,200,000
1%
1.00%
21.00%
150,000
1,050,000
1,200,000
2%
2.00%
18.00%
300,000
900,000
1,200,000
3%
3.00%
15.00%
450,000
750,000
1,200,000
4%
4.00%
12.00%
600,000
600,000
1,200,000
5%
5.00%
9.00%
750,000
450,000
1,200,000
6%
6.00%
6.00%
900,000
300,000
1,200,000
7%
7.00%
3.00%
1,050,000
150,000
1,200,000
8%
8.00%
0.00%
1,200,000
0
1,200,000
Note that the inverse floater now starts at a higher rate with Libor equal to 0% and decreases at a faster rate than before.
Problem 20-6
See below:
IRR of IO
IRR of PO
Prepayment
0.00%
30.00%
5.38%
5.00%
25.32%
6.57%
10.00%
20.62%
7.90%
15.00%
15.91%
9.37%
20.00%
11.19%
10.94%
25.00%
6.45%
12.59%
30.00%
1.70%
14.30%
IRR on IO and PO vs. Prepayment
0%
5%
10%
15%
20%
25%
30%
35%
0% 5% 10% 15% 20% 25% 30%
Prepayment rate
IRR
IO
PO
Problem 20-7
Solutions to Problems – Chapter 20 Appendix
The Secondary Mortgage Market: CMOs and Derivative Securities
Problem 20A-1
(a) Interest Payments of a Corporate Bond = $10,000
Duration Calculation of a Corporate Bond:
Weighting
Present
Weighted PV
Period
Payment
Factor
Value
of Payment
0
1
10,000
1.0
0.9091
14,795
2
10,000
2.0
0.8264
26,901
3
10,000
3.0
0.7513
36,693
4
10,000
4.0
0.6830
44,464
5
10,000
5.0
0.6209
50,527
5
100,000
5.0
0.6209
191,532
Total
$364,903
Duration = Total weighted present value of payments
(b) New price for corporate bond if interest rate falls from 10% to 7%
New price for mortgage bond if interest rate falls from 10% to 7%
Difference = 10% –7% = 3.00%