Transcript Mortgage-Backed Securities

Chapter 11
Securitization and
Mortgage-Backed Securities
Securitization
• Securitization refers to a process in which the
assets of a corporation or financial institution are
pooled into a package of securities backed by the
assets.
• The process starts when an originator, who owns
the assets (e.g., mortgages or accounts
receivable), sells them to an issuer.
• The issuer then creates a security backed by the
assets called an asset-backed security or passthrough that he sells to investors.
Securitization
• The securitization process often involves a thirdparty trustee who ensures that the issuer complies
with the terms underlying the asset-backed
security.
• Many securitized assets are backed by credit
enhancements, such as a third-party guarantee
against the default on the underlying assets.
• The most common types of asset-backed
securities are those secured by mortgages,
automobile loans, credit card receivables, and
home equity loans.
Securitization
• By far the largest type and the one in which the
process of securitization has been most
extensively applied is mortgages.
• Asset-backed securities formed with mortgages
are called mortgage-backed securities, MBSs, or
mortgage pass-throughs.
Mortgage-Backed Securities: Definition
• Mortgage-Backed Securities (MBS) or
Mortgage Pass-Throughs (PT) are claims
on a portfolio of mortgages.
• The securities entitle the holder to the cash
flows from a pool of mortgages.
Mortgage-Backed Securities: Creation
• Typically, the issuer of a MBS buys a portfolio or
pool of mortgages of a certain type from a
mortgage originator, such as a commercial bank,
savings and loan, or mortgage banker.
• The issuer finances the purchase of the mortgage
portfolio through the sale of the mortgage passthroughs, which have a claim on the portfolio's
cash flow.
• The mortgage originator usually agrees to
continue to service the loans, passing the
payments on to the MBS holders.
Agency Pass-Throughs
• There are three federal agencies that buy certain types of
mortgage loan portfolios (e.g., FHA- or VA-insured
mortgages) and then pool them to create MBSs to sell to
investors:
– Federal National Mortgage Association (FNMA)
– Government National Mortgage Association (GNMA)
– Federal Home Loan Mortgage Corporation (FHLMC)
• Collectively, the MBSs created by these agencies are
referred to as agency pass-throughs.
• Agency pass-throughs are guaranteed by the agencies, and
the loans they purchase must be conforming loans,
meaning they meet certain standards.
Agency Pass-Throughs
Government National Mortgage Association, GNMA
• GNMA mortgage-backed securities or pass-throughs are
formed with FHA- or VA-insured mortgages.
• They are put together by an originator (bank, thrift, or
mortgage banker), who presents a block of FHA and VA
mortgages to GNMA.
• If GNMA finds them in order, they will issue a guarantee
and assign a pool number that identifies the MBS that is
to be issued.
• The originator will transfer the mortgages to a trustee, and
then issue the pass-throughs, often selling them to
investment bankers for distribution.
Agency Pass-Throughs
Government National Mortgage Association, GNMA
• The mortgages underlying GNMA’s MBSs are very
similar (e.g., single-family, 30-year maturity, and fixed
rate), with the mortgage rates usually differing by no more
than 50 basis point from the average mortgage rate.
• GNMA does offer programs in which the underlying
mortgages are more diverse.
• Note: Since GNMA is a federal agency, its guarantee of
timely interest and principal payments is backed by the
full faith and credit of the U.S. government -- the only
MBS with this type of guarantee.
Agency Pass-Throughs
Federal Home Loan Mortgage Corporation, FHLMC
• The FHLMC (Freddie Mac) issues MBSs that they refer
to as participation certificates (PCs).
• The FHLMC has a regular MBS (also called a cash PC),
which is backed by a pool of either conventional, FHA, or
VA mortgages that the FHLMC has purchased from
mortgage originators.
• They also offer a pass-though formed through their
Guarantor/Swap Program. In this program, mortgage
originators can swap mortgages for a FMLMC passthrough.
Agency Pass-Throughs
Federal Home Loan Mortgage Corporation
• Unlike GNMA’s MBSs, Freddie Mac's MBSs are
formed with more heterogeneous mortgages.
• Like GNMA, the Federal Home Loan Mortgage
Corporation backs the interest and principal
payments of its securities, but the FHLMC's
guarantee is not backed by the U.S. government.
Agency Pass-Throughs
Federal National Mortgage Association, FNMA
• FNMA (Fannie Mae) offers several types of passthroughs, referred to as FNMA mortgage-backed
securities.
• Like FHLMC’s pass-throughs, FNMA’s securities
are backed by the agency, but not by the
government.
• Like the FHLMC, FNMA buys conventional,
FHA, and VA mortgages, and offers a SWAP
program whereby mortgage loans can be swapped
for FNMA-issued MBSs.
Agency Pass-Throughs
Federal National Mortgage Association
• Like the FHLMC, FNMA's mortgages are more
heterogeneous than GNMA's mortgages, with
mortgage rates in some pools differing by as
much as 200 basis points from the portfolio's
average mortgage rate.
Conventional Pass-Throughs
• Conventional pass-throughs are sold by
commercial banks, savings and loans, other
thrifts, and mortgage bankers.
• These nonagency pass-throughs, also called
private labels, are often formed with
nonconforming mortgages; that is, mortgages that
fail to meet size limits and other requirements
placed on agency pass-throughs.
Conventional Pass-Throughs
• The larger issuers of conventional MBSs
include:
–
–
–
–
–
Citicorp Housing
Countrywide
Prudential Home
Ryland/Saxon
G.E. Capital Mortgage
Conventional Pass-Throughs
• Conventional pass-throughs are often guaranteed
against default through external credit
enhancements, such as:
– Guarantee of a corporation
– Bank letter of credit
– Private insurance from such insurers as the Financial
Guarantee Insurance Corporation (FGIC), the Capital
Markets Assurance Corporation (CAPMAC), or the
Financial Security Assurance Company (FSA)
Conventional Pass-Throughs
• Some conventional pass-throughs are guaranteed
internally through the creation of senior and subordinate
classes of bonds with different priority claims on the
pool's cash flows in the case some of the mortgages in the
pool default.
• Example: A conventional pass-through, known as an A/B
pass-through, consists of two types of claims on the
underlying pool of mortgages - senior and subordinate.
– The senior claim is backed by the mortgages, while the
subordinate claim is not.
– The more subordinate claims sold relative to senior,
the more secured the senior claims.
Conventional Pass-Throughs
• Conventional MBSs are rated by Moody's
and Standard and Poor's.
• They must be registered with the SEC
when they are issued.
Conventional Pass-Throughs
• Most financial entities that issue private-labeled MBSs or
derivatives of MBSs are legally set up so that they do not
have to pay taxes on the interest and principal that passes
through them to their MBS investors.
• The requirements that MBS issuers must meet to ensure
tax-exempt status are specified in the Tax Reform Act of
1983 in the section on trusts referred to as Real Estate
Mortgage Investment Conduits, REMIC.
• Private-labeled MBS issuers who comply with these
provisions are sometimes referred to as REMICs.
Market
• Primary Market: Investors buy MBSs issued by
agencies or private-label investment companies
either directly or through dealers. Many of the
investors are institutional investors. Thus, the
creation of MBS has provided a tool for having
real estate financed more by institutions.
• Secondary Market:
– Existing MBS are traded by dealers on the
OTC
Cash Flows
• Cash flows from MBSs are generated from the
cash flows from the underlying pool of
mortgages, minus servicing and other fees.
• Typically, fees for constructing, managing, and
servicing the underlying mortgages (also referred
to as the mortgage collateral) and the MBSs are
equal to the difference between the rates
associated with the mortgage pool and the rates
paid on the MBS (pass-through (PT) rate).
Cash Flows: Terms
• Weighted Average Coupon Rate, WAC: Mortgage
portfolio's (collateral’s) weighted average rate.
• Weighted Average Maturity, WAM: Mortgage
portfolio's weighted average maturity.
• Pass-Through Rate, PT Rate: Interest rate paid
on the MBS; PT rate is lower than WAC -- the
difference going to MBS issuer.
• Prepayment Rate or Speed: Assumed prepayment
rate.
Prepayment
• A number of prepayment models have been
developed to try to predict the cash flows from a
portfolio of mortgages.
• Most of these models estimate the prepayment
rate, referred to as the prepayment speed or
simply speed, in terms of four factors:
–
–
–
–
Refinancing incentive
Seasoning (the age of the mortgage)
Monthly factors
Prepayment burnout
Prepayment
Refinancing Incentive
• The refinancing incentive is the most important
factor influencing prepayment.
• If mortgage rates decrease below the mortgage
loan rate, borrowers have a strong incentive to
refinance.
• This incentive increases during periods of falling
interest rates, with the greatest increases
occurring when borrowers determine that rates
have bottomed out.
Prepayment
Refinancing Incentive
• The refinancing incentive can be measured by the
difference between the mortgage portfolio's WAC
and the refinancing rate, Rref.
• A study by Goldman, Sachs, and Company found
that the annualized prepayment speed, referred to
as the conditional prepayment rate, CPR, is
greater the larger the positive difference between
the WAC and Rref.
Prepayment
Seasoning
• A second factor determining prepayment is the
age of the mortgage, referred to as seasoning.
• Prepayment tends to be greater during the early
part of the loan, then stabilize after about three
years. The figures on the next slide depicts a
commonly referenced seasoning pattern known as
the PSA model (Public Securities Association).
Prepayment
• PSA Model:
CPR (%)
9.0
6.0
30
.



150 PSA
100 PSA
50 PSA
0.2
0 1
30
360
Month
Prepayment
Seasoning
• In the standard PSA model, known as 100 PSA,
the CPR starts at .2% for the first month and then
increases at a constant rate of .2% per month to
equal 6% at the 30th month; then after the 30th
month the CPR stays at a constant 6%. Thus for
any month t, the CPR is
 t 
CP R.06 , if t  30,
 30 
CP R  .06, if t  30
Prepayment
Seasoning
• Note that the CPR is quoted on an annual
basis.
• The monthly prepayment rate, referred to
as the single monthly mortality rate, SMM,
can be obtained given the annual CPR by
using the following formula:
SMM  1  [1  CPR]
1/12
Prepayment
Seasoning
• The 100 PSA model is often used as a benchmark. The
actual aging pattern will differ depending on where
current mortgage rates are relative to the WAC.
• Analysts often refer to the applicable pattern as being a
certain percentage of the PSA.
• For example:
– If the pattern is described as being 200 PSA, then the
prepayment speeds are twice the 100 PSA rates.
– If the pattern is described as 50 PSA, then the CPRs
are half of the 100 PSA rates (see figures on previous
slide).
Prepayment
Seasoning
• A current mortgage pool described by a 100 PSA
would have a annual prepayment rate of 2% after
10 months (or a monthly prepayment rate of
SMM = .00168), and a premium pool described
as a 150 PSA would have a 3% CPR (or SMM =
.002535) after 10 months.
Prepayment
Monthly Factors
• In addition to the effect of seasoning, mortgage
prepayment rates are also influenced by the
month of the year, with prepayment tending to be
higher during the summer months.
• Monthly factors can be taken into account by
multiplying the CPR by the estimated monthly
multiplier to obtain a monthly-adjusted CPR.
PSA provides estimates of the monthly
multipliers.
Prepayment
Burnout Factors
• Many prepayment models also try to capture what
is known as the burnout factor.
• The burnout factor refers to the tendency for
premium mortgages to hit some maximum CPR
and then level off.
• For example, in response to a 2% decrease in
refinancing rates, a pool of premium mortgages
might peak at a 40% prepayment rate after one
year, then level off at approximately 25%.
Cash Flow from a Mortgage Portfolio
• The cash flow from a portfolio of
mortgages consists
– Interest payments
– Scheduled principal
– Prepaid principal
Cash Flow from a Mortgage Portfolio: Example 1
• Example 1: Consider a bank that has a pool
of current fixed rate mortgages that are
– worth $100 million (Par, F)
– yield a WAC of 8%, and
– have a WAM of 360 months.
Cash Flow from a Mortgage Portfolio: Example 1
• For the first month, the portfolio would
generate an aggregate mortgage payment of
$733,765:
$100,000,000
p
 $733,765
360
11 /(1(.08 / 12)) 


.
08
/
12


F0 
M
 (1  (R
t 1
p
A
/ 12))t
11 /(1(R A / 12))M 
F0  p 

A
R
/
12


F0
p 
11 /(1(R A / 12))M 


R A / 12


Cash Flow from a Mortgage Portfolio: Example 1
• From the $733,765 payment, $666,667 would go
towards interest and $67,098 would go towards
the scheduled principal payment:
 RA 
 .08 
 F0  
Int erest  
$100,000,000  $666,667
 12 
 12 
Scheduled P r incipal P ayment p  Int erest  $733,765  $666,667  $67,098
Cash Flow from a Mortgage Portfolio: Example 1
• The projected first month prepaid principal can
be estimated with a prepayment model. Using the
100% PSA model, the monthly prepayment rate
for the first month (t = 1) is equal to SMM =
.0001668:
 1 
CPR   .06  .002
 30 
SMM  1  [1.002]1/12  .0001668
Cash Flow from a Mortgage Portfolio: Example 1
• Given the prepayment rate, the projected prepaid
principal in the first month is found by
multiplying the balance at the beginning of the
month minus the scheduled principal by the
SMM.
• Doing this yields a projected prepaid principal of
$16,671 in the first month:
prepaid principal  SMM [F0  Scheduled principal]
prepaid principal  .0001668
[$100,000,000 $67,098]  $16,671
Cash Flow from a Mortgage Portfolio: Example 1
• Thus, for the first month, the mortgage portfolio
would generate an estimated cash flow of
$750,435, and a balance at the beginning of the
next month of $99,916,231:
CF  Interest Scheduled principal prepaid principal
CF  $666,666  $ 67,098  $16,671  $750,435
Beginning Balance for Month 2  F0  Scheduled principal prepaid principal
Beginning Balance for Month 2  $100,000,000  $67,098  $16,671  $99,916,231
Cash Flow from a Mortgage Portfolio: Example 1
• In the second month (t = 2), the projected
payment would be $733,642 with $666,108
going to interest and $67,534 to scheduled
principal:
$99,916,231
p
 $733,642
359
11 /(1(.08 / 12)) 


.
08
/
12


 .08 
Interest   ($99,916,231)  $666,108
 12 
Scheduled principal $733,642  $666,108  $67,534
Cash Flow from a Mortgage Portfolio: Example 1
• Using the 100% PSA model, the estimated
monthly prepayment rate is .000333946, yielding
a projected prepaid principal in month 2 of
$33,344:
 2
CPR   .06  .004
 30 
SMM  1  [1.004]1/12  .000333946
prepaid principal .000333946
[$99,916,231  $67,534]  33,344
Cash Flow from a Mortgage Portfolio: Example 1
• Thus, for the second month, the mortgage
portfolio would generate an estimated cash flow
of $766,986 and have a balance at the beginning
of month three of $99,815,353:
CF  $666,108  $ 67,534  $33,344  $766,986
Beginning Balance for Month 3  $99,916,231  $67,534  $33,344
 $99,815,353
Cash Flow from a Mortgage Portfolio: Example 1
•
The exhibit on the next slide summarizes the
mortgage portfolio's cash flow for the first two
months and other selected months.
•
In examining the exhibit, two points should be
noted:
1. Starting in month 30 the SMM remains constant at
.005143; this reflects the 100% PSA model's
assumption of a constant CPR of 6% starting in
month 30.
2. The projected cash flows are based on a static
analysis in which rates are assumed fixed over the
time period.
Cash Flow from a Mortgage Portfolio: Example 1
Period
1
2
3
4
5
6
7
23
24
25
26
27
28
29
30
31
32
110
111
112
113
114
115
357
358
359
360
Balance
100000000
100000000
99916231
99815353
99697380
99562336
99410252
99241170
94291147
93847732
93389518
92916704
92429495
91928105
91412755
90883671
90341088
89801183
54900442
54533417
54168153
53804641
53442873
53082839
496620
371778
247395
123470
Interest
p
Sch. Prin.
SMM
Prepaid Prin.
CF
666667
666108
665436
664649
663749
662735
661608
628608
625652
622597
619445
616197
612854
609418
605891
602274
598675
366003
363556
361121
358698
356286
353886
3311
2479
1649
823
733765
733642
733397
733029
732539
731926
731190
703012
700259
697394
694420
691336
688146
684849
681447
677943
674456
451112
448792
446484
444188
441903
439631
126231
125582
124936
124293
67098
67534
67961
68380
68790
69191
69582
74405
74607
74798
74975
75140
75292
75430
75556
75669
75781
85109
85236
85363
85490
85617
85745
122920
123103
123287
123470
0.0001668
0.0003339
0.0005014
0.0006691
0.0008372
0.0010055
0.0011742
0.0039166
0.0040908
0.0042653
0.0044402
0.0046154
0.0047909
0.0049668
0.005143
0.005143
0.005143
0.005143
0.005143
0.005143
0.005143
0.005143
0.005143
0.005143
0.005143
0.005143
0.005143
16671
33344
50011
66664
83294
99892
116449
369010
383607
398017
412234
426250
440059
453653
467027
464236
461459
281916
280028
278148
276278
274417
272565
1922
1279
638
0
750435
766986
783409
799694
815833
831817
847639
1072023
1083866
1095411
1106653
1117586
1128204
1138502
1148475
1142179
1135915
733028
728820
724632
720466
716320
712195
128153
126861
125574
124293
Cash Flow from a MBS: Example 2
Example 2
• The next exhibit shows the monthly cash
flows for a MBS issue constructed from a
$100M mortgage pool with the following
features
–
–
–
–
–
Current balance = $100M
WAC = 8%
WAM = 355 months
PT rate = 7.5%
Prepayment speed equal to 150% of the
standard PSA model: PSA = 150
Cash Flow from a MBS: Example 2
Period
1
2
3
4
5
6
20
21
22
23
24
25
26
27
28
29
30
31
32
33
100
101
102
103
200
201
353
354
355
Balance
100000000
100000000
99779252
99533099
99261650
98965033
98643396
91641550
90975181
90288672
89582468
88857032
88112838
87350375
86593968
85843572
85099137
84360619
83627969
82901143
82180094
44933791
44515680
44100923
43689493
16163713
15978416
148527
98569
49061
Interest
p
625000
623620
622082
620385
618531
616521
572760
568595
564304
559890
555356
550705
545940
541212
536522
531870
527254
522675
518132
513626
280836
278223
275631
273059
101023
99865
928
616
307
736268
735154
733855
732371
730702
728850
684341
679910
675324
670587
665702
660671
655499
650368
645277
640225
635213
630240
625307
620411
366433
363564
360718
357894
166983
165676
50171
49778
49388
Scheduled
Principal
69601
69959
70301
70627
70936
71227
73398
73408
73399
73370
73321
73253
73164
73075
72986
72897
72809
72721
72632
72544
66874
66793
66712
66631
59225
59153
49181
49121
49061
SMM
0.0015125
0.0017671
0.0020223
0.0022783
0.002535
0.0027925
0.0064757
0.0067447
0.0070144
0.0072849
0.0075563
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
0.0078284
Prepaid
Principal
151147
176194
201148
225990
250701
275262
592971
613101
632804
652066
670873
689211
683243
677322
671448
665621
659840
654106
648416
642772
351237
347965
344718
341498
126073
124623
778
387
0
Principal
CF
220748
246153
271449
296617
321637
346489
666369
686510
706204
725436
744194
762463
756406
750397
744434
738519
732649
726826
721049
715317
418111
414758
411430
408129
185298
183776
49958
49508
49061
845748
869773
893531
917002
940168
963011
1239128
1255105
1270508
1285327
1299550
1313169
1302346
1291609
1280957
1270388
1259903
1249501
1239181
1228942
698947
692981
687061
681188
286321
283641
50887
50124
49368
Cash Flow from a MBS: Example 2
•
Notes:
The first month's CPR for the MBS issue reflects a fivemonth seasoning in which t = 6, and a speed that is
150% greater than the 100 PSA. For the MBS issue, this
yields a first month SMM of .0015125 and a constant
SMM of .0078284 starting in month 25.
•
The WAC of 8% is used to determine the mortgage
payment and scheduled principal, while the PT rate of
7.5% is used to determine the interest.
•
The monthly fees implied on the MBS issue are equal to
.04167% = (8% - 7.5%)/12 of the monthly balance.
Cash Flow from a MBS: Example 2
• First Month’s Payment:
$100M
p 
 $736,268
355
11 /(1(.08 / 12)) 


.
08
/
12


WAC
Cash Flow from a MBS: Example 2
• From the $736,268 payment, $625,000 would go
towards interest and $69,601 would go towards
the scheduled principal payment:
PT Rate
WAC
 RA 
 .075
F0  
Interest  
$100,000,000  $625,000
12
12




Scheduled P r incipal Payment p  Interest $736,268 [(.08 / 12)($100,000,000)]
 $69,601
Cash Flow from a MBS: Example 2
• Using 150% PSA model and seasoning of 5
months the first month SMM = .0015125:
 6 
CPR  1.50 .06  .018
 30 
SMM  1  [1.018]1/12  .0015125
Cash Flow from a MBS: Example 2
• Given the prepayment rate, the projected prepaid
principal in the first month is $151,147
prepaid principal  SMM [F0  Scheduled principal]
prepaid principal  .0015125
[$100,000,000 $69,601]  $151,147
Allow for slight rounding differences
Cash Flow from a MBS: Example 2
• Thus, for the first month, the MBS would
generate an estimated cash flow of $845,748 and
a balance at the beginning of the next month of
$99,779,252:
CF  Interest Scheduled principal prepaid principal
CF  $625,000  $ 69,601  $151,147  $845,748
Beginning Balance for Month 2  F0  Scheduled principal prepaid principal
Beginning Balance for Month 2  $100,000,000  $69,601  $151,147  $99,779,252
Allow for slight rounding differences
Cash Flow from a MBS: Example 2
• Second Month: Payment, Interest, Scheduled Principal,
Prepaid Principal, and Cash flow:
p 
$99,779,252
 $735,154
11 /(1(.08 / 12))354 


.08 / 12


 RA 
 .075
F0  
Int erest  
$99,779,252  $623,620
12
12




Scheduled P r incipal P ayment p  Int erest  $735,154 [(.08 / 12)($99,779,252)]  $69,959
 7
CPR  1.50 .06  .021
 30 
SMM  1  [1.021]1/12  .0017671
prepaid principal  SMM [F0  Scheduled principal]
prepaid principal  .0017671[$99,779,252 $69,959]  $176,194
CF  Interest Scheduled principal prepaid principal
CF  $623,620  $69,959  $176,194  $869,773
Allow for slight rounding differences
Market
•
As noted, investors can acquire newly issued
mortgage-backed securities from the agencies,
originators, or dealers specializing in specific
pass-throughs.
•
There is also a secondary market consisting of
dealers who operate in the OTC market as part
of the Mortgage-Backed Security Dealers
Association.
•
These dealers form the core of the secondary
market for the trading of existing pass-throughs.
Market
•
Mortgage pass-throughs are normally sold in
denominations ranging from $25,000 to
$250,000, although some privately-placed issues
are sold with denominations as high as $1
million.
Price Quotes
•
•
•
The prices of MBSs are quoted as a percentage
of the underlying MBS issue’s balance.
The mortgage balance at time t, Ft, is usually
calculated by the servicing institution and is
quoted as a proportion of the original balance,
F0.
This proportion is referred to as the pool factor,
pf:
Ft
pft 
F0
Price Quotes
• Example: A MBS backed by a mortgage pool
originally worth $100M, a current pf of .92, and
quoted at 95 - 16 (Note: 16 is 16/32) would have a
market value of $87.86M:
Ft  (pft )F0
 (.92)($100M)  $92M
Market Value  (.9550)($92M)  $87.86M
Price Quotes
• The market value is the clean price; it does not take
into account accrued interest, ai.
• For MBS, accrued interest is based on the time
period from the settlement date (two days after the
trade) and the first day of the next month.
• Example: If the time period is 20 days, the month is
30 days, and the WAC = 9%, then ai is $.46M:
 20  .09 
ai   
 $92M  $0.46M
 30  12 
Price Quotes
• The full market value would be $88.32M:
Full Mkt Value  $87.86M  $0.46M
 $88.32M
Price Quotes
• The market price per share is the full market value
divided by the number of shares.
• If the number of shares is 400, then the price of
the MBS based on a 95 - 16 quote would be
$220,800:
$88.32M
MBS price per share 
400
 $220,800
Extension Risk
•
Like other fixed-income securities, the value of
a MBS is determined by the MBS's future cash
flow (CF), maturity, default risk, and other
features germane to fixed-income securities.
M
VMBS
VMBS
CFt
 
t
(
1

R
)
t 1
 f (CFt , R )
Extension Risk
•
In contrast to other bonds, MBSs are also
subject to prepayment risk.
•
Prepayment affects the MBS’s CF.
•
Prepayment, in turn, is affected by interest rates.
•
Thus, interest rates affects the MBS’s CFs.
VMBS  f (CF, R)
CF  f (R )
Extension Risk
• With the CF a function of rates, the value of
a MBS is more sensitive to interest rate
changes than those bonds whose CFs are
not.
• This sensitivity is known as extension risk.
Extension Risk
•
If interest rates decrease, then the prices of
MBSs, like the prices of most bonds, increase as
a result of the lower discount rates.
•
However, the decrease in rates will also augment
prepayment speed, causing the earlier cash flow
of the mortgages to be larger which, depending
on the level of rates and the maturity remaining,
could also contribute to increasing the MBS’s
price.
Extension Risk
• Rate Decrease
like most bonds
if R   lower discount rate  VM 
if R   Increases prepayment Earlier CFs   VM  or 
Extension Risk
• If interest rates increase, then the prices of
MBSs will decrease as a result of higher
discount rates and possibly the smaller
earlier cash flow resulting from lower
prepayment speeds.
Extension Risk
• Rate Increase
like most bonds
if R   greater discount rate  VM 
if R   Decreases prepayment Earlier CFs   VM  or 
Average Life
•
The average life of a MBS or mortgage portfolio
is the weighted average of the security’s time
periods, with the weights being the periodic
principal payments (scheduled and prepaid
principal) divided by the total principal:
1 T
Average Life 
t

12 t 1
 principal received at

total principal

t


Average Life
•
The average life for the MBS issue with WAC = 8%,
WAM = 355, PT Rate = 7.5%, and PSA = 150 is
9.18 years
1  1($220,748)  2($246,153)      355($49,061
Average Life  

12 
$100,000,000

 9.18 years
Average Life
• The average life of a MBS depends on
prepayment speed:
– If the PSA speed of the $100M MBS issue
were to increase from 150 to 200, the MBS’s
average life would decrease from 9.18 to 7.55,
reflecting greater principal payments in the
earlier years.
– If the PSA speed were to decrease from 150 to
100, then the average life of the MBS would
increase to 11.51.
Average Life and Prepayment Risk
•
For MBSs and mortgage portfolios, prepayment risk can be
evaluated in terms of how responsive a MBS's or mortgage
portfolio’s average life is to changes in prepayment speeds:
Average Life
prepayment risk 
PSA
•
A MBS with an average life that did not change with PSA
speeds, in turn, would have stable principal payments over
time and would be absent of prepayment risk.
 Av life
 0  Zero prepayment risk
PSA
MBS Derivatives
•
One of the more creative developments in the
security market industry over the last two decades
has been the creation of derivative securities
formed from MBSs and mortgage portfolios that
have different prepayment risk characteristics,
including some that are formed that have average
lives that are invariant to changes in prepayment
rates.
•
The most popular of these derivatives are
–
–
Collateralized Mortgage Obligations, CMOs
Stripped MBS
Collateralized Mortgage Obligations
•
Collateralized mortgage obligations, CMOs, are
formed by dividing the cash flow of an
underlying pool of mortgages or a MBS issue into
several classes, with each class having a different
claim on the mortgage collateral and with each
sold separately to different types of investors.
Collateralized Mortgage Obligations
• The different classes making up a CMO are
called tranches or bond classes.
• There are two general types of CMO
tranches:
– Sequential-Pay Tranches
– Planned Amortization Class Tranches, PAC
Sequential-Pay Tranches
•
A CMO with sequential-pay tranches, called a
sequential-pay CMO, is divided into classes with
different priority claims on the collateral's
principal.
•
The tranche with the first priority claim has its
principal paid entirely before the next priority
class, which has its principal paid before the third
class, and so on.
•
Interest payments on most CMO tranches are
made until the tranche's principal is retired.
Sequential-Pay Tranches
•
Example: A sequential-pay CMO is shown in the
next exhibit.
•
This CMO consist of three tranches, A, B, and C,
formed from the collateral making up the $100M
MBS in the previous example: F = $100M, WAM
= 355, WAC = 8%, PT Rate = 7.5%, PSA = 150.
– Tranche A = $50M
– Tranche B = $30M
– Tranche C = $20M
Sequential-Pay Tranches
•
In terms of the priority disbursement rules:
– Tranche A receives all principal payment from the collateral
until its principal of $50M is retired. No other tranche's
principal payments are disbursed until the principal on A is
paid.
– After tranche A's principal is retired, all principal payments
from the collateral are then made to tranche B until its
principal of $30M is retired.
– Finally, tranche C receives the remaining principal that is
equal to its par value of $20M.
– Note: while the principal is paid sequentially, each tranche
does receive interest each period equal to its stated PT rate
(7.5%) times its outstanding balance at the beginning of each
month.
Sequential-Pay Tranches
Period
Month
1
2
3
4
5
85
86
87
88
89
90
91
92
178
181
182
183
184
353
354
355
Par = $100M Rate = 7.5%
Collateral Collateral
Balance
Interest
100000000
100000000
625000
99779252
623620
99533099
622082
99261650
620385
98965033
618531
51626473
322665
51154749
319717
50686799
316792
50222595
313891
49762107
311013
49305305
308158
48852161
305326
48402646
302517
20650839
129068
19990210
124939
19773585
123585
19558729
122242
19345627
120910
148527
928
98569
616
49061
307
Collateral
Principal
220748
246153
271449
296617
321637
471724
467949
464204
460488
456802
453144
449515
445915
222016
216625
214856
213101
211360
49958
49508
49061
A: Par = $50M Rate = 7.5%
Tranch A
A
A
Balance
Interest
Principal
50000000
50000000
312500
220748
49779252
311120
246153
49533099
309582
271449
49261650
307885
296617
48965033
306031
321637
1626473
10165
471724
1154749
7217
467949
686799
4292
464204
222595
1391
222595
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
B: Par =$30M
Tranche B
Balance
30000000
30000000
30000000
30000000
30000000
30000000
30000000
30000000
30000000
30000000
29762107
29305305
28852161
28402646
650839
0
0
0
0
0
0
0
Rate = 7.5%
B
Principal
B
Interest
0
0
0
0
0
0
0
0
237893
456802
453144
449515
445915
222016
0
0
0
0
0
0
0
187500
187500
187500
187500
187500
187500
187500
187500
187500
186013
183158
180326
177517
4068
0
0
0
0
0
0
0
C: Par = $20M Rate = 7.5%
Tranche C
C
C
Balance
Principal Interest
20000000
0
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
20000000
0
125000
19990210
216625
124939
19773585
214856
123585
19558729
213101
122242
19345627
211360
120910
148527
49958
928
98569
49508
616
49061
49061
307
Sequential-Pay Tranches
•
Given the assumed PSA of 150, the first month
cash flow for tranche A consist of a principal
payment (scheduled and prepaid) of $220,748
and an interest payment of $312,500:
[(.075/12)($50M) = $312,500]
•
In month 2, tranche A receives an interest
payment of $311,120 based on the balance of
$49.779252M and a principal payment of
$246,153.
Sequential-Pay Tranches
•
Based on the assumption of a 150% PSA speed, it takes
88 months before A's principal of $50M is retired.
•
During the first 88 months, the cash flows for tranches B
and C consist of just the interest on their balances, with no
principal payments made to them.
•
Starting in month 88, tranche B begins to receive the
principal payment.
•
Tranche B is paid off in month 180, at which time
principal payments begin to be paid to tranche C.
•
Finally, in month 355 tranche C's principal is retired.
Sequential-Pay Tranches
Features of Sequential-Pay CMOs
• By creating sequential-pay tranches, issuers
of CMOs are able to offer investors
maturities, principal payment periods, and
average lives different from those defined
by the underlying mortgage collateral.
Sequential-Pay Tranches
•
Features of Sequential-Pay CMOs
Maturity:
–
–
–
–
•
Window: The period between the beginning and ending principal
payment is referred to as the principal pay-down window:
–
–
–
–
•
Collateral's maturity = 355 months (29.58 years)
Tranche A’s maturity = 88 months (7.33 years)
Tranche B's maturity = 180 months (15 years)
Tranche C’s maturity = 355 months (29.58 years)
Collateral’s window = 355 months
Tranche A’s window = 87 months
Tranche B's window = 92 months
Tranche C's window =176 months
Average Life:
–
–
–
–
Collateral's average = 9.18 years
Tranche A’s average life = 3.69 years
Tranche B’s average life = 10.71 years
Tranche C’s Average life = 20.59 years
Sequential-Pay Tranches
Tranche
Maturity
Window Average Life
A
88 Months 87 Months 3.69 years
B
179 Months 92 Months 10.71 years
C
355 Months 176 Months 20.59 years
Collateral 355 Months 355 Months 9.18 years
Sequential-Pay Tranches
•
•
•
Note: A CMO tranche with a lower average life is still
susceptible to prepayment risk.
The average lives for the collateral and the three tranches
are shown below for different PSA models
Note that the average life of each of the tranches still varies
as prepayment speed changes.
PSA
50
100
150
200
300
Collateral
14.95
11.51
9.18
7.55
5.5
Tranche A
7.53
4.92
3.69
3.01
2.26
Tranche B
19.4
14.18
10.71
8.51
6.03
Tranche C
26.81
23.99
9.18
17.46
12.82
Sequential-Pay Tranches
• Note: Issuers of CMOs are able to offer a
number of CMO tranches with different
maturities and windows by simply creating
more tranches.
Different Types of Sequential-Pay Tranches
• Sequential-pay CMOs often include traches
with special features. These include:
– Accrual Bond Trache
– Floating-Rate Tranche
– Notional Interest-Only Tranche
Accrual Tranche
•
Many sequential-pay CMOs have an accrual
bond class.
•
Such a tranche, also referred to as the Z bond,
does not receive current interest but instead has it
deferred.
•
The Z bond's current interest is used to pay down
the principal on the other tranches, increasing
their speed and reducing their average life.
Accrual Tranche
•
Example: suppose in our preceding equential-pay
CMO example we make tranche C an accrual
tranche in which its interest of 7.5% is to paid to
the earlier tranches and its principal of $20M and
accrued interest is to be paid after tranche B's
principal has been retired
•
The next exhibit shows the principal and interest
payments from the collateral and Tranches A, B,
and Z.
Accrual Tranche
Period
Month
1
2
3
4
5
68
69
70
71
72
122
123
125
126
354
355
Par = $100M Rate = 7.5%
Collateral Collateral Collateral
Balance
Interest
Principal
100000000
100000000
625000
220748
99779252
623620
246153
99533099
622082
271449
99261650
620385
296617
98965033
618531
321637
60253239
376583
540668
59712571
373204
536352
59176219
369851
532069
58644150
366526
527821
58116329
363227
523605
36470935
227943
350111
36120824
225755
347292
35429038
221431
341719
35087319
219296
338966
98569
616
49508
49061
307
49061
A: Par =
A
Balance
50000000
50000000
49654252
49283099
48886650
48465033
1878239
1212571
551219
0
0
0
0
0
0
0
0
$50M Rate = 7.5%
A
A
Interest
Principal
312500
310339
308019
305542
302906
11739
7579
3445
0
0
0
0
0
0
0
0
345748
371153
396449
421617
446637
665668
661352
551219
0
0
0
0
0
0
0
0
B: Par = $30M Rate = 7.5%
B
B
Balance
Principal
30000000
30000000
0
30000000
0
30000000
0
30000000
0
30000000
0
30000000
0
30000000
0
30000000
105850
29894150
652821
29241329
648605
1345935
475111
870824
472292
0
0
0
0
0
0
0
0
B
Interest
187500
187500
187500
187500
187500
187500
187500
187500
186838
182758
8412
5443
0
0
0
0
Z: Par = $20M Rate = 7.5%
Z
Z
Z
Bal.+Cum Int Principal Interest
20000000
0
20000000
0
0
20125000
0
0
20250000
0
0
20375000
0
0
20500000
0
0
28375000
0
0
28500000
0
0
28625000
0
0
28750000
0
0
28875000
0
0
35125000
0
0
35250000
0
0
35429038
341719
221431
35087319
338966
219296
98569
49508
616
49061
49061
307
Accrual Tranche
•
Since the accrual tranche's current interest of $125,000 is
now used to pay down the other classes' principals, the
other tranches now have lower maturities and average
lives.
•
For example, the principal payment on tranche A is
$345,748 in the first month ($220,748 of scheduled and
projected prepaid principal and $125,000 of Z's interest);
in contrast, the principal is only $220,748 when there is
no Z bond.
•
As a result of the Z bond, trache A's window is reduced
from 87 months to 69 months and its average life from
3.69 years to 3.06 years.
Accrual Tranche
Tranche
A
B
Window
Average Life
69 Months 3.06 Years
54 Months 8.23 Years
Floating-Rate Tranche
•
In order to attract investors who prefer variable
rate securities, CMO issuers often create floatingrate and inverse floating-rate tranches.
•
The monthly coupon rate on the floating-rate
tranche is usually set equal to a reference rate
such as the London Interbank Offer Rate, LIBOR,
while the rate on the inverse floating-rate tranche
is determined by a formula that is inversely
related to the reference rate.
Floating-Rate Tranche
•
Example: Sequential-pay CMO with a
floating and inverse floating tranches
Tranche
A
FR
IFR
Z
Total
•
Par Value
$50M
$22.5M
$7.5M
$20M
$100
PT Rate
7.5%
LIBOR +50BP
28.3 – 3 LIBOR
7.5%
7.5%
Note: The CMO is identical to our preceding CMO, except that
tranche B has been replaced with a floating-rate tranche, FR, and an
inverse floating-rate tranche, IFR.
Floating-Rate Tranche
•
The rate on the FR tranche, RFR, is set to the
LIBOR plus 50 basis points, with the maximum
rate permitted being 9.5%.
•
The rate on the IFR tranche, RIFR, is determined
by the following formula:
RIFR = 28.5 - 3 LIBOR
•
This formula ensures that the weighted average
coupon rate (WAC) of the two tranches will be
equal to the coupon rate on tranche B of 7.5%,
provided the LIBOR is less than 9.5%.
Floating-Rate Tranche
•
For example, if the LIBOR is 8%, then the rate on the FR
tranche is 8.5%, the IFR tranche's rate is 4.5%, and the
WAC of the two tranches is 7.5%:
LIBOR  8%
R FR  LIBOR  50BP  8.5%
R IFR  28.5  3 LIBOR  4.5%
WAC  .75R FR  .25R IFR  7.5%
Notional Interest-Only Class
•
Each of the fixed-rate tranches in the previous CMOs
have the same coupon rate as the collateral rate of 7.5%.
•
Many CMOs, though, are structured with tranches that
have different rates. When CMOs are formed this way, an
additional tranche, known as a notional interest-only (IO)
class, is often created.
•
This tranche receives the excess interest on the other
tranches’ principals, with the excess rate being equal to
the difference in the collateral’s PT rate minus the
tranches’ PT rates.
Notional Interest-Only Class
•
Example: A sequential-pay CMO with a Z bond
and notional IO tranche is shown in the next
exhibit.
•
This CMO is identical to our previous CMO with
a Z bond, except that each of the tranches has a
coupon rate lower than the collateral rate of 7.5%
and there is a notional IO class.
Notional Interest-Only Class
•
The notional IO class receives the excess interest
on each tranche's remaining balance, with the
excess rate based on the collateral rate of 7.5%.
•
In the first month, for example, the IO class
would receive interest of $87,500:
 .075 .06 
 .075 .065
Interest 
$
50
,
000
,
000



$30,000,000
12
 12 


Interest $62,500  $25,000  $87,500
Notional Interest-Only Class
Period
Month
1
2
3
4
5
70
71
72
122
123
124
125
126
127
353
354
355
Collateral:
Collateral
Balance
100000000
100000000
99779252
99533099
99261650
98965033
59176219
58644150
58116329
36470935
36120824
35773533
35429038
35087319
34748353
148527
98569
49061
Par = $100M Rate = 7.5%
Collateral
Collateral
Interest
Principal
625000
623620
622082
620385
618531
369851
366526
363227
227943
225755
223585
221431
219296
217177
928
616
307
220748
246153
271449
296617
321637
532069
527821
523605
350111
347292
344494
341719
338966
336235
49958
49508
49061
Tranche A:
A
Balance
50000000
50000000
49654252
49283099
48886650
48465033
551219
0
0
0
0
0
0
0
0
0
0
0
Par = $50M Rate = 6%
Tranche B: Par = $30M Rate = 6.5%
Tranche Z: Par = $20M Rate = 7%
Notional Par = $15.333M
A
A
Notional
B
B
B
Notional
Z
Z
Z
Notional
Notional
Interest
Principal Interest
Balance
Principal
Interest
Interest Balance
Principal
Interest Interest
Total CF
0.06
0.015
30000000
0.065
0.01
20000000
0
0.07
0.005
250000
345748
62500
30000000
0
162500
25000
20000000
0
0
0
87500
248271
371153
62068
30000000
0
162500
25000
20125000
0
0
0
87068
246415
396449
61604
30000000
0
162500
25000
20250000
0
0
0
86604
244433
421617
61108
30000000
0
162500
25000
20375000
0
0
0
86108
242325
446637
60581
30000000
0
162500
25000
20500000
0
0
0
85581
2756
551219
689
30000000
105850
162500
25000
28625000
0
0
0
25689
0
0
0
29894150
652821
161927
24912
28750000
0
0
0
24912
0
0
0
29241329
648605
158391
24368
28875000
0
0
0
24368
0
0
0
1345935
475111
7290
1122
35125000
0
0
0
1122
0
0
0
870824
472292
4717
726
35250000
0
0
0
726
0
0
0
398533
398533
2159
332
35375000
-54038
206354
14740
15072
0
0
0
0
0
0
0
35429038
341719
206669
14762
14762
0
0
0
0
0
0
0
35087319
338966
204676
14620
14620
0
0
0
0
0
0
0
34748353
336235
202699
14478
14478
0
0
0
0
0
0
0
148527
49958
866
62
62
0
0
0
0
0
0
0
98569
49508
575
41
41
0
0
0
0
0
0
0
49061
49061
286
20
20
Notional Interest-Only Class
•
The IO class is described as paying 7.5% interest
on a notional principal of $15,333,333.
•
This notional principal is determined by summing
each tranche's notional principal.
•
A tranche's notional principal is the number of
dollars that makes the return on the tranche's
principal equal to 7.5%.
Notional Interest-Only Class
•
The notional principal for tranche A is
$10,000,000, for B, $4,000,000, and for Z,
$1,333,333, yielding a total notional principal of
$15,333,333:
($50,000,000)(.075 .06)
A ' s Not ionalprincipal
 $10,000,000
.075
$30,000,000)(.075 .065)
B' s Not ionalprincipal
 $4,000,000
.075
($20,000,000)(.075 .07)
Z' s Not ionalprincipal
 $1,333,333
.075
T ot alNot ionalP r incipal  $15,333,333
Planned Amortization Class, PAC
•
A CMO with a planned amortization class, PAC,
is structured such that there is virtually no
prepayment risk.
•
In a PAC-structured CMO, the underlying
mortgages or MBS (i.e., the collateral) is divided
into two general tranches:
– The PAC (also called the PAC bond)
– The support class (also called the support
bond or the companion bond)
Planned Amortization Class, PAC
•
The two tranches are formed by generating two
monthly principal payment schedules from the
collateral:
–
One schedule is based on assuming a relatively low
PSA speed – lower collar.
– The other schedule is based on assuming a relatively
high PSA speed – upper collar.
•
The PAC bond is then set up so that it will receive
a monthly principal payment schedule based on
the minimum principal from the two principal
payments.
Planned Amortization Class, PAC
• The PAC bond is designed to have no
prepayment risk provided the actual
prepayment falls within the minimum and
maximum assumed PSA speeds.
• The support bond, on the other hand,
receives the remaining principal balance
and is therefore subject to prepayment risk.
Planned Amortization Class, PAC
•
To illustrate, suppose we form PAC and support
bonds from the $100M collateral that we used to
construct our sequential-pay tranches:
–
•
Underlying MBS = $100M, WAC = 8%, WAM = 355
months, and PT rate = 7.5%
To generate the minimum monthly principal
payments for the PAC, assume:
–
Minimum speed of 100% PSA; lower collar = 100
PSA
– Maximum speed of 300% PSA; upper collar = 300
PSA
Planned Amortization Class, PAC
•
The next exhibit shows the cash flows for the
PAC, collateral, and support bond. The exhibit
shows:
–
In columns 2 and 3 the principal payments (scheduled
and prepaid) for selected months at both collars.
– In the fourth column the minimum of the two
payments.
•
For example, in the first month the principal
payment is $170,085 for the 100% PSA and
$374,456 for the 300% PSA; thus, the principal
payment for the PAC would be $170,085.
Planned Amortization Class, PAC
Period
Month
1
2
3
4
5
98
99
100
101
102
201
202
203
204
205
206
354
355
Pac
Pac
Low PSA Pr high PSA Pr
100
300
170085
374456
187135
425190
204125
475588
221048
525572
237895
575064
381871
386139
380032
379499
378204
372970
376384
366552
374575
360242
235460
61932
234395
60806
233336
59699
232283
58611
231235
57542
230193
56492
124660
2559
124203
2493
Pac
Min. Principal
170085
187135
204125
221048
237895
381871
379499
372970
366552
360242
61932
60806
59699
58611
57542
56492
2559
2493
Par = 63777030
Pac
Int
0.075
398606
397543
396374
395098
393716
135237
132851
130479
128148
125857
19312
18925
18545
18172
17806
17446
32
16
Pac
CF
568692
584678
600499
616147
631612
517108
512349
503449
494700
486099
81245
79731
78244
76783
75348
73938
2591
2509
Collateral Collateral Collateral Collateral
Support
Balance
Interest Prncipal
CF
Principal
100000000
Col Pr - PAC Pr
100000000
625000
220748
845748
50662
99779252
623620
246153
869773
59018
99533099
622082
271449
893531
67324
99261650
620385
296617
917002
75568
98965033
618531
321637
940168
83742
45780181
286126
424898
711025
43028
45355283
283471
421491
704962
41993
44933791
280836
418111
698947
45141
44515680
278223
414758
692981
48205
44100923
275631
411430
687061
51188
15978416
99865
183776
283641
121844
15794640
98716
182266
280982
121460
15612374
97577
180768
278345
121069
15431606
96448
179282
275729
120671
15252325
95327
177807
273134
120265
15074517
94216
176344
270560
119852
98569
616
49508
50124
46948
49061
307
49061
49368
46568
Par = 36222970
Support
Balance
36222970
36172308
36113290
36045966
35970398
24142190
24099163
24057170
24012029
23963824
12888435
12766592
12645131
12524062
12403392
12283127
93517
46568
Support
Interest
0.075
226394
226077
225708
225287
224815
150889
150620
150357
150075
149774
80553
79791
79032
78275
77521
76770
584
291
Support
CF
277056
285095
293032
300856
308557
193916
192613
195498
198281
200962
202396
201251
200101
198946
197786
196622
47533
46859
Planned Amortization Class, PAC
• Note: For the first 98 months, the minimum
principal payment comes from the 100%
PSA model, and from month 99 on the
minimum principal payment comes from
the 300% PSA model.
Planned Amortization Class, PAC
• Based on the 100-300 PSA range, a PAC
bond can be formed that would promise to
pay the principal based on the minimum
principal payment schedule shown in the
exhibit.
• The support bond would receive any excess
monthly principal payment.
Planned Amortization Class, PAC
•
The sum of the PAC's principal payments is
$63.777M. Thus, the PAC can be described as
having:
–
–
–
–
•
Par value of $63.777M
Coupon rate of 7.5%
Lower collar of 100% PSA
Upper collar of 300% PSA
The support bond, in turn, would have a par value
of $36.223M ($100M - $63.777M) and pay a
coupon of 7.5%.
Planned Amortization Class, PAC
•
The PAC bond has no prepayment risk as long as
the actual prepayment speed is between 100 and
300.
•
This can be seen by calculating the PAC's average
life given different prepayment rates.
•
The next exhibit shows the average lives for the
collateral, PAC bond, and support bond for
various prepayment speeds ranging from 50%
PSA to 350% PSA.
Planned Amortization Class, PAC
PSA
50
100
150
200
250
300
350
Collateral
14.95
11.51
9.18
7.55
6.37
5.50
4.84
Average Life
PAC
7.90
6.98
6.98
6.98
6.98
6.98
6.34
Support
21.50
19.49
13.05
8.55
5.31
2.91
2.71
Planned Amortization Class, PAC
•
Note:
The PAC bond has an average life of 6.98 years
between 100% PSA and 300% PSA; its average
life does change, though, when prepayment
speeds are outside the 100-300 PSA range.
•
In contrast, the support bond's average life
changes as prepayment speed changes.
•
Changes in the support bond's average life due to
changes in speed are greater than the underlying
collateral's responsiveness.
Other PAC-Structured CMOs
•
The PAC and support bond underlying a CMO
can be divided into different classes. Often the
PAC bond is divided into several sequential-pay
tranches, with each PAC having a different
priority in principal payments over the other.
•
Each sequential-pay PAC, in turn, will have a
constant average life if the prepayment speed is
within the lower and upper collars.
•
In addition, it is possible that some PACs will
have ranges of stability that will increase beyond
the actual collar range, expanding their effective
collars.
Other PAC-Structured CMOs
•
A PAC-structured CMO can also be formed with
PAC classes having different collars.
•
Some PACs are formed with just one PSA rate.
These PACs are referred to as targeted
amortization class (TAC) bonds.
•
Different types of tranches can also be formed out
of the support bond class. These include
sequential-pay, floating and inverse-floating rate,
and accrual bond classes.
Stripped MBS
•
Stripped MBSs consist of two classes:
1. Principal-only (PO) class that receives only
the principal from the underlying mortgages.
2. Interest-only (IO) class that receives just the
interest.
Principal-Only Stripped MBS
•
The return on a PO MBS is greater with greater
prepayment speed.
•
For example, a PO class formed with $100M of
mortgages (principal) and priced at $75M would
yield an immediate return of $25M if the
mortgage borrowers prepaid immediately. Since
investors can reinvest the $25M, this early return
will have a greater return per period than a $25M
return that is spread out over a longer period.
Principal-Only Stripped MBS
•
Because of prepayment, the price of a PO MBS tends to
be more responsive to interest rate changes than an
option-free bond.
•
That is, if interest rates are decreasing, then like the price
of most bonds, the price of a PO MBS will increase. In
addition, the price of a PO MBS is also likely to increase
further because of the expectation of greater earlier
principal payments as a result of an increase in
prepayment caused by the lower rates.
(1)  prepayment
  ret urn   VPO 
R
(2)  lower discount rat e  VPO 
Principal-Only Stripped MBS
•
In contrast, if rates are increasing, the price of a
PO MBS will decrease as a result of both lower
discount rates and lower returns from slower
principal payments.
(1)  prepayment
  ret urn   VPO 
R
(2)  great er discount rat e  VPO 
Principal-Only Stripped MBS
•
Thus, like most bonds, the prices of PO
MBSs are inversely related to interest
rates, and, like other MBSs with embedded
principal prepayment options, their prices
tend to be more responsive to interest rate
changes.
VPO
0
R
Interest-Only Stripped MBS
•
Cash flows from an I0 MBS come from the
interest paid on the mortgages portfolio’s
principal balance.
•
In contrast to a PO MBS, the cash flows and the
returns on an IO MBS will be greater, the slower
the prepayment rate.
Interest-Only Stripped MBS
•
If the mortgages underlying a $100M, 7.5% MBS with PO
and IO classes were paid off in the first year, then the IO
MBS holders would receive a one-time cash flow of $7.5M:
$7.5M = (.075)($100M)
•
If $50M of the mortgages were prepaid in the first year and
the remaining $50M in the second year, then the IO MBS
investors would receive an annualized cash flow over two
years totaling $11.25M:
$11.25M = (.075) ($100M) + (.075)($100M-$50M)
•
If the mortgage principal is paid down $25M per year, then
the cash flow over four years would total $18.75M:
$18.75M = (.075)($100M) + (.075)($100M-$25M)
+ (.075)($75M-$25M) + (.075)($50M-$25M))
Interest-Only Stripped MBS
•
Thus, IO MBSs are characterized by an inverse
relationship between prepayment speed and returns: the
slower the prepayment rate, the greater the total cash
flow on an IO MBS.
•
Interestingly, if this relationship dominates the price and
discount rate relation, then the price of an IO MBS will
vary directly with interest rates.
(1)  prepayment  ret urn   VIO 
R
(2)  great er discount rat e  VIO 
If 1st effect  2nd effect, t hen
VIO
0
R
IO and PO Stripped MBS
•
An example of a PO MBS and an IO MBS
are shown in next exhibit.
•
The stripped MBSs are formed from
collateral with
–
–
–
–
–
Mortgage Balance = $100M
WAC = 8%
PT Rate = 8%
WAM = 360
PSA = 100
IO and PO Stripped MBS
Period
Month
1
2
3
4
5
100
101
200
201
358
359
360
Collateral
Balance
100000000
100000000
99916231
99815353
99697380
99562336
58669646
58284486
27947479
27706937
371778
247395
123470
Collateral
Interest
666667
666108
665436
664649
663749
391131
388563
186317
184713
2479
1649
823
Collateral
Scheduled
Principal
67098
67534
67961
68380
68790
83852
83977
97308
97453
123103
123287
123470
Collateral
Prepaid
Principal
16671
33344
50011
66664
83294
301307
299326
143234
141996
1279
638
0
Collateral Collateral
Total
CF
Principal
83769
750435
100878
766986
117973
783409
135044
799694
152084
815833
385159
776290
383303
771866
240542
426858
239449
424162
124382
126861
123925
125574
123470
124293
Stripped
PO
Stripped
IO
83769
100878
117973
135044
152084
385159
383303
240542
239449
124382
123925
123470
666667
666108
665436
664649
663749
391131
388563
186317
184713
2479
1649
823
IO and PO Stripped MBS
•
The table shows the values of the collateral, PO
MBS, and IO MBS for different discount rate
and PSA combinations of 8% and 150, 8.5% and
125, and 9% and 100.
•
Note: The IO MBS is characterized by a direct
relation between its value and rate of return.
Discount
Rate
8%
8.50%
9.00%
PSA
150
125
100
Value of
PO
$54,228,764
$49,336,738
$44,044,300
Value of
IO
$47,426,196
$49,513,363
$51,795,188
Value of
Collateral
$101,654,960
$98,850,101
$95,799,488
Other Asset-Backed Securities
•
•
•
MBSs represent the largest and most extensively
developed asset-backed security. A number of other
asset-backed securities have been developed.
The three most common types are those backed by
– Automobile Loans
– Credit Card Receivables
– Home Equity Loans
These asset-backed securities are structured as passthroughs and many have tranches.
Other Asset-Backed Securities
•
Automobile Loan-Backed Securities
Automobile loan-backed securities are often referred to
as CARS (certificates for automobile receivables).
•
The automobile loans underlying these securities are
similar to mortgages in that borrowers make regular
monthly payments that include interest and a scheduled
principal.
•
Like mortgages, automobile loans are characterized by
prepayment. For such loans, prepayment can occur as a
result of car sales, trade-ins, repossessions and
subsequent resales, wrecks, and refinancing when rates
are low.
•
Some CARS are structured as PACS.
Other Asset-Backed Securities
•
Automobile Loan-Backed Securities
CARS differ from MBSs in that
–
–
–
they have much shorter lives
their prepayment rates are less influenced by
interest rates than mortgage prepayment rates
they are subject to greater default risk.
Other Asset-Backed Securities
•
•
Credit-Card Receivable-Backed Securities
Credit-card receivable-backed securities are
commonly referred to as CARDS (certificates for
amortizing revolving debts).
In contrast to MBSs and CARS, CARDS
investors do not receive an amortorized principal
payment as part of their monthly cash flow.
Other Asset-Backed Securities
Credit-Card Receivable-Backed Securities
• CARDS are often structured with two periods:
– In one period, known as the lockout period, all
principal payments made on the receivables
are retained and either reinvested in other
receivables or invested in other securities.
–
In the other period, known as the principalamortization period, all current and
accumulated principal payments are paid.
Other Asset-Backed Securities
•
Home Equity Loan-Backed Securities
Home-equity loan-backed securities are referred
to as HELS.
•
They are similar to MBSs in that they pay a
monthly cash flow consisting of interest,
scheduled principal, and prepaid principal.
•
In contrast to mortgages, the home equity loans
securing HELS tend to have a shorter maturity
and different factors influencing their
prepayment rates.
Websites
•
For more information on the mortgage industry,
statistics, trends, and rates go to www.mbaa.org
•
Mortgage rates in different geographical areas
can be found by going to www.interest.com.
•
For historical mortgage rates go to
http://research.stlouisfed.org/fred2 and click on
“Interest Rates.”
Websites
•
Agency information: www.fanniemae.com,
www.ginniemae.gov, www.freddiemac.com
•
For general information on MBS go to www.ficc.com
•
For information on the market for mortgage-backed
securities go to www.bondmarkets.com and click on
“Research Statistics and “Mortgage-Backed Securities.”
For information on links to other sites click on “Gateway
to Related Links.”
•
For information on the market for asset-backed securities
go to www.bondmarkets.com and click on “Research
Statistics and “Asset-Backed Securities.”
Evaluating Mortgage-Backed
Securities
Monte Carlo Simulation
• Objective: Determine the MBS’s theoretical value
• Steps:
1. Simulation of interest rates: Use a binomial interest rate
tree to generate different paths for spot rates and
refinancing rates.
2. Estimate the cash flows of a mortgage portfolio, MBS, or
tranche for each path given a specified prepayment model
based on the spot rates
3. Determine the present values of each path.
4. Calculate the average value – theoretical value.
Step 1: Simulation of Interest Rate
• Determination of interest rate paths from a binomial
interest rate tree.
• Example:
– Assume three-period binomial tree of one-year
spot rates (S) and refinancing rates, (Rref) where:
S0  6%
u  1.1
d  .9091  1 / 1.1
R 0ref  8%
u  1.1
d  .9091  1 / 1.1
– With three periods, there are four possible rates
after three periods (years) and there are eight
possible paths.
Step 1: Binomial Tree for Spot
and Refinancing Rates
Suuu  7.9860
u = 1.10, d = .9091
Suu  7.260
R ref
uuu  10.648
R ref
uu  9.680
Su  6.600
Suud  6.600
R  8.800
ref
u
R ref
uud  8.800
S0  6.00%
Sud  6.00
R 0ref  8.00%
R ref
ud  8.00
Sd  5.4546
Sudd  5.4546
R dref  7.2728
R ref
udd  7.2728
Sdd  4.9588
R ddref  6.6117
Sddd  4.5080
ref
R ddd
 6.0107
Step 1: Interest Rate Paths
Path 1
Path 2
Path 3
Path 4
6.0000%
5.4546
4.9588
4.5080
6.0000%
5.4546
4.9588
5.4546
6.0000%
5.4546
6.0000
5.4546
6.0000%
6.6000
6.0000
5.4546
Path 5
Path 6
Path 7
Path 8
6.0000%
5.4546
6.0000
6.6000
6.0000%
6.6000
6.0000
6.6000
6.0000%
6.6000
7.2600
6.6000
6.0000%
6.6000
7.2600
7.9860
Step 2: Estimating Cash Flows
• The second step is to estimate the cash flow for
each interest rate path.
• The cash flow depends on the prepayment rates
assumed.
• Most analysts use a prepayment model in which
the conditional prepayment rate (CPR) is
determined by the seasonality of the mortgages,
and by a refinancing incentive that ties the interest
rate paths to the proportion of the mortgage
collateral prepaid.
Step 2: Estimating Cash Flows
• To illustrate, consider a MBS formed from a
mortgage pool with a par value of $1M, WAC =
8%, and WAM = 10 years.
• To fit this example to the three-year binomial tree
assume that
– the mortgages in the pool all make annual cash flows
(instead of monthly)
– all have a balloon payment at the end of year 4
– the pass-through rate on the MBS is equal to the WAC
of 8
Step 2: Estimating Cash Flows
• The mortgage pool can be viewed as a four-year
asset with a principal payment made at the end of
year four that is equal to the original principal
less the amount paid down.
• As shown in the next exhibit, if there were no
prepayments, then the pool would generate cash
flows of $149,029M each year and a balloon
payment of $688,946 at the end of year 4.
Step 2: Estimating Cash Flows
•
Mortgage Portfolio:
– Par Value = $1M, WAC = 8%, WAM = 10 Yrs,
– PT Rate = 8%, Balloon at the end of the 4th Year
Year
Balance
P
Interest
Scheduled
Principal
1
2
3
4
$1,000,000
$930,971
$856,419
$775,903
$149,029
$149,029
$149,029
$149,029
$80,000
$74,478
$68,513
$62,072
$69,029
$74,552
$80,516
$86,957
Balloon  Balance( yr4)  Sch .prin( yr4)
 $775,903$86,957$688,946
CF4  Balloon  p
$688,946$149,029$837,975
CF4  Balance( yr4)  Int erest
$775,903$62,072$837,975
Cash
Flow
$149,029
$149,029
$149,029
$837,975
Step 2: Estimating Cash Flows
• Such a cash flow is, of course, unlikely
given prepayment. A simple prepayment
model to apply to this mortgage pool is
shown in next exhibit.
Step 2: Estimating Cash Flows
•
The prepayment model assumes:
1. The annual CPR is equal to 5% if the mortgage pool rate is at
a par or discount (that is, if the current refinancing rate is
equal to the WAC of 8% or greater).
2. The CPR will exceed 5% if the rate on the mortgage pool is
at a premium.
3. The CPR will increase within certain ranges as the premium
increases.
4. The relationship between the CPRs and the range of rates is
the same in each period; that is, there is no seasoning factor.
Step 2: Estimating Cash Flows
Range
CPR
X = WAC – Rref
X 0
0.0% < X  0.5%
0.5% < X  1.0%
1.0% < X  1.5%
1.5% < X  2.0%
2.0% < X  2.5%
2.5% < X  3.0%
X > 3.0%
5%
10%
20%
30%
40%
50%
60%
70%
Step 2: Estimating Cash Flows
• With this prepayment model, cash flows can
be generated for the eight interest rate paths.
• These cash flows are shown in Exhibit A
(next two slides).
Exhibit A
Path 1
Year
1
2
3
4
Path 2
Year
1
2
3
4
Path 3
Year
1
2
3
4
Path 4
Year
1
2
3
4
1
Rref
2
Balance
0.072728 1000000
0.066117 744776
0.060107 479594
260703
1
Rref
2
Balance
0.072728 1000000
0.066117 744776
0.072728 479594
347604
1.000000
2
Rref
Balance
0.072728 1000000
0.080000 744776
0.072728 650878
471749
1
Rref
2
Balance
0.088000 1000000
0.080000 884422
0.072728 772918
560202
WAC
0.08
0.08
0.08
0.08
WAC
0.08
0.08
0.08
0.08
WAC
0.08
0.08
0.08
0.08
WAC
0.08
0.08
0.08
0.08
3
4
Interest Sch. Prin.
80000
69029
59582
59641
38368
45089
20856
3
4
Interest Sch. Prin.
80000
69029
59582
59641
38368
45089
27808
3
4
Interest Sch. Prin.
80000
69029
59582
59641
52070
61192
37740
3
4
Interest Sch. Prin.
80000
69029
70754
70824
61833
72666
44816
5
CPR
0.20
0.30
0.40
5
CPR
0.20
0.30
0.20
5
CPR
0.20
0.05
0.20
5
CPR
0.05
0.05
0.20
6
Prepaid Prin.
186194
205540
173802
6
Prepaid Prin.
186194
205540
86901
6
Prepaid Prin.
186194
34257
117937
6
Prepaid Prin.
46549
40680
140050
7
CF
335224
324764
257259
281560
7
CF
335224
324764
170358
375413
7
CF
335224
153480
231200
509489
7
CF
195578
182258
274550
605018
8
Z1,t-1
9
Zt0
0.080000
0.074546
0.069588
0.065080
0.080000
0.077270
0.074703
0.072289
Value =
8
Z1,t-1
9
Zt0
0.080000
0.074546
0.069588
0.074546
0.080000
0.077270
0.074703
0.074664
Value =
8
Z1,t-1
0.080000
0.074546
0.080000
0.074546
8
Z1,t-1
0.080000
0.086000
0.080000
0.074546
9
Zt0
0.080000
0.077270
0.078179
0.077270
Value =
9
Zt0
0.080000
0.082996
0.081996
0.080129
Value =
10
Value
310392
279846
207255
212972
1010465
11
Prob.
0.5
0.5
0.5
10
Value
310392
279846
137245
281461
1008945
11
Prob.
0.5
0.5
0.5
10
Value
310392
132253
184465
378301
1005411
11
Prob.
0.5
0.5
0.5
10
Value
181091
155393
216742
444494
997720
11
Prob.
0.5
0.5
0.5
0.125
0.125
0.125
0.125
Exhibit A
Path 5
Year
1
2
3
4
Path 6
Year
1
2
3
4
Path 7
Year
1
2
3
4
Path 8
Year
1
2
3
4
1
Rref
2
Balance
0.072728 1000000
0.080000 744776
0.088000 650878
560202
1
Rref
2
Balance
0.088000 1000000
0.080000 884422
0.088000 772918
665240
1
Rref
2
Balance
0.088000 1000000
0.096000 884422
0.088000 772918
665240
1
Rref
2
Balance
0.088000 1000000
0.096000 884422
0.106480 772918
665240
WAC
0.08
0.08
0.08
0.08
WAC
0.08
0.08
0.08
0.08
WAC
0.08
0.08
0.08
0.08
WAC
0.08
0.08
0.08
0.08
3
4
Interest Sch. Prin.
80000
69029
59582
59641
52070
61192
44816
3
4
Interest Sch. Prin.
80000
69029
70754
70824
61833
72666
53219
3
4
Interest Sch. Prin.
80000
69029
70754
70824
61833
72666
53219
3
4
Interest Sch. Prin.
80000
69029
70754
70824
61833
72666
53219
5
CPR
0.20
0.05
0.05
5
CPR
0.05
0.05
0.05
5
CPR
0.05
0.05
0.05
5
CPR
0.05
0.05
0.05
6
Prepaid Prin.
186194
34257
29484
6
Prepaid Prin.
46549
40680
35013
6
Prepaid Prin.
46549
40680
35013
6
Prepaid Prin.
46549
40680
35013
7
CF
335224
153480
142747
605018
7
CF
195578
182258
169512
718459
7
CF
195578
182258
169512
718459
7
CF
195578
182258
169512
718459
8
Z1,t-1
0.080000
0.074546
0.080000
0.086000
8
Z1,t-1
0.080000
0.086000
0.080000
0.086000
8
Z1,t-1
0.080000
0.086000
0.092600
0.086000
8
Z1,t-1
0.080000
0.086000
0.092600
0.099860
9
Zt0
0.080000
0.077270
0.078179
0.080129
Value =
9
Zt0
0.080000
0.082996
0.081996
0.082996
Value =
9
Zt0
0.080000
0.082996
0.086188
0.086141
Value =
9
Zt0
0.080000
0.082996
0.086188
0.089590
Value =
10
Value
310392
132253
113892
444494
1001031
11
Prob.
0.5
0.5
0.5
10
Value
181091
155393
133820
522269
992574
11
Prob.
0.5
0.5
0.5
10
Value
181091
155393
132277
516247
985008
11
Prob.
0.5
0.5
0.5
10
Value
181091
155393
132277
509741
978502
11
Prob.
0.5
0.5
0.5
0.125
0.125
0.125
0.125
Wt. Value $997,457
Step 2: Estimating Cash Flows
Path 1
• The cash flows for path 1 (the path with three
consecutive decreases in rates) consist of
– $335,224 in year 1 (interest = $80,000, scheduled
principal = $69,029.49, and prepaid principal =
$186,194.10, reflecting a CPR of .20)
– $324,764 in year 2, with $205,540 being prepaid
principal (CPR = .30)
– $257,259 in year 3, with $173,802 being prepaid
principal (CPR = .40)
– $251,560 in year 4
• The year 4 cash flow with the balloon payment is
equal to the principal balance at the beginning of
the year and the 8% interest on that balance.
Step 2: Estimating Cash Flows
•
•
•
•
Calculations for CF for Path 1:
$335,224 in year 1:
Interest = $80,000
scheduled principal = $69,029.49
prepaid principal = $186,194.10, reflecting a CPR of .20
$1,000,000
p
 $149,029
10
1  (1 /(1.08)
.08
int erest  .08($1,000,000)  $80,000
scheduled principal  $149,029 $80,000  $69,029
prepaid principal  .20($1,000,000 $69,029)  $186,194
CF1  $80,000  $69,029  $186,194  $335,224
Allow for slight rounding differences
Step 2: Estimating Cash Flows
•
•
•
•
Calculations for CF for Path 1:
$324,764 in year 2:
Interest = $59,582
scheduled principal = $59,641
prepaid principal = $205,540, reflecting a CPR of .30
Balance  $1,000,000  ($69,029 $186,194)  $744,776
$744,776
p
 $119,223
9
1  (1 /(1.08)
.08
int erest  .08($744,776)  $59,582
scheduled principal  $119,223 $59,582  $59,641
prepaid principal  .30($744,776 $59,641)  $205,540
CF2  $59,582 $59,641  $205,540  $324,764
Allow for slight rounding differences
Step 2: Estimating Cash Flows
•
•
•
•
Calculations for CF for Path 1:
$257,259 in year 3:
Interest = $38,368
scheduled principal = $45,089
prepaid principal = $173,802, reflecting a CPR of .40
Balance  $744,776  ($59,641 $205,540)  $479,594
$479,594
p
 $83,456
8
1  (1 /(1.08)
.08
int erest  .08($479,594)  $38,368
scheduled principal  $83,456 $38,368  $45,089
prepaid principal  .40($479,594 $45,089)  $173,802
CF3  $38,368  $45,089  $173,802  $257,259
Allow for slight rounding differences
Step 2: Estimating Cash Flows
•
•
Calculations for CF for Path 1:
$281,560 in year 4:
The year 4 cash flow with the balloon payment is equal to
the principal balance at the beginning of the year and the
8% interest on that balance.
Balance  $479,594  ($45,089 $173,802)  $260,703
int erest  .08($260,703)  $20,856
CF4  $260,703  $20,856  $281,560
Allow for slight rounding differences
Step 2: Estimating Cash Flows
Path 8
• In contrast, the cash flows for path 8 (the path with
three consecutive interest rate increases) are
smaller in the first three years and larger in year 4,
reflecting the low CPR of 5% in each period.
Step 3: Valuing Each Path
•
Like any bond, a MBS or CMO tranche should be valued
by discounting the cash flows by the appropriate riskadjusted spot rates.
•
For a MBS or CMO tranche, the risk-adjusted spot rate, zt,
is equal to the riskless spot rate, St, plus a risk premium.
•
If the underlying mortgages are insured against default,
then the risk premium would only reflect the additional
return needed to compensate investors for the prepayment
risk they are assuming.
•
As noted in Chapter 4, this premium is referred to as the
option-adjusted spread (OAS).
Step 3: Valuing Each Path
• If we assume no default risk, then the riskadjusted spot rate can be defined as
z t  St  k t
where: k = OAS
Step 3: Valuing Each Path
• The value of each path can be defined as
T
Path
i
V
CF3
CFM
CF1
CF2
CFT




  
M
2
3
T
(
1

z
)
1

z
(
1

z
)
(
1

z
)
(
1

z
)
M 1
M
1
2
3
T
where:
• i = ith path
• zM = spot rate on bond with M-year maturity
• T = maturity of the MBS
Step 3: Valuing Each Path
• For this example, assume the optionadjusted spread is 2% greater than the oneyear, risk-free spot rates.
Step 3: Valuing Each Path
Binomial Tree for
Discount Rates
9.9860%
9.26%
8.6%
8 .6 %
8%
8%
7.4546%
7.4546%
69588%
.
65080%
.
Z10
Z11
Z12
Z13
Step 3: Valuing Each Path
• From these current and future one-year spot rates,
the current 1-year, 2-year, 3-year, and 4-year
equilibrium spot rates can be obtained for each
path by using the geometric mean:
z M  (1  z10 )(1  z11 )   (1  z1,M1 )
1/ M
1
Step 3: Valuing Each Path
• The set of spot rates z1, z2, z3, and z4 needed
to discount the cash flows for path 1 would
be:
z1  .08
z 2  (1  z10 )(1  z11 )  1
1/ 2
 (1.08)(1.074546)  1  .07727
1/ 2
z 3  (1  z10 )(1  z11 )(1  z12 )  1
1/ 3
1/ 3


 (1.08)(1.074546)(1.069588) 1  .074703
1/ 4
z 4  (1  z10 )(1  z11 )(1  z12 )(1  z13 )   1
1/ 4
 (1.08)(1.074546)(1.069588)(1.06508)  1  .072289
Step 3: Valuing Each Path
• Using these rates, the value of the MBS
following path 1 is $1,010,465:
Path
1
V
$335,224 $324,764 $257,259
$281,560




 $1,010,465
2
3
4
1.08
(1.07727) (1.074703) (1.072289)
The spot rates and values of each of the eight paths are
shown in columns 9 and 10 of Exhibit A.
Step 4: Theoretical Path
• The theoretical value of the MBS is defined as the
average of the values of all the interest rate paths:
1 N path
V   Vi
N i 1
• In this example, the theoretical value of the MBS
issue is $997,457 or 99.7457% of its par value (see
bottom of Exhibit A).
Step 4: Theoretical Path
• The theoretical value along with the standard
deviation of the path values are useful measures in
evaluating a MBS or CMO tranche relative to
other securities.
• A MBS's theoretical value can also be compared to
its actual price to determine if the MBS is over or
underpriced.
• For example, if the theoretical value is 98% of par
and the actual price is at 96%, then the mortgage
security is underpriced, '$2 cheap', and if it is
priced at par, then it is considered overpriced, '$2
rich.'
Option-Adjusted Spread
• Instead of determining the theoretical value of the
MBS or tranche given a path of spot rates and
option-adjusted spreads, analysts can use a Monte
Carlo simulation to estimate the mortgage
security's rate of return given its market price.
Option-Adjusted Spread
• Since the security's rate of return is equal to
a riskless spot rate plus the OAS (assuming
no default risk), many analysts use the
simulation to estimate just the OAS.
• From the simulation, the OAS is determined
by finding that OAS that makes the
theoretical value of the MBS equal to its
market price.
Option-Adjusted Spread
• This spread can be found by iteratively
solving for the k that satisfies the following
equation:
 T

T

CF(1) M
CF( 2) M
CF( N ) M
1   T
Market Price 

  
      

M
M
M
N  M1 (1  S(1) M  k)  M1 (1  S( 2) M  k) 
M1 (1  S( N) M  k)  
where: N = number of paths
Effective Duration and Convexity
• Effective duration and convexity can be used with a
binomial tree to measure the duration of a MBS.
Duration 
P  P  2(P0 )
P  P
; Convexity 
2(P0 )y
(P0 )(y) 2
where :
P  price associat ed with a small decrease in rat es
P  price associat ed with a small increase in rat es
Effective Duration and Convexity
•
Steps for using the binomial tree to estimate duration and
convexity:
1. Take yield curve estimated with bootstrapping and value the
MBS (theoretical value), P0, using the calibration approach.
2. Let the yield curve estimated with bootstrapping decrease by
a small amount and then estimate the price of the MBS using
the calibration approach -- P-.
3. Let the yield curve estimated with bootstrapping increase by
a small amount and then estimate the price of the MBS using
the calibration approach -- P+.
4. Calculate effective duration and convexity.
P  P  2(P0 )
P  P
D
; Convexity
2(P0 )y
(P0 )(y) 2
Yield Analysis
• Yield analysis involves calculating the yields on
MBSs or CMO tranches given different prices and
prepayment speed assumptions or alternatively
calculating the values on MBSs or tranches given
different rates and speeds.
Yield Analysis
• For example, suppose an institutional investor is
interested in buying a MBS issue that has a par
value of $100M, WAC = 8, WAM = 355 months,
and a PT rate of 7.5%.
• The value, as well as average life, maturity,
duration, and other characteristics of this security
would depend on the rate the investor requires on
the MBS and the prepayment speed she estimates.
Yield Analysis
•
If the investor’s required return on the MBS is 9% and her
estimate of the PSA speed is 150, then she would value the
MBS issue at $93,702,142.
•
At that rate and speed, the MBS would have an average life
of 9.18 years. Whether a purchase of the MBS issue at
$93,702,142 to yield 9% represents a good investment
depends, in part, on rates for other securities with similar
maturities, durations, and risk, and in part, on how good the
prepayment rate assumption is.
•
For example, if the investor felt that the prepayment rate
should be 100% PSA and her required rate with that level
of prepayment is 9%, then she would price the MBS issue
at $92,732,145 and the average life would be 11.51 years.
Yield Analysis
• In general, for many institutional investors the
decision on whether or not to invest in a particular
MBS or tranche depends on the price the
institution can command.
• For example, based on an expectation of a 100%
PSA, our investor might conclude that a yield of
9% on the MBS would make it a good investment.
In this case, the investor would be willing to offer
no more than $92,732,145 for the MBS issue.
Yield Analysis
•
One common approach used in conducting a yield analysis
is to generate a matrix of different yields by varying the
prices and prepayment speeds.
•
The next exhibit shows the different values for our
illustrative MBS given different required rates and different
prepayment speeds.
•
Using this matrix, an investor could determine, for a given
price and assumed speed, the estimated yield, or determine,
for a given speed and yield, the price. Using this approach,
an investor can also evaluate for each price the average
yield and standard deviation over a range of PSA speeds.
Yield and Vector Analysis
Mortgage Portfolio = $100M, WAC = 8%, WAM = 355 Months, PT Rate = 7.5
Rate/PSA
50
100
150
7%
8%
9%
10%
Value
$106,039,631
$98,251,269
$91,442,890
$85,457,483
Value
$105,043,489
$98,526,830
$92,732,145
$87,554,145
Value
$104,309,207
$98,732,083
$93,702,142
$89,146,871
Average Life
14.95
11.51
9.18
Rate
7%
8%
9%
10%
Vector
Month Range: PSA
1-50: 200
51-150: 250
151-250: 150
251-355: 200
Value
$103,729,227
$98,893,974
$94,465,328
$90,395,704
Vector
Month Range: PSA
1-50: 200
51-150: 300
151-250: 350
251-355: 400
Value
$103,473,139
$98,964,637
$94,794,856
$90,929,474
Vector
Month Range: PSA
1-50: 200
51-150: 150
151-250: 100
251-355: 50
Value
$104,229,758
$98,756,370
93,826,053
89,,364,229
Yield Analysis
• One of the limitations of the above yield analysis
is the assumption that the PSA speed used to
estimate the yield is constant during the life of the
MBS.
• In fact, such an analysis is sometimes referred to
as static yield analysis.
• In practice, prepayment speeds change over the
life of a MBS as interest rates change in the
market.
Vector Analysis
• A more dynamic yield analysis, known as vector
analysis, can be used.
• In applying vector analysis, PSA speeds are
assumed to change over time.
• In the above case, a matrix of values for different
rates can be obtained for different PSA vectors
formed by dividing the total period into a number
of periods with different PSA speeds assumed for
each period.
• A vector analysis example is also shown at the
bottom of the last exhibit.