Transcript of "Securitization and Mortgage-Backed Securities "
1.
Chapter 11
Securitization and
Mortgage-Backed Securities
2.
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 pass-
through that he sells to investors.
3.
Securitization
• The securitization process often involves a third-
party 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.
4.
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.
5.
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.
6.
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 pass-
throughs, 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.
7.
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.
8.
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.
9.
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.
10.
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 pass-
through.
11.
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.
12.
Agency Pass-Throughs
Federal National Mortgage Association, FNMA
• FNMA (Fannie Mae) offers several types of pass-
throughs, 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.
13.
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.
14.
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.
15.
Conventional Pass-Throughs
• The larger issuers of conventional MBSs
include:
– Citicorp Housing
– Countrywide
– Prudential Home
– Ryland/Saxon
– G.E. Capital Mortgage
16.
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)
17.
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.
18.
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.
19.
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.
20.
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
21.
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).
22.
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.
23.
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
24.
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.
25.
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.
26.
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).
28.
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
CPR .06 , if t 30,
30
CPR .06, if t 30
29.
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
30.
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).
31.
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.
32.
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.
33.
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%.
34.
Cash Flow from a Mortgage Portfolio
• The cash flow from a portfolio of
mortgages consists
– Interest payments
– Scheduled principal
– Prepaid principal
35.
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.
36.
Cash Flow from a Mortgage Portfolio: Example 1
• For the first month, the portfolio would
generate an aggregate mortgage payment
of $733,765:
M
p
$100,000,000 F0 (1 (R A
/ 12)) t
p $733,765 t 1
11 /(1(.08 / 12))
360
11 /(1(R A / 12)) M
.08 / 12 F0 p
R A / 12
F0
p
11 /(1( R A / 12)) M
R A / 12
37.
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
Interest F0
12 $100,000,000 $666,667
12
Scheduled Pr incipal Payment p Interest $733,765 $666,667 $67,098
38.
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
39.
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
40.
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
41.
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
11 /(1(.08 / 12))
359
.08 / 12
.08
Interest ($99,916,231) $666,108
12
Scheduled principal $733,642 $666,108 $67,534
42.
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
43.
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
44.
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.
46.
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
48.
Cash Flow from a MBS: Example 2
Notes:
• The first month's CPR for the MBS issue reflects a five-
month 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.
49.
Cash Flow from a MBS: Example 2
• First Month’s Payment:
$100M
p $736,268
11 /(1(.08 / 12))
355
.08 / 12
WAC
50.
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
Interest F0
12 $100,000,000 $625,000
12
Scheduled Pr incipal Payment p Interest $736,268 [(.08 / 12)($100,000,000)]
$69,601
51.
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
52.
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 difference s
53.
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 difference s
54.
Cash Flow from a MBS: Example 2
• Second Month: Payment, Interest, Scheduled Principal,
Prepaid Principal, and Cash flow:
$99,779,252
p $735,154
11 /(1(.08 / 12))354
.08 / 12
RA .075
Interest F0
12 $99,779,252 $623,620
12
Scheduled Pr incipal Payment p Interest $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 difference s
55.
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.
56.
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.
57.
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
pf t
F0
58.
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 (pf t )F0
(.92)($100M ) $92M
Market Value (.9550)($92M ) $87.86M
59.
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
60.
Price Quotes
• The full market value would be $88.32M:
Full Mkt Value $87.86M $0.46M
$88.32M
61.
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
62.
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
CFt
VMBS
t 1 (1 R ) t
VMBS f (CFt , R )
63.
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 )
64.
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.
65.
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.
66.
Extension Risk
• Rate Decrease
like most bonds
if R lower discount rate VM
if R Increases prepayment Earlier CFs VM or
67.
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.
68.
Extension Risk
• Rate Increase
like most bonds
if R greater discount rate VM
if R Decreases prepayment Earlier CFs VM or
69.
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 principal received at t
Average Life t
12 t 1
total principal
70.
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
71.
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.
72.
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
73.
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
74.
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.
75.
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
76.
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.
77.
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
78.
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.
80.
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.
81.
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.
82.
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.
83.
Sequential-Pay Tranches
Features of Sequential-Pay CMOs
• Maturity:
– 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)
• Window: The period between the beginning and ending principal
payment is referred to as the principal pay-down window:
– 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
84.
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
85.
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 Collateral Tranche A Tranche B Tranche C
50 14.95 7.53 19.4 26.81
100 11.51 4.92 14.18 23.99
150 9.18 3.69 10.71 9.18
200 7.55 3.01 8.51 17.46
300 5.5 2.26 6.03 12.82
86.
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.
87.
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
88.
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.
89.
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.
91.
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.
92.
Accrual Tranche
Tranche Window Average Life
A 69 Months 3.06 Years
B 54 Months 8.23 Years
93.
Floating-Rate Tranche
• In order to attract investors who prefer variable
rate securities, CMO issuers often create floating-
rate 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.
94.
Floating-Rate Tranche
• Example: Sequential-pay CMO with a floating
and inverse floating tranches
Tranche Par Value PT Rate
A $50M 7.5%
FR $22.5M LIBOR +50BP
IFR $7.5M 28.3 – 3 LIBOR
Z $20M 7.5%
Total $100 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.
95.
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%.
96.
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%
97.
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.
98.
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.
99.
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
101.
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%.
102.
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 Notional principal $10,000,000
.075
$30,000,000)(.075 .065)
B' s Notional principal $4,000,000
.075
($20,000,000)(.075 .07)
Z' s Notional principal $1,333,333
.075
Total Notional Pr incipal $15,333,333
103.
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)
104.
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.
105.
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.
106.
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
107.
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.
109.
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.
110.
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.
111.
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%.
112.
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.
114.
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.
115.
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.
116.
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.
117.
Stripped MBS
• Stripped MBSs consist of two classes:
1. Principal-only (PO) class that receives only the
principal from the underlying mortgages.
3. Interest-only (IO) class that receives just the
interest.
118.
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.
119.
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 return VPO
R
(2) lower discount rate VPO
120.
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 return VPO
R
(2) greater discount rate VPO
121.
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
122.
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.
123.
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))
124.
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 return VIO
R
(2) greater discount rate VIO
VIO
If 1st effect 2nd effect, then 0
R
125.
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
127.
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 PSA Value of Value of Value of
Rate PO IO Collateral
8% 150 $54,228,764 $47,426,196 $101,654,960
8.50% 125 $49,336,738 $49,513,363 $98,850,101
9.00% 100 $44,044,300 $51,795,188 $95,799,488
128.
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 pass-
throughs and many have tranches.
129.
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.
130.
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.
131.
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.
132.
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 principal-
amortization period, all current and
accumulated principal payments are paid.
133.
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.
134.
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.”
135.
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.”
137.
Monte Carlo Simulation
• Objective: Determine the MBS’s theoretical value
• Steps:
3. Simulation of interest rates: Use a binomial interest rate tree
to generate different paths for spot rates and refinancing
rates.
5. Estimate the cash flows of a mortgage portfolio, MBS, or
tranche for each path given a specified prepayment model
based on the spot rates
7. Determine the present values of each path.
9. Calculate the average value – theoretical value.
138.
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:
S 0 6% R 0 8%
ref
u 1.1 u 1 .1
d .9091 1 / 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.
139.
Step 1: Binomial Tree for Spot
and Refinancing Rates
Su u u 7.9860
u = 1.10, d = .9091
R ref u 10.648
uu
Su u 7.260
R ref 9.680
uu
Su 6.600
Su u d 6.600
R ref
8.800
u R refd 8.800
uu
S0 6.00% Su d 6.00
ref
R 0 8.00% R ref 8.00
ud
Sd 5.4546 Su d d 5.4546
R d 7.2728
ref
R ref d 7.2728
ud
Sd d 4.9588
R d d 6.6117
ref
Sd d d 4.5080
R d d d 6.0107
ref
141.
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.
142.
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
143.
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.
144.
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 Cash
Principal Flow
1 $1,000,000 $149,029 $80,000 $69,029 $149,029
2 $930,971 $149,029 $74,478 $74,552 $149,029
3 $856,419 $149,029 $68,513 $80,516 $149,029
4 $775,903 $149,029 $62,072 $86,957 $837,975
Balloon Balance( yr4) Sch.prin( yr4)
$775,903$86,957 $688,946
CF4 Balloon p
$688,946$149,029 $837,975
CF4 Balance( yr4) Interest
$775,903$62,072 $837,975
145.
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.
146.
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).
3. The CPR will exceed 5% if the rate on the mortgage pool is at
a premium.
5. The CPR will increase within certain ranges as the premium
increases.
7. The relationship between the CPRs and the range of rates is
the same in each period; that is, there is no seasoning factor.
147.
Step 2: Estimating Cash Flows
Range CPR
X = WAC – Rref
X 0 5%
0.0% < X 0.5% 10%
0.5% < X 1.0% 20%
1.0% < X 1.5% 30%
40%
1.5% < X 2.0%
50%
2.0% < X 2.5% 60%
2.5% < X 3.0% 70%
X > 3.0%
148.
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).
151.
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.
152.
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
1 (1 /(1.08)10
.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 difference s
153.
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
1 (1 /(1.08) 9
.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 difference s
154.
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
1 (1 /(1.08) 8
.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 difference s
155.
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 difference s
156.
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.
157.
Step 3: Valuing Each Path
• Like any bond, a MBS or CMO tranche should be valued by
discounting the cash flows by the appropriate risk-adjusted
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).
158.
Step 3: Valuing Each Path
• If we assume no default risk, then the risk-
adjusted spot rate can be defined as
z t St k t
where: k = OAS
159.
Step 3: Valuing Each Path
• The value of each path can be defined as
T
CFM CF1 CF2 CF3 CFT
V i
Path
M 1 (1 z M ) M
1 z1 (1 z 2 ) (1 z 3 )
2 3
(1 z T ) T
where:
• i = ith path
• zM = spot rate on bond with M-year maturity
• T = maturity of the MBS
160.
Step 3: Valuing Each Path
• For this example, assume the option-
adjusted spread is 2% greater than the one-
year, risk-free spot rates.
161.
Step 3: Valuing Each Path
Binomial Tree for
9.9860%
Discount Rates
9.26%
8.6% 8.6%
8% 8%
7.4546% 7.4546%
6.9588%
6.5080%
Z10 Z11 Z12 Z13
162.
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 z 10 )(1 z 11 ) (1 z 1, M 1 )
1/ M
1
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