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FIGURE 4.14 Nominal, forward, and option-adjusted spreads.


However, there can be very different option-related dynamics between
a bundle of mortgages packaged into a single security (called a mortgage-
backed security, or MBS) and a callable bond. Indeed, there are a variety of
structure types between a callable bond and an MBS. The variations can be
explained largely by option-related differences, as shown next.


VARIATIONS IN OPTIONALITY AMONG BOND PRODUCTS
An MBS is comprised of a portfolio of individual mortgages that are pack-
aged together into a single security and sold to investors. The security is a
coupon-bearing instrument, and it has a principal component as well. The
funds used to pay the coupons of an MBS come directly from the monthly
interest payments made by homeowners. The payments made by home-
owners are passed through a servicing agent, who sends along appropriate
payments directly to holders of the MBS. Accordingly, an MBS is sometimes
called a pass-through security (or pass-thru), or an asset-backed security since
its cash flows come from a bundle of assets (namely the home mortgages
that are bundled together). An MBS also is sometimes called a securitized



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Financial Engineering



asset, for the same reason. All else being equal, investors like the idea of a
bond that is physically backed by (supported by) assets that they can ana-
lyze and understand. In contrast with a more generic bond (debenture) that
is backed by an issuer™s overall credit rating or general financial standing,
an asset-backed security provides investors with things they can “touch and
feel””not in a literal sense, but in the sense of bringing some form and def-
inition to what they are buying.4
When homeowners make their monthly mortgage payment, a portion
of that payment goes to paying the interest on the mortgage and a portion
goes to paying the principal. In the early phase of the typically 30-year mort-
gage life, the largest portion of the monthly payment goes toward payment
of interest. A growing portion of the monthly payment goes toward princi-
pal, and in the same way that interest payments are passed along to MBS
holders as coupons, principal payments are passed along to MBS holders as
principal. Herein lies a key difference between a traditional bond and a tra-
ditional pass-thru; the former pays 100 percent of its principal at maturity,
while the latter pays out its principal over the life of the security as it is
received and passed along to investors. Payments of principal and interest
may not always be predictable; homeowners can refinance their mortgages
if they want to, which involves paying down the principal remaining on their
existing mortgage. This act of paying off a loan prior to its natural matu-
rity (even if the purpose is to take on a new loan) is called prepaying, and
prepayments can be attributable to many things, including a sudden decline
in interest rates5 (so that investors find it more cost-effective to obtain a new
lower-cost loan), a natural disaster that destroys homes, changes in personal
situations, and so forth.
Most MBSs are rated triple A. How is this possible unless every home-
owner with a mortgage that is in the bundle has a personal credit rating that
is comparable to a triple-A profile? One way to achieve this is by overcollat-
eralizing (providing more collateralization than a 1:1 ratio of face value of
security relative to underlying asset). The MBS is collateralized (backed by)
mortgages. To overcollateralize an MBS, the originator of the MBS puts in
more mortgages than the face value of the MBS. For example, if originators
want to issue $10 million face amount of MBS that will be sold to investors,
they put more than $10 million face amount of underlying mortgages into the


4
Some larger investors do actively request and analyze detailed data underlying
various asset-backed instruments.
5
This decline in interest rates gives value to the long call option that homeowners
have embedded in their mortgage agreement; the option (or choice) to refinance the
mortgage at a lower rate has economic value that is realized only by refinancing
the existing mortgage to secure new and lower monthly payments.




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136 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT



bundle that comprises the MBS. Accordingly, if some homeowners happen to
default on their mortgages, the excess supply of mortgages in the bundle will
help to cover that event. Another way that MBS products are able to secure
a triple-A rating is by virtue of their being supported by federal agencies. The
three major agencies of the United States involved with supporting mortgages
include Ginnie Mae, Fannie Mae, and Freddie Mac.6 The key purpose of these
governmental organizations is to provide assurance and confidence in the mar-
ket for MBSs and other mortgage products.
Table 4.3 summarizes key differences between an MBS and a callable
bond.
The most dramatic differences between MBSs and callable bonds are that
the options embedded with the former are continuous while the single option
embedded in the latter tends to be discrete, and the multiple options within
an MBS can be triggered by many more variables.
Figure 4.15 shows how an MBS™s cash flows might look; none of the
cash flow boxes is solid because none of them can be relied on with 100
percent certainty. While less-than-100% certainty might be due partly to the
vagaries of what precisely is meant by saying that the federal agencies issu-
ing these debt types are “supported by” the federal government, more of the
uncertainty stems from the embedded optionality. Although it may very well
be unlikely, it is theoretically possible that an investor holding a mortgage-
backed security could receive some portion of a principal payment in one
of the very first cash flows that is paid out. This would happen if a home-


TABLE 4.3 MBS versus Callable Bond Optionality
Mortgage Callable

Callability Continuous Discrete (sometimes continuous)
Call period Immediately Eligible after the passage of some time
Call trigger Level of yields Level of yields, other cost considerations
Homeowner defaults
Homeowner sells property
for any reason
Property is destroyed as
by natural disaster



6
Ginnie Mae pass-thrus are guaranteed directly by the U.S. government regarding
timely payment of interest and principal. Fannie Mae and Freddie Mac pass-thrus
carry the guarantee of their respective agency only; however, both agencies can
borrow from the Treasury, and it is not considered to be likely that the U.S.
government would allow any of these agencies to default.




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Financial Engineering



Cash Flow

+
p2 p4 p6 p8 p 719

p1 p3 p5 p7 p 720
O Time




FIGURE 4.15 MBS cash flows over time.



owner decides or is forced to sell the home almost immediately after pur-
chase and pays off the full principal of the loan. In line with what we would
generally expect, principal payments will likely make their way more mean-
ingfully into the mix of principal-coupon cash flows after some time passes
(or, in the jargon of the marketplace, with some seasoning).
How can probabilities be assigned to the mortgage product™s cash flows
over time? While we can take the view that we adopted for our callable
debenture”that at the start of the game every uncertain cash flow has a
50/50 chance of being paid”this type of evenly split tactic may not be very
practical or realistic for mortgage products. For example, a typical home
mortgage is a 30-year fixed-rate product. This type of product has been
around for some time, and some useful data have been collected to allow
for the evaluation of its cash flows over a variety of interest rate and eco-
nomic environments. In short, various patterns can and do emerge with the
nature of the cash flows. Indeed, a small cottage industry has grown up for
the creation and maintenance of models that attempt to divine insight into
the expected nature of mortgage product cash flows. It is sufficient here
merely to note that no model produces a series of expected cash flows from
year 1 to year 30 with a 50/50 likelihood attached to each and every pay-
out. Happily, this conforms to what we would expect from more of an intu-
itive or common sense approach.
Given the importance of prepayment rates when valuing an MBS, sev-
eral models have been developed to forecast prepayment patterns. Clearly,
investors with a superior prepayment model are better equipped to identify
fair market value.
In an attempt to impose a homogeneity across prepayment assumptions,
certain market conventions have been adopted. These conventions facilitate
trades in MBSs since respective buyers and sellers know exactly what
assumptions are being used to value various securities.



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138 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT



One commonly used method to proxy prepayment speeds is the constant
prepayment rate (CPR). A CPR is the ratio of the amount of mortgages pre-
paid in a given period to the total amount of mortgages in the pool at the
beginning of the period. That is, the CPR is the percentage of the principal
outstanding at the beginning of a period that will prepay over the follow-
ing period. For example, if the CPR for a given security in a particular month
is 10.5, then the annualized percentage of principal outstanding at the begin-
ning of the month that will repay during the month is 10.5 percent. As the
name implies, CPR assumes that prepayment rates are constant over the life
of the MBS.
To move beyond the rather limiting assumption imposed by a CPR”
that prepayments are made at a constant rate over the life of an MBS”the
industry proposed an alternative measure, the Public Securities Association
(PSA) model. The PSA model posits that any given MBS will prepay at an
annualized rate of 0.2 percent in the first month that an MBS is outstand-
ing, and prepayments will increase by 0.2 percent per month until month
30. After month 30, it is assumed that prepayments occur at a rate of 6 per-
cent per year for all succeeding months.
Generally speaking, the PSA model provides a good description of pre-
payment patterns for the first several years in the life of an MBS and has
proven to be a standard for comparing various MBSs. Figure 4.16 shows
theoretical principal and coupon cash flows for a 9 percent Ginnie Mae MBS
at 100 percent PSA. When an MBS is quoted at 100 percent PSA, this means
that prepayment assumptions are right in line with the PSA model, above.
An MBS quoted at 200 percent PSA assumes prepayment speeds that are
twice the PSA model, and an MBS quoted at 50 percent PSA assumes a
slower prepayment pattern.


9% 30-year Ginnie Mae, 100% PSA
$1,000s

140 Interest
120 Principal


100
80
40
20
Month
60 120 180 240 300 360

FIGURE 4.16 The relationship between pay-down of interest and principal for a pass-
thru.




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139
Financial Engineering



Another important concept linked to MBS is that of average life. As
depicted in Figure 4.17, average life is the weighted average time to the return
of a dollar of principal. It is often used as a measure of the investment life
of an MBS and is typically compared against a Treasury with a final matu-
rity that approximates the average life of the MBS. In short, it is a way to
help fence in the nature of MBS cash flows to allow for some comparabil-
ity with non-pass-thru type structures.
Since the principal or face value of an MBS is paid out over the life of
the MBS and not in one lump sum at maturity, this is reflected in the price
formula provided below. Accordingly, as shown, the MBS price formula has
an F variable alongside every C variable. Further, every C and every F has
its own unique probability value.

C p1&F p2 C p3&F p4
11 11
Price
Y>22 1 Y>22 2
C p5&F p6
11
$1,000
...
Y>22 3

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