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where
p1 probability-weighted first coupon
p2 probability-weighted first receipt of principal
p3 probability-weighted second coupon
p4 probability-weighted second receipt of principal,
. . . and so forth.
Average life
(years)

25


20


15


10


5

10 20 30 40 50 60 70

Prepayment rate (%)

FIGURE 4.17 Average life vs. prepayment rate.



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“Probability-weighted coupon” means the statistical likelihood of receiv-
ing a full coupon payment (equivalent to 100 percent of F times C). As prin-
cipal is paid down from par, the reference amount of coupon payment
declines as well (so that when principal is fully paid down, a coupon pay-
ment is equal to zero percent of F times C, or zero).
“Probability-weighted principal” means the statistical likelihood of
receiving some portion of principal™s payment.
As is the case with a callable debenture, the initial price of an MBS is
par, and Y C. However, unlike our callable debenture, there is no formal
lockout period with an MBS. While we might informally postulate that prob-
ability values for F should be quite small in the early stages of an MBS™s life
(where maturities can run as long as 15 or 30 years), this is merely an edu-
cated guess. The same would be true for postulating that probability values
for C should be quite large in the early stages of an MBS™s life. Because an
MBS is comprised of an entire portfolio of short call options (with each one
linked to an individual mortgage), in contrast with the single short option
embedded in a callable debenture, the modeling process for C and F is more
complex; hence the existence and application of simplifying benchmark mod-
els, as with the CPR approach.
At this stage we have pretty much defined the two extremes of option-
ality with fixed income products in the U.S. marketplace. However, there
are gradations of product within these two extremes. For example, there are
PACs, or planned amortization class securities.
Much like a Thanksgiving turkey, an MBS can be carved up in a vari-
ety of ways. At Thanksgiving, some people like drumsticks and others pre-
fer the thigh or breast. With bonds, some people like predictable cash flows
while others like a higher yield that comes with products that behave in less
predictable ways. To satisfy a variety of investor appetites, MBS pass-thrus
can be sliced in a variety of ways. For example, classes of MBS can be cre-
ated. Investors holding a Class A security might be given assurances that they
will be given cash distributions that conform more to a debenture than a
pass-thru. Investors in a Class B security would have slightly weaker assur-
ances, those in a Class C security would have even weaker assurances, and
so forth. As a trade-off to these levels of assurances, the class yield levels
would be progressively higher.
A PAC is a prime example of a security type created from a pool of mort-
gages. What happens is that the cash flows of an MBS pool are combined
such that separate bundles of securities are created. What essentially distin-
guishes one bundle from another is the priority given for one bundle to be
assured of receiving full and timely cash flows versus another bundle.
For simplicity, let us assume a scenario where a pool of mortgages is
assembled so as to create three tranches of cash flow types. In tranche 1,
investors would be assured of being first in line to receive coupon cash flows



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141
Financial Engineering



generated by the underlying mortgages. In tranche 2, investors would be sec-
ond in line to receive coupon cash flows generated by the underlying mort-
gages. If homeowners with mortgages in this pool decide to pay off their
mortgage for whatever reason, then over time tranche 2 investors would not
expect to receive the same complete flow of payouts relative to tranche 1
investors. If only for this reason, the tranche 2 investors should not expect
to pay the same up-front price for their investment relative to what is paid
by tranche 1 investors. They should pay less. Why? Because tranche 2
investors do not enjoy the same peace of mind as tranche 1 investors of being
kept whole (or at least “more whole”) over the investment horizon. And
finally, we have tranche 3, which can be thought of as a “residual” or “clean-
up” tranche. The tranche 3 investors would stand last in line to receive cash
flows, only after tranche 1 and tranche 2 investors were paid. And consis-
tent with the logic presented above for tranche 2, tranche 3 investors should
not expect to pay the same up-front price for their investment as tranche 1
or 2 investors; they should pay less.
Note that tranche 1 investors are not by any means guaranteed of receiv-
ing all cash flows in a complete and timely matter; they only are the first in
line as laying priority to complete and timely cash flows. In the unlikely event
that every mortgage within the pool were to be paid off at precisely the same
time, then each of the three tranches would simply cease to exist. This com-
ment helps to reinforce the idea that tranche creation does not create new cash
flows where none existed previously; tranche creation simply reallocates
existing cash flows in such a way that at one end of a continuum is a security
type that at least initially looks and feels like a more typical bond while at the
other end is a security type that exhibits a price volatility in keeping with its
more uncertain place in the pecking order of all-important cash flow receipts.
This illustration is a fairly simplified version of the many different ways
in which products can be created out of mortgage pools. Generally speak-
ing, PAC-type products are consistent with the tranche 1 scenario presented.
Readers can refer to a variety of texts to explore this kind of product cre-
ation methodology in considerable detail. From PACs to TACs to A, B, C,
and Z tranches (and much, much more), there is much to keep an avid mort-
gage investor occupied.
Figure 4.18 applies the PAC discussion to our cash flow diagram.
Notice that the cash flow boxes in the early part of the PAC™s life are
drawn in with solid lines. PACs typically come with preannounced lockout
periods. Here, lockout refers to that period of time when the PAC is pro-
tected from not receiving complete and timely cash flow payments owing to
option-related phenomena. The term of lockouts varies, though is generally
5 to 10 years. Again, the PAC is protected in this lockout period because it
stands first in line to receive cash flows out of the mortgage pool. Many times
a PAC is specified as being protected only within certain bandwidths of



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The p™s represent probability
values that are assigned to each
cash flow after purchase.

Post lockout

Cash Flow

+ Lockout period

p1 p2

O Time



FIGURE 4.18 Applying the PAC discussion to the cash flow diagram.


option-related activity. Typically the activity of homeowners paying off their
mortgages is referred to as prepayment speed (or speed). Moreover, a con-
vention exists for how these speeds are quoted. Accordingly, often PAC band-
widths define an upper and lower bound within which speeds may vary
without having any detrimental effect on the PAC™s cash flows. The wider
the bandwidth, the pricier the PAC compared to PACs with narrower bands.
Once a particular PAC has experienced a prepayment speed that falls out-
side of its band, it is referred to as a “busted PAC.” A PAC also is “busted”
once its lockout period has passed. Not surprisingly, once “busted,” a PAC™s
value tends to cheapen.
As perhaps the next logical step from a PAC, we have DUS, or delegated
underwriting and servicing security. In brief, a DUS carries significant pre-
payment penalties, so borrowers do not have a great incentive to prepay their
loans. Accordingly, a DUS can be thought of as having significant lockout
protection.
The formula for a PAC or DUS or a variety of other products created
from pass-thru might very well look like our last price formula, and it is
repeated below. What would clearly differ, however, are the values we insert
for probability. While large bond fund investors might perform a variety of
complex analyses to calibrate precise probability values across cash flows,
other investors might simply observe whether respective yield levels appear
to be in line with one another. That is, we would expect a 10-noncall-five
to trade at a yield below a 10-year DUS, a 10-year DUS to trade at a yield
below a 10-year PAC with a lockout of five years, and so forth.




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143
Financial Engineering



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

Figure 4.19 summarizes the yield relationship to the different callable
bond structures presented in this section. Each successive layer represents a
different and higher-yielding callable product.
For another perspective on the relationships among products, cash
flows, and credit, consider Figure 4.20, which plots the price volatility of a
triple-A-rated pass-thru against four 10-year final maturity bonds. One of the
bonds is a bullet, while the other three are different types of callables. Each




Mortgage-Backed Security
Prepayment penalties are comparable with PACs, but there are no
bands to limit exposure to changes in prepayment activity, and these
uncertain changes contribute to the uncertainty in timing of both
coupon and principal payments.


Planned Amortization Class Security
Prepayment penalties are not as severe as with DUS, and
although there are bands intended to limit exposure to changes
in prepayment activity, these changes are nonetheless uncertain
Layers of increasing
and thus contribute to the uncertainty in timing of both coupon
option-related risks
and principal payments.
(on top of credit risk
and market risk)
DUS
Although relatively severe penalties exist for early
prepayments, there is uncertainty associated with the
timing of both coupon and principal payments.



Callable Non-Treasury Coupon-Bearing Bond
After an initial lockout period, there is uncertainty
of timing of final coupon and principal.



Non-Treasury Coupon-Bearing Bond Credit risk



Coupon-Bearing Treasury Bond Market risk



FIGURE 4.19 Summary of the yield relationship to callable bond structures.



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Price volatility
1 Deep in-the-money
2 At-the-money The intersection of
3 Deep out-of-the-money the price volatility of
4 10-year bullet bond a triple-A rated 10-
noncall-2 and a
double-A rated 10-
year bullet bond.



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