Financial Instruments Toolbox™ supports collateralized mortgage obligations (CMOs) to provide investors with a greater range of risk and return characteristics than mortgage-backed securities (MBS). In contrast to an MBS, which simply redirects principal and interest cash flows to investors on a pro rata basis, a CMO structures cash flows to different tranches, or slices, to create securities that are better tailored to specific investors.

For example, banks might be primarily concerned with *extension
risk*, or the risk that their investment lengthens in time
due to increasing interest rates, given that they typically have short-term
deposits as liabilities. Insurance companies and pension funds might
be concerned primarily with *contraction risk*,
or the risk that their investment will pay off too soon, with liabilities
that have much longer lives. A CMO structure addresses the interest-rate
risk of extension or contraction with a blend of short-term and long-term
CMO securities, called tranches.

Prepayment risk is the risk that the term of the security varies according to differing rates of repayment of principal by borrowers (repayments from refinancings, sales, curtailments, or foreclosures). In a CMO, you can structure the principal (and associated coupon) stream from the underlying mortgage pool collateral to allocate prepayment risk. If principal is prepaid faster than expected (for example, if mortgage rates fall and borrowers refinance), then the overall term of the mortgage pool collateral shortens.

You cannot remove prepayment risk, but you can reallocate it among CMO tranches so that some tranches have some protection against this risk, and other tranches will absorb more of this risk. To facilitate this allocation of prepayment risk, CMOs are structured such that prepayments are allocated among tranches using a fixed set of rules. The most common schemes for prepayment tranching are:

Sequential tranching, with or without, Z-bond tranching

Schedule bond tranching

Planned amortization class (PAC) bonds

Target amortization class (TAC) bonds

Financial Instruments Toolbox supports these schemes for prepayment tranching for CMOs and tools for pricing and scheduling cash flows between the tranches, as well as analyzing the price and yield for CMOs. Financial Instruments Toolbox functionality for CMOs does not model credit risk. Therefore, this functionality is most appropriate for CMOs where credit risk is not an issue (for example, agency CMOs where the underlying mortgage pool collateral is insured for default by the agency Government-Sponsored Enterprises (GSEs), such as Fannie Mae and Freddie Mac).

All available principal and interest payments go to the first sequential tranche, until its balance decrements to zero, then to the second, and so on. For example, consider the following example where all principal and interest from the underlying mortgage pool is repaid on tranche A first, then tranche B, then tranche C. Interest is paid on each tranche as long as the principal for the tranche has not been retired.

The Z-bond, also called an accural bond, is a type of interest and principal pay rule. The Z-bond tranche supports other sequential pay tranches by not receiving an interest payment. The interest payment that would have accrued to the Z-bond tranche pays off the principal of other bonds, and the principal of the Z-bond tranche increases. The Z-bond tranche starts receiving interest and principal payments only after the other tranches in the CMO have been fully paid. The Z-bond tranche is used in a sequential-pay structure to accelerate the principal repayments of the sequential-pay bonds.

A Z-bond differs from other CMO instruments because it is not tranching principal but interest. The Z-bond receives no cash flows until all other securities have been paid off. In the interim, the interest that is owed to the Z-bond is accrued to its principal. The following chart demonstrates the difference between a Z-bond and a normal sequential pay tranche. The C tranche pays off sooner with the Z-bond, because the interest cash flows to the Z-bond are being used to pay down the principal of the C tranche.

For comparison, the following graphic is the same sequential CMO with no Z-bond.

Planned amortization class (PAC) bonds help reduce the effects of prepayment risk. They are designed to produce more stable cash flows by redirecting prepayments from the underlying mortgage collateral to other classes (tranches) called companion or support classes. PAC bonds have a principal payment rate over a predetermined period of time. The PAC bond payment schedule is determined by two different prepayment rates, which together form a band (also called a collar). Early in the life of the CMO, the prepayment at the lower PSA yields a lower prepayment. Later in its life, the principal in the higher PSA declines enough that it yields a lower prepayment. The PAC tranche receives whichever rate is lower, so it will change prepayment at one PSA for the first part of its life, then switch to the other rate. The ability to stay on this schedule is maintained by a support bond, which absorbs excess prepayments, and receives fewer prepayments to prevent extension of average life.

However, the PAC is only protected from extension to the amount that prepayments are made on the underlying MBSs. If there is a sustained period of fast prepayments, then that might completely eliminate a PAC bond's outstanding support class. When the principal of the associated PAC bond is exhausted, the CMO is called a "busted PAC", or "busted collar". Alternatively, in times of slow prepayments, amortization of the support bonds is delayed if there is not enough principal for the currently paying PAC bond. This extends the average life of the class.

A PAC bond protects against both extension and contraction risk by:

Specifying a schedule of principal payments for the PAC bond

Including support tranches that are allocated prepayments inside a specified prepayment band

PAC bonds typically specify a band expressed using the PSA model. A PAC bond with a range of 100–250% has this principal schedule.

The principal repayment schedule is the minimum principal payment as Region 1 shows. This is the principal payment schedule as long as the actual prepayment stays within the prepayment band of 100–250% PSA.

For example, for different prepayment speeds of 125%, 175%, and 225% PSA, the actual principal payments are shown in the following graphs. At higher prepayment speeds, the support tranche is allocated principal earlier while the principal timing for the other tranches remains constant.

Target amortization class (TAC) bonds are similar to PAC bonds, but they do not provide protection against extension of average life. Create the schedule of principal payments by using just a single PSA. TAC bonds pay a "targeted" principal payment schedule at a single, constant prepayment speed. As long as the underlying mortgage collateral does not prepay at a rate slower than this speed, the TAC bond payment schedule is met. TAC bonds can protect against increasing prepayments and early retirement of the TAC bond investment. If the principal cash flow from the mortgage collateral exceeds the TAC schedule, the excess is allocated to TAC companion (support) classes. Alternatively, if prepayments fall below the speed necessary to maintain the TAC schedule, the weighted average life of the TAC is extended. The TAC bond does not protect against low prepayment rates.

For example, here is a TAC structure rated for 125%, 175%, and 450% PSA.

For prepayments below 175% PSA, the TAC bond extends like a normal sequential pay CMO. TAC bonds are appealing because they offer higher yields than comparable PAC bonds. The unaddressed risk from low prepyament rates generally does not concern investors as much as risk from high prepayment rates.

In general, the CMO workflow is:

Calculate underlying mortgage cash flows.

Define CMO tranches

If using a PAC or TAC CMO, calculate the principal schedule.

Calculate cash flows for each tranche.

Analyze the CMO by computing price, yield, spread of CMO cash flows.

Underlying mortgage pool pass-through cash flows are calculated
by the existing function `mbspassthrough`

.
The CMO cash flow functions require the principal payments (including
prepayments) calculated from existing functions `mbspassthrough`

or `mbscfamounts`

.

principal = 10000000; coupon = 0.06; terms = 360; psa = 150; [principal_balance, monthly_payments, sched_principal_payments,... interest_payments, prepayments] = mbspassthrough(principal,... coupon, terms, terms, psa, []); principal_payments = sched_principal_payments.' + prepayments.';

After determining principal payments for the underlying mortgage
collateral, you can generate cash flows for a sequential CMO, with
or without a Z-bond, by using `cmoseqcf`

.
For a PAC or TAC CMO, the cash flows are generated using `cmoschedcf`

Define CMO tranche; for example, define a CMO with two tranches:

TranchePrincipals = [500000; 500000]; TrancheCoupons = [0.06; 0.06];

Calculate the PAC/TAC principal balance schedule based on a
band of PSA speeds. For scheduled CMOs (PAC/TAC), the CMO cash flow
functions additionally take in the principal balance schedule calculated
by the CMO schedule function `cmosched`

.

```
speed = [100 300];
[balanceSchedule, initialBalance] = cmosched(principal, coupon,...
terms, terms, speed, TranchePrincipals(1));
```

You can reuse the output from the cash flow generation functions
to further divide the cash flows into tranches. For example, the output
from `cmoschedcf`

for a PAC tranche
can be divided into sequential tranches by passing the principal cash
flows of the PAC tranche into the `cmoschedcf`

function.
The output of the CMO cash flow functions are the principal and interest
cash flows, as well as the principal balance.

```
[principal_balances, principal_cashflows, interest_cashflows] = cmoschedcf(principal_payments,...
TranchePrincipals, TrancheCoupons, balanceSchedule);
```

The outputs from the CMO functions (`cmoseqcf`

and `cmoschedcf`

) are cash flows. The functions
used to analyze a CMO are based on these cash flows. To that end,
you can use `cfbyzero`

, `cfspread`

, `cfyield`

,
and `cfprice`

to compute prices,
yield, and spreads for the CMO cash flows. In addition, using the
following, you can calculate a weighted average life (WAL) for each
tranche in the CMO:

$$WAL={\displaystyle \sum _{i=1}^{n}\frac{{P}_{i}}{P}}{t}_{i}$$

where:

*P* is the total principal.

*P _{i}* is the principal
repayment of the coupon

$$\frac{{P}_{i}}{P}$$ is the fraction of the principal
repaid in coupon *i*.

*t _{i}* is the time in
years from the start to coupon

`cmosched`

| `cmoschedcf`

| `cmoseqcf`

| `mbscfamounts`

| `mbspassthrough`

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