In this piece, we will seek to describe the nature of the calculations and avoid wading into the actual mathematics. We will outline what factors are included and excluded from the calculations, and address the different conventions for quoting speeds. We’ll close with a quick discussion of the relationships between historical and projected prepayment speeds, and address the different options traders can utilize to calculate yields, performance, and risks.

**Measuring Prepayment Speeds**

The standard measure of prepayment speeds is the “constant prepayment rate” or CPR. The most commonly used CPRs are 1-month CPRs (or CPR1 in Eikon) and are based on a single month’s experience. (CPRs can also be generated for 3-, 6-, and 12-month horizons, as well as over the life of a security.) For all horizons, CPRs represent an annualized rate of prepayment speed or, put differently, are generated by annualizing the periodic calculation.

As we noted in the previous article, prepayment speeds are calculated based on the amount of unscheduled principal returned to investors, removing scheduled principal payments from the computations. The first step in calculating the one-month speed for a pool is to obtain the following information:

- the initial balance (i.e., the balance as of the prior record date)
- the ending balance; and
- the amount of scheduled principal received.

The first step is to calculate the total principal received by subtracting the ending balance from the beginning one; the total prepaid principal is then computed by subtracting the scheduled principal paid. The monthly prepayment speed, referred to as the Single Monthly Mortality (borrowing an insurance term) or SMM is then calculated by dividing the prepaid principal by the starting balance MINUS the scheduled principal payment. Below is a simple example of calculating an SMM for a pool with an original balance of $10,000,000.

- Month 9 balance: $9,719,777
- Month 10 balance: $9,672,195
- Total principal paid: $47,582 (#2 minus #1)
- Scheduled principal paid: $14,622 (most often obtained from the servicer records)
- Prepaid principal: $32,959 (#3 minus #4)
- SMM: 0.34% (#5 divided by [#1 minus #4]).

The SMM is converted to a CPR (i.e., annualized) as follows:

*CPR = 1-(1-SMM) 12 *

In the example, an SMM of 0.34% equates to an annualized rate of 4.0% CPR. (Remember to treat the SMM as a percentage.) A 3-month CPR (i.e., CPR3 in Eikon) would be calculated similarly, but you would first need to calculate a quarterly mortality (using the starting and ending balances for the 3-month period, as well as total amortizations paid over the quarter) and then annualize it with the above formula having the power of 4 (rather than 12).

**Prepayment Ramps and the “PSA Model”**

As noted, the “C” in CPR stands for “constant,” which implies that the prepayment speed of the security in question is constant or, more accurately, that the timing of prepayments doesn’t adhere to defined patterns. For example, calculating a pool’s yield at 8% CPR means that the pool is projected to prepay at a constant 8% CPR for life.

However, both logic and empirical research suggest that prepayments follow a fairly predictable pattern based on the age of the loan (or more to the point, the time since origination). Borrowers that have just taken out a loan typically don’t move immediately after closing, nor are they inclined to instantly refinance for anything short of a major rate reduction. As time elapses, however, borrowers are increasingly inclined to either move or undergo the loan application process again, and that behavior is represented by a fairly linear increase in prepayment speeds. At some point in time, however, prepayment speeds tend to flatten out and remain fairly constant thereafter.

These insights led to the creation of the “PSA Model” from the precursor organization of today’s SIFMA. (PSA stood for Public Securities Association; it was also called the Bond Market Association for a number of years.) The PSA Model divides a pool’s life into two separate periods. The initial period assumes that a pool’s prepayment speed (quoted in CPR1s) increases linearly over the first thirty months; this period is also referred to as “the ramp,” and pools that are “on the ramp” are less than 30 months old. After month 30, the pool is considered to be “off the ramp” and the model assumes that the speed remains constant for the pool’s remaining life.

The base PSA Model (or, more accurately, 100% of the model or 100% PSA) assumes the following:

- a speed in month 1 of 0.2% CPR;
- the prepayment speed increases 0.2% CPR per month until reaching 6.0% CPR in month 30; and
- the pool prepays at 6.0% CPR for its remaining life.

PSAs are quoted as percentages of the base model. For example, 250% PSA implies that for every month, the base model’s CPR is multiple by 2.5. (Since 100% PSA implies a 1.0% CPR in month 5, 250% PSA implies 2.5% CPR for that month.) The chart below illustrates prepayment speeds at different percentages of the PSA model for a newly-issued pool. (Note that while the PSA model is widely use, other prepayment conventions are also utilized that incorporate the concept of a prepayment ramp; most of these are deal-specific and defined within the transaction’s offering documents.)

There are a few things to note when considering the PSA Model. The model was developed in the early 1980s when both the mortgage and bond markets were very different. There was very little media attention given to mortgage rates at the time; in addition, the concept of the “mortgage banker” (i.e., an entity whose business model is based on originating and distributing loans into the capital markets) was fairly new. The subsequent growth and maturation of the mortgage market shortened the actual duration of the ramping behavior from 30 months to somewhere between 12 and 20 months. In addition, the interplay in the PSA Model between age and CPR distorts the model’s output, especially during periods of very fast speeds (i.e., “refi waves”) when borrowers seek to refinance their loans more quickly and aggressively. For example, in the early 1990s traders began pricing premium securities to what they thought were very fast PSAs. However, they didn’t account for the age effect embedded into PSAs; for example, pricing a 30 month old pool at 1,000% PSA (which equates to 60% CPR) is very different from valuing a 5 month old pool at the same 1,000% PSA (or 10% CPR). This led to traders overpricing new securities that were prepaying must faster than indicated by the PSA model. Diagram 2 illustrates this effect, showing one-month CPRs over time for pools with different initial ages at 250% PSA. In the diagram, 250% PSA for a 24-month old pool equates to 15% CPR within a few months; over the same time, the CPRs on a 0-WALA pool only ramp up to around 2% CPR over the same period.