PSA speed is the concept in the MBS prepayment model developed originally by PSA. It's widely used in MBS industry.

The idea is that CPR grows linearly every month until 30th month, when it becomes constant. I checked several books, Googled and read the requirements in one of our systems and found out that all sources describe PSA model as in this Wiki article. The formula is:

[1] CPR = min(age,30) * 0.2% * PSA / 100

I found this formula (explicitely or implicetely) in every source on MBS prepayments. You can also find a graph corresponding to the formula.

The issue is that if we use it as it's shown, then for PSA speeds higher than 10000/6 ~ 1666.67, CPR becomes greater than 100%. CPR is an annualized prepayment rate. If it's higher than 100%, then we can't compute SMM (single month mortaility), because the formula for SMM is:

[2] SMM = 100 * (1 - (1 - CPR/100))^1/12

If CPR is greater than 100%, then SMM becomes a complex number. It doesn't make any sense.

I think that the matter is that when PSA, then Public Securities Association, developed the prepayment model it meant to say:

[3] CPR = min(100%, min(age,30) * 0.2% * PSA / 100) ,

i.e. the same formula as [1], but capped at 100%. I couldn't find a confirmation to this statement, but it's the only logical conclusion. I'm trying to get Standard Formulas book by Bond Market Association. Hopefully, it will confirm my theory.

Bloomberg's service returns PSA speeds greater than 1666 for many securities. If we used the first formula [1], then we'd have problems with CPR greater than 100% and inability to compute SMM. We actually had this problem and were told to "clip" PSA speeds (at 1666) in such cases. I think it's not right. CPR should be clipped at 100%, not PSA speeds.

On the other hand, with formula [3], high PSA speeds would simply lead to quicker saturation of CPR, say, in 10 monthsfor PSA speed of 5000 instead of "standard" 30 months for lower PSA speeds. They wouldn't cause any issues with SMM computation.

UPDATE: I logged on to Bloomberg desktop and computed CPR and SMMs for PSA equal to 1666 and 2000. Results confirm my conclusion that CPR should be capped at 100%, but not PSA speed at 1666.

## Wednesday, July 26, 2006

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