Fixed Income - SMM - CPR twice, why, understand

ftwcfa

Does anyone understand the logic of this formula?
SMM = 1 – (1 – CPR)^1/12
why do they use 1 - (minus) twice here?

hassan

Let’s say you have $1 is mortgage balances AFTER scheduled payments of interest and principal.
The SMM seeks to get a reduced number B which says if we regularly have SMM rate of PREPAYMENTS then the $1 reduces to B
Assume that the annualized CPR is given in:
SMM = 1 – (1 – CPR)^1/12
i.e.
CPR= 1 - (1 – SMM )^12
The B=(1-SMM)^12 is an annualized result of monthly prepayments at the SMM rate. This reduces the $1 ( after scheduled payments ) balance of the mortgage geometrically throughout the year.
Then B is a number very close to 1
And CPR is the difference between 1 and B.
CPR=1-B
i.e. CPR represents how much is prepaid on an annualized percentage basis
Say B=0 , then CPR is 100% , all the balance ( after scheduled payments ) is paid off in 1 year.
If B=1 , then CPR is 0 , and the homeowners are paying nothing beyond the scheduled payments ( a huge extension risk)