Appendix 2: Old Annualizing Formula

This appendix shows how unusually large annualized prepayment rates could be obtained under the old formulas. For simplicity, we will only consider total prepayments. Let u be the constant monthly prepayment rate. Under the old standard, u is annualized using the formula:

(1+u)¹²-1

This is an appropriate annualizing formula in the situation where the principal grows over time, as is the case with compound interest. However, the formula is not appropriate for MBS prepayment, a situation where principal decays over time. For example, if an MBS experiences total monthly prepayments at rate u, the balance remaining after one month is, ignoring scheduled principal reductions,

B_k+1=B_k(1-u)

Similarly, the balance after two months is:

B_k+2=B_k(1-u)· (1-u)

It follows that the balance after one year is:

B_k+12=B_k(1-u)¹²

The annualized prepayment rate, UPP, is therefore calculated thus:

(1-UPP)=(1-u)¹²
UPP=1-(1-u)¹²

The following table compares the results of the old and new annualizing formulas for different values of u, the total monthly prepayment rate:

As noted previously, the new method of annualizing better reflects the actual principal paid down, but otherwise does not affect the MBS pricing calculations.