The standard formulas use separate annualized rates of partial prepayment and liquidation, PPR and LQR. Though this approach works in practice, it involves a minor technical inconsistency.
Let p be the constant monthly partial prepayment rate, and q the constant monthly liquidation rate. For simplicity, we ignore scheduled principal payments. The remaining principal balance after one month is:
and after two months is:
Similarly, the remaining balance after one year is:
We can thus write the total annual prepayment rate, UPP:
Strictly speaking, there is no way to separate the expression (1-p)12 from the expression (1-q)12 and thereby obtain separate expressions for the annual rates PPR and LQR. The two expressions are inter-dependent (for this reason, the expression UPP is provided in the standard formulas).
However, because partial prepayments, p, tend to be low and relatively invariant, the standard formulas
reasonably approximate the reductions in principal that result when constant rates p and q are applied in the standard formulas for one year. Thus, the Canadian investment dealers have agreed that the expressions PPR and LQR are reasonable in practice.