Appendix 3: Appropriateness of PPR and LQR

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:

B_k+1=B_k(1-q)·(1-p)

and after two months is:

B_k+2=B_k(1-q)·(1-p) ·(1-q)·(1-p)

Similarly, the remaining balance after one year is:

B_k+12=B_k(1-q)¹² ·(1-p)¹²

We can thus write the total annual prepayment rate, UPP:

(1-UPP)=(1-q)¹² ·(1-p)¹²
UPP=1-(1-q)¹² ·(1-p)¹²

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

PPR=1-(1-p)¹²
LQR=1-(1-q)¹²

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.