On numerical stability of recursive present value computation method
Argyn Kuketayev
TL;DR
This paper examines numerical stability in recursive present value computations, comparing forward and backward recursions under positive discount rates. It derives an error-propagation model showing forward recursion yields exponential growth of absolute error with increasing schedule length, while backward recursion avoids this instability by recasting the recurrence as $PV_{k-1}=\frac{PV_k+C_k}{1+r}$. Theoretical analysis supplemented by amortization-table experiments demonstrates the robustness of the backward approach for positive IRR, suggesting a practical shift in financial software toward backward recursion for carrying amounts and IRR-based discounting. The work highlights the impact of finite-precision arithmetic on PV calculations and provides guidance for more stable numerical implementations in finance.
Abstract
We analyze numerical stability of a recursive computation scheme of present value (PV) amd show that the absolute error increases exponentially for positive discount rates. We show that reversing the direction of calculations in the recurrence equation yields a robust PV computation routine.