Exponentially-Fitted Simpson’s Rule for Present Value Estimation

Introduction to Computational Mathematics — April 2026, with Robby Enright and Luke Baker

The problem

Present value calculations require integrating a cash flow stream against a discount factor, \(PV = \int_0^T C(t) \cdot e^{-rt} \, dt\). Classical numerical integration methods like the Trapezoidal and Simpson’s rules approximate this integrand using generic polynomial bases — they don’t take advantage of the fact that every present-value integrand shares the same exponential discount structure, regardless of the cash flow shape.

My approach

We derived a modified version of Simpson’s rule that swaps the standard quadratic polynomial basis for one that explicitly includes the exponential discount term itself, so the quadrature rule is built around the same structure the problem actually has. We:

Derived the modified quadrature weights by solving a 3×3 linear system fit to the endpoints and midpoint of each integration panel.
Proved the method recovers standard Simpson’s rule exactly as the discount rate approaches zero, and integrates exponential-basis functions exactly regardless of step size.
Benchmarked the method against the Trapezoidal and standard Simpson’s rules across three cash flow types (exponential, smooth, and oscillatory), measuring convergence behavior across a range of panel counts.

Results

The modified method outperformed standard Simpson’s rule by 10–20x in accuracy at the same computational cost, with the largest gains on smooth and exponential-shaped cash flows — exactly where the method’s underlying assumptions hold best. Performance narrowed on oscillatory cash flows, where the periodic component falls outside the fitted basis, which we treat as the method’s natural boundary rather than a flaw.

Skills used

Numerical analysis · Quadrature methods · MATLAB/Python · Linear algebra · Technical writing