Polynomial Expansion of the Future Value Factor for an Annuity Due : Generalization via the Hockey-stick Identity and Binomial Theorem

This paper derives an m-th order approximation formula, denoted by Sm(r) for the future value annuity factor (FVAF) of an annuity due and analyzes its mathematical properties. While the FVAF is typically defined as the sum of a geometric series, polynomial approximation is highly effective in practical simulations when the interest rate is small. This paper presents two distinct mathematical approaches to derive a general approximation formula Sm(r) for the FVAF. The first approach applies the binomial theorem to the expanded form (A) of the FVAF and consolidates the resulting coefficients using the hockey-stick identity. The second approach starts from the closed-form expression (B) of the FVAF and achieves the same result as the first approach by integrating terms through algebraic operations and Pascal’s rule. By demonstrating that these two methods converge to the same general approximation formula, this paper provides a robust mathematical foundation for the validity of the general approximation formula for the FVAF. The results of this research contribute to an intuitive understanding of the close relationship between combinatorics and financial mathematics.