Sincov's and other functional equations and negative interest rates

TL;DR

This paper explores solutions to functional equations related to negative interest rates, extending classical investment models to include cases where the future value can be zero or negative, thus addressing new financial realities.

Contribution

It provides a complete characterization of solutions to the multiplicative Sincov equation allowing zero values, linking these solutions to investment theory under negative interest rates.

Findings

Solved the Sincov equation with zero values

Connected solutions to investment models with negative interest rates

Extended classical financial models to non-positive interest scenarios

Abstract

Investigating the future value $F(K,s,t)$ of a capital $K$ invested at date $S$ at date $t$ the "natural" condition $F(K,s,t)\geq K$ has lost its naturality because of the strange fact of negative interest rates. This leads to the task of describing the possible solutions of the multiplicative Sincov equation $f(s,u)=f(s,t)f(t,u)$ for $s\leq t\leq u$, where $f(s,t)=0$ may happen. In this paper we solve this task and discuss connections to the theory of investments.