Chaotic time series in financial processes consisting of savings with piecewise constant monthly contributions

TL;DR

This paper analyzes a deterministic financial model with periodic contributions and negative interest, revealing conditions under which the resulting time series are either periodic or chaotic with sensitive dependence on initial conditions.

Contribution

It demonstrates a dichotomy in the behavior of the financial process, showing when it converges to periodic sequences or exhibits chaos with a Cantor attractor.

Findings

Financial time series are either asymptotic to periodic sequences or have a Cantor set of limit points.

Explicit parameters can induce chaos with sensitive dependence on initial conditions.

The model provides a mathematical framework for understanding complex dynamics in savings processes.

Abstract

We investigate the time series generated by an elementary and deterministic financial process that consists in making monthly contributions to a savings account subjected to the devaluation by a monthly negative real interest rate. The monthly contribution is a piecewise constant function of the account balance. We show that a dichotomy holds for such a financial time series: either the financial time series are asymptotic to finitely many periodic sequences or the financial time series have an uncountable (Cantor) set of {\omega}-limit points. We also provide explicit parameters for which the financial process is chaotic in the sense that the financial time series have sensitive dependence on initial conditions at points of a Cantor attractor.