Ponzi system dynamics: time to bankruptcy, optimal bailout time and a condition for survival

Every Ponzi ‘investment’ scheme is fraudulent and it is destined for eventual collapse. When a Ponzi scheme starts, the early investors make unreasonably high profits from receiving unrealistically high (and ‘guaranteed’) returns. However, unbeknown to the unsuspecting investors, these returns are made possible not by the profits of a successful business (as claimed by the Ponzi operator), but by the deposits of the later investors. Unfortunately for the later investors, either the money runs out, or the operator disappears. In this paper we analyse the time-dependent progress of the Ponzi scheme using a system of two difference (and later, differential) equations. We estimate the time to bankruptcy (and optimal bailout time for the operator) under several assumptions, including constant, time-varying, random deposit amounts by the investors. An optimal control model to maximise the final time cash balance is also included. We provide a simple (but unusual) condition under which the Ponzi scheme may never go bankrupt. Our results could be of potential benefit to investors to warn them not to be fooled by the promises of a Ponzi scheme fraudster; and if they have invested in such a scheme, to cash out before the scheme collapses or the promoter disappears.