As I wrote a few days ago, one of my ongoing projects is to devise mathematical models of Ponzi schemes. On my previous post I briefly explained what a Ponzi scheme is, so I will now focus on building a first model which, though somewhat simplistic and crude, will hopefully give us some insight.
For starters, let us introduce some variables:
- : cash in the scammer’s pockets at time .
- : cash influx rate at time . Note that this is a rate! Thus, the amount of cash that flows into the scammer’s pockets in a time interval of infinitesimal duration is . You may wonder why I chose letter to denote influx: the reason is that I have used letter to denote influx since I started using it years ago to denote heat flow. Habits are hard to change.
- : lock-up period (). The scammer promises to return the money to the investors after a lock-up period of units of time.
- : promised return on investment (). The scammer promises to return to the investors times the money they invested, units of time after they invested.
Keeping track of the influx and outflow of cash, we can build a conservation law.
Money Dynamics and the Law of Conservation of Cash
We can now write the “law of conservation of cash”: if more money comes in than it goes out, then the scammer’s pile of cash will be increasing (and so will his debt). We will assume that the scammer does not deposit the cash in a bank to earn interest on it, though we may consider that possibility in a more elaborate model. For the time being, our “law of conservation of cash” is given by the following differential equation
in which is the function to be determined, and is the forcing function. Note that the forcing function includes a delayed forcing term. Let us try to interpret this differential equation:
- the rate at which cash flows out is higher than the rate at which money flows in. This is NOT good! Imagine a wash basin in which the water influx from the tap is lower than the outflux going down the drain. That’s right, the basin will soon be empty, and that is precisely why Ponzi schemes are unsustainable: either you increase the cash influx over time, or you will soon be out of money. Since the cash influx cannot grow without bound, at some point the cash influx will be insufficient to pay off the debt and the scammer will be owing money to a lot of very angry creditors.
- cash flows out with a delay equivalent to the lock-up period.
We can now solve the differential equation.
General solution of the differential equation
If we denote the initial amount of cash by and solve the differential equation above, we obtain
Note that for
and that for
and therefore the general solution can be written as
This might look a bit complicated, but in fact it is very simple: the amount of cash the scammer will have in his pockets at time will be given by his initial amount of cash, plus the amount of cash his creditors lent him, minus the amount of cash he had to pay to his creditors (which is “amplified” by ) over time interval . Note that the scammer only starts returning money to the investors at time .
The aforementioned general solution can also be written as
Before we try out some forcing functions, let’s get the notation straight for the remainder of this post:
- the Dirac delta function will be denoted by .
- the Heaviside step function will be denoted by . The Heaviside step function is the integral of the Dirac delta function.
- the ramp function will be denoted by . The ramp function is the integral of the Heaviside step function.
This is the usual notation in signal processing and control systems engineering. To understand the aforementioned general solution, let’s now try out a couple of forcing functions.
Example: single discretionary cash influx at
In this case, an amount of cash equal to is credited to the scammer’s pockets at time . The solution will thus be
which is the Ponzi scheme’s impulse response (borrowing another signal processing expression), so to say. Note that for , while for .
Example: constant cash influx rate
In this case, a constant influx of cash flows to the scammer’s pockets. Unfortunately for the scammer, he will have to repay this cash with interest units of time later. The solution of the differential equation is now
which is the Ponzi scheme’s step response (once again borrowing a signals & systems expression). The cash will increase linearly with time for , a maximum is reached at (maximum: ), and then the debt-paying time starts and the cash will decrease linearly with slope until the amount of cash reaches zero at . Making we have
which is the “bankruptcy time”. For , the scammer will have to go into debt to pay off to his investors.
On this post, I presented a simple continuous-time model of a simplified Ponzi scheme. In the two examples I presented, one can conclude that the scammer’s best strategy is to run away at moment , when he has the most cash in his pockets (and owes the most to his creditors).
However, if we think beyond this model’s assumptions and limitations, it is clear that running away just before the lock-up period has expired might not be the “best strategy”. Note that if the scammer returns the money to the earliest investors, these might believe that the scammer truly has a Midas touch and decide to re-invest their money. In addition to that, if the scammer honors the early contracts and returns the money to his earlist investors, these investors may tell their friends and thus serve as “viral marketers” of the scammer’s investing “ingenuity”. Hence, it would then be natural to expect an increase in the money influx after the scammer returns the money to the earliest investors.
I will be writing more on this topic on future posts.
Disclaimer: I am NOT an accountant, I am an electrical engineer. I have NEVER studied any accounting at all. Please bear with me if I misused technical terms, or if I invented silly new ones. Thank you.
Earlier posts in this series: