The Math of Ponzi Schemes

A first order linear differential equation is often used to describe the dynamics of an investment fund, a.k.a. Ponzi Scheme. This type of fraud is named after its creator—Carlo Ponzi, who collected $9.8 million from 10.550 people and then paid out $7.8 million in just 8 months in 1920 Boston by offering  profits of 50%  return on investment in postal coupons, every 45 days. A Financial Ponzi scheme is a particular kind of pyramid scheme.

Pyramid Schemes

As in Ponzi schemes, the money collected from newer victims (investors) of the fraud is paid to earlier victims to provide a veneer of legitimacy. The profit is earned, by the sale of new distributorships and it is not created by the success of the business venture but instead is derived fraudulently from the capital contributions of other investors. Emphasis on selling franchises rather than the product leads to a point where the supply of potential investors is exhausted; when the operator flees with all of the proceeds or when a sufficient number of new investors cannot be found to allow the continued payment of “dividends”, the scheme falls apart and the pyramid collapses, because the underlying asset upon which the investment was based either never existed, or was overvalued.

The way that it works is that you create some kind of “investment fund“. You promise some kind of great payoff and convince people to invest. Ponzi schemes promise high financial returns or dividends over a short period of time, while still maintaining some semblance of credibility.

Instead of investing the funds of victims, the con artist (a con artist is an individual who is skilled and experienced at devising and executing scams and other fraudulent schemes) pays “dividends” to initial investors, using the funds of subsequent investors. As you get new investors, you use their investments to pay off the previous investors. The regular payment of dividends induces investors to bring friends, family members, or business colleagues (commonly targeted victims) into the scheme and to put up additional funds themselves now that they are convinced of its veracity.

 Several distinctions between Ponzi schemes and pyramid selling schemes: Pyramid schemes require active participants who will bring in more participants. Ponzi schemes require passive investors without any responsibility to promote the opportunity.


We assume that the fund starts at time t=0

  • c(t): cash in initial deposit at time t \in [0,\infty).
  • \lambda(t): cash influx rate at time t \in [0,\infty). The amount of cash that flows in the deposit scam in an infinitesimal time interval [t, t + dt] is \lambda(t)dt.
  • T: lock-up period (T>0). The con artist promises to return the money to the investors after a lock-up for T units of time.
  • R: promised rate on investment (R>0). It’s called Ponzi rate.


The Law of Conservation of Investment Value

 Money doesn’t evaporate, when it surprisingly dissapears.  It follows the same laws of conservation of energy. It may just transform: be it coffee or book or iphone.

If you spend few minutes in the day, you’ll check what you’ve spent.  So, at the end of month, you could see where your money was transformed. An advanced method is to identify your sources of money inflow and track how it flows out, that way you can determine if your inflow matches up with your outflow.
We will assume that the com artist does not deposit the cash in a bank to earn interest on it.

A first order linear differential equation is used to describe the dynamics of an investment fund.  The following differential equation:

\displaystyle\dot{c}(t) = \displaystyle \lambda (t) - (1+R) \lambda (t-T),

in which c: [0,\infty) \rightarrow \mathbb{R} is the function to be determined, and \lambda: [0,\infty) \rightarrow \mathbb{R} is the forcing function.

The rate at which the cash flows (Cash flow is the movement of cash into or out of a business, financial product) out is higher than the rate at which money flows in. After a large cash influx: since the cash influx cannot grow without bound, the cash influx will be insufficient to pay off the debt.

Solving the differential equation:

If we denote the initial amount of cash by c(0) = c_0 and solve the differential equation above, we obtain:

c(t) = c_0 +\displaystyle\int_{0}^{t} \lambda(\tau - T)d\tau - (1+R)\displaystyle\int_{0}^{t} \lambda (\tau - T) d\tau

Fort \in [0,T):

\displaystyle\int_{0}^{t} \lambda (\tau - T) d\tau = 0,

and for t>T is:

\displaystyle\int_{0}^{t} \lambda (\tau - T) d\tau = \displaystyle\int_{0}^{t -T} \lambda (\tau ) d\tau.


\displaystyle\int_{0}^{t} \lambda (\tau - T) d\tau = \displaystyle\int_{0}^{\max(t-T,0)} \lambda (\tau) d\tau,

and the general solution can be written:

c(t) = c_0 +\displaystyle\int_{0}^{t} \lambda(\tau - T)d\tau - (1+R)\displaystyle\int_{0}^{\max(t-T,0)} \lambda (\tau) d\tau.

The amount of cash at time t will be given by initial amount of cash, plus the amount of cash his creditors lent him, minus the amount of cash the con artist had to pay to his creditors (which is “amplified” by R) over time interval (t-T, t). The con artist starts returning money to the investors at time t=T.

The general solution can also be written as

c(t) = c_0 +\int_{\max(t-T,0)}^{t } \lambda (\tau)  d\tau -R\int_{0}^{\max(t-T,0)} \lambda (\tau) d\tau .

Example: Cash influx at t=0

\displaystyle \lambda(t) = \displaystyle \lambda_0 \delta(t)

In this case, an amount of cash equal to \lambda_0 is credited to the con artist’s pockets at time t=0. The solution will be:

\displaystyle c(t) = \displaystyle c_0 + \lambda_0 u(t) - (1+R) \lambda_0 u(t-T),

which is the Ponzi scheme’s impulse response (impulse response is response of system when input is provided to it is impulse; yet another explanation: the impulse response function is the inverse Laplace transform of the system transfer function H(s)). Note that c(t) = c_0 - R \lambda_0 for t \in (0, T), while c(t) = c_0 - R \lambda_0 for t>T.


Example: constant cash influx rate

\displaystyle \lambda(t) = \displaystyle \lambda_0 u(t)

A constant influx of cash flows to the con artist’s pockets. He will have to repay this cash with interest T units of time later. The solution of the differential equation is:

c(t) = \displaystyle c_0 + \lambda_0 s(t) - (1+R) \lambda_0 s(t-T),

which is the Ponzi scheme’s step response (the step response is defined as the response of a system to a step input; look here: ). The cash will increase linearly with time for t \in (0,T), a maximum is reached at t=T (maximum: c(T) = c_0 + \lambda_0 T), and then the debt-paying time starts and the cash will decrease linearly with slope - R \lambda_0 until the amount of cash reaches zero at  t = t_B > T. Making  c(t_B) = 0, we have:

c_0 +\lambda_0 t_B - (1+R)\lambda_0 (t_b - T) = 0,

t_B = \frac{c_0}{R \lambda_0} + (1 +\frac{1}{R}) T,

which is the “bankruptcy time” ( bankruptcy is a legal status of a person or an organization that cannot repay the debts it owes to its creditors). For t > t_B, the con artist will have to go into debt to pay off to his investors.
The bankruptcy proceedings:

  1. free you from overwhelming debts so you can make a fresh start, subject to some restrictions;
  2. make sure your assets are shared out fairly among your creditors;

An individual can be made bankrupt either in one of three ways:

  • Voluntarily – By the debtor themselves;
  • Involuntarily – By the creditor owed money;
  • The supervisor or anyone bound by an IVA.



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