Modeling Ponzi Schemes II

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.

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Model Variables

For starters, let us introduce some variables:

• $c(t)$: cash in the scammer’s pockets at time $t \in [0,\infty)$.
• $q(t)$: cash influx rate at time $t \in [0,\infty)$. 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 $[t, t + dt]$ is $q(t) dt$. You may wonder why I chose letter $q$ to denote influx: the reason is that I have used letter $q$ to denote influx since I started using it years ago to denote heat flow. Habits are hard to change.
• $T$: lock-up period ($T > 0$). The scammer promises to return the money to the investors after a lock-up period of $T$ units of time.
• $R$: promised return on investment ($R > 0$). The scammer promises to return to the investors $1+R$ times the money they invested, $T$ units of time after they invested.

Keeping track of the influx and outflow of cash, we can build a conservation law.

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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

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

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

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General solution of the differential equation

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

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

Note that for $t \in [0,T)$

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

and that for $t > T$

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

Therefore

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

and therefore the general solution can be written as

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

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 $t$ 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 $R$) over time interval $(t-T, t)$. Note that the scammer only starts returning money to the investors at time $t=T$.

The aforementioned general solution can also be written as

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

Before we try out some forcing functions, let’s get the notation straight for the remainder of this post:

• the Heaviside step function will be denoted by $u(t)$. The Heaviside step function is the integral of the Dirac delta function.
• the ramp function will be denoted by $s(t)$. 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.

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Example: single discretionary cash influx at $t=0$

$\displaystyle q(t) = \displaystyle q_0 \delta(t)$

In this case, an amount of cash equal to $q_0$ is credited to the scammer’s pockets at time $t=0$. The solution will thus be

$\displaystyle c(t) = \displaystyle c_0 + q_0 u(t) - (1+R) q_0 u(t-T)$,

which is the Ponzi scheme’s impulse response (borrowing another signal processing expression), so to say. Note that $c(t) = c_0 + q_0$ for $t \in (0, T)$, while $c(t) = c_0 - R q_0$ for $t > T$.

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Example: constant cash influx rate

$\displaystyle q(t) = \displaystyle q_0 u(t)$

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 $T$ units of time later. The solution of the differential equation is now

$c(t) = \displaystyle c_0 + q_0 s(t) - (1+R) q_0 s(t-T)$,

which is the Ponzi scheme’s step response (once again borrowing a signals & systems expression). The cash will increase linearly with time for $t \in (0,T)$, a maximum is reached at $t=T$ (maximum: $c(T) = c_0 + q_0 T$), and then the debt-paying time starts and the cash will decrease linearly with slope $- R q_0$ until the amount of cash reaches zero at $t = t_B > T$. Making $c(t_B) = 0$ we have

$\displaystyle c_0 + q_0 t_B - (1+R) q_0 (t_b - T) = 0$,

which yields

$t_B = \displaystyle\frac{c_0}{R q_0} + \displaystyle\left(1 + \frac{1}{R}\right) T$,

which is the “bankruptcy time”. For $t > t_B$, the scammer will have to go into debt to pay off to his investors.

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Concluding remarks

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 $t = T^{-}$, 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.

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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.

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Earlier posts in this series:

10 Responses to “Modeling Ponzi Schemes II”

1. Alexandru Says:

I think another way you can model a Ponzi scheme is by using continuous time Markov chains with rewards (or money), where the constant rates are investments. More precise one can use also inhomogeneous continuous time Markov chains where transition rates depend on time. Which means that investment rate changes over time.

2. Rod Carvalho Says:

Alexandru,

Indeed I thought of using Markov chains, but I am a bit rusty on that topic, so I didn’t exactly have great ideas on what to do with them. What would the discrete states on the Markov chain represent? How do you implement the “rewards”? This reminds me of Markov Decision Processes, an area which I heard about but never really did study in any depth.

Would you please care to elucidate? If you could use $\LaTeX$ and write a short comment explaining your idea, I would be most grateful. My brain has trouble processing abstract info, so I need to see some equations to get the gears running ;-)

3. Alexandru Says:

Rod,

One way to model a Ponzi scheme is to use a Continuous Time Markov Chain (CTMC) which will represent a birth-death process. More formally assume a CTMC given by a collection of random variables $\{X_{t}\vert t\in \mathbb{R}_{+}\}$ on the state space $\mathbb{N}$ (states represent the number of investments).

A transition $(i)\rightarrow(i+1)$ from state $i$ to state $i+1$, $i \in \mathbb{N}$ will hold with some constant rate $\lambda$ (investment rate). Also there is a transition $(i)\rightarrow(i-1)$ from state $i$ to state $i-1$ with some constant rate $\mu$ (return rate to the investors). The initial state of CTMC is $i=0$.

For each transition $(i)\rightarrow(i+1)$ define an investment cost $r_i$ and for each $(i)\rightarrow(i-1)$ define a payment cost of $-r_p$. More precisely the costs are called impulse rewards.

Assume that the $n$‘th transition occurs at time $\tau_n$, $n=1,2,\dots$. Then the $n$‘th transition will be given by the pair of states $\sigma_n=(X_{\tau_{n-1}},X_{\tau_{n}})$. A path $\rho$ is a sequence of state where $\rho[\sigma_{n}]$ is the cost of transition which occur at the moment of time $\tau_n$.

The cumulative impulse reward during at time $t$:

$I(t) = \displaystyle\sum_{i=1}^{N(t)}\rho[\sigma_{n}]$,

$N(t)$ is the number of transition in the interval of time $(0,t)$ and $I(t), N(t)$ are random variables. One has to compute the expected reward $E[I(t)]$ at time $t$.

4. kreso bilan Says:

Hi Rod,

Just wanted to say that I find the idea very interesting. Chance favors the prepared mind.

5. iamreddave Says:

Great post. Intuitively the problem seems similar to that of spreading a disease. How fast you can spread. How long the person stays infected. How many people are suceptible. The chance of one person spreading the disease or Ponzi scheme to someone they are in contact with, etc.

• Rod Carvalho Says:

That’s precisely my long-term goal. The model discussed in this post is rather simplistic, and it requires knowledge of the system’s input in order to produce any results. I see it as a first step towards something more interesting. I would like to have a more “structured” input, which could be attained by some sort of feedback mechanism. I don’t claim that such endeavour would be an easy one.

What I envision is a model based on social networks, in which the Ponzi scheme is propagated from node to node, until the entire social network (or a “strongly connected” sub-network, i.e., a clique) is “infected”. When such saturation phenomenon occurs, the influx of cash dies off and the Ponzi scheme collapses.

• iamreddave Says:

Is there something peculiar about Ponzi schemes? Do they spread in a different way to trends and fashions? To good ideas like adopting a particular new piece of technology?

Say Ponzi schemes do spread differently to “good” idea. Do the ideas that spread bubbles “tech stock will go up forever”, “house prices will go up forever” spread like Ponzi schemes, or like good ideas such as “wash hands before dealing with a patient”.

Fascinating area.

• Rod Carvalho Says:

I don’t think that Ponzi schemes spread differently from other fads. They do spread differently from diseases in the sense that for a disease to be propagated, one must have some sort of “geographical proximity” (one can’t infect someone on the other side of the world by sneezing). By contrast, a Ponzi scheme can be spread via email or blogging (an enthusiastic early investor might send an email to all his friends because he thinks that the Ponzi scheme is a great investment opportunity and doesn’t want the others to miss the boat).

Fads are (by definition?) irrational. A habit such as washing one’s hands is a good practise that is rational, and that can be derived logically from a few simple axioms such as:

i) germs are bad for one’s health

ii) washing one’s hands kills germs

and, hence, assuming that people are interested in their health, they will wash their hands.

This is completely different from, say, starting a company called pets.com that sells dog food online. Has the idea been tested? No! Is there a market? Nobody knows! One can’t quite conclude whether the idea is good by using logical reasoning, one can only find out by trying out and seeing if it works.

6. George Habash Says:

Hello everybody!

I have a question. In the formulation proposed above, nothing is said about T. In particular the relationship between t and T.
Can T be greater than t?

• George Habash Says:

…by the way, I noticed something about that in the solution of the differential equation. My question, what does t > T economically mean? and when t < T what does that mean, of course from an economic point of view. Many thanks