The Mathematics of a Ponzi scheme

A Ponzi Scheme is a form of investment where profits are paid to early investors with funds from later investors. It often involves fraud and as a result it is of course illegal.
This article will explore the mathematics of a Ponzi Scheme which has been transparent about its intentions. Suppose someone were to set up a fund that aims to pay investors a return of \(\rho\) (rho).
I.e. for each unit of currency invested, it will aim to return \(1+\rho\).
The fund pays out on a first in, first out basis. Early investor money is returned when (or if) later money arrives.
Suppose, at a point in time, the cumulative money invested is N.
The money that's been paid out is P and the fund has remaining money R.
We known that
(Investment in) = (Payouts made) + (Remaining money)
\[N = P + R\] We can classify the investment so far into two categories. There is the money that has already been settled S, i.e. the early investors that have already received their funds returned with a positive return \(\rho\) .
And there is the balance outstanding B, that remains unsettled as yet.
So we can write:
(Investment in) = (Settled amount) + (Balance outstanding) \[N = S + B\]
The early investors who have received their settlement all got a return of \(\rho\).
So we know that the payout: \[P=S(1+\rho)\] The fund administrator has guaranteed that the maximum loss that could be imposed on an investor is \(\lambda\) (lambda).
So for each unit of currency invested, the worst case scenario for the investor is that \((1-\lambda )\) is returned.
Hence we require the remaining funds (R) to be able to cover a payment of the balance (B) with an imposed loss. \[R= B( 1 - \lambda )\] We now have 4 equations and 4 unknowns (P,R,S,B) along with three knowns (N, \(\rho\), \(\lambda\) ).
If we do the algebra, we can find our unknows:

Settled amount:	\(S = N \frac{\lambda}{\lambda + \rho} \)
Balance outstanding:	\(B=N-S\)
Payouts made:	\(P=S (1+ \rho) \)
Remaining money:	\(R=N-P \)

We can try that out with some numbers:

Inputs
Total invested funds (N)		units of currency
Investment return (\(\rho\))		%
Acceptable loss (\(\lambda\))		%
Outputs
Settled amount (S)	-
Balance outstanding (B)	-
Payouts made (P)	-
Remaining money (R)	-

With such a scheme, any investor would be allowed to demand an immediate repayment of funds at any time, though in that case a loss of \(\lambda\) would be imposed. In other words, for each unit of currency invested, \(1-\lambda\) would be returned.

One problem for early investors is that they don't know how long they will have to wait until they'll be repaid their investment with the positive return (\(\rho\)). But it is not just a question of when they'll be paid, it is uncertain if they'll be paid. When later investors don't arrive, the early investors just wait and wait, until they eventually give up and request a return of funds, in that case they'll just have to accept a loss.

It is also worth noting that this model above ignores any administrative fees that may be imposed on funds on the way in and or the way out.