ponzi schemes
Bernard Madoff ran a huge Ponzi scheme. As the former head of NASDAQ he had exceptional credentials and connections. Therefore he did not have any trouble in developing a loyal customer base that included some of the most sophisticated institutional investors and wealthy individuals.
The annual returns of his fund was suspiciously stable. Nevertheless this consistency did not trigger any scepticisms. The returns did not draw the inquisitive spotlights, because they were not that outrageously high when compared to the performances of other top hedge funds. While typical Ponzi schemes are based on a contagious hysteria, his was built upon a calculated, slow-expanding exclusivity.
The fraud was detected only after a large number of clients attempted to draw their money at the same time. However, even if the financial crisis had not occurred, Madoff would sooner or later be caught. At some point, the incoming fresh capital would not be sufficient to cover the promised dividends to the already existing clients.
Question: Are all Ponzi schemes doomed to collapse?
Consider a simple scheme with the following properties:
- Investors are promised an annual return of 10%.
- Dividends are paid out at the beginning of each year.
- Clients are allowed to enter or exit the fund only at the beginning of each year.
- Net annual inflow of capital diminishes at a constant rate "R" where "R" is a real number between 0 and 1. (e.g. If this year's net inflow is 100, then the next year's will be R*100.)
- Dividends for year "n" are paid entirely out of the net inflow for year "n".
- The portion of net inflow that remains after dividend payments is confiscated and spent in a "non-recoverable" fashion.
This scheme collapses when the net annual inflow for year "n" becomes insufficient to cover the dividends for year "n".
Assume that net inflow for year zero is 100. Let's consider what happens at the beginning of year "n":
Due to property 4, the net inflow is 100*(R^n). The accumulation of net inflows from the previous periods is 100*[1+R+R^(2)+...+R^(n-1)]=100*[(1-R^n)/(1-R)]. The dividends are paid out on this accumulated amount. Hence, for the survival of the scheme, the net inflow needs to be greater than (10%)*100*[(1-R^n)/(1-R)]. This is true only for the (R,n) pairs that fall outside the black region:
No matter what R is, the fraud will be exposed within 10 years. If R=80%, then the fund collapses on its fifth year. If R=100%, then it survives till the beginning of year 10.
Question: What if R is greater than 100%? (i.e. What if the net inflow is perpetually increasing?)
If R assumes a value between 100% and 110%, then the Ponzi scheme will still collapse. However, in this region, there is no upper limit on the year of collapse. In fact, as R gets arbitrarily closer to 110%, downfall of the fund gets postponed to infinity.
Once R transgresses 110%, the scheme becomes collapse-proof.
Question: What if the investors are promised an annual return of D where D is a real number between 0 and 1? Can the results above be generalized beyond Property 1?
Yes, they can be.
When R=100%, the year of collapse is found by solving for "n" in the following equation: 100*(R^n)=D*100*[1+R+R^(2)+...+R^(n-1)]. So n=(1/D).
Hence, no matter what value R assumes between 0 and 1, the fraud will be exposed in (1/D) years.
In order to figure out when the scheme becomes collapse-proof, we need to solve the following problem:
Here we used the fact that 1+R+R^(2)+...+R^(n-1)=[(R^n)-1]/(R-1).
By a single application of the L'Hopital's Rule, it is easily seen that the limit of [(R^n)*(R-1)]/[(R^n)-1] as "n" goes to infinity is (R-1). Therefore, the solution to our minimization problem needs to be larger than (1+D).
But (1+D) obviously satisfies the inequality in the curly brackets for all "n". Hence, by the previous remark, it has to be the solution itself. (i.e. If R transgresses (1+D), the scheme becomes collapse-proof.)
If R assumes a value between 1 and (1+D), then the scheme still collapses. However, in this region, there is no strict upper bound on the year of collapse.
Here is a graphical summary:
As D gets smaller, the collapse-proof region expands, and the average survival duration for schemes that face strictly decreasing net inflow becomes greater.
P.S. The net inflow for year zero was completely inconsequential for our analysis. Assuming a value other than 100 would have made no difference.
P.S. Guess what is the biggest Ponzi Scheme on earth?
