Ponzi Scheme Maths

By now the US$50 billion Bernie Madoff scandal is pretty much old news, but like many other people I have been wondering how he was able to sustain his giant Ponzi scheme for over 20 years.

In stark contrast, good old Charles Ponzi himself could not even make his scheme last a year. He started taking in money from investors in early 1920 through his "Securities Exchange Company".

By July 1920 he had become a millionaire.

By November he was in jail.

Of course, no Ponzi scheme can last - there is no influx of funds from outsiders and the money just gets redistributed from new investors to existing investors. Since money is constantly drawn out of the system (especially the scam artist himself) the entire operation will collapse once recruitment starts slowing down.

However I wanted to get a feel for just how aggressive the recruitment will have to be in order to keep the scheme running, and I'm also curious about the difference in scale between Ponzi's operation and Madoff's.

So I hunted the Intertubes for a mathematical model of the Ponzi scheme, ideally an applet that allows you to plug in key variables such as the rate of return.

Unfortunately the models available on the Net either totally ignore the recruitment aspect (Bernie Madoff Calculator) or rely on difficult mathematics such as matrices and calculus.

Thus I have no choice but to work it out myself. I wish I could turn it into an automated applet, but my maths and programming skills are not good enough. If you are able to make that, please drop us a link.

Here's a simple, arithmetic model of the Ponzi Scheme:

1. Let's start with Charles Ponzi's version. He promised a 50% rate of return after an investment period of 45 days.

For the purposes of simplicity, let's restrict investment to one $1000 lot per person, synchronize their investment periods, and allow the investors to withdraw interest payments only at the end of each 45-day cycle.

My model is sort of a worst case scenario for Ponzi, in that he must have the capital to pay off the maximum potential interests due at the end of each cycle. (Note that if all the investors also demand their principal back, then the scheme would immediately end).

In reality he need not have so much cash at hand, since he can simply issue bogus profit statements to his investors and encourage them not to make withdrawals.

I also assume that Ponzi himself expects to make the same rate of return as his investors, which clearly isn't the case in real life.

Day 0

Ponzi starts the ball rolling with a capital of $1000.

He has to potentially pay himself $500 (0.5x1000) interest on Day 45.

If he doesn't withdraw his interest payments, it will accumulate to $1250[ (1.5x1.5x1000)-(1000)]on Day 90, which will make his cash flow negative $250 (1000-1250) and end his scheme.

So he must find another investor on Day 45.

Day 45

Ponzi plus one investor. Total capital = $2000

By Day 90 he has to potentially pay out (1250+500) = $1750
By Day 135 he has to potentially pay out (1.5x1.5x1.5x1000)-1000+1250) = $3635 (cash flow = negative $1635)

So he must find additional investors on Day 90.

Day 90

Four more investors. Total investors = 6. Total capital = $6000

By Day 135 he has to potentially pay out [2375+1250+(4x500)] = $5625
By Day 170 he has to potentially pay out [4062.5+2375+(4x1250)] = $11437.5 (cash flow = negative $5437.5)

Needs more investors again.

Day 135

11 more investors. Total investors = 17. Total capital = $17000

Day 180

Total investors = 52.

Day 225

Total investors = 155.

And so on...

As you can see, the number of investors required to sustain the scheme goes up exponentially.
















This is due to the compound effect of the rate of return, which causes the capital to appreciate exponentially, as shown in the chart below.

















50% return over 45 days actually represents a monstrous annual rate of return of 2563%!

This means that an initial investment of $1000 will grow to exceed a million dollars in only 2 years, an insane rate that is clearly impossible to sustain for long.

2. Now let's look at the figures for Madoff's version. Madoff promised a steady 12% return annually.

Year 0

Madoff also starts off with $1000.

By Year 1 he has to potentially pay himself $120
By Year 2 he needs $254.40
By Year 3 he needs $404.93
By Year 4 he needs $573.52
By Year 5 he needs $762.34
By Year 6 he needs $973.82
By Year 7 he needs $1210.68 (negative cash flow - scheme ends)

Needs to find an investor in Year 6.

Year 6

Madoff plus one investor. Total capital = $2000

By Year 7 he needs (1210.68+120) = $1330.68
By Year 8 he needs (1475.96+254.40) = $1730.36
By Year 9 he needs (1773.08+404.93) = $2178.01 (negative cash flow - scheme ends)

Needs another bloke.

Year 8

Total investors = 3. Total capital = $3000

By Year 9 he needs (1773.08+404.93+120) = $2298.01
By Year 10 he needs (2105.85+573.52+254.40) = $2933.77
By Year 11 he needs (2478.55+762.34+404.93) = $3645.82 (negative cash flow - scheme ends)

Add another bloke.

Year 10

Total investors = 4. Total capital = $4000

By Year 11 he needs (2478.55+762.34+404.93+120) = $3765.82
By Year 12 he needs (2895.98+973.82+573.52+254.40) = $4697.72 (negative cash flow - scheme ends)

Year 11

Total investors = 5.

Year 12

Total investors = 6.

And so on...

You can immediately see that there is much less pressure to recruit, especially during the early days of the scheme.

After five rounds of recruitment, Ponzi needed 154 more investors to keep going, whereas Madoff only needed five more.

In both cases, the money required to maintain the scheme is going up exponentially, but in the Madoff's version it is going up so slowly that the recruitment rate looks almost linear.
















It would take many years before this increase is significant enough to be noticeable, which could be part of the reason why he remained undetected for so long.