Ever since I wrote this, I am getting emails from friends with news of other ponzi schemes being found out.  

But there is one thing that I saw on TV that totally blew me away.  Apparently in 2006 Madoff was told that he was being tracked and had to register -- or something like that.  At that point his fund was only $16 Billion.  By the end of 2008 it had balooned to $50 Billion.  Around the same point many peole were predicting that things were going to fall off the endge -- including the covservative economist from Harvard Martin Feldstein.

This tells me that Madoff saw the end coming, and figured he might as well hord more money since he will get the same sentece -- if ever!!!  

The way he was turned in was a suspect too.  He told his sons, who were working for him, and then they turned him in, claiming he had no idea what their dad was doing!!!  

Ya right.

 

Mathematics of Sustaining a Ponzi Scheme

I really don't have time to read about Bernard Madoff and his $50B fund, now fraud.  But he is hard to miss on WSJ and NYTimes.  

Driving from Menlo Park to Sunnyvale, I did some math in my head.  It turns out that the mathematics of a ponzi scheme is actually one of compound interest rate, very simple, but remarkably the scheme can be made sustainable for a long time.  

Lets assume we start with the capital C, and for simplicity we want to return an interest rate i every year to the investors -- we will get to the compound interest in a second.  So the question is how much do we have to raise each year to keep the ponzi scheme going.  

Well in year one we have to raise C*i. Now we owe our investors C*(1+i).  The by the end second year we need to raise C*(1+i)*i.  and now we owe: 

C*(1+i)+C*(1+i)*i = C *(1+i)^2

where ^ means to the power of (and it takes precendence over multiplication -- i.e.,* ).  

Clearly this is like compound interest and by the end of year N we have to raise:

C * {(1+i)^(N-1)} * i  

and we end up owing:

C * (1+i)^N.

Wait for it, wait for it !!!  But here is the interesting part, this scheme is remarkably flexible, making it even more sustainable.  If our investors reinvest their money for compound returns, even better, we have to raise less (or spend more).  Of course, if someone asks for their money back, we just have to raise that much more.  And remember, we still have C which we probably invested, and can use that as a cushion.  Obviously the interest, i, does not have to be constant, and we do not have to raise just the amount we need (it adds or subtracts from the money we owe).  Moreover, we can pay ourselves by returning r less than i to the investors, and pay ourselves with the rest.  The mathematics does not change.  We just have a i-r front cost for the funds held and are giving ourselves a raise of i% every year!  

But here is the really interesting part.  C * (1+i)^N * i  is amazingly sustainable.  In fact, the next time we have to raise the amount C for capital in one year is:

  -log(i) / log(1+i) years away.  So for 10% rate of return we are talking 

1/log(1.1) =1/{log(11)-log(10)}  which is 24 years 2 months later.  In other words at year end 24 we need to raise approximately C.  

For a decent 8% we have 32 years 10 months.

Now it turns out that by year log(2)/log(1.1) - 1 = 6.27  that is 6 years 3 months we have raised a total of C in capital (this is just geometric series -- i + i*(1+i) +i*(1+i)^2...).  Which is still remarkable, because it means that we can raise 16% of the C for the first 6 years and we will still be OK.

So the ponzi scheme is remarkably sustainable -- except if everyone starts asking for their money back.  This is exactly what I think happened to Madoff .  People asked for their money back to cover other losses.  Moreover, this  was compounded by him getting too greedy and not being to raise $50B * interest + the capital outflow in a bad economy.  Madoff probably had invested the excess money he had raised (at much less returns) and then had lost a large part of it -- may be in Financials ;-)

I don't mean to be glib about it.  In fact, I am not at all surprised if this happens without bad intensions on the part of some funds, where they take the money that came in, leverage it to invest, and then take parts or (all) of what they raised to pay off their investors (OK so it may work a bit different).

I don't need to put in a disclaimer here saying that I hope this does not become a study in defrauding people.  I did the math in my head -- it doesn't take a math genius to know this already.

But here is the thing -- how would you stop this?  Look at the portfolio that was held (may be entries and exit values), and if the returns make sense.  This is simple java program (Excel + web crawling for the accuntants).   As for secrecy -- then SEC is or should be doing this.

Merry Christmas,

E