Ponzi Schemes: The Structure and the Inevitable End

In a Ponzi structure, the payments made to earlier participants come from money brought in by later ones rather than from anything the arrangement produces. Funding the payments that way makes the ending arithmetic rather than misfortune. Sustaining it needs a base that grows without limit, no such base exists, and the share of participants who lose is settled by the structure before anybody joins.

The early payments are real, and their reality is the part that confuses everybody. Money genuinely arrives, in full, on the day it was described, and the people who received it are telling the truth when they say so. Real early payments are not a flaw in the deception. Real early payments are the deception, and the whole distance between recognising the structure and being reassured by it runs straight through that one sentence.

What is a Ponzi structure, once the story is taken off it?

Every arrangement of this kind arrives wrapped in a story, and the stories are all different. One is about trading. One is about property. One is about importing something and selling it on. One is about a machine, or a contract, or a connection somebody has that other people do not. The stories are interesting and usually detailed, and attention goes to them first. Holding the attention is exactly the story's purpose.

Take the story off and what is left is a shape, and the shape is very simple: the money paid out to people who joined earlier is money paid in by people who joined later. The shape is the entire structural fact. A Ponzi structureAn arrangement in which the payments made to earlier participants come from money brought in by later participants rather than from anything produced. is not defined by what it claims to do, by how sophisticated it sounds or by whether anybody involved intends harm. A Ponzi structure is defined by where the money that reaches a participant came from, and there are only ever two answers to that question.

Here is the smallest version of it, at a scale that fits in the head. Suppose somebody tells ten cousins at a wedding that if each of them hands over Rs 1,000/-, they will be handed back Rs 1,500/- in three months. Three of them agree. Three months later the promiser owes Rs 4,500/- and holds Rs 3,000/-. So four more cousins are found, Rs 4,000/- is taken from them, the first three are paid their Rs 4,500/- and the arrangement keeps going. The first three cousins are delighted. The first three were paid in full and on time, they say so at the next wedding, and now more cousins want in. Nothing about their delight is faked. Nothing about their money is fake. The Rs 4,500/- is real currency and it is in their hands.

But notice what did not happen anywhere in that sequence. Nothing was bought and sold at a margin. Nothing was made. Nothing was lent to a borrower who paid interest on it. Nothing was let out to a tenant. No work was done for anybody who paid for the work. The Rs 4,500/- that reached the first three cousins is, to the last rupee, money the next four cousins handed over. The money did not grow. It moved.

One more thing follows immediately from money moving rather than growing, and the rest of the arithmetic turns on it. If every payment out is funded by a payment in, then continuing to pay requires a continuing supply of people paying in. The baseThe number of participants in the arrangement. Payments continue only while that number keeps growing. has to grow, and it has to keep growing, and it has to keep growing at whatever rate the promised payments demand. A base that must keep growing is not a risk attached to the arrangement. A base that must keep growing is the arrangement.

Where does the money actually come from?

Asked of anything that pays, this question has one of two answers, never a third. Either something happened in the world which created the money, or somebody else handed it over. There is no other source. Money does not appear because a document says it should, and it does not appear because a person is confident, well dressed or persuasive.

Ashok Bhosale runs a tailoring counter, so the first answer is in plain sight every day of the week. Cloth and thread go in. A day of cutting and stitching goes in. A garment that did not exist that morning comes out. Somebody who wants that garment pays for it, and the money that reaches the counter reaches it because a specific thing was made for a specific person who valued it more than the money. If the counter shut tomorrow, that money would stop, and it would stop for a reason anybody could point at.

In a Ponzi structure, the money that reaches a participant reaches them because somebody else has just paid in, and following it backwards never arrives at anything being made, sold, lent or let out. Following the money backwards arrives at another participant. Followed back from them, it arrives at another one. The chain is long, it can run for years, and it terminates at people rather than at production. A chain that ends at people is a different kind of thing from a tailoring counter, and the difference is not a matter of degree.

Why are the early payments genuinely real?

The reality of the early payments is the question the whole subject turns on. Nothing about the early payments is faked, delayed or partial, and expecting them to be is why one of these arrangements goes unrecognised when it is standing in front of somebody.

The early payments are real. The money leaves one account and lands in another. It clears. The money can be withdrawn, spent on a wedding, put towards a two-wheeler, handed to a contractor. Somebody who received it and says so is telling the truth, in full, without exaggeration. Asked for proof, they will show proof, and the proof will be genuine.

Paying the early participants is not a trick played inside the structure; it is the only thing the structure actually does, and it does it perfectly. Every rupee that comes in is available to be paid out, minus whatever the arrangement retains, and paying it out is what produces the next wave of people wanting to come in. Nobody would recommend a structure that failed to pay its early participants, so such a structure would collapse in its second round. The payments are the recruitment.

So when somebody at a wedding says they have been paid every quarter for two years, they are not lying and they are not confused. The people who say so are describing, accurately, a structure doing the one thing it is built to do. SustainingContinuing to make the payments that have been described. In this structure that requires new money coming in rather than anything being produced. the payments is the whole activity, and it can go on for as long as the base keeps growing at the rate the payments require. How long the base can keep growing at that rate can be worked out, and the answer turns out not to depend on anybody involved.

What does the base have to do for the payments to continue?

Take a structure built out of a single rule. Every step of the arithmetic can be checked on the back of an envelope, and a claim that has been checked is worth more than a claim somebody asserted.

The rule is this. Every participant who joins must bring two more people for the arrangement to continue. That is all. No rates, no periods, no returns, no fees. Just: each person brings two.

If each participant brings two, then each roundOne stage of recruitment. After it, the number of participants must have grown again for the payments to continue. of recruitment must be exactly twice the size of the round before it, so the one rule fixes everything else. Round 1 brings in 2 people. Each of those 2 people must bring 2, so round 2 brings in 4. Each of those 4 must bring 2, so round 3 brings in 8. Round 4 brings in 16. The requirement is not growing because the arrangement got greedy or because somebody decided to expand. The requirement is growing because the rule says each person brings two, and doubling is what that sentence means once it is written down.

The running total behaves in a way worth noticing on its own. After round 1, two people have joined. After round 2, six. After round 3, fourteen. After round 4, thirty. Each running total is two short of double the one before it, and each one is two short of double the round that has just finished. The pattern is not a coincidence, and it decides the share of participants who lose, so hold on to it: at every point in the life of this structure, the round that has just happened is roughly as large as every round before it put together.

Sixteen circles already crowd the width of the drawing, and sixteen people is nothing at all: it is a small wedding party, a classroom, the households on one floor of one building. Sixteen circles is round 4 of a structure that is about to be run on to round 31.

What do the numbers look like, round by round?

Every count below can be checked on the back of an envelope, and it is worth doing. Doubling rests on numbers a school child can verify rather than on anything that has to be taken on trust from anybody.

The second column, read downwards, is where intuition breaks. Round 10 needs 1,024 people: a large housing society, entirely believable. Round 20 needs 10,48,576, the population of a city, and round 20 is only ten rounds later. Ten more rounds after that, round 30 needs 1,07,37,41,824 people in that round alone, and round 31 needs 2,14,74,83,648.

Round 31 needs more people than live in this country, so round 31 does not happen, and there is no round 32, no round 33, no round 34 and no round 35. Round 31 is not a prediction about the structure and not a statement about anybody's competence. Round 31 is a statement about the number of human beings who exist. There is no promoter skilled enough, no market condition favourable enough and no amount of enthusiasm at any wedding anywhere that reaches a number of people who are not there.

Where does the arithmetic stop, and why must it stop?

The structure stops where the requirement crosses the supply of people, and that crossing point is fixed by the doubling rule and by nothing else. In the structure worked here, it falls between round 30 and round 31. Round 30 asks for 1,07,37,41,824 new participants, fewer than the people who live in this country. Round 31 asks for 2,14,74,83,648, more than the people who live in this country. Somewhere in that gap the structure runs out of the only raw material it consumes.

In practice it stops well before that, and it stops for ordinary reasons: recruitment slows in one town, a season is bad, people ask for their money back at the same time, somebody moves away. But the reason it stops is not the reason it had to stop, and confusing those two is the single most common error made about these arrangements. Whatever the immediate trigger, the structure was already walking towards a wall that was drawn before it opened, and the trigger only decided which round it happened to be standing in when it arrived.

Draw the requirement on a scale where each step upwards is one doubling and the picture becomes very plain. The line is straight. The line does not accelerate, it does not curve away at the end, and it does not do anything unexpected at round 28 that it was not already doing at round 4. The line rises at exactly the same rate the entire way, and it crosses the number of people who exist at a place that was determined the moment somebody wrote down the words each participant brings two.

A straight line is why the honest description of the ending is arithmetic rather than misfortune. Misfortune is a thing that might have gone otherwise. A doubling structure could not have gone otherwise, here or in any structure of this shape, and the only open question was ever which round it would be standing in when it stopped.

How many participants lose, and can that share change?

When the structure ends is only half the arithmetic, and the other half is the one a household actually needs. The share who lose is about who is standing where when it ends, and the answer is fixed in a way that surprises almost everybody who meets it for the first time.

Go back to the running totals. After round 5, thirty two people have just joined and sixty two people have joined in total. The newest round of thirty two is 51.6 per cent of everybody. After round 20, 10,48,576 people have just joined out of 20,97,150 in total, or 50.0 per cent to one decimal place. After round 27 it is 50.0 per cent again. The proportion barely moves, and it never falls below half.

The steadiness of that proportion is a property of doubling and nothing else. If each round is twice the round before, then the round that has just happened is the same size as every previous round put together, less the two participants the structure started with. So the newest round is always about half the total, whatever the total happens to be. Add the round before it, half as large again, and the last two rounds together are about three quarters of everybody who ever joined.

Whenever the structure stops, the people standing in its final roundThe stage at which no further participants arrive. In a doubling structure that stage always holds about half of everybody who ever joined. are at least half of everybody who ever joined, and the last two rounds together are at least three quarters, and neither of those figures depends on when it stopped or why. Stop it at round 5 and half of sixty two people are in the final round. Stop it at round 20 and half of 20,97,150 are. Stop it at round 27 and half of 26,84,35,454 are. The structure is bigger, the number of people affected is larger, and the proportion is the same.

The arithmetic and the structure are only numbers while they are kept apart, so put them together. The people in the final round have paid their money in. For them to be paid, there has to be a round after them. Final round means there is no round after them. So at least half of everybody who ever joins a structure of this shape loses everything they put in, and that is not a risk they are running, it is a fact about the structure they joined.

The ordinary way of putting it is much weaker than the truth, so say that back to yourself in the way it would sound if somebody said it aloud at a wedding. The ordinary way is: these things might fail, so be careful. The accurate way is: at least one person in every two who ever joins is certain, arithmetically, to lose their money, that proportion was settled before the first participant arrived, and nobody inside can see which of the two they are.

Why does nothing anybody does inside it move that share?

Because the share is not produced by anything that happens inside. The share is produced by the rule, and the rule was written before anybody joined.

Think about what a very good year inside one of these arrangements actually looks like. Recruitment goes well in three towns. Every payment goes out early and in full. Nobody asks for their money back. Word spreads, and the next round fills faster than the one before it. Every one of those is a real event, and a participant watching from inside would reasonably describe the arrangement as doing well.

Now ask what any of it did to the proportion. Recruitment going well means the base grew, and the base growing means there is now a larger round which is, once again, about half of a larger total. Payments going out early means earlier participants were paid, and paying them is the structure functioning exactly as described. Word spreading means more people joined, so more people are standing in a round that will turn out to be the last one. The doubling puts every new participant on the side of the line that loses, so every event that looks like the arrangement doing well is an event that adds people there.

The fixed shareA proportion settled by the shape of the structure before anybody joined, rather than by anything that happens once it is running. is why the language people reach for around these arrangements is so misleading. Somebody says it went wrong at the end. The arrangement did not go wrong at the end. The arrangement did at the end exactly what it was built to do from the beginning, and the ending was in doubt as to timing rather than as to fact.

Why do the early payments look like evidence when they are not?

Because they are evidence. Genuine evidence is what makes this so difficult, and calling the payments suspicious gets the structure exactly backwards. The payments are evidence of something. The question is what.

An unbroken recordA history of payments made in full and on time. A structure of this shape produces exactly that while it is running. of payments made in full and on time is evidence, precisely and only, that money arrived. The record is a true record of a true thing. The one thing a record cannot do is say where that money came from, and where the money came from is the only question that decides anything.

In this structure, money arriving is the whole of what happens, faithfully, every single round, right up until the round it cannot. A record of payments made is exactly what a functioning structure of this shape produces, and it carries no information at all about whether the next round exists. Two years of perfect payments is consistent with an arrangement that produces something, and it is equally consistent with a doubling structure standing in round 8 of a life that ends at round 30. In both cases the payments were made, so the record does not separate them.

Did somebody who joined early judge it better than somebody who joined late?

The comparison between an early joiner and a late one matters most of all, and it lands hardest on a household that has already lost money in something shaped like this. The arithmetic proves what people usually only assert.

Picture two people. One joined in round 4 and was paid, in full, three times. The other joined in round 20 and lost everything. Now ask what was different about them.

Both heard the same description. Both saw the same kind of record, a run of payments made in full and on time, longer in the second case than in the first. Both asked the same questions, and got the same answers, and those answers were consistent with everything they could check. Both reasoned identically. By round 20 the run of successful payments was sixteen rounds longer, so the one in round 20 arguably had more evidence in front of them.

The difference between them is a position in a sequenceWhere somebody happened to arrive in the order of rounds. In this structure that position decides the outcome entirely. and not a difference in judgement. No event inside the structure moves the share who lose. If nothing inside moves it, then nothing anybody inside did moved it either, including everything either of these two people did. Round 4 was paid because there was a round 5. Round 20 was not paid because there was no round 21. The presence or absence of a next round is the entire difference.

What does this arithmetic mean for somebody deciding today?

The arithmetic means the thing they are uncertain about is not the thing they think they are uncertain about.

Somebody weighing one of these arrangements almost always frames the question as: is this going to work, or is it going to fall over. The doubling rule already answered it. Whether it will work is a reasonable question about most things and the wrong question here. A structure of this shape stops. A structure of this shape stops for the same reason a staircase with 31 steps has no 35th step. The ending is not the open question. Treating the ending as the open question leaves somebody with nothing to think about except how convincing the arrangement looks, and looking convincing is the one thing it is designed to do.

The open question is which half a participant is joining. An answer that exists and cannot be seen makes for a different kind of question entirely. Somebody joining in round 8 of a structure that will stop at round 30 is in the half that gets paid. Somebody joining in round 8 of a structure that will stop at round 9 is not. Same person, same decision, same evidence, and the two situations are indistinguishable from where they are standing.

Here is what that person can actually see: how long it has been running, that payments were made in full, that people they know were paid, and that whoever runs it is organised and answers questions. Here is what would actually decide their outcome: which round they are joining, how many rounds are left, where the money reaching them came from, and whether anybody will join after them. Nothing on the second list is visible from inside, and everything on the first list looks the same in round 8 of thirty as it does in round 8 of nine.

Nobody can say in advance which round is the last one. A description that states an outcome as certain, that names what money will become by a particular date, or that treats a payment as settled before it has happened is therefore a thing to recognise and never a thing to act on. Everything certain in the arithmetic is certain about the structure and not about a person: at least half of all participants lose, and which half any one of them is standing in is exactly the thing the arithmetic proves nobody can know from inside.

How does anybody use this arithmetic in front of a real offer?

By replacing a question that cannot be answered with one that can. The question is this: where does the money that reaches me come from, and what happens to it if nobody new arrives after me.

The question is not clever and it is not technical, and it is the same question a lender asks when deciding whether to lend to a business. A lender does not primarily ask whether the last twelve repayments were made. Twelve repayments made is a fact about the past that a borrower with a cheque book can produce in either case. The lender asks what the repayments are coming out of: takings, receipts, work billed and collected. If the answer is that the next repayment depends on raising money from somebody new, the lender is looking at a very different arrangement from one where it depends on customers buying things, and the past twelve repayments look identical in both.

An analyst reading a set of accounts does the same separation with a different vocabulary, splitting cash the activity generated from cash raised from people putting money in. The two arrive in the same bank account, spend identically, and mean completely different things about next year. Every professional treatment of this subject, in lending, in accounts and in supervision, comes back to the same move: separate money that was produced from money that was handed over. Only the first kind has any bearing on whether it happens again.

For a household the same question is even shorter, and the Bhosale household is a fair example of why it is worth having. The household holds two accounts, a recurring deposit, a public provident fund account, some gold and a two-wheeler. There are no shares, no fund and no monthly investment plan, and nobody there has ever taken advice from anybody. Nobody in the household is in a position to analyse a set of accounts. But it can ask one question at a wedding, and the question costs nothing: where does the money that would come back come from. If the honest answer is from people who join later, the arithmetic above applies in full, and the record of payments already made does not soften it by a single round.

What can arithmetic on a stated rule prove, and what can it not?

Arithmetic on a stated rule proves whatever the rule forces, and nothing else. The rule is: each participant brings two. From that one sentence come the rounds, the counts, the stopping point between round 30 and round 31, the ledger of eight quarters and the two participants in round 4 and round 20.

A pattern is not an accusation, and keeping the two apart is not a legal formality. Naming a scheme, a promoter or a platform is a claim about what particular people actually did, and a claim of that kind is a matter for evidence and for the authorities named above. Arithmetic on a stated rule settles the shape of a structure and settles nothing about any particular party.

The fate of money that has gone into such an arrangement, what an authority does about it, what a court would find and whether anything is ever returned are all matters of fact in a particular case, decided in that case and by the authorities named above. The Bhosale household did not get the Rs 18,000/- back that it lost on 14 September. Its buffer fell from Rs 31,320/- to Rs 13,320/-, and from 0.73 months of the Rs 42,770/- that leaves the household each month to 0.31 months. Most of the time money does not come back, and an ending in which it did would be a comfort written at the expense of every reader whose money did not return.

The Bhosale household, Meghna Bhosale, Ashok Bhosale, Ira Bhosale and the doubling structure worked here are invented.
Educational material. Not advice on any investment, tax, budget or market position.