Ponzi Maths–Part 3

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This is get­ting a bit like Star Wars, but I promise–this will be the last post in this series. In the pre­vi­ous two, I con­structed a model of a pure credit econ­omy in which the money sup­ply and eco­nomic activ­ity can expand smoothly of time.

Of course, that’s not the real world. As we know from the bit­ter expe­ri­ence of the finan­cial cri­sis that is this blog’s rai­son d’etre, finance char­ac­ter­is­ti­cally desta­bilises an oth­er­wise healthy econ­omy. Part of the rea­son for that is the exis­tence of Ponzi Financing–something that the recent Bernie Mad­off scan­dal has thrown into $50 bil­lion high relief.

In real­ity, given our cur­rent finan­cial sys­tem and how assets are val­ued and defined, there is a Ponzi poten­tial that the sys­tem almost inevitably suc­cumbs to. But it is a “Type II” Ponzi Scheme: given the poten­tial to make unearned prof­its by buy­ing and sell­ing assets on a ris­ing mar­ket, the game of lever­aged asset price spec­u­la­tion seems to inevitably take hold–and in our epoch, this has reached heights that have never before been seen.

But this series of posts is about a pure “Type I” Ponzi Scheme: one in which a pro­moter accepts “invest­ments” and pro­vides fan­tas­tic returns from the sim­ple expe­di­ent of return­ing the prin­ci­pal (with a time lag) and call­ing it inter­est. So long as the inflow of funds con­tin­ues to grow, such a Scheme seem like an easy road to riches, which nor­mally inspires “investors” to rein­vest their “profits”–and pos­si­bly even more–as the Scheme con­tin­ues. But ulti­mately the Scheme must come a-cropper, as either the inflow becomes insuf­fi­cient to finance the level of promised out­flows, or redemp­tions rapidly empty the Scheme of its triv­ial actual reserves.

It’s the lat­ter mech­a­nism that I model here.

Firstly we need our legit­i­mate finan­cial sys­tem, as described in the last table of the pre­vi­ous post:

Type 1 0 –1 –1 –1
Account Firm Loan (FL) Bank Reserves (BR) Firm Deposit (FD) Bank Deposit (BD) Worker Deposit (WD)
Inter­est on Loan +A        
Inter­est on Deposit     +B –B  
Pay Inter­est on Loan –C   –C +C  
Pay Wages     –D   +D
Inter­est on Deposit       –E +E
Con­sume     +F+G –F –G
Repay Loan –H +H –H    
Relend Reserves +I –I +I    
Extend Credit +J   +J    

To this we need to add one more col­umn to record the actual deposits of a Ponzi Scheme:

Type 1 0 –1 –1 –1 –1
Account Firm Loan (FL) Bank Reserves (BR) Firm Deposit (FD) Bank Deposit (BD) Worker Deposit (WD) Ponzi Deposit (PD)
Inter­est on Loan +A          
Inter­est on Deposit     +B –B    
Pay Inter­est on Loan –C   –C +C    
Pay Wages     –D   +D  
Inter­est on Deposit       –E +E  
Con­sume     +F+G –F –G  
Repay Loan –H +H –H      
Relend Reserves +I –I +I      
Extend Credit +J   +J      

The first action in this extended table is a trans­fer from firms (for now I’ll pre­tend that firms do the Ponzi investing–it’s actu­ally cap­i­tal­ists them­selves who tend to do it, as the Mad­off Scan­dal has made clear, but that would add yet another col­umn to an already com­pli­cated table) of funds to invest in the Ponzi Scheme. This adds an addi­tional row in which the sum K is trans­ferred from the firm’s deposit account to the Ponzi’s.

We now need a sec­ond table to record the actions of the Ponzi Scheme itself. Here there are only two columns: one mir­ror­ing the Ponzi Deposit above, which records the actual money the Ponzi Scheme has; the other a col­umn show­ing the returns the Ponzi Scheme alleges it is mak­ing from its fan­tas­tic scheme (the orig­i­nal Charles Ponzi promised to make the money by exploit­ing arbi­trage on postage coupons), but which are in fact entirely fictional.

The essence of a Ponzi Scheme is to return prin­ci­pal to the investors, and call it inter­est. In fact, the only inter­est the Ponzi is mak­ing is com­ing from the banks’s pay­ments of inter­est on the Ponzi’s deposit account at the bank–so the two entries in this sec­ond row (L for the actual return and M for the fic­tional) are very different.

It’s impor­tant for a Ponzi pro­moter to appear fab­u­lously wealthy, so there is a sub­stan­tial con­sump­tion from that is shown to come from both the actual and the fic­tional. This is the deduc­tion N in the third row.

An essen­tial step in a suc­cess­ful Ponzi Scheme is to give investors what they expect; so the next row shows the Ponzi Scheme mak­ing a pay­ment from its account to the firm’s account, where the rate of return is much higher than that on stan­dard finan­cial prod­ucts; this is the deduc­tion O in the fourth row.

The final step is the rein­vest­ment of these prof­its by the firms with the Ponzi Scheme–and even more funds, given how “suc­cess­ful” the Ponzi Scheme is in com­par­i­son to other invest­ments. A trans­fer of P occurs

All these steps in a Ponzi process are mir­rored in the accounts kept by the banks–except for the fic­tional bal­ance of course, as again the Mad­off scan­dal has indi­cated. So our table for legit­i­mate finance now is:

Type 1 0 –1 –1 –1 –1
Account Firm Loan (FL) Bank Reserves (BR) Firm Deposit (FD) Bank Deposit (BD) Worker Deposit (WD) Ponzi Deposit (PD)
Inter­est on Loan +A          
Inter­est on Deposit     +B –B    
Pay Inter­est on Loan –C   –C +C    
Pay Wages     –D   +D  
Inter­est on Deposit       –E +E  
Con­sume     +F+G –F –G  
Repay Loan –H +H –H      
Relend Reserves +I –I +I      
Extend Credit +J   +J      
Ponzi Invest     –K     +K
Ponzi Scheme       –L   +L
Ponzi Con­sume     +N     –N
Ponzi Return     +O     –O
Ponzi Rein­vest       –P   +P

and the table for the Ponzi Scheme is:

Type 1 –1
Account Ponzi Deposit (PD) Ponzi Scheme (PS)
Ponzi Invest +K +K
Ponzi Scheme +L +M (where M»L)
Ponzi Con­sume –N –N
Ponzi Return –O –O
Ponzi Rein­vest +P +P (where P>O)

One impor­tant aspect of the Scheme is that, from the bank’s point of view, every­thing seems legit­i­mate. All flows from the Scheme on the bank’s books are balanced–and the Ponzi investor is a major source of demand for the out­put of the firm sec­tor too.

As Mad­off showed, a Scheme like this can go on for a very long time, so long as the returns being promised are not too out­ra­geously good. His deliv­ery of effec­tively 12% a year through good times and bad was much more sus­tain­able than Charles Ponzi’s promise of a 50% return in 45 days.

But trou­ble comes when either the inflow needed to sus­tain the (lagged) out­flow from the Scheme exceeds the capac­ity of the legit­i­mate econ­omy, where actual mon­e­tary prof­its are being made, or when trou­ble in the econ­omy mean that investor stop putting their funds into the Scheme–or worse still, as hap­pened to Mad­off, start to make with­drawals from the Scheme to meet debt com­mit­ments else­where in the economy.

I model this with a sim­ple change in para­me­ter val­ues: when a cri­sis hits, the inflow and rein­vest­ment in the Ponzi Scheme drop to zero (in prac­tice, as we also know from Mad­off, “investors” actu­ally with­draw their funds–something a Ponzi Scheme has to allow to avoid sus­pi­cion, in the hope that it will sur­vive the downturn.

No such luck. The ques­tion every­one was ask­ing after Mad­off failed was “where did the money go?” The answer is (a) that the amount of money that was actu­ally “there” was a lot less than Mad­off claimed–he didn’t make the returns he claimed to be mak­ing, so the size of the fund itself (con­sist­ing largely of “rein­vested” “prof­its”) was much smaller than he claimed; and (b) it was repaid to investors who hung on to it–and spent it, or used it to pay other debts–rather than rein­vest­ing it. So it evap­o­rates in a com­par­a­tive flash:

It is also the case that, for its entire his­tory, the Ponzi is effec­tively insol­vent. The accu­mu­lated funds it claims to have are its debts to its “investors”; the funds it has on hand at the bank are its actual assets. The for­mer far exceeds the lat­ter, either side of the cri­sis that brings the Scheme to an end:

So that’s the basic math­e­mat­ics of a Ponzi Scheme. In my forth­com­ing book (still a long way off!), this will be gen­er­alised to include a process by which the shift in sen­ti­ment occurs that brings a Ponzi Scheme to a crash­ing halt. This how­ever is the bare bones of how such Schemes work, and how they fail.

Appen­dix: the actual mathematics

I hope the above wasn’t too intim­i­dat­ing to non-mathematical read­ers! In my Feb­ru­ary Debt­watch, I’ll go much more slowly and explain why a model like this–which allows for the endoge­nous expan­sion of credit inde­pen­dent of any manip­u­la­tion of the money mul­ti­plier rela­tion by the cen­tral bank–is needed to explain the empir­i­cal data on money.

Below are the full the full equa­tions for the model (in tab­u­lar lay­out). The mod­els are sim­u­lated in Math­cad (though they could eas­ily be done in any math­e­mat­ics pro­gram that sim­u­lates sys­tems of Ordi­nary Dif­fer­en­tial Equa­tions, such as Math­e­mat­ica or Mat­lab). I’ll gladly share them with any researcher inter­ested in explor­ing the tech­nique and the mod­els. For­mat­ting equa­tions in Word­Press is a bit tricky, and I haven’t mas­tered the tables yet, so to make them look not too dread­ful I’ve put line breaks inside some of the terms. I’ll explain all this in more detail in February–for now I’m bug­gered and it’s time I got back to some aca­d­e­mic papers!

Legit­i­mate Finance

 

Type 1 0 –1 –1 –1 –1
Account Firm Loan (FL) Bank Reserves (BR) Firm Deposit (FD) Bank Deposit (BD) Worker Deposit (WD) Ponzi Deposit (PD)
Inter­est on Loan +rL×FL          
Inter­est on Deposit     +rD×FD –rD×FD    
Pay Inter­est on Loan –rL×FL   –rL×FL +rL×FL    
Pay Wages     -(1-s)/tF×FD   +(1-s)/tF×FD  
Inter­est on Deposit       –rD×WD +rD×WD  
Con­sume     +BD/tB+WD/tW –BD/tB –WD/tW  
Repay Loan –FL/tLR +FL/tLR –FL/tLR      
Relend Reserves +mR×BR –mR×BR +mR×BR      
Extend Credit +nM×FD   +nM×FD      
Ponzi Invest     –pI×FD     +pI×FD
Ponzi Scheme       –rD×PD   +rD×PD
Ponzi Con­sume     +PS/tP     –PS/tP
Ponzi Return     +rP× PS     –rP×PS
Ponzi Rein­vest       –pR×rP  

×PS

  +pR×rP  

×PS

Ponzi Scheme

 

Type 1 –1
Account Ponzi Deposit (PD) Ponzi Scheme (PS)
Ponzi Invest +pI × FD +pI × FD
Ponzi Scheme +rD × PD +rP × PS
Ponzi Con­sume –PS/tP –PS/tP
Ponzi Return –rP × PS –rP × PS
Ponzi Rein­vest +pR × rP × PS +pR × rP × PS

About Steve Keen

I am a professional economist and a long time critic of conventional economic thought. As well as attacking mainstream thought in Debunking Economics, I am also developing an alternative dynamic approach to economic modelling. The key issue I am tackling here is the prospect for a debt-deflation on the back of the enormous debts accumulated in Australia, and our very low rate of inflation.
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One Response to Ponzi Maths–Part 3

  1. Is this a 6 vari­able sys­tem or a 7 vari­able one?

    Do you need dPD/dt as well as dPS/dt?

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