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:

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

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:

and the table for the Ponzi Scheme is:

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

Ponzi Scheme