5.4 Arbitrage in the Multi-period Model
5.4.6 Ponzi Schemes and Bubbles
Up to now we have left the choice of time horizon in our multi-period model open:
there could have been finitely many periods (T <1) or infinitely many (“TD 1”).
There are, however, some interesting theoretical issues arising in the infinite time horizon settings that we discuss in this section.
The first is the so-calledPonzi scheme. A Ponzi scheme is probably the only theoretical concept in economics that is named after an outright criminal, Charles Ponzi. In 1920 he attracted enormous investments and paid huge interest for it – but only by using newly arriving investments, thus getting deeper and deeper into debt.
At the same time he financed a luxurious life from this and even bought a private bank. Finally, the whole scheme broke down, and he ended in prison for many years.
The idea of using the money collected from subsequent investors to repay the initial investors and their interest was not invented by him, but since his was probably the most famous case, the scheme was named after him. In economics, Ponzi schemes are defined in a more abstract way: borrowing money from the future to repay debts now and, at the same time, finance consumption. To model this more precisely, we consider a set of short-lived bondsDtwith payoff
Dt.!/D
(1 if!2˝t; 0 otherwise:
With their help we can construct the following consumption path:
ctDwtC1for alltD0; 1; 2; : : : ;T,
wherewt is the exogenous income. Taking a closer look, this looks very attractive:
we get something extra ineveryperiod in time! It seems impossible to finance such a consumption stream without a large initial endowment, but in fact we can do this without any initial endowment: define the investment strategytD0;1;:::;Tas
0D.R; 0; : : : /; : : : ; tD.0; : : : ; 0;
Xt iD1
Ri; 0; : : : / ;
wheretis different from zero in itst-th component (Fig.5.7).
This means nothing else than that consumption in periodt D 0is financed by going short on the first bond with a total amount ofR. To finance consumption in periodt D 1and to pay back the first bond one issues a second bond with a total amount ofRCR2, etc. Formally, we get in periodtD0:
01q10Cc0D R1
RCc0Dw0:
5.4 Arbitrage in the Multi-period Model 233 Fig. 5.7 Illustration of the
Ponzi scheme
Analogously for any arbitrary periodt, ttC1qtCt 1CctD
tC1
X
iD1
Ri1
RCctDtt 1CwtD Xt iD1
RiCwt:
Thereby we getctD wtC1for alltas mentioned above. This leads to immediate problems. For instance, there does not exist any maximum utility, since the choice set becomes unbounded. Whatever we have, we can still get more. An arbitrage opportunity arises out of seemingly innocent short-term bonds!
There are two ways of solving this problem:
• We can restrict our multi-period model to a finite time horizonT<1.
• We can impose atransversality condition:
t!1lim Rt XK kD1
tkqkt
! 0 ;
wherePK
kD1tkqkt is the total value of the portfolio.
Are Ponzi schemes nowadays extinct? Not really: there are still many fraudulent schemes like this, disguised under various, often creative names. How long can a Ponzi scheme run? Since the input of money is bounded in reality, such schemes are doomed to collapse. Charles Ponzi’s scheme broke down after several months – the interest rates he promised were just too high. The lower the interest rates promised, the longer such a scheme can live.
The most recent example was the now infamous hedge fund run by Bernard Madoff, former chairman of the NASDAQ stock exchange, from the 1980s to December 2008. His returns were always around 10 %. Nothing outrageously high, but still substantially above the risk-free rate and even above the average performance of stocks, yet seemingly without much variance. Even in times of stock market declines like after the end of the internet bubble (2000–2002) or during the financial crisis (2007–2008), his hedge fund still generated steady returns. From December 1990 to May 2005 there were in fact only seven months (out of 174) with negative returns!
That Madoff was quite silent about how he generated these returns was not remarkable: no hedge fund would share its secrets and thus destroy the sole base of its business. What was maybe more remarkable is that even though the hedge fund got checked a few times by the SEC and other regulatory commissions, there seemed to have been found nothing wrong with it. In the end, 50 billion US dollars were invested into the fund, some of it directly, some of it via a network of hedge funds and investment companies worldwide that either knew or didn’t know what was really going on (we might never know!): the biggest Ponzi scheme of all times!
The way Madoff had marketed his fund was fundamentally different from Ponzi’s. This allowed the fund to grow over such a long time and to such a large size.
The end, however, was similar: some investors wanted to withdraw their money, but there was none left, and Madoff had to admit that everything had been a giant Ponzi scheme. Not only rich private investors that knew Madoff personally and trusted him as a respectable person had lost their money (and the money of several charity funds), but also various hedge funds and even larger banks had lost billions.
Are all Ponzi schemes designed by criminals collecting money of investors by pretending to invest it? Or are there other phenomena that can be interpreted as Ponzi schemes in the economic sense? Let us consider the following prominent real life example: credit cards. To finance consumption some people obtain a credit by getting a credit card. A second credit card is later used to repay the debt on the first credit card and also to finance new consumption, etc. In praxis such strategies are used by a substantial number of people, in particular in the USA where the average credit card debts is particularly high.
But not only Ponzi schemes can pose a problem for an infinite horizon model.
Bubbles are a further challenge. Similar to a Ponzi scheme, a bubble relies on expectations of large profits in the future. However in this case no debts are accumulated. Consider an asset withDtD1for allt(a “console”). We can compute its price attD0:
q0 D X1
tD1
.!t/ :
Using
.!t/Dıtu0.ct/ u0.c0/;
which we know from the two-period model (Sect.4.1), we get:
q0D X1
tD1
ıtu0.ct/ u0.c0/:
5.4 Arbitrage in the Multi-period Model 235
Analogously we compute attD1: q1D
X1 tD2
.!t/D X1 tD2
ıt1u0.ct/ u0.c1/ Dı1u0.c0/
u0.c1/ X1
tD2
ıtu0.ct/ u0.c0/ Dı1u0.c0/
u0.c1/
q0ıu0.c1/ u0.c0/
:
For an arbitraryt, we get
qtC1Dı1 u0.ct/ u0.ctC1/
qtıu0.ctC1/ u0.ct/
:
Hence,
qtDıu0.ctC1/
u0.ct/ Cıu0.ctC1/ u0.ct/ qtC1
DWMtC1CMtC1qtC1: We defineM0tWDQt
D1Mand obtain by iteration
q0DM1CM1q1DM1CM1.M2CM2q2/D D D
X1 tD1
M0t
„ƒ‚…
fundamental
C lim
T!1M0TqT
„ ƒ‚ …
bubble
:
Whereas the first component is the classical “fundamental value” price of the asset, the second component only depends on expected later prices and is therefore in fact a “bubble” component. This decomposition implies in particular that the price is not uniquely determined anymore. But even worse, a bubble can lead to a positive value for an asset without any dividend or payment! LetDt D0for allt, then
q0D lim
T!1MT0qT
can be positive (compare [Bew80]). One could argue that money, having no dividend payment nor interest, has just a value because it is a bubble, but maybe this point of view is taking it a bit too far. . .
For many applications, we would like to exclude bubbles. This can be achieved by either restricting the analysis to a finite time horizon or by imposing the condition
T!1lim M0TqTD0 :
It is, however, interesting to know that bubbles occur naturally in an infinite horizon model, given that they also form on real financial markets.