What are "Financial Bubbles": Approaches and Definitions

Dimitriadi G.G.fgdimitriadi@yahoo.com ) The Institute for System Analysis of the Russian Academy of Sciences

1. Introduction

In the press some messages about different financial bubbles, about investors deceived take place from time to time. A natural question arises: what are financial bubbles? How journalists, the population of a country and, at last, researches-economists understand them?

But really there are several approaches towards mathematical description of financial bubbles. In the sections 2-5 the main modern approaches are described; their variety is shown. Later in the sections 6-7 a definition of a financial bubbles in terms of scenario approach is introduced.

2. Rational Financial Bubbles

The paper [1] is deduced to an investigation of the question what financial bubbles are, if they can exist, which forms can they take in case of rational behavior of all market participants and what tests can be used to detect them.

Not giving a definition of a bubble the authors mentioned that financial bubbles are "movements in the price, apparently unjustified by information available at the time, taking the form of a rapid increase followed by a burst or at least a sharp decline".

In the paper the effectiveness of the market is assumed and a quite strong assumption is made: all market participants have one and the same information after "announcement" of prices. Yield of the

p — p + x

ownership of the asset is Rt = 't+1 ^ 1 ^ . Here p is a price of the asset, xt is a benefit of

pt

possessing of the asset at the moment t, referred to as "dividend". In the paper it is assumed that E (R |fit) = r, E (pt+1 |fit) — pt + xt = rpt, where r is given constant, Qt is an information,

accessible to all market participants at the moment t. This condition is satisfied not only by the fundamental of the asset, defined by the formula:

p = Yj0'+1E(xt+i |Qt) ,where 0 = (1 + r)—1 < 1, -

i =0

but by any other values of the following form:

pt = p"t + ct, where E (ct+1 |Qt) = 0—ct.

Later in [1] three examples of such ct, being financial bubbles according to the definition introduced, are given.

After giving a definition and examples of financial bubbles the authors consider more complex questions. For example, they obtain that bubbles can exist in an individual perception of concrete market participants if the condition of equal common information is eliminated, because fundamentals of the asset are different from the point of view of different persons in this case.

Later it is shown that financial bubbles cannot exist in case when there are only infinitely lived market participants. Also the authors assert that bubbles are more probable in markets, having difficulty defined fundamentals, for example, in the gold market, and bubbles are less probable in markets, having clearly defined fundamentals as blue chips stocks or perpetuities.

In the next section the following question is investigated: how financial bubbles can influence on fundamentals? It is important that presence of a bubble causes changes in fundamental of the asset.

Besides that it is shown that a bubble on the price on one asset can cause change (diminishing) of the prices on many other ones, i.e. it influences on other markets. According to the authors there are general equilibrium models when a bubble takes place in the market of only one asset.

2627

The last part of the paper [1] is devoted to design tests to detect financial bubbles using statistical data. As it is impossible to describe all bubbles, the authors use the hypothesis of the absence as the null hypothesis.

At first, considering prices p and dividends x as known ones, bounders of variances for rational financial bubbles are obtained, and their violation at real statistics is shown. However this violation can be caused, for example, by irrationality of market participants etc.; therefore one can say nothing exactly about presence of bubbles in case in question.

Later tests, when only prices p are known, were introduced and used for the statistics: runs tests and tail tests. Results of their usage give presents of bubbles in the gold market. However these criteria have less power then the ones mentioned above.

3. Financial Bubbles as Ponzi Games

At the beginning of the paper [2] the authors define a financial bubble as a private company, which attracts deposits of private persons and promises unlikely high interest rates, which are many times more than market rates. For some period of time the firm upholds a reputation and fulfills its commitments, but later it becomes a bankrupt.

At the same time existences of financial bubbles cannot be explained by the point of view of rational behavior of individuals. Different authors use various models, dealing with violation of rationality, to explain this phenomenon. In paper [2] the game approach is used.

A financial bubble is being modeled as stochastic incomplete information game (as this term is understood in the game theory) between the Ponzi firm and a population of individuals.

It is supposed that the Ponzi firm knows all its moves and moves of the population, but individuals know only their own moves and of several their acquaintances. The Markov's property of the history is assumed. It allows to introduce probabilistic transition functions from one state into another one. At each step the Ponzi firm plays not on every concrete individual but on their totality and every of them plays only on the firm. Nobody invests into the financial bubble after its burst. In the paper [2] other assumptions are made too.

In contrast to other approaches in [2] a population is not considered homogeneous; that is important achievement of the authors. All amount of real and potentials investors is divided by the them into two large groups: naïve and sophisticated individuals. Sophisticated individuals know that a financial bubble is predeterminantly finished. But they are sure they can invest into the bubble such that it gives them profit. They are substantionally rational, i.e. they maximize their profit on the bases of all the information known and current preferences. Naturally their preferences are changing in time. In contrast to them naïve investors do not understand, what game they play, and invest because others have made the same before and have been profitable. It is assumed the amount of the investors among naïve individuals mainly grows proportionally to the amount of successful bubble investors, known by them.

The authors consider strategies of sophisticated individuals. An idea of phasic incomplete information game, i.e. such a game which takes place at one step between an individual and the Ponzi firm, is introduced. Let us emphasize that really the firm plays on the totality of individuals but not on every of them; but an individual see only the subtracted game. In the paper it is obtained that in the phasic subtracted game in strategic form the equilibrium components ('O+C, I) and

(O~D,W) contain perfect successive own entity equilibriums only if the firm considers the

strategies C and D as optimal ones accordingly and an individual believes that this events have high probabilities. In spite of the fact, that according to this statement there are two equilibriums in the subtracted (i.e. perceived by an individual) game, the following statement is true: the only equilibrium in the stochastic incomplete information Ponzi game is the trivial equilibrium in every phasic game.

At first sight this statement looks paradoxical, however it is strictly proved. Its truth and existence of real financial bubbles is connected with the fact that the phasic game, mentioned

2628

above, does not reflect the whole game (the Ponzi firm plays on the population, not on the concrete individual) and with the incomplete information, which is known by an individual.

The authors of [2] consider the dynamics of the amount of naïve and sophisticated individuals and the dynamics of the amount of investors among them. Naïve individuals choose their own strategy starting from their current observations about the most profitable strategy and the amount of acquaintance, who are successful investors. Also deposits can be withdrawn to pay unforeseen bills. To model such behavior of naïve individuals the authors use equations of replicator dynamics in discrete form, which allows to take into account teaching rules, described above.

Also some assumptions about a behavior of sophisticated individuals are introduced. The urgent moment is that they nevertheless take part in a Ponzi game, so they do not have full information about the whole game. Among their persuasions there is a probability of the event that a financial bubble bursts at the next step according to their opinion. This probability is rising in time, trending at the limit to the unity. Using this and other assumptions in the paper equations of dynamics of naïve, sophisticated individuals, their united amount and the amount of investors among each of these groups are obtained.

The main conclusion of the paper is: "DO NOT PLAY PONZI GAMES!"

4. Scenario Approach to Bubbles Description

In the papers [3, 4] there is a model of a financial bubble, which is designed to describe qualitatively a mechanism of Russian crisis of 1998.

In [3, 4] S. V. Dubovsky introduced a model of a financial bubble (so called scenario approach to bubbles description), in which different cases of the growth of the outstanding total face-value of the bubble securities in circulation or total value of current sales are given as scenarios, which are monotone increasing functions of time. A bubble exists without reinvestment of funds gathered and only by the moment, when current expenses for paying to off the securities issued before become more than the current revenue from sales of new securities. This moment is a moment of the financial bubble burst, because continuation of its existence is not lucrative for the Organizer. From this moment in case of the continuation the total revenue gathered by the Organizer stops to increase and begins decreasing. So, in the model of S. V. Dubovsky the hypothesis that a bubble bursts, when almost all potential investors have already become real ones, is not necessarily; though the scenario of growth of total values of sales of bubble securities, where the power of growth slowly decreases, can be given, for example, using the equation of the logistic curve. Three formal relationships (an equation of bubble growth, a condition of stopping of its existence, a scenario of growth of the outstanding total face-value of the bubble securities in circulation or total value of current sales) allow to calculate main parameters of a bubble: the breakeven lifetime, the total revenue, gathered by the Organizer, benefits and losses of investors.

In [3, 4] analytical expressions for breakeven lifetime under conditions of two scenarios (the linear and the quadratic growth of the outstanding total face-value of the financial bubble securities on time) are obtained. It proved that the breakeven lifetimes depend only on periods of fulfillment commitments and interest rate to be paid to investors. The gradient of the line of the linear growth and the parameters of the quadratic growth influence on profits and losses and does not influence on the bubble lifetime. To three main parameters S. V. Dubovsky added a two-parameter demand function of investors on bubble securities. The demand is rising when profitability for investors is rising; it is infinite at zero price; it is equal to zero at price, which is equal to the price-value. Including of such function into a scenario of growth allowed to calculate the optimal price, which gives the maximum of the revenue, obtained by the bubble Organizer.

dG

The financial bubble growth is described by the author using the equation: — = g(t) - g(t - <).

dt

The condition of existence is written as cg(t) - g(t -<) > 0.

2629

Here G is the outstanding total face-value of the bubble; g(t) is a total face-value of the bubble

securities sold exactly at the moment t; cg is a price (noted in parts of the face-value) of security

sales; 0 > 0 is a fixed time period, after which the moment of commitments fulfillment takes place; it is counted off from the moment of their sale.

The equation, written above, causes: because always cg < 1 a financial bubble is knowingly

unprofitable for the Organizer if it is not rising.

The author obtained the analytical formulae for the bubble lifetime for scenarios of the linear and the quadratic growth: g(t) - g(t -p) = p and

g (t) - g (t -p) = P(t - to).

For example, if investors are paid of 100% per annum (p = 12 months, cg = 0.5), the "linear"

financial bubble exists during 24 months, the "quadratic" one exists during 35 months.

With the help of such models and other speculations the author of [3, 4] gives a quality explanation of the Russian crisis of 1998.

5. Discussion of the Approaches Described

The descriptions, given above, show that the term "financial bubble" can be interpreted as conceptually similar, but quite different objects (phenomena).

In the approach of "rational financial bubbles" [1] they are understood as "movements in the price, apparently unjustified by information available at the time, taking the form of a rapid increase followed by a burst or at least a sharp decline", i.e. as one of types of market behavior.

In the paper [2] it is declared that a financial bubble is a private company, which attracts deposits of private persons and promises unlikely high interest rates, many times more than market rates. For some period the firm upholds a reputation and fulfills its commitments, but later it becomes a bankrupt. Such comprehension is usual for bubbles description in the press.

To be more exact, in terms of the game approach a financial bubble is understood as one of types of economic behavior, as Ponzi game (named by the Italian researcher, who firstly described a financial bubble in scientific literature).

In terms of scenario approach of S. V. Dubovsky the author [3, 4] does not gives a clear definition. Evidently he interprets a bubble as such behavior of its Organizer that fulfillment of commitments undertaken takes place using only attracting of new borrowings. So, there is no common approach to the idea of a "financial bubble". Below a more detailed definition of a bubble in terms of scenario approach is introduced.

6. Development of the Scenario Approach

From the review, given above, and from the press it is clear the though there is some common understanding of "financial bubbles" they can take completely different forms.

For example there are schemes, where each next participant should attract not less than 4 new ones, or, schemes, connected with creation of a holding company, active advertising of its businesses included and sale of its shares at the moment of the maximum price.

Below there is an effort if not to cover all situations (for example, evidently, some cases taken into account in the "rational bubbles" approach will not be covered), but in any case to abstract from an organizational form and to give a definition as general as possible.

Definition 1. A financial bubble is such policy of external debts management of an organization1 that during long time period fulfillment of commitments undertaken takes place using only attracting of new borrowings.

Such definitions suits more for an external observer, because usage of a model of a bubble to describe debts of organizations is entirely natural, since behavior of the company management (with

1 Using the terms "organization", "Organizer", "firm" they understand a person (juridical or physical), which is factually organizes a financial bubble and can essentially influence on its activity.

2630

respect to external borrowings) as a bubble is a pessimistic way of a firm activity. For an external observer it is important to clarify if the behavior of an organization can be regarded as a bubble, and if yes, what parameters it has.

Let us give one more variant of the same definition, which is more comfortable for usage from the point of view of the Organizer.

Definition 2. A financial bubble is the following financial scheme. The bubble Organizer sells its commitments. According to them it engages to pay some concrete amount of money in the future. Their fulfillment takes place using only the revenue of new commitments cells.

It is assumed that the Organizer meets all commitments up to the moment called "the bubble

Only the period of a financial bubble existence before its burst is considered. A main feature of the approach is that the bubble burst is caused not by the lack of investors, when almost all potential investors have already become real ones, or by other similar reasons, but by the fact, that at a certain moment of time it may prove that the bubble Organizer does not have a sufficient amount of funds to meet its commitments.

Such definition has already been implicitly used in developing of the scenario approach in the papers [5-9], but when it is clearly formulated it allows to implement the results obtained to more wide scope of situations.

7. Main Ideas of the Scenario Approach

To illustrate the ideas abovementioned let us state the main ideas of the scenario approach. Without any loss of generality let us suppose that the Organizer commitments are bonds without coupon ("zero" bonds); let us call the sum of commitments of every bond its "face-value". Without any loss of generality let us believe the face-value is equal to unity.

Here the following notations are used. Let us assume that the bubble begins at the moment t = 0. G (t) is the outstanding total face-value of the bubble securities in circulation by the moment t. V (t) is the financial bubble Organizer's benefit. W (t) is the financial bubble investors' benefit.

g (t) is the total face-value of the bubble securities sold exactly at the moment t. 0 > 0 is the fixed period of time, after which the moment of commitments fulfillment takes place; it is counted off from the moment of their sale (i.e. a period of the bonds run).

So, it is supposed that the bubble Organizer securities are zero-coupon bonds having equal time to run 0 counted off from the moment of their sale.

g(t — 0) is the total face-value of the bubble securities issued by the Organizer at the moment t — 0 and retired at the moment t.

cg(t) e [0;1] is the price (expressed in parts of the face-value) which is the real price of security

sales at the moment t.

The Organizer meets its commitments to investors using only the revenue of securities sales within the bubble.

Starting from this assumption it is clear that if during some time periods "too small" amount of the Organizer securities is being sold, at some moment the Organizer won't have enough funds to fulfill its commitments, because the revenue, received by the Organizer at the moment t cg (t)g(t)2,

is less than the obligations g(t) undertaken.

Starting from economical meaning of quantities introduced and assumptions made one can write:

2 Bearing market reality in mind it is easy to understand that sales of the Organizer zero-coupon bonds will take place only if the price cg (t) (expressed in parts of face-value) is less than 1, when there is no non-economical means of pressure.

burst".

(1)

2631

d^ > ^ , V(0) = 0, (2)

dt \cg(t)g(t)-g(t-ff),t W ' W

dW fo,t <0

— = , W (0) = 0. (3) dt [[1 - cg (t-ff)] g(t-ff), t >ff KJ KJ

*i*Не можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

Using these formulae one can calculate functions G (t), V (t) and W (t) at any moment of time using the known function g (t).

In writing these formulae constancy of price is not assumed. It (as other quantities), generally speaking, depends on time: cg = cg (t).

Besides that, in the model a demand function on the bubble Organizer bonds is introduced (it is described in details in [7]).

dV

Then — = cg (t)g(t) - g(t-ff), where g(0) = 0 at t e [-ff,0) and g(t) = go®(cg (t))f(t) at

t > 0, is a dependence of the bubble Organizer revenue on time. The function g(t) is expanded into production of 2 functions: the demand function O(cg), which depends on the bubble Organizer commitments price, and the given growth function f (t) (i.e. a priori scenario of the bubble investments inflow depending on time). g0 is a constant of proportionality for units chosen.

As an evolution of the model its variant was introduced. It takes into account expenses for an advertising campaign, which allows to stimulate sales of the bubble Organizer bonds. Let us suppose that the bubble advertising campaign gives additional expenses and an additional inflow of investors as well [8].

To take advertising into account one more variable (which is a control variable) is introduced:

s e [0,1] is a part of investments made into advertising campaign of the bubble (part of the

current revenue cgg (t) of the Organizer securities cells).

In case when we take an advertising campaign of the Organizer commitments into account the equations (1)-(3) turn to:

dG = [g(t),t <ff , G(0) = 0,

dt \g(t) - g(t-ff), t W '

dv =ffcg(t) s(t) g (tXt <ff = 0

dt \cg(t)s(t)g(t)-g(t-ff),t >0' () ,

dW f0,t <0

— = \r i , W (0) = 0. dt [[1 -cg(t-ff)]g(t-ff),t W

In the model when we take an advertising campaign into account g(t) = goeYsO(cg)f (t) accordingly when t > 0: the function g(t) is expanded into production of 3 functions: the function of effectiveness of investments into an advertising campaign eYs, which depends on part of investments in advertising, the demand function O(cg), which depends on the bubble Organizer

commitments price, and the given growth function f (t) (i.e. a priori scenario of the bubble investments inflow depending on time). g0 is a constant.

Different goals of the financial bubble Organizer, which concretize its behavior, are discussed in detail in the paper [9]. It is shown that extreme alternative goals are the following ones.

Goal 1. The goal of the financial bubble Organizer is to maximize its own revenue V(T) at the moment of the bubble burst T . Let us assume after this moment the Organizer does not fulfill its commitments.

2632

Such goal is often purposed by small companies, private well-known persons etc. However, Organizers, who explicitly declare the goal 1, are able to attract large funds as it is demonstrated by the history.

Goal 2. The goal of the financial bubble Organizer is to maximize the bubble lifetime, i.e. the quantity T, which is a moment of the bubble burst. Let us assume after this moment the Organizer does not fulfill its commitments.

Goals of type 2 usually correspond to large organizations or state structures having authorities to implement state budget and to manage public debts.

For example, a large company, who purposes the aim of type 2, thereby (due to predictability of its behavior to external observers and existence of a liquid market of its commitments) diminishes for itself expenses of external market borrowings and obtains other indirect benefits. At the same time a part of funds obtained can be used to cover budget deficit.

Another example is an authorized state structure whose goal may be long-term issue of public debt commitments. This goal may be caused by two reasons: covering of (part of) state budget deficit and / or creating of the liquid market of public securities to increase of the liquidity of the market as whole and to increase its stability.

In [9] a statement about a comparison of the goals is obtained.

8. Concluding notes

In the article the main approaches to modeling of financial bubblers are described; it is shown that this term can be interpreted as quite different phenomena.

A definition of financial bubble, stated to cover the most part of situations, is introduced. Such definition has been implicitly used in developing of the scenario approach, but when it is clearly formulated it allows to implement the results obtained to more wide scope of situations. To illustrate this fact the main ideas of the scenario approach are stated.

As a result one can say that financial bubbles can be interpreted in different ways; there is no any common definition. And they cannot be unambiguously regarded as a positive or negative phenomenon.

References

1. O.-J. Blanchard and M. Watson. Bubbles, rational expectations and financial markets. In P. Wachtel. Crises in economic and financial structure. Lexington (MA). 1982.

2. A. V. Belyanin, O. G. Isupova. Bubbles in transitional economy from point of view of the game theory in Russian Program of Economic Investigations. Paper # 2000/10 (www.eerc.ru, http://195.28.33.75/). (in Russian)

3. S. V. Dubovsky. Prediction of inflation and rate of exchange of ruble in Russian unsteady economy. Moscow: Editorial URSS, 2001. (in Russian)

4. S. V. Dubovsky. Rate of exchange of ruble as a result of money emission, foreign trade and wandering financial flows // "Economics and Mathematical Methods" journal, 2002, vol. 38, # 2, p. 84-96. (in Russian)

5. G. G. Dimitriadi. Review of models of public debt. Moscow: Editorial URSS Press, 2002, and site www.mirkin.ru. (in Russian)

6. G. G. Dimitriadi. Mathematical models of bubbles // Electronic journal "Investigated in Russia", 83, p.p. 929-936, 2002. http:Wzhurnal.ape.relarn.ru/articles/2002/083.pdf (in Russian)

7. G. G. Dimitriadi. Models of bubbles: the deterministic approach. Moscow: Yeditorial URSS Press, 2002, and site www.mirkin.ru. (in Russian)

8. G. G. Dimitriadi. Deterministic approach to the description of bubbles considering advertizing campaign // "Mathematical modeling" journal, 2003, № 4, vol.15, p.p. 23-33. (in Russian)

9. G. G. Dimitriadi. Deterministic approach to bubbles description: goals of the bubble Organizer. Electronic journal "Investigated in Russia", 175e, p.p. 2125-2132, 2003. http://zhurnal.ape.relarn.ru/articles/2003/175e.pdf

2633