Financial bubbles in terms of the scenario approach: bubble "step" and its lifetime

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

1. Introduction

The article [1] is devoted to answering the question what financial bubbles are. There are several different understandings of this term and accordingly variety of approaches towards description and modeling of bubbles.

Here financial bubbles will be interpreted in terms of the scenario approach. Let us remember, that according to the approach a bubble is a financial scheme, in which commitments of the Organizer are equivalently expressed as zero-coupon bonds having one and the same period to run (see details in the next section).

Let us consider a question naturally arisen: how does the lifetime (i.e. the time of breakeven existence of a financial bubble) depends on this period, called the "step" of a bubble? Intuitive speculations cause that there is a monotone increasing dependence. The article is devoted to answering this question strictly.

In the sections 2-4 the main ideas of the scenario approach is briefly introduced in compliance with the papers [2-4]. In the sections 5-6 results of the investigation of the dependence of the breakeven lifetime of a financial bubble on its "step" are described.

2. Model of a Financial Bubble

In terms of the scenario approach a financial bubble is understood as 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 sales (see [2-4]). It is assumed that the Organizer meets all its commitments up to the moment called "the bubble burst".

Without any loss of generality let us suppose that the Organizer commitments are bonds without coupon (zero-coupon 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 the unity.

The following notations are used. A financial bubble begins at the moment t = 0. V (t) is the financial bubble Organizer's benefit; g (t) is the total face-value of the bubble securities sold exactly at the moment t; Q > 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 to run); 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.

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

The main equation of the dynamics of a bubble looks like:

dV (cg (t)g(t), t < Q

dt I Cg (t)g(t) - g(t -6), t >6

, V(0) = 0.

Besides that, in the model a demand function on the bubble Organizer bonds is introduced (it is described in detail in [4]).

Then d: = Cg (t)g(t) - g(t-Q), where g(0) = 0 at t e [-Q,0) and g(t) = goO(Cg (t))f(t) at t > 0, is a dependence of the bubble Organizer revenue on time. The function g(t) is expanded into

the production of 2 functions: the demand function O(cg), which depends on the financial 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.

3. Model When We Take an Advertising Campaign of the Organizer Commitments into Account

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 a bubble advertising campaign gives additional expenses and an additional inflow of investors as well [4].

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

In case when we take an advertising campaign of the Organizer's commitments into account the main equation of the dynamics of a bubble turns to:

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 the production of 3 functions: the function of effectiveness of investments into an advertising campaign eYs, which depends on the 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.

4. Goals of the Organizer of a Financial Bubble

Different goals of the financial bubble Organizer, which concretize its behavior, are discussed in detail in the papers [3,4]. 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.

Goal 2. The goal of the financial bubble Organizer is to maximize the breakeven lifetime of a financial bubble, 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.

In [3, 4] a statement about a comparison of the goals is obtained.

5. Dependence of the Breakeven Lifetime of a Financial Bubble on its "Step" 0

In this section we will consider the model without an advertising campaign, as such consideration is enough to investigate qualitative characteristics of a bubble.

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

This time 0 is a part of all formulae for the time of the breakeven lifetime of a bubble (see, for example, [2] and [4, §3.1]).

It is easy to see that according to these formulae for the case of deterministic investors inflow in time (i.e. in the case of the absence of the demand function on Organizer's commitments; this case can be regarded as the main one, "standard" for the following comparisons) the time of breakeven existence is monotone increasing when the parameter 0 is increasing.

So an impression appears that the regularity detected takes place always; even more it corresponds to intuitive assumption: the longer the time period we borrow for, the longer time we will repay after and the longer bubble existence. Let us prove an appropriate statement.

Statement. Monotone increasing dependence of the time of breakeven existence of a financial bubble on its "step" Q .

Let us assume that for the Organizer, achieving the goal 1, a bubble having a dependence of the sales volume on timeg(t), price of the sales cg(t) and the "step" Q, exists up to the moment

T > 0. Than a bubble, which differs from the previous one only in the "step " aQ, a > 1, exists up to the same moment T.

An analogous statement is true for the Organizer, achieving the goal 2.

(Here it is assumed that g(t) = 0 when t < 0.)

Proof.

In the period [0, aQ) the second bubble exists, because there are no fulfillment of the Organizer's commitments during this time. Let us prove its existence at t e [aQ, T].

1) For the Organizer, achieving the goal 1.

cg (t)g(t) - g(t - aQ) = (cg (t)g(t) - g(t - 0)) + (g(t - 0) - g(t - a)).

The expression in the first brackets is non-negative, because it is a condition of breakeven existence of the first bubble during the period in question.

The expression in the second brackets is non- negative because of the necessary condition of breakeven existence of the first bubble1 as t-Q> t-aQ . So, cg(t)g(t)-g(t-aQ) > 0. Q.E.D.

2) For the Organizer, achieving the goal 2.

t t t t-aQ t t-6

J cg (%)g(%)d%-J g^-aQ)di = \ cg (g)g(g)dZ- J g(n)dn>\cg (£)g(£)d{ - J g(£)d{ > 0. 0 0 0 0 0 0 When writing the first equality an substitution n = % -aQ is made and the condition g(t) = 0 at t < 0 is taken into account. When writing the first inequality the expressions t-Q> t-aQ and g(t) > 0 are taken into account. The last inequality is the necessarily condition of breakeven existence of the first bubble. The statement is proved.

6. Discussion

The statement derived causes the following conclusion:

To increase the time of breakeven existence of a financial bubble one should increase at the most the period to run Q of the Organizer's zero-coupon bonds.

So, the conclusion is in accordance with intuitively clear idea that the postponement of the repay of debts increases the lifetime of a scheme of holdings.

7. Concluding Notes

In the first part of the article the main ideas of the scenario approach towards financial bubbles description are briefly introduced. Later in terms of this model the question about dependence of the bubble breakeven lifetime on the "step" of a bubble is considered. Here the "step" of a bubble is a period for the Organizer's bonds to run (let us remember that the following simplifying assumption

1 If a bubble exists up to the moment t0 for the Organizer, whose goal is the goal 1, than for the bubble it is true

that: g (t) g (t - 9), t e [9, t0 ] (see [4, statement 4.1.5]). So, g (t) > g (t - 9), because cg (t) e [0; 1].

cg (t)

Hence, g (t) is a monotone increasing function when t e [9, t0 ].

is made in the model: all zero-coupon bonds of the Organizer have one and the same period to run 9).

It is strictly shown that for the Organizer, achieving different goals, the time of breakeven existence of a financial bubble is increasing when its "step" is increasing, which is in accordance with intuitive speculations.

As a result one can say that the longer commitments the Organizer of a financial bubble issues, the longer the time of breakeven existence of a financial bubble the Organizer can provide under other equal conditions.

References

1. G. G. Dimitriadi. What are "Financial Bubbles": Approaches and Definitions // Electronic journal "Investigated in Russia", 245e, p.p. 2627-2633, 2004. http://zhurnal.ape.relarn.ru/articles/2004/245e.pdf

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

3. 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

4. G. G. Dimitriadi. Public and Corporate Debt Instruments as bubbles. Modeling and Forecasting. Moscow: Moscow Humanitarian University Press, 2004. (in Russian)