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Finite-time singularities in the dynamics of Mexican financial crises
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Physica A 331 (2004) 253 – 268
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Finite-time singularities in the dynamics of Mexican (nancial crises Jose Alvarez-Ramirez∗ , Carlos Ibarra-Valdez1 Division de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F. 09340, Mexico Received 3 February 2003; received in revised form 17 July 2003
Abstract Historically, symptoms of Mexican (nancial crises have been strongly re4ected in the dynamics of the Mexican peso to the dollar exchange currency market. Speci(cally, in the Mexican (nancial crises during 1990’s, the peso su9ered signi(cant depreciation processes, which has important impacts in the macro- and micro-economical environment. In this paper, it is shown that the peso depreciation growth was greater than an exponential and that these growth rates are compatible with a spontaneous singularity occurring at a critical time, which signals an abrupt transition to new dynamical conditions. As in the major 1990’s (nancial crisis in 1994 –1995, some control actions (e.g., increasing the USA dollar supply) are commonly taken to decelerate the degree of abruptness of peso depreciation. Implications of these control actions on the crisis dynamics are discussed. Interestingly, by means of a simple model, it is demonstrated that the time at which the control actions begin to apply is critical to moderate the adverse e9ects of the (nancial crisis. c 2003 Published by Elsevier B.V. PACS: 64.60.Ak Keywords: Econophysics; Financial crisis; Finite time singularity
1. Introduction During the 1990’s, Mexico has been plagued with recurrent economic and (nancial crises. Instability sources involved both internal (e.g, political, social, etc.) and ∗
Corresponding author. Fax: +52-5-8044900. E-mail address: [email protected] (J. Alvarez-Ramirez).
1 The authors are also at Programa de Investigacion en Matematicas Aplicadas y Computacion, Instituto Mexicano del Petroleo.
c 2003 Published by Elsevier B.V. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.09.058
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Fig. 1. Peso–dollar exchange rate. Some important world (nancial events are highlighted.
external (e.g., contagion from foreign crises, etc.) factors. Historically, signals of Mexican (nancial crises have been strongly re4ected in the dynamics of the USA dollar–Mexican peso exchange currency market, leading to signi(cant peso depreciation (see Fig. 1). If a major economic and (nancial crises can be stated when at least 50% in4ation or currency depreciation in one year is produced [1], at least two major Mexican (nancial crises can be recognized during the 1990’s. The (rst one lasted around one year (Tequila e9ect), beginning on December 20, 1994, and produced about a 100% depreciation of the peso with respect to dollar. The dynamics of this crisis was temporarily disturbed by attempts to reduce the fall down of the peso (see the chronology in Section 3). The e9ect was a momentary (around 15 days) decrement in the peso exchange currency slip, although it continued a slower depreciation to attain its “fundamental” value stabilizing about 7.70 by November, 1995. The e9ects of the Tequila crisis on the Mexican economy and society were very adverse, including large goods price increments and high unemployment rates. Besides, the GNP fall (9.3% in annualized rate basis. The second major crisis seems to be triggered by the Asian Paci(c crisis by around August, 1997. In fact, due to a complex interrelated trading network in the Paci(c region, the (nancial crisis of the Asian Paci(c countries (Indonesian, Malaysia, etc.) infected the dynamics of the Mexican (nancial system. This second crisis lasted about one year, and produced a 60% peso depreciation. The main adverse e9ect of this crisis was that it has been delaying signi(cantly the recovery of the macro- and micro-economic conditions degraded by the Tequila crisis. To
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date, Mexico seems to have recovered similar macro-economic conditions present before December, 1994. However, from the micro-economic viewpoint, the sequels of the two major 1990’s crises on the Mexican society and economy cannot be underscored. To date, employment and wage are unstable and crime in large cities has achieved dangerous levels. From both social and economic viewpoint, it would be desirable to dispose of methodologies to predict the advent of (nancial and economic crises, in order to take corrective actions at appropriate times. In principle, a systematic understanding of (nancial crises leading to hyperin4ation-type dynamics would help to design suitable control policies and prevent to some extent their adverse e9ects [2]. Recently, Econophysics researchers have taken some steps towards this endeavour. Johansen and Sornette [3] detected that world population and some economic indices cannot be described as a (Malthusian) exponential growth function of time. A key concept to explain such explosive behavior was the presence of a singularity at a critical time tc , at which the population or economic signal approaches an unbounded value. Mizuno et al. [4] analyzed historical data of price indices. They found that, under hyperin4ation conditions, the logarithm of the price can be approximated by a double-exponential function of time. A generalization of Cagan’s model for description of in4ation was used to justify theoretically the double-exponential behavior. Based on a continuous-time version of the Cagan’s model developed by Mizuno et al. [4], Sornette et al. [5] found that hyperin4ation dynamics can be characterized by a power-law singularity culminating at a critical time tc . In this way, Mizuno et al.’s double-exponential function can be seen as a discrete-time approximation of the reported continuous-time model. Historical hyperin4ation data from many countries were used to illustrate the goodness of the power-law description. Also, in several papers the technique of stretched exponentials has been used for studying (nancial crashes [6]. The aim of this paper is twofold: To show that the power-law singularity formalism can be used to describe the dynamics of recent (nancial and economic Mexican crises. In this way, Mexican (nancial crises can be explained by a mechanism of in4ationary-type expectation (or positive feedbacks) between realized growth rate and people’s expected growth rate. To show that the oscillatory decoration on the (monotonous) power-law singularity function can be explained, at least partially, as the e9ects of control (e.g., monetary) actions aimed to correct hyperin4ationary-type e9ects. To this end, a simple control model is developed. The analysis and discussion are based on historical peso–dollar exchange currency data. 2. Peso–dollar exchange rate data In our analysis, the peso–dollar exchange currency rate data are taken from the Banco de Mexico web page (www.banxico.org.mx). The data describe the dollar exchange currency, in peso to the dollar, within the period from December 20, 1994 to January
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7, 2003. Before December 20, 1994, the peso to the dollar value was regulated by the regime of bands (i.e., minimum and maximum exchange currency levels were imposed). On December 20, 1994, an economic and (nancial crash, popularly denominated as “Tequila e9ect”, disturbed signi(cantly the peso to the dollar market. After the crash, the peso was allowed to 4uctuate freely. Currently, the level of the peso to the dollar value, known as FIX, is estimated daily by a sort of an auction procedure made by the Central Bank of Mexico (Banxico), with the participation of the main private banks and some other intermediaries. No data point is supposed to be missing. In other words, when applied to (nancial data, breaks due to holidays and weekends are disregarded. Nevertheless, the time units are said to be in business days (b-days) in the following, a week has often 5 days, a month about 23 days, and a year about 250 days. December 20, 1994 was taken as the (rst day. Fig. 1 shows the peso value p(ti ) and some signi(cant world economic events a9ecting, to some extent, the exchange market dynamics. Dates of (nancial crises and events in Fig. 1 state only approximately the moment at which the peso was signi(cantly a9ected. 3. Chronology of Mexican crises in the 1990’s To gain some insights on the micro- and macro-economic conditions leading to (nancial Mexican crises, in the following we provide a brief chronological description of major event during the 1990’s decade (see, for instance, [7]). December 1994: The crash followed the attempt on December 20th to devalue by 15%, which was met with a fresh attack which exhausted the reserves. The peso had to be allowed to 4oat: its value fell from 3.10 pesos to the dollar before the 20th to just a little short of 6.50. The government was unable to roll over its short-term debt at any reasonable rate, and the private sector also became unable to borrow. January 1995: On January 2, President Clinton announced Plan A to win Congress approval for 40 billion of USA loan guarantees for Mexico. On January 26, the IMF announced readiness to provide Mexico with a $7.76 billion standby credit, the IMF’s largest ever. On January 30, the White House turned to Plan B: in lack of Congress approval for loan guarantees, President Clinton launched a new support package centering on 20 billion of (nance from the USA Exchange Stabilization Fund (ESF). That money would be provided on Clinton’s own authority. On January 31, the White House announced new 50 billion package, including a Bank for International Settlements (BIS), standby credit and an extended IMF credit. February 1995: The USA.20 billion dollar emergency package was available by February 1st, and its immediate e9ect was an apparent recovery of the peso. In fact, during February and the (rst days of March, the peso value raised from 6.50 to around 5.00 pesos to the dollar. Throughout 1995: The bene(cial e9ects of the (rst loan on the peso–dollar exchange currency stability dissipated quickly. In fact, the peso continued to drop further in later months to achieve a value of around 8.00 pesos to the dollar at the end of the year. Mexico faced the consequences of the massive devaluation. In4ation and, interest rates
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soared to reach levels of 100% in some months. Out4ow of portfolio investment totaled 60 billion dollars. A regrettable e9ect of the rescue package was the Mexican Society feeling of a lack of sovereignty, since the collateral assurance (guarantee) for the whole loan were the Mexican oil receipts directed through the New York Federal Reserve Bank. January 1996–September 1997: The economy went through a stabilization period, where the peso 4uctuated freely within an interval from 7.4 to 8.0 pesos to the dollar as lower and upper bounds. Basically, the peso did not su9er a major depreciation. Interestingly, in addition to apparently random daily 4uctuations, the current exchange value displayed a cyclic behavior with cycle period of around a quarter. This so regular cyclic behavior has been explained mainly by the activity of closing and opening contracts for import and export of goods with the USA [8]. October 1997–November 1998: When the Mexican economy started to achieve a stable evolution, an external event disturbed it. In fact, between the months of June and December 1997, a group of newly industrialized Asian Paci(c countries underwent a severe (nancial crisis. The stock markets tumbled and the exchange rate depreciated deeply. Due to its strong trading interaction with the Asian Paci(c countries, the Mexican economy su9ered a severe contagion. By the mid-October, a sudden depreciation from 7.7 to 8.25 peso to the dollar was induced to try counteracting the adverse domestic e9ects of the Asian crisis. Such control action had no important e9ects since the exchange rate started a deep depreciation processes, attaining its peak by November 1998 where the exchange rate was 10.55 peso to the dollar. Some attempts directed to stop, or at least reduce, the depreciation process were taken. Basically, such control actions consisted in increasing the dollar supply, so to equilibrate the exchange market. As one can observe in Fig. 1, the result was an exchange rate evolution leading to a severe peso depreciation, decorated with three oscillations associated with such control actions. December 1998–March 2002: After the Asian Paci(c crisis shock, the Mexican economy started an important peso to the dollar appreciation. Out of a short transient (around 10 days) induced by the Brazilian (nancial crisis and a largest peak of 11.40 peso to the dollar, which apparently did not disturb signi(cantly the exchange rate dynamics, the Mexican economy evolved within a stable process with well tuned macro-economic indices and a stable peso to the dollar currency value 4uctuating around 9.50. April 2002–date: By 2002, USA recognized that its economy was entering a recession period. Since about 80% foreign Mexican trading depends on the American economy, Mexico was not immune to the USA deceleration process. At this moment, Mexican economy is under a recession process with the exchange rate depreciating slowly. However, it is not clear if this process is only a deceleration dynamics or can pave the way to a major economic and (nancial crisis. Summarizing, during the 1990’s the Mexican economy was subjected to alternating periods of severe (nancial crisis and stabilization attempts where the peso to the dollar exchange rate oscillated around a stable trajectory. As discussed in the Introduction, the matter of this paper is to gain some insights on the dynamics of major (nancial crisis. In this way, two major Mexican (nancial crises during the 1990’s can be easily
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identi(ed. Crisis I started on December 1994 and ended by November 1995, while Crisis II, induced by Asian Paci(c (nancial crisis, started by October 1997 and ended by November 1998. 4. Finite-time singularity description of %nancial crisis One can see that major (nancial crisis dynamics are characterized by (1) an induction period with a slow depreciation process, and (2) a fast-growth process. By considering only a crisis period (e.g., October 1997–November 1998), it is not hard to see that the dynamics cannot be explained as an exponential-growth process. In fact, one can easily show that ln(p) versus t plot cannot be adjusted by a straight line. This simple observation led to the conclusion that this hyperin4ation-type dynamics display a larger-than-exponential growth dynamics. From a discrete-time framework, Mizuno et al. [4] used an enhanced version of the Cagan’s model of in4ation to show that hyperin4ation processes can be described with a double-exponential model. In a recent paper, Sornette et al. [5] used a continuous-time approach to generalize the Cagan’s model. They found that hyperin4ations can be characterized by a power-law singularity culminating at a critical time tc . In this way, Mizuno et al. [4]’s double-exponential function can be seen as a discrete time step approximation of the more general nonlinear ODE formulation of the price dynamics which exhibits a (nite-time singular behavior. A feature of Sornette et al.’s formalism is that hyperin4ation description is made on ln p(t) rather than on the price itself p(t). At least in the Mexican exchange market during a (nancial crisis, people’s averaged expectation is based on the exchange rate p(t). This is not a serious drawback, since slight modi(cation to the generalized Cagan’s model proposed by Sornette et al. [5] can be made in order to (t into this circumstance. For the sake of completeness in presentation, in the following a brief description of modi(cations introduced to Sornette et al. [5]’s approach are given. Let p(t) be the market price and let p∗ (t) be the people’s averaged expectation price. These two prices are thought to evolve due to a positive feedback mechanisms where an upward change in market price p(t) in a unit time Qt induces a rise in the people’s expectation price, and such anticipation pulls up the market price. Let r(t) = p(t) − p(t − Qt)
(1)
be the growth rate at time t. According to the generalization introduced by Mizuno et al. [4], the expected growth rate r ∗ (t) at time t is proportional to the past realized growth rate r(t − Qt) in the following form: r ∗ (t) = r(t − Qt) + ar(t − Qt)m :
(2)
The (rst term in Eq. (2) expresses that the expected growth rate is at least the past realized growth rate r(t − Qt). On the other hand, the second term expresses nonlinear feedback e9ects induced by re-enforcement phenomena during large return periods. That is, when the realized growth rate becomes signi(cant, people’s expectations start
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to amplify these realized growth rates, which can lead to an unbounded expected growth rate. In principle, the aim of the market actors is to achieve the expected return r ∗ (t) within the next period. This can be expressed as follows: r(t) = r ∗ (t − Qt) :
(3)
Putting Eq. (2) together with Eq. (3) leads to r(t + Qt) = r(t − Qt) + ar(t − Qt)m . Taking the continuous-time limit, the above equation becomes dr = ar m ; dt
(4)
where a is a positive coeScient. The solution exhibits a (nite-time singularity 1=(m−1) tc r(t) = ar(0) ; tc − t
(5)
where the critical time tc = (m − 1)=r(0)m−1 is determined by the initial condition r(0) and the exponent m. When Eq. (1) is considered in the continuous-time limit, one obtains that the price p(t) is the integral of r(t) and also exhibits a (nite-time singularity at the same critical value tc . The time dependence of the market price p(t) exhibits the following two regimes: (i) Finite-time singularity in the price itself: p(t) = A + B(tc − t)−
with =
2−m ¿ 0 and B ¿ 0; for 1 ¡ m ¡ 2 : m−1
(6)
This solution is a genuine divergence of p(t) in (nite time at the critical time tc , (ii) Finite-time singularity in the derivative of the price: p(t) = A + B (tc − t)−
with =
m−2 ¿ 0 and B ¿ 0; for 2 ¡ m : m−1
(7)
As time approaches tc , the price accelerates with an in(nite slope reached at tc , while remaining (nite at the value A . The constants A and A arise from the integration of the growth rate r(t) given by Eq. (5). In the next section, we will show that expressions (6) and (7) can be used to describe the dynamics of Crisis I and Crisis II discussed before. The following should be stressed. As mentioned before, Mizuno et al. [4] proposed the double-exponential model p(t) ˙ exp(C1 exp(C2 t)) to describe dynamics with larger-than-exponential growth. It has been shown that this expression is actually a solution of the discrete-time model (2) r(t) = r(t − Qt) + ar(t − Qt)m . Since the model (4) can be seen as the continuous-time limit of the discrete-time model (2), the double-exponential expression corresponds to the limit of the singular function (6) and (7). In practice, one could use either the double-exponential or the singular functions to describe price dynamics during crises scenarios. However,
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in this paper we follow the Sornette’s setting and use the continuous-time description of crises phenomena. The advantage of using singular functions (6) and (7) over the double exponential function is that good estimates of singular times can be obtained. In turn, these singular time have an interesting interpretation within an economical and (nancial discussion.
5. Results Consider the Crisis I dynamics starting by December 1994 and ending by November 1995 (see Fig. 1). One can note that the USA–IMF rescue package disturbed signi(cantly the evolution of the exchange rate depreciation. In fact, by the (rst days of March the peso to the dollar parity decreased by about 25%. However, after around one month the e9ects of the injected (nancial resources seem to be dissipated and the peso continued its depreciation evolution. In this way, Crisis I can be separated into two periods: (i) Crisis I.bl corresponding to the before-loan process, and (ii) Crisis I.al corresponding to after-loan depreciation. The Crisis I.bl period was considered from December 23, 1994 to March 9 30, 1995, when the exchange rate attained a peak value of 7.59. On the other hand, Crisis I.al period was taken from May 3, 1995 to November 3, 1995, when a 8.13 peso to the dollar exchange rate was reached. Fig. 2 shows the dynamics of Crisis I.bl. These dynamics can be
Fig. 2. Dynamics and (tting functions for Crisis I.bl.
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Fig. 3. Dynamics and (tting functions for Crisis I.al.
described by the singular function p(t) = 6:9424(57:5 − t)−0:0654 (R2 = 0:9785 and = 0:01521). One can question if the dynamics can be described also with as an exponential growth function. In Fig. 2 is also shown our best exponential growth (tting given by p(t) = 5:53156 + 4:51 × 10−5 exp(t=5:3876) (R2 = 0:6927 and = 0:07081). Initially, the exponential growth function is able to provide a good description of the price dynamics. However, at later times close to the apparent singularity, the exponential growth function presents signi(cant deviations from the data. Besides, after the escape time located at about 57.5, the exponential growth function predicts prices that are signi(cantly below (around −15%) the actual ones. The lack of the exponential growth function to provide an acceptable description of the price dynamics is re4ected in the quite low correlation value R2 = 0:6927. We have tried to (t the price dynamics with second-order exponential growth dynamics like p(t) = A0 + A1 exp(t=1 ) + A2 exp(t=2 ), obtaining similar poor (tting results. In this way, it seems the singularity is an authentic property of Crisis I.bl dynamics. Analogous results are found for Crisis I.al and Crisis II. Fig. 3 shows the dynamics of Crisis I.al, which can be also described by the singular expression p(t) = 7:4(221:25 − t)−0:0423 . It is noted that the exponent of Crisis I.bl is larger than that of Crisis I.al. This means that Crisis I.bl was more severe than Crisis I.al in the sense that the velocity of peso depreciation was larger in the former case. In this way, it seems that the e9ect of the USA–IMF package was to delay the crossover to a saturation of the economic and (nancial system.
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Fig. 4. Dynamics and (tting functions for Crisis II.
The period of the Crisis II dynamics was taken from November 5, 1997 (just after the sudden depreciation induced by the Asian Paci(c crisis) to November 15, 1998, then the peso to the dollar value reached a historical peak value of 10.58. Fig. 4 shows the dynamics of Crisis II. The continuous line shows that, on the average, the Crisis II dynamics can be described by the expression p(t) = 10:975(936:25 − t)−0:0538 . The exponent of this crisis is between the exponents of Crisis I.bl and Crisis I.al, demonstrating that Crisis II was as severe as 1994 –1995 (nancial crisis. The above results seem to demonstrate that Mexican (nancial crisis in the 1990’s can be explained by a mechanism of in4ationary-type expectation between realized growth rate and people’s expected growth rate. That is, after an adverse information (the 15% government-induced sudden depreciation in December 20, 1994, and the Asian Paci(c (nancial crisis in the mid-1997), negative expectation of a further depreciation of people is reinforced, leading to an increased dollar demand to protect their wealth. Indeed, this process produces a severe shock to the domestic economy. 6. Control actions and %nancial crisis We have shown that, on the average, Mexican (nancial crisis can be well (tted with a power-law singularity. This approach provides a nice description of (nite-time escapes and monotonic peso depreciation. However, one notes that Crisis I.al and Crisis II
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are decorated with large oscillations, which disappear during the escape regime. For instance, in the (rst part of Crisis II one can (nd two large oscillations, with periods of around one quarter. The aim of this section is to provide a qualitative explanation to the oscillatory decorations on the (monotonic) power-law singularity function. During a (nancial crisis, governments and (nancial authorities and institutions take measures that can be seen as control actions to try counteracting, and even stopping, the adverse e9ects of a severe depreciation. The most immediate control action is taken by central banks by increasing the currency (dollar) supply. In this way, it is expected that n the market reaches a saturation level by satisfying most agent demands. Let u(t) = j=1 uj (t) be the control action function taken during a (nancial crisis, where uj (t) are individual control actions. In a very elusive way, u(t) can be seen as modeling (nancial resources injected into the market. From the viewpoint of the modeling framework described in Section 4, the aim of u(t) is to counteract the reinforcement dynamics leading to a singularity-type behavior. In this way, model (4) can be modi(ed as follows: dr = ar m − u : dt
(8)
The control action u(t) is assumed to be bounded, re4ecting the realistic fact that the injected resources are (nite. Non-negativity is also assumed to recast the fact that the control action reverts, to some extent, the monotonous increasing escape of the growth rate r(t). Our intention is not to give a precise modeling of u(t), which is a very complex problem. Instead, our aim is to use a simple model to gain some insight on the source of the oscillatory phenomenon accompanying the power-law singularity behavior. In this way, as a (rst approach we use the following expression for individual control actions: 0 if t ¡ tj ; uj (t) = (9) umax; j e−t=j if t ¿ tj ; where tj is the time at which the jth control action starts, umax is the maximum control action, and j is a dissipation time. The idea behind the model (9) is the following: At time tj , resources umax; j are injected into the market, which are absorbed by the market agents with a rate −1 j u(t). In other words, j is the mean time the control action u(t) a9ects the market dynamics. In the following, we will concentrate only on the Crisis I because it is the most interesting. The discussion and qualitative result apply vis-Ta-vis to oscillations in Crisis II. Unfortunately, for uj (t) given by Eq. (9), it is apparent that Eq. (8) have no explicit global solution, so that our discussion will be based on numerical simulations. The model (8), (9) was integrated with a Runge–Kutta method with a time step of 1:0 × 10−3 . Fig. 5 shows the simulations for the following set of parameters: m = 16, which corresponds approximately to the exponential values of the (tting function (6) for Crisis I, a = 2:5 × 10−9 , n = 2, umax; 1 = 155, umax; 2 = 55, t1 = 60, t2 = 110, 1 = 2:1 and 2 = 2:2. It should be stressed that for simulation purposes the values umax; j does not correspond to actual rescue package values. Instead, such values re4ect only qualitatively the relative magnitude of the corresponding control actions. On the other
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Fig. 5. Qualitative numerical simulations for Crisis I including two control actions as speci(ed by Eq. (9).
hand, the time when the control action u(t) is applied corresponds exactly with the times when the rescue packages are delivered. It is noted that the modeling approach (8), (9) is able to describe qualitatively the oscillations in Crisis I. In turn, this implies that such oscillations are induced, at least partially, by control actions directed to counteract the excessive dollar demand. It seems that the large peak in Fig. 5 was induced by the USA–IMF rescue package, which was available to the Mexican (nancial and economical systems by mid-February, 1995. At this point, two interesting question can be posed: How opportune and eScient was the control action induced by the 50 billion dollar rescue package? Does the same e9ect could be obtained if a smaller rescue package was available at earlier days of the crisis? To address these questions from the modeling approach described above, let us consider the model (9) with only the larger control action u1 (t) (i.e., u2 (t) = 0 for all t). This numerical simulation is presented in Fig. 6. Assume that the control action u1 (t) was applied several days after. Dotted line in Fig. 6 presents the numerical simulation results for t1 = 45. It is noted that, although the (nite-time singularity is not removed, it is moved far away by about 1400 days (approx. 5 years). One can conclude that the control action u(t) delays the e9ects of the (nite-time singularity. Fig. 7 presents the e9ect of applying a lower control action at an earlier time. It is
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Fig. 6. Qualitative numerical simulations for Crisis I showing the e9ect of the control time t1 .
observed that, despite the fact that umax; 1 has been lowered to about one half, control actions at earlier times of crisis can induce more bene(cial e9ects that larger control action applied at later times. Summarizing, the time at which the control action u(t) is applied plays a central role in the future dynamics of the (nite-time singularity. 7. Discussion The main conclusion from the above simulations is that, under a severe (nancial crisis showing escape behavior similar to a (nite-time singularity, the times at which control actions are applied (e.g., (nancial resources, political actions, etc.) are crucial in the stabilization process. Our simple simulations have shown that the application of 50 billion package in Crisis I was not opportune. That is, although such resources were necessary to reduce the adverse e9ects, the opportune application of the loan, maybe from 15 to 20 days earlier, would have delayed considerably the catastrophic peso depreciation (see Fig. 6). By delaying by at least one year the larger-than-exponential depreciation process, Mexican government and international (nancial institutions (e.g., IMF, WB, etc.) would have a larger period to make some corrective actions and to roll over the debts. Corrective actions would include reforming the (nancial sector to overcome its weaknesses and 4exibilizing the decision-making schemes to a changing economic situation.
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Fig. 7. Qualitative numerical simulations for Crisis I showing the combined e9ect the control action amount umax; 1 and the control time t1 .
Regarding the government debt during December 1994 and January 1995, Cole and Kehoe [9] have used a self-ful(lling debt modeling approach to explain the crash. The author’s analysis suggested that for a country, like Mexico, with a very short maturity structure of debt, there is an interval of debt for which the government, although it (nds it optimal to repay old debt if it can sell new debt, goes to default if it cannot sell new debt. In this way, a delaying of the (nancial crisis peak would allow Mexican government to sell new debt with a larger maturity structure. For an alternative, more comprehensive view of the debt crisis, see [10]. It is now a historical fact that the 20 billion loan was available to the Mexican (nancial and economical system by mid-February, 1995. The peso obtained a transient recovery to take again (with a slower velocity) its depreciation processes. Our analysis suggests that if a control action is not applied at early stages of the crisis (i.e., before the crisis reach a larger-than-exponential process), it is useless. In this way, it seems that the 20 billion control action was not so opportune. In fact, the (nancial crisis was highly excited and dissipated in a short period the possible bene(cial e9ects of the fresh resources. The government could not roll over its short-term debt. All this happened despite infusions of funds from the USA and the IMF totaling 50 billion dollars. A posteriori, one could conclude that, since the loan package was not available at the just time, a possible stabilization procedure could be the following: (a) Not to
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inject the 20 billion loan, and wait until the economic and (nancial systems attain their saturation limit, and (b) Once the economical and (nancial indices showed a saturation level, inject the fresh resources to reactivate and stabilize the economy. Of course, this is a very simplistic view. In practice, such control action could lead to some political and social instabilities, as one can witness in recent Argentine crisis. The possible bene(cial e9ects this type of control action is that it could lead to elimination of ineScient and spurious components of the (nancial and economical system. A general approach based upon this kind of considerations belong to the strategy 4exibility approach for management, which has crystallized in the recent “Real Options” theory [11]. We believe that a theory of dynamics and control of trading mechanisms in (nancial markets based on 4exibility and option theory is very promising. On the other hand, during a severe crisis, governments and (nancial institutions (e.g., central banks) are forced to do something before hyperin4ation-type behavior reaches unsustainable levels. On the other hand, (nancial crises are very costly. Not only the direct costs to governments for the resolution of the crisis, but the cost of economy recovery and social costs of poorer groups are high. Recovery of the precrisis economic growth rates in past crises has taken on the average of two to three years [12]. The best prescription for a (nancial crisis is therefore to prevent it. The analysis in this paper and recent results by Sornette et al. [5] are directed to this objective. However, further analysis with the use of more involved nonlinear dynamics should lead to better characterization of positive feedback mechanism. In principle, this should help to establish and implement an e9ective signaling system to warn the imminence of a (nancial crisis. A crucial step to confront a severe (nancial crisis is the suScient, opportune and feasible implementation of control actions to prevent crisis blow up. Tools from (optimal and feedback) control theories can help on the designing of timely policy actions. The preliminary results in this paper have shown that not only the quality but the timing of the policy action was essential for a successful response. 8. Conclusions We have analyzed the dynamics of the recent Mexican (nancial crises. On the basis of a hyperin4ation-type model based on positive feedback between realized return and agent’s expected returns proposed by Sornette et al. [5], we have shown that Mexican crisis can be described by a power-law singularity. We have extended previous analysis reported by Mizuno et al. [4] and Sornette et al. [5] on hyperin4ation-type dynamics, to account for control actions aimed to reduce the hyper-in4ationary-type e9ects. In this way, by modeling the infusion of (nancial resources as a dissipative signal, we have been able to reproduce qualitatively certain oscillations decorating the averaged crash dynamics. Namely, such oscillations have been partially explained as control actions that governments and (nancial institutions make to counteract the adverse e9ects of the crisis. In the main streamline, the results in this paper seem to reinforce the idea that during a (nancial and economical crisis, nonlinear feedback e9ects plays a central role. In this way, as Sornette et al. [5] have also suggested, our results point to the use of nonlinear
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analysis to understand the mechanisms leading to (nancial crises and crashes. Future work should be directed along this direction. Another potentially useful tool that can be explored is robust control and related issues directed towards attenuation and/or rejection of exogenous disturbances. This could be specially interesting in connection with Asian Paci(c crisis e9ects on Mexico Forex market. References [1] P. Cagan, The monetary dynamics of hyperin4ation, in: M. Friedman (Ed.), Studies in the Quantity Theory of Money, University of Chicago Press, Chicago, 1956. [2] R.T. Froyen, R.N. Waud, The determinants of federal reserve policy actions: a reexamination, J. Macroeconomics 24 (2002) 456–477. [3] A. Johansen, D. Sornette, Finite-time singularities in (nance, population and rupture, Physica A 294 (2002) 465–502. [4] T. Mizuno, M. Takayasu, H. Takayasu, The mechanism of double-exponential growth in hyperin4ation, Physica A 308 (2002) 411–419. [5] D. Sornette, H. Takayasu, W.-X. Zhou, Finite-time singularity signature of hyperin4ation, arXiv:physics/ 0301007 v1, 2003. [6] A. Johansen, D. Sornette, Endogeneous versus exogeneous crashes in (nancial markets, arXiv:condmat/0210509 v1, 23 Oct 2002. [7] I. Tower, Mexico 1994 versus Thailand 1997, in: B. Fuller, R. Suwanraks (Eds.), TDRI Quarterly Review, Vol. 12, 1997, pp. 9 –14. [8] J. Alvarez-Ramirez, Characteristic time-scales in the Mexpeso/USAdollar exchange currency market, Physica A 309 (2002) 157–170. [9] H.L. Cole, T.J. Kehoe, Federal Reserve Bank of Minneapolis, Research Department Sta9 Report, Vol. 210, April 1996. [10] F. Gil-Diaz, The origin of Mexico’s (nancial crisis, Cato J. 17 (1998) 1–7. [11] A.K. Dixit, R.S. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, New Jersey, 1994. [12] M. Goldstein, C. Reinhart, Forecasting Financial Crisis: Early Warning Signals for Emerging Markets. Policy Analysis in International Economics, Institute for International Economics, Washington, 1998.
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www.elsevier.com/locate/physa
Finite-time singularities in the dynamics of Mexican (nancial crises Jose Alvarez-Ramirez∗ , Carlos Ibarra-Valdez1 Division de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F. 09340, Mexico Received 3 February 2003; received in revised form 17 July 2003
Abstract Historically, symptoms of Mexican (nancial crises have been strongly re4ected in the dynamics of the Mexican peso to the dollar exchange currency market. Speci(cally, in the Mexican (nancial crises during 1990’s, the peso su9ered signi(cant depreciation processes, which has important impacts in the macro- and micro-economical environment. In this paper, it is shown that the peso depreciation growth was greater than an exponential and that these growth rates are compatible with a spontaneous singularity occurring at a critical time, which signals an abrupt transition to new dynamical conditions. As in the major 1990’s (nancial crisis in 1994 –1995, some control actions (e.g., increasing the USA dollar supply) are commonly taken to decelerate the degree of abruptness of peso depreciation. Implications of these control actions on the crisis dynamics are discussed. Interestingly, by means of a simple model, it is demonstrated that the time at which the control actions begin to apply is critical to moderate the adverse e9ects of the (nancial crisis. c 2003 Published by Elsevier B.V. PACS: 64.60.Ak Keywords: Econophysics; Financial crisis; Finite time singularity
1. Introduction During the 1990’s, Mexico has been plagued with recurrent economic and (nancial crises. Instability sources involved both internal (e.g, political, social, etc.) and ∗
Corresponding author. Fax: +52-5-8044900. E-mail address: [email protected] (J. Alvarez-Ramirez).
1 The authors are also at Programa de Investigacion en Matematicas Aplicadas y Computacion, Instituto Mexicano del Petroleo.
c 2003 Published by Elsevier B.V. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.09.058
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Fig. 1. Peso–dollar exchange rate. Some important world (nancial events are highlighted.
external (e.g., contagion from foreign crises, etc.) factors. Historically, signals of Mexican (nancial crises have been strongly re4ected in the dynamics of the USA dollar–Mexican peso exchange currency market, leading to signi(cant peso depreciation (see Fig. 1). If a major economic and (nancial crises can be stated when at least 50% in4ation or currency depreciation in one year is produced [1], at least two major Mexican (nancial crises can be recognized during the 1990’s. The (rst one lasted around one year (Tequila e9ect), beginning on December 20, 1994, and produced about a 100% depreciation of the peso with respect to dollar. The dynamics of this crisis was temporarily disturbed by attempts to reduce the fall down of the peso (see the chronology in Section 3). The e9ect was a momentary (around 15 days) decrement in the peso exchange currency slip, although it continued a slower depreciation to attain its “fundamental” value stabilizing about 7.70 by November, 1995. The e9ects of the Tequila crisis on the Mexican economy and society were very adverse, including large goods price increments and high unemployment rates. Besides, the GNP fall (9.3% in annualized rate basis. The second major crisis seems to be triggered by the Asian Paci(c crisis by around August, 1997. In fact, due to a complex interrelated trading network in the Paci(c region, the (nancial crisis of the Asian Paci(c countries (Indonesian, Malaysia, etc.) infected the dynamics of the Mexican (nancial system. This second crisis lasted about one year, and produced a 60% peso depreciation. The main adverse e9ect of this crisis was that it has been delaying signi(cantly the recovery of the macro- and micro-economic conditions degraded by the Tequila crisis. To
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date, Mexico seems to have recovered similar macro-economic conditions present before December, 1994. However, from the micro-economic viewpoint, the sequels of the two major 1990’s crises on the Mexican society and economy cannot be underscored. To date, employment and wage are unstable and crime in large cities has achieved dangerous levels. From both social and economic viewpoint, it would be desirable to dispose of methodologies to predict the advent of (nancial and economic crises, in order to take corrective actions at appropriate times. In principle, a systematic understanding of (nancial crises leading to hyperin4ation-type dynamics would help to design suitable control policies and prevent to some extent their adverse e9ects [2]. Recently, Econophysics researchers have taken some steps towards this endeavour. Johansen and Sornette [3] detected that world population and some economic indices cannot be described as a (Malthusian) exponential growth function of time. A key concept to explain such explosive behavior was the presence of a singularity at a critical time tc , at which the population or economic signal approaches an unbounded value. Mizuno et al. [4] analyzed historical data of price indices. They found that, under hyperin4ation conditions, the logarithm of the price can be approximated by a double-exponential function of time. A generalization of Cagan’s model for description of in4ation was used to justify theoretically the double-exponential behavior. Based on a continuous-time version of the Cagan’s model developed by Mizuno et al. [4], Sornette et al. [5] found that hyperin4ation dynamics can be characterized by a power-law singularity culminating at a critical time tc . In this way, Mizuno et al.’s double-exponential function can be seen as a discrete-time approximation of the reported continuous-time model. Historical hyperin4ation data from many countries were used to illustrate the goodness of the power-law description. Also, in several papers the technique of stretched exponentials has been used for studying (nancial crashes [6]. The aim of this paper is twofold: To show that the power-law singularity formalism can be used to describe the dynamics of recent (nancial and economic Mexican crises. In this way, Mexican (nancial crises can be explained by a mechanism of in4ationary-type expectation (or positive feedbacks) between realized growth rate and people’s expected growth rate. To show that the oscillatory decoration on the (monotonous) power-law singularity function can be explained, at least partially, as the e9ects of control (e.g., monetary) actions aimed to correct hyperin4ationary-type e9ects. To this end, a simple control model is developed. The analysis and discussion are based on historical peso–dollar exchange currency data. 2. Peso–dollar exchange rate data In our analysis, the peso–dollar exchange currency rate data are taken from the Banco de Mexico web page (www.banxico.org.mx). The data describe the dollar exchange currency, in peso to the dollar, within the period from December 20, 1994 to January
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7, 2003. Before December 20, 1994, the peso to the dollar value was regulated by the regime of bands (i.e., minimum and maximum exchange currency levels were imposed). On December 20, 1994, an economic and (nancial crash, popularly denominated as “Tequila e9ect”, disturbed signi(cantly the peso to the dollar market. After the crash, the peso was allowed to 4uctuate freely. Currently, the level of the peso to the dollar value, known as FIX, is estimated daily by a sort of an auction procedure made by the Central Bank of Mexico (Banxico), with the participation of the main private banks and some other intermediaries. No data point is supposed to be missing. In other words, when applied to (nancial data, breaks due to holidays and weekends are disregarded. Nevertheless, the time units are said to be in business days (b-days) in the following, a week has often 5 days, a month about 23 days, and a year about 250 days. December 20, 1994 was taken as the (rst day. Fig. 1 shows the peso value p(ti ) and some signi(cant world economic events a9ecting, to some extent, the exchange market dynamics. Dates of (nancial crises and events in Fig. 1 state only approximately the moment at which the peso was signi(cantly a9ected. 3. Chronology of Mexican crises in the 1990’s To gain some insights on the micro- and macro-economic conditions leading to (nancial Mexican crises, in the following we provide a brief chronological description of major event during the 1990’s decade (see, for instance, [7]). December 1994: The crash followed the attempt on December 20th to devalue by 15%, which was met with a fresh attack which exhausted the reserves. The peso had to be allowed to 4oat: its value fell from 3.10 pesos to the dollar before the 20th to just a little short of 6.50. The government was unable to roll over its short-term debt at any reasonable rate, and the private sector also became unable to borrow. January 1995: On January 2, President Clinton announced Plan A to win Congress approval for 40 billion of USA loan guarantees for Mexico. On January 26, the IMF announced readiness to provide Mexico with a $7.76 billion standby credit, the IMF’s largest ever. On January 30, the White House turned to Plan B: in lack of Congress approval for loan guarantees, President Clinton launched a new support package centering on 20 billion of (nance from the USA Exchange Stabilization Fund (ESF). That money would be provided on Clinton’s own authority. On January 31, the White House announced new 50 billion package, including a Bank for International Settlements (BIS), standby credit and an extended IMF credit. February 1995: The USA.20 billion dollar emergency package was available by February 1st, and its immediate e9ect was an apparent recovery of the peso. In fact, during February and the (rst days of March, the peso value raised from 6.50 to around 5.00 pesos to the dollar. Throughout 1995: The bene(cial e9ects of the (rst loan on the peso–dollar exchange currency stability dissipated quickly. In fact, the peso continued to drop further in later months to achieve a value of around 8.00 pesos to the dollar at the end of the year. Mexico faced the consequences of the massive devaluation. In4ation and, interest rates
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soared to reach levels of 100% in some months. Out4ow of portfolio investment totaled 60 billion dollars. A regrettable e9ect of the rescue package was the Mexican Society feeling of a lack of sovereignty, since the collateral assurance (guarantee) for the whole loan were the Mexican oil receipts directed through the New York Federal Reserve Bank. January 1996–September 1997: The economy went through a stabilization period, where the peso 4uctuated freely within an interval from 7.4 to 8.0 pesos to the dollar as lower and upper bounds. Basically, the peso did not su9er a major depreciation. Interestingly, in addition to apparently random daily 4uctuations, the current exchange value displayed a cyclic behavior with cycle period of around a quarter. This so regular cyclic behavior has been explained mainly by the activity of closing and opening contracts for import and export of goods with the USA [8]. October 1997–November 1998: When the Mexican economy started to achieve a stable evolution, an external event disturbed it. In fact, between the months of June and December 1997, a group of newly industrialized Asian Paci(c countries underwent a severe (nancial crisis. The stock markets tumbled and the exchange rate depreciated deeply. Due to its strong trading interaction with the Asian Paci(c countries, the Mexican economy su9ered a severe contagion. By the mid-October, a sudden depreciation from 7.7 to 8.25 peso to the dollar was induced to try counteracting the adverse domestic e9ects of the Asian crisis. Such control action had no important e9ects since the exchange rate started a deep depreciation processes, attaining its peak by November 1998 where the exchange rate was 10.55 peso to the dollar. Some attempts directed to stop, or at least reduce, the depreciation process were taken. Basically, such control actions consisted in increasing the dollar supply, so to equilibrate the exchange market. As one can observe in Fig. 1, the result was an exchange rate evolution leading to a severe peso depreciation, decorated with three oscillations associated with such control actions. December 1998–March 2002: After the Asian Paci(c crisis shock, the Mexican economy started an important peso to the dollar appreciation. Out of a short transient (around 10 days) induced by the Brazilian (nancial crisis and a largest peak of 11.40 peso to the dollar, which apparently did not disturb signi(cantly the exchange rate dynamics, the Mexican economy evolved within a stable process with well tuned macro-economic indices and a stable peso to the dollar currency value 4uctuating around 9.50. April 2002–date: By 2002, USA recognized that its economy was entering a recession period. Since about 80% foreign Mexican trading depends on the American economy, Mexico was not immune to the USA deceleration process. At this moment, Mexican economy is under a recession process with the exchange rate depreciating slowly. However, it is not clear if this process is only a deceleration dynamics or can pave the way to a major economic and (nancial crisis. Summarizing, during the 1990’s the Mexican economy was subjected to alternating periods of severe (nancial crisis and stabilization attempts where the peso to the dollar exchange rate oscillated around a stable trajectory. As discussed in the Introduction, the matter of this paper is to gain some insights on the dynamics of major (nancial crisis. In this way, two major Mexican (nancial crises during the 1990’s can be easily
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identi(ed. Crisis I started on December 1994 and ended by November 1995, while Crisis II, induced by Asian Paci(c (nancial crisis, started by October 1997 and ended by November 1998. 4. Finite-time singularity description of %nancial crisis One can see that major (nancial crisis dynamics are characterized by (1) an induction period with a slow depreciation process, and (2) a fast-growth process. By considering only a crisis period (e.g., October 1997–November 1998), it is not hard to see that the dynamics cannot be explained as an exponential-growth process. In fact, one can easily show that ln(p) versus t plot cannot be adjusted by a straight line. This simple observation led to the conclusion that this hyperin4ation-type dynamics display a larger-than-exponential growth dynamics. From a discrete-time framework, Mizuno et al. [4] used an enhanced version of the Cagan’s model of in4ation to show that hyperin4ation processes can be described with a double-exponential model. In a recent paper, Sornette et al. [5] used a continuous-time approach to generalize the Cagan’s model. They found that hyperin4ations can be characterized by a power-law singularity culminating at a critical time tc . In this way, Mizuno et al. [4]’s double-exponential function can be seen as a discrete time step approximation of the more general nonlinear ODE formulation of the price dynamics which exhibits a (nite-time singular behavior. A feature of Sornette et al.’s formalism is that hyperin4ation description is made on ln p(t) rather than on the price itself p(t). At least in the Mexican exchange market during a (nancial crisis, people’s averaged expectation is based on the exchange rate p(t). This is not a serious drawback, since slight modi(cation to the generalized Cagan’s model proposed by Sornette et al. [5] can be made in order to (t into this circumstance. For the sake of completeness in presentation, in the following a brief description of modi(cations introduced to Sornette et al. [5]’s approach are given. Let p(t) be the market price and let p∗ (t) be the people’s averaged expectation price. These two prices are thought to evolve due to a positive feedback mechanisms where an upward change in market price p(t) in a unit time Qt induces a rise in the people’s expectation price, and such anticipation pulls up the market price. Let r(t) = p(t) − p(t − Qt)
(1)
be the growth rate at time t. According to the generalization introduced by Mizuno et al. [4], the expected growth rate r ∗ (t) at time t is proportional to the past realized growth rate r(t − Qt) in the following form: r ∗ (t) = r(t − Qt) + ar(t − Qt)m :
(2)
The (rst term in Eq. (2) expresses that the expected growth rate is at least the past realized growth rate r(t − Qt). On the other hand, the second term expresses nonlinear feedback e9ects induced by re-enforcement phenomena during large return periods. That is, when the realized growth rate becomes signi(cant, people’s expectations start
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to amplify these realized growth rates, which can lead to an unbounded expected growth rate. In principle, the aim of the market actors is to achieve the expected return r ∗ (t) within the next period. This can be expressed as follows: r(t) = r ∗ (t − Qt) :
(3)
Putting Eq. (2) together with Eq. (3) leads to r(t + Qt) = r(t − Qt) + ar(t − Qt)m . Taking the continuous-time limit, the above equation becomes dr = ar m ; dt
(4)
where a is a positive coeScient. The solution exhibits a (nite-time singularity 1=(m−1) tc r(t) = ar(0) ; tc − t
(5)
where the critical time tc = (m − 1)=r(0)m−1 is determined by the initial condition r(0) and the exponent m. When Eq. (1) is considered in the continuous-time limit, one obtains that the price p(t) is the integral of r(t) and also exhibits a (nite-time singularity at the same critical value tc . The time dependence of the market price p(t) exhibits the following two regimes: (i) Finite-time singularity in the price itself: p(t) = A + B(tc − t)−
with =
2−m ¿ 0 and B ¿ 0; for 1 ¡ m ¡ 2 : m−1
(6)
This solution is a genuine divergence of p(t) in (nite time at the critical time tc , (ii) Finite-time singularity in the derivative of the price: p(t) = A + B (tc − t)−
with =
m−2 ¿ 0 and B ¿ 0; for 2 ¡ m : m−1
(7)
As time approaches tc , the price accelerates with an in(nite slope reached at tc , while remaining (nite at the value A . The constants A and A arise from the integration of the growth rate r(t) given by Eq. (5). In the next section, we will show that expressions (6) and (7) can be used to describe the dynamics of Crisis I and Crisis II discussed before. The following should be stressed. As mentioned before, Mizuno et al. [4] proposed the double-exponential model p(t) ˙ exp(C1 exp(C2 t)) to describe dynamics with larger-than-exponential growth. It has been shown that this expression is actually a solution of the discrete-time model (2) r(t) = r(t − Qt) + ar(t − Qt)m . Since the model (4) can be seen as the continuous-time limit of the discrete-time model (2), the double-exponential expression corresponds to the limit of the singular function (6) and (7). In practice, one could use either the double-exponential or the singular functions to describe price dynamics during crises scenarios. However,
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in this paper we follow the Sornette’s setting and use the continuous-time description of crises phenomena. The advantage of using singular functions (6) and (7) over the double exponential function is that good estimates of singular times can be obtained. In turn, these singular time have an interesting interpretation within an economical and (nancial discussion.
5. Results Consider the Crisis I dynamics starting by December 1994 and ending by November 1995 (see Fig. 1). One can note that the USA–IMF rescue package disturbed signi(cantly the evolution of the exchange rate depreciation. In fact, by the (rst days of March the peso to the dollar parity decreased by about 25%. However, after around one month the e9ects of the injected (nancial resources seem to be dissipated and the peso continued its depreciation evolution. In this way, Crisis I can be separated into two periods: (i) Crisis I.bl corresponding to the before-loan process, and (ii) Crisis I.al corresponding to after-loan depreciation. The Crisis I.bl period was considered from December 23, 1994 to March 9 30, 1995, when the exchange rate attained a peak value of 7.59. On the other hand, Crisis I.al period was taken from May 3, 1995 to November 3, 1995, when a 8.13 peso to the dollar exchange rate was reached. Fig. 2 shows the dynamics of Crisis I.bl. These dynamics can be
Fig. 2. Dynamics and (tting functions for Crisis I.bl.
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Fig. 3. Dynamics and (tting functions for Crisis I.al.
described by the singular function p(t) = 6:9424(57:5 − t)−0:0654 (R2 = 0:9785 and = 0:01521). One can question if the dynamics can be described also with as an exponential growth function. In Fig. 2 is also shown our best exponential growth (tting given by p(t) = 5:53156 + 4:51 × 10−5 exp(t=5:3876) (R2 = 0:6927 and = 0:07081). Initially, the exponential growth function is able to provide a good description of the price dynamics. However, at later times close to the apparent singularity, the exponential growth function presents signi(cant deviations from the data. Besides, after the escape time located at about 57.5, the exponential growth function predicts prices that are signi(cantly below (around −15%) the actual ones. The lack of the exponential growth function to provide an acceptable description of the price dynamics is re4ected in the quite low correlation value R2 = 0:6927. We have tried to (t the price dynamics with second-order exponential growth dynamics like p(t) = A0 + A1 exp(t=1 ) + A2 exp(t=2 ), obtaining similar poor (tting results. In this way, it seems the singularity is an authentic property of Crisis I.bl dynamics. Analogous results are found for Crisis I.al and Crisis II. Fig. 3 shows the dynamics of Crisis I.al, which can be also described by the singular expression p(t) = 7:4(221:25 − t)−0:0423 . It is noted that the exponent of Crisis I.bl is larger than that of Crisis I.al. This means that Crisis I.bl was more severe than Crisis I.al in the sense that the velocity of peso depreciation was larger in the former case. In this way, it seems that the e9ect of the USA–IMF package was to delay the crossover to a saturation of the economic and (nancial system.
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Fig. 4. Dynamics and (tting functions for Crisis II.
The period of the Crisis II dynamics was taken from November 5, 1997 (just after the sudden depreciation induced by the Asian Paci(c crisis) to November 15, 1998, then the peso to the dollar value reached a historical peak value of 10.58. Fig. 4 shows the dynamics of Crisis II. The continuous line shows that, on the average, the Crisis II dynamics can be described by the expression p(t) = 10:975(936:25 − t)−0:0538 . The exponent of this crisis is between the exponents of Crisis I.bl and Crisis I.al, demonstrating that Crisis II was as severe as 1994 –1995 (nancial crisis. The above results seem to demonstrate that Mexican (nancial crisis in the 1990’s can be explained by a mechanism of in4ationary-type expectation between realized growth rate and people’s expected growth rate. That is, after an adverse information (the 15% government-induced sudden depreciation in December 20, 1994, and the Asian Paci(c (nancial crisis in the mid-1997), negative expectation of a further depreciation of people is reinforced, leading to an increased dollar demand to protect their wealth. Indeed, this process produces a severe shock to the domestic economy. 6. Control actions and %nancial crisis We have shown that, on the average, Mexican (nancial crisis can be well (tted with a power-law singularity. This approach provides a nice description of (nite-time escapes and monotonic peso depreciation. However, one notes that Crisis I.al and Crisis II
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are decorated with large oscillations, which disappear during the escape regime. For instance, in the (rst part of Crisis II one can (nd two large oscillations, with periods of around one quarter. The aim of this section is to provide a qualitative explanation to the oscillatory decorations on the (monotonic) power-law singularity function. During a (nancial crisis, governments and (nancial authorities and institutions take measures that can be seen as control actions to try counteracting, and even stopping, the adverse e9ects of a severe depreciation. The most immediate control action is taken by central banks by increasing the currency (dollar) supply. In this way, it is expected that n the market reaches a saturation level by satisfying most agent demands. Let u(t) = j=1 uj (t) be the control action function taken during a (nancial crisis, where uj (t) are individual control actions. In a very elusive way, u(t) can be seen as modeling (nancial resources injected into the market. From the viewpoint of the modeling framework described in Section 4, the aim of u(t) is to counteract the reinforcement dynamics leading to a singularity-type behavior. In this way, model (4) can be modi(ed as follows: dr = ar m − u : dt
(8)
The control action u(t) is assumed to be bounded, re4ecting the realistic fact that the injected resources are (nite. Non-negativity is also assumed to recast the fact that the control action reverts, to some extent, the monotonous increasing escape of the growth rate r(t). Our intention is not to give a precise modeling of u(t), which is a very complex problem. Instead, our aim is to use a simple model to gain some insight on the source of the oscillatory phenomenon accompanying the power-law singularity behavior. In this way, as a (rst approach we use the following expression for individual control actions: 0 if t ¡ tj ; uj (t) = (9) umax; j e−t=j if t ¿ tj ; where tj is the time at which the jth control action starts, umax is the maximum control action, and j is a dissipation time. The idea behind the model (9) is the following: At time tj , resources umax; j are injected into the market, which are absorbed by the market agents with a rate −1 j u(t). In other words, j is the mean time the control action u(t) a9ects the market dynamics. In the following, we will concentrate only on the Crisis I because it is the most interesting. The discussion and qualitative result apply vis-Ta-vis to oscillations in Crisis II. Unfortunately, for uj (t) given by Eq. (9), it is apparent that Eq. (8) have no explicit global solution, so that our discussion will be based on numerical simulations. The model (8), (9) was integrated with a Runge–Kutta method with a time step of 1:0 × 10−3 . Fig. 5 shows the simulations for the following set of parameters: m = 16, which corresponds approximately to the exponential values of the (tting function (6) for Crisis I, a = 2:5 × 10−9 , n = 2, umax; 1 = 155, umax; 2 = 55, t1 = 60, t2 = 110, 1 = 2:1 and 2 = 2:2. It should be stressed that for simulation purposes the values umax; j does not correspond to actual rescue package values. Instead, such values re4ect only qualitatively the relative magnitude of the corresponding control actions. On the other
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Fig. 5. Qualitative numerical simulations for Crisis I including two control actions as speci(ed by Eq. (9).
hand, the time when the control action u(t) is applied corresponds exactly with the times when the rescue packages are delivered. It is noted that the modeling approach (8), (9) is able to describe qualitatively the oscillations in Crisis I. In turn, this implies that such oscillations are induced, at least partially, by control actions directed to counteract the excessive dollar demand. It seems that the large peak in Fig. 5 was induced by the USA–IMF rescue package, which was available to the Mexican (nancial and economical systems by mid-February, 1995. At this point, two interesting question can be posed: How opportune and eScient was the control action induced by the 50 billion dollar rescue package? Does the same e9ect could be obtained if a smaller rescue package was available at earlier days of the crisis? To address these questions from the modeling approach described above, let us consider the model (9) with only the larger control action u1 (t) (i.e., u2 (t) = 0 for all t). This numerical simulation is presented in Fig. 6. Assume that the control action u1 (t) was applied several days after. Dotted line in Fig. 6 presents the numerical simulation results for t1 = 45. It is noted that, although the (nite-time singularity is not removed, it is moved far away by about 1400 days (approx. 5 years). One can conclude that the control action u(t) delays the e9ects of the (nite-time singularity. Fig. 7 presents the e9ect of applying a lower control action at an earlier time. It is
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Fig. 6. Qualitative numerical simulations for Crisis I showing the e9ect of the control time t1 .
observed that, despite the fact that umax; 1 has been lowered to about one half, control actions at earlier times of crisis can induce more bene(cial e9ects that larger control action applied at later times. Summarizing, the time at which the control action u(t) is applied plays a central role in the future dynamics of the (nite-time singularity. 7. Discussion The main conclusion from the above simulations is that, under a severe (nancial crisis showing escape behavior similar to a (nite-time singularity, the times at which control actions are applied (e.g., (nancial resources, political actions, etc.) are crucial in the stabilization process. Our simple simulations have shown that the application of 50 billion package in Crisis I was not opportune. That is, although such resources were necessary to reduce the adverse e9ects, the opportune application of the loan, maybe from 15 to 20 days earlier, would have delayed considerably the catastrophic peso depreciation (see Fig. 6). By delaying by at least one year the larger-than-exponential depreciation process, Mexican government and international (nancial institutions (e.g., IMF, WB, etc.) would have a larger period to make some corrective actions and to roll over the debts. Corrective actions would include reforming the (nancial sector to overcome its weaknesses and 4exibilizing the decision-making schemes to a changing economic situation.
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Fig. 7. Qualitative numerical simulations for Crisis I showing the combined e9ect the control action amount umax; 1 and the control time t1 .
Regarding the government debt during December 1994 and January 1995, Cole and Kehoe [9] have used a self-ful(lling debt modeling approach to explain the crash. The author’s analysis suggested that for a country, like Mexico, with a very short maturity structure of debt, there is an interval of debt for which the government, although it (nds it optimal to repay old debt if it can sell new debt, goes to default if it cannot sell new debt. In this way, a delaying of the (nancial crisis peak would allow Mexican government to sell new debt with a larger maturity structure. For an alternative, more comprehensive view of the debt crisis, see [10]. It is now a historical fact that the 20 billion loan was available to the Mexican (nancial and economical system by mid-February, 1995. The peso obtained a transient recovery to take again (with a slower velocity) its depreciation processes. Our analysis suggests that if a control action is not applied at early stages of the crisis (i.e., before the crisis reach a larger-than-exponential process), it is useless. In this way, it seems that the 20 billion control action was not so opportune. In fact, the (nancial crisis was highly excited and dissipated in a short period the possible bene(cial e9ects of the fresh resources. The government could not roll over its short-term debt. All this happened despite infusions of funds from the USA and the IMF totaling 50 billion dollars. A posteriori, one could conclude that, since the loan package was not available at the just time, a possible stabilization procedure could be the following: (a) Not to
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inject the 20 billion loan, and wait until the economic and (nancial systems attain their saturation limit, and (b) Once the economical and (nancial indices showed a saturation level, inject the fresh resources to reactivate and stabilize the economy. Of course, this is a very simplistic view. In practice, such control action could lead to some political and social instabilities, as one can witness in recent Argentine crisis. The possible bene(cial e9ects this type of control action is that it could lead to elimination of ineScient and spurious components of the (nancial and economical system. A general approach based upon this kind of considerations belong to the strategy 4exibility approach for management, which has crystallized in the recent “Real Options” theory [11]. We believe that a theory of dynamics and control of trading mechanisms in (nancial markets based on 4exibility and option theory is very promising. On the other hand, during a severe crisis, governments and (nancial institutions (e.g., central banks) are forced to do something before hyperin4ation-type behavior reaches unsustainable levels. On the other hand, (nancial crises are very costly. Not only the direct costs to governments for the resolution of the crisis, but the cost of economy recovery and social costs of poorer groups are high. Recovery of the precrisis economic growth rates in past crises has taken on the average of two to three years [12]. The best prescription for a (nancial crisis is therefore to prevent it. The analysis in this paper and recent results by Sornette et al. [5] are directed to this objective. However, further analysis with the use of more involved nonlinear dynamics should lead to better characterization of positive feedback mechanism. In principle, this should help to establish and implement an e9ective signaling system to warn the imminence of a (nancial crisis. A crucial step to confront a severe (nancial crisis is the suScient, opportune and feasible implementation of control actions to prevent crisis blow up. Tools from (optimal and feedback) control theories can help on the designing of timely policy actions. The preliminary results in this paper have shown that not only the quality but the timing of the policy action was essential for a successful response. 8. Conclusions We have analyzed the dynamics of the recent Mexican (nancial crises. On the basis of a hyperin4ation-type model based on positive feedback between realized return and agent’s expected returns proposed by Sornette et al. [5], we have shown that Mexican crisis can be described by a power-law singularity. We have extended previous analysis reported by Mizuno et al. [4] and Sornette et al. [5] on hyperin4ation-type dynamics, to account for control actions aimed to reduce the hyper-in4ationary-type e9ects. In this way, by modeling the infusion of (nancial resources as a dissipative signal, we have been able to reproduce qualitatively certain oscillations decorating the averaged crash dynamics. Namely, such oscillations have been partially explained as control actions that governments and (nancial institutions make to counteract the adverse e9ects of the crisis. In the main streamline, the results in this paper seem to reinforce the idea that during a (nancial and economical crisis, nonlinear feedback e9ects plays a central role. In this way, as Sornette et al. [5] have also suggested, our results point to the use of nonlinear
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analysis to understand the mechanisms leading to (nancial crises and crashes. Future work should be directed along this direction. Another potentially useful tool that can be explored is robust control and related issues directed towards attenuation and/or rejection of exogenous disturbances. This could be specially interesting in connection with Asian Paci(c crisis e9ects on Mexico Forex market. References [1] P. Cagan, The monetary dynamics of hyperin4ation, in: M. Friedman (Ed.), Studies in the Quantity Theory of Money, University of Chicago Press, Chicago, 1956. [2] R.T. Froyen, R.N. Waud, The determinants of federal reserve policy actions: a reexamination, J. Macroeconomics 24 (2002) 456–477. [3] A. Johansen, D. Sornette, Finite-time singularities in (nance, population and rupture, Physica A 294 (2002) 465–502. [4] T. Mizuno, M. Takayasu, H. Takayasu, The mechanism of double-exponential growth in hyperin4ation, Physica A 308 (2002) 411–419. [5] D. Sornette, H. Takayasu, W.-X. Zhou, Finite-time singularity signature of hyperin4ation, arXiv:physics/ 0301007 v1, 2003. [6] A. Johansen, D. Sornette, Endogeneous versus exogeneous crashes in (nancial markets, arXiv:condmat/0210509 v1, 23 Oct 2002. [7] I. Tower, Mexico 1994 versus Thailand 1997, in: B. Fuller, R. Suwanraks (Eds.), TDRI Quarterly Review, Vol. 12, 1997, pp. 9 –14. [8] J. Alvarez-Ramirez, Characteristic time-scales in the Mexpeso/USAdollar exchange currency market, Physica A 309 (2002) 157–170. [9] H.L. Cole, T.J. Kehoe, Federal Reserve Bank of Minneapolis, Research Department Sta9 Report, Vol. 210, April 1996. [10] F. Gil-Diaz, The origin of Mexico’s (nancial crisis, Cato J. 17 (1998) 1–7. [11] A.K. Dixit, R.S. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, New Jersey, 1994. [12] M. Goldstein, C. Reinhart, Forecasting Financial Crisis: Early Warning Signals for Emerging Markets. Policy Analysis in International Economics, Institute for International Economics, Washington, 1998.