Scale invariant properties of public debt growth

Public debt is one of the important economic variables that quantitatively describes a nation's economy. Because bankruptcy is a risk faced even by institutions as large as governments (e.g. Iceland), national debt should be strictly controlled with respect to national wealth. Also, the problem of eliminating extreme poverty in the world is closely connected to the study of extremely poor debtor nations. We analyze the time evolution of national public debt and find "convergence": initially less-indebted countries increase their debt more quickly than initially more-indebted countries. We also analyze the public debt-to-GDP ratio R, a proxy for default risk, and approximate the probability density function P(R) with a Gamma distribution, which can be used to establish thresholds for sustainable debt. We also observe "convergence" in R: countries with initially small R increase their R more quickly than countries with initially large R. The scaling relationships for debt and R have practical applications, e.g. the Maastricht Treaty requires members of the European Monetary Union to maintain R < 0.6.Comment: 9 pages, 8 figure

Systemic risk in dynamical networks with stochastic failure criterion

Complex non-linear interactions between banks and assets we model by two time-dependent Erd\H{o}s Renyi network models where each node, representing bank, can invest either to a single asset (model I) or multiple assets (model II). We use dynamical network approach to evaluate the collective financial failure---systemic risk---quantified by the fraction of active nodes. The systemic risk can be calculated over any future time period, divided on sub-periods, where within each sub-period banks may contiguously fail due to links to either (i) assets or (ii) other banks, controlled by two parameters, probability of internal failure $p$ and threshold $T_h$ ("solvency" parameter). The systemic risk non-linearly increases with $p$ and decreases with average network degree faster when all assets are equally distributed across banks than if assets are randomly distributed. The more inactive banks each bank can sustain (smaller $T_h$), the smaller the systemic risk---for some $T_h$ values in I we report a discontinuity in systemic risk. When contiguous spreading becomes stochastic (ii) controlled by probability $p_2$---a condition for the bank to be solvent (active) is stochastic---the systemic risk decreases with decreasing $p_2$. We analyse asset allocation for the U.S. banks.Comment: 7 pages, 7 figure

Cross-correlations between volume change and price change

In finance, one usually deals not with prices but with growth rates $R$, defined as the difference in logarithm between two consecutive prices. Here we consider not the trading volume, but rather the volume growth rate $\tilde R$, the difference in logarithm between two consecutive values of trading volume. To this end, we use several methods to analyze the properties of volume changes $|\tilde R|$, and their relationship to price changes $|R|$. We analyze $14,981$ daily recordings of the S\&P 500 index over the 59-year period 1950--2009, and find power-law {\it cross-correlations\/} between $|R|$ and $|\tilde R|$ using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these cross-correlations. Motivated by the relationship between $| R|$ and $|\tilde R|$, we estimate the tail exponent ${\tilde\alpha}$ of the probability density function $P(|\tilde R|) \sim |\tilde R|^{-1 -\tilde\alpha}$ for both the S\&P 500 index as well as the collection of 1819 constituents of the New York Stock Exchange Composite index on 17 July 2009. As a new method to estimate $\tilde\alpha$, we calculate the time intervals $\tau_q$ between events where $\tilde R>q$. We demonstrate that $\bar\tau_q$, the average of $\tau_q$, obeys $\bar \tau_q \sim q^{\tilde\alpha}$. We find $\tilde \alpha \approx 3$. Furthermore, by aggregating all $\tau_q$ values of 28 global financial indices, we also observe an approximate inverse cubic law.Comment: 7 pages, 5 figure

Random matrix approach in search for weak signals immersed in background noise

We present new, original and alternative method for searching signals coded in noisy data. The method is based on the properties of random matrix eigenvalue spectra. First, we describe general ideas and support them with results of numerical simulations for basic periodic signals immersed in artificial stochastic noise. Then, the main effort is put to examine the strength of a new method in investigation of data content taken from the real astrophysical NAUTILUS detector, searching for the presence of gravitational waves. Our method discovers some previously unknown problems with data aggregation in this experiment. We provide also the results of new method applied to the entire respond signal from ground based detectors in future experimental activities with reduced background noise level. We indicate good performance of our method what makes it a positive predictor for further applications in many areas.Comment: 15 pages, 16 figure

A universal mechanism for long-range cross-correlations

Cross-correlations are thought to emerge through interaction between particles. Here we present a universal dynamical mechanism capable of generating power-law cross-correlations between non-interacting particles exposed to an external potential. This phenomenon can occur as an ensemble property when the external potential induces intermittent dynamics of Pomeau-Manneville type, providing laminar and stochastic phases of motion in a system with a large number of particles. In this case, the ensemble of particle-trajectories forms a random fractal in time. The underlying statistical self-similarity is the origin of the observed power-law cross-correlations. Furthermore, we have strong indications that a sufficient condition for the emergence of these long-range cross-correlations is the divergence of the mean residence time in the laminar phase of the single particle motion (sporadic dynamics). We argue that the proposed mechanism may be relevant for the occurrence of collective behaviour in critical systems

Bankruptcy risk model and empirical tests

We analyze the size dependence and temporal stability of firm bankruptcy risk in the US economy by applying Zipf scaling techniques. We focus on a single risk factor-the debt-to-asset ratio R-in order to study the stability of the Zipf distribution of R over time. We find that the Zipf exponent increases during market crashes, implying that firms go bankrupt with larger values of R. Based on the Zipf analysis, we employ Bayes's theorem and relate the conditional probability that a bankrupt firm has a ratio R with the conditional probability of bankruptcy for a firm with a given R value. For 2,737 bankrupt firms, we demonstrate size dependence in assets change during the bankruptcy proceedings. Prepetition firm assets and petition firm assets follow Zipf distributions but with different exponents, meaning that firms with smaller assets adjust their assets more than firms with larger assets during the bankruptcy process. We compare bankrupt firms with nonbankrupt firms by analyzing the assets and liabilities of two large subsets of the US economy: 2,545 Nasdaq members and 1,680 New York Stock Exchange (NYSE) members. We find that both assets and liabilities follow a Pareto distribution. The finding is not a trivial consequence of the Zipf scaling relationship of firm size quantified by employees-although the market capitalization of Nasdaq stocks follows a Pareto distribution, the same distribution does not describe NYSE stocks. We propose a coupled Simon model that simultaneously evolves both assets and debt with the possibility of bankruptcy, and we also consider the possibility of firm mergers.Comment: 8 pages, 8 figure

Common scaling behavior in finance and macroeconomics

In order to test whether scaling exists in finance at the world level, we test whether the average growth rates and volatility of market capitalization (MC) depend on the level of MC. We analyze the MC for 54 worldwide stock indices and 48 worldwide bond indices. We find that (i) the average growth rate $\langle$ r $\rangle$ of the MC and (ii) the standard deviation $\sigma(r)$ of growth rates r decrease both with MC as power laws, with exponents $\alpha_w$ = 0.28 ± 0.09 and $\beta_w$ = 0.12 ± 0.04. We define a stochastic process in order to model the scaling results we find for worldwide stock and bond indices. We establish a power-law relationship between the MC of a country's financial market and the gross domestic product (GDP) of the same countr

Anti-correlation and subsector structure in financial systems

With the random matrix theory, we study the spatial structure of the Chinese stock market, American stock market and global market indices. After taking into account the signs of the components in the eigenvectors of the cross-correlation matrix, we detect the subsector structure of the financial systems. The positive and negative subsectors are anti-correlated each other in the corresponding eigenmode. The subsector structure is strong in the Chinese stock market, while somewhat weaker in the American stock market and global market indices. Characteristics of the subsector structures in different markets are revealed.Comment: 6 pages, 2 figures, 4 table

Common Scaling Patterns in Intertrade Times of U. S. Stocks

We analyze the sequence of time intervals between consecutive stock trades of thirty companies representing eight sectors of the U. S. economy over a period of four years. For all companies we find that: (i) the probability density function of intertrade times may be fit by a Weibull distribution; (ii) when appropriately rescaled the probability densities of all companies collapse onto a single curve implying a universal functional form; (iii) the intertrade times exhibit power-law correlated behavior within a trading day and a consistently greater degree of correlation over larger time scales, in agreement with the correlation behavior of the absolute price returns for the corresponding company, and (iv) the magnitude series of intertrade time increments is characterized by long-range power-law correlations suggesting the presence of nonlinear features in the trading dynamics, while the sign series is anti-correlated at small scales. Our results suggest that independent of industry sector, market capitalization and average level of trading activity, the series of intertrade times exhibit possibly universal scaling patterns, which may relate to a common mechanism underlying the trading dynamics of diverse companies. Further, our observation of long-range power-law correlations and a parallel with the crossover in the scaling of absolute price returns for each individual stock, support the hypothesis that the dynamics of transaction times may play a role in the process of price formation.Comment: 8 pages, 5 figures. Presented at The Second Nikkei Econophysics Workshop, Tokyo, 11-14 Nov. 2002. A subset appears in "The Application of Econophysics: Proceedings of the Second Nikkei Econophysics Symposium", editor H. Takayasu (Springer-Verlag, Tokyo, 2003) pp.51-57. Submitted to Phys. Rev. E on 25 June 200