Araújo, Tanya

Louçã, Francisco

English

Universidade Técnica de Lisboa

Trouble Ahead - The Subprime Crisis as Evidence of a New Regime in the Stock Market

In August 2007, the crisis of the subprime mortgage industry stormed the financial systems of several countries. As a response, hundreds of billions of dollars were injected by the authorities into the market. Nevertheless, this was not enough to avoid the second wave of the subprime crisis, as revealed by the fall of the S&P500 stock index since January 2008. In spite of the important achievements obtained in finance theory, the conventional wisdom on the inexistence of structure in the evolution of stock markets still prevails. The Efficient Market Hypothesis view ([1],[2]) assesses the evolution of financial markets as the result of a Brownian process. More recently, the contributions of econophysicists ([3],[4],[5]) have challenged the dominance of randomness. Using a stochastic geometry technique, here we show that the dynamics of the S&P500 set of stocks defines a structure, which is specifically shaped by the occurrence of crises. A new coefficient is defined in order to capture the structural changes occurring on the S&P500 set of stocks.
This coefficient highlights an important modification of the dynamics of the 253 firms represented in the S&P500 and acting in the market for the period August 1988-January 2008, and situates the turbulence since the Summer 2007 as replica of a larger structural change going on for a decade

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        Carlos Pestana Barros & Nicolas Peypoch    A Comparative Analysis of Productivity Change in Italian and Portuguese Airports       WP 006/2007/DE _________________________________________________________    Tanya Araújo and Francisco Louçã    Trouble Ahead – The Subprime Crisis as Evidence of a New Regime in the Stock Market    WP 26/2008/DE/UECE Department of Economics WORKING PAPERS  ISSN Nº 0874-4548 School of Economics and Management TECHNICAL UNIVERSITY OF LISBON TROUBLE AHEAD- The Subprime Crisis as Evidence of a NewRegime in the Stock MarketTanya Araújo ∗ and Francisco LouçãISEG (School of Economics and Management)Technical University of Lisbon (TULisbon)Rua do Quelhas, 6 1200-781 Lisboa PortugalResearch Unit on Complexity in Economics (UECE) †April 22, 2008In August 2007, the crisis of the subprime mortgage industrystormed the financial systems of several countries. As a response,hundreds of billions of dollars were injected by the authorities intothe market. Nevertheless, this was not enough to avoid the secondwave of the subprime crisis, as revealed by the fall of the S&P500stock index since January 2008. In spite of the important achieve-ments obtained in finance theory, the conventional wisdom on theinexistence of structure in the evolution of stock markets still pre-vails. The Efficient Market Hypothesis view ([1],[2]) assesses theevolution of financial markets as the result of a Brownian process.More recently, the contributions of econophysicists ([3],[4],[5]) havechallenged the dominance of randomness. Using a stochastic ge-ometry technique, here we show that the dynamics of the S&P500set of stocks defines a structure, which is specifically shaped by theoccurrence of crises. A new coefficient is defined in order to cap-ture the structural changes occurring on the S&P500 set of stocks.∗corresponding author: tanya@iseg.utl.pt†Acknowledgement: This research was supported by FCT Portugal, projectPDCT/EGE/60193/20041This coefficient highlights an important modification of the dynam-ics of the 253 firms represented in the S&P500 and acting in themarket for the period August 1988-January 2008, and situates theturbulence since the Summer 2007 as replica of a larger structuralchange going on for a decade.In order to identify the structure in the market we proceed as follows.Define pt as the price of a given stock and the stock price return rt as thedaily change of the logarithm of stock price, rt = ln(pt)− ln(pt−1). Pick therepresentative set of N stocks and their historical data of returns over a timeinterval (window). From the returns data, compute the matrix of distancesbetween the N stocks (as in references [3] and [4]).dkl =√2 (1− Ckl) (1)being Ckl the correlation coefficient of the returns. From the matrixof distances, compute coordinates for the stocks in an Euclidean space ofdimension N − 1. The stocks are now represented by a set {xi} of points inRN−1. To this cloud of points apply the standard analysis of reduction of theircoordinates to the center of mass and the computation of the eigenvectors ofthe inertial tensor. The directions for which the eigenvalues are significantlydifferent from those obtained applying the same technique to surrogate data(as obtained by independent time permutation for each stock) are identifiedas the market systematic variables. The number of systematic variablesdefine the effective dimensions of the market space (as in [6], [7]).It was empirically found that markets of different sizes, ranging from 70to 424 stocks, across different time windows (from one year to 35 years)and also from different market indexes1 have only six effective dimensions.These six-dimensional spaces define the reduced subspaces which carry thesystematic information related to the correlation structures of the markets.These dimensions capture the structure of the deterministic correlations andeconomic trends that are driving the market, whereas the remainder of themarket space may be considered as being generated by random fluctuations.Moreover, the application of our stochastic geometry technique over time win-dows of one year (w = 250) shows that the quite ’universal’ six-dimensionalreduction also holds for turbulent and normal periods. Both the years includ-ing the most extreme events and the years where ’business as usual’ prevailsfit into a six-dimensional space.1stocks from the S&P500 and Dow Jones indexes were considered2However, observing the evolution of the S&P500 market space along itsleading effective dimensions, important differences appeared in the marketshape (Fig.1). The most remarkable differences depend on whether the yearlyperiod corresponds to a normal or to a turbulent period. If the yearly pe-riod contains relevant crashes, the geometric object defining the dynamics ofthe market is distorted, acquiring prominences in some particular directions.The index S ([7], [8]) computes that distortion effect, namely, the lack ofuniformity along the S&P500 six effective dimensions.St = Σ6i=1λt(i)−λ′t(i)λ′t(i)= Σ6i=1λt(i)λ′t(i)− 1 (2)where λt(1), λt(2), ..., λt(6) are the six largest eigenvalues of the marketspace and λ′t(1), λ′t(2), ..., λ′t(6) are the largest six eigenvalues obtained fromsurrogate data. In computing S, at a given time t, both λt and λ′t areobtained over the same time window and for the same set of stocks. Lookingfor relevant distortions in the shape of the S&P500 market space through thelast 20 years, we found that amongst the highest values of the index S arethose computed for some important dates, as October 1997 and September2001, as Fig.2 shows.But if the geometric object defined by the dynamics of the market isdistorted whenever a crisis occurs, it may be caused by groups of stock thatevolve in strong synchronicity. To characterize this additional informationon the structure of the market spaces, here we define the coefficient R, whichquantifies the distribution of the correlation strengths between stocks presentin the S&P500 market space along the last 20 years.From the matrix of distances between stocks (equation 1) computed in thereduced six dimensional space (D6) over a time window of 22 days, we applythe hierarchical clustering process to construct the minimal spanning tree(MST) that connects the N securities. Then the boolean graph BD6 is definedby setting b(i, j) = 1 if d6(i, j) ≤ LD6 and b(i, j) = 0 if d6(i, j) > LD6 , whereLD6 is the smallest threshold distance value d6(i, j) that insures connectivityof the whole network in the hierarchical clustering process. This allows fordefining the coefficient R, which captures the relative distribution of thedistance values below and above the smallest threshold distance value (LD6)that insures connectivity of the whole network.Rt =∑d6t (i,j)≤LD6d6t (i,j)∑d6t (i,j)>LD6d6t (i,j)(3)3Results (Fig.3) show that the amount of strong correlated (short-distant)fluctuations in the network of stocks is very large for some particular crises(1997 second Black Monday, 2003 general recession and 2007-2008 SubprimeCrisis). These networks display a large amount of distances whose valuesare below the endogenous threshold value. This is due to the emergence of arelevant set of highly correlated fluctuations of the stock returns during mar-ket shocks forcing several weak correlated fluctuations to leave this category.Although the values of the overall network distances decrease with crashes,the emergence of highly correlated groups of stocks occupying the promi-nences in the market distorted shape, leads to an increase of the value of theendogenous threshold LD6 . As a consequence, the number of distances belowLD6 tend to be much higher than the number of those that remain above theendogenous threshold, leading to a significant increase of the values of R.During the Subprime Crisis, R reaches 1.4 (Fig.3), while the same co-efficient computed for normal periods rests below 0.5 (computing R fromsurrogate data yields typical values around 0.025). The evolution of R con-firm our previous results, identifying the major crashes in the period anddetecting how peculiar it is the Winter 2008 crisis. The Subprime Crisisconstitutes the highest peaks in the evolution of R for the period under con-sideration. The results reveal that a major change is occurring for the lastdecade, imposing a new dynamic structure marked by frequent crashes. AsFig.2 shows, the crashes concentrate in the period after 1997. This differencein the empirically described evolution suggests that in the period of the easyinterest rates, the ’Internet boom’ and the housing bubble, a new regime wasgenerated, giving birth to a new phase of turbulence in the financial mar-kets. Distortion effects occurring on the market effective dimensions provideuseful insight on the structure of the markets as it is revealed under shocks.Here we show that the characterization of the distribution of the correlationstrengths between stocks provides additional informational on the mutationin the structure of the market, suggesting that the prominences emerging inthe market shape during crisis correspond to groups of companies that movefollowing sectoral dynamics, in an even stronger synchronization.The results shed a new light on the subprime crisis since August 2007through the Winter 2008, and suggests that it is not a simple speculativeshock. As the data and the stochastic geometry suggest, this ongoing tur-bulence is part of a mutation in the structure of the market since around1997. The new structure has generated deeper and more frequent episodesof crises. The trouble is ahead.4References[1] Samuelson, P. (1995), Proof that properly anticipated prices fluctuaterandomly, Industrial Management Review 6.[2] Fama, E. F. (1998), Market efficiency, long-term returns and behavioralfinance, Journal of Financial Economics 49.[3] Mantegna R.N. (1999), Hierarchical structure in financial markets, Eu-ropean Physics Journal B 11.[4] Mantegna, R.N., Stanley, H.E. (2000), An Introduction to Econophysics:Correlations and Complexity in Finance, Cambridge, Cambridge Uni-versity Press.[5] Gopikrishnan P., Rosenow B., Plerou V. and Stanley HE (2001), Quan-tifying and Interpreting Collective Behavior in Financial Markets, Phys.Rev. E 64.[6] Vilela Mendes R., Araújo T. and Louçã F. (2003), Reconstructing anEconomic Space from a Market Metric, Physica A, 323.[7] Araújo T., Louçã, F. (2007), The Geometry of Crashes - A Measure ofthe Dynamics of Stock Market Crises, Quantitative Finance, 7(1).[8] Araújo T., Louçã, F. (2008), The Seismography of Crashes in FinancialMarkets, Physics Letters A, Vol. 372.5−0.500.5−0.500.5−0.500.52001  253 stockse2e3−0.5 00.5−0.500.5−0.500.5e12007  253 stockse3e2−0.500.5−0.500.5−0.500.5e11992  253 stockse3e2−0.500.5−0.500.5−0.500.5e1Surrogate  Data   253 stockse3e2Figure 1: The evolution of the stock market for one year (or for a randompermutation of data). The first two plots show the spherical configurationof both surrogate (time permuted) and a year of ’business as usual’ data,whereas the last two plots show the distorted shapes of the same marketspace in years of turbulence. While in 1992 there is no relevant difference inrelation to time permuted data, in 2001 and 2007 a new shape emerges.688 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08024681012141618253 stocks        A new regime after 1997SS>5Figure 2: The evolution of the structure index S - using a moving windowof one month of daily returns (w = 22) - corresponds to the evidence that,during financial crises, the geometric object describing the stock market isdistorted along definite directions.87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 0800.511.5253 stocks         From Jan.1988 to Mar.2008RR>1Figure 3: The evolution of the coefficient R (with w = 22) captures theemergence of highly correlated groups of stocks and detects how peculiar itis the Winter 2008 crisis.7