Measuring critical transitions in financial markets

Tipping points in complex systems are structural transitions from one state to another. In financial markets these critical points are connected to systemic risks, which have led to financial crisis in the past. Due to this, researchers are studying tipping points with different methods. This paper introduces a new method which bridges the gap between real-world portfolio management and statistical facts in financial markets in order to give more insight into the mechanics of financial markets.

the inverse of rebalancing need as the energy barrier, which prevents an investor from moving to the next energy basin and a new preferred state. This paper aims to measure these critical transition points in time.

Results
Extending the approach by Münnix et al. 38 , who investigated the similarity between correlation matrices at different times, we define the similarity between two points in time t, t′ as the L 1 -norm similarity.
(1) 1 between two portfolios P(t), P(t′). The entries of the similarity matrix ζ effectively measure the transaction costs an investor has to pay in order to rebalance the portfolio from one point in time to another. We use the daily closing prices of the S&P 500 components from 2000 until the end of 2016 as input data for the creation of mean-variance portfolios with a time window ω of 3 years. Choosing the parameter ω to be 3 years ensures that the ratio between the number of data-points and the number of assets Q is greater than 1.5 and therefore the correlation matrix is always well behaved. Figure (1) shows the similarity matrix ζ for the minimal variance portfolios.
In this matrix, one can identify two clusters of high similarity. The first one spans from October 2008 to September 2011 with an average similarity ζ = 0.58. The next visible cluster ranges from September 2011 until July 2014 and has an average similarity of ζ = 0.48. Before October 2008, the high similarity clusters are overlapping each other and do not show such a sharp structural change as in October 2008, September 2011 and July 2014. One notices smaller transitions, for example in September 2001, but these occur within a moderate similarity level.
In order to get more detailed information on the market phase duration depicted by similarity clusters, we use eigenvalue decomposition on the similarity matrix ζ.
The resulting normed eigenvalues λ λ λ >...> >...> k k 0 max correspond to the importance of the eigenvector u k in decreasing order.
These unit eigenvectors are orthogonal to each other and describe the data with a new set of basis vectors. In our case, the eigenvectors have the following interpretation: Each component of an eigenvector corresponds to a point in time where its absolute value signals whether at time t there is a significant contribution for describing a certain level of similarity in the matrix ζ. Since the eigenvectors correspond to a time-series, successive high values in the eigenvector imply a similar level of similarity. Therefore by following the temporal evolvement of u kt , one follows a direction of similarity level. By taking the fourth power of every entry u kt 4 , unimportant entries become negligible small while the rest are amplified, which is usually done when calculating the inverse participation ratio 39,40 . This allows us to capture the participation of the eigenvector to a market state at that point in time.
In Fig. (2) the first twelve eigenvectors to the fourth power are shown, which cover approximately 82% of the information.
The eigenvectors k = 1, 3 and 4 are needed to explain most of the similarities between 2000 and 2008 and represent 28% of the overall information. The eigenvectors u 2 and u 5 are representing 17% of the similarity matrix. One notices the separation in October 2008. After this point in time a new subspace is needed to describe the properties of the similarity matrix ζ. This is a sign of structural change in the investor's minimal-variance portfolio. From October 2008 on, an investor would have had to completely change his portfolio in order to reposition himself in the new market environment. The next six eigenvectors show shorter clusters. In the subspace spanned by the directions of k = 6, 8, 10 one can find a cluster right before the financial crisis of 2008. This phase ranges Since this analysis relies on visually inspecting the temporal development of the eigenvectors, we automatized this procedure by using an algorithm (see methods), which maps, from a chosen set of eigenvectors, the significant ones at a time t to a state.
In In order to verify whether there is an impact on the S&P 500 by a state change, we performed a Granger causality test 41 between the changes in overall trading volume within a week and the absolute weekly state changes.  The results in Table 1 show that the week after a state change, with more than two eigenvectors, is linked to the volume changes in the S&P 500. For lags longer than a week, there is no Granger causality on a 0.05 significance level performed with a F-test.

Discussion
The approximation of a risk aware investor by the mean-variance model aims to close the gap between stylized facts (volatility clustering, fat tails) and the connection between systemic risks and correlation matrices.
The S&P 500 analysis shows similar transition points to the pure PCA analysis of a correlation matrix 22,24,42 . That implies that one can measure the transition point t c of a financial system by applying eigenvalue decomposition on the similarity matrix ζ and determine whether t′ belongs to the same similarity subspace as t < t′. If at t′ another set of eigenvectors is needed, then t′ is a transition point t c . By increasing the number of eigenvectors and therefore gaining descriptive information of the similarity matrix, more transition points can be identified and more subtle distinctions between states can be found. These points in time highlight the investment shifts in the market, which can be small or larger structural changes as it was the case in 2008 and 2011. These critical transitions are linked to volume changes in the S&P 500.
In summary, our method to find critical transition points within a financial market is based on the similarity between mean-variance portfolios at different times. The resulting matrix is analysed by eigenvalue decomposition, which uncovers the temporal development of different similarity levels. As a last step, the subspaces formed by the eigenvectors are mapped to unique states along the time axis. This method allows to find two new aspects of financial markets: Market phases must be classifiable by an investor approximation, times of systemic risk fall into transition phases. These results are complementary to the known stylized facts and should be incorporated when constructing new general financial market models.

Methods
In order to approximate an investors risk aversion, we use the mean-variance model by Markowitz 43,44 in its classical form. This is an optimisation problem, where one has to minimise the variance and maximise the return of a portfolio p. Moreover, by restricting the portfolio components p i to be positive for all available assets N, we only allow long positions. The resulting selection problem is solved by minimising the cost-function where μ i is the expected return of asset i and C the pearson covariance matrix. The parameter γ is used to balance the trade-off between risk and return. A value of γ = 0 would result in only minimising the variance, while a value of γ = 1 would cause portfolios to be only optimized for maximum return. We implemented a coordinate descent algorithm as used by Friedman et al. 45,46 for fitting generalised linear models in order to generate efficient portfolios for a specific γ. As proposed by Altenbuchinger et al. 47 , one can incorporate the equality constraint ∑ = p 1 i i in the coordinate descent algorithm by substituting = − ∑ ≠ p p 1 k i i k i . By calculating the partial derivative and solving for p k one obtains an extreme value for p k and p s . With the help of the second derivative and curve sketching, one can fulfill the constraint p i ≥ 0 and a coordinate descent step can be constructed. Iterating this update step in combination with active set cycling 46,48,49 , allows the generation of thousands of portfolios in a reasonable time frame.
The algorithm for mapping a given set of eigenvectors to a state works in 4 steps: 1. Calculate the participation = u U kt 4 and normalize each column to the max-Norm. 2. Map U to a state matrix S = U > τ, where τ is a given threshold. τ is set to 0.01 in this paper. Each row of S now represents a state. 3. These states are now mapped to a state number S → s between 1 and the number of unique states found in S, where state 1 is the state combined from the eigenvectors with the lowest eigenvalues. 4. The state time-series s is returned.
The data we used to create the mean-variance portfolios was downloaded from the WIKI Quandl database 50 . The financial time-series data for the S&P 500 was downloaded with the QUANDL50 data interface from the WIKI database.