Discrete Dynamics in Nature and Society

Volume 2015, Article ID 263908, 8 pages

http://dx.doi.org/10.1155/2015/263908

## Does Expectation of Correlation Breakdown in Financial Market Fulfill Itself?

^{1}Department of Quantitative Methods, University of Brescia, 25121 Brescia, Italy^{2}Department of Statistics and Quantitative Methods, University of Milano-Bicocca, 20126 Milano, Italy

Received 10 November 2014; Revised 14 July 2015; Accepted 15 July 2015

Academic Editor: Cengiz Çinar

Copyright © 2015 Paolo Falbo and Rosanna Grassi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper develops a model appeared in the literature whose focus was the way rational risk averse investors anticipate the correlation breakdowns of asset returns in periods of excess demand. That model analysed the dynamics of the “expected” returns of the risky asset, and their consistency with empirical evidence. However, the same model did not provide any evidence on actual correlation generated by the dynamics of returns. A model to link asset returns to excess demand is required to analyse the implied correlation between the securities traded. In this work we estimate such a model. Results confirm that the expected and ex-post correlation tend to move closely. In other words a self-fulfilling prophecy about correlation breakdown can take place, even when rational agents dominate the financial market.

#### 1. Introduction

Several studies in the literature document the so-called* correlation breakdown* phenomenon (also reported as* run to unity*) that is a sudden convergence of the correlation between the returns of traded assets to the value of 1. In particular this effect appears during periods of financial crisis, but there is also evidence during market euphoria. In Bertero and Mayer [1] and King and Wadhwani [2] authors show an increase in the correlation of stock returns at the time of the 1987 crash. Calvo and Reinhart [3] give evidence of correlation change associated with the Mexican crisis, and Baig and Goldfajn [4] find an increase of correlation in several East-Asian markets and currencies during the East-Asian crisis. Also Longin [5], Hartmann et al. [6], and Bae et al. [7] propose models based on extreme value theory, whereas Ramchand and Susmel [8], Ang and Bekaert [9], and Chesney and Jondeau [10] analyze Markov switching models. Loretan and English [11] identify in the “breakdowns of historical correlations” the origin of the Russian default in August 1998 and Karolyi and Stulz [12] show the existence of factors influencing joint movements in the US-Japan markets using regression methods. Situations, where the “historical” correlation among the assets traded on the same and even on different markets breaks down and asset suddenly shows a “run to unity” of this correlation, represent clearly a major problem for investors, since the risk protection usually guaranteed by the diversification of their portfolios is lost; besides these events tend to show exactly in the moment of major need.

The study of financial contagion was developed mostly around the notion of correlation breakdown (see [1–3, 13, 14]), so this paradigm helps to explain important dynamics of financial markets such as financial crises and speculative bubbles.

In this work we analyze the model of Falbo and Grassi [15] where the price dynamics of two assets (a high-risk and a low-risk ones) is subject to a time varying correlation. In particular, the authors consider the case where rational investors use expectation of excess demand to forecast the correlation between the assets to compose their investment portfolio. Their model explains several market dynamics, including market crashes, creation of rational bubbles, or cycles of diverse periods. Such results depend on the initial conditions as well as the percentage composition of the market between rational and irrational agents and their attitude to respond more or less aggressively to shocks in the excess demand. At the heart of their model there is the hypothesis that rational agents update the estimates of variance and covariance functions, depending on the excess demand of the risky asset.

Despite its interesting results, that model is missing an important component. In particular it does not include an equation on the returns of the low-risky asset and therefore it cannot provide clear insights into the origin of the time varying correlation. Besides, the absence of a conclusive evidence that the model does generate time varying correlation as a function of the excess demand of the risky asset weakens the hypothesis that rational agents are correct in linking their estimation of the correlation to the excess demand.

The main objective of this paper is therefore to complete the analysis of the model of Falbo and Grassi. To this purpose, we introduce an equation where the returns of the low-risk asset depend on the excess demand of the risky asset. This model is inspired by the observation of empirical data. This is relevant because next to supporting the theoretical consistency of the entire model such equation gives it also an empirical support. It is worth pointing out that, to the best of our knowledge, the literature on time varying correlation and correlation breakdown is entirely based on empirical and econometric analysis, so this work is a first attempt to provide a theoretical framework to explain and model the origin of this relevant feature of financial markets.

This work is organized as follows. Section 2 summarizes the main equations of the original model, with the purpose of making this paper self-contained. In Section 3 the equation for the low-risk asset is introduced and its empirical evidence is discussed. Section 4 develops the transition dynamics of the model integrated with the new equation of the low-risk assets. The results are used to discuss the consistency of the model with its central hypothesis on how rational agents estimate the correlation between the traded assets. Section 5 concludes.

#### 2. A Contagion Model with Rational and Speculative Investors

Moving from the settings in Falbo and Grassi [15] that we partially recover here to introduce the notation and to make this paper minimally self-contained, we consider a market with two types of agents, rational investors and speculators. These agents interact in a discrete time framework by trading two kinds of assets, a stock and a consol bond , with different risk levels and different expected returns, and , respectively.

Both types of investors look at excess demand, but with different premises.

Speculators are not informed of the true value of the risky asset at time and they place their investing decision on the basis of the excess demand of the previous period. In particular they will introduce an excess demand for the risky asset based on the formula . Parameter or describes, respectively, momentum or contrarian strategies.

Rational investors behave differently: at the beginning of each period , they update their estimate of the expected return of the high-risk asset comparing its current price with its true value (this knowledge is their information advantage):Observe that this formula describes a mean reverting dynamic of the expected return, where is a coefficient of reversion speed. In general, the expected return differs from the actual one, since it depends on the excess demand:

Rational agents diversify their investment through a Markowitz portfolio model estimating the correlation between the two assets depending on the excess demand. In particular they believe that during market phases with high or low excess demand, correlation between the assets tends to unity. So excess demand influences their portfolio decision as we will discuss later on.

A feedback develops at this point, since rational agents also influence the excess demand. The equation modeling the excess demand generated by the rational agents is , where is the optimal quantity of the stock in the Markowitz portfolio. Rational excess demand is a (linear) function of the difference between and , modulated by a sensitivity parameter , the share of speculators in the market.

The total market excess demand is then a convex combination of those generated by the two kinds of agents, , with being the share of speculators in the market.

The two-dimensional system is then the following:which fully describes the evolution of the fundamental variables of this market, and . The optimal quantity in the first equation of (3) is calculated by informed investors solving a portfolio optimization problem. In the Appendix, the expression of the solution is reported.

In such solution the central assumption consists in letting rational agents estimate the variance and correlation matrix of the two assets through the following equations:These equations tell us that rational agents update, respectively, their estimate of the variance and correlation, observing how the market is overbuying or overselling. In particular when the current excess demand gets very large, they expect that a correlation breakdown is going to develop ( as for ). As a consequence of the theory of Markowitz, the portfolio of rational agents will be largely impacted in this case. The response of rational agents when they fear that the correlation gets close to 1 has in turn a large impact on . In this way no obvious market dynamics originates, motivating the present study.

Of course, other functional forms for can be assumed. The key feature to be saved, to keep the economic structure of this model, is that as The parameter in the variance equation (4) can be either positive or negative; assuming that is positive amounts to estimating (on the side of the rational agents) that as , whereas as for negative This is a slight modification of the original model of Falbo and Grassi [15] which generalizes possible empirical applications.

It is worth observing that the hypotheses in (4) and (5) are just concerned with estimation of rational investors. They are well distinguished from speculators, since they do not use the excess demand to work out new estimates for the expected returns.

The model of Falbo and Grassi [15] generates returns of the stock price distributed in a way consistent with the variance function in (4). However, with respect to the correlation, that paper did not say anything conclusive since the model was missing an equation linking the low-risk asset returns to the excess demand . It is a major purpose of this paper to bridge this gap. By introducing an equation for the low-risk asset, we study under which conditions the model is able to generate a correlation between the two assets and check the internal consistency of the hypothesis in (5). Proving this consistency is of primary theoretical relevance. If it is indeed confirmed, the model would supply a robust explanation of the phenomenon of the correlation breakdowns, which, to the best of our knowledge, is still missing in the literature. In particular the model explains the origin of the correlation breakdown attributing it to two (possibly concurrent) causes: the herding behavior of momentum speculators and the procyclical attempt of rational investors to protect against the correlation breakdown.

The modeling contribution of this paper consists of including in the study an equation describing the price dynamics of the low-risk asset (). Through this modification the model driving the market becomes a three-dimension system:where and are suitable parameters. Let us observe that the new equation implies that in the absence of excess demand the realized returns of the low-risk assets coincide with their expected value. Moreover, adding such equation to system (3) does not introduce a feedback and so it does not alter the theoretical properties of that model, which can be still studied as a two-dimensional system.

#### 3. An Empirical Model for the Low-Risk Asset

In their empirical application Falbo and Grassi [15] develop a proxy of the excess demand of a stock index. We recover the same equations, since that variable is also needed here to model the returns of low-risky assets. Of course excess demand is not a directly observable variable. Besides, the definition itself of excess demand is not precise, given that it would require a clear notion of what “regular” or “normal level of” demand is, which is not at all obvious. Nevertheless the concept of excess demand in the financial markets is familiar and it is often used, both in the financial literature and in the comments of specialized magazines, as an explanation of large price movements, when markets take a clear direction either increasing or decreasing.

The proxy for the excess demand for a stock index can be obtained counting the number of times that the daily returns take a plus or minus sign over a given period in the stock market. The reason why the prevalence of a sign in the returns (in a given period of time) can be considered as a proxy for the excess demand is related to contagion arguments. Indeed it can be supposed that the longer a given sign prevails on the stock market, the higher the probability that agents are sharing a common sentiment in the same period. The following two expressions count the number of times that positive and negative returns are observed from back to :where is the indicator function and is the return observed on a stock market index at time . These expressions are actually slightly modified with respect to the original proposal in [15], extending the summations to 6 instead of 5 to capture more extreme cases. The authors argue that if, at any time , the counter of either sign exceeds the other, the market is expressing a “consensus” that can be taken as a measure of excess demand. In particular this consensus variable is defined as

This variable takes values in normal conditions (i.e., no stop of the market in the period to ) in the range Usually does not take the values , given that in the period to such values would be exceeded by their opposite counter (e.g., supposing that, from to , 4 negative and 3 positive returns have been observed, we would obtain , , and then ). However it is occasionally possible that, apart from the Saturdays and Sundays (that are preliminarily excluded from the analysis), the market stops for celebrations or some other reasons. For example, it can occasionally happen that the market works only 5 days between and . Then a possible result could be , , and then .

##### 3.1. Data

We observe the daily series of a US stock index calculated by Datastream (which is adjusted for various stock splits, mergers, etc., of individual companies) and the daily series of prices of the US Treasury bond, 30 years’ maturity, issued on September 1986. The period of observation ranges from January 1, 1993, to March 31, 2008. The following graph in Figure 1 shows the frequency distribution of the proxy of the excess demand as in (8). It can be observed that the cases where is equal to −6 or 6 are rare but not negligible.