Complex Systems: Solutions and Challenges in Economics, Management and Engineering pp 411-422 | Cite as

# Persistent Correlations in Major Indices of the World Stock Markets

## Abstract

Time-dependent cross-correlation functions have been calculated between returns of the major indices of the world stock markets. One-, two-, and three-day shifts have been considered. Surprisingly high and persistent-in-time correlations have been found among some of the indices. Part of those correlations can attributed to the geographical factors, for instance, strong correlations between two major Japanese indices have been observed. The reason for other, somewhat exotic correlations, appear to be as much accidental as it is apparent. It seems that the observed correlations may be of practical value in the stock market speculations.

### Keywords

Stock market indices Correlation functions Pearson correlation Technical analysis## 1 Introduction

As is well known, the time-evolution of the stock markets, and, in particular, the stock market indices, exhibit considerable short-time correlations. These are sufficient to disprove claims of the reliability of the random-walk approximations to that evolution or strong versions of the efficient-market hypothesis [1, 2]. On the other hand, there exist a set of usually quite simple computational and visualization techniques, called technical analysis [3, 4], which aims to obtain the approximate predictions of trends and their corrections in market the data. It is so even though the data appear as realizations of a *random* process. It is to be noticed that some recent publications, e.g. [5, 6, 7, 8] have lead to considerable revision of the previously ultra-critical stand of the many experts regarding technical analysis.

One of the many possible strategies of “beating the market” which are close to the spirit of technical analysis consists of identification of correlated pairs of stocks. In fact, if we find that, during a specific, long, time interval, the increase (decrease) of the value of one stocks has been followed by a corresponding change in another stock, we may attempt to guess that this relation may persist, at least for short additional time.

To quantify the above intuitive remarks, we have computed the time-dependent cross-correlation functions of returns of the major world stock market indices. Among the latter, the following indices have been included: ALL ORDINARIES (ALL-ORD), AMEX MAJOR (AMEX-MAJ), BOVESPA, B-SHARES, BUENOS, BUX, CAC40, DAX, DJIA, DJTA, DJUA, EOE, FTSE100, HANGSENG, MEXICIPS, NASDAQ, NIKKEI, RUSSELL, SASESLCT, SMI, SP500, TOPIX, and TSE300. Let us notice that in one of our previous work a similar research but involving *normalized values* has been reported. Here, however, we deal with correlations of daily *returns*.

The main body of this work is organized as follows. In Sect. 2 we describe our procedure; rhe description as it is, in fact, very simple, and involves well-known quantities. Section 3 is devoted to the presentation of results. Finally, Sect. 4 comprises some concluding remarks.

## 2 Correlation Functions of World Stock Market Indices

*n*of a stock market index (

*a*), and let \(Z_{n}^{(a)}\) denote the corresponding relative return, i.e. \(Z_{n}^{(a)} = (K_{n}^{(a)} - K_{n-1}^{(a)})/K_{n-1}^{(a)}\). We define the cross-correlation function of returns of two indices (

*a*) and (

*b*) as:

*t*denotes the time interval over which the averaging in the calculations of the variances and covariance has been performed (unfortunately, we have no ensemble-averaging in our disposal here).

Thus, we have defined the correlation function in terms of the Pearson R coefficient [9] made of two sequences of the length *t*. A similar definition can be given in terms of, e.g., Spearman coefficient.

It is to be noted that \(C^{(a,b)} (n, M, t)\) is a function of three time variables. In the following, we have considered only three values of *M*, \(M = 1, 2, 3\). The averaging time *t* has been varied from \(t = 60\) to \(t = N_{0}\), with \(N_{0}\) being the largest value of the trading sessions common to all indices. \(N_{0}\) has been equal to 3452 enclosing the time interval from 10th of October, 2001 to 31st of October, 2016). The data for stock market indices have been downloaded from [11]. The correlation functions have been computed using pearsonr function from the Python (2.7) module *scipy.stats*, version 0.14.0 [10].

## 3 Results and Discussion

We have started our study with *n* equal to *t*. Firstly, we have considered Pearson’s R coefficient using all available data.

^{1}we have provided pairs of indices with largest values of the correlation functions for \(n = t = u\), \(u = N_{0} - 4\), for \(M = 1, 2, 3\).

List of indices with largest correlation functions for \(n = t = N_0 - 4\), \(M = 1\). In all cases p-value has been smaller than 0.05

Index (a) | Index (b) | \(C^{(a,b)}(u, 1, u)\) |
---|---|---|

NIKKEI | TOPIX | 0.785 |

NASDAQ | SP500 | 0.694 |

NASDAQ | RUSSELL | 0.690 |

NASDAQ | DJTA | 0.651 |

DJIA | NASDAQ | 0.474 |

DJIA | RUSSELL | 0.411 |

*t*significantly smaller than

*u*a quite interesting structure in the

*n*-dependence of the correlation function occurs, please see below.

List of indices with largest correlation functions for \(n = t = N_0 - 4\), \(M = 2\). In all cases p-value has been smaller than 0.05

Index (a) | Index (b) | \(C^{(a,b)}(u, 2, u)\) |
---|---|---|

DJIA | SP500 | 0.388 |

DJIA | DJUA | 0.366 |

DJIA | DJTA | 0.335 |

DJIA | RUSSELL | 0.325 |

DJIA | MEXICIPC | 0.256 |

*not*appear in the list of highly correlated pairs for the shift \(M \ne 2\). Please see also further comments following Fig. 2.

List of indices with largest correlation functions for \(n = t = N_0 - 4\), \(M = 3\). In all cases p-value has been smaller than 0.05

Index (a) | Index (b) | \(C^{(a,b)}(u, 3, u)\) |
---|---|---|

SMI | TSE-300 | 0.159 |

FTSE100 | ALL-ORD | 0.135 |

DJIA | MEXICIPC | 0.105 |

EOE | SMI | 0.092 |

SP500 | TSE-300 | 0.091 |

AMEX-MAJ | TSE-300 | 0.090 |

List of indices with largest correlation functions for \(n = t = N_0 - 4\), \(M = 1\) with the numbers of correct and incorrect predictions of the behavior one the second-column indices based on the change of the corresponding first-column ones

Index (a) | Index (b) | No. of correct preditions | No. of incorrect predictions |
---|---|---|---|

NIKKEI | TOPIX | 2852 | 596 |

NASDAQ | SP500 | 2296 | 1089 |

NASDAQ | RUSSELL | 2368 | 1080 |

NASDAQ | DJTA | 2301 | 1147 |

DJIA | NASDAQ | 2359 | 1089 |

DJIA | RUSSELL | 2281 | 1167 |

For \(M = 3\) we have not observed any large values of the *R* coefficient, for \(M > 3\) they become even smaller. Let us also report that we have not obtained any negative *R* coefficient with significant absolute value for any *M*. The indices definitely tend to be correlated rather than anti-correlated.

*a*) at time

*t*and index (

*b*) at time \(t+M\) is positive, we say that the prediction is correct; otherwise the prediction is incorrect.

List of indices with largest correlation functions for \(n = t = N_0 - 4\), \(M = 1\) with the numbers of correct and incorrect predictions of the behavior one the second-column indices based on the change of the corresponding first-column ones

Index (a) | Index (b) | No. of correct preditions | No. of incorrect predictions |
---|---|---|---|

DJIA | SP500 | 2086 | 1362 |

DJIA | DJUA | 1953 | 1495 |

DJIA | DJTA | 2042 | 1406 |

DJIA | RUSSELL | 1953 | 1495 |

DJIA | MEXICIPC | 1887 | 1561 |

List of indices with largest correlation functions for \(n = t = N_0 - 4\), \(M = 3\) with the numbers of correct and incorrect predictions of the behavior one the second-column indices based on the change of the corresponding first-column ones

Index (a) | Index (b) | No. of correct preditions | No. of incorrect predictions |
---|---|---|---|

SMI | TSE-300 | 1833 | 1516 |

FTSE100 | ALL-ORD | 1790 | 1658 |

DJIA | MEXICIPC | 1828 | 1620 |

EOE | SMI | 1777 | 1671 |

SP500 | TSE-300 | 1885 | 1563 |

AMEX-MAJ | TSE-300 | 1813 | 1635 |

The content of the Tables 4, 5 and 6 largely agrees with the conclusions one can draw from three previous tables. For the first five rows of Table 4 we can realize that the number of correct predictions is at least two times larger that the incorrect ones. Quite obviously, even such a ratio could not guarantee any successful trading on any index-based financial instruments. On the other hand, even the smallest advantage like that provided by Table 6 may sometimes be sufficient to get the edge on trading competitors.

Let us notice that the difference between correct and incorrect predictions in the case of EOE-SMI pair is so small that p-value in the G-test is \({\approx }\)0.07. Thus, one cannot exclude the zeroth hypothesis that the difference between the numbers of predictions if purely accidental. In all other cases we have had p-value in the G-test smaller than 0.05.

We have also performed calculations for the values of *n* and *t* different from *u*. For instance, we have obtained the *R* coefficient from the last 480 trading sessions (approximately two years) preceding 31st of October, 2016. For the pair CAC40—EOE a record value of *R* equal to 0.94 has been obtained with the percent of correct predictions equal to 0.875. For \(M = 2\) the most considerable correlation has been obtained between the DAX and SMI indices; *R* has been equal to 0.526, and the ratio of correct to incorrect predictions has almost exactly been equal to 2:1.

We have not, however, been satisfied with the above results for it has not been clear precisely how the correlation function depends on the time *n* if the averaging window *t* is smaller than *n*.

*M*and

*t*as functions of

*n*. These have been compared with the time evolution of the indices themselves. Note that it is the closing values of the index (not the returns) which have been shown in the central and lower parts of each figure.

*R*coefficient calculated during the last 14 years was built during the relatively short time interval 2006–2010. Recently, DJIA-SP500 have become almost completely “unreliable”, so to say.

## 4 Concluding Remarks

In this work we have computed cross-correlation function of time series generated by the returns of important world stock market indices. We have identified pairs of indices such that correlations are both strong and persistent in time. Preliminary assessments of predictive power of the correlations have been performed. We believe that, with sufficient care, the knowledge of correlations between indices and individual stocks may be used in practice, possibly even to enhance the predictive power of technical indicators for trading purposes. Needless to say, extreme caution is required in any such attempts. In fact, they can break down at any time. For instance, when we used historical data up to the middle summer of 2016, we have observed the highest correlation function for \(M = 3\), \(n = t = u - 3\) in the pair MEXICIPC—SMI. During the subsequent mounts that correlation deteriorated quite spectacularly. One might think that it would be a valuable enterprise to find any indicators which could suggest that the correlations have just started to build up or are just about to get ruined, provided that such indicators exist. Finally, one may reasonably argue that the above very simple correlation analysis can and should be combined with the analysis of cointegration to have deeper insight into connection between shifted (i.e. with \(M \ne 0\)) and unshifted time series.^{2}

## Footnotes

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