# Time Scales and Characteristics of Stock Markets in Different Investment Horizons

- Department of Physics, NIT Sikkim, South Sikkim, India

Investors adopt varied investment strategies depending on the time scales (*τ*) of short-term and long-term investment time horizons (ITH). The nature of the market is very different in various investment *τ*. Empirical mode decomposition *(EMD) *based Hurst exponents (*H*) and normalized variance (NV) techniques have been applied to identify the *τ* and characteristics of the market in different time horizons. The values of *H* and NV have been estimated for the decomposed intrinsic mode functions (IMF) of the stock price. We obtained *τ* ranging from a few days to 3 months and *τ* from a few days to 3 months; b) long-term time series

## Introduction

The stock market is a complex dynamical system where the evolution of the dynamics depends on the participation of different types of investors or traders [1–3]. Investors/traders participate in the stock market to gain profit implementing different investment and trading strategies depending on investment time horizons (ITH) [45]. The participation of diversified investors, reaction to the information, and short-term and long-term investment approaches play crucial roles in the movement of stock prices [4].

In the stock markets, there are mainly two types of investors: short-term investors who invest for short-term gain and long-term investors who invest for long-term gain [67]. Studies show that the *τ*) of short-term and long-term *τ* from stock price time series using a well defined technique may be helpful for both the short-term and long-term investors.

As the market is mean reversing in short-term

In the short-term *τ* of short-term and long-term investment horizon in an arbitrary manner [98]. Recently, we used structural break study to show that the *τ* for the short-term is usually less than a few months [9]. The separation of the short- and long-term dynamics in terms of *τ* plays a vital role in the prediction of future price movement. Hence, detailed studies are required to find the correlation of the stock price with fundamental variables and to identify the *τ* of market dynamics in the short-term

In this article, we estimated the *τ* of the stock price in the short-term and long-term *H*) analysis. We have reconstructed short-term and long-term time series based on the *H*. Finally, we estimated the correlation coefficient between long-term time series and fundamental variables. Herein we establish that short-term

The remaining part of this paper is organized as follows: In Section 2, we introduce the method of analysis, while Section 3 presents the data analyzed. Results and discussion and conclusion are delineated in Sections 4 and 5, respectively.

## Method of Analysis

A nonlinear two-step technique—EMD followed by Hilbert–Huang Transform (*τ* and the important trends and components present in the data [3].

The *τ* by preserving the nonstationarity and nonlinearity of the data. These oscillatory modes are termed intrinsic mode functions (*τ* of each

The *i*) the number of extrema and the number of zero crossing must be equal or differ by one; and

a. Lower envelope

b. Mean value of the envelope

c. Repeat the processes (a) and (b) by considering *h* as a new data set until the *i* and

Once the conditions are satisfied, the process terminates, and *h* is stored as the first

where *i*th

*τ*. The *τ*, the *τ*, and so on. Hence, *τ* from the signal. The characteristic *τ* of each

where *P* is the Cauchy principle value, and *H*.

Rescaled-range (R/S) analysis is a technique to estimate the correlation present in a time series by calculating *H* [26–28]. Details of the R/S technique are described below. Let us consider a time series of length *L* and divided into *p* subseries of length *l*. Each subseries is denoted as *t* = 1, 2, 3, …, *p*. Mean and standard deviation of the subseries

and

respectively. Mean adjusted series is calculated as

for *j* = 1, 2, 3, …, *l*. Cumulative time series is given by

for *j* = 1, 2, 3, …, *l.*

Range of the series has been calculated as

Individual subseries range can be rescaled or normalized by dividing the standard deviation. So, R/S is written as

The ratio of each subseries of length *l* is expressed as *H* is the Hurst exponent. *H* can be estimated from the slope of ln(R/S) vs. ln(l). For a random time series, *H* is around 0.5, and for correlated and anticorrelated time series, *H* is greater than 0.5 and less than 0.5, respectively.

Normalized variance (*i*th *i*th

where *q* is the total number of

## Data Analyzed

We have analyzed the stock indices and stock prices of a few companies of different countries from December 1995 to July 2018, namely, 1) S&P 500 (USA), 2) Nikkei 225 (Japan), 3) CAC 40 (France), 4) IBEX 35 (Spain) 5) HSI (Hong Kong), 6) SSE (China), 7) BSE SENSEX (India), 8) IBOVESPA (Brazil), 9) BEL 20 (Euro-Next Brussels), 10) IPC (Mexico), 11) Russell 2000 (USA), and 12) TA125 (Israel), and stock prices of the companies 1) IBM (USA), 2) Microsoft (USA), 3) Tata Motors (India), 4) Reliance Communication (RCOM) (India), 5) Apple Inc. (USA), and 6) Reliance Industries Limited (RIL) (India). Stock indices and price data were downloaded from yahoo finance, and the analysis was carried out using MATLAB software.

## Results and Discussions

The stock market shows different behavior in different investment horizon. *H* and

Figures 1A–J show the *τ*, and it gradually increases with the increase in

**FIGURE 1**. The plots **(A)–(I)** represent the **(J)** represents residue of the S&P 500 index.

$EMD$ Based *H* and $NV$ Analysis

*H* has been calculated for all the *H* versus *τ* of all the indices and companies. We obtained single *H* from *H*, namely, *τ* ranging from a few days (*D*) to 3 months (*M*). The value of *M*. It gradually increases for *τ* ranging from 1 year (*Y*) to 12 *Y*. *τ* of *D*, 7–10 *D*, 15–18 *D*, 1–1.5 *M*, and 2.5–3 *M*, respectively.

**FIGURE 2**. **(A)** shows the Hurst exponents (*τ* of all the *τ* of around 3 *M*. The value of *D*), (*M*), and (*Y*) in the *x*-axis represent the day, month, and year, respectively. **(B)–(D)** represent the normalized variance

To further validate the robustness of the proposed method, analysis of the decomposed time series has been carried out using

### Reconstruction of Short-Term and Long-Term Time Series

In order to analyze the market dynamics in short-term

We have reconstructed a time series *τ* ranging from a few days to 3 months. Hence, *τ* is in the range of a few days to 3 months.

**FIGURE 3**. **(A)** represents the daily price movement of Apple Inc. from April 2007 to March 2018. **(B)**,**(C)** represent the reconstructed short-term time series

Higher-order

Table 1 shows that the correlation coefficient (*J*) between *J* for a few stocks. These stocks show a small *J* for the following two possible reasons: a) the stock price of a company with strong growth prospect increases even though sale or NP decreases temporarily; b) the stock price of a company with weak growth prospect decreases even though sale or NP increases temporarily. Hence, for long-term investment, the fundamental variables are the most crucial variables for the prediction of the future price. In the future, we would like to study the correlation between stock price and other fundamental variables of companies.

**TABLE 1**. Correlation coefficient

## Conclusion

In this paper, we have studied the stock market using the empirical mode decomposition (*H*) analysis and normalized variance (*H* and

The analysis yielded a value of

A detailed study of the market in the long-term

## Data Availability Statement

Publicly available datasets were analyzed in this study. These data can be found here: https://in.finance.yahoo.com, https://www.screener.in, https://www.macrotrends.net

## Author Contributions

All the authors have equally contributed to preparing the manuscript.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

The authors acknowledge the help of Taraknath Kundu and suggestions of the anonymous reviewers in preparing and improving the manuscript.

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Keywords: empirical mode decomposition, Hurst exponent, short-term investment time horizon, long-term investment time horizon, time scale, normalized variance

Citation: Mahata A and Nurujjaman M (2020) Time Scales and Characteristics of Stock Markets in Different Investment Horizons. *Front. Phys.* 8:590623. doi: 10.3389/fphy.2020.590623

Received: 02 August 2020; Accepted: 29 September 2020;

Published: 12 November 2020.

Edited by:

Anirban Chakraborti, Jawaharlal Nehru University, IndiaReviewed by:

Suchetana Sadhukhan, National Autonomous University of Mexico, MexicoSunil Kumar, University of Delhi, India

Copyright © 2020 Nurujjaman and Mahata. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Md. Nurujjaman, jaman_nonlinear@yahoo.co.in