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Научни трудове на Съюза на учените в България-Пловдив, серия Б. Естествени и хуманитарни науки, т. XVIII, ISSN 1311-9192 (Print), ISSN 2534-9376 (On-line), 2018. Scientific researches of the Union of Scientists in Bulgaria-Plovdiv, series B. Natural Sciences and the Humanities, Vol. XVШ, ISSN 1311-9192 (Print), ISSN 2534-9376 (On-line), 2018.
PERSISTENCE IN MOMENTUM TRADING STRATEGY: AN EMPIRICAL EVIDENCE FRO M THE BULGARIAN STOCK
EXCHANGE
Boryana Bogdanova, Bozhidar Nedev Sofia University “Sit. Kliment Ohrid ski”, Faculty of Economics and
Business Administration
Abstract
This paper studies the statistical properties of raw profit series of momentum trading strategy applied to the Bulgarian stock exchange for a period spanning from Jan-2004 to Jul-2017. In particular, we apply a number of estimation procedures and tests so as to document if persistent behavior is observed. Furthermore, we split our sample into three non-overlapping periods in order to study closely the changes in the strength of persistency that took place during and after the 2008 Financial crisis.
Key words: persistence, long memory, momentum effect, Bulgarian Stock Exchange
1. Introduction
The concept of momentum effect has been introduced for the first time by (Jagadeesh & Titman, 1993). By its essence, the momentum effect is a market anomaly that is inconsistent with the Efficient Market Hypothesis (EMH). It represents short-term return predictability based on past performance of assets over a period of less than 12 months. In particular, momentum trading strategy assumes that past winners, i.e. best performing stocks, tend to perform well in subsequent periods and past losers, i.e. worst performing stocks, tend to perform poorly in the upcoming periods. Furthermore, as argued by (Jegadeesh & Titman, 2011) momentum might be considered as the strongest evidence against the EMH. To be specific, since best performers appear to be less risky than the worst performers, standard risk adjustments tend to increase instead of decrease the return spread.
Even though, currently a huge body of literature is engaged with the study of this phenomenon1, the aforementioned facts amplify the importance of gaining deeper knowledge on momentum. Apart from documenting its presence at financial markets and identifying the drivers behind it, the understudying of momentum effect might be reinforced through an in-depth analysis of its statistical properties. This paper contributes to the issue by studying momentum raw profits at the Bulgarian stock exchange for presence of persistence. We motivate our research choice by the fact that significant body of literature in the field of empirical finance evidences presence of persistent behavior in stock market returns2. The latter has important implications for practical purposes.
1 See, for example, the papers of (Asness, et al., 2013), (Foltice & Langer, 2015), (Narayan, et al., 2015), and (Andrei & Cujean, 2017) as well as the reference therein.
2 The reader might find a review of literature on this issue in (Bogdanova & Ivanov, 2014), for example.
2. Methodology
The nature of persistence might be easily explained in terms of the definition of a fractal process3. Essencially, the fractal process is a generalization of the Brownian motion and it is defined
with the introduction of the Hurst exponent, H £ (0,1). An important feature of the fractal process is given by Eq. (1):
22H = 2 + 2p (- 1/2 < p < 1) (1)
It states that the correlation coefficient p between any two successive increments, AX(t) and AX(t + At), does not depend on the time change At but it is a function of the Hurst exponent. When the Hurst exponent is greater than 1/2, it might be seen that the correlation coefficient p is positive. In this case, the process exhibits persistence, which means that positive values are likely to be followed by positive values and negative values, are likely to be followed by negative values. Different approaches are available for the estimation of the Hurst exponent. The most popular amongst them are the R/S analysis, originally proposed by (Hurst, 1951) and later on developed by (Mandelbrot & Willis, 1969), the modified R/S analysis of Lo (Lo, 1991) and the spectral regression method, first proposed by (Geweke & Porter-Hudak, 1983). The idea of spectral regression is extended into wavelet regression. The latter is developed by (Arby & Veitch, 1998) and (Jensen, 1999), and later by (Arby, et al., 2003). Similar to the spectral regression, the wavelet regression is based on the presence of linear relationship on the log-log diagram between the wavelet variance and the level of decomposition.
We would apply the R/S approach and the wavelet regression of (Arby, et al., 2003) in order to estimate the Hurst exponent for the raw profit series of a momentum trading strategy applied to the Bulgarian stock exchange for the period Jan-2004 - Jul-2017. Furthermore, we use the test of Lo (Lo, 1991) to support the significance of our results. In order to obtain a times series of momentum raw profits, we use the procedure developed by (Alphonse & Nguyen, 2013), which is briefly described below and might be found in a greater detail at pp. 188-189 of (Alphonse & Nguyen, 2013).
In week t stocks are divided into quintiles according to their average lagged returns over the past K weeks. The stocks in the highest and in the lowest quintile are called “winners” and “losers” respectively. Then all component stocks in both of the portfolios are weighted equally. For the reasons outlined in (Foltice & Langer, 2015) we would analyze further only the winners portfolios. The raw profits for winners portfolios, formed k weeks ago (k = 1,..., J), are denoted by: R^t. We examine overlapping portfolios, since at week t there are J winners portfolios, that have been formed in week t-1, t-2,..., t-J. Their profits in a given calendar week t are equally averaged:
0R% = jSLr^t. (2)
In (Nedev & Bogdanova, 2017) the profits delivered by eq. (2) are averaged over the pre-crisis (Jan-2004 - Dec-2007), the crisis4 (Jan-2008 - Dec-2012) and the post-crisis period (Jan-2013 -Jul-2017). Only for the pre-crisis period is identified significant momentum. For this period the momentum trading strategy, based on the lagged 26 week returns reports the highest average profit for all holding periods. Furthermore, the authors found that among them highest profits are registered for a holding period of 8 weeks. Therefore, for the purposes of this study we set К = 26 and ] = 8 and apply eq. (2) so as to deliver a time series of averaged profits which is subject of further research via the R/S analysis and the wavelet regression.
3. Empirical findings
For the purpose of our study, we take past price records on weekly basis for all stocks traded at the Bulgarian stock exchange for the period Jan-2004 - Jul-2017. As of 31-Jul-2017 their total number amounts to 90. The raw data is downloaded from http://www.infostock.bg. Since some of the
3 The reader is referred to (Mandelbrot & Van Ness, 1968) for a detailed exhibition on fractal processes.
4 We undertook this particular sample division, motivated by the results of (Mileva, 2014).
stocks are characterized by long periods of missing values, we use the applied algorithm of (Nedev & Bogdanova, 2017). It relies on spline interpolation and a careful aftermath analysis assuring that data properties are preserved in accordance with the guidelines defined in (Lazarov, 2013). After the algorithm is applied, the number of retained stocks varies from 35 in earlier periods to 69 for the later periods of operation of the Bulgarian stock exchange. We apply eq. (2) to the returns series of the retained stocks and obtain a time series of averaged raw profits on the winners portfolio that is visually presented at Figure 1.
______ _______ftf—l Partfcjp___________ ________________________ 1^чи ■ F4gtfo>H __________
Figure 1. The left panel presents the time series of averaged raw profits on the winners portfolio, while the right panel summarizes its frequency distribution with a superimposed Gaussian probability density function so as to enhance comparison.
Figure 1 suggests that the pre-crisis period is dominated by positive raw profits on the winners portfolio. As summarized in Table 1, the raw profit for the pre-crisis period is 2.31% per week on average and the corresponding t-statistics suggests this figure is significant. Therefore, momentum-based trading strategies are profitable during this period. However, during the 2008 Financial crisis the raw profit on the winners portfolio becomes negative and insignificant. The crisis’ aftermath period is also characterized by insignificant average raw profit.
Table 1. Average raw profit on the winners portfolio for the pre-crisis, crisis, post-crisis, and the entire sample period for the Bulgarian stock exchange.
Average raw profit t-stat p-value
Jan-2004 - Dec-2007 0.0231 6.7986 0.00%
Jan-2008 - Dec-2012 -0.0015 -0.9007 36.87%
Jan-2013 - Jul-2017 0.0040 1.8426 6.69%
Jan-2004 - Jul-2017 0.0077 5.0702 0.00%
We deepen the understanding of the observed phenomenon through persistence analysis. Its main results are summarized in Table 2. The first two rows of the table report estimates of the Hurst exponent delivered through the R/S analysis and the wavelet regression. Both of the approaches suggests that over the entire period under study (Jan-2004 - Jul-2017) the Hurst exponent is close to 65%. This number translates as follows. After a week with a positive (negative) raw profit, the chance for the next week to end up with a positive (negative) raw profit on the winners portfolio is close to 65%. The significance of this result is proven through the Lo’s test. The last three rows of
Table 2 provide the Lo’s modified R/S test statistics5. It might be seen that for the entire period under study the identified persistence is statistically significant. Another interesting finding is that the major source of this persistence is the pre-crisis period, where significant profitability of the momentum trading strategy has been documented.
Table 2. Summary results on the estimates of the Hurst exponent and the Lo’s test statistics.
Jan-04 - Dec-07 Jan-08 - Dec-12 Jan-13 - Jul-17 Jan-04 - Jul-17
R/S method 0.7525 0.5515 0.5599 0.6726
wavelet regression 0.6451 0.5489 0.6186 0.6492
Lo's R/S (q=0) 1.4567 1.8938 1.1190 3.3576
Lo'sR/S (q=1) 1.2902 1.7288 1.0999 2.9979
Lo's R/S (q=5) 1.1587 1.5989 1.1409 2.5389
4. References
Alphonse, P. & Nguyen, T., 2013. Momentum effect: Evidence from the Vietnamese stock market. Asian Journal of Finance & Accounting, 5(2), pp. 183-202.
Andrei, D. & Cujean, J., 2017. Information percolation, momentum and reversal. Journal of Financial Economics, Volume 123, pp. 617-45.
Arby, P., Flandrin, P., Taqqu, M. & Vietch, D., 2003. Self-similarity and long-range dependence through the wavelet lens. Theory and Applications of Long-range Dependence, pp. 527-56.
Arby, P. & Veitch, D., 1998. Wavelet analysis of long-range-dependent traffic. IEEE Transactions on Information Theory, 44(1), pp. 2-15.
Asness, C., Moskowitz, T. & Pedersen, L., 2013. Value and momentum everywhere. The Journal of Finance, 68(3), pp. 929-85.
Bogdanova, B. & Ivanov, I., 2014. Adaptive and relative efficiency of stock markets from Southeastern Europe: a wavelet approach. Applied Financial Economics, 24(10), pp. 705-22.
Foltice, B. & Langer, T., 2015. Profitable momentum trading strategies for individual investors. Financial Markets and Portfolio Management, 29(2), pp. 85-113.
Geweke, J. & Porter-Hudak, S., 1983. The estimation and application of long memory time series models. Journal of Time Series Analysis, Volume 4, pp. 221-38.
Hurst, E., 1951. Long term storage capacity of reservoirs. Transactions of American Society of Civil Engineers, Volume 116, pp. 770-79.
Jagadeesh, N. & Titman, S., 1993. Returns to buying winners and selling losers: Implications for stock market efficiency. The Journal of Finance, XLVIII(1), pp. 65-91.
Jegadeesh, N. & Titman, S., 2011. Momentum. Annual Review of Financial Economics, 3(1), pp. 493-509. Jensen, M., 1999. Using wavlets to obtain a consistent ordinary least squares estimator of the fractional differencing parameter. Journal of Forecasting, Volume 18, pp. 17-32.
Lazarov, D., 2013. Simmulations with missing values. Vanguard Scientific Instruments in Management, 2(7), pp. 10-46.
Lo, A., 1991. Long-term memory in stock market prices. Econometrica, Volume 59, pp. 1279-1313. Mandelbrot, B. & Van Ness, J., 1968. Fractional Brownian motions, fractional noises and applications. SIAM Review, Volume 10, pp. 422-37.
Mandelbrot, B. & Willis, J., 1969. Some long-run properties of geophysical records. Water Resourses, Volume 5, pp. 321-340.
Narayan, P., Ahmed, H. & Narayan, S., 2015. Do momentum-based trading strategies work in the commodity futures market?. The Journal of Futures Markets, 35(9), pp. 868-91.
Nedev, B. & Bogdanova, B., 2017. A study on the momentum effect of the Bulgarian Stock Exchange: Some practical issues of applied importance. The paper is accepted for publishing in Vanguard Scientific Instruments in Management (ISSN 1314-0582).
5 The region of acceptance of the null hypothesis of the Lo’s test is [0.809, 1.862].
