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Analysis on Runs of Daily Returns in Istanbul Stock Exchange

Analysis on Runs of Daily Returns in Istanbul Stock Exchange The aim of this paper is to obtain some statistical properties about runs of daily returns of ISE30, ISE50 and ISE100 indices and compare these results with the empirical stylized facts of developed stock markets. In this manner, all time historical daily closing values of these indices are studied and the following observations are obtained: Exponential law fits pretty well for the distribution of both run length and magnitude of run returns. Market is equally likely to go up or go down every day. Market depth has improved over recent years. Large magnitudes of run returns are more likely to be seen in positive runs. As in the developed stock markets, daily returns in Istanbul Stock Exchange don't have significant autocorrelations but absolute values (i.e. magnitudes) of daily returns exhibit strong and slowly decaying autocorrelations up to several weeks suggesting volatility clustering. Similar to the absolute daily returns, absolute value of run returns display strong and slowly decaying autocorrelations which again supporting the existence of volatility clustering. Unlike magnitudes of run returns, lengths of runs don't have significant autocorrelations. Keywords: Stylized Facts; Return Runs; Autocorrelation; Volatility Clustering; Stock Market Efficiency JEL Classification: C10, C50, G14 1. Introduction In recent decades, empirical studies on financial time series indicate that if we examine these series from a statistical point of view, the seemingly random variations of asset prices do share some quite nontrivial statistical properties. Such properties, common across a wide range of instruments, markets and time periods are called stylized empirical facts (Cont 2001). Researchers have now come to agree on several stylized facts about financial markets: heavy tails in asset return distributions, absence of autocorrelations in asset returns, volatility clustering and asymmetry between rises and falls... (Cont (2001); Engle, and Patton (2001); Abergel et al. (2009); Mantegna, and Stanley (2000); Bouchaud, and Potters (2003)) Most of the time, these studies mainly focus on analyzing daily or weekly individual returns of the assets, but sometimes just the sign of these returns can be a useful tool for understanding the market structure (Marumo et al. (2002)). Moreover, instead of individual returns, considering the cumulative returns of specific sequences may give us nontrivial information about the market or even help us to reveal some stylized facts about stock price movements. In this paper, we will conduct a detailed runs analysis similar to work of Gao, and Li (2006) on Dow Jones Industrial; first we will analyze the distributions of run lengths and run returns of ISE30, ISE50 and ISE100 indices then we will talk about some of the stylized facts observed in Istanbul Stock Exchange, and finally we will investigate the time correlation of the run lengths and magnitudes of run returns. 2. Analysis In financial markets, a run is a consecutive series of price movements without a sign reversal, hence a positive (negative) run is an uninterrupted sequence of positive (negative) returns and this run continues until a negative (positive) return comes out. For example, consider daily closing values of ISE30 index for 12 days from 27.01.1997 to 13.02.1997. These values are: The views and opinions in the studies belong to the author and do not necessarily reflect those of the Istanbul Stock Exchange management and/or its departments , which give us the following daily returns: , , , . Signs of these returns generate the sequence which contains three positive and three negative runs. The lengths of the three negative runs are and similarly lengths of the three positive runs are: . The cumulative returns (which we will call run returns) obtained in the positive runs are and the cumulative returns obtained in the negative runs are: .2 Runs are simple constructs, but little research has been done on them in finance. Most of these researches aim to examine the informal efficiency of stocks (however this paper does not have such a purpose) because of the distinctive3 run length of a random walk. Fama (1965) investigated the runs of several stocks, and found little evidence for violations of efficiency based on serial dependence in returns. Similar researches have been done by Moore (1978), and Grafton (1981) to test the efficient market hypothesis. Easley, and others (1997) used runs to examine dependence in intra-day data. We consider daily closing values of ISE30, ISE50 and ISE100 indices from the day they have been introduced to the date 24.04.2012. The daily return of an index is found by where is the index' closing value of day . Distribution of the Run Length. Using and the definition of a run, we obtain several information from the empirical data. Tables 1.a and Tables 1.b show us the longest positive and negative runs, their corresponding date periods and their returns and Table 2 shows the frequencies of all runs with different lengths; Table 1.a. All time longest positive runs of ISE30, ISE50 and ISE100 Longest Positive Runs 02.09.1997 ­ 17.09.1997 12 days 14.01.1997 ­ 27.01.1997 10 days 02.11.1999 ­ 15.11.1999 10 days 18.08.2005 ­ 01.09.2005 10 days 13.02.1989 ­ 02.03.1989 12 days 15.09.1989 ­ 04.10.1989 12 days 13.08.1993 ­ 31.08.1993 12 days Returns 0,22338 0,601792 0.292414 0,158245 0,62534 0,29406 0,39589 ISE30 ISE50 ISE100 Table 1.b. All time longest negative runs of ISE30, ISE50 and ISE100 The possibility is very small but if there happens to be a day with zero return, it is omitted For pure random walks, average run length is two 152 Longest Negative Runs 09.01.2008 ­ 23.01.2008 - 11 days 14.11.2011 ­ 24.11.2011 - 9 days 09.01.2008 ­ 23.01.2008 - 11 days 16.08.1988 ­ 01.09.1988 - 12 days 23.06.1988 ­ 07.07.1988 - 11 days 18.04.1994 ­ 02.05.1994 - 11 days 09.01.2008 ­ 23.01.2008 - 11 days Table 2. Frequencies of runs with different lengths 1 465 483 375 390 633 697 2 223 233 188 194 354 344 3 136 135 102 111 202 210 4 65 59 55 45 108 97 5 33 25 26 17 56 34 6 19 8 18 8 35 18 LENGTH 7 5 7 3 6 12 14 8 2 2 2 4 4 9 4 1 2 6 1 10 3 2 9 11 1 1 3 12 1 3 1 TOTAL 954 954 773 774 1422 1423 Returns -0,19799 -0,12934 -0,19919 -0,08515 -0,17495 -0,37174 -0,20093 ISE30 ISE50 ISE10 0 Positive Run Negative Run Positive Run Negative Run Positive Run Negative Run Considering run length distributions (obtained from Table 2) in Figure 1.a, 1.b and 1.c; we suggest that the number of observations of a run with length can be expressed as the following exponential form; Figure. 1.a: Run length distribution of ISE 30 Figure. 1.b: Run length distribution of ISE 50 Figure. 1.c: Run length distribution of ISE 100 For positive and negative runs of each index, fitting an exponential form of gives us the following results; Table 3.a POSITIVE RUNS adjusted 0,66098 0,65982 0,58594 0,99778 0,99936 0,99953 Table 3.b NEGATIVE RUNS adjusted 0,69224 0,68613 0,66049 0,99836 0,99808 0,99758 0,99816 0,99781 0,99731 0,99751 0,99928 0,99948 to the data in Table 2 confidence interval for (0,61654 , 0,70542) (0,63561 , 0,68403) (0,56924 , 0,60266) confidence interval for (0,65154 , 0,73294) (0,63846 , 0,73379) (0,61576 , 0,70522) As we see from table 3.a and 3.b; and values show that for both positive and negative runs of each index, exponential law fits pretty well for the frequency of run lengths. Consider a simple random process with two equally likely outcomes; in such a process the probability density function of run length should follow an exponential distribution of the form . We see that for negative runs of ISE 30, ISE 50 and ISE 100 and positive runs of ISE 30 indices, is in the %95 confidence interval for estimated but for positive runs of ISE 50 and ISE 100, it is not in the %95 confidence interval (for ISE 50 it is pretty close though; see table 3.a) hence we can roughly conclude that, ignoring the magnitudes and considering just the signs of the daily returns, market is equally likely to go up or go down every day. 2.2 Distribution of the run returns Tables 4.a and Tables 4.b give some historical information about largest negative and positive run returns of ISE 30, ISE 50 and ISE 100 indices; Table 4.a. Largest negative run returns of ISE 30, ISE 50 and ISE 100 LARGEST NEGATIVE RUN RETURNS Return Duration -0,36078 23.11.2000 ­ 04.12.2000 - 8 days -0,27329 21.03.2001 ­ 29.03.2001 - 7 days -0,26727 28.05.2002 ­ 03.06.2002 - 5 days -0,36568 23.11.2000 ­ 04.12.2000 - 8 days -0,22093 06.07.2001 ­ 11.07.2001 - 4 days -0,20937 09.09.2008 ­ 18.09.2008 - 7 days -0,37174 18.04.1994 ­ 02.05.1994 - 11 days 154 -0,36565 -0,31315 23.11.2000 ­ 04.12.2000 - 8 days 21.01.1994 ­ 18.01.1994 - 6 days Table 4.b. Largest positive run returns of ISE30, ISE50 and ISE100 LARGEST POSITIVE RUN RETURNS Return Duration 0,60179 14.01.1997 ­ 27.01.1997 - 10 days 0,42027 05.12.2000 ­ 06.12.2000 - 2 days 0,33490 09.12.1999 ­ 13.12.1999 - 3 days 0,41564 05.12.2000 ­ 06.12.2000 - 2 days 0,24112 22.02.2001 ­ 26.02.2001 - 3 days 0,23099 26.04.2001 ­ 30.04.2001 - 3 days 0,62534 15.09.1989 ­ 04.10.1989 - 12 days 0,49633 14.01.1997 ­ 27.01.1997 - 12 days 0,41718 05.12.2000 ­ 06.12.2000 - 2 days Before starting the analysis, an interesting observation is (also as suggested by Tables 1.a, 1.b, 4.a and 4.b) as getting close to present day we still observe considerable amount of long runs, but in these long runs, magnitudes of run returns seem significantly smaller compared to those of the long runs in earlier dates suggesting that market depth has improved over recent years. To analyze the distribution of run returns, we consider the frequency distributions as the following: First we create intervals with 0,01 increments as:.... , [-0,02 , -0,01) , [-0,01 , 0) , [0 , 0,01) , [0,01 , 0,02) , .... and for each index, we count the number of observed run returns belonging to each interval (see LHS of Figure 2.a, 2.b and 2.c). Then we take absolute values of the observed run returns and count the number of these absolute values belonging to each interval mentioned above (see RHS of Figure 2.a, 2.b and 2.c). First thing to notice here is one of the stylized facts of financial markets: just like the distribution of the daily returns, distribution of the run returns display heavy tails and sharp peaks. Figure. 2.a. Frequency distributions of run returns and absolute run returns of ISE 30 Figure. 2.b. Frequency distributions of run returns and absolute run returns of ISE50 155 Figure 2.c. Frequency distributions of run returns and absolute run returns of ISE 100 Theory suggests the idea of fitting normal distributions and exponential distributions to the data in the LHS and RHS of Figure 2.a, 2.b and Figure 2.c. respectively. To understand which one would be a better fit, we compare the actual frequency and fitted frequency values with the same returns (see Figure 3.a, 3.b and 3.c; dashed lines have unit slope and pass through the origin). Figure 3.a. actual freq. vs. exponential fit freq. Figure. 3.b. actual freq. vs. exponential fit freq Figure 3.c. actual freq. vs. exponential fit freq. We observe that for each index, exponential distribution fits better than normal distribution in explaining the relationship between magnitudes of run returns and their frequency. and values of these fits also suggest us a similar idea (see Table 5); Table 5. and values of exponential and normal fit adjusted Exponential Fit: Normal Fit: Exponential Fit: Normal Fit: Exponential Fit: Normal Fit: 0,98096 0,97265 0,97912 0,97635 0,99047 0,96420 0,98033 0,97160 0,97825 0,97527 0,99022 0,96308 2.3 Asymmetry of gains and losses in a run Using the frequency data obtained in the last subsection, for each index we compare the number of positive and negative runs according to their return magnitudes. Figure 4.a Figure 4.b Figure 4.c 157 Just like in the previous case, dashed lines pass through the origin with unit slope and as it is easily understood magnitude of run returns gets larger as points get closer to the origin. We see that for each index there is a significant asymmetry, suggesting that large magnitudes of run returns is more likely to be seen in positive runs, in other words It is more likely to see big gain rather than big loss in a run. 2.4 Autocorrelation Analysis 2.4.1 Daily returns One of the stylized facts of liquid markets is that daily returns do not exhibit any significant autocorrelation and ISE is no exception as we see from Figure. 5; Figure 5. Autocorrelations of daily returns The absence of significant autocorrelations in asset returns has been studied in detail (Fama 1971; Pagan 1996) and it is usually used to support random walk models in which the returns are considered to be independent random variables (Fama 1991)). But as Cont (2001) states "independence implies that any nonlinear function of returns will also have no significant autocorrelation: For example, absolute values or squares of daily returns should also have no significant autocorrelation" but various empirical studies (Bollerslev et al. 1992; Comte, and Renault 1996; Bouchaud et al. 1997; Cont 1998; Ding et al. 1983; Ding, and Granger 1994; Engle 1995) show that in this case, autocorrelations remains significantly positive for several weeks and decays slowly 4. This situation is usually interpreted as there is a correlation in volatility of returns but not the returns themselves. This is a quantitative manifestation of one of the stylized facts in financial markets called volatility clustering.5 In order to see if we observe this stylized fact in Istanbul Stock Exchange, we consider the autocorrelations of absolute daily returns of our indices and obtain very similar results (see Figure. 6) 2.4.2 Run returns In the next step, we investigate if there exists significant autocorrelations in magnitudes of run returns. In this case, the autocorrelation function is defined as; Sometimes this slow decay is considered as an indicator of long - range dependence in volatility, but arguments still continue on whether it should imply long time memory of financial time series (Cont 2007; Taqqu et al. 1999) 5 In finance, volatility clustering refers to the observation that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes." (Mandelbrot 1963) Figure 6. Autocorrelations of absolute daily returns where is the return of run and absolute daily returns. (See Figure 7) denotes the lag. Analyzing this case gives us a similar result as of Figure 7. Autocorrelations of absolute run returns As it is seen from Figure 7, there exists strong correlation decaying slowly (persisting up to several months) which again suggesting volatility clustering that also can be directly observed from progress of run returns in Figure 8. 2.4.3. Run Lengths We ask ourselves if we can we find a similar result for run lengths. Here the autocorrelation function is defined as; where is the length of run. We see that the autocorrelation function fluctuates around zero in the %95 confidence interval meaning there does not exist significant time correlation in run lengths. (See Figure 9) Figure 8. Run returns Figure 9. Autocorrelations of run lengths Conclusion In this work, we conducted a detailed analysis on runs of daily returns of three popular Istanbul Stock Exchange indices (ISE30, ISE50 and ISE100) hoping to find some meaningful properties. As a result we have the following observations: Exponential law fits pretty well for the distribution of both length and return magnitude of the runs. Market is equally likely to go up or go down every day. Market depth has improved over recent years. It is more likely to see big gain rather than big loss in a run. Just like in most of the developed stock markets, in Istanbul Stock Exchange there is an absence of significant autocorrelations in daily returns but the autocorrelations of absolute daily returns are strong and slowly decaying (persisting up to several months) suggesting volatility clustering. Similarly, significant correlation exists in the absolute run returns which also support the same deduction. Hoping to find a similar relation, we investigated the autocorrelations of runs length but in this case there seems no significance. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Advanced Studies in Finance de Gruyter

Analysis on Runs of Daily Returns in Istanbul Stock Exchange

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Abstract

The aim of this paper is to obtain some statistical properties about runs of daily returns of ISE30, ISE50 and ISE100 indices and compare these results with the empirical stylized facts of developed stock markets. In this manner, all time historical daily closing values of these indices are studied and the following observations are obtained: Exponential law fits pretty well for the distribution of both run length and magnitude of run returns. Market is equally likely to go up or go down every day. Market depth has improved over recent years. Large magnitudes of run returns are more likely to be seen in positive runs. As in the developed stock markets, daily returns in Istanbul Stock Exchange don't have significant autocorrelations but absolute values (i.e. magnitudes) of daily returns exhibit strong and slowly decaying autocorrelations up to several weeks suggesting volatility clustering. Similar to the absolute daily returns, absolute value of run returns display strong and slowly decaying autocorrelations which again supporting the existence of volatility clustering. Unlike magnitudes of run returns, lengths of runs don't have significant autocorrelations. Keywords: Stylized Facts; Return Runs; Autocorrelation; Volatility Clustering; Stock Market Efficiency JEL Classification: C10, C50, G14 1. Introduction In recent decades, empirical studies on financial time series indicate that if we examine these series from a statistical point of view, the seemingly random variations of asset prices do share some quite nontrivial statistical properties. Such properties, common across a wide range of instruments, markets and time periods are called stylized empirical facts (Cont 2001). Researchers have now come to agree on several stylized facts about financial markets: heavy tails in asset return distributions, absence of autocorrelations in asset returns, volatility clustering and asymmetry between rises and falls... (Cont (2001); Engle, and Patton (2001); Abergel et al. (2009); Mantegna, and Stanley (2000); Bouchaud, and Potters (2003)) Most of the time, these studies mainly focus on analyzing daily or weekly individual returns of the assets, but sometimes just the sign of these returns can be a useful tool for understanding the market structure (Marumo et al. (2002)). Moreover, instead of individual returns, considering the cumulative returns of specific sequences may give us nontrivial information about the market or even help us to reveal some stylized facts about stock price movements. In this paper, we will conduct a detailed runs analysis similar to work of Gao, and Li (2006) on Dow Jones Industrial; first we will analyze the distributions of run lengths and run returns of ISE30, ISE50 and ISE100 indices then we will talk about some of the stylized facts observed in Istanbul Stock Exchange, and finally we will investigate the time correlation of the run lengths and magnitudes of run returns. 2. Analysis In financial markets, a run is a consecutive series of price movements without a sign reversal, hence a positive (negative) run is an uninterrupted sequence of positive (negative) returns and this run continues until a negative (positive) return comes out. For example, consider daily closing values of ISE30 index for 12 days from 27.01.1997 to 13.02.1997. These values are: The views and opinions in the studies belong to the author and do not necessarily reflect those of the Istanbul Stock Exchange management and/or its departments , which give us the following daily returns: , , , . Signs of these returns generate the sequence which contains three positive and three negative runs. The lengths of the three negative runs are and similarly lengths of the three positive runs are: . The cumulative returns (which we will call run returns) obtained in the positive runs are and the cumulative returns obtained in the negative runs are: .2 Runs are simple constructs, but little research has been done on them in finance. Most of these researches aim to examine the informal efficiency of stocks (however this paper does not have such a purpose) because of the distinctive3 run length of a random walk. Fama (1965) investigated the runs of several stocks, and found little evidence for violations of efficiency based on serial dependence in returns. Similar researches have been done by Moore (1978), and Grafton (1981) to test the efficient market hypothesis. Easley, and others (1997) used runs to examine dependence in intra-day data. We consider daily closing values of ISE30, ISE50 and ISE100 indices from the day they have been introduced to the date 24.04.2012. The daily return of an index is found by where is the index' closing value of day . Distribution of the Run Length. Using and the definition of a run, we obtain several information from the empirical data. Tables 1.a and Tables 1.b show us the longest positive and negative runs, their corresponding date periods and their returns and Table 2 shows the frequencies of all runs with different lengths; Table 1.a. All time longest positive runs of ISE30, ISE50 and ISE100 Longest Positive Runs 02.09.1997 ­ 17.09.1997 12 days 14.01.1997 ­ 27.01.1997 10 days 02.11.1999 ­ 15.11.1999 10 days 18.08.2005 ­ 01.09.2005 10 days 13.02.1989 ­ 02.03.1989 12 days 15.09.1989 ­ 04.10.1989 12 days 13.08.1993 ­ 31.08.1993 12 days Returns 0,22338 0,601792 0.292414 0,158245 0,62534 0,29406 0,39589 ISE30 ISE50 ISE100 Table 1.b. All time longest negative runs of ISE30, ISE50 and ISE100 The possibility is very small but if there happens to be a day with zero return, it is omitted For pure random walks, average run length is two 152 Longest Negative Runs 09.01.2008 ­ 23.01.2008 - 11 days 14.11.2011 ­ 24.11.2011 - 9 days 09.01.2008 ­ 23.01.2008 - 11 days 16.08.1988 ­ 01.09.1988 - 12 days 23.06.1988 ­ 07.07.1988 - 11 days 18.04.1994 ­ 02.05.1994 - 11 days 09.01.2008 ­ 23.01.2008 - 11 days Table 2. Frequencies of runs with different lengths 1 465 483 375 390 633 697 2 223 233 188 194 354 344 3 136 135 102 111 202 210 4 65 59 55 45 108 97 5 33 25 26 17 56 34 6 19 8 18 8 35 18 LENGTH 7 5 7 3 6 12 14 8 2 2 2 4 4 9 4 1 2 6 1 10 3 2 9 11 1 1 3 12 1 3 1 TOTAL 954 954 773 774 1422 1423 Returns -0,19799 -0,12934 -0,19919 -0,08515 -0,17495 -0,37174 -0,20093 ISE30 ISE50 ISE10 0 Positive Run Negative Run Positive Run Negative Run Positive Run Negative Run Considering run length distributions (obtained from Table 2) in Figure 1.a, 1.b and 1.c; we suggest that the number of observations of a run with length can be expressed as the following exponential form; Figure. 1.a: Run length distribution of ISE 30 Figure. 1.b: Run length distribution of ISE 50 Figure. 1.c: Run length distribution of ISE 100 For positive and negative runs of each index, fitting an exponential form of gives us the following results; Table 3.a POSITIVE RUNS adjusted 0,66098 0,65982 0,58594 0,99778 0,99936 0,99953 Table 3.b NEGATIVE RUNS adjusted 0,69224 0,68613 0,66049 0,99836 0,99808 0,99758 0,99816 0,99781 0,99731 0,99751 0,99928 0,99948 to the data in Table 2 confidence interval for (0,61654 , 0,70542) (0,63561 , 0,68403) (0,56924 , 0,60266) confidence interval for (0,65154 , 0,73294) (0,63846 , 0,73379) (0,61576 , 0,70522) As we see from table 3.a and 3.b; and values show that for both positive and negative runs of each index, exponential law fits pretty well for the frequency of run lengths. Consider a simple random process with two equally likely outcomes; in such a process the probability density function of run length should follow an exponential distribution of the form . We see that for negative runs of ISE 30, ISE 50 and ISE 100 and positive runs of ISE 30 indices, is in the %95 confidence interval for estimated but for positive runs of ISE 50 and ISE 100, it is not in the %95 confidence interval (for ISE 50 it is pretty close though; see table 3.a) hence we can roughly conclude that, ignoring the magnitudes and considering just the signs of the daily returns, market is equally likely to go up or go down every day. 2.2 Distribution of the run returns Tables 4.a and Tables 4.b give some historical information about largest negative and positive run returns of ISE 30, ISE 50 and ISE 100 indices; Table 4.a. Largest negative run returns of ISE 30, ISE 50 and ISE 100 LARGEST NEGATIVE RUN RETURNS Return Duration -0,36078 23.11.2000 ­ 04.12.2000 - 8 days -0,27329 21.03.2001 ­ 29.03.2001 - 7 days -0,26727 28.05.2002 ­ 03.06.2002 - 5 days -0,36568 23.11.2000 ­ 04.12.2000 - 8 days -0,22093 06.07.2001 ­ 11.07.2001 - 4 days -0,20937 09.09.2008 ­ 18.09.2008 - 7 days -0,37174 18.04.1994 ­ 02.05.1994 - 11 days 154 -0,36565 -0,31315 23.11.2000 ­ 04.12.2000 - 8 days 21.01.1994 ­ 18.01.1994 - 6 days Table 4.b. Largest positive run returns of ISE30, ISE50 and ISE100 LARGEST POSITIVE RUN RETURNS Return Duration 0,60179 14.01.1997 ­ 27.01.1997 - 10 days 0,42027 05.12.2000 ­ 06.12.2000 - 2 days 0,33490 09.12.1999 ­ 13.12.1999 - 3 days 0,41564 05.12.2000 ­ 06.12.2000 - 2 days 0,24112 22.02.2001 ­ 26.02.2001 - 3 days 0,23099 26.04.2001 ­ 30.04.2001 - 3 days 0,62534 15.09.1989 ­ 04.10.1989 - 12 days 0,49633 14.01.1997 ­ 27.01.1997 - 12 days 0,41718 05.12.2000 ­ 06.12.2000 - 2 days Before starting the analysis, an interesting observation is (also as suggested by Tables 1.a, 1.b, 4.a and 4.b) as getting close to present day we still observe considerable amount of long runs, but in these long runs, magnitudes of run returns seem significantly smaller compared to those of the long runs in earlier dates suggesting that market depth has improved over recent years. To analyze the distribution of run returns, we consider the frequency distributions as the following: First we create intervals with 0,01 increments as:.... , [-0,02 , -0,01) , [-0,01 , 0) , [0 , 0,01) , [0,01 , 0,02) , .... and for each index, we count the number of observed run returns belonging to each interval (see LHS of Figure 2.a, 2.b and 2.c). Then we take absolute values of the observed run returns and count the number of these absolute values belonging to each interval mentioned above (see RHS of Figure 2.a, 2.b and 2.c). First thing to notice here is one of the stylized facts of financial markets: just like the distribution of the daily returns, distribution of the run returns display heavy tails and sharp peaks. Figure. 2.a. Frequency distributions of run returns and absolute run returns of ISE 30 Figure. 2.b. Frequency distributions of run returns and absolute run returns of ISE50 155 Figure 2.c. Frequency distributions of run returns and absolute run returns of ISE 100 Theory suggests the idea of fitting normal distributions and exponential distributions to the data in the LHS and RHS of Figure 2.a, 2.b and Figure 2.c. respectively. To understand which one would be a better fit, we compare the actual frequency and fitted frequency values with the same returns (see Figure 3.a, 3.b and 3.c; dashed lines have unit slope and pass through the origin). Figure 3.a. actual freq. vs. exponential fit freq. Figure. 3.b. actual freq. vs. exponential fit freq Figure 3.c. actual freq. vs. exponential fit freq. We observe that for each index, exponential distribution fits better than normal distribution in explaining the relationship between magnitudes of run returns and their frequency. and values of these fits also suggest us a similar idea (see Table 5); Table 5. and values of exponential and normal fit adjusted Exponential Fit: Normal Fit: Exponential Fit: Normal Fit: Exponential Fit: Normal Fit: 0,98096 0,97265 0,97912 0,97635 0,99047 0,96420 0,98033 0,97160 0,97825 0,97527 0,99022 0,96308 2.3 Asymmetry of gains and losses in a run Using the frequency data obtained in the last subsection, for each index we compare the number of positive and negative runs according to their return magnitudes. Figure 4.a Figure 4.b Figure 4.c 157 Just like in the previous case, dashed lines pass through the origin with unit slope and as it is easily understood magnitude of run returns gets larger as points get closer to the origin. We see that for each index there is a significant asymmetry, suggesting that large magnitudes of run returns is more likely to be seen in positive runs, in other words It is more likely to see big gain rather than big loss in a run. 2.4 Autocorrelation Analysis 2.4.1 Daily returns One of the stylized facts of liquid markets is that daily returns do not exhibit any significant autocorrelation and ISE is no exception as we see from Figure. 5; Figure 5. Autocorrelations of daily returns The absence of significant autocorrelations in asset returns has been studied in detail (Fama 1971; Pagan 1996) and it is usually used to support random walk models in which the returns are considered to be independent random variables (Fama 1991)). But as Cont (2001) states "independence implies that any nonlinear function of returns will also have no significant autocorrelation: For example, absolute values or squares of daily returns should also have no significant autocorrelation" but various empirical studies (Bollerslev et al. 1992; Comte, and Renault 1996; Bouchaud et al. 1997; Cont 1998; Ding et al. 1983; Ding, and Granger 1994; Engle 1995) show that in this case, autocorrelations remains significantly positive for several weeks and decays slowly 4. This situation is usually interpreted as there is a correlation in volatility of returns but not the returns themselves. This is a quantitative manifestation of one of the stylized facts in financial markets called volatility clustering.5 In order to see if we observe this stylized fact in Istanbul Stock Exchange, we consider the autocorrelations of absolute daily returns of our indices and obtain very similar results (see Figure. 6) 2.4.2 Run returns In the next step, we investigate if there exists significant autocorrelations in magnitudes of run returns. In this case, the autocorrelation function is defined as; Sometimes this slow decay is considered as an indicator of long - range dependence in volatility, but arguments still continue on whether it should imply long time memory of financial time series (Cont 2007; Taqqu et al. 1999) 5 In finance, volatility clustering refers to the observation that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes." (Mandelbrot 1963) Figure 6. Autocorrelations of absolute daily returns where is the return of run and absolute daily returns. (See Figure 7) denotes the lag. Analyzing this case gives us a similar result as of Figure 7. Autocorrelations of absolute run returns As it is seen from Figure 7, there exists strong correlation decaying slowly (persisting up to several months) which again suggesting volatility clustering that also can be directly observed from progress of run returns in Figure 8. 2.4.3. Run Lengths We ask ourselves if we can we find a similar result for run lengths. Here the autocorrelation function is defined as; where is the length of run. We see that the autocorrelation function fluctuates around zero in the %95 confidence interval meaning there does not exist significant time correlation in run lengths. (See Figure 9) Figure 8. Run returns Figure 9. Autocorrelations of run lengths Conclusion In this work, we conducted a detailed analysis on runs of daily returns of three popular Istanbul Stock Exchange indices (ISE30, ISE50 and ISE100) hoping to find some meaningful properties. As a result we have the following observations: Exponential law fits pretty well for the distribution of both length and return magnitude of the runs. Market is equally likely to go up or go down every day. Market depth has improved over recent years. It is more likely to see big gain rather than big loss in a run. Just like in most of the developed stock markets, in Istanbul Stock Exchange there is an absence of significant autocorrelations in daily returns but the autocorrelations of absolute daily returns are strong and slowly decaying (persisting up to several months) suggesting volatility clustering. Similarly, significant correlation exists in the absolute run returns which also support the same deduction. Hoping to find a similar relation, we investigated the autocorrelations of runs length but in this case there seems no significance.

Journal

Journal of Advanced Studies in Finance – de Gruyter

Published: Dec 1, 2012

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