TASI Sectors’ Normality Assumption

As observed in the previous post, the normality assumption in TASI index is not valid. In this post, I’ll explore the normality assumption in all the sectors in Saudi stock market. As you’ll see, all the sectors violate the normality assumption.

The normality assumption is not valid due to the presence of skewness and large positive excess kurtosis. This causes the number of observations below the lower bound of 99% confidence interval to be higher than expected. For example, TASI index has a skewness of -0.8486 (expected zero) and an excess kurtosis of 9.903 (expected zero) which results in 55 observation (expected 23) below the the lower bound of 99% confidence internal.

In the table below, I show the skewness, kurtosis and number of observation belows the lower bound of 99% CI for all the sectors.

Note: the sectors names are abbreviated for readability. Refer to my earlier post for sectors names’ abbreviations.

Sector Skewness vs 0 expected Excess kurtosis vs 0 expected Obs. below 99% CI vs 23 expected
TASI -0.8486 9.903 55
TASI.BFS -0.1863 8.269 46
TASI.PCI -0.6701 7.078 52
TASI.CMT -0.6123 13.622 45
TASI.RTL -0.7570 9.699 48
TASI.EU +0.0872 8.605 45
TASI.AFI -0.6024 7.611 53
TASI.TIT -0.4697 8.224 54
TASI.INS -0.8096 4.082 60
TASI.MUI -0.8107 6.041 62
TASI.INI -0.9271 6.758 60
TASI.BDC -1.0291 7.505 66
TASI.RED -0.5750 6.827 59
TASI.TRA -0.4645 6.138 57
TASI.MAP -0.0686 4.439 67
TASI.HTT +0.1061 7.969 51

Notice the following:

  • Building and Construction (TASI.BDC) has the highest negative skewness.
  • Only two sectors show positive skewness; i.e. Energy & Utilities (TASI.EU) and Hotels & Tourism (TASI.HTT). But despite that, the same two sectors show high positive kurtosis and this result in having large number of observations below the lower bound of 99% CI; i.e. 45 and 51, respectively.
  • All sectors show high excess kurtosis resulting in large number of observations below the lower bound of 99% CI.

Conclusion: analysis demonstrated that all sectors in the Saudi stock market violate the assumption of normality.

To visualize the skewed returns in the distribution of the sectors, I made the following box plots. A box plot consists of a box and whiskers. The box starts from the 25th percentile to the 75th percentile of the data, known as the inter-quartile range (IQR). In the box, the line in the middle indicates the median (i.e., the 50th percentile) and the diamond shape indicates the mean. The whiskers start from from the edge of the box and extend to the furthest data point that is 1.5 times the IQR. If there are any data points that are farther than the end of the whiskers, they are considered outliers and indicated with dots.

Sectors_bwplot1
Box plot of the returns of Banks & Financial Services, Petrochemical Industries and Cement sectors.
Sectors_bwplot2
Box plot of the returns of Retail, Energy & Utilities and Agriculture & Food Industries sectors.
Sectors_bwplot3
Box plot of the returns of Telecommunication and Information Technology, Insurance and Multi-Investment sectors.
Sectors_bwplot4
Box plot of the returns of Industrial Investment, Building & Construction and Real Estate Development sectors.
Sectors_bwplot5
Box plot of the returns of Transport, Media & Publishing and Hotel & Tourism sectors.

Thank you.

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TASI’s Normality Assumption

In this post, I explore normality assumption of the Saudi stock market; i.e. Tadawul. Mainly, I attempt to answer the question, “Is it valid to assume that Tadawul shows characteristics of normality in its returns?” The short answer is a resounding NO. Why? You need to continue reading this post.

Normality assumption is a choice made by financial analysts and risk managers to simplify their understanding of the financial markets.

Normality assumption implies that each stock/portfolio return is an independent realization from the same normal distribution; i.e. returns are i.i.d normal. It also implies that the returns distribution can be completely characterized by only two parameters: the mean and the variance. The skewness and (excess) kurtosis should be zero. However, this is rarely the case in financial markets.

Now let us consider the daily returns of TASI index for the period from 7-Jan-2007 to 27-Dec-2015 (illustrated and described below).

R_Lab_4_1
Daily log returns on TASI index from 7-Jan-2007 to 27-Dec-2015. Notice the extreme volatility in 2007 and 2008.

Analysis of TASI index returns, 7-Jan-2007 to 27-Dec-2015

Observations 2241
Mean return -6.123602e-05
Standard deviation per day 1.46%
Volatility 23.06%
Skewness -0.8486 vs. 0 expected
Excess kurtosis 9.903 vs. 0 expected
Observations below the lower bound of 99% CI 55 vs. 23 expected

As shown in the last line in the above table, the number of observations below the lower bound of 99% confidence interval (marked red in the plot below) is more than expected due mainly to the negative skewness and positive excess kurtosis.

R_Lab_4_1a
Observations below the lower bound of 99% CI are marked red. Notice the clustering and calendar dates of these observations.

Conclusion: this analysis demonstrated that TASI violates the assumption of normality.

In the next blog, I’ll explore the normality assumption of all the sectors in TASI.

Thank you.

 

TASI sectors correlation

In this post, I explore the correlations between the different sectors in the Saudi stock market, TASI. The data set used is from 6-Jan-2007 to 27-Dec-2015.

R Lab 2_5
TASI sectors’ correlation matrix

First, some explanation about the above correlation matrix is in order. For readability, I have abbreviated the long sectors names; for example, TASI.BFS is the Banks & Financial Services sector. The full list of the sectors names and their abbreviations is listed at the end of this post.

Correlation is a number between -1 (negative correlation) and +1 (positive correlation). For readability, I use a scale between 1 (weak correlation) to 10 (strong correlation) to represent the strength of correlation between two sectors. Note that in the Saudi stock market the correlation between sectors are all positive.

Data Analysis

As mentioned above, correlations between sectors in the Saudi stock market are all observed to be positive correlations (i.e. not negative correlations between sectors).

The strongest positive correlation (around .85) is observed between the sector Industrial InvestmentĀ (TASI.INI) and the sector Building & Construction (TASI.BDC). Below is a plot of the correlation between the daily log returns of both sectors. As you can see, the points in the plot are very close to each other.

R Lab 2_5 cor1
The strongest correlation is between TASI.INI and TASI.BDC

The weakest positive correlation (around 0.35) is observed between the sector Energy & Utilities (TASI.EU) and the sector Media and Publishing (TASI.MAP). Below is a plot of the correlation between the daily log returns of both sectors. As you can see, the points in the plot are very dispersed. However, regardless of the weak correlation, extreme price changes happens at the same time.

R Lab 2_5 cor2
The weakest correlation is between TASI.EU and TASI.MAP

That’s it for now. I’ll present more observations in the next post.

Thank you.

TASI Sectors Abbreviation

TASI.BFS Banks & Financial Services
TASI.PCI Petrochemical Industries
TASI.CMT Cement
TASI.RTL Retail
TASI.EU Energy & Utilities
TASI.AFI Agriculture & Food Industries
TASI.TIT Telecommunication & Information Technology
TASI.INS Insurance
TASI.MUI Multi-Investment
TASI.INI Industrial Investment
TASI.BDC Building & Construction
TASI.RED Real Estate Development
TASI.TRA Transport
TASI.MAP Media and Publishing
TASI.HTT Hotel & Tourism

 

Introduction

Greetings,

My name is Thamir K. AlHashemi. I am a Quant.

The purpose of this blog is to publish quantitative research on the GCC countries’ stock markets as well as the Forex market. I have access to enormous quantities of data and I will use power methods for extracting quantitative information, particularly about volatility and risk. I am using R for computations and graphics. I am planning to cover advanced topics such as multivariate distribution, copulas, Bayesian computations, VaR, expected shortfall and cointegration.

The prerequisites to understand the subject I am going to blog present are basic statistics and probability, matrices and linear algebra, and calculus. Some exposure to finance is helpful.

Thank you for dropping by.

Regards.