Using Heavy-Tailed Distributions with TASI: Student t Distribution

In this post, I continue trying to fit the daily log returns of TASI index using heavy-tailed distributions. In the previous post, I used Pareto distribution to model TASI index’s left tail. In this post, I use Student t distribution.

Recently, Student t distribution has been used widely by financial engineers as models for heavy-tailed distribution such as the distribution of financial markets.The t-distribution is characterized by degrees of freedom v (shape parameter) in addition to mean and variance. Both the kurtosis and weight of the tails increase as the v degrees of freedom decreases. When v ≤ 4, the tail weight is so high that the kurtosis is infinite. Thus, v degrees of freedom is also interpreted as the tail index α.

The following figure contains Quantile-Quantile plots of the normal and t probability with 1, 2, 4, 8, and 15 degrees of freedom. The data used is TASI Index daily log returns from Jan 7, 2007 to Feb 2, 2016.

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As shown, none of the plots looks exactly linear, but the t plot with 2 degrees of freedom is almost straight through the bulk of the data. There are approximately two returns in the left tail and one in the right tail that are not along the line, but these can be ignored considering the sample size of 2270. Nevertheless, it is worthwhile to keep in mind that the historical data have more extreme outliers than a t-distribution.

Conclusion: Extreme returns from TASI index are more likely than the probability given by a t-distribtuion with 2 degrees of freedom.

Thank you.

Using Heavy-Tailed Distributions with TASI: Pareto Distribution

As established in a previous post, Tadawul All Shares Index (TASI) of the Saudi stock market has high excess kurtosis (9.903). The high kurtosis indicates that TASI has heavy tails. This means that the probability of extremely large negative returns is higher compared to a normal distribution. In this post, I use Pareto distribution to model TASI’s left tail.

Pareto distribution is used in modeling excesses over a predefined threshold. Pareto distribution is characterized by two parameters: a minimum value (scale parameter) and the tail index α (shape parameter). For the minimum value parameter, I decided to use the daily normal value-at-risk (VaR) at 95% confidence level (note: normal VaR was computed using the normal distribution). The tail index α is estimated to be 0.2637. The smaller the value of the tail index α, the heavier the tail. The value of the tail index α must be larger than 0.

In the plot below, I fit returns from TASI’s left tail using Pareto distribution using the minimum threshold value at -0.0244 and the tail index α at 0.2637.

TASI Pareto
Using Pareto distribution to model the left tail of TASI returns

Conclusion: Returns from TASI’s left tail fit Pareto distribution. This clearly indicates that TASI has heavy tails; i.e. large negative returns have high probability.

Thank you.

Comparing the distribution of TASI vs other GCC Markets

World markets have been extremely volatile recently. TASI changed -12.68% in 1-month, -25% in 3-months and -31.73% in 12-months. How do the changes in TASI compare to other GCC stock markets?

In this post, I use QQ plots for comparing the log returns distribution of TASI index versus other GCC stock markets indexes. In particular, I want to see how TASI index’s extreme outliers compare with other GCC stock markets. In almost all cases, TASI returns have more extreme outliers than other GCC markets. There is one exception though!

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QQ plot of TASI vs Dubai Financial Market (DFM)

TASI vs  Dubai Financial Market (DFM): in the plot above, the distribution of the Interquartile Range (IQR) seems to be the same. However, TASI returns have more extreme outliers than DFM returns.

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QQ plot of TASI vs Abu Dhabi Securities Exchange (ADX)

TASI vs  Abu Dhabi Securities Exchange (ADX): in the plot above, the distribution of the Interquartile Range (IQR) seems to be the same. However, TASI returns have more extreme outliers than ADX returns.

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QQ plot of TASI vs Kuwait Stock Exchange (KSE)

TASI vs  Kuwait Stock Exchange (KSE): in the plot above, the distribution of the Interquartile Range (IQR) seems to be the same. However, TASI returns have more extreme outliers than KSE returns.

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QQ plot of TASI vs Bahrain Stock Exchange (BSE)

TASI vs  Bahrain Stock Exchange (BSE): in the plot above, the distribution of the Interquartile Range (IQR) seems to be the same. However, TASI returns have more extreme outliers than BSE returns.

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QQ plot of TASI vs Muscat Securities Market (MSM)

TASI vs  Muscat Securities Market (MSM): in the plot above, the distribution of TASI returns seems to be similar to MSM returns even for outliers.

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QQ plot of TASI vs Qatar Securities Market (QE)

TASI vs  Qatar Securities Market (QE): in the plot above, the distribution of the Interquartile Range (IQR) seems to be the same. However, QE is the only GCC market that have more extreme outliers than TASI!

Conclusion: changes in TASI returns is far more extreme when compared to other GCC stock markets returns except Qatar which is even more extreme than TASI.

Thank you.

Updated on Jan 18, 2016: as if the markets is confirming the above stated conclusion, the article Mid-East Massacre: Qatar Crashes, Saudi Stocks Plunge Most Since Black Monday at Zero Hedge reported that “Saudi Arabia’s Tadawul Index is down 5.4% on the day – the worst since August’s collapse and has lost over 50% since its exuberant peak in 2014.” Also, the same article reported that “But Qatar was carnaged… (down over 8%).”

Is the Density of TASI Unimodal?

I established in my previous posts that the normality assumption for the log returns of TASI index and its sectors is false. In this post, I’ll use the Normal probability plot to investigate the distributions of the data. In particular, I want to know if the log returns of the TASI index is unimodal or not. The short answer is TASI index log returns is not unimodal. But then what?

Below are the density plot and the normal probability plot for TASI index log returns.

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Density (left) and Normal probability (right) plots for TASI index log returns

The normal plot of a random sample coming from a normal distribution must be close enough to a straight line. But as you can see, the normal plot of TASI index log returns is far from being linear (see the plotted normal line). This confirms of course that the data is not coming from a normal distribution but also reveals other informations.

The convexity and concavity in the normal plot worth checking. The alteration between concavity and convexity in the normal plot for TASI index log returns indicates a complex behavior which can be investigated by the density plot (left). Studying the normal plot (right) shows that convexity is changed three times; concave to convex to concave to convex. Further studying the density plot (left) shows that TASI log returns has a multimodal distribution.

One last point worth mentioning, The Shapiro-Wilks test uses the normality plot to test normality. For TASI index log returns, the Shapiro-Wilk test rejects the null hypothesis of normality with a p-value less than 2.2e-16; W is 0.84611.

Thank you.

TASI Sectors Density Estimation

As in the previous post; i.e. Why Normality Assumption, I estimate the density of all TASI sectors’ log returns using the histogram and the kernel density estimation (KDE) then compare with the normal density.The figures below are from the period starting from 7-Jan-2007 to 31-Dec-2015.

Notice that the highest density of returns is in the middle of each plot causing the high kurtosis. Also, comparison between the KDE and normal density in each plot reveals that TASI Insurance sector’s density closely resembles a normal curve. In fact, TASI Insurance sectors has the lowest (excess) kurtosis among other sectors in TASI; i.e. 4.082.

Enjoy!

Why Normality Assumption

As shown in a previous post; i.e. TASI’s Normality Assumption, TASI violates the normality assumption due mainly to the presence of negative skewness (i.e. -0.8486) and high positive kurtosis (i.e. 9.903). But why then assume normality? Because estimation of the probability density function (PDF) for TASI log return suggests that TASI’s PDF is a normal density. This is very convenient to financial modelers because using a parametric statistical model such as the normal density function would simplify their understanding of the financial markets.

In this post, I’ll try to estimate the density of TASI log returns using the histogram and the kernel density estimation (KDE) then I’ll compare them with the normal density.

The histogram is a simple and well-known estimator of probability density. The figure below is a histogram of TASI index log returns from 7-Jan-2007 to 31-Dec-2015 using 30 cells. The vertical dashes at the bottom represents the data points. Notice the outliers around -.10. Notice that the histogram roughly resembles a normal density. Notice also the high density of returns in the middle causing the histogram to appear more like a city skyline than a density curve.

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Histogram of TASI log returns.

The histogram is a crude estimator of density. A better estimator is the kernel density estimator (KDE). In the figure blow, KDE is mush closer in estimating the true density of TASI log returns which resembles a normal density curve. Using KDE requires some thought about the variance-bias trade off. Considering that, I am using a bandwidth of 1.5 in the plot below so that the KDE curve doesn’t overfit nor underfit the data.

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Histogram and Kernel Density Estimation of TASI log returns.

Finally, in the figure below I plot both the histogram and the KDE compared to a normal density curve having the same mean and standard deviation.

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Comparison of TASI log returns histogram and KDE with the normal curve.

Having the normal density curve overlaid on the same plot, it is easy to notice the high peak corresponding to the high positive kurtosis and the negative skewness.

Notice that although the histogram and the KDE resemble a normal density curve, the fact is, both are not a normal density curve. Thus, the distribution of TASI returns is not normal although both the histogram and the KDE suggest it.

Thank you.

 

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.