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.

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.

]]>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.

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.

]]>The matrix below shows the correlation between the prices of crude oil (using WTI as proxy to OPEC basket prices) and GCC Banks and Financial Services in the period from June 20, 2014 to Jan 11, 2016. This period was chosen specifically because it marks the beginning of the drop in crude oil prices.

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). However, for readability, I used a scale between 1/-1 (weak positive/negative correlation) to 10/-10 (strong positive/negative correlation) to represent the magnitude of correlation.

Referring to the first column in the correlation matrix above, it is evident that the correlation between the prices of crude oil and GCC Banks and Financial Services is very weak; ranging from -1 to 0 to 1. Meaning, it seems that share prices of GCC Banks and Financial Services are not coupled to the prices of crude oil since the start of the drop in crude oil prices from a high of US$107.95 in June 20, 2014 to a low of US$31.42 in Jan 11, 2016. This of course does not rule out the possibility of an emerging correlation between the prices of crude oil and shares of GCC Banks and Financial Sectors.

Thank you.

TASI.BFS | Saudi Stock Exchange – Banks & Financial Services |

DFM.BAN | Dubai Financial Market – Banks |

DFM.IFS | Dubai Financial Market – Investment & Financial Services |

ADX.BAN | Abu Dhabi Securities Exchange – Banks |

ADX.IFS | Abu Dhabi Securities Exchange – Investment & Financial Services |

KSE.BAN | Kuwait Stock Exchange – Banks |

KSE.FIS | Kuwait Stock Exchange – Financial Services |

BAH.CBN | Bahrain Stock Exchange – Commercial Banks |

MSM.BKI | Muscat Securities Market – Banks and Investment |

QE.BFS | Qatar Exchange – Banks & Financial Services |

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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!

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.

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.

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.

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.

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.

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%).”

Below are the density plot and the normal probability plot 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.

]]>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!

]]>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.

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.

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.

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.

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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.

Thank you.

]]>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).

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.

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.

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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.

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 Investment (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.

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.

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

Thank you.

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 |

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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.

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