Joined: Sep 2010
Mon Sep 20, 10 12:57 PM
These are the notes I have gotten together on Kurtosis and Skewness. Feel free to critique them.
What do these measures mean broadly?
These measures are an indication of the deviation of the data series from a normal distribution where the mean, median and mode are the same.
Skewness simply is about the symmetry of the distribution and Kurtosis about the peakedness or flatness of the distribution.
What does that mean in terms of application?
For a normal distribution the Skewness is 0. A +ve skewness means the mean has been pulled to the right of the median and there are more values stacked on the lower end of the scale and a longer tail of positive values. A -ve Skewness means the mean has been pulled to the left of the median and there are more values stacked on the higher end of the scale and a longer tail of negative values. When reporting the skew the median of the distribution should also be reported.
For a normal distribution Kurtosis is 3. A Kurtosis higher than 3 means the distribution is leptokurtic and there are more outliers away from the mean while a Kurtosis less than 3 means the distribution is platykurtic and there less outliers away from mean.
Examples of when returns are skewed?
A portfolio containing short put options is negatively skewed while portfolio containing long call options is positively skewed.
Examples of when returns have kurtosis?
Imagine going to a high school cafeteria during lunchtime. If you were to survey the age of each person eating in this cafeteria, you would get ages in a very specific range, with few outliers. This is an example of a platykurtic distribution.
Conducting the same study at a restaurant like Garfunkel's during early dinnertime, when parents bring their children, and when more older folks are present, would result in an age distribution curve that has more outliers. It is very probable that this distribution would be leptokurtic. It is certain that the kurtosis would be more positive here than in the first case.
What about the range of these measures?
There is no standard range as such for these measures. But we can still depending on the value of the Skew or Kurtosis make some conclusions on whether the skew or kurtosis is high or low or can be ignored.
If we are calculating the skew of the whole population then:
If Skewness is less than -1 or greater than +1, the distribution is highly skewed
If Skewness is between -1 an -1/2 or 1/2 and 1 the distibution is moderately skewed
If Skewness is between -1/2 and +1/2 the distribution is approximately symmetric.
If we are calculating the Skew of the sample then the above 3 conditions apply to that sample but you cannot infer the same for the whole population. To make some inferences about the population one needs to calculate the SES which is
SES = SQRT(6/N) or more accurately SQRT(6N(N-1)/(N-2)(N+1)(N+3)) where N = Sample Size (Note SQRT(6/N) applies for sample sizes of greater than 150)
Now if Skew of Sample/SES is < -2 population is very likely skewed negatively though you don't know by how much
If its is between -2 and +2 then you cant reach any conclusion bout the skewness of the population. It might be symmetric or it might be skewed in either direction.
If it is greater than 2 population is very likely skewed positively though you don't know by how much
If the Skew of Sample/SES is quite small one can give for the population a 95% confidence of skewness as about Sample Skew +/- 2 SES
The smallest possible Kurtosis is 1 i.e excess Kurtosis is -2.
If we are calculating the Kurtosis of the whole population or sample then:
If Excess Kurtosis is less than -1 or greater than +1, the distribution has significant Kurtosis
If Excess Kurtosis is between -1 an -1/2 or 1/2 and 1 the distibution has moderate Kurtosis
If Excess Kurtosis is between -1/2 and +1/2 the distribution has approximately no Kurtosis.
If we are calculating the Kurtosis of the sample then we cannot infer the same for the whole population. To make some inferences about the population one needs to calculate the SEK which is
SEK = SQRT(24/N) or more accurately 2*SES*SQRT(N*N-1/(N-3)(N+5)) where N = Sample Size (Note SQRT(24/N) applies for sample sizes of greater than 1000)
Now if Kurtosis of Sample/SEK is < -2 population very likely has negative excess Kurtosis(Kurtosis-3) though you don't know by how much
If its is between -2 and +2 then you cant reach any conclusion bout the Kurtosis of the population. Excess Kurtosis might be positive, negative or zero.
If it is greater than 2 population very likely has positive excess Kurtosis(Kurtosis-3) though you don't know by how much
What are the minimum data points required for these measurements to be valid?
Alongwith the conditions described above there should be atleast 30 data points available. This figure of 30 comes from the Central Limit Theorem. The general rule of thumb with these measures though is the more the data points available the more accurate these measures are.
What are the calculations used here for these measures?
Fisher's formula for calculating Kurtosis and Skewness is being considered here.