Mar 31

Skew & Kurtosis - Spreadsheet with Google’s data

by David Harper, CFA, FRM, CIPM

FRM | CFA |

image

Below is a copy of the spreadsheet used in my screencast review of the manual calculation of skew and kurtosis. This calculates the skew and kurtosis of Google's (GOOG) daily returns for calendar 2007. Here are the formulas:

skew

kurt

The distribution's "moment" is the numerator; e.g., the 3rd moment is the expected value of the cubed deviation. Skew and kurtosis are not the moments per se, they are standardized moments because the moments are divided by the denominator:

  • Variance = 2nd moment (i.e., the expected squared deviation. If daily returns, the expected squared return)
  • Skew = 3rd moment divided by cubed standard deviation,
  • kurtosis = 4th moment divided by squared variance.

In the spreadsheet, I compared the to Excel's built-in functions, =SKEW() and =KURT(). The =KURT() gives an answer which needs to be adjusted because Excel's kurtosis is an excess kurtosis. If normal kurtosis = 3, then excess kurtosis is kurtosis greater than 3.

Further, my calculations are consistent with Excel because they calculate sample skew and sample kurtosis (see page 72 of FRM assigned reading Gujarati). Just like sample variance, the sample skew and sample kurtosis adjust for a "small sample bias." For example, a population skew is based on a population moment: the third moment (numerator) is the sum of cubed deviations divided by the number of observations (251 trading days in the GOOG example). But the sample skew derives a slightly larger moment: the third moment (numerator) is the sum of cubed deviations divided by a number smaller than the number of observations (248, in the case of GOOG).

In the case of GOOG's 251 observations (daily periodic returns), the sample skew is -0.43 which indicates left skew (a longer tail to the left). The kurtosis = 4.6 which is the same as "excess kurtosis = 1.6." We typically say that's fat tails. However, technically kurtosis refers to peakedness, so positive excess kurtosis implies a higher than normal peak. Usually this creates more density in the tails, but it may depend on the distribution. For example, Gujarati's example is a single six-sided die: a normal distribution with a kurtosis of 1.73 (platykurtic). Kurtosis is < 3 because it's flat, it has no peak. But note the uniform distribution also has fat tails!

Here is the EditGrid spreadsheet:

Comments

  1. phone 18 May 2008

    Why 248 ; it should be 250 I think ...correct me If I am wrong

  2. David Harper, CFA, FRM, CIPM 18 May 2008

    I think my language is sloppy, but i meant: if you look at the ‘adjust for small sample bias’ it is for skew = n/[(n-1)(n-2)]. Like for kurt, = n(n+1)/[(n-1)(n-2)(n-3)]. I think those are right. Of course the idea for sample skew is that it divides a sample 3rd moment (i.e., n-1 in denominator) by the cube of a sample standard deviation. Admittedly, I trusted this to be correct as it matches Excel’s function…

  3. phone 18 May 2008

    Dear Sir,
    I cannot find this adjustment newhr in Gujrati can you tell me on which page this adjustment is there ? in my copy at page 72 only formula mentioned is Sum(deviation^k)/(n-1) and not as the one mentioned by you....srry to bug u with these silly problems..Thnx

  4. David Harper, CFA, FRM, CIPM 18 May 2008

    phone,

    yours formula is correct, that is the sample third moment. I don’t think Gujarati, even in the question sets, ever goes further with an example. (in doing so, he skirts the nuance at play here).

    So, please don’t let my spreadsheet confuse you: the point in the XLS is to reconcile with Excel’s built-in SKEW() function.

    From a conceptual standpoint, the sample skew = sample third moment divided by sample standard deviation. This is the relevant def of UNITLESS SKEW. There are other definitions of skew and, frankly, there is more than one way to adjust for small sample.

    Yours is fine, it would result in a sample skew which i doubt would be different at 2 decimals. David

  5. David Harper, CFA, FRM, CIPM 18 May 2008

    phone,

    please note i added two rows to the XLS above: Gujarati’s sample skew and sample variance. This, I believe, reflects his approach (while differing with Excel). Let me know what you think, thanks, David

  6. phone 18 May 2008

    Ya thank you sir for explaining it so clearly....

Leave a Comment