YouTube T2-7 Kurtosis of a probability distribution

[Nicole Seaman]

Kurtosis is the standardized fourth central moment and is a measure of tail density; e.g., heavy or fat-tails. Heavy-tailedness also tends to correspond to high peakedness. Excess kurtosis (aka, leptokurtosis) is given by (kurtosis-3). We subtract three because the normal distribution has kurtosis of three; in this way, kurtosis implicitly compares to the normal distribution and "positive excess kurtosis" means "tails are heavier than the normal" or "extreme outcomes are MORE likely than under the normal."

Here is David's XLS: http://trtl.bz/121817-yt-kurtosis-xls

 

[Sixcarbs]

Good morning Professor and friends,

I haven't been around for some time. Have been working full time, preparing for CFA, and getting yelled out for not making family time.

I am trying to replicate this demonstration in Excel as a piece of a larger question. But instead of using n=4 I am using n=100. I am getting some strange numbers and they do not line up with the Excel built in functions.

For n=100, p=.5 (I would hope this would be close to normal), I get:

Average 50
Std. Dev 5
Skew 0.000
Kurtosis 2.98

But the excel functions show:

Std deviation 1.075
Skew 2.236
Kurtosis 3.73

For my argument I am using the Xf(X) column.

For p=.3

Average 30
Std. Dev 4.583
Skew 0.08729
Kurtosis 2.9876

But the excel functions show:

Std deviation 0.6801
Skew 2.3875
Kurtosis 4.481

What am I doing wrong here? Is it units or something?

If anyone is bored on a Saturday and would like to chime in here I would greatly appreciate it.

Thank you,

Sixcarbs

 

[Amierul]

Hi @David Harper CFA FRM

I think what u shared above is no the same as in the schweser 2018 book. Can you explain to me.. i might be wrong as leptokurtic has and excess kurtosis equal to 3 no -3.?



Thank you very much

 

[David Harper CFA FRM]

Hi @Amierul What is the exact difference that you perceive? My only criticism of the above text is the first sentence: "excess kurtosis" is (trivial) adjusted kurtosis that substracts 3, such that excess kurtosis = kurtosis - 3. While there are esoteric reasons to work with the excess kurtosis, we retrieve the kurtosis (you'll see throughout any of my XLS) by calculating fourth moments, i.e., E(X - μ)^4. Mechanically, it has the same pattern as the variance, E(X - μ)^2. Notice this value, as a (squared) square, must be positive, even after we standardize this fourth moment by dividing by a necessarily positive σ^4 = (σ^2)^2. In summary, kurtosis must be positive, such that the lower bound on excess kurtosis is -3.

The first sentence you've quoted is nonsensical because all probability distributions have kurtosis or excess kurtosis measures. A true statement, instead, is (e.g.) "A distribution with kurtosis > 3, or equivalently excess kurtosis > 0, is leptokurtic; aka, heavy-tailed. A distribution with kurtosis < 3 or equivalently excess kurtosis < 0 is platykurtic; aka, light-tailed." I prefer heavy- or light-tailed because leptokurtic distributions do tend to exhibit tall peaks (including the student's t) but it is not necessarily the case: the mathematical definition decides and it is a feature of the tail. I hope that's helpful,

 

[Amierul]

Hi @David Harper CFA FRM, than you for the very clear explanation . I really appreciate it!

At the moment only ur explanation can enlighten me and suit my needs .