Kurtosis and description of tails

[fullofquestions]

Please refer to 2009_Notes_2Quantitative_v1a.pdf, page 31 at the bottom and page 32 at the top. Hopefully we can put this topic to rest once and for all because I have seen various websites and resources with conflicting descriptions of kurtosis and how it relates to the tails of a distribution...

Page 31 bottom, it reads:
"Kurtosis greater than three (>3), which is the same thing as saying “excess kurtosis > 0,” indicates high peaks and fat tails (leptokurtic). Kurtosis less than three (<3), which is the same thing as saying “excess kurtosis < 0,” indicates lower peaks.
Financial asset returns are typically considered leptokurtic (i.e., heavy or fat-tailed)"

Page 32 top, it reads:
"For example, the logistic distribution exhibits leptokurtosis (heavy-tails; kurtosis > 3.0):

GRAPH

A probability distribution with “thicker tails” or “heavier tails” than the normal distribution has kurtosis > 3 and it called leptokurtic.
When a distribution is less peaked than the normal distribution, it is said to be platykurtic. This distribution is characterized by less probability in the tails than the normal distribution. It will have a kurtosis that is less than 3 or, equivalently, an excess kurtosis that is negative."

In the GRAPH listed we have the standard normal distribution in blue. Then we have a platykurtic distribution, one with lower peak, in purple. The platykurtic distribution CLEARLY has more probability(or samples rather) in the tails than the normal distribution. Can someone please explain the following:

1. what is a fat tail? Is it a tail with more *meat* on it, i.e. more height per unit area, much like a platykurtic distribution or is it somehow how much farther the tail extends ouT from the mean? In my opinion:

a. Leptokurtic distribution clearly has highest peak but quickly contours very close to the X axis. Therefore I see it as a 'thin' tail
b. Mesokurtic distribution, i.e. standard distribution, has a medium sized peak and the tails, in either direction contour closer to the X axis slower than the leptokurtic example. In this case there are more probabilities in the tails than in the leptokurtic case.
c. Platykurtic distribution, has the lowest peak and the tails contour to the X axis way more gradually and therefore has more probabilities in the tails. The tails are taller than compared to the leptokurtic and mesokurtic distribution.

2. is a fat tail any different from a heavy tail?

Please advise and correct where I've mistaken. Thank you.

 

[David Harper CFA FRM]

I think the most precise term is "heavy tail" because kurtosis is about the *density* of the tail. Kurtosis > 3 implies more area "under the curve" in the tail of a PDF/PMF compared to the normal.

While it's true that a single-humped (unimodal) distribution that is leptokurtotic has a higher peak and therefore heavier tails, I think it leads to confusion if you try to characterize kurtosis in terms of "peakedness" or fatness (as you suggest, leptokurtosis can appear as a long skinny tails). Also confusing if you try to think in terms of vertical height in a PDF (e.g., y axis varies). But, for a PDF, leptokurtosis means there is more "area under the curve" in the extreme tail, compared to a normal, and that means the odds are higher of ending up in the tail. (if you want to think only vertically, then you can use: for the CDF, such a distribution has a lower y value, cumulative probability, for extreme outcomes...)

Leptokurtosis means that the probability of an extreme tail outcome is more likely than it would be under the normal. The student's t is a good example: its graph will maybe not help so much if you think vertically, but student's t is always leptokurtotic. It's excess kurtosis > 0, as it's excess kurtosis = 6/ (df - 4). So, I prefer "heavy tail" (which, yes is synonomous with fat tail) and this is what it means:

if you look in the tail, what is the CDF probability of ending up there?

say we are looking at 3 sigma, then normal = NORMSDIST(-3) = 2.275%
i.e., for a standard normal, our odds of ending up more than 3 SD are 2.275%

a leptokurtos distribution has a higher probability of ending up more than 3 SD. In the case of the student's t:
=TDIST(3, d.f.,1 tail) = will always be greater than 2.275%, although converging to 2.275% as d.f. increase

so, leptokurtosis implies more area in the tail vis a vis a normal (i.e., heavier density).

and fortunately, this mathematical view (i.e., higher CDF P[X<x] compared to normal), rather than graphical, view fits our risk concern: we may prefer a leptokurtotic distr (e.g., Levy) because it means that our odds of an extreme tail outcome are greater.

David

 

[fullofquestions]

Thank you for the explanation Dave. I am still a bit unclear although I think I'm getting there. Perhaps it would help to discuss the following details:

1. In describing a distribution the following phrases are common:
-heavy tailed
-fat tails
-fat tailed
-heavy tails

So my question is, when talking about 'tail' is it possible that in some cases we are refering to the hump/peak while in others we referring to the ends of the distribution, you know, the ones that stretch to - infinity and + infinity? Both could be construed as 'tail.'

I understand that a leptokurtic distribution such as a student's t has the following characteristics
- kurtosis > 3
- higher hump/peak than a normal distribution (this apparently is NOT always the case, take minute 38:30 from your video 2ai_p2_quant_iPod)
- longer, skinnier tails (- infinity and + infinity). This, in proper statistical terms, is referred to as 'heavier tails.' The reason being that the tails, although they look skinny, they extend further out to -/+ infinity ultimately carrying more weight/area, and therefore, probabilities in the tails.
Please correct me.

I think the following image explains it very well although it would help if you could zoom in more (http://en.wikipedia.org/wiki/File:Standard_symmetric_pdfs.png). In it, it is clear that the leptokurtic distributions carry more weight in the tails. I think that in many cases (take wikipedia or investopedia for example), leptokurtosis is referred to as 'higher peaks,' and this is NOT always the case. I swear I have seen examples dealing with kurtosis where the answer lies not in describing the tails but the peaks. If and when I encounter them I will collect them because I think these are causing unnecessary confusion.

 

[David Harper CFA FRM]

fullofquestions,

(sorry for delay). What i have read (somewhere, I can't remember) is that higher peaks for a unimodal (single humped) distribution imply heavier tails. I frankly can't grab the intution. I am "with you" in regard to higher peaks but please note it can be misleading if the Y-axis differs. for years, i thought the student's t was an "exception" (i.e., shorter peak but heavier tail) but someone pointed out that's just a y-axis issue....

...the peakedness description doesn't personally work for me (or i have not heard a compelling explanation). What works for me is: tail density or tail heaviness. And mathematically, this is simply that, for a given "cutoff" (+ standard deviations to the right), the "rejection region" has more density (a larger % of the entire 100% probability). Since kurtosis is about the tail, it seems to me the peak (being more body) is incidental anyway

David