Could someone please explain the difference between a Leptokurtic and Platykurtic distribution in relation to returns?
Why does a Leptokurtic distribution have “more frequent extremely large deviations from the mean than a nomal distribution?”
I would have thought a Platykurtic distribution would have extremely large deviations from the mean as the observations are more spread out over a larger area than a Leptokurtic distribution?
Huge appreciation for anyone that can shed light on this!
Kurtosis is a measure of “peakedness”, that is, a the curve is placed higher relative to the normal distribution if it has more kurtosis. This means that the curve’s tails at both sides are fatter as a result of it being higher. A leptokurtic distribution has a higher level kurtosis than a platykurtic distribution so its tails are fatter, hence implying that it has more extreme deviations from the mean.
This is somewhat inaccurate: it isn’t possible for the entire curve to be higher than a normal distribution because the area under each curve (and above the x-axis) has to be 1.
A leptokurtic distribution is higher than a normal distribution near the center (say from -1σ to +1σ), and at the ends (say, below -3σ and above +3σ), but in the midranges (between -3σ and -1σ, and between +1σ and +3σ) the leptokurtic distribution is lower than a normal distribution.
Thus, in a leptokurtic distribution, you are more likely to see values near the mean or at the extremes, but less likely to see values moderately above or below the mean (compared to a normal distribution).
I wrote an article on kurtosis that may help (it has pictures): http://financialexamhelp123.com/kurtosis/.
Great article S2000. Very clear explanation, thanks
Thanks.
Glad to be of help.
You’re welcome.
Thanks S2000! The article and your explanation above really helps!
So from an investor’s perspective, generally which distribution would be preferred if there was excess kurtosis?
Would it be a Platykurtic distribution as the values are more moderately spread out than a Leptokurtic distribution?
My pleasure. I’m happy it helped.
A lot would depend on the skewness of the distribution. If the returns are positively skewed (generally rare), leptokurtosis is probably preferable; if the returns are negatively skewed (more common), platykurtosis is probably preferable. When the distribution is not skewed (or only mildly skewed), it depends more on the risk tolerance of the investor. I would require a higher risk premium for higher kurtosis.
Very interesting, thanks!
lepto = fatter tails = more extreme values = more risks
platy = the other way around.
btw, where did you find this question?
So when it refers to “fatter tails”, this is referring to the very extreme deviations that are not visible in most graphs of a leptokurtic distribution? I’ve been having difficulty with this concept, because in most visualizations of a leptokurtic, normal, and platykurtic distributions, it looks like the platykurtic has the fattest tails, and then normal, and then leptokurtic. Maybe I am misunderstanding what “fat tails” means. Thanks for any help you can provide!
You’re correct.
When you get to the tails, the probabilities are so low that it’s hard to see which distribution has the higher (or lower) probabilities.
Leptokurtic distributions generally have higher probabilities in the tails; platykurtic distributions generally have lower probabilities in the tails.
Thank you.
You’re welcome.
Thus, in a leptokurtic distribution, you are more likely to see values near the mean or at the extremes, but less likely to see values moderately above or below the mean (compared to a normal distribution)???
waleeed
Correct.
I am a bit late to this thread. Can you please elaborate further the above scenarios? Why would an investor prefer leptokurtosis when it is positively skewed (or vice versa). I have some difficulty with this concept. My other confusion is why an investor would prefer positive sknewness over negative sknewness? CFA Institute level 1 indicates: “If a distribution is positively skewed with a mean greater than its median, then more than half of the deviations from the mean are negative and less than half are positive.” – why do we want a portfolio with more than half negative?
A positively skewed (right tailed) return distribution means that you will have more small losses but more big gains. So if you look at the tail of a positively skewed return distribution, there will be many small losses clustered around the mean while there will be a few big gains which causes the right tail and mean to move to the right. And so the the positive skew pulls the average to the right (resulting in better return expectations). Think of it this way. You would prefer to own a stock that has, lets say five -1% returns and one +15% return. In this case, you have more negative deviations but would still be in a good position. Unfortunately, you rarely see distributions that is right tailed.
For a leptokurtic distribution, you are more likely to see values near the mean or at the extremes (big deviations). If you consider this with the fact that a positively skewed distribution reflects more large gains than large losses, it is an advantages position. You will have a lot of returns close to the mean and more large positive deviations than large negative deviations from the mean.