Why Kurtosis Does Not Simply Measure Distributional Peakedness

# Rationale for describing kurtosis as "peakedness"?


# Rationale for describing kurtosis as "peakedness"?












Despite plenty of evidence to the contrary, many quantitative finance sources of information, including teaching resources such as CFA prep, persist in defining kurtosis as a measure of "peakedness." Can anyone give a logical rationale for this characterization in terms of distributions of asset returns?

## Answer by BigBendRegion (score 5)

https://quant.stackexchange.com/a/45344

Since there are upvotes on Alex C's answer, it seems that people on this site are inclined to believe it. It is a bad idea to let bad memes go unchecked. Science, after all, should be self-correcting. So here is the mathematical logic correcting Alex C's answer.

Alex C's comment:

So "peakedness in the centre" and "fat in the tails" describe exactly the same thing. You can't have one without the other.

This statement is wrong. Take the beta(.5,1) distribution. It is infinitely peaked yet lighter tailed than the normal distribution.

Now consider a very fat tailed distribution, a mixture involving the Cauchy (an extremely fat tailed distribution). Take a U(0,1) distribution, and mix it with the Cauchy, with .00001 mixing probability on the Cauchy. The distribution appears perfectly flat over 99.999% of the potentially observable data, yet has extremely fat tails. These two counterexamples provide ample evidence that the statement "peak and tails go hand in hand" is simply wrong.

But they are not the only counterexamples; there are infinitely many. Another counterexample is given here: math.stackexchange.com/a/2510884/472987 . Within that family of distributions (1) the probability in the center stays constant (.5) as kurtosis increases to infinity, (2) the distributions become more flat-topped as kurtosis increases to infinity, and (3) the probability in the tails beyond two standard deviations from the mean decreases as kurtosis increases to infinity.

Yet another counterexample, one that completely destroys the "peak and tails are related" thought, is given in the paper "Kurtosis inversely related to peakedness," here: https://share.google/bChjShGxN0v40KW4r. That paper describes a family of distributions where a higher peak causes thinner tails, and a lower, flatter peak causes heavier tails.

A common thread of the incorrect peaked/tails connection is the incorrect notion that probability in the tails increases with higher kurtosis; the counterexample shows that this is not true: It is not the probability in the tails that increases; it is the extension of the tails.

Here is a better way to understand it: Find the distribution (theoretical or empirical) of the standardized values, each taken to the fourth power (the $z^4$-values). Graph this distribution. Place a fulcrum at 3.0, the kurtosis of the normal distribution. If the distribution of $z^4$ "falls to the right," then the return (or whatever) distribution has higher kurtosis than the normal distribution. Now, does it fall to the right because of the tails ($z^4$ values greater than 3) or because of the "peak" ($z^4$ values less than 3.0)? The answer is obviously, "tails". That particular visualization also explains why it is increased tail extension rather than increased tail probability that causes higher kurtosis. It is a question of leverage, not mass. You can move a large object with little mass on the lever if the lever extends far enough.

I am surprised that no one on this site has mentioned the following financial scenario that leads to higher kurtosis: Include a substantial amount of fixed return assets in your portfolio. What does this do? It increases the "peakedness" of the portfolio return distirbution, and also increases the kurtosis. So there you have it: More peakedness implies more kurtosis.

All fine and good. However, the question is about what does higher kurtosis indicate, not what a higher peak indicates. Just because higher peak implies greater kurtosis does not imply that greater kurtosis implies a higher peak. (A perfect analogy: Just because all bears are mammals does not imply that all mammals are bears.) Here is a simple finance counterexample: Take the same portfolio, and spike it with a very small percentage of lottery tickets of various types. The small percentage means that the peak will look the same. But the occasional lottery wins (and even possible multiple wins) increase the kurtosis greatly. So in this case larger kurtosis corresponds with virtually no change in the shape of the peak of the portfolio return distribution, whether flat, U-shaped, wavy, or bell-shaped.

So isn't it time to revise your thinking in regard to kurtosis? Especially as regards the CFA and other test preps - it seems criminal to force students to regurgitate things that are simply wrong.

## Answer by Alex C (score 3)

https://quant.stackexchange.com/a/45217

Quailtatively a (zero skewness) Leptokurtic distribution, after being standardized to have zero mean and unit variance shows three features when you plot the density and compare it to a standard normal N(0,1) distribution: higher peak, higher (fatter) tails, and lower mid-range(*). All three properties go together, even though people sometimes mention only one of them ("a fat tailed distribution"). After all there has to be an area of 1 under the curve, and a variance of 1, so a deficit in the mid-range has to be made up by an excess elsewhere, i.e. in the tails and near the centre. Otherwise it would not be a probability distribution or would not have unit variance. The peakedness in the centre "balances" the thickness in the tails while staying with a unit variance.

So "peakedness in the centre" and "fat in the tails" describe exactly the same thing. You can't have one without the other.

(*) By mid-range is meant the two areas, one on each side, located approximately one standard deviation from the centre.

## Answer by Dave Harris (score 1)

https://quant.stackexchange.com/a/45358

Let me begin with the fact that economists and those in finance have been struggling with the source of kurtosis since Mandelbrot published “On the Variation of Certain Speculative Prices,” in 1963. There was a serious attempt to properly found finance on solid mathematical ground in the 1960s, but it failed. I believe it failed for three reasons and this is the source of the error you observe.

First, if you read Markowitz’s 1952 paper on what would become mean-variance finance, it was non-rigorous in the extreme. The paper has to explain to the field what variance is. So that is the starting point for the field. Mandelbrot’s paper attempted two things. The first was to construct a “rules of the road” and second to say “if that is your theory, then this cannot be your data, and this is your data.”

The second problem is that economists only solve inverse problems. When an economist sees someone buying an orange, they view that as a solution to someone’s problem. The economist, in a generalized way, is attempting to reverse engineer why an orange is a solution to a problem as opposed to an apple, or a trip to Disneyworld. Mandelbrot’s efforts would have stranded economics in a world where $d=\frac{1}{2}gt^2+at+c$ which follows from first principles deductively. The economist knows $d$ and $c$ and cannot know the interim path. Mandelbrot’s efforts would have been great if economists already knew how everything worked. They were useless for inverse problems.

On the other hand, the mean-variance problems were using rules that resulted in an understandable system. If it was wrong but close enough, then why worry? Of course, the mountain of papers on anomalies shows it isn’t close enough. Mandelbrot tried to help economists as if they were physicists or chemists with controlled experiments. It would be great if they could solve the direct case rather than the indirect case.

The third reason was that if Mandelbrot were correct, the math would have had to have been solved with punchcard computers and that was infeasible. The distributions involved lack sufficient point statistics. Everything wonderful with Pearson and Neyman’s method become challenging, fast, without a mean or a variance to work with in 1965.

So that brings us to kurtosis and your observation about myths. I am a social scientist with degrees in three academic fields and who has performed research in two of them. I have been tracking down the mathematical assumptions for the principal models going back to Cardano, Fermat and Maria Agnesi on the math side; and, Regnault and Edgeworth on the economics side regarding finance.

The source of the kurtosis is that returns are $$r_t=\frac{p_{t+1}}{p_t}\times\frac{q_{t+1}}{q_t}-1.$$ Because of the math used, the parameters are assumed to be known, but if you drop that assumption, then what you get is the product of two ratio distributions. Because $q$ can change in the event of a stock-for-stock merger, become 0 in bankruptcy, be converted to cash in a cash-for-stock merger, be multiplied in a split or stock dividend, or simply remain the same the distribution is a rather ugly mixture distribution. Furthermore, because there is a 100% chance that someone would accept 100 shares of IBM for 0 dollars per share and a zero percent chance of them being willing to pay an infinite amount there is skew in the distribution due to the intertemporal budget constraint.

You can see this in the distribution of daily rewards minus dividends for Carnival Cruise Line stock. The distribution used was $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]^{-1}\frac{\sigma}{\sigma^2+(r_t-\mu)^2}.$$

The field doesn’t accept this as an explanation. Because of this, it uses mechanisms that don’t work in practice when viewed out-of-sample. This leads to myths. I have been tracking them down for several years now. I hadn’t heard that one before, but I thought maybe there is a different way you could think about it.

It is true that describing kurtosis with respect to peakedness is incorrect and generally unhelpful, but within the narrow context of finance, it is not “wrong.” It is better to have a rigorous understanding of the math, but finance isn’t going to run into a uniform distribution mixed with a Cauchy distribution as a rule of thumb.

To be honest, most of the component distributions lack a defined kurtosis, so it still isn’t a good way to talk about it. Your question is in error. With respect to finance, the distributions have no kurtosis as they have no defined moments other than the trivial ones. The larger issue is talking of scale parameters as if they were "volatility," particularly when thought about as $s^2_{\Delta{t}}$.

I'll be proposing a new stochastic calculus to price options shortly that should account for this, soon.