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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?

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  • $\begingroup$ what is the "evidence to the contrary" ? $\endgroup$ – Alex C Apr 20 at 17:03
  • $\begingroup$ Not trying to be a jerk here, but "Peakedness," "Tailedness" who cares? Is the characterization even important? For any metric, understanding use and interpretation are what matters. $\endgroup$ – amdopt Apr 20 at 21:17
  • $\begingroup$ Downvoted for not citing this evidence. $\endgroup$ – Bob Jansen Apr 21 at 9:12
  • $\begingroup$ Wikipedia and sources therein. $\endgroup$ – Peter Westfall Apr 21 at 13:43
  • $\begingroup$ Who cares? I would think any financial analyst who cares about risk would want to know how "higher kurtosis" implies "higher risk." So yes, understanding use and interpretation are what matters. "Peakedness" is not only wrong as a characterization of kurtosis, it also impedes understanding of "kurtosis risk." $\endgroup$ – Peter Westfall May 8 at 11:30
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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.

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  • $\begingroup$ I understand where you are coming from but I think this is fruitless. Distributions are functions, i.e. they have infinite degrees of freedom. Were your statement still true if the distribution is multimodal or had point masses? $\endgroup$ – g g Apr 20 at 18:40
  • $\begingroup$ Fat tails are a problem because so many of the models we use assume that the data are Gaussian IID. Fat tails means an increased probablility of events that we considered rare. For example the Black Scholes option pricing model assume your data are Gaussian. If they have a high kurtosis, this problematic. $\endgroup$ – Jacques Joubert Apr 20 at 22:14
  • $\begingroup$ These comments are simply not true. It has been known since Kaplansky (1945) that higher peak does not correspond with higher kurtosis. Also, distributions such as beta(.5,1) have infinite peaks but negative excess kurtosis. (More support for my "evidence to the contrary" statement which for some reason got me downvoted.) Also, if you take a uniform(0,1) and mix it with a N(0,1000000), with .0001 probability on the latter, you get a distribution that is perfectly flat-topped over 99.99% of the observable data but has very high kurtosis. So the argument connecting tail to peak is not logical. $\endgroup$ – Peter Westfall Apr 21 at 13:48
  • $\begingroup$ But my question is not about the math, since that is already settled. My question is whether there is a finance rationale for the "peakedness" characterization, say in terms of return distributions of assets or portfolios. $\endgroup$ – Peter Westfall Apr 21 at 14:22
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    $\begingroup$ are you simply looking to argue semantics? equity returns are pretty widely held to display excess kurtosis, with a peak around the mean and fatter than normal tails. peaked-ness is often used as a stand-in description for this, though references to fat tails are also made. $\endgroup$ – Chris Apr 22 at 7:37
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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 hane 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.

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.

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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 $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]^{-1}\frac{\sigma}{\sigma^2+(r_t-\mu)^2}.$$

empirical_cauchy

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.

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  • $\begingroup$ Here is the defined kurtosis: Using the standard Pearson defintion, kurtosis = $E\{(R - \mu_R)/\sigma_R\}^4$. It's not clear what you mean by not defined, and also what you mean by my "question is in error." $\endgroup$ – Peter Westfall Apr 28 at 20:54
  • $\begingroup$ @PeterWestfall I know the definition of kurtosis. Apply the formula to the above density, you will find that the integral diverges. Finance is made up of ratio distributions. Under mild circumstances, the distribution will either be the ratio of normals or Gumbels. You are in error as there can be no question of kurtosis with respect, at least, to equity securities any more than you can have questions regarding noses with respect to trees. $\endgroup$ – Dave Harris Apr 28 at 21:14
  • $\begingroup$ Ok, fine for infinite Pearson kurtosis, substitute a quantile variant. And let my question be universally applicable to sample data, which is necessarily bounded and always has defined kurtosis (Expectation operator then applies to empirical distribution, which has all moments finite.) Undefined kurtosis is a red herring argument. $\endgroup$ – Peter Westfall Apr 28 at 21:17
  • $\begingroup$ @PeterWestfall With the exception of single period discount bonds, life annuities where the payments are consumed and a few interesting side cases, the distribution of returns is always some deformation of the Cauchy distribution. Instead of the kurtosis being undefined, one could speak of it as infinite, but I don't like that way of talking about the missing moments. I think it confuses the discussion with most people. $\endgroup$ – Dave Harris Apr 28 at 21:17
  • $\begingroup$ @PeterWestfall I am a bit busy today, but I will try to include the broader conceptualization of kurtosis in an edit. $\endgroup$ – Dave Harris Apr 28 at 21:19

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