# Kurtosis in asset logarithmic returns

Assets such as stocks usually display kurtosis in their logarithmic returns. However, their logarithmic returns in a time interval $$n$$ are the sum of smaller logarithmic returns in $$1/n$$ time intervals. In other words, their return distribution can always be decomposed as the sum of many other distributions.

Asset return time series, under the Efficient Market Hypothesis, are martingales: they possess no autocorrelation.

By the Central Limit Theorem, return distributions of such assets should be normal. Nonetheless, they exhibit kurtosis, which is absent in a normal-distributed variate since the gaussian distribution is zero-valued in all moments beyond the second. Indeed, the presence of kurtosis has been famously pointed as a flaw in Black-Scholes.

The question is: why is there kurtosis in asset returns?

• " gaussian distribution is zero-valued in all moments beyond the second" is not true. Look here for central and non-central moments: en.wikipedia.org/wiki/Normal_distribution
– Ric
Sep 24 '15 at 8:30

Generally Kurtosis measures the degree to which a distribution is more or less peaked than a normal distribution.

• Positive kurtosis indicates a relatively peaked distribution.
• Negative kurtosis indicates a relatively flat distribution.

In time series we can encounter high kurtosis which is caused by "fat tails" (higher frequencies of outcomes) at the extreme negative and positive ends of the distribution curve.
They are usually caused by ** a very strong price reaction** to a postive or negative information(bankruptcy, war, ceo resignation).

A distribution of returns exhibiting high kurtosis tends to overestimate the probability of achieving the mean return.

And why Kurtosis exists in log time series?
It exists because many researchers claim that we cannot assume that market is efficient. They lean towards Fractal Market Hypothesis which explains a investor behavior throughout a market cycle, including booms and busts (which cause the "fat tails" in the data series).
Secondly, FMH says that market can have show some dependiences hidden in the time series but they are unstable and often small change in initial conditions of a discovered model can lead to big changes in a forecast. To sum up the deterministic nature of these systems does not make them predictable (Chaos Theory).
If you want to learn more about the chaos in the financial markets I can recommend you a book by Edgar E. Peters "Fractal Market Analysis: Applying Chaos Theory to Investment and Economics"

Perhaps an answer coming from a different angle and giving you some perspective: The typical approach taken by statistics is top-down: Just looking at the data and finding patterns and stylized facts (like excess volatility, volatility clustering, fat tails, no autocorrelation in returns but significant autocorrelation in absolute returns etc.) The problem is that you have lots of simplifying assumptions (e.g. iid-ness, CLTs etc.) which make the results hard to explain (in your case "why kurtosis?"...)

Another approach is to have a look at the underlying data generating process (traders interacting in markets!), which would be bottom-up. One very fascinating area of research are multi-agent simulations. Even when you create a comparatively simple artificial market with only two types of traders (i.e. technical traders = basically trend-followers and fundamental traders = basically mean-reversion traders) you will be able to replicate all (!) of the abovementioned stylized facts!

One way to go would be to really get into the micro-structure of this simplified model and see where and why kurtosis is created (I have no good answer for you at the moment, but when we are not even able to understand such very basic models I guess there is no hope for understanding real markets).

The paper that is mentioned there can be replicated quite easily (which I have done) - and then you can start your own experiments! It is well worth the effort...

I think there are a few conflating ideas here.

1. With respect to the sum of logs idea, I think you're thinking about infinitely divisible distributions (https://en.wikipedia.org/wiki/Infinite_divisibility_(probability)). These ideas are indeed used to build more complicated models (i.e. Levy processes) for asset returns.

2. With regards to the Efficient Market Hypothesis, I think most people think of them as asset prices --- under an appropriate measure change --- are martingales (i.e. past history information cannot be used to predict tomorrow's asset price), and not returns.

3. I'm not why you'd think the central limit theorem here would necessary hold. While there are tons of CLT's out there, having a random variable of the form $S_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - E X_i)$ does not necessarily mean that as $n \to \infty$, $S_n$ will converge (in distribution, or some other strong sense of convergence) to a Gaussian random variable. Next, the absence of kurtosis in the Black-Scholes is not surprising, especially since the asset prices there follow a geometric Brownian motion (so effectively, each small time interval is Gaussian). But if one replaces the GBM asset price assumption to something else (i.e. Levy models and/or the current literature favorite flavor of Ito semimartingales), then one will definitely generate kurtosis and other higher moment properties. The downside, of course, is that more sophisticated asset pricing models inevitably implies it'll be a challenge to model derivative prices, and/or econometric inference on the underlying parameters.

But going back to the fundamental question of --- "why is there kurtosis in asset prices / returns" --- this I believe, is at heart a question that's too broad to answer. From a statistical perspective, the response is simply: "it is what it is" and people just build models that build kurtosis into it. From an economics and finance perspective, there are simply too many different theories out there; perhaps the representative agent cares about disaster risks in consumption, or that he has a preference that's state dependent and such macro state variable has some higher order effects. Really, who knows.

At what scale do you see kurtosis? Daily data? Single stocks or indices?

Let us not look at single stock data, because you always find crazy stocks whose price process breaks all rules.

Talking about daily data of indices: they could be thought of the sum of hourly returns or other returns of high frequency (minute returns, milliseconds ...). What are the assumptions of "the" (or most other forms of) Central Limits theorem? You need independent summands.

They might be uncorrelated, but this does not imply that they are independent. If you look at $$r_t^2$$ - the square of returns - then they are highly correlated (volatility clustering). This implies that returns themselves are not independent.

Finally no reason for the CTL to hold true. However what we find is "aggreational Gaussianity" meaning that if you go to lower frequencies you get more and more Gaussian looking returns. In my experience this is only true for weekly to monthly returns.

• Well, to be fair, there are tons of CLT's out there that require something much less of independent summand increments (i.e. Lindeberg-Levy CLT)? But then I'd be too picky :P Sep 24 '15 at 7:13
• This is the first hit on wikipedia and there ii says that the sequence is iid ... right? "iid" stands for "indepdendent ...". :P
– Ric
Sep 24 '15 at 7:15
• Oh my bad. Maybe you do need to maintain some weak form of independence (say Lyapunov needs only a triangular sense of iid). You could go to ergodic theory and get something more there, I suppose? But I digress... Sep 24 '15 at 7:19
• Say this will also work too: en.wikipedia.org/wiki/Martingale_central_limit_theorem Sep 24 '15 at 7:21

The subordinate return process for log returns is normal (or Gaussian).

The kurtosis stems from the "activity rate" of events that move asset prices. When we measure in "clock time" we see kurtosis. However, when we measure in "event times" or "business times" the distribution is normal. The "event time" is a subordinator.

Substitute "event time" for "clock time" and the subordinate process based on the time change removes the kurtosis. For a literature review and applications see:

• But how can we measure, let alone predict, this “alternative” time? Sep 3 '19 at 7:01

We know from psychology about “prospect theory”, that a human’s reaction to gain and loss is wildly asymmetric, and powerful enough to change our behaviour when the same payoff structure is presented differently. Good news is not equivalent to bad news; so why should asset returns be any different.

Nor is it unreasonable to suppose that a violation of the assumption of costless and frictionless hedging produces a bias towards frequent near-zero and rare extreme outcomes. If you will, there is some “grey noise” on top of the dominant Gaussian white noise.

None of this would be controversial to the vast majority of market participants. But they’re a different crowd to an academic statistician!

However, here’s a simple mechanism for you. You only have to accept the existence of “greed” and “fear” as distinct emotional states that influence human behaviour in markets.

Then take a long-run series of eg equity market returns, as the archetypal risk asset. Run a two-state Hidden Markov Model on these returns. The outputs will suggest one state with, say, a 70% probability of mean +3% a month with a Gaussian stdev of 4%. Versus a rarer 30% frequency of -5% +/-8%. The analogy with “greed” and “fear” or “calm” and “chaos” are hopefully obvious.

These in aggregate will have a mu and sigma of 7% and 19% per annum respectively, in line with the market’s historical record. Both components are perfectly Gaussian. However, the combination of the two will have left-skew and positive kurtosis, exactly like market history.

You simply have to believe there is more than one “regime” that can drive markets at any point in time. Accept that, and the higher moments will naturally follow.