In finance, a leptokurtic distribution shows that the investment returns may be prone to extreme values on either side. Therefore, an investment whose returns follow a leptokurtic distribution is considered to be risky. Excess kurtosis is an important tool in finance and, more specifically, in risk management. With excess kurtosis, any event in question is prone to extreme outcomes. It is an important consideration to take when examining historical returns from a particular stock or portfolio.

A positively skewed kurtosis is indicated by the term leptokurtic. It is characterized by huge tails on either side with large outliers. For investors, this could mean that the result would be an extreme of positive or negative. Thus, this graph could indicate a risky pattern for investors to make investment on either side of the distribution. The term excess kurtosis refers to a metric used in statistics and probability theory comparing the kurtosis coefficient with that of a normal distribution. Kurtosis is a statistical measure that is used to describe the size of the tails on a distribution.

1. For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see variance.
2. In this statistical tool there is sometimes a case of confusion when distribution peak occurs.
3. And is commonly denoted
   (Abramowitz and Stegun 1972, p. 928) or .
4. Distributions with low kurtosis have fewer tail data, which appears to push the tails of the bell curve away from the mean.
5. Recall that an indicator random variable is one that just takes the values 0 and 1.
6. In each case, note the shape of the probability density function in relation to the calculated moment results.

A stock with a leptokurtic distribution generally depicts a high level of risk but the possibility of higher returns because the stock has typically demonstrated large price movements. The frequency distribution (shown by the gray bars) doesn’t follow a normal distribution (shown by the dotted green curve). Instead, it approximately follows a uniform distribution (shown by the purple curve). A kurtosis greater than three will indicate Positive Kurtosis.

This is really the excess kurtosis, but most software packages just call it kurtosis. In addition to this, the discrete probability distribution from a single flip of a coin is platykurtic. Kurtosis measures how much of the data in a probability distribution are centered around the middle (mean) vs. the tails. Skewness instead measures the relative symmetry of a distribution around the mean. The Sharpe ratio is used by investors to better understand whether the level of returns they are receiving are commiserate with the level of risk incurred. While kurtosis analyzes the distribution of a dataset, the Sharpe ratio more commonly is used to evaluate investment performance.

Graphical examples

Excess kurtosis helps determine how much risk is involved in a specific investment. It signals that the probability of obtaining an extreme outcome or value from the event in question is higher than would be found in a probabilistically normal distribution of outcomes. For many distributions encountered in practice, a positive corresponds to a sharper peak with higher tails than
if the distribution were normal (Kenney and Keeping 1951, p. 54). This observation
is likely the reason kurtosis excess was historically
(but incorrectly) regarded as a measure of the “peakedness” of a distribution. Kurtosis is a measure of the combined weight of a distribution’s tails relative to the center of the distribution curve (the mean). However, when high kurtosis is present, the tails extend farther than the three standard deviations of the normal bell-curved distribution.

Kurtosis

A normal distribution is a continuous probability distribution for a random variable. A random variable is a variable whose value depends on the outcome of a random event. For example, flipping a coin will give you either heads or tails at random.

What Is Kurtosis?

The tails of ranges with low kurtosis are often less severe than the tails of a normal circulation. Kurtosis is a statistical measure which defines how the tails of your data distribution differ from the tails of a normal distribution. High kurtosis indicates you have more outliers and have longer tails than a normal distribution. Kurtosis, in statistics, a measure of how much of a variable distribution can be found in the tails. The term kurtosis is derived from kurtos (Greek for “convex” or “humpbacked”). A prevalent misconception is that kurtosis measures the “peakedness” of a distribution; however, the contribution of a central peak or range to kurtosis is often small.

You also take a look at how different values of skewness and kurtosis affect the distribution. Since kurtosis is defined in terms of an even power of the standard score, it’s invariant under linear transformations. Prior to doing some statistical analysis on a manufacturing process, the company Six Sigma Black Belt (BB) tested her data for normality. Below is her result showing the current process data has a significant degree of kurtosis. You can see that with the long tails and a kurtosis value of over 4. If you use the above equation, the kurtosis for a normal distribution is 3.

The kurtosis of a sample is an estimate of the kurtosis of the population. A trick to https://1investing.in/ remember the meaning of “platykurtic” is to think of a platypus with a thin tail.

Indicator Variables

Run the simulation 1000 times and compare the empirical density function to the probability density function. The most frequently occurring type of data and probability distribution is the normal distribution. However, under the influence of significant causes, the normal distribution too can get distorted. This distortion can be calculated using skewness and kurtosis. In this tutorial titled ‘The Simplified and Complete Guide to Skewness and Kurtosis’, you will be exploring some of the different types of distortion that can occur in a normal curve.

Alpha measures excess return relative to a benchmark index. While kurtosis measures the nature of the peak or flatness of the distribution, alpha measures the skewness or asymmetry of the distribution. It might seem natural to calculate a sample’s kurtosis as the fourth moment of the sample divided by its standard deviation to the fourth power. In this statistical tool there is sometimes a case of confusion when distribution peak occurs.

Kurtosis measures how fat a distribution’s tail is when compared to the center of the distribution. The tails of a distribution measure the number of events that occurred outside of the normal range. Unlike skewness, kurtosis measures either tail’s extreme values.

Rather, it means the distribution produces fewer and/or less extreme outliers than the normal distribution. An example of a platykurtic distribution is the uniform distribution, which does not produce outliers. Distributions with a positive excess kurtosis are said to be leptokurtic. It is common practice to use excess kurtosis, which is defined as Pearson’s kurtosis minus 3, to provide a simple comparison to the normal distribution. Some authors and software packages use “kurtosis” by itself to refer to the excess kurtosis. For clarity and generality, however, this article explicitly indicates where non-excess kurtosis is meant.

The values of excess kurtosis can be either negative or positive. When the value of an excess kurtosis is negative, the distribution is called platykurtic. This kind of distribution has a tail that’s thinner than a normal distribution. Open the special distribution simulator and select the normal distribution.

Since normal distributions have a kurtosis of three, excess kurtosis can be calculated by subtracting kurtosis by three. The types of kurtosis are determined by the excess kurtosis define kurtosis of a particular distribution. The excess kurtosis can take positive or negative values, as well as values close to zero. Kurtosis risk differs from more commonly used measurements.