## Leptokurtic distribution with low variance

through August 12, 2022

**Leptokurtic stocks with low variance:**

**Positive Net Income & Positive Cash Flow from Operations**

The stocks that are leptokurtic and low variance:

This is the minimum Leptokurtic score and variance of all stocks: 5.25 0.0001925

['ACC', 'AMGN', 'EBF', 'GIS', 'HRL', 'HSY', 'KO', 'MDLZ', 'PNM', 'VZ', 'WSBF'] 11

The stocks that are leptokurtic and up to 30% higher variance:

This is the minimum Leptokurtic score and variance of all stocks: 5.25 0.0002502

['ACC', 'AIZ', 'AMGN', 'BF.A', 'BF.B', 'CAG', 'CHD', 'CLBK', 'CPB', 'DGICA', 'DX', 'EBF', 'FCN', 'GIS', 'HRL', 'HSY', 'IBM', 'INGR', 'JJSF', 'KO', 'MDLZ', 'MDU', 'MGRC', 'MKC', 'MRK', 'PNM', 'PNW', 'SJM', 'UVV', 'VZ', 'WMT', 'WSBF'] 32

**Negative Net Income**

The stocks that are leptokurtic and low variance:

This is the minimum Leptokurtic score and variance of all stocks: 5.25 0.0001860

['AMGN', 'EBF', 'GIS', 'HRL', 'HSY', 'KO', 'MDLZ', 'NECB', 'PCSB', 'PNM', 'VZ', 'WSBF'] 12

The stocks that are leptokurtic and up to 30% higher variance:

This is the minimum Leptokurtic score and variance of all stocks: 5.25 0.0002418

['ACNB', 'AIZ', 'AMGN', 'ARTNA', 'CAG', 'CHD', 'CLBK', 'CPB', 'DGICA', 'DX', 'EBF', 'FCN', 'GIS', 'HRL', 'HSY', 'IBM', 'INGR', 'KO', 'KTCC', 'MDLZ', 'MDU', 'MKC', 'NECB', 'ORRF', 'PCSB', 'PNM', 'PNW', 'SJM', 'UVV', 'VZ', 'WSBF'] 31

This is the minimum Leptokurtic score and variance of all stocks: 5.25 0.0001860

['AMGN', 'EBF', 'GIS', 'HRL', 'HSY', 'KO', 'MDLZ', 'NECB', 'PCSB', 'PNM', 'VZ', 'WSBF'] 12

The stocks that are leptokurtic and up to 30% higher variance:

This is the minimum Leptokurtic score and variance of all stocks: 5.25 0.0002418

['ACNB', 'AIZ', 'AMGN', 'ARTNA', 'CAG', 'CHD', 'CLBK', 'CPB', 'DGICA', 'DX', 'EBF', 'FCN', 'GIS', 'HRL', 'HSY', 'IBM', 'INGR', 'KO', 'KTCC', 'MDLZ', 'MDU', 'MKC', 'NECB', 'ORRF', 'PCSB', 'PNM', 'PNW', 'SJM', 'UVV', 'VZ', 'WSBF'] 31

## Overview of Kurtosis and Variance

Kurtosis is the fourth central movement of a distribution. It is one way to measure whether a random variable has a normal distribution. We use it to better understand expected (or future) stock price changes.

Normally distributed random variables can be thought of as having a bell shaped curve that visually describes the percentage probability of having future readings within a certain number of standard deviations from the mean. The variance defines the standard deviation, which is the width or spread of the distribution. A normally distributed random variable is very well behaved around those standard deviations and has a non-standardized kurtosis score of 3. Kurtosis indicates whether the distribution has a different (fatter & flatter or thinner & taller) shape of the curve.

There are four centralized moments in a random distribution of price changes (e.g., stock price changes tomorrow).

1. The mean, which is the central tendency, or average daily price change.

2. The variance, which is the width or size of the standard deviation.

3. The skewness, which indicates the symmetry of the randomized points around the mean. If there is a skew to the data, the bell appears to be fatter (or longer) on one side.

4. The kurtosis, which is the degree that randomized points cluster around the mean.

We use the kurtosis of a year's worth of daily stock price changes, along with the variance of those changes, to better understand how a stock might perform in the future.

What is a leptokurtic distribution stock with low variance?

A leptokurtic distribution has fewer expected observations far away from the mean. It is more centralized within fewer standard deviations. It has a tighter spread within the standard deviations. We expect to see future observations closer to the mean. It visually looks like the bell shaped curve has thinner tails and taller peaks. By example, an infinitely leptokurtic distribution would only yield the mean, which is like flipping a one-sided coin.

A stock that is leptokurtic and low variance appears safer with fewer and smaller price changes when compared to the mean price change. It appears safer, and so options premia may be smaller and this might be a 'flight to safety' stock in highly volatile market situations. We see utility stocks in this category.

A leptokurtic distribution has fewer expected observations far away from the mean. It is more centralized within fewer standard deviations. It has a tighter spread within the standard deviations. We expect to see future observations closer to the mean. It visually looks like the bell shaped curve has thinner tails and taller peaks. By example, an infinitely leptokurtic distribution would only yield the mean, which is like flipping a one-sided coin.

We define leptokurtic stocks as those with a non-standardized kurtosis score of 4.5 or greater. We define low variance stocks as those with a variance less than a multiple of all stocks analyzed that day, in that run, held together evenly, in a highly diversified portfolio. The multiple is often less than one. Our list of leptokurtic stocks have a non-standardized kurtosis of 4.5 or greater and lower variance than a fully diversified portfolio of stocks in that run.

What is a platykurtic distribution stock with high variance?

A stock that is platykurtic and high variance is more wild and seemingly more random than a normal distribution. It has observations that are located more standard deviations away from the mean. It has more volatility and more risk than a normal distribution. We expect to see future observations further away from the mean, or evenly spaced away from the mean. It visually looks like the bell shaped curve has fatter tails and a shorter peak. By example, an infinitely platykurtic distribution would look like a table-top with random observations anywhere on the range of expected observations. It is like flipping an infinitely sided coin.

We define platykurtic stocks as those with a non-standardized kurtosis score of 1.5 or less. We define high variance stocks as those with a variance greater than a multiple of all stocks analyzed that day, in that run, held together evenly, in a highly diversified portfolio. The multiple is greater than one. Our list of platykurtic stocks have a non-standardized kurtosis of 1.5 or less and higher variance than a fully diversified portfolio of stocks in that run.

When we look at the behavior of platykurtic and high variance stocks, we see risky stocks such as growth stocks that are tied to commodity prices (e.g., crypto miners and natural gas exporters). These stocks may not be in demand in high-risk market scenarios, and may be a welcome source of variance in a calm market.

This analysis of stock behavior is based on original research performed by US Advanced Computing Infrastructure, Inc. We have traded on this information profitably in 2021 and have been providing this data to clients for over a year. It is up to you to decide whether these stocks will provide you with opportunities in your portfolio, either for the common stock or options.

Here are pictures that represent leptokurtic distributions (on the left) and platykurtic distributions (on the right). GLTA.

**NEW CLIENTS WELCOME**