Kurtotic Stock Distribution - Additional Details

• Can you show us an example of a stock with a leptokurtic distribution?

The most leptokurtic stock in the US-listed stocks we analyzed on that day (best out of 3,020 stocks) is PNM or New Mexico's public utility. It has a kurtosis score of 32.0 (vs. 3 being a normal kurtosis score), and a variance that is 6.6 x 10-6 (which is also very low). This stock has risen significantly over the past few years, and consistently shows well in our model.

Here is a discussion of a leptokurtic stock with real data (Source: Yahoo Finance on Sept 15, 2023).

We post charts to show this.
Chart 1: Price over the measurement period of one year
Chart 2: Time-series of price changes per day
Chart 3: Frequency Distribution (this is the bell shaped curve with the narrow body we were looking for).
Chart 4: Summary frequency data

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The initial chart looks pretty normal, if not pretty risky. Maybe a bad earnings call, but then the stock tried to close the gap and failed, falling further.

The time-series of prices shows the few days when the stock moved more than 1.5% in a day (which is not a big move for a US stock) out of 250 price change measurements. You can see the two moves that blew out the chart. Even after those big moves, the stock fell back into the pattern of small daily price moves.

The third chart shows how the stock rarely moved more than 1%, only doing this 8 times out of 250 days. The variance of this stock is low, which allows us to see the kurtotic nature very clearly. Most moves are within 0.5%, either up or down.

The final chart shows the raw data, so you can recreate the chart. We did divide the six days of no change into three up and three down less than 0.5%.

The idea of a leptokurtic and low variance stock is that it allows the investor to rest easy most days, but there is that real, and non-normal 'aggressive' probability of a big move on occasion. In this case, the stock moved less than 1% almost every day (242 out of 250 days), but on one day it fell dramatically.

Investor Expectations & Possible Trading Implications

Investors may expect a normal distribution of stock prices.  Stocks go up and down seemingly randomly along with expectedly random new information. However, with a stock with a leptokurtic distribution, the investor sees typical observations closer to the mean. There is also more of a chance that a future observation will be an outlier. When plotted, the peak is taller and narrower (typical observations closer to the mean) and the tails rise up from the x-axis (more frequent outliers).

This provides information for a unique trading opportunity in stocks and stock options.

​This way, these stocks will typically have smaller movements, and lower implied volatility, which reduces the cost of stock options, but still have the ability to surprise with a large standard deviation move. We believe the combination of leptokurtic and low variance gives an options trader a chance to invest less money buying an option, predict a direction of surprise, and earn profits as the stock eventually makes its big more.

At least, this is our hypothesis on how it should work. No trading strategy is fool-proof. We have felt foolish on a few of our leptokurtic options bets where the options seemed cheap, but we bet on a move in the wrong direction.

This church represents a leptokurtic distribution. The peak is high and narrow, and the tails are wide and sloping.

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Platykurtic Stock Details

An example of a stock with a platykurtic stock price distribution:

​If you can visualize it, the distribution is flattening the bell-shaped curve (or frequency distribution) down onto the X-axis, giving less room in the tails for observations greater than 2 standard deviations from the mean. Put another way, a fully platykurtic stock distribution has no tails, no outliers, no surprises. In some cases, the tails seem longer, but there are fewer outliers in the expected observations. Another visualization is that the center of the bell is stretched out to be fatter and shorter than a bell shaped curve, which seems to spread out the observations around the mean.

A stock that is platykurtic and low variance are safer when compared to a normally distributed stock.  Options premia may be higher than fair value since it prices in a normal occurance of outliers. This is actually a safer stock in highly volatile market situations. We see energy services companies, healthcare, financial services and industrial companies that are not economically sensitive, nor do they seem to be subject to normal business cycles in this category.
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Stocks that are platykurtic, or that have a platykurtic distribution with a standardized kurtosis score </= a parameter and a variance that is </= the variance of the total population in this run (multiplied by a parameter ~ 1.0). In other words, stocks that are platykurtic with a low variance.

A platykurtic distribution of daily stock price changes have fewer results in the tails (or > 2 standard deviations) than a normal distribution. They have a flatter frequency distribution than a normal distribution, so their typical results are spread across the high potential scores and are not as frequently returning the expected return, or mean.

A perfect example of a platykurtic distribution is the results of rolling a fair, 6-sided dice, or flipping a fair coin. There are no outliers because there are few, knowable scores (discrete scores), and no possible outliers. These are a uniform distribution where each value of a fair dice roll has a 1/6 probability.

For stocks, this could be returns that are in the typical range (say within one standard deviation), and fewer surprises.

Investor Expectations & Possible Trading Implications
Investors may expect a normal distribution of stock prices.  Stocks go up and down seemingly randomly along with expectedly random new information. However, with a stock with a platykurtic distribution, the investor sees typical observations grouped into a typical range, although that range may be slightly wider than a normal distribution. There is also a smaller chance that a future observation will be an outlier. When plotted, the peak is shorter and wider (typical observations spread more widely around the mean) and the tails are pushed flat into the x-axis (less frequent outliers).

This provides information for a unique trading opportunity in stocks and stock options.

​These stocks will typically have a wider range of daily price changes, and potentially higher actual and implied volatility, which increases the cost of stock options. However, it will be less likely to surprise with a large standard deviation move. 

We believe the combination of platykurtic and low variance gives an options trader a chance to sell options and have less chance of making a big payout from any one daily move.

At least, this is our hypothesis on how it should work. No trading strategy is fool-proof, and this discussion does not take into account fundamental performance of the company or the markets...only the distribution of 1 year of daily price changes. Do your own due diligence.

This mushroom represents a platykurtic distribution. The peak is low and wide, and the tails are non-existent.

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