Distribution Curves and Investing
I recently released an indicator called the Cumulative Distribution Density of Dataset indicator. One of the main highlights of this indicator is its, at the time of writing, the only indicator available on Tradingview/Pinescript that assesses the degree of normality as well as the type of distribution of a ticker, index or economic variable. Before this, you would need to export data into a statistical package such as Excel, SPSS, R or SAS to perform such an analysis. So I figured its probably time to talk about the bell curve again.
Some of you may remember, I released an educational video called “Trading Using Bell Curves”:
In this video, I discuss the implications of using bell-curves for trading.
However, I want to reel it back and talk more specifically about distributions and trading, and why you, the investor and/or trader, should be paying attention to them. This is something I honestly have never seen talked about and really, you are doing yourself a huge disservice as a trader AND an investor for ignoring it. so let’s get into it! But before we start, I won't review the basics of the bell curve, but if you are interested, consider watching the video above.
Alright, now on with the math!
Understanding stock distributions, which come in various forms such as leptokurtic, platykurtic, and more, will provide you with valuable insights into market behavior and risk management. Did you know that certain distribution types can alert you that a stock generally has an unstable trajectory? And by looking at the distributions, you can also tell which stocks are more prone to aggressive crashes and which are more stable?
Well you can, and I am going to teach you how! So let’s go over the main types of distributions in stocks and their implications for you as a trader.
Types of Stock Distributions
Normal Distribution
This is probably the one you hear talked about a lot. A normal distribution, also known as Gaussian distribution or bell curve, is characterized by its symmetrical shape. In a normal distribution, the mean, median, and mode are all equal, and data points are evenly distributed around the central value. This distribution tends to be common in nature but tends to be not all that common in long term stocks. There are some exceptions; however. For example,
BAC (Bank of America) actually has a normally distributed dataset from initial listing to now:
When stock returns follow a normal distribution, it becomes easier to predict future price movements and assess risk more easily. One way to do this is by using the cumulative distribution function (or CDF). Which is a mathematical function that provides the probability that a random variable takes on a value less than or equal to a specific value. For example, if we have 10 students with various test scores, we can plot all test scores using CDF and determine what the probability is that a random student’s test score will be above 90% or below 20%.
We can visualize this onBAC by having the indicator plot the CDF for BAC:
The image above plots the CDF distribution forBAC on the monthly timeframe since its IPO. Because BAC is normally distributed, we can place a high level of confidence in the results of the CDF. We can also use the CDF to our advantage. How? By planning where we could buy.
We should buy when the price is at a level where 50% to 60% or more of the time the price will fall above. Turning back to our BAC example, we can display this with a simple trendline:
We can also operate on the assumption thatBAC is likely to go lower from here. Why? Because the normal distribution is not yet invalidated. As of right now, BAC retains a normal distribution. Thus, we can expect BAC to cycle back down to bring its CDF back towards 50% and 60%. We can see another example below, 
XLE:
Key Points for Tickers that are Normally Distributed:
But what about other distributions? Let’s talk about them.
Leptokurtic Distribution
A leptokurtic distribution is characterized by a higher peak and fatter tails than a normal distribution. In this distribution, extreme events, such as market crashes or rapid price spikes, are more likely to occur compared to a normal distribution. From my experience, most stocks fit this description, but one of major note is
BA:
Leptokurtic distributions indicate higher volatility and a higher likelihood of extreme price movements. In general, you need to be more cautious with leptokurtic distributions because there is generally heightened volatility. A CDF on a leptokurtic distribution is not as clean, as we can see from plotting BA’s CDF:
Because BA’s distribution is not normal, the CDF becomes slightly unreliable and we cannot employ the 60% rule. So can we still use the distribution to help us gauge entries? Yes! We can! However, it’s a bit more nuanced with leptokurtic distributions.
The first thing to remember with leptokurtic distributions is… they crash… a lot. We can see this with BA:
The flags in this chart represent areas BA has crashed. Crashes in leptokurtic distributions are usually characterized by a drop on the CDF of the probability a stock will go lower to around 75% to 85%. We can see this if we overlay the CDF for BA with the chart:
These are the dips you would want to buy in a leptokurtic distribution. If we take a look at another example, AMD:
Key Takeaways from Leptokurtic Tickers:
Platykurtic Distribution
A platykurtic distribution has a flatter and wider shape compared to a normal distribution. In this case, the data points are more spread out, and extreme events are less likely to occur.
As such, platykurtic distributions suggest lower volatility and a more stable market environment. However, it is important to know that prolonged periods of low volatility can be followed by sudden spikes, leading to unexpected market movements.
These are extremely rare distributions that I have not observed in any of the tickers I have traded. However, theoretically, platykurtic distributions would come in smooth waves up and down. We can visualize this if we look at SPY’s January 2022 highs till its October lows. This was a platykurtic, negative distribution (indicating a stable downtrend):
Because platykurtic distributions are cyclical, you long on the bullish peaks when the probability of higher prices is >= 90% and short on the bearish peaks when the probability of downside is >= 90%:
However, this is not at all prevalent or observed in stocks ever, so you would be lucky to find a platykurtic distribution!
Key take aways from Platykurtic distributions:
Skewed Distribution
A skewed distribution is asymmetric, with a longer tail on one side. Positive skewness means the tail is on the right (indicating more extreme positive values), while negative skewness implies a left tail (indicating more extreme negative values).
Skewed distributions can signal a bias in market sentiment. For example, positive skewness may indicate a bullish bias, while negative skewness may suggest a bearish bias. While many people look to EMAs or trendlines to identify long-standing bull or bear markets, its actually not necessary, you can ascertain this simply from the distribution. If we take a look at SPY:
This is SPY’s distribution since the IPO. We can see that it has a positive skewness (right tail), with extreme outliers. This signals to us, the investors, that SPY has been in a bull run since its IPO. Despite multiple corrections and bear markets, SPY retains the distribution characteristic of a bullish stock. In fact, SPY frequently experiences extremely positive outliars (outliars to the upside) more often than extremely negative outliars (crashes to the downside). This is observed with its positive skewedness.
Planning entries on a positively or negatively skewed ticker is a bit more difficult. Crashes substantial enough to bring the probability of going higher to 50% or more tend to be rare (see image below):
So when you are dealing with a positively skewed stock, its best to apply alternative, complementary strategies to determine entries, such as using regression channels, longer running EMAs or time series modelling. You can still use CDFs, but you will need to focus on a narrower timeframe. For example, if we plot SPY from its January high to the current day, we can see the data is normally distributed and thus can refer to our parameters for entry on a normal distribution:
Key Takeaways from the Skewed Distribution:
Conclusion
While I didn’t cover all possible distributions, I did cover the main ones to pay attention to. However, I hope you now have a better understanding and appreciation for the importance of paying attention to the distributions of stocks. The importance of this is often underestimated but it is, in fact, a crucial aspect of successful investing and trading. Various distribution types, such as normal, leptokurtic, platykurtic, and skewed, provide valuable insights into market behavior and risk assessment. Investors and traders who take the time to understand these distributions can make more informed decisions, manage risk effectively, and enhance their overall success.
Thanks so much for reading and hopefully you learned something!
Safe trades and, as always, feel free to share your questions and comments below :-).
By the way, the indicator is linked below if you would like!
Some of you may remember, I released an educational video called “Trading Using Bell Curves”:
In this video, I discuss the implications of using bell-curves for trading.
However, I want to reel it back and talk more specifically about distributions and trading, and why you, the investor and/or trader, should be paying attention to them. This is something I honestly have never seen talked about and really, you are doing yourself a huge disservice as a trader AND an investor for ignoring it. so let’s get into it! But before we start, I won't review the basics of the bell curve, but if you are interested, consider watching the video above.
Alright, now on with the math!
Understanding stock distributions, which come in various forms such as leptokurtic, platykurtic, and more, will provide you with valuable insights into market behavior and risk management. Did you know that certain distribution types can alert you that a stock generally has an unstable trajectory? And by looking at the distributions, you can also tell which stocks are more prone to aggressive crashes and which are more stable?
Well you can, and I am going to teach you how! So let’s go over the main types of distributions in stocks and their implications for you as a trader.
Types of Stock Distributions
Normal Distribution
This is probably the one you hear talked about a lot. A normal distribution, also known as Gaussian distribution or bell curve, is characterized by its symmetrical shape. In a normal distribution, the mean, median, and mode are all equal, and data points are evenly distributed around the central value. This distribution tends to be common in nature but tends to be not all that common in long term stocks. There are some exceptions; however. For example,
BAC (Bank of America) actually has a normally distributed dataset from initial listing to now:When stock returns follow a normal distribution, it becomes easier to predict future price movements and assess risk more easily. One way to do this is by using the cumulative distribution function (or CDF). Which is a mathematical function that provides the probability that a random variable takes on a value less than or equal to a specific value. For example, if we have 10 students with various test scores, we can plot all test scores using CDF and determine what the probability is that a random student’s test score will be above 90% or below 20%.
We can visualize this on
BAC by having the indicator plot the CDF for BAC:The image above plots the CDF distribution for
BAC on the monthly timeframe since its IPO. Because BAC is normally distributed, we can place a high level of confidence in the results of the CDF. We can also use the CDF to our advantage. How? By planning where we could buy.We should buy when the price is at a level where 50% to 60% or more of the time the price will fall above. Turning back to our BAC example, we can display this with a simple trendline:
We can also operate on the assumption that
BAC is likely to go lower from here. Why? Because the normal distribution is not yet invalidated. As of right now, BAC retains a normal distribution. Thus, we can expect BAC to cycle back down to bring its CDF back towards 50% and 60%. We can see another example below, XLE:Key Points for Tickers that are Normally Distributed:
- They tend to be more cyclical, having periods of sustained decline, followed by periods of sustained rise.
- They are the most stable and predictable type of ticker to invest or trade in, but tend to be general underperformers (because of their cyclical behaviour of decline and then rise). However, this is not always the rule, the advantage to a normally distributed ticker is you can calculate your likely returns to a high degree of accuracy!
- Some examples of stocks that have a normally distributed history are
T (AT&T), BAC (Bank of America), XLE (Energy ETF), T-Mobile and BABA. - You will generally notice that, if a ticker in one industry is normally distributed, chances are other tickers in the same sector is as well, even international tickers in the identical sector. For example, T-Mobile (TMUS), T (AT&T) and TSX:T (Telus) all are telecommunication providers and all have normally distributed data.
- They respond very well to log-linear and linear regression methods.
But what about other distributions? Let’s talk about them.
Leptokurtic Distribution
A leptokurtic distribution is characterized by a higher peak and fatter tails than a normal distribution. In this distribution, extreme events, such as market crashes or rapid price spikes, are more likely to occur compared to a normal distribution. From my experience, most stocks fit this description, but one of major note is
BA:Leptokurtic distributions indicate higher volatility and a higher likelihood of extreme price movements. In general, you need to be more cautious with leptokurtic distributions because there is generally heightened volatility. A CDF on a leptokurtic distribution is not as clean, as we can see from plotting BA’s CDF:
Because BA’s distribution is not normal, the CDF becomes slightly unreliable and we cannot employ the 60% rule. So can we still use the distribution to help us gauge entries? Yes! We can! However, it’s a bit more nuanced with leptokurtic distributions.
The first thing to remember with leptokurtic distributions is… they crash… a lot. We can see this with BA:
The flags in this chart represent areas BA has crashed. Crashes in leptokurtic distributions are usually characterized by a drop on the CDF of the probability a stock will go lower to around 75% to 85%. We can see this if we overlay the CDF for BA with the chart:
These are the dips you would want to buy in a leptokurtic distribution. If we take a look at another example, AMD:
Key Takeaways from Leptokurtic Tickers:
- They are among the most unstable tickers and experience among the most crashes. Your risk as an investor is heightened on any ticker that is leptokurtic.
- They do not respond well at all to log-linear or linear regression methods.
- Unlike normal distributions, leptokurtic distributions don’t generally follow sectors and they tend to be company specific tickers (which explains their proneness to crashing and volatility).
- Some major examples of leptokurtic distributions are BA,
AMD, 
MSFT,
Platykurtic Distribution
A platykurtic distribution has a flatter and wider shape compared to a normal distribution. In this case, the data points are more spread out, and extreme events are less likely to occur.
As such, platykurtic distributions suggest lower volatility and a more stable market environment. However, it is important to know that prolonged periods of low volatility can be followed by sudden spikes, leading to unexpected market movements.
These are extremely rare distributions that I have not observed in any of the tickers I have traded. However, theoretically, platykurtic distributions would come in smooth waves up and down. We can visualize this if we look at SPY’s January 2022 highs till its October lows. This was a platykurtic, negative distribution (indicating a stable downtrend):
Because platykurtic distributions are cyclical, you long on the bullish peaks when the probability of higher prices is >= 90% and short on the bearish peaks when the probability of downside is >= 90%:
However, this is not at all prevalent or observed in stocks ever, so you would be lucky to find a platykurtic distribution!
Key take aways from Platykurtic distributions:
- Playkurtic distributions, theoretically, are cyclical like the normal distribution, which make them more stable.
- They would be similar to normally distributed tickers in their under-performance, but superior in their ability to not generally experience equal rises and declines.
Skewed Distribution
A skewed distribution is asymmetric, with a longer tail on one side. Positive skewness means the tail is on the right (indicating more extreme positive values), while negative skewness implies a left tail (indicating more extreme negative values).
Skewed distributions can signal a bias in market sentiment. For example, positive skewness may indicate a bullish bias, while negative skewness may suggest a bearish bias. While many people look to EMAs or trendlines to identify long-standing bull or bear markets, its actually not necessary, you can ascertain this simply from the distribution. If we take a look at SPY:
This is SPY’s distribution since the IPO. We can see that it has a positive skewness (right tail), with extreme outliers. This signals to us, the investors, that SPY has been in a bull run since its IPO. Despite multiple corrections and bear markets, SPY retains the distribution characteristic of a bullish stock. In fact, SPY frequently experiences extremely positive outliars (outliars to the upside) more often than extremely negative outliars (crashes to the downside). This is observed with its positive skewedness.
Planning entries on a positively or negatively skewed ticker is a bit more difficult. Crashes substantial enough to bring the probability of going higher to 50% or more tend to be rare (see image below):
So when you are dealing with a positively skewed stock, its best to apply alternative, complementary strategies to determine entries, such as using regression channels, longer running EMAs or time series modelling. You can still use CDFs, but you will need to focus on a narrower timeframe. For example, if we plot SPY from its January high to the current day, we can see the data is normally distributed and thus can refer to our parameters for entry on a normal distribution:
Key Takeaways from the Skewed Distribution:
- The Skewed distribution are going to net you your returns (assuming, of course, the ticker is POSITIVELY skewed). These are the tickers that tend to experience exponential growth and returns.
- Skewed distributions tend to outperform other distribution types, but not without risk.
- Skewed distributions have an inherent tendency to see dramatic outliars either up or down.
- Unlike a leptokurtic distribution which is more prone to crashes, a positively skewed distribution is more likely to experience extreme outliars to the upside (meaning bull runs) than to the downside. However, a negatively skewed distribution is more likely to experience more frequent and dramatic drops to the downside than to the upside. So pay attention to the skewness! If it is negative, the risk of a downturn is greatly augmented.
- Skewed distributions respond reasonably well to log-linear and linear regression methods.
- Famous example is
SPY of course!
Conclusion
While I didn’t cover all possible distributions, I did cover the main ones to pay attention to. However, I hope you now have a better understanding and appreciation for the importance of paying attention to the distributions of stocks. The importance of this is often underestimated but it is, in fact, a crucial aspect of successful investing and trading. Various distribution types, such as normal, leptokurtic, platykurtic, and skewed, provide valuable insights into market behavior and risk assessment. Investors and traders who take the time to understand these distributions can make more informed decisions, manage risk effectively, and enhance their overall success.
Thanks so much for reading and hopefully you learned something!
Safe trades and, as always, feel free to share your questions and comments below :-).
By the way, the indicator is linked below if you would like!
For real-time updates and premium indicators, consider joining my group at: patreon.com/steversteves
Related publications
Disclaimer
The information and publications are not meant to be, and do not constitute, financial, investment, trading, or other types of advice or recommendations supplied or endorsed by TradingView. Read more in the Terms of Use.