# Geometric vs Arithmetic Mean: Unveiling the Power of the Geometric Mean (Updated 2023)

Are you ready to dive into something new and exciting?

Then let's talk about the battle between geometric and arithmetic mean in trading.

But first, what are these means exactly?

The arithmetic mean is simply the average of a set of numbers.

It's easy to calculate, but it doesn't account for compounding returns.

On the other hand, the geometric mean does consider compounding returns, making it a more accurate measure of investment growth over time.

It depends on your goals and preferences.

If you're looking for quick gains, then the arithmetic mean may be more suitable for you.

However, if you're playing the long game and want to maximize your returns over time, then using the geometric mean would be a smarter choice.

But don't take our word for it!

In this article, we'll explore both options in-depth and provide examples to help guide your decision-making process.

So whether you're an experienced trader or just starting out, join us as we delve into this fascinating topic.

Then dive into our article now!

## Overview: Geometric vs Arithmetic Mean in Trading

The arithmetic mean is calculated by adding up all the values and dividing by the number of observations.

On the other hand, the geometric mean takes into account compounding returns over time.

When it comes to trading analysis, research has shown that while arithmetic means are more commonly used, geometric means may be more relevant to trading strategies.

This is because they provide a better representation of long-term performance and can help traders avoid overestimating their returns.

To calculate the geometric mean, you need to multiply all the numbers in a series and then take the nth root of the product, where n is the number of observations.

Let's take an example to understand this better.

Suppose you invested \$100 in a stock that grew to \$260.25 over a period of 5 years.

The annual growth rate would be (260.25/100)^(1/5) - 1 = 39.5%.

The average growth rate would be (1 + 39.5%) / 2 - 1 = 19.75%.

The arithmetic average return would be (0% + 39.5%)/2 = 19.75%.

The geometric average return would be (1 + 39.5%)^(1/2) - 1 = 18.5%.

As you can see, the geometric mean provides a more accurate representation of the stock's performance over time.

Real-world examples also demonstrate how one mean may be more appropriate than the other.

For instance, if you're looking at a stock's historical performance over several years, using an arithmetic mean would give equal weight to each year's return.

However, if you're interested in calculating your overall return on investment for that period, using a geometric mean would take into account compounding returns.

As a trader, it's important to understand when to use each type of mean effectively.

By utilizing both methods in your decision-making process, you can gain a more comprehensive understanding of your portfolio's performance.

So next time you're analyzing your portfolio's performance or considering new investments, remember to calculate both the arithmetic and geometric means for a well-rounded perspective on your financial situation.

## Understanding the Arithmetic Average in Trading

Arithmetic mean is used as a statistical tool in trading that helps traders analyze market trends and make informed decisions.

It is calculated by adding up all the values in a set of data and dividing it by the number of values.

To better understand the difference between the arithmetic mean and geometric mean, let's take a look at a comparison table.

Suppose we have the following numbers: 2, 4, 6, 8, and 10.

The arithmetic mean of this data set is (2+4+6+8+10)/5 = 6.

The geometric mean, on the other hand, is calculated by taking the product of these numbers and finding the fifth root of that product: (2 x 4 x 6 x 8 x 10)^(1/5) = 4.38.

As a trader, it's important to understand the difference between these two means and their applications in risk management.

While arithmetic mean is preferred in trading due to its simplicity and effectiveness in managing risks, geometric mean can be a useful tool for analyzing investments with a compound interest rate.

By understanding the formula for geometric mean and its compounding effect, traders can gain valuable insights into market trends that will help them make informed decisions about their investments.

## The Formula for Calculating Geometric Mean in Trading

Geometric mean is used to calculate investment returns over multiple periods.

The geometric mean, also known as the geometric average, is a type of mean that is calculated by multiplying all the numbers in a series together and then taking the nth root of the product, where n is the length of the series.

Research has shown that using geometric mean provides a more accurate representation of investment performance over time.

This is because it takes into account compounding returns, which can significantly impact long-term growth.

The geometric mean is also known as the growth rate, which is the average rate of return of an investment over a period of time.

It is important to note that the geometric mean is the inverse of the length of the series of returns, while the arithmetic mean is the sum of the returns divided by the length of the series.

To calculate geometric mean for a portfolio or investment strategy, you'll need to use a specific formula.

This involves taking the product of all returns (including dividends) and then taking the nth root of that product (where n is the number of periods).

The logarithmic mean is another type of mean that is used in finance, which is the average of the logarithms of a series of numbers.

You may also have a look at the following articles to gain a deeper understanding of the difference between arithmetic mean and the geometric mean.

So next time you're analyzing your portfolio or considering new investments, be sure to take into account the power of geometric mean.

## Effect of Outliers on Mean in Trading

Outliers are extreme values that can skew data and distort results.

In trading, outliers can be caused by unexpected events such as market crashes or sudden price changes.

When calculating returns using arithmetic mean, outliers have a greater impact on the final result because they are included in every calculation.

On the other hand, geometric mean reduces the effect of outliers by taking into account compounding returns.

It's important to identify and handle outliers in trading to avoid misleading results that could lead to poor investment decisions.

For example, if an outlier is not properly identified and removed from calculations using arithmetic mean, it could give a false impression of higher returns than what was actually earned.

When it comes to calculating moving averages, traders can use either arithmetic or geometric mean.

Arithmetic mean is the simpler of the two, as it only requires adding up the values and dividing by the number of data points.

However, it can be heavily influenced by outliers.

Geometric mean, on the other hand, is more complex as it involves multiplying the values and raising them to the power of the multiplicative inverse of the number of values.

This method is less affected by outliers and is often used in finance to calculate returns over multiple periods.

In addition, traders may use multiples of 10 to describe the difference between two moving averages.

For example, if the 50-day moving average is higher than the 200-day moving average, traders may say that the 50-day moving average is "two multiples of 10" higher than the 200-day moving average.

This method provides a quick and easy way to describe the difference between two moving averages.

Knowing how geometric and arithmetic means differ in handling outliers is crucial for accurate return calculations in trading.

By identifying and handling outliers appropriately through proper statistical analysis techniques such as trimming or winsorizing data points outside certain thresholds or ranges), traders can make more informed investment decisions based on reliable data rather than skewed results caused by extreme values.

Traders can use either arithmetic or geometric mean to calculate moving averages, and may use multiples of 10 to describe the difference between two moving averages.

## Volatility and Moving Average: Arithmetic vs Geometric

As a trader, you know that volatility is a crucial factor in determining your profits.

The choice of mean can have a significant impact on how you measure volatility.

Arithmetic mean is the simple average of a set of numbers, while geometric mean is the nth root of the product of these numbers.

In trading, arithmetic mean is commonly used to calculate moving averages because it gives equal weight to each data point.

On the other hand, geometric mean gives more weight to recent data points and less weight to older ones.

When calculating arithmetic mean, you add up all the numbers in the data set and divide the sum by the number of numbers in the series.

This method is useful for finding the average of positive and negative numbers.

However, it may be more prone to outliers, which are extreme values that can skew the results.

Geometric mean, on the other hand, involves multiplying all the numbers in the series and raising it to the inverse of the number of numbers in the data set.

The result gives the nth root of the product of these numbers.

This method is useful for finding the average of positive numbers and can be less affected by outliers.

Studies have shown that using geometric mean for moving averages can result in smoother trends and better performance during trending markets.

However, during choppy markets with frequent reversals, arithmetic mean may be more effective.

It's important to note that both means have their advantages and disadvantages when it comes to trading strategies.

While arithmetic mean may be simpler to calculate and interpret, it may also be more prone to outliers and less responsive to changes in market conditions.

Geometric mean may provide better signals during trending markets but may also lag behind during sudden price movements.

Awareness of the differences between geometric and arithmetic means can help traders make informed decisions when measuring volatility and calculating moving averages.

By considering their strengths and weaknesses in different market conditions, traders can develop effective strategies that maximize their profits while minimizing risks.

## The Difference between Arithmetic and Geometric Mean

Now, let's talk about the difference between geometric and arithmetic mean in trading.

As a trader, you're always looking for ways to analyze investment returns and make informed decisions.

One way to do this is by calculating the arithmetic or geometric mean of your investments.

Arithmetic mean is simply the simple average return of a set of numbers over a certain period of time.

It is calculated by adding up all the numbers and dividing by the total number of numbers.

However, arithmetic mean may not be the best choice when there are significant outliers or large fluctuations in returns.

This is because it doesn't take into account the effect of compounding returns.

On the other hand, geometric mean takes into account the compounding returns and gives a more accurate representation of long-term growth.

It is calculated by taking the nth root of the product of all the numbers in the set.

Geometric mean is particularly useful for analyzing investments with high volatility or those that are held for longer periods of time.

For example, let's say you start with 100 and your investments have returns of 10%, -5%, and 20% over three years.

The arithmetic mean would be (10% - 5% + 20%) / 3 = 8.33%.

However, the geometric mean would be taking the product of (1 + 10%) x (1 - 5%) x (1 + 20%) = 1.138, and then taking the cube root of that number, which gives you a geometric mean of 3.57%.

Real-world examples show that traders use both arithmetic and geometric means to make investment decisions depending on their goals and investment strategies.

If you're looking for short-term gains or analyzing stocks with low volatility, arithmetic mean may be sufficient.

However, if you're looking at long-term growth or analyzing stocks with high volatility, geometric mean would be a better choice.

Familiarity with the differences between these two types of means is crucial for traders to make informed decisions when analyzing investment returns.

By considering factors such as volatility and compounding returns, traders can choose which method best suits their needs and goals in trading.

### Q: What is the difference between geometric mean and arithmetic mean?

The geometric mean, calculated by taking the nth root of the product of n numbers, and the arithmetic mean, obtained by summing up all values and dividing by the total number of values, are two different ways of calculating averages.

### Q: When should I use the geometric mean vs arithmetic mean?

When dealing with ratios or growth rates, like compound annual growth rates or population growth rates, it is appropriate to use the geometric mean. On the other hand, the arithmetic mean is useful for finding the typical value of a dataset, such as average test scores.

### Q: What is the Kelly Criterion?

A: The Kelly Criterion is a mathematical formula developed by mathematician John Kelly in 1956. It provides a method for determining the optimal betting or investment size based on the known expected returns. Unlike traditional investment strategies that consider the arithmetic average, the Kelly Criterion utilizes the expected geometric return, which takes into account the logarithmic scale of returns. By maximizing the expected value while considering the risk of ruin and losses, the Kelly Criterion aims to guide optimal capital allocation decisions.

### Q: What is the volatility tax?

A: The term "volatility tax" is derived from Mark Spitznagel's book, "Safe Haven – Investing For Financial Storms." It refers to the hidden impact of compounding and drawdowns in investment returns. Unlike the arithmetic average, which may mask the effects of drawdowns, the geometric average considers the multiplicative dynamics of compounding and reveals the true impact of losses. The volatility tax highlights how significant losses require disproportionately larger gains to recover. For instance, a 33% loss requires a subsequent 49% return to break even, while a 50% loss necessitates a 100% gain for recovery. Minimizing drawdowns during unfavorable times becomes crucial to mitigate the effects of the volatility tax.

## Conclusion: Arithmetic Mean and Geometric Mean

Arithmetic mean is always the most commonly used mean in trading analysis.

It calculates the average of a set of numbers by adding them up and dividing by the number of values.

This is a straightforward way to calculate the average, but it may not be the best option when dealing with volatile numbers.

The arithmetic mean gives equal weight to each data point, which can be problematic when there is volatility in the data.

On the other hand, geometric mean calculates the average growth rate over a period of time.

It is often used for long-term investments.

This method takes into account the compounding effect, which is crucial when dealing with long-term investments.

The values are divided by the number of periods, and then the nth root of the product is taken.

This way to calculate the average is more complex, but it can provide a more accurate representation of the data.

So, which one should you use?

If you're using a short-term trading strategy, arithmetic mean may be more appropriate as it gives equal weight to each data point.

However, if you're looking at long-term investments or compounding returns over time, using the geometric mean may be more useful as it takes into account the effect of compounding.

Real-life examples highlight the importance of selecting the appropriate mean to accurately assess investment returns.

Consider the following scenario: if an individual invested \$1000 at an annual return rate of 10% for 3 years and subsequently experienced a 10% loss in the fourth year, the arithmetic mean return would be calculated by summing all the numbers (10%, 10%, 10%, -10%) and dividing by the total number of years (4), resulting in a value of 7.5%.

However, the actual return would only amount to 2.1% due to the compounding effects associated with investing.

In this case, utilizing the geometric mean, which considers the product of all the numbers divided by the total number of years, would provide a more accurate representation of the returns.

The arithmetic mean, used to calculate the average, may be misleading when evaluating investment returns over multiple periods due to the compounding nature of investments.

In the given example, the sum of all the numbers (10%, 10%, 10%, -10%) divided by the total number of years (4) yields an arithmetic mean return of 7.5%.

However, when accounting for the compounding effects, the actual return amounts to only 2.1%.

To obtain a more precise representation of the returns, it is advisable to employ the geometric mean, which considers the product of all the numbers divided by the total number of years.

This approach takes into account the compounding effects and provides a more accurate assessment of investment performance.

In trading, it is important to consider the volatility in the data and choose the appropriate mean to calculate the average.

By understanding the differences between arithmetic and geometric means and applying them appropriately to your trades, you can maximize your profits and minimize risk.