# What Is The Difference Between Arithmetic And Geometric Averages? (Solved)

Geometric mean is the calculation of mean or average of series of values of product which takes into account the effect of compounding and it is used for determining the performance of investment whereas arithmetic mean is the calculation of mean by sum of total of values divided by number of values.

## What is the difference between the arithmetic average return and the geometric average return?

Arithmetic returns are the everyday calculation of the average. The geometric mean is calculated by multiplying all the (1+ returns), taking the n-th root and subtracting the initial capital (1). The result is the same as compounding the returns across the years.

## Should I use geometric or arithmetic mean?

Each mean is appropriate for different types of data; for example: If values have the same units: Use the arithmetic mean. If values have differing units: Use the geometric mean. If values are rates: Use the harmonic mean.

## What is the relationship between arithmetic mean and geometric mean?

Let A and G be the Arithmetic Means and Geometric Means respectively of two positive numbers m and n. Then, we have A = m + n/2 and G = ±√mn. Since, m and n are positive numbers, hence it is evident that A > G when G = -√mn.

## How do you find arithmetic and geometric average returns?

Geometric Average Return Example If you were to calculate this using the arithmetic mean return, you would add the rates together and divide them by three, giving you an average of 6%. Using this method the ending balance of 6% a year for three years would be \$5,955.08.

## What is the difference between arithmetic mean and mean?

Average, also called the arithmetic mean, is the sum of all the values divided by the number of values. Whereas, mean is the average in the given data.

## Which is mathematical average?

A mathematical average is calculated by taking the sum of a group of values and dividing it by the number of values in the group. It is also known as an arithmetic mean.

## What is the difference between geometric and arithmetic growth?

Under arithmetic growth, successive population totals differ from one another by a constant amount. Under geometric growth they differ by a constant ratio. In other words, the population totals for successive years form a geometric progression in which the ratio of adjacent totals remains constant.

## For what type of data is the geometric mean used?

Geometric Mean Definition and Formula The geometric mean is a type of average, usually used for growth rates, like population growth or interest rates. While the arithmetic mean adds items, the geometric mean multiplies items. Also, you can only get the geometric mean for positive numbers.

## What is the relation between arithmetic mean am and geometric mean GM?

If AM and GM are the arithmetic mean and the geometric mean of two positive integers a and b, respectively, then, AM>GM. Hence proved that the arithmetic mean of two positive numbers is always greater than their GM. This is also called the arithmetic mean – geometric mean (AM-GM) inequality.

## How does the arithmetic and geometric mean compare on random integers?

Geometric Mean is known as the Multiplicative Mean. Arithmetic Mean is known as Additive Mean. The geometric mean is always lower than the arithmetic means due to the compounding effect. The arithmetic mean is always higher than the geometric mean as it is calculated as a simple average.

## How do you find arithmetic average?

One method is to calculate the arithmetic mean. To do this, add up all the values and divide the sum by the number of values. For example, if there are a set of “n” numbers, add the numbers together for example: a + b + c + d and so on. Then divide the sum by “n”.

## How do you calculate geometric average?

Basically, we multiply the ‘n’ values altogether and take out the nth root of the numbers, where n is the total number of values. For example: for a given set of two numbers such as 8 and 1, the geometric mean is equal to √(8×1) = √8 = 2√2.

## How do you find geometric average?

In order to find the geometric mean, multiply all of the values together before taking the nth root, where n equals the total number of values in the set. You can also use the logarithmic functions on your calculator to solve the geometric mean if you want.

## The Difference Between the Arithmetic Mean and Geometric Mean

There are several methods for evaluating the performance of a financial portfolio and determining whether or not an investment plan is effective. The geometric average, often known as the geometric mean, is frequently used by investment experts to make decisions.

### Key Takeaways:

• In the case of series that display serial correlation, the geometric mean is the most appropriate choice. Specifically, this is true for investment portfolios because the majority of financial returns are connected, such as bond yields, stock returns, and market risk premiums, among other things. Compounding becomes increasingly crucial with increasing time horizon, and the usage of the geometric mean becomes more acceptable. Because it takes into account year-over-year compounding, the geometric average gives a significantly more accurate representation of the underlying return for volatile values.

Due to the compounding that happens from period to period, the geometric mean differs from thearithmetic mean, or arithmetic mean, in how it is determined. Investors generally believe that the geometric mean is a more accurate gauge of returns than the arithmetic mean as a result of this phenomenon.

## The Formula for Arithmetic Average

The portfolio returns for periodnn are represented by the numbers a1, a2,., ann where: a1, a2,., ann=Portfolio returns for periodnn=Number of periods begin A = fracsum_ n a i = fractextbfa 1, a 2, dotso, a n=textn n=textn n=textn n=textn n=textn n=textn n=textn n=textn n=textn n=textn n=textn n=textn n=textn n=textn n=textn n=text A=n1 i=1n ai =na1 +a2 +.+an where:a1,a2,.,an =Portfolio returns for periodnn=Number of periodsnn=Number of periodsnn=Number of periodsnn=Number of periodsnn=Number of periods ​

## How to Calculate the Arithmetic Average

An arithmetic average is the product of the sum of a series of numbers divided by the number of numbers in that series. If you were asked to calculate the class (arithmetic) average of test results, you would simply add up all of the students’ test scores and divide that total by the number of students in the class. For example, if five students completed an exam and had scores of 60 percent, 70 percent, 80 percent, 90 percent, and 100 percent, the average for the arithmetic class would be 80 percent.

• It is because each score is an independent event that we use an arithmetic average to calculate test scores instead of a simple average.
• It is not uncommon in the field of finance to find that the arithmetic mean is not an acceptable way for determining an average.
• Consider the following scenario: you have been investing your funds in the financial markets for five years.
• With the arithmetic average, the average return would be 12 percent, which looks to be a substantial amount at first glance—but it is not totally correct in this case.
• They are interdependent.

Our goal is to arrive at an accurate calculation of your actual average yearly return over a five-year period. To do so, we must compute the geometric average of your investment returns.

## The Formula for Geometric Average

x1,x2,=Portfolio returns for each periodn=Number of periodsbeginleft(prod_ n x i ) = sqrttextbfx 1, x 2, the number of periods is equal to the sum of the returns on the portfolio for each period (x1, x2,.) and the number of periods is equal to the sum of the returns on the portfolio for each period (n1). ​

## How to Calculate the Geometric Average

It is possible to determine the geometric mean for a series of integers by multiplying the product of these values by the inverse of the length of the series. In order to do this, we add one to each of the numbers (to avoid any problems with negative percentages). In the next step, add up all of the numbers and elevate their product to a power of one divided by the number of numbers in the series. Then we take one away from the final result. When expressed in decimal form, the formula looks somewhat like this: [(1+R1)×(1+R2)×(1+R3)…×(1+Rn)] beginning with the number 1 and ending with the number 1 where R=Returnn=Count of the numbers in the series starting with the number 1 and ending with the number 1 – 1 textbftext= textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = textn = text 1 where: R = Returnn = Number of numbers in the series 1 where: R = Returnn = Number of numbers in the series Although the formula seems complicated, it is not as tough as it appears on paper.

Using our previous example, we can calculate the geometric average as follows: The percentages of returns we received were 90 percent, 10 percent, 20 percent, 30 percent, and -90 percent, therefore we entered them into the calculation as: (1.91.11.21.30.1).

The figure obtained by applying the geometric average is far worse than the 12 percent arithmetic average we obtained previously, and regrettably, it is also the number that best depicts reality in this particular instance as well.

## Geometric Mean vs Arithmetic Mean

In the case of a product, the geometric mean is calculated as the mean or average of a series of values that takes into account the effect of compounding and is used to determine the performance of the investment, whereas the arithmetic mean is calculated as the mean as the sum of the sum of the total of values divided by the number of values in the series of values. You are allowed to use this image on your website, in templates, or in any other way you see fit. Please credit us by include a link to this page.

As an illustration: Geometric Mean vs.

The Arithmetic Mean is simply the average, and it is computed by putting all of the numbers together and dividing the total number of numbers by the total number of numbers in that series.

### Geometric Mean vs. Arithmetic Mean Infographics

In the case of a product, the geometric mean is calculated as the mean or average of a series of values that takes into account the effect of compounding and is used to determine the performance of the investment, whereas the arithmetic mean is calculated as the mean as the sum of the sum of the total of values divided by the number of values in the series of values in the product You are allowed to use this picture on your website, in templates, or in any other way you see fit.

As an illustration, Geometric Mean vs.

It is simple to determine the arithmetic mean, which is obtained by adding up all of the numbers and dividing by the number of times that sequence of numbers has been encountered.

### Key Differences

• In the case of a product, the geometric mean is calculated as the mean or average of a series of values that takes into account the effect of compounding and is used to determine the performance of the investment, whereas the arithmetic mean is calculated as the mean as the sum of the total of the values divided by the total number of values. You are welcome to use this picture on your website, in templates, or in any other way you see fit. However, please attribute us by including a link to this page. Hyperlinking the article link As an illustration, consider the following: Geometric Mean versus Arithmetic Mean (Source) (wallstreetmojo.com) The geometric mean of a sequence of numbers is computed by taking the product of the numbers in the series and increasing it to the inverse length of the series. The Arithmetic Mean is simply the average, and it is computed by adding all of the numbers together and dividing the total by the total number of numbers in that series.
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### Comparative Table

Basis Geometric Mean Arithmetic Mean
Meaning Geometric Mean is known as the Multiplicative Mean. Arithmetic Mean is known as Additive Mean.
Formula – 1 (Return1 + Return2 + Return3 + Return4)/ 4
Values The geometric mean is always lower than the arithmetic means due to the compounding effect. The arithmetic mean is always higher than the geometric mean as it is calculated as a simple average.
Calculation Suppose a dataset has the following numbers – 50, 75, 100. Geometric mean is calculated as cube root of (50 x 75 x 100) = 72.1 Similarly, for a dataset of 50, 75, and 100, arithmetic mean is calculated as (50+75+100)/3 = 75
Dataset It is applicable only to only a positive set of numbers. It can be calculated with both positive and negative sets of numbers.
Usefulness Geometric mean can be more useful when the dataset is logarithmic. The difference between the two values is the length. This method is more appropriate when calculating the mean value of the outputs of a set ofindependent eventsIndependent event refers to the set of two events in which the occurrence of one of the events doesn’t impact the occurrence of another event of the set.read more.
Effect of Outlier The effect of outliers on the Geometric mean is mild. Consider the dataset 11,13,17 and 1000. In this case, 1000 is the outlier. Here, the average is 39.5 The arithmetic mean has a severe effect of outliers. In the dataset 11,13,17 and 1000, the average is 260.25
Uses The geometric mean is used by biologists, economists, and also majorly byfinancial analysts. It is most appropriate for a dataset that exhibits correlation. The arithmetic mean is used to represent average temperature as well as for car speed.

### Conclusion

If you are dealing with percentage changes, volatile figures or data that exhibits correlation, the geometric mean is the best option. This is especially true for investment portfolios. If you make a portfolio investment instead of a single asset, you are investing in a group of assets (equity or debt), with the goal of earning returns that are proportional to the investor’s risk profile. Portfolio investments can include anything from stocks to bonds to mutual funds to derivatives to bitcoins.

The majority of financial returns, such as stock returns, bond yields, and premiums, are connected.

While arithmetic means is more suited for independent data sets because it is straightforward to use and comprehend, it is less appropriate for dependent data sets.

### Recommended Articles

This article served as a comparison of the Geometric Mean and the Arithmetic Mean. With the use of infographics and a comparison table, we’ll go through the top nine distinctions between Geometric Mean and Arithmetic Mean in this article. Check out the following articles as well if you want to learn more:

• The power of compounding
• The formula for the weighted mean
• The difference between the mean and the median
• The annualized rate of return

## Geometric Average vs. Arithmetic Average: Which is Correct For Investment Returns? – Arbor Asset Allocation Model Portfolio (AAAMP) Value Blog

A sequence is a collection of numbers, referred to as terms, that are organized in a specified order.

### Arithmetic Sequences

There is a common distinction between arithmetic sequences. In other words, the same amount of money is added or withdrawn from one period to the next. For example: 1, 3, 5, 7, 9, 11 (the most frequent difference is +2) or 25, 20, 15, 10, 5 (the most common difference is +2). (common difference is -5)

### Geometric Sequences

Geometric sequences have a common ratio in common with one another. As a result, the ratio between each chronological term in the series remains constant from one to the next. For example, 50, 100, 200, 400, 800 (common ratio is 2) or 800, 400, 200, 100, 50 (common ratio is 1/2) are examples of common ratios in math. In order to be correct, investment average returns must be calculated as a geometric average.

This is due to the fact that, as a result of compounding, each consecutive term is reliant on the prior conclusion. Arithmetic averages are accurate when used to calculate investment returns only when there is no volatility in the underlying asset class (i.e. 5 percent return each period).

### Volatility Lowers Investment Returns

Generally speaking, geometric sequences have the same ratio as well. In other words, the ratio between each chronological term in the series is the same for all of the chronological terms. For example, 50, 100, 200, 400, 800 (common ratio is 2) or 800, 400, 200, 100, 50 (common ratio is 1/2) are examples of common ratios in mathematics. If you want to get reliable results from your investments, you should use a geometric average. For this reason, because of compounding, each consecutive term is reliant on the outcome of the preceding term, as seen in Figure 1.

5 percent return each period).

## Geometric Mean vs Arithmetic Mean

In the realm of finance, the Arithmetic mean and the Geometric mean are the two most generally used techniques for calculating the returns on investment for investment portfolios, respectively. When people report larger returns, they do it using mathematics, which is not the most accurate way of estimating the return on investment. Because the return on investment for a portfolio over time is reliant on the returns in prior years, calculating the return on investment for a certain time period using the Geometric mean is the right method of calculating returns.

• Use of Geometric Mean vs.
• Let’s look at an example of return on investment for a \$100 investment over a period of two years.
• Calculating the average return using the arithmetic mean will result in a return of 0 percent (Arithmetic mean = (-50 percent + 50 percent) /2 = 0 percent).
• A deeper look at the situation, on the other hand, paints a completely different image of the situation.
• As a result, the investor does not achieve breakeven on its investment as predicted by the arithmetic mean average, but instead suffers a loss of \$25 on its investment after two years.
• Following two years, the following is the investment position: Consequently, the Geometric mean reveals the actual image of investment, which is that there has been a loss in investment with an annualized negative return of -13.40 percent compared to the historical data.
• 2.

It is possible to determine the average of a student’s marks for five topics using the arithmetic mean, because the scores of the student in different courses are independent of one another.

### Head to Head Comparison between Geometric Mean vs Arithmetic Mean (Infographics)

The following are the main eight distinctions between Geometric Mean and Arithmetic Mean:

### Key Differences between Geometric Mean vs Arithmetic Mean

Let’s have a look at some of the most significant distinctions between Geometric Mean and Arithmetic Mean:

• Neither the Geometric Mean nor the Arithmetic Mean are the techniques used to determine the returns on investment in finance, although they are both utilized in other areas such as economics and statistics. The arithmetic mean is determined by dividing the sum of the numbers by the total number of numbers in the sample size. Geometric methods, on the other hand, take into consideration the compounding impact during the calculation. In order to accurately evaluate the return on investment over a specified time period, the geometric mean should be used. Because the returns on investment for a portfolio over time are interconnected, it is important to understand how they work. The Arithmetic mean, on the other hand, is more appropriate in situations when the variables being utilized for computation are not dependent on one another. Because the arithmetic mean is more helpful and accurate when used to compute the average of a data collection with numbers that are not skewed or dependant on one another, it is more commonly utilized. Geometric means, on the other hand, are more effective and accurate in situations when there is a lot of volatility in a data set. In compared to the geometric mean, which is somewhat hard to compute and use, the arithmetic mean is comparatively simple to calculate and apply. When it comes to the realm of finance, the geometric mean is quite popular, especially when it comes to the computation of portfolio returns. However, when it comes to return calculation, an Arithmetic mean is not an effective instrument to utilize. Whenever two numbers are compared, the Arithmetic mean of the numbers is always greater than the Geometric mean of the numbers.

### Geometric Mean vs Arithmetic Mean Comparison Table

Now, let’s have a look at the top 8 comparisons between the geometric mean and the algebraic mean.

 The Basis Of Comparison Arithmetic Mean Geometric Mean Definition The arithmetic average of a series of numbers is the sum of all the numbers in the series divided by the counts of the total number in the series. Geometric means takes into account the compounding effect during the calculation period. This is calculated by multiplying the numbers in a series and taking the nth root of the multiplication. Where n is the numbers count in series. Formula If there are two numbers X and Y in the series thanArithmetic mean = (X+Y)/2 If there are two numbers X and Y in the series thanGeometric mean = (XY)^(1/2) Suitability of Use Arithmetic means shall be used in a situation wherein the variables are not dependent on each other, and data sets are not varying extremely. Such as calculating the average score of a student in all the subjects. Geometric mean shall be used to calculate the mean where the variables are dependent on each other. Such as calculating the annualized return on investment over a period of time. Effect of Compounding The arithmetic mean does not take into account the impact of compounding, and therefore, it is not best suited to calculate the portfolio returns. The geometric mean takes into account the effect of compounding, therefore, better suited for calculating the returns. Accuracy The use of Arithmetic means to provide more accurate results when the data sets are not skewed and not dependent on each other. Where there is a lot of volatility in the data set, a geometric mean is more effective and more accurate. Application The arithmetic mean is widely used in day to day simple calculations with a more uniform data set. It is used in economics and statistics very frequently. The geometric mean is widely used in the world of finance, specifically in calculating portfolio returns. Ease of Use The arithmetic mean is relatively easy to use in comparison to the Geometric mean. The geometric mean is relatively complex to use in comparison to the Arithmetic mean. Mean for the same set of numbers The arithmetic mean for two positive numbers is always higher than the Geometric mean. The geometric mean for two positive numbers is always lower than the Arithmetic mean.

### Conclusion

Geometric Mean and Arithmetic Mean are both used in many fields such as economics, finance, statistics, and other related fields depending on their applicability. When the variables are dependent and highly skewed, the geometric mean is a more appropriate method of finding the mean since it produces more accurate results. When the variables are not interdependent, an Arithmetic mean is used to determine the average, however when they are, an arithmetic mean is employed. As a result, in order to get the optimum effects, these two should be employed in an appropriate context.

### Recommended Articles

This has served as a reference to the most significant distinction between Geometric Mean and Arithmetic Mean. In this section, we also highlight the fundamental distinctions between Geometric Mean and Arithmetic Mean, using infographics and a comparison table. You may also want to have a look at the following articles for further information.

1. This article has served as a guide to the most significant differences between Geometric Mean and Arithmetic Mean in mathematical calculations. Also included are infographics and a comparison chart to demonstrate the fundamental distinctions between Geometric and Arithmetic Mean. Check out the following articles if you want to find out more more.

## Differences in Arithmetic & Geometric Mean

This article has served as a guide to the most significant differences between Geometric Mean and Arithmetic Mean. We also cover the fundamental distinctions between Geometric Mean and Arithmetic Mean, as illustrated via infographics and a comparison table. You may also want to have a look at the following articles for further information.

## Formulas for Calculation

The method by which the arithmetic mean and the geometric mean for a data set are determined is the most visible distinction between the two measures. It is possible to compute the arithmetic mean of a data collection by adding up all of the numbers in that data set and then dividing the result by the total number of data points. In this example, the arithmetic mean of 11, 13, 17, and 1,000 is calculated as follows: (11 + 13 + 17 + 1,000) / 4 = 261.25 It is possible to determine the geometric mean of a data set by multiplying each integer in the data set and then calculating the nth root of the result, where “n” represents the total number of data points in the data set.

## The Effect of Outliers

When you compare the results of arithmetic mean and geometric mean calculations, you’ll find that the influence of outliers is significantly reduced in the geometric mean calculations. What exactly does this imply? The number 1,000 is considered a “outlier” in the data set consisting of the numbers 11, 13, 17, and 1,000 since its value is much greater than all of the other numbers. It is found that the arithmetic mean is 260.25 when the result is computed. It is important to note that there is no number in the data set that is even somewhat near to 260.25, indicating that the arithmetic mean is not representative in this situation.

The influence of the outlier has been overstated in this study. The geometric mean, with a value of 39.5, performs a better job of demonstrating that the majority of the values in the data set fall between 0 and 50.

## Uses

Look at the results of arithmetic mean and geometric mean computations and you will find that the influence of outliers is much reduced in the geometric mean. Why should I care about this? A “outlier” is a number that is much higher than all the other numbers in a data set consisting of the numbers 11, 13, 17, and 1,000. 260.25 is the result of calculating the arithmetic mean of the numbers given. Because there is no number in the data set that is even somewhat near to 260.25, the arithmetic mean in this situation is not representative.

In this data set, the geometric mean (39.5) performs a better job of demonstrating that majority of the values fall between 0 and 50.

## The difference between arithmetic and geometric investment returns

The historical average investment return is the figure that is most frequently referenced in the financial world. The question is, however, whether we are talking about mathematical or geometric means of those returns, and if this is universal across the sector and whether it matters. It is critical to comprehend this in order to properly analyze previous outcomes. To begin, here are some definitions:

• Arithmetic returns are the averages that are calculated on a daily basis. It is necessary to take a series of returns (in this example, annual numbers) and add them all up, after which you must divide the sum by the number of returns in the series. Geometric returns (also known as compound returns) include significantly more difficult mathematical calculations. By multiplying all of the (1+ returns) together, taking the n-th root, and removing the original capital, we get the geometric mean (1). The result is the same as if the returns were compounded over a period of time.
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It is impossible for the arithmetic mean to be smaller than the geometric mean. Using the numbers 2 and 8 as examples, it is easy to see how the two numbers differ. The arithmetic average is 5, since (2 + 8)/2 = 10/2 = 5 is the sum of the two numbers. The geometric mean, on the other hand, is 4: this is exactly 20% lower than the geometric mean. This may be determined using the formula v(2 x 8) = v16 = 4. Returns on investment may be seen by looking over the last 33 years of the S P/ASX 200 accumulation index, which shows: In terms of annualized returns, the arithmetic mean of these returns is 13.9 percent.

If the investment return averaged 11.6 percent per year from 1980 to the present, and you compounded the results, you would have increased your wealth by the same amount as the index during the same period (ignoring cash flows, taxes, fees and so on).

When compounded, the index would have concluded 2012 at 73,330.2, nearly double the actual value of 37,134.5, if it had increased at the rate of 13.9 percent each year during the previous year.

#### Volatility, risk and average returns

Volatility is to blame for the chasm. As the stream of investment returns becomes more variable, the gap between the two measurements becomes wider. Let’s look at three possible investment return scenarios and see how much of a disparity there is over two years:

• Two years of zero returns (0, 0)
• An increase of 10% in the first year and a decrease of 10% in the second year (+10, -10)
• An increase of 20% in the first year and a decrease of 20% in the second year (+20, -20)
• And two years of zero returns (0, 0).

0 percent per annum is the arithmetic average of each of these situations when taken together (over-weighting the effect of gains and under-weighting the effect of losses).

Each has a distinct geometric mean, which is as follows:

• 0 percent per annum is the arithmetic average of each of these situations when combined (over-weighting the effect of gains and under-weighting the effect of losses). It differs from the other in that the geometric mean of each is:

When it comes to stock investments, where the standard deviation of volatility can reach as high as 20 percent per year, the difference between arithmetic and geometric means can be substantial. After factoring in the market’s volatility of 20 percent every year, an annualized rate of return of 7.5 percent will translate into a compound rate of 5.9 percent (ie what actually ends up in your pocket over the longer term). Volatility costs the investor money because of this chasm.

It is important not to leap to the erroneous conclusion and believe that the gap is a component of the risk premium. If you want to compare the returns on a hazardous asset with the returns on a risk-free asset, you must take the end outcome into account for both assets; in other words, you must utilize the geometric return. Any risk premium, which the investor requires to be paid in exchange for accepting risk, must be greater than the compound return of the risk-free investment. Also, when calculating historical risk premiums (i.e., what was really received), it is important to remember to employ compounding rather than simply taking an average.

Assume that you are looking at the same 33-year period from December 1979 to December 2012, but you are looking at the return on Australian bond investments instead.

Arithmetic average returns were 9.9 percent each year for the course of the investment.

This is the same as calculating the equity risk premium from our 33-year sample period as 4.0 percent per annum (from 13.9 – 9.9 = 4.0) when it is only 2.0 percent per annum, using the geometric returns (11.6 – 9.6 = 2.0) when it is only 2.0 percent per annum.

#### The right set of scales for ‘weighing’ returns

Self-directed investors would be wise to inquire of themselves or their financial advisors about the performance of their investments over the previous 12 months, as well as over longer periods of three, five, and ten years (or longer). Consider the situation of an SMSF trustee who does not have access to the appropriate counsel or resources. What method do they use to do this? Chances are that they will overestimate their returns if they construct a spreadsheet containing each return from the relevant periods and then just average them.

However, it is simple enough for a self-directed investor to average these 12-month measurements improperly (e.g., using a simple arithmetic mean) over numerous periods and, even worse, to use the resulting estimate of future wealth increase as a basis for making investment decisions.

It is the geometric mean that is the most appropriate set of scales for this task, at least until you look at the difference between time-weighted and money-weighted returns, but that is a matter for an other discussion.

In his current position at Challenger Limited, Aaron Minney is the Head of Retirement Income Research. His essay was written with the assistance of Liam McCarthy, a Senior Research Analyst at the University of Michigan.

## Difference Between Arithmetic Mean and Geometric Sequence (With Table) – Ask Any Difference

Inquiring about the performance of their assets over the previous 12 months, as well as over longer periods of three, five, and ten years, is a good idea for self-directed investors (or longer). Take for example a trustee of an SMSF who does not have access to the appropriate counsel or tools. What is their method of accomplishing this? Chances are that they will overestimate their returns if they construct a spreadsheet with each return from the relevant periods and then just average the results.

For a self-directed investor, however, it is simple enough to average these 12-month measurements improperly (i.e., by applying a simple arithmetic mean) over successive periods and, worse, to use the resulting estimate of future wealth increase as a guide.

It is the geometric mean that is the most appropriate set of scales for this task, at least until you consider the difference between time-weighted and money-weighted returns, which is a topic for another day.

Liam McCarthy, a Senior Research Analyst, supported him in putting together this piece.

## Arithmetic Mean vs Geometric Sequence

There is a significant distinction between Arithmetic Mean and Geometric Sequence in that arithmetic mean is used to discover the average of a collection of numbers, but geometric sequence is just a collection of numbers with a fixed ratio (as opposed to arithmetic mean). Arithmetic mean or simply average is a collection of numbers divided by the total number of numbers, whereas geometric sequence is a collection of terms formed by dividing or multiplying a constant term by the number of terms in the collection As opposed to an average calculated from a sequence of numbers, the word “arithmetic mean” refers to the average produced from an organized collection of terms in a repeated pattern.

The words ‘arithmetic mean’ and ‘geometric sequence’ are two mathematical concepts that are frequently used to describe the methodical structuring of terms that have been discovered.

When these terms are present in a definite ratio, the ratio is determined by the common ratio.

## Comparison Table Between Arithmetic Mean and Geometric Sequence (in Tabular Form)

Parameters of Comparison Arithmetic mean Geometricsequence
Definition The arithmetic mean is the average of the collection of numbers in a given sequence. Thegeometric sequenceis the collection of terms with the difference in the ratio of two consecutive terms being constant.
Determined by It can be determined by dividing the sum of the collection of numbers by the total count of numbers. It can be determined by multiplying or dividing aconstantto the preceding term.
Form This is expressed as an average of the collection. This sequence is usually expressed in the exponential form.
Common formula A= (a1 + a2+. + an)/n (where a is the 1st digit and n is the total number of digits we can find the mean A through this formula) tn = t1. r(n – 1) (where r is the common ratio and tn is the nth term, t1 is the first term)
Uses Thearithmeticmean or the average is used in the observational and experimental studies to get a brief idea of the large sample size because mean then becomes the central tendency of the data. ageometric sequenceis used in various sectors such as financial and economic sectors to calculate the growth rates, savings, costs, etc.

arithmetic mean is the average of a series of words that may or may not be separated by the common difference, as defined by the definition above. To calculate the mean, we divide the sum of a collection of terms by the total number of numbers included in the collection. Due to the fact that the’mean’ of any given data is always the central tendency of that data, the average or arithmetic mean is the most straightforward and easy approach of sizing down a big sample size. If you are conducting experimental research or observational studies, you may compute the mean by summing the total number of observations divided by the total number of observations, which is expressed as: The Arithmetic Mean is defined as (the sum of all the observations divided by the number of observations) (total number of observations) When the data is presented in the form of a sequence, the average of any sequence may be calculated using the following formula: An is equal to (a1 + a2+.

+ an)/n.

This may be accomplished with relative ease using the aforementioned formula: (2+4+6+8+10)/5= 6.

The average is extremely important in the domains of anthropology, history, statistics, and the calculation of per capita income, among other things.

It is not possible to use average as a formula for computations in financial data when each figure of a term is important; hence, average cannot be employed.

## What is Geometric Sequence?

A geometric sequence is a succession of integers in which the words that follow each other have the same ratio. Simply put, when a progression is multiplied or divided by the same, non-zero number, the sequence that results is referred to as a geometric sequence. This sequence can be represented by the letters a, ar, ar 2, ar 3, ar4, and so on (where a is the 1 stterm and r is the common ratio) As an illustration: 3, 9, 27, 81, and are all prime numbers. The geometric sequence is stated in the exponential form by the formula:t n= t 1.r (n – 1), where t is the number of steps in the series (wheret nis thenth term, t 1is the first term and d is the common ratio) Geometrical sequences appear to be a little more difficult to figure out than the arithmetic mean, yet they have several applications in everyday life, such as estimating growth rates, stock markets, interest rates, and other financial indicators.

## Main Differences Between Arithmetic Mean and Geometric Sequence

1. Arithmetic mean is calculated by dividing the total of the collection of words by the number of terms in the collection
2. Conversely, geographic sequence is defined as a series of successive terms with the same common ratio. It is possible to acquire the arithmetic mean by multiplying the collection of terms together and dividing them by the number of terms present, whilst the geometric sequence may be derived by multiplying or dividing the constant non-zero term from the preceding number
3. Arithmetic mean represents the center tendency of a data set, whereas geometric sequence represents the exponential variance of a data set. The arithmetic mean is often utilized in observational studies, experiments, and other similar endeavors, but the geometric sequence is widely used in the financial markets, to compute savings, costs, and other similar endeavors. It is also used to shrink down enormous amounts of data in order to obtain an estimated clue of the outcomes, whereas geometric sequence is the sequence of accurate data. It is consequently not always possible to obtain accurate findings using the arithmetic mean.

When it comes to numbers, an arithmetic mean is the average of a collection of numbers where the common difference between the successive terms may or may not be defined by an integer, whereas a Geometric Sequence is simply a sequence of numbers where the successive terms must have a common ratio defined by the integer ‘r’. Unlike the arithmetic mean, which is calculated by dividing the total number of terms in the series by the sum of all of the terms in the series, the geometric sequence is computed by multiplying or dividing the consecutive terms with the common ratio.

If we take a look around us, we will notice that both the arithmetic mean and geometric sequences have applications in our everyday lives.

## References

As soon as Sam thinks of the word “average,” the first thing that comes to mind is his recollection of calculating averages on his high school examinations. He’s thinking in terms of an average of this kind: if his past five exam scores were 95 percent, 80 percent, 99 percent, 86 percent, and 90 percent, his average would be 90 percent. He’s thinking in terms of a 95 percent average. Arithmetic average is calculated using the formula (a+b+c+d+e)/n, wheren is the number of data points in the average and is the sum of the data points in each category.

For example, using the same technique for calculating Sam’s returns on his investments, we might conclude that if Sam’s returns for the previous five years were -3 percent, 5 percent, 10 percentage points, 2 percentage points, and 20 percentage points, the arithmetic average would be 6 percent.

In other words, Sam’s performance on each exam has absolutely nothing to do with his performance on the preceding test.

Returns on investments, on the other hand, are extremely dependant on how the investment has fared in the past. As a result, the arithmetic average may be deceiving, so let’s have a look at what happens when we employ a method that takes historical performance into consideration.

## The Geometric Average

When computing the average return on an investment, the geometric average takes into consideration how the investment has fared in the past. Consider the following case in point: After making a 100 percent return on an investment in year one and a negative 50 percent return in year two, the arithmetic average for that investment would be 25 percent. The total cost, on the other hand, remains at \$100. With inflation, Sam has really lost money, as opposed to making any real gains from his investments.

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## Geometric Vs Arithmetic Return Example

Both arithmetic return and geometric return are ways of calculating the yield on a particular investment that are extensively employed in the financial industry. In reality, it is the geometric return that matters, not the arithmetic return, which is the most important. Analysts who have a thorough awareness of the differences between the two approaches of calculating returns are better positioned to make sound investment decisions. A prudent investment considers the volatility, or, more accurately, the risk associated with a particular investment.

## Arithmetic Mean Return

In order to compute the yield on a particular investment, both the arithmetic return and the geometric return are regularly utilized approaches. The geometric return, rather than the arithmetic return, is the one that counts most in this case. Analysts can make more informed decisions about their investments if they grasp the differences between the two approaches of estimating returns. Investments that are prudent take into consideration the volatility or, more accurately, the risk associated with the investments they make.

1. The compounding of returns, as well as the fluctuations in the percentage return gained from year to year, are important considerations.

#### Example: Arithmetic Mean

For our purposes, let us assume that we have a 6-year series of investment returns that looks like this: Solution The arithmetic mean return is just the total of all of the returns divided by the number of returns, in this example, which is six. \$\$ text=cfrac=0.05 text5 percent \$\$ text=cfrac=0.05 text5 percent \$\$

## Geometric Mean Return

In order to compute the geometric mean return, we must first go through the stages given here:

1. First, multiply each return by one. Negative values should be avoided at all costs
2. All returns should be multiplied
3. Product raised to power of 1 divided by number of returns ‘n’
4. And lastly, one should be subtracted from the final result.

#### Example: Geometric Mean

To begin, double each result by one hundredth. The secret is to prevent issues caused by negative values; to multiply all of the returns in the sequence; to raise the product to the power of 1 divided by the number of returns ‘n’; and, lastly, to remove one from the final result.

### Arithmetic vs. Geometric Mean Returns

When the media and investing institutions utilize the mathematical return wrong, they have the potential to deceive an investor. Taking the preceding example into consideration, a fund manager will most likely quote a return of 5 percent. Unfortunately, this does not represent the true return! The real return is a negative percentage (a loss). Perhaps it would be easier for you to grasp the concept if we work using real-world figures. Suppose you invested in an emerging mutual fund and received 100% in the first year, followed by a 50% loss in the second year.

There will be a 25 percent arithmetic mean return, which is equal to (100 – 50)/2.

Take, for example, a \$1,000 investment that grows to \$2,000 at the end of year 1, and then shrinks to \$1,000 at the end of year 2.

For the reasons stated above, the geometric return is always a superior measure of investment success when compared to the arithmetic return, unless there is no volatility in the return on investment.

Then, when the volatility increases, the discrepancy between the arithmetic mean return and the geometric mean return grows larger.

## Why Arithmetic And Geometric Averages Differ In Trading And Investing (Position Sizing And The Kelly Criterion) – Quantified Strategies

Posted on December 18, 2021 by The distinction between arithmetic and geometric averages might be confusing. Albert Einstein is credited as claiming that compounding is the eighth wonder of the world. But what if he’s completely wrong? Is it possible that multiplicative compounding is the most damaging power known to mankind? The sequence of returns and the many potential pathways all contribute to determining your geometrical average, which may be significantly different from your arithmetic average in some cases.

• There are important distinctions between the mathematical and geometric averages, means, and returns used in trading and investing.
• It is possible to end up with losses – even financial ruin – even though your arithmetic average is positive.
• The cumulative consequences of compounding may cause you to incur losses from which you may never be able to recoup.
• As a result, we conclude this essay by discussing the Kelly Criterion, which is concerned with determining the ideal position and betting sizes.
• Due to the path/sequence and compound average growth rate, this is the cause behind this (CAGR).

## What is the arithmetic average?

Suppose you have an automated system that generates 10 transactions with the following percentage results: 11, 33, 6, -5, 7, 21, 19, 9, 29, and -24 (in percentage terms). 5 percent is obtained by adding all of the numbers together and dividing them by the number of observations (10). In other words, the average gain per deal was 5 percent on an annual basis. This is known as the arithmetic mean. Nevertheless, one disadvantage of using the arithmetic average is that it does not reflect your compounded return on your transactions or your final outcome.

## What is the geometric average?

Now, if you start with \$100,000 and add or subtract the ten deals shown above, you will end up with \$139,922. This equates to a compound annual growth rate (CAGR) of 3.35 percent, which is much lower than the arithmetic average of 5 percent. In finance, the compound annual growth rate (CAGR) is the rate of return on an investment over a certain time period.

Because of this, it varies from the arithmetic mean and median. It takes into consideration the compounding from the beginning to the end of the process. In other words, the arithmetic average and the geometric average are two fundamentally distinct concepts.

## A rithmetic vs geometric averages:Why do the arithmetic and geometrical averages differ?

There is a distinction between the two because the arithmetic average is mostly an abstract number, but the geometrical average is what you receive in real life based on the sequence of returns. For example, Mark Spitznagel argues in his Safe Haven book that you receive what you get, rather than what you anticipate. Because of the sequence and order in which the dice are rolled, it is possible that the theoretical average will not be obtained when the dice are rolled. When you roll the dice, you may obtain the numbers 1, 4, and 3, but the following sequence might be 2, 6, and 1.

We’ll demonstrate this with an example lower down the page.

When we use the geometric return, our outcomes are poorer.

Due to the volatility tax, this has happened:

## What is the volatility tax?

Over the course of my 25 years in the game of quantitative investing, and after being immersed in equations and models, I’ve zeroed down on the only arithmetic that really matters: minimizing drawdowns during adverse periods of market performance. Positive returns during good times are multiplicative in nature, which takes care of the remainder of the work. Wayne Himelsein may be found on Twitter. The term “volatility tax” comes from Mark Spitznagel’s book “Safe Haven – Investing for Financial Storms,” which he wrote on investing for financial storms.

• It is possible to avoid paying the volatility tax by using the arithmetic average.
• A 33 percent loss in one year necessitates a 49 percent gain in the next year in order to break even again.
• This is why Spitznagel refers to it as a “volatility tax” in his book.
• This is why you must keep your losses under control as well as your position size under control.

## An example of arithmetic vs geometric average

In his book, Safe Haven Investing, Mark Spitznagel provides a highly clear illustration of what he is talking about. He concocted a game in which he would roll dice and see which of the three possible results he would get:

• If the number 1 is drawn, you will lose half of your equity. If the numbers 2, 3, 4, and 5 show up, you will receive a 5 percent increase in your equity. If the number 6 is drawn, you will receive a 50% return on your investment.

It is possible to lose 50% of your equity if the number 1 is called. It is possible to gain 5% of your equity if the numbers 2, 3, 4, and 5 show up. It is possible to gain 50% of your equity if the number 6 is drawn.

## What is the Kelly Criterion?

An American mathematician called John Kelly developed a method in 1956 for determining the ideal betting size when the predicted returns are known. He did it using the difference between the arithmetic and geometric averages. The predicted geometric return, rather than the arithmetic average, is used to determine the criteria.

The Kelly Criterion is based on the logarithmic scale of the geometric anticipated return, which is used to determine the probability of a positive return. It seeks to maximize the expected value while taking into account the risk of disaster and losses.

#### How to calculate the optimal betting size by using the Kelly Criterion

The Kelly Criterion is simple to compute since it only requires two inputs to determine the optimal betting size: the number of bets and the amount of money wagered.

1. In trading, the win/loss ratio (R) is the proportion of deals that are profitable compared to those that are not (calculated by dividing the total profits of the winning trades by the total loss of the losing trades). This is the win ratio of the trading strategy (W), which is defined as the number of deals that resulted in a profit divided by the total number of trades that resulted in a loss.

You will need to include the following two variables in this formula: Kelly percent is equal to W – Consider the following scenario: you have a successful out-of-sample backtest of a trading technique that you wish to implement. The results of the backtest were as follows:

• The total profit from the successful transactions was \$9 229 137. The total loss incurred by the losing transactions is 3 206 730 (resulting in a win/loss ratio of 2.88)
• In this case, the win ratio is 74 percent (i.e., 26 percent of the transactions resulted in a loss)

When you plug this into the formula above, you get the following result: Kelly percent = 0.74 –= 0.6497 = 0.74 As a result, according to the Kelly method, the ideal betting size is 0.65 of your total equity. Of course, we cannot predict the success of a trading strategy in the future, thus the Kelly Criterion should be utilized with caution. Also, keep in mind that the worst is yet to come in terms of the drop in value. Always trade for a smaller size than you’d want, as we’ve stressed several times.

## Why betting size is extremely important

Let’s go back to Spitznagel’s example of the dice-rolling strategy. As an alternative to wagering 100 percent of your equity on each roll, you wager only 40 percent of your equity on each roll, with the remaining 60 percent lying idle and yielding nothing in the process. Because of this, your arithmetic average has dipped to 1.32 percent. However, an amusing thing occurs in relation to your anticipated final result: the geometric average improves from a negative 1.5 percent to a positive 0.6 percent!

#### The optimal betting size in a coin toss

According to a recent behavioral research conducted by experts, a group of 61 people was given 25 USD and instructed to wager equal money on a game that would result in heads 60% of the time, and they were requested to do so. With a time restriction of 30 minutes and the coin tossing 10 times each minute, the participants may put a maximum of 300 bets in that time frame. The total value of the awards was limited at 250 USD. What is the optimum bet size in this situation? According to the Kelly Criterion, the ideal betting size is around 20%, which corresponds to a gain of approximately 2% each coin flip.

What was the performance of the participants in this game when they had a high level of expectation of success?

Only 21% of the participants were able to attain the maximum number of participants.

If everyone had adhered to the Kelly Criterion’s recommended betting size, almost 95% of the participants would have attained the maximum amount of profit possible.

The backtest report for a trading strategy in the S P 500 is shown below. The time period covered by the backtest is from the commencement of SPY in 1993 to October 2021. The following statistics were obtained from the backtest:

• 533 transactions
• A compound annual growth rate of 15.4 percent if 100 percent of the equity is allocated (buy and hold was 10.4 percent)
• In the average deal, the trader gained 0.8 percent. The winning percentage was 74 percent. The average winner received 1.68 percent of the vote
• The average loser received a negative 1.72 percent of the vote. Despite the fact that theprofit factor was a strong 2.88, the Sharpe Ratio was 2

Amibroker provided the following findings for a simulated Monte Carlo study if we allocate 100 percent of our equity to each transaction for this strategy: Even with such excellent backtest data, the technique has a very high likelihood of failing (10 percent ). The following values are obtained if the allocation is reduced to 70 percent of the equity, with the optional allocation ranging between 65 and 70 percent: In all simulations, the yearly return is smaller because we allocate less money; nonetheless, we have a 0% danger of going bankrupt in every scenario.

Everything else is a secondary consideration.

## Conclusion – arithmetic vs geometric averages:

The arithmetic average should not be taken for granted: The arithmetic and geometric averages are not the same thing. The sequence of returns might result in geometrical returns and averages that are radically different from one another. In real life, there is only one road you can choose, and you best make sure you have a safety net in case that path turns out to be incorrect. You are given just one road to survival; you are not given the average of all possible paths. If you manage to get on the right track with multiplicative compounding, it may be quite beneficial; but, if you do it wrong, it can be extremely destructive.

We’ll round up the essay by highlighting our key trading tip once more: always trade smaller than you’d like to be.