The Arithmetic Average Return is calculated by **adding the rate of returns of “n” sub-periods and then dividing the result by “n”**. In other words, the returns of “n” sub-periods are added and then divided by “n” to find the value of the average return.The Arithmetic Average Return is calculated by **adding the rate of returns of “n” sub-periods and then dividing the result by “n”**. In other words, the returns of “n” sub-periods are added and then divided by “n” to find the value of the average return.

Contents

- 1 What is the arithmetic average return calculator?
- 2 How do you find the arithmetic and geometric return?
- 3 How do you find the arithmetic mean percentage?
- 4 What is arithmetic rate?
- 5 How do you calculate rate of return over years?
- 6 How do I calculate average return in Excel?
- 7 How to calculate Arithmetic Average Return?
- 8 Annualized Return vs. Average Return
- 9 Calculating Average Return Using Arithmetic Mean
- 10 Average Return vs. Geometric Average
- 11 The Difference Between the Arithmetic Mean and Geometric Mean
- 12 The Formula for Arithmetic Average
- 13 How to Calculate the Arithmetic Average
- 14 The Formula for Geometric Average
- 15 How to Calculate the Geometric Average
- 16 Arithmetic Average Return
- 17 Formula
- 18 Example
- 19 How to Calculate Arithmetic Average
- 20 Why Use Arithmetic Average
- 21 Arithmetic Average Calculation
- 22 Calculating Arithmetic Average in Excel
- 23 Limitations of Arithmetic Average
- 24 Arithmetic Returns Vs. Geometric Returns
- 25 Arithmetic Average Return Calculator – Rate of Return Expert
- 26 The difference between arithmetic and geometric investment returns
- 27 Arithmetic Return Definition and Tutorial for Investment Modeling
- 28 Arithmetic Return for investment performance analysis
- 29 What’s Next?
- 30 Arithmetic and Geometric Mean, Total Return, Return Relative, Currency Conversions, and Risk Measurement
- 31 Calculating Investment Returns Involving Foreign Currency
- 32 Calculating the Average of Investment Returns
- 33 Cumulative Wealth Index
- 34 Measuring Investment Risk
- 35 Related Links
- 36 Geometric Average vs. Arithmetic Average: Which is Correct For Investment Returns? – Arbor Asset Allocation Model Portfolio (AAAMP) Value Blog
- 37 Arithmetic Mean Return – Excel Template • 365 Financial Analyst
- 38 Average Return

## What is the arithmetic average return calculator?

The Arithmetic Average Return Calculator is used to calculate the Arithmetic Average Return of an investment, given the initial value of the investment and the value of the investment at the end of each period.

## How do you find the arithmetic and geometric return?

A simple way to explain the difference is by taking the numbers 2 and 8. The arithmetic average is 5, being (2 + 8)/2 = 10/2 = 5. The geometric mean, on the other hand, is 4: exactly 20 per cent lower. This is calculated as v(2 x 8) = v16 = 4.

## How do you find the arithmetic mean percentage?

Calculate the percentage average To find the average percentage of the two percentages in this example, you need to first divide the sum of the two percentage numbers by the sum of the two sample sizes. So, 95 divided by 350 equals 0.27. You then multiply this decimal by 100 to get the average percentage.

## What is arithmetic rate?

The arithmetic mean is simply the sum of the all of the returns. divided by the number of periods over which the sum total is calculated. This is also called the average, or average return.

## How do you calculate rate of return over years?

ROI is calculated by subtracting the initial value of the investment from the final value of the investment (which equals the net return), then dividing this new number (the net return) by the cost of the investment, then finally, multiplying it by 100.

## How do I calculate average return in Excel?

Returns the average (arithmetic mean) of the arguments. For example, if the range A1:A20 contains numbers, the formula =AVERAGE(A1:A20) returns the average of those numbers.

## How to calculate Arithmetic Average Return?

This figure is produced by summing the rates of returns of”n”sub-periods and then dividing the result by the number of sub-periods in question. To put it another way, the returns of “n” sub-periods are combined together and then divided by “n” to determine the amount of the average return on investment. It is frequently referred to as “Arithmetic Average Return” since it is a technique that involves determining the average of a sequence of values. For those who are interested, below is the formula to compute Arithmetic Average Return (mathrm).

It is also used to establish the company’s range of products and services.

## Annualized Return vs. Average Return

When comparing “annualized returns” versus “average returns,” there are some distinctions. Returns on an annualized basis are computed on a year-to-year basis, and they are often compounded over a period of time. Average returns, on the other hand, are not compounded and are instead stated as simple interest in the computations. When calculating the return on equity investments, the average yearly return is utilized as a benchmark. The fact that yearly returns are compounded means that they are not regarded an ideal calculating approach, and as a result, they are only utilized in limited circumstances to determine the worth of changing returns.

## Calculating Average Return Using Arithmetic Mean

When using the arithmetic average approach, calculating the average return is straightforward. Take a look at the returns over the next five years. As a result, the arithmetic average return will be $$$mathrm=8.8$$.

## Average Return vs. Geometric Average

When assessing average previous returns, the geometric average is the best option. It takes into account the actual worth of the money that has been put in stocks or any other type of investment instrument. When examining the performance of a single investment across a number of time periods, the computation simply takes into account the return values and applies a comparison model to the data. Outliers caused by cash inflows and withdrawals over a period of time are taken into account when calculating the geometric average return.

Cash inflows and outflows are taken into consideration as well as the time and magnitude of cash inflows and outflows.

However, unlike the internal rate of return, the Money-Weighted Rate of Return (MWRR) has a net current value of zero, but the internal rate of return does.

- Questions that are related Answers
- To determine the average, a MongoDB query is used. Using MongoDB, compute the average value
- What is the Accounting Rate of Return and how do you calculate it? In MongoDB, how to calculate the average of a certain field is explained. Methods for calculating the Expected Rate of Return (ERR)
- Instructions on how to compute the Geometric Mean of Return
- In MySQL, learn how to determine an average value over all of the records in a database. In R, how to determine the monthly average for a time series object is explained. Using arrays, a C++ program to calculate the average of a set of numbers is written. What is the procedure for calculating the two-period moving average for vector elements in R? Average of integers is calculated using a Java program written in Java. The average of the numbers in a MySQL query column is calculated. To find out the average of a row of data, use the following MySQL query: Create an average from JSON data based on numerous filters by combining them together. JavaScript
- The average of the numbers in a given list is calculated using a Golang program

## 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 order to establish if an investment plan is effective, it is necessary to evaluate the performance of a financial portfolio in many ways. The geometric average, also known as the geometric mean, is frequently used by financial experts in their investments.

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.

## Arithmetic Average Return

HomeFinanceRisk and Return on Investment Average Arithmetic Return (AAR) Risk and Return in the HomeFinance average return calculated through arithmetic

## Formula

The arithmetic average return may be determined with the help of the following equation: It may be determined with the help of the Excel AVERAGE function.

## Example

Your university has established a $100 million endowment to support financial assistance programs that are offered on the basis of merit and need. Endowment returns for the first five years were 5 percent, 8 percent, -2 percent, 12, and 9 percent, according to the numbers provided by the fund manager. Consider the possibility that 100% of the return is in the form of capital gains. Arithmetic average return will be 6.4 percent, which is equivalent to (5% plus 8% plus -2% + 12 percent plus 9%)/5 (5 percent plus 8 percent + -2 percent)/5 According to the following calculations, the investment value after 5 years will be $135.67 million: Endowment after 5 years = $100 million (1 +5%) (1 +8%) (1 +2%) (1 +12%) (1 +9%)= $135.67 million.

5=$145.09 million (five million dollars).

If, for example, the endowment experienced a 2 percent decline in a year in which it had grown by 5 percent and 8 percent the year before, the arithmetic average return does not account for the compounding effect.

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## How to Calculate Arithmetic Average

The arithmetic average, sometimes known as the arithmetic mean, or simply the mean, is the most fundamental statistical metric. When it comes to general level of values in a data set, it is quick and simple to obtain this information. It is one of the measures of the central tendency of a population.

## Why Use Arithmetic Average

When you have a set of data, it can be difficult to tell what the values are in general (you can’t see the forest for the trees) because the data is arranged in columns. Consider the following scenario: you have ten stocks. Last year, they earned 11 percent, a 5% annual return on their initial investment, a 1% annual return on their initial investment in the following years: 2017, 1, 9, 21, 4, 6, 6, 7, and -1 percent in the previous year. The information supplied in this form will supply you with the specifics, but you will need to do some thinking in order to gain an idea of the yearly return of this group of stocks as a whole from the information provided.

In the previous year, the mean (arithmetic average) return of our basket of ten stocks was 4 percent.

## Arithmetic Average Calculation

Calculating the arithmetic average is a basic procedure. You add up all of the values and then divide the total by the number of values in the total. As an illustration of how to calculate the mean, let us look at the following example:

- To begin, add up the yearly returns: 11 percent plus (-5 percent) plus 17 percent plus 1 percent plus (-9 percent) plus 21 percent plus 4 percent plus (-6 percent) plus 7 percent plus (-1 percent) = 40 percent
- Next, subtract the annual returns from the total. Then you divide the sum (which in this case is +40 percent) by the number of observations (which in this case is ten), and you get the arithmetic average, which in this case is +4 percent.

## Calculating Arithmetic Average in Excel

Despite the fact that the procedure is straightforward, it may be tedious and prone to mistakes when dealing with huge amounts of data (imagine calculating the average return of the 500 stocks in S P500 like this). Computers do the arithmetic average calculation for humans. The AVERAGE function in Microsoft Excel may be used to calculate averages. The parameter refers to the area of the cells in which the individual values are contained.

## Limitations of Arithmetic Average

The arithmetic average is a simple and elegant way to get the first quick information about a data set, but it has some limitations, and it is sometimes preferable to use one of the other measures of central tendency, such as the geometric average, the weighted average, the middle of the distribution, or the mode. Furthermore, simply knowing the overall level of values is not always sufficient. There are a variety of other characteristics that you may want to measure, such as volatility or dispersion (using standard deviation or variance).

## Arithmetic Returns Vs. Geometric Returns

In the course of evaluating the returns on financial assets, we will frequently consider the returns from a variety of different holding periods. For example, if a person holds an asset for five years, the asset may have generated total returns of 150 percent over a seven-year period. The interpretation of these returns, on the other hand, is difficult because we cannot compare them to the returns on other assets. We will have to aggregate these returns over the same time period, such as daily returns, monthly returns, or yearly returns, in order to make meaningful comparisons.

- Depending on which method we use to calculate the aggregate returns, our return measure will differ when calculating the aggregate returns.
- Let’s look at an example to better comprehend each of these approaches.
- Arithmetic Has Its Payoff In order to compute the arithmetic average, we take the simple average of the five-year returns and divide it by five.
- As a result, the amount of the initial investment is considered to be the same for each period.
- Our metric, if it is based on mathematical returns, may be very wrong.
- Assume that the investment’s value increases to $200 in the first year.
- In year 2, the investment is reduced to $100, resulting in a return of -50 percent in the second year.
- This difficulty may be handled by computing geometric returns, which takes into account the compounding impact of the investment.

The geometric returns in our case can be computed in the following ways: Geometric returns are equal to (1/5) – 1= 10%. Clearly, geometric return is lower than arithmetic return, and it is a preferable strategy for aggregating returns over different holding periods, as shown in the example.

## Arithmetic Average Return Calculator – Rate of Return Expert

Given an investment’s beginning value and the value at the end of each period, this page calculates its Arithmetic Average Return, which is expressed as a percentage. For more instructions on how to use this tool, please see the Help tab. With the help of the Arithmetic Average Return Calculator, you can figure out what your investment’s Arithmetic Average Return will be based on its original value as well as its value at the conclusion of every period. Initial Value- Enter the value of the investment’s initial capitalization in this area.

- This number must be in the positive direction.
- Value at the beginning of each period- Enter the value of the investment at the beginning of each period.
- To add another period, use the plus sign (+) button.
- To compute the Arithmetic Average Return, click on theCalculatebutton in the toolbar.
- If any of the values have been entered improperly, an error notice will be shown next to the field where the wrong value was input.
- Simply click on theClearbutton to remove all information from the form and start again.

## 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.

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 different 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 per 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 demands 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

Don’t make the mistake of assuming that the gap is a component of the risk premia. The geometric return should be used in the comparison of the returns on a hazardous asset and the returns on a risk-free asset. If you are comparing the returns on two dangerous assets, you should consider the final outcome for both assets. There must be a difference between any risk premium that an investor wants to be paid and the compound return on the risk-free asset. Also, when calculating historical risk premiums (i.e., what was really received), it is important to remember to employ compounding rather than simply taking the mean.

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.

Arithmetic average returns were 9.9 percent each year during the course of the study.

This is the same as assessing 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 1.0 percent per annum.

Assuming our sample period of 33 years is representative, this 2 percent is the risk premium that will accumulate over time.

## Arithmetic Return Definition and Tutorial for Investment Modeling

Understanding the three techniques of determining return may be a game changer in one’s professional life. Here’s the shortest and most straightforward version.

- The Arithmetic Return must be defined. Calculation and interpretation- Examine how it is computed and understood. Use it in a sentence to put it into context. Video- Watch the video and read the transcript. Take a quiz to see how much you know

Originally published on August 25, 2016. The most recent update was made on February 16, 2021. Because it does not entail compounding, the arithmetic return is the most straightforward. See how to do so below. OutlineBackTipNext

## Arithmetic Return for investment performance analysis

Computing return across many time periods with BeginnerArithmetic Return is one of three approaches for calculating return. The term is frequently heard in the context of forecasting. After adding all of the returns together, you obtain what is known as the “total arithmetic return.” In turn, when you divide that amount by the number of observations, you get what is known as the “arithmetic return on average.” This is a time-uncertain transaction, which means that the order in which the returns are received does not important.

In this scenario, arithmetic returns are employed for forecasting since, while looking into the future, the investment modeler does not know in what order the returns will come in, and so compounding is not required in this case.

### How it is Calculated?

As an illustration, suppose the final value of an investment was $11 and the starting value was $10. This is known as a capital gain. In order to achieve a 10 percent outcome, the Excel formula would be =(11/10)-1. In mathematics and statistics, the difference between discrete and continuous data is frequently made. The log return is the more theoretical continuous variant of the log function. In the real world, however, most people view of returns as being divided into distinct intervals rather than as a continuous stream.

- It is possible to get an arithmetic return in one period that is not compounded and is discrete. Multiple period, compounded, and discrete geometric returns are all possible. There is no limit to the number of times a logarithmic return may be computed.

It is possible to get an arithmetic return in one period that is not compounded and is discrete; Multiple period, compounded, and discrete geometric returns are available. There is no limit to the number of logarithmic periods that can be used.

### In a Sentence

Doc: 0 percent or 25 percent, according to the mathematical return of 100 percentage points followed by a -50 percentage point Mia: I understand! Yes, I understand! Doc gets a quarter of the credit since math ignores compounding.

### Video

This movie may be accessed in a new window or App by clicking here, visiting our YouTube Channel, or by scrolling down to the bottom of this page. The concept of Arithmetic Return in the context of investment modeling (4:24)

### Video Script

We will illustrate and show the computation of arithmetic return in two portions of the script, which are included in this document.

#### Visualize

This is an excerpt from our boot camp course, which we’re working on right now in Microsoft Excel. Let’s break this down into two periods, although the concept applies to three and beyond. We have two different versions of the arithmetic return. The arithmetic total return, which is also known as the arithmetic return, is calculated by adding all of the returns together. And to get the average, also known as the arithmetic mean, add up all of the observations and divide by the number of observations.

In this case, I placed the total in parenthesis since it’s similar to what you’d do in Microsoft Excel. Look at a chart for a moment since the eye often notices something that you would otherwise miss in a table. Okay, everything appears to be in order.

#### Demonstrate

Consider the total and average of two hypothetical stocks across three periods, as shown in the chart below. Ten percent returns were received by ABC, followed by a -11 percent return, and then another ten percent return. XYZ had returns of 100 percent, followed by a drop to -50 percent, and finally a return of zero percent. Logic dictates that the whole arithmetic return calculation be performed. You may either add them one by one manually, as shown here, or use the =SUM() method, with the range of returns enclosed in parentheses, as shown here.

- Additionally, for the arithmetic average, you might manually divide the number by two as seen below.
- You could wonder, when would you pick the arithmetic approach over the other methods.
- While it is impossible to anticipate the sequence of returns in the future, it is possible to predict averages over lengthy periods of time, which is usual.
- Allow me to demonstrate why.
- It is more accurate in reflecting the genuine client experience over a longer period of time.
- 3,67 percent compared to 2,5 percent, and 16,67 percent in comparison to 0 percent Hmmm.

### Quiz

To find out the solution, click on the box. When the average arithmetic return is positive, it is always the same as or larger than the average geometric return. | True or False? True. For the same location, Geometric returns compound at a lower rate than the other methods. The average arithmetic return of 10 percent, -10 percent, and 0 percent (0 percent or -0.3 percent) is what percentage of the total return? 0 percent of the population.

#### Questions or Comments?

Are you still unsure about Arithmetic Aeturn? Fill out the form below to ask a question in the YouTube comments area. Also, visit the Quant 101Series’ teaching page and video on stock return computation methodologies for more information. There, we go through the differences between arithmetic, geometric, and log returns, as well as provide examples in Excel.

### Related Terms

Other keywords in the category return math were discovered by our trained humans, which you may find useful.

## What’s Next?

Subscribe to our YouTube channel so that you may be alerted whenever we release new videos. Additionally, our no-spam email list is only utilized infrequently.

- To see all of the terms in the Glossary, select Outline from the drop-down menu. To read more about Arbitrage Pricing Theory, go back to the previous page. Let’s not make this uncomfortable, so click Tip. Following the word Autocorrelation, click on the next button.

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## Arithmetic and Geometric Mean, Total Return, Return Relative, Currency Conversions, and Risk Measurement

To make money from investing, you must first understand how much money you are making or losing. You must be able to compare the returns of several assets in order to determine which ones to purchase. Several investments earn the majority of their return through the payment of dividends, distributions, or interest, which is often expressed as ayield, which is a percentage of the asset’s value or the purchase price; other investments, such as growth stocks, earn a return solely through changes in their price, which is known as a capital gain or a capital loss.

Total Return FormulaTotal Return | = | End Price – Initial Price + Income Received
Initial Price |

In the absence of income or a change in the price of an investment, the return will be defined by the other component of the total return, which is obvious. The term “return relative” (RR) is also frequently used to compute investment returns, which is just the sum of the whole end value of the investment plus income divided by the starting value of the investment, as in the example below. As a result, it has the following return in relation to the beginning price:

Return Relative FormulaReturn Relative | = | End Value of Investment+ Income Received
Initial Price |

Principal: | $1,000 |

Coupon Rate: | 6% |

Initial Price: | $945 |

Sale Price: | $1,005 |

Total Return: | 12.70% |

Return Relative: | 1.1270 |

Initial Price: | $100 | |

Sale Price: | $94 | |

Dividend: | $4 | |

Total Return: | -2.00% | |

Return Relative: | 0.9800 | = (Income + End Price) ÷ Initial Price |

Using the return relative calculation method has the primary advantage of avoiding negative values, which are incompatible with various computations, such as the arithmetic or geometric mean, or the cumulative wealth index.

## Calculating Investment Returns Involving Foreign Currency

Additionally, the return relative may be used to convert a return paid in a foreign currency to a return paid in the local currency. In the case of a foreign investment, fluctuations in the foreign exchange rate will either boost or decrease the overall return on the investment in terms of the local currency. The following formula may be used to compute the total return in the home currency: multiply the rate of return relative by the end value of a foreign currency divided by the starting value, then subtract 1:

Total Return Formula in Domestic Currencyin Terms of Foreign CurrencyTotal Return in Domestic Currency | = | Return Relative | × | Foreign Exchange Rate at End of Term
Foreign Exchange Rate at Beginning of Term |
– | 1 |

Buy European Stock in Euros | 100 | |

Dollar per Euros at Initial Investment | 1.25 | |

Investment Value When Sold in Euros | 200 | |

Dollar per Euros at End of Investment | 1.35 | |

Return Relative for Investment | 2 | =End Investment Value/Start Investment Value |

Total Return in USD | 1.16 | =Return Relative×End Foreign Currency Value / Start Foreign Currency Value- 1 |

## Calculating the Average of Investment Returns

Calculating the average of investment returns may be done in a number of ways, the most common of which are the arithmetic mean and the geometric mean. Arithmetic mean is just the total of the returns for each year divided by the number of years in the period under consideration. However, because it does not take into account compounding, the arithmetic mean is rarely correct over a series of holding periods. When the returns on investments are negative in some years, the variation from the real average investment return can be significant, as in the case of the stock market.

Normally, an investor would expect an average annual return of (100 percent – 50 percent)/2 = 25 percent, but in this case it is really 0 percent, because the stock is worth exactly what it was at at the beginning of the holding period.

Because it takes into account compounding, the geometric mean is more accurate than the arithmetic mean: The Geometric Mean is equal to 1/n–1.

As a result, the geometric mean for the $10 stock in the example above would be (1 + 1) – (1 -.5)– 1 = 2 – 1 = 1 – 1 = 0 percent.

Example: Calculating the Arithmetic and Geometric Mean for the S P 500 Index from 2000 to 2006Year | Return | Return Relative |

2000 | -10.14% | 89.86% |

2001 | -13.04% | 86.96% |

2002 | -23.37% | 76.63% |

2003 | 26.38% | 126.38% |

2004 | 8.99% | 108.99% |

2005 | 3.00% | 103.00% |

2006 | 13.62% | 113.62% |

Total of Returns | 5.44% | |

Arithmetic Mean = | 0.78% | |

Geometric Mean = | -0.50% | |

Index at Start of 2000 | 1469.25 | |

Index at End of 2006 | 1418.30 | |

Predicted Ending Index using Arithmetic Mean = | 1551.06 | =Initial Value* (1 +Average Return Rate) n |

Predicted Ending Index using Geometric Mean = | 1418.20 | =Initial Value* (1 +Geometric Mean) n |

n = Number of Compounding Periods |

Because the S P example contains several years with negative returns, the disparity between the arithmetic mean and the geometric mean is rather considerable compared to the other examples. The geometric mean will almost always produce the right and accurate answer, however the example was a bit off due to rounding mistakes in the data. It is also possible to adjust total returns for inflation by dividing the total return over a certain period of time by the inflation rate during that same period of time, which is typically one year.

Inflation-Adjusted Total Return FormulaTR a | = | 1 + TR
1 + IR |
– | 1 |

- The total return after accounting for inflation is denoted by the letter TR a. IR is an abbreviation for Inflation Rate.

TR a | = | 1 + 10%
1 + 3% |
– 1 | = | 1.1
1.03 |
– 1 | ≈ 6.8% |

- The total return before adjusting for inflation is 10%
- The inflation rate is 3%
- The total return after adjusting for inflation is 10%.

## Cumulative Wealth Index

The cumulative wealth index (CWI) is essentially the return earned by a specific beginning amount of money over a number of years, represented as a decimal multiple of the initial amount. The following is a typical computation that utilizes $1 as the original investment, with the returns compounded annually: CWI n= WI 0 + (1 + TR 1), (1 + TR 2),., (1 + TR n). CWI n= WI 0 + (1 + TR 1), (1 + TR 2),., (1 + TR n). Assuming that the beginning wealth (WI 0) is set at one dollar, the cumulative wealth index simplifies to:CWI n= (1 + TR 1) (1 + TR 2), and so on until (1 + TR n).

Make a note of the fact that if you had started with a $1 investment, the cumulative wealth index would have been 1.26; thus, to determine the cumulative wealth index for any amount, simply multiply that amount by the cumulative wealth index.

## Measuring Investment Risk

An investment risk is the possibility that investment returns will be lower than expected or that investments will suffer losses in excess of those anticipated. The greater the chance of anything happening, the greater the danger. It is reasonable to expect that a higher risk investment may possibly provide a higher return; otherwise, the investor would never take on the risk if there was no chance of earning more in the long run. Because the normal distribution curve can usually be used to represent the dispersions of investment returns, another way of looking at this is to say that the probability of any losses equals the probability of any gains, because the area under the normal distribution curve before the mean equals the area under the distribution curve after the mean (and vice versa).

Because the investment returns of riskier assets have a wider dispersion — meaning that investment returns fluctuate more widely than those of less risky assets — the variance and standard deviation are employed to assess this dispersion and, consequently, the risk associated with such investments.

When calculating the variance of investment returns, these values are taken into consideration.

If an asset always returned the mean, there would be no dispersion and, as a result, no risk in making the investment because it would always pay the projected rate of return, as long as it always returned the mean.

As a result, the United States Treasury bonds pay the lowest rate of interest. The following equation may be used to calculate variance:

- X k is the total return for the k th sample asset
- X is the sample mean
- N is the sample size
- And 2 is the variance.

To calculate the total return for each sample asset, first divide it by the sample mean; then divide by the sample size; and then divide by the variance of the total return.

## Related Links

- Nominal and real interest rates
- Compounded and continuously compounded interest rate formulas
- Nominal and real interest rates
- Nominal and real interest rate formulas Calculating the time value of money
- Determining the present and future value of a dollar, using formulas and examples The Present Value and Future Value of an Annuity, as well as the Net Present Value, are discussed in detail, with formulas and examples.

## 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 accurate, 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.

5 percent return each period).

### Volatility Lowers Investment Returns

Let us look at a simple example to demonstrate how volatility reduces the profits on your investments: As an illustration, consider a 50 percent gain and a 50 percent loss. It makes no difference whether sequence the steps are taken in; the effects are the same. Arithmetic Average = (.50 + (-.50)) / 2 = 0 Arithmetic Average Many people are under the impression that they have reached break even. In truth, they have suffered a 25 percent decrease in capital! If an investor starts with $100 and loses 50% of his or her investment, he or she will have $50.

The findings are the same whether the returns are returned in the reverse order or not.

Their capital will be reduced to $75 as a result of the 50 percent decrease.

The geometric average (which I will demonstrate momentarily) will reveal the real findings, which are as follows: geometrical average = -1 =.1339745962 or -13.4 percent geometrical average Proof: Year 1: $100.0 – 13.4 percent = $86.6 in the first year.

## Arithmetic Mean Return – Excel Template • 365 Financial Analyst

Arithmetic Mean Return is a way for evaluating the return on an investment over a period of time that is greater than one. It is computed by adding up all of the returns for all of the sub-periods and then dividing that amount by the number of periods in the whole period. Other similar subjects that you might be interested in learning more about includeGeometric Mean andHolding Period Return, among others. XLSX format is an open-access Excel template that will be valuable for anybody aspiring to a career in the fields of statistics, financial analysis, data analysis, or portfolio management.

You can now obtain a free copy of the Excel template by clicking here. When studying for the CFA Level 1 Curriculum, Arithmetic Mean Return is one topic that will be covered in the Portfolio Management module.

### 3-Statement Model – Excel Template

Calculating the return on an investment over a period of time is done using the Arithmetic Mean Return approach. It is computed by adding up all of the returns for all of the sub-periods and then dividing that amount by the number of periods in the overall period. Other similar subjects that you might be interested in learning more about include Geometric Mean and Holding Period Return, to name a couple. Everyone who wishes to work in the fields of statistics, financial analysis, data analysis, or portfolio management will find this open-access Excel template in XLSX format to be quite useful.

When studying for the CFA Level 1 Curriculum, Arithmetic Mean Return is one topic that will be covered in the Portfolio Management Module.

### 3-Statement Model – Excel Template

The profit and loss statement, the balance sheet, and the cash flow statement are all interconnected. The profit and loss account (P L) transfers net income to the liabilities and equity sides of the balance sheet. A cash asset is also created by adding up the bottom-line results of the Cash flow statement for each year since the preceding year. Read on to find out more

### Cash Flow – Excel Template

The cash flow statement depicts how a company generated and spent money over the course of a specified time period. This is a significant fact.

### Cash Flow – Excel Template

The cash flow statement depicts how a corporation produced and spent money over the course of a specified time period. An essential fact that novice business owners usually overlook is the fact that profit does not always equate to cash. Every business owner and management must have a thorough understanding of their company’s cash flow. Read on to find out more

### Balance Sheet – Excel Template

If the profit and loss statement demonstrates how lucrative a firm was over a specified period of time, we may argue that the company was profitable.

### Balance Sheet – Excel Template

The profit and loss statement reveals how successful a company was over a certain period of time, and the balance sheet is like a snapshot of the firm’s position at the time of preparation, we may compare the two statements. The balance sheet indicates what a corporation possesses (assets) and what it owes (liabilities) to other people and organizations. Read on to find out more

## Average Return

The arithmetic average of a series of returns that have accrued over time is known as the average return. In its most basic form, average return is the sum of all returns during a period of time divided by the number of periods in the period.

### Summary

- The average return is a statistic that calculates the value of a sequence of returns that have accumulated over time by taking a mathematical average
- In order to determine the average growth rate, which measures the rise or decrease in value of an investment over a specific period, the average return is needed. Because of the numerous problems in the method of calculating the internal rate of return, investors and analysts prefer to utilize money-weighted returns as an alternate method.

The average return, just like the basic average, is determined by combining a collection of values together to get a single total. Although there are numerous ideas that may be utilized to determine the average return, the arithmetic average return is calculated by dividing the entire sum of numbers in the series by the total number of numbers in the series, as shown by the formula below: The average return is used by investors and market experts to determine the historical returns for stockStock.

An individual who holds stock in a corporation is referred to as a shareholder, and he or she is entitled to receive a portion of the corporation’s leftover assets and earnings (should the company ever be dissolved).

orsecurity Security A security is a financial instrument, which may be defined as any financial asset that can be exchanged on a stock exchange.

The nature of what may and cannot be referred to as a security is determined by the jurisdiction in which the assets are being exchanged, in most cases. A company’s portfolio yields are determined by calculating the average return over a period of time.

### Annualized Return vs. Average Return

When showing historical returns, the annualized return is compounded, however the average return does not take into account compounding. When evaluating the performance of equity investments, the average annual return is widely utilized. Annual average return, on the other hand, is often not regarded a suitable analytical measure due to the fact that it compounds; as a result, it is seldom used to examine fluctuating returns. In addition, the annualized return is calculated using a regularmean distribution.

It is frequently employed as a parameter.

### Calculating Average Return Using Arithmetic Mean

The simple arithmetic mean is a common illustration of an average rate of return. Consider the case of a mutual investment that pays the following every year for six years in a row, as indicated in the chart below. It is possible to compute the average return for six years by adding up the yearly returns and dividing by six, which is expressed as follows: The annual average return is calculated as follows: Return on an annual basis = (15 percent plus 17.50 percent plus 3 percent plus 10 percent plus 5 percent plus 8 percent) / 6 =9.75 percent If we look at hypothetical returns for Wal-Mart (NYSE: WMT) between 2012 and 2017, we get a different picture.

Table 1 shows the company’s return on investment, which includes the following figures: The average return for Wal-Mart over a period of six years is computed using the same methodology.

### Computing Return From Value Growth

In order to determine whether the value of an investment has increased or decreased over a period of time, the average growth rate is utilized. It is possible to calculate the growth rate by utilizing the growth rate formula: Consider the following scenario: an investor made a $100,000 investment in an investment product, and the stock price varied between $100 and $250. When the average return is calculated using the formula above, the following results are obtained: This means that the returns will now be $160,000 because the growth rate is equal to ($250 – $150) / $250 =60 percent.

### Average Return vs. Geometric Average

When assessing historical returns on average, the geometric average appears to be the most effective method. What causes the geometric mean to be set? Geometric Mean (Geometric Mean) The geometric mean is the average growth rate of an investment calculated by multiplying n variables and then calculating the square root of the resultant number. It is the average return, with the exception of the fact that it assumes the real value of the investment. When examining the performance of more than one investment over a period of time, computation only considers the return values and use a comparison concept to make comparisons.

As a result, it is referred to as the time-weighted rate of return in some circles (TWRR).

As a result, the TWRR becomes a more exact measure of returns on a portfolio that has seen withdrawals or other activities – such as the receipt of interest payments and deposits – in the last year.

The money-weighted rate of return (MWRR) is the same as the internal rate of return (IRR), where zero represents the net current value of the investment.

### Limitations of Average Return

Despite its popularity as a simple and effective measure of internal returns, the average return has a number of drawbacks that must be considered. It does not take into consideration the fact that various projects may need varying capital expenditures. In the same spirit, it makes no consideration for future costs that may have an impact on profit; rather, it simply considers expected cash flows as a consequence of a capital investment. Furthermore, average return does not take into account the rate of reinvestment; rather, it implicitly implies that future cash flows may be recreated at rates equivalent to the internal rates of return.

Investors and analysts prefer to utilize money-weighted return or geometric mean as alternative metrics for analysis as a result of these shortcomings.

### More Resources

Get your Commercial BankingCredit Analyst (CBCA)TM certification from CFI, the official supplier of the program. Program Page – CBCAGet your Commercial BankingCredit Analyst certification from CFI, the official provider of the program. Enroll in one of our certification programs or take one of our courses to boost your profession. Anyone may benefit from this certification program, which is meant to turn them into world-class financial analysts. We strongly advise you to continue studying and enhancing your understanding of financial analysis by using the extra resources listed below:

- Total Return on an Annualized Basis Total Return on an Annualized Basis An annualized total return is the return on an investment that is received on a yearly basis. It is calculated as a geometric average of the returns earned in each year over a period of time. Obtaining a Return on Investment (ROI) Obtaining a Return on Investment (ROI) Profitability (also known as Return on Investment (ROI)) is a performance indicator used to analyze the returns on an investment or to compare the efficiency of other investments. Annualized Growth Rate on a Percentage Basis Annualized Growth Rate on a Percentage Basis The average annual growth rate (AAGR) is the average yearly increase in the value of an investment asset, portfolio, or cash flow over a certain period of time. Annualized Rate of Return (ARR) is a measure of how profitable a business is on an annual basis. Annualized Rate of Return (ARR) is a measure of how profitable a business is on an annual basis. Calculating investment returns on an annual level is accomplished through the use of annualized rate of return (ARR). When we invest, we are frequently interested in knowing how much money we are making.