## Definition

In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the th root of the product of numbers, i.e., for a set of numbers, the geometric mean is defined as.

### What is the 'Geometric Mean'

The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically defined as "the 'n'th root product of 'n' numbers." The geometric mean must be used when working with percentages, which are derived from values, while the standard arithmetic mean works with the values themselves.

### Explaining 'Geometric Mean'

The main benefit to using the geometric mean is the actual amounts invested do not need to be known; the calculation focuses entirely on the return figures themselves and presents an "apples-to-apples" comparison when looking at two investment options over more than one time period.

### Geometric Mean

If you have \$10,000 and get paid 10% interest on that \$10,000 every year for 25 years, the amount of interest is \$1,000 every year for 25 years, or \$25,000. However, this does not take the interest into consideration. That is, the calculation assumes you only get paid interest on the original \$10,000, not the \$1,000 added to it every year. If the investor gets paid interest on the interest, it is referred to as compounding interest, which is calculated using the geometric mean. Using the geometric mean allows analysts to calculate the return on an investment that gets paid interest on interest. This is one reason portfolio managers advise clients to reinvest dividends and earnings.

### Geometric Mean Calculation

To calculate compounding interest using the geometric mean, the investor needs to first calculate the interest in year one, which is \$10,000 multiplied by 10%, or \$1,000. In year two, the new principal amount is \$11,000, and 10% of \$11,000 is \$1,100. The new principal amount is now \$11,000 plus \$1,100, or \$12,100. In year three, the new principal amount is \$12,100, and 10% of \$12,100 is \$1,210. At the end of 25 years, the \$10,000 turns into \$108,347.06, which is \$98,347.05 more than the original investment. The shortcut is to multiply the current principal by one plus the interest rate, and then raise the factor to the number of years compounded. The calculation is \$10,000 × (1+0.1) 25 = \$108,347.06.

Uncertain linguistic hybrid geometric mean operator and its application to group decision making under uncertain linguistic environment
www.worldscientific.com [PDF]
… UNCERTAIN LINGUISTIC HYBRID GEOMETRIC MEAN OPERATOR AND ITS APPLICATION TO GROUP DECISION MAKING UNDER UNCERTAIN LINGUISTIC ENVIRONMENT. GUI-WU WEI. GUI-WU WEI. Department of Economics and Management, Chongqing University of … Geometric mean reversion: formulas for the equilibrium density and analytic moment matching
papers.ssrn.com [PDF]
… UNCERTAIN LINGUISTIC HYBRID GEOMETRIC MEAN OPERATOR AND ITS APPLICATION TO GROUP DECISION MAKING UNDER UNCERTAIN LINGUISTIC ENVIRONMENT. GUI-WU WEI. GUI-WU WEI. Department of Economics and Management, Chongqing University of … Geometric or arithmetic mean: A reconsideration
www.tandfonline.com [PDF]
… UNCERTAIN LINGUISTIC HYBRID GEOMETRIC MEAN OPERATOR AND ITS APPLICATION TO GROUP DECISION MAKING UNDER UNCERTAIN LINGUISTIC ENVIRONMENT. GUI-WU WEI. GUI-WU WEI. Department of Economics and Management, Chongqing University of … A monte carlo investigation of characteristics of optimal geometric mean portfolios
www.jstor.org [PDF]
… UNCERTAIN LINGUISTIC HYBRID GEOMETRIC MEAN OPERATOR AND ITS APPLICATION TO GROUP DECISION MAKING UNDER UNCERTAIN LINGUISTIC ENVIRONMENT. GUI-WU WEI. GUI-WU WEI. Department of Economics and Management, Chongqing University of … Mean-variance approximations to the geometric mean
www.worldscientific.com [PDF]
… UNCERTAIN LINGUISTIC HYBRID GEOMETRIC MEAN OPERATOR AND ITS APPLICATION TO GROUP DECISION MAKING UNDER UNCERTAIN LINGUISTIC ENVIRONMENT. GUI-WU WEI. GUI-WU WEI. Department of Economics and Management, Chongqing University of … A requiem for the use of the geometric mean in evaluating portfolio performance
www.tandfonline.com [PDF]
… UNCERTAIN LINGUISTIC HYBRID GEOMETRIC MEAN OPERATOR AND ITS APPLICATION TO GROUP DECISION MAKING UNDER UNCERTAIN LINGUISTIC ENVIRONMENT. GUI-WU WEI. GUI-WU WEI. Department of Economics and Management, Chongqing University of … The “fallacy” of maximizing the geometric mean in long sequences of investing or gambling
www.pnas.org [PDF]
… UNCERTAIN LINGUISTIC HYBRID GEOMETRIC MEAN OPERATOR AND ITS APPLICATION TO GROUP DECISION MAKING UNDER UNCERTAIN LINGUISTIC ENVIRONMENT. GUI-WU WEI. GUI-WU WEI. Department of Economics and Management, Chongqing University of … Estimating cost of capital in firm valuations with arithmetic or geometric mean–or better use the Cooper estimator?
www.tandfonline.com [PDF]
… UNCERTAIN LINGUISTIC HYBRID GEOMETRIC MEAN OPERATOR AND ITS APPLICATION TO GROUP DECISION MAKING UNDER UNCERTAIN LINGUISTIC ENVIRONMENT. GUI-WU WEI. GUI-WU WEI. Department of Economics and Management, Chongqing University of …

#### How do you calculate a geometric mean?

To find a geometric mean, take the product of all values and then take that result and raise it to some power (usually two). Then divide by how many values there are in total.

#### What does the geometric mean need to be used for?

The geometric mean must be used when working with percentages, which are derived from values, while the standard arithmetic mean works with the values themselves.

#### What does "arithmetic" mean in "arithmetic mean"?

Arithmetic means without any form of statistical manipulation.

#### What is the geometric mean?

The geometric mean is used to measure rates of change over time or ratios between quantities. It can be thought of as a type of average rate rather than an average amount.

#### What is the arithmetic mean?

The arithmetic mean, also known as the average, is a number that indicates the central or typical value of a set of numbers.

#### How does one calculate an arithmetic mean?

To find an arithmetic mean, add up all values and divide by how many values there are.

#### Why do you need to use compounding interest when calculating returns on investments that get paid interest on interest?

To calculate compounding interest using the geometric mean, investors first need to calculate the interest in year one which is 1 multiplied by 1 or 1. In year two, new principal amount is 11 times 1 plus 1 times 11 or 12 plus 11 or 22. In year three, new principal amount is 12 times 1 plus 21 times 11 or 123 plus 121 or 244 and so forth until 25 years later where it becomes 18347.6 which equals 98347.5 more than original investment after 25 years """.

#### How does the geometric mean work?

It focuses on return figures themselves and presents an "apples-to-apples" comparison when looking at two investment options over more than one time period.