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# How to Calculate the Geometric Mean of Return

## Geometric Mean Return

The geometric mean return, also called the geometric average return, is a way to calculate the average compounding rate of return on the investments. It considers the compound interests multiplied by the interest over the number of periods.

The geometric mean return is a good measure above the arithmetic return that calculates the interests in a simple arithmetic measure. In case of arithmetic returns, all interests of sub-periods are added and then the total is divided by the total number of sub-periods. The arithmetic average return is misleading in case of long-tenured investments because it overstates the true return. That is why arithmetic returns are used only in case of returns of shorter time periods.

While calculating interests for a longer period of time, the geometric average return (GAR) is a better formula that takes into consideration the order of the return and the compounding effect applied on the investment.

**Note** − Geometric average return is a rate of return for a series of terms using the products of the terms.

## Geometric Average Return Formula

The most commonly used formula to calculate the Geometric Average Return is −

$$\mathrm{[(1 + 𝑅_{1}) × (1 + 𝑅_{2}) × (1 + 𝑅_{3}) × … × (1 + 𝑅_{n})]^{\frac{1}{n}} − 1}$$

Where,

- R = rate of return
- n = number of periods

The geometric mean return formula is helpful for investors looking for an “apples to apples” approach of comparison when the investors consider multiple investment options and is specifically useful for investments that are compounded.

The formula allows one to calculate the holding period return, or the total return on the investment across multiple sub-periods.

**Note** − Geometric mean is more applicable over longer periods of time, and it is a better option than arithmetic mean too.

## Geometric Average Return Analysis

The geometric mean is called by many names, such as the compounded annual growth rate (CAGR), the geometric average, or the time-weighted rate of return (TWRR). It represents the rate of the average return for a set of values.

The CAGR takes ‘n’ numerous values (the interest return rates), multiplies all of them together, and puts them to the$(\frac{1}{n})^{th}$ power.

The best use of geometric mean return is for longer time periods, which means multiplying a lot more rates that are compounding at several sub time-periods. Therefore, the use of an Excel spreadsheet or calculator is evident for these calculations.

**Note** − Usually, Arithmetic mean return overstates and overestimates the average.

One of the important benefits of using geometric mean is that it doesn’t need the investment data. The calculation can be done using just the returns figures themselves. This is the reason why it is called an “apples-to-apples” comparison when considering numerous different investment options.

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