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Both arithmetic return and geometric return are methods commonly used to calculate the yield on a given investment. However, the return that really matters is the geometric return, not the arithmetic return. A good understanding of the difference between the two methods of calculating returns helps analysts to invest wisely. A wise investment takes the volatility or, rather, the risk attached to an investment into account. For instance, is the return from year to year fixed or purely volatile?

Most companies report returns in the form of an arithmetic average because it is usually the highest average that can be announced. However, the arithmetic return is actually misleading unless the return earned is fixed for the entire investment period. Two factors cause the inaccuracy:

- the compounding of returns; and
- the fluctuation in the percentage return earned from year to year.

Assume that we have a 6-year sequence of investment returns as follows:

{40% -30% 40% -30% 40% -30%}

**Solution**

The arithmetic mean return is simply the sum of all the returns divided by the number of returns, ‘n’ (6 in this case):

$$ \text{Arithmetic mean return} =\cfrac {(0.4 – 0.3 + 0.4 – 0.3 + 0.4 – 0.3 + 0.4 – 0.3)}{6} = 0.05 \text{ or } 5\% $$

To calculate the geometric mean return, we follow the steps outlined below:

- first, add 1 to each return. The trick is to avoid problems posed by negative values;
- multiply all the returns in the sequence;
- raise the product to the power of 1 divided by the number of returns ‘n’; and
- finally, subtract 1 from the final result.

Again, assume that we have a 6-year sequence of investment returns as follows:

{40% -30% 40% -30% 40% -30%}

**Solution**

$$ \text{Geometric mean return} =(1.4 × 0.7 × 1.4 × 0.7 × 1.4 × 0.7)^{\frac {1}{6}} – 1 = -0.01 \text{ or } -1\% $$

The media and investment institutions can mislead an investor if they incorrectly use the arithmetic return.

Considering the above example, a fund manager will most likely quote the 5% return. Unfortunately, this is not the real return! The actual return is -1% (a loss).

Perhaps you will understand the idea better if we work with actual figures.

For example, suppose you invested in an emerging mutual fund and earned 100% in the first year, followed by a 50% loss in the second year. The arithmetic mean return will be 25%, i.e., (100 – 50)/2.

Applying the geometric mean return formula in the case outlined above will give you a mean return of zero! For example, if you start with $1,000, you will have $2,000 at the end of year 1, which will be reduced to $1,000 by the end of year 2. Thus, you earn a return of zero over the 2-year period.

In conclusion, the geometric return is always a better measure of investment performance compared to the arithmetic return, **unless there is no volatility of returns**. Then, the difference between the arithmetic mean return and the geometric mean return increases as the volatility increases.