How is the Geometric Mean calculated for compounded returns?

The geometric mean is a crucial measure for calculating compounded returns, as it accurately reflects the average rate of return over multiple periods when considering the effect of compounding. The formula for the geometric mean of a series of compounded returns takes the form of e raised to the power of the arithmetic mean of the natural logarithms of the returns, often expressed as e^(arithmetic mean) - 1.

This calculation is important because it accounts for the compounding effect by transforming the returns into their logarithmic form, which neutralizes the volatility that can occur with simple averages. As a result, it provides a more accurate representation of the compounded return over a specified period.

In contrast, the other choices do not appropriately represent the method to calculate the geometric mean for compounded returns. The arithmetic mean simply averages the returns without considering the compounding effect; summing returns divided by n does not apply to compounded growth. Products of returns divided by n is not a valid method for averaging returns, and taking the square root of the sum of returns neglects the proper consideration of compounding as well. Thus, the correct answer underscores the significance of using the exponential function in conjunction with the arithmetic mean for accurate compounded return calculations.