What condition is true when the arithmetic mean and geometric mean are the same?

The condition where the arithmetic mean and geometric mean are the same occurs specifically when all values being averaged are identical. This is particularly evident in the realm of investment returns, where the arithmetic mean is calculated by summing all values and dividing by the number of values, while the geometric mean takes into account the compounding effect of returns.

When sub-period returns are identical, each period contributes equally to the overall performance, leading both means to yield the same value. For example, if an investment grows by the same rate each year (e.g., a consistent return of 5% year after year), both the arithmetic mean and geometric mean will reflect this uniform growth as 5%.

In contrast, market volatility tends to cause fluctuations in returns, resulting in differences between the arithmetic and geometric means, as the geometric mean would factor in the compounding effect of varying returns over time. Multiple investments may result in diverse returns which further differentiate the two means. The declaration of dividends does not directly influence the equality of these means as it pertains more to cash flows than to the underlying rate of return consistency. Thus, identical sub-period returns is the definitive condition for the arithmetic and geometric means to be equal.