Choose the best response about the Rate of Return and Mean Rate of a Market Index:

• A. The Rate of Return is always higher than the Mean Rate.
• B. The Means rate is irrelevant to the Market Index.
• C. The Means Rate is always lower than the Rate of Return.
• D. Mean rate will always equal, or be greater than, the Rate of Return of a given Market Index.

Answer: (D)

The key idea is the difference between an arithmetic average of periodic returns and the compound (geometric) rate of return of the index over time. Over several periods, the index’s rate of return reflects compounding and is best represented by the geometric mean of the periodic returns. The mean rate, calculated as the simple arithmetic average of those periodic returns, tends to be at least as large as the geometric rate, and they are equal only if every period yields the same return. This happens because volatility in returns boosts the arithmetic average relative to the compounded growth rate.

For example, if one period returns 10% and another -5%, the arithmetic mean is 2.5%, while the geometric rate is about 2.23%. The arithmetic mean exceeds the geometric rate, illustrating why the mean rate will equal or be greater than the rate of return over the same horizon.