Question

Suppose a portfolio had an arithmetic average return of 8 percent for a 4-year period. Which one of these statements must be true regarding this portfolio for the period?

A. At least one of the four years produced an annual rate of return of 8 percent.

B. If the standard deviation of the portfolio is greater than zero, then the geometric average portfolio return is less than 8 percent.

C. The standard deviation of the portfolio must be lower than the standard deviation of a comparable portfolio that had an arithmetic average return of 9 percent.

D. If the standard deviation of the portfolio is zero, then the geometric average return must also be zero.

F. The holding period return must be less than 8 percent.

Answer 1

The correct answer is option B. If the standard deviation of the portfolio is greater than zero, then the geometric average portfolio return is less than 8 percent.

Step-by-step explanation:

The correct statement regarding the portfolio with an arithmetic average return of 8 percent over a 4-year period is option B: If the standard deviation of the portfolio is greater than zero, then the geometric average portfolio return is less than 8 percent. This is because the geometric average will always be lower than the arithmetic mean whenever there are variations in the periodic returns, which is indicated by a standard deviation greater than zero.

To further explain, the arithmetic average simply sums up the annual returns and divides by the number of years, not considering the compounding effect. However, the geometric average takes into account the cumulative effect of the returns over each period, effectively 'smoothing' out the impact of significantly negative or positive years.

Thus, in any portfolio where there is volatility (as indicated by a nonzero standard deviation), the geometric mean will be less than the arithmetic mean.