Question

Calculate the annual arithmetic mean and geometric mean return on the following security, purchase price =$41; first-year dividend =$3; price after one year =$46; second-year dividend =$3; selling price after two years =$39. (Round intermediate calculations to 3 decimal places, e.g. 15.251% and the final answers to 2 decimal places, e.g. 15.25\%.) State which method is more appropriate for the situation.

Answer 1

The annual arithmetic mean return is 5.41%, while the annual geometric mean return is 4.91%. The geometric mean is more appropriate for assessing the performance of an investment over multiple periods since it accounts for compounding effects.

Step-by-step explanation:

To calculate the annual arithmetic mean return and geometric mean return for the security in question, we need to consider both the dividends received and the change in price of the security. For the arithmetic mean return, we first find the total return for each year and then average these returns. For the geometric mean return, we use the compounded return over the two-year period.

Year 1: You bought the security for $41 and received a dividend of $3 and the price after one year was $46. The return for the first year is (($46 + $3) - $41) / $41 = 0.1951 or 19.51% when rounded to two decimal places.

Year 2: The price at the beginning of Year 2 is $46, you received another dividend of $3, and sold the security for $39. The return for the second year is (($39 + $3) - $46) / $46 = -0.0869 or -8.69%.

The arithmetic mean of these two returns is (19.51% + (-8.69%)) / 2 = 5.41%.

The geometric mean, which accounts for compounding, is the square root of the product of (1 + 0.1951) * (1 - 0.0869) minus 1, which equates to a two-year compounded return. This returns a geometric mean of 4.91% after rounding to two decimal places.

The geometric mean is generally more appropriate for evaluating returns over multiple periods because it accounts for the effect of compounding.