Arithmetic and Geometric Rates of Return

Explanation

The mean equals (5% + (-6%) + 8% + 3%)/4 = 2.5%

The geometric mean equals (1.05 * 0.94 * 1.08 * 1.03)^(1/4) = 2.36%

Note that the geometric mean is less than the mean.

To calculate the arithmetic and geometric rates of return over the given 4-year period, we'll first calculate the annual arithmetic rate of return.

Arithmetic Rate of Return: The arithmetic rate of return is simply the average of the annual returns.

Annual Returns: +5%, -6%, +8%, +3%

To calculate the arithmetic rate of return, we sum up the annual returns and divide by the number of years.

Arithmetic Rate of Return = (5% - 6% + 8% + 3%) / 4 = 10% / 4 = 2.5%

So, the arithmetic rate of return over the 4 years is 2.5%.

Geometric Rate of Return: The geometric rate of return is the compounded average rate of return over the given period.

To calculate the geometric rate of return, we first need to convert the annual returns to their respective growth rates. We add 1 to each annual return and then take the geometric mean of these growth rates.

Growth Rates: (1 + 0.05), (1 - 0.06), (1 + 0.08), (1 + 0.03)

Next, we take the geometric mean of these growth rates. We multiply them together and raise the product to the power of 1 divided by the number of years, and then subtract 1.

Geometric Rate of Return = [(1 + 0.05) * (1 - 0.06) * (1 + 0.08) * (1 + 0.03)] ^ (1/4) - 1

Calculating the expression inside the brackets: (1 + 0.05) * (1 - 0.06) * (1 + 0.08) * (1 + 0.03) = 1.05 * 0.94 * 1.08 * 1.03 = 1.103678

Geometric Rate of Return = 1.103678 ^ (1/4) - 1 = 1.024620 - 1 = 0.024620

So, the geometric rate of return over the 4 years is approximately 2.46% (rounded to two decimal places).

Therefore, the correct answer is D. 2.5%; 2.36%.