Final answer:
To calculate the average annual rate of return over a three-year period, we use the geometric mean formula. The average annual rate of return realized over the three-year period is 9.5%. For the second question, with investments and withdrawals, the average annual return is 4.68% or 0.0468.
Step-by-step explanation:
To calculate the average annual rate of return over a three-year period, we need to find the geometric mean of the rates of return for each year. The formula for calculating the geometric mean is:
Geometric mean = (1 + r1)(1 + r2)(1 + r3) ... (1 + rn) - 1
For the given rates of return, the average annual rate of return can be calculated as follows:
Geometric mean = (1 + 1)(1 + (-0.5))(1 + 0.3) - 1 = 0.095 or 9.5%
Therefore, the average annual rate of return realized over the three-year period is 9.5%.
For the second question, to calculate the average annual return realized over the three-year period with investments and withdrawals, we can consider each year separately. In the first year, there is a 100% return on the $100 investment, resulting in a value of $200. In the second year, there is a -50% return, resulting in a value of $100. In the third year, there is a 30% return, resulting in a value of $130. The average annual return can be calculated as follows:
Average annual return = ((ending value / starting value)^(1/years)) - 1
Average annual return = ((130 / 200)^(1/3)) - 1 = 4.68% or 0.0468
Therefore, the average annual return realized over the three-year period with investments and withdrawals is 4.68% or 0.0468.