Final Answer:
The geometric average return for the stock with returns of 9.18 percent, -6.66 percent, 22.24 percent, and 14.96 percent over the past four years is 10.41 percent. Therefore, the correct option is a) 10.41 percent.
Step-by-step explanation:
The geometric average return is a measure of the compound annual growth rate (CAGR) for an investment over a specific period. Unlike the arithmetic average return, which adds all the returns and divides by the number of returns, the geometric average return takes into account the compounding effect of reinvested earnings.
To calculate the geometric average return, we follow these steps:
1. Calculate the value of the investment at the end of year n using the formula: Ending Value = Starting Value * (1 + Annual Return)^n
2. Calculate the ending value of the investment after n years using the formula above for each year, and then find the product of all these ending values.
3. Divide this product by the starting value to get the final value after n years.
4. Take the nth root of this final value to get the geometric average return over n years.
Let's apply these steps to our example:
Year 1: Starting Value = $100, Annual Return = 9.18%
Ending Value = $100 * (1 + 0.0918) = $109.18
Year 2: Starting Value = $109.18, Annual Return = -6.66%
Ending Value = $109.18 * (1 - 0.0666) = $102.57
Year 3: Starting Value = $102.57, Annual Return = 22.24%
Ending Value = $102.57 * (1 + 0.2224) = $125.37
Year 4: Starting Value = $125.37, Annual Return = 14.96%
Ending Value = $125.37 * (1 + 0.1496) = $145.73
Final Value after four years = $145.73 - Starting Value = $45.73
Geometric Average Return over four years = [Final Value / Starting Value]^(1/4) - 1 = 0.1041 or 10.41% per year (rounded to two decimal places). Therefore, the correct option is a) 10.41 percent.