The returns from an investment are 2% in Year 1, 7% in Year 2, and 9.8% in the first half of Year 3. Calculate the annualized return for the entire period. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
To calculate the annualized return, we need to sum up the returns from each year and then find the compound annual growth rate (CAGR).
The returns from each year are as follows:
Year 1: 2%
Year 2: 7%
First half of Year 3: 9.8%
To find the compound annual growth rate, we use the formula:
CAGR = (Ending Value ÷ Beginning Value) ^ (1 ÷ Number of Years) - 1
We can assume that the initial investment is $100.
For Year 1, the return is 2% of $100, which is $2. So the Ending Value is $100 + $2 = $102.
For Year 2, the return is 7% of $102, which is $7.14. So the Ending Value is $102 + $7.14 = $109.14.
For the first half of Year 3, the return is 9.8% of $109.14, which is $10.68. So the Ending Value is $109.14 + $10.68 = $119.82.
Using the compound annual growth rate formula:
CAGR = ($119.82 ÷ $100) ^ (1 ÷ 2.5) - 1
Calculating the values inside the parenthesis:
CAGR = 1.1982 ^ (1 ÷ 2.5) - 1
CAGR = 1.1982 ^ 0.4 - 1
CAGR = 1.0837 - 1
CAGR = 0.0837
Converting to a percentage by multiplying by 100:
CAGR = 0.0837 * 100
CAGR = 8.37%
Therefore, the annualized return for the entire period is 8.37%.
To calculate the annualized return for the entire period, we need to use the compound annual growth rate formula.
First, let's calculate the total return for the entire period:
1 + total return = (1 + return in Year 1) * (1 + return in Year 2) * (1 + return in the first half of Year 3)
1 + total return = (1 + 0.02) * (1 + 0.07) * (1 + 0.098)
1 + total return = 1.02 * 1.07 * 1.098
1 + total return = 1.2287326
total return = 1.2287326 - 1
total return = 0.2287326
Next, let's calculate the annualized return:
annualized return = (1 + total return)^(1 / number of years) - 1
annualized return = (1 + 0.2287326)^(1 / 2.5) - 1
annualized return = 1.1680238 - 1
annualized return = 0.1680238
Finally, rounding the final answer to 2 decimal places:
Annualized return for the entire period = 0.17 or 17.00%