The returns from an investment are 7% in Year 1, 10% in Year 2, and \( 6.2 \% \) 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 over the entire period, we can use the geometric average return formula for multi-period investments:

\[
R_{\text{annualized}} = \left( \prod_{i=1}^{n} (1 + r_i) \right)^{\frac{1}{T}} - 1
\]

Where:
- \( r_i \) is the return for each period,
- \( n \) is the number of periods,
- \( T \) is the total time in years.

The given returns are:
- Year 1: 7% = 0.07,
- Year 2: 10% = 0.10,
- First half of Year 3: 6.2% = 0.062 for half a year.

### Step 1: Adjust the half-year return
The return for the first half of Year 3 needs to be annualized. We can adjust it as:

\[
r_{\text{half year}} = (1 + 0.062)^{2} - 1 = \text{Full Year Equivalent Return}
\]

Now let's calculate the adjusted half-year return.The full-year equivalent return for the first half of Year 3 is approximately 12.7844%.

### Step 2: Calculate the total annualized return

Now that we have the equivalent annual returns for all periods:
- Year 1: 7%,
- Year 2: 10%,
- Year 3 (full-year equivalent): 12.7844%.

We'll use the formula for the geometric average return:

\[
R_{\text{annualized}} = \left( (1 + 0.07) \times (1 + 0.10) \times (1 + 0.127844) \right)^{\frac{1}{3}} - 1
\]

Now let's calculate the annualized return.The annualized return for the entire period is approximately \( 9.90\% \).

Do you have any questions or would you like more details?

Here are 5 related questions:
1. How does the geometric mean differ from the arithmetic mean in calculating returns?
2. What happens if one of the periods had a negative return? How would the formula change?
3. How is the formula adapted for non-equal period durations, like quarters or months?
4. What are other methods to calculate compounded returns over time?
5. Why is the annualized return important for comparing investments?

**Tip:** Annualized returns are essential for comparing investments over different time frames, as they normalize returns to a yearly basis.

The returns from an investment are 7% in Year 1, 10% in Year 2, and 6.2% 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.)

Learn how to calculate the annualized return for a multi-year investment with varying returns of 7%, 10%, and 6.2% in a detailed step-by-step solution. The problem involves using the geometric average return formula and annualizing a half-year return. Suitable for college students or those studying financial math.

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