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.