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### Part (a): Compute the **annual rates of return**.

The formula for **annual rate of return** is:
\[
\text{Rate of Return} = \frac{\text{Ending Price} - \text{Beginning Price}}{\text{Beginning Price}} \times 100
\]

Let’s apply this formula to each firm.

#### **Harris Stock Annual Returns:**
1. **From Time 1 to 2:**  
   \[
   \frac{6 - 10}{10} \times 100 = -40.00\%
   \]
2. **From Time 2 to 3:**  
   \[
   \frac{13 - 6}{6} \times 100 = 116.67\%
   \]
3. **From Time 3 to 4:**  
   \[
   \frac{16 - 13}{13} \times 100 \approx 23.08\%
   \]

#### **Pinwheel Stock Annual Returns:**
1. **From Time 1 to 2:**  
   \[
   \frac{33 - 21}{21} \times 100 \approx 57.14\%
   \]
2. **From Time 2 to 3:**  
   \[
   \frac{28 - 33}{33} \times 100 \approx -15.15\%
   \]
3. **From Time 3 to 4:**  
   \[
   \frac{23 - 28}{28} \times 100 \approx -17.86\%
   \]

---

### Part (b): **Arithmetic and Geometric Mean Rates of Return**

1. **Arithmetic Mean:**
   \[
   \text{Arithmetic Mean} = \frac{\sum(\text{Annual Returns})}{n}
   \]

   - For **Harris**:  
     \[
     \frac{-40.00 + 116.67 + 23.08}{3} \approx 33.25\%
     \]
   - For **Pinwheel**:  
     \[
     \frac{57.14 - 15.15 - 17.86}{3} \approx 8.71\%
     \]

2. **Geometric Mean:**
   The geometric mean rate of return is calculated as:
   \[
   \left( \prod (1 + \text{Annual Return}) \right)^{\frac{1}{n}} - 1
   \]
   where the product involves each year's rate of return as a decimal.

---

### Part (c): **Three-Year Rate of Return**
You calculate the total three-year return by comparing the stock value at **Time 4** with the value at **Time 1**:
\[
\text{Three-Year Return} = \frac{\text{Price at Time 4} - \text{Price at Time 1}}{\text{Price at Time 1}} \times 100
\]

- **Harris**:  
  \[
  \frac{16 - 10}{10} \times 100 = 60\%
  \]
- **Pinwheel**:  
  \[
  \frac{23 - 21}{21} \times 100 \approx 9.52\%
  \]

---

### Part (d): **Annual Rate from Three-Year Rate**
We need to convert the total three-year return into an annualized rate. Use this formula:
\[
\left(1 + \text{Three-Year Rate}\right)^{\frac{1}{3}} - 1 = \text{Annual Rate}
\]

- **Harris**:
  \[
  \left(1 + 0.60\right)^{\frac{1}{3}} - 1 \approx 0.1693 = 16.93\%
  \]
- **Pinwheel**:
  \[
  \left(1 + 0.0952\right)^{\frac{1}{3}} - 1 \approx 0.0308 = 3.08\%
  \]

---

Let me know if you need help completing the missing tables or further steps. 

---

Here are **5 follow-up questions**:
1. How does the arithmetic mean differ from the geometric mean in this context?
2. Can a negative rate of return still result in a positive three-year return?
3. How do compounding effects influence the geometric mean?
4. Why is it important to annualize multi-year returns?
5. How would dividends (if included) change these calculations?

**Tip:** Always interpret arithmetic and geometric means carefully—they give different insights into investment performance.

Compute the annual rates of return for both Harris and Pinwheel stocks, then calculate the arithmetic and geometric mean rates of return, and compute the three-year rate of return and convert it into an annual rate.

Learn how to calculate the annual and three-year rates of return for Harris and Pinwheel stocks using arithmetic and geometric mean methods. This guide covers essential formulas and provides step-by-step calculations for stock performance.

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