Let's walk through the tasks based on the instructions and data provided in your image.

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.