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Let's go through the problem and solve it step by step.

### a. Calculate the annual rate of return for each year:

We use the formula for annual rate of return:
\[
\text{Rate of Return} = \frac{P_{\text{end}} - P_{\text{begin}}}{P_{\text{begin}}} \times 100
\]
where \( P_{\text{end}} \) is the price at the end of the period, and \( P_{\text{begin}} \) is the price at the beginning of the period.

1. **Year 1 to Year 2:**
   \[
   \text{Rate of Return} = \frac{12 - 8}{8} \times 100 = 50.00\%
   \]

2. **Year 2 to Year 3:**
   \[
   \text{Rate of Return} = \frac{16 - 12}{12} \times 100 = 33.33\%
   \]

3. **Year 3 to Year 4:**
   \[
   \text{Rate of Return} = \frac{7 - 16}{16} \times 100 = -56.25\%
   \]

4. **Year 4 to Year 5:**
   \[
   \text{Rate of Return} = \frac{8 - 7}{7} \times 100 = 14.29\%
   \]

### b. Arithmetic Average Rate of Return:
The arithmetic average is the sum of the individual annual rates divided by the number of periods:
\[
\text{Arithmetic Average} = \frac{50.00 + 33.33 - 56.25 + 14.29}{4} = 10.34\%
   \]

### c. Geometric Average Rate of Return:
The geometric average rate of return is calculated using the following formula:
\[
\text{Geometric Average} = \left( \prod_{i=1}^{n} (1 + r_i) \right)^{\frac{1}{n}} - 1
\]
where \( r_i \) is the rate of return for each year, and \( n \) is the number of periods.

1. Convert the percentage returns to decimals:
   - Year 2: \( 1 + 0.50 = 1.50 \)
   - Year 3: \( 1 + 0.3333 = 1.3333 \)
   - Year 4: \( 1 - 0.5625 = 0.4375 \)
   - Year 5: \( 1 + 0.1429 = 1.1429 \)

2. Multiply the terms:
   \[
   (1.50) \times (1.3333) \times (0.4375) \times (1.1429) = 0.8233
   \]

3. Apply the geometric average formula:
   \[
   \text{Geometric Average} = (0.8233)^{\frac{1}{4}} - 1 \approx -0.1767 = -17.67\%
   \]

### d. Which average rate of return better describes the performance?

The **geometric average rate of return** better describes the performance over the period because it accounts for the compounding effect of returns over time, especially when there are both positive and negative returns, as in this case. The arithmetic average may overestimate the actual return when there is significant volatility.

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### Related Questions:
1. How is the arithmetic average rate of return affected by volatility?
2. What is the significance of using the geometric mean in finance?
3. How do you interpret a negative geometric average rate of return?
4. How would the overall return change if the stock price had not dropped in Year 4?
5. Can you explain the difference between nominal and real rates of return?

**Tip:** When dealing with volatile returns, the geometric mean gives a more accurate measure of actual performance over time.

a. Calculate the annual rate of return for each year from the above information. b. What is the arithmetic average rate of return earned by investing in Marsh's stock over this period? c. What is the geometric average rate of return earned by investing in Marsh's stock over this period? d. Considering the beginning and ending stock prices for the five-year period are the same, which type of average rate of return (the arithmetic or geometric) better describes the average annual rate of return earned over the period?

Solve a step-by-step financial problem involving calculating the annual, arithmetic, and geometric average rates of return for Marsh's stock over a five-year period. Explore the significance of the geometric mean when assessing volatile returns.

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