To calculate the required rates of return, we will approach the problem step by step.

Part (a): Annual Rate of Return

The formula for the annual rate of return for year $t$ is given by: $R_t = \frac{P_t - P_{t-1}}{P_{t-1}} \times 100$ where:

• $P_t$ is the stock price at the end of year $t$,
• $P_{t-1}$ is the stock price at the end of the previous year $t-1$,
• $R_t$ is the annual rate of return.

The stock prices for Marsh Inc. over the five years are:

• Year 1: $88$
• Year 2: $1212$
• Year 3: $1616$
• Year 4: $77$
• Year 5: $88$

Now, we calculate the annual return for each year:

1. From Year 1 to Year 2: $R_2 = \frac{1212 - 88}{88} \times 100 = \frac{1124}{88} \times 100 = 1277.27\%$

2. From Year 2 to Year 3: $R_3 = \frac{1616 - 1212}{1212} \times 100 = \frac{404}{1212} \times 100 = 33.33\%$

3. From Year 3 to Year 4: $R_4 = \frac{77 - 1616}{1616} \times 100 = \frac{-1539}{1616} \times 100 = -95.23\%$

4. From Year 4 to Year 5: $R_5 = \frac{88 - 77}{77} \times 100 = \frac{11}{77} \times 100 = 14.29\%$

The annual rates of return are:

• Year 2: 1277.27%
• Year 3: 33.33%
• Year 4: -95.23%
• Year 5: 14.29%

Part (b): Arithmetic Average Rate of Return

The arithmetic average rate of return is simply the sum of the annual returns divided by the number of years:

$\text{Arithmetic Average} = \frac{1277.27\% + 33.33\% + (-95.23\%) + 14.29\%}{4}$ $= \frac{1229.66\%}{4} = 307.42\%$

Part (c): Geometric Average Rate of Return

The geometric average rate of return is calculated using the following formula:

$\text{Geometric Average} = \left( \prod_{t=1}^{n} (1 + R_t) \right)^{\frac{1}{n}} - 1$ where $R_t$ is in decimal form, and $n$ is the number of years.

First, we convert the rates of return to decimal form:

• Year 2: 12.7727
• Year 3: 0.3333
• Year 4: -0.9523
• Year 5: 0.1429

Now, applying the formula: $\text{Geometric Average} = \left( (1 + 12.7727) \times (1 + 0.3333) \times (1 - 0.9523) \times (1 + 0.1429) \right)^{\frac{1}{4}} - 1$ $= \left( 13.7727 \times 1.3333 \times 0.0477 \times 1.1429 \right)^{\frac{1}{4}} - 1$ $= \left( 1.0006 \right)^{\frac{1}{4}} - 1 \approx 0.00015 \text{ or } 0.015\%$

Part (d): Comparing the Arithmetic and Geometric Averages

Since the beginning and ending stock prices are the same, the geometric average rate of return (0.015%) provides a better description of the average annual return over the period. This is because it takes into account the compounding effect and reflects the fact that the stock ended at the same price it started. The arithmetic average, being much higher, does not accurately represent the overall performance due to the extreme fluctuations.

Would you like further clarifications on any part of the solution? Here are some related questions for further exploration:

1. How does the geometric average handle negative returns differently than the arithmetic average?
2. What would the outcome be if dividends were paid each year?
3. How would the geometric average change if the stock price at the end of Year 5 was higher?
4. How do extreme price movements affect the arithmetic average?
5. Can the geometric average ever exceed the arithmetic average?

Tip: The geometric average is more appropriate for multi-period returns as it accurately reflects compound growth over time.