​(Related **to Checkpoint**​ 7.2)​ (Calculating **the geometric and arithmetic average rate of**​ return)**** Marsh Inc. had the following​ end-of-year stock prices over the last five years and paid no cash​ dividends:

 **Time**

**Marsh**

 1

​$88

 2

1212

 3

1616

 4

77

 5

88

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**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?

Question content area bottom

Part 1

**a.** The annual rate of return at the end of year 2 is 

enter your response here​%. 

**** ​(Round to two decimal​ places.)

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.

(Related to Checkpoint 7.2) (Calculating the geometric and arithmetic average rate of return) Marsh Inc. had the following end-of-year stock prices over the last five years and paid no cash dividends:

Time
Marsh
1 - $88
2 - $1212
3 - $1616
4 - $77
5 - $88

(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?

Step-by-step guide to calculating annual, arithmetic, and geometric average rates of return for Marsh Inc. stock over five years. Includes detailed calculations for each annual rate of return based on end-of-year stock prices: $88, $1212, $1616, $77, and $88.

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