Ten annual returns are listed in the following​ table: LOADING....
a. What is the arithmetic average return over the​ 10-year period?
b. What is the geometric average return over the​ 10-year period?
c. If you invested ​$100.00 at the​ beginning, how much would you have at the​ end?
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Part 1
a. What is the arithmetic average return over the​ 10-year period?
The arithmetic average return over the​ 10-year period is 
  
enter your response here​%.  ​(Round to two decimal​ places.)

To solve this, we need the 10 annual returns to compute both the arithmetic and geometric average. However, since the table isn't provided, I can explain how to approach this problem once we have the dataTo solve this, we need the 10 annual returns to compute both the arithmetic and geometric average. However, since the table isn't provided, I can explain how to approach this problem once we have the 

### a. Arithmetic Average Return
The **arithmetic average return** is the simple average of the returns over the 10 years. The formula is:

\[
\text{Arithmetic Average Return} = \frac{\sum_{i=1}^{n} R_i}{n}
\]

Where:
- $ R_i $ are the annual returns,
- $ n $ is the number of years (in this case, $ n = 10 $).

You simply sum up all the annual returns and divide by 10.

### b. Geometric Average Return
The **geometric average return** takes compounding into account and is given by the formula:

\[
\text{Geometric Average Return} = \left( \prod_{i=1}^{n} (1 + R_i) \right)^{\frac{1}{n}} - 1
\]

Where:
- $ R_i $ are the annual returns expressed as decimals (so 10% = 0.10),
- $ n $ is the number of years (10 in this case).

### c. Investment Growth Calculation
To calculate how much a $100 investment grows over the 10-year period:

\[
\text{Ending Value} = 100 \times \prod_{i=1}^{n} (1 + R_i)
\]

This product reflects how the $100 grows each year by being multiplied by (1 + the return rate for that year).

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If you can provide the 10 annual returns, I can calculate each part for you. Let me know if you'd like to proceed with these explanations!

### Follow-up questions:
1. What is the difference between arithmetic and geometric average returns?
2. How does compounding affect long-term returns?
3. Why is the geometric average typically lower than the arithmetic average for volatile returns?
4. How do negative returns affect the geometric average?
5. How does time period length influence geometric return calculations?

**Tip:** The geometric average is a better measure of long-term performance because it accounts for compounding, whereas the arithmetic average can be misleading for volatile returns.

Ten annual returns are listed in the following​ table: LOADING.... a. What is the arithmetic average return over the​ 10-year period? b. What is the geometric average return over the​ 10-year period? c. If you invested ​$100.00 at the​ beginning, how much would you have at the​ end?

Learn how to calculate the arithmetic and geometric average returns over a 10-year period using key mathematical formulas. Discover how a $100 investment grows based on annual returns. This problem covers essential finance and investment concepts for college-level students.

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