What are the arithmetic and geometric average returns for a stock with annual returns of 4 percent, 9 percent, −5 percent, and 17 percent?

Multiple Choice

6.25%; 5.95%

8.63%; 5.95%

5.95%; 6.25%

6.25%; 8.63%

8.63%; 6.25%

To solve for the arithmetic and geometric average returns, let’s go through each one:

### Arithmetic Average Return
The arithmetic average return is simply the sum of the returns divided by the number of periods.

Given returns: \( 4\%, 9\%, -5\%, 17\% \)

\[
\text{Arithmetic Average} = \frac{4 + 9 - 5 + 17}{4} = \frac{25}{4} = 6.25\%
\]

### Geometric Average Return
The geometric average return is calculated as the \(n\)-th root of the product of \( (1 + r_i) \) values, minus 1, where \( r_i \) is each annual return and \( n \) is the number of periods.

\[
\text{Geometric Average} = \left( (1 + 0.04)(1 + 0.09)(1 - 0.05)(1 + 0.17) \right)^{\frac{1}{4}} - 1
\]

Calculating this step-by-step:

1. \( 1 + 0.04 = 1.04 \)
2. \( 1 + 0.09 = 1.09 \)
3. \( 1 - 0.05 = 0.95 \)
4. \( 1 + 0.17 = 1.17 \)

Now, multiplying them:

\[
1.04 \times 1.09 \times 0.95 \times 1.17 \approx 1.2387
\]

Taking the fourth root:

\[
1.2387^{\frac{1}{4}} \approx 1.0595
\]

Subtracting 1:

\[
\text{Geometric Average} \approx 0.0595 = 5.95\%
\]

### Answer
The correct answer is:

**6.25%; 5.95%**

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Would you like further details or have any questions?

### Related Questions:
1. How do you interpret the difference between arithmetic and geometric average returns?
2. Why might an investor prefer the geometric mean over the arithmetic mean?
3. How would the averages change if there was a negative return greater than 17%?
4. What would the effect of compounding be on these averages over more years?
5. How do arithmetic and geometric returns differ in interpreting stock volatility?

**Tip:** Arithmetic returns are useful for estimating average short-term performance, while geometric returns are better for understanding compounded growth over time.

What are the arithmetic and geometric average returns for a stock with annual returns of 4 percent, 9 percent, −5 percent, and 17 percent?

Learn how to calculate the arithmetic and geometric average returns for a stock with annual returns of 4%, 9%, -5%, and 17%. This guide covers key finance concepts and formulas useful for understanding average returns in stock performance. Perfect for students in finance or business studies.

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