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Part 1

Consider the following five monthly​ returns: 

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.

**a.** Calculate the arithmetic average monthly return over this period. 

**b.** Calculate the geometric average monthly return over this period.

**c.** Calculate the monthly variance over this period.

**d.** Calculate the monthly standard deviation over this period.

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Part 1

**a.** Calculate the arithmetic average monthly return over this period. 

The arithmetic average is 

enter your response here​%. 

​ (Round to two decimal​ places.)

Part 2

**b.** Calculate the geometric average monthly return over this period.

The geometric average is 

enter your response here​%. 

​ (Round to two decimal​ places.)

Part 3

**c.** Calculate the monthly variance over this period.

The monthly variance is 

enter your response here. 

 ​(Round to five decimal​ places.)

Part 4

**d.** Calculate the monthly standard deviation over this period.

The monthly standard deviation is 

enter your response here​%. 

 ​(Round to two decimal​ places.)

The arithmetic average is(Round to two decimal places.)The geometric average is(Round to two decimal places.)The monthly variance is(Round to five decimal places.)The monthly standard deviation is(Round to two decimal places.)Monthly Return	0.03	-0.04	0.05	0.10	-0.05

Let's break down the steps to solve this problem based on the provided monthly returns: 0.03, -0.04, 0.05, 0.10, and -0.05.

### **Part 1: Arithmetic Average Monthly Return**

The arithmetic average return is calculated by summing the monthly returns and dividing by the number of periods.

\[
\text{Arithmetic Average} = \frac{0.03 + (-0.04) + 0.05 + 0.10 + (-0.05)}{5}
\]

\[
\text{Arithmetic Average} = \frac{0.03 - 0.04 + 0.05 + 0.10 - 0.05}{5}
\]

\[
\text{Arithmetic Average} = \frac{0.09}{5} = 0.018
\]

So, the arithmetic average monthly return is **1.80%**.

### **Part 2: Geometric Average Monthly Return**

The geometric average return is calculated as:

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

Where \(R_i\) are the returns for each period. For the given returns:

\[
\text{Geometric Average} = \left( (1 + 0.03) \times (1 - 0.04) \times (1 + 0.05) \times (1 + 0.10) \times (1 - 0.05) \right)^{\frac{1}{5}} - 1
\]

\[
\text{Geometric Average} = \left( 1.03 \times 0.96 \times 1.05 \times 1.10 \times 0.95 \right)^{\frac{1}{5}} - 1
\]

Let's calculate this.

### **Part 3: Monthly Variance**

Variance is calculated as:

\[
\text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (R_i - \bar{R})^2
\]

Where \(\bar{R}\) is the arithmetic average return, and \(R_i\) are the individual returns. We'll compute this by finding the squared differences from the arithmetic mean.

### **Part 4: Monthly Standard Deviation**

The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\]

I'll now calculate the geometric average, variance, and standard deviation.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through the steps to manually calculate each part if you'd like. Feel free to ask if you want to proceed that way!

Here are some related questions to deepen your understanding:

1. How is the geometric average return different from the arithmetic average?
2. Why is the geometric average return often preferred for analyzing investment performance?
3. What does the variance of a return tell you about the risk involved?
4. How would increasing the number of returns in the dataset impact the standard deviation?
5. How do you interpret the relationship between standard deviation and risk?

**Tip**: Always check if returns are compounded or simple when calculating average returns to avoid mistakes.

Consider the following five monthly returns: 0.03, -0.04, 0.05, 0.10, -0.05.

Part 1: Calculate the arithmetic average monthly return over this period.
Part 2: Calculate the geometric average monthly return over this period.
Part 3: Calculate the monthly variance over this period.
Part 4: Calculate the monthly standard deviation over this period.

Learn how to calculate the arithmetic and geometric average monthly returns, variance, and standard deviation for a set of five monthly returns: 0.03, -0.04, 0.05, 0.10, and -0.05. This detailed breakdown covers essential statistical concepts used in finance.

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