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.