The last four years of returns for a stock are as​ follows: 

LOADING...

.

**a.** What is the average annual​ return?

**b.** What is the variance of the​ stock's returns?

**c.** What is the standard deviation of the​ stock's returns?

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Part 1

**a.** What is the average annual​ return?

The average annual return is 

enter your response here​%. 

 ​(Round to two decimal​ places.)

Part 2

**b.** What is the variance of the​ stock's returns?

The variance of the​ stock's returns is 

enter your response here. 

** ​(Round to five decimal​ places.)

Part 3

**c.** What is the standard deviation of the​ stock's returns?

The standard deviation of the​ stock's returns is 

enter your response here​%. 

 ​(Round to two decimal​ places.)Year	1	2	3	4

Return (%)	-3.69	28.44	12.38	4.48

To solve this, we will first define the given stock returns for four years:

- Year 1: -3.69%
- Year 2: 28.44%
- Year 3: 12.38%
- Year 4: 4.48%

### **Step-by-Step Breakdown**

### **a. Average Annual Return**
The average annual return (\(\bar{r}\)) is the arithmetic mean of the returns over the four years. We calculate it as:

\[
\bar{r} = \frac{r_1 + r_2 + r_3 + r_4}{4}
\]

Where:
- \(r_1 = -3.69\%\)
- \(r_2 = 28.44\%\)
- \(r_3 = 12.38\%\)
- \(r_4 = 4.48\%\)

### **b. Variance of Stock Returns**
The variance (\(\sigma^2\)) measures how much the returns deviate from the average return. It is calculated as:

\[
\sigma^2 = \frac{(r_1 - \bar{r})^2 + (r_2 - \bar{r})^2 + (r_3 - \bar{r})^2 + (r_4 - \bar{r})^2}{4}
\]

### **c. Standard Deviation of Stock Returns**
The standard deviation (\(\sigma\)) is the square root of the variance:

\[
\sigma = \sqrt{\sigma^2}
\]

Let me calculate the values for you.### **a. Average Annual Return**
The average annual return is **10.40%** (rounded to two decimal places).

### **b. Variance of the Stock's Returns**
The variance of the stock's returns is **140.73412** (rounded to five decimal places).

### **c. Standard Deviation of the Stock's Returns**
The standard deviation of the stock's returns is **11.86%** (rounded to two decimal places).

Let me know if you would like further details or explanations. Here are five related questions to expand your understanding:

1. How does the average return differ from the geometric mean return?
2. How would an increase in stock volatility affect the standard deviation?
3. What are the implications of a high variance in stock returns?
4. How is the variance of returns different from the covariance between two stocks?
5. What other methods can be used to measure the risk of a stock's returns?

**Tip**: The standard deviation is a key measure of risk, helping investors understand the volatility of an investment.

The last four years of returns for a stock are as follows: LOADING... a. What is the average annual return? b. What is the variance of the stock's returns? c. What is the standard deviation of the stock's returns? Return (%) for Year 1: -3.69, Year 2: 28.44, Year 3: 12.38, Year 4: 4.48

Learn how to calculate the average annual return, variance, and standard deviation for a stock's returns over four years. This example covers key statistical concepts like arithmetic mean, variance, and standard deviation, with the stock returns for Years 1 to 4: -3.69%, 28.44%, 12.38%, and 4.48%. Suitable for students in Grades 10-12.

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