Consider the following 6 months of returns for 2 stocks and a portfolio of those 2 stocks: 

Jan

Feb

Mar

Apr

May

Jun

Stock A

2%

5%

- 6%

3%

- 2%

4%

Stock B

1%

- 2%

9%

0%

5%

- 1%

Portfolio

1.5%

1.5%

1.5%

1.5%

1.5%

1.5%.  

What is the expected return and standard deviation of returns for each of the two stocks?


To calculate the expected return and standard deviation of returns for each stock, we’ll use the following formulas:

1. **Expected Return (Mean)**: The average return over the 6 months.
   \[
   \text{Expected Return} = \frac{\sum \text{Returns}}{\text{Number of Periods}}
   \]

2. **Standard Deviation of Returns**: This measures the variability of returns. It is calculated using:
   \[
   \sigma = \sqrt{\frac{\sum (\text{Return} - \text{Mean})^2}{N}}
   \]
   where \(N\) is the number of periods.

Let's calculate these for both Stock A and Stock B based on the given data.

### Stock A
Returns for Stock A:
- \(2\%\), \(5\%\), \(-6\%\), \(3\%\), \(-2\%\), \(4\%\)

#### Expected Return for Stock A
\[
\text{Mean of Stock A} = \frac{2 + 5 - 6 + 3 - 2 + 4}{6} = \frac{6}{6} = 1\%
\]

#### Standard Deviation for Stock A
1. Calculate each deviation from the mean, square them, and sum them up.
2. Divide by the number of observations (6), and take the square root.

#### Step-by-Step Calculation
\[
\text{Deviations: } (2 - 1)^2 = 1, \, (5 - 1)^2 = 16, \, (-6 - 1)^2 = 49, \, (3 - 1)^2 = 4, \, (-2 - 1)^2 = 9, \, (4 - 1)^2 = 9
\]
\[
\sum \text{Squared Deviations} = 1 + 16 + 49 + 4 + 9 + 9 = 88
\]
\[
\text{Variance} = \frac{88}{6} \approx 14.67
\]
\[
\text{Standard Deviation} = \sqrt{14.67} \approx 3.83\%
\]

### Stock B
Returns for Stock B:
- \(1\%\), \(-2\%\), \(9\%\), \(0\%\), \(5\%\), \(-1\%\)

#### Expected Return for Stock B
\[
\text{Mean of Stock B} = \frac{1 - 2 + 9 + 0 + 5 - 1}{6} = \frac{12}{6} = 2\%
\]

#### Standard Deviation for Stock B
1. Calculate each deviation from the mean, square them, and sum them up.
2. Divide by the number of observations (6), and take the square root.

#### Step-by-Step Calculation
\[
\text{Deviations: } (1 - 2)^2 = 1, \, (-2 - 2)^2 = 16, \, (9 - 2)^2 = 49, \, (0 - 2)^2 = 4, \, (5 - 2)^2 = 9, \, (-1 - 2)^2 = 9
\]
\[
\sum \text{Squared Deviations} = 1 + 16 + 49 + 4 + 9 + 9 = 88
\]
\[
\text{Variance} = \frac{88}{6} \approx 14.67
\]
\[
\text{Standard Deviation} = \sqrt{14.67} \approx 3.83\%
\]

### Summary

- **Stock A**:
  - Expected Return = \(1\%\)
  - Standard Deviation = \(3.83\%\)
  
- **Stock B**:
  - Expected Return = \(2\%\)
  - Standard Deviation = \(3.83\%\)

Would you like to see a more detailed breakdown of each step or have any questions on this?

---

Here are some related questions you might find helpful:

1. How is portfolio standard deviation calculated when combining two assets?
2. What is the role of covariance in portfolio standard deviation?
3. How does expected return differ from actual returns in investments?
4. Why is standard deviation a critical measure in finance?
5. How would the addition of more stocks impact the portfolio's overall risk?

**Tip:** The expected return represents the central tendency of returns, while the standard deviation gives insight into return volatility, helping in assessing the risk level.

Consider the following 6 months of returns for 2 stocks and a portfolio of those 2 stocks: Jan, Feb, Mar, Apr, May, Jun. Stock A: 2%, 5%, -6%, 3%, -2%, 4%. Stock B: 1%, -2%, 9%, 0%, 5%, -1%. Portfolio: 1.5%, 1.5%, 1.5%, 1.5%, 1.5%, 1.5%. What is the expected return and standard deviation of returns for each of the two stocks?

Calculate the expected return and standard deviation for Stock A and Stock B over six months of returns. This solution covers essential statistical concepts in finance, including the expected return and standard deviation formulas, ideal for finance and statistics students.

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