For parts 7(d) and 7(e). You could, but need not, show that 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚(𝑋𝑋) = 16.5, 𝑠𝑠𝑠𝑠(𝑋𝑋) = 18.9143,
𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚(𝑌𝑌) =−1.8, 𝑠𝑠𝑠𝑠(𝑌𝑌) = 4.0694, and 𝑑𝑑𝑏𝑏𝐹𝐹𝐹𝐹(𝑋𝑋, 𝑌𝑌) = −0.8614. Consider a portfolio that invests 60% in
Stock A and 40% in Stock B. This portfolio’s return rate can be expressed as 𝑅𝑅 = 0.6 𝑋𝑋 + 0.4 𝑌𝑌.
7(d). Compute the expected return rate for this portfolio. That is, compute 𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎(𝑹𝑹).

To find the expected return rate for this portfolio, we calculate \( \text{E}(R) \), or \( \text{mean}(R) \), which is the expected value of the portfolio return rate \( R = 0.6X + 0.4Y \).

Given:
- \( \text{mean}(X) = 16.5 \)
- \( \text{mean}(Y) = -1.8 \)

The expected return \( \text{E}(R) \) for the portfolio can be calculated using the linearity of expectation:

\[
\text{E}(R) = 0.6 \cdot \text{E}(X) + 0.4 \cdot \text{E}(Y)
\]

Substituting the given values:

\[
\text{E}(R) = 0.6 \cdot 16.5 + 0.4 \cdot (-1.8)
\]

Calculating each part:

1. \( 0.6 \cdot 16.5 = 9.9 \)
2. \( 0.4 \cdot (-1.8) = -0.72 \)

Thus,

\[
\text{E}(R) = 9.9 - 0.72 = 9.18
\]

### Answer:
The expected return rate for this portfolio, \( \text{mean}(R) \), is \( 9.18 \).

Would you like a further breakdown of these calculations?

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Here are 5 related questions:
1. How would the expected return change if the portfolio weights were 50% in each stock?
2. How does the portfolio’s expected return compare to investing solely in Stock A?
3. What would happen to the expected return if the mean return of Stock B changed to a positive value?
4. How would changes in the weights impact the risk (standard deviation) of the portfolio?
5. What is the interpretation of a negative mean for Stock B's returns in the context of this portfolio?

**Tip:** In portfolio theory, diversification helps in managing risk by combining assets with varying correlations and expected returns.

For parts 7(d) and 7(e). You could, but need not, show that 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚(𝑋𝑋) = 16.5, 𝑠𝑠𝑠𝑠(𝑋𝑋) = 18.9143, 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚(𝑌𝑌) = −1.8, 𝑠𝑠𝑠𝑠(𝑌𝑌) = 4.0694, and 𝑑𝑑𝑏𝑏𝐹𝐹𝐹𝐹(𝑋𝑋, 𝑌𝑌) = −0.8614. Consider a portfolio that invests 60% in Stock A and 40% in Stock B. This portfolio’s return rate can be expressed as 𝑅𝑅 = 0.6 𝑋𝑋 + 0.4 𝑌𝑌. 7(d). Compute the expected return rate for this portfolio. That is, compute 𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎(𝑹𝑹).

Learn how to calculate the expected return rate for a portfolio investing 60% in Stock A and 40% in Stock B, with given statistics. This example uses the linearity of expectation and covers essential concepts in portfolio theory, suitable for undergraduate finance students.

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