To solve this problem, we will use the concept of the optimal risky portfolio and the two-asset portfolio theory, where you combine a risky asset (your portfolio) with a risk-free asset (the T-bills).

Given:

• The expected return of the risky portfolio $r_p = 15\% = 0.15$
• The standard deviation of the risky portfolio $\sigma_p = 31\% = 0.31$
• The risk-free rate $r_f = 5\% = 0.05$
• The constraint on the overall portfolio’s standard deviation is $\sigma_{\text{total}} = 20\% = 0.20$

Step 1: Find the proportion of the risky portfolio to invest in (y)

The total portfolio's standard deviation $\sigma_{\text{total}}$ can be written as:

$\sigma_{\text{total}} = y \cdot \sigma_p$

where $y$ is the proportion of the portfolio invested in the risky asset. We can now solve for $y$ using the constraint that the overall portfolio’s standard deviation should not exceed 20%.

$0.20 = y \cdot 0.31$

Solving for $y$:

$y = \frac{0.20}{0.31} \approx 0.645$

So, the proportion $y$ invested in the risky portfolio is approximately 0.65 (or 65%).

Step 2: Calculate the expected return on the overall portfolio

The expected return on the overall portfolio $r_{\text{total}}$ is a weighted average of the returns on the risky portfolio and the risk-free asset. The formula is:

$r_{\text{total}} = y \cdot r_p + (1 - y) \cdot r_f$

Substituting the known values:

$r_{\text{total}} = 0.645 \cdot 0.15 + (1 - 0.645) \cdot 0.05$ $r_{\text{total}} = 0.09675 + 0.01775 = 0.1145$

So, the expected return on the overall portfolio is approximately 11.45%.

Final Answers:

• The proportion $y$ to invest in the risky portfolio is 65%.
• The expected rate of return on the overall portfolio is 11.45%.

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Would you like to go through any specific part of the solution in more detail? Here are some related questions that might interest you:

1. How does changing the standard deviation constraint affect the investment proportion in the risky portfolio?
2. What would happen if the risk-free rate increased to 6%? How would that affect the investment proportion and expected return?
3. How do you calculate the risk of a portfolio that contains both risky and risk-free assets?
4. What role does diversification play in reducing the standard deviation of a portfolio?
5. How can you use the Capital Market Line (CML) to visualize this problem?

Tip: To maximize returns while controlling risk, balancing the proportion of the risky asset and the risk-free asset is essential. As the risk-free rate rises, more of the portfolio can be allocated to the risk-free asset for a similar expected return.