To find the average beta of the new stocks to be added, we follow these steps:

Step 1: Understand the portfolio's required return

The required return ($R_p$) of a portfolio is given by the Capital Asset Pricing Model (CAPM):

$R_p = R_f + \beta_p \cdot (R_m - R_f)$

Where:

• $R_p$ is the portfolio's required return (15% or 0.15)
• $R_f$ is the risk-free rate (4.5% or 0.045)
• $R_m - R_f$ is the market risk premium (7% or 0.07)
• $\beta_p$ is the portfolio's beta

Step 2: Find the beta of the portfolio after the addition of funds

The total portfolio beta is a weighted average of the betas of the existing portfolio and the new stocks:

$\beta_p = \frac{W_{existing} \cdot \beta_{existing} + W_{new} \cdot \beta_{new}}{W_{existing} + W_{new}}$

Where:

• $W_{existing}$ is the value of the existing portfolio ($20M)
• $W_{new}$ is the value of the new investment ($5M)
• $\beta_{existing}$ is the beta of the existing portfolio (1.7)
• $\beta_{new}$ is the beta of the new stocks to be determined

Step 3: Solve for $\beta_{new}$

Rearranging the weighted average beta equation:

$\beta_p \cdot (W_{existing} + W_{new}) = W_{existing} \cdot \beta_{existing} + W_{new} \cdot \beta_{new}$

$\beta_{new} = \frac{\beta_p \cdot (W_{existing} + W_{new}) - W_{existing} \cdot \beta_{existing}}{W_{new}}$

First, find $\beta_p$ using the required return equation. Substituting $R_p = 0.15$, $R_f = 0.045$, and $R_m - R_f = 0.07$:

$0.15 = 0.045 + \beta_p \cdot 0.07$

$\beta_p = \frac{0.15 - 0.045}{0.07} = 1.5$

Now substitute $\beta_p = 1.5$, $W_{existing} = 20M$, $W_{new} = 5M$, and $\beta_{existing} = 1.7$ into the $\beta_{new}$ equation:

$\beta_{new} = \frac{1.5 \cdot (20 + 5) - 20 \cdot 1.7}{5}$

$\beta_{new} = \frac{37.5 - 34}{5} = \frac{3.5}{5} = 0.7$

Final Answer:

The average beta of the new stocks added to the portfolio should be 0.7.

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Do you want additional details or further clarification?

Here are 5 related questions to deepen understanding:

1. What is the significance of beta in a portfolio, and how does it affect the required return?
2. How does the risk-free rate influence the portfolio’s required return?
3. What would happen to the portfolio's required return if the market risk premium changed?
4. Can the beta of a portfolio ever be negative? If so, what would it imply?
5. How do weighted averages affect portfolio management in practice?

Tip: Always reassess the portfolio's risk profile after changes to ensure alignment with investment goals.