You have $110,000 to invest in a portfolio containing Stock X, Stock Y, and a risk-free asset. You must invest all of your money. Your goal is to create a portfolio that has an expected return of 10 percent and that has only 74 percent of the risk of the overall market. If X has an expected return of 30 percent and a beta of 2.0, Y has an expected return of 20 percent and a beta of 1.2, and the risk-free rate is 4 percent, how much money will you invest in Stock Y?

We need to determine how much money to invest in Stock Y under the given conditions. Let's start by defining the variables and break down the problem.

### Variables:
- Total amount to invest: $ 110,000 $
- Expected return of the portfolio: 10% or 0.10
- Portfolio's risk (beta): 74% of the overall market, so $ \beta_p = 0.74 \times 1 = 0.74 $
- Stock X: 
  - Expected return: 30% or 0.30
  - Beta: 2.0
- Stock Y: 
  - Expected return: 20% or 0.20
  - Beta: 1.2
- Risk-free asset:
  - Return: 4% or 0.04
  - Beta: 0 (since it's risk-free)
  
### Portfolio Setup:
The portfolio contains Stock X, Stock Y, and the risk-free asset. Let:
- $ w_X $: the weight of Stock X in the portfolio
- $ w_Y $: the weight of Stock Y in the portfolio
- $ w_f $: the weight of the risk-free asset in the portfolio

We know:
\[
w_X + w_Y + w_f = 1
\]
since the total investment must sum to 100% of the portfolio.

### Step 1: Expected Return of the Portfolio
The expected return of the portfolio is the weighted average of the returns of the assets:
\[
E(R_p) = w_X \cdot E(R_X) + w_Y \cdot E(R_Y) + w_f \cdot R_f
\]
Substituting the known values:
\[
0.10 = w_X \cdot 0.30 + w_Y \cdot 0.20 + w_f \cdot 0.04
\]

### Step 2: Beta of the Portfolio
The portfolio beta is the weighted average of the betas of the assets:
\[
\beta_p = w_X \cdot \beta_X + w_Y \cdot \beta_Y + w_f \cdot 0
\]
Since the beta of the risk-free asset is 0, the equation simplifies to:
\[
0.74 = w_X \cdot 2.0 + w_Y \cdot 1.2
\]

### Step 3: Solving the System of Equations
We now have two equations:
1. $ 0.10 = 0.30w_X + 0.20w_Y + 0.04w_f $
2. $ 0.74 = 2.0w_X + 1.2w_Y $

Since $ w_X + w_Y + w_f = 1 $, we can solve these equations for $ w_Y $.

Let me calculate this.The solution indicates that $ w_X \approx 5.80 $ and $ w_Y \approx -9.05 $. These weights are not feasible in a practical sense, since weights cannot be negative (especially for stocks) or larger than 1 in this context. A negative weight for $ w_Y $ suggests short selling, but this likely doesn't align with your objective of purely allocating the $110,000 between stocks and the risk-free asset.

It seems we need to reconsider or refine the assumptions, likely examining the constraints on allocation more closely. If you'd like, we can explore alternative solutions like allowing short-selling or double-check the problem's risk and return constraints.

Would you like more details, or should we discuss other aspects?

Here are 5 follow-up questions:
1. Should we allow short-selling in this portfolio construction?
2. Are there other assets you'd like to consider?
3. What would happen if the expected return were adjusted?
4. How would a change in beta affect the solution?
5. Would rebalancing the portfolio over time improve results?

**Tip:** If you're working with financial portfolios, sometimes introducing constraints like no short-selling changes how assets are allocated significantly.

Learn how to calculate the investment in Stock Y in a portfolio with $110,000, given target expected returns and risk parameters. This solution uses key concepts from portfolio theory, expected return formulas, and the CAPM. Suitable for advanced finance students.

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