1. Expected Return on an Equally Weighted Portfolio

For an equally weighted portfolio, the weight of each stock is $w = \frac{1}{3} = 0.3333$. The expected return ($E(R_p)$) is calculated using:

$E(R_p) = \sum P_i \cdot R_p$

Where:

• $P_i$: Probability of the state.
• $R_p$: Portfolio return in that state.

Portfolio Return in Each State:

$R_p = w_A R_A + w_B R_B + w_C R_C$

• Boom State Portfolio Return: $R_{p, \text{Boom}} = (0.3333)(0.11) + (0.3333)(0.05) + (0.3333)(0.31) = 0.1533$

• Bust State Portfolio Return: $R_{p, \text{Bust}} = (0.3333)(0.20) + (0.3333)(0.26) + (0.3333)(-0.11) = 0.1167$

Expected Return Calculation:

$E(R_p) = (0.73)(0.1533) + (0.27)(0.1167)$ $E(R_p) = 0.111909 + 0.031509 = 0.143418 \text{ (or } 14.34\%\text{)}$

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2. Variance of the Portfolio with Custom Weights

Given weights:

• $w_A = 0.28$
• $w_B = 0.28$
• $w_C = 0.44$

Portfolio Return Formula:

$R_p = w_A R_A + w_B R_B + w_C R_C$

• Boom State Portfolio Return: $R_{p, \text{Boom}} = (0.28)(0.11) + (0.28)(0.05) + (0.44)(0.31) = 0.1852$

• Bust State Portfolio Return: $R_{p, \text{Bust}} = (0.28)(0.20) + (0.28)(0.26) + (0.44)(-0.11) = 0.0920$

Expected Return:

$E(R_p) = (0.73)(0.1852) + (0.27)(0.0920) = 0.135396 + 0.024840 = 0.160236$

Variance Formula:

$\text{Variance} = \sum P_i \cdot (R_{p, i} - E(R_p))^2$

• Boom State Deviation: $R_{p, \text{Boom}} - E(R_p) = 0.1852 - 0.160236 = 0.024964$

• Bust State Deviation: $R_{p, \text{Bust}} - E(R_p) = 0.0920 - 0.160236 = -0.068236$

• Variance Calculation: $\text{Variance} = (0.73)(0.024964^2) + (0.27)(-0.068236^2)$ $\text{Variance} = (0.73)(0.0006232) + (0.27)(0.004656)$ $\text{Variance} = 0.0004549 + 0.0012571 = 0.001712$

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Final Results:

1. Expected Return (Equally Weighted Portfolio): $\mathbf{14.34\%}$
2. Variance (Custom Weighted Portfolio): $\mathbf{0.00171}$

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Would you like further clarification or details on these calculations?

Related Questions:

1. How does portfolio weight allocation impact variance and expected return?
2. What is the significance of the expected return in portfolio optimization?
3. How does the variance measure portfolio risk?
4. What are the advantages of equally weighted portfolios?
5. How would changing the weight of Stock C affect the portfolio variance?

Tip:

Variance is a key measure of portfolio risk. Lower variance often indicates a more stable investment, but diversification plays a critical role in achieving that balance.