Part 1: Expected Return on an Equally Weighted Portfolio

An equally weighted portfolio means each stock has an equal weight of $\frac{1}{3}$ (33.33%).

The expected return is calculated as: $E(R_p) = \sum P(s) \times \left[ w_A \cdot R_A + w_B \cdot R_B + w_C \cdot R_C \right]$

Where:

• $P(s)$: Probability of each state
• $R_A, R_B, R_C$: Returns for Stock A, B, and C
• $w_A, w_B, w_C = \frac{1}{3}$

Calculation:

1. For Boom State ($P(s) = 0.73$): $w_A \cdot R_A + w_B \cdot R_B + w_C \cdot R_C = \frac{1}{3} \cdot 0.11 + \frac{1}{3} \cdot 0.05 + \frac{1}{3} \cdot 0.31 = 0.15667$

2. For Bust State ($P(s) = 0.27$): $w_A \cdot R_A + w_B \cdot R_B + w_C \cdot R_C = \frac{1}{3} \cdot 0.20 + \frac{1}{3} \cdot 0.26 + \frac{1}{3} \cdot (-0.11) = 0.11667$

3. Expected Return: $E(R_p) = 0.73 \cdot 0.15667 + 0.27 \cdot 0.11667 = 0.14436 \, (or \, 14.44\%)$

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Part 2: Variance of a Portfolio with Weights 28% in A, 28% in B, and 44% in C

The variance of a portfolio is calculated as: $\sigma_p^2 = \sum P(s) \left[ w_A \cdot R_A + w_B \cdot R_B + w_C \cdot R_C - E(R_p) \right]^2$

Where $E(R_p)$ is the portfolio's expected return.

Calculation:

Weights:

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

1. Expected Return: $E(R_p) = \sum P(s) \cdot \left[ w_A \cdot R_A + w_B \cdot R_B + w_C \cdot R_C \right]$

   • Boom State: $w_A \cdot R_A + w_B \cdot R_B + w_C \cdot R_C = 0.28 \cdot 0.11 + 0.28 \cdot 0.05 + 0.44 \cdot 0.31 = 0.1958$
   • Bust State: $w_A \cdot R_A + w_B \cdot R_B + w_C \cdot R_C = 0.28 \cdot 0.20 + 0.28 \cdot 0.26 + 0.44 \cdot (-0.11) = 0.1116$
   • Expected Return: $E(R_p) = 0.73 \cdot 0.1958 + 0.27 \cdot 0.1116 = 0.17186$

2. Variance: $\sigma_p^2 = \sum P(s) \cdot \left[ w_A \cdot R_A + w_B \cdot R_B + w_C \cdot R_C - E(R_p) \right]^2$

   • Boom State: $\text{Deviation} = 0.1958 - 0.17186 = 0.02394$ $\text{Squared Deviation} = (0.02394)^2 = 0.000573$ $\text{Weighted Contribution} = 0.73 \cdot 0.000573 = 0.000418$
   • Bust State: $\text{Deviation} = 0.1116 - 0.17186 = -0.06026$ $\text{Squared Deviation} = (-0.06026)^2 = 0.003631$ $\text{Weighted Contribution} = 0.27 \cdot 0.003631 = 0.000981$
   • Total Variance: $\sigma_p^2 = 0.000418 + 0.000981 = 0.001399$

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Final Results:

1. Expected Return (Equally Weighted Portfolio): $14.44\%$

2. Variance (Weighted Portfolio): $0.00140$ (rounded to 5 decimal places)

Would you like further explanations or assistance with related topics?

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Additional Questions:

1. How does changing portfolio weights affect expected returns and variance?
2. What is the interpretation of variance in portfolio performance?
3. How can covariance between stocks influence portfolio variance?
4. What are the limitations of this approach in real-world scenarios?
5. How can diversification reduce portfolio risk?

Tip:

Always verify portfolio weights sum to 1 to ensure correct calculations!