To calculate the standard deviation of the portfolio, we will follow these steps:

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Step 1: Calculate the Expected Portfolio Return

The portfolio weights are:

• Stock X = 20% = 0.20
• Stock Y = 41% = 0.41
• Stock Z = 39% = $1 - 0.20 - 0.41 = 0.39$

The expected return for each state of the economy is: $R_p = w_X R_X + w_Y R_Y + w_Z R_Z$

Normal State:

$R_p^{\text{Normal}} = (0.20 \times 0.095) + (0.41 \times 0.029) + (0.39 \times 0.119) = 0.06018 = 6.018\%$

Boom State:

$R_p^{\text{Boom}} = (0.20 \times 0.168) + (0.41 \times 0.248) + (0.39 \times 0.163) = 0.20111 = 20.111\%$

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Step 2: Calculate the Expected Portfolio Return

Using the probabilities: $\mathbb{E}[R_p] = P_{\text{Normal}} \cdot R_p^{\text{Normal}} + P_{\text{Boom}} \cdot R_p^{\text{Boom}}$

$\mathbb{E}[R_p] = (0.79 \times 0.06018) + (0.21 \times 0.20111) = 0.09039 = 9.039\%$

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Step 3: Calculate the Variance of the Portfolio

The variance formula is: $\text{Var}(R_p) = \sum P_i \cdot (R_p^i - \mathbb{E}[R_p])^2$

Normal State Contribution:

$(0.06018 - 0.09039)^2 = (-0.03021)^2 = 0.0009127$ $\text{Weighted: } 0.79 \cdot 0.0009127 = 0.000721$

Boom State Contribution:

$(0.20111 - 0.09039)^2 = (0.11072)^2 = 0.012259$ $\text{Weighted: } 0.21 \cdot 0.012259 = 0.002574$

Total Variance:

$\text{Var}(R_p) = 0.000721 + 0.002574 = 0.003295$

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Step 4: Calculate the Standard Deviation

$\text{StdDev}(R_p) = \sqrt{\text{Var}(R_p)} = \sqrt{0.003295} = 0.0574 = 5.74\%$

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Step 5: Match Closest Answer

The closest option is 4.95%.

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Would you like further clarification on any step? Here are related questions for you to explore:

1. How is portfolio variance calculated with more than three stocks?
2. What is the role of correlation in calculating portfolio variance?
3. How does changing probabilities affect the portfolio standard deviation?
4. Can you explain how weights impact the risk of the portfolio?
5. What methods exist to reduce portfolio standard deviation?

Tip: Always confirm portfolio weights sum to 1 before beginning calculations.