Let's solve **Question 4: What is the standard deviation of expected returns for the portfolio?** ### **Step 1: Calculate Portfolio Expected Returns in Each State** Let: - $50,000 is invested in Goldcorp. - $40,000 is invested in Enzo Biochem. - Total portfolio = $90,000. Weights: - Goldcorp: \( w_G = \frac{50,000}{90,000} = .5556 \) - Enzo Biochem: \( w_E = \frac{40,000}{90,000} = .4444 \) For each state, portfolio return (\( R_p \)) is: \[ R_p = w_G \cdot R_G + w_E \cdot R_E \] | State | Probability | Goldcorp (\(R_G\)) | Enzo Biochem (\(R_E\)) | Portfolio Return (\(R_p\)) | |-----------------|-------------|--------------------|-------------------------|----------------------------| | Strong Growth | .20 | 30% | 10% | \(.5556 \times 30\% + .4444 \times 10\% = 22.22\%\) | | Moderate Growth | .50 | 10% | 7% | \(.5556 \times 10\% + .4444 \times 7\% = 8.33\%\) | | Recession | .30 | -10% | 1% | \(.5556 \times -10\% + .4444 \times 1\% = -5.56\%\) | **Explanation:** We computed the weighted portfolio return for each economic state using the weights and expected returns. --- ### **Step 2: Calculate Portfolio Expected Return and Standard Deviation** #### **A. Portfolio Expected Return (\(E[R_p]\)):** \[ E[R_p] = (.20 \times 22.22\%) + (.50 \times 8.33\%) + (.30 \times -5.56\%) \] \[ = 4.444\% + 4.165\% - 1.668\% = 6.941\% \] #### **B. Portfolio Standard Deviation (\(\sigma_p\)):** \[ \sigma_p = \sqrt{\sum P_i (R_{p,i} - E[R_p])^2} \] - For Strong Growth: \( (22.22\% - 6.941\%) = 15.279\% \) - For Moderate Growth: \( (8.33\% - 6.941\%) = 1.389\% \) - For Recession: \( (-5.56\% - 6.941\%) = -12.501\% \) Now, square these differences, multiply by probability, and sum: \[ \begin{align*} &.20 \times (.15279^2) = .20 \times .02336 = .004672 \\ &.50 \times (.01389^2) = .50 \times .000193 = .0000965 \\ &.30 \times (-.12501^2) = .30 \times .015627 = .004688 \\ \end{align*} \] \[ \text{Sum} = .004672 + .0000965 + .004688 = .0094565 \] \[ \sigma_p = \sqrt{.0094565} \approx .0972 = 9.72\% \] **Explanation:** We calculated the expected return and then the standard deviation using the probability-weighted squared deviations from the mean. --- ## **Final Answer:** **The standard deviation of expected returns for the portfolio is approximately \(9.72\%\).**