1. **State the problem:** We have three stocks X, Y, and Z with given expected returns, standard deviations, and betas. Fund Q invests equally in these stocks. We need to find: - The market risk premium - The beta of Fund Q - The required return of Fund Q 2. **Given data:** - Stock X: $E(R_X) = 9\%$, $\beta_X = 0.8$ - Stock Y: $E(R_Y) = 10.75\%$, $\beta_Y = 1.2$ - Stock Z: $E(R_Z) = 12.5\%$, $\beta_Z = 1.6$ - Risk-free rate: $R_f = 5.5\%$ - Fund Q invests equally: weights $w_X = w_Y = w_Z = \frac{1}{3}$ 3. **Formula used:** The Capital Asset Pricing Model (CAPM) states: $$E(R_i) = R_f + \beta_i (E(R_m) - R_f)$$ where $E(R_m) - R_f$ is the market risk premium. 4. **Find the market risk premium:** Since the market is in equilibrium, required returns equal expected returns. Using Stock X: $$9 = 5.5 + 0.8 \times (E(R_m) - 5.5)$$ Rearranged: $$9 - 5.5 = 0.8 \times (E(R_m) - 5.5)$$ $$3.5 = 0.8 \times (E(R_m) - 5.5)$$ Divide both sides by 0.8: $$\frac{3.5}{0.8} = E(R_m) - 5.5$$ $$4.375 = E(R_m) - 5.5$$ Add 5.5 to both sides: $$E(R_m) = 9.875\%$$ Market risk premium: $$E(R_m) - R_f = 9.875 - 5.5 = 4.375\%$$ 5. **Check with Stock Y:** $$10.75 = 5.5 + 1.2 \times 4.375$$ $$10.75 = 5.5 + 5.25 = 10.75\%$$ (matches) 6. **Calculate beta of Fund Q:** $$\beta_Q = w_X \beta_X + w_Y \beta_Y + w_Z \beta_Z = \frac{1}{3}(0.8 + 1.2 + 1.6) = \frac{3.6}{3} = 1.2$$ 7. **Calculate required return of Fund Q:** Using CAPM: $$E(R_Q) = R_f + \beta_Q (E(R_m) - R_f) = 5.5 + 1.2 \times 4.375 = 5.5 + 5.25 = 10.75\%$$ **Final answers:** - Market risk premium = $4.375\%$ - Beta of Fund Q = $1.2$ - Required return of Fund Q = $10.75\%$