1. **Problem Statement:** Determine the total ending wealth under two possible outcomes, the expected utility of the risky investment, and the optimal amount to invest in the risky asset for an investor with a negative exponential utility function $U(x) = -e^{-Aw}$. 2. **Given:** - Current wealth $W_0 = 100$ - Risky asset returns: +20% or -10% with equal probability 0.5 - Utility function: $U(w) = -e^{-Aw}$ - Risk aversion coefficient $A = 0.00231$ - Investment amount in risky asset: $\alpha$ 3. **(i) Total ending wealth under two outcomes:** - If return is +20%, wealth is: $$W_1 = 100 - \alpha + 1.2\alpha = 100 + 0.2\alpha$$ - If return is -10%, wealth is: $$W_2 = 100 - \alpha + 0.9\alpha = 100 - 0.1\alpha$$ 4. **(ii) Expected utility of the risky investment:** Expected utility is the average of utilities weighted by probabilities: $$EU(\alpha) = 0.5 \times U(W_1) + 0.5 \times U(W_2) = 0.5 \times (-e^{-A(100 + 0.2\alpha)}) + 0.5 \times (-e^{-A(100 - 0.1\alpha)})$$ Simplify: $$EU(\alpha) = -0.5 e^{-100A} e^{-0.2A\alpha} - 0.5 e^{-100A} e^{0.1A\alpha} = -0.5 e^{-100A} \left(e^{-0.2A\alpha} + e^{0.1A\alpha}\right)$$ 5. **Maximizing expected utility:** Maximize $EU(\alpha)$ is equivalent to minimizing: $$f(\alpha) = e^{-0.2A\alpha} + e^{0.1A\alpha}$$ Take derivative: $$f'(\alpha) = -0.2A e^{-0.2A\alpha} + 0.1A e^{0.1A\alpha}$$ Set $f'(\alpha) = 0$: $$-0.2A e^{-0.2A\alpha} + 0.1A e^{0.1A\alpha} = 0$$ Divide both sides by $A$ (nonzero): $$-0.2 e^{-0.2A\alpha} + 0.1 e^{0.1A\alpha} = 0$$ Rearranged: $$0.1 e^{0.1A\alpha} = 0.2 e^{-0.2A\alpha}$$ Divide both sides by 0.1: $$e^{0.1A\alpha} = 2 e^{-0.2A\alpha}$$ Multiply both sides by $e^{0.2A\alpha}$: $$e^{0.3A\alpha} = 2$$ Take natural log: $$0.3A\alpha = \ln 2$$ Solve for $\alpha$: $$\alpha = \frac{\ln 2}{0.3A}$$ 6. **(iii) Calculate $\alpha$ for $A=0.00231$:** $$\alpha = \frac{\ln 2}{0.3 \times 0.00231} = \frac{0.6931}{0.000693} \approx 1000$$ **Interpretation:** The investor should invest approximately 1000 units in the risky asset, which is more than current wealth, indicating leverage or borrowing to invest. **Final answers:** - Total wealth outcomes: $W_1 = 100 + 0.2\alpha$, $W_2 = 100 - 0.1\alpha$ - Expected utility maximized at $\alpha = \frac{\ln 2}{0.3A}$ - For $A=0.00231$, $\alpha \approx 1000$