Consider a risk-averse investor with the power preferences (where y denotes the risk aver- sion magnitude)...

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Consider a risk-averse investor with the power preferences (where y denotes the risk aver- sion magnitude) and an initial fund value of W = 250,000 at date t. The investment environment offers two alternatives to the investors including: (i) a safe asset with a fixed risk-free rate RF = 1+r per invested unit between datest and t + 1; and (ii) common equity share of a private company. The equity return (RE) involves risk depending on the future company's performance. Consider that the company's dividend per share value summarises the performance and follows a Normal distribution with Dt+i ~ N(, ) for any future date i > 1. Q.1. Assume that the investor intends to invest all of their fund value where the fraction w is allocated to the risky investment and the remaining 1 - w is allocated to the risk-free investment such that Rw = (1 w).RF +w.RE. Derive the pricing expres- sion for the common equity (price per share Pt) according to the first fundamental asset pricing equation. Denote all additional terms you used and define the stochastic discount factor in this context. (0.5 page) Q.2. Suppose the investor considers two scenarios where w is pre-determined: the equity share is priced based on the stochastic discount factor obtained in the previous part when w= O and second when the stochastic discount factor includes w = 1. Under both scenarios, compute the present value of the equity share (from the perspective date t). Suppose that rF = 5.25%, = 1.00, = 0.50 and y = 2. (include brief derivations and final numerical answers: 0.5-1 page) Q.3. Provide an asset pricing argument to explain the difference between valuations. (2-3 lines) Q.4. Consider an alternative scenario where w is not pre-determined therefore the valu- ation depends upon the optimal value of the allocation to each asset. Assume the no-arbitrage condition and discuss the steps the investors undertakes to compute the optimal value w*. The answer is expected to identify main considerations through- out the pricing procedure and provide methodological approaches used to address the considerations. Structure the answers under separate bullet points. (0.5-1 page) Q.5. What is optimal allocation of funds to the risky investment? Your answer should only provide the fraction or overall fund allocated. (1 line) Clearly separate and number answers to each of the parts Q1-Q5 in the submitted docu- ment. Consider a risk-averse investor with the power preferences (where y denotes the risk aver- sion magnitude) and an initial fund value of W = 250,000 at date t. The investment environment offers two alternatives to the investors including: (i) a safe asset with a fixed risk-free rate RF = 1+r per invested unit between datest and t + 1; and (ii) common equity share of a private company. The equity return (RE) involves risk depending on the future company's performance. Consider that the company's dividend per share value summarises the performance and follows a Normal distribution with Dt+i ~ N(, ) for any future date i > 1. Q.1. Assume that the investor intends to invest all of their fund value where the fraction w is allocated to the risky investment and the remaining 1 - w is allocated to the risk-free investment such that Rw = (1 w).RF +w.RE. Derive the pricing expres- sion for the common equity (price per share Pt) according to the first fundamental asset pricing equation. Denote all additional terms you used and define the stochastic discount factor in this context. (0.5 page) Q.2. Suppose the investor considers two scenarios where w is pre-determined: the equity share is priced based on the stochastic discount factor obtained in the previous part when w= O and second when the stochastic discount factor includes w = 1. Under both scenarios, compute the present value of the equity share (from the perspective date t). Suppose that rF = 5.25%, = 1.00, = 0.50 and y = 2. (include brief derivations and final numerical answers: 0.5-1 page) Q.3. Provide an asset pricing argument to explain the difference between valuations. (2-3 lines) Q.4. Consider an alternative scenario where w is not pre-determined therefore the valu- ation depends upon the optimal value of the allocation to each asset. Assume the no-arbitrage condition and discuss the steps the investors undertakes to compute the optimal value w*. The answer is expected to identify main considerations through- out the pricing procedure and provide methodological approaches used to address the considerations. Structure the answers under separate bullet points. (0.5-1 page) Q.5. What is optimal allocation of funds to the risky investment? Your answer should only provide the fraction or overall fund allocated. (1 line) Clearly separate and number answers to each of the parts Q1-Q5 in the submitted docu- ment.