Consider a risk$-$averse investor with the power preferences $($where $$ denotes the risk aversion magnitude$)$ and an initial fund value of $W_t=250,000$ at date $t.$ The investment environment offers two alternatives to the investors including: $($i$)$ a safe asset with a fixed risk$-$free rate $R_F=1+r_F$ per invested unit between dates $t$ and $t+1$; and $($ii$)$ common equity share of a private company. The equity return $(R_E)$ 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 $D_{t+i}N(,{}^2)$ for any future date $i1.$

Q$\mathrm{.}4.$ Consider an alternative scenario where $$ is not pre$-$determined therefore the valuation 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 ${}^{**}.$ The answer is expected to identify main considerations throughout the pricing procedure and provide methodological approaches used to address the considerations. Structure the answers under separate bullet points. Drive briefly$)$

Q$\mathrm{.}5.$ What is optimal allocation of funds to the risky investment? Your answer should only provide the fraction or overall fund allocated. $($ Give exact numerical solution$)$