Consider a risk$-$averse invest or with the power preferences $($where $$ denotes the risk aver$-$

sion magnitude$)$ and an initial fund value of $W_t=250,000$ at date $t.$ The investment

envir onment offers two alternatives to the investor including: $($i$)$ a safe asset with a fixed

risk$-$free rate $R_F=1+r_F$ per invested unit between dat es $t$ 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$\mathrm{.}1.$ Assume that the investor intends to invest all of their fund value where the fraction

$$ is allocated to the risky investment and the remaining $1-$ is allocated to the

risk$-$free investment such that $R_W=(1-)*R_F+*R_E.$ Derive the pricing expres$-$

sion for the common equity $($price per share $P_t)$ according to the first fundamental

asset pricing equation. Denote all additional terms you used and define the stochastic

discount fact or in this context. $(0\mathrm{.}5$ page$)$