Consider a risk$-$averse investor 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 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$\mathrm{.}1.$ 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$)$

Q$.2.$ Suppose the invest or considers two scenarios where $$ is pre$-$det ermined: the equity

share is priced based on the st ochastic discount factor obtained in the previous part

when $=0$ and second when the stochastic discount factor includes $=1.$ Under

both scenarios, compute the present value of the equity share $($from the perspective

date $t).$ Suppose that $r_F=5\mathrm{.}25\%,=1\mathrm{.}00,=0\mathrm{.}50$ and $=2.($include brief

derivations and final numerical answers: $0\mathrm{.}5-1$ page$)$

Q$\mathrm{.}3.$ Provide an asset pricing argument to explain the difference between valuations. $(2-3$

lines$)$

Q$\mathrm{.}4.$ Consider an alternative scenario where $$ is not pre$-$det ermined therefore the valu$-$

ation depends upon the optimal value of the allocation to each asset. Assume the

no$-$arbitrage condition and discuss the st eps the invest ors undertakes to compute the

optimal value ${}^{**}.$ The answer is expected to identify main consider ations through$-$

out the pricing procedure and provide methodological approaches used to address

the considerations. Structure the answers under separate bullet points. $(0\mathrm{.}5-1$ page$)$

Q$\mathrm{.}5.$ 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 Q$1-$Q$5$ in the submitted docu$-$

ment.