The Arrow$-$Debreu model has three states and three assets: two are risky and one is risk$-$free. The initial price vector and the

payoff matrix are

$S=\left[\begin{array}{c}2\\ 3\\ 2\end{array}\right]$ and $D=\left[\begin{array}{ccc}8& 1& 7\\ 1& 9& 1\\ 6& 6& 6\end{array}\right]$

Consider a payoff vector $x=\left[\begin{array}{ccc}-35& 28\mathrm{.}5& -30\mathrm{.}5\end{array}\right]$

$($a$)$ Find a portfolio $$ replicating the given payoff vector:

$($b$)$ Find its initial value:

$($c$)$ Assuming all states are equilikely, what is the rate of return on this portfolio $($as a percentage$)?$