$(8$ points$)$ Portfolio i$\text{'}$s returns are described by:

$r_i=r_{f}1(r_m-r_f)2(r_c-r_f)0\mathrm{.}5$

where $r_m$ is the return on the market index, $r_c$ is the return on the commodity index and $r_f$ is the risk$-$free rate.

a$.(6$ points$)$ Construct a pure arbitrage trade using the market index, a commodity index, $t-$bills and portfolio i$.$ What percent of the mimicking portfolio should be invested in each index and the t$-$bills? Should you go long or short the mimicking portfolio and long or short Portfolio i$?$