Consider a single$-$factor model economy. Portfolio M has a beta of $1\mathrm{.}0$ on the factor and portfolio P has a beta of $0\mathrm{.}5$ on the factor. The expected returns on portfolios M and P are $19\%$ and $10\mathrm{.}5\%,$ respectively. The risk$-$free rate is $6\%.$ Short selling is possible. There are no transaction costs or other frictions.

i$.$ Calculate the alpha of a zero$-$beta portfolio Z$.$

ii$.$ Design a trading strategy that takes advantage of the alpha from part $($i$).$ Calculate the expected profit of your strategy, assuming that your short position has a value of $200,000$