Exercise: Mutual Fund vs Hedge Fund

The risk$-$free interest rate is zero.

The mutual funds claim to deliver the following gross returns:

$R_t^{\text{p}\text{a}\text{s}\text{s}\text{i}\text{v}\text{e}\text{f}\text{u}\text{n}\text{d}\text{b}\text{e}\text{f}\text{o}\text{r}\text{e}\text{f}\text{e}\text{e}\text{s}}=R_t^M=3\%+u_t$

$R_t^{\text{a}\text{c}\text{t}\text{i}\text{v}\text{e}\text{f}\text{u}\text{n}\text{d}\text{b}\text{e}\text{f}\text{o}\text{r}\text{e}\text{f}\text{e}\text{e}\text{s}}=1\%+1\mathrm{.}5R_t^M+{}_t$

where the error terms are independent over time and of each other, have zero means $E(u_t)=E({}_t)=0,$ and volatilities $(u_t)=10\%$ and $({}_t)=3\%.$

The hedge fund uses the same strategy as the active mutual fund, but implements the strategy as a long$-$short hedge fund, applying $5$ times of leverage, generating the following return before fees:

$R_t^{\text{h}\text{e}\text{d}\text{g}\text{e}\text{f}\text{u}\text{n}\text{d}\text{b}\text{e}\text{f}\text{o}\text{r}\text{e}\text{f}\text{e}\text{e}\text{s}}=5(R_t^{\text{a}\text{c}\text{t}\text{i}\text{v}\text{e}\text{f}\text{u}\text{n}\text{d}\text{b}\text{e}\text{f}\text{o}\text{r}\text{e}\text{f}\text{e}\text{e}\text{s}}-1\mathrm{.}5R_t^M)$

Question: What are the sharp ratios of the mutual fund and active fund?

$36$