$1.)$The historical mean return to S&P$500$ is $10\%.$ A hedge fund charges $2$ plus $20\%.$ The incentive fee is paid on the net return $($if positive$)$ after management fees have been subtracted. Use this information to respond to questions $1-3.$ What return should the fund earn, before fees, to provide investors with the same rate of return as S&P$500?$

$2.)$ If the excess return to above hedge fund returns, defined by its returns minus the returns to the S&P$500,$ are normally distribution with a mean of $2\%$ and a standard deviation of $4\%.$ What is the probability that the investors in the fund earn a return $($net all fees$)$ that is worse than simply buying S&P$500?$

a$.73\mathrm{.}4\%$

b$.26\mathrm{.}6\%$

c$.94\mathrm{.}5\%$

d$.5\mathrm{.}5\%$

$3.)$ Over the past $4$ years, the fund gross returns $($before fees$)$ have been $-5\%,15\%,20\%,10\%.$ Compute the geometric return net all fees to this fund. Hint: first compute the year$-$by$-$year overall returns, and then compute the geometric mean of those returns.

a$.5\mathrm{.}33\%$

b$.6\mathrm{.}13\%$

c$.6\mathrm{.}05\%$

d$.5\mathrm{.}73\%$