Greta has a risk aversion of A $=3$ and a $1-$year investment horizon. She is pondering two portfolios, the S&P $500$ and a hedge fund, as well as a number of $1-$year strategies. $($All rates are annual and continuously compounded.$)$ The S&P $500$ risk premium is estimated at $8\%$ per year, with a standard deviation of $23\%.$ The hedge fund risk premium is estimated at $10\%$ with a standard deviation of $38\%.$ The returns on these portfolios in any particular year are uncorrelated with their own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P $500$ and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim.

Calculate Greta$$s capital allocation using an annual correlation of $0\mathrm{.}3.$

Here are my incorrect answers thus far:

$50\mathrm{.}41\%$

$23\mathrm{.}08\%$

$26\mathrm{.}51\%$

$47\mathrm{.}40\%$

$16\mathrm{.}90\%$

$63\mathrm{.}10\%$