Show me the steps to solve Problem $7-23$

Greta has risk aversion of $A=3$ when applied to return on wealth over

a one$-$year 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 $6\%$ per year, with a standard deviation of $21\%.$ The hedge

fund risk premium is estimated at $9\%$ with a standard deviation of

$36\%.$ The returns on both of these portfolios in any particular year are

uncorrelated with its 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.

a$-1.$ Assuming the correlation between the annual returns on the two

portfolios is indeed zero, what would be the optimal asset allocation?

$($Do not round intermediate calculations. Enter your answers as

decimals rounded to $4$ places.$)$

a$-2.$ What is the expected risk premium on the portfolio? $($Do not

round intermediate calculations. Enter your answer as decimals

rounded to $4$ places.$)$