Greta has 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 $6\%$ 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.

Required:

a$-1.$ Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation?

a$-2.$ What is the expected risk premium on the portfolio?

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What is the expected risk premium on the portfolio?

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Expected risk premium