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The following information applies to Problems $22$ through $27$: Greta has risk aversion of $A=3$ when applied to return on wealth over a one$-$year horizon. She is pondering two portfolios, the SThe following information applies to Problems $22$ through $27$: 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

The following information applies to Problems $22$ through $27$: 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 one$-$year strategies. $($All rates are annual and continuously compounded.$)$ The S&P $500$ risk premium is estimated at $5\%$ per year, with a standard deviation of $20\%.$ The hedge fund risk premium is estimated at $10\%$ with a standard deviation of $35\%.$ 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.

$22.$ Compute the estimated annual risk premiums, standard deviations, and Sharpe ratios for the two portfolios.

$23.$ Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation?

$24.$ What should be Greta's capital allocation?

$25.$ If the correlation coefficient between annual portfolio returns is actually $\mathrm{.}3,$ what is the covariance between the returns?