I think I've managed to solve this one, but there's a part or two I'm unsure of due to some of the question's wording. I'll first list the question.

Your parents find out that you are taking financial economics and come to you for financial advice. A family friend has asked them to buy equity in a gold mine that she is developing in northern Ontario. The expected return of the gold mine is 9% and its standard deviation is 40%. Currently your parents divide their savings between a risk-free asset (Government of Canada bonds) and a large, well-diversified portfolio of stocks. The expected return of the risk-free asset is 3%, and the expected return of the portfolio of risky stocks is 9% with a standard deviation of 25%. The gold mine’s returns are independent of your parents’ stock portfolio. They are considering investing 5% of their money in the gold mine. Would you advise them to undertake this investment?
Here's my answer:
$\displaystyle \mu_G=9%$, $\displaystyle \sigma_G=40%$, $\displaystyle \mu_P=9%$, $\displaystyle \sigma_P=25%$, $\displaystyle r_f=3%$, $\displaystyle w=5%$
$\displaystyle \sigma_{PG}=0$ (this is something I'm unsure of, but I'm assuming the covariance is zero because the question says the gold mine's returns are independent of the stock portfolio)
I first compute the Sharpe ratio of the portfolio:
$\displaystyle S_P=\frac{\mu_P-r_f}{\sigma_P}=\frac{0.09-0.03}{0.25}=24$
Then, for the Sharpe ratio of the new portfolio, we need the mean and variance of said new portfolio.
$\displaystyle \mu_{NP}=w\mu_G+(1-w)\mu_P=0.09$
$\displaystyle \sigma_{NP}^2=w^2\sigma_G^2+(1-w)^2\sigma_P^2+2w(1-w)\sigma_{PG}=0.5680625$
This is less than $\displaystyle \sigma_P^2=0.625$
$\displaystyle \sigma_{NP}=0.238340617$
I now compute the Sharpe ratio of the new portfolio:
$\displaystyle S_{NP}=\frac{\mu_{NP}-r_f}{\sigma_{NP}}=\frac{0.09-0.03}{0.23834}=25.174$

Since the Sharpe ratio of the new portfolio is higher than that of the original portfolio, it would be advisable to undertake the investment.


I'm not sure if I made a mistake somewhere, but if I did, could you show/correct it?