# Financial Economics: Optimal Risky Portfolio question - involves Sharpe ratio

• Oct 13th 2011, 01:59 PM
Runty
Financial Economics: Optimal Risky Portfolio question - involves Sharpe ratio
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.

Quote:

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?
$\displaystyle \mu_G=9%$, $\displaystyle \sigma_G=40%$, $\displaystyle \mu_P=9%$, $\displaystyle \sigma_P=25%$, $\displaystyle r_f=3%$, $\displaystyle w=5%$
$\displaystyle \sigma_{PG}=0$ (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?