Question

1. please include calculations/explanations to all answersYou are currently only invested in the Natasha Fund (aside from risk-free securities). It has an expected return of 14% with a volatility of 20%. Currently, the risk-free rate of interest is 3.8%. Your broker suggests that you add Hannah Corporation to your portfolio. Hannah Corporation has an expected return of 20%, a volatility of 60%, and a correlation of 0 with the Natasha Fund.

1. Is your broker right?
2. You follow your broker’s advice and make a substantial investment in Hannah stock so that, considering only your risky investments, 60% is in the Natasha Fund and 40% is in Hannah stock. When you tell your finance professor about your investment, he says that you made a mistake and should reduce your investment in Hannah. Is your finance professor right?
3. You decide to follow your finance professor’s advice and reduce your exposure to Hannah. Now Hannah represents 15% of your risky portfolio, with the rest in the Natasha fund. Is this the correct amount of Hannah stock to hold?
4. Calculate the Sharpe ratio of each of the three portfolios. What portfolio weight in Hannah stock maximizes the Sharpe ratio?

Answer 1

Considering that 60% funds is in Natasha Fund and 40% in Hannah Stock the Weighted average return of the Portfolio would be

Return on Portfolio = Weight of Natasha Fund x Return on Natasha Fund + Weight of Hannah Corp x Return on Hannah Corp

= 60% x 14% + 20% x 40%

= 8.4%+8%

=16.4%

Volatility of Portfolio = (Weight of Natasha Fund^2 x Volatility of Natasha Fund^2+Weight of Hannah Corp ^2 x Volatility of Hannah Corp^2+2 x Weight of Natasha Fund x Weight of Hannah Corp x Covariance of stocks x Volatility of Natasha Fund x Volatility of Hannah Corp)^0.5

= (60%^2*20%^2+40%^2*60%^2+2*60%*40%*0*20%*60%)^0.5

= 26.83%

(Note: Since correlation is zero, covariance is also zero. Covariance = Correlation xSD of Natasha Fund x SD of Hannah = 0%x20%x 60% = 0% )

If now instead of 40% in Hannah stock, 15% is invested, the returns and Volatility would be as follows

Return on Portfolio = Weight of Natasha Fund x Return on Natasha Fund + Weight of Hannah Corp x Return on Hannah Corp

= 85% x 14% + 20% x 15%

= 11.9%+3%

=14.9%

Volatility of Portfolio = (Weight of Natasha Fund^2 x Volatility of Natasha Fund^2+Weight of Hannah Corp ^2 x Volatility of Hannah Corp^2+2 x Weight of Natasha Fund x Weight of Hannah Corp x Correlation between them x Volatility of Natasha Fund x Volatility of Hannah Corp)^0.5

= (85%^2*20%^2+15%^2*60%^2+2*85%*15%*0*20%*60%)^0.5

= 19.24%

Now from the above it is clear that

a) Broker has suggested higher investment in Hannah Corp. This leads to higher returns for the portfolio and at the same time higher volatility for the portfolio.

Sharpe Ratio of three portfolio is as follows

1) Only Investment in Natasha fund

= (Expected Return of Natasha Fund - Risk free rate)/ Standard deviation of Natasha Fund.

= (14%-3.8%)/20%

=10.2%/20%

=0.51

2) Investment in Natasha fund @ 60% and Investment in Hannah Corp @ 40%

= (Expected Return of Portfolio - Risk free rate)/ Standard deviation of Natasha Fund.

= (16.4%-3.8%)/26.83%

=12.6%/26.83%

=0.47

3) Investment in Natasha fund @ 85% and Investment in Hannah Corp @ 15%

= (Expected Return of Portfolio - Risk free rate)/ Standard deviation of Natasha Fund.

= (14.9%-3.8%)/19.24%

= 11.1%/19.24%

= 0.58

Now from the above it is clear that

a) Broker has suggested higher investment in Hannah Corp and 40% is invested in Hannah Corp. This leads to higher returns for the portfolio and at the same time higher volatility for the portfolio..Since the Sharpe ratio of this portfolio is lesser than the Sharpe Ratio of Natasha Fund, it signifies the investment is not attractive.

b) Professor is right when he says that I have made a mistake,since the Sharpe ratio of Natasha Fund was better than the portfolio with 60% in natasha fund and 40% in Hannah Corp.

c) Now based on professor advice 15% is invested in Hannah Corp and 85% in natasha Fund. This portfolio leads to higher Sharpe ratio as against 100% investment in natasha Fund or 60% in natasha Fund and 40% in Hannah Corp. Hence this is the correct level of investment. A higher Sharpe ratio signifies attractive risk adjusted investment.

d) Among all the portfolios as worked out, A portfolio weight of 15% in Hannah Corp maximises the Sharpe Ratio