Question

1. You currently have $100,000 invested in a portfolio that has an expected return of 12% and a volatility of 8%. Suppose the risk-free rate is 5%, and there is another portfolio that has an expected return of 20% and a volatility of 12%.

1. How do you construct a new portfolio that has a higher expected return than your current portfolio but with the same volatility?   
2. How do you construct a new portfolio that has a lower volatility than your current portfolio but with the same expected return?   

To get the full marks of this question, you need to specify the dollar amount that you invest in the new portfolios in (i) and (ii)

Answer 1

1.
Let w be the weight invested in new portfolio and 1-w be the weight in risk free rate
w*12%=8%
=>w=8%/12%=0.666666667

Expected return=0.666666667*20%+(1-0.666666667)*5%=15.0000%

This has higher expected return

Amount in new portfolio=0.666666667*100000=66666.6667
Amount in risk free rate=100000-66666.6667=33333.3333

2.
Let w be the weight invested in new portfolio and 1-w be the weight in risk free rate

w*20%+(1-w)*5%=12%
=>w=(12%-5%)/(20%-5%)=0.466666667

Volatility=Standard deviation=0.466666667*12%=5.6000%

This has lower volatility

Amount in new portfolio=0.466666667*100000=46666.6667

Amount in risk free rate=100000-46666.6667=53333.3333