Question

You are considering investing $2,100 in a complete portfolio. The complete portfolio is composed of Treasury bills that pay 5% and a risky portfolio, P, constructed with two risky securities, X and Y. The optimal weights of X and Y in P are 75% and 25% respectively. X has an expected rate of return of 12%, and Y has an expected rate of return of 9%. To form a complete portfolio with an expected rate of return of 8%, you should invest approximately __________ in the risky portfolio. This will mean you will also invest approximately __________ and __________ of your complete portfolio in security X and Y, respectively.

rev: 01_31_2017_QC_CS-76878

0%; 75%; 25%

50%; 38%; 13%

48%; 36%; 12%

21%; 63%; 16%

Answer 1

The expected return on the risky portfolio of assets X and Y is given by: w*E[R_x] + (1-w)*E[R_y]; where w is the proportion of the risky portfolio invested in asset X. E[R_x] and E[R_y] are the expected returns on assets X and Y respectively.
Thus, expected return on the risky portfolio = 0.75*12% + 0.25*9% = 11.25%.

Now, the expected return on the complete portfolio is given by: W*(Expected Return on risky portfolio) + (1-W)*(Return on risk-free portfolio); where W is the proportion of the complete portfolio invested in the risky portfolio.
Since the expected return on the complete portfolio is 8%, we get: 8% = W*11.25% + (1-W)*(5%) = 5% + W*6.25%.
This yields: W = 48%.

This 48% is further divided in the ratio 0.75:0.25 among assets X and Y respectively, which means 36% and 12% of the complete portfolio is invested in assets X and Y respectively.

Hence, the final answer is: 48%, 36%, 12%.

Please feel free to comment if you require any further clarification.