Problem 27

Intro

The risk-free asset pays a return of r_F=0.5%. There are 3 risky assets: A, B and C. The expected returns and variances of the risky assets are as follows:

Asset	A	B	C
Expected return	0.01	0.02	0.03
Variance 2	0.00001	0.0004	0.0036

The covariances are: cov_AB=0.0002 cov_AC=0 cov_BC=-0.0002

Combining A,B and C, we create four risky portofolios, called 1,2,3 and 4. The shares of assets A, B and C in the portfolios are as follows:

Portfolio	wA	wB	wC
1	0.6	0.2	0.2
2	0.2	0.4	0.4
3	0.3	0.2	0.5
4	0.2	0.6	0.2

Attempt 1/10 for 10 pts.

Part 1

What is the expected return of portfolio 1?

Attempt 1/10 for 12 pts.

Part 2

What is the standard deviation of portfolio 1?

Attempt 1/8 for 10 pts.

Part 3

Calculate the expected return and standard deviation for all portfolios. Which of the risky portfolios clearly does not belong to the efficient frontier?

Portfolio 4

Portfolio 2

Portfolio 1

Portfolio 3

Correct

To answer this question, we need to calculate the expected return and standard deviation for each portfolio:

Portfolio	wA	wB	wC	E(ri)	i
1	0.6	0.2	0.2	0.016	0.01399
2	0.2	0.4	0.4	0.022	0.02467
3	0.3	0.2	0.5	0.022	0.03001
4	0.2	0.6	0.2	0.02	0.01698

The efficient frontier with more than two risky assets is obtained by minimizing the volatility of the risky portfolio for any given value of the portfolio expected return.

Both porfolios 2 and 3 yield an expected return of 0.022. However the standard deviation of portfolio 3 is 0.03001, while the standard deviation of porfolio 2 is 0.02467. This implies that, for a given expected return of 0.022, portfolio 3 does not minimize the volatility and thus it does not belong to the efficient frontier.

Attempt 2/8 for 8 pts.

Part 4

Which one of the risky portfolios is closest to the optimal risky portfolio?

Hints:

• You do not need to consider the portfolio that does not belong to the efficient frontier.
• The optimal risky portfolio maximizes the Sharpe ratio.

Portfolio 4

Portfolio 2

Portfolio 1

Portfolio 3

Correct

The Sharpe ratio of portfolio i is: S_i=(E(r_i)-r_F)/_i

Portfolio	wA	wB	wC	E(ri)	i	Si
1	0.6	0.2	0.2	0.016	0.01399	0.787
2	0.2	0.4	0.4	0.022	0.02467	0.689
3	0.3	0.2	0.5	0.022	0.03001	0.566
4	0.2	0.6	0.2	0.02	0.01698	0.883

The portfolio with the highest Sharpe ratio is portfolio 4.

Attempt 1/10 for 10 pts.

Part 5

Now we combine the risky portfolio found in part 4 with the risk-free asset to form a complete portfolio to achieve an expected return of 0.8%.

Let w_F be the share of the risk-free asset in the complete portfolio and 1-w_F the share of the risky portfolio found in part 4.

What should be the share of the risk-free asset, w_F, in the complete portfolio?

Attempt 1/10 for 10 pts.

Part 6

Finally, what is the share of asset A in the complete portfolio that you determined in part 5?