Question

Suppose you live in a world with only two risky assets: Amazon and Facebook (you can label these assets A and F, respectively). Furthermore, suppose these assets have the following expected returns and standard deviations:

	Amazon (A)	Facebook (F)
Er	14%	5%
σ	10%	4%

a)If the correlation of Amazon and Facebook returns is equal to 0.3, what is the covariance of their returns?

Use Excel to calculate the expected return and standard deviation of the risky portfolio for different combinations of weights for A and F. In other words, set up your Excel spreadsheet to calculate the following:

wA	wF	ErP	Std dev P
-0.5	1.5
-0.4	1.4
-0.3	1.3
-0.2	1.2
-0.1	1.1
0	1
0.1	0.9
0.2	0.8

etc… (for additional combinations of weights from w_A = -0.5 to w_A = 1.5)

Please include a screenshot of the filled-in table with your problem set solutions.

a)What is the optimal risky portfolio assuming r_f = 2% as in part e? Label this point P on your graph from part d.(Hint: Use Solver to choose the risky portfolio weights that maximize the Sharpe Ratio of the CAL created by combining the risky portfolio and the risk-free asset.)

b) Draw in the CAL through the optimal portfolio from part a.

c) What is the expected return and standard deviation of the complete portfolio that invests y=0.4 (i.e. 40%) in the optimal risky portfolio and 1-y = 0.6 (i.e 60%) in the risk-free asset.

d) If your utility from investing in the market is given by U = Er - 12σ² – 10σ⁴ , what is your optimal complete portfolio? (Hint: Use Solver to find the % of your complete portfolio in the risky asset (i.e. y) that maximizes your utility.)

e)What is a possible intuition of including the σ⁴ term in the utility function in part b?

Answer 1

a) Covariance = Varianceof amazon*Variance of Fb* correlation

=0.1*0.4*0.3=0.012

Assuming Rf=0, the optimal portfolio is one with 30% Amazon and 70%facebook

Rp = 7.7% and ?p = 4.68%

a) Based on above graph,

the optimal risky portfolio consists of risk-free asset = 0% and  30% Amazon and 70%facebook

Rp = 7.7% and ?p = 4.68%

c) Expected return= 0.4*7.7%+0.6*2% = 4.28%

Standard deviation= 0.4*4.68% = 1.87%

d) Based on analysis, optimal complete portfolio; risk-free asset = 0% and  30% Amazon and 70%facebook