Your (super) discount retail stock broker allows you to trade the following equities in the Toronto Stock Exchange:

Royal Bank of Canada RY.TO

Suncor Energy SU.TO

Canadian National Railway Company CNR.TO

Thomson Reuters Company TRI.TO

Telus Corporation T.TO

Fortis Inc. FTS.TO.

a) Find weekly returns for each of these equities for 2012-01-01 to 2017-01-01. Using these weekly returns, calculate and present the expected return and standard deviation for each of the equities. Also, show the covariances among all the equity pairs. Hint, create and populate a variance-covariance matrix.

b) Find the optimal risky portfolio for these risky assets. Show all your work. If your work is done numerically, your submission should include a (clearly explained) spreadsheet showing your work. If you solve for the optimal weights using calculus, show me your derivation of the optimal holdings for all six equities. Present the portfolio weights for each security, the expected return and risk of the portfolio, and the Sharpe ratio. The average interest rate quoted for 3-month Canadian Treasury bills over this period was (approximately) 1% per year.

c) In an E[r]- graph, show all securities and the optimal risky portfolio. Does it make sense where the optimal risky portfolio is relative to the equities? Why or why not?

d) Briefly discuss what would happen to the characteristics of the optimal risky portfolio if the covariances of the equities were lower and if they were higher. What would happen if the interest rate for the T-bills were higher and lower.

e) We want to get a general sense of the minimum variance frontier. To do this, it would be helpful to have another point on the frontier. Lets use the global minimum variance portfolio. Find the global minimum variance portfolio. Report the equity portfolio weights and the portfolios expected return, standard deviation, and Sharpe ratio. Add it to your E[r]- graph as well.

f) Your broker now informs you that it does not allow short-selling. If this is a problem for your optimal risky portfolio, please adjust your holdings to satisfy the no-short-selling requirement. Report equity portfolio weights and the new optimal risky portfolios expected return, standard deviation, and Sharpe ratio. Additionally, add the no-short-selling optimal risky portfolio to your graph.

g) Did the no-short-selling requirement have a large effect on your optimal risky portfolio? Compare the two portfolios characteristics. Now that youve chosen a risky portfolio, you need to make your capital allocation decision. You have $1 million available to invest that you need to split between your risky portfolio and the risk-free asset. You take some time to think about your preferences regarding risk aversion (by comparing bets, potential investments, etc.) and figure out that your preferences fit a mean-variance utility function with a risk aversion coefficient of 15.

h) Determine what proportion of your wealth should go into the (no-short-selling) risky portfolio and what proportion should go into the (risk-free) 3-month Canadian T-bills. Based on this capital allocation, how much of your wealth goes into T-bills and how much goes into each equity? What is the expected return and risk on your combined portfolio?

Your broker informs you that it is considering allowing clients like you to place short sale orders. The broker intends to charge for the ability to place short sale orders by taking a percent of clients returns. It wants to know how much you would be willing to pay for this feature.

i) First, lets figure out your capital allocation if there were no short sale restriction. What are the characteristics (risky allocation, expected return, risk, portfolio weights) of your combined portfolio with no short selling restriction?

j) What utility do you derive from the no-short-sale portfolio? What utility do you derive from the short-sale-allowed portfolio? k) How much lower could the return be for your short-sale-allowed portfolio without lowering its utility below that of the no-short-sale portfolio? That difference in percentage is the maximum rate youd be willing to pay for the right to short sell.

Spreadsheet Model

Sec	E[r]		SR
A	0.1108	0.1295	0.7011583
B	0.0536	0.1466	0.22919509
C	0.0837	0.1741	0.36588168
rf	0.02
	A	B	C
A	1	0.64	0.54
B	0.64	1	0.91
C	0.54	0.91	1
Covariance matrix
Weights		1.27386825	-1.0557531	0.78188488
		A	B	C
1.274	A	0.01677025	0.01215021	0.01217481	0.0229994	=A18*SUMPRODUCT(C$16:E$16,C18:E18)
-1.056	B	0.01215021	0.02149156	0.02322598	-0.0115584	=A19*SUMPRODUCT(C$16:E$16,C19:E19)
0.782	C	0.01217481	0.02322598	0.03031081	0.01148414	=A20*SUMPRODUCT(C$16:E$16,C20:E20)
1	Wght sum	=SUM(A18:A20)
0.15	E[r]	=SUMPRODUCT(A18:A20,B4:B6)
0.151		=SUM(F18:F20)^0.5
0.859	Sharpe ratio	=(A22-B7)/A23
Portfolios
		Min var
	MVP	E[r] = 0.09	E[r] = 0.11	E[r] = 0.13	E[r] = 0.15	Max SR
E[r]	0.08761035	0.09	0.11	0.13	0.15	0.20030767
	0.12318117	0.12322732	0.12716875	0.13693773	0.15141046	0.2011615
SR	0.54886915	0.56805589	0.70772102	0.8032848	0.85859327	0.89633286