Question

Assume the following information for two stocks, A and B, and a risk-free asset.

	Expected Return	Standard Deviation	beta	Correlation
				Stock A	Stock B
Stock A	10%	14%	1.4	1	0.4
Stock B	7%	19%	0.8		1
Risk-free asset	3%	0%

(a) Suppose you have a portfolio with investment of $200 in stock A, $400 in stock B, and $400 in the risk-free asset. Compute the expected return and standard deviation of this portfolio.

(b) Consider a portfolio that consists of only stocks A and B (but not the risk-free asset). If the expected return of this portfolio is 8%, what is the amount invested in Stock A?

(c) Explain why investing in a portfolio with both Stocks A and B is more preferable than investing in either Stock A, or Stock B only. Provide TWO reasons.

(d) Suppose the market portfolio return is 8%. Draw the security market line (SML) on a graph with clear labels on X and Y axis. You must plot the values of the equation of the line on the axis.

(e) Determine where Stock A and Stock B lies on the graph of SML and whether they are correctly priced, underpriced or overpriced.

Answer 1

a). Total invested amount = 200 + 400 + 400 = 1,000

weight of A (wA) = 200/1,000 = 20%; weight (wB) = 400/1,000 = 40%; weight of risk-free asset (wRF) = 400/1,000= 40%

Portfolio return = sum of weighted returns = (20%*10%) + (40%*7%) + (40%*3%) = 6.00%

Portfolio variance = (wA*SDA)^2 + (wB*SDB)^2 + (2*wA*wB*Correlation(A,B)*SDA*SDB)

(Since standard deviation for the risk-free asset is 0.)

= (20%*14%)^2 + (40%*19%)^2 + (2*20%*40%*0.4*14%*19%) = 0.0082624

Portfolio standard deviation = variance^0.5 = (0.0082624)^0.5 = 9.09%

b). Let the weight of A be x. Then weight of B is (1-x).

Portfolio return = (x*10%) + (1-x)*7%

8% = 10%x - 7%x + 7%

1% = 3%x

x = 1/3

Amount to be invested in A should be 1/3 rd of the total amount, so if 1,000 is invested then amount in A should be 333.33 and amount in B should be 666.67

c). A portfolio of A and B is a better investment than a single investment in either A or B because the risk decreases and because the correlation between A and B is not very much, both offset each other to some degree.

d). For plotting the SML, we need to calculate the optimal market portfolio (maximum Sharpe ratio) comprising of A and B. We maximize the Sharpe ratio using the variance-covariance matrix for A and B, and Solver.

The SML equation becomes expected return = risk-free rate + beta*excess return

expected return = 0.03 + beta*0.06945

The SML graph is:

e). As can be seen, stocks A and B lie on the SML and so, are correctly priced.