In: Finance
Problem 1. Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%.
a) In light of the apparent inferiority of gold with respect to both mean return and volatility, would anyone hold gold? If so, demonstrate graphically why one would do so.
b) Given the data above, reanswer a) with the additional assumption that the correlation coefficient between gold and stocks equals 1. Draw a graph illustrating why one would or would not hold gold in one’s portfolio. Could this set of assumptions for expected returns, standard deviations, and correlation represent an equilibrium for the security market?
Problem 2. Consider the following properties of the returns of stock 1, the returns of stock 2 and the returns of the market portfolio (m):
Standard deviation of stock 1 σ1 = 0.30
Standard deviation of stock 2 σ2 = 0.30
Correlation between stock 1 and the market portfolio ρ1, m = 0.2
Correlation between stock 2 and the market portfolio ρ2, m = 0.5
Standard deviation of the market portfolio σm = 0.2
Expected return of stock 1 E (r1) = 0.08
Suppose further that the risk-free rate is 5%.
a) According to the Capital Asset Pricing Model, what should be the expected return on the market portfolio and the expected return of stock 2?
b) Suppose that the correlation between the return of stock 1 and the return of stock 2 is 0.5. What is the expected return, the beta, and the standard deviation of the return of a portfolio that has a 50% investment in stock 1 and a 50% investment in stock 2?
c) Is the portfolio you constructed in part b) an efficient portfolio? Assuming the CAPM is true, could you build a combination of the market portfolio and the portfolio of part b) to increase the expected return of the market portfolio without changing the variance of the combined portfolio.
Problem 1:
a) Gold might seem dominated by stocks but it is still an attractive asset to hold as a part of the portfolio if not alone. If the correlation between gold and stocks is very low, it can be hold as a part in portfolio. In the above graph it can be seen that the graph for the portfolio of stocks and gold touch the optimal allocation line at the optimal tangency portfolio P
b) if the correlation between gold and stocks equal one, then hold should not be held. the set of risk and return combination of stock and gold would plot as a straight line with a negative slope/refer to the graph above. It shows that in this case any portfolio that contains any gold is dominated by only stock portfolio and this holding gold is not an option. If explained in a different manner the CAL for only stock portfolio is steeper than any other CAL passing through all other possible portfolio.
Problem 2:
(a) Computation of the expected return on the market portfolio and the expected return of stock 2 using capital asset pricing model.We have,
Step1: Computation of the beta.We have,
Beta = Correlation between stock 2 and the market portfolio x standard deviation of stock 2 / standard deviation of market
Beta = 0.5 x 0.3/0.2 = 0.75
Hence,the beta of stock 2 is 0.75
Beta of Stock1:
Beta = Correlation between stock 1 and the market portfolio x standard deviation of stock 1 / standard deviation of market
Beta = 0.2 x 0.3/0.2 = 0.30
Hence,the beta of stock 1 is 0.30
Step2: Computation of the expected return of stock 2.We have,
Expected return = Risk-free return + Beta ( Market return - Risk-free return)
For Stock1:
0.08 = 0.05 + 0.3( Market return - 0.05)
0.3( Market return - 0.05) = 0.03
Market return = 0.1 + 0.05 = 0.15*100 = 15 %
Hence, the expected market return is 15%.
For stock2:
Expected return = 0.05 + 0.75( 0.15 - 0.05)
Expected return = 0.05 + 0.075 = 0.125*100 = 12.50 %
Hence,the expected return of stock 2 is 12.50 %
(b-1) Computation of the expected return of portfolio of stock A & B.We have,
Expected return = (Weight of Stock 1 x Expected return of stock 1) + (Weight of stock 2 x Expected return of stock 2)
Expected return = ( 0.50 x 0.08) + (0.50 x 0.1250)
Expected return = 0.04 + 0.0625 = 0.1025*100 = 10.25 %
Hence,the expected return of the portfolio is 10.25 %.
(b-2) Computation of the standard deviation of the portfolio.We have,
Standard deviation = [ ] 1/2
Standard deviation = [ (0.5)2 (0.3)2 + (0.5)2 (0.3)2 + 2x 0.3 x 0.3 x 0.5x0.5]1/2
Standard deviation = [ 0.0225 + 0.0225 + 0.045 ]1/2
Standard deviation = 0.30*100 = 30 %
Hence,the standard deviation of the portfolio of stock 1 and stock 2 shall be 30%.
(c) Computation of the weight of stock 1 & 2 using minimum variance portfolio.We have,
W1 =
W1 = (0.3)2 - 0.3 x 0.3 x 0.5 / (0.3)2 + (0.3)2 - 2 x 0.3 x 0.3 x 0.5
W1 = 0.09 - 0.045 / ( 0.09 + 0.09 - 0.09)
W1 = 0.045 / 0.09 = 0.5
W1 = 0.50
W2 = 1 - W1 = 1 - 0.50 = 0.50
Hence, the weight of stock 1 & 2 shall be 0.5 and 0.5 by using minimum variance portfolio. Therefore, the portfolio constructed in part(b) is the efficient portfolio and expected return of the market portfolio without changing the variances of the combined portfolio are 10.25 % and there is no change from part(b)