a. Suppose asset A has an expected return of 10 percent and a standard deviation of 20 percent. Asset B has an expected return of 16 percent and a standard deviation of 40 percent. If the correlation between A and B is 0.4, what are the expected return and standard deviation for a portfolio comprised of 30 percent asset A and 70 percent asset B? Answer: %. 2 . 14 142 . 0 ) 16 . 0 ( 7 . 0 ) 1 . 0 ( 3 . 0 rˆ ) w 1 ( rˆ w rˆ B A A A P 309 . 0 ) 4 . 0 )( 2 . 0 )( 4 . 0 )( 7 . 0 )( 3 . 0 ( 2 ) 4 . 0 ( 7 . 0 ) 2 . 0 ( 3 . 0 ) W 1 ( W 2 ) W 1 ( W 2 2 2 2 B A AB A A 2 B 2 A 2 A 2 A p b. Plot the attainable portfolios for a correlation of 0.4. Now plot the attainable portfolios for correlations of +1.0 and -1.0. Answer: AB = +0.4: Attainable Set of Risk/Return Combinations 0% 5% 10% 15% 20% 0% 10% 20% 30% 40% Risk, p Expected return Mini Case: 5 - 8 MINI CASE
AB = +1.0: Attainable Set of Risk/Return Combinations 0% 5% 10% 15% 20% 0% 10% 20% 30% 40% Risk, p Expected return AB = -1.0: Attainable Set of Risk/Return Combinations 0% 5% 10% 15% 20% 0% 10% 20% 30% 40% Risk, p Expected return Mini Case: 5 - 9
c. Suppose a risk-free asset has an expected return of 5 percent. By definition, its standard deviation is zero, and its correlation with any other asset is also zero. Using only asset A and the risk-free asset, plot the attainable portfolios. Answer: Attainable Set of Risk/Return Combinations with Risk-Free Asset 0% 5% 10% 15% 0% 5% 10% 15% 20% Risk, p Expected return Mini Case: 5 - 10
d. Construct a reasonable, but hypothetical, graph which shows risk, as measured by portfolio standard deviation, on the x axis and expected rate of return on the y axis. Now add an illustrative feasible (or attainable) set of portfolios, and show what portion of the feasible set is efficient. What makes a particular portfolio efficient? Don't worry about specific values when constructing the graph— merely illustrate how things look with "reasonable" data. Answer: The figure above shows the feasible set of portfolios. The points B, C, D, and E represent single securities (or portfolios containing only one security). All the other points in the shaded area, including its boundaries, represent portfolios of two or more securities. The shaded area is called the feasible, or attainable, set . The boundary AB defines the efficient set of portfolios, which is also called the efficient frontier . Portfolios to the left of the efficient set are not possible because they lie outside the attainable set. Portfolios to the right of the boundary line (interior portfolios) are inefficient because some other portfolio would provide either a higher return with the same degree of risk or a lower level of risk for the same rate of return. e. Now add a set of indifference curves to the graph created for part B. What do these curves represent? What is the optimal portfolio for this investor? Finally, add a second set of indifference curves which leads to the selection of a different optimal portfolio. Why do the two investors choose different portfolios?