STOCHASTIC

among two risky assets and a riskless asset having zero net return (cash). The two risky
assets have mean returns pt and p2 and variance-covariance matrix of returns
j, = / a h a \
We allow short sales and margin loans and suppose that the investment allocations satisfy
2 g Xi £ — 1, and 1 £ £ *i £ 0. Assume that brokerage costs are negligible and that investors
retain other assets which may be used to offset margin calls.
(a) Suppose there is perfect positive correlation, i.e., a\2 = Cii 0.05, or if not, to (—1,0,2). If 0.05 < r < 0.10, the frontier
passes from (0,0,1) to (—1,0,2) and is linear between these points. If r S 0.10, the efficient
surface terminates at (0,0,1). Plot this efficient surface, and also the efficient surface
corresponding to the constraints £ *i = 1 > *i £ 0. Compare these plots paying particular
attention to the number of admissible efficient portfolios.
(b) Suppose p2 = 0.06 and all other parameters are as in (a). Show that cash dominates
the riskless portfolio (i, — 1,$). Show that the portfolio (0,1, —0.67) is an efficient riskless
portfolio having expected return 0.10. Show that the next set of efficient portfolios
lies on a curve between (0,1, —0.67) and (0,1,0), the following set along the straight line
from (0,1,0) to (0,0,1), and the final set is again linear between (0,0,1) and (-1,0,2).
Plot this efficient surface and compare it with the conventional efficient surface as in (a).
(c) Suppose there is zero correlation, i.e.,